AN ENGINEERING STUDY OF BACTERIAL

KINETICS AND ENERGETICS

A . A . Esener

ár

o o •o

Delft University of Technology

AN ENGINEERING STUDY OF BACTERIAL

KINETICS AND ENERGETICS

Proefschrift

ter verkrijging van de graad van doctor in de technische wetenschappen

aan de Technische Hogeschool Delft, op gezag van de rector magnificus

prof. ir. B.P.Th. Veltman, voor een commissie aangewezen

door het college van dekanen te verdedigen op donderdag 1 oktober te 14.00 uur

door A l i A y d i n Esener

Chemical Engineer B.Sc. M.Sc. geboren te Ankara

Delft University Press, 1981

Dit proefschrift is goedgekeurd door de promotoren PROF. DR. IR. N.W.F. KOSSEN PROF. IR. J.A. ROELS

On the f r o n t cover d e v i a t i o n s between the unstructured model p r e d i c t i o n s and the experimental r e s u l t s are shown f o r oxygen uptake and carbon d i o x i d e production rates during fed-batch growth (part of F i g . 3 of Chapter 4)

Dedicated to my parents

In the completion of t h i s t h e s i s I g r a t e f u l l y acknowledge:

Professors Kossen and Roels f o r t h e i r e x c e p t i o n a l guidance and encouragement as my supervisors

Dr. I r . J.C. van Suijdam f o r many di s c u s s i o n s and t r a n s l a t i n g the summary to Dutch

I r . G.C. van Eybergen f o r trou b l e shooting i n many computer programs

Messrs. C. Ras and G. van der Steen f o r chemical a n a l y s i s of samples

Messrs. J.Ph. Bronkhorst, A.L. de Graaf and B.J.T. K e r k d i j k f o r help i n the handling and maintenance of the b i o r e a c t o r s and a u x i l i a r y equipment

Messrs. F. Bolmann and C. Warnaar f o r drawings and photographs

My students, J . Roozenburg, G.M. Bol and T. Veerman f o r t h e i r c o n t r i b u t i o n s to A p p l i c a t i o n 3, Chapter 5 and Chapter 7, r e s p e c t i v e l y

G i s t Brocades N.V. of D e l f t f o r f i n a n c i a l l y supporting me during t h i s work

Kraus-Uithof Fonds f o r t h e i r f i n a n c i a l c o n t r i b u t i o n towards the p r i n t i n g costs of t h i s t h e s i s

TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION

CHAPTER

I Aim and scope I I Organization of t h i s t h e s i s

ON THE THEORY AND APPLICATIONS OF UNSTRUCTURED GROWTH MODELS I Development of m i c r o b i a l energetics I I Macroscopic methods i n the study of energetics I I I An i n t r o d u c t i o n to the modelling of m i c r o b i a l growth IV A simple unstructured model f o r m i c r o b i a l growth V Discussion of unstructured models and energetics

with reference to experimental r e s u l t s VI A d i s c u s s i o n on the concept of maintenance VII Nomenclature and references

5 7 9 10

13 21 22

CHAPTER 3 MATERIALS AND METHODS I D e s c r i p t i o n of the experimental system and

a n a l y t i c a l methods 25 I I Developed and used t o o l s and methods 27 I I I A p p l i c a t i o n of the s t a t i s t i c a l techniques i n the

study of m i c r o b i a l k i n e t i c s and energetics 32 IV Nomenclature and references 39

CHAPTER 4

CHAPTER 5

FED-BATCH CULTURE ; MODELLING AND APPLICATIONS IN THE STUDY OF MICROBIAL ENERGETICS I Summary 41 I I I n t r o d u c t i o n 41 I I I Model 42 IV Determination of b i o k i n e t i c and energetic parameters 44 V M a t e r i a l s and Methods 45 VI Results and d i s c u s s i o n 46 VII Conclus ions 55 V I I I Appendix 56 IX Nomenclature and references 57

GROWTH OF MONO AND MIXED CULTURES IN SALINE ENVIRONMENT I A b s t ract 59 I I I n t r o d u c t i o n 59 I I I M a t e r i a l s and Methods 60 IV Results 61 V Discu s s i o n 63 VI Conclusions 66 VII References 67

CHAPTER 6 THE INFLUENCE OF TEMPERATURE ON THE KINETICS AND ENERGETICS I I n t r o d u c t i o n 69 I I Model 69 I I I M a t e r i a l s and Methods 70 IV Results and Dis c u s s i o n 71 V Nomenclature and references 73 VI Addendum

infl u e n c e of temperature on k s 75 infl u e n c e of temperature on energetic parameters 75 consequences f o r engineering operations and design 77

VII References 78

CHAPTER 7 A STRUCTURED MODEL FOR BACTERIAL GROWTH I I n t r o d u c t i o n 79 I I T h e o r e t i c a l development of the general s t r u c t u r e d

model 80 I I I D e s c r i p t i o n of the two compartmental system 81 IV D e r i v a t i o n of the balance equations 83 V Ev a l u a t i o n of the v a l i d i t y of the model 84 VI D i s c u s s i o n 85 VII Nomenclature and references 88

CHAPTER 8 APPLICATIONS I Comments on the d e s c r i p t i o n of maintenance

metabolism during anaerobic growth w i t h product formation 89

I I Bioenergetic c o r r e l a t i o n of COD to BOD 95 I I I D e s c r i p t i o n of m i c r o b i a l growth behaviour during

the wash-out phase; determination of the maximum s p e c i f i c growth rate 101

IV On the s t a t i s t i c a l a n a l y s i s of batch data 109 V Carbon dioxide hold-up as a source of e r r o r

i n batch c u l t u r e c a l c u l a t i o n s 117

SUMMARY 121 SAMENVATTING 123 OZET 125

VI

ERRATA

should read and not

P .

p -P .

P .

P .

P .

P .

P .

P .

19 Table 1,4 l i n e 4 0.698(0.689-19 idem t l i n e 5 -4.631).10" 2

22 4- l i n e 8 10 times lower 44 81 82 83 83

110 112 113

0.698(0.869--4.631) 10 times higher

^ x » Ks

dx/dt = dX/dt = 7 K should be replaced by G (2x)

Mg » Kg + l i n e 4 eq(3)

t 1ine.6 eq(16) 2 nd term of eq(19) should be d i v i d e d by ( k K Y^q)2

and Y S 7 > should be Ygjj f l i n e 4 cannot can

the l a s t term of eq(5) should be - 0Lj)2} eq(8) unsubscripted ¥ s should be Y s.

/

1 I s

CHAPTER 1

INTRODUCTION

I AIM AND SCOPE

In t h i s t h e s i s some aspects of b a c t e r i a l k i n e t i c s and energetics are studied with reference to engineering a p p l i c a t i o n s .

For engineering a p p l i c a t i o n s the v e r b a l model presented i n Figure 1 provides a good approximation to r e a l i t y i n the d e s c r i p t i o n of b a c t e r i a l growth and primary product metabolism.

product synthesis

use of substrate

A T P

pool

synthesis of biomass precursors

biomass synthesis

maintenance

Fig. 1: Distribution of substrate energy in microbial metabolism (from Roels and Kossen, 19 78, see Chapter 2, ref.16).

Here b a s i c a l l y three processes are i d e n t i f i e d ; i ) b i o s y n t h e t i c process during which precursors are formed from the substrate followed by the po l y m e r i z a t i o n of them i n t o biomass, i i ) product formation and i i i ) maintenance processes. The energy input i n t o the system i n the form of chemical energy i s d i s t r i b u t e d between these processes. Often there are i n t e r a c t i o n s between these processes as i n d i c a t e d by the two way arrows i n F i g . 1.

An ambition of the b i o t e c h n o l o g i s t i s to understand how the input energy and

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mass i n the form of substrate are d i s t r i b u t e d between these processes, i n an attempt to manipulate t h i s d i s t r i b u t i o n as to minimize the o v e r a l l cost of the d e s i r e d product. This can best be achieved by d e s c r i b i n g the whole process by a mathematical model and o p t i m i z i n g i t according to an o b j e c t i v e f u n c t i o n .

With these considerations i n mind the f o l l o w i n g questions were set and studied i n an e f f o r t to develop a sound strategy f o r research, design and c o n t r o l of m i c r o b i a l processes.

1. How can a m i c r o b i a l system be modelled based on the e x i s t i n g knowledge?

2. How can the experimental and computational methods be improved and data processed i n order to o b t a i n more optimal information?

3. To what extent are the k i n e t i c and energetic parameters influ e n c e d by selected environmental changes ( s a l i n i t y and temperature)?

4. Does the mode of c u l t i v a t i o n (batch, fed-batch, continuous) i n f l u e n c e the energetic behaviour of the system?

5. How can models considering i n t e r n a l changes i n the microorganisms be formulated? What are t h e i r prospects?

I I ORGANIZATION OF THIS THESIS

Contents of the chapters are b r i e f l y o u t l i n e d i n the f o l l o w i n g .

Chapter 2, s t a r t s w i t h an i n t r o d u c t i o n to the current s t a t e of m i c r o b i a l en e r g e t i c s . The t h e o r i e s to be used l a t e r on, are developed here. M i c r o b i a l energetics and k i n e t i c s are discussed i n connection w i t h the f o r m u l a t i o n of unstructured models. The choice of k i n e t i c and energetic r e l a t i o n s i s discussed. The d i s c u s s i o n s are i l l u s t r a t e d , supported and/or tested w i t h the batch and continuous c u l t u r e data obtained during t h i s study. F i n a l l y , a short comment on the concept of maintenance i s given.

Chapter 3, c o n s i s t s of three sub-sections. The f i r s t describes the experimental system and the a n a l y t i c a l methods used. The second sub-section o u t l i n e s some of the techniques developed f o r o b t a i n i n g more optimal information from the experimental data. The l a s t sub-section shows how the use of s t a t i s t i c a l pro­cedures can improve the e f f i c i e n c y of experimentation and the r e l i a b i l i t y of the data obtained.

Chapter 4, describes b a c t e r i a l growth i n fed-batch mode. I t i s shown that the unstructured model presented e a r l i e r , breaks down during the t r a n s i t i o n from exponential to substrate l i m i t e d growth phase. Use of fed-batch c u l t i v a t i o n i n the study of m i c r o b i a l k i n e t i c s and energetics i s a l s o shown and discussed.

In Chapters 5 and 6, the i n f l u e n c e s of s e l e c t e d environmental changes on the substrate energy d i s t r i b u t i o n are studied. F i r s t (Chapter S) the i n f l u e n c e of the presence of NaCl i s evaluated at d i f f e r e n t concentrations. The concentra­t i o n range i s from 0 to 40 kg/m3. The data obtained i s a l s o compared w i t h those reported f o r a c t i v a t e d sludge c u l t u r e s under the same c o n d i t i o n s . In Chapter 6, the i n f l u e n c e of temperature i s studied i n fed-batch c u l t u r e s . K i n e t i c data are used f o r the e s t i m a t i o n of thermodynamic parameters i n an Arrhenius type of model extended to describe a l s o the superoptimal temperature range. Temperature e f f e c t s on energetic parameters are a l s o presented.

2

Chapter 7, i s an attempt to formulate and t e s t a simple str u c t u r e d model i . e . , a model d e s c r i b i n g the i n t e r n a l s t r u c t u r e of the organism i n a d d i t i o n to macroscopic v a r i a b l e s . The simple two compartmental model developed i s shown to describe biomass and substrate p r o f i l e s w e l l . Extensive t e s t s with i n t e r n a l composition data i n d i c a t e s the weakness of the model. Prospects of these type of models and the c o r r e c t approach to t h e i r formulation and v e r i f i c a t i o n are stressed.

Chapter 8, c o n s i s t s of f i v e short p u b l i c a t i o n s which are a l l more or l e s s a p p l i c a t i o n s of the considerations presented i n Chapter 2 and 3. The f i r s t two i l l u s t r a t e the use of macroscopic methods i n data a n a l y s i s and c o r r e l a ­t i o n . The others are on the a p p l i c a t i o n of s t a t i s t i c s on batch data, estima­t i o n of the maximum s p e c i f i c growth rate by the wash-out technique and estimation of the carbon d i o x i d e r e t a i n e d i n broth during batch c u l t i v a t i o n .

Klebsiella pneumoniae NCTC 418 formerly known as Klebsiella aerogenes i s chosen as the experimental organism, since i t i s a t y p i c a l s o i l bacterium o f t e n also present i n waste waters and i s capable of growing a e r o b i c a l l y and a n a e r o b i c a l l y .

A l l u n i t s i n v o l v i n g biomass dry weight are expressed on ash-free b a s i s . An equivalent of any compound i s defined as that amount containing 12 grams of elemental carbon. For the biomass formulae used here, an equivalent of biomass i s the same as one mole of biomass. The y i e l d of biomass on substrate i s sometimes expressed as C-equiv/C-equiv ( same as C-mole/C-mole ),since t h i s i s a more fundamentel u n i t then molar or mass u n i t s . I t i n d i c a t e s d i r e c t l y the f r a c t i o n a l conversion of substrate carbon to biomass carbon.

This thesis has been carried out within the Biotechnology Group of the delft University of Technology. Postal address: Department of Chemical Engineering, Biotechnology Group, SBR, Jaffalaan 9, TH., Delft 2600, The Netherlands

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CHAPTER 2

ON THE THEORY AND APPLICATIONS OF UNSTRUCTURED GROWTH MODELS: KINETIC AND ENERGETIC ASPECTS

I DEVELOPMENT OF MICROBIAL ENERGETICS

Development of q u a n t i t a t i v e b a c t e r i a l energetics can be assumed to have com­menced wi t h the work of Monod'. Monod has defined the macroscopic y i e l d of b i o -mass on substrate as the r a t i o of the biomass produced to substrate consumed. He has produced h i s w e l l known k i n e t i c expression d e s c r i b i n g the dependence of growth rate on the concentration of the growth l i m i t i n g substrate'. F o l l o w i n g the i n t r o d u c t i o n of continuous c u l t i v a t i o n techniques, Herbert (1958) has pre­sented evidence that i n C - l i m i t e d continuous c u l t u r e s , the macroscopic growth y i e l d , Y s x was not constant but decreased as the d i l u t i o n r a t e decreased.* Herbert a t t r i b u t e d t h i s e f f e c t to what he c a l l e d the endogeneous metabolisrn. In e f f e c t the f i r s t attempt was a curve f i t t i n g e x e r c i s e . Then p h y s i o l o g i c a l con­s i d e r a t i o n s were attached to t h i s observation. I t was suggested t h a t , endoge­neous metabolism proceeds at constant r a t e at a l l p o s s i b l e growth r a t e s . The i n t r o d u c t i o n of t h i s concept modified Monod's expression to a new form:

U - U g { C s / ( K s + C g )} - U e (1)

Here i s the r a t e of endogeneous metabolism. When C s >> K s , p -> P m a x and hence Vmax = Pg ~ Ve • When C s = 0 \i eguals to - p e , i . e . , negative growth i s achieved. This i s equivalent to s e l f d e s t r u c t i o n . The observed y i e l d f o r a continuous c u l t u r e system can now be shown to be given by;3

y = y m a X {D / ( D + u )} (2) sx sx e

P i r t (1965) considered the substrate requirement f o r growth associated and non-associated functions separately and postulated h i s w e l l known r e c i p r o c a l / l i n e a r r e l a t i o n ^ :

1 / Y = 1 / Y m a x + m / y (3) S X S X s v '

He has fo r m a l l y introduced the maintenance c o e f f i c i e n t ms, .and a t t r i b u t e d i t to

5

the s o - c a l l e d maintenance functions which i n c l u d e ; turn over of c e l l m a t e r i a l s , osmotic work to maintain concentration g r a d i e n t s , c e l l m o t i l i t y e t c . Eq.(3) p r e d i c t s a s t r a i g h t l i n e f o r 1/Y s x vs. 1/u . In a number of cases, however, s t r a i g h t l i n e s could not be obtained and t h i s was shown to be due to the i n f l u ­ence of the growth rate on the fermentation p a t t e r n and ATP y i e l d of the par­t i c u l a r organism. Based on t h i s observation, Stouthamer and Bettenhaussen modified eq.(3) by considering the general energy currency, ATP, and obtained the f o l l o w i n g form^:

qATP = y 1 ATP + mATP ( A )

Later on, i n an attempt to account f o r the discrepancy between the t h e o r e t i c a l and experimental growth y i e l d s , the below r e l a t i o n has been proposed6;

q,™ - U / ( YIJ1 ) . . - + m \i + m (5) ATP M ATP ' t h e o r e t i c a l g e

This equation has found l i m i t e d a p p l i c a t i o n since no means were o f f e r e d f o r the determination of the growth associated (nig) and independent (me) maintenance c o e f f i c i e n t s . Furthermore, the Y ^ p values can only be c a l c u l a t e d f o r anaerobic systems i f the metabolic pathway and the associated stoichiometry are e x a c t l y known. The r e s p i r a t o r y chain of b a c t e r i a d i f f e r widely and depend on the growth c o n d i t i o n s . Thus, f o r aerobic systems one must know the s o - c a l l e d P/0 r a t i o i n order to c a l c u l a t e the Y^ Tp values, or v i c e versa. In some cases ^ATP v a l u e s obtained from anaerobic studies were used for the c a l c u l a t i o n of P/0 r a t i o s and Y ^ p values. This approach may not be v a l i d since aerobic and anaerobic sysems are q u i t e d i f f e r e n t e n e r g e t i c a l l y . Recently van Verseveld^ has reviewed the methods a v a i l a b l e f o r determining the P/0 r a t i o i n b a c t e r i a l sys­tems. From h i s account and l i t e r a t u r e i t becomes c l e a r that there i s yet no r e l i a b l e method f o r the e s t i m a t i o n of the P/0 r a t i o s . Thus more of t e n than not, one has to work i n terms of macroscopic y i e l d s and hence eq.(3) s t i l l f i n d s wide a p p l i c a t i o n , p a r t i c u l a r l y f o r aerobic growth with no by-product formation.

Under a v a r i e t y of c o n d i t i o n s the growth y i e l d s observed were much lower than expected. Senez^ studied t h i s phenomenon and introduced the term unbalanced growth implying that the two major processes i n the microorganisms; anabolism and catabolism are sometimes not i n tune w i t h each other and considerable amount of ATP produced could be wasted. Recently, N e i j s s e l and Tempest^i'O have demonstrated the occurence of t h i s phenomenon i n a number of systems and considered energy s p i l l i n g r e a c t i o n s as an i n t e g r a l part of the evolutionary competition c a p a b i l i t i e s of microorganisms. These authors have demonstrated that the presence of uncouplers, excess energy and C-source and forced t r a n ­s i e n t s enhance the extent of energy spillage.9-11 For a comprehensive a n a l y s i s of the current s i t u a t i o n i n m i c r o b i a l energetics the reader i s r e f e r ­red to recent r e v i e w s ^ - 1 4

Although an overwhelming body of information e x i s t s i n l i t e r a t u r e , the s t a t e of m i c r o b i a l energetics i s s t i l l not advanced enough at the fundamental l e v e l to allow engineering a p p l i c a t i o n s to be based on them. A d d i t i o n a l l y , as has been pointed out by Stouthamer'3 enough a t t e n t i o n has not been paid by many workers to t h e i r energetic c a l c u l a t i o n s and t h i s could be one of the reasons fo r the accumulation of i n c o n s i s t e n t data over the years. M i c r o b i a l energetics being at an impasse at the fundamental l e v e l , r e c e n t l y much work has been done on the macro-energetic behaviour. Since these studies r e l y on balancing methods and p r i n c i p l e s of thermodynamics, they are favoured f o r q u a n t i t a t i v e technolo­g i c a l a p p l i c a t i o n s .

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I I MACROSCOPIC METHODS IN THE STUDY OF MICROBIAL ENERGETICS

Information obtained by the a p p l i c a t i o n of elemental and energy balances and entropy i n e q u a l i t i e s can be c l a s s i f i e d as macroscopic information. Although such information provides u s e f u l t o o l s i n engineering a p p l i c a t i o n s , no micros­copic d e t a i l s are provided. These techniques, nevertheless, provide the tech­n o l o g i s t w i t h a strong s t a r t i n g point i n i n d u s t r i a l a p p l i c a t i o n s . I f s t r u c t u r e d information i s supplemented and checked for consistency by the a p p l i c a t i o n of macroscopic methods very u s e f u l information can be obtained w i t h q u a n t i t a t i v e confidence.

The subject has been advanced by many workers i n the recent years. Recent advancement of the subject i s due to Roels and Kossen'^, Roels'^, E r i c k s o n et a l l 8 - 2 0 and Heijnen and Roels21. In t h i s s e c t i o n only the relevant r e l a t i o n s from these p u b l i c a t i o n s w i l l be given without proof. These r e l a t i o n s w i l l be applied to experimental and t h e o r e t i c a l analyses, l a t e r on i n t h i s work.

Assuming that C, H, N and 0 are the only elements exchanged i n n o n - n e g l i g i b l e amounts i n the system, the f o l l o w i n g s t o i c h i o m e t r i c growth equation can be w r i t t e n f o r growth on a s i n g l e C and energy source. This source i s assumed to be growth l i m i t i n g .

0„C H, 0 N, + <I>0„ + $.NH„ 2 b j Cj 5 2 4 3

substrate

>.C H, 0 Nj 1 b, c, d,

biomass

$ QC H, 0 N, + $,C0„ + $7H.0 (6) 3 a^ b 3 c 3 d^ 6 2 7 2

product

The macroscopic y i e l d f a c t o r i s now defined as:

Y s x = I $2 I ^ a2 (C-equiv/C-equiv) (7)

Y s x simply i n d i c a t e s the degree of transformation of the substrate carbon i n t o biomass. Hence i t seems to be a more fundamental parameter than y i e l d values expressed on mass or molar b a s i s . The above equation has 7 flows i n v o l v i n g 4 elements. Thus s p e c i f i c a t i o n of any 3 flows f i x e s the system a l g e b r a i c a l l y , i . e . , any 4 unknown flows can be c a l c u l a t e d from the knowledge of any 3 flows at steady s t a t e .

S t a r t i n g from the p r i n c i p l e of the conservation of atomic species the f o l l o w i n g balances can be shown to hold f o r the system described:

r = r - r - r (8) c s x p

r = 1/4 ( y r - y r - v r ) (9) o 's s 'x x 'p p r.T = -d, r + d r + d r (10) N 2 s 1 x 3 p

Here, Y x , y s Y p a r e defined by the f o l l o w i n g :

7

= 4 + b l " 2 C, - 3 d . (11)

= 4 + b 2 " 2 c 2 - 3 d 2 (12)

Y P = 4 + b 3 " 2 c 3 " 3 d 3 (13)

Y i s a l i n e a r combination of elemental balances. I t can also be derived from a degree of reduction balance as defined by Er i c k s o n and coworkers.'8>'* I t must als o be noted that Y defined here only holds f o r NH3 being the N-source. Roels'? has introduced the generalized degree of reduction concept which can be appl i e d to growth with any N source.

Eq.(3) introduced p r e v i o u s l y does not consider product formation. Humphrey and J e f f e r i s * ^ and l a t e r on Roels and Kossen'^ have included the c o n t r i b u t i o n of product formation process and modified eq.(3) to :

, ..max , „max ,, .. r = r / Y + r / Y + m C (14) s x sx p sp s x

S i m i l a r forms of the above equation can be derived f o r the conversion rates of carbon dioxide and oxygen. By combining eqs.(8), (9) and (14) the f o l l o w i n g can be given :

r = (1/ Y m a x - 1) r + (1/ Y m a x - 1) r + m C (15) c sx x sp p S X

r„ 1/4 {(Y / Y m 3 X - Y ) r + (Y / Y m a X - Y ) r + Y m C } (16) S S X X x s sp p p s s x

From eqs.(15) and (16) a number of u s e f u l r e l a t i o n s can be obtained. A few are shown below. A more comprehensive l i s t has been given r e c e n t l y by Heijnen and Roels.21

1 / Y m a x = Y M (1/ Y m a x ) - 7 / 4 (17) ox S S X X

m = m (18) c s

m = (y /4) m (19) o s s

Eric k s o n et a l . ' ^ have shown methods f o r data a n a l y s i s and checking the con­s i s t e n c y by using r e l a t i o n s of t h i s s o r t . Another advantage of these t o o l s l i e s i n the formula t i o n of t h e o r e t i c a l l i m i t s to the e f f i c i e n c y of conversion pro­cesses. Roels'' has c a l c u l a t e d the maximum p o s s i b l e y i e l d values allowed by the second Law of Thermodynamics. C a l c u l a t i o n of t h i s allows the d e f i n i t i o n of the thermodynamic e f f i c i e n c y , l ^ h ' o r t^ l e growth process.

\ h = Y s x / U f • ( 2 0 )

Other n parameters have als o been defined based on oxygen and e l e c t r o n balances and i n e q u a l i t i e s . For growth w i t h NH3 as the ammonia source, however, there are no s i g n i f i c a n t d i f f e r e n c e s between the various r e l a t i o n s . For the system described the f o l l o w i n g u s e f u l l i m i t s have been shown to hold:

Y < sx f o r Y (21)

8

(22)

Y < Y / Y (23) sx s x

A p p l i c a t i o n of macroscopic p r i n c i p l e s can a l s o play an important r o l e i n pro­cess c o n t r o l , where the c o n t r o l parameter cannot be determined d i r e c t l y e.g., e s t i m a t i o n of biomass concentration i n broth w i t h suspended p a r t i c u l a t e sub­s t r a t e , l i k e s t a r c h . E s t i m a t i o n of heat output, a very important task i n pro­cess design and c o n t r o l , can a l s o be done, based on the measurement of a few o n - l i n e determined parameters.

The macroscopic t o o l s were applied to the experimental data obtained during t h i s work f o r checking data consistency and f o r the e s t i m a t i o n of energetic parameters. A d d i t i o n a l a p p l i c a t i o n s are presented i n Chapter 8.

I l l AN INTRODUCTION TO THE MODELLING OF MICROBIAL GROWTH

In general, b i o l o g i c a l systems are sub-sets of chemical systems. A wealth of information e x i s t s on chemical k i n e t i c s and dynamics. Thus, one can at l e a s t i n theory, expect to be able to describe b i o l o g i c a l systems i n terms of the dynamic behaviour of i t s c o n s t i t u e n t s ; chemical systems. In p r a c t i c e , however, t h i s approach i n e v i t a b l y f a i l s due to two reasons:

i . F i r s t l y , f o r an exact d e s c r i p t i o n of m i c r o b i a l metabolism one has to consider a l l the concentrations of the chemical substances i n the imme­di a t e environment of microorganisms ( a - b i o t i c phase) as w e l l as i n the organism i t s e l f ( b i o t i c phase). Considering that E. ooli has more than 2000 d i f f e r e n t p r o t e i n s , t h i s becomes an impossible task even at the age of f a s t computers,

i i . Secondly, although biochemical k i n e t i c s has b a s i c a l l y the same tasks as chemical k i n e t i c s i . e . , i d e n t i f i c a t i o n of r e a c t i o n s between molecules, determination of the rates of chemical r e a c t i o n s and the development of the r e l e v a n t t h e o r i e s , i t has to consider more complex kin d of i n t e r a c ­t i o n s , such as r e a c t i o n s between molecules and c e l l s , molecules and o r g a n e l l e s , c e l l s and c e l l s e t c . In other words, b i o l o g i c a l k i n e t i c s i s not r e s t r i c t e d to the study of r e a c t i o n s between e n t i t i e s belonging to a s i n g l e l e v e l of o r g a n i z a t i o n but a l s o belonging to d i f f e r e n t l e v e l s . 2 4

25 • Recently,. Savageau has given a general growth equation that i s based upon the nature of the elemental mechanisms i n complex systems. The r e s u l t i n g set of d i f f e r e n t i a l equations would be, however, very large and complex f o r a complete system d e s c r i p t i o n . Moreover, as shown by Prigogine26 these type of equations are not only s p e c i f i c to b i o l o g i c a l systems but are a p p l i c a b l e to any system, u n i v e r s a l l y . Since t h i s type of complete d e s c r i p t i o n has proven to be not p o s s i b l e , one aims f o r s i m p l i f i c a t i o n s through j u s t i f i a b l e assumptions.

S i g n i f i c a n t s i m p l i f i c a t i o n s become p o s s i b l e v i a a study of the r e l a x a t i o n times of the various processes taking place i n s i d e and the outside of the b i o t i c phase. Thus one has to consider and compare the time constants of the environmental changes and those of mechanisms i n s i d e the organism which f a c i l i t a t e the adaptation of the organism to these environmental changes. In two cases s i m p l i f i c a t i o n become p o s s i b l e : i . For processes which are c h a r a c t e r i z e d by very l a r g e r e l a x a t i o n times,

compared with that of the growth process, the mechanism and thus the concentration of the compounds i t r e g u l a t e s , do not change s i g n i f i c a n t l y . Thus these mechanisms and t h e i r e f f e c t s on the t o t a l , system behaviour may

9

be conveniently neglected, i i . For processes which are ch a r a c t e r i z e d by very short r e l a x a t i o n times,

b i o t i c mechanisms f o l l o w and respond t i g h t l y to the environmental changes and again the concentrations of the b i o t i c components that are associated w i t h these p a r t i c u l a r mechanisms can be c a l c u l a t e d from the a - b i o t i c concentrations. The st a t e of the mechanism can be r i g o r o u s l y described using only environmental concentration i . e . , concentration i n the a - b i o t i c phase.

Based on these considerations i . e . , most changes among the components of a system occur much f a s t e r than the rate of the growth f o r the system as a whole, Savageau25 concluded that t h i s mathematically implies a small number of r e l a t i o n s representing the slowest phenomena determine the temporal response of the e n t i r e system. A l l other r e l a t i o n s representing the f a s t e r phenomena can be assumed to have reached a pseudo-steady s t a t e w i t h time d e r i v a t i v e s equal to zero.

P r a c t i c a l growth models are u s u a l l y expressed i n terms of a rather a b s t r a c t u n i t s of l i f e , that i s , i n terms of populations. This approach considers the population as an e n t i t y homogeneously d i s t r i b u t e d i n space and time, and thus avoids complications that might a r i s e due to the s t o c h a s t i c phenomena a s s o c i a ­ted w i t h the existence of i n d i v i d u a l organisms. I t i s , however, important to note that t h i s approach i s only v a l i d when the number of organisms i n the system i s very large. This was the case f o r experiments to be reported i n t h i s work.

Qcowth i s the production of new biomass by a population when i t consumes a s u i t a b l e l i v i n g or non l i v i n g substrate from i t s environment and incorporates some of t h i s substance i n t o i t s own.24 Reproduction i s the increase i n the number of d i s c r e t e independent c e l l s of a population. Growth and reproduction are obviously coupled processes, however, the degree of coupling may be d i f f e ­rent f o r each case. In t h i s study these two processes w i l l not be considered separately but the t o t a l e f f e c t i s summed w i t h the dry weight measurements.

One of the most general approaches f o r d e s c r i b i n g growth, was provided by Powell27. In h i s approach, the current s p e c i f i c growth rate of a population i s assumed to depend not only on the current s t a t e of the a - b i o t i c phase but also on the e n t i r e h i s t o r y of the a - b i o t i c phase seen by the b i o t i c phase. In other words Powell expressed the s p e c i f i c growth r a t e at any i n s t a n t to be a 'fu n c t i o n a l ' o f the s t a t e of the a - b i o t i c phase. In p r a c t i c e , however, t h i s approach i s d i f f i c u l t to apply and p a r t i c u l a r l y i n the choice of f u n c t i o n a l s . A simpler approach would be to assume that the current growth rat e s are funct i o n s of the current s t a t e of the a - b i o t i c and b i o t i c phases.

The most rigorous s i m p l i f i c a t i o n done i n the development of population models i s the assumption that the t o t a l amount of the biomass i n the c u l t u r e i s s u f f i c i e n t to s p e c i f y the a c t i v i t i e s of the microorganisms. Model based on t h i s assumption i . e . , i n which the v a r i a t i o n i n the biomass composition i s t o t a l l y ignored, are c a l l e d UNSTRUCTURED WDELS.

IV A SIMPLE UNSRUCTURED MODEL FOR MICROBIAL GROWTH

The most popular k i n e t i c expression used today i s the Monod r e l a t i o n . Although t h i s r e l a t i o n i s an homologue of the Michaelis-Menten equation, Monod a r r i v e d at i t e m p i r i c a l l y . That i s , h i s r e l a t i o n provided good f i t f o r h i s experimen­t a l data. A l t e r n a t i v e l y , one can t r y to provide a mechanistic foundation by reasoning that one enzymic r e a c t i o n taking part i n a long sequence might be the

10

bottleneck and thus rate l i m i t i n g .

Now, consider a constant volume c u l t i v a t i o n system i n which the t o t a l m i c r o b i a l a c t i v i t y i s q u a n t i f i e d by the amount of biomass (biomass w i l l imply dry weight throughout t h i s work) and there i s a s i n g l e l i m i t i n g substrate (C and energy source). To describe the system, changes of C s and C x and t h e i r i n t e r ­dependence have to be evaluated. For the general case the f o l l o w i n g balances can be formulated:

d C / dt (24)

d C / dt s

r + s

(25)

I f the r e l a t i o n of l i n e a r substrate consumption i s chosen f o r r e l a t i n g r x to r s 2,4 ( f o r the no-product case):

r / Y x sx

m C s x (26)

Eqs. (24) (25) and (26) w i l l be s u f f i c i e n t to describe simple systems such as batch, continuous and fed-batch. $ i s the net flow term to the system and i s f i x e d by the mode of operation, e.g., f o r continuous c u l t i v a t i o n $ i s described by the f o l l o w i n g :

D ( C (27)

* = x D C (28)

For batch c u l t i v a t i o n , $ are zero, since i t i s a closed system as f a r as the non-gaseous phases are concerned. Thus the f o l l o w i n g p a i r of eqs. describe the system:

dC /dt = u C C / (K + C ) x max s x s s

dC /dt = - y C C . / {(K + C ) Y m a X } - m C s max s x s s sx s x

(29)

(30)

No a n a l y t i c a l s o l u t i o n i s p o s s i b l e f o r t h i s set and hence numerical methods ' were used f o r s i m u l a t i o n purposes during t h i s study. Most of the time during batch growth organisms grow at or near U m a x • Since the e f f e c t of maintenance requirements are e f f e c t i v e l y minimized at nigh u , a convenient s i m p l i f i c a t i o n can be introduced by n e g l e c t i n g the ms term i n the above model. In t h i s case an a n a l y t i c a l s o l u t i o n i s p o s s i b l e and can be shown to be given by:

ln(C /C ) + K Y /(Y C + C ) In {(C /C ) / ( l + C / (Y C ) -x xo s sx sx so xo X xo xo sx so

C/(Y C ))} = y t (31) x sx so max

The i m p l i c i t nature of t h i s expression gives problems i n parameter e s t i m a t i o n from experimental data by nonlinear r e g r e s s i o n . Further s i m p l i f i c a t i o n s are po s s i b l e by considering various experimental c o n d i t i o n s , e.g., i f C >> C Y and C >> K , the model reduces down to: so s

1 1

C / C = exp ( U t ) (32) x xo max

Growth behaviour i n fed-batch c u l t u r e s can a l s o be described by t h i s general unstructured model. This i s presented i n Chapter 4.

I t i s important to note that an u n r e a l i s t i c f e a t u r e ^ 0 f the l i n e a r r e l a t i o n i s that i t p r e d i c t s substrate uptake even a f t e r substrate has been exhausted. P a r t i c u l a r l y i n numerical simulations the experimenter must consider t h i s point c a r e f u l l y as t h i s might lead to the c a l c u l a t i o n of negative substrate concentration values. When C s = 0, the concept of endogeneous metabolism becomes handy. Expression (1) p r e d i c t s zero growth r a t e when:

or,

C = K y / ( y ~ P ) (33) s s e g e

C a K u / y (34) s s e g

This expression can a l s o a l l o w f o r negative growth; a n a t u r a l phenomenon which can be observed experimentally, when Cs i s smaller than the r i g h t hand side of eq.(33) .

The d i f f e r e n c e between P i r t and Herbert r e l a t i o n s stem from t h e i r d i f f e r e n t ways of i n t e r p r e t i n g the f u n c t i o n i n g of the maintenance processes.

The simple unstructured model has been a p p l i e d to some p r a c t i c a l systems, s u c c e s s f u l l y , p a r t i c u l a r l y f o r batch growth where the growth i s not l i m i t e d by substrate,and f o r substrate l i m i t e d growth i n continuous c u l t u r e s .

One wonders i f the Monod' r e l a t i o n i s the only s u i t a b l e k i n e t i c expression f o r modelling. As has been shown by Roels28 the d e t a i l e d nature of the k i n e t i c equation i s only of s l i g h t improtance f o r substrate l i m i t e d growth. This i s because of the very low C g under substrate l i m i t i n g c o n d i t i o n s . At such low C s values a pseudo-steady s t a t e hypothesis w i t h respect to C s holds. Under these c o n d i t i o n s , i t can be shown that the s t a t e equation f o r C x i s only defined by the energetic and experimental parameters and not by the k i n e t i c r e l a t i o n . Since C x vs. time, p r o f i l e i s not i n f l u e n c e d much by the rate equation, one should not expect to o b t a i n accurate information on the nature of the rate expression from the biomass-time data. C s , however, may w e l l provide u s e f u l information f o r the v e r i f i c a t i o n or r e j e c t i o n of the rate expression. Unfortunately, C s data obtained under substrate l i m i t a t i o n s u f f e r from large u n c e r t a i n i t i e s . Thus d i s c r i m i n a t i o n between the various k i n e t i c models become a d i f f i c u l t task. In non-substrate l i m i t e d systems e.g., batch systems, g e n e r a l l y the rate of growth increases w i t h i n c r e a s i n g C g up to a p o i n t , t h e r e a f t e r r x remains constant e.g., l i k e s a t u r a t i o n k i n e t i c s . In t h i s case only data from the t r a n s i e n t phase from exponential to s t a t i o n a r y phase can be used f o r model d i s c r i m i n a t i o n . However, t h i s t r a n s i t i o n i s u s u a l l y q u i t e abrupt, since at the point when the r e s i d u a l s u b s t r a t e , C g, i s low (K s - C s) the high c e l l c oncentration r a p i d l y u t i l i z e s the remaining s u b s t r a t e .

In t h e i r review, Roels and Kossen'6 s t u i e d a number of unstructured models and have shown that almost any observation can be modelled by any of them. Thus the choice of the k i n e t i c expression remains to be rather a r b i t r a r y . Therefore, throughout t h i s work Monod r e l a t i o n w i l l be used without any comparative j u s t i f i c a t i o n . Two other k i n e t i c expressions w i l l be compared w i t h that of Monod :

12

T i e s s i e r Model u = U { 1 - exp(-C / K ) } max s (35)

where K = K / In 2 s

Blackman Model: u f o r max C > u A s — max

C / A f o r C < u A s s — max

(36)

Many more expressions have been reported and claimed to be superior under c e r t a i n cases. For a comprehensive l i s t recent reviews can be consulted!6,24,25,31 i n general a l l the proposed models are e m p i r i c a l or semi-e m p i r i c a l and have more or le s s the same p r o p e r t i e s . I t has r e c e n t l y been shown i n l i t e r a t u r e that most of these models can i n f a c t be ge n e r a l i z e d i n to one form, each model having d i f f e r e n t parameters.^5,31

In view of these considerations most workers favour Monod r e l a t i o n and do not give any f u r t h e r a t t e n t i o n to other r e l a t i o n s . In some cases, however, other equations might be pr e f e r r e d from the point of mathematical convenience. For instance, the use of Blackman k i n e t i c s allows a n a l y t i c a l s o l u t i o n of fed-batch models, while t h i s i s not p o s s i b l e w i t h Monod k i n e t i c s .

V DISCUSSION OF UNSTRUCTURED MODELS AND MICROBIAL ENERGETICS WITH REFERENCE TO EXPERIMENTAL RESULTS

Batch Cultures

K. •pneumoniae (aerogenes) was c u l t i v a t e d i n batch mode, to study the k i n e t i c and energetic behaviour. Inocula used were always a c t i v e l y growing and conse­quently no lags were encountered. A t y p i c a l experiment i s shown i n F i g . 1 where C s and C x p r o f i l e s are shown as fun c t i o n s of time. A d d i t i o n a l l y the si m u l a t i o n p r o f i l e s by using the Monod, Blackman and T i e s s i e r models i n combi­n a t i o n w i t h the l i n e a r r e l a t i o n f o r substrate consumption(eq.(26)), are presen­ted ( s o l i d l i n e s ) .

Fig. 1: A typical batch experiment; C^ and C g vs, time p r o f i l e s and simulataion.

13

The broken l i n e becomes a c o n t i n u a t i o n of l i n e 'a' i f the l i n e a r r e l a t i o n i s replaced by Herbert's endogeneous metabolism d e s c r i p t i o n . The parameters used f o r simualtions were obtained from continuous and batch c u l t u r e data. Parameters f o r Blackman and T i e s s i e r models were estimated roughly. Even then, F i g . 1 shows that a l l three models can describe the experimental observations i n a more or l e s s i d e n t i c a l way, provided a l l parameters are estimated w i t h equal care. The endogeneous metabolism model, i n a d d i t i o n to p r e d i c t i n g e x a c t l y the same behaviour • as Monod+linear r e l a t i o n model, describes the e a r l y decay phase w e l l . Thus as f a r as t h i s system i s concerned, Herbert's model seems to provide a more comprehensive d e s c r i p t i o n of the r e a l behaviour.

Having shown the r e l a t i v e s i m i l a r i t y of the presented models, Monod + l i n e a r r e l a t i o n model w i l l now be subjected to a s e n s i t i v i t y a n a l y s i s w i t h respect to i t s parameters. This i s a four parameter model ( K s, P m a x , Y™ x

x, mg ). In F i g s . 2,3,and 4 r e s u l t s of simulations c a r r i e d out by changing one parameter at a time, are shown.

Fiq. 2 : Sensitivity of the batch model to variations in u v J J max

t(min)

Fig. S: Sensitivity of the batch model to changes in m .

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J71CUC Fig. 4: Sensitivity of the batch model to variations in s x

The parameter values were v a r i e d around the estimated true values. The case of K s i s discussed i n Chapter 4.From these p l o t s one can c l e a r l y conclude that the k i n e t i c d e s c r i p t i o n of batch growth i n terms of C x vs. time p r o f i l e , i s not inf l u e n c e d by considerable changes i n energetic parameters, ms and Y m a x . However, the k i n e t i c parameter, y max l s shown to be of great importance. As pr e v i o u s l y discussed, t h i s parameter i s the most important f o r batch growth as i t r i g i d l y f i x e s the growth behaviour. Another conclusion can be drawn from F i g 4 i n r e l a t i o n to parameter e s t i m a t i o n . That i s , i f Y m a. x i s to be estimated from batch data,use of C s p r o f i l e would be more accurate.

Since mg has no s i g n i f i c a n t i n f l u e n c e on the outcome of batch simulations one can see that the s i m p l i f i c a t i o n of the general unstructured model (eqs.(29), (30) to (31) ) has no s i g n i f i c a n t drawbacks. Consequently, i t can be concluded that m values cannot be determined from batch data a c c u r a t e l y . More­over, i f the experimental batch data i s p l o t t e d on l o g - l i n e a r axes, one can see that a f a i r l y good s t r a i g h t l i n e i s obtained. That i s even a very simple expression l i k e that given by eq.(32) i s s u f f i c i e n t to describe most of the batch growth. Thus the use of complicated expressions l i k e that given by eq.(31) may introduce unnecessary complications p a r t i c u l a r l y i n the e s t i m a t i o n of model parameters, i n which case i m p l i c i t nonlinear r e g r e s s i o n i s necessary.

As pointed out p r e v i o u s l y , unstructured models do not consider changes i n c e l l u l a r composition. Hence they are expected to be s u c c e s s f u l at steady states or during t r a n s i e n t states where the c e l l u l a r composition remains the same. For batch growth i t has been shown that f o r the composition to remain the same, each c o n s t i t u e n t compartment i n the c e l l must grow e x p o n e n t i a l l y at the same rate as the t o t a l biomass. i . e . , steady s t a t e w i t h respect to weight f r a c t i o n s of the components. When t h i s c o n d i t i o n i s s a t i s f i e d growth i s c a l l e d balanced.32 Thus when growth i s balanced i t i s exponential too. The reverse i s however, not true i . e . , exponential growth need not be balanced.

In an attempt to check the v a l i d i t y of t h i s mathematical statement, macro-molecular composition was determined during exponential growth. As shown i n F i g . 5, v i s u a l a n a l y s i s cannot r e j e c t the hypothesis of balanced growth. However, t h i s might be a s i m p l i f i e d p i c t u r e , s i n c e , f o r instance the p r o t e i n

15

composition might change w h i l e the t o t a l measurable amount remains the same.

Fig. 5: Macromolecular composition during batch growth.

In F i g . 6, the gas exchange data f o r a d i f f e r e n t batch experiment i s shown together w i t h the s i m u l a t i o n p r e d i c t i o n s ( s o l i d l i n e s ) . Here, a discrepancy e x i s t s between the simulated and experimental behaviour towards the end of the exponential phase. The experimental data i n d i c a t e s i n c r e a s i n g Y o x and Y c x

values. No s a t i s f y i n g explanation f o r t h i s behaviour could be o f f e r e d . This i s f u r t h e r discussed i n Chapter 4.

Fig. 6: Gas exchange profiles during batch growth; solid line simulation.

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Continuous cultures

During growth i n continuous mode at steady s t a t e the biomass composition remains the same. Hence an unstructured model has a good chance of success. However, such a model a l s o assumes the b i o t i c composition to remain the same at d i f f e r e n t d i l u t i o n r a t e s . As shown i n F i g . 7 the macromolecular composition, p a r t i c u l a r l y the RNA f r a c t i o n , changes as a f u n c t i o n of the growth r a t e . Moreover, the elemental composition of biomass a l s o changes. S t a t i s t i c a l a n a l y s i s c a r r i e d out f o r 9 elemental composition determinations revealed that v a r i a t i o n i n C, H and Ncontents are s i g n i f i c a n t . Based on t h i s a n a l y s i s the elemental formula of biomass can be approximated as a f u n c t i o n of the growth r a t e , by :

C H N 0 where b = (7.33 - 0.50 y)/z 1 b c d c = (12.33 + 3.40 y)/(14z)

d = 26.97/(16z) z = (53.61 - 3.74 y)/12

For most c a l c u l a t i o n s an average formula at y = 0.5 h r _ 1 i s taken;(y x = 4.16, molecular weight = 23.16, C H „N 9->oOn ,„,)•

Fig. 7: Micvomolecular composition as a function of the growth rate, w 3 RNA = 0.11 wV

n 3 Protein = 0.71 W]i~0 ' CaJb°Hld-Ta~te = 0.065

The s e n s i t i v i t y a n a l y s i s presented f o r batch growth model w i l l not be repeated f o r continuous c u l t i v a t i o n . The conclusion w i l l simply be stated as: growth behaviour described by the general unstructured model, except near the wash­out region, i s r i g i d l y f i x e d by the energetic parameters, Y ™ X

X > ms -

Data obtained from a continuous c u l t u r e experiment were f i t t e d by the l i n e a r r e l a t i o n , as shown i n F i g s . 8 and 9. These p l o t s i n d i c a t e that growth i n continuous c u l t u r e can indeed be described by the presented model.Fig.8 shows a good s t r a i g h t l i n e f i t . However, small but d i s t i n c t d e v i a t i o n s can be seen fo r data at low growth r a t e s .

17

Fig. 8: Specific rate of svbstrate consumption as a fuction of the growth rate.

Fig. 9: Specific OUR and CPR as fuations of the growth rate.

The energetic parameters may be obtained from continuous c u l t u r e data by per­forming l i n e a r r e g r e s s i o n v i a the use of equation:

q = y I Y m a X + m (38) s sx s

or by performing nonlinear r e g r e s s i o n v i a the use of the f o l l o w i n g equation:

Y = y Y m a x / ( y + m Y m a x ) (39) sx sx s sx

If the experimental measurements are er r o r f r e e , both methods should give e x a c t l y the same parameters. I f there are associated e r r o r s these approaches may r e s u l t i n d i f f e r e n t parameter estimates. An a n a l y s i s of the two procedures was c a r r i e d out and i t was found out that i n a l l three cases ( f o r s u b s t r a t e , oxygen, carbon d i o x i d e data vs. growth rate) nonlinear r e g r e s s i o n gave a b e t t e r f i t f o r the experimental data. This has been assessed by c a l c u l a t i n g the scaled sum of residuals (see Table I ) . The d i f f e r e n c e might stem from the fa c t that q s i s not a d i r e c t l y measurable q u a n t i t y , but i s c a l c u l a t e d from q s = y / Y s x . Thus i t may have a d i f f e r e n t e r r o r s t r u c t u r e . Moreover, i n a q s vs. y p l o t one has y i m p l i c i t l y i n both axes and t h i s may be q u i t e undesi­r a b l e from a mathematical point of view. The dangers of t h i s e x e r c i s e i . e . , i n c l u d i n g the same v a r i a b l e i n both axes i s discussed by Himmelblau.33

The experimental data have f i r s t been i n d i s c r i m a n e n t l y processed by l i n e a r and nonlinear r e g r e s s i o n procedures. The r e s u l t s are presented i n Table I .

If the r e s i d u a l s are examined, one can detect a trend ( F i g . 8,9). This i s par­t i c u l a r l y apparent i n the q c vs. y p l o t . Here the r e s i d u a l s change t h e i r s i gn

18

Table I: Parameter estimates obtained from continuous and batch c u l t u r e data.

Continuous Culture data: Nonlinear Regression Linear Regresión

A l l data p o i n t s , n=27 / ( Z i e e r r n t i i r n

Y m a x 0.698 (0.869 - 0.707) 0.710 (0.701 - 0.720) sx Y ™ a x 1.583 (1.542 - 1.624) 1.613 (1.570 - 1.659) Y m a x 2.391 (2.289 - 2.493) 2.465 (2.367 - 2.570) cx m 3.342 (3.059 - 3.623).10" 2 4.072 (3.290 - 4.852).10 - 2

s m 3.740 (3.365 - 4.115).10 - 2 4.065 (3.373 - 4.757).10 - 2

o m 3.668 (3.177 - 4.159).10 _ 2 3.998 (3.321 - 4.675).10 - 2

Data c o l l e c t e d above , y > 0.1, n=21

Y m a x 0.710 (0.699 - 0.721) 0.7* 0.719 (0.706 - 0.732) 1.4* sx Y m a x 1.620 (1.563 - 1.677) 2.5* 1.640 (1.570 - 1.701) 3.0* ox

Y ™ a x 2.513 (2.376 - 2.650) 5.3* 2.561 (2.428 - 2.709) 6.8* ms 4.241 (3.627 - 4.855).10 - 2 4.997 (3.846 - 6.148).10 - 2

m0 4.320 (3.627 - 5.012).10~ 2 4.609 (3.517 - 5.701).10~ 2

mc 4.530 (3.756 - 5.305).10 - 2 4.819 (3.835 - 5.803).10~ 2

Thermodynamic e f f i c i e n c y versus growth r a t e data, y> 0.1, n=63

Y m a x 0.700 (0.693 - 0.707) m g

X 4.146 (3.661 - 4.631). lö~1,B errrttlJ

Batch Culture data (average of three experiments):

Y™£ x 0.705 a l l Y m a x i n C-eq/C-eq, C-eq/mole Y™|x 1.544 a l l m i n c-eq/C-eq/hr, mole/C-eq/hr Y ™ a x 2.313

* scaled sum of r e s i d u a l s x 10 2

f i g u r e s i n parentheses are the 95 % confidence l i m i t s .

i n the time sequence only three times. Whereas i f they had been randomly d i s t ­r i b u t e d the expected number of s i g n change would have been (n-I)/2= 13. The d i s t i n c t d e v i a t i o n at low y can be thought to be due to the reduced v i a b i l i t y of the organisms. Since above y=0.1, v i a b i l i t y i s more than 95 %^4 a n o t h e r set of parameters were obtained only by processing data c o l l e c t e d above y=0.1, (n=21, see Table I ) . Note that there are s i g n i f i c a n t d i f f e r e n c e s i n the m value obtained by the two procedures. For the second s e t , the goodness of f i t i s a l s o shown f o r l i n e a r and nonlinear r e l a t i o n s . For data above y=0.1 r e s i d u a l s of the q s vs. y r e l a t i o n changes sign 11 times, q Q vs. y,9 times and q c vs. y, 6 times ( ( n - l ) / 2 = 10 ). Thus w i t h the exception of q c data,

19

e x c l u s i o n of data c o l l e c t e d at very low growth r a t e s , restored the l i n e a r i t y of the data i n r e l a t i o n to the l i n e a r law. Thus i t can be concluded that above p=0.1 the l i n e a r r e l a t i o n i s a reasonable d e s c r i p t i o n of the continu­ous c u l t u r e e n e r g e t i c s .

As described p r e v i o u s l y Y or m values c a l c u l a t e d from one experimental respon­se can be converted to one based on another, by the use of macroscopic methods Thus the most optimal estimate of the parameters, Y and m can be obtained by consid e r i n g data obtained from a l l responses, simultaneously i . e . , when every measurement c o n t r i b u t e s to the r e s u l t . This can be done by processing I] vs. H data, where r| i s c a l c u l a t e d from Y s x , Y o x and Y c x data (n=63) . The parameters obtained i n t h i s way are a l s o given i n Table I . From t h i s t a b l e i t can be seen that the 95 % confidence l i m i t s of m values are q u i t e l a r g e when compared w i t h those of Y m a x values. Since these parameters are determined simultaneously, t h e i r estimates can be c o r r e l a t e d . A b e t t e r p i c t u r e can be obtained about the accuracy of these parameters by c a l c u l a t i n g t h e i r approximate locus of the j o i n t confidence l i m i t s ( F i g . 10). From t h i s f i g u r e i t can be seen that the estimates of Y ™ x

x and mg are s l i g h t l y c o r r e l a t e d ; the p r i n c i p a l axes of the e l l i p s e are at an angle to the coordinate axes. I t has to be emphasized that the estimates of Y m

xx and ms may l i e outside t h e i r i n d i v i d u a l confidence

levels.From t h i s d i s c u s s i o n i t can be concluded that the value of Y m g x can be determined w i t h reasonable c e r t a i n i t y , w h i l e that of ms can only be deter­mined w i t h a large u n c e r t a i n i t y . Wide confidence l e v e l s r e s t r i c t s one to draw f i r m conclusions from experimental work concerning maintenance metabolism. Such wide confidence l e v e l s may be one of the reasons f o r the wide range of m values reported i n l i t e r a t u r e , f o r the same or s i m i l a r system(s).

With reference to Table I , as f a r as the maximal y i e l d s are concerned, one can al s o conclude that there i s no s i g n i f i c a n t d i f f e r e n c e between the energetics of m i c r o b i a l growth i n batch and continuous modes.

Fig. 10: The loons of the joint confidence limits for energetic parameters as determined by the linear relation for substrate consumption; sq.(S&). (for data shewn in Fig.8)

20

VI A DISCUSSION ON THE CONCEPT OF MAINTENANCE

S p e c i f i c maintenance functions are now believed to in c l u d e : turnover of c e l l 1 m a t e r i a l s , osmotic work to maintain concentration g r a d i e n t s , c e l l m o t i l i t y , membrane e n e r g e t i z a t i o n etc. When the l i n e a r r e l a t i o n i s assumed to be v a l i d , a l l non-growth associated energy expenditure i s a u t o m a t i c a l l y assumed to have been channelled to maintenance metabolism. ms i s a l s o assumed to be constant. However, some of the maintenance functions may w e l l be influ e n c e d by the growth r a t e e.g., i t i s known that the surface area to volume r a t i o of b a c t e r i a i s a f u n c t i o n of the growth r a t e . At low growth rates the r a t i o i s high, thus the organism i s expected to spend more energy to keep the undesir­able solutes out. Therefore such maintenance requirements may be expected to be growth rate dependent. However, i n a number of systems the l i n e a r r e l a t i o n seems to give a good approximation over a wide range of growth r a t e s . That i s , constant maintenance hypothesis i s d i f f i c u l t to r e j e c t . However, one must note that s t r a i g h t l i n e s can be obtained even i f the ms i s a f u n c t i o n of y but the o v e r a l l e f f e c t i s too small or the a c t u a l phenomenon i s i n t e r a c t i n g with others, e.g., i f the e f f i c i e n c y of o x i d a t i v e phosphorylation or the degree of coupling are al s o functions of the growth r a t e , i t would be impossib­l e to f i l t e r out conclusions regarding the v a r i a t i o n of maintenance demands.

An observation shared by many workers i s the s i g n i f i c a n t systematic d e v i a t i o n s observed from the l i n e a r r e l a t i o n at low y ( t h i s work,34,35) _ j f Y m a x can be assumed to be a b i o l o g i c a l l y meaningful parameter and as a constant, the above mentioned d e v i a t i o n s imply reduced maintenance requirements at low growth r a t e s . This e f f e c t can be a t t r i b u t e d to phenotypic changes (since the time to reach steady state at low y i s very long) and/or loss of v i a b i l i t y . The pre­sented data are not s u f f i c i e n t f o r an accurate assessment of t h i s observation. F o r t u n a t e l y , i n l i t e r a t u r e a set of data c o l l e c t e d at very low y has been reported.34 The data was processes and p l o t t e d as q s vs. y i n F i g . 11. A s i g n i f i c a n t d e v i a t i o n can be seen below y=0.06. Since the v i a b i l i t y data were als o reported , the f o l l o w i n g a n a l y s i s can be c a r r i e d out: Assuming that only v i a b l e c e l l s consume s i g n i f i c a n t q u a n t i t i e s of substrate, the q s values can be corrected f o r v i a b l e c e l l s , as shown by hollow c i r c l e s i n the F i g . 11. From t h i s f i g u r e i t i s c l e a r that v i a b i l i t y alone, cannot account f u l l y f o r the observed d e v i a t i o n s at low growth r a t e s . The r e a l p i c t u r e may a l s o include dormant c e l l s , which show up v i a b l e when c u l t u r e d i n r i c h media. I t i s als o i n t e r e s t i n g that the d e v i a t i o n from the l i n e a r r e l a t i o n does not occur over the e n t i r e y range but develops i n a small range.

Fig. 11: q versus V data:(recalculated from ref: 34).

21

Recently Pipyn and Verstraete35 have reported that i n waste water systems ms values determined are about 10 times lower than those f o r l a b o r a t o r y pure c u l t u r e systems. Based on t h i s observation and experiments they have reasoned that the explanation must be sought i n the f a c t that ms decreases w i t h dec­reasing \l values. An observation s i m i l a r to that of Pipyn and V e r s t r a e t e was also made during t h i s study. For a c t i v a t e d sludge c u l t i v a t i o n i n f i l l and draw mode, a maintenance c o e f f i c i e n t of 5.Ox 10 _3 (C-eq/C-eq/hr) was c a l c u l a t e d . This

Zie erratumvalue i s about 10 times higher than that obtained from a continuous c u l t u r e study with a mono c u l t u r e (4.15xl0~2). S i m i l a r observations have al s o been reported by Gaudy and Gaudy.36

In l i t e r a t u r e sometimes remarkable claims concerning the maintenance concept are made. For instance, i t has been claimed that maintenance requirements could be made zero by merely manipulation of the medium composition.37 This claim i s discussed i n d e t a i l i n Chapter 8. However, a simple experiment against zero maintenance c l a i m s , w i l l be c i t e d here. In 1970 Gaudy et a l . ^ reported an experimental a c t i v a t e d sludge system i n which a l l the b i o l o g i c a l s o l i d s were recy c l e d back to the a e r a t i o n tank a f t e r being separated by a c e n t r i f u g e , continuously. During the 2 nd and 3 rd years of operation the amount of biomass i n the system and i t s composition remained more or l e s s constant and the system r e t a i n e d i t s 90 % COD removal capacity i . e . , no growth but substrate uptake. This type of substrate expenditure i s by d e f i n i t i o n f o r maintenance metabolism. I n f a c t one need not to perform experiments to confirm the presence of non-growth associated energy expenditure. In thermodynamics i t has long been known that energy i s needed to keep an open system i n i t s ordered s t a t e . This f a c t was i t e r a t e d Schrodinger 39 f o r b i o l o g i c a l systems as e a r l y as 1944. Schrodinger wrote ( c i t ) " l i v i n g matter evades e q u i l i b r i u m by feeding upon i t s negative entropy produced by i t s metabolism (Greek word f o r exchange)" or l e s s p a r a d o x i c a l l y " the organism succeeds i n f r e e i n g i t s e l f from the entropy i t cannot help producing to remain a l i v e " .

VII NOMENCLATURE

A constant i n Blackman equation ATP adenosinetriphosphate C concentration (kg/m3) (C-eq/m^) C H k j O ^ N j j biomass elemental formula

<-'a2^b2^C2Nd2 s u b s t r a t e elemental formula

Ca^H^O^N^ product elemental formula K s Monod s a t u r a t i o n constant (kg/m^) (C-eq/m-*) K constant i n T i e s s i e r equation D d i l u t i o n r a t e ( h r - ' ) m maintenance c o e f f i c i e n t (C-eq/C-eq/hr) (mole/C-eq/hr) r^ r a t e of consumption or production of the i ' t h component

(C-eq/m3/hr) (mole/m3/hr) q i s p e c i f i c rate of consumption or.production of the i ' t h component

(C-eq/C-eq/hr) (mole/C-eq/hr) Y^ x y i e l d of biomass on the i ' t h component (C-eq/C-eq) (C-eq/mole) ymax Y £ x corrected f o r maintenance (C-eq/C-eq)(C-eq/mole) Y Sp y i e l d of product on substrate (C-eq/C-eq) W£ weight f r a c t i o n of the i ' t h component

sub s c r i p t s x biomass 22

p N g

c o

s substrate product n i t r o g e n source growth associated oxygen carbon dioxide

Greek symbols degree of reduction as defined by eq.(11)—(13) thermodynamic e f f i c i e n c y of the growth process (dimensionless) net flow of component i to the system (kg/m3/hr) (C-eq/m^/hr) maximum p o s s i b l e y i e l d f o r the d e f i n i t i o n of n (C-eq/C-eq) s p e c i f i c growth r a t e (hr~1) endogeneous metabolism r a t e constant ( h r - ' ) maximum growth r a t e i n the absence of endogeneous metabolism ( h r - ' )

V I I I REFERENCES

1. J. Monod, Recherches sur l a Croissance des Cultures Bacteriennes, 2 nd ed., (Hermann, P a r i s ) ( 1 9 4 2 ) .

2. D. Herbert, Symp. I n t . Congr. M i c r o b i o l . , 6,381(1958), 3. D. Herbert, Continuous Culture,6,1(1976) . 4. S.J. P i r t , Proc. Roy. Soc. Londob B, 163,224(1965). 5. A.H. Stouthamer and C.W. Bettenhaussen, Biochim-Biophys . Acta,301 ,53 (1 973) . 6. A.H. Stouthamer and C.W. Bettenhaussen, Arch. Microbiol.,11,21(1976). 7. H.W. van Verseveld, Ph.D t h e s i s , Free U n i v e r s i t y of Amsterdam,(1979). 8. J.C. Senez, B a c t e r i o l . Rev.,26.95(1 962) . 9. O.M. N e i j s s e l , Ph.D. t h e s i s , U n i v e r s i t y of Amsterdam,(1 977).

10. O.M. N e i j s s e l and D.W. Tempest, Soc. Gen. M i c r o b i o l . Symp.,29,53(I 979). 11. D.W. Tempest, Paper presented to the 2 nd Eur. Symp. on B i o t e c h n o l . held at

Eastbourne, England (1981). 12. A.H. Stouthamer, Symp. Soc. Gen. Microbiol.,27,285(1977). 13. A.H. Stouthamer, I n t . Rev. Biochem. M i c r o b i o l . Biochem., 21(1979). 14. A.H. Stouthamer, Y i e l d Studies i n Microorganisms,(Meadowfield, Durham)1976. 15. S. Nagai, Advances i n Biochemical Engineering,11,48(1979) . 16. J.A. Roels and N.W.F. Kossen, Progress i n I n d u s t r i a l M i c r o b i o l o g y , M.J.

B u l l , ed. ( E l s e v i e r , Amsterdam 1978)vol.14,p.95. 17. J.A. Roels, B i o t e c h n o l . Bioeng.,22,2457(1980). 18. L.E. E r i c k s o n , I.G. Minkevich and V.K. E r o s h i n , B i o t e c h n o l . Bioeng.,20,1595

(1978). 19. L.E. E r i c k s o n , S.E. Selga and U.E. V i e s t u r s , B i o t e c h n o l . Bioeng.,20,1623

(1978). 20. L.E. E r i c k s o n , B i o t e c h n o l . Bioeng.,21,725(1979) . 21. J . J Heijnen and J.A. Roels, B i o t e c h n o l . Bioeng.,23 ,739(1 981) . 22. J.G. Minkevich and V.K. E r o s h i n , F o l i o Microbiol.,18,376(I 973). 23. A.E. Humphrey and P.R. J e f f e r i s , IV GIAM meeting, Sao Paulo, B r a z i l ( 1 9 7 3 ) . 24. A.G. Frederickson and H.M. Tsuchiya, i n Chemical Reactor Theory, Lapidus

and Amundson, eds.,(Prentice Hall,New Jersey, 1977)p.405. 25. M.A. Savageau, Math. Biosci.,48,267(1980). 26. I . P r i g o g i n e , Thermodynamics of i r r e v e r s i b l e processes(John Wiley, New York

1955). 27. E.O. Powell, i n M i c r o b i a l Physiology and Continuous c u l t u r e , Powell et a l

eds.,(H.M. S t a t i o n a r y O f f i c e London, 1967)p.34. 28. J.A. Roels, Biochemical Engineering; k i n e t i c s and e n e r g e t i c s , a book to be

published by E l s e v i e r , Amsterdam.

23

29. G. T i e s s i e r , Rev. S e i . E x t r a i t du 3208,209(194-2). 30. F.F Blackman, Ann. Bot.,19,281(1905). 31. F. Kargi and M.L. Schuler, B i o t e c h n o l . Bioeng.,21,1871(1979). 32. A.G. Frederickson, R.D. Megee and H.M. Tsuchiya,Adv. Appl. M i c r o b i o l . , 1 3 ,

419(1970). 33. D.M. Himmelblau, Process A n a l y s i s by S t a t i s t i c a l Methods, (John Wiley, New

York, 1970). 34. D.W. Tempest, D. Herbert and P.J. Phipps, i n M i c r o b i a l Physiology and

Continuous Culture (H.M. S t a t i o n a r y O f f i c e , London, 1967)p.240. 35. P.Pipyn and W. V e r s t r a e t e , B i o t e c h n o l . Bioeng.,20,1883(1978). 36. A. Gaudy and E.Gaudy , Microbiology f o r Environmental S c i e n t i s t s and Engi­

neers , (McGraw Hill,New York, 1980). 37. S. Cromie and H.W. D o e l l e , B i o t e c h n o l . Lett.,2(8),357(1980). 38. A. Gaudy, P.Y. Yang and A.W. Obayashi, J . Wat. P o l l u t . Control Fed.,43,40,

(1971). 39. E. Schrodinger, What i s L i f e ? (Cambridge U n i v e r s i t y Press, Cambridge,1944)

p.75.

24

CHAPTER 3

MATERIALS AND METHODS

I DESCRIPTION OF THE EXPERIMENTAL SYSTEM AND ANALYTICAL METHODS

Experimental set-up

A l l batch and fed-batch experiments with mono c u l t u r e s were performed i n a B i o l a f i t t e 15 S b i o r e a c t o r with 11.10 _3 m 3 working volume. Continuous c u l t u r e experiments were c a r r i e d out i n a New Brunswick reac t o r w i t h 3 . 10 _3 m3 working volume. In a l l cases, except otherwise s t a t e d , pH and temperature were set and c o n t r o l l e d at 6.8 ± 0.05 and 308 ± 0.5 K, r e s p e c t i v e l y .

A c i d / a l k a l i added f o r pH c o n t r o l was monitored by a Servo-chem Dose monitor DM1 and recorded. Data obtained a s s i s t e d the v e r i f i c a t i o n of steady states and the c o r r e c t i o n of the d i l u t i o n r a te during continuous c u l t i v a t i o n and p a r t i c u l a r l y f o r the c o r r e c t i o n of volume balances i n batch and fed-batch experiments.

A i r flow rate to the fermentor was c o n t r o l l e d by a thermal mass flow meter (Brooks 5811) at about 60 m3/m3/hr at STP. The e r r o r introduced by the mass flow meter was determined by checking i t against a p r e c i s i o n wet gas flow meter and was found to be l e s s than 3%. The gas stream out of the fermentor was passed through a condensor which had r e f r i g e r a n t c i r c u l a t i o n at 268 K on the c o o l i n g s i d e . For t r a n s i e n t experiments the o u t l e t gas stream to the carbon d i o x i d e analyser could not be passed through s i l i c a g el d r i e r s due to the a f f i n i t y of CO2 to s i l i c a g e l (adsorption/desorption) which d i s t o r t s the observed dynamic response of the system. Therefore, the o u t l e t gas stream was d r i e d by Permeation Distillation technique which introduced v i r t u a l l y no time lags or other i n t e r f e r i n g e f f e c t s (Perma Pure D r i e r PD-750-24P). Gas phase oxygen concentration was measured by a twin channel paramagnetic oxygen analyser (Taylor Servomex OA 184) coupled to a r a t i o box, which measured the concentration i n the o u t l e t gas stream as a f r a c t i o n of the i n l e t concentration. This c a p a b i l i t y , e liminated the n e c e s s i t y f o r c o r r e c t i o n s due to changes i n the atmospheric pressure. Carbon d i o x i d e concentration was measured by an i n f r a r e d analyser (Beckman 865). A l l tubes used f o r gas t r a n s p o r t a t i o n were made of e i t h e r b u t y l rubber or aluminium, to avoid i n t e r f e r e n c e s by d i f f u s i o n of gases i n and/or out. Experiments w i t h a c t i v a t e d sludge were performed mostly i n a l e s s s o p h i s t i c a t e d 8. 10 _3 m3 working volume New Brunswick fermentor operated i n batch , f i l l and draw or continuous modes. These experiments were c a r r i e d out at a temperature of 293 K and pH of 6.8 .

25

Organism

For mono -culture experiments Klebsiella pneumoniae NCTC 418; formerly known as Klebsiella aerogenes, was used throughout t h i s study. For mixed c u l t u r e expe­riments a c t i v a t e d sludge was obtained form the P i l o t P l a n t Operated by the Department of C i v i l Engineering of the D e l f t U n i v e r s i t y of Technology. This sludge was f i r s t a c c l i m a t i z e d to g l y c e r o l as the sole carbon and energy source. A f t e r eight weeks of adaptation a w e l l s e t t l i n g yellowish-brown sludge was obtained.

Cultivation Methods

Growth medium was prepared according to the f o r m u l a t i o n given by Evans et a l ' . G l y c e r o l was used as the l i m i t i n g substrate i n a l l cases (except f o r wash-out experiments where l a c t i c a c i d was the s u b s t r a t e ) . Water a c t i v i t y of the stan­dard medium was about 0.996. For a c t i v a t e d sludge c u l t u r e s the b a s a l medium rec i p e was taken from Harder.^

I n i t i a l substrate concentration was 10 kg/m3 for batch and continuous c u l t u r e experiments and adjusted according to the f i n a l d e s i r e d biomass concentration and feeding p r o f i l e i n fed-batch experiments. The medium was s t e r i l i z e d by membrane f i l t r a t i o n through a 0.2 micron pore diameter membrane (S a r t o r i u s 11307). This circumvented complications causedby heat s t e r i l i z a t i o n l i k e p r e c i p i t a t i o n , pH d r i f t , evaporation e t c . I n t e g r i t y of the s t e r i l i z e d f i l t e r was tested p r i o r to each f i l t r a t i o n by the Bubble point test.

A t t e n t i o n was paid to o b t a i n an a c t i v e l y growing inoculum and i n most cases the inoculum used was very small to avoid the p o s s i b i l i t y of unbalanced growth?

A n a l y t i c a l Methods

Substrate : Reagent q u a l i t y g l y c e r o l (BDH Chemicals) was used and assayed enzymically. (Boehringer UV method, 14270) Detection l i m i t of the assay was estimated to be about 10 .10"^ kg/m3. In general e r r o r s of the order of 5 % or l e s s were experienced. This value included e r r o r c o n t r i b u t i o n s of sampling and i n t e r f e r e n c e s from other components present i n the c u l t u r e supernatant. When very low concentrations have to be estimated the method of Standard Additions was used to increase the r e l i a b i l i t y of the a n a l y s i s . Samples were cooled down to about 278-280 K while being taken, by an o n - l i n e exchanger designed and manufactured inour department. T y p i c a l residence time i n the exchanger was about 5-10 seconds.

Nitrogen : Nitrogen content of the feed/medium and the c u l t u r e supernatant was determined by an auto K j e l d a h l - N i t r o g e n analyser.

TC, T0C : O c c a s i o n a l l y T o t a l carbon (TC) and T o t a l organic carbon (TOC) mea­surements were performed to check the accuracy of the enzymic assay and a l s o to look f o r the presence of unexpected C-containing compounds.

Dry Weight: Dry weights were determined as described by de V r i e s and Stouthamer.* In continuous c u l t u r e experiments w i t h C x i n the order of 3-5 kg/m3, the average e r r o r i n dry weight d i d not exceed 2 %. In dynamic e x p e r i ­ments, p a r t i c u l a r l y i n batch runs, the e r r o r s involved were considerably higher.

26

Biomass Analysis

Elemental Composition : Three elements were determined i n the twice washed and d r i e d biomass powder by a computer coupled element analyser (C, N, H), (Perkin Elmer 240). Ash content was determined separately and a l l r e s u l t s r e ­ported i n t h i s t h e s i s are expressed on Ash Free Basis. Oxygen content was c a l c u l a t e d from the d i f f e r e n c e . The consistency of the method was checked and found to be good by comparing with the r e s u l t s of an independent lab o r a t o r y s p e c i a l i z e d i n these analyses, f o r the same samples.

RNAj DNA : RNA and DNA contents of the freeze d r i e d biomass were determined a f t e r e x t r a c t i o n w i t h p e r c h l o r i c a c i d , by the o r c i n o l and diphenylamine proce­dures, r e s p e c t i v e l y , as described by Herbert et a l . 5

Protein : C e l l p r o t e i n content was estimated by the B i u r e t method as described by Herbert et a l .

Total Carbohydrate : Anthrone method, as described by Herbert et a l . 5 was used.

A n a l y s i s of the macromolecular composition of freeze d r i e d biomass was one of the most tedious and l e a s t accurate a n a l y s i s reported i n t h i s work, because of the d i f f i c u l t i e s w e l l documented elsewhere.-* Therefore these r e s u l t s must be treated w i t h care. P r e c i s i o n was good w i t h i n the samples analysed i n one s e t . Between s e t s , however, the same p r e c i s i o n could not be achieved.

I I DEVELOPED AND USED TOOLS AND METHODS

Quantification of Gas Exchange

Assumptions: a. Steady s t a t e operation w i t h respect to the concerned gases. b. No CO2 and/or N2 f i x a t i o n . c. Only O2 and CO2 exchanged. fermentor

condensor mass flow meter a i r input pump perma pure d r i e r humidity measurement s i l i c a g e l column dry gas generation c y c l e O2 analyser r e f e r e n ­ce channel O2 analyser sample channel s i l i c a g e l d r i e r s CO2 analyser heat exchanger sampling port

s. sample gas l i n e r. reference gas l i n e

Fig. 1: Experimental set-up; gas processing and analysis flow diagram.

27

With reference to F i g . 1, where the gas flows are i n d i c a t e d , a balance on gaseous N2 , which passes through the system unchanged, provides the s t a r t i n g p o i n t . Therefore, a mass balance reads:

E N„ - i n = Z N„ - out and N^ n (b. = N ° U t i> 2 2 2 i n 2 out or . A A

N 2 n = *out ( 1 - C ° 2 - ° 2 > / * i n ( 1 )

Here note that <|>in and '('out r e f e r to dry volumetric gas flows at STP and that they are not n e c e s s a r i l y equal to each other except when RQ equals one. In p r a c t i c a l systems u s u a l l y <j>£n i s measured and/or c o n t r o l l e d , therefore ( J ) o u t

can be c a l c u l a t e d i f the exhaust gas stream i s analysed f o r O2 and CO2 , i . e ,

*out = * i n ' ( 1 " C°2 - °2 > (2)

Once the outflow '('out '""s c a i c u l a t e d , OUR, CPR can be e a s i l y c a l c u l a t e d by the expressions ;

OUR = ( 0.21 6. - 4> ) / 0.0224 (mole 0„/m3/hr) (3) i n 2 out 2

CPR = ( CO^ <f> - CO" <}). ) / 0.0224 (mole C0„/m3/hr) (4) 2 out 2 i n 2

Then RQ can be given by :

RQ = CPR / OUR (5)

For f r e s h a i r , N|n , 0^ n , and CO^1 can be taken as 0.79, 0.21 and 0.033, r e s ­p e c t i v e l y . In the v i c i n i t y of our labo r a t o r y COj 1 1 was found to be around 0.0360. From eq.(2) i t f o l l o w s that at each moment ^out w i l l depend on the extent of gas exchange and f o r the sake of accuracy i t has to be c a l c u l a t e d f o r each measurement po i n t . During t h i s study a simple computer program was developed which, when provided w i t h the raw gas exchange data, c a l c u l a t e d tyout r o r e a c n point and OUR, CPR and RQ, subsequently. The program a l s o i n t e g ­rated the OUR and CPR p r o f i l e s n u merically to f i n d the cumulatives required f o r balances and y i e l d c a l c u l a t i o n s .

In the a n a l y s i s presented above the gas flows were assumed to be p r a c t i c a l l y dry. I f t h i s cannot be assumed, the water vapour content of the flows can be estimated by measuring the gas dry and wet bulb temperatures and using standard c o r r e l a t i o n s . 6 , 7

Volume Balancing for Fermentation Processes

Volume changes do occur i n fermentation systems. U s u a l l y such changes are con­sidered to be i n s i g n i f i c a n t f o r batch and continuous c u l t i v a t i o n systems. In fed-batch systems, volume changes can be expected to be s i g n i f i c a n t and there­fore an equation d e s c r i b i n g the change i n the volume of the c u l t u r e i s r e ­quired. In the f o l l o w i n g , the general case, capable of d e s c r i b i n g any mode of operation, w i l l be considered.

For a b i o l o g i c a l system, r e l e v a n t to t h i s study, the f o l l o w i n g diagram defines the major input and output flows. 28

Of feed O carbon dioxide feed O O oxygen

oxygen Ckg/m3/hr) effluent 1 •

<l>c <Dgut

<De

Fig. 2: Major input and output flaws for a fermentation process.

In a d d i t i o n to the flows i n d i c a t e d , one may think of a c i d / a l k a l i a d d i t i o n asso­c i a t e d w i t h the pH c o n t r o l system, net water vapour flow and loss of organic vapours v i a the gas phase. These flows, however, under c a r e f u l l y designed and conducted experiments, can be minimized to l e v e l s which would i n s i g n i f i c a n t compared w i t h the s o - c a l l e d major flows.

Mass, to a very good approximation, i s a conserved quantity i n the absence of nuclear transformations. Therefore, a t o t a l mass balance f o r the system p r o v i ­des a convenient s t a r t i n g point f o r the present d e r i v a t i o n . The general mass balance reads:

n d (mass)/ dt = Z Net transport of mass (6)

i For the system described i n F i g . 2 t h i s equation gives;

d (mass)/ dt = $ p + - $ e - $ C 0 2 - ( ? )

Since <PQ2 - $§2' defines the net oxygen uptake and ®C02 t' i e c a r ^ o n d i o x i d e production r a t e (kg/m^/hr), f o r these c o n t r i b u t i o n s the more g e n e r a l l y used molar flows can be s u b s t i t u t e d i f the accumulation of CO2 and O2 i n the l i q u i d phase can be assumed to be n e g l i g i b l e compared w i t h t h e i r production and consumption terms,respectively. Moreover, s u b s t i t u t i n g p s V s f o r the t o t a l mass the f o l l o w i n g form i s obtained:

d ( p V )/ dt = * - $ + 0.032 r - 0.044 r (8) s s F e o c

where $ i s the mass flow (kg/m^/hr) r a t e and r , the net molar flow r a t e (mole/ m-Vhr). P s (kg/m3) and V s (m3) are the system d e n s i t y and volume,respective­l y . Expanding the d i f f e r e n t i a l and s u b s t i t u t i n g the r e l a t i o n ;

RQ = r / r (9) c o

we ob t a i n the r e l a t i o n ;

dV /dt - {$ - $ + r (0.032/RQ - 0.044) - V (dp / d t ) } / p (10) s F e c s s s

Although t h i s equation does not include a l l of the p o s s i b l e c o n t r i b u t i o n s to volume change, e.g. volume of d i l u t i o n e t c . , i t includes the most s i g n i f i c a n t f a c t o r s and c l e a r l y shows that the volume change during a c u l t i v a t i o n process i n any mode, i s not only influenced by the tra n s p o r t of matter i n l i q u i d phase as u s u a l l y f i x e d by the experimenter, but a l s o by exchange i n gas phase, the magnitude of which depends e n t i r e l y on the metabolic state'of the b i o l o g i c a l

29

System as r e f l e c t e d by i t s RQ value. Further s i m p l i f i c a t i o n of the above equa­t i o n can be obtained by assuming the time d e r i v a t i v e of p s as zero. This assumption i s j u s t i f i a b l e i n most p r a c t i c a l cases. Considering the case f o r continuous c u l t i v a t i o n , both time d e r i v a t i v e s can be set to zero f o r steady s t a t e operation. This a c t u a l l y means that at steady s t a t e the incoming and outgoing flows would not be equal to each other on volume b a s i s and that the d i f f e r e n c e would be a f u n c t i o n of the metabolic s t a t e . For aerobic growth with moderate RQ values the gas exchange term i s u s u a l l y a small f r a c t i o n of the l i q u i d flows and hence can be neglected f o r most purposes. However, the cont­r i b u t i o n of the gas exchange can be s i g n i f i c a n t f o r some anaerobic processes?

The Declining Feed Fed-Batch System

Modelling and experimental study of m i c r o b i a l growth i n fed-batch mode led to the development of a powerful novel technique for the determination of mainte­nance c o e f f i c i e n t . This method w i l l be r e f e r r e d to as the Declining Feed, Fed-Batch Technique.9 The method can be described with the a i d of the f o l l o w i n g f i g u r e ( F i g . 3).

Fig. 3: Schematic representation of the Declining Feed, Fed-Batch Technique.

The procedure i s as f o l l o w s : i . S t a r t as batch or maintain exponential growth and determine q s , P max

and Y s x f o r the exponential phase. i i . Continue as fed-batch; decrease the r a t e of substrate a d d i t i o n according

to a p r e - f i x e d p a t t e r n , say l i n e a r l y . This w i l l b r i n g the system to zero growth rate s t a t e , at l e a s t momentarily.

i i i . C a l c u l a t e ms from a mass balance d i r e c t l y , i . e . , from F(t)= mg M x

when dM x/dt = 0 (see the model presented i n Chapter 4.) i v . Having determined q s , y m a x , Y s x and mg c a l c u l a t e Y m

xx from :

Y m* x = y / ( q - m ) (11) sx max s s

The i m p l i c i t assumption i n t h i s c o n s i d e r a t i o n i s that when dMj^/dt =0, dM g/dt can a l s o be assumed to be zero. This assumption i s q u i t e j u s t i f i a b l e i f one considers the time constants of the major processes involved. Since the time constant of the b i o s y n t h e t i c process i s much l a r g e r than that of the substrate

30

consumption process, dM s/dt can be taken as zero whenever dM x/dt approximates to zero. Supporting evidence f o r t h i s reasoning has been provided by Minkevich and Utkina'O. Even f o r conservative changes i n the growth r a t e , provided the c u l t u r e i s substrate l i m i t e d , dM s/dt w i l l be n e g l i g i b l e . I f , however, there i s s t i l l appreciable substrate l e f t i n the fermentor, then the net flow of substrate to the m i c r o b i a l system must be determined. This might happen when, for instance, growth ceases due to the accumulation of i n h i b i t o r s i n the system.

The technique has f i r s t been t r i e d w ith constant feeding i . e . , F ( t ) = constant, and was found to be u n s a t i s f a c t o r y . Cultures fed w i t h constant supply kept on growing at a very small but s i g n i f i c a n t growth r a t e f o r long periods of time during which the c e l l morphology and v i a b i l i t y changed considerably. Under these c o n d i t i o n s i t becomes extremely d i f f i c u l t to i n t e r p r e t the data obtained and apply the conventional r e l a t i o n s .

I f performed as described the experimenter can t h e o r e t i c a l l y determine y m a x , k s, ms and Y m a x from a s i n g l e experiment. In p r a c t i c e however, the method r e s u l t s i n k g values higher than those reported i n l i t e r a t u r e . The p r a c t i c a l use of t h i s method, as i l l u s t r a t e d w ith experimental and s i m u l a t i o n data , i s provided i n Chapter 4.

Advantages of the new method: i . no m„ constant hypothesis. , s , *% , . , max i l . e v a l u a t i o n of mg independent of Y g x . i i i . problems associated w i t h v i a b i l i t y and adaptation can be reduced to t h e i r

minima. i v . three parameters ( y m a x , Y g x , mg ) can be obtained from a r e l a t i v e l y

short experiment. v. sampling i n the substrate l i m i t e d phase can be reduced to a minimum by

f o l l o w i n g the RQ p r o f i l e . Many p o s s i b i l i t i e s f o r computer a p p l i c a t i o n s .

Disadvantages : i . mg i s determined only at zero growth r a t e , i . e . , cannot check i f m g=f(li). i i . r e q u i res c a r e f u l experimental design and more s o p h i s t i c a t e d equipment

(a programmer). i i i . s e n s i t i v e to sampling; best to perform i n large s c a l e to minimize e r r o r s

introduced by sampling.

Obviously t h i s methods determines the maintenance values under t r a n s i e n t condi­t i o n s whereas the values determined i n continuous c u l t u r e studies r e f e r to steady s t a t e c o n d i t i o n s . Hence d i f f e r e n c e s may be expected as the c u l t u r e s are e s s e n t i a l l y under d i f f e r e n t environmental c o n d i t i o n s and thus have d i f f e r e n t p h y s i o l o g i e s , as pointed out by H a r r i s o n . ' ' However, t h i s technique may be expected to r e s u l t i n more r e a l i s t i c estimates of ms f o r most i n d u s t r i a l fermentations as these are mainly c a r r i e d out i n batch and fed-batch modes.

Further i n v e s t i g a t i o n s i n t o the use of t h i s t r a n s i e n t technique seems to be d e s i r a b l e . A p p l i c a t i o n of the c o n t r o l theory and p a r t i c u l a r l y a study of whether or not the rate of approach to the zero growth s t a t e has any e f f e c t on the maintenance value determined, would be u s e f u l .

Determination of the Maximum Specific Growth Rate

'max c a n ^ e conveniently determined by many techniques. The most conventional one i s to determine i t i n batch c u l t u r e s during the exponential growth phase. T h e o r e t i c a l l y U m a x can a l s o be determined from the OUR and CPR curves i f the

31

r e s p e c t i v e y i e l d s are assumed to be constant.'2,13 However, i n our batch expe­riments s i g n i f i c a n t systematic d i f f e r e n c e s were observed i n the U m a x values obtainedfrom the three responses determined i . e . , C x, OUR and CPR data. These r e s u l t s are presented and discussed i n Chapter 4. The wash-out technique, however, r e s u l t e d i n co n s i s t e n t u values from va r i o u s responses.

max

I I I APPLICATION OF STATISTICAL TECHNIQUES IN THE STUDY OF MICROBIAL KINETICS AND ENERGETICS

Notes on the use of Statistical Techniques; regression analysis

One can determine the confidence to be placed on the experimental r e s u l t s , by sub j e c t i n g the experimental data to s t a t i s t i c a l a n a l y s i s . Most of the parameters reported i n t h i s work were c a l c u l a t e d by standard l i n e a r or nonlinear regress­ion procedures. V a l i d i t y of r e l a t i o n s were tested whenever p o s s i b l e . The non­l i n e a r r e g r e s s i o n program used was developed i n the Chemical Engineering Department of the D e l f t U n i v e r s i t y of Technology and was based on Marquart's method.'^

In l i t e r a t u r e e x c e l l e n t accounts of e f f i c i e n t experimental design and model b u i l d i n g processes have been given, p a r t i c u l a r l y by Johnson and Berthouex' >' and Boyle and Berthouex17 w i t h waste water processes i n mind. Advanced s t a t i s ­t i c a l techniques have a l s o been made a v a i l a b l e f o r data r e d u c t i o n , a n a l y s i s fo r c o r r e l a t i o n , parameter es t i m a t i o n from multi-response data e t c . by many a u t h o r s . ' ' However, i t i s q u i t e d i s s a p p o i n t i n g that the a p p l i c a t i o n s of these a v a i l a b l e t o o l s are s t i l l scarce i n the study of m i c r o b i a l e n e r g e t i c s .

To the knowledge of the author, the only rigorous attemp i n t h i s d i r e c t i o n has come from de Kwaadsteniet et a l . ^ 2 For the k i n e t i c s and modelling side much more work has been done and reported. However, there are s t i l l problems to be solved. One of the most c h a l l e n g i n g one i s to estimate s e v e r a l parameters from multi-response data i n t r a n s i e n t range. In some cases t h i s i s q u i t e d i f f i c u l t to perform due to the h i g h l y nonlinear and i m p l i c i t nature of the d i f f e r e n t i a l equations d e s c r i b i n g the b i o l o g i c a l systems.

The use of nonlinear r e g r e s s i o n during t h i s study, proved to be superior to l i n e a r r e g r e s s i o n . This d i f f e r e n c e stems from the f a c t t h a t , i n performing l i n e a r r e g r e s s i o n , often transformations of v a r i a b l e s are made. When there are n o n - n e g l i g i b l e e r r o r s attached to the measured v a r i a b l e s , a f t e r trans­formation, some complex f u n c t i o n of the e r r o r s r e s u l t rather than the e r r o r being added to the transformed v a r i a b l e . 2 3

This can best be i l l u s t r a t e d by a r e a l i s t i c example; F i g . 4 i s a s i m u l a t i o n of an exponential growth phase i n a batch experiment w i t h parameters s i m i l a r to those reported i n t h i s study ( p m a x =1.0). Here, the dry weight p r o f i l e i s shown.The associated e r r o r i s assumed to be a =0.05 kg/m3 . Based on t h i s data the curve i . e . , C x vs. time data was p l o t t e d . The u n c e r t a i n i t i e s of data points are assumed to be equal a s i n d i c a t e d on the l i n e a r s c a l e but appear to be unequal on the l o g a r i t h m i c s c a l e . By f i t t i n g the l i n e a r equation instead of the expo­n e n t i a l r e l a t i o n the experimenter assumes that the u n c e r t a i n i t i e s are equal on the exponential s c a l e and therefore underestimates the e r r o r s f o r small values of C x. In other words small values of C x i n f l u e n c e the outcome of the exercise most. Therefore, where p o s s i b l e use of nonlinear r e g r e s s i o n should be p r e f e r r e d i n order to avoid v a r i a b l e transformation. I f the er r o r s involved are exceedingly small there should be no d i f f e r e n c e between the two procedures.

32

Fig. 4: Simulation of exponential growth; error structure for linear and nonlinear regression procedures (see text).

The conventional l i n e a r r e g r e s s i o n procedure was a l s o found to be i n f e r i o r to the nonlinear r e g r e s s i o n procedure i n the determination of energetic parame­t e r s . The double r e c i p r o c a l p l o t i . e . , 1/Y vs. 1/y p l o t , s u f f e r s badly from the uneven weighting of the data points and the data c o l l e c t e d at moderate y values tend to c l u s t e r near the o r i g i n .

Continuous c u l t u r e data obtained during t h i s study were found to be f i t t e d best by n o n l i n e a r r e g r e s s i o n technique when assessed by the scaled sum of r e s i d u a l s . Therefore, parameters obtained by nonlinear r e g r e s s i o n were used i n the c a l ­c u l a t i o n s .

I t must be reminded that both methods, l i n e a r and n o n l i n e a r , are founded upon the assumption of the randomness of e r r o r s . This may not be j u s t i f i e d always. I t i s important to r e a l i z e that the nature of some b i o t e c h n o l o g i c a l experiments are such that s i g n i f i c a n t systematic e r r o r may be introduced. A n a l y s i s of r e s i d u a l s can be very u s e f u l i n t h i s context. These t o o l s have been provided by Smith and Draper'" among others. A more d e t a i l e d d i s c u s s i o n and a p p l i c a t i o n to experimental systems, of the systematic e r r o r a n a l y s i s i s provided i n Ch. 8.

Assessment of Error Propagation; Calculation of the Maximal and Probable Errors

When the f u n c t i o n a l r e l a t i o n s h i p connecting the f i n a l r e s u l t w i t h the v a r i a b l e components, as w e l l as the e r r o r s t r u c t u r e of these components are known, one can evaluate how the v a r i a t i o n i n vari o u s components are transmitted to the f i n a l r e s u l t . For a l i n e a r f u n c t i o n of the form ;

a Xj + b X 2 + c X 3 + (12)

where a, b, c,... are constants and X j , X^, X^, . . . . random v a r i a b l e s , the maximal numerical e r r o r i n W, f o r small changes i n X's i s given by24 :

AW = |a| AX + |b| AX 2 + |c| AX 3 + (13)

where AW i s the maximal e r r o r i n W. From a s t a t i s t i c a l p oint of view t h i s procedure i s p e s s i m i s t i c ; the combined p r o b a b i l i t y that the e r r o r s i n X^, X , X^, are such as to produce the maximum p o s s i b l e e r r o r i n W i s very small.

33

A more r e a l i s t i c procedure i s to consider the p r o b a b i l i t y d i s t r i b u t i o n of the t o t a l e r r o r i n W. In t h i s case the f o l l o w i n g r e l a t i o n s h i p holds between the variances of the r e s u l t and the components:23,24

Var (W) = a 2 Var (Xj) + b 2 Var(X 2) + c 2 Var(X 3> + (14)

I f the f u n c t i o n a l r e l a t i o n s h i p between the v a r i a b l e s i s not l i n e a r , the func­t i o n must f i r s t be l i n e a r i z e d i n order to use eq.(14). L i n e a r i z a t i o n can be achieved by expansion i n t o a Taylor s e r i e s about a reference point and by neg l e c t i n g the second and higher order terms. For a f u n c t i o n of sev e r a l v a r i a b l e s , the truncated Taylor s e r i e s i s given by Himmelblau23 a s ;

EC x , x^, x^, > x^ ) - f ( X j , x 2 , x^, (15)

E {3f( X X X i=l

, X )/ 3X.

x° x° x° V X 2 ' V ' " ' n

} ( X . 0

where the s u p e r s c r i p t '0' r e f e r s to the reference s t a t e . Furthermore, by assuming that the random v a r i a b l e s are independent, the variance i s given by:

Var { f ( X , X , X„ )} n E { 1=1

3f( X x_ )/ St } Var(X i)

X j , ^ 2 ' •

I f the v a r i a b l e s are c o r r e l a t e d however, t h i s approach w i l l only be able to provide an i n d i c a t i o n of the order of the magnitude of the e r r o r s .

Carbon and Nitrogen Balances ; Error analysis

Most energetic and k i n e t i c studies are c a r r i e d out i n C - l i m i t e d c u l t u r e s where only one substrate i s the source of both carbon and energy. Construction of C balances i s therefore of prime importance i n such s t u d i e s . The a n a l y s i s to be reported here considers the case of continuous c u l t i v a t i o n w i t h one l i m i t i n g substrate. However, the procedure can be extended and applied to any system at steady or pseudo-steady s t a t e .

3 Considering a continuous c u l t u r e system of u n i t volume (1 m ) and the f o l l o w i n g assumptions: i . Steady s t a t e operation i i . No C O 2 or N 2 f i x a t i o n i i i . Carbon l i m i t e d c u l t u r e and no C-containing products i n gas phase other

than C O 2 and only one metabolic product i v . E f f i c i e n t condensation and drying operation f o r fermentation gases For t h i s system a t o t a l C-balance should read:

E C - E C - out

F. C . w + S. CO. 12/22.4 = F ( C w + C w + C w ) + i n s i s i n 2 out s s x x p p

|> „ CO^ 12/22.4 out 2 (17)

34

Due to experimental e r r o r s , however, the above equation does not always hold and we can therefore define the Carbon Recovery Index, CRI, to assess the consistency of the C-balances:

CRI = IC -out / I c - i n (18)

or the % C-recovery as 100 x CRI. I f , F£ n - F o u t , CRI can be w r i t t e n as (see s e c t i o n on volume balancing f o r the j u s t i f i c a t i o n of t h i s assumption):

CRI F- (C w + C w + C w ) + <j> „ C0_ 12/22.4 l n s s x x P p out 2 (19) F. C . w + A. C0^ n 12/22.4 i n s i s i n 2

Therefore ; CRI = f ( F. , C , C , C , w , c|> . <)>. , C0^, CO", C .)

i n s x p x out i n 2 2 s i w and w are not included i n the above expression s i n c e , they are the carbon f r a c t i o n ! of substrate and product, r e s p e c t i v e l y , and hence can be assumed to be constant f o r compounds of known formulae. To estimate Var(CRI), p a r t i a l d e r i v a t i v e s of CRI with respect to i t s v a r i a b l e s i n d i c a t e d above, were evaluated and the maximal and probable e r r o r i n CRI were c a l c u l a t e d . The f o l l o w i n g example r e f e r s to continuous c u l t u r e data obtained during t h i s study (sample no 9). The raw data at the p o i n t of l i n e a r i z a t i o n and the corresponding estimates of e r r o r s involved are l i s t e d i n Table I , together w i t h t h e i r c a l c u l a t e d c o n t r i b u t i o n s to the t o t a l maximal e r r o r i n CRI and the t o t a l variance.

Table I : E r r o r assessment i n C-balance f o r a continuous c u l t u r e experiment.

„ i g C R I | . v % t o t a l s a CRI -,2 5 % e r r o r • a=—AX. . . — } Var(X.).10 S X . 1 i maximal aX. l ° l l e r r o r

F 0 54 3 0 0092 8 8.46 C s

0 09 5 0 0044 4 1 .94 C

X

5 02 2 0. 0130 1 1 16.90 w

X

0 51 3 0 0194 17 37.64 C P

0 0 0 0 0 0. <tWt 67 70 3 0 0062 5 3.84 COA 0 0187 5 0 0156 13 24.34 C . s i

10 20 5 0 0480 41 230.04

$in 68 10 3 0 0001 2 0. 1 0.0014 C0in 3 6x10~ 4 10 0 0006 0.5 0.036

Sum 0 1 165 99.6 0.00324xl0~ 5

35

Using the values l i s t e d , CRI was c a l c u l a t e d to be. 0.95, i . e . , 95 % C-recovery. The maximum e r r o r involved was estimated to be 12 % while the probable e r r o r , 6 %.

The c o n t r i b u t i o n s of the major sources of e r r o r s to the t o t a l maximal e r r o r are shown diagrammatically i n F i g . 5. The d i s t r i b u t i o n s of the e r r o r s , however, can be q u i t e d i f f e r e n t i n other s i t u a t i o n s e.g., during anaerobic growth w i t h product formation. Therefore every d i f f e r e n t s i t u a t i o n should be studied on i t s own m e r i t s . This type of est i m a t i o n analyses can be very u s e f u l i f c a r r i e d out p r i o r to experimentation as t h i s would all o w the experimenter to choose the r i g h t equipment, the optimal number of a n a l y t i c a l determinations necessary and the optimal experimentation range to keep the t o t a l discrepancy i n C or any other r e l e v a n t balance(s) below the maximum acceptable l e v e l . The present a n a l y s i s shows that during the course of t h i s work substrate analyses c o n t r i ­buted most to the t o t a l e r r o r . I t i s als o apparent, f o r instance that any improvement i n the c o n t r o l of a i r flow r a t e or feed i n l e t stream i s hardly necessary.

A s i m i l a r a n a l y s i s was als o c a r r i e d out f o r N-balances. The r e s u l t s f o r the same set of data are summarized diagrammatically i n F i g . 5. In t h i s case NRI, nitrogen recovery index defined s i m i l a r l y , was c a l c u l a t e d to be 1.01 i . e . , 101 % recovery. The corresponding t o t a l maximal and probable e r r o r s were 11 % and 6 %, r e s p e c t i v e l y .

Fig. 5: Contribution of the major sources to the total maximal error in C- and N-balances.

F i n a l l y an assessment of the o r i g i n a l C- and N-balances f o r the continuous c u l t u r e experiment reported i n t h i s study, i s presented i n Table I I .

The r e s u l t s show b e t t e r recoveries than those p r e d i c t e d by the e r r o r analyses. This was qui t e expected because most experimental data points were repeated and averaged. Also most of the a n a l y t i c a l determinations were performed i n d u p l i ­cates and gas concentration measurements were time averaged over long periods of time. As can be seen C- and N- re c o v e r i e s were good. In both cases a small f r a c t i o n of the input could not be accounted f o r . S i m i l a r observations were made f o r batch and fed-batch experiments. Recoveries i n these cases were not as good. The apparent l o s s of C might be due to the formation of unnoticed by-products or l o s s of C as d i s s o l v e d carbon d i o x ide or carbonate-bicarbonate i n the l i q u i d phase. The l a t t e r p o s s i b i l i t y was found to be p a r t i c u l a r l y important f o r t r a n s i e n t experiments, (see Chapter 8)

N-losses may be due to v o l a t i l i z a t i o n of ammonia and e x i t v i a the gas phase.

36

A l t e r n a t i v e l y systematic e r r o r s might have been introduced by one or more of the components and/or a n a l y s i s involved i n the experimental system.

Table I I : Assessment of the o r i g i n a l C- and N-balances f o r continuous c u l t u r e .

Average recoveries C- balance no of data CRI x l O 2 a (min max)

27 97 94 4.08 90.0 104.6 23 + 99 15 2.97 94.5 104.6

N- balance NRI 2 xlO

27 97 61 6.77 83.0 108. 1 *

24 98 78 5.59 91.8 108. 1

+ i f 4 data points are excluded as o u t l i e r s : CRI of 0.90, 0.91 , 0.92,0.90 * i f 3 data points are excluded as o u t l i e r s : NRI of 0.83, 0.84, 0.85 a i s the standard e r r o r of the average recovery

System Overdetermination in Elemental Balancing

Overdetermined systems a r i s e i n experimental and computational work where more r e s u l t s are generated than would be required i f p r e c i s i o n were a t t a i n a b l e . In a sense a number of inexact and sometimes c o n f l i c t i n g information becomes a substitude of a few p e r f e c t r e s u l t s and one has to f i l t e r out b e t t e r estimates from the inconsistencies.25-28

Considering the system studied i n t h i s work the f o l l o w i n g s t o i c h i o m e t r i c equation can be w r i t t e n i . e . , f o r f u l l y aerobic growth w i t h no by-product formation.

aC,H o0 o + b0„ + cNH 0 — C H 0 DN + dC 0. + eH„ 0 (20) 3 8 3 2 3 a B y 2 2

For t h i s system, the knowledge of any two steady s t a t e flows i s s u f f i c i e n t to estimate the r e s t , as four equations can be w r i t t e n f o r the four elements involved and there are s i x flows a l l t o g e t h e r . During the experiments however, substrate, oxygen, biomass, carbon d i o x i d e and sometimes the ammonia flows were measured i . e . , more flows were measured than were minimally needed to c a l c u l a t e the remaining ones. Such a surplus of data were not wasted but used to o b t a i n the more optimal estimates of a l l measured and unknown flo w ( s ) (water). The mathematical background to the problem has been given by a number of authors.25-32 ^he procedure used i n t h i s work has already been described i n the l i t e r a t u r e 3 2 a n j j_s s i m i l a r to that reported by Madron et £1.29-31 The reader i s r e f e r r e d to the o r i g i n a l a r t i c l e s f o r a d e t a i l e d d e s c r i p t i o n . Here only a b r i e f i l l u s t r a t i o n w i l l be provided as an example since a l l t h e steady and pseudo-steady s t a t e flows and hence the y i e l d values have been correc t e d by the use of t h i s powerful s t a t i s t i c a l technique.

In F i g . 6 c o r r e c t i o n s brought about by the a p p l i c a t i o n of t h i s technique to the raw gas exchange data of a continuous c u l t u r e run (see Chapter 2, F i g s . 8,9 ) i s shown. RQ was chosen f o r t h i s i l l u s t r a t i o n as i t i s one of the most s e n s i t i v e response v a r i a b l e s and hence s i g n i f i c a n t c o r r e c t i o n s may be introduced. Generally the a p p l i c a t i o n of t h i s procedure to continuous c u l t u r e y i e l d data r e s u l t e d i n l e s s dramatic c o r r e c t i o n s than those shown i n F i g . 6.

37

An impression of the r e l a t i v e magnitude of the c o r r e c t i o n s can be obtained from Table I I I . Inspection of t h i s Table re v e a l s that w i t h the exception of CO2 flow, a l l flow c o r r e c t i o n s show some b i a s but these can be neglected when compared w i t h the standard d e v i a t i o n s of the concerned measurements, which were approximately 5,10,7,3 and 5% f o r substrate, oxygen, ammonia, biomass and carbon dioxide flow, r e s p e c t i v e l y . Corrections introduced f o r batch and f e d -batch ( t r a n s i e n t ) data were l a r g e r than the continuous c u l t u r e data. In an i d e a l case i . e . , when a l l e r r o r s are random, the average c o r r e c t i o n f o r each flow should approach to zero f o r a large number of samples.

 R Q

0.9-raw data

• corrected data c

1

Fig. 6: Rao and statistically corrected RQ data for a continuous culture run.

Table I I I : Corrections applied to raw continuous c u l t u r e data (27 points)

Average c o r r e c t i o n applied to the net flow of : % a min - max

substrate -0 78 2 04 -5 6 2.6 oxygen 1 AÍ 2 46 -3 2 5.4 ammonia -1 23 9 22 -16 5 17.2 biomass -0 44 1 76 -4 0 2.8 carbon dioxide -0 04 1 40 -3 1 2.2

a standard d e v i a t i o n of the average c o r r e c t i o n

F i n a l l y a word of caution must be s a i d about the r e l i a b i l i t y of t h i s procedure. The method i s founded on the assumption of the randomness of the e r r o r s . I f through ignorance or otherwise the above method i s used f o r c o r r e c t i n g data containing large systematic e r r o r s , t h e method may b r i n g about serious devia­t i o n s from the r e a l i t y . Thus the experimenter must always be aware of the l i m i t a t i o n s and the i m p l i c a t i o n s of the method used, to avoid biased conclu­sions through what he regards as l e g i t i m a t e s t a t i s t i c a l procedures.

38

IV NOMENCLATURE

C concentration (kg/m3) C 0 2 _ i n CO2 mole f r a c t i o n i n the i n l e t gas (—) C02 _A CO2 mole f r a c t i o n i n the dry o u t l e t g a s ( a n a l y s e r ) ( — ) CPR CO2 production rate (mole/m3/hr) CRI carbon recovery index (—) F volumetric l i q u i d flow (m3/m3/hr) F( t ) r a t e of substrate a d d i t i o n (kg/hr) k s Monod s a t u r a t i o n constant (kg/m3) ms maintenance c o e f f i c i e n t on substrate (kg/kg/hr) Mj t o t a l amount of j i n the fermentor (kg) N2 N2 mole f r a c t i o n i n gas streams (—) NRI ni t r o g e n recovery index (—) O2-A O2 mole f r a c t i o n i n dry o u t l e t gas (analyser) (—) 0 2 ~ i n O2 mole f r a c t i o n i n the i n l e t gas (—) OUR oxygen uptake rate (mole/m3/hr) q s s p e c i f i c r a te of substrate consumption (kg/kg/hr) r Q OUR r c CPR RQ r e s p i r a t o r y quotient (—) V g c u l t u r e volume (irw) w carbon weight f r a c t i o n (—) ^sx y i e l d of biomass on substrate (kg/kg) Y s x-max maximal Y s x

u s p e c i f i c growth rate ( h r ~ l ) p s c u l t u r e density (kg/m3) a standard d e v i a t i o n $j mass flow of the j ' t h substance (kg/m3/hr) <j> volumetric flow of i n and out going dry gas streams at STP (m3/m3/hr) su b s c r i p t s x biomass s substrate p product i i n l e t F feed

V REFERENCES

1. C.G.T. Evans, D. Herbert and D.W. Tempest,in Methods i n Mic r o b i o l o g y , J.R. No r r i s and D.W. Ribbons, eds,(Academic, London, 1970)vol.2,313.

2. A. Harder, Ph.D. t h e s i s , U n i v e r s i t y of Wageningen,(1979). 3. J.P. Barford and R.J. H a l l , B i o t e c h n o l . Bioeng.,21,609,(1979). 4. W. de V r i e s and A.H. Stouthamer, J . Bacteriol.,96,472(1968). 5. D. Herbert, P.J. Phipps and R.E. Strange, i n Methods i n M i c r o b i o l o g y , J.R.

N o r r i s and D.W. Ribbons,eds,(Academic, London, 1971)vol.5b,209. 6. R.E. Treybal, Mass Transfer Operation, 2 nd ed.(McGraw H i l l , New York,

1968)p.192. 7. J.H. Perry, Chemical Engineers Handbook, 4 th ed.,(McGraw H i l l , New York,

1963)p.15.2. 8. S.Cromie amd H.W. Do e l l e , B i o t e c h n o l . Lett.,2(8),357(1980). 9. A.A. Esener, N.W.F. Kossen and J.A. Roels, paper presented to the 2 nd

European Congress i n Biotechnology, Eastbourne(1981). 10. I.G. Minkevich and L . I . U t k i n a , B i o t e c h n o l . Bioeng.,21,357(1979). 11. D.E.F. Ha r r i s o n , d i s c u s s i o n during the open forum session of the 2 nd

European Congress on Biotechnology, Eastbourne(1981).

39

12. D.T. Boyle, B i o t e c h n o l . Bioeng.,19,297(1977). 13. D.T. Boyle, B i o t e c h n o l . Bioeng.,20,1101(1978). 14. D.W. Marquart, J . Soc. Indust. Appl. Math.,2,431(1963). 15. D.B. Johnson and P.M. Berthouex, B i o t e c h n o l . Bioeng.,17,557 (1975). 16. D.B. Johnson and P.M. Berthouex, Biotec h n o l . Bioeng.,17,571(1975). 17. W.C. Boyle and P.M. Berthouex, B i o t e c h n o l . Bioeng.,16,1139(1974) 18. G.E.P. Box and N.R. Draper, Biometrika.52,355(1965). 19. N.R. Draper and H. Smith, Applied Regression A n a l y s i s , ( W i l e y , New York,

1967) . 20. P.M. Berthouex and W.G. Hunter, J . Sa n i t . Eng. Div. Amer. Soc. C i v i l Eng.,

97,393(1971). 21. P.M. Berthouex and W.C. Hunter, J . S a n i t . Eng. Div. Amer. Soc. C i v i l End.,

97,333(1971) 22. J.W. de Kwaadsteniet, J.C. Jaeger and A.H. Stouthamer, J . Theor. B i o l . ,

57,103(1976). 23. D.M. Himmelblau, Process A n a l y s i s by S t a t i s t i c a l Methods,(Wiley, New York,

1970)p.36. 24. C.G. Paradine and B.H.P. R i v e t t , S t a t i s t i c a l Methods f o r Technologists,

(The E n g l i s h Univ. Press, 1972)p.l61. 25. F.Scheid, Numerical Analysis,(Schaum O u t l i n e S e r i e s , McGraw H i l l , New York

1968) p.357. 26. A.K.S. Murthy, Ind. Eng. Chem. Process Des. Develop.,12(3),246(1973). 27. A.K.S. Murthy, Ind. Eng. Chem. Process Des. Develop.,13(4),347(1974). 28. P.M.E.M. van Grinten and J.M.H. Le n o i r , S t a t i s c h e Process Behersing,

(Prisma Technica, Utrecht,1973)p.336. 29. F. Madron and V.Vacenek, C l l . Czech. Chem. Commun.,42,1805(1977). 30. F. Madron, V, Veverka and V. Vacenek, J.AIChE.,23(4),482(1977). 31. F. Madron, Bi o t e c h n o l . Bioeng.,21,1487(1979). 32. H.E. de Kok and J.A. Roels, B i o t e c h n o l . Bioeng.,22,1097(1980).

40'

CHAPTER 4

FED-BATCH CULTURE: MODELLING AND APPLICATIONS IN THE STUDY OF MICROBIAL ENERGETICS *

A. A. Esener, J . A. Roels and N. W. F. Kossen

SUMMARY

M i c r o b i a l growth i n fed-batch mode i s described by a simple unstructured model. The model i s found to be i n good agreement w i t h the experimental observations, except under h i g h l y t r a n s i e n t c o n d i t i o n s . Extensive experimental data were c o l l e c t e d and the energetics of the bacterium K l e b s i e l l a pneumoniae i s evaluated. I t i s shown that the fed-batch c u l t i v a t i o n i s a powerful experimen­t a l t o o l i n the study of m i c r o b i a l k i n e t i c s and energetics simultaneously. Methods f o r determining the maintenance requirements are shown and evaluated. The maintenance c o e f f i c i e n t s determined from fed-batch data are s y s t e m a t i c a l l y smaller than those reported f o r continuous c u l t u r e systems. Results suggest a decrease i n maintenance demands at low s p e c i f i c growth r a t e s .

INTRODUCTION

The theory of fed-batch c u l t i v a t i o n of microorganisms has been studied q u i t e e x t e n s i v e l y i n l i t e r a t u r e . Several workers a p p l i e d mathematical and modeling techniques f o r d e s c r i b i n g t h i s process.1-12 i n m o s t cases, however, authors r e s t r i c t e d the use and a p p l i c a b i l i t y of t h e i r models by imposing severe r e s t r i c t i o n s on the behaviour of the system under c o n s i d e r a t i o n . These include the assumptions of constant volume, substrate l i m i t a t i o n , quasi-steady s t a t e e t c . Moreover i n many cases, these studies were confined to t h e o r e t i c a l c onsid­er a t i o n s w i t h l i t t l e or no experimental data; hence one i s not able to check the v a l i d i t y of the models and to get an accurate i n s i g h t i n t o the growth phenomena under these c o n d i t i o n s .

In an attempt to f i l l t h i s gap, extensive data and m a t e r i a l balances f o r the fed-batch c u l t i v a t i o n of the bacterium K l e b s i e l l a pneumoniae are presented i n t h i s paper. A simple unstructured model w i t h a minimum number of assumptions based on Monod k i n e t i c s ' ^ a n d the l i n e a r law of substrate consumption'^ i s also developed as an extension of the model, reported e a r l i e r by Roels and Kossen.'^ The outcomes of model simulations are compared w i t h the experimental r e s u l t s . Furthermore, i t i s shown that fed-batch c u l t u r i n g technique can be used as a

* Accepted f o r p u b l i c a t i o n i n B i o t e c h n o l . Bioeng'. (1981)

41

powerful t o o l f o r the determination of the b i o k i n e t i c parameters and y i e l d and maintenance c o e f f i c i e n t s w i t h minimal e f f o r t compared w i t h continuous c u l t u r e methods. Here, one of the experiments to determine maintenance c o e f f i c i e n t s a l s o provided severe experimental c o n d i t i o n s (rate of substrate a d d i t i o n decreased l i n e a r l y ) , which enabled the experimenter to put the developed model i n jeopardy. I t i s e s p e c i a l l y important to note that the maintenance c o e f f i ­c i e n t s can be obtained d i r e c t l y and a c c u r a t e l y from w e l l designed fed-batch experiments without the use of transformation of v a r i a b l e s . F i n a l l y , an e f f e c t i v e method f o r volume balancing derived from a mass balance i s introduced.

MODEL

To describe a fed-batch process b a s i c a l l y three r e l a t i o n s are needed. These are balance equations f o r biomass and the l i m i t i n g substrate and an equation f o r the volume of the c u l t u r e . Often, however, s i g n i f i c a n t changes i n c u l t u r e volume occur i n a l e s s c o n t r o l l a b l e way, due to evaporation, a c i d a n d / a l k a l i a d d i t i o n e t c . Therefore, i n some cases i t i s more convenient to avoid the volume equation. This i s p o s s i b l e i f the t o t a l mass of b i o m a t e r i a l and substrate are considered instead of t h e i r c o n c e n t r a tions.^ In such a case two sta t e v a r i a b l e s w i l l s u f f i c e to describe the system. Introducing them, they are

M = V C x x and

M = V C s s where V i s the c u l t u r e volume, C x biomass concentration, and C s the l i m i t i n g s ubstrate concentration, the balance f o r biomass can be given by

d(M )/dt = R (1) x x

where R x i s the rate of biomass production and i s assumed to be a f u n c t i o n of the l i m i t i n g substrate concentration. Balance f o r the l i m i t i n g substrate can be expressed by

d(M )/dt = F ( t ) - R g (2)

where F ( t ) i s the substrate feeding r a t e to the fermentor and R s i s the rate of substrate consumption i n the fermentor. By d e f i n i t i o n

R = y(s) M (3) x x

where y i s the s p e c i f i c growth r a t e and i s a f u n c t i o n of the l i m i t i n g substrate concentration. Furthermore, i f Monod k i n e t i c s ' - ^ and the l i n e a r law of substrate consumption'^ are assumed to be a p p l i c a b l e to the system under c o n s i d e r a t i o n , one can w r i t e

and

42

^ ( S ) = V x M s / ( M s + K s } ( 4 )

R s = ( V C * ) + ™s M* (5)

Here Ygx i s t':le y i e l d 0 1 1 substrate corrected f o r maintenance, ms the mainte­nance c o e f f i c i e n t on s ubstrate, and K s i s defined as

K = V k s s

where k g i s the Monod s a t u r a t i o n constant. I t must be stressed that K s was defined i n t h i s manner only f o r convenience and treated as a parameter i n the model (see f i g . 1 the flow diagram of the mathematical model). To a f i r s t approximation K s was taken as the product of Monod s a t u r a t i o n constant and the average volume of the c u l t u r e during the fermentation. This assumption d i d not cause any s i g n i f i c a n t d i f f e r e n c e s i n the s i m u l a t i o n r e s u l t s .

S u b s t i t u t i n g f o r R x , R s and y(s) i n balance e q s . ( l ) and ( 2 ) , one obtains

d ( M„ )/dt = U M M / ( M + K ) (6) x max s x s s

d ( M )/dt = F ( t ) - ( R / Y m a X + m M ) (7) s x sx s x

Equations (6) and (7) cannot be solved a n a l y t i c a l l y except f o r s p e c i a l cases. S i m p l i f i c a t i o n s based on various assumptions can be found i n the l i t e r a t u r e . 1-5 ,10,15

I f , however, one wishes to work i n concentrations rather than mass, d i f f e r e n ­t i a l s i n eqs,(6) and (7) can be expanded and reduced to

d ( C )/dt = u C C /( C + k ) - C / V (dV/dt) (8) x max s x s s x

d ( C )/dt = F ( t ) / V - r - C /V (dV/dt) (9) s s s

where R s = V r s . In t h i s case one needs to know dV/dt to solve the equations. The required volume balance can best be derived from a t o t a l mass balance. Considering a mass balance f o r the c u l t u r e , one can w r i t e

(dW/dt) = 6(t) + (0™ - 0° U t) + (CC> - C0° U t)

+ ( H 2 0 i n - H 20° U t) + (rate of a c i d / a l k a l i a d d i t i o n ) (10)

- (escape r a t e of organic vapors)

where 9(t) i s the rate of feed s o l u t i o n a d d i t i o n , and F ( t ) = (weight f r a c t i o n of substrate i n feed s o l u t i o n ) x 9 ( t ) .

However, i t i s known that the density of the broth changes l i t t l e during the course of fermentation. Furthermore, the c o n t r i b u t i o n of humidity, l o s s of organic vapors, and a c i d / a l k a l i c o r r e c t i o n s can be assumed or made to be i n s i g n i f i c a n t under p r a c t i c a l operating c o n d i t i o n s . I f they can be neglected e q . ( l l ) provides the required volume balance :

43

p„(dV/dt) = 6(t) + (32 r - 44 r ) V / 1000 B o c (11)

Here pg i s the d e n s i t y of the broth , r Q oxygen uptake, and r c carbon dioxide production rate per u n i t volume of the c u l t u r e . Since the r e s p i r a t o r y quotient i s defined by

RQ = r / r (12) c o

Eq.(11) f o r the aerobic case becomes

p„(dV/dt) = 9(t) + 44 V r { 32/(44 RQ) - 1 } / 1000 (13) B c

Therefore, when

RQ > 32/44 , dV/dt < 9 ( t ) / p

and when

RQ < 32/44 , dV/dt > 9(t)/p„ D

This d e r i v a t i o n shows that the volume change during the fed-batch fermentation i s not only dependent on the volumetric feeding rate but a l s o on the metabolic s t a t e of the c u l t u r e as described by RQ.

In most aerobic fermentations the second term on the right-hand side of eq.(13) can be regarded as i n s i g n i f i c a n t when compared with the value of 9 ( t ) . In anaerobic fermentations, however, m a t e r i a l exchange v i a the gas phase can be more s i g n i f i c a n t and hence may have to be allowed f o r .

For l a b o r a t o r y s c a l e fermentations the t o t a l volume of the samples taken can al s o be a s i g n i f i c a n t f r a c t i o n of the i n i t i a l c u l t u r e volume. This causes discrepancies i n m a t e r i a l balances. A method f o r c o r r e c t i n g m a t e r i a l balances i s provided i n the appendix.

DETERMINATION OF BIOKINETIC PARAMETERS AND YIELD AND MAINTENANCE COEFFICIENTS

I f designed c o r r e c t l y , b i o k i n e t i c parameters and y i e l d and maintenance c o e f f i ­c i e n t s can be determined from a s i n g l e fed-batch experiment. Considering eqs.(6) and (7) when

erratum M » K jj -»• ]i _Xj s max

and hence d(M )/dt - u M (14) x max x

Therefore during the exponential part of the fed-batch experiment y m a x can be

44

obtained from a p l o t of ln(M x) vs. time. During the t r a n s i t i o n from exponential to substrate l i m i t e d phase,y, the s p e c i f i c growth rate w i l l decrease from i t s maximum value to i t s new value as determined by the rate of substrate a d d i t i o n . I f t h i s new value i s l e s s than V m a x/2 > at one stage during t h i s t r a n s i t i o n p e r iod, y w i l l assume a value of y /2. The substrate concentration at t h i s moment i s by d e f i n i t i o n equal to the value of k s, the Monod s a t u r a t i o n constant. U s u a l l y the biomass curve has a smooth t r a n s i t i o n i n t h i s period and a p o l y ­nomial can be f i t t e d to dry weight data, which, when d i f f e r e n t i a t e d and made equal to y m a x / 2 y i e l d s the time where c s = k s. The major advantage of f e d -batch c u l t u r m g technique, however, i s i n the determination of maintenance c o e f f i c i e n t s . When R x = dM x/dt = 0, dM s/dt may be assumed to be equal to zero, since the substrate removal process has a much smaller time constant than that f o r biomass production.'" This assumption was checked and j u s t i f i e d l a t e r on, experimentally. I n s e r t i n g zeros f o r R x and dM s/dt i n eq.(7), ms can now be given by ms = F ( t ) / M x . S i m i l a r l y , m0 = R 0/M x and mc = R c/M x . Therefore, i f the c u l t u r e i s allowed to grow w i t h constant feed of l i m i t i n g s u b s t r a t e , the t o t a l biomass M x w i l l reach a l e v e l at which the feed input can only s a t i s f y the maintenance requirements.''' By t h i s method various maintenance c o e f f i c i e n t s can be obtained d i r e c t l y . Transformation of v a r i a b l e s , e x t r a ­p o l a t i o n of data, and above a l l , the assumption of constant maintenance (independent of the s p e c i f i c growth rate) are not necessary.

Various y i e l d c o e f f i c i e n t s can a l s o be obtained from fed-batch data. What i s important here i s that v a r i a t i o n s i n y i e l d due to changes i n growth r a t e can al s o be observed, i . e . with a c a r e f u l l y planned experiment v a r i a t i o n i n various y i e l d s from y = 0 to p = U m a x can be determined. At t h i s stage, however, the experimenter must be aware of the f a c t that a fed-batch c u l t u r e i s never i n a true steady s t a t e and state of the c u l t u r e at any i n s t a n t i s i n f l u e n c e d by i t s h i s t o r y . Therefore the behaviour of the c u l t u r e may not f o l l o w the exact trend shown i n steady state continuous c u l t u r e s at the same growth r a t e s . There i s already some evidence f o r s i g n i f i c a n t lags i n the c e l l metabolism. For instance, i t has been observed i n our l a b o r a t o r y that RNA l e v e l s l a g s i g n i f i ­c a n t l y behind the growth rate i n fed-batch c u l t u r e s during the decreasing growth r a t e period.

MATERIALS AND METHODS

Organism

Klebsiella pneumoniae NCTC 418, formerly known as Klebsiella aerogenes ,was used throughout t h i s study.

Cultivation Methods

Growth medium was prepared according to the f o r m u l a t i o n given by Evans et a l . ' ^ G l y c e r o l was used as the l i m i t i n g s ubstrate. I n i t i a l substrate concentration was adjusted according to the d e s i r e d f i n a l biomass concentra­t i o n . The medium was s t e r i l i z e d by membrane f i l t r a t i o n through a 0.2 ymmembrane f i l t e r i n t o a s t e r i l e fermentor. A t t e n t i o n was paid to o b t a i n an a c t i v e l y growing inoculum and i n most experiments the inoculum used was very small ( i n i t i a l concentration l e s s than 0.05 % of the f i n a l biomass concentrations) to avoid the p o s s i b i l i t y of an unbalanced growth of the organism.'9

Equipment

A l l experiments were c a r r i e d out i n an 11 x 10"^ m working volume fermentor maintained at 308 ± 0.5 K. The pH was c o n t r o l l e d at 6.8 ± 0.05. A i r flow to

45

the fermentor was c o n t r o l l e d by a thermal mass flow meter(Brooks 5811) at about 0.77 kg dry a i r / h . S p e c i a l a t t e n t i o n was paid f o r the accurate determination of gas flows and concentrations. A l l flows were corrected f o r humidity and volumetric changes. Gas phase oxygen and carbon dioxide concentrations were determined by a twin channel paramagnetic oxygen analyzer (Taylor Servomex OA 184) and an i n f r a r e d carbon dioxide analyzer (Beckman 864), r e s p e c t i v e l y . Feed flow to the fermentor was r e a l i z e d by a p r e c i s i o n p e r i s t a l t i c (LKB 2120) pump. For v a r i a b l e feed r a t e experiments the feed flow was decreased according to a predetermined l i n e a r f u n c t i o n by an analog gradient programmer (Joens type PG) coupled to the p r e c i s i o n pump. The feed b o t t l e was placed on a balance and a continuous read-out of the decrease of i t s weight was obtained.

Analytical Methods

A 99 % pure reagent q u a l i t y g l y c e r o l was used and assayed enzymatically (Boehringer UV method,148270). The d e t e c t i o n l i m i t of the assay was estimated to be 10 mg g l y c e r o l / 1 . Dry weights were determined by the method of de V r i e s and Stouthamer.20 Biomass was c o l l e c t e d on a 0.2 um pore diameter f i l t e r ( S a r t o r i u s 11370), washed with d i s t i l l e d water, and d r i e d to constant weight at 378 K. Elemental composition of biomass was determined by a computer-coupled element analyzer (Perkin Elmer 240). Ash content of biomass was determined separately and the composition was expressed on ash free b a s i s (Table I ) .

Table I : Avarage elemental composition and formula of K. pneumoniae.

Experiment C H N 0 Formula

FB 830 50.22 6.71 13.44 29.64 C H1.60 N0.23°0.44 FB 21 1 50.75 6.84 14.07 28.35 C H1.62 N0.24°0.42

In percent ash-free dry weight ;average ash content was 8 %.

Chromatographic a n a l y s i s of the c u l t u r e supernatant revealed that no by-product was present at a l e v e l that could be s i g n i f i c a n t . A l l samples were cooled during sampling down to about 278-280 K by an o n - l i n e heat exchanger manufactured i n the workshop of t h i s department. T y p i c a l residence time i n the heat exchanger was about 5-10 seconds.

Computation of the Results

A l l values used and reported i n t h i s paper are expressed on ash-free b a s i s . The average ash content of dry biomass was about 8 %. Y i e l d values p l o t t e d , have been corrected s t a t i s t i c a l l y by a computer program which c a l c u l a t e d the most probable values of m a t e r i a l flows i n the system as described e a r l i e r . ^ ' The method used f o r the computer program i s s i m i l a r to that reported r e c e n t l y by Madron and others.22-26 ^-Q simulations were c a r r i e d out with an IBM 360/65 computer system using the continuous systems modeling program (CSMP). Amodified rectangular method was used f o r i n t e g r a t i o n . Corrections f o r sampling was incorporated i n t o the program as described i n the Appendix.

RESULTS AND DISCUSSION

Experiment FB 830 was chosen as re p r e s e n t a t i v e of a c l a s s i c a l fed-batch e x p e r i ­ment. Here, the i n i t i a l substrate was about 3.5 kg/m3. With i n o c u l a t i o n the feed pump was also s t a r t e d . The substrate feeding rate was kept constant

46

M S = M „ „ + t

r d M s

rzi [Tmâx m S • , V , X

t r d M x _ M m a « - M S - M X

dt M = +K Q

1 i t fdMx

1x = M x o + / _

d ^ — •

Fig. 1: Simplified block diagram of the mathematical model used for simulations.

throughout the experiment at 8.807 x l O - ^ kg g l y c e r o l / h r . In Figure 2. the t o t a l biomass, M x and substrate, M s, are p l o t t e d as functions of the fermentation time, together with the s i m u l a t i o n r e s u l t s c a r r i e d out p r i o r to experimenta­t i o n with parameters estimated from our previous batch experiments and l i t e r a t u r e . 2 7 Here, good agreement can be observed w i t h the experiment and the model developed. The most s i g n i f i c a n t d e v i a t i o n occurs during the t r a n s i t i o n from the exponential to substrate l i m i t e d phase where the growth i s expected to be h i g h l y unbalanced. Even f o r t h i s period the maximum d e v i a t i o n observed between the experimental and simulated M x p r o f i l e s remain w i t h i n the l i m i t of acceptable v a r i a t i o n of operation f o r most i n d u s t r i a l a p p l i c a t i o n s .

Fig. 2: Experiment FB 830, biomass and substrate profiles : (9) biomass; (M) substrate; ( ) model simulation. Parameters: Wiax - 1.05 hr~l; Ks = 1.0 kg; F i t ) = 8.807. 10~3 kg/hr; Mxo = 7.4 .10''5 kg; Mso - 34.62 .1CT2 kg; - 0.54 kg/kg; ms = 7 .10~2 kg/kg/hr

When the OUR (oxygen uptake rate) and CPR (carbon d i o x i d e production rate) p l o t s ( F i g . 3) are examined, however, i t w i l l be seen that the concerned d e v i a t i o n s i n these v a r i a b l e s are much l a r g e r . This f i n d i n g i s not very s u r p r i s i n g as

47

these v a r i a b l e s respond to changes much f a s t e r than M x, since M x i s an inte g r a t e d quantity. These p l o t s make i t c l e a r that during t h i s h i g h l y t r a n s i e n t phase the unstructured model f a i l s to hold.

Fig. 3: Experiment FB 830, gas exchange rate profiles: (9) oxygen uptake rate; (0) carbon dioxide production rate; (—) model simulation. Parameters ^oxX - 36-15 • l c r 2 kg/mole; Y^E° = 54.12 . 10~2 kg/mole; m0 = 1.54 mole/kg/hr; mc = 1.32 mole/kg/nr.; other parameters same as shewn in Fig. 2.

Fig. 4: Experiment FB 830: change of yields during the fermentation. Yield of biomass on (0) substrate; (O) oxygen.

In F i g . 4. the y i e l d s on substrate and oxygen are p l o t t e d . These f i n d i n g s are i n good agreement w i t h the g e n e r a l l y accepted p a t t e r n of change throughout the fermentation. A decrease of y i e l d values w i t h the c u l t i v a t i o n time, i . e . w i t h decreasing growth r a t e , i s observed.

The other fed-batch experiment FB 211, reported f u l l y i n t h i s paper, was performed under unconventional c o n d i t i o n s . Here, the medium was inocul a t e d and the c u l t u r e was allowed to grow i n the batch mode. Before the termination of batch growth the feed pump was switched on. The rate of feed a d d i t i o n was decreased l i n e a r l y v i a a programmer. The behaviour of the c u l t u r e during the 48

fed-batch mode i s shown i n F i g . 5. together with the outcome of the s i m u l a t i o n of the presented model. The r e s u l t s of the s i m u l a t i o n can be s a i d to be i n f a i r l y good agreement with the experimental data, up to about t = 865 min. A s i g n i f i c a n t systematic d e v i a t i o n i s observed t h e r e a f t e r . These f i n d i n g s suggest a change i n maintenance requirements around t = 865 min, i f the model i s to be v a l i d . The observed RQ during the fed-batch growth mode increased almost l i n e a r l y . The rate of increase i n RQ changed suddenly a f t e r the biomass peak was reached. The RQ p r o f i l e observed can i n f a c t be f i t t e d w i t h two s t r a i g h t l i n e s ( F i g . 5.). The i n t e r s e c t i o n of these must have a p h y s i o l o g i c a l l y important meaning. P o s s i b l y up to t h i s p o i n t the substrate a d d i t i o n r a t e was enough f o r s a t i s f y i n g b i o s y n t h e s i s and maintenance demands. Thereafter, the incoming substrate could not even supply the maintenance requirements of the c u l t u r e and the c e l l s s t a r t using up t h e i r i n t e r n a l storage polymers and autolyse. In f a c t the presented model should not be a p p l i e d to t h i s decay phase. Here, the use of a k i n e t i c expression which allows f o r maintenance requirements through c e l l a u t o l y s i s i s p r e f e r a b l e . Herbert's model based on the concept of 'endogene-ous metabolism' can be used f o r t h i s s i t u a t i o n . ^ 7

Fig. 6: Experiment FB 211, biomass and RQ profiles during substrate limited phase: (*) biomass; ($) RQ; ( ) model simulation of biomass p r o f i l e Parameters same as given in Fig. 2. F i t ) varied according to F i t ) -(85 - 0.064. time). 10~z kg/hr, where t > 49 2 min.

In F i g . 6. y i e l d and RQ values obtained from raw and s t a t i s t i c a l l y c o r r e c t e d data are p l o t t e d together w i t h the t h e o r e t i c a l r e l a t i o n determined as described by E r i c k s o n et al.^° Using t h e i r n o t a t i o n and s i m p l i f y i n g f o r the no-product case, RQ i s given by

RQ = {1 - ( a b / a s ) Y^} / { ( Y s / 4 ) . ( . - <Vb'Vs)Y«x» ° 5 )

For t h i s experiment, using the values of aj, = 0.508, a s - 0.391, Ys = 4.67 and Yt, = 4.06, the above equation provides the t h e o r e t i c a l r e l a t i o n between the RQ

and Y s x. This p l o t also serves as a check of consistency of the raw and processed data. The raw data shows some s c a t t e r about the t h e o r e t i c a l l i n e . S t a t i s t i c a l l y corrected data of course gives a b e t t e r f i t . Based on t h i s c r i t e r i a , p r o c e s s e d data can be assumed to have no s i g n i f i c a n t systematic d i s c r e p a n c i e s .

49

Fig. 6. Consistency check for data obtained from experiment FB 211: ( ) theoretical relation between RQ and Ysx , eq. (15) in text; f • ) raD experimental data; (*) statistically corrected experimental data.

Estimation of the Maximum Specific Growth Rate

From the experimental data presented, an attempt was made to determine the maximum s p e c i f i c growth r a t e of t h i s organism. During the exponential part of the experiments the growth rate may be assumed to be equal to i t s maximum as shown p r e v i o u s l y . Furthermore, assuming constant y i e l d s on oxygen and carbon dioxide i t i s also p o s s i b l e to c a l c u l a t e U m a x based on the rates of oxygen uptake and carbon d i o x i d e production. A non-linear r e g r e s s i o n technique was used f o r M m a x c a l c u l a t i o n s . The computer program used was developed i n the Department of Chemical Engineeering, D e l f t U n i v e r s i t y of Technology and based on Marquart's method.29,30

The maximum s p e c i f i c growth rates f o r d i f f e r e n t experiments as determined from dry weight, oxygen uptake (OUR) and carbon d i o x i d e production (CPR) rate data are presented i n Table I I together with the 95 % confidence l e v e l s . As can be seen c l e a r l y from t h i s t a b l e there are s i g n i f i c a n t systematic d i f f e r e n c e s i n the measured values of y max depending on the v a r i a b l e upon which the determi­n a t i o n was based. Lower y m a x values from OUR and CPR data imply e i t h e r

1) y i e l d s on oxygen and carbon d i o x i d e are not constant but change during the course of the fermentation , or

2) some holding mechanism delays the output (release) of oxygen and carbon dioxide from the system at the rate w i t h which they are processed i n the system. A s i g n i f i c a n t hold-up of carbon d i o x i d e has r e c e n t l y been reported.31,32

A combination of the above two mechanisms i s , of course, al s o p o s s i b l e . I f the p l o t s of OUR and CPR ( F i g . 3.) f o r experiment FB 830 are to be reexamined and compared w i t h the s i m u l a t i o n r e s u l t s , one can see that d e v i a t i o n s between the predi c t e d and experimentally observed values get higher during the exponential part of the experiment. Moreover, there are also d i f f e r e n c e s , however s l i g h t , between the simulated and experimental peak times. The experimental curves are found to be somewhat t i l t e d and before the peak they show negative d e v i a t i o n s from the simulated curves, whereas a f t e r the peak, the de v i a t i o n s are p o s i t i v e .

50

Table I I : Maximum s p e c i f i c growth rate of K. pneumoniae c a l c u l a t e d from d i f f e r e n t v a r i a b l e s .

^max c a l c u l a t e d from data (hr" Experiment OUR CPR

FB 830 FB 21 1 B AV b

1 .064(1

1.070(1

.020-1.108) a

.061-1.076)

0.811(0.809-0 0.848(0.788-0

813) 908)

0.710(0 0.745(0

705-0 661-0

716) 829)

a F i g u r e s i n parentheses are the 95 % confidence l i m i t s . ^B AV - average value obtained from three batch experiments.

This c l e a r l y i m plies the presence of a b i o l o g i c a l and/or p h y s i c a l delay mechanism. However, one cannot accept t h i s hypothesis r i g h t away since the amount of these h y p o t h e t i c a l l y delayed q u a n t i t i e s are not found equal to the ones released a f t e r the peak, when i n t e g r a t e d . One can, t h e r e f o r e , assume a combination of various phenomena ta k i n g place. The c e l l s seem to be g e t t i n g more e f f i c i e n t throughout the fermentation. One might, of course, propose a hypothesis of the production of an intermediate energy-rich compound during the i n i t i a l part of the exponential phase, which i s f u r t h e r metabolized, r e q u i r i n g l e s s oxygen, etc . f o r c e l l b i o s y n t h e s i s . The p o s s i b i l i t y of the existence of such a phenomenon has a l s o been studied but could not be accepted. Based on these c o n s i d e r a t i o n s , the maximum s p e c i f i c growth r a t e can best be determined from dry weight data. However, one should not forget that biomass concentration (dry weight) i s an i n t e g r a t e d q u a n t i t y and i s a f f e c t e d by the h i s t o r y of i t s production and the course of the fermentation. Therefore i t i s much l e s s s e n s i t i v e to changes during the fermentation compared with oxygen uptake and carbon d i o x i d e production r a t e s .

Determination of ks

From Figure 2 , the growth rate can be c a l c u l a t e d based on dry weight data. When growth rate y i s p l o t t e d against experimentally measured substrate concentration, a value of 1.43 kg g l y c e r o l / m 3 has been c a l c u l a t e d f o r k s, i . e . g l y c e r o l concentration at y = P m a x / 2 . I t has already been reported that k s

values measured i n batch c u l t u r e s may be ten times higher than those measured i n continuous cultures.33 Even then to check the s i g n i f i c a n c e of t h i s determined parameter, simulations were c a r r i e d out f o r the same experimental co n d i t i o n s w i t h the experimentally determined value of k s. The r e s u l t s are shown i n Figure 7. From t h i s f i g u r e , i t becomes obvious that what has been measured as k s has no b i o l o g i c a l and/or q u a n t i t a t i v e s i g n i f i c a n c e . Therefore, i t can be concluded that the Monod r e l a t i o n by which k s i s defined i s not v a l i d f o r t h i s h i g h l y t r a n s i e n t period during which the observed y passes through the value of y m a x / 2 . Thus, fed-batch c u l t u r e data, when based on a simple unstructured Monod type model, do not provide a meaningful value of k s. Yamane and Hirano have als o expressed t h e i r doubts about the a p p l i c a b i l i t y of Monod model i n fed-batch systems.' The value of k s they determined was 28 times higher than that obtained from continuous c u l t u r e data.

Determination of Maintenance Coefficients

From the r e s u l t s of experiment FB 211 c o e f f i c i e n t s of maintenance were deter­mined by the f o l l o w i n g methods.

A) Biomass peak. As p r e v i o u s l y explained, when the amount of biomass reaches i t s maximum,i.e. when the growth rate i s zero, a l l substrate input must be going to

51

60 M x »10-(kg)

50

40 k s C k g / m 3 )

20

30 ® 0.1 ® 0.5 © 1.0 ® '5

10

0 0 300 600

Fig. 7: Experiment FB 830, ks evaluation (*) biomass; ( ) simulation. Parameters other than ks same as in Fig. 2.

maintenance metabolism. Therefore, ms can be obtained from a point determina­t i o n of the t o t a l biomass i n the fermentor and the rate of substrate input at that i n s t a n t . I f the gas exchange data are also a v a i l a b l e m0 and mc can also be obtained s i m i l a r l y .

B) RQ. I f a l l the incoming substrate i s used f o r maintenance processes, w i t h no products being produced, the f o l l o w i n g s t o i c h i o m e t r i c equation holds

From t h i s i t f o l l o w s that at true maintenance the RQ of the system must be equal to 0.86. Hence from F i g . 5. the time at which RQ becomes 0.86 can be determined and the values of ms, m 0 and mc can be c a l c u l a t e d from the co r r e s ­ponding data. Furthermore, i f the RQ data i s f i t t e d by two s t r a i g h t l i n e s , i t w i l l be seen that the i n t e r s e c t i o n l i e s approximately on the RQ = 0.86 l i n e ( F i g . 5.).

The above mentioned point determination methods have some drawbacks f o r labor a t o r y systems. F i r s t , the fed-batch system i s always i n a t r a n s i e n t state and i t i s d i f f i c u l t to r e a l i z e a state at which the growth rate i s e x a c t l y equal to zero. As the growth rate decreases i t takes longer to reach a steady s t a t e . A second problem i s caused by frequent sampling of the c u l t u r e . I t must be r e a l i z e d t h a t , whenever a sample i s taken from the fermentor, the growth rate would increase and some of the incoming substrate w i l l be used f o r b i o ­s y n thesis. With the RQ method another major disadvantage i s introduced due to the p o s s i b l e carbon d i o x i d e hold-up i n broth. D i f f e r e n c e s can e x i s t between the measured and true RQ values.

C) Balancing. To overcome the d i f f i c u l t i e s concerned w i t h the above mentioned methods, i t i s best to construct a m a t e r i a l balance f o r the r e l a t i v e l y steady s t a t e , observed as the biomass peak i n Figure 5. By t h i s method the substrate spent f o r b i o s y s t h e s i s to compensate f o r the biomass l o s t i n samples can a l s o be allowed f o r . This i s done i n the f o l l o w i n g way. I f during a time i n t e r v a l of At, the increase i n M x i s AMX, where AMX << M x, ms can be c a l c u l a t e d by

+ 7/2 0 2 — 3C 0 2 + 4H 20 (16)

52

m = {(substrate fed during At) - (AM /Y )} / {At(M + AM /2)} (17)

For an accurate determination of ms the second term i n the nominator should be small compared with the f i r s t . I t i s important to note here that Y™^ x i s assumed to be constant and known. This method has been ap p l i e d to the experimental data of FB 211 f o r the period of 825 < time(min) < 1105.

Table I I I : Maintenance values c a l c u l a t e d from the experimental data.

Experiment Method m s.10 2

kg/kg/hr ra° , mole/kg/hr mc

mole/kg/hr

FB 830 C) Balancing SC a 3.13 1.37 (1.19) 1.10 (1.02) FB 016 b B) RQ = 0.86 RP C 7.52 1.55 (2.86) 1.33 (2.45)

O Balancing Rd 3.56 1.00 (1.35) 0.99 (1.16) SC 3.15 1.13 (1.20) 0.96 (1.03)

FB 21 1 A) Biomass peak RP 5.47 1.14 (2.08) 1.23 (2.43) B) RQ = 0.86 RP 6.78 1.38 (2.58) 1.18 (2.21) C) Balancing R 4.53 1.25 (1.72) 1.12 (1.48)

SC 3.53 1.34 (1.24) 1.16 (1.06)

References (values reported i n l i t e r a t u r e )

34 9.21 2.67 (3.50) 35 7.55 3.46 (2.87) 36 7.68 2.54 (2.92)

aSC- c a l c u l a t e d from s t a t i s t i c a l l y corrected data. ^Experiment FB 016 was a r e p l i c a t e of FB 211 and not reported i n t h i s paper. CRP- c a l c u l a t e d from raw point data. R- c a l c u l a t e d from raw data. The values i n parentheses are c a l c u l a t e d from stoichiometry (eq.(16)) and the experimentally determined ms values.

In Table I I I , ms, m0 and mc values determined by the above l i s t e d methods are presented and compared w i t h those reported i n l i t e r a t u r e as c i t e d by Heijnen and Roels.34 When the data presented i n Table I I I are examined f o r consistency, e.g., i f the experimentally determined m0 and mc values are compared w i t h those c a l c u l a t e d from ms and stoichiometry ( i n parentheses) i t w i l l be c l e a r l y noted that the methods A and B y i e l d e d i n c o n s i s t e n t estimates of maintenance c o e f f i c i e n t s . Method C, however, gave r e s u l t s which show good consistency. The d i f f e r e n c e between the maintenance values obtained from raw and corrected data can stem from the f a c t that carbon recoveries of 95, 93 and 93 % were obtained f o r experiments FB 830, FB 016 and FB 211, r e s p e c t i v e l y . The missing carbon was probably r e t a i n e d i n broth as c o 2 being trapped i n c e l l s and/or p a r t i c i p a t i n g i n the carbonate-bicarbonate b u f f e r system. The s t a t i s t i c a l l y c orrected data y i e l d e d the most c o n s i s t e n t and therefore r e l i a b l e estimates of maintenance c o e f f i c i e n t s .

The mg values c a l c u l a t e d and found to be more co n s i s t e n t are much smaller than those reported i n l i t e r a t u r e . 3 4 1 3 5 , 3 6 j n p i g u r e g j ^ data f o r FB 211 i s given together w i t h the r e s u l t s of simulations c a r r i e d out wi t h d i f f e r e n t ms

values. As can be seen, up to t = 865 min (sample 7), the model w i t h an ms

value taken from l i t e r a t u r e f i t s the experimental data w e l l ; t h e r e a f t e r ,

53

a smaller mg value gives a b e t t e r f i t . I f samples 6 and 7 can be regarded as o u t l i e r s , a smooth curve f i t s the M x data. However, a s i m i l a r behaviour has also been observed i n experiment FB 016. This almost abrupt change suggests a s h i f t i n c e l l metabolism, p o s s i b l y t r i g g e r e d by a c o n t r o l mechanism which i s capable of i d e n t i f y i n g a p o t e n t i a l s t a r v a t i o n . The c e l l s seem to become more e f f i c i e n t .Consumption of i n t e r n a l carbohydrate storage m a t e r i a l may provide an explanation f o r t h i s s h i f t . Unfortunately, no s i g n i f i c a n t changes i n the elemental composition or carbohydrate content of the biomass could be detected. The i n t e r e s t i n g thing to note i s that ms values reported i n l i t e r a t u r e were a l l obtained from continuous c u l t u r e data c o l l e c t e d at U > 0.05 h r - ' a f t e r a hazardous e x t r a p o l a t i o n . This may be the reason f o r the s i g n i f i c a n t d i f f e r e n c e s observed i n maintenance values (see Table I I I )

The experimental ms value c a l c u l a t e d by balancing around the r e l a t i v e l y steady Mj peak al s o f a i l s to give a good f i t . (see Figure 8)

Fig. 8: Experiment FB 211, simulation of the experiment with various ms values. Other parameters same as before.

To i n v e s t i g a t e the behaviour of the system f u r t h e r , m a t e r i a l balances were constructed f o r eight time i n t e r v a l s and ms and mQ were c a l c u l a t e d as described p r e v i o u s l y by method C. The new values of Y m a x and Y m a x used f o r these balances were 0.524 kg/kg and 35.57 x 10 J kg/mole, r e s p e c t i v e l y . These were c a l c u l a t e d from from our previous batch data, assuming true values of ms and m 0 as 3.53 x 1 0 - 2 kg/kg/hr and 1.34 mole/kg/hr, r e s p e c t i v e l y . mg and m 0

values c a l c u l a t e d f o r these time i n t e r v a l s are p l o t t e d i n Figure 9. Although there i s considerable s c a t t e r , the trends i n these p l o t s suggest decreasing maintenance r a t e s . A s t a t i s t i c a l t e s t r e j e c t s the hypothesis of zero slope at 95 % l e v e l f o r both p l o t s .

N e i j s s e l - ^ has already speculated about the p o s s i b i l i t y of maintenance being a f u n c t i o n of the growth r a t e . Pipyn and Verstraete^S have reported lower ms

values f o r the a c t i v a t e d sludge growing at very low growth rates compared w i t h those grown at higher growth r a t e s . Recently, van Verseveld39 has mentioned t h i s p o s s i b i l i t y and speculated that maintenance i s most probably a l i n e a r f u n c t i o n of the s p e c i f i c growth r a t e . The r e s u l t s reported i n t h i s work add to the newly accumulating speculations of reduced maintenance requirements at very low growth r a t e s . Whether the a c t u a l reduction of the apparent maintenance requirements at low growth rates i s a consequence of adaptation or not has s t i l l to be studied. In view of t h i s s t a t e of the accumulating i n f o r m a t i o n , 54

Fig. 9: Experiment FB 211: Maintenance coefficients as calculated by balancing for different time intervals during the fed-batch growth phase. Coefficients of maintenance on (O) substrate; (0) oxygen.

f u r t h e r i n v e s t i g a t i o n s i n t o the growth phenomenon at low growth rates under ac c u r a t e l y c o n t r o l l e d environmental c o n d i t i o n s are necessary. Fed-batch c u l t u r i n g technique lends i t s e l f f o r t h i s purpose qu i t e s u c c e s s f u l l y . Further research i n t h i s area i s needed.

CONCLUSIONS

1) The fed-batch c u l t i v a t i o n technique i s a u s e f u l t o o l f o r the determination of b i o k i n e t i c parameters and the study of b i o e n e r g e t i c s , simultaneously.

2) The unstructured model presented describes the behaviour of the system f a i r l y w e l l during the exponential phase and the pseudo-steady s t a t e . However, the model f a i l s to hold during the t r a n s i t i o n p e r i o d . A s t r u c t u r e d model should be t r i e d f o r a b e t t e r d e s c r i p t i o n of the system during t h i s p e r i o d .

3) The model presented can s a f e l y be used f o r i n d u s t r i a l design purposes as i t p r e d i c t s conservative estimates f o r oxygen uptake and carbon d i o x i d e production r a t e s ; i . e . , f o r a p a r t i c u l a r a p p l i c a t i o n the model demands a mass t r a n s f e r capacity higher than a c t u a l l y r e q u i r e d .

4) Maintenance requirements determined i n fed-batch c u l t u r e s were found to be l e s s than those reported i n the l i t e r a t u r e which were obtained from c o n t i ­nuous c u l t u r e data. This f i n d i n g may have important p h y s i o l o g i c a l and economical i m p l i c a t i o n s .

5) Further i n v e s t i g a t i o n s i n t o the growth behaviour of slow growing c u l t u r e s are necessary, as there i s accumulating s p e c u l a t i v e evidence about maintenance requirements being lower at low growth r a t e s .

55

APPENDIX

Volume changes during a fed-batch experiment can of t e n be s i g n i f i c a n t . At labora t o r y s c a l e , changes i n c u l t u r e volume due to feed a d d i t i o n , samples taken evaporation, a c i d / a l k a l i a d d i t i o n e t c . can amount up to 15 % of the i n i t i a l volume. When the r e s u l t s of such experiments are to be compared w i t h the outcomes of the mathematical s i m u l a t i o n s , one of the sets of data has to be corrected. In the f o l l o w i n g two procedures f o r c o r r e c t i o n are described.

Method 1: Say at t = t s i , a sample of volume V s] and composition C x j and C s] i s taken. At t = t s j , t h e r e f o r e , the t o t a l amount of biomass and substrate removed from the fermentor are V s i . C x] and V S ] . C S ] , r e s p e c t i v e l y . Therefore at any time t = t g n , the t o t a l amount of biomass taken out of the fermentor t i l l then can be given by

n

S i m i l a r l y f o r substrate n

v . c . I—I S I S I

i These cumulatives can be p l o t t e d as a f u n c t i o n of fermentation time. I f the curves obtained i n t h i s manner are added/subtracted to the experimental data, the "no sampling" case can be obtained as a f i r s t approximation. This method of c o r r e c t i o n i s p a r t i c u l a r l y u s e f u l f o r m a t e r i a l balancing. However, such a c o r r e c t i o n w i l l d i s t o r t the k i n e t i c s shown by the experimental data. For comparison purposes, i t i s therefore b e t t e r to substract the c o r r e c t i o n curves obtained from the curves p r e d i c t e d by model s i m u l a t i o n w i t h no sampling.

R e a d t s U) .V s ( I ) .C x ( I )

r E H

M s ( t ) = M s ( t ) - V S ( I ) .C S (U M x ( t ) = M x ( t ) - V S(I).C X(I)

Fig. 10: Correction procedure for sampling in the simulation program.

56

Method 2: A more accurate approach i s to allow f o r sampling w i t h i n the simula­t i o n program. This would y i e l d the most accurate r e s u l t s since the i n t e g r a t i o n s are c a r r i e d out w i t h the corrected values a f t e r each sampling. The simu l a t i o n s presented i n t h i s study have been executed i n t h i s manner. A s i m p l i f i e d schema­t i c flow diagram of t h i s c o r r e c t i o n procedure i s shown i n Figure 10.

NOMENCLATURE

C x biomass concentration (dry weight) (kg/m^) C s l i m i t i n g substrate concentration(kg/m3) F ( t ) rate of substrate input (kg/hr) k s Monod s a t u r a t i o n constant (kg/m3) K s product of k s and c u l t u r e volume (kg) ms maintenance c o e f f i c i e n t on substrate (kg/kg/hr) mD maintenance c o e f f i c i e n t on oxygen (mole/kg/hr) mc maintenance c o e f f i c i e n t on carbon d i o x i d e (mole/kg/hr) M x t o t a l mass of biomass i n the fermentor (kg) M s t o t a l mass of substrate i n the fermentor (kg) r s rate of substrate consumption (kg/m3/hr) r D rate of oxygen uptake (mole/m^/hr) r c rate of carbon dioxide production (mole/m3/hr) r x rate of biomass production (kg/m3/hr) R x t o t a l biomass production i n the fermentor (kg/hr) R s t o t a l substrate consumption i n the fermentor (kg/hr) R 0 t o t a l oxygen consumption i n the fermentor (mole/hr) R c t o t a l carbon d i o x i d e production i n the fermentor (mole/hr) RQ r e s p i r a t o r y quotient (dimensionless) V volume of the c u l t u r e (m3) W t o t a l mass of the c u l t u r e (kg) Y s x biomass y i e l d on substrate (kg/kg) Y o x biomass y i e l d on oxygen (kg/mole) Y c x biomass y i e l d on carbon d i o x i d e (kg/mole) OTax maximal y i e l d of biomass on i (kg/kg) (kg/mole) Y D degree of re d u c t i o n of biomass(equiv a v a i l a b l e e l e c t r o n s / g . atom) Y s degree of reduction of substrate (equiv a v a i l a b l e electrons/g. atom) \i s p e c i f i c growth rate ( h r - ' ) Pmax maximum s p e c i f i c growth r a t e ( h r - ' ) 8(t) r a t e of feed s o l u t i o n input (kg/hr) a D carbon weight f r a c t i o n i n biomass (dimensionless) Os carbon weight f r a c t i o n i n substrate (dimensionless) p B d e n s i t y of the broth (kg/m3)

REFERENCES

1. S.J. P i r t , J.Appl. Chem. Bi o t e c h n o l . , 24,415(1974). 2. F. Yoshida, T. Yamane and K. Nakamoto, B i o t e c h n o l . Bioeng.,15,257(1973) 3. R. K e l l e r and I . J . Dunn, J . Appl. Chem. Biotechnol.,28,508(1978) 4. R. K e l l e r and I . J . Dunn, J . Appl. Chem. Biotechnol.,28,784(1978) 5. S. Aiba, S. Nagai and Y. Nishizewa, Biotechnol.Bioeng.,18,1001(1976) 6. T. Yamane and S. Hirano, J . Ferment. Technol.,55,156(1977) 7. T. Yamane and S. Hirano, J . Ferment. Technol.,55,380(1977) 8. T. Yamane, T. Kume, E. Sada and T. Takamatsu, J . Ferment. Technol.,55,587

(1977) 9. T. Yamane, E. Sada and T. Takamatsu, B i o t e c h n o l . Bioeng.,21,111(1979)

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10. S.J. P i r t , Ann. N.Y. Acad. S e i . ,0077-8223/0326-119(1979) 11. I . J . Dunn, S. Shioya and R. K e l l e r , Ann. N.Y. Acad. Sei.,0077-8923/0326-

0127(1979) 12. H.C. Lim, B.J. Chen and C. Creagen, B i o t e c h n o l . Bioeng., 19,425(1977) 13. J . Monod, Recherches sur l a croissance des c u l t u r e s bacteriennes (Hermann,

Paris,1942) 14. A.H. Stouthamer and C. Bettenhaussen, Biochim. Biophys. Acta,301,53(1973) 15. J.A. Roels and N.W.F. Kossen,"On the modelling of m i c r o b i a l metabolism,"in

Progress i n I n d u s t r i a l M i c r o b i o l o g y , M.J. B u l l , E d . ( E l s e v i e r , Amsterdam, 1978)

16. I.G. Minkevich and L . I . Utkina, B i o t e c h n o l . Bioeng.,21,357(1979) 17. A.G. Marr, E.H. N i l s o n and D.J. C l a r k , Ann. N.Y. Acad. Sei.,102,536(1963) 18. C.G.T. Evans, D. Herbert and D.W. Tempest, i n Methods i n M i c r o b i l o g y , J.R.

No r r i s and W.W.Ribbons, Eds.(Academic press, London, 1970)vol.2,p.313. 19. J.P. Barford and R.J. Hall,Exp. C e l l . Res., 102,276(1976) 20. W. de V r i e s and A.H. Stouthamer, J . Bacteriol.,96,472(1968) 21. H.E. de Kok and J.A. Roels, B i o t e c h n o l . Bioeng.,22,1097 (1980) 22. F. Madron and V. Vanecek, C o l l e c t . Czech. Chem. Commun.,42,1805(1977) 23. F. Madron, V. Veverka and V. Vanecek, AIChE J.,23,482(1977) 24. F. Madron, Biotec h n o l . Bioeng.,21,1478(1979) 25. A.K.S. Murthy, Ind. Eng. Chem. Proc. Res. Dev.,12,246(1973) 26. A.K.S. Murthy, Ind. Eng. Chem. Proc. Res. Dev.,13,347(1974) 27. D. Herbert, Continuous C u l t u r e , vol.6,(1975) 28. L.E. E r i c k s o n , I.G. Minkevich and V.K. Ero s h i n , B i o t e c h n o l . Bioeng.,20,1595

(1978) 29. D.M. Himmelblau, Process A n a l y s i s by S t a t i s t i c a l Methods(Wiley-Interscience

New York,1979) 30. D.W. Marquart.J. Soc. Ind. Appl. Math.,2,431(1963) 31. A.A. Esener, J.A. Roels and N.W.F. Kossen, B i o t e c h n o l . Bioeng.,22,1979(1980) 32. J.P.Barford and R.J. H a l l , B i o t e c h n o l . Bioeng.,21,609(1979) 33. S.W. F i t z p a t r i c k , P h . D . t h e s i s , U.M.I.S.T.,1977 34. J.J.Heijnen and J.A. Roels, B i o t e c h n o l . Bioeng.,23,739(1981) 35. D. Herbert, Symp. I n t . Congr. Microbiol.,6,38(1958) 36. A.H. Stouthamer, Symp. Soc. Gen. Microbiol.,27,285(1977) 37. O.M. N e i j s s e l , Ph.D. Thesis, Amsterdam U n i v e r s i t y , 1976 38. P. Pipyn and W. V e r s t r a e t e , B i o t e c h n o l . Bioeng.,20,1883(1978) 39. H. van Verseveld, Ph.D. t h e s i s , Amsterdam Free University,1979

58

CHAPTER 5

GROWTH OF MONO AND MIXED CULTURES IN SALINE ENVIRONMENT *

A.A. Esener, N.W.F. Kossen and J.A. Roels

ABSTRACT

The e f f e c t s of s a l i n i t y on the k i n e t i c s and energetics of mono c u l t u r e s were studied i n batch. Results were compared with those reported f o r a c t i v a t e d sludge. I t was shown that the response of both mono and mixed c u l t u r e s to increased s a l i n i t y followed a s i m i l a r p a t t e r n but the magnitudes of the e f f e c t s d i f f e r e d s i g n i f i c a n t l y . S a l i n i t y was shown to be an important parameter i n f l u e n c i n g the k i n e t i c s and energetics of b i o l o g i c a l systems.

KEYWORDS

S a l i n i t y ; mono c u l t u r e ; mixed c u l t u r e ; e n e r g e t i c s ; k i n e t i c s ; b i o l o g i c a l waste water treatment.

INTRODUCTION

There are s e v e r a l processes that produce b r i n e s . These processes i n c l u d e d i s t i l l a t i o n of sea water, production of s a l t s , water s o f t e n i n g by i o n exchange, reverse osmosis, e l e c t r o d i a l y s i s , p i c k l i n g , canning, cheese and f i s h -meal manufacturing e t c . Some of the wastes from these processes and/or combina­t i o n of them w i t h domestic waste waters present a s p e c i a l case f o r the conven­t i o n a l b i o l o g i c a l waste water treatment f a c i l i t i e s . The major load of s a l i n e waste water, however, a r i s e from the use of sea water f o r domestic purposes and as a c a r r i e r of domestic and i n d u s t r i a l waste at c o a s t a l l o c a t i o n s w i t h l i m i t e d supply of fr e s h water. S a l i n e wastes are also generated and has to be treated on board of marine v e s s e l s and off-shore i n s t a l l a t i o n s . Discharge of waste water i n t o seas and s a l t lakes i s also common. Study of m i c r o b i a l growth i n s a l i n e environment i s therefore of p r a c t i c a l importance.

* Paper presented to the Second I n t e r n a t i o n a l Symposium on Waste Treatment and U t i l i z a t i o n , held at Waterloo, Canada(June 1980)

59

The e f f e c t s of the presence of inorganic s a l t s on m i c r o b i a l growth have f i r s t been studied by m i c r o b i o l o g i s t s , p a r t i c u l a r l y f o r the formulation of s u i t a b l e growth media f o r exacting organisms and mammalian c e l l (Ingram, 1939; P i r t and Tackeray, 1964; Sc o t t , 1953; Wodzinski and F r a z i e r , 1960) . From these studies i t can be concluded that a d d i t i o n of s a l t s , i n most cases NaCl, increased the r e s p i r a t i o n r ate up to a s p e c i f i c s a l t concentration, t h e r e a f t e r a decrease was observed. Ingram (1939)concluded that c a t i o n of the s a l t s was of importance i n determining the r e s p i r a t i o n r a t e , but t h i s f i n d i n g was not confirmed by others. I t was g e n e r a l l y agreed that maximum adaptation was obtained when the s a l i n i t y of the medium was r a i s e d g r a d u a l l y .

In waste water f i e l d the e f f e c t of NaCl on a c t i v a t e d sludge and t r i c k l i n g f i l t r a t i o n processes have been studied (Burnett, 1974;Imai, Endoh and Kobayashi 1979a, 1979b; Kessick and Manchen, 1976; Kincannon and Gaudy, 1966, 1968; Kincannon, Gaudy and Gaudy, 1966; Lawton and Eggert,1957; Ludzack and Noran, 1965; Tokuz and Eckenfelder, 1979). The r e s u l t s of most of these studies confirm that high concentrations and/or shocks of NaCl have adverse e f f e c t s on the performance of the treatment p l a n t s . Small shocks or gradual increases i n s a l t l e v e l s had l i t t l e e f f e c t on the system performance and u s u a l l y the system ret a i n e d i t s o r i g i n a l a c t i v i t y a f t e r some adaptation time. Studies w i t h mixed c u l t u r e s , however, a l l suffered from the same phenomena ; adaptation, s e l e c t i o n and predominance of species prevented the experimenter to draw pure cause-response r e l a t i o n s h i p s from t h e i r experiments e.g. Kincannon and Gaudy (1968) have observed a tremendous increase i n biomass y i e l d (75 %) i n the presence of 8 kg/m3 NaCl but were not able to i d e n t i f y whether t h i s increase was due to a change i n the e f f i c i e n c y of m i c r o b i a l metabolism or s e l e c t i o n of s a l t t o l e r a n t species. Moreover, there i s l i t t l e data on the k i n e t i c s of m i c r o b i a l growth i n s a l i n e waters.

A more fundamental approach has, t h e r e f o r e , been adopted i n t h i s study. F i r s t the growth behaviour of Klebsiella pneumoniae (aevogenes); a bacterium commonly present i n s o i l and waste waters, was studied i n batch c u l t u r e s . With t h i s approach the pure response of one of the p o s s i b l e species present i n a c t i v a t e d sludge communities was determined. The r e s u l t s obtained were then compared wi t h those reported f o r a c t i v a t e d sludge growing under s i m i l a r c o n d i t i o n s (Imai, Endoh and Kobayashi, 1979a). I t i s shown that although the responses of mono and mixed c u l t u r e s followed a s i m i l a r p a t t e r n , the magnitude of responses changed considerably. P r e d i c t i o n of the amount of biomass which i s to be expected due to breakdown of organic matter and the corresponding oxygen demand i s of great importance i n the design and operation of treatment p l a n t s . Therefore various y i e l d s were evaluated as functions of NaCl concentration i n the growth environment. P r a c t i c a l i m p l i c a t i o n s of these f i n d i n g s are discussed.

MATERIALS AND METHODS

Klebsiella pneumoniae (aevogenes) NCTC 418 was c u l t i v a t e d a e r o b i c a l l y i n simple s a l t s medium w i t h g l y c e r o l being the only carbon source. A l l experiments were performed i n a 11 x 10~3 m3 working volume fermentor i n batch mode. Tempera­ture and pH were set and c o n t r o l l e d at 308 K and 6.8, r e s p e c t i v e l y . Inoculum was provided by an e x p o n e n t i a l l y growing c u l t u r e i n NaCl free medium. S p e c i a l a t t e n t i o n was paid f o r the accurate determination of gas flows and concentra­t i o n s . Elemental composition of biomass was determined by an element analyser. Oxygen content was found by d i f f e r e n c e . Ash content was determined separately. No carbon containing by-product could be detected at s i g n i f i c a n t q u a n t i t i e s . Amount of biomass co n t a i n i n g 12 grams of carbon was defined as 1 mole of b i o ­mass. Results were treated and corrected by a s t a t i s t i c a l procedure as reported elsewhere (de Kok and Roels,1980).

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RESULTS

Experimental r e s u l t s of mono c u l t u r e s were obtained i n t h i s l a b o r a t o r y . Results of a c t i v a t e d sludge, named as mixed c u l t u r e h e r e a f t e r , were obtained from a p u b l i c a t i o n by Imai, Endoh and Kobayashi (1979a). These authors studied the response of a c t i v a t e d sludge to i n c r e a s i n g concentrations of NaCl i n a respirometer. They used c h l o r i d e concentration as a parameter, therefore, t h e i r r e s u l t s were converted by us i n order to ob t a i n NaCl dependence. The experiements were performed i n 0 - AO kg/m3 NaCl range which also covered the sea water case (approx. 33 kg/m3) .

M m a x

H m a x

0.5

0 0 10 20 30 40

»- [ N a C l ] C k g / m 3 )

Fig. 1: Maximum specific growth rate vs. NaCl concentration.

In F i g . 1, the normalized U m a x , the maximum s p e c i f i c growth rate as defined by Monod k i n e t i c s , i s shown as a f u n c t i o n of NaCl concentration i n the growth medium together with the 95 % confidence l e v e l s (the s u p e r s c r i p t 0 denotes the value obtained i n the NaCl free medium. y m a x was determined from the dry weight data by nonlinear r e g r e s s i o n . Imai, Endoh and Kobayashi observed a maximum y m a x i n the low NaCl range. Such a maximum was not observed i n t h i s study. However, i t must be noted that the f i r s t two data points have over­lapping confidence i n t e r v a l s .

Table Is Average elemental composition of K. pneumoniae.

c H N 0 Formula

49.68 6.71 13.45 30. 16 C H l . 6 2 N0.23 °0.46

% ash f r e e dry weight. Average ash content 8.63%.

In F i g . 2, l a g time as defined by Dean and Hinshelwood (1966) i s shown as a f u n c t i o n of the NaCl concentration. The inoculum concentration was not deter­mined f o r each experiment but was of the same order of magnitude. The observed lag times have, therefore been assumed to be only a f u n c t i o n of the NaCl concentration. The same assumption was als o adopted by Imai, Endoh and

61

Kobayashi (1979a).

1 1—

- L a g T i m e Chr)

1 1

- m o n o /

- A Imai et.al. /

/ mixecC*^

/ ^ • I l l • i i i

) 10 20 30 4 0

[ N a C l ] ( k g / m 3 )

Fig. 2: Observed lag time vs. NaCl concentration.

Respiratory a c t i v i t y as the t o t a l oxygen uptake, q Q , i s p l o t t e d i n F i g . 3. A maximum q 0 reported by Imai et a l . (1979a) was not observed i n t h i s study.

Fig. 3: Oxygen uptake rate vs. NaCl concentration.

No s i g n i f i c a n t change i n the elemental composition of biomass could be detected i n the experimental range and an average formula as shown i n Table I , was used f o r y i e l d c a l c u l a t i o n s .

Y i e l d s of biomass on sub s t r a t e , Y s x , oxygen Y o x and carbon d i o x i d e , Y c x , are presented i n F i g . 4. A l l y i e l d s are expressed as mole/mole. S l i g h t l y d i s t i n c t optima i n y i e l d values can be observed at about 5 kg/m3 NaCl. The presence of a corresponding minimum i n the r e s p i r a t o r y q u o t i e n t , RQ, ensures the existence

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of an optimal y i e l d range ( F i g . 5.). I t therefore becomes evident that the optimal y i e l d i s not achieved at the highest growth r a t e but at about 0.9u where the NaCl concentration i s 5 kg/m.3. m

Fig. 4: Influence of NaCl concentrations on biomass yields.

Respiratory Quotient

|RQ

Fig. 5: Respiratory quotient as affected by NaCl concentration.

DISCUSSION

Comparison of the Behaviour of Mono and Mixed Cultures

From data presented i n F i g s . l , 2 and 3, i t can be concluded that both mono and mixed c u l t u r e s show s i m i l a r behaviour i n s a l i n e environment. Mono c u l t u r e s , however, are shown to be much more s e n s i t i v e to i n c r e a s i n g NaCl concentrations. From 0 to 40 kg/m3 N a C l , y m a x of the mono c u l t u r e decreased by about 10 times, whereas that of mixed c u l t u r e by 5. Lag time and r e s p i r a t i o n r a t e data a l s o showed a s i m i l a r p a t t e r n . The functions adversely a f f e c t e d by the presence of NaCl showed a smooth l o s s of a c t i v i t y i n mixed c u l t u r e s , throughout the experimental range.Mono c u l t u r e s , however, seemed to have a c r i t i c a l s a l i n i t y range above which the c u l t u r e a c t i v i t y slowed down d r a s t i c a l l y (15-20 kg/m.3) . P o s s i b l y above t h i s l e v e l s i g n i f i c a n t damage i s done to the c e l l membrane by the osmotic pressure of i t s e x t e r i o r . The r e l a t i v e success of .mixed c u l t u r e s i n

63

s a l i n e environment i s most p o s s i b l y due to adaptation and predominance of s a l t t o l e r a n t species. This p o s s i b i l i t y has already been pointed at by Kincannon and Gaudy (1968). These authors have reported an increase i n biomass y i e l d (about 75 %) at 8 kg/m3. However, i n t h i s work with mono c u l t u r e s only a s l i g h t increase i n the y i e l d was observed (< 5 %) at the same NaCl range. Based on these observation i t i s u n l i k e l y that any other species could increase i t s y i e l d at such amounts. This means that increase i n y i e l d observed by Kincannon and Gaudy (1968) was mainly due to changes i n the predominance of species.

Energetic Considerations

In F i g . 6 , the thermodynamic e f f i c i e n c y of the growth process, n , i s shown as a f u n c t i o n of the s a l t concentration i n the medium.

Fig. 6: Thermodynamic efficiency of the growth process as a function of NaCl concentration.

T\ i s given by Roels (1980) as:

n = Y g x / Y < (.)

where Y g x i s the biomass y i e l d on substrate (mole/mole) and Y j i s the maximum p o s s i b l e value of Y s x c o n s i s t e n t w i t h the second law of thermo­dynamics, n describes the e f f i c i e n c y of the growth process by consi d e r i n g i t s r e v e r s i b i l i t y . I t i s important to note that the knowledge of n allows the st r a i g h t f o r w a r d estimation of the oxygen demand by :

Y o x = ( 4 / Y x } • n / ( i - n ) (2) where y x i s the degree of reduction of biomass (E r i c k s o n , Minkevich and Erosh i n , 1978; Roels, 1980)

From the data presented i t i s c l e a r that b i o s y n t h e s i s gets l e s s e f f i c i e n t at high NaCl l e v e l s . Although d e t a i l e d biochemical and p h y s i o l o g i c a l reasons f o r t h i s decrease are s t i l l not c l e a r , i t i s b e l i e v e d that the c e l l s i n s a l i n e environment have to do extr a work to maintain concentration gradients and v i a b i l i t y . This type of energy expenditure i s c u r r e n t l y accounted f o r , by the s o - c a l l e d "maintenance energy" requirements; a term which includes energy 64

expenditure f o r other functions too. S i g n i f i c a n t increases i n ms requirements i n h i g h l y s a l i n e media has already been reported i n l i t e r a t u r e (Stouthamer and Bettenhaussen, 1973; Watson, 1970). Unfortunately the current s t a t e of m i c r o b i a l energetics does not allow a p r e c i s e assessment of the d i r e c t c o n t r i b u t i o n of energy spent f o r osmotic work to maintenance requirements. Therefore, only a rough a n a l y s i s w i l l be reported here.

Assuming the l i n e a r r e l a t i o n f o r substrate consumption i s v a l i d , q s (mole/mole / h r ) , s p e c i f i c consumption r a t e , i s given by:

q = V I Y m 3 X + m (3) s sx s

where Y m a x i s the maximal y i e l d on substrate (mole/mole) and ms i s the mainte­nance c o e f f i c i e n t (mole/mole/hr). Here, both of these parameters may be influenced by s a l i n i t y . I f they are assumed constants, s t a t i s t i c a l a n a l y s i s of the experimental data y i e l d s the values of 2.09 (1.97 - 2.21) and 0.032 (0.023 - 0.040) f o r Y m a x and ms, r e s p e c t i v e l y ( 95 % confidence l i m i t s i n p a r a n t h e s i s ) . The maintenance value determined i s about 50 % higher than that reported i n l i t e r a t u r e by Herbert (1958) f o r the same organism i n s a l t f r e e medium. Although ms i s expected to be a f u n c t i o n of NaCl concentration the present data does not a l l o w the r e j e c t i o n of constant ms hypothesis. However, i t must be noted that the value of ms obtained i n t h i s study i s i n f l u e n c e d most by data c o l l e c t e d at low growth rates i . e . , at high s a l i n i t i e s .

% Carbon input I ICOn I I biomass

100-

50-

N a C l ( k g / m 3 )

5 10 15 20 25 30 40

Fig. 7: Effect of NaCl level on the distribution of substrate carbon.

Rewriting equation (3) as :

r / r = 1 / Y m a x + m / y (4) S X sx s

where r s and r x are the net flows of substrate and biomass to and from the system, r e s p e c t i v e l y , one can see the e f f e c t of i n c r e a s i n g NaCl l e v e l on the d i s t r i b u t i o n of the substrate input. I f Y ™ a x i s constant, e l e v a t i o n of NaCl content increases ms and decreases y ; thus the p o r t i o n of substrate used f o r non-growth associated functions get higher. This i s a l s o apparent from F i g . 7

65

where the d i s t r i b u t i o n of the substrate carbon input i n the system i s shown.

P r a c t i c a l Aspects

Generally high sludge production i s undesirable i n treatment i n s t a l l a t i o n s . Removal and d i s p o s a l of sludge may present f i n a n c i a l and environmental problems. I t i s , t h e r e f o r e , d e s i r a b l e to operate an a c t i v a t e d sludge plant at low growth rates as to increase maintenance requirements of the c u l t u r e , thereby reducing the amount of sludge formed. From t h i s point of view, high s a l i n i t y i s a d e s i r a b l e property. However, from F i g . 6 one must remember that the thermodynamic e f f i c i e n c y i s g r e a t l y reduced at high s a l i n i t i e s . A consequent decrease i n Y o x , as p r e d i c t e d by eq. (2.) , w i l l then c a l l f o r higher a e r a t i o n c a p acity. Thus the engineer w i l l be faced w i t h an o p t i m i z a t i o n problem, the s o l u t i o n of which, of course, depends on the r e l a t i v e costs i n v o l ­ved. In F i g . 9, Y s x shows the amount of sludge formed per mole of substrate consumed and Y s x / Y o x , i s the amount of oxygen taken up per mole of substrate consumed. Therefore i t i s d e s i r a b l e to minimize both.

Fig. 9: Graphical representation of the optimization problem.

CONCLUSIONS

1. Mono and mixed c u l t u r e s show s i m i l a r responses to increase i n s a l i n i t y i n t h e i r environments. Mono c u l t u r e s , however, are much more s e n s i t i v e and therefore l e s s e f f i c i e n t i n such environments.

2. S a l i n i t y has s i g n i f i c a n t e f f e c t s on the energetics and k i n e t i c s of growth process and therefore i s an important parameter.

3. Design c a l c u l a t i o n s based on data obtained from mono c u l t u r e s should a l l o w f o r the d i f f e r e n c e s observed between the s e n s i t i v i t y of mono and mixed c u l t u r e s .

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4. For e f f i c i e n t operation an o p t i m i z a t i o n must be c a r r i e d out to minimize sludge production while maximizing y i e l d on oxygen.

REFERENCES

Burnett, W (1974). The e f f e c t of s a l i n i t y v a r i a t i o n s on the a c t i v a t e d sludge process., Wat. Sewage Wks., 37-55.

Dean, A.C.R., and C. Hinshelwood(1966). Growth Function and Regulation i n B a c t e r i a l C e l l s . , Clarendon Press, Oxford, pp. 55-68.

E r i c k s o n , L.E., I.G. Minkevich, and V.K. Eroshin (1978). A p p l i c a t i o n of mass and energy balance r e g u l a r i t i e s i n fermentation., B i o t e c h n o l . Bioeng.,20, 1595-1621.

Herbert, D. (1958) Some p r i n c i p l e s of continuous c u l t u r e . In G. Tunewall (Ed), Recent Progress i n Microbiology, Almquvist and W i n k s e l l , Stockholm,p.381.

Imai, H., K. Endoh and J . Kobayashi (1979a).Effects of high s a l i n i t y on the r e s p i r a t i o n c h a r a c t e r i s t i c s of a c t i v a t e d sludge. J . Ferment. Technol.,57(4) 333-340.

Imai, H., K. Endoh, and C.Kobayashi (1979b) Respiratory a c t i v i t y and sludge volume index of a c t i v a t e d sludge during a c c l i m a t i o n to s a l i n e water., J . Ferment. Thechnol.,57(4), 453-459.

Ingram, M. (1939)The endogeneous r e s p i r a t i o n of B. cereus.,J. B a c t e r i o l . , 3 8 , 613-629.

Kessick, M.A., and K.L. Manchen (1976) S a l t water domestic waste treatment. , J. Wat. P o l l u t . C o n t r o l . Fed., 48(9), 2131-2136.

Kincannon, D.F., and A.G. Gaudy (1966) Some e f f e c t s of high s a l t c o ncentration on a c t i v a t e d sludge. J . Wat. P o l l u t . C o n t r o l . Fed.,38(7), 1148-1159.

Kincannon, D.F., A.F. Gaudy and A.G. Gaudy (1966) Sequential substrate removal by a c t i v a t e d sludge. B i o t e c h n o l . Bioeng.,8,371-378.

Kincannon, D.F., and A.G. Gaudy (1968) Response of b i o l o g i c a l waste water t r e a t ­ment systems to changes i n s a l t concentrations. B i o t e c h n o l . Bioeng., 10,483-496.

de Kok, H.E., and J.A. Roels (1980) Method f o r the s t a t i s t i c a l treatment of elemental and energy balances. Biote c h n o l . Bioeng.,22, 1097-1104.

Lawton, G.W., and Eggert (1957) E f f e c t of high sodium c h l o r i d e concentration on t r i c k l i n g f i l t e r slimes. Sewage and Ind. Wastes, 29(11), 1228.

Ludzack, F.J., and D.K. Noran (1965) Tolerance of high s a l i n i t i e s by conven­t i o n a l waste water treatment processes.,J. Wat. P o l l u t . C o n t r o l . Fed., 37(10), 1404-1416.

P i r t , S.J., and Thackeray (1964). Environmental i n f l u e n c e s on the growth of erk mammalian c e l l s i n mono laye r c u l t u r e s . Exp. C e l l Res.,33, 396-405.

Roels, J.A. (1980) Bioengineering r e p o r t , A p p l i c a t i o n of macroscopic p r i n c i p l e s to m i c r o b i a l metabolism, B i o t e c h n o l . Bioeng., 22, 2457

Scott, W.J. (1953) Water r e l a t i o n s of S. aeureus at 30OC, Aust. J . B i o l . S c i . 6, 549-564.

Stouthamer, A.H. and C. Bettenhaussen (1973) U t i l i z a t i o n of energy f o r growth and maintenance i n continuous and batch c u l t u r e s of microorganisms. Biochim Biophys. Acta.,301, 54-69.

Tokuz, R.Y. and W.W. Eckenfelder (1978) The e f f e c t of inorganic s a l t s on the ac t i v a t e d sludge process performance. Water Res.,13, 99-104.

Watson, T.G. (1970) E f f e c t s of sodium c h l o r i d e on steady s t a t e growth and metabolism of S. c e r e v i s i a e . , J . Gen. M i c r o b i o l . , 64,91-98.

Wodzinski, R.J., and W.C. F r a z i e r (1960) Moisture requirements of b a c t e r i a . J . B a c t e r i o l . , 79, 572-578.

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I CHAPTER 6

THE INFLUENCE OF TEMPERATURE ON THE MAXIMUM SPECIFIC GROWTH RATE OF KLEBSIELLA PNEUMONIAE *

A.A. Esener, J.A. Roels and N.W.F. Kossen

INTRODUCTION Temperature i s an important environmental parameter f o r m i c r o b i a l growth. Mic­roorganisms, u n l i k e higher organisms l i k e mammals, do not possess the c a p a b i l ­i t y of r e g u l a t i n g t h e i r i n t e r n a l temperature. In m i c r o b i a l c u l t u r e s the c e l l temperature must become equal to the environmental temperature. Therefore, a l l the biochemical reactions taking place i n the c e l l are a f f e c t e d by the tempe­ra t u r e . I t has long been known that the temperature influences the nature of metabolism, the n u t r i t i o n a l requirements and the biomass composition i n a d d i t i o n to i t s primary e f f e c t ; changing the r e a c t i o n r a t e s .

I f the maximal s p e c i f i c growth r a t e of microorganisms i s l i m i t e d by the r e a c t i o n rate of one s p e c i f i c enzymatic r e a c t i o n i n a complex sequence as postulated by the Monod^ model of growth, a mathematical r e l a t i o n between the absolute tem­perature and the maximal growth rate can be sought a f t e r . This has already been studied by many authors and Arrhenius type expressions have been derived but found to be a p p l i c a b l e w i t h i n only a l i m i t e d range.4-7

In t h i s study the i n f l u e n c e of the temperature on the maximum s p e c i f i c growth rate of the bacterium Klebsiella pneumoniae was studied i n fed-batch mode. Results were used f o r the determination of the thermodynamic parameters i n an Arrhenius type model extended to describe al s o the high temperature range where the maximum s p e c i f i c growth rate d e c l i n e s w i t h i n c r e a s i n g temperature.

MODEL

Assuming that the b a c t e r i a l growth i s an end product of a number of enzymatic re a c t i o n s and that one s p e c i f i c r e a c t i o n determines the o v e r a l l r e a c t i o n r a t e , temperature dependence of the maximum growth rate can be assumed to f o l l o w an Arrhenius r e l a t i o n of the f o l l o w i n g type :

r = A exp(AH*/RT) E C (1) x,max r I x

where r x i s the r e a c t i o n r a t e , C x i s the biomass concentration, E i s the weight f r a c t i o n of the s p e c i f i c enzyme i n biomass, AH, i s the a c t i v a t i o n enthalpy of

* Published i n Biotec h n o l . Bioeng. , 23 ,1 401 (1 981 >

69

the r a t e l i m i t i n g r e a c t i o n , A i s a constant, R i s the u n i v e r s a l gas constant and T i s the absolute temperature. I f the maximum s p e c i f i c growth rate,y i s defined i n the usual way i t follows from e q . ( l ) :

y = A exp(-AH* / RT) E (2) max ]

This r e l a t i o n s h i p has been shown to describe w e l l the experimental observations i n a l i m i t e d range, below the s o - c a l l e d optimal temperature.' At temperatures higher than optimum a negative c o r r e l a t i o n has been observed between the tempe­rature and y m a x . This phenomenon can be allowed f o r i n a generalized model i f the i n f l u e n c e of temperature on the a c t i v i t y of the enzyme involved i n the growth l i m i t i n g r e a c t i o n i s taken i n t o account.^ Assuming that t h i s enzyme can e x i s t i n two p o s s i b l e c o n f i g u r a t i o n s , an a c t i v e and an i n a c t i v e form i n e q u i l i b r i u m with each other, the e f f e c t of temperature on enzyme a c t i v i t y can be evaluated by con s i d e r i n g the a c t i v a t i o n - i n a c t i v a t i o n r e a c t i o n . I f the inac -t i v a t i o n r e a c t i o n i s f a s t , the f o l l o w i n g e q u i l i b r i u m r e l a t i o n s h i p can be formulated:

f x = f A K exp(-AH 2 / RT) (3)

where f A and f j are the f r a c t i o n s of the t o t a l amount of the enzyme being a c t i v e and i n a c t i v e , r e s p e c t i v e l y , A H 2 i s the enthalpy change of the i n a c t i v a t i o n r e a c t i o n and K i s a constant. From eq. (3), since f j + f A = 1, one obtains:

f A = {1 + K exp(-AH 2 / RT)}"' (4)

Furthermore, i f only the a c t i v e f r a c t i o n of the enzyme i s engaged i n the growth l i m i t i n g r e a c t i o n , from eq.(3) i t f o l l o w s t h a t :

r = A exp(-AH* / RT) f. E C (5) x, max 1 A x

then from eqs.(4) and (5) i n combination w i t h the d e f i n i t i o n of u the f o l l o w -, . max ing can be w r i t t e n :

A' exp (-AH* / RT) V = (6)

m a X 1 + K exp(-AH / RT)

MATERIALS AND METHODS

The organism, Klebsiella pneumoniae NCTC 418, was c u l t i v a t e d i n s y n t h e t i c medium as described by Evans et al.9 G l y c e r o l was used as the only carbon and energy source. I t was assayed en z y m a t i c a l l y . Dry weights were determined i n d u p l i c a t e s by the method of de V r i e s and Stouthamer.'0 Biomass was c o l l e c t e d on a 0.2 ym pore diameter membrane f i l t e r ( S a r t o r i u s SM 11307), washed with d i s t i l l e d water and d r i e d to constant weight at 378 K. Medium was s t e r i l i z e d by membrane f i l t r a t i o n through a 0.2 ym membrane f i l t e r and inoc u l a t e d with an inoculum a c t i v e l y growing at the experimental temperature to e l i m i n a t e any lag phase and the p o s s i b i l i t y of unbalanced growth.

Experiments were c a r r i e d out i n a 11 x 10 _3 m working volume fermentor. The

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pH was c o n t r o l l e d at 6.8 ± 0.05. The a i r flow rate to the fermentor was cont­r o l l e d by a thermal mass flow meter (Brooks 5811) at about 0.77 kg dry a i r / h r .

The c u l t u r e supernatant was checked f o r the presence of products other than biomass, carbon dioxide and water, but none could be detected at any s i g n i f i c a n t l e v e l . A l l samples were cooled during sampling to about 278 K by an o n - l i n e heat exchanger manufactured i n our workshop. The t y p i c a l residence time i n the exchanger was about 5-10 sees.

COMPUTATIONS

The maximum s p e c i f i c growth rate U m a x w a s determined from the dry weight data

c o l l e c t e d during the exponential phase. The upper boundary of the exponential phase was determined by p l o t t i n g the oxygen uptake rate data and e v a l u a t i n g the time at which the maximum i s reached. Then y m a x was determined by perform­ing nonlinear r e g r e s s i o n based on Marquarts algorithm.'' The same computer program was used f o r e v a l u a t i n g the parameters of the temperature- y m a x model, i . e . , eq.(6).

RESULTS AND DISCUSSION

Experimental r e s u l t s are shown i n Figure 1. together w i t h the 95 % confidence l e v e l s . The continuous l i n e represents the model evaluated w i t h the parameters given i n Table I.

T 1 1 r

Fig. 1: Maximum specific growth rate vs. temperature. (•) experiment; ( ) model equation (6) evaluated with parameters given in Table I.

In Figure 2 the n a t u r a l logarithm of y m a x i s p l o t t e d against the r e c i p r o c a l of temperature. Since there are no abrupt changes i n the slope of t h i s b e l l shaped curve, one can conclude that there has been no s i g n i f i c a n t changes i n the c e l l metabolism and that the same enzymatic r e a c t i o n remains the r a t e l i m i t i n g one throughout the experimental range. A supporting observation f o r t h i s was the absence of metabolic products at a l l temperatures.

From Figure 2 i t i s evident that a simple Arrhenius type expression can only

71

Table I : Calculated and reported model parameters.

* 95 % confidence l e v e l s

AH 86.40 kJ/mole (44.19 - 128.64) AH 2 287.78 kJ/mole (188.33 - 387.23) A' 5.69 . 1 0 1 4 1/hr

48 K 1.38 .10 (—)

Range

AH^ 343 kJ/mole 147 828

a From Morowitz, average f o r 20 p r o t e i n s .

Fig. 2: Natural logarithm of V>max vs. the reciprocal of absolute temperature.

describe a part of the experimental observations. I f such a simple expression i s assumed to hold up to the optimal temperature f o r f a s t e s t growth, the magnitude of the a c t i v a t i o n enthalpy, AH*, may be underestimated s i g n i f i c a n t l y . This i s why workers who d i d not allow f o r thermal enzyme i n a c t i v a t i o n process have reported low AH] values.'2

From eq.(6) the optimal c u l t i v a t i o n temperature ( i f the f a s t e s t growth i s the only con s i d e r a t i o n s ) can be shown to be given by:

T = AH 2 / {R ln(K (AH^AH* - 1))} (7)

A r e s u l t which i s obtained by s e t t i n g the f i r s t d e r i v a t i v e of y wi t h M -, J O M A X respect to T equal to zero.

I n s e r t i n g the parameters obtained one gets 36.9°C as the optimal temperature f o r t h i s system. An i n f l e c t i o n p o i n t of 31.7°C was c a l c u l a t e d by ev a l u a t i n g the

72

temperature at which the second d e r i v a t i v e ( d2y m a x/dT2) becomes zero. Up to t h i s p o i n t , increase i n V^ax i-s mainly determined by the simple Arrhenius type of expression, i.e.,the nominator of eq.(6). At higher temperatures the c o n t r i b u t i o n of the thermal enzyme i n a c t i v a t i o n process becomes s i g n i f i c a n t . This can best be v i s u a l i z e d from Figure 3 where the a c t i v e f r a c t i o n of the enzyme i s p l o t t e d as a f u n c t i o n of the growth temperature. As can be seen from

Fig. 3: Active fraction of the enzyme taking part in the growth limiting reaction as a function of temperature. Computed according to the model equation (4).

t h i s p l o t , at low temperatures almost a l l of the enzyme remains a c t i v e w h i l e a f t e r about 32°C a dramatic decrease i n the a c t i v i t y i s c a l c u l a t e d . In the range 33 < T < 38°C, changes r e l a t i v e l y l i t t l e as the c o n t r i b u t i o n s of the two processes balance each other. Therefore, the optimal operation temperature f o r an i n d u s t r i a l process can be chosen i n t h i s range f o r f a s t biomass production.

Morowitz has compiled a l i s t of enthalpy change values f o r the thermal i n a c t i ­v a t i o n of some 20 p r o t e i n s . x h e value of A H j c a l c u l a t e d i n t h i s study, 287 kJ/mole, compares w e l l w i t h h i s average of 343 kJ/mole (Table I ) . These r e s u l t s i n d i c a t e that the model can be used as a u s e f u l approximation to describe the temperature-reaction rate r e l a t i o n s h i p of m i c r o b i a l growth.

NOMENCLATURE

A, A' constants(hr~') Cx concentration (kg/m3) E weight f r a c t i o n of the s p e c i f i c enzyme i n jiomass (dimensionless) *A a c t i v e f r a c t i o n of the s p e c i f i c enzyme A H * a c t i v a t i o n enthalpy f o r the growth l i m i t i n g reaction(kJ/mole) A H 2 enthalpy change f o r enzyme i n a c t i v a t i o n r e a c t i o n (kJ/mole) K a constant(dimensionless) r x rate of r e a c t i o n (kg/m3/hr)

73

R T max

gas constant (kj/kg.mole) temperature (K) maximum s p e c i f i c growth rate ( h r - ' )

REFERENCES

1. S.J. P i r t , P r i n c i p l e s of Microbe and C e l l C u l t i v a t i o n , ( B l a c k w e l l , London, 1975)

2. D.W. Tempest, i n ' M i c r o b i a l Growth' 19 th Symp. Soc. Gen. M i c r o b i o l . , (Cambridge U.P., Cambridge,1969),p. 87.

3. J . Monod, Recherches sur l a Croissance des Cultures Bacteriennes(Hermann, P a r i s , 1942).

4. H.Topiwala and C. G. S i n c l a i r , B i o t e c h n o l . Bioeng.,13,795(1971) . 5. F. Watanable and S. Okada, J . C e l l . Biol.,32,309(1967). 6. A.C.R. Dean and C. Hinshelwood, Growth Function and Regulation i n B a c t e r i a l

C e l l s , ( C l a r e n d o n , Oxford, 1966). 7. A. Prokop and A. E. Humphrey,'Kinetics of D i s i n f e c t i o n ' , i n D i s i n f e c t i o n ,

M.A. Benarde, ed.,(Macel Dekker, New York, 1970). 8. H. E y r i n g and D.W. Urry, 'Thermodynamics and Chemical K i n e t i c s ' , i n

T h e o r e t i c a l and Mathematical Biology, T.H. Waterman and H.J. Morowitz, e d s . , ( B l a i n d e l l , New York, 1965).

9. C.G.T. Evans, D.Herbert and D.W. Tempest,in Methods i n Mic r o b i o l o g y , J.R. N o r r i s and W.W. Ribbons, eds.,(Academic, London, 1970), vol.2.,p.313.

10. de V r i e s and A.H. Stouthamer, J . B a c t e r i d ., 96,472, (1 968) . 11. D.W. Marquart, J . Soc. Ind. Appl. Mat.,2,431(1963). 12. J.H. Lee, D. Williamson and P.L. Rogers, B i o t e c h n o l . Lettr.,2(4),83(1980). 13. H.J. Morowitz, Energy Flow i n Biology,(Academic, New York, 1968),p.ll4.

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Addendum to Chapter 6:

Influence of Temperature on feg

Very l i t t l e has been published on the i n f l u e n c e of environmental f a c t o r s on the value of k s ; Monod s a t u r a t i o n constant. There i s by no means an agreement between the reported f i n d i n g s as to whether k s increases or decreases w i t h temperature.

Topiwala and S i n c l a i r have reported an inverse r e l a t i o n f o r k s with the environmental temperature.'^ They have obtained an associated a c t i v a t i o n energy of -49.4 kJ/mole by p l o t t i n g -In k s vs. 1/T(K). In contrast to t h e i r f i n d i n g s Marr et a l ' 5 have assumed k s to increase w i t h temperature i n accordance with the Arrhenius r e l a t i o n and estimated an E value of 20.9 kj/mole. In the waste water f i e l d Novakl6 has reviewed the temperature-substrate i n t e r a c t i o n i n f o r ­mation ; here k s-temperature r e l a t i o n s h i p was also expressed i n the Arrhenius form f o r a r e s t r i c t e d temperature range. For aerobic systems, k g increased w i t h temperature. In anaerobic systems, however, an inverse r e l a t i o n was observed (see Table I I ) .

Table I I : Temperature dependence of k s ; the a c t i v a t i o n energy values.

Organism substrate E (k s) kJ/mole T°C range r e f .

E. coli glucose 20.9 15 - 30 15 A. aerogenes glucose -49.4 25 - 40 14 Ac t i v a t e d sludge glucose 22.3 20 - 40 #

A c t i v a t e d sludge waste water 61 .2 Ac t i v a t e d sludge s y n t h e t i c waste 71.1 15 - 26 * A c t i v a t e d sludge l i n o e l i c a c i d 83.7 20 - 30 Anaerobic sludge a c e t i c a c i d -130.5 25 - 35 Anaerobic sludge complex waste -51.5 20 - 35 #

* C a l c u l a t e d from data c o l l e c t e d and reported by Novak'6.

I f the Monod s a t u r a t i o n constant i s assumed to be a r a t i o of the k i n e t i c constants, as i s assumed i n the homologous Michealis-Menten r e l a t i o n , i t may increase.or decrease with i n c r e a s i n g temperature depending on the s p e c i f i c r e a c t i o n .

Influence of lemperature on the Energetic Parameters

From a macro-energetic point of view, temperature may b a s i c a l l y be expected to i n f l u e n c e the observed y i e l d s through i t s e f f e c t on the maximal y i e l d s and maintenance c o e f f i c i e n t s . Provided that no s i g n i f i c a n t changes i n the metabo­l i c routes take place, Y m a x cannot be expected to be a strong f u n c t i o n of tem­perature. The r e s u l t s obtained i n t h i s study r e a f f i r m s t h i s statement(see F i g . 4) f o r the range 25 - 39° C. Considering the experimental e r r o r s ( r e f e r back to the j o i n t confidence area f o r Y ™ a x and mg ; Chapter 2, F i g . 1 0 ) Y m a x

can be assumed to be constant and independent of temperature. S i m i l a r f i n d i n g s have als o been reported by sev e r a l workers f o r mesophilic organisms.'4,17-20 Mainzer and Hempling'^ w i t h E. coli have found only s l i g h t changes i n Y m a x i n the range 17.5 - 32°C. S i m i l a r l y Y m a x values of 52.3 and 50.9 g.dry weight/mole g l y c e r o l , were reported f o r E. coli at 20.3 and 30 °C,

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r e s p e c t i v e l y . 1 " At 40 °C however, Y m a x was 35.3.'8 A s i m i l a r drop i n Y m a x was also observed i n t h i s study as shown i n F i g . 4. This may be due to some s i g n i f i c a n t change i n the b i o s y n t h e t i c pathways or due to uncoupling of the energy producing and consuming processes. 2'» 2 2 Based on t h e i r experimental studies and l i t e r a t u r e survey Farmer and Jones '8 generalized that i n E.coli and s e v e r a l other mesophilic b a c t e r i a , decreases i n growth e f f i c i e n c y occur p r i n c i p a l l y at temperatures which are roughly equal or higher than those which support the f a s t e s t growth. Results reported i n t h i s chapter g e n e r a l l y support t h i s statement.

/naze Fig. 4: Variation of 1^ with the growth temperature.

Fig. 5: Normalized maintenance coefficient versus the reciprocal of the absolute temperature.

Following the above d i s c u s s i o n , i t can be concluded that the v a r i a t i o n of the apparent y i e l d b a s i c a l l y i s a consequence of the strong dependence of mainte­nance requirements on temperature. In F i g . 5 maintenance c o e f f i c i e n t s norma­l i z e d f o r that at 25°C are shown as a f u n c t i o n of the c u l t i v a t i o n temperature. These experimental ( fed-batch experiments as described i n Chapter 4) f i n d i n g s can w e l l be described by an Arrhenius type of rate-temperature model as shown by the s t r a i g h t l i n e . Energy of a c t i v a t i o n associated w i t h maintenance f u n c t ­ions was c a l c u l a t e d to be 56.9 kj/mole (combined estimate) w i t h a 95 % confidence l e v e l of(50.2 - 63.6).From t h i s p l o t i t can c l e a r l y be seen that ms i s a strong f u n c t i o n of temperature and increases four f o l d f o r a 20°C increase i n temperature. Values of E (ms) are presented i n Table I I I f o r com­parison. These data i n d i c a t e that i n a l l cases ms i s a f u n c t i o n of T, however, the degree of dependence changes considerably f o r d i f f e r e n t systems. Another reason f o r such a wide range of r e s u l t s could be the e r r o r s introduced i n the c a l c u l a t i o n of m values. Unfortunately i n l i t e r a t u r e no information was provided about the accuracy or the 95 % confidence l e v e l s of the estimates reported. In t h e i r recent a r t i c l e Heijnen and R o e l s 2 ^ c o l l e c t e d a vast number of ms data and a l i n e a r r e g r e s s i o n a n a l y s i s y i e l d e d an E(m s) value of 39 kJ/mole w i t h 95 % confidence l e v e l of (21 - 57). Their value i s smaller than that reported here but the confidence l e v e l s are s t i l l overlapping.From t h i s d i s c u s s i o n and Table I I I one can conclude that the E(m s) values reported can only be used to obtain estimations of the order of magnitude of the ms values at d i f f e r e n t temperatures.

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Table I I I : Temperature dependence of m values; the a c t i v a t i o n energy data.

Organism substrate E (ms) kJ/mole T°C range r e f . E. coli g l y c e r o l 101 3 20 - 45 18 E. coli 284 6 18 C. lipolytica hexane 82 5 18 - 30 19 C. lipolytica hexane 86 2* 18 - 30 1 9 E. coli glucose 83 7 15 - 30 1 5 A. aerogenes glucose 37 7 25 - 40 14 K. pneumoniae g l y c e r o l 58 2 25 - 43 t h i s study K. pneumoniae g l y c e r o l 54 0* 25 - 43 t h i s study 11. polymovpha me thanol 117 2 35 - 43 20 B. caIdotenax glucose 66 5 50 - 60 24

*obtained from oxygen y i e l d data.

F i n a l l y , two s p e c i a l cases have to be pointed out; thermophilic b a c t e r i a seem to d i s p l a y temperature dependent behaviour, d i f f e r e n t from that reported above. Recent work of Kuhn et al.24 i n d i c a t e s that f o r thermophilic B. caldotenax, Y m a x and Y ™ a x are a l s o functions of temperature w i t h E's of 8.3 and 23.8 k J /

O X , O \ • «

mole, r e s p e c t i v e l y (T range of 50 - 60 C). With mixed c u l t u r e s , p a r t i c u l a r l y f o r aerobic a c t i v a t e d sludge an a n a l y s i s of temperature dependence i s not so c l e a r cut due to unavoidable changes i n the c u l t u r e , l i k e changes i n the pre­dominance of species e t c . Although there i s wide agreement on the temperature dependence of m f o r mixed c u l t u r e s , data on Y ™ a x shows no d e f i n i t e trend of change. In most reports some so r t of maximum i s reached at about 20°C.25

Consequences for Engineering Processes and Design

Temperature i s an important engineering parameter which can be c o n t r o l l e d q u i t e a c c u r a t e l y i n some processes e.g., productive fermentation, and not i n others, e.g., b i o l o g i c a l waste water treatment processes. In a l l of these pro­cesses i t s i n f l u e n c e i s important and o f t e n have i n t e r a c t i n g consequences. Few w i l l be c i t e d here : - From a process operation view p o i n t , the higher the fermentation temperature

the e a s i e r to remove the heat produced by metabolism and thus to c o n t r o l the b i o r e a c t i o n . This i s because of the temperature c o n s t r a i n t brought out by the c o o l i n g water. Cooling problems may become d i f f i c u l t to solve p a r t i c ­u l a r l y when sc a l i n g - u p , where the fermentor volume increases i n p r o p o r t i o n to the cube of i t s r a d i u s , whereas the a v a i l a b l e heat t r a n s f e r area increases i n p r o p o r t i o n to the square of the fermentor r a d i u s .

- In productive fermentations c a r r i e d out i n batch mode, where the d e s i r e d product i s biomass e.g. SCP , Bakers yeast processes, one aims to maximize the p r o d u c t i v i t y , i . e . , amount of biomass produced per u n i t substrate con­sumed per u n i t time, which i s a f u n c t i o n of the product of the maximum spe­c i f i c growth rate and the y i e l d value. In the suboptimal temperature range Umax increase with T but the y i e l d drops due to increased maintenance requirements. Therefore the optimal operating point has to be obtained by performing a cost o p t i m i z a t i o n a n a l y s i s .

- In continuous fermentation operations, the economics w i l l mainly be d i c t a t e d by the yield-temperature r e l a t i o n , provided that the process i s operated below y m a x - Influence of temperature i n continuous c u l t u r e w i l l be evident near the wash-out range, as the growth r a t e w i l l equal to the d i l u t i o n r a te below t h i s range.

- In contrast to the processes w i t h primary aim of biomass production, pro­cesses i n which the d e s i r e d end product i s c l o s e l y associated with the main-

77

tenance metabolism, e.g., ethanol production by Z. mob His, high temperatu­res would be advantageous. A d d i t i o n a l l y , a high fermentor temperature may f a c i l i t a t e e a s i e r separation of ethanol from the broth. Recently, much a t t e n t i o n has therefore been focused on t h i s aspect and p a r t i c u l a r l y on the s e l e c t i o n and use of thermophilic organisms.

- In the f i e l d of waste water treatment, temperature has received l i t t l e a t t e n t i o n , probably because of the lack of an economical method f o r chan­ging or c o n t r o l l i n g waste water temperatures. However, i n recent years c o n s i d e r a t i o n has been given to the f e a s i b i l i t y of using waste heat from power s t a t i o n s f o r c o n t r o l l i n g or at l e a s t i n c r e a s i n g the temperatures of waste streams. T h e o r e t i c a l l y , due to increased maintenance demands, high temperatures w i l l favour formation of l e s s sludge thus reducing the costs and problems associated w i t h the d i s p o s a l of i t . Increased temperature i n turn w i l l c a l l f o r increased a e r a t i o n capacity to s a t i s f y the increased r e s p i r a t i o n demands and due to the reduced s o l u b i l i t y of oxygen. Therefore, the problem w i l l again be one of o p t i m i z a t i o n f o r the desired o b j e c t i v e . The i n f l u e n c e of the treatment temperature on the s e t t l i n g p r o p e r t i e s and rates have yet to be stud i e d . C o l l i n s et a l . ^ 5 reported a comprehensive study of the in f l u e n c e of increased temperatures on the performance and e f f i c i e n c y of the a c t i v a t e d sludge process as a whole.

REFERENCES

14. H. Topiwala and C.G. S i n c l a i r , B i o t e c h n o l . Bioeng.,13,795(1971). 15. A.G. Marr, E.H. N i l s o n and D.J. Clark, Ann. N. Y. Acad. Sci.,102(3),536

(1963). 16. J.T. Novak, J . Water P o l l u t . Cont. Fed.,46,1984(1974). 17. S.E. Mainzer and W.P. Hempling, J . Bacterid.,126(1),251(1966). 18. I.S. Farmer and C.W. Jones, FEBS Letters,67(3),359(1976). 19. R. M o l e t t a , G. Goma and G. Durand, Arch. Microbiol.,118,293(1978). 20. C.L. Cooney and A. Makiguchi, i n Continuous culture,6,A.C.R. Dean, D.C.

Elwood, C.G.T. Evans and J . M e l l i n g , eds., ( E l l i s Horwood, Chichester, 1975)p.146.

21. J.C. Senez, B a c t e r i o l . Rev.,26,95(1962). 22. A.H. Stouthamer, I n t . Rev. Biochem. M i c r o b i a l . Biochem.,21,1(1979). 23. J . J . Heijnen and J.A. Roels, B i o t e c h n o l . Bioeng.,23,739(1981). 24. H.J. Kuhn, S. Cometta and A. F i e c h t e r , Eur. J . Appl. M i c r o b i o l . Biotechnol.

10(4),303(1980). 25. C.E. C o l l i n s , F.P. Incropera and J.P.L. Grady, Water Res.,12,547(1978).

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CHAPTER 7

A STRUCTURED MODEL FOR BACTERIAL GROWTH

I INTRODUCTION

In Chapters 2 and 4, theory and a p p l i c a t i o n s of unstructured models have been discussed. I t has been shown that unstructured models, as f o r the system described, provide good approximations f o r d e s c r i b i n g growth i n exponential phase or at steady or pseudo steady s t a t e s . In Chapter 2 i t has a l s o been shown that both the elemental and macromolecular composition of the organisms are f u n c t i o n s of the growth r a t e . Hence i t w i l l be f o r m a l l y c o r r e c t to use s t r u c t u r e d models f o r d e s c r i b i n g s i t u a t i o n s i n which the change i n biomass composition i n response to environmental changes, i s s i g n i f i c a n t . The unstruc­tured models are expected to f a i l p a r t i c u l a r l y i f the time constants f o r e n v i ­ronmental changes are of the same order of magnitude as those a s s o c i a t e d adap­t a t i o n mechanisms i n s i d e the b i o t i c phase.

Structured models by d e f i n i t i o n are those models which describe the a c t i v i t y of the organisms by more than one v a r i a b l e . These models i d e a l l y consider the p h y s i o l o g i c a l s t a t e of the organism and express i t not only as a f u n c t i o n of the present environmental s t a t e but a l s o on the e n t i r e past h i s t o r y of the system i . e . , the s t a t e of the environment the c e l l s have seen t i l l the present s t a t e . In t h i s way the s t r u c t u r e d models not only provide information on the q u a n t i t y of the c e l l s but also on t h e i r q u a l i t y .

In s t r u c t u r e d model b u i l d i n g one has to s e l e c t the parameters which are the most rel e v a n t f o r the d e s c r i p t i o n of the p h y s i o l o g i c a l s t a t e of the organism. Information from molecular chemistry and microbiology i s required f o r t h i s purpose. A l o g i c a l f i r s t choice would be to s e l e c t RNA, DNA, carbohydrate and p r o t e i n contents of the c e l l s f o r d e s c r i b i n g t h e i r p h y s i o l o g i c a l s t a t e . A l l these v a r i a b l e s can be experimentally determined and t h e i r dependence on the growth r a t e i s w e l l documented at l e a s t f o r the steady s t a t e continuous c u l ­t i v a t i o n . ' I t has to be noted, however, that even the c o n s i d e r a t i o n of these four relevant components i s not s u f f i c i e n t to describe the a c t i v i t i e s of the organisms f u l l y e.g., no information can be obtained about the geometrical s t r u c t u r e of the c e l l s nor on the dependence of d i f f u s i o n a l processes on such structures.2,3 Thus a s t r u c t u r e d model should be constructed such that i t only gives information about the most relevant processes and v a r i a b l e s . For a comprehensive d e s c r i p t i o n too many parameters would have to be incorporated i n t o the model. Such complicated models are mathematically very complex to manipulate and most of the parameters often lose t h e i r b i o l o g i c a l meaning.

Simpler models can be obtained by considering a few v a r i a b l e s , as an extension of the unstructured models, which consider one b i o t i c v a r i a b l e . These models,

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i n which the a c t i v i t y of the biomass i s s p e c i f i e d by more than one and up to 3 or 4 v a r i a b l e s , a r e c a l l e d Compartmental Models. These models have moderate mathematical complexity and are e a s i e r to v e r i f y experimentally.

The f i r s t compartmental model i s due to Williams^ 1. Williams has shown that even a simple model w i t h two compartments could describe most of the e x p e r i ­mentally observed phenomena, at l e a s t q u a l i t a t i v e l y . Some i n c o n s i s t e n c i e s of h i s model were corrected and been modified by Roels and KossenS and Roels^.

7 5 . Ramkrishna et a l . ' and F r e d r i c k s o n et a l . have co n t r i b u t e d g r e a t l y to the t h e o r e t i c a l aspects of s t r u c t u r e d models. Very l i t t l e has so f a r been done wi t h experimental systems. Work that has been reported i n the l i t e r a t u r e include the modelling of d i a u x i c growth^, fungal alpha-amylase production""*, growth i n a c t i v a t e d sludge system'', product formation by B. lioheniformis^^, n u t r i e n t dynamics of phytoplankton growing i n n i t r a t e l i m i t e d environment'3 etc. Recent reviews on the s t a t e of the a r t of c o n s t r u c t i n g s t r u c t u r e d models and c r i t i c a l l i t e r a t u r e surveys can be consulted f o r f u r t h e r information.0»14 In t h i s chapter a simple two compartmental model developed by Roels^ w i l l be c r i t i c a l l y evaluated with experimental data. Advantages, disadvantages and f u t u r e prospects of s t r u c t u r e d modelling w i l l be discussed.

I I THEORETICAL DEVELOPMENT OF THE GENERAL STRUCTURED MODEL

The b u i l d i n g of chemically s t r u c t u r e d models c o n s i s t s of ^ : i . s e t t i n g up mass balances f o r relevant components of the b i o m a t e r i a l over

an a p p r o p r i a t e l y chosen system, i i . p o s t u l a t i o n of the relevant k i n e t i c expressions f o r r e a c t i o n s taking

place w i t h i n the b i o t i c phase and at i t s boundaries. i i i . e v a l u a t i o n of the c o n s t r a i n t s imposed by elemental balances and by the

a p p l i c a t i o n of thermodynamic p r i n c i p l e s . The general d e s c r i p t i o n to be presented here f o l l o w s the work by Roels^. For a complete d e r i v a t i o n the reader i s r e f e r r e d to the o r i g i n a l work.

Let C be the chemical s t a t e v e c t o r of dimension n, f o r the system i n c o n s i ­d e r a t i o n . Two phases can now be i d e n t i f i e d : a. B i o t i c phase : Concentrations of a l l the components i n the b i o t i c phase are

c o l l e c t e d i n an m dimensional v e c t o r , x , on a kg/m3 of c u l t u r e volume b a s i s . Based on t h i s d e f i n i t i o n note that C x i s given by :

m C = E x.

X i - 1 1

b. A - b i o t i c phase : A l l r e a c t i o n s taking place i n the a - b i o t i c phase are c o n t r o l l e d by the concentrations of the reactants i n that phase. Since i n most cases the volume occupied by the b i o t i c phase i s very small compared w i t h that of the a - b i o t i c phase, these concentrations may be expressed per u n i t c u l t u r e volume. The v e c t o r , Y , d e s c r i b i n g these concentrations i s of dimension n - m.

Thus, C =[ x Y ] (1)

However, the rates of r e a c t i o n s taking place i n the b i o t i c phase are c o n t r o l l e d by the i n t r i n s i c concentration of the reactants i . e . , mass f r a c t i o n s i n the b i o t i c phase. De f i n i n g a new v e c t o r , X , f o r these i n t r i n s i c concentrations, one obtains :

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X. 1 m

x. / E x . 1 1-1 1

and £ X. i= l 1

In a homogeneous c u l t u r e of constant volume, a general mass balance reads :

dC / dt = r . a + 0

or s p l i t t i n g f o r the two phases :

(2)

7<o erratum d'X / dt = r . a + $ (3) — — —x —x

dY / dt = r . a i (4) - - -y + -y

Eq.(4) can be used to describe the time e v o l u t i o n of the s t a t e of the a - b i o t i c phase. For the t o t a l biomass i n the system the f o l l o w i n g balance equation can be shown to hol d :

dC / dt = (r . a x) . 1 + $ v (5) x — — A — — K

Thus (r . o^) . 1 can be i d e n t i f i e d as r x , the r a t e of biomass production. Furthermore, a s t a t e equation f o r the i n t r i n s i c concentrations can be derived as :

dX / dt = {r . Ox - ({r . Ox). J_) X} / C x (6)

Eqs(4),(5) and (6) give the t o t a l d e s c r i p t i o n of the system i . e . the a - b i o t i c and b i o t i c components and the biomass concentration. Note, however, that the d e r i v a t i o n depends on the r e s t r i c t i o n that only biomass of the same composi­t i o n can leave or enter i n t o the system. Also note that eq.(6) describes the i n t e r n a l s t a t e of the biomass and i s independent of the mode of operation, i n order to o b t a i n a workable form, r e l a t i o n s have to be formulated f o r J ^ , J>y and r .

I l l DESCRIPTION OF THE TWO COMPARTMENTAL SYSTEM

With reference to F i g . 7 i n Chapter 2, i t can be concluded that RNA and carbohydrate f r a c t i o n s change most i n response to changes i n growth r a t e . Thus i t might be d e s i r a b l e to consider these components e x p l i c i t l y i n a st r u c t u r e d model.Protein content changes only s l i g h t l y over the same range. Moreover, i t would be d i f f i c u l t to consider carbohydrate compartment separately as i t s absolute i n t r i n s i c c oncentration i s very low (about 5 % dry weight) and hence v a r i a t i o n s i n i t are d i f f i c u l t to detect a n a l y t i c a l l y . Therefore i n the model to be presented, RNA, carbohydrate and other small molecules are lumped i n t o one compartment, c a l l e d the K-compartment. Since RNA i s the main component, K may be expected to be p r o p o r t i o n a l to the RNA content. The other compartment i s c a l l e d the G-compartment and contains the s t r u c t u r a l and genetic m a t e r i a l i . e . , p r o t e i n s , DNA, etc. F i g . 1 gives the block diagram of the model.

Here, e x t e r n a l substrate i s assumed to be converted to K, from which G forms w i t h a constant y i e l d . A t h i r d r e a c t i o n i n v o l v e s the turn over of macromole-cules i . e . , G m a t e r i a l back to K m a t e r i a l (small molecules). This allows f o r

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K G K G K G K G

Fig. 1: Block diagram of the model.

the maintenance requirements to be incorporated i n t o the model. The maintenan­ce process i s accounted f o r only by the depolymerization r e a c t i o n of the macromolecules w i t h no mass l o s s i n the b i o t i c phase. Thus the f o l l o w i n g reac­t i o n s can be formulated:

YSK K

K — YKG G r a t e r„

rate r

(7)

(8)

(9)

Now, the rate equations have to be postulated f o r rg , r ^ and r m .

The rate of substrate uptake i s assumed to be of s a t u r a t i o n type:

rs ' ks CS C x 1 ( KS + CS > (10)

Here kg represents the maximum rg . Such s a t u r a t i o n type r e a c t i o n s are charac­t e r i s t i c of enzymic or m i c r o b i a l r e a c t i o n s , r ^ i s suggested to be described by

r R = k K K G C x (1.)

= k K K ( 1 - K ) C x

since K + G = 1. Here note that when G =0 e q . ( l l ) p r e d i c t s r^, = 0 i . e . , no " iK formation. This has a b i o l o g i c a l e xplanation i . e . , when G = 0 there are no

enzyme and DNA necessary f o r K formation. B e erratum J * . \

The r a t e of turnover of G m a t e r i a l back to K i s tought to be a f i r s t order r e a c t i o n w i t h a y i e l d of 1.

r = m„ G C (12) m G x

= m ( 1 - K ) C Lj x

Here mg i s the maintenance f a c t o r f o r the G compartment.

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IV DERIVATION OF THE BALANCE EQUATIONS

The vector of r e a c t i o n rates r , i s now given by :

r = r r„ r„ r 1 - L S K mJ

The a - b i o t i c s t a t e vector and the stoichiometry matrix read :

i T a [ -1 0 0 ]

S i m i l a r l y f o r the b i o t i c phase :

X = [ K G ] ( Y 0 SK

KG 1 -1

Thus f o r the two compartment system described, the f o l l o w i n g can be deri v e d :

dC g / dt

dC / dt x

-r + <P S S

Y r + (Y SK s KG K x

(13)

(14)

dK / dt = {(1 - K) Y S K r g - r K (1 * ( Y R G - 1) K) + r,} / ^

SK S K m x x (15)

A s i m i l a r equation need not be derived f o r the dynamics of G, since dG/dt = -dK/dt.

When the mode operation i s known the balances are f i x e d , e.g., f o r continuous c u l t u r e operation :

J) = D ( C . - C ) Yx< s i s

D C

(16)

(17)

Steady s t a t e r e l a t i o n s can be obtained f o r Cg, C x and K by p u t t i n g t h e i r r e s p e c t i v e time d e r i v a t i v e s equal to zero, i . e . , eqs(13), (14) and (15). For instance K* can be shown to be given by:

K* = (D + m G ) / ( k R Y R G ) (18)

Another i n t e r e s t i n g r e l a t i o n to be discussed l a t e r on, can be given f o r the steady s t a t e as:

rS - V Y s x + ( V Y S K ) ( D + m G ) ( k K \ G - ( D + m G ) ) ( 1 " YKG> °x ° 9 )

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V EVALUATION OF THE VALIDITY OF THE MODEL

The model presented has s i x parameters; 2 s t o i c h i o m e t r i c c o e f f i c i e n t s ( Y K Q , Yg K) , 3 rate constants (kg, kj^ , mG) and one Michealis-Menten type constant Kg . The model w i l l now be tested w i t h the continuous and fed-batch data presented p r e v i o u s l y .

Model parameters were estimated from fed-batch and batch data and used f o r the si m u l a t i o n of continuous c u l t u r e data. The parameters are l i s t e d i n Table I.

Table I : Model parameters.

Parameter ks 1 96 hr-> kK 5 0 h r H

mG 0 06 h r " 1

YSK 0 73 kg/kg YKG 0 66 kg/kg KS 0 07 kg/m3

A rough s e n s i t i v i t y a n a l y s i s of the model to i t s parameters was c a r r i e d out i n a manner s i m i l a r to that explained i n Chapter 2 f o r batch growth modelling. Results are shown i n Table I I .

Table I I : Model s e n s i t i v i t y to i t s parameters.

Model output Parameters kg k K m G Y S K Y K G Kg

Biomass + + + - - + + + -Substrate + + + - - + + + -K compartment + + - + +

(++) very s e n s i t i v e (-) i n s e n s i t i v e

I t must be reminded that t h i s a n a l y s i s was c a r r i e d out f o r fed-batch e x p e r i ­ment FB 830, and i s therefore s p e c i f i c to i t .

With parameters l i s t e d i n Table I , steady s t a t e continuous c u l t u r e behaviour was simulated w i t h the two compartment model ( F i g . 2). There seems to be a very good agreement between the model and the experiment. A s l i g h t systematic d e v i a t i o n of 2 - 3 % i s evident at the plateau of the C x p r o f i l e . Moreover, the model p r e d i c t s wash-out at y = i.05 h r - ' which i s i n very good agreement with the experimentally determined Umax values. In F i g . 3 the pred i c t e d K and experimentally determined RNA f r a c t i o n s are shown as functions of y. Both sets of data are w e l l described by s t r a i g h t l i n e s . However, the slopes and in t e r c e p t s are markedly d i f f e r e n t .

In order to check whether the model could p r e d i c t overshoot phenomenon during adaptation phase f o l l o w i n g a s h i f t , a wash-out experiment was simulated.

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C x (kg/m3)

Fig. 2: Simulation of continuous culture behaviour with the model.

Fig. 3: Predicted K profile and experimentally determined RNA profile. (- - -) straight line fitted to RNA data by linear regression.

A step i n d i l u t i o n r a t e from D = 0.1 to D = 1.2 was a p p l i e d ( F i g . 4). Here y increases very f a s t and overshoots the y m a x value before reaching the new steady s t a t e value.

In F i g . 5 the model s i m u l a t i o n f o r fed-batch experiment FB 830 i s shown together w i t h the experimental data. Here, the s t r u c t u r e d model provides a b e t t e r d e s c r i p t i o n than the unstructured model (compare w i t h F i g . 2 of Chapter 4). In f a c t s t r u c t u r e d models would be expected to be superior to unstructured models under h i g h l y t r a n s i e n t c o n d i t i o n s . Thus i t would be much b e t t e r to study and t e s t them under p r e c i s e l y c o n t r o l l e d t r a n s i e n t environmental condi­t i o n s e.g., step-up and down experiments or s i n u s o i d a l feeding schemes. Unfortunately, such data i s l a c k i n g f o r model t e s t i n g .

VI DISCUSSION

From the foregoing one can i n general conclude that the presented simple str u c t u r e d model can describe most of the phenomena observed i n p r a c t i c e ,

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1.2-

Ol I I I 1 1 1 1 1 I I M I 1 1 1 1 I I I I I 10° 10' io 2 to3

Fig. 4: Response of the growth rate to a step change in the dilution rate from D=0.1 to 1.2 .

Fig. 5: Model simulation of the fed-batch experiment FB 830.

at l e a s t q u a l i t a t i v e l y . The model, i n a d d i t i o n to q u a n t i t a t i v e l y p r e d i c t i n g everything an unstructured model can, provides information about the i n t e r n a l s t r u c t u r e of the biomass and on adaptation and overshooting phenomena. The model a l s o allows f o r maintenance turnover to be a f u c t i o n of the growth rate as can be seen from eq.(19) i f i t i s compared w i t h the l i n e a r r e l a t i o n f o r substrate consumption given i n Chapter 2. I t can be seen that the model would p r e d i c t decreasing maintenance requirements at low growth rates which might be expected on t h e o r e t i c a l grounds.

The model f a i l s to give an accurate d e s c r i p t i o n of the RNA f r a c t i o n . This can be f u r t h e r analysed by e v a l u a t i n g the r a t i o of the i n t e r c e p t to the slope of the r e l a t i o n f o r K* vs. D 6,11 (see eq.(18)). This r a t i o equals mg . I f W.Q i s determined from the experimental RNA data i n t h i s manner, a value of 0.5 h r - ' i s obtained which i s f a r too high. A r e a l i s t i c value should be 0.03 -0.05 . Thus one can conclude that the exact q u a n t i t a t i v e nature of the depen­dence of RNA f r a c t i o n to the growth r a t e i s not described by the model. As Roels^ has noted, t h i s i s h a r d l y s u r p r i s i n g as no c o n s i d e r a t i o n has been given to any r e g u l a t o r y mechanism operating i n s i d e the biomass. The system has simply been modelled i n analogy w i t h chemical k i n e t i c s i . e . , assumed to be only chemically s t r u c t u r e d .

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Another shortcoming of the model i s that i t p r e d i c t s maintenance requirements as the turnover of only the G compartment i . e . , no maintenance f u n c t i o n i s associated with the p r e s e r v a t i o n of the K compartment.

In l i t e r a t u r e some authors proposed s t r u c t u r e d models w i t h compartments of abst r a c t f u n c t i o n s . These models could never be tested and therefore remain as mathematical e x e r c i s e s . A s t r u c t u r e d model should best be tested f o r i t s p r e d i c t i o n capacity of the i n t e r n a l composition i n a d d i t i o n to k i n e t i c and energetic behaviour. I f i n t e r n a l composition i s not checked qu i t e misleading conclusions may be drawn, e.g., the model presented can be accepted as a good approximation to r e a l i t y based on the C x p r o f i l e s . However, from the K com­partment and RNA data a n a l y s i s i t was concluded that the model was not j u s t a f t e r a l l .

U nfortunately, more b i o l o g i c a l l y meaningful models c a l l f o r e x t r a compartments and t h i s u s u a l l y brings about mathematical complexity. The experimenter must aim f o r a comprimise between b i o l o g i c a l sense and mathematical complexity. He should, whenever p o s s i b l e , check mechanisms and obtain parameters from independent experiments.

Recently i n l i t e r a t u r e a model claimed to be one of the most comprehensive so f a r , has a p p e a r e d ^ I t has '72' parameters and a s i z a b l e p r o p o r t i o n of the parameters were obtained by curve f i t t i n g procedures or estimated a r b i t r a ­r i l y . From a mathematical point of view i t i s very hard not to assume the whole e x e r c i s e as one of curve f i t t i n g and programming.

Structured modelling i s s t i l l i n i t s infancy.However, there i s no doubt that as the understanding of m i c r o b i o l o g i c a l s y s t e m s , a n a l y t i c a l techniques and computational procedures get b e t t e r , s t r u c t u r e d models w i l l r e place the unstructured ones. There seems to be great scope f o r the development of these models much e f f o r t i s needed i n t h i s d i r e c t i o n .

Acknowledgement :

The author wishes to thank student T. Veerman f o r h i s c o n t r i b u t i o n to t h i s chapter.

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VII NOMENCLATURE

C chemical s t a t e vector c concentration (kg/kg) D d i l u t i o n r a t e ( h r ~ l ) G mass f r a c t i o n of G compartment i n dry biomass (kg/kg) K mass f r a c t i o n of K compartment i n dry biomass (kg/kg) K S

s a t u r a t i o n constant (kg/m^) k r a t e constant ( h r - ' ) mG maintenance f a c t o r f o r G compartment ( h r - 1 ) M S t o t a l substrate (kg) Mx t o t a l biomass (kg) r r e a c t i o n r a t e (kg/m3/hr) rm maintenance r a t e (kg/m3/hr) X mass f r a c t i o n of b i o t i c component (kg/kg) Y K G y i e l d of G compartment on K compartment (kg/kg)

y i e l d of K compartment on substrate (kg/kg) ot stoichiometry matrix $ flow r a t e (kg/m3/hr) u s p e c i f i c growth rate ( h r - ' ) subs c r i p t s S substrate i i n i t i a l x biomass (or b i o t i c phase) y a - b i o t i c phase

V I I I REFERENCES

1. D. Herbert, Symp. Soc. Gen. M i c r o b i o l . , 1 I,391 (1 961 ). 2. N.W.F. Kossen, Symp. Soc. Gen. Microbiol.,29,327(1979). 3. A.G. Fr e d r i c k s o n , B i o t e c h n o l . Bioeng.,18,1481(1976). 4. F.M. Williams , J . Theor. B i o l . , 15,190(1967). 5. J.A. Roels and N.W.F. Kossen, i n Progress i n I n d u s t r i a l M i c r o b i o l o g y , 14,

95(1978). 6. J.A. Roels, Biochemical Engineering; Energetics and K i n e t i c s , a book to

be published by E l s e v i e r . 7. D. Ramkrishna, A.G. Fre d r i c k s o n and H.M. Tsuchiya, B i o t e c h n o l . Bioeng.,9,

129(1967) . 8. A.G. Fre d r i c k s o n , R.D. Megee and H,M. Tsuchiya, Advances i n Applied

Microbiology,13,419(1970). 9. G. van Dedem and M.Moo-Young, Bi o t e c h n o l . Bioeng.,17,1301(1975). 10. S.T. F i t z p a t r i c k , Ph.D. Thesis, U.M.I.S.T.,(1977). 11. A. Harder, Ph.D. Thesis, A g r i c u l t u r a l U n i v e r s i t y of Wageningen,(1979). 12. R. Larsen and L. Kjaergaard, Eur.J. Appl. M i c r o b i o l . Biotechnol.,5,177

(1978) . 13. W.J. Grenney, D.A. B e l l a and H.C. C u r l , B i o t e c h n o l . Bioeng.,15,331(1973). 14. A. Harder and J.A. Roels, to be published i n Advances i n Biochemical Engi­

neering. 15. M.L. Schuler, S. Leung and C.C. Dick, Ann. N. Y. Acad. Sci.,326,35(1979).

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CHAPTER 8/APPLICATION 1

COMMENTS ON THE DESCRIPTION OF MAINTENANCE METABOLISM DURING ANAEROBIC GROWTH WITH PRODUCT FORMATION *

A. A. Esener, J . A. Roels and N. W. F. Kossen

SUMMARY

Product formation during anaerobic growth i n the absence of e x t e r n a l e l e c t r o n acceptors has been shown to be l i n k e d to the energy production processes. I t has been demonstrated that when t h i s f a c t i s not taken i n t o account, e x p e r i ­mental data r e s u l t s i n an i n c o r r e c t d e s c r i p t i o n of the maintenance metabolism

RESULTS AND DISCUSSION

Maintenance metabolism during anaerobic growth has r e c e n t l y received much atten t i o n i n the l i t e r a t u r e . Some of these a r t i c l e s , i n our opinio n have drawn a very misleading p i c t u r e of the energetic status of anaerobic systems (Cromie and Doelle,1980; Goma et.al.,1979). Based on i n c o r r e c t t h e o r e t i c a l treatments and biased estimates, i t has been claimed that the maintenance requirements could be reduced e f f e c t i v e l y to zeroby merely v a r y i n g the chemical composition of the c u l t i v a t i o n medium (Cromie and Doelle,1980). We therefore f e e l o bliged to express our opinion about t h i s apparently i l l - t r e a t e d system i n order to promote a c o n s t r u c t i v e d i s c u s s i o n which may be u s e f u l f o r f u r t h e r research and a b e t t e r understanding of the maintenance metabolism.

In t h i s work, f i r s t , i t w i l l be shown that i n anaerobic systems w i t h product formation i n the absence of e x t e r n a l e l e c t r o n acceptors, energy generation i s coupled to the product formation process and secondly, examples of m i s i n t e r p r e ­t a t i o n of data from such systems w i l l be pointed out. The mathematical t r e a t ­ment to be used f o l l o w s from the work of Roels(1980) and Roels and Kossen (1978). The reader i s r e f e r r e d to the o r i g i n a l a r t i c l e s f o r a d e t a i l e d a n a l y s i s

S t a r t i n g from macro-balancing p r i n c i p l e s , a general s t o i c h i o m e t r i c equation can be w r i t t e n f o r m i c r o b i a l growth on a s i n g l e carbon and energy source w i t h ammonia as the n i t r o g e n source. For s i m p l i c i t y only one product w i l l be considered.

E>2C H 0 N + $ 50„ + <KNH,, a 2 b 2 c 2 d 2 2 3

$iC H, 0 N, + $ 3C PL 0 N, + $ 6C 0„ + *7H.0 (1) b, c, d, a 3 b 3 c 3 d 3 2 2

* Published i n Biotechnology L e t t e r s , V ol.3, No.]-, 15-20 (1981)

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For a system at steady s t a t e , i f the above d e s c r i p t i o n holds, flows of a l l components can be expressed i n terms of any three known flows. The l i n e a r law of substrate consumption, introduced a f t e r the p o s t u l a t i o n of the maintenance mechanism (Herbert, 1958), can be generalized to account f o r the product formation during growth (Humphrey and J e f f e r i s , 1 9 7 3 ; Roels,I980; Roels and Kossen,1978).

r = r / Y + r / Y + m . C (2) s x sx p sp S X ,

A s i m i l a r expression can now be shown to hold f o r the oxygen consumption rate

r = r / Y + r / Y + m . C (3) o x ox p op o x

Equations ( l ) - ( 3 ) describe the general case. The anaerobic case, which i s a s p e c i a l case , can a l s o be described by t h i s set of equations when the oxygen uptake rate i s put equal to zero. i . e . when r ^ = 0, from eq.(3) one obtains :

r = Y ( - r / Y - m . C ) (4) p op X O X O X

S u b s t i t u t i n g t h i s r e s u l t i n eq(2) and rearrenging, the f o l l o w i n g expression i s obtained f o r the substrate consumption r a t e .

r = (1/ Y - Y /(Y .Y ) ) . r + ( m - m .Y / Y ). C (5) s sx op ox sp x s o op sp X

The terms given i n paranthesis i n the above equation are a l l constants and hence i t becomes c l e a r that there remains no separate c o n t r i b u t i o n f o r r _ , i n the substrate consumption equation. For the case of anaerobic growth w i t h product formation, r s can hence be expressed by :

r = r / Y' + m' . C (6) s x sx s x

Combining t h i s r e s u l t w i t h a degree of r e d u c t i o n balance (Erickson et.al.,1978; Roels,1980), of the form :

Y . r = Y - r + Y - r (7)

s s x x p p

an expression f o r the r a t e of product formation can be formulated :

r = (( y - Y -Y' )/( Y -Y' ) ) . r + ( y / y ). m' . C (8) p S X sx p S X X s p S X

Equation (6) shows t h a t , f o r t h i s s p e c i a l case (anaerobic growth w i t h the e x c r e t i o n of one product or a mixture of them i n constant r a t i o ) no separate term f o r rp appears i n the r s expression. This i m p l i e s that the energy generation i s d i r e c t l y coupled to the product formation. Equation (8) shows that the rate of product formation thus contains two c o n t r i b u t i o n s ; the f i r s t p r o p o r t i o n a l to the biomass production, and the second to the amount of biomass present. This type of expressions have already been used f o r some processes (Roels,1980). Therefore, i f rp i s expressed simply as :

r = r / Y + m . C (9) p x px p X

the f o l l o w i n g r e l a t i o n s h i p s can be obtained by comparing eqs. (8) and (9) :

Y = Y • Y' / ( Y -Y • Y' ) (10) px p S X S X S X

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m = ( Y / Y ) • m' p s p s (11)

These r e l a t i o n s h i p s are shown as s o l i d l i n e s i n f i g u r e s 1. and 2., assuming average values of Y

x = 4.2 and the molecular weight of the organism as 25.0, for the case of ethanol production from glucose a n a e r o b i c a l l y . I t can be noticed that the s o l i d l i n e i n f i g . 1 . i s w e l l approximated by the s t r a i g h t l i n e Y p x = ( Y P 1 Y s ) • Y s x s i n c e Y x - Y s x « Y s (see eq.(10)) .

Y (C-eq/C-eq) px

0.04 -

Y' (C-eq/C-eq) s x i |

m .10 (C-eq/C-eq/hr) P

5.0

0.01 0.02 0.03

2.5 -

0.005 0.010

fig. 1:1 vs. 1' :( ) eq.(10); ( • ) experimental data (Cromie and Doelle) pec sec fig. 2. m vs. m'

J - p s : (- -) eq.(11); ( • ) experimental data (Cromie and Doelle)

Parameters Y s x , Yp x , ms and nipWere determined by performing l i n e a r r e g r e s s i o n a n a l y s i s on the data reported by Cromie and Doelle (1980) using equations (6) and (9). Although there i s considerable s c a t t e r i n f i q . 2 , g e n e r a l l y the above presented argument seems to hold f o r t h i s system, i . e . product formation i s indeed associated with the energy generation i n the system.

This type of energy generation was not taken i n t o account by Goma e t . a l (1979) who have c a l c u l a t e d zero maintenance during ethanol production by Saceharomyoes oerevisiae growing on glucose, a n a e r o b i c a l l y . Using eq. (2) these workers have c a l c u l a t e d mg to be zero towards the end of the fermentation. However, t h e i r a n a l y s i s obviously takes no account of the energy produced during product formation. Energy produced i n the form of ATP during ethanol production can n a t u r a l l y be channelled to s a t i s f y the maintenance demands. A simple block diagram ( f i g . 3) i l l u s t r a t e s the p o s s i b l e mechanisms f o r the d i s t r i b u t i o n of the substrate energy.

Recently another claim of zero maintenance came from Cromie and Doelle (1980). These authors claimed that they have succeeded i n reducing the maintenance requirements e f f e c t i v e l y to zero by changing the chemical composition of the c u l t i v a t i o n medium they have used f o r ethanol production by Zymomonas mobilis, a n a e r o b i c a l l y . However, an a n a l y s i s of t h e i r o r i g i n a l data by the above presented procedure revealed that t h e i r conclusions were i n c o r r e c t . When f u r t h e r analyses were c a r r i e d out i t became obvious that t h e i r conclusions were e n t i r e l y based on t h e i r biased parameter e s t i m a t i o n technique. When t h e i r data were subjected to l i n e a r r e g r e s s i o n a n a l y s i s , i t was found out that a l l t h e i r parameter estimations w i t h the exception of one (experiment no:10, Table I) were i n great e r r o r s , i . e . see F i g . 4. For the two cases f o r which Cromie and

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Doelle (1980) claimed zero maintenance a l i n e a r r e g r e s s i o n procedure y i e l d e d the values of 1.85 and -1.25 g/g/hr f o r mg (experiment no: 6 and 9 i n Table I ) . A negative value f o r ms obviously does not make any sense, however, i t does not allo w the experimenter to assume and report i t zero. A l l that can be said i s t h a t , e i t h e r the model does not hold f o r t h i s s i t u a t i o n or/and the experimental data are not accurate.

product synthesis

i i

use of substrate

A T P

pool

synthesis —*~ of biomass

precursors

i

biomass synthesis

maintenance

fig. 3: Distribution of the substrate energy in microbial metabolism (Roels and Kossen, 1978)

Table I: Reported and c a l c u l a t e d maintenance c o e f f i c i e n t s .

Exp. no: N Mg ms (Cromie & Doelle mg ( l i n e a r r e g r e s s i o n ) * 1980) g/g/hr. g/g/hr ( t h i s study)

1 0 0 0 15 5.9 1 55 2 0 0 0 30 2.9 1 05 3 0 0 0 45 3.2 2 35 4 0 05 0 15 3.9 5 80 5 0 05 0 30 5.8 2 95 6 0 05 0 45 0.0 1 85 7 0 10 0 15 1 .8 6 75 8 0 10 0 30 1 .0 4 05 9 0 10 0 45 0.0 -1 25

10 0 20 0 30 3.0 3 00 1 1 0 20 0 45 5.4 5 00

N = (NH ) SO, %, Mg = MgSO % , *From the o r i g i n a l data reported by Cromie and Doelle (1980).

Since i t has been shown that the product formation was d i r e c t l y coupled to energy generation an energy balance f o r t h i s system can now be given. Considering ATP as the energy currency, ATP consumption, i n terms of the l i n e a r law, i s given by ( Stouthamer and Bettenhaussen,1973) :

qATP = " I YATP + mATP ( 1 2 )

Moreover, as one mole of ATP i s produced f o r every two moles of ethanol produced v i a the Enthner-Doudorof pathway ( f o t the case of Z. mobilis ) :

92

q E t h a n o l ( M / Y ATP + mATP } (13)

I f t h i s equation i s f i t t e d to the data of experiment no:6 ( f o r which a zero maintenance c l a i m has been made) an m A Tp of 10 mmoles ATP/g.biomass/hr can be c a l c u l a t e d ( F i g . 5). This value i s q u i t e high when compared w i t h some of the data c o l l e c t e d and reported by Stouthamer (1977) as shown i n Table I I . One can therefore conclude that the maintenance demands were never reduced to zero as claimed by Cromie and Doelle (1980). In f a c t i t i s q u i t e probable that t h e i r organism, Z. mobilis , i s a good ethanol producer because of i t s high mainte­nance requirements and since i t cannot generate s u f f i c i e n t energy v i a any other metabolic route , to s a t i s f y them.

^ATP (mmole/g.biomass/hr)

0.05 0.10

fig. 4: Regression line for experiment no:l.

fig. 5: m ATP determination for experiment no:6 by eq.(12).

Table I I : values f o r some organisms grown a n a e r o b i c a l l y i n chemostat .

Organism Growth l i m i t i n g substance n* ATP

L. casei glucose 1 .5 E. aerogenes glucose (minimal) 6.8

glucose (complex) 9.9 tryptophan 38.7

E. coli glucose 18.9 S. cerevisiae glucose 0.5

glucose 0.25 S. oerevisiae glucose 0.70 c. paraplosis glucose 0.21 z. mobilis glucose 10.3**

* References as quoted by Stouthamer (1977) ** C a l c u l a t e d from data reported by Cromie and Doelle (1980) f o r exp. no:6.

CONCLUSIONS

Product formation (a s i n g l e product or a constant r a t i o mixture) during anaerobic growth i n the absence of e x t e r n a l e l e c t r o n acceptors, i s l i n k e d to the energy producing processes. This f a c t must be taken i n t o account when data obtained from such systems are to be i n t e r p r e t e d . The statement of Cromie and Doelle (1980) " In v a r y i n g not only the d i l u t i o n r a t e but a l s o the ammonium

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sulphate (nitrogen source) and magnesium sulphate concentrations, the mainte­nance c o e f f i c i e n t m, was s u c c e s s f u l l y reduced to zero at a l l d i l u t i o n rates " i s based on t h e i r i n c o r r e c t conceptual and mathematical treatment. In f a c t i t can be shown that a high maintenance value f o r an ethanol producing organism v i a t h i s route, i s a d e s i r a b l e property as t h i s w i l l increase the e f f i c i e n c y of conversion of the substrate to the product.

NOMENCLATURE

C H, 0 N, ^1 C! d l

biomass elemental formula

C H, 0 Nj substrate elemental formula a 2 b 2 c 2 d 2

C H, 0 N product elemental formula ^3 3 *"3 3

C x biomass concentration C- eq. carbon equivalent m i maintenance c o e f f i c i e n t of the i ' t h component Y i , j maximal y i e l d of j on the i ' t h component H flow of the i ' t h component Y i degree of red u c t i o n of the i ' t h component

Y i = 4a + b - 2c - 3d y s p e c i f i c growth rate s u b s c r i p t s 0 oxygen s substrate X biomass

REFERENCES

Cromie, S. and D o e l l e , H.W. (1980) B i o t e c h n o l . Lett.,2,(8),357. E r i c k s o n , L.E., Minkevich, I.G. and Eros h i n , V.K. (1078) B i o t e c h n o l . Bioeng.,

20,1595. Goma, G. M o l e t t a , R., and Novak, M. (1979) B i o t e c h n o l . Lett.,1(10),415. Herbert, D. (1958) i n 'Recent Progress i n M i c r o b i o l o g y ' Symp.7 th I n t . Congr.

M i c r o b i a l . , E d . Tunewall, G.,p.381, Almqvist, Stockholm. Humphrey, A.E., and J e f f e r i s , P.K.,(1973) GIAM meeting, Sao Paulo, B r a s i l ,

vol.4,p.767. Roels, J.A.,(1980) Biotechnol.Bioeng.,22,2457. Roels, J.A., and Kossen, N.W.F. (1978) i n Progress i n I n d u s t r i a l M i c r o b i o l o g y ,

Ed. B u l l , M.J.,14,p95, E l s e v i e r , Amsterdam. Stouthamer, A.H. (1977) Symp. Soc. Gen. M i c r o b i o l . , 27, 287. Stouthamer, A.H., and Bettenhaussen, C. (1973) Biochim. Biophys. Acta.,301,53.

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APPLICATION 2

BIOENERGETIC CORRELATION OF COD TO BOD *

A.A. Esener, J.A. Roels and N.W.F. Kossen

SUMMARY A simple expression has been derived to demonstrate that the r a t i o of COD to BOD i s not constant but a f u n c t i o n of the thermodynamic e f f i c i e n c y of the growth process. An a n a l y s i s of data reported i n l i t e r a t u r e i n d i c a t e s that t h i s r a t i o can assume a wide range values f o r d i f f e r e n t s u bstrates. The i n f l u e n c e of a selected change i n the environmental c o n d i t i o n s on the r a t i o , COD/BOD, has been evaluated experimentally, f o r a b a c t e r i a l system.

INTRODUCTION

BOD and COD are the most commonly used parameters f o r the c h a r a c t e r i z a t i o n of waste waters. Of these, BOD i s the most widely used measure of the biodegradable organic content and i s u s u a l l y expressed as the 5-day/20 C(B0D5) value (Schroeder, 1977). COD on the other hand, q u a n t i f i e s the t o t a l amount of o x i d i z a b l e m a t e r i a l i n the sample. Both of these u n i t s have advantages and disadvantages and the choice u s u a l l y depends on many f a c t o r s such as the r e p r o d u c i b i l i t y of the determinations, time period r e q u i r e d , l o c a t i o n of the t e s t , e t c . Since both of these u n i t s are used and reported i n l i t e r a t u r e i t would be very u s e f u l i f one could c o r r e l a t e one to the other. For sewage often a r a t i o of 2 - 2.5 i s used as an approximation to COD/BOD (Nat. Swedish Env. Prot.Brd.', 1972).The use of such constants however, can be very hazardous under c e r t a i n c o n d i t i o n s . As w i l l be shown i n the f o l l o w i n g , the r a t i o i s a fu n c t i o n of the e f f i c i e n c y of m i c r o b i a l growth. The COD of any waste i s a constant since i t s determination involves a c a r e f u l l y c o n t r o l l e d chemical r e a c t i o n . However, the BOD value determined f o r the same sample may vary under d i f f e r e n t environmental c o n d i t i o n s because the e f f i c i e n c y of m i c r o b i a l growth process may be af f e c t e d s i g n i f i c a n t l y by d i f f e r e n t c o n d i t i o n s . Since COD i s constant and BOD v a r i a b l e , the r a t i o COD/BOD cannot be regarded as a constant.

The e f f i c i e n c y of m i c r o b i a l growth i s r e f l e c t e d i n the s t o i c h i o m e t r i c c o e f f i c i e n t s of the growth equation. Several workers i n the waste water f i e l d have r e a l i z e d that the r e a c t i o n stoichiometry was not constant, but v a r i e d , f o r

* Published i n Biotechnology L e t t e r s , Vol.3 No.A 193-198 (1981)

9 5

instance , with the growth r a t e . They have completed balances at d i f f e r e n t d i s c r e t e growth rat e s (Sherrard and Schroeder, 1975; Sherrard, 1980). However, a generalized r e l a t i o n has so f a r not been reported. In t h i s work, u t i l i z i n g the r e c e n t l y developed macro-balancing p r i n c i p l e s and energetics of m i c r o b i a l processes (Minkevich and Eros h i n , 1973; Er i c k s o n et a l , 1978; Roels, 1980) a r e l a t i o n f o r the r a t i o COD/BOD i s developed i n terms of the thermo­dynamic e f f i c i e n c y of the growth process, i . e . change i n r e a c t i o n stoichiometry i s accounted f o r , by co n s i d e r i n g the thermodynamic e f f i c i e n c y . For a d e t a i l e d d e s c r i p t i o n of the above mentioned t o o l s to be used here, the reader i s r e f e r r e d to the o r i g i n a l p u b l i c a t i o n s (Minkevich and Eros h i n , 1973; Er i c k s o n et a l , 1978; Roels, 1980).

THEORY A general s t o i c h i o m e t r i c r e l a t i o n f o r aerobic m i c r o b i a l growth on a s i n g l e carbon and energy source with ammonia as the n i t r o g e n source, w i l l be considered. For s i m p l i c i t y by-product formation w i l l not be included.

$„C H, 0 N, + $ 0 + <J> NH — • $ CH, 0 N + $CC0„ + $,H„0 (1) 2 a2 D2 C2 d2 4 2 3 3 1 b] cj d] 5 2 6 2

Here, i t i s assumed that carbon, hydrogen, oxygen and n i t r o g e n constitude a l l the elements present i n the system i n n o n - n e g l i g i b l e q u a n t i t i e s . The s t o i c h i o m e t r i c c o e f f i c i e n t shown by $, represent the net steady-state flows of the r e s p e c t i v e components. Using the concept of the degree of r e d u c t i o n , y, the thermodynamic e f f i c i e n c y , T], f o r the m i c r o b i a l growth process has been approximated by the f o l l o w i n g : (Erickson et a l , 1978; Roels, 1980)

n = Y" (y /y ) (2) sx X s where, Y> i s the degree of re d u c t i o n and i s defined by the r e l a t i o n :

Y = 4a + b - 2c - 3d (3) and Y" i s the y i e l d of biomass on substrate (C-equiv/C-equiv) and i s given by:

S X Ysx= *1 1 (*2 ' a2> <4> Moreover i t has been shown that once,n , i s known, oxygen demand can be estimated by (Roels, 1980) :

Y Q X = ( 4 / Y x) (0 / (1 -0 )) (5)

Where Y o x i s the y i e l d of biomass on oxygen (C-equiv/mole). For u n i t amount of substrate consumption, BOD, the biochemical oxygen demand can be shown to be given by :

BOD = Y" / Y sx ox

or from eq.(2) and (4), by: BOD = (Y s / 4) (1 -n ) (6)

Furthermore, COD, defined as the amount of oxygen required f o r the complete combustion of u n i t s u b s t r a t e , can r e a d i l y be given by the expression :

COD = (1/4) Y s (7) Thus the r a t i o , C0D/B0D can now be given simply as :

C0D/B0D = 1 / (1 -n ) (8)

In f i g . l the r a t i o , C0D/B0D i s p l o t t e d against the thermodynamic e f f i c i e n c y , according to equation (8). Here, at n=0 C0D/B0D becomes equal to 1. i . e . no

96

biomass i s produced and a l l the incoming substrate carbon i s converted i n t o carbon d i o x i d e . The other end of the scale i . e . r\ = 1 , corresponds to a s i t u a t i o n where a l l the incoming chemical energy stored i n the substrate i s incorperated i n t o the biomass, r e v e r s i b l y . Although n can t h e o r e t i c a l l y assume any value between 0 and 1, i n p r a c t i c e the maximum e f f i c i e n c y determined has never been found to exceed 0.7. The p l o t has therefore been drawn to n =0.7 which represents the p r a c t i c a l maximum f o r n. As the r a t i o of COD/BOD i s e n t i r e l y dependent on n, i t i s important to evaluate the most important f a c t o r s which i n f l u e n c e the fj. X] as defined by equation (2) i s a f u n c t i o n of the composition of the biomass and substrate as w e l l as the molar growth y i e l d . I t i s known that the elemental composition of biomass i s u s u a l l y not very s e n s i t i v e to environmental changes. The composition of subs t r a t e , however, i s of prime importance. Growth on some substrates i s more e f f i c i e n t than others, ( g l y c e r o l ; n =0.57, ox a l a t e ; n =0.26) Roels (1980) has compiled a number of l i t e r a t u r e data and computedn f o r various substrates and organisms. The values ranged from 0.26 f o r oxalate to 0.67 f o r gluconate. These values as i n d i c a t e d i n f i g . 1. correspond to a change i n COD/BOD of more than two f o l d .

I t must be emphasized that these r e s u l t s were obtained w i t h mono c u l t u r e s u s u a l l y operated at optimum condi t i o n s f o r maximum p r o d u c t i v i t y . In an a c t u a l waste water s i t u a t i o n however, ( i . e . a p o l l u t e d r i v e r ) the condit i o n s may be f a r from optimum and hence n much lower. The presence of i n h i b i t o r s or the absence of e s s e n t i a l growth c o - f a c t o r s can f u r t h e r i n f l u e n c e the y i e l d and consequently the growth e f f i c i e n c y . Furthermore i t i s known that d i f f e r e n t organisms d i s p l a y d i f f e r e n t thermodynamic e f f i c i e n c i e s when grown on the same substrate. For growth on methanol f o r example, a range of n , from 0.23 to 0.46 has been reported w i t h v a r i o u s organisms (Roels, 1980). The above presented treatment can also be extended to cases w i t h N-sources other than ammonia, by u t i l i z i n g the concept of "generalized degree of redu c t i o n " , introduced by Roels (1980) instead of equation (3). I t i s to be noted, however, that the treatment does not consider the c o n t r i b u t i o n of the po s s i b l e oxygen demand f o r the n i t r i f i c a t i o n process. Furthermore a l l o x i d i z a b l e m a t e r i a l has been assumed to be biodegradable.

MATERIALS AND METHODS

K l e b s i e l l a pneumoniae (aerogenes) NCTC 418 was c u l t i v a t e d a e r o b i c a l l y i n simple s a l t s medium w i t h g l y c e r o l being the only carbon source. A l l experiments were performed i n a 11 x 10~3 m3 working volume b i o r e a c t o r i n batch mode. Temperature and pH were set and c o n t r o l l e d at 308 K and 6.8, r e s p e c t i v e l y . Elemental composition of the biomass was determined by a computer coupled element analyser. Average elemental composition of the biomass was c a l c u l a t e d to be C H]_62No.23 <-l0.46 » y i e l d i n g a degree of reduction of 4.01 .

RESULTS AND DISCUSSION

In a set of experiments c a r r i e d out i n batch mode, the environmental c o n d i t i o n s were changed by adding sodium c h l o r i d e i n t o the growth medium, which brought a change i n the thermodynamic e f f i c i e n c y of the growth process. A l l other experimental c o n d i t i o n s were kept i d e n t i c a l to e l i m i n a t e ri being e f f e c t e d by any parameter other than the i n h i b i t o r (NaCl). A d d i t i o n of sodium c h l o r i d e reduced the growth r a t e and y i e l d s . Using eqs. (2) and (8) the r a t i o , COD/BOD was evaluated f o r each experiment and p l o t t e d against the NaCl concentration and the maximum s p e c i f i c growth rate determined f o r f o r each batch , i n f i g s . 2. and 3., r e s p e c t i v e l y . From these p l o t s i t i s c l e a r that-the r a t i o COD/BOD

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• Na CI (kg/m3)

Fig. 1: COD'/BOD versus the thermodynamic efficiency of the growth process.

Fig. 2: COD/BOD versus the Sodium Chloride concentration in growth medium.

C O D B O D

Fig. 3: COD/BOD versus the growth rate as influenced by the NaCl concentration.

changed considerably throughout the experimental range. The growth rat e s determined were much higher than those normally a t t a i n e d i n waste water treatment i n s t a l l a t i o n s therefore the r a t i o may change l e s s d r a m a t i c a l l y i n those a p p l i c a t i o n s . Nevertheless the r a t i o COD/BOD must be considered as a fu n c t i o n of the growth e f f i c i e n c y which can be a f f e c t e d s i g n i f i c a n t l y by the environmental c o n d i t i o n s . Presence of an i n h i b i t o r , f o r instance NaCl ( i n t h i s case), can therefore a l t e r the r a t i o . This may have important consequences when i n d u s t r i a l waste waters are d i l u t e d to d i f f e r e n t extents p r i o r to laboratory BOD determinations. Hence i t can be concluded that a constant value f o r COD/BOD cannot be assumed unless under i d e n t i c a l environmental c o n d i t i o n s .

REFERENCES

Schroeder, E.D., Water and waste water treatment,p.219, McGraw H i l l , New York (1977)

N a t i o n a l Sweedish Environmental P r o t e c t i o n Board, S t a t i s t i c a l C o r r e l a t i o n Between A n a l y s i s Data f o r COD and BOD, (1972) as quoted by Johnson COD meter pamphlet.

Sherrard, J.H. and Schroeder, E.D.,(1975) Proceedings of Purdue conference,p- 14 Minkevich, I.G. and Erosh i n , V.K., (1973) F o l i o Microbiol.,18,p•376. E r i c k s o n , L.E., Minkevich, I.G. and Erosh i n , V.K.,(1978) Bi o t e c h n o l . Bioeng.,

20, p. 1595. Roels, J.A.,(1980) B i o t e c h n o l . Bioeng., 22, p. 2457. Esener, A.A., Kossen, N.W.F. and Roels, J.A. (1980) Paper presented to the

second i n t e r n a t i o n a l symposium on Waste Treatment and U t i l i z a t i o n , held at Waterloo, Canada.

NOMENCLATURE

BOD biochemical oxygen demand C H b ] 0 C l N d l biomass elemental formulae Ca 2Hb 2°c 2Nd 2

substrate elemental formulae COD chemical oxygen demand Y" biomass y i e l d on substrate sx

biomass y i e l d on substrate Y ox

biomass y i e l d on oxygen 0. l f l ow of the re s p e c t i v e component Y i degree of red u c t i o n of the i th component n thermodynamic e f f i c i e n c y f a c t o r f o r m i c r o b i a l growth V s p e c i f i c growth rate

s u b s c r i p t s s substrate X biomass

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1

APPLICATION 3

DESCRIPTION OF MICROBIAL GROWTH BEHAVIOUR DURING THE WASH-OUT PHASE ; DETERMINATION OF THE MAXIMUM SPECIFIC GROWTH RATE *

A. A. Esener, J . A. Roels and N. W. F. Kossen

SUMMARY

M i c r o b i a l growth during the wash-out phase has been described by an u n s t r u c t u ­red model based on Monod k i n e t i c s and the r e l a t i o n of l i n e a r substrate consump­t i o n . The optimal experimentation range and procedure have been evaluated f o r accurate estimation of the maximum s p e c i f i c growth r a t e s .

INTRODUCTION

The maximum s p e c i f i c growth rate i s an important parameter i n the d e s c r i p t i o n of m i c r o b i a l growth. A convenient method of i t s determination involves washing-out of a c u l t u r e , as f i r s t described by P i r t and Callow (1960). The procedure f o r t h i s method i s as f o l l o w s :

i . b r i n g a continuous c u l t u r e system to steady s t a t e at D < u m ax- Therefore the y m a x has to be estimated roughly to p r i o r to experimentation,

i i . step-up D to D + AD at time t=0, such that D + AD > U m a x where the organism i s expected to grow at i t s p o s s i b l e maximum r a t e ,

i i i . measure response(s) of the system, u s u a l l y the dry weight oxygen uptake and carbon dioxide production rates as func t i o n s of time.

Although the method has been reported and used widely, a d e t a i l e d t h e o r e t i c a l and experimental e v a l u a t i o n of i t has not been undertaken ( P i r t and Callow, 1960; Tempest, 1970; P i r t , 1975; Kuhn et a l , 1980). In t h i s work, growth behaviour during the wash-out phase has been studied w i t h two o b j e c t i v e s i n mind:

a. to e s t a b l i s h a s u i t a b l e experimental range and procedure f o r the optimal estimation of the b i o k i n e t i c parameter, V max-

b. to t e s t the v a l i d i t y of Monod model under these c o n d i t i o n s .

*Submitted f o r p u b l i c a t i o n , ( 1 9 8 1 ) .

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THEORY

For a constant volume continuous c u l t u r e system the f o l l o w i n g balance equations can be formulated f o r biomass and the l i m i t i n g substrate concentrations:

d C / d t = r - D C (1) X X X

d C / dt = D (C . - C ) - r (2) S S I s s

Here, r x , the rate of biomass production may be given by the usual Monod r e l a t i o n :

r = y {C / (C + k )} C (3) x max s s s x

I f , however, C g becomes much greater than k g during the wash-out phase, r x can be approximated by the f o l l o w i n g :

r = y C (4) x max x

In t h i s case the balance f o r biomass, a f t e r i n t e g r a t i o n y i e l d s :

C = C exp {( y - D ) t} (5) x xo max

or when transformed:

In C = In C + ( y - D) t (5a) x xo max

Thus Vmax can e a s i l y be c a l c u l a t e d by performing nonlinear or l i n e a r r e g r e s s i o n on the C x (In C x) versus time data obtained, by the use of eqs.(5) and (5a), r e s p e c t i v e l y . Moreover, i f the l i n e a r substrate consumption r e l a t i o n (Herbert, 1958 ; P i r t , 1965) can be assumed to be a p p l i c a b l e , the f o l l o w i n g r e l a t i o n s can be derived f o r the oxygen uptake rate (OUR) and carbon dioxide production rate (CPR), provided the r e s p e c t i v e accumulation f o r both gases i n the system can be neglected. Also note that t h i s a n a l y s i s assumes n e g l i g i b l e time con­stants f o r the adaptation of the growth rate and the subsequent adaptation of the substrate concentration.

OUR = OUR exp {(y - D ) t} (6)

where,

CPR = CPR exp {(y - D ) t} (7) o max

OUR = ( y / Y m a x + m ) C (8) o max ox o xo

CPR = ( y / Y m a X + m ) C (9) o max cx c xo

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Furthermore, at the maximum s p e c i f i c growth rate the i n f l u e n c e of the mainte­nance terms w i l l be n e g l i g i b l e i n comparison w i t h the c o n t r i b u t i o n of the growth terms. Hence the maintenance terms can be conveniently dropped out.

The substrate accumulation term cannot be ignored however, and under these cond i t i o n s C g p r o f i l e can be given as a f u n c t i o n of time by the f o l l o w i n g expression which can be derived from eqs.(2) and (5) i n combination w i t h the l i n e a r r e l a t i o n f o r substrate consumption:

dC /dt = D (C . - C )-(y /Y + m ) C exp {(y - D) t} s s i s max sx s xo max

An a n a l y t i c a l s o l u t i o n f o r t h i s equation can be shown to be given by:

C = {(C - C . + z ) - z exp(y t ) } / exp(D t) + C . (9) s so s i max s i where, , ,„max . „ , z = (y /Y + m ) C /y

max sx s xo max

The s u b s c r i p t o r e f e r s to state at time t=0, i . e . , commencement of the step increase i n the d i l u t i o n r a t e . Hence four responses can be monitored i n a w e l l designed experiment (C , C g, OUR, CPR) and the most optimal estimate of y max can be obtained by f i t t i n g the experimental data to the above presented r e l a t i o n s , simultaneously.

I t must be noted that i n the above presented a n a l y s i s the organism i s assumed to adapt to the new d i l u t i o n rate instantaneously and begin to grow at i t s maximum r a t e . This assumption seems to be reasonable f o r only small steps i n D (Pir t , 1 9 7 5 ) . For b i g steps the organism needs a considerable adaptation time to reach balanced growth and exert maximum growth r a t e . During such an adapta­t i o n period the c e l l s w i l l wash-out f a s t e r than expected r e s u l t i n g i n the c a l c u l a t i o n of a Umax value smaller than a c t u a l . Too a small step i n D, on the other hand might r e s u l t i n a prolonged s t a t e of substrate l i m i t a t i o n or u n j u s t i f y the assumption of C s » k s, f o r a considerable period of time f o l l o w i n g the step-up. To avoid t h i s c o mplication Tempest (1970) recommends the use of a large step increase i n D to a c u l t u r e which has been kept at steady s t a t e very close to the true U m a x . Therefore, the experimental condi­t i o n s must be chosen w i t h care.

Apart from p h y s i o l o g i c a l c o n s i d e r a t i o n s , e r r o r a n a l y s i s can f a c i l i t a t e the determination of a s u i t a b l e experimental range. From a s t a t i s t i c a l p o i n t of view the optimal range can be determined by evalua t i n g the t o t a l variance of Umax a n d f i n d i n g the co n d i t i o n s under which i t can be minimized. Considering the simplest case of a wash-out experiment w i t h one response v a r i a b l e , C x, the probable e r r o r i n the estimated Umax c a n be approximated by c a l c u l a t i n g i t s t o t a l variance a f t e r l i n e a r i z i n g the r e l a t i o n used i . e . , eq.(5) as described by Himmelblau (1970).

Var(u ) = {(y - D ) / ( l n C - In C ) } 2 Var(C )/C 2 + Var(D) (10) max max x xo x x

l f Var(C ) = Var(C ) and 1/C2 « 1/C2

X O X X O X

This expression reveals that Var(u max) w i l l be minimized i f ( y m a x ~ D) i - s

minimized (or l n C x - l n C X 0 i s maximized) i n a d d i t i o n to ' m i n i m i z a t i o n of the

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e r r o r s involved i n the measurement of C x and D. This means that wash-out c a r r i e d out close to Umax . o r i n other words an experiment which takes a long time f o r s i g n i f i c a n t decrease i n biomass concentration would be more accurate. From eq.(10) one can also deduce that i t would be d e s i r a b l e to operate at high C x values as the variance of Umaxis i n v e r s e l y p r o p o r t i o n a l to the square of the biomass concentration. However, a p r a c t i c a l c o n s t r a i n t i s u s u a l l y imposed on the value of C x by the oxygen t r a n s f e r capacity of the experimental system.

MATERIALS AND METHODS

Wash-out experiments were c a r r i e d out i n a one l i t e r working volume fermentor ( B i o l a f i t t e ) with a mono c u l t u r e of Klebsiella pneumoniae (aerogenes) NCTC 418. A s y n t h e t i c medium w i t h l a c t i c a c i d as the sole carbon and energy source was used. Temperature and pH were set and c o n t r o l l e d at 35°C and 6.8, r e s p e c t i v e ­l y . Batch experiments were c a r r i e d out i n an 11 l i t e r working volume fermentor. A l l other experimental d e t a i l s were as described p r e v i o u s l y (Esener et a l , 1980).

RESULTS AND DISCUSSION

Six experiments were performed w i t h d i f f e r e n t step s i z e s i n the d i l u t i o n r a t e . The Umax values c a l c u l a t e d from d i f f e r e n t responses are presented i n Table I . The values of the wash-out D were not s i g n i f i c a n t l y d i f f e r e n t f o r these experiments w i t h the exception of experiment no. 1.

Table I : Umax values c a l c u l a t e d from various responses.

Run no: Step i n D (hr 1) Umax c a l c u l a t e d from ( h r ~ ' ) : from to C x data OUR data CPR data

1 0.16 0.976 0.390 0.628 0.748 2 0.78 1 .095 0.922 0.944 0.968 3 0.71 1 .023 0.934 0.939 0.963 4 0.86 1 .050 0.918 0.966 0.966 5 0.88 1 .036 0.948 0.959 0.966 6 0.97 1 .087 0.985 1 .038 1 .038

Raw data f o r a t y p i c a l wash-out experiment are shown i n F i g . 1. From t h i s f i g u r e i t can be seen that the system cannot be described by eq.(5) immediately a f t e r the step-up i n the d i l u t i o n r a t e . This was probably due to the substrate l i m i t a t i o n c o mplication mentioned e a r l i e r and/or due to the i n a b i l i t y of the organism to adapt to the new conditi o n s instanteneously. S i m i l a r response patterns can be seen f o r both C x and OUR. During the i n i t i a l period the organism seems to wash-out f a s t e r . A f t e r t h i s p e r i o d , which l a s t e d about three h y d r a u l i c residence time, both responses can be reasonably w e l l described by eq. (5) .

In F i g . 2 , c a l c u l a t e d Umax values are shown as functions of the step s i z e applied to achieve wash-out. Since the f i n a l D was s i m i l a r f o r each experiment the step s i z e i n D can be regarded as a measure of p h y s i o l o g i c a l shock. In a l l cases the same general trend has been observed. That i s , the smaller the p h y s i o l o g i c a l shock the higher the value of Umax > c a l c u l a t e d . An extreme s i t u a t i o n i s represented by run no:l where the a p p l i c a t i o n of a very large step r e s u l t e d i n the c a l c u l a t i o n of a Umax value much smaller than a c t u a l . Also f o r

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, 0 0F OUR Cmole/m3/hr)

50-

20L

Cx (kg/m3) 2V à

_1 1 I I I I I L_

Fig. 1: Wash-out experiment no:4. C and OUR profiles.

C v data

O U R data

CPR data

1.0

0.3 1.0-

0.5 1.0

0.5 0.5

AD Chr"') 1.0

Fig. 2: Calculated y values as a function of the step-size. a max J J r

t h i s case there was no good agreement between the y m a x values c a l c u l a t e d from d i f f e r e n t responses ( C x , OUR, CPR).

From these r e s u l t s i t seems that the most optimal estimate of y m a x can be obtained by applying an i n f i n i t e s i m a l step i n D at steady s t a t e achieved at almost the a c t u a l y m a x . Such an i n f i n i t e s i m a l step i s not p o s s i b l e i n p r a c t i c e however, from the experimental data presented a good approximation to that i d e a l s t a t e seems to be obtainable by e x t r a p o l a t i n g the determined y m a x

versus step s i z e data, to zero step s i z e . By t h i s procedure many experiments c a r r i e d out at d i f f e r e n t step s i z e s c o n t r i b u t e to the c a l c u l a t i o n of the true p m a x . Here, as a f i r s t approximation s t r a i g h t l i n e s are f i t t e d to data presented i n f i g . 2 . u m a x values obtained by t h i s procedure are shown i n Table I I . They compare w e l l w i t h the y m a x value c a l c u l a t e d from the normalized data

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obtained from s i x independent batch experiments and a l l are w i t h i n the 95 % confidence l e v e l of the batch r e s u l t s ( F i g . 3).

Table I I : Umax values c a l c u l a t e d by e x t r a p o l a t i o n to AD = 0.

u c a l c u l a t i o n based on (hr ): C data OUR data CPR data x 1.121 1.088 1.060 A 0.987 1.046 1.029 B

From batch experiments (see F i g . 3 ) : 1.087 (0.950 - 1.223) nonlinear r e g r e s s i o n 1.047 (0.987 - 1.106) l i n e a r r e g r e s s i o n

A e x t r a p o l a t i o n w i t h a l l data points B e x t r a p o l a t i o n excluding data of run no : l Figures i n paranthesis are the 95 % confidence l e v e l s

Fig. 3: Normaized data for six batch experiments.

In view of the above presented r e s u l t s i t can be concluded that the optimal e s t i m a t i o n of Umax wi t h the wash-out method should i n v o l v e more than one experiment c a r r i e d out with d i f f e r e n t p h y s i o l o g i c a l step s i z e s . A l o g i c a l experimental sequence would be to perform the f i r s t experiment w i t h a large step i n d i l u t i o n r a t e ( s i n c e no estimate of Umax l s a v a i l a b l e at t h i s stage) and g r a d u a l l y to reduce the step s i z e as much as p r a c t i c a b l e i n an attempt to avoid e x t r a p o l a t i o n over a large step range. Once a reasonably accurate

106

estimate of M m a x i s obtained i t might als o be u s e f u l to perform one e x t r a experiment i n which a large step i n D i s app l i e d to a c u l t u r e maintained at steady s t a t e very close to t h i s estimate. This large step i n D w i l l s t i l l be a small p h y s i o l o g i c a l step but w i l l avoid disturbances due to p o s s i b l e substrate l i m i t a t i o n , mentioned e a r l i e r .

The f a c t that d i f f e r e n t U m a x values obtained under d i f f e r e n t c o n d i t i o n s , i n d i c a t e s that Monod model f a i l s to hold i n these instances.

The wash-out method has major experimental advantages. Since OUR and CPR can be recorded o n - l i n e and that there i s good agreement between the Umax values obtained from d i f f e r e n t responses the whole experiment can be c a r r i e d out without a s i n g l e sample being taken out. Moreover, i f C x p r o f i l e i s to be determined t h i s can be done by sampling v i a the fermentor o u t l e t stream and hence without d i s t u r b i n g the system. C x p r o f i l e can a l s o be measured conveni­e n t l y by connecting an o n - l i n e spectrophotometer to the o u t l e t l i n e . For small steps no s i g n i f i c a n t p h y s i o l o g i c a l changes are expected; thus no c o r r e c ­t i o n s f o r changes i n c e l l s i z e , colour due to pigment production e t c . would be necessary.

I t can therefore be concluded t h a t , provided the p h y s i o l o g i c a l , experimental and s t a t i s t i c a l c o n siderations are given a t t e n t i o n t o , the wash-out method i s a powerful technique i n the study of m i c r o b i a l behaviour and p a r t i c u l a r l y f o r the est i m a t i o n of the maximum s p e c i f i c growth r a t e s .

SYMBOLS AND ABBREVIATION

concentration (kg/m^) carbon dioxide production r a t e (mole/m^/hr) d i l u t i o n r a te ( h r - ' ) Monod s a t u r a t i o n constant (kg/m^) maintenance c o e f f i c i e n t on the i th substance (kg/kg/hr)or(mole/kg/hr) oxygen uptake r a t e (mole/m-Vhr) ra t e of consumption or production of i . (kg/m3/hr) time (hr) maximal y i e l d on i (kg/kg)or(kg/mole) s p e c i f i c growth rate ( h r - ' )

s u b s c r i p t s i i n l e t o i n i t i a l s t a t e (t=0) s substrate x biomass

REFERENCES

Esener A A , Roels J A , Kossen N W F (1980) Carbon d i o x i d e hold-up as a source of e r r o r i n batch c u l t u r e c a l c u l a t i o n . B i o t e c h n o l . Bioeng 22:1979-1983.

Herbert D (1958) Some p r i n c i p l e s of continuous c u l t u r e , i n Recent Progress i n Microbiology, Almqvist and Winksel Stockholm, p.381.

Himmelblau D H (1970)Process A n a l y s i s by S t a t i s t i c a l Methods, John Wiley, New York.

Kuhn H J , Cometta S, F i e c h t e r A (1980) E f f e c t of growth temperature on maximal s p e c i f i c growth r a t e , y i e l d , maintenance and death r a t e i n glucose l i m i t e d

C CPR D k s

mi OUR r i t max

I i x U

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continuous c u l t u r e of the thermophilic B a c i l l u s caldetonax. Eur J Appl M i c r o b i o l Biotechnol 10: 303-315.

P i r t S J , Callow D S (1960) Studies of the growth of P e n i c i l l i u m chyrogenum i n continuous flow c u l t u r e w i t h reference to p e n i c i l l i n production. J Appl B a c t e r i o l 23(1): 87-98.

P i r t S J (1965)The maintenance energy of b a c t e r i a i n growing c u l t u r e s . Proc Roy Soc B 163:224-234.

P i r t S J (1975) i n P r i n c i p l e s of Microbe and C e l l C u l t i v a t i o n , B l a c k w e l l , Oxford.

Tempest D W (1970) Theory of chemostat, i n Methods i n M i c r o b i o l o g y , N o r r i s J R Ribbons D W (eds) v o l . 2 , p.268, Academic, London.

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APPLICATION 4

ON THE STATISTICAL ANALYSIS OF BATCH DATA *

A.A. Esener, J.A. Roels and N.W.F. Kossen

INTRODUCTION

In Biotechnology, a v a r i e t y of processes are studied i n batch mode. Examples include batch c u l t i v a t i o n of microorganisms, enzymatic h y d r o l y s i s and conver­sions. The rates of these processes are u s u a l l y determined by applying the conventional s t a t i s t i c a l techniques and p a r t i c u l a r l y the r e g r e s s i o n techniques.

In a study of k i n e t i c s and energetics of b a c t e r i a l growth i n batch mode, we have n o t i c e d that the a p p l i c a t i o n of the l i n e a r r e g r e s s i o n procedure r e s u l t e d i n a serious overestimation of the p r e c i s i o n of the rate constant (the maximum s p e c i f i c growth r a t e ) . Rate constants c a l c u l a t e d from repeated experiments r e s u l t e d i n a range of values sometimes extending outside the 95 % confidence i n t e r v a l s determined f o r i n d i v i d u a l experiments. S t a r t i n g from t h i s observation f u r t h e r a n a l y s i s of the c o l l e c t e d data was i n i t i a t e d and i t was found out that i n most cases the e r r o r s were not independent but h i g h l y c o r r e l a t e d . This obviously v i o l a t e s the assumption of the independence of e r r o r s upon which the ordinary l i n e a r r e g r e s s i o n techniques are based.

In t h i s paper consequences of these f i n d i n g s w i l l be shown. A simple method developed by Mandel , to overcome these d i f f i c u l t i e s w i l l a l s o be presented and i t s use and r e s u l t s w i l l be compared with those of the conventional l i n e a r r e g r e s s i o n technique.

THEORY

I t has already been shown that when growth i s balanced i n batch mode, i t has to be exponential t o o . 2 Therefore, l i n e a r r e g r e s s i o n a n a l y s i s can be performed to o b t a i n the best estimate of the required parameter; the maximum s p e c i f i c growth r a t e . The l i n e a r form of the exponential growth equation can be obtained by i n t e g r a t i o n and then tranformation:

In C = constant + u . t (1) x max

* Accepted f o r p u b l i c a t i o n i n Bi o t e c h n o l . Bioeng. (1981)

109

where C x i s the biomass concentration at time t .

I t i s w e l l known that the assumption of independence of the process e r r o r s i s by f a r the most c r i t i c a l one i n the a n a l y s i s of data by r e g r e s s i o n procedures. L i k e many other s i t u a t i o n s , experimental measurements taken at progressive stages of batch c u l t u r e fermentations are e s s e n t i a l l y c a r r i e d out on the same subject of experimentation, i . e . , a l l samples are withdrawn and analysed f o r the d e s i r e d components, o r i g i n a t e s from the same bulk. This i n v a l i d a t e s the assumption of the independence of the e r r o r s . For measurements to be a c t u a l l y independent, s e v e r a l batch runs must be performed simultane­ously, one measurement being made from each, at d i f f e r e n t run times. In such a ( c o s t l y ) experiment the e r r o r s can be regarded as independent and the l i n e a r r e g r e s s i o n procedure can be used s a f e l y .

When parameter and e r r o r e s t i m a t i o n are c a r r i e d out on data obtained from a s i n g l e batch, the s i t u a t i o n i s q u i t e d i f f e r e n t and the a p p l i c a t i o n of the conventional r e g r e s s i o n procedures can r e s u l t i n a serious overestimation of the p r e c i s i o n of the estimated slope. Data obtained at progressive stages of a process c a r r i e d out on the same subject of experimentation are subject to e r r o r s of two types: a. Measurement- e r r o r , introduced by a n a l y t i c a l methods, instrumental readings

etc. These are u s u a l l y independent.

b. In a d d i t i o n to e r r o r s of type a., u n c e r t a i n i t i e s can a r i s e due to f l u c t u ­a t i o n s i n the process of fermentation. These e r r o r s are always present because of the f l u c t u a t i o n s i n the experimental c o n d i t i o n s . No matter how a c c u r a t e l y the environmental c o n d i t i o n s , l i k e pH, temperature, a w ,DOT e t c . are c o n t r o l l e d , there w i l l always be d e v i a t i o n s from the set p o i n t s . These type of e r r o r s are c a l l e d Process errors. I t i s important to note that process e r r o r s are cumulative i . e . , each e r r o r during the course of fermen­t a t i o n not only a f f e c t s the next measurement but a l l the subsequent measu­rements. In other words each i n d i v i d u a l e r r o r e n , includes a l l process e r r o r s preceeding i t . i . e . ,

n E = E e. n i i

This suggests the use of d i f f e r e n c e s between the successive measurements i n data a n a l y s i s , as t h i s w i l l avoid the i n c l u s i o n of process e r r o r s . Since i n t h i s case e r r o r s w i l l be :

{e,}, {( e.+ e 9 ) - E }, ,{(E + e ) -( e )} l I l I n-l n n-1

or E I ' £ 2 £ n

However, these new observations w i l l no longer be homoscedastic i n general, since the variance of each d i f f e r e n c e w i l l depend on the corresponding i n t e r v a l .

"' - i . . . f '

In general p a r t i t i o n i n g can be made i n t o known proportions of independent and cumulative components and one has to consider the data as belonging to one or the other type. I f the data are predominantly cumulative, conven­t i o n a l l i n e a r regression technique can no longer be used.

1 10

- How to determine which of the two type of e r r o r s ; measurement or process e r r o r , i s predominant ?

For large n (number of observations) the experimental points tend to be sc a t t e r e d above and below the f i t t e d o r d inary r e g r e s s i o n l i n e , i f the e r r o r s are of predominantly independent type. For process e r r o r s , however, the e x p e r i ­mental points tend to remain on one side of the r e g r e s s i o n l i n e f o r long sequences of points ( F i g . 1.) A more q u a n t i t a t i v e d i s c r i m i n a t i o n procedure can be obtained i f one computes the a u t o c o r r e l a t i o n f u n c t i o n f o r the r e s i d u a l s obtained by the a p p l i c a t i o n of the ordinary r e g r e s s i o n procedure. For data w i t h independent e r r o r s the a u t o c o r r e l a t i o n c o e f f i c i e n t , r ^ , should o s c i l l a t e above and below r ^ = 0, at random. For data w i t h predominantly cumulative e r r o r s (Process e r r o r s ) the s i g n i f i c a n c e of the dependence of r e s i d u a l s can r e a d i l y be assessed at various lags from the r ^ p l o t .

Regression f o r data with predominant process e r r o r s (cumulative e r r o r s )

The s t a t i s t i c a l treatment f o l l o w s the work of Mandel'. The reader i s r e f e r r e d to the o r i g i n a l a r t i c l e f o r the d e r i v a t i o n of the r e l a t i o n s to be used here. Once the data i s concluded to contain predominant process e r r o r s , the below model holds f o r the simple case of a s t r a i g h t l i n e through the o r i g i n :

n Y. = g x. + I e. (2) l l . i l

The corresponding model f o r independent e r r o r s , i . e . , the ordinary l i n e a r r e g r e s s i o n model, i s given by :

Y. = g x. + e. (2a) l 1 1

The best unbiased l i n e a r estimator of the slope f o r model (2) i s obtained by minimizing the weighted sum of squares. The r e s u l t i n g estimator i s of the form:

n n B = ( H . )/( 2 L. ) (3)

5-i J i - i J

where Zj = Yj -

or Z = 6 ( X j - X j _ j ) + e. ( j = 1, n : X Q = 0)

and L. = (x. - *,_,)

Furthermore, E. i s assumed to be independent f o r non-overlapping t e s t i n t e r v a l s and :

Var(e.) = f ( L . ) = c. L. c = constant J i J

i . e . , the variance of i s assumed to be a f u n c t i o n of the corresponding t e s t i n t e r v a l . Here, i t i s i n t e r e s t i n g to note t h a t , i f the t e s t i n t e r v a l s repre­sented by Lj form an uninterrupted sequence, equation (3) becomes simply:

1 1 1

6 = Y / x n n

i . e . , the best estimate of the slope of the cumulative model i s made by t a k i n g the l a s t value of the dependent v a r i a b l e and d i v i d i n g i t by the l a s t value of the independent v a r i a b l e . The intermediate r e s u l t s might seem u s e l e s s , however, they are indeed e s s e n t i a l f o r : a. d e c i d i n g whether the process can best be described by an independent model

or not, b. e s t i m a t i o n of the standard e r r o r of the slope, which i s given by :

n Var (6) = c / Z L. (4)

5=1 J

when c i s not known, the unbiased estimate of the variance i s given by : a-, -.• .

e s t . Var (B) = ( Jy~ ) ( l—r ) Z f - i - (Z. - P, L.) 2} (5) I L. n - 1 L. 1 r j J J

Note that the e s t i m a t i o n i n v o l v e s a l l measurements despite the f a c t that the e s t i m a t i o n of slope requires only the f i r s t and the l a s t .

Furthermore, Mandel' has shown that i f the experiment i s best represented by a cumulative model but through ignorance or otherwise i s treated as being represented by an independent model, the r a t i o of variances i s given by :

Var (%) _ 6 2n 2 + 2n + 1 (6) Var (B) 5 (n + 1) (2n + 1)

where the t i l d e (y) denotes an estimate i n c o r r e c t l y c a l c u l a t e d . As n •+• °°, the r a t i o approaches to 1.2 or f o r n= 10, i t i s 1.15 . Thus the i n c o r r e c t estimate of B i s only s l i g h t l y d i f f e r e n t than the c o r r e c t estimate.

However, a large e r r o r i s introduced when data with predominantly cumulative e r r o r s are treated by l i n e a r r e g r e s s i o n as i f independent, and the standard e r r o r of the B i s to be estimated. In t h i s case the r a t i o becomes:

est. Var (&) n + 2 2(2n2 + 2n + 1) Var (B)

This r a t i o i n d i c a t e s that f o r n = 100

(7)

est. standard e r r o r of (B) = 1/19.9 s.e. (6)

i . e . , the standard e r r o r of the slope i s underestimated by 20 times. This underestimation f a c t o r can conveniently be approximated by 2/n f o r n > 10.

112

R E S U L T S A N D D I S C U S S I O N

Dry weight measurements are ge n e r a l l y time consuming and therefore only a l i m i t e d number of them can be made during the exponential growth phase. For the experiment reported f u l l y here, the oxygen uptake rate (OUR) w i l l be considered f i r s t since a continuous readout of t h i s v a r i a b l e i s a v a i l a b l e . The o r i g i n a l OUR data i s p l o t t e d against time as shown i n F i g . 1 together w i t h the conv e n t i o n a l l y f i t t e d r e g r e s s i o n l i n e (n = 40).

1 10 20 30 40

Fig. 1: Oxygen uptake rate data and the conventionally fitted straight line for a batch fermentation.

When the r e s i d u a l s are evaluated and p l o t t e d against the same x-axis ( F i g . 2.), a c l e a r trend can be observed. Approximately during the f i r s t and l a s t 1/4 of the experimental range the r e s i d u a l s were p o s i t i v e and f o r the r e s t negative. This suggests that the consecutive r e s i d u a l s were c o r r e l a t e d . When the auto­c o r r e l a t i o n of these r e s i d u a l s i s approximated f o r l a g up to 30, according to the formula;

n-k n r k = {( £ (Y. - Y ) ( Y . + k - Y)J / Z(Y. - Y ) Z (8)

i i k = 1,2, , k

the trend indeed shows considerable c o r r e l a t i o n at lags 1 and 2. ( F i g . 3.). At lag 1, r ^ i s 0.63, which i s more than four times higher than the standard d e v i a t i o n of the expected mean, i f the data were independent. For independent data a l l r ^ are expected to be of s i m i l a r i n s i g n i f i c a n t value because the presence of predominant measurement e r r o r s would have introduced randomness i n t o the d i s t r i b u t i o n of r ^ . Since t h i s i s not the case w i t h the reported data. ( F i g . 3.) we can conclude that the cumulative e r r o r s are predominant.In Table I y m a x values c a l c u l a t e d by assuming constant y i e l d on oxygen are shown f o r three methods o u t l i n e d e a r l i e r . The d i f f e r e n t i a l a n a l y s i s i s also included s i n c e the

113

data p o i n t s are e q u a l l y spaced i n time. I t i s important to note that i n t h i s a n a l y s i s , e r r o r s i n the transformed form are al s o assumed to be normally d i s t r i b u t e d and independent of the l e v e l of the measurement.

Residual

( V observed - v r e g r e s s i o n )» 1Û 3

10 B 20* 30 — r - n 40

Fig. 2: Behaviour of the residuals obtained by the application of the conventional linear regression procedure.

Fig. 3: Autocorrelation function for the resiuduals shewn in Fig. 2.

Table I : P m a x c a l c u l a t i o n by various methods based on OUR data.

Method of a n a l y s i s n Mmax( h r ') a 0 % of y m a x

independent 40 0.913 0 0032 0.35 d i f f e r e n t i a l 39 0.939 0 0338 3.60 cumulative 40 0.921 0 0346 3.75

a = standard d e v i a t i o n

From Table I . i t can be observed that the three methods used a l l r e s u l t e d i n s i m i l a r values of the slope ( U m a x ) , however, the f i r s t method (independent) r e s u l t e d i n a serious overestimation of the p r e c i s i o n of the slope, i n t h i s case by about 10 times. Using r e l a t i o n (7) an underestimation f a c t o r of 12.5 can be c a l c u l a t e d f o r n = 40 , which i s close to the c a l c u l a t e d value of 10 (see Table I ) . Dry weight data not reported here, were a l s o treated by the independent and dependent models and the outcomes compared i n Table I I . Since only a few data p o i n t s could be obtained a r e s i d u a l a n a l y s i s cannot be c a r r i e d out. I t can be n o t i c e d from Tables I and I I that as n increases the d i f f e r e n c e between the estimates of the standard e r r o r s c a l c u l a t e d by the two methods becomes much more s e r i o u s . Both methods, however, r e s u l t e d i n s i m i l a r values of P m a x . I t can hence be concluded that i f data from batch processes are to be analysed one must f i r s t t e s t the experimental data f o r p o s s i b l e v i o l a t i o n of the hypothesis of independence. I f the e r r o r s are found to be cumulative, the model o u t l i n e d above should be used. In our experience most batch fermentations are indeed s i g n i f i c a n t l y i n f l u e n c e d by the presence of the s o - c a l l e d "process e r r o r s " . I t must be emphasized, however, that f o r experiments with very inaccurate a n a l y t i c a l measurements, e.g., RNA,protein, enzyme a c t i v i t y etc

114

the independent e r r o r s may be predominating. As the assumption of a cumulative model f o r an a c t u a l l y independent case w i l l a l s o r e s u l t i n serious e r r o r s , the safest procedure w i l l be to evaluate and t e s t the dependence/independence of each process on i t s own mer i t s .

Table I I : y c a l c u l a t i o n f o r three batches based on dry weight data, max Method of a n a l y s i s

exp. no : n independent cumulative u max

I 0 V

max c

0 C . I

a 1 a

0530 6 1.011 0.0160 0.996 0 0494 3. 1 0517 6 0.968 0.0267 0.970 0 0826 3. 1 0405 6 1 .098 0.0404 1 .106 0 0807 2.0

REFERENCES

1. J . Mandel, J . Amer. St a t . Assn., 52,552(1957). 2. A.G. Frederickson, D. Ramkrishna and H.M. Tsuchiya, Chem. Eng. Prog. Symp.

S e r i e s , 108(67),53(1971).

1 15

APPLICATION 5

CARBON DIOXIDE HOLD-UP AS A SOURCE OF ERROR IN BATCH CULTURE CALCULATIONS *

A. A. Esener, N. W. F. Kossen and J . A. Roels

INTRODUCTION

Consruction of m a t e r i a l balances i s of prime importance i n the study of b i o -en e r g e t i c s . The consistency of experimental data should always be checked v i a the known r e g u l a r i t i e s of fermentation processes and elemental balances. This subject has r e c e n t l y r e c i v e d much a t t e n t i o n i n the l i t e r a t u r e 1 - ^ . However, extensive data have so f a r appeared only f o r continuous c u l t u r e systems operating at steady s t a t e . In batch c u l t u r e s of Klebsiella pneumoniae studied i n t h i s l a b o r a t o r y , systematic losses i n carbon balances were observed. This discrepancy was then traced to the presence of carbon d i o x i d e i n the c u l t u r e broth at concentrations much higher than a n t i c i p a t e d . A s i m i l a r observation has r e c e n t l y been reported f o r baker's yeast by Barford and H a l l ^ .

An attempt was therefore made to qua n t i f y the amount of missing carbon dioxide and to evaluate the i n f l u e n c e of t h i s discrepancy on the c a l c u l a t i o n of r e s p i ­r a t o r y quotient (RQ). Therefore the amount of carbon d i o x i d e remaining i n the broth was determined during the exponential growth phase and the measured o v e r a l l RQ was corrected. Measured and corrected RQ values were als o compared wi t h the t h e o r e t i c a l RQ which can be c a l c u l a t e d from b i o e n e r g e t i c considerations i f the experimental y i e l d on s u b s t r a t e , degree of reduction and the carbon contents of biomass and substrate, are known. The corrected RQ was found to be close to the t h e o r e t i c a l l y c a l c u l a t e d RQ whereas the measured RQ deviated s i g n i f i c a n t l y . This phenomenon has therefore important i m p l i c a t i o n s f o r the i n t e r p r e t a t i o n and use of metabolic data, p a r t i c u l a r l y f o r computer-controlled batch or any other system operating at unsteady s t a t e , where the c o n t r o l parameter i s obtained from the instantaneous gas exchange data ^,7 .

MATERIALS AND METHODS

The organism, Klebsiella pneumoniae NCTC 418, was c u l t u r e d i n a s y n t h e t i c medium as described by Evans et a l G l y c e r o l was used as substrate and assayed enzymatically (Boehringer UV method No.148270). Dry weights were determined by the method of de V r i e s and Stouthamer ^. Biomass was c o l l e c t e d on a 0.2 pm pore diameter f i l t e r ( S a r t o r i u s ) , washed with d i s t i l l e d water and

* Published i n B i o t e c h n o l . Bioeng.,Vol.22, 1979-1983 (1980) 1 1 7

d r i e d to constant weight at 378 K. Medium was s t e r i l i z e d by membrane f i l t r a t i o n through a 0.2 ym f i l t e r and inoculated by an a c t i v e l y growing inoculum to e l i m i n a t e any l a g phase and the p o s s i b i l i t y of unbalanced growth. The elemental composition of the biomass (Table I) was determined by a computer coupled element analyzer (Perkin Elmer 240) .

Experiments were c a r r i e d out i n a 11 x 10~3 m 3 working volume fermentor maintained at 308 K. pH was c o n t r o l l e d at 6.80 ± 0.05 . The a i r flow rate to the fermentor was c o n t r o l l e d by a thermal mass flow meter (Brooks 5811) at about 0.77 kg dry a i r / h r . S p e c i a l a t t e n t i o n was paid f o r the accurate determi­nat i o n of gas flows and concentrations. A l l flows were corrected f o r humidity and volumetric changes. Gas phase oxygen and carbon d i o x i d e concentrations were determined by a twin channel paramagnetic oxygen analyzer (Servomex OA 184) and an i n f r a r e d carbon d i o x i d e analyzer (Beckman 864), r e s p e c t i v e l y . Complete aerobic growth was ensured by maintaining the d i s s o l v e d oxygen tension w e l l above the l i m i t i n g range. No products other than biomass, carbon d i o x i d e , and water could be detected at l e v e l s that can be s i g n i f i c a n t .

Table I : Elemental composition* and Formula of K. pneumoniae grown on g l y c e r o l at i t s maximum growth r a t e .

c H N 0 Formula 50.96 6.92 14.27 27.86 C H1.63N0.24°0.41 * I n % dry weight ; ash f r e e b a s i s .

CARBON DIOXIDE DETERMINATION

For the system used carbon d i o x i d e can be assumed to be the only source of inorganic carbon. Therefore f o r the c u l t u r e broth at any i n s t a n t

TC = TIC + TOC (1)

where TC i s the t o t a l carbon (kg/m3), TIC i s the t o t a l inorganic carbon and TOC i s the t o t a l organic carbon. Of these TC and TOC were measured by a Dohrmann DC-50 carbon analyzer. Carbon d i o x i d e not accounted f o r , by the gas phase a n a l y s i s , i . e . , carbon dioxide remaining i n the b r o t h , can then be c a l c u l a t e d by the f o l l o w i n g expression :

C0 2 (broth) =44/12 . (TC - TOC) (kg/m3) (2)

TC and TOC determinations were c a r r i e d out immediately a f t e r sampling. A l l samples were cooled during sampling down to about 278 K by an o n - l i n e heat exchanger manufactured i n our workshop. The t y p i c a l residence time i n the heat exchanger was about 5 seconds.

RESULTS AND DISCUSSION

Experimental r e s u l t s are shown i n f i g u r e s 1 and 2. The maximum s p e c i f i c growth rate of t h i s organism i s qu i t e high (1.07 ± 0.02 h r ~ ' ) , t h e r e f o r e i t was not p o s s i b l e to obtain more extensive data due to sampling, processing, and a n a l y s i s times r e q u i r e d . From f i g u r e 2 i t can be seen that the amount of unaccounted carbon dioxide increases during the exponential growth phase. Few

118

fig. 1: Biomass vs. fermentation time.

fig. 2: Carbon dioxide retained in the broth during exponential growth.

mechanisms that can maintain t h i s carbon d i o x i d e i n the broth can be thought of. These i n c l u d e entrapment by c e l l s , pure p h y s i c a l absorption by supernatant, or p a r t i c i p a t i o n i n the carbon dioxide-carbonate b u f f e r system. At t h i s stage however, the r e l a t i v e importance of these mechanisms i s not c l e a r . Therefore only a macro a n a l y s i s i s reported here.

Table I I : O v e r a l l carbonbalance and RQ c a l c u l a t i o n s f o r the exponential growth of K. pneumoniae.

per m3 of c u l t u r e (kg)

Substrate carbon used up 4.05 Carbon recovered i n biomass 2.66 Carbon recovered i n gas phas e 1 .06 Carbon recovered i n broth 0.27 Y i e l s on substrate (ash f r e e basis) 0.50

without w i t h c o r r e c t i o n c o r r e c t i o n D i f f e r e n c e (%)

C recovery (%) 92 99 7 RQ 0.57 0.71 18 RQ c a l c u l a t e d independently from y i e l d 0.69

on substrate by eq.(3)

In Table I I the o v e r a l l carbon balance f o r the exponential growth phase i s presented. These c a l c u l a t i o n s i n d i c a t e that carbon dioxide corresponding to 7% of the t o t a l substrate carbon input remained i n the broth at the end of exponential phase. This r e s u l t s i n a large e r r o r i n the measured value of RQ. Assuming complete aerobic growth w i t h no products other than biomass, carbon d i o x i d e and water, RQ can al s o be c a l c u l a t e d from the experimentally determined y i e l d on substrate by the mass-energy balance method f i r s t developed by Minkevich and Eroshin 3, Using t h e i r n o t a t i o n and rearrenging, RQ i s given by:

119

RQ = {1 - Y s ( a b / a s ) } / { ( Y s / 4 ) . ( 1 - Y s ( V b / a s Y s ) ) } (3)

For 0^=0.51, as=0.39 , Y s=4.67 and Y g =0.50 the above expression gives an RQ

value of 0.69 which i s i n c l o s e agreement w i t h the RQ value corrected f o r the carbon d i o x i d e r e t a i n e d i n the c u l t u r e and hence unaccounted f o r i n the gas phase (Table I I ) . The measured RQ, however, deviates s i g n i f i c a n t l y from the corrected and independently c a l c u l a t e d RQ values. These r e s u l t s i n d i c a t e that unless carbon balances f i t t i g h t l y or broth phase C O 2 measurements are made, measured RQ values can be i n great e r r o r . This can lead to serious problems i n i n d u s t r i e s where batch, fed-batch or any other fermentation e s s e n t i a l l y not at steady state i s monitored and c o n t r o l l e d by o n - l i n e measurement of the RQ.

NOMENCLATURE

RQ r e s p i r a t o r y quotient(dimensionless) TC t o t a l carbon i n broth(kg/m 3) T0C t o t a l organic carbon i n broth(kg/m 3) TIC t o t a l inorganic carbon i n broth(kg/m 3) Y s substrate y i e l d ( d i m e n s i o n l e s s ) 0 D carbon weight f r a c t i o n i n biomass(dimensionless) a s carbon weight f r a c t i o n i n substrate(dimensionless) Yfo reductance degree of biomass(equiv electrons/g.atom carbon) Y s reductance degree of substrate(equiv electrons/g.atom carbon)

REFERENCES

1. C. L. Cooney, H.Y. Wang and D.I.C. Wang, Biotechnol.Bioeng.,19,55(1977). 2. J.A. Roels and N.W.F. Kossen,"In the modelling of m i c r o b i a l metabolism",in

Progress i n I n d u s t r i a l M i c r o b i o l o g y , M.J. B u l l , E d . ( E l s e v i e r , Amsterdam, 1978).

3. L.E. E r i c k s o n , I.G. Minkevich and V.K. E r o s h i n , B i o t e c h n o l . Bioeng.,20,1595 (1978)

4. J.A. Roels, B i o t e c h n o l . Bioeng.,22,2457(1980) 5. J.P. Barford and R.J. H a l l , B i o t e c h n o l . Bioeng., 21,609(1979) 6. H.Y. Wang, C.L. Cooney and D.I.C. Wang, Bi o t e c h n o l . Bioeng.,19,69(1977) 7. S. Aiba, S. Nagai and Y. Nishizawa, B i o t e c h n o l . Bioeng.,18,1001(1976) 8. C.G.T. Evans, D. Herbert and D.W. Tempest, i n Methods i n M i c r o b i o l o g y , J.R.

N o r r i s and W.W. Ribbins.Eds. (Academic Press, London,1970),vol.2,p.313. 9. W. de V r i e s and A.H. Stouthamer, J . Bacteriol.,96,472(1968)

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S U M M A R Y

In t h i s t h e s i s an attempt i s made to evaluate the current s t a t e of m i c r o b i a l energetics from an engineering point of view. Some shortcomings of the current experimental and t h e o r e t i c a l p r a c t i c e are stressed and improvements developed are described. The work i s aimed to help the b i o t e c h n o l o g i s t i n the t r a n s l a t i o n and f i l t r a t i o n of m i c r o b i o l o g i c a l data to a q u a n t i t a t i v e form with s p e c i f i e d c e r t a i n i t y , upon which engineering design and operations can be based.

The theory and a p p l i c a t i o n s of unstructured models are discussed i n Chapter 2. A simple unstructured model based on Monod k i n e t i c s and the l i n e a r substrate consumption r e l a t i o n i s developed. I t i s shown that during growth i n batch mode the behaviour of the system i s r i g i d l y f i x e d by the k i n e t i c parameter; the maximum s p e c i f i c growth r a t e . The energetic parameters have minimal i n f l u ­ence. In continuous c u l t i v a t i o n the reverse i s true i . e . , the behaviour i s f i x e d by the energetic parameters; maximal Y s x and ms . I t i s a l s o shown that the choice of the k i n e t i c expression i s not c r i t i c a l and that almost any of the r e l a t i o n s reported i n l i t e r a t u r e , given the same a t t e n t i o n f o r parameter e s t i m a t i o n , i s capable of d e s c r i b i n g the experimentally observed behaviour. Linear r e l a t i o n f o r substrate consumption i s tested w i t h continuous c u l t u r e data. I t i s shown that the s i g n i f i c a n t d e v i a t i o n s at low growth rates cannot be f u l l y accounted f o r by the loss of v i a b i l i t y .

In Chapter 3 engineering t o o l s are applied to the study of b a c t e r i a l k i n e t i c s and e n e r g e t i c s . S t r a t e g i e s f o r more e f f i c i e n t experimentation, parameter e s t i ­mation and s t a t i s t i c a l a n a l y s i s are developed.

In Chapter 4, the unstructured model developed i s extended f o r the d e s c r i p t i o n of growth behaviour i n fed-batch mode. Extensive data and m a t e r i a l balances are presented. The unstructured model i s found to provide a reasonably good d e s c r i p t i o n f o r the exponential and pseudo-steady s t a t e s . The model f a i l s to hold during the t r a n s i t i o n phase. I t i s a l s o shown that fed-batch c u l t i v a t i o n technique i s a u s e f u l t o o l i n the study of b i o k i n e t i c s and energetics s i m u l ­taneously.

In Chapters 5 and 6, e f f e c t s of two s e l e c t e d environmental changes on the parameters of the unstructured model are s t u d i e d . The i n f l u e n c e s of s a l i n i t y are studied i n Chapter 5. Results are a l s o compared w i t h those reported f o r mixed c u l t u r e s ( a c t i v a t e d sludge). I t i s shown that the mixed c u l t u r e s can adapt much f a s t e r to changes i n s a l i n i t y . The other parameter more r e l e v a n t to i n d u s t r i a l operations i s temperature. Chapter 6 presents r e s u l t s obtained i n fed-batch c u l t u r e s . An Arrhenius type enzyme a c t i v a t i o n - i n a c t i v a t i o n model extended to describe the superoptimal temperature range i s used to c o r r e l a t e the change i n the maximum s p e c i f i c growth r a t e w i t h temperature. Of the energetic parameters, Y m

xx i s found to be roughly the same over the tempe­

ratur e range of 25 - 40 °C. Above 40 °C i t decreased abruptly. Growth could not be sustained at 50 °C. The maintenance c o e f f i c i e n t increased e x p o n e n t i a l l y over the same range.

In view of the l i m i t e d p r e d i c t i o n c a p a b i l i t y of the unstructured model, a simple s t r u c t u r e d model i s developed i n Chapter 7. When tested w i t h fed-batch and continuous c u l t u r e data the model seems to p r e d i c t everything an

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unstructured model can. A d d i t i o n a l l y , i t provides information about the time dependent behaviour of the i n t e r n a l composition of the c e l l s . Tests i n v o l v i n g experimentally obtained RNA data i n d i c a t e that the model i s not r e a l i s t i c i n t h i s respect and thus has to be r e j e c t e d . Nevertheless, an important conclu­s i o n i s drawn from t h i s e x e r c i s e ; that i s ,the c r i t i c a l t e s t s f o r the v e r i f i ­c a t i o n of s t r u c t u r e d models should include the i n t e r n a l composition data obtained from c a r e f u l l y planned t r a n s i e n t experiments.

In Chapter 8, a number of t o p i c s are d e a l t w i t h i n short u n i t s ( a p p l i c a t i o n s ) . These are a l l a p p l i c a t i o n s of the considerations developed i n the previous chapters. A p p l i c a t i o n 1 and 2 demonstrate the use of macroscopic p r i n c i p l e s i n the study of e n e r g e t i c s . In A p p l i c a t i o n 1, i t i s shown that product formation i n the absence of e x t e r n a l e l e c t r o n acceptors i s l i n k e d to energy generation process. In A p p l i c a t i o n 2, a simple expression r e l a t i n g COD to BOD i s derived i n terms of the thermodynamic e f f i c i e n c y of the growth process. In A p p l i c a t i o n 3, growth during the wash-out phase i s described. Optimal experimentation range and procedure f o r the e s t i m a t i o n of the maximum s p e c i f i c growth rate are e s t a b l i s h e d . A p p l i c a t i o n 4, i s on the s t a t i s t i c a l treatment of batch data. I t i s shown that batch data may have c o r r e l a t e d e r r o r s and that t h i s r e s u l t s i n the overestimation of the p r e c i s i o n of the estimated parameters. F i n a l l y , A p p l i c a t i o n 5 gives evidence on the hold-up of carbon d i o x i d e i n broth during batch c u l t i v a t i o n . The i n f l u e n c e of carbon dioxide hold-up on the energetic c a l c u l a t i o n s and C-balances are assessed q u a n t i t a t i v e l y .

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SAMENVATTING

Met d i t p r o e f s c h r i f t wordt beoogd vanuit de optiek van de ingenieursweten­schappen een o v e r z i c h t te geven op het t e r r e i n van de energetische aspekten van mikrobiële g r o e i . Van de ontwikkelingen t o t op heden worden zowel op ex­perimenteel a l s op t h e o r e t i s c h gebied enige tekortkomingen onderstreept en verbeteringen hiervoor aangedragen. De studie poogt verder een ondersteuning te z i j n voor een biotechnoloog ten behoeve van het v e r t a l e n en s e l e k t e r e n van mikrobiologische gegevens i n een kw a n t i t a t i e v e vorm, d i e bruikbaar i s voor het ontwikkelen en uitvoeren van mikrobiële processen.

In Hoofdstuk 2 worden de theo r e t i s c h e achtergronden en de toepassingen van on-gestruktureerde groeimodellen besproken. Tevens wordt een eenvoudig ongestruk-tureerd model ontwikkeld, dat gebaseerd i s op een k i n e t i e k volgens Monod en de l i n e a i r e groeiwet. Er wordt aangetoond dat i n een batch k u i t u r e het gedrag van het systeem s t e r k wordt bepaald door één k i n e t i s c h e parameter: de maximale s p e c i f i e k e g r o e i s n e l h e i d u m . De energetische parameters b l i j k e n een minimale in v l o e d u i t te oefenen. Het tegendeel b l i j k t waar te z i j n i n een kontinu k u i ­t u r e ; h i e r i n wordt het gedrag j u i s t bepaald door de energetische parameters, te weten de maximale ' y i e l d ' van biomassa op substraat Y g x en de 'maintenance' koëfficient mg. Vervolgens wordt gedemonstreerd dat de keuze van de b e s c h r i j ­v i n g van de k i n e t i e k n i e t van erg groot belang i s . Gegeven een goede parame­t e r s c h a t t i n g , voldoen b i j n a a l l e i n de l i t e r a t u u r vermelde r e l a t i e s aan het experimenteel vastgestelde gedrag. De l i n e a i r e groeiwet toegepast op sub-s t r a a t v e r b r u i k werd getoetst aan gegevens d ie z i j n verkregen i n een kon t i n u k u i t u u r . B i j lage groeisnelheden z i j n s i g n i f i k a n t e a f w i j k i n g e n van deze wet gevonden, d ie n i e t v o l l e d i g konden worden v e r k l a a r d door een v e r l i e s aan l e ­vensvatbaarheid van de c e l l e n .

In Hoofdstuk 3 worden enige technieken aangegeven om de k i n e t i e k en de ener­getische aspekten van de g r o e i van mikro-organismen te bestuderen. V e r v o l ­gens wordt er een s t r a t e g i e ontwikkeld voor een efficiëntere proefopzet, pa­rameterschatting en s t a t i s t i s c h e analyse.

Het i n Hoofdstuk 2 beschreven ongestruktureerde model wordt i n Hoofstuk 4 u i t ­gebreid ten einde de g r o e i i n een 'fed-batch' k u i t u r e te b e s c h r i j v e n . Om d i t model te toetsen worden u i t g e b r e i d meetgegevens en massabalansen gepresen­teerd. Het ongestruktureerde model b l i j k t een adequate b e s c h r i j v i n g te z i j n voor het gedrag t i j d e n s exponentiële g r o e i en gedurende een pseudo s t a t i o ­n a i r e toestand. Het model i s echter n i e t meer g e l d i g t i j d e n s overgangstoe­standen. Tot s l o t wordt i n d i t hoofdstuk aangetoond dat een fed-batch k u i t u r e een zeer bruikbaar hulpmiddel i s voor het t e g e l i j k e r t i j d bestuderen van zowel de k i n e t i e k a l s de energetische aspekten van mikrobiële g r o e i .

De Hoofdstukken 5 en 6 handelen over het e f f e k t van wisselende omgevingskondi-t i e s , te weten het zoutgehalte en de temperatuur, op de parameters van het on­gestruktureerde model. De i n v l o e d van het zoutgehalte wordt i n Hoofdstuk 5 be­schreven. In d i t Hoofdstuk worden de r e s u l t a t e n ook vergeleken met d i e i n de l i t e r a t u u r worden vermeld voor mengkultures ( a k t i e f s l i b ) . H i e r u i t kan de kon-k l u s i e worden getrokken dat mengkultures z i c h v e e l s n e l l e r aanpassen aan ve r ­anderingen i n het zoutgehalte van de k u i t u u r v l o e i s t o f . In Hoofstuk 6 wordt ingegaan op de in v l o e d van de temperatuur a l s mogelijke s t u u r v a r i a b e l e voor

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industriële processen. Er worden i n d i t Hoofdstuk r e s u l t a t e n weergegeven die z i j n verkreken i n een fed-batch k u i t u r e . Een model van het type Arrhenius-v e r g e l i j k i n g word gebruikt om de exyme a c t i v a t i e - d e a k t i v a t i e te beschrijven, die ten grondslag l i g t aan de temperatuurafhankelijkheid van de maximale groeisn e l h e i d b i j superoptimale temperaturen. Van de energetische parameters bleek Y ™ X

X v r l J w e l konstant te b l i j v e n binnen het temperatuurgebied van 25-40°C. Boven de 40°C nam de maximale s p e c i f i e k e g r o e i s n e l h e i d abrupt af. Boven de 50°C trad er z e l f s helemaal geen groei meer op. Over hetzelfde temperatuur-t r a j e c t nam de maintenance koëfficient exponentieel toe.

In Hoofdstuk 7 wordt gepoogd de beperkt voorspellende waarde van het onge-struktureerde model te verbeteren door een gestruktureerd model te ontwikke­len. Een toets van d i t model uitgevoerd met behulp van gegevens u i t fed-batch kultures toont aan dat het gestruktureerde model a l l e s voorspelt wat het on-gestruktureerde model kan v o o r s p e l l e n . Bovendien verschaft het model informa­t i e over het t i j d a f h a n k e l i j k e gedrag van de samenstelling van de celinhoud. Een toets met experimenteel verkregen RNA meetgegevens geeft echter aan dat het model wat d i t b e t r e f t n i e t r e a l i s t i s c h i s en derhalve dient te worden verworpen. Niettemin kan er een b e l a n g r i j k e konklusie worden verbonden aan d i t werk: k r i t i s c h e toetsen voor de experimentele v e r i f i k a t i e van gestruktureerde modellen behoren ook de celsamenstelling i n ogenschouw te nemen die wordt ge­meten b i j zorgvuldig uitgezochte overgangstoestand experimenten.

In Hoofdstuk 8 worden een aantal toepassingen behandeld van de i n de voorgaan­de Hoofdstukken gegeven beschouwingen. De eerste twee toepassingen b e t r e f f e n het gebruik van makroskopische p r i n c i p e s om de energetische aspekten van mi-krobiële groei te onderzoeken. De eerste l a a t z i e n dat produktvorming i n a f ­wezigheid van externe elektron akseptoren gekoppeld i s aan de energie l e v e r ­ende processen. In de tweede toepassing wordt met behulp van het thermodyna-misch rendement van een groeiproces een eenvoudige r e l a t i e a f g e l e i d , die het chemisch zuurstofverbruik COD koppelt aan het b i o l o g i s c h zuurstofverbruik BOD. De derde toepassing b e s c h r i j f t de groei t i j d e n s een zogenaamd'wash-out' expe­riment. H i e r u i t a f g e l e i d worden de grenzen aangegeven van het optimale proef-opzetgebeid en een procedure voor de bepaling van de maximale s p e c i f i e k e g r o e i s n e l h e i d . De vierde toepassing handelt over de s t a t i s t i s c h e verwerking van batch g i s -tingsgegevens. Er wordt aangetoond dat batch gegevens onderling gekorreleerde fouten kunnen hebben, hetgeen tot gevolg kan hebben dat de nauwkeurigheid van de geschatte parameters wordt overschat. Tenslotte wordt i n de l a a t s t e toepassing het ophopen van C O n i n het beslag t i j d e n s een batch kweek gememoreerd. De invloed van deze C O 2 ophoping op de energetische berekeningen en koolstofbalansen wordt op kwantitatieve wijze aangegeven.

I 24

ÔZET

Bu tezde mikrópsal e n e r j e t i g i n bugünkü durumu mühendislik gbrüs açxsindan i n c e l e n m i s t i r . Mevcut deneysel ve t e o r i k uygulamalarxn e k s i k l i k l e r i n e i s a r e t e d i l m i s , g e l i s t i r i l m i s olan y e n i yontemler açxklanmxstxr. Çalxsmanxn amacx biyoteknologa m i k x o b i y o l o j i k v e r i l e r d e n mühendislik tasarxm ve i s l e m l e r i n i n d a y a n d x r x l a b i l e c e g i n i c e l v e r i l e r i n eldesinde yardxmcx o l m a k t i r .

Yapxsxz (unstructuxed) m o d e l l e r i n t e o r i ve uygulamalarx Solum 2 de t a r t x s x l -mxstxr. Monod k i n e t i g i n e ve dogrusal s u b s t r a t tüketimi bagxntxsxna dayanan b a s i t b i r yapxsxz model g e l i s t : L r i l m i s t i r . K e s i k l i üretimde sistem davranxsxnx k i n e t i k b i r paramètre olan maksimum bzgül çogalma hxzxnxn k e s i n o l a r a k sapta-dxgx, e n e r j e t i k parametrelerin i s e az e t k i l e d i g i g o s t e r x l m i s t i r . Sürekli üretimde i s e bu durumun t e r s i geçerli bulumnustur; y a n i sistem davranxsx ener­j e t i k parametreler olan and ms tarafxndan saptanmáktadxr. Aynx zamanda k i n e t i k bagxntx seçiminin f a z l a onemli olmadxgx ve aynx d i k k a t l e parametre-l e r i saptandxgx ta k d i r d e literatürde v e r i l e n herhangi b i r bagxntxnxn deneysel davranxsx t a r i f e d e b i l e c e g i g o s t e r x l m i s t i r . Dogrusal s u b s t r a t tüketimx bagxn-t x s x , sürekli kültur v e r i l e r i i l e sxnanmxstxr. Düsük üreme hxzlarxnda g b z l e - s

nen b e l i r g i n sapmalarxn sadeoe üreme k a p a s i t e s i n d e k i ( v i a b i l i t y ) düsüse b a g l a -namxyacagx gbsterxlmi§tir.

Bolüm 3 de b a k t e r i s e l k i n e t i k ve e n e r j e t i k çalxsmalarxna mühendislik ybntemle-r i n i n n a s x l u y g u l a n a b i l e c e g i g b s t e r i l m i s t i r . Baha v e r i m l i deney yapma, para-metre t a y i n etme ve i s t a t i s t i k çbzumleme ybntemleri g e l i s t i r i l m i s t i r .

Bolüm 4 de, ew e l o e kxorulm'us olan yapxsxz model b e s l e m e l i k e s i k l i kültürdeki davranxsxda kapsamak üzere g e n i s l e t i l m i s t x r . Genis kapsamlx v e r i l e r ve mater-y e l d e n k l i k l e r e sunulmustur. Yapxsxz model, üssel ve yalancx-yatxskxn durum-l a r x y e t e r l i b i r s e k i l d e t a r i f e t m i s t i r . Model geçis doneminde geçerliligini k a y b e t m i s t i r . B e s l e m e l i kültür t e k n i g i n i n b i y o k i n e t i k ve e n e r j e t i k çalxsma-l a r x n b i r arada yürütülebilecegi fa y d a l x b i r yontem oldugu da g o s t e r x l m i s t i r .

Bolüm 5 ve 6 da, seçilmis olan i k i çevresel degisimin yapxsxz model paramet-r e l e r e üzerindeki e t k i s i i n c e l e n m i s t i r . T/uzlulugun e t k i s i Bolüm 5 de a r a s t x -r x l m x s t x r . B u l g u l a r karma kültürler ( a k t i f çamur) için elde e d i l e n l e r i l e k a r s x l a s t x r x l m x s t x r . Karma kültürlerin t u z l u l u k d e g i s i k l i k l e r i n e daha çabuk uyarlama (adaptation) g o s t e r d i g i i z l e n m i s t i r . Mühendislik i s l e m l e r i n d e d i g e r onemli b i r paramètre i s e s x c a k l x k t x r . B e s l e m e l i k e s i k l i kültürde elde e d i l e n sxcaklxk d e n e y l e r i n i n b u l g u l a r x Bolüm 6 da sunulmu§tur. Maksimum b'zgül üreme hxzx i l e sxcaklxk arasxndaki i l i s k i y i t a r i f etmek üzere Arrhenius t i p i b i r enzim aktivasyon-deaktivasyon modeli optimal üzeri sxcaklxk aral^xjpxLxda kapsayacak s e k i l d e g e l i s t i r i l m i s t i r . E n e r j e t i k parametrelerden 1 25-40 C sxcaklxk aralxgxnda yaklasxk olarak degxsmemistir. 40°C n i n üzerxnde bu paramètre birden azalmxstxr. 50°C de üreme saglanamamxstxr. Bakxm katsayxsx (maintenance c o e f f i c i e n t ) i s e aynx sxcaklxk aralxgxnda üssel olarak artmxstxr.

Yapxsxz modelin ondeyi k a p a s i t e s i n i n k x s x t l x bulunmasx karsxsxnda Bolüm 7 ¿Le b a s i t b i r yapxlx model g e l i s t i r i l m i s t i r . Model b e s l e m e l i k e s i k l i kültür ve sürekli kültür v e r i l e r i i l e t e s t e d i l d i g i n d e , yapxsxz model b n d e y i l e r i n i n h e p s i n i v e r e b i l e c e k kapasitede bulunmustur. Buna ek ol a r a k yapxlx model hücre i c i b i l e s i m i n i n zamana baglx dxnamik durumunu t a r i f etmektedir, Ancak deneysel RNA v e r i l e r i i l e yapxlan t e s t , modelin bu bakxmdan geçerle olmadxgxnx ve dogrulanamxyacagxnx g b s t e r m i s t i r . Yinede bu çalxsmadan- b'nemli b i r sonuç

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c x k a r x l a b i l i r ; yapxlx model dogrulamasx l o i n yapxlacak k r i t i k t e s t i e r d i b k a t l e tasarlanmxs dinamik deneylerden elde e d i l e n hücre i c i b i l e g i m i v e r i -l e r i i l e olmalxdxr.

Bölüm 8 de b i r kac konii kxsa u n i t e l e r (uygulamalar) §eklinde t a r t x s x l m x g t x r . Bunlarxn h e p s i e w e l k i 'bölümlerde g e l i g t i r i l m i s , olan düsüncelerin uygulama-l a r i d i r . Uygulama 1 ve 2 makroskopik i l k e l e r i n e n e r j e t i k galxsmalarxndaki faydalarxnx vurgulamaktadxr. Uygulama 1 de hücre dxsx e l e k t r o n a l x c x l a r x n x n yoklugunda ürün yapxmxnxn e n e r j i üretim süreci i l e bagxntxlx oldugu gösteril-mx§tir. Uygulama 2 de COD (kimyasal o k s i j e n g e r e k s i n i m i ) ve BOD (biyokimyasal o k s i j e n g e r e k s i n i m i ) arasxnda b i r bagxntx, üreme sürecinin termodinamik v e r i m i göz önüne alxnarak, türetilmigtir. Uygulama 3 de üreme süreci s i l i n m e (wash-out) durumunda i n c e l e n m i s t i r , Bu dönemde maksimum özgül üreme h i z x n i saptamak i c i n en uygun deneysel a r a l x k ve yöntem geli§tirilmi§tir. Uygulama 4» k e s i k l i kültür v e r i l e r i n i n i s t a t i s t i k i g lenmesi üzerinedir. Burada k e s i k l i kültür v e r i l e r i n d e k i h a t a l a r x n i l i s k i l i ( c o r r e l a t e d ) o l a b i l e -c e g i ve bu. durumun saptanan pa r a m e t r e l e r i n k e s i n l i g i n i ( p r e c i s i o n ) h a t a l x olarak a r t t x r a b i l e c e g i gösterilmigtir. Uygulama 5» k e s i k l i süreclerde karbon d i o k s i t b i r i k m e s i n i n olduguna i§axet etmektedir. Bu durumun e n e r j e t i k hesaplamalarxn ve karbon d e n k l i k l e r i üzerindeki e t k i l e r i n i t e l o l arak i n c e l e n m i s t i r .

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STELLINGEN

1. Product formation during anaerobic growth i n the absence of e x t e r n a l e l e c t r o n acceptors, i s l i n k e d to energy production.

This thesis, Chapter 8, Application 1.

2. The r a t i o of COD to BOD i s not a constant but a simple f u n c t i o n of the thermodynamic e f f i c i e n c y of the growth process.

This thesis, Chapter 8, Application 2.

3. Unstructured models provide good approximation to r e a l i t y during balanced growth or at pseudo-steady s t a t e s . They f a i l during h i g h l y t r a n s i e n t s i t u a t i o n s .

This thesis, Chapter 2.

4. Simple s t r u c t u r e d models w i l l no doubt replace the popular unstructured models i n the near f u t u r e . Much e f f o r t i s needed f o r the development and v e r i f i c a t i o n of these models.

5. The 74 parameter model reported by Schuler et. a l . , d e s c r i b i n g the growth of a s i n g l e b a c t e r i a l c e l l , i s not r e a l i s t i c and seems to be only an e x e r c i s e i n mathematical model b u i l d i n g and computer programming.

M.L. Schuler, S. Leung and C.C. Dick, Ann. NeV York Acad. Sci.,326, 35(1979).

6. A B i o t e c h n o l o g i c a l model can be s a i d to be r e a l i s t i c when i t i s used by i n d u s t r i a l producers f o r process design and c o n t r o l .

7. Much open c r i t i s i s m i s needed i n Biotechnology. The procedures employed by A.I.C.E. , i.e.,opening and p u b l i s h i n g d i s c u s s i o n s f o r every paper they p u b l i s h i n t h e i r j o u r n a l , and that by I.A.W.P.R., i . e . , a s s i g n i n g an o f f i c i a l d i s c u s s e r to every paper presented, would be u s e f u l i n Biotechnology.

8. The number of d i f f e r e n t hypothesis erected to e x p l a i n a given b i o l o g i c a l phenomenon i s i n v e r s e l y p r o p o r t i o n a l to the a v a i l a b l e knowledge.

Erdingston Lao, in Murphy's Law, A. Bloch (Price, Los Angeles) 19 78.

9. The statement made by Gaden, " t h a t , these days some academics are i n v e n t i n g problems that they can solve, rather than t a c k l i n g the new problems or the e x i s t i n g ones" can be generalized to other f i e l d s than Biotechnology and p a r t i c u l a r l y to P o l i t i c s .

E.L. Gaden, Opening Address to the Sixth International Fermentation Symposium, held at London, Canada (1980).

10. Man and the computer belong to d i f f e r e n t genus and therefore the computers should not be made to perform c e r t a i n tasks.

A. A. Esener, 1 st October 1981.