Qualitative Modeling in Computational Systems Biology

Qualitative Modeling in ComputationalSystems Biology

Applied to Vascular Aging

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op dinsdag 18 september 2007 om 16.00 uur

door

Mark Wilhelmus Johannes Maria Musters

geboren te Breda

Dit proefschrift is goedgekeurd door de promotor:

prof.dr.ir. P.P.J. van den Bosch

Copromotor:

dr.ir. N.A.W. van Riel

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Musters, Mark W.J.M.

Qualitative modeling in computational systems biology : applied to vascular aging / by

Mark Wilhelmus Johannes Maria Musters. - Eindhoven : Technische Universiteit

Eindhoven, 2007.

Proefschrift. - ISBN 978-90-386-1564-6

NUR 954

Trefw.: nietlineaire differentiaalvergelijkingen / kunstmatige intelligentie /

regelsystemen ; parameterschatting / fysieke veroudering.

Subject headings: nonlinear differential equations / piecewise linear techniques /

parameter estimation / ageing.

This thesis was prepared by using the LATEX typesetting system.

Cover design by Christoph Brach [email protected], http://www.nutsdesign.net

Printed by Gildeprint drukkerijen, Enschede.

Qualitative Modeling in ComputationalSystems Biology

Applied to Vascular Aging

Samenstelling kerncommissie:

prof. dr. ir. P.P.J. van den Bosch promotor TU/e

dr. ir. N.A.W. van Riel copromotor TU/e

prof. dr. P.A.J. Hilbers lid kerncommissie TU/e

dr. ir. H. de Jong lid kerncommissie INRIA Rhone-Alpes

prof. dr. ir. C. Th. Verrips lid kerncommissie UU

Contents

1 Introduction 1

1.1 Challenges in Systems Biology . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Top-Down versus Bottom-Up . . . . . . . . . . . . . . . . . . . . . 4

1.2 Aging of the Vascular System . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Extending Longevity in Mythology . . . . . . . . . . . . . . . . . . 5

1.2.2 Understanding Aging: the Scientific Approach . . . . . . . . . . . . 5

1.2.3 Changes in Biochemical Networks during Vascular Aging . . . . . . 6

1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Project Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Analysis of Bistable Systems 11

2.1 Basic Knowledge about Feedback Loops, Circuits and Systems . . . . . . . 11

2.2 Nonlinear Dynamics in Biology . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Mathematical Model of ECM Remodeling . . . . . . . . . . . . . . . . . . 14

2.4 Graphical Study of Bistability . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Breaking the Feedback Loop . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Solving the Steady-States Symbolically . . . . . . . . . . . . . . . . 16

2.4.3 Deriving Restrictions on Parameter Values . . . . . . . . . . . . . . 17

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Qualitative Analysis of Nonlinear Biochemical Networks 19

3.1 General Description of the Procedure . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Approximation of Nonlinear Function with PWA Functions . . . . . 20

3.1.2 Selection of PWA Parameters . . . . . . . . . . . . . . . . . . . . . 25

3.1.3 Detection of Equilibrium Points and Performing Stability Analysis . 27

3.1.4 Construction of Qualitative Transition Graphs . . . . . . . . . . . . 29

3.2 Example: an Artificial Biochemical Network . . . . . . . . . . . . . . . . . 29

3.2.1 PWA Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.2 Determination of the Equilibrium Points . . . . . . . . . . . . . . . 31

3.2.3 Dynamical Behavior at the Equilibrium Points and Stability Analysis 33

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CONTENTS

3.3 Transition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Analysis of the Transforming Growth Factor-β1 pathway 41

4.1 Physiology of the TGF-β1 Signaling Pathway . . . . . . . . . . . . . . . . . 41

4.1.1 Isolation of the R-SMAD Loop . . . . . . . . . . . . . . . . . . . . 42

4.2 Qualitative Analysis of the Transforming Growth Factor-β1 Pathway . . . 44

4.2.1 Model Reduction of the TGF-β1 Pathway . . . . . . . . . . . . . . 44

4.2.2 Quasi-Steady-State Approximation of the TGF-β1 Model . . . . . . 48

4.2.3 From Nonlinear to Piecewise-Affine . . . . . . . . . . . . . . . . . . 51

4.2.4 Equilibria and Stability Analysis . . . . . . . . . . . . . . . . . . . 52

4.2.5 Transition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Signal Transduction of the Unfolded Protein Response 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Protein Folding of the von Willebrand Factor . . . . . . . . . . . . . . . . 65

5.2.1 Translation and Translocation . . . . . . . . . . . . . . . . . . . . . 66

5.2.2 Protein Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 The Unfolded Protein Response . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3.1 Signal Transduction in the UPR . . . . . . . . . . . . . . . . . . . . 67

5.4 Mathematical Model of Signal Transduction during the UPR . . . . . . . . 69

5.5 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5.1 From Nonlinear to Piecewise-Affine . . . . . . . . . . . . . . . . . . 72

5.5.2 Equilibrium Points in the UPR Model . . . . . . . . . . . . . . . . 74


5.5.4 Comparison with Experimental Data . . . . . . . . . . . . . . . . . 80

5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 System Identification with Parameter Constraints 87

6.1 The Biochemical Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2 Qualitative Phase Space Analysis . . . . . . . . . . . . . . . . . . . . . . . 90

6.2.1 Nonlinear to PWA Conversion . . . . . . . . . . . . . . . . . . . . . 90


6.2.3 Constrained Nonlinear Parameter Estimation . . . . . . . . . . . . 92

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7 Hybrid System Identification 97

7.1 General Identification Procedure . . . . . . . . . . . . . . . . . . . . . . . . 97

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Contents

7.1.1 Model Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.1.2 Identification and Classification of a Hybrid Model . . . . . . . . . 98

7.2 PWA Identification of the Biochemical Oscillator . . . . . . . . . . . . . . 99

7.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8 Conclusion and Discussion 103

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8.2 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A Nomenclature 107

A.1 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

A.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.2.1 Latin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.2.2 Greek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Summary 127

Samenvatting 129

Dankwoord 131

About the Author 133

vii

Contents

viii

1Introduction

KNOWLEDGE of health and the need of solutions for addressing diseases have

always fascinated humanity. Understanding the biochemical processes within

cells,“the building blocks of life”, has become indispensable. During the 20th century,

our understanding of cellular biology has increased at an astonishing rate. Over the last

decades, it can mainly be attributed to breakthroughs in the research field of molecular

biology. This research field deals with the use of techniques from various research areas

on solving biological problems. For example, molecular biology provided the necessary

high throughput methods for unraveling the complete human genome in 2001 [96, 173].

This was an important step towards a better comprehension of the “blueprint of life” at

that time, but it has recently become clear that dynamical information provides more

insights about human physiology and pathological phenomena. A human can be viewed

as a system with 1014 cells [72], each containing approximately 25, 000 genes [96, 173]

and intertwined signaling networks operating over distinct spatio-temporal scales [132].

These data emphasize that comprehension of life’s complexity is impossible by intuitive

reasoning alone; computational approaches that integrate the available information into

a single framework have therefore become indispensable [15, 91].

A mathematical model is a description of a system in terms of mathematical equa-

tions [25]. A plethora of mathematical formalisms exists to describe physiological pro-

cesses: discrete, continuous, deterministic, stochastic and combinations of these model-

ing frameworks. Some of these frameworks are suitable for specific situations. For in-

stance, stochastic models are used for processes that are in essence dominated by random

events [156], e.g. binding of a substrate to a receptor; it can result in different outcomes

for the same initial conditions. Since the deterministic approximation of the biochemical

reaction systems becomes more difficult for reactions with a few number of molecules,

stochastic models are therefore more appropriate for describing these biochemical reac-

tions. They provide a thorough description, based on several fundamental properties

from physics. Unfortunately, computational complexity increases exponentially for sys-

1

Chapter 1

tems with more substrates. For a detailed overview of the wide variety of computational

methods, the reader is referred to a review of de Jong [26].

At a cellular level, mathematical models are frequently deterministic descriptions as

they provide a good balance between computational effort and accuracy. Tools from

system identification enable building mathematical models of a dynamic system based on

measured data [170]. The parameters of a given model are subsequently adjusted until the

predicted dynamics coincide as good as possible with the measured signals. To acquire

mathematical models of biochemical networks and to obtain system-level understanding of

the biochemical interactions of these networks have gained much popularity in the research

community over the last few decades [157] and has been called systems biology [85]. Large-

scale and mechanism-based models have gradually earned a central role in this research

field. It has resulted in a variety of models, ranging from signal transduction [31, 68, 147]

to genetic networks [29, 67, 70].

1.1 Challenges in Systems Biology

In their quest for the identification and validation of drug targets and biomarkers, phar-

maceutical companies like AstraZeneca, Bayer AG, Eli Lilly, Merck, Novartis, Organon,

Pfizer and Roche have become interested in the use of the in silico1 concept. However,

there are several challenges that system biologists have to face when constructing com-

puter models of biochemical systems from both the biological and technical perspective:

Experimental Issues

1. The amount of quantitative information is limited in biology. In Fig. 1.1 typi-

cal biological data is shown for which the intensity of each band represent the

qualitative level of a given species in arbitrary units during a certain amount of

time. These data have limited quantitative significance, since they do not represent

physiologically-relevant quantities. Even if these data could be linked to substrate

concentrations, the intensities are extremely sensitive to external influences during

the experiments. Another example is the large pool of qualitative -omics data2

in biology, valuable information which is usually omitted in traditional parameter

estimation. Although by improving measuring techniques much progress has been

made in obtaining data on a cellular level, accurate quantitative information is

only available for a few well-characterized systems [159].

1in silico refers to “performed on computer or via computer simulations”.2-omics data represent data obtained by transcriptomics, proteomics and metabolomics. These are

methods to derive information from messenger-RNA (mRNA), proteins and metabolic analysis profiles,respectively, in a massive parallel way (i.e., currently dozens of metabolites/proteins and thousands of

2

Section 1.1

Figure 1.1 – Typical nuclear extracts (a. and b.) obtained from electrophoretic mobilityshift array (EMSA) of the IκB-NF-κB signaling module [70].

2. The large variations in quantitative measurements due to differences between spe-

cies, individuals and experimental procedures [117, 118, 119]. Experimental data

are also polluted with noise which yields extremely large standard deviations,

thus relatively low reliability. Furthermore, generating experimental data is labor-

intensive and expensive with complicated experimental protocols, which limits the

number and reproducibility of data points considerably.

Technical Challenges

1. Interactions between components are inherently nonlinear due to the chemical law

of mass action [37], which hampers system analysis considerably.

2. Nonlinear system identification requires initial estimates for the unknown param-

eters which has to be relatively close to the “true values”3 to avoid converging to

a wrong solution. During parameter estimation, the parameters of a given model

are varied and the dynamics of the model are simulated. The simulation results

are compared with experimental data to see whether the parameters result in the

most accurate approximation of the experimental for the given model structure.

If not, the procedure is repeated for a different set of parameter values, until the

error between simulated results and experimental results have been minimized to

the lowest error, i.e., the global minimum. However, parameter estimation can

sometimes converge to a suboptimal solution (local minimum) which should be

avoided.

How do Researchers Deal with these Challenges?

Several strategies have been developed to tackle these issues, each with their advantages

and disadvantages:

mRNA levels).3“true value” is a parameter that would produce the best simulated model fit, compared to the

experimental data.

3

Chapter 1

• Mathematical models are frequently implemented with arbitrary parameter values

to obtain a certain realization of the system behavior [82, 97, 135, 137]. Practi-

cal relevance of these models is of marginal importance since a different choice in

parameter values can result in completely different system dynamics [162].

• Mathematical models have been constructed for various biochemical networks [22,

23, 51, 70, 82, 143, 167]. These are, in general, large and complex nonlinear struc-

tures and have been fitted on insufficient data points which produces models with

low predictive power [176].

• A principal part of experimental data is obtained from bacteria and yeast which

are less complex structures than mammalian cells and hence, extrapolating these

data to mammals, is therefore not always allowed. Even if parameters are obtained

from mammalian cells, parameters can vary several orders of magnitude depending

on the experimental procedures [119].

• A specific class of biochemical networks can be studied which shows some mathemat-

ically advantageous properties for which mathematical analysis can be performed.

For example, genetic regulatory networks contain switch-like behavior. Under these

conditions, Boolean abstractions are a suitable modeling class [161, 164]. However,

most biochemical networks do not contain these hard switches. The large disconti-

nuities in discrete formulations introduce system behavior which is not observed in

smoother approximations.

1.1.1 Top-Down versus Bottom-Up

Systems biology makes a distinction between two modeling approaches: top-down and

bottom-up.4 In a top-down modeling approach, the organism is analyzed as a whole and

broken down into smaller, computable entities [66]. This makes modeling and simulation

of multicellular systems feasible, but the sparsity in data forms a difficulty (Fig. 1.2).

Bottom-up modeling is based on reductionism and reconstructs pathways of basic units

like proteins and genes. The post-genomic era has contributed much to the amount of

qualitative information, but the numerous pathways and their complexity in a (single)

cell hamper the translation of results to a physiological multicellular level. The challenge

in human systems biology is therefore to combine information from -omics experiments

(bottom-up) with qualitative information of the physiology (top-down).

We may conclude that computer modeling of biochemical networks is not a trivial task.

Meanwhile it has become more and more important in the post-genomic era, dominated

by qualitative information. This thesis shows that qualitative information can be utilized

to describe and analyze biochemical networks. Biochemical reactions involved in vascular

4Different definitions of bottom-up and top-down appear in literature, see [170] for an overview.

4

Section 1.2

Organism

Tissues

Cells

Proteins

Genes

phenotypes

TOP-DOWN

BOTTOM-UP

-omics profiling

available data

Figure 1.2 – Graphical representation of the top-down and bottom-up approach withrespect to the level of biological organization [93].

aging, strongly related to cardiovascular diseases, will be studied to verify our developed

procedures.

1.2 Aging of the Vascular System

1.2.1 Extending Longevity in Mythology

Mythical stories and history show a profound interest in extending longevity by means of

food or drinking. Ancient Greek mythology is interlarded with tales about divine food,

called ambrosia, that conferred immortality on whoever it consumes. In Asia, the Chinese

emperor Qin Shi Huang sent his alchemist Xu Fu to find the elixir of life, but he discovered

Japan instead of completing his mission. On the American continent, the Incas believed

that the creator god Viracoca rose up from the sacred lake Titicaca in Peru and created

the world. One of its islands, Isla del Sol, contains three separate springs which is believed

to represent a fountain of eternal youth. In the same region, the Spanish conquistador

Juan Ponce de Leon (Fig. 1.3) started his journey in 1512 to find the Fountain of Youth in

the New World. Celtic mythology tells about the quest of the Knight of the Round Table

to search for the Holy Grail, which was believed to give an immortal life to the finder.

There are indications that overproducing certain enzymes involved in nutrient withdrawal

assists in prolonging life in yeast [77] and worms [165] significantly. Besides increasing

life span, quality of life is an additional aspect that should be considered. It is therefore

an intriguing research question whether healthy aging could be pursued for mammals as

well by means of nutrients.

1.2.2 Understanding Aging: the Scientific Approach

Aging is the accumulation of changes responsible for the sequential alterations that ac-

company advancing age and the associated increases in the chance of disease and death

[6]. Although this process concerns the whole senior world population, little is known

5

Chapter 1

Figure 1.3 – Juan Ponce de Leon during his quest [41].

about the effects of aging on the cellular level and how aging cells lead to aging of the or-

ganism. In Western society, the main causes of death are cardiovascular diseases that are

related to aging processes, e.g. atherosclerosis and heart failure. For a more healthy aging,

it is therefore of great importance to understand the highly complex processes involved

in vascular aging. It is a fact that more than 300 theories circulate about aging [108].

General agreement exists over the crucial role of oxidative stress in aging [65], also known

as reactive oxygen species (ROS). Unfortunately, the primary mechanism that underlies

stress-induced aging processes has not been revealed yet. Over the last few decades, sev-

eral aging theories have been proposed in the literature which include telomere shortening

[64, 128, 129], hormones [32], DNA damage/repair [57, 100], caloric restriction [104], mito-

chondria [112] and protein quality control [9, 73, 134, 180]. Identifying and disentangling

the crucial biochemical networks in vascular aging is therefore one of the challenges in the

research field of aging.

1.2.3 Changes in Biochemical Networks during Vascular Aging

Alterations in cellular processes during aging have been considered as a possible indicator

for pathological phenomena [93]. Based on experimental evidence, three aging-related

biochemical networks were selected

Extracellular Matrix

The extracellular matrix (ECM) is a complex network of biomolecules like polysaccharides

and proteins secreted by the cell and serves as a structural element in tissues. The link

between vascular aging and a change in the ECM structure has been reported in litera-

6

Section 1.3

ture [75], but has also experimentally been demonstrated by disturbed levels of mRNA

encoding for proteases, which are enzymes that degrade the ECM mesh. Proteases are

assumed to play a role in cell-controlled composition changes of the ECM (“remodel-

ing”) [97], as will be shown in chapter 2.

Transforming Growth Factor-β1

From experimental data [17], the Transforming Growth Factor-β1 (TGF-β1) pathway is

most likely involved in vascular aging. TGF-β1 is a cytokine that binds to receptors on the

endothelial cells. Many cellular processes have been imputed to this substrate [30], e.g.

the cell cycle [182], vasodilatation [144] and ECM formation [123, 160]. More physiological

details of this network are given in chapter 4.

Unfolded Protein Response

Proteins that are secreted outside the cells need to be folded in the endoplasmic reticulum

(ER) for proper functioning. The unfolded protein response (UPR) is the quality control

mechanism that monitors the folding procedures. However, during aging the UPR seems

to fail in performing its task which leads to an accumulation of misfolded proteins [168]

that can form the basis of various diseases [148]. A general description of the UPR in

mammalian cells is provided in chapter 5.

1.3 Problem Statement

1.3.1 Project Description

This thesis is a result of a project in which Unilever Research Vlaardingen, Aurion, Uni-

versity of Utrecht and Eindhoven University of Technology have cooperated. The aim of

the project was to increase our knowledge of vascular aging by applying genomics, pro-

teomics and metabolomics, which involve data from the genome, proteins and metabolic

products, respectively. The contribution of Eindhoven University of Technology within

this project was to integrate the experimental data in computer models and evaluate the

outcome, providing valuable insights in the development and screening of new compo-

nents for functional foods by means of systems biology. Eventually this would lead to a

selection of natural ingredients that could slow down vascular aging processes, which con-

tributes to healthy aging. The research has financially been supported by the Netherlands

Organization for Scientific Research, grant R 61-594, and by Senter grant TSGE1028.

7

Chapter 1

1.3.2 Research Goals

As mentioned in section 1.1, analysis of biological aging processes with computer models

requires alternative approaches due to the limited amount of quantitative information.

This leads to the main research question of this thesis:

Primary Goal − Develop mathematical procedures to extract information from

typical nonlinear biochemical models that contain little quantitative information.

Main purpose: assistance in (qualitative) system analysis and improved parameter

estimation.

The problem has been reformulated even more strictly: if we assume that initially no

quantitative information is available, how much valuable information can be extracted by

means of (qualitative) analysis of the system dynamics given the topology of the network

and the type of kinetic interactions? And could qualitative analysis be used to analyze

the global dynamics of the system? In this thesis, two symbolic strategies have been

advocated for the analysis of nonlinear biochemical networks:

1. Graphical analysis of monotone systems that exhibit bistability, i.e., dynamics with

two stable steady-states or equilibrium points. This method was initially developed

for the determination of bistable behavior [3], but could be adapted to constrain

specific parameters. This method is the subject of chapter 2.

2. General qualitative piecewise-affine (PWA) analysis. PWA systems are a type

of hybrid systems, i.e., a mathematical modeling paradigm that combines both

discrete and continuous features into one unifying framework. We used this PWA

procedure to approximate a biological process by a number of simpler functions,

which results in a more refined modeling framework for qualitative analysis [141].

This method is applicable to a wide range of biochemical networks and will be

introduced in chapter 3.

These two strategies contribute to a reduction in parameter search space and assist in

system identification.

As the overall project was aimed at a better understanding of vascular aging processes,

the system analysis tools were applied to various biochemical processes involved in vascular

aging. Therefore another goal has been defined:

Secondary Goal − Apply the developed methodologies to typical biochemical

networks that are involved in vascular aging processes.

The graphical analysis was applied to a model of ECM remodeling (chapter 2). The

qualitative PWA analysis will be applied to analyze the TGF-β1 pathway (chapter 4)

8

Section 1.4

and UPR module (chapter 5). Furthermore, this methodology also facilitated parameter

estimation of a putative biochemical oscillator in chapter 6.

1.4 Related Work

The project encompasses several research areas. An overview of work related to the topics

in this thesis will be given below.

Modeling Aging Processes

Studying vascular aging with computer models is a recent development. Top-down ap-

proaches are limited to scaling laws [158, 177, 178, 179] and telomere shortening [128,

129, 174], whereas the bottom-up method is more popular in biochemical networks, like

cytosolic protein folding [135], and the “network theory of aging” [84]. The UPR module

of Saccharomyces Cerevisiae has been modeled by [106], but the mammalian UPR is

more complex [103]. Similar to the UPR in the ER is the heat shock response in the

cytosol of the cell, for which a stochastic model was created [135]. Most parameter values

in this model were chosen arbitrarily and, consequently, the physiological significance is

low. The TGF-β1 pathway is relatively new. Recently a mathematical model of this

pathway was proposed with parameters varying over several ranges [24]. However, the

inhibitory influence of specific substrates [30] was not included in this model. Therefore

mathematical models of the TGF-β1 pathway and UPR response were constructed from

scratch by performing an extensive literature search and translating the obtained infor-

mation in differential equations [117]. To our knowledge, such models have not yet been

developed.

Nonlinear System Identification and Parameter Estimation

System identification of nonlinear systems requires an initial estimate of the parameter

values [102] and these are, as a consequence, often arbitrarily chosen. The parameters

of the model are tuned by Levenberg-Marquardt, Gauss-Newton or multiple shooting

method [95, 116] algorithms by minimizing the cost function J(θ) of the error between

the true parameter set θ and the model fit θ. The solution with the lowest cost function

J(θ), the global minimum, results in the best model fit. Due to poor quality of the data

and erratic initial estimates, the solution can also converge to a local minimum. Another

common way to study the effect of parameters on the output of a model is to apply

sensitivity analysis. The parameters are varied across a range of parameter values to check

the effect of each individual parameter on the system. However, this method demands

a proper estimate of the parameter values as well. Defining upper and lower bounds on

9

Chapter 1

the parameter values could provide a good solution, but becomes computationally too

complex for large variations in the parameter values.

Hybrid Systems in Biology

Although piecewise-linear functions have been used in biology for decades [50], the appli-

cation of hybrid systems in biology is currently reviving this re. Applications are limited

to specific processes with clear switching characteristics [2, 12, 14, 20, 29, 35, 48, 99, 181]

and therefore primarily aimed at genetic regulatory networks. Here we show that hy-

brid systems are not limited to these mathematical functions, but can also be applied to

systems that display “soft switching” such as Michaelis-Menten kinetics. Hybrid system

identification of biochemical networks is in its infancy and has primarily been focussed on

genetic regulatory networks [35] or requires initial estimates of the parameter values [171].

Hybrid system identification can be performed in different ways, see [76] for an overview.

In chapter 6, a hybrid system identification procedure is presented which makes use of a

linear least-squares methodology and therefore no initial estimate of the parameter values

is required.

1.5 Thesis Outline

This thesis develops and significantly extends existing methods to cope with the limita-

tions in biological modeling due to scarcity of biological data. In chapter 2 the class to

model nonlinear biochemical networks is defined and graphical analysis is applied to a

nonlinear model of the ECM. A mathematical model of the formation and degradation

of ECM is derived from the literature [97]. An improved version of the graphical analysis

procedure is proposed and used to derive the so-called Michaelis constants of the ECM

model. Although graphical analysis yields constraints for specific parameters, the class

of systems is limited to monotonic networks that satisfy certain criteria [3]. Chapter 3

introduces a PWA method that is more general applicable. In chapter 4 the TGF-β1

pathway is tested with this procedure and the predicted qualitative behavior agrees with

the experimental data. A model of the UPR, a bit larger in complexity, is developed and

analyzed similarly in chapter 5. In chapter 6, a biochemical oscillator is introduced [16, 54]

and analyzed with the qualitative procedure from chapter 3. It yields a set of constraints

on the parameter values, which have been incorporated in nonlinear system identification.

This method results in better estimates of the parameter values compared to traditional

unconstrained system identification. In chapter 7, a hybrid system identification frame-

work is developed that can provide accurate initial estimates of the parameter values.

This thesis concludes with the main conclusions and future perspectives.

10

2Analysis of Bistable Systems

Based upon experiments, it has been suggested that the Transforming Growth Factor-β1

(TGF-β1) pathway plays a significant role in the pathological phenomena of vascular ag-

ing [75]. One of the processes controlled by TGF-β1 is the “remodeling” of the extracellular

matrix (ECM), an extensive network of biomolecules like glycoproteins and proteoglycans.

The ECM is a layer that surrounds cells and is essential for inter- and intracellular com-

munication, serves as a protective layer and contributes to cell-cell adhesion. Changes

in TGF-β1 expression imply disturbances in the ECM structure as well. Disturbances in

ECM remodeling have directly been related to cardiovascular diseases [75].

2.1 Basic Knowledge about Feedback Loops, Circuits

and Systems

Before a model of the ECM is introduced, some basics on systems theory are required for

understanding the structure of biochemical networks. In general, biochemical networks

contain one or multiple feedback loops, i.e., some proportion of an output signal of a

system is passed (fed back) to the input. This is a common way to control the dynamical

behavior. A single feedback loop consists of a single element and has an enhancing or

inhibiting effect on the input signal. Regulatory elements A, B and C form a feedback

circuit if the level of A exerts an influence on the rate of production of B, whose level

influences the rate of production of C, whose level in turn influences the rate of production

of A [162]. Two classes of feedback circuits exist. Either each element in the circuit exerts

a positive action on its own future evolution (activation), or the sum of all elements in

the circuit exerts a negative action on this evolution (repression); a circuit is positive

or negative if the parity of the number of negative elements in the circuit is even or

odd, respectively [162]. Negative feedback circuits are common in biology. This circuit

class is mainly responsible for stability of an attractor, i.e., a set in the phase space

11

Chapter 2

which is asymptotically approached in the course of dynamics. Positive feedback circuits

have been observed in biological switches [8, 40], the cell cycle [153], and “memory”

effects [101]. Adding several negative and/or positive feedback circuits together results in

negative- or positive-feedback systems, depending on the number of negative and positive

feedback circuits [164]. The graphical study in this chapter is tailored for positive-feedback

systems. These systems have in common that they have multiple steady-states, also called

multistability. Depending on the history of the system and a given set of initial conditions,

the system can converge to a different attractor. A specific subset of multistability is

bistability, in which two stable nodes are present.

2.2 Nonlinear Dynamics in Biology

Biochemical networks are comprised of several biochemical processes, in which special

proteins, called enzymes, enhance the rate of these reactions [46]. For a simple irreversible

conversion of a substrate S into a product P, facilitated by an enzyme E, the chemical

reaction is given by

E + Sk1

GGGGGBFGGGGG

k−1

ESk2

GGGA E + P, (2.1)

ES enzyme-substrate complex,

k1 association rate constant of E with S,

k−1 dissociation rate constant,

k2 conversion rate constant of ES into P.

Making the assumptions that all parameters are larger than zero, ES quickly reaches a

constant value and that the total amount of enzyme ET in the system is fixed, leads to a

single expression for the enzymatic conversion rate f(x):

f(x) =k2ETx

k−1+k2

k1+ x

=Vmaxx

Km + x, (2.2)

x substrate concentration,

Vmax maximal rate of conversion,

Km Michaelis constant.

This equation is also known as the Michaelis-Menten equation [111], its graph is shown

in Fig. 2.1a. Sometimes binding of a substrate molecule to a single enzyme can increase

its affinity to other substrate molecules. This positive cooperativity has been described by

the Hill equation

f(x) =Vmaxx

r

Krm + xr

, (2.3)

12

Section 2.3

00

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

r = 2r = 5

r = 10r = 100

Kmx x

Km

Vmax

f x( ) f x( )

Vmax

Vmax

2

(a) (b)

Figure 2.1 – (a) Michaelis-Menten curve for Vmax = 1 and Km = 0.1 and (b) Hill kineticsfor various cooperativity coefficients (r = 2, 5, 10 or 100), Vmax = 1 and Km = 0.5.

r cooperativity coefficient.

Note that if r = 1, one obtains the standard Michaelis-Menten equation, and for r À 1,

the system resembles a biological switch, see also Fig. 2.1(b). Michaelis-Menten and Hill

kinetics are typical monotonically increasing functions. This property is a necessity for

the graphical analysis procedure, which will be applied to a mathematical model of ECM

remodeling that is composed of ordinary differential equations (ODEs) [122, 124]. This

type of mathematical models has been widely used to simulate concentrations of substrates

by time-dependent variables in order to describe the dynamics of various biochemical

networks [26]. They have the following mathematical form

dxi

dt=

∑j

fj(x), i = 1, . . . , Nx, j = 1, . . . , Nf , (2.4)

x [x1, . . . , xNx ]T ≥ 0, vector with state variables,

fj(x) ≥ 0, rate equation,

Nx total number of states,

Nf total number of rate equations,

t time.

It falls outside the scope of this thesis to provide an extensive overview of modeling dy-

namical systems with ODEs. More information about this topic can be found in standard

engineering textbooks, e.g. [25].

13

Chapter 2

2.3 Mathematical Model of ECM Remodeling

It was hypothesized that there exists a delicate balance between ECM construction and

degradation, which can be simplified to a two- or three-state model, which are defined as

the closed loop and open loop model, respectively [97]. In this chapter, we will focus on

the dimensionless closed loop model (Fig. 2.2) that is composed of so-called bisubstrate

Michaelis-Menten (see chapter 3 for more details) and Hill equations:

dx1

dt=

(c0 − x1)x2

1 + c0 − x1

− k1x1

Km1 + x1

, (2.5)

dx2

dt=

k2xr1

Krm2 + xr

1

− k3x22

Km3 + x2

, (2.6)

x1 concentration of proteolysis fragments,

x2 concentrations protease,

c0 maximal initial ECM concentration,

k1 maximal rate of ECM formation,

k2 maximal rate of protease production,

k3 maximal rate of proteolysis activity,

Km1 Michaelis constant of ECM formation,

Km2 Michaelis constant of protease production,

Km3 Michaelis constant of proteolysis activity.

Numerical exploration of the phase space by varying the parameter values in Eqs. 2.5 and

2.6 has shown that the closed model can generate mono- or bistable behavior, depending

on the parameter values. The closed mode contains indeed a positive feedback circuit

(Fig. 2.2), namely the stimulation of protease production which results in a bistable

system that depends on the choice of parameter values, see Fig. 2.3.

2.4 Graphical Study of Bistability

A method to study system dynamics is by numerically exploring all parameter combina-

tions to check whether bistability occurs. Additionally, a graphical procedure has been

developed to verify whether a given system, composed of monotone functions, exhibits

bistable behavior [3]. A more in-depth view of the mathematical aspects of the graphical

analysis of monotone functions can be found in [4]. The graphical analysis consists of

several steps which are listed below.

14

Section 2.4

Protease

Extracellularmatrix

Proteolysisfragments

c0 - x1 x1

x2

η

ω

Substrate

Degradationproducts

Reaction

Enzymaticstimulation

k1

k2

k3

Figure 2.2 – Closed model of ECM formation and degradation [97]. The scissors indicatethe place where the feedback circuit is cut for analysis purposes. Input of the system isgiven by ω, output of the system is η, the strength of feedback is ν = ω

η . Graphical notationis in agreement with the proposed style of Kitano [86].

0 0.2 0.4 0.6 0.8 1 1.20

0.02

0.04

0.06

0.08

0.1

stable node

stable node

saddle

dx dt1/ = 0

dx dt2/ = 0

0 0.2 0.4 0.6 0.8 1 1.20

0.02

0.04

0.06

0.08

0.1dx dt2/ = 0

dx dt1/ = 0

stable node

(a)

x1

x2 x2

x1

k K1 m2= 0.05, = 0.1

(b)

k K1 m2= 0.06, = 0.1

Figure 2.3 – Distinct dynamics for different sets of parameter values in the ECM model ofLarreta-Garde and Berry [97]: (a) Bistable dynamics if k1 = 0.05, but (b) only one stablenode if k1 = 0.06. The other parameter values are c0 = 0.1, k2 = k3 = 0.1, Km1 = Km2 =0.1, and Km3 = 1.

15

Chapter 2

0 0.2 0.4 0.6 0.8 1 1.20

0.02

0.04

0.06

0.08

0.1

x2

x1

k K1 m2= 0.05, = 0.1081

saddle

stable nodes

0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1

0.12

ω

η

ν = 0.925

ν = 1

dx dt2/ = 0

dx dt1/ = 0

(a) (b)

Figure 2.4 – (a) Graphical representation of g(ω). The dashed and dotted line indicateν = 1 and ν = 0.925, respectively. (b) Bistable dynamics for the limit value of Km2.

2.4.1 Breaking the Feedback Loop

First the feedback loop of the model is cut open, see Fig. 2.2. The ECM model is

transformed into an open loop system with input ω and output η. For mathematical

reasons [4], this open loop system has to satisfy two critical properties to allow graphical

analysis:

1. It has a monostable steady-state response to constant inputs (a well-defined steady-

state input/output characteristic),

2. No possible negative feedback circuits are present in the system.1

Rewriting Eqs. 2.5 and 2.6 in terms of input ω yields

dx1

dt=

(c0 − x1)x2

1 + c0 − x1

− k1x1

Km1 + x1

, (2.7)

dx2

dt=

k2ωr

Krm2 + ωr

− k3x22

Km3 + x2

. (2.8)

2.4.2 Solving the Steady-States Symbolically

Next step in the procedure is to derive a mathematical expression x1 = η = g(ω) in steady-

state. In Eqs. 2.7 and 2.8, this is done by solving dx1

dt= dx2

dt= 0 and subsequently deriving

η = g(ω). This leads to a complicated symbolical function g(ω), the corresponding graph

is calculated for the parameter set in the caption of Fig. 2.3. The results are shown in

Fig. 2.4(a).

1Especially this property is a rather strong claim in practice, as most biochemical networks do containnegative feedback circuits. This will be one of the discussion points in section 2.5.

16

Section 2.5

In the previous subsection the closed loop system was converted into an open loop.

Recovering the closed loop system from the open loop description can be performed by

putting ω = η. However, one could also study the effect of a feedback law ω = ν×η, with

ν: the influence of output η on input ω. In Fig. 2.2, ν can be coupled to the contribution of

proteolysis fragments on the formation of protease. Possible bistability within this system

can be derived graphically: if the line η = ων

has three intersections with g(ω), bistability

is guaranteed [3]. Note that the three intersections in this model structure indicate the

position of two stable nodes and one saddle. Fig. 2.4(a) shows that bistability is present

for unitary feedback (ν = 1). For ν < 1, the slope of the line becomes steeper; for

ν < 0.925 only one intersection point can be found, which corresponds to a single stable

node. Consequently ν ≥ 0.925 is a necessary constraint for this specific model to ensure

bistability, for parameter values as stated in the caption of Fig. 2.3.

2.4.3 Deriving Restrictions on Parameter Values

Bistability and related limit cycles are imputed properties of the ECM for the normal

homeostatic situation [97]. Graphical analysis in the previous subsection has shown that

for ν < 0.925 no bistable behavior is observed. The graphical method of [3] is therefore

improved by using this information to put bounds on the parameter value Km2. Rewriting

the nonlinear function with input ω in Eq. 2.8 in terms of η and ν yields

k2ωr

Krm2 + ωr

=k2(νη)r

Krm2 + (νη)r

=k2η

r

(Km2

ν

)r+ ηr

. (2.9)

Hence, after closing the open loop model, the feedback law only influences the value of

Km2. The graph in Fig. 2.4(a) shows that bistability is present for 0.925 < ν < ∞. This

implies that a bistable system can be found for 0 < Km2 ≤ 0.10.925

= 0.1081. Numerical

validation, displayed in Fig. 2.4(b), confirms these results for k1 = 0.05, c0 = 0.1, k2 =

k3 = 0.1, Km1 = 0.1, Km2 = 0.1081, and Km3 = 1.

2.5 Discussion

The graphical method is designed for nonlinear monotone functions that frequently arise

in biochemical networks and can be applied to define limits on the Michaelis constant of

the feedback loop. It is a graphical alternative for standard methods to explore each equi-

librium point numerically [3, 4]. The novelty in this chapter is the extension of an existing

graphical procedure to derive bounds on the parameter values to guarantee bistability.

By means of trial-and-error, the location to open the feedback system turns out to be es-

sential for applying the graphical procedure successfully. In the ECM model, for instance,

17

Chapter 2

only one cut can be analyzed properly, other open loop models were insoluble due to

the limitations in solving mathematical equations symbolically. Despite this restriction,

a five-state model of the mitogen-activated protein kinase (MAPK) cascade was analyzed

in the paper of Angeli and Sontag [3] to illustrate the basics of their graphical method.

However, the negative feedback circuit within the MAPK pathway [82] was replaced by a

positive feedback circuit in order to satisfy the properties listed in subsection 2.4.1. Bio-

chemical networks without negative feedback circuits are more the exception to the rule,

because negative feedback usually induces stability. This underlines the specificity of the

graphical procedure. Another drawback of the graphical method is that one still requires

numerical values for most parameters, like the majority of the classical mathematical

models. Therefore, a more general approach for the analysis of biochemical networks,

that relies only on qualitative information, is desired. This will be the subject of the next

chapter.

18

3Qualitative Analysis of Nonlinear Biochemical

NetworksAnalysis of nonlinear biochemical networks with little quantitative information is not

restricted to systems with positive feedback circuits as shown in chapter 2. Various

methods have been proposed to analyze biochemical networks in general, but the majority

requires quantitative information. In general, qualitative modeling has been common in

the field of artificial intelligence for several decades [94, 126, 141, 150, 163, 164], but

has been limited to second order systems. Expansion of this theory to larger, multi-

dimensional, biological models has been complicated. Therefore applications on biological

examples forced the qualitative research community towards special model classes with

beneficial features [12, 50, 28, 109, 130]. Excluding genetic regulatory networks, most

biochemical networks do not fulfill these strict requirements. Here we elaborate on the

qualitative modeling approach to analyze nonlinear biochemical regulatory networks, by

extending it to a significantly larger class of systems. This novel procedure contributes

to our understanding of nonlinear biochemical networks by means of qualitative system

analysis (chapters 4 and 5) and parameter estimation (chapter 6).

3.1 General Description of the Procedure

The deterministic modeling approach with ordinary differential equations (ODEs), as

introduced in the previous chapter, was used as basic principle of simulating biochemical

networks:dxi

dt=

∑j

fj(x), i = 1 . . . Nx, j = 1 . . . Nf , (3.1)

x [x1, . . . , xNx ]T ≥ 0, vector with state variables,

fj(x) ≥ 0, rate equations,

Nx total number of states,

Nf total number of rate equations,

t time.

19

Chapter 3

The procedure consists of three consecutive steps.

1. Approximation of nonlinear functions with piecewise-affine (PWA) functions. Non-

linear functions are common in biochemical models, which complicates system iden-

tification and analysis. PWA functions are a collection of linear functions separated

by switching planes. PWA functions can be used to approximate the nonlinearities.

2. Detection of equilibrium points and performing stability analysis. Equilibrium

points give an indication of the qualitative behavior of the system.

3. Construction of qualitative transition graphs. A transition graph contains all pos-

sible trajectories of a system and provides valuable knowledge about the dynamics.

Biochemical networks are often large systems that operate at different time scales. Reduc-

ing the complexity of these networks by means of model reduction contributes to a more

compact and comprehensible representation of the original model. As a consequence,

specific parts of the biochemical network can be studied in more detail. Model reduction

will be explained in the next chapter, when this procedure is applied to a model of the

Transforming Growth Factor-β1 (TGF-β1) pathway.

3.1.1 Approximation of Nonlinear Function with PWA Func-

tions

The rate equations fj(x) in Eq. 3.1 are linear and nonlinear functions of the state vector

x. The nonlinear functions are based on Michaelis-Menten kinetics. Such models are

generally too complicated to analyze, so we concentrated on a PWA approximation of the

nonlinear functions [94, 126, 141].

Michaelis-Menten Kinetics

Fig. 3.1 shows an example of how a classical Michaelis-Menten function is approximated

by two PWA segments. In reference [141] an iterative method was presented to select the

correct number of segments, but it primarily applies to complex trigonometric functions.

Nonlinear functions in biochemical networks are quite often monotonic, which are less

complex functions. Therefore two segments, separated by a switching plane α, were

assumed to be sufficient to capture the nonlinear behavior of these functions.1 Note

that using multiple segments would lead to a more refined approximation, but would

simultaneously increase the computational burden. For example, the Michaelis-Menten

equation, f(x), is mathematically defined by

f(x) =Vmaxx

Km + x, (3.2)

1The only exception on this assumption will be bisubstrate Michaelis-Menten inhibition, which willbe approximated by three segments.

20

Section 3.1

Figure 3.1 – Standard Michaelis-Menten equation (solid line) and its PWA approximation(dashed line). The first segment of the PWA function for 0 < x < α is linear; the secondsegment is constant for α < x < ∞. The functions f(x) and ϕ(x) intersect at xIP. Theshaded area between these functions represents the area of the cost functions J1, J2 and J3.

x substrate concentration,

Vmax maximal rate of conversion,

Km Michaelis constant.

The PWA function, ϕ(x), is a two-segment approximation of f(x)

ϕ(x) =

Vmaxxα

if x < α,

Vmax if x ≥ α,(3.3)

α switching plane.

Hill Kinetics

The Hill equation is an extension of Michaelis-Menten kinetics [46]

f(x) =Vmaxx

r

Krm + xr

, (3.4)


Ramp functions have recently been used as a gentle approach to approximate Hill functions

with low cooperativity coefficients (r < 10) [11]. A three-segment PWA function would be

necessary to generate a ramp function. For large values of the cooperativity coefficient r,

21

Chapter 3

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Vmax

φ( )x

f x( )

f x( )

Km

φ( )x

x

Figure 3.2 – Hill function f(x) for r = 100 (solid line) and its PWA approximation (dashedline).

the Hill function shows the characteristics of a biological switch, see Fig. 3.2. In this thesis,

only systems with large cooperativity coefficients are considered for the sake of simplicity,

so a two-segment approximation is adequate. The following piecewise-constant function

is therefore an appropriate description

ϕ(x) =

0 if x < Km,

Vmax if x ≥ Km.(3.5)

Inhibition

Inhibition of biochemical processes can be modeled as well with a Michaelis-Menten de-

scription

f(x) =VmaxKI

KI + x, (3.6)

KI Michaelis constant of the inhibitor.

The maximum rate of this function is Vmax and decreases monotonically towards zero for

x →∞, see Fig. 3.3. The corresponding PWA approximation is

ϕ(x) =

Vmax

(1− x

α

)if x < α,

0 if x ≥ α.(3.7)

Bisubstrate Michaelis-Menten

Conversion of nonlinear to PWA functions is not only confined to monosubstrate reactions.

Bisubstrate reactions can be approximated with planes. A Michaelis-Menten equation

22

Section 3.1

0 1 α 3 4 50

0.2

0.4

0.6

0.8

Vmax

φ( )x

f x( )

x0 KI α

f x( )

φ( )x

Figure 3.3 – Inhibited Michaelis-Menten function (solid line) and its PWA approximation(dashed line). The circle indicates the point of intersection between f(x) and ϕ(x).

with two substrates, x1 and x2, can be derived and is formulated as (if x1 has a linear

effect and x2 Michaelis-like kinetics)

f(x1, x2) =Vmaxx1x2

Km + x2

. (3.8)

Inspection of the 3D-plot (Fig. 3.4) shows that Eq. 3.8 is a combination of linear and

Michaelis-Menten dynamics. Two planes are therefore assumed to be sufficient for a

PWA description of this function.

ϕ(x1, x2) =

Vmaxx1 if αx1 − x2 < 0,

Vmaxx2

αif αx1 − x2 ≥ 0.

(3.9)

Competitive Inhibition

Some ligands could function as inhibitor of Michaelis-Menten kinetics. A special form

of Michaelis-Menten inhibition is competitive inhibition, in which an inhibitor molecule

binds reversibly to the enzyme at the same site as the substrate. It is mathematically

formulated as

f(x1, x2) =Vmaxx1

Km

(1 + x2

KI

)+ x1

, (3.10)

x1 substrate concentration,

x2 concentration of the inhibiting substrate,

KI Michaelis constant of the inhibition reaction.

23

Chapter 3

fx

x(

,)

12

φ(

,)

xx

12

(a) (b)

Figure 3.4 – (a) Plot of bisubstrate Michaelis-Menten kinetics, and (b) its two-segmentPWA approximation.

A three-segment PWA function is assumed to be a reasonable approximation of Eq. 3.10

ϕ(x1, x2) =

Vmaxx1

α1if x1

α1+ x2

α2< 1 ∧ x2 < α2,

Vmax

(1− x2

α2

)if x1

α1+ x2

α2≥ 1 ∧ x2 < α2,

0 if x2 ≥ α2,

(3.11)

α1 switching plane of the Michaelis-Menten function,

α2 switching plane of the inhibition reaction.

Fig. 3.5 shows the plots of the rate equations. For a more detailed description of the

equations above and other enzymatic kinetic reactions, we refer to standard biochem-

istry literature, e.g. [46]. The nonlinear functions in this section are a small selection of

nonlinearities in biochemical networks which can be approximated with PWA functions.

Extending this PWA approximation to other nonlinear functions is certainly possible

with the theory presented here. However, one should bare in mind that complex non-

linear functions demand a more refined segmentation and, consequently, requires more

computational effort.

24

Section 3.1

Vmax

Km

Vmax

KI

α1

α2x2

x2

x1 x1

fx

x(

,)

12

φ(

,)

xx

12

(a) (b)

Figure 3.5 – (a) Nonlinear function of competitive inhibition, and (b) its three-segmentPWA approximation.

3.1.2 Selection of PWA Parameters

To our knowledge, no research has established a link between the physiological parameters

of a nonlinear model and the parameters of its PWA approximation. Therefore, the most

optimal location of the switching planes with respect to the nonlinear parameters to

obtain the best PWA approximation has to be determined. A minimal difference between

the nonlinear function f(x) and its PWA approximation ϕ(x), must be derived. The

integral of the difference between these functions, the so-called cost function J , has to

be minimized. Cost functions are, as a rule, quadratic functions for penalizing large

errors between two functions [102]. However, an explicit solution for α as function of the

parameters of the nonlinear representation is computationally impossible to derive from a

quadratic cost function. Therefore the modulus of a linear cost function is chosen instead.

The nonlinear Michaelis-Menten equation from Eq. 3.2 and its PWA approximation in

Eq. 3.3 are selected to illustrate the derivation. The total cost function, Jtot, of f(x) and

ϕ(x) is mathematically described by

Jtot =

∫ xmax

0

∣∣∣f(x)− ϕ(x)∣∣∣dx, (3.12)

xmax maximum value of x.

25

Chapter 3

The function f(x) intersects with ϕ(x) at x = xIP and can be subdivided in three separate

cost functions

Jtot = J1 + J2 + J3. (3.13)

The three cost functions are

• For 0 < x < xIP: the nonlinear function f(x) is larger than the linear segment in

PWA function ϕ(x),

J1 =

∫ xIP

0

(Vmaxx

Km + x− Vmaxx

α

)dx, (3.14)

• For xIP < x < α: f(x) is smaller than the linear segment of ϕ(x),

J2 =

∫ α

xIP

(Vmaxx

α− Vmaxx

Km + x

)dx, (3.15)

• For α < x < xmax: f(x) is smaller than the constant segment of ϕ(x),

J3 =

∫ xmax

α

(Vmax − Vmaxx

Km + x

)dx. (3.16)

Fig. 3.1 depicts the graphs of f(x) and ϕ(x). An analytical expression for xIP is required

to solve Eq. 3.13. It is derived by calculating the intersection point between f(x) and the

linear segment of ϕ(x)

VmaxxIP

Km + xIP=

VmaxxIP

α⇒ xIP = α−Km. (3.17)

Combining Eq. 3.13 and 3.17 gives an analytical expression for Jtot

Jtot =1

2Vmax

(α− 2K2

m

α+ . . .

. . . 2Km

(log

Km

α− log

α

α + Km

+ logKm + xmax

α + Km

)), (3.18)

Eq. 3.18 is subsequently minimized as function of the PWA parameter α with Mathematica

arg minα

Jtot ⇒ dJtot

dα=

(α2 − 4αKm + 2K2m) Vmax

2α2= 0. (3.19)

26

Section 3.1

Solving this quadratic expression results in two solutions: α =(2 − √

2)Km and α =(

2 +√

2)Km. With the assumption that α > Km, one solution remains

α =(2 +

√2)

Km, (3.20)

or Km =(1−

√2

2

)α. A plot of f(x) and its PWA approximation ϕ(x) with α =

(2 +

√2)Km is given in Fig. 3.1. These derivations were repeated for the other nonlinear

functions as well. An overview of the nonlinear functions, their PWA approximations and

the link between the auxiliary parameters (α, α1, α2) and their nonlinear counterparts

(Km and KI) are listed in Table 3.1.

3.1.3 Detection of Equilibrium Points and Performing Stability

Analysis

A hybrid approximation Φ(x) of the original model in Eq. 3.1 can be formulated after the

nonlinear functions have been substituted for the PWA approximations. The switching

planes of the PWA approximation Φ(x) divide the phase space in discrete states, or modes,

denoted by q1, . . . , qNq with Nq: total number of modes. Each mode is governed by its own

characteristic set of continuous, linear ODEs and has an invariant associated to it, which

describes the conditions that the continuous state has to satisfy at this mode. Symbolic

expressions of the equilibria can be derived by solving Φ(x) = 0, for each individual mode

q. The equilibrium points have to satisfy the invariants and biochemical constraints, like

x > 0 (positive systems). This procedure leads to a collection of qualitative equilibrium

points that have to satisfy certain existence conditions.

The dynamical behavior at the equilibrium points are given by the eigenvalues λ

of the Jacobian matrix Jm, i.e., a matrix of all first order partial derivatives of the state

vector [81]. The eigenvalues λ give an indication of the dynamical behavior (see Table 3.2)

at a given equilibrium point and can be calculated with standard linear algebra [69, 81, 92]:

det(λINx − Jm) = 0, (3.21)

INx Nx ×Nx identity matrix.

Next step is to verify whether the hybrid system under study is stable. Even if stability is

guaranteed in a certain mode, this does not automatically imply that the complete hybrid

system is stable [19]. Global stability can be calculated by means of finding a Lyapunov

function, see [10, 98] for a thorough description of the methodology. We remark that

topological classification of the equilibria and Lyapunov stability can only be performed

27

Chapter 3

Tab

le3.1

–O

verviewof

nonlinearfunctions

andtheir

correspondingP

WA

approximations.

Nam

ef(x

)ϕ(x

)Lin

kf(x

)an

dϕ(x

)

Mich

aelis-M

enten

Vm

axx

Km

+x

{Vm

axx

αif

x<

α

Vm

ax

ifx≥

αα

=(2

+√

2 )K

m

Inhib

itionVm

axK

I

KI +

x

{V

max

(1−xα )

ifx

<α

0if

x≥α

α=

4.09KI

Hill

Vm

axx

r

Krm

+x

r

{0

ifx

<K

m

Vm

ax

ifx≥

Km

Direct

link

forlarge

r

Bisu

bstrate

M-M

Vm

axx1x2

Km

+x2

{V

max x

1if

αx

1 −x

2<

0Vm

axx2

αif

αx

1 −x

2 ≥0

α=

(2+√

2 )K

m

Com

petitive

Inhib

ition

Vm

axx1

Km (

1+

x2

KI )

+x1

Vm

axx1

α1

ifx1

α1

+x2

α2

<1∧

x2

<α

2

Vm

ax (

1−x2

α2 )

ifx1

α1

+x2

α2 ≥

1∧x

2<

α2

0if

x2 ≥

α2

α1

=(2

+√

2 )K

m

α2

=4.09K

I

28

Section 3.2

Table 3.2 – Eigenvalues of a two-state system and its relation to system behavior.

Eigenvalues Topological classification

Re(λ1,2) < 0 Stable node

Re(λ1,2) > 0 Unstable node

Re(λ1) < 0 < Re(λ2) Saddle point

λ1,2 = a± b i with a = 0 Limit cycle

λ1,2 = a± b i with a < 0 Stable focus

λ1,2 = a± b i with a > 0 Unstable focus

on relatively simple systems due to the complexity of solving symbolic equations. To

understand the system dynamics for larger systems, qualitative transition graphs become

therefore more important.

3.1.4 Construction of Qualitative Transition Graphs

If the motion of the continuous state would lead to violation of the conditions given by the

invariant, a transition must take place to a mode that satisfies this motion. A transition

graph contains all possible trajectories of a system and provides valuable knowledge about

the dynamics. It can be derived by calculating all possible mode transitions. Mode

transitions occur if the inner product of the tangent t(x) of the continuous state with the

normal of the switching plane n is larger than zero [141]

t(x) · n > 0. (3.22)

Hence, the transitions are expressed as symbolic inequalities, assigned to the variable Γ.

Since the dynamics are linear in each mode, Eq. 3.22 can be evaluated at the vertices of

the switching planes [58, 59, 88]. This leads to relatively simple inequalities for all vertices

of the switching planes. Since the dynamics in each mode are linear, this is sufficient to

predict the dynamics of the complete phase space. It yields multiple qualitative transition

graphs that have to satisfy a set of Γs.

3.2 Example: an Artificial Biochemical Network

To illustrate the above procedure, an artificial biochemical network with Michaelis-Menten

kinetics has been selected. The biochemical network consists of two substrates, x1 and

x2, which mutually induce each others production (Fig. 3.6).

29

Chapter 3

x1

Substrate

Degradationproducts

Reaction


x2

f1

f2

k2

k4

Figure 3.6 – Graphical interaction scheme of two substrates, x1 and x2, that mutuallyinduce each others production. Notation taken from [86].

3.2.1 PWA Approximation

This system can be described by a nonlinear model of two coupled differential equations:.

dx1

dt= f1(x2)− k2x1, (3.23)

dx2

dt= f2(x1)− k4x2, (3.24)

with Michaelis-Menten functions

f1(x2) =k1x2

Km1 + x2

, (3.25)

f2(x1) =k3x1

Km2 + x1

, (3.26)

x1, x2 substrate concentrations,

k1 maximal rate of x2, induced by x1,

k2 degradation rate constant of x1,

k3 maximal rate of x1, induced by x2,


Km1 Michaelis constant of x1 production,

Km2 Michaelis constant of x2 production.

30

Section 3.2

α2

α1

x1

x2

0 x1

max

x2

max

q1 q2

q3 q4

Figure 3.7 – Phase plane diagram, divided in four different modes. The dotted lines arethe switching planes.

This artificial biochemical network is a second order model. The PWA approximations of

the nonlinear functions f1(x2) and f2(x1) are ϕ1(x2) and ϕ2(x1), respectively:

ϕ1(x2) =

k1x2

α1if x2 < α1,

k1 if x2 ≥ α1,(3.27)

ϕ2(x1) =

k3x1

α2if x1 < α2,

k3 if x1 ≥ α2,(3.28)

α1 switching plane of f1(x),

α2 switching plane of f2(x).

3.2.2 Determination of the Equilibrium Points

Switching planes x1 = α2 and x2 = α1 divide the phase space into four distinct modes

(q1, · · · , q4), as shown in Fig. 3.7. The mathematical description of the hybrid approxi-

mation Φ(x) of the nonlinear system is

Φ(x) =dx

dt=

[k1x2

α1− k2x1;

k3x1

α2− k4x2]

T for mode q1,

[k1x2

α1− k2x1; k3 − k4x2]

T for mode q2,

[k1 − k2x1;k3x1

α2− k4x2]

T for mode q3,

[k1 − k2x1; k3 − k4x2]T for mode q4,

(3.29)

31

Chapter 3

with invariants

q1 : 0 ≤ x1 < α2 ∧ 0 ≤ x2 ≤ α1,

q2 : α2 ≤ x1 ≤ xmax1 ∧ 0 ≤ x2 ≤ α1,

q3 : 0 ≤ x1 < α2 ∧ α1 ≤ x2 ≤ xmax2 ,

q4 : α2 ≤ x1 ≤ xmax1 ∧ α1 ≤ x2 ≤ xmax

2 .

(3.30)

xmax1 maximum concentration of x1,

xmax2 maximum concentration of x2.

For each mode, an analytical expression of the equilibrium points can be derived by

putting the derivatives in Eq. 3.29 to zero. Pseudo equilibrium points are considered as

well. These equilibrium points are located outside the invariant of the corresponding

mode [141]. The positions of the (pseudo) equilibrium points are symbolic expressions of

the parameters k1, k2, k3, k4, α1, and α2. For example, consider an equilibrium point in

mode q1. The equilibrium point in the differential equations of mode q1 in Eq. 3.29 are

put to zero to calculate the position of the equilibrium point

dx1

dt=

k1x2

α1

− k2x1 = 0 ⇔ x1 =k1x2

α1k2

, (3.31)

dx2

dt=

k3x1

α2

− k4x2 = 0 ⇔ x2 =k3x1

α2k4

. (3.32)

Hence, (x1, x2) = (0, 0) is a real equilibrium point in mode q1 and satisfies the invariant

of mode q1, see Eq. 3.30. This procedure is repeated for the equilibria in modes q2, q3 and

q4, the results are listed in Table 3.3.

Table 3.3 – Dynamical behavior of the PWA system for different parameter constraints.

Mode Equilibrium point Classification Existence conditions

q1 (0, 0)Stable node k1k3

α1α2k2k4< 1

Saddle point k1k3

α1α2k2k4> 1

q2

(k1k3

α2k2k4, k3

k4

)Stable node k1k3

α1α2k2k4> 1 ∧ α1 ≥ k3

k4∧ α2 < k1

k2

q3

(k1

k2, k1k3

α1k2k4

)Stable node k1k3

α1α2k2k4> 1 ∧ α1 < k3

k4∧ α2 ≥ k1

k2

q4

(k1

k2, k3

k4

)Stable node k1k3

α1α2k2k4> 1 ∧ α1 < k3

k4∧ α2 < k1

k2

32

Section 3.3

3.2.3 Dynamical Behavior at the Equilibrium Points and Sta-

bility Analysis

First we consider mode q1 again. The Jacobian matrix for mode q1 is defined by

Jq1 =

(−k2

k1

α1

k3

α2−k4

). (3.33)

Eq. 3.33 yields two symbolic expression for the eigenvalues:

λ1,2 =−(k2 + k4)±

√D

2with D = (k2 + k4)

2 − 4

(k2k4 − k1k3

α1α2

). (3.34)

Symbolic calculations show that the equilibrium point at (0, 0) is a stable node fork1k3

α1α2k2k4< 1 and a saddle point for k1k3

α1α2k2k4> 1, see Table 3.2. Similarly, the proce-

dure is repeated for q2, q3 and q4 . The equilibria have to satisfy the invariants from

Eq. 3.30 and biochemical constraints. These impose additional constraints on the pa-

rameter values. An overview of the equilibria and corresponding constraints are listed in

Table 3.3. The next step is to check the stability of the complete hybrid system. Let the

Lyapunov function [10] be defined as

V =1

2

((x1 − xeq

1 )2 + (x2 − xeq2 )2

), (3.35)

with inequality constraint

dV

dt= (x1 − xeq

1 )dx1

dt+ (x2 − xeq

2 )dx2

dt< 0, (3.36)

xeq1 x1 evaluated at equilibrium point,

xeq2 x2 evaluated at equilibrium point,

to guarantee global stability. For mode q1, Eq. 3.36 is evaluated at equilibrium point (0, 0)

for all modes. The result shows that the inequality in Eq. 3.36 is always satisfied for all

(positive) parameter values. Hence, global stability has been proven for this system. The

stability of the equilibria in the other modes was also confirmed according to the same

procedure.

3.3 Transition Analysis

Qualitative transition analysis is required to derive the direction of the trajectories within

the system and to construct the possible transition graphs. In Fig. 3.8, the nullclines are

drawn for the four modes. The nullclines divide the total phase space in different flow

33

Chapter 3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) (b)

(c) (d)

stable node in q1 stable node in q2

stable node in q3 stable node in q4

Figure 3.8 – Nullclines (solid lines) and vector fields of the trajectories. Filled circle: stablenode; open circle: saddle point. The vector field is obtained by simulating the PWA modelin Eq. 3.29 with parameters (a) k1 = k3 = 1, k2 = k4 = 4, (b) k1 = 7, k2 = 8, k3 = 4,k4 = 10, (c) k1 = 4, k2 = 10, k3 = 9, k4 = 10, (d) k1 = k3 = 3, k2 = k4 = 4. The otherparameter values were in all simulations: α1 = α1 = 0.5, xmax

1 = xmax2 = 1.

34

Section 3.3

domains. The sign of the derivatives of the trajectories is the same in each individual flow

domain. For example, Fig. 3.8(a) is composed of three flow domains, whereas Fig. 3.8(b)

- (d) consist of four flow domains each. The directions of the trajectories in each flow

domain can be calculated on the vertices of the switching planes α1 and α2. For example,

mode q1 in Fig. 3.7 is separated from modes q3 and q2 by x1 = α2 and x2 = α1, respectively.

A trajectory in q1, directed towards q2, crosses x1 = α2 with normal n1→2 = [1, 0]T . The

tangent in q1, Φ1(x), is a mathematical function of the parameters and states

Φ1(x) =

(−k2x1 + k1x2

α1

k3x1

α2− k4x2

). (3.37)

One can deduce from Eqs. 3.22 that trajectories in q1 directed towards q3 are only present

ifk1x2

α1k2

> x1. (3.38)

The same procedure has been applied to switching plane x2 = α1 with n1→3 = [0, 1]T ,

which resulted in the inequality constraint

k3x1

α2k4

> x2. (3.39)

Solving Eqs. 3.38 and 3.39 on their respective switching planes results in complicated and

generally unsolvable algebraic equations [120]. The solutions were therefore calculated

on the vertices of the switching plane [11, 13, 14, 58]. This is sufficient to capture the

complete dynamics on the switching plane as the state equations are linear [59]. The

vertices of switching plane x1 = α2 are located at p1 = (α2, 0) and p3 = (α2, α1), see

Fig. 3.9(a). Substitution of these vertex coordinates in Eq. 3.38 and 3.39 yields the

following constraints on the parameters

Γ0 = k2α2 < 0, (3.40)

Γ1 =k1

k2

> α2, (3.41)

for (x1, x2) = (α2, 0) and (x1, x2) = (α2, α1), respectively. Eq. 3.40 can never be fulfilled

for positive parameter values. No trajectories can therefore traverse from mode q1 to q2

at p1 = (α2, 0), irrespective of the parameter value. Trajectories on vertex p3 can traverse

from q1 towards q2, but have to satisfy Γ1 in Eq. 3.41. As a consequence, trajectories

from q2 directed towards q1 have to meet the complementary constraints. These are

Γ∗0 = k2α2 > 0 and Γ∗1 = k1

k2< α2 for p1 and p3 , respectively. The asterisk indicates the

complementary set, i.e., Γ∗ = NOT(Γ). As Γ∗0 is always satisfied, trajectories at p1 can

only move from q2 to q1 and not vice versa.

35

Chapter 3

Table 3.4 – Inequality constraints, belonging to the mode transitions.

Variable Inequality set

Γ0 k2α2 < 0 ⇒ neverΓ∗0 k2α2 > 0 ⇒ alwaysΓ1 α2 < k1

k2

Γ∗1 α2 > k1

k2

Γ2 α1 < k3

k4

Γ∗2 α1 > k3

k4

Table 3.5 – Mode transitions at the vertices of the artificial biochemical network andcorresponding inequality sets.

Transition p1 p2 p3 p4 p5

q1 → q2 Γ0 × Γ1 × ×q1 → q3 × Γ0 Γ2 × ×q2 → q4 × × Γ2 Γ2 ×q3 → q4 × × Γ1 × Γ1

The complete phase space of the model is analyzed according to the procedure de-

scribed above. This yields six different sets of constraints: Γ0, Γ∗0, Γ1, Γ∗1, Γ2 and Γ∗2. The

results are summarized in Fig. 3.9(b) and Tables 3.4 and 3.5.

Not all sets are simultaneously present in the same transition graph. The sets Γ1 and

Γ2 are complementary to Γ∗1 and Γ∗2, respectively. This gives four unique combinations

of Γs and, hence, four possible transition schemes for the model, displayed in Fig. 3.10.

These transition graphs are qualitatively identical to the vector field in Fig. 3.8.

3.4 Discussion

In this chapter, a procedure has been presented to describe and analyze nonlinear models

as a hybrid system. The focus is on nonlinear processes that are typically observed in

biochemical networks. The methodology is based on three consecutive steps. To demon-

strate the procedure, a model of an artificial biochemical network is chosen. This second

order model is small and can be analyzed with standard phase plane analysis, but in this

case it has an exemplary role. It shows that qualitative analysis can predict the nature

of the transition graphs very well. However, stability analysis and determination of the

dynamical behavior at the equilibria can become complicated for larger systems (more

than three state variables), but are not a requirement for system analysis: qualitative

transition analysis can already provide some essential information about the system dy-

namics. Stability analysis might be performed by exploiting the monotonicity of nonlinear

36

Section 3.4

α2

α1

x1

x2

0

Γ1

Γ1*

Γ2

Γ2*

Γ2

Γ2*

q3

Γ1*

Γ1q4

q2q1

α2

α1

x1

x2

0

q3

q2q1

q4

x

x

n1 2 Γ0*

Γ0*

x x

x

p1

p2 p3 p4

p5

xx x

x

x

(a) (b)

Figure 3.9 – (a) Normal vector of switching plane, directed from q1 towards q2. The verticesof this plane are marked with ×. (b) Complete transition graph of the artificial biochemicalnetwork. The arrows show the direction of the trajectories at all vertices, the constraintsare indicated as well.

functions or using tools like Surface Lyapunov Functions [52, 53], in which exponential

stability can be proven for a large class of systems.

The main advantage of qualitative PWA analysis is the practical relevance in the field

of systems biology. Most methods require models with numerical parameter values for

system analysis [51, 82, 116] or are designed for biochemical networks with advantageous

properties [3, 27]. Contrary to these methods, qualitative PWA analysis is a general pro-

cedure that covers an extensive class of nonlinear biochemical networks, without requiring

quantitative information. The dynamics in each mode is linear, thus studying dynamics

at the vertices of the switching planes is sufficient to make statements about the dynam-

ics of the complete system. Vertex analysis yields simple inequalities. Even for larger

biochemical networks, the complexity does not increase exponentially, which contributes

to the practical relevance of qualitative PWA analysis. To further enhance the practical

usefulness, the symbolic calculations have been automated: determination of the loca-

tions of the equilibrium points and transition graph reconstruction are fully automated in

Mathematica. Quantifier elimination [166] or the algorithms of other qualitative modeling

programs like Robust Verification of Gene Networks (RoVerGeNe) [11] and the Genetic

Network Analyzer (GNA) [27] might be of great value to improve our implementation.

An artificial network is analyzed in this chapter, but it can be used for physiolog-

ically relevant biochemical networks as well. In the next chapter, this method will be

37

Chapter 3

x x

x

x

x

x x

x

x

x

x x

x

x

x

x x

x

x

x

(a) (b)

(c) (d)

q2 q2

q2 q2

q3

q3q3

q3

Figure 3.10 – All possible transitions schemes of the model for inequality sets (a) Γ∗1 ∧Γ∗2,(b) Γ1∧Γ∗2, (c) Γ∗1∧Γ2, and (d) Γ1∧Γ2. We remark that these transition graphs qualitativelycorrespond with the vector fields in Fig. 3.8.

38

Section 3.4

demonstrated for model of the TGF-β1 pathway.

39

Chapter 3

40

4Analysis of the Transforming Growth

Factor-β1 pathwayThis chapter was based on an extended version of paper [120]

The Transforming Growth Factor-β1 (TGF-β1) pathway plays a crucial role in prolifera-

tion, differentiation, apoptosis and motility [7]. It can be targeted by 42 known ligands,

which leads to transciptional control of more than 300 target genes [79]. As already men-

tioned in chapter 1, significant alterations in the TGF-β1 pathway have been observed in

aging endothelial cells. Also the experimental findings of other research groups suggest

the implication of TGF-β1 in vascular wall aging [55, 56, 107] and pathogenesis of chronic

vascular diseases [78, 110]. The qualitative analysis procedure from the previous chapter

will be applied to this biologically relevant example.

4.1 Physiology of the TGF-β1 Signaling Pathway

TGF-β1 is an extracellular cytokine that binds to membrane receptors and initiates the

activation of receptor-regulated SMAD proteins (R-SMADs) [30]. The signal transduction

pathway of TGF-β1 is initiated upon binding of the active form of TGF-β1 to the TGF-β1

receptor [30]. This leads to dimerization of two TGF-β1 receptors (RI and RII), which

are constitutively internalized by the formation of endosomes (membrane-bound compart-

ments inside cells) [33, 113]. The fate of these endosomes depends on whether a complex of

the protein SMAD ubiquitination regulatory factor 2 (SMURF2) and inhibitory-SMADs

(I-SMADs) is bound to the activated receptors, which targets the endosomes for degra-

dation [115]. If the endosomes are not degraded, they contribute to the phosphorylation

of R-SMADs, which are present in the cytosol of the cell (internal fluid of the cell). This

process is enzymatically stimulated by the activated receptors. Phosphorylated R-SMADs

quickly form homodimers, which subsequently bind to a common-mediator SMAD (co-

SMAD) to produce a oligotrimer: the so-called SMAD-complex [105, 154]. This complex

is transported via the nuclear pores into the nucleus in which it promotes the outward

41

Chapter 4

transport of nuclear I-SMADs [30, 74]. The SMAD-complex binds to specific regions on

the DNA to promote gene transcription [30], including the ones involved in ECM forma-

tion [160], the reduction in cell cycle time [182], the regulation of nitric oxide [144], the

production of latent-TGF-β1 [83, 169] and I-SMADs [60]. The latter two can be seen

as auto-induction (positive feedback circuit) and -inhibition (negative feedback circuit)

of the TGF-β1 pathway, respectively [114]. The phosphorylated R-SMAD complexes be-

come dephosphorylated after their regulatory role is accomplished. These substrates are

subsequently transported back as individual SMAD proteins into the cytosol to complete

the R-SMAD loop.

A putative positive feedback in the TGF-β1 pathway is the auto-induction loop with

latent-TGF-β1. This protein is the inactive form of TGF-β1 and present in the ECM. It

is converted into the active form when exposed to radiation, oxidative stress, hormones

or other factors [142]. It is still unknown whether the secreted latent-TGF-β1 is used by

the cell itself or transported to the ECM of neighboring cells. Its affinity to the TGF-β1

receptor might play a role in the effect of latent-TGF-β1 on the cellular response, as was

demonstrated for the endothelial growth factor (EGF) [31].

Negative feedback is regulated by I-SMADs in this biochemical network. Normally,

I-SMADs are located inside the nucleus and are transported into the cytosol upon R-

SMAD activation. In the cytosol, they inhibit the R-SMAD signaling loop by binding

to the activated TGF-β1 receptor. This reduces the phosporylation rate of R-SMADs

by competitive inhibition and degrades the receptor. I-SMADs have a limited survival

time and are eventually degraded. A second form of inhibition is the binding of I-SMADs

to SMURF2. This complex binds to activated receptors in endosomes and target these

vesicles for degradation. The interaction graph of the total TGF-β1 pathway is displayed

in Figure 4.1, following the conventions of [86].

4.1.1 Isolation of the R-SMAD Loop

Fig. 4.1 shows that a plethora of biochemical processes are involved in the TGF-β1 path-

way. Timeseries data of R-SMAD profiles have been the only set of experimental data

we have at our disposal. Therefore the study of the TGF-β1 pathway will be centered

around the R-SMAD loop and additional assumptions are made to prepare the TGF-β1

model for model reduction.

• The positive feedback loop with latent TGF-β1 is discarded, since we assume that

this process is negligible. Also controversy exists about the contribution of auto-

induction in the TGF-β1 pathway. Instead a constant input of active TGF-β1 has

been postulated.

• The assumption is made that the constant presence of active TGF-β1 generates a

42

Section 4.1

TGF- β1

TGF- β1

RI/RII RI RII

TGF- β1

RI/RII

TGF- β1

RI/RII

SMURF2/I-SMAD

SMURF2/I-SMAD

SMURF2/I-SMAD

SMURF2/I-SMAD

SMURF2/I-SMAD

SMURF2

I-SMAD

R-SMAD

R-SMAD

P

co-SMAD

R-SMAD

P

co-SMADSMAD-complex

P

R-SMAD

P

co-SMADSMAD-complex

P

I-SMAD

latentTGF- β1

extracellular space

Substrate

Receptor

Phosphor

Degradationproducts

Geneticregulation

Reaction

Inhibition


P

R-SMADco-SMADSMAD-complex

Figure 4.1 – Schematical representation of the TGF-β1 pathway in the vascular endothelialcell.

43

Chapter 4

continuous flow of endosomes.

• The exact role of co-SMAD is unrevealed yet. Most likely it plays a role in genetic

regulation. Exclusion of co-SMAD would produce a simpler model structure and is

not rate-limiting in the whole process, as shown by [24].

These assumptions lead to a reduced model, see Fig. 4.2. This will be the basis for the

quasi-steady-state approximation in the next section.

4.2 Qualitative Analysis of the Transforming Growth

Factor-β1 Pathway

The procedure for qualitative analysis follows the same steps as for the toy example in

chapter 3. In this case model reduction has been applied. Although reducing the number

of state equations is not required for the analysis of this model, it decreases the complexity

of the model significantly by capturing the system dynamics within the time window of

interest. The dynamics of available experimental data, see Fig. 4.3 [125], show that

these operate in a minutes/hours time range. The preliminary experimental data [125] in

Fig. 4.3 on primary mouse hepatocytes [87] might contain a limit cycle. Analysis of the

frequency spectrum confirms this observation (Fig. 4.4).

To focus on a time window of minutes and hours, the adapted procedure for the

qualitative analysis of the TGF-β1 pathway will become:

1. Perform model reduction by means of quasi-steady-state approximation. Providing

additional information on the qualitative parameter values reduces the number of

possibilities in dynamical behavior.

2. Derive the PWA model of the system.

3. Analyze the dynamical behavior in each mode. Determine the position of the

equilibrium points located and what symbolic inequalities do these equilibria have

to fulfill. In addition, stability of the PWA model can be performed.

4. Construct all possible qualitative phase planes on basis of the information above,

yielding the qualitative constraints that they have to satisfy.

4.2.1 Model Reduction of the TGF-β1 Pathway

A nonlinear mathematical model can be formulated on basis of the biochemistry of the

TGF-β1 interaction graph [46], see Fig. 4.2. The nonlinear deterministic model of the

44

Section 4.2

TGF- β1

RI/RII

I-SMAD

R-SMAD

R-SMAD

P

co-SMADSMAD-complex

P

extracellular space

Substrate

Receptor

Phosphor

Degradationproducts

Geneticregulation

Reaction

Inhibition


I-SMAD

R-SMAD

P

co-SMADSMAD-complex

P

R-SMADco-SMADSMAD-complex

INPUT

y1 y2

y3

y4

y5y6

y7

P

u

k1

k2

k3

k4

k5

k6k7

k8

k8

k9

Figure 4.2 – Interaction graph of the TGF-β1 pathway with the focus on the R-SMADloop.

45

Chapter 4

0

0.5

1

1.5

0 100 200 300 400 500 600

Phosphorylated R-SMADE

xp

ress

ion

(ar

bit

rary

un

its)

time (min)

Figure 4.3 – Experimental data of phosphorylated R-SMADs. Expression data (diamonds)show that phosphorylated R-SMAD levels oscillate when a constant extracellular TGF-β1

load is applied [125].

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

frequency (Hz)

peak = oscillations

mag

nit

ude

Figure 4.4 – The peak in the graph shows that the experimental data in Fig. 4.3 can exhibitoscillatory behavior.

46

Section 4.2

TGF-β1 pathway consists of seven coupled ordinary differential equations [120]:

dy1

dt= u− k1y1 − k2y1y7

Km1 + y1

, (4.1)

dy2

dt= k3y5 − k4y1y2

Km2

(1 + y7

KI

)+ y2

, (4.2)

dy3

dt=

k4y1y2

Km2

(1 + y7

KI

)+ y2

− k5y3, (4.3)

dy4

dt= k5y3 − k6y4, (4.4)

dy5

dt= k6y4 − k3y5, (4.5)

dy6

dt=

k7yr4

Krm3

+ yr4

− (k8 + k9) y6, (4.6)

dy7

dt= k9y6 − k8y7, (4.7)

y1 activated TGF-β1 receptor concentration,

y2 unphosphorylated R-SMAD concentration,

y3 phosphorylated R-SMAD concentration,

y4 phosphorylated R-SMAD complex concentration in the nucleus,

y5 unphosphorylated R-SMAD complex concentration in the nucleus,

y6 I-SMAD concentration in the nucleus,

y7 I-SMAD concentration in the cytosol,

u input of TGF-β1 supply,

k1 normal receptor decay constant,

k2 maximal decay of I-SMAD induced receptor degradation,

k3 transport rate constant of R-SMADs out of the nucleus,

k4 maximal phosphorylation rate of R-SMADs,

k5 translocation rate constant of phosphorylated R-SMAD complexes into the

nucleus,

k6 dephosphorylation rate constant of SMAD complex,

k7 maximal rate of I-SMAD production,

k8 I-SMAD decay constant,

k9 translocation rate constant of nuclear I-SMAD to cytosolic I-SMAD,

Km1 Michaelis constant of I-SMAD induced receptor degradation,

Km2 Michaelis constant of phosphorylated R-SMAD complex formation,

Km3 Michaelis constant of I-SMAD production,

KI Michaelis constant of I-SMAD binding to the receptor,


47

Chapter 4

Table 4.1 – Timescales for various biochemical processes.

Biochemical process Time (in sec)

Molecular motion 10−15

Translation (per protein) 10−5 − 10−3

Kinase/phosphatase reactions 10−3

Protein conformational changes 10−3

Cell-scale protein diffusion (active) < 100

Protein folding 100

Biomolecular binding 10−1 − 101

Cell-scale protein diffusion (passive) 100 − 101

Cell migration 100 − 102

Receptor internalization 102

Transcriptional control 102

Amino acids → proteins in ER 103

DNA replication 103

Protein secretion by Golgi apparatus 103 − 104

Cellular growth > 104

4.2.2 Quasi-Steady-State Approximation of the TGF-β1 Model

Like the TGF-β1 pathway, typical systems biology models consist of large sets of coupled

differential equations. Frequently these are too large given the amount and quality of

available data from experiments to guarantee a reliable parameter estimation [176]. Re-

ducing the model size is desired, but under the requirement that it captures the essential

processes. Model reduction tools have been widely used, but primarily for linear models [5]

or systems with quantitative information [89, 127]. For nonlinear biochemical networks

with a lack in quantitative information, none of these model reduction techniques is re-

ally suitable with the exception of one: the quasi-steady-state approximation [42, 151]. It

makes use of the differences in time scales of various processes [131, 145, 175]. Table 4.1

gives an overview of the various time scales in which biochemical processes operate [132];

our time window of interest covers the minute and hour range. This a priori knowledge

enables a distinction between slow and fast varying (state) variables. By assuming that

the fast dynamics are instantaneously in equilibrium, only the relatively slow dynam-

ics are preserved and, consequently, the model size will be reduced. A more thorough

explanation of such quasi-steady-state approach will be given below.

48

Section 4.2

From Eqs. 4.1 to 4.7, the following SMAD pools can be defined:

unphosphorylated R-SMAD pool yuRS = y2 + y5, (4.8)

phosphorylated R-SMAD pool ypRS = y3 + y4, (4.9)

total R-SMAD pool ytRS = yuRS + ypRS, (4.10)

total I-SMAD pool ytIS = y6 + y7, (4.11)

The pools ytRS and ytIS are assumed to be constant, based on mass conservation laws.

Model reduction can be applied to Eqs. 4.1 - 4.7, the relatively slow dynamics will be

preserved. First, we consider the state equations of y2 and y5 in Eqs. 4.2 and 4.5, respec-

tively. A distinction between slow and fast varying rates on basis of a priori knowledge

is made by dividing the large rates by ε, given that ε ¿ 1; the rate constants of these

functions are indicated with an apostrophe. In addition, the reasonable assumption is

made that translocation rate constants of SMADs across the nuclear membrane (k3, k5

and k9) are large compared to the other rates

dy2

dt=

k′3y5

ε− k4y1y2

Km2

(1 + y7

KI

)+ y2

, (4.12)

dy5

dt=k6y4 − k′3y5

ε. (4.13)

The dynamics of yuRS are

dyuRS

dt= k6y4 − k4y1y2

Km2

(1 + y7

KI

)+ y2

. (4.14)

Ask′3y5

εÀ k6y4, Eq. 4.13 can be rewritten as y5

dt≈ −k′3y5

ε, preserving the slow dynamics

limε→0

dy5

dt= −k′3y5

ε⇒ y5 = 0

Eq. 4.8−−−−→ yuRS = y2. (4.15)

Combining Eqs. 4.14 and 4.15 results in a reduced expression for the dynamics of yuRS

dyuRS

dt≈ dy2

dt= k6y4 − k4y1y2

Km2

(1 + y7

KI

)+ y2

. (4.16)

Similarly, expressions for the slow dynamics ofdypRS

dtand dytIS

dtcan be derived

dypRS

dt≈ dy4

dt=

k4y1y2

Km2

(1 + y7

KI

)+ y2

− k6y4, (4.17)

49

Chapter 4

I-SMAD

extracellular space

Substrate

Phosphor

Degradationproducts

Reaction

Inhibition

P

R-SMAD

P

co-SMADSMAD-complex

Px4

x7

k8

k9

k4

k6

Figure 4.5 – The components and their interactions of the reduced model of the TGF-β1

pathway.

dytIS

dt≈ dy7

dt=

k8yr4

Krm3

+ yr4

− k9y7. (4.18)

Eqs. 4.16 and 4.17 are combined with Eq. 4.10. By assuming that the TGF-β1 receptor

sustains constant activity (y1 = constant), the complete model is reduced to two states.

The reduced model is converted into a dimensionless system by defining x4 = y4

ytRSand

x7 = y7

ytIS.

dx4

dt= k∗4

1− x4

K∗m2

(1 + x7

K∗I

)+ 1− x4

− k∗6x4, (4.19)

dx7

dt= k∗8

xr4

Krm3

+ xr4

− k∗9x7. (4.20)

The asterisk-superscript of the parameters is left out in the remainder of this chapter for

simplicity reasons. Figure 4.5 shows the interaction graph of the reduced model.

50

Section 4.2

4.2.3 From Nonlinear to Piecewise-Affine

The nonlinear expressions in Eqs. 4.19 and 4.20 are defined as

f1(x4, x7) =k4(1− x4)

1− x4 + Km2

(1 + x7

KI

) , (4.21)

f2(x4) =k8x

r4

Krm3

+ xr4

, (4.22)

respectively, and will be converted into piecewise continuously differentiable functions.

A nonlinear function can be approximated by a PWA function of two or more segments

[141], as shown in chapter 3. The nonlinear function f1(x4, x7) represents competitive

inhibition of R-SMAD phosphorylation. Fig. 3.5(a) gives an impression of the 2D-plot of

this function for some arbitrary selected parameter values. The parameter k4 determines

the maximum of this surface. A PWA approximation of f1(x4, x7) is

ϕ1(x4, x7) =

k4(1− x7

α2) if x4 − α1

α2x7 ≤ 1− α1 ∧ x7 < α2,

k4

α1(1− x4) if x4 − α1

α2x7 > 1− α1 ∧ x7 < α2,

0 if x7 ≥ α2.

(4.23)

Fig. 3.5(a) and (b) display typical graphs of the nonlinear function and its PWA coun-

terpart, respectively. The function f2(x4) is the rate of I-SMAD (x7) expression, induced

by the SMAD complex (x4). Genetic regulation is assumed to function like a switch [2].

Its PWA description ϕ2(x4) is therefore

ϕ2(x4) =

0 if x4 < Km3 ,

k8 if x4 ≥ Km3 .(4.24)

The final model equations Φ(x) can be constructed by replacing the nonlinear functions

f1(x4, x7) and f2(x4) in Eqs. 4.21 and 4.22 with ϕ1(x4, x7) and ϕ2(x4), respectively. The

resulting PWA model dxdt

= Φ(x) consists of maximally six discrete modes (q1, . . . , q6) of

two linear state equations (dx4

dtand dx7

dt), in matrix form given by

Φi(x) =dx

dt= Aix + Bi, (4.25)

Ai, Bi the coefficient matrices of mode qi, listed in Table 4.2.

51

Chapter 4

Table 4.2 – Coefficient matrices of the TGF-β1 model.

Mode A-matrix B-matrix

q1 A1 =

(−k6 − k4

α2

0 −k9

)B1 =

(k4

0

)

q2 A2 =

(−(k6 + k4

α1) 0

0 −k9

)B2 =

(k4

α1

0

)

q3 A3 =

(−k6 00 −k9

)B3 =

(00

)

q4 A4 =

(−k6 − k4

α2

0 −k9

)B4 =

(k4

k8

)

q5 A5 =

(−(k6 + k4

α1) 0

0 −k9

)B5 =

(k4

α1

k8

)

q6 A6 =

(−k6 00 −k9

)B6 =

(0k8

)

The invariants of the six modes are

if q1 : x4 − α1

α2

x7 ≤ 1− α1 ∧ x7 < α2 ∧ x4 < Km3 , (4.26)

if q2 : x4 − α1

α2

x7 > 1− α1 ∧ x7 < α2 ∧ x4 < Km3 , (4.27)

if q3 : x7 ≥ α2 ∧ x4 < Km3 , (4.28)

if q4 : x4 − α1

α2

x7 ≤ 1− α1 ∧ x7 < α2 ∧ x4 ≥ Km3 , (4.29)

if q5 : x4 − α1

α2

x7 > 1− α1 ∧ x7 < α2 ∧ x4 ≥ Km3 , (4.30)

if q6 : x7 ≥ α2 ∧ x4 ≥ Km3 . (4.31)

The switching planes are x4 − α1

α2x7 = 1− α1 and x4 = Km3.

4.2.4 Equilibria and Stability Analysis

The PWA representation in Eq. 4.25 enables qualitative analysis. For Km3 > 1− α1, the

system is divided in exactly six modes by the switching planes, see Fig. 4.6. We remark

that three modes are present for Km3 < 1− α1, but only the more complicated situation

with six modes will be analyzed here.

Proposition 4.2.1. The model contains a single steady-state or has a limit cycle, de-

52

Section 4.2

x4

x7

1-α1 Km3

α2

q1

q2

q3

q4

q5

q6

x

x

x

x x x

x

p1 p2

p3

p4 p5 p6

p7

Figure 4.6 – Modes of the TGF-β1 model. The switching planes divide the model in sixmodes. The vertices of the switching planes are numbered (p1, . . . , p7) and indicated withcrosses.

pending on the choice of parameter values.

Symbolic expressions for all equilibrium points in each mode (q1, ..., q6) have been

derived (see chapter 3 for the procedure). The equilibrium points have to satisfy their

invariants, see Eqs. 4.26 - 4.31. For example, the equilibrium point in mode q1 is derived

by solving Eq. 4.25 for mode q1

Φ1(x) = A1x + B1 = 0, (4.32)

with x = [x4, x7]T , which leads to (x4, x7) = (k4

k6, 0). As the equilibrium is located in q1, it

has to satisfy the invariant x4− α1

α2x7 ≤ 1−α1∧x7 < α2∧x4 < Km3 and this consequently

leads to the existence condition k6 ≥ k4

1−α1. This procedure is repeated for al modes and

confirmed by numerical studies, see Fig. 4.7. A summary of all equilibrium points and

their associated existence conditions is listed in Table 4.3. Note that modes q3 and q6

have pseudo equilibrium points and for the set

k6 <k4(1−Km3)

α1Km3

∧ k9 ≤ k4k8

α2(k4 − k6Km3), (4.33)

no equilibrium point is present. Numerical verification shows that the model exhibits limit

cycle behavior under the parameter constraints in Eq. 4.33, see Fig. 4.7(e). Furthermore,

the existence conditions for each individual equilibrium point are complementary to each

53

Chapter 4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

stable node in q1

stable node in q5

x4 x4

x4 x4

x7

x7 x7

x7

α2

1-α1 Km3

α2

1-α1 Km3

α2

1-α1 Km3

α2

1-α1 Km3

stable node in q2

stable node in q4

(a) (b)

(c) (d)

Figure 4.7 – Vector field of the PWA model with parameter values that satisfy the existenceconditions for a stable node (dot) in (a) q1: k4 = 0.8, k6 = 5, k8 = 0.5 and k9 = 0.4 ; (b) q2:k4 = 0.8, k6 = 2.5, k8 = 0.5 and k9 = 0.4; (c) q4: k4 = 0.8, k6 = 0.5, k8 = 0.5 and k9 = 1.6;and (d) q5: k4 = 0.8, k6 = 0.5, k8 = 0.5 and k9 = 10. In addition, these vector fields werecomputed with parameter values α1 = 0.75, α2 = 0.5, and Km3 = 0.5. The direction of thesolution trajectories are visualized with arrows.

54

Section 4.2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x4

x7

α2

1-α1 Km3

limit cycle

(e)

¿ continuation of Figure 4.7 À (e) Phase plane of the PWA model with parameter valuesthat satisfy the existence conditions for a limit cycle: k4 = 0.8, k6 = 0.6, k8 = 0.5 andk9 = 0.4.

other, i.e., the existence conditions of the equilibrium points in Table 4.3) do not overlap.

This implies that no multiple equilibrium points can occur.

Proposition 4.2.2. If the parameter values and states satisfy the biochemically induced

constraints given in Table 4.3, provided that it has an equilibrium point in one of the

modes, the TGF-β1 model converges to a single stable node that is globally uniformly

asymptotically stable (GUAS).

As shown in Proposition 4.2.1, a single equilibrium point can be present in modes q1,

q2, q4 or q5. The exact location of this equilibrium is dependent on the specific existence

conditions for parameter values (Table 4.3). The stability is guaranteed if a Lyapunov

function V (x) can be found for all modes. This is an essential requirement to guarantee

GUAS. Let the Lyapunov function be defined as V = 12((x4 − xeq

4 )2 + (x7 − xeq7 )2), with

xeq4 and xeq

7 : equilibrium value for x4 and x7, respectively. Hence, dVdt

= (x4 − xeq4 )dx4

dt+

(x7 − xeq7 )dx7

dt< 0 is required to guarantee stability. For q1, this inequality can be solved

given the equilibrium point of q1: (xeq4 , xeq

7 ) = (k4

k6, 0). This condition is always satisfied in

mode q1; stability is therefore guaranteed. The procedure above is applied to the other

modes as well, which shows stability with a common Lyapunov function in all modes.

Similarly, stability of the equilibrium points in modes q2, q4, and q5 can also be proven.

Consequently, the complete system is guaranteed to be GUAS, regardless of the parameter

values.

55

Chapter 4

Table 4.3 – Equilibrium points and corresponding existence conditions.

Mode Equilibrium point Existence condition

q1

(k4

k6, 0

)k6 ≥ k4

1−α1

q2

(k4

k4+α1k6, 0

)k4(1−Km3)

α1Km3≤ k6 < k4

1−α1

q3 (0, 0) invalid

q4

(k4(α2k9−k8)

α2k6k9, k8

k9

)k6 < k4(1−Km3)

α1Km3∧ k4k8

α2(k4−k6Km3)< k9 ≤ k8(k4+α1k6)

α2(k4+k6(α1−1))

q5

(k4

k4+α1k6, k8

k9

)k6 < k4(1−Km3)

α1Km3∧ k9 > k8(k4+α1k6)

α2(k4+k6(α1−1))

q6

(0, k8

k9

)invalid

4.2.5 Transition Analysis

Proposition 4.2.3. The TGF-β1 model contains 14 mode transitions for which 14 dif-

ferent sets of constraints are valid.

The transition q1 → q2, for example, is only feasible if the solution of Φ1(x) in

Eq. 4.25 traverses the switching plane x4 − α1

α2x7 = 1 − α1 from q1 → q2 with normal

n1→2 = [1 − α1

α2]T The dot product of n1→2, and Φ1(x) yields the inequality k4 − k6x4 −

(k4−α1k9

α2)x7 > 0. Subsequent substitution of the vertex coordinates p1 = (1 − α1, 0)

and p3 = (Km3,α2(Km3+α1−1)

α1) in this inequality gives Γ1 : k6 < k4

1−α1and Γ2 : k6 <

k4(1−Km3)+α1k9(Km3+α1−1)α1Km3

, respectively. This procedure is repeated for all transitions, for

which the results have been summarized in Table 4.4 (description of the inequalities) and

Table 4.5 (at which vertices these inequalities hold). This yielded a collection of seven

different sets of constraints on the parameter values (Γ0, · · · , Γ6). However, there are also

seven complementary sets (Γ∗0, · · · , Γ∗6). For example, Γ∗1 is described by k6 > k4

1−α1. One

can conclude that 14 inequalities are therefore required to describe all 14 mode transitions.

Proposition 4.2.4. Transitions q3 → q1 at vertices p4 and p5, transition q4 → q1 at

vertex p5 and transition q6 → q3 at vertices p5 and p7 are always possible.

Transitions q1 → q3, transition q1 → q4, and q3 → q6 have the same set of inequality

constraints on the parameters: Γ0. In other words, these transitions are not possible for

the vertices mentioned here. This automatically implies that the reverse mode transitions

should be valid.

Proposition 4.2.5. Given the existence conditions for the equilibrium points and the

transitions, all possible transition graphs of the TGF-β1 model are shown in Fig. 4.8.

56

Section 4.2

Table 4.4 – Inequality sets of the TGF-β1 model.

Name Inequality set

Γ0 α2k9 < 0 ∨ k6Km3 < 0 ⇒ never

Γ1 k6 < k4

1−α1

Γ2 k6 < k4(1−Km3)+α1k9(Km3+α1−1)α1Km3

Γ3 Km3 < k4

k4+α1k6

Γ4 k6 <α1(k9− k8

α2)+k9(Km3−1)+

k4(1−Km3)α1

Km3

Γ5 k6 < α1(k9 − k8

α2)

Γ6 α2 < k8

k9

Table 4.5 – Transitions in the TGF-β1 model at the vertices.

Transition p1 p2 p3 p4 p5 p6 p7

q1 → q2 Γ1 × Γ2 × × × ×q1 → q3 × × × Γ0 Γ0 × ×q1 → q4 × × Γ3 × Γ0 × ×q2 → q5 × Γ3 Γ3 × × × ×q3 → q6 × × × × Γ0 × Γ0

q4 → q5 × × Γ4 × × Γ5 ×q4 → q6 × × × × Γ6 Γ6 ×

57

Chapter 4

x4

x7

q1

q2

q3

q4

q5

q6

x x

x

x

{Γ5, }Γ6*

x

{Γ5* 6, }Γ

x

{Γ5* 6*, }Γ

x

{Γ2, }Γ4

x

{Γ2, }Γ4*

x

{Γ2* 4*, }Γ

x

9 combinations of inequalities:

Γ0*

Γ1*Γ3*Γ3*

Γ3*

Γ3*

Γ0*

x

{Γ0* 6, }Γ

x

{Γ0* 6*, }Γ

Γ0* Γ0*

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4 * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4 * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4* * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4* * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2* 3* 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2* 3* 4* * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2* 3* 4* * 6*, , , , , , }

stable node in mode q1

(a)

Figure 4.8 – (a) Transition graph of all possibilities that satisfy the constraints for anequilibrium point in mode q1. The location of the mode containing the equilibrium point iscolored gray. The sets between curly brackets, for example {Γ1, Γ2} is shorthand notationfor Γ1∧Γ2. If a vertex has multiple mode transitions, the various possibilities are displayed.

58

Section 4.2


x4

x7

q1

q2

q3

q4

q5

q6

x x

x

x

{Γ5, }Γ6*

x

{Γ5* 6, }Γ

x

{Γ5* 6*, }Γ

x

{Γ2, }Γ4

x

{Γ2, }Γ4*

x

{Γ2* 4*, }Γ

x

Γ3* Γ3*

Γ3*

Γ0*

Γ1 Γ3*

Γ0*

x

{Γ0* 6, }Γ

x

{Γ0* 6*, }Γ

Γ0* Γ0*


{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4 * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4 * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4* * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4* * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3* 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3* 4* * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3* 4* * 6*, , , , , , }

(b)

¿ continuation of Figure 4.8 À (b) Transition graph of all possibilities that satisfy theconstraints for a stable node in mode q2.

59

Chapter 4


x4

x7

q1

q2

q3

q4

q5

q6

x x

x

x

x

x

{Γ5, }Γ6*

x

{Γ5* 6*, }Γ

{Γ2, }Γ4

x

{Γ2, }Γ4*

x

Γ3Γ3

Γ1 Γ3

Γ0* Γ6*

Γ0*

Γ0*


{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6*, , , , , , }

(c)

¿ continuation of Figure 4.8 À (c) Transition graph of all possibilities that satisfy theconstraints for a stable node in mode q4.

60

Section 4.2


x4

x7

q1

q2

q3

q4

q5

q6

x x

x

x

{Γ5, }Γ6*

x

{Γ5* 6*, }Γ

x

{Γ2, }Γ4

x

{Γ2, }Γ4*

x

{Γ2* 4*, }Γ

x

Γ3

Γ0*

Γ1 Γ3

Γ0*


{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* * 6*, , , , , , }

Γ3

Γ3

x

Γ0*

Γ6*

(d)

¿ continuation of Figure 4.8 À (d) Transition graph of all possibilities that satisfy theconstraints for a stable node in mode q5.

61

Chapter 4

limit cycle

x4

x7

q1

q2

q3

q4

q5

q6

x x

x

x

{Γ5, }Γ6*

x

{Γ5* 6, }Γ

x

{Γ5* 6*, }Γ

x

Γ0*

Γ1 Γ3

Γ0*

x

{Γ0* 6, }Γ

x

{Γ0* 6*, }Γ

Γ0* Γ0*


{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* 6*, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* * 6, , , , , , }

{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* * 6*, , , , , , }

{Γ2, }Γ4

x

{Γ2, }Γ4*

x

{Γ2* 4*, }Γ

x

Γ3 Γ3

Γ3

(e)

¿ continuation of Figure 4.8 À (e) Transition graph of all possibilities that satisfy theconstraints for a limit cycle.

62

Section 4.3

Each equilibrium point has to satisfy specific existence conditions (Table 4.3). Simi-

larly, each transition has been constrained as well (Tables 4.4 and 4.5). A combination of

these inequalities provides information whether a transition is feasible for a given equilib-

rium point. To illustrate this, consider an equilibrium point in mode q1. The existence

condition for an equilibrium point in q1 is k6 ≥ k4

1−α1, see Table 4.3. The vertex p1 of

the switching plane between q1 and q2 has to obey the inequality k6 < k4

1−α1, which is

incompatible with the existence conditions for the equilibrium point in mode q1. So the

transition q1 → q2 is not feasible if an equilibrium point is present in mode q1, the tran-

sition q2 → q1 is always valid in this case. This verification process is automated and has

been repeated for all equilibrium point and transitions. There are seven sets of inequalities

and an identical number of complementary sets. Γ0 is never fulfilled, while Γ∗0 is present

in all transition graphs. Excluding Γ0 reduces to six sets of inequalities per transition

graph. The maximum number of combinations with such amount of inequalities is equal

to the number of subsets, namely 26 = 64 combinations are possible in theory. However,

not all subsets are valid in practice as will be shown for an equilibrium point in q1. The

inequality constraints of the equilibrium points in Table 4.3 rule out specific transitions.

For example, the inequality constraints of an equilibrium point in mode q1 exclude Γ1

from the transition graphs. Transitions are verified for all equilibrium points with these

extra restrictions and resulted in multiple transition graphs per equilibrium point. All

possible combinations are displayed in Fig. 4.8. Some vertices contain multiple transitions

which can be directed in opposite direction. To take this into account, the resultant of

both arrows is taken. Some mode transitions are therefore not set perpendicular to the

switching plane the trajectories cross.

4.3 Discussion

A model of the TGF-β1 pathway was developed from scratch and analyzed with the

qualitative analysis procedure of biochemical networks. In the literature, another model of

the TGF-β1 has been reported: Clarke and co-workers [24] created a complex deterministic

model of the SMAD signaling pathway, but they excluded negative feedback by I-SMADs.

The troubling tendency of such large quantitative models is that many parameters values

are needed. These parameter values are often obtained by in vitro evidence or arbitrarily

chosen, which make the results automatically less reliable. Our model shows that a

simple second order model can exhibit both a stable equilibrium or a limit cycle. Unlike

conventional methods, no quantitative information is demanded to do these observations.

Fig. 4.4 can be interpreted as a limit cycle which suggests that the parameters should

obey Eq. 4.33. There are indications that during aging, the TGF-β1 pathway becomes

insensitive to external TGF-β1 influences [107], which would correspond to an equilibrium

63

Chapter 4

with low amounts of phosphorylated SMAD-complexes, e.g. a stable node in mode q1. A

shift from limit cycle to a stable node in q1 can be achieved by adapting the parameters

in Eq. 4.33 until these satisfy the existence condition of mode q1 (see Table 4.3). In this

case, increasing Km2 (deduced from α1), Km3, and/or k6 leads to a stable node, but a

sufficient decrease in k4 provides the same answer. These restrictions assist in parameter

estimation by reducing the parameter search space. Additional experimental data of

I-SMAD (to determine the behavior of x7) would be desirable, but are unfortunately

not available. More details about the TGF-β1 pathway, e.g. the trajectories in the

phase space, would rule out even more parameters by imposing more inequalities on the

parameter values. As shown above, qualitative PWA analysis enables the researcher to do

predictions of a system without having quantitative information. Researchers can profit

from this by designing new experiments to explore specific parts of the dynamics. Besides

assisting parameter estimation and experimental design, a third advantage is the analysis

of pathological phenomena. Diseases can lead to alterations in the dynamics of a system.

If a healthy situation is compared with the diseased state, one can verify which parameters

are changed in the system and examine those in more detail.

The nonlinear functions have been approximated with continuous PWA functions, ex-

cept for Hill kinetics. The piecewise-constant approach to model Hill functions is a rough

approximation and could result in some discrepancies at the vertices. Smoother continu-

ous (but not necessarily differentiable) functions could prevent these phenomena [11]. In

this chapter, a non-continuous PWA function was incorporated to keep the analysis of the

TGF-β1 pathway as simple as possible. Therefore, the choice has been made to approxi-

mate the Hill function with two piecewise-constant segments instead of the three-segment

PWA approach.

64

5Signal Transduction of the Unfolded Protein

Response

5.1 Introduction

Proteins are large biomolecules and built from various types of amino acids, which are

organic molecules that contain both an amino and a carboxyl group. The formation of

proteins takes place by annealing amino acids on specialized particles in the cell, called

ribosomes. The sequence of amino acids determines the primary shape of the protein (con-

formation), but sometimes the protein needs additional folding and other post-processing

steps before it can be used. The fate of proteins determines the location of folding and

post-processing: proteins that remain inside the cell are folded in the cytosol, extracellular

proteins are processed in the endoplasmic reticulum (ER). The ER contains a plethora of

specialized proteins that assist in protein folding, but are highly sensitive to reactive oxy-

gen species (ROS) [168]. This makes protein folding in the ER a possible target for ROS

and could even result in programmed cell death (apoptosis) [138]. It has therefore been

suggested that aging could have a significant impact on protein folding in the ER as well.

This chapter will study protein folding of the von Willebrand factor (vWF). The vWF is

a protein that mediates adhesion of platelets to sites of vascular injury [133]. Secretion

of the vWF is an important physiological function of endothelial cells. The focus will be

on the control part of the protein folding process that initiates processes to reduce stress

in the ER: the unfolded protein response (UPR). First we propose a mathematical model

of the UPR. This model is characterized by the many unknown parameters. Therefore

qualitative analysis can be applied to do some statements about the dynamics.

5.2 Protein Folding of the von Willebrand Factor

One of the cell types that produces the vWF is the endothelial cell. The vWF is also

folded in the ER of its cell. After correct folding, additional post-processing takes place

65

Chapter 5

in the Golgi apparatus. The Golgi apparatus is a system of stacked, membrane-bounded,

flattened sacs involved in modifying, sorting, and packaging macromolecules for secre-

tion [1]. Several pro-vWF molecules are linked here in a tail-to-tail configuration to a

single vWF protein. After the Golgi apparatus, the vWF is stored in Weibel-Palade

bodies which secrete the vWF in response to external signals. The individual steps of

post-translational processing of the vWF will be described in more detail below.

5.2.1 Translation and Translocation

Proteins are formed on a single or multiple ribsomes with an annealing rate of 20 amino

acids per second per ribosome [1]. This process is called translation. The pro-vWF

molecules contain a special signal peptide, which is a short sequence of amino acids that

determines the eventual location of a protein in the cell during translation. This signal

peptide is recognized by a signal-recognition particle (SRP) which is bound to a special

SRP receptor on the ER. The translated protein is subsequently transported (translocated)

in the ER. Immediately after transport, the protein Binding Protein (BiP) binds to the

hydrophobic regions of the surface of an unfolded protein (uPr) to maintain the protein

in a folding-competent state. After addition of adenosine 5’-triphosphate (ATP, principal

carrier of chemical energy in cells), uPr is released from BiP [34, 47] which triggers the

successive steps in protein folding.

5.2.2 Protein Folding

Protein folding starts with adding a oligosaccharide structure, a collection of sugar resi-

dues, to the uPr. This so-called N -glycosylation has four roles [38]:

1. It defines the attachment area for the surface of the protein,

2. The attachment area is shielded from surrounding proteins,

3. It stabilizes the conformation,

4. It provides a biological timer on the folding status of the protein.

Two sugar residues are clipped off the uPr before it enters the CNX/CRT cycle, which

folds the protein in the right configuration. Protein Disulfide Isomerase (PDI) catalyze

several folding steps in this procedure [49, 136].1 After pro-vWF is folded, it is monitored

whether the protein is folded correctly. If not, the enzyme uridine diphosphate (UDP)-

glucose:glycoprotein glucosyl transferase (UGGT) adds new sugar residues to the protein.

This targets the pro-vWF for another folding cycle in the CNX/CRT machinery. After

several cycles the protein is folded correctly. Misfolded or incompletely assembled proteins

are retained in the ER, either bound to BiP [90, 146] or in aggregates [34] that are

1Disulfide bonds are important for stabilization of tertiary structure and for their assembly into mul-timeric structures [152]. PDI is essential in facilitating disulfide bond-dependent folding [139].

66

Section 5.3

subsequently degraded. Therefore, pro-vWF exit from the ER can only occur when folding

and the pro-vWF assembly are successfully completed. The next step is to link two pro-

vWF molecules to construct a dimer, a structure composed of two protein of the same

kind. Again the CNX/CRT cycle is involved. The correctly folded dimer is transported to

the Golgi apparatus in which it forms the final product: the vWF. Fig. 5.1 is a graphical

representation of the folding of the vWF [133].

5.3 The Unfolded Protein Response

Two quality control mechanisms are incorporated in the folding process to ensure high

yields of correctly folded proteins [155]:

1. The protein folding procedure is only continued if sufficient ATP is present to

release BiP from the uPr [43, 44].

2. Folding occurs in several cycles. If the uPr has passed the CNX/CRT cycle several

times, without proper folding, the uPr is targeted for ER associated degradation

(ERAD).

Errors in these two processes are detected and trigger the UPR. The UPR acts on various

processes to reduce ER stress, including promotion of ERAD, increased expression of BiP,

decrease in transcription and translation of proteins, and stimulation of the antioxidant

response.

5.3.1 Signal Transduction in the UPR

The UPR is activated upon deficiency of free BiP, i.e., BiP not bound to uPrs, which trig-

gers the stress sensors. BiP maintains these stress sensors in the inactive state [63, 80],

so that BiP insufficiency initiates the UPR. In the mammalian UPR, three stress sensors

are involved: inositol requiring kinase-1 family (Ire1α and Ire1β), activating transcrip-

tion factor 6 (ATF6), and (PKR)-like endoplasmic reticulum kinase (PERK) [148, 149].

The latter two stress sensors are predominant in mammals under normal physiological

conditions [103]. Although the stress sensors both respond to BiP deficiency, there is a

difference in lag time before they become fully activated [140]. First, activation of the

PERK pathway leads to the phosphorylation of PERK. This activated PERK phosphory-

lates translation initiation factor eIF2α [62] that inhibits the translation of uPr. Another

role of eIF2α is the formation of the protein GADD34 which assists in the dephosphory-

lation of eIF2α [61] and can be seen as negative feedback. Second, ATF6 is translocated

to the Golgi complex and cleaved. The cleaved ATF6 contributes to the transcriptional

activation of BiP [148, 149]. Fig. 5.2 shows graphically how signal transduction in the

mammalian UPR takes place. Note that the precise interaction between BiP and uPr

67

Chapter 5

amino acids

ribos

ome

receptor

BiP

BiPtrans

latio

n

UPR

BiP

UPR

sign

altr

ansd

ucti

onUPR

+ATP

G M

N-glycosylated uPr

Erp

57

CN

X

G M

foldingcycl

e+

PDI

aggregate

G

cor rectly folded

ER

AD

UPR

Proteindegradation

+ATP

Golgipost-processing

correctlyfolded protein

Figure 5.1 – Schematical overview of protein folding and post-processing of the von Wille-brand factor.

68

Section 5.4

is lumped in the production and degradation of uPr. ERAD and folding by means of

BiP/PDI are not separately included in this model.

5.4 Mathematical Model of Signal Transduction dur-

ing the UPR

We can conclude from the previous chapter that the UPR is indispensable for accurate

protein folding. An interesting part of this regulation process is the signal transduction of

the UPR and how it affects the protein folding process. Two related modeling attempts

have been reported. First, a model of the UPR in yeast (Saccharomyces Cerevisiae)

was created [106]. The UPR response in this organism differs considerably from the

mammalian UPR [103]. Namely, S. cerevisiae controls the UPR pathway mainly by one

”stress sensing” branch: the Ire1p-pathway. Second, a stochastic model of the heat shock

process in the cytosol was created by [135], which is closely related to the UPR. Most

parameter values in this model were chosen arbitrarily, so the results were of limited

physiological significance. A kinetic deterministic model was constructed of the signal

transduction pathways in the UPR, based on the physiology of subsection 5.3.1:

dx1

dt= f1(x4)− f2(x1, x7), (5.1)

dx2

dt= f3(x1)− f4(x2), (5.2)

dx3

dt= −f5(x2, x3) + f6(x4, x5), (5.3)

dx4

dt= f5(x2, x3)− f6(x4, x5), (5.4)

dx5

dt= f7(x4)− f8(x5), (5.5)

dx6

dt= f9(x1)− f10(x6), (5.6)

dx7

dt= f10(x6)− f11(x7), (5.7)

x1 uPr,

x2 PERK,

x3 unphosphorylated eIF2α,

x4 phosphorylated eIF2α,

x5 GADD34,

x6 ATF6,

x7 spliced ATF6.

69

Chapter 5

ATF6

unfoldedprotein

PERK

eIF2α eIF2α

GADD34

P

ATF6

Substrate

Splicedsubstrate

Phosphor

Degradationproducts

Transcriptional control

Reaction

Transcriptionalinhibition


Delayedtranslation

P

x1

x2

x3 x4

x5

x6

x7

f1 f2

f3

f4

f5

f6

f7f8

f9

f10

f11

Figure 5.2 – Interaction graph of the PERK and ATF6 branches in the mammalian UPR.The molecular interaction between BiP and the uPr is omitted for sake of simplicity; theinfluence of attenuation of translation by PERK and the transcriptional activation of BiPby ATF6 are represented by inhibition of uPr formation and promotion of uPr degradation,respectively.

70

Section 5.4

The corresponding rates are

f1(x4) =k1KI

KI + x4

, (5.8)

f2(x1, x7) =k2x1x7

Km1 + x7

, (5.9)

f3(x1) = k3x1, (5.10)

f4(x2) = k4x2, (5.11)

f5(x2, x3) =k5x2x3

Km2 + x2

, (5.12)

f6(x4, x5) =k6x4x5

Km3 + x5

, (5.13)

f7(x4) = k7x4, (5.14)

f8(x5) = k8x5, (5.15)

f9(x1) = k9x1, (5.16)

f10(x6) = k10x6, (5.17)

f11(x7) = k11x7, (5.18)

k1 maximal rate of uPr production,

k2 maximal degradation rate of uPr,

k3 induction rate constant of uPr on PERK activation,

k4 degradation rate constant of PERK,

k5 maximal phosphorylation rate of eIF2α,

k6 maximal dephosphorylation rate of phosphorylated eIF2α,

k7 induction rate constant of phosphorylated eIF2α on GADD34,

k8 degradation rate constant of GADD34,

k9 induction rate constant of uPr on ATF6 activation,

k10 ATF6 splicing rate constant,

k11 degradation rate constant of ATF6,

KI Michaelis constant of inhibiting uPr production,

Km1 Michaelis constant of uPr degradation,

Km2 Michaelis constant of eIF2α phosphorylation,

Km3 Michaelis constant of eIF2α dephosphorylation.

The original model consists of seven state equations and should be reduced to make it

more comprehensible for analysis. Therefore we assumed the following:

• Activation of GADD34 effect takes place (k7) after ATF6 has been activated [140]; it

only plays a role for prolonged activation of the UPR, under severe stress conditions.

71

Chapter 5

Therefore a constant rate of eIF2α dephosphorylation was assumed. Thus dx5

dt= 0

and f adapted6 (x4) = k]

6x4.

• Total eIF2α pool remains constant, which is mathematically represented by x3 +

x4 = 1 for normalized concentrations.

• uPr activates ATF6 splicing directly, but slowly, with a rate k]2.

• Association and dissociation rates of BiP with PERK is fast [149], so x2 is replaced

by a linear function of x1.

These assumptions reduced the state equations with the quasi-steady-state approximation

and were subsequently normalized to a third-order model.2

dx∗1dt

=k∗1K

∗I

K∗I + x∗4

− k∗2x∗1x∗7

K∗m1 + x∗7

(5.19)

dx∗4dt

=k]

5x∗1(1− x∗4)

K∗m2 + x∗1

− k]6x∗4 (5.20)

dx∗7dt

= k]2x∗1 − k∗11x

∗7 (5.21)

In remainder of this chapter, the ∗ and ] were omitted to make the text more readable.

Fig. 5.3 visualizes the modes of the PWA approximation.

5.5 Qualitative Analysis

The complete qualitative analysis will not be discussed in detail, since this has already

been done in chapters 3 and 4.

5.5.1 From Nonlinear to Piecewise-Affine

Eqs. 5.19 - 5.21 contain three nonlinear equations, i.e.,

f adapted1 (x4) =

k1KI

KI + x4

, (5.22)

f adapted2 (x1, x7) =

k2x1x7

Km1 + x7

, (5.23)

f adapted5 (x1, x4) =

k5x1(1− x4)

Km2 + x1

, (5.24)

2Remark that the intermediate steps of this model reduction has been left out as the same proceduredescribed in chapter 4 is followed.

72

Section 5.5

x

x

x x =7 2α

x =4 1α

x =1 3α

x

x

x

xx

x

x

x

x

x

x

x

x

p1

p3

p15

p4

p5

p6

p7p8

p9

p10

p11 p12p13

p14

p16

p2

Figure 5.3 – Phase space of the PWA approximation of the UPR model. The phasespace is divided by three switching planes: x4 = α1 (dash), α2x1 − x7 = 0 (dot), andx1 + α3x4 = α3 (dash-dot). The vertices of the various modes are marked with crosses andnumbered (p1, . . . , p16).

73

Chapter 5

which were reformulated as the following PWA functions

ϕ1(x4) =

k1(1− x4

α1) if x4 < α1,

0 if x4 ≥ α1,(5.25)

ϕ2(x1, x7) =

k2x1 if α2x1 − x7 < 0,

k2x7

α2, if α2x1 − x7 ≥ 0,

(5.26)

ϕ3(x1, x4) =

k5x1

α3if x1 + α3x4 < α3,

k5(1− x4) if x1 + α3x4 ≥ α3,(5.27)

respectively. The PWA model is in matrix form given by Eq. 4.25. The coefficient matrices

Ai and Bi are given in Table 5.1. The invariants of the eight modes (q1, . . . , q8) are listed

in Table 5.2 and visualized in Fig. 5.4.

5.5.2 Equilibrium Points in the UPR Model

Determining the position of the equilibrium points in the model is the next step in the

procedure.

Proposition 5.5.1. The UPR model has a single steady-state in mode q1, q3, q5 or q7.

The steady-state solutions for all modes have been calculated, see Table 5.3. The equi-

librium points have to satisfy the invariants in Table 5.2, yielding four possible equilibrium

points in modes q1, q3, q5 and q7 with existence conditions

α2 <k9

k11

∧((

α1 ≤ k5

k5 + k6

)∨

(α1 >

k5

k5 + k6

∧ α3 >k1(α1(k5 + k6)− k5)

α1k2k6

)),

(5.28)

α2 >k9

k11

∧((

α1 ≤ k5

k5 + k6

)∨

(α1 >

k5

k5 + k6

∧ α3 >α2k1k11(α1(k5 + k6)− k5)

α1k2k6k9

)),

(5.29)

α2 <k9

k11

∧ α1 >k5

k5 + k6

∧ α3 <k1(α1(k5 + k6)− k5)

α1k2k6

, (5.30)

α2 >k9

k11

∧ α1 >k5

k5 + k6

∧ α3 <α2k1k11(α1(k5 + k6)− k5)

α1k2k6k9

, (5.31)

respectively. The sets of these existence conditions cover all parameter combinations and

74

Section 5.5

Table 5.1 – Coefficient matrices of the eight modes of the UPR model.

Mode A-matrix B-matrix

q1 A1 =

−k2 − k1

α10

k5

α3−k6 0

k9 0 −k11

B1 =

k1

00

q2 A2 =

−k2 0 0

k5

α3−k6 0

k9 0 −k11)

B2 =

000

q3 A3 =

0 − k1

α1− k2

α2k5

α3−k6 0

k9 0 −k11

B3 =

k1

00

q4 A4 =

0 0 − k2

α2k5

α3−k6 0

k9 0 −k11

B4 =

000

q5 A5 =

−k2 − k1

α10

0 −(k5 + k6) 0k9 0 −k11

B5 =

k1

k5

0

q6 A6 =

−k2 0 00 −(k5 + k6) 0k9 0 −k11

B6 =

0k5

0

q7 A7 =

0 − k1

α1− k2

α2

0 −(k5 + k6) 0k9 0 −k11

B7 =

k1

k5

0

q8 A8 =

0 0 − k2

α2

0 −(k5 + k6) 0k9 0 −k11

B8 =

0k5

0

75

Chapter 5

Mode q1 Mode q2

Mode q3 Mode q4

(a) (b)

(c) (d)

Figure 5.4 – Modes of the UPR model: (a) mode q1, (b) mode q2, (c) mode q3, and (d)mode q4.

76

Section 5.5

Mode q5 Mode q6

Mode q7 Mode q8

(e) (f)

(g) (h)

¿ continuation of Fig. 5.4 À (e) mode q5, (f) mode q6, (g) mode q7, and (h) mode q8.

77

Chapter 5

Table 5.2 – Invariants of all modes.

Mode Invariant

q1 x4 < α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 < α3

q2 x4 ≥ α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 < α3

q3 x4 < α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 < α3

q4 x4 ≥ α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 < α3

q5 x4 < α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 ≥ α3

q6 x4 ≥ α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 ≥ α3

q7 x4 < α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 ≥ α3

q8 x4 ≥ α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 ≥ α3

have no overlap; no multiple equilibrium points can coexist. Calculation of Lyapunov

stability and symbolic eigenvalues could not be performed on this system. Therefore no

statements could be made about the nature and stability of these equilibria.


Proposition 5.5.2. Transition analysis shows that 12 transition graphs are feasible in

UPR model, which yields 9 qualitatively different transition graphs.

Three switching planes with normal vectors n1→2 = [0, 1, 0]T , n1→3 = [α2, 0,−1]T

and n1→5 = [1, α3, 0]T divide the phase space into eight different modes and 16 ver-

tices (p1, . . . , p16, see Fig. 5.3). The mode transitions at each vertex is calculated with

Eq. 3.22 and the coefficient matrices in Table 5.1. The transitions and their parameter

constraints were deduced similarly. The results are summarized in Tables 5.4 and 5.5; the

*-superscript in Table 5.5 indicates the complementary set of a specific set.

Table 5.4 shows that five inequality sets (Γ1, . . . , Γ5) are required to derive all possible

transition graphs, but some inequality sets belong to a subset of another inequality con-

straint. For example, Γ1 ⊂ Γ2 means that Γ1 is only valid if Γ2 is satisfied. This implies

that {Γ1, Γ2}, {Γ∗1, Γ2}, and {Γ∗1, Γ∗2} are possible combinations. Other subsets are:

Γ5 ⊂ Γ1 ⇒ {Γ5, Γ1}, {Γ∗5, Γ1}, {Γ∗5, Γ∗1}, (5.32)

Γ5 ⊂ Γ2 ⇒ {Γ5, Γ2}, {Γ∗5, Γ2}, {Γ∗5, Γ∗2}, (5.33)

Γ4 ⊂ Γ3 ⇒ {Γ3, Γ4}, {Γ∗3, Γ4}, {Γ∗3, Γ∗4}. (5.34)

These subsets reduce the number of possible combinations from 32 = (25) to 12. The

transition graphs have been plotted in Fig. 5.5 and show that the 12 inequality constraints

78

Section 5.5

Tab

le5.

3–

Equ

ilibr

ium

poin

tsin

the

mod

esof

the

UP

Rm

odel

.

Mode

Equilib

rium

poi

nt

(x1,x

4,x

7)

q 1

(α

1α

3k1k6

k1k5+

α1α

3k2k6,

α1k1k5

k1k5+

α1α

3k2k6,

α1α

3k1k6k9

k11(k

1k5+

α1α

3k2k6)

)

q 2in

valid

q 3

(α

1α

2α

3k1k6k11

α2k1k5k11+

α1α

3k2k6k9,

α1α

2k1k5k11

α2k1k5k11+

α1α

3k2k6k9,

α1α

2α

3k1k6k9

α2k1k5k11+

α1α

3k2k6k9

)

q 4in

valid

q 5

( k1(α

1(k

5+

k6)−

k5)

α1k2(k

5+

k6)

,k5

k5+

k6,

k1k9(α

1(k

5+

k6)−

k5)

α1k2k11(k

5+

k6)

)

q 6in

valid

q 7

( α2k1k11(α

1(k

5+

k6)−

k5)

α1k2k9(k

5+

k6)

,k5

k5+

k6,

α2k1(α

1(k

5+

k6)−

k5)

α1k2(k

5+

k6)

)

q 8in

valid

79

Chapter 5

Table 5.4 – Inequality sets that determine the transitions.

Name Inequality set

Γ0 k1 < 0 ⇒ infeasible

Γ1 α2 > α3

k1

Γ2 α3 < k1

Γ3 α1 < α3−k5

α3−k5−k6∧

((k5 < 1 ∧ α3 < k5) ∨ (k5 > 1 ∧ α3 < 1)

)

Γ4 k5 > 1 ∧ α1 < k5−1k5+k6

Γ5 α2 > 1k1

lead to 9 qualitatively different transition graphs. The arrows in Fig. 5.5 represent the

mode transitions. For example, Fig. 5.5(a) shows that trajectories from q6 can only leave

mode q6 by entering q2. If the parameter constraints of Fig. 5.5(c) are taken, trajectories

can also move in mode q5.

Proposition 5.5.3. Mode transitions q2 → q1, q4 → q2, q4 → q3, q6 → q2, q8 → q4, and

q8 → q6 are always present.

Table 5.5 shows that the inequality sets at the vertices of mode transitions q1 → q2,

q2 → q4, q2 → q6, q3 → q4, q4 → q8, and q6 → q8 are all equal to Γ0 which cannot

be satisfied. Consequently, the mode transitions in the opposite direction are therefore

always valid.

5.5.4 Comparison with Experimental Data

Experimental results of the mammalian UPR are scarce and have recently been obtained

by [36]. Various types of ER stress were applied to suddenly increase and maintain a high

level of uPr (x1) to induce the UPR. The data show that ER stress leads to a quick response

of the PERK branch (x4) if stress-inducing agents like thapsigargin (Tg) or dithiothreitol

(DTT) are used. Experiments with tunicamycin (Tm) and Tg show a delayed response of

the ATF6 pathway (x7) , see Fig. 5.6. This lag in time is in complete agreement with work

of Rutkowski and co-workers [140]. The experimental findings can be matched with the

transition graphs. A large initial amount of uPr (x1) and low activity levels of eIF2α (x4)

and spliced ATF6 (x7) can be taken as point of departure for the model. Mode q7 would

therefore be the most appropriate initial point to simulate the experiment of DuRose

[36]. According to Fig. 5.6, x4 should increase prior to a rise in x7, which corresponds to

80

Section 5.5

Tab

le5.

5–

Tra

nsit

ions

atth

eve

rtic

esan

dco

rres

pond

ing

para

met

erco

nstr

aint

s.

Tra

nsi

tion

p 1p 2

p 3p 4

p 5p 6

p 7p 8

p 9p 1

0p 1

1p 1

2p 1

3p 1

4p 1

5p 1

6

q 1→

q 2×

××

××

Γ0×

×Γ

0×

Γ0

Γ0

××

××

q 1→

q 3Γ∗ 0×

Γ1×

×Γ

0×

×Γ

0×

××

××

××

q 1→

q 5×

×Γ

2×

Γ2×

××

Γ0×

×Γ

0×

××

×q 2→

q 4×

××

××

Γ0×

×Γ

0×

××

×Γ

0×

×q 2→

q 6×

××

××

××

×Γ

0×

×Γ

0×

Γ0

×Γ

0

q 3→

q 4×

××

××

Γ0

Γ0×

Γ0×

××

××

××

q 3→

q 7×

Γ2

Γ2×

××

Γ0×

Γ0×

××

××

××

q 4→

q 8×

××

××

×Γ

0×

Γ0×

××

×Γ

0×

×q 5→

q 6×

××

××

××

×Γ

3Γ

4×

Γ3

Γ4

××

×q 5→

q 7×

×Γ

1Γ

5×

××

×Γ

0Γ

0×

××

××

×q 6→

q 8×

××

××

××

×Γ

0Γ

0×

××

Γ0

Γ0

×q 7→

q 8×

××

××

×Γ

3Γ

4Γ

3Γ

4×

××

××

×

81

Chapter 5

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , , *

q1

q2

q3

q4

q5

q6

q7

q8

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , ,

(a)

Figure 5.5 – Transition graph for a given set of inequality constraints.

a transition of q7 → q8. Figs. 5.5(c), (f) and (i) do not have this mode transition so the

parameter constraints with Γ∗3∧Γ∗4 are not valid. From a biological perspective, one expects

that the main aim of the UPR is to reduce x1, x4 and x7 to a minimum which corresponds

to mode q3, see Fig.5.5(c). However, the sustained ER stress in the experiments of [36] by

artificially increasing the uPr levels creates a different response. The equilibrium point

in mode q5 has the most similarities with large levels of uPr, ATF6 and phosphorylated

PERK, although mode q6 would intuitively have been more appropriate if an equilibrium

point had been located in that mode. An equilibrium point in q5 can only be obtained if

the parameters satisfy the restrictions in Eq. 5.30, which reduces the number of possible

parameter values. The parameter constraints in Eq. 5.30 also rule out inequality set Γ4 in

Table 5.4, implying that the qualitative transition graphs in Fig. 5.5(a), (b) and (e) are no

valid options. As mentioned before, one of the roles of the UPR is to reduce the amount

of uPr in the ER and, as a consequence, the levels of phosphorylated PERK and spliced

ATF6 to a minimum. In the PWA model, this is represented by mode q3. Experimental

data [36] do not reproduce this physiologically relevant situation due to limitations in the

current measuring techniques. Therefore qualitative analysis can give additional insights

by exploring the qualitative transition graphs and to predict the outcome. Selection

of the correct initial conditions on basis of these qualitative transition graphs assists in

experimental design.

82

Section 5.6

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, , , , *{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , *, *,

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , *, *, *

q1

q2

q3

q4

q5

q6

q7

q8

q1

q2

q3

q4

q5

q6

q7

q8

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , *,

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , *, * { *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, *, , , *

q1

q2

q3

q4

q5

q6

q7

q8

q1

q2

q3

q4

q5

q6

q7

q8

(b) (c)

(d) (e)

¿ continuation of Fig. 5.5 À The gray arrows emphasize the change in transition directionscompared to Fig. 5.5(a).

83

Chapter 5

q1

q2

q3

q4

q5

q6

q7

q8

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, , *, *, *

q1

q2

q3

q4

q5

q6

q7

q8

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, , , *, *

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, *, , *, *

q1

q2

q3

q4

q5

q6

q7

q8

q1

q2

q3

q4

q5

q6

q7

q8

{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, *, *, *, *

(f) (g)

(h) (I)

¿ continuation of Fig. 5.5 À The gray arrows emphasize the change in transition directionscompared to Fig. 5.5(a).

84

Section 5.6

0

20

40

60

80

100

0 1 2 3 4 5

time (hours)

%cl

eaved

AT

F6

Tg

Tm

DTT

0

20

40

60

80

100

0 1 2 3 4 5

time (hours)

%P

ER

Kphosp

hory

late

d

Tg

Tm

DTT

(a)

(b)

Figure 5.6 – Expression profiles of (a) phosphorylated PERK (indication of phosphorylatedeIF2α, x4) and (b) cleaved ATF6 (x7) [36]. Alternate types of ER stress were applied foreach branch of the UPR. Thapsigargin (Tg): knocks out the energy requirements for proteinfolding; tunicamycin (Tm) inhibits protein glycosylation; and dithiothreitol (DTT) disruptsor prevents protein disulfide bonding.

85

Chapter 5

5.6 Discussion

The UPR is a collection of processes that monitors the quality of protein folding in the

ER and, if necessary, takes appropriate action if this quality tends to diminish. A math-

ematical model of the signaling cascade of the mammalian UPR is created and analyzed

with the qualitative procedure from chapter 3. The analysis of the UPR model yields 9

different types of qualitative transition graphs. Several transition graphs showed qualita-

tively good resemblance with experimental data [36], while others were less likely. This

information could be used to select specific inequality sets of the parameter values, which

reduces the parameter search space during system identification, as will be shown in

chapter 6. Current methods for determining the stability and typological classification

of equilibria are inadequate for multidimensional qualitative systems, but graphical anal-

ysis of the transitions in the phase space provide some indications about the nature of

the system. Still, a solid theoretic framework would be desired. We believe that two

approaches are very promising. First the use of Surface Lyapunov functions (SuLF), in

which the stability of any PWA system can be verified by analyzing a hybrid system with

specialized surfaces [52, 53] might be an option. Second, the monotonic characteristics of

the nonlinear functions in biochemical networks have some advantageous properties, as

was shown in chapter 2, and could help in developing mathematical proofs for stability.

86

6System Identification with Parameter

ConstraintsSo far qualitative information has been used to analyze the system dynamics. However,

qualitative information is not restricted to system analysis alone. In this chapter, qualita-

tive information will be applied to improve parameter estimation by putting constraints on

the parameter values, obtained by qualitative analysis (chapter 3). These restrictions can

be included in a constrained nonlinear optimization procedure to reduce the parameter

search space for parameter estimation. A mathematical model of a biochemical oscillator

was used as a test case [16, 121].

6.1 The Biochemical Oscillator

A simple model of a biochemical oscillator was introduced by Goodwin [54], which de-

scribes the genetically regulated enzymatic conversion of a substrate into a product, see

Fig. 6.1:

dx1

dt=

k1KI

KrI + xr

3

− k2x1, (6.1)

dx2

dt= k3x1 − k4x2, (6.2)

dx3

dt= k5x2 − k6x3, (6.3)

x1 mRNA concentration,

x2 enzyme concentration,

x3 product concentration,

k1 maximum rate of mRNA production,


k3 rate constant of enzyme induction,

87

Chapter 6

DNA mRNA enzyme

substrate

product

Figure 6.1 – Model of enzymatic substrate conversion, as proposed by Goodwin [54]. Thefirst step is mRNA transcription from DNA, which leads to the formation of an enzyme.This enzyme catalyzes the conversion of a substrate into a product that inhibits the mRNAtranscription of the enzyme (bar). Note that the degradation of mRNA, enzyme and productare omitted from this graph. The degradation of the product was assumed to be enzyme-mediated [16].


k5 rate constant of product formation,


KI Michaelis constant of mRNA inhibition,


For r → ∞, the Hill equation shows ideal relay characteristics. Eqs. 6 - 8 describe

a basic oscillator, but periodic cycles are only obtained for r > 8 [39]. It has been

argued that this large value is less likely in this situation from a physiological point of

view [16]. Goodwin’s model [54] was therefore slightly modified by Bliss et al. [16]. The

latter assumed a low cooperativity coefficient (r = 1) and introduced enzyme-catalyzed

degradation of the product, described by Michaelis-Menten kinetics [111]. This resulted

in the following differential equations

dx1

dt=

k1KI

KI + x3

− k2x1, (6.4)

dx2

dt= k3x1 − k4x2, (6.5)

dx3

dt= k5x2 − k6x3

Km + x3

, (6.6)

k6 maximum rate of x3 degradation,

Km Michaelis constant of x3 degradation.

This model was assumed to represent the actual system and will be used as in silico test

case. To provide experimental data for the identification procedure, the nonlinear model

in Eqs. 6.4-6.6 was simulated over a time span of 150 units. Within this period, 50 samples

(N = 50) were collected for x1, x2 and x3 at equidistant discrete time instants. Normally

88

Section 6.2

0 50 100 1500

50

100

150

200

250

300

350

mRNA

enzyme

product

x1

σ = 2.5

x2

x3

t

Figure 6.2 – Simulated experiment of the biochemical oscillator with parameter valuesk1 = 150, KI = 1, k2 = 0.1, k3 = 0.1, k4 = 0.1, k5 = 0.1, k6 = 10, and Km = 1.5. Whitenoise with standard deviation σ = 2.5 was added to the output. The response of mRNA(x1), enzyme (x2) and product concentration (x3) are plotted as function of time t.

distributed white noise e(k) was added to each sample. This resulted in the following

equation for the measurement data (k = 1, . . . , N)

y(k) = x(k) + e(k), (6.7)

y(k): measured output.

The parameter values of the simulated model are k1 = 150, KI = 1, k2 = 0.1, k3 = 0.1,

k4 = 0.1, k5 = 0.1, k6 = 10, and Km = 1.5. Four simulations were performed with an

varying noise level that have standard deviations σ = 0, 1, 2.5 and 5. A typical plot of

the dynamics for σ = 2.5 (Fig. 6.2) shows that the model, for the given set of parameter

values, oscillates. Although the data feigns damped oscillations in Fig. 6.2, the trajectories

converge to a limit cycle over a longer time period.

89

Chapter 6

6.2 Qualitative Phase Space Analysis

6.2.1 Nonlinear to PWA Conversion

The nonlinear model in Eqs. 6.4-6.6 contains two nonlinearities that are functions of x3:

f1(x3) =k1KI

KI + x3

(6.8)

f2(x3) =k6x3

Km + x3

. (6.9)

These functions are approximated with two PWA functions (see chapter 3):

ϕ1(x3) =

k1

(1− x3

α1

)if x3 < α1,

0 if x3 ≥ α1,(6.10)

ϕ2(x3) =

k6x3

α2if x3 < α2,

k6 if x3 ≥ α2.(6.11)

After normalization, these approximations yield the following set of state equations

Φi(x) =dx

dt= Aix + Bi, (6.12)

with x = [x∗1, x∗2, x

∗3]

T of which the coefficient matrices Ai and Bi are listed in Table 6.1.1

The parameters of the normalized PWA approximation are k∗1 = k1

xmax1

, K∗I = KI

xmax3

, k∗2 = k2,

k∗3 =k3xmax

1

xmax2

, k∗4 = k4, k∗5 =k5xmax

2

xmax3

, k∗6 = k6

xmax3

, and K∗m = Km

xmax3

. A graphical representation

of the state space is displayed in Fig. 6.3(a).


The next step is to determine the transitions between the three modes. As the modes are

stacked on one another (Fig. 6.3), maximal four mode transitions are possible: q1 → q2,

q2 → q3, q2 → q1 and q3 → q2. Transition analysis (chapter 3) shows that three inequality

constraints are sufficient to describe all transitions at the vertices of the modes. These

constraints are listed in Tables 6.2 and 6.3 and visualized in Fig. 6.3(b). It can be deduced

from Fig. 6.3(b) that the trajectories are always directed towards q1 at vertices p1, . . . , p4.

Graphical analysis of Fig. 6.3(b) shows that a limit cycle is feasible if inequalities Γ1 and

Γ2 are satisfied.

1The additional assumption was made that α1 < α2; simulations showed that changing the order ofthese switching planes did not have any qualitative effect on the end results.

90

Section 6.2

Table 6.1 – Coefficient matrices and invariants of the three-mode biochemical oscillatormodel.

Mode A-matrix B-matrix Invariant

q1 A1 =

−k∗2 0 − k∗1

α1

k∗3 −k∗4 0

0 k∗5 − k∗6α2

B1 =

k∗100

x∗3 < α1 < α2

q2 A2 =

−k∗2 0 0k∗3 −k∗4 0

0 k∗5 − k∗6α2

B2 =

000

α1 < x∗3 < α2

q3 A3 =

−k∗2 0 0k∗3 −k∗4 00 k∗5 0

B3 =

00−k∗6

α1 < α2 < x∗3

x

x

x

x

x

x

x

x

p1

p2

p3 p4

p5 p6

p7

x1

x2

x3

x1max

x2max

x3max

p8

Γ1

Γ1

Γ2

Γ2

Γ0*

Γ0*

Γ0*

Γ0*

x1

x2

x3

x1max

x2max

x3max

q3 α1

α2

q2

q1

(a) (b)

*

*

*

*

*

*

Figure 6.3 – (a) 3D phase space of a model of the biochemical oscillator. The switchingplanes divide the phase space in three modes (q1, q2, and q3) (b) Mode transitions withcorresponding sets of constraints on the parameter values.

91

Chapter 6

Table 6.2 – Mode transitions at the vertices and corresponding symbolic constraints.

Transition p1 p2 p3 p4 p5 p6 p7 p8

q1 → q2 Γ0 Γ0 × × Γ1 Γ1 × ×q2 → q3 × × Γ0 Γ0 × × Γ2 Γ2

Table 6.3 – Inequality sets of the biochemical oscillator model.

Name Inequality set PWA Inequality set nonlinear

Γ0 absent absent

Γ1α1

α2<

k∗5k∗6

1.20KI

Km<

xmax2 k5

k6

Γ2 k∗5 > k∗6 xmax2 k5 > k6

6.2.3 Constrained Nonlinear Parameter Estimation

Transition analysis provides the inequality constraints on the parameter values to guar-

antee a limit cycle. These constraints are converted in parameters of the nonlinear model

(see Table 6.3) with the information from subsection 6.2.1 and chapter 3 to make the

qualitative information compatible for the nonlinear estimation procedure. The values of

xmax2 can roughly be obtained from Fig. 6.2 as a slightly larger value than the maximum

of x2: 200. The nonlinear function is discretized to a data set x(k) in such a way that the

cost function J(θ) can be defined as

J(θ) =N∑

k=1

(x(k, θ)− y(k)

)2

, (6.13)

θ set of estimated parameter values,

N number of measurements,

y(k) experimental data.

92

Section 6.3

Table 6.4 – Results of the parameter identification procedure with the traditional approachand the adapted one that uses qualitative information.

Parameter Real Traditional Qualitative procedure

k1 150 136.25± 40.03 149.96± 11.83

KI 1 0.992± 0.36 1.161± 0.42

k2 0.1 0.101± 0.025 0.110± 0.022

k3 0.1 0.147± 0.081 0.116± 0.035

k4 0.1 0.151± 0.096 0.116± 0.038

k5 0.1 0.094± 0.039 0.107± 0.018

k6 10 9.27± 4.65 10.85± 2.06

Km 1.5 1.27± 0.73 1.62± 0.19

The qualitative constraints in Table 6.3 have been incorporated in the following optimiza-

tion problem

θ = arg minθ>0

J(θ) subject to

k6 − 200k5 < 0,

1.20KI

Km

− 200k5

k6

< 0.

(6.14)

Problem 6.14 is solved in MATLAB with the function fmincon. This parameter iden-

tification procedure iss compared with a traditional unconstrained nonlinear estimation

procedure (lsqnonlin). Both methods require an initial estimate of the parameter values

and therefore several sets of initial estimates θini are selected as input, ranging from 0.5×to 1.5× θ. For each θini, the optimal solution for θ has been calculated. A data set with

σ = 2.5 is used to test both identification approaches. The results are listed in Table 6.4;

estimated k1’s are plotted in Fig. 6.4 for various θini. The parameter values are, in gen-

eral, well estimated for both methods if the initial estimate of the parameters were chosen

close to the actual parameter values (see Fig. 6.4). However, the constrained procedure

performed better if this estimate was less accurate, yielding a smaller standard deviation

on most parameter values (see Table 6.4). We therefore may conclude that incorporating

additional qualitative information in the parameter estimation procedure in this example

outperforms traditional nonlinear identification procedures in terms of accuracy.

93

Chapter 6

0.5 1.50

20

40

60

80

100

120

140

160

180

200

real

traditional

novel

initial value of k1

esti

mat

edk 1

k1true

k1true

k1true

Figure 6.4 – Parameter estimation of k1 for various initial estimates on a data set withσ = 2.5. On the x-axis, the initial estimation is chosen as a fraction of the true value ktrue

1 .Note that the precise estimate cannot be found, which can probably be ascribed to the noisein the data set.

94

Section 6.3

6.3 Discussion

In this chapter, qualitative PWA analysis is applied to a model of a hypothetical biochem-

ical oscillator. Qualitative analysis of the PWA model showed that the approximation

of this results in a specific set of parameter restrictions for which limit cycle behavior is

guaranteed. This extra information fulfills an assisting role in the nonlinear parameter

estimation procedure and contributes to more accurate parameter estimations compared

to conventional nonlinear estimation procedures without constraints, but both procedures

still require a reasonably well initial estimate of the parameter values. In the next chapter,

it is shown that hybrid system identification can provide a fairly good initial estimate,

which can improve parameter estimation even more.

95

Chapter 6

96

7Hybrid System Identification

This chapter was based on paper [121]

In the previous chapter it was shown that the initial estimate of the parameter values

influences identification (see Fig. 6.4). An appropriate initial estimate of the param-

eters is therefore desired, but we do not know which one is appropriate. We develop a

PWA parameter estimation procedure that approximates the nonlinearities with two PWA

functions. Parameter estimation is performed by shifting the switching plane of the PWA

model, thereby classifying the data to a specific mode by minimizing the cost function of

the estimation procedure. The advantage of this method is that no initial estimation of

the parameter values is needed, unlike traditional nonlinear least-squares methods [102]

or the multiple shooting method [18] that require this information. Furthermore, this

novel parameter estimation procedure requires considerably less calculation time than

nonlinear least-squares, multiple shooting methods, and a previously proposed method

with one-step ahead prediction [171]. Again, the mathematical model of a biochemical

oscillator was used as a test case [121].

7.1 General Identification Procedure

7.1.1 Model Class

We consider the following class of discrete time systems:

x(k + 1) =

g1(x(k), θ1) if xi(k) < α,

g2(x(k), θ2) if xi(k) ≥ α,(7.1)

where x ∈ Rn, xi ∈ R denotes the ith component of the state vector x, and the switching

plane α ∈ R. Functions g1(x(k), θ1) and g2(x(k), θ2) are assumed to be smooth nonlinear

97

Chapter 7

functions of the parameters x and θ. The system in Eq. 7.1 is a bimodal hybrid system, in

which the currently active mode is determined by the value of one of the state components.

All states are measured. We assume that both modes are excited in the data set. The

identification problem for Eq. 7.1 consists of determining values of parameters (θ1, θ2 and

α) on basis of measurements x(k), with k = 1, . . . , N . The difficulty of the identification

problem stems from the fact that the switching threshold is not known a priori.

7.1.2 Identification and Classification of a Hybrid Model

For a given α we can define two sets of data:

χ1(α) = {x(k)|xi(k) < α}, (7.2)

χ2(α) = {x(k)|xi(k) ≥ α}. (7.3)

We consider the cost function J of the form:

J(α, θ1, θ2) =∑

x(k)∈χ1(α)

‖x(k + 1)− g1(x(k), θ1)‖2 +

∑

x(k)∈χ2(α)

‖x(k + 1)− g2(x(k), θ2)‖2 . (7.4)

The identification problem can now be formulated as:

{α, θ1, θ2} = arg min{α,θ1,θ2}>0

J(α, θ1, θ2). (7.5)

Note that J(α, θ1, θ2) is not continuous in α. Hence, minimization methods that require

computation of the gradient of J (i.e., steepest descent) cannot be applied. Also note

that if the value of α = αknown is known, the optimization problem in Eq. 7.5 reduces to

{θ1, θ2} = arg min{θ1,θ2}>0

J(αknown, θ1, θ2), (7.6)

which is a smooth nonlinear least squares problem, and can be solved (locally) for θ1 and

θ2 using classical methods. This automatically classifies each data point to mode q1 and

q2.

One way to solve Eq. 7.5 is by “gridding”: we choose a set αmin = α1 < α2 < . . . <

αm = αmax of values (“grid”) for α, and solve Eq. 7.6 for every value in this set. We

chose αmin = mink xi(k) and αmax = maxk xi(k). The advantage of this method is that

the optimal value of the parameter α can always be found with the required precision for

a suitably selected grid. The disadvantage is a larger computational burden. However,

98

Section 7.2

f x1 3( )

f x2 3( )

β11

β

β12

β γ21

-γ11

-γ12

q2

x3

f

β11

β

β12

β

-γ11

-γ12

q1 q2

φ1

3(

)x

α x3

q2q1 q2

φ2

3(

)x

α x3

γ22β2

2β2

β21

(a) (b) (c)

Figure 7.1 – Conversion of nonlinear to PWA for hybrid identification. (a) The two non-linear functions f1(x3) and f2(x3) were approximated by two PWA functions, namely (b)PWA approximation of f1(x3), and (c) PWA approximation of f2(x3). We assumed that theswitching plane α was located at the same position for both functions. The PWA model hasconsequently two modes: q1 and q2. The superscript of the parameters refer to the modenumber.

the data set is relatively small, so gridding can be performed.

7.2 PWA Identification of the Biochemical Oscillator

In previous chapters the PWA approximation of nonlinear functions consisted of a linear

and a constant part. This rough approximation is suitable to explore the complete phase

space, but is too rough for describing the dynamics in a certain region of the phase space.

A more refined approach is required for PWA identification and therefore we approximate

the nonlinear functions f1(x3) and f2(x3) in Eq. 6.8 and Eq. 6.9, respectively, with two

linear segments. In previous chapter, the order of threshold was not important. Therefore

the switching plane x3 = α was assumed to be identical for both functions, also to simplify

the identification and classification methods, which divide the PWA approximation in two

modes. The mode with invariant x3 < α was classified as mode q1; mode q2 was assigned

to the invariant set x3 ≥ α, see Fig. 7.1.

The corresponding continuous-time PWA model is given by the following equations:

dx

dt=

A1x + B1 if x3(t) < α,

A2x + B2 if x3(t) ≥ α,(7.7)

with x: [x1, x2, x3]T ; the coefficient matrices A1, A2, B1 and B2 are listed in Table 7.1.

99

Chapter 7

Table 7.1 – Coefficient matrices and invariant sets of the biochemical oscillator model oftwo modes.

Mode A-matrix B-matrix Invariant

q1 A1 =

−k2 0 −γ1

1

k3 −k4 00 k5 −γ1

2

B1 =

β11

0−β1

2

x3 < α

q2 A2 =

−k2 0 −γ2

1

k3 −k4 00 k5 −γ2

2

B2 =

β21

0−β2

2

x3 ≥ α

7.2.1 Methods

Since the system was composed of two linear modes, a linear least-squares problem was

solved with the MATLAB command lsqlin. The estimated parameters of the PWA

model were collected in vectors θ1 and θ2 as follows:

θ1 = [β11 , γ

11 , β

12 , γ

12 , k2, k3, k4, k5], (7.8)

θ2 = [β21 , γ

21 , β

22 , γ

22 , k2, k3, k4, k5]. (7.9)

The bootstrap method (100 iterations) [18] was applied to determine the variance of these

parameters. The grid has the following dimensions: αmin = 0, αmax = 146.6, and step size

= 0.1.

7.2.2 Results

The experimental data of the simulated nonlinear biochemical oscillator with varying

noise levels were classified and identified with the hybrid identification method. The

results are displayed in Fig. 7.2 for σ = 0 and 5, the estimated parameter values for σ =

0, 1, 2.5 and 5 are listed in Table 7.2. The classification of the data to q1 and q2 satisfies

the expectations: the measurements are assigned to q1 for low x3 values, whereas the

other data correspond to q2. Since a PWA approximation was applied, not all estimated

parameter values could be verified with the original nonlinear parameters used in the

model of biochemical oscillator, i.e., k1, KI, k6, and Km. The estimated parameter values

of k2, k3, k4 and k5 in Table 7.2 were in good agreement with the true parameter values

(Table 6.4).

100

Section 7.3

0

200

400

0

100

200

300

0 50 100 150

0

100

200

x1

x2

x3

0

200

400

0

100

200

300

0 50 100 150

0

100

200

(a)

(b)

x1

x2

x3

t

t

Figure 7.2 – Results of the hybrid identification method for noise levels of (a) σ = 0, and(b) σ = 5.

101

Chapter 7

Table 7.2 – Estimated parameter values for the four data sets with varying noise levels.

Parameter σ = 0 σ = 1 σ = 2.5 σ = 5

β11 55.077 55.236± 1.6 55.944± 4.5 52.035± 4.9

γ11 2.915 2.921± 0.2 3.097± 0.1 2.368± 0.7

β12 5.566 5.574± 0.3 5.615± 0.8 6.236± 0.9

γ12 0.220 0.218± 0.04 0.215± 0.1 0.095± 0.1

β21 7.502 7.607± 0.34 7.645± 0.9 7.734± 1.6

γ21 0.052 0.053± 0.002 0.053± 0.005 0.052± 0.01

β22 9.274 9.313± 0.1 9.289± 0.3 9.235± 0.4

γ22 0.005 0.005± 0.001 0.005± 0.002 0.004± 0.004

k2 0.111 0.111± 0.003 0.111± 0.009 0.114± 0.01

k3 0.103 0.103± 0.001 0.102± 0.002 0.102± 0.003

k4 0.103 0.103± 0.001 0.102± 0.002 0.102± 0.003

k5 0.100 0.1± 0.001 0.099± 0.002 0.098± 0.003

α 12.367 12.122± 1.2 12.510± 2.6 14.339± 4.1

7.3 Discussion

A hybrid system identification procedure was applied: a two-segment PWA approxima-

tion of the nonlinear model is fitted on the data to estimate the parameters in the two

modes. The estimates of the parameters are quite accurate and can be used for nonlinear

parameter estimation procedures. The parameters which are not linked to the nonlinear

functions can be used as initial estimate of the parameters for standard nonlinear param-

eter estimation. For the hybrid identification procedure, we assume that all states are

observable, which is not always the case. As future work, it needs to be verified whether

parameter values of the PWA approximation can be used for estimating the parameter

values of the original nonlinear functions. The challenge is to apply this technique to

larger biological systems with multiple modes.

102

8Conclusion and Discussion

For a thorough understanding of biomedical phenomena it is essential to understand

the underlying cellular dynamics. Mathematical models have shown to integrate data

and information from various sources to solve numerous biomedical research questions

[71, 85]. The expectation is that the role of such models will become more important in

the near future. Performing accurate quantitative measurements in mammalian cells re-

mains the main bottleneck in present research. Reliable data are essential for formulating

mathematical models with high predictive power. In addition, mathematical models of

bioregulatory networks often contain nonlinear functions to describe the various biologi-

cal processes. System analysis and parameter estimation of such models is a cumbersome

task, especially due to the scarcity of quantitative data. Therefore, systems biologists are

exploring alternative ways to improve system analysis and parameter estimation, also to

assist the experimentalists to design most informative experiments [95, 116, 170, 172]. So

far, existing methods are primarily applied to the analysis of a few very specific biochem-

ical networks for which experimental data are available.

8.1 Conclusions

Two research goals have been defined in the introduction.

Primary Goal − Develop mathematical procedures to extract information from

typical nonlinear biochemical models that contain little quantitative information.

Main purpose: assistance in (qualitative) system analysis and improved parameter

estimation.

103

Chapter 8

Graphical Analysis of Bistable Systems

Nonlinear ordinary differential equations (ODEs) are used to describe the dynamical be-

havior of various biochemical networks. In chapter 2, a graphical study of a specific class

of monotone systems is adapted to yield constraints on parameter values associated with

a certain desired (experimentally observed) dynamic behavior. This info could subse-

quently be used to improve parameter estimation. This graphical procedure is tested on

an existing model of extracellular matrix (ECM) remodeling [97]. The closed-loop ECM

model is converted to an open-loop system. Reclosing the loop and adding an adaptable

feedback component puts bounds on a specific parameter value to guarantee bistability,

which is expected behavior allowing constraints on the parameter values. This strategy

is limited to a restricted class of systems that contains only positive feedback circuits,

which limits the applicability of this methodology in practice.

Qualitative PWA Analysis

A qualitative method to analyze a general class of biochemical regulatory networks as

piecewise-affine (PWA) functions has been developed. Nonlinear functions are approxi-

mated with two or three PWA functions, which enables qualitative analysis of the system.

The main contribution of qualitative analysis is its practical relevance to analyze nonlinear

deterministic networks with little quantitative information. Recent work on qualitative

analysis of large biochemical networks is so far limited to nonlinear functions approxi-

mated by piecewise-constant [12, 28, 29, 99] or, very recently, ramp functions [11]. The

work in this thesis extends qualitative analysis with (multidimensional) PWA approxima-

tions; virtually all nonlinear, biochemical, deterministic models can be approximated as a

collection of PWA functions, including metabolic networks and signal transduction path-

ways. The PWA approximation divides the phase space of the nonlinear model in several

modes by means of switching planes. Qualitatively different types of dynamic behavior

appear depending on if and how the state trajectories of the system move through the

modes of the phase space. Analysis of the dynamics at the vertices of the switching planes

leads to a set of, relatively simple, inequalities for the parameters to ensure that certain

mode transitions can occur. The complexity of the analysis is primarily determined by

the number of PWA approximations (i.e., the number of nonlinearities in the model) and

the number of segments used, not by the number of differential equations. The procedure

was demonstrated by using a second order model of TGF-β1. This low order model al-

lowed easy graphical representation. General qualitative methods developed in the area of

artificial intelligence are only applicable to 2D models [126, 141, 164, 163]. Our approach

can be applied to larger systems, as was shown for the UPR model in chapter 5.

Qualitative PWA analysis enables qualitative sensitivity analysis by studying the ef-

104

Section 8.1

fect of changing a specific parameter. It provides valuable information when studying

pathological phenomena to verify what parameters can cause an observed change in sys-

tem dynamics. Traditional sensitivity analysis requires quantitative parameter estimates,

which are usually not available.

Furthermore, qualitative PWA analysis provides (relative) bounds on the parameter

values. This information can be used to reduce the parameter search space and lead to

more accurate parameter estimates than conventional methods, as shown in chapter 6.

Initial estimates can influence the outcome of parameter estimation. Therefore a hybrid

parameter estimation is developed in chapter 7 to provide a rough initial estimate of the

parameter values, which is subsequently applied to a model of the biochemical oscillator.

In short, we can conclude that qualitative PWA analysis is currently the most general

(all types of kinetics and large-scale models) method for qualitative analysis of nonlinear

biochemical networks.

Secondary Goal − Apply the developed methodologies to typical biochemical

networks that are involved in vascular aging processes.

There is a wide variety of biochemical processes that can be linked to aging in gen-

eral [21, 108] or, more specifically, vascular aging [45]. Based on limited experimental

evidence within the project, three relevant biochemical networks have been studied:

1. Remodeling of the ECM (chapter 2), for which a mathematical model has been

derived from the literature [97]. Bistability is expected for this model and only

possible if certain constraints are satisfied.

2. The TGF-β1 signaling pathway (chapter 4). An extensive literature search re-

sulted in a complex kinetic model with positive and negative feedback circuits. By

means of the quasi-steady-state approximation, this can be reduced to a second

order model. System analysis shows that, depending on the choice of parameter

values, a single stable node or a limit cycle is present. Experimental findings in

hepatocytes [125] demonstrate that oscillatory behavior can be observed. During

aging, the TGF-β1 pathway of endothelial cells become less sensitive to external

TGF-β1 [107], which can be explained by an increase in the Michaelis constants

for phosphorylation R-SMAD complex formation, Michaelis constant of I-SMAD

production, dephosphorylation rate constant of SMAD complex or a decrease in

maximal phosphorylation rate of R-SMADs. These are interesting targets for in-

depth analysis.

3. The mammalian UPR in the ER (chapter 5). A kinetic model has been build on

basis of the literature. The PERK and ATF6 branch are isolated as these signaling

pathways indirectly control the regulation of protein formation and degradation.

105

Chapter 8

Equilibrium points were identified by the qualitative analysis procedure. The time

span of the experimental validation [36] has most likely been too short to draw

conclusions about the system behavior, although model simulations coincided quite

well with the experimental data of DuRose et al. [36] for certain conditions of the

parameter values.

8.2 Future Perspectives

• It is shown in genetic regulatory networks that important system features can be

extracted from qualitative information [12, 27, 161], although the focus has primarily

been on systems with piecewise constant functions. The qualitative analysis in

this thesis is not restricted to piecewise-constant, but can deal with PWA systems

as well. Automated procedures to analyze qualitative models, like GNA [27] and

RoVerGeNe [11], are indispensable for larger systems that are common in biology.

So far the procedure has been automated, but should be extended with an automatic

verification and visualization procedure to make data from large, multidimensional

systems easier to grasp.

• The theory about the stability of PWA systems is another point of attention. Cur-

rent approaches for stability analysis of hybrid systems is very limited, and aimed

at hybrid systems with abundant quantitative information. Surface Lyapunov func-

tions (SuLF), in which the stability of any PWA system can be verified by analyzing

a hybrid system with specialized surfaces, could be a promising and solid framework

to analyze the limit cycles and equilibrium points in this thesis [52, 53]. Also exploit-

ing the monotonicity of the nonlinearities in biochemical networks can contribute

to make mathematical statements about stability.

• Sensitivity analysis is frequently applied in systems biology: each parameter is

checked for its impact on the global system dynamics by varying it within a given

parameter range. For nonlinear models, this behavior is highly dependent on the

base value of all parameters (the operating point) and how large the change is.

Qualitative analysis can assist in this process by excluding parameters that have

less significance.

• The development of the qualitative method in this thesis was motivated by biochem-

ical networks, but complicated nonlinear behavior in other engineering fields can

have the same type of functions and lack in quantitative information. Qualitative

PWA analysis can therefore be applied to other research fields as well.

106

Appendix A

Nomenclature

A.1 List of Abbreviations

ADP adenosine 5’-diphosphate

ATF6 activating transcription factor 6

ATP adenosine 5’-triphosphate

BiP binding protein

CNX calnexin

co-SMAD common-mediator SMAD

CRT calreticulin

DNA deoxyribonucleic acid

DTT dithiothreitol

ECM extracellular matrix

eIF2α eukaryotic initiation factor 2α

EMSA electrophoretic mobility shift array

ER endoplasmic reticulum

ERAD ER associated degradation

GADD34 growth-arrest and DNA damage-inducible protein

GNA genetic network analyzer

IRE1 inositol requiring kinase-1

I-SMAD inhibitory SMAD

IκB inhibitor of κB

MAPK mitogen-activated protein kinase

mRNA messenger ribonucleic acid

NF-κB nuclear factor κB

ODE ordinary differential equation

PDE partial differential equation

PDI protein disulfide isomerase

PERK (PKR)-like endoplasmic reticulum kinase

PKR protein kinase regulated by RNA

107

Nomenclature

pRS phosphorylated R-SMAD pool

PWA piecewise-affine

ROS reactive oxygen species

RoVerGeNe robust verification of gene networks

R-SMAD receptor-regulated SMADs

SMURF2 SMAD ubiquitination regulatory factor 2

SRP signal recognition particle

SuLF surface Lyapunov functions

Tg thapsigargin

TGF-β1 Transforming Growth Factor-β1

Tm tunicamycin

tIS total I-SMAD pool

tPA tissue-type plasminogen activator

tRS total R-SMAD pool

UGGT uridine diphosphate-glucose:glycoprotein glucosyl transferase

uPA urokinase-type plasminogen activator

UPR unfolded protein response

uPr unfolded protein

uRS unphosphorylated R-SMAD pool

vWF von Willebrand factor

A.2 Symbols

A.2.1 Latin

A coefficient matrix of the linear part of a state space model

B coefficient matrix of the constant part of a state space model

f(x) rate equation

Jm Jacobian matrix

J(θ) cost function

k rate constant

Km Michaelis constant

KI Michaelis constant of an inhibitor

n normal at a certain switching plane

N total number of samples

Nf total number of rate equations

Nq total number of modes

108

Nomenclature

Nx total number of states

p vertex on switching threshold

q mode of a hybrid system

r cooperativity coefficient

t time

t(x) tangent of a switching threshold

V (x) Lyapunov function

Vmax conversion rate at maximal substrate concentration

x dimensionless state variable

xeq state variable evaluated at equilibrium point

xIP intersection point of f(x) with ϕ(x)

xmax maximal value of x

y regular state variable

y(k) measured output data

A.2.2 Greek

α switching threshold

β constant in PWA approximation

γ slope of PWA approximation

Γ set of constraints

Γ∗ complementary set of Γ

ε small factor, required for the quasi-steady-state approximation

η output of a system

λ eigenvalue

ν feedback strength = ωη

σ standard deviation

θ set of true parameter values

θ set of estimated parameter values

θini initial estimate of the parameter values

ϕ(x) PWA approximation of f(x)

Φ(x) set of PWA state equations

χ data set

ω input of a system

109

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arrest requires the activity of the proteasome pathway. Exp. Cell Res., 281:190–196,

2002.

125

Summary

126

Summary

The human body is composed of a large collection of cells,“the building blocks of life”. In

each cell, complex networks of biochemical processes contribute in maintaining a healthy

organism. Alterations in these biochemical processes can result in diseases. It is therefore

of vital importance to know how these biochemical networks function. Simple reasoning

is not sufficient to comprehend life’s complexity. Mathematical models have to be used

to integrate information from various sources for solving numerous biomedical research

questions, the so-called systems biology approach, in which quantitative data are scarce

and qualitative information is abundant.

Traditional mathematical models require quantitative information. The lack in ac-

curate and sufficient quantitative data has driven systems biologists towards alternative

ways to describe and analyze biochemical networks. Their focus is primarily on the anal-

ysis of a few very specific biochemical networks for which accurate experimental data are

available. However, quantitative information is not a strict requirement. The mutual

interaction and relative contribution of the components determine the global system dy-

namics; qualitative information is sufficient to analyze and predict the potential system

behavior. In addition, mathematical models of biochemical networks contain nonlinear

functions that describe the various physiological processes. System analysis and parame-

ter estimation of nonlinear models is difficult in practice, especially if little quantitative

information is available.

The main contribution of this thesis is to apply qualitative information to model and

analyze nonlinear biochemical networks. Nonlinear functions are approximated with two

or three linear functions, i.e., piecewise-affine (PWA) functions, which enables qualitative

analysis of the system. This work shows that qualitative information is sufficient for the

analysis of complex nonlinear biochemical networks. Moreover, this extra information can

be used to put relative bounds on the parameter values which significantly improves the

parameter estimation compared to standard nonlinear estimation algorithms. Also a PWA

parameter estimation procedure is presented, which results in more accurate parameter

estimates than conventional parameter estimation procedures. Besides qualitative analysis

with PWA functions, graphical analysis of a specific class of systems is improved for

a certain less general class of systems to yield constraints on the parameters. As the

applicability of graphical analysis is limited to a small class of systems, graphical analysis

is less suitable for general use, as opposed to the qualitative analysis of PWA systems.

The technological contribution of this thesis is tested on several biochemical networks

127

Summary

that are involved in vascular aging. Vascular aging is the accumulation of changes respon-

sible for the sequential alterations that accompany advancing age of the vascular system

and the associated increase in the chance of vascular diseases. Three biochemical networks

are selected from experimental data, i.e., remodeling of the extracellular matrix (ECM),

the signal transduction pathway of Transforming Growth Factor-β1 (TGF-β1) and the

unfolded protein response (UPR).

The TGF-β1 model is constructed by means of an extensive literature search and con-

sists of many state equations. Model reduction (the quasi-steady-state approximation)

reduces the model to a version with only two states, such that the procedure can be visual-

ized. The nonlinearities in this reduced model are approximated with PWA functions and

subsequently analyzed. Typical results show that oscillatory behavior can occur in the

TGF-β1 model for specific sets of parameter values. These results meet the expectations

of preliminary experimental results. Finally, a model of the UPR has been formulated and

analyzed similarly. The qualitative analysis yields constraints on the parameter values.

Model simulations with these parameter constraints agree with experimental results.

128

Samenvatting

Het menselijk lichaam bestaat uit een grote hoeveelheid cellen, “de bouwstenen van het

leven”. In iedere cel dragen complexe netwerken van biochemische processen bij aan de

handhaving van een gezond organisme. Veranderingen in deze biochemische processen

kunnen ziektes tot gevolg hebben. Het is daarom van groot belang te achterhalen hoe

deze biochemische netwerken functioneren. Eenvoudig redeneren is niet voldoende om de

complexiteit van het leven te doorgronden. Wiskundige modellen zijn nodig om informatie

van verschillende bronnen te integreren, de zogenaamde systeembiologie aanpak, waarin

kwantitatieve informatie schaars en kwalitatieve informatie meer voor handen is.

Traditionele wiskundige modellen vereisen kwantitatieve informatie. Het gebrek aan

voldoende en nauwkeurige kwantitatieve data hebben systeembiologen gedwongen alter-

natieve methodes te verkennen om biochemische netwerken te beschrijven en te analyseren.

Hun focus is vooral gericht op de analyse van slechts enkele, zeer specifieke biochemische

netwerken waarbij nauwkeurige experimentele data beschikbaar zijn. Kwantitatieve infor-

matie is echter niet strict noodzakelijk om het potentiele systeemgedrag te analyseren. De

onderlinge interactie en de relatieve bijdrage van de individuele componenten bepalen de

globale dynamica van een systeem. Kwalitatieve informatie is soms voldoende. Daarnaast

bevatten wiskundige modellen van biochemische netwerken vaak niet-lineaire functies die

de verschillende fysiologische processen beschrijven. In de praktijk bemoeilijkt dit systee-

manalyse en parameterschatting aanzienlijk, zeker als er weinig kwantitatieve informatie

beschikbaar is.

De belangrijkste bijdrage van dit proefschrift is kwalitatieve informatie toe te passen

om niet-lineaire biochemische netwerken te modelleren en te analyseren. De niet-lineaire

functies worden benaderd met twee of drie lineaire functies, d.w.z. stuksgewijs lineaire

functies, die kwalitatieve analyse van het systeem mogelijk maken. Dit werk toont aan dat

kwalitatieve informatie vaak voldoende is voor systeemanalyse van complexe niet-lineaire

biochemische netwerken. Bovendien kan deze extra informatie worden gebruikt om de

relatieve grenzen van de parameters op te stellen. Dat verbetert de parameterschatting

significant t.o.v. standaard niet-lineaire parameterschatting methodes. Daarnaast is er

ook een stuksgewijs lineaire parameterschatting procedure ontwikkeld die eveneens betere

parameterschattingen oplevert dan traditionele methodes. Behalve kwalitatieve analyse

met stuksgewijs lineaire functies, is er een grafische analyse methode aangepast die ook het

parameterschatten verbetert. De toepasbaarheid van deze grafische methode is beperkt

tot een kleine klasse van systemen wat deze minder geschikt maakt voor algemeen gebruik.

129

Samenvatting

De bijdrage van dit proefschrift is toegepast op verschillende biochemische netwerken

die betrokken zijn bij vaatwandveroudering. Vaatwandveroudering is de verandering in het

vasculaire systeem die samengaat met een toename in leeftijd, met de daarbij behorende

toenemende kans om ziektes van de vaatwand te krijgen. Drie biochemische netwerken zijn

geselecteerd waarvan experimentele data bekend was: modellering van de extracellulaire

matrix (ECM), de signaaltransductie route van Transforming Growth Factor-β1 (TGF-

β1) en de unfolded protein response (UPR). Grafische analyse is uitgevoerd op het ECM

modellering proces, zodat er grenzen aan een parameterwaarde gesteld kunnen worden

om het verwachte systeemgedrag te garanderen. Het TGF-β1 model is opgesteld d.m.v.

een uitgebreid literatuuronderzoek. Model reductie (de quasi-steady-state benadering)

reduceert dit model naar een versie met twee toestanden. De niet-lineariteiten in dit

gereduceerde model zijn benaderd met stuksgewijs lineaire functies en vervolgens wordt

dit model geanalyseerd. De resultaten tonen aan dat oscillaties kunnen ontstaan in het

TGF-β1 model voor specifieke parameterwaarden. Deze resultaten zijn in overeenstem-

ming met voorlopige experimentele resultaten. Tenslotte is het UPR model opgesteld en

geanalyseerd op een soortgelijke manier. De kwalitatieve analyse brengt restricties op

de parameterwaarden voort. Simulaties met deze restricties komen overeen met experi-

mentele resultaten.

130

Dankwoord

Vier jaar onderzoek verrichten zit er op en staat op papier. Dus mag ik eindelijk het

vrolijkste en meest gelezen hoofdstuk van dit proefschrift schrijven: het dankwoord. Ten

eerste wil ik mijn copromotor bedanken. Natal, onze samenwerking begon in 2002 toen

ik vetzuurtransport in de hartspiercel begon te bestuderen in Seattle, daarna afstuderen

bij CS en vervolgens promoveren. Bij jou kon ik altijd terecht als ik even een goed inzicht

kon gebruiken. Daarnaast heb je me vrij gelaten in het onderzoek, waardoor ik deze

vier jaar met plezier heb beleefd. Paul, beste promotor, je kritische en eerlijke houding

hebben ervoor gezorgd dat ik dit proefschrift op tijd heb kunnen voltooien en dat ik ook

tevreden kan terugkijken op het eindresultaat. Gedurende mijn promotie heb ik voor

enkele maanden onderzoek gedaan bij de Helix-groep, INRIA Rhone-Alpes, Frankrijk.

Hidde, hartelijk dank voor je gastvrijheid, grote interesse in dit project en de vele hulp

die ik heb mogen ontvangen gedurende die periode. I would like to thank the people

at the Helix group for introducing me in the French life. I especially want to mention

Samuel and Delphine (hopefully Quentin is doing well) as my helpful roommates and

for their hospitality, Philippe for the nice diners and the badminton evenings in Crolles.

Finally a big thank you for Thomas: the amusing bike trips, the weekend in Saint Etienne,

and the French cursing lessons are things I will never forget. Merci! I hope to welcome

you and Gaelle in the Netherlands in the near future. Axel and Jan, we had some nice

talks in Italy. I will certainly visit you soon in Germany. Verder wil ik Theo Verrips,

Jan Andries Post, Branko Braam en Arie Verkleij bedanken voor hun enthousiasme en

bruikbare ideeen over de unfolded protein response aan het begin van dit project. Peter

Hilbers, bedankt voor het lezen van mijn proefschrift en voor de suggesties. Jorn, we

hebben slechts kort samengewerkt voordat je naar de VS emigreerde, maar bedankt voor

de hulp die je me hebt gegeven in het begin van mijn promotie. Daniel, jouw stage en

afstudeerwerk hebben mij geınspireerd om kwalitatief modelleren beter te bestuderen.

Daarnaast zijn de stafleden en (oud-)promovendi onmisbaar geweest voor de goede sfeer

op de afdeling: Aleksandar, (SpongeBob) Bart, Femke, Heico, Jasper (professor Preisig),

John, Leo, Michal, Michiel, Nelis (het is zo stil in mij...), Patrick (Patty), Patricia, en

skate maharadja Satya. Mircea, thank you for being an excellent office mate. Barbara,

bij vragen over de Engelse taal, keuvelen over Britse seriemoordenaars, vakanties, en

problemen met m’n reisdeclaraties kon ik altijd bij je terecht. Udo, computerproblemen

werden door jou snel verholpen, waardoor ik mijn onderzoek kon blijven doen. Andrej

en paranimf Maarten N., de laatste loodjes wegen het zwaarst... maar ze worden een

131

Dankwoord

stuk lichter als je met drie man aan het overwerken bent op een maandagnacht met harde

trance op de achtergrond. Erg bedankt voor deze en vele andere leuke momenten in de

laatste maanden van mijn promotie.

Naast promoveren kon ik voor een gezonde hoeveelheid sport bij BC Bavel terecht.

Ik wil iedereen bij BC Bavel bedanken voor een erg leuke tijd. Helaas zal ik verhuizen

en daarom afscheid moeten nemen van BC Bavel, maar ik kom zeker nog een paar keer

langs. Joep, collega-promovendus, sportief dat je me keer op keer uitdaagde voor een potje

badminton in Geldrop en de vele flessen La Chouffe die ik daaraan heb overgehouden.

Jeroen, dankzij jou ben ik me meer gaan interesseren voor fitness. Onze gesprekken over

jouw fotografie-droom zullen me bijblijven.

Gedurende mijn studie en promotie heb ik in het leukste studentenhuis van Eindhoven

gewoond. De laatste jaren bruiste het van de gezelligheid en was het heerlijk om thuis te

komen. Het gaat te ver om iedereen te bedanken, maar toch wil ik er een paar uitlichten.

Remco en Tjitske, bedankt voor de leuke spelletjesavonden, halve marathon “wandelinget-

jes”, jullie bezoek in Grenoble en de ontelbare mooie momenten samen; Virginie en Bart,

de koffie smaakt altijd uitstekend, de wintersport was super en jullie waren altijd be-

trokken bij de gezellige feesten in het huis; Andrea, dankje voor jouw oprechte interesse

en de mini-trip naar Oostenrijk; Lex, Lieke, Marijke, en Pascal: ik heb altijd genoten

van jullie gesprekken; Christoph, dankjewel voor het ontwerpen van het meest creatieve

gedeelte van dit proefschrift: de kaft. Halve huisgenoot en paranimf Maarten v.d. V., we

hebben veel meegemaakt in de afgelopen vier jaar en ook vele goede gesprekken gehad,

waarvoor ik je enorm dankbaar ben. Ik heb er veel aan gehad. Daarnaast wil ik nog al

mijn vrienden bedanken voor de leuke tijd samen.

Anouk & Jochem, Petra & Patrick, Esther & Gijs, de enkele keren dat ik jullie zag

vond ik het erg gezellig. Wim en Anneke, door jullie warmte en steun heb ik voor mijn

gevoel een extra thuis erbij gekregen. Jan, Lia en Toon en de kids, jullie betrokkenheid

was enorm. Jullie hebben altijd voor me klaargestaan en gestimuleerd om door te zetten.

Nu maar hopen dat een flinke dosis wijsheid bij die doctors-titel wordt geleverd. Roland,

broertje, de dagelijkse mailtjes met updates over ons hectische leven heeft ons in die vier

jaar meer naar elkaar toegebracht....het is dus toch nog goed gekomen met ons en dat zal

zeker nog lang blijven als het aan mij ligt. Lieve Vivien, tenslotte wil ik jou als laatste

bedanken. Ondanks de afstand voelde je door onze gesprekken precies aan wanneer ik

jouw hulp kon gebruiken tijdens mijn promotie. Bovendien heb je mij laten inzien dat

ontspanning ook nodig is, vooral in de laatste paar zware maanden. Hierdoor kijk ik met

een zeer goed gevoel terug naar die periode en verwacht ik zeker dat er nog een ontzettend

mooie tijd voor ons samen te wachten staat.

Iedereen bedankt voor zijn/haar bijdrage aan dit proefschrift!

132

About the Author

Mark Musters was born on May 20th, 1980 in Breda,

The Netherlands. He studied Biomedical Engineering

at Eindhoven University of Technology (TU/e) from

1998 - 2003. In 2002 he did an internship for three

months at the bioengineering group of Jim Bassingth-

waighte, University of Washington, Seattle, USA, for

which he received the Dr. E. Dekker grant from the

Netherlands Heart Foundation. The subject of this

internship was to develop a computational model of fatty acid transport across the sar-

colemma of the cardiomyocyte. Mark continued this work in a graduation project, which

resulted in a book chapter. After obtaining his Master of Science degree in Biomedical

Engineering, Mark pursued his PhD degree at the Department of Electrical Engineering,

TU/e. The project was a cooperation between the University of Utrecht, Unilever Re-

search, Aurion and TU/e. The role of the TU/e was to provide computational assistance

in understanding vascular aging processes, the so-called systems biology approach. Espe-

cially the lack of quantitative information formed the main computational challenge in this

project. A qualitative approach was therefore chosen to analyze the mathematical models

of the biochemical networks involved in vascular aging. To become more acquainted with

qualitative modeling, Mark was a visiting PhD student from February - May 2006 in the

group of Hidde de Jong, INRIA Rhone-Alpes, Saint Ismier cedex, France (funded by an

NWO grant).

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Qualitative Modeling in Computational Systems Biology

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