Salakhieva, D., Sadreev, I., Chen, M., Umezawa, Y., Evstifeev, A., Welsh,G., & Kotov, N. (2016). Kinetic regulation of multi-ligand binding proteins.BMC Systems Biology, 10(1), [32]. 10.1186/s12918-016-0277-0

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RESEARCH ARTICLE Open Access

Kinetic regulation of multi-ligand bindingproteinsDiana V. Salakhieva1†, Ildar I. Sadreev2†, Michael Z. Q. Chen3*, Yoshinori Umezawa4, Aleksandr I. Evstifeev5,Gavin I. Welsh6 and Nikolay V. Kotov5

Abstract

Background: Second messengers, such as calcium, regulate the activity of multisite binding proteins in aconcentration-dependent manner. For example, calcium binding has been shown to induce conformationaltransitions in the calcium-dependent protein calmodulin, under steady state conditions. However, intracellularconcentrations of these second messengers are often subject to rapid change. The mechanisms underlyingdynamic ligand-dependent regulation of multisite proteins require further elucidation.

Results: In this study, a computational analysis of multisite protein kinetics in response to rapid changes in ligandconcentrations is presented. Two major physiological scenarios are investigated: i) Ligand concentration is abundantand the ligand-multisite protein binding does not affect free ligand concentration, ii) Ligand concentration is of thesame order of magnitude as the interacting multisite protein concentration and does not change. Therefore, bufferingeffects significantly influence the amounts of free ligands. For each of these scenarios the influence of the number ofbinding sites, the temporal effects on intermediate apo- and fully saturated conformations and the multisite regulatoryeffects on target proteins are investigated.

Conclusions: The developed models allow for a novel and accurate interpretation of concentration and pressurejump-dependent kinetic experiments. The presented model makes predictions for the temporal distribution of multisiteprotein conformations in complex with variable numbers of ligands. Furthermore, it derives the characteristic time andthe dynamics for the kinetic responses elicited by a ligand concentration change as a function of ligand concentrationand the number of ligand binding sites. Effector proteins regulated by multisite ligand binding are shown to dependon ligand concentration in a highly nonlinear fashion.

Keywords: Ligand-receptor binding, Transient kinetics, Calcium, Calmodulin

BackgroundA wide variety of intracellular events are initiated viatemporal change of ligand concentrations. One of themost important ligands in many cells is calcium (Ca2+).Calcium interacts with and regulates the activities of alarge number of calcium-binding proteins as well asnumerous effectors. The number of functional calciumbinding sites within these proteins can range from oneor two to ten or more [1–3]. The most common numberof calcium binding sites is four: as observed in the most

ubiquitous protein, calmodulin as well as troponinand other EF-hand containing proteins [4, 5]. Tem-poral elevation of intracellular free Ca2+ is the keyregulatory factor of the Ca2+-dependent protein activ-ity [6–10]. The characteristics of the induced signalare not fully understood. It remains to be determinedhow a single ligand is able to govern numerous intra-cellular properties.Whilst the multisite ligand binding is not limited to

Ca2+ signaling, Ca2+ is probably the most versatile ion,regulating the largest number of cellular events. SeveralCa2+-binding proteins can be considered as examples ofmultisite ligand protein interactions. Structural biologyinvestigations of calcium binding proteins in complexeswith target protein peptides have suggested that thespecificity in Ca2+-CaM binding protein-dependent

* Correspondence: mzqchen@gmail.comDiana V. Salakhieva and Ildar I. Sadreev are first authors.†Equal contributors3Department of Mechanical Engineering, The University of Hong Kong,Pokfulam Road, Hong Kong, ChinaFull list of author information is available at the end of the article

© 2016 Salakhieva et al. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

Salakhieva et al. BMC Systems Biology (2016) 10:32 DOI 10.1186/s12918-016-0277-0

target activation arises from the diversity of interactioninterfaces between the Ca2+-regulated protein and itstarget proteins [5, 11–21]. The most ubiquitous protein,calmodulin (CaM), consists of two globular domains,each domain containing a pair of helix-loop-helix Ca2+-binding motifs called EF-hands [1, 3, 5, 16, 17]. In earl-ier studies the authors demonstrated that in addition tothe diversity of CaM-target interfaces; the CaM selectiv-ity emerges from its target specific Ca2+-affinity; thenumber of Ca2+ ions bound and the target specific coop-erativity [22–24].Another major factor that contributes to the selectiv-

ity of seemingly simultaneous regulation of severalmultisite Ca2+ binding proteins and Ca2+-mediatedprocesses is the temporal alterations of Ca2+ [25–27].The remarkable variety of Ca2+ signals in cells, rangingfrom infrequent spikes to sustained oscillations andplateaus, requires an understanding of how fast intracellu-lar calcium changes regulate the kinetics of multiplemultisite Ca2+ binding proteins. Therefore mathematicalmodeling of Ca2+ jump induced responses could prove tobe invaluable in the interpretation of transient kineticexperiments.Cooperative binding is a special case of molecular in-

teractions where ligand binding to one site of a mol-ecule depends on the ligand binding to the other sites.The first quantitative determination of the dynamicproperties of cooperative binding was proposed by [28].In this work the authors emphasized the significance ofthe cooperativity by studying the fast dynamics of Ca2+

binding to calretenin (CR), which has one independentand four cooperative binding sites. The investigation ofcooperative effects of Ca2+ binding to CR was per-formed both experimentally and using mathematicalmodeling. The authors employed the simplified versionof the Adair-Klotz model [29, 30] to describe the dy-namics of the interactions involved in Ca2+ binding toCR. This approach was then extended to the binding ofCa2+ to CaM [31]. The models proposed in these stud-ies [28, 31] demonstrated excellent fitting results to theexperimental data, in comparison with the previouslypublished models. However, the described approach israther limited, as it describes fitting instead of provid-ing a mechanistic description. An alternative method-ology offered by [28, 31] is not directly applicable fromthe physical and chemical point of view because theAdair-Klotz model for sequential ligand binding wasutilized [29, 30] whereas binding of Ca2+ to EF-handproteins [32–35] is non-sequential [22]. Given the im-portance of studying fast Ca2+ binding kinetics and thelack of understanding of the underlying mechanisms,we developed a detailed mathematical model for ligandbinding to multisite proteins with both cooperative andindependent binding sites.

Mathematical modelling of multisite protein kinetics inresponse to rapid ligand changes presented in this paperprovide new insights into the mechanism of conform-ational kinetics of multisite proteins in complex withvariable number of bound ligands for the two distinctphysiological situations described:

i) When the ligand concentration significantly exceedsprotein concentration,

ii) When the total amount of ligand is conserved andcomparable with the protein concentration.

In the first case, the buffering effects are negligiblewhereas in the latter, the ligand-protein interactionshave a significant impact on the amount of availableligand and the binding kinetics. In this work, the equa-tions for the dependence of the characteristic timeconstants and the temporal distribution of individualconformations as a function of the ligand concentra-tion, the number of binding sites and the bindingaffinities have been derived. The impact of the numberof binding sites, temporal effects on conformations,and regulation by multisite proteins of their effectorproteins have been investigated by employing the devel-oped models. The analysis of the ligand-multisite pro-tein mediated regulation of effector proteins suggeststhat significant degree of selectivity in regulation can beachieved by a single ligand by employing mechanismsdescribed in this study.

ResultsA new model for multisite protein ligand binding kineticsThe majority of studies of the activation of multisite pro-teins consider only the ligand concentration-dependentprofiles. One of the interesting questions about thesemultisite proteins is how temporal alterations of ligandconcentration contribute to their function. The shape ofdistribution of multisite proteins in complex with variablenumbers of bound ligand is known or can be experimen-tally elucidated in many cases [4, 5, 16, 22–24, 36–38].However, the role of temporal transitions caused by fastalteration of ligand concentration on multisite proteinsand on multisite protein-regulated target proteins remainsunclear. It is reasonable to assume that there can be atleast two distinct mechanisms of fast ligand alteration-mediated effects exhibited in two distinct system scenar-ios: i) the ligand concentration is significantly greater thanthe multisite protein concentration, ii) the ligand is com-parable with the multisite protein concentration. In thefirst scenario ligand binding to multisite protein leads toinsignificant changes of free ligand, whereas in the secondcase, free ligand concentration can vary substantially whenligand molecules bind to the multisite proteins. This paper

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 2 of 19

describes the development of the two models, which ad-dress these distinct physiological situations.

The model for abundant ligand concentrationIn this model we describe physiological situations wherethe ligand concentration significantly exceeds the multi-site protein concentration. In a previous study [22, 24]the authors analyzed functions for the probability of anindividual site being in the bound or non-bound stateand a function giving the probability of a multisite pro-tein being in a complex with different number of boundligands (Eqs. (2) and (3) in Methods) [29, 39, 40]. Toinvestigate the kinetics of the multisite protein ligandinteractions the present study extends the previousmodel to consider the ligand concentration as a func-tion of time (Eqs. (4) and (5) in Methods). The solu-tions for the individual sites to be in a particular statewere obtained for those cases where ligands are subjectto rapid changes between steady-states (Eqs. (6) and (7)in Methods). Due to the large number of sites involved,knowledge of the state probability distribution for indi-vidual binding sites allows accurate estimation of thedynamics of the total concentration of bound ligand inresponse to a jump in free ligand concentration (Eq.(11) in Methods).In order to gain more insights into the distribution

of the intermediate protein conformations (complexeswith variable number of bound ions) we investigatedthe case of a multisite protein with identical bindingsites (Eq. (12) in Methods). While this case is a rela-tively rare occurrence in living cells, it enables insightinto the role that the number of binding sites plays incellular signalling. There are several examples of pro-tein families that have variable number of ligand bind-ing sites either due to their structural properties or bythem forming large tertiary complexes. For examplemembers of the Ca2+ family of binding proteins candiffer in the number of ligand binding sites [41, 42].The most ubiquitous Ca2+-binding protein, calmodulin(CaM), contains four Ca2+ binding sites as does tropo-nin (TnC) [18] and calcineurin phosphatase (CaN)[43]. However the number of functional Ca2+ bindingsites can vary from two to ten as in the protease, cal-pain [1] or even more in other cases [2]. To investigatethe role that the number of ligand binding sites playsin multisite kinetics, the ligand concentrations, at whichthe intermediate conformations reach their maximumvalues, (Eq. (13) in Methods) and the corresponding mag-nitudes for those conformations (Eq. (14) in Methods)were estimated.Figure 1 shows the maximum protein conformations

in complex with one, two and three ligands as a functionof the number of binding sites (Eq. (14) in Methods).The graph demonstrates that the magnitude of the

ligand-multisite complexes decreases dramatically as thenumber of binding sites increases. The presented resultssuggest that the relative magnitude of individual inter-mediate conformations decreases as the number of bind-ing sites increases. This in turn results in subtlerregulatory effects of those proteins with larger numberof ligand binding sites. For example, in CaM that hasfour binding sites for calcium [18], the presence of foursites leads to the increased multifunctionality of this pro-tein due to the additional regulatory properties of inter-mediate conformations [24]. However, as an exceptionsome forms of CaM have six binding sites [44, 45]. Inthis case, according to Fig. 1 the magnitude of inter-mediate conformations significantly decreases comparedto the case of four binding sites, resulting in decreasedregulatory properties of the protein. The presence ofmore than one binding site results in increased multi-functionality of the protein but at the same time leadsthe decrease of the regulatory effects. Thus, it seems thatthere is an “optimal” number of binding sites, whichhave been developed during the evolution, for instancefour calcium binding sites in CaM.Next, the ligand concentrations for half maximum ef-

fective ligand concentrations, U00.5 and Un

0.5, for theapo- and saturated multisite protein conformationswhen the protein species equals 50 % of the total con-centration were estimated (Eqs. (15) in Methods). Thissolution shows that the ligand concentration for thehalf maximal protein activity, known as EC50, would beequal to the equilibrium dissociation constant K (EC50

= K) for proteins with one binding site only (n = 1).Figure 2a shows the dependence of U0

0.5/K and Un0.5/K,

Fig. 1 The effect of the number of binding sites on intermediateconformations. The maximum magnitude of protein conformationsin complex with one, two and three ligand molecules are shown asa function of the total number of binding sites. The relative amountof ligand binding by conformations bound to a specific number ofsites clearly diminishes as the number of sites grow

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 3 of 19

on the number of binding sites. The model predictsthat there is a significant change in the required ligandconcentration Un

0.5/K for the fully bound conformation,while U0

0.5/K does not change with time.In the previous study the authors reported on the regu-

latory importance of the distribution of individual multi-site protein conformations [22, 24]. Here calculations(Eqs. (15) in Methods) are presented for the half-width be-tween the half-maximal effective ligand concentrationsU0.5/K as a function of the ligand concentration for theintermediate conformations (one bound ligand) of theproteins with different number of binding sites (Fig. 2b).The observation of the half-width for the concentra-tions of intermediate conformations that have a bell-shaped dependence on ligand concentration, enablesthe range of physiologically plausible concentrationsof ligand, where protein functions can be regulated byintermediate conformations to be obtained. For ex-ample, in Fig. 2b the half-width range of calcium con-centrations is approximately from -1 to 1 on thelogarithmic scale, which corresponds to 10-7M-10-5Mdue to the fact that the affinity of calcium bindingsites in CaM is approximately 10-6M [46, 47]. Inter-estingly, the range 10-7M-10-5M corresponds to thephysiological range of intracellular calcium concentra-tions in cardiac muscle cells [48]. Within this range

of calcium concentrations, the switching between cal-cium channel opening and closure takes place [49].The difference An between ligand concentrations

U0.9/K and U0.1/K for the saturated multisite proteinconformations, when the protein species are equal to90 % and 10 % of the total concentration, as a functionof the ligand concentration for proteins with differentnumber of binding sites is shown in Fig. 2c and d re-spectively. The determination of An allows an under-standing as to how an increase of the number ofbinding sites n affects the steepness of the dose-response curve (Fig. 2c). It can be seen from Fig. 2cand d that with an increase of n, the width An decreaseswhile the steepness increases and shifts to the range ofhigher ligand concentrations U/K. The greater steep-ness of the dose-response curve caused by the presenceof increasing number of binding sites (Fig. 2c) may re-sult in the switch-like response and ultrasensitivity ofthe protein activation [50]. For instance, the steepnessof CaM activation defines the threshold properties forthe switching of erythrocyte aggregation and deform-ability from one steady-state to another [51].Equation (17) for the total amount of bound ligand

in the case of multisite protein with identical bindingsites, can be used to estimate the amount of bound lig-and when the ligand concentration is equal to the

Fig. 2 Model predictions for the half-maximal effective ligand concentrations as a function of the number of binding sites and ligand concentration.a. The dependence of the half-maximal effective ligand concentration, U0

0.5/K and Un0.5/K, for the apo- and saturated multisite protein conformations

respectively, on the number of binding sites. The effect of the increasing of the amount of binding sites is negligible for the fully bound conformation.b. Calculations for the half-width between the half-maximal effective ligand concentrations as a function of the ligand concentration for proteins withtwo, three, four and five binding sites. c. The difference between ligand concentrations for the saturated multisite protein conformations when theprotein species equal to 90 % and 10 % of the total concentration as a function of the ligand concentration for the proteins with one to six bindingsites. d. The difference between ligand concentrations for the saturated multisite protein conformations when the protein species equal to 90 % and10 % of the total concentration as a function of the number of binding sites up to six

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 4 of 19

equilibrium dissociation constant (U = K). Our modelpredicts that the ligand concentration for the half maximalprotein activation, EC50, is equal to the equilibrium dis-sociation constant K for any number of bound sites for amultisite protein with identical binding sites.Figure 3 shows the calculations for the temporal char-

acteristics of the apo- and fully bound species. Figure 3aand b show that the temporal shapes of the apo- andfully bound conformations (Eqs. (19) in Methods) in re-sponse to a ligand change are similar to the steady-statedependence of the same conformations on ligand con-centration [24]. The kinetic parameters, τ0

0.5 and τ40.5 can

be estimated as the time required to reach 50 % of thetotal concentration (Fig. 3a and b). The described kineticparameters have been investigated as a function of theinitial and final ligand concentrations (Fig. 3c, d andEqs. (24) in Methods). Our analysis reveals a reductionof the time constant, τ0

0.5, of the apo- conformation withthe reduction of the initial ligand concentration and anincrease of the final ligand concentration. However, thedependence of the characteristic time τ4

0.5 (Fig. 3d)showed an unexpected bell shaped dependence on thefinal ligand concentration compared to the simplermonotonic dependence for τ0

0.5 (Fig. 3c). The model pre-dicts that there is an “optimal” ligand concentration forthe saturation effect to take the longest time (Fig. 3d).

The bell shaped dependence of τ40.5 for a protein with

four binding sites, for example CaM [18], on final ligandconcentration shown in Fig. 3d appears to be related tothe presence of intermediate conformations with one,two and three bound sites. In a single site molecule(n = 1), for example crystalline bovine β-trypsin [52]that is characterized by the absence of intermediateforms, τ4

0.5 depends on U1/K monotonically, i.e. thereis no bell shaped dependence, as is evident from Eqs.(24). In CaM (n = 4), the activation of intermediateconformations takes extra time, which affects τ4

0.5.This activation precedes the activation of the satu-rated form in time, and also as U1/K increases, i.e.the intermediate conformations are more prevalentfor smaller values of U1/K, and the stationary distri-bution shifts towards the saturated form for largerU1/K. As a result, an increase of U1/K leads to anincrease of the contribution of the kinetics of theintermediate complexes to the overall dynamics, andhence to an increase of τ4

0.5. For larger U1/K the roleof the intermediate conformations is less importantand overall speed-up dominates, hence τ4

0.5 de-creases. Thus the model predicts that there is anintermediate ligand concentration, at which the kin-etics of the fully saturated form, represented by τ4

0.5,is the slowest.

Fig. 3 Temporal characteristics of the apo- and fully bound species in response to a ligand jump. The dynamics of the concentration of proteins wasinvestigated for apo- (a) and fully bound (b) forms in response to the non-dimensional ligand concentration change from U0/K = 0.1 to U1/K = 10 as afunction of non-dimensional time η = t ⋅ k−. The dotted lines indicate the time, τ0

0.5, required for the non-dimensional concentration of apo-conformation, N0/LT, to reach half of the fall in concentration and the period of time, τ40.5, that takes for the fully saturated protein species,N4/LT, to gain half of the growth in concentration. These kinetic parameters then were subject to the investigation as a function of theinitial (c) and final (d) ligand concentrations. The presented analysis clearly demonstrates the bell shaped dependence of τ40.5 on the finalligand concentration

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 5 of 19

The affinities of binding sites differentially affect thekinetic responses of intermediate conformationsThe proposed model has been employed to investigatethe ligand jump-dependent kinetics of both saturatedand non-saturated conformations. Initially, an idealizedmodel of a multisite protein with identical binding siteswas used to investigate the impact of ligand concentra-tions on the multisite protein kinetics. However, in livingcells there are very few proteins (if any) that have identi-cal ligand binding sites. Therefore the model was ex-tended to examine the implications of variations inbinding site affinities on the predicted concentration-response profiles.Figure 4 shows the dependence of the time point

τmmaxk− when the intermediate protein conformationsreach the maximum as a function of magnitude of ligandjump (Eqs. (26) in Methods). It can be seen from Eqs.(26) that τ1

maxk−, τ2maxk− and τ3

maxk− do not exist for U1=

K < 13 , U1/K < 1 and U1/K < 3, respectively. Under these

special cases, where the ligand concentration U1/K isnot sufficient for the concentrations of the intermediateconformations to reach their maximal values, these con-centrations monotonously grow to their respectivesteady-state levels. According to Eq. (26), the values U1=

K < 13 ; U1/K < 1 and U1/K < 3 correspond to the three

individual intermediate conformations with one, twoand three bound sites respectively.To investigate the impact of the dissociation constants

of individual binding sites we employed the multisiteprotein model (please see subsection “Multisite proteins

with four different ligand binding sites” in Methods)with marginally (Fig. 5) and significantly different associ-ation constants (Fig. 6). The main result that followsfrom the analysis of the intermediate conformationcurves is that the affinities of the different binding sitesmainly affect the magnitudes of corresponding proteinconformation. For example, the conformation of a multi-site protein corresponding to the one ligand bound stateis present in lower concentration if the affinity of thebinding centre is lower. However the overall shape ofthe concentration dependent profile has not changed.This property is very similar to the case of steady-statedependence on the ligand concentration. The only dif-ference is that the bell shape dependence on time duringthe kinetic response is partially skewed. However, signifi-cant variation in affinities changes the magnitude, andalso leads to the asynchronous kinetics of the intermedi-ate conformations.

The effects of cooperativity in Ca2+ binding to CaMIn order to investigate the influence of cooperativity, wechose a well-characterized protein, CaM, as the modelobject. The CaM protein contains two independent EF-hand globular domains, with two binding sites [1, 3, 5,16, 17]. The sites within each of the domains coopera-tively influence each other. It has been reported that co-operative binding occurs between two neighbouring siteswithin the N- and C- terminal domains of CaM [22, 53,54]. Figure 7 shows the model predictions for CaMwhere we assume that the molecule has two independ-ent domains, with two identical cooperative sites. In thefirst domain the affinity of one site changes from K1 =0.9 μM to K1

c = 0.2 μM if the other site is occupied andin the second domain the affinity changes from K2 =0.8 μM to K2

c = 0.1 μM (Eq. (28) in Methods) [22].Figure 7a and b show the influence of cooperativity onthe steady-state concentrations of CaM with certainnumber of bound sites. The presence of cooperativityshifts the dose-response characteristics along the ligandconcentration axis and changes the magnitude of inter-mediate conformations allowing more developed select-ive regulation of the activity of CaM. The investigationof the dynamic properties of co-operativity in CaM(Fig. 7c and d) for intermediate, apo- and saturated spe-cies revealed that the cooperativity influences the magni-tudes of time-dependent characteristics. The proposedmodel predicts that the cooperative binding leads tomore pronounced selective effects for intermediate con-formations and higher differences between the initialand steady-state levels for the apo- and saturated forms.The results of the present analysis suggest that coopera-

tivity plays an important role in the regulation of the activ-ity of multisite proteins by allowing wider possibilities forselectivity. However, the presence of cooperativity leads to

Fig. 4 Characteristic time required for intermediate conformationsto reach their maximum levels as a function of the step changemagnitude. The analysis shows that the non-dimensional time(ηm

max = τmmaxk−, where m = 1, 2 and 3) required for reaching the

maximum level of the intermediate species, is inversely proportionalto the concentration of the applied ligand U1/K. This effect is due tothe growing abundance of the free ligand concentration availablefor faster interaction with the multisite protein

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 6 of 19

Fig. 5 Kinetics predictions for multisite protein species with marginally different association constants. The kinetics of multisite protein specieswas investigated for the intermediate (N1/LT, N2/LT and N3/LT in a as well as the apo- and fully bound conformations (N0/LT and N4/LT respectivelyin b in response to step change of ligand from U0/K = 0.001 to U1/K = 1.43 for marginally different association constants h1 = 1, h2 = 0.9, h3 = 0.8,h4 = 0.7 and the same dissociation constants h1

− = h2− = h3

− = h4− = 1. Similar analysis was also performed when step change was U0/K = 0.001, U1/K = 100

for apo- (c) and fully bound (d) forms. The calculations show that the final level of the multisite protein species are defined by theligand concentration after the step change. It is very clear that the fully bound species are not saturated and most of the ligand is distributed amongspecies bound to fewer ligands. However, step change application of ligand with much higher concentration from U0/K = 0.001 to U1/K = 100 forapo- (c) and fully bound (d) species demonstrate that the application of higher concentrations of ligand causes fully saturates the protein

Fig. 6 Kinetics of multisite protein species alterations for a protein with significantly different association constants. The kinetics of multisiteprotein species was investigated in response to step change of ligand from U0/K = 0.001 to U1/K = 8 and to U1/K = 400 for the intermediate (N1/LT,N2/LT and N3/LT in a, c) as well as apo- and fully bound conformations (N0/LT and N4/LT in b, d), respectively, in the case of significantly differentassociation h1 = 1, h2 = 0.6, h3 = 0.2, h4 = 0.1 and the same dissociation constants h1

− = h2− = h3

− = h4− = 1. The comparison with the kinetics of the

protein with slightly different association constants suggests that in this case the species acquire a degree of asynchronous dynamics

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 7 of 19

quantitative rather than qualitative changes in the system.The introduction of cooperative binding is crucial for theexperimental data fitting but at the same time brings fur-ther complexity to the system, which does not necessarilylead to a better understanding of the underlying mecha-nisms. As a result of this, further analysis was carried outwithout considering this effect.

The model for comparable ligand and proteinconcentrationsThe previous sections considered the physiological casewhere the ligand concentration was above saturation levelmeaning that the ligand-multisite protein interactions didnot affect the availability of the ligand. Here the case whenthe amount of ligand is limited is considered. This canoccur in cases when the ligand concentration level is com-parable to the multisite protein availability. The modelpredictions show the relative redistribution of the ligandin the free and bound states. The responses of a multisiteprotein with four identical binding sites to ligand concen-tration step change (Eqs. (44) and (45) in Methods)for two different protein concentrations, LT/K = 2 andLT/K = 50, were studied (Fig. 8). Instead of consideringabsolute ligand concentrations, the approach consideredratios of the ligand concentration to the affinities of the

binding sites. A borderline case where the free ligand con-centration is barely affected by interaction was considered(Fig. 8a and b) as well as a smaller total ligand concentrationwhere the free ligand is nearly exhausted as a result of buff-ering by the multisite protein (Fig. 8c and d). The compari-son of the free ligand concentration (Eqs. (43) and(44) in Methods) for these two cases is shown inFig. 8e. The model predicts that the strongest effectof the ligand availability can be observed for the mul-tisite protein conformations with three and four (fullysaturated) bound ligands. A possible explanation forthis phenomenon may be that the multisite proteinconformations, which form complexes with smallernumber of ligand molecules by definition, do not re-quire significant amount of ligand and as a result arenot strongly affected under conditions when the freeligand is limited. Whereas the multisite protein inter-actions with the larger number of ions occur after thesignificant amount of ligand is “used up” to form theintermediate conformations, however is still requiredfor conformations with larger number of ions. As aresult the final levels of the conformations with threeand four ions are affected. It can also be seen fromFig. 8 that the shapes of the intermediate conforma-tions time lines are skewed.

Fig. 7 Comparative analysis of cooperative versus non-cooperative Ca2+ binding to CaM. Two mathematical models for Ca2+-Cam interactions arecompared under the assumptions for the presence and absence of cooperative binding. The comparison between the two scenarios was performed understeady state conditions (a), (b) and in response to a step change in Ca2+ concentration (c), (d). The model predicts that the cooperativity influences themaximums of the concentrations for the intermediate forms (N1/LT, N2/LT and N3/LT) as well as the steady-state levels of apo- (N0/LT) and fully saturatedforms (N4/LT). However, the difference observed in the distribution of the conformation species in the presence and absence of the cooperative binding isquantitative while the overall shape of the distributions remains unchanged. Due to this finding the following model analysis was performed withoutcooperative binding assumptions

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 8 of 19

Figure 9a shows the model predictions for the charac-teristic time required for intermediate protein conforma-tions with one, two and three bound ligands to reachtheir highest concentrations in response to a step changein ligand concentration (Eq. (50) in Methods). Themodel predicts that the characteristic time τ0

0.5, requiredfor the apo- form to reach its half growth level mono-tonically decreases with the increase of the total ligandconcentration. However, the characteristic time constantτ40.5, which represents the saturated conformation revealsa distorted bell shaped dependence on ligand concentra-tion (Fig. 9b and Eqs. (52) in Methods). This bell shapeddependence, which was also observed in Fig. 3d, can also

be explained by the presence of intermediate confor-mations. The distortion of the bell shape in Fig. 9bappears to be due to the ligand consumption that isincluded into the consideration in this section andwas not considered in Fig. 3d. This result may be sig-nificant for the dynamics of CaM activation as ourmodel predicts that with an increase of the total lig-and concentration, the limited amount of ligand leadsto an additional increase of τ4

0.5 compared to the casewithout the ligand consumption. The model, there-fore, predicts possible transient differences in multi-site protein signal transduction in response to fasttransient kinetics of multisite proteins.

Fig. 8 The comparison between the cases where the free ligand concentration is barely affected by interaction and exhausted as a result ofbuffering. The kinetics of multisite protein species alterations in response to step change in ligand concentration from UT0/K = 0.01 to UT1/K = 200for two different ratios of the protein concentration to the affinities of the binding sites LT/K = 2 a for the intermediate species N1/LT, N2/LT andN3/LT, b for the apo- and fully bound species N0/LT and N4/LT respectively) and LT/K = 50 c for the intermediate species N1/LT, N2/LT and N3/LT, d forthe apo- and fully bound species N0/LT and N4/LT respectively). The model predicts that due to the lack of available ligand and buffering by themultisite protein in the case of limited amount of ligand, the multisite protein is unable to become fully saturated after the step change inligand, and the majority of the ligand becomes distributed among the intermediate species. e. The comparison of the dynamics of free ligandconcentration U/K after step change in ligand. The amount of available ligand is barely altered for LT/K = 2, and exhausted when the ratio of totalprotein concentration to the binding constant is LT/K = 50

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 9 of 19

DiscussionIn this paper we analysed multisite protein kinetics inresponse to rapid changes in ligand concentrations. Themodel for multisite protein kinetics for variable numberof bound ligands was developed for two physiologicalcases: when the concentration of ligand is much higher[55, 56] and when it is comparable with the concentra-tion of the multisite protein [57]. The results obtainedby the proposed model allow for an accurate interpret-ation of the experimental data for the concentration ofmultisite proteins such as CaM, TnC, CaN and otherCa2+ dependent secondary messengers regulated by Ca2+

ions. The kinetic effects in response to ligand binding[13, 58, 59], required for understanding of pathway regu-lation, can be interpreted by the presented model.The approach developed in this project is applicable to

a number of areas of kinetic experiments. One of themis widely used techniques to study chemical kineticsusing the pressure jump technique [12, 60, 61]. Accord-ing to our model, the effects induced by rapid change inpressure leading to the change in protein-ligand interac-tions [12, 60, 61], can be interpreted and explained bythe alteration in the affinities of the binding constants ofmultisite proteins. The time dynamics of the individualmultisite protein species can offer new insights into theunderlying biophysical mechanism of ligand-protein in-teractions in response to fast change.Our model shows that the concentration of intermedi-

ate conformations as a function of time representsskewed bell shapes. We found that as the protein con-centration rises, free ligand concentration becomesexhausted [13]. This result is consistent with experimen-tally observed Ca2+-CaM-dependent inhibition of my-osin motor function [62]. The results obtained by thismodel increase our understanding of differential activa-tion of protein phosphatase 2B (PP2B) [59, 63] and cal-cium/calmodulin-dependent protein kinase II (CaMKII)[64, 65] kinetics. PP2B binding increases the affinity ofCaM for its targets [59] and, therefore, is likely be acti-vated by low amounts of calcium.

The presented model offers a new tool for the interpret-ation of transient kinetics experiments performed by theflash photolysis and stopped-flow techniques [66–68].Availability of a caged analogue of the necessary reactantis considered as one of the biggest problems of flash pho-tolysis [66], which can be overcome by employing the pro-posed model with the low amount of protein. A lessobvious area of application for this methodology is thekinetics of proteins with multiple phosphorylation sites[69–72]. This work shows that the highly versatile intra-cellular multifunctionality of multisite proteins is achievednot only by the order of ligand-protein interaction and thenumber of bound ligands, but also by temporal regulation.

ConclusionsThe developed in this study models for the kinetics ofthe multisite ligand-receptor binding for the two physio-logical cases, where the ligand concentration is abundantand comparable with the protein concentration, makeuniversal predictions for the temporal distribution ofmultisite protein conformations in complex with variablenumbers of ligands. The strongest effect of the ligandavailability is observed for the multisite protein confor-mations with larger numbers of bound ligands. The twomodels show that the concentration of individual multi-site protein conformations changes with time nonli-nearly and that the temporal distribution for theconcentrations of the intermediate conformations repre-sents skewed bell shapes. The models derive the charac-teristic times and the dynamics for the kinetic responseselicited by a ligand concentration change as a functionof ligand concentration and the number of ligand bind-ing sites. The developed models allow for a novel and ac-curate interpretation of concentration and pressure jump-dependent kinetic experiments. Our models are applied tostudy the kinetics of calmodulin, however the models alsoprovide universal predictions and allow us to extend theunderstanding of a large number of multisite binding-regulated circuits and the mechanisms underlying dy-namic ligand-dependent regulation of multisite proteins.

Fig. 9 Model predictions for the time required for multisite protein conformations to reach their maximal and half growth concentrations. Thenon-dimensional characteristic times, τmmaxk− (m = 1, 2, 3) for intermediate (a), τ00.5k− for apo- and τ40.5k− for fully bound (b) multisite proteinconformations are shown as a function of the step change ligand concentration from UT0/K = 0.1 to UT1/K and LT/K = 3

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 10 of 19

MethodsKinetic properties of multisite proteins with n bindingsitesHere, we describe mathematical equations used to de-scribe dynamic interactions of ligand molecules withmultisite proteins. The model described in this sectionextends our previous analysis for multisite protein inter-actions under steady-state conditions [22–24].

Multisite protein with independent ligand binding sitesThe kinetic scheme for such interaction can be repre-sented as follows:

L0i þ U⇄kþi

k−iL1i ; i ¼ 0;…; n−1; ð1Þ

where Li0 is the ith binding site of the multisite protein

in unbound state, U is a ligand molecule, Li1 is the ith

binding site of the multisite protein being occupied,ki+ and ki

− are the association and dissociation ratesrespectively.The probabilities for the ith binding site to be not

occupied or occupied as a function of ligand concentra-tion U are given by:

p0i Uð Þ ¼ Ki

Ki þ U;

p1i Uð Þ ¼ UKi þ U

;ð2Þ

where Ki ¼ k−ikþi:

There are two possible states of a binding site: occu-pied or not occupied. Since the number of sites in themolecule is n, there are 2n possible molecular forms, i.e.the states characterized by combinations of bound andfree sites.The probability for a multisite protein with independ-

ent binding sites to be in a particular molecular form isgiven by multiplications of probabilities of ligand bind-ing at each site:

Pj Uð Þ ¼Yn−1i¼0

pci jð Þi Uð Þ; j ¼ 0;…; 2n−1; ð3Þ

where j ¼Xn−1i¼0

2ici jð Þ is the number of possible molecular

form, ci(j) = 0 or ci(j) = 1 for free or occupied bindingsite, respectively.

The kinetics of multisite protein interactions withabundant ligand concentrations (UT > > LT)In this case we assume the ligand concentration onlychanges at a given time point (t = 0) in a step fashionfrom U0 to U1 and remains constant otherwise.

The probability of a multisite protein being occupiedby ligand molecules as it was shown in Eq. (3) as a func-tion of time can be reformulated as follows:

Pj U ; tð Þ ¼Yn−1i¼0

pci jð Þi U ; tð Þ; j ¼ 0;…; 2n−1: ð4Þ

pci jð Þi U ; tð Þ can be determined by considering the ori-ginal set of ordinary differential equations for single siteinteraction of a protein with a ligand according to Eq.(1) [24]:

dL0i U ; tð Þdt

¼ −kþi ⋅L0i U ; tð Þ⋅U1 þ k−i ⋅L

1i U ; tð Þ;

dL1i U ; tð Þdt

¼ kþi ⋅L0i U ; tð Þ⋅U1−k−i ⋅L

1i U ; tð Þ;

L0i U ; tð Þ þ L1i U ; tð Þ ¼ LT ;

ð5Þ

where Li0(U, t) is the concentration of a free binding site,

Li1(U, t) is the concentration of a bound ligand molecule

and LT is the total number of the protein molecules.Here we use steady-state solutions L0i Uð Þ ¼ LT

KiKiþU

and L1i Uð Þ ¼ LT UKiþU as initial conditions for the ligand

concentration jump from U0 to U1. A particular solutionfor the system of differential Eqs. (5) in response to theligand concentration shift from U0 to U1 is given by:

L0i U ; tð Þ ¼ LT ⋅Ki

Ki þU1−

Ki

Ki þ U1−

Ki

Ki þ U0

� �⋅ exp −

tτ U1ð Þ

� �� �;

L1i U ; tð Þ ¼ LT ⋅U1

Ki þU1−

U1

Ki þ U1−

U0

Ki þ U0

� �⋅ exp −

tτ U1ð Þ

� �� �;

ð6Þ

where τ U1ð Þ ¼ Kik−i ⋅ U1þKið Þ ;Ki ¼ k−i

kþi:

The normalisation of the solution (6) by the total pro-tein concentration allows the definition of probability ofthe ith binding site to be in an occupied pi

1(U, t) or un-occupied pi

0(U, t) state, respectively, at a given ligandconcentration:

p0i U ; tð Þ ¼ p0i U1ð Þ− p0i U1ð Þ−p0i U0ð Þ� �⋅ exp −

tτ U1ð Þ

� �;

p1i U ; tð Þ ¼ p1i U1ð Þ− p1i U1ð Þ−p1i U0ð Þ� �⋅ exp −

tτ U1ð Þ

� �:

ð7ÞAt any given time the total concentration of the pro-

tein, LT, is conserved and the sum of the probabilities (7)equals 1:

L0i U ; tð Þ þ L1i U ; tð Þ ¼ LT ;p0i U ; tð Þ þ p1i U ; tð Þ ¼ 1

ð8Þ

In the most general case, the concentration of thejth molecular form, Mj(U, t), of a molecule with n differ-ent binding sites in response to a step in ligand

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 11 of 19

concentration is given by the product of probabilities (7)according to Eq. (4):

Mj U ; tð Þ ¼ LT ⋅Yn−1i¼0

pсi jð Þi U ; tð Þ; j ¼ 0;…; 2n−1; ð9Þ

where ci(j) = 0 or ci(j) = 1 for free or occupied ith bindingsite respectively.The probabilities for individual molecular forms in

steady-state can be obtained from Eq. (9) by settingt→∞:

Mj Uð Þ ¼ LT ⋅Yn−1i¼0

pci jð Þi Uð Þ; j ¼ 0;…; 2n−1; ð10Þ

The kinetics of the amount of ligand bound to a multi-site protein with n binding sites can be written asfollows:

S U ; tð Þ ¼X2n−1j¼0

Mj U ; tð ÞXn−1i¼0

ci jð Þ: ð11Þ

Multisite proteins with identical ligand binding sitesThe multisite proteins that contain n identical bindingsites with equilibrium dissociation constant equal K

¼ k−

kþ can then be considered. The steady-state con-centration of one of the molecular forms of multisiteprotein with n identical binding sites for m boundsites, as a function of ligand concentration [22, 24] isgiven by:

Nm Uð Þ ¼ LT ⋅Um⋅Kn−m

K þUð Þn ; m ¼ 0; 1;…; n: ð12Þ

For intermediate forms the N1(U),…,Nn − 1(U) speciesdependence on ligand concentration represents a bellshape with one molecular form “magnitude” at a par-ticular ligand concentration Um

max.Differentiating Eq. (12) with respect to U and solving

dN/dU = 0 for U gives:

Umaxm ¼ K ⋅m

n−m; m ¼ 1; 2;…; n−1: ð13Þ

The magnitudes Nmmax of the intermediate conforma-

tions Nm corresponding to Ummax values are:

Nmaxm ¼ LT ⋅

mm

nnn−mð Þn−m; m ¼ 1; 2;…; n−1: ð14Þ

The multisite protein conformations in the apo-,N0, and in the fully saturated states, Nn, wouldreach their maximum that equal to the total multi-site protein concentration LT under conditions ofvery low and very high ligand concentrations,respectively.

Equation (12) can be used to estimate the halfmaximal effective ligand concentration (EC50), U0

0.5

and Un0.5, for the apo- and saturated multisite

protein conformations respectively, when the pro-tein species equal 50 % of the total concentrationLT:

U0:50 ¼ K ⋅

1−ffiffiffiffiffiffiffi0:5n

pffiffiffiffiffiffiffi0:5n

p ;

U0:5n ¼ K ⋅

ffiffiffiffiffiffiffi0:5n

p

1−ffiffiffiffiffiffiffi0:5n

p ≈K ⋅ 1:44⋅n−0:44ð Þ:ð15Þ

It can be noted that the half maximal effective ligandconcentration is equal to equilibrium dissociation con-stant (EC50 = K) in proteins with only one binding site(n = 1).Equations (15) can be solved with respect to n:

n ¼ 0:693⋅ lnK þ U0:5

n

U0:5n

� �� �−1

ð16Þ

Equation (11) can be used to derive the full amount ofligand bound to multisite protein for U = K:

S K ; tð Þ ¼ 14⋅S Usat; tð Þ; ð17Þ

where Usat is the ligand concentration for the case whenall binding sites are occupied. The amount of boundligand for U =Usat is given by S(Usat, t) = n ⋅ LT and forU = K is given by

S K ; tð Þ ¼ LT2n

⋅12⋅n⋅2n−1: ð18Þ

The dynamic alterations of intermediate conformationNm(U, t) in response to ligand concentration jump fromU0 to U1 according to Eqs. (6) and (7) are given by:

Nm U ; tð Þ ¼ LT ⋅ p1 U ; tð Þð Þm⋅ p0 U ; tð Þð Þn−m;

p0 U ; tð Þ ¼ KK þ U1

−K

K þU1−

KK þ U0

0@

1A⋅ exp −

tτ U1ð Þ

0@

1A;

p1 U ; tð Þ ¼ U1

K þ U1−

U1

K þU1−

U0

K þ U0

0@

1A⋅ exp −

tτ U1ð Þ

0@

1A;

τ U1ð Þ ¼ Kk−⋅ U1 þ Kð Þ :

ð19Þ

Differentiating Eq. (19) with respect to t and solvingfor dNm/dt = 0 yields the time τm

max when the multisiteprotein forms Nm with m bound molecules of ligandreach their maximal values Nm

max:

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 12 of 19

τmaxm ¼ τ U1ð Þ⋅ ln n⋅K ⋅ U1−U0ð Þ

K þ U0ð Þ⋅ n−mð Þ⋅U1−m⋅Kð Þ� �

;

m ¼ 1; 2;…; n−1:ð20Þ

The substitution of τmmax into Eq. (19) gives the max-

imal values of intermediate multisite protein conforma-tions reached at τm

max:

Nmaxm ¼ LT ⋅

mm

nnn−mð Þn−m; m ¼ 1; 2;…; n−1: ð21Þ

Comparison of Eqs. (14) and (21) suggests thatthe steady-state maximum values of intermediatemultisite protein conformations steady-state aresimilar to those transiently reached during the dy-namic response to the step in ligand concentration.According to Eq. (19) Nm(U, t) for the apo- and fully

saturated forms when t = 0:

N0 U ; 0ð Þ ¼ LT ⋅K

K þ U0

� �n

;

Nn U ; 0ð Þ ¼ LT ⋅U0

K þU0

� �n

:

ð22Þ

According to Eq. (12) steady state levels for the apo-and fully saturated forms when t→∞:

N0 U ;∞ð Þ ¼ LT ⋅K

K þ U1

� �n

;

Nn U ;∞ð Þ ¼ LT ⋅U1

K þ U1

� �n

:

ð23Þ

Equation (19) is further used to define the time,τ00.5, required for the apo- form, N0, to reach half of

the growth concentration N0 U;∞ð ÞþN0 U ;0ð Þ2 and the time

period, τn0.5, required for the fully saturated protein

species to gain half of the growth concentrationNn U;∞ð ÞþNn U ;0ð Þ

2 (Fig. 3):

Multisite proteins with four identical ligand binding sitesThis study next considered the kinetic properties of aprotein with 4 binding sites. This allows 24 = 16 molecu-lar forms, each with potentially unique biochemicalproperties. There are 4 possible combinations of proteinspecies bound to one or to three ligands, and 6 possibledistinct molecular forms with two sites occupied. In theprevious work we described the steady-state dependenceof the individual multisite conformations on lig-and concentration [22–24]. Here we analyse thekinetic transition of the individual species con-centrations in response to the step in ligandconcentration.The dynamical alterations of intermediate conform-

ation Nm(U, t) in response to a step in ligand concentra-tion from U0 to U1 according to Eq. (19) are given by:

Nm u; ηð Þ ¼ LT ⋅ p1 u; ηð Þð Þm⋅ p0 u; ηð Þð Þ4−m;p0 u; ηð Þ ¼ 1

1þ u1−

11þ u1

−1

1þ u0

� �⋅ exp −η⋅ u1 þ 1ð Þð Þ;

p1 u; ηð Þ ¼ u11þ u1

−u1

1þ u1−

u01þ u0

� �⋅ exp −η⋅ u1 þ 1ð Þð Þ;

ð25Þ

where: u0 ¼ U0K , u1 ¼ U1

K are non-dimensional ligandconcentrations, and η = t ⋅ k− is non-dimensional time.k− and K ¼ k−

kþ are the dissociation and equilibriumdissociation constants for ligand binding, respectively.The maximum values of Nm

max are reached atthe following ligand concentrations: umax

1 ¼ K3 ,

u2max = K, u3

max = 3K, for the multisite protein spe-cies with one, two and three bound ions, respect-ively according to Eq. (13) and equal N1

max =0.105LT, N2

max = 0.063LT, N3max = 0.105LT, according

to Eq. (14).Differentiating Eq. (25) with respect to η and solv-

ing dNm/dη = 0 for η gives the non-dimensional time

τ0:50 ¼ τ U1ð Þ⋅ ln K ⋅U1−U0

K þ U0ð Þ⋅ K þ U1ð Þ⋅ 0:5⋅K

K þ U0

� �nþ K

K þ U1

� �n� �� �1n−K

0B@

1CA

0BBBBBBBBB@

1CCCCCCCCCA;

τ0:5n ¼ τ U1ð Þ⋅ ln K ⋅U1−U0

K þ U0ð Þ⋅ U1− K þ U1ð Þ⋅ 0:5⋅U0

K þ U0

� �nþ U1

K þU1

� �n� �� �1n

0B@

1CA

0BBBBBBBBB@

1CCCCCCCCCA:

ð24Þ

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 13 of 19

ηmmax = τm

maxk− when the intermediate species reachtheir maximum:

ηmax1 ¼ τmax

1 k− ¼ 1u1 þ 1

ln4⋅ u1−u0ð Þ

1þ u0ð Þ⋅ 3u1−1ð Þ� �

;

ηmax2 ¼ τmax

2 k− ¼ 1u1 þ 1

ln2⋅ u1−u0ð Þ

1þ u0ð Þ⋅ u1−1ð Þ� �

;

ηmax3 ¼ τmax

3 k− ¼ 1u1 þ 1

ln4⋅ u1−u0ð Þ

1þ u0ð Þ⋅ u1−3ð Þ� �

:

ð26Þ

Equations (26) suggest that η1max, η2

max and η3max are

undefined when u1 < 13 ; u1 < 1 and u1 < 3; respectively

(q1, q2 and q3 asymptotes). Under these conditions, theintermediate species do not reach the maximum in re-sponse to ligand step, instead their relative number in-crease in a monotonous manner.

Multisite proteins with four different ligand binding sitesIn this section we analyse the kinetic properties of amultisite protein with four different binding sites. Thesteady-state analysis can be found in the authors previ-ous investigation [22, 24].It is assumed that all association k1

+, k2+, k3

+, k4+ and dis-

sociation k1−, k2

−, k3−, k4

− rates are unique for each bindingcentre. Then assuming for example that k1

+ > k2+ > k3

+ > k4+

and k1− = k2

− = k3− = k4

−. The non-dimensional concen-tration u ¼ U

K and non-dimensional constants h1 = 1,

h2 ¼ kþ2kþ1, h3 ¼ kþ3

kþ1, h4 ¼ kþ4

kþ1, h1

− = 1, h−2 ¼ k−2k−1¼ 1, h−3 ¼ k−3

k−1

¼ 1, h−4 ¼ k−4k−1¼ 1 can be introduced..

The set of Eqs. (25) can then be employed to calculatethe dynamics of the multisite protein conformationsbound to a different number of ligand molecules.

Multisite proteins with two pairs of cooperative bindingsitesThe molecule contains two independent domains A andB, with two identical cooperative binding sites. Thedomain A is described as follows:

A00 þ U⇄kþ1

k−1A10

A00 þ U⇄kþ1

k−1A01

A10 þ U⇄kсþ1

kс−1A11

A01 þ U⇄kсþ1

kс−1A11

ð27Þ

The ODEs for the scheme (27) is given by:

dA00 U ; tð Þdt

¼ −2kþ1 ⋅A00 U ; tð Þ⋅U þ k−1 ⋅A10 U ; tð Þþk−1 ⋅A01 U ; tð Þ;

dA10 U ; tð Þdt

¼ kþ1 ⋅A00 U ; tð Þ⋅U−k−1 ⋅A10 U ; tð Þ−kcþ1�A10 U ; tð Þ⋅U þ kc−1 ⋅A11 U ; tð Þ;

dA01 U ; tð Þdt

¼ kþ1 ⋅A00 U ; tð Þ⋅U−k−1 ⋅A01 U ; tð Þ−kcþ1�A01 U ; tð Þ⋅U þ kc−1 ⋅A11 U ; tð Þ;

dA11 U ; tð Þdt

¼ kcþ1 ⋅A01 U ; tð Þ⋅U þ kcþ1 ⋅A10 U ; tð Þ�U−2kc−1 ⋅A11 U ; tð Þ;

ð28Þ

The total number of species that follows from Eqs.(28) is given by:

A00 U ; tð Þ þ A10 U ; tð Þ þ A01 U ; tð Þ þ A11 U ; tð Þ ¼ AT :

ð29Þ

The steady-state solutions of the system (28) are givenby:

A00 Uð Þ ¼ AT ⋅K 1⋅Kc

1

U2 þ 2⋅Kc1⋅U þ K1⋅Kc

1

;

A10 Uð Þ ¼ AT ⋅U⋅Kc

1

U2 þ 2⋅Kc1⋅U þ K1⋅Kc

1

;

A01 Uð Þ ¼ AT ⋅U⋅Kc

1

U2 þ 2⋅Kc1⋅U þ K1⋅Kc

1

;

A11 Uð Þ ¼ AT ⋅U2

U2 þ 2⋅Kc1⋅U þ K1⋅Kc

1

;

ð30Þ

where K1 ¼ k−1kþ1

and Kc1 ¼ kc−1

kcþ1.

One can re-write Eqs. (30) as follows:

a0 Uð Þ ¼ K1⋅Kc1

U2 þ 2⋅Kc1⋅U þ K1⋅Kc

1

;

a1 Uð Þ ¼ U⋅Kc1

U2 þ 2⋅Kc1⋅U þ K1⋅Kc

1

;

a2 Uð Þ ¼ U2

U2 þ 2⋅Kc1⋅U þ K1⋅Kc

1

;

ð31Þ

where a0 Uð Þ ¼ A00 Uð ÞAT

; a1 Uð Þ ¼ A10 Uð ÞAT

¼ A01 Uð ÞAT

; a2 Uð Þ ¼A11 Uð ÞAT

are the probabilities for the domain to be in a

particular conformation due to the bound ligandmolecules.The probabilities for the other domain, which also

contains a pair of cooperative binding sites, are given by:

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 14 of 19

b0 Uð Þ ¼ K 2⋅Kc2

U2 þ 2⋅Kc2⋅U þ K2⋅Kc

2

;

b1 Uð Þ ¼ U⋅Kc2

U2 þ 2⋅Kc2⋅U þ K2⋅Kc

2

;

b2 Uð Þ ¼ U2

U2 þ 2⋅Kc2⋅U þ K2⋅Kc

2

;

ð32Þ

These probabilities were derived for the two domainsof molecule (Eqs. (31) and (32) respectively). The prob-abilities of the molecule to be in a certain conformationwith 0, 1 or 2 bound ligands in each of the domains, areas follows:

p0;0 Uð Þ ¼ a0 Uð Þ⋅b0 Uð Þ;p0;1 Uð Þ ¼ a0 Uð Þ⋅2b1 Uð Þ;p0;2 Uð Þ ¼ a0 Uð Þ⋅b2 Uð Þ;p1;0 Uð Þ ¼ 2a1 Uð Þ⋅b0 Uð Þ;p1;1 Uð Þ ¼ 2a1 Uð Þ⋅2b1 Uð Þ;p1;2 Uð Þ ¼ 2a1 Uð Þ⋅b2 Uð Þ;p2;0 Uð Þ ¼ a2 Uð Þ⋅b0 Uð Þ;p2;1 Uð Þ ¼ a2 Uð Þ⋅2b1 Uð Þ;p2;2 Uð Þ ¼ a2 Uð Þ⋅b2 Uð Þ;

ð33Þ

where pi,j(U) is the probability of the protein conform-ation with i bound sites in the first domain and j boundsites in the other. We use the sum of probabilities forthe case of 1 bound site in a domain since we assumethat all the sites in each domain are identical.The concentrations of the molecular forms of the pro-

tein with certain number of bound sites are given by:

N0 Uð Þ ¼ LT ⋅p0;0 Uð Þ;N1 Uð Þ ¼ LT ⋅ p0;1 Uð Þ þ p1;0 Uð Þ

� �;

N2 Uð Þ ¼ LT ⋅ p0;2 Uð Þ þ p1;1 Uð Þ þ p2;0 Uð Þ� �

;

N3 Uð Þ ¼ LT ⋅ p1;2 Uð Þ þ p2;1 Uð Þ� �

;

N4 Uð Þ ¼ LT ⋅p2;2 Uð Þ;

ð34Þ

One can rewrite Eq. (34) as follows:

N0 Uð ÞLT

¼ a0 Uð Þ⋅b0 Uð Þ;N1 Uð ÞLT

¼ 2⋅ a0 Uð Þ⋅b1 Uð Þ þ a1 Uð Þ⋅b0 Uð Þð Þ;N2 Uð ÞLT

¼ a0 Uð Þ⋅b2 Uð Þ þ 4⋅a1 Uð Þ⋅b1 Uð Þ þ a2 Uð Þ⋅b0 Uð Þ;N3 Uð ÞLT

¼ 2⋅ a1 Uð Þ⋅b2 Uð Þ þ a2 Uð Þ⋅b1 Uð Þð Þ;N4 Uð ÞLT

¼ a2 Uð Þ⋅b2 Uð Þ;

ð35Þ

Next we find the kinetic solution of system (28) forU =U1:

dA00 U ; tð Þdt

¼ −2kþ1 ⋅A00 U ; tð Þ⋅U1 þ k−1 ⋅A10 U ; tð Þþk−1 ⋅A01 U ; tð Þ;

dA10 U ; tð Þdt

¼ kþ1 ⋅A00 U ; tð Þ⋅U1−k−1 ⋅A10 U ; tð Þ−kcþ1�A10 U ; tð Þ⋅U1 þ kc−1 ⋅A11 U ; tð Þ;

dA01 U ; tð Þdt

¼ kþ1 ⋅A00 U ; tð Þ⋅U1−k−1 ⋅A01 U ; tð Þ−kcþ1�A01 U ; tð Þ⋅U1 þ kc−1 ⋅A11 U ; tð Þ;

dA11 U ; tð Þdt

¼ kcþ1 ⋅A01 U ; tð Þ⋅U1þkcþ1 ⋅A10 U ; tð Þ�U1−2kc−1 ⋅A11 U ; tð Þ;

ð36Þ

The kinetics of multisite protein interactions withconstant ligand concentrationsIn this section we consider the mechanism of multisitebinding for the case when the total ligand concentrationis conserved. Under this assumption the system of differ-ential equations for ligand binding to a molecule withsingle binding site (5) needs to be complemented by thelaw of ligand conservation:

dL0 U ; tð Þdt

¼ −kþ⋅L0 U ; tð Þ⋅U tð Þ þ k−⋅L1 U ; tð Þ;

dL1 U ; tð Þdt

¼ kþ⋅L0 U ; tð Þ⋅U tð Þ−k−⋅L1 U ; tð Þ;

L0 U ; tð Þ þ L1 U ; tð Þ ¼ LT ;U tð Þ þ L1 U ; tð Þ ¼ UT ;

ð37Þ

where L0 is the concentration of the free site, L1 is theconcentration of the occupied site, UT and LT are thetotal concentrations of ligand and protein molecules,respectively.There is only one positive steady-state solution of the

system (37):

L0 UTð Þ ¼ K2⋅

LTK

−UT

K−1þ F UTð Þ

� �;

L1 UTð Þ ¼ K2⋅

LTK

þ UT

Kþ 1−F UTð Þ

� �;

U ¼ UT−K2⋅

LTK

þ UT

Kþ 1−F UTð Þ

� �;

ð38Þ

where K ¼ k−

kþ and F UTð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUTK − LT

K

� �2 þ 2⋅ UTK þ LT

K

� �þ 1q

:

We use the steady-state solutions (38) as initial condi-tions to find the particular solution. The solution of thesystem (37) in response to the ligand concentrationjump from UT0 to UT1 is given by:

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 15 of 19

L0 UT ; tð Þ ¼ LT ⋅�

K2⋅LT

⋅�LTK

−UT1

K−1þ F UT1ð Þ

�C UTð Þ⋅ exp t⋅k−⋅F UT1ð Þð Þ þ 1C UTð Þ⋅ exp t⋅k−⋅F UT1ð Þð Þ−1

��;

L1 UT ; tð Þ ¼ LT ⋅�1−

K2⋅LT

⋅�LTK

−UT1

K−1þ F UT1ð Þ

�C UTð Þ⋅ exp t⋅k−⋅F UT1ð Þð Þ þ 1C UTð Þ⋅ exp t⋅k−⋅F UT1ð Þð Þ−1

��;

U tð Þ ¼ UT1−LT ⋅�1−

K2⋅LT

⋅�LTK

−UT1

K−1þ F UT1ð Þ

�C UTð Þ⋅ exp t⋅k−⋅F UT1ð Þð Þ þ 1C UTð Þ⋅ exp t⋅k−⋅F UT1ð Þð Þ−1

��:

ð39Þ

where C UTð Þ ¼ F UT0ð ÞþF UT1ð ÞþUT1K −UT0

K

F UT0ð Þ−F UT1ð ÞþUT1K −UT0

K

:

The normalisation of the solution (39) by the totalprotein concentration allows the definition of probabilityof the binding site to be in an occupied p1(UT, t) or un-occupied p0(UT, t) state, respectively, at a given total lig-and concentration:

p0 UT ; ηð Þ ¼ K2⋅LT

⋅

LTK

−UT1

K−1þ F UT1ð Þ

� C2 UTð Þ⋅ exp η⋅F UT1ð Þð Þ þ 1C2 UTð Þ⋅ exp η⋅F UT1ð Þð Þ−1

!;

p1 UT ; ηð Þ ¼ K2⋅LT

⋅

LTK

þ UT1

Kþ 1−F UT1ð Þ

� C2 UTð Þ⋅ exp η⋅F UT1ð Þð Þ þ 1C2 UTð Þ⋅ exp η⋅F UT1ð Þð Þ−1

!;

U ηð ÞK

¼ UT1

K−12⋅

LTK

þ UT1

Kþ 1−F UT1ð Þ

� C2 UTð Þ⋅ exp η⋅F UT1ð Þð Þ þ 1C2 UTð Þ⋅ exp η⋅F UT1ð Þð Þ−1

!;

ð40Þwhere η = t ⋅ k−.The concentration of the jth molecular form,

Mj(UT, t), of a protein with n independent bindingsites in response to a step change in ligand concen-tration is given by the product of probabilities accord-ing to Eq. (9):

Mj UT ; tð Þ ¼ LT ⋅Yn−1i¼0

pсi jð Þ UT ; tð Þ; j ¼ 0;…; 2n−1

ð41Þwhere ci(j) equals 0 or 1 for free and occupied sites,

respectively and j ¼Xn−1i¼0

2ici jð Þ.

The kinetics of the amount of ligand bound to a multi-site protein with n independent binding sites can bewritten as follows:

S UT ; tð Þ ¼X2n−1j¼0

Mj UT ; tð ÞXn−1i¼0

ci jð Þ: ð42Þ

The concentration of free ligand can be writtenas the difference between the total ligand con-centration and the bound ligand concentration(42):

U tð Þ ¼ UT1−S UT ; tð Þ; ð43ÞThe probabilities of the ith site to be free or occupied

respectively for the molecule with n independent bind-ing sites in this case:

p0i UT ; ηð Þ ¼ Ki

2⋅n⋅LT⋅

n⋅LTKi

−UT1

Ki−1þ F UT1ð Þ

� C UTð Þ⋅ exp η⋅F UT1ð Þð Þ þ 1C UTð Þ⋅ exp η⋅F UT1ð Þð Þ−1

!;

p1i UT ; ηð Þ ¼ Ki

2⋅n⋅LT⋅

n⋅LTKi

þ UT1

Kiþ 1−F UT1ð Þ

� C UTð Þ⋅ exp η⋅F UT1ð Þð Þ þ 1C UTð Þ⋅ exp η⋅F UT1ð Þð Þ−1

!;

ð44Þ

where F UTð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUTKi

− n⋅LTKi

� �2þ 2⋅ UT

Kiþ n⋅LT

Ki

� �þ 1

rand

C UTð Þ ¼ F UT0ð ÞþF UT1ð ÞþUT1Ki

−UT0Ki

F UT0ð Þ−F UT1ð ÞþUT1Ki

−UT0Ki

:

The dynamic alterations of the molecularforms with m bound out of n independentidentical binding sites Nm(UT, t) in response tothe total ligand concentration jump from UT0

to UT1 according to Eqs. (19) and (44) aregiven by:

Nm UT ; tð Þ ¼ LT ⋅ p1 UT ; tð Þ� �m⋅ p0 UT ; tð Þ� �n−m

: ð45ÞThe concentration of multisite protein conformations,

Nm, bound to m ligand molecules as a function of ligandconcentration in steady-state is given by:

Nm UTð Þ ¼ LTK

2nLT

nLTK

þUT

Kþ 1−F UTð Þ

� �� �m

� K2nLT

nLTK

−UT

K−1þ F UTð Þ

� �� �n−m

ð46ÞThe ligand concentrations, Um

max, for the maximalvalues of intermediate protein conformations, Nm(Um

max),can be found by differentiating Eq. (46) with respect toUT and solving dNm/dUT = 0 for UT:

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 16 of 19

Umaxm ¼ m⋅ LT þ K

n−m

� �ð47Þ

The corresponding maximal magnitudes for the inter-

mediate conformations are given by:

Nm Umaxm

� � ¼ LT ⋅mm

nn⋅ n−mð Þn−m ð48Þ

The half maximal effective ligand concentration, U00.5

and Un0.5, when the protein species equal half of the total

concentration LT, for the apo- and saturated multisiteprotein conformations respectively, in this case are givenby:

U0:50 ¼ �1− ffiffiffiffiffiffiffi

0:5np Þ⋅

n⋅LT þ K

ffiffiffiffiffiffiffi0:5

n

q !;

U0:5n ¼

ffiffiffiffiffiffiffi0:5n

p⋅

n⋅LT þ K

1−ffiffiffiffiffiffiffi0:5n

p!:

ð49Þ

Differentiating Eq. (45) with respect to t andsolving for dNm/dt = 0 yields the time τm

max whenthe concentration of multisite protein conforma-tions, Nm, bound to m ligand molecules ismaximal:

τmaxm ¼ H V mð Þð Þ; ð50Þ

where H xð Þ ¼ln

1þUT1K þF UT1ð Þþx

C UTð Þ⋅ 1þUT1K −F UT1ð Þþxð Þ

� �F UT1ð Þ⋅k− and V mð Þ ¼

LTK ⋅ n−2mð Þ:Equation (50) has an asymptote UT1

K for 1þ UT1K −F

UT1ð Þ þ x ¼ 0:

qm ¼ m⋅LTK

þ 1n−m

� �; ð51Þ

Equation (45) is further used to define thetime, τ0

0.5, required for the apo- form, N0, toreach half of the growth concentration and thetime period, τn

0.5, required for the fully saturatedprotein species to gain half of the growthconcentration:

τ0:50 ¼ H Wð Þ;τ0:5n ¼ H Yð Þ; ð52Þ

where

W ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−UT1

Kþ LT

K⋅nþ F UT1ð Þ−1

� �n

þ −UT0

Kþ LT

K⋅nþ F UT0ð Þ−1

� �n

2

n

vuuut−LTK

⋅n;

Y ¼ LTK

⋅n−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUT1

Kþ LT

K⋅n−F UT1ð Þ þ 1

� �n

þ UT0

Kþ LT

K⋅n−F UT0ð Þ þ 1

� �n

2

n

vuuut:

Ethics (and consent to participate)Not applicable.

Consent to publishNot applicable.

Availability of data and materialsThe data supporting the findings of this work are con-tained within the manuscript.

AbbreviationsCa2+: calcium ion; CaM: calmodulin; CaMKII: calcium/calmodulin-dependentprotein kinase II; CaN: calcineurin; CR: calretenin; PP2B: protein phosphatase2B, or calcineurin; TnC: troponin C.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsDVS and IIS developed and implemented the project under the supervisionof NVK, GIW, AIE, YU and MZQC. All authors contributed to the analysis ofthe model. YU and GIW provided biological interpretations of the results. Allauthors contributed to the writing of the final manuscript. All authors haveread and approved the final version of the manuscript.

AcknowledgementsWe would like to thank Professor Vadim Biktashev (University of Exeter), DrDaniel Roper and Dr Timur Abdullin (Kazan Federal University) for helpfulcomments and discussion.

FundingThis work was carried out under Severnside Alliance for TranslationalResearch (SARTRE) grant (GIW) and NSFC grant 61374053 (MZQC). This workwas co-funded by the subsidy allocated to the Kazan Federal University forstate assignment in the sphere of scientific activities and performed accord-ing to the ‘Russian Government Program of Competitive Growth’ of theKazan Federal University.

Author details1Kazan (Volga Region) Federal University, 18 Kremlyovskaya St., 420008 Kazan,Russia. 2Centre for Systems, Dynamics and Control, College of Engineering,Mathematics and Physical Sciences, University of Exeter, Harrison Building,North Park Road, Exeter EX4 4QF, UK. 3Department of MechanicalEngineering, The University of Hong Kong, Pokfulam Road, Hong Kong,China. 4Department of Dermatology, The Jikei University School of Medicine,3-25-8 Nishishimbashi, Minato-ku, Tokyo 105-8461, Japan. 5Biophysics &Bionics Lab, Institute of Physics, Kazan Federal University, Kazan 420008,Russia. 6Academic Renal Unit, School of Clinical Sciences, University of Bristol,Dorothy Hodgkin Building, Whitson Street, Bristol BS1 3NY, UK.

Received: 20 October 2015 Accepted: 13 April 2016

References1. Maki M, Maemoto Y, Osako Y, Shibata H. Evolutionary and physical linkage

between calpains and penta-EF-hand Ca2 + -binding proteins. FEBS J. 2012;279:1414–21.

2. Chillakuri CR, Sheppard D, Lea SM, Handford PA. Notch receptor-ligandbinding and activation: insights from molecular studies. Semin Cell Dev Biol.2012;23:421–8.

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 17 of 19

3. Grabarek Z. Insights into modulation of calcium signaling by magnesium incalmodulin, troponin C and related EF-hand proteins. Biochim Biophys Acta.2011;1813:913–21.

4. Yap KL, Ames JB, Swindells MB, Ikura M. Diversity of conformational statesand changes within the EF-hand protein superfamily. Proteins Struct FunctGenet. 1999;37:499–507.

5. Kawasaki H, Nakayama S, Kretsinger RH. Classification and evolution of EF-hand proteins. Biometals. 1998;11:277–95.

6. Canepari M, Maffei M, Longa E, Geeves M, Bottinelli R. Actomyosin kineticsof pure fast and slow rat myosin isoforms studied by in vitro motility assayapproach. Exp Physiol. 2012;97:873–81.

7. Oz S, Benmocha A, Sasson Y, Sachyani D, Almagor L, et al. Competitive andNon-competitive Regulation of Calcium-dependent Inactivation in Ca(V)1.2 L-type Ca2+ Channels by Calmodulin and Ca2 + -binding Protein 1. J BiolChem. 2013;288:12680–91.

8. Wei XY, Pan S, Lang WH, Kim HY, Schneider T, et al. Molecular Determinantsof Cardiac Ca2+ Channel Pharmacology - Subunit Requirement for theHigh-Affinity and Allosteric Regulation of Dihydropyridine Binding. J BiolChem. 1995;270:27106–11.

9. McCarron JG, Chalmers S, Olson ML, Girkin JM. Subplasma Membrane Ca2+Signals. Iubmb Life. 2012;64:573–85.

10. Kotov NV, Bates DG, Gizatullina AN, Gilaziev B, Khairullin RN, et al.Computational modelling elucidates the mechanism of ciliary regulation inhealth and disease. BMC Syst Biol. 2011;5:143.

11. Fuchs F, Grabarek Z. The Ca/Mg Sites of Troponin C Can ModulateCrossbridge-Mediated Thin Filament Activation in Rat Cardiac Myofibrils.Biophys J. 2011;100:112–3.

12. Pearson DS, Swartz DR, Geeves MA. Fast pressure jumps can perturbcalcium and magnesium binding to troponin C F29W. Biochemistry. 2008;47:12146–58.

13. Shifman JM, Choi MH, Mihalas S, Mayo SL, Kennedy MB. Ca2+/calmodulin-dependent protein kinase II (CaMKII) is activated by calmodulin with twobound calciums. Proc Natl Acad Sci U S A. 2006;103:13968–73.

14. Guo Q, Shen Y, Lee YS, Gibbs CS, Mrksich M, et al. Structural basis for theinteraction of Bordetella pertussis adenylyl cyclase toxin with calmodulin.EMBO J. 2005;24:3190–201.

15. Robison J C, RJ. Calcium/Calmodulin-Dependent Protein Kinases. In: WJLennarz ML, editor. Encyclopedia of Biological Chemistry: Elsevier Inc.Cambridge, Massachusetts: Academic Press; 2004. pp. 281-286.

16. Bhattacharya S, Bunick CG, Chazin WJ. Target selectivity in EF-hand calciumbinding proteins. Biochim Biophys Acta. 2004;1742:69–79.

17. Nelson MR, Thulin E, Fagan PA, Forsen S, Chazin WJ. The EF-hand domain: aglobally cooperative structural unit. Protein Sci. 2002;11:198–205.

18. Hoeflich KP, Ikura M. Calmodulin in action: diversity in target recognitionand activation mechanisms. Cell. 2002;108:739–42.

19. Schumacher MA, Rivard AF, Bachinger HP, Adelman JP. Structure of thegating domain of a Ca2 + -activated K+ channel complexed with Ca2+/calmodulin. Nature. 2001;410:1120–4.

20. Rodney GG, Moore CP, Williams BY, Zhang JZ, Krol J, et al. Calcium bindingto calmodulin leads to an N-terminal shift in its binding site on theryanodine Receptor. J Biol Chem. 2001;276:2069–74.

21. Yap KL, Ames JB, Swindells MB, Ikura M. Diversity of conformational states andchanges within the EF-hand protein superfamily. Proteins. 1999;37:499–507.

22. Valeyev NV, Bates DG, Heslop-Harrison P, Postlethwaite I, Kotov NV.Elucidating the mechanisms of cooperative calcium-calmodulin interactions:a structural systems biology approach. BMC Syst Biol. 2008;2:48.

23. Valeyev NV, Heslop-Harrison P, Postlethwaite I, Gizatullina AN, Kotov NV,et al. Crosstalk between G-protein and Ca2+ pathways switches intracellularcAMP levels. Mol Biosyst. 2009;5:43–51.

24. Valeyev NV, Heslop-Harrison P, Postlethwaite I, Kotov NV, Bates DG. Multiplecalcium binding sites make calmodulin multifunctional. Mol Biosyst. 2008;4:66–73.

25. Hyde JR, Kezunovic N, Urbano FJ, Garcia-Rill E. Spatiotemporal properties ofhigh speed calcium oscillations in the pedunculopontine nucleus. J ApplPhysiol. 2013;115:1402-1414.

26. Bading H. Nuclear calcium signalling in the regulation of brain function. NatRev Neurosci. 2013;14:593–608.

27. Semenov I, Xiao S, Pakhomova ON, Pakhomov AG. Recruitment of theintracellular Ca(2+) by ultrashort electric stimuli: The impact of pulseduration. Cell Calcium. 2013;54:145–50.

28. Faas GC, Schwaller B, Vergara JL, Mody I. Resolving the fast kinetics ofcooperative binding: Ca2+ buffering by calretinin. PLoS Biol. 2007;5:e311.

29. Adair GS. The hemoglobin system. VI. The oxygen dissociation curve ofhemoglobin. J Biol Chem. 1925;63:529–45.

30. Klotz IM, Hunston DL. Protein interactions with small molecules.Relationships between stoichiometric binding constants, site bindingconstants, and empirical binding parameters. J Biol Chem. 1975;250:3001–9.

31. Faas GC, Raghavachari S, Lisman JE, Mody I. Calmodulin as a direct detectorof Ca2+ signals. Nat Neurosci. 2011;14:301–4.

32. Andre I, Kesvatera T, Jonsson B, Akerfeldt KS, Linse S. The role ofelectrostatic interactions in calmodulin-peptide complex formation. BiophysJ. 2004;87:1929–38.

33. Andre I, Kesvatera T, Jonsson B, Linse S. Salt enhances calmodulin-targetinteraction. Biophys J. 2006;90:2903–10.

34. Zhang M, Tanaka T, Ikura M. Calcium-induced conformational transition revealedby the solution structure of apo calmodulin. Nat Struct Biol. 1995;2:758–67.

35. Chattopadhyaya R, Meador WE, Means AR, Quiocho FA. Calmodulinstructure refined at 1.7 A resolution. J Mol Biol. 1992;228:1177–92.

36. Aloy P, Russell RB. Structural systems biology: modelling proteininteractions. Nat Rev Mol Cell Biol. 2006;7:188–97.

37. Vetter SW, Leclerc E. Novel aspects of calmodulin target recognition andactivation. Eur J Biochem. 2003;270:404–14.

38. Nelson MR, Chazin WJ. An interaction-based analysis of calcium-inducedconformational changes in Ca2+ sensor proteins. Protein Sci. 1998;7:270–82.

39. Hill AV. The possible effects of the aggregation of the molecules ofhemoglobin on its dissociation curves. J Physiol. 1910;40:4–7.

40. Weiss JN. The Hill equation revisited: uses and misuses. FASEB J. 1997;11:835–41.41. Yang W, Lee HW, Hellinga H, Yang JJ. Structural analysis, identification, and

design of calcium-binding sites in proteins. Proteins. 2002;47:344–56.42. Friedberg F. Calcium Binding Protein Families: The ‘E-F Hand’ Family.

Biochem Educ. 1988;16:35–6.43. Kissinger CR, Parge HE, Knighton DR, Lewis CT, Pelletier LA, et al. Crystal-

Structures of Human Calcineurin and the Human Fkbp12-Fk506-CalcineurinComplex. Nature. 1995;378:641–4.

44. Kilhoffer MC, Roberts DM, Adibi AO, Watterson DM, Haiech J. Investigationof the mechanism of calcium binding to calmodulin. Use of anisofunctional mutant with a tryptophan introduced by site-directedmutagenesis. J Biol Chem. 1988;263:17023–9.

45. Holzer H. Metabolic Interconversion of Enzymes 1980 International TitiseeConference October 1st - 5th, 1980. Proceedings in Life Sciences, 0172-6625. Berlin, Heidelberg: Springer; 1981. p. 1. online resource.

46. Klee CB, Crouch TH, Krinks MH. Calcineurin: a calcium- and calmodulin-binding protein of the nervous system. Proc Natl Acad Sci U S A. 1979;76:6270–3.

47. Crouch TH, Klee CB. Positive cooperative binding of calcium to bovine braincalmodulin. Biochemistry. 1980;19:3692–8.

48. Klabunde RE. Cardiovascular physiology concepts, vol. xi. Philadelphia:Lippincott Williams & Wilkins/Wolters Kluwer; 2012. p. 243.

49. Stein WD, Lieb WR. Transport and diffusion across cell membranes, vol. xvii.Orlando: Academic; 1986. p. 685.

50. Ryerson S, Enciso GA. Ultrasensitivity in independent multisite systems. JMath Biol. 2014;69:977–99.

51. Bazanovas AN, Evstifeev AI, Khaiboullina SF, Sadreev II, Skorinkin AI, et al.Erythrocyte: A systems model of the control of aggregation anddeformability. Biosystems. 2015;131:1–8.

52. Bode W, Schwager P. The single calcium-binding site of crystallin bovinbeta-trypsin. FEBS Lett. 1975;56:139–43.

53. Minowa O, Yagi K. Calcium binding to tryptic fragments of calmodulin. JBiochem. 1984;96:1175–82.

54. Linse S, Helmersson A, Forsen S. Calcium binding to calmodulin and itsglobular domains. J Biol Chem. 1991;266:8050–4.

55. Cimmperman P, Baranauskiene L, Jachimoviciute S, Jachno J, Torresan J,et al. A quantitative model of thermal stabilization and destabilization ofproteins by ligands. Biophys J. 2008;95:3222–31.

56. Almagor H, Levitzki A. Analytical determination of receptor-liganddissociation constants of two populations of receptors from displacementcurves. Proc Natl Acad Sci U S A. 1990;87:6482–6.

57. Kragh-Hansen U. Graphical analysis of competitive binding of comparableconcentrations of ligand, inhibitor and protein. Ligand binding to serumalbumin. Biochem Pharmacol. 1983;32:2679–81.

58. Heeley DH, Belknap B, White HD. Maximal activation of skeletal musclethin filaments requires both rigor myosin S1 and calcium. J Biol Chem.2006;281:668–76.

Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 18 of 19

59. Stefan MI, Edelstein SJ, Le Novere N. An allosteric model of calmodulinexplains differential activation of PP2B and CaMKII. Proc Natl Acad Sci U S A.2008;105:10768–73.

60. Nishiyama M, Kimura Y, Nishiyama Y, Terazima M. Pressure-induced changesin the structure and function of the kinesin-microtubule complex. Biophys J.2009;96:1142–50.

61. Pearson DS, Holtermann G, Ellison P, Cremo C, Geeves MA. A novelpressure-jump apparatus for the microvolume analysis of protein-ligand andprotein-protein interactions: its application to nucleotide binding toskeletal-muscle and smooth-muscle myosin subfragment-1. Biochem J.2002;366:643–51.

62. Zhu T, Beckingham K, Ikebe M. High affinity Ca2+ binding sites ofcalmodulin are critical for the regulation of myosin Ibeta motor function. JBiol Chem. 1998;273:20481–6.

63. Rusnak F, Mertz P. Calcineurin: form and function. Physiol Rev. 2000;80:1483–521.

64. Lu HE, MacGillavry HD, Frost NA, Blanpied TA. Multiple spatial and kineticsubpopulations of CaMKII in spines and dendrites as resolved by single-molecule tracking PALM. J Neurosci. 2014;34:7600–10.

65. Lee SJ, Escobedo-Lozoya Y, Szatmari EM, Yasuda R. Activation of CaMKIIin single dendritic spines during long-term potentiation. Nature. 2009;458:299–304.

66. Shaikh TR, Barnard D, Meng X, Wagenknecht T. Implementation of a flash-photolysis system for time-resolved cryo-electron microscopy. J Struct Biol.2009;165:184–9.

67. Frieden C, Hoeltzli SD, Ropson IJ. NMR and protein folding: equilibrium andstopped-flow studies. Protein Sci. 1993;2:2007–14.

68. Nagerl UV, Novo D, Mody I, Vergara JL. Binding kinetics of calbindin-D(28 k)determined by flash photolysis of caged Ca(2+). Biophys J. 2000;79:3009–18.

69. Plaza-Menacho I, Barnouin K, Goodman K, Martinez-Torres RJ, Borg A, etal. Oncogenic RET kinase domain mutations perturb theautophosphorylation trajectory by enhancing substrate presentation intrans. Mol Cell. 2014;53:738–51.

70. Sadreev II, Chen MZ, Welsh GI, Umezawa Y, Kotov NV, et al. A systemsmodel of phosphorylation for inflammatory signaling events. PLoS One.2014;9:e110913.

71. Cutillas PR, Geering B, Waterfield MD, Vanhaesebroeck B. Quantification ofgel-separated proteins and their phosphorylation sites by LC-MS usingunlabeled internal standards: analysis of phosphoprotein dynamics in a Bcell lymphoma cell line. Mol Cell Proteomics. 2005;4:1038–51.

72. Zorba A, Buosi V, Kutter S, Kern N, Pontiggia F, et al. Molecular mechanismof Aurora A kinase autophosphorylation and its allosteric activation by TPX2.Elife. 2014;3:e02667.

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