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

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RESEARCH ARTICLE Open Access Kinetic regulation of multi-ligand binding proteins Diana 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 a concentration-dependent manner. For example, calcium binding has been shown to induce conformational transitions in the calcium-dependent , under steady state conditions. However, intracellular concentrations of these second messengers are often subject to rapid change. The mechanisms underlying dynamic 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 ligand concentrations is presented. Two major physiological scenarios are investigated: i) Ligand concentration is abundant and the ligand-multisite protein binding does not affect free ligand concentration, ii) Ligand concentration is of the same order of magnitude as the interacting multisite protein concentration and does not change. Therefore, buffering effects significantly influence the amounts of free ligands. For each of these scenarios the influence of the number of binding sites, the temporal effects on intermediate apo- and fully saturated conformations and the multisite regulatory effects on target proteins are investigated. Conclusions: The developed models allow for a novel and accurate interpretation of concentration and pressure jump-dependent kinetic experiments. The presented model makes predictions for the temporal distribution of multisite protein conformations in complex with variable numbers of ligands. Furthermore, it derives the characteristic time and the dynamics for the kinetic responses elicited by a ligand concentration change as a function of ligand concentration and the number of ligand binding sites. Effector proteins regulated by multisite ligand binding are shown to depend on ligand concentration in a highly nonlinear fashion. Keywords: Ligand-receptor binding, Transient kinetics, Calcium, Calmodulin

Background ubiquitous protein, calmodulin as well as troponin A wide variety of intracellular events are initiated via andotherEF-handcontainingproteins[4,5].Tem- temporal change of ligand concentrations. One of the poral elevation of intracellular free Ca2+ is the key most important ligands in many cells is calcium (Ca2+). regulatory factor of the Ca2+-dependent protein activ- Calcium interacts with and regulates the activities of a ity [6–10]. The characteristics of the induced signal large number of calcium-binding proteins as well as are not fully understood. It remains to be determined numerous effectors. The number of functional calcium how a single ligand is able to govern numerous intra- binding sites within these proteins can range from one cellular properties. or two to ten or more [1–3]. The most common number Whilst the multisite ligand binding is not limited to of calcium binding sites is four: as observed in the most Ca2+ signaling, Ca2+ is probably the most versatile ion, regulating the largest number of cellular events. Several 2+ * Correspondence: [email protected] Ca -binding proteins can be considered as examples of Diana V. Salakhieva and Ildar I. Sadreev are first authors. multisite ligand protein interactions. Structural biology † Equal contributors investigations of calcium binding proteins in complexes 3Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China with target protein peptides have suggested that the 2+ Full list of author information is available at the end of the article specificity in Ca -CaM binding protein-dependent

© 2016 Salakhieva et al. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the 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 Page 2 of 19

target activation arises from the diversity of interaction Mathematical modelling of multisite protein kinetics in interfaces between the Ca2+-regulated protein and its response to rapid ligand changes presented in this paper target proteins [5, 11–21]. The most ubiquitous protein, provide new insights into the mechanism of conform- calmodulin (CaM), consists of two globular domains, ational kinetics of multisite proteins in complex with each domain containing a pair of helix-loop-helix Ca2 variable number of bound ligands for the two distinct +-binding motifs called EF-hands [1, 3, 5, 16, 17]. In earl- physiological situations described: ier studies the authors demonstrated that in addition to the diversity of CaM-target interfaces; the CaM selectiv- i) When the ligand concentration significantly exceeds ity emerges from its target specific Ca2+-affinity; the protein concentration, number of Ca2+ ions bound and the target specific coop- ii) When the total amount of ligand is conserved and erativity [22–24]. comparable with the protein concentration. Another major factor that contributes to the selectiv- ity of seemingly simultaneous regulation of several In the first case, the buffering effects are negligible multisite Ca2+ binding proteins and Ca2+-mediated whereas in the latter, the ligand-protein interactions processesisthetemporalalterationsofCa2+ [25–27]. have a significant impact on the amount of available The remarkable variety of Ca2+ signals in cells, ranging ligand and the binding kinetics. In this work, the equa- from infrequent spikes to sustained oscillations and tions for the dependence of the characteristic time plateaus, requires an understanding of how fast intracellu- constants and the temporal distribution of individual lar calcium changes regulate the kinetics of multiple conformations as a function of the ligand concentra- multisite Ca2+ binding proteins. Therefore mathematical tion, the number of binding sites and the binding modeling of Ca2+ jump induced responses could prove to affinities have been derived. The impact of the number be invaluable in the interpretation of transient kinetic of binding sites, temporal effects on conformations, experiments. and regulation by multisite proteins of their effector Cooperative binding is a special case of molecular in- proteins have been investigated by employing the devel- teractions where ligand binding to one site of a mol- oped models. The analysis of the ligand-multisite pro- eculedependsontheligandbindingtotheothersites. tein mediated regulation of effector proteins suggests The first quantitative determination of the dynamic that significant degree of selectivity in regulation can be properties of cooperative binding was proposed by [28]. achieved by a single ligand by employing mechanisms In this work the authors emphasized the significance of described in this study. the by studying the fast dynamics of Ca2+ binding to calretenin (CR), which has one independent and four cooperative binding sites. The investigation of Results cooperative effects of Ca2+ binding to CR was per- A new model for multisite protein ligand binding kinetics formed both experimentally and using mathematical The majority of studies of the activation of multisite pro- modeling. The authors employed the simplified version teins consider only the ligand concentration-dependent of the Adair-Klotz model [29, 30] to describe the dy- profiles. One of the interesting questions about these namics of the interactions involved in Ca2+ binding to multisite proteins is how temporal alterations of ligand CR. This approach was then extended to the binding of concentration contribute to their function. The shape of Ca2+ to CaM [31]. The models proposed in these stud- distribution of multisite proteins in complex with variable ies [28, 31] demonstrated excellent fitting results to the numbers of bound ligand is known or can be experimen- experimental data, in comparison with the previously tally elucidated in many cases [4, 5, 16, 22–24, 36–38]. published models. However, the described approach is However, the role of temporal transitions caused by fast rather limited, as it describes fitting instead of provid- alteration of ligand concentration on multisite proteins ing a mechanistic description. An alternative method- and on multisite protein-regulated target proteins remains ology offered by [28, 31] is not directly applicable from unclear. It is reasonable to assume that there can be at the physical and chemical point of view because the least two distinct mechanisms of fast ligand alteration- Adair-Klotz model for sequential ligand binding was mediated effects exhibited in two distinct system scenar- utilized [29, 30] whereas binding of Ca2+ to EF-hand ios: i) the ligand concentration is significantly greater than proteins [32–35] is non-sequential [22]. Given the im- the multisite protein concentration, ii) the ligand is com- portance of studying fast Ca2+ binding kinetics and the parable with the multisite protein concentration. In the lack of understanding of the underlying mechanisms, first scenario ligand binding to multisite protein leads to we developed a detailed mathematical model for ligand insignificant changes of free ligand, whereas in the second binding to multisite proteins with both cooperative and case, free ligand concentration can vary substantially when independent binding sites. ligand molecules bind to the multisite proteins. This paper Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 3 of 19

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

The model for abundant ligand concentration In this model we describe physiological situations where the ligand concentration significantly exceeds the multi- site protein concentration. In a previous study [22, 24] the authors analyzed functions for the probability of an individual site being in the bound or non-bound state and a function giving the probability of a multisite pro- tein being in a complex with different number of bound ligands (Eqs. (2) and (3) in Methods) [29, 39, 40]. To investigate the kinetics of the multisite protein ligand interactions the present study extends the previous model to consider the ligand concentration as a func- tion of time (Eqs. (4) and (5) in Methods). The solu- Fig. 1 The effect of the number of binding sites on intermediate tions for the individual sites to be in a particular state conformations. The maximum magnitude of protein conformations in complex with one, two and three ligand molecules are shown as were obtained for those cases where ligands are subject a function of the total number of binding sites. The relative amount to rapid changes between steady-states (Eqs. (6) and (7) of ligand binding by conformations bound to a specific number of in Methods). Due to the large number of sites involved, sites clearly diminishes as the number of sites grow knowledge of the state probability distribution for indi- vidual binding sites allows accurate estimation of the dynamics of the total concentration of bound ligand in ligand-multisite complexes decreases dramatically as the response to a jump in free ligand concentration (Eq. number of binding sites increases. The presented results (11) in Methods). suggest that the relative magnitude of individual inter- In order to gain more insights into the distribution mediate conformations decreases as the number of bind- of the intermediate protein conformations (complexes ing sites increases. This in turn results in subtler with variable number of bound ions) we investigated regulatory effects of those proteins with larger number the case of a multisite protein with identical binding of ligand binding sites. For example, in CaM that has sites (Eq. (12) in Methods). While this case is a rela- four binding sites for calcium [18], the presence of four tively rare occurrence in living cells, it enables insight sites leads to the increased multifunctionality of this pro- into the role that the number of binding sites plays in tein due to the additional regulatory properties of inter- cellular signalling. There are several examples of pro- mediate conformations [24]. However, as an exception tein families that have variable number of ligand bind- some forms of CaM have six binding sites [44, 45]. In ing sites either due to their structural properties or by this case, according to Fig. 1 the magnitude of inter- them forming large tertiary complexes. For example mediate conformations significantly decreases compared members of the Ca2+ family of binding proteins can to the case of four binding sites, resulting in decreased differ in the number of ligand binding sites [41, 42]. regulatory properties of the protein. The presence of The most ubiquitous Ca2+-binding protein, calmodulin more than one results in increased multi- (CaM), contains four Ca2+ binding sites as does tropo- functionality of the protein but at the same time leads nin (TnC) [18] and calcineurin phosphatase (CaN) the decrease of the regulatory effects. Thus, it seems that [43]. However the number of functional Ca2+ binding there is an “optimal” number of binding sites, which sites can vary from two to ten as in the protease, cal- have been developed during the evolution, for instance pain [1] or even more in other cases [2]. To investigate four calcium binding sites in CaM. the role that the number of ligand binding sites plays Next, the ligand concentrations for half maximum ef- 0.5 0.5 in multisite kinetics, the ligand concentrations, at which fective ligand concentrations, U0 and Un ,forthe the intermediate conformations reach their maximum apo- and saturated multisite protein conformations values, (Eq. (13) in Methods) and the corresponding mag- when the protein species equals 50 % of the total con- nitudes for those conformations (Eq. (14) in Methods) centration were estimated (Eqs. (15) in Methods). This were estimated. solution shows that the ligand concentration for the Figure 1 shows the maximum protein conformations half maximal protein activity, known as EC50, would be in complex with one, two and three ligands as a function equal to the equilibrium dissociation constant K (EC50 of the number of binding sites (Eq. (14) in Methods). = K) for proteins with one binding site only (n =1). 0.5 0.5 The graph demonstrates that the magnitude of the Figure 2a shows the dependence of U0 /K and Un /K, Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 4 of 19

Fig. 2 Model predictions for the half-maximal effective ligand concentrations as a function of the number of binding sites and ligand concentration. 0.5 0.5 a. The dependence of the half-maximal effective ligand concentration, U0 /K and Un /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 with two, three, four and five binding sites. c. The difference between ligand concentrations for the saturated multisite protein conformations when the protein species equal to 90 % and 10 % of the total concentration as a function of the ligand concentration for the proteins with one to six binding sites. d. The difference between ligand concentrations for the saturated multisite protein conformations when the protein species equal to 90 % and 10 % of the total concentration as a function of the number of binding sites up to six

on the number of binding sites. The model predicts of calcium concentrations, the switching between cal- that there is a significant change in the required ligand cium channel opening and closure takes place [49]. 0.5 concentration Un /K for the fully bound conformation, The difference An between ligand concentrations 0.5 0.9 0.1 while U0 /K does not change with time. U /K and U /K for the saturated multisite protein In the previous study the authors reported on the regu- conformations, when the protein species are equal to latory importance of the distribution of individual multi- 90 % and 10 % of the total concentration, as a function site protein conformations [22, 24]. Here calculations of the ligand concentration for proteins with different (Eqs. (15) in Methods) are presented for the half-width be- number of binding sites is shown in Fig. 2c and d re- tween the half-maximal effective ligand concentrations spectively. The determination of An allows an under- U0.5/K as a function of the ligand concentration for the standing as to how an increase of the number of intermediate conformations (one bound ligand) of the binding sites n affects the steepness of the dose- proteins with different number of binding sites (Fig. 2b). response curve (Fig. 2c). It can be seen from Fig. 2c The observation of the half-width for the concentra- and d that with an increase of n,thewidthAn decreases tions of intermediate conformations that have a bell- while the steepness increases and shifts to the range of shaped dependence on ligand concentration, enables higher ligand concentrations U/K. The greater steep- the range of physiologically plausible concentrations ness of the dose-response curve caused by the presence of ligand, where protein functions can be regulated by of increasing number of binding sites (Fig. 2c) may re- intermediate conformations to be obtained. For ex- sult in the switch-like response and of ample, in Fig. 2b the half-width range of calcium con- the protein activation [50]. For instance, the steepness centrations is approximately from -1 to 1 on the of CaM activation defines the threshold properties for logarithmic scale, which corresponds to 10-7M-10-5M the switching of erythrocyte aggregation and deform- due to the fact that the affinity of calcium binding ability from one steady-state to another [51]. sites in CaM is approximately 10-6M [46, 47]. Inter- Equation (17) for the total amount of bound ligand estingly, the range 10-7M-10-5M corresponds to the in the case of multisite protein with identical binding physiological range of intracellular calcium concentra- sites, can be used to estimate the amount of bound lig- tions in cardiac muscle cells [48]. Within this range and when the ligand concentration is equal to the Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 5 of 19

0.5 equilibrium dissociation constant (U = K). Our model The bell shaped dependence of τ4 for a protein with predicts that the ligand concentration for the half maximal four binding sites, for example CaM [18], on final ligand protein activation, EC50, is equal to the equilibrium dis- concentration shown in Fig. 3d appears to be related to sociation constant K for any number of bound sites for a the presence of intermediate conformations with one, multisite protein with identical binding sites. twoandthreeboundsites.Inasinglesitemolecule Figure 3 shows the calculations for the temporal char- (n = 1), for example crystalline bovine β-trypsin [52] acteristics of the apo- and fully bound species. Figure 3a that is characterized by the absence of intermediate 0.5 and b show that the temporal shapes of the apo- and forms, τ4 depends on U1/K monotonically, i.e. there fully bound conformations (Eqs. (19) in Methods) in re- is no bell shaped dependence, as is evident from Eqs. sponse to a ligand change are similar to the steady-state (24). In CaM (n = 4), the activation of intermediate 0.5 dependence of the same conformations on ligand con- conformations takes extra time, which affects τ4 . 0.5 0.5 centration [24]. The kinetic parameters, τ0 and τ4 can This activation precedes the activation of the satu- be estimated as the time required to reach 50 % of the rated form in time, and also as U1/K increases, i.e. total concentration (Fig. 3a and b). The described kinetic the intermediate conformations are more prevalent parameters have been investigated as a function of the for smaller values of U1/K, and the stationary distri- initial and final ligand concentrations (Fig. 3c, d and bution shifts towards the saturated form for larger Eqs. (24) in Methods). Our analysis reveals a reduction U1/K.Asaresult,anincreaseofU1/K leads to an 0.5 of the time constant, τ0 , of the apo- conformation with increase of the contribution of the kinetics of the the reduction of the initial ligand concentration and an intermediate complexes to the overall dynamics, and 0.5 increase of the final ligand concentration. However, the hence to an increase of τ4 .ForlargerU1/K the role 0.5 dependence of the characteristic time τ4 (Fig. 3d) of the intermediate conformations is less important 0.5 showed an unexpected bell shaped dependence on the and overall speed-up dominates, hence τ4 de- final ligand concentration compared to the simpler creases. Thus the model predicts that there is an 0.5 monotonic dependence for τ0 (Fig. 3c). The model pre- intermediate ligand concentration, at which the kin- 0.5 dicts that there is an “optimal” ligand concentration for etics of the fully saturated form, represented by τ4 , the saturation effect to take the longest time (Fig. 3d). 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 was

investigated for apo- (a) and fully bound (b) forms in response to the non-dimensional ligand concentration change from U0/K =0.1toU1/K =10asa − 0.5 function of non-dimensional time η = t ⋅ k . The dotted lines indicate the time, τ0 , required for the non-dimensional concentration of apo- 0.5 conformation, N0/LT, to reach half of the fall in concentration and the period of time, τ4 , 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 the 0.5 initial (c) and final (d) ligand concentrations. The presented analysis clearly demonstrates the bell shaped dependence of τ4 on the final ligand concentration Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 6 of 19

The affinities of binding sites differentially affect the with four different ligand binding sites” in Methods) kinetic responses of intermediate conformations with marginally (Fig. 5) and significantly different associ- The proposed model has been employed to investigate ation constants (Fig. 6). The main result that follows the ligand jump-dependent kinetics of both saturated from the analysis of the intermediate conformation and non-saturated conformations. Initially, an idealized curves is that the affinities of the different binding sites model of a multisite protein with identical binding sites mainly affect the magnitudes of corresponding protein was used to investigate the impact of ligand concentra- conformation. For example, the conformation of a multi- tions on the multisite protein kinetics. However, in living site protein corresponding to the one ligand bound state cells there are very few proteins (if any) that have identi- is present in lower concentration if the affinity of the cal ligand binding sites. Therefore the model was ex- binding centre is lower. However the overall shape of tended to examine the implications of variations in the concentration dependent profile has not changed. binding site affinities on the predicted concentration- This property is very similar to the case of steady-state response profiles. dependence on the ligand concentration. The only dif- Figure 4 shows the dependence of the time point ference is that the bell shape dependence on time during max − τm k when the intermediate protein conformations the kinetic response is partially skewed. However, signifi- reach the maximum as a function of magnitude of ligand cant variation in affinities changes the magnitude, and jump (Eqs. (26) in Methods). It can be seen from Eqs. also leads to the asynchronous kinetics of the intermedi- max − max − max − (26) that τ1 k , τ2 k and τ3 k do not exist for U1= ate conformations. K < 1 U K U K 3 , 1/ < 1 and 1/ < 3, respectively. Under these The effects of cooperativity in Ca2+ binding to CaM special cases, where the ligand concentration U1/K is not sufficient for the concentrations of the intermediate In order to investigate the influence of cooperativity, we conformations to reach their maximal values, these con- chose a well-characterized protein, CaM, as the model centrations monotonously grow to their respective object. The CaM protein contains two independent EF- hand globular domains, with two binding sites [1, 3, 5, steady-state levels. According to Eq. (26), the values U1= 16, 17]. The sites within each of the domains coopera- K < 1 ; U /K < 1 and U /K < 3 correspond to the three 3 1 1 tively influence each other. It has been reported that co- individual intermediate conformations with one, two operative binding occurs between two neighbouring sites and three bound sites respectively. within the N- and C- terminal domains of CaM [22, 53, To investigate the impact of the dissociation constants 54]. Figure 7 shows the model predictions for CaM of individual binding sites we employed the multisite where we assume that the molecule has two independ- protein model (please see subsection “Multisite proteins ent domains, with two identical cooperative sites. In the first domain the affinity of one site changes from K1 = c 0.9 μMtoK1 = 0.2 μM if the other site is occupied and in the second domain the affinity changes from K2 = c 0.8 μMtoK2 = 0.1 μM (Eq. (28) in Methods) [22]. Figure 7a and b show the influence of cooperativity on the steady-state concentrations of CaM with certain number of bound sites. The presence of cooperativity shifts the dose-response characteristics along the ligand concentration axis and changes the magnitude of inter- mediate conformations allowing more developed select- ive regulation of the activity of CaM. The investigation of 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 proposed model predicts that the cooperative binding leads to Fig. 4 Characteristic time required for intermediate conformations more pronounced selective effects for intermediate con- to reach their maximum levels as a function of the step change magnitude. The analysis shows that the non-dimensional time formations and higher differences between the initial max max − (ηm = τm k , where m = 1, 2 and 3) required for reaching the and steady-state levels for the apo- and saturated forms. maximum level of the intermediate species, is inversely proportional The results of the present analysis suggest that coopera- U K to the concentration of the applied ligand 1/ . This effect is due to tivity plays an important role in the regulation of the activ- the growing abundance of the free ligand concentration available ity of multisite proteins by allowing wider possibilities for for faster interaction with the multisite protein selectivity. However, the presence of cooperativity leads to Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 7 of 19

Fig. 5 Kinetics predictions for multisite protein species with marginally different association constants. The kinetics of multisite protein species

was 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 respectively in b in response to step change of ligand from U0/K =0.001toU1/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 forapo-(c) and fully bound (d) forms. The calculations show that the final level of the multisite protein species are defined by the ligand concentration after the step change. It is very clear that the fully bound species are not saturated and most of the ligand is distributed among

species bound to fewer ligands. However, step change application of ligand with much higher concentration from U0/K = 0.001 to U1/K = 100 for apo- (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 multisite

protein 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 different − − − − association 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 8 of 19

Fig. 7 Comparative analysis of cooperative versus non-cooperative Ca2+ binding to CaM. Two mathematical models for Ca2+-Cam interactions are compared under the assumptions for the presence and absence of cooperative binding. The comparison between the two scenarios was performed under steady state conditions (a), (b) and in response to a step change in Ca2+ concentration (c), (d). The model predicts that the cooperativity influences the

maximums of the concentrations for the intermediate forms (N1/LT, N2/LT and N3/LT)aswellasthesteady-statelevelsofapo-(N0/LT) and fully saturated forms (N4/LT). However, the difference observed in the distribution of the conformation species in the presence and absence of the cooperative binding is quantitative while the overall shape of the distributions remains unchanged. Due to this finding the following model analysis was performed without cooperative binding assumptions

quantitative rather than qualitative changes in the system. binding sites. A borderline case where the free ligand con- The introduction of cooperative binding is crucial for the centration is barely affected by interaction was considered experimental data fitting but at the same time brings fur- (Fig. 8a and b) as well as a smaller total ligand concentration ther complexity to the system, which does not necessarily where the free ligand is nearly exhausted as a result of buff- lead to a better understanding of the underlying mecha- ering by the multisite protein (Fig. 8c and d). The compari- nisms. As a result of this, further analysis was carried out son of the free ligand concentration (Eqs. (43) and without considering this effect. (44) in Methods) for these two cases is shown in Fig. 8e. The model predicts that the strongest effect The model for comparable ligand and protein of the ligand availability can be observed for the mul- concentrations tisite protein conformations with three and four (fully The previous sections considered the physiological case saturated) bound ligands. A possible explanation for where the ligand concentration was above saturation level this phenomenon may be that the multisite protein meaning that the ligand-multisite protein interactions did conformations, which form complexes with smaller not affect the availability of the ligand. Here the case when number of ligand molecules by definition, do not re- the amount of ligand is limited is considered. This can quiresignificantamountofligandandasaresultare occur in cases when the ligand concentration level is com- not strongly affected under conditions when the free parable to the multisite protein availability. The model ligand is limited. Whereas the multisite protein inter- predictions show the relative redistribution of the ligand actions with the larger number of ions occur after the in the free and bound states. The responses of a multisite significant amount of ligand is “used up” to form the protein with four identical binding sites to ligand concen- intermediate conformations, however is still required tration step change (Eqs. (44) and (45) in Methods) for conformations with larger number of ions. As a for two different protein concentrations, LT/K =2 and result the final levels of the conformations with three LT/K = 50, were studied (Fig. 8). Instead of considering and four ions are affected. It can also be seen from absolute ligand concentrations, the approach considered Fig. 8 that the shapes of the intermediate conforma- ratios of the ligand concentration to the affinities of the tions time lines are skewed. Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 9 of 19

Fig. 8 The comparison between the cases where the free ligand concentration is barely affected by interaction and exhausted as a result of

buffering. The kinetics of multisite protein species alterations in response to step change in ligand concentration from UT0/K = 0.01 to UT1/K = 200 for two different ratios of the protein concentration to the affinities of the binding sites LT/K =2a for the intermediate species N1/LT, N2/LT and N3/LT, b for the apo- and fully bound species N0/LT and N4/LT respectively) and LT/K =50c for the intermediate species N1/LT, N2/LT and N3/LT, d for the 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 the multisite protein in the case of limited amount of ligand, the multisite protein is unable to become fully saturated after the step change in ligand, and the majority of the ligand becomes distributed among the intermediate species. e. The comparison of the dynamics of free ligand concentration 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 total protein concentration to the binding constant is LT/K =50

Figure 9a shows the model predictions for the charac- be explained by the presence of intermediate confor- teristic time required for intermediate protein conforma- mations. The distortion of the bell shape in Fig. 9b tions with one, two and three bound ligands to reach appears to be due to the ligand consumption that is their highest concentrations in response to a step change included into the consideration in this section and in ligand concentration (Eq. (50) in Methods). The was not considered in Fig. 3d. This result may be sig- 0.5 model predicts that the characteristic time τ0 , required nificant for the dynamics of CaM activation as our for the apo- form to reach its half growth level mono- model predicts that with an increase of the total lig- tonically decreases with the increase of the total ligand and concentration, the limited amount of ligand leads 0.5 concentration. However, the characteristic time constant to an additional increase of τ4 compared to the case 0.5 τ4 , which represents the saturated conformation reveals without the ligand consumption. The model, there- a distorted bell shaped dependence on ligand concentra- fore, predicts possible transient differences in multi- tion (Fig. 9b and Eqs. (52) in Methods). This bell shaped site protein signal transduction in response to fast dependence, which was also observed in Fig. 3d, can also transient kinetics of multisite proteins. Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 10 of 19

Fig. 9 Model predictions for the time required for multisite protein conformations to reach their maximal and half growth concentrations. The max − 0.5 − 0.5 − non-dimensional characteristic times, τm k (m = 1, 2, 3) for intermediate (a), τ0 k for apo- and τ4 k for fully bound (b) multisite protein conformations are shown as a function of the step change ligand concentration from UT0/K = 0.1 to UT1/K and LT/K =3

Discussion The presented model offers a new tool for the interpret- In this paper we analysed multisite protein kinetics in ation of transient kinetics experiments performed by the response to rapid changes in ligand concentrations. The flash photolysis and stopped-flow techniques [66–68]. model for multisite protein kinetics for variable number Availability of a caged analogue of the necessary reactant of bound ligands was developed for two physiological is considered as one of the biggest problems of flash pho- cases: when the concentration of ligand is much higher tolysis [66], which can be overcome by employing the pro- [55, 56] and when it is comparable with the concentra- posed model with the low amount of protein. A less tion of the multisite protein [57]. The results obtained obvious area of application for this methodology is the by the proposed model allow for an accurate interpret- kinetics of proteins with multiple phosphorylation sites ation of the experimental data for the concentration of [69–72]. This work shows that the highly versatile intra- multisite proteins such as CaM, TnC, CaN and other cellular multifunctionality of multisite proteins is achieved Ca2+ dependent secondary messengers regulated by Ca2+ not only by the order of ligand-protein interaction and the ions. The kinetic effects in response to ligand binding number of bound ligands, but also by temporal regulation. [13, 58, 59], required for understanding of pathway regu- lation, can be interpreted by the presented model. Conclusions The approach developed in this project is applicable to The developed in this study models for the kinetics of a number of areas of kinetic experiments. One of them the multisite ligand-receptor binding for the two physio- is widely used techniques to study chemical kinetics logical cases, where the ligand concentration is abundant using the pressure jump technique [12, 60, 61]. Accord- and comparable with the protein concentration, make ing to our model, the effects induced by rapid change in universal predictions for the temporal distribution of pressure leading to the change in protein-ligand interac- multisite protein conformations in complex with variable tions [12, 60, 61], can be interpreted and explained by numbers of ligands. The strongest effect of the ligand the alteration in the affinities of the binding constants of availability is observed for the multisite protein confor- multisite proteins. The time dynamics of the individual mations with larger numbers of bound ligands. The two multisite protein species can offer new insights into the models show that the concentration of individual multi- underlying biophysical mechanism of ligand-protein in- site protein conformations changes with time nonli- teractions in response to fast change. nearly and that the temporal distribution for the Our model shows that the concentration of intermedi- concentrations of the intermediate conformations repre- ate conformations as a function of time represents sents skewed bell shapes. The models derive the charac- skewed bell shapes. We found that as the protein con- teristic times and the dynamics for the kinetic responses centration rises, free ligand concentration becomes elicited by a ligand concentration change as a function exhausted [13]. This result is consistent with experimen- of ligand concentration and the number of ligand bind- tally observed Ca2+-CaM-dependent inhibition of my- ing sites. The developed models allow for a novel and ac- osin motor function [62]. The results obtained by this curate interpretation of concentration and pressure jump- model increase our understanding of differential activa- dependent kinetic experiments. Our models are applied to tion of protein phosphatase 2B (PP2B) [59, 63] and cal- study the kinetics of calmodulin, however the models also cium/calmodulin-dependent protein kinase II (CaMKII) provide universal predictions and allow us to extend the [64, 65] kinetics. PP2B binding increases the affinity of understanding of a large number of multisite binding- CaM for its targets [59] and, therefore, is likely be acti- regulated circuits and the mechanisms underlying dy- vated by low amounts of calcium. namic ligand-dependent regulation of multisite proteins. Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 11 of 19

Methods The probability of a multisite protein being occupied Kinetic properties of multisite proteins with n binding by ligand molecules as it was shown in Eq. (3) as a func- sites tion of time can be reformulated as follows: Here, we describe mathematical equations used to de- Yn−1 scribe dynamic interactions of ligand molecules with ciðÞj n PjðÞ¼U; t pi ðÞU; t ; j ¼ 0; …; 2 −1: ð4Þ multisite proteins. The model described in this section i¼0 extends our previous analysis for multisite protein inter- ciðÞj actions under steady-state conditions [22–24]. pi ðÞU; t can be determined by considering the ori- ginal set of ordinary differential equations for single site Multisite protein with independent ligand binding sites interaction of a protein with a ligand according to Eq. The kinetic scheme for such interaction can be repre- (1) [24]: sented as follows: dL0ðÞU; t i þ 0 − 1 kþ ¼ −ki ⋅Li ðÞU; t ⋅U1 þ ki ⋅Li ðÞU; t ; i dt 0 1 Li þ U ⇄ Li ; i ¼ 0; …; n−1; ð1Þ dL1ðÞU; t ð Þ k− i þ − 5 i ¼ k ⋅L0ðÞU; t ⋅U −k ⋅L1ðÞU; t ; dt i i 1 i i 0 0 1 where Li is the ith binding site of the multisite protein Li ðÞþU; t Li ðÞ¼U; t LT ; U L1 i in unbound state, is a ligand molecule, i is the th 0 where Li (U, t) is the concentration of a free binding site, binding site of the multisite protein being occupied, 1 + − Li (U, t) is the concentration of a bound ligand molecule ki and ki are the association and dissociation rates and LT is the total number of the protein molecules. respectively. 0 K i Here we use steady-state solutions Li ðÞ¼U LT The probabilities for the ith binding site to be not K iþU 1 U L ðÞ¼U LT occupied or occupied as a function of ligand concentra- and i K iþU as initial conditions for the ligand U tion are given by: concentration jump from U0 to U1. A particular solution K for the system of differential Eqs. (5) in response to the p0ðÞ¼U i ; U U i K þ U ligand concentration shift from 0 to 1 is given by: i  ð2Þ K i K i K i t U L0ðÞ¼U; t L ⋅ − − ⋅ − ; p1ðÞ¼U ; i T K þ U K þ U K þ U exp τðÞU i i 1 i 1 i 0 1 K i þ U U U U t L1ðÞ¼U; t L ⋅ 1 − 1 − 0 ⋅ − ; i T K þ U K þ U K þ U exp τðÞU − i 1 i 1 i 0 1 ki where K i ¼ þ : ð Þ ki 6 There are two possible states of a binding site: occu- k− τðÞ¼U K i ; K ¼ i : pied or not occupied. Since the number of sites in the where 1 k−⋅ðÞU þK i kþ n n i 1 i i molecule is , there are 2 possible molecular forms, i.e. The normalisation of the solution (6) by the total pro- the states characterized by combinations of bound and tein concentration allows the definition of probability of free sites. 1 the ith binding site to be in an occupied pi (U, t) or un- The probability for a multisite protein with independ- 0 occupied pi (U, t) state, respectively, at a given ligand ent binding sites to be in a particular molecular form is concentration: given by multiplications of probabilities of ligand bind-  ing at each site: ÀÁt p0ðÞ¼U; t p0ðÞU − p0ðÞU −p0ðÞU ⋅ − ; i i 1 i 1 i 0 exp τðÞU Yn−1 1 ciðÞj n ÀÁt PjðÞ¼U p ðÞU ; j ¼ ; …; − ; ð Þ 1 1 1 1 i 0 2 1 3 pi ðÞ¼U; t pi ðÞU1 − pi ðÞU1 −pi ðÞU0 ⋅ exp − : i¼0 τðÞU1 ð Þ Xn−1 7 i where j ¼ 2 ciðÞj is the number of possible molecular At any given time the total concentration of the pro- i¼0 tein, LT, is conserved and the sum of the probabilities (7) form, ci(j)=0 or ci(j) = 1 for free or occupied binding site, respectively. equals 1: L0ðÞþU; t L1ðÞ¼U; t L ; i i T ð Þ The kinetics of multisite protein interactions with 0 1 8 pi ðÞþU; t pi ðÞ¼U; t 1 abundant ligand concentrations (UT >>LT) In this case we assume the ligand concentration only In the most general case, the concentration of the changes at a given time point (t = 0) in a step fashion jth molecular form, Mj(U, t), of a molecule with n differ- from U0 to U1 and remains constant otherwise. ent binding sites in response to a step in ligand Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 12 of 19

concentration is given by the product of probabilities (7) Equation (12) can be used to estimate the half 0.5 according to Eq. (4): maximal effective ligand concentration (EC50), U0 0.5 and Un , for the apo- and saturated multisite Yn−1 сiðÞj n protein conformations respectively, when the pro- MjðÞ¼U; t LT ⋅ pi ðÞU; t ; j ¼ 0; …; 2 −1; ð9Þ i¼0 tein species equal 50 % of the total concentration LT: where ci(j)=0orci(j) = 1 for free or occupied ith binding pffiffiffiffiffiffiffi site respectively. − n : U0:5 ¼ K⋅1pffiffiffiffiffiffiffi0 5 ; The probabilities for individual molecular forms in n 0 0:5 steady-state can be obtained from Eq. (9) by setting pffiffiffiffiffiffiffi ð Þ n : 15 t → ∞: 0:5 0 5 Un ¼ K⋅ pffiffiffiffiffiffiffi ≈K⋅ðÞ1:44⋅n−0:44 : 1− n 0:5 Yn−1 ciðÞj n MjðÞ¼U LT ⋅ pi ðÞU ; j ¼ 0; …; 2 −1; ð10Þ i¼0 It can be noted that the half maximal effective ligand concentration is equal to equilibrium dissociation con- The kinetics of the amount of ligand bound to a multi- stant (EC = K) in proteins with only one binding site site protein with n binding sites can be written as 50 (n = 1). follows: Equations (15) can be solved with respect to n: X2n−1 Xn−1 − SUðÞ¼; t MjðÞU; t ciðÞj : ð11Þ 0:5 1 K þ Un j¼0 i¼0 n ¼ : ⋅ ð Þ 0 693 ln 0:5 16 Un

Multisite proteins with identical ligand binding sites Equation (11) can be used to derive the full amount of The multisite proteins that contain n identical binding ligand bound to multisite protein for U = K: sites with equilibrium dissociation constant equal K k− 1 sat ¼ þ can then be considered. The steady-state con- SKðÞ¼; t ⋅SUðÞ; t ; ð17Þ k 4 centration of one of the molecular forms of multisite sat protein with n identical binding sites for m bound where U is the ligand concentration for the case when sites, as a function of ligand concentration [22, 24] is all binding sites are occupied. The amount of bound sat sat given by: ligand for U = U is given by S(U , t)=n ⋅ LT and for Um⋅K n−m U = K is given by N mðÞ¼U LT ⋅ n ; m ¼ 0; 1; …; n: ð12Þ ðÞK þ U L T 1 n−1 SKðÞ¼; t n ⋅ ⋅n⋅2 : ð18Þ For intermediate forms the N1(U), …, Nn − 1(U) species 2 2 dependence on ligand concentration represents a bell The dynamic alterations of intermediate conformation shape with one molecular form “magnitude” at a par- max Nm(U, t) in response to ligand concentration jump from ticular ligand concentration Um . U to U according to Eqs. (6) and (7) are given by: Differentiating Eq. (12) with respect to U and solving 0 1 dN/dU = 0 for U gives: m n−m NmðÞ¼U; t LT ⋅ðÞp1ðÞU0; t ⋅ðÞp0ðÞU; t 1; 0 1 max K⋅m K K K t Um ¼ ; m ¼ 1; 2; …; n−1: ð13Þ p0ðÞ¼U; t −@ − A⋅ @− A; n−m K þ U K þ U K þ U exp τðÞU 1 0 1 01 0 1 1 Nmax The magnitudes m of the intermediate conforma- U U U t max 1 1 @ 1 0 A @ A tions Nm corresponding to Um values are: p ðÞ¼U; t − − ⋅ exp − ; K þ U1 K þ U1 K þ U0 τðÞU1 mm K N max ¼ L ⋅ ðÞn−m n−m; m ¼ ; ; …; n− : ð Þ τðÞ¼U : m T n 1 2 1 14 1 − n k ⋅ðÞU1 þ K The multisite protein conformations in the apo-, ð19Þ N0, and in the fully saturated states, Nn,would reach their maximum that equal to the total multi- Differentiating Eq. (19) with respect to t and solving max site protein concentration LT under conditions of for dNm/dt = 0 yields the time τm when the multisite very low and very high ligand concentrations, protein forms Nm with m bound molecules of ligand max respectively. reach their maximal values Nm : Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 13 of 19

 max n⋅K⋅ðÞU1−U0 Multisite proteins with four identical ligand binding sites τm ¼ τðÞU1 ⋅ ln ; ðÞK þ U0 ⋅ðÞðÞn−m ⋅U1−m⋅K This study next considered the kinetic properties of a 4 m ¼ 1; 2; …; n−1: protein with 4 binding sites. This allows 2 = 16 molecu- ð20Þ lar forms, each with potentially unique biochemical properties. There are 4 possible combinations of protein max The substitution of τm into Eq. (19) gives the max- species bound to one or to three ligands, and 6 possible imal values of intermediate multisite protein conforma- distinct molecular forms with two sites occupied. In the max tions reached at τm : previous work we described the steady-state dependence m of the individual multisite conformations on lig- max m n−m N ¼ LT ⋅ ðÞn−m ; m ¼ ; ; …; n− : ð Þ and concentration [22–24].Hereweanalysethe m nn 1 2 1 21 kinetic transition of the individual species con- Comparison of Eqs. (14) and (21) suggests that centrations in response to the step in ligand the steady-state maximum values of intermediate concentration. multisite protein conformations steady-state are The dynamical alterations of intermediate conform- similar to those transiently reached during the dy- ation Nm(U, t) in response to a step in ligand concentra- namic response to the step in ligand concentration. tion from U0 to U1 according to Eq. (19) are given by: According to Eq. (19) Nm(U, t) for the apo- and fully m −m t 1 0 4 saturated forms when =0: NmðÞ¼u; η LT ⋅ðÞp ðÞu; η ⋅ðÞp ðÞu; η ;  1 1 1 K n p0ðÞ¼u; η − − ⋅ ðÞ−η⋅ðÞu þ ; þ u þ u þ u exp 1 1 N 0ðÞ¼U; 0 LT ⋅ ; 1 1 1 1 1 0 K þ U0 u u u n ð22Þ p1ðÞ¼u; η 1 − 1 − 0 ⋅ ðÞ−η⋅ðÞu þ ; U þ u þ u þ u exp 1 1 N ðÞ¼U; L ⋅ 0 : 1 1 1 1 1 0 n 0 T K þ U 0 ð25Þ

According to Eq. (12) steady state levels for the apo- U U u ¼ 0 u ¼ 1 and fully saturated forms when t → ∞: where: 0 K , 1 K are non-dimensional ligand η t ⋅ k− n concentrations, and = is non-dimensional time. K k− K ¼ k− N ðÞ¼U; ∞ L ⋅ ; and kþ are the dissociation and equilibrium 0 T K þ U 1 ð Þ dissociation constants for ligand binding, respectively. U n 23 Nmax N ðÞ¼U; ∞ L ⋅ 1 : The maximum values of m are reached at n T K þ U umax ¼ K 1 the following ligand concentrations: 1 3 , max max u = K, u =3K, for the multisite protein spe- Equation (19) is further used to define the time, 2 3 τ0.5 N cies with one, two and three bound ions, respect- 0 ,requiredfortheapo-form, 0, to reach half of Nmax N ðÞþU;∞ N ðÞU; ively according to Eq. (13) and equal 1 = the growth concentration 0 0 0 and the time max max 2 0.105LT, N2 =0.063LT, N3 = 0.105LT,according τ0.5 period, n , required for the fully saturated protein to Eq. (14). species to gain half of the growth concentration Differentiating Eq. (25) with respect to η and solv- N nðÞþU;∞ N nðÞU;0 dN dη η 2 (Fig. 3): ing m/ =0 for gives the non-dimensional time 0 1 B C B C B C B C B U −U C τ0:5 ¼ τðÞU ⋅ BK⋅ 0 1 0 1C; 0 1 lnB C B 1 C B B K n K n n CC @ ðÞK þ U0 ⋅@ðÞK þ U1 ⋅ 0:5⋅ þ −KAA K þ U0 K þ U1 0 1 ð24Þ B C B C B C B C B U −U C τ0:5 ¼ τðÞU ⋅ lnBK⋅ 0 1 0 1C: n 1 B 1 C B n n C B B U0 U1 nCC @ ðÞK þ U0 ⋅@U1−ðÞK þ U1 ⋅ 0:5⋅ þ AA K þ U0 K þ U1 Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 14 of 19

max max − ηm = τm k when the intermediate species reach dA ðÞU; t their maximum: 00 ¼ − kþ⋅A ðÞU; t ⋅U þ k−⋅A ðÞU; t dt 2 1 00 1 10 þk−⋅A ðÞU; t ;  1 01 − 1 4⋅ðÞu −u dA ðÞU; t ηmax ¼ τmaxk ¼ ln 1 0 ; 10 ¼ kþ⋅A ðÞU; t ⋅U−k−⋅A ðÞU; t −kcþ 1 1 u þ ðÞþ u ⋅ðÞu − 1 00 1 10 1 1 1 1 0 3 1 1 dt − ÂA10ðÞU; t ⋅U þ kc ⋅A11ðÞU; t ; − 1 2⋅ðÞu −u 1 ηmax ¼ τmaxk ¼ ln 1 0 ; dA ðÞU; t 2 2 u þ ðÞþ u ⋅ðÞu − 01 þ − þ 1 1 1 0 1 1 ¼ k ⋅A ðÞU; t ⋅U−k ⋅A ðÞU; t −kc dt 1 00 1 01 1 − 1 4⋅ðÞu −u − ηmax ¼ τmaxk ¼ 1 0 : ÂA01ðÞU; t ⋅U þ kc ⋅A11ðÞU; t ; 3 3 u þ ln ðÞþ u ⋅ðÞu − 1 1 1 1 0 1 3 dA ðÞU; t 11 ¼ kcþ⋅A ðÞU; t ⋅U þ kcþ⋅A ðÞU; t ð26Þ dt 1 01 1 10 ÂU− kc−⋅A ðÞU; t ; 2 1 11 max max max ð28Þ Equations (26) suggest that η1 , η2 and η3 are 1 undefined when u1 < ; u1 < 1 and u1 < 3; respectively 3 The total number of species that follows from Eqs. (q1, q2 and q3 asymptotes). Under these conditions, the intermediate species do not reach the maximum in re- (28) is given by: sponse to ligand step, instead their relative number in- crease in a monotonous manner. A00ðÞþU; t A10ðÞþU; t A01ðÞþU; t A11ðÞ¼U; t AT : ð29Þ Multisite proteins with four different ligand binding sites In this section we analyse the kinetic properties of a multisite protein with four different binding sites. The The steady-state solutions of the system (28) are given steady-state analysis can be found in the authors previ- by: ous investigation [22, 24]. c + + + + K ⋅K k k k k 1 1 It is assumed that all association 1, 2, 3, 4 and dis- A ðÞ¼U AT ⋅ ; k− k− k− k− 00 U2 þ ⋅K c ⋅U þ K ⋅K c sociation 1, 2, 3, 4 rates are unique for each binding 2 1 1 1 + + + + c centre. Then assuming for example that k1 > k2 > k3 > k4 U⋅K − − − − 1 k k k k A10ðÞ¼U AT ⋅ ; and 1 = 2 = 3 = 4. The non-dimensional concen- U2 þ 2⋅K c ⋅U þ K ⋅K c u ¼ U h 1 1 1 tration K and non-dimensional constants 1 =1, U⋅K c ð30Þ kþ kþ kþ − − k− − k− A ðÞ¼U A ⋅ 1 ; h ¼ 2 h ¼ 3 h ¼ 4 h h ¼ 2 ¼ h ¼ 3 01 T c c 2 kþ , 3 kþ , 4 kþ , 1 =1, 2 k− 1, 3 k− U2 þ ⋅K ⋅U þ K ⋅K 1 1 1 1 1 2 1 1 1 − k− ¼ h ¼ 4 ¼ 2 1, 4 k− 1 can be introduced.. U 1 A ðÞ¼U AT ⋅ ; 11 U2 þ ⋅K c ⋅U þ K ⋅K c The set of Eqs. (25) can then be employed to calculate 2 1 1 1 the dynamics of the multisite protein conformations bound to a different number of ligand molecules. − − k c kc K ¼ 1 K ¼ 1 where 1 kþ and 1 kcþ. 1 1 Multisite proteins with two pairs of cooperative binding One can re-write Eqs. (30) as follows: sites The molecule contains two independent domains A and K ⋅K c a ðÞ¼U 1 1 ; B, with two identical cooperative binding sites. The 0 c c U2 þ 2⋅K ⋅U þ K ⋅K domain A is described as follows: 1 1 1 U⋅K c 1 a1ðÞ¼U c c ; ð Þ U2 þ 2⋅K ⋅U þ K ⋅K 31 kþ 1 1 1 1 2 A00 þ U⇄A10 U k− a ðÞ¼U ; 1 2 2 c c kþ U þ 2⋅K ⋅U þ K ⋅K 1 1 1 1 A00 þ U⇄A01 k− 1 ð Þ kсþ 27 A ðÞU A ðÞU A ðÞU 1 00 10 01 where a0ðÞ¼U A ; a1ðÞ¼U A ¼ A ; a2ðÞ¼U A10 þ U⇄A11 T T T kс− A ðÞU 1 11 kсþ AT are the probabilities for the domain to be in a 1 A01 þ U⇄A11 particular conformation due to the bound ligand kс− 1 molecules. The probabilities for the other domain, which also The ODEs for the scheme (27) is given by: contains a pair of cooperative binding sites, are given by: Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 15 of 19

K ⋅K c 2 2 b0ðÞ¼U ; dA ðÞU; t U2 þ 2⋅K c ⋅U þ K ⋅K c 00 ¼ − kþ⋅A ðÞU; t ⋅U þ k−⋅A ðÞU; t 2 2 2 dt 2 1 00 1 1 10 U⋅K c þk−⋅A ðÞU; t ; b ðÞ¼U 2 ; 1 01 1 U2 þ ⋅K c ⋅U þ K ⋅K c ð32Þ dA ðÞU; t 2 2 2 2 10 ¼ kþ⋅A ðÞU; t ⋅U −k−⋅A ðÞU; t −kcþ dt 1 00 1 1 10 1 U2 ÂA ðÞU; t ⋅U þ kc−⋅A ðÞU; t ; 10 1 1 11 b2ðÞ¼U ; U2 þ 2⋅K c ⋅U þ K ⋅K c dA ðÞU; t 2 2 2 01 ¼ kþ⋅A ðÞU; t ⋅U −k−⋅A ðÞU; t −kcþ dt 1 00 1 1 01 1 ÂA ðÞU; t ⋅U þ kc−⋅A ðÞU; t ; These probabilities were derived for the two domains 01 1 1 11 dA ðÞU; t of molecule (Eqs. (31) and (32) respectively). The prob- 11 ¼ kcþ⋅A ðÞU; t ⋅U þkcþ⋅A ðÞU; t dt 1 01 1 1 10 abilities of the molecule to be in a certain conformation ÂU − kc−⋅A ðÞU; t ; 1 2 1 11 with 0, 1 or 2 bound ligands in each of the domains, are ð36Þ as follows: p ðÞ¼U a ðÞU ⋅b ðÞU ; 0;0 0 0 p ðÞ¼U a ðÞU ⋅ b ðÞU ; 0;1 0 2 1 The kinetics of multisite protein interactions with p ðÞ¼U a ðÞU ⋅b ðÞU ; 0;2 0 2 constant ligand concentrations p ðÞ¼U a ðÞU ⋅b ðÞU ; 1;0 2 1 0 In this section we consider the mechanism of multisite p ðÞ¼U a ðÞU ⋅ b ðÞU ; ð Þ 1;1 2 1 2 1 33 binding for the case when the total ligand concentration p ðÞ¼U a ðÞU ⋅b ðÞU ; 1;2 2 1 2 is conserved. Under this assumption the system of differ- p ðÞ¼U a ðÞU ⋅b ðÞU ; 2;0 2 0 ential equations for ligand binding to a molecule with p ðÞ¼U a ðÞU ⋅ b ðÞU ; 2;1 2 2 1 single binding site (5) needs to be complemented by the p ðÞ¼U a ðÞU ⋅b ðÞU ; 2;2 2 2 law of ligand conservation: where pi,j(U) is the probability of the protein conform- i j dL0ðÞU; t ation with bound sites in the first domain and bound ¼ −kþ⋅L0ðÞU; t ⋅UtðÞþk−⋅L1ðÞU; t ; sites in the other. We use the sum of probabilities for dt the case of 1 bound site in a domain since we assume dL1ðÞU; t ¼ kþ⋅L0ðÞU; t ⋅UtðÞ−k−⋅L1ðÞU; t ; ð37Þ that all the sites in each domain are identical. dt The concentrations of the molecular forms of the pro- 0 1 tein with certain number of bound sites are given by: L ðÞþU; t L ðÞ¼U; t LT ; 1 UtðÞþL ðÞ¼U; t UT ; N ðÞ¼U L ⋅p ðÞU ; 0 T 0;0 N ðÞ¼U L ⋅ p ðÞþU p ðÞU ; 0 1 1 T 0;1 1;0 where L is the concentration of the free site, L is the N ðÞ¼U L ⋅ p ðÞþU p ðÞþU p ðÞU ; ð Þ concentration of the occupied site, UT and LT are the 2 T 0;2 1;1 2;0 34  total concentrations of ligand and protein molecules, N ðÞ¼U L ⋅ p ðÞþU p ðÞU ; 3 T 1;2 2;1 respectively. N ðÞ¼U L ⋅p ðÞU ; 4 T 2;2 There is only one positive steady-state solution of the system (37): One can rewrite Eq. (34) as follows:  N ðÞU K LT UT 0 L0ðÞ¼U ⋅ − − þ FUðÞ; ¼ a0ðÞU ⋅b0ðÞU ; T 1 T LT 2 K K N ðÞU K L U 1 ¼ ⋅ðÞa ðÞU ⋅b ðÞþU a ðÞU ⋅b ðÞU ; 1 T T 2 0 1 1 0 L ðÞ¼UT ⋅ þ þ 1−FUðÞT ; ð38Þ LT 2 K K N ðÞU  2 ¼ a ðÞU ⋅b ðÞþU 4⋅a ðÞU ⋅b ðÞþU a ðÞU ⋅b ðÞU ; K LT UT L 0 2 1 1 2 0 U ¼ UT − ⋅ þ þ 1−FUðÞT ; T 2 K K N 3ðÞU ¼ 2⋅ðÞa1ðÞU ⋅b2ðÞþU a2ðÞU ⋅b1ðÞU ; LT N ðÞU qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁÀÁ 4 k− U L 2 U L ¼ a2ðÞU ⋅b2ðÞU ; K ¼ FUðÞ¼ T − T þ ⋅ T þ T þ : LT where kþ and T K K 2 K K 1 ð35Þ We use the steady-state solutions (38) as initial condi- tions to find the particular solution. The solution of the Next we find the kinetic solution of system (28) for system (37) in response to the ligand concentration U = U1: jump from UT0 to UT1 is given by: Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 16 of 19

  K L U The kinetics of the amount of ligand bound to a multi- L0ðÞ¼U ; t L ⋅ ⋅ T − T1 − þ FUðÞ T T ⋅L K K 1 T1 site protein with n independent binding sites can be 2 T  − written as follows: CUðÞT ⋅ expðÞþt⋅k ⋅FUðÞT1 1  − ; CUðÞT ⋅ expðÞt⋅k ⋅FUðÞT −1 2Xn−1 Xn−1   1 K L U SUðÞ¼T ; t MjðÞUT ; t ciðÞj : ð42Þ 1 T T1 L ðÞ¼UT ; t LT ⋅ 1− ⋅ − −1 þ FUðÞT1 j¼0 i¼0 2⋅LT K K −  CUðÞT ⋅ expðÞþt⋅k ⋅FUðÞT1 1 The concentration of free ligand can be written  − ; CUðÞT ⋅ expðÞt⋅k ⋅FUðÞT −1 as the difference between the total ligand con-   1 K L U centration and the bound ligand concentration UtðÞ¼U −L ⋅ − ⋅ T − T1 − þ FUðÞ T1 T 1 ⋅L K K 1 T1 (42): 2 T  CUðÞ⋅ ðÞþt⋅k−⋅FUðÞ T exp T1 1 UtðÞ¼UT1−SUðÞT ; t ; ð43Þ Â − : CUðÞT ⋅ expðÞt⋅k ⋅FUðÞT −1 1 The probabilities of the ith site to be free or occupied ð39Þ respectively for the molecule with n independent bind-

U U ing sites in this case: FUðÞþFUðÞþT1− T0 T0 T1 K K CUðÞ¼T U U : where T1 T0 FUðÞT0 −FUðÞþT1 K − K K n⋅L U 0 i T T1 pi ðÞ¼UT ; η ⋅ − −1 þ FUðÞT1 The normalisation of the solution (39) by the total 2⋅n⋅LT K i K i ! protein concentration allows the definition of probability CUðÞT ⋅ expðÞþη⋅FUðÞT 1 1 Â 1 ; of the binding site to be in an occupied p (UT, t) or un- CUðÞT ⋅ expðÞη⋅FUðÞT −1 0 1 occupied p (UT, t) state, respectively, at a given total lig- K n⋅L U 1 i T T1 and concentration: pi ðÞ¼UT ; η ⋅ þ þ 1−FUðÞT1 2⋅n⋅LT K i K i ! K L U CUðÞT ⋅ expðÞþη⋅FUðÞT1 1 p0ðÞ¼U ; η ⋅ T − T1 − þ FUðÞ Â ; T 1 T1 CUðÞ⋅ ðÞη⋅FUðÞ− 2⋅LT K K ! T exp T1 1 C ðÞU ⋅ ðÞþη⋅FUðÞ ð Þ Â 2 T exp T1 1 ; 44 C ðÞU ⋅ ðÞη⋅FUðÞ− rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T exp T1 1  2 UT n⋅LT UT n⋅LT where FUðÞ¼T − þ 2⋅ þ þ 1 and K LT UT1 K i K i K i K i p1ðÞ¼UT ; η ⋅ þ þ 1−FUðÞT 1 UT UT ⋅LT K K FUðÞþT FUðÞþT 1− 0 2 ! 0 1 Ki Ki CUðÞ¼T U U : T1 T0 C ðÞUT ⋅ ðÞþη⋅FUðÞT FUðÞT0 −FUðÞþT1 − Â 2 exp 1 1 ; Ki Ki C2ðÞUT ⋅ expðÞη⋅FUðÞT1 −1 The dynamic alterations of the molecular m n UðÞη U L U forms with bound out of independent T1 1 T T1 N U t ¼ − ⋅ þ þ 1−FUðÞT identical binding sites m( T, )inresponseto K K 2 K K 1 ! the total ligand concentration jump from UT0 C ðÞUT ⋅ expðÞþη⋅FUðÞT 1 U Â 2 1 ; to T1 according to Eqs. (19) and (44) are C2ðÞUT ⋅ expðÞη⋅FUðÞT1 −1 given by: ÀÁÀÁ ð Þ 1 m 0 n−m 40 N mðÞ¼UT ; t LT ⋅ p ðÞUT ; t ⋅ p ðÞUT ; t : ð45Þ − where η = t ⋅ k . The concentration of multisite protein conformations, The concentration of the jth molecular form, Nm, bound to m ligand molecules as a function of ligand Mj(UT, t), of a protein with n independent binding concentration in steady-state is given by: sites in response to a step change in ligand concen- m K nLT UT tration is given by the product of probabilities accord- N mðÞ¼UT LT þ þ 1−FUðÞT ing to Eq. (9): 2nLT K K n−m K nLT UT Yn−1 Â − −1 þ FUðÞT сiðÞj n 2nLT K K MjðÞ¼UT ; t LT ⋅ p ðÞUT ; t ; j ¼ 0; …; 2 −1 i¼0 ð46Þ ð41Þ max The ligand concentrations, Um , for the maximal N Umax where ci(j) equals 0 or 1 for free and occupied sites, values of intermediate protein conformations, m( m ), Xn−1 can be found by differentiating Eq. (46) with respect to i respectively and j ¼ 2 ciðÞj . UT and solving dNm/dUT = 0 for UT: i¼0 Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 17 of 19

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  un n u UT LT UT LT un 1 0 K − þ ⋅n þ FUðÞT1 −1 þ − þ ⋅n þ FUðÞT0 −1 max t K K K K LT Um ¼ m⋅ LT þ ð47Þ W ¼ − ⋅n; n−m 2 K vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u n n UT1 LT UT0 LT un þ ⋅n−FUðÞþ þ þ ⋅n−FUðÞþ L t K K T1 1 K K T0 1 The corresponding maximal magnitudes for the inter- Y ¼ T ⋅n− : K 2 mediate conformations are given by:

ÀÁ mm Ethics (and consent to participate) N Umax ¼ L ⋅ ⋅ðÞn−m n−m ð Þ m m T nn 48 Not applicable.

0.5 Consent to publish The half maximal effective ligand concentration, U0 0.5 and Un , when the protein species equal half of the total Not applicable. concentration LT, for the apo- and saturated multisite protein conformations respectively, in this case are given Availability of data and materials by: The data supporting the findings of this work are con- tained within the manuscript. ! À pffiffiffiffiffiffiffi qffiffiffiffiffiffiffi : n Abbreviations U0 5 ¼ − : Þ⋅ n⋅L þ K n ; 2+ 0 1 0 5 T : Ca : calcium ion; CaM: calmodulin; CaMKII: calcium/calmodulin-dependent 0 5 protein kinase II; CaN: calcineurin; CR: calretenin; PP2B: protein phosphatase ! 2B, or calcineurin; TnC: troponin C. pffiffiffiffiffiffiffi K 0:5 n Un ¼ 0:5⋅ n⋅LT þ pffiffiffiffiffiffiffi : − n : Competing interests 1 0 5 The authors declare that they have no competing interests.

ð49Þ Authors’ contributions DVS and IIS developed and implemented the project under the supervision Differentiating Eq. (45) with respect to t and of NVK, GIW, AIE, YU and MZQC. All authors contributed to the analysis of max the model. YU and GIW provided biological interpretations of the results. All solving for dNm/dt = 0 yields the time τm when authors contributed to the writing of the final manuscript. All authors have the concentration of multisite protein conforma- read and approved the final version of the manuscript. N m tions, m,boundto ligand molecules is Acknowledgements maximal: We would like to thank Professor Vadim Biktashev (University of Exeter), Dr Daniel Roper and Dr Timur Abdullin (Kazan Federal University) for helpful max comments and discussion. τm ¼ HVmðÞðÞ; ð50Þ  Funding U This work was carried out under Severnside Alliance for Translational þ T1þFU þx 1 K ðÞT1 ln U Research (SARTRE) grant (GIW) and NSFC grant 61374053 (MZQC). This work CUðÞ⋅ þ T1−FUðÞþx T ðÞ1 K T1 was co-funded by the subsidy allocated to the Kazan Federal University for where HxðÞ¼ − and VmðÞ¼ FUðÞT1 ⋅k state assignment in the sphere of scientific activities and performed accord- LT ing to the ‘Russian Government Program of Competitive Growth’ of the K ⋅ðÞn−2m : Kazan Federal University. UT1 UT1 Equation (50) has an asymptote K for 1 þ K −F Author details ðÞþUT1 x ¼ 0: 1Kazan (Volga Region) Federal University, 18 Kremlyovskaya St., 420008 Kazan,  Russia. 2Centre for Systems, Dynamics and Control, College of Engineering, LT 1 Mathematics and Physical Sciences, University of Exeter, Harrison Building, q ¼ m⋅ þ ; ð Þ 3 m K n−m 51 North Park Road, Exeter EX4 4QF, UK. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China. 4Department of Dermatology, The Jikei University School of Medicine, 5 Equation (45) is further used to define the 3-25-8 Nishishimbashi, Minato-ku, Tokyo 105-8461, Japan. Biophysics & τ0.5 N Bionics Lab, Institute of Physics, Kazan Federal University, Kazan 420008, time, 0 , required for the apo- form, 0,to Russia. 6Academic Renal Unit, School of Clinical Sciences, University of Bristol, reach half of the growth concentration and the Dorothy Hodgkin Building, Whitson Street, Bristol BS1 3NY, UK. 0.5 time period, τn , required for the fully saturated Received: 20 October 2015 Accepted: 13 April 2016 protein species to gain half of the growth concentration: References 1. Maki M, Maemoto Y, Osako Y, Shibata H. Evolutionary and physical linkage τ0:5 ¼ HWðÞ; 0 ð Þ between calpains and penta-EF-hand Ca2 + -binding proteins. FEBS J. 2012; 0:5 52 – τn ¼ HYðÞ; 279:1414 21. 2. Chillakuri CR, Sheppard D, Lea SM, Handford PA. Notch receptor-ligand binding and activation: insights from molecular studies. Semin Cell Dev Biol. where 2012;23:421–8. Salakhieva et al. BMC Systems Biology (2016) 10:32 Page 18 of 19

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