### Model 1. CaM with independent Ca^{2+} binding sites

In this model we assume that at very basic level of Ca

^{2+}-CaM interactions, Ca

^{2+} binding sites can be considered independent. We therefore derive the equations for the case of independent binding. We assume that CaM undergoes a conformational transition upon Ca

^{2+} binding and adopts a unique conformation according to the number of bound ions. We will denote CaM conformations by

*cm*
_{
j
},

*j* = 0,1, ..., 4.

*cm*
_{0} is a conformation with no bound ligand molecules and

*cm*
_{1} is a conformation with one bound ligand molecule. The concentration of CaM conformation with a given number of bound ligand molecules as a function of ligand concentration is given by:

where *cm*0 is the total concentration of CaM, *u* is normalised Ca^{2+} concentration,
is the probability of binding site *i* not being occupied and
is the probability of binding site *i* being bound. *c*
_{
i
} equals 1 if a binding site is occupied and 0 if it is not. The probability of CaM being in a particular bound state is equal to the product of the probabilities of each individual binding site.

The probabilities for a binding site to be not occupied or occupied as a function of Ca

^{2+} concentration are given by:

where the *K* and *u* are the microscopic equilibrium dissociation constant and the ligand concentration, respectively. Effectively these are Michaelis-Menten equations for a protein in a complex with and without a ligand molecule, but normalized by the total protein concentration.

The multiplication of probabilities from (2) for occupied sites gives an equation for a fully bound protein with

*n* binding sites:

where the *K* and *u* are the equilibrium dissociation constant and the ligand concentration, respectively.

Other well known models to describe Ca

^{2+} binding to CaM are the Hill [

42] and the modified Adair [

11,

52] equations:

A complete mathematical description of the relationship between macroscopic constants derived from the Adair equation and the proposed model is provided in the Supplementary Materials Section (Additional File 1).

In the most general case, the concentration of any multisite protein

*L*
_{
i
} (

*u*) with

*n* ligand binding sites in a particular ligand-bound state is given by the multiplication of probabilities (2):

where *L*0 is the total protein concentration and *L*
_{
i
} is the concentration of conformation *i*, *K*
_{
j
} are the equilibrium dissociation constants of each binding site, and *u* is the ligand concentration.

If a protein has identical binding sites, equation 5 simplifies to the following formula:

The conformations *L*
_{1}(*u*), ..., *L*
_{n-1}(*u*) of a multisite protein are all bell-shaped curves, the conformation *L*
_{0}(*u*) is the apo state of a multisite protein, whereas the *L*
_{
n
}(*u*) is the fully bound multisite protein. If equation (5) is divided by *L*0, then instead of predicting protein concentrations in specific ligand-bound conformations, it predicts the probability of a particular conformation to be in that state as a function of ligand concentration.

The formula predicting the concentration of CaM in complex with target protein

*N* as a function of Ca

^{2+} concentration is given by:

where *cm*
_{
i
}(*u*) is substituted from equation 5, and *K*
_{d} is the dissociation constant for CaM-target interactions, *N*0 is the total concentration of target protein.

### Model 2. Cooperative Ca^{2+}-CaM interactions

While the previous model provides predictions for the number of Ca^{2+}-CaM complexes as a function of Ca^{2+} concentration with a reasonable accuracy, it does not capture the effects of the cooperative influence of Ca^{2+} binding sites. There are several possible ways to incorporate these cooperative mechanisms into the model. In order to derive a model that illustrates what contribution cooperativity makes to the distribution of concentration profiles of Ca^{2+}-CaM complexes, we assume that in the *N*-terminal domain, the first centre is cooperatively bound to the second, and in the *C*-terminal, the third is cooperatively bound to the fourth. In this case, we will define the dissociation constants as *K*
_{1}, *K*
_{2},
, *K*
_{3}, *K*
_{4},
, where
and
are the cooperatively influenced dissociation constants for the second and fourth centres when the ligand is bound to the first and third binding sites, correspondingly.

The probabilities for the first binding site to be free

or occupied

are given by:

where *K*
_{1} and *u* are the equilibrium dissociation constant for the first centre and the ligand concentration, respectively.

The probabilities for the second centre to be in a particular state are:

where
is the probability for both the first and the second centres to be free,
is the probability for the first site to be free and the second to be occupied,
is the probability for the first site to be bound and the second to be free, and
is the probability for both sites to be ligand bound. *K*
_{2} and
are the dissociation and the cooperatively modified dissociation constants for the second Ca^{2+} binding site.

The probabilities for the third centre are:

where *K*
_{3} is a dissociation of the third binding site.

The probabilities for the fourth site are given by:

where
is the probability for both the third and the fourth centres to be free,
is the probability for the third site to be free and the fourth to be occupied,
is the probability for the third site to be bound and the fourth to be free, and
is the probability for both sites to be ligand bound. *K*
_{4} and
are the dissociation and the cooperatively modified dissociation constants for the second Ca^{2+} binding site.

The probabilities of

*cm*
_{i} conformations to be in a particular conformation with i bound ligand molecules as a function of ligand concentration are:

where *k*
_{i} = 0, if the binding site *i* is not occupied and *k*
_{i} = 1 if the centre *i* is occupied by a ligand molecule.

The distribution of intermediate Ca^{2+}-CaM complexes with 1, 2 and 3 Ca^{2+} ions with and without cooperativity is shown in Figure 5A.

We next compare the amount of bound ligand in the presence and in the absence of the cooperative mechanism. In the absence of any cooperativity the multisite protein binds ligand molecules according to the equation:

In the case of pairs of cooperatively interacting centres, the amount of bound ligand is given by:

Figure 5B shows the total amount of ligand bound to CaM in the presence (1) and absence (2) of cooperative binding. The line (3) shows the difference in the level of bound ligand between the two types of binding mechanisms.

### Model 3. The Ca^{2+}-CaM-target protein complex assembly

In the previous model we have assumed that the second binding site was cooperatively dependent on the first, but the first site was not dependent on the second. Similar assumptions were made for the

*C*-terminal domain. A more precise description would also assume the first Ca

^{2+} binding site cooperatively depends on the second and the third Ca

^{2+} binding site depends on the fourth binding sites. For two Ca

^{2+} binding sites in the

*N*-terminal domain, the more realistic case is described by the following system of differential equations:

where *cm*
_{00}, *cm*
_{01}, *cm*
_{10}, *cm*
_{11} are CaM molecules without Ca^{2+} ions, with one Ca^{2+} ion bound to the *N*-terminal domain, with one Ca^{2+} ion bound to the *N*-terminal domain, and CaM species with two bound Ca^{2+} ions at each terminal domain, respectively. *k*
_{1} and *k*
_{2} are the association constants and *kc*
_{1} and *kc*
_{2} are the cooperatively modified association constants for the *N*- terminal binding sites of CaM, respectively. Similarly, *d*
_{1}, *d*
_{2}, *dc*
_{1}, *dc*
_{2} are the dissociation and cooperatively modified dissociation constants for the *N*- and terminal binding sites, respectively. Note that a similar system of differential equations can be developed for the *C*-terminal.

The conservation law gives:

*cm*_{00} +*cm*_{10} + *cm*_{01} + *cm*_{11} = 1

In steady-state, the matrix for the system (15) with the last equation substituted by (16) is given by:

The matrix (17) can be solved using Cramer's method. The determinant of the matrix is given by:

where
,
. *k*
_{1} = *k*
_{2} = *kc*
_{1} = *kc*
_{2} is a simplifying assumption that allows analytical solution of the current system.

The matrices for the individual species of the

*N*-terminal domain are given by:

where *N*
_{00}, *N*
_{01}, *N*
_{10} and *N*
_{11} correspond to the *N*-terminal domain without any Ca^{2+} ions bound, with one Ca^{2+} ion bound to one or another binding site and to the fully bound state, respectively.

The determinants of the matrices (19–22) are given by:

The probabilities for the

*N*-terminal to be in a particular state with variable numbers of Ca

^{2+} ions are given by:

where
is the probability for the *N*-terminal to be in a Ca^{2+} state,
and
describe the probabilities for a complex with one Ca^{2+} ion, and
is the probability function for the *N*-terminal to be occupied by Ca^{2+}.

In order to derive the steady-state dependence of the

*C*-terminal state as a function of Ca

^{2+} concentration, a similar procedure can be applied. Similarly to (17) and (18), the determinant for the

*C*-terminal is given by:

Where

Similarly to (19–23), the solution for the

*C*-terminal states is given by:

The probabilities for the

*C*-terminal to be in a particular state with variable number of Ca

^{2+} ions are given by:

where
is the probability for the *N*-terminal to be in a Ca^{2+} state,
and
describe the probabilities for a complex with one Ca^{2+} ion, and
is the probability function for the *N*-terminal occupied by Ca^{2+}.

The combination of (24) and (27) provides the solution for individual CaM species with variable numbers of Ca

^{2+} ions:

where
are CaM species in apo state,
is fully bound CaM, and
and
are CaM species with fully bound *N*- and *C*-terminals, respectively.
and
are CaM species with one Ca^{2+} ion bound to *N*- and *C*-terminals, respectively. Comparison of numerical solutions of this system with the available experimental data [9, 15, 33] allows us to propose that CaM molecules can be represented as a pair of two independent *N*- and *C*-terminal globular domains, each containing two symmetrical and cooperatively bound EF-hand Ca^{2+} binding sites.

The equations (28) are specific to Ca^{2+}-CaM interactions and incorporate the pairwise cooperative interactions between the EF-hand binding sites within the *N*- and *C*-terminals, whereas the *N*- and *C*-terminal domains are considered to be independent of each other. Note that the equations developed here for the Ca^{2+}-CaM complexes are essentially different from the Michelis-Menten, Hill and Adair models, and also differ from models with independent binding sites (5) or with limited amounts of cooperativity (11) and (12).

The equations for the Ca

^{2+}-CaM complexes (28) have been further applied to calculate binding with the target peptides and proteins:

where *T*0 is the total concentration of a target protein or a peptide, *K*
_{
d
} is the equilibrium dissociation constant between CaM and a target protein, and
is a single Ca^{2+} complex or the sum of several CaM complexes.

The combinations of Ca^{2+}-CaM complexes have been varied simultaneously with the dissociation constants to fit the experimental data. This analysis allows us to predict the Ca^{2+}-CaM complexes required for activation of specific protein targets. The fitting of dissociation constants of Ca^{2+} binding sites on CaM molecules to the experimental dose-response curves reveals the impact of the target protein on the Ca^{2+}-CaM interactions. The dissociation constants calculated based on the cooperative **Model 3,** which also takes into account the impact of target proteins, are compared with the dissociation constants calculated using the Hill and modified Adair equation in the original experimental publications for the same data in Table 1.