Handling complexity in biological interactions

Abstract

Biological processes rely on interactions between many binding partners. Binding results in the modulation of the conformational landscape of the interacting molecules, a phenomenon rooted in folding and binding cooperativity underlying the allosteric functional regulation of biomacromolecules. The conformational equilibrium of a protein and the binding equilibria of different interacting and cooperative ligands are coupled giving rise to a complex scenario in which protein function can be finely tuned and modulated. Binding cooperativity and allostery add additional levels of complexity in protein function regulation. Here we will review some important concepts associated with binding, cooperativity and allostery in protein interactions, illustrated with several representative protein-dependent biological systems related to drug discovery and physiological mechanisms characterization and studied by isothermal titration calorimetry.

Appendix

The overall association constants β _i can be expressed as a function of β ₁

It can be demonstrated that the overall stoichiometric association constant β_si for the formation of complex P_sA_i can be expressed as a function of the total number of binding sites, n, the number of ligand A molecules bound per macromolecule, i, the intrinsic site-specific association constant for ligand A, β_s1 (that is, the overall stoichiometric association constant for the formation of complex P_sA), and the cooperativity constant accounting for homotropic cooperativity effects for ligand A binding, α_si:

$$ \beta_{\text{si}} = \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)n^{ - \text{i}} \beta_{{{\text{s}}1}}^{ \text{i}} \alpha_{\text{si}} $$

(44)

If a protein conformational state P has n identical and independent binding sites for ligand A, the binding polynomial factorizes into identical terms, each one representing one of the ligand binding sites as a subsystem:

$$ Z = \mathop \sum \limits_{{{\text{i}} = 0}}^{n} \beta_{\text{i}} \left[ A \right]^{\text{i}} = \left( {1 + K\left[ A \right]} \right)^{\text{n}} $$

(45)

where K is the intrinsic site-specific association constant for each of the ligand binding sites. From that expression, it is obvious that:

$$ \beta_{\text{i}} = \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)K^{\text{i}} $$

(46)

with:

$$ \beta_{1} = nK $$

(47)

and from that:

$$ \beta_{\text{i}} = \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)n^{ - i} \beta_{1}^{i} $$

(48)

Then, potential homotropic cooperative effects can be accounted for by introducing an additional factor α_i, with α₁ = 1, because there is no homotropic effect for the first bound ligand:

$$ \beta_{\text{i}} = \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)n^{ - \text{i}} \beta_{1}^{\text{i}} \alpha_{\text{i}} $$

(49)

This expression is completely general and it can be applied to any situation (i.e., identical or nonidentical binding sites, cooperative or non-cooperative sites) by decomposing β₁ and α_i into additional factors [8]. Such decomposition will help in reducing the number of binding parameters required in the model [61].

From Eq. (46), the binding polynomial can be factorized into identical terms if, for all i’s, it is fulfilled that [1]:

$$ \frac{{\frac{{\beta_{\text{i}} }}{{\left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)}}}}{{\left( {\frac{{\beta_{{{\text{i}} - 1}} }}{{\left( {\begin{array}{*{20}c} n \\ {i - 1} \\ \end{array} } \right)}}} \right)^{{{\raise0.7ex\hbox{$\text{i}$} \!\mathord{\left/ {\vphantom {i {i - }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\text{i} - 1}$}}}} }} = 1 $$

(50)

And it can be factorized into nonidentical terms if, for all i’s, it is fulfilled that [1]:

$$ \frac{{\frac{{\beta_{\text{i}} }}{{\left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)}}}}{{\left( {\frac{{\beta_{{{\text{i}} - 1}} }}{{\left( {\begin{array}{*{20}c} n \\ {i - 1} \\ \end{array} } \right)}}} \right)^{{{\raise0.7ex\hbox{$\text{i}$} \!\mathord{\left/ {\vphantom {i {i - 1}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\text{i} - 1}$}}}} }} < 1 $$

(51)

Therefore, the ligand binding sites are identical and independent if:

$$ \frac{{\alpha_{\text{i}} }}{{\left( {\alpha_{{{\text{i}} - 1}} } \right)^{{{\raise0.7ex\hbox{$\text{i}$} \!\mathord{\left/ {\vphantom {i {i - 1}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\text{i} - 1}$}}}} }} = 1 $$

(52)

and the ligand binding sites are nonidentical and independent if:

$$ \frac{{\alpha_{\text{i}} }}{{\left( {\alpha_{{{\text{i}} - 1}} } \right)^{{{\raise0.7ex\hbox{$\text{i}$} \!\mathord{\left/ {\vphantom {i {i - 1}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\text{i} - 1}$}}}} }} < 1 $$

(53)

Thus, the independent and identical nature of the ligand binding sites can be judged by comparing the overall association constants (β’s) or the homotropic cooperativity parameters (α’s).

The extent of the heterotropic effect between ligand A and ligand B is fully reciprocal

The heterotropic effect between two ligands is fully reciprocal or symmetric. Because of the energy conservation principle, if ligand B modifies the intrinsic affinity for ligand A by a factor α_ij, then ligand A causes the same effect on ligand B. The constant β_ij can be split into two factors: β_ij = β_i0β_j/i, where β_i0 is the overall association constant for P + A_i ↔ PA_i and β_j/i is the overall association constant for PA_i + B_j ↔ PA_iB_j. Alternatively, β_ij can be split into two factors: β_ij = β_0jβ_i/j, where β_oj is the overall association constant for P + B_j ↔ PB_j and β_i/j is the overall association constant for PB_j + A_i ↔ PA_iB_j. Furthermore, the constants β_j/i and β_i/j can be factorized as: β_j/i = β_0jα_ij and β_i/j = β_i0α_ij.

The maximal allosteric effect occurs at concentrations around the dissociation constant of the ligand

From Eq. 20, if, for the sake of simplicity, only a single binding site for ligand B is considered:

$$ \frac{{\partial \ln \gamma_{\text{s}}^{\text{app}} }}{\partial \ln \left[ B \right]} = \Delta n_{\text{LB,s,B}} = \frac{{\beta_{\text{s}} \left[ B \right]}}{{1 + \beta_{\text{s}} \left[ B \right]}} - \frac{{\beta_{0} \left[ B \right]}}{{1 + \beta_{0} \left[ B \right]}} $$

(54)

Δn_LB,s,B will be zero for zero and infinite ligand B concentration, and for β_s = β₀. Deriving with respect to ligand B concentration:

$$ \frac{{\partial \Delta n_{\text{LB,s,B}} }}{\partial \ln \left[ B \right]} = \frac{{\beta_{\text{s}} \left[ B \right]}}{{\left( {1 + \beta_{\text{s}} \left[ B \right]} \right)^{2} }} - \frac{{\beta_{0} \left[ B \right]}}{{\left( {1 + \beta_{0} \left[ B \right]} \right)^{2} }} $$

(55)

and a maximal value for Δn_LB,s,B will be achieved when:

$$ \beta_{\text{s}} - \beta_{0} - \beta_{\text{s}} \beta_{0} \left( {\beta_{\text{s}} - \beta_{0} } \right)\left[ B \right]^{2} = 0 $$

(56)

whose solutions are: (1) β_s = β₀ and any value of [B] (trivial case); or (2) β_s ≠ β₀ and [B] = (β_sβ₀)^−1/2, the inverse of the geometric mean of the overall association constants, which is the concentration of ligand B for a maximal allosteric effect on the apparent conformational constant $\gamma_{\text{s}}^{\text{app}}$ for P_s.

The maximal heterotropic effect occurs at concentrations around the dissociation constant of the secondary ligand

From Eq. 25, if there are several (i = 1…n) binding sites for ligand A and, for the sake of simplicity, a single binding site for ligand B:

$$ \frac{{\partial \ln \beta_{\text{i}}^{\text{app}} }}{\partial \ln \left[ B \right]} = \Delta n_{\text{LB,i,B}} = \frac{{\beta_{01} \alpha_{{{\text{i}}1}} \left[ B \right]}}{{1 + \beta_{01} \alpha_{{{\text{i}}1}} \left[ B \right]}} - \frac{{\beta_{01} \left[ B \right]}}{{1 + \beta_{01} \left[ B \right]}} $$

(57)

Δn_LB,i,B will be zero for zero and infinite ligand B concentration, and for α_i1 = 1. Deriving with respect to ligand B concentration:

$$ \frac{{\partial \Delta n_{\text{LB,i,B}} }}{\partial \ln \left[ B \right]} = \frac{{\beta_{01} \alpha_{{{\text{i}}1}} \left[ B \right]}}{{\left( {1 + \beta_{01} \alpha_{{{\text{i}}1}} \left[ B \right]} \right)^{2} }} - \frac{{\beta_{01} \left[ B \right]}}{{\left( {1 + \beta_{01} \left[ B \right]} \right)^{2} }} $$

(58)

and a maximal value for Δn_LB,B will be achieved when:

$$ \alpha_{{{\text{i}}1}} - 1 - \beta_{01}^{2} \alpha_{{{\text{i}}1}} \left( {\alpha_{{{\text{i}}1}} - 1} \right)\left[ B \right] = 0 $$

(59)

whose solutions are: (1) α_i1 = 1 and any value of β₀₁ and [B] (trivial case); or (2) α_i1 ≠ 1 and [B] = β⁻¹₀₁α^−1/2_i1, which is the concentration of ligand B for a maximal heterotropic effect on the apparent overall association constant $\beta_{\text{i}}^{\text{app}}$ for PA_i.

Enthalpy changes associated with equilibrium constants are calculated by applying the van’t Hoff equation

Every equilibrium constants has an associated enthalpy change (besides changes in other thermodynamic potentials). The corresponding enthalpy change can be calculated by applying the van’t Hoff equation:

$$ \Delta H = RT^{2} \frac{{\partial \ln K_{\text{eq}} }}{\partial T} $$

(60)

From that, if a given equilibrium constant is a function of other equilibrium constants, then, the corresponding enthalpy can be calculated by applying Eq. 53:

$$ \begin{aligned} K_{\text{eq}} & = f\left( {\left\{ {K_{\text{eq,r}} } \right\}} \right) \\ \Delta H & = RT^{2} \frac{{\partial \ln K_{\text{eq}} }}{\partial T} = RT^{2} \frac{{\partial \ln f\left( {\left\{ {K_{{{\text{eq}},r}} } \right\},\left\{ {\Delta H_{r} } \right\}} \right)}}{\partial T} \\ \end{aligned} $$

(61)