5.5. Surface Partition Model with Electrostatic Contributions

In a typical ITC experiment, when LUVs are injected into the peptide solution, the total concentration of the peptide is the sum of the concentration of bound peptides and the free peptides. The apparent binding constant K_app is defined as X_b = K_appC_f, where X_b is the extent of peptide binding per mole of lipid and C_f is the free peptide concentration of the solution. When the charged peptide adsorbs, the membrane becomes changed. Hence, further adsorption of peptide onto the membrane is restricted by the electrostatic repulsion. Therefore, K_app is no longer a constant but rather changes with C_f. Therefore, the most relevant binding parameter would be the intrinsic binding constant K_int (X_b = K_intC_M), which is assumed to be directly proportional to the surface concentration of the peptide in the membrane. The model essentially calculates X_b and C_M for each injection of the ITC experiment. As the C_M of the peptide is governed by the electrostatic contribution, it is determined by the Boltzmann distribution.

where z_p is the effective peptide charge. The maximum charge of NK-2 is +10. F is the Faraday constant (= charge of 1 mol of electron = electronic charge × Avogadro’s number = 96 485 C/mol). ψ₀ is the surface potential. R is the universal gas constant and T is the absolute temperature. As the ψ₀ is related to surface charge density, σ, of the peptide, it can be estimated from the well-known Gouy–Chapman theory. As the ζ potential can be a good approximation to the surface potential, we have taken ψ₀ same as the ζ potential. Now we need to know the ζ potential for each peptide-to-lipid molar ratio obtained after every injection. However, we have measured the ζ potential for few peptide-to-lipid ratios. To obtain the ζ potential for all peptide-to-lipid ratios required for the calculation of C_M, we have fitted the ζ potential data to a Hill equation originally used to describe cooperative binding of ligand to macromolecules.³⁵ Hence, we have obtained ζ potentials for all required concentrations from interpolation or extrapolation of the fitted curve. Now, for each data point, the concentration of the free peptide and ψ₀ have been determined. Once C_f and ψ₀ are known, C_M can be calculated. We now discuss briefly how X_b and C_f have been estimated from an ITC experiment.

In an ITC experiment (lipids into peptide injection), the X_b and the enthalpy change ΔH per mole of peptides can be measured directly. The ΔH has been estimated from the sum over all heat per injection divided by the number of moles of peptide in the calorimeter cell.

where δh_i is the heat per injection in the ITC experiment. C_p⁰ is the initial molar concentration of peptide in the ITC cell, and V_cell is the volume of the ITC cell. Now, the extent of peptide binding, X_b, per mole of lipid after completion of ith is given by

where V_inj is the volume of each injection with lipid of concentration C_L. Now, the fraction of bound peptide after ith injection is given by

where the free peptide concentration C_f can be obtained from

Now, the K_app can be obtained from the X_b and C_f. Therefore, C_M and K_int can be estimated for each pair of X_b and C_f, obtained from experimental data, as discussed above.