Computational chemistry methods in structural biology by Christo Christov (Eds.)

By Christo Christov (Eds.)

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3). Consider, for example, Eq. (18). The largest contributions to the integral come from regions where the binding PMF DF(zL) is large and favorable and exp[À bDF(zL)] is large compared to 1, the value obtained in regions where the receptor and the ligand are not significantly interacting. If the minima of DF(zL) are deep and localized, that is binding is strong and specific, the choice of the domain of integration has a small effect on the value of the integral as long as it covers all the regions where DF(zL) is deep.

Assuming that the set of macrostates collectively covers all possible conformations of the complex (which is always possible by including a ‘‘catch-all’’ macrostate), we can express p0(u) as a marginal of p0(u, i): X X p0 ðuÞ ¼ p0 ðu; i Þ ¼ P0 ði Þp0 ðuji Þ; ð52Þ i i where we have introduced the conditional distribution p0(u|i) and the population P0(i) of macrostate i in the uncoupled reference state and used the relationship p0(u, i) ¼ P0(i)p0(u|i) between the joint and conditional distributions.

23) has the same form as in Eq. (11) expressed in terms of differences of effective potential energies u ðxL ; zL ; xR Þ ¼ Ueff ðxR ; xL ; zL Þ À Ueff ðxR Þ À Ueff ðxL Þ: ð24Þ It is straightforward to show, from the definition of the solvent PMF (Eq. (21)), that the effective binding energy is the interaction free energy MODELING OF PROTEIN-LIGAND BINDING AFFINITIES 37 with explicit solvation (Eq. (16)) for a fixed conformation (xL, zL, xR) of the complex. Eq. (23) then expresses a combination rule to obtain the total interaction free energy for binding by averaging over the ensemble of the conformations of the uncoupled state of the complex.

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