320 Free energy calculations
f(λ) andg(λ), which means that eqn. (8.2.7) always yields the reversible work via
the free energy difference. The flexibility in the choice of theλ-path, however, can be
exploited to design adiabatic switching algorithms of greater efficiency than can be
achieved with the simple choicef(λ) = 1−λ,g(λ) =λ.
In practice, the thermodynamic integration formula is implemented as follows: A
set ofMvalues ofλis chosen from the interval [0,1], and at each chosen valueλka
full molecular dynamics or Monte Carlo calculation is carried out in order to generate
the average〈∂U/∂λk〉λk. The resulting values of〈∂U/∂λk〉λk,k= 1,...,M, are then
substituted into eqn. (8.2.7), and the result is integrated numerically to produce the
free energy difference ∆AAB. The selected values{λk}can be evenly spaced, for
example, or they could be a set of Gaussian quadrature nodes, depending on the
anticipated variation ofA(N,V,T,λ) withλfor particularf(λ) andg(λ).
Let us now consider an example of a particular type of free energy calculation of
particular relevance, specifically, the binding of a lead drug candidate to the active
site of an enzyme E. The purpose of the drug candidate is to inhibit the catalytic
mechanism of an enzyme used, for instance, by a virus to attack a host cell, hijack its
cellular machinery, or replicate itself. We will refer to the candidate drug compound as
“I” (inhibitor). The efficacy of the compound as an inhibitor of the enzyme is measured
by an equilibrium constant known as theinhibition constantKi= [E][I]/[EI], where
[E], [I], and [EI] refer to the concentrations in aqueous solution of the uncomplexed
enzyme, uncomplexed inhibitor, and enzyme–inhibitor complex EI, respectively. Since
Kiis an equilibrium constant, it is also related to the Gibbs free energy ofbinding
∆Gb, which is the free energy of the reaction E(aq) + I(aq)⇀↽EI(aq). That is,Ki=
exp(∆Gb/kT). For the purposes of this discussion, we will assume that the binding
Helmholtz free energy ∆Abis approximately equal to the Gibbs free energy, so that
the former can be reasonably used to estimate the inhibition constant. If we wish to
determineKifor a given drug candidate by calculating the binding free energy, a
technical complication immediately arises. In principle, we can let the potentialUA
contain all interactions except that between the enzyme and the inhibitor and then
let this excluded interaction be included inUB. Now consider placing an enzyme and
inhibitor in a bath of water molecules in order to perform the calculation. First, in
order to sample the unbound state, the enzyme and inhibitor need to be separated by
a distance large enough that both are fully solvated. If we then attempt to let them
bind by turning on the enzyme–inhibitor interaction in stages, the probability that
they will “find” each other and bind properly under this interaction issmall, and the
calculation will be inefficient. For this reason, a more efficient thermodynamic path
for this problem is a three-stage one in which the enzyme and the inhibitor are first
desolvated by transferring them from solution to vacuum. Followingthis, the enzyme
and inhibitor are allowed to bind in vacuum, and finally, the complex EI is solvated
by transferring it back to solution. Fig. 8.1 illustrates the direct andindirect paths.
Since free energy is a state function, the final result is independent of the path taken.
The advantage of the indirect path, however, is that once desolvated, the enzyme
and inhibitor no longer need to be at such a large separation. Hence,we can start
them much closer to each other in order to obtain the vacuum bindingfree energy.
Moreover, this part of the calculation will have a low computational overhead because