8.6.1.1 Thermodynamic integration

Thermodynamic integration is the method of obtaining a Helmholtz free energy change, A F, between two states where the difference is determined from averages that are accessible from MD simulations. If a path between two states can be defined and states along that path can be defined by a parameter, X є (X0, X1), then the following statistical mechanical relationship can be derived,

dFk Id UA dX d X jX

where UX is the potential energy at the state defined by X, and the average is over all configurations visited by the system during an MD simulation of the system with constant X. Then using simple integration schemes, integrating from initial state X0 to final state X1, for simplicity we choose X0 = 0 and X1 = 1,

Although there are some very delicate issues regarding the endpoints of this method, the beauty is that one can run a few points between each of the endpoints, and using Gaussian quadrature, integrate to high precision the averages of the potential energy derivatives to yield the A F. In addition to the simplicity of the method, the ability to choose any path from state 0 to state 1 including creating and annihilating atoms. A typical application is to use a thermodynamic cycle to find the change in binding free energy, AAFbmdmg, for two ligands binding to a substrate without calculating the binding free energy of either one. Free energies of binding are particularly hard to calculate. The method can be illustrated by the following free energy cycle, whose total free energy change will be zero since the beginning and ending states are the same state. In this illustration, Figure 8.4, the two ligands are
phenol and toluene, which differ by the constituent group being hydroxyl or methyl. The question to answer is which binds to cellulose [100] surface more strongly, as judged by the difference in free energy of binding, or AAFbmdmg, or the difference in A F between the binding processes, 4 and 2. The free energy change for the sum of reactions 1 and 2 must be equal to the free energy change for the sum of reactions 4 and 3 since they have the same initial and final states. This relationship is expressed, for the reactions going in the directions indicated in the figure,

A F4 + A F3 = A F1 + A F2

which can be rearranged to yield the difference in binding free energies desired,

AF4 — AF2 = AF1 — AF3 = AAFbinding (8.9)

With this relation, we can determine the difference between two processes that are very hard to compute using two processes that are straightforward and relatively easy. In particular, both processes involve only changing CH3 to OH, and process 1 does not even involve the cellulose since there is no change in its solvated state in process 1. Calculation of the
significant entropic contribution to the binding process is eliminated in this method since the difference of the two binding processes largely cancels out the entropic contributions. The simulations do not have to correctly simulate the large changes associated with desolvating both the cellulose and the ligand, only the small solvation and structural entropy changes associated with changing a small functional group.