Computational Drug Discovery and Design

histogram analysis method (WHAM) [11, 24]. MBAR has the

lowest variance among the methods discussed here, and is likely

the most reliable estimator for the type of free energy calculations

described in this chapter [24, 25].

EXP and BAR are available in popular simulation packages such

as Gromacs [26], Amber [27], and NAMD [28]. A python imple-

mentation of the MBAR estimator is instead provided by the

authors of the original publication at https://github.com/

choderalab/pymbar.

2.2.2 Thermodynamic
Integration

In the thermodynamic integration (TI) approach, rather than

potential energy differences, the data needed to estimate the free

energy difference is the derivative of the potential energy with

respect to the coupling parameterλ, a continuous variable that

describes the series of intermediates states between the two end

statesλ¼0 andλ¼1. From this observable, it is possible to recover

the free energy difference with the following formula.

ΔG 0 , 1 ¼

ðλ¼ 1

λ¼ 0

∂UðÞλ;x

∂λ λ

ð 8 Þ

The ensemble averageh∂U/∂λiis obtained from an equilib-

rium simulation performed at a certain value ofλ. Numerical inte-

gration is then needed to recover the free energy difference

betweenλ¼0 andλ¼1. The trapezoidal rule is often used for

simplicity, but any numerical integration method can be employed.

Whileλis a continuous variable, only discrete values of it are

sampled, so that there will be a bias in the estimate that depends

on how well the chosen values ofλallow for an accurate quadrature.

Therefore, while the accuracy of perturbation approaches depends

on the overlap of energy distributions, the accuracy of TI depends

on the smoothness of the integrand [29, 30].

TI is a popular method for the estimation of free energy differ-

ences, as it is robust and accurate while also easy to implement if the

simulation code provides∂U/∂λvalues. Thealchemical analysis

tool [30] available at https://github.com/MobleyLab/alchemi

cal-analysisalready implements the automated analysis of simula-

tion data collected with Gromacs [26], Amber [31], or Sire

(http://siremol.org/), and the estimation of free energy differ-

ences using TI, EXP, BAR, and MBAR.

2.3 The
Thermodynamic Cycle

For both perturbation and thermodynamic integration methods,

several intermediate states are needed in order to obtain a reliable

estimate of their free energy difference. In fact, when the two end

states are the protein-bound and protein-unbound ligand, it is not

possible to obtain a reliable binding free energy estimate by simu-

lating only these two, and a pathway of intermediate states is

needed. The computation of a free energy difference thus involves

Absolute Alchemical Free Energy 205