Introduction to Molecular Dynamics: Theory and Applications in Biomolecular Modeling - Computational Modeling of Biological Systems - page 23

If is a parameter in the potential energy function, we have

P exp. ˇV/ @ @

@V
@

dA
d D

P exp. ˇV/ D

: (36)

The free energy difference between state 0 and state 1 is obtained by integrating
dA=d in the range of 0 to 1 .

@V
@

Z 1

A D : (37)

0

The derivative of the potential energy V with respect to can be performed
analytically in many cases, and the calculation of dA=d amounts to obtain the
numerical value of dV=d for each configuration sampled at a particular ,andthen
calculate the ensemble average dV=d i by summing over all the configurations.

The integration in ( 37 ) is performed numerically, often using either the trapezoidal
or the Simpson's rule. The performance of these two integration methods has been
compared in a recent study, which shows that the Simpson's rule tends to generate
smaller systemic errors in the results [ 60 ].
An important development that significantly improves the accuracy of alchemical
transformation calculations is the soft-core potential [ 12 , 126 ], which can be used
in combination with both the TI and FEP methods. Such a potential effectively
removes the singularities in the potential energy function when the distance between
the ligand and surrounding atoms approaches zero during the creation or annihila-
tion process. Since the numerical accuracy of the result is significantly improved,
the soft-core potential should always be used in alchemical transformations.

h

6.5 Umbrella Sampling and Other Techniques

Apart from FEP and TI, another commonly used free energy calculation method is
the umbrella sampling (US) approach. This method divides the transition from 0
to 1 into multiple windows, and uses a biasing potential to restrain the system at a
particular in each window. The probability distribution along in each window is
collected as a histogram, which is combined to give the complete free energy profile
using the weighted histogram analysis method (WHAM) [ 74 , 97 ].
The US method is often used as a benchmark to evaluate the performance of
new free energy calculation techniques. Recently, several such techniques have
been developed, including metadynamics [ 75 ] and adaptive biasing force (ABF)
method [ 30 , 31 , 53 ]. Both of them have been applied to biomolecular systems
and demonstrated superior performance than the US method. Another recent
development that significantly improves the efficiency of protein-ligand binding
affinity calculation is the enveloping distribution sampling (EDS) method [ 26 , 91 ],