Computational Drug Discovery and Design

3.5 Umbrella
Sampling (US) Protocol
for Unbinding Process

3.5.1 Bias and
Parameter Set

In this methodology, the main idea is to force the sampling of the

system in a particular direction, with the aim to sample an event of

interest. To do this, the system is constrained through the addition

of a harmonic potential energy (Ebias) to the energy of the system

(ESys)(seeNote 15).ζis the restrained variable along which the bias

is applied and can be of different nature (distance, angle, dihedral,

geometrical combination, etc.) depending on the studied biological

event. Alongζseveral windows are defined. One window is asso-

ciated to a specific value of the constrained variable (ζth), and each

window consists in a short molecular dynamic simulation with

Ebiasdefined with to the considered value ofζth. The energy of

the system is then defined by:

EUS¼ESysþEbias

withEbias¼

k

2

ðÞζζth^2

ζrepresents the value of the restrained variable sampled in the

current system state,kis the force constant of the harmonic poten-

tial, andζthis the fixed value of the constrained variable in the

considered window. For each window, the value ofζthis different.

The main advantage of this type of protocol is the possibility to

unbias the simulation, in order to get the variation of energy

associated to the event that was sampled, called the Potential of

Mean Force (PMF) [40]. This PMF can be associated to a free

energy variation when the convergence of each window is reached,

meaning that the system is equilibrated around the corresponding

value ofζth. The sampling ofζin each window has to be extracted

from these simulations, and the Weighted Histogram Analysis

Method (WHAM) [41] is used to unbias the simulation and to

Fig. 3PC1–PC2 plots of the backbone calculated from the last 100 ns of the aMD simulation in green and the
crystallographic structures in red. (a) Before reweighting. (b) Two-dimensional free energy profile projection
(kcal mol^1 ) after reweighting using Maclaurin series to the tenth order

Enhanced Molecular Dynamics 417