Computational Approaches in Peptide and Protein Design: An Overview - Peptide and Protein Design for Biopharmaceutical Applications - page 23

compared with that of the old state. If it is the same or lower, the new
configuration is kept and becomes the basis for calculation of the next
configuration. If the new configuration has a higher energy than the
previous one then it is either kept or discarded depending on the energetic
difference between the two states (DE) and a random number, x, chosen
between 0 and 1. If exp( DE/kT) x then the new configuration is
accepted. In other words, there is a unitary probability of accepting a
move which results in an energy decrease, and an exponential probability
based on the Boltzmann factor of accepting a move with a higher energy
(the so-called Metropolis technique [54]). This procedure generates
ensembles of configurations weighted in accord with the canonical
Boltzmann distribution, and the average thermodynamic properties of
the system can be calculated simply by averaging the thermodynamic
properties associated with each configuration.
At the same time, non-Boltzmann Monte Carlo sampling may be
instrumental in accessing high-energy states that are infrequently reached
in the Boltzmann simulations. For instance, energetics associated with a
conformational transition from conformer A to conformer B would
involve one or more transition states, which are of higher energy and
would not be frequently sampled in an unconstrained simulation. In this
case, one can perform a series of individual simulations with specifically
selected constraints to focus on discrete regions of the reaction coordinate
of the transition from A to B. This procedure is called umbrella sampling
[55], and the effects of the different constraining potentials for each
individual simulation can be mathematically removed to generate a
potential of mean force for the transition coordinate that estimates the
activation energy.
One aspect shared by Monte Carlo methods and molecular dynamics is
the ability to cross barriers between local minima. In the case of Monte
Carlo, barrier crossing occurs by random change and acceptance of
higher energy states is a function of temperature. In the case of molecular
dynamics, the ability of crossing the barriers depends mostly on the
initial velocities of atoms, or, in effect, on the average temperature.
Because it is difficult to simulate systems (especially with explicit solvent)
for long enough to allow conformational transitions, there will always be
a concern that sampling of the potential surface was insufficient. One
approach to this problem is to do multiple runs from different starting
configurations of the system. One can examine convergence of the ensem-
ble averaged properties of each run to determine if adequate sampling has
occurred. Obviously, the ability of the system to make transitions over
activation energy barriers depends on the temperature. In order to