# Monte Carlo Simulations

Simulated annealing (SA) (Kirkpatrick et al. 1983) is probably the most popular conformational search method. SA is general in that it is easy and straightforward to apply to any kind of optimization problem. In SA, one typically applies the Metropolis MC algorithm to generate a series of conformational states following the canonical Boltzmann energy distribution for a given temperature. SA initially executes high temperature MC simulation, followed by a series of simulations subject to a temperature-lowering schedule, hence the name simulated annealing. As much as SA is simple, its conformational search efficiency is not so impressive compared to other more sophisticated methods discussed below.

When the energy landscape of the system under investigation is rugged (due to numerous energy barriers), MC simulations are prone to get stuck in meta-stable states that will distort the distribution of sampled states by breaking the ergodicity of sampling. To avoid this malfunction, many simulation techniques have been developed, and one of the successful approaches is based on the generalized ensemble approach in contrast to the usual canonical ensemble. This kind of method was initially called by different names including multi-canonical ensemble (Berg and Neuhaus 1992) and entropic ensemble (Lee 1993). The underlying idea is to expedite the transition between states separated by energy barriers by modifying the transition probability so that the final energy distribution of sampling becomes more or less flat rather than bell-shaped. A popular method similar in this spirit is the replica exchange MC method (REM) (Swendsen and Wang 1986) where a set of many canonical MC simulations with temperatures distributed in a selected range are simultaneously carried out. From time to time one attempts to exchange structures (or equivalently temperatures) from neighboring simulations to sample states in a wide range of energy spectrum as the means to overcome energy barriers. Parallel hyperbolic sampling (PHS) (Zhang et al. 2002) further extends the REM by dynamically deforming energy using an inverse hyperbolic sine function to lower the energy barrier.

Monte Carlo with minimization (MCM), proposed by Li and Scheraga (1987), was successfully applied for the conformational search by several structure prediction programs (Simons et al. 1997). In MCM, one performs MC moves between local energy minima after local energy minimization of each perturbed protein structure. For a given local energy minimum structure A, a trial structure B is generated by random perturbation of A and is subsequently subject to local energy minimization. The usual Metropolis algorithm is used to determine the acceptance of B over A by calculating the energy difference between the two.