Monte Carlo Calculations (Molecular Biology)

Computer simulation calculations of average macromolecular properties by statistical mechanical approaches require one to evaluate the Boltzmann-weighted average of the given property over the entire conformation space (see equation (1) of Free Energy Calculations). Fortunately, it is possible to estimate such an average without actually sampling the entire enormous conformational space by exploiting the fact that points with high energy do not contribute significantly to the overall average. A powerful and systematic approach of doing so is the so-called Monte Carlo method devised by Metropolis et al. (1) This approach generates random conformations by moving several atoms or torsional angles and uses the corresponding value of the potential surface to accept or reject each new conformation. If the energy of the new conformation (n) is lower than the previous one (n-1), the new conformation is accepted. If the energy of the new conformation is higher than that of the previous one, a random number, Rn, between 0 and 1 is generated and compared to the Boltzmann factor.

If the random number is larger than P the move is rejected. If Rn is smaller than P the conformation is accepted. The desired average is then simply evaluated by

This random procedure generates a probability that corresponds to the proper Boltzmann distribution of the entire system if enough points are evaluated. In principle, one needs an enormous number of points, but in practice one can obtain reasonable results with a reasonable amount of computer time (if the property has similar values at different conformations). In fact, Monte Carlo approaches do not involve complete random searches, because they are usually implemented with umbrella sampling or free energy perturbation approaches that bias the search toward important regions in the conformational space (see Free Energy Calculations).

Average properties that can be evaluated by molecular dynamics (MD) techniques can also be evaluated by Monte Carlo approaches. Thus, the selection of the proper method depends on the problem at hand. The advantage of Monte Carlo approaches is the fact that they do not require the evaluation of the first derivatives of the potential functions, which is a prerequisite for MD approaches. Thus, one can use Monte Carlo approaches to nonphysical moves in studies of folding processes and related problems while using lattice models. On the other hand, MD approaches are much "smarter" in that they use derivatives that allow, in fact, much more systematic local searches. Thus, one can state that MD does a much better job in exploring local problems while Monte Carlo allows more uniform global searches. This is useful, for example, in particle insertion methods (2). Of course, one can combine the benefits of both approaches, using Monte Carlo calculations to generate starting conformations and MD for local exploration. Approaches that exploit this and related ideas have been developed (eg, Refs. 3 and 4). In fact, Brownian dynamics is a form of a smart Monte Carlo approach.