Monte Carlo (MC) Simulation

Instead of evaluating forces to determine incremental atomic motions, Monte Carlo simulation simply imposes relatively large motions on the system and determines whether or not the altered structure is energetically feasible at the temperature simulated. The system jumps abruptly from conformation to conformation, rather than evolving smoothly through time. It can traverse barriers without feeling them; all that matters is the relative energy of the conformations before and after the jump. Because MC simulation samples conformation space without a true `time' variable or a realistic dynamics trajectory, it cannot provide time-dependent quantities. However, it may be much better than MD in estimating average thermodynamic properties for which the sampling of many system configurations is important.

Monte Carlo makes use of Boltzmann probabilities, not forces.

When the potential energy V and observables to be calculated from the simulation are velocity-independent (as is typical), an MC simulation need only compare potential energies V, not total energies E (see `Calculating Equilibrium Averages' above). Two conformations, $\vec{R}$ and $\vec{R^\prime}$, are compared and updated as shown below [13]. $RAND$ is a random number uniformly distributed on [0,1].

Metropolis Monte Carlo.

For simple systems, the structural modifications are often tuned so that about 50% of the $\vec{R^\prime}$ conformations are accepted. For macromolecular systems, this acceptance ratio can be much smaller, e.g. when dihedral angles are modified by large amounts. It is then generally expedient to bias the random moves in favor of known structural preferences such as side chain rotamers (`biased probability Monte Carlo'). In searching for low-energy local minima, it can be advantageous to minimize the energy before evaluating the energy $V^\prime$ (`Monte Carlo-minimization', or MCM [14]). Simulated annealing has also been performed prior to accepting or rejecting the new conformation in a `Monte Carlo-minimization/annealing' (MCMA) protocol [15]. Because explicit water molecules can hinder the acceptance of new conformations, Monte Carlo (or MCM, MCMA) simulations of macromolecules generally use an implicit model of solvation, as in references [15,16]. That is, a term is added to the empirical potential energy function that mimics the effects of water and, in some cases, counter ions.