Calculating free energies using scaled-force molecular dynamics algorithm

One common objective of molecular simulations in chemistry and biology is to calculate the free energy difference between different states of the system of interest. Examples of problems that have such an objective are calculations of receptor-ligand or proteindrug interactions, associations of molecules in response to hydrophobic, and electrostatic interactions or partition of molecules between immiscible liquids. Another common objective is to describe evolution of the system towards a low energy (possibly the global minimum energy), “native” state. Perhaps the best example of such a problem is folding of proteins or short RNA molecules. Both types of problems share the same difficulty. Often, different states of the system are separated by high energy barriers, which implies that transitions between these states are rare events. This, in turn, can greatly impede exploration of phase space. In some instances this can lead to “quasi non-ergodicity”, whereby a part of phase space is inaccessible on timescales of the simulation. A host of strategies has been developed to improve efficiency of sampling the phase space. For example, some Monte Carlo techniques involve large steps which move the system between low-energy regions in phase space without the need for sampling the configurations corresponding to energy barriers (J-walking). Most strategies, however, rely on modifying probabilities of sampling low and high-energy regions in phase space such that transitions between states of interest are encouraged. Perhaps the simplest implementation of this strategy is to increase the temperature of the system. This approach was successfully used to identify denaturation pathways in several proteins, but it is clearly not applicable to protein folding. It is also not a successful method for determining free energy differences. Finally, the approach is likely to fail for systems with co-existing phases, such as water-membrane systems, because it may lead to spontaneous mixing. A similar difficulty may be encountered in any method relying on global modifications of phase space. A new, promising technique is the multicanonical Monte Carlo method. In this algorithm, sampling of energy proportional to the Boltzmann factor is substituted by sampling from the uniform energy probability distribution. This means that multicanonical simulation corresponds to a random walk in one-dimensional energy space and, therefore, the system does not experience energy barriers. Since multicanonical weights are not known initially, they have to be estimated in the first step of the simulations. The multicanonical Monte Carlo method appears to be particularly suitable to study helixcoil transition in proteins because a single simulation can provide information about both the low-temperature, helical state and the high-temperature, disordered state. Many problems of chemical or biological interest can be formulated in terms of evo-