The potential of mean force (PMF) is used to get information about how the free energy of a system changes as it moves along a particular trajectory, which could represent a conformational change or chemical reaction. The key relationship is

P(x) a e-A(x)/kbT (8.10)

where P (x) is the probability of state x, A(x) is the Helmholtz free energy of state x, kb is Boltzman’s constant, and T is the temperature. State x is defined as a configuration or reaction coordinate and is constrained to a particular place in configuration space while the rest of the system is averaged over all accessible states. The PMF is useful in following the change in Helmholtz free energy as the system moves along the trajectory coordinate, most often to find the change in free energy between initial state, the transition and final state. In some cases, this probability can be arrived at by a simple MD simulation, run for sufficiently long time that all states x are sampled frequently enough to yield valid probabilities. The method of straight MD breaks down for this method when the barriers are high enough that the states around the transition state do not get visited even for very long simulations. The answer to this problem is to use some method of enhanced sampling, such as umbrella sampling (66). Umbrella sampling is implemented by applying a biasing potential, usually harmonic similar in shape to an inverse umbrella, centered at various points along the defined trajectory or reaction path and constructed such that the umbrellas are large enough and the points are close enough that the sampling within each umbrella overlaps with the adjacent umbrella sampling. The system will sample within the umbrella potential and give information about the probabilities of states within the umbrella, though they are biased. These probabilities can be unbiased but still will contain an unknown constant. The pieces can be spliced directlyback together lining up the overlap regions; more commonly the more exact method, weighted histogram analysis method (WHAM) (67), is used to produce the full PMF for the trajectory. Other methods, not discussed here, which are useful for the same problems as umbrella sampling and can often be used in cases where umbrella sampling does not work, are: locally enhanced sampling (68), replica exchange (69), and lambda dynamics (70).