umbrellasampling

Umbrella sampling is a computational technique used to enhance the sampling of rare events in molecular simulations by applying biasing potentials along a chosen reaction coordinate. It is primarily used to estimate free energy profiles, such as the potential of mean force, by enabling the system to explore regions of phase space that are infrequently visited in unbiased simulations.

In practice, a reaction coordinate or collective variable ξ is identified to describe progress along a transition or reaction. The range of ξ is divided into a series of overlapping windows. In each window, a biasing potential, commonly a harmonic restraint W_i(ξ) = ½ k_i (ξ − ξ_i)^2, is imposed to keep sampling near a chosen center ξ_i. Molecular dynamics or Monte Carlo simulations are run independently for each window, yielding biased histograms of ξ. Adequate overlap between neighboring windows is crucial to allow a reliable reconstruction of the unbiased distribution.

Analysis typically uses reweighting methods such as the Weighted Histogram Analysis Method (WHAM) or multiparameter reweighting (MBAR) to combine the biased distributions and recover the unbiased probability P0(ξ). From P0(ξ), the potential of mean force F(ξ) is obtained as F(ξ) = −kT ln P0(ξ) + constant, with the constant fixed by convention (e.g., setting a reference point to zero). MBAR can offer improved statistical efficiency over WHAM in some cases.

Applications of umbrella sampling span chemical reactions, conformational transitions in biomolecules, and binding or diffusion processes along a coordinate. Limitations include dependence on an appropriate reaction coordinate, sufficient overlap between windows, and the computational cost of multiple concurrent simulations.