Weighted least square analysis method for free energy calculations

Free energy calculation is an efficient way for studying rare event dynamics. For a complex rare event dynamics, multiple reaction coordinates may be required to describe the transition path between equilibrium states. Theoretically, a one-dimensional sampling along the transition path can provide sufficient information to calculate the potential of mean force (PMF) along the transition path. In the widely used free energy analysis method Wham, the sample data are divided into a series of bins to calculate PMF. However, bin segmentation in Wham is coupled with the umbrella potentials applied in each window, because each umbrella potential is assumed to have a close value for all sample points in each bin. This coupling makes it difficult to perform one-dimensional bin segmentation along the transition path when multivariable umbrella potentials are used in sampling. Here, we develop a weighted least square analysis method (Welsam) to take the place of Wham for free energy analysis. In the new method Welsam, bin segmentation is decoupled from application of umbrella potentials. As a result, it becomes very convenient to perform one-dimensional bin segmentation and calculate one-dimensional PMF along the transition path. Our simulation results suggest that Welsam has a comparable statistical error with Wham. Furthermore, Welsam can be used to reduce waste of sample data obtained during exploration of reaction coordinates.