Principal component analysis of nonequilibrium molecular dynamics simulations

Principal component analysis (PCA) represents a standard approach to identify collective variables $\{x_i\}\!=\!\boldsymbol{x}$, which can be used to construct the free energy landscape $\Delta G(\boldsymbol{x})$ of a molecular system. While PCA is routinely applied to equilibrium molecular dynamics (MD) simulations, it is less obvious how to extend the approach to nonequilibrium simulation techniques. This includes, e.g., the definition of the statistical averages employed in PCA, as well as the relation between the equilibrium free energy landscape $\Delta G(\boldsymbol{x})$ and energy landscapes $\Delta{\cal G} (\boldsymbol{x})$ obtained from nonequilibrium MD. As an example for a nonequilibrium method, `targeted MD' is considered which employs a moving distance constraint to enforce rare transitions along some biasing coordinate $s$. The introduced bias can be described by a weighting function $P(s)$, which provides a direct relation between equilibrium and nonequilibrium data, and thus establishes a well-defined way to perform PCA on nonequilibrium data. While the resulting distribution ${\cal P}(\boldsymbol{x})$ and energy $\Delta{\cal G} \propto \ln {\cal P}$ will not reflect the equilibrium state of the system, the nonequilibrium energy landscape $\Delta{\cal G} (\boldsymbol{x})$ may directly reveal the molecular reaction mechanism. Applied to targeted MD simulations of the unfolding of decaalanine, for example, a PCA performed on backbone dihedral angles is shown to discriminate several unfolding pathways. Although the formulation is in principle exact, its practical use depends critically on the choice of the biasing coordinate $s$, which should account for a naturally occurring motion between two well-defined end-states of the system.