Enhanced Sampling of Configuration and Path Space in a Generalized Ensemble by Shooting Point Exchange

The computer simulation of many molecular processes is complicated by long timescales caused by rare transitions between long-lived states. Here, we propose a new approach to simulate such rare events, which combines transition path sampling with enhanced exploration of configuration space. The method relies on exchange moves between configuration and trajectory space, carried out based on a generalized ensemble. This scheme substantially enhances the efficiency of the transition path sampling simulations, particularly for systems with multiple transition channels, and yields information on thermodynamics, kinetics and reaction coordinates of molecular processes without distorting their dynamics. The method is illustrated using the isomerization of proline in the KPTP tetrapeptide.

Figure 1

Schematic representation (a) and pseudocode (b) of the SPEx algorithm.

Figure 2

Shooting point exchange for sampling the two reaction channels in a two-dimensional double well model. (a) Potential energy and state definitions of the model system. (b) Free energy as a function of $x^{(0)}$ for different barrier heights. (c) Average switching time between states A and B, ${\tau }_{\mathrm{A}\leftrightarrow \mathrm{B}}$, for stand-alone metadynamics and metadynamics with SPEx, starting the sampling from a converged bias potential. Methods are tested with $x^{(0)}$ and the polar angle (Polar CV) as the reaction coordinate. (d),(e) Root mean squared error of the fraction of paths in the upper reaction channel as a function of simulation length for standalone TPS and SPEx. Each curve is for a specific barrier height and is estimated from 2500 independent sampling runs. (f) Number of switches between the upper and lower reaction channel $N_s$ per force evaluation $n_F$ as a function of the barrier height for different path sampling methods.

Figure 3

Isomerization of proline in the tetrapeptide KPTP. (a) Structure of the peptide (relevant torsion angle definitions in the inset). (b) Scheme of the cis to trans isomerization and the transition state geometry. (c) Schematic overview of the committor learning process. (d) Free energy from metadynamics including shooting point exchanges reconstructed from the converged metadynamics bias potential. (e) Fraction of paths in each reaction channel as a function of the number of trials based on ten independent shooting point exchange simulations. (f) Training data for the committor prediction on top of the free energy surface. Circles show the state definitions ($c=cis$, $t=trans$). (g) Comparison of the sampled committor from fleeting trajectories and the predicted committor. The dashed black line shows the ideal correspondence while the orange line and shaded area show the average and standard deviation of the sampled committor in a given window of the predicted committor. (h) Attributions corresponding to the ten most important input features of the neural network. (i) Free energy along the two most relevant collective variables from (h). Black lines show isolines of the committor function obtained using symbolic regression. Crosses and circles indicate if a trajectory starting from that point reached the cis or trans state first. Representative reactive paths are shown in the same color scheme as in (e).