Generalized Path Reweighting and History-Dependent Free Energies

TL;DR

This work advances path-based rare-event sampling by coupling Infinity-RETIS with a generalized path reweighting framework, producing WHAM-like weights that unify observables across multiple ensembles under biased sampling and fractional counts. It introduces history-dependent conditional free energy surfaces, which encode kinetic effects such as mass and friction and can reveal barriers invisible to standard thermodynamic FE profiles, while remaining robust to suboptimal reaction coordinates. The approach is demonstrated across multiple 1D and 2D Langevin models and a biomembrane permeation system, showing that conditional FEs provide nuanced insight into transition dynamics and that unconditional FEs can be reconstructed from the conditional information. Overall, the method offers a scalable, accurate toolkit for characterizing complex molecular transitions with strong relevance to reaction mechanism elucidation and reaction-coordinate optimization.

Abstract

Transition interface sampling (TIS) and replica exchange TIS (RETIS) are powerful methods for computing rates of rare events inaccessible to straightforward molecular dynamics (MD) simulations. Path reweighting extends their output, enabling the evaluation of diverse thermodynamic and kinetic quantities, including reaction prediction metrics, activation barriers, committor functions, and free energies. The recently developed Infinity-RETIS algorithm boosts parallel efficiency through asynchronous replica exchanges in the infinite-swap limit, eliminating the wall-time bottlenecks of conventional RETIS. This approach introduces fractional samples and biased sampling distributions, requiring a generalized path reweighting framework, for which we derive expressions demonstrating how it can be used to compute exact dynamic and thermodynamic variables. We then focus on a special class of free energy surfaces defined by history-dependent conditions, whose values are influenced by kinetic factors such as particle mass and friction, unlike standard unconditional free energy surfaces. These conditional free energies can reveal kinetically relevant barriers even with suboptimal reaction coordinates and therefore provide a rigorous and versatile tool for characterizing complex molecular transitions.

Figures (4)

• Figure 1: (a)--(d): Conditional free energy $F_\mathcal{A}(\lambda)$ for the flat potential (a,b) and the cosine bump (c,d). The inset in (a) depicts $\varrho_\mathcal{A}(\lambda)$ for $\gamma=1$ and 500 for the flat potential. $u_\text{\rm cos}(x)$ is drawn with a black dashed line. Panels (a) and (c) show results for various friction $\gamma$ values with fixed mass $m=1$, while panels (b) and (d) show results for various mass values $m$ with fixed $\gamma=20$. (e)--(f): $\Delta F_\mathcal{A}$ as a function of $\gamma$ and $m$.
• Figure 2: Double-well potential $u_{\rm dw}(x)$ of Eq. \ref{['eq:dw']}. Reduced units. (a): The unconditional free energy $F(x)$ (dashed black line) and the conditional free energy $F_\mathcal{A}(x)$ for $\gamma=0.3$ and $\gamma=10$. The utilized interfaces are shown in vertical gray/black in (a) and (d). (b) and (c): Heatmap of the $x$ and velocity conditional free energy $F_\mathcal{A}(x,v)$ for $\gamma=0.3$ and $\gamma=10$, with example reactive (blue), nonreactive (orange), and state A (green) trajectories also shown. (d): The crossing probability $P_A(\lambda|\lambda_A)$ for the two $\gamma$ simulations.
• Figure 3: The 2D potential $u_\text{2D}$ (Eq. \ref{['eq:2dangle']}) is shown by the black contour lines in sub-panels (a) and (c). The 1D $x$ and $y$ projections are shown by the black solid ($F(x)$) and dashed ($F(y)$) lines in (d), (e) and (f). The computed 2D conditional free energies $F_\mathcal{A}(x,y)$ and $F_\mathcal{B}(x,y)$ are shown in (a) and (c) as a heatmap (with maximum values $\approx24$ and 34 $k_BT$). The 1D conditional free energy profiles $F_\mathcal{A}(x)$ (blue) and $F_\mathcal{A}(y)$ (orange) for the forward transition are shown as lines in (d), and similarly for $F_\mathcal{B}(x)$ (blue) and $F_\mathcal{B}(y)$ (orange) for the backward transition in (f). Other colored lines (blue shade to orange shade) in (d)-(e)-(f) are $F_\mathcal{A}(\chi;\theta)$, $F(\chi;\theta)$, and $F_\mathcal{B}(\chi;\theta)$ for a series of parameter values $\theta$, i.e. for changing definitions of the $\chi$-axis. The conditional zero free energy and the unconditional $y$ projection in (d) is shifted to the local left minima in (d). The vertical red dashed lines in (d), (e) and (f) serve as reference points for determining the barrier heights $\Delta F(\theta)$ shown in (b).
• Figure 4: a) (Un)conditional free energies of the drug molecule (5-ALA) permeating through the DPPC membrane as a function of the order parameter ($\lambda$). b) Comparison of the unconditional free energies obtained from $\infty$RETIS and umbrella sampling simulations starting from different initial trajectories. All free energy profiles are shifted to zero in the water phase. The PMF profiles obtained from the SMD trajectory and from Genheden and Eriksson genheden2016estimation were symmetrized to span the full transition from state $A$ to state $B$.