Estimating Full Path Lengths and Kinetics from Partial Path Transition Interface Sampling Simulations

TL;DR

Rare-event kinetics in molecular dynamics are difficult to obtain directly, motivating the development of REPPTIS which samples partial paths to accelerate trajectories. The authors introduce a Markov state model (MSM) that treats overlapping REPPTIS partial-path ensembles as states connected by local crossing probabilities, yielding closed-form expressions for the global crossing probability $P_A(\lambda_B|\lambda_A)$, MFPTs, flux $f_A$, and rate $k_{AB}$. Validation on 1D potentials and all-atom KCl dissociation shows accurate replication of kinetics compared to RETIS, while trypsin-benzamidine dissociation tests reveal limitations and the need for careful path initialization and force-field considerations. Overall, the MSM framework provides a robust theoretical and practical means to extract time-dependent kinetic information from computationally efficient partial-path simulations, broadening the applicability of REPPTIS to biomolecular kinetics.

Abstract

Assessing the time scale of biological processes using molecular dynamics (MD) simulations with sufficient statistical accuracy is a challenging task, as processes are often rare and/or slow events, which may extend largely beyond the time scale of what is accessible with modern day high performance computational infrastructure. Recently, the replica exchange partial path transition interface sampling (REPPTIS) algorithm was developed to study rare and slow events involving metastable states along their reactive pathways. REPPTIS is a path sampling method where paths are cut short to reduce the computational cost, while combining this with the efficiency offered by replica exchange between the partial path ensembles. However, REPPTIS still lacks a formalism to extract time-dependent properties, such as mean first passage times, fluxes, and rates, from the short partial paths. In this work, we introduce a Markov state model (MSM) framework to estimate full path lengths and kinetic properties from the overlapping partial paths generated by REPPTIS. The framework results in newly derived closed formulas for the REPPTIS crossing probability, mean first passage times (MFPTs), flux, and rate constant. Our approach is then validated using simulations of Brownian and Langevin particles on a series of one-dimensional potential energy profiles as well as the dissociation of KCl in solution, demonstrating that REPPTIS accurately reproduces the exact kinetics benchmark. The MSM framework is further applied to the trypsin-benzamidine complex to compute the dissociation rate as a test case of a biological system, albeit the computed rate underestimates the experimental value. In conclusion, our MSM framework equips REPPTIS simulations with a robust theoretical and practical foundation for extracting kinetic information from computationally efficient partial paths.

Figures (10)

• Figure 1: Both RETIS and REPPTIS use a set of $N+1$ interfaces $\lambda_i$, i.e. _A$\lambda_{A}$ , _1$\lambda_{1}$ , …, _B$\lambda_{B}$ . Left: free energy $F$ as a function of $\lambda$. Right: A long MD trajectory (shown twice) is cut into segments of RETIS and REPPTIS path ensembles. State $A$ is sampled in the [0^-] $\left[0^{-}\right]$ ensemble (light blue). For RETIS, paths that reach up to _i$\lambda_{i}$ are element of the $[0^+],\ldots,[i^+]$ ensembles, indicated by multicoloring. E.g., segments that did not reach _1$\lambda_{1}$ are only a part of the [0^+] $\left[0^{+}\right]$ ensemble (red), while the final segment reaches _B$\lambda_{B}$ and is in all [i^+] $\left[i^{+}\right]$ ensembles (red, green and blue). For REPPTIS, paths of ensemble [i^±] $\left[i^{\pm}\right]$ are restricted in $\left[\lambda{i-1}$λ_i-1$, \lambda{i+1}$λ_i+1$\right]$ (exceptions for the ensembles around _A$\lambda_{A}$ , see main text).
• Figure 2: Illustration highlighting the transition between partial and full paths. Short PPTIS or REPPTIS paths should be recombined to form full paths when kinetics properties such as flux, rate, and mean first passage time are sought after.
• Figure 3: (A) Paths of the TIS [i^+] $\left[i^{+}\right]$ ensemble start at _A$\lambda_{A}$ , cross _i$\lambda_{i}$ before recrossing _A$\lambda_{A}$ , and end in _A$\lambda_{A}$ or _B$\lambda_{B}$ . (B) Paths of the PPTIS [i^±] $\left[i^{\pm}\right]$ ensemble cross _i$\lambda_{i}$ , while starting and ending from either _i-1$\lambda_{i-1}$ or _i+1$\lambda_{i+1}$ . The [0^±] $\left[0^{\pm}\right]$ ensemble is a special case.
• Figure 4: (A) A long MD path decomposed into its (overlapping) PPTIS path segments ($U_0, U_1, \ldots$). (B) Ensemble [i^±] $\left[i^{\pm}\right]$ ($i=1,..,N-2$) contains 4 path types (red) that each can transition to 2 possible path types in a neighboring ensemble (black dotted) with a probability given by PPTIS local crossing probabilities. (C) Path lengths $\tau$ are decomposed into three parts: the part $\tau_{(1)}$ before the first crossing of _i$\lambda_{i}$ (red), the part $\tau_{(2)}$ after the last crossing of _i$\lambda_{i}$ (green), and the part $\tau_{(m)}$ in between (blue). Some parts can be zero; e.g. the second path has no middle part.
• Figure 5: The trajectory is colored according to the overall state (blue in $\mathcal{A}$, orange in $\mathcal{B}$). Different shades of blue are used to distinguish between $[0^-]$ and $[0^+]$. The flux is the inverse of the average time a path spends in $\mathcal{A}$ (in $[0^-]$ and $[0^+]$) between two subsequent positive crossings of $\lambda_A$ (purple triangles), as written in Eq. \ref{['eq:flux']}. The rate constant is the inverse of the average time $\tau_{\mathcal{A},1}$ of one visit to state $\mathcal{A}$, i.e. between first entering into state $A$ (green circle) and exiting into state $B$ (red circle), as written in Eq. \ref{['eq:k_1visit']}. This example has 1 entrance into A, 4 positive crossings of $\lambda_A$, and 1 exit into state B.