Analysis and optimization of weighted ensemble sampling

We give a mathematical framework for weighted ensemble (WE) sampling, a binning and resampling technique for efficiently computing probabilities in molecular dynamics. We prove that WE sampling is unbiased in a very general setting that includes adaptive binning. We show that when WE is used for stationary calculations in tandem with a coarse model, the coarse model can be used to optimize the allocation of replicas in the bins.Comment: 22 pages, 3 figure

Markov State Models Based on Milestoning

Markov State Models (MSMs) have become the tool of choice to analyze large amounts of molec- ular dynamics data by approximating them as a Markov jump process between suitably predefined states. Here we investigate ”Core Set MSMs”, a new type of MSMs that builds on metastable core sets acting as milestones for tracing the rare event kinetics. We present a thorough analysis of Core Set MSMs based on the existing milestoning framework, Bayesian estimation methods and Transi- tion Path Theory (TPT). As a result, Core Set MSMs can now be used to extract phenomenological rate constants between the metastable sets of the system and to approximate the evolution of certain key observables. The performance of Core Set MSMs in comparison to standard MSMs is analyzed and illustrated on a model potential and the torsion angle dynamics of Alanine dipeptide

Kinetics and Free Energy of Ligand Dissociation Using Weighted Ensemble Milestoning

We consider the recently developed weighted ensemble milestoning (WEM) scheme [J. Chem. Phys. 152, 234114 (2020)], and test its capability of simulating ligand-receptor dissociation dynamics. We performed WEM simulations on the following host-guest systems: Na$^+$/Cl$^-$ ion pair and 4-hydroxy-2-butanone (BUT) ligand with FK506 binding protein (FKBP). As proof or principle, we show that the WEM formalism reproduces the Na$^+$/Cl$^-$ ion pair dissociation timescale and the free energy profile obtained from long conventional MD simulation. To increase accuracy of WEM calculations applied to kinetics and thermodynamics in protein-ligand binding, we introduced a modified WEM scheme called weighted ensemble milestoning with restraint release (WEM-RR), which can increase the number of starting points per milestone without adding additional computational cost. WEM-RR calculations obtained a ligand residence time and binding free energy in agreement with experimental and previous computational results. Moreover, using the milestoning framework, the binding time and rate constants, dissociation constant and the committor probabilities could also be calculated at a low computational cost. We also present an analytical approach for estimating the association rate constant ($k_{\text{on}}$) when binding is primarily diffusion driven. We show that the WEM method can efficiently calculate multiple experimental observables describing ligand-receptor binding/unbinding and is a promising candidate for computer-aided inhibitor design

Approximating First Hitting Point Distribution in Milestoning for Rare Event Kinetics

Milestoning is an efficient method for rare event kinetics calculation using short trajectory parallelization. The phase space is partitioned into small compartments, and interfaces of compartments are called milestones. Local kinetics between milestones are aggregated to compute the flux through the entire reaction space. Besides the accuracy of force fields, the accuracy of Milestoning crucially depends on two factors: the initial distribution of a short trajectory ensemble and statistical adequacy of trajectory sampling. The latter can be improved by increasing the number of trajectories while the true initial distribution, i.e., first hitting point distribution (FHPD), has no analytic expression in the general case. Here, we propose two algorithms, local passage time weighted Milestoning (LPT-M) and Bayesian inference Milestoning (BI-M), to accurately and efficiently approximate FHPD in Milestoning for systems at equilibrium condition, leading to accurate mean first passage time (MFPT) computation. Starting from equilibrium Boltzmann distribution on milestones, we calculate the proper weighting factor for FHPD approximation and consequently trajectories. The method is tested on two model examples for illustration purpose. The results show that LPT-M is especially advantageous in terms of computational costs and robustness with respect to the increasing number of intermediate milestones. The MFPT estimation achieves the same accuracy as a long equilibrium trajectory simulation while the consumed wall-clock time is as cheap as the widely used classical Milestoning. BI-M also improves over classical Milestoning and covers the directional Milestoning method as a special case in the deterministic Hamiltonian dynamics. Iterative correction on FHPD can be further performed for exact MFPT calculation

On Markov State Models for Metastable Processes

We consider Markov processes on large state spaces and want to find low-dimensional structure-preserving approximations of the process in the sense that the longest timescales of the dynamics of the original process are reproduced well. Recent years have seen the advance of so-called Markov state models (MSM) for processes on very large state spaces exhibiting metastable dynamics. It has been demonstrated that MSMs are especially useful for modelling the interesting slow dynamics of biomolecules (cf. Noe et al, PNAS(106) 2009) and materials. From the mathematical perspective, MSMs result from Galerkin projection of the transfer operator underlying the original process onto some low-dimensional subspace which leads to an approximation of the dominant eigenvalues of the transfer operators and thus of the longest timescales of the original dynamics. Until now, most articles on MSMs have been based on full subdivisions of state space, i.e., Galerkin projections onto subspaces spanned by indicator functions. We show how to generalize MSMs to alternative low-dimensional subspaces with superior approximation properties, and how to analyse the approximation quality (dominant eigenvalues, propagation of functions) of the resulting MSMs. To this end, we give an overview of the construction of MSMs, the associated stochastics and functional-analysis background, and its algorithmic consequences. Furthermore, we illustrate the mathematical construction with numerical examples

Numerical multiscale methods: from homogenization to milestoning

The dissertation focuses on addressing the challenges posed by multiscale problems in applied mathematics, which stem from the intricate interplay between microscales and the computational demands of resolving fine details. To alleviate this burden, numerical homogenization and averaging methods are favored. This study explores three interconnected topics related to numerical techniques for handling multiscale problems in both spatial and temporal domains. In the first part, we establish the equivalence principle between time averaging and space homogenization. This principle facilitates the application of various numerical averaging techniques, such as FLAVORS, Seamless, and HMM, to boundary value problems. Moreover, we introduce the dilation operator as a decomposition-free approach for numerical homogenization in higher dimensions. Additionally, we utilize the Synchrosqueezing transform as a preprocessing step to extract oscillatory components, crucial for the structure-aware dilation method. The second part extends the Deep Ritz method to multiscale problems. We delve into the scale convergence theory to derive the [Gamma]-limit of energy functionals exhibiting oscillatory behavior. The resulting limit object, formulated as a minimization problem, captures spatial oscillations and can be tackled using existing neural network architectures. In the third part, we lay the groundwork for the milestoning algorithm, a successful tool in computational chemistry for molecular dynamics simulations. We adapt this algorithm to a domain-decomposition-based framework for coarse-grained descriptions and establish the well-posedness of primal and dual PDEs. Additionally, we investigate the convergence rate and optimal milestone placements. We illustrate this framework through the understanding of the Forward Flux algorithm as a specific example.Mathematic