Jan-Hendrik Prinz

Markov models of molecular kinetics: Generation and validation

Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This approach has many appealing characteristics compared to straightforward molecular dynamics simulation and analysis, including the potential to mitigate the sampling problem by extracting long-time kinetic information from short trajectories and the ability to straightforwardly calculate expectation values and statistical uncertainties of various stationary and dynamical molecular observables. In this paper, we summarize the current state of the art in generation and validation of MSMs and give some important new results. We describe an upper bound for the approximation error made by modeling molecular dynamics with a MSM and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast to previous practice, it becomes clear that the best MSM is not obtained by the most metastable discretization, but the MSM can be much improved if non-metastable states are introduced near the transition states. Moreover, we show that it is not necessary to resolve all slow processes by the state space partitioning, but individual dynamical processes of interest can be resolved separately. We also present an efficient estimator for reversible transition matrices and a robust test to validate that a MSM reproduces the kinetics of the molecular dynamics data.

PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models

Markov (state) models (MSMs) and related models of molecular kinetics have recently received a surge of interest as they can systematically reconcile simulation data from either a few long or many short simulations and allow us to analyze the essential metastable structures, thermodynamics, and kinetics of the molecular system under investigation. However, the estimation, validation, and analysis of such models is far from trivial and involves sophisticated and often numerically sensitive methods. In this work we present the open-source Python package PyEMMA ( http://pyemma.org ) that provides accurate and efficient algorithms for kinetic model construction. PyEMMA can read all common molecular dynamics data formats, helps in the selection of input features, provides easy access to dimension reduction algorithms such as principal component analysis (PCA) and time-lagged independent component analysis (TICA) and clustering algorithms such as k-means, and contains estimators for MSMs, hidden Markov models, and several other models. Systematic model validation and error calculation methods are provided. PyEMMA offers a wealth of analysis functions such that the user can conveniently compute molecular observables of interest. We have derived a systematic and accurate way to coarse-grain MSMs to few states and to illustrate the structures of the metastable states of the system. Plotting functions to produce a manuscript-ready presentation of the results are available. In this work, we demonstrate the features of the software and show new methodological concepts and results produced by PyEMMA.

Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules

Markov state models (MSMs) have been successful in computing metastable states, slow relaxation timescales and associated structural changes, and stationary or kinetic experimental observables of complex molecules from large amounts of molecular dynamics simulation data. However, MSMs approximate the true dynamics by assuming a Markov chain on a clusters discretization of the state space. This approximation is difficult to make for high-dimensional biomolecular systems, and the quality and reproducibility of MSMs has, therefore, been limited. Here, we discard the assumption that dynamics are Markovian on the discrete clusters. Instead, we only assume that the full phase-space molecular dynamics is Markovian, and a projection of this full dynamics is observed on the discrete states, leading to the concept of Projected Markov Models (PMMs). Robust estimation methods for PMMs are not yet available, but we derive a practically feasible approximation via Hidden Markov Models (HMMs). It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs/PMMs. The new framework is applicable to both, simulation and single-molecule experimental data. We demonstrate its versatility by applications to educative model systems, a 1 ms Anton MD simulation of the bovine pancreatic trypsin inhibitor protein, and an optical tweezer force probe trajectory of an RNA hairpin.

Mechanisms of Protein-Ligand Association and Its Modulation by Protein Mutations

Protein-ligand interactions are essential for nearly all biological processes, and yet the biophysical mechanism that enables potential binding partners to associate before specific binding occurs remains poorly understood. Fundamental questions include which factors influence the formation of protein-ligand encounter complexes, and whether designated association pathways exist. To address these questions, we developed a computational approach to systematically analyze the complete ensemble of association pathways. Here, we use this approach to study the binding of a phosphate ion to the Escherichia coli phosphate-binding protein. Various mutants of the protein are considered, and their effects on binding free-energy profiles, association rates, and association pathway distributions are quantified. The results reveal the existence of two anion attractors, i.e., regions that initially attract negatively charged particles and allow them to be efficiently screened for phosphate, which is subsequently specifically bound. Point mutations that affect the charge on these attractors modulate their attraction strength and speed up association to a factor of 10 of the diffusion limit, and thus change the association pathways of the phosphate ligand. It is demonstrated that a phosphate that prebinds to such an attractor neutralizes its attraction effect to the environment, making the simultaneous association of a second phosphate ion unlikely. This study suggests ways in which structural properties can be used to tune molecular association kinetics so as to optimize the efficiency of binding, and highlights the importance of kinetic properties.

Dynamical reweighting: Improved estimates of dynamical properties from simulations at multiple temperatures

Dynamical averages based on functionals of dynamical trajectories, such as time-correlation functions, play an important role in determining kinetic or transport properties of matter. At temperatures of interest, the expectations of these quantities are often dominated by contributions from rare events, making the precise calculation of these quantities by molecular dynamics simulation difficult. Here, we present a reweighting method for combining simulations from multiple temperatures (or from simulated or parallel tempering simulations) to compute an optimal estimate of the dynamical properties at the temperature of interest without the need to invoke an approximate kinetic model (such as the Arrhenius law). Continuous and differentiable estimates of these expectations at any temperature in the sampled range can also be computed, along with an assessment of the associated statistical uncertainty. For rare events, aggregating data from multiple temperatures can produce an estimate with the desired precision at greatly reduced computational cost compared with simulations conducted at a single temperature. Here, we describe use of the method for the canonical (NVT) ensemble using four common models of dynamics (canonical distribution of Hamiltonian trajectories, Andersen thermostatting, Langevin, and overdamped Langevin or Brownian dynamics), but it can be applied to any thermodynamic ensemble provided the ratio of path probabilities at different temperatures can be computed. To illustrate the method, we compute a time-correlation function for solvated terminally-blocked alanine peptide across a range of temperatures using trajectories harvested using a modified parallel tempering protocol.

Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics.

Parallel tempering (PT) molecular dynamics simulations have been extensively investigated as a means of efficient sampling of the configurations of biomolecular systems. Recent work has demonstrated how the short physical trajectories generated in PT simulations of biomolecules can be used to construct the Markov models describing biomolecular dynamics at each simulated temperature. While this approach describes the temperature-dependent kinetics, it does not make optimal use of all available PT data, instead estimating the rates at a given temperature using only data from that temperature. This can be problematic, as some relevant transitions or states may not be sufficiently sampled at the temperature of interest, but might be readily sampled at nearby temperatures. Further, the comparison of temperature-dependent properties can suffer from the false assumption that data collected from different temperatures are uncorrelated. We propose here a strategy in which, by a simple modification of the PT protocol, the harvested trajectories can be reweighted, permitting data from all temperatures to contribute to the estimated kinetic model. The method reduces the statistical uncertainty in the kinetic model relative to the single temperature approach and provides estimates of transition probabilities even for transitions not observed at the temperature of interest. Further, the method allows the kinetics to be estimated at temperatures other than those at which simulations were run. We illustrate this method by applying it to the generation of a Markov model of the conformational dynamics of the solvated terminally blocked alanine peptide.

Efficient Computation, Sensitivity, and Error Analysis of Committor Probabilities for Complex Dynamical Processes

In many fields of physics, chemistry, and biology, the characterization of rates and pathways between certain states or species is of fundamental interest. The central mathematical object in such situations is the committor probability—a generalized reaction coordinate that measures the progress of the process as the probability of proceeding to the target state rather than relapsing to the source state. Here, we conduct a numerical analysis of the committor. First, it is shown that committors can be expressed by the stationary eigenfunctions of a modified dynamical operator, thus relating the committors to the dominant eigenfunctions of the original operator. Based on this reformulation, committors can be efficiently computed for systems with large state spaces. Moreover, a sensitivity analysis of the committor is conducted, which allows its statistical uncertainty from estimation to be quantified within a Bayesian framework. The methods are illustrated on two examples of diffusive dynamics: a two-dimens...

Dynamic neutron scattering from conformational dynamics. II. Application using molecular dynamics simulation and Markov modeling.

Neutron scattering experiments directly probe the dynamics of complex molecules on the sub pico- to microsecond time scales. However, the assignment of the relaxations seen experimentally to specific structural rearrangements is difficult, since many of the underlying dynamical processes may exist on similar timescales. In an accompanying article, we present a theoretical approach to the analysis of molecular dynamics simulations with a Markov State Model (MSM) that permits the direct identification of structural transitions leading to each contributing relaxation process. Here, we demonstrate the use of the method by applying it to the configurational dynamics of the well-characterized alanine dipeptide. A practical procedure for deriving the MSM from an MD is introduced. The result is a 9-state MSM in the space of the backbone dihedral angles and the side-chain methyl group. The agreement between the quasielastic spectrum calculated directly from the atomic trajectories and that derived from the Markov state model is excellent. The dependence on the wavevector of the individual Markov processes is described. The procedure means that it is now practicable to interpret quasielastic scattering spectra in terms of well-defined intramolecular transitions with minimal a priori assumptions as to the nature of the dynamics taking place.

Spectral Rate Theory for Two-State Kinetics.

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality. The reaction coordinate quality can be bounded from below by the directly computable observation quality, thus providing a measure allowing the reaction coordinate quality to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force-probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, focusing here on the case of two-state kinetics.

Markov Model Theory

This section reviews the relation between the continuous dynamics of a molecular system in thermal equilibrium and the kinetics given by a Markov State Model (MSM). We will introduce the dynamical propagator, an error-less, alternative description of the continuous dynamics, and show how MSMs result from its discretization. This allows for an precise understanding of the approximation quality of MSMs in comparison to the continuous dynamics. The results on the approximation quality are key for the design of good MSMs. While this section is important for understanding the theory of discretization and related systematic errors, practitioners wishing only to learn how to construct MSMs may skip directly to the discussion of Markov model estimation.