Frank Noé

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.

Constructing the equilibrium ensemble of folding pathways from short off-equilibrium simulations

Characterizing the equilibrium ensemble of folding pathways, including their relative probability, is one of the major challenges in protein folding theory today. Although this information is in principle accessible via all-atom molecular dynamics simulations, it is difficult to compute in practice because protein folding is a rare event and the affordable simulation length is typically not sufficient to observe an appreciable number of folding events, unless very simplified protein models are used. Here we present an approach that allows for the reconstruction of the full ensemble of folding pathways from simulations that are much shorter than the folding time. This approach can be applied to all-atom protein simulations in explicit solvent. It does not use a predefined reaction coordinate but is based on partitioning the state space into small conformational states and constructing a Markov model between them. A theory is presented that allows for the extraction of the full ensemble of transition pathways from the unfolded to the folded configurations. The approach is applied to the folding of a PinWW domain in explicit solvent where the folding time is two orders of magnitude larger than the length of individual simulations. The results are in good agreement with kinetic experimental data and give detailed insights about the nature of the folding process which is shown to be surprisingly complex and parallel. The analysis reveals the existence of misfolded trap states outside the network of efficient folding intermediates that significantly reduce the folding speed.

Transition networks for modeling the kinetics of conformational change in macromolecules

The kinetics and thermodynamics of complex transitions in biomolecules can be modeled in terms of a network of transitions between the relevant conformational substates. Such a transition network, which overcomes the fundamental limitations of reaction-coordinate-based methods, can be constructed either based on the features of the energy landscape, or from molecular dynamics simulations. Energy-landscape-based networks are generated with the aid of automated path-optimization methods, and, using graph-theoretical adaptive methods, can now be constructed for large molecules such as proteins. Dynamics-based networks, also called Markov State Models, can be interpreted and adaptively improved using statistical concepts, such as the mean first passage time, reactive flux and sampling error analysis. This makes transition networks powerful tools for understanding large-scale conformational changes.

Hierarchical analysis of conformational dynamics in biomolecules: Transition networks of metastable states

Molecular dynamics simulation generates large quantities of data that must be interpreted using physically meaningful analysis. A common approach is to describe the system dynamics in terms of transitions between coarse partitions of conformational space. In contrast to previous work that partitions the space according to geometric proximity, the authors examine here clustering based on kinetics, merging configurational microstates together so as to identify long-lived, i.e., dynamically metastable, states. As test systems microsecond molecular dynamics simulations of the polyalanines Ala(8) and Ala(12) are analyzed. Both systems clearly exhibit metastability, with some kinetically distinct metastable states being geometrically very similar. Using the backbone torsion rotamer pattern to define the microstates, a definition is obtained of metastable states whose lifetimes considerably exceed the memory associated with interstate dynamics, thus allowing the kinetics to be described by a Markov model. This model is shown to be valid by comparison of its predictions with the kinetics obtained directly from the molecular dynamics simulations. In contrast, clustering based on the hydrogen-bonding pattern fails to identify long-lived metastable states or a reliable Markov model. Finally, an approach is proposed to generate a hierarchical model of networks, each having a different number of metastable states. The model hierarchy yields a qualitative understanding of the multiple time and length scales in the dynamics of biomolecules.

Spatiotemporal control of endocytosis by phosphatidylinositol-3,4-bisphosphate

Phosphoinositides serve crucial roles in cell physiology, ranging from cell signalling to membrane traffic. Among the seven eukaryotic phosphoinositides the best studied species is phosphatidylinositol-4,5-bisphosphate (PI(4,5)P2), which is concentrated at the plasma membrane where, among other functions, it is required for the nucleation of endocytic clathrin-coated pits. No phosphatidylinositol other than PI(4,5)P2 has been implicated in clathrin-mediated endocytosis, whereas the subsequent endosomal stages of the endocytic pathway are dominated by phosphatidylinositol-3-phosphates(PI(3)P). How phosphatidylinositol conversion from PI(4,5)P2-positive endocytic intermediates to PI(3)P-containing endosomes is achieved is unclear. Here we show that formation of phosphatidylinositol-3,4-bisphosphate (PI(3,4)P2) by class II phosphatidylinositol-3-kinase C2α (PI(3)K C2α) spatiotemporally controls clathrin-mediated endocytosis. Depletion of PI(3,4)P2 or PI(3)K C2α impairs the maturation of late-stage clathrin-coated pits before fission. Timed formation of PI(3,4)P2 by PI(3)K C2α is required for selective enrichment of the BAR domain protein SNX9 at late-stage endocytic intermediates. These findings provide a mechanistic framework for the role of PI(3,4)P2 in endocytosis and unravel a novel discrete function of PI(3,4)P2 in a central cell physiological process.

Crystal structure of nucleotide-free dynamin

Dynamin is a mechanochemical GTPase that oligomerizes around the neck of clathrin-coated pits and catalyses vesicle scission in a GTP-hydrolysis-dependent manner. The molecular details of oligomerization and the mechanism of the mechanochemical coupling are currently unknown. Here we present the crystal structure of human dynamin 1 in the nucleotide-free state with a four-domain architecture comprising the GTPase domain, the bundle signalling element, the stalk and the pleckstrin homology domain. Dynamin 1 oligomerized in the crystals via the stalks, which assemble in a criss-cross fashion. The stalks further interact via conserved surfaces with the pleckstrin homology domain and the bundle signalling element of the neighbouring dynamin molecule. This intricate domain interaction rationalizes a number of disease-related mutations in dynamin 2 and suggests a structural model for the mechanochemical coupling that reconciles previous models of dynamin function.

Markov state models of biomolecular conformational dynamics

It has recently become practical to construct Markov state models (MSMs) that reproduce the long-time statistical conformational dynamics of biomolecules using data from molecular dynamics simulations. MSMs can predict both stationary and kinetic quantities on long timescales (e.g. milliseconds) using a set of atomistic molecular dynamics simulations that are individually much shorter, thus addressing the well-known sampling problem in molecular dynamics simulation. In addition to providing predictive quantitative models, MSMs greatly facilitate both the extraction of insight into biomolecular mechanism (such as folding and functional dynamics) and quantitative comparison with single-molecule and ensemble kinetics experiments. A variety of methodological advances and software packages now bring the construction of these models closer to routine practice. Here, we review recent progress in this field, considering theoretical and methodological advances, new software tools, and recent applications of these approaches in several domains of biochemistry and biophysics, commenting on remaining challenges.

On the Approximation Quality of Markov State Models

We consider a continuous-time Markov process on a large continuous or discrete state space. The process is assumed to have strong enough ergodicity properties and to exhibit a number of metastable sets. Markov state models (MSMs) are designed to represent the effective dynamics of such a process by a Markov chain that jumps between the metastable sets with the transition rates of the original process. MSMs have been used for a number of applications, including molecular dynamics, for more than a decade. Their approximation quality, however, has not yet been fully understood. In particular, it would be desirable to have a sharp error bound for the difference in propagation of probability densities between the MSM and the original process on long timescales. Here, we provide such a bound for a rather general class of Markov processes ranging from diffusions in energy landscapes to Markov jump processes on large discrete spaces. Furthermore, we discuss how this result provides formal support or shows the limitations of algorithmic strategies that have been found to be useful for the construction of MSMs. Our findings are illustrated by numerical experiments.

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.

Protein conformational plasticity and complex ligand-binding kinetics explored by atomistic simulations and Markov models

Understanding the structural mechanisms of protein–ligand binding and their dependence on protein sequence and conformation is of fundamental importance for biomedical research. Here we investigate the interplay of conformational change and ligand-binding kinetics for the serine protease Trypsin and its competitive inhibitor Benzamidine with an extensive set of 150 μs molecular dynamics simulation data, analysed using a Markov state model. Seven metastable conformations with different binding pocket structures are found that interconvert at timescales of tens of microseconds. These conformations differ in their substrate-binding affinities and binding/dissociation rates. For each metastable state, corresponding solved structures of Trypsin mutants or similar serine proteases are contained in the protein data bank. Thus, our wild-type simulations explore a space of conformations that can be individually stabilized by adding ligands or making suitable changes in protein sequence. These findings provide direct evidence of conformational plasticity in receptors.