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- A (21)
he generalized Langevin equation is useful for modeling a wide
range of physical processes. Unfortunately its parameters,
especially the memory function, are difficult to determine for
nontrivial processes. In this paper, relations between a
time-discrete generalized Langevin model and discrete multivariate
autoregressive (AR) or autoregressive moving average models (ARMA)
are established. This allows a wide range of discrete linear
methods known from time series analysis to be applied. In
particular, the determination of the memory function {\it via} the
order of the respective AR or ARMA model is addressed. The method
is illustrated on a one-dimensional test system and subsequently
applied to the molecular dynamics of a biomolecule which exhibits
an interesting relationship between the solvent method used, the
molecular conformation and the depth of the memory.
While studies of protein-ligand association have mostly focused on the native complex and its
stability (binding affinity), relatively little attention has been paid on the association process
that precedes the formation of the complex. Here we review approaches to study the kinet-
ics of association and association mechanisms, i.e. the probability distribution of association
pathways. Selected methods are described that allow these properties to be calculated quan-
titatively from simulation models. We summarize some applications of these methods and
finally propose a model mechanism by which proteins may efficiently screen potential ligands
for those that can be natively bound.
Protein-ligand interactions are essential for nearly all biological processes, and yet the bio-
physical 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 path-
ways exist. In this article we introduce a computational approach to systematically analyze
the complete ensemble of association pathways and to thus investigate these questions. This
approach is employed here 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 specifically bound subsequently. 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 pre-binds to such an attractor neutralizes its attraction
effect to the environment, making the simultaneous association of a second phosphate ion
unlikely. Our study suggests ways how 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 fingerprints of macromolecules obtained from experiments often seem to indicate two- or
three state kinetics while simulations typically reveal a more complex picture. Markov state models of
molecular conformational dynamics can be used to predict these dynamical fingerprints and to reconcile
experiment with simulation. This is illustrated on two model systems: a one-dimensional energy surface
and a four-state model of a protein folding equilibrium. We show that (i) there might be no process
which corresponds to our notion of folding, (ii) often the experiment will be insensitive to some of the
processes present in the system, (iii) with a suitable combination the observable and initial conditions in
a relaxation experiment one can selectively measure specific processes. Furthermore, our method can be
used to design experiments such that specific processes appear with large amplitudes. We demonstrate
that for a fluorescence quenching experiment of the MR121-G9-W peptide.
Dynamical averages based on functionals of dynamical trajectories, such as time-correlation func-
tions, 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 prop-
erties 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
of 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 us-
ing 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.
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 article, 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 an 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 an MSM reproduces the kinetics of the molecular dynamics data.
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.
Resolving the apparent gap in complexity between
simulated and measured kinetics of biomolecules
(2012)
Molecular simulations of biomolecules often reveal a complex picture of the their kinetics,
whereas kinetic experiments typically seem to indicate considerably simpler two- or three-state
kinetics. Markov state models (MSM) provide a tool to link between simulation and experi-
ment, and to resolve this apparent contradiction.
In many fields of physics, chemistry and biology the characterization of dynamical processes
between states or species is of fundamental interest. The central mathematical function in such sit-
uations is the committor probability - a generalized reaction coordinate that measures the progress
of the process of interest as the probability of proceeding towards the target state rather than re-
lapsing to the source state. Here, we present methodology for the efficient computation of com-
mittor probabilities for large-scale systems, such as, for example simuations of biomolecular fold-
ing. A method is derived for computing the committor for discrete state spaces using eigenvectors
with expressions for the sensitivity and a Bayesian error model for the committor. The concepts
are illustrated on two examples of diffusive dynamics with a very large number of states: a two-
dimensional model potential with three minima, and a three-dimensional model representing
protein-ligand binding. The method can finally be used to compute committor probabilities in-
cluding error estimations for medium and large system sizes allowing access to the apparatus of
transition path theory and its applications.
Diffusion processes are relevant for a variety of phenomena in the natural sciences, including
diffusion of cells or biomolecules within cells, diffusion of molecules on a membrane or surface,
diffusion of a molecular conformation within a complex energy landscape. Many experimental
tools exist now to track such diffusive motions in single cells or molecules, including high-resolution
light microscopy, optical tweezers, fluorescence quenching, and Förster resonance energy transfer
(FRET). Experimental observations are most often indirect and incomplete: (1) They do not
directly reveal the potential or diffusion constants that govern the diffusion process, (2) they have
limited time and space resolution, and (3) the highest-resolution experiments do not track the
motion directly but rather probe it stochastically by recording single events, such as photons,
whose properties depend on the state of the system under investigation.
Here, we propose a general Bayesian framework to model diffusion processes with nonlinear
drift based on incomplete observations as generated by various types of experiments. A maximum
penalized likelihood estimator is given as well as a Gibbs sampling method that allows to estimate
the trajectories that have caused the measurement, the nonlinear drift or potential function and
the noise or diffusion matrices, as well as uncertainty estimates of these properties. The approach
is illustrated on numerical simulations of FRET experiments where it is shown that trajectories,
potentials and diffusion constants can be efficiently and reliably estimated even in cases with little
statistics or non-equilibrium measurement conditions.
Large-scale stochastic models are relevant in many different fields such as com- putational biology, finance, social sciences, communication and traffic networks. In order to both efficiently simulate and analyze such models and to understand the essential properties of the sys- tem, it is desirable to have model reduction techniques that much reduce the dimensionality of the model while at the same time preserving the system’s essential dynamical properties. In this paper, a general model reduction technique for the class of discrete space and time Hidden Markov Models is presented, thereby also including the more special class discrete Markov Chains. The method is illustrated on some model applications.
Markov (state) models (MSMs) have attracted a lot of interest recently as they (1) can probe
long-term molecular kinetics based on short-time simulations, (2) offer a way to analyze great
amounts of simulation data with relatively little subjectivity of the analyst, (3) provide insight into
microscopic quantities such as the ensemble of transition pathways, and (4) allow simulation data
to be reconciled with measurement data in a rigorous and explicit way. Here we sketch our current
perspective of Markov models and explain in short their theoretical basis and assumptions. We
describe transition path theory which allows the entire ensemble of protein folding pathways to be
investigated and that combines naturally with Markov models. Experimental observations can be
naturally linked to Markov models with the dynamical fingerprint theory, by which experimentally
observable timescales can be equipped with an understanding of the structural rearrangement
processes that take place at these timescales. The concepts of this paper are illustrated by a
simple kinetic model of protein folding.
Discrete-state Markov (or master equation) models provide a useful simplified representation for
characterizing the long-time statistical evolution of biomolecules in a manner that allows direct
comparison with experiments as well as the elucidation of mechanistic pathways for an inherently
stochastic process. A vital part of meaningful comparison with experiment is the characterization of
the statistical uncertainty in the predicted experimental measurement, which may take the form of
an equilibrium measurement of some spectroscopic signal, the time-evolution of this signal following
a perturbation, or the observation of some statistic (such as the correlation function) of the equilib-
rium dynamics of a single molecule. Without meaningful error bars (which arise due to the finite
quantity of data used to construct the model), there is no way to determine whether the deviations
between model and experiment are statistically meaningful. Previous work has demonstrated that
a Bayesian method that enforces microscopic reversibility can be used to characterize the correlated
uncertainties in state-to-state transition probabilities (and functions thereof) for a model inferred from
molecular simulation data. Here, we extend this approach to include the uncertainty in observables
that are functions of molecular conformation (such as surrogate spectroscopic signals) characteriz-
ing each state, permitting the full statistical uncertainty in computed spectroscopic experiments to be
assessed. We test the approach in a simple model system to demonstrate that the computed uncer-
tainties provide a useful indictor of statistical variation, and then apply it to the computation of the
fluorescence autocorrelation function measured for a dye-labeled peptide previously studied by both
experiment and simulation.
Optimal Identification of Semi-Rigid Domains in Macromolecules from Molecular Dynamics Simulation
(2012)
Biological function relies on the fact that biomolecules can switch between different conformations and aggregation states.
Such transitions involve a rearrangement of parts of the biomolecules involved that act as dynamic domains. The reliable
identification of such domains is thus a key problem in biophysics. In this work we present a method to identify semi-rigid
domains based on dynamical data that can be obtained from molecular dynamics simulations or experiments. To this end
the average inter-atomic distance-deviations are computed. The resulting matrix is then clustered by a constrained
quadratic optimization problem. The reliability and performance of the method are demonstrated for two artificial peptides.
Furthermore we correlate the mechanical properties with biological malfunction in three variants of amyloidogenic
transthyretin protein, where the method reveals that a pathological mutation destabilizes the natural dimer structure of the
protein. Finally the method is used to identify functional domains of the GroEL-GroES chaperone, thus illustrating the
efficiency of the method for large biomolecular machines.
In this paper, we present a Gaussian Markov random field (GMRF) model for the transition
matrices (TMs) of Markov chains (MCs) by assuming the existence of a neighborhood relationship
between states, and develop the maximum a posteriori (MAP) estimators under different obser-
vation conditions. Unlike earlier work on TM estimation, our method can make full use of the
similarity between different states to improve the estimated accuracy, and the estimator can be
performed very efficiently by solving a convex programming problem. In addition, we discuss the
parameter choice of the proposed model, and introduce a Monte Carlo cross validation (MCCV)
method. The numerical simulations of a diffusion process are employed to show the effectiveness
of the proposed models and algorithms.
Hybrid systems are often used to describe many complex dynamic phenomena by combining multiple modes of
dynamics into whole systems. In this paper, we present a flat Dirichlet process switching (FDPS) model that defines
a prior on mode switching dynamics of hybrid systems. Compared with the classical Markovian jump system (MJS)
models, the FDPS model is nonparametric and can be applied to the hybrid systems with an unbounded number of
potential modes. On the other hand, the probability structure of the new model is simpler and more flexible than the
recently proposed hierarchical Dirichlet process (HDP) based MJS. Furthermore, we develop a Markov chain Monte
Carlo (MCMC) method for estimating the states of hybrid systems with FDPS prior. And the numerical simulations
of a hybrid system in different conditions are employed to show the effectiveness of the proposed approach.
Actin is a major structural protein of the eukaryotic cytoskeleton and enables cell motility.
Here, we present a model of the actin filament (F-actin) that incorporates the global structure
of the recently published model by Oda et al. but also conserves internal stereochemistry. A
comparison is made using molecular dynamics simulation of the model with other recent F-
actin models. A number of structural determents such as the protomer propeller angle, the
number of hydrogen bonds and the structural variation among the protomers are analyzed.
The MD comparison is found to reflect the evolution in quality of actin models over the last
six years. In addition, simulations of the model are carried out in states with both ADP or
ATP bound and local hydrogen-bonding differences characterized. The results point to the
significance of a direct interaction of Gln137 with ATP for activation of ATPase activity after
the G-to-F-actin transition.
Single-molecule force spectroscopy has proven to be a powerful tool for studying the kinetic be- havior of biomolecules. Through application of an external force, conformational states with small or transient populations can be stabilized, allowing them to be characterized and the statistics of in- dividual trajectories studied to provide insight into biomolecular folding and function. Because the observed quantity (force or extension) is not necessarily an ideal reaction coordinate, individual ob- servations cannot be uniquely associated with kinetically distinct conformations. While maximum- likelihood schemes such as hidden Markov models have solved this problem for other classes of single-molecule experiments by using temporal information to aid in the inference of a sequence of distinct conformational states, these methods do not give a clear picture of how precisely the model parameters are determined by the data due to instrument noise and finite-sample statistics, both sig- nificant problems in force spectroscopy. We solve this problem through a Bayesian extension that allows the experimental uncertainties to be directly quantified, and build in detailed balance to fur- ther reduce uncertainty through physical constraints. We illustrate the utility of this approach in characterizing the three-state kinetic behavior of an RNA hairpin in a stationary optical trap.
While seemingly straightforward in principle, the reliable estimation of rate constants is seldom easy in practice. Numerous issues, such as the complication of poor reaction coordinates, cause obvious approaches to yield unreliable estimates. When a reliable order parameter is available, the reactive flux theory of Chandler allows the rate constant to be extracted from the plateau region of an appropriate reactive flux correlation function. However, when applied to real data from single- molecule experiments or molecular dynamics simulations, the reactive flux correlation function requires the numerical differentiation of a noisy empirical correlation function, which can result in an unacceptably poor estimate of the rate and pathological dependence on the sampling interval. We present a modified version of this theory which does not require numerical derivatives, allowing rate constants to be robustly estimated from the time-correlation function directly. We illustrate the approach using single-molecule passive force spectroscopy measurements of an RNA hairpin.
Folding and conformational changes of macromolecules often require the generation of large amounts of simulation data that are difficult to ana- lyze. Markov state models (MSMs) address this challenge by providing a systematic way to decompose the state space of the molecular system into substates and to estimate a transition matrix containing the transi- tion probabilities between these substates. This transition matrix can be analyzed to reveal the metastable, i.e. long-living, states of the system, its slowest relaxation timescales and transition pathways and rates e.g. from unfolded to folded, or from dissociated to bound states. To reduce the technical burden of constructing such MSMs we provide the software framework EMMA (available at https://simtk.org/home/emma) to con- struct, validate and analyse such Markov State Models.
The slow processes of metastable stochastic dynamical systems are difficult to access by direct numerical simulation due the sampling problem. Here, we suggest an approach for modeling the slow parts of Markov processes by approximating the dominant eigenfunctions and eigenvalues of the propagator. To this end, a variational principle is derived that is based on the maximization of a Raleigh coefficient. It is shown that this Raleigh coefficient can be estimated from statistical observables that can be obtained from short distributed simulations starting from different parts of state space. The approach forms a basis for the development of adaptive and efficient computational algorithms for simulating and analyzing metastable Markov processes while avoiding the sampling problem. Since any stochastic process with finite memory can be transformed into a Markov process, the approach is applicable to a wide range of processes relevant for modeling complex real-world phenomena.