Calculation of the distribution of eigenvalues and eigenvectors in Markovian state models for molecular dynamics

Markovian state models (MSMs) are a convenient and efficient means to compactly describe the kinetics of a molecular system as well as a formalism for using many short simulations to predict long time scale behavior. Building a MSM consists of grouping the conformations into states and estimating the transition probabilities between these states. In a previous paper, we described an efficient method for calculating the uncertainty due to finite sampling in the mean first passage time between two states. In this paper, we extend the uncertainty analysis to derive similar closed-form solutions for the distributions of the eigenvalues and eigenvectors of the transition matrix, quantities that have numerous applications when using the model. We demonstrate the accuracy of the distributions on a six-state model of the terminally blocked alanine peptide. We also show how to significantly reduce the total number of simulations necessary to build a model with a given precision using these uncertainty estimates for the blocked alanine system and for a 2454-state MSM for the dynamics of the villin headpiece.     

Using path sampling to build better Markovian state models: predicting the folding rate and mechanism of a tryptophan zipper beta hairpin

We propose an efficient method for the prediction of protein folding rate constants and mechanisms. We use molecular dynamics simulation data to build Markovian state models (MSMs), discrete representations of the pathways sampled. Using these MSMs, we can quickly calculate the folding probability (P(fold)) and mean first passage time of all the sampled points. In addition, we provide techniques for evaluating these values under perturbed conditions without expensive recomputations. To demonstrate this method on a challenging system, we apply these techniques to a two-dimensional model energy landscape and the folding of a tryptophan zipper beta hairpin.     

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.     

Dynamical integration of a Markovian web: a first passage time approach

In this work we address the dynamics of Markovian systems by tracking the evolution of the probability distribution, utilizing mean first passage time theory to augment the set of states considered. The method is validated on a lattice system and is applied, in conjunction with landscape analysis (saddle point searches) and multidimensional transition-state theory, to an atomistic model of glassy atactic polystyrene, in order to follow its time evolution over more than ten orders of magnitude on the time scale, from less than 10(-15) up to 10(-5) s. Frequencies extracted from the eigenvalues of the rate constant matrix are in favorable agreement with experimental measurements of subglass relaxation transitions at 250 K.     

Using massively parallel simulation and Markovian models to study protein folding: examining the dynamics of the villin headpiece

We report on the use of large-scale distributed computing simulation and novel analysis techniques for examining the dynamics of a small protein. Matters addressed include folding rate, very long time scale kinetics, ensemble properties, and interaction with water. The target system for the study, the villin headpiece, has been of great interest to experimentalists and theorists both. Sampling totaled nearly 500 mus-the most extensive published to date for a system of villin's size in explicit solvent with all atom detail-and was in the form of tens of thousands of independent molecular dynamics trajectories, each several tens of nanoseconds in length. We report on kinetics sensitivity analyses that, using a set of short simulations, probed the role of water in villin's folding and sensitivity to the simulation's electrostatics treatment. By constructing Markovian state models (MSMs) from the collected data, we were able to propagate dynamics to times far beyond those directly simulated and to rapidly compute mean first passage times, long time kinetics (tens of microseconds), and evolution of ensemble property distributions over long times, otherwise currently impossible. We also tested our MSM by using it to predict the structure of villin de novo.     

Markov state model reveals folding and functional dynamics in ultra-long MD trajectories

Two strategies have been recently employed to push molecular simulation to long, biologically relevant time scales: projection-based analysis of results from specialized hardware producing a small number of ultralong trajectories and the statistical interpretation of massive parallel sampling performed with Markov state models (MSMs). Here, we assess the MSM as an analysis method by constructing a Markov model from ultralong trajectories, specifically two previously reported 100 μs trajectories of the FiP35 WW domain (Shaw, D. E. Science 2010, 330, 341-346). We find that the MSM approach yields novel insights. It discovers new statistically significant folding pathways, in which either beta-hairpin of the WW domain can form first. The rates of this process approach experimental values in a direct quantitative comparison (time scales of 5.0 μs and 100 ns), within a factor of ～2. Finally, the hub-like topology of the MSM and identification of a holo conformation predicts how WW domains may function through a conformational selection mechanism.     

Laser control in open molecular systems: STIRAP and Optimal Control

We examine the effect of dissipation on the laser control of a process that transforms a state into a superposed state. We consider a two-dimensional double well of a single potential energy surface. In the context of reactivity, the objective of the control is the localization in a given well, for instance the creation of an enantiomeric form whereas for quantum gates, this control corresponds to one of the transformation of the Hadamard gate. The environment is either modelled by coupling few harmonic oscillators (up to five) to the system or by an effective interaction with an Ohmic bath. In the discrete case, dynamics is carried out exactly by using the coupled harmonic adiabatic channels. In the continuous case, Markovian and non-Markovian dynamics are considered. We compare two laser control strategies: the Stimulated Raman Adiabatic Passage (STIRAP) method and the optimal control theory. Analytical estimations for the control by adiabatic passage in a Markovian environment are also derived.     

Deflation techniques in quantum chemistry: Excited states from ground states

Applications of deflation techniques to the study of excited states of quantum systems are analyzed. It is demonstrated how these methods allow us to transform the excited state problem of one Hamiltonian, into the ground state problem of an auxiliary one. As an example, potential application in the density functional treatment of excited states is discussed. The inclusion of approximations in this scheme, such as the solution of the proposed model within a finite basis set is discussed. An extension of the Hartree–Fock (HF) method to excited states is presented. This new treatment includes previous self consistent field extensions to excited states and provides us with a way to obtain the HF extension to excited states of any ground state method. These results make the excited states of a system accessible through all ground state theoretical techniques. © 2013 Wiley Periodicals, Inc.     

A comparison of methods to compute the potential of mean force.

Most processes occurring in a system are determined by the relative free energy between two or more states because the free energy is a measure of the probability of finding the system in a given state. When the two states of interest are connected by a pathway, usually called reaction coordinate, along which the free-energy profile is determined, this profile or potential of mean force (PMF) will also yield the relative free energy of the two states. Twelve different methods to compute a PMF are reviewed and compared, with regard to their precision, for a system consisting of a pair of methane molecules in aqueous solution. We analyze all combinations of the type of sampling (unbiased, umbrella-biased or constraint-biased), how to compute free energies (from density of states or force averaging) and the type of coordinate system (internal or Cartesian) used for the PMF degree of freedom. The method of choice is constraint-bias simulation combined with force averaging for either an internal or a Cartesian PMF degree of freedom.     

Ergodicity breaking and conformational hysteresis in the dynamics of a polymer tethered at a surface stagnation point

We study the dynamics of long chain polymer molecules tethered to a plane wall and subjected to a stagnation point flow. Using a combination of theory and numerical techniques, including Brownian dynamics (BD), we demonstrate that a chain conformation hysteresis exists even for freely draining (FD) chains. Hydrodynamic interactions (HI) between the polymer and the wall are included in the BD simulations. We find qualitative agreement between the FD and HI simulations, with both exhibiting simultaneous coiled and stretched states for a wide range of fixed flow strengths. The range of state coexistence is understood by considering an equivalent projected equilibrium problem of a two state reaction. Using this formalism, we construct Kramers rate theory (from the inverse mean first passage time for a Markov process) for the hopping transition from coil to stretch and stretch to coil. The activation energy for this rate is found to scale proportionally to chain length or Kuhn step number. Thus, in the limit of infinite chain size the hopping rates at a fixed value of the suitably defined Deborah number approach zero and the states are "frozen." We present the results that demonstrate this "ergodicity breaking."     

Optimized ensemble Monte Carlo simulations of dense Lennard-Jones fluids

We apply the recently developed adaptive ensemble optimization technique to simulate dense Lennard-Jones fluids and a particle-solvent model by broad-histogram Monte Carlo techniques. Equilibration of the simulated fluid is improved by sampling an optimized histogram in radial coordinates that shifts statistical weight towards the entropic barriers between the shells of the liquid. Interstitial states in the vicinity of these barriers are identified with unprecedented accuracy by sharp signatures in the quickly converging histogram and measurements of the local diffusivity. The radial distribution function and potential of mean force are calculated to high precision.     

Validation of Markov state models using Shannon's entropy

Markov state models are kinetic models built from the dynamics of molecular simulation trajectories by grouping similar configurations into states and examining the transition probabilities between states. Here we present a procedure for validating the underlying Markov assumption in Markov state models based on information theory using Shannon's entropy. This entropy method is applied to a simple system and is compared with the previous eigenvalue method. The entropy method also provides a way to identify states that are least Markovian, which can then be divided into finer states to improve the model.     

Exploring reaction pathways with transition path and umbrella sampling: application to methyl maltoside

The transition path sampling (TPS) method is a powerful approach to study chemical reactions or transitional properties on complex potential energy landscapes. One of the main advantages of the method over potential of mean force methods is that reaction rates can be directly accessed without knowledge of the exact reaction coordinate. We have investigated the complementary nature of these two differing approaches, comparing transition path sampling with the weighted histogram analysis method to study a conformational change in a small model system. In this case study, the transition paths for a transition between two rotational conformers of a model disaccharide molecule, methyl beta-D-maltoside, were compared with a free energy surface constrained by the two commonly used glycosidic (phi,psi) torsional angles. The TPS method revealed a reaction channel that was not apparent from the potential of mean force method, and the suitability of phi and psi as reaction coordinates to describe the isomerization in vacuo was confirmed by examination of the transition path ensemble. Using both transition state theory and transition path sampling methods, the transition rate was estimated. We have estimated a characteristic time between transitions of approximately 160 ns for this rare isomerization event between the two conformations of the carbohydrate. We conclude that transition path sampling can extract subtle information about the dynamics not apparent from the potential of mean force method. However, in calculating the reaction rate, the transition path sampling method required 27.5 times the computational effort than was needed by the potential of mean force method.     

Enhanced Wang Landau sampling of adsorbed protein conformations

Using computer simulations to model the folding of proteins into their native states is computationally expensive due to the extraordinarily low degeneracy of the ground state. In this paper, we develop an efficient way to sample these folded conformations using Wang Landau sampling coupled with the configurational bias method (which uses an unphysical "temperature" that lies between the collapse and folding transition temperatures of the protein). This method speeds up the folding process by roughly an order of magnitude over existing algorithms for the sequences studied. We apply this method to study the adsorption of intrinsically disordered hydrophobic polar protein fragments on a hydrophobic surface. We find that these fragments, which are unstructured in the bulk, acquire secondary structure upon adsorption onto a strong hydrophobic surface. Apparently, the presence of a hydrophobic surface allows these random coil fragments to fold by providing hydrophobic contacts that were lost in protein fragmentation.     

The variational iteration method for solving systems of third-order Emden-Fowler type equations

In this paper, we employ the variational iteration method (VIM) for solving systems of nonlinear equations of Emden-Fowler type of third-order which arise in many scientific applications. The third-order Emden-Fowler equation is characterized by two models, where the shape factor appears twice in the first model, and once in the second. The VIM handles these kinds of shape factors, and overcomes the singularity at the origin. We will use the proper Lagrange multiplier for each model. We solve several complex numerical examples obtaining a rapidly convergent sequence of approximations as the solution. In all examples investigated, we achieved approximations with a high level, thus confirming that only a few variational iterations can provide an accurate approximation.     

Length dependent folding kinetics of phenylacetylene oligomers: structural characterization of a kinetic trap

Using simulation to study the folding kinetics of 20-mer poly-phenylacetylene (pPA) oligomers, we find a long time scale trapped kinetic phase in the cumulative folding time distribution. This is demonstrated using molecular dynamics to simulate an ensemble of over 100 folding trajectories. The simulation data are fit to a four-state kinetic model which includes the typical folded and unfolded states, along with an intermediate state, and most surprisingly, a kinetically trapped state. Topologically diverse conformations reminiscent of alpha helices, beta turns, and sheets in proteins are observed, along with unique structures in the form of knots. The nonhelical conformations are implicated, on the basis of structural correlations to kinetic parameters, to contribute to the trapped kinetic behavior. The strong solvophobic forces which mediate the folding process and produce a stable helical folded state also serve to overstabilize the nonhelical conformations, ultimately trapping them. From our simulations, the folding time is predicted to be on the order of 2.5-12.5 mus in the presence of the trapped kinetic phase. The folding mechanism for these 20-mer chains is compared with the previously reported folding mechanism for the pPA 12-mer chains. A linear scaling relationship between the chain length and the mean first passage time is predicted in the absence of the trapped kinetic phase. We discuss the major implications of this discovery in the design of self-assembling nanostructures.     

Integral tau methods for stiff stochastic chemical systems

Tau leaping methods enable efficient simulation of discrete stochastic chemical systems. Stiff stochastic systems are particularly challenging since implicit methods, which are good for stiffness, result in noninteger states. The occurrence of negative states is also a common problem in tau leaping. In this paper, we introduce the implicit Minkowski-Weyl tau (IMW-τ) methods. Two updating schemes of the IMW-τ methods are presented: implicit Minkowski-Weyl sequential (IMW-S) and implicit Minkowski-Weyl parallel (IMW-P). The main desirable feature of these methods is that they are designed for stiff stochastic systems with molecular copy numbers ranging from small to large and that they produce integer states without rounding. This is accomplished by the use of a split step where the first part is implicit and computes the mean update while the second part is explicit and generates a random update with the mean computed in the first part. We illustrate the IMW-S and IMW-P methods by some numerical examples, and compare them with existing tau methods. For most cases, the IMW-S and IMW-P methods perform favorably.