Characterization of protein folding by dominant reaction pathways

Dynamics Sampling in Transition Pathway Space

The minimum energy pathway contains important information describing the transition between two states on a potential energy surface (PES). Chain-of-states methods were developed to efficiently calculate minimum energy pathways connecting two stable states. In the chain-of-states framework, a series of structures are generated and optimized to represent the minimum energy pathway connecting two states. However, multiple pathways may exist connecting two existing states and should be identified to obtain a full view of the transitions. Therefore, we developed an enhanced sampling method, named as the direct pathway dynamics sampling (DPDS) method, to facilitate exploration of a PES for multiple pathways connecting two stable states as well as addition minima and their associated transition pathways. In the DPDS method, molecular dynamics simulations are carried out on the targeting PES within a chain-of-states framework to directly sample the transition pathway space. The simulations of DPDS could be regulated by two parameters controlling distance among states along the pathway and smoothness of the pathway. One advantage of the chain-of-states framework is that no specific reaction coordinates are necessary to generate the reaction pathway, because such information is implicitly represented by the structures along the pathway. The chain-of-states setup in a DPDS method greatly enhances the sufficient sampling in high-energy space between two end states, such as transition states. By removing the constraint on the end states of the pathway, DPDS will also sample pathways connecting minima on a PES in addition to the end points of the starting pathway. This feature makes DPDS an ideal method to directly explore transition pathway space. Three examples demonstrate the efficiency of DPDS methods in sampling the high-energy area important for reactions on the PES.

Augmenting the anisotropic network model with torsional potentials improves PATH performance, enabling detailed comparison with experimental rate data

PATH algorithms for identifying conformational transition states provide computational parameters-time to the transition state, conformational free energy differences, and transition state activation energies-for comparison to experimental data and can be carried out sufficiently rapidly to use in the "high throughput" mode. These advantages are especially useful for interpreting results from combinatorial mutagenesis experiments. This report updates the previously published algorithm with enhancements that improve correlations between PATH convergence parameters derived from virtual variant structures generated by RosettaBackrub and previously published kinetic data for a complete, four-way combinatorial mutagenesis of a conformational switch in Tryptophanyl-tRNA synthetase.

Folding Mechanism of Proteins Im7 and Im9: Insight from All-Atom Simulations in Implicit and Explicit Solvent

Im7 and Im9 are evolutionary related proteins with almost identical native structures. In spite of their structural similarity, experiments show that Im7 folds through a long-lived on-pathway intermediate, while Im9 folds according to two-state kinetics. In this work, we use a recently developed enhanced path sampling method to generate many folding trajectories for these proteins, using realistic atomistic force fields, in both implicit and explicit solvent. Overall, our results are in good agreement with the experimental ϕ values and with the result of ϕ-value-restrained molecular dynamics (MD) simulations. However, our implicit solvent simulations fail to predict a qualitative difference in the folding pathways of Im7 and Im9. In contrast, our simulations in explicit solvent correctly reproduce the fact that only protein Im7 folds through a on-pathway intermediate. By analyzing our atomistic trajectories, we provide a physical picture which explains the observed difference in the folding kinetics of these chains.

A modified PATH algorithm rapidly generates transition states comparable to those found by other well established algorithms

PATH rapidly computes a path and a transition state between crystal structures by minimizing the Onsager-Machlup action. It requires input parameters whose range of values can generate different transition-state structures that cannot be uniquely compared with those generated by other methods. We outline modifications to estimate these input parameters to circumvent these difficulties and validate the PATH transition states by showing consistency between transition-states derived by different algorithms for unrelated protein systems. Although functional protein conformational change trajectories are to a degree stochastic, they nonetheless pass through a well-defined transition state whose detailed structural properties can rapidly be identified using PATH.

Computing Reaction Pathways of Rare Biomolecular Transitions using Atomistic Force-Fields

The Dominant Reaction Pathway (DRP) method is an approximate variational scheme which can be used to compute reaction pathways in conformational transitions undergone by large biomolecules (up to ~10(3) amino-acids) using realistic all-atom force fields. We first review the status of development of this method. Next, we discuss its validation against the results of plain MD protein folding simulations performed by the DE-Shaw group using the Anton supercomputer. Finally, we review a few representative applications of the DRP approach to study reactions which are far too complex and rare to be investigated by plain MD, even on the Anton machine.

Serpin latency transition at atomic resolution

Protease inhibition by serpins requires a large conformational transition from an active, metastable state to an inactive, stable state. Similar reactions can also occur in the absence of proteases, and these latency transitions take hours, making their time scales many orders of magnitude larger than are currently accessible using conventional molecular dynamics simulations. Using a variational path sampling algorithm, we simulated the entire serpin active-to-latent transition in all-atom detail with a physically realistic force field using a standard computing cluster. These simulations provide a unifying picture explaining existing experimental data for the latency transition of the serpin plasminogen activator inhibitor-1 (PAI-1). They predict a long-lived intermediate that resembles a previously proposed, partially loop-inserted, prelatent state; correctly predict the effects of PAI-1 mutations on the kinetics; and provide a potential means to identify ligands able to accelerate the latency transition. Interestingly, although all of the simulated PAI-1 variants readily access the prelatent intermediate, this conformation is not populated in the active-to-latent transition of another serpin, α1-antitrypsin, which does not readily go latent. Thus, these simulations also help elucidate why some inhibitory serpin families are more conformationally labile than others.

Microscopically computing free-energy profiles and transition path time of rare macromolecular transitions

We introduce a rigorous method to microscopically compute the observables which characterize the thermodynamics and kinetics of rare macromolecular transitions for which it is possible to identify a priori a slow reaction coordinate. In order to sample the ensemble of statistically significant reaction pathways, we define a biased molecular dynamics (MD) in which barrier-crossing transitions are accelerated without introducing any unphysical external force. In contrast to other biased MD methods, in the present approach the systematic errors which are generated in order to accelerate the transition can be analytically calculated and therefore can be corrected for. This allows for a computationally efficient reconstruction of the free-energy profile as a function of the reaction coordinate and for the calculation of the corresponding diffusion coefficient. The transition path time can then be readily evaluated within the dominant reaction pathways approach. We illustrate and test this method by characterizing a thermally activated transition on a two-dimensional energy surface and the folding of a small protein fragment within a coarse-grained model.

Quasi-Continuous Interpolation Scheme for Pathways between Distant Configurations

A quasi-continuous interpolation (QCI) scheme is introduced for characterizing physically realistic initial pathways from which to initiate transition state searches and construct kinetic transition networks. Applications are presented for peptides, proteins, and a morphological transformation in an atomic cluster. The first step in each case involves end point alignment, and we describe the use of a shortest augmenting path algorithm for optimizing permutational isomers. The QCI procedure then employs an interpolating potential, which preserves the covalent bonding framework for the biomolecules and includes repulsive terms between unconstrained atoms. This potential is used to identify an interpolating path by minimizing contributions from a connected set of images, including terms corresponding to minima in the interatomic distances between them. This procedure detects unphysical geometries in the line segments between images. The most difficult cases, where linear interpolation would involve chain crossings, are treated by growing the structure an atom at a time using the interpolating potential. To test the QCI procedure, we carry through a series of benchmark calculations where the initial interpolation is coupled to explicit transition state searches to produce complete pathways between specified local minima.

Dominant folding pathways of a WW domain

We investigate the folding mechanism of the WW domain Fip35 using a realistic atomistic force field by applying the Dominant Reaction Pathways approach. We find evidence for the existence of two folding pathways, which differ by the order of formation of the two hairpins. This result is consistent with the analysis of the experimental data on the folding kinetics of WW domains and with the results obtained from large-scale molecular dynamics simulations of this system. Free-energy calculations performed in two coarse-grained models support the robustness of our results and suggest that the qualitative structure of the dominant paths are mostly shaped by the native interactions. Computing a folding trajectory in atomistic detail only required about one hour on 48 Central Processing Units. The gain in computational efficiency opens the door to a systematic investigation of the folding pathways of a large number of globular proteins.

Fluctuations in the ensemble of reaction pathways

The dominant reaction pathway is a rigorous framework to microscopically compute the most probable trajectories, in nonequilibrium transitions. In the low-temperature regime, such dominant pathways encode the information about the reaction mechanism and can be used to estimate nonequilibrium averages of arbitrary observables. On the other hand, at sufficiently high temperatures, the stochastic fluctuations around the dominant paths become important and have to be taken into account. In this work, we develop a technique to systematically include the effects of such stochastic fluctuations, to order k(B)T. This method is used to compute the probability for a transition to take place through a specific reaction channel and to evaluate the reaction rate.

Molecular dynamics at low time resolution

The internal dynamics of macromolecular systems is characterized by widely separated time scales, ranging from fraction of picoseconds to nanoseconds. In ordinary molecular dynamics simulations, the elementary time step Δt used to integrate the equation of motion needs to be chosen much smaller of the shortest time scale in order not to cut-off physical effects. We show that in systems obeying the overdamped Langevin equation, it is possible to systematically correct for such discretization errors. This is done by analytically averaging out the fast molecular dynamics which occurs at time scales smaller than Δt, using a renormalization group based technique. Such a procedure gives raise to a time-dependent calculable correction to the diffusion coefficient. The resulting effective Langevin equation describes by construction the same long-time dynamics, but has a lower time resolution power, hence it can be integrated using larger time steps Δt. We illustrate and validate this method by studying the diffusion of a point-particle in a one-dimensional toy model and the denaturation of a protein.

Communications: Ab initio dynamics of rare thermally activated reactions

We introduce a framework to investigate ab initio the dynamics of rare thermally activated reactions, which cannot be studied using the existing techniques. The electronic degrees of freedom are described at the quantum-mechanical level in the Born-Oppenheimer approximation, while the nuclear degrees of freedom are coupled to a thermal bath, through a classical Langevin equation. This method is based on the path integral representation for the stochastic dynamics and yields the time evolution of both nuclear and electronic degrees of freedom, along the most probable reaction pathways, without spending computational time to explore metastable states. As a first illustrative application, we characterize the dominant pathway in the cyclobutene-->butadiene reaction, using the semiempirical Parametrized Model 3 (PM3) approach.

Simulating stochastic dynamics using large time steps

We present an approach to investigate the long-time stochastic dynamics of multidimensional classical systems, in contact with a heat bath. When the potential energy landscape is rugged, the kinetics displays a decoupling of short- and long-time scales and both molecular dynamics or Monte Carlo (MC) simulations are generally inefficient. Using a field theoretic approach, we perform analytically the average over the short-time stochastic fluctuations. This way, we obtain an effective theory, which generates the same long-time dynamics of the original theory, but has a lower time-resolution power. Such an approach is used to develop an improved version of the MC algorithm, which is particularly suitable to investigate the dynamics of rare conformational transitions. In the specific case of molecular systems at room temperature, we show that elementary integration time steps used to simulate the effective theory can be chosen a factor approximately 100 larger than those used in the original theory. Our results are illustrated and tested on a simple system, characterized by a rugged energy landscape.

Dominant reaction pathways in high-dimensional systems

This paper is devoted to the development of a theoretical and computational framework denominated dominant reaction pathways (DRPs) to efficiently sample the statistically significant thermally activated reaction pathways, in multidimensional systems. The DRP approach is consistently derived from the Langevin equation through a systematic expansion in the thermal energy, k(B)T. Its main advantage with respect to existing simulation techniques is that it provides a natural and rigorous framework to perform the path sampling using constant displacement steps, rather than constant time steps. In our previous work, we have shown how to obtain the set of most probable reaction pathways, i.e., the lowest order in the k(B)T expansion. In this work, we show how to compute the corrections to the leading order due to stochastic fluctuations around the most probable trajectories. We also discuss how to obtain predictions for the evolution of arbitrary observables and how to generate conformations, which are representative of the transition state ensemble. We illustrate how our method works in practice by studying the diffusion of a point particle in a two-dimensional funneled external potential.

Insights from coarse-grained Gō models for protein folding and dynamics

Exploring the landscape of large scale conformational changes such as protein folding at atomistic detail poses a considerable computational challenge. Coarse-grained representations of the peptide chain have therefore been developed and over the last decade have proved extremely valuable. These include topology-based Gō models, which constitute a smooth and funnel-like approximation to the folding landscape. We review the many variations of the Gō model that have been employed to yield insight into folding mechanisms. Their success has been interpreted as a consequence of the dominant role of the native topology in folding. The role of local contact density in determining protein dynamics is also discussed and is used to explain the ability of Gō-like models to capture sequence effects in folding and elucidate conformational transitions.