Barrier-crossing Events Research Articles

An interoperable implementation of collective-variable based enhanced sampling methods in extended phase space within the OpenMM package.

Collective variable (CV)-based enhanced sampling techniques are widely used today for accelerating barrier-crossing events in molecular simulations. A class of these methods, which includes temperature accelerated molecular dynamics (TAMD)/driven-adiabatic free energy dynamics (d-AFED), unified free energy dynamics (UFED), and temperature accelerated sliced sampling (TASS), uses an extended variable formalism to achieve quick exploration of conformational space. These techniques are powerful, as they enhance the sampling of a large number of CVs simultaneously compared to other techniques. Extended variables are kept at a much higher temperature than the physical temperature by ensuring adiabatic separation between the extended and physical subsystems and employing rigorous thermostatting. In this work, we present a computational platform to perform extended phase space enhanced sampling simulations using the open-source molecular dynamics engine OpenMM. The implementation allows users to have interoperability of sampling techniques, as well as employ state-of-the-art thermostats and multiple time-stepping. This work also presents protocols for determining the critical parameters and procedures for reconstructing high-dimensional free energy surfaces. As a demonstration, we present simulation results on the high dimensional conformational landscapes of the alanine tripeptide in vacuo, tetra-N-methylglycine (tetra-sarcosine) peptoid in implicit solvent, and the Trp-cage mini protein in explicit water.

Slow Folding of a Helical Protein: Large Barriers, Strong Internal Friction, or a Shallow, Bumpy Landscape?

The rate at which a protein molecule folds is determined by opposing energetic and entropic contributions to the free energy that shape the folding landscape. Delineating the extent to which they impact the diffusional barrier-crossing events, including the magnitude of internal friction and barrier height, has largely been a challenging task. In this work, we extract the underlying thermodynamic and dynamic contributions to the folding rate of an unusually slow-folding helical DNA-binding domain, PurR, which shares the characteristics of ultrafast downhill-folding proteins but nonetheless appears to exhibit an apparent two-state equilibrium. We combine equilibrium spectroscopy, temperature-viscosity-dependent kinetics, statistical mechanical modeling, and coarse-grained simulations to show that the conformational behavior of PurR is highly heterogeneous characterized by a large spread in melting temperatures, marginal thermodynamic barriers, and populated partially structured states. PurR appears to be at the threshold of disorder arising from frustrated electrostatics and weak packing that in turn slows down folding due to a shallow, bumpy landscape and not due to large thermodynamic barriers or strong internal friction. Our work highlights how a strong temperature dependence on the pre-exponential could signal a shallow landscape and not necessarily a slow-folding diffusion coefficient, thus determining the folding timescales of even millisecond folding proteins and hints at possible structural origins for the shallow landscape.

Dissecting the Energetics of Subunit Rotation in the Ribosome.

The accurate expression of proteins requires the ribosome to efficiently undergo elaborate conformational rearrangements. The most dramatic of these motions is subunit rotation, which is necessary for tRNA molecules to transition between ribosomal binding sites. While rigid-body descriptions provide a qualitative picture of the process, obtaining quantitative mechanistic insights requires one to account for the relationship between molecular flexibility and collective dynamics. Using simulated rotation events, we assess the quality of experimentally accessible measures for describing the collective displacement of the ∼4000-residue small subunit. For this, we ask whether each coordinate is able to identify the underlying free-energy barrier and transition state ensemble (TSE). We find that intuitive structurally motivated coordinates (e.g., rotation angle, interprotein distances) can distinguish between the endpoints, though they are poor indicators of barrier-crossing events, and they underestimate the free-energy barrier. In contrast, coordinates based on intersubunit bridges can identify the TSE. We additionally verify that the committor probability for the putative TSE configurations is 0.5, a hallmark feature of any transition state. In terms of structural properties, these calculations implicate a transition state in which flexibility allows for asynchronous rearrangements of the bridges, as the ribosome adopts a partially rotated orientation. This provides a theoretical foundation, upon which experimental techniques may precisely quantify the energy landscape of the ribosome.

Accurate Structure-Based Coarse Graining Leads to Consistent Barrier-Crossing Dynamics.

Structure-based coarse graining of molecular systems offers a systematic route to reproduce the many-body potential of mean force. Unfortunately, common strategies are inherently limited by the molecular mechanics force field employed. Here, we extend the concept of multisurface dynamics, initially developed to describe electronic transitions in chemical reactions, to accurately sample the conformational ensemble of a classical system in equilibrium. In analogy to describing different electronic configurations, a surface-hopping scheme couples distinct conformational basins beyond the additivity of the Hamiltonian. The incorporation of more surfaces leads systematically toward improved cross-correlations. The resulting models naturally achieve consistent long-time dynamics for systems governed by barrier-crossing events.

Rare-event sampling of epigenetic landscapes and phenotype transitions.

Stochastic simulation has been a powerful tool for studying the dynamics of gene regulatory networks, particularly in terms of understanding how cell-phenotype stability and fate-transitions are impacted by noisy gene expression. However, gene networks often have dynamics characterized by multiple attractors. Stochastic simulation is often inefficient for such systems, because most of the simulation time is spent waiting for rare, barrier-crossing events to occur. We present a rare-event simulation-based method for computing epigenetic landscapes and phenotype-transitions in metastable gene networks. Our computational pipeline was inspired by studies of metastability and barrier-crossing in protein folding, and provides an automated means of computing and visualizing essential stationary and dynamic information that is generally inaccessible to conventional simulation. Applied to a network model of pluripotency in Embryonic Stem Cells, our simulations revealed rare phenotypes and approximately Markovian transitions among phenotype-states, occurring with a broad range of timescales. The relative probabilities of phenotypes and the transition paths linking pluripotency and differentiation are sensitive to global kinetic parameters governing transcription factor-DNA binding kinetics. Our approach significantly expands the capability of stochastic simulation to investigate gene regulatory network dynamics, which may help guide rational cell reprogramming strategies. Our approach is also generalizable to other types of molecular networks and stochastic dynamics frameworks.

A dichotomy for sampling barrier-crossing events of random walks with regularly varying tails

AbstractWe study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are independent and identically distributed with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index α of the right tail lies in the interval (1, 3/2).

Correlated Movement of tRNA and the Ribosome during Elongation

The ribosome is a large RNA-protein complex, and it is the sole producer of protein in a cell. Accommodation is a key conformational change that allows movement of tRNA into the ribosome. To understand the accommodation process, it is central to quantify the energetic and structural content of the transition state ensemble. Previous studies have shown that the elbow-to-elbow distance between U8 of the P-site tRNA and U60 of aa-tRNA can capture the correct number of barrier-crossing events and movement along this coordinate is diffusive. Previous simulations have shown that the free-energy barrier during accommodation is predominantly due to the shape of the accommodation corridor. To explore the relationship between tRNA barrier-crossing events and fluctuations in the ribosome, we tested the degree to which these fluctuations are predictive of accommodation events. From this analysis, we have found that specific fluctuations in the corridor are strongly correlated with the process of aa-tRNA accommodation. We also found that the barrier-crossing events are accompanied by bending and contraction of the aa-tRNA. The coordinate analysis methods employed here can be applied to study correlated movements that are involved in other bimolecular processes, e.g. tRNA translocation. Furthermore, the identified coordinates can potentially be used in the experimental studies to indirectly monitor the process of aa-tRNA accommodation, and possible allow for precise control of the dynamics.

A chain-of-states acceleration method for the efficient location of minimum energy paths.

We describe a robust and efficient chain-of-states method for computing Minimum Energy Paths (MEPs) associated to barrier-crossing events in poly-atomic systems, which we call the acceleration method. The path is parametrized in terms of a continuous variable t ∈ [0, 1] that plays the role of time. In contrast to previous chain-of-states algorithms such as the nudged elastic band or string methods, where the positions of the states in the chain are taken as variational parameters in the search for the MEP, our strategy is to formulate the problem in terms of the second derivatives of the coordinates with respect to t, i.e., the state accelerations. We show this to result in a very simple and efficient method for determining the MEP. We describe the application of the method to a series of test cases, including two low-dimensional problems and the Stone-Wales transformation in C60.

Vibrationally Coherent Preparation of the Transition State for Photoisomerization of the Cyanine Dye Cy5 in Water

Femtosecond pump-continuum probe spectroscopy with impulsive excitation was employed to observe coherent wavepacket motions of the cyanine dye Cy5 in water that promote photoisomerization after optical preparation of the first excited singlet state, S1. The chief component in the excited-state vibrational coherence is a resonance Raman-inactive, 273 cm(-1) mode of mixed carbon-carbon bond length alternation and out-of-plane or twisting character. The ultrafast (30 fs) damping of these motions arises from trajectories that irreversibly cross the transition state barrier; after several recurrences to the transition state region, vibrational cooling traps a significant fraction of the excited-state molecules in the planar, Franck-Condon region of the potential energy surface. Motion in the 273 cm(-1) promoting mode is apparently launched by a change in conformation of the conjugated polyene backbone during the first few vibrations of the high-frequency C-C and C═C bond length alternation coordinates that principally contribute to the initial displacement from the Franck-Condon structure. To our knowledge, this work provides the first direct observations of the intramolecular redistribution of excited-state potential energy into reactive motions that are rapidly damped by transition state barrier-crossing events leading to photoisomerization in a conjugated polyene. These results provide insight into the vibrational dynamics that contribute to the photoisomerization of retinal protonated Schiff bases in the rhodopsins and to the formation of intramolecular charge transfer character in carotenoids in photosynthetic light-harvesting proteins.

Capturing Transition Paths and Transition States for Conformational Rearrangements in the Ribosome

To reveal the molecular determinants of biological function, one seeks to characterize the interactions that are formed in conformational and chemical transition states. In other words, what interactions govern the molecule’s energy landscape? To accomplish this, it is necessary to determine which degrees of freedom can unambiguously identify each transition state. Here, we perform simulations of large-scale aminoacyl-transfer RNA (aa-tRNA) rearrangements during accommodation on the ribosome and project the dynamics along experimentally accessible atomic distances. From this analysis, we obtain evidence for which coordinates capture the correct number of barrier-crossing events and accurately indicate when the aa-tRNA is on a transition path. Although a commonly used coordinate in single-molecule experiments performs poorly, this study implicates alternative coordinates along which rearrangements are accurately described as diffusive movements across a one-dimensional free-energy profile. From this, we provide the theoretical foundation required for single-molecule techniques to uncover the energy landscape governing aa-tRNA selection by the ribosome.

Elastic and Conformational Softness of a Globular Protein

Protein molecules in their native states, although highly structured, have a degree of flexibility (sometimes called ‘softness’) required for the biological function. Protein flexibility has been often characterized by the thermal fluctuations of atomic positions on short time scales (≤ 1 ns) as determined using simulation and scattering techniques. However, on these time scales, flexibility is partly conformational, involving barrier-crossing events, and partly “elastic”, i.e., involving fluctuations confined in energy wells [1-3]. The experimental separation of the contributions of these two types of motion to the protein atomic displacement is a considerable challenge. The present work addresses this challenge by combining molecular dynamics simulation (MD) with incoherent neutron scattering and light scattering on a globular protein. It is shown that the intra-well fluctuations, corresponding to the elastic flexibility, are essentially determined by the experimental elastic modulus at GHz-THz frequencies. This finding enables a simple and direct experimental separation of the conformational and elastic fluctuations in the protein. [1] H. Frauenfelder, S. G. Sligar and P. G. Wolynes, Science 254 (1991). [2] L. Hong, N. Smolin, B. Lindner, A.P. Sokolov and J.C. Smith, Phys. Rev. Lett., 107, 148102 (2011). [3] L. Hong, X. Cheng, D. C. Glass, and J.C. Smith, Phys. Rev. Lett., 108, 238102 (2012).

How does Water Boost the Protein Dynamics?

Protein dynamics is crucial for its function and folding, and depends on various external parameters, among which, hydration is of particular importance. Enzymes lose their bio-activity when they are dry and the so-called dynamical transition, manifesting as an anharmonic onset in the mean squared atomic displacement around 180 ∼ 230 K, occurs in proteins only when certain hydration level is achieved. Despite decades of study, the microscopic mechanism for how water boosts the protein dynamics and function is still lacking.By combining the neutron scattering and molecular dynamics simulation, the present work decomposes the internal motions of lysozyme on the ps to ns time scales into three components: localized diffusion, methyl group rotation and random jumps. In terms of the energy landscape, the first component is ascribed to be intra-well motion, while the other two correspond to transitions among wells, i.e., barrier-crossing events. The analysis reveals that the major change of protein dynamics on these time scales induced by hydration arises mainly from the intra-well motion “localized diffusion”, manifesting as increase of the diffusion size, via broadening the well. This result is contradictory to the traditional expectation that the hydration-boosted protein internal motions are barrier-crossing events through lowering the transition barriers, and will make a great impact on the future study of protein dynamics and function.

Optimum action method for the study of barrier-crossing events

Optimum action method for the study of barrier-crossing events

Precision shooting: Sampling long transition pathways

The kinetics of collective rearrangements in solution, such as protein folding and nanocrystal phase transitions, often involve free energy barriers that are both long and rough. Applying methods of transition path sampling to harvest simulated trajectories that exemplify such processes is typically made difficult by a very low acceptance rate for newly generated trajectories. We address this problem by introducing a new generation algorithm based on the linear short time behavior of small disturbances in phase space. Using this "precision shooting" technique, arbitrarily small disturbances can be propagated in time, and any desired acceptance ratio of shooting moves can be obtained. We demonstrate the method for a simple but computationally problematic isomerization process in a dense liquid of soft spheres. We also discuss its applicability to barrier-crossing events involving metastable intermediate states.

Simplified and improved string method for computing the minimum energy paths in barrier-crossing events

We present a simplified and improved version of the string method, originally proposed by E et al. [Phys. Rev. B 66, 052301 (2002)] for identifying the minimum energy paths in barrier-crossing events. In this new version, the step of projecting the potential force to the direction normal to the string is eliminated and the full potential force is used in the evolution of the string. This not only simplifies the numerical procedure, but also makes the method more stable and accurate. We discuss the algorithmic details of the improved string method, analyze its stability, accuracy and efficiency, and illustrate it via numerical examples. We also show how the string method can be combined with the climbing image technique for the accurate calculation of saddle points and we present another algorithm for the accurate calculation of the unstable directions at the saddle points.

Time-Trimming Tricks for Dynamic Simulations: Splitting Force Updates to Reduce Computational Work

Time-Trimming Tricks for Dynamic Simulations: Splitting Force Updates to Reduce Computational Work

Quantum Dissipative Dynamics: A Numerically Exact Methodology

A fully quantum mechanical methodology for simulating the time evolution of low-dimensional systems in harmonic dissipative environments is presented. The key features of the method are the numerical construction of accurate propagators based on physically motivated reference Hamiltonians and the decomposition of the path integral into a series of shorter time operations, which leads to an iterative algorithm. Illustrative applications to barrier-crossing events and biological electron transfer are presented.

Smart walking: A new method for Boltzmann sampling of protein conformations

A new Monte Carlo algorithm is presented for the efficient sampling of protein conformation space called the Smart-Walking (S-Walking) method. The method is implemented using a hybrid Monte Carlo protocol. The S-Walking method is closely related to the J-Walking method proposed by Frantz et al. (J. Chem. Phys. 93, 2769, 1990). Like the J-Walking method, the S-Walking method runs two walkers, one at the temperature of interest, the other at a higher temperature which more efficiently generates ergodic distributions. Instead of sampling from the Boltzmann distribution of the higher temperature walker as in J-Walking, S-Walking first approximately minimizes the structures being jumped into, and then uses the relaxed structures as the trial moves at the low temperature. By jumping into a relaxed structure, or a local minimum, the jump acceptance ratio increases dramatically, which makes the protein system easily undergo barrier-crossing events from one basin to another, thus greatly improving the ergodicity of the sampling. The method approximately preserves detailed balance provided the time between jumps is large enough to allow effective sampling of conformations in each local basin.