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Abstract: Various methods to parameterize an exciton model of amide I vibrations are suggested and analyzed. In the spirit of a systematic fragmentation scheme, small model peptides are employed as building-blocks to construct the vibrations of a polypeptide. As an example, extensive density functional theory (DFT) calculations at the B3LYP/6-31+G(d) theoretical level of glycine tripeptide are presented, which serve as reference data for testing various approximate schemes. The combination of a DFT description of next-neighbor interactions via dipeptide building blocks combined with an electrostatic model to account for the long-range interactions appears as a promising approach to achieve spectroscopic accuracy at low computational cost. [Copyright &y& Elsevier]

Nonfibrillar soluble oligomers, which are intermediates in the transition from monomers to amyloid fibrils, may be the toxic species in Alzheimer's disease. To monitor the early events that direct assembly of amyloidogenic peptides we probe the dynamics of formation of (Aβ16-22)n by adding a monomer to a preformed (Aβ16-22)n-1 (n = 4–6) oligomer in which the peptides are arranged in an antiparallel β-sheet conformation. All atom molecular dynamics simulations in water and multiple long trajectories, for a cumulative time of 6.9 μs, show that the oligomer grows by a two-stage dock-lock mechanism. The largest conformational change in the added disordered monomer occurs during the rapid (≈50 ns) first dock stage in which the β-strand content of the monomer increases substantially from a low initial value. In the second slow-lock phase, the monomer rearranges to form in register antiparallel structures. Surprisingly, the mobile structured oligomers undergo large conformational changes in order to accommodate the added monomer. The time needed to incorporate the monomer into the fluid-like oligomer grows even when n = 6, which suggests that the critical nucleus size must exceed six. Stable antiparallel structure formation exceeds hundreds of nanoseconds even though frequent interpeptide collisions occur at elevated monomer concentrations used in the simulations. The dock-lock mechanism should be a generic mechanism for growth of oligomers of amyloidogenic peptides. [ABSTRACT FROM AUTHOR]

Abstract: Femtosecond time-resolved experiments on photoswitchable peptides provide a new and promising way to study the folding and unfolding of biomolecules in real time and unprecedented detail. To obtain an appropriate theoretical description of these experiments, a computational strategy is presented that aims to extend well-established molecular dynamics simulation techniques to the description of photoinduced conformational dynamics in peptides. Adopting a bicyclic azobenzene octapeptide as a representative example for a photoswitchable biomolecule, detailed nonequilibrium molecular dynamics studies are performed in which (i) the laser-induced initial state of the molecule is represented by a suitable nonequilibrium phase-space distribution that is sampled by an ensemble of many trajectories and (ii) the time-dependent mean values of the system are calculated from these trajectories by an ensemble average. To establish the applicability and the accuracy of the methodology, it is investigated to what extent the photoinduced conformational dynamics depends on the details of the nonequilibrium method, including the sampling of the initial state, the initially assumed excess energy, and the coupling of the system to a temperature bath. Furthermore, the photoinduced conformational dynamics is analyzed and the results are discussed in the light of recent time-resolved infrared experiments. [Copyright &y& Elsevier]

Abstract: Nonadiabatic photoisomerization dynamics in a condensed-phase environment is studied within the framework of Redfield theory. Considering several measures of decoherence and dissipation, the relaxation behavior of various models of nonadiabatic cis–trans photoisomerization is investigated. Several levels of relaxation theory are compared: The full Redfield theory (including all terms of the relaxation tensor), the secular approximation to it (including only the resonant terms) and the popular Bloch model (which also neglects terms that are resonant by accidence). Although these approximations are shown to work well for the single-mode model, they are found to lead to significant deviations for various two-mode models. The latter behavior is expected to be generic for multidimensional systems, which typically exhibit numerous near-degeneracies of the level spacings. To correctly describe the relaxation dynamics, while still retaining the advantageous N2 scaling of the Bloch model, a “nonsecular” algorithm is proposed that systematically includes the most important nonsecular terms. The algorithm correctly reproduces the results of the full Redfield theory, while the numerical effort is reduced by typically an order of magnitude for the examples considered. [Copyright &y& Elsevier]

A 100 ns molecular dynamics simulation of penta-alanine in explicit water is performed to study the reversible folding and unfolding of the peptide. Employing a standard principal component analysis (PCA) using Cartesian coordinates, the resulting free-energy landscape is found to have a single minimum, thus suggesting a simple, relatively smooth free-energy landscape. Introducing a novel PCA based on a transformation of the peptide dihedral angles, it is found, however, that there are numerous free energy minima of comparable energy (≲ 1 kcal/mol), which correspond to well-defined structures with characteristic hydrogen-bonding patterns. That is, the true free-energy landscape is actually quite rugged and its smooth appearance in the Cartesian PCA represents an artifact of the mixing of internal and overall motion. Well-separated minima corresponding to specific conformational structures are also found in the unfolded part of the free energy landscape, revealing that the unfolded state of penta-alanine is structured rather than random. Performing a connectivity analysis, it is shown that neighboring states are connected by low barriers of similar height and that each state typically makes transitions to three or four neighbor states. Several principal pathways for helix nucleation are identified and discussed in some detail. Proteins 2005. © 2004 Wiley-Liss, Inc. [ABSTRACT FROM AUTHOR]

Principal component analysis (PCA) represents a standard approach to identify collective variables {xi} = x, which can be used to construct the free energy landscape ΔG(x) of a molecular system. While PCA is routinely applied to equilibrium molecular dynamics (MD) simulations, it is less obvious as to how to extend the approach to nonequilibrium simulation techniques. This includes, e.g., the definition of the statistical averages employed in PCA as well as the relation between the equilibrium free energy landscape ΔG(x) and the energy landscapes Δ G (x) obtained from nonequilibrium MD. As an example for a nonequilibrium method, "targeted MD" is considered which employs a moving distance constraint to enforce rare transitions along some biasing coordinate s. The introduced bias can be described by a weighting function P(s), which provides a direct relation between equilibrium and nonequilibrium data, and thus establishes a well-defined way to perform PCA on nonequilibrium data. While the resulting distribution P (x) and energy Δ G ∝ ln P will not reflect the equilibrium state of the system, the nonequilibrium energy landscape Δ G (x) may directly reveal the molecular reaction mechanism. Applied to targeted MD simulations of the unfolding of decaalanine, for example, a PCA performed on backbone dihedral angles is shown to discriminate several unfolding pathways. Although the formulation is in principle exact, its practical use depends critically on the choice of the biasing coordinate s, which should account for a naturally occurring motion between two well-defined end-states of the system. [ABSTRACT FROM AUTHOR]

Studies the subpisecond conformational dynamics of small peptides probed by two-dimensional vibrational spectroscopy. Importance of conformational fluctuations for proteins and peptides; Molecular structure of trialanine; Transient spectra showing the absorption change as a function of probe frequency for a range of mixing times.

Adimensionality reduction method for high-dimensional circular data is developed, which is based on a principal component analysis (PCA) of data points on a torus. Adopting a geometrical view of PCA, various distance measures on a torus are introduced and the associated problem of projecting data onto the principal subspaces is discussed. The main idea is that the (periodicity-induced) projection error can be minimized by transforming the data such that the maximal gap of the sampling is shifted to the periodic boundary. In a second step, the covariance matrix and its eigendecomposition can be computed in a standard manner. Adopting molecular dynamics simulations of two well-established biomolecular systems (Aib9 and villin headpiece), the potential of the method to analyze the dynamics of backbone dihedral angles is demonstrated. The new approach allows for a robust and well-defined construction of metastable states and provides low-dimensional reaction coordinates that accurately describe the free energy landscape. Moreover, it offers a direct interpretation of covariances and principal components in terms of the angular variables. Apart from its application to PCA, the method of maximal gap shifting is general and can be applied to any other dimensionality reduction method for circular data. [ABSTRACT FROM AUTHOR]

A recently proposed semiclassical approach to the description of nonadiabatic quantum dynamics [G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 (1997), X. Sun and W. H. Miller, J. Chem. Phys. 106, 916 (1997)] is applied to the S[sub 1]-S[sub 2] conical intersection in pyrazine. This semiclassical method is based on a transformation of discrete quantum variables to continuous variables, thereby bypassing the problem of a classical treatment of discrete quantum degrees of freedom such as electronic states. Extending previous work on small systems, we investigate the applicability of the semiclassical method to larger systems with strong vibronic coupling. To this end, we present results for several pyrazine models of increasing dimensionality and complexity. In particular, we discuss the quality and performance of the semiclassical approach when the number of nuclear degrees of freedom is increased. Comparison with quantum-mechanical calculations and experimental results shows that the semiclassical method is able to describe the ultrafast dynamics in this system. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

To interpret molecular dynamics simulations of complex systems, systematic dimensionality reduction methods such as principal component analysis (PCA) represent a well-established and popular approach. Apart from Cartesian coordinates, internal coordinates, e.g., backbone dihedral angles or various kinds of distances, may be used as input data in a PCA. Adopting two well-known model problems, folding of villin headpiece and the functional dynamics of BPTI, a systematic study of PCA using distance-based measures is presented which employs distances between Ca-atoms as well as distances between inter-residue contacts including side chains. While this approach seems prohibitive for larger systems due to the quadratic scaling of the number of distances with the size of the molecule, it is shown that it is sufficient (and sometimes even better) to include only relatively few selected distances in the analysis. The quality of the PCA is assessed by considering the resolution of the resulting free energy landscape (to identify metastable conformational states and barriers) and the decay behavior of the corresponding autocorrelation functions (to test the time scale separation of the PCA). By comparing results obtained with distance-based, dihedral angle, and Cartesian coordinates, the study shows that the choice of input variables may drastically influence the outcome of a PCA. [ABSTRACT FROM AUTHOR]

Allostery represents a fundamental mechanism of biological regulation that is mediated via long-range communication between distant protein sites. Although little is known about the underlying dynamical process, recent time-resolved infrared spectroscopy experiments on a photoswitchable PDZ domain (PDZ2S) have indicated that the allosteric transition occurs on multiple timescales. Here, using extensive nonequilibrium molecular dynamics simulations, a time-dependent picture of the allosteric communication in PDZ2S is developed. The simulations reveal that allostery amounts to the propagation of structural and dynamical changes that are genuinely nonlinear and can occur in a nonlocal fashion. A dynamic network model is constructed that illustrates the hierarchy and exceeding structural heterogeneity of the process. In compelling agreement with experiment, three physically distinct phases of the time evolution are identified, describing elastic response (≲0.1 ns), inelastic reorganization (∼100 ns), and structural relaxation (≳1μs). Issues such as the similarity to downhill folding as well as the interpretation of allosteric pathways are discussed. [ABSTRACT FROM AUTHOR]

The accurate definition of suitable metastable conformational states is fundamental for the construction of a Markov state model describing biomolecular dynamics. Following the dimensionality reduction in a molecular dynamics trajectory, these microstates can be generated by a recently proposed density-based geometrical clustering algorithm [F. Sittel and G. Stock, J. Chem. Theory Comput. 12, 2426 (2016)], which by design cuts the resulting clusters at the energy barriers and allows for a data-based identification of all parameters. Nevertheless, projection artifacts due to the inevitable restriction to a low-dimensional space combined with insufficient sampling often leads to a misclassification of sampled points in the transition regions. This typically causes intrastate fluctuations to be mistaken as interstate transitions, which leads to artificially short life times of the metastable states. As a simple but effective remedy, dynamical coring requires that the trajectory spends a minimum time in the new state for the transition to be counted. Adopting molecular dynamics simulations of two well-established biomolecular systems (alanine dipeptide and villin headpiece), dynamical coring is shown to considerably improve the Markovianity of the resulting metastable states, which is demonstrated by Chapman-Kolmogorov tests and increased implied time scales of the Markov model. Providing high structural and temporal resolution, the combination of density-based clustering and dynamical coring is particularly suited to describe the complex structural dynamics of unfolded biomolecules. [ABSTRACT FROM AUTHOR]

Molecular dynamics simulations of biomolecular processes are often discussed in terms of diffusive motion on a low-dimensional free energy landscape F(x). To provide a theoretical basis for this interpretation, one may invoke the system-bath ansatz á la Zwanzig. That is, by assuming a time scale separation between the slow motion along the system coordinate x and the fast fluctuations of the bath, a memory-free Langevin equation can be derived that describes the system's motion on the free energy landscape F(x), which is damped by a friction field and driven by a stochastic force that is related to the friction via the fluctuation-dissipation theorem. While the theoretical formulation of Zwanzig typically assumes a highly idealized form of the bath Hamiltonian and the system-bath coupling, one would like to extend the approach to realistic data-based biomolecular systems. Here a practical method is proposed to construct an analytically defined global model of structural dynamics. Given a molecular dynamics simulation and adequate collective coordinates, the approach employs an ?empirical valence bond?-type model which is suitable to represent multidimensional free energy landscapes as well as an approximate description of the friction field. Adopting alanine dipeptide and a three-dimensional model of heptaalanine as simple examples, the resulting Langevin model is shown to reproduce the results of the underlying all-atom simulations. Because the Langevin equation can also be shown to satisfy the underlying assumptions of the theory (such as a delta-correlated Gaussiandistributed noise), the global model provides a correct, albeit empirical, realization of Zwanzig's formulation. As an application, the model can be used to investigate the dependence of the system on parameter changes and to predict the effect of site-selective mutations on the dynamics. [ABSTRACT FROM AUTHOR]

Abstract: The mapping formulation of nonadiabatic quantum dynamics is applied to obtain a classical description of the ultrafast dynamics and time-resolved spectroscopy of a photochemical reaction. Adopting a previously studied dissipative two-state two-mode model of nonadiabatic cis–trans photoisomerization, classical mapping simulations are compared to quantum-mechanical reduced density matrix calculations. Overall, the simple classical method is found to reproduce the quantum reference calculations quite well. In particular, it is studied if the classical approach yields the correct long-time cis/trans localization of the wave packet and therefore the correct quantum yield of the photoreaction. As the long-time behavior of the classical mapping formulation suffers from the well-known zero point energy problem of classical mechanics, a new practical method is proposed to determine a zero point energy correction. Employing a second-order Franck–Condon-type approximation, the capability of the classical method to simulate time- and frequency-resolved pump–probe spectra of the nonadiabatic photoreaction is studied. The potential of the classical approach as a practical method to describe condensed-phase photoreactions is discussed. [Copyright &y& Elsevier]

We examine vibrational energy flow in dehydrated and hydrated villin headpiece subdomain HP36 by master equation simulations. Transition rates used in the simulations are obtained from communication maps calculated for HP36. In addition to energy flow along the main chain, we identify pathways for energy transport in HP36 via hydrogen bonding between residues quite far in sequence space. The results of the master equation simulations compare well with all-atom non-equilibrium simulations to about 1 ps following initial excitation of the protein, and quite well at long times, though for some residues we observe deviations between the master equation and all-atom simulations at intermediate times from about 1-10 ps. Those deviations are less noticeable for hydrated than dehydrated HP36 due to energy flow into the water. [ABSTRACT FROM AUTHOR]

Based on a given time series, the data-driven Langevin equation (dLE) estimates the drift and the diffusion field of the dynamics, which are then employed to reproduce the essential statistical and dynamical features of the original time series. Because the propagation of the dLE requires only local information, the input data are neither required to be Boltzmann weighted nor to be a continuous trajectory. Similar to a Markov state model, the dLE approach therefore holds the promise of predicting the long-time dynamics of a biomolecular system from relatively short trajectories which can be run in parallel. The practical applicability of the approach is shown to be mainly limited by the initial sampling of the system's conformational space obtained from the short trajectories. Adopting extensive molecular dynamics simulations of the unfolding and refolding of a short peptide helix, it is shown that the dLE approach is able to describe microsecond conformational dynamics from a few hundred nanosecond trajectories. In particular, the dLE quantitatively reproduces the free energy landscape and the associated conformational dynamics along the chosen five-dimensional reaction coordinate. [ABSTRACT FROM AUTHOR]

While allostery is of paramount importance for protein regulation, the underlying dynamical process of ligand (un)binding at one site, resulting time evolution of the protein structure, and change of the binding affinity at a remote site are not well understood. Here the ligand-induced conformational transition in a widely studied model system of allostery, the PDZ2 domain, is investigated by transient infrared spectroscopy accompanied by molecular dynamics simulations. To this end, an azobenzene derived photoswitch is linked to a peptide ligand in a way that its binding affinity to the PDZ2 domain changes upon switching, thus initiating an allosteric transition in the PDZ2 domain protein. The subsequent response of the protein, covering four decades of time, ranging from ∼1 ns to ∼10 µs, can be rationalized by a remodeling of its rugged free-energy landscape, with very subtle shifts in the populations of a small number of structurally well-defined states. It is proposed that structurally and dynamically driven allostery, often discussed as limiting scenarios of allosteric communication, actually go hand-in-hand, allowing the protein to adapt its free-energy landscape to incoming signals. [ABSTRACT FROM AUTHOR]

Allosteric regulation plays an important role in many biological processes, such as signal transduction, transcriptional regulation, and metabolism. Allostery is rooted in the fundamental physical properties of macromolecular systems, but its underlying mechanisms are still poorly understood. A collection of contributions to a recent interdisciplinary CECAM (Center Européen de Calcul Atomique et Moléculaire) workshop is used here to provide an overview of the progress and remaining limitations in the understanding of the mechanistic foundations of allostery gained from computational and experimental analyses of real protein systems and model systems. The main conceptual frameworks instrumental in driving the field are discussed. We illustrate the role of these frameworks in illuminating molecular mechanisms and explaining cellular processes, and describe some of their promising practical applications in engineering molecular sensors and informing drug design efforts. A collection of contributions to a recent interdisciplinary CECAM (Center Européen de Calcul Atomique et Moléculaire) workshop offers an insightful overview of the understanding of the mechanistic foundations of allostery, gained from computational and experimental analyses of real protein systems and model systems. Various practical applications are illustrated. [ABSTRACT FROM AUTHOR]

Performing comprehensive quantum-chemical calculations, a vibrational Hamiltonian of acetylbenzonitrile is constructed, on the basis of which a quantum-mechanical “tier model” is developed that describes the vibrational dynamics following excitation of the CN stretch mode. Taking into account 36 vibrational modes and cubic and quartic anharmonic couplings between up to three different modes, the tier model calculations are shown to qualitatively reproduce the main findings of the experiments of Rubtsov and coworkers (2011), including the energy relaxation of the initially excited CN mode and the structure-dependent vibrational transport. Moreover, the calculations suggest that the experimentally measured cross-peak among the CN and CO modes does not correspond to direct excitation of the CO normal mode but rather reflects excited low-frequency vibrations that anharmonically couple to the CO mode. Complementary quasiclassical trajectory calculations are found to be in good overall agreement with the quantum calculations. [ABSTRACT FROM AUTHOR]

Abstract: A quantum-classical description of the amide I vibrational spectrum of peptides in aqueous solution is given, which is concerned with the effects of nonadiabatic couplings between vibrational eigenstates. It consists of a classical molecular dynamics simulation of the conformational distribution of the system, density functional theory calculations of the conformation-dependent and solvent-induced frequency fluctuations, and a semiclassical description of the vibrational line shapes. The study shows that the adiabatic approximation usually employed in semiclassical line shape theory is generally not valid, because it assumes a time scale separation between the dynamics of the amide I mode (with a period of ≈20fs) and the motion of the solvent and the peptide (which also exhibits sub-100fs dynamics). A practical and general computational scheme is presented, which allows for the calculation of spectroscopic response functions by directly solving the nonadiabatically coupled time-dependent Schrödinger equation. Adopting trialanine and heptaalanine as representative examples, it is shown that nonadiabatic interactions may considerably change the overall shape as well as local details of the amide I spectrum. [Copyright &y& Elsevier]

While the linear Pearson correlation coefficient represents a well-established normalized measure to quantify the inter-relation of two stochastic variables X and Y, it fails for multidimensional variables, such as Cartesian coordinates. Avoiding any assumption about the underlying data, the mutual information I(X, Y) does account for multidimensional correlations. However, unlike the normalized Pearson correlation, it has no upper bound (I ∈ [0, ∞)), i.e., it is not clear if say, I = 0.4 corresponds to a low or a high correlation. Moreover, the mutual information (MI) involves the estimation of high-dimensional probability densities (e.g., six-dimensional for Cartesian coordinates), which requires a k nearest-neighbor algorithm, such as the estimator by Kraskov et al. [Phys. Rev. E 69, 066138 (2004)]. As existing methods to normalize the MI cannot be used in connection with this estimator, a new approach is presented, which uses an entropy estimation method that is invariant under variable transformations. The algorithm is numerically efficient and does not require more effort than the calculation of the (un-normalized) MI. After validating the method by applying it to various toy models, the normalized MI between the Cα-coordinates of T4 lysozyme is considered and compared to a correlation analysis of inter-residue contacts., (© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).)

A novel mixed quantum-classical approach to simulating nonadiabatic dynamics of molecules at metal surfaces is presented. The method combines the numerically exact hierarchical equations of motion approach for the quantum electronic degrees of freedom with Langevin dynamics for the classical degrees of freedom, namely, low-frequency vibrational modes within the molecule. The approach extends previous mixed quantum-classical methods based on Langevin equations to models containing strong electron-electron or quantum electronic-vibrational interactions, while maintaining a nonperturbative and non-Markovian treatment of the molecule-metal coupling. To demonstrate the approach, nonequilibrium transport observables are calculated for a molecular nanojunction containing strong interactions., (© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).)

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. D. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape and can, in simple cases, be interpreted in terms of the barriers of the staircase landscape., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

To sample rare events, dissipation-corrected targeted molecular dynamics (dcTMD) applies a constant velocity constraint along a one-dimensional reaction coordinate s , which drives an atomistic system from an initial state into a target state. Employing a cumulant approximation of Jarzynski's identity, the free energy Δ G ( s ) is calculated from the mean external work and dissipated work of the process. By calculating the friction coefficient Γ( s ) from the dissipated work, in a second step, the equilibrium dynamics of the process can be studied by propagating a Langevin equation. While so far dcTMD has been mostly applied to study the unbinding of protein-ligand complexes, here its applicability to rare conformational transitions within a protein and the prediction of their kinetics are investigated. As this typically requires the introduction of multiple collective variables { x j } = x , a theoretical framework is outlined to calculate the associated free energy Δ G ( x ) and friction Γ ( x ) from dcTMD simulations along coordinate s . Adopting the α-β transition of alanine dipeptide as well as the open-closed transition of T4 lysozyme as representative examples, the virtues and shortcomings of dcTMD to predict protein conformational transitions and the related kinetics are studied.

Adopting a 300 μs long MD trajectory of the folding of villin headpiece (HP35) by D. E. Shaw Research, we recently constructed a Markov state model (MSM) based on inter-residue contacts. The model reproduces the folding time and predicts that the native basin and unfolded region consist of metastable substates that are structurally well-characterized. Recognizing the need to establish well-defined benchmark problems, we study to what extent and in what sense this MSM can be employed as a reference model. Hence, we test the robustness of the MSM by comparing it to models that use alternative combinations of features, dimensionality reduction methods, and clustering schemes. The study suggests some main characteristics of the folding of HP35 that should be reproduced by other competitive models. Moreover, the discussion reveals which parts of the MSM workflow matter most for the considered problem and illustrates the promises and pitfalls of state-based models for the interpretation of biomolecular simulations.

Markov state models represent a popular means to interpret molecular dynamics trajectories in terms of memoryless transitions between metastable conformational states. To provide a mechanistic understanding of the considered biomolecular process, these states should reflect structurally distinct conformations and ensure a time scale separation between fast intrastate and slow interstate dynamics. Adopting the folding of villin headpiece (HP35) as a well-established model problem, here we discuss the selection of suitable input coordinates or "features", such as backbone dihedral angles and interresidue distances. We show that dihedral angles account accurately for the structure of the native energy basin of HP35, while the unfolded region of the free energy landscape and the folding process are best described by tertiary contacts of the protein. To construct a contact-based model, we consider various ways to define and select contact distances and introduce a low-pass filtering of the feature trajectory as well as a correlation-based characterization of states. Relying on input data that faithfully account for the mechanistic origin of the studied process, the states of the resulting Markov model are clearly discriminated by the features, describe consistently the hierarchical structure of the free energy landscape, and─as a consequence─correctly reproduce the slow time scales of the process.

Protein-ligand (un)binding simulations are a recent focus of biased molecular dynamics simulations. Such binding and unbinding can occur via different pathways in and out of a binding site. Here, we present a theoretical framework on how to compute kinetics along separate paths and on how to combine the path-specific rates into global binding and unbinding rates for comparison with experimental results. Using dissipation-corrected targeted molecular dynamics in combination with temperature-boosted Langevin equation simulations [S. Wolf et al., Nat. Commun. 11, 2918 (2020)] applied to a two-dimensional model and the trypsin-benzamidine complex as test systems, we assess the robustness of the procedure and discuss the aspects of its practical applicability to predict multisecond kinetics of complex biomolecular systems.

Photoproteins such as bacteriorhodopsin (bR) and rhodopsin (Rho) need to effectively dissipate photoinduced excess energy to prevent themselves from damage. Another well-studied seven transmembrane (TM) helices protein is the β 2 adrenergic receptor (β 2 AR), a G protein-coupled receptor for which energy dissipation paths have been linked with allosteric communication. To study the vibrational energy transport in the active and inactive states of these proteins, a master equation approach [ J. Chem. Phys. 2020 , 152 , 045103] is employed, which uses scaling rules that allow us to calculate energy transport rates solely based on the protein structure. Despite their overall structural similarity, the three 7TM proteins reveal quite different strategies to redistribute excess energy. While bR quickly removes the energy using the TM7 helix as a "lightning rod", Rho exhibits a rather poor energy dissipation, which might eventually require the hydrolysis of the Schiff base between the protein and the retinal chromophore to prevent overheating. Heating the ligand adrenaline of β 2 AR, the resulting energy transport network of the protein is found to change significantly upon switching from the active state to the inactive state. While the energy flow may highlight aspects of the inter-residue couplings of β 2 AR, it seems not particularly suited to explain allosteric phenomena.

While allostery is of paramount importance for protein signaling and regulation, the underlying dynamical process of allosteric communication is not well understood. The PDZ3 domain represents a prime example of an allosteric single-domain protein, as it features a well-established long-range coupling between the C-terminal α 3 -helix and ligand binding. In an intriguing experiment, Hamm and co-workers employed photoswitching of the α 3 -helix to initiate a conformational change of PDZ3 that propagates from the C-terminus to the bound ligand within 200 ns. Performing extensive nonequilibrium molecular dynamics simulations, the modeling of the experiment reproduces the measured time scales and reveals a detailed picture of the allosteric communication in PDZ3. In particular, a correlation analysis identifies a network of contacts connecting the α 3 -helix and the core of the protein, which move in a concerted manner. Representing a one-step process and involving direct α 3 -ligand contacts, this cooperative transition is considered as the elementary step in the propagation of conformational change.

Given nonstationary data from molecular dynamics simulations, a Markovian Langevin model is constructed that aims to reproduce the time evolution of the underlying process. While at equilibrium the free energy landscape is sampled, nonequilibrium processes can be associated with a biased energy landscape, which accounts for finite sampling effects and external driving. When the data-driven Langevin equation (dLE) approach [ Phys . Rev . Lett . 2015 , 115 , 050602] is extended to the modeling of nonequilibrium processes, an efficient way to calculate multidimensional Langevin fields is outlined. The dLE is shown to correctly account for various nonequilibrium processes, including the enforced dissociation of sodium chloride in water, the pressure-jump induced nucleation of a liquid of hard spheres, and the conformational dynamics of a helical peptide sampled from nonstationary short trajectories.

Vibrational energy transfer (VET) is essential for protein function. It is responsible for efficient energy dissipation in reaction sites, and has been linked to pathways of allosteric communication. While it is understood that VET occurs via backbone as well as via non-covalent contacts, little is known about the competition of these two transport channels, which determines the VET pathways. To tackle this problem, we equipped the β-hairpin fold of a tryptophan zipper with pairs of non-canonical amino acids, one serving as a VET injector and one as a VET sensor in a femtosecond pump probe experiment. Accompanying extensive non-equilibrium molecular dynamics simulations combined with a master equation analysis unravel the VET pathways. Our joint experimental/computational endeavor reveals the efficiency of backbone vs. contact transport, showing that even if cutting short backbone stretches of only 3 to 4 amino acids in a protein, hydrogen bonds are the dominant VET pathway.

We report on a study that combines advanced fluorescence methods with molecular dynamics (MD) simulations to cover timescales from nanoseconds to milliseconds for a large protein. This allows us to delineate how ATP hydrolysis in a protein causes allosteric changes at a distant protein binding site, using the chaperone Hsp90 as test system. The allosteric process occurs via hierarchical dynamics involving timescales from nano- to milliseconds and length scales from Ångstroms to several nanometers. We find that hydrolysis of one ATP is coupled to a conformational change of Arg380, which in turn passes structural information via the large M-domain α-helix to the whole protein. The resulting structural asymmetry in Hsp90 leads to the collapse of a central folding substrate binding site, causing the formation of a novel collapsed state (closed state B) that we characterise structurally. We presume that similar hierarchical mechanisms are fundamental for information transfer induced by ATP hydrolysis through many other proteins., Competing Interests: There are no conflicts to declare., (This journal is © The Royal Society of Chemistry.)

Markov processes provide a popular approach to construct low-dimensional dynamical models of a complex biomolecular system. By partitioning the conformational space into metastable states, protein dynamics can be approximated in terms of memory-less jumps between these states, resulting in a Markov state model (MSM). Alternatively, suitable low-dimensional collective variables may be identified to construct a data-driven Langevin equation (dLE). In both cases, the underlying Markovian approximation requires a propagation time step (or lag time) δt that is longer than the memory time τ M of the system. On the other hand, δt needs to be chosen short enough to resolve the system timescale τ S of interest. If these conditions are in conflict (i.e., τ M > τ S ), one may opt for a short time step δt = τ S and try to account for the residual non-Markovianity of the data by optimizing the transition matrix or the Langevin fields such that the resulting model best reproduces the observables of interest. In this work, rescaling the friction tensor of the dLE based on short-time information in order to obtain the correct long-time behavior of the system is suggested. Adopting various model problems of increasing complexity, including a double-well system, the dissociation of solvated sodium chloride, and the functional dynamics of T4 lysozyme, the virtues and shortcomings of the rescaled dLE are discussed and compared to the corresponding MSMs.