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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 τM of the system. On the other hand, δt needs to be chosen short enough to resolve the system timescale τS of interest. If these conditions are in conflict (i.e., τM > τS), one may opt for a short time step δt = τS 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. [ABSTRACT FROM AUTHOR]

Recent time-resolved experiments and accompanying molecular dynamics simulations allow us to monitor the flow of vibrational energy in biomolecules. As a simple means to describe these experimental and simulated data, Buchenberg et al. [J. Phys. Chem. Lett. 7, 25 (2016)] suggested a master equation model that accounts for the energy transport from an initially excited residue to some target residue. The transfer rates of the model were obtained from two scaling rules, which account for the energy transport through the backbone and via tertiary contacts, respectively, and were parameterized using simulation data of a small α-helical protein at low temperatures. To extend the applicability of the model to general proteins at room temperature, here a new parameterization is presented, which is based on extensive nonequilibrium molecular dynamics simulations of a number of model systems. With typical transfer times of 0.5–1 ps between adjacent residues, backbone transport represents the fastest channel of energy flow. It is well described by a diffusive-type scaling rule, which requires only an overall backbone diffusion coefficient and interatom distances as input. Contact transport, e.g., via hydrogen bonds, is considerably slower (6–30 ps) at room temperature. A new scaling rule depending on the inverse square contact distance is suggested, which is shown to successfully describe the energy transport in the allosteric protein PDZ3. Since both scaling rules require only the structure of the considered system, the model provides a simple and general means to predict energy transport in proteins. To identify the pathways of energy transport, Monte Carlo Markov chain simulations are performed, which highlight the competition between backbone and contact transport channels. [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]

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]

The statistical analysis of molecular dynamics simulations requires dimensionality reduction techniques, which yield a low-dimensional set of collective variables (CVs) {xi} = x that in some sense describe the essential dynamics of the system. Considering the distribution P(x) of the CVs, the primal goal of a statistical analysis is to detect the characteristic features of P(x), in particular, its maxima and their connection paths. This is because these features characterize the low-energy regions and the energy barriers of the corresponding free energy landscape ΔG(x) = −kBT ln P(x), and therefore amount to the metastable states and transition regions of the system. In this perspective, we outline a systematic strategy to identify CVs and metastable states, which subsequently can be employed to construct a Langevin or a Markov state model of the dynamics. In particular, we account for the still limited sampling typically achieved by molecular dynamics simulations, which in practice seriously limits the applicability of theories (e.g., assuming ergodicity) and black-box software tools (e.g., using redundant input coordinates). We show that it is essential to use internal (rather than Cartesian) input coordinates, employ dimensionality reduction methods that avoid rescaling errors (such as principal component analysis), and perform density based (rather than k-means-type) clustering. Finally, we briefly discuss a machine learning approach to dimensionality reduction, which highlights the essential internal coordinates of a system and may reveal hidden reaction mechanisms. [ABSTRACT FROM AUTHOR]

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]

The structure, dynamics, and thermal stability of the structurally similar 14-mer RNA hairpins uCACGg and cUUCGg are studied by using replica-exchange molecular dynamics simulations. The two RNA hairpins exhibited similar structures at room temperature, but the UUCG loop was found to be more flexible than the CACG loop.

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 [(A[[beta].sub.16-22]).sub.n] by adding a monomer to a preformed [(A[[beta].sub.16-22]).sub.n-1] (n = 4-6) oligomer in which the peptides are arranged in an antiparallel [beta]-sheet conformation. All atom molecular dynamics simulations in water and multiple long trajectories, for a cumulative time of 6.9 [micro]s, 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 ([approximately equal to] 50 ns) first dock stage in which the [beta]-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.

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]

Adopting trialanine as a paradigm of a small peptide, a comprehensive comparison of six popular force fields is presented, including the recent versions of AMBER, CHARMM, GROMOS, and OPLS models. For all force fields under consideration a 20ns long MD simulation is performed to study in detail the structure and conformational dynamics of the solvated peptide.

The fluorescence and resonance Raman emission of molecular systems undergoing ultrafast nonadiabatic isomerization is discussed. The validity of the approximation is studied by employing a two-mode, two-state model of photoisomerization of retinal.

A technique for the photoisomerization of retinal rhodopsin using a vibronically coupled electronic two-state system is described.

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]

Based on extensive ab initio calculations and the time-propagation of the nuclear Schrödinger equation, we study the vibrational relaxation dynamics and resulting spectral signatures of the OH stretch vibration of a hydrogen-bonded complex, HCO2- · H2O. Despite their smallness, it has been shown experimentally by Johnson and coworkers that the gas-phase infrared spectra of these types of complexes exhibit much of the complexity commonly observed for hydrogen-bonded systems. That is, the OH stretch band exhibits a significant red shift together with an extreme broadening and a pronounced substructure, which reflects its very strong anharmonicity. Employing an adiabatic separation of time scales between the three intramolecular high-frequency modes of the water molecule and the three most important intermolecular low-frequency modes of the complex, we calculate potential energy surfaces (PESs) of the ground and the first excited states of the high-frequency modes and identify a vibrational conical intersection between the PESs of the OH stretch fundamental and the HOH bend overtone. By performing a time-dependent propagation of the resulting system, we show that the conical intersection affects a coherent population transfer between the two states, the first step of which being ultrafast (60 fs) and irreversible. The subsequent relaxation of vibrational energy into the HOH bend and ground state occurs incoherently but also quite fast (1 ps), although the corresponding PESs are well separated in energy. Owing to the smaller effective mass difference between light and heavy degrees of freedom, the adiabatic ansatz is consequently less significant for vibrations than in the electronic case. Based on the model, we consider several approximations to calculate the measured Ar-tag action spectrum of HCO2- · H2O and achieve semiquantitative agreement with the experiment. [ABSTRACT FROM AUTHOR]

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]

When developing coarse‐grained models of complex processes out of equilibrium, one encounters the non‐stationary generalized Langevin equation. The most important feature of this equation is the presence of a non‐stationary memory kernel. Here, a method is presented to infer this memory kernel from MD simulation data in non‐equilibrium processes. The method provides an improvement of a previously published numerical scheme, the applicability of which is limited by a truncation problem. As an illustration, the method is applied to ion dissociation of NaCl in water, for which non‐trivial dampened oscillations are observed in the memory kernel. [ABSTRACT FROM AUTHOR]

Principal component analysis of molecular dynamics simulations is a popular method to account for the essential dynamics of the system on a low-dimensional free energy landscape. Using Cartesian coordinates, first the translation and overall rotation need to be removed from the trajectory. Since the rotation depends via the moment of inertia on the molecule's structure, this separation is only straightforward for relatively rigid systems. Adopting millisecond molecular dynamics simulations of the folding of villin headpiece and the functional dynamics of BPTI provided by D. E. Shaw Research, it is demonstrated via a comparison of local and global rotational fitting that the structural dynamics of flexible molecules necessarily results in a mixing of overall and internal motion. Even for the small-amplitude functional motion of BPTI, the conformational distribution obtained from a Cartesian principal component analysis therefore reflects to some extend the dominant overall motion rather than the much smaller internal motion of the protein. Internal coordinates such as backbone dihedral angles, on the other hand, are found to yield correct and well-resolved energy landscapes for both examples. The virtues and shortcomings of the choice of various fitting schemes and coordinate sets as well as the generality of these results are discussed in some detail. [ABSTRACT FROM AUTHOR]

Based on a given time series, the data-driven Langevin equation proposed by Hegger and Stock [J. Chem. Phys. 130, 034106 (2009)] aims to construct a low-dimensional dynamical model of the system. Adopting various simple model problems of biomolecular dynamics, this work presents a systematic study of the theoretical virtues and limitations as well as of the practical applicability and performance of the method. As the method requires only local information, the input data need not to be Boltzmann weighted in order to warrant that the Langevin model yields correct Boltzmann-distributed results. Moreover, a delay embedding of the state vector allows for the treatment of memory effects. The robustness of the modeling with respect to wrongly chosen model parameters or low sampling is discussed, as well as the treatment of inertial effects. Given sufficiently sampled input data, the Langevin modeling is shown to successfully recover the correct statistics (such as the probability distribution) and the dynamics (such as the position autocorrelation function) of all considered problems. [ABSTRACT FROM AUTHOR]