Realistic master equation modeling of relaxation on complete potential energy surfaces: Kinetic results

Principal component analysis of potential energy surfaces of large clusters: allowing the practical calculation of the master equation

The number of variables in many-particle systems is typically unmanageably large; some way to reduce that number and still retain access to the important information about the system of interest is one of the great challenges in the broad topic of complexity. Principal components and principal coordinates provide a powerful means to extract--from unwieldy, large data sets--a reduced collection of variables that provide the information one needs, in a relatively efficient way and useful form. We investigate the application of principal components to the analysis of kinetics of the atomic motions in atomic clusters, particularly of clusters that are large enough so that a full description in terms of the entire high-dimensional potential surface is entirely impractical. A specific application is the use of principal components linking minima with their adjacent saddles, permitting the evaluation of rate coefficients (in the context of transition state theory) as ratios of partition functions of only one or two key variables.

Determination, prediction, and understanding of structures, using the energy landscapes of chemical systems – Part II

In the past decade, new theoretical approaches have been developed to determine, predict and understand the struc-ture of chemical compounds. The central element of these methods has been the investigation of the energy landscape of chemical systems. Applications range from extended crystalline and amorphous compounds over clusters and molecular crystals to proteins. In this review, we are going to give an introduction to energy landscapes and methods for their investigation, together with a number of examples. These include structure prediction of extended and mo-lecular crystals, structure prediction and folding of proteins, structure analysis of zeolites, and structure determination of crystals from powder diffraction data.

Pathways for conformational change in nitrogen regulatory protein C from discrete path sampling

Pathways corresponding to the conformational change in nitrogen regulatory protein C are calculated using the CHARMM19 force field with an implicit solvation model. Our analysis employs the discrete path sampling approach to grow a database of local minima and transition states from the potential energy surface that contains kinetically relevant pathways. The pathways with the largest contribution to the phenomenological two-state rate constants are found to exhibit a number of structural features that agree with experimental observations. Further details of the calculated pathways for conformational change may therefore provide useful predictions of how this large-scale motion is achieved.

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.

Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics

To meet the challenge of modeling the conformational dynamics of biological macromolecules over long time scales, much recent effort has been devoted to constructing stochastic kinetic models, often in the form of discrete-state Markov models, from short molecular dynamics simulations. To construct useful models that faithfully represent dynamics at the time scales of interest, it is necessary to decompose configuration space into a set of kinetically metastable states. Previous attempts to define these states have relied upon either prior knowledge of the slow degrees of freedom or on the application of conformational clustering techniques which assume that conformationally distinct clusters are also kinetically distinct. Here, we present a first version of an automatic algorithm for the discovery of kinetically metastable states that is generally applicable to solvated macromolecules. Given molecular dynamics trajectories initiated from a well-defined starting distribution, the algorithm discovers long lived, kinetically metastable states through successive iterations of partitioning and aggregating conformation space into kinetically related regions. The authors apply this method to three peptides in explicit solvent-terminally blocked alanine, the 21-residue helical F(s) peptide, and the engineered 12-residue beta-hairpin trpzip2-to assess its ability to generate physically meaningful states and faithful kinetic models.

Time autocorrelation function analysis of master equation and its application to atomic clusters

We derive the energy fluctuation Delta(2)E, and the time autocorrelation kappa(tau) and its Fourier transformation--the fluctuation spectra S(omega)--of the master-equation transition matrix. The contribution from each eigenmode of the transition matrix to these fluctuation quantities reveals the relevant importance of the individual mode in the relaxation processes. The time scales associated with these relaxation processes are determined by the corresponding eigenvalues. Unlike traditional time evolution analysis, the autocorrelation function and fluctuation spectra analysis does not involve an arbitrary initial population. It is also more suitable for analyzing the underlying dynamic, kinetic behavior near the equilibrium and the behavior of the long-time-scale rare events. We utilize our technique to analyze the solid-liquid phase coexistence of the 13-atom Morse cluster and the fcc-to-icosahedral structure transition of the 38-atom Lennard-Jones cluster. For the processes studied, the fluctuation spectra from the master equation simplify the analysis of the transition matrix, and the important relaxation modes are easily extracted.

Nonequilibrium theory of polymer stretching based on the master equation

We present a model for fast polymer-stretching experiments. We use the master equation and argue that the end-to-end extension of a polymer molecule can be used as a stochastic variable after appropriate coarse graining. The main effect of increasing pulling speed or force loading rate is a marked hysteresis in the force-extension curve as well as an overall shift of the curve to higher forces when compared to the equilibrium curve. This can be understood in terms of the moments of the transition probability in the master equation. An analysis of the fluctuations and relaxation times is also given in the framework of our theory.

Activated dynamics and timescale separation within the landscape paradigm: signature of complexity, diversity and glassiness

The landscape paradigm has become a widespread picture within the realm of complex systems. Complex systems include a great variety of systems, ranging from glasses to biopolymers, which display a common dynamical behavior. Within this framework, the dynamics of a such a system can be envisioned as the search it performs on its (potential energy) landscape. This approach rests on the belief that the relaxation behavior depends only on generic features, irrespective of specific details and lies on the validity of a timescale separation scenario computationally corroborated but not properly validated yet form first principles. In this work we shall show that the prevalence of activated dynamics over other kinds of mechanisms determines the emergence of complex dynamical behavior. Thus, complexity and diversity are not intrinsic properties of a system but depend on the kind of exploration of the landscape. We shall focus mainly on an ample generic context (complex hierarchical systems which have been used as models of glasses, spin glasses and biopolymers) and a specific one (model glass formers). For the last case we shall be able to reveal (in mechanistic terms) the microscopic rationale for the occurrence of timescale separation. Furthermore, we shall explore the connections between these two up to now mostly unrelated contexts and the relation to a variational principle, and we shall reveal the conditions for the applicability of the landscape approach.

Kinetics of model energy landscapes: an approach to complex systems

An idealized potential energy surface (PES), simply a PES-like network of stationary points, is demonstrated to be a useful tool to study kinetic relaxation of complex energy landscapes. Combined with a master equation, we show that if constructed with proper regularity, the kinetics of the PES is easy to predict and understand by carefully examining the eigenmodes of the master equation. By modifying the idealized PES model to make it more and more complicated, we demonstrate a systematic method to study the complex kinetics on large PES. The idealized PES model is used to explore the feasibility and the robustness of statistical sampling of large PES. We develop several sampling strategies, such as the "rough topography sampling" and the "low barrier saddle sampling" in the idealized PES model and find they are also applicable to a realistic PES of the 13-atom Morse cluster with range parameter rho= 6. To measure the robustness of the sampling methods, we compare the eigenvalue spectra, the eigenvector similarity and the relaxation times of the total energy of the full and sample PESs.

Top-down free-energy minimization on protein potential energy landscapes

The hierarchical properties of potential energy landscapes have been used to gain insight into thermodynamic and kinetic properties of protein ensembles. It also may be possible to use them to direct computational searches for thermodynamically stable macroscopic states, i.e., computational protein folding. To this end, we have developed a top-down search procedure in which conformation space is recursively dissected according to the intrinsic hierarchical structure of a landscape's effective-energy barriers. This procedure generates an inverted tree similar to the disconnectivity graphs generated by local minima-clustering methods, but it fundamentally differs in the manner in which the portion of the tree that is to be computationally explored is selected. A key ingredient is a branch-selection algorithm that takes advantage of statistically predictive properties of the landscape to guide searches down the tree branches that are most likely to lead to the physically relevant macroscopic states. Using the computational folding of a beta-hairpin-forming peptide as an example, we show that such predictive properties indeed exist and can be used for structure prediction by free-energy global minimization.

Solvent effects on the energy landscapes and folding kinetics of polyalanine

The effect of a solvation on the thermodynamics and kinetics of polyalanine (Ala(12)) is explored on the basis of its energy landscapes in vacuum and in an aqueous solution. Both energy landscapes are characterized by two basins, one associated with alpha-helical structures and the other with coil and beta-structures of the peptide. In both environments, the basin that corresponds to the alpha-helical structure is considerably narrower than the basin corresponding to the beta-state, reflecting their different contributions to the entropy of the peptide. In vacuum, the alpha-helical state of Ala(12) constitutes the native state, in agreement with common helical propensity scales, whereas in the aqueous medium, the alpha-helical state is destabilized, and the beta-state becomes the native state. Thus solvation has a dramatic effect on the energy landscape of this peptide, resulting in an inverted stability of the two states. Different folding and unfolding time scales for Ala(12) in hydrophilic and hydrophobic chemical environments are caused by the higher entropy of the native state in water relative to vacuum. The concept of a helical propensity has to be extended to incorporate environmental solvent effects.

Structural relaxation in atomic clusters: master equation dynamics

The role of the potential energy landscape in determining the relaxation dynamics of model clusters is studied using a master equation. Two types of energy landscape are examined: a single funnel, as exemplified by 13-atom Morse clusters, and the double funnel landscape of the 38-atom Lennard-Jones cluster. Interwell rate constants are calculated using Rice-Ramsperger-Kassel-Marcus theory within the harmonic approximation, but anharmonic model partition functions are also considered. Decreasing the range of the potential in the Morse clusters is shown to hinder relaxation toward the global minimum, and this effect is related to the concomitant changes in the energy landscape. The relaxation modes that emerge from the master equation are interpreted and analyzed to extract interfunnel rate constants for the Lennard-Jones cluster. Since this system is too large for a complete characterization of the energy landscape, the conditions under which the master equation can be applied to a limited database are explored. Connections are made to relaxation processes in proteins and structural glasses.