CiteULike: Diego_Prada's library [95 articles]

Complexity of free energy landscapes of peptides revealed by nonlinear principal component analysis

Phuong Nguyen — 2006-11-29T16:29:23-00:00

Proteins: Structure, Function, and Bioinformatics, Vol. 65, No. 4. (2006), pp. 898-913.

Employing the recently developed hierarchical nonlinear principal component analysis (NLPCA) method of Saegusa et al. (Neurocomputing 2004;61:57-70 and IEICE Trans Inf Syst 2005;E88-D:2242-2248), the complexities of the free energy landscapes of several peptides, including triglycine, hexaalanine, and the C-terminal ?-hairpin of protein G, were studied. First, the performance of this NLPCA method was compared with the standard linear principal component analysis (PCA). In particular, we compared two methods according to (1) the ability of the dimensionality reduction and (2) the efficient representation of peptide conformations in low-dimensional spaces spanned by the first few principal components. The study revealed that NLPCA reduces the dimensionality of the considered systems much better, than did PCA. For example, in order to get the similar error, which is due to representation of the original data of ?-hairpin in low dimensional space, one needs 4 and 21 principal components of NLPCA and PCA, respectively. Second, by representing the free energy landscapes of the considered systems as a function of the first two principal components obtained from PCA, we obtained the relatively well-structured free energy landscapes. In contrast, the free energy landscapes of NLPCA are much more complicated, exhibiting many states which are hidden in the PCA maps, especially in the unfolded regions. Furthermore, the study also showed that many states in the PCA maps are mixed up by several peptide conformations, while those of the NLPCA maps are more pure. This finding suggests that the NLPCA should be used to capture the essential features of the systems. Proteins 2006. © 2006 Wiley-Liss, Inc.

Energy landscapes of conformationally constrained peptides

Yaakov Levy — 2008-07-04T16:32:53-00:00

The Journal of Chemical Physics, Vol. 114, No. 2. (2001), pp. 993-1009.

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Dynamics of hierarchical folding on energy landscapes of hexapeptides

Yaakov Levy — 2008-07-04T16:30:21-00:00

The Journal of Chemical Physics, Vol. 115, No. 22. (2001), pp. 10533-10547.

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Effect of Conformational Constraints on the Topography of Complex Potential Energy Surfaces

Yaakov Levy — 2008-07-04T16:15:25-00:00

Physical Review Letters, Vol. 81, No. 5. (1998), 1126.

On mean residence and first passage times in finite one-dimensional systems

Arie Haim — 2008-05-10T10:46:08-00:00

The Journal of Chemical Physics, Vol. 109, No. 13. (1998), pp. 5187-5193.

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Conformational Study of the Alanine Dipeptide at the MP2 and DFT Levels

R Vargas — 2008-05-07T14:37:53-00:00

J. Phys. Chem. A, Vol. 106, No. 13. (4 April 2002), pp. 3213-3218.

Abstract: Conformations of an important model system, the alanine dipeptide, have been calculated by using high-level, ab initio electronic structure theory. A Ramachandran plot, with the angle in the range -180 to 90 and the angle in the range -60 to 180, was generated by using density functional theory with the generalized-gradient BLYP functional and a polarized triple- basis set (TZVP+). Six conformers, C7eq, C5, C7ax, 2, L, and ', have been identified in this region of the Ramachandran plot. A second derivative (frequency) analysis showed that all conformers are stable at this level of theory. These structures were used as starting points for geometry optimizations at the MP2/aug-cc-pVDZ level. Single-point energies were calculated at the MP2/aug-cc-pVTZ and MP2/aug-cc-pVQZ levels at the final MP2/aug-cc-pVDZ structures and together with the MP2/aug-cc-pVDZ results were used in extrapolations to the complete basis set limit. The N-H···O, N-H···N, and C-H···O hydrogen bond interactions that are key to the energetics are discussed. In general, the results obtained at the BLYP/TZVP+, MP2/aug-cc-pVDZ, MP2/aug-cc-pVTZ//aug-cc-pVDZ, and MP2/aug-cc-pVQZ//aug-cc-pVDZ levels are in reasonable agreement with each other, except for the 2 conformation for which there are significant differences in the structures. Although the same stability order is obtained at all levels of theory that were used, there are significant differences in the magnitude of the relative conformational energies.

Some further applications of discrete path sampling to cluster isomerization

David Wales — 2008-05-07T14:33:35-00:00

Molecular Physics, pp. 891-908.

The discrete path sampling approach is applied to analyse the dynamics of several atomic and molecular clusters. Permutational isomerization rates are first calculated for icosahedral atomic clusters containing 13 and 55 atoms. The transformation between decahedral and icosahedral morphologies of a 75-atom cluster is then investigated, for which the potential energy surface has double funnel character. The final system considered is a cluster of twenty water molecules treated using a rigid molecule pair potential. Detailed analysis of the database of stationary points produced by the initial sampling is used to investigate the accuracy of the two-state description in each case. A clear deviation from two-state behaviour occurs for (H2O)20, where low-lying intervening minima exist.

Discrete path sampling

DJ Wales — 2008-05-07T14:31:13-00:00

Molecular Physics, pp. 3285-3305.

A theoretical framework is developed for the calculation of rate constants by sampling connected pathways composed of local minima and transition states that link them together. The theory is applicable to two-state or effective two-state systems and is applied to permutational or morphological isomerization in a two-dimensional cluster of seven Lennard-Jones atoms, water clusters containing eight and nine molecules, and a cluster of 38 Lennard-Jones atoms, which exhibits a double funnel energy landscape.

Annealed importance sampling of dileucine peptide

Edward Lyman — 2008-04-21T09:15:27-00:00

(3 Apr 2007)

Annealed importance sampling is a means to assign equilibrium weights to a nonequilibrium sample that was generated by a simulated annealing protocol. The weights may then be used to calculate equilibrium averages, and also serve as an “adiabatic signature” of the chosen cooling schedule. In this paper we demonstrate the method on the 50-atom dileucine peptide, showing that equilibrium distributions are attained for manageable cooling schedules. For this system, as naively implemented here, the method is modestly more efficient than constant temperature simulation. However, the method is worth considering whenever any simulated heating or cooling is performed (as is often done at the beginning of a simulation project, or during an NMR structure calculation), as it is simple to implement and requires minimal additional CPU expense. Furthermore, the naive implementation presented here can be improved.

Demonstrated convergence of the equilibrium ensemble for a fast united-residue protein model

Marty Ytreberg — 2008-04-21T09:12:42-00:00

(24 Mar 2007)

Due to the time-scale limitations of all-atom simulation of proteins, there has been substantial interest in coarse-grained approaches. Some methods, like "Resolution Exchange," [E. Lyman et al., Phys. Rev. Lett. 96, 028105 (2006)] can accelerate canonical all-atom sampling, but require properly distributed coarse ensembles. We therefore demonstrate that full sampling can indeed be achieved in a sufficiently simplified protein model, as verified by a recently developed convergence analysis. The model accounts for protein backbone geometry in that rigid peptide planes rotate according to atomistically defined dihedral angles, but there are only two degrees of freedom (phi and psi dihedrals) per residue. Our convergence analysis indicates that small proteins (up to 89 residues in our tests) can be simulated for more than 50 "structural decorrelation times" in less than a week on a single processor. We show that the fluctuation behavior is reasonable, as well as discussing applications, limitations, and extensions of the model.

Free energy landscapes of model peptides and proteins

David Evans — 2008-04-11T19:03:52-00:00

The Journal of Chemical Physics, Vol. 118, No. 8. (2003), pp. 3891-3897.

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Network and graph analyses of folding free energy surfaces

Amedeo Caflisch — 2006-02-14T20:14:38-00:00

Current Opinion in Structural Biology, Vol. 16, No. 1. (February 2006), pp. 71-78.

Protein folding is governed by a complex free energy surface whose entropic contributions are relevant because of the large number of degrees of freedom involved. Such complexity, in particular the conformational heterogeneity of the denatured state, is hidden in projections onto one or two order parameters (e.g. fraction of native contacts and/or radius of gyration), which usually results in relatively smooth surfaces. Recent approaches borrowed from network and graph theory have yielded quantitative unprojected representations of the free energy surfaces of a [beta]-hairpin and a three-stranded [beta]-sheet peptide using equilibrium folding-unfolding molecular dynamics simulations. Interestingly, the network and graph analyses of these structured peptides have revealed a very heterogeneous denatured state ensemble. It includes high-enthalpy, high-entropy conformations with fluctuating non-native secondary structure, as well as low-enthalpy, low-entropy traps.

Graph transformation method for calculating waiting times in Markov chains

Semen Trygubenko — 2008-04-11T18:39:51-00:00

The Journal of Chemical Physics, Vol. 124, No. 23. (2006)

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Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states.

F Noé — 2008-01-25T08:41:43-00:00

J Chem Phys, Vol. 126, No. 15. (21 April 2007)

Molecular dynamics simulation generates large quantities of data that must be interpreted using physically meaningful analysis. A common approach is to describe the system dynamics in terms of transitions between coarse partitions of conformational space. In contrast to previous work that partitions the space according to geometric proximity, the authors examine here clustering based on kinetics, merging configurational microstates together so as to identify long-lived, i.e., dynamically metastable, states. As test systems microsecond molecular dynamics simulations of the polyalanines Ala(8) and Ala(12) are analyzed. Both systems clearly exhibit metastability, with some kinetically distinct metastable states being geometrically very similar. Using the backbone torsion rotamer pattern to define the microstates, a definition is obtained of metastable states whose lifetimes considerably exceed the memory associated with interstate dynamics, thus allowing the kinetics to be described by a Markov model. This model is shown to be valid by comparison of its predictions with the kinetics obtained directly from the molecular dynamics simulations. In contrast, clustering based on the hydrogen-bonding pattern fails to identify long-lived metastable states or a reliable Markov model. Finally, an approach is proposed to generate a hierarchical model of networks, each having a different number of metastable states. The model hierarchy yields a qualitative understanding of the multiple time and length scales in the dynamics of biomolecules.

Calculation of the distribution of eigenvalues and eigenvectors in Markovian state models for molecular dynamics

Nina Hinrichs — 2008-03-30T15:17:00-00:00

The Journal of Chemical Physics, Vol. 126, No. 24. (2007)

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Statistically optimal analysis of samples from multiple equilibrium states

Michael Shirts — 2008-02-19T12:59:09-00:00

(9 Jan 2008)

We present a maximum likelihood estimator for computing free energy differences and thermodynamic expectations of physical quantities, as well as their uncertainties, from samples drawn from multiple equilibrium states. The resulting estimator is similar to the weighted histogram analysis method (WHAM), but is derived without the need to invoke histograms. The estimator is asymptotically efficient; in the large sample limit, it is unbiased and has the lowest variance of any estimator previously derived utilizing a given set of sampled data.

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

JD Chodera — 2007-06-04T12:06:07-00:00

J Chem Phys, Vol. 126, No. 15. (21 April 2007)

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.

Thermally activated processes in polymer dynamics

Lorenzo Bongini — 2008-03-23T21:01:14-00:00

Physical Review E, Vol. 68, No. 6. (31 December 2003), 061111.

Jumps between neighboring minima in the energy landscape of both homopolymeric and heteropolymeric chains are numerically investigated by determining the average escape time from different valleys. The numerical results are compared to the theoretical expression derived by Langer [J.S. Langer; Ann. Phys. (N.Y.) 54 ; 258 (1969)] with reference to a 2 N -dimensional space. Our simulations indicate that the dynamics within the native valley is well described by a sequence of thermally activated process up to temperatures well above the folding temperature. At larger temperatures; systematic deviations from the Langer’s estimate are instead observed. Several sources for such discrepancies are thoroughly discussed.

Extended least action principle for steady flows under a prescribed flux

G Wolansky — 2008-03-05T17:29:25-00:00

(12 Jul 2005)

The extended principle of minimal action is described in the presence of prescribed source and sink points. Under the assumption of zero net flux, it leads to an optimal Monge-Kantorovich transport problem of metric type. We concentrate on action corresponding to a mecahnical Lagrangian. The optimal solution turns out to be a measure supprted on a graph composed of geodesic arcs connecting pairs of sources and sinks.

Maximum entropy change and least action principle for nonequilibrium systems

Qiuping Wang — 2006-09-06T18:34:50-00:00

(8 Sep 2004)

A path information is defined in connection with different possible paths of irregular dynamic systems moving in its phase space between two points. On the basis of the assumption that the paths are physically differentiated by their actions, we show that the maximum path information leads to a path probability distribution in exponentials of action. This means that the most probable paths are just the paths of least action. This distribution naturally leads to important laws of normal diffusion. A conclusion of this work is that, for probabilistic mechanics or irregular dynamics, the principle of maximization of path information is equivalent to the least action principle for regular dynamics. <br />We also show that an average path information between the initial phase volume and the final phase volume can be related to the entropy change defined with natural invariant measure of dynamic system. Hence the principles of least action and maximum path information suggest the maximum entropy change. This result is used for some chaotic systems evolving in fractal phase space in order to derive their invariant measures.

Stochastic embedding of dynamical systems

Jacky Cresson — 2008-03-05T17:20:44-00:00

(30 Sep 2005)

Most physical systems are modelled by an ordinary or a partial differential equation, like the n-body problem in celestial mechanics. In some cases, for example when studying the long term behaviour of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modelled or even known. One way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects, that are eventually stochastic. In this paper, we develop a theory for the stochastic embedding of ordinary differential equations. We apply this method to Lagrangian systems. In this particular case, we extend many results of classical mechanics namely, the least action principle, the Euler-Lagrange equations, and Noether's theorem. We also obtain a Hamiltonian formulation for our stochastic Lagrangian systems. Many applications are discussed at the end of the paper.

Maximum path information and the principle of least action for chaotic system

QA Wang — 2008-03-05T17:03:27-00:00

Chaos, Solitons & Fractals, Vol. 23, No. 4. (February 2005), pp. 1253-1258.

A path information is defined in connection with the different possible paths of chaotic system moving in its phase space between two cells. On the basis of the assumption that the paths are differentiated by their actions, we show that the maximum path information leads to a path probability distribution as a function of action from which the well known transition probability of Brownian motion can be easily derived. An interesting result is that the most probable paths are just the paths of least action. This suggests that the principle of least action, in a probabilistic situation, is equivalent to the principle of maximization of information or uncertainty associated with the probability distribution.

Computing communities in large networks using random walks

Matthieu Latapy — 2008-02-29T17:19:53-00:00

(14 Dec 2004)

Dense subgraphs of sparse graphs (communities), which appear in most real-world complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between vertices based on random walks which has several important advantages: it captures well the community structure in a network, it can be computed efficiently, it works at various scales, and it can be used in an agglomerative algorithm to compute efficiently the community structure of a network. We propose such an algorithm which runs in time O(mn^2) and space O(n^2) in the worst case, and in time O(n^2log n) and space O(n^2) in most real-world cases (n and m are respectively the number of vertices and edges in the input graph). Experimental evaluation shows that our algorithm surpasses previously proposed ones concerning the quality of the obtained community structures and that it stands among the best ones concerning the running time. This is very promising because our algorithm can be improved in several ways, which we sketch at the end of the paper.

Network Brownian Motion: A New Method to Measure Vertex-Vertex Proximity and to Identify Communities and Subcommunities

Haijun Zhou — 2008-02-29T17:17:49-00:00

Computational Science - ICCS 2004 (2004), pp. 1062-1069.

The networks considered here consist of sets of interconnected vertices, examples of which include social networks, technological networks, and biological networks. Two important issues are to measure the extent of proximity between vertices and to identify the community structure of a network. In this paper, the proximity index between two nearest-neighboring vertices of a network is measured by a biased Brownian particle which moves on the network. This proximity index integrates both the local and the global structural information of a given network, and it is used by an agglomerative hierarchical algorithm to identify the community structure of the network. This method is applied to several artificial or real-world networks and satisfying results are attained. Finding the proximity indices for all nearest-neighboring vertex pairs needs a computational time that scales as O( N 3), with N being the total number of vertices in the network.

Distance, dissimilarity index, and network community structure

Haijun Zhou — 2008-02-29T17:15:11-00:00

Physical Review E, Vol. 67, No. 6. (10 June 2003), 061901.

We address the question of finding the community structure of a complex network. In an earlier effort [H. Zhou; Phys. Rev. E 67 ; 041908 (2003)]; the concept of network random walking is introduced and a distance measure defined. Here we calculate; based on this distance measure; the dissimilarity index between nearest-neighboring vertices of a network and design an algorithm to partition these vertices into communities that are hierarchically organized. Each community is characterized by an upper and a lower dissimilarity threshold. The algorithm is applied to several artificial and real-world networks; and excellent results are obtained. In the case of artificially generated random modular networks; this method outperforms the algorithm based on the concept of edge betweenness centrality. For yeast’s protein-protein interaction network; we are able to identify many clusters that have well defined biological functions.

Network landscape from a Brownian particle's perspective

Haijun Zhou — 2008-02-29T17:10:20-00:00

Physical Review E, Vol. 67, No. 4. (21 April 2003), 041908.

Given a complex biological or social network; how many clusters should it be decomposed into? We define the distance d i ; j from node i to node j as the average number of steps a Brownian particle takes to reach j from i . Node j is a global attractor of i if d i ; j < ~ d i ; k for any k of the graph; it is a local attractor of i if j ∈ E i (the set of nearest neighbors of i ) and d i ; j < ~ d i ; l for any l ∈ E i . Based on the intuition that each node should have a high probability to be in the same community as its global (local) attractor on the global (local) scale; we present a simple method to uncover a network’s community structure. This method is applied to several real networks and some discussion on its possible extensions is made.

Comparing community structure identification

Leon Danon — 2005-10-20T00:27:23-00:00

(18 Oct 2005)

We compare recent approaches to community structure identification in terms of sensitivity and computational cost. The recently proposed modularity measure is revisited and the performance of the methods as applied to ad hoc networks with known community structure, is compared. We find that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes. The work is intended as an introduction as well as a proposal for a standard benchmark test of community detection methods.

Detecting Fuzzy Community Structures in Complex Networks with a Potts Model

Jörg Reichardt — 2007-09-03T05:02:40-00:00

Physical Review Letters, Vol. 93, No. 21. (2004)

A fast community detection algorithm based on a q-state Potts model is presented. Communities (groups of densely interconnected nodes that are only loosely connected to the rest of the network) are found to coincide with the domains of equal spin value in the minima of a modified Potts spin glass Hamiltonian. Comparing global and local minima of the Hamiltonian allows for the detection of overlapping ("fuzzy") communities and quantifying the association of nodes with multiple communities as well as the robustness of a community. No prior knowledge of the number of communities has to be assumed.

From the Cover: Modularity and community structure in networks

MEJ Newman — 2006-06-07T05:40:22-00:00

PNAS, Vol. 103, No. 23. (6 June 2006), pp. 8577-8582.

Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as "modularity" over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets. 10.1073/pnas.0601602103

Finding and evaluating community structure in networks

MEJ Newman — 2007-01-15T21:33:05-00:00

Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 69, No. 2. (2004)

We propose and study a set of algorithms for discovering community structure in networks—natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using any one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems.

Community structure in social and biological networks.

M Girvan — 2005-01-21T16:03:08-00:00

Proc Natl Acad Sci U S A, Vol. 99, No. 12. (11 June 2002), pp. 7821-7826.

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known--a collaboration network and a food web--and find that it detects significant and informative community divisions in both cases.

Energy landscapes and properties of biomolecules

David Wales — 2005-11-10T03:09:17-00:00

Physical Biology, Vol. 2, No. 4. (December 2005), S86.

Importance of Metastable States in the Free Energy Landscapes of Polypeptide Chains

Stefan Auer — 2008-02-19T10:52:56-00:00

Physical Review Letters, Vol. 99, No. 17. (2007)

We show that the interplay between excluded volume effects, hydrophobicity, and hydrogen bonding in a tubelike representation of a polypeptide chain gives rise to free energy landscapes that, in addition to a clear global minimum, are characterized by the general presence of a small number of metastable minima, which correspond to common structural motifs observed in proteins. The complexity of the landscape increases only moderately with the length of the chain. Analysis of the temperature dependence of these landscapes reveals that the stability of specific metastable states is maximal at a temperature close to the midpoint of folding. These mestastable states are therefore likely to be of particular significance in determining the generic tendency of proteins to aggregate into potentially pathogenic agents.

Free energy disconnectivity graphs: Application to peptide models

Sergei Krivov — 2008-02-19T10:28:22-00:00

The Journal of Chemical Physics, Vol. 117, No. 23. (2002), pp. 10894-10903.

Potential Energy Surfaces and Conformational Transitions in Biomolecules: A Successive Confinement Approach Applied to a Solvated Tetrapeptide

Sergei Krivov — 2008-02-19T10:26:24-00:00

Physical Review Letters, Vol. 88, No. 3. (2 January 2002), 038101.

A simple approach for the efficient exploration of the potential energy surface of a many-body system is presented. The method uses Langevin dynamics trajectories that are successively confined in the various basins of the potential energy surface. The approach is illustrated by determining the potential energy surface; and the thermodynamic and kinetic properties of a solvated model for the alanine tetrapeptide; the shortest peptide that can form an α-helical turn. All possible cis isomers are sampled; even though the barriers separating them are as high as 25 kcal/mole. Comparisons with conventional Langevin dynamics confirm the greater efficacy of the approach.

One-Dimensional Free-Energy Profiles of Complex Systems: Progress Variables that Preserve the Barriers.

SV Krivov — 2006-07-04T02:11:48-00:00

J Phys Chem B Condens Matter Mater Surf Interfaces Biophys, Vol. 110, No. 25. (29 June 2006), pp. 12689-12698.

We show that the balanced minimum-cut procedure introduced in PNAS 2004, 101, 14766 can be reinterpreted as a method for solving the constrained optimization problem of finding the minimum cut among the cuts with a particular value of an additive function of the nodes on either side of the cut. Such an additive function (e.g., the partition function of the reactant region) can be used as a progress coordinate to determine a one-dimensional profile (FEP) of the free-energy surface of the protein-folding reaction as well as other complex reactions. The algorithm is based on the network (obtained from an equilibrium molecular dynamics simulation) that represents the calculated reaction behavior. The resulting FEP gives the exact values of the free energy as a function of the progress coordinate; i.e., at each value of the progress coordinate, the profile is obtained from the surface with the minimal partition function among the surfaces that divide the full free-energy surface between two chosen end points. In many cases, the balanced minimum-cut procedure gives results for only a limited set of points. An approximate method based on p(fold) is shown to provide the profile for a more complete set of values of the progress coordinate. Applications of the approach to model problems and to realistic systems (beta-hairpin of protein G, LJ(38) cluster) are presented.

Hidden complexity of free energy surfaces for peptide (protein) folding.

SV Krivov — 2006-02-16T20:20:53-00:00

Proc Natl Acad Sci U S A, Vol. 101, No. 41. (12 October 2004), pp. 14766-14770.

An understanding of the thermodynamics and kinetics of protein folding requires a knowledge of the free energy surface governing the motion of the polypeptide chain. Because of the many degrees of freedom involved, surfaces projected on only one or two progress variables are generally used in descriptions of the folding reaction. Such projections result in relatively smooth surfaces, but they could mask the complexity of the unprojected surface. Here we introduce an approach to determine the actual (unprojected) free energy surface and apply it to the second beta-hairpin of protein G, which has been used as a model system for protein folding. The surface is represented by a disconnectivity graph calculated from a long equilibrium folding-unfolding trajectory. The denatured state is found to have multiple low free energy basins. Nevertheless, the peptide shows exponential kinetics in folding to the native basin. Projected surfaces obtained from the present analysis have a simple form in agreement with other studies of the beta-hairpin. The hidden complexity found for the beta-hairpin surface suggests that the standard funnel picture of protein folding should be revisited.

Exploration of the free-energy surface of a three-helix peptide with stochastic optimization methods

T Herges — 2008-02-19T10:19:01-00:00

International Journal of Quantum Chemistry, Vol. 99, No. 5. (2004), pp. 854-863.

We report our recent efforts to develop all-atom forcefields that can be used to predict the three-dimensional, tertiary structure of proteins using stochastic optimization methods. We have analyzed an approximate free-energy surface of the 36-residue headpiece of the villin protein and found configurations that were lower in energy than the NMR configuration. We adjusted the parameters of the solvent model to stabilize the NMR structure using a decoy approach. We arrived at a free-energy surface that is characterized by a deep folding funnel populated by different three-helix structures, one of which is very similar to the NMR structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

From A to B in free energy space

Davide Branduardi — 2007-02-08T12:56:34-00:00

The Journal of Chemical Physics, Vol. 126, No. 5. (2007)

The authors present a new method for searching low free energy paths in complex molecular systems at finite temperature. They introduce two variables that are able to describe the position of a point in configurational space relative to a preassigned path. With the help of these two variables the authors combine features of approaches such as metadynamics or umbrella sampling with those of path based methods. This allows global searches in the space of paths to be performed and a new variational principle for the determination of low free energy paths to be established. Contrary to metadynamics or umbrella sampling the path can be described by an arbitrary large number of variables, still the energy profile along the path can be calculated. The authors exemplify the method numerically by studying the conformational changes of alanine dipeptide. ©2007 American Institute of Physics

The Meaning of Component Analysis: Decomposition of the Free Energy in Terms of Specific Interactions

S Boresch — 2008-02-19T10:11:29-00:00

Journal of Molecular Biology, Vol. 254, No. 5. (15 December 1995), pp. 801-807.

Free energy simulations are of particular interest for the interpretation of macroscopic data in terms of microscopic interactions. This can be done by expressing calculated free energies as a sum of components that correspond to the contributions of different energy terms or different parts of the system. Since the resulting components depend on the integration path, care is required in their use. We show that a linear coupling scheme for the alchemical creation of a chemical identity corresponds to a particularly useful path because it leads to a symmetric decoupling of the free energy components. The path dependence also provides an additional degree of freedom that can be used to study different processes. This latter point is illustrated by a reinterpretation of a recent simulation on wild-type and mutant azurin by Mark and van Gunsteren.

Free-energy landscape of the villin headpiece in an all-atom force field.

T Herges — 2005-09-08T14:22:15-00:00

Structure (Camb), Vol. 13, No. 4. (April 2005), pp. 661-668.

We investigate the landscape of the internal free-energy of the 36 amino acid villin headpiece with a modified basin hopping method in the all-atom force field PFF01, which was previously used to predictively fold several helical proteins with atomic resolution. We identify near native conformations of the protein as the global optimum of the force field. More than half of the twenty best simulations started from random initial conditions converge to the folding funnel of the native conformation, but several competing low-energy metastable conformations were observed. From 76,000 independently generated conformations we derived a decoy tree which illustrates the topological structure of the entire low-energy part of the free-energy landscape and characterizes the ensemble of metastable conformations. These emerge as similar in secondary content, but differ in tertiary arrangement.

Escaping free-energy minima.

A Laio — 2005-08-03T11:22:20-00:00

Proc Natl Acad Sci U S A, Vol. 99, No. 20. (1 October 2002), pp. 12562-12566.

We introduce a powerful method for exploring the properties of the multidimensional free energy surfaces (FESs) of complex many-body systems by means of coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates. A characteristic feature of these dynamics is the presence of a history-dependent potential term that, in time, fills the minima in the FES, allowing the efficient exploration and accurate determination of the FES as a function of the collective coordinates. We demonstrate the usefulness of this approach in the case of the dissociation of a NaCl molecule in water and in the study of the conformational changes of a dialanine in solution.

Design of amino acid sequences to fold into C[sub alpha]-model proteins

A Amatori — 2008-02-18T18:48:06-00:00

The Journal of Chemical Physics, Vol. 123, No. 5. (2005)

Low Frequency Motion in Proteins

P Dauber-Osguthorpe — 2008-02-18T18:43:16-00:00

Journal of Computational Physics, Vol. 151 (May 1999), pp. 169-189.

The motion of a chymotrypsin-like serine protease,SGPA, has been studied by torsion space normal mode analysis and by Cartesian space molecular dynamics, and the results have been compared. The molecular dynamics trajectory was analyzed using digital signal processing techniques to provide a set of characteristic modes that can be compared directly with the normal modes. The results were also compared with the motion implied by the crystallographic temperature factors. We find that in spite of the radically different approximations used in the two methods, agreement between the resulting motions and with the experimental data is surprisingly high. We conclude that this agreement probably reflects an underlying robustness in the motion, dictated primarily by van der Waals packing. In contrast to other proteins, there are no large amplitude inter-domain motions. Rather, the low frequency, high amplitude motions are concentrated in three surface hairpin loops. The movement of one these loops, the specificity loop, appears to facilitate substrate binding.

Vibrational Dynamics of Folded Proteins: Significance of Slow and Fast Motions in Relation to Function and Stability

Ivet Bahar — 2006-11-07T01:05:25-00:00

Physical Review Letters, Vol. 80, No. 12. (23 March 1998), 2733.

A single-parameter harmonic Hamiltonian based on local packing density and contact topology is proposed for studying residue fluctuations in native proteins. The internal energy obeys an equipartition law; and free energy changes result from entropy fluctuations only. Frequencyâwave-number maps show communication between residues involved in slow and fast modes. Fast modes are strongly localized; resulting from the geometric irregularity of the structure. Comparison with experiments shows that slow and fast modes are associated; respectively; with function and stability. Specifically; domain motions and folding cores of HIV-1 protease are accurately identified.

Important fluctuation dynamics of large protein structures are preserved upon coarse-grained renormalization

Pemra Doruker — 2008-02-18T18:38:28-00:00

International Journal of Quantum Chemistry, Vol. 90, No. 2. (2002), pp. 822-837.

The fluctuations and important motions of three large proteins - hemaglutinin, xanthine dehydrogenase, and ?-galactosidase - have been considered with a range of models having various levels of detail to represent the structures. Because the slowest modes of motion are the largest contributors to the total motions, and because these motions depend mainly on the shapes of the structures rather than their details, it is possible to replace the real structures with significantly fewer points and still retain the essential features of the structure for these modes of motion. We obtain excellent results, both for the magnitudes of the individual motions as well as for the molecular changes occurring during these motions. Similar results are obtained with another completely different approach where the coarse graining is based on invariant regions of structure found by comparing two structures of the same protein, given as an example here for myosin. Results confirm the important coupling of local functional motions with the large-scale motions, implying important functional roles for the entire protein structure. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Large Amplitude Elastic Motions in Proteins from a Single-Parameter, Atomic Analysis

Monique Tirion — 2006-07-19T15:10:13-00:00

Physical Review Letters, Vol. 77, No. 9. (1996), 1905.

Normal mode analysis (NMA) is a leading method for studying long-time dynamics and elasticity of biomolecules. The method proceeds from complex semiempirical potentials characterizing the covalent and noncovalent interactions between atoms. It is widely accepted that such detailed potentials are essential to the success of NMA's. We show that a single-parameter potential is sufficient to reproduce the slow dynamics in good detail. Costly and inaccurate energy minimizations are eliminated; permitting direct analysis of crystal coordinates. The technique can be used for new applications; such as mapping of one crystal form to another by means of slow modes; and studying anomalous dynamics of large proteins and complexes.

Comparison of mode analyses at different resolutions applied to nucleic acid systems.

AW Van Wynsberghe — 2006-01-06T14:31:17-00:00

Biophys J, Vol. 89, No. 5. (November 2005), pp. 2939-2949.

More than two decades of different types of mode analyses has shown that these techniques can be useful in describing large-scale motions in protein systems. A number of mode analyses are available and include quasiharmonics, classical normal mode, block normal mode, and the elastic network model. Each of these methods has been validated for protein systems and this variety allows researchers to choose the technique that gives the best compromise between computational cost and the level of detail in the calculation. These same techniques have not been systematically tested for nucleic acid systems, however. Given the differences in interactions and structural features between nucleic acid and protein systems, the validity of these techniques in the protein regime cannot be directly translated into validity in the nucleic acid realm. In this work, we investigate the usefulness of the above mode analyses as applied to two RNA systems, i.e., the hammerhead ribozyme and a guanine riboswitch. We show that classical normal-mode analysis can match the magnitude and direction of residue fluctuations from the more detailed, anharmonic technique, quasiharmonic analysis of a molecular dynamics trajectory. The block normal-mode approximation is shown to hold in the nucleic acid systems studied. Only the mode analysis at the lowest level of detail, the elastic network model, produced mixed results in our calculations. We present data that suggest that the elastic network model, with the popular parameterization, is not best suited for systems that do not have a close packed structure; this observation also hints at why the elastic network model has been found to be valid for many globular protein systems. The different behaviors of block normal-mode analysis and the elastic network model, which invoke similar degrees of coarse-graining to the dynamics but use different potentials, suggest the importance of applying a heterogeneous potential function in a robust analysis of the dynamics of biomolecules, especially those that are not closely packed. In addition to these comparisons, we briefly discuss insights into the conformational space available to the hammerhead ribozyme.

Harmonic and anharmonic aspects in the dynamics of BPTI: A normal mode analysis and principal component analysis

S Hayward — 2008-02-18T18:14:55-00:00

Protein Sci, Vol. 3, No. 6. (1 June 1994), pp. 936-943.

Global ribosome motions revealed with elastic network model

Yongmei Wang — 2005-08-10T08:37:46-00:00

Journal of Structural Biology, Vol. 147, No. 3. (September 2004), pp. 302-314.

The motions of large systems such as the ribosome are not fully accessible with conventional molecular simulations. A coarse-grained, less-than-atomic-detail model such as the anisotropic network model (ANM) is a convenient informative tool to study the cooperative motions of the ribosome. The motions of the small 30S subunit, the larger 50S subunit, and the entire 70S assembly of the two subunits have been analyzed using ANM. The lowest frequency collective modes predicted by ANM show that the 50S subunit and 30S subunit are strongly anti-correlated in the motion of the 70S assembly. A ratchet-like motion is observed that corresponds well to the experimentally reported ratchet motion. Other slow modes are also examined because of their potential links to the translocation steps in the ribosome. We identify several modes that may facilitate the E-tRNA exiting from the assembly. The A-site t-RNA and P-site t-RNA are found to be strongly coupled and positively correlated in these slow modes, suggesting that the translocations of these two t-RNAs occur simultaneously, while the motions of the E-site t-RNA are less correlated, and thus less likely to occur simultaneously. Overall the t-RNAs exhibit relatively large deformations. Animations of these slow modes of motion can be viewed at http://ribosome.bb.iastate.edu/70SnKmode.