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Simulations of protein folding and protein association happen on timescales that are orders of magnitude larger than what can typically be covered in all-atom molecular dynamics simulations. Use of low-resolution models alleviates this problem but may reduce the accuracy of the simulations. We introduce a replica-exchange-based multiscale sampling technique that combines the faster sampling in coarse-grained simulations with the potentially higher accuracy of all-atom simulations. After testing the efficiency of our Resolution Exchange with Tunneling (ResET) in simulations of the Trp-cage protein, an often used model to evaluate sampling techniques in protein simulations, we use our approach to compare the landscape of wild type and A2T mutant Abeta 1-42 peptides. Our results suggest a mechanism by that the mutation of a small hydrophobic Alanine (A) into a bulky polar Threonine (T) may interfere with the self-assembly of Abeta - fibrils.

Collapse of the polypeptide backbone is an integral part of protein folding. Using polyglycine as a probe, we explore the nonequilibrium pathways of protein collapse in water. We find that the collapse depends on the competition between hydration effects and intra-peptide interactions. Once intra-peptide van der Waal interactions dominate, the chain collapses along a nonequilibrium pathway characterized by formation of pearl-necklace-like local clusters as intermediates that eventually coagulate into a single globule. By describing this coarsening through the contact probability as a function of distance along the chain, we extract a time-dependent length scale that grows in linear fashion. The collapse dynamics is characterized by a dynamical critical exponent $z=0.5$ that is much smaller than the values of $z=1-2$ reported for non-biological polymers. This difference in the exponents is explained by the instantaneous formation of intra-chain hydrogen bonds and local ordering that may be correlated with the observed fast folding times in proteins., Comment: 14 pages, 4 figures

Using a variant of parallel tempering, we study the changes in sampling within a simulation when the all-atom model is coupled to a Go-like potential. We find that the native structure is not the lowest-energy configuration in the all-atom force field. Adding a Go-term deforms the energy landscape in a way that the native configuration becomes the global minimum., Comment: 9 pages, 8 figures

We implement a well-established concept to consider dispersion effects within a Poisson-Boltzmann approach of continuum solvation of proteins. The theoretical framework is particularly suited for boundary element methods. Free parameters are determined by comparison to experimental data as well as high-level quantum mechanical reference calculations. The method is general and can be easily extended in several directions. The model is tested on various chemical substances and found to yield good-quality estimates of the solvation free energy without obvious indication of any introduced bias. Once optimized, the model is applied to a series of proteins, and factors such as protein size or partial charge assignments are studied.

We study the dynamics of parallel tempering simulations, also known as the replica exchange technique, which has become the method of choice for simulation of proteins and other complex systems. Recent results for the optimal choice of the control parameter discretization allow a treatment independent of the system in question. Analyzing mean first passage times across control parameter space, we find an expression for the optimal number of replicas in simulations covering a given temperature range. Our results suggest a particular protocol to optimize the number of replicas in actual simulations., Comment: 4 pages, 2 figures, submitted

In this short note we sketch the statistical physics framework of the replica exchange technique when applied to molecular dynamics simulations. In particular, we draw attention to generalized move sets that allow a variety of optimizations as well as new applications of the method., Comment: 4 pages, 3 figures; revised version (1 figure added), PRE in press

The authors study the short-time dynamics of helix-forming polypeptide chains using an all-atom representation of the molecules and an implicit solvation model to approximate the interaction with the surrounding solvent. The results confirm earlier observations that the helix-coil transition in proteins can be described by a set of critical exponents. The high statistics of the simulations allows the authors to determine the exponents values with increased precision and support universality of the helix-coil transition in homopolymers and (helical) proteins., Comment: 11 pages, 7 figures

In order to study the relation between backbone and side chain ordering in proteins, we have performed multicanonical simulations of deka-peptide chains with various side groups. Glu10, Gln10, Asp10, Asn10, and Lys10 were selected to cover a wide variety of possible interactions between the side chains of the monomers. All homopolymers undergo helix-coil transitions. We found that peptides with long side chains that are capable of hydrogen bonding, i.e. Glu10, and Gln10, exhibit a second transition at lower temperatures connected with side chain ordering. This occurs in gas phase as well as in solvent, although the character of the side chain structure is different in each case. However, in polymers with short side chains capable of hydrogen bonding, i.e. Asp10 and Asn10, side chain ordering takes place over a wide temperature range and exhibits no phase transition like character. Moreover, non-backbone hydrogen bonds show enhanced formation and fluctuations already at the helix-coil transition temperature, indicating competition between side chain and backbone hydrogen bond formation. Again, these results are qualitatively independent of the environment. Side chain ordering in Lys10, whose side groups are long and polar, also takes place over a wide temperature range and exhibits no phase transition like character in both environments. Reasons for the observed chain length threshold and consequences from these results for protein folding are discussed., Comment: 12 pages,11 figures

From the underlying Master equations we derive one-dimensional stochastic processes that describe generalized ensemble simulations as well as tempering (simulated and parallel) simulations. The representations obtained are either in the form of a one-dimensional Fokker-Planck equation or a hopping process on a one-dimensional chain. In particular, we discuss the conditions under which these representations are valid approximate Markovian descriptions of the random walk in order parameter or control parameter space. They allow a unified discussion of the stationary distribution on, as well as of the stationary flow across each space. We demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space, and discuss the consequences of our results for optimizing simulations. Finally, we point out the limitations of these representations under conditions of broken ergodicity., Comment: 11 pages Latex, 2 eps figures, revised version, typos corrected, PRE in press

We apply a recently developed adaptive algorithm that systematically improves the efficiency of parallel tempering or replica exchange methods in the numerical simulation of small proteins. Feedback iterations allow us to identify an optimal set of temperatures/replicas which are found to concentrate at the bottlenecks of the simulations. A measure of convergence for the equilibration of the parallel tempering algorithm is discussed. We test our algorithm by simulating the 36-residue villin headpiece sub-domain HP-36 wherewe find a lowest-energy configuration with a root-mean-square-deviation of less than 4 Angstroem to the experimentally determined structure., Comment: 22 pages, 7 figures

A generalized-ensemble technique, multicanonical sampling, is used to study the folding of a 34-residue human parathyroid hormone fragment. An all-atom model of the peptide is employed and the protein-solvent interactions are approximated by an implicit solvent. Our results demonstrate that generalized-ensemble simulations are well suited to sample low-energy structures of such large polypeptides. Configurations with a root-mean-square deviation (rmsd) to the crystal structure of less than one \AA are found. Finally, we discuss limitations of our implicit solvent model., Comment: To appear in J. Chem. Phys

Segments with the amino acid sequence EKAYLRT appear in natural occurring proteins both in $\alpha$-helices and $\beta$-sheets. For this reason, we have use this peptide to study how secondary structure formation in proteins depends on the local environment. Our data rely on multicanonical Monte Carlo simulations where the interactions among all atoms are taken into account. Results in gas phase are compared with that in an implicit solvent. We find that both in gas phase and solvated EKAYLRT forms an $\alpha$-helix when not interacting with other molecules. However, in the vicinity of a $\beta$-strand, the peptide forms a $\beta$-strand. Because of this change in secondary structure our peptide may provide a simple model for the $\alpha \to \beta$ transition that is supposedly related to the outbreak of Prion diseases and similar illnesses., Comment: to appear in Physical Review E

We study helix-coil transitions in an all-atom model of polyalanine. Molecules of up to length 30 residues are investigated by multicanonical simulations. Results from two implicit solvent models are compared with each other and with that from simulations in gas phase. While the helix-coil transition is in all three models a true thermodynamic phase transition, we find that its strength is reduced by the protein-solvent interaction term. The order of the helix-coil transition depends on the details of the solvation term., Comment: 19 pages, 11 figures

We study the relation between $\alpha$-helix formation and folding for a simple artificial peptide, Ala$_{10}$-Gly$_5$-Ala$_{10}$. Our data rely on multicanonical Monte Carlo simulations where the interactions among all atoms are taken into account. The free-energy landscape of the peptide is evaluated for various temperatures. Our data indicate that folding of this peptide is a two-step process: in a first step two $\alpha$-helices are formed which afterwards re-arrange themselves into a U-like structure., Comment: 15 pages, with 9 eps figures

We compare two recently proposed methods for the characterization of phase transitions in small systems. The validity and usefulness of these approaches are studied for the case of the q=4 and q=5 Potts model, i.e. systems where a thermodynamic limit and exact results exist. Guided by this analysis we discuss then the helix-coil transition in polyalanine, an example of structural transitions in biological molecules., Comment: 16 pages and 7 figures

A simple spin system is studied as an analog for proteins. We investigate how the introduction of randomness and frustration into the system effects the designability and stability of ground state configurations. We observe that the spin system exhibits protein-like behavior in the vicinity of the transition between ferromagnet and spin glass. Our results illuminate some guiding principles in protein evolution., Comment: 12 pages, 4 figures

We study the thermodynamics and kinetics of folding for a small peptide. Our data rely on Monte Carlo simulations where the interactions among all atoms are taken into account. Monte Carlo kinetics is used to study folding of the peptide at suitable temperatures. The results of these canonical simulations are compared with that of a generalized-ensemble simulation. Our work demonstrates that concepts of folding which were developed in the past for minimalist models hold also for this peptide when simulated with an all-atom force field.

We calculate the Yang-Lee zeros for characteristic temperatures of the helix-coil transition in a continuum model of polyalanine. The distribution of these zeros differs from predictions of the Zimm-Bragg theory and supports recent claims that polyalanine exhibits a true phase transition. New estimates for critical exponents are presented and the relation of our results to the Lee-Yang theorem is discussed., Comment: 15 pages and 5 figures

The density of states for the three-dimensional Ising model is calculated with high-precision from multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L=32. Combining previous statistics for smaller lattice sizes, we have evaluated the correction to scaling and the critical exponent $\nu$ through out an analysis of a multi-parameter fit and of Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known., Comment: 22 pages, Latex, package graphicx, with 5 eps figures

The similarity in the thermodynamic properties of two completely different theoretical models for the helix-coil transition is examined critically. The first model is an all-atomic representation for a poly-alanine chain, while the second model is a minimal helix-forming model that contains no system specifics. Key characteristics of the helix-coil transition, in particular, the effective critical exponents of these two models agree with each other, within a finite-size scaling analysis., Comment: Latex, to appear in Eur. Phys. J. B

We report on multicanonical simulations of the helix-coil transition of a polypeptide. The nature of this transition was studied by calculating partition function zeros and the finite-size scaling of various quantities. Estimates for critical exponents are presented., Comment: RevTex, 4 eps-files; to appear in Phys. Rev. Let

A modified version of stochastic tunneling, a recently introduced global optimization technique, is introduced as a new generalized-ensemble technique and tested for a benchmark peptide, Met-enkephalin. It is demonstrated that the new technique allows to evaluate folding properties of this peptide, Comment: Latex, 4 figures (ps-files)

The free-energy landscape of two peptides is evaluated at various temperatures and an estimate for its fractal dimension at these temperatures calculated. We show that monitoring this quantity as a function of temperature allows to determine the glass transition temperature., Comment: 10 pages, Latex, package graphicx, with four eps figures. Submitted to Int. J. Mod. Phys. C

We develop a formulation for molecular dynamics, Langevin, and hybrid Monte Carlo algorithms in the recently proposed generalized ensemble that is based on a physically motivated realisation of Tsallis weights. The effectiveness of the methods are tested with an energy function for a protein system. Simulations in this generalized ensemble by the three methods are performed for a penta peptide, Met-enkephalin. For each algorithm, it is shown that from only one simulation run one can not only find the global-minimum-energy conformation but also obtain probability distributions in canonical ensemble at any temperature, which allows the calculation of any thermodynamic quantity as a function of temperature., Comment: to appear in Chem. Phy. Lett

We perform a generalized-ensemble simulation of a small peptide taking the interactions among all atoms into account. From this simulation we obtain thermodynamic quantities over a wide range of temperatures. In particular, we show that the folding of a small peptide is a multi-stage process associated with two characteristic temperatures, the collapse temperature T_{\theta} and the folding temperature T_f. Our results give supporting evidence for the energy landscape picture and funnel concept. These ideas were previously developed in the context of studies of simplified protein models, and here for the first time checked in an all-atom Monte Carlo simulation., Comment: Latex, 6 Figures

We have performed multicanonical Monte Carlo simulations of C-peptide of ribonuclease A. It is known by CD and NMR experiments that this peptide has high alpha-helix content in aqueous solution and that the side-chain charges of residues Glu-2$^-$ and His-12$^+$ play an important role in the stability of the alpha-helix. In order to confirm these experimental implications, we have used two analogues of the peptide with charged and neutral side chains of Glu-2 and His-12. Two dielectric functions, distance-dependent and constant, are considered to study the effects of solvent contributions. All the simulations were started from random initial conformations. Various thermodynamic quantities such as average helicity as a function of residue number and average distance between two side chains as a function of temperature are calculated. The results are found to be in accord with the implications of CD and NMR experiments. The lowest-energy conformation obtained has an alpha-helix from Ala-4 to Gln-11 in complete agreement with the corresponding structure deduced from an X-ray crystallography experiment of ribonuclease A. It is shown that the salt bridge between the side chains of Glu-2$^-$ and Arg-10$^+$, which is known to exist from both NMR and X-ray experiments, is formed only when the side chains are properly charged. Its formation is greatly enhanced when the distance-dependent dielectric function is used., Comment: Latex, 3 Figures

Applied to statistical physics models, the random cost algorithm enforces a Random Walk (RW) in energy (or possibly other thermodynamic quantities). The dynamics of this procedure is distinct from fixed weight updates. The probability for a configuration to be sampled depends on a number of unusual quantities, which are explained in this paper. This has been overlooked in recent literature, where the method is advertised for the calculation of canonical expectation values. We illustrate these points for the $2d$ Ising model. In addition, we proof a previously conjectured equation which relates microcanonical expectation values to the spectral density., Comment: Various minor changes, appendix added, Fig. 2 dropped

We present a novel Monte Carlo algorithm which enhances equilibrization of low-temperature simulations and allows sampling of configurations over a large range of energies. The method is based on a non-Boltzmann probability weight factor and is another version of the so-called generalized-ensemble techniques. The effectiveness of the new approach is demonstrated for the system of a small peptide, an example of the frustrated system with a rugged energy landscape., Comment: Latex; ps-files included

The effectiveness of a new algorithm, parallel tempering, is studied for numerical simulations of biological molecules. These molecules suffer from a rough energy landscape. The resulting slowing down in numerical simulations is overcome by the new method. This is demonstrated by performing simulations with high statistics for one of the simplest peptides, Met-enkephalin. The numerical effectiveness of the new technique was found to be much better than traditional methods and is comparable to sophisticated methods like generalized ensemble techniques., Comment: Latex, ps-files included; to appear in Chem. Phys. Lett

We describe a simple ansatz to approximate the low temperature behavior of proteins and peptides by a mean-field-like model which is analytically solvable. For a small peptide some thermodynamic quantities are calculated and compared with numerical results of an all-atoms simulation. Our approach can be used to determine the weights for a multicanonical simulation of the molecule under consideration., Comment: 11 pages, Latex, 4 Postscript figures, to appear in Int. J. Mod. Phys. C (1997)

We compare numerically the effectiveness of three recently proposed algorithms, multicanonical simulations, simulations in a 1/k-ensemble, and simulated tempering for the protein folding problem. For this we perform simulations with high statistics for one of the simplest peptides, Met-enkephalin. While the performances of all three approaches is much better than traditional methods, we find that the differences among the three are only marginal., Comment: Latex, no figures include, for reprints send e-mail to authors

The multicanonical ansatz is used to study variations in the energy landscape of a small peptide, Met-enkephalin, under a change from the ECEPP/2 force field to ECEPP/3. Local minima with energies up to 5 kcal/mol higher than the global minima are sampled and classified according to H-bridges and backbone angles. The distribution and relative weight for various temperatures of the minima are calculated and compared for the two force fields. We demonstrate that while there are small differences in the energy landscape our results at relevant temperatures are robust under changes between ECEPP/2 to ECEPP/3., Comment: Latex, no figures included. For reprints send e-mail to authors

We demonstrate that the multicanonical approach is not restricted to Monte Carlo simulations, but can also be applied to simulation techniques such as molecular dynamics, Langevin, and hybrid Monte Carlo algorithms. The effectiveness of the methods are tested with an energy function for the protein folding problem. Simulations in the multicanonical ensemble by the three methods are performed for a penta peptide, Met-enkephalin. For each algorithm, it is shown that from only one simulation run one can not only find the global-minimum-energy conformation but also obtain probability distributions in canonical ensemble at any temperature, which allows the calculation of any thermodynamic quantity as a function of temperature., Comment: Latex, no figures included, for reprints send e-mail to authors

We discuss the use of Tsallis generalized mechanics in simulated annealing algorithms. For a small peptide it is shown that older implementations are not more effective than regular simulated annealing in finding ground state configurations. We propose a new implementation which leads to an improvement over regular simulated annealing., Comment: Latex

A general numerical method is presented to locate the partition function zeros in the complex beta plane for large lattice sizes. We apply this method to the 2D Ising model and results are reported for square lattice sizes up tp L=64. We also propose an alternative method to evaluate corrections to scaling which relies only on the leading zeros. This method is illustrated with our data., Comment: 9 pages, Latex, 3 figures. To appear in Int. J. Mod. Phys. C

Thermodynamics of helix-coil transitions in amino-acid homo-oligomers are studied by the recently proposed multicanonical algorithms. Homo-oligomers of length 10 are considered for three characteristic amino acids, alanine (helix former), valine (helix indifferent), and glycine (helix breaker). For alanine other lengths (15 and 20) are also considered in order to examine the length dependence. From one multicanonical production run with completely random initial conformations, we have obtained the lowest-energy conformations and various thermodynamic quantities (average helicity, Zimm-Bragg $s$ and $\sigma$ parameters, free energy differences between helix and coil states, etc.) as functions of temperature. The results confirm the fact that alanine is helix-forming, valine is helix-indifferent, and glycine is helix-breaking., Comment: 35 pages, no figures included; to appear in J.Phys.Chem

We comment on a recent article by Hao and Scheraga., Comment: TEX 3 pages

We compare a few variants of the recently proposed multicanonical method with the well known simulated annealing for the effectiveness in search of the energy global minimum of a biomolecular system. For this we study in detail Met-enkephalin, one of the simplest peptides. We show that the new method not only outperforms simulated annealing in the search of the energy groundstate but also provides more statistical-mechanical information about the system., Comment: to be published in Physica A, LATEX 32 pages, figures available on request

We study zero--temperature properties of the 3d Edwards--Anderson Ising spin glass on finite lattices up to size $12^3$. Using multicanonical sampling we generate large numbers of groundstate configurations in thermal equilibrium. Finite size scaling with a zero--temperature scaling exponent $y = 0.74 \pm 0.12$ describes the data well. Alternatively, a descriptions in terms of Parisi mean field behaviour is still possible. The two scenarios give significantly different predictions on lattices of size $\ge 12^3$., Comment: LATEX 9pages,figures upon request ,SCRI-93

We studied the performance of the multicanonical algorithm by simulating the van Hemmen spin glass model and reproduced the exact results for this mean field model. Physical quantities such as energy density, specific heat, susceptibility and order parameters are evaluated at all temperatures., Comment: sorry: no figures included, 5 pages, FSU-SCRI-93-32

We test the effectiveness of the multicanonical algorithm for the tertiary structure prediction of peptides and proteins. As a simple example we study Met-enkephalin. The lowest-energy conformation obtained agrees with that determined by other methods such as Monte Carlo simulated annealing. But unlike to simulated annealing the relationship to the canonical ensemble remains exactly controlled. Thermodynamic quantities at various temperature can be calculated from one run., Comment: 5 pages, Latex file with attached postscript files, FSU-SCRI-93C-34

We performed numerical simulations of 2D and 3D Edwards-Anderson spin glass models by using the recently developed multicanonical ensemble. Our ergodicity times increase with the lattice size approximately as $V^3$. The energy, entropy and other physical quantities are easily calculable at all temperatures from a single simulation. Their finite size scalings and the zero temperature limits are also explored., Comment: no figures included, 5 pages, FSU-SCRI-93-31

We apply a recently developed method, multicanonical algorithm, to the problem of tertiary structure prediction of peptides and proteins. As a simple example to test the effectiveness of the algorithm, Met-enkephalin is studied and the ergodicity problem, or multiple-minima problem, is shown to be overcome by this algorithm. The lowest-energy conformation obtained agrees with that determined by other efficient methods such as Monte Carlo simulated annealing. The superiority of the present method to simulated annealing lies in the fact that the relationship to the canonical ensemble remains exactly controlled. Once the multicanonical parameters are determined, only one simulation run is necessary to obtain the lowest-energy conformation and furthermore the results of this one run can be used to calculate various thermodynamic quantities at any temperature. The latter point is demonstrated by the calculation of the average potential energy and specific heat as functions of temperature., Comment: 13 pages