Rough Energy Landscape Research Articles

Multicanonical molecular dynamics algorithm employing an adaptive force-biased iteration scheme

We present one effective multicanonical molecular dynamics (MCMD) algorithm accelerating the convergence of rough energy landscapes simulations via an adaptive force-biased iteration scheme. Our method utilizes several short MCMD simulations with dynamically updated weights and combines them to estimate the density of states via multiple histogram technique. The key step of our algorithm is the adaptive refinement for the derivative of multicanonical weight, which allows the system to enlarge the sampling energy range maintaining the statistical accuracy. The performance of our method has been validated for atomic Lennard-Jones clusters.

Exploring the energy landscape of GFP by single-molecule mechanical experiments.

We use single-molecule force spectroscopy to drive single GFP molecules from the native state through their complex energy landscape into the completely unfolded state. Unlike many smaller proteins, mechanical GFP unfolding proceeds by means of two subsequent intermediate states. The transition from the native state to the first intermediate state occurs near thermal equilibrium at approximately 35 pN and is characterized by detachment of a seven-residue N-terminal alpha-helix from the beta barrel. We measure the equilibrium free energy cost associated with this transition as 22 k(B)T. Detachment of this small alpha-helix completely destabilizes GFP thermodynamically even though the beta-barrel is still intact and can bear load. Mechanical stability of the protein on the millisecond timescale, however, is determined by the activation barrier of unfolding the beta-barrel out of this thermodynamically unstable intermediate state. High bandwidth, time-resolved measurements of the cantilever relaxation phase upon unfolding of the beta-barrel revealed a second metastable mechanical intermediate with one complete beta-strand detached from the barrel. Quantitative analysis of force distributions and lifetimes lead to a detailed picture of the complex mechanical unfolding pathway through a rough energy landscape.

Coarse-grained models of materials with non-convex free-energy: two case studies

Bridging across length scales is one of the fundamental challenges in the computational modelling of material systems whose mechanical response is driven by rough energy landscapes. The typical feature of such systems is that of exhibiting fine scale microstructures. Two case studies, namely, nematic elastomers and ferromagnetic shape memory alloys, are presented to illustrate the use of modern techniques from (non-convex) calculus of variations in developing coarse-grained models of microstructure-driven material response.

Generalized simulated tempering realized on expanded ensembles of non-Boltzmann weights.

A generalized version of the simulated tempering operated in the expanded ensembles of non-Boltzmann weights has been proposed to mitigate a quasiergodicity problem occurring in simulations of rough energy landscapes. In contrast to conventional simulated tempering employing the Boltzmann weight, our method utilizes a parametrized, generalized distribution as a workhorse for stochastic exchanges of configurations and subensembles transitions, which allows a considerable enhancement for the rate of convergence of Monte Carlo and molecular dynamics simulations using delocalized weights. A feature of our method is that the exploration of the parameter space encouraging subensembles transitions is greatly accelerated using the dynamic update scheme for the weight via the average guide specific to the energy distribution. The performance and characteristic feature of our method have been validated in the liquid-solid transition of Lennard-Jones clusters and the conformational sampling of alanine dipeptide by taking two types of Tsallis [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] expanded ensembles associated with different parametrization schemes.

Replica-exchange extensions of simulated tempering method.

In this paper we consider combinations of two well-known generalized-ensemble algorithms, namely, simulated tempering and replica-exchange method. We discuss two examples of such combinations. One is the replica-exchange simulated tempering and the other is the simulated tempering replica-exchange method. In the former method, a short replica-exchange simulation is first performed and the simulated tempering weight factor is obtained by the multiple-histogram reweighting techniques. This process of simulated tempering weight factor determination is faster and simpler than that in the usual iterative process. A long simulated tempering production run is then performed with this weight factor. The latter method is a further extension of the former in which a simulated tempering replica-exchange simulation is performed with a small number of replicas. These algorithms are particularly useful for studying frustrated systems with rough energy landscape. We give the formulations of these two methods in detail and demonstrate their effectiveness taking the example of the system of a 17-residue helical peptide.

Heterogeneous Folding of the trpzip Hairpin: Full Atom Simulation and Experiment

The β-hairpin trpzip2 can be tuned continuously from a two-state folder to folding on a rough energy landscape without a dominant refolding barrier. At high denaturant concentration, this extremely stable peptide exhibits a single apparent “two-state” transition temperature when monitored by different spectroscopic techniques. However, under optimal folding conditions the hairpin undergoes an unusual folding process with three clusters of melting transitions ranging from 15 °C to 160 °C, as monitored by 12 different experimental and computational observables. We explain this behavior in terms of a rough free energy landscape of the unfolded peptide caused by multiple tryptophan interactions and alternative backbone conformations. The landscape is mapped out by potentials of mean force derived from replica-exchange molecular dynamics simulations. Implications for deducing cooperativity from denaturant titrations, for the origin of folding cooperativity, and for the folding of thermophilic proteins are pointed out. trpzip is an excellent small tunable model system for the glass-like folding transitions predicted by landscape theory.

Kinetics of the coil-to-helix transition on a rough energy landscape.

The kinetics of folding of a fully atomic seven-residue polyalanine peptide in an implicit solvent are studied using molecular dynamics simulations. The use of an implicit solvent is found to dramatically increase the frustration of the energy landscape relative to simulations performed in an explicit solvent [Phys. Rev. Lett. 85, 2637 (2000)]. While the native state in both implicit and explicit solvent simulations is an alpha-helix, the kinetics of the coil-to-helix transition differ significantly. In contrast to the explicit solvent simulations, the native state in the implicit solvent simulations is not kinetically accessible at temperatures where it is thermodynamically stable and could not be brought into equilibrium with other conformational states. At temperatures where statistical equilibrium was achieved, the conformational diffusion folding mechanism, found earlier to be adequate for this peptide in an explicit solvent [Phys. Rev. Lett. 85, 2637 (2000)], is met with only limited success. Issues relating to the evaluation of the quality of implicit solvent models on the basis of thermodynamic criteria only are reexamined.

A Novel Approach to Decoy Set Generation: Designing a Physical Energy Function Having Local Minima with Native Structure Characteristics

We suggest a new approach to the generation of candidate structures (decoys) for ab initio prediction of protein structures. Our method is based on random sampling of conformation space and subsequent local energy minimization. At the core of this approach lies the design of a novel type of energy function. This energy function has local minima with native structure characteristics and wide basins of attraction. The current work presents our motivation for deriving such an energy function and also tests the derived energy function. Our approach is novel in that it takes advantage of the inherently rough energy landscape of proteins, which is generally considered a major obstacle for protein structure prediction. When local minima have wide basins of attraction, the protein's conformation space can be greatly reduced by the convergence of large regions of the space into single points, namely the local minima corresponding to these funnels. We have implemented this concept by an iterative process. The potential is first used to generate decoy sets and then we study these sets of decoys to guide further development of the potential. A key feature of our potential is the use of cooperative multi-body interactions that mimic the role of the entropic and solvent contributions to the free energy. The validity and value of our approach is demonstrated by applying it to 14 diverse, small proteins. We show that, for these proteins, the size of conformation space is considerably reduced by the new energy function. In fact, the reduction is so substantial as to allow efficient conformational sampling. As a result we are able to find a significant number of near-native conformations in random searches performed with limited computational resources.

Surface-Enhanced Resonance Raman Spectroscopy Signals from Single Myoglobin Molecules

The extremely large cross-section available from metallic surface enhancement has been exploited to investigate the Raman spectrum of heme myoglobin adsorbed on silver colloidal nanoparticles at very low concentrations. The study has been performed on particles both in solution and immobilized onto a polymer-coated glass surface. In both the cases, we have observed striking temporal fluctuations in the surface-enhanced resonance Raman spectroscopy (SERRS) spectra collected at short times. A statistical analysis of the temporal intensity fluctuations and of the associated correlations of the Raman signals has allowed us to verify that the single molecule limit is approached. The possible connections of these fluctuations with the entanglement of the biomolecule within the local minima of its rough energy landscape is discussed.

Sequential vs. parallel protein-folding mechanisms: experimental tests for complex folding reactions

The recent emphasis on rough energy landscapes for protein folding reactions by theoreticians, and the many observations of complex folding kinetics by experimentalists provide a rationale for a brief literature survey of various empirical approaches for validating the underlying mechanisms. The determination of the folding mechanism is a key step in defining the energy surface on which the folding reactions occurs and in interpreting the effects of amino acid replacements on this reaction. Case studies that illustrate methods for differentiating between sequential and parallel channel folding mechanisms are presented. The ultimate goal of such efforts is to understand how the one-dimensional information contained in the amino acid sequence is rapidly and efficiently translated into three-dimensional structure.

Optimized Monte Carlo analysis for generalized ensembles

We present a new, maximum-likelihood based method to combine data from a multiple number of Monte Carlo simulations performed within any type of ensemble. The method offers an efficient iterative scheme to obtain the density of states of a wide range of energies as well as of other macroscopic variables. It should in particular be useful for the study of systems with a rough energy landscape.

Using novel variable transformations to enhance conformational sampling in molecular dynamics.

One of the computational "grand challenges" is to develop methodology capable of sampling conformational equilibria in systems with rough energy landscapes. Here, a significant step forward is made by combining molecular dynamics with a novel variable transformation designed to enhance sampling by reducing barriers without introducing bias and, thus, to preserve, perfectly, equilibrium properties.

Two–dimensional modelling of soft ferromagnetic films

We examine the response of a soft ferromagnetic film to an in–plane applied magnetic field by means of both theory and experiment. In the thin–film limit, we uncover a separation of scales in the rough energy landscape of micromagnetics: leading–order terms generate constraints which eliminate degrees of freedom, terms of second order in the film thickness lead to a (new) reduced variational model, higher–order terms are related to wall, vortex and anisotropy energies. We propose a new strategy to compute low–energy domain patterns, which proceeds in two steps: we determine first the magnetic charge density by solving a convex variational problem, then we construct an associated magnetization field using a robust numerical method. Experimental results show good agreement with the theory. Our analysis is consistent with prior work by van den Berg and by Bryant and Suhl, but it goes much further; in particular it applies even for large fields which penetrate the sample.

Rough energy landscapes in protein folding: dimeric E. coliTrp repressor folds through three parallel channels

The folding mechanism of the dimeric Escherichia coli Trp repressor (TR) is a kinetically complex process that involves three distinguishable stages of development. Following the formation of a partially folded, monomeric ensemble of species, within 5 ms, folding to the native dimer is controlled by three kinetic phases. The rate-limiting step in each phase is either a non-proline isomerization reaction or a dimerization reaction, depending on the final denaturant concentration. Two approaches have been employed to test the previously proposed folding mechanism of TR through three parallel channels: (1) unfolding double-jump experiments demonstrate that all three folding channels lead directly to native dimer; and (2) the differential stabilization of the transition state for the final step in folding and the native dimer, by the addition of salt, shows that all three channels involve isomerization of a dimeric species. A refined model for the folding of Trp repressor is presented, in which all three channels involve a rapid dimerization reaction between partially folded monomers followed by the isomerization of the dimeric intermediates to yield native dimer. The ensemble of partially folded monomers can be captured at equilibrium by low pH; one-dimensional proton NMR spectra at pH 2.5 demonstrate that monomers exist in two distinct, slowly interconverting conformations. These data provide a potential structural explanation for the three-channel folding mechanism of TR: random association of two different monomeric forms, which are distinguished by alternative packing modes of the core dimerization domain and the DNA-binding, helix-turn-helix, domain. One, perhaps both, of these packing modes contains non-native contacts.

Extended state-space Monte Carlo methods.

In this paper various extensions of the parallel-tempering algorithm are developed and their properties are analyzed. The algorithms are designed to alleviate quasiergodic sampling in systems which have rough energy landscapes by coupling individual Monte Carlo chains to form a composite chain. As with parallel tempering, the procedures are based upon extending the state space to include parameters to encourage sampling mobility. One of the drawbacks of the parallel-tempering method is the stochastic nature of the Monte Carlo dynamics in the auxiliary variables which extend the state space. In this work, the possibility of improving the sampling rate by designing deterministic methods of moving through the parameter space is investigated. The methods developed in this article, which are based upon a statistical quenching and heating procedure similar in spirit to simulated annealing, are tested on a simple two-dimensional spin system (xy model) and on a model in vacuo polypeptide system. In the coupled Monte Carlo chain algorithms, we find that the net mobility of the composite chain is determined by the competition between the characteristic time of coupling between adjacent chains and the degree of overlap of their distributions. Extensive studies of all methods are carried out to obtain optimal sampling conditions. In particular, the most efficient parallel-tempering procedure is to attempt to swap configurations after very few Monte Carlo updates of the composite chains. Furthermore, it is demonstrated that, contrary to expectations, the deterministic procedure does not improve the sampling rate over that of parallel tempering.

Efficient, multiple-range random walk algorithm to calculate the density of states.

We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy, and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape.

Replica-exchange simulated tempering method for simulations of frustrated systems

We propose a new method for the determination of the weight factor for the simulated tempering method. In this method a short replica-exchange simulation is performed and the simulated tempering weight factor is obtained by the multiple-histogram reweighting techniques. The new algorithm is particularly useful for studying frustrated systems with rough energy landscape where the determination of the simulated tempering weight factor by the usual iterative process becomes very difficult. The effectiveness of the method is illustrated by taking an example for protein folding.

Enhanced sampling of rare events

We present a method to enhance sampling of a given reaction coordinate by projecting part of the random thermal noise along a preferential direction. The approach is promising to study rough energy landscapes and highly activated barriers that can be overcome by increasing the attempt frequency. Furthermore it allows us to rescale a given reaction coordinate without biasing the configurational properties of the system. A major advantage of the method is its simplicity in the analytical derivation and numerical implementation.

Catalytic tempering: A method for sampling rough energy landscapes by Monte Carlo.

A new Monte Carlo algorithm is presented for the efficient sampling of the Boltzmann distribution of configurations of systems with rough energy landscapes. The method is based on the introduction of a fictitious coordinate y so that the dimensionality of the system is increased by one. This augmented system has a potential surface and a temperature that is made to depend on the new coordinate y in such a way that for a small strip of the y space, called the "normal region," the temperature is set equal to the temperature desired and the potential is the original rough energy potential. To enhance barrier crossing outside the "normal region," the energy barriers are reduced by truncation (with preservation of the potential minima) and the temperature is made to increase with ||y ||. The method, called catalytic tempering or CAT, is found to greatly improve the rate of convergence of Monte Carlo sampling in model systems and to eliminate the quasi-ergodic behavior often found in the sampling of rough energy landscapes.

Replica-exchange multicanonical algorithm and multicanonical replica-exchange method for simulating systems with rough energy landscape

We propose two efficient algorithms for configurational sampling of systems with rough energy landscape. The first one is a new method for the determination of the multicanonical weight factor. In this method, a short replica-exchange simulation is performed and the multicanonical weight factor is obtained by the multiple histogram reweighting techniques. The second one is a further extension of the first in which a replica-exchange multicanonical simulation is performed with a small number of replicas. These new algorithms are particularly useful for studying the protein folding problem.