We present results of Monte Carlo computer simulations of a coarse-grained hydrophobic-polar Go-like heteropolymer model and discuss thermodynamic properties and kinetics of an exemplified heteropolymer, exhibiting two-state folding behavior. It turns out that general, characteristic folding features of realistic proteins with a single free-energy barrier can also be observed in this simplified model, where the folding transition is primarily driven by the hydrophobic force.
Applying multicanonical simulations we investigated folding properties of off-lattice heteropolymers employing a mesoscopic hydrophobic-polar model. We study for various sequences folding channels in the free-energy landscape by comparing the equilibrium conformations with the folded state in terms of an angular overlap parameter. Although all investigated heteropolymer sequences contain the same content of hydrophobic and polar monomers, our analysis of the folding channels reveals a variety of characteristic folding behaviors known from realistic peptides.
Folding channels and free-energy landscapes of hydrophobic-polar heteropolymers are discussed on the basis of a minimalistic off-lattice coarse-grained model. We investigate how rearrangements of hydrophobic and polar monomers in a heteropolymer sequence lead to completely different folding behaviors. Studying three exemplified sequences with the same content of hydrophobic and polar residues, we can reproduce within this simple model two-state folding, folding through intermediates, as well as metastability.
We have performed parallel tempering Monte Carlo simulations using a simple continuum heteropolymer model for proteins. All ten heteropolymer sequences which we have studied have shown first-order transitions at low temperature to ordered states dominated by single chain conformations. These results are in contrast with the theoretical predictions of the random energy model for heteropolymers, from which we would expect continuous transitions to glassy behavior at low temperatures.
We present the results of a self-consistent, unified molecular dynamics study of simple model heteropolymers in the continuum with emphasis on folding, sequence design and the determination of the interaction parameters of the effective potential between the amino acids from the knowledge of the native states of the designed sequences.
We present a statistical mechanics approach to the protein folding problem. We first review some of the basic properties of proteins, and introduce some physical models to describe their thermodynamics. These models rely on a random heteropolymeric description of these non random biomolecules. Various kinds of randomness are investigated, and the connection with disordered systems is discussed. We conclude by a brief study of the dynamics of proteins.
The kinetic behavior of a three-dimensional off-lattice heteropolymer model is studied in terms of the time dependence of the average mean-square displacement between configurations. It is found that at short time-scales similar behavior is obtained even for sequences with very different thermodynamic properties. Furthermore, the degree of cooperativity in the folding process is examined by studying the residual number of degrees of freedom, obtained from an eigenvalue analysis of the correlation matrix, contributing to the structural fluctuations. In the compact state, a gradual decrease in t…
Simple coarse-grained hydrophobic-polar models for heteropolymers as the lattice HP and the off-lattice AB model allow a general classification of characteristic behaviors for hydrophobic-core based tertiary folding. The strongly reduced computational efforts enable one to reveal systematically the thermodynamic properties of comparatively long sequences in a wide temperature range of conformational activity. Based on a suitable cooperativity parameter, characteristic folding channels and free-energy landscapes, which have strong similarities with realistic folding paths, can be analysed.
The statistical mechanics of heteropolymer structure formation is studied in the context of RNA secondary structures. A designed RNA sequence biased energetically towards a particular native structure (a hairpin) is used to study the transition between the native and molten phase of the RNA as a function of temperature. The transition is driven by a competition between the energy gained from the polymer's overlap with the native structure and the entropic gain of forming random contacts. A simplified Go-like model is proposed and solved exactly. The predicted critical behavior is verified via …
The dynamical chaos in Lennard-Jones toy models of heteropolymers is studied by molecular dynamics simulations. It is shown that two nearby trajectories quickly diverge from each other if the heteropolymer corresponds to a random sequence. For good folders, on the other hand, two nearby trajectories may initially move apart but eventually they come together. Thus good folders are intrinsically non-chaotic. A choice of a distance of the initial conformation from the native state affects the way in which a separation between the twin trajectories behaves in time. This observation allows one to d…
Within the frame of an effective, coarse-grained hydrophobic-polar protein model, we employ multicanonical Monte Carlo simulations to investigate free-energy landscapes and folding channels of exemplified heteropolymer sequences, which are permutations of each other. Despite the simplicity of the model, the knowledge of the free-energy landscape in dependence of a suitable system order parameter enables us to reveal complex folding characteristics known from real bioproteins and synthetic peptides, such as two-state folding, folding through weakly stable intermediates, and glassy metastability…
What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for proteinlike calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for proteinlike thermodynamics. Kinetically, however, eve…
To what extent do general features of folding/unfolding kinetics of small globular proteins follow from their thermodynamic properties? To address this question, we investigate a new simplifed protein chain model that embodies a cooperative interplay between local conformational preferences and hydrophobic burial. The present four-helix-bundle 55mer model exhibits proteinlike calorimetric two-state cooperativity. It rationalizes native-state hydrogen exchange observations. Our analysis indicates that a coherent, self-consistent physical account of both the thermodynamic and kinetic properties …
We demonstrate that the recently proposed interacting growth walk (IGW) model, modified for generating self-avoiding heteropolymers, proves to be a simpler alternative to the other Monte Carlo methods available in the literature for obtaining minimum energy conformation of lattice proteins. In fact, this simple growth algorithm seems to be capable of quickly leading to low energy states for all the three dimensional bench mark HP-sequences investigated.
In this paper we extend the Gaussian self-consistent method to permit study of the equilibrium and kinetics of conformational transitions for heteropolymers with any given primary sequence. The kinetic equations earlier derived by us are transformed to a form containing only the mean squared distances between pairs of monomers. These equations are further expressed in terms of instantaneous gradients of the variational free energy. The method allowed us to study exhaustively the stability and conformational structure of some periodic and random aperiodic sequences. A typical phase diagram of a…
We demonstrate that the recently proposed pruned-enriched Rosenbluth method PERM (P.~Grassberger, Phys.~Rev.~{\bf E 56} (1997) 3682) leads to very efficient algorithms for the folding of simple model proteins. We test it on several models for lattice heteropolymers, and compare to published Monte Carlo studies of the properties of particular sequences. In all cases our method is faster than the previous ones, and in several cases we find new minimal energy states. In addition to producing more reliable candidates for ground states, our method gives detailed information about the thermal spectr…
We have performed multicanonical simulations of hydrophobic-hydrophilic heteropolymers with a simple effective, coarse-grained off-lattice model to study the structure and the topology of the energy surface. The multicanonical method samples the whole rugged energy landscape, in particular the low-energy part, and enables one to better understand the critical behaviors and visualize the folding pathways of the considered protein model.
By means of continuous space Monte Carlo simulation we study conformational structures formed by star and comb heteropolymers during kinetics of folding from the coil to the globule, as well as the corresponding equilibrium states on going from the good to the poor solution. Particular examples of combs with hydrophobic backbone and hydrophilic side-groups (and vice versa), as well as stars with flexible and semi-stiff arms are studied. It is interesting to note that star-like conformations naturally appear for a comb polymer with a strongly hydrophobic backbone. We emphasise the crucial diffe…
Monte Carlo simulations of proteins are hindered by the system's high density which often makes local moves ineffective. Here we devise and test a set of long range moves that work well even when all sites of a lattice simulation are filled. We demonstrate that for a 27-mer cube, the ground state of random heteropolymers can quickly be reached. We discuss results for 48-mer systems where the ground state is known exactly. For ten sequences that were examined, the introduction of long range moves speeds up the search for the ground state by about one order of magnitude. The method is compared t…
We describe a class of growth algorithms for finding low energy states of heteropolymers. These polymers form toy models for proteins, and the hope is that similar methods will ultimately be useful for finding native states of real proteins from heuristic or a priori determined force fields. These algorithms share with standard Markov chain Monte Carlo methods that they generate Gibbs-Boltzmann distributions, but they are not based on the strategy that this distribution is obtained as stationary state of a suitably constructed Markov chain. Rather, they are based on growing the polymer by succ…
Simple two-state folding kinetics of many small single-domain proteins are characterized by chevron plots with linear folding and unfolding arms consistent with a two-state description of equilibrium thermodynamics. This phenomenon is hereby recognized as a nontrivial heteropolymer property capable of providing fundamental insight into protein energetics. Many current protein chain models, including common lattice and continuum Gō models with explicit native biases, fail to reproduce this generic protein property. Here we show that simple two-state kinetics is obtainable from models with a coo…
We study a simple heteropolymer model containing sequence-independent local interactions on both square and triangular lattices. Sticking to a two-letter code, we investigate the model for varying strength $κ$ of the local interactions; $κ=0$ corresponds to the well-known HP model [K.F. Lau and K.A. Dill, Macromolecules 22, 3986 (1989)]. By exhaustive enumerations for short chains, we obtain all structures which act as a unique and pronounced energy minimum for at least one sequence. We find that the number of such designable structures depends strongly on $κ$. Also, we find that the number of…
Coarse-grained (lattice-) models have a long tradition in aiding efforts to decipher the physical or biological complexity of proteins. Despite the simplicity of these models, however, numerical simulations are often computationally very demanding and the quest for efficient algorithms is as old as the models themselves. Expanding on our previous work [T. Wüst and D. P. Landau, Phys. Rev. Lett. 102, 178101 (2009)], we present a complete picture of a Monte Carlo method based on Wang-Landau sampling in combination with efficient trial moves (pull, bond-rebridging and pivot moves) which is partic…
In this review, we describe applications of the pruned-enriched Rosenbluth method (PERM), a sequential Monte Carlo algorithm with resampling, to various problems in polymer physics. PERM produces samples according to any given prescribed weight distribution, by growing configurations step by step with controlled bias, and correcting "bad" configurations by "population control". The latter is implemented, in contrast to other population based algorithms like e.g. genetic algorithms, by depth-first recursion which avoids storing all members of the population at the same time in computer memory. …
