Lattice Model Of Protein Folding Research Articles

Structure optimisation by thermal cycling for the hydrophobic-polar lattice model of protein folding

The function of a protein depends strongly on its spatial structure. Therefore the transition from an unfolded stage to the functional fold is one of the most important problems in computational molecular biology. Since the corresponding free energy landscapes exhibit huge numbers of local minima, the search for the lowest-energy configurations is very demanding. Because of that, efficient heuristic algorithms are of high value. In the present work, we investigate whether and how the thermal cycling (TC) approach can be applied to the hydrophobic-polar (HP) lattice model of protein folding. Evaluating the efficiency of TC for a set of two- and three-dimensional examples, we compare the performance of this strategy with that of multi-start local search (MSLS) procedures and that of simulated annealing (SA). For this aim, we incorporated several simple but rather efficient modifications into the standard procedures: in particular, a strong improvement was achieved by also allowing energy conserving state modifications. Furthermore, the consideration of ensembles instead of single samples was found to greatly improve the efficiency of TC. In the framework of different benchmarks, for all considered HP sequences, we found TC to be far superior to SA, and to be faster than Wang-Landau sampling.

Effect of increasing the energy gap between the two lowest energy states on the mixing time of the Metropolis algorithm

In order to understand what makes natural proteins fold rapidly, Šali, Shakhnovich and Karplus (1994) [6,7] had used the Metropolis algorithm to search for the minimum energy conformations of chains of beads in the lattice model of protein folding. Based on their computational experiments, they concluded that the Metropolis algorithm would find the minimum energy conformation of a chain of beads within an acceptable time scale if and only if there is a large gap between the energies of the minimum energy conformation and that of the second minimum. Clote (1999) [1] attempted to support this conclusion by a proof that the mixing time of the underlying Markov chain would decrease as the gap in energies of the minimum energy conformation and that of the second minimum increased. He was able to show that an upper bound on the mixing time does indeed decrease as the energy gap increases. We show in this paper that the mixing time itself, however, is a non-decreasing function of the value of the energy gap. Therefore, our result contradicts what Clote had attempted to prove.

Finding low-energy conformations of lattice protein models by quantum annealing.

Lattice protein folding models are a cornerstone of computational biophysics. Although these models are a coarse grained representation, they provide useful insight into the energy landscape of natural proteins. Finding low-energy threedimensional structures is an intractable problem even in the simplest model, the Hydrophobic-Polar (HP) model. Description of protein-like properties are more accurately described by generalized models, such as the one proposed by Miyazawa and Jernigan (MJ), which explicitly take into account the unique interactions among all 20 amino acids. There is theoretical and experimental evidence of the advantage of solving classical optimization problems using quantum annealing over its classical analogue (simulated annealing). In this report, we present a benchmark implementation of quantum annealing for lattice protein folding problems (six different experiments up to 81 superconducting quantum bits). This first implementation of a biophysical problem paves the way towards studying optimization problems in biophysics and statistical mechanics using quantum devices.

The Origin of Nonmonotonic Complex Behavior and the Effects of Nonnative Interactions on the Diffusive Properties of Protein Folding

We present a method for calculating the configurational-dependent diffusion coefficient of a globular protein as a function of the global folding process. Using a coarse-grained structure-based model, we determined the diffusion coefficient, in reaction coordinate space, as a function of the fraction of native contacts formed Q for the cold shock protein (TmCSP). We find nonmonotonic behavior for the diffusion coefficient, with high values for the folded and unfolded ensembles and a lower range of values in the transition state ensemble. We also characterized the folding landscape associated with an energetically frustrated variant of the model. We find that a low-level of frustration can actually stabilize the native ensemble and increase the associated diffusion coefficient. These findings can be understood from a mechanistic standpoint, in that the transition state ensemble has a more homogeneous structural content when frustration is present. Additionally, these findings are consistent with earlier calculations based on lattice models of protein folding and more recent single-molecule fluorescence measurements.

A study of density of states and ground states in hydrophobic-hydrophilic protein folding models by equi-energy sampling

We propose an equi-energy (EE) sampling approach to study protein folding in the two-dimensional hydrophobic-hydrophilic (HP) lattice model. This approach enables efficient exploration of the global energy landscape and provides accurate estimates of the density of states, which then allows us to conduct a detailed study of the thermodynamics of HP protein folding, in particular, on the temperature dependence of the transition from folding to unfolding and on how sequence composition affects this phenomenon. With no extra cost, this approach also provides estimates on global energy minima and ground states. Without using any prior structural information of the protein the EE sampler is able to find the ground states that match the best known results in most benchmark cases. The numerical results demonstrate it as a powerful method to study lattice protein folding models.

Fast Tree Search for A Triangular Lattice Model of Protein Folding

Using a triangular lattice model to study the designability of protein folding, we overcame the parity problem of previous cubic lattice model and enumerated all the sequences and compact structures on a simple two-dimensional triangular lattice model of size 4+5+6+5+4. We used two types of amino acids, hydrophobic and polar, to make up the sequences, and achieved 223+212 different sequences excluding the reverse symmetry sequences. The total string number of distinct compact structures was 219,093, excluding reflection symmetry in the self-avoiding path of length 24 triangular lattice model. Based on this model, we applied a fast search algorithm by constructing a cluster tree. The algorithm decreased the computation by computing the objective energy of non-leaf nodes. The parallel experiments proved that the fast tree search algorithm yielded an exponential speed-up in the model of size 4+5+6+5+4. Designability analysis was performed to understand the search result.

Towards realistic description of collective motions in the lattice protein folding models

Collective motions and the formation of clusters of residues play an important role in the folding of real proteins. However, existing Monte Carlo (MC) techniques of the protein folding simulations based on highly popular lattice models provide only a schematic representation of collective motions, which is rather far from physical reality. The Clustering Monte Carlo (CMC) algorithm was developed with particular aim to provide a realistic description of collective motions on the lattice. CMC allows modeling the cluster dynamics and the effects of the solvent viscosity, which is impossible in conventional algorithms. In this study two 2D lattice peptides, with the ground states of hierarchical and non-hierarchical design, were investigated comparatively using three methods: Metropolis MC with the local move set, Metropolis MC with unspecific rigid rotations and the CMC algorithm. We present evidence that the folding pathways and kinetics of hierarchically folding clustered sequence are not adequately described in conventional MC simulations, and the account for cluster dynamics provided by CMC allows to capture essential features of the folding process. Our data suggest that the methods, which enable specific cluster motions, such as CMC, should be used for a more realistic description of protein folding.

Thermodynamic control and dynamical regimes in protein folding

Monte Carlo simulations of a simple lattice model of protein folding show two distinct regimes depending on the chain length. The first regime well describes the folding of small protein sequences and its kinetic counterpart appears to be single exponential in nature, while the second regime is typical of sequences longer than 80 amino acids and the folding performance achievable is sensitive to target conformation. The extent to which stability, as measured by the energy of a sequence in the target, is an essential requirement and affects the folding dynamics of protein molecules in the first regime is investigated. The folding dynamics of sequences whose design stage was restricted to a certain fraction of randomly selected amino acids shows that while some degree of stability is a necessary and sufficient condition for successful folding, designing sequences that provide the lowest energy in the target seems to be a superfluous constraint. By studying the dynamics of under annealed but otherwise freely designed sequences we explore the relation between stability and kinetic accessibility. We find that there is no one-to-one correspondence between having low energy and folding quickly to the target, as only a small fraction of the most stable sequences were also found to fold relatively quickly.

Lattice models of protein folding permitting disordered native states

Self-avoiding lattice walks are often used as minimalist models of proteins. Typically, the polypeptide chain is represented as a lattice walk with each amino acid residue lying on a lattice point, and the Hamiltonian being a sum of interactions between pairs of sequentially nonadjacent residues on adjacent points. Interactions depend on the types of the two residues, and there are usually two or more types. A sequence is said to fold to a particular “native” conformation if the ground state is nondegenerate, i.e., that native conformation is the unique global energy minimum conformation. However, real proteins have some flexibility in the folded state. If this is permitted in a lattice model, the most stably and cooperatively folding sequences have very disordered native states unless the Hamiltonian either favors only a few specific interactions or includes a solvation term. The result points the way toward qualitatively more realistic lattice models for protein folding.

Fast tree search for enumeration of a lattice model of protein folding

Using a fast tree-searching algorithm and a Pentium cluster, we enumerated all the sequences and compact conformations (structures) for a protein folding model on a cubic lattice of size 4×3×3. We used two types of amino acids—hydrophobic (H) and polar (P)—to make up the sequences, so there were 236≈6.87×1010 different sequences. The total number of distinct structures was 84 731 192. We made use of a simple solvation model in which the energy of a sequence folded into a structure is minus the number of hydrophobic amino acids in the “core” of the structure. For every sequence, we found its ground state or ground states, i.e., the structure or structures for which its energy is lowest. About 0.3% of the sequences have a unique ground state. The number of structures that are unique ground states of at least one sequence is 2 662 050, about 3% of the total number of structures. However, these “designable” structures differ drastically in their designability, defined as the number of sequences whose unique ground state is that structure. To understand this variation in designability, we studied the distribution of structures in a high dimensional space in which each structure is represented by a string of 1’s and 0’s, denoting core and surface sites, respectively.

Strange kinetics and complex energy landscapes in a lattice model of protein folding

Non-exponential relaxation in a simplified lattice model of folding is studied with Monte Carlo (MC) calculation. As folding proceeds, population of the native conformation approaches its equilibrium value with the stretched exponential form. As temperature increases, relaxation becomes less stretched, and for 2 sequences out of 5 tested ones, the relaxation becomes faster than exponential at high temperature. Energy landscape of the model is analyzed and flow of trajectories is followed to explain temperature dependence of kinetics. Measurement of stretched or shrunken kinetics of folding should help to understand nature of intermediates and ruggedness of the landscape.

Secondary-structure-favored hydrophobic-polar lattice model of protein folding

Protein folding is studied using a two-dimensional lattice model with the Hamiltonian including both hydrophobic interactions and main chain hydrogen bond interactions of amino acids. Since compact conformations have different designabilities and only highly designable conformations can act as native structural candidates [H. Li, R. Helling, C. Tang, and N. Wingreen, Science 273, 666 (1996)], it is shown that hydrophobic interaction alone is insufficient to explain the appearance of a high proportion of regular secondary structures, especially beta sheets whose content decreases with increasing designability, but interactions of main chain hydrogen bonds can account for this. Thus the emergence of only a small number of structure types (folds) among all possible structures can be understood to some extent.

Population analyses of kinetic partitioning in protein folding.

A simple lattice model of protein folding is studied in order to analyze the kinetic partitioning phenomena in the energy landscape perspective. By restricting the area of conformational space, it becomes possible to follow many Monte Carlo trajectories until they reach equilibrium. Alteration of population of trajectories is monitored and the relations between the energy landscape and kinetics are examined. Kinetic partitioning phenomena are categorized into different types in terms of characteristic time constants and partitioning ratio. In a specific partitioning process, refolding proceeds along the parallel pathways; the time constants have a temperature dependence similar to that observed in hen lysozyme. High-energy conformations are classified into groups according to the probability that the trajectories starting from those conformations will reach each energy valley. The partitioning ratio is determined by the way in which the conformational space is organized into these groups.

1C0945 格子モデルを用いたタンパク質巻き取り過程におけるキネティクな中間状態蓄積の時間スケールと蓄積量を決める要素の探求

1C0945 格子モデルを用いたタンパク質巻き取り過程におけるキネティクな中間状態蓄積の時間スケールと蓄積量を決める要素の探求

On the role of conformational geometry in protein folding

Using a lattice model of protein folding, we find that once certain native contacts have been formed, folding to the native state is inevitable, even if the only energetic bias in the system is nonspecific, homopolymeric attraction to a collapsed state. These conformations can be quite geometrically unrelated to the native state (with as low as only 53% of the native contacts formed). We demonstrate these results by examining the Monte Carlo kinetics of both heteropolymers under Go interactions and homopolymers, with the folding of both types of polymers to the native state of the heteropolymer. Although we only consider a 48-mer lattice model, our findings shed light on the effects of geometrical restrictions, including those of chain connectivity and steric excluded volume, on protein folding. These effects play a complementary role to that of the rugged energy landscape. In addition, the results of this work can aid in the interpretation of experiments and computer simulations of protein folding performed at elevated temperatures.

Stretching lattice models of protein folding.

A new class of experiments that probe folding of individual protein domains uses mechanical stretching to cause the transition. We show how stretching forces can be incorporated in lattice models of folding. For fast folding proteins, the analysis suggests a complex relation between the force dependence and the reaction coordinate for folding.

Thermodynamics of model prions and its implications for the problem of prion protein folding

Prion disease is caused by the propagation of a particle containing PrPSc, a misfolded form of the normal cellular prion protein (PrPC). PrPCcan re-fold to form PrPScwith loss of α-helical structure and formation of extensive β-sheet structure. Here, we model this prion folding problem with a simple, low-resolution lattice model of protein folding. If model proteins are allowed to re-fold upon dimerization, a minor proportion of them (up to ∼17 %) encrypts an alternative native state as a homodimer. The structures in this homodimeric native state re-arrange so that they are very different in conformation from the monomeric native state. We find that model proteins that are relatively less stable as monomers are more susceptible to the formation of alternative native states as homodimers. These results suggest that less-stable proteins have a greater need for a well-designed energy landscape for protein folding to overcome an increased chance of encrypting substantially different native conformations stabilized by multimeric interactions. This conceptual framework for aberrant folding should be relevant in Alzheimer’s disease and other disorders associated with protein aggregation.

Emergence of preferred structures in a simple model of protein folding.

Protein structures in nature often exhibit a high degree of regularity (for example, secondary structure and tertiary symmetries) that is absent from random compact conformations. With the use of a simple lattice model of protein folding, it was demonstrated that structural regularities are related to high "designability" and evolutionary stability. The designability of each compact structure is measured by the number of sequences that can design the structure-that is, sequences that possess the structure as their nondegenerate ground state. Compact structures differ markedly in terms of their designability; highly designable structures emerge with a number of associated sequences much larger than the average. These highly designable structures possess "proteinlike" secondary structure and even tertiary symmetries. In addition, they are thermodynamically more stable than other structures. These results suggest that protein structures are selected in nature because they are readily designed and stable against mutations, and that such a selection simultaneously leads to thermodynamic stability.

Lattice models of protein folding.

Lattice models of protein folding.

Sensitivity Analysis of a Two-Dimensional Square Lattice Model of Protein Folding

Sensitivity Analysis of a Two-Dimensional Square Lattice Model of Protein Folding