Lattice Protein Research Articles

Exact methods for lattice protein models

Abstract Lattice protein models are well-studied abstractions of globular proteins. By discretizing the structure space and simplifying the energy model over regular proteins, they enable detailed studies of protein structure formation and evolution. However, even in the simplest lattice protein models, the prediction of optimal structures is computationally difficult. Therefore, often, heuristic approaches are applied to find such conformations. Commonly, heuristic methods find only locally optimal solutions. Nevertheless, there exist methods that guarantee to predict globally optimal structures. Currently, only one such exact approach is publicly available, namely the constraint-based protein structure prediction method and variants. Here, we review exact approaches and derived methods. We discuss fundamental concepts like hydrophobic core construction and their use in optimal structure prediction, as well as possible applications like combinations of different energy models.

Stabilization of the β-hairpin in Mason-Pfizer monkey virus capsid protein- a critical step for infectivity

BackgroundFormation of a mature core is a crucial event for infectivity of retroviruses such as Mason-Pfizer monkey virus (M-PMV). The process is triggered by proteolytic cleavage of the polyprotein precursor Gag, which releases matrix, capsid (CA), and nucleocapsid proteins. Once released, CA assembles to form a mature core – a hexameric lattice protein shell that protects retroviral genomic RNA. Subtle conformational changes within CA induce the transition from the immature lattice to the mature lattice. Upon release from the precursor, the initially unstructured N-terminus of CA is refolded to form a β-hairpin stabilized by a salt bridge between the N-terminal proline and conserved aspartate. Although the crucial role of the β-hairpin in the mature core assembly has been confirmed, its precise structural function remains poorly understood.ResultsBased on a previous NMR analysis of the N-terminal part of M-PMV CA, which suggested the role of additional interactions besides the proline-aspartate salt bridge in stabilization of the β-hairpin, we introduced a series of mutations into the CA sequence. The effect of the mutations on virus assembly and infectivity was analyzed. In addition, the structural consequences of selected mutations were determined by NMR spectroscopy. We identified a network of interactions critical for proper formation of the M-PMV core. This network involves residue R14, located in the N-terminal β-hairpin; residue W52 in the loop connecting helices 2 and 3; and residues Q113, Q115, and Y116 in helix 5.ConclusionCombining functional and structural analyses, we identified a network of supportive interactions that stabilize the β-hairpin in mature M-PMV CA.Electronic supplementary materialThe online version of this article (doi:10.1186/s12977-014-0094-8) contains supplementary material, which is available to authorized users.

Effect of single-site mutations on hydrophobic-polar lattice proteins.

We developed a heuristic method for determining the ground-state degeneracy of hydrophobic-polar (HP) lattice proteins, based on Wang-Landau and multicanonical sampling. It is applied during comprehensive studies of single-site mutations in specific HP proteins with different sequences. The effects in which we are interested include structural changes in ground states, changes of ground-state energy, degeneracy, and thermodynamic properties of the system. With respect to mutations, both extremely sensitive and insensitive positions in the HP sequence have been found. That is, ground-state energies and degeneracies, as well as other thermodynamic and structural quantities, may be either largely unaffected or may change significantly due to mutation.

Visualization of protein folding funnels in lattice models.

Protein folding occurs in a very high dimensional phase space with an exponentially large number of states, and according to the energy landscape theory it exhibits a topology resembling a funnel. In this statistical approach, the folding mechanism is unveiled by describing the local minima in an effective one-dimensional representation. Other approaches based on potential energy landscapes address the hierarchical structure of local energy minima through disconnectivity graphs. In this paper, we introduce a metric to describe the distance between any two conformations, which also allows us to go beyond the one-dimensional representation and visualize the folding funnel in 2D and 3D. In this way it is possible to assess the folding process in detail, e.g., by identifying the connectivity between conformations and establishing the paths to reach the native state, in addition to regions where trapping may occur. Unlike the disconnectivity maps method, which is based on the kinetic connections between states, our methodology is based on structural similarities inferred from the new metric. The method was developed in a 27-mer protein lattice model, folded into a 3×3×3 cube. Five sequences were studied and distinct funnels were generated in an analysis restricted to conformations from the transition-state to the native configuration. Consistent with the expected results from the energy landscape theory, folding routes can be visualized to probe different regions of the phase space, as well as determine the difficulty in folding of the distinct sequences. Changes in the landscape due to mutations were visualized, with the comparison between wild and mutated local minima in a single map, which serves to identify different trapping regions. The extension of this approach to more realistic models and its use in combination with other approaches are discussed.

Effects of knot type in the folding of topologically complex lattice proteins.

The folding properties of a protein whose native structure contains a 52 knot are investigated by means of extensive Monte Carlo simulations of a simple lattice model and compared with those of a 31 knot. A 52 knot embedded in the native structure enhances the kinetic stability of the carrier lattice protein in a way that is clearly more pronounced than in the case of the 31 knot. However, this happens at the expense of a severe loss in folding efficiency, an observation that is consistent with the relative abundance of 31 and 52 knots in the Protein Data Bank. The folding mechanism of the 52 knot shares with that of the 31 knot the occurrence of a threading movement of the chain terminus that lays closer to the knotted core. However, co-concomitant knotting and folding in the 52 knot occurs with negligible probability, in sharp contrast to what is observed for the 31 knot. The study of several single point mutations highlights the importance in the folding of knotted proteins of the so-called structural mutations (i.e., energetic perturbations of native interactions between residues that are critical for knotting but not for folding). On the other hand, the present study predicts that mutations that perturb the folding transition state may significantly enhance the kinetic stability of knotted proteins provided they involve residues located within the knotted core.

Rare-event trajectory ensemble analysis reveals metastable dynamical phases in lattice proteins.

We explore the dynamical large deviations of a lattice heteropolymer model of a protein by means of path sampling of trajectories. We uncover the existence of nonequilibrium dynamical phase transitions in ensembles of trajectories between active and inactive dynamical phases, whose nature depends on the properties of the interaction potential. We consider three potentials: two heterogeneous interaction potentials and a homogeneous Gō potential. When preserving the full heterogeneity of interactions due to a given amino acid sequence, either in a fully interacting model or in a native contacts interacting model (heterogeneous Gō model), the observed dynamic transitions occur between equilibrium highly native states and highly native but kinetically trapped states. A native activity is defined that allows us to distinguish these dynamic phases. In contrast, for the homogeneous Gō model, where all native interaction energies are uniform and the amino acid sequence plays no role, the dynamical transition is a direct consequence of the static bistability between the unfolded and the native state. In the two heterogeneous interaction models the native-active and native-inactive states, despite their thermodynamic similarity, have widely varying dynamical properties, and the transition between them occurs even in lattice proteins whose sequences are designed to make them optimal folders.

Wang–Landau sampling of the interplay between surface adsorption and folding of HP lattice proteins

Generic features associated with the adsorption of proteins on solid surfaces are reviewed within the framework of the hydrophobic-polar (HP) lattice protein model. The thermodynamic behaviour and structural properties of various HP protein sequences interacting with attractive surfaces have been studied using extensive Wang–Landau sampling with different types of surfaces, each of which attracts either: all monomers, only hydrophobic (H) monomers or only polar (P) monomers, respectively. Consequently, different types of folding behaviour occur for varied surface strengths. Analysis of the combined patterns of various structural observables, e.g. the derivatives of the number of interaction contacts, together with the specific heat, leads to the identification of fundamental categories of folding and transition hierarchies. We also inferred a connection between the transition categories and the relative surface strengths, i.e. the ratios of the surface attractive strengths to the intra-chain attraction among H monomers. Thus, we believe that the folding hierarchies and identification scheme are generic for different HP sequences interacting with attractive surfaces, regardless of the chain length, sequence or surface attraction.

Solving Lattice Protein Folding Problems by Discrete Particle Swarm Optimization

Using computer programs to predict protein structures from a mass of protein sequences is promising for discovering the relationship between the protein construction and their functions. In the area of computational protein structure analysis, the hydrophobic-polar (HP) model is one of the most commonly applied models. The protein folding problem based on HP model has been shown as NP-hard, to handle such an NP-hard problem, this paper proposes a discrete particle swarm optimization algorithm (DPSO HP ) to solve various 2D and 3D HP lattice models-based protein folding problems. The discrete particle swarm optimization method used in DPSO HP is based on the set concept and the possibility theory from a set-based PSO (S-PSO). A selection strategy incorporating heuristic information and possibilities is adopted in DPSO HP . A particle’s positions in the algorithm are defined as a set of elements and the velocities of a particle are defined as a set of elements associated with possibilities. The experimental results on a series of 2D and 3D protein sequences show that DPSO HP is promising and performs better than various competitive state-of-the-art evolutionary algorithms.

Can a Protein's Evolutionary Fate be Predicted from its Structure?

A fundamental question in evolution is the role of a protein's 3D structure in determining its evolvability. We observe that different protein structures vary widely in their ability to respond adaptively to simulated selective pressures, even with the aid of folding chaperones. This variation that cannot be explained by factors such as stability, that are already understood to promote evolvability. In these simulations, lattice proteins evolve in a crowded, cell-like environment with complex dynamics between protein structure, protein-protein interactions, and cellular fitness, while remaining exactly solvable. We develop new structural metrics predictive of a protein's evolvability and corroborate the conclusions with analysis of protein folds in nature.View Large Image | View Hi-Res Image | Download PowerPoint Slide

Energy landscape and dynamics of an HP lattice model of proteins —The role of anisotropy

We present the results of exact numerical studies of the energy landscape and the dynamics of a 12-monomer chain comprised of two types of amino acids called the HP model. We benchmark our findings against the corresponding results of previous studies of a Go model, which encodes the native state conformation. We also show how the energy landscape gets modified dramatically and improves the folding properties on incorporating the inherent anisotropy of a chain, albeit in a simplified manner.

Designing sequences with varied flexibility and stability through pair mutations

Intrinsic conformational flexibility is a consequence of the rugged energy landscape of the protein which is encoded in its amino acid sequence. In this work, a self-consistent mean-field based theory is developed and applied on lattice and real proteins to design sequences with varying foldability and desired flexibility. Monte Carlo simulations are used to evaluate the conformational flexibility of the designed sequences in terms of the relative propensity of the sequence to choose the native state among the ensemble of near-native conformations. The mutation of specific residue pairs in the designed sequences affects their conformational flexibility. Site pairs which exhibit a greater deviation from the corresponding random values of the residue probabilities may be preferentially selected for pair mutation of residues to modulate the flexibility. The mutation of such residue pairs due to evolution may lead to significantly altered conformational flexibility and stability while retaining its sequence identity. This result is supported by experimental evidence [Lehmann et al., Protein Eng., 2000, 13, 49], which suggests that the conserved residues in a set of homologous protein sequences contribute more to the stability of the protein. Mutations of such residue pairs by decreasing the foldability lead to higher flexibility and may mimic protein evolution [Tomatis et al., Proc. Natl. Acad. Sci. U. S. A., 2008, 105, 20605]. The spatial distance between any two residues is minimally correlated to the generation of the mutations. However, long ranged correlation between residue pairs may also be important for specific residue pairs. A higher correlation between residue pairs is noted to be sufficient but not a necessary condition for accommodating higher generations of mutation.

Energy-landscape paving for prediction of face-centered-cubic hydrophobic-hydrophilic lattice model proteins

Protein structure prediction (PSP) is a classical NP-hard problem in computational biology. The energy-landscape paving (ELP) method is a class of heuristic global optimization algorithm, and has been successfully applied to solving many optimization problems with complex energy landscapes in the continuous space. By putting forward a new update mechanism of the histogram function in ELP and incorporating the generation of initial conformation based on the greedy strategy and the neighborhood search strategy based on pull moves into ELP, an improved energy-landscape paving (ELP+) method is put forward. Twelve general benchmark instances are first tested on both two-dimensional and three-dimensional (3D) face-centered-cubic (fcc) hydrophobic-hydrophilic (HP) lattice models. The lowest energies by ELP+ are as good as or better than those of other methods in the literature for all instances. Then, five sets of larger-scale instances, denoted by S, R, F90, F180, and CASP target instances on the 3D FCC HP lattice model are tested. The proposed algorithm finds lower energies than those by the five other methods in literature. Not unexpectedly, this is particularly pronounced for the longer sequences considered. Computational results show that ELP+ is an effective method for PSP on the fcc HP lattice model.

Quinary lattice model of secondary structures of polymers

In the standard approach to lattice proteins models based on nearest neighbor interaction are used. In this kind of model it is difficult to explain the existence of secondary structures—special preferred conformations of protein chains.In the present paper a new lattice model of proteins is proposed which is based on non-local cooperative interactions. In this model the energy of a conformation of a polymer is equal to the sum of energies of conformations of fragments of the polymer chain of length five.It is shown that this quinary lattice model is able to describe at a qualitative level secondary structures of proteins: for this model all conformations with minimal energy are combinations of lattice models of alpha-helix and beta-strand. Moreover for lattice polymers of length not longer that 38 monomers we can describe all conformations with minimal energy.

Multicanonical simulation of the Domb-Joyce model and the Gō model: new enumeration methods for self-avoiding walks

We develop statistical enumeration methods for self-avoiding walks using a powerful sampling technique called the multicanonical Monte Carlo method. Using these methods, we estimate the numbers of the two dimensional N-step self-avoiding walks up to N = 256 with statistical errors. The developed methods are based on statistical mechanical models of paths which include self-avoiding walks. The criterion for selecting a suitable model for enumerating self-avoiding walks is whether or not the configuration space of the model includes a set for which the number of the elements can be exactly counted. We call this set a scale fixing set. We selected the following two models which satisfy the criterion: the Gō model for lattice proteins and the Domb-Joyce model for generalized random walks. There is a contrast between these two models in the structures of the configuration space. The configuration space of the Gō model is defined as the universal set of self-avoiding walks, and the set of the ground state conformation provides a scale fixing set. On the other hand, the configuration space of the Domb-Joyce model is defined as the universal set of random walks which can be used as a scale fixing set, and the set of the ground state conformation is the same as the universal set of self-avoiding walks. From the perspective of enumeration performance, we conclude that the Domb-Joyce model is the better of the two. The reason for the performance difference is partly explained by the existence of the first-order phase transition of the Gō model.

An Effective Exact Algorithm and a New Upper Bound for the Number of Contacts in the Hydrophobic-Polar Two-Dimensional Lattice Model

Protein Structure Prediction (PSP) is the problem of predicting the three-dimensional native structure of a protein given its primary structure, i.e., the corresponding sequence of amino acids. Different approaches have been proposed to model this problem, and this research explores the prediction of optimal structures using the well studied simplified lattice Hydrophobic and Polar (HP) model--in particular, on the 2D square lattice. We present a twofold result. First, we devise a new upper bound for the number of contacts achievable by an HP sequence, and show that it is in several cases more stringent than the upper bound previously known in literature. Then, we present an innovative algorithm that outperforms the state of the art in exact approaches for the prediction of optimal structures in lattice protein model, for 2D square lattices. The algorithm, called minwalk and based on a heavily pruned exhaustive search, also outperforms the state of the art in non-exact approaches in several cases. Due to this algorithm, it is now possible to prove optimal results in the square 2D lattice, for standard HP sequences of size up to 80 elements, which were only best-known-results previously. Furthermore, we provide the degeneracy (i.e. all optimal solutions) of such benchmark sequences, which was unknown in literature. These results can be a useful tool to foster advances in further research.

Construction of an intermediate-resolution lattice model and re-examination of the helix-coil transition: a dynamic Monte Carlo simulation

In protein modeling, spatial resolution and computational efficiency are always incompatible. As a compromise, an intermediate-resolution lattice model has been constructed in the present work. Each residue is decomposed into four basic units, i.e. the α-carbon group, the carboxyl group, the imino group, and the side-chain group, and each basic coarse-grained unit is represented by a minimum cubic box with eight lattice sites. The spacing of the lattice is about 0.56 Å, holding the highest spatial resolution for the present lattice protein models. As the first report of this new model, the helix-coil transition of a polyalanine chain was examined via dynamic Monte Carlo simulation. The period of formed α-helix was about 3.68 residues, close to that of a natural α-helix. The resultant backbone motion was found to be in the realistic regions of the conformational space in the Ramachandran plot. Helix propagation constant and nucleation constant were further determined through the dynamic hydrogen bonding process and torsional angle variation, and the results were used to make comparison between classical Zimm-Bragg theory and Lifson-Roig theory based on the Qian-Schellman relationship. The simulation results confirmed that our lattice model can reproduce the helix-coil transition of polypeptide and construct a moderately fine α-helix conformation without significantly weakening the priority in efficiency for a lattice model.

Generic, Hierarchical Framework for Massively Parallel Wang-Landau Sampling

We introduce a parallel Wang-Landau method based on the replica-exchange framework for MonteCarlo simulations. To demonstrate its advantages and general applicability for simulations of complex systems, we apply it to different spin models including spin glasses, the Ising model, and the Potts model, lattice protein adsorption, and the self-assembly process in amphiphilic solutions. Without loss of accuracy, the method gives significant speed-up and potentially scales up to petaflop machines.

Long-term evolution is surprisingly predictable in lattice proteins

It has long been debated whether natural selection acts primarily upon individual organisms, or whether it also commonly acts upon higher-level entities such as lineages. Two arguments against the effectiveness of long-term selection on lineages have been (i) that long-term evolutionary outcomes will not be sufficiently predictable to support a meaningful long-term fitness and (ii) that short-term selection on organisms will almost always overpower long-term selection. Here, we use a computational model of protein folding and binding called 'lattice proteins'. We quantify the long-term evolutionary success of lineages with two metrics called the k-fitness and k-survivability. We show that long-term outcomes are surprisingly predictable in this model: only a small fraction of the possible outcomes are ever realized in multiple replicates. Furthermore, the long-term fitness of a lineage depends only partly on its short-term fitness; other factors are also important, including the 'evolvability' of a lineage-its capacity to produce adaptive variation. In a system with a distinct short-term and long-term fitness, evolution need not be 'short-sighted': lineages may be selected for their long-term properties, sometimes in opposition to short-term selection. Similar evolutionary basins of attraction have been observed in vivo, suggesting that natural biological lineages will also have a predictive long-term fitness.

On the Characterization and Software Implementation of General Protein Lattice Models

models of proteins have been widely used as a practical means to computationally investigate general properties of the system. In lattice models any sterically feasible conformation is represented as a self-avoiding walk on a lattice, and residue types are limited in number. So far, only two- or three-dimensional lattices have been used. The inspection of the neighborhood of alpha carbons in the core of real proteins reveals that also lattices with higher coordination numbers, possibly in higher dimensional spaces, can be adopted. In this paper, a new general parametric lattice model for simplified protein conformations is proposed and investigated. It is shown how the supporting software can be consistently designed to let algorithms that operate on protein structures be implemented in a lattice-agnostic way. The necessary theoretical foundations are developed and organically presented, pinpointing the role of the concept of main directions in lattice-agnostic model handling. Subsequently, the model features across dimensions and lattice types are explored in tests performed on benchmark protein sequences, using a Python implementation. Simulations give insights on the use of square and triangular lattices in a range of dimensions. The trend of potential minimum for sequences of different lengths, varying the lattice dimension, is uncovered. Moreover, an extensive quantitative characterization of the usage of the so-called “move types” is reported for the first time. The proposed general framework for the development of lattice models is simple yet complete, and an object-oriented architecture can be proficiently employed for the supporting software, by designing ad-hoc classes. The proposed framework represents a new general viewpoint that potentially subsumes a number of solutions previously studied. The adoption of the described model pushes to look at protein structure issues from a more general and essential perspective, making computational investigations over simplified models more straightforward as well.

Generic folding and transition hierarchies for surface adsorption of hydrophobic-polar lattice model proteins

The thermodynamic behavior and structural properties of hydrophobic-polar (HP) lattice proteins interacting with attractive surfaces are studied by means of Wang-Landau sampling. Three benchmark HP sequences (48mer, 67mer, and 103mer) are considered with different types of surfaces, each of which attract either all monomers, only hydrophobic (H) monomers, or only polar (P) monomers, respectively. The diversity of folding behavior in dependence of surface strength is discussed. Analyzing the combined patterns of various structural observables, such as, e.g., the derivatives of the numbers of surface contacts, together with the specific heat, we are able to identify generic categories of folding and transition hierarchies. We also infer a connection between these transition categories and the relative surface strengths, i.e., the ratio of the surface attractive strength to the interchain attraction among H monomers. The validity of our proposed classification scheme is reinforced by the analysis of additional benchmark sequences. We thus believe that the folding hierarchies and identification scheme are generic for HP proteins interacting with attractive surfaces, regardless of chain length, sequence, or surface attraction.