Lattice Protein Research Articles

Replica Exchange Wang—Landau Simulation of Lattice Protein Folding Funnels

The resolution of Levinthal’s paradox concerning the ability of proteins to fold rapidly postulates the existence of a rough “folding funnel” in free energy space that guides the protein to its lowest free energy, native state. To study the folding of the protein ribonuclease A we have mapped it onto a 124 monomer, coarse-grained HP lattice model and onto an H0P model that also includes “neutral” 0-mers in addition to the hydrophobic H-mers and polar P-mers. Using Replica Exchange Wang-Landau sampling, we determined the density of states, g(E), which we then used to calculate the free energy of the protein vs end-to-end distance as a function of temperature. At low temperature the HP model shows a rather shallow and at free energy minimum, while the H0P model maintains a rough free energy funnel. Unlike the common, schematic figures, we find an asymmetric folding funnel that also changes shape substantially as the temperature decreases. Even the location of the free energy minimum shifts as the temperature decreases.

Graph's Topology and Free Energy of a Spin Model on the Graph.

In this Letter we investigate a direct relationship between a graph's topology and the free energy of a spin system on the graph. We develop a method of separating topological and energetic contributions to the free energy, and find that considering the topology is sufficient to qualitatively compare the free energies of different graph systems at high temperature, even when the energetics are not fully known. This method was applied to the metal lattice system with defects, and we found that it partially explains why point defects are more stable than high-dimensional defects. Given the energetics, we can even quantitatively compare free energies of different graph structures via a closed form of linear graph contributions. The closed form is applied to predict the sequence-space free energy of lattice proteins, which is a key factor determining the designability of a protein structure.

MBAR-enhanced lattice Monte Carlo simulation of the effect of helices on membrane protein aggregation

We study the effect of helical structure on the aggregation of proteins using a simplified lattice protein model with an implicit membrane environment. A recently proposed Monte Carlo approach, which exploits the proven statistical optimality of the MBAR estimator in order to improve simulation efficiency, was used. The results show that with both two and four proteins present, the tendency to aggregate is strongly expedited by the presence of amphipathic helix (APH), whereas a transmembrane helix (TMH) slightly disfavours aggregation. When four protein molecules are present, partially aggregated states (dimers and trimers) were more common when the APH was present, compared with the cases where no helices or only the TMH is present.

Characterizing folding funnels with replica exchange Wang-Landau simulation of lattice proteins.

We have studied the folding of ribonuclease A by mapping it onto coarse-grained lattice protein models. With replica exchange Wang-Landau sampling, we calculated the free energy vs end-to-end distance as a function of temperature. A mapping to the famous hydrophobic-polar (HP) model shows a relatively shallow folding funnel and flat free energy minimum, reflecting the high degeneracy of the ground state. In contrast, extending the HP model with an additional "neutral" monomer type (i.e., a mapping to the three-letter H0P model) has a well developed, rough free energy funnel with a low degeneracy ground state. In both cases, folding funnels are asymmetric with temperature dependent shape.

Direct coevolutionary couplings reflect biophysical residue interactions in proteins.

Coevolution of residues in contact imposes strong statistical constraints on the sequence variability between homologous proteins. Direct-Coupling Analysis (DCA), a global statistical inference method, successfully models this variability across homologous protein families to infer structural information about proteins. For each residue pair, DCA infers 21 × 21 matrices describing the coevolutionary coupling for each pair of amino acids (or gaps). To achieve the residue-residue contact prediction, these matrices are mapped onto simple scalar parameters; the full information they contain gets lost. Here, we perform a detailed spectral analysis of the coupling matrices resulting from 70 protein families, to show that they contain quantitative information about the physico-chemical properties of amino-acid interactions. Results for protein families are corroborated by the analysis of synthetic data from lattice-protein models, which emphasizes the critical effect of sampling quality and regularization on the biochemical features of the statistical coupling matrices.

Initial Graphite Disorder of Carbon Lattice Structures Increase Surface Hydrophilicity and Protein Adsorption

Initial Graphite Disorder of Carbon Lattice Structures Increase Surface Hydrophilicity and Protein Adsorption

Lattice model for amyloid peptides: OPEP force field parametrization and applications to the nucleus size of Alzheimer's peptides.

Coarse-grained protein lattice models approximate atomistic details and keep the essential interactions. They are, therefore, suitable for capturing generic features of protein folding and amyloid formation at low computational cost. As our aim is to study the critical nucleus sizes of two experimentally well-characterized peptide fragments Aβ16-22 and Aβ37-42 of the full length Aβ1-42 Alzheimer's peptide, it is important that simulations with the lattice model reproduce all-atom simulations. In this study, we present a comprehensive force field parameterization based on the OPEP (Optimized Potential for Efficient protein structure Prediction) force field for an on-lattice protein model, which incorporates explicitly the formation of hydrogen bonds and directions of side-chains. Our bottom-up approach starts with the determination of the best lattice force parameters for the Aβ16-22 dimer by fitting its equilibrium parallel and anti-parallel β-sheet populations to all-atom simulation results. Surprisingly, the calibrated force field is transferable to the trimer of Aβ16-22 and the dimer and trimer of Aβ37-42. Encouraged by this finding, we characterized the free energy landscapes of the two decamers. The dominant structure of the Aβ16-22 decamer matches the microcrystal structure. Pushing the simulations for aggregates between 4-mer and 12-mer suggests a nucleus size for fibril formation of 10 chains. In contrast, the Aβ37-42 decamer is largely disordered with mixed by parallel and antiparallel chains, suggesting that the nucleus size is >10 peptides. Our refined force field coupled to this on-lattice model should provide useful insights into the critical nucleation number associated with neurodegenerative diseases.

Benchmarking Inverse Statistical Approaches for Protein Structure and Design with Exactly Solvable Models.

Inverse statistical approaches to determine protein structure and function from Multiple Sequence Alignments (MSA) are emerging as powerful tools in computational biology. However the underlying assumptions of the relationship between the inferred effective Potts Hamiltonian and real protein structure and energetics remain untested so far. Here we use lattice protein model (LP) to benchmark those inverse statistical approaches. We build MSA of highly stable sequences in target LP structures, and infer the effective pairwise Potts Hamiltonians from those MSA. We find that inferred Potts Hamiltonians reproduce many important aspects of ‘true’ LP structures and energetics. Careful analysis reveals that effective pairwise couplings in inferred Potts Hamiltonians depend not only on the energetics of the native structure but also on competing folds; in particular, the coupling values reflect both positive design (stabilization of native conformation) and negative design (destabilization of competing folds). In addition to providing detailed structural information, the inferred Potts models used as protein Hamiltonian for design of new sequences are able to generate with high probability completely new sequences with the desired folds, which is not possible using independent-site models. Those are remarkable results as the effective LP Hamiltonians used to generate MSA are not simple pairwise models due to the competition between the folds. Our findings elucidate the reasons for the success of inverse approaches to the modelling of proteins from sequence data, and their limitations.

Consistent Treatment of Hydrophobicity in Protein Lattice Models Accounts for Cold Denaturation.

The hydrophobic effect stabilizes the native structure of proteins by minimizing the unfavorable interactions between hydrophobic residues and water through the formation of a hydrophobic core. Here, we include the entropic and enthalpic contributions of the hydrophobic effect explicitly in an implicit solvent model. This allows us to capture two important effects: a length-scale dependence and a temperature dependence for the solvation of a hydrophobic particle. This consistent treatment of the hydrophobic effect explains cold denaturation and heat capacity measurements of solvated proteins.

Alzheimer's Disease: Insights into Amyloid Fibril Formation from Lattice Monte Carlo Simulations

The neurodegenerative Alzheimer's disease is affecting more than 30 million people worldwide and is linked to the aggregation of the amyloid-β (Aβ) proteins of 40/42 amino acids. Despite extensive experimental studies, the mechanism of formation of amyloid fibrils and plaques is still unclear. To complement experiments, computational studies based on all-atom simulations are very often used. They are however limited in the sampling problem for small systems and short time scales (ns-µs).Our primary aim is to develop a coarse-grained protein lattice model for amyloid proteins. This allows us to determine structures, thermodynamics and dynamics of very large amyloid systems, with the focus on the characterization of oligomers prior to nucleation, and the mechanisms leading to amyloid fibril.After presenting our model and the force field, we show the results of extensive Monte Carlo simulations of oligomers formed by the (Aβ)16-22 and (Aβ)37-42 fragments of the Alzheimer's peptide Aβ. Considering different system sizes, we show that the nucleus size is larger than 10 chains.

On the Entropy of Protein Families

Proteins are essential components of living systems, capable of performing a huge variety of tasks at the molecular level, such as recognition, signalling, copy, transport, ... The protein sequences realizing a given function may largely vary across organisms, giving rise to a protein family. Here, we estimate the entropy of those families based on different approaches, including Hidden Markov Models used for protein databases and inferred statistical models reproducing the low-order (1-and 2-point) statistics of multi-sequence alignments. We also compute the entropic cost, that is, the loss in entropy resulting from a constraint acting on the protein, such as the fixation of one particular amino-acid on a specific site, and relate this notion to the escape probability of the HIV virus. The case of lattice proteins, for which the entropy can be computed exactly, allows us to provide another illustration of the concept of cost, due to the competition of different folds. The relevance of the entropy in relation to directed evolution experiments is stressed.

Folding in a semi-flexible lattice model for Crambin

Using the Replica-Exchange Wang-Landau sampling method, we investigated and compared three different coarse-grained lattice protein models for the small, hydrophobic protein Crambin. We show that slight extensions of the HP lattice protein model, including the stiffness of bonds can lead to a significant decrease in ground-state degeneracies (up to 5 orders of magnitudes). Moreover, the ground-state structures begin to bear resemblance to native structures observed in real Crambin.

Study on collapse and folding transitions of a lattice protein using exact enumeration

We study the conformational transitions of proteins by using the hydrophobic-polar (HP) model on a square lattice. In contrast with previous studies that relied on sampling techniques, we conducted an exhaustive enumeration of all possible conformations to obtain the density of states so that exact physical quantities could be computed. We study the conformational transitions of three sequences with varying lengths and observe both the collapse and folding transitions. The transitions exhibit distinct characteristics that depend on the sequence.

Exact enumeration of conformations for two and three dimensional lattice proteins

We report an efficient methodology for exactly enumerating conformations of lattice proteins, taking into account the self-avoiding nature. These self-avoiding walks in square and simple cubic lattices take into account, the detailed paths by which a destination site can be reached from a starting site. The strategy employing JAVA programming enables the exact enumeration for very large lattice sizes. The estimation of these conformations is helpful in designing the protein sequences pertaining to Hydrophobic-Polar models. Program summaryProgram title: Exact enumeration of conformations in lattice proteinsNo of lines in distributed program: 1027No of bytes in distributed program: 51698Programming language: JAVAComputer: Tested on Intel®Core(TM) i5-4570. Will function on any computer with JAVA compilers.Operating system: Tested on Microsoft Windows 7 Professional. Should run on any platform with JAVAHas the code been vectorized or parallelized? : NoRAM: Depends on the system size.Nature of the problem: The problem involves the enumeration of the conformation of lattice proteins for various amino acid chain lengths by considering them as self-avoiding walks. The methodology is illustrated for square and simple cubic lattices.Solution method: The code was implemented using the nodes as in a tree which was arranged in the stack. Each element is visited once and is moved to the next node. The process is repeated till all nodes are visited and the process terminates when a site is revisited.Restrictions: Runs only in JAVA compilersRunning time: Depends on the system size. Using 4 processors, the output leading to entries in Tables 1 and 2 was generated within one minute.The listing of the codes and the output of the program are provided in the Supporting Information.

Effect of surface attractive strength on structural transitions of a confined HP lattice protein

We investigate the influence of surface attractive strength on structural transitions of a hydrophobic-polar (HP) lattice protein confined in a slit formed by two parallel, attractive walls. We apply Wang-Landau sampling together with efficient Monte Carlo updates to estimate the density of states of the system. The conformational transitions, namely, the debridging process and hydrophobic core formation, can be identified by analyzing the specific heat together with several structural observables, such as the numbers of surface contacts, the number of hydrophobic pairs, and radii of gyration in different directions. As temperature decreases, we find that the occurrence of the debridging process is conditional depending on the surface attractive strength. This, in turn, affects the nature of the hydrophobic core formation that takes place at a lower temperature. We illustrate these observations with the aid of a HP protein chain with 48 monomers.

Massively parallel sampling of lattice proteins reveals foundations of thermal adaptation.

Evolution of proteins in bacteria and archaea living in different conditions leads to significant correlations between amino acid usage and environmental temperature. The origins of these correlations are poorly understood, and an important question of protein theory, physics-based prediction of types of amino acids overrepresented in highly thermostable proteins, remains largely unsolved. Here, we extend the random energy model of protein folding by weighting the interaction energies of amino acids by their frequencies in protein sequences and predict the energy gap of proteins designed to fold well at elevated temperatures. To test the model, we present a novel scalable algorithm for simultaneous energy calculation for many sequences in many structures, targeting massively parallel computing architectures such as graphics processing unit. The energy calculation is performed by multiplying two matrices, one representing the complete set of sequences, and the other describing the contact maps of all structural templates. An implementation of the algorithm for the CUDA platform is available at http://www.github.com/kzeldovich/galeprot and calculates protein folding energies over 250 times faster than a single central processing unit. Analysis of amino acid usage in 64-mer cubic lattice proteins designed to fold well at different temperatures demonstrates an excellent agreement between theoretical and simulated values of energy gap. The theoretical predictions of temperature trends of amino acid frequencies are significantly correlated with bioinformatics data on 191 bacteria and archaea, and highlight protein folding constraints as a fundamental selection pressure during thermal adaptation in biological evolution.

Effective temperature of mutations.

Biological macromolecules experience two seemingly very different types of noise acting on different time scales: (i)point mutations corresponding to changes in molecular sequence and (ii)thermal fluctuations. Examining the secondary structures of a large number of microRNA precursor sequences and model lattice proteins, we show that the effects of single point mutations are statistically indistinguishable from those of an increase in temperature by a few tens of kelvins. The existence of such an effective mutational temperature establishes a quantitative connection between robustness to genetic (mutational) and environmental (thermal) perturbations.

Sequence Determines Degree of Knottedness in a Coarse-Grained Protein Model

Knots are abundant in globular homopolymers but rare in globular proteins. To shed new light on this long-standing conundrum, we study the influence of sequence on the formation of knots in proteins under native conditions within the framework of the hydrophobic-polar lattice protein model. By employing large-scale Wang-Landau simulations combined with suitable MonteCarlo trial moves we show that even though knots are still abundant on average, sequence introduces large variability in the degree of self-entanglements. Moreover, we are able to design sequences which are either almost always or almost never knotted. Our findings serve as proof of concept that the introduction of just one additional degree of freedom per monomer (in our case sequence) facilitates evolution towards a protein universe in which knots are rare.

Identification of peptide adsorbates for strong nanoparticle–nanoparticle binding by lattice protein simulations

Nanoparticles with short peptides adsorbed to their surfaces often assemble into interesting structures when suspended solution. In this paper, the interaction between a peptide adsorbed to a surface and another peptide adsorbed to an opposing surface is studied using the lattice protein model. It is found that the interaction strength between the two peptides generally increases as the hydrophobicity of peptides increases. Moreover, there is a preference for hydrophobic amino acids to be neighboring hydrophilic amino acids in cases which yield strong peptide–peptide interactions. These results provide insights for tailoring the strength of nanoparticle–nanoparticle binding by engineering of peptide adsorbates.

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.