A test of lattice protein folding algorithms

Globular Proteins and Where to Find Them within a Polymer Brush-A Case Study

Protein adsorption by polymerized surfaces is an interdisciplinary topic that has been approached in many ways, leading to a plethora of theoretical, numerical and experimental insight. There is a wide variety of models trying to accurately capture the essence of adsorption and its effect on the conformations of proteins and polymers. However, atomistic simulations are case-specific and computationally demanding. Here, we explore universal aspects of the dynamics of protein adsorption through a coarse-grained (CG) model, that allows us to explore the effects of various design parameters. To this end, we adopt the hydrophobic-polar (HP) model for proteins, place them uniformly at the upper bound of a CG polymer brush whose multibead-spring chains are tethered to a solid implicit wall. We find that the most crucial factor affecting the adsorption efficiency appears to be the polymer grafting density, while the size of the protein and its hydrophobicity ratio come also into play. We discuss the roles of ligands and attractive tethering surfaces to the primary adsorption as well as secondary and ternary adsorption in the presence of attractive (towards the hydrophilic part of the protein) beads along varying spots of the backbone of the polymer chains. The percentage and rate of adsorption, density profiles and the shapes of the proteins, alongside with the respective potential of mean force are recorded to compare the various scenarios during protein adsorption.

An Introduction to the Stickers-and-Spacers Framework as Applied to Biomolecular Condensates

Cellular organization is determined by a combination of membrane-bound and membrane-less biomolecular assemblies that range from clusters of tens of molecules to micrometer-sized cellular bodies. Over the last decade, membrane-less assemblies have come to be referred to as biomolecular condensates, reflecting their ability to condense specific molecules with respect to the remainder of the cell. In many cases, the physics of phase transitions provides a conceptual framework and a mathematical toolkit to describe the assembly, maintenance, and dissolution of biomolecular condensates. Among the various quantitative and qualitative models applied to understand intracellular phase transitions, the stickers-and-spacers framework offers an intuitive yet rigorous means to map biomolecular sequences and structure to the driving forces needed for higher-order assembly. This chapter introduces the fundamental concepts behind the stickers-and-spacers model, considers its application to different biological systems, and discusses limitations and misconceptions around the model.

Folding Coarse-Grained Oligomer Models with PyRosetta

Non-biological foldamers are a promising class of macromolecules that share similarities to classical biopolymers such as proteins and nucleic acids. Currently, designing novel foldamers is a non-trivial process, often involving many iterations of trial synthesis and characterization until folded structures are observed. In this work, we aim to tackle these foldamer design challenges using computational modeling techniques. We developed CG PyRosetta, an extension to the popular protein folding python package, PyRosetta, which introduces coarse-grained (CG) residues into PyRosetta, enabling the folding of toy CG foldamer models. Although these models are simplified, they can help explore overarching physical hypotheses about how oligomers can form. Through systematic variation of CG parameters in these models, we can investigate various folding hypotheses at the CG scale to inform the design process of new foldamer chemistries. In this study, we demonstrate CG PyRosetta's ability to identify minimum energy structures with a diverse structural search over a range of simple models, as well as two hypothesis-driven parameter scans investigating the effects of side-chain size and internal backbone angle on secondary structures. We are able to identify several types of secondary structures from single- and double-helices to sheet-like and knot-like structures. We show how side-chain size and backbone bond angle both play an important role in the structure and energetics of these toy models. Optimal side-chain sizes promote favorable packing of side chains, while specific backbone bond angles influence the specific helix type found in folded structures.

Modeling protein structure as a stable static equilibrium

We present evidence that the protein structure can be modeled as a stable static equilibrium, determined mainly by compressive supports in the nonpolar interior. That is, protein structures derive their structural strength through the same mechanical principles as do conventional structures like bridges and buildings. This is based on the observation that the experimentally elucidated structural determinants, the interior nonpolar side chains, are engaged in strong compressions in static terms. At the same time, major substructures in proteins, helices and h-bonded strands, because of their geometry, inherently leave gaps in the space they occupy. Under the compressive force, nonpolar side chains from one substructure can protrude into the gaps of another neighboring substructure and block its motion. As a result, interlocking of substructures can form, which builds up the nonpolar core assembly. The native structure then is the one with the structurally most stable core assembly. While intuitively appealing, this is a radical departure from the prevailing thinking that protein native structure is determined by global energy minimum, which is founded on thermodynamic hypothesis. Furthermore, to develop an effective model for analyzing protein structure with conventional tools, a proper mechanical representation must be established. By proving that the stability of the equilibrium in compressive interactions is conditioned on a form of mechanical energy minimum, we show that our notion of native structure can be equally consistent with the thermodynamic hypothesis. By mathematically treating the blocking action, an interaction, as a bar, a physical object, we succeed in representing and analyzing the core assembly as truss, a conventional structure. In this paper we define and expound step-by-step increasingly integrated interlocking patterns. We then analyze the core assemblies of a large set of diverse protein database structures. A native structure can be distinguished from decoys by comparing the composition and strength of their core assemblies. We show the results for two sets of native structures vs decoys.

Transferable and Polarizable Coarse Grained Model for Proteins─ProMPT

The application of classical molecular dynamics (MD) simulations at atomic resolution (fine-grained level, FG), to most biomolecular processes, remains limited because of the associated computational complexity of representing all the atoms. This problem is magnified in the presence of protein-based biomolecular systems that have a very large conformational space, and MD simulations with fine-grained resolution have slow dynamics to explore this space. Current transferable coarse grained (CG) force fields in literature are either limited to only peptides with the environment encoded in an implicit form or cannot capture transitions into secondary/tertiary peptide structures from a primary sequence of amino acids. In this work, we present a transferable CG force field with an explicit representation of the environment for accurate simulations with proteins. The force field consists of a set of pseudoatoms representing different chemical groups that can be joined/associated together to create different biomolecular systems. This preserves the transferability of the force field to multiple environments and simulation conditions. We have added electronic polarization that can respond to environmental heterogeneity/fluctuations and couple it to protein's structural transitions. The nonbonded interactions are parametrized with physics-based features such as solvation and partitioning free energies determined by thermodynamic calculations and matched with experiments and/or atomistic simulations. The bonded potentials are inferred from corresponding distributions in nonredundant protein structure databases. We present validations of the CG model with simulations of well-studied aqueous protein systems with specific protein fold types─Trp-cage, Trpzip4, villin, WW-domain, and β-α-β. We also explore the applications of the force field to study aqueous aggregation of Aβ 16-22 peptides.

Thermodynamics of hydrophobic-polar model proteins on the face-centered cubic lattice

The HP model, a coarse-grained protein representation with only hydrophobic (H) and polar (P) amino acids, has already been extensively studied on the simple cubic (SC) lattice. However, this geometry severely restricts possible bond angles, and a simple improvement is to instead use the face-centered cubic (fcc) lattice. In this paper, the density of states and ground state energies are calculated for several benchmark HP sequences on the fcc lattice using the replica-exchange Wang-Landau algorithm and a powerful set of Monte Carlo trial moves. Results from the fcc lattice proteins are directly compared with those obtained from a previous lattice protein folding study with a similar methodology on the SC lattice. A thermodynamic analysis shows comparable folding behavior between the two lattice geometries, but with a greater rate of hydrophobic-core formation persisting into lower temperatures on the fcc lattice.

Understanding the role of hydrophobic patches in protein disaggregation

Protein folding is a very complex process and, so far, the mechanism of folding still intrigues the research community. Despite a large conformational space available (O(10⁴⁷) for a 100 amino acid residue), most proteins fold into their native state within a very short time. While small proteins fold relatively fast (a few microseconds) large globular proteins may take as long as several milliseconds to fold. During the folding process, the protein synthesized in the ribosome is exposed to the crowded environment of the cell and is easily prone to misfolding and aggregation due to interactions with other proteins or biomacromolecules present within the cell. These large proteins, therefore, rely on chaperones for their folding and repair. Chaperones are known to have hydrophobic patchy domains that play a crucial role in shielding the protein against misfolding and disaggregation of aggregated proteins. In the current article, Monte Carlo simulations carried out in the framework of the hydrophobic-polar (H-P) lattice model indicate that hydrophobic patchy domains drastically reduce the inter-protein interactions and are efficient in disaggregating proteins. The effectiveness of the disaggregation depends on the size and distribution of these patches on the surface and also on the strength of the interaction between the protein and the surface. Further, our results indicate that when the patch is complementary to the exposed hydrophobic patch of the protein, protein disaggregation is accompanied by stabilization of the protein even relative to its bulk behavior due to favorable protein-surface interactions. We believe that these findings shed light on the role of the class of chaperones known as heat shock proteins (Hsps) on protein disaggregation and refolding.

A Novel Branch-and-Bound Algorithm for the Protein Folding Problem in the 3D HP Model

The protein folding problem (PFP) is an important issue in bioinformatics and biochemical physics. One of the most widely studied models of protein folding is the hydrophobic-polar (HP) model introduced by Dill. The PFP in the three-dimensional (3D) lattice HP model has been shown to be NP-complete; the proposed algorithms for solving the problem can therefore only find near-optimal energy structures for most long benchmark sequences within acceptable time periods. In this paper, we propose a novel algorithm based on the branch-and-bound approach to solve the PFP in the 3D lattice HP model. For 10 48-monomer benchmark sequences, our proposed algorithm finds the lowest energies so far within comparable computation times than previous methods.

Effects of lattice constraints in coarse-grained protein models

We compare and contrast folding behavior in several coarse-grained protein models, both on- and off-lattice, in an attempt to uncover the effect of lattice constraints in these kinds of models. Using modern, extended ensemble Monte Carlo methods-Wang-Landau sampling, multicanonical sampling, replica-exchange Wang-Landau sampling, and replica-exchange multicanonical sampling, we investigate the thermodynamic and structural behavior of the protein Crambin within the context of the hydrophobic-polar, hydrophobic-"neutral"-polar (H0P), and semi-flexible H0P model frameworks. We uncover the folding process in all cases; all models undergo, at least, the two major structural transitions observed in nature-the coil-globule collapse and the folding transition. As the complexity of the model increases, these two major transitions begin to split into multi-step processes, wherein the lattice coarse-graining has a significant impact on the details of these processes. The results show that the level of structural coarse-graining is coupled to the level of interaction coarse-graining.

Theoretical and computational advances in protein misfolding

Misfolded proteins escape the cellular quality control mechanism and fail to fold properly or remain correctly folded leading to a loss in their functional specificity. Thus misfolding of proteins cause a large number of very different diseases ranging from errors in metabolism to various types of complex neurodegenerative diseases. A theoretical and computational perspective of protein misfolding is presented with a special emphasis on its salient features, mechanism and consequences. These insights quantitatively analyze different determinants of misfolding, that may be applied to design disease specific molecular targets.

The role of chain-stiffness in lattice protein models: A replica-exchange Wang-Landau study

Using Monte Carlo simulations, we investigate simple, physically motivated extensions to the hydrophobic-polar lattice protein model for the small (46 amino acid) protein Crambin. We use two-dimensional replica-exchange Wang-Landau sampling to study the effects of a bond angle stiffness parameter on the folding and uncover a new step in the collapse process for particular values of this stiffness parameter. A physical interpretation of the folding is developed by analysis of changes in structural quantities, and the free energy landscape is explored. For these special values of stiffness, we find non-degenerate ground states, a property that is consistent with behavior of real proteins, and we use these unique ground states to elucidate the formation of native contacts during the folding process. Through this analysis, we conclude that chain-stiffness is particularly influential in the low energy, low temperature regime of the folding process once the lattice protein has partially collapsed.

Protein folding prediction in the HP model using ions motion optimization with a greedy algorithm

Background

The function of a protein is determined by its native protein structure. Among many protein prediction methods, the Hydrophobic-Polar (HP) model, an ab initio method, simplifies the protein folding prediction process in order to reduce the prediction complexity.

Results

In this study, the ions motion optimization (IMO) algorithm was combined with the greedy algorithm (namely IMOG) and implemented to the HP model for the protein folding prediction based on the 2D-triangular-lattice model. Prediction results showed that the integration method IMOG provided a better prediction efficiency in a HP model. Compared to others, our proposed method turned out as superior in its prediction ability and resilience for most of the test sequences. The efficiency of the proposed method was verified by the prediction results. The global search capability and the ability to escape from the local best solution of IMO combined with a local search (greedy algorithm) to the new algorithm IMOG greatly improve the search for the best solution with reliable protein folding prediction.

Conclusion

Overall, the HP model integrated with IMO and a greedy algorithm as IMOG provides an improved way of protein structure prediction of high stability, high efficiency, and outstanding performance.

How Many Protein Sequences Fold to a Given Structure? A Coevolutionary Analysis

Quantifying the relationship between protein sequence and structure is key to understanding the protein universe. A fundamental measure of this relationship is the total number of amino acid sequences that can fold to a target protein structure, known as the "sequence capacity," which has been suggested as a proxy for how designable a given protein fold is. Although sequence capacity has been extensively studied using lattice models and theory, numerical estimates for real protein structures are currently lacking. In this work, we have quantitatively estimated the sequence capacity of 10 proteins with a variety of different structures using a statistical model based on residue-residue co-evolution to capture the variation of sequences from the same protein family. Remarkably, we find that even for the smallest protein folds, such as the WW domain, the number of foldable sequences is extremely large, exceeding the Avogadro constant. In agreement with earlier theoretical work, the calculated sequence capacity is positively correlated with the size of the protein, or better, the density of contacts. This allows the absolute sequence capacity of a given protein to be approximately predicted from its structure. On the other hand, the relative sequence capacity, i.e., normalized by the total number of possible sequences, is an extremely tiny number and is strongly anti-correlated with the protein length. Thus, although there may be more foldable sequences for larger proteins, it will be much harder to find them. Lastly, we have correlated the evolutionary age of proteins in the CATH database with their sequence capacity as predicted by our model. The results suggest a trade-off between the opposing requirements of high designability and the likelihood of a novel fold emerging by chance.

Searching for the Pareto frontier in multi-objective protein design

The goal of protein engineering and design is to identify sequences that adopt three-dimensional structures of desired function. Often, this is treated as a single-objective optimization problem, identifying the sequence-structure solution with the lowest computed free energy of folding. However, many design problems are multi-state, multi-specificity, or otherwise require concurrent optimization of multiple objectives. There may be tradeoffs among objectives, where improving one feature requires compromising another. The challenge lies in determining solutions that are part of the Pareto optimal set-designs where no further improvement can be achieved in any of the objectives without degrading one of the others. Pareto optimality problems are found in all areas of study, from economics to engineering to biology, and computational methods have been developed specifically to identify the Pareto frontier. We review progress in multi-objective protein design, the development of Pareto optimization methods, and present a specific case study using multi-objective optimization methods to model the tradeoff between three parameters, stability, specificity, and complexity, of a set of interacting synthetic collagen peptides.

Biophysics of protein evolution and evolutionary protein biophysics

The study of molecular evolution at the level of protein-coding genes often entails comparing large datasets of sequences to infer their evolutionary relationships. Despite the importance of a protein's structure and conformational dynamics to its function and thus its fitness, common phylogenetic methods embody minimal biophysical knowledge of proteins. To underscore the biophysical constraints on natural selection, we survey effects of protein mutations, highlighting the physical basis for marginal stability of natural globular proteins and how requirement for kinetic stability and avoidance of misfolding and misinteractions might have affected protein evolution. The biophysical underpinnings of these effects have been addressed by models with an explicit coarse-grained spatial representation of the polypeptide chain. Sequence-structure mappings based on such models are powerful conceptual tools that rationalize mutational robustness, evolvability, epistasis, promiscuous function performed by 'hidden' conformational states, resolution of adaptive conflicts and conformational switches in the evolution from one protein fold to another. Recently, protein biophysics has been applied to derive more accurate evolutionary accounts of sequence data. Methods have also been developed to exploit sequence-based evolutionary information to predict biophysical behaviours of proteins. The success of these approaches demonstrates a deep synergy between the fields of protein biophysics and protein evolution.

Distance-dependent hydrophobic-hydrophobic contacts in protein folding simulations

Successful prediction of protein folding from an amino acid sequence is a challenge in computational biology. In order to reveal the geometric constraints that drive protein folding, highlight those constraints kept or missed by distinct lattices and for establishing which class of intra- and inter-secondary structure element interactions is the most relevant for the correct folding of proteins, we have calculated inter-alpha carbon distances in a set of 42 crystal structures consisting of mainly helix, sheet or mixed conformations. The inter-alpha carbon distances were also calculated in several lattice "hydrophobic-polar" models built from the same protein set. We found that helix structures are more prone to form "hydrophobic-hydrophobic" contacts than beta-sheet structures. At a distance lower than or equal to 3.8 Å (very short-range interactions), "hydrophobic-hydrophobic" contacts are almost absent in the native structures, while they are frequent in all the analyzed lattice models. At distances in-between 3.8 and 9.5 Å (short-/medium-range interactions), the best performing lattice for reproducing mainly helix structures is the body-centered-cubic lattice. If protein structures contain sheet portions, lattice performances get worse, with few exceptions observed for double-tetrahedral and body-centered-cubic lattices. Finally, we can observe that ab initio protein folding algorithms, i.e. those based on the employment of lattices and Monte Carlo simulated annealings, can be improved simply and effectively by preventing the generation of "hydrophobic-hydrophobic" contacts shorter than 3.8 Å, by monitoring the "hydrophobic-hydrophobic/polar-polar" contact ratio in short-/medium distance ranges and by using preferentially a body-centered-cubic lattice.

Stability of proteins on hydrophilic surfaces

The physical and chemical properties of solid substrates or surfaces critically influence the stability and activity of immobilized proteins such as enzymes. Reports of increased stability and activity of enzymes near/on surfaces as compared with those in solution abound; however, a mechanistic understanding is wanting. Simulations and experiments are used here to provide details toward such a mechanistic understanding. Experiments demonstrate increased activity of alcohol dehydrogenase (ADH) inside moderate hydrophilic mesopourous silica (SBA-15) pores but drastically decreased activity inside very hydrophilic NH2-SBA-15 surfaces as compared with that in solution. Also, the temperature stability of ADH was increased over that in solution when immobilized in a cavity with a mildly hydrophilic surface. Simulations confirm these experimental findings. Simulations calculated in the framework of a hydrophobic-polar (H-P) lattice model show increased thermal stability of a model 64-mer peptide on positive and zero curvature surfaces over that in solution. Peptides immobilized inside negative curvature cavities (concave) with hydrophilic surfaces exhibit increased stability only inside pores that are only 3-4 nm larger than the hydrodynamic radius of the peptide. Peptides are destabilized, however, when the surface hydrophilic character inside very small cavities/pores becomes large.

Frustration in biomolecules

Biomolecules are the prime information processing elements of living matter. Most of these inanimate systems are polymers that compute their own structures and dynamics using as input seemingly random character strings of their sequence, following which they coalesce and perform integrated cellular functions. In large computational systems with finite interaction-codes, the appearance of conflicting goals is inevitable. Simple conflicting forces can lead to quite complex structures and behaviors, leading to the concept of frustration in condensed matter. We present here some basic ideas about frustration in biomolecules and how the frustration concept leads to a better appreciation of many aspects of the architecture of biomolecules, and especially how biomolecular structure connects to function by means of localized frustration. These ideas are simultaneously both seductively simple and perilously subtle to grasp completely. The energy landscape theory of protein folding provides a framework for quantifying frustration in large systems and has been implemented at many levels of description. We first review the notion of frustration from the areas of abstract logic and its uses in simple condensed matter systems. We discuss then how the frustration concept applies specifically to heteropolymers, testing folding landscape theory in computer simulations of protein models and in experimentally accessible systems. Studying the aspects of frustration averaged over many proteins provides ways to infer energy functions useful for reliable structure prediction. We discuss how frustration affects folding mechanisms. We review here how the biological functions of proteins are related to subtle local physical frustration effects and how frustration influences the appearance of metastable states, the nature of binding processes, catalysis and allosteric transitions. In this review, we also emphasize that frustration, far from being always a bad thing, is an essential feature of biomolecules that allows dynamics to be harnessed for function. In this way, we hope to illustrate how Frustration is a fundamental concept in molecular biology.

A Parallel Framework for Multipoint Spiral Search in ab Initio Protein Structure Prediction

Protein structure prediction is computationally a very challenging problem. A large number of existing search algorithms attempt to solve the problem by exploring possible structures and finding the one with the minimum free energy. However, these algorithms perform poorly on large sized proteins due to an astronomically wide search space. In this paper, we present a multipoint spiral search framework that uses parallel processing techniques to expedite exploration by starting from different points. In our approach, a set of random initial solutions are generated and distributed to different threads. We allow each thread to run for a predefined period of time. The improved solutions are stored threadwise. When the threads finish, the solutions are merged together and the duplicates are removed. A selected distinct set of solutions are then split to different threads again. In our ab initio protein structure prediction method, we use the three-dimensional face-centred-cubic lattice for structure-backbone mapping. We use both the low resolution hydrophobic-polar energy model and the high-resolution 20 × 20 energy model for search guiding. The experimental results show that our new parallel framework significantly improves the results obtained by the state-of-the-art single-point search approaches for both energy models on three-dimensional face-centred-cubic lattice. We also experimentally show the effectiveness of mixing energy models within parallel threads.

Surface-mediated protein disaggregation

Preventing protein aggregation is of both biological and industrial importance. Interprotein interactions between the hydrophobic residues of the protein are known to be the major driving force for protein aggregation. In this article, we show how surface chemistry and curvature can be tuned to mitigate these interprotein interactions. Our results calculated in the framework of the Hydrophobic-Polar (HP) lattice model show that interprotein interactions can be drastically reduced by increasing the surface hydrophobicity to a critical value corresponding to the adsorption transition of the protein. At this value of surface hydrophobicity, proteins lose interprotein contacts to gain surface contacts, and thus the surface helps to reduce the interprotein interactions. Furthermore, we show that the adsorption of the proteins inside hydrophobic pores of optimal sizes are most efficient at both reducing interprotein contacts and simultaneously retaining most of the native contacts probably as a result of confinement-induced stabilization.

Quantitative theory of hydrophobic effect as a driving force of protein structure

Various studies suggest that the hydrophobic effect plays a major role in driving the folding of proteins. In the past, however, it has been challenging to translate this understanding into a predictive, quantitative theory of how the full pattern of sequence hydrophobicity in a protein shapes functionally important features of its tertiary structure. Here, we extend and apply such a phenomenological theory of the sequence-structure relationship in globular protein domains, which had previously been applied to the study of allosteric motion. In an effort to optimize parameters for the model, we first analyze the patterns of backbone burial found in single-domain crystal structures, and discover that classic hydrophobicity scales derived from bulk physicochemical properties of amino acids are already nearly optimal for prediction of burial using the model. Subsequently, we apply the model to studying structural fluctuations in proteins and establish a means of identifying ligand-binding and protein-protein interaction sites using this approach.

A firefly-inspired method for protein structure prediction in lattice models

We introduce a Firefly-inspired algorithmic approach for protein structure prediction over two different lattice models in three-dimensional space. In particular, we consider three-dimensional cubic and three-dimensional face-centred-cubic (FCC) lattices. The underlying energy models are the Hydrophobic-Polar (H-P) model, the Miyazawa–Jernigan (M-J) model and a related matrix model. The implementation of our approach is tested on ten H-P benchmark problems of a length of 48 and ten M-J benchmark problems of a length ranging from 48 until 61. The key complexity parameter we investigate is the total number of objective function evaluations required to achieve the optimum energy values for the H-P model or competitive results in comparison to published values for the M-J model. For H-P instances and cubic lattices, where data for comparison are available, we obtain an average speed-up over eight instances of 2.1, leaving out two extreme values (otherwise, 8.8). For six M-J instances, data for comparison are available for cubic lattices and runs with a population size of 100, where, a priori, the minimum free energy is a termination criterion. The average speed-up over four instances is 1.2 (leaving out two extreme values, otherwise 1.1), which is achieved for a population size of only eight instances. The present study is a test case with initial results for ad hoc parameter settings, with the aim of justifying future research on larger instances within lattice model settings, eventually leading to the ultimate goal of implementations for off-lattice models.

Balancing bond, nonbond, and gō-like terms in coarse grain simulations of conformational dynamics

Characterization of the protein conformational landscape remains a challenging problem, whether it concerns elucidating folding mechanisms, predicting native structures or modeling functional transitions. Coarse-grained molecular dynamics simulation methods enable exhaustive sampling of the energetic landscape at resolutions of biological interest. The general utility of structure-based models is reviewed along with their differing levels of approximation. Simple Gō models incorporate attractive native interactions and repulsive nonnative contacts, resulting in an ideal smooth landscape. Non-Gō coarse-grained models reduce the parameter set as needed but do not include bias to any desired native structure. While non-Gō models have achieved limited success in protein coarse-graining, they can be combined with native structured-based potentials to create a balanced and powerful force field. Recent applications of such Gō-like models have yielded insight into complex folding mechanisms and conformational transitions in large macromolecules. The accuracy and usefulness of reduced representations are also revealed to be a function of the mathematical treatment of the intrinsic bonded topology.

Funneling and frustration in the energy landscapes of some designed and simplified proteins

We explore the similarities and differences between the energy landscapes of proteins that have been selected by nature and those of some proteins designed by humans. Natural proteins have evolved to function as well as fold, and this is a source of energetic frustration. The sequence of Top7, on the other hand, was designed with architecture alone in mind using only native state stability as the optimization criterion. Its topology had not previously been observed in nature. Experimental studies show that the folding kinetics of Top7 is more complex than the kinetics of folding of otherwise comparable naturally occurring proteins. In this paper, we use structure prediction tools, frustration analysis, and free energy profiles to illustrate the folding landscapes of Top7 and two other proteins designed by Takada. We use both perfectly funneled (structure-based) and predictive (transferable) models to gain insight into the role of topological versus energetic frustration in these systems and show how they differ from those found for natural proteins. We also study how robust the folding of these designs would be to the simplification of the sequences using fewer amino acid types. Simplification using a five amino acid type code results in comparable quality of structure prediction to the full sequence in some cases, while the two-letter simplification scheme dramatically reduces the quality of structure prediction.

Stability of proteins inside a hydrophobic cavity

We study the effects of confinement and hydrophobicity of a spherical cavity on the structural and thermal stability of proteins in the framework of a hydrophobic-polar (HP) lattice model. We observe that a neutral confinement stabilizes the folded state of the protein by eliminating many of the open-chain conformations of the unfolded state. Hydrophobic confinement always destabilizes the protein because of protein-surface interactions. However, for moderate surface hydrophobicities, the protein remains stabilized relative to its state in free solution because of the dominance of entropic effects. These results are consistent with our experimental findings of (a) enhanced activity of alcohol dehydrogenase (ADH) when immobilized inside the essentially cylindrical pores of hydrophilic mesoporous silica (SBA-15) and (b) unaffected activity when immobilized inside weakly hydrophobic pores of methacrylate resin compared to its activity in free solution. In the same vein, our predictions are also consistent with the behavior of lysozyme and myoglobin in hydrophilic and hydrophobic SBA-15, which show qualitatively the same trends. Apparently, our results have validity across these very different enzymes, and we therefore suggest that confinement can be used to selectively improve enzyme performance.

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.

Spiral search: a hydrophobic-core directed local search for simplified PSP on 3D FCC lattice

Background

Protein structure prediction is an important but unsolved problem in biological science. Predicted structures vary much with energy functions and structure-mapping spaces. In our simplified ab initio protein structure prediction methods, we use hydrophobic-polar (HP) energy model for structure evaluation, and 3-dimensional face-centred-cubic lattice for structure mapping. For HP energy model, developing a compact hydrophobic-core (H-core) is essential for the progress of the search. The H-core helps find a stable structure with the lowest possible free energy.

Results

In order to build H-cores, we present a new Spiral Search algorithm based on tabu-guided local search. Our algorithm uses a novel H-core directed guidance heuristic that squeezes the structure around a dynamic hydrophobic-core centre. We applied random walks to break premature H-cores and thus to avoid early convergence. We also used a novel relay-restart technique to handle stagnation.

Conclusions

We have tested our algorithms on a set of benchmark protein sequences. The experimental results show that our spiral search algorithm outperforms the state-of-the-art local search algorithms for simplified protein structure prediction. We also experimentally show the effectiveness of the relay-restart.

Limiting assumptions in molecular modeling: electrostatics

Molecular mechanics attempts to represent intermolecular interactions in terms of classical physics. Initial efforts assumed a point charge located at the atom center and coulombic interactions. It is been recognized over multiple decades that simply representing electrostatics with a charge on each atom failed to reproduce the electrostatic potential surrounding a molecule as estimated by quantum mechanics. Molecular orbitals are not spherically symmetrical, an implicit assumption of monopole electrostatics. This perspective reviews recent evidence that requires use of multipole electrostatics and polarizability in molecular modeling.

Enumerating Designing Sequences in the HP Model

The hydrophobic/polar HP model on the square lattice has been widely used toinvestigate basics of protein folding. In the cases where all designing sequences (sequences with unique ground states) were enumerated without restrictions on the number of contacts, the upper limit on the chain length N has been 18-20 because of the rapid exponential growth of thenumbers of conformations and sequences. We show how a few optimizations push this limit by about 5 units. Based on these calculations, we study the statistical distribution of hydrophobicity along designing sequences. We find that the average number of hydrophobic and polar clumps along the chains is larger for designing sequences than for random ones, which is in agreement with earlier findings for N ≤ 18 and with results for real enzymes. We also show that this deviation from randomness disappears if the calculations are restricted to maximally compact structures.

The road not taken: retreat and diverge in local search for simplified protein structure prediction

BACKGROUND

Given a protein's amino acid sequence, the protein structure prediction problem is to find a three dimensional structure that has the native energy level. For many decades, it has been one of the most challenging problems in computational biology. A simplified version of the problem is to find an on-lattice self-avoiding walk that minimizes the interaction energy among the amino acids. Local search methods have been preferably used in solving the protein structure prediction problem for their efficiency in finding very good solutions quickly. However, they suffer mainly from two problems: re-visitation and stagnancy.

RESULTS

In this paper, we present an efficient local search algorithm that deals with these two problems. During search, we select the best candidate at each iteration, but store the unexplored second best candidates in a set of elite conformations, and explore them whenever the search faces stagnation. Moreover, we propose a new non-isomorphic encoding for the protein conformations to store the conformations and to check similarity when applied with a memory based search. This new encoding helps eliminate conformations that are equivalent under rotation and translation, and thus results in better prevention of re-visitation.

CONCLUSION

On standard benchmark proteins, our algorithm significantly outperforms the state-of-the art approaches for Hydrophobic-Polar energy models and Face Centered Cubic Lattice.

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.

Heuristic energy landscape paving for protein folding problem in the three-dimensional HP lattice model

The protein folding problem, i.e., the prediction of the tertiary structures of protein molecules from their amino acid sequences is one of the most important problems in computational biology and biochemistry. However, the extremely difficult optimization problem arising from energy function is a key challenge in protein folding simulation. The energy landscape paving (ELP) method has already been applied very successfully to off-lattice protein models and other optimization problems with complex energy landscape in continuous space. By improving the ELP method, and subsequently incorporating the neighborhood strategy with the pull-move set into the improved ELP method, a heuristic ELP algorithm is proposed to find low-energy conformations of 3D HP lattice model proteins in the discrete space. The algorithm is tested on three sets of 3D HP benchmark instances consisting 31 sequences. For eleven sequences with 27 monomers, the proposed method explores the conformation surfaces more efficiently than other methods, and finds new lower energies in several cases. For ten 48-monomer sequences, we find the lowest energies so far. With the achieved results, the algorithm converges rapidly and efficiently. For all ten 64-monomer sequences, the algorithm finds lower energies within comparable computation times than previous methods. Numeric results show that the heuristic ELP method is a competitive tool for protein folding simulation in 3D lattice model. To the best of our knowledge, this is the first application of ELP to the 3D discrete space.

Comparative modeling and protein-like features of hydrophobic-polar models on a two-dimensional lattice

Lattice models of proteins have been extensively used to study protein thermodynamics, folding dynamics, and evolution. Our study considers two different hydrophobic-polar (HP) models on the 2D square lattice: the purely HP model and a model where a compactness-favoring term is added. We exhaustively enumerate all the possible structures in our models and perform the study of their corresponding folds, HP arrangements in space and shapes. The two models considered differ greatly in their numbers of structures, folds, arrangements, and shapes. Despite their differences, both lattice models have distinctive protein-like features: (1) Shapes are compact in both models, especially when a compactness-favoring energy term is added. (2) The residue composition is independent of the chain length and is very close to 50% hydrophobic in both models, as we observe in real proteins. (3) Comparative modeling works well in both models, particularly in the more compact one. The fact that our models show protein-like features suggests that lattice models incorporate the fundamental physical principles of proteins. Our study supports the use of lattice models to study questions about proteins that require exactness and extensive calculations, such as protein design and evolution, which are often too complex and computationally demanding to be addressed with more detailed models.

On lattice protein structure prediction revisited

Protein structure prediction is regarded as a highly challenging problem both for the biology and for the computational communities. In recent years, many approaches have been developed, moving to increasingly complex lattice models and off-lattice models. This paper presents a Large Neighborhood Search (LNS) to find the native state for the Hydrophobic-Polar (HP) model on the Face-Centered Cubic (FCC) lattice or, in other words, a self-avoiding walk on the FCC lattice having a maximum number of H-H contacts. The algorithm starts with a tabu-search algorithm, whose solution is then improved by a combination of constraint programming and LNS. The flexible framework of this hybrid algorithm allows an adaptation to the Miyazawa-Jernigan contact potential, in place of the HP model, thus suggesting its potential for tertiary structure prediction. Benchmarking statistics are given for our method against the hydrophobic core threading program HPstruct, an exact method which can be viewed as complementary to our method.

What determines the activity of antimicrobial and cytolytic peptides in model membranes

We previously proposed three hypotheses relating the mechanism of antimicrobial and cytolytic peptides in model membranes to the Gibbs free energies of binding and insertion into the membrane [Almeida, P. F., and Pokorny, A. (2009) Biochemistry 48, 8083-8093]. Two sets of peptides were designed to test those hypotheses, by mutating of the sequences of δ-lysin, cecropin A, and magainin 2. Peptide binding and activity were measured on phosphatidylcholine membranes. In the first set, the peptide charge was changed by mutating basic to acidic residues or vice versa, but the amino acid sequence was not altered much otherwise. The type of dye release changed from graded to all-or-none according to prediction. However, location of charged residues in the sequence with the correct spacing to form salt bridges failed to improve binding. In the second set, the charged and other key residues were kept in the same positions, whereas most of the sequence was significantly but conservatively simplified, maintaining the same hydrophobicity and amphipathicity. This set behaved completely different from predicted. The type of release, which was expected to be maintained, changed dramatically from all-or-none to graded in the mutants of cecropin and magainin. Finally, contrary to the hypotheses, the results indicate that the Gibbs energy of binding to the membrane, not the Gibbs energy of insertion, is the primary determinant of peptide activity.