Lattice model for rapidly folding protein-like heteropolymers

Evolutionary conservation of amino acids contributing to the protein folding transition state

The question of whether amino acids critical to protein folding kinetics are evolutionarily conserved has been investigated intensively in the past, but no consensus has yet been reached. Recently, we have demonstrated that the transition state, dictating folding kinetics, is characterized as the state of maximum dynamic cooperativity, i.e., the state of maximum correlations between amino acid contact formations. Here, we investigate the evolutionary conservation of those amino acids contributing significantly to the dynamic cooperativity. We find a strong indication of a new kind of relationship-necessary but not sufficient causality-between the evolutionary conservation and the dynamic cooperativity: larger contributions to the dynamic cooperativity arise from more conserved residues, but not vice versa. This holds for all the protein systems for which long folding simulation trajectories are available. To our knowledge, this is the first systematic demonstration of any kind of evolutionary conservation of amino acids relevant to folding kinetics.

Pseudo-Improper-Dihedral Model for Intrinsically Disordered Proteins

We present a new coarse-grained C_α-based protein model with a nonradial multibody pseudo-improper-dihedral potential that is transferable, time-independent, and suitable for molecular dynamics. It captures the nature of backbone and side-chain interactions between amino acid residues by adapting a simple improper dihedral term for a one-bead-per-residue model. It is parameterized for intrinsically disordered proteins and applicable to simulations of such proteins and their assemblies on millisecond time scales.

Disordered peptide chains in an α-C-based coarse-grained model

We construct a one-bead-per-residue coarse-grained dynamical model to describe intrinsically disordered proteins at significantly longer timescales than in the all-atom models. In this model, inter-residue contacts form and disappear during the course of the time evolution. The contacts may arise between the sidechains, the backbones or the sidechains and backbones of the interacting residues. The model yields results that are consistent with many all-atom and experimental data on these systems. We demonstrate that the geometrical properties of various homopeptides differ substantially in this model. In particular, the average radius of gyration scales with the sequence length in a residue-dependent manner.

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.

Biomolecular dynamics: order-disorder transitions and energy landscapes

While the energy landscape theory of protein folding is now a widely accepted view for understanding how relatively weak molecular interactions lead to rapid and cooperative protein folding, such a framework must be extended to describe the large-scale functional motions observed in molecular machines. In this review, we discuss (1) the development of the energy landscape theory of biomolecular folding, (2) recent advances toward establishing a consistent understanding of folding and function and (3) emerging themes in the functional motions of enzymes, biomolecular motors and other biomolecular machines. Recent theoretical, computational and experimental lines of investigation have provided a very dynamic picture of biomolecular motion. In contrast to earlier ideas, where molecular machines were thought to function similarly to macroscopic machines, with rigid components that move along a few degrees of freedom in a deterministic fashion, biomolecular complexes are only marginally stable. Since the stabilizing contribution of each atomic interaction is on the order of the thermal fluctuations in solution, the rigid body description of molecular function must be revisited. An emerging theme is that functional motions encompass order-disorder transitions and structural flexibility provides significant contributions to the free energy. In this review, we describe the biological importance of order-disorder transitions and discuss the statistical-mechanical foundation of theoretical approaches that can characterize such transitions.

Computational Cellular Dynamics Based on the Chemical Master Equation: A Challenge for Understanding Complexity

Modern molecular biology has always been a great source of inspiration for computational science. Half a century ago, the challenge from understanding macromolecular dynamics has led the way for computations to be part of the tool set to study molecular biology. Twenty-five years ago, the demand from genome science has inspired an entire generation of computer scientists with an interest in discrete mathematics to join the field that is now called bioinformatics. In this paper, we shall lay out a new mathematical theory for dynamics of biochemical reaction systems in a small volume (i.e., mesoscopic) in terms of a stochastic, discrete-state continuous-time formulation, called the chemical master equation (CME). Similar to the wavefunction in quantum mechanics, the dynamically changing probability landscape associated with the state space provides a fundamental characterization of the biochemical reaction system. The stochastic trajectories of the dynamics are best known through the simulations using the Gillespie algorithm. In contrast to the Metropolis algorithm, this Monte Carlo sampling technique does not follow a process with detailed balance. We shall show several examples how CMEs are used to model cellular biochemical systems. We shall also illustrate the computational challenges involved: multiscale phenomena, the interplay between stochasticity and nonlinearity, and how macroscopic determinism arises from mesoscopic dynamics. We point out recent advances in computing solutions to the CME, including exact solution of the steady state landscape and stochastic differential equations that offer alternatives to the Gilespie algorithm. We argue that the CME is an ideal system from which one can learn to understand "complex behavior" and complexity theory, and from which important biological insight can be gained.

Statistical geometry of lattice chain polymers with voids of defined shapes: sampling with strong constraints

Proteins contain many voids, which are unfilled spaces enclosed in the interior. A few of them have shapes compatible to ligands and substrates and are important for protein functions. An important general question is how the need for maintaining functional voids is influenced by, and affects other aspects of proteins structures and properties (e.g., protein folding stability, kinetic accessibility, and evolution selection pressure). In this paper, we examine in detail the effects of maintaining voids of different shapes and sizes using two-dimensional lattice models. We study the propensity for conformations to form a void of specific shape, which is related to the entropic cost of void maintenance. We also study the location that voids of a specific shape and size tend to form, and the influence of compactness on the formation of such voids. As enumeration is infeasible for long chain polymer, a key development in this work is the design of a novel sequential Monte Carlo strategy for generating large number of sample conformations under very constraining restrictions. Our method is validated by comparing results obtained from sampling and from enumeration for short polymer chains. We succeeded in accurate estimation of entropic cost of void maintenance, with and without an increasing number of restrictive conditions, such as loops forming the wall of void with fixed length, with additionally fixed starting position in the sequence. Additionally, we have identified the key structural properties of voids that are important in determining the entropic cost of void formation. We have further developed a parametric model to predict quantitatively void entropy. Our model is highly effective, and these results indicate that voids representing functional sites can be used as an improved model for studying the evolution of protein functions and how protein function relates to protein stability.

A model study of protein nascent chain and cotranslational folding using hydrophobic-polar residues

To study protein nascent chain folding during biosynthesis, we investigate the folding behavior of models of hydrophobic and polar (HP) chains at growing length using both two-dimensional square lattice model and an optimized three-dimensional 4-state discrete off-lattice model. After enumerating all possible sequences and conformations of HP heteropolymers up to length N = 18 and N = 15 in two and three-dimensional space, respectively, we examine changes in adopted structure, stability, and tolerance to single point mutation as the nascent chain grows. In both models, we find that stable model proteins have fewer folded nascent chains during growth, and often will only fold after reaching full length. For the few occasions where partial chains of stable proteins fold, these partial conformations on average are very similar to the corresponding parts of the final conformations at full length. Conversely, we find that sequences with fewer stable nascent chains and sequences with native-like folded nascent chains are more stable. In addition, these stable sequences in general can have many more point mutations and still fold into the same conformation as the wild type sequence. Our results suggest that stable proteins are less likely to be trapped in metastable conformations during biosynthesis, and are more resistant to point-mutations. Our results also imply that less stable proteins will require the assistance of chaperone and other factors during nascent chain folding. Taken together with other reported studies, it seems that cotranslational folding may not be a general mechanism of in vivo protein folding for small proteins, and in vitro folding studies are still relevant for understanding how proteins fold biologically.

Protein sequence design based on the topology of the native state structure

Computational design of sequences for a given structure is generally studied by exhaustively enumerating the sequence space or by searching in such a large space, which is prohibitively expensive. However, we point out that the protein topology has a wealth of information, which can be exploited to design sequences for a chosen structure. In this paper, we present a computationally efficient method for ranking the residue sites in a given native-state structure, which enables us to design sequences for a chosen structure. The premise for the method is that the topology of the graph representing the energetically interacting neighbours in a protein plays an important role in the inverse-folding problem. While our previous work (which was also based on topology) used eigenspectral analysis of the adjacency matrix of interactions for ranking the residue sites in a given chain, here we use a simple but effective way of assigning weights to the nodes on the basis of secondary connections, along with primary connections. This indirectly accounts for the edge weight in the graph and removes degeneracy in the degree. The new scheme needs only a few multiplications and additions to compute the preferred ranking of the residue sites even for structures of real proteins of sizes of a few hundred amino acid residues. We use HP lattice model examples (for which exhaustive enumeration of sequences is practical) to validate our ranking approach in obtaining sequences of lowest energy for any H-P residue composition for a given native-state structure. Some examples of native structures of real proteins are also included. Quantitative comparison of the efficacy of the new scheme with the earlier schemes is made. The new scheme consistently performs better and with much lower computational cost. An optimization procedure is added to work with the new scheme in a few rare cases wherein the new scheme fails to provide the best sequence, an optimization procedure is added to work with the new scheme.

Topological information embodied in local juxtaposition geometry provides a statistical mechanical basis for unknotting by type-2 DNA topoisomerases

Topoisomerases may unknot by recognizing specific DNA juxtapositions. The physical basis of this hypothesis is investigated by considering single-loop conformations in a coarse-grained polymer model. We determine the statistical relationship between the local geometry of a juxtaposition of two chain segments and whether the loop is knotted globally, and ascertain how the knot/unknot topology is altered by a topoisomerase-like segment passage at the juxtaposition. Segment passages at a "free" juxtaposition tend to increase knot probability. In contrast, segment passages at a "hooked" juxtaposition cause more transitions from knot to unknot than vice versa, resulting in a steady-state knot probability far lower than that at topological equilibrium. The reduction in knot population by passing chain segments through a hooked juxtaposition is more prominent for loops of smaller sizes, n, but remains significant even for larger loops: steady-state knot probability is only approximately 2%, and approximately 5% of equilibrium, respectively, for n=100 and 500 in the model. An exhaustive analysis of approximately 6000 different juxtaposition geometries indicates that the ability of a segment passage to unknot correlates strongly with the juxtaposition's "hookedness". Remarkably, and consistent with experiments on type-2 topoisomerases from different organisms, the unknotting potential of a juxtaposition geometry in our polymer model correlates almost perfectly with its corresponding decatenation potential. These quantitative findings suggest that it is possible for topoisomerases to disentangle by acting selectively on juxtapositions with "hooked" geometries.

Are residues in a protein folding nucleus evolutionarily conserved?

Protein is the working molecule of the cell, and evolution is the hallmark of life. It is important to understand how protein folding and evolution influence each other. Several studies correlating experimental measurement of residue participation in folding nucleus and sequence conservation have reached different conclusions. These studies are based on assessment of sequence conservation at folding nucleus sites using entropy or relative entropy measurement derived from multiple sequence alignment. Here we report analysis of conservation of folding nucleus using an evolutionary model alternative to entropy-based approaches. We employ a continuous time Markov model of codon substitution to distinguish mutation fixed by evolution and mutation fixed by chance. This model takes into account bias in codon frequency, bias-favoring transition over transversion, as well as explicit phylogenetic information. We measure selection pressure using the ratio omega of synonymous versus non-synonymous substitution at individual residue site. The omega-values are estimated using the PAML method, a maximum-likelihood estimator. Our results show that there is little correlation between the extent of kinetic participation in protein folding nucleus as measured by experimental phi-value and selection pressure as measured by omega-value. In addition, two randomization tests failed to show that folding nucleus residues are significantly more conserved than the whole protein, or the median omega value of all residues in the protein. These results suggest that at the level of codon substitution, there is no indication that folding nucleus residues are significantly more conserved than other residues. We further reconstruct candidate ancestral residues of the folding nucleus and suggest possible test tube mutation studies for testing folding behavior of ancient folding nucleus.

Solvation effects and driving forces for protein thermodynamic and kinetic cooperativity: how adequate is native-centric topological modeling?

What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for protein-like calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for protein-like thermodynamics. Kinetically, however, even models with both contact and local native-centric energies do not produce simple two-state chevron plots. Thus a model protein's thermodynamic cooperativity is not sufficient for simple two-state kinetics. The models tested appear to have increasing internal friction with increasing native stability, leading to chevron rollovers that typify kinetics that are commonly referred to as non-two-state. The free energy profiles of these models are found to be sensitive to the choice of native contacts and the presumed spatial ranges of the contact interactions. Motivated by explicit-water considerations, we explore recent treatments of solvent granularity that incorporate desolvation free energy barriers into effective implicit-solvent intraprotein interactions. This additional feature reduces both folding and unfolding rates vis-à-vis that of the corresponding models without desolvation barriers, but the kinetics remain non-two-state. Taken together, our observations suggest that interaction mechanisms more intricate than simple Gō-like constructs and pairwise additive solvation-like contributions are needed to rationalize some of the most basic generic protein properties. Therefore, as experimental constraints on protein chain models, requiring a consistent account of protein-like thermodynamic and kinetic cooperativity can be more stringent and productive for some applications than simply requiring a model heteropolymer to fold to a target structure.

Residues participating in the protein folding nucleus do not exhibit preferential evolutionary conservation

To what extent does natural selection act to optimize the details of protein folding kinetics? In an effort to address this question, the relationship between an amino acid's evolutionary conservation and its role in protein folding kinetics has been investigated intensively. Despite this effort, no consensus has been reached regarding the degree to which residues involved in native-like transition state structure (the folding nucleus) are conserved. Here we report the results of an exhaustive, systematic study of sequence conservation among residues known to participate in the experimentally (Phi-value) defined folding nuclei of all of the appropriately characterized proteins reported to date. We observe no significant evidence that these residues exhibit any anomalous sequence conservation. We do observe, however, a significant bias in the existing kinetic data: the mean sequence conservation of the residues that have been the subject of kinetic characterization is greater than the mean sequence conservation of all residues in 13 of 14 proteins studied. This systematic experimental bias gives rise to the previous observation that the median conservation of residues reported to participate in the folding nucleus is greater than the median conservation of all of the residues in a protein. When this bias is corrected (by comparing, for example, the conservation of residues known to participate in the folding nucleus with that of other, kinetically characterized residues) the previously reported preferential conservation is effectively eliminated. In contrast to well-established theoretical expectations, both poorly and highly conserved residues are apparently equally likely to participate in the protein-folding nucleus.

A method for parameter optimization in computational biology

Models in computational biology, such as those used in binding, docking, and folding, are often empirical and have adjustable parameters. Because few of these models are yet fully predictive, the problem may be nonoptimal choices of parameters. We describe an algorithm called ENPOP (energy function parameter optimization) that improves-and sometimes optimizes-the parameters for any given model and for any given search strategy that identifies the stable state of that model. ENPOP iteratively adjusts the parameters simultaneously to move the model global minimum energy conformation for each of m different molecules as close as possible to the true native conformations, based on some appropriate measure of structural error. A proof of principle is given for two very different test problems. The first involves three different two-dimensional model protein molecules having 12 to 37 monomers and four parameters in common. The parameters converge to the values used to design the model native structures. The second problem involves nine bumpy landscapes, each having between 4 and 12 degrees of freedom. For the three adjustable parameters, the globally optimal values are known in advance. ENPOP converges quickly to the correct parameter set.

Evolutionary conservation in protein folding kinetics

The sequence and structural conservation of folding transition states have been predicted on theoretical grounds. Using homologous sequence alignments of proteins previously characterized via coupled mutagenesis/kinetics studies, we tested these predictions experimentally. Only one of the six appropriately characterized proteins exhibits a statistically significant correlation between residues' roles in transition state structure and their evolutionary conservation. However, a significant correlation is observed between the contributions of individual sequence positions to the transition state structure across a set of homologous proteins. Thus the structure of the folding transition state ensemble appears to be more highly conserved than the specific interactions that stabilize it.

A simple model for evolution of proteins towards the global minimum of free energy

BACKGROUND

Proteins seem to have their native structure in a global minimum of free energy. No mechanism is known, however, for ensuring this property. Furthermore, computational complexity studies suggest that such a mechanism is not feasible. These seemingly contradictory observations can be reconciled by the suggestion that evolutionary selection can yield proteins whose native conformation is in the global minimum of free energy. The aim of this study is to investigate such evolutionary processes in a simple model of protein folding.

RESULTS

Three possible evolutionary processes are explored.First, if the free energy of the chain is kept below a fixed threshold there is no improvement towards the global minimum. Second, if free energy is minimized directly, sequences emerge whose native conformation is in the global minimum of free energy. Third, when evolutionary pressure is applied within a small set of close homologs, sequences emerge whose functional conformation is in the global minimum of free energy.

CONCLUSIONS

Although minimizing free energy does select for sequences whose functional conformation is in the global free energy minimum, we argue that for most proteins, which typically have free energy values of only 5-15 kcal/mol, such evolutionary pressure cannot be considered biologically plausible. In contrast, by repeatedly forcing sequences to avoid drifting towards competing "non-native" conformations, sequences emerge whose native conformation becomes very close to the global minimum of free energy. We argue that such a mechanism is both efficient and biologically plausible.

Folding funnels and frustration in off-lattice minimalist protein landscapes

A full quantitative understanding of the protein folding problem is now becoming possible with the help of the energy landscape theory and the protein folding funnel concept. Good folding sequences have a landscape that resembles a rough funnel where the energy bias towards the native state is larger than its ruggedness. Such a landscape leads not only to fast folding and stable native conformations but, more importantly, to sequences that are robust to variations in the protein environment and to sequence mutations. In this paper, an off-lattice model of sequences that fold into a beta-barrel native structure is used to describe a framework that can quantitatively distinguish good and bad folders. The two sequences analyzed have the same native structure, but one of them is minimally frustrated whereas the other one exhibits a high degree of frustration.

Structure-based design of model proteins

A structure-based, sequence-design procedure is proposed in which one considers a set of decoy structures that compete significantly with the target structure in being low energy conformations. The decoy structures are chosen to have strong overlaps in contacts with the putative native state. The procedure allows the design of sequences with large and small stability gaps in a random-bond heteropolymer model in both two and three dimensions by an appropriate assignment of the contact energies to both the native and nonnative contacts. The design procedure is also successfully applied to the two-dimensional HP model.

Interaction potentials for protein folding

We outline a general strategy for determining the effective coarse-grained interactions between the amino acids of a protein from the experimentally derived native-state structures. The method is, in principle, free from any adjustable or empirically determined parameters, and it is tested on simple models and compared with other existing approaches.

Protein folding in the landscape perspective: chevron plots and non-Arrhenius kinetics

We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have "chevron plot" dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponential behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process.

Theory of protein folding: the energy landscape perspective

The energy landscape theory of protein folding is a statistical description of a protein's potential surface. It assumes that folding occurs through organizing an ensemble of structures rather than through only a few uniquely defined structural intermediates. It suggests that the most realistic model of a protein is a minimally frustrated heteropolymer with a rugged funnel-like landscape biased toward the native structure. This statistical description has been developed using tools from the statistical mechanics of disordered systems, polymers, and phase transitions of finite systems. We review here its analytical background and contrast the phenomena in homopolymers, random heteropolymers, and protein-like heteropolymers that are kinetically and thermodynamically capable of folding. The connection between these statistical concepts and the results of minimalist models used in computer simulations is discussed. The review concludes with a brief discussion of how the theory helps in the interpretation of results from fast folding experiments and in the practical task of protein structure prediction.

Protein folding: optimized sequences obtained by simulated breeding in a minimalist model

For a minimalist model of protein folding, which we introduced recently, we investigate various methods to obtain folding sequences. A detailed study of random sequences shows that, for this model, such sequences usually do not fold to their ground states during simulations. Straight-forward techniques for the construction of folding sequences, based solely on the target structure, fail. We describe in detail an optimization algorithm, based on genetic algorithms, for the "simulated breeding" of folding sequences in this model. We find that, for any target structure studied, there is not only a single folding sequence but a patch of sequences in sequence space that fold to this structure. In addition, we show that, much as in real proteins, nonhomologous sequences may fold to the same target structure.

Cell dynamics of folding in two-dimensional model proteins

BACKGROUND

Functionally useful proteins are sequences of amino acids that fold rapidly under appropriate conditions into their native states. It is believed that rapid folders are sequences for which the folding dynamics entail the exploration of restricted conformations-the phase space can be thought of as a folding funnel. While there are many experimentally accessible predictions pertaining to the existence of such funnels and a coherent picture of the kinetics of folding has begun to emerge, there have been relatively few simple studies in the controlled setting of well-characterized lattice models.

RESULTS

We design rapidly folding sequences by assigning the strongest couplings to the contacts present in a target native state in a two-dimensional model of heteropolymers. Such sequences have large folding transition temperatures and low glass transition temperatures. The dependence of median folding times on temperature is investigated. The pathways to folding and their dependence on the temperature are illustrated via a study of the cell dynamics-a mapping of the dynamics into motion within the space of the maximally compact cells.

CONCLUSIONS

Folding funnels can be defined operationally in a coarse-grained sense by mapping the states of the system into maximally compact conformations and then by identifying significant connectivities between them.

From Levinthal to pathways to funnels

While the classical view of protein folding kinetics relies on phenomenological models, and regards folding intermediates in a structural way, the new view emphasizes the ensemble nature of protein conformations. Although folding has sometimes been regarded as a linear sequence of events, the new view sees folding as parallel microscopic multi-pathway diffusion-like processes. While the classical view invoked pathways to solve the problem of searching for the needle in the haystack, the pathway idea was then seen as conflicting with Anfinsen's experiments showing that folding is pathway-independent (Levinthal's paradox). In contrast, the new view sees no inherent paradox because it eliminates the pathway idea: folding can funnel to a single stable state by multiple routes in conformational space. The general energy landscape picture provides a conceptual framework for understanding both two-state and multi-state folding kinetics. Better tests of these ideas will come when new experiments become available for measuring not just averages of structural observables, but also correlations among their fluctuations. At that point we hope to learn much more about the real shapes of protein folding landscapes.

Exploring the energy landscapes of molecular recognition by a genetic algorithm: analysis of the requirements for robust docking of HIV-1 protease and FKBP-12 complexes

Energy landscapes of molecular recognition are explored by performing "semi-rigid" docking of FK-506 and rapamycin with the Fukisawa binding protein (FKBP-12), and flexible docking simulations of the Ro-31-8959 and AG-1284 inhibitors with HIV-1 protease by a genetic algorithm. The requirements of a molecular recognition model to meet thermodynamic and kinetic criteria of ligand-protein docking simultaneously are investigated using a family of simple molecular recognition energy functions. The critical factor that determines the success rate in predicting the structure of ligand-protein complexes is found to be the roughness of the binding energy landscape, in accordance with a minimal frustration principle. The results suggest that further progress in structure prediction of ligand-protein complexes can be achieved by designing molecular recognition energy functions that generate binding landscapes with reduced frustration.

How optimization of potential functions affects protein folding

The relationship between the optimization of the potential function and the foldability of theoretical protein models is studied based on investigations of a 27-mer cubic-lattice protein model and a more realistic lattice model for the protein crambin. In both the simple and the more complicated systems, optimization of the energy parameters achieves significant improvements in the statistical-mechanical characteristics of the systems and leads to foldable protein models in simulation experiments. The foldability of the protein models is characterized by their statistical-mechanical properties--e.g., by the density of states and by Monte Carlo folding simulations of the models. With optimized energy parameters, a high level of consistency exists among different interactions in the native structures of the protein models, as revealed by a correlation function between the optimized energy parameters and the native structure of the model proteins. The results of this work are relevant to the design of a general potential function for folding proteins by theoretical simulations.