Comparing folding codes for proteins and polymers

Key aspects of the past 30 years of protein design

Proteins are the workhorse of life. They are the building infrastructure of living systems; they are the most efficient molecular machines known, and their enzymatic activity is still unmatched in versatility by any artificial system. Perhaps proteins' most remarkable feature is their modularity. The large amount of information required to specify each protein's function is analogically encoded with an alphabet of just ∼20 letters. The protein folding problem is how to encode all such information in a sequence of 20 letters. In this review, we go through the last 30 years of research to summarize the state of the art and highlight some applications related to fundamental problems of protein evolution.

Coarse-Grained Simulations of Protein Folding: Bridging Theory and Experiments

Computational coarse-grained models play a fundamental role as a research tool in protein folding, and they are important in bridging theory and experiments. Folding mechanisms are generally discussed using the energy landscape framework, which is well mapped within a class of simplified structure-based models. In this chapter, simplified computer models are discussed with special focus on structure-based ones.

Quantifying the Mutational Robustness of Protein-Coding Genes

We use large-scale mutagenesis data and computer simulations to quantify the mutational robustness of protein-coding genes by taking into account constraints arising from protein function and the genetic code. Analyses of the distribution of amino acid substitutions from 18 mutagenesis studies revealed an average of 45% of neutral variants; while mutagenesis data of 12 proteins artificially designed under no other constraints but stability, reach an average of 60%. Simulations using a lattice protein model allow us to contrast these estimates to the expected mutational robustness of protein families by generating unbiased samples of foldable sequences, which we find to have 30% of neutral variants. In agreement with mutagenesis data of designed proteins, the model shows that maximally robust protein families might access up to twice the amount of neutral variants observed in the unbiased samples (i.e. 60%). A biophysical model of protein-ligand binding suggests that constraints associated to molecular function have only a moderate impact on robustness of approximately 5 to 10% of neutral variants; and that the direction of this effect depends on the relation between functional performance and thermodynamic stability. Although the genetic code constraints the access of a gene's nucleotide sequence to only 30% of the full distribution of amino acid mutations, it provides an extra 15 to 20% of neutral variants to the estimations above, such that the expected, observed, and maximal robustness of protein-coding genes are approximately 50, 65, and 75%, respectively. We discuss our results in the light of three main hypothesis put forward to explain the existence of mutationally robust genes.

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.

The amino acid alphabet and the architecture of the protein sequence-structure map. I. Binary alphabets

The correspondence between protein sequences and structures, or sequence-structure map, relates to fundamental aspects of structural, evolutionary and synthetic biology. The specifics of the mapping, such as the fraction of accessible sequences and structures, or the sequences' ability to fold fast, are dictated by the type of interactions between the monomers that compose the sequences. The set of possible interactions between monomers is encapsulated by the potential energy function. In this study, I explore the impact of the relative forces of the potential on the architecture of the sequence-structure map. My observations rely on simple exact models of proteins and random samples of the space of potential energy functions of binary alphabets. I adopt a graph perspective and study the distribution of viable sequences and the structures they produce, as networks of sequences connected by point mutations. I observe that the relative proportion of attractive, neutral and repulsive forces defines types of potentials, that induce sequence-structure maps of vastly different architectures. I characterize the properties underlying these differences and relate them to the structure of the potential. Among these properties are the expected number and relative distribution of sequences associated to specific structures and the diversity of structures as a function of sequence divergence. I study the types of binary potentials observed in natural amino acids and show that there is a strong bias towards only some types of potentials, a bias that seems to characterize the folding code of natural proteins. I discuss implications of these observations for the architecture of the sequence-structure map of natural proteins, the construction of random libraries of peptides, and the early evolution of the natural amino acid alphabet.

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.

Evolutionary dynamics on protein bi-stability landscapes can potentially resolve adaptive conflicts

Experimental studies have shown that some proteins exist in two alternative native-state conformations. It has been proposed that such bi-stable proteins can potentially function as evolutionary bridges at the interface between two neutral networks of protein sequences that fold uniquely into the two different native conformations. Under adaptive conflict scenarios, bi-stable proteins may be of particular advantage if they simultaneously provide two beneficial biological functions. However, computational models that simulate protein structure evolution do not yet recognize the importance of bi-stability. Here we use a biophysical model to analyze sequence space to identify bi-stable or multi-stable proteins with two or more equally stable native-state structures. The inclusion of such proteins enhances phenotype connectivity between neutral networks in sequence space. Consideration of the sequence space neighborhood of bridge proteins revealed that bi-stability decreases gradually with each mutation that takes the sequence further away from an exactly bi-stable protein. With relaxed selection pressures, we found that bi-stable proteins in our model are highly successful under simulated adaptive conflict. Inspired by these model predictions, we developed a method to identify real proteins in the PDB with bridge-like properties, and have verified a clear bi-stability gradient for a series of mutants studied by Alexander et al. (Proc Nat Acad Sci USA 2009, 106:21149–21154) that connect two sequences that fold uniquely into two different native structures via a bridge-like intermediate mutant sequence. Based on these findings, new testable predictions for future studies on protein bi-stability and evolution are discussed.

Proteins are essential molecules for performing a majority of functions in all biological systems. These functions often depend on the three-dimensional structures of proteins. Here, we investigate a fundamental question in molecular evolution: how can proteins acquire new advantageous structures via mutations while not sacrificing their existing structures that are still needed? Some authors have suggested that the same protein may adopt two or more alternative structures, switch between them and thus perform different functions with each of the alternative structures. Intuitively, such a protein could provide an evolutionary compromise between conflicting demands for existing and new protein structures. Yet no theoretical study has systematically tackled the biophysical basis of such compromises during evolutionary processes. Here we devise a model of evolution that specifically recognizes protein molecules that can exist in several different stable structures. Our model demonstrates that proteins can indeed utilize multiple structures to satisfy conflicting evolutionary requirements. In light of these results, we identify data from known protein structures that are consistent with our predictions and suggest novel directions for future investigation.

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.

Evolvability and single-genotype fluctuation in phenotypic properties: a simple heteropolymer model

Experiment showed that the response of a genotype to mutation, i.e., the magnitude of mutational change in a phenotypic property, can be correlated with the extent of phenotypic fluctuation among genetic clones. To address a possible statistical mechanical basis for such phenomena at the protein level, we consider a simple hydrophobic-polar lattice protein-chain model with an exhaustive mapping between sequence (genotype) and conformational (phenotype) spaces. Using squared end-to-end distance, R(N)(2), as an example conformational property, we study how the thermal fluctuation of a sequence's R(N)(2) may be predictive of the changes in the Boltzmann average R(N)(2) caused by single-point mutations on that sequence. We found that sequences with the same ground-state (R(N)(2))(0) exhibit a funnel-like organization under conditions favorable to chain collapse or folding: fluctuation (standard deviation sigma) of R(N)(2) tends to increase with mutational distance from a prototype sequence whose R(N)(2) deviates little from its (R(N)(2))(0). In general, large mutational decreases in R(N)(2) or in sigma are only possible for some, though not all, sequences with large sigma values. This finding suggests that single-genotype phenotypic fluctuation is a necessary, though not sufficient, indicator of evolvability toward genotypes with less phenotypic fluctuations.

Computational exploration of the network of sequence flow between protein structures

We investigate small sequence adjustments (of one or a few amino acids) that induce large conformational transitions between distinct and stable folds of proteins. Such transitions are intriguing from evolutionary and protein-design perspectives. They make it possible to search for ancient protein structures or to design protein switches that flip between folds and functions. A network of sequence flow between protein folds is computed for representative structures of the Protein Data Bank. The computed network is dense, on an average each structure is connected to tens of other folds. Proteins that attract sequences from a higher than expected number of neighboring folds are more likely to be enzymes and alpha/beta fold. The large number of connections between folds may reflect the need of enzymes to adjust their structures for alternative substrates. The network of the Cro family is discussed, and we speculate that capacity is an important factor (but not the only one) that determines protein evolution. The experimentally observed flip from all alpha to alpha + beta fold is examined by the network tools. A kinetic model for the transition of sequences between the folds (with only protein stability in mind) is proposed. Proteins 2010. (c) 2009 Wiley-Liss, Inc.

Competition between native topology and nonnative interactions in simple and complex folding kinetics of natural and designed proteins

We compared folding properties of designed protein Top7 and natural protein S6 by using coarse-grained chain models with a mainly native-centric construct that accounted also for nonnative hydrophobic interactions and desolvation barriers. Top7 and S6 have similar secondary structure elements and are approximately equal in length and hydrophobic composition. Yet their experimental folding kinetics were drastically different. Consistent with experiment, our simulated folding chevron arm for Top7 exhibited a severe rollover, whereas that for S6 was essentially linear, and Top7 model kinetic relaxation was multiphasic under strongly folding conditions. The peculiar behavior of Top7 was associated with several classes of kinetic traps in our model. Significantly, the amino acid residues participating in nonnative interactions in trapped conformations in our Top7 model overlapped with those deduced experimentally. These affirmations suggest that the simple ingredients of native topology plus sequence-dependent nonnative interactions are sufficient to account for some key features of protein folding kinetics. Notably, when nonnative interactions were absent in the model, Top7 chevron rollover was not correctly predicted. In contrast, nonnative interactions had little effect on the quasi linearity of the model folding chevron arm for S6. This intriguing distinction indicates that folding cooperativity is governed by a subtle interplay between the sequence-dependent driving forces for native topology and the locations of favorable nonnative interactions entailed by the same sequence. Constructed with a capability to mimic this interplay, our simple modeling approach should be useful in general for assessing a designed sequence's potential to fold cooperatively.

Probing possible downhill folding: native contact topology likely places a significant constraint on the folding cooperativity of proteins with approximately 40 residues

Experiments point to appreciable variations in folding cooperativity among natural proteins with approximately 40 residues, indicating that the behaviors of these proteins are valuable for delineating the contributing factors to cooperative folding. To explore the role of native topology in a protein's propensity to fold cooperatively and how native topology might constrain the degree of cooperativity achievable by a given set of physical interactions, we compared folding/unfolding kinetics simulated using three classes of native-centric C(alpha) chain models with different interaction schemes. The approach was applied to two homologous 45-residue fragments from the peripheral subunit-binding domain family and a 39-residue fragment of the N-terminal domain of ribosomal protein L9. Free-energy profiles as functions of native contact number were computed to assess the heights of thermodynamic barriers to folding. In addition, chevron plots of folding/unfolding rates were constructed as functions of native stability to facilitate comparison with available experimental data. Although common Gō-like models with pairwise Lennard-Jones-type interactions generally fold less cooperatively than real proteins, the rank ordering of cooperativity predicted by these models is consistent with experiment for the proteins investigated, showing increasing folding cooperativity with increasing nonlocality of a protein's native contacts. Models that account for water-expulsion (desolvation) barriers and models with many-body (nonadditive) interactions generally entail higher degrees of folding cooperativity indicated by more linear model chevron plots, but the rank ordering of cooperativity remains unchanged. A robust, experimentally valid rank ordering of model folding cooperativity independent of the multiple native-centric interaction schemes tested here argues that native topology places significant constraints on how cooperatively a protein can fold.

Population dynamics simulations of functional model proteins

In order to probe the fundamental principles that govern protein evolution, we use a minimalist model of proteins to provide a mapping from genotype to phenotype. The model is based on physically realistic forces of protein folding and includes an explicit definition of protein function. Thus, we can find the fitness of a sequence from its ability to fold to a stable structure and perform a function. We study the fitness landscapes of these functional model proteins, that is, the set of all sequences mapped on to their corresponding fitnesses and connected to their one mutant neighbors. Through population dynamics simulations we directly study the influence of the nature of the fitness landscape on evolution. Populations are observed to move to a steady state, the distribution of which can often be predicted prior to the population dynamics simulations from the nature of the fitness landscape and a quantity analogous to a partition function. In this paper, we develop a scheme for predicting the steady-state population on a fitness landscape, based on the nature of the fitness landscape, thereby obviating the need for explicit population dynamics simulations and providing some insight into the impact on molecular evolution of the nature of fitness landscapes. Poor predictions are indicative of fitness landscapes that consist of a series of weakly connected sublandscapes.

An asymmetric underlying rule in the assignment of codons: possible clue to a quick early evolution of the genetic code via successive binary choices

Aminoacyl-tRNA synthetases (aaRSs) are responsible for creating the pool of correctly charged aminoacyl-tRNAs that are necessary for the translation of genetic information (mRNA) by the ribosome. Each aaRS belongs to either one of only two classes with two different mechanisms of aminoacylation, making use of either the 2'OH (Class I) or the 3'OH (Class II) of the terminal A76 of the tRNA and approaching the tRNA either from the minor groove (2'OH) or the major groove (3'OH). Here, an asymmetric pattern typical of differentiation is uncovered in the partition of the codon repertoire, as defined by the mechanism of aminoacylation of each corresponding tRNA. This pattern can be reproduced in a unique cascade of successive binary decisions that progressively reduces codon ambiguity. The deduced order of differentiation is manifestly driven by the reduction of translation errors. A simple rule can be defined, decoding each codon sequence in its binary class, thereby providing both the code and the key to decode it. Assuming that the partition into two mechanisms of tRNA aminoacylation is a relic that dates back to the invention of the genetic code in the RNA World, a model for the assignment of amino acids in the codon table can be derived. The model implies that the stop codon was always there, as the codon whose tRNA cannot be charged with any amino acid, and makes the prediction of an ultimate differentiation step, which is found to correspond to the codon assignment of the 22nd amino acid pyrrolysine in archaebacteria.

Selective adsorption of block copolymers on patterned surfaces

Adsorption of copolymers on patterned surfaces is studied using lattice modeling and multiple Markov chain Monte Carlo methods. The copolymer is composed of alternating blocks of A and B monomers, and the adsorbing surface is composed of alternating square blocks containing C and D sites. Effects of interaction specificity on the adsorbed pattern of the copolymer and the sharpness of the adsorption transition are investigated by comparing three different models of copolymer-surface interactions. Analyses of the underlying energy distribution indicate that adsorption transitions in our models are not two-state-like. We show how the corresponding experimental question may be addressed by calorimetric measurements as have been applied to protein folding. Although the adsorption transitions are not "first order" or two-state-like, the sharpness of the transition increases when interaction specificity is enhanced by either including more attractive interaction types or by introducing repulsive interactions. Uniformity of the pattern of the adsorbed copolymer is also sensitive to the interaction scheme. Ramifications of the results from the present minimalist models of pattern recognition on the energetic and statistical mechanical origins of undesirable nonspecific adsorption of synthetic biopolymers in cellular environments are discussed.

Criteria for downhill protein folding: Calorimetry, chevron plot, kinetic relaxation, and single-molecule radius of gyration in chain models with subdued degrees of cooperativity

Recent investigations of possible downhill folding of small proteins such as BBL have focused on the thermodynamics of non-two-state, "barrierless" folding/denaturation transitions. Downhill folding is noncooperative and thermodynamically "one-state," a phenomenon underpinned by a unimodal conformational distribution over chain properties such as enthalpy, hydrophobic exposure, and conformational dimension. In contrast, corresponding distributions for cooperative two-state folding are bimodal with well-separated population peaks. Using simplified atomic modeling of a three-helix bundle-in a scheme that accounts for hydrophobic interactions and hydrogen bonding-and coarse-grained C(alpha) models of four real proteins with various degrees of cooperativity, we evaluate the effectiveness of several observables at defining the underlying distribution. Bimodal distributions generally lead to sharper transitions, with a higher heat capacity peak at the transition midpoint, compared with unimodal distributions. However, the observation of a sigmoidal transition is not a reliable criterion for two-state behavior, and the heat capacity baselines, used to determine the van't Hoff and calorimetric enthalpies of the transition, can introduce ambiguity. Interestingly we find that, if the distribution of the single-molecule radius of gyration were available, it would permit discrimination between unimodal and bimodal underlying distributions. We investigate kinetic implications of thermodynamic noncooperativity using Langevin dynamics. Despite substantial chevron rollovers, the relaxation of the models considered is essentially single-exponential over an extended range of native stabilities. Consistent with experiments, significant deviations from single-exponential behavior occur only under strongly folding conditions.

Accuracy of sequence alignment and fold assessment using reduced amino acid alphabets

Reduced or simplified amino acid alphabets group the 20 naturally occurring amino acids into a smaller number of representative protein residues. To date, several reduced amino acid alphabets have been proposed, which have been derived and optimized by a variety of methods. The resulting reduced amino acid alphabets have been applied to pattern recognition, generation of consensus sequences from multiple alignments, protein folding, and protein structure prediction. In this work, amino acid substitution matrices and statistical potentials were derived based on several reduced amino acid alphabets and their performance assessed in a large benchmark for the tasks of sequence alignment and fold assessment of protein structure models, using as a reference frame the standard alphabet of 20 amino acids. The results showed that a large reduction in the total number of residue types does not necessarily translate into a significant loss of discriminative power for sequence alignment and fold assessment. Therefore, some definitions of a few residue types are able to encode most of the relevant sequence/structure information that is present in the 20 standard amino acids. Based on these results, we suggest that the use of reduced amino acid alphabets may allow to increasing the accuracy of current substitution matrices and statistical potentials for the prediction of protein structure of remote homologs.

Conservation of statistical results under the reduction of pair-contact interactions to solvation interactions

We show that the hydrophobicity of sequences is the leading term in Miyazawa-Jernigan interactions. Being the source of additive (solvation) terms in pair-contact interactions, they were used to reduce the energy parameters while resulting in a clear vector manipulation of energy. The reduced (additive) potential performs considerably successful in predicting the statistical properties of arbitrary structures. The evaluated designabilities of the structures by both models are highly correlated. Suggesting geometrically nondegenerate vectors (structures) as proteinlike structures, the additive model is a powerful tool for protein design. Moreover, a crossing point in the log-linear diagram of designability ranking shows that about 1/e of the structures have designabilities above the average, independent on the used model.

Predicting the tolerance of proteins to random amino acid substitution

We have recently proposed a thermodynamic model that predicts the tolerance of proteins to random amino acid substitutions. Here we test this model against extensive simulations with compact lattice proteins, and find that the overall performance of the model is very good. We also derive an approximate analytic expression for the fraction of mutant proteins that fold stably to the native structure, Pf(m), as a function of the number of amino acid substitutions m, and present several methods to estimate the asymptotic behavior of Pf(m) for large m. We test the accuracy of all approximations against our simulation results, and find good overall agreement between the approximations and the simulation measurements.

Comparing folding codes in simple heteropolymer models of protein evolutionary landscape: robustness of the superfunnel paradigm

Understanding the evolution of biopolymers is a key element in rationalizing their structures and functions. Simple exact models (SEMs) are well-positioned to address general principles of evolution as they permit the exhaustive enumeration of both sequence and structure (conformational) spaces. The physics-based models of the complete mapping between genotypes and phenotypes afforded by SEMs have proven valuable for gaining insight into how adaptation and selection operate among large collections of sequences and structures. This study compares the properties of evolutionary landscapes of a variety of SEMs to delineate robust predictions and possible model-specific artifacts. Among the models studied, the ruggedness of evolutionary landscape is significantly model-dependent; those derived from more protein-like models appear to be smoother. We found that a common practice of restricting protein structure space to maximally compact lattice conformations results in (i.e., "designs in") many encodable (designable) structures that are not otherwise encodable in the corresponding unrestrained structure space. This discrepancy is especially severe for model potentials that seek to mimic the major role of hydrophobic interactions in protein folding. In general, restricting conformations to be maximally compact leads to larger changes in the model genotype-phenotype mapping than a moderate shifting of reference state energy of the model potential function to allow for more specific encoding via the "designing out" effects of repulsive interactions. Despite these variations, the superfunnel paradigm applies to all SEMs we have tested: For a majority of neutral nets across different models, there exists a funnel-like organization of native stabilities for the sequences in a neutral net encoding for the same structure, and the thermodynamically most stable sequence is also the most robust against mutation.

Imprint of evolution on protein structures

We attempt to understand the evolutionary origin of protein folds by simulating their divergent evolution with a three-dimensional lattice model. Starting from an initial seed lattice structure, evolution of model proteins progresses by sequence duplication and subsequent point mutations. A new gene's ability to fold into a stable and unique structure is tested each time through direct kinetic folding simulations. Where possible, the algorithm accepts the new sequence and structure and thus a "new protein structure" is born. During the course of each run, this model evolutionary algorithm provides several thousand new proteins with diverse structures. Analysis of evolved structures shows that later evolved structures are more designable than seed structures as judged by recently developed structural determinant of protein designability, as well as direct estimate of designability for selected structures by thermodynamic sampling of their sequence space. We test the significance of this trend predicted on lattice models on real proteins and show that protein domains that are found in eukaryotic organisms only feature statistically significant higher designability than their prokaryotic counterparts. These results present a fundamental view on protein evolution highlighting the relative roles of structural selection and evolutionary dynamics on genesis of modern proteins.

Side-chain control of beta-peptide secondary structures

As one of the most important families of non-natural polymers with the propensity to form well-defined secondary structures, the beta-peptides are attracting increasing attention. The compounds incorporating beta-amino acid residues have found various applications in medicinal chemistry and biochemistry. The conformational pool of beta-peptides comprises several periodic folded conformations, which can be classified as helices, and nonpolar and polar strands. The latter two are prone to form pleated sheets. The numerous studies that have been performed on the side-chain dependence of the stability of the folded structures allow certain conclusions concerning the principles of design of the beta-peptide foldamers. The folding propensity is influenced by local torsional, side-chain to backbone and long-range side-chain interactions. Although beta-peptide foldamers are sensitive to solvent, the systematic choice of the side-chain pattern and spatiality allows the design of the desired specific secondary structure. The application of beta-peptide foldamers may open up new directions in the synthesis of highly organized artificial tertiary structures with biochemical functions.

Relevance of structural segregation and chain compaction for the thermodynamics of folding of a hydrophobic protein model

The relevance of inside-outside segregation and chain compaction for the thermodynamics of folding of a hydrophobic protein model is probed by complete enumeration of two-dimensional chains of up to 18 monomers in the square lattice. The exact computation of Z scores for uniquely designed sequences confirms that Z tends to decrease linearly with sigma square root of N, as previously suggested by theoretical analysis and Monte Carlo simulations, where sigma, the standard deviation of the number of contacts made by different monomers in the target structure, is a measure of structural segregation and N is the chain length. The probability that the target conformation is indeed the unique global energy minimum of the designed sequence is found to increase dramatically with sigma, approaching unity at maximal segregation. However, due to the huge number of conformations with sub-maximal values of sigma, which correspond to intermediate, only mildly discriminative, values of Z, in addition to significant oscillations of Z around its estimated value, the probability that a correctly designed sequence corresponds to a maximally segregated conformation is small. This behavior of Z also explains the observed relation between sigma and different measures of folding cooperativity of correctly designed sequences.

Anti-cooperativity and cooperativity in hydrophobic interactions: Three-body free energy landscapes and comparison with implicit-solvent potential functions for proteins

Potentials of mean force (PMFs) of three-body hydrophobic association are investigated to gain insight into similar processes in protein folding. Free energy landscapes obtained from explicit simulations of three methanes in water are compared with that predicted by popular implicit-solvent effective potentials for the study of proteins. Explicit-water simulations show that for an extended range of three-methane configurations, hydrophobic association at 25 degrees C under atmospheric pressure is mostly anti-cooperative, that is, less favorable than if the interaction free energies were pairwise additive. Effects of free energy nonadditivity on the kinetic path of association and the temperature dependence of additivity are explored by using a three-methane system and simplified chain models. The prevalence of anti-cooperativity under ambient conditions suggests that driving forces other than hydrophobicity also play critical roles in protein thermodynamic cooperativity. We evaluate the effectiveness of several implicit-solvent potentials in mimicking explicit water simulated three-body PMFs. The favorability of the contact free energy minimum is found to be drastically overestimated by solvent accessible surface area (SASA). Both the SASA and a volume-based Gaussian solvent exclusion model fail to predict the desolvation barrier. However, this barrier is qualitatively captured by the molecular surface area model and a recent "hydrophobic force field." None of the implicit-solvent models tested are accurate for the entire range of three-methane configurations and several other thermodynamic signatures considered.

Geometric and statistical properties of the mean-field hydrophobic-polar model, the large-small model, and real protein sequences

Lattice models, for their coarse-grained nature, are best suited for the study of the "designability problem," the phenomenon in which most of the about 16 000 proteins of known structure have their native conformations concentrated in a relatively small number of about 500 topological classes of conformations. Here it is shown that on a lattice the most highly designable simulated protein structures are those that have the largest number of surface-core switchbacks. A combination of physical, mathematical, and biological reasons that causes the phenomenon is given. By comparing the most foldable model peptides with protein sequences in the Protein Data Bank, it is shown that whereas different models may yield similar designabilities, predicted foldable peptides will simulate natural proteins only when the model incorporates the correct physics and biology, in this case if the main folding force arises from the differing hydrophobicity of the residues, but does not originate, say, from the steric hindrance effect caused by the differing sizes of the residues.

Recombinatoric exploration of novel folded structures: a heteropolymer-based model of protein evolutionary landscapes

The role of recombination in evolution is compared with that of point mutations (substitutions) in the context of a simple, polymer physics-based model mapping between sequence (genotype) and conformational (phenotype) spaces. Crossovers and point mutations of lattice chains with a hydrophobic polar code are investigated. Sequences encoding for a single ground-state conformation are considered viable and used as model proteins. Point mutations lead to diffusive walks on the evolutionary landscape, whereas crossovers can "tunnel" through barriers of diminished fitness. The degree to which crossovers allow for more efficient sequence and structural exploration depends on the relative rates of point mutations versus that of crossovers and the dispersion in fitness that characterizes the ruggedness of the evolutionary landscape. The probability that a crossover between a pair of viable sequences results in viable sequences is an order of magnitude higher than random, implying that a sequence's overall propensity to encode uniquely is embodied partially in local signals. Consistent with this observation, certain hydrophobicity patterns are significantly more favored than others among fragments (i.e., subsequences) of sequences that encode uniquely, and examples reminiscent of autonomous folding units in real proteins are found. The number of structures explored by both crossovers and point mutations is always substantially larger than that via point mutations alone, but the corresponding numbers of sequences explored can be comparable when the evolutionary landscape is rugged. Efficient structural exploration requires intermediate nonextreme ratios between point-mutation and crossover rates.

Randomness, Structural Uniqueness, Modularity and Neutral Evolution in Sequence Space of Model Proteins

The genotype-phenotype map for short chains of a protein-like hetero-polymer model has been characterised [9, 12]. Hydrophobic-Polar (HP) sequences on a square lattice, their structures and partition functions have been exhaustively enumerated and analysed. Homologous sequences folding uniquely into the same structure are interconnected by point mutations. These

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

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

Energetic components of cooperative protein folding

A new lattice protein model with a four-helix bundle ground state is analyzed by a parameter-space Monte Carlo histogram technique to evaluate the effects of an extensive variety of model potentials on folding thermodynamics. Cooperative helical formation and contact energies based on a 5-letter alphabet are found to be insufficient to satisfy calorimetric and other experimental criteria for two-state folding. Such proteinlike behaviors are predicted, however, by models with polypeptidelike local conformational restrictions and environment-dependent hydrogen-bondinglike interactions.

Polymer principles of protein calorimetric two-state cooperativity

The experimental calorimetric two-state criterion requires the van't Hoff enthalpy DeltaH(vH) around the folding/unfolding transition midpoint to be equal or very close to the calorimetric enthalpy DeltaH(cal) of the entire transition. We use an analytical model with experimental parameters from chymotrypsin inhibitor 2 to elucidate the relationship among several different van't Hoff enthalpies used in calorimetric analyses. Under reasonable assumptions, the implications of these DeltaH(vH)'s being approximately equal to DeltaH(cal) are equivalent: Enthalpic variations among denatured conformations in real proteins are much narrower than some previous lattice-model estimates, suggesting that the energy landscape theory "folding to glass transition temperature ratio" T(f) /T(g) may exceed 6.0 for real calorimetrically two-state proteins. Several popular three-dimensional lattice protein models, with different numbers of residue types in their alphabets, are found to fall short of the high experimental standard for being calorimetrically two-state. Some models postulate a multiple-conformation native state with substantial pre-denaturational energetic fluctuations well below the unfolding transition temperature, or predict a significant post-denaturational continuous conformational expansion of the denatured ensemble at temperatures well above the transition point, or both. These scenarios either disagree with experiments on protein size and dynamics, or are inconsistent with conventional interpretation of calorimetric data. However, when empirical linear baseline subtractions are employed, the resulting DeltaH(vH)/DeltaH(cal)'s for some models can be increased to values closer to unity, and baseline subtractions are found to correspond roughly to an operational definition of native-state conformational diversity. These results necessitate a re-assessment of theoretical models and experimental interpretations.

Modeling protein density of states: additive hydrophobic effects are insufficient for calorimetric two-state cooperativity

A well-established experimental criterion for two-state thermodynamic cooperativity in protein folding is that the van't Hoff enthalpy DeltaH(vH) around the transition midpoint is equal, or very nearly so, to the calorimetric enthalpy DeltaH(cal) of the entire transition. This condition is satisfied by many small proteins. We use simple lattice models to provide a statistical mechanical framework to elucidate how this calorimetric two-state picture may be reconciled with the hierarchical multistate scenario emerging from recent hydrogen exchange experiments. We investigate the feasibility of using inverse Laplace transforms to recover the underlying density of states (i.e., enthalpy distribution) from calorimetric data. We find that the constraint imposed by DeltaH(vH)/DeltaH(cal) approximately 1 on densities of states of proteins is often more stringent than other "two-state" criteria proposed in recent theoretical studies. In conjunction with reasonable assumptions, the calorimetric two-state condition implies a narrow distribution of denatured-state enthalpies relative to the overall enthalpy difference between the native and the denatured conformations. This requirement does not always correlate with simple definitions of "sharpness" of a transition and has important ramifications for theoretical modeling. We find that protein models that assume capillarity cooperativity can exhibit overall calorimetric two-state-like behaviors. However, common heteropolymer models based on additive hydrophobic-like interactions, including highly specific two-dimensional Gō models, fail to produce proteinlike DeltaH(vH)/DeltaH(cal) approximately 1. A simple model is constructed to illustrate a proposed scenario in which physically plausible local and nonlocal cooperative terms, which mimic helical cooperativity and environment-dependent hydrogen bonding strength, can lead to thermodynamic behaviors closer to experiment. Our results suggest that proteinlike thermodynamic cooperativity may require a cooperative interplay between local and nonlocal interactions. The prospect of using calorimetric data to constrain Z-scores of knowledge-based potentials is discussed.

De novo design and structural characterization of proteins and metalloproteins

De novo protein design has recently emerged as an attractive approach for studying the structure and function of proteins. This approach critically tests our understanding of the principles of protein folding; only in de novo design must one truly confront the issue of how to specify a protein's fold and function. If we truly understand proteins, it should be possible to design receptors, enzymes, and ion channels from scratch. Further, as this understanding evolves and is further refined, it should be possible to design proteins and biomimetic polymers with properties unprecedented in nature.

Origin of the native driving force for protein folding

We derive an expression with four adjustable parameters that reproduces well the 20x20 Miyazawa-Jernigan potential matrix extracted from known protein structures. The numerical values of the parameters can be approximately computed from the surface tension of water, water-screened dipole interactions between residues and water and among residues, and average exposures of residues in folded proteins.

Mean-field HP model, designability and alpha-helices in protein structures

Analysis of the geometric properties of a mean-field HP model on a square lattice for protein structure shows that structures with a large number of switchbacks between surface and core sites are chosen favorably by peptides as unique ground states. Global comparison of model (binary) peptide sequences with concatenated (binary) protein sequences listed in the Protein Data Bank and the Dali Domain Dictionary indicates that the highest correlation occurs between model peptides choosing the favored structures and those portions of protein sequences containing alpha helices.