Improvement of statistical potentials and threading score functions using information maximization

QMEANDisCo-distance constraints applied on model quality estimation

Motivation

Methods that estimate the quality of a 3D protein structure model in absence of an experimental reference structure are crucial to determine a model’s utility and potential applications. Single model methods assess individual models whereas consensus methods require an ensemble of models as input. In this work, we extend the single model composite score QMEAN that employs statistical potentials of mean force and agreement terms by introducing a consensus-based distance constraint (DisCo) score.

Results

DisCo exploits distance distributions from experimentally determined protein structures that are homologous to the model being assessed. Feed-forward neural networks are trained to adaptively weigh contributions by the multi-template DisCo score and classical single model QMEAN parameters. The result is the composite score QMEANDisCo, which combines the accuracy of consensus methods with the broad applicability of single model approaches. We also demonstrate that, despite being the de-facto standard for structure prediction benchmarking, CASP models are not the ideal data source to train predictive methods for model quality estimation. For performance assessment, QMEANDisCo is continuously benchmarked within the CAMEO project and participated in CASP13. For both, it ranks among the top performers and excels with low response times.

Availability and implementation

QMEANDisCo is available as web-server at https://swissmodel.expasy.org/qmean. The source code can be downloaded from https://git.scicore.unibas.ch/schwede/QMEAN.

Supplementary information

Supplementary data are available at Bioinformatics online.

Reduced alphabet of prebiotic amino acids optimally encodes the conformational space of diverse extant protein folds

Background

There is wide agreement that only a subset of the twenty standard amino acids existed prebiotically in sufficient concentrations to form functional polypeptides. We ask how this subset, postulated as {A,D,E,G,I,L,P,S,T,V}, could have formed structures stable enough to found metabolic pathways. Inspired by alphabet reduction experiments, we undertook a computational analysis to measure the structural coding behavior of sequences simplified by reduced alphabets. We sought to discern characteristics of the prebiotic set that would endow it with unique properties relevant to structure, stability, and folding.

Results

Drawing on a large dataset of single-domain proteins, we employed an information-theoretic measure to assess how well the prebiotic amino acid set preserves fold information against all other possible ten-amino acid sets. An extensive virtual mutagenesis procedure revealed that the prebiotic set excellently preserves sequence-dependent information regarding both backbone conformation and tertiary contact matrix of proteins. We observed that information retention is fold-class dependent: the prebiotic set sufficiently encodes the structure space of α/β and α + β folds, and to a lesser extent, of all-α and all-β folds. The prebiotic set appeared insufficient to encode the small proteins. Assessing how well the prebiotic set discriminates native vs. incorrect sequence-structure matches, we found that α/β and α + β folds exhibit more pronounced energy gaps with the prebiotic set than with nearly all alternatives.

Conclusions

The prebiotic set optimally encodes local backbone structures that appear in the folded environment and near-optimally encodes the tertiary contact matrix of extant proteins. The fold-class-specific patterns observed from our structural analysis confirm the postulated timeline of fold appearance in proteogenesis derived from proteomic sequence analyses. Polypeptides arising in a prebiotic environment will likely form α/β and α + β-like folds if any at all. We infer that the progressive expansion of the alphabet allowed the increased conformational stability and functional specificity of later folds, including all-α, all-β, and small proteins. Our results suggest that prebiotic sequences are amenable to mutations that significantly lower native conformational energies and increase discrimination amidst incorrect folds. This property may have assisted the genesis of functional proto-enzymes prior to the expansion of the full amino acid alphabet.

A Membrane Burial Potential with H-Bonds and Applications to Curved Membranes and Fast Simulations

We use the statistics of a large and curated training set of transmembrane helical proteins to develop a knowledge-based potential that accounts for the dependence on both the depth of burial of the protein in the membrane and the degree of side-chain exposure. Additionally, the statistical potential includes depth-dependent energies for unsatisfied backbone hydrogen bond donors and acceptors, which are found to be relatively small, ∼2 RT. Our potential accurately places known proteins within the bilayer. The potential is applied to the mechanosensing MscL channel in membranes of varying thickness and curvature, as well as to the prediction of protein structure. The potential is incorporated into our new Upside molecular dynamics algorithm. Notably, we account for the exchange of protein-lipid interactions for protein-protein interactions as helices contact each other, thereby avoiding overestimating the energetics of helix association within the membrane. Simulations of most multimeric complexes find that isolated monomers and the oligomers retain the same orientation in the membrane, suggesting that the assembly of prepositioned monomers presents a viable mechanism of oligomerization.

Assessing the local structural quality of transmembrane protein models using statistical potentials (QMEANBrane)

Motivation: Membrane proteins are an important class of biological macromolecules involved in many cellular key processes including signalling and transport. They account for one third of genes in the human genome and >50% of current drug targets. Despite their importance, experimental structural data are sparse, resulting in high expectations for computational modelling tools to help fill this gap. However, as many empirical methods have been trained on experimental structural data, which is biased towards soluble globular proteins, their accuracy for transmembrane proteins is often limited.

Results: We developed a local model quality estimation method for membrane proteins (‘QMEANBrane’) by combining statistical potentials trained on membrane protein structures with a per-residue weighting scheme. The increasing number of available experimental membrane protein structures allowed us to train membrane-specific statistical potentials that approach statistical saturation. We show that reliable local quality estimation of membrane protein models is possible, thereby extending local quality estimation to these biologically relevant molecules.

Availability and implementation: Source code and datasets are available on request.

Contact:torsten.schwede@unibas.ch

Supplementary Information:Supplementary data are available at Bioinformatics online.

Deriving high-resolution protein backbone structure propensities from all crystal data using the information maximization device

The most informative probability distribution functions (PDFs) describing the Ramachandran phi-psi dihedral angle pair, a fundamental descriptor of backbone conformation of protein molecules, are derived from high-resolution X-ray crystal structures using an information-theoretic approach. The Information Maximization Device (IMD) is established, based on fundamental information-theoretic concepts, and then applied specifically to derive highly resolved phi-psi maps for all 20 single amino acid and all 8000 triplet sequences at an optimal resolution determined by the volume of current data. The paper shows that utilizing the latent information contained in all viable high-resolution crystal structures found in the Protein Data Bank (PDB), totaling more than 77,000 chains, permits the derivation of a large number of optimized sequence-dependent PDFs. This work demonstrates the effectiveness of the IMD and the superiority of the resulting PDFs by extensive fold recognition experiments and rigorous comparisons with previously published triplet PDFs. Because it automatically optimizes PDFs, IMD results in improved performance of knowledge-based potentials, which rely on such PDFs. Furthermore, it provides an easy computational recipe for empirically deriving other kinds of sequence-dependent structural PDFs with greater detail and precision. The high-resolution phi-psi maps derived in this work are available for download.

Absolute quality evaluation of protein model structures using statistical potentials with respect to the native and reference states

In protein structure prediction, it is crucial to evaluate the degree of native-likeness of given model structures. Statistical potentials extracted from protein structure data sets are widely used for such quality assessment problems, but they are only applicable for comparing different models of the same protein. Although various other methods, such as machine learning approaches, were developed to predict the absolute similarity of model structures to the native ones, they required a set of decoy structures in addition to the model structures. In this paper, we tried to reformulate the statistical potentials as absolute quality scores, without using the information from decoy structures. For this purpose, we regarded the native state and the reference state, which are necessary components of statistical potentials, as the good and bad standard states, respectively, and first showed that the statistical potentials can be regarded as the state functions, which relate a model structure to the native and reference states. Then, we proposed a standardized measure of protein structure, called native-likeness, by interpolating the score of a model structure between the native and reference state scores defined for each protein. The native-likeness correlated with the similarity to the native structures and discriminated the native structures from the models, with better accuracy than the raw score. Our results show that statistical potentials can quantify the native-like properties of protein structures, if they fully utilize the statistical information obtained from the data set.

Fold homology detection using sequence fragment composition profiles of proteins

The effectiveness of sequence alignment in detecting structural homology among protein sequences decreases markedly when pairwise sequence identity is low (the so-called "twilight zone" problem of sequence alignment). Alternative sequence comparison strategies able to detect structural kinship among highly divergent sequences are necessary to address this need. Among them are alignment-free methods, which use global sequence properties (such as amino acid composition) to identify structural homology in a rapid and straightforward way. We explore the viability of using tetramer sequence fragment composition profiles in finding structural relationships that lie undetected by traditional alignment. We establish a strategy to recast any given protein sequence into a tetramer sequence fragment composition profile, using a series of amino acid clustering steps that have been optimized for mutual information. Our method has the effect of compressing the set of 160,000 unique tetramers (if using the 20-letter amino acid alphabet) into a more tractable number of reduced tetramers (approximately 15-30), so that a meaningful tetramer composition profile can be constructed. We test remote homology detection at the topology and fold superfamily levels using a comprehensive set of fold homologs, culled from the CATH database that share low pairwise sequence similarity. Using the receiver-operating characteristic measure, we demonstrate potentially significant improvement in using information-optimized reduced tetramer composition, over methods relying only on the raw amino acid composition or on traditional sequence alignment, in homology detection at or below the "twilight zone".

All-atom knowledge-based potential for RNA structure prediction and assessment

MOTIVATION

Over the recent years, the vision that RNA simply serves as information transfer molecule has dramatically changed. The study of the sequence/structure/function relationships in RNA is becoming more important. As a direct consequence, the total number of experimentally solved RNA structures has dramatically increased and new computer tools for predicting RNA structure from sequence are rapidly emerging. Therefore, new and accurate methods for assessing the accuracy of RNA structure models are clearly needed.

RESULTS

Here, we introduce an all-atom knowledge-based potential for the assessment of RNA three-dimensional (3D) structures. We have benchmarked our new potential, called Ribonucleic Acids Statistical Potential (RASP), with two different decoy datasets composed of near-native RNA structures. In one of the benchmark sets, RASP was able to rank the closest model to the X-ray structure as the best and within the top 10 models for ∼93 and ∼95% of decoys, respectively. The average correlation coefficient between model accuracy, calculated as the root mean square deviation and global distance test-total score (GDT-TS) measures of C3' atoms, and the RASP score was 0.85 and 0.89, respectively. Based on a recently released benchmark dataset that contains hundreds of 3D models for 32 RNA motifs with non-canonical base pairs, RASP scoring function compared favorably to ROSETTA FARFAR force field in the selection of accurate models. Finally, using the self-splicing group I intron and the stem-loop IIIc from hepatitis C virus internal ribosome entry site as test cases, we show that RASP is able to discriminate between known structure-destabilizing mutations and compensatory mutations.

AVAILABILITY

RASP can be readily applied to assess all-atom or coarse-grained RNA structures and thus should be of interest to both developers and end-users of RNA structure prediction methods. The computer software and knowledge-based potentials are freely available at http://melolab.org/supmat.html.

CONTACT

fmelo@bio.puc.cl; mmarti@cipf.es

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Extending the PRIME model for protein aggregation to all 20 amino acids

We extend PRIME, an intermediate-resolution protein model previously used in simulations of the aggregation of polyalanine and polyglutamine, to the description of the geometry and energetics of peptides containing all 20 amino acid residues. The 20 amino acid side chains are classified into 14 groups according to their hydrophobicity, polarity, size, charge, and potential for side chain hydrogen bonding. The parameters for extended PRIME, called PRIME 20, include hydrogen-bonding energies, side chain interaction range and energy, and excluded volume. The parameters are obtained by applying a perceptron-learning algorithm and a modified stochastic learning algorithm that optimizes the energy gap between 711 known native states from the PDB and decoy structures generated by gapless threading. The number of independent pair interaction parameters is chosen to be small enough to be physically meaningful yet large enough to give reasonably accurate results in discriminating decoys from native structures. The most physically meaningful results are obtained with 19 energy parameters.

Information-theoretic analysis of the reference state in contact potentials used for protein structure prediction

Using information-theoretic concepts, we examine the role of the reference state, a crucial component of empirical potential functions, in protein fold recognition. We derive an information-based connection between the probability distribution functions of the reference state and those that characterize the decoy set used in threading. In examining commonly used contact reference states, we find that the quasi-chemical approximation is informatically superior to other variant models designed to include characteristics of real protein chains, such as finite length and variable amino acid composition from protein to protein. We observe that in these variant models, the total divergence, the operative function that quantifies discrimination, decreases along with threading performance. We find that any amount of nativeness encoded in the reference state model does not significantly improve threading performance. A promising avenue for the development of better potentials is suggested by our information-theoretic analysis of the action of contact potentials on individual protein sequences. Our results show that contact potentials perform better when the compositional properties of the data set used to derive the score function probabilities are similar to the properties of the sequence of interest. Results also suggest to use only sequences of similar composition in deriving contact potentials, to tailor the contact potential specifically for a test sequence.

New statistical potential for quality assessment of protein models and a survey of energy functions

Background

Scoring functions, such as molecular mechanic forcefields and statistical potentials are fundamentally important tools in protein structure modeling and quality assessment.

Results

The performances of a number of publicly available scoring functions are compared with a statistical rigor, with an emphasis on knowledge-based potentials. We explored the effect on accuracy of alternative choices for representing interaction center types and other features of scoring functions, such as using information on solvent accessibility, on torsion angles, accounting for secondary structure preferences and side chain orientation. Partially based on the observations made, we present a novel residue based statistical potential, which employs a shuffled reference state definition and takes into account the mutual orientation of residue side chains. Atom- and residue-level statistical potentials and Linux executables to calculate the energy of a given protein proposed in this work can be downloaded from http://www.fiserlab.org/potentials.

Conclusions

Among the most influential terms we observed a critical role of a proper reference state definition and the benefits of including information about the microenvironment of interaction centers. Molecular mechanical potentials were also tested and found to be over-sensitive to small local imperfections in a structure, requiring unfeasible long energy relaxation before energy scores started to correlate with model quality.

Effective knowledge-based potentials

Empirical or knowledge-based potentials have many applications in structural biology such as the prediction of protein structure, protein-protein, and protein-ligand interactions and in the evaluation of stability for mutant proteins, the assessment of errors in experimentally solved structures, and the design of new proteins. Here, we describe a simple procedure to derive and use pairwise distance-dependent potentials that rely on the definition of effective atomic interactions, which attempt to capture interactions that are more likely to be physically relevant. Based on a difficult benchmark test composed of proteins with different secondary structure composition and representing many different folds, we show that the use of effective atomic interactions significantly improves the performance of potentials at discriminating between native and near-native conformations. We also found that, in agreement with previous reports, the potentials derived from the observed effective atomic interactions in native protein structures contain a larger amount of mutual information. A detailed analysis of the effective energy functions shows that atom connectivity effects, which mostly arise when deriving the potential by the incorporation of those indirect atomic interactions occurring beyond the first atomic shell, are clearly filtered out. The shape of the energy functions for direct atomic interactions representing hydrogen bonding and disulfide and salt bridges formation is almost unaffected when effective interactions are taken into account. On the contrary, the shape of the energy functions for indirect atom interactions (i.e., those describing the interaction between two atoms bound to a direct interacting pair) is clearly different when effective interactions are considered. Effective energy functions for indirect interacting atom pairs are not influenced by the shape or the energy minimum observed for the corresponding direct interacting atom pair. Our results suggest that the dependency between the signals in different energy functions is a key aspect that need to be addressed when empirical energy functions are derived and used, and also highlight the importance of additivity assumptions in the use of potential energy functions.

Fast optimization of statistical potentials for structurally constrained phylogenetic models

Background

Statistical approaches for protein design are relevant in the field of molecular evolutionary studies. In recent years, new, so-called structurally constrained (SC) models of protein-coding sequence evolution have been proposed, which use statistical potentials to assess sequence-structure compatibility. In a previous work, we defined a statistical framework for optimizing knowledge-based potentials especially suited to SC models. Our method used the maximum likelihood principle and provided what we call the joint potentials. However, the method required numerical estimations by the use of computationally heavy Markov Chain Monte Carlo sampling algorithms.

Results

Here, we develop an alternative optimization procedure, based on a leave-one-out argument coupled to fast gradient descent algorithms. We assess that the leave-one-out potential yields very similar results to the joint approach developed previously, both in terms of the resulting potential parameters, and by Bayes factor evaluation in a phylogenetic context. On the other hand, the leave-one-out approach results in a considerable computational benefit (up to a 1,000 fold decrease in computational time for the optimization procedure).

Conclusion

Due to its computational speed, the optimization method we propose offers an attractive alternative for the design and empirical evaluation of alternative forms of potentials, using large data sets and high-dimensional parameterizations.

Fragment-based local statistical potentials derived by combining an alphabet of protein local structures with secondary structures and solvent accessibilities

General and transferable statistical potentials to quantify the compatibility between local structures and local sequences of peptide fragments in proteins were derived. In the derivation, structure clusters of fragments are obtained by clustering five-residue fragments in native proteins based on their conformations represented by a local structure alphabet (de Brevern et al., Proteins 2000;41:271-287), secondary structure states, and solvent accessibilities. On the basis of the native sequences of the structurally clustered fragments, the probabilities of different amino acid sequences were estimated for each structure cluster. From the sequence probabilities, statistical energies as a function of sequence for a given structure were directly derived. The same sequence probabilities were employed in a database-matching approach to derive statistical energies as a function of local structure for a given sequence. Compared with prior models of local statistical potentials, we provided an integrated approach in which local conformations and local environments are treated jointly, structures are treated in units of fragments instead of individual residues so that coupling between the conformations of adjacent residues is included, and strong interdependences between the conformations of overlapping or neighboring fragment units are also considered. In tests including fragment threading, pseudosequence design, and local structure predictions, the potentials performed at least comparably and, in most cases, better than a number of existing models applicable to the same contexts indicating the advantages of such an integrated approach for deriving local potentials and suggesting applicability of the statistical potentials derived here in sequence designs and structure predictions.

Automated alphabet reduction for protein datasets

BACKGROUND

We investigate automated and generic alphabet reduction techniques for protein structure prediction datasets. Reducing alphabet cardinality without losing key biochemical information opens the door to potentially faster machine learning, data mining and optimization applications in structural bioinformatics. Furthermore, reduced but informative alphabets often result in, e.g., more compact and human-friendly classification/clustering rules. In this paper we propose a robust and sophisticated alphabet reduction protocol based on mutual information and state-of-the-art optimization techniques.

RESULTS

We applied this protocol to the prediction of two protein structural features: contact number and relative solvent accessibility. For both features we generated alphabets of two, three, four and five letters. The five-letter alphabets gave prediction accuracies statistically similar to that obtained using the full amino acid alphabet. Moreover, the automatically designed alphabets were compared against other reduced alphabets taken from the literature or human-designed, outperforming them. The differences between our alphabets and the alphabets taken from the literature were quantitatively analyzed. All the above process had been performed using a primary sequence representation of proteins. As a final experiment, we extrapolated the obtained five-letter alphabet to reduce a, much richer, protein representation based on evolutionary information for the prediction of the same two features. Again, the performance gap between the full representation and the reduced representation was small, showing that the results of our automated alphabet reduction protocol, even if they were obtained using a simple representation, are also able to capture the crucial information needed for state-of-the-art protein representations.

CONCLUSION

Our automated alphabet reduction protocol generates competent reduced alphabets tailored specifically for a variety of protein datasets. This process is done without any domain knowledge, using information theory metrics instead. The reduced alphabets contain some unexpected (but sound) groups of amino acids, thus suggesting new ways of interpreting the data.

Information and discrimination in pairwise contact potentials

We examine the information-theoretic characteristics of statistical potentials that describe pairwise long-range contacts between amino acid residues in proteins. In our work, we seek to map out an efficient information-based strategy to detect and optimally utilize the structural information latent in empirical data, to make contact potentials, and other statistically derived folding potentials, more effective tools in protein structure prediction. Foremost, we establish fundamental connections between basic information-theoretic quantities (including the ubiquitous Z-score) and contact "energies" or scores used routinely in protein structure prediction, and demonstrate that the informatic quantity that mediates fold discrimination is the total divergence. We find that pairwise contacts between residues bear a moderate amount of fold information, and if optimized, can assist in the discrimination of native conformations from large ensembles of native-like decoys. Using an extensive battery of threading tests, we demonstrate that parameters that affect the information content of contact potentials (e.g., choice of atoms to define residue location and the cut-off distance between pairs) have a significant influence in their performance in fold recognition. We conclude that potentials that have been optimized for mutual information and that have high number of score events per sequence-structure alignment are superior in identifying the correct fold. We derive the quantity "information product" that embodies these two critical factors. We demonstrate that the information product, which does not require explicit threading to compute, is as effective as the Z-score, which requires expensive decoy threading to evaluate. This new objective function may be able to speed up the multidimensional parameter search for better statistical potentials. Lastly, by demonstrating the functional equivalence of quasi-chemically approximated "energies" to fundamental informatic quantities, we make statistical potentials less dependent on theoretically tenuous biophysical formalisms and more amenable to direct bioinformatic optimization.

A knowledge-based potential with an accurate description of local interactions improves discrimination between native and near-native protein conformations

The correct discrimination between native and near-native protein conformations is essential for achieving accurate computer-based protein structure prediction. However, this has proven to be a difficult task, since currently available physical energy functions, empirical potentials and statistical scoring functions are still limited in achieving this goal consistently. In this work, we assess and compare the ability of different full atom knowledge-based potentials to discriminate between native protein structures and near-native protein conformations generated by comparative modeling. Using a benchmark of 152 near-native protein models and their corresponding native structures that encompass several different folds, we demonstrate that the incorporation of close non-bonded pairwise atom terms improves the discriminating power of the empirical potentials. Since the direct and unbiased derivation of close non-bonded terms from current experimental data is not possible, we obtained and used those terms from the corresponding pseudo-energy functions of a non-local knowledge-based potential. It is shown that this methodology significantly improves the discrimination between native and near-native protein conformations, suggesting that a proper description of close non-bonded terms is important to achieve a more complete and accurate description of native protein conformations. Some external knowledge-based energy functions that are widely used in model assessment performed poorly, indicating that the benchmark of models and the specific discrimination task tested in this work constitutes a difficult challenge.

Reduced C(beta) statistical potentials can outperform all-atom potentials in decoy identification

We developed a series of statistical potentials to recognize the native protein from decoys, particularly when using only a reduced representation in which each side chain is treated as a single C(beta) atom. Beginning with a highly successful all-atom statistical potential, the Discrete Optimized Protein Energy function (DOPE), we considered the implications of including additional information in the all-atom statistical potential and subsequently reducing to the C(beta) representation. One of the potentials includes interaction energies conditional on backbone geometries. A second potential separates sequence local from sequence nonlocal interactions and introduces a novel reference state for the sequence local interactions. The resultant potentials perform better than the original DOPE statistical potential in decoy identification. Moreover, even upon passing to a reduced C(beta) representation, these statistical potentials outscore the original (all-atom) DOPE potential in identifying native states for sets of decoys. Interestingly, the backbone-dependent statistical potential is shown to retain nearly all of the information content of the all-atom representation in the C(beta) representation. In addition, these new statistical potentials are combined with existing potentials to model hydrogen bonding, torsion energies, and solvation energies to produce even better performing potentials. The ability of the C(beta) statistical potentials to accurately represent protein interactions bodes well for computational efficiency in protein folding calculations using reduced backbone representations, while the extensions to DOPE illustrate general principles for improving knowledge-based potentials.

Nonbonded terms extrapolated from nonlocal knowledge-based energy functions improve error detection in near-native protein structure models

The accurate assessment of structural errors plays a key role in protein structure prediction, constitutes the first step of protein structure refinement, and has a major impact on subsequent functional inference from structural data. In this study, we assess and compare the ability of different full atom knowledge-based potentials to detect small and localized errors in comparative protein structure models of known accuracy. We have evaluated the effect of incorporating close nonbonded pairwise atom terms on the task of classifying residue modeling accuracy. Since the direct and unbiased derivation of close nonbonded terms from current experimental data is not possible, we extrapolated those terms from the corresponding pseudo-energy functions of a nonlocal knowledge-based potential. It is shown that this methodology clearly improves the detection of errors in protein models, suggesting that a proper description of close nonbonded terms is important to achieve a more complete and accurate description of native protein conformations. The use of close nonbonded terms directly derived from experimental data exhibited a poor performance, demonstrating that these terms cannot be accurately obtained by using the current data and methodology. Some external knowledge-based energy functions that are widely used in model assessment also performed poorly, which suggests that the benchmark of models and the specific error detection task tested in this study constituted a difficult challenge. The methodology presented here could be useful to detect localized structural errors not only in high-quality protein models, but also in experimental protein structures.

Effects of amino acid composition, finite size of proteins, and sparse statistics on distance-dependent statistical pair potentials

Statistical distance dependent pair potentials are frequently used in a variety of folding, threading, and modeling studies of proteins. The applicability of these types of potentials is tightly connected to the reliability of statistical observations. We explored the possible origin and extent of false positive signals in statistical potentials by analyzing their distance dependence in a variety of randomized protein-like models. While on average potentials derived from such models are expected to equal zero at any distance, we demonstrate that systematic and significant distortions exist. These distortions originate from the limited statistical counts in local environments of proteins and from the limited size of protein structures at large distances. We suggest that these systematic errors in statistical potentials are connected to the dependence of amino acid composition on protein size and to variation in protein sizes. Additionally, atom-based potentials are dominated by a false positive signal that is due to correlation among distances measured from atoms of one residue to atoms of another residue. The significance of residue-based pairwise potentials at various spatial pair separations was assessed in this study and it was found that as few as approximately 50% of potential values were statistically significant at distances below 4 A, and only at most approximately 80% of them were significant at larger pair separations. A new definition for reference state, free of the observed systematic errors, is suggested. It has been demonstrated to generate statistical potentials that compare favorably to other publicly available ones.

Methods for optimizing the structure alphabet sequences of proteins

Protein structure prediction based on fragment assemble has made great progress in recent years. Local protein structure prediction is receiving increased attention. One essential step of local protein structure prediction method is that the three-dimensional conformations must be compressed into one-dimensional series of letters of a structural alphabet. The traditional method assigns each structure fragment the structure alphabet that has the best local structure similarity. However, such locally optimal structure alphabet sequence does not guarantee to produce the globally optimal structure. This study presents two efficient methods trying to find the optimal structure alphabet sequence, which can model the native structures as accuracy as possible. First, a 28-letter structure alphabet is derived by clustering fragment in Cartesian space with fragment length of seven residues. The average quantization error of the 28 letters is 0.82 A in term of root mean square deviation. Then, two efficient methods are presented to encode the protein structures into series of structure alphabet letters, that is, the greedy and dynamic programming algorithm. They are tested on PDB database using the structure alphabet developed in Cartesian coordinates space (our structure alphabet) and in torsion angles space (the PB structure alphabet), respectively. The experimental results show that these two methods can find the approximately optimal structure alphabet sequences by searching a small fraction of the modeling space. The traditional local-optimization method achieves 26.27 A root mean square deviations between the reconstructed structures and the native one, while the modeling accuracy is improved to 3.28 A by the greedy algorithm. The results are helpful for local protein structure prediction.

A maximum likelihood framework for protein design

Background

The aim of protein design is to predict amino-acid sequences compatible with a given target structure. Traditionally envisioned as a purely thermodynamic question, this problem can also be understood in a wider context, where additional constraints are captured by learning the sequence patterns displayed by natural proteins of known conformation. In this latter perspective, however, we still need a theoretical formalization of the question, leading to general and efficient learning methods, and allowing for the selection of fast and accurate objective functions quantifying sequence/structure compatibility.

Results

We propose a formulation of the protein design problem in terms of model-based statistical inference. Our framework uses the maximum likelihood principle to optimize the unknown parameters of a statistical potential, which we call an inverse potential to contrast with classical potentials used for structure prediction. We propose an implementation based on Markov chain Monte Carlo, in which the likelihood is maximized by gradient descent and is numerically estimated by thermodynamic integration. The fit of the models is evaluated by cross-validation. We apply this to a simple pairwise contact potential, supplemented with a solvent-accessibility term, and show that the resulting models have a better predictive power than currently available pairwise potentials. Furthermore, the model comparison method presented here allows one to measure the relative contribution of each component of the potential, and to choose the optimal number of accessibility classes, which turns out to be much higher than classically considered.

Conclusion

Altogether, this reformulation makes it possible to test a wide diversity of models, using different forms of potentials, or accounting for other factors than just the constraint of thermodynamic stability. Ultimately, such model-based statistical analyses may help to understand the forces shaping protein sequences, and driving their evolution.