Predicting protein secondary structure with probabilistic schemata of evolutionarily derived information.

We demonstrate the applicability of our previously developed Bayesian probabilistic approach for predicting residue solvent accessibility to the problem of predicting secondary structure. Using only single-sequence data, this method achieves a three-state accuracy of 67% over a database of 473 non-homologous proteins. This approach is more amenable to inspection and less likely to overlearn specifics of a dataset than "black box" methods such as neural networks. It is also conceptually simpler and less computationally costly. We also introduce a novel method for representing and incorporating multiple-sequence alignment information within the prediction algorithm, achieving 72% accuracy over a dataset of 304 non-homologous proteins. This is accomplished by creating a statistical model of the evolutionarily derived correlations between patterns of amino acid substitution and local protein structure. This model consists of parameter vectors, termed "substitution schemata," which probabilistically encode the structure-based heterogeneity in the distributions of amino acid substitutions found in alignments of homologous proteins. The model is optimized for structure prediction by maximizing the mutual information between the set of schemata and the database of secondary structures. Unlike "expert heuristic" methods, this approach has been demonstrated to work well over large datasets. Unlike the opaque neural network algorithms, this approach is physicochemically intelligible. Moreover, the model optimization procedure, the formalism for predicting one-dimensional structural features and our previously developed method for tertiary structure recognition all share a common Bayesian probabilistic basis. This consistency starkly contrasts with the hybrid and ad hoc nature of methods that have dominated this field in recent years.     

Assessing the Performance of Neural Encoding Models in the Presence of Noise

An analytical method is introduced for evaluating the performance of neural encoding models. The method addresses a critical question that arises during the course of the development and validation of encoding models: is a given model near optimal in terms of its accuracy in predicting the stimulus-elicited responses of a neural system, or can the predictive accuracy be improved significantly by further model development? The evaluation method is based on a derivation of the minimum mean-square error between actual responses and modeled responses. It is formulated as a comparison between the mean-square error of the candidate model and the theoretical minimum mean-square error attainable through an optimal model for the system. However, no a priori information about the nature of the optimal model is required. The theoretically minimum error is determined solely from the coherence function between pairs of system responses to repeated presentations of the same dynamic stimulus. Thus, the performance of the candidate model is judged against the performance of an optimal model rather than against that of an arbitrarily assumed model. Using this method, we evaluated a linear model for neural encoding by mechanosensory cells in the cricket cercal system. At low stimulus intensities, the best-fit linear model of encoding by single cells was found to be nearly optimal, even though the coherence between stimulus-response pairs (a commonly used measure of system linearity) was low. In this low-stimulus-intensity regime, the mean square error of the linear model was on the order of the power of the cell responses. In contrast, at higher stimulus intensities the linear model was not an accurate representation of neural encoding, even though the stimulus-response coherence was substantially higher than in the low-intensity regime.     

Determination of eukaryotic protein coding regions using neural networks and information theory.

Our previous work applied neural network techniques to the problem of discriminating open reading frame (ORF) sequences taken from introns versus exons. The method counted the codon frequencies in an ORF of a specified length, and then used this codon frequency representation of DNA fragments to train a neural net (essentially a Perceptron with a sigmoidal, or "soft step function", output) to perform this discrimination. After training, the network was then applied to a disjoint "predict" set of data to assess accuracy. The resulting accuracy in our previous work was 98.4%, exceeding accuracies reported in the literature at that time for other algorithms. Here, we report even higher accuracies stemming from calculations of mutual information (a correlation measure) of spatially separated codons in exons, and in introns. Significant mutual information exists in exons, but not in introns, between adjacent codons. This suggests that dicodon frequencies of adjacent codons are important for intron/exon discrimination. We report that accuracies obtained using a neural net trained on the frequency of dicodons is significantly higher at smaller fragment lengths than even our original results using codon frequencies, which were already higher than simple statistical methods that also used codon frequencies. We also report accuracies obtained from including codon and dicodon statistics in all six reading frames, i.e. the three frames on the original and complement strand. Inclusion of six-frame statistics increases the accuracy still further. We also compare these neural net results to a Bayesian statistical prediction method that assumes independent codon frequencies in each position. The performance of the Bayesian scheme is poorer than any of the neural based schemes, however many methods reported in the literature either explicitly, or implicitly, use this method. Specifically, Bayesian prediction schemes based on codon frequencies achieve 90.9% accuracy on 90 codon ORFs, while our best neural net scheme reaches 99.4% accuracy on 60 codon ORFs. "Accuracy" is defined as the average of the exon and intron sensitivities. Achievement of sufficiently high accuracies on short fragment lengths can be useful in providing a computational means of finding coding regions in unannotated DNA sequences such as those arising from the mega-base sequencing efforts of the Human Genome Project. We caution that the high accuracies reported here do not represent a complete solution to the problem of identifying exons in "raw" base sequences. The accuracies are considerably lower from exons of small length, although still higher than accuracies reported in the literature for other methods. Short exon lengths are not uncommon.(ABSTRACT TRUNCATED AT 400 WORDS)     

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.     

A Bayesian extension of phylogenetic generalized least squares: Incorporating uncertainty in the comparative study of trait relationships and evolutionary rates

Phylogenetic comparative methods use tree topology, branch lengths, and models of phenotypic change to take into account nonindependence in statistical analysis. However, these methods normally assume that trees and models are known without error. Approaches relying on evolutionary regimes also assume specific distributions of character states across a tree, which often result from ancestral state reconstructions that are subject to uncertainty. Several methods have been proposed to deal with some of these sources of uncertainty, but approaches accounting for all of them are less common. Here, we show how Bayesian statistics facilitates this task while relaxing the homogeneous rate assumption of the well-known phylogenetic generalized least squares (PGLS) framework. This Bayesian formulation allows uncertainty about phylogeny, evolutionary regimes, or other statistical parameters to be taken into account for studies as simple as testing for coevolution in two traits or as complex as testing whether bursts of phenotypic change are associated with evolutionary shifts in intertrait correlations. A mixture of validation approaches indicates that the approach has good inferential properties and predictive performance. We provide suggestions for implementation and show its usefulness by exploring the coevolution of ankle posture and forefoot proportions in Carnivora.     

QBES: predicting real values of solvent accessibility from sequences by efficient, constrained energy optimization

Solvent accessibility, one of the key properties of amino acid residues in proteins, can be used to assist protein structure prediction. Various approaches such as neural network, support vector machines, probability profiles, information theory, Bayesian theory, logistic function, and multiple linear regression have been developed for solvent accessibility prediction. In this article, a much simpler quadratic programming method based on the buriability parameter set of amino acid residues is developed. The new method, called QBES (Quadratic programming and Buriability Energy function for Solvent accessibility prediction), is reasonably accurate for predicting the real value of solvent accessibility. By using a dataset of 30 proteins to optimize three parameters, the average correlation coefficients between the predicted and actual solvent accessibility are about 0.5 for all four independent test sets ranging from 126 to 513 proteins. The method is efficient. It takes only 20 min for a regular PC to obtain results of 30 proteins with an average length of 263 amino acids. Although the proposed method is less accurate than a few more sophisticated methods based on neural network or support vector machines, this is the first attempt to predict solvent accessibility by energy optimization with constraints. Possible improvements and other applications of the method are discussed.     

Positive Darwinian Selection at Single Amino Acid Sites Conferring Plant Virus Resistance

Explicit evaluation of the accuracy and power of maximum likelihood and Bayesian methods for detecting site-specific positive Darwinian selection presents a challenge because selective consequences of single amino acid changes are generally unknown. We exploited extensive molecular and functional characterization of amino acid substitutions in the plant gene eIF4E to evaluate the performance of these methods in detecting site-specific positive selection. We documented for the first time a molecular signature of positive selection within a recessive resistance gene in plants. We then used two statistical platforms, Phylogenetic Analysis Using Maximum Likelihood and Hypothesis Testing Using Phylogenies (HyPhy), to look for site-specific positive selection. Their relative power and accuracy are assessed by comparing the sites they identify as being positively selected with those of resistance-determining amino acids. Our results indicate that although both methods are surprisingly accurate in their identification of resistance sites, HyPhy appears to more accurately identify biologically significant amino acids using our data set. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users. J. R. Cavatorta and A. E. Savage have contributed equally to this work.     

A novel representation of protein sequences for prediction of subcellular location using support vector machines

As the number of complete genomes rapidly increases, accurate methods to automatically predict the subcellular location of proteins are increasingly useful to help their functional annotation. In order to improve the predictive accuracy of the many prediction methods developed to date, a novel representation of protein sequences is proposed. This representation involves local compositions of amino acids and twin amino acids, and local frequencies of distance between successive (basic, hydrophobic, and other) amino acids. For calculating the local features, each sequence is split into three parts: N-terminal, middle, and C-terminal. The N-terminal part is further divided into four regions to consider ambiguity in the length and position of signal sequences. We tested this representation with support vector machines on two data sets extracted from the SWISS-PROT database. Through fivefold cross-validation tests, overall accuracies of more than 87% and 91% were obtained for eukaryotic and prokaryotic proteins, respectively. It is concluded that considering the respective features in the N-terminal, middle, and C-terminal parts is helpful to predict the subcellular location.     

Support vector machine for discrimination of thermophilic and mesophilic proteins based on amino acid composition

The identification of the thermostability from the amino acid sequence information would be helpful in computational screening for thermostable proteins. We have developed a method to discriminate thermophilic and mesophilic proteins based on support vector machines. Using self-consistency validation, 5-fold cross-validation and independent testing procedure with other datasets, this module achieved overall accuracy of 94.2%, 90.5% and 92.4%, respectively. The performance of this SVM-based module was better than the classifiers built using alternative machine learning and statistical algorithms including artificial neural networks, Bayesian statistics, and decision trees, when evaluated using these three validation methods. The influence of protein size on prediction accuracy was also addressed.     

Generation and evaluation of dimension-reduced amino acid parameter representations by artificial neural networks

In order to process data of proteins, a numerical representation for an amino acid is often necessary. Many suitable parameters can be derived from experiments or statistical analysis of databases. To ensure a fast and efficient use of these sources of information, a reduction and extraction of relevant information out of these parameters is a basic need. In this approach established methods like principal component analysis (PCA) are supplemented by a method based on symmetric neural networks. Two different parameter representations of amino acids are reduced from five and seven dimensions, respectively, to one, two, three, or four dimensions by using a symmetric neural network approach alternatively with one or three hidden layers. It is possible to create general reduced parameter representations for amino acids. To demonstrate the ability of this approach, these reduced sets of parameters are applied for the ab initio prediction of protein secondary structure from primary structure only. Artificial neural networks are implemented and trained with a diverse representation of 430 proteins out of the PDB. An essentially faster training and also prediction without a decrease in accuracy is obtained for the reduced parameter representations in comparison with the complete set of parameters. The method is transferable to other amino acids or even other molecular building blocks, like nucleic acids, and therefore represents a general approach.Electronic Supplementary Material available.     

The independent distribution of amino acid near neighbor pairs into polypeptides

With the assumption that individual amino acids are independently distributed into naturally occurring polypeptide chains, it is shown that amino acid pairs with 0–2 arbitrary intervening residues are also independently distributed, with a few possible exceptions. This is not true of N- and C-terminal amino acids.     

Bayesian neural networks

A neural network that uses the basic Hebbian learning rule and the Bayesian combination function is defined. Analogously to Hopfield's neural network, the convergence for the Bayesian neural network that asynchronously updates its neurons' states is proved. The performance of the Bayesian neural network in four medical domains is compared with various classification methods. The Bayesian neural network uses more sophisticated combination function than Hopfield's neural network and uses more economically the available information. The naive Bayesian classifier typically outperforms the basic Bayesian neural network since iterations in network make too many mistakes. By restricting the number of iterations and increasing the number of fixed points the network performs better than the naive Bayesian classifier. The Bayesian neural network is designed to learn very quickly and incrementally.     

Posterior predictive checks of coalescent models: P2C2M,an R package

Bayesian inference operates under the assumption that the empirical data are a good statistical fit to the analytical model, but this assumption can be challenging to evaluate. Here, we introduce a novel r package that utilizes posterior predictive simulation to evaluate the fit of the multispecies coalescent model used to estimate species trees. We conduct a simulation study to evaluate the consistency of different summary statistics in comparing posterior and posterior predictive distributions, the use of simulation replication in reducing error rates and the utility of parallel process invocation towards improving computation times. We also test P2C2M on two empirical data sets in which hybridization and gene flow are suspected of contributing to shared polymorphism, which is in violation with the coalescent model: Tamias chipmunks and Myotis bats. Our results indicate that (i) probability‐based summary statistics display the lowest error rates, (ii) the implementation of simulation replication decreases the rate of type II errors, and (iii) our r package displays improved statistical power compared to previous implementations of this approach. When probabilistic summary statistics are used, P2C2M corroborates the assumption that genealogies collected from Tamias and Myotis are not a good fit to the multispecies coalescent model. Taken as a whole, our findings argue that an assessment of the fit of the multispecies coalescent model should accompany any phylogenetic analysis that estimates a species tree.     

Detection of convergent and parallel evolution at the amino acid sequence level

Adaptive evolution at the molecular level can be studied by detecting convergent and parallel evolution at the amino acid sequence level. For a set of homologous protein sequences, the ancestral amino acids at all interior nodes of the phylogenetic tree of the proteins can be statistically inferred. The amino acid sites that have experienced convergent or parallel changes on independent evolutionary lineages can then be identified by comparing the amino acids at the beginning and end of each lineage. At present, the efficiency of the methods of ancestral sequence inference in identifying convergent and parallel changes is unknown. More seriously, when we identify convergent or parallel changes, it is unclear whether these changes are attributable to random chance. For these reasons, claims of convergent and parallel evolution at the amino acid sequence level have been disputed. We have conducted computer simulations to assess the efficiencies, of the parsimony and Bayesian methods of ancestral sequence inference in identifying convergent and parallel-change sites. Our results showed that the Bayesian method performs better than the parsimony method in identifying parallel changes, and both methods are inefficient in identifying convergent changes. However, the Bayesian method is recommended for estimating the number of convergent-change sites because it gives a conservative estimate. We have developed statistical tests for examining whether the observed numbers of convergent and parallel changes are due to random chance. As an example, we reanalyzed the stomach lysozyme sequences of foregut fermenters and found that parallel evolution is statistically significant, whereas convergent evolution is not well supported.      

Robustness of Phylogenetic Inference to Model Misspecification Caused by Pairwise Epistasis

Likelihood-based phylogenetic inference posits a probabilistic model of character state change along branches of a phylogenetic tree. These models typically assume statistical independence of sites in the sequence alignment. This is a restrictive assumption that facilitates computational tractability, but ignores how epistasis, the effect of genetic background on mutational effects, influences the evolution of functional sequences. We consider the effect of using a misspecified site-independent model on the accuracy of Bayesian phylogenetic inference in the setting of pairwise-site epistasis. Previous work has shown that as alignment length increases, tree reconstruction accuracy also increases. Here, we present a simulation study demonstrating that accuracy increases with alignment size even if the additional sites are epistatically coupled. We introduce an alignment-based test statistic that is a diagnostic for pairwise epistasis and can be used in posterior predictive checks.     

A Bayesian Alternative to Mutual Information for the Hierarchical Clustering of Dependent Random Variables

The use of mutual information as a similarity measure in agglomerative hierarchical clustering (AHC) raises an important issue: some correction needs to be applied for the dimensionality of variables. In this work, we formulate the decision of merging dependent multivariate normal variables in an AHC procedure as a Bayesian model comparison. We found that the Bayesian formulation naturally shrinks the empirical covariance matrix towards a matrix set a priori (e.g., the identity), provides an automated stopping rule, and corrects for dimensionality using a term that scales up the measure as a function of the dimensionality of the variables. Also, the resulting log Bayes factor is asymptotically proportional to the plug-in estimate of mutual information, with an additive correction for dimensionality in agreement with the Bayesian information criterion. We investigated the behavior of these Bayesian alternatives (in exact and asymptotic forms) to mutual information on simulated and real data. An encouraging result was first derived on simulations: the hierarchical clustering based on the log Bayes factor outperformed off-the-shelf clustering techniques as well as raw and normalized mutual information in terms of classification accuracy. On a toy example, we found that the Bayesian approaches led to results that were similar to those of mutual information clustering techniques, with the advantage of an automated thresholding. On real functional magnetic resonance imaging (fMRI) datasets measuring brain activity, it identified clusters consistent with the established outcome of standard procedures. On this application, normalized mutual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering.     

Phylogeny reconstruction: increasing the accuracy of pairwise distance estimation using Bayesian inference of evolutionary rates

Distance-based methods for phylogeny reconstruction are the fastest and easiest to use, and their popularity is accordingly high. They are also the only known methods that can cope with huge datasets of thousands of sequences. These methods rely on evolutionary distance estimation and are sensitive to errors in such estimations. In this study, a novel Bayesian method for estimation of evolutionary distances is developed. The proposed method enables the use of a sophisticated evolutionary model that better accounts for among-site rate variation (ASRV), thereby improving the accuracy of distance estimation. Rate variations are estimated within a Bayesian framework by extracting information from the entire dataset of sequences, unlike standard methods that can only use one pair of sequences at a time. We compare the accuracy of a cascade of distance estimation methods, starting from commonly used methods and moving towards the more sophisticated novel method. Simulation studies show significant improvements in the accuracy of distance estimation by the novel method over the commonly used ones. We demonstrate the effect of the improved accuracy on tree reconstruction using both real and simulated protein sequence alignments. An implementation of this method is available as part of the SEMPHY package.     

Monte Carlo simulation studies on the prediction of protein folding types from amino acid composition

A number of methods to predicting the folding type of a protein based on its amino acid composition have been developed during the past few years. In order to perform an objective and fair comparison of different prediction methods, a Monte Carlo simulation method was proposed to calculate the asymptotic limit of the prediction accuracy [Zhang and Chou (1992),Biophys. J. 63, 1523–1529, referred to as simulation method I]. However, simulation method I was based on an oversimplified assumption, i.e., there are no correlations between the compositions of different amino acids. By taking into account such correlations, a new method, referred to as simulation method II, has been proposed to recalculate the objective accuracy of prediction for the least Euclidean distance method [Nakashimaet al. (1986),J. Biochem. 99, 152–162] and the least Minkowski distance method [Chou (1989),Prediction in Protein Structure and the Principles of Protein Conformation, Plenum Press, New York, pp. 549–586], respectively. The results show that the prediction accuracy of the former is still better than that of the latter, as found by simulation method I; however, after incorporating the correlative effect, the objective prediction accuracies become lower for both methods. The reason for this phenomenon is discussed in detail. The simulation method and the idea developed in this paper can be applied to examine any other statistical prediction method, including the computersimulated neural network method.     

Hodgkin-Huxley type ion channel characterization: an improved method of voltage clamp experiment parameter estimation

The Hodgkin-Huxley formalism for quantitative characterization of ionic channels is widely used in cellular electrophysiological models. Model parameters for these individual channels are determined from voltage clamp experiments and usually involve the assumption that inactivation process occurs on a time scale which is infinitely slow compared to the activation process. This work shows that such an assumption may lead to appreciable errors under certain physiological conditions and proposes a new numerical approach to interpret voltage clamp experiment results. In simulated experimental protocols the new method was shown to exhibit superior accuracy compared to the traditional least squares fitting methods. With noiseless input data the error in gating variables and time constants was less than 1%, whereas the traditional methods generated upwards of 10% error and predicted incorrect gating kinetics. A sensitivity analysis showed that the new method could tolerate up to approximately 15% perturbation in the input data without unstably amplifying error in the solution. This method could also assist in designing more efficient experimental protocols, since all channel parameters (gating variables, time constants and maximum conductance) could be determined from a single voltage step.     

A pedagogical walkthrough of computational modeling and simulation of Wnt signaling pathway using static causal models in MATLAB

The discrete coefficient of determination (CoD) measures the nonlinear interaction between discrete predictor and target variables and has had far-reaching applications in Genomic Signal Processing. Previous work has addressed the inference of the discrete CoD using classical parametric and nonparametric approaches. In this paper, we introduce a Bayesian framework for the inference of the discrete CoD. We derive analytically the optimal minimum mean-square error (MMSE) CoD estimator, as well as a CoD estimator based on the Optimal Bayesian Predictor (OBP). For the latter estimator, exact expressions for its bias, variance, and root-mean-square (RMS) are given. The accuracy of both Bayesian CoD estimators with non-informative and informative priors, under fixed or random parameters, is studied via analytical and numerical approaches. We also demonstrate the application of the proposed Bayesian approach in the inference of gene regulatory networks, using gene-expression data from a previously published study on metastatic melanoma.