Scott C. Schmidler

Bayesian Segmentation of Protein Secondary Structure

We present a novel method for predicting the secondary structure of a protein from its amino acid sequence. Most existing methods predict each position in turn based on a local window of residues, sliding this window along the length of the sequence. In contrast, we develop a probabilistic model of protein sequence/structure relationships in terms of structural segments, and formulate secondary structure prediction as a general Bayesian inference problem. A distinctive feature of our approach is the ability to develop explicit probabilistic models for alpha-helices, beta-strands, and other classes of secondary structure, incorporating experimentally and empirically observed aspects of protein structure such as helical capping signals, side chain correlations, and segment length distributions. Our model is Markovian in the segments, permitting efficient exact calculation of the posterior probability distribution over all possible segmentations of the sequence using dynamic programming. The optimal segmentation is computed and compared to a predictor based on marginal posterior modes, and the latter is shown to provide significant improvement in predictive accuracy. The marginalization procedure provides exact secondary structure probabilities at each sequence position, which are shown to be reliable estimates of prediction uncertainty. We apply this model to a database of 452 nonhomologous structures, achieving accuracies as high as the best currently available methods. We conclude by discussing an extension of this framework to model nonlocal interactions in protein structures, providing a possible direction for future improvements in secondary structure prediction accuracy.

Conditions for rapid and torpid mixing of parallel and simulated tempering on multimodal distributions

We obtain upper bounds on the convergence rates of Markov chains constructed by parallel and simulated tempering. These bounds are used to provide a set of sucien t conditions for torpid mixing of both techniques. We apply these conditions to show torpid mixing of parallel and simulated tempering for three examples: a normal mixture model with unequal covariances in R M and the mean-eld Potts model with q 3, regardless of the number and choice of temperatures, and the meaneld Ising model when an insucien t set of temperatures is chosen. The latter result contrasts with the rapid mixing of parallel and simulated tempering on the meaneld Ising model with a linearly increasing set of temperatures as shown previously.

Regression analysis of multiple protein structures

A general framework is presented for analyzing multiple protein structures using statistical regression methods. The regression approach can superimpose protein structures rigidly or with shear. Also, this approach can superimpose multiple structures explicitly, without resorting to pairwise superpositions. The algorithm alternates between matching corresponding landmarks among the protein structures and superimposing these landmarks. Matching is performed using a robust dynamic programming technique that uses gap penalties that adapt to the given data. Superposition is performed using either orthogonal transformations, which impose the rigid-body assumption, or affine transformations, which allow shear. The resulting regression model of a protein family measures the amount of structural variability at each landmark. A variation of our algorithm permits a separate weight for each landmark, thereby allowing one to emphasize particular segments of a protein structure or to compensate for variances that differ at various positions in a structure. In addition, a method is introduced for finding an initial correspondence, by measuring the discrete curvature along each protein backbone. Discrete curvature also characterizes the secondary structure of a protein backbone, distinguishing among helical, strand, and loop regions. An example is presented involving a set of seven globin structures. Regression analysis, using both affine and orthogonal transformations, reveals that globins are most strongly conserved structurally in helical regions, particularly in the mid-regions of the E, F, and G helices.

Hydration and Conformational Mechanics of Single, End-Tethered Elastin-like Polypeptides

We investigated the effect of temperature, ionic strength, solvent polarity, and type of guest residue on the force-extension behavior of single, end-tethered elastin-like polypeptides (ELPs), using single molecule force spectroscopy (SMFS). ELPs are stimulus-responsive polypeptides that contain repeats of the five amino acids Val-Pro-Gly-Xaa-Gly (VPGXG), where Xaa is a guest residue that can be any amino acid with the exception of proline. We fitted the force-extension data with a freely jointed chain (FJC) model which allowed us to resolve small differences in the effective Kuhn segment length distributions that largely arise from differences in the hydrophobic hydration behavior of ELP. Our results agree qualitatively with predictions from recent molecular dynamics simulations and demonstrate that hydrophobic hydration modulates the molecular elasticity for ELPs. Furthermore, our results show that SMFS, when combined with our approach for data analysis, can be used to study the subtleties of polypeptide-water interactions and thus provides a basis for the study of hydrophobic hydration in intrinsically unstructured biomacromolecules.

Adaptive Markov Chain Monte Carlo for Bayesian Variable Selection

We describe adaptive Markov chain Monte Carlo (MCMC) methods for sampling posterior distributions arising from Bayesian variable selection problems. Point-mass mixture priors are commonly used in Bayesian variable selection problems in regression. However, for generalized linear and nonlinear models where the conditional densities cannot be obtained directly, the resulting mixture posterior may be difficult to sample using standard MCMC methods due to multimodality. We introduce an adaptive MCMC scheme that automatically tunes the parameters of a family of mixture proposal distributions during simulation. The resulting chain adapts to sample efficiently from multimodal target distributions. For variable selection problems point-mass components are included in the mixture, and the associated weights adapt to approximate marginal posterior variable inclusion probabilities, while the remaining components approximate the posterior over nonzero values. The resulting sampler transitions efficiently between models, performing parameter estimation and variable selection simultaneously. Ergodicity and convergence are guaranteed by limiting the adaptation based on recent theoretical results. The algorithm is demonstrated on a logistic regression model, a sparse kernel regression, and a random field model from statistical biophysics; in each case the adaptive algorithm dramatically outperforms traditional MH algorithms. Supplementary materials for this article are available online.

Preserving the Boltzmann ensemble in replica-exchange molecular dynamics.

We consider the convergence behavior of replica-exchange molecular dynamics (REMD) [Sugita and Okamoto, Chem. Phys. Lett. 314, 141 (1999)] based on properties of the numerical integrators in the underlying isothermal molecular dynamics (MD) simulations. We show that a variety of deterministic algorithms favored by molecular dynamics practitioners for constant-temperature simulation of biomolecules fail either to be measure invariant or irreducible, and are therefore not ergodic. We then show that REMD using these algorithms also fails to be ergodic. As a result, the entire configuration space may not be explored even in an infinitely long simulation, and the simulation may not converge to the desired equilibrium Boltzmann ensemble. Moreover, our analysis shows that for initial configurations with unfavorable energy, it may be impossible for the system to reach a region surrounding the minimum energy configuration. We demonstrate these failures of REMD algorithms for three small systems: a Gaussian distribution (simple harmonic oscillator dynamics), a bimodal mixture of Gaussians distribution, and the alanine dipeptide. Examination of the resulting phase plots and equilibrium configuration densities indicates significant errors in the ensemble generated by REMD simulation. We describe a simple modification to address these failures based on a stochastic hybrid Monte Carlo correction, and prove that this is ergodic.

Simultaneous Bayesian Estimation of Alignment and Phylogeny under a Joint Model of Protein Sequence and Structure

For sequences that are highly divergent, there is often insufficient information to infer accurate alignments, and phylogenetic uncertainty may be high. One way to address this issue is to make use of protein structural information, since structures generally diverge more slowly than sequences. In this work, we extend a recently developed stochastic model of pairwise structural evolution to multiple structures on a tree, analytically integrating over ancestral structures to permit efficient likelihood computations under the resulting joint sequence-structure model. We observe that the inclusion of structural information significantly reduces alignment and topology uncertainty, and reduces the number of topology and alignment errors in cases where the true trees and alignments are known. In some cases, the inclusion of structure results in changes to the consensus topology, indicating that structure may contain additional information beyond that which can be obtained from sequences. We use the model to investigate the order of divergence of cytoglobins, myoglobins, and hemoglobins and observe a stabilization of phylogenetic inference: although a sequence-based inference assigns significant posterior probability to several different topologies, the structural model strongly favors one of these over the others and is more robust to the choice of data set.

Bayesian Protein Structure Prediction

An important role for statisticians in the age of the Human Genome Project has developed in the emerging area of “structural bioinformatics”. Sequence analysis and structure prediction for biopolymers is a crucial step on the path to turning newly sequenced genomic data into biologically and pharmaceutically relevant information in support of molecular medicine. We describe our work on Bayesian models for prediction of protein structure from sequence, based on analysis of a database of experimentally determined protein structures. We have previously developed segment-based models of protein secondary structure which capture fundamental aspects of the protein folding process. These models provide predictive performance at the level of the best available methods in the field (Schmidler et al., 2000). Here we show that this Bayesian framework is naturally generalized to incorporate information based on non-local sequence interactions. We demonstrate this idea by presenting a simple model for β-strand pairing and a Markov chain Monte Carlo (MCMC) algorithm for inference. We apply the approach to prediction of 3-dimensional contacts for two example proteins.

Regression analysis of multiple protein structures.

A general framework is presented for analyzing multiple protein structures using statistical regression methods. The regression approach can superimpose protein structures rigidly or with shear. Also, this approach can superimpose multiple structures explicitly, without resorting to pairwise superpositions. The algorithm alternates between matching corresponding landmarks among the protein structures and superimposing these landmarks. Matching is performed using a robust dynamic programming technique that uses gap penalties that adapt to the given data. Superposition is performed using either orthogonal transformations, which impose the rigid-body assumption, or affine transformations, which allow shear. The resulting regression model of a protein family measures the amount of structural variability at each landmark. A variation of our algorithm permits a separate weight for each landmark, thereby allowing one to emphasize particular segments of a protein structure or to compensate for variances that differ at various positions in a structure. In addition, a method is introduced for finding an initial correspondence, by measuring the discrete curvature along each protein backbone. Discrete curvature also characterizes the secondary structure of a protein backbone, distinguishing among helical, strand, and loop regions. An example is presented involving a set of seven globin structures. Regression analysis, using both affine and orthogonal transformations, reveals that globins are most strongly conserved structurally in helical regions, particularly in the mid-regions of the E, F, and G helices.

A Stochastic Evolutionary Model for Protein Structure Alignment and Phylogeny

We present a stochastic process model for the joint evolution of protein primary and tertiary structure, suitable for use in alignment and estimation of phylogeny. Indels arise from a classic Links model, and mutations follow a standard substitution matrix, whereas backbone atoms diffuse in three-dimensional space according to an Ornstein-Uhlenbeck process. The model allows for simultaneous estimation of evolutionary distances, indel rates, structural drift rates, and alignments, while fully accounting for uncertainty. The inclusion of structural information enables phylogenetic inference on time scales not previously attainable with sequence evolution models. The model also provides a tool for testing evolutionary hypotheses and improving our understanding of protein structural evolution.