Statistical Estimation of Statistical Mechanical Models: Helix-Coil Theory and Peptide Helicity Prediction

Abstract

Analysis of biopolymer sequences and structures generally adopts one of two approaches: use of detailed biophysical theoretical models of the system with experimentally-determined parameters, or largely empirical statistical models obtained by extracting parameters from large datasets. In this work, we demonstrate a merger of these two approaches using Bayesian statistics. We adopt a common biophysical model for local protein folding and peptide configuration, the helix-coil model. The parameters of this model are estimated by statistical fitting to a large dataset, using prior distributions based on experimental data. L(1)-norm shrinkage priors are applied to induce sparsity among the estimated parameters, resulting in a significantly simplified model. Formal statistical procedures for evaluating support in the data for previously proposed model extensions are presented. We demonstrate the advantages of this approach including improved prediction accuracy and quantification of prediction uncertainty, and discuss opportunities for statistical design of experiments. Our approach yields a 39% improvement in mean-squared predictive error over the current best algorithm for this problem. In the process we also provide an efficient recursive algorithm for exact calculation of ensemble helicity including sidechain interactions, and derive an explicit relation between homo- and heteropolymer helix-coil theories and Markov chains and (non-standard) hidden Markov models respectively, which has not appeared in the literature previously.

Statistical Prediction and Molecular Dynamics Simulation

“…Similarly, the purpose of a molecular simulation is rarely done simply to predict a single quantity such as helicity; and when such predictions are desired, statistical models may often be developed that are significantly more accurate across a wider range of input molecules (8). The advantage of a simulation is the ability to examine many different ensemble quantities calculated from a single simulation output.…”

“…We have run extensive molecular dynamics simulations for eight helical peptides, some naturally occurring and some designed, which have been previously studied experimentally by CD and shown to have measurable helicity (mean u 222 ellipticity) in solution. Table 1 shows the peptides studied along with their original experimental characterization; these peptides were selected from a database of helical peptides (8) to obtain a range of helicities among native peptides at physiological pH.…”

“…3. We do use the estimated value s exp ¼ 0.07, from Schmidler et al (8), for convenience; more formally s exp could be estimated or integrated out to obtain the marginal posterior distribution (37).…”

“…A grand challenge of macromolecular simulation is the simulation of protein folding (6). Adequate ensemble-averaging for large proteins remains beyond current computational doi: 10.1529/biophysj.108.131623 resources; here we study helical peptide folding, which has been widely studied as a model system for protein folding both experimentally (e.g., (7), and the large body of subsequent literature summarized in (8)) and computationally (in (9)(10)(11)(12)(13)(14)(15)(16)). For short helical peptides, using modern simulation algorithms and a cluster of computer processors, we are able to adequately address the sampling issue.…”

Nonadditivity in the alpha‐helix to coil transition

“…Unfortunately, for any mLRM that is relatively simple using a few parameters per amino acid type, the parameters are found not to be transferable 15. However, Herculean versions16–18 that account for myriad effects using many hundreds of parameters to describe arbitrary heterogeneous chains have predictability power. Although many more parameters are used, they are characterizing local effects.…”