Subtle Monte Carlo Updates in Dense Molecular Systems

Although Markov chain Monte Carlo (MC) simulation is a potentially powerful approach for exploring conformational space, it has been unable to compete with molecular dynamics (MD) in the analysis of high density structural states, such as the native state of globular proteins. Here, we introduce a kinetic algorithm, CRISP, that greatly enhances the sampling efficiency in all-atom MC simulations of dense systems. The algorithm is based on an exact analytical solution to the classic chain-closure problem, making it possible to express the interdependencies among degrees of freedom in the molecule as correlations in a multivariate Gaussian distribution. We demonstrate that our method reproduces structural variation in proteins with greater efficiency than current state-of-the-art Monte Carlo methods and has real-time simulation performance on par with molecular dynamics simulations. The presented results suggest our method as a valuable tool in the study of molecules in atomic detail, offering a potential alternative to molecular dynamics for probing long time-scale conformational transitions

Iterative estimation of a PMF.

<p>For each of the eight hydrogen bond categories (see text), the black bar to the right denotes the fraction of occurrence in the native structure of protein G. The gray bars denote the fractions of the eight categories in samples from each iteration; the first iteration is shown to the left in light gray. In the last iteration (iteration 6; dark gray bars, right) the values are very close to the native values for all eight categories. Note that hydrogen bonds between -strands are nearly absent in the first iteration (category ).</p

A PMF based on the radius of gyration.

<p>The goal is to adapt a distribution – which allows sampling of local structures – such that a given target distribution is obtained. For , we used the amino acid sequence of ubiquitin. Sampling from alone results in a distribution with an average of about 27 (triangles). Sampling using the correct expression (open circles), given by Eq. 8, results in a distribution that coincides with the target distribution (solid line). Not taking the reference state into account results in a significant shift towards higher (black circles).</p

Highest probability structures for each iteration.

<p>The structures with highest probability out of 50,000 samples for all six iterations (indicated by a number) are shown as cartoon representations. The N-terminus is shown in blue. The figure was made using PyMOL <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0013714#pone.0013714-Delano1" target="_blank">[64]</a>.</p

Illustration of the central idea presented in this article.

<p>In this example, the goal is to sample conformations with a given distribution for the radius of gyration , and a plausible local structure. could, for example, be derived from known structures in the Protein Data Bank (PDB, left box). is a probability distribution over local structure , typically embodied in fragment library (right box). In order to combine and in a meaningful way (see text), the two distributions are multiplied and divided by (formula at the bottom); is the probability distribution over the radius of gyration for conformations sampled solely from the fragment library (that is, ). The probability distribution will generate conformations with plausible local structures (due to ), while their radii of gyration will be distributed according to , as desired. This simple idea lies at the theoretical heart of the PMF expressions used in protein structure prediction.</p

General statistical justification of PMFs.

<p>The goal is to combine a distribution over a fine grained variable (top right), with a probability distribution over a coarse grained variable (top left). could be, for example, embodied in a fragment library (), a probabilistic model of local structure () or an energy function (); could be, for example, the radius of gyration, the hydrogen bond network, or the set of pairwise distances. usually reflects the distribution of in known protein structures (PDB), but could also stem from experimental data (). Sampling from results in a distribution that differs from . Multiplying and does not result in the desired distribution for either (red box); the correct result requires dividing out the signal with respect to due to (green box). The <i>reference</i> distribution in the denominator corresponds to the contribution of the reference state in a PMF. If is only approximately known, the method can be applied iteratively (dashed arrow). In that case, one attempts to iteratively sculpt an energy funnel. The procedure is statistically rigorous provided and are proper probability distributions; this is usually not the case for conventional pairwise distance PMFs.</p