Systematic Testing of Belief-Propagation Estimates for Absolute Free Energies in Atomistic Peptides and Proteins

Abstract

Motivated by the extremely high computing costs associated with estimates of free energies for biological systems using molecular simulations, we further the exploration of existing "belief propagation" (BP) algorithms for fixed-backbone peptide and protein systems. The precalculation of pairwise interactions among discretized libraries of side-chain conformations, along with representation of protein side chains as nodes in a graphical model, enables direct application of the BP approach, which requires only ∼1 s of single-processor run time after the precalculation stage. We use a "loopy BP" algorithm, which can be seen as an approximate generalization of the transfer-matrix approach to highly connected (i.e., loopy) graphs, and it has previously been applied to protein calculations. We examine the application of loopy BP to several peptides as well as the binding site of the T4 lysozyme L99A mutant. The present study reports on (i) the comparison of the approximate BP results with estimates from unbiased estimators based on the Amber99SB force field; (ii) investigation of the effects of varying library size on BP predictions; and (iii) a theoretical discussion of the discretization effects that can arise in BP calculations. The data suggest that, despite their approximate nature, BP free-energy estimates are highly accurate-indeed, they never fall outside confidence intervals from unbiased estimators for the systems where independent results could be obtained. Furthermore, we find that libraries of sufficiently fine discretization (which diminish library-size sensitivity) can be obtained with standard computing resources in most cases. Altogether, the extremely low computing times and accurate results suggest the BP approach warrants further study.

The binding and specificity of chemokine binding proteins, through the lens of experiment and computation

“…The routine calculation of accurate protein–protein binding free energies by FEP is hampered by the size and flexibility of protein ligands, but recent, carefully validated studies demonstrate that accurate results are achievable [ 84 ]. In addition to free energy perturbation methods, several promising new techniques for calculating the binding free energies are being developed, including fragment-based methods [ 85 ] and machine learning methods [ 86 ] that offer the promise that protein–protein binding energy calculations will become routine and accurate.…”

Making high-dimensional molecular distribution functions tractable through Belief Propagation on Factor Graphs

“…II C). 8,13 In order to assess the benefits of this factor graph-based approach we apply it to simulations of two well-studied small peptides [14][15][16] Ala 3 and Aib 9 peptide that display rich conformational dynamics in spite of their small sizes (Fig. 1), and can be described using their φ and ψ dihedral angles.…”

Making High-Dimensional Molecular Distribution Functions Tractable through Belief Propagation on Factor Graphs

“…To do so, we propose a framework using methods from probabilistic graphical models to construct a factorization of input degrees of freedom. Among other works in this area, our work is inspired by ref . Given our eventual interest in sampling slow, representative degrees of freedom, we take the liberty to call these degrees of freedom as order parameters (OPs).…”

“…These functions are then learned using the belief propagation (BP) algorithm (Section ). , To assess the benefits of this factor graph-based approach, we apply it to simulations of two well-studied small peptides − Ala 3 and Aib 9 that display rich conformational dynamics in spite of their small sizes (Figure ) and can be described using their ϕ and ψ dihedral angles. The factor graphs are validated using a range of additional simulations in Section , which include (i) an intervention protocol in the sense of Pearl that directly confirms our predictions of conditional dependency between different degrees of freedom and (ii) a demonstration of how this knowledge can lead to up to 3 orders of magnitude enhanced sampling along all degrees of freedom, with hysteresis-free back-and-forth movement between different metastable states.…”

“…Among other works in this area, our work is inspired by ref 9. 10 Given our eventual interest in sampling slow, representative degrees of freedom, we take the liberty to call these degrees of freedom as order parameters (OPs). We chose to use probabilistic graphical models, taking inspiration from their successes in a variety of fields ranging from identifying disaster victims 11 to decoding messages 12 where the relationships between unknowns are leveraged to perform an inference task with levels of efficiency that would not be possible otherwise.…”