A new distributed algorithm for side-chain repacking

A new distributed algorithm for side-chain repacking in protein-protein association 1

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Mohammad Moghadasi , Dima Kozakov , Pirooz Vakili , Sandor Vajda and Ioannis Ch. Paschalidis 1 Center of Information & Systems Engineering and Division of Systems Engineering, Boston University 2 Department of Biomedical Engineering, Boston University

Abstract

Side-Chain Repacking (SCR)

Our Distributed Algorithm

Side-chain repacking (SCR) is an important component of computational protein docking methods. Existing SCR methods and available software have been designed for protein folding applications where side-chain positioning is also important. We propose a new algorithm which poses SCR as a Maximum Weighted Independent Set (MWIS) problem on an appropriately constructed graph. We develop an approach which solves a relaxation of the MWIS and then rounds the solution to obtain a high-quality feasible solution to the problem. The algorithm is fully distributed and can be executed on a large network of processing nodes requiring only local information and message-passing between neighboring nodes. Motivated by the special structure in docking, we establish optimality guarantees for a certain class of graphs.

SCR : given a receptor-ligand complex with fixed backbones and flexible side chains, the goal is to choose one rotamer for each side chain such that the overall energy of the complex is minimized.

 Phase I. Gradient Projection (GP): iterativey solves the LP-relaxation of (1) and its dual concurrently (x and θ: primal and dual variables)

Protein Structure

• Repeat until convergence: at iteration n for all i ∈ V: (n) (n−1) ♦ mij = xi :: node i sends to all neighbors P (n) (n−1) (n−1) ♦ θj = [θj − γ(1 − k:Cj ∈S,k∈Cj xk )]+ where:

xi (θ) =

SCR as an MWIS Problem

(0) θj

:= maxi:i∈Cj {wi } and to update:

r  2 2 4  1− a (θ) +sgn(ai (θ)) (a (θ))2 +1 i

i

2



1/2,

• Clique-Constrained MWIS: PN max Pi=1 wi xi s.t. i∈Cj xi ≤ 1, xi ∈ {0, 1},

where ai (θ) = wi − ∀j : Cj ∈ S, i = 1, . . . , N.

P

j:Cj ∈S,i∈Cj

, if ai (θ) 6= 0 if ai (θ) = 0

θj .

(1)

 Phase II. Greedy Estimation: to construct a feasible solution to (2) from Phase I outputs.

• Example: Weighted G of a system of 2 residues r and s with sets of rotamers: Ur = {r1 , r2 , r3 } and Us = {s1 , s2 }:

B GP algorithm is exact for Perfect Graphs B The algorithm is fully distributed and requires only message-passing between neighboring nodes B It is designed to be run over multiple processors B The distributed fashion of the algorithm allows us to solve SCP for huge protein interface sets

Partitioning the Interface

Peptide ← Residue ← Side-Chain ← Rotamer

Protein Docking • Protein-ligand docking: to predict the position and orientation of a ligand when it is bound to a protein receptor or enzyme. • Our protein docking refinement procedure:

Computational Results

MWIS Graph [G(V, E)] • Node type I. ri sj for pairs of rotamers i and j which belong to two different residues r and s with associated interaction energy Eri ,sj • Node type II. ri ri for each rotamer i which belongs to residue r with associated self energy Eri • Weights: wri sj = M −Eri ,sj and wri ri = M −Eri B M chosen such that all weights are nonnegative • Edges: (ri sj , tk wl ) ∈ E if the choice of rotamers {ri , sj , tk , wl } has a conflict

• We partition the residue set into non-overlapping clusters, i.e. I = I1 ∪ · · · ∪ IM • We run our algorithm on each cluster Ii independently and in parallel • 2-residue clusters G is perfect and GP is exact • A heuristic to avoid enumerating all cliques: considering biophysically important cliques only, i.e. single-residue cliques and pair-residue cliques

References Including Unbound Conformers • Unbound protein structure carries substantial information about the side-chains in the bound state • Including the unbound side-chain structures in the set of probable conformations considerably improves the prediction quality.

[1] M. Moghadasi, D. Kozakov, P. Vakili, S. Vajda and I.C. Paschalidis, “A New Distributed Algorithm for Side-Chain Positioning in the Process of Protein Docking”, submitted, proceedings of 52nd IEEE Conference on Decision and Control, Firenze, Italy, 2013.