Multiple Protein Structure Alignment by Deterministic Annealing

Although many algorithms have been developed so far for structure alignment problem based on either distancebased methods or vector-based methods, such as iterative dynamical programming, Monte Carlo simulation, graph theory, and mean field equations, only a few methods are published for multiple structure alignment, such as progressive structure alignment and Monte Carlo optimization. In this paper, we propose a novel method for solving multiple structure alignment problem, based on mean field annealing technique. We define the structure alignment as a mixed integer-programming (MIP) problem with the inter-atomic distances between two or more structures as an objective function[1]. The integer variables represent the marchings among structures whereas the continuous variables are translation vectors and rotation matrices with each protein structure as a rigid body. By exploiting the special structure of continuous partial problem, we transform the MIP into a nonlinear optimization problem (NOP) with a nonlinear objective function and linear constraints, based on mean field equations. To optimize the NOP, a mean field annealing procedure is adopted with a modified Potts spin model[2]. Since all linear constraints are embedded in the mean field equations, we do not need to add any penalty terms of the constraints to the error function. In other words, there is no ”soft constraint” in our mean field model and all constraints are automatically satisfied during the annealing process, thereby not only making the optimization more efficiently but also eliminating unnecessary parameters of penalty that usually require careful tuning dependent on the problems. 2 Pairwise Structure Alignment