Approximating net interactions among rigid domains

Many physical simulations aim at evaluating the net interaction between two rigid bodies, resulting from the cumulative effect of pairwise interactions between their constituents. This is manifested particularly in biomolecular applications such as hierarchical protein folding instances where the interaction between almost rigid domains directly influences the folding pathway, the interaction between macromolecules for drug design purposes, self-assembly of nanoparticles for drug design and drug delivery, and design of smart materials and bio-sensors. In general, the brute force approach requires quadratic (in terms of the number of particles) number of pairwise evaluation operations for any relative pose of the two bodies, unless simplifying assumptions lead to a collapse of the computational complexity. We propose to approximate the pairwise interaction function using a linear predictor function, in which the basis functions have separated forms, i.e. the variables that describe local geometries of the two rigid bodies and the ones that reflect the relative pose between them are split in each basis function. Doing so replaces the quadratic number of interaction evaluations for each relative pose with a one-time quadratic computation of a set of characteristic parameters at a preprocessing step, plus constant number of pose function evaluations at each pose, where this constant is determined by the required accuracy of approximation as well as the efficiency of the used approximation method. We will show that the standard deviation of the error for the net interaction is linearly (in terms of number of particles) proportional to the regression error, if the regression errors are from a normal distribution. Our results show that proper balance of the tradeoff between accuracy and speed-up yields an approximation which is computationally superior to other existing methods while maintaining reasonable precision.