Computational aspects of aggregation in biological systems

Abstract

Many biologically relevant dynamical systems are aggregable, in the sense that one can divide their microvariables x1,..., xn into several (k) nonintersecting groups and find functions y1,..., yk (k < n) from these groups (macrovariables) whose dynamics only depend on the initial state of the macrovariable. For example, the state of a population genetic system can be described by listing the frequencies xi of different genotypes, so that the corresponding dynamical system describe the effects of mutation, recombination, and natural selection. The goal of aggregation approaches in population genetics is to find macrovariables ya,..., yk to which aggregated mutation, recombination, and selection functions could be applied. Population genetic models are formally equivalent to genetic algorithms, and are therefore of wide interest in the computational sciences. Another example of a multi-variable biological system of interest occurs in ecology. Ecosystems contain many interacting species, and because of the complexity of multi-variable nonlinear systems, it would be of value to derive a formal description that reduces the number of variables to some macrostates that are weighted sums of the densities of several species. In this chapter, we explore different computational aspects of aggregability for linear and non-linear systems. Specifically, we investigate the problem of conditional aggregability (i.e., aggregability restricted to modular states) and aggregation of variables in biologically relevant quadratic dynamical systems. © 2008 Springer-Verlag Berlin Heidelberg.