An Analytically Solvable Model for Rapid Evolution of Modular Structure

Biological systems often display modularity, in the sense that they can be decomposed into nearly independent subsystems. Recent studies have suggested that modular structure can spontaneously emerge if goals (environments) change over time, such that each new goal shares the same set of sub-problems with previous goals. Such modularly varying goals can also dramatically speed up evolution, relative to evolution under a constant goal. These studies were based on simulations of model systems, such as logic circuits and RNA structure, which are generally not easy to treat analytically. We present, here, a simple model for evolution under modularly varying goals that can be solved analytically. This model helps to understand some of the fundamental mechanisms that lead to rapid emergence of modular structure under modularly varying goals. In particular, the model suggests a mechanism for the dramatic speedup in evolution observed under such temporally varying goals. Author Summary: Biological systems often display modularity, in the sense that they can be decomposed into nearly independent subsystems. The evolutionary origin of modularity has recently been the focus of renewed attention. A series of studies suggested that modularity can spontaneously emerge in environments that vary over time in a modular fashion—goals composed of the same set of subgoals but each time in a different combination. In addition to spontaneous generation of modularity, evolution was found to be dramatically accelerated under such varying environments. The time to achieve a given goal was much shorter under varying environments in comparison to constant conditions. These studies were based on computer simulations of simple model systems such as logic circuits and RNA secondary structure. Here, we take this a step forward. We present a simple mathematical model that can be solved analytically and suggests mechanisms that lead to the rapid emergence of modular structure.