Evolution in random environment and structural instability

Abstract

We consider stability and evolution of complex biological systems in particular, genetic networks. We focus our attention on supporting of homeostasis in these systems with respect to fluctuations of an external medium (the problem is posed by M. Gromov, A.Carbone [32]). Using a measure of stochastic stability we show that a generic system with fixed parameters is unstable, i.e., the probability to support homeostasis converges to zero as time T → ∞. However, if we consider a population of unstable systems, which are capable to evolve (change their parameters), then such a population can be stable as T → ∞. This means that the probability to survive may be non-zero as T → ∞.Evolution algorithms, that provide stability of populations, are not trivial. We show that the mathematical results on evolution algorithms are consistent with experimental data on genetic evolution.

Instability, complexity, and evolution

“…In this paper we extend our previous estimates( [65]), which show that the survival probability of each circuit (2.7) of a fixed structure tends to zero as T → ∞. Therefore, "homeostasis"…”

“…To describe mathematically biological systems, we consider classical main models of mathematical biology (circuits, reaction -diffusion equations). This paper develops our previous results [67,64,65] and uses basic concepts proposed by M. Gromov-A. Carbone and L. Van Valen ( [70], [27]).…”

“…For example, for them one can define natural complexity characteristic [67,64,65]. They can simulate all Turing machines [39], generate any structurally stable dynamics [69] and spatio-temporal patterns [68,63,67].…”

“…In previous works [67,64,65] we have considered circuit models important in biology, chemistry, economics and physics. We first find some properties of stable evolution algorithms for these circuits.…”

“…The contemporary overview of this approach can be found, for example, in [20,22], some results in [23,24,25,26] and applications for biology in [64,65,67]. By this approach we introduce a new class of random dynamical systems which contain many previously studied important systems (neural and genetic circuits) and, on the other hand, for such systems we can prove some general results on instability.…”