Punctuated equilibrium and shock waves in molecular models of biological evolution

Abstract

We consider the dynamics in infinite population evolution models with a general symmetric fitness landscape. We find shock waves, i.e., discontinuous transitions in the mean fitness, in evolution dynamics even with smooth fitness landscapes, which means that the search for the optimal evolution trajectory is more complicated. These shock waves appear in the case of positive epistasis and can be used to represent punctuated equilibria in biological evolution during long geological time scales. We find exact analytical solutions for discontinuous dynamics at the large-genome-length limit and derive optimal mutation rates for a fixed fitness landscape to send the population from the initial configuration to some final configuration in the fastest way.

An optimal strategy to solve the Prisoner’s Dilemma

“…The above assumptions can be translated into the mathematical problem of maximizing a functional which consists of two terms (one corresponding to selection at the level of individuals, and one corresponding to selection at the level of populations) subject to a dynamical constraint set by the standard RE. In order to find the best strategy to realize this task, we need to solve a maximization problem (similar ideas have been put forward in 37 – 39 ). This is the arena of Optimal Control Theory (OCT) 40 – 43 .…”

“…This subject is also fundamentally linked to the fact that evolution is similar to an optimization process: a population evolves in order to adapt as much as possible to some external conditions (which in turn may change in time). Some works from different perspectives have been recently proposed in this context 37 – 39 . In particular, in 38 it has been argued that living systems adapt to the environment by performing an optimal control.…”

Thermodynamics and evolutionary biology through optimal control

“…This means that the corresponding parameters f a start to follow the bang-bang protocol, with possible jumps between the maximum and minimum allowed values. It is tempting to speculate that the first phase can be associated with Darwinism, meaning that changes are smooth and dictated only by natural selection, while the second phase can be associated with punctuated equilibria, meaning that the fitness of each species (and the very presence of the species) can change abruptly, with the dynamics not completely determined by the RE [36][37][38][39]. In this sense our optimization principle may provide a unified dynamical scheme for these two complementary theories of evolution.…”

“…Moreover, we interpret our principle in terms of the maximization of Fisher's information proposed by Frank [22,23] and we show that Price equation emerges naturally in this context. Finally, we comment on the fact that such optimal control strategies can describe experimentally observed data from protein reactions [20], as well as the emergence of cooperation in social dilemmas [35], and we speculate that in some cases the different dynamical phases stemming from the optimal control could be used to provide a unified dynamical scheme for the different ("Darwinian" and "punctuated equilibria") phases of natural evolution [36][37][38][39].…”