Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

On the existence and stability of equilibrium probability measures in Gaussian mutator models with environmental stress within Fisher's geometric framework.
Michał Piotr Startek, Arnaud Le Rouzic, Anna Gambin

Last modified: 2014-04-01


Fisher's geometric model is a well-established (class of) models in population genetics, wherein the organism's phenotype is represented in a geometric setting, as a vector of real-valued parameters. Selection is based on a fitness function (dependant on the phenotype), and random mutations are modelled by random changes to the organism's phenotype. The population is represented as a probability measure over the space of possible phenotypes, representing the proportion of organisms with given phenotype. The applications of these models range from studying adaptation to an environment, to simulating the evolutionary effects of pleiotropy.

In a mutator model with environmental stress, the organisms are differentiated also by their mutation rate, which in some scenarios is allowed to vary as well. Environmental stress is reflected by moving the optimum of the selection function in each generation.

In our study we assume a Gaussian selection function, as well as Gaussian mutation operator. Exploiting the various properties of the class of Gaussian functions, such as its stability and closure under various operations, allows us to analytically derive the equilibrium probability measure (with respect to the moving optimum) describing the stable state of a population chasing a shifting (with constant speed) phenotypic optimum. Furthermore, convergence of a population described by any probability measure (be it continuous or singular) to the aforementioned equilibrium may be studied and analytically proven. We have performed studies of interdependence of the model's variables and the ability of the modelled population to survive (as even if there is a theoretical equilibrium state it is not realistic for a population which loses a too big fraction of its headcount in each generation to sustain itself).

The selection of Gaussian mutation and selection has allowed to explicitly derive the closed-form formulas for equilibrium state, as well as has enabled the derivation of closed-form exact formulas for various traits of the population, such as genetic variance or average fitness, as a function of the parameters of the model. We feel that the generality of this model, as well as the mathematical elegance of obtained results (and also of their derivation) yields itself well to the model being used in various diverse population genetic settings. The model may be easily extended as well, we plan on providing a preliminary look (as a case study) on the extension of the model towards studying the impact of mobile elements (such as transposons) on the evolution, and the interplay of the activity of mobile elements, environmental stress and evolution.


Fisher's geometric model, population genetics, mutator, Gaussian selection, stability, equilibrium distribution, probability measure