Two models of population genetics

Abstract

The production of proteins in a cell is a regulated process. This means that the cell will only produce a type of protein when that type is needed. A fundamental step where this regulation occurs is at gene transcription. It is observed that transcription is regulated differently for different genes, and the question is therefore asked: why has evolution come up with different modes of transcriptional regulation for different genes?

Mathematical models of biological evolution are important for two reasons: 1) aiding researchers in understanding how complex biological systems have emerged and 2) enabling modelers to predict future outcomes of evolution. In this work, models of evolution of natural populations are applied to better understand the mechanisms of gene regulation in E. coli by investigating two predictor arguments of gene regulatory mode, namely the demand rule and the rule of minimal error load.

Two models of population genetics are derived: the Wright-Fisher model and the Moran model. These discrete stochastic models are approximated to continuous stochastic models and to continuous deterministic mean field models. The continuous stochastic models are used to investigate the demand rule, while the continuous deterministic models are used to investigate the rule of minimal error load.

In the continuous limits it is found that both discrete Wright-Fisher and Moran models can be described by the same equations. Two special cases are investigated in the model derivations: variable population size for the Wright-Fisher model and non-zero selection coefficients for continuous approximation of the Moran model. The models show that the demand rule describes well the evolution for the most basic mode of gene regulation, and that the rule of minimal error load describes the evolution for a larger group of gene regulation modes.

It is concluded that one should use the rule of minimal error load to investigate advanced systems of gene regulation. The demand rule is correct only as a special case for the most basic mode of gene regulation.