The Application Of Population Genetics Theory To The Morphological Evolution And The Fixation Probability Of Beneficial Mutations

Population genetics is a field of biology that studies the genetic composition of biological populations and the variety in genetic composition under various factors.In the contemporary population genetics,design of mathematical models to illustrate the evolution process is still the dominant research methodology.Through the mathematical methods,we can explore many questions about evolution in a quantitative and rigorous way.In this study,we focus on two important questions in evolutionary biology by means of population genetics theory.The first question is the waiting time until the morphological stability in a population in a constant environment and the effect of stochasticity and various evolutionary forces on morphological evolution.The second question is the fixation probability of beneficial mutations in changing environments.By studying these issues,we can deeply understand the evolutionary mechanismWe first adopt a population genetics model,the Wright-Fisher model,to describe a morphology(cell size)evolution observed from a long-term evolution experiment with Escherichia coli by Lenski et al.We calculate the waiting time to the ultimate stasis in cell size in the experiment by numerical simulations under the Wright-Fisher model.We compare this estimate with the experimental data,and find that they are consistent,which demonstrates the effectiveness of this model.Further,we show how the mutation rate,the selective advantage and the population size influence this waiting time to morphological stability.The results indicate that the selection plays a prominent role in this morphological evolution,whereas all other evolutionary forces are less influentialTo quantitatively analyze the morphology(cell size)evolution in the long-term evolution experiment with Escherichia coli and obtain an analytical expression for the waiting time until the stability of cell size,we present three mathematical approximations to this morphological evolution under the Wright-Fisher model.Due to the large population size of the Escherichia coli populations,we neglect all the random factors and firstly give a deterministic approximation(DA).However,we find that the results by DA do not coincide with the experimental data.Therefore,the DA fails to predict the evolutionary dynamics of cell size,which certifies the significance of stochasticity even in the very large populations Then,we develop a stochastic approximation(SA),and derive an approximate expression for the average waiting time to reach the final stasis in cell size.The calculated result under the parameter values is in good accordance with the experimental data,which shows that the SA is valid for this morphological evolution.Through this analytic expression,we determine that the selection has the greatest influence on the cell size evolution in the experiment.Thirdly,in order to test the conclusion obtained by the SA,we give a multistep process(MP)for the Wright-Fisher model of cell size evolution.And we acquire the median waiting time to the stability of cell size,which verifies the dominant effect of selectionFor the comparative study,we employ a two-stage clonal expansion model to depict the cell size evolution in the experimental Escherichia coli populations.Using this model,we derive the incidence function of the appearance of stability for cell size,the approximately analytic formula of waiting time until this morphological stability,and the conditional and unconditional probabilities of morphological stability.After assessing the parameter values,we verify that the calculated waiting time compares well with the experimental results,demonstrating the validity of this model.According to the relative contributions of parameters to the incidence function and the waiting time,we find that the clonal expansion rate of selectively advantageous organisms,i.e.,selection,has the largest impact on the cell size evolution.We compare the Wright-Fisher model and the two-stage clonal expansion model for the description of this morphological evolution.The comparison results indicate that the conclusions given by these two models are consistentThe second question we focus on in this article is to extend the classical result of the fixation probability of beneficial mutations obtained by Haldane,and to estimate the fixation probability of a beneficial mutation with a reduced generation time in a variable environment Assuming that the selective advantage is very small,we concentrate all the changing factors of environment on a single quantity:effective selective advantage.Using a time-dependent branching process,we get the analytic approximation for the fixation probability of beneficial mutations that decrease the generation time.Then,we apply this approximation to two interesting biological case.In these two instances,this approximation we obtained is in good agreement with the exact value,which shows that our result is effective.