Single Peak Gaussian Distribution To Adapt To The Study Of The Quasispecies Evolution

This paper is an introduction to theoretical models for biological evolution at microscopic (organism) level, especially for those asexual populations with quick replication rate and high mutation rate. The theoretical models are based on a variety of simplification and assumption, and this article places emphasis on the description of the Eigen quasi-species model. On the basis of the concept of fitness landscape, we can get two significant features from the Eigen model: the quasi-species distribution at low mutation rate and the transition to random distribution at critical mutation rate, i.e. error threshold. The object of natural selection is the quasi-species, and the existence of the error threshold places a limit on the genome length that the organisms carry. The two phenomena are significant for application and have been observed in experiments. The methods in the field are very similar to those used in statistical physics. The text has four parts. The first part deals with the Darwinian system and Eigen model, including the concept of fitness landscape, Fisher's theory, and the analytical solution, perturbation approximation and numerical solution for quasi-species model. The second part gives the statistical description of the quasi-species model in which the error threshold is similar to the phase transition in statistical physics, and then discusses several simple and complex fitness landscapes. The third part studies the time-dependent fitness landscape, especially discusses the generally phenomena in periodic fitness landscape. Finally, we present the single peak Gaussian distributed fitness model. Considering various uncontrollable factors in the living environment, we use the ensemble average to give the simulation of quasi-species and error threshold. Our results show that the error threshold should locate in a certain range for any limited sequence length instead of a single point, and the upper bound of the error threshold shifts to a larger value compared with the deterministic case.