The speed of evolution in large asexual populations

We consider an asexual biological population of constant size $N$ evolving in discrete time under the influence of selection and mutation. Beneficial mutations appear at rate $U$ and their selective effects $s$ are drawn from a distribution $g(s)$. After introducing the required models and concepts of mathematical population genetics, we review different approaches to computing the speed of logarithmic fitness increase as a function of $N$, $U$ and $g(s)$. We present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid. We then discuss approximate approaches to the finite population problem, distinguishing between the case of a single selection coefficient, $g(s) = \delta(s - s_b)$, and a continuous distribution of selection coefficients. Analytic estimates for the speed are compared to numerical simulations up to population sizes of order $10^{300}$.