Biological evolution through mutation, selection, and drift: An introductory review

Motivated by present activities in (statistical) physics directed towards biological evolution, we review the interplay of three evolutionary forces: mutation, selection, and genetic drift. The review addresses itself to physicists and intends to bridge the gap between the biological and the physical literature. We first clarify the terminology and recapitulate the basic models of population genetics, which describe the evolution of the composition of a population under the joint action of the various evolutionary forces. Building on these foundations, we specify the ingredients explicitly, namely, the various mutation models and fitness landscapes. We then review recent developments concerning models of mutational degradation. These predict upper limits for the mutation rate above which mutation can no longer be controlled by selection, the most important phenomena being error thresholds, Muller's ratchet, and mutational meltdowns. Error thresholds are deterministic phenomena, whereas Muller's ratchet requires the stochastic component brought about by finite population size. Mutational meltdowns additionally rely on an explicit model of population dynamics, and describe the extinction of populations. Special emphasis is put on the mutual relationship between these phenomena. Finally, a few connections with the process of molecular evolution are established.Comment: 62 pages, 6 figures, many reference

Critical mutation rate has an exponential dependence on population size in haploid and diploid populations

Understanding the effect of population size on the key parameters of evolution is particularly important for populations nearing extinction. There are evolutionary pressures to evolve sequences that are both fit and robust. At high mutation rates, individuals with greater mutational robustness can outcompete those with higher fitness. This is survival-of-the-flattest, and has been observed in digital organisms, theoretically, in simulated RNA evolution, and in RNA viruses. We introduce an algorithmic method capable of determining the relationship between population size, the critical mutation rate at which individuals with greater robustness to mutation are favoured over individuals with greater fitness, and the error threshold. Verification for this method is provided against analytical models for the error threshold. We show that the critical mutation rate for increasing haploid population sizes can be approximated by an exponential function, with much lower mutation rates tolerated by small populations. This is in contrast to previous studies which identified that critical mutation rate was independent of population size. The algorithm is extended to diploid populations in a system modelled on the biological process of meiosis. The results confirm that the relationship remains exponential, but show that both the critical mutation rate and error threshold are lower for diploids, rather than higher as might have been expected. Analyzing the transition from critical mutation rate to error threshold provides an improved definition of critical mutation rate. Natural populations with their numbers in decline can be expected to lose genetic material in line with the exponential model, accelerating and potentially irreversibly advancing their decline, and this could potentially affect extinction, recovery and population management strategy. The effect of population size is particularly strong in small populations with 100 individuals or less; the exponential model has significant potential in aiding population management to prevent local (and global) extinction events

Nonlinear deterministic equations in biological evolution

We review models of biological evolution in which the population frequency changes deterministically with time. If the population is self-replicating, although the equations for simple prototypes can be linearised, nonlinear equations arise in many complex situations. For sexual populations, even in the simplest setting, the equations are necessarily nonlinear due to the mixing of the parental genetic material. The solutions of such nonlinear equations display interesting features such as multiple equilibria and phase transitions. We mainly discuss those models for which an analytical understanding of such nonlinear equations is available.Comment: Invited review for J. Nonlin. Math. Phy

Maximum principle and mutation thresholds for four-letter sequence evolution

A four-state mutation-selection model for the evolution of populations of DNA-sequences is investigated with particular interest in the phenomenon of error thresholds. The mutation model considered is the Kimura 3ST mutation scheme, fitness functions, which determine the selection process, come from the permutation-invariant class. Error thresholds can be found for various fitness functions, the phase diagrams are more interesting than for equivalent two-state models. Results for (small) finite sequence lengths are compared with those for infinite sequence length, obtained via a maximum principle that is equivalent to the principle of minimal free energy in physics.Comment: 25 pages, 16 figure

Mutation-Selection Balance: Ancestry, Load, and Maximum Principle

We show how concepts from statistical physics, such as order parameter, thermodynamic limit, and quantum phase transition, translate into biological concepts in mutation-selection models for sequence evolution and can be used there. The article takes a biological point of view within a population genetics framework, but contains an appendix for physicists, which makes this correspondence clear. We analyze the equilibrium behavior of deterministic haploid mutation-selection models. Both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that it determines the sensitivity of the equilibrium mean fitness to changes in the fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to (error) threshold phenomena caused by the interplay of selection and mutation. They lead to a clarification of concepts, as well as criteria for the existence of thresholds.Comment: 54 pages, 15 figures; to appear in Theor. Pop. Biol. 61 or 62 (2002

A generalized model of mutation-selection balance with applications to aging

A probability model is presented for the dynamics of mutation-selection balance in a haploid infinite-population infinite-sites setting sufficiently general to cover mutation-driven changes in full age-specific demographic schedules. The model accommodates epistatic as well as additive selective costs. Closed form characterizations are obtained for solutions in finite time, along with proofs of convergence to stationary distributions and a proof of the uniqueness of solutions in a restricted case. Examples are given of applications to the biodemography of aging, including instabilities in current formulations of mutation accumulation.Comment: 20 pages Updated to include more historical comment and references to the literature, as well as to make clear how our non-linear, non-Markovian model differs from previous linear, Markovian particle system and measure-valued diffusion models. Further updated to take into account referee's comment