# Feller’s Contributions to Mathematical Biology

Feller’s Contributions to Mathematical Biology

Ellen Baake, Anton Wakolbinger
(Submitted on 21 Jan 2015)

This is a review of William Feller’s important contributions to mathematical biology. The seminal paper [Feller1951] “Diffusion processes in genetics” was particularly influential on the development of stochastic processes at the interface to evolutionary biology, and interesting ideas in this direction (including a first characterization of what is nowadays known as “Feller’s branching diffusion”) already shaped up in the paper [Feller 1939] (written in German) “The foundations of a probabistic treatment of Volterra’s theory of the struggle for life”. Feller’s article “On fitness and the cost of natural selection” [Feller 1967] contains a critical analysis of the concept of “genetic load”.

# Rise and fall of asexual mutators in adapted populations

Rise and fall of asexual mutators in adapted populations
Ananthu James, Kavita Jain
Subjects: Populations and Evolution (q-bio.PE)

In an adapted population in which most mutations are deleterious, the mutation rates are expected to be low. Indeed, in recent experiments on adapted populations of asexual mutators, beneficial mutations that lower the mutation rates have been observed to get fixed. Using a multitype branching process and a deterministic argument, we calculate the time to fix the wildtype mutation rate in an asexual population of mutators, and find it to be a ${\tt U}$-shaped function of the population size. In contrast, the fixation time for mutators is known to increase with population size. On comparing these two time scales, we find that a critical population size exists below which the mutators prevail, while the mutation rate remains low in larger populations. We also discuss how our analytical results compare with the experiments.

# The SMC’ is a highly accurate approximation to the ancestral recombination graph

The SMC’ is a highly accurate approximation to the ancestral recombination graph

Peter R. Wilton, Shai Carmi, Asger Hobolth
(Submitted on 12 Jan 2015)

Two sequentially Markov coalescent models (SMC and SMC’) are available as tractable approximations to the ancestral recombination graph (ARG). We present a model of coalescence at two fixed points along a pair of sequences evolving under the SMC’. Using our model, we derive a number of new quantities related to the pairwise SMC’, thereby analytically quantifying for the first time the similarity between the SMC’ and ARG. We use our model to show that the joint distribution of pairwise coalescence times at recombination sites under the SMC’ is the same as it is marginally under the ARG, demonstrating that the SMC’ is the canonical first-order sequentially Markov approximation to the pairwise ARG. Finally, we use these results to show that population size estimates under the pairwise SMC are asymptotically biased, while under the pairwise SMC’ they are approximately asymptotically unbiased.

# Response of polygenic traits under stabilising selection and mutation when loci have unequal effects

Response of polygenic traits under stabilising selection and mutation when loci have unequal effects

Kavita Jain, Wolfgang Stephan
(Submitted on 9 Jan 2015)

We consider an infinitely large population under stabilising selection and mutation in which the allelic effects determining a polygenic trait vary between loci. We obtain analytical expressions for the stationary genetic variance as a function of the distribution of effects, mutation rate and selection coefficient. We also study the dynamics of the allele frequencies, focussing on short-term evolution of the phenotypic mean as it approaches the optimum after an environmental change. We find that when most effects are small, the genetic variance does not change appreciably during adaptation, and the time until the phenotypic mean reaches the optimum is short if the number of loci is large. However, when most effects are large, the change of the variance during the adaptive process cannot be neglected. In this case, the short-term dynamics may be described by that of a single locus of large effect. Our results may be used to understand polygenic selection driving rapid adaptation.

# The effect of the dispersal kernel on isolation-by-distance in a continuous population

The effect of the dispersal kernel on isolation-by-distance in a continuous population

Tara N. Furstenau, Reed A. Cartwright
Comments: 18 pages (main); 4 pages (supp)
Subjects: Populations and Evolution (q-bio.PE)

Under models of isolation-by-distance, population structure is determined by the probability of identity-by-descent between pairs of genes according to the geographic distance between them. Well established analytical results indicate that the relationship between geographical and genetic distance depends mostly on the neighborhood size of the population, $N_b = 4{\pi}{\sigma}^2 D_e$, which represents a standardized measure of dispersal. To test this prediction, we model local dispersal of haploid individuals on a two-dimensional torus using four dispersal kernels: Rayleigh, exponential, half-normal and triangular. When neighborhood size is held constant, the distributions produce similar patterns of isolation-by-distance, confirming predictions. Considering this, we propose that the triangular distribution is the appropriate null distribution for isolation-by-distance studies. Under the triangular distribution, dispersal is uniform within an area of $4{\pi}{\sigma}^2$ (i.e. the neighborhood area), which suggests that the common description of neighborhood size as a measure of a local panmictic population is valid for popular families of dispersal distributions. We further show how to draw from the triangular distribution efficiently and argue that it should be utilized in other studies in which computational efficiency is important

# The evolutionarily stable distribution of fitness effects

The evolutionarily stable distribution of fitness effects

Daniel P. Rice, Benjamin H. Good, Michael M. Desai
doi: http://dx.doi.org/10.1101/013052

The distribution of fitness effects of new mutations (the DFE) is a key parameter in determining the course of evolution. This fact has motivated extensive efforts to measure the DFE or to predict it from first principles. However, just as the DFE determines the course of evolution, the evolutionary process itself constrains the DFE. Here, we analyze a simple model of genome evolution in a constant environment in which natural selection drives the population toward a dynamic steady state where beneficial and deleterious substitutions balance. The distribution of fitness effects at this steady state is stable under further evolution, and provides a natural null expectation for the DFE in a population that has evolved in a constant environment for a long time. We calculate how the shape of the evolutionarily stable DFE depends on the underlying population genetic parameters. We show that, in the absence of epistasis, the ratio of beneficial to deleterious mutations of a given fitness effect obeys a simple relationship independent of population genetic details. Finally, we analyze how the stable DFE changes in the presence of a simple form of diminishing returns epistasis.

# Stationary solutions for metapopulation Moran models with mutation and selection

Stationary solutions for metapopulation Moran models with mutation and selection

George W. A. Constable, Alan J. McKane
(Submitted on 19 Dec 2014)

We construct an individual-based metapopulation model of population genetics featuring migration, mutation, selection and genetic drift. In the case of a single `island’, the model reduces to the Moran model. Using the diffusion approximation and timescale separation arguments, an effective one-variable description of the model is developed. The effective description bears similarities to the well-mixed Moran model with effective parameters which depend on the network structure and island sizes, and is amenable to analysis. Predictions from the reduced theory match the results from stochastic simulations across a range of parameters. The nature of the fast-variable elimination technique we adopt is further studied by applying it to a linear system, where it provides a precise description of the slow-dynamics in the limit of large timescale separation.

# Imperfect drug penetration leads to spatial monotherapy and rapid evolution of multi-drug resistance

Imperfect drug penetration leads to spatial monotherapy and rapid evolution of multi-drug resistance

Stefany Moreno-Gamez, Alison L Hill, Daniel I.S. Rosenbloom, Dmitri A. Petrov, Martin A Nowak, Pleuni Pennings
doi: http://dx.doi.org/10.1101/013003

Infections with rapidly evolving pathogens are often treated using combinations of drugs with different mechanisms of action. One of the major goals of combination therapy is to reduce the risk of drug resistance emerging during a patient’s treatment. While this strategy generally has significant benefits over monotherapy, it may also select for multi-drug resistant strains, which present an important clinical and public health problem. For many antimicrobial treatment regimes, individual drugs have imperfect penetration throughout the body, so there may be regions where only one drug reaches an effective concentration. Here we propose that mismatched drug coverage can greatly speed up the evolution of multi-drug resistance by allowing mutations to accumulate in a stepwise fashion. We develop a mathematical model of within-host pathogen evolution under spatially heterogeneous drug coverage and demonstrate that even very small single-drug compartments lead to dramatically higher resistance risk. We find that it is often better to use drug combinations with matched penetration profiles, although there may be a trade-off between preventing eventual treatment failure due to resistance in this way, and temporarily reducing pathogen levels systemically. Our results show that drugs with the most extensive distribution are likely to be the most vulnerable to resistance. We conclude that optimal combination treatments should be designed to prevent this spatial effective monotherapy. These results are widely applicable to diverse microbial infections including viruses, bacteria and parasites.