Mathematical Modelling of Natural Selection

Question

I'm a math undergrad looking for some papers on modelling the process of natural selection. The only paper I've been able to find is by the pre-eminent mathematician Herbert Wilf from 2010,

There's Plenty of Time for Evolution

Unfortunately, Wilf's model is extremely simplistic - he calculates the number of 'generations' required to spell out a 'word', if we allow each letter to 'mutate' with certain probability every generation, and we stop mutating a letter once it is correct (this is the 'selective' feature of the model). So to spell the word 'Evolution' by randomly placing scrabble tiles would require 5.4 trillion generations, but if we keep the correctly placed letters each generation and only allow incorrect letters to 'mutate', Wilf calculates we'd only need about 57 generations on average.

Wilf's model is a good first step towards modelling natural selection, but it's clearly only a first step. In particular, the fact that nature seems to know in advance exactly what letters it needs to keep in particular places and what it needs to throw out to construct a complex genetic 'word' is dubious at best.

As a young maths student the idea of the incredible complexity and diversity of life developing by a directed stochastic process gets me shamelessly excited :) I have been very surprised at just how little mathematical literature there seems to be on this topic, as I say, Wilf's one super-simplistic model is all I've been able to find. Can someone direct me to any other theoretical analyses of the power of natural selection?

Thanks

Answer 1

Mathematical and computational modeling of evolution is a huge field. To start you off with some broad strokes, there are two main approaches to evolutionary modeling with their own communities: frequency-independent and frequency-dependent models. Of course, in a real biological setting, the truth lies somewhere in between, but models are idealizations of key principles, and so the two extremes are good to study.

For frequency-independent selection, they key concept is the fitness landscape -- a way to map each genotype to a fitness. The population then lives in this landscape and agents with higher fitness reproduce better and so the population slowly movers over the lattice. Under reasonable macroevolutionary assumptions such as very rare mutations and asexual populations, the population will tend to be homogenous and can be just modeled as a single point in the fitness landscape with properly chosen selective sweeps to move the point from vertex to vertx (Gillespie, 1983; 1984).

The most popular concrete approach for frequency-independent selection is [Kauffman's NK model of rugged fitness landscapes] (Kauffman & Weinberger, 1989; Kauffman, 1993). Computer simulations tend to favor this sort of model. Wilf's model would be an NK model with K = 0, and fitness of 0 for incorrect letters and 1 for correct letters (to get the shorter walk). Related questions:

• Is local equilibrium a reasonable assumption for evolutionary processes?
• Structure of fitness landscapes in the NK model

The frequency-dependent selection approach is dominated by evolutionary game theory. A typical question for this field is the evolution of cooperation, and the most basic approach is to use replicator dynamics and look at evolutionary stable strategies. However, the field is in general saturated with all kinds of exciting models (both analytic and computational). A very friendly book-length intro is Nowak's Evolutionary Dynamics. For a genetle intro and connections to evolutionary psychology look at the slides in this post. A brief and sophisticated introduction aimed at mathematicians can be found in Hofbauer & Sigmund (2003). Related questions:

• Sources for Algorithmic Evolutionary Game Theory
• Why is 'Grudger' an evolutionary stable strategy?
• How exactly are game theoretical evolutionary models described during implementation for computer simulations?
• When does weak selection produce qualitatively different results from strong selection?

References

Gillespie, J.H. (1983). A simple stochastic gene substitution model. Theor. Pop. Biol. 23, 202.

Gillespie, J.H. (1984). Molecular evolution over the mutational landscape. Evolution 38, 1116.

Hofbauer, J., & Sigmund, K. (2003). Evolutionary game dynamics. Bulletin of the American Mathematical Society, 40(4), 479-519.

Kauffman, S. (1993). The origins of order: Self organization and selection in evolution. Oxford University Press, USA.

Kauffman, S. & Weinberger, E. (1989) The NK Model of rugged fitness landscapes and its application to the maturation of the immune response. Journal of Theoretical Biology, 141(2): 211-245

Nowak, M. A. (2006). Evolutionary dynamics: exploring the equations of life. Harvard University Press.

Answer 2

Modeling evolutionary processes has a long history in evolutionary biology. Initially, these models were only theoretical or back-of-the-envelope quantitative (e.g., Simpson's Tempo and Mode in Evolution), but with the advent of cheap and accessible computing power, interest in stochastic models has greatly expanded.

The primary literature is vast, so I recommend a few books, which will give you a sampling of approaches and, more importantly, pointers to the relevant literature. If you are at a university, there is a good chance that your library will have one or more of these:

• Otto SP and T Day. 2007. A biologist's guide to mathematical modeling in ecology and evolution. Princeton University Press.
• Bolker BM. 2008. Ecological models and data in R. Princeton University Press.
• Roff DA. 2010. Modeling evolution: An introduction to numerical methods. Oxford University Press.

And want to try your own models and you use R, there is an entire page of evolutionary biology packages. Many of these have built-in functions to simulate evolutionary processes:

• CRAN Task View: Phylogenetics, Especially Comparative Methods

One thing to keep in mind is that simple models, like the one by Wilf and Ewens, often encapsulate the question at hand very well. Even though relatively simple, they are nonetheless very powerful. Wilf and Ewens wanted to prove a point about mutation, and they do that very well. It's easy to imagine way to extend their model to be more realistic, but as a start this is a very sound approach.

A rule of thumb for modeling is to make the model as simple as possible at first, including only the essential process or processes. Then gradually add complexity.

Answer 3

You might also have a look at some theoretical population genetics, which traditionally models selection as a diffusion process. The best introduction I know is chapter 15 of Karlin & Taylor's Second Course in Stochastic Processes (which, despite being only one chapter, makes up about half the book). From this, you can get results on hitting times, fixation times and other related distributions. Similar results are available in Durret's book, but he goes a bit deeper into the biological interpretations of his results.