If the change of frequencies of alleles in a population is not brought about through selection, it has to be due to other mechanisms. One is genetic drift (the random sampling of alleles from one generation to form the next generation), another is migration, where new individuals come from outside the system and change the frequencies of the various alleles inside the system. These two mechanisms, along with mutation which creates new alleles in the first place, are especially interesting to those working in nonequilibrium statistical mechanics and the theory of stochastic processes, since their study involves many of the tools and techniques from these fields. Neutral theories are models of such selection-free processes. They appear in several different areas: I have been involved in their study in the fields of population genetics, ecology and language evolution, where they seem to have a far greater realm of validity than might naively be expected. Many of the models also seem to be exactly soluble, or generally amenable to analysis. A paper JSTAT P07018 (2007) based on seminars given by Richard Blythe and myself at the Newton Institute in Cambridge, gives some more background and information on neutral theories.
The concept of neutral theories had its origins in population genetics and so it is here that the idea is most well developed. As someone trained in theoretical physics, my preference is for starting from "microscopic" models, formulated as either Markov chains or master equations (continuous time Markov chains) and then deriving the corresponding "macroscopic" model, which in this case are Fokker-Planck equations, frequently known as Kolmogorov equations in population genetics. Derivations of this kind are discussed in detail in JSTAT P07018 (2007). It turns out that, in an application to be discussed below, the case where more than two alleles of a particular gene exists needed to be studied, and so Gareth Baxter, Richard Blythe and myself looked at the Fokker-Planck equation describing genetic drift and mutation when M alleles were present. This is a non-trivial M-dimensional partial differential equation, which remarkably is separable, and so exactly soluble. It is interesting to note that when Kimura first studied these equations in the 1950s, he speculated that "additional techniques would be required to make the mathematical manipulations manageable" for an arbitrary number of alleles. The details are given in Math. Biosci. 209, 124-170 (2007), where many other quantities are calculated exactly, including probabilities of fixation (all alleles but one have become extinct), mean time to the rth extinction, probability of a particular sequence of extinctions, and so on. Care needs to be taken with the boundary conditions of the Fokker-Planck equation, since the diffusion matrix is state dependent and vanishes at the boundaries. An alternative method of implementing the boundary conditions, consisting of working in Fourier space, was worked out with David Waxman, is described in Jour. Theor. Biol. 247, 849-858 (2007) and gives a systematic way of approaching these problems.
A second field in which neutral theories have become popular is ecology. The main proponent of a neutral theory of ecology is Stephen Hubbell, especially through his book The Unified Neutral Theory of Biodiversity and Biogeography. In Hubbell's neutral theory, genes map into individuals and alleles into species, thus no one species is any "fitter" than another, and the time evolution of an ecological community of trophically similar species is primarily due to random processes of birth, death and migration. Not surprisingly, this is a controversial claim, however the species abundance distribution predicted from Hubbell's theory is in surprisingly good agreement with data. I, together with David Alonso and Ricard Sole were able to calculate this species abundance distribution analytically, where previously it had only been obtained numerically. We achieved this by formulating the model as a master equation and mapping it on to a previous model of ours which we had also solved exactly. The details are given in Theor. Popul. Biol. 65, 67-73 (2004). A review article, Trends Ecol. Evol. 21, 451-457 (2006), written together with David Alonso and Rampal Etienne discusses the merits of neutral theory in an ecological context. Neutral theories have a rich mathematical structure which is explored in two other papers of mine. In the first, Ecol. Lett. 7, 901-910 (2004), the fact that it is a sampling theory is studied and in the second Jour. Theor. Biol. 248, 522-536 (2007), the consequences of relaxing the zero-sum assumption (i.e. fixed population size) is investigated.Ideas of neutrality can also provide a framework for understanding language change and a group of us having been working with Bill Croft, formally in the Department of Linguistics at the University of Manchester, but now at the University of New Mexico, in constructing a mathematical model of the evolution of language. The model is of a generalised evolutionary type, where alleles are mapped to linguistic variants, which are "different ways of saying the same thing". The interaction between speakers, who may use one variant at a higher frequency as a result, is analogous to the migration of alleles from one island to another. The analogy is exhibited in more detail in JSTAT P07018 (2007). The model may be cast into the form of a Fokker-Planck equation which is identical to that found in multi-allelic population genetics with genetic drift and migration. As discussed in Phys. Rev. E73, 046118 (2006), the model may be studied through a mixture of analysis and computer simulation, leading to a prediction for the mean time to fixation (the mean time for all but one linguistic variant to die out). If no speaker is more influential than any other (i.e. the theory is neutral), then surprisingly it turns out that this mean time to fixation is independent of the network structure, so long as the network is not too anomalous. This result is demonstrated in Phys. Rev. Lett. 101, 258701 (2008). Gareth Baxter, Richard Blythe, Bill Croft and myself are currently applying these ideas to understanding the timescales involved in the emergence of New Zealand English. Back to my home page.