Coalescent theory

In genetics, coalescent theory states that all genes or alleles in a given population are ultimately inherited from a single ancestor shared by all members of the population, known as the most recent common ancestor. If the inheritance relationships are written in the form of a phylogenetic tree, termed a gene genealogy, the gene or allele of interest is said to undergo coalescence to the common ancestor (sometimes termed the concestor to emphasize the coalescent relationship^[1]). Coalescence theory assumes that genes do not undergo recombination and models genetic drift as a Markov process^[2]. Because the process of gene fixation due to genetic drift is a crucial component of coalescence theory, it is most useful when the genetic locus under study is not under natural selection.

Probability of fixation

Under conditions of genetic drift alone, every finite set of genes or alleles has a "coalescent point" at which all descendants converge to a single ancestor. This fact can be used to derive the rate of gene fixation of a neutral allele (that is, one not under any form of selection) for a population of varying size (provided that it is finite and nonzero). Because the effect of natural selection is stipulated to be negligible, the probability at any given time of an allele becoming fixed is just its frequency $p$ in the population at that time. For a diploid population of size $N$ and (neutral) mutation rate $\mu$, the initial frequency of a novel mutation is simply ${\frac {1}{2N}}$ and the number of new mutations per generation is $2N\mu$. Since the fixation rate is the rate of novel neutral mutation multiplied by their probability of fixation, the overall fixation rate is $2N\mu \times {\frac {1}{2N}}=\mu$. Thus the rate of fixation for a mutation not subject to selection is simply the rate of introduction of such mutations.

Time to coalescence

A useful analysis based on coalescence theory seeks to predict the amount of time elapsed between the introduction of a mutation and a particular allele or gene distribution in a population. This time period is equal to how long ago the most recent common ancestor existed.

The probability that two lineages coalesce in the immediately preceding generation is the probability that they share a parent. In a diploid population with $2N$ total loci, there are $2N$ "potential parents" in the previous generation, so the probability that two alleles share a parent is ${\frac {1}{2N}}$ and correspondingly, the probability that they do not coalesce is $1-{\frac {1}{2N}}$.

At each successive preceding generation, the probability of coalescence is geometrically distributed - that is, it is the probability of noncoalescence at the $t-1$ preceding generations multiplied by the probability of coalescence at the generation of interest:

$P_{c}=\left(1-{\frac {1}{2N}}\right)^{t-1}\left({\frac {1}{2N}}\right)$

For sufficiently large values of $t$, this distribution is well approximated by the continuously defined exponential distribution

$P_{c}={\frac {1}{2N}}e^{-{\frac {t}{2N}}}$

The standard exponential distribution has both the expectation value and the standard deviation equal to $2N$ - therefore, although the expected time to coalescence is $2N$, actual coalescence times have a wide range of variation.

Neutral variation

Coalescence theory can also be used to model the amount of variation in DNA sequences expected from genetic drift alone. This value is termed the mean heterozygosity, represented as ${\bar {H}}$. Mean heterozygosity is calculated as the probability of a mutation occurring at a given generation divided by the probability of any "event" at that generation (either a mutation or a coalescence). The probability that the event is a mutation is the probability of a mutation in either of the two lineages: $2\mu$. Thus the mean heterozygosity is equal to:

${\bar {H}}={\frac {2\mu }{2\mu +{\frac {1}{2N}}}}$

${\bar {H}}={\frac {4N\mu }{1+4N\mu }}$

For $4N\mu >>1$, the vast majority of allele pairs have at least one difference in nucleotide sequence.

External links

• Book chapters by John Wakeley, available free online
• Coalescence ntroduction by Magnus Nordborg

References

1. ^ Dawkins R. (2004). The Ancestor's Tale: A Pilgrimage to the Dawn of Evolution. Houghton Mifflin: New York, NY.
2. ^ Rice SH. (2004). Evolutionary Theory: Mathematical and Conceptual Foundations. Sinauer Associates: Sunderland, MA. See esp. ch. 3 for detailed derivations.