The Ancestry of a Gene

Introduction Gene fixation in the sense that there is a single ancestor from which all the base pairs in all the copies of a gene in the population are descended only occurs in small (N < 1000) populations. In large populations (N > 1 000 000) crossing over (recombination) within the gene provides that there is an ancestral pool rather than a single ancestor of the gene. In the absence of recombination there is a common ancestor, such as the mitochondrial Eve or the y-chromosome Adam. Wiuf and Hein (1997) have provided an estimate for the size of the ancestral pool for chromosome 20, and Chang (1999) has provided an upper bound for the size of the ancestral pool for the entire genome. This paper presents upper and lower bounds for the probability of existence of a common ancestor, and the expected size of the ancestral pool, for a gene. The Model We assume a diploid population of N individuals (haploid size 2N , the analysis is haploid), the effective population size is the same as the actual population size. A gene is defined as 1000 contiguous base pairs. The probability of recombination between adjacent base pairs is 10−8. Coalescence decreases the number of ancestors as modelled by Kingman (1982a,b); only coalescence of the entire population is considered. Crossing over increases the number of ancestors in accordance with the recombination model of Hudson and Kaplan (1985) as employed by Wiuf and Hein (1997). Existence of a MRCA Whether all the base pairs share the same most recent common ancestor (MRCA) is measured by the probability that a cross over event occurred between a member of the coalescent and an individual not descended from the MRCA of the coalescent (in Fig. 1, x1 and x2 are in the coalescent, x3 is outside the coalescent but is descended from the MRCA, and x4 is outside the coalescent and is not descended from the MRCA. Calculations are based on the number of individuals in the coalescent t generations after the MRCA (1 + 1/2N − t/4N)−1 and the rate of increase of the number of descendants of the MRCA 1− (k−1)/(2N −1) (k is the number of descendants). Results in Table 1 show that in populations smaller than 1000 all the base pairs in a gene share the same MRCA more than .77 of the time, but in populations larger than 1 000 000 they share the same MRCA less than .10 of the time. Size of Ancestral Pool An upper bound for the expected size of an ancestral pool (and a lower bound for the asymptotic probability of there being a single ancestor in a given generation) is obtained from the inequality E[k(k − 1)/(4N)] ≤ E[k × 10−5] where k is the size of the ancestral pool, the left hand side is the probability a coalescent event occurs, and the right hand side is an upper bound for the probability a cross over event occurs. With convexity, this yields the inequality E[k] ≤ 1 + 4N × 10−5. A lower bound for the expected size of the ancestral pool (and an upper bound for the asymptotic probability of there being a single ancestor in a given generation) is obtained from the infinite stochastic matrix