COMMENTS ON THE LURIA-DELBRUCK DISTRIBUTION

The long-tailed Luria-Delbruck distribution arises in connection with the 'random mutation' hypothesis (whereas the 'directed adaptation' hypothesis is thought to give a Poisson distribution). At time t the distribution depends on the parameter m = gN(t)/(a + g) where N(t) is the current population size and g/(a + g) is the relative mutation rate (assumed constant). The paper identifies three models for the distribution in the existing literature and gives a fourth model. Ma et al. (1992) recently proved that there is a remarkably simple recursion relation for the Luria-Delbruck probabilities p(n) and found that asymptotically p(n) almost-equal-to c/n2; their numerical studies suggested that c = 1 when the parameter m is unity. Cairns et al. (1988) had previously argued and shown numerically that P(n) = SIGMA(j greater-than-or-equal-to n) p(j) almost-equal-to m/n. Here we prove that n(n + 1) p(n) < m(1 + 11m/30) for n = 1, 2,..., and hence prove that as n becomes large n(n + 1)p(n) almost-equal-to m; the result mP(n) almost-equal-to m follows immediately.