One of the central results of the Neutral Theory of evolution (Kimura and Ohta 1971; Kimura 1983) states that the rate *k* of allele substitution (rate of evolution) at neutral loci is unaffected by fluctuations in population size and is simply equal to the mutation rate. The explanation behind this result goes as follows. The number of mutants that enter a haploid population of size *N* is equal to ; the number of individuals born in the last generation times the mutation rate at the locus considered. Conversely, the probability that any new mutant allele reaches ultimate fixation (i.e., replaces the other alleles at this locus) corresponds to its initial frequency *1/N* in the population. Because the substitution rate *k* is the product of the number of mutants () and the fixation probability (*1/N*), *k* becomes independent of *N*: *k* =μ. This result provides a justification for the assumption of a molecular clock (Zuckerkandl and Pauling 1962; Margoliash 1963), which allows dating evolutionary events such as host jumps, age of infectious disease outbreaks, and speciation events (e.g., Ingman et al. 2000; Korber et al. 2000; Nübel et al. 2010).

It is well known that for alleles under natural selection the substitution rate is not independent of effective population size, which is itself affected by past population sizes (Kimura 1983; Otto and Whitlock 1997; Orr 2000; Bromham and Penny 2003; Waples 2010). However, effective population size is generally difficult to estimate accurately for natural populations. As such, a frequent assumption in population genetics and phylogenetics is that the genetic markers under study are strictly neutral, which, in practice, corresponds to using genetic markers deemed largely neutral, such as synonymous mutations in sequence data. Under such conditions, the result that the substitution rate is independent of population size fluctuations can be applied and is often invoked in molecular population biology. However, the range of biological assumptions under which the substitution rate is independent of population size fluctuations may not have been fully explored.

Our attention has been drawn to a possible dependency of substitution rate on demography by empirical work on the plague (*Yersinia pestis*). Morelli et al. (2010) observed an excess of apparently neutral mutations reaching fixation following population size expansion during new plague epidemics. Additional empirical work on 118 complete genomes points to repeated episodes of acceleration of the molecular clock at apparently strictly neutral genes during episodes of population expansion of the plague and qualitatively similar patterns could be generated with stochastic simulations of serial outbreaks of plague lineages with overlapping generations (Cui Y. et al. unpubl. ms.). These results suggest that even for strictly neutral genes, the substitution rate may depend on population size fluctuations.

Indeed, one may expect that the combination of overlapping generations and fluctuations in population size may affect the product of the number of new alleles entering the population through mutation and the probability of each such new mutation reaching eventual fixation. Assuming that all mutations happen at reproduction (replication), the number of new mutants entering the system varies under overlapping generations and fluctuating demography. Thus, in a growing population, there will be a relatively large number of new offspring, but this number will not necessarily correspond to the inverse of population size if some adults are surviving. Conversely, in a declining population there will be little or no space left for newborn individuals carrying new mutations to enter the population if adults have a high survival probability. Here again, the number of new offspring is unlikely to correspond to the inverse of population size.

In this article, we present an analytical model for the substitution rate under the joint effect of overlapping generations and population size fluctuations. Although the two effects are hardly ever considered together in the same population genetics model, they represent both most reasonable assumptions. Population fluctuations and overlapping generations represent the norm rather than the exception throughout natural populations and their dynamic interactions are eagerly studied in evolutionary demography (Tuljapurkar 1989). The analytical model we develop allows us to show that the substitution rate at neutral genes does depend on demography in populations with overlapping generations and population size fluctuations. Moreover, the quantitative deviation can be strong depending on the demography considered.