The spatial Muller’s ratchet: surfing of deleterious mutations during range expansion

Abstract

During a range expansion, deleterious mutations can "surf" on the colonisation front. The resultant decrease in fitness is known as expansion load. An Allee effect is known to reduce the loss of genetic diversity of expanding populations, by changing the nature of the expansion from "pulled" to "pushed". We study the impact of an Allee effect on the formation of an expansion load with a new model, in which individuals have the genetic structure of a Muller's ratchet. A key feature of Muller's ratchet is that the population fatally accumulates deleterious mutations due to the stochastic loss of the fittest individuals, an event called a click of the ratchet. We observe fast clicks of the ratchet at the colonization front owing to small population size, followed by a slow fitness recovery due to migration of fit individuals from the bulk of the population, leading to a transient expansion load. For large population size, we are able to derive quantitative features of the expansion wave, such as the wave speed and the frequency of individuals carrying a given number of mutations. Using simulations, we show that the presence of an Allee effect reduces the rate at which clicks occur at the front, and thus reduces the expansion load. Impact of an Allee effect.A population exhibits an Allee effect if its maximal per-capita growth rate is achieved at intermediate population density rather than at low population density. Allee effects arise in many biological contexts, including for example the presence of cooperation, or the difficulty in finding mates for reproduction [33,45]. In the context of range expansion, an Allee effect shifts the location of the individuals with the highest per-capita growth rate towards the bulk of the population, see Figure 2 panel (a). In the absence of an Allee effect, individuals far in the front, where the population density is the lowest, have the highest growth rate. The wave is pulled by these individuals in the front. Conversely, if the Allee effect is strong enough, individuals with the highest growth rate are located midway between the front and the bulk of the population. The wave is then pushed by the bulk of the population. The distinction between pushed and pulled waves is a well-established paradigm of the reaction-diffusion literature, see [46,48].

Functional central limit theorems for epidemic models with varying infectivity

“…However, that is largely unrealistic since it does not allow for partial immunity or gradual waning of immunity. Various models have been developed to study the effects of the waning of immunity and the associated vaccination policies [15,16,30,1,9,4,28,7,27,21,13,10]. All the models except [4,13] start from an ODE model with the additional waning immunity characteristic.…”

“…Various models have been developed to study the effects of the waning of immunity and the associated vaccination policies [15,16,30,1,9,4,28,7,27,21,13,10]. All the models except [4,13] start from an ODE model with the additional waning immunity characteristic. In particular, El Khalifi and Britton [21] recently studied an extension of the ODE for the classical SIRS model with a linear or exponential waning function.…”

“…It can of course also be regarded as a generalized SEIRS model. We also mention the recent work [13], where a similar stochastic model of varying infectivity and waning immunity with vaccination is studied, where the focus is on the effect of vaccination policies to prevent endemicity. We notice the difference from our modeling approach besides the vaccination aspect: the random susceptibility function and varying infectivity function are taken independently in each infection.…”