Invasion and fixation of microbial dormancy traits under competitive pressure

Abstract

Microbial dormancy is an evolutionary trait that has emerged independently at various positions across the tree of life. It describes the ability of a microorganism to switch to a metabolically inactive state that can withstand unfavourable conditions. However, maintaining such a trait requires additional resources that could otherwise be used to increase e.g. reproductive rates. In this paper, we aim for gaining a basic understanding under which conditions maintaining a seed bank of dormant individuals provides a “fitness advantage” when facing resource limitations and competition for resources among individuals (in an otherwise stable environment). In particular, we wish to understand when an individual with a “dormancy trait” can invade a resident population lacking this trait despite having a lower reproduction rate than the residents. To this end, we follow a stochastic individual-based approach employing birth-and-death processes, where dormancy is triggered by competitive pressure for resources. In the large-population limit, we identify a necessary and sufficient condition under which a complete invasion of mutants has a positive probability. Further, we explicitly determine the limiting probability of invasion and the asymptotic time to fixation of mutants in the case of a successful invasion. In the proofs, we observe the three classical phases of invasion dynamics in the guise of Coron et al. (2017, 2019).

Introduction

Dormancy is an evolutionary trait that has emerged independently at various positions across the tree of life. In the present article, we are in particular interested in microbial dormancy (cf. [27] and [30] for recent overviews of this subject). Microbial dormancy describes the ability of a microorganism to switch to a metabolically inactive state in order to withstand unfavourable conditions (such as resource scarcity and competitive pressure or extreme environmental fluctuations), and this seems to be a highly effective (yet costly) evolutionary strategy. In certain cases, for example in marine sediments, simulation studies indicate that under oligotrophic conditions, the fitness of an organism is determined to a large degree by its ability to simply stay alive, rather than to grow and reproduce (cf. [9]). Indeed, maintaining a dormancy trait requires additional resources in comparison to individuals lacking this trait, resulting in significant trade-offs such as e.g. a lower reproduction rate.

In this paper, we aim at gaining a basic rigorous understanding for the conditions under which maintaining a dormancy trait can be beneficial. We investigate the particular question whether an individual with a dormancy trait can invade a resident population lacking this trait, even if maintaining dormancy reduces its reproduction rate compared to the rate of the residents, under otherwise stable environmental conditions. To this end, we follow a stochastic individual-based approach employing birth-and-death processes (a classic set-up underlying much of adaptive dynamics, as outlined e.g. in [8]), where dormancy is triggered in response to competitive pressure for limited resources. In the large-population limit, we identify a necessary and sufficient condition under which the invasion of mutants, despite having a lower reproduction rate than the resident population, has a positive probability. Further, we explicitly determine the limiting probability of invasion and the asymptotic time of fixation of mutants in the case of a successful invasion.

To be more explicit, in our model the total population evolves according to a continuous time Markov chain. Initially, there is a fit resident population, which we assume to be close to its equilibrium population size, featuring (random) reproduction, natural death (“death by age”), and death by competition. This results in a stochastically evolving population with logistically regulated drift fluctuating around a constant carrying capacity (reflecting a stable yet limited supply of resources). We assume that environmental conditions are also stable and do not affect reproduction, death or competition rates. In this situation, we then assume that a single “mutant” (or “migrant”) with “dormancy trait” appears in the population, who on the one hand is still fit enough to survive in absence of the residents (however with a strictly lower reproduction rate), but on the other hand is able to switch to a dormant state at a rate proportional to the “competitive pressure” exerted on her due to crowding and limited resource availability. That is, for some $0<p<1$, “competition events” that would normally cause death for an ordinary resident individual kill a mutant individual only with probability $1-p$. Otherwise, with probability $p$, the mutant individual affected by competition will persist and switch to the dormant state. Finally, dormant mutant individuals neither reproduce nor are affected by competitive pressure for resources while they are still to some degree exposed to natural death (at a rate typically smaller than for active individuals). We assume that at a constant “resuscitation rate” , they switch back to the active state.

Our main results show that the mutants will invade the resident population with positive probability under a suitable condition on the parameters of the model. This condition has the following interpretation: the advantage of the resident population caused by its higher reproduction rate needs to be over-compensated by the advantage of the mutant population resulting from being able to escape competitive deaths due to overcrowding by switching into dormancy. This condition can be made entirely transparent in terms of the parameters of the model, see (5) resp. Section 3. Under this condition, we characterize the probability of invasion (that is, the mutants completely replace the residents and reach their own equilibrium carrying capacity), and we identify the expected time of invasion on a logarithmic scale in the large-population limit. With high probability, a successful mutation follows the three classical phases exhibited in basic adaptive dynamics models (which were introduced in [11, Section 3]; see e.g. [8, Section 4.1] for a slightly more general picture, but in particular [15], [16] for work in a closely related context that inspired our analysis and provides many of the necessary tools): (1) mutant growth until reaching a population size comparable to the carrying capacity, while during the same time period the resident population stays close to its equilibrium size, (2) a phase where all sub-populations are large and the dynamics of the frequency process can be approximated by a deterministic dynamical system, (3) extinction of the resident population, while the mutant population remains close to its equilibrium size.

Note that for our results it is essential that switching into dormancy is induced by competitive pressure. Indeed, if instead this switching happens at a constant rate (“spontaneous” or “stochastic switching”, cf. e.g. [27]), the mutants will never be able to invade the resident population unless their birth rate is higher than that of the residents (in which case their invasion would also be possible without a dormancy trait, and the assumption that dormancy is a costly trait would be violated). Further, mutants cannot make the residents go extinct unless they are fit enough to survive on their own; thus, evolutionary suicide, as observed e.g. in [4], does not occur in our model. Long-term coexistence of residents and mutants is also excluded in our modelling set-up.

Let us note that while dormancy was recently investigated in several mathematical works in the area of population genetics and coalescent theory (see e.g. [5], [6], [7], [24], [26]), in the field of adaptive dynamics we are not aware of prior work involving dormancy. The present paper takes a first step in this direction, analysing the invasion dynamics in a simple toy model. In order to make this model more realistic, one could e.g. incorporate further mutations in the spirit of adaptive dynamics. In the regime of very rare mutations introduced by Champagnat (cf. [3], [11], [13]), we expect that the model behaves similarly to the case of no further mutation. Recently, in [14], a regime of still rare but more frequent mutations was considered, with the additional effect of horizontal gene transfer. Here, mutation rates are large enough so that small sub-populations can have macroscopic effects on the whole population. It should be interesting to study the additional effects of dormancy traits in this regime. As a further step, one could also introduce spatiality in the model, which is relevant in modelling the trait space (see e.g. [2]) or the environment of the populations (see e.g. [12], [20]). Finally, the resuscitation rate, which is assumed constant in the present paper, could also be made dependent on the strength of competition.

Note that related scenarios involving “phenotypic switches”, arising e.g. in cancer modelling, have been analysed recently by [2], [23]. For dormancy and switching models in fluctuating environments, dynamical systems and branching process models have been investigated in [17], [29]. Here, as in the competition setup of the present paper, the basis of a rigorous understanding for the evolutionary advantages of seed banks seems to be emerging. It seems fair to say that dormancy in its many forms, and its interplay with other evolutionary and ecological forces, will provide many interesting future research challenges in mathematical biology.

The remainder of this paper is organized as follows. In Section 2 we introduce our model and state our main results. Next, in Section 3 we discuss some strongly related questions. Finally, in Section 4 we prove the main results. Each of these sections starts with a description about its internal organization.