Mathematical models of evolutionary rescue

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Félix Geoffroy, Mario Santer


When a population faces a fast detrimental environmental change, it can escape extinction via genetic adaptation. This scenario is often referred to as “evolutionary rescue” and has been extensively studied both empirically and theoretically. Understanding evolutionary rescue, and the conditions that favor it, is of great importance in many practical applications. In conservation biology, for instance, it is helpful for issuing policies that prevent the extinction of populations of interest. On the other hand, in medicine or agriculture, it is needed to prevent the emergence of mutant pathogens that are resistant to drug treatment. In particular, the evolutionary rescue framework has been extensively used for understanding tumor evolution and the evolution of antibiotic resistance. Over the past few years, a great deal of theoretical work has been done to build more realistic models that can address the diversity of biological contexts. Mathematical models have been proposed that take into account population or spatial structure, mating systems, migration or different mutation effects. In addition to these various biological questions, the mathematical approaches that are used in the study of evolutionary rescue are diverse. The two most common techniques to address the stochastic nature of rescue are the branching process and the diffusion process. Besides, under some circumstances, the modalities of rescue are better understood in a deterministic framework. The speakers of this minisymposium will present recent theoretical works in the field of evolutionary rescue and they cover a wide range of both biological questions and mathematical approaches.

Stephan Peischl

University of Berne
"The effect of gene flow on evolutionary rescue"
It seems certain that a substantial fraction of our planet’s current biodiversity will be lost to extinction as species’ habitats change at an accelerating rate. Some species, however, may be able to escape that fate by adapting, shifting their geographical ranges, or both. This leads to the questions of when, where and how might adaptation allow species to survive, leading to ‘evolutionary rescue’. Some basic answers to those questions come from theory. Experimental and theoretical studies have highlighted the impact of gene flow on the probability of evolutionary rescue. Mathematical modelling and simulations of evolutionary rescue in spatially or otherwise structured populations showed that intermediate migration rates can often maximise the probability of rescue in gradually or abruptly deteriorating habitats. In this talk, I present several mathematical approaches to studying evolutionary rescue in spatial or otherwise structured populations with gene flow between sub-populations, using discrete or continuous space models. I present simple conditions for when gene flow facilitates evolutionary rescue as compared to isolated populations, investigate the role of long-distance dispersal, as well as the role of phenotypic variation in dispersal traits.

Robert Noble

ETH Zurich
"The logic of containing tumours"
Challenging the paradigm of maximum tolerated dose, evolutionary theory suggests that the emer- gence of resistance to cancer therapy may be prevented or delayed by exploiting competitive ecological interactions between drug-sensitive and resistant tumour sub-clones. Recent studies have shown that a treatment strategy aiming for containment, not elimination, can control tumour burden more effectively than more aggressive approaches in vitro, in mouse models, and in the clinic, but theoretical understand- ing of these outcomes is underdeveloped. I will present a new, mathematically rigorous framework for understanding tumour containment that unifies and generalizes previous formulations. Results obtained within this framework provide timely guidance for empirical research including the design of clinical trials.

Mario Santer

Max Planck Institute for Evolutionary Biology
"Evolutionary Rescue and Drug Resistance on Multicopy Plasmids"
Bacteria often carry 'extra DNA' in form of plasmids in addition to their chromosome. Many plasmids have a copy number greater than one such that the genes encoded on these plasmids are present in multiple copies per cell. This has evolutionary consequences by increasing the mutational target size, by prompting the (transitory) co-occurrence of mutant and wild-type alleles within the same cell, and by allowing for gene dosage effects. We present a mathematical model for bacterial adaptation to harsh environmental change if adaptation is driven by beneficial alleles on multicopy plasmids. Successful adaptation depends on the availability of advantageous alleles and on their establishment probability. The establishment process involves the segregation of mutant and wild-type plasmids to the two daughter cells, allowing for the emergence of mutant homozygous cells over the course of several generations. To model this process, we use the theory of multi-type branching processes, where a type is defined by the genetic composition of the cell. Both factors – the availability of advantageous alleles and their establishment probability – depend on the plasmid copy number, and they often do so antagonistically. We find that in the interplay of various effects, a lower or higher copy number may maximize the probability of evolutionary rescue. The decisive factor is the dominance relationship between mutant and wild-type plasmids and potential gene dosage effects. Results from a simple model of antibiotic degradation indicate that the optimal plasmid copy number may depend on the specific environment encountered by the population.

Jacek Miękisz

University of Warsaw
"Evolution of populations with strategy- dependent time delays"
We address the issue of the stability of coexistence of two strategies with respect to time delays in evolving populations. It is well known that time delays may cause oscillations. Here we report a novel behavior. We show that a microscopic model of evolutionary games with a unique mixed evolutionarily stable strategy (a globally asymptotically stable interior stationary state in the standard replicator dynamics) and with strategy-dependent time delays leads to a new type of replicator dynamics. It describes the time evolution of fractions of the population playing given strategies and the size of the population. Unlike in all previous models, stationary states of such dynamics depend on time delays. Moreover, at certain time delays, an interior stationary state may disappear or there may appear another interior stationary state. This shows that effects of time delays are much more complex then it was previously thought.

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