Evolutionary dynamics of invasion and escape

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Abstract

Whenever life wants to invade a new habitat or escape from a lethal selection pressure, some mutations may be necessary to yield sustainable replication. We imagine situations like (i) a parasite infecting a new host, (ii) a species trying to invade a new ecological niche, (iii) cancer cells escaping from chemotherapy, (iv) viruses or microbes evading anti-microbial therapy, and also (v) the repeated attempts of combinatorial chemistry in the very beginning of life to produce self-replicating molecules. All such seemingly unrelated situations have a common structure in terms of Darwinian dynamics: a replicator with a basic reproductive ratio less than one attempts to find some mutations that allow indefinite survival. We develop a general theory, based on multitype branching processes, to describe the evolutionary dynamics of invasion and escape.

Introduction

At the very beginning of life (on earth or elsewhere), organic chemistry produced some macromolecules with random sequences. According to the RNA first hypothesis (Eigen and Schuster 1977, Eigen and Schuster 1982; Cairns-Smith, 1982; Szathmary and Maynard Smith, 1997; Orgel, 1998), these polymers had a limited ability of self-replication. Only a small subset of the random sequences had a basic reproductive ratio, R, greater than one, which means that one sequence could produce on average more than one offspring per lifetime. The overwhelming majority of random sequences had R<1, and therefore lead to lineages that would eventually go to extinction. Hence the physical and chemical environment of this world generated many short sequences capable of replication. We want to calculate the probability that one or many of those sub-critical lineages will mutate to a sustainable sequence with R>1.

Imagine a virus of one host species that is transferred to another host species, such as recent epidemics like HIV or SARS. In the new host, the virus has a basic reproductive ratio less than one (Anderson and May, 1992). Some mutations may be required to generate a virus mutant that can lead to an epidemic in the new host species. There will be repeated attempts to invade the new host. We want to calculate the probability that such an attempt succeeds in producing a mutant virus that initiates a new epidemic.

Suppose a successful HIV vaccine is found. Vaccinated hosts become exposed to a viral quasispecies. If the vaccine is effective, then most virus mutants will have a basic reproductive ratio less than one in the vaccinated host (Nowak and May, 2000). There will be some mutants, however, that can break through the protective immunity of the vaccine. We want to calculate the probability that a virus quasispecies of a given size finds (or already contains) an escape mutant that establishes an infection and thereby causes vaccine failure.

Cancer therapy often involves surgery or radiation to remove the main tumor followed by chemotherapy to eliminate remaining cancer cells. In the case of effective chemotherapy the majority of those cancer cells have a basic reproductive ratio less than one. Those cells are sensitive to this particular therapy. Genetic heterogeneity in the population of cancer cells could mean that some mutants have a basic reproductive ratio in excess of one. Those cells are resistant. Furthermore, sensitive cells could mutate to give rise to resistant cells. For example, resistance to Gleevec is caused by point mutations in the Bcr-Abl oncogene (Gorre et al., 2001; Sawyers, 2001). Other resistance mutations involve the inactivation of p53 or other tumor suppressor genes (Ozols, 1989; Keshelava et al., 2000; Kigawa et al., 2001). We want to calculate the probability of success or failure of anti-cancer therapy.

These are the main applications we have in mind when constructing our theory. More generally we describe situations where a genetically heterogeneous population of replicators is under selection to invade a new niche or repopulate a niche under a major selection pressure (such as vaccination or chemotherapy). Our approach is based on the theory of multi-type branching processes (Athreya and Ney, 1972; Seneta, 1970). The main assumption is that lineages behave independently of each other. Thus the fitnesses of individual genotypes are constant and not frequency or density dependent. Recombination as a mutational mechanism has to be excluded because its effective rate is intrinsically frequency dependent.

In Section 2 we outline the basic theory calculating the probability of non-extinction/escape for lineages starting from single individuals. We perform the calculation for various cases of increasing complexity ranging from single types to sequential mutations, networks and quasi-species (Section 3). In Section 4 we calculate the non-extinction probability when starting with heterogeneous populations.

Section snippets

Non-extinction of lineages

Consider a continuous-time branching process. Individual replicators undergo reproduction or death at random times (Fig. 1). There are different mutants with different fitnesses. In accordance with the fundamental assumption of branching processes, individuals behave independently of each other: there is no frequency- or density-dependent fitness. All birth and death events occur independently of each other.

Populations

We can also calculate the probability of non-extinction starting from a heterogeneous population of size N. Let us revisit the example of Section 2.5.1 with four genotypes denoted by 00, 01, 10 and 11. Suppose the relative abundances of the individual types in the initial population are given by x00,x01,x10 and x11. The probability of non-extinction of the population isP=1−exp[−N(x00ξ00+x01ξ01+x10ξ10+x11ξ11)].

The initial distribution of genotypes could be the consequence of a mutation selection

(Almost) Neutral intermediate steps

We have analysed cases where all replicators could be clearly separated into escape mutants with Ri>1 and non-escape mutants with Ri<1. The calculations are valid if all basic reproductive ratios are either less than 1−Ou or greater than 1+Ou (see Appendix A for detail). Under this assumption, all escape mutants have escape probability ξi=1−1/Ri, which is determined only by their own reproductive ratio and is independent of mutation rates. For non-escape mutants, we trace the shortest paths

Discussion

Branching process models have been used in a number of different contexts in biology. In ecology, for example, branching processes have been developed to calculate the success of invasion of a species into a new habitat, but without consideration of evolutionary change. Environmental fluctuations can lead to time dependent fitness (Iwasa and Mochizuki, 1988). If the environment fluctuates according to a stationary process, then the population size is described by a doubly stochastic process,

Acknowledgements

The work was done during Y.I.'s membership in Institute for Advanced Study, Princeton, in the academic year 2002–2003. Support from the Leon Levy and Shelby White Initiatives Fund, the Ambrose Monell Foundation, the David and Lucile Packard Foundation, and Jeffrey Epstein is gratefully acknowledged. We thank the following people for very useful comments: Steve Ellner, Steve Frank, Natasha Komarova, Akira Sasaki, and Karl Sigmund.

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