Edited by: Luis Raul Comolli, Lawrence Berkeley National Laboratory, USA
Reviewed by: Chiara Mocenni, University of Siena, Italy; Franz Luef, NTNU Trondlheim, Norway
*Correspondence: Robert M. Sinclair, Mathematical Biology Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan e-mail:
This article was submitted to Terrestrial Microbiology, a section of the journal Frontiers in Microbiology.
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Killing is perhaps the most definite form of communication possible. Microbes such as yeasts and gut bacteria have been shown to exhibit killer phenotypes. The killer strains are able to kill other microbes occupying the same ecological niche, and do so with impunity. It would therefore be expected that, wherever a killer phenotype has arisen, all members of the population would soon be killers or dead. Surprisingly, (1) one can find both killer and sensitive strains in coexistence, both in the wild and in
The Purpose of Computing Is Insight, Not Numbers (Hamming,
It will be useful to begin with a clear statement of what this work is really about. The central question being addressed is whether killer and sensitive phenotypes could in theory coexist in the same environment even if the absolute fitness penalty for the killer phenotype were arbitrarily small. The question is motivated by experimental and field observations to be described below, but is to be approached here from a theoretical point of view. This immediately implies that what is needed is a theoretical framework which has two apparently contradictory properties: First, it must capture important aspects of the relevant biology. Second, it must be simple enough that the question can be answered. This theoretical framework will therefore necessarily be a compromise between realism and tractability. Furthermore, what is actually needed is only a single affirmative result, since that proves the theoretical possibility of coexistence. The phrase “arbitrarily small” excludes some standard approaches, and it is best to make this point now rather than later, because it is this compact phrase which drags us into what will be unfamiliar territory for many readers. If the theoretical framework were to be in the form of a standard numerical simulation of a model, then the best one could do would be to show that coexistence is possible for given small absolute fitness penalties for the killer phenotype. This is not the same as “arbitrarily small”: if a simulation were to show coexistence to be possible for a fitness penalty of 0.1, it would still leave open the question of whether coexistence could be possible for a fitness penalty of 0.01 and so on ad infinitum. We are thus motivated, by the question we have chosen to investigate, to search outside of the box of standard scientific computing tools until a truly suitable approach is found. The field of mathematical analysis (Ross,
The phenomenon of killer phenotypes, which possess the ability to kill conspecifics while being themselves immune (Marquina et al.,
Rather surprisingly, it has been shown that the cost of toxin production can be negligible (Garbeva et al.,
We describe here an explicitly solvable model of yeast population dynamics on an infinite number of patches, in which killer and sensitive strains can coexist. Our model includes killer and sensitive strains only. In the following, we will use the example of a killer yeast in our verbal descriptions of the model. A full mathematical treatment would not be appropriate here. We will instead provide what may be called a sketch of the model and our analysis of it. The Supplementary Material contains details of the most important part of the mathematical analysis, but it is also best described as a sketch rather than a formal proof.
Each patch is intended to represent a single piece of fruit. A patch can be colonized by spores from any patch. If a patch is colonized only by spores of the sensitive yeast strain, then the patch will emit only spores of the sensitive strain. If a killer yeast spore lands on a patch, then any sensitive yeast colony will be eradicated, and the patch will emit only spores of the killer yeast. If a sensitive yeast spore lands on a patch colonized by killer yeast, it will not survive nor influence the (killer yeast) spore production of the patch. The number of spores emitted by a patch depends only upon the type of yeast that has successfully colonized it. If no spores have landed on a patch then that patch will emit no spores. Sporulation occurs in all patches simultaneously, leaving all patches barren and ready for the next cycle, initialized by the dispersal of the spores.
Let
The dynamics of the killer yeast strain is not in any way influenced or restricted by the sensitive strain, and so can be treated independently. The probability of a given patch not being reached by any killer strain spore is
The reason for this can be understood by first considering a finite number of patches, and then taking the limit as that number goes to infinity. Let
This will be, for an infinite number of patches, the fraction of patches which are not reached by any spore. On the other hand, the fraction of patches which are reached by a spore must be the remainder, or 1 −
According to standard theory, the map (Equation 1) has an unstable fixed point at
The dynamics of the sensitive strain is governed by the same equations in the complete absence of spores of the killer strain. The reason for this assumption is the observation (discussed above) that the difference in absolute fitness between the killer and sensitive strains can be very small. In the presence of an established killer strain population occupying a fixed fraction (
One can construct (details are in the Supplementary Material and see also Figure
Given a stable subpopulation of the killer strain, a necessary condition for establishment of a subpopulation of the sensitive strain from a finite number of sensitive spores is
The total fraction of patches stably colonized by either strain is
Can any ratio of killer to sensitive phenotypes be achieved in this model? Furthermore, can any total fraction of patches be stably colonized? Since
As a numerical example, if
Two direct consequences of the model are (1) that killer and sensitive strains can coexist in any given proportion, and (2) that the presence of a sensitive subpopulation increases the total population size of yeast (including both strains) without reducing the population size of the virus population maintained by the killer yeast strain. Taking a broader point of view, the second consequence means that the species benefits from having both sensitive and killer strains.
Since our model is explicitly solvable, we are able to perform a mathematical analysis which showed (see the Equation 3 and the associated comments above) that coexistence is possible for any extra fitness cost of the killer phenotype,
As a numerical example, if
Using the explicit formulae from our analysis,
Therefore, we are able to construct pairs (
It is not intuitively obvious that sensitive strains can survive in the presence of killers, given that our model has no fixed barriers to prevent the sensitive strains from being eradicated by encounters with killers. The value of our model lies not only in this prediction, which is consistent with other, related, models (the semi-analytical configuration-field approximations for the one- and two-species SCA models of Czárán and Hoekstra,
We have been able to prove that killer-sensitive coexistence is possible for any fitness penalty of the killer phenotype,
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The Supplementary Material for this article can be found online at: