Diffusive Public Goods and Coexistence of Cooperators and Cheaters on a 1D Lattice

Citation: Scheuring I ( Diffusive Public Goods and Coexistence of Cooperators and Cheaters on a 1D Lattice Istva n Scheuring 0 Attila Csikasz-Nagy, Fondazione Edmund Mach, Research and Innovation Centre, Italy 0 MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group, Department of Plant Systematics, Ecology and Theoretical Biology, Pa zma ny P. se ta ny 1/c H- 1117 , Budapest , Hungary Many populations of cells cooperate through the production of extracellular materials. These materials (enzymes, siderophores) spread by diffusion and can be applied by both the cooperator and cheater (non-producer) cells. In this paper the problem of coexistence of cooperator and cheater cells is studied on a 1D lattice where cooperator cells produce a diffusive material which is beneficial to the individuals according to the local concentration of this public good. The reproduction success of a cell increases linearly with the benefit in the first model version and increases non-linearly (saturates) in the second version. Two types of update rules are considered; either the cooperative cell stops producing material before death (death-production-birth, DpB) or it produces the common material before it is selected to die (production-death-birth, pDB). The empty space is occupied by its neighbors according to their replication rates. By using analytical and numerical methods I have shown that coexistence of the cooperator and cheater cells is possible although atypical in the linear version of this 1D model if either DpB or pDB update rule is assumed. While coexistence is impossible in the non-linear model with pDB update rule, it is one of the typical behaviors in case of the non-linear model with DpB update rule. - Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. Relevant data are included within the paper and its Supporting Information files. Funding: The work is supported by the Hungarian National Science Foundation (OTKA, Grant no. 100296 and 100299). The production cost of the paper is covered by Hungarian Academy of Science (MTA). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The author has declared that no competing interests exist. The evolutionary stability of cooperation has been in the focus of theoretical biology for decades [15]. On the one hand, cooperation is widespread and frequent in nature. More importantly in all major transitions of evolution it is connected with cooperation of subunits of the newly evolved replication unit [6]. The spread of cooperative behavior seems to be surprising for the first sight since cooperation is often costly, and cheaters which do not cooperate do not bear the cost of it, and thus can exploit cooperators. Consequently, we might think that cheaters have greater fitness than cooperators, which leads to the extinction of cooperators from the population. Motivated by this discrepancy between field observations and verbal reasoning presented above, many theoretical explanations were given on the origin and evolutionary stability of cooperative act, e.g. [2,3]. Knowing that the concept of cooperation covers behaviors from extracellular enzyme production of bacteria [79] through cooperative hunting [10,11] to eusocial insects [12], it is not surprising that the explanatory mechanisms have great diversity as well. However, alternative explanations are present even within the more narrower areas. For example, there is no consensus as to the main mechanism explaining the evolutionary stability of producing extracellular enzymes (or any other molecules as public good) by microorganisms. One of the widespread explanations is based on slow cell motion and local interactions among densely packed cells. Thus, cooperator and cheater cells distribute in patches. Cooperator cells interact other cooperators (producers) with higher probability than with cheaters (non producers) if their motion is slow and progenies are distributed in the vicinity of the mother cells, so their average fitness will be higher than it would be in a well mixed system. It is shown that if cooperation is not too costly and this assortative pairing is strong enough then cooperator cells can coexist with cheaters even if Prisoners Dilemma (PD) (the strongest dilemma of cooperation) is considered as the basic model of interaction [3,1315]. The alternative view emphasizes that local interaction and limited motion lead to positive genetic correlation among interacting individuals, thus kin selection can easily explain the benefit of cooperators [4,8,16,17]. Although these two explanations seem to be only two sides of the same coin, it has been shown recently that they are not always completely identical [12,18,19]. However, one certainly important point is generally neglected in the strategic models mentioned above, namely that the produced material, which is a common good for everyone, is frequently a diffusive molecule. So, while cells move slowly on the surface they live, the molecules for which competition takes place disperse much faster. More elaborated models should consider this effect. The simplest way to build up this effect into the models is that if the interaction range of individuals is larger than the competition well by using two types of death-birth update rules. The first rule assumes that a cell stops to produce common material before death, thus concentration distribution is computed without this producer. This rule is termed as death-production-birth and is denoted by DpB. The second rule assumes that a producer cell synthesizes the common material before it dies, so the concentration distribution of the diffusive material computed before this death event, and birth success depend on this concentration. So this will be called the production-death-birth rule denoted by pDB. I have shown that independently to the used update rule coexistence is possible in the linear model (fitness increases linearly with local concentration of the common material) although this behavior is rather atypical. However, coexistence is a robust behavior in non-linear model (if fitness is a saturating function of local concentration of the common material) if the pDB update rule is used while coexistence is impossible if the DpB update rule is applied. In the last section, I compare them with results of previous similar models of bacterial cooperation. Consider an 1D lattice of lattice size N where every lattice point is occupied by a P (producer) or an NP (non-producer) cell. The grid is 1-dimensional, thus every cell has two nearest neighbors except the ends of the lattice where cells have only one neighbor. I choose this boundary condition, since it follows the biological situation and makes the calculations simpler. P cells are considered to be point sources of diffusive materials which is a (...truncated)