Publisher Correction: Programmed cell death can increase the efficacy of microbial bet hedging

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if at the end of a thousand rounds it makes up a higher percentage of the total population. 112 Simulations were coded in the language python and are available in the Supplementary (1)

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There is a direct cost to PCD because cells are killed. If all cells were replaced by the same 116 strain then this would remove the cost. However, if replacement does not occur with perfect 117 assortment (i.e. r < 1) then some of the cells that die will be replaced by members of the 118 competing strain. We call the total number of a strain that are replaced by a competing 119 strain the cost of PCD. Since our model has disasters interspersed with rounds of growth 120 and PCD, we can calculate the expected cost of PCD during the period between disasters. (1-r) r and G2 switch between phenotypes A and B with a probabilities p 1 and p 2 , respectively.
Organisms undergo PCD with probabilities c 1 and c 2 such that c 1 (A 1 + B 1 ) is the expected "cost" of PCD experienced by genotype G1. The population then regrows back to the carrying capacity N such that each genotype repopulates a fraction of its own cells that underwent PCD determined by the value of r. The repopulation is partitioned among phenotypes according to their relative frequency such that The "gain" corresponds to the number of cell reproductive events reallocated to each genotype.
This factor has three terms which determine its magnitude: c, t, and r. for diversification between disasters (for example, strains with a very low rate of stochastic 140 switching), higher diversification rates can reduce a strain's possibility of extinction. We 141 will focus on this second manifestation because it is more crucial to the survival of the 142 PCD strain-failure to diversify can result in extinction. 143 We assume that one phenotype, say B, has just experienced a disaster and is no longer There is a narrow band where PCD is beneficial corresponding to cmp = const. Outside of this area, the switching rate is too high or too low to incur a benefit to PCD genotypes.
B) A contour plot shows the log 10 relative benefit versus cost of PCD as expressed in Eq.
5 for a range of PCD probabilities (c) and number of cells (m) with r = .5 and p = 10 −6 .
The blue area corresponds to a greater cost while red areas correspond to a greater benefit.
Since the plot is transformed by log 10 the benefit is many orders of magnitude (> 10) greater when the number of cells is larger than 10. where replacement was not perfectly assortative (r < 1).

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One advantageous situation for the PCD strain would be if it produced a B phenotype 156 but the non-PCD strain did not. The probability of this event is shown in Eqn. 3.

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( The probability increases with the structure parameter r, reaching a maximum at r = 1.

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It is highest at p = 1 − (1 − r) 1 c 1 mr which for r << 1 is approximately one expected 159 B phenotype produced by the PCD strain (see Figure 2A). At slower switching rates and 160 lower rates of PCD, the PCD strain is not likely to diversify. At higher switching rates and 161 higher rates of PCD the non-PCD strain is likely to diversify along with the PCD strain, 162 thereby reducing the relative advantage of PCD.

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The probabilistic formulation of the benefit of PCD can also be compared with a prob-164 abilistic formulation of the cost. We put the cost into a similar currency by considering 165 the probability that the PCD strain goes extinct because of PCD (shown in Eqn. 4). This 166 requires all m cells to undergo PCD and be replaced by the competing strain.
In this formulation, the PCD strain faces the cost of going extinct from PCD but has 168 the benefit of diversifying when the competition does not. The relative probability of the 169 benefit to the cost is shown in Eqn. 5.

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( This relationship is low when p is very high (p = 1) or when p is very low p = 10 −6 and r 171 is small (r = .5). If we assume a low p, say p = 10 −6 , and an unbiased r, i.e. r = .5, then 172 we can compare the ratio of probabilities for a range of values of c and m (see Figure 2B).

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Effectively, if m is larger than 10 then the benefit is more likely than the cost by over 10 174 orders of magnitude. i.e. c < .1. For PCD rates above .1, the cost is too high to compensate for any benefit.

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For high switching probabilities, the benefit of PCD is diminished because it is likely that 185 the non-PCD strain will always diversify. Similarly, if the switching probability is too low then the PCD strain cannot adequately diversify. For more structured environments, with 187 say r = .9 shown in Figure 3B, the PCD strain wins over a much larger area of parameter 188 space. This is because the higher value of r reduces the cost to PCD. G1 t , and find that this depends only on the PCD rates, the structure parameter r, and the 200 initial amounts of each genotype (see Eqn. 6).
One interesting consequence of Eqn. 2 is that the higher PCD genotype, G1, can actually of a disaster, the genotypes approach the equilibrium shown in Eqn. 7 (assuming c 1 < 1 208 and r > −1).
We note that the equilibrium does not depend on the value of the structure parameter r 210 as long as r > 0. Thus, as long as there is some chance that the higher PCD genotype 211 replaces some of the population it lost to cell death, then this equilibrium will be reached. 212 We note that in this calculation (in the absence of disasters) if G2 does not engage in PCD, 213 i.e. c 2 = 0, then G1 will eventually go extinct. If, however, both genotypes engage in PCD, 214 then they can coexist-at least until a disaster comes.

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The above calculations show the possibility of coexistence in the absence of disasters 216 and stochasticity. We might expect that the combination of disasters and stochastic events 217 (phenotype switching) may interfere with long-term coexistence. Figure 4 shows an exam-   (Figures 5b and 5c). Overall, the relative performance of the PCD strain 255 increases with the frequency of disasters. Not only do disasters select for rapid phenotypic 256 diversification, but they also reduce the population to a relatively sparse state. Since the 257 effects of birth on assortment/structure outweigh those of migration at low population 258 densities, spatial structure (characterized by r) increases following a disaster. Thus, more 259 frequent disasters increase the relative benefit of phenotypic diversity as well as the mean  by supplying resources to starving survivors [48]. Although PCD by a subpopulation 320 may conceivably benefit surviving cells in the event of an abrupt shift back to favorable 321 conditions [49], our model suggests that increased diversification may be another benefit 322 of such population cycling.

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Understanding the evolution of cellular suicide will require a plurality of approaches, of PCD are only supported when the population is structured, but we also find that simple 331 processes of death and clonal reproduction easily generate the necessary structure. While 332 much work remains before we have a complete understanding of altruistic suicide, it is well 333 worth the effort. Not only is it a topic of fundamental biological importance, it also has 334 the potential to help generate novel therapeutic interventions [23,50].