Crystal Toxins and the volunteer’s dilemma in bacteria

The growth and virulence of the bacteria Bacillus thuringiensis depends on the production of Cry toxins, which are used to perforate the gut of its host. Successful invasion of the host relies on producing a threshold amount of toxin, after which there is no benefit from producing more toxin. Consequently, the production of Cry toxin appears to be a different type of social problem compared with the public goods scenarios that bacteria often encounter. We show that selection for toxin production is a volunteer’s dilemma. We make the specific predictions that: (1) selection for toxin production depends upon an interplay between the number of bacterial cells that each host ingests, and the genetic relatedness between those cells; (2) cheats that do not produce toxin gain an advantage when at low frequencies, and at high bacterial density, allowing them to be maintained in a population alongside toxin producing cells. More generally, our results emphasise the diversity of the social games that bacteria play.

approach assumes only small variations in toxin production (weak selection), and looks 48 for a single equilibrium. In contrast, in nature there is large variation in toxin production, 49 between cells that produce (cooperators) or do not produce (cheats) Cry toxin (Raymond 50 et al., 2010(Raymond 50 et al., , 2012Deng et al., 2015). Furthermore, factors such as population density and 51 cooperator frequency can fluctuate over short timescales (Schoener, 2011;Raymond et al., 52 2012; Gokhale and Hauert, 2016), and studies of the density of spores in the wild have 53 shown that group sizes are very low suggesting that stochastic effects could be important 54 (Maduell et al., 2002;Collier et al., 2005;Raymond et al., 2010). Therefore, our second 55 approach is to model the dynamics of a system that contains both co-operators and cheats, 56 to examine how these dynamics are influenced by bacterial density, and the frequency of We use a game theoretic approach to express the fitness of a bacterial cell as a function 60 of: the probability it infects a host, β(z); and, the number of spores it generates, f (y) - 61 where, z is the group average strategy and y is the individual cells strategy. We assume an 62 infinitely sized population of bacteria distributed into finitely sized patches of n bacteria. 63 There are non-overlapping generations and the bacterial spores disperse randomly to other 64 patches. 65 We assume that the probability that a bacteria in a group of n cells successfully infects 66 a host, β, is a function of their average investment, z. We model this probability using a 67 sigmoidal curve as a continuously differentiable approximation of a step function: where, the group production of toxin nz is compared to k, which is the threshold at 69 which the chance of infection would be 0.5 (Cornforth et al., 2012). When the total 70 toxin production is low (nz << k) then the chance of infection is close to 0 as toxin 71 production increases 0 ≤ nz ≤ k then the function is accelerating and then past the 72 threshold (k < cz) the function is decelerating and asymptotes to 1.
The fitness function of a focal bacterium will be the product of the probability it invades 77 a host and the growth of the bacterium once it has successfully invaded (β(z) · f (y)): 78 ω(y, z) = 1 − ay 1 + e −(nz−k) . ( Equation (3) illustrates that producing the Cry toxin has a cost to the individual by 79 reducing its growth, f (y). However, it is beneficial to the group, including our focal 80 individual, as it increases the chance of successful invasion, β(z). 81 We seek an evolutionarily stable strategy (ESS), which is the individual strategy at 82 fixation which cannot be invaded by some rare alternative strategy. Following Taylor and 83 Frank (1996), we construct an expression for the change in inclusive fitness, ∆ω IF , and 84 solve for a monomorphic population that is at equilibrium: where, W is a Lambert-W function which is strictly positive (see B) and r is the relatedness 86 between the different bacterial cells infecting the host. We define r as the probability that 87 two individuals share the same gene at a locus relative to the population average (Grafen, 88 1985). This measure is obtained by replacing the regression of the recipients phenotype 89 on the focal individuals genotype (R in Taylor and Frank (1996)) with: R = 1 n + n−1 n r.

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Where 1/n is the chance the other individual is oneself and n − 1/n is the chance of a 91 social partner with other's only relatedness r to the focal individual (Pepper, 2000).

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The equilibrium at z * is a maximum however it may be unreachable. To test whether 93 a population under weak selection would converge to equilibrium (convergence stability), 94 we examined whether the second order terms at the equilibrium were negative (Otto and 95 Day, 2011). We found that: if: a > 0, 0 ≤ r ≤ 1, n ≥ 1, and W ≥ 0 .
So the equilibrium at z * is a candidate ESS. To determine uninvadibility we implement an 97 extension to the Taylor and Frank (1996) approach, by interpreting the second derivative 98 of the fitness equation in terms of inclusive fitness effects, therefore establishing a condition 99 for the candidate equilibrium to be a local maximum (Cooper and West, 2018). In A we 100 show that z * is an uninvaidable strategy as well as being convergently stable. We found that increasing relatedness (r) increases individual toxin production. Examining 103 the derivative of the equilibrium toxin production (z * ) with respect to relatedness (r) we 104 found that: So as relatedness (r) within the group increases the ESS of toxin also increases (z * ) 106 (C). Increasing relatedness increases the indirect benefit from toxin production as the 107 group chance of invasion, β(z), has a greater chance of being shared with kin. However, 108 even when relatedness is low (r = 0) toxin production is favoured as it is essential to 109 reproductive success ( fig. 1). As groups increase in size individual toxin production initially peaks and then declines 112 -when relatedness is non-zero ( fig. 1a). This is due to the efficiency gained when close 113 to the accelerating section of the sigmoidal β(z) function (near the threshold). As the 114 benefits (β(z)) are accelerating, small increases in toxin production lead to large increases 115 in infection chance. Past the peak toxin production, the greater number of individuals in 116 the patch allow for individual bacteria to reduce their investment but the group remains 117 at a high chance of successfully invading (see D). The derivative of toxin production, z * , with respect to the threshold is always positive or 120 zero: Therefore, in the absence of other limits if more toxin is required to invade the host (higher 122 k) individuals will be selected to increase their toxin production (z * ).  parameters.

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We assume a population of bacteria whose spores freely mix and are taken up at 144 random by a host. We assume that the host ingests P bacterial spores. In the environment From eq. (3) given i cooperators in a group the payoff, π, for the focal bacteria producing 148 y toxin will be: Therefore, the overall fitness of a focal bacteria producing y toxin in a population of 150 cooperators producing z toxin will be: This allows us to express the fitness of a cooperator in the population as ω(z, z) and that 152 of a cheat as ω(0, z). The relative fitness of cheats to cooperators in the population is:

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However, in our model we find that the relative fitness of a cheat is frequency depen-167 dent. This is because we relax both of the assumptions made by Ross-Gillespie et al. causes higher order terms of the relative fitness to matter and these higher order terms will include frequency dependent terms (Hamilton, 1964;Ross-Gillespie et al., 2007).

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These two effects lead to a frequency dependent relative fitness found here -unlike 181 the frequency independence found in earlier models (Ross-Gillespie et al., 2007). The 182 synergistic game causes the first order term of the Taylor expansion to be frequency 183 dependent. The strong selection causes higher order terms to become more substantial.

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These two effects are sufficient but not necessary conditions for frequency dependence to 185 arise.

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So given that the patch is started by a cooperator then the distribution of number of 203 cooperators among such patches is: and similarly for cheats: The binomial coefficient is C(P −1, i−1) for cooperators as the founder individual counts δ 2 ), in terms of φ, c and d, we ensure that the sum over both distributions is equal to one, 215 and the terms are weighted probabilities.

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The distribution of the number of cooperators in a patch is weighted by the fitness of 217 the focal individual in such a group (the sum of the above two distributions), giving: and from this we calculate a structured relative fitness: ν DS = ω S (0, z)/ω S (z, z)

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At maximum aggregation (φ = 1) cheats will do very poorly against cooperators as benefit, which means that even when looking at first order terms frequency is present as 276 a variable (Rousset, 2004). Secondly, in our models we assume that the cheater produces 277 no toxin and the cooperator produces a large quantity, leading to strong selection, which 278 means that linearising the relative fitness is no longer appropriate as higher order terms In the paper we then analyse the behaviour of ∆ω IF and ∆ω IF to characterise the equilib-403 rium as maximal and convergent. Cooper and West (2018) method is used to determine 404 if the equilibrium is unavailable. In brief, we consider the second derivative of the total 405 derivative taken to obtain the inclusive fitness effects (Taylor and Frank, 1996). This ex-406 pands into a long chain rule where we drop all higher order terms (∂g 2 etc.) as negligible 407 and substitute the regression coefficients as before, leaving us with: When this expression is less than zero we can say the equilibrium found is uninvaidable.

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W is a Lambert-W function which is strictly positive. The Lambert-W function or Pro-411 ductLog function is the inverse of the functions in form Xe X : In this case the function in full is: From the above we can see that -assuming: a > 0, 0 ≤ r ≤ 1, n ≥ 1, k ≥ 0 -then the 414 function within the brackets will be positive and therefore the value of the function will 415 be a positive real number. W (a, n, k, r)) The expression obtained in eq. (20) is indeed always greater than or equal to zero forall 418 values of r in the internal [0, 1]. We can see this by first remembering that the function 419 W is always positive for any parameter set which is biologically reasonable -a > 0, 0 ≤ 420 r ≤ 1, n ≥ 1, k ≥ 0. We then see that the first term is positive in the denominator (a In the paper we assume a cost of two-thirds and a threshold of two in all scenarios. This 429 was done so that in the case of two individuals total investment by both is necessary to 430 reach the threshold value in β(z). The reason that the cost was set to 2 3 was to represent 431 the fact that the tradeoff is against future investment not current investment. In fig. 5 we 432 can see a greater range of parameters which are presented here to show that the patterns 433 found are generally true across a reasonable range of parameter space.

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( Peña et al., 2014Peña et al., , 2015. This allows us to draw general conclusions about the shape 437 and behaviour of this function by looking at a simple gain function. In essence we can 438 calculate a i as the payoff for cooperating when i others cooperate, and b i as the payoff 439 when defecting and i others cooperate.
These are used to generate a measure of the gain from switching given i cooperators: Which gives a gain sequence: Now the purpose of this process is that the signs of the elements in the gain sequence, d,   Figure 1: The equilibrium toxin production depends on group size (n) and relatedness (r). a) When r > 0, as we increase group size, toxin production initially increases and then decreases. b) The total amount of toxin produced by the group, nz * , increases with group size, therefore, the chance of infecting the host is always higher in larger groups. These graphs assume k = 2 and a = 2 3 (D).  Figure 2: (a) The relative fitness of cheats is negatively frequency dependent as cheats become more common they are more often aggregated together and so suffer in relative fitness to cooperators. (b) As group size increases there is a positive density dependent effect on cheat fitness, the larger the group the more chance that sufficient toxin is produced by the group  Figure 3: Graphs of ω S , eq. (14), using parameters: k = 2, a = 2/3 and z = 0.17. (a) When group size is low, P = 5 increasing aggregation leads to decreasing relative fitness for cheats regardless of the initial cooperator frequency (b) At higher group sizes (P = 10) the pattern is also decreasing at high cooperator frequencies however at middling and low densities we see a non monotonic pattern with an intermediate aggregation causing a maximum relative fitness in cheats.   Group Size (n) Cooperator toxin production (z) Equilibrium Type Defector Mixed Cooperator Figure 6: This figure shows the dynamics of a population of cooperators and defectors as described by eq. (10). Each point represents a poulation with n group size and cooperators that produce z toxin. Using the criteria for the gain sequence for each population we cklassify it as either a defector only equilibrium a cooperator only one or a mixed equilibrium where the two strategies coexist. The graph was drawn using the parameters, a = 2 3 and k = 2