Abstract
By nature of their small size, dense growth and frequent need for extracellular metabolism, microbes face persistent public goods dilemmas. Genetic assortment is the only general solution stabilizing cooperation, but all known mechanisms structuring microbial populations depend on the availability of free space, an often unrealistic constraint. Here we describe a class of selforganization that operates within densely packed bacterial populations. Through mathematical modelling and experiments with Vibrio cholerae, we show how killing adjacent competitors via the Type VI secretion system (T6SS) precipitates phase separation via the ‘Model A’ universality class of orderdisorder transition mediated by killing. We mathematically demonstrate that T6SSmediated killing should favour the evolution of public goods cooperation, and empirically support this prediction using a phylogenetic comparative analysis. This work illustrates the twin role played by the T6SS, dealing death to local competitors while simultaneously creating conditions potentially favouring the evolution of cooperation with kin.
Introduction
Microbes are fundamentally social organisms^{1,2,3,4,5}. They often live in dense, surfaceattached communities, and participate in a range of social behaviours mediated through the production and consumption of extracellular proteins and metabolites. Paradigmatic examples include the cooperative production of digestive enzymes^{6}, metal chelators^{7}, signalling molecules^{6} and the structural components of biofilms^{8}. Many of these extracellular compounds are susceptible to social exploitation, in which nonproducing ‘cheats’ gain an evolutionary advantage. If unchecked, this social exploitation can lead to the extinction of cooperative genotypes^{9,10}.
It is widely recognized that the spatial segregation of cooperative microbes away from cheats can solve this cooperative dilemma by ensuring that the investment of cooperators goes to other adjacent cooperative individuals^{1,5,10,11,12}. Mechanisms creating assortment when organisms expand their ranges via growth into free space have recently received much attention^{13,14,15,16,17,18}, where robust patterns of genetic segregation can occur via stochastic bottlenecking. However, this mechanism cannot generate genetic segregation within dense, wellmixed communities displaying no net growth, despite the clear ecological relevance of such communities.
One mechanism that has been proposed to potentially generate spatial structure in dense communities is antagonistic interactions among genotypes^{1,19,20,21,22,23}. If different genotypes interact antagonistically then wherever a genotype is in the minority they will be killed by competitors at a high rate, resulting in genetically homogenous patches. While mechanisms via which individuals can recognize and kill nonkin have been extensively studied, the consequences of such interactions for the spatial structure of communities have not been explored in detail.
The Type VI secretion system (T6SS) is a potent mechanism of bacterial aggression that can deliver effector proteins directly into eukaryotic cells to mediate virulence by cellular disruption, and into adjacent bacteria to mediate competition by killing nonkin while leaving kin with corresponding protective immunity proteins unscathed^{24,25}. In Vibrio cholerae, T6proficient strains utilize the T6SS to intoxicate T6deficient eukaryotic predators and diverse proteobacteria, as well as other more closely related V. cholerae isolates that lack identical effector immunity pairs^{26,27,28,29,30,31,32}. T6mediated segregation occurs during coculture of T6proficient V. cholerae with T6deficient E. coli. Segregation was also predicted to occur between two mutually antagonistic T6proficient strains^{33}, and recently demonstrated at the single cell level in cocultures of V. cholerae and Aeromonas hydrophila^{20}.
Here we examine the causes and consequences of neighbour killing via the T6SS on the physical structure of microbial communities. Using a V. cholerae experimental system and mathematical modelling, we show that T6SSmediated killing causes an initially wellmixed population of mutually antagonistic bacteria to phase separate, forming clonal patches that grow larger through time. This phase separation belongs to the ‘Model A’ class of orderdisorder transitions, which is described by the AllenCahn equation. We mathematically demonstrate that the spatial structure generated as a consequence of T6SSmediated killing can favour the evolution of publicgoods cooperation by limiting the potential for unrelated ‘cheats’ to access secreted products. Finally, we bioinformatically show that bacteria with more T6SS systems and effectors dedicate a larger fraction of their genomes to secreted products. While it is too early to rule out alternative hypotheses, this correlation is consistent with general predictions from social evolutionary theory that spatially structured environments favour the evolution of cooperation.
Results
Mutual antagonism drives phase separation
Our system illustrates the profound effect of T6SSmediated killing on emergent spatial patterning of a surface attached population. Mathematical modelling suggests that an initially wellmixed population of mutual killers should rapidly undergo phase separation due to ‘selfish herd’ dynamics^{34}, as the cells within genetically uniform groups no longer risk T6SSmediated death. Indeed, we observe rapid phase separation in three distinct classes of models, all starting with a randomly seeded population on a twodimensional lattice (Fig. 1a). We first developed an individualbased model (IBM; Fig. 1b; Supplementary Movie 1) that simulates bacterial growth, the killing of adjacent competitors and reproduction into empty patches through time. IBMs are appealing, in that they offer an intuitive simulation of discretized, interacting individuals. However, IBMs often lack mathematical transparency, limiting generalization. We thus modelled our system using two distinct, mathematically defined approaches: an ecologically based partial differential equation model in order to gain analytical insight into the dynamics (PDE; Fig. 1c; Supplementary Figs 1 and 2, Supplementary Movie 2), and the ecologicallybased Ising spin model in order to relate our results to classical modelling of phase separation in statistical mechanics^{35} (Fig. 1d; Supplementary Movie 3). In all three modelling frameworks, initially well mixed populations rapidly underwent phase separation. Similarly, initiallywell mixed populations of two Vibrio cholerae strains (C6706 and 692–79; Supplementary Table 1) capable of mutual T6SSmediated killing (Supplementary Fig. 3) underwent phase separation (Fig. 1f,i,j). Nonkilling controls (ΔvasK, that is, T6SS^{−}; Supplementary Fig. 3) and T6SS^{+} mutual killers cultured at low temperatures which impede T6SS activity^{36} remained wellmixed (Fig. 1e,g,h).
Spatial analysis
To determine whether our models and experiments undergo the same type of orderdisorder transition, we quantitatively examined the dynamics of phase separation in each. We first computed the Fouriertransformed structure factor, S(q). The characteristic wavenumber of clonal groups is , and the height of the peak is related to how often it occurs in the lattice (that is, the strength of patterning at that length scale). At early timesteps (Fig. 2a, Supplementary Fig. 4), or for nonkilling controls (Fig. 2b), S(q) is relatively flat, as expected for a wellmixed population lacking a characteristic length scale. T6SSmediated killing causes S(q) to increase at smaller values of q (longer length scales) as the population grows increasingly structured. This progression of S(q) is a hallmark of phase separation^{37}. For Model A, q_{m} scales as q_{m}∝1/, while S(q_{m}) scales as (ref. 38). It is ambiguous how to relate simulation time to experimental time; instead, we plot S(q_{m}) versus q_{m}. All models (IBM, PDE and Ising) and experiments fall on the same line (S(q_{m})∝1/) (Fig. 2c), a relationship consistent with the ‘Model A’ orderdisorder phase separation process^{39}, developed to explain the interaction of atomic ‘spins’ in systems that lack conservation, and described by the AllenCahn equation =η−μ, which relates the change in local concentration, ϕ, over time to diffusion and the chemical potential, μ and stochastic fluctuations (see Methods)^{40}. To demonstrate this equivalence across wavenumbers, we plot q_{m}^{2}S(q) versus q/q_{m} (Fig. 2d). This collapses all data onto one master curve. In fact, due to the universality of nonconserved domain growth, this collapse could have been expected. Importantly, this universality, shown in Fig. 2c,d, demonstrates that while initial conditions—such as the initial number ratio of the two competing strains—may influence the timing of phase separation, they do not influence how phase separation occurs, or that the characteristic clonal group size always grows as . Cellular mobility has a surprising effect on phase separation: rather than impeding phase separation, it accelerates it by enhancing killing at the borders of clonal patches (Supplementary Methods, Supplementary Fig. 5 and Supplementary Movie 4).
To provide biological context for this process of phase separation, we calculated clonal assortment (r), for the IBM (Fig. 2e) and the Vibrio experiments (Fig. 2f). Assortment, which can be thought of as analogous to Hamiltonian relatedness^{17,41,42}, describes the extent to which clonemates spatially colocalize after accounting for their frequency in the population (see Methods for details). T6SSmediated killing resulted in the creation of highly structured populations with high assortment over long length scales (Fig. 2e,f). Such assortment can protect diffusible public goods from consumption by competing strains^{43,44}.
Spatial assortment supports cooperation
To explore the effect of T6SSmediated killing on the evolutionary stability of public goods cooperation, we introduced a diffusible cooperative good into our model. Because all three of our modelling frameworks displayed similar dynamics, we chose the PDE framework because it is the most amenable to analytical investigation. We considered two competing strains: a cooperator that secretes an exoproduct into its environment at an individual cost, and a nonproducing cheat that, all else equal, grows faster than the cooperator as it does not pay the cost of production. In this model, cellular growth rates for both strains depend on the local concentration of the diffusible exoproduct. We find that T6SSmediated killing protects cooperation in two different ways. In a nonspatial (that is, constantly mixed) environment, T6SSmediated killing can allow cooperators to resist invasion by rare cheats owing to the cooperators’ numerical dominance in antagonistic interactions (that is, it creates positive frequencydependence (Fig. 3c), while without T6SSmediated killing (either because strains lack T6SS, or because the cheat is of the same T6SS type as the cooperator) cheats outcompete cooperators at all starting frequencies (Fig. 3a). However, in a spatially defined environment, phase separation driven by T6SSmediated killing physically separates producers from cheats, expanding the conditions favouring cooperation and allowing them to invade a population of cheats from rarity (Fig. 3d,e and Supplementary Movie 5; see proof in Supplementary Methods), while cheats ultimately win in the absence of T6SSmediated killing (Fig. 3b).
Our models and experiments demonstrate that T6SSmediated killing can generate favourable conditions for the evolution of publicgoods cooperation^{1,12,15,45}. This can occur in two ways. First, T6SSmediated killing induces positive frequencydependent selection, allowing cooperators to resist rare cheats. Second, T6SSmediated killing precipitates selforganized structuring of microbial communities, reducing the diversity of cheats that can exploit cooperators. In addition to the predictions of our models previous experimental work has suggested that, via pleiotropic linkage to quorum sensing communication systems, the T6SS can also act as a policing mechanism protecting against the evolution of quorum sensing cheats^{46}. Altogether, this suggests that there are three complementary paths via which T6SSmediated killing could favour the evolution of cooperation.
Association between T6SS and secreted product evolution
Does T6SSmediated killing have a similar effect in the real world, where ephemeral resources, physical disturbance and intense competition may impede these mechanisms? We approach this question phylogenetically, examining the relationship between the proportion of each genome coding for potentially exploitable secreted proteins and its T6SS complexity, with the rationale that microbes possessing a greater number of T6SSs may face less social exploitation by living in more structured communities. All else equal, genotypes that possess a greater number of T6SSs should form more highly structured patches (higher r), phase separating with a greater proportion of competitor genotypes (that is, those with noncomplementary effector/immunity proteins). As a result, we hypothesize that genotypes with more T6SSs should experience less pressure from social cheating imposed by distantly related competitors. There is, of course, an important caveat to this scenario: while T6SSbased phase separation should effectively exclude competitors, it does not address the de novo evolution of cheating from within the clone (for example, Fig. 3a,b).
As a firstorder proxy for cooperativity, we measured the proportion of a clone’s genome dedicated to secreted proteins (henceforth referred to as ‘secretome size’). While many of these secretions may have antagonistic effects on other microbes, they can still be seen as cooperative from the producing cell’s perspective, as their kin can benefit from the reduced competition that they create^{47,48}. We constructed a Bayesian phylogenetic mixed model of T6SScontaining Proteobacteria and Bacteroidetes (Fig. 4a, Supplementary Fig. 6) using 439 genomes from 26 genera. Secretome size is positively correlated with both the number of T6SSs (Fig. 4b,d, Supplementary Table 2) and T6SS effector proteins (Fig. 4c,e, Supplementary Table 2) present, and the model shows an excellent overall fit to the data, explaining 99% of the variance in secretome size (Fig. 4f). These results are also robust in univariate analyses (Supplementary Tables 3 and 4) and to the inclusion of genome size as a predictor (Supplementary Table 5). As our analyses include many closely related strains (for example, many Helicobacter pylori, Fig. 4a), most (91%) of the variance in secretome size is explained by the phylogenetic relationships among strains. Nonetheless, the number of T6 secretion systems and T6SS effectors are important predictors of secretome size, explaining 8% of the total, and 90% of the nonphylogenetic variance in secretome size.
While the above analysis is consistent with the predictions of our mathematical model showing that phase separation should favour the evolution of cooperation (Fig. 3), it is not conclusive. As with any broadscale phylogenetic analysis, alternative mechanisms explaining this correlation cannot be ruled out. For example, some unknown aspect of bacterial ecology may independently select for both investment in T6SSmediated killing and exoproduct production. By controlling for phylogenetic variation, our analysis should capture some of this ecological variance, through it does not eliminate it entirely. Alternatively, increases in the presence of dead competitors owing to T6SSmediated killing can create additional opportunities for horizontal gene transfer^{49}, which could allow for increased acquisition of genes coding for secretions. However, such an explanation relies on horizontal gene transfer being biased towards genes encoding extracellular secretions. While there is evidence that this bias towards secretions is the case for plasmids and other mobile genetic elements^{50}, whether this bias occurs when sampling the genes of dead competitors remains to be determined. Detailed analysis of the phylogenetic dynamics of T6SS genes and secretions in individual clades may allow the relative contributions of these hypotheses to our observed correlation to be disentangled.
Discussion
Phase separation is wellknown to drive pattern formation in biology^{51,52}, but has mainly been investigated using either Turing activatorinhibitor feedbacks^{53,54}, or positive densitydependent movement, described by the CahnHilliard equation^{51,55,56,57}. In this paper, we describe a third general mechanism of selforganized pattern formation: targeted killing of nonkin competitors. This drives a ‘Model A’ phase separation; the kinetics of this coarsening process—described by the AllenCahn equation—only depend on a few cellular details. While we restrict our analysis in this paper to the T6SS, the role of antagonistic interactions in structuring biological communities it is probably far more general, applying to diffusible compounds that kill adjacent nonkin in both microorganisms (for example, antibiotics) and macroorganisms (for example, allelopathy in plants^{58} and animals^{59}). However, while ‘Model A’ coarsening is universal, the realization of such dynamics in a densely packed, immobile, athermal system is likely unique to biology.
Physically, this system bears similarities to active matter^{18,51,56,57}; phase separation has also been observed in these far from equilibrium active systems, wherein constituents expend energy to move. Phase separation in these systems typically occurs due to differences in mobility as a function of density; constituents move slowly through crowded regions, and quickly through low density regions. Mobilityinduced phase separation has been observed (or predicted) in systems as varied as swimming bacteria^{60}, selfpropelled colloids^{61,62}, mussels^{51}, granular rods^{63}, active filaments^{64,65}, rotating particles^{66}, among other systems^{52}. In the current system, activity is derived from reproduction and killing events at high density rather than constituent mobility^{67}, leading to a ‘Model A’ transition.
Model A coarsening captures the behaviour of a broad array of phase transitions that lack conservation. This transition was originally developed to model magnetization in ferromagnetic materials via the Ising model. Ferromagnetic spins have minimum energy when they align; they do so via Glauber spin flips, leading to a change in the overall magnetization. The physical universality of this transition may be reflected in the strong correlation between secretome and T6SS effectors and apparatuses seen in Fig. 4. The microscopic details of the system do not strongly affect coarsening, so long as densely packed cells are equipped with T6SS.
In recent years, there has been a growing appreciation that many microbial behaviours requiring extracellular metabolism are susceptible to social exploitation. Here we show how simple cellcell aggression can, as a consequence, create a structured population favourable to cooperation. Clearly, many factors contribute to the structure and function of microbial communities^{1,4,15,16,19,20,21,22,42,47}. However, because T6SSs are common (found in ∼25% of Gramnegative bacteria^{68}), and microbes often live in dense communities, phaseseparation driven by contactmediated killing may have a fundamental role in defining the genetic composition and ecosystemlevel functionality of microbial communities worldwide.
Methods
Bacterial strains and culture conditions
The bacterial strains used in this study are shown in Supplementary Table 1. Vibrio cholerae fluorescence reporter constructs were chromosomally integrated and gene deletions and promoter replacements were constructed by allelic exchange as described and verified by Sanger sequencing^{69,70,71}. Vibrio cholerae was routinely grown at 30 °C in LuriaBertani (LB) liquid medium supplemented with 50 μg ml^{−1} of kanamycin or 150 μg ml^{−1} spectinomycin when appropriate. For confocal microscopy experiments, overnight cultures were mixed and 0.5 μl was inoculated onto LuriaBertani agar (1.5%) pads on glass slides, and incubated at 17, 25 or 30 °C for 24 h. C6706 and 692–79 were inoculated at a 1:6 initial ratio, as T6SS^{+} C6706 is more competitive than T6SS^{+} 692–79 under our assay conditions (it grew from an inoculation ratio of 16.6% to constitute an average of 40–62% of each colony at all three temperatures). To visualize lessadvanced stages of phase separation, we used a 1:8 initial ratio of strain C6706 to 692–79. For all images, we show C6706 in red and 692–79 in blue. As expected^{69}, phase occurred similarly when the fluorescence reporters were swapped between strains (Supplementary Fig. 7).
Microscopy and image analysis
Laser fluorescence confocal microscopy was performed with a Nikon A1R. The filters used were fluorescein isothiocyanate (for detecting mTFP1, cyan) and tetramethylrhodamine (TRITC) (for detecting mKO, orange). Full colony images were captured in one zplane using the 20 × Plan Apo objective lens and a 2 × internal multiplier was applied to capture closeup images. The Galvano scanner was used to scan 2,048 × 2,048 pixels on all images in order to maximize resolution. For every sample, the top and bottom of the colony was located, and a plane in the middle was imaged. The images were stitched and channels were merged using NIS Elements software. To eliminate issues with red–green colourblindness, we present green fluorescence in images as blue.
To calculate the structure factor, S(q), we start with an image from a simulation or experiment, I(x,y). S(qx,qy) is the absolute value of the Fourier Transform of I(x,y), squared , where q_{x} and q_{y} are spatial frequencies in the x and y directions, respectively. We then radially average S(q_{x},q_{y}), .
To calculate the assortment (r) of the genotype over interaction radius h, we again start out with a binarized image from a simulation or experiment I(x,y) in which we set values of the focal strain g to 1 and the competitor strain c to −1. We first convolved I(x,y) with a kernel in which all positions other than the center were set to 1, and the center set to −((2h+1)^{2}−1), generating the transformed matrix C(x,y). For example, the kernel for distance h of 1 would be . Edges within distance h were trimmed. For each interaction radius h (which ranged from 1–36), we calculated the assortment r of the focal strain g as r_{g}= . r is thus the mean frequency of g within interaction radius h, relative to frequency of g in the population as a whole. r, which ranges from −1 to 1, describes the spatial association of each genotype above or below what would be expected from random associations (r=0). This definition of assortment is commonly used in social evolution studies, and is conceptually analogous to Hamiltonian relatedness^{17,41,42}. Similarly, we calculated the assortment of the competitor strain c as r_{c}=.
Phylogenetic comparative analysis
We gathered data on the presence of putative T6SSs and effectors across proteobacterial genomes from the SecReT6 database^{72}. We restricted our analysis to genera in which there has been experimental verification of the presence of at least one T6SS in at least one strain in SecReT6 (ref. 72). This gave data for a total of 439 genomes from the Proterobacteria and Bacteroidetes of the genera Acidovorax (N=5), Acinetobacter (N=19), Aeromonas (N=4), Agrobacterium (N=4), Azoarcus (N=2), Bacteroides (N=9), Bordetella (N=10), Burkholderia (N=38), Campylobacter (N=24), Citrobacter (N=2), Edwardsiella (N=4), Enterobacter (N=11), Escherichia (N=59), Flavobacterium (N=5), Francisella (N=19), Helicobacter (N=59), Methylomonas (N=1), Myxococcus (N=3), Pectobacterium (N=5), Proteus (N=2), Pseudomonas (N=53), Ralstonia (N=10), Salmonella (N=41), Serratia (N=9), Vibrio (N=22) and Yersinia (N=19). For each genome we also recorded the genome size and secretome size (number of genes coding for secreted proteins) from PSORTdb^{73}. Any T6SS effectors identified in SecReT6 were removed from secretome size counts from PSORTdb to avoid creating a spurious correlation owing to double counting of effectors. Scatterplots of all raw data are shown in Supplementary Fig. 6. To control for the phylogenetic relationships among strains we used the SUPERFAMILY phylogeny^{74}, which we ultrametricised using the chronpl function in ape^{75}. The data and phylogeny used are included in Supplementary Data 1 and 2.
We used a Bayesian phylogenetic mixed model (BPMM) approach to test for an evolutionary association between T6SSs and secretome size. Analyses were implemented in R using the package MCMCglmm^{76}. We treated secretome size as a binomial response variable, expressing it as a proportion of genome size. In all models, we included phylogeny as a random effect to control for the shared evolutionary history of strains, and also included a residual random effect to account for overdispersion. For fixed effects we used an uninformative normally distributed prior with mean 0 and variance of 10^{8}. For the phylogenetic and residual variances, we used an uninformative inverse gamma prior with shape and scale both set to 0.001. We ran all models for 6,000,000 iterations with a burnin of 1,000,000, and thinning interval of 1,000 iterations. We used visual inspection of traces, as well as the GelmanRubin test^{77,78} on three independent chains to assess model convergence. In all cases, the potential scale reduction factor was <1.03. We first fit a model including both the number of T6SSs and number of T6SS effectors as fixed effects (Supplementary Table 2). To test the sensitivity of our results we also fit univariate models with number of T6SSs and number of T6SS effectors as fixed effects in isolation (Supplementary Tables 3 and 4). Finally, to control for the potential of nonlinear scaling of secretome size with genome size we ran a model with number of T6SSs, number of T6SS effectors, and the log of genome size as fixed effects (Supplementary Table 5). In all cases, both the numbers of T6SSs and T6SS effectors show significant associations with secretome size. Statistics quoted are posterior modes, 95% credible intervals, and p_{MCMC} a Bayesian equivalent to the frequentist P value, which is set as twice whichever is smaller of the proportion of posterior samples above or below zero. Values for proportion of variance explained (Rsquared) were calculated following the approach of Nakagawa and Schielzeth^{79} as the proportion of total variance attributable to the variable in question on the link (logit) scale, but removing the term for the intrinsic variance of the binomial distribution as we are interested in prediction at the level of proportion of the genome dedicated to secretions, rather than prediction of whether individual genes code for secretions. Total variance was calculated as the posterior mode of the sum of the residual, phylogenetic and fixed effect variance, with the proportional contributions of each component then determined. In addition, we calculated the proportion of nonphylogenetic variance accounted for by the fixed effects by repeating the calculations while neglecting the phylogenetic variance estimates.
Individual based simulation model
We randomly seeded a 500 × 500 lattice with an equal number of red and blue cells. Every time step, 5% of the cells were randomly chosen to activate their T6SS systems, killing any adjacent (eight cells surrounding the focal cell) cells of the opposite colour. Similarly, 5% of the cells in the landscape were randomly chosen to attempt to reproduce, filling up to one adjacent unoccupied patch with a cell of its colour. Rates of killing and reproduction were chosen to provide sufficient temporal resolution of population dynamics while still being computationally efficient. Reproduction was aborted if all neighbouring patches were occupied. Within each time step, model updates were propagated sequentially across rows, starting with the first position in the upper left corner. This model was coded in Python and is available upon request.
Ising spin model
We randomly seeded a 500 × 500 lattice with an equal number of ‘1’ and ‘−1’ entries, representing ‘up’ and ‘down’ magnetic spins. The eight nearest neighbours of each point in the lattice are summed and multiplied by the entry they circumscribe—if an entry is surrounded by neighbours of its own type, the resultant quantity (the Hamiltonian ‘H’) is 8; if surrounded by neighbours opposite its own type, −8. We randomly choose a spin in the lattice and calculate H_{old} and H_{new}, where H_{new} is the Hamiltonian if the chosen spin were to flip, and H_{old} the Hamiltonian if the spin were to remain unflipped.
Each spin flips independently, so flipping one spin does not change the sum of its eight nearest neighbours. Because the Hamiltonian for each entry, however, is the product of that sum with the entry itself, the net effect is simply to add a minus sign. Thus, the difference between the energy for the flipped spin and the unflipped spin is
With this quantity calculated for our randomly chosen spin, we finally calculate , and compare it to a random number between 0 and 1—if and only if it is larger, the spin will flip^{80}. This process is applied iteratively, and after 500^{2} possible flips, one timestep is said to have passed. In the above discussion, the Hamiltonian is expressed in units of k_{B}*T. This proportionality can be adjusted to modify the speed of convergence to the phaseseparated state; our simulation used a ratio of unity. No external field was applied. This model was coded in Mathematica and is available upon request.
ODE model for wellmixed environment
We will consider the interaction of two strains with different T6SS effectorimmunity pairs so that each strain can kill the other strain upon cell–cell contact. For simplicity, we assume that both strains differ only in their T6SSs, having the same basal growth rate r and density dependent mortality rate s. We allow for asymmetric killing between the strains so that strain A kills strain B at rate α_{BA} and strain B kills strain A at rate α_{AB}. From these assumptions, we can write the change in density of strains A and B in a wellmixed environment as
There are four possible equilibria for this system: a bacteriafree equilibrium at A=0, B=0; two singlestrain equilibria at A=r/s, B=0 and A=0, B=r/s; and a coexistence equilibrium with both strains present at A=α_{AB}r/(s(α_{AB}+α_{BA})+α_{AB}α_{BA}), B=α_{BA}r/(s(α_{AB}+α_{BA})+α_{AB}α_{BA}). Note that at the coexistence equilibrium the ratio of strain A to strain B simply depends on their relative rates of killing (A/B=α_{AB}/α_{BA}). Analysing the Jacobian (J) of the system around the equilibrium we can see that the bacteriafree equilibrium is unstable as long as r>0 (tr(J)=2r,J=r^{2}), while the single strain equilibria are always stable for our assumption of positive rates of killing (tr(J)=−r(s+α_{BA})/s, J=α_{BA}r^{2}/s for A=r/s, B=0, and tr(J) =−r(s+α_{AB})/s, J=α_{AB}r^{2}/s for A=0, B=r/s). Analysing the coexistence equilibrium we can see that both the trace and the determinant at the equilibrium are strictly negative
meaning that this is a saddle node, and is hence unstable. This means that in a wellmixed environment killing by T6SS will always lead to one strain coming to dominate the environment, with the critical ratio of strains A to B at which A eventually dominates decided by its relative rate of killing (A dominates if A/B>α_{AB}/α_{BA}) and vice versa for domination by strain B (Supplementary Fig. 1).
PDE model for spatially extended environment
In order to study the dynamics of strains with different T6SS effectorimmunity pairs in a spatial environment, we use the following system of PDEs
The dynamics are given as before but with a Laplacian operator for the diffusion/dispersal of cells through space, where d is the dispersal/diffusion rate for both strains. We will first consider the stability of a homogenous coexistence equilibrium where the average densities of the strains are
We will consider the effect of a fluctuation in the strain composition in onedimensional space of the form
This fluctuation leads to a change in density of strain A at location x of a sin βx with a corresponding change of −a sin βx in the density of strain B, where a is the amplitude of the fluctuation (which e will assume to be infinitesimally small) and β is the angular frequency. As we are concerned with changes in composition we will denote the density/volume fraction of strain A as ϕ_{A}=A/(A+B) and using equations (3a3b), the rate of change in the density fraction is
Substituting in equations (4a, 4b, 5a5b), , we get
For the compositions to diverge δϕ_{A}/δt must have the same sign as the fluctuation sin βx, as this leads to A increase in frequency in regions of positive fluctuation and B to increase in frequency in regions of negative fluctuation. At any point of zero compositional fluctuation (x=nπ/β, where ), δϕ_{A}/δt=0 and this condition cannot be fulfilled. However, the condition for δϕ_{A}/δt and sin βx to have the same sign in a region of nonzero fluctuation (x≠nπ/β, where ) is
Assuming that the amplitude of the compositional fluctuation is infinitesimally small (a → 0), inequality 8 simplifies to
Therefore, the strain compositions will diverge, with A increasingly dominating in regions of positive compositional fluctuation and B coming to dominate in regions of negative compositional fluctuation, if the fluctuations occur on a sufficiently wide scale relative to bacterial dispersal/diffusion (that is, sufficiently low d and/or β, Supplementary Fig. 2). It is also worth noting that this condition is more easily satisfied for higher rates of killing by both strains (higher α_{AB} and α_{AB}), and for higher values of the basal growth rate (higher r) and lower values of densitydependent mortality (lower s). This highlights that the local demography of killing by T6SS pushes the system towards decomposition even for fluctuations on short length scales, while higher diffusion/dispersal of cells mean that fluctuations of longer length scales are required for decomposition to occur.
Relationship of the PDE model to the AllenCahn equation
The AllenCahn equation governs the density of strains A and B during phase separation absent the conservation of strains A and B. Assuming that the effective diffusion rate d_{Eff} doesnot vary through space (as will be approximately the case when oscillations in composition are of vanishingly small amplitude, a → 0) we can write the AllenCahn equation for strain A as
Where f(ϕ_{A}) is a function of ϕ_{A}. The δf(ϕ_{A})/δϕ_{A} term drives ϕ_{A} to the free energy minimum over time, and can work with or against diffusion. Thus, the effective diffusion is uphill whenever δϕ_{A}/δt and are of opposite sign (that is, whenever the change in volume fraction is of opposite sign to the Laplacian). The Laplacian, is given by
The sign of is, therefore, given by the sign of −sin βx, and it follows that δϕ_{A}/δt and will be of opposite sign, and effective diffusion will be uphill, in the limit of infinitesimally small amplitude fluctuations (a → 0) whenever inequality 9 is satisfied. This highlights the two counteracting forces governing whether decomposition occurs—the diffusion/dispersal of cells favours downhill diffusion, while the local demographic effects of killing mimic uphill diffusion by a strain’s relative growth rate being increased in regions in which it is in the majority.
Adding public goods to the model
To consider the effects of T6SSmediated phase separation on the evolution of cooperation we extend our model so that strain A produces a diffusable public good secretion at rate ρ, while strain B does not invest in its production. We assume that strain A pays a growth rate cost c for production of the secretion. We also assume that the secretion increases each strains growth rate by amount b per unit concentration S by increasing nutrient availability (for example, an exoenzyme digesting a substrate or siderophores binding insoluble iron). From these assumptions, we write the dynamics of the two strains and the secretion concentration as
for a wellmixed, nonspatial system and as
for a spatially extended system, where λ is the decay rate of the secretion, D is its diffusion coefficient, and all other variables are as previously defined. We numerically explore both the nonspatial and spatial systems in the presence and absence of killing in the Fig. 3, and here we analytically explore these four scenarios.
Nonspatial model in the absence of killing
In order to gain analytical insight into our model, we will make the assumption that the dynamics of the public good occur on a much faster timescale than the ecological dynamics so that we may treat the public good as being at equilibrium for any given ecological state of the model. For the nonspatial model this gives us
Substituting this into equation (10) and setting the dynamics of the two strains are
We will assume throughout that s>ρb/λ, such that in the absence of cheaters a population of cooperators has a finite stable equilibrium. Solving for the equilibrium of equations (15a15b), we can see that the only stable equilibrium is B=r/s so long as c>0. This means that, as long as there is a cost of cooperation, cheaters will always outcompete cooperators in a nonspatial environment in the absence of killing.
Nonspatial model with killing
Again using the assumption that public goods dynamics play out on a faster timescale than the ecological dynamics but now setting (that is, assuming symmetric killing) the dynamics of the two strains are
For this system there are two single strain equilibria, one for a pure population of cooperators at
and one for a pure population of cheaters at
The cooperator equilibrium is stable so long as α>c(λs−ρb)/(λr−λc), while the cheater equilibrium is always stable. There is a an unstable saddle point at
which exists whenever α>c(λs−ρb)/(λr−λc). This means that in the presence of killing cooperation can potentially still be a stable outcome. However, the location of the saddle point given is always at a higher value of A than of B (that is, A is always larger than B in equations (19a19b), whenever c>0), meaning that the basin of attraction for strain A is smaller than that for strain B. Thus, though killing can help protect cooperation from cheaters in a nonspatial environment, they are still disfavoured compared with cheaters. More generally, the condition for the cooperative strain to increase in frequency is
Here we see that, in a nonspatial environment killing can help protect cooperators owing to the positive frequency dependence that it induces. However, cooperators are still disfavoured compared with cheaters as the basin of attraction for cheaters is larger (equations (19a19b), ).
Spatial model in the absence of killing
We will consider a onedimensional spatial environment with a fluctuation in the strain composition as given in equations (5a5b), . Using this spatial distribution of strains and assuming that the public goods dynamics happen on a much faster timescale than the ecological dynamics, we can solve for the equilibrium concentration of the public good across space as
Note here that as the diffusion rate of the public good approaches infinity (D → ∞) this simplifies to S(x)=ρA_{0}/λ, and thus has a constant concentration through space, while as the diffusion rate of the public good approaches zero it simplifies to S(x)=ρ(A_{0}+a sin βx)/λ, and is thus simply proportional to the local density of the cooperator strain. Using equations (11, 18a18b), and setting we get
We can evaluate the fitness (per cell growth rate) of each strain as
We can then evaluate when cooperators are favoured by evaluating the inequality w_{A}>w_{B}, which gives
as the condition for cooperators to increase in frequency. This condition is more easily favoured for lower costs of cooperation (low c), higher benefits of cooperation (high b), and with spatial fluctuations that are of large amplitude (high a) and over a wide spatial scale (low β).
Inequality 24 shows that cooperation can be favoured by spatial variance in the population composition. However, in the absence of outside forces or stochastic effects will such spatial fluctuations be maintained? To answer this, we first evaluate an expression for the change in the density/volume fraction of the cooperator strain giving
which is always negative, meaning that the cooperator strain is always locally decreasing in frequency, even when globally increasing in frequency. This means that the change in cooperator frequency cannot match the sign of the spatial fluctuation , and thus in the absence of external forces any spatial fluctuations in composition will be lost. As we can see from inequality 24 as the amplitude of the fluctuation decays to zero (a approaches 0) cooperation cannot be favoured, and thus without killing, external perturbations to the system or stochastic effects are required to maintain the spatial structure necessary to maintain cooperation.
Spatial model with killing
We now set and evaluate the consequences of the combination of killing and a spatial environment of the dynamics of cooperation. From our assumptions, the dynamics are now given by
We can again evaluate the fitness of each strain as in equations (23a23b), , and derive the condition for the cooperative strain to be favoured (w_{A}>w_{B}) as
Note that if the cooperator and cheater strains are at equal average density (A_{0}=B_{0}) this simplifies to the inequality given in 24. As the amplitude of the spatial fluctuation must be less than the average density of the less abundant of the two strains the condition given in inequality 27 is more easily satisfied than that in inequality 24 whenever A_{0}>B_{0} and less easily satisfied whenever A_{0}<B_{0}. This occurs owing to the positive frequency dependence introduced by killing. However, as we will show, unlike in a spatial environment without killing, killing in a spatial environment can increase structuring, thus further favouring cooperation.
Comparing inequality 27 (condition for cooperation to increase in frequency in a spatial environment with killing) with inequality 20 (condition for cooperation to increase in frequency in a nonspatial environment with killing), we can also see that the presence of spatial heterogeneity means that the cooperator can be favoured even when numerically less abundant in a spatial environment with killing (that is, inequality 27 can be satisfied when inequality 20 is not). This occurs as spatial heterogeneity can allow cooperators to disproportionately gain the benefits of cooperation compared with cheaters.
Finally, we will show that killing causes phase separation in a spatial environment when one strain is a cooperator and the other a cheat. We follow the same approach as before and consider whether a spatial fluctuation around the homogenous coexistence equilibrium given in equations (17a17b), will be amplified, which can be evaluated by considering if δϕ_{A}/δt has the same sign as the fluctuation sin βx. At any point of zero compositional fluctuation (x=nπ/β, where ), δϕ_{A}/δt=0 and this condition cannot be fulfilled. However, the condition for δϕ_{A}/δt and sin βx to have the same sign in a region of nonzero fluctuation (x≠nπ/β, where ) in the limit of an infinitesimally small amplitude for the compositional fluctuation (a→0) is
This shows that killing promotes phase separation of cooperator and cheater strains in a spatial environment which favours cooperators of cheats as shown in inequality 27.
Taken together, these results show that: (1) Killing in a nonspatial environment can protect cooperators from cheats when the cooperator is at higher abundance. However, cooperators cannot invade from rarity; (2) Heterogeneity in a spatial environment without killing can favour cooperators over cheats. However, in the absence of any external forces or stochastic effects (for example., bottlenecking during range expansion) this structure will ultimately be lost, allowing cheaters to win; and (3) In a spatial environment with killing phase separation can occur, protecting cooperators from cheats and potentially allowing them to invade from rarity.
Parameter values for numerical simulations of PDE model
Equations were numerically evaluated in R. Parameter values for figures and videos are as follows: Figs 1 and 2 and Supplementary Movie 2: r=2, s=2, α_{AB}=0.5, α_{BA}=0.5, d=0.01; Supplementary Fig. 5: r=2, s=2, α_{AB}=0.5, α_{BA}=0.5; Supplementary Movie 4: r=2, s=2, α_{AB}=0.5, α_{BA}=0.5, and d as indicated in the panels; Fig. 3 and Supplementary Movie 5: r=2, s=2, α_{AB}=0.5, α_{BA}=0.5, d=0.01, b=1.9, c=0.1, D=0.1, λ=100, ρ=100.
Code availability
All code is available from the authors upon request.
Data availability
All data sets generated or analysed during this study are included in this published article (and its Supplementary Information files) or are available from the corresponding authors upon request.
Additional information
How to cite this article: McNally, L. et al. Killing by Type VI secretion drives genetic phase separation and correlates with increased cooperation. Nat. Commun. 8, 14371 doi: 10.1038/ncomms14371 (2017).
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Acknowledgements
This work was supported by NSF grant #DEB1456652 to W.R. and NASA Exobiology grant #NNX15AR33G to W.R.; by the Gordon and Betty Moore Foundation grant #4308.07 to B.H., NSF grant #MCB1149925 to B.H. and Wellcome Trust grant #WT095831 to S.B. L.M. was supported by HFSP grant #RGP0011/2014 to S.B. We would like to thank David Yanni, Jonathan Michel, Shane Jacobeen and Wouter Ellenbroek for helpful comments.
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All authors actively participated in the planning of this research. E.B., J.T. and B.H. constructed the Vibrio mutants and performed all bacterial experiments. E.B., J.T., P.J.Y. and A.K. performed and analysed the microscopy. L.M. and P.J.Y. wrote and analysed the PDE model. L.M. performed the phylogenetic analysis. W.R. wrote the individualbased model and script to calculate assortment, P.J.Y. and A.K. wrote the Ising spin model and script to calculate S(q). W.R., L.M., E.B. and P.J.Y. wrote the first draft of the paper, all authors contributed to revision.
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Supplementary information
Supplementary Information
Supplementary Figures, Supplementary Tables and Supplementary References (PDF 1088 kb)
Supplementary Data 1
Data used in phylogenetic comparative analysis. (ZIP 6 kb)
Supplementary Data 2
Phylogenetic tree for Figure 4a. (TXT 27 kb)
Supplementary Movie 1
Timelapse of the individual based model. Both strains kill with equal probability. Each timestep, 5% of cells attempt to kill neighboring nonkin, and 5% of cells attempt to reproduce into an adjacent open patch. (MOV 16244 kb)
Supplementary Movie 2
Timelapse of the PDE model. Cellular mobility, d, is 0.001. Other parameter values are r = 2, s = 2, α_{AB} = 0.5, α_{BA} = 0.5. (MOV 2958 kb)
Supplementary Movie 3
Timelapse of the Ising spin model. Initial conditions correspond to infinite temperature, or completely random spins. Thermodynamic beta was set to unity and the eight nearest neighbors were considered. One generation is equivalent to an average of one spinflip opportunity per site. (MOV 10262 kb)
Supplementary Movie 4
Variation in cellular mobility, d, within the PDE model. Higher rates of mobility delay the formation of clonal patches, but increase the rate of phase separation. (MOV 4063 kb)
Supplementary Movie 5
Public goods cooperation in the PDE model. In the leftmost panel cooperator frequency of 0 is represented by red and frequency of 1 by blue. In the center panel the concentration of public goods (produced by the cooperator) is indicated by intensity of yellow. In the rightmost panel the cell density is indicated by intensity of green. Parameter values are r = 2, s = 2, α_{AB} = 0.5, α_{BA} = 0.5, d = 0.01, b = 1.9, c = 0.1, D = 0.1, λ = 100, ρ = 100. (MOV 10624 kb)
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McNally, L., Bernardy, E., Thomas, J. et al. Killing by Type VI secretion drives genetic phase separation and correlates with increased cooperation. Nat Commun 8, 14371 (2017). https://doi.org/10.1038/ncomms14371
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