According to the principle of the ‘survival of the fittest’, selfish individuals should be better off compared to peers that cooperate with each other. Indeed, even though a population of organisms benefits from working together, selfish members can exploit the cooperative behavior of others without doing their part. These ‘cheaters’ then use their advantage to reproduce faster and take over the population.

Yet, social cooperation is widespread in the natural world, and occurs in creatures as diverse as bacteria and whales. How can it arise and persist then? One idea is that when individuals form distinct groups, the ones with cheaters will perish. Even though a selfish individual will fare better than the rest of its team, overall, cooperating groups will survive more and reproduce faster; ultimately, they will be favored by evolution. This is called group selection.

Here, Uppal and Vural examine how the physical properties of the environment can influence the evolution of social interactions between bacteria. To this end, mathematical models are used to simulate how bacteria grow, evolve and drift in a flowing fluid. These are based on equations worked out from the behavior of real-life populations.

The results show that flow patterns in a fluid habitat govern the social behavior of bacteria. When different regions of the fluid are moving at different speeds, ‘shear forces’ are created that cause bacterial colonies to distort and occasionally break apart to form two groups. As such, cooperative groups will rapidly form new cooperating colonies, whereas groups with cheaters will reproduce slower or perish.

Furthermore, results show that when different areas of the fluid have different shear forces, social cooperation will only prevail in certain places. This makes it possible to use flow patterns to fine tune social evolution so that cooperating bacteria will be confined in a certain region. Outside of this area, these bacteria would be taken over by cheaters and go extinct.

Bacteria are both useful and dangerous to humans: for example, certain species can break down pollutants in the water, when others cause deadly infections. These results show it could be possible to control the activity of these microorganisms to our advantage by changing the flow of the fluids in which they live. More broadly, the simulations developed by Uppal and Vural can be applied to a variety of ecosystems where microscopic organisms inhabit fluids, such as plankton flowing in oceanic currents.

Cooperation is the cement of biological complexity. A combined investment brings larger returns. However, while cooperating populations are fitter, individuals have evolutionary incentive to cheat by taking advantage of available public goods without contributing their own. Avoiding the cost of these goods allow larger reproduction rates, causing cheaters to proliferate until the lack of public goods compromise the fitness of the entire population. In other words, while cooperating populations are fitter than non–cooperating ones, cooperation is not evolutionarily stable. How then can social behavior emerge and persist in microbial colonies?

The evolution of cooperation is an active field of research, with multiple theories resolving this dilemma (Axelrod and Hamilton, 1981; Sachs et al., 2004; Sachs and Simms, 2006; Nowak, 2006). According to (Fletcher and Doebeli, 2009) the fundamental mechanism is assortment. That is, in order for cooperation to evolve, cooperators and cheaters must experience different interaction environments.

How this assortment is achieved is a central question. Possibilities include positive and negative reciprocity (Trivers, 1971; Clutton-Brock and Parker, 1995; El Mouden et al., 2010), where cooperators are rewarded later by others, or where cheaters are inflicted a cost, via policing or reputation. For example, quorum signals reveal whether available public goods add up to the population density. In this case, altruists cut back public good production to eliminate cheaters (albeit with collateral damage) (Allen et al., 2016; Sandoz et al., 2007; Diggle et al., 2007). Another idea is group selection (Wynne-Edwards, 1962; Haldane, 1932; Traulsen and Nowak, 2006; Wilson, 1975) and its modern incarnation, multi-level selection, (Wilson and Sober, 1994) which propose that cooperating groups (or groups of groups) will reproduce faster than non-cooperating ones and prevail. Kin-selection theory (Hamilton, 1964a, 1964b; Williams, 1966; Smith, 1964; Hamilton, 1975; Lion et al., 2011) provides a mechanism that arises from individual level dynamics. Kin-selection proposes that individuals cooperate with those to which they are genetically related, and thus, a cooperative genotype is really cooperating with itself.

Hamilton conjectured that kin selection should promote cooperation if the population is viscous, that is when the mobility of the population is limited (Hamilton, 1964a; Hamilton, 1964b). This helps ensure that genetically related individuals cooperate with each other. However, competition within kin can inhibit altruism (Taylor, 1992; Wilson et al., 1992). One solution to this is if individuals disperse as groups, also known as budding dispersal. This was shown to promote cooperation theoretically by Gardner and West (2006) and demonstrated experimentally by (Kümmerli et al., 2009). Budding dispersal has also been studied by (Pollock, 1983; Goodnight, 1992; Kelly, 1994) and by (Wilson et al., 1992) from a group selection perspective.

There may be multiple and overlapping mechanisms underlying assortment. There is much debate in the literature over which theories best explain the evolution of cooperation and under which conditions each theory may be applicable. There is still not agreement, for example, on whether kin-selection and group selection can be viewed as equivalent theories (Lion et al., 2011; Kramer and Meunier, 2016). According to (Simon et al., 2012; Simon et al., 2013), since relatedness need not impact certain group level selection events, such as various games between groups, group selection is distinct from kin selection. Also, individual and group level selection events are generally asynchronous in nature and therefore cannot be equivalent. However, the debate still goes on (Kramer and Meunier, 2016; West et al., 2007; Gardner, 2015; Goodnight, 2015).

Typically, evolution of cooperation is quantitatively analyzed with the aid of game theoretic models applied to well-mixed populations, networks and other phenomenological spatial structures (Szabó and Fáth, 2007; Allen et al., 2013; Nowak and Sigmund, 2004; Vural et al., 2015). While there are few models that take into account spatial proximity effects, (Medvinsky et al., 2002; Nadell et al., 2010; Nadell et al., 2013; Dobay et al., 2014; Driscoll and Pepper, 2010) and the influence of decay and diffusion of public goods (Dobay et al., 2014; Wakano et al., 2009; Hauert et al., 2008), how advective fluid flow influences social evolution remains mostly unexplored. The present study aims to fill this gap.

A flowing habitat can have a drastic effect on population dynamics (Tél et al., 2005; Nickerson et al., 2004; Koshel' and Prants, 2006; Sandulescu et al., 2008). For example, a flowing open system can allow the coexistence of species despite their differential fitness (KarolyiKárolyi et al., 2000). Interactions between fluid shear and bacterial motility has been shown to lead to shear trapping (Rusconi et al., 2014; Berke et al., 2008) which causes preferential attachment to surfaces (Berke et al., 2008; Li et al., 2011). Turbulent flows can also lead to a trade-off in nutrient uptake and the cost of locomotion due to chemotaxis (Taylor and Stocker, 2012), and can drastically effect the population density (Pigolotti et al., 2012; Perlekar et al., 2010). Most importantly, the reproductive successes of species (and individuals within a single species) are coupled over distance, through the secretion of toxins, goods, and signals (Mimura et al., 2000; Allison, 2005; Hibbing et al., 2010). The spatial distributions of all such fitness altering intermediaries depend on flow. Indeed, the experimental study by Drescher et al. (2014) has shown that flow can help promote cooperation in bacterial biofilms. Thus, we are motivated to find out how flow plays a role in the evolution of cooperation.

Here we theoretically study how fluid dynamics molds the social behavior of a planktonic microbial population. Qualitatively stated, our evolutionary model has three assumptions: (1) Individuals secrete one waste compound and one public good. The former has no cost, whereas the latter does. (2) Mutations can vary the public good secretion rates of microbes, thereby producing a continuum of social behavior. (3) Microbes and their secretions diffuse and flow according to the laws of fluid dynamics.

Under these assumptions, we find, through computer simulations and analytical theory, that bacteria self organize and form patterns of spots, which then exhibit an interesting form of budding dispersal when sheared by ambient fluid flow. The dispersal process preserves the group structure, thereby enabling evolutionarily stable social behavior.

Our model is applicable to a wide variety of social ecosystems ranging from phytoplankton flowing in oceanic currents to opportunistic bacteria colonizing blood or industrial pipelines. Our findings imply that greater social complexity amongst planktonic species would be observed in regions of large shear, such as by rocks and river banks. We might even speculate that multicellularity may have originated near fluid domains with large shear flow, rather than the bulk of oceans or lakes.

This paper is organized as follows: We first establish that under certain conditions our physical model gives rise to spatially organized cooperative structures. The structures are a natural byproduct of the dynamics of the system. Furthermore, these social structures reproduce in whole to form new identical structures. Variants of such structures have already been studied in ecological settings (Tian et al., 2011; Camara, 2011; Baurmann et al., 2007; Wilson et al., 2003) and growth patterns of microbial populations havebeen explored (Ben-Jacob et al., 1994; Chang-Li et al., 1988). We then study the effects of mutation. We first start with a simplified model with only two phenotypes: cheater and altruist. We then generalize to a continuum of public good secretion rates. In both cases, we observe that above a certain mutation rate, cheating strains take over groups which leads to total extinction. The latter finding is consistent with other empirical (Rainey and Rainey, 2003; Diggle et al., 2007) and theoretical studies (Nowak and May, 1992).Through the fragmentation of social groups, and death of cheating groups, we recover the results of Simpson’s paradox (Chuang et al., 2009) where individual groups may decrease in sociality, but the population as a whole becomes more social.

After setting up the stage for naturally forming social groups, we present our central and novel finding, that flow shear can lead to evolutionarily stable cooperative behavior within the population. Specifically, we demonstrate and study the evolution ofsociality of a microbial population (1) subjected to constant shear, (2) embedded in a cylindrical laminar flow and (3) in a Rankine vortex. We find that in all three cases population density and cooperative behavior scales with flow shear.

The mechanism of action is that shear distortion enhances the fragmentation of cooperative clusters, thereby increasing the group fragmentation rate and limiting the spread of cheaters. If the shear is large enough that groups are torn apart at a larger rate than the mutation rate, then cooperation will prevail. Otherwise, groups will become dominated by cheaters, and eventually die out (Figure 2, Videos 1–3).

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We study a physically realistic spatial model of microorganisms, where fluid dynamical forces contribute significantly to the evolution of their social behavior. Our analysis consists of simulations and analytical formulas.

We simulate microbes as discrete particles subject to stochastic physical and evolutionary forces, and the compounds secreted by microbes as continuous fields. In contrast, our analytical expressions are derived from analogous equations that are entirely continuous and deterministic. In general, we should not expect the discrete simulations to perfectly be described by the continuous set of partial differential equations. Nevertheless, the continuous system of equations do allow us to obtain relevant quantities such as group size and group fragmentation rate to a good approximation (cf. appendix 1 and Figure 3).

In our model, the microorganisms secrete two types of diffusive molecules that influence each other’s fitness (Figure 1). The first molecule, the concentration of which is denoted by $c_1(x,t)$, is a beneficial public good that increases the fitness of those exposed, whereas the second, $c_2(x,t)$, is a waste compound or toxin that has the opposite effect, and effectively acts as a volumetric carrying capacity. The continuous equations that represent our system are

Here $n$ is a shorthand for $n(x,t,s_1)$, the number density of microbes at time $t$ and position $x$ that produce the public good at a rate of $s_1$. These microbes pay a fitness cost of ${\beta }_1{s}_1$ per unit time. The production rate of waste $s_2$ on the other hand, is assumed constant for all, and has no fitness cost. Waste limits the number of individuals that a unit volume can carry. Microbes secreting public goods at a rate $s_1$ replicate to produce others with the same secretion rate. This reproduction rate is given by the square bracket. However, the production rate $s_1$ can change due to mutations. This is described by the last term of Equation 1. Mutations can be thought as diffusion in $s_1$ space.

Figure 1

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In all three equations the first two terms describe diffusion and advection, while the last two terms of Equation 2 and Equation 3 describe the production and decay of chemicals. The first two terms in the square bracket describe the effect of the secreted compounds on fitness. This saturating form is experimentally established and well understood (Monod, 1949). The crucial third term in the square bracket describes the cost of producing the public good, which increases linearly.

Social groups as turing patterns

Diffusion can cause an instability that leads to the formation of intriguing patterns (Turing, 1990), which among other fields, have been investigated in ecological context (Tian et al., 2011; Camara, 2011; Baurmann et al., 2007; Wilson et al., 2003). These so called Turing patterns typically form when an inhibiting agent has a diffusion length greater than that of an activating agent. For our model system, the waste compound and public good play the role of inhibiting and activating agents, and patterns manifest as cooperating microbial clusters Figure 2. The size and reproduction rate of these clusters, in terms of system parameters, can be estimated from a Turing analysis (cf. appendix 1). Figure 3 shows the values of diffusion constants that gives rise to Turing patterns, as well as the size of the groups.

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In the homogeneous phase, the system is evolutionarily very unstable, since as soon as one cheating mutant emerges, it quickly takes over the entire population, ultimately causing the population to go extinct (Video 1). The group phase tolerates cheaters better, since once a cheater emerges it will take over and compromise the fitness of only one group, while the others will live on. However, in the absence of group fragmentation, novel cheating mutations will ultimately emerge in all groups, and annihilate the population one group at a time (Video 2).

The main contribution of this paper is to demonstrate that stable social cooperation can be induced or enhanced by fluid flow gradients. Specifically, shear forces induced by advective flow distorts and fragments microbial clusters, leading to a kind of budding dispersal, which in turn enables evolutionary stable cooperation (Figure 2, Video 3).

It is well known that evolutionary outcomes can depend on individuals being discrete (Durrett and Levin, 1994). In our model, having a continuous population density can allow for the existence of ‘micro-mutant populations’ which can spread easier between adjacent groups. The discreteness further separates the clusters of microbes from each other, since there cannot exist fractional individuals. In reality microbes are quantized, and we thus expect a discrete simulation to better model the biology. In Figure 3 we present the phase diagram of the system, as obtained by analytical theory, discrete agent based simulations (where microbes are discrete, self-replicating brownian particles), and continuous simulations (where Equations 1,2,3 are solved numerically). In order not to obfuscate the biology, we report our detailed mathematical treatment in the appendix.

Evolution of sociality in constant shear for a binary phenotype

We first look at a simplified system with just two phenotypes, cheater and altruist, to gain a basic understanding of the mechanism involved. Mutations can cause a switch in social behavior.

To see the effect of shear on social evolution, we introduced Couette flow to the microbial habitat. In this case, the flow velocity takes the form

${\displaystyle \mathbf{v}\left(\mathbf{x}\right)=v_{max}\frac{r}{R}\hat{z},}$

where $R$ is the radius of the pipe, and $\widehat{z}$ is the longitudinal direction.

The shear rate is the derivative of the flow velocity and is related to the maximum flow rate $v_{\text{max}}$,

$\sigma =\frac{dv}{dr}=\frac{v_{\text{max}}}{R}.$

We ran simulations for various shear rates and diffusion constants and observed that shear does not significantly influence the region of parameter space that gives rise to cooperating groups. However if the system parameters are conducive to the formation of groups, shear tears groups apart and increases the rate at which spatially distinct cooperative clusters form.

We find that the group fragmentation rate $\omega \left(\sigma \right)$, depends linearly on the shear rate $\sigma$ (Figure 4)

$\omega \left(\sigma \right)=m\sigma +{\omega }_0,$

where ${\omega }_0$ is the fragmentation rate solely due to microbial diffusion and can approximately be given by the Turing eigenvalue ${\omega }_0\approx {\Lambda }_{\text{max}}$ (see appendix). The constant of proportionality $m$ is given empirically from our simulations and depends on diffusion lengths and group density. This holds in the low density regime. Once the population density becomes large, group-group interactions slow the group reproduction rate and the population saturates.

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Larger shear corresponds to a faster rate of group fragmentation, thus enabling or enhancing social behavior in the microbial population. Given the group size and fragmentation rate as a function of shear, we can calculate roughly where the critical shearrate is for sociality. We also find that the group population $N$ increases linearly with shear,

$N\left(\sigma \right)=n\sigma +N_0.$

where $n$ and $N_0$ are the slope and intercept of the line in Figure 4B. This also influences where the critical shear rate will be.

Cooperation is stable if a group is able to fragment before a cheating strain emerges and proliferates in the group. Therefore, for stability, we need the take-over time to exceed the time it takes for a group to reproduce. A mutant emerges at a rate of $\mu N\left(\sigma \right)$ (we emphasize that $\mu$ is not the generic mutation rate, but the rate at which a particular social gene mutates). Once a mutant emerges, it takes some time ${\tau }_d$ to spread to where the daughter group forms. ${\tau }_d$ will depend on where the mutant first emerges. Assuming a uniform distribution, and taking the diffusion time in two dimensions as a function of radius $r$, ${\tau }_d\left(r\right)=r^2/4d_b$, we obtain

$<{\tau }_d>={\displaystyle \underset{0}{\overset{R}{\int }}}\frac{r^2}{4d_b}\frac{2r}{R^2}\text{d}r=\frac{R^2}{8d_b},$

where $R$ is the group radius. The take-over rate is then given by taking the inverse of the total take-over time, and the critical shear rate ${\sigma }_c$ necessary for social cooperation is given by equating the take-over rate with the reproduction rate,

(4) $\omega \left({\sigma }_c\right)={{\displaystyle \left[\frac{1}{\mu N\left({\sigma }_c\right)}+<{\tau }_d>\right]}}^{-1}.$

The critical shear rate ${\sigma }_c$ above which the system can maintain stable cooperation is then given by the positive root,

(5) ${\sigma }_c=\frac{-b_{\sigma }+\sqrt{b_{\sigma }^2-4a_{\sigma }{c}_{\sigma }}}{2a_{\sigma }}$

where ${\displaystyle a_{\sigma }=<{\tau }_d>\mu mn,b_{\sigma }=<{\tau }_d>\mu n{\omega }_0+<{\tau }_d>\mu m{N}_0+m-\mu n,}$ and $c_{\sigma }=<{\tau }_d>\mu N_0{\omega }_0+{\omega }_0-\mu N_0$.

The values obtained from Equation 5 is indicated by the vertical dashed lines in Figure 4 and agrees with the computationally observed critical shear reasonably well. It may be possible to improve this formula further by taking into account additional factors, such as the non-uniform spatial distribution of population within a group and the elongation of groups with shear. Furthermore, as the mutants increase in numbers it becomes more likely that one of them crosses over the daughter group, thereby reducing further the expected $<{\tau }_d>$. We see better agreement with analytical theory and simulations at lower mutation rates, since these corrections are mainly to the diffusion time $<{\tau }_d>$, and become more significant at higher mutation rates, where $<{\tau }_d>\gg 1/\mu N$, (cf. Equation 4).

If shear is below the critical value (Equation 5), the system will be in a non-social state. Ultimately, cheaters will take over, and wipe out all groups. When shear is increased above the critical value however, the system will transition to a stable social state, thereby maintaining its fitness and dense population indefinitely. Figure 4 shows the long-time population of the system versus the shear rate. The population goes extinct under larger mutation rates unless the shear rate is above the critical value.

When is shear necessary, and when is it just a sufficient condition for cooperation? By setting ${\sigma }_c=0$ in Equation 5 and solving for $\mu$ we can also obtain the critical mutation rate above which shear is necessary in order to have social cooperation. We get ${\mu }_c=6.9\times {10}^{-7}$ analytically and our simulations show a critical mutation rate around ${\mu }_c=5.5\times {10}^{-7}$.

Evolution of sociality in constant shear for a continuum of phenotypes

As we will see, we obtain similar results when the available phenotypes include a continuum of social behaviors. In this case, a mutation changes the secretion rate of a microbe by a uniformly chosen random number between 0 and 1 ${\text{s}}^{-1}$.

We observe from our simulations, that mutations that increase the secretion rate of a microbe do not fixate, since the microbe now pays a higher cost and is less fit than its neighbors. However, once a mutation that lowers the secretion rate of a microbe occurs within a group, it quickly takes over the entire group, leaving individual groups homogeneous in secretion rate.

We show the diversity of social behaviors across groups and within individual groups in greater detail in Appendix 1—figure 1. Since less cooperative phenotypes always dominate more cooperative phenotypes, we find no diversity of social behavior within a group. However we do see a large variation across groups, which increases with shear.

Groups with different secretion rates reproduce at different rates. Groups with too low of a secretion rate are not stable and die off. In general, the system will evolve to a distribution of groups with secretion rates centered around the value of secretion rate that maximizes the group reproduction. For larger mutation rates, the system will tend towards lower average secretion rate and/or go extinct. The average secretion rate of the population can be maintained at a higher value by introducing shear flow, Figure 5.

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We therefore have the same qualitative result as in the two phenotype case, if shear is below some critical value, the system will be in a non-social state. Ultimately, cheaters will take over, and wipe out all groups. As before, the social state of the population can transition from a non-cooperative state to a cooperative one with increased flow shear.

Evolution of sociality in a flowing pipe

We now further generalize our results by looking at laminar flow with fixed boundaries, and with a continuum of public good secretion rates. For Hagen-Poiseuille flow, the shear rate varies linearly with the radius, taking its maximum value adjacent to the boundaries, when $r=R$. The flow and shear profiles are given as,

${\displaystyle v=v_{max}\left(1-\frac{r^2}{R^2}\right),\frac{\text{d}v}{\text{d}r}=-2v_{max}\frac{r}{R^2}.}$

We therefore expect to see groups fragment quicker at the boundary, leading to larger cooperation, higher average secretion rate, and larger population, which is indeed what we do see (Figure 6, Video 4).

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Earlier studies have proposed and shown that shear trapping due to the interaction between bacterial motility and fluid shear can result in preferential attachment to surfaces, (Rusconi et al., 2014; Berke et al., 2008; Li et al., 2011). In a similar spirit, we suggest that inhabiting surfaces may have the additional advantage of enhanced sociality, due to shear driven group fragmentation and dispersal.

Evolution of sociality in vortices

In a vortex, the region above the critical shear value constitutes an annulus. Thus, we expect social behavior to be localized. Any point in the fluid outside this annulus will be taken over and destroyed by cheaters. In our simulations, at steady state we indeed see clusters whirling around exclusively within annulus, neither too near, nor too far from the vortex core (Video 5). Life cannot exist outside this annulus, as cheaters kill these groups.

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The Rankine vortex in two dimensions is characterized by a vortex radius $R$ and a rotation rate $\Gamma$. The shear rate acting on a group acts tangential to the flow. The velocity profile and shear magnitude are given as,

${\displaystyle \mathbf{v}=\{\begin{array}{cc}\frac{\mathrm{\Gamma }r}{2\pi R^2}\hat{\theta },& r\le R\\ \frac{\mathrm{\Gamma }}{2\pi r}\hat{\theta },& r>R\end{array}\sigma =\{\begin{array}{cc}0,& r\le R\\ \frac{\mathrm{\Gamma }}{2\pi r^2},& r>R\end{array}}$

where $r^2=x^2+y^2$.

The shear rate is then a maximum at the minimum value of $r$ which occurs at the vortex radius $R$. We therefore expect to see the largest concentration of groups at the vortex radius, which is what we observe in our simulations (Figure 6).

It is well known that spatial structure is crucial in the evolution of cooperation, (Wilson et al., 1992; Taylor, 1992; Lion and Baalen, 2008). Many of these studies introduce these mechanisms ‘manually’, for example density regulation and migration are enforced by applying carrying capacities and migration rates to groups. In this study we have distanced ourselves from the typical game theoretic abstractions used to investigate evolution of cooperation. Instead we adopted a mechanical point of view. We investigated in detail, the fluid dynamical forces between microbes and their secretions, to understand how cooperation evolves among a population of planktonic microbes inhabiting in a flowing medium. In our first principles model, the spatial structuring and dispersion occur naturally from the physical dynamics.

We found that under certain conditions, microbes naturally form social communities, which then procreate new social communities of the same structure. More importantly, we discovered that regions of a fluid with large shear can enhance the formation of such social structures. The mechanism behind this effect is that fluid shear distorts and tears apart microbial clusters, thereby limiting the spread of cheating mutants. Our proposed mechanism can be seen as shear flow enhanced budding dispersal. This can also be viewed under the phenomenon of Simpson’s paradox (Chuang et al., 2009) where individual groups may decrease in sociality, but the population as a whole becomes more social.

In our investigation, we found only certain regions of the fluid domain admits life, social or otherwise, as governed by the domain geometry and flow rate. From this perspective, it appears that evolution of sociality is a mechanical phenomenon.

In our physics-based model, groups emerge from individual-level dynamics and selection. Groups with cheaters are negatively selected, and give way to those without cheaters. On the other hand the ensemble of groups do not exhibit any variation in their propensity to progenerate cheaters, neither is such propensity heritable. Rather, the progeneration of cheating is a (non-genetic) symptom that inevitably manifests in every group that has been around long enough. In this sense, it might be appropriate to view the emergence and spread of cheaters in a microbial population as a phenomenon of ‘aging’, in the non-evolutionary and mechanical sense, that any system consisting of a large number of interdependent components will inevitably and with increasing likelihood, fall apart (Vural et al., 2014).

The analytical conclusions we derive from our system (Equations 1,2,3) has been guided and supplemented by an agent based stochastic simulation. Videos of simulations are provided in Videos 1–5. Our simulation algorithm is as follows: at each time interval, $\Delta t$, the microbes (1) diffuse by Brownian motion, with step size $\delta$ derived from the diffusion constant and a bias dependent on the flow velocity, $\delta =\sqrt{4d_b\Delta t}+v\Delta t$, (2) secrete chemicals locally that then diffuse and advect using a finite difference scheme, and (3) reproduce or die with a probability dependent on their local fitness given by $f=\Delta t\left[{\alpha }_1\frac{c_1}{c_1+k_1}-{\alpha }_2\frac{c_2}{c_2+k_2}-{\beta }_1{s}_1\right]$. If $f$ is negative, the microbes die with probability 1, if $f$ is between 0 and 1 they reproduce with probability $f$, and if $f$ is larger than 1, they produce number of offspring given by the integer part of $f$ and another with probability given by the decimal part of $f$. Upon reproduction, random mutations may alter the secretion rate of the public good –and thus the reproduction rate– of the microbes. Mutations occur with probability $\mu$ and can change the secretion rate by a random number between $0$ and $s_1$. The secretion rate is assumed to be heritable, and constant in time. Numerical simulations for figures were performed by implementing the model described above using the Matlab programming language and simulated using Matlab (Mathworks, Inc.). The source code for discrete simulations is provided in Source code 1 and the source code for continuous simulations used in Figure 3B is provided in Source code 2.

A summary of the system parameters is given in Table 1, along with typical ranges for their values used in the simulations. The relevant ratios of parameters are consistent with those observed experimentally (Kim, 1996; Ma et al., 2005).

Here we describe a number of simplifying assumptions and following mathematical analysis that allow us to make sense of our simulation results, both qualitatively and quantitatively. Our approach is the standard Turing analysis commonly used to describe pattern formation in reaction diffusion systems. We first obtain a steady state solution, that is find out the population density and concentration of the public good and waste compound that leads to an equilibrium state. We then linearize the system around this equilibrium, and find out what kind of perturbations destabilize this equilibrium. The spotty patterns that form upon destabilization biologically correspond to cooperating microbial communities, the size of which we obtain in terms of system parameters. Our analytical outcomes are compared to discrete simulations in Figure 3.

Our simulations start with the whole population having a given secretion rate. The Turing analysis conducted below is then to find the group size and reproduction rate with this secretion rate.

Appendix 1—figure 1

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We also observe in our simulations, that groups are typically homogeneous in terms of secretion rate; that is once a cheating mutation occurs in a group, it fixates and quickly takes over the group, on a time scale much quicker than the group fragmentation (Appendix 1—figure 1). In other words, while the populations in different groups may have different secretion rates, the secretion rate within one group is approximately uniform. We can therefore simplify our equations to determine the spatial structure of a group in terms of the secretion rate of its constituents, even after mutations occur. To determine the native group size and reproduction rate (i.e. when there is no shear and mutation) we can now write down a simplified set of equations,.

Through some rescaling, we can non-dimensionalize the system. If we define the rescaled variables as

then our equations become,

We will now obtain steady states and conditions for linear stability. We first obtain the steady states in the absence of diffusion and investigate the stability of the system. The steady states are given by

This gives, either the trivial solution $n^{\ast }=c_1^{\ast }=c_2^{\ast }=0$, or the solutions:

where ${\displaystyle a=\left({\alpha }_1-{\alpha }_2-{\beta }_1{s}_1\right)s_1{s}_2,b={\alpha }_1{s}_1-{\alpha }_2{s}_2-{\beta }_1{s}_1\left(s_1+s_2\right),}$ and $c=-{\beta }_1{s}_1$. For this to be a sensible solution, we require $n^{\ast }$ to be real and positive. This also imposes conditions on the system parameters.

Next, we establish the local stability of this solution by perturbing the system away from the steady state and expand up to first order. We take the perturbation $w={(n,c_1,c_2)}^T-{(n^{\ast },c_1^{\ast },c_2^{\ast })}^T$, and substitute it into our system to get the linear system

With the stability matrix, $A$, given as,

 where

The system is stable if the eigenvalues, $\Lambda$ of this matrix have a negative real part. The characteristic polynomial for the eigenvalues is given as ${\Lambda }^3+A_0{\Lambda }^2+B_0\Lambda +C_0=0$, where

Since the first two coefficients of the characteristic equation are positive, by Descartes’ rule of signs, in order to get only negative real part eigenvalues, we need $B_0$ and $C_0$ to be positive as well. This is a requirement for linear stability. Thus, we have the conditions

Next we include diffusion and analyze the instability caused by diffusion. Fourier expanding the solution

and plugging this into our equations, we get the eigenvalue equation, $(-k^{2}D+A)W=\Lambda W$, where

Solving for the eigenvalues again gives a characteristic equation of the form ${\Lambda }^3+A{\Lambda }^2+B\Lambda +C=0$, where now

and the $k^2$ dependent functions are,

We can again use Descartes’ rule of signs, this time looking for an instability, which will happen when $\Psi \left(k^2\right)$ is sufficiently negative. To be precise, the range of unstable wavenumbers satisfy,

At the critical values for the diffusion parameters, the function $\Psi \left(k^2\right)+C_0$ only vanishes at one point, the local minumum of $\Psi \left(k^2\right)$. This occurs at the critical $k^2$ value where $\text{d}\Psi /\text{d}{k}^2=0$, giving

 where ${\displaystyle a_k=3d_1{d}_2,b_k=2\left[d_1\sigma +d_2\right],}$ and $c_k=\sigma -d_2{f}_1{s}_1-d_1{f}_2{s}_2\sigma .$ We take the positive root in 33 corresponding to the physical, positive $k^2$.

For all combinations of parameters giving rise to stable groups, we observe in our simulations that the group size approximated well by $2\pi /k_{\text{fast}}$, where, $k_{\text{fast}}$ is the wave number corresponding to the fastest growing mode (i.e. the value that maximizes $\Lambda \left(k^2\right)$). The size of the microbial clusters, as obtained by analytical theory $2\pi /k_{\text{fast}}$ and stochastic simulations are shown in the top row of 3. The color indicates the size of microbialgroups.

Roughly speaking, if we view a microbial cluster as the cause of perturbation at a nearby location, the Turing instability will manifest as group reproduction. We should strongly caution that as the instability proceeds, the system moves away from the initial unstable fixed point around which it was linearized, and thus the exponential dependence in 24 should eventually break down. Nevertheless, the eigenvalues in the exponents still provide us with an approximate estimation of the group reproduction rate, within a factor of two near the phase boundary.

All data generated or analysed during this study are included in the manuscript and supporting files

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Author details

1. Gurdip Uppal

   Department of Physics, University of Notre Dame, Notre Dame, United States

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing—original draft, Writing—review and editing

   For correspondence

   guppal@nd.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3957-256X

2. Dervis Can Vural

   Department of Physics, University of Notre Dame, Notre Dame, United States

   Contribution

   Conceptualization, Resources, Supervision, Funding acquisition, Investigation, Visualization, Methodology, Writing—original draft, Project administration, Writing—review and editing

   For correspondence

   dvural@nd.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-0495-8086

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