Flowerlike patterns in multispecies bacterial colonies
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Abstract
Diverse interactions among species within bacterial colonies lead to intricate spatiotemporal dynamics, which can affect their growth and survival. Here, we describe the emergence of complex structures in a colony grown from mixtures of motile and nonmotile bacterial species on a soft agar surface. Timelapse imaging shows that nonmotile bacteria 'hitchhike' on the motile bacteria as the latter migrate outward. The nonmotile bacteria accumulate at the boundary of the colony and trigger an instability that leaves behind striking flowerlike patterns. The mechanism of the front instability governing this pattern formation is elucidated by a mathematical model for the frictional motion of the colony interface, with friction depending on the local concentration of the nonmotile species. A more elaborate twodimensional phasefield model that explicitly accounts for the interplay between growth, mechanical stress from the motile species, and friction provided by the nonmotile species, fully reproduces the observed flowerlike patterns.
eLife digest
Communities of bacteria and other microbes live in every ecosystem on Earth, including in soil, in hydrothermal vents, on the surface of plants and in the human gut. They often attach to solid surfaces and form dense colonies called biofilms. Most biofilms found in nature are comprised of many different species of bacteria. How the bacteria interact shapes the internal structures of these communities.
Many previous studies have focused on the molecules that bacteria use to relate to each other, for example, some bacteria exchange nutrients or release toxins that are harmful to their neighbors. However, it is less clear how direct physical contacts between bacteria affect the whole community.
Escherichia coli is a rodshaped bacterium that is a good swimmer, but has a hard time moving on solid surfaces. Therefore, when a droplet of liquid containing these bacteria is placed in a Petri dish containing a jellylike substance called agar, the droplet barely expands over a 24hour period. On the other hand, a droplet containing another rodshaped bacterium known as Acinetobacter baylyi expands rapidly on agar because these bacteria are able to crawl using microscopic “legs” called pili.
Here, Xiong et al. set out to investigate how a colony containing both E. coli and A. baylyi developed on a solid surface. The experiments showed that when a droplet of liquid containing both species was placed on agar, both species grew and spread rapidly, as if the E. coli hitchhiked on the highly motile A. baylyi cells. Furthermore, the growing colony developed a complex flowerlike shape. Xiong et al. developed mathematical models that took into account how quickly each species generally grows, their ability to move, the friction between cells and the agar, and other physical properties. The models predicted that the E. coli cells that accumulate at the expanding boundary of the colony make the boundary unstable, leading to the flowerlike patterns.
Further analysis suggested that similar patterns may form in other situations when motile and nonmotile species of bacteria are together. These findings may help us understand the origins of the complex structures observed in many naturally occurring communities of bacteria.
Introduction
Microbial communities inhabit every ecosystem on Earth, from soil to hydrothermal vents to plants to the human gut (Moyer et al., 1995; Gill et al., 2006; Fierer and Jackson, 2006). They often form dense biofilms, whose structures are shaped by biological, chemical, and physical factors (Stoodley et al., 2002; Flemming et al., 2016; Stubbendieck et al., 2016). In the wild, most biofilms are comprised of multiple bacterial strains. They feature a diverse repertoire of social interactions, including cooperation (BenJacob et al., 2000; Griffin et al., 2004), competition (Hibbing et al., 2010), and predation (Jürgens and Matz, 2002). Bacteria often signal, sense, and respond to each other through secondary metabolites (Traxler et al., 2013) or antibiotic compounds (Garbeva et al., 2014), and cocultures can even exhibit different motility from either species on its own (McCully et al., 2019). These interactions may lead to the emergence of complex spatial structures, which can have a profound effect on bacteria survival and function, and promote biodiversity by optimizing the division of labor within the biofilm (Nadell et al., 2016). Spatial structure can also enhance horizontal gene transfer among different species (Cooper et al., 2017).
In addition to biochemical interactions, mechanical forces also play an important role in shaping the structure of bacterial communities. In dense colonies, bacteria push against each other due to growth and motility. Bacteria can exploit these mechanical interactions to adapt to the environment. For example, mechanical stresses cause buckling in Bacillus subtilis biofilms that allows them to improve nutrient transport and consumption (Asally et al., 2012; Trejo et al., 2013; Wilking et al., 2013). Although the role of mechanical interactions in singlespecies colonies has been studied previously (Volfson et al., 2008; Xavier et al., 2009; Kearns, 2010; Boyer et al., 2011; Persat et al., 2015), dynamics of multispecies communities driven by mechanical forces have received much less attention. Since bacterial strains can have significant differences in their growth and motility characteristics, one can expect the development of highlyheterogeneous mechanical stress distribution, which in turn can result in a complex spatiotemporal dynamics of the colony.
To study the interactions between bacterial species with distinct biological and physical properties, we choose Acinetobacter baylyi, a gramnegative bacterium that easily moves on soft surfaces using twitching motility (Harshey, 2003; Bitrian et al., 2013; Leong et al., 2017), and an Escherichia coli strain that is almost nonmotile on soft agar. Additionally, wildtype A. baylyi possesses a Type VI Secretion System (T6SS) that enables them to kill other bacteria (including E. coli) on direct contact (Schwarz et al., 2010; Cooper et al., 2017). We found that when these two strains are mixed together and inoculated on an agar surface, growing colonies develop intricate flowerlike structures that are absent when either species is grown by itself.
To shed light on the mechanism behind this intricate pattern formation, we tested whether biological cellcell communication or mechanical interaction between strains with different motilities played the key role. Experiments with A. baylyi mutants lacking T6SS showed that the pattern formation did not rely on this system. On the other hand, genetically impairing A. baylyi motility eliminated the patterns entirely. We also demonstrated that agar concentration affects cell motility and plays an important role in pattern formation. These findings suggested that the mechanical interactions between species are indeed primarily responsible for the pattern formation.
We then formulated and analyzed two models: a geometrical model of the colony boundary motion and a 2D phasefield model of the entire colony, to describe the mechanical interactions between two species. Our results show that growth and cell motility differences are sufficient to explain the emerging patterns. Since the mechanism of flowerlike pattern formation is rather general, it may be broadly generalizable to other multispecies colonies.
Results
Flowerlike patterns in mixtures of A. baylyi and E. coli on nutrientrich soft agar
We inoculated a mixture of E. coli and A. baylyi cells with an initial density ratio of 10:1 at the center of a Petri dish filled with soft LB agar (0.5% agar). To distinguish the two strains, we labeled E. coli with constitutively expressed mTFP. After growing at 37 °C for 3 days, this colony developed an intricate flowerlike pattern (Figure 1a). To see how such patterns form, we tracked the colony growth with timelapse imaging (Figure 1b, Video 1). Up to 8 hr after inoculation, the expanding colony remained nearly uniform and circular. Then the colony front began to visibly undulate. As the colony expanded further, the undulations grew and formed cusps that in turn would leave behind tracks (or ‘branches’). These branches then merged, following the movement of cusps along the interface as the colony continued to expand. The branches were visible even in brightfield imaging, but they were also bright in the teal fluorescence channel, indicating that branches consisted of relatively more E. coli cells (Figure 1—figure supplement 1).
To test whether these flowerlike patterns originate from interactions between the two species, we grew each species separately on the same 0.5% LB agar surface. The E. coli motility on agar is small, and the colony size remained relatively unchanged after 16 hr of growth (Figure 1c, left). After the same time, a colony of highly motile A. baylyi reached the edge of the plate (Figure 1c, right). In neither case did patterns emerge, showing that the flowerlike pattern formation was a result of interspecies interaction. We measured the sizes of mixed, pure E. coli and pure A. baylyi colonies at different times after inoculation (Figure 1d). After an initial growth period in which cells filled the surface in a complete monolayer, the colony began to expand (an example is shown in Figure 1—figure supplement 2). The expansion speed of mixed colonies fell between those of pure A. baylyi and pure E. coli colonies, and the speed did not change much once the colonies began expanding.
E. coli destabilize colony front by hindering A. baylyi expansion
To observe the pattern formation at higher resolution, we modified the experimental setup to fit under a fluorescence microscope (see Materials and methods). After 24 hr of growth, a droplet of 1:1 mixture of E. coli (expressing mTFP) and A. baylyi (expressing mCherry) grew into a clearlyvisible flowerlike pattern (Figure 2a). By zooming in on the front of the expanding colony, we were able to track the formation and merging of branches that gave rise to the flowerlike structure of the patterns (Figure 2b, Video 2). While A. baylyi killed most E. coli via T6SS within the center of the inoculum, a significant number of E. coli managed to survive at the periphery where they were not in direct contact with A. baylyi. E. coli also has a higher growth rate (1.53 ± 0.11 h^{1}, n = 3) than A. baylyi (1.13 ± 0.01 h^{1}, n = 3), so by the time the colony began to expand, E. coli cells had already grown near the colony boundary which resulted in a band of E. coli around the expanding colony of mostly A. baylyi (Figures 2b, 11h).
As the colony kept expanding, in regions with more E. coli cells near the front, the expansion was slower, so the interface began to curve inward (Figures 2b, 13h). As the undulations grew bigger, the E. coli in the regions lagging behind became more concentrated, thus slowing down the local front advance even more. Eventually, the front folded onto itself near these stagnant regions and formed narrow ‘branches’ that continued to grow outward with the expanding colony front (Figures 2b, 15h, 17h). Later, the front with the branches folded again, and the previous branches merged inside the new fold (Figures 2b, 19h, 21h). Since E. coli continued to grow at the expanding colony front, new undulations and branches constantly appeared, and eventually a macroscopic, flowerlike pattern of growing and converging branches formed. From Figure 2—figure supplement 1), it can be seen that the branches predominantly consisted of E. coli cells.
To quantify the effect of local E. coli concentration on the colony expansion, we analyzed the timelapse images in Figure 1b (see Materials and methods). We adapted a boundary tracking program for eukaryotic cells (Skoge et al., 2010) to track the boundary of the bacterial colony. The colony boundary was parameterized by 300 virtual ‘nodes’ connected by springs (Machacek and Danuser, 2006). For each node, we measured local brightness (a proxy for E. coli concentration), front speed and front curvature. To offset the nonuniformity of the illumination and the overall change in speed and curvature for a growing colony, we detrended the data. The kymographs of these quantities for each node are shown in Figure 2c. Then we computed correlations between these quantities within the time window when the pattern began to form (about 9.5–11.5 hr after inoculation). As shown in Figure 2d (left), the brightness and expansion speed show strong anticorrelation (Pearson coefficient ρ=−0.67). This result confirms that higher E. coli density slows down the front propagation. Variations in the front speed lead to variations of the local curvature, and the scatter plot between brightness and curvature indeed shows significant anticorrelation (Figure 2d right, Pearson coefficient ρ=−0.43).
Robustness of flowerlike patterns to perturbations
First, we explored the effect of the initial A. baylyi:E. coli (A:E) density ratio on the resulting pattern. We varied the ratio of A. baylyi to E. coli in the inoculum while maintaining the same total density of bacteria. We found that when the starting ratios are low (A:E = 1:100 and 1:10), flowerlike patterns emerged, while at high ratios (10:1 and 100:1) the E. coli were completely eliminated and no patterns formed (Figure 3a). At the intermediate ratio 1:1, A. baylyi dominated significantly at the center of the colony by killing E. coli, but the flowerlike structure still developed at the colony periphery.
Second, we wondered whether T6SSdependent killing played a role in the formation of these patterns when E. coli were not completely eliminated. We tested this by knocking out T6SS in A. baylyi (see Materials and methods for details). The growth rate of T6SS^{− }A. baylyi (1.09 ± 0.01 h^{1}, n=3) was not significantly different from the wild type, but their motility was slightly lower as determined by colony expansion rate. Still, their motility remained much higher than E. coli (Figure 3—figure supplement 1 and Figure 3—figure supplement 2). We inoculated mixtures of T6SS^{− }A. baylyi and E. coli with different initial ratios on 0.75% LB agar, and observed that the colony formed an outer ring of E. coli (Figure 3—figure supplement 3) and subsequently developed front instability, branches of E. coli, and a flowerlike pattern in all cases (Figure 3b). The only qualitative difference between the T6SS^{−} and T6SS^{+} cases was that in the nonkilling case, E. coli remained at a high concentration within the area of the initial inoculum. We measured the average radius of the colonies with different initial density ratios 16 hr after inoculations (Figure 3c, n = 3). In the case of a mixture of T6SS^{− }A. baylyi and E. coli, the more E. coli in the inoculum, the slower the colony expanded, which is consistent with our hypothesis that E. coli hinders the overall colony expansion. However, the trend is not as significant for the T6SS^{+} case, likely because T6SS^{+} A. baylyi kill most E. coli at the early stage, which increases and stabilizes the effective A:E ratio. We also counted the number of branches as they first emerged, when their circumferences were roughly the same, and found more branches in colonies seeded with less E. coli (Figure 3d, n = 3). In general, the overall structure of the patterns remained unchanged in the mixture of T6SS^{− }A. baylyi and E. coli. Thus, we concluded that the T6SS did not play a major role in the formation of flowerlike patterns.
Third, the fact that twospecies colonies expanded much more quickly than pure E. coli colonies strongly suggested that the high motility of A. baylyi is primarily responsible for the colony expansion. To test this hypothesis, we knocked out the pilTU locus of T6SS^{+} A. baylyi, which is required for the pilusbased twitching motility of A. baylyi (Zhan et al., 2012; Leong et al., 2017). As expected, colonies of pilTU^{−} A. baylyi cells did not expand significantly (Figure 3e, top) and did not form branching patterns when mixed with E. coli cells on 0.75% LB agar (Figure 3e, bottom). The results were the same when the colonies grew on other concentrations of LB agar (Figure 3—figure supplement 4). This demonstrates that the high A. baylyi motility plays a crucial role in the flowerlike pattern formation.
Finally, we tested the pattern formation in mixtures of motile and nonmotile A. baylyi (see Appendix 3). We found that flowerlike patterns emerged in this case as well, which confirms the key role of the difference in motility for pattern formation. The patterns were less pronounced, but this can be probably explained by the fact that other physical parameters of nonmotile A. baylyi (such as growth rates and effective friction) are more similar to motile A. baylyi than E. coli.
Patternforming instability originates at the colony interface
Experiments showed that the formation of flowerlike patterns appears to be preceded and caused by growing undulations of the colony front, where E. coli cells concentrate and locally slow expansion. To mechanistically understand how a ring of lowmotility bacteria surrounding an expanding core of highlymotile bacteria can create such patterns, we turned to mathematical modeling. We adapted a onedimensional ‘geometrical’ model of front dynamics (Brower et al., 1983; Brower et al., 1984) that casts the motion of the interface $\mathbf{\mathbf{x}}(\sigma ,t)$ in natural, referenceframe independent variables of curvature $\kappa $ and metric $g$ as a function of its arclength $s$ and time $t$ (see Appendix 1):
In the overdamped limit, the velocity functional $\mathcal{F}=({F}_{0}{F}_{s})/\mu (c)$ is determined by the balance of a constant outward force ${F}_{0}$ due to A. baylyi motility, surface tension ${F}_{s}=\gamma \kappa $ proportional to the interface curvature, and the resistance (friction) force ${F}_{r}=\mu (c)v$ that is proportional to the local velocity $v(s,t)$ with the friction coefficient $\mu (c)$ that in turn is proportional to the concentration of E. coli on the interface $c(s,t)$. Note that, in principle, nutrient depletion in the agar under the growing colony and chemotaxis towards the developing nutrient gradient may also contribute to the outward force ${F}_{0}$, however it should not change the mechanism of the patternforming instability we are discussing here. All these forces are assumed to be normal to the interface (Figure 4a). For simplicity, in this interface model we ignore E. coli growth and leakage from the boundary into the interior and assume that the local interface concentration of E. coli is only changed by stretching or contraction of the interface, therefore $c$ should be inversely proportional to the square root of the metric $g$. A straightforward linear stability analysis demonstrates that the interface is indeed unstable to a broad spectrum of initial perturbations (for more details see Appendix 1).
To simulate the interface dynamics beyond the linear regime, we also constructed a discrete model of the continuous interface by replacing it with a closed chain of nodes connected by straight links (Figure 4a bottom). Each node carries a fixed amount of E. coli, so the local density of nodes per unit length of the interface corresponds to the local density of E. coli. Nodes are driven by a constant outwards expansion force ${F}_{0}$, surface tension, and a friction force that is proportional to the windowweighted average density of nodes per unit length. Additionally, we introduced shortrange repulsive forces between nodes and between nodes and links, to prevent selfcrossing of the interface. Detailed description of this model is also given in Appendix 1.
As an initial condition, we assumed that the chain forms a circle with nodes slightly perturbed from equidistant positions. Figure 4b shows timelapse snapshots of the interface in a sample simulation (also see Video 3). Figure 4c shows the aggregate image of the interface during the colony expansion, with the color of a point corresponding to inverse local density of nodes when the interface passed through that point (also see Video 4). Assuming that a fixed fraction of E. coli is left behind the interface, this interface ‘fossil record’ should roughly correspond to the density of E. coli inside the colony. At the beginning, the interface remains nearly circular, but initial perturbations quickly grow as the colony expands, producing large front undulations. Regions with lower node density expand more quickly because they experience less friction, and this expansion stretches the chain and further reduces the node density per unit length, creating a positive feedback loop. Concave regions, on the contrary, accumulate nodes and thus move outward more slowly. Eventually, cusps are formed in these lagging regions that have very high node density and therefore move very slowly, if at all. The regions on both sides of the cusp continue to expand toward each other and eventually ‘collide’. After collision they form ‘doublelayers’ that remain nearly static and only increase in length as the overall interface expands further. Thus, ‘branches’ with high concentration of E. coli form. As the front continues to expand, the interface already containing branches continues to undulate and form new cusps. This causes the earlier branches to merge, similar to what we observed in experiments (Figure 2). These simulation results suggest that indeed branch formation and merging can be explained by mechanics of a resistive ring surrounding a colony, which is stretched by the colony expansion. However, since this model neglects E. coli growth, the average density of nodes per unit length gradually decays, and eventually, the front instability ceases, in divergence with experimental results. To account for cell growth as well as for the diffusive leakage of E. coli from the interface into the bulk of the expanding colony, we developed a more elaborate 2D model of the growing multispecies colony.
Phasefield model of flowerlike pattern formation
We also developed a more detailed twodimensional, multicomponent model of the expanding bacterial colony that is conceptually similar to the phasefield models used for description of eukaryotic cell motility and migration (Shao et al., 2010; Shao et al., 2012; Camley et al., 2013) (Figure 5a). It is based on PDEs for the densities of A. baylyi ${\rho}_{A}$ and E. coli ${\rho}_{E}$, together with an equation that describes the velocity field $\mathbf{\mathbf{u}}$ of the colony. This velocity field drives the expansion of the colony and is generated by a combination of stress due to cell growth and motility, viscosity, and bottom friction that is dependent on local E. coli density. The resulting free boundary problem is solved using the phasefield method, which introduces another PDE for an auxiliary field $\varphi $ that changes continuously from 1 inside the colony to 0 outside (see Appendix 2 for the detailed formulation of the model). The boundary is then automatically defined as $\varphi =1/2$ and can thus be computed without explicit tracking techniques.
When we initialized the model with small circular domains of either pure E. coli or A. baylyi, the colony boundaries remained circular, and no patterns emerged (Figure 5b). Consistent with the experiments, the E. coli colony only slightly expanded, while the A. baylyi colony expanded rapidly (Figure 5c). When we initialized the model with a mixture of A. baylyi and E. coli, the colony grew at an intermediate speed (Figure 5c), as in the experiments (Figure 1d). The mixed colony simulations also exhibited front instability leading to formation of branches of E. coli (Figure 5d, the snapshots of A. baylyi are shown in Figure 5—figure supplement 1, also see Video 5). As the colony grew, the branches merged and expanded, and a flowerlike pattern developed. The E. coli density, colony boundary curvature and expansion speed can be analyzed using the same method we used for experimental data shown in Figure 2c,d, which also shows the anticorrelation between E. coli density and local speed (Figure 5—figure supplement 2).
Agar concentration is known to have a strong effect on the motility of bacteria (Harshey, 2003) and their adhesion to the agar surface (Kolewe et al., 2015), so we reasoned that in our phasefield model changing agar concentration could be simulated by changing friction parameters. The frictional force in our model consists of two contributions: a small basal friction (characterized by parameter $\xi $) and stronger contribution proportional to the local E. coli concentration with coefficient $\beta $. Thus, to mimic different agar concentrations, we varied both $\xi $ and $\beta $. The leftmost panel in Figure 5e shows the colony snapshot at $t=16$ for the same parameter values as the timelapse sequence in Figure 5d. The next panel corresponds to larger $\xi $ and $\beta $ (presumably, higher agar concentration), where as expected, the colony expanded slower. The third panel shows the snapshot for smaller $\xi $ and $\beta $ (lower agar concentration), in which case the colony expands fast, but no patterns emerge. However, for the same low $\xi $ and $\beta $, when we started a simulation from 10x higher E. coli density, the friction provided by E. coli increased, and patterning reemerged (Figure 5e, fourth panel).
These numerical predictions were fully validated by experiments in which we varied the agar concentration and the initial density ratio of E. coli and T6SS^{−} A. baylyi. The leftmost panel in Figure 5f shows the snapshot of the colony started from 1:1 mixture after 16 hr of growth on 0.75% agar surface. When we increased the agar concentration to 1% (Figure 5f, second panel), the colony expanded slower but the flowerlike pattern emerged. Conversely, for low agar concentration (0.5%), colony grew fast but patterns were completely eliminated (Figure 5f, third panel). However, for the same 0.5% agar concentration but A:E = 1:100 initial density ratio, the flowerlike pattern formation was rescued (Figure 5f, fourth panel).
Discussion
Motility plays a key role in the local spread of bacteria. In this paper, we studied the structure of growing colonies comprised of two bacterial species, E. coli and A. baylyi, with very different motilities. Not only did the highlymotile species (A. baylyi) accelerate the spread of the slow species (E. coli), but the structure of the expanding colony quickly became highly heterogeneous and eventually produced very intricate, flowerlike patterns.
Bacterial colonies can expand on a surface in a variety of ways, assisted by volumetric pressure from cell growth and division, multiple types of motility (Harshey, 2003), chemotaxis (Golding et al., 1998; Ben Amar, 2013), osmotic pressure gradients from the extracellular matrix (Seminara et al., 2012; Dilanji et al., 2014; Srinivasan et al., 2019), secretion of surfactants that assist wetting (Kearns, 2010; Trinschek et al., 2017), etc, and these mechanisms are not mutually exclusive. In our case, we found that the key, necessary driver for expansion of mixed A. baylyi/E. coli colonies is the motility of A. baylyi. The expansion force appears to be mediated by cells physically bumping into and pushing each other, as colonies do not begin to expand outward until they reach a near confluent monolayer density (Figure 1—figure supplement 2). Before this point, motility in the interior can simply result in cell rearrangement, but once a confluent monolayer is reached, growth combined with motility begins to push the boundary outward. In our models, the effective expansion and friction forces are physically and experimentally motivated, but it is unclear to what extent the effective forces result from true friction, wetting forces, etc. In future work, it would be interesting to explore the detailed mechanistic underpinnings of these forces.
Pattern formation in growing colonies of single bacterial species has been studied extensively (Fujikawa and Matsushita, 1989; Budrene and Berg, 1991; Golding et al., 1998; Matsushita et al., 1998), and branching patterns were often found in these experiments. The emergence of these patterns is usually driven by nutrient limitation and ensuing chemotaxis, with agar concentration also having a strong effect on their morphology. For example, colonies expand homogeneously on soft agar rich with nutrients, but under nutrient limitation and in semisolid agar, complex patterns emerge (Budrene and Berg, 1991; Matsushita et al., 1998; Golding et al., 1998). In our system, however, we used rich LB media, and singlespecies colonies in the same conditions did not produce patterns, suggesting that the mechanism of pattern formation here is different.
Cell killing via the T6SS is an important ecological interaction, but it did not appear to play a major role in the formation of these patterns. We found no significant differences in pattern formation with T6SS^{+} and T6SS^{−} strains of A. baylyi. In fact, we did not observe noticeable killing of E. coli by T6SS^{+} A. baylyi after a short initial period (Video 2). We believe that an extracellular matrix may have played a role here, as recent studies showed that it protected bacteria from T6SS attacks from other species (Toska et al., 2018; MolinaSantiago et al., 2018). Overall, our experiments and modeling provided strong evidence in favor of the mechanical nature of the patternforming instability, arising from the interplay between outward pressure generated by the growth and high motility of A. baylyi, and the friction provided by sessile E. coli that adhere to the agar surface.
Ecologically, one of the primary challenges for any species is to maximize its geographic dispersal. Motility enables bacteria to escape from local stresses, move to locations with more nutrients, or invade host tissue (Harshey, 2003). However, motility, especially on hard surfaces, requires additional gene expression which could be a metabolic burden (Kearns, 2010). So some bacteria take advantage of other species with larger motility to colonize new niches. For example, by hitchhiking on zooplankton, waterborne bacteria can reach places that are otherwise inaccessible for them due to density gradients (Grossart et al., 2010). Nonmotile staphylococcal species hitchhike on swimming bacteria such as Pseudomonas aeruginosa (Samad et al., 2017). Motile swarming Paenibacillus vortex was shown to transport nonmotile Xanthomonas perforans (Hagai et al., 2014) or E. coli (Finkelshtein et al., 2015) on agar surfaces. In our system, A. baylyi cells move by twitching instead of swarming, and our results suggest that slowmoving bacteria might take advantage of fastmoving twitching species by hitchhiking, or ‘surfing’ along the expanding boundary, and thus spread farther. This can be seen clearly from the experiment in which E. coli and A. baylyi were inoculated separately at a small distance on agar surface (Video 6). The A. baylyi colony expanded and pushed E. coli to places where E. coli alone could not reach.
The flowerlike patterns appear to require a combination of several factors: motility of one of the two species, hitchhiking of the nonmotile species with the motile one, and sufficiently strong effective friction from the nonmotile strain. Indeed, no patterns form without motility of one of the species, see Figure 3e. Hitchhiking appears to be a necessary but not sufficient condition for flowerlike pattern formation. Indeed, without hitchhiking E. coli would simply be left behind and not present in the expanding colony of A. baylyi. On the other hand, we observed that E. coli also hitchhiked in round colonies (e.g. Figure 5f, third panel), where patterns did not form, presumably because E. coli did not exert sufficiently strong effective friction, due to properties of the agar or too low cell density. In the phasefield model, lowering E. colidependent friction corresponds to reducing parameter $\beta $, and indeed, for small $\beta $, patterns do not form. Additionally, higher growth rate of the nonmotile strain facilitates formation of a dense ring around the expanding colony, which makes patterns more robust. We also observed flowerlike patterns when E. coli was replaced with nonmotile A. baylyi strain, but they were less robust, presumably because the growth rate of the nonmotile A. baylyi strain was lower than that of the motile one, although it is also possible that the effective friction of nonmotile A. baylyi could be less than that of E. coli (Appendix 3—figure 1).
Although E. coli and A. baylyi may not necessarily find themselves in the same ecological niche, bacteria with different motilities are ubiquitous in the environment (Harshey, 2003). Therefore, the mechanisms of codependent motility and pattern formation described here are likely to be broadly applicable in natural habitats or even have implications in the transmission of pathogenic microbes. For example, Acinetobacter baumannii, an increasing threat in hospitals due to multidrug resistance (Dijkshoorn et al., 2007), is closely related to A. baylyi (Touchon et al., 2014), also has twitching motility (Eijkelkamp et al., 2011; Clemmer et al., 2011), and coexists with E. coli in at least one known niche, namely hospitals. Thus, the generic patternformation and hitchhiking described here may be quite common in diverse environments.
Materials and methods
Strains
We used E. coli MG1655 and A. baylyi ADP1 (ATCC #33305). The E. coli strain carried a plasmid that constitutively expressed mTFP and a kanamycin resistance gene. A. baylyi had a kanamycin resistance gene and the mCherry gene integrated in the genome. We also constructed a T6SS^{− }A. baylyi (Δhcp) mutant by first fusing the tetracycline resistance marker (TetA) from pTKS/CS to approximately 400 bp homology arms amplified from either side of hcp (ACIAD2689) in the A. baylyi genome, and mixing the donor oligo with naturally competent A. baylyi. The pilTU^{−} strain was constructed similarly to delete the genes ACIAD09110912. All A. baylyi strains used in this study retain their endogenous immunity genes to T6SS attack.
Culture conditions and image capturing
Request a detailed protocolE. coli and A. baylyi cells were taken from −80 °C glycerol stocks, inoculated in LB with appropriate antibiotics (kanamycin for E. coli and T6SS^{+} A. baylyi, tetracycline for T6SS^{−} A. baylyi) and grown at 37 °C separately. When their OD600 reached about 0.3, both E. coli and A. baylyi were concentrated to OD = 1, still separately. They were then mixed at specified volume ratios, and 3 μL was inoculated on the surface of 10 mL LB agar in the center of an 8.5 cm Petri dish. The plate was incubated at 37 °C. The images were taken using a custom ‘milliscope’ fluorescence imaging device unless indicated otherwise.
When the colony development was to be observed under a microscope, a 5.5 cm Petri dish was used with 15 mL 1% base agar (without LB) and top 10 mL LB agar (1% agar). After the cell culture was inoculated and dried, it was put on the stage of an inverted, epifluorescence microscope (Nikon TI2). The magnification was 4X. Fluorescent images were acquired using a 4X objective and a Photometrics CoolSnap cooled CCD camera in a 37 °C chamber. The microscope and accessories were controlled using the Nikon Elements software.
The bacteria growth rates were measured in a Tecan plate reader.
Colony tracking
Request a detailed protocolWe adapted the method and the MATLAB code from Skoge et al. (2010) to track the colony boundary. The brightfield images were first segmented to identify the colony using an active contour method (Chan et al., 2000). The segmentation result is illustrated in Video 7. Then the colony boundary pixels were interpolated by a closed cubic spline and the boundary was parameterized by 300 virtual nodes, which were evolved in time as a coupled spring system (Figure 2—figure supplement 2) (Machacek and Danuser, 2006). For each node, three quantities were measured: brightness, extension speed and curvature. Brightness at each node was defined as the median of the neighboring pixels assigned to each node (see Skoge et al., 2010). Extension speed was computed by the displacement of a node from t to t+50 min. Curvature was calculated by taking derivatives of the spline contour. Then the time series of these quantities were detrended as following: At each time point, fast Fourier transform (FFT) is carried out for each variable across all nodes and in the resulting transform, the first few low frequencies are set to zero. Then inverse FFT is carried out to obtain the detrended values for each variable at each node. After detrending, all variables can be negative at certain nodes. An example of these quantities for all nodes at a particular time point is shown in Figure 2—figure supplement 3. In Figure 2d, we sampled 7 time points with 20 min interval from 9.5 hr to 11.5 hr and for each time point we plotted 100 nodes.
Mathematical models
Request a detailed protocolDetailed description of the two models is given in Appendices 1 and 2.
Appendix 1
Interface model
Continuous interface dynamics
To describe the motion of the interface separating the growing bacterial colony from the environment, we can use the framework originally proposed by Brower et al. (1983); Brower et al. (1984) for solidification patterns. We assume that the motion of the interface is a result of the local balance of the ‘pushing force’ and the frictional force that is linearly proportional to the local interface velocity. The 1D interface (a closed line) at time $t$ is specified by the position vector $\mathbf{x}(t,\sigma )$ where $0\le \sigma \le 1$ is the variable parametrizing the interface such that $\mathbf{x}(t,0)=\mathbf{x}(t,1)$. Using the ‘orthogonal gauge’ assumption that the velocity $d\mathbf{x}/dt$ is orthogonal to the tangent vector $\tau =\mathrm{\partial}\mathbf{x}/\mathrm{\partial}\sigma$, the equation of motion for the interface in the overdamped limit can be written in the general form
where $\widehat{\mathbf{\mathbf{n}}}$ is the unit vector normal to the interface at $\mathbf{\mathbf{x}}$ (perpendicular to $\tau$), $F$ is the normal force that generally may depend on the overall interface position and other parameters, and $\mu $ is the friction coefficient. As Brower et al. (1984) demonstrated, this equation can be transformed to the referenceframe independent local equations of motion for the local curvature $\kappa $ and the curve metric $g=\tau \cdot \tau$ as a function of arclength $s$ and time $t$:
where $\mathcal{F}=F/\mu $, the arclength $s$ is given by
and the curvature is defined by
Now we need to specify the driving force $F$ and the friction coefficient μ for our system in which a growing colony is surrounded by the thin band of highly frictional E. coli that hinders the colony expansion. We assume that $F$ depends only on the local curvature $\kappa $ in the following simple form:
This assumption will be violated if/when the interface will develop large folds and will attempt to ‘collide’ with each other, then nonlocal terms in $F$ become essential. We confine our continuous description here to sufficiently early times before this nonlocal interaction occurs. We postulate that the friction coefficient is a linear function of the local concentration of E. coli, $c$,
where without loss of generality we take $\mu =1$ at $c=0$. Under the additional simplifying assumption that the total amount of E. coli on the interface is conserved and neglecting their diffusion along the interface, the local concentration of E. coli will be inversely proportional to the square root of metric $g$, $c={c}_{0}/\sqrt{g}$. In reality, of course, E. coli also grows and is left behind in the bulk of the colony, but we ingore these effects in this simple model (see the phasefield model below where these effects are taken into consideration). Thus, the closedform model for the interface expansion has the following form
We can perform a linear stability analysis of a flat interface $(\kappa =0,g=1$) by substituting ansatz
in Equations (8), (9). The Jacobian of the linearized system reads
For positive $\gamma ,\alpha $, one of the two eigenvalues of this Jacobian is always positive. At small wavenumbers $k$, it increases linearly with $k$,
and for large $k$ it reaches the maximum value
Since the growth rate is positive for all values of $k$, this instability may lead to singularities in curvature (cusps). This is indeed what is found in numerical simulations of the discrete analog of this model (see the next section). These singularities correspond to the origins of ‘branches’ of E. coli that the interface leaves behind during the flower pattern growth.
Flexiblechain interface model
The interface dynamics beyond linear instability stage can be analyzed numerically. Unfortunately, it is difficult to implement selfavoidance of the interface in the framework of the continuum model described in the previous section. Thus, we implemented a discrete flexiblechain model that is analogous to the continuum model described above but contains additional interaction terms between the nodes that prevent selfintersection of the chain. Specifically, we represent the interface as a closed chain of $N$ nodes with coordinates ${\mathbf{x}}_{i},i=1,...,N$. Let us introduce the vectors connecting node $i1$ to node $i$ (we assume that node 0 is the same as node $N$): $\mathbf{\Delta}}_{i}={\mathbf{x}}_{i}{\mathbf{x}}_{i1$. Each node is driven by the ‘expansion force’ ${F}_{0}$ that acts along the unit vector $\hat{\mathbf{n}}}_{i$ that is directed outwards along the bisectrix of two adjacent edges, $\mathbf{\Delta}}_{i$ and $\mathbf{\Delta}}_{i+1$. It is counteracted by the ‘friction’ force that is directed along ${\hat{\mathbf{n}}}_{i}$ and is proportional to the local density of E. coli ${c}_{i}$ associated with node $i$ and by the surface tension force that is proportional to the local curvature of the interface ${\kappa}_{i}$. In addition, we introduce repulsion forces between all nodes and edges that prevent the interface from selfintersecting. The equation of motion in the overdamped limit can be written as follows:
The discrete analog of the local curvature at node $i$ is defined as follows,
where ${\mathrm{\Delta}}_{i}={\mathbf{\Delta}}_{i}$.
We assume that each node carries the fixed ‘amount’ of E. coli $c$, and the local concentration of E. coli ${c}_{i}$ is defined as the average amount of $c$ per unit length of the interface. In the simplest case, it can be computed as $c/{L}_{i}$ where ${L}_{i}$ is the halfsum of lengths of two edges attached to node $i$, ${L}_{i}={\textstyle \text{textonehalf}}({\mathrm{\Delta}}_{i}+{\mathrm{\Delta}}_{i+1})/2$, however in simulations we typically used longer averaging over two adjacent edges on both sides,
with $K=2$.
The last two terms in the r.h.s. of Equation (15) represents the vector sum of possible repulsive forces acting on the node $i$ from other nodes ($\mathbf{f}}_{ij}^{nn$) or edges ($\mathbf{f}}_{ij}^{ne$) of the chain. The nodenode force acts along the vector connecting nodes $i$ and $j$, $\mathbf{x}}_{i}{\mathbf{x}}_{j$. We assume that the nodeedge force acts perpendicular to the orientation of the $j$th link, $\mathbf{\Delta}}_{j$. We assume that the nodenode force $\mathbf{f}}_{ij}^{nn$ is zero if $d}_{ij}^{nn}={\mathbf{x}}_{i}{\mathbf{x}}_{j}>{d}_{0$ and varies as ${F}_{m}{(1{d}_{ij}^{nn}/{d}_{0})}^{4}$ for ${d}_{ij}^{nn}<{d}_{0}$ with small ${F}_{m}\ll {F}_{0}$. Similarly, the nodeedge force $\mathbf{f}}_{ij}^{ne$ is zero if the distance between the node $i$ and the edge $j$, ${d}_{ij}^{ne}>{d}_{0}$ and varies as ${F}_{m}{(1{d}_{ij}^{ne}/{d}_{0})}^{4}$ for ${d}_{ij}^{ne}<{d}_{0}$.
Parameters
We used parameters below (Appendix 1—table 1) unless specified otherwise.
Appendix 2
Phasefield model
Model description
In this more elaborate 2D model of a twostrain colony, we consider it as a growing mass of compressible twocomponent fluid. A convenient way to describe a compact expanding colony is to use a phasefield approach where the phase $\varphi $ changes smoothly from 0 outside the colony to 1 inside. The evolution of phase field $\varphi $ is described in earlier work (Shao et al., 2012). $\varphi $ is given by the equation:
where $\mathbf{\mathbf{u}}$ is the velocity field, $\mathrm{\Gamma}$ is a Lagrange multiplier, $\kappa =\nabla \cdot (\nabla \varphi /\nabla \varphi )$ is the local interface curvature, and $\u03f5$ characterizes the interface width. The first term on the righthand side is the advection term. The second one is the surface energy. In the third term $G(\varphi )=18{\varphi}^{2}{(1\varphi )}^{2}$ is included to force the bistable dynamics of $\varphi $ field with two stable fixed points at 0 and 1. The last term is added to cancel the surface energy and stablize the phasefield interface, as detailed in Biben and Misbah (2003) and Biben et al. (2005). Note that in the interface model, we include the surface tension term $\gamma \kappa $ to stabilize the system, otherwise Equation (13) holds for all $k$ and $\lambda $ goes to infinity when $k$ increases.
Close inspection of the growing colony showed that the velocities of the two strains in close proximity are very similar, since the mixture of E. coli, A. baylyi and the (presumable) extracellular matrix is dense, liquidlike, and miscible. Therefore, we use a single local velocity, which represents the actual velocity of the bacterial cells, to describe the movement of two species.
The dynamics of the A. baylyi cells density ${\rho}_{A}$ within the colony is described by
The second term in the lefthand side is the advection term while the two terms in the righthand side are diffusion and growth terms respectively. ${D}_{A}$ and ${\alpha}_{A}$ are the diffusion constant and growth rate of A. baylyi respectively. The growth term follows logistic form and we assume that the growth can be saturated when the total density of A. baylyi (${\rho}_{A}$) and E. coli (${\rho}_{E}$) reaches 1. Note that the densities of two species are already scaled here.
Similarly, the dynamics for E. coli cells density ${\rho}_{E}$ is described by
where ${D}_{E}$ and ${\alpha}_{E}$ are the diffusion rate and growth rate of E. coli. Note that the advection of the phase field and both cell densities is provided by the same velocity field $\mathbf{\mathbf{u}}$.
The system is treated as a viscous Newtonian fluid (Rubinstein et al., 2009; Shao et al., 2012). The velocity field can be determined by the overdamped Stokes equation:
where $\nu (\varphi )={\nu}_{0}\varphi $ is the viscosity, ${\sigma}_{A}=\eta \varphi {\rho}_{A}\mathbf{\mathbf{I}}$ is the stress provided by motile A. baylyi cells ($\mathbf{\mathbf{I}}$ is the identity matrix). $\chi $ is a random number uniformly distributed between $1\pm \mathrm{\Delta}$, which adds noise to the stress driven by A. baylyi. Because pure E. coli colony expands very slowly and pure A. baylyi colony expands fast, we assume that the stress provided by E. coli is negligible compared to A. baylyi. Our experiments with mixtures of E. coli and A. baylyi show that regions where there are more E. coli move outward more slowly, so we assume that E. coli cells provide friction to prevent colony from expanding fast. This is described by the last term in which $\xi $ is the basal friction constant and $f({\rho}_{E})={\rho}_{E}$ determines how the friction is modulated by E. coli cells. Here we assume it is simply proportional to ${\rho}_{E}$.
In reality, as the colony expands, the nutrients in the media are expected to get depleted over time at the center of the colony. However, in the experiment, where we use rich LB media, interesting dynamics mainly happen at the colony boundary, and the pattern inside the colony does not change once it forms. Therefore, we do not include the nutrient diffusion and uptake in our model.
Parameters
Parameters of simulations on 0.75% LB agar are shown in Appendix 2—table 1. Some of these parameters (such as growth rates ${\alpha}_{E}$ and ${\alpha}_{A}$) are known from experiments, while others had to be plausibly hypothesized. For example, the diffusion constants for bacterial motion are only known very roughly (Budrene and Berg, 1991; Kim, 1996), but since A. baylyi is motile and E. coli is not, we chose the diffusion constant of A. baylyi to be two orders of magnitude higher than that of E. coli.
Note that if parameters ${\nu}_{0},\eta ,\xi ,\beta $ are multiplied by the same constant factor, the velocity as determined by Equation (21) will not change. So we set arbitrarily $\xi =1$ and chose other parameters ${\nu}_{0},\eta ,\beta $ relative to $\xi $. Based on the presence of sharp kinks in the developing front structure, we concluded that viscosity plays a minor role in the dynamics, so we chose the viscosity coefficient to be small. The value of $\beta $ is chosen based on fitting the average expansion rates of colonies of A. baylyi and E. coli mixtures.
Our simulations showed that diffusion and viscosity terms did not play significant roles in the dynamics. Changing ${D}_{A}$ had little effect on the colony expansion speed and the pattern formation (Appendix 2—figure 1). The reduction of ${\nu}_{0}$ makes colony expand faster but the flowerlike pattern still forms (Appendix 2—figure 1). On the contrary, the stress and friction terms play major roles in our model. For the stress term, $\eta $ is chosen to make the expansion speed of pure A. baylyi colony similar to experimental measurement. We also added white uniformlydistributed noise (with magnitude $\mathrm{\Delta}$) to the stress term to break the circular symmetry and induce the front instability. When $\mathrm{\Delta}$ is small, the colony front instability also occurs, but at a later time point and merging of branches is not obvious (Appendix 2—figure 1, first row), so we choose $\mathrm{\Delta}=0.3$ in our simulations.
To model changes in the agar concentration (Figure 5e), we varied $\xi $ and $\beta $ while keeping ${\nu}_{0}$ and $\eta $ the same. As shown in Figure 5e, for the simulation in 0.5% LB agar, $\xi =0.5,\beta =1$ and for the simulation in 1% LB agar, $\xi =2,\beta =35$. The colony radii after 14 hr in simulations are illustrated in Appendix 2—figure 2 which can be compared to Figure 3—figure supplement 1. Note that in Figure 3—figure supplement 1, we show the experimental data after 16 hr of growth because in experiments, the colonies only begin to expand 2 to 3 hr after inoculation, while in simulations the colonies begin to expand immediately.
Numerical algorithm
The numerical algorithm is similar to Camley et al. (2013). For the initial conditions, we set $\varphi =0.5+0.5\mathrm{tanh}[3({r}_{0}r)/\u03f5]$ where ${r}_{0}=0.2$ cm and $r$ is the distance from the center of the simulation domain, so that $\varphi $ is 1 inside and 0 outside of the colony. Initial ${\rho}_{A}$ and ${\rho}_{E}$ are proportional to $\varphi $. We use periodic boundary conditions in the simulations.
We aim to solve Equations (18)(21) with uniform spatial grid sizes $\mathrm{\Delta}x,\mathrm{\Delta}y$ and fixed time step $\mathrm{\Delta}t$ from initial conditions ${\varphi}^{0},{\rho}_{A}^{0},{\rho}_{E}^{0},{\mathbf{\mathbf{u}}}^{0}$. The system variables at time $t=n\mathrm{\Delta}t$ are denoted as ${\varphi}^{n},{\rho}_{A}^{n},{\rho}_{E}^{n},{\mathbf{\mathbf{u}}}^{n}$.
We first solve Equation (18) by forward Euler scheme:
with ${\kappa}^{n}$ calculated by ${\kappa}^{n}=\nabla \cdot (\nabla {\varphi}^{n}/\nabla {\varphi}^{n})$ when $\nabla {\varphi}^{n}>0.05$, and set to 0 otherwise.
The reactiondiffusionadvection equations for ${\rho}_{A}$ and ${\rho}_{E}$ are discretized using the forward Euler scheme:
where ${\varphi}^{n+1}$ is obtained from the above step, and ${\rho}^{n+1}$ is only updated when ${\varphi}^{n}>{10}^{4}$. The advection term is calculated by
and for the diffusion term
where $\mathbf{\mathbf{u}}=(u,v)$, ${\varphi}_{i\pm 1/2,j}=({\varphi}_{i\pm 1,j}+{\varphi}_{i,j})/2$, ${\varphi}_{i,j\pm 1/2}=({\varphi}_{i,j\pm 1}+{\varphi}_{i,j})/2$, and we used the same definitions for $\rho $, $u$ and $v$ between collocation points. Then we can calculate ${\rho}^{n+1}$ from Equation (22).
The Stokes equation Equation (21) is integrated by the semiimplicit Fourier spectral method (Chen and Shen, 1998; Camley et al., 2013) (to stabilize the scheme, we subtract the term ${\nu}_{0}{\varphi}_{0}{\nabla}^{2}\mathbf{\mathbf{u}}$ from both sides of Stokes equation with large constant ${\varphi}_{0}$, e.g. ${\varphi}_{0}=200$):
To obtain ${\mathbf{\mathbf{u}}}^{n+1}$, we set ${\mathbf{\mathbf{u}}}_{0}^{n+1}={\mathbf{\mathbf{u}}}^{n}$ and solve the equation below iteratively using spectral Fourier method:
where $k=0,1,2,\mathrm{\cdots}$ are iteration steps. In simulations, we constrain the error by iterating the above process until
or until ${k}_{max}=200$, and the final ${\mathbf{\mathbf{u}}}^{n+1}={\mathbf{\mathbf{u}}}_{m}^{n+1}$.
Appendix 3
Mixtures of motile and nonmotile A. baylyi
For an additional test of our hypothesis that the difference in motility between the two strains is indeed the key factor of the pattern formation, we mixed motile A. baylyi (T6SS^{−} or T6SS^{+}) with nonmotile (pilTU^{−}) mutant of A. baylyi and inoculated them on 0.75% LB agar. Note that all A. baylyi strains used in this study have their endogenous T6SS immunity genes intact, so they do not kill each other (see Materials and methods).
In these experiments we also observed complex flowerlike structures (see Appendix 3—figure 1 and Appendix 3—video 1), however the pattern formation was less robust than in the case of A. baylyi/E. coli mixtures. In particular, patterns were observed in the narrower range of initial density ratios for mixtures of motile T6SS^{− }A. baylyi and pilTU^{−} A. baylyi: Welldeveloped flowerlike patterns were observed for initial density ratio $R$ = 1:10 (motile:nonmotile), however, unlike the case of T6SS^{−} A. baylyi and E. coli mixtures, no patterns were observed for $R$ = 1:1, and only weak patterning was observed for $R$ = 1:100 (see Appendix 3—figure 1, panels ac). We hypothesize that the main reason for these differences is that in this case the nonmotile strain did not have a faster growth rate. Nonmotile A. baylyi has a significantly smaller growth rate (1.03 ± 0.12h^{1}, n = 3) than our E. coli strain (1.53 ± 0.11h^{1}, n = 3). The nonmotile A. baylyi growth rate was actually even smaller than the growth rate of our motile strain, which may have been due to metabolic burden from the highly expressed tetA gene used to select them. Thus, for large $R$, motile A. baylyi ‘outruns’ the nonmotile strain, which does not grow fast enough to first form a nonmotile band around the colony. It is also possible that nonmotile A. baylyi provide less friction (less adhesion to the agar surface) and that this also contributes to the differences in pattern formation with the case of A. baylyi/E. coli mixtures. This hypothesis is confirmed by the simulations of phasefield model (Appendix 3—figure 2). When the nonmotile strain growth rate and nonmotile straindependent friction are large, the pattern occurs. When one of the two parameters decreases, the patterns still persist while the pattern disappears if both growth rate and friction drop.
We also found significant differences in patterning between mixtures of nonmotile A. baylyi with T6SS^{+} or T6SS^{−} motile A. baylyi. When motile T6SS^{−} A. baylyi and nonmotile A. baylyi are mixed with initial density ratio 1:100 (Appendix 3—figure 1, panel c), the nonmotile strain dominates the colony and only weak patterns are observed, which are different from the earlier flowerlike structures. However, when T6SS^{+} motile A. baylyi and nonmotile A. baylyi are mixed, even with initial density ratio 1:100, T6SS^{+} motile A. baylyi dominate the colony (Appendix 3—figure 1, panel d). In this case, streaks of the nonmotile strain (similar to those in flowerlike patterns) can be observed, but they do not merge as in the earlier flowerlike patterns. We believe that these differences are caused by the fact that our T6SS^{+} motile A. baylyi has larger growth rate and motility than T6SS^{−}, likely due to metabolic burden from the selection marker.
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Decision letter

Detlef WeigelSenior Editor; Max Planck Institute for Developmental Biology, Germany

Aleksandra M WalczakReviewing Editor; École Normale Supérieure, France

Agnese SeminaraReviewer; French National Centre for Scientific Research, France
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Acceptance summary:
The paper describes the morphology of a twospecies bacterial colony on soft agar, composed of twitching Acinetobacter bayli and nonmotile Escherichia coli. When cocultured, these colonies exhibit a flowerlike pattern that is absent in pure cultures. The authors find that the type IV secretion system of A. baylyi is not essential for the observed morphology, whereas agar concentration strongly affects the pattern. Using continuous models of colony morphology they conclude differences in cell motility and growth are sufficient to explain the observed pattern.
Decision letter after peer review:
Thank you for submitting your article "Flowerlike patterns in multispecies bacterial colonies" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Detlef Weigel as the Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Agnese Seminara (Reviewer #2).
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
The paper describes the morphology of a twospecies bacterial colony on soft agar, composed of twitching Acinetobacter bayli and nonmotile Escherichia coli. When cocultured, these colonies exhibit a flowerlike pattern that is absent in pure cultures. The authors find that the type IV secretion system of A. baylyi is not essential for the observed morphology, whereas agar concentration strongly affects the pattern. Using continuous models of colony morphology they conclude differences in cell motility and growth are sufficient to explain the observed pattern.
Overall, all the reviewers where very positive about the work. However, they raise a number of both experimental (mix motile/nonmotile A. baylyi) and theoretical questions. Addressing all of these will greatly help the readers. I am attaching the reviews in full; please respond to all the points. As you see many of the theoretical points are similar between reviewers.
Reviewer #1:
In this manuscript, Xiong et al. describe how complex pattern formation arises within a 2species bacterial biofilm. Specifically, they show that combining E. coli and A. baylyi results in flowerlike patterns while neither species individually produces this effect. They use genetic perturbations to show that this effect requires motility (mixed biofilms with nonmotile A. baylyi do not produce patterns) and does not depend on T6SS. They go on to reproduce the patterns using a mathematical model that captures the differences in motility between the 2 strains.
Overall, I thought this was a very nice and clearly presented study. The phenotype is quite interesting and the authors do a good job of providing a plausible explanation (mechanical effect resulting from differences in motility). I have only one experimental suggestion to confirm this explanation. Can the authors mix motile and nonmotile A. baylyi and produce the patterns in a single species A. baylyi biofilm? If the floral patterns are really a result of motility differences (and not other unknown interspecies effects) than this should be possible. They have already created the nonmotile A. baylyi mutant so it should be straightforward to try this experiment. If they observe this result, then I am enthusiastic for publishing the manuscript.
Reviewer #2:
I find the results interesting and convincing; I have a series of comments that I would like the authors to address before I can recommend this manuscript for publication. I am mainly concerned with justifying the choices made for the physical modeling and provide ideas to corroborate hitchhiking as a relevant evolutionary determinant. Below I detail my concerns, and suggest ways to improve clarity and strength of the results.
The experiments describing the evolution of the flower like pattern are clear and convincing. E. coli accumulates at boundaries and causes the colony to bend inward and fold onto this decelerating region creating cusps and branches. The speed of the boundary is anticorrelated with E. coli concentration and to curvature, corroborating the picture.
Flower like patterns are recovered by a simple model tracking the shape of the leading edge, assuming that each portion of the interface expands by a balance of friction, surface tension and an expanding force due to motility. Friction is proportional to concentration of (nonmotile) E Coli, which causes instability.
1) It would be helpful to discuss the origin of the expansion force, since only part of the colony is motile. Are twitching cells pushing the other strain through celltocell contact? (Can the authors provide a highresolution image to show that cells are highly packed?) Or – given that an extracellular matrix is mentioned – is it entropically driven? (Is an extracellular matrix knockout available to the authors?)
2) Are wetting forces negligible? An orderofmagnitude estimate of the different forces or experimental evidence would justify choice of these and not other forces.
3) In the more refined model, 2D concentrations of the two strains and of the phase field are evolved in time. I would imagine the two strains move relative to one another (E. coli remains attached to the substrate whereas A baylyi twitches). A twofluid mixture appears a natural choice; the authors consider instead a single velocity field carrying all cells, and I am unclear what this velocity represents. Is there a way to relate it to actual velocities measurable experimentally, e.g. a weighted average of velocities of the two strains? Because I am confused about the meaning of v, I do not have an intuition why it follows Stokes equation.
4) The authors predict that at low friction, the colony expands quicker with no flower like pattern, unless a large concentration of E. coli is inoculated. Assuming agar concentration affects friction, these predictions are verified experimentally. A discussion is missing about the choice of ξ and β, which appears to me quite arbitrary. Also: is the value of η obtained by fitting the expansion rate a plausible value for a bacterial colony?
5) Hitchhiking is an appealing evolutionary advantage. The discussion would benefit from two additional points. First: the authors imply that flowerlike patterns are directly related to hitchhiking. However from Figure 3A it looks like E. coli hitchhikes even in round colonies maybe less so than in flowerlike colonies? Could you quantify this aspect? Second: what are the parameters that switch hitchhiking on/off in the model? Are there evidences that these parameters vary in different strains? Could genes control these parameters? This would help elucidate whether hitchhiking may be actively controlled (e.g. strains turning off motility would benefit from correspondingly turning on hitchhiking)
Reviewer #3:
The authors studied interactions between a motile and a nonmotile bacterial species in growing colonies, and observed the development of complex patterns that were not present in colonies consisting only of one of these species. These patterns included an undulated interface of the colony that correlated with branched structures inside of the colony. The authors provided several lines of evidence that the speed of colony expansion correlates negatively with the local concentration of the nonmotile bacteria. Based on these findings, they showed that a quite simple model describing the dynamics of the 1D colony interface can account for the formation of undulations and branches. This model essentially describes the advection and dilution of the nonmotile species with the motion of the front of the motile bacteria, where a local friction increases with the concentration of nonmotile bacteria. To additionally account for growth and diffusion of the bacteria, the authors further introduce a more complex 2D phase field model. Similar to the 1D interface model, patterns similar to the experimentally observed ones could be qualitatively reproduced.
While I am not an expert on biofilms, I found this manuscript interesting and well written. I found particularly intriguing that pattern formation based on the collective motion of two different bacteria species can be at least qualitatively accounted for by a simple 1D model.
In the following, I will focus on discussing the mechanical modeling part, which falls more in my area of expertise. While the interface model has appeal due to its simplicity, there are a few things that need to be better clarified if not corrected about it (see below). The phase field modeling seems to be mostly appropriate with respect to the assumptions made, but also here at least some clarifications would be good, mostly related to the field Φ (see below).
More detailed comments:
1) A brief discussion on possibilities of where the force F_{0} and the "active pressure" in Equation 22 could originate would be good (also in the initial phase where the colony is not expanding (Figure 1D)). In subsection “Patternforming instability originates at the colony interface” the authors suggest some kind of active pressure created by the A. Baylyi motility. However, it is not clear to me how this alone could explain an initial phase where the colony is not expanding at all (Figure 1D). The latter seems to be more consistent with a picture of nutrient consumption combined with chemotaxis (as proposed in earlier work, Discussion section paragraph two), which could also effectively create such an F_{0}.
2) There are problems or at least a lack of clarity related to the way the friction force F_{r} is included in both 1D models describing the interface behavior.
a) The authors assume F_{r} to be proportional to the concentration of E. coli, c, but independent of velocity. As a consequence, if the concentration is high enough, the surface would move inward (even without surface tension F_{s}=0), driven by this friction force. How realistic is such a friction force?
b) The normal velocity F[κ, g] is computed directly as a difference of normal forces (Equation 8). Apparently, the authors have implicitly assumed some additional, velocitydependent friction here (and set the friction coefficient to one). It would be good to comment on how this additional friction is motivated.
c) As an alternative, one could assume F_{r} itself to depend linearly on velocity, a common assumption for motile cells on a substrate, which is also used in the authors' phasefield model (Equation 22). In this case and without additional friction, through force balance (F_{0}F_{s}F_{r}=0) the normal velocity F[κ, g] would be given by a quotient between F_{0}F_{s} and a friction coefficient that is a function of c.
3) Even though the phase field approach is similar to previous work, some explanations, in particular related to the field Φ would make the manuscript more selfcontained.
a) In subsection “Phasefield model of flowerlike pattern formation” the authors state that Φ is introduced to avoid computational difficulties of dealing with the boundary. However, these difficulties and hence the necessity of Φ is nowhere explained in more detail. Why are ρ_{A} and ρ_{E} not sufficient?
b) Explanations on Equation 19 can be expanded on. In particular, brief comments on the second, diffusionlike term and the last term on the righthand side could be helpful for readers.
c) The last term in Equation 19 is known to cancel the surface tension effect created by the second term (e.g. Biben and Misbah, 2003). Why do the authors explicitly remove surface tension in their phase field model while it is present in their interface models?
https://doi.org/10.7554/eLife.48885.sa1Author response
Reviewer #1:
In this manuscript, Xiong et al. describe how complex pattern formation arises within a 2species bacterial biofilm. Specifically, they show that combining E. coli and A. baylyi results in flowerlike patterns while neither species individually produces this effect. They use genetic perturbations to show that this effect requires motility (mixed biofilms with nonmotile A. baylyi do not produce patterns) and does not depend on T6SS. They go on to reproduce the patterns using a mathematical model that captures the differences in motility between the 2 strains.
Overall, I thought this was a very nice and clearly presented study. The phenotype is quite interesting and the authors do a good job of providing a plausible explanation (mechanical effect resulting from differences in motility). I have only one experimental suggestion to confirm this explanation. Can the authors mix motile and nonmotile A. baylyi and produce the patterns in a single species A. baylyi biofilm? If the floral patterns are really a result of motility differences (and not other unknown interspecies effects) than this should be possible. They have already created the nonmotile A. baylyi mutant so it should be straightforward to try this experiment. If they observe this result, then I am enthusiastic for publishing the manuscript.
Thank you for a very interesting suggestion that we followed. We carried out additional experiments mixing nonmotile A. baylyi with motile ones and observed interesting spatial structures similar to the flowerlike patterns in mixtures of E. coli and motile A. baylyi. Therefore, it appears that indeed our proposed mechanism of pattern formation is still applicable to this situation, although the patterns in mixtures of two A. baylyi strains were not as pronounced and robust as in mixtures of E. coli and A. baylyi. There could be several factors responsible for this: (i) the growth rate of nonmotile A. baylyi is less than that of E. coli (actually, it is even smaller than that of the motile A. baylyi due to burden from the selection marker), so the band of nonmotile A. baylyi at the perimeter of the colony is not as thick as E. coli’s; (ii) the effective friction coefficient of nonmotile A. baylyi may be smaller than that of E. coli and close to that of motile A.baylyi. We added a new Appendix 3 (Mixtures of motile and nonmotile A. baylyi) to discuss these new experimental results. We also added Video 8 to show the development of patterns in a mixture of motile and nonmotile A. baylyi.
Reviewer #2:
1) It would be helpful to discuss the origin of the expansion force, since only part of the colony is motile. Are twitching cells pushing the other strain through celltocell contact? (can the authors provide a high resolution image to show that cells are highly packed?) Or – given that an extracellular matrix is mentioned – is it entropically driven? (Is an extracellular matrix knockout available to the authors?)
Both mechanisms (entropicallydriven expansion and pushing each other through direct cellcell contacts) are not mutually exclusive. Based on our experimental findings that A. baylyi colonies expand but E. coli ones don’t, despite having comparable growth rates, we believe that the core reason for the colony expansion is motility, or effective temperature, i.e. it is entropically driven. Colonies do not begin to expand until they have completely covered the interior surface in a monolayer (see new Figure 1—figure supplement 2). Therefore, we believe the entropic expansion acts through direct cellcell contacts, when cells constantly bump and push each other. Because bacteria continuously grow and divide, the cell density remains high throughout the colony expansion. ECM has been shown to entropically drive phase separation and dispersion in nonmotile, singlespecies colonies (Seminara et al., 2011, Dilanji et al., 2014, Ghosh et al., PNAS 2015), but this does not appear to be a driving factor here, given the dependence on both motility (Figure 3E) and a mixture of two species (Figure 1C). We have now added a discussion of the origin of the expansion force in the Discussion section.
2) Are wetting forces negligible? An orderofmagnitude estimate of the different forces or experimental evidence would justify choice of these and not other forces.
We are not sure about the importance of wetting forces, however the drastic difference between the expansion of motile A. baylyi colonies compared with E. coli colonies and nonmotile mutant A.baylyi colonies (that presumably should have similar wetting properties), leads us to believe that passive wetting forces probably do not play an important role in the process. It leaves us with the (entropic by nature) expansion force and the resistance (bottom friction, viscosity) forces that comprise the force balance in the form of Stokes equation. We added such discussion in the Discussion section.
It is true that wetting forces could play a role in expansion of the colony onto new regions of agar. In principle, A. baylyi could generate more surface wetting, locally aiding expansion, and E. coli could generate less, or even inhibit wetting. This could indeed be part of the effective friction forces associated with the different cell types – with A. baylyi having less effective friction and E. coli having more. The expansion and friction forces are left here as largely phenomenological, though experimentally motivated. It would certainly be worthwhile to further explore the physical underpinnings of these forces, but we believe that is an area for future work, although worth including in the Discussion, which we have done in the revised version.
3) In the more refined model, 2D concentrations of the two strains and of the phase field are evolved in time. I would imagine the two strains move relative to one another (E. coli remains attached to the substrate whereas A. baylyi twitches). A twofluid mixture appears a natural choice; the authors consider instead a single velocity field carrying all cells, and I am unclear what this velocity represents. Is there a way to relate it to actual velocities measurable experimentally, e.g. a weighted average of velocities of the two strains? Because I am confused about the meaning of v, I do not have an intuition why it follows Stokes equation.
Our close inspection of the growing colony showed that the velocities of the two strains in close proximity are very similar, which is not very surprising since the mixture of E. coli and A. baylyi is dense (new Figure 1—figure supplement 2), liquidlike, and miscible. We saw no evidence of the two strains expanding at different velocities: E. coli were carried with the surrounding A. baylyi (hitchhiking) or pushed by it in the front of the colony. Thus, we decided that it would make more sense to use a single local velocity, which represents the actual velocity of the bacterial cells. Since the singlevelocity model already appears sufficient in capturing the salient features of the phenomenon, we did not see a reason to make it more complex than necessary. We mention this rationale in the revised version of Appendix 2.
4) The authors predict that at low friction, the colony expands quicker with no flower like pattern, unless a large concentration of E. coli is inoculated. Assuming agar concentration affects friction, these predictions are verified experimentally. A discussion is missing about the choice of ξ and β, which appears to me quite arbitrary. Also: is the value of η obtained by fitting the expansion rate a plausible value for a bacterial colony?
Indeed, the parameters of the model (friction and viscosity coefficients in particular) depend on a number of physical and biochemical properties of the system (the agar composition, the properties of the extracellular matrix, density of pili on the surface of bacterial cells, etc.). Thus, we selected the parameters based on fitting the observed colony dynamics, as we explain in more detail in Appendix 2. One of the parameters in the stationary Stokes equation (22) can be scaled out without affecting the velocity, and we chose the basal friction coefficient of pure A. baylyi, \xi=1. Base on the presence of sharp kinks in the developing front structure, we concluded that viscosity plays a minor role in the dynamics, so we chose the viscosity coefficient to be small. The remaining two parameters (η and β) were chosen based on fitting the average expansion rates of colonies of pure A. baylyi (η) and different A. baylyi / E. coli mixtures (β). We provided a more detailed discussion of the model parameters in revised Appendix 2.
5) Hitchhiking is an appealing evolutionary advantage. The discussion would benefit from two additional points. First: the authors imply that flowerlike patterns are directly related to hitchhiking. However from Figure 3A it looks like E. coli hitchhikes even in round colonies, maybe less so than in flowerlike colonies? Could you quantify this aspect? Second: what are the parameters that switch hitchhiking on/off in the model? Are there evidences that these parameters vary in different strains? Could genes control these parameters? This would help elucidate whether hitchhiking may be actively controlled (e.g. strains turning off motility would benefit from correspondingly turning on hitchhiking)
Hitchhiking appears to be a necessary but not sufficient condition for flowerlike pattern formation. Indeed, we observed that E. coli also hitchhiked in round colonies (e.g. in Figure 5F third panel), however patterns did not form, presumably because E. coli did not exert significant friction in those conditions. In the phasefield model that would correspond to reducing parameter \β that controls E. colidependent increase in friction. For small β, patterns indeed don’t form, while hitchhiking fully persists.
Reviewer #3:
In the following, I will focus on discussing the mechanical modeling part, which falls more in my area of expertise. While the interface model has appeal due to its simplicity, there are a few things that need to be better clarified if not corrected about it (see below). The phase field modeling seems to be mostly appropriate with respect to the assumptions made, but also here at least some clarifications would be good, mostly related to the field Φ (see below).
More detailed comments:
1) A brief discussion on possibilities of where the force F0 and the "active pressure" in Equation 22 could originate would be good (also in the initial phase where the colony is not expanding (Figure 1D)). In subsection “Patternforming instability originates at the colony interface” the authors suggest some kind of active pressure created by the A. Baylyi motility. However, it is not clear to me how this alone could explain an initial phase where the colony is not expanding at all (Figure 1D). The latter seems to be more consistent with a picture of nutrient consumption combined with chemotaxis (as proposed in earlier work, Discussion section paragraph two), which could also effectively create such an F0.
The origin of the expansion force F_{0} is explained in the answer to the question 1 of reviewer 2. Briefly, it is related to the motility of A. baylyi so when they move randomly and bump into each other, it creates pressure field that leads to the colony expansion. However, when bacterial cells are inoculated, the cell density is low (see new Figure 1—figure supplement 2) and cells are not pushing against each other. That is the reason the colonies do not expand initially (Figure 1D). From Author response image 1 which shows the time course of the colony radius (of a mixture of E. coli and A. baylyi) obtained from numerical simulations of the phase field model with low initial density of A. baylyi, it can be seen that initially the colony does not expand, but once the cell density becomes large (order of 1), the colony starts to expand.
2) There are problems or at least a lack of clarity related to the way the friction force F_{r} is included in both 1D models describing the interface behavior.
a) The authors assume F_{r} to be proportional to the concentration of E. coli, c, but independent of velocity. As a consequence, if the concentration is high enough, the surface would move inward (even without surface tension F_{s}=0), driven by this friction force. How realistic is such a friction force?
The reviewer is correct, in the original formulation of the interface model the friction force F_{r} was assumed to be independent of velocity. We neglected to mention that we enforced the condition that the friction force never exceeds F_{0}, so the total normal force acting on the interface was always nonnegative. Thus, the interface could stall but would never move backwards. We apologize for this omission.
b) The normal velocity F[κ, g] is computed directly as a difference of normal forces (Equation 8). Apparently, the authors have implicitly assumed some additional, velocitydependent friction here (and set the friction coefficient to one). It would be good to comment on how this additional friction is motivated.
The reviewer is also correct here; in the original formulation of the interface model we assume an overdamped motion of the interface in which the normal velocity is proportional to the balance of the pushing force and the velocityindependent friction force acting on it. Indeed, this implied that there was an additional interface resistance force proportional to the interface velocity.
c) As an alternative, one could assume F_{r} itself to depend linearly on velocity, a common assumption for motile cells on a substrate, which is also used in the authors' phasefield model (Equation 22). In this case and without additional friction, through force balance (F0F_{s}F_{r}=0) the normal velocity F[κ, g] would be given by a quotient between F0F_{s} and a friction coefficient that is a function of c.
This is an excellent suggestion, and we decided to implement it in order to make the interface model more consistent with the phasefield model where both basal and E. colidependent friction forces are proportional to the local velocity. Thus, we assumed that the normal velocity of the interface is proportional to the sum of the constant normal force F_{0}, curvaturedriven surface tension and interface selfrepulsion forces divided by the “friction coefficient” that has a constant (basal) term and concentrationdependent term. It only required changing two lines of code, and we did not have to change any of the model parameters. As we expected, this modification of the model did not change its stability properties, and the resultant flowerlike shape of the interface remained qualitatively the same. We changed the description of the model in the corresponding section of Appendix 1, and Figure 4B,C. We are grateful to the reviewer for this valuable input that made our model more reasonable.
3) Even though the phase field approach is similar to previous work, some explanations, in particular related to the field Φ would make the manuscript more selfcontained.
a) In subsection “Phasefield model of flowerlike pattern formation” the authors state that Φ is introduced to avoid computational difficulties of dealing with the boundary. However, these difficulties and hence the necessity of Φ is nowhere explained in more detail. Why are ρ_{A} and ρ_{E} not sufficient?
We appreciate the comment of the reviewer. Please note that the evolution of the boundary of the colony is determined by the velocity field u, which, in turn, depends on ρ_{A} and ρ_{E}. Thus, it is not sufficient to have only equations for ρ_{A} and ρ_{E} but need an additional equation that governs the motion of the colony. Tracking a moving boundary using traditional techniques is challenging. For this reason, we have chosen to use the phase field method to simulate our growing colony. We have now added additional text to explain this better.
Specifically, we now write in the main text:
“It is based on PDEs for the densities of A. baylyi ρ_{A} and E. coli ρ_{E}, together with an equation that describes the velocity field u of the colony. This velocity field drives the motion of the boundary of the colony and is generated by a combination of stress due to cell growth and motility, viscosity, and bottom friction that is dependent local E. coli density. The resulting free boundary problem is solved using the phasefield method, which introduces another PDE for an auxiliary field phi that changes continuously from 1 inside the colony to 0 outside (see Appendix 2 for the detailed formulation of the model). The boundary is then automatically defined as Φ=1/2 and can thus be computed without explicit tracking techniques.”
b) Explanations on Equation 19 can be expanded on. In particular, brief comments on the second, diffusionlike term and the last term on the righthand side could be helpful for readers.
Our phase field description follows our earlier work on cell migration (Shao et al., 2012). To clarify this, we have added additional explanation for the different terms in Equation 19. Please note that the last term in this equation is added to stabilize the interface. This term is a computational technique, which was indeed described by Biben and Misbah, 2003 and Biben et al., 2005. We have now also added these references to the text in Appendix 3.
c) The last term in Equation 19 is known to cancel the surface tension effect created by the second term (e.g. Biben and Misbah, 2003). Why do the authors explicitly remove surface tension in their phase field model while it is present in their interface models?
The required restoring force in the interface model is provided by surface tension. Without such term, from Jacobian matrix (12), the positive eigenvalue λ(k) is proportional to k and goes to infinity as k increases. This restoring force ensures that the interface does not expand indefinitely. In our phase field model, the friction force constrains the expansion, resulting in finite expansion rates. Thus, surface tension, which can in principle be added to the phase field as a separate force term, is not necessary. We added such discussion in Appendix 3.
https://doi.org/10.7554/eLife.48885.sa2Article and author information
Author details
Funding
National Institutes of Health (R01GM069811)
 Lev Tsimring
 Liyang Xiong
 Robert Cooper
 Jeff Hasty
National Science Foundation (PHY1707637)
 Yuansheng Cao
 WouterJan Rappel
National Institutes of Health (San Diego Center for Systems Biology (P50GM085764))
 Lev Tsimring
 Liyang Xiong
 Robert Cooper
 Jeff Hasty
Office of Naval Research (N000141612093)
 Lev Tsimring
 Liyang Xiong
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank Megan Dueck for building the custom ‘milliscope’ used in our study, Philip Bittihn for helpful discussions, and Kit Pogliano lab for providing the original E. coli strain. This work was supported by the National Institutes of Health (grant R01GM069811), the National Science Foundation (grant PHY1707637), San Diego Center for Systems Biology (NIH grant P50GM085764) and the DOD Office of Naval Research (grant N000141612093).
Senior Editor
 Detlef Weigel, Max Planck Institute for Developmental Biology, Germany
Reviewing Editor
 Aleksandra M Walczak, École Normale Supérieure, France
Reviewer
 Agnese Seminara, French National Centre for Scientific Research, France
Publication history
 Received: May 29, 2019
 Accepted: November 16, 2019
 Version of Record published: January 14, 2020 (version 1)
Copyright
© 2020, Xiong et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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