A multiphase theory for spreading microbial swarms and films
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Abstract
Bacterial swarming and biofilm formation are collective multicellular phenomena through which diverse microbial species colonize and spread over waterpermeable tissue. During both modes of surface translocation, fluid uptake and transport play a key role in shaping the overall morphology and spreading dynamics. Here we develop a generalized twophase thinfilm model that couples bacterial growth, extracellular matrix swelling, fluid flow, and nutrient transport to describe the expansion of both highly motile bacterial swarms, and sessile bacterial biofilms. We show that swarm expansion corresponds to steadystate solutions in a nutrientrich, capillarity dominated regime. In contrast, biofilm colony growth is described by transient solutions associated with a nutrientlimited, extracellular polymer stress driven limit. We apply our unified framework to explain a range of recent experimental observations of steady and unsteady expansion of microbial swarms and biofilms. Our results demonstrate how the physics of flow and transport in slender geometries serve to constrain biological organization in microbial communities.
https://doi.org/10.7554/eLife.42697.001eLife digest
Bacteria can grow and thrive in many different environments. Although we usually think of bacteria as singlecelled organisms, they are not always solitary; they can also form groups containing large numbers of individuals. These aggregates work together as one supercolony, allowing the bacteria to feed and protect themselves more efficiently than they could as isolated cells.
These colonies move and grow in characteristic patterns as they respond to their environment. They can form swarms, like insects, or biofilms, which are thin, flat structures containing both cells and a filmlike substance that the cells secrete. Availability of food and water influences the way colonies spread; however, since movement and growth are accompanied by mechanical forces, physical constraints are also important. These include the ability of the bacteria to change the water balance and their local mechanical environment, and the forces they create as they grow and move.
Previous research has used a variety of experimental and theoretical approaches to explain the dynamics of bacterial swarms and biofilms as separate phenomena. However, while they do differ biologically, they also share many physical characteristics.
Srinivasan et al. wanted to exploit these similarities, and use them to predict the growth and shape of biofilms and bacterial swarms under different conditions. To do this, a unified mathematical model for the growth of both swarms and biofilms was created. The model accounted for various factors, such as the transport of nutrients into the colony, the movement of water between the colony and the surface on which it grew, and mechanical changes in the environment (e.g. swelling/softening). The theoretical results were then compared with results from experimental measurements of different bacterial aggregates grown on a soft, hydrated gel. For both swarms and biofilms, the model correctly predicted how fast the colony expanded overall, as well as the shape and location of actively growing regions.
Biofilms and other bacterial aggregates can cause diseases and increase inflammation in tissues, and also hinder industrial processes by damage to submerged surfaces, such as ships and waterpipes. The results described here may open up new approaches to restrict the spreading of bacterial aggregates by focusing on their physical constraints.
https://doi.org/10.7554/eLife.42697.002Introduction
Bacteria employ sophisticated surface translocation machinery to actively swarm, twitch, glide or slide over solid surfaces (Kearns, 2010; Mattick, 2002; Spormann, 1999; Hölscher and Kovács, 2017). Collectively, they also aggregate into multicellular communities on hydrated surfaces and exhibit largescale coordinated movement (Verstraeten et al., 2008). Surface motility in macroscopic colonies on hydrated surfaces such as gels occurs primarily via two distinct modes: either by rapid flagellamediated swarming expansion (Harshey, 1994; Harshey, 2003), or alternatively by slow biofilm expansion driven by extracellular polymer matrix production (HallStoodley et al., 2004). In both cases, an interplay between mechanical constraints and biological organization sets limits on the overall colony morphology and expansion dynamics (Persat et al., 2015). The forces driving colony expansion are generated by nonhomogeneous patterns of biological activity, originating from spatial localizations in cell growth and division (Hamouche et al., 2017), extracellular polymer matrix production (Seminara et al., 2012; Yan et al., 2017; Srinivasan et al., 2018), osmolyte secretion (Ping et al., 2014) and active stresses (Farrell et al., 2013; Delarue et al., 2016). Conversely, the formation of localized biologically active zones is tightly coupled to the heterogeneity of the environment, including the diffusion and transport of nutrients (Wang et al., 2017), accumulation of metabolic byproducts (Liu et al., 2015; Gozzi et al., 2017) and presence of quorum sensing and signaling agents that regulate celldifferentiation and development.
Consequently, the dynamics of colony growth requires a mechanistic description that accounts for spatiotemporal inhomogeneities in biological activity, emergent forces, and flows that transport metabolic agents. In bacterial swarming, cells within the colony are actively propelled by the rotation of flagella in a thin layer of fluid extracted from the underlying soft tissue or gel (Kearns, 2010). In contrast, bacterial biofilms are surface aggregates of sessile bacteria embedded in a selfgenerated extracellular polymer matrix (Flemming and Wingender, 2010). Despite marked differences in regulatory genetic pathways, morphology and cell function (Verstraeten et al., 2008), physical characteristics such as the fluidization of the substrate/tissue, gradients in nutrient availability, the lowaspectratio geometry and the existence of multiple phases (i.e. cells, biopolymer and fluid) are common to both bacterial film and swarm colonies. Motivated by these similarities, we present a unified multiphase framework that couples mechanics, hydrodynamics and transport to explain the dynamics of bacterial swarm and film expansion.
Experimental background
Bacterial swarms
Experiments on swarming colonies of E. coli (Darnton et al., 2010; Wu and Berg, 2012; Ping et al., 2014), S. enterica (Harshey and Matsuyama, 1994; Butler et al., 2010; Kalai Chelvam et al., 2014; Chen et al., 2007) and P. aeruginosa (Yang et al., 2017) reveal certain reproducible features associated with this modality of collective behavior. For example, E. coli swarms on agarose gels have a steady front shape that propagates radially at a uniform speed (Wu and Berg, 2012). In these swarms, measurements of the osmotic pressure profiles were found to be consistent with the active secretion of wetting agents in regions of high cell density that serve to fluidize the swarm by extracting water from the underlying tissue, thus allowing it to spread (Ping et al., 2014). These observations are not unique to E. coli; indeed our experiments with B. subtilis swarms, following (Kearns and Losick, 2003), indicate the same phenomena, that is a steadystate front shape and speed, as shown in Figure 1A–1E. Close to the spreading front, we observe a multilayer region of width $W$ = 195 µm ± 35 µm, indicated by the dashed white lines in Figure 1B and 1C. The multilayer region correlates with increased colony thickness and local bacterial density (Wu and Berg, 2012). At the edge, and in the interior, there is just a monolayer of cells. The swarm radial expansion velocity is constant at $V$ = 2 mm/hr (see Figure 1D) and the swarm front maintains a steadystate profile during expansion (see Figure 1E). These observations raise a number of natural questions associated with the steadystate velocity and profile of the swarm colony. Given the observations of osmotic gradientdriven flow in the vicinity of the growing front (Ping et al., 2014), coupled with variations in the thickness and activity of bacteria, any framework to explain these requires a consideration of a dynamic bacterial population interacting with ambient fluid, necessitating a multiphase description.
Bacterial films
In contrast with bacterial swarms, the spreading of bacterial biofilms is faciliated by the extracellular polymeric substance (EPS) matrix that expands via osmotic fluid influx, for example in B. subtilis (Seminara et al., 2012) and V. cholerae (Yan et al., 2017) biofilm colonies. However, EPS synthesis is not homogeneous, and depends on the local nutrient concentration and environmental heterogeneities experienced by cells within the same biofilm (Vlamakis et al., 2008; Berk et al., 2012). Recently, it was shown that the EPS matrix production is localized to cells in the propagating front of B. subtilis biofilms (Srinivasan et al., 2018). In Figure 1F–1J, we show the results of repeating these experiments, but now focusing on a peripheral region of a biofilm colony using a B. subtilis strain (MTC832) that harbors the ${P}_{\text{\mathit{t}\mathit{a}\mathit{p}\mathit{A}}}\text{cfp}$ construct as a reporter for matrix production activity (Wang et al., 2016; Srinivasan et al., 2018). This highlights a $\sim 1$ mm zone of matrix production activity at the periphery, seen in Figure 1G and H; indeed plots of averaged matrix production reporter intensity exhibit a distinct peak at the periphery, as shown in Figure 1J. The dynamics of radial expansion shows the existence of an initial acceleration regime followed by a transition to a second regime characterized by a monotonic decrease in expansion velocity, as plotted in Figure 1I. This transient mode of biofilm spreading driven by EPS production and swelling is quite different from that of bacterial swarming, and suggests that we might need a fundamentally different way to address its origins. However, if we now consider the EPS matrix and fluid as distinct phases (Cogan and Keener, 2004; Cogan and Keener, 2005; Winstanley et al., 2011; Seminara et al., 2012), with the bacterial population being relatively small, we are again led to a multiphase description of the system, but with a different dominant balance relative to that seen in bacterial swarms, which we now turn to.
Theoretical framework
Recent theoretical approaches have considered specific physical factors such as the wettability of the biofilm (Trinschek et al., 2016; Trinschek et al., 2017), osmotic pressure in the EPS matrix (Winstanley et al., 2011; Seminara et al., 2012), or Marangoni stresses associated with the swarm fluid (Fauvart et al., 2012), as reviewed by Allen and Waclaw (2019). However, a description that captures the experimental observations described in Figure 1 remains lacking. Here, given the similarities between the bacterial swarming and biofilm systems, we provide a unified description of their spreading dynamics by recognizing that in both cases we need to consider large slender microbial colonies with $H/R\ll 1$, where $H$ is the colony thickness and $R$ is the radius. This approximation results in a quasi2dimensional, twophase model (assuming axisymmetry) of a colony that spreads along the xaxis, with a varying thickness, as shown in Figure 2. The subscript i = (1,2) denotes the actively growing phase and passive phase, respectively. Within the swarm colonies, the highly motile cells constitute the actively growing phase whereas the fluid comprises the passive phase. Similarly, in biofilms, the EPS matrix constitutes the active phase, and the aqueous fluid is the passive phase.
In both cases, colony growth occurs over a semipermeable soft gel substrate, as shown in Figure 2. We develop a continuous description of colony expansion in terms of variables which are coarsegrained depth integrated averages (Drew, 1983; Ishii and Hibiki, 2011), The averaged height of the colony interface is $h(x,t)$, the volume fraction of the active phase (i.e., swarmer cells or polymer matrix) is ${\varphi}_{1}=\varphi (x,t)$ and the volume fraction of the fluid phase is ${\varphi}_{2}=1\varphi (x,t)$. The 1D substrate depthaveraged nutrient concentration field within the substrate is $c(x,t)$. As detailed in Appendix 2, combining mass and momentum balances yields the following generalized set of partial differential equations that governs the dynamics of both expanding swarms and biofilms,
where, ${(\cdot )}_{x}=\partial (\cdot )/\partial x$, etc. Here, ${Q}_{1}(x)$ is the horizontal flux in the active phase, ${Q}_{2}(x)$ is the horizontal flux in the fluid phase and ${V}_{0}(x)$ is the osmoticallydriven net vertical fluid influx per unit length across the permeable substrate. Furthermore, ${g}_{1}(h,c,\varphi )$ is the depth integrated active phase growth rate within the bacterial colony, and ${g}_{2}(h,c,\varphi )$ is the depth integrated nutrient uptake rate. The dynamics of swarms and biofilms differ in the details of the expressions for ${Q}_{1},{Q}_{2}$, ${V}_{0}$, which are provided in Table 1. While a full derivation of each term is provided in Appendix 2, a direct comparison of the terms listed in Table 1 reveals a number of structural similarities and differences.
Nutrient uptake
For both swarms and biofilms, the active phase (i.e., swarm cells or the EPS matrix) is generated within the bacterial colony by converting nutrient in the underlying substrate to biomass. The rate of change of nutrient concentration within the substrate depends on diffusion and nutrient uptake (see Equation (3) and Equation (A2)), and is derived in Appendix 2. When the substrate concentration is scaled by the initial concentration ${c}_{0}$, the nutrient depletion rate depends on $\mathrm{\Gamma}/{c}_{0}$, the ratio of the specific nutrient consumption rate to the initial concentration. Bacterial swarming is typically associated with nutrient rich conditions, where ${c}_{0}\gg \mathrm{\Gamma}$. As a result, the nutrient uptake term can be neglected in bacterial swarming as ${g}_{2}\to 0$, and the concentration $c\approx {c}_{0}$ throughout swarm expansion. In contrast, biofilm growth occurs under nutrient limited conditions where $\mathrm{\Gamma}/{c}_{0}\sim O(1)$, resulting in a corresponding uptake term shown in Table 1. Therefore, biofilm expansion is necessarily unsteady and driven by the dynamics of the transient nutrient field.
Growth
In both swarms and biofilms, the generation of the active phase drives colony expansion and is described by the growth term in Equation (1) using a logistic function ${g}_{1}={g}_{0}h\varphi (1h\varphi /(H{\varphi}_{0}))$ to model the active phase growth, where $H{\varphi}_{0}$ is the limiting thickness, and ${g}_{0}$ indicates a specific growth rate. In bacterial swarms, ${g}_{0}$ is independent of the nutrient concentration (as $c\approx {c}_{0}$ during swarm expansion). Therefore, the spreading swarm films have a steadystate structure that exhibits a central spatial plateau about $h\varphi =H{\varphi}_{0}$. In contrast, biofilm growth corresponds to a nutrient poor environment. We model the biofilm growth dependence on nutrient concentration via a minimal MichaelisMenten form ${g}_{0}=Gc/(K+c)$, . Unlike in nutrient rich conditions associated with swarms, this implies that biofilm growth is fundamentally transient; once the nutrient field at the interior is depleted as $c\to 0$, biofilm growth term in that region is arrested and ${g}_{1}\to 0$ independently of the vertical thickness (i.e., even if $h\varphi \ne H{\varphi}_{0}$). As a result, the biofilm does not give form a central plateau and the dynamics of the biofilm rim is fixed by the dynamics of nutrient depletion. Eventually the effect of the finitesize of the system (the petri dish) also becomes important it determines the overall dynamics of nutrient depletion.
Active and passive fluxes
The terms ${Q}_{1}(x)$ and ${Q}_{2}(x)$ that represent the horizontal flux of the active and passive phases are obtained by depth integrating the momentum balance equations in the thinfilm lubrication limit, as described in Appendix 2 (c.f. Equations (A9)(A11)). Within bacterial swarms, the passive aqeuous fluid phase is modeled as a Newtonian liquid with viscosity ${\mu}_{2}$. The first term of ${Q}_{1}(x)$ and ${Q}_{2}(x)$ in Table 1 for swarms is generated by viscous and capillary stresses within the swarm fluid. The active swarmer cells are treated as inviscid and subjected to a hydrodynamic frictional drag force. Specifically, we assume that individual bacteria within the swarm are undergoing a random walk process with zero net displacement (upon averaging over sufficiently large timeintervals). Even though there is no overall displacement, there is a net timeaveraged drift that arises from viscous stokes drag interaction between the fluid and the active bacteria. The second term for ${Q}_{1}(x)$ in Table 1 represents this timeaveraged drift arising from frictional drag interaction of the bacteria with the swarm fluid.
In biofilms, the EPS matrix phase constitutes an active viscous hydrogel network with viscosity ${\mu}_{1}$, whereas the passive aqueous fluid phase is treated as a solvent with viscosity ${\mu}_{2}$. The dominant stress within the EPS phase in the biofilm model arises from a FloryHuggins swelling pressure in the polymer chains (Cogan and Guy, 2010; Winstanley et al., 2011). In the fluid phase, the pressure ${p}_{f}$ is set by surface tension and curvature of the swarm fluid. Both these stresses contribute to the effective EPS phase pressure term $\mathrm{\Pi}(x)$, as described in Appendix 2. Consequently, the first term for ${Q}_{1}(x)$ and ${Q}_{2}(x)$ in Table 1 for biofilms is related to the gradient of the effective pressure. Moreover, following Winstanley et al. (2011), we assume that the capillary and viscous stresses in the swarm fluid are negligible when compared to the frictional drag due to flow between water and the EPS polymer chain network in the biofilm model. Therefore, the second term for ${Q}_{2}(x)$ in Table 1 represents a Darcytype flow of the aqueous phase within the EPS matrix. The osmotic influx terms are considered separately in the following sections when describing the equations governing swarm and biofilm expansion.
Bacterial swarms
Species of bacteria that swarm on hydrated surfaces are known to secrete distinct wetting agents. For example, B. subtilis secretes the lipopeptide surfactin, whereas P. aeruginosa secrets rhamnolipids as the wetting agent. Consequently, existing thinfilm models to describe bacterial swarming assume that gradients in wetting agent activity generate Marangoni stresses that drives swarming motility (Fauvart et al., 2012; Trinschek et al., 2018). However, E. coli exhibits swarming behavior despite the absence of lipopeptides or other agents that act as surfactants. Moreover, recent experiments (Yang et al., 2017) demonstrates that P. aeruginosa swarms robustly even after exogenously eliminating gradients in surfactant concentration within the swarm fluid, eliminating Marangoni flows as the principal mechanism that drives swarming. Here, we take a different approach based on experiments that show that steadystate swarm colony expansion maybe mediated by secretion of agents that are osmotically active (Wu and Berg, 2012). As we will see, this leads to fluid being extracted from the substrate near the front, then driven into the colony by capillary and viscous stresses, and eventually returns into the substrate in the interior of the swarm.
Within the bacterial swarms, the dominant phases are the swarmer cell phase, and the viscous aqueous phase, as shown in the bottom panel of Figure 2A. Fluid uptake from the substrate is regulated by the secretion of osmotically active agents by the swarmer cells (Ping et al., 2014). We represent the osmotic agent in the fluid by a concentration field, ${c}_{\text{osm}}(\varphi )$ that is proportional to the local volume fraction of cells such that $c\phantom{\rule{thinmathspace}{0ex}}\propto \phantom{\rule{thinmathspace}{0ex}}\varphi /(1\varphi )$, and gives rise to an osmotic pressure described by van’t Hoff’s law as (van't Hoff, 1887), $\mathrm{\Delta}\mathrm{\Psi}=\left({\mathrm{\Psi}}_{0}\varphi /(1\varphi ){\mathrm{\Psi}}_{\text{eq}}\right)$, that drives the fluid intake. Here, ${\mathrm{\Psi}}_{0}$ is the osmotic pressure scale in the swarm fluid and ${\mathrm{\Psi}}_{\text{eq}}$ is the equilibrium osmotic pressure within the underlying tissue/gel substrate. Away from the front, in the interior of the swarm colony, there is no net fluid influx (Ping et al., 2014). Therefore, the equilibirium volume fraction of the swarm cells at the interior is, ${\varphi}_{0}={\mathrm{\Psi}}_{\text{eq}}/({\mathrm{\Psi}}_{0}+{\mathrm{\Psi}}_{\text{eq}})$. At the front itself, the difference in osmotic pressure results in a net Darcytype fluid influx into the swarm, ${V}_{0}(x)$, expressed as,
where ${Q}_{0}$ is a velocity scale associated with fluid inflow from the substrate. Measurements of cell replication within swarms reveals that growth is restricted to swarmer cells at the periphery (Hamouche et al., 2017), which we model using a modified logistic growth term ${g}_{1}(h,\varphi )$ as listed in Table 1, that localizes all cell division to the periphery. Here, $H{\varphi}_{0}$ is the limiting thickness of the swarmercell phase at the interior, and ${g}_{0}$ is an effective specific growth rate, related to true specific cell growth rate by a geometric factor (see discussion in Appendix. [2]).
Parameters and scaling laws for bacterial swarms
To make sense of the scales in the problem, we use the dimensionless variables $\widehat{x}=x/L$, $\widehat{z}=z/H$ and $\widehat{t}=t{g}_{0}$ where H is the vertical length scale, L is a horizontal length scale and $1/{g}_{0}$ is the timescale associated with bacterial growth. The resultant horizontal velocity scale in the swarm colony is $U=L{g}_{0}$. Swarm expansion is fluid driven, and therefore balancing the viscous stresses generated in the swarm fluid, with the curvature pressure due to surface tension (Levich and Landau, 1942) results in ${\mu}_{2}U/{H}^{2}\sim \gamma H/{L}^{3}$, where ${\mu}_{2}$ is the viscosity and $\gamma $ is the surface tension of the aqueous phase. As a result, the natural horizontal length scale is $L=H{(\text{\mathit{C}\mathit{a}})}^{1/3}$, where $\text{\mathit{C}\mathit{a}}=({\mu}_{2}U/\gamma )$ is a capillary number associated with the microbial swarm fluid. Consequently, in our model the expansion speed of the swarm colony, $V=dR/dt$, is determined by the product of the horizontal length scale and an effective growth rate, and is predicted to scale as,
whereas, the swarm front itself is analogous to a capillary ridge in thin fluid film with a width $W$ that is predicted to scale as,
where, ${C}_{1}$ and ${C}_{2}$ are dimensionless prefactors that require a detailed numerical calculation, and are discussed later. There are two important dimensionless parameters that describe swarm colony expansion. The first dimensionless parameter, ${\alpha}_{1}$, relates the magnitude of capillary forces to the viscous drag acting on cells within the swarm and is defined as ${\alpha}_{1}=(\gamma H/{L}^{2})/(\zeta LU)$. Here, $\zeta ={\zeta}_{c}/{V}_{c}$ where ${\zeta}_{c}$ is the friction coefficient of a single swarmer cell and ${V}_{c}$ is its volume. The second dimensionless parameter ${\alpha}_{2}$ is defined as the ratio of a vertical fluid influx velocity ${Q}_{0}$, to a thickness velocity scale $H{g}_{0}$ associated with bacterial growth as ${\alpha}_{2}={Q}_{0}/(H{g}_{0})$.
The vertical length scale and equilibrium fluid volume fraction are estimated from the interior monolayer region as $H$ = 0.5 μm and ${\varphi}_{0}=0.5$ (Wu and Berg, 2012). We assume values of ${\mu}_{2}={10}^{3}$ Pa.s for the (aqeuous) swarm fluid viscosity, and $\gamma ={10}^{2}$ N/m as its surface tension. The friction coefficient of a single cell is estimated from Stokes law as ${\zeta}_{c}=3\pi {\mu}_{2}a$, and its volume is approximated as ${V}_{c}=\pi {a}^{3}/6$, where $a$ = 1 μm is the cell diameter. Therefore, the friction coefficient is $\zeta ={\zeta}_{c}/{V}_{c}\approx 18{\mu}_{2}/{a}^{2}$. As a result of substituting the values of known parameters above, the dimensionless parameter ${\alpha}_{1}$ reduces to a constant geometric ratio, ${\alpha}_{1}\approx 2{a}^{2}/{H}^{2}\approx 2/9\approx 0.22$.
The value of ${\alpha}_{2}$ depends on the ratio ${Q}_{0}/{g}_{0}$. Direct experimental measurements of the vertical influx fluid velocity profile ${V}_{0}(x)$ and the spatial profiles of cell division in swarm colonies remain scarce (Hamouche et al., 2017). In order to make progress in validating our model with real experimental data, the vertical fluid influx velocity scale is chosen as ${Q}_{0}={10}^{2}\phantom{\rule{thinmathspace}{0ex}}\mu$m/s. Consequently, we have chosen ${g}_{0}$ as the only fitting parameter in our study, as detailed in Appendix 2. As an example, in the following section we will show that a choice of ${g}_{0}=0.013$ s^{1} in our model reproduces the experimental swarm expansion speed shown in Figure 1D, and leads to a horizontal length scale of $L=H(\mathit{\text{Ca}}{)}^{1/3}=100\phantom{\rule{thinmathspace}{0ex}}\mu$m, velocity scale of $U=L{g}_{0}=1.3\phantom{\rule{thinmathspace}{0ex}}\mu$m s^{1}, $\text{\mathit{C}\mathit{a}}=1.3\times {10}^{7}$ and a value of ${\alpha}_{2}\approx 1.5$. A complete set of parameters for three experimental measurements of swarm expansion in B. subtilis, and two existing measurements in E. coli previously reported by Darnton et al. (2010) and Wu and Berg (2012) are summarized in Appendix 2.
Steady state swarms
With these assumptions, and assuming that the nutrient concentration is constant, Equations (1–3) reduce to the following scaled equations in the swarming limit,
To complete the formulation of the problem, we need five boundary conditions which are ${\widehat{h}}_{\widehat{x}}(0)={\widehat{h}}_{\widehat{x}}({R}_{P})=0$, ${\widehat{h}}_{\widehat{x}\widehat{x}\widehat{x}}(0)={\widehat{h}}_{\widehat{x}\widehat{x}\widehat{x}}({R}_{P})=0$, and $\varphi (0)={\varphi}_{0}$, where ${R}_{P}$ is the dimensionless size of the petridish and is set much larger than the colony size ($\widehat{R}=150$) in our simulations. The initial condition corresponds to a circularly inoculated swarm colony, along with a thin prewetting film where no bacterial growth occurs (see Appendix 3—figure 1).
Solving Equations (7) and (8) with the prescribed initial and boundary conditions numerically results in a steady state solution that advances at a constant speed (see Figure 3). In Figure 3, we plot a representative steady state solution in the frame of the advancing front for ${\alpha}_{1}=0.2$, ${\alpha}_{2}=1.5$ and ${\varphi}_{0}=0.5$. At the interior of the swarm, the average cell volume fraction is $\varphi \approx {\varphi}_{0}$. Near the leading edge of the swarm, there is a region of enhanced thickness as indicated by the red line in Figure 3A. Immediately behind the leading edge, where the cell concentration is highest, so is the osmolyte concentration leading to fluid extraction from the substrate, while further behind, fluid is reabsorbed, as indicated by the arrows in Figure 3A. In Figure 3B, we show the steadystate osmotic flow solution and see that it correlates well with the experimentally measured osmotic pressure profile by Ping et al. (2014) in E. coli swarms. As shown in Appendix 3—figure 3, our numerical horizontal flow profiles are also consistent the scaled radial fluid velocity measurements of Wu and Berg (2012). In Figure 3C, we see that the radial expansion velocity scales as $H{g}_{0}$ and shows quantitative agreement with experiments and is insensitive to the fluid influx velocity scale when ${Q}_{0}\gg {g}_{0}H$. Note that our model uses a coarsegraining procedure and represents the swarm thickness field using a continuum approximation. As a consequence, we are not able to quantitatively capture the decreasing height of the swarm (i.e., of the order of a few cells), that is experimentally observed over hundreds of micron towards the interior (see Figure 1E).
Furthermore, we corroborate our scaling law in Equation (5) by fitting our model to five independent experimental measurements of swarm expansion velocities for different systems, as shown in Figure 3D. These include measurements in B. subtilis swarms in this work, and in E. coli swarms previously reported by Darnton et al. (2010) and Wu and Berg (2012) that are summarized in Table A2 in Appendix 2. The expansion velocity follows the $1/3$ exponent predicted by Equation (5) for $Ca$ varying from ∼ 5 × 10^{8} to 10^{6}. For each experiment, we have fit our theoretical model using the effective growth rate ${g}_{0}$ as the fitting parameter and find that the numerical prefactor ${C}_{1}\approx 0.42$. However, as shown in Appendix 3—figure 5 in the Appendix, the measured multilayer width does not follow the predicted scaling. From an experimental point of view, the width of the multilayer region is not sharply defined in Figure 1E, and will depend on the choice of threshold. However, our multiphase model is able to describe the zone of cellular and osmolyte activity near the leading edge that drives the advancing swarm front. This leads to a picture wherein the combination of a fluidfilled substrate and swarm front work together like a localized active circulatory system, quantitatively rationalizing the experimental observations of Wu and Berg (2012) and Ping et al. (2014).
Bacterial films
In bacterial biofilms, the EPS matrix secreted by bacteria constitutes the active phase and undergoes swelling, drawing in the fluid that acts as the passive phase. As shown in Figure 2B, the EPS is initially synthesized in a partially swollen, outofequilibrium state at the periphery. The polymer chains gradually relax to an equilibrium fullyswollen configuration by the generation of a swelling pressure $\mathrm{\Psi}$ within the biofilm, and via fluid uptake ${V}_{0}(x)$ from the substrate. As discussed in Appendix 2, the swelling pressure is $\mathrm{\Psi}(\varphi )=\psi (\varphi )/\varphi $, where $\psi (\varphi )={\psi}_{0}\times {\varphi}^{3}$ is the osmotic pressure in the EPS matrix using the FloryHuggins model for a polymer network in a $\theta $solvent (Rubinstein and Colby, 2003), where ${\psi}_{0}=kT/({b}^{3})$ is the osmotic pressure scale, $kT$ is the product of the Boltzmann constant with the temperature and $b$ is the approximate size of the monomer unit. The net effective pressure term driving biofilm expansion is, $\mathrm{\Pi}={\psi}_{0}{\varphi}^{3}+{p}_{f}$, where ${p}_{f}$ is the capillary pressure, so that the water influx across the substrate is
where ${Q}_{0}$ is the influx fluid velocity scale, ${\varphi}_{0}={({\mathrm{\Psi}}_{\text{eq}}/{\mathrm{\Psi}}_{0})}^{1/3}$ is the fullyswollen EPS polymer volume fraction and ${\mathrm{\Psi}}_{\text{eq}}$ is the osmotic pressure of the substrate over which the colony grows. Finally, nutrient uptake is modeled by a Monod growth law, while the synthesis of the EPS matrix is modeled by a logistic term as listed in Table 1.
Parameters and scaling estimates for bacterial films
We consider dimensionless variables $\widehat{x}=x/L$, $\widehat{z}=z/H$, $\widehat{t}=tG$, $\widehat{\varphi}=\varphi /{\varphi}_{0}$ and $\widehat{c}=c/{c}_{0}$, where $H$ is now the maximum biofilm thickness, $G$ is the rate of EPS production, and ${c}_{0}$ is the initial nutrient concentration in the substrate. As biofilm growth is nutrient limited (Liu et al., 2015), the dimensionless length scale is determined from Equation (3) and is expected to scale as $L={(D/G)}^{1/2}$ and the corresponding velocity scale is $U={(DG)}^{1/2}$.
Using these scales, we can define the ratio of osmotic stresses relative to viscous stress in the EPS phase in terms of the dimensionless parameter, ${\beta}_{1}=({\mathrm{\Psi}}_{0}/L)/({\mu}_{1}U/{H}^{2})$, the ratio of capillary stresses relative to the EPS viscous stress in terms of another parameter, ${\beta}_{2}=(\gamma H/{L}^{3})/({\mu}_{1}U/{H}^{2})$, the ratio of capillary stress to the interfacial drag in the aqueous fluid phase, ${\beta}_{3}=(\gamma H/{L}^{2})/(\zeta UL)$, and the ratio of the fluid influx velocity to the EPS swelling velocity, ${\beta}_{4}={Q}_{0}/(HG)$. As shown in Appendix 2, the effective nutrient uptake rate is $S=(\mathrm{\Gamma}H{\varphi}_{0})/({c}_{0}d)$, where $\mathrm{\Gamma}$ is the nutrient consumption rate per unit concentration and $d$ is the substrate thickness. Consequently, we define ${\beta}_{5}=S/G$ as the ratio of the effective nutrient uptake rate to the EPS production rate.
We set the EPS production timescale as $G=1/40$ min${}^{1}$, resulting in a horizontal length scale of $L={(D/G)}^{1/2}=1.1$ mm and velocity scale $U=(DG{)}^{1/2}=0.5\phantom{\rule{thinmathspace}{0ex}}\mu$m/s. The effective nutrient uptake rate is estimated as $S=1/25$ min${}^{1}$, where we have taken $d=7$ mm as the substrate thickness (Srinivasan et al., 2018), $\mathrm{\Gamma}={10}^{2}$ mM/s as the nutrient uptake rate (Zhang et al., 2010), and ${c}_{0}=35$ mM as the initial concentration of the carbon source. The friction coefficient is $\zeta \sim {\mu}_{2}/{\xi}^{2}$, where the EPS mesh size is $\xi =50$ nm (Yan et al., 2017). Using measured estimates of the biofilm viscosity ${\mu}_{1}={10}^{5}$ Pa.s (Stoodley et al., 2002; Lau et al., 2009), fluid phase viscosity ${\mu}_{2}={10}^{3}$ Pa.s, surface tension $\gamma ={10}^{2}$ N/m, an osmotic scale ${\mathrm{\Psi}}_{0}=2100$ Pa (Yan et al., 2017) (i.e., ${\varphi}_{0}=0.04$), biofilm thickness $H=400\phantom{\rule{thinmathspace}{0ex}}\mu$m, and nutrient diffusivity in agarose gels of $D=5\times {10}^{10}{m}^{2}$/s (Zhang et al., 2010) implies that ${\beta}_{1}\approx 7$, ${\beta}_{2}\approx 0.01$, ${\beta}_{3}\approx 0.02$, ${\beta}_{4}\approx 1$ and ${\beta}_{5}\approx 2$. Consequently, within the context of our model, it is evident that osmotic stresses, fluid influx and biomass growth are the dominant forces that drive colony expansion. Moreover, in the nutrient limited regime, our model predicts the transient maximum biofilm expansion velocity to scale as,
whereas, the width of the propagating fronts of EPS production experimentally observed by Srinivasan et al. (2018) is predicted to scale according to,
where ${C}_{3}$ and ${C}_{4}$ are once again dimensionless prefactors that require a detailed numerical calculation, as discussed later.
Transient biofilm solutions
With the above scaling assumptions, Equations (1–3) now reduces to the following partial differential equations that describe biofilm colony expansion,
where $\kappa (\widehat{\varphi})=(1{\varphi}_{0}\widehat{\varphi})/({\varphi}_{0}\widehat{\varphi})$ is a volume fraction dependent permeability term. The eight boundary conditions associated with Equations (12)–(14) are the symmetry boundary conditions ${\widehat{h}}_{\widehat{x}}(0)={\widehat{h}}_{\widehat{x}}({R}_{P})=0$, ${\widehat{h}}_{\widehat{x}\widehat{x}\widehat{x}}(0)={\widehat{h}}_{\widehat{x}\widehat{x}\widehat{x}}({R}_{P})=0$, ${\widehat{\varphi}}_{\widehat{x}}(0)={\widehat{\varphi}}_{\widehat{x}}({R}_{P})=0$, ${\widehat{\varphi}}_{\widehat{x}\widehat{x}\widehat{x}}(0)={\widehat{\varphi}}_{\widehat{x}\widehat{x}\widehat{x}}({R}_{P})=0$ and ${\widehat{c}}_{\widehat{x}}(0)={\widehat{c}}_{\widehat{x}}({R}_{P})=0$, where the dimensionless petridish size is chosen as ${R}_{P}=16$ to match the size of typical 35 mm diameter petri dishes used in experiments (Srinivasan et al., 2018). In Figure 4A, we plot the time evolution of the shape and nutrient concentration field for a biofilm colony of initial radius ${\widehat{R}}_{0}=0.5$ and thickness ${\widehat{h}}_{\text{in}}=0.06$.
Unlike in the case of swarms, the solutions to Equations (12)–(14) are transient, and exhibit two distinct expansion regimes: initial acceleration phase until ${\widehat{t}}_{c}=5$, followed by a decelerating phase beyond. For $\hat{t}\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}{\hat{t}}_{c}$, colony expansion arises as the microbes rapidly consumes locally available nutrient at the interior and synthesize fresh EPS matrix, generating spatial gradients in nutrient availability (see Figure 4A). In Figure 4B, we show that the newly synthesized EPS generates a large osmotic pressure differential between the biofilm and the substrate, and osmotic fluid influx gradually relaxes the biofilm matrix to a swollen configuration. For $t\phantom{\rule{thinmathspace}{0ex}}>\phantom{\rule{thinmathspace}{0ex}}{t}_{c}$, the localized zone of EPS production near the film front propagates with a fixed shape as shown in Figure 4C, consistent with the observed spatial localization in tapA gene activity (see Figure 1J and Srinivasan et al., 2018). Moreover, the radial colony expansion profile in Figure 4D is also consistent with the nonmonotonic front speed observed experimentally (Srinivasan et al., 2018). For the specific experimental conditions we consider, our detailed theory allows us to estimate the prefactors in the scaling laws Equations (10)–(11) so that ${C}_{3}\approx 0.2$ and ${C}_{4}\approx 1.8$.
These results are hallmarks of a transition from a bulk to an edge biofilm growth mode, triggered by nutrient limitation (Pirt, 1967). In the deceleration regime, diffusive transport of nutrients from a region external to the colony continues to sustain EPS production at the biofilm periphery, analogous to Stefanlike problems in solidification. Our generalized multiphase model is thus able to quantitatively rationalize the expansion curves, transition time and localized biological activity observed experimentally, and demonstrates that nutrient availability and diffusive transport governs the dynamics of Bacillus subtilis macrocolonies grown on agar.
Discussion
Analysis of collective microbial expansion in thin film geometries often prioritizes biological mechanisms, such as genetic regulation, developmental programs and cellular signaling/competition, over the role of the heterogeneous physical microenvironments. Here we have presented a multiphase theory that quantitatively describes the expansion dynamics of microbial swarms and biofilms and considers variations in the colony thickness, an aspect of colony expansion that has often been overlooked in many theories (Korolev et al., 2012; Ghosh et al., 2015; Wang et al., 2017). The resulting unified description of both steadystate swarms and transient biofilm spreading leads to simple estimates and scaling laws for the colony expansion rate that are validated via comparison with experimental measurements for different systems. In swarms, exudation of water from the permeable substrate via bacterial osmolyte secretion facilitates steady state colony expansion. Numerical solutions of our model demonstrate that the shape of the swarm front is determined by capillarity, and its expansion speed by celldivision and growth, leading to scaling laws validated by comparison with previous experiments. In contrast, transient biofilm macrocolony expansion on agar is driven by osmotic polymer stresses generated via EPS matrix production in a spatially localized zone at the periphery. Nutrient transport and depletion leads to the formation of these heterogenous zones, and results in two regimes in biofilm expansion.
However our depthintegrated theory also has certain limitations. For example, we are unable to capture discrete thickness variations of the order of a few cells, which might require an agentbased approach. For bacterial swarms, our model is unable to quantitatively account for the region of enhanced thickness (i.e., the multilayer region in Figure 1C and E), likely because the multilayer width is difficult to experimentally ascertain, owing to the large tail distribution seen in the mean intensity trace in Figure 1E, and the arbitrariness in the choice of threshold in Appendix 3—figure 5. Similarly, in the context of biofilm colony expansion, our model does not account for sliding and frictional contact between the cells/EPS matrix and the substrate (Farrell et al., 2013). More generally, our meanfield picture neglects fluctuationdriven effects during colony expansion, such as the formation multicellular raft structures (Kearns, 2010) and synchronized longrange interactions (Chen et al., 2017).
Natural next steps of our approach include (i) adding threedimensional effects by allowing for spatial variations in the mechanical stresses, flows and nutrient fields in the vertical direction, (ii) accounting for orientational order in the bacterial swarms and films, and (iii) accounting for interfacial tension on the stability of the growing swarm/biofilmfluid interface, especially in the context of fingering instabilities in microbial colonies Trinschek et al. (2018).
A rigorous multiphase approach may also be relevant in revisiting pattern formation phenomena in microbial colony expansion (Matsushita et al., 1999), that so far been addressed primarily using various nonlinear diffusion models (Golding et al., 1998; Allen and Waclaw, 2019) that ignore the third dimension. Finally, from an experimental and theoretical perspective, our results naturally raise the question of controlling biofilm and swarm expansion by manipulating water and nutrient availability, complementing the better studied approaches of manipulating colonies by the genetic regulation of EPS production, cell division, and chemical signaling in microbial colonies.
Appendix 1
Experiment
Strains
In this study, we used two B. subtilis strains, MTC822 and MTC832, that were both previously constructed from a wildtype NCIB 3610 B. subtilis strain using a standard transformation protocol (Sinha, 2013). The MTC822 strain is used for fluorescence visualization in the swarming experiment, where the mkate2 red fluorescent protein reports on the activity of the constitutive hyperspank promotor via the amyE::Phyperspankmkate2 construct. The MTC832 strain was used in the biofilm experiments in order to visualize localized matrix production activity and harbors the amyE::PtapAcfp construct. In the MTC832 strain, the cfp cyan fluorescent protein reports on the activity of the tapA gene that is associated with exopolysaccharide production activity.
Materials and methods
Swarm plates were prepared using 0.5 wt% agarose gel (A1296, Sigma) infused with 25 ml of LuriaBertani (Miller) medium (i.e. 10 g/L Tryptone 10 g/L NaCl 5 g/L Yeast Extract, Sigma) and 25 μg/ml Chloramphenicol. Biofilm plates were prepared using 1.5 wt% agarose gel (A1296, Sigma) infused with the standard MSgg biofilminducing growth medium (Branda et al., 2001) (i.e. 50 μM MnCl_{2}, 5 mM KH_{2}PO_{4}, 1 μM ZnCl_{2}, 50 μM FeCl_{3}, 2 mM MgCl_{2}, 700 μM CaCl_{2}, 50 μg/ml threonine, 50 μg/ml tryptophan, 50 μg/ml phenylalanine, 0.5 wt% glutamate, 0.5 wt% glycerol, 2 μM thiamine and 100 mM MOPS (pH 7)) and 50 μg/ml Spectinomycin. Note that all plates underwent an identical drying protocol prior to use. Freshly poured plates were initially dried with the lid open under a laminar flow hood for 15 min. Subsequently, the lid was closed and the dish was cooled at 25 C overnight for a period of 10 hr. All strains were initially grown in fresh LuriaBertani (Miller) broth medium (Sigma) until midexponential phase in a shaker/incubator at 37C. The cultures were diluted to $O{D}_{650}=0.1$ and ~ 1 μl drop was deposited onto the corresponding swarm (for MTC822) or biofilm (for MTC832) plates. The petriplates were transfered to a 30C incubator chamber during growth. Fluorescence imaging was performed using a Zeiss Axiozoom.V16 microscope with a PlanNeoFluar Z 1.0x objective (NA 0.25), with a Zeiss 63 HE filter to image the red mkate2 protein, and a Zeiss 47 HE filter to image the cyan cfp protein. For swarm profile measurements, images of the advancing swarm front were captured every 10 s over a period of 10 min. For biofilm colonies, expansion velocities were measured every 10 min over a period of 72 hr following the protocol described in Srinivasan et al. (2018).
Appendix 2
Theory
Consider a quasi2D expanding swarm or biofilm colony where, $x$ denotes the horizontal direction and $z$ denotes the vertical (thickness) direction. The macroscopic colony is described by (i) the thickness field $h(x,t)$, (ii) the volume fraction of the active phase (i.e. swarmer cells or EPS matrix $\varphi (x,t)$ and (iii) the averaged concentration field in the substrate $c(x,t)$.
Nutrient transport
We assume that $\stackrel{~}{c}(x,z,t)$ is the nutrient concentration in the substrate denoted by region $d\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}z\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}0$, where $z=0$ is the substrate/colony interface and $d$ is the substrate thickness. The time evolution of $\stackrel{~}{c}(x,z,t)$ is governed by the diffusion equation.
where $D$ is the diffusivity of the nutrient in the agarose gel. Integrating the diffusion equation through the substrate thickness results in,
where $c(x,t)=(1/d){\int}_{d}^{0}\stackrel{~}{c}\mathit{d}z$ is the mean depthaveraged nutrient concentration in the substrate, $\mathrm{\Gamma}$ is the specific nutrient consumption rate, and $K$ is the half velocity constant at which the specific growth rate is one half the maximum value. In obtaining Equation (A2), we have balanced nutrient efflux from the substrate with nutrient consumption within the colony as ${D\left(\partial \stackrel{~}{c}/\partial z\right)}_{d}=\mathrm{\Gamma}h\varphi c/(K+c)$. Equation (A2) describes nutrient transport within the permeable tissue/gel substrate underneath the colony, and holds when the diffusion time scale is much smaller than the flow time scale within the lowaspectratio substrate. Upon nondimensionalizing Equation (A2), the effective nutrient consumption rate $S$ used in Table 1 in the main text is given by $S=\mathrm{\Gamma}{h}^{*}{\varphi}_{\text{eq}}/({c}_{0}d)$, where $h*$ is the colony thickness scale, ${\varphi}_{\text{eq}}$ is the active phase volume fraction scale and ${c}_{0}$ is the initial nutrient concentration in the substrate.
Continuity
For a volume element within the microbial colony, the averaged conservation of mass (Ishii and Hibiki, 2011) for the active phase and the aqueous phase is expressed as,
where $\varphi $ is the volume fraction of the active phase (i.e. swarmer cells or biomass), ${\mathbf{u}}_{\mathbf{1}}=({u}_{1},{w}_{1})$ is the averaged velocity of the active phase, ${\mathbf{u}}_{\mathbf{2}}=({u}_{2},{w}_{2})$ is the averaged velocity of the fluid phase along the $x$ (horizontal) and $z$ (vertical) directions respectively. Integrating Equations (A3) and (A4) through the colony thickness leads to,
The boundary conditions in Equations (A5) and (A6) for the vertical velocities ${w}_{1}$ and ${w}_{2}$ are,
where the noflux condition is applied to active phase at $z=0.$ For the aqueous phase, at $z=0$, there is a spatial fluid influx described by ${V}_{0}(x)$ due to the osmotic pressure difference between the substrate and the microbial colony. Both phases at the upper interface at $z=h$ obey the kinematic boundary condition. In Equations (A2), (A5) and (A6), the five unknown fields are $h$, $\varphi $, $c$, ${u}_{1}$ and ${u}_{2}$. The remaining two equations for closure are obtained from an averaged momentum balance.
Momentum balance
The averaged momentum balances for each phase can be expressed as (Ishii and Hibiki, 2011; Drew, 1983),
where ${\mathit{\bm{\sigma}}}_{\mathrm{\U0001d7cf}}$ is the averaged stress tensor acting on a volume element of the swarm cell/EPS matrix phase, ${\mathit{\bm{\sigma}}}_{\mathrm{\U0001d7d0}}$ is the averaged stress tensor in a volume element of the fluid phase and $\mathit{\bm{M}}$ denotes the total momentum transfer between the swarm cell/EPS matrix and the fluid phase. The interfacial momentum transfer term is expressed as (Ishii and Hibiki, 2011; Drew, 1983),
The first term in Equation (A11) denotes the force ${p}_{f}\nabla \varphi $ due to the averaged interfacial pressure (Drew, 1983; Ishii and Hibiki, 2011) on the cells/EPS matrix by the surrounding fluid. Note that the momentum transfer terms in Equations (A9)(A10) balance each other as the averaged pressures at the twophase interface are equal, leading to a vanishing net buoyancy. The second term corresponds to the net viscous stokes drag, where $\zeta $ is a friction coefficient. While Equations (A9–A11) are generally applicable in describing both swarms and films, particular expressions that describe ${\mathit{\bm{\sigma}}}_{\mathrm{\U0001d7cf}},{\mathit{\bm{\sigma}}}_{\mathrm{\U0001d7d0}}$ and $\zeta $ are unique to swarming and biofilm expansion and are discussed below for each case.
Bacterial swarms
Within microbial swarms, the fluid phase is modeled as a Newtonian liquid with viscosity ${\mu}_{2}$ whereas the swarmer cells are treated as inviscid with an isotropic stress equal to the surrounding fluid pressure ${p}_{f}$. The averaged constitutive laws for the swarmer cell phase and the fluid phase are expressed as,
Substituting Equations (A12) and (A13) in Equations (A9–A11), we obtain
where ${\tau}_{\mathbf{2}}={\mu}_{2}(\mathrm{\nabla}{\mathbf{u}}_{\mathbf{2}}+\mathrm{\nabla}{\mathbf{u}}_{\mathbf{2}}^{\mathbf{T}})$ is the deviatoric stress tensor. In bacterial swarms, the friction coefficient is $\zeta ={\zeta}_{c}/{V}_{c}\approx 18{\mu}_{2}/{a}^{2}$ where $a$ is the diameter of the cell, ${\zeta}_{c}\approx 3\pi {\mu}_{2}a$ is the friction coefficient of a single swarmer cell from Stokes’s law, ${V}_{c}=\pi {a}^{3}/6$ is the cell volume.
Thinfilm lubrication limit for swarms
We consider the limit of $h\ll R$, where $R$ is the colony radius. In the thinfilm lubrication limit, combining Equations (A14) and (A15) results in the equation governing the mean horizontal fluid velocity to leading order,
where the fluid pressure is assumed to vary only in the horizontal direction, and is set by the local curvature of the swarm colony and the fluid surface tension as,
where ${p}_{0}$ is a constant (atmospheric) pressure. Integrating Equation (A16) twice and using the boundary conditions ${u}_{2}(0)=0$ and ${(d{u}_{2}/dz)}_{z=h}=0$ lead to the expression for the averaged horizontal fluid phase velocity profile as,
The mean horizontal swarmer cell velocity is determined from the Darcytype equation in Equation (A14) as,
Growth rate of bacterial swarms
We note that bacterial swarming is typically associated with nutrient rich environments, where the initial concentration level ${c}_{0}\gg \mathrm{\Gamma}$, the specific nutrient consumption rate. Paradoxically, although the rate of bacterial cell division and volume expansion are exponential, the rate of swarm expansion is constant. Therefore, despite abundant nutrient availability, cell division must be halted at the swarm interior through nonnutrient meditated regulatory/signaling mechanisms, such that only a subpopulation of swarmer cells undergo cell division. In our study, we account for this effect by considering a simple logistic model for the growth term of the form,
where $H$ is the swarm colony thickness at the interior, and ${g}_{0}$ is an effective growth rate that accounts for spatial localization in cell division. More specifically, if $L(x)$ describes the spatial profile of cell growth within a swarm colony, then ${g}_{0}={\int}_{0}^{R}L(x)\mathit{d}x/{\int}_{0}^{R}\varphi [1(h\varphi /(H{\varphi}_{0}))]\mathit{d}x$, where $R$ is the radius of the swarm colony. Measurements of the spatial distribution of cell growth rates within the colony during swarming remain lacking. Consequently, in our model, we determine the value of ${g}_{0}$ by fitting it to the experimental data. We use a nonlinear leastsquares solver to match steadystate expansion speeds obtained from solving Equations (A21)(A22) to the experimental data for steadystate B. subtilis swarms (see Figure 1D in main text).
Swarm equations
Combining Equations (A5)(A8) with Equations (A17)(A19) results in the dimensional thickness averaged equations for swarm colonies,
The five boundary conditions are ${(\partial h/\partial x)}_{x=0}={(\partial h/\partial x)}_{x\to \mathrm{\infty}}=0$, ${({\partial}^{3}h/\partial {x}^{3})}_{x=0}={({\partial}^{3}h/\partial {x}^{3})}_{x\to \mathrm{\infty}}=0$, and $\varphi (0)={\varphi}_{0}$, where ${\varphi}_{0}$ is the volume fraction of the swarmer cells at the interior where there is no flow. As discussed in the main text, we impose the farfield boundary condition by choosing a domain much larger than the colony size.
Bacterial films
In our model, the EPS matrix constitutes an active viscous hydrogel network with a polymer volume fraction $\varphi (x,t)$, while the fluid phase is considered as a freely moving solvent of volume fraction $1\varphi (x,t)$. The averaged constitutive laws for the EPS matrix phase and the fluid phase are expressed as,
where ${\mu}_{1}$ is the viscosity of the EPS matrix, ${\mu}_{2}$ is the aqueous phase viscosity, $\mathrm{\Psi}$ represents an effective swelling pressure in the biofilm EPS hydrogel and ${p}_{f}$ is the fluid pressure. Substituting Equations (A23) and (A24) in Equations (A9–A11), we obtain
where ${\tau}_{\mathbf{1}}={\mu}_{1}(\mathrm{\nabla}{\mathbf{u}}_{\mathbf{1}}+\mathrm{\nabla}{\mathbf{u}}_{\mathbf{1}}^{\mathbf{T}})$ and ${\tau}_{\mathbf{2}}={\mu}_{2}(\mathrm{\nabla}{\mathbf{u}}_{\mathbf{2}}+\mathrm{\nabla}{\mathbf{u}}_{\mathbf{2}}^{\mathbf{T}})$ are the deviatoric stresses in the EPS phase and aqueous phase, respectively. In our model, we assume that the contribution of stress from the viscosity of the aqueous phase is negligible compared to the frictional drag due to flow between water and EPS polymer chain network. Consequently, Equation (A26) reduces to a Darcytype law of the form,
where $\kappa (\varphi )=(1\varphi )/\varphi $ is a volume fraction dependent permeability and the fricton coefficient is $\zeta ={\mu}_{2}/{\xi}^{2},$ provided that $\xi \sim 50$ nm is the polymer mesh network length scale. Substituting Equation (A27) in Equation (A25) results in,
Thinfilm lubrication limit for bacterial films
In the thinfilm lubrication limit for biofilms when $h\ll R$, Equation (A28) reduces to
where $\mathrm{\Pi}=\varphi \mathrm{\Psi}+{p}_{f}$ is treated as an effective EPS phase pressure, as it is $d\mathrm{\Pi}/dx$ that drives the viscous EPS flow. In Equation (A29), the effective pressure is assumed to vary only in the horizontal direction, and the fluid pressure is once again set by the curvature and fluid surface tension according to Equation (A17). Moreover, the swelling pressure is expressed as $\mathrm{\Psi}=\psi (\varphi )/\varphi $, where $\psi (\varphi )={\psi}_{0}\times {\varphi}^{3}$ is the osmotic pressure of the biofilm EPS matrix using the FloryHuggins model for a polymer network in a $\theta $solvent (Cogan and Guy, 2010; Winstanley et al., 2011). The osmotic pressure scale is ${\psi}_{0}=kT/{b}^{3}$, where $b\sim 0.5$ nm is the approximate size of the monomer unit in the EPS matrix. Note that although we refer to $\mathrm{\Pi}$ as an effective pressure, in Equation (A23) the mechanical pressure acting on the EPS phase is instead $\psi (\varphi )/\varphi +{p}_{f}$. Integrating Equation (A29) twice using the boundary conditions ${u}_{1}(0)=0$ and ${(d{u}_{1}/dz)}_{z=h}=0$ results in,
The mean horizontal fluid velocity is determined from the Darcytype equation in Equation (A27) as,
Growth rate of bacterial films
In biofilms, the growth term ${g}_{1}(h,\varphi ,c)$ is expressed by the modified logistic term,
where $G$ is the specific EPS production rate, $H$ is the maximum overall biofilm colony thickness and ${\varphi}_{0}$ is the volume fraction of the EPS matrix in the fully swollen state. Note that, in contrast to microbial swarming, the growth rate term in expanding biofilms is strongly coupled to the local nutrient concentration field via a Monodtype term in Equation (A32).
Biofilm equations
Combining Equations (A5)(A8) with Equations (A30)(A31) results in the dimensional thinfilm governing equations for biofilm colonies,
The eight boundary conditions are the symmetry conditions ${(\partial h/\partial x)}_{x=0}={(\partial h/\partial x)}_{x=R}=0$, ${(\partial {h}^{3}/\partial {x}^{3})}_{x=0}={({\partial}^{3}h/\partial {x}^{3})}_{x=R}=0$, ${(\partial \varphi /\partial x)}_{x=0}={(\partial \varphi /\partial x)}_{x=R}=0$, and ${(\partial c/\partial x)}_{x=0}={(\partial c/\partial x)}_{x=R}=0$ where $R$ is now the size of the petri dish that is infused with nutrients.
Numerical computation
Numerical solutions to Equations (A21)(A22) and Equations (A33)(A35) were implemented using the COMSOL 5.0 finite element package and have been provided as source code files. We use a fixed 1D domain of size $x\in [0,{L}_{1}]$ where $L}_{1$ = 16 and 150 for the biofilm and swarm simulations respectively. We use quintic Lagrange basis functions with element sizes below 0.04 and use the general form PDE solver. To handle the moving contact line, we introduce a precursor film of thickness ${h}_{p}/H$ = 0.0125, where $H$ is the vertical length scale (see Appendix 3—figure 1). We follow the regularization described in Trinschek et al. (2016) to introduce a minimum threshold for growth and a stable fixed point in the precursor film. Specifically, the growth terms in (A2), (A20) and (A32) are multiplied by a factor $F=[1\mathrm{exp}\left(5\left({\widehat{h}}_{p}\widehat{h}\widehat{\varphi}\right)\right)]\times \left(1{\widehat{h}}_{p}/(\widehat{h}\widehat{\varphi})\right)$, where $F\approx 1$ everywhere except near the precursor film where $F=0$.
Experimental data
Appendix 3
Figures
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Decision letter

Raymond E GoldsteinReviewing Editor; University of Cambridge, United Kingdom

Arup K ChakrabortySenior Editor; Massachusetts Institute of Technology, United States

Alexandre PersatReviewer; EPFL, Switzerland
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
Thank you for submitting your article "Dynamics of spreading microbial swarms and films" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Arup Chakraborty as the Senior Editor. The following individual involved in review of your submission has agreed to reveal his identity: Alex Persat (Reviewer #1).
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
Summary:
Srinivasan and coauthors propose a generalized model to describe the expansion of bacterial communities growing on agar plates in two distinct modes: swarms and biofilms. They demonstrate that the interplay between biological mechanisms, such as osmolyte secretion, extracellular matrix production and spatial localization in cell growth, and physics of the microbial environment, such as hydrodynamics, mechanics, osmotic flux and nutrient diffusion, controls the expansion dynamics of microbial swarms and biofilms. They show that swarming colonies undergo steady state expansion, while biofilms expand transiently, going from an initial acceleration phase to a decelerating one. These observations are in agreement with a few experiments. In the end, they prove that despite the differences in biological mechanisms of films and swarms, a unified twophase model of a spreading colony can describe both systems.
The reviewers found the paper wellwritten and the theoretical and numerical analysis was performed very carefully. The phasefield theory developed in the paper goes substantially beyond earlier models as recently reviewed by Allen and Waclaw, 2018. The detailed parameter list in the Appendix can become a valuable reference for future studies. The manuscript thus offers a significant theoretical advance.
Title:
The reviewers thought that the current title sounds like an area of research rather than the title of a paper. Perhaps adding the term "model" or "theory" or any other more specific/descriptive statements would more clearly reflect the content of the paper.
Essential revisions:
The reviewers found that the manuscript lacks a bit of clarity with respect to conclusions and agreement with experimental data:
1) There is a lack of comparison between scaling and experiments. I think the beauty of this work is to highlight scaling of the front velocity and width of the front. However, the authors limit their validation as fitting a single data point to obtain prefactors (e.g. subsection “Steady state swarms”, last paragraph). I would expect that the authors would make an effort to measure velocities for a few capillary numbers for example, and verify the 1/3 exponent behavior. It looks like the authors can actually do these experiments since they already have some data shown in Figure 1. This would absolutely strengthen the paper and the model (independently of whether the model fits the data).
2) The aim of the model is to describe both swarming and biofilm expansion and weigh in on their physical differences. However, describing swarming and biofilm separately makes it difficult to have a direct comparison between the two. It would be useful to directly compare the two (in concomitance with Table 1), explaining, for example, the choices of different activepassive phases, osmotic flux, growth term and nutrient uptake. For example, it would be useful to know why in swarmer cells nutrient uptake is not considered (is it related to nutrient diffusion in different agar substrate?). Why isn't surface tension considered in the biofilm model? This lack of discussion weakens the argument of a unifying model.
3) In biofilms, it is not clear to me why the active phase consists of only the EPS and cell growth is not considered as a driver of expansion.
4) I would imagine that one important factor limiting growth particularly in the biofilm case is the friction between the colony and the agar substrate. This is pointed out in Farell, 2013, but it seems to be vastly overlooked by the authors. Can they incorporate this in their fluidic model?
5) They state that their model reveals that bacterial swarm colonization corresponds to a fluid regulated limit, whereas the dynamics of biofilm expansion is governed by nutrient transport (subsection “Conclusions”). However, these are actually the assumptions used to describe the system, so it is not so surprising. Can the authors comment on this point?
One reviewer identified himself as "eviewer A" of the previous submission to PNAS for this manuscript (i.e. the positive reviewer), and stated that the authors had responded to his comments in their "response to reviewers" document, which they attached to the eLife submission.
This reviewer did note the following:
6) It is still slightly worrisome that 2 out of the 4 figures are introductory/motivational, as pointed out in my major comment 1 of the PNAS report.
7) Regarding my previous comment 2, about limitations of the model. I appreciate that the authors have added the sentence about shortcomings of the model after I pointed out one issue. Given that eLife does not have any length restrictions, I think it would be appropriate to add a new section to the main text where general limitations of the model are discussed, and where the authors believe the model is failing. I think such a section would strengthen the manuscript, provide additional context, and it would be helpful to the readers.
https://doi.org/10.7554/eLife.42697.022Author response
As per the reviewers’ recommendation, we have changed the title to “A multiphase theory for spreading microbial swarms and films”.
Essential revisions:
1) There is a lack of comparison between scaling and experiments. I think the beauty of this work is to highlight scaling of the front velocity and width of the front. However, the authors limit their validation as fitting a single data point to obtain prefactors (e.g. subsection “Steady state swarms”, last paragraph). I would expect that the authors would make an effort to measure velocities for a few capillary numbers for example, and verify the 1/3 exponent behavior. It looks like the authors can actually do these experiments since they already have some data shown in Figure 1. This would absolutely strengthen the paper and the model (independently of whether the model fits the data).
We thank the reviewers for their comment. Following the reviewers’ suggestion, we have now included additional comparisons with experiments that verify some of the predictions of our model. Specifically, we now fit our model and compare the scaling behavior using five independent experimental measurements of swarm expansion velocities. These include two new experimental measurements of swarm expansion in B. subtilis, and two existing measurements in E. coli previously reported by Darton et al., 2010, and Wu and Berg, 2012. The comparison between the various experiments and fitting to our model is described in the main text in Appendix 2—table 2 and Figure 3D.
As seen in the figure for the velocity scaling, the measured swarm expansion velocities (dR/dt) indeed follows a 1/3 exponent with the Capillary number. We have introduced a new figure in the main text (i.e., Figure 3D) to emphasize the verification of the 1/3 scaling. We have rewritten the introduction of the time and length scales, that clarifies the estimation of the various parameters.
“There are two key dimensionless parameters that describes swarm colony expansion. […] A complete set of parameters for three experimental measurements of swarm expansion in B. subtilis, and two existing measurements in E. coli previously reported by Darton et al., 2010, and Wu and Berg, 2012, are summarized in Appendix [3].”
Finally, upon considering the new dataset, the width of the swarm front does not follow the scaling we had originally proposed, as seen in Appendix 3—figure A5.
For each experiment, we fit our model using the effective growth rate g0 as the fitting parameter. We have also included a discussion of this fitting, and the overall comparison between model and theory with the new experimental measurements:
“Furthermore, we corroborate our scaling law in Equation 5 by fitting our model to five independent experimental measurements of swarm expansion velocities, as shown in Figure 3D. [...] From an experimental point of view, the width of the multilayer region is not sharply defined in Figure 1E, and will depend on the choice of threshold.”
As the reviewer indicates, we think these results strengthen the paper and our model, especially the agreement of the scaling law with the experimental velocities. We are once again grateful and thank the reviewer for their valuable insight towards improving the manuscript.
2) The aim of the model is to describe both swarming and biofilm expansion and weigh in on their physical differences. However, describing swarming and biofilm separately makes it difficult to have a direct comparison between the two. It would be useful to directly compare the two (in concomitance with Table 1), explaining, for example, the choices of different activepassive phases, osmotic flux, growth term and nutrient uptake. For example, it would be useful to know why in swarmer cells nutrient uptake is not considered (is it related to nutrient diffusion in different agar substrate?). Why isn't surface tension considered in the biofilm model? This lack of discussion weakens the argument of a unifying model.
We thank the reviewers for their comment. As per the reviewers’ suggestion, we have now added a number of paragraphs (immediately after Table 1), that directly compares the various terms in our unifying model across swarms and biofilms. We have now described the origin of the differences in the nutrient uptake, growth term and the active and passive fluxes in parallel with Table 1.
Our Discussion directly addresses some of the questions the reviewers raise, viz.
1) The overall rate of change of nutrient concentration in the substrate depends on nutrient diffusion and uptake as described by Appendix 2—equation A2, i.e.
∂c∂tD∂2c∂x2=ΓhϕdcK+c
where, c(x,t) is the mean depthaveraged nutrient concentration in the substrate, Γis the specific nutrient consumption rate per unit concentration, k is the half velocity constant at which the specific growth rate is one half the maximum value and d is the substrate thickness. When the substrate concentration is scaled by the initial concentrationc0, the depletion term on the RHS of Equation 18 depends on Γ/c0, the ratio of the specific nutrient consumption rate to the initial concentration.
Bacterial swarming is typically associated with nutrient rich conditions, wherec0≫Γ. As a result, the nutrient uptake term can be neglected in bacterial swarming asg2→0, and the concentration c≈c0 throughout swarm expansion. In contrast, biofilm growth occurs under nutrient limited conditions whereΓc0∼O1, resulting in a corresponding uptake term as shown in Table 1.
2) Surface tension is indeed included in the biofilm model, and sets the capillary fluid pressure aspf∼γ∂2h/∂x2. Furthermore, as shown in Appendix 2—equations A28 and A294445, the gradient of the effective EPS phase pressure Π drives the horizontal fluxes Q1(x) and Q2(x) of the EPS matrix and fluid phase in Table 1. Note that Π is a sum of the FloryHuggins osmotic pressure and the capillary fluid pressure. In this manner, surface tension is indeed considered in the biofilm model.
3) In biofilms, it is not clear to me why the active phase consists of only the EPS and cell growth is not considered as a driver of expansion.
Our model considers the growth of mature, macroscopic biofilm colonies. In these colonies, the biomass almost entirely consists of the EPS matrix. For example, as stated in Flemming and Wingender, 2010.
“In most biofilms, the microorganisms account for less than 10% of the dry mass, whereas the matrix can account for over 90%. The matrix is the extracellular material, mostly produced by the organisms themselves, in which the biofilm cells are embedded.”
Moreover, the total biomass itself constitutes only ~5% of the biofilm by mass (with the fluid being the predominant phase). Therefore, we treat the biofilm as a highly viscous and hydrated gel. Our model considers the osmotic stress and swelling pressure in freshly produced EPS matrix as the major contributor to biofilm expansion.
It only at the very early stages of biofilm colony growth (when EPS production is minimal) where cellcell contact is the major contributor to growth pressure. An example of the latter scenario is microcolonies that consist of only ~100 – 1000s of cells. Models of microcolonies do indeed consider stress contributions from growth, for example, as in the work of Farrell, 2013.
4) I would imagine that one important factor limiting growth particularly in the biofilm case is the friction between the colony and the agar substrate. This is pointed out in Farell, 2013, but it seems to be vastly overlooked by the authors. Can they incorporate this in their fluidic model?
In our model, we consider the both the active EPS matrix phase, and the passive aqueous phase as Newtonian fluids. Consequently, friction between the colony and the agar substrate is accounted for in our model by the noslip boundary condition, u1z=0=0 and u2z=0=0, where u1 and u2 are the horizontal components of the EPS phase and fluid phase velocities.
Therefore, our assumption of a noslip boundary condition corresponds to the limit of an infinite friction coefficient in the discrete cell 2D model of Farrell, 2013. There are two key factors that differentiates our model from that of Farrell, 2013: (i) the presence of the EPS matrix, and (ii) the consideration of a third (i.e., vertical thickness) dimension.
Variations in the biofilm thickness allow the EPS matrix to sustain shear gradients in velocity. As a result, the biofilm colony “flows” over the agar substrate as a fluid in our model, while individual bacterial cells expand by “sliding” as a 2D dimensional colony by sliding in the model of Farrell et al. Similarly, in our model EPS production and growth in the third dimension generates the osmotic stresses and swelling pressure that drives colony expansion.
Note also that Farrell et al. consider the very early microcolony stage of biofilms (< 2hrs), where cells are present as a monolayer and both EPS production and variations in the third dimension are absent.
One possibility towards incorporating the effect of frictional sliding in macroscopic biofilms would be the introduction of a Navier sliplength parameter in our model, where the sliplength is tuned to account for sliding of EPS matrix on the agar substrate. We have added this point to the Discussion, where we discuss limitations and scope of our model, by the following line:
“Similarly, during biofilm colony expansion, we do not account for sliding and frictional contact between the cells/EPS matrix and the substrate (Farrell, 2013).”
5) They state that their model reveals that bacterial swarm colonization corresponds to a fluid regulated limit, whereas the dynamics of biofilm expansion is governed by nutrient transport (subsection “Conclusions”). However, these are actually the assumptions used to describe the system, so it is not so surprising. Can the authors comment on this point?
We thank the reviewer for this observation. We agree, our model does not reveal this statement and is indeed an assumption used to describe the system. We have rephrased the sentence to emphasize that our model merely “rationalizes the assumption”, rather than reveals the statement.
One reviewer identified himself as "reviewer A" of the previous submission to PNAS for this manuscript (i.e. the positive reviewer), and stated that the authors had responded to his comments in their "response to reviewers" document, which they attached to the eLife submission.
This reviewer did note the following:
6) It is still slightly worrisome that 2 out of the 4 figures are introductory/motivational, as pointed out in my major comment 1 of the PNAS report.
We thank the reviewer for the comment. We have now included an additional subpanel in Figure 3D to show the scaling of the velocity with the capillary number. We also note that despite having 2 figures, each figure has a number of subpanels that could easily be expanded as separate figures. In the interest of brevity and condensing the results of the data and model in one location for both swarms and biofilms, we have opted to use the subfigure approach, rather than expanding each panel into separate figures.
7) Regarding my previous comment 2, about limitations of the model. I appreciate that the authors have added the sentence about shortcomings of the model after I pointed out one issue. Given that eLife does not have any length restrictions, I think it would be appropriate to add a new section to the main text where general limitations of the model are discussed, and where the authors believe the model is failing. I think such a section would strengthen the manuscript, provide additional context, and it would be helpful to the readers.
As the reviewer has suggested, we have now added paragraphs under the conclusions section where we discuss the limitations of our model and scope for expansion:
“Analysis of collective microbial expansion in thin film geometries often prioritizes biological mechanisms, such as genetic regulation, developmental programs and cellular signaling/competition, over the role of the heterogeneous physical microenvironments. […] Finally, from an experimental and theoretical perspective, our results naturally raise the question of controlling biofilm and swarm expansion by manipulating water and nutrient availability, complementing the better studied approaches of manipulating colonies by the genetic regulation of EPS production, cell division, and chemical signaling in microbial colonies.”
https://doi.org/10.7554/eLife.42697.023Article and author information
Author details
Funding
National Science Foundation (DMR 1420570)
 L Mahadevan
MacArthur Foundation
 L Mahadevan
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank M Cabeen, R Losick for providing strains, S Rubinstein for access to the fluorescent imaging and microbial culture facility, and H Berg, L Ping and A Pahlavan for helpful discussions. This work was supported by the Harvard MRSEC DMR 1420570 and the MacArthur Foundation (LM).
Senior Editor
 Arup K Chakraborty, Massachusetts Institute of Technology, United States
Reviewing Editor
 Raymond E Goldstein, University of Cambridge, United Kingdom
Reviewer
 Alexandre Persat, EPFL, Switzerland
Publication history
 Received: October 9, 2018
 Accepted: March 14, 2019
 Version of Record published: April 30, 2019 (version 1)
Copyright
© 2019, Srinivasan 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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