Role of metabolic spatiotemporal dynamics inregulating biofilm colony expansionFederico Boccia,b, Yoko Suzukic, Mingyang Lud, and José N. Onuchica,b,e,f,1

aCenter for Theoretical Biological Physics, Rice University, Houston, TX 77005-1827; bDepartment of Chemistry, Rice University, Houston, TX 77005-1827;cDepartment of Physics, School of Science and Engineering, Meisei University, Tokyo 191-8506, Japan; dThe Jackson Laboratory, Ben Harbor, ME 04609;eDepartment of Physics and Astronomy, Rice University, Houston, TX 77005-1827; and fDepartment of Biosciences, Rice University, Houston, TX 77005-1827

Edited by William Bialek, Princeton University, Princeton, NJ, and approved March 12, 2018 (received for review May 5, 2017)

Cell fate determination is typically regulated by biological net-works, yet increasing evidences suggest that cell−cell communica-tion and environmental stresses play crucial roles in the behaviorof a cell population. A recent microfluidic experiment showed thatthe metabolic codependence of two cell populations generates acollective oscillatory dynamic during the expansion of a Bacillussubtilis biofilm. We develop a modeling framework for the spatio-temporal dynamics of the associated metabolic circuit for cells in acolony. We elucidate the role of metabolite diffusion and the needof two distinct cell populations to observe oscillations. Uniquely,this description captures the onset and thereafter stable oscilla-tory dynamics during expansion and predicts the existence ofdamping oscillations under various environmental conditions. Thismodeling scheme provides insights to understand how cells inte-grate the information from external signaling and cell−cell com-munication to determine the optimal survival strategy and/ormaximize cell fitness in a multicellular system.

biofilm expansion | phenotypic differentiation | metabolic codependence |oscillations | reaction−diffusion system

Great efforts have been spent in recent years toward un-derstanding the role of gene regulatory networks in modu-

lating the gene expression dynamics of cells (1–4). In thesestudies, the dynamic behavior of gene expression for a cell can bemodeled by a set of coupled chemical rate equations that rep-resent the combinatorial interactions among genes. Cell−cellcommunication and cellular microenvironment can, however,substantially affect this single-cell picture, such as in the cases ofbacterial chemotaxis (5, 6), Notch signaling in normal developmentand tumorigenesis (7, 8), and the interplay among cell populationsin synthetic systems (9).The oscillatory expansion of bacterial colonies recently ob-

served in a biofilm system exemplifies how intercellular com-munication plays a central role (10). Liu et al. (10) placed acolony of the Gram-positive organism Bacillus subtilis in a growthchamber that confines biofilm expansion within a 2D space,while a microfluidic device constantly provides glutamate andwashes away metabolic products diffusing outside of the biofilm.Interestingly, the biofilm initially undergoes rapid and steadyexpansion until it reaches a certain size, when it exhibits oscil-latory growth rate with periodic halting. Detailed analysis revealedthat the spatial regulation of the biochemical pathways for cellmetabolism is critical for the formation of this specific dynamics.Certainly, bacterial colonies must develop efficient communicationmechanisms to react effectively to various environmental stressesand optimize the overall colony fitness.The biofilm dynamics can be discerned by examining the

glutamate/glutamine synthesis pathway in bacterial cells (Fig. 1).Since glutamate is the only available nitrogen source, the pro-liferation is supported by glutamine, which is synthesized fromboth glutamate and ammonium (11). Without ammonium in thegrowth media, cells need to produce ammonium from glutamatecatalyzed by the glutamate dehydrogenase (GDH) enzyme (12). Toexplain the oscillatory dynamics of biofilm expansion, Liu et al. (10)

proposed a model of metabolic codependence. According to theirexperimental observations, bacterial cells have the following twospatially distinct metabolic phenotypes. Cells at the internal regionof the colony (interior) are responsible for ammonium synthesis bymeans of glutamate diffusing from periphery. Cells at the externalpart of the biofilm (periphery) do not produce ammonium by theirown due to a futile cycle: if ammonium was synthesized in theperiphery, it would rapidly diffuse to the growth media, making theprocess especially inefficient and thus unlikely (13, 14).Therefore, peripheral cells exploit the ammonium produced in

the interior and the glutamate diffused from the media to syn-thesize glutamine and support cell growth. The proliferation ofperipheral cells consumes substantial amounts of glutamate andammonium, leaving interior cells without substrate for ammo-nium synthesis. Since the interior stops ammonium production,peripheral cells do not have sufficient nutrients for growth. Whengrowth halts, the external glutamate from the media becomes againaccessible to the internal cells. Thus, the production of ammoniumis restarted, leading to another round of growth.To test this proposed mechanism, Liu et al. (10) established a

quantitative model by considering two discrete spatial compo-nents: the internal cell population and the external cell pop-ulation. The dynamics of the biofilm were described by a set ofordinary differential equations governing the time evolution ofthe size of the cell populations and the levels of the metabolites

Significance

Oscillatory dynamics commonly arises in a variety of multicel-lular biological systems. Bacterial colonies exploit such oscilla-tions to control the interplay between growth and resourceavailability. We model a recent experiment that observes theoscillations of the growth rate during the expansion of a bac-terial colony of Bacillus subtilis, elucidating the origin of theoscillations in terms of the spatiotemporal dynamics of themetabolic interactions between cells within the biofilm. Fur-ther, different bacterial cell populations are required in thismodel for the oscillatory behavior to arise, therefore demon-strating they are necessary for the colony survival. This ap-proach provides a platform to model a large class of biologicalphenomena involving the formation of large aggregates ofcells and/or a heterogeneous cell population.

Author contributions: F.B., Y.S., M.L., and J.N.O. designed research; F.B., Y.S., and M.L.performed research; F.B., Y.S., and M.L. analyzed data; and F.B., Y.S., M.L., and J.N.O.wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

Data deposition: The code used for the numerical solution of the model is freely availableat https://github.com/federicobocci91.1To whom correspondence should be addressed. Email: jonuchic@rice.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1706920115/-/DCSupplemental.

Published online April 2, 2018.

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within these two regions. This design can capture the essentialtraits of the biofilm dynamics and reproduce the experimentallyobserved oscillations in the growth rate of the colony. It alsocorrectly predicted the dynamics of biofilm growth under variousconditions of growth media.Much remains to be understood, however, regarding how

these metabolic reactions are regulated by the spatial organiza-tion of the organisms inside the community. Without a directcharacterization of the diffusion of the nutrient molecules in thebiofilm, the original model is not capable of explaining why os-cillations are detected only when the size of the colony exceeds acertain threshold and, more generally, how the metabolic inter-actions are regulated in different regions of the biofilm. Thesequestions are hard to address without a more comprehensiveframework that describes, at different spatial locations, both thetemporal dynamics of the metabolic activities and the proliferationof the cells.We introduced an improved scheme to study both temporal

and spatial dynamics of a biofilm. Compared with the descriptionproposed by Liu et al. (10), this model considers the diffusion ofsmall metabolites by incorporating the spatial dynamics of thebacterial colony. The biofilm is described as a continuous 2Dsystem where the more external layer represents bacteria withthe peripheral phenotype and the more internal layer representsbacteria with the internal phenotype. These improvements allowus to investigate the spatial organization of cells with differentphenotypes, and directly explore the repercussions of the gluta-mate/glutamine synthesis pathway on the biofilm development.The proposed model can explain the recurrent oscillatory cyclesof the growth rate in terms of the space-dependent interplaybetween the internal and peripheral phenotypes and reproducethe observed growth dynamics in the presence of altered conditionsof the growth media. Moreover, the occurrence of oscillations isinsensitive to the radius of the biofilm; instead, it is sensitive tothe width of the peripheral layer. These findings suggest that theonset of the oscillation of the biofilm expansion rate is due to theswitch of the bacterial cells from interior to peripheral phenotype,and it is specifically triggered when the peripheral layer increasesits width to a certain level. Finally, various types of growth dynamics,

including dampened, stable, dissipating dynamics, are revealed byvarying the ratio between interior and peripheral cells and mod-ifying the biofilm’s external conditions.

ResultsA Spatial−Temporal Model of Metabolic Interplay in the Biofilm. Inthe original experiment by Liu et al. (10), a special microfluidicdevice was designed to limit biofilm expansion to two dimensions.With a growth media containing nutrients such as glutamate, B.subtilis grows and eventually forms a colony that is approximatelycentrosymmetric as no shear was applied to the system. There-fore, we described the biofilm dynamics in a 2D space, but thecurrent framework can be naturally generalized to a 3D system.Due to the symmetry, we further simplified the model with apolar coordinate reference frame centered in the biofilm andconsidered the spatial dynamics along the radial coordinate only,hence reducing the biofilm to a 1D system.We implemented the glutamate metabolic pathway via a

reaction−diffusion system that extends the previous modeling ofLiu et al. (10) and describes the spatiotemporal dynamics ofammonium, glutamate, GDH, and housekeeping proteins (Eqs.1–4). Ammonium and glutamate can diffuse through the biofilmand its boundaries, while GDH and housekeeping proteins arenot diffusible, because they represent the specific levels insidebacterial cells. Specifically, ammonium diffuses faster than glu-tamate (15, 16). Nutrient availability in the growth media wasmodeled via boundary conditions at the biofilm boundary. Thecell metabolic phenotype is determined by a sigmoid signal thatdepends on the radial distance of the cell from the biofilm center(Eq. 5 and SI Appendix, Fig. S1). Cells within the inner layer, theinterior, can activate the GDH enzyme and therefore catalyzeammonium production when there is sufficient glutamate. Con-versely, cells in the external layer, the periphery, cannot synthesizethe nutrient, due to their futile cycle of ammonia.

Metabolic Oscillations in the Biofilm. Although our main objectivewas to investigate the growth process of the biofilm, we studied asimplified “nongrowth model” first to understand the governingprinciples of the glutamate metabolic pathway. This reduced modelconsidered the dynamics of the chemical species and increasednutrients’ consumptions for fast-growing cells, but disregardedbiofilm expansion.First, we discovered that the nongrowth model can exhibit

different oscillatory dynamics by varying the sharpness of thetransition between internal and peripheral phenotypes and thediffusion coefficients of the nutrients (SI Appendix, Fig. S2).Specifically, tuning the diffusion coefficient of ammonia andglutamate to measured values (15, 16) led to metabolic oscillationsspread through the whole biofilm (Fig. 2 A and B). Further, suchoscillations presented the typical traits of nutrients exchange andmanagement reported by Liu et al. (10). The concentration ofammonium was higher in the interior, where the metabolite isproduced, while the glutamate level was larger in the periphery dueto the high external concentration (Fig. 2 C and D). Housekeepingproteins synthesized from glutamate and ammonium were mostlyobserved in the periphery (SI Appendix, Fig. S3).Notably, a smooth interior−periphery phenotypic transition

suppressed oscillations, strongly suggesting the need for two distinctcell phenotypes. Also, oscillations were insensitive to biofilmsize, but required an intermediate size of the peripheral layer (SIAppendix, Fig. S4). Further, the system was robust against localvariation of biochemical interaction parameters due to the archi-tecture of the metabolic circuit (17) (SI Appendix, Figs. S4 and S5).Lastly, the oscillation period increased with the peripheral layer’swidth, while it decreased with the glutamate diffusion coefficient(SI Appendix, Fig. S5), hence indicating that how fast glutamatediffuses through the periphery determines the timescale of metabolicoscillation.

Fig. 1. Schematic of the glutamate metabolic pathway in a B. subtilis col-ony. Glutamate diffuses from the growth media through the periphery ofthe biofilm (blue-shaded area) and then toward the interior (yellow-shadedarea), where it activates the GDH, leading to the synthesis of ammonium.Biomass production requires both glutamate (diffused from the exterior)and ammonium (produced and diffused from the interior). The feedbackmechanism of the new biomass synthesis on glutamate and ammoniumrepresents the increased metabolism resulted from cell growth. Metabolicproducts that diffuse outside the biofilm are washed away from the growthchamber.

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The Onset of Biofilm’s Oscillations. Next, we developed a theoreti-cal framework for the growth of the biofilm (Eqs. 6–8). In theproposed scheme, the production of new biomass correlates tothe level of housekeeping proteins. Specifically, we hypothesizedthat a basal level of proteins is employed in ordinary cellularactivities and it is thus not available for the growth. The biofilmgrowth was coupled with the metabolism of glutamate by solvingthe reaction−diffusion equations of the glutamate biochemicalnetwork on the expanding biofilm area. To simulate the creationof new cells, the growth algorithm first computed the newlyproduced biomass and then updated the radius of the biofilmaccordingly (see Methods for details).Furthermore, we assumed that the peripheral layer grows

proportionally to the radius of the biofilm when the colony issmall but saturates to a constant value when the colony is large(Fig. 3A and Eq. 9). This hypothesis ensures that there areenough peripheral cells to effectively sustain the growth at allstages of biofilm development and prevents the oscillationquenching observed for an excessively large periphery.The model correctly reproduced the onset of oscillations and

replicated the stable oscillations observed afterward (Fig. 3B). Inthe small biofilm, the glutamate diffuses through the thin peripherallayer and guarantees an adequate nutrient supply to interior cells.Under these circumstances, the growth rate does not oscillate. Oncethe critical colony diameter is exceeded, the trajectory of the growth

rate presents a continuous transition to oscillations whose amplitudeevolves from zero to a nearly steady value while the diameter ofthe biofilm—and thus the size of the peripheral layer—increases.In this regime, the time required by glutamate to diffuse throughthe peripheral layer is too long to directly supply the interior, asthis nutrient is also employed in the periphery to synthesizehousekeeping proteins. The possible starvation of interior cells isresolved through periodic decreases of the growth rate, the oscillationminima, that allow some glutamate to diffuse to the interior beforebeing consumed by peripheral cells. Once the internal level ofammonium has been restored, the biofilm can start a new stage offast growth, which in turn consumes the interior ammonium andsets the conditions for a new cycle. As the growth continues, thetrajectory approaches a limit cycle and the growth rate oscillateswith a nearly constant amplitude and a period around 2.5 h (Fig.3C), in agreement with experiments (10).We further tracked the growth rate in different points of the

biofilm and showed that peripheral cells give the larger contri-bution to biofilm expansion (Fig. 3 D–F).The growth trajectory was robust against local variation of the

diffusion constants and the parameters of the growth model (SIAppendix, Fig. S6 A–D). Additionally, stress factors such as en-vironmental pollution, pH variation, or waste accumulationcould modulate cellular growth (18, 19). We modeled such fac-tors as (i) decrease of the cellular growth rate constant or (ii)increase of housekeeping proteins threshold for cellular growth.Both approaches resulted in a slower growth but with robustoscillatory behavior (SI Appendix, Fig. S6 E and F).

Oscillations Depend on the Peripheral Layer. Different growth dy-namics were observed (Fig. 4) by changing the maximum widthof the peripheral layer (the parameter D0 of Fig. 3A). This

A B

C D

Fig. 2. The metabolic network generates oscillations in the biofilm. (A) Radialdistribution of ammonia in the nongrowth biofilmmodel. The continuous blackline shows the average level, while dotted black lines and color fill depict theoscillation range. The vertical red line at r = 850 μm highlights the transitionbetween the interior region and peripheral region. Ammonia level is nearlyconstant in the interior, increases slightly in the transitional region, and dropsquickly approaching the biofilm boundary. (B) Radial distribution of glutamate.The periphery exhibits larger levels of glutamate, due to the higher exposure tothe external media. (C) Time trajectories of the concentration of ammonia inone point in the interior (red) and one point in the periphery (black). The twopoints are equidistant from the interior−periphery transition line (radial coor-dinates for interior point: r = 800 μm; peripheral point: r = 900 μm). Oscillationsare observed in both points, but the concentration is higher in the interior. (D)Time trajectories of the concentration of glutamate in one point in the interiorand one in the periphery. The level of glutamate is much higher in the pe-riphery. For this simulation, the radius of the biofilm was S=1,000  μm and thewidth of the periphery was d = 150 μm.

(d) (e)

(f)

A

C

D E F

B

Fig. 3. The growth model correctly reproduces the expansion dynamics ofthe biofilm. (A) The peripheral layer expands linearly for a small biofilmradius (S < S0) and saturates to the limiting value D0 at a large biofilm size(S > S0). (B) Radial growth rate as a function of biofilm diameter. Oscillationsarise around 550 μm to 600 μm. (C) Radial growth rate as a function of time.The oscillation period gradually increases and relaxes toward ∼2.5 h. (D–F)The spatial (radial) distributions of growth rate at different time points: (D)small biofilm, (E) maximal growth rate, and (F) minimal growth rate accordingto arrows in C.

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parameter controls the fraction of cells with a peripheral phe-notype in the biofilm. If the periphery did not reach the transi-tion width for oscillation onset (vertical dashed line of Fig. 4A),the growth rate did not exhibit oscillations (Fig. 4 A and B). Forintermediate values of D0, the biofilm attained a stable oscillatinglimit cycle with constant amplitude, that was maintained even whenthe colony was very large (Fig. 4 A and C). As the parameter D0was further increased, the oscillations underwent a partial damp-ing. In the damping region, the growth rate is considerably lower,as the glutamate diffusing from growth media is consumed byperipheral cells, preventing the necessary supply to interior cells(Fig. 4 A and D). At even larger values, the oscillations couldcompletely stop after some oscillatory cycles (Fig. 4 A and E).According to the experimental observations of Liu et al. (10),

the oscillations remain stable for a long period (more than a day),suggesting that an intermediate value of the parameter yields stableoscillations such as in Fig. 4C. This regime is characterized by acompetition between the maximal growth and the effective protectionof the interior cells, and supports the interesting interpretation ofthe oscillations as a trade-off between colony growth and shelteringof the interior population proposed by Liu et al. (10).

Strategies to Control Biofilm Growth via Metabolic Interplay. Am-monium administration in the growth media quenched oscilla-tions as the periodical dearth of ammonium in the periphery wasbalanced by environmental supply (Fig. 5A). This finding agreeswell with experimental observations (10). Interestingly, simulating

biofilm growth for different ammonium levels at the biofilm boundary(AEXT) showed a continuous, first-order transition of the oscillationamplitude (Fig. 5B). Further, a low value of AEXT allowed growthoscillations but increased the oscillation onset (Fig. 5C). Similarly,GDH induction in the periphery quenched the oscillations viaremoving phenotypic diversification, as observed experimentally(10) (SI Appendix, Fig. S7).A higher concentration of glutamate in the growth media

(GEXT) provides interior cells with more nutrient and shifts theoscillation onset to a larger biofilm size (20). Therefore, we assumedthat the transition biofilm radius S0 (Fig. 3A) for periphery expan-sion is a linear function of glutamate (Eq. 10). We further hypoth-esized that the deeper glutamate penetration inside the biofilm couldallow a larger peripheral cell population, and therefore assumed thatthe peripheral layer could expand more (Eq. 11). Under these as-sumptions, we observed faster biofilm growth and larger oscillationamplitude upon glutamate increase on a twofold change around theoriginal concentration of 30 mM (15 mM to 60 mM) (Fig. 5D). Theincreased nutrient availability shifted the oscillation onset to alarger biofilm size (Fig. 5E), in good agreement with experiments(20). Intriguingly, the model predicted a maximal oscillation periodat intermediate GEXT (Fig. 5F). This finding can be interpreted asa balance of the following processes: a larger GEXT implies alarger peripheral layer, which increases the oscillation period, butalso a deeper glutamate penetration, therefore supplying interiorcells more rapidly. All features of growth rate, onset, and oscil-lation period were conserved upon variation of the parameterscontrolling the peripheral layer’s dependence on GEXT (SI Ap-pendix, Fig. S8).

DiscussionWe introduced a mathematical description to investigate theglutamate metabolic pathway in cells and explain the oscillatorygrowth of bacterial colonies of B. subtilis recently observed by

A

B C

D E

Fig. 4. Peripheral layer variation generates different growth dynamics.(A) Average growth rate (dots) and oscillation amplitude (error bars) as afunction of the saturation value of the peripheral layer (D0). The vertical redline depicts the transition to oscillatory behavior. Each data point representsa different biofilm growth trajectory. The growth rate and oscillation amplitudewere always computed at a diameter of 1,000 μm. (B−E) Growth rate as afunction of biofilm diameter for (B) D0 = 50 μm (no oscillations), (C) D0 =150 μm (stable oscillations), (D) D0 = 250 μm (partial dampening), and(E) D0 = 350 μm (complete quenching).

A

D

B C

E

F

Fig. 5. Biofilm growth dynamics under various perturbations. (A) Growth rateas a function of time before and after the administration of 1 mM ammonium(orange shading) in the biofilm growth media. (B) Growth rate maximum(blue) and minimum (red) 10 h after the administration of external ammoniumat different levels (AEXT). For AEXT > 0.1 mM, oscillations disappear (black). (C)Oscillation onset as a function of AEXT. In this panel, external ammonium AEXT

was present since the beginning of the simulation. (D) Growth rate (dots) andoscillation amplitude (error bars + color shade) as a function of the externalglutamate. Growth rate and oscillation amplitude were computed at a diameterof 1,500 μm. (E) Oscillation onset as a function of GEXT . (F) Oscillation period as afunction of GEXT computed at a diameter of 1,500 μm. In B–F, each dot repre-sents a different growth trajectory with a different AEXT or GEXT condition.

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Liu et al. (10). Improving the previous modeling scheme, thiswork regarded the biofilm as a continuum domain and investi-gated the spatiotemporal regulation of the metabolic interac-tions with a set of coupled reaction−diffusion equations. Weconsidered a biofilm composed of two distinct cell phenotypes.The interior population accounts for ammonium production inthe presence of active GDH enzymes, while the peripheralpopulation utilizes the ammonium produced in the interior andthe external glutamate to sustain colony growth. The phenotypictransition was based on the futile cycle of ammonia loss thatmakes ammonium homeostasis costly for peripheral cells. Thechoice of a space-dependent framework was advantageous as itallowed us to explicitly describe the diffusion of metabolites inthe colony. Secondly, the metabolic interactions could be eval-uated in different parts of the system, allowing us to preciselyaddress the phenotypic distribution and/or heterogeneity of thebiofilm. Therefore, this modeling scheme provides a perspectiveon how external signaling and cell−cell communication contributeto the dynamics of intracellular biological networks.The spatiotemporal model can reproduce the observed growth

dynamics of a B. subtilis biofilm. Specifically, improving thepreexisting scheme, this model can predict the onset of the os-cillations. When the size of the biofilm is small, the growth doesnot present oscillations, as diffusion ensures nutrient availabilityeverywhere in the colony. As the biofilm expands, the dual roleof glutamate, required both for ammonium synthesis in the in-terior and new biomass creation in the periphery, generates aconflict that requires an ad hoc management of the availableresources. These competitive processes are resolved with oscil-lations of the growth rate, where the periodic halting of the ex-pansion allows the glutamate to diffuse in the biofilm interior.The variation of the peripheral layer size led to a rich ensembleof growth dynamics, including stable growth, stable oscillations,and partial and complete damping. Currently available experi-ments suggest that oscillations can remain stable for a longperiod (10). It has been suggested that the metabolic interplaybetween interior and peripheral populations represents an op-timal trade-off that can resolve the conflict between shelteringinterior cells and promote growth of the colony at once (10, 21).This model’s predictions strengthen this hypothesis, since thestable oscillations correspond to an intermediate size of theperipheral layer which balances a maximal growth rate, achievedwith a small periphery, and an effective protection of interiorcells, accomplished with a large periphery. The validation of thesepredictions would further improve our understanding of the orga-nization and survival strategies of bacterial communities.Biofilm proliferation is certainly regulated by biophysical pro-

cesses such as local variations of pH, accumulations of waste pro-ducts, or pollution (18, 19) that we modeled as simple rescaling ofthe growth parameters. However, understanding how these pro-cesses impact the biofilm dynamics represents an exciting challengeat both the experimental and theoretical levels.Seemingly counterintuitive, providing excessive nutrients sup-

presses the metabolic interplay of the colony: the supplement ofammonium allowed peripheral cells to grow autonomously, hencedecoupling them from the interior. This “overfeeding” strategyresults in strong nutrient consumption in the periphery and aconsequent starvation of the interior. Although sporulation can bea response to growth slowdown and nutrient shortage (22, 23), suchan energy-consuming process can be quite challenging in the densebiofilm environment if the level of nutrients diminishes too rapidly.As suggested by Liu et al. (10), this could thus represent an effectivestrategy to suppress the interior population of the colony. Investi-gating, both experimentally and theoretically, cell fate dynamics inresponse to overfeeding could play a major role in developing newtherapies to fight biofilm infections. Further, a low exposure toammonium conserved oscillations but could increase the oscillationonset compared with the ammonium-free biofilm.

Increasing the glutamate level in the growth media resulted ina larger growth rate. Nonetheless, metabolic oscillations werestill observed, because ammonium was still self-produced insidethe biofilm. Additionally, the oscillation onset increased withthe external glutamate, in agreement with experiments (20),while the oscillation period was maximized at an intermediateexposure. Quantitatively validating these predictions would requiretracking biofilm growth for different nutrients conditions, which, inturn, would offer further data to calibrate the mathematical model.This work presented a theoretical formalism that can go be-

yond the expansion of bacterial biofilm and may be applied to abroader class of biological systems that involve the competitionover limited resources, the development of large aggregates ofcells, and/or a heterogeneous cell population. Cells within thebiofilm integrate the information from cell−cell signaling andmicrocellular environment to design the optimal survival strategyand maximize cell fitness. In the considered resource-limitedenvironment, the bacterial community resolves the shortage ofnutrients through collective metabolic oscillations. Cell fate de-termination is influenced by intrinsic fluctuations in gene expres-sion dynamics and extrinsic noise resulting from the surroundingmicroenvironment at a single-cell level (24–26). Signaling mecha-nisms among neighboring cells represent a driving force, as well, inestablishing phenotypic expression (21, 27), especially when ex-ternal conditions force microorganisms to cooperate to maximizethe overall fitness level (28, 29). The traditional population-basedapproach already provided important insights in the system biologymodeling over the last decades (30, 31). This approach, however,needs to be expanded to include the spatial dynamics to capture theheterogeneity of the cell−environment and cell−cell interactions(32, 33). Here, the spatiotemporal modulation of the glutamatemetabolism in B. subtilis was addressed within a multicellular system,thus taking a step further toward understanding the connectionbetween cell−cell signaling, metabolism, and biofilm expansion.

MethodsThe reaction scheme presented in Fig. 1 is implemented via a reaction−diffusionsystem that describes the spatial and temporal dynamics of the concentrationsof the four considered variables: ammonium (A), glutamate (G), GDH (H), andhousekeeping proteins (R), as a function of the radial position r and time t,

∂A∂t

=DA∇2A+ αGH− δAAR, [1]

∂G∂t

=DG∇2G− αGH− δGGR, [2]

∂H∂t

= βHGn

Gn +KHn fðrÞ− γHH, [3]

∂R∂t

= βRAG–γRR. [4]

Ammonium is synthesized from glutamate, and the reaction is catalyzed bythe enzyme GDH, so the production rate of ammonium is expressed as αGH.GDH is activated once glutamate reaches a threshold level, which can bemodeled by a Hill function, Gn=ðGn +KH

nÞ, where KH is the glutamatethreshold and n is the Hill coefficient. Additionally, GDH activation dependson the colony spatial organization through a position-dependent sigmoidfunction fðrÞ, expressed by

fðrÞ= 1−1

1+ expð−ðr − riÞ=l0Þ, [5]

where ri is the threshold position for the switching function (hence, theradius of the biofilm interior) and l0 represents the distance of the spatialtransition between internal and peripheral cells; the smaller the l0, thesharper the transition. Therefore, the width of the peripheral layer isd = S− ri, where S is the total biofilm radius. The production of housekeepingproteins is a second-order kinetic process that depends on both ammonium andglutamate (βRAG). The process is accompanied by the consumption of bothammonium (δAAR) and glutamate (δGGR). The model also includes the de-activation of GDH (γHH) and the degradation of housekeeping proteins (γRR).

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The conditions at biofilm boundary (r = S) are GðSÞ=GEXT , AðSÞ=AEXT

(Dirichlet boundary conditions). In Figs. 2–4, GEXT = 30 mM and AEXT = 0 mMaccording to the experimental setup of Liu et al. (10). Conversely, the pa-rameter values were varied in Fig. 5.

The growth rate of biofilm area (Ω= πS2) depends on the amount ofhousekeeping proteins in the colony, defined as an integral over the wholebiofilm,

dΩdt

= kZΩ

gðr, ϑÞrdr   dϑ, [6]

where k is the growth area per unit time and per mole of proteins and theintegration is carried over the biofilm area Ω. The function gðr, ϑÞ representsthe amount of housekeeping proteins used for colony expansion for cells atthe position ðr, ϑÞ,

gðr,ϑÞ=�Rðr, ϑÞ−R0 Rðr, ϑÞ>R0

0 Rðr, ϑÞ≤R0, [7]

where R0 is the basal concentration of housekeeping proteins. The azi-muthal symmetry yields a factor 2π in the integration of the angular com-ponents. Consequently, the growth rate of the biofilm radius S is

dSdt

=kS

Zs

0

gðrÞrdr. [8]

The thickness of the peripheral layer d is a function of the biofilm radius Sand the external glutamate GEXT ,

d =D0ðGEXT Þtanh�

SS0ðGEXT Þ

�. [9]

It expands linearly for small S and saturates to a limiting value D0ðGEXT Þ forlarge S [tanhðS=S0ðGEXTÞÞ]. The linear expansion−saturation switch point in-creases linearly with GEXT ,

S0ðGEXT Þ= s0GEXT , [10]

where s0 is a constant. Further, the limiting size of the periphery at verylarge radius S � s0GEXT increases with GEXT according to a Monod-like sat-uration curve,

D0ðGEXT Þ=d0GEXT

GEXT +G0[11]

with glutamate threshold G0. Therefore, d0 is the maximal peripheral sizeachieved at GEXT � G0 and S � s0GEXT . In Figs. 3 and 4, D0ðGEXT Þ andS0ðGEXT Þ are indicated as constants (D0, S0), because GEXT is fixed. This satu-ration value G0 is a very likely possibility, but the experimental data cannotprobe this regime yet.

The algorithm for the growth of the biofilm includes three steps. First, thenewly produced biomass is computed by solving Eqs. 1–4 while the radius ofthe biofilm S is kept fixed. Next, the increment of S is computed using Eqs. 6–8. Lastly, the width of the peripheral layer d corresponding to the new S iscomputed using Eqs. 9–11, and the interior radius ri is updated as well. Thegrowth step increases the area of the discrete elements of the 2D grid whereEqs. 1–4 are solved. Therefore, the concentrations of chemicals are rescaledto the new biofilm area to ensure the same amount of total nutrients in thebiofilm area before and after the growth step; Xnew = Xold Sold

2/Snew2, where

X = A, G, H, R, and S is the radius of the biofilm. The code used for thenumerical solution of the model is freely available on the github page of F.B.(https://github.com/federicobocci91).

The numerical values of the parameters and further details on themathematical model are presented in SI Appendix, sections S1–S5.

ACKNOWLEDGMENTS. We have benefited from fruitful conversationswith Gurol Suel and Jintao Liu. The work at the Center for TheoreticalBiological Physics was sponsored by the National Science Foundation(Grants PHY-1427654 and MCB-1241332). M.L. is partially supported by theNational Cancer Institute of the National Institute of Health under AwardP30CA034196.

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