Morphological optimization for access to dual oxidantsin biofilmsChristopher P. Kempesa,b,c,d,1, Chinweike Okegbee, Zwoisaint Mears-Clarkee, Michael J. Followsd,and Lars E. P. Dietriche,1

aExobiology Branch, National Aeronautics and Space Administration Ames Research Center, Moffett Field, CA 94035; bControl and Dynamical Systems,California Institute of Technology, Pasadena, CA 91125; cSETI Institute, Mountain View, CA 94034; dDepartment of Earth, Atmospheric and Planetary Sciences,Massachusetts Institute of Technology, Cambridge, MA 02139; and eDepartment of Biological Sciences, Columbia University, New York, NY 10027

Edited by James H. Brown, University of New Mexico, Albuquerque, NM, and approved November 8, 2013 (received for review August 16, 2013)

A major theme driving research in biology is the relationshipbetween form and function. In particular, a longstanding goal hasbeen to understand how the evolution of multicellularity conferredfitness advantages. Here we show that biofilms of the bacteriumPseudomonas aeruginosa produce structures that maximize cellularreproduction. Specifically, we develop a mathematical model of re-source availability and metabolic response within colony features.This analysis accurately predicts the measured distribution of twotypes of electron acceptors: oxygen, which is available from theatmosphere, and phenazines, redox-active antibiotics produced bythe bacterium. Using thismodel,we demonstrate that the geometryof colony structures is optimal with respect to growth efficiency.Because our model is based on resource dynamics, we also can an-ticipate shifts in feature geometry based on changes to the avail-ability of electron acceptors, including variations in the externalavailability of oxygen and genetic manipulation that renders thecells incapable of phenazine production.

Adesire to understand the relationship between form andfunction motivates many lines of inquiry in biology, in-

cluding the study of multicellular morphologies and the evolu-tion of these features. Cells grow and survive in populations inmany types of natural systems. These multicellular populationsinclude bacterial biofilms, such as Pseudomonas aeruginosa micro-colonies in the lungs of cystic fibrosis patients (1) and cyanobacterialcolonies and mats in marshes, lakes, and oceans (2, 3), and complexeukaryotic macroorganisms, such as plants and animals. In manyof these contexts, enhanced survival arises from advantagesassociated with multicellularity at multiple scales. Recent workdemonstrated that simple cooperation within aqueous microbialbiofilms allows groups of genetically similar cells to form tallmushroom-like structures that reach beyond local depletionzones into areas of fresh resources (4–12). Other work has shownthat basic colonial growth (e.g., that of budding yeast) and theevolution of complex multicellular life are accompanied by en-hanced growth efficiency and several energetic advantages (13–15).The specific organization of and relationship between cells liv-

ing within a population are important characteristics that de-termine whether organisms benefit from the multicellular lifestyle.For example, in mammals the metabolic rate of individual cells isregulated by the size of the whole organism, an effect that confersgreater efficiency (15). Cells have dramatically elevated metabolicrates when living as individuals in culture compared with whenthey survive and grow as members of a multicellular organism(15). This regulation is considered a natural consequence of re-source supply within hierarchical vascular networks (15–17) andhighlights the importance of morphology and structure indictating cellular physiology for multicellular systems. Variationsin gross morphology also may provide information about howdifferent species are adapted to their environments. For exam-ple, differences in leaf venation patterns among diverse speciesof plants are fundamentally related to tradeoffs associated withcarbon assimilation and transpiration rates, water supply rates,and overall mass investment (18). This variation in venation net-work structure is affected by and may influence the mechanisms

and efficiency with which resources are supplied to individual cells.Gradients of resources, particularly oxygen, are similarly impor-tant in modulating the development of embryos, lungs, hearts, andtumors (19–24).Thus, one of the dominant effects of multicellularity is to alter

the environment experienced by the individual cell. This raisesquestions such as (i) how are morphology, metabolism, and en-vironmental conditions interconnected and (ii) how do theserelationships govern the fundamental benefits and tradeoffs ofmulticellularity? Understanding these connections may help usunderstand the general trajectory by which simple and complexmulticellular life evolved and allow us to better identify andquantify morphologically complex structures that may representpaleobiological populations (25).Bacteria form structurally diverse biofilms in different envi-

ronments and are well-suited for the study of relationships amongthe morphology, metabolism, and chemical heterogeneity ofmulticellular populations (26). Most laboratory studies address-ing biofilm development are performed in flow-cell systems, inwhich bacteria are grown on a glass slide and exposed to a con-tinuous flow of a liquid growth medium. In the regime of biofilmssubmersed in liquid, representative Gram-negative bacteria, in-cluding P. aeruginosa, form mushroom-like structures in responseto external nutrient gradients (4–12). We grow P. aeruginosabiofilms on agar-solidified growth media in a controlled atmo-sphere. A standard protocol, the colony morphology assay, is usedto generate these biofilms (27). Ten microliters of a cell suspen-sion is pipetted onto an agar plate, and the development of thecolony at room temperature and constant humidity is monitoredover several days. In this model system, we can modify the

Significance

Principle questions in the field of biology concern the evolutionaryadvantages conferred by the multicellular lifestyle and by theformation of specific multicellular structures. In microbial biofilms,many cells cooperate to form distinct macroscopic patterns. Herewe develop and apply a mathematical model, which incorporatesempirical values, to show that the tall “ridges” produced byPseudomonas aeruginosa colony biofilms maximize communitygrowth via enhanced access to oxygen. Production of endogenousredox-active compounds, which mediate electron transfer fromcells to oxygen, allows the formation of wider ridges. Externallysupplied and endogenously produced oxidant availability con-trols feature geometry in a manner consistent with the modelpredictions for optimal geometry.

Author contributions: C.P.K., M.J.F., and L.E.P.D. designed research; C.P.K., C.O., Z.M.-C.,and L.E.P.D. performed research; C.P.K. and C.O. contributed new reagents/analytic tools;C.P.K., C.O., and L.E.P.D. analyzed data; and C.P.K., M.J.F., and L.E.P.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. E-mail: ckempes@gmail.com or LDietrich@columbia.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1315521110/-/DCSupplemental.

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environmental and biological conditions and examine the re-sulting shifts in structure and chemistry. Colony biofilms that formunder these conditions exhibit spatial patterning (Fig. 1A) andenvironmental sensitivity and constitute an ideal model systemfor exploring morphology as a metabolic adaptation (28).Although numerous processes, including extracellular matrix

production, osmotic pressure gradients, cellular motility, andspatially heterogeneous cellular death (e.g., refs. 4, 9, 12, 29, 30),have been shown to be involved in the production of biofilmpatterning, it is not yet fully understood why or how feature for-mation has evolved, especially in the context of air-exposed bio-films. We have reported that redox-active signaling moleculescalled phenazines have dramatic effects on population behavior.Whereas wild-type colonies remain smooth for the first 2 d ofgrowth, phenazine-null mutant (Δphz) colonies undergo a majormorphotypic transformation (31). At day 2, Δphz colonies begin tospread over the surface of the agar plate. Two types of wrinklestructures form: those within the area of the original droplet, whichappear to be more “disorganized,” at the colony center and thosethat radiate from the center toward the colony edge (referred tohere as “ridges”). Both types of features are vertical structures (Fig.1). Wild-type colonies also form wrinkles in the center, but onlyafter 3 d of growth, and a mutant that overproduces phenazinesremains smooth when monitored for 6 d. These strains thereforedemonstrate an inverse correlation between increased colonysurface area and phenazine production (31).More recently, we showed that P. aeruginosa colony wrinkling is

a strategy to increase oxygen accessibility (28). We have proposedthat phenazines attenuate this mechanism because they act aselectron acceptors for cells in anoxic regions of the biofilm andshuttle electrons to the well-aerated regions, allowing cells to bal-ance their internal redox state (28, 32, 33). We proposed that a re-duced cellular redox status triggers a morphological change at thepopulation level. Consistent with this model, adding nitrate (analternate electron acceptor) to the agar, or increasing atmosphericoxygen concentrations (to 40%) prevents spreading of the Δphzcolony and discourages wrinkle formation. Decreasing externaloxygen availability (to 15%) increases wrinkle formation as well asspreading of the colony (28). Oxygen profiling of the coloniesindicates that the Δphz mutant is well oxygenated within wrinkles,suggesting that its morphotype is an adaptation for enhanced oxy-gen uptake (28). It is noteworthy that wrinkles reach a final, staticwidth while continuing to grow taller. The wrinkle width achievedis correlated with the concentration of oxygen provided in the

atmosphere. Additionally, eliminating the ability to wrinkle (bydeleting the genes for extracellular matrix production) has beenshown to lead to a redox imbalance, as indicated by an elevatedNADH/NAD+ ratio (28).The sensitivity of colony morphogenesis to oxygen, nitrate, and

phenazines suggests a significant role for oxidative capacity inmodulating morphology. Here we develop a simple mathematicalmodel, rooted in established physical laws and physiology, thatdemonstrates and confirms this significance. Combining this modelwith experimental verification enables us to make predictions andinterpretations that would be infeasible using modeling or experi-mentation alone. The model incorporates the response of growthrate to oxygen availability and accurately predicts the measuredoxygen distribution within colony features. Because it is based onresource dynamics, it also anticipates shifts in feature geometrybased on external oxygen availability. Finally, quantitative experi-ments verify our methodology. We demonstrate that biofilmsproduce features with geometries optimal for the growth effi-ciency of the entire colony and sensitive to the redox environ-ment, as determined by factors such as oxygen availability andphenazine production.

Modeling Biofilm Metabolism and Oxygen DynamicsSeveral key features of the colony biofilm system inform our un-derstanding of the processes governing morphology and form thefoundation for our modeling framework. (i) Wrinkling in thecolony system is a basic feature of the wild type that occurs late indevelopment as a response to electron acceptor limitation (28).(ii) Overall colony wrinkling, as well as the geometry of individualfeatures (width of wrinkles), can be modified easily by altering theredox conditions (e.g., increasing oxygen availability or knockingout the ability to produce phenazines) (see ref. 28 for P. aerugi-nosa and ref. 34 for similar results in Bacillus subtilis). (iii) Colonywrinkles reach a constant width early in development but con-tinue to grow taller at a nearly linear rate. Taking these last twoobservations together, we assume that electron acceptor ratherthan nutrient supply is the primary limitation faced during colonydevelopment and is responsible for feature geometry. If nutrientsupply from the agar below a feature were controlling width, wewould expect to see a decrease in the overall growth of a feature asthe constant width is reached. We would not expect to observe thelinear increase in ridge height; this would only increase nutrientlimitation as chemicals are forced to diffuse over a narrow andincreasingly long distance.

A B

C

Fig. 1. Horizontal (A) and vertical (B) structures ofP. aeruginosa colony biofilms. The colony was grownair-exposed on 1% tryptone, 1% agar for 5 d. Thescale bar represents 0.5 cm. (C) The simulated con-centration of oxygen within a ridge.

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To study the basic dynamics of resources and geometric featureformation, we first empirically characterize and model colonies ofthe phenazine-deficient mutant (Δphz) in a context in which oxygenis the only available terminal electron acceptor. We later expandexperiments and the model to consider wild-type colonies, whichhave the sole addition of redox-active phenazines produced bythe cells.We base our mathematical model on the simplest hypothesis for

feature regulation: the dynamic feedback between metabolism, re-source availability, and physical diffusion (e.g., refs. 6, 8, 35–43). Weuse the Pirt model, which interprets the linear relationship betweenpopulation metabolism and growth rate as a metabolic partitioningbetween growth and maintenance (44) and is consistent with thephysiology of a broad range of bacterial and eukaryotic species (see,e.g., refs. 13, 45 for a review). It is combined with the Monod model,which parameterizes population growth rate as a saturatingfunction of the external concentration of a limiting resource(46). This combination of parameterizations has been usedwidely (e.g., refs. 6, 35, 41, 43) and here provides an equation forthe oxygen consumption as a function of available oxygen:

Q=μmax

Y½O2�

ks + ½O2�+P; [1]

where Q is a consumption rate per unit mass of a limiting re-source that provides energy and/or structural material to thepopulation of cells (mol resource·s−1·g cells−1), μ is the specificgrowth rate (s−1), Y is a yield coefficient (g cells·mol resource−1),P is a maintenance term (mol resource·s−1·g cells−1), μmax is themaximum growth rate approached as the substrate concentrationis taken to infinity, and ks is the half-saturation constant.Depletion of oxygen within the biofilm creates a gradient with the

atmosphere and drives a diffusive flux of oxygen into the colony.The time-dependent spatial concentration of oxygen may be de-scribed by

∂½O2�∂t

= D∇2½O2�−�μmax

Y½O2�

ks + ½O2�+P�a; [2]

where a is the density of cells in the colony (g cells ·m−3). Similarmodels have been used widely in biofilm and broad ecologicalmodeling (e.g., ref. 35). This equation can be conveniently non-dimensionalized:

∂½O2�p∂tp

=∇2½O2�p −� ½O2�p1+ ½O2�p

+ g�; [3]

where the temporal and spatial scales, respectively, have been

normalized by the factors tfac =aμmaxksY

and xfac =�aμmaxksYD

�1=2. Oxygen

concentrations have been divided by ks, and the nondimensionalmaintenance term is g = YP/μmax. For steady-state solutions,the model relies only on the concentration of oxygen at thecolony surface, which is determined by the atmospheric mix-ing ratio and pressure, and the two parameters xfac and ks. Wedetermined the value for both parameters by first compiling pub-lished measurements and then constructing the mean of everycombination within this compilation (Table S1). Here we use thePirt–Monod diffusion model framework to interpret the communitybenefit of specific geometrical features of the colony and to inter-pret mechanisms underlying the architectural optimization withinbiofilms. We also model the dynamics and benefits associated withmultiple oxidative resources and show that biogenic redox-activemolecules are beneficial to community growth and survival.

ResultsObservations of Internal Oxygen Distribution. To investigate howthese mechanisms regulate morphology, we study a single featureof a P. aeruginosa colony biofilm, i.e., the ridge (Fig. 1), and itssurrounding geometry.We take oxygen profiles of the colony “base” (Fig. 1B), where

biofilm geometry is simplest, and we use these profiles to examineour literature-based parameter estimates and calibrate our model.First, we measure oxygen availability with increasing depth (Fig. 2).There is some variation in the decay rate and the depth at whichoxygen no longer can be detected (Figs. S1 and S2). Most of thisobserved variation may be accounted for by rescaling the depth axisby a simple factor, after which the profiles share a similar shape.This normalization highlights common overall biological activityand variation in the physiological parameter values.We fit the observed variation in vertical 1D profiles by varying xfac

and solving for the steady-state oxygen concentration in our modelfor an isolated base geometry while keeping ks fixed. Fig. 2A showsthat steady-state solutions of our model in the base capture therange of observed oxygen profiles. The variation in profiles alsoallows us to calibrate the range of xfac to be bounded by 7.3 ×104 and 1.6 × 105 with a mean value of 9.1 × 104. This rangecompares well to our estimate from the literature of 7.7 × 104 μm(Supporting Information). Given our definition of xfac the observedvariation could be due to differences in physiological param-eters, such as maximum growth rate or substrate yield, or to dif-ferences in the physical parameters, such as density, wetness, andoverall oxygen diffusivity of the colony.Experimentally, we find that oxygen availability decreases with

increasing depth into the colony. However, this effect is muchmore dramatic in the base than in the ridge. This likely is theresult of increased surface area for a reduced cellular mass.To interpret this phenomenon, we use the calibrated model tosimulate the steady-state distribution of oxygen availability withina ridge and the surrounding base, and we do this for variousgeometries of the ridge (Fig. S3A).

A B

Fig. 2. Oxygen profiles of 4-d-old colony biofilmsfor (A) the base region and (B) the ridge. Measuredvalues are given by points, and simulations are rep-resented by the solid curves. We took measurementsfor 37 base profiles and 27 ridge profiles over 5 d ofgrowth, and the data in A and B represent the var-iation within the observed profiles. Arrows indicatethe top of the biofilm surface. Data from ref. 28.

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Our model also captures the general decay rate and shape ofthese ridge profiles (Fig. 2B). These profiles were created usingthe mean value of the parameter calibration and the generalvariation in ridge width because it is experimentally challenging tomeasure the width and oxygen profile simultaneously.Our ability to anticipate the general shape and variation of oxy-

gen profiles within the ridge gives us confidence in our calibrationand the model’s predictive capacity. It should be noted that oncethese calibrations have been made, all other results in this paperfollow directly as predictions.

The Impact of Colony and Feature Geometry. The increased depthof oxygen penetration in the ridge region (Fig. 2B) illustrates theimportance of geometry in enhancing oxygen availability withinthe colony. The question then becomes how changes in this ge-ometry affect oxygen availability within the colony. To explorethis, we simulate the simplest geometric variation by preservingthe overall shape of the ridge but varying the width of the ridgefor a fixed height. Fig. S3A (Supporting Information) shows thedistribution of oxygen within ridges of different widths. From theinternal oxygen distribution and Monod equation (SupportingInformation), we can calculate the local growth rate and we findthat ridges generally have an enhanced growth rate relative toflatter regions of the colony (Fig. S3B). This is because tall, thinfeatures increase the total surface area for a relatively smallamount of biomass, which enhances the oxygen flux per totalconsumption, resulting in increased oxygen penetration. How-ever, we also see in Fig. S3 (Supporting Information) that asridges grow wider, the effect of enhanced growth is diminishedand the layer of growing cells within the ridge region becomessimilar in thickness to that of the base regions.This effect implies a tradeoff for population metabolism: Al-

though thin structures are entirely oxygenated, allowing cells to growquickly, their relatively small size limits the number of cells theycontain. Thick features house many more cells, but the region ofenhanced oxygen penetration and growth is smaller. This tradeoffmay be summarized using the total reproductive rate, R, of thecolony, where the derivative of R with respect to mass describeshow much added growth the colony will experience given an in-vestment in additional mass. This derivative gives the growth ef-ficiency of the colony in that it represents the return on massinvestment in terms of total reproductive capacity. We calculate Rfor an ensemble of simulated features at a fixed height but withdifferent widths, and find that R reaches a maximum for ridgeswith a specific width. This represents the optimal width for effi-cient cell proliferation within the colony and also corresponds tothe width at which all cells within a feature can at least meet theirminimum metabolic maintenance requirements. More impor-tantly, the optimal width found in our model accurately predictsthe observed average width of colony features along with theobserved variation in widths for colonies growing on an agar plate

exposed to 21%external oxygen (Fig. 3). This prediction ismade bysolving for the optimal width using the minimum, maximum, andaverage parameter values from our calibration to base profiles.Similarly, for simulations of the base, we find that R approaches

a constant value when the height of the base reaches the maximumdepth to which oxygen can penetrate. Thus, any additional baseheight does not produce additional growth for the colony. Ex-perimentally, we find that the average base thickness correspondsto the average depth at which oxygen no longer is detectable.The observed colony features are such that if features grow

wider (or taller in the case of the base) than this optimal size, thecolony has invested a large amount of mass that will not growquickly. If the features are smaller than this optimum, the colonyhas invested a small amount of mass, but there will be less overallgrowth as a consequence. We suggest that natural selection hasfavored cellular behavior that gives rise to features with emer-gent geometries that maximally benefit the population, as wasproposed previously in similar contexts (e.g., ref. 26).It should be noted that we observe that during early colony

growth, the ridges maintain a constant width while height continuesto increase. This implies that nutrients are not limiting, and it then issurprising that in the outermost layer of cells, growth is arresteddespite the abundance of oxygen and capacity for further growth.This raises the possibility that cellular growth within the ridges isregulated to optimize the growth efficiency of the population.

Morphological Response to Oxygen Availability. We have shownevidence of oxygen control driving the optimization of colony form.Feature geometry is important because of the impact it has on theavailability of oxygen within the colony. Given this interconnectionbetween geometry and oxygen, we would expect that changes in theexternal availability of oxygen would result in shifts in colony form.To test this hypothesis, we exposed colonies to elevated concen-trations of external oxygen (40% compared with the standard 21%).To quantify these shifts, we again calibrate the parameters of our

model. We do not have the appropriate experimental setup tomeasure oxygen profiles in colonies grown in our hyperoxia cham-bers. As an alternative, we use colony base heights as proxies foroxygen penetration distances (base heights correspond to the depthat which oxygen is depleted). These measurements are sufficient forcalibrating xfac to the 40% external oxygen conditions.The observed widths in 40% external oxygen are wider than in

21%. Running our simulations with 40% external oxygen, and thecalibrated range of parameter values, we again successfully predictthe mean, upper bound, and lower bound on observed ridge widths(Fig. 3). Through our model, we interpret the wider ridge width asthe result of deeper oxygen penetration, which extends the width atwhich a diminishing return on mass investment occurs. We alsopredict, in good agreement with data, that in 15% external oxygen,ridge width will be reduced (Fig. S4).

A

B

C D

Fig. 3. The average width of colony ridges as a function of time for Δphz (A–C) and wild-type (D) colonies. (A and B) Histograms for all ridge widthmeasurements at 21% and 40% oxygen. (A–D) The solid red (21% oxygen) and blue (40% oxygen) lines represent our predictions for optimal ridge width, andthe dashed lines represent our predictions for the bounds of variation in ridge width.

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Morphological Response to Other Oxidants. Our observations andmodeling thus far have concerned a P. aeruginosa Δphz mutant,which cannot produce phenazines. We can predict the significanteffects of oxygen availability on feature development and theoptimal ridge width in this scenario, in which oxygen is the onlyelectron acceptor the bacteria can use to balance the intracellularredox state. However, it is important to test whether our con-ceptual framework may be used to determine the effects thatphenazines, as endogenously produced oxidants, have on featuregeometry (28). To address this challenge, we examine aspects ofthe wild-type colony in which we have developed a model thataccounts for the dual use of oxygen and phenazines as electronacceptors. We have developed a mathematical model in which weallow both oxygen and phenazines to diffuse and be reducedwithin the colony. In this model, oxygen is consumed directly bycells and also is used for reoxidizing phenazines, and cells reduceboth substrates to support their metabolic rate (see SupportingInformation for a detailed presentation of the model).This model accurately predicts the decay of oxygen with in-

creasing depth into the base region of the colony. Near thesurface, this decay rate is observed to be nearly identical to thatof the Δphz colonies. However, deep in wild-type colonies, cellscease to consume oxygen, which in the context of our model, maybe interpreted as the point at which phenazines are used as al-ternate electron acceptors. Experimentally, cells expressing YFPfrom a constitutive promoter (suggesting metabolic activity) havebeen observed at a depth of 100 μm in wild-type colonies com-pared with a depth of only 60 μm in Δphz mutant colonies (28).It appears that the presence of phenazines allows an oxidative

potential to extend deeper into the colony. In our model, we seethat this is the consequence of phenazines being oxidized in thesurface layer and diffusing into the deep layers, where they areconsumed when oxygen reaches a low enough level.Using this model, we again can evaluate the effect of ridge width

on colony growth and predict optimal ridge width when phenazinesare present. Our predictions agree well with observations for thewild type growing in 21% oxygen (Fig. 3C). The observed andpredicted features are significantly wider than the Δphz ridgesgrown under the same conditions. The wild type is effectively aΔphz mutant with phenazines added, and the effect of phenazineaddition is qualitatively the same as that of increasing oxygenavailability for the Δphzmutant. This highlights that the availabilityof oxidative resources strongly controls feature geometry and thatthese effects can be modeled predictably, provided that the redoxchemistry and physiology of the colony are well defined.

DiscussionThe study of biofilms has diverse applications. For example, bio-films may act as model ecological systems to investigate the natureof cooperative and competitive behavior and dynamics (e.g., refs.4, 12). To understand biofilm physiology in a variety of contexts, itis critical to define the connections between population structureand access to resources. We have developed a mathematicalmodel of substrate availability and cellular metabolism withincolonies. The predicted oxygen distribution and its effects oncolony morphology have been verified empirically using a standardbiofilm assay. We have shown that standard physiological char-acterizations of biofilms may be combined with an analysis ofcommunity growth as a function of geometry to predict optimalbiofilm morphology. Furthermore, we have illustrated that theseoptima are influenced greatly by external redox conditions and theability of biofilms to produce redox-active molecules.One of the major findings of this paper is that the shapes of

biofilms may allow optimal interaction with the environment forresource availability and therefore optimal total growth and returnon mass investment. The ridges of P. aeruginosa colonies maintainfairly uniform width even as they grow taller. Previous studiesshowed that steady-state biofilm feature shape results from externalresource limitation and occurs when the environment no longer cansupply substrates for further growth (e.g., refs. 4, 5, 7, 12, 47). Incontrast to the model systems used for these studies, our colony

biofilm is exposed to the open atmosphere, in which external re-source gradients effectively are nonexistent. Thus, cells in theoutermost layer of a colony are not resource limited, and theyhave the capacity for further growth, as evidenced by the oxygenprofiles and the observation that they grow taller. The ridgeshave the capacity for horizontal growth, and the simplest passivemodel of the dynamics driven solely by growth rate wouldpredict increasing ridge width over time. Therefore, unknownmechanisms must maintain the constant ridge width that ap-parently is advantageous to the overall colony.Although wrinkling is a common response of the biofilm system,

we have seen that ridge geometry is affected by both environmental(external oxygen availability) and physiological (the inability of theΔphz mutant to produce phenazines) alterations. Furthermore, ithas been observed that no wrinkling occurs in a mutant unable toproduce the Pel polysaccharide [a critical component of matrixstructure (27, 28)], and our experience with a wide variety of singlegene knockouts indicates that many physiological changes alter thedetails of spatial patterning and wrinkle formation. This suggeststhat wrinkling is the result of many physiological processes, as al-ready highlighted by the variety of single mechanisms shown to berelated to pattern formation in biofilms (4, 12, 29). Given thecomplexity of cellular physiology and response, there are countlesssystems of cellular behaviors, interactions, and feedbacks that con-ceivably might evolve to produce diverse biofilm structures. Manyof these structures would not be optimal for the average re-productive success of cells within the colony. This suggests thatthe specific set of mechanisms in our biofilm system may havebeen selected to produce emergent patterns that we observeto be optimally beneficial to the colony. Studies recently in-dicated that biofilm structures are underpinned by diverse cellbehaviors, including chemotaxis, extracellular matrix production,chemical signaling, and selective cellular death as an inductionfor mechanical buckling in thin films (4, 12, 29). Whether dis-ruptions in these mechanisms produce suboptimal biofilm struc-tures remains to be investigated. In addition, features may servemultiple roles in different contexts or species, implying a variety ofselective pressures. For example, a recent study reported that thechannels that form within wrinkles in B. subtilis biofilms facilitateliquid movement through the structure (48).In considering the possibility that biofilm patterning has been

selected for as a stable adaptive trait, it is important to recognize thetradeoffs faced by individual cells, including the potential for“cheaters” to disrupt patterning and the associated benefits to thepopulation. For example, phenazines are a community resourceproduced at a cost. Once excreted, these compounds may be takenup and participate in repeated redox cycling by both phenazine-producing and nonproducing cells (32). In the biofilm context,strategies may be used to avoid the likely benefit experienced bycheaters that catalyze redox cycling without contributing to thephenazine pool. Recent work has shown that extracellular matrixformation is a strategy that allows only cooperating cells to gainthe benefit of the tall structures built by the community, so long asthe cooperators exist in high enough local abundance and matrixproperties (production rate and density) fall within a predictablerange (4, 12).Characterization of the relationships among metabolism, chem-

ical gradients, and morphogenesis of bacterial biofilms providesinformation relevant to diverse fields. Bacterial biofilms may ex-hibit complex spatial patterning at scales comparable to those ofputative microbial fossils (25, 47). Our biofilm system has allowedus to test the effects of different atmospheric oxygen concen-trations directly and to demonstrate that the width of featureschanges predictably. These findings may provide a basis for veri-fying microbial fossils and using them as reporters of the ancientenvironment. For example, in aquatic environments, the diffusiveboundary layer for oxygen that forms around the biofilm is whatinfluences feature geometry (4–12) in contrast to the open-atmo-sphere biofilms studied here, in which the important oxidative gra-dients occur inside the biofilm. These differences provide avenues for

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interpreting which environments are capable of producing an ob-served morphology.Biofilm development is a critical component of bacterial coloni-

zation of and persistence within human and other eukaryotic hosts.It contributes to the establishment and maintenance of various typesof P. aeruginosa infections. Here we find that biofilms produce op-timal structures to enhance growth and that the success of the colonydepends on the ability to build these structures effectively. As welearn more about the complex set of cellular behaviors responsiblefor the emergent regulation of these features, we become betterequipped to take new approaches toward controlling biofilms inclinical and industrial settings.

MethodsBacterial Strains and Growth Conditions. Wild-type P. aeruginosa PA14 anda mutant (Δphz) deficient in phenazine production were used in this study (49).Bacterial cultures were grown routinely at 37 °C in lysogeny broth, shaking at250 rpm overnight. For all oxygen profiling experiments, colony biofilms weregrown on 1% tryptone (Teknova)/1% agar (Teknova) plates amended with 20μg/mL Coomassie blue (EMD) and 40 μg/mL Congo red (EMD). Colony biofilmswere grown at room temperature (22–25 °C) and at high humidity (>90%). Sixtymilliliters of the medium was poured per 10-cm2 plate (D210-16; Simport) andallowed to dry with closed lids at room temperature for 24 h. Ten microliters ofthe overnight culture was spotted on these agar plates, and colony biofilms weregrown for 6 d. Colonies were grown under hypoxic (15% oxygen) or hyperoxicconditions (40% oxygen) as described earlier (28). Briefly, we incubated agarplates in C-Chambers (C274; BioSpherix). Oxygen concentrations were regulated

by mixing pure nitrogen and oxygen (Tech Air), using the gas controller ProOxP110 (BioSpherix).

Colony Geometry Measurements. Width and height of ridges within colonybiofilms were measured with a digital microscope (Keyence VHX-1000).

Oxygen Depth Profiles. Oxygen profiling was done using a miniaturized Clark-type oxygen sensor (Unisense; 10-μm tip diameter) on a motorized microma-nipulator (Unisense) for depth control. The electrode was connected to apicoampere amplifier multimeter (Unisense) and polarized at −800 mV. Thesensor was calibrated using a two-point calibration system at atmosphericoxygen and zero oxygen. The atmospheric oxygen reading was obtained byplacing the electrode in a calibration chamber (Unisense) containing well-aerated deionized water. Complete aeration was achieved by constantlybubbling the water with air. The zero reading was obtained by bubblingwater in the calibration chamber with ultra–high-purity nitrogen gas (TechAir). All calibration readings and profile measurements were obtained usingSensorTrace Pro 2.0 software (Unisense).

ACKNOWLEDGMENTS. We thank Alexa Price-Whelan for insightful discus-sion regarding this project and the manuscript. We thank Daniel Bellin forcalculating the phenazine diffusion constant. This work was supported bya National Science Foundation Graduate Research Fellowship (C.P.K.), theGordon and Betty Moore Foundation (C.P.K. and M.J.F.), the NationalAeronautics and Space Administration (M.J.F.), the National Science Foun-dation (M.J.F.), a Gilliam Fellowship from the Howard Hughes MedicalInstitute (C.O.), a startup fund from Columbia University, and Research Grant1 R01 AI103369-01A1 from the National Institute of Allergy and InfectiousDiseases (to L.E.P.D.).

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Kempes et al. PNAS | January 7, 2014 | vol. 111 | no. 1 | 213

BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY

Supporting InformationKempes et al. 10.1073/pnas.1315521110Model for Oxygen Use Within the ColonyAs described in the main text, our model is based on the com-bination of Monod kinetics and the Pirt model, which is a com-mon approach in biofilm modeling (1–11). Here we provide moredetails for the foundation of these two perspectives and theircombination into the model.For organisms living in a continuous culture, it has been dem-

onstrated that many species exhibit a linear relationship betweentheir total metabolism and growth rate (12–15). The classic Pirtmodel (12) describes this relationship in terms of a metabolicpartitioning between maintenance and growth:

Q=μYO2

+PO2 ; [S1]

where Q is a consumption rate per unit mass of a limiting re-source that provides energy and/or structural material to thepopulation of cells (mol resource·s−1·g cells−1), μ is the specificgrowth rate (s−1), Y is a yield coefficient (g cells·mol resource−1),and PO2 is a maintenance term (mol resource·s−1·g cells−1).Maintenance metabolism is defined as the consumption rate atzero growth, or the minimal requirement for survival.The metabolic activity of a respiring population depends on the

availability of an electron acceptor, such as oxygen. We add thisprocess to Eq. S1 by considering the dependence of growth rateon the local oxygen concentration. For Pseudomonas aeruginosaand many other species, growth saturates at increasing concen-trations of many potentially limiting substrates, including oxygen(e.g., ref. 16). This sensitivity often is described using Monodkinetics (17):

μ= μmax½O2�

kO2 + ½O2�; [S2]

where μmax is the maximum growth rate approached as the sub-strate is taken to infinity and kO2 is the half-saturation constant.Substituting Eq. S2 into Eq. S1 gives oxygen consumption asa saturating function of available oxygen:

Q=μmax

YO2

½O2�kO2 + ½O2�+PO2 : [S3]

As discussed in the main text, this consumption rate of oxygen,which depends on local oxygen availability, creates gradients inoxygen concentration that drive diffusion. Combining these con-cepts, the time-dependent spatial concentration of oxygen maybe described by

∂½O2�∂t

=D∇2½O2�−�μmax

YO2

½O2�kO2 + ½O2�+PO2

�a; [S4]

where a is the density of cells in the colony (g cells·m−3). Steadystate occurs when the consumption of oxygen, which depends onthe local oxygen concentration via the Monod equation, balancesthe diffusive supply of oxygen, which is driven by gradients pro-duced by biological consumption. It also should be noted that atzero oxygen concentration, the cells either die or become dor-mant; thus, we take the maintenance term PO2 to be zero in theabsence of oxygen. The nondimensionalization of Eq. S4 then isgiven by

∂½O2�*∂t*

=∇2½O2�* −

½O2�*1+ ½O2�* + g

!; [S5]

where, again, the temporal and spatial scales, respectively, havebeen normalized by the factors

tfac =aμmax

kO2YO2

; xfac =�

aμmax

kO2YO2D

�1=2

; [S6]

oxygen concentrations have been divided by kO2 , and the non-dimensional maintenance term is g=YO2PO2=μmax.

Model for Dual Use of Oxygen and Phenazines as OxidantsAbove and in the main text we provide a model for the metabolicresponse to, and the consumption of, oxygen along with itsphysical diffusion into and within the colony. This model, how-ever, is appropriate only for the phenazine-null mutant (Δphz),for which oxygen is the only available electron acceptor. Thewild-type strain produces phenazines, which are redox active.For P. aeruginosa, oxidized phenazines have been shown to act aselectron acceptors (18–20). When reduced phenazines are ex-posed to an oxidizing agent, they are thought to act as “shuttles”for electrons; that is, phenazines freely diffuse between cells,where they are reduced, and a distant electron acceptor, wherethey again are oxidized. Long-term anaerobic survival has beendemonstrated for P. aeruginosa cells incubated in suspensionwith phenazines provided as the sole electron acceptor and anelectrode poised at a potential that oxidizes the phenazines forrepeated use by the cells (18). It also has been shown that theproduction of phenazines may help restore redox balance andreduce oxygen consumption rates as oxygen levels decrease (seeFig. 4C of ref. 19). Taking all these observations into consider-ation, we can model the oxygen consumption and reduction ofphenazines as part of the redox demands of the colony in thefollowing manner: (i) Oxygen freely diffuses into and within thecolony and (ii) has two sinks: direct consumption by cells withinthe colony and oxidation of reduced phenazines. (iii) Phenazinesfreely diffuse within the colony in two forms, reduced and oxi-dized. (This is a simplification but captures the dominant first-order effects.) (iv) Phenazines are oxidized by oxygen accordingto the representative reaction

½Phzr�+ 2½O2�→ ½Phzo�+ 2�O−

2

�: [S7]

Regarding the order of kinetics for this reaction, it has beenshown using pseudo-order kinetics that the exponent for phena-zines should be linear (21). However, the exponent for oxygen hasnot, to our knowledge, been calculated, and here we leave theexponent for oxygen as a free parameter to be determined laterusing measurements of the colony system. Thus, we assume thatoxidized phenazines are produced at a rate of kR[Phzr][O2]

α. (v)Phenazines are reduced by cells at a constant rate. This assump-tion follows from the observation that cells can survive but notgrow on phenazines (18, 19), and within the Pirt framework (al-ready used for oxygen consumption in Eqs. 1–3 of the main text),this is equivalent to assuming that phenazines have zero yield butcan meet the minimal maintenance requirement, and thus

Qphz =Pphz: [S8]

(vi) We assume that cells commit to using either oxygen orphenazines, but not both simultaneously, as an electron acceptor

Kempes et al. www.pnas.org/cgi/content/short/1315521110 1 of 7

because it may be costly to switch between metabolic pathways.In addition, we assume that cells have a strong preference foroxygen directly as an electron acceptor and will use oxygen unlesslevels are below a critical value of availability. (vi) We assume thatthe phenazine concentration within the colony is constant and thatany loss terms (e.g., diffusion into and binding to the agar) arebalanced by production terms (e.g., secretion by the cells). Oursimulations concern the steady-state concentrations for a snapshotof the colony. On such a timescale, this balance is justified, as anyproduction or loss dynamics would be negligible.These processes are summarized by the following system of

equations:If [O2] ≥ s, then

∂½O2�∂t

=DO2∇2½O2�−

�μmax

YO2

½O2�kO2 + ½O2�+PO2

�a− 2kR½Phzr�½O2�α

[S9]

∂½Phzo�∂t

=Dphz∇2½Phzo�+ kR½Phzr�½O2�α [S10]

∂½Phzr�∂t

=Dphz∇2½Phzr�− kR½Phzr�½O2�α; [S11]

if [O2] < s, then

∂½O2�∂t

=DO2∇2½O2�− 2kR½Phzr�½O2�α [S12]

∂½Phzo�∂t

=Dphz∇2½Phzo�− aPphz + kR½Phzr�½O2�α [S13]

∂½Phzr�∂t

=Dphz∇2½Phzr�+ aPphz − kR½Phzr�½O2�α; [S14]

where s is the minimum concentration of oxygen that can sup-port cells; [Phzo] and [Phzr] are the concentrations of oxidizedand reduced phenazines, respectively; kR is the reaction ratecoefficient for phenazine oxidation by oxygen; D again denotesthe diffusivity of either oxygen or phenazines within the colony;Y and P again represent the yield coefficient and maintenancemetabolism for growth on either phenazines or oxygen; and kO2

and kphz are the half-saturation constants for growth on oxygenand phenazines, respectively.These equations can be nondimensionalized as:if [O2] ≥ s, then

∂½O2�p∂t p

=∇*2½O2�p −� ½O2�p1+ ½O2�p

+ g1

�− 2R1½Phzr� p ½O2�*α

[S15]

∂½Phzo�p∂t p

=D1∇*2½Phzo�p +R1½Phzr�p ½O2�*α [S16]

∂½Phzr�p∂t p

=D1∇*2½Phzr�p −R1½Phzr�p ½O2�*α; [S17]

if [O2]* < s*, then

∂½O2�p∂t p

=∇*2½O2�p − 2R1½Phzr�p ½O2�*α [S18]

∂½Phzo�p∂tp

=D1∇*2½Phzo�p − g2 +R1½Phzr�p ½O2�*α [S19]

∂½Phzr�p∂tp

=D1∇*2½Phzr�p + g2 −R1½Phzr�p ½O2�*α; [S20]

where we have nondimensionalized time, spatial scales, and con-centrations as

t* ≡ tfact [S21]

fx * ; y * ; z * g≡ xfacfx; y; zg [S22]

½O2�p ≡ ½O2�=kO2 [S23]

s*≡ s=kO2 [S24]

½Phzo�p ≡ ½Phzo�=kO2 [S25]

½Phzr�p ≡ ½Phzr�=kO2 [S26]

with

tfac ≡aμmax

kO2YO2

[S27]

xfac ≡�

aμmax

kO2YO2DO2

�1=2=�tfacDO2

�1=2[S28]

g1≡YO2PO2

μmax[S29]

g2 ≡YO2Pphz

μmax[S30]

D1 ≡Dphz

DO2

[S31]

R1 ≡kRkα+1O2

YO2

aμmax: [S32]

Parameter Values of the Mathematical ModelsMany of the key parameters from our model for oxygen con-sumption and/or phenazine reduction can be estimated fromearlier studies (19, 21).Using the data from figure 4A of ref. 19, we calculate growth

rate as a function of oxygen concentration and determine theMonod kinetics for oxygen-limited growth. We calculate thespecific growth rate according to μ = d ln(OD)/dt, and we convertoxygen percentages to concentration by multiplying by the sat-uration concentration of oxygen in water. We find that themaximum growth rate is μmax = 2.27 × 10−4 (divisions s−1) andthat the half-saturation constant is kO2 = 0:0124 (mol O2·m

−3).Using figure 2 of ref. 19, which provides a time series for both

optical density and phenazine reduction rates by cells in liquidculture, we calculate the maintenance coefficient for survival onphenazines. We used the reduction rates to calculate Qphz, thephenazine consumption rate per gram of cell (mol Phzo·g

−1·s−1),by noting that the sample volume is 1 mL and using the con-versions found below. Under our assumptions for phenazineconsumption, this is equivalent to the maintenance coefficient.We make these calculations for the data at 27 h, as this corre-sponds to the maximum phenazine reduction rate reached after

Kempes et al. www.pnas.org/cgi/content/short/1315521110 2 of 7

the culture has entered stationary phase. We estimate that Pphz =5.03 × 10−7 (mol Phzo·g cells−1·s−1), which is roughly four timeslarger than the value for O2.The oxidizing reaction rate of reduced phenazines and oxygen

may be calculated using the data in ref. 21 and solving for kR = Ro/([Phzr][O2]

α), where Ro is the observed oxidation rate. We have thatfor 0.005 (mol·m−3) of phenazines and roughly 0.022 (mol·m−3)of oxygen (2% oxygen) that the reaction rate for pyocyanin is1.75 × 10−5 (mol·m−3·s−1), and thus kR = 1.75 × 10−5/(0.005 ×0.22α) (mol−2·m−3·s−1).The critical value for O2 consumption, s, can be estimated from

figure 4C of ref. 19. It is observed that as oxygen levels are drawndown, there is a sudden increase in the production of phena-zines. Following the increase in phenazine production, oxygenlevels are observed to rise in the culture, indicating that oxygenconsumption has been reduced and that some of the cells havestarted using phenazines rather than oxygen. These transitionsoccur for a concentration s ∼ 0.0087 (mol O2·m

−3), which may beinterpreted as the critical oxygen concentration determiningwhether wild-type cells are using oxygen or phenazines. It shouldbe noted that in ref. 19, our estimate for s also corresponds to themaximum redox imbalance (as measured by the ratio of NADHand NAD+ values), and after phenazine production increases,this redox imbalance decreases. These observations agree withthe perspective that s is the critical concentration at which cellsrespond by producing phenazines and switching between elec-tron acceptors.Finally, the total pool of phenazines within the colony is es-

timated from figure 4C of ref. 19. We use the maximum con-centration of pyocyanin, 0.27 (mol·m−3), as the concentration inthe colony biofilm. It is possible that this value should be scaledby differences in density of cells in the liquid culture comparedwith the colony, but we find that measurements of pyocyaninconcentration in the biofilm and agar are comparable to theconcentration in liquid culture. We take D1 = .3, which agreeswith empirical observations.Many parameters of our model, such as the maximum growth

rate μmax, often are measured in the literature. Thus, forcomparison with our calculated values, we searched the liter-ature for reasonable ranges of these parameters as summarizedin Table S1. When we nondimensionalize the mathematicalmodel, all these parameters are summarized by three normal-ization factors and one dimensionless parameter: xfac, tfac, kO2 ,g1, g2, D1, and R1. Each of these terms is determined bya combination of the underlying parameters. To obtain a bestestimate for each factor, we first calculated each factor fromevery combination of measured parameter values (Table S1),then found the mean of all of the combinations. Doing this, wefind that the mean value of xfac is 76,542.9 s−1. The mean es-timate for g is 0.012. Note that tfac is not critical to the analysisin which we examine steady states, but it may be found easilyfrom the parameters presented here.

Converting Between Optical Density and Mass UnitsIn ref. 27, the conversion between OD and cfu per cubic meter isgiven by

cfu=m3 =�2:0× 1014

�OD+ 4:0× 1012: [S33]

Similarly, taking the average of the reviewed literature in ref. 27,the mass per number of cells is 2.14 ± 3.9 × 10−12 (g per cell),which implies that the conversion between OD and grams percubic meter is given by

g=m3 = 428:3 OD+ 8:6: [S34]

Modeling ApproachTo determine the spatial distribution of oxygen in the Δphzcolony, we numerically solved Eq. 3 from the main text for 2Dwrinkles (one vertical dimension and one horizontal dimension).We held the colony surface to a constant value of atmosphericoxygen concentration and used circular horizontal boundaryconditions to represent the repeating pattern of wrinkles thatemanate outward from the center of the colony. We used anexplicit second-order Runge–Kutta finite difference method forthe numerical solutions.For the geometry of biofilm features, we use a “Gaussian” shape

with an elliptical tip, in which we adjust the width of the base tomatch cross-sections of the colony wrinkles given the observedheight of these wrinkles. We found that various mathematicalforms that approximate the shape of the wrinkles do not alter ourresults significantly.Steady-state solutions for the profiles of oxygen and phenazines

in the wild-type colonies are solved by using Eqs. S17–S24, usingthe same numerical techniques described above. It should benoted that each model may be used within either a 1D or 2Dcontext. For example, the Δphz model is used in a 1D context tosolve for the ideal base thickness, whereas the optimal wrinklewidth is found using the full 2D geometry of the wrinkle, asdescribed above and in the next section.

Geometric Effects on Oxygen Availability and ColonyGrowthOur model can capture the decrease in oxygen availability withincreasing depth into the colony, and we find that this decay rate iscontrolled largely by local geometry. For example, the wrinkleshave a much deeper penetration depth of oxygen resulting fromincreased local surface area per unit of cellular mass. As discussedin the main text, we have observed that wrinkles reach a constantwidth while growing taller and that this width is sensitive to theavailability of oxygen (and the presence or absence of phena-zines). To interpret this phenomenon, and the significance ofgeometry, we simulate the steady-state distribution of oxygenavailability within a wrinkle and the surrounding base for variousgeometries of the wrinkle (Fig. S3A). From the distribution ofoxygen availability within the colony, we also can calculate thespatial distribution of growth rates (Fig. S3B). It is from theseresults that we calculate the total growth rate of the colony asa function of wrinkle width and find that there is a wrinkle widththat optimizes colony growth per mass investment.

Model Calibration and Metabolic ObservationsIn the main text, we used measured profiles of the oxygen con-centration as a function of depth into the colony to verify ourmodel and compare parameter values to our estimate from theliterature. Fig. S1 provides all the measured profiles for both thebase and wrinkle regions of Δphz grown in 21% external oxygen.These data also were used for analysis in ref. 20. Similarly, Fig.S2 shows all the wild-type oxygen profiles for the base grown in40% oxygen.Ref. 20 provides measurements for constitutively expressed

YFP in the base regions of wild type and the Δphz mutant. TheYFP signal can be interpreted as a proxy for some aspect ofoverall metabolic activity (protein expression). In the Δphz mu-tant oxygen availability, base thickness, and the expression ofYFP coincide. In contrast, a wild-type base is significantly thickerthan the distance oxygen can penetrate into the colony, creatingan oxic and apparent anoxic zone. Interestingly, YFP expressionextends into the anoxic zone (YFP expressed in the anoxic zonematures and fluoresces after the thin section is exposed to oxy-gen, post fixation). We find that the thickness of the base cor-responds to the optimum for cellular reproduction, and thelevels of YFP in the wild type that extend deeper than oxygen

Kempes et al. www.pnas.org/cgi/content/short/1315521110 3 of 7

indicate that cells are at least meeting their maintenance re-quirement at these depths. Thus, the depth at which oxygen is nolonger detectable cannot simply be that at which growth ceasesbut instead should be considered as the site of a transition inmetabolic activity. Using our wild-type model (calibrated byoxygen profiles) we solve for the depth at which phenazines areno longer able to support basic maintenance requirements andwe predict that this depth should be 103.7 μm for 21% externaloxygen. This prediction compares well with the measured YFPdepth of 99.1 ± 15.0 (20). This analysis supports our treatment ofmetabolic switching and the value for s and provides anotheravenue for model calibration using the YFP signals.Thus, experimental data from reference (20) allowed us to

calibrate oxygen and phenazine parameters. Base oxygen profilesverified the simulated oxygen profile for both wild-type and Δphzcolonies. The maximum YFP depth verified the wild-typephenazine profile given that we interpret this depth to corre-spond to the point at which oxidized phenazine concentrationscan no longer support the basic maintenance requirement of thecells. Our free parameter of the phenazine model (compared toxfac for oxygen) is the exponent for oxygen, α, in the phenazineoxidation rate equation. We find that for α = 1 our modelmatches the oxygen profiles and the depth at which oxidizedphenazines are negligible compares well to the maximum depthat which YFP is detectable.For situations where we do not have the appropriate experi-

mental setup to directly measure oxygen profiles in Δphz (40% or15% external oxygen) we used total base thickness or the YFPdepth, both proxies for the depth at which oxygen is no longerdetectable, to calibrate the modeled oxygen profiles. The mini-mum base thickness in 40% oxygen is 98.2 μm, the maximum is169.8 μm, and the mean is 138.8 ± 27.2 μm where the errorrepresents the standard deviation.For all the base calculations, we find solutions of the Δphz and

wild-type models in one dimension in which we enforce a no-fluxboundary condition at the agar and hold the surface of the base

to atmospheric oxygen concentrations. (For the Δphz model, no-flux boundary conditions are imposed for phenazines at all sur-faces.) This is equivalent to a base extending infinitely in all di-rections and is a good approximation for the base regionbetween wrinkles, as the wrinkles do not alter the chemicalconcentrations in the base region beyond a distance of roughly40 μm (Fig. S3). The optimal base thickness may be found byobserving the saturation in the total growth R as a function ofbase thickness.

Predictions for Geometry in Δphz Colonies Grown in 15%External OxygenIn the main text, we provide our predictions for the wrinkle widthsof Δphz grown in 21% and 40% external oxygen. Fig. S4 givesour predictions for wrinkles grown in 15% oxygen. We find thatin this condition, wrinkles are narrower compared with 21%external oxygen conditions. Our results show that wrinkles willrespond to external oxygen by becoming thicker in elevated(40%) oxygen conditions and narrower in lower oxygen con-ditions (15%). The reduced oxygen conditions are importantfor understanding the response of the biofilm to common hyp-oxic environments.

Predictions for Geometry in Wild-Type Colonies Grown in21% External OxygenFig. 3D in the main text provides our predictions for the optimalwidth of wrinkles in the wild-type biofilm. These were foundusing Eqs. S17–S24. In our analysis of the Δphz system, we foundthat optimizing growth for a 1D plate of different thicknessesgave approximately the same predicted width as the full 2D so-lution within the wrinkle geometry. This 1D setup is equivalentto an infinitely tall ridge with both sides held to atmosphericoxygen concentration. We found a simple scaling factor, ofroughly 1.1, relating the 1D prediction to the full 2D prediction.For the wild-type predictions, we solve Eqs. S17–S24 in one di-mension and multiply by this same scaling factor.

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Fig. S1. All the measured oxygen profiles for the base region (A) and wrinkles (B) of Δphz colonies grown in 21% external oxygen. Note that the wide rangeof wrinkle profile decay rates, as well as the occasional increase in oxygen with increasing depth, is a result of the thin geometry of the wrinkle and theresulting experimental challenge of keeping the oxygen probe centered in the wrinkle as it moves deeper into the colony. Data from ref. 20.

Fig. S2. All the measured oxygen profiles for the base region of wild-type colonies grown in 40% oxygen. Data from ref. 20.

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Fig. S3. The modeled internal availability of oxygen (A) along with the spatial distribution of growth rate (B) within a biofilm for features of different sizes.

0 2 4 6 80

100

200

300

400

Day of colony growth

ridge

wid

th (µ

m)

Fig. S4. The average width (points) of Δphz colony wrinkles grown in 15% external oxygen as a function of time. The solid line represents our predictions foroptimal wrinkle width, and the dashed lines are our predictions for the bounds of variation in wrinkle width. The data were published previously in ref. 20.

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Table S1. Reported values for key biological and physical parameters

Parameter (symbol) Value Ref.

Half-saturation constant (kO2 ), mol O2·m−3 0.0124 (19)

0.0078 (22)0.0014 (23)0.0125 (24)0.0369 (24)

Maximum growth rate (μmax), s−1 2.27 × 10−4 (19)

1.11 × 10−4 (25)7.89 × 10−5 (26)2.78 × 10−5 (26)6.11 × 10−5 (23)5.28 × 10−5 (24)8.89 × 10−5 (24)2.22 × 10−4 (5)

Yield coefficient for O2 (YO2 ), g cells·mol O−12 20.80 (23)

27.20 (5)20.32 (24)

Maintenance coefficient for O2 (PO2 ), mol O2·g cells−1·s−1 1.22 × 10−7 (24)Maintenance coefficient for phenazines (Pphz), mol Phzo·g cells −1·s−1 5.03 × 10−7 (19)Oxidation rate for phenazines by O2 (kR), mol−2·m−3·s−1 0.16 (21)Diffusivity for O2 in the colony (DO2 ), m

2·s−1 1.76 × 10−9 (22)1.53 × 10−9 (5)

Biofilm cell density (a), g cells·m−3 12,000 (5)Critical value of oxygen (s), mol O2·m

−3 0.0087 (19)

Bold text indicates situations in which we used a value calculated here rather than the average of thepublished values.

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