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Local growth rules can maintain metabolically efficientspatial structure throughout growthYipei Guoa,b,1, Mikhail Tikhonova,c, and Michael P. Brennera

aJohn A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; bProgram in Biophysics, Harvard University,Boston, MA 02115; and cDepartment of Applied Physics, Stanford University, Stanford, CA 94305

Edited by Boris I. Shraiman, University of California, Santa Barbara, CA, and approved February 21, 2018 (received for review February 2, 2018)

A ubiquitous feature of bacterial communities is the existenceof spatial structures. These are often coupled to metabolism,whereby the spatial organization can improve chemical reactionefficiency. However, it is not clear whether or how a desiredcolony configuration, for example, one that optimizes some over-all global objective, could be achieved by individual cells that donot have knowledge of their positions or of the states of all othercells. By using a model which consists of cells producing enzymesthat catalyze coupled metabolic reactions, we show that simple,local rules can be sufficient for achieving a global, community-level goal. In particular, even though the optimal configurationvaries with colony size, we demonstrate that cells regulating theirrelative enzyme levels based solely on local metabolite concen-trations can maintain the desired overall spatial structure duringcolony growth. We also show that these rules can be very sim-ple and hence easily implemented by cells. Our framework alsopredicts scenarios where additional signaling mechanisms maybe required.

spatial structure | metabolism | local rules

Recent years have shown tremendous advances in our knowl-edge of the microbial communities that underlie diverse

physiological and natural processes, ranging from the gut micro-biome involved in digestion (1, 2), to those facilitating wastewa-ter treatment (3, 4), to the microbes responsible for dental caries(5, 6) or for the fermentation of food products (7). A strikinglyubiquitous feature of these communities is the existence of well-defined spatial structures; a common motif is the layered archi-tecture, where different microbial species are stacked on top ofeach other with increasing distance from a nutrient source. Acrucial question is to discover the rules for building these spatialarchitectures and to understand how these rules are encoded inthe genomes of individual cells.

Although spatial organization undoubtedly has diverse func-tionality, there are many contexts in which spatial organizationimproves chemical reaction efficiency (Fig. 1) (8). Within cells,enzymes in the same metabolic pathway are often packagedtogether into complexes (8); a striking example is the bacterialcarboxysome, whose protein shell self-assembles around a corewith two differentially localized enzymes required for carbon fix-ation (Fig. 1A) (9, 10). The spatial localization of ribosomes andmRNA at different regions within the bacterial cell has also beenstudied (11). In microbial colonies, spatial structures often cou-ple to metabolism, most commonly through cross-feeding, whereone species lives off the products of another (12). Such metabolicinteractions between cells are often crucial for the survival of thecolony (13). Examples include methanogenic consortia such assludge granules found in wastewater treatment reactors (Fig. 1B)(14–16) or the consortium that carries out anaerobic oxidation ofmethane (Fig. 1C) (17, 18). In such consortia, the ordering of thedifferent layers suggests that observed spatial structures couldbe adapted to the overall reaction carried out by the consortium(19). Spatial structures in microbial communities also arise evenwithout any exchange of intermediates. In microbial mats (Fig.1D) (20) and in water columns (Fig. 1E), oxygen concentration

decreases with depth due to depletion by bacteria (21), givingrise to oxic and anoxic zones (22). For more general multispeciesbiofilms (Fig. 1F), it is often difficult to understand specific spa-tial structures, but since cell behavior is strongly influenced bylocal environmental conditions, which depend on the activitiesof other cells, the metabolic interactions (direct or indirect) arelikely to play a crucial role (23).

The rules for creating spatial organization must somehow beencoded into the individual components. A natural mechanismis self-assembly, in which the constituents bind to each other sothe desired configuration is a free-energy minimum. This is likelythe mechanism for the assembly of multiprotein complexes andalso has been widely discussed in developmental biology underthe so-called “selective adhesion hypothesis” (24–26). It is clearthat a cell in a growing colony has more tools in its repertoirethan selective adhesion: The cell can move, it can control itsgrowth rate and expression levels of different genes, and so on.Whichever strategy the cell uses, it must make its decision basedon information that it can measure from its local environment.The fundamental question is to understand how strategies canbe so encoded.

In this paper, we study the spatial structure that can developsolely from the cells reacting to the local chemical environ-ment. Our model considers cells producing enzymes that catalyzecoupled metabolic reactions, with the enzyme activities depen-dent on the respective substrate and product concentrations.We demonstrate that it is possible to maintain a desired spa-tial structure during colony growth when cells regulate their rela-tive enzyme levels based solely on local chemical concentrationsof metabolites or additional signaling molecules. Local rules aretherefore sufficient for achieving a global, community-level goal.Surprisingly, these local rules can be very simple and hence easilyimplemented by cells.

Significance

In many biological systems, efficient implementation of amultistep chemical reaction is achieved by an ordered spa-tial arrangement of individual reaction units. How can inde-pendently acting agents assemble into such a structure, andhow much communication between them is required? We con-sider a model of a metabolically coupled multispecies bacte-rial colony and show that surprisingly simple threshold-basedresponses to local environment are sufficient to assemble awide variety of structures. We also predict when additionalsignaling mechanisms between cells may be required.

Author contributions: Y.G., M.T., and M.P.B. designed research, performed research, andwrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 To whom correspondence should be addressed. Email: yipeiguo@g.harvard.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1801853115/-/DCSupplemental.

Published online March 19, 2018.

www.pnas.org/cgi/doi/10.1073/pnas.1801853115 PNAS | April 3, 2018 | vol. 115 | no. 14 | 3593–3598

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A B C D E F2 32

Biomass

Organic Acids

Hydrogen

Methane

4 + 42−

+

Oxygen

Wat

er D

epth

Salt Concentra on

Aerobes

Anaerobes

Fig. 1. Spatial structure is ubiquitous. (A) Spatial structure exists within cells. Carboxysomes are examples of microcompartments within cyanobacteria cellswhich house the two enzymes required for carbon fixation. (B and C) Spatial structure in communities can be driven by cross-feeding between multiple celltypes, e.g., (B) sludge granules (adapted with permission from ref. 15) and (C) anaerobic oxidation of methane (AOM) consortium (adapted with permissionfrom ref. 17). (D and E) Spatial structure can also be driven by environmental conditions which are shaped by the activities of all cells in the community,e.g., (D) microbial mats (photo by John R. Spear and Norman R. Pace) and (E) water columns. In other cases such as (F) multispecies biofilms (adapted withpermission from ref. 23), spatial structure is less well understood but metabolic interactions are likely to play an equally important role.

ResultsThe Model. Suppose a substrate S is converted to a product Pthrough a series of enzyme-catalyzed reactions, all of which arenecessary for the formation of the final product (Fig. 2A). Weconsider a spherical colony with a single type of cell that hasthe ability to express any of these enzymes (Fig. 2B). [A one-dimensional planar colony yields similar results (Model: Extend-ing to Other Geometries)]. By regulating enzyme expressions, cellscan have different enzyme densities {ni} under the constraint ofa limited enzyme budget nT =

∑Ni=1 ni ≤nmax . In the rest of this

paper, without loss of generality, we set nmax to be 1 since it isthe enzyme fraction ni/nmax that is the quantity of interest. Sincereactions are thermodynamically more favorable when there is ahigh level of substrate and a low level of product, the activity ofeach enzyme µi = fµ(cSi , cPi ) increases with increasing substrateand decreasing product concentrations.

For simplicity, we assume that the colony maintains sphericalsymmetry throughout its growth and that the rate of cell growth ismuch slower than both the reaction rate and the diffusion rate, asis typically the case. Given the radius of the colony R and densityprofiles of the enzymes n(r), the chemical concentrations sat-isfy the steady-state reaction–diffusion equations. The enzymesthemselves are not subject to diffusion, as they are assumed tobe present only within or on the surface of cells. For an interme-diate k produced by enzymes Ai and consumed by enzymes Bi

(Fig. 2C), we have

Dk1

r

d

dr

(rdckdr

)+∑i

nAiµAi (cXi , ck )

−∑i

nBiµBi (ck , cPi ) = 0,[1]

where Dk is the diffusion constant of k, and all cs and ns arefunctions of position r . Consistent with the examples of Fig. 1,we consider the nutrient S to be provided at infinity: cS→ s∞as r→∞. Placing the food source at the origin instead [e.g., fora biofilm growing on a food particle (27)] yields similar conclu-sions (Model: Extending to Other Geometries). We also demandthe vanishing of the other chemicals far from the colony (ck (r→∞)→ 0) since they are not provided externally, as well as regu-larity at the origin

(dcSdr

= dckdr

= 0 at r = 0).

Local Growth Rules Can Be Stored in Lookup Tables. The spatiallyvarying enzyme densities ~n(r ,R) encode the spatial structure ofthe colony. How should cells regulate their enzyme expressionlevels? It is sometimes plausible to assume that evolution mightseek to optimize the overall growth rate of the colony. Similarly,an engineer designing a synthetic consortium might want to max-imize the production of some chemical. Since enzyme activities

depend on chemical concentrations and therefore on positionwithin the colony, both problems take the form of optimizingsome global quantity:

G(~µ(~c(r)), ~n(r)) =

∫ R

0

g(~µ(~c(r), ~n(r))4πr2dr . [2]

In addition to a fixed enzyme budget, this optimization must besolved under the constraints of K reaction–diffusion equations(Eq. 1), where K is the number of different metabolites in thepathway. Denoting these constraints as Γj (r) = 0, j = 1, 2, ...K ,the problem reduces to minimizing the function,

L(~µ(~c(r)), ~n(r)) =−G −∑j

∫ R

0

λj (r)Γj (r)dr , [3]

where λj (r) is the Lagrange multiplier for the j th constraintand can be found by solving a second-order differential equation(Optimal Colony Structure for Activity Functions with a Finite HillCoefficient). The optimal configuration is obtained by requiringthat ∂δL

∂δ~nvanishes. For any colony size, the optimal solution for

~n(r) relates the position of a cell to how much of each enzymeshould be produced. If each cell could solve such a math prob-lem, the colony would perform optimally. This requires that the

A CBS

P

1

2

k

A1 A2

B1 B2

X1 X2

Y2Y1

colony

cell{ 1, 2, … , }

=1≤

Fig. 2. Model. (A) We assume that cells can take up a substrate S and con-vert it to P through multiple reactions. All of the intermediate reactions arenecessary; i.e., if an enzyme can take in multiple chemicals, all of them mustbe present for the enzyme to be active. (B) The spherical colony increases insize radially as cells grow and divide. Each cell in the colony can regulate therelative enzyme levels with a fixed enzyme budget. (C) Part of a metabolicnetwork with chemical k produced by enzymes Ai and taken up by enzymesBi . The chemical concentration ck is found by solving the reaction–diffusionequation (Eq. 1).

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cell must (i) have the capability to continuously track the entirestate of the system during colony growth, including the size ofthe colony and its position within the colony as well as the stateof the other cells, and (ii) have the computational power to con-tinuously solve a set of nonlinear partial differential equationsthroughout colony growth. Indeed, in human-designed roboticswarms, groups of robots have been shown to be able to self-organize according to a global rule when endowed with thesetwo capabilities through the help of wireless sensors and pow-erful central processing units (CPUs) (28, 29).

However, a cell in the middle of the colony has neither ofthese capabilities. Individual cells have limited computationalpower, and they do not have access to global quantities such asthe colony size and their relative position within it. How can acell regulate its enzyme density, if it can react only to its localenvironment? Every ~n(r) that maximizes G has a correspond-ing ~c(r); this means there exists a lookup table that maps thelocal concentrations ~c(r) to ~n(r). As the colony grows, the opti-mal ~n(r) and ~c(r) continuously shift, such that the optimal solu-tion traces out a surface in the N-dimensional concentrationspace with a corresponding surface in the N-dimensional den-sity space (Fig. 3B). If the objective function G is such that theoptimal surface in concentration space does not self-intersect (orif it does, the corresponding ns are the same), there exist localrules that ensure optimal colony structure throughout growth.Since a generic 2D surface in a high-dimensional space is non–self-intersecting, this general argument suggests that the seem-ingly limited toolkit of local decisions should allow assemblinga wide range of structures, as long as the cells follow a suitablelookup table.

In practice, however, such a lookup table could be very com-plex. It is not obvious whether or how it could be encodedbiochemically. Therefore, the question that emerges is whetherthere exist local growth rules that can not only maintain optimalstructure during growth, but also be easily implemented by cells.For the rest of this paper, we consider G to be the total reactionflux in the colony; the same analysis can be done for other Gs.

Simple Threshold-Based Decision Rules Can Maintain Close to OptimalConfiguration. Let activity functions be sigmoidal with Hill’s coef-ficient m . In this section, we show that threshold-based expres-sion rules, where enzymes are produced when a function of theconcentrations is above or below a threshold, are sufficient forthe colony to maintain close to the optimal configuration. Simi-lar rules are commonly used for a wide variety of functions (e.g.,quorum sensing, gene regulation through binary switches, etc.).We first discuss the m =∞ case, where the structure achieved bythreshold rules is exactly optimal, returning to the more generalcase later.

A B

( , )colony

cell

1R

r1

R

r

Fig. 3. Local rules that maintain optimal colony structure can exist in theform of lookup tables. (A) An individual cell does not know the size of thecolony (R) or its position within it (r). It must regulate its enzyme densitiesbased only on local information, namely the local chemical concentrations~c.(B) Solving for optimal reaction flux gives a mapping between the optimalcolony structure ~n(r) and the corresponding~c. This map can be interpretedas a lookup table, specifying the optimal activity level given a certain exter-nal chemical environment.

For the infinite Hill coefficient, simple local rules can maintainoptimality. For m =∞, activity functions take a step-like form,so an enzyme is active only if its substrate concentration is abovea critical value (cS ≥Scrit ) and its product concentration is belowanother critical value (cP ≤Pcrit ) (Fig. 4A). For this simple formof the activity functions, the spatial configuration of enzymes thatmaximizes total reaction flux can be found analytically for differ-ent reaction schemes (Simple Local Rules Can Maintain Optimal-ity for Step-Like Activity Functions). The solution is a multilayeredball with different layers containing different subsets of enzymes.We find that as long as the reaction scheme does not contain anynonessential intermediates, a simple local strategy can be used bycells to maintain exactly maximal colony metabolic flux (SimpleLocal Rules Can Maintain Optimality for Step-Like Activity Func-tions). We now illustrate this in detail for the two-step and N-stepreactions.

Two-step reaction. As our first example, we consider the two-step reaction: S A−→ I

B−→P . Here, the optimal structure is a two-layer ball with the shell consisting of only enzyme A and the innercore consisting ofA andB in a fixed ratio ensuring that both reac-tions occur at the same total rate nAµA =nBµB (Fig. 4B) (Sim-ple Local Rules Can Maintain Optimality for Step-Like ActivityFunctions). This structure can be realized by a simple decision rulewhere cells produce as muchB as possible when cI ≥ Icrit and pro-duce only A when cI < Icrit . Specifically, we take the rule to be

nB

=0, if cI < Icrit

=n∗B = µAµA+µB

, if cI = Icrit

>n∗B , if cI > Icrit

. [4]

At an optimal configuration, cI ≤ Icrit and all cells follow rule1 or 2 in Eq. 4. Rule 3, which applies only when cI > Icrit ,allows the colony to reach this optimal structure from some otherstate. These rules represent a simple mapping between concen-trations and enzyme densities that is independent of R and henceis a local rule that maintains optimal configuration throughoutgrowth. In fact, they perform optimally even in the presence ofexternal factors changing the steady-state concentration profiles,such as other competitors acting as extra sinks (Simple LocalRules Can Maintain Optimality for Step-Like Activity Functions).

Compared with the unregulated strategy of all cells always hav-ing some fixed value of nB , this threshold-based strategy substan-tially improves the total reaction flux (Fig. 4C), especially duringthe initial stages of growth when the shell ofA takes up a large frac-tion of the colony (Fig. 4D) (Spatially Nonuniform Colony Struc-ture Can Significantly Outperform a Spatially Uniform Strategy).

N-step reaction. The two-step reaction easily generalizes to an

N -step reaction, SE1−−→ I1

E2−−→ I2E3−−→ ...

EN−−→P , an example beingthe four-step reduction of nitrate to dinitrogen (30). Here, thedesired structure is an N-layer ball with the innermost coreconsisting of all enzymes, the next layer consisting of the firstN − 1 enzymes, and so on, with the outermost layer consisting ofonly enzyme E1 (Simple Local Rules Can Maintain Optimality forStep-Like Activity Functions). In regions with multiple enzymes,whenever both the producer and consumer of an intermediateare present, optimality requires its total consumption rate to beequal to its total production rate. As before, this optimal alloca-tion can be achieved with a threshold-based rule that is indepen-dent of colony size, whereby cells produce as much of an enzymeas possible whenever the concentration of the respective sub-strate exceeds a threshold.

The same simple strategy works for any reaction scheme,even those that include branching, as long as there are nononessential intermediates (Simple Local Rules Can MaintainOptimality for Step-Like Activity Functions). A simple example ofsuch a branched metabolic pathway involves the conversion of

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A B

Inactive

Active

==

C D

Fig. 4. Concrete example of how local rules can be used to maintain opti-mal configuration during growth. (A) Here we assume that the enzymeactivities depend on chemical concentrations in a step-like manner. (B) Forthe two-step reaction, the optimal configuration is a two-layered ball withcells in the shell producing only enzyme A and cells in the core producingboth enzymes. This configuration reflects a specific mapping between con-centrations and enzyme densities that stays the same throughout growth.(C) The threshold-based local rule substantially improves the performanceof the colony compared with using the strategy of producing a constant,fixed amount of B. Here, ζ= G

Goptis the ratio of the total flux through B

for a configuration to the maximum possible flux (obtained by the optimalconfiguration) (other parameters: µA = 1, µB = 1, TI = 1, DI = 1, nmax = 1).(D) The thickness of the shell relative to the size of the colony decreases asthe colony grows.

glutamate to the intermediate ammonium, both of which arerequired by the enzyme producing glutamine (31).Simple, local rules are also effective when the Hill coefficient isfinite. The activity of real enzymes is better modeled with a finiteHill coefficient m , so that µB (cI ) =µ

(max)B cmI /(T

mI + cmI ), with

µ(max)B being the maximum possible activity of B for the two-

step reaction. By numerically solving for the optimal nB (r) usingEqs. 1 and 2, we obtain the globally optimal configuration for anycolony size (Optimal Colony Structure for Activity Functions witha Finite Hill Coefficient). The solution to this nonlinear optimiza-tion problem is more complex and cannot be achieved exactly bysimple, threshold-like rules.

Nonetheless, for all sigmoidal activity functions (m ≥ 2), wefind that the optimal colony closely resembles a core–shell struc-ture, with a uniform core containing both enzyme types sur-rounded by a shell containing only enzyme A (Fig. 5A). (Form = 1, the core–shell structure is preserved, but the shell con-tains only B; this special case is discussed separately in OptimalColony Structure for Activity Functions with a Finite Hill Coef-ficient.) Furthermore, the slight deviations from the core–shellstructure do not significantly change the metabolic flux. Defin-ing the efficiency of a given configuration ζ = G

Goptas the ratio of

the total flux through B for that configuration to the maximumpossible flux obtained at true optimum, we find that ζ ≈ 1 for allcolony sizes. Thus, the best solution in the two-parameter familyof core–shell structures performs almost as well as the true opti-mum (Fig. 5B). We can therefore use the same class of threshold-based rules, yet incur virtually no performance penalty. By con-

sidering various strategies that give rise to core–shell structuresand calculating their efficiencies ζ during growth, we now showthat even in this case, simple local rules are sufficient for main-taining a close-to-optimal metabolic efficiency (Fig. 6).

For the initial growth of the colony, we find that an effec-tive, simple strategy would be for cells to secrete an extra signal-ing molecule M and produce enzyme B only when cM

cIexceeds

a critical value (Fig. 6A). In this regime, the optimal RcoreR

isapproximately constant such that both cI (r =Rcore) and cM (r =Rcore) scale with R in approximately the same way (OptimalColony Structure for Activity Functions with a Finite Hill Coeffi-cient). Therefore, imposing a threshold on cM

cIto be the desired

cM (r=Rcore )cI (r=Rcore)

allows the colony to maintain optimal Rcore duringgrowth. In comparison, the best spatially homogeneous colonyperforms significantly worse (Fig. 6A). This strategy is anotherexample of a local, simple rule that can be easily implementedby cells.

At large colony sizes, we find that cells can achieve a close-to-optimal core–shell structure simply by imposing a thresholdon cI . In this limit, the optimal shell thickness is small (OptimalColony Structure for Activity Functions with a Finite Hill Coeffi-cient), implying that the presence of a shell is of little importanceto the overall performance of the colony. As a result, adoptinga strategy where all cells produce the same amount of B canalready give close to maximal yield. However, using a strategybased on the cI threshold increases the range of R at which theoptimal flux is maintained (Fig. 6B).

Each of the above strategies works in a different range of radii.One could imagine a cell switching between the two strategiesbased on other signals that contain information of the colonysize. Curiously, depending on the life cycle of a typical colony,this switching may not even be necessary: In some systems, whenthe colony becomes too large, a chunk of cells breaks off, suchthat the colony size is always between two radii, and growth fromzero is rare (17).These local growth rules may be naturally implemented by indi-vidual cells. Besides being easily implementable, another aspectof local rules being simple is the question of how easily theserules are selected for during evolution. It is therefore importantto stress that depending on the energetics of the reaction steps,the rules we find can take a purely “selfish” form, maximizingthe yield (e.g., amount of ATP from the enzymatic reactions) ofan individual cell (Simple Local Rules Can Maintain Optimalityfor Step-Like Activity Functions). Under such conditions, there

1 2 30

0.5

nB

r

0 100.9

1

R

A

B

Fig. 5. Optimal colony configuration that maximizes total reaction flux canbe approximated by a core–shell structure. (A) The optimal spatial profileof B (blue) has a core that consists of B surrounded by a shell region withonly A. It resembles the optimal core–shell structure (red). (B) The total fluxthrough B for the optimal core–shell structure is within 1% of that obtainedfrom the full optimization. (Parameters: m = 3, µ(max)

B = 10, µA = 8, TI = 8,DI = 3, nmax = 1.)

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0 50

0.5

1

R

unregulatedcM/cI threshold

0 100

0.5

1

R

unregulatedcI threshold

A B

Fig. 6. Simple, local rules allow the colony to maintain optimal configu-ration in the different stages of colony growth. (A) For the initial growthstage, an effective strategy involves cells in the core secreting an additionalsignaling molecule M, with cells producing B only when cM

cIexceeds some

threshold (red line) ( cMcI

threshold = 0.22, nB,core = 0.4). In comparison, if all

cells were to produce a constant amount of nB regardless of its environ-ment, the best uniform colony for small colony sizes does not perform aswell (blue line) (nB = 0.17). [Other parameters: m = 5, µ(max)

B = 10, µA = 8,TI = 8, DI = 3, nmax = 1.] (B) The strategy based on cI thresholds enablesthe colony to maintain optimal structure when the colony grows beyonda certain size (red line) (cIthreshold = 8, nB,core = 0.4). When colony sizeis sufficiently large that the optimal shell layer becomes negligible, highmetabolic efficiency can be achieved with all cells having the same nB (redline) (nB = 0.4).

would be no need for cells to sacrifice their own yield to achievethe best overall colony configuration.

Our Framework Predicts Scenarios Where Additional Signaling IsRequired. There are situations where local rules based onmetabolite concentrations cannot maintain a global optimumduring growth. Most prominently, when the same chemical con-centrations correspond to different points in the space of opti-mal enzyme densities (Fig. 3B), local rules necessarily cannotwork. One way this could happen is when there are competingpathways within the metabolic network such that there are mul-tiple activity or rate maxima with their relative peaks changingas the colony grows. The critical colony size then corresponds tothe point when the optimal strategy shifts from one activity max-imum to another.

As a simple example, consider the scenario where there aretwo possible pathways converting a substrate to a product, withone pathway being more efficient, but requiring a large accu-mulation of an intermediate. At small colony sizes the pathwaywith a lower threshold for the intermediate is preferred, but atlarge colony sizes, the more efficient pathway would give a highermetabolic flux [Scenarios Where Additional Mechanisms (e.g., Sig-naling) Would Be Necessary for Maintaining Optimality]. Since theoptimal decision of cells depends on whether the colony is aboveor below a critical size, additional signaling must be present forcells to choose the right strategy. One way such a switch betweentwo strategies could happen is for cells to secrete some othersignaling molecule M1 whose concentration at the center of thecolony correlates with the colony size and for cells to secrete asecond signaling molecule M2 when the concentration of M1 isabove a threshold. The presence (absence) ofM2 provides all cellsin the colony with the information that the colony is above (below)a critical size. This is analogous to how bacteria cells in biofilmsknow to die when the colony becomes large enough that channelsfor nutrient transportation become necessary (32). Similar mech-anisms could also be useful for improving the performance of thehomogeneous colony in Fig. 4C by allowing cells to switch betweena few different values of nB , depending on the colony size.

Another place where local rules may break down concerns ourassumption that all cells have the ability to express all of therelevant enzymes. Real colonies may be composed of multiple

species, each producing only a subset of the enzymes of the over-all pathway. In these situations, simple local rules may need to besupplemented by other mechanisms, such as differential adhe-sion to counteract demixing (daughter cells tend to stay next totheir parent cells upon division) or additional signaling to regu-late the relative growth rates of cells in different regions of thecolony [Scenarios Where Additional Mechanisms (e.g., Signaling)Would Be Necessary for Maintaining Optimality].

Therefore, our approach of understanding when local rules aresufficient provides insight into when local rules fail and whenmore complex mechanisms are required. If a certain observedcolony spatial structure happens to optimize the net reaction rateof some metabolic pathway, this approach can be used to inferwhether it is possible that such an arrangement arises throughlocal rules based on chemical concentrations or whether theremust exist some more evolved form of communication strategysuch as differential adhesion between cells. Hence this frame-work can potentially allow us to classify structures into those thatare easily or inevitably formed and those that would require morecomplex regulation strategies.

DiscussionIn this work, we used optimization merely as an example of aglobal constraint. There are many other natural and syntheticexamples of multiagent systems that accomplish a community-level task. These are usually achieved through complex coordi-nation between individuals. In contrast, we have shown that cellscan solve a global problem without the need to know what othercells are doing. Our local rules can not only substantially improvecolony performance throughout growth, but also take on verysimple and easily implementable forms, for example, threshold-like responses to external concentrations or ratios of concentra-tions. These simple threshold rules are also reasonable approxi-mations of mechanisms known to be relevant in real colonies, inresponse to both metabolites [e.g., when oxygen is depleted in abiofilm (33)] and messenger molecules [e.g., in quorum sensing(34, 35)].

Our analysis shows that such rules could serve as a powerfulorganizing principle shaping complex structures in multispeciescolonies. The framework explored here has long been one of thecore mechanisms studied in developmental biology, for example,the French flag model in Drosophila (36). During organogenesis,the emerging spatial structure results in different cell types in dif-ferent regions of an organ, even though these cell types all arosefrom a single cell. For example, the liver consists of different zoneswhere cells in these different zones have different functions, andthese zones are dictated by their local chemical environment. Ifcells within the group are coupled metabolically, our work mightbe relevant to the understanding of such processes.

A similar perspective on multispecies biofilms formation is anexciting developing field (37–39). In fact, a layered spatial struc-ture observed in Bacillus subtilis biofilms has been shown to becoupled to metabolism—cells in the core carry out a differentpart of the metabolic pathway from that of cells at the periphery(31). Although such patterning is similar to that described here,whether our framework can explain and predict structures of nat-ural multispecies colonies requires experimental investigation.Such experiments might probe the states of all cells through-out colony growth in different nutrient conditions, while measur-ing or inferring metabolite concentrations with, for example, thecombination of FISH and microautoradiography (40). To distin-guish such rules from other potential mechanisms by which lay-ered patterns could arise such as mechanical cell–cell signaling,passing of cell state from parent to daughter cell, or response toother environmental factors like variations in pH or temperature,one could also experimentally control the strength of cell–celladhesion, track mother–daughter pairs and measure correlations

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between their states, repeat the experiments in different externalenvironments, or even isolate an individual cell and watch howits state varies with environmental conditions.

Why do these local rules based on chemical concentrationswork? One crucial factor distinguishing our problem from otherscenarios of collective decision making is that even when bacte-rial cells seem to make decisions independently, they inevitablyinfluence one another through chemical fields. The local concen-trations sensed by the cells carry information about what othercells in the colony are collectively doing. This can therefore beconsidered as a form of diffusion-mediated communication. Thefact that information is contained in the external chemical envi-ronment also implies that the ability to measure concentrationswell is important for local rules to work. However, cells are notperfect sensors. The implications of noise in sensing are an activearea of research (41), and how this would change the optimalstrategy is an interesting question that we leave for future work.

The framework introduced here can also be useful when think-ing about self-assembly in general. The typical self-assembly pro-gram examines how specific interaction rules can lead to theassembly of desired structures from individual building blocks.This is a natural approach when looking at macromolecularstructures held together by molecular interactions. However,spatially structured colonies emerge from cell growth ratherthan the minimization of some energy function. Still, we canthink of cell growth as being analogous to allowing buildingblocks to replicate according to diffusion-mediated growth rulesin the colony’s self-assembly program. Understanding when localgrowth rules are sufficient is hence especially useful when we areinterested in creating or assembling a desired spatially structuredcommunity through growth.

ACKNOWLEDGMENTS. This research was funded by the National ScienceFoundation through Grants DMS-1715477, MRSEC DMR-1420570, and ONRN00014-17-1-3029. M.P.B. is an investigator of the Simons Foundation.

1. Tlaskalova-Hogenova H, et al. (2004) Commensal bacteria (normal microflora),mucosal immunity and chronic inflammatory and autoimmune diseases. Immunol Lett93:97–108.

2. Cummings J, Macfarlane G (1991) The control and consequences of bacterial fermen-tation in the human colon. J Appl Microbiol 70:443–459.

3. Nicolella C, Van Loosdrecht M, Heijnen J (2000) Wastewater treatment with particu-late biofilm reactors. J Biotechnol 80:1–33.

4. Henze M, van Loosdrecht MC, Ekama GA, Brdjanovic D (2008) Biological WastewaterTreatment (IWA Publishing, London).

5. Marcotte H, Lavoie MC (1998) Oral microbial ecology and the role of salivaryimmunoglobulin A. Microbiol Mol Biol Rev 62:71–109.

6. Hojo K, Nagaoka S, Ohshima T, Maeda N (2009) Bacterial interactions in dentalbiofilm development. J Dental Res 88:982–990.

7. Leroy F, De Vuyst L (2004) Lactic acid bacteria as functional starter cultures for thefood fermentation industry. Trends Food Sci Technol 15:67–78.

8. Agapakis CM, Boyle PM, Silver PA (2012) Natural strategies for the spatial optimiza-tion of metabolism in synthetic biology. Nat Chem Biol 8:527–535.

9. Bonacci W, et al. (2012) Modularity of a carbon-fixing protein organelle. Proc NatlAcad Sci USA 109:478–483.

10. Iancu CV, et al. (2007) The structure of isolated synechococcus strain wh8102 car-boxysomes as revealed by electron cryotomography. J Mol Biol 372:764–773.

11. Castellana M, Li SHJ, Wingreen NS (2016) Spatial organization of bacterial transcrip-tion and translation. Proc Natl Acad Sci USA 113:9286–9291.

12. Phelan VV, Liu WT, Pogliano K, Dorrestein PC (2012) Microbial metabolic exchange—The chemotype-to-phenotype link. Nat Chem Biol 8:26–35.

13. de Vos MG, Zagorski M, McNally A, Bollenbach T (2017) Interaction networks, ecolog-ical stability, and collective antibiotic tolerance in polymicrobial infections. Proc NatlAcad Sci USA 114:10666–10671.

14. Gonzalez-Gil G, et al. (2001) Cluster structure of anaerobic aggregates of anexpanded granular sludge bed reactor. Appl Environ Microbiol 67:3683–3692.

15. Sekiguchi Y, Kamagata Y, Nakamura K, Ohashi A, Harada H (1999) Fluorescence insitu hybridization using 16S rRNA-targeted oligonucleotides reveals localization ofmethanogens and selected uncultured bacteria in mesophilic and thermophilic sludgegranules. Appl Environ Microbiol 65:1280–1288.

16. Satoh H, Miura Y, Tsushima I, Okabe S (2007) Layered structure of bacterial andarchaeal communities and their in situ activities in anaerobic granules. Appl EnvironMicrobiol 73:7300–7307.

17. Nauhaus K, Albrecht M, Elvert M, Boetius A, Widdel F (2007) In vitro cell growthof marine archaeal-bacterial consortia during anaerobic oxidation of methane withsulfate. Environ Microbiol 9:187–196.

18. Scheller S, Ermler U, Shima S (2017) Catabolic pathways and enzymes involvedin anaerobic methane oxidation. Anaerobic Utilization of Hydrocarbons, Oils, andLipids, ed Boll M (Springer, New York), pp 1–29.

19. Guiot S, Pauss A, Costerton J (1992) A structured model of the anaerobic granuleconsortium. Water Sci Technol 25:1–10.

20. Ley RE, et al. (2006) Unexpected diversity and complexity of the Guerrero Negrohypersaline microbial mat. Appl Environ Microbiol 72:3685–3695.

21. Fenchel T, Finlay B (2008) Oxygen and the spatial structure of microbial communities.Biol Rev 83:553–569.

22. Labrenz M, Jost G, Jurgens K (2007) Distribution of abundant prokaryotic organismsin the water column of the central Baltic sea with an oxic–anoxic interface. AquatMicrob Ecol 46:177–190.

23. Welch JLM, Rossetti BJ, Rieken CW, Dewhirst FE, Borisy GG (2016) Biogeography ofa human oral microbiome at the micron scale. Proc Natl Acad Sci USA 113:E791–E800.

24. Steinberg MS (1963) Reconstruction of tissues by dissociated cells. Science 141:401–408.

25. Foty RA, Steinberg MS (2005) The differential adhesion hypothesis: A direct evalua-tion. Dev Biol 278:255–263.

26. Steinberg MS (2007) Differential adhesion in morphogenesis: A modern view. CurrOpin Genet Dev 17:281–286.

27. Datta MS, Sliwerska E, Gore J, Polz MF, Cordero OX (2016) Microbial interactionslead to rapid micro-scale successions on model marine particles. Nat Commun 7:11965.

28. Rubenstein M, Cornejo A, Nagpal R (2014) Programmable self-assembly in athousand-robot swarm. Science 345:795–799.

29. Werfel J, Petersen K, Nagpal R (2014) Designing collective behavior in a termite-inspired robot construction team. Science 343:754–758.

30. Lilja EE, Johnson DR (2016) Segregating metabolic processes into different microbialcells accelerates the consumption of inhibitory substrates. ISME J 10:1568–1578.

31. Liu J, et al. (2015) Metabolic co-dependence gives rise to collective oscillations withinbiofilms. Nature 523:550–554.

32. Wilking JN, et al. (2013) Liquid transport facilitated by channels in Bacillus subtilisbiofilms. Proc Natl Acad Sci USA 110:848–852.

33. McDougald D, Rice SA, Barraud N, Steinberg PD, Kjelleberg S (2012) Should we stayor should we go: Mechanisms and ecological consequences for biofilm dispersal. NatRev Microbiol 10:39–50.

34. Waters CM, Bassler BL (2005) Quorum sensing: Cell-to-cell communication in bacteria.Annu Rev Cell Dev Biol 21:319–346.

35. Dobretsov S, Teplitski M, Paul V (2009) Mini-review: Quorum sensing in the marineenvironment and its relationship to biofouling. Biofouling 25:413–427.

36. Wolpert L (1969) Positional information and the spatial pattern of cellular differenti-ation. J Theor Biol 25:1–47.

37. Shapiro JA (1998) Thinking about bacterial populations as multicellular organisms.Annu Rev Microbiol 52:81–104.

38. Vlamakis H, Aguilar C, Losick R, Kolter R (2008) Control of cell fate by the formationof an architecturally complex bacterial community. Genes Dev 22:945–953.

39. Stewart PS, Costerton JW (2001) Antibiotic resistance of bacteria in biofilms. Lancet358:135–138.

40. Lee N, et al. (1999) Combination of fluorescent in situ hybridization and microauto-radiography—A new tool for structure-function analyses in microbial ecology. ApplEnviron Microbiol 65:1289–1297.

41. Balazsi G, van Oudenaarden A, Collins JJ (2011) Cellular decision making and biolog-ical noise: From microbes to mammals. Cell 144:910–925.

3598 | www.pnas.org/cgi/doi/10.1073/pnas.1801853115 Guo et al.

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