Continuum Models and Discrete Systems (CMDS9)Proceedings of the 9th International SymposiumJune 29{July 3, 1998, Istanbul, Turkey. Editors E. Inan & K. Z. Markovc 1999 World Scienti�c Publishing Co., pp. 00{00CONTINUOUS AND DISCRETE MODELS OF COOPERATIONIN COMPLEX BACTERIAL COLONIESI. COHEN, E. BEN-JACOB, I. GOLDING and Y. KOZLOVSKYSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of ExactSciences, Tel Aviv University,Tel Aviv 69978, IsraelWe study the e�ect of discreteness on various models for patterning in bacterialcolonies. In a bacterial colony with branching pattern, there are discrete entities {bacteria { which are only two order of magnitude smaller than the elements of themacroscopic pattern. We present two types of models. The �rst is the Commu-nicating Walkers model, a hybrid model composed of both continuous �elds anddiscrete entities { walkers, which are coarse-graining of the bacteria. Models ofthe second type are systems of reaction di�usion equations, where the branchingof the pattern is due to non-constant di�usion coe�cient of the bacterial �eld.The di�usion coe�cient represents the e�ect of self-generated lubrication uid onthe bacterial movement. We implement the discreteness of the biological systemby introducing a cuto� in the growth term at low bacterial densities. We demon-strate that the cuto� does not improve the models in any way. Its only e�ect isto decrease the e�ective surface tension of the front, making it more sensitive toanisotropy. We compare the models by introducing food chemotaxis and repulsivechemotactic signaling into the models. we �nd that the growth dynamics of theCommunication Walkers model and the growth dynamics of the Non-Linear di�u-sion model are are a�ected in the same manner. From such similarities and fromthe insensitivity of the Communication Walkers model to implicit anisotropy weconclude that the increased discreteness, introduced by the coarse-graining of thewalkers, is small enough to by neglected.Keywords. bacteria, bacterial colonies, bacterial communication, chemotaxis,discreteness cuto�, non-linear di�usion, random-walk, reaction-di�usion equations,signaling chemotaxis . 1

1 IntroductionThe endless array of patterns and shapes in nature has long been a sourceof joy and wonder to layman and scientists alike [1, 2, 3, 4]. During the lastdecade, there were exciting developments in the understanding of pattern de-termination in non-living systems [?]. The attention of many researchers isnow shifting towards living systems, in a hope to employ the new insights intoprocesses of pattern formation [?, to mention just a few]. Bacterial colonies of-fer a suitable subject for such research. In some senses they are similar enoughto non-living systems so as their study can bene�t from the knowledge aboutnon-living systems, yet their building blocks (bacteria) are complex enough toensure ever so new surprises.In �gure 1 we show representative branching patterns of bacterial colonies.These colonies are made up of about 1010 bacteria of the type Paenibacillusdendritiformis var. dendron (see [5, 6] for �rst reference in the literature and[7] for identi�cation). Each colony is grown in a standard petri-dish on athin layer of agar (semi-solid jelly). The bacteria cannot move on the drysurface and cooperatively they produce a layer of lubrication uid in whichthey swim (Fig. 2). Bacterial swimming is a random-walk-like movement, inwhich the bacteria propel themselves in nearly straight runs separated by brieftumbling. The bacteria consume nutrient from the media, nutrient which aregiven in limited supply. The growth of a colony is limited by the di�usion ofnutrients towards the colony { the bacterial reproduction rate that determinesthe growth rate of the colony is limited by the level of nutrients available forthe cells. Note, however, that a single bacterium put alone on the agar canreproduce, grow in numbers and make a new colony.Bacterial colonies entangle entities in many length scales: the colony asa whole is the range of several cm; the individual branches are of width inthe range of mm and less; the individual bacteria are in the range of �m, sois the width of the colony's boundary; and chemicals in the agar such as theconstitutes of the nutrient are on the molecular length scale.Kessler and Levine [9] studied discrete pattern forming systems, usingreaction-di�usion models with linear di�usion and various growth terms. Theyshowed that the ability of the system to form two-dimensional patterns de-pend on the derivative of the growth term (reaction term) at zero densities.With a negative derivative, the system can form branching pattern; with apositive derivative, the system can form only compact patterns with circularenvelope. They accounting for the discreteness of the system by introducinga low-densities-cuto� in the growth term. Doing so to a growth term withpositive derivative at zero can introduce bumps to the pattern, which is a2

Fig. 1: Observed branchingpatterns of colonies of Paenibacillus dendritiformis var. dendrongrown on 2% agar concentration. The nutrient level is, from left to right, 0.25 gram peptoneper liter, 0.5g=l and 5g=l. On the right wide branches, much wider than the gaps betweenthem. In the middle less ordered, fractal-like pattern, similar to patterns seen in electro-chemical deposition and DLA simulations [8, 2]. As the nutrient level is farther decreasedthe pattern become denser again, with pronounced circular envelope (on the left).manifestation of a di�usive instability in the two-dimensional front (the �rststep towards branching pattern).We present here three models for growth of the bacterial colonies. The �rstis the Communicating Walkers model (Sec. 2) which includes discrete entitiesto describe the bacteria, continuous �elds to describe chemicals in the agar andan explicit free boundary for the colony's edge. The second model is a contin-uous one, a reaction-di�usion model that couples the bacterial movement toa �eld of lubrication uid (Sec. 3). The di�usion coe�cients of the bacterial�eld and the lubrication �eld depend on the lubrication uid, resulting in aspontanuous formation of a sharp boundary to the colony. The third modeltries to simplify the former model and dispose of the lubrication �eld by intro-ducing a density-dependent di�usion of the bacterial �eld (Sec. 4). We discussthe e�ect of a cuto� in the growth term in the two continuous models. Wethan turn our attention to various features of the observed bacterial patternsand see similarities in the di�erent models' ability to reproduce this features(Sec. 5).2 The Communicating Walkers Model : An Hybrid ModelThe CommunicatingWalkers model [10] was inspired by the di�usion-transitionscheme used to study solidi�cation from supersaturated solutions [11, 12, 13].The former is a hybridization of the \continuous" and \atomistic" approachesused in the study of non-living systems. The di�usion of the chemicals is han-3

Fig. 2: Closer look on a branch of a bacterial colony. The left �gure shows the lubrication uid in which the bacteria are immersed. On the right, the individual bacteria can be seen.Each dot in the branch is a 1�2�m bacterium. The dots outside the branch are not bacteriabut deformations of the agar.dled by solving a continuous di�usion equation (including sources and sinks)on a tridiagonal lattice with a lattice constant a0. The bacterial cells are repre-sented by walkers allowing a more detailed description. In a typical experimentthere are 109 � 1010 cells in a petri-dish at the end of the growth. Hence it isimpractical to incorporate into the model each and every cell. Instead, eachof the walkers represents about 104� 105 cells so that we work with 104� 106walkers in one numerical \experiment".The walkers perform an o�-lattice random walk on a plane within an en-velope representing the boundary of the wetting uid. This envelope is de�nedon the same triangular lattice where the di�usion equations are solved. To in-corporate the swimming of the bacteria into the model, at each time step eachof the active walkers (motile and metabolizing, as described below) moves astep of size d < a0 at a random angle �. Starting from location ~ri, it attemptsto move to a new location ~r0i given by:~r0i = ~ri + d(cos�; sin�) : (1)If ~r0i is outside the envelope, the walker does not move. A counter on thesegment of the envelope which would have been crossed by the movement~ri ! ~r0i is increased by one. When the segment counter reaches a speci�ednumber of hits Nc, the envelope propagates one lattice step and an additionallattice cell is added to the colony. This requirement of Nc hits represents thecolony propagation through wetting of unoccupied areas by the bacteria. Notethat Nc is related to the agar dryness, as more wetting uid must be produced(more \collisions" are needed) to push the envelope on a harder substrate.4

Motivated by the presence of a maximal growth rate of the bacteria evenfor optimal conditions, each walker in the model consumes food at a constantrate c if su�cient food is available. We represent the metabolic state of thei-th walker by an 'internal energy' Ei. The rate of change of the internal energyis given by dEidt = �Cconsumed � Em�R ; (2)where � is a conversion factor from food to internal energy (� �= 5 � 103cal=g)and Em represent the total energy loss for all processes over the reproductiontime �R, excluding energy loss for cell division. Cconsumed is Cconsumed �min (C ;0C) ; where 0C is the maximal rate of food consumption as limitedby the locally available food [14]. When su�cient food is available,Ei increasesuntil it reaches a threshold energy. Upon reaching this threshold, the walkerdivides into two. When a walker is starved for long interval of time, Ei drops tozero and the walker \freezes". This \freezing" represents entering a pre-sporestate (starting the process of sporulation, see section 5).We represent the di�usion of nutrients by solving the di�usion equationfor a single agent whose concentration is denoted by n(~r; t):@n@t = Dnr2C � bCconsumed ; (3)where the last term includes the consumption of food by the walkers (b is theirdensity). The equation is solved on the tridiagonal lattice. The simulationsare started with inoculum of walkers at the center and a uniform distributionof the nutrient.Results of numerical simulations of the model are shown in �gure 3. As inthe case of real bacterial colonies, the patterns are compact at high nutrientlevels and become fractal with decreasing food level. For a given nutrient level,the patterns are more rami�ed as the agar concentration increases. The re-sults shown in �gure 3 do capture some features of the experimentally observedpatterns. However, at this stage the model does not account for some criticalfeatures, such as the ability of the bacteria to develop organized patterns atvery low nutrient levels. Ben-Jacob et al. [26, 27, 28, 3] suggested that chemo-tactic signaling must be included in the model to produce these features (seesection 5).3 A Layer of LubricationThe Lubrication-Bacteria model is a reaction-di�usion model for the bacterialcolonies [15]. This model includes four coupled �elds. One �eld describes the5

Fig. 3: Colonial pattern of the Communicating Walkers model. Here Nc = 20 and n0 is 6,8, 10 and 30 from left to right respectively.Fig. 4: Closer look on simulated colonies. On the right: a tip of a branch in the Commu-nicating Walkers model. The boundary of the branch and walkers can be seen. On the left:lubrication at a tip of a branch in the Lubrication-Bacteria model.bacterial density b(~x; t), the second describe the height of lubrication layer inwhich the bacteria swim l(~x; t), third �eld describes the nutrients n(~x; t) andthe fourth �eld is the stationary bacteria that \freeze" and begin to sporulates(~x; t) (see section 5).The time evolution of the bacterial �eld b consist of two parts; a di�u-sion term which is coupled to the lubrication �eld and a reaction part whichcontains terms for reproduction and death. Following the same arguments pre-sented for the Communicating Walkers model, we get a reaction term of theform (�bmin(C ; n)�Emb=�R). Assuming that nutrient is always the factorlimiting the bacterial growth we get, upon rescaling, the growth term bn� �b(� constant).We now turn to the bacterial movement. In a uniform layer of liquid,bacteria swimming is a random walk with variable step length and can beapproximated by di�usion. The layer of lubrication is not uniform, and itsheight a�ects the bacterial movement. An increase in the amount of lubrication6

decreases the friction between the bacteria and the agar surface. The term'friction' is used here in a very loose manner to represent the total e�ect of anyforce or process that slows down the bacteria. As the bacterial motion is over-damped, the local speed of the bacteria is proportional to the self-generatedpropulsion force divided by the friction. It can be shown that variation of thespeed leads to variation of the di�usion coe�cient, with the di�usion coe�cientproportional to the speed to the power of two. We assume that the frictionis inversely related to the local lubrication height through some power law:friction � l and < 0. The bacterial ux is:~Jb = �Dbl�2 rb (4)The lubrication �eld l is the local height of the lubrication uid on theagar surface. Its dynamics is given by:@l@t = �r � ~Jl + �bn(lmax � l) � �l (5)where ~Jl is the uid ux (to be discussed), � is the production rate and � isthe absorption rate of the uid by the agar. � is inversly related to the agardryness.The uid production is assumed to depend on the bacterial density. As theproduction of lubrication probably demands substantial energy, it also dependson the nutrients level. We assume the simplest case where the productiondepends linearly on the concentrations of both the bacteria and the nutrients.The lubrication uid ows by di�usion and by convection caused by bac-terial motion. A simple description of the convection is that as each bacteriummoves, it drags along with it the uid surrounding it.~Jl = �Dl l�rl + j ~Jb (6)where Dl is a lubrication di�usion constant, ~Jb is the bacterial ux and j isthe amount of uid dragged by each bacterium. The di�usion term of the uiddepends on the height of the uid to the power � > 0 (the nonlinearity inthe di�usion of the lubrication, a very complex uid, is motivated by hydro-dynamics of simple uids). The nonlinearity causes the uid to have a sharpboundary at the front of the colony, as is observed in the bacterial colonies(Fig. 4).The complete model for the bacterial colony is:@b@t = Dbr � (l�2 rb) + bn� �b (7)7

Fig. 5: Growth patterns of the Lubrication-Bacteria model, for di�erent values of initialnutrient levels n0. The apparent (though weak) 6-fold anisotropy is due to the underlyingtridiagonal lattice.@n@t = Dnr2n� bn@l@t = r � (Dll�rl + jDbl�2 rb) + �bn(lmax � l) � �l@s@t = �bThe second term in the equation for b represents the reproduction of the bacte-ria. The reproduction depends on the local amount of nutrient and it reducesthis amount. The third term in the equation for b reprsents the process ofbacterial \freezing". For the initial condition, we set n to have uniform dis-tribution of level n0, b to have compact support at the center, and the other�elds to be zero everywhere.Preliminary results show that the model can reproduce branching pat-terns, similar to the bacterial colonies (Fig. 5). At low values of absorptionrate, the model exhibits dense �ngers. At higher absorption rates the modelexhibits �ner branches. We also obtain �ner branches if we change other pa-rameters that e�ectively decrease the amount of lubrication. We can relatethese conditions to high agar concentration.We can now check the e�ect of bacterial discreteness on the observed colo-nial patterns. Following Kessler and Levine [9], we introduce the discretenessof the system into the continuous model by repressing the growth term at lowbacterial densities (\half a bacterium cannot reproduce"). The growth termis multiplied by a Heaviside step function �(b � �), where � is the thresholddensity for growth. In �gure 6 we show the e�ect of various values of � onthe pattern. High cuto� values make the model more sensitive to the implicitanisotropy of the underlying tridiagonal lattice used in the simulation. Theresult is dendritic growth with marked 6-fold symmetry of the pattern. In-8

Fig. 6: The e�ect of a cuto� on the growth patterns in the Lubrication-Bacteria model.Aside from the cuto�, the conditions are the same as in the middle pattern of �gure 5, wherethe maximal value of b was about 0.025. The values of the cuto� � are, from left to right,10�6, 10�5 and 3 �10�5. The 6-fold symmetry is due to anisotropy of the underlying latticewhich is enhanced by the cuto�.creased values of cuto� also decrease the maximal values of b reached in thesimulations (and the total area occupied by the colony).The reason for the pattern turning dendritic is as follows: the di�erencebetween tip-splitting growth and dendritic growth is the relative strength ofthe e�ect of anisotropy and an e�ective surface tension [2]. In the Lubrication-Bacteria model there is no explicit anisotropy and no explicit surface tension.The implicit anisotropy is related to the underlaying lattice, and the e�ectivesurface tension is related to the width of the front. The cuto� prevents thegrowth at the outer parts of the front, thus making it thinner, reduces thee�ective surface tension and enables the implicit anisotropy to express itself.We stress that it is possible to �nd a range of parameters in which thegrowth patterns resembles the bacterial patterns, in spite a high value of cut-o�. Yet the cuto� does not improve the model in any sense, it introduces anadditional parameter, and it slows the numerical simulation. We believe thatthe well-de�ned boundary makes the cuto� (as a representation of the bacterialdiscreteness) unnecessary.4 Non-linear di�usionIt is possible to introduce a simpli�ed model, where the uid �eld is not in-cluded, and is replaced by a density-dependent di�usion coe�cient for thebacteria Db � bk [16, 17]. For this purpose, a few assumptions are neededabout the dynamics at low bacterial and lubrication density:1. The production of lubricant is proportional to the bacterial density to thepower � > 0.2. There is a sink in the equation for the time evolution of the lubrication9

n0=1.0 n0=1.5 n0=2.0

Fig. 2: 2D growth patterns (b+ s) of the Kitsunezaki model, for di�erent values of initialnutrient level n0. Parameters are: D0 = 0:1; k = 1; � = 0:15. The apparent 6-fold anisotropyis due to the underlying tridiagonal lattice.�eld, e.g. absorption of the lubricant into the agar. This sink is proportionalto the lubrication density to the power � > 0.3. Over the bacterial length scale, the two processes above are much fasterthan the di�usion process, so the lubrication density is proportional to thebacterial density to the power of �=�.4. The friction is proportional to the lubrication density to the power < 0.Given the above assumptions, the lubrication �eld can be removed from thedynamics and be replaced by a density dependent di�usion coe�cient. This co-e�cient is proportional to the bacterial density to the power k � �2 �=� > 0A model of this type is o�ered by Kitsunezaki[18]:@b@t = r(D0bkrb) + nb� �b (8)@n@t = r2n� bn (9)@s@t = �b (10)For k > 0 the 1D model gives rise to a front \wall", with compact support(i.e. b = 0 outside a �nite domain). For k > 1 this wall has an in�niteslope. The propagation velocity in this case is determined by the conditionat the front, not at in�nity [16, 19]. We therefore expect a Mullins-Sekerkainstability in 2D (as is claimed in [18]). Indeed, the model exhibits branchingpatterns for suitable parameter values and initial conditions, as depicted inFig. 4.Increasing the initial nutrient level is seen to make to colony more dense,10

n0=2.0n0=1.5

Fig. 2: 2D growth patterns (b+s) of the Kitsunezaki model, with a cuto� correction. Cuto�value � = 0:1, all other parameters as in Fig. 4. The apparent 6-fold anisotropy is due tothe underlying tridiagonal lattice.similarly to what happens in the discrete model. The hexagonal shape ofthe colony envelope is an artifact of numerical simulation, stemming from theanisotropy of the underlying lattice.Adding the \Kessler and Levine correction" to the model, i. e. makingthe growth term disappear for b < �, does not seem to make the patterns\better", or closer to the experimental observations. On the contrary, thebranching patterns now seen are quite unsimilar to any colony shape we know(Fig. 4).5 ChemotaxisSo far, we have tested the models for they ability to reproduce macroscopicpatterns and microscopic dynamics of the bacterial colonies. All succeededequally well, reproducing some aspects of the microscopic dynamics and thepatterns in some range of nutrient level and agar concentration, but so can doother models [15, and reference there in]. We will now extend the Commu-nicating Walkers model and the Non-Linear Di�usion model to test for theirsuccess in describing other aspects of the bacterial colonies involving chemo-taxis and chemotactic signaling (which are beleived to by used by the bacteria[26, 27, 28, 3]). Chemotaxis means changes in the movement of the cell inresponse to a gradient of certain chemical �elds [20, 21, 22, 23]. The move-ment is biased along the gradient either in the gradient direction or in theopposite direction. Usually chemotactic response means a response to an ex-ternally produced �eld, like in the case of chemotaxis towards food. However,the chemotactic response can be also to a �eld produced directly or indirectlyby the bacterial cells. We will refer to this case as chemotactic signaling. The11

bacteria sense the local concentration r of a chemical via membrane recep-tors binding the chemical's molecules [20, 22]. It is crucial to note that whenestimating gradients of chemicals, the cells actually measure changes in thereceptors' occupancy and not in the concentration itself. When put in contin-uous equations [29, 15], this indirect measurement translates to measuring thegradient @@x r(K + r) = K(K + r)2 @r@x: (11)where K is a constant whose value depends on the receptors' a�nity, the speedin which the bacterium processes the signal from the receptor, etc. This meansthat the chemical gradient times a factor K=(K + r)2 is measured, and it isknown as the \receptor law" [29].When modeling chemotaxis performed by walkers, it is possible to modu-late the periods between tumbling (without changing the speed) in the sameway the bacteria do. It can be shown that step length modulation has thesame mean e�ect as keeping the step length constant and biasing the direc-tion of the steps (higher probability to move in the preferred direction). Asthis later approach is numerically simpler, this is the one implemented in theCommunicating Walkers model.In a continuous model, we incorporate the e�ect of chemotaxis by intro-ducing a chemotactic ux ~Jchem:~Jchem � �(�)�(r)rr (12)�(r)rr is the gradient sensed by the cell (with �(r) having the units of 1 overchemical's concentration). �(r) is usually taken to be either constant or the\receptor law". �(�) is the bacterial response to the sensed gradient (havingthe same units as a di�usion coe�cient). In the Non-Linear Di�usion modelthe bacterial di�usion is Db = D0bk, and the bacterial response to chemotaxisis �(b) = �0b �D0bk� = �0D0bk+1. �0 is a constant, positive for attractivechemotaxis and negative for repulsive chemotaxis.Ben-Jacob et al. argued [26, 27, 28, 3] that for the colonial adaptive self-organization the bacteria employ three kinds of chemotactic responses, eachdominant in di�erent regime of the morphology diagram. One response is thefood chemotaxis mentioned above. It is expected to be dominant for only arange of nutrient levels (see the \receptor law" below). The two other kindsof chemotactic responses are signaling chemotaxis. One is long-range repulsivechemotaxis. The repelling chemical is secreted by starved bacteria at the innerparts of the colony. The second signal is a short-range attractive chemotaxis.The length scale of each signal is determined by the di�usion constant of thechemical agent and the rate of its spontaneous decomposition.12

Fig. 9: The e�ect of chemotaxis on growth in the Communicating Walkers model. On theleft: chemotaxis towards food is added to the model. The conditions are the same as in �gure3, second from right pattern. The pattern is essentially unchanged by food chemotaxis, butthe growth velocity is almost doubled. On the right: repulsive chemotactic signaling is addedto the model. The conditions are the same as in �gure 3, left pattern. The pattern is of �neradial branches with circular envelope, like in �gure 1, left pattern.Ampli�cation of di�usive Instability Due to Nutrients Chemotaxis: In non-living systems, more rami�ed patterns (lower fractal dimension) are observedfor lower growth velocity. Based on growth velocity as function of nutrient leveland based on growth dynamics, Ben-Jacob et al. [10] concluded that in thecase of bacterial colonies there is a need for mechanism that can both increasethe growth velocity and maintain, or even decrease, the fractal dimension.They suggested food chemotaxis to be the required mechanism. It providesan outward drift to the cellular movements; thus, it should increase the rateof envelope propagation. At the same time, being a response to an external�eld it should also amplify the basic di�usion instability of the nutrient �eld.Hence, it can support faster growth velocity together with a rami�ed patternof low fractal dimension.The above hypothesis was tested in the Communicating Walkers modeland in the Non-Linear Di�usion model. In �gures 9 and 10 it is shown thatas expected, the inclusion of food chemotaxis in both models led to a consid-erable increase of the growth velocity without signi�cant change in the fractaldimension of the pattern.Repulsive chemotactic signaling: We focus now on the formation of the�ne radial branching patterns at low nutrient levels. From the study of non-living systems, it is known that in the same manner that an external di�usion�eld leads to the di�usion instability, an internal di�usion �eld will stabilizethe growth. It is natural to assume that some sort of chemotactic agent pro-duces such a �eld. To regulate the organization of the branches, it must be13

Fig. 10: Growth patterns of the Non-Linear Di�usionmodel with food chemotaxis (left) andrepulsive chemotactic signaling (right) included. �0f = 3; �0r = 1;Dr = 1;�r = 0:25;r =0;�r = 0:001. Other parameters are the same as in �gure 7. The apparent 6-fold symmetryis due to the underlying tridiagonal lattice.a long-range signal. To result in radial branches it must be a repulsive chem-ical produced by cells at the inner parts of the colony. The most probablecandidates are the bacteria entering a pre-spore stage.If nutrient is de�cient for a long enough time, bacterial cells may enter aspecial stationary state { a state of a spore { which enables them to survivemuch longer without food. While the spores themselves do not emit any chem-icals (as they have no metabolism), the pre-spores (sporulating cells) do notmove and emit a very wide range of waste materials, some of which uniqueto the sporulating cell. These emitted chemicals might be used by other cellsas a signal carrying information about the conditions at the location of thepre-spores. Ben-Jacob et al. [10, 30, 27] suggested that such materials arerepelling the bacteria ('repulsive chemotactic signaling') as if they escape adangerous location.The equation describing the dynamics of the chemorepellent contains termsfor di�usion, production by pre-spores, decomposition by active bacteria andspontaneous decomposition:@r@t = Drr2r + �rs � rbr � �rr (13)where Dr is the di�usion coe�cient of the chemorepellent, �r is the emissionrate of repellent by pre-spores, r is the decomposition rate of the repellentby active bacteria, and �r is the rate of self decomposition of the repellent. Inthe Communicating Walkers model b and s are replaced by active and inactivewalkers, respectively.In �gures 9 and 10 the e�ect of repulsive chemotactic signaling is shown.14

In the presence of repulsive chemotaxis the patterns in both models becomemuch denser with a smooth circular envelope, while the branches are thinnerand radially oriented.6 ConclusionsWe show here a pattern forming system, bacterial colony, whose discrete el-ements, the bacteria, are big enough to raise the question of modeling dis-crete systems. We study two types of models. The Communicating Walkersmodel has explicit discrete units to represent the bacteria. The ratio betweenthe walkers' size and the pattern's size is even bigger than the ratio in thebacterial colony. The second type of models is continuous reaction-di�usionequations. Non-linear di�usion causes a sharp boundary to appear in thesemodels. Following Kessler and Levine [?], we account for the discreteness ofthe bacteria by including a cuto� in the bacterial growth term. The cuto�does not improve the models' descriptive power. The main e�ect of such cuto�is to decrease the width of the colony's front, making the growth pattern moresensitive to e�ects such as implicit anisotropy. We conclude that the presenceof a boundary cancels the need for explicit treatment of discreteness.In order to assess the similarity between the discrete CommunicatingWalk-ers model and the continuous Non-Linear Di�usion model, we incorporate foodchemotaxis and repulsive chemotactic signaling into the models (both are ex-pected to exist in the bacterial colonies). Both models respond to such changesin the same way, exhibiting altered patterns and altered dynamics, similar tothose observed in the bacterial colonies. From this similarity we conclude thatto some extent inferences from one model can be applied to the other. Speci�-cally we focus on insensitivity of the CommunicatingWalkers model to implicitanisotropy and on the sensitivity a cuto� imposes on the continuous models.From the two facts combined we conclude that the magni�ed discreteness inthe Communicating Walkers model is still small enough to be neglected.Acknowledgements. We have bene�ted from many discussions on the presentedstudies with H. Levine. IG wishes to thank R. Segev for fruitful discussions.Identi�cations of the Paenibacillus dendritiformis var. dendron and geneticstudies are carried in collaboration with the group of D. Gutnick. Presentedstudies are supported in part by a grant from the Israeli Academy of Sciencesgrant no. 593/95 and by the Israeli-US Binational Science Foundation BSFgrant no. 00410-95. 15

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