Bursts of sectors in expanding bacterial colonies as a possible model for tumor growth and...

Available online at www.sciencedirect.com

Physica A 320 (2003) 485–496www.elsevier.com/locate/physa

Bursts of sectors in expanding bacterialcolonies as a possible model for tumor

growth and metastasesIlan G. Rona;b;∗, Ido Goldinga ;1, Beatrice Lifsitz-Mercerc,

Eshel Ben-JacobaaSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences,

Tel-Aviv University, Tel-Aviv 69978, IsraelbDepartment of Oncology, Tel-Aviv Sourasky Medical Center, The Sackler Faculty of Medicine,

Tel-Aviv University, Tel-Aviv 69978, IsraelcDepartment of Pathology, Tel-Aviv Sourasky Medical Center, The Sackler Faculty of Medicine,

Tel-Aviv University, Tel-Aviv 69978, Israel

Received 24 September 2002

Abstract

All kinds of hypotheses have been proposed to explain the mechanism of tumorigenesis. Upuntil now, we have not had any generally acknowledged model that helps us understand theprocess. However, it is well accepted that cancer development and progression is dictated by aseries of alterations in genes such as oncogenes, tumor suppressor genes, DNA replication genesand others. Segregation of cell populations is a key question in evolution theory. One importantaspect when observing cell proliferation in general or bacterial colonies in particular is therelation between spatial organization and the composition of the populations. Here we study aspeci:c example—sectors in expanding bacterial colonies. Such sectors are spatially segregatedsub-populations of mutants. Bursts of sectors are observed during compact growth and duringbranching growth. For theoretical studies of these bursts we employ two mathematical models.Using these models we investigate the amount of segregation achieved by a neutral mutation, aswell as by mutations having some advantage over the wild type.c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Tumor growth; Colonial sectors; Expanding bacterial colonies

∗ Corresponding author. Tel.: +972-3-6973494; fax: +972-3-6974828.E-mail address: [email protected] (I.G. Ron).

1 Current address: Department of Molecular Biology, Princeton University, Princeton, NJ 08544.

0378-4371/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0378 -4371(02)01547 -9

mailto:[email protected]

486 I.G. Ron et al. / Physica A 320 (2003) 485–496

1. Introduction

In nature, microorganisms must often cope with hostile environmental conditions. Todo so they have developed sophisticated cooperative behavior and intricate communica-tion capabilities, such as: direct cell–cell physical interactions mediated by cell surfacestructures, collective production of extracellular “wetting” Huid for movement on hardsurfaces, long range chemical signaling such as quorum sensing and chemotactic (biasof movement according to gradient of chemical agent) signaling, collective activationand deactivation of genes and even exchange of genetic material. In some cases, thesecapabilities are exploited by the colonies to develop complex spatio-temporal patternsin response to adverse growth conditions.EIcient adaptation of the colony to adverse growth conditions requires self-

organization on all levels which can only be achieved via cooperative behavior ofthe individual cells [1–3].The cooperative behavior is observed in a diverse range of microorganisms [4–12].

They all possess an unexpectedly broad repertoire of chemical and physical signal-ing mechanisms. Escherichia coli, gut bacteria, and Salmonella typhimurium use at-tractive chemotaxis to call their peers for aid in detoxi:cation of their environment[6,12–14]. Paenibacillus dendritiformis [15] and Serratia marcescens cooperativelyproduce lubrication Huid in which they swim [16,17]. Bacillus circulans and Paeni-bacillus vortex utilize cell–cell physical interaction for response to external signals[18,12,19].Shapiro and coworkers have begun to characterize the morphologic developments

of these colonies [20,21] as well as the genetic mechanisms at play during diLeren-tiation that give rise to wedged-shaped sectors [22,23]. It appears that these pheno-typically recognizable sectors arise from a single ancestor cell in which genetic rear-rangements, such as base substitution, transposition, and frame shift may have takenplace.Ben-Jacob et al. have also shown bursts of sectors during growth of branching

bacterial colonies (Fig. 1). Actually, the phenomenon is more general. Boschke andBley show in [24] emerging sectors in a yeast colony. If the mutants have the samegrowth dynamics as the “normal”, wild-type bacteria, they will usually go unnoticed,unless some property such as coloring distinguishes them. DiLerent coloring may resultfrom diLerent enzymatic activity (natural coloring) or from a diLerent response to astaining process (arti:cial coloring). In both cases the mutation is not neutral in thestrictest sense, but it is neutral as far as the dynamics is concerned. If the mutantshave diLerent growth dynamics, a distinguished sector with a diLerent growth patternmight indicate their presence.Following Shapiro we proposed that the study of bursts of sectors during colonial

development might shed light on the burst of tumors, the complexity of their growthpatterns (Fig. 2) and cells organization [25]. It is motivated in part in the similarity ofthe morphogenesis and in the changes in rate of cancer cell division and communicationrelative to normal cell.

I.G. Ron et al. / Physica A 320 (2003) 485–496 487

Fig. 1. Emerging sectors in branching colonies of P. dendritiformis, obtained at 5 g=l peptone and 1.75%agar, in the presence of the antibiotic material Stromocine.

2. Experimental background

In this paper, we study the subject of expression of genetic changes: either changes inthe sequence or in the activity and expression of the genome in bacterial colonies. Forsimplicity we shall refer all these cases as mutations. Besides being an interesting andimportant subject by itself, the bacterial colonies might serve as a model system for thestudy of expression of mutations in a spreading population. Ben-Jacob et al. [18,1,2]conducted experiments to see how adaptive bacterial colonies could be in the presenceof external pressure, here in the form of a limited nutrient supply and hard surface. Thework was done with a newly identi:ed species, Paenibacillus dendritiformis [26]. Thisspecies is motile on the hard surface and its colonies exhibit branching patterns. Duringthe last few years we have studied this strain experimentally as well as theoretically.There is a well-known observed phenomenon of bursts of new sectors of mutantsduring the growth of bacterial colonies [27,28,24]. If the mutants have the same growthdynamics as the normal they usually go unnoticed . If, however, the mutants havediLerent growth dynamics, a distinguished sector with a diLerent growth pattern mightindicate their presence. The bursting of sectors can be observed both during compactand branching growth. Sectors emerging during branching growth have a greater varietyof structure and shapes than those emerging from compact colonies.

3. Results

The typical growth pattern on semi-solid agar is a branching pattern. The structureof the branching pattern varies for diLerent growth conditions. However, at very lowagar concentrations or very high nutrient concentrations the colonies exhibit compactpatterns.


Fig. 2. (a) Emerging sectors in benign tumor of the large-intestine (polipoid growth adenoma oflarge-intestine). (b) Emerging sectors in malignant tumors. Left: sesile sector growth of skin cancer(cutaneous malignant mmelanoma). Right: sectors of emerging adeno-carcinoma of large-intestine (in:l-trating poliopoid pattern of growth). All :gures are with ×10 magni:cation.


Fig. 3. Emerging sectors in compact colonies of P. dendritiformis. (a) T morphotype, 10 g=l peptone, 0.5%agar. (b) C morphotype, 1:5 g=l peptone, 0.4% agar. (c) C morphotype, 10 g=l peptone, 0.4% agar.

Fig. 4. Emerging sectors in branching colonies of P. dendritiformis, obtained at 1 g=l peptone and 1.75%agar (a), 0:8 g=l peptone and 2% agar (b).

The bursting of sectors can be observed both during compact and branching growth.Sectoring takes the form of wedge shaped sectors with a phenotype strikingly diLerentfrom that of the rest of the colony. Examples of burst of sectors during compact growthare shown in Fig. 3, while examples of sectors emerging from branching growth areshown in Figs. 1 and 4. As can be seen from these pictures, a greater variety ofstructures and shapes exist in the branching growth sectors.The term mutant throughout this paper describes the population in the emerging

sectors. The existence of genetic diLerence between the bacteria in the sector andthose in the rest of the colony has not been veri:ed. We have, however, veri:ed thatthe phenotypic diLerence between the tow populations is inheritable, using inoculation.


4. Numerical studies

To study theoretically the sectors in the bacterial colonies we use two diLerentmodels [29]:1. The non-linear diLusion model [30,31], in which the bacteria are represented by

their 2D density, and a reaction–diLusion equation of this density describes their timeevolution. This equation is coupled to the other reaction–diLusion equations for :eldsof chemicals, such as nutrient.2. The discrete “communicating walkers” model of Ben- Jacob et al. [32,18,33], in

which the bacteria are represented by discrete entities (walkers) which can consumenutrient, reproduce, perform random or biased movement, and produce or respondto chemicals. The time evolution of the nutrient and the chemicals is described byreaction–diLusion equations.

4.1. The continuous model

The bacterial population is described by two :elds, for the densities of the wild-typebacteria (“type 1”) and the mutants (“type 2”). We allow some probability of transitionfrom wild-type to mutants. In the absence of chemotaxis, the equations for the bacterialdensity of the two strains will be written (with subscript denoting bacteria type):

@b1@t

=∇(D1(b)∇b1) + j1nb1(b− )− �1b1 − F12 ; (1)

@b2@t

=∇(D2(b)∇b2) + j2nb2(b− )− �2b2 + F12 ; (2)

where D1;2 =D01; 2bk (b=b1 +b2). Each of these equations describes the rate of change

of bacterial density at a particular point in space and time. The terms on the right-handside of each equations are:1. A non-linear diLusion term, representing random motion of bacteria within a layer

of liquid that they themselves produce.2. A growth term proportional to food consumption and to bacterial density. The

term is multiplied by a step function which makes it non-zero only if bacterial densityis above a threshold . This feature represents the discreteness of bacteria [34].3. A death term describing the proccess of sporulation at a rate �.4. A mutation term (F12) giving the rate of transition from wild-type to mutant. This

process is modeled by adding an amount of to b2(x; t) when the local growth rateof type 1, j1b1n is larger than some threshold �.We also write the time evolution equations for the stationary cells density, and the

nutrient density:

@n@t

= Dn∇2n− (j1b1 + j2b2)n(b− ) ; (3)

@si@t

= �ib : (4)


Chemotactic response can be incorporated by adding a chemotactic Hux for eachstrain:

J̃ chemi = �i(b)�i(r)∇r (5)

with �i(r) = �0iK=(K + r)2 and �i(b) = bi · bk . A similar expression is written for foodchemotaxis. The two strains can have diLerent values of �0i, representing diLerentsensitivity to chemotaxis (either repulsive or food chemotaxis).The equations are integrated numerically on a two-dimensional lattice, starting with

an initial population of wild-type alone, concentrated at a small area in the middle ofthe “plate”.

4.2. The communicating walkers model

As an additional means to support our theoretical study, we have also performedseveral simulations using the “Communicating Walkers” model of Ben-Jacob et al.(see [32,8,18] for a description of the model). When dealing with two bacterial strains,each walker in the simulation belongs to either “type 1” (wild-type) or “type 2” (mu-tant), which may diLer in their various biological parameters, such as step length (i.e.,motility) or sensitivity to chemotaxis. The colony is initialized with an inoculationof wild-type walkers, which—when multiplying—have some :nite probability of giv-ing birth to a mutant. The two populations then co-evolve according to the dynamicsdescribed in the references above.

5. Results of simulations

5.1. Sector formation during compact growth

We start with simulations of sector formation of neutral mutations. We expect thatsuch mutations will not form a segregated sector. Indeed the simulations show nosegregation in this case. Next we study the more interesting case of superior mutants.In this case one observes a sector which grows faster than the rest of the colony. In thesimulations, a sharp segregation is obtained when the mutant is endowed with a highergrowth rate � or a higher motility D0, as shown in Fig. 5). The :gure also depicts theresults obtained by simulations of the Communicating Walker model. As can be seen,both models exhibit a fan-like sector of mutants, very similar to the one observed inthe experiments. The “mixing area”, where both strains are present, is narrow, and itswidth is related to the width of the propagating front of the colony (the region wherethe bacteria are active).

5.2. Sector formation during branching growth

We now turn to the case of sectoring during branching growth. In this case the mod-els show a slow process of segregation even for a neutral mutation. This might resultfrom the fact that a particular branch may stem from a small number of bacteria, thus


Fig. 5. Mutants in a compact colony: Results of numerical simulation. (a) Mutant with a higher growth rate(continuous model). The mutant (dark) erupts in a fan-like sector from the wildtype (light) colony. Modelparameters for the wild-type: D01 = 0:2, k = 0, �1 = 1:0, = 0:0001, �1 = 0:1, mutant has �2 = 5:0. Initialnutrient level n0 = 1:0. (b) Mutant with a higher motility (continuous model). wild-type as in (a), mutanthas D02 = 0:3. (c) Mutant with a higher motility (larger step size in the communicating walker model).

Fig. 6. Mutant with a higher growth rate in a branching colony: Numerical simulation of the continuousmodel. The mutant (dark) erupts in a fan-like sector from the wildtype (light) colony. Model parameters forthe wild-type: D01 = 0:1, k = 1, �1 = 1:0, = 0:0001, �1 = 0:15, mutant has �2 = 5:0. Initial nutrient leveln0 = 1:0.

allowing an initially insigni:cant number of mutants quickly to become the majorityin some branches, and therefore in some area of the colony.Mutants superior in motility or growth rate (Fig. 6) form segregated fan-like sectors

which burst out of relatively slow advancing colony.

5.3. The e:ect of chemotaxis

As we have mentioned earlier, an additional important feature of the bacterial move-ment is chemotaxis. We :rst studied neutral mutations in the case of a colony which


Fig. 7. Mutant with a higher sensitivity to repulsive chemotactic signaling, in a branching colony: Numericalsimulation of the continuous model. The mutant erupts in a fan-like sector from the colony of wildtype bac-teria. However, the sector is not a segregated area, and contains wildtype bacteria as well. Model parametersfor the wild-type: D01 = 0:1, k = 1, �1 = 1:0, = 0:0001, �1 = 0:2, �01 = 0:5, mutant has �02 = 1:0 . Initialnutrient level n0 = 1:0.

employs repulsive chemotactic signaling. Neutral mutants increase their relative part ofthe total population in a sector of the colony. Segregation in this case might be sup-ported by the fact that branches are thinner in the presence of repulsive chemotaxis,and by the reduced mixing of bacteria because of the directed motion.A fan-like sector appears when the mutant has a higher sensitivity to the chemotactic

signals (�0 in the model), as shown in Fig. 7.Fig. 8 displays the result of Communicating Walker simulation for this case. Note

the similarity of both models’ results with experimental observations.The inHuence of chemotaxis towards food on the sectors is similar to that of repulsive

chemotaxis.

6. Discussion

Following our experimental observations and the numerical studies, we are able nowto appreciate what factors favor the segregation of the mutant population. These factorsinclude:

1. Expansion of the colony towards area of high nutrient concentration.2. Branching patterns, where the population in each branch is much smaller than the

colony’s population.


Fig. 8. Mutant with a higher sensitivity to repulsive chemotactic signaling, in a branching colony: Results ofthe communicating walker model. As in the continuous model, the mutant erupts in a fan-like sector fromthe colony of wildtype bacteria.

3. Chemotaxis: Food chemotaxis and repulsive chemotactic signaling cause the bacterialmotion to become less random and more directed.

4. An advantageous mutant, one who has a higher motility or a faster reproductionrate, will probably conquer a sector of its own and quickly become segregated.This sector will be the one bursting out of the colony, i.e., having the ability tometastasise.

Beyond the study of sectoring in bacterial colonies, intuition about the basic mech-anisms of segregation of populations might be useful for other problems, such as thegrowth of populations on a macroscopic scale. Going in this direction, we have inmind, being able to relate, biophysical assays of cell to cell adhesion and measure-ments of the motive force generated by the cytoskeleton, possibly in response to anextracellular signal, to actual motion of cells in multicellular environment. Modelswhich could accomplish this task, even if only semi-quantitative, could be used forthe study of cell migration during embryogenesis, the dynamic of wound healing andthe metastatic spread of cancerous tumors. It seems that the bursting out of sectorsin a mutant colony behave in an equivalent mechanism to the multistage carcinogen-esis. The multistage process of carcinogenesis involves the progressive acquisition ofmutations, and epigenetic abnormalities in the expression of multiple genes that havehighly diverse functions. Cancer development and progression is dictated by a seriesof alterations in genes such as oncogenes, tumor suppressor genes, DNA repair genesand others.The :nal clinical manifestation of cancer is a result of a complex series of changes

in a single cell. The emphasis is on cancer as a disease of the stem cells within atissue that undergoes initiation as a result of mutational insult to one or more genesthat are critical for cell growth. During the second stage (promotion stage) the initiated


cells (the new sectors) acquire aproliferative capacity due to epigenetic changes, i.e.,altered expression of genes whose products play a central role in signal changes. Thefollowing stage is the stage of progression, where the neoplastic lesions acquire addi-tional genetic alteration and become clinically manifestable malignant neoplasms. Thisstage is characterized by the four factors as the factorial mutant population: (1) abilityto grow and expand locally and regionally; (2) change in the pattern of the growth fromnormal to new pattern, i.e., abnormal; (3) higher motility mainly due to faster repro-duction rate, and (4) chemotaxis or aInity to grow and to seed themselves elsewhere,i.e., to metastasize.

Acknowledgements

We are thankful to I. Brainis and R. Segev for technical assistance.

References

[1] E. Ben-Jacob, H. Shmueli, O. Shochet, A. Tenenbaum, Adaptive self-organization during growth ofbacterial colonies, Physica A 187 (1992) 378–424.

[2] E. Ben-Jacob, A. Tenenbaum, O. Shochet, O. Avidan, Holotransformations of bacterial colonies andgenome cybernetics, Physica A 202 (1994) 1–47.

[3] E. Ben-Jacob, O. Shochet, A. Tenenbaum, Bakterien schlieRen sich zu bizarren formationen zusammen,in: A. Deutsch (Ed.), Muster des Ledendigen: Faszination inher Entstehung und Simulation, VerlagVieweg, Braunschweig, 1994.

[4] M. Dworkin, Developmental Biology of the Bacteria, Benjamin/Cummings Publishing Company,Reading, MA, 1985.

[5] J.A. Shapiro, Bacteria as multicellular organisms, Sci. Am. 258 (6) (1988) 62–69.[6] E.O. Budrene, H.C. Berg, Complex patterns formed by motile cells of Esherichia coli, Nature 349

(1991) 630–633.[7] J.A. Shapiro, M. Dworkin (Eds.), Bacteria as Multicellular Organisms, Oxford University Press, New

York, 1997.[8] E. Ben-Jacob, From snowHake formation to the growth of bacterial colonies. Part II: Cooperative

formation of complex colonial patterns, Contemp. Phys. 38 (1997) 205–241.[9] E. Ben-Jacob, H. Levine, The artistry of microorganisms, Sci. Am. 279 (4) (1998) 82–87.[10] J.A. Shapiro, Thinking about bacterial populations as multicellular organisms, Ann. Rev. Microbiol. 52

(1998) 81–104.[11] E. Ben-Jacob, I. Cohen, H. Levine, Cooperative self-organization of microorganisms. Adv. Phys. 49

(2000) 395.[12] E. Ben-Jacob, I. Cohen, D. Gutnick, Cooperative organization of bacterial colonies: from genotype to

morphotype, Annu. Rev. Microbiol. 52 (1998) 779–806.[13] Y. Blat, M. Eisenbach, Tar-dependent and -independent pattern formation by Salmonella typhimurium,

J. Bacteriol. 177 (7) (1995) 1683–1691.[14] E.O. Budrene, H.C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,

Nature 376 (1995) 49–53.[15] M. Tcherpakov, E. Ben-Jacob, D.L. Gutnick, Paenibacillus dendritiformis sp. nov., proposal for a new

pattern-forming species and its localization within a phylogenetic cluster, Int. J. Syst. Bacteriol. 49(1999) 239–246.

[16] M.A. Marahiel, M.M. Nakano, P. Zuber, Regulation of peptide antibiotic production in Bacillus, Mol.Microbiol. 7 (1993) 631–636.


[17] J.D. Desai, I.M. Banat, Microbial production of surfactants and their commercial potential, Microbiol.Mol. Biol. Rev. 61 (1997) 47–64.

[18] E. Ben-Jacob, I. Cohen, A. CzirUok, T. Vicsek, D.L. Gutnick, Chemomodulation of cellular movementand collective formation of vortices by swarming bacteria and colonial development, Physica A 238(1997) 181–197.

[19] M. Tcherpakov, E. Ben-Jacob, I. Cohen, D.L. Gutnick, Paenibacillus vortex sp. nov., proposal for anew pattern-forming species and its localization within a phylogenetic cluster, Int. J. Syst. Bacteriol. 49(1999) 239.

[20] J.A. Shapiro, Organization of developing Escherichia coli colonies viewed by scanning electronmicroscopy, J. Bacteriol. 169 (1989) 142–156.

[21] J.A. Shapiro, C. Hsu, Escherichia coli K-12 cell–cell interactions seen by time-lapse video, J. Bacteriol.171 (1989) 5963–5974.

[22] N.P. Higgins, Death and trans:guration among bacteria, Trends. Biol. Sci. 17 (1992) 207–211.[23] J.A. Shapiro, DiLerential action and diLerential expression of DNA polymerase I during Escherichia

coli colony development, J. Bacteriol. 174 (1992) 7262–7272.[24] E. Boschke, Th. Bley, Growth patterns of yeast colonies depending on nutrient supply, Acta Biotechnol.

18 (1) (1998) 17–27.[25] L.M. Sander, T.S. Deisboeck, Growth pattern of microscopic brain tumors, Phys. Rev. E (2002), in

press.[26] I. Rafols, Formation of concentric rings in bacterial colonies, M.Sc. Thesis, Chuo University, Japan,

1998.[27] J.A. Shapiro, Adaptive mutation: Who’s really in the garden, Science 268 (1995) 373–374.[28] A. Grondin, H.C. Jarell, L.R. Berube, Discontinuous expansion linked to sector formation in

Pseudomonas aeruginosa colonies, Arch. Microbiol. 172 (1999) 59–62.[29] I. Golding, I. Cohen, E. Ben-Jacob, Studies of sector formation in expanding bacterial colonies,

Europhys. Lett. 48 (5) (1999) 587–593.[30] I. Golding, Y. Kozlovsky, I. Cohen, E. Ben-Jacob, Studies of bacterial branching growth using

reaction-diLusion models of colonial development, Physica A 260 (3-4) (1998) 510–554.[31] S. Kitsunezaki, Interface dynamics for bacterial colony formation, J. Phys. Soc. Jpn. 66 (5) (1997)

1544–1550.[32] E. Ben-Jacob, O. Shochet, A. Tenenbaum, I. Cohen, A. CzirUok, T. Vicsek, Generic modeling of

cooperative growth patterns in bacterial colonies, Nature 368 (1994) 46–49.[33] E. Ben-Jacob, I. Cohen, O. Shochet, A. CzirUok, T. Vicsek, Cooperative formation of chiral patterns

during growth of bacterial colonies, Phys. Rev. Lett. 75 (15) (1995) 2899–2902.[34] D.A. Kessler, H. Levine, Fluctuation-induced diLusive instabilities, Nature 394 (6) (1998) 556–558.

Bursts of sectors in expanding bacterial colonies as a possible model for tumor growth and...

Documents

Transcript of Bursts of sectors in expanding bacterial colonies as a possible model for tumor growth and...