Theory of periodic swarming of bacteria: Application to Proteus mirabilis

PHYSICAL REVIEW E, VOLUME 63, 031915

Theory of periodic swarming of bacteria: Application to Proteus mirabilis

A. Czirok,1,2,* M. Matsushita,2,3 and T. Vicsek1,2

1Department of Biological Physics, Eo¨tvos University, 1117 Budapest, Pa´zmany stny 1A, Hungary2Institute for Advanced Study, Collegium Budapest, 1014 Budapest, Szentha´romsag u 2, Hungary

3Department of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan~Received 28 July 2000; published 27 February 2001!

The periodic swarming of bacteria is one of the simplest examples for pattern formation produced by theself-organized collective behavior of a large number of organisms. In the spectacular colonies ofProteusmirabilis ~the most common species exhibiting this type of growth!, a series of concentric rings are developedas the bacteria multiply and swarm following a scenario that periodically repeats itself. We have developed atheoretical description for this process in order to obtain a deeper insight into some of the typical processesgoverning the phenomena in systems of many interacting living units. Our approach is based on simpleassumptions directly related to the latest experimental observations on colony formation under various condi-tions. The corresponding one-dimensional model consists of two coupled differential equations investigatedhere both by numerical integrations and by analyzing the various expressions obtained from these equationsusing a few natural assumptions about the parameters of the model. We determine the phase diagram corre-sponding to systems exhibiting periodic swarming, and discuss in detail how the various stages of the colonydevelopment can be interpreted in our framework. We point out that all of our theoretical results are inexcellent agreement with the complete set of available observations. Thus the present study represents one ofthe few examples where self-organized biological pattern formation is understood within a relatively simpletheoretical approach, leading to results and predictions fully compatible with experiments.

DOI: 10.1103/PhysRevE.63.031915 PACS number~s!: 87.10.1e, 87.18.Hf

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I. INTRODUCTION

To gain insight into the development and dynamicsvarious multicellular assemblies, we must understand hcellular interactions build up the structure and result in ctain functions at the macroscopic, multicellular level. Microrganism colonies are one of the simplest systems consisof many interacting cells and exhibiting a nontrivial macrscopic behavior. Therefore, a number of recent studiescused on experimental and theoretical aspects of colonymation and the related collective behavior of microorganis@1,2#.

Theswarmingcycles exhibited by many bacterial specienotablyProteus (P.) mirabilis, have been known for overcentury@3#. WhenProteuscells are inoculated on the surfacof a suitable hard agar medium, they grow as short‘‘vegeta-tive’’ rods. After a certain time, however, cells start to dferentiate at the colony margin into long‘‘swarmer’’ cellspossessing up to 50 times more flagella per unit cell surfarea. These swarmer cells migrate rapidly away fromcolony until they stop, and revert by a series of cell fissiointo the vegetative cell form, in a process termedconsolida-tion. The resulting vegetative cells grow normally for a timthen swarmer cell differentiation is initiated in the outermozone~terrace!, and the process continues in periodic cycresulting in a colony with concentric zonation depicted in F1. A similar cyclic behavior was observed in an increasnumber of Gram-negative and Gram-positive genera incing Proteus, Vibrio, Serratia, Bacillus, andClostridium ~forreviews, see Refs.@4–6#!.

*Email address: [email protected]

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The reproducibility and regularity of swarming cycles, tgether with the finding that its occurrence is not limited tosingle species, suggest that periodic swarming phenomcan be understood and quantitatively explained on the bof mathematical models. In this paper we first give an ovview of the relevant experimental findings related to tswarming ofP. mirabilis, then construct a simple model wittwo limit densities. We then investigate the behavior of tmodel as a function of the control parameters and compato experimental results.

II. OVERVIEW OF EXPERIMENTAL FINDINGS

A. Differentiation

As reviewed in Refs.@5,6#, the differentiation of vegeta-tive cells is accompanied by specific biochemical changSwarmer cells enhance the synthesis of flagellar proteextracellular polysaccharides, proteases, and virulencetors, while they exhibit reduced overall protein and nucleacid synthesis and oxygen uptake. These findings mayexplained by arguing that the production and operationflagella is expensive and may require the repression of nessential biosynthetic pathways. The largely~10–30 fold!elongated swarmer cells develop by a specific inhibitioncell fission which seemsnot to affect the doubling time ofthe cell mass or DNA.

The differentiation process is initiated by a numberexternal stimuli, including specific signaling molecules aphysicochemical parameters of the environment. As anample of the latter, theviscosityof the surrounding mediumis presumably sensed by the hampered rotation of the flag@7,8#. Neither the signal molecules that initiate the differetiation nor the involved intracellular signaling pathways ha

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A. CZIROK, M. MATSUSHITA, AND T. VICSEK PHYSICAL REVIEW E 63 031915

been identified, but a corresponding transmembrane recewas found recently@9#. The structure of this receptor, together with other findings reviewed in Ref.@6#, suggest a‘‘quorum sensing’’ regulatory pathway@10#, characteristic ofmany, cell density dependent collective bacterial behavlike sporulation, luminescence, production of antibiotics,virulence factors@11#.

B. Migration of swarmer cells

It is well established@5# that swarmer cell migration doenot require exogenous nutrient sources, since swarmerreplated onto media devoid of nutrients continue to migrnormally. The ability of migration depends on the locswarmer cell density as isolated single swarmer cells canmove, while a group of them can. It was also demonstra@12# that the mechanism by which bacteria swarm outwinvolved neither repulsive nor attractive chemotaxis. Ttypical swimming velocity~i.e., in a liquid environment! ofswarmer cells is'100 mm/h@13# which is also their maxi-mal swarming speed and rate of colony expansion onagar plates@5#. In typical experimental conditions for investigating swarming colony formation, the front advances wa speed of 0.5–10 mm/h@14,9,15,13#. Unfortunately, inthese cases there is no information available on the veloof individual swarmer cells, but it must be between tcolony expansion speed and the swimming velocity.

C. Consolidation

The molecular mechanisms of consolidation, i.e.,downregulation of the gene activity responsible for swaring behavior@16#, is even less known than that of differentiation. If the swarming motility utilizes intracellular energreserves as has been suggested@5#, then swarmer cells mushave afinite lifetime. In addition to the septation of swarmecells taking place at the outermost terrace, inside the colthe differentiation process, i.e., the supply of fresh swarmcells must also be shut off. The cessation of swarmerproduction does not seem to be due to severe nutrient detion since vegetative cells keep dividing~although with de-

FIG. 1. TypicalProteus mirabiliscolony. It was grown on thesurface of a 2.0% agar substrate for two days. The inner diametthe petri dish is 8.8 cm. Gray shades are proportional to thedensity: the cyclic modulation is apparent.

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creasing growth rate! well inside the colony for many hourafter the last swarmer cells were produced in that reg@14#.

D. Colony formation

The cycle time~total length of the migration and consoldation periods! has been found@14# to be rather stable('4 h) for a wide range of nutrient and agar content of tmedium. The size of the terraces and the duration ofmigration phases were strongly influenced~up to an order ofmagnitude, and up to a factor of 3, respectively! by the agarhardness. The nutrient concentration did not have an obsable effect on these quantities from 0.01% up to 1% glucconcentrations. There was, however a remarkable poscorrelation between the cycle time and the doubling ti~ranging from 0.7 up to 1.8 h) of the cells.

Two interfacing colonies inoculated with a time diffeence of a few hours, and therefore being in different phaof the migration-consolidation cycle, were found to maintatheir characteristic phases@14,15#. Thus, the control of theswarming cycle must be sufficiently local.

E. Cycle rescheduling

A few experiments investigated the cell density depedence of the duration of quiescent growth~lag phase! priorto the first migration phase. These studies clearly reveathat vegetative cells have to reach a threshold density totiate swarming@14,15#, in accord with the suggested quorusensing molecular pathway of the initiation of swarmer cdifferentiation.

Agar cutting experiments demonstrated that a cut insthe inner terraces does not influence the swarming acti@15#. However, when the cut was made just behindswarming front, the duration of the swarming phase wshortened and consolidation was lengthened by up to 4@15#.

Even more interestingly,mechanical mixingof the cellpopulations before the expected beginning of consolidaexpands the duration of the swarming phase considerablyup to 50% @13#. This finding, together withreplicaprintingexperiments@13# demonstrates that at the beginning of tconsolidation phase a large pool of swarmer cells still exiand seems to be ‘‘trapped’’ at the rear of the outermostrace.

III. MODEL

Taking into account the above described experimenfindings, here we construct a model which is capable ofplaining most of the observed features of colony expansthrough swarming cycles.

~i! The model is based on vegetative and swarmerpopulation densitiesonly, denoted byr0 and r* , respec-tively. These values are defined on the basis of cell minstead of cell number; therefore, one unit of swarmer celltransformed into one unit of vegetative cells during consodation.

~ii ! Vegetative cells grow and divide with a constant ra

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THEORY OF PERIODIC SWARMING OF BACTERIA: . . . PHYSICAL REVIEW E63 031915

r 0'1 h21 @14,15#. This will later allow us to establish adirect correspondence between cell density increaseelapsed time.

~iii ! Usually, swarmer cell differentiation is initiated whethe local density of the vegetative cells exceeds a thresvalue (rmin

0 '1022 cells/mm2 @14,15#!. ~Before the begin-ning of the first swarming phase, experiments indicatedpresence of an extra time periodt l which is probably associated with the biochemical changes required to developability of the swarming transition. This effect is present onat the seeding of the colony, thust l50 otherwise.! Whenr05rmin

0 at time t0, some of the vegetative cells enter thdifferentiation process, modeled by introducing a rater.Since the biomass production rate is assumed to bechanged during the differentiation process, the rate of pducing new vegetative cells isr 02r , and the differentiatingcells elongate with the normal growth rater 0.

~iv! The full development of swarmer cells, i.e., a typic20-fold increase in length needs a time (td' ln 20/r 0'3 h)comparable with, or even longer than the length of a consdation period, hence cannot be neglected. Therefore, the‘‘real’’ swarmer cells, which are able to move appear onlyt01td .

~v! The production of swarmer cells is limited in timeand the lengtht of the time interval during which vegetativcells can enter the differentiation process is another phenenological parameter of our model. As here we focus onperiodicity of colony expansion, we do not consider whhappens in densely populated regions aftert01t. Specifi-cally, in our model any activity of the vegetative cells ceasin these parts of the system.

~vi! Swarmer cells can migrate only if their density eceeds a threshold densityrmin* . Above this threshold,swarmer cells are assumed to move randomly with a dision constantD0. Both of these parameters are thought todetermined by the quality of the agar substrate. The finlifetime of swarmer cells is incorporated into the modthrough a constant rate (r * ) decay.

Unfortunately there are no good estimates on theserameters in the literature. According to recent experimenobservations@17#, rmin* is less than 10% of the value ofr0

prior to the beginning of the swarming phase, i.0.1rmin

0 er 0td'1022 cells/mm2. D0 may be estimated av0

2tp/2 with v0 being the typical speed of swarming cells atp being the persistence time of their motion. Asv0 is ap-proximately 30 mm/h~see Sec II B! andtp is in the order ofminutes,D0 is estimated to be in the order of 10 mm2/h.

The above considerations lead to the set of equations

r0~ t !5r 0r0~ t !1G~ t !2G* ~ t !,~1!

r* ~ t !52G~ t !1G* ~ t2td!er 0td1¹D~r* !¹r* ,

where the consolidation (G) and differentiation (G* ) termsare given by

G5r * r* and G* 5r ~r0,t !r0. ~2!

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Equations~1! can be significantly further simplified by making use of the possibility to measure elapsed time withincrease inr0. In particular, neglecting correction terms rlated to consolidation in areas wherer0.rmin

0 , i.e., whereprimarily differentiation takes place, we can castG* (t2td)er 0td in the form

rr0~ t2td!er 0td5rrmin0 e(r 02r )(t2td2t0)er 0td

5rrmin0 er 0(t2t0)e2r (t2t02td)5rr0~ t !ert d

~3!

for t.t01td[t1. Let us introduce a transformed populatiodensityr as

r~ t !5H r0~ t ! for r0,rmin0 , i.e., for t,t0

r0~ t !er (t2t0) for r0.rmin0 and t,t1

r0~ t !ert d for r0.rmin0 and t.t1 ,

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which is in fact the total density of the vegetative and tdifferentiating, but not yet fully differentiated swarmer cel~see Fig. 2!. With this notation, using Eq.~3! and similar

FIG. 2. Schematic representation of the model for swarmcolony formation.~a! The two basic quantities are the vegetatiand swarmer cell densities, which can be transformed into eother, and changed by proliferation and motility. The dotted linrepresent regulatory~threshold! effects: The rate of differentiationis assumed to be dependent on the vegetative cell density, anmotility of swarmer cells is also determined by their local densi~b! Notations and typical time courses of vegetative and swarcell densities prior to the beginning of the first migration phaset,t0, the densityr ~or ro) grows at a constant rater 0. After reach-ing the density thresholdrmin

o , vegetative cells keep growing onlwith a rater 02r , but the total density (r) of the vegetative cellsand the differentiating swarmer cells continues to grow with a rr 0. After the time required for the full elongation of a swarmer c~whenr reachesrmin), r* becomes positive, and forr * 50 wouldgrow asymptotically with a rater.

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considerations fort0,t,t01td , Eqs.~1! can be written intosimple, not retarded forms

r5r 0r1r * r* 2r ~r!r,~5!

r* 52r * r* 1r ~r!r1¹D~r* !¹r* ,

where

r ~r!5H r for rmin,r,rmax

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with rmin5ert drmin0 '1021 cells/mm2 and rmax(t)

5e(r 02r )trmin . Thus t and rmax mutually dermine eachother. If the G!(r 02r )r condition does not hold in the@ t0 ,t1# time interval, thenrmin must also be treated asdynamical variable. This case will not be considered herethe following we user for the characterization of vegetativcell density, andrmax instead oft.

IV. RESULTS

A. Numerical method in one dimension

The model defined through Eqs.~5! has the followingseven parameters: the ratesr 0 , r, andr * ; threshold densitiesrmin , rmax ~or t), andrmin* ; and diffusivity D0. However,this number can be reduced to four by casting the equatin a dimensionless form using 1/r 0'1 h as time unit,rmin

'0.1 cells/mm2 as density unit andx05AD0 /r 0'3 mmas the unit length. The resulting control parameters arer /r 0 ,r * /r 0 , rmax/rmin5exp@(r02r)t#, andrmin* /rmin . To obtaincontinuous density profiles, the step-function dependencD on r* was replaced by

D~r* !5D0

2 F11tanh 2ar* 2rmin*

rmin* G , ~7!

with a510 providing a rather steep, but continuous croover. The following results do not depend on the particuchoice of Eq.~7!.

Representative examples of the time development ofmodel are shown in Fig. 3. The production of swarmer ceis localized, and determined by the density profile of vegetive cells at the end of migration periods. In this particumodelr(x) is decreasing toward the colony edge; therefoin the migration phases the source of swarmer cells is ming outward. The front of swarmer cells is expanding frothe inside of the last terrace. Because of the decay termG,cells become nonmotile first at the colony edge.

B. Phase diagram

Each of the dimensionless control parameters can havimportant effect on the dynamics of the system. As anample, if the durationt of swarmer cell production is increased, then the consecutive swarming cycles are not srated and a continuous expansion takes place with damoscillations@Fig. 3~b!#. To map the behavior of the systema function of the control parameters, the following proced

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was applied. Migration periods were identifed by requirimaxx r* (x).rmin* . For a given set of parameters we detemined the lengths$t i% of the consecutive migration periodsand the system was classified asperiodic if the three largestvalues of the set$t i% were the same within 20%. Otherwisthe expansion was classified ascontinuousas long as maxi$ti%was large enough: comparable with the total duration ofsimulated expansion.

The behavior of the model is summarized in Fig. 4, whethe boundaries of the various regimes are plotted for thdifferent values ofr * /r 0. We found, thatr andrmax can becombined into one relevant parameter, the swarmer cell pduction density, as

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r 02r~rmax2rmin!, ~8!

which quantity does not depend on the choice of positionx.As the inset demonstrates, for a givenP, the actual values ofr or rmax are irrelevant to this kind of classification in thparameter regime investigated. The general structure ofphase diagram was found to be similar for various valuesr * . For large enoughP or low enoughrmin* a continuousexpansion takes place, while for too smallP or largermin* theexpansion of the system is finite. For intermediate valuesthese parameters an oscillating growth develops, exhibivery distinguishable consolidation and migration phases.the lifetime of the swarmer cells is increased, the paramregime, in which periodic behavior is exhibited, shrinks ais moved toward lowerP values.

One can easily estimate the position of the boundarythe nongrowing phase based on that~i! the width w of theterraces is small~this assumption is justified later, in Fig. 7!;thus ~ii ! the time required for the diffusive expansion of thswarmer cells is much shorter than their lifetime, which,turn, is ~iii ! shorter than the duration of a swarming cycr * /r 0;1. The amount of swarmer cells produced in operiod is Pw. Neglecting the decay during expansion, twidth w8 of the next, new terrace can be determined fromconservation of cell number as

2rmin* w85w~P1r r* 2rmin* !, ~9!

wherer r* denotes the swarmer cell density remaining frothe previous swarming cycle and the symmetric expansiothe released swarmers was also taken into account.achieve a sustainable growthw8>w is required, resulting ina condition 3rmin* <P1r r* . If r r* !P, as one can expect for * /r 0;1, for the boundary of the nongrowing phase we otain

P53rmin* , ~10!

which, as Fig. 4 demonstrates, is indeed in good agreemwith the numerical data.

C. Terrace formation

The average lengthT of a full swarming cycle was calculated by determining the position of the peak in the pow

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FIG. 3. Time development of the model obtained by numerical integration of the equations, starting from a localized ‘‘inoculumt50 andx50. The continuous line represent the colony boundary@maximal value ofx for which r(x)1r* (x).0#. The filled gray and blackareas are regions where swarmer cells are motile, and where swarmer cells are produced, respectively. Forrmax51.3, rmin* 50.01, r 50.3,andr * 51.0, the expansion of the system is clearly periodic~a!. If we increase the production of swarmer cells by increasingrmax to 2.0 thenthe periodicity is gradually lost and a continuous expansion takes place~b!. The density profiles of the vegetative and swarmer cpopulations are plotted for consecutive time points in~c! and~d!, respectively. CurvesA–F correspond tor 0t544,45, . . . 49,i.e., a completeswarming cycle.

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total number of swarmer cells in the system. As Fig. 5 deonstrates, the dimensionless cycle time values are widspread between values of 3 and 12. However,T is only sen-sitive to changes inr 0 , r * , andrmin* /rmin ; hence it does nodepend onrmax or t.

The average expansion speedv and terrace sizew werealso calculated in the parameter regime resulting oscillaexpansion of the colony. First we determined the timet1/3when the system reached one-third its maximal simulaexpansionRmax5R(tmax), with R(t) being the position ofthe expanding colony edge andtmax is the total duration ofthe simulation. The average speed was then calculated fotime interval betweentmin5max(t1/3,tmax25T) and tmax:for the last 5T long time interval, or for the last two-thirds othe total expansion, depending on which was smaller. Aobtaining v as @Rmax2R(tmin)#/(tmax2tmin), the averageterrace width was calculated asw5vT. Figure 6 shows thedependence of these parameters on the swarmer cell pro

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tion densityP and migration density thresholdrmin* . In gen-eral, decreasingrmin* or increasingP results in an increase inboth w and v. As Fig. 7 demonstrates, for a givenr * , therelevant parameter controllingw is P/rmin* .

The results on the cycle timeT ~Fig. 5! can be interpretedas follows. Asrmin* is a good estimate on the densityswarmer cells in the expanding front, at the end of migratphase the vegetative cell density within the new terracegiven by r * rmin* T* , with T* being the duration of the mi-gration phase. Now the length of the consolidation phaT2T* , is determined by the requirement thatr must reachrmin :

rmin5r * rmin* T* er 0(T2T* )5r * rmin*T*

er 0T* er 0T. ~11!

As T* '1/r 0, estimate~11! is simplified to

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FIG. 4. Phase diagram of the system as the function of cumtive swarmer cell production densityP and migration thresholdrmin*for various values of swarmer cell decay rater * . If the productionis high enough or the motility threshold is low enough, then cotinuous expansion can be observed. On the other hand, if theduction is too low or the motility threshold is too high, then nexpansion takes place. In an intermediate regime periodic grocan be observed. The boundaries of this parameter regime areted for r * /r 051.0 ~thick continuous line!, 0.4 ~thick dashed line!,and 0.1~thick dotted line!. The thin continuous line represents aapproximate upper bound~10! for cyclic colony expansion. Theinset demonstrates that the fourth parameter of the model,r, isirrelevant: forr * /r 051 andr /r 050.01 (h), 0.3 (s), 0.5 (n) andvarious values ofrmax andrmin* the type of colony expansion waclassified. Open symbols correspond to cyclic growth, filled sybols correspond to continuous growth, and dots denote no exsion. Note that the corresponding regions completely overlap ispective of the value ofr.

FIG. 5. The cycle timeT as a function of the approximate consolidation rater * rmin* . The data collapse indicates that the impaon T of the other parameters (rmax andr ) is negligible. The varioussymbols correspond to different values ofr * /r 0 as 0.1 (1), 0.2(3), 0.3 (h), 0.4 (s), 0.5 (n), 0.7 (,), and 1.0 (L). Thedashed line is a plot of Eq.~12!. Circles mark out the assumeparameter values characteristic of four differentP. mirabilisstrains.

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r 0T5 lnr 0rmine

r * rmin*, ~12!

which gives a rather accurate fit to the numerically detmined data~Fig. 5!.

D. Lag phase

Since the duration (TL) of the lag phase~the time periodbefore the first migration phase! has been in the focus omany recent experiments, we now turn our attention towthis quantity. At least four processes determineTL . First,there is a timet l associated with the biochemical changrequired to switch into swarming mode. As discussed in SII A, these processes take place only prior the first swarmphase, and are presumably related to sensing the alterevironmental conditions. Second, the cell population mreach the threshold densityrmin

0 ~at timet0). Third, td time is

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FIG. 6. Terrace size~a! and expansion speed~b! of the systemfor r * /r 051 as a function of the dimensionless swarmer cell pduction densityP. The connected points correspond to various vues of the migration thresholdrmin* /rmin . In general, decreasingrmin* or increasingP results in an increase in bothw and v. Onlyparameters resulting in periodic expansion were investigaCircles mark out the assumed parameter values characteristfour differentP. mirabilis strains.

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required to produce fully differentiated swarmer cells~attime t1). Finally, the density of the swarmer cells must reathe migration thresholdrmin* . Let us investigate how thesparameters depend on the initial inoculum densityr0.

As r grows with a rater 0 until the appearance of swarmecells,

r 0t15max@ t l1td , ln~rmin /r0!#. ~13!

The time development ofr* can be estimated by the integration of Eq.~5! with r * 50 ~i.e., assumingG!G* ), yield-ing

r* ~ t1t1!5r

r 02rr0er 0t1@e(r 02r )t21#. ~14!

Therefore,t* 5TL2t1 is given by

r 0t* 51

12r /r 0lnS r 02r

r

rmin*

r0er 0t111D , ~15!

an expression usually giving a minor correction tot1.Figure 8~a! shows the above calculatedr 0TL vs r0 for t l

50. The increase in length of the swarmer cells was assuto be 20-fold; thusr 0td5 ln 20'3, which value can be seefor r0@rmin . In the opposite limit, whenr0!rmin , we haver 0TL'2 ln r01const. These relations allow the determintion of both r 0 and r ~using the known value ofrmin /rmin* )from the experimental data onTL(r0).

E. Comparison with experiments

Most of the published experimental data are related toaverage period lengthT and terrace sizew. From these pa-rameters the average expansion speed can be calculatv5w/T, i.e.,v is not an independent quantity. As we sawSec. IV D, from the density dependence of the lag phaseparametersr 0 , t l1td , andr can be estimated. Note that thestimate onr 0 is in principle different from the value ob

FIG. 7. Terrace sizew vs P/rmin* for various values ofr * /r 0.Note the collapse of the data presented in Fig. 6~a!. The terrace sizevanishes approaching the parameter regime where no sustaiexpansion of the system is possible.

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tained by the usual methods based on densitometry in liqcultures. Technically,rmin* could be also determined@14#,but such measurements are not yet published.

There are fourProteusstrains studied systematically iexperiments: the PRM1, PRM2, BB2000, and BB22strains~see Table I!. To extract the values of the model’parameters the following procedure was applied.~i! We es-timated r 0 based on lag phase length measurements.~ii !From the calculatedr 0T values thermin* /rmin ratio was esti-mated~assumingr * /r 051) based on Eq.~12!; see Fig. 5.~iii ! Using Eqs.~14! and ~15!, by a nonlinear fitting proce-dure @Levenberg-Marquardt method@18#; see Fig. 8~b!#,rmin , r, andt l1td were determined.~The latter value is notrelevant in respect the periodicity of the behavior.! ~iv!Knowing rmin* and rmin , from the experimental terracwidth datax0 and P can be estimated using Fig. 6.~v! Fi-nally, t is given by

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FIG. 8. ~a! Length TL of the lag phase vs the initial inoculumdensityr0 for t l50, r 0td53, and various values ofr /r 0 ~0.3, 0.1,0.03, 0.01, 0.003, and 0.001!. The transition between the regimeseparated byrmin

o becomes more smooth for decreasingr /r 0. ~b!The experimentally calculatedr 0TL values vs the estimated inoculum density~based on 5 and 7-mm inoculum droplet sizes for tRPM and BB strains, respectively! and the corresponding fits usinEqs.~13! and ~15!.

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TABLE I. Summary of experimental data for four differentP. mirabilis strains under various experimental conditions. The value ofr 0 was determined from growth monitoring in liquid cultures.

Strain PRM1 PRM2 BB2000 BB2235

Experimental Temperature 32 °C 32 °C 32 °C 37 °C 22 °C 32 °C 37 °C 37 °condition Agar 2.0% 2.45% 2.0% n.a. n.a. 2.0% n.a. n.a.

Reference @14# @14# @15# @14# @14# @14# @9# @9#

T ~h! 4.7 4.7 4.0 3.5 8.5 6.0 3.0 3.1Colony level v ~mm/h! 1.7 0.6 1.0 n.a. n.a. n.a. 3.3 3.3

w ~mm! 8.0 3.0 3.8 n.a. n.a. n.a. 10 10

Cellular level r 0 ~1/h! 0.6 0.6 n.a. 1.0 0.4 n.a. n.a. n.a.

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The parameter values of the model are summarized in TII, together with the predictions onT, v, andw. An excellentagreement can be achieved with biologically relevant pareter values.

Two classes of model parameters should be distinguis~a! those which are related to the growth and differentiatof the cells@r 0 , r * , rmin , andP(r ,t)#, and~b! those whichdepend on agar softness (D0 andrmin* ). For a given strain weexpect that a change in the agar concentration influen

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only the latter group, while changes in temperature mayfect both, but primarilyr 0. In fact, as Table II demonstratesby changingD0 and keeping all the other, growth-relateparameters constant, we couldquantitativelyreproduce thecolony behavior observed on various agar concentrationssimilar statement holds for the temperature effects as wwhere the only parameter we changed was the growthr 0.

V. DISCUSSION

Periodic bacterial growth patterns have been in the foof research in the last few years. Since a colony can

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TABLE II. Model parameters and the corresponding results for the strains and experimental conditions specified in Table I. Thhas seven microscopic parameters, the ratesr 0 , r, andr * , the threshold densitiesrmin , rmin* , andrmax, and the diffusivityD0. For each ofthe strainsr * 5r 0 was assumed. For comparison, other~derived! microscopic parameters are also included. The calculated period lenterrace sizes, and expansion speeds are also presented together with the corresponding experimental values~in parentheses!. The valuesmarked by an asterisk (*) were derived from the lag phase length data based on Eqs.~13! and~15!. Note the similarity between the PRMand BB2235, and also between the PRM2 and BB2000 strains.

Strain PRM1 PRM2 BB2000 BB2235

Experimental Temperature 32 °C 32 °C 32 °C 37 °C 22 °C 32 °C 37 °C 37 °condition Agar 2.0% 2.45% 2.0% n.a. n.a. 2.0% n.a. n.a.

Microscopic r 0 ~1/h! 0.53* ~0.6! 0.53* ~0.6! 0.7* 1.0 0.4 1.0* 2.5* 1.5*parameters rmin (cell/mm2) 0.06 0.6 2.0 0.2~independent! rmin* (cell/mm2) 631023 631023 431023 431023

rmax (cell/mm2) 0.6 240 3000 2.2r /r 0 1021 1024 231025 331022

D0 (mm2/h) 20 3.2 6 – – – 60 40

~derived! rmin0 (cell/mm2) 331023 331022 1021 1022

P/rmin 0.9 0.04 0.03 0.3t/T 0.7 0.9 0.9 0.3

rmin* /rmin 1021 1022 231023 231022

x0 ~mm! 6 2.3 3.0 – – – 5.0 5.0v0 ~mm/h! 50 20 27 – – – 85 70

Macroscopic T ~h! 5.5 ~4.7! 5.5 ~4.7! 4.7 ~4.0! 3.3 ~3.5! 8.2 ~8.5! 5.6 ~6.0! 2.9 ~3.0! 3.2 ~3.1!behavior v ~mm/h! 1.4 ~1.7! 0.5 ~0.6! 0.8 ~1.0! n.a. n.a. n.a. 3.4~3.3! 3.1 ~3.3!

w ~mm! 7.8 ~8.0! 3.0 ~3.0! 3.9 ~3.8! n.a. n.a. n.a. 10~10! 10 ~10!

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viewed as a system where diffusing nutrients are conveinto diffusing bacteria, one may not be surprised byemergence of spatial structures@19#. However, the periodicpatterns of bacterial colonies are qualitatively different frothe Liesegang rings~for a recent review, see Ref.@20#! de-veloping in reaction-diffusion systems: the spacing betwthe densely populated areas is uniform and independenthe concentration of the other diffusing species, i.e., thetrients. The Turing instability is also well known for produing spatial structures@21#, but in this case the patteremerges simultaneously in the whole system. It is aknown that bacteria can aggregate in steady concentricstructures as a consequence of chemotactic interac@22,23#, but, as discussed in Sec. II., it is established tswarming ofP. mirabilis does not involve chemotaxis communication. Therefore, none of the well known generic ptern forming schemes can explain the colony structureswarming bacteria.

As we mentioned in Sec. I, oscillatory growth is also ehibited by other bacterial species. One of these,Bacillus sub-tilis, has been the subject of systematic studies on colformation, and a number of models have been constructeexplain the observed morphology diagram~for recent re-views, see Refs.@24,25#!. Only one model addressed thproblem of migration and consolidation phases: Mimuet al. @26# set up a reaction-diffusion system in which thdecay rate of the bacteria was dependent both on theircentration and the locally available amount of nutrients. Tperiodic behavior is then a consequence of the followcycle: if nutrients are used up locally, then the bacterial dsity starts to decay, preventing the further expansion ofcolony. Nutrients diffuse to the colony and accumulate dto the reduced consumption of the already decreased plation. The increased nutrient concentration gradually alloan increase in population density and an expansion ofcolony, which starts the cycle from the beginning. While thcan be a sound explanation forB. subtilis, as we discussed inSec. II, the nutrient limitation clearly cannot explain neiththe differentiation nor the consolidation ofP. mirabilisswarmer cells.

Another recent study@27# focused on the swarming oSerratia liquefaciens. In this case the structure of the molecular feedback loops are better explored, and were resoin the model. The production of a wetting agent was initiaby high concentrations of specific signalling molecules. Tcolony expansion was considered to be a direct consequof the flow of the wetting fluid film, in which process thonly effect of bacteria~besides the aforementioned prodution! was changing the effective viscosity of the fluid. Thwetting agent production was downregulated through a netive feedback loop involving swarmer cell differentiatioThis scenario is certainly not applicable toP. mirabilis,where swarmer cells actively migrate outward and their ris quite the opposite: enhancing the expansion of the colo

The first theoretical analysis focusing onP. mirabilis wasperformed by Esipov and Shapiro~ES! in Ref. @28#. Theirmodel was constructed based on assumptions similar to oand could reproduce the alternating migration and consoltion phases during the colony expansion. However, the c

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plexity of the ES model involves a rather large numbermodel parameters, which practically impedes both themapping of the parameter space and a quantitative compson of the model results with experimental findings. The mjor differences between our model and ES’s can be sumrized as follows: ~i! We do not resolve theage of theswarmer population. Instead, we have a density measurea constant decay rate, implying an exponential lifetime dtribution on the~unresolved! level of individual cells. Sincethe available microbiological observations@5,13# suggestonly that the lifetime is finite, there is no reason for prefering any specific distribution.~ii ! We did not incorporate intoour model an unspecified ‘‘memory field’’ with abuilt-inhysteresis. Instead, we implemented adensity-dependent motility of the swarmer cells, which behavior was indeed oserved @4–6#. ~iii ! In our model the fully differentiatedswarmer cellsdo not grow, which assumption is probablynot fundamental for the reported behavior, but it seems tomore realistic because of the repression of many biosynthpathways@5#. ~iv! Finally, we do not consider any specifiinteraction between the motility of swarmer cells and tnon-motile vegetative cell population. Although such inter-actions probably exist, they are undocumented, and asdemonstrated, are not required for the formation of perioswarming cycles. However, such effects can be importanthe actual determination of the density profiles.

With these differences, which are not compromising tbiological relevance of the model, we were able to mapphase diagramcompletely, establish approximate analyticaformulas, and estimate the value ofall model parameters inthe case of four different strains. In addition, experimendata measured under various conditions could be explawith one particular parameter setting in the case ofPRM1 strain, indicating the predictive power of our aproach. Our model is aminimalmodel in the sense that all othe explicitly considered effects~thresholds, diffusion, etc.!were required to produce the oscillatory behavior; thuscannot be simplified further. Such minimal models can seas a comparison baseline for later investigations of varispecific interactions.

The values of the microscopic parameters of the mocan be either measured directly~like r 0 , rmin* , rmin , rmax,or r * ! or can be determined indirectly from experimendata~asrmin

o and r ). Most of these measurements have nyet been performed; we hope that our work will motivasuch experiments further examining the validity of our asumptions. In fact, one of the parameters,r * /r 0 was set to 1during the fitting processes, as currently there is no availadata to estimate its value. Our numerical results suggestit is probably larger than 0.3, and it is unlikely to be largthan 2 ~meaning an average lifetime less than 30 mi!.Within this range our qualitative conclusions are valid, whthe numerical values of the parameter estimates can chup to a factor of 3.

The behavior of ‘‘precocious’’ swarming mutants reported in Ref.@9# deserves special attention. First we woulike to comment on the huge difference found in the valuethe transition rater ~see Table II!. We emphasize that this inot an arbitrary output of a multiparameter fitting proce

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First, we have reasons to believe that the motility threshoof the two BB strains are rather similar. Knowing the growrates and the cycle times, Eq.~12! shows us that the difference in the values ofrmin cannot exceed one order of manitude. Assuming then this maximal difference inrmin , rremains theonly free variable in Eqs.~13!–~15!, and thefitting can be performed unambigously. Thus, we are quconfident that such a large difference exists inr showing thatthe rsbA gene~in which these strains differ! influences notonly the cell density threshold, but the rate of differentiatias well. It is also interesting to note that in Fig. 8~b! thebehavior of the PRM2 and PRM1 strains reflect a relatvery similar to that of the BB2000 and BB2235 strains. Fnally, our calculations predicted a slightly longer cycle timfor the precocious swarming mutant BB2235, which is ain accord with the actual experimental findings~see Fig 2 ofRef. @9#!.

In our model the assumed functional form of the densdependence of the diffusion coefficient is somewhat differfrom the one most often considered@24,26#, namely,

D~r!;rk. ~17!

The advantage of Eq.~17! is that it allows analytic solutionsfor certain cases@21#; however, it describes an unlimitedarbitrarily fast diffusion inside the colony where the densis high. In contrast, in real colonies the diffusion of cellscertainly bounded, and the expansion of the boundary caoften limited by the supply of cells from behind@14#. There-fore we believe that our thresholded formulation@Eq. ~7!# isa better approximation of what is taking place inside the rcolonies.

Finally, we would like to comment on the role of nutrienin the swarming behavior ofP. mirabilis. In our model there

u

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is a phenomenological parameter (t) determining how longthe swarmer cells are produced at a given position incolony. When investigating the dependence of the cycle ton this parameter, as Fig. 5 demonstrates, we found antremely weak effect. Thus, at least within the frameworkthis model, there is no contradiction between the assumpthat the swarmer cellproductionceases due to nutrient~oraccumulated waste! limitations, and the seemingly nutrienindependent cyclic behavior. In fact, this idea can be devoped further. By increasingt ~or decreasing the motilitythresholdrmin* ) we arrive at a regime where the migratioand consolidation phases are not clearly separable, as atile swarmer cell population exists even when the expansof the colony is slower. Experiments mapping the morphogy diagram ofP. mirabilis ~Fig. 2 of Ref.@29#! showed thatthere are certain values of agar hardness and nutrient contration, for which the expansion of the colony is still osclating, but the periodic density changes are smeared outto the presence of motile swarmer cells in the consolidatperiods. If one associates the increasing agar hardnessincreasingrmin* , and the nutrient concentration witht, thenone can qualitatively reproduce those~i.e., thePr and Ph)regions of the morphology diagram.

ACKNOWLEDGMENTS

One of the authors~M.M.! is grateful to T. Matsuyamaand H. Itoh for many stimulating discussions on experimetal results. This work was supported by funds OTKT019299, F026645; FKFP 0203/197 and by Grant N09640471 and 11214205 from the Ministry of EducatioScience and Culture of Japan.

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Documents

Theory of periodic swarming of bacteria: Application to Proteus mirabilis