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Self-Organized Pattern Formation of a Bacteria Colony Modeledby a Reaction Diffusion System and Nucleation Theory
Joe Y. Wakano,* Shinya Maenosono, Atsushi Komoto, Noriko Eiha, and Yukio Yamaguchi
Department of Chemical System Engineering, University of Tokyo, Hongo 7-3-1, Tokyo, Japan(Received 15 October 2002; published 23 June 2003)
Self-organized pattern formation is observed in bacterial colony growth. The recently reported
knotted-branching pattern of the Bacillus circulans colony consists of the trajectories of aggregates
which grow, move, and reproduce simultaneously. We modeled these processes by combining a reaction
diffusion system of nutrient dynamics, nucleation theory for aggregate generation, and individual based
dynamics of motion and growth of aggregates. The branching pattern produced by computer simulationshows great similarity with experiments. Response to the initial nutrient concentration is also
consistent with the experiments.
DOI: 10.1103/PhysRevLett.90.258102

PACS numbers: 87.18.Hf, 47.20.Hw, 87.23.Cc

Self-organized pattern formation is observed in bacterial colony growth [1–6]. Such cooperative behavior
can be considered as an adaptive response under unfavorable environments [1]. Reaction diffusion equations
have been applied to study the branching pattern formation [7–10]. However, in most studies, bifurcation of abranch is reproduced just as a result of numerical calculation and the mechanism is still of great interest. It isrecently reported that Bacillus circulans forms a knottedbranching pattern (KBP) which consists of the trajectories of aggregates which grow, move, and reproduce
simultaneously [2]. By observing growth, movement,and generation processes of an aggregate, the underlying
mechanism of the branching pattern formation might beunderstood. In this paper, we briefly introduce recent
experimental results and their implications. Then, ourhypothesis for the aggregate dynamics is explained, followed by mathematical formulation. Results of the computer simulation are posed and discussed.
When a droplet containing bacteria is inoculated at thecenter of a Petri dish filled with agarose gel medium, B.
circulans forms various colony patterns [Figs. 1(a) –1(d)].
Hardness of the medium (agar concentration) and nutrient
concentration are major factors affecting the pattern.
Some patterns are characterized by aggregates of bacteria. The observed pattern is considered as trajectories ofmoving aggregates [11]. When hardness of the medium is
so low that bacteria can move inside individually, only asimple and typical disklike pattern is produced [Fig. 1(d)].
Experimental study shows that formation of an aggregatecaused by a high density of bacteria is a key which
triggers the formation of KBP [2]. Once aggregates areformed, the colony expanding rate increases drastically.
A radius of the aggregate linearly increases with time
while its velocity increases sharply and then decreases[11]. The generation process, which is essential to thebifurcation of a branch, had been left untouched. Froma microscopic view, the generation process is a creation ofa new aggregate. Bacteria with the size of a few micro-meters form an aggregate with the size of a few millimeters. We focus upon the similarity between this selforganized phenomenon and nucleation. Nucleation theory
developed in physical chemistry is applied to model ageneration process.
Aggregate formation of bacterial systems has been
considered as a consequence of attractive chemotaxis[12 –14]. Such long-distance force plays an important rolein the first step for globally distributed bacteria to aggregate. When the average distance between cells becomesas small as the cell size ( 10 m), the short-distanceforce becomes dominant. Surprisingly, the existence of258102-1

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FIG. 1. Colony patterns of B. circulance. (a) A typicalknotted-branching pattern (KBP). (b) When nutrient concentration is denser. The pattern is more isotropic with morebranches and less gaps. (c) When agar medium is harder.
This branching pattern is called an Eden-like pattern in whichknots are not clearly distinguished. (d) When agar medium issofter. No spatial structure is formed.

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van der Waals attractive force between cells had been
experimentally verified [15]. There is also an experimental study showing that the bacterial surface is negatively
charged and that electrostatic repulsive force works between cells [16]. These studies suggest that a critical stepin aggregation of bacteria should be considered withcolloid theory [17]. The Derjaguin-Landau-VerweyOverbeek theory in physical chemistry predicts nucleation in such a system [18–20]. Actually, many bacteria
coagulate above the critical coagulation concentration[21]. An aggregate of B. circulans is also very closely
packed with uniform direction [11] and might be regarded
as a nucleus. Therefore, in the present study, we regard
aggregate formation as the nucleation process.
Individual bacterium can hardly move in hard agar.
Aggregation is considered to be an adaptive response as it
reduces viscous drag, allowing bacteria to move toward
nutrient concentration. Moreover, some bacterial species
are known to secrete lubricant fluid around the aggregates
so that they can move on the hard and dry agar surface[Fig. 2(a)]. Ben-Jacob et al. called the lubricant an envelope [1]. As an aggregate moves, the envelope is formed
along with the trajectory. Bacteria can disperse in theenvelope, i.e., the envelope is considered as a solvent ofbacteria. Dispersed bacteria keep reproducing in the envelope after an aggregate goes away and the branch isformed [11]. As the thickness of the envelope is limited,
reproduction of bacteria easily leads to supersaturation
followed by nucleation. The initial nuclei size (i.e., critical nuclei size expected from classical nucleation theory)is much smaller than the observed aggregate size. We
assume that viscous drag force, which is a major obstacle
for minute particles to move quickly, prevents the initial
nuclei from escaping the parent envelope. As the nuclei
grow, viscous drag becomes smaller than propulsion generated by the flagella of all active bacteria. When the size
reaches the critical escape size, it escapes the parent
envelope to move toward nutrient concentration (nutrient
chemotaxis). At the beginning of a movement, nutrient
concentration is locally small but it gradually increases asit leaves the parent nucleus. High nutrient concentration
results in a high reproduction rate. The model of reproduction is illustrated in Fig. 2(b). Experimental study
shows that a nucleus grows larger even after it stops,
which suggests that the stop is not due to the lack ofnutrient but due to the existence of frictional force [11].
As a radius becomes very large, friction between thenucleus and the agar surface becomes dominant and thenucleus stops. All these dynamics are globally governedby spatiotemporal dynamics of nutrient concentration.
For simplicity, we assume a nucleus’s shape remains
similar as it grows characterized by the radius r. The
surface area S and the volume V are written as S 2 r2 and V  3 r3 . A unit of weight is rescaled so thatthe weight of a nucleus m  V. A nucleus’s position isrepresented by the coordinates in two-dimensional space.
Bacteria on the surface of the nucleus are active but
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FIG. 2. Schematic illustration of the model. (a) An aggregate(nucleus) swims in the surrounding lubricant, which stands still
due to friction. The lubricant secreted by an aggregate also
plays the role of a solvent of dispersed bacteria. (b) Bacteria
reproduce in both the envelope and the active layer of a nucleus.
As the envelope is supersaturated and adsorption (nucleusgrowth) is dominant, net adsorption rate is considered in themodel.

bacteria at the inner part are inactive spores [22] which
do not reproduce nor propel a nucleus because they aredensely packed and nutrient concentration is low [11].
Thus, the number of active bacteria generating propulsion
is proportional to the surface area. The equation of motionof a nucleus is formulated asm~nv~dv~r:
pS  mg~jv~ jdt
jrnj

(1)

The first term on the right-hand side represents the nutrient chemotaxis. The second term represents friction
between a nucleus and an agar surface. Apparently, as
the radius increases, a propulsion term and a friction term
become dominant in that order.
Each nucleus is surrounded by the lubricant envelopewith constant thickness Ed . In the envelope, dispersed
bacteria are active, the number of which being b. The
supersaturation level is denoted by   b=SEd  (  1when saturated). The reproduction process is denoted by
the following differential equations:258102-2

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b  ka   1S;
dt Ks  n(2)dknV aS  ka   1S;
dtKs  n
where V is the number of bacteria in a nucleus withvolume V. We adopt a Mechaelis-Menten–type equation
for bacterial reproduction. aS represents the volume of theactive layer; ka represents the net adsorption rate. If thenutrient concentration is very large and the supersaturation level is almost 1 at the earlier stage of nuclei growth,a radius becomes a linear function of time. By putting
this result into Eq. (1) and integrating it, we get  X lnr  Yr  Z. Both results agree with the experimental
observation [11].
For the nucleation process, we adopt a classical nucleation theory. The difference in free energy between Nmonomers and a N-mer nucleus is GN   N  S’,
where is the difference of the chemical potentials of amonomer at supersaturation and that at equilibrium with anucleus. ’ represents the surface free energy of a N-merper unit area. The nucleation rate is denoted by J ! exp G =kT where G is a maximum of
GN . The generated nucleus stays in the parent envelope
until it reaches critical escape size %. So we assume anucleus with size % is generated at a rate J. Each nucleus is generated with the envelope saturated with thedispersed bacteria, i.e.,   1. Hence, the number ofbacteria in the parent envelope is decreased by % 
1=Ed 2 %=3 2=3 to keep the mass balance. Thedaughter nucleus is put randomly near the parent.
Spatiotemporal dynamics of nutrient concentration isrepresented byX@nkn Dr2 n  )x~  x~ i b  aSi ;@tKs  n ii
where the subscript i represents the ith nucleus. A nucleus
loses activeness (sporulates) when experiencing nutrient
concentration that is less than . Once it sporulates, itdoes not reproduce, move, nor recover activeness.
In computer simulation, nutrient dynamics in a twodimensional agar surface is calculated. Growth, motion,and generation are also simulated for all nuclei at various
points on the surface. The history of the resulting colony
growth dynamics is stored in a computer memory and thetrajectory of all nuclei including spores is considered as acolony pattern.
Computer simulations proved that the model can reproduce essential characteristics of KBP (Figs. 3 and 4).
Our scenario of the formation of KBP is as follows. Anucleus size is monotonically increasing while its velocity increases and then decreases. Thus, thickness of abranch increases as a nucleus moves outward. Finally, anucleus stops but reproduction does not stop until it exhausts the nutrient, resulting in a large nucleus at the end
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FIG. 3. Simulation results. The final patterns when all nuclei
become inactive are shown. The shaded area represents the sizeof the nucleus when it passes the corresponding point of thebranch. For appropriate parameters, KBP is reproduced (a).
When the initial nutrient concentration is increased, the number of branches increases and the pattern becomes more isotropic (b). When a friction coefficient is large, whichcorresponds to a hard agar condition, a nucleus can hardly
move and thus nutrient concentration is expected to be almost
the same among nuclei. However, simulation reproduced abranching pattern (c). All these results are consistent withcorresponding experimental results (Fig. 1).

of the branch. This is a knot. In a rough approximation,
the number of bacteria in the envelope grows exponentially while the radius increases linearly. This will resultin an explosive increase in the supersaturation level in thelatter period of growth, which causes the generation of a
new nucleus through nucleation. Nucleation decreases at asuper saturation level but it may recover the nucleationthreshold as long as nutrient concentration is high.
Therefore, a few branches may be produced by the parent
nucleus. Note that nutrient concentration is expected to behigh when the parent is isolated or located at a projection.
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suspect that the tendency is due to vertical diffusion inagar. If a super saturation level exceeds the nucleationthreshold, rich nutrient is consumed to produce daughter
nuclei. In order for a nucleus to grow large without too
frequent nucleation, nutrient concentration must stay at aproper level. A continuous supply of nutrient from thebottom half of agar may enable this. As our model considers only two-dimensional diffusion, the tendency isnot reproduced. However, preliminary experiments (datanot shown) show the dependence of the colony pattern on
agar thickness. The effect of three-dimensional diffusionof nutrient is left for future work.
The knotted-branching pattern has a distinct structureoffering insight into underlying biological and physicalmechanisms. Based on this feature, our model reproduces
the pattern. The present study shows that a complex
colony pattern can be explained by the combination ofwell-known physical laws and biological observation.
FIG. 4. Enlarged pictures of simulation (left) and experimental (right) results. The great similarity is observed.

As a chain of the branches grows concentrically outward,
an aggregate at the head of the chain becomes moreisolated. Then, multiple generations will occur to fillgaps, which will result in a beautifully branching colony
pattern. Despite many complex factors in both the modeland the experiment, the diffusion of nutrient governs thepattern formation. The branching mechanism is similar
to a diffusion limited aggregation formation [23].
The colony pattern of B. circulans is similar to thatof Paenibacillus vortex in a point that rotating aggregates (vortex) migrate to draw patterns. For the lattersystem, a combined model of vortex formation and migration of the vortex has been proposed [14]. In theirmodel, the vortex is formed by short range attractive
chemotaxis while the global pattern of the vortex trajectory is governed by long range repulsive chemotaxis. On
the other side, in our model, aggregate formation is due tofundamental (or physicochemical) forces between cells,
which are experimentally verified and global dynamics ofaggregates is due to nutrient chemotaxis, which is also
experimentally well known. Our model does not explain
rotational motion of the aggregate while assumptions ofour model are more strongly related to experimental
facts. The introduction of the mechanism of vortex formation into the present study would be the next important
step. Hereafter, the fruits achieved by the physical chemistry of microorganisms will provide new hypotheses andnew possibilities to the study of pattern formation ofmicroorganisms.
The major inconsistency with the experiment is thetendency for an aggregate size to increase as the initialnutrient concentration increases [Figs. 1(b) and 3(b)]. We

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