ARTICLE IN PRESS

Journal of Theoretical Biology 225 (2003) 91–97

Growth dynamics of Bacillus circulans colony
Atsushi Komotoa, Ken-ichi Hanakib, Shinya Maenosonoa, Joe Yuichiro Wakanoa,
Yukio Yamaguchia, Kenji Yamamotoa,b,*a

Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japanb
Department of Medical Ecology and Informatics, Research Institute, International Medical Center of Japan, 1-21-1 Toyama,
Shinjuku-ku, Tokyo 162-8655, Japan
Received 8 July 2002; received in revised form 13 May 2003; accepted 21 May 2003

Abstract
We have investigated the growth dynamics of Bacillus circulans colony exhibiting the knotted-branching pattern by swarming on ahard agar medium. The knotted-branching pattern consists of many circular clusters, so-called subcolonies, and their trajectories.
We analysed the processes of a subcolony because they are presumably the key elements for the formation of knotted-branchingpattern. It was found that a subcolony has three processes, i.e. ‘‘generation’’, ‘‘growth’’, and ‘‘migration’’ by microscopic and timeresolved observations. An embryonic small subcolony (child subcolony) formed around an existing subcolony (parent subcolony)
grows larger and migrates away from the parent subcolony. We proposed a simple model to explain the migration and the growth
processes. It is assumed that the internal part of the subcolony is unfavorable for the bacteria and that the motion of the childsubcolony on the agar medium can be modeled using a frictional force. The experimental data were quantitatively analysed in order
to compare with models. Our models are consistent with the experimental results on following three points: (1) the radius of a
subcolony increases linearly with the incubation time, (2) a subcolony stops just after formation and then starts to migrate suddenly,and (3) the trajectory of a subcolony predicted by the model agrees with the experimental one.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Growth dynamics; Bacillus circulans colony; Knotted-branching pattern; Prediction of subcolony trajectory

1. Introduction
Recently, the mechanism of spontaneous patternformation, i.e. self-organization, has been intensively
investigated from both the fundamental and practicalpoints of view (Sachs et al., 2001; Emmerich et al., 1999;
Jin et al., 2001). On the other hand, self-organization is
one of the characteristic features of biological systems(Kondo and Asai, 1995). It is thus important to study
self-organizing pattern formation observed in biologicalsystems. In many cases, however, the self-organization
mechanisms in biological system are quite complex
because there are a large number of elements playing arole in the given system (Kondo and Asai, 1995; Franks
and Tofts, 1994). Through studying bacterial colonies,
the diverse pattern formation mechanism can be*Corresponding author. Department of Medical Ecology andInformatics, Research Institute, International Medical Center ofJapan, 1-21-1 Toyama, Shinjuku-ku, Tokyo 162-8655, Japan. Tel.:+81-3-3202-7181x2856; fax: +81-3-3202-7364.
E-mail address: backen@ri.imcj.go.jp (K. Yamamoto).
0022-5193/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0022-5193(03)00224-8

understood. It has been studied during the last few
decades because it is simple as compared with otherbiological systems (Budrene and Berg, 1991; Itoh et al.,
1999; Ben-Jacob et al., 1997; Mimura et al., 2000; Cohenet al., 2000). Moreover, various kinds of mathematicalmodels based on reaction–diffusion equations have been
proposed (Mimura et al., 2000; Cohen et al., 2000;
Ben-Jacob and Cohen, 2000 for the review article).
In particular, Bacillus subtilis (B. subtilis) (Matsushita,1997) and Proteus mirabilis (P. mirabilis) (Watanabe
et al., 2002) are well investigated, where the bacteria
swarms on the agar plate and shows various colony
patterns on the agar medium depending on theenvironmental conditions (i.e., the agar and nutrient
concentration in the medium). There have been many
reports on B. subtilis branching-patterned colonies(Matsushita, 1997; Ohgiwari et al., 1992). The similar
behavior is also observed in the other bacterial strains(for example, see Ben-Jacob and Cohen, 2000 for thereview article). The dynamics of the B. subtilis colony
growth was also investigated (Wakita et al., 2001). It

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A. Komoto et al. / Journal of Theoretical Biology 225 (2003) 91–97

was reported that B. subtilis could move on the hard
agar surface due to secretion of lubricant and acooperative behavior (Ben-Jacob et al., 1994). In sucha case, the bacterial mobility on the hard agar surface isan increasing function of the bacterial density. On theother hand, P. mirabilis forms a concentric-patternedcolony (Watanabe et al., 2002; Matsuyama et al., 2000).
On the hard agar surface, P. mirabilis forms a swarmer,
which is a multinucleate and hyperﬂagellated cell, toincrease the mobility (Belas, 1997). A number ofmathematical models have been proposed to elucidate
the pattern formation of bacterial colonies based onthese experimental observations (Golding et al., 1998;
Arouh, 2001). However, the validity of these models is
limited due to the lack of quantitative experimental data
related to the growth dynamics.
We have reported that Bacillus circulans (B. circulans)also exhibits diverse colony patterns depending on agar
concentration (CA) and nutrient concentration (CN) as
shown in Fig. 1(a) (Eiha et al., 2002). Unlike B. subtilis,
B. circulans colony shows the knotted-branching pattern, which is the more complicated pattern containing
many knots consisting of bacterial aggregates (described
as a ‘‘subcolony’’), on the hard agar surface. The
knotted-branching pattern is a result of the growth andmovement of the subcolonies, i.e. the trajectories ofsubcolonies. By forming the subcolony, B. circulans
presumably acquires higher mobility on the hard agar
surface as cooperative behavior.
Our ultimate goal is to understand the pattern
dynamics exhibited in the colony growth of B. circulans
and to build a mathematical model based on theexperimental observations. In this paper, we analyse
the elemental process for the colony growth consideringthat the subcolony is an important element for theformation of the knotted-branching pattern.

B1

30

CN [g/L]

92

20
B2
10

A
B3

0
(a)

(b)

0

1.0

2.0

3.0

CA [wt%]1cm

Fig. 1. (a) Morphology diagram of B. circulans. The colony patterns
diversely change depending on the agar concentration CA and thenutrient concentration CN. These patterns are classiﬁed as follows:(A) disk-like, (B1) concentric-ring, (B2) knotted-branching, and (B3)
Eden-like pattern. (b) The knotted-branching pattern of B. circulans.
The photograph was taken after 5-day incubation. The colors wereinverted by Adobe PhotoShop (Adobe Systems). The agar conditions
are CA=1.5 wt% and CN=15 g/l.

2. Colony formation dynamics of B: circulans
Fig. 1(b) shows a colony of B. circulans colony on thehard agar medium (concentration of agar CA=1.5 wt%,
and nutrient concentration CN=15 g/l). The details of themedium and the experimental procedures are described in
the appendix. One can see many small circles in theinterior of the colony which we call ‘‘subcolonies’’. Fig. 2
shows the transmission optical micrograph of an innerpart of the subcolony taken by differential interference
contrast microscopy. As seen in Fig. 2, the bacteria areclosely packed and lose mobility. The internal part of thesubcolony is presumably an unfavorable environment forthe bacteria due to the high population density. The
bacteria placed in the periphery of the subcolony might
be active. The dynamics of the growth and the movementof subcolony are observed by time-resolved optical
microscopy as shown in Fig. 3. The newborn subcolonies

10 µm
Fig. 2. Micrograph of the internal part of the subcolony. The whitelines are guides for eye indicating individual bacteria strains. As seen in
the image, the bacteria are closely packed.

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93

child subcolony
parent subcolony
child subcolony

(a)

1mm

(b)

1mm

(c)

1mm

(d)

1mm

(e)

1mm

(f)

1mm

Fig. 3. Time series of enlarged photographs focusing on subcolonies. The arrows in Figs. 3(c)–(f) indicate child subcolonies migrating away from theparent subcolony. Each photograph was taken at every 30 min.

(child subcolonies) are formed at the periphery of theparent subcolony as seen in Figs. 3(a) and (b). After theformation of child subcolonies, they gradually grow andmove away from the parent subcolony as shown in
Figs. 3(c)–(f). We summarise that three elemental processes
dominate the subcolony dynamics, i.e. ‘‘generation’’,‘‘growth’’, and ‘‘migration’’. In this paper, we focus on
two elemental processes: ‘‘growth’’ and ‘‘migration’’.

3. Modelfor simplicity that the shape of subcolony is disk-likeand the bacterial density in the subcolony is spatially
uniform constant. The cell-division rate for an individual bacterial cell is constant assuming that the nutrient
concentration around the subcolony does not decreasesigniﬁcantly. However, the multiplication due to cell
division occurred only in the outermost part of thesubcolony from our observations. Thus, it is consideredthat the total increase in the distribution of bacteria is
proportional to the perimeter of the subcolony. The
mass balance is given by

3.1. Growthdrpr2 ¼ 2prk;
dt

From the experimental observation, the subcolony
grows larger with incubation time. Here let us assume

where r, r, and k are bacterial density, radius of subcolony, and the reproduction rate constant, respectively.

ð1Þ

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A. Komoto et al. / Journal of Theoretical Biology 225 (2003) 91–97

By assuming that the bacterial density is constant (dr/dt=0), we obtain from Eq. (1)
 2pk
d
pr2 ¼
r:
ð2Þ
dt
r
Consequently, the area velocity of the subcolony isproportional to its radius. Thus we obtaindr k¼ ¼ K  const:;
dt r

ð3Þ

where K is a constant. Eq. (3) predicts that the radius of
subcolony linearly relates to the incubation time.
3.2. Migration
The subcolony sinuates in the migration process andthe resulting trajectory of the subcolony is not usuallystraight (see Fig. 1(b)). It is presumable that the
migration direction is determined by attractive or
repulsive chemotaxis, which is the bias of the bacterial
movement according to the concentration gradient ofchemical species. For example, such a phenomenon
was experimentally studied in Escherichia coli colony(Budrene and Berg, 1995). However, the migration
direction is not discussed in our model. We mainly
discuss the one-dimensional mobility of the subcolony
and describe a model based on the equation of motion.
The mass of the subcolony is proportional to the area
because it is assumed that the subcolony is disk-likeshaped and its bacterial density is spatially uniform.
That ismppr2 ;

ð4Þ

where m represents the mass of the subcolony.
Next, we deﬁne impelling force. We previously
conﬁrmed that individual B. circulans cell swim insideof the soft agar medium (agar concentration o0.8 wt%)
(Eiha et al., 2002). In our study, the colony growth rate
on a hard agar medium (agar concentration=1.5%),
which was experimentally observed, was analysed by aset of reaction–diffusion equations and the diffusion
coefﬁcient of the bacteria was estimated. The estimateddiffusion coefﬁcient was much larger than that of aspherical particle of the same size in agarose gel. We
suggested that it is because B. circulans has ﬂagella andacquired much larger mobility than in the case of simple
Brownian motion. It is natural to assume that the
impelling force is an increasing function of the numberof active ﬂagella contained in the subcolony. Here wetake two kinds of model into account to express theimpelling force.
If we assume that the ﬂagella located in the internalpart of the subcolony are inactive, then the impelling
force is proportional to the perimeter of the subcolony.
If we assume that all ﬂagella in the subcolony are active,
the impelling force is proportional to the area of thesubcolony. Thus, the impelling force Fi is described as
Fi prlðl ¼ 1; 2Þ:ð5Þ

The subcolony ﬁnally stops migrating after the
movement. However, the subcolony grows larger
and child subcolonies are formed at the periphery as
aforementioned after the subcolony stops migrating.
This fact suggests that the stop of the subcolony is notdue to the lack of nutrient but due to the existence ofresistive force. We assume that the resistive force iscaused by friction between the subcolony and the agar
surface. Thus the resistive force Fr is proportional to themass of the subcolony, described as
Fr ppr2 :ð6Þ

From Eqs. (4)–(6), the equation of motion for thesubcolony is given by
A 1 r2dv¼ A 2 rl  A 3 r2dtðl ¼ 1; 2Þ;

ð7Þ

where A1, A2, and A3 are positive constants and v
represents the velocity of the subcolony.
3.3. Integration of the two models
Here we integrate the two aforementioned models, i.e.
growth and migration processes, to derive the timedependence of the subcolony velocity v. Combining
Eqs. (3) and (7), we obtain the relations between thevelocity and the radius.
l=1:v ¼ B1 ln r  B2 r þ B3 :

ð8Þ

l=2:
v ¼ ðB1  B2 Þr þ B3 ;

ð9Þ

where B1=A2/(A1K), B2=A3/(A1K), and B3 are constants. Eq. (8), which corresponds to the case of theimpelling force proportional to the perimeter of thesubcolony, indicates that the velocity of the subcolony
initially increases just after its formation because theimpelling force is larger than the frictional force.
Subsequently, the frictional force starts to dominate
the migration as the subcolony grows, i.e. the radiusof the subcolony increases. In short, it is predicted thatthe subcolony velocity v initially increases and thendecreases with the subcolony growth. Eq. (9), whichcorresponds to the case of the impelling force proportional to the area of the subcolony, evidently indicatesa linear dependence of the subcolony radius on thevelocity. Since the velocity must ﬁnally reach zero, thesign of the pre-factor in Eq. (9) should be negative(B1B2o0).

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4. Results and discussion

1.2

r  ri ¼ Kðt  ti Þ:ð10ÞHere ti and ri are the time and the radius of thesubcolony when the child subcolony becomes visible
to the naked eye, respectively. Fig. 4 shows the linear
time dependence of r(t). This result is consistent with
Eq. (10). The separation between the experimental data
and the theoretical ﬁt at late time could be due to thedecrease in the reproduction rate caused by the nutrient
starvation.
Here we compare the experimental data with Eqs. (8)and (9) regarding the migration process. The migration
velocity of the subcolony was experimentally measured
by chasing the position of the subcolony with using
close-up pictures as shown in Fig. 3. However, the initialstage dynamics were observed by optical microscopy
because the subcolony is too small to observe by the
naked eye at the initial stage. According to themicroscopic observation, it was conﬁrmed that the childsubcolony stands still for about 30 min after thegeneration and then it gradually starts to migrate.
Fig. 5 shows the relation between the radius and thevelocity of the child subcolony by analysing both theclose-up observation and the microscopic observation(see the appendix for detail data). It was clariﬁed thatthe velocity increases with the radius at small r(t). As aresult, we could conclude that the impelling force iscreated by the only ﬂagella located at perimeter of thesubcolony, i.e. l=1.

0.8

r-ri [mm]

0.6

0.4

0.2

0

2

4

6

8

10 12 14 16

t-ti [h]Fig. 4. Radius of the subcolony plotted versus incubation time. Here ti
and ri are the time when the child subcolony becomes visible by the
naked eye and the radius of the subcolony, respectively. The ﬁlled
circles and the solid line represent experimental data and theoretical ﬁt
based on Eq. (10), respectively. The value of K obtained by ﬁtting ofexperimental data.

microscopic
close-upmodel (l=1)
model (l=2)

1v [mm/h]We experimentally measured the time evolution of theaverage radius of the subcolony r(t) regarding they were
circular-shaped. From Eq. (3), we obtain

0

95

0.8
0.6
0.4
0.2
0

0

0.2

0.4

0.6

0.8

r [mm]
Fig. 5. Migration velocity of the subcolony plotted versus radius of thesubcolony. Open circles and ﬁlled circles represent the experimental
data from the microscopic and the close-up observation, respectively.
Dashed and solid lines correspond to theoretical ﬁts based on Eqs. (8)and (9), respectively. In the case of l=1, the values of each constant are
B1=1.1 mm/h, B2=4.2 h1, and B3=3.2 mm/h, respectively. In thecase of l=2: B1B2=1.4 h1 and B3=1.4 mm/h.

Combining Eqs. (3) and (8), we can obtain theequation for the trajectory of the subcolony is given by
x ¼ C1 r ln r  C2 r2 þ C3 r þ C4 :

ð11ÞHere x is the migration distance and C1=B1/K, C2=B2/(2 K), and C3=(B3B1)/K are constants. We compared
the trajectory of the subcolony which migrated almost
radially with Eq. (11). In Figs. 6(a) and (b), thecalculated and the experimentally observed trajectories
are shown, respectively. This result also shows that ourmodel is quantitatively consistent with the experimental
result.
B. circulans exhibits the knotted-branching patternon the hard agar medium. The origin of the knottedbranching pattern is the existence of the subcolonies. We
believe that the subcolony is the key element for theB. circulans colony growth and analysed the dynamics
of the subcolony formation process. According to theexperimental and the theoretical analysis, it is possible
to summarize that the subcolony formation could be
divided into three processes: ‘‘generation’’, ‘‘growth’’,and ‘‘migration’’. We consider, in this paper, growthand migration processes and propose a simple physicalmodel to elucidate the formation mechanism of thesubcolony. Based on our analysis, the ‘‘active layer’’
exists at the periphery of the subcolony where themultiplication due to cell division occurs and the ﬂagellaare active. The model is very simple; however, it canquantitatively explain the shape of a branch, which is an
important element of the knotted-branching pattern.
The details of the ‘‘generation’’ process is unknown, but
if we assume a small child sub-colony is generated near
the parent and that sub-colony tends to move toward

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96

4

x [mm]

3

2

1

0
-2
(a)

-1

0

1

2

r [mm](b)

1mm

Fig. 6. The calculated (a) and the experimentally observed (b)trajectory of the subcolony. The values of each constant, which wereused to draw the trajectory shown in Fig. 8(a), are C1=14,
C2=29 mm1, C3=30, and C4=0.61 mm, respectively.

the nutrient rich locations, then the knotted-branchingpattern is expected. To analyse it would be the fertile
ground for the future studies.

Acknowledgements
We are grateful to Noriko Eiha for her help. This
work was supported by the grants from the Ministryof Health, Labor and Welfare and the Ministry of
Education, Science and Technology.

Appendix A
A.1. Experimental procedures
We used Bacillus circulans, which is a rod-shaped
bacterium with a few ﬂagella. The bacterial cells wereincubated on the agar medium, which contains 10 g/l of
Bacto Tryptone Peptone (Difco), 5 g/l of Bacto Yeast

Extract (Difco), 1.5 wt% of Bacto agar (Difco) and10 g/l of sodium chloride dissolved in pure water (thecomponents of the liquid medium are the same as thoseof the agar medium without agar). This solution was
autoclaved at temperature 120 C and pressure 2-atmpressure for 15 min, and then kept at 70 C in a waterbath. Fifteen milliliters of the solution was poured into a
Petri dish whose diameter was 86 mm and was cooled
down to room temperature for about 1 hour. After thegelatin, the Petri dish was overturned and the agar
medium was dried for 3 days. Then the weight of thedish decreased by 1.2–1.5 g. The liquid culture, whosebacterial density was about 108 cell/ml, was preparedby incubating bacteria at 37 C for 15–18 h. The droplet(1–2 ml) taken from the solution was inoculated at thecenter of the agar medium. Then the Petri dish wassealed with its cover by plastic tape to prevent the agar
medium from drying. It was incubated at 37 C. The
dynamics of the bacterial colony was observed by taking
photographs every 10 min by a digital camera Nikon
D1X (Nikon) equipped with a close-up lens (MicroNikon 60 mm lens, Nikon). Too small, and thus invisible
to the naked eye, the subcolony was photographed by
the digital camera through a microscope Olympus IX71
(Olympus) with the phase contrast ﬁlter. The size andthe position of the subcolony were analysed from thesephotographs. The radius of the subcolony was estimatedfrom its area. On the other hand, the migration velocityof the subcolony was experimentally measured bychasing the position of the subcolony. The averagevelocity of subcolony was plotted versus radius by
converting original data into a discrete distribution. The
class intervals for the radius are 50 mm for close-up and
20 mm for microscopic data. The averages of radius andvelocity were calculated.

Table 1
Experimental data of Fig. 5r (mm)

sr (mm)v (mm/h)sv (mm/h)

(a)
0.084
0.098
0.116
0.139

0.006
0.005
0.004
0.003

0.131
0.079
0.322
0.432

0.113
0.113
0.101
0.102

(b)
0.171
0.225
0.273
0.324
0.377
0.425
0.478
0.523
0.573
0.625
0.673

0.017
0.018
0.013
0.010
0.016
0.015
0.022
0.016
0.015
0.014
0.016

0.813
0.724
0.641
0.626
0.498
0.342
0.412
0.290
0.214
0.090
0.040

0.298
0.173
0.175
0.265
0.161
0.085
0.085
0.214
0.168
0.075
0.042

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A.2. Experimental data
We show the experimental data, which compose
Fig. 5. Tables 1(a) and (b) show the experimental datafrom the microscopic observation and from the close-up
observation, respectively. In the table, r and v represent
the averages of radius and velocity. And sr and sv
represent the standard deviations of radius and velocity,respectively.
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