Physica A 313 (2002) 609 – 624

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The mode transition of the bacterial colony
Noriko Eihaa;∗ , Atsushi Komotoa , Shinya Maenosonoa ,Joe Yuichiro Wakanoa , Kenji Yamamotob , Yukio Yamaguchiaa 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 12 April 2002

Abstract
The colony patterns of Bacillus circulans on agar medium were experimentally investigated to
study about the growth mode transition. From the optical microscopic observation, the bacteria
distribute inside the medium, that is, the colony grows three-dimensionally (3D) in the soft agar
condition (CA ¡ 0:9 wt%). On the other hand, the bacteria distribute on the surface, that is, thecolony grows two-dimensionally (2D) in the hard agar condition (CA ¿ 0:9 wt%). It indicatesthat the mode transition occurs by changing the hardness of the medium. The mode transition
is con8rmed by the discontinuous change in the pattern diagram and in the colony growth rate,
which is de8ned as the expanding speed of the colony. Under the softer agar condition in therange of the 2D mode (0:9 6 CA 6 1:2 wt%), the induction time exists. Before the induction
time, the colony grows in 3D mode. The length of the induction time decreases with increasing
the bacterial number density. In addition, bacterial aggregation plays a key role on 2D growth.
These results suggest that the increasing bacterial number density causes bacterial aggregation
c 2002 Elsevier Science B.V. All rights reserved.
resulting in the mode transition. 
PACS: 05.−65.+b; 87.18.Ed; 87.18.Hf
Keywords: Self-organized systems; Aggregation and other collective behavior;
Spatio-temporal pattern formation; Growth mode transition; Colony growth rate

1. Introduction
During the last few decades, much attention was paid to self-assembly pattern formation in various 8elds from various viewpoints [1–3]. Above all, pattern formation∗ Corresponding author. Tel.: +81-3-5841-7309; fax: +81-3-5841-7279.

E-mail address: eiha@chemsys.t.u-tokyo.ac.jp (N. Eiha).

c 2002 Elsevier Science B.V. All rights reserved.
0378-4371/02/$ - see front matter 
PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 9 8 3 - 4

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in biological systems is most sophisticated and interesting. To understand the complex
mechanism of such pattern formation is our ultimate goal in spite of the complexity. The pattern of populations of simple biological objects which is dominated bypurely physical conditions may be a good example for the starting point. Here wewill pay attention to the colony formation of bacteria, which are one of the simplest
biological objects. It is known that some kinds of bacteria such as Bacillus subtilis(B. subtilis) exhibit various colony patterns on agar medium by changing the incubation condition, i.e. the agar and the nutrient concentration in the medium, the former
corresponds to the hardness of the medium [4,5]. Recently, various kinds of mathematical models were proposed to understand the mechanism of colony pattern formation
using reaction–diJusion equations, and many of the patterns were successfully reproduced by numerical simulations [6 –10]. However, some problems still remain. First, it
is reported that the colony forming rate strongly depends on the colony pattern [10,11].
But no model can reproduce the drastic change. Second, there is no model to be applied
for the transition between diJerent colony patterns. Finally, in these models the colony
is assumed to be formed on the surface of the medium. However, it is reported thatthe bacteria distribute inside the medium in the case of the soft agar [12]. Thus, thegrowth mode transition is expected in the bacterial colony by changing the hardness of
the agar. Observing and modeling such mode transition will be able to explain these
problems. However, there are few researches about the mode transition [12].
The purpose of our study is to investigate the mechanism of the mode transition. We observe the inLuence of the mode transition on the colony pattern and therate of the colony formation to propose a hypothesis of the mechanism of the modetransition.
2. Materials and methods
2.1. Bacterial strain
Bacillus circulans (B. circulans), which are used in this study, are rod-shaped
gram-positive bacteria with Lagella having the length of about 3–8 m and thediameter of 0.75 –1:0 m (Fig. 1). The species are moderate thermophilic and aerobic
bacillus, so they usually distribute in hot springs. B. circulans sporulate during
nutritional starvation [13] and lose their activity in motion and division. B. circulans
are known to move collectively in some conditions [14]. The aggregate of B. circulans slowly moves with rotation along the nutrient concentration gradient. B. circulans
form various characteristic and complicated colony patterns, but much less researches ofB. circulans have been reported than that of B. subtilis [15,16]. The reason seems tobe partly due to too much complexity of B. circulans colony pattern.
2.2. Agarose
Bacto Agar (Difco, Detroit, MI, USA) is agarose containing ash of concentration
less than 3:6 wt% and sulfate of 1:8 wt%. Agarose is a polysaccharide with alternating

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Fig. 1. The transmission electron micrograph of B. circulans.

copolymer of 1,4-linked 3,6-anhydro--L-galactose and 1,3-linked -D-galactose and it◦
is extracted from marine red algae. The agarose gel is stable and melts about 85 C.
The agarose gels consist of thick bundles of agarose chains and large pores 8lled withwater [17]. The average pore size is varied by the agarose concentration [18–20] andby cooling rate [20,21].
2.3. Growth culture
The culture medium on which B. circulans were cultured, was prepared as follows.
One liter of water solutions containing 10 g sodium chloride, and the nutrient of different concentration were prepared. Two kinds of nutrition were used: Bacto Yeast
Extract (YE) and Bacto Tryptonepeptone (TP). Both YE and TP are the mixture ofvitamin, amino acid, minerals, and carbon sources, such as sugar and protein. The
weight ratio between TP and YE is kept constant at TP=YE = 2. The total nutrient
concentration CN (TP + YE) is varied from 3 to 90 g=l. Bacto Yeast Extract and Bacto
Tryptonepeptone were purchased from Difco.
To solidify the liquid culture medium, Bacto Agar was added to the nutrient solution.
The concentration of the agar CA was varied from 0.4 to 3:0 wt%. The mixture was then◦autoclaved for 15 min at temperature 120 C and pressure 2 atm. After the autoclave,◦the mixture was cooled down to 80 C in the autoclave. Then a mixture of 15 ml was
poured into a circular plastic dish of diameter 86 mm, and the lid was put on the dish.

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The dish was cooled down to room temperature in ambient conditions. The agarose
solution gelled about 1 h after being poured.
After the gelation, the dish was turned upside down, and it was kept at room temperature for 2 days to remove excess water. The weight loss of the agar medium was
1.3–1:4 g during drying. As a result, the thickness of the agar medium was about 2 mm.
2.4. Inoculation and incubation
The bacteria were pre-incubated in the standard liquid Luria–Bertani (LB) medium◦
for about 20 h at 37 C. After the pre-incubation, the number density of the bacteria
was the order of 108 m l−3 . The bacterial suspension of 1 l was then inoculated on
the center of the agar medium by a needle (stick mate P, Ono, Japan). The diameterof the inoculated area was 3 mm. To prevent from drying, the dish with the lid wassealed with paraOn sheet and packed into the polyvinyl bag. The inoculated dish was◦
incubated for 3–10 days at 37 C in the incubator. The photographs of the bacterialcolony were regularly taken once a day by a digital camera. The area of the colony
was measured from the photograph by using an image analysis software (Image Pro).
3. Results and discussion
3.1. The growth mode transition
The morphology diagram of the colony patterns shows the existence of two typicalmodes in Fig. 2, that is, the pattern on the soft agar, and on the hard agar. The agar
concentration dependence is discussed in the case of CN = 15 g=l unless otherwise
stated.
Fig. 3 shows the cross-sectional optical micrographs of the colony in the case of thesoft and the hard agar. In the case of the soft agar (Fig. 3a), the bacteria uniformly distribute in the agar medium, i.e. the colony grows in the three-dimensional (3D) mode.
On the other hand, in the case of the hard agar (Fig. 3b), the bacteria distribute on
the agar surface, i.e. the colony grows in the two-dimensional (2D) mode. The growthmode transition from 3D to 2D mode occurs by increasing the agar concentration. In
the case of 3D mode (soft agar condition), the colony pattern named as a disk-likepattern is not aJected by CA and CN . On the other hand, in the case of 2D mode (hardagar condition), the colony patterns are aJected by CA and CN (Fig. 2).
Under the condition between two modes (oblique region in Fig. 2, 0:9 6 CA 6
1:2 wt%), the colony pattern sensitively depends on the agar condition as shown in
Fig. 4. Moreover, the cooling rate of the agar during gelation process and the wettabilityof the agar surface also aJect on the colony pattern (not shown). Strictly under thesame condition, the colony almost grows to the same pattern. However, the pattern in
the region is less reproducible than that of in the other region.
In this region, the colony abruptly expands during the growth process. Fig. 5 shows
the time evolution of the colony pattern. The colony gradually grows for several days
after the inoculation at 8rst. In this period, the growth is almost isotropic and theN. Eiha et al. / Physica A 313 (2002) 609 – 624

613

Fig. 2. The morphology diagram of B. circulans colony. Two typical phases A and B exist along CA . In
the phase A and B, the colony growth is three- and two-dimensional, respectively. These patterns are called
disk-like (A), knotted-branching (B1), concentric-ring (B2), and Eden-like (B3) pattern, respectively. The
oblique region represents the transition region and is shown in Fig. 4.

growth rate is quite low. We call the length of this period as induction time Tind . Aftert = Tind (t = 0 at the inoculation), the colony bursts at a few points with high growthrate, although the growth rate except for burst parts is close to zero. The colony results
in anisotropic pattern as seen in Fig. 5c.
The bacteria are observed inside the agar medium by using the optical microscope,
before Tind . However, the bacteria are observed only on the surface after Tind . Before
Tind the colony grows in 3D mode as well as in the soft agar condition, and after Tindthe colony grows in 2D mode as well as in the hard agar condition. Since, this growthmode transition from 3D to 2D mode is irreversible, 2D mode is more stable than the
3D mode. The transition region is de8ned as the region in which the induction time
exists. This region is identical to the oblique region in Fig. 2. The transition regionslightly shifts to lower agar concentration condition with increasing CN as shown in
Fig. 2.
The colony growth rate is measured to characterize the mode transition. The colony
growth rate is de8ned as the increase rate of the average radius of an expanding
colony. To estimate the colony radius, the area of a colony is measured from its
photograph (for the details of the estimating method of the growth rate, see
Appendix A).
The time dependence of the colony growth rate is shown in Fig. 6. In the transition
region, the growth rate discontinuously increases at about 130 h for CA = 1:0 wt%,

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Fig. 3. The cross-sectional optical micrograph and the schematic illustration of the colony formed on soft
agar (CA = 0:6 wt%) (a), and on hard agar (CA = 1:5 wt%) (b). The nutrient concentration is 8xed at
CN = 15 g=l.

which corresponds to the induction time. The dependence of CA on the growth rateis shown in Fig. 7. In the case of 3D mode, the average of the growth rate is usedbecause the growth rate does not change with time (Fig. 6). On the other hand, in thecase of 2D mode, the maximum of the growth rate is used because the growth rate
decreases with time.
In the case of the soft and the hard agar conditions, the growth rate decreases withincreasing CA . The colony growth rate discontinuously increases at CA = 0:9 wt%.
In the transition region, two growth rates are plotted. The lower and higher values
correspond to 3D growth (before Tind ) and 2D growth (after Tind ), respectively. In
general, such discontinuous changes and the existence of the induction time are thetypical characteristics of the phase transition.
3.2. The mechanism of the mode transition
We propose a following hypothesis as a mechanism of the mode transition. The
number of large pores which bacteria can pass through decreases with increasing theagar concentration. As the number of large pores decreases, the probability that twoneighboring places are linked by the passable pores decreases. The percolation theory
tells that such system produces phase transition. At a threshold of agar concentration,

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615

Fig. 4. The morphology diagram at the transition region. The photographs of colony patterns are taken in
the 8nal states, that is, no more signi8cant change is observed by additional incubation.

the growth rate of 3D mode abruptly drops to zero. The hypothesis seems to explain
the abrupt mode transition observed on our study.
Fig. 8 shows the microstructure of the agar in the diJerent agar concentration conditions taken by transmission electron microscopy (TEM). The average pore size isestimated at about, 2:0 m (Fig. 8a, CA = 0:5 wt%), 0:6 m (Fig. 8b, CA = 1:0 wt%),
and 0:3 m (Fig. 8c, CA = 1:5 wt%) from TEM micrographs. Since the average poresize is almost equal to the diameter of B. circulans of about 1:0 m at the transition
region, the bacteria cannot move inside the medium in the case of the hard agar.
The second characteristic in the transition region is the existence of Tind . The dependence of the Tind on CA and CN is shown in Fig. 9. Tind decreases with increasing
both CA and CN . The hypothesis can explain the dependence of Tind on CA because
the mobility of bacteria inside the media decreases with increasing CA . But the hypothesis cannot explain the dependence of Tind on CN because the gel microstructure isnot signi8cantly aJected by CN [17]. And it cannot explain the shift of the transition
region to lower agar concentration with increasing CN (Fig. 2). These results suggest
the existence of another factor.
The optical micrographs of the morphology of bacteria in the case of 3D and 2D
growth mode are shown in Fig. 10. The bacteria are almost isolated in the case of 3Dmode (Fig. 10a), on the other hand, the bacterial aggregates are observed in the caseof 2D mode (Fig. 10b). In general, at the hard agar condition, most species of bacteria
cannot expand on the hard surface because the mobility of a single bacteria on the616

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Fig. 5. The time evolution of the colony in the transition region. Each image is taken 6 (a), 7 (b), 8 (c)days after the inoculation (d). The schematic illustration for the measurement of the growth rate.

Fig. 6. The time dependence of the colony growth rate at each agar condition. Circle, square, and triangle
correspond to the growth rate at the soft agar (CA = 0:6 wt%), the moderate agar (CA = 1:0 wt%), and thehard agar condition (CA = 1:5 wt%), respectively.

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617

Fig. 7. The dependence of the average growth rate on the agar concentration CA . Filled circle and 8lled
square correspond to 2D and 3D mode, respectively.

Fig. 8. TEM micrographs of the agarose gels in diJerent CA : (a) CA = 0:5 wt%; (b) CA = 1:0 wt%; and(c) CA = 1:5 wt%. The black 8ber corresponds to the agarose bundle. The TEM samples are preparedaccording to the method of Ref. [20].

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Fig. 9. The inLuence of CA (a) and CN (b) on the induction time Tind .

agar surface is quite small [22]. Only some bacterial strain, such as Proteus mirabilis(P. mirabilis) [23], B. subtilis [7,24], and some kinds of Paenibacillus [12,25], form
the large 2D colony on the hard agar utilizing the cooperative intercellular interaction,e.g., forming a swarmer cell, secreting lubricant Luid, and forming a bacterial aggregate.
On the other hand, the collective motion of B. circulans is known to form a motile
sub-colony [13]. According to our aforementioned experimental results and the previous
study, it can be concluded that to form the aggregates is necessary for 2D growth in
the case of B. circulans.
As another hypothesis, it is proposed that the increasing bacterial number density
causes the bacterial aggregation like the phase transition from the gas phase to theliquid phase. In general, the aggregation of particles relates to its number density. In
the dispersion system, e.g. colloidal suspension, nucleation takes place and the resulting aggregation is universally observed at the high number density of particles bothexperimentally [26] and theoretically [27]. Similarly, most bacteria coagulate above thecritical coagulation concentration (CCC) in liquid culture [28]. In agarose gel culture,
the aggregate should be formed in the same way. The aggregate moves around on
the surface to produce 2D pattern. This hypothesis can explain both the abrupt modetransition and the mechanism of 2D growth.
To testify the hypothesis, it is necessary to show that increasing CA and CN cause
higher bacterial number density. The dependence of CN on the bacterial number density would be self-evident. Let us show the relation between CA and the number
density. Because the mobility of bacteria in the medium decreases with increasing CA(Fig. 8), we investigate the relation between the bacterial mobility in the medium andthe bacterial number density by mathematical modeling. In the case of 3D growthmode, because the colony pattern and the colony growth rate are not signi8cantly affected by nutrient concentration, it is suggested that the colony growth rate in this mode
is limited by the mobility of the bacteria in the agar (Fig. 11). Let us assume that thebacterial movement could be regarded as the simple diJusion of particle and use thereaction–diJusion model. The reaction–diJusion equations for the bacterial density b

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619

Fig. 10. The optical micrograph of bacteria at the expanding periphery of the colony of the disk-like pattern(CA = 0:6 wt%; CN = 15 g=l) (a) and the knotted-branching pattern (CA = 1:5 wt%; CN = 15 g=l) (b). The
sample for the optical microscope observation is prepared as follows. The bacteria at the periphery of thecolony are picked by needle and then dispersed in about 200 l water solution of NaCl of concentration
0:18 wt%. The dispersion was dried on slide glass and then the bacteria was stained by safranine. The
aggregates of bacteria is also observed in the dispersed solution.

and the nutrient concentration n are described as@bknb;= Db ∇ 2 b +@t
KS + n

(1)

@n
kn
b;
= Dn ∇ 2 n −@tKS + n(2)where Db and Dn are the diJusion coeOcient of bacteria and nutrient, respectively.
Here k; KS , and  are the reaction rate constant, Michaelis constant, and the nutrient
eOciency, respectively. Second terms on the right-hand side in Eqs. (1) and (2) correspond to the reproduction of bacteria and the consumption of nutrition, respectively.
To perform quantitative analysis, the values of the parameters are estimated by ourexperiments or quoted from the previous experimental studies (see Appendices B and
C for details).

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Fig. 11. The dependence of the colony growth rate on CN at the soft agar condition. The agar concentration
is 8xed at CA = 0:5 wt%.

Fig. 12. The spatio-temporal distribution of the bacterial number density calculated from Eqs. (1) and (2).
Solid and dashed lines represent Db =1:3×10−4 cm2 =h (at t=200, 250, and 300 h) and Db =1:3×10−5 cm2 =h(at t = 700, 750, and 800 h), respectively.

The spatio-temporal bacterial number density distribution is calculated (Fig. 12). The
bacterial number density distribution obeys a traveling wave solution. It is consistent
with the experimental results in the point that the growth rate is constant during the
growth process and that the density distribution is uniform. As diJusitivity decreases,
the velocity of the traveling wave decreases and the number density increases. Thus, theincrease of CA causes the larger bacterial number density due to the smaller diJusion

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621

coeOcient. These results support the second hypothesis that increasing bacterial density
causes formation of the bacterial aggregate, which migrates on to the surface of themedium. The induction time would be the time until the bacterial density in the medium
reaches certain critical density.
4. Conclusion
The mode transition from 3D to 2D mode by increasing the agar concentration isobserved in B. circulans colony. The mode transition relates to the change of the colony
pattern and to the discontinuous change in the colony growth rate. In the transition
region, the growth mode is changed from 3D to 2D during the growth process andas a result, the colony growth rate increases discontinuously. From the dependenceof the induction time on the agar and the nutrient concentration, it is suggested thatthe increasing bacterial density and resultant bacterial aggregation causes the modetransition.
Acknowledgements
We thank Dr. Shimada of National Institute of Infectious Diseases for providing uswith the bacterial strain.
Appendix A. The method of calculating colony growth rate
In the case of the isotropic growth (except for the case after induction time in thetransition state), in which the colony can be regarded as a circle, the colony growthrate dR=dt is
1 d √
dR
=√
(A.1)
( A) ;dt dt
where R and A are the average radius and the area of the colony, respectively. The
colony area A is measured from the photograph of the colony. In the case of the hardagar, the average radius R is estimated smaller than the apparent radius because oflower fractal dimension of the colony pattern.
In the case of the anisotropic growth, which corresponds to the case after the induction time in the transition state, the colony growth rate is calculated as


dR2
2
d
(A.2)
=
(A − R21 ) + R21 ;dt
dt
where R1 ; R2 , and  are the radius of the isotropic disk-like part formed before theinduction time, the radius of the anisotropic fan-shaped part formed after the induction
time, and the spreading angle of the anisotropic part, respectively (see Fig. 5d). A and 
are measured from the photograph of the colony. Since the growth rate of the isotropic
part is quite low, R1 is regarded as a constant and the average radius measured just

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before the mode transition is used as R1 . The estimated growth rate is always smaller
than the actual rate because the colony pattern is not a complete sector shape and the
trajectory is not straight.
Appendix B. The details of the calculating conditions
B.1. The values of the parameters used in the calculation
To perform quantitative analysis, the values of the parameters are estimated by ourexperiments or quoted from the previous experimental studies. The relationship between
the cell division time and the nutrient concentration is 8tted to the Mechaelis–Menten
relation using the experimental data in liquid culture. As a result, k and KS are estimatedas k = 0:6 h−1 and KS = 1:5 g=l, respectively. The nutrient eOciency  is in the order of
10−9 g. The diJusion coeOcients of the nutrient molecules such as glucose and small
protein are not aJected by agar concentration in the rage of CA ¡ 7:0 g=l [22]. Thus
Dn = 2:3 × 10−2 cm2 =h is regarded as a constant, which is the value in the water [22].
B.2. Boundary conditions and other calculating conditions
The equations are calculated numerically in one-dimensional form. The length of thespace L is set in 4:3 cm, which is equal to the dish radius, and the space is cut by
700 meshes. At the initial condition, n is homogeneously distributed with n = CN andbacteria is inoculated at x = 0. Dirichlet boundary condition (n|x=L = CN ) and Neumannboundary condition


 
9b 
9n 
= 0;
=0
9x x=09x x=0; L
is used.
Appendix C. The relation between the agar concentration CA and the bacterial
di#usion coe$cient on Db
To see the dependency of CA on Db quantitatively, the colony growth rate of theexperimental data (Fig. 7) is compared to that of the calculated results. The colony
growth rate in the model is de8ned as the proceeding velocity of the traveling wave.
The dependence of Db on the colony growth rate is calculated as shown in Fig. 13a.
Fig. 13b shows the estimated relationship between CA and Db . Db decreases withincreasing CA . The diJusion coeOcient of the bacteria in CA = 0 is extrapolated
in the order of 10−2 cm2 =h. On the other hand, the theoretical diJusion coeOcientof the particle of diameter 1 m in water is in the order of 10−6 cm2 =h from the
Stokes–Einstein relation. The diJerence could be due to the mobility of bacteria. Fromthe published data [29], the diJusion coeOcients of some other motile bacteria such
as E. coli are measured experimentally, and they are in the range from 10−2 toN. Eiha et al. / Physica A 313 (2002) 609 – 624

623

Fig. 13. (a) The dependence of the diJusion coeOcient of bacteria Db on the colony growth rate estimatedfrom Eqs. (1) and (2). (b) The relation between CA and Db . Filled circle represents the data estimated bycomparing the experimental data with the calculated result.

10−3 cm2 =h in water. Because of the good agreement with the previous study andour calculated result, it can be regarded that the calculated diJusion coeOcient ofbacteria in the agar medium is reliable.
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