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Periodic growth of bacterial colonies
Physica D 205 (2005) 136–153
Periodic growth of bacterial colonies Yoshihiro Yamazakia , Takemasa Ikedab , Hirotoshi Shimadab , Fumiko Hiramatsub , Naoki Kobayashib , Jun-ichi Wakitab , Hiroto Itohb , Sayuri Kurosub , Michio Nakatsuchib , Tohey Matsuyamac , Mitsugu Matsushitab,∗ a Department of Physics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan Department of Physics, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan c Department of Bacteriology, Niigata University School of Medicine, Niigata 951-8510, Japan
b
Available online 12 February 2005
Abstract The formation of concentric ring colonies by bacterial species Bacillus subtilis and Proteus mirabilis has been investigated experimentally, focusing our attention on the dependence of local cell density upon the bacterial motility. It has been confirmed that these concentric ring colonies reflect the periodic change of the bacterial motility between motile cell state and immotile cell state. We conclude that this periodic change is macroscopically determined neither by biological factors (i.e., biological clock) nor by chemical factors (chemotaxis as inhibitor). And our experimental results strongly suggest that the essential factor for the change of the bacterial motility during concentric ring formation is the local cell density. © 2005 Elsevier B.V. All rights reserved. Keywords: Bacterial colony; Concentric ring pattern; Bacterial motility; B. subtilis; P. mirabilis; Cell density; Replica-printing
1. Introduction In order to understand pattern formation in nature by physics, it is important to clarify the dynamics of the formation of spatiotemporal structure characterizing the pattern and to extract its universality [1]. For this purpose, we usually focus our attention on the pattern formation in two-dimensional systems, such as mineral dendrites on the surface of rocks [2] and water imbibition in a thin paper sheet [3], for the reason that the pattern formation in two dimensions is easily reproducible experimentally and analyzed both numerically and theoretically. For example, when we pour solution of some chemicals which exhibit Belousov–Zhabotinsky (BZ) reaction into a Petri dish thinly, time evolution of target and rotating spiral patterns originating from spatiotemporal change of ∗
Corresponding author. Tel.: +81 3 3817 1787; fax: +81 3 3817 1792. E-mail addresses: [email protected] (Y. Yamazaki), [email protected] (M. Matsushita). URL: http://www.phys.chuo-u.ac.jp/labs/matusita. 0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2004.12.013
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ion concentrations essential for these pattern formations can be observed [4]. The pattern formation in the BZ reaction is produced due to both the oscillatory or excitable properties of the chemical reactions and spatial inhomogeneity of the solution in a Petri dish. Therefore, the formation of target and spiral patterns in two-dimensional system of the BZ reaction has been modeled by constructing reaction–diffusion equations of the ion concentrations. As another system for the pattern formation in two-dimensional system, much attention has been paid to bacterial colony growth on the surface of an agar plate [5]. In general, in order to know the whole types of patterns experimentally obtained in a certain system, we often produce a morphological diagram of the patterns by identifying control parameters for the environmental condition of the system and by varying them. With respect to the growth of bacterial colonies on the surface of an agar plate, important control parameters are agar concentration Ca and nutrient concentration Cn in the agar plate, because the bacterial proliferation and its motility are affected by Ca and Cn . Therefore, we can summarize the types of the colonies of a bacterial species in a morphological diagram as a function of both Ca and Cn . One of the important points in studying the bacterial colony growth from physical viewpoint is that spatiotemporal behavior of bacterial growth and motion can be readily observed in a wide range of scales with optical and stereo microscopes. In microscopic scale, which is the order of a micron, multiplication of bacteria and their individual motion are visible. In mesoscopic scale, which is the order of a tenth of a millimeter, the collective motion of bacteria is observed. And in macroscopic scale, which is the order of a centimeter, the whole morphology formation of bacterial colonies is characterized. In practice, the process of bacterial colony formation has been studied experimentally with a variety of bacterial species such as Bacillus (B.) subtilis [5], Proteus (P.) mirabilis [6], and Serratia (S.) marcescens and so forth [7]. If we consider a group of bacteria on the surface of an agar plate as their local population density in a mesoscopic grid whose size is about 100 m typically, we can construct a model for the time evolution of the bacterial population density in each grid. In terms of model construction, there exists a remarkable similarity between the bacterial colony formation and BZ reaction. In other words, it is possible to construct a reaction–diffusion like model for the bacterial colony growth by considering local bacterial population, bacterial proliferation and its motility as ion concentration, chemical reaction and diffusion, respectively. So far, we have carried out a lot of experiments about the colony growth of bacterial species B. subtilis and P. mirabilis [5,6]. As for B. subtilis, the wild-type strain OG-01, which was used in our experiment, is rod-shaped (0.5–1.0 m in diameter, 2–5 m in length) with flagella and is motile in water by collectively rotating the flagella. When the environmental condition is unfavorable to the bacteria, such as on a nutrient-poor or dry agar plate, the bacterial cells become spores. On the other hand, P. mirabilis is a flagellated rod-like bacteria and takes two states. One is a normally flagellated vegetative swimmer cell (1.5–2.0 m in length), having 6–10 peritrichous flagella. The other is a hyperflagellated long swarmer cell (10–80 m in length) with 103 − 104 peritrichous flagella. These two states are transformed into each other alternately through cell differentiation and dedifferentiation processes only when P. mirabilis lives on the surface of substrate medium such as agar gels. Each bacterial species used for the study of colony formation has different structures in physiological and genetic properties such as cell wall, DNA sequence, and so on. However, the bacterial species we have treated have the similarities that they are rod-shaped cell with flagella and possess the motility to expand their colony actively. We consider these similarities and investigate the mechanism of bacterial colony formation irrespective of their detailed biological features. In this paper, we briefly explain the morphological diagrams of B. subtilis and P. mirabilis first. Then, by focusing our attention on the concentric ring pattern of bacterial colony obtained by both species, we report the results of our experiments about the formation of the concentric ring colonies. We conclude that the periodic change of bacterial motility in time is the essential factor for the formation of the concentric ring colonies and argue that the origin of its periodic change strongly depends on the local population density at the growth front of a colony. Finally, we show the results of our recent experiment about the colony formation of S. marcescens.
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2. Colony growth and morphological diagram The experimental setup and procedure for producing bacterial colonies of B. subtilis and P. mirabilis can be summarized as the following four processes: (i) preparation of an agar plate for the observation of colony growth, (ii) preparation of liquid culture of sample bacteria, (iii) inoculation of the liquid culture on the agar plate, and (iv) incubation of the bacteria inoculated on the agar plate. The detailed explanation for these processes were described elsewhere [8–10]. Macroscopic colony patterns and their growth were observed through a camera and a CCD camera and recorded on photographs, video tapes or hard discs. The microscopic and mesoscopic movement of individual bacterial cells were observed through an optical microscope and recorded on photographs, video tapes or hard discs. Fig. 1 shows the typical morphology diagram of B. subtilis obtained so far as a function of both Ca and Cn [8,9]. As shown in Fig. 1, the morphology diagram is divided into five regions by the pattern characteristics of bacterial colonies: region A (diffusion-limited aggregation (DLA)-like), B (Eden-like), C (concentric ring-like), D (homogeneously spreading disk-like), and E (dense branching morphology (DBM)-like). This fact suggests the existence of the physical mechanism for the bacterial colony formation irrelevant to the detail of their biological properties. In addition, it is noted that the growth rates were also very different among these five morphologies. Actually, it has been found that growth to the size of about 5 cm (about half a diameter of Petri dishes) required about a month in the region A, a week in the region B, a day in the regions C and E, and half a day in the region D. Regarding the growth of the bacterial colony, the following two features are important: (i) Lowering of nutrient concentration Cn causes a tip-splitting of the growth front of a bacterial colony such as DLA-like pattern (region A) and DBM-like pattern (region E). (ii) The softness of the agar has a great influence on the motility of bacterial cells. In regions A and B where agar plates are hard, individual cells do not move actively. And cells are connected with each other and form a bundle of chains. Therefore, the colony expansion is done in the way cells, which are located in the outermost part of a colony and have access to nutrient easily, proliferate by cell division. Meanwhile, cells in the inner part change to spores and enter into a resting phase. On the other hand, the colony expansion in the regions D and E, where agar plates are softer, takes place due to a random walk-like movement of individual active cells [9,11]. The growth of the bacterial colony in the region C between the regions B and D is quite different from that in the other four regions: A concentric ring-like pattern in this region involves a rhythmic growth having a characteristic period. It has been found from a mesoscopic observation that a concentric ring-like pattern is formed by alternating the region D-like colony expansion (very fast) and the region B-like growth (very slow) [12]. The similar morphological diagram has been obtained for P. mirabilis. From Fig. 2 it is found that colony patterns can be classified mainly into three types [6]; disk-like spreading growth (region P), diffusion-limited growth (region Q), and three-dimensional growth inside the agar medium (region R). The patterns observed in the regions Q and
Fig. 1. Morphology diagram of B. subtilis as a function of nutrient concentration Cn and the inverse of agar concentration Ca .
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Fig. 2. Morphology diagram of P. mirabilis as a function of nutrient concentration Cn and the inverse of agar concentration Ca .
R are considered to be typical ones in the limiting cases where Cn is low and Ca is high, and where Ca is low, respectively. On the other hand, the region P is an intermediate one, where the motility and the transformation of the cell body are apparently considered to strongly influence the pattern formation. Actually the morphologies in the region P is classified into three subgroups: concentric ring pattern (Pr ), homogeneous pattern (Ph ), and transient spatio-temporal pattern (Ps ) [13]. It is well-known that the concentric-rings observed in the region Pr are well correlated circumferentially. And in the region Ps one observes target and rotating spiral patterns [14] which quite resemble those in chemical reaction system [15], liquid crystal system [16] and so on. From the above two cases of concentric ring patterns, the temporal change of the bacterial motility is considered to be important for the periodic growth into concentric ring patterns. Then, in the following sections, we summarize the observed findings of the formation of concentric ring colony patterns. We also show some experimental ideas for the investigation of the origin of the periodic growth.
3. Observed findings Fig. 3 is a typical concentric ring pattern of B. subtilis obtained in the region C. We have confirmed that the gray scale of the concentric rings corresponds to the change of the collective motion of bacteria in time during the colony expansion. In order to understand the origin of these spatially periodic profiles, we have mesoscopically observed the motility of bacterial cells during the colony formation and obtained the following findings [12]: (i) When the liquid culture is inoculated at the center spot on the surface of the agar plate, there is a time lag until the colony begins to expand (the existence of lag phase). During the lag phase, the bacteria proliferate by cell division at the point-inoculated center spot without any migration. (ii) After the lag phase, each bacterial cell becomes active and seems to move individually at random. Bacterial cells go outward collectively by forming a single layer of cells, and the colony expands fast. The motion of bacterial cells during the colony expansion is similar to that in the region D. We call this period for colony expansion migration phase. (iii) When migration phase continues for a period of time, the expansion of the colony stops rather abruptly. The mesoscopic observation showed that bacterial cells, meanwhile, proliferate without active motion and form a bundle of chains by connecting with each other. The behavior of bacterial cells in this period after migration phase is quite similar to that of Eden-like pattern formation in the B region. It is noted that the local density
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Fig. 3. Concentric ring-like pattern formed by B. subtilis when Ca = 6.8 g/l and Cn = 40 g/l. The photograph was taken 2 days after inoculation. The inner diameter of Petri dish is 8.8 cm.
of bacterial cells in the vicinity of the colony front increases drastically. We call the period for proliferation without active bacterial motion consolidation phase. (iv) The consolidation phase also finishes after a while and the migration phase as stated in (ii) starts and the colony expands again. Then, the concentric ring pattern is found to be formed due to the periodic change of the bacterial motility in alternate migration phase and consolidation phase. (v) From the shape of the region C in the morphology diagram (Fig. 1), it is found that the formation of concentric ring patterns is quite sensitive to the softness of the agar plate, while it is insensitive to the richness of the nutrient. In fact, with respect to the influence of both the softness of agar plate and the richness of the nutrient upon migration time and consolidation time which characterize the periodicity of the colony formation, we have already obtained the following results shown in the previous paper [12]: migration time is a weakly decreasing function of Ca , but it seems to be independent of Cn . Consolidation time is a weakly increasing function of Ca , but it seems to be independent of Cn . Cycle time which is the sum of migration time and consolidation time seems to be constant, independently of both Ca and Cn . (vi) B. subtilis secretes a surfactant called surfactin while forming a colony. The surfactin relaxes the effect of the surface tension of water on the bacterial motion and enhances the motility of the active bacterial cells. And the colony morphology in the presence of surfactin changes drastically in comparison with the morphology in its absence. It is then considered that surfactin may play a crucial role for the periodic change of bacterial motility in the region C. However, in the experiment by using the mutant cells which are obtained from OG-01 by nitrosoguanidine-mutagenesis and are defective in producing surfactin, it has been confirmed that a concentric ring pattern is also reproducible. Therefore, the secretion and existence of surfactin is considered not to be essential for the appearance of a concentric ring colony pattern. P. mirabilis also forms a concentric ring pattern [10], as shown in Fig. 4 [17]. As in the case of B. subtilis, the migration–consolidation mechanism is also essential for the formation of a concentric ring pattern by P. mirabilis. As seen in Fig. 4, the circumference of each circle by P. mirabilis is much smoother than that by B. subtilis. From the microscopic viewpoint, there are two cell states for P. mirabilis cultivated on the surface of an agar plate: swimmer
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Fig. 4. Concentric ring-like pattern formed by P. mirabilis. The photograph was taken 2 days after inoculation. The inner diameter of Petri dish is 8.8 cm.
cell and swarmer cell as described above. The typical feature is that P. mirabilis executes a cyclic transition of differentiation and dedifferentiation between these two cell states. With respect to the formation of a concentric ring pattern in the case of P. mirabilis, migration phase corresponds to the expansion period of colony growth by swarmer cells, and consolidation phase to the period when the expansion of a colony stops by shortening the cell length, i.e. by becoming swimmer cells. Interestingly, it is known that the dependencies of migration time and consolidation time for B. subtilis upon both Cn and Ca explained above have also been confirmed by P. mirabilis, which means that the cycle time of P. mirabilis is also constant, independently of Cn and Ca [10,17]. Furthermore, from mesoscopic observations, one or more internal waves were sometimes observed to advance toward the growth front from the inner terrace during migration phase in both species. They were seen with naked eyes, too. In the case of B. subtilis, the internal waves consisted of many active bacterial cells, which originated around the edge of the inner terrace. On the other hand, in the case of P. mirabilis, many internal waves also advance successively towards the outermost terrace front, even after the front stopped moving, as shown in Fig. 5. Each internal wave in a concentric ring colony of P. mirabilis consists of a superimposed monolayer of swarming cells moving towards the growth front. 4. Experimental ideas So far, many studies on the origin of a concentric ring pattern formation of both P. mirabilis and B. subtilis have been done. Among them, there exist experimental studies with physical approaches which are not concerned with the detail of genetic and biological information. Here we summarize several experimental ideas about the investigation of the periodic change of bacterial motility [10,17–19]. (i) Some chemicals secreted by bacteria during their proliferation may have the ability to affect their motion and then their colony morphology. Especially, on the analogy of target patterns observed in the reaction–diffusion system such as the BZ reaction, there is the possibility that a concentric ring pattern in the bacterial colony is formed in the same way as the above target pattern in the BZ reaction. If so, the local cell density of bacteria
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Fig. 5. Snapshot of internal waves in the case of P. mirabilis. The right edge shows the growth front of the colony. Internal waves advance toward the growth front from inside. The width of the snapshot is about 1.3 cm.
corresponds to an activator. Moreover, it is necessary for the formation of the concentric ring pattern that there are pacemaker cells which secrete some chemical periodically at the center of the concentric ring colony. Then, the following experiment can be considered in order to check whether any chemical as an inhibitor and pacemaker cells for the appearance of concentric ring exist or not: Cut a colony together with an agar plate along the dotted lines with a surgical knife in Fig. 6(b) during its growth. Fig. 6(a) shows an intact, reference colony. Then the part of the colony represented by the darker region in Fig. 6(b) is isolated from cells at the center spot which are considered to be a pacemaker for the concentric ring pattern formation. If some chemical and pacemaker cells are needed, this isolated part of the colony must grow and produce a colony whose morphology is different from the previous one. (ii) The formation of the concentric ring pattern in bacterial colonies results from periodic oscillation of the motility of bacteria. It has been revealed that the cycle time characterizing the periodic growth does not depend on Cn and Ca for both B. subtilis and P. mirabilis. Hence in order to investigate whether the origin of the stable periodicity
Fig. 6. Schematic representation of experimental procedure to check whether some chemical as an inhibitor and pacemaker cells are necessary for concentric ring pattern formation or not. (a) Intact, reference colony. (b) A colony together with an agar plate is cut along the dotted lines.
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in the cycle time is physical or biological, the following experiment has been done: after point-inoculation of the liquid culture at a spot on the surface of an agar plate, we additionally inoculate again at a different spot apart from the previous one with some time interval. The time interval in the point-inoculation at the two spots brings the phase difference between two concentric ring colonies. If this periodic change of motility is derived from biological properties of bacteria, e.g. from the collective synchronization of biological clocks in bacterial cells, the period of the formation of concentric rings in the two colonies must be also synchronized after two colonies collide. (iii) The measurement of lag phase time is important to understanding the beginning of a colony growth and the emergence of migration phase. Especially, by varying the initial concentration of liquid culture inoculated at the center spot on the surface of an agar plate, this measurement has a clue to clarifying the dependence of the onset of migration phase on the cell density. (iv) From the result of (iii), the periodicity in the bacterial motility is considered to reflect physical properties as the group of bacterial cells. Then we can focus our attention on the dependence of the local cell density on their motility. In practice, we sector a part of the outermost ring of a colony together with an agar plate during the migration phase of the colony growth, as shown in Fig. 7. And we investigate the variation of ring formation for removing the influence from the inner concentric rings. (v) It is found from the above result that bacterial motility in migration phase depends strongly on the cell density of active bacteria. In order to understand the collective behavior of active bacteria in a colony, there is a simple idea that a sectorial part of a colony is excised and immediately the surface of the sectorial part is attached onto another fresh agar plate and separated (replica-printing). By this method, bacteria in migration and consolidation states of a colony can be picked up on fresh agar plate, and we can observe the motility of bacteria in each state. Consequently, we are able to investigate the change of the position in a colony where bacteria move actively by varying the time when replica-printing is carried out.
5. Results In this section, we report the experimental results with respect to the appearance of concentric ring colony patterns by B. subtilis [18] and P. mirabilis [17,19], based on the ideas as stated in the previous section.
Fig. 7. Schematic explanation for the investigation of the influence of the consolidation terrace just behind the growth front in migration phase upon the colony expansion. (a) Intact, reference colony. (b) A colony together with an agar plate is cut along the dotted lines.
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5.1. Existence of a pacemaker As in the Section 4 (i), we examined whether the chemical as an inhibitor and the pacemaker for the formation of concentric ring patterns by B. subtilis are necessary or not. Fig. 8 shows two snapshots of the experimental result. The left photo (a) shows the colony immediately after removing the inner sectorial part of the colony. The right (b) was taken 7 h after the left. It is found from these snapshots that the formation of a concentric ring colony is not influenced even if the inner sectorial part including the center spot of the colony is removed. Therefore, we can conclude that both the chemical and the pacemaker do not exist for the concentric ring formation by B. subtilis. In the case of P. mirabilis, it has also been confirmed that the colony on the isolated part grows into almost the same concentric ring pattern as the colony on the rest part, as shown in Fig. 9. Therefore, it is concluded that macroscopically any chemicals and the pacemaker cells are not necessary for the formation of the macroscopic concentric ring colonies. 5.2. Phase entrainment It has been known from the previous study that the cycle time of B. subtilis is about 7 h. Then we carried out the following experiment as introduced in the Section 4 (ii): We inoculated the liquid culture at a spot on the surface of an agar plate. After 2 h from the first inoculation, we additionally inoculated the liquid culture at the spot 2.5 cm apart from the previous spot. This procedure introduces phase difference in the formation cycle of concentric rings between two colonies. When these two colonies expand, they collide as shown in Fig. 10. We find that two phases of the cycle of the concentric ring formation are not synchronized with each other even after the collision of these colonies. In fact, Fig. 10 shows us that cells at the growth front of the colony A are in the fourth consolidation phase, whereas that of the colony B are still in the fourth migration phase. Furthermore, the boundary of the growth front between the colonies A and B as shown in Fig. 10(b) is not blurred even if the colonies expand. From these results, it is concluded that the phase entrainment due to biological properties of the bacterial cells such as biological clocks, if any, is limited only in the local region where bacteria of colony A and B mix together. In the case of P. mirabilis, it has also been found that the period of concentric ring in the two colonies is not synchronized even after they collide, and that concentric rings in the two colonies are produced independently of each other, keeping their own phases. Fig. 11 shows the experimental result for P. mirabilis. In this case, we inoculated the liquid culture at a spot on the surface of an agar plate. After 2 h from the first inoculation, we additionally inoculated the liquid culture at the spot
Fig. 8. Experimental results for the necessity of a pacemaker in the concentric ring-like pattern formation of B. subtilis. The diameter of these figures is 8.8 cm.
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Fig. 9. Experimental results for the necessity of a pacemaker in the concentric ring-like pattern formation of P. mirabilis.
1 cm apart from the previous spot. As seen in this figure, two colliding colonies showed the same periodicity and remained out of phase with each other during their expansion. Therefore, the biological properties are considered not to macroscopically affect the formation of the concentric ring pattern. 5.3. Lag phase According to the idea explained in the Section 4 (iii), we investigated the relation between lag phase time and the initial concentration of the liquid culture for B. subtilis. Fig. 12 shows the dependence of the initial cell density on the lag phase time. The cell density at 100 on the abscissa in the figure corresponds to that of our standard inocula, 104 cells/l. It is found from this figure that the lag phase time is a monotonically decreasing function of the initial
Fig. 10. Experimental results for the possibility of phase entrainment between two colonies with different phases of periodic colony expansion in the case of B. subtilis.
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Fig. 11. Experimental results for the possibility of phase entrainment between two colonies with different phases of periodic colony expansion in the case of P. mirabilis.
cell density up to about 101 , and that the lag phase time seems to saturate to a fixed value (about 7 h in this case) over 101 . The similar experiments for P. mirabilis have revealed that lag phase time is also a monotonically decreasing function of the initial density below some value of the density, and that above this value it saturates to a constant value. Fig. 13 shows the lag phase time as a function of the initial concentration of the liquid culture for P. mirabilis. It is found from these figures that the cell density has to increase to a certain value for the beginning of a colony growth and that there exists the threshold density value for the migration phase. 5.4. Cutting In order to investigate the dependence of the cell density upon the end of migration phase for B. subtilis, we carried out the following experiment based on the idea explained in the Section 4 (iv): we divided a colony together
Fig. 12. Lag phase time as a function of the initial concentration of the liquid culture of B. subtilis. The unity 100 corresponds to the standard cell concentration of liquid culture (104 cells/l) for inoculation.
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Fig. 13. Lag phase time as a function of the initial concentration of the liquid culture of P. mirabilis. The unity 100 corresponds to the standard cell concentration of liquid culture (106 cells/l) for inoculation.
with an agar plate into two halves, and we cut and removed the first terrace from the one half of the colony 15 min after the beginning of the second migration phase. The other half of the colony remained intact as reference. We compared the growth of the case of cutting with that of the reference. It should be noted that in the case of cutting there exist only active bacteria which came from inside the first terrace. We confirmed that the width of the second terrace in the case of cutting is smaller than that in the reference part. From the third terrace on, however, the width of terrace is almost the same in both halves. Fig. 14 shows the time needed for second, third and fourth cycles of each half. (The suffix “r” at the terrace number in the figure signifies “reference”.) The cycle time is defined as the sum of migration time and consolidation time, and was obtained by the average of 10 trials in our experiment. It is found from this figure that the cycle time seems to be insensitive to the cutting of the first terrace. However, there is a tendency that the migration time and consolidation time of the second terrace are shorter and longer than those of the second reference terrace, respectively. In the following, we consider the time difference in migration time and consolidation time between the case of cutting and the case of reference of each terrace. Fig. 15 shows our result that the migration time and consolidation time of the second terrace which grew just after the removal of the 1st terrace are about 40 min shorter
Fig. 14. The effect on cycle time (the sum of the migration time and the consolidation time) of B. subtilis for each terrace when the first terrace of a colony was cut and removed 15 min after the beginning of the second migration phase. The results of the terrace number with the suffix “r” represent the case where the first terrace remains intact for reference.
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Fig. 15. Time difference of the migration time (circle) and the consolidation time (triangle) for B. subtilis between the case of cutting and the reference for each terrace.
and longer than those of the reference half, respectively. On the other hand, the time differences of the third and fourth terraces become small. It has also been confirmed that the colony formation of P. mirabilis has the same tendency that migration phase and consolidation phase of the separated part of the ring are shorter and longer, respectively, than the reference part, as shown in Figs. 16 and 17. In this case, we cut the third terrace 15 min after the beginning of the fourth migration phase. Furthermore, Fig. 18 shows that the width of the fourth terrace becomes smaller than that of the terrace denoted by “4r”. This change seems to be due to the fact that the cell density of the separated part in the fourth migration is decreased.
Fig. 16. The effect on cycle time (the sum of the migration time and the consolidation time) of P. mirabilis for each terrace when the third terrace of a colony was cut and removed 15 min after the beginning of the fourth migration phase. The results of the terrace number with the suffix “r” represent the case where the third terrace remains intact for reference.
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Fig. 17. Time difference of the migration time (circle) and the consolidation time (triangle) for P. mirabilis between the case of cutting and the reference for each terrace.
Therefore, we consider from these facts the following scenario with respect to the dependence of the cell density on the bacterial motility: The outermost terrace supplies active bacteria to the growth front. And the lowering the active cell density due to cutting behind causes shortening of the migration period and prolongation of the consolidation period to reach the threshold value for the restart of the next migration. 5.5. Replica-printing By using replica-printing method as explained in the Section 4 (v), we investigated the collective behavior of the active cells of B. subtilis at several time stages in migration and consolidation phases.
Fig. 18. Snapshot of the growth of a concentric colony for P. mirabilis. The colony was cut and divided into two halves by a scalpel and then the edge of the third terrace of right-half colony was cut 15 min after the beginning of the fourth migration phase. The width of the figure is about 1.3 cm.
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Fig. 19. Collective behavior of a replica-printed active bacteria. The sectorial part of a colony in the second migration phase was replica-printed onto a fresh agar medium. These two photographs show the expansion of the replica-printed colony (a) just after and (b) 1 h after the replicaprinting. The two black points indicate the edges of the initial growth front (the front of the second migration terrace) of the replica-printed colony.
Fig. 19 shows that the expansion of a colony which was replica-printed with a sectorial part excised from the original colony in the second migration phase (a) just after and (b) 1 h after the replica-printing. The two black points in Fig.19 indicate the edges of the initial growth front of the replica-printed colony. It is found that the colony forms disk-like shape and its center is at the front of the first consolidation terrace. We also found that the growth front of the initial replica-printed colony does not move for about the first 30–60 min. The reason for these facts are considered as follows: by the procedure of replica-printing, not all of active bacteria in the second migration phase are printed on a fresh agar plate. Then the printed active bacteria are isolated from one another, and the motility of the isolated individual bacterial cells gets low. On the other hand, it was observed that active bacteria continuously emerge and collectively move outward from the front of the first consolidation terrace. That is to say, the replica-printed colony expands from the front of the first consolidation terrace. Therefore, the supply of active bacteria from the first consolidation terrace is essential for the expansion of a concentric ring colony. From the observation of the growth of a colony which is replica-printed in migration phase, the contribution of the inner consolidation terrace to the collective behavior of active bacteria out there was found to be quite important. Hence we carried out the observation of the growth of a colony which is replica-printed at three different time stages (initial, intermediate, and late stages) in the consolidation phase. As a result, at all the three time stages in the consolidation phase the expansion of a replica-printed colony by the collective motion of active bacteria began immediately after the replica-printing. It should be noted that this result is in contrast to that in the case of replica-printing in the migration phase. It was also found that the region of the colony expansion by the collective motion of active bacteria depends on the time stage of replica-printing. Fig. 20 is a set of the snapshots of an expanding colony 1 h after replica-printing which was done at the three different time stages in the consolidation phase of the original colony: Fig. 20(a) shows an expanding colony which was replica-printed at the initial time stage in the second consolidation phase. It is found that active bacteria emerge out at the front of the first terrace. Fig. 20(b) is an expanding colony replica-printed 1.5 h after the beginning of the second consolidation phase. This case corresponds to replica-printing at the intermediate time stage. We find that active bacteria expand their region from the lateral cut line of the second terrace, i.e., the central part of the second terrace. Fig. 20(c) is an expanding colony which was replica-printed 3 h after the beginning of the second consolidation phase. This case corresponds to replica-printing at the late time stage of the consolidation phase. It
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Fig. 20. Collective behavior of a replica-printed active bacteria. The sectorial part of a colony in the second consolidation phase was replicaprinted onto a fresh agar medium. These three photographs show the expansion of the replica-printed colony 1 h after the replica-printing which was done at three different time stages in the second consolidation phase of the original colony: (a) initial, (b) intermediate, and (c) late time stages. In practice, the replica-printing was done (a) just after, (b) 1.5 h after, and (c) 3 h after the beginning of the second consolidation phase.
is found that active bacteria emerge out from the front of the second terrace, namely from the growth front of the original colony. In the case of P. mirabilis, it has also been found that the behavior of active bacteria was found to depend on the time of excision: When the sectorial part of a colony is printed at the early migration phase, active bacteria expand their region from the front of a colony. On the other hand, when the sectorial part is printed at the early consolidation phase, active bacteria do not migrate from the consolidation front at which bacterial cells remained in consolidation state. Instead they begin to expand their region from the lateral cut line of the previous ring.
Fig. 21. An example of concentric ring patterns of S. marcescens. The diameter of the pattern is 8.8 cm.
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From these observations, therefore, it was experimentally confirmed that during the consolidation phase in the concentric ring-like colony formation by both B. subtilis and P. mirabilis, active bacteria move collectively from inside to outside of the outermost terrace. 6. Summary and discussion In this paper, we have shown the experimental results of the investigation on the periodic change of the motility of B. subtilis and P. mirabilis in forming a concentric ring-like colonies. From our results (Sections 5.1 and 5.2), we conclude that the periodic change is determined neither by biological factors (i.e., biological clock) nor by chemical factors (chemotaxis as inhibitor). It may be possible to adopt nutrient as a candidate for inhibitor. However, our previous results tell us that migration time, consolidation time and the width of consolidation terrace are insensitive to the variation of the amount of nutrient [12]. It is noted that these experimental results are essentially different from those obtained from numerical simulations of models previously proposed on the basis of reaction–diffusion dynamics [20,21]. Especially, there is a crucial difference between the experiments and numerical results of the reaction–diffusion model since the location of region C in the morphological diagram obtained from the reaction– diffusion model is different from that obtained from our experiments. Hence the periodic change of bacterial motility is considered to take place due to other physical factors. As one plausible physical factor, we focused on the local density of bacterial cells, and we obtained the following conclusions: (i) From the result regarding the dependence of the lag phase time upon the initial cell density (Section 5.3), there is an upper density threshold where migration phase starts. (ii) On the other hand, a clear density threshold was not confirmed when migration phase stops. However, as explained in Section 5.4, the motility of B. subtilis and P. mirabilis has a tendency to change from active to inactive earlier than usual when we cut off the consolidation terrace just after the migration phase starts. Furthermore, from Section 5.5, we have also shown that motile bacterial cells are supplied from the outermost consolidation terrace. Therefore, it is strongly suggested that the bacterial motility depends on the cell density of the motile bacteria, and that there exists a lower density threshold of the motile bacteria regarding the start of consolidation phase (the stop of migration phase). We expect that there also exists the similar lower threshold about local bacterial density at the front of migration phase. Further investigation must be done in the future in order to clearly confirm the existence of the lower density threshold. Moreover, it is noted that even if the existence of local cell density threshold of the bacterial motility is strongly suggested, the reason why the cycle time seems to be independent of both Ca and Cn remains poorly understood. However, our experimental results show the importance of local cell density for the bacterial motility and the concentric-ring colony formation. And it is found from the results of our observation that the formation of concentric-ring colony is also closely connected to the internal wave of active bacteria. Therefore, additional investigation of the dependence of the motility and the proliferation of active bacteria on the cell density inside the colony is needed for understanding the relationship between the cycle time and both Ca and Cn . Recently, we have carried out the colony formation of Serratia (S.) marcescens, and we found this species also produced a concentric ring pattern, as shown in Fig. 21 [22]. In this pattern, the gray scale corresponds to the concentration of the pigment prodigiosin which is produced in the cells of S. marcescens. Therefore, this concentric pattern is different from the cases of B. subtilis and P. mirabilis. Nevertheless, we have obtained some experimental results suggesting that this concentric pattern also originates from the periodic change of the motility of S. marcescens [22]. We expect that there exists a universal mechanism of the bacterial colony growth into concentric ring pattern irrespective of bacterial species. Acknowledgements The present study was supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 09640471, 11214205 and 15340126).
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
P. Ball, The Self-made Tapestry: Pattern Formation in Nature, Oxford University Press, New York, 1998. B. Chopard, H.J. Herrmann, T. Vicsek, Nature 353 (1991) 409. A.L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University, New York, 1995. A.T. Winfree, The Geometry of Biological Time, Springer, Berlin, 1980. M. Matsushita, J.A. Shapiro, M. Dworkin (Eds.), Bacteria as Multicellular Organisms, Oxford University Press, New York, 1997, pp. 363–393. A. Nakahara, Y. Shimada, J. Wakita, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 65 (1996) 2700. T. Vicsek, Fluctuations and Scaling in Biology, Oxford University Press, New York, 2001. M. Ohgiwari, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 61 (1992) 816. J. Wakita, K. Komatsu, A. Nakahara, T. Matsuyama, M. Matasushita, J. Phys. Soc. Jpn. 63 (1994) 1205. O. Rauprich, M. Matsushita, C.J. Weijer, F. Siegert, S.E. Esipov, J.A. Shapiro, J. Bacteriol. 178 (1996) 6525. J. Wakita, I. R`afols, H. Itoh, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 67 (1998) 3630. J. Wakita, H. Shimada, H. Itoh, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 70 (2001) 911. Y. Shimada, A. Nakahara, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 64 (1995) 1896. K. Watanabe, J. Wakita, H. Itoh, H. Shimada, S. Kurosu, T. Ikeda, Y. Yamazaki, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 71 (2002) 650. J. Boissonade, E. Dulos, P. De Kepper, R. Kapral, K. Showalter (Eds.), Chemical Waves and Patterns, Kluwer, Dordrecht, 1995, p. 221. M. Sano, H. Kokubo, B. Janiaud, K. Sato, Prog. Theor. Phys. 90 (1993) 1. H. Itoh, J. Wakita, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 68 (1999) 1436. H. Shimada, T. Ikeda, J. Wakita, H. Itoh, S. Kurosu, F. Hiramatsu, M. Nakatsuchi, Y. Yamazaki, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 73 (2004) 1082. T. Matsuyama, Y. Takagi, Y. Nakagawa, H. Itoh, J. Wakita, M. Matsushita, J. Bacteriol. 182 (2000) 385. M. Mimura, H. Sakaguchi, M. Matsushita, Physica A 282 (2000) 283. N. Kobayashi, T. Sato, Y. Yamazaki, M. Matsushita, J. Phys. Soc. Jpn. 72 (2003) 970. F. Hiramatsu, Master Thesis, Chuo University, 2004.
Periodic growth of bacterial colonies Yoshihiro Yamazakia , Takemasa Ikedab , Hirotoshi Shimadab , Fumiko Hiramatsub , Naoki Kobayashib , Jun-ichi Wakitab , Hiroto Itohb , Sayuri Kurosub , Michio Nakatsuchib , Tohey Matsuyamac , Mitsugu Matsushitab,∗ a Department of Physics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan Department of Physics, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan c Department of Bacteriology, Niigata University School of Medicine, Niigata 951-8510, Japan
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Available online 12 February 2005
Abstract The formation of concentric ring colonies by bacterial species Bacillus subtilis and Proteus mirabilis has been investigated experimentally, focusing our attention on the dependence of local cell density upon the bacterial motility. It has been confirmed that these concentric ring colonies reflect the periodic change of the bacterial motility between motile cell state and immotile cell state. We conclude that this periodic change is macroscopically determined neither by biological factors (i.e., biological clock) nor by chemical factors (chemotaxis as inhibitor). And our experimental results strongly suggest that the essential factor for the change of the bacterial motility during concentric ring formation is the local cell density. © 2005 Elsevier B.V. All rights reserved. Keywords: Bacterial colony; Concentric ring pattern; Bacterial motility; B. subtilis; P. mirabilis; Cell density; Replica-printing
1. Introduction In order to understand pattern formation in nature by physics, it is important to clarify the dynamics of the formation of spatiotemporal structure characterizing the pattern and to extract its universality [1]. For this purpose, we usually focus our attention on the pattern formation in two-dimensional systems, such as mineral dendrites on the surface of rocks [2] and water imbibition in a thin paper sheet [3], for the reason that the pattern formation in two dimensions is easily reproducible experimentally and analyzed both numerically and theoretically. For example, when we pour solution of some chemicals which exhibit Belousov–Zhabotinsky (BZ) reaction into a Petri dish thinly, time evolution of target and rotating spiral patterns originating from spatiotemporal change of ∗
Corresponding author. Tel.: +81 3 3817 1787; fax: +81 3 3817 1792. E-mail addresses: [email protected] (Y. Yamazaki), [email protected] (M. Matsushita). URL: http://www.phys.chuo-u.ac.jp/labs/matusita. 0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2004.12.013
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ion concentrations essential for these pattern formations can be observed [4]. The pattern formation in the BZ reaction is produced due to both the oscillatory or excitable properties of the chemical reactions and spatial inhomogeneity of the solution in a Petri dish. Therefore, the formation of target and spiral patterns in two-dimensional system of the BZ reaction has been modeled by constructing reaction–diffusion equations of the ion concentrations. As another system for the pattern formation in two-dimensional system, much attention has been paid to bacterial colony growth on the surface of an agar plate [5]. In general, in order to know the whole types of patterns experimentally obtained in a certain system, we often produce a morphological diagram of the patterns by identifying control parameters for the environmental condition of the system and by varying them. With respect to the growth of bacterial colonies on the surface of an agar plate, important control parameters are agar concentration Ca and nutrient concentration Cn in the agar plate, because the bacterial proliferation and its motility are affected by Ca and Cn . Therefore, we can summarize the types of the colonies of a bacterial species in a morphological diagram as a function of both Ca and Cn . One of the important points in studying the bacterial colony growth from physical viewpoint is that spatiotemporal behavior of bacterial growth and motion can be readily observed in a wide range of scales with optical and stereo microscopes. In microscopic scale, which is the order of a micron, multiplication of bacteria and their individual motion are visible. In mesoscopic scale, which is the order of a tenth of a millimeter, the collective motion of bacteria is observed. And in macroscopic scale, which is the order of a centimeter, the whole morphology formation of bacterial colonies is characterized. In practice, the process of bacterial colony formation has been studied experimentally with a variety of bacterial species such as Bacillus (B.) subtilis [5], Proteus (P.) mirabilis [6], and Serratia (S.) marcescens and so forth [7]. If we consider a group of bacteria on the surface of an agar plate as their local population density in a mesoscopic grid whose size is about 100 m typically, we can construct a model for the time evolution of the bacterial population density in each grid. In terms of model construction, there exists a remarkable similarity between the bacterial colony formation and BZ reaction. In other words, it is possible to construct a reaction–diffusion like model for the bacterial colony growth by considering local bacterial population, bacterial proliferation and its motility as ion concentration, chemical reaction and diffusion, respectively. So far, we have carried out a lot of experiments about the colony growth of bacterial species B. subtilis and P. mirabilis [5,6]. As for B. subtilis, the wild-type strain OG-01, which was used in our experiment, is rod-shaped (0.5–1.0 m in diameter, 2–5 m in length) with flagella and is motile in water by collectively rotating the flagella. When the environmental condition is unfavorable to the bacteria, such as on a nutrient-poor or dry agar plate, the bacterial cells become spores. On the other hand, P. mirabilis is a flagellated rod-like bacteria and takes two states. One is a normally flagellated vegetative swimmer cell (1.5–2.0 m in length), having 6–10 peritrichous flagella. The other is a hyperflagellated long swarmer cell (10–80 m in length) with 103 − 104 peritrichous flagella. These two states are transformed into each other alternately through cell differentiation and dedifferentiation processes only when P. mirabilis lives on the surface of substrate medium such as agar gels. Each bacterial species used for the study of colony formation has different structures in physiological and genetic properties such as cell wall, DNA sequence, and so on. However, the bacterial species we have treated have the similarities that they are rod-shaped cell with flagella and possess the motility to expand their colony actively. We consider these similarities and investigate the mechanism of bacterial colony formation irrespective of their detailed biological features. In this paper, we briefly explain the morphological diagrams of B. subtilis and P. mirabilis first. Then, by focusing our attention on the concentric ring pattern of bacterial colony obtained by both species, we report the results of our experiments about the formation of the concentric ring colonies. We conclude that the periodic change of bacterial motility in time is the essential factor for the formation of the concentric ring colonies and argue that the origin of its periodic change strongly depends on the local population density at the growth front of a colony. Finally, we show the results of our recent experiment about the colony formation of S. marcescens.
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2. Colony growth and morphological diagram The experimental setup and procedure for producing bacterial colonies of B. subtilis and P. mirabilis can be summarized as the following four processes: (i) preparation of an agar plate for the observation of colony growth, (ii) preparation of liquid culture of sample bacteria, (iii) inoculation of the liquid culture on the agar plate, and (iv) incubation of the bacteria inoculated on the agar plate. The detailed explanation for these processes were described elsewhere [8–10]. Macroscopic colony patterns and their growth were observed through a camera and a CCD camera and recorded on photographs, video tapes or hard discs. The microscopic and mesoscopic movement of individual bacterial cells were observed through an optical microscope and recorded on photographs, video tapes or hard discs. Fig. 1 shows the typical morphology diagram of B. subtilis obtained so far as a function of both Ca and Cn [8,9]. As shown in Fig. 1, the morphology diagram is divided into five regions by the pattern characteristics of bacterial colonies: region A (diffusion-limited aggregation (DLA)-like), B (Eden-like), C (concentric ring-like), D (homogeneously spreading disk-like), and E (dense branching morphology (DBM)-like). This fact suggests the existence of the physical mechanism for the bacterial colony formation irrelevant to the detail of their biological properties. In addition, it is noted that the growth rates were also very different among these five morphologies. Actually, it has been found that growth to the size of about 5 cm (about half a diameter of Petri dishes) required about a month in the region A, a week in the region B, a day in the regions C and E, and half a day in the region D. Regarding the growth of the bacterial colony, the following two features are important: (i) Lowering of nutrient concentration Cn causes a tip-splitting of the growth front of a bacterial colony such as DLA-like pattern (region A) and DBM-like pattern (region E). (ii) The softness of the agar has a great influence on the motility of bacterial cells. In regions A and B where agar plates are hard, individual cells do not move actively. And cells are connected with each other and form a bundle of chains. Therefore, the colony expansion is done in the way cells, which are located in the outermost part of a colony and have access to nutrient easily, proliferate by cell division. Meanwhile, cells in the inner part change to spores and enter into a resting phase. On the other hand, the colony expansion in the regions D and E, where agar plates are softer, takes place due to a random walk-like movement of individual active cells [9,11]. The growth of the bacterial colony in the region C between the regions B and D is quite different from that in the other four regions: A concentric ring-like pattern in this region involves a rhythmic growth having a characteristic period. It has been found from a mesoscopic observation that a concentric ring-like pattern is formed by alternating the region D-like colony expansion (very fast) and the region B-like growth (very slow) [12]. The similar morphological diagram has been obtained for P. mirabilis. From Fig. 2 it is found that colony patterns can be classified mainly into three types [6]; disk-like spreading growth (region P), diffusion-limited growth (region Q), and three-dimensional growth inside the agar medium (region R). The patterns observed in the regions Q and
Fig. 1. Morphology diagram of B. subtilis as a function of nutrient concentration Cn and the inverse of agar concentration Ca .
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Fig. 2. Morphology diagram of P. mirabilis as a function of nutrient concentration Cn and the inverse of agar concentration Ca .
R are considered to be typical ones in the limiting cases where Cn is low and Ca is high, and where Ca is low, respectively. On the other hand, the region P is an intermediate one, where the motility and the transformation of the cell body are apparently considered to strongly influence the pattern formation. Actually the morphologies in the region P is classified into three subgroups: concentric ring pattern (Pr ), homogeneous pattern (Ph ), and transient spatio-temporal pattern (Ps ) [13]. It is well-known that the concentric-rings observed in the region Pr are well correlated circumferentially. And in the region Ps one observes target and rotating spiral patterns [14] which quite resemble those in chemical reaction system [15], liquid crystal system [16] and so on. From the above two cases of concentric ring patterns, the temporal change of the bacterial motility is considered to be important for the periodic growth into concentric ring patterns. Then, in the following sections, we summarize the observed findings of the formation of concentric ring colony patterns. We also show some experimental ideas for the investigation of the origin of the periodic growth.
3. Observed findings Fig. 3 is a typical concentric ring pattern of B. subtilis obtained in the region C. We have confirmed that the gray scale of the concentric rings corresponds to the change of the collective motion of bacteria in time during the colony expansion. In order to understand the origin of these spatially periodic profiles, we have mesoscopically observed the motility of bacterial cells during the colony formation and obtained the following findings [12]: (i) When the liquid culture is inoculated at the center spot on the surface of the agar plate, there is a time lag until the colony begins to expand (the existence of lag phase). During the lag phase, the bacteria proliferate by cell division at the point-inoculated center spot without any migration. (ii) After the lag phase, each bacterial cell becomes active and seems to move individually at random. Bacterial cells go outward collectively by forming a single layer of cells, and the colony expands fast. The motion of bacterial cells during the colony expansion is similar to that in the region D. We call this period for colony expansion migration phase. (iii) When migration phase continues for a period of time, the expansion of the colony stops rather abruptly. The mesoscopic observation showed that bacterial cells, meanwhile, proliferate without active motion and form a bundle of chains by connecting with each other. The behavior of bacterial cells in this period after migration phase is quite similar to that of Eden-like pattern formation in the B region. It is noted that the local density
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Fig. 3. Concentric ring-like pattern formed by B. subtilis when Ca = 6.8 g/l and Cn = 40 g/l. The photograph was taken 2 days after inoculation. The inner diameter of Petri dish is 8.8 cm.
of bacterial cells in the vicinity of the colony front increases drastically. We call the period for proliferation without active bacterial motion consolidation phase. (iv) The consolidation phase also finishes after a while and the migration phase as stated in (ii) starts and the colony expands again. Then, the concentric ring pattern is found to be formed due to the periodic change of the bacterial motility in alternate migration phase and consolidation phase. (v) From the shape of the region C in the morphology diagram (Fig. 1), it is found that the formation of concentric ring patterns is quite sensitive to the softness of the agar plate, while it is insensitive to the richness of the nutrient. In fact, with respect to the influence of both the softness of agar plate and the richness of the nutrient upon migration time and consolidation time which characterize the periodicity of the colony formation, we have already obtained the following results shown in the previous paper [12]: migration time is a weakly decreasing function of Ca , but it seems to be independent of Cn . Consolidation time is a weakly increasing function of Ca , but it seems to be independent of Cn . Cycle time which is the sum of migration time and consolidation time seems to be constant, independently of both Ca and Cn . (vi) B. subtilis secretes a surfactant called surfactin while forming a colony. The surfactin relaxes the effect of the surface tension of water on the bacterial motion and enhances the motility of the active bacterial cells. And the colony morphology in the presence of surfactin changes drastically in comparison with the morphology in its absence. It is then considered that surfactin may play a crucial role for the periodic change of bacterial motility in the region C. However, in the experiment by using the mutant cells which are obtained from OG-01 by nitrosoguanidine-mutagenesis and are defective in producing surfactin, it has been confirmed that a concentric ring pattern is also reproducible. Therefore, the secretion and existence of surfactin is considered not to be essential for the appearance of a concentric ring colony pattern. P. mirabilis also forms a concentric ring pattern [10], as shown in Fig. 4 [17]. As in the case of B. subtilis, the migration–consolidation mechanism is also essential for the formation of a concentric ring pattern by P. mirabilis. As seen in Fig. 4, the circumference of each circle by P. mirabilis is much smoother than that by B. subtilis. From the microscopic viewpoint, there are two cell states for P. mirabilis cultivated on the surface of an agar plate: swimmer
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Fig. 4. Concentric ring-like pattern formed by P. mirabilis. The photograph was taken 2 days after inoculation. The inner diameter of Petri dish is 8.8 cm.
cell and swarmer cell as described above. The typical feature is that P. mirabilis executes a cyclic transition of differentiation and dedifferentiation between these two cell states. With respect to the formation of a concentric ring pattern in the case of P. mirabilis, migration phase corresponds to the expansion period of colony growth by swarmer cells, and consolidation phase to the period when the expansion of a colony stops by shortening the cell length, i.e. by becoming swimmer cells. Interestingly, it is known that the dependencies of migration time and consolidation time for B. subtilis upon both Cn and Ca explained above have also been confirmed by P. mirabilis, which means that the cycle time of P. mirabilis is also constant, independently of Cn and Ca [10,17]. Furthermore, from mesoscopic observations, one or more internal waves were sometimes observed to advance toward the growth front from the inner terrace during migration phase in both species. They were seen with naked eyes, too. In the case of B. subtilis, the internal waves consisted of many active bacterial cells, which originated around the edge of the inner terrace. On the other hand, in the case of P. mirabilis, many internal waves also advance successively towards the outermost terrace front, even after the front stopped moving, as shown in Fig. 5. Each internal wave in a concentric ring colony of P. mirabilis consists of a superimposed monolayer of swarming cells moving towards the growth front. 4. Experimental ideas So far, many studies on the origin of a concentric ring pattern formation of both P. mirabilis and B. subtilis have been done. Among them, there exist experimental studies with physical approaches which are not concerned with the detail of genetic and biological information. Here we summarize several experimental ideas about the investigation of the periodic change of bacterial motility [10,17–19]. (i) Some chemicals secreted by bacteria during their proliferation may have the ability to affect their motion and then their colony morphology. Especially, on the analogy of target patterns observed in the reaction–diffusion system such as the BZ reaction, there is the possibility that a concentric ring pattern in the bacterial colony is formed in the same way as the above target pattern in the BZ reaction. If so, the local cell density of bacteria
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Fig. 5. Snapshot of internal waves in the case of P. mirabilis. The right edge shows the growth front of the colony. Internal waves advance toward the growth front from inside. The width of the snapshot is about 1.3 cm.
corresponds to an activator. Moreover, it is necessary for the formation of the concentric ring pattern that there are pacemaker cells which secrete some chemical periodically at the center of the concentric ring colony. Then, the following experiment can be considered in order to check whether any chemical as an inhibitor and pacemaker cells for the appearance of concentric ring exist or not: Cut a colony together with an agar plate along the dotted lines with a surgical knife in Fig. 6(b) during its growth. Fig. 6(a) shows an intact, reference colony. Then the part of the colony represented by the darker region in Fig. 6(b) is isolated from cells at the center spot which are considered to be a pacemaker for the concentric ring pattern formation. If some chemical and pacemaker cells are needed, this isolated part of the colony must grow and produce a colony whose morphology is different from the previous one. (ii) The formation of the concentric ring pattern in bacterial colonies results from periodic oscillation of the motility of bacteria. It has been revealed that the cycle time characterizing the periodic growth does not depend on Cn and Ca for both B. subtilis and P. mirabilis. Hence in order to investigate whether the origin of the stable periodicity
Fig. 6. Schematic representation of experimental procedure to check whether some chemical as an inhibitor and pacemaker cells are necessary for concentric ring pattern formation or not. (a) Intact, reference colony. (b) A colony together with an agar plate is cut along the dotted lines.
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in the cycle time is physical or biological, the following experiment has been done: after point-inoculation of the liquid culture at a spot on the surface of an agar plate, we additionally inoculate again at a different spot apart from the previous one with some time interval. The time interval in the point-inoculation at the two spots brings the phase difference between two concentric ring colonies. If this periodic change of motility is derived from biological properties of bacteria, e.g. from the collective synchronization of biological clocks in bacterial cells, the period of the formation of concentric rings in the two colonies must be also synchronized after two colonies collide. (iii) The measurement of lag phase time is important to understanding the beginning of a colony growth and the emergence of migration phase. Especially, by varying the initial concentration of liquid culture inoculated at the center spot on the surface of an agar plate, this measurement has a clue to clarifying the dependence of the onset of migration phase on the cell density. (iv) From the result of (iii), the periodicity in the bacterial motility is considered to reflect physical properties as the group of bacterial cells. Then we can focus our attention on the dependence of the local cell density on their motility. In practice, we sector a part of the outermost ring of a colony together with an agar plate during the migration phase of the colony growth, as shown in Fig. 7. And we investigate the variation of ring formation for removing the influence from the inner concentric rings. (v) It is found from the above result that bacterial motility in migration phase depends strongly on the cell density of active bacteria. In order to understand the collective behavior of active bacteria in a colony, there is a simple idea that a sectorial part of a colony is excised and immediately the surface of the sectorial part is attached onto another fresh agar plate and separated (replica-printing). By this method, bacteria in migration and consolidation states of a colony can be picked up on fresh agar plate, and we can observe the motility of bacteria in each state. Consequently, we are able to investigate the change of the position in a colony where bacteria move actively by varying the time when replica-printing is carried out.
5. Results In this section, we report the experimental results with respect to the appearance of concentric ring colony patterns by B. subtilis [18] and P. mirabilis [17,19], based on the ideas as stated in the previous section.
Fig. 7. Schematic explanation for the investigation of the influence of the consolidation terrace just behind the growth front in migration phase upon the colony expansion. (a) Intact, reference colony. (b) A colony together with an agar plate is cut along the dotted lines.
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5.1. Existence of a pacemaker As in the Section 4 (i), we examined whether the chemical as an inhibitor and the pacemaker for the formation of concentric ring patterns by B. subtilis are necessary or not. Fig. 8 shows two snapshots of the experimental result. The left photo (a) shows the colony immediately after removing the inner sectorial part of the colony. The right (b) was taken 7 h after the left. It is found from these snapshots that the formation of a concentric ring colony is not influenced even if the inner sectorial part including the center spot of the colony is removed. Therefore, we can conclude that both the chemical and the pacemaker do not exist for the concentric ring formation by B. subtilis. In the case of P. mirabilis, it has also been confirmed that the colony on the isolated part grows into almost the same concentric ring pattern as the colony on the rest part, as shown in Fig. 9. Therefore, it is concluded that macroscopically any chemicals and the pacemaker cells are not necessary for the formation of the macroscopic concentric ring colonies. 5.2. Phase entrainment It has been known from the previous study that the cycle time of B. subtilis is about 7 h. Then we carried out the following experiment as introduced in the Section 4 (ii): We inoculated the liquid culture at a spot on the surface of an agar plate. After 2 h from the first inoculation, we additionally inoculated the liquid culture at the spot 2.5 cm apart from the previous spot. This procedure introduces phase difference in the formation cycle of concentric rings between two colonies. When these two colonies expand, they collide as shown in Fig. 10. We find that two phases of the cycle of the concentric ring formation are not synchronized with each other even after the collision of these colonies. In fact, Fig. 10 shows us that cells at the growth front of the colony A are in the fourth consolidation phase, whereas that of the colony B are still in the fourth migration phase. Furthermore, the boundary of the growth front between the colonies A and B as shown in Fig. 10(b) is not blurred even if the colonies expand. From these results, it is concluded that the phase entrainment due to biological properties of the bacterial cells such as biological clocks, if any, is limited only in the local region where bacteria of colony A and B mix together. In the case of P. mirabilis, it has also been found that the period of concentric ring in the two colonies is not synchronized even after they collide, and that concentric rings in the two colonies are produced independently of each other, keeping their own phases. Fig. 11 shows the experimental result for P. mirabilis. In this case, we inoculated the liquid culture at a spot on the surface of an agar plate. After 2 h from the first inoculation, we additionally inoculated the liquid culture at the spot
Fig. 8. Experimental results for the necessity of a pacemaker in the concentric ring-like pattern formation of B. subtilis. The diameter of these figures is 8.8 cm.
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Fig. 9. Experimental results for the necessity of a pacemaker in the concentric ring-like pattern formation of P. mirabilis.
1 cm apart from the previous spot. As seen in this figure, two colliding colonies showed the same periodicity and remained out of phase with each other during their expansion. Therefore, the biological properties are considered not to macroscopically affect the formation of the concentric ring pattern. 5.3. Lag phase According to the idea explained in the Section 4 (iii), we investigated the relation between lag phase time and the initial concentration of the liquid culture for B. subtilis. Fig. 12 shows the dependence of the initial cell density on the lag phase time. The cell density at 100 on the abscissa in the figure corresponds to that of our standard inocula, 104 cells/l. It is found from this figure that the lag phase time is a monotonically decreasing function of the initial
Fig. 10. Experimental results for the possibility of phase entrainment between two colonies with different phases of periodic colony expansion in the case of B. subtilis.
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Fig. 11. Experimental results for the possibility of phase entrainment between two colonies with different phases of periodic colony expansion in the case of P. mirabilis.
cell density up to about 101 , and that the lag phase time seems to saturate to a fixed value (about 7 h in this case) over 101 . The similar experiments for P. mirabilis have revealed that lag phase time is also a monotonically decreasing function of the initial density below some value of the density, and that above this value it saturates to a constant value. Fig. 13 shows the lag phase time as a function of the initial concentration of the liquid culture for P. mirabilis. It is found from these figures that the cell density has to increase to a certain value for the beginning of a colony growth and that there exists the threshold density value for the migration phase. 5.4. Cutting In order to investigate the dependence of the cell density upon the end of migration phase for B. subtilis, we carried out the following experiment based on the idea explained in the Section 4 (iv): we divided a colony together
Fig. 12. Lag phase time as a function of the initial concentration of the liquid culture of B. subtilis. The unity 100 corresponds to the standard cell concentration of liquid culture (104 cells/l) for inoculation.
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Fig. 13. Lag phase time as a function of the initial concentration of the liquid culture of P. mirabilis. The unity 100 corresponds to the standard cell concentration of liquid culture (106 cells/l) for inoculation.
with an agar plate into two halves, and we cut and removed the first terrace from the one half of the colony 15 min after the beginning of the second migration phase. The other half of the colony remained intact as reference. We compared the growth of the case of cutting with that of the reference. It should be noted that in the case of cutting there exist only active bacteria which came from inside the first terrace. We confirmed that the width of the second terrace in the case of cutting is smaller than that in the reference part. From the third terrace on, however, the width of terrace is almost the same in both halves. Fig. 14 shows the time needed for second, third and fourth cycles of each half. (The suffix “r” at the terrace number in the figure signifies “reference”.) The cycle time is defined as the sum of migration time and consolidation time, and was obtained by the average of 10 trials in our experiment. It is found from this figure that the cycle time seems to be insensitive to the cutting of the first terrace. However, there is a tendency that the migration time and consolidation time of the second terrace are shorter and longer than those of the second reference terrace, respectively. In the following, we consider the time difference in migration time and consolidation time between the case of cutting and the case of reference of each terrace. Fig. 15 shows our result that the migration time and consolidation time of the second terrace which grew just after the removal of the 1st terrace are about 40 min shorter
Fig. 14. The effect on cycle time (the sum of the migration time and the consolidation time) of B. subtilis for each terrace when the first terrace of a colony was cut and removed 15 min after the beginning of the second migration phase. The results of the terrace number with the suffix “r” represent the case where the first terrace remains intact for reference.
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Fig. 15. Time difference of the migration time (circle) and the consolidation time (triangle) for B. subtilis between the case of cutting and the reference for each terrace.
and longer than those of the reference half, respectively. On the other hand, the time differences of the third and fourth terraces become small. It has also been confirmed that the colony formation of P. mirabilis has the same tendency that migration phase and consolidation phase of the separated part of the ring are shorter and longer, respectively, than the reference part, as shown in Figs. 16 and 17. In this case, we cut the third terrace 15 min after the beginning of the fourth migration phase. Furthermore, Fig. 18 shows that the width of the fourth terrace becomes smaller than that of the terrace denoted by “4r”. This change seems to be due to the fact that the cell density of the separated part in the fourth migration is decreased.
Fig. 16. The effect on cycle time (the sum of the migration time and the consolidation time) of P. mirabilis for each terrace when the third terrace of a colony was cut and removed 15 min after the beginning of the fourth migration phase. The results of the terrace number with the suffix “r” represent the case where the third terrace remains intact for reference.
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Fig. 17. Time difference of the migration time (circle) and the consolidation time (triangle) for P. mirabilis between the case of cutting and the reference for each terrace.
Therefore, we consider from these facts the following scenario with respect to the dependence of the cell density on the bacterial motility: The outermost terrace supplies active bacteria to the growth front. And the lowering the active cell density due to cutting behind causes shortening of the migration period and prolongation of the consolidation period to reach the threshold value for the restart of the next migration. 5.5. Replica-printing By using replica-printing method as explained in the Section 4 (v), we investigated the collective behavior of the active cells of B. subtilis at several time stages in migration and consolidation phases.
Fig. 18. Snapshot of the growth of a concentric colony for P. mirabilis. The colony was cut and divided into two halves by a scalpel and then the edge of the third terrace of right-half colony was cut 15 min after the beginning of the fourth migration phase. The width of the figure is about 1.3 cm.
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Fig. 19. Collective behavior of a replica-printed active bacteria. The sectorial part of a colony in the second migration phase was replica-printed onto a fresh agar medium. These two photographs show the expansion of the replica-printed colony (a) just after and (b) 1 h after the replicaprinting. The two black points indicate the edges of the initial growth front (the front of the second migration terrace) of the replica-printed colony.
Fig. 19 shows that the expansion of a colony which was replica-printed with a sectorial part excised from the original colony in the second migration phase (a) just after and (b) 1 h after the replica-printing. The two black points in Fig.19 indicate the edges of the initial growth front of the replica-printed colony. It is found that the colony forms disk-like shape and its center is at the front of the first consolidation terrace. We also found that the growth front of the initial replica-printed colony does not move for about the first 30–60 min. The reason for these facts are considered as follows: by the procedure of replica-printing, not all of active bacteria in the second migration phase are printed on a fresh agar plate. Then the printed active bacteria are isolated from one another, and the motility of the isolated individual bacterial cells gets low. On the other hand, it was observed that active bacteria continuously emerge and collectively move outward from the front of the first consolidation terrace. That is to say, the replica-printed colony expands from the front of the first consolidation terrace. Therefore, the supply of active bacteria from the first consolidation terrace is essential for the expansion of a concentric ring colony. From the observation of the growth of a colony which is replica-printed in migration phase, the contribution of the inner consolidation terrace to the collective behavior of active bacteria out there was found to be quite important. Hence we carried out the observation of the growth of a colony which is replica-printed at three different time stages (initial, intermediate, and late stages) in the consolidation phase. As a result, at all the three time stages in the consolidation phase the expansion of a replica-printed colony by the collective motion of active bacteria began immediately after the replica-printing. It should be noted that this result is in contrast to that in the case of replica-printing in the migration phase. It was also found that the region of the colony expansion by the collective motion of active bacteria depends on the time stage of replica-printing. Fig. 20 is a set of the snapshots of an expanding colony 1 h after replica-printing which was done at the three different time stages in the consolidation phase of the original colony: Fig. 20(a) shows an expanding colony which was replica-printed at the initial time stage in the second consolidation phase. It is found that active bacteria emerge out at the front of the first terrace. Fig. 20(b) is an expanding colony replica-printed 1.5 h after the beginning of the second consolidation phase. This case corresponds to replica-printing at the intermediate time stage. We find that active bacteria expand their region from the lateral cut line of the second terrace, i.e., the central part of the second terrace. Fig. 20(c) is an expanding colony which was replica-printed 3 h after the beginning of the second consolidation phase. This case corresponds to replica-printing at the late time stage of the consolidation phase. It
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Fig. 20. Collective behavior of a replica-printed active bacteria. The sectorial part of a colony in the second consolidation phase was replicaprinted onto a fresh agar medium. These three photographs show the expansion of the replica-printed colony 1 h after the replica-printing which was done at three different time stages in the second consolidation phase of the original colony: (a) initial, (b) intermediate, and (c) late time stages. In practice, the replica-printing was done (a) just after, (b) 1.5 h after, and (c) 3 h after the beginning of the second consolidation phase.
is found that active bacteria emerge out from the front of the second terrace, namely from the growth front of the original colony. In the case of P. mirabilis, it has also been found that the behavior of active bacteria was found to depend on the time of excision: When the sectorial part of a colony is printed at the early migration phase, active bacteria expand their region from the front of a colony. On the other hand, when the sectorial part is printed at the early consolidation phase, active bacteria do not migrate from the consolidation front at which bacterial cells remained in consolidation state. Instead they begin to expand their region from the lateral cut line of the previous ring.
Fig. 21. An example of concentric ring patterns of S. marcescens. The diameter of the pattern is 8.8 cm.
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From these observations, therefore, it was experimentally confirmed that during the consolidation phase in the concentric ring-like colony formation by both B. subtilis and P. mirabilis, active bacteria move collectively from inside to outside of the outermost terrace. 6. Summary and discussion In this paper, we have shown the experimental results of the investigation on the periodic change of the motility of B. subtilis and P. mirabilis in forming a concentric ring-like colonies. From our results (Sections 5.1 and 5.2), we conclude that the periodic change is determined neither by biological factors (i.e., biological clock) nor by chemical factors (chemotaxis as inhibitor). It may be possible to adopt nutrient as a candidate for inhibitor. However, our previous results tell us that migration time, consolidation time and the width of consolidation terrace are insensitive to the variation of the amount of nutrient [12]. It is noted that these experimental results are essentially different from those obtained from numerical simulations of models previously proposed on the basis of reaction–diffusion dynamics [20,21]. Especially, there is a crucial difference between the experiments and numerical results of the reaction–diffusion model since the location of region C in the morphological diagram obtained from the reaction– diffusion model is different from that obtained from our experiments. Hence the periodic change of bacterial motility is considered to take place due to other physical factors. As one plausible physical factor, we focused on the local density of bacterial cells, and we obtained the following conclusions: (i) From the result regarding the dependence of the lag phase time upon the initial cell density (Section 5.3), there is an upper density threshold where migration phase starts. (ii) On the other hand, a clear density threshold was not confirmed when migration phase stops. However, as explained in Section 5.4, the motility of B. subtilis and P. mirabilis has a tendency to change from active to inactive earlier than usual when we cut off the consolidation terrace just after the migration phase starts. Furthermore, from Section 5.5, we have also shown that motile bacterial cells are supplied from the outermost consolidation terrace. Therefore, it is strongly suggested that the bacterial motility depends on the cell density of the motile bacteria, and that there exists a lower density threshold of the motile bacteria regarding the start of consolidation phase (the stop of migration phase). We expect that there also exists the similar lower threshold about local bacterial density at the front of migration phase. Further investigation must be done in the future in order to clearly confirm the existence of the lower density threshold. Moreover, it is noted that even if the existence of local cell density threshold of the bacterial motility is strongly suggested, the reason why the cycle time seems to be independent of both Ca and Cn remains poorly understood. However, our experimental results show the importance of local cell density for the bacterial motility and the concentric-ring colony formation. And it is found from the results of our observation that the formation of concentric-ring colony is also closely connected to the internal wave of active bacteria. Therefore, additional investigation of the dependence of the motility and the proliferation of active bacteria on the cell density inside the colony is needed for understanding the relationship between the cycle time and both Ca and Cn . Recently, we have carried out the colony formation of Serratia (S.) marcescens, and we found this species also produced a concentric ring pattern, as shown in Fig. 21 [22]. In this pattern, the gray scale corresponds to the concentration of the pigment prodigiosin which is produced in the cells of S. marcescens. Therefore, this concentric pattern is different from the cases of B. subtilis and P. mirabilis. Nevertheless, we have obtained some experimental results suggesting that this concentric pattern also originates from the periodic change of the motility of S. marcescens [22]. We expect that there exists a universal mechanism of the bacterial colony growth into concentric ring pattern irrespective of bacterial species. Acknowledgements The present study was supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 09640471, 11214205 and 15340126).
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
P. Ball, The Self-made Tapestry: Pattern Formation in Nature, Oxford University Press, New York, 1998. B. Chopard, H.J. Herrmann, T. Vicsek, Nature 353 (1991) 409. A.L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University, New York, 1995. A.T. Winfree, The Geometry of Biological Time, Springer, Berlin, 1980. M. Matsushita, J.A. Shapiro, M. Dworkin (Eds.), Bacteria as Multicellular Organisms, Oxford University Press, New York, 1997, pp. 363–393. A. Nakahara, Y. Shimada, J. Wakita, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 65 (1996) 2700. T. Vicsek, Fluctuations and Scaling in Biology, Oxford University Press, New York, 2001. M. Ohgiwari, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 61 (1992) 816. J. Wakita, K. Komatsu, A. Nakahara, T. Matsuyama, M. Matasushita, J. Phys. Soc. Jpn. 63 (1994) 1205. O. Rauprich, M. Matsushita, C.J. Weijer, F. Siegert, S.E. Esipov, J.A. Shapiro, J. Bacteriol. 178 (1996) 6525. J. Wakita, I. R`afols, H. Itoh, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 67 (1998) 3630. J. Wakita, H. Shimada, H. Itoh, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 70 (2001) 911. Y. Shimada, A. Nakahara, M. Matsushita, T. Matsuyama, J. Phys. Soc. Jpn. 64 (1995) 1896. K. Watanabe, J. Wakita, H. Itoh, H. Shimada, S. Kurosu, T. Ikeda, Y. Yamazaki, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 71 (2002) 650. J. Boissonade, E. Dulos, P. De Kepper, R. Kapral, K. Showalter (Eds.), Chemical Waves and Patterns, Kluwer, Dordrecht, 1995, p. 221. M. Sano, H. Kokubo, B. Janiaud, K. Sato, Prog. Theor. Phys. 90 (1993) 1. H. Itoh, J. Wakita, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 68 (1999) 1436. H. Shimada, T. Ikeda, J. Wakita, H. Itoh, S. Kurosu, F. Hiramatsu, M. Nakatsuchi, Y. Yamazaki, T. Matsuyama, M. Matsushita, J. Phys. Soc. Jpn. 73 (2004) 1082. T. Matsuyama, Y. Takagi, Y. Nakagawa, H. Itoh, J. Wakita, M. Matsushita, J. Bacteriol. 182 (2000) 385. M. Mimura, H. Sakaguchi, M. Matsushita, Physica A 282 (2000) 283. N. Kobayashi, T. Sato, Y. Yamazaki, M. Matsushita, J. Phys. Soc. Jpn. 72 (2003) 970. F. Hiramatsu, Master Thesis, Chuo University, 2004.