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Diffusion-limited growth in bacterial colony formation
Physica A 168 (1990) 498-506 North-Holland
D I F F U S I O N - L I M I T E D G R O W T H IN B A C T E R I A L C O L O N Y FORMATION Mitsugu M A T S U S H I T A a and Hiroshi F U J I K A W A b
aDepartment of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112, Japan bTokyo Metropolitan Research Laboratory of Public Health, Hyakunin-cho, Shinjuku-ku, Tokyo 169, Japan Colonies of bacterial species called Bacillus subtilis have been found to grow twodimensionally and self-similarly on agar plates through diffusion-limited processes in a nutrient concentration field. We obtained a fractal dimension of the colony patterns of D = 1.73 --+0.02, very close to that of the two-dimensional DLA model, and confirmed the existence of the screening effect of protruding main branches against inner ones in a colony, the repulsion between two neighboring colonies and the tendency to grow toward nutrient. These effects are all characteristic of the pattern formation in a Laplacian field. This finding implies the importance of physical properties of the environment for the morphology of bacterial colonies in general.
1. Introduction T h e f o r m a t i o n of r a n d o m fractals and their structural properties is a p r o b l e m of considerable current interest [1-3]. A m o n g m a n y growth models p r o p o s e d so far to describe it, such as the E d e n [4] and the ballistic aggregation [5, 6] models, particular attention has b e e n paid to the diffusion-limited aggregation ( D L A ) m o d e l [7]. It not only generates ffactal structures with a fractal dimension D clearly less than the spatial dimension d (D ~ 1.7 and 2.5 for d = 2 and 3, respectively), but also describes the growth processes in a wide variety of physical and chemical p h e n o m e n a such as electrodeposition, dielectric b r e a k d o w n , viscous fingering, dendritic crystal growth and chemical dissolution [8, 9]. It is n o w well k n o w n [8] that D L A patterns have o u t w a r d l y o p e n and r a n d o m l y b r a n c h e d structures with no characteristic length scales except their whole size and b r a n c h thickness (aggregated particle size). A l t h o u g h f r o m the physicist's viewpoint biological g r o w t h generally leads to the f o r m a t i o n of t o o c o m p l e x patterns, biology m a y still have great potentiality for the application of fractal concepts. In fact, there are a wide variety of biological patterns which seem to be self-similar, such as trees, bronchial trees and n e t w o r k s of nerves and b l o o d vessels. This has already b e e n confirmed by ffactal analyses for blood vessel patterns in the chicken e m b r y o [10], the 0378-4371/90/$03.50 © 1990-Elsevier Science Publishers B.V. (North-Holland)
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
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structure of the bronchial tree [11, 12], the cerebral surface of the normal human brain [13], the vascular heterogeneity in the heart [14, 15], the neuronal arborization [16] and the human retinal vessel [17] patterns. Unfortunately, however, they are rather "static" analyses to obtain the fractal dimension of "snapshot" patterns. One cannot fully characterize given patterns by their fractal dimensions alone. For instance, DLA and invasion percolation patterns have almost the same fractal dimensions (D ~2.5) for d = 3, but they look quite different from each other. The observation of the growth itself, i.e., the dynamical behavior, is clearly needed to characterize patterns and identify more elementary growth processes. Meakin [18] demonstrated by extensive computer simulations that when biological growth is governed by diffusion-limited processes, the growing patterns show characteristic features such as screening, repulsion, and so on. Conversely, the existence of these effects enables one to confirm the diffusion-limited growth in biological systems. No experiments have, however, been reported so far. In this paper, we present our recent results on the fractal growth in the formation of bacterial colonies. They clearly indicate that the growth of bacterial colonies on agar plates is governed by DLA processes of the nutrient contained in the agar plates. To the best of our knowledge this is the first unambiguous identification of fractal growth processes in biological systems. Moreover, this implies the important revelation that the formation of bacterial colonies is strongly affected by the physical conditions of the environment as well as the proliferation way of the bacteria. Only the latter aspect has so far been taken into account in the morphological study of bacterial colonies. This paper is organized as follows: Section 2 briefly describes the experimental procedures. A full description of the experimental results follows in section 3. Conclusions and discussions of the results are presented in section 4. Preliminary results of this work have been reported in ref. [19].
2. Experimental procedures Experiments are quite simple, as shown below. Careful attention should be paid to avoiding contamination by miscellaneous bacteria and fungi. First of all, we chose Bacillus subtilis as the sample bacterial species. The reason for the choice is as follows. Bacteria are unicellular organisms, and the species Bacillus subtilis is rod-shaped with about 0.7 ~m diameter and 2 Ixm length, and proliferates through the simplest cell division. This enables us to study most clearly the effect of environmental conditions (nutrient concentration, agar moisture, temperature, and so on) on the colony formation. One more advantage to use this bacterial species is that it is hydrophobic and tends
500
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
to grow two-dimensionally on the surface of agar plates, which enables us to analyze colony patterns much more easily. The bacterial strain was isolated from food and identified as Bacillus subtilis. We used the same strain throughout the present experiments. A solution of 5 g of sodium chloride, 5 g of potassium phosphate dibasic and a certain amount of Bactopeptone (Difco Laboratories, Detroit, USA) as nutrient was made in 1 liter of distilled water, and adjusted at p H 7.1 by adding 6 N hydrochloric acid. The solution was then mixed with 15 g of Bactoagar (Difco). The mixture was heated at 121°C for 15 min to autoclave it and then 20 ml of it was poured into each plastic petri dish of 88 mm diameter. After being kept at room temperature overnight, the agar plates were dried at 50°C for about 40 min. The thickness of the agar plates thus prepared is about 3 mm. The bacterial strain was inoculated on the agar plate surface at the center of the dish. The plates were stored in a humidified box at 35°C. Bacterial colonies incubated and grown on agar plates for a while (typically three weeks) were photographed through transmitted light. The photos were then analyzed digitally by a personal computer through an image scanner to obtain the fractal dimension by means of the box-counting method.
3. Experimental results The colony patterns grow two-dimensionally on the agar plates, since the species used in the present experiments (Bacillus subtilis) is hydrophobic. The effect of nutrient concentration was investigated first. Fig. 1 shows colony patterns incubated three weeks after inoculation at various initial nutrient concentrations. Maximum colony size in the figure (fig. lc) is about 2.5 cm. An increase of the nutrient concentration is found to enhance the colony growth. Particularly, the growth does not take place without nutrient, as seen in fig. la. This means that the local bacterial proliferation (local growth process) at the interface of colonies is governed primarily by the presence of nutrient. It should be noted that already in fig. lc the colony pattern exhibits an outwardly open and randomly branched structure, clearly reminiscent of a two-dimensional D L A pattern. From now on we fix the initial nutrient concentration at I g/l. (Note that this concentration is much less than usual, e.g., 15 g/l.) In fig. 2 a typical example is shown of colony patterns incubated three weeks after inoculation at this nutrient concentration. It really looks like a two-dimensional D L A pattern. In fact, its fractal dimension measured by the box-counting method is about 1.72. Averaged over 14 samples from the same strain which look more or less similar, we obtained a fractal dimension
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
501
b
~i
¸ i
,
Fig. 1. The effect of nutrient concentrations on the colony growth. The photographs were taken three weeks after inoculation. The initial peptone concentrations were (a) 0 g/l, (b) 0.25 g/I and (c) 0.5 g/l (the colony size is about 2.5 cm). Note in particular that no growth occurs without nutrient.
502
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
Fig. 2. A typical example of DLA-like colony patterns incubated at 35°C for three weeks after inoculation on the surface of agar plates containing initially 1 g/l of peptone as nutrient. This pattern has a fractal dimension of D ~ 1.72.
D = 1.73 _+ 0.02. This is in good agreement with that of two-dimensional D L A patterns [8]. As described above, this result does not immediately mean that bacterial colonies are f o r m e d by the D L A mechanism; m o r e evidence is needed. The next problem is whether these colony patterns are really growing through D L A processes or not. In order to examine this, we first tried to observe the existence of the screening effect during the colony growth. This effect is characteristic for the pattern formation in a Laplacian field. As clearly seen in fig. 3, many interior branches were found to stop growing afterwards in spite of their open neighborhood during the incubation period. This evidences the existence of the screening effect of protruding main branches against interior ones in a colony. We next observed the behavior of two neighboring colonies. In fig. 4 the colony pattern is shown incubated three weeks after having inoculated at two points 1 cm apart. As clearly seen in the figure, two neighboring colonies repel each other and never fuse together. This behavior is also another feature of the pattern formation in a Laplacian field. Taking all these experimental results into account, it is now clear that the present bacterial colonies grow through D L A processes. Then, a remaining problem is: What makes the Laplacian field, or what diffuses? In the present case there are two possibilities: (1) Nutrient diffuses in toward the colonies, a n d / o r (2) some waste material discharged by bacteria diffuses out from the colonies. Both cases are equally possible to produce the same D L A patterns.
M. Matsushita and H. Fufikawa
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Diffusion-limited bacterial colony growth
503
"
Fig. 3. The growth of a DLA-like colony pattern photographed (a) 8, (b) 13 and (c) 19 days after inoculation. Note that many inner branches, typical examples of which are pointed by triangles, are seen to stop growing afterwards, indicating the screening effect.
504
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
Fig. 4. The repulsion of two neighboring colonies. Note that two colonies inoculated initially at two points I cm apart repel each other and never fuse together.
In o r d e r to d e t e r m i n e which o n e is r e a l l y t a k i n g p l a c e , w e p u t t h e n u t r i e n t p e p t o n e at a c o r n e r o f a dish ( e l s e w h e r e n o n u t r i e n t ) , a n d i n o c u l a t e d b a c t e r i a at t h e c e n t e r . A s s e e n in fig. 5, t h e c o l o n y s h o w e d a c l e a r t e n d e n c y to g r o w t o w a r d w h e r e t h e n u t r i e n t was initially p l a c e d . This result i n e v i t a b l y l e a d s us to t a k e t h e first p o s s i b i l i t y a n d c o n c l u d e t h a t t h e n u t r i e n t c o n c e n t r a t i o n field is essential for t h e c o l o n y p a t t e r n to g r o w t h r o u g h t h e D L A p r o c e s s e s .
Fig. 5. Bacterial colony showing a clear tendency to grow toward the nutrient. The nutrient peptone was initially put only at the right corner of the dish in the photo.
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
505
4. Discussion
We have inoculated bacteria called Bacillus subtilis on the surface of agar plates which initially contained a small amount of peptone as nutrient, and incubated them for a while. We have found that bacterial colonies grow two-dimensionally on the agar plates and have an outwardly open and randomly branched structure, clearly reminiscent of two-dimensional DLA patterns. We then obtained a fractal dimension of the colony patterns of D = 1.73-+ 0.02, which is very close to that of DLA. We have, in addition, confirmed the existence of the screening effect of protruding main branches against inner ones in a colony, the repulsion between two neighboring colonies and the tendency of colony growth toward the nutrient. All these results clearly indicate that the bacterial colonies grow through the DLA mechanism in a nutrient concentration field. DLA essentially consists of two ingredients [20]: (i) a Laplacian field surrounding a cluster, and (ii) a local growth mechanism at the cluster surface. A diffusion field instead of the Laplacian one due to, e.g., a high concentration of diffusing particles, or the addition of a drift field such as sedimentation gives rise to a morphological change in growing patterns [8]. The modification of the local growth mechanism such as the introduction of anisotropy and surface tension effects is also known to induce a morphological change [8]. Usual bacterial colony patterns do not look like DLA; nevertheless, our present results may be generalized as follows: For the bacterial colony growth in general, the nutrient concentration constitutes the most important field surrounding a colony and the way of bacterial proliferation corresponds to the essential local growth mechanism. Sometimes, the nutrient concentration may be too high, which results in rather compact colony patterns, instead of DLA-like ramified ones. And also, the style of the bacterial proliferation may change from species to species and from genus to genus. Suppose, for instance, that some bacteria tend to proliferate a little bit left-handed. Then the colony may exhibit something like a left-handed spiral galaxy. This kind of colony patterns are actually seen for some bacteria. In other words, although they look different from DLA, it is plausible that in the limit of low nutrient concentrations the growth of bacterial colonies is generally controlled by diffusion-limited processes. Making one more step forward, one may insist that the intrinsic morphology of bacterial colonies can only be obtained in the limit of low nutrient concentrations, since only then the local growth process inherent to a bacterial species may produce some colony pattern inherent to the species. We have also studied the effect of high nutrient concentration and agar plate moisture on the colony morphology [19]. Particularly interesting is the fact that
506
M. Matsushita and H. Fufikawa / Diffusion-limited bacterial colony growth
c o l o n y p a t t e r n s t e n d to show the so-called d e n s e - b r a n c h e d m o r p h o l o g y w h e n they grow o n moist agar plates. M o r e extensive e x p e r i m e n t a l i n v e s t i g a t i o n is, h o w e v e r , n e e d e d o n the m o r p h o l o g i c a l c h a n g e in bacterial colonies, a n d we h o p e to r e p o r t o u r results o n this p r o b l e m in the n e a r future.
Acknowledgements T h e a u t h o r s are grateful to Prof. T. M a t s u y a m a for v a l u a b l e suggestions a n d discussions. T h a n k s are also d u e to J. T e r a d a a n d K. M u r a t a for their assistance in the fractal analysis.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). J. Feder, Fractals (Plenum, New York, 1988). T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989). M. Eden, in: Proc. 4th Berkeley Symp. Math. Stat. and Prob., vol. 4, F. Neyman, ed. (Univ. of California Press, Berkeley, 1961) p. 233. M.J. Void, J. Colloid Sci. 18 (1963) 684. D.N. Sutherland, J. Colloid Interface Sci. 25 (1967) 373. T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. P. Meakin, in: Phase Transitions and Critical Phenomena, vol. 12, C. Domb and J.L. Lebowitz, eds. (Academic Press, New York, 1988) p. 335. See, e.g., review articles by M. Matsushita, G. Daccord and H. Van Damme, in: The Fractal Approach to Heterogeneous Chemistry, D. Avnir, ed. (Wiley, Chichester, 1989). A.A. Tsonis and P.A. Tsonis, Perspect. Biol. Med. 30 (1987) 355. B.J. West and A.L. Goldberger, J. Appl. Physiol. 60 (1986) 189; Am. Sci. 75 (1987) 354. B.J. West, Bull. Am. Phys. Soc. 34 (1989) 716. S. Majumdar and R.R. Prasad, Comp. Phys. 2 (Nov/Dec 1988) 69. J.H.G.M. van Beek, J.B. Bassingthwaighte and R.B. King, Biophys. J. 53 (Suppl.) (1988) 401. J.B. Bassingthwaighte, Bull. Am. Phys. Soc. 34 (1989) 715. H.E. Stanley, Bull. Am. Phys. Soc. 34 (1989) 716. F. Family, B.R. Masters and D.E. Platt, Physica D 38 (1989) 98. P. Meakin, J. Theor. Biol. 118 (1986) 101. H. Fujikawa and M. Matsushita, J. Phys. Soc. Jpn. 58 (1989) 3875. M. Matsushita, K. Honda, H. Toyoki, Y. Hayakawa and H. Kondo, J. Phys. Soc. Jpn. 55 (1986) 2618.
D I F F U S I O N - L I M I T E D G R O W T H IN B A C T E R I A L C O L O N Y FORMATION Mitsugu M A T S U S H I T A a and Hiroshi F U J I K A W A b
aDepartment of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112, Japan bTokyo Metropolitan Research Laboratory of Public Health, Hyakunin-cho, Shinjuku-ku, Tokyo 169, Japan Colonies of bacterial species called Bacillus subtilis have been found to grow twodimensionally and self-similarly on agar plates through diffusion-limited processes in a nutrient concentration field. We obtained a fractal dimension of the colony patterns of D = 1.73 --+0.02, very close to that of the two-dimensional DLA model, and confirmed the existence of the screening effect of protruding main branches against inner ones in a colony, the repulsion between two neighboring colonies and the tendency to grow toward nutrient. These effects are all characteristic of the pattern formation in a Laplacian field. This finding implies the importance of physical properties of the environment for the morphology of bacterial colonies in general.
1. Introduction T h e f o r m a t i o n of r a n d o m fractals and their structural properties is a p r o b l e m of considerable current interest [1-3]. A m o n g m a n y growth models p r o p o s e d so far to describe it, such as the E d e n [4] and the ballistic aggregation [5, 6] models, particular attention has b e e n paid to the diffusion-limited aggregation ( D L A ) m o d e l [7]. It not only generates ffactal structures with a fractal dimension D clearly less than the spatial dimension d (D ~ 1.7 and 2.5 for d = 2 and 3, respectively), but also describes the growth processes in a wide variety of physical and chemical p h e n o m e n a such as electrodeposition, dielectric b r e a k d o w n , viscous fingering, dendritic crystal growth and chemical dissolution [8, 9]. It is n o w well k n o w n [8] that D L A patterns have o u t w a r d l y o p e n and r a n d o m l y b r a n c h e d structures with no characteristic length scales except their whole size and b r a n c h thickness (aggregated particle size). A l t h o u g h f r o m the physicist's viewpoint biological g r o w t h generally leads to the f o r m a t i o n of t o o c o m p l e x patterns, biology m a y still have great potentiality for the application of fractal concepts. In fact, there are a wide variety of biological patterns which seem to be self-similar, such as trees, bronchial trees and n e t w o r k s of nerves and b l o o d vessels. This has already b e e n confirmed by ffactal analyses for blood vessel patterns in the chicken e m b r y o [10], the 0378-4371/90/$03.50 © 1990-Elsevier Science Publishers B.V. (North-Holland)
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
499
structure of the bronchial tree [11, 12], the cerebral surface of the normal human brain [13], the vascular heterogeneity in the heart [14, 15], the neuronal arborization [16] and the human retinal vessel [17] patterns. Unfortunately, however, they are rather "static" analyses to obtain the fractal dimension of "snapshot" patterns. One cannot fully characterize given patterns by their fractal dimensions alone. For instance, DLA and invasion percolation patterns have almost the same fractal dimensions (D ~2.5) for d = 3, but they look quite different from each other. The observation of the growth itself, i.e., the dynamical behavior, is clearly needed to characterize patterns and identify more elementary growth processes. Meakin [18] demonstrated by extensive computer simulations that when biological growth is governed by diffusion-limited processes, the growing patterns show characteristic features such as screening, repulsion, and so on. Conversely, the existence of these effects enables one to confirm the diffusion-limited growth in biological systems. No experiments have, however, been reported so far. In this paper, we present our recent results on the fractal growth in the formation of bacterial colonies. They clearly indicate that the growth of bacterial colonies on agar plates is governed by DLA processes of the nutrient contained in the agar plates. To the best of our knowledge this is the first unambiguous identification of fractal growth processes in biological systems. Moreover, this implies the important revelation that the formation of bacterial colonies is strongly affected by the physical conditions of the environment as well as the proliferation way of the bacteria. Only the latter aspect has so far been taken into account in the morphological study of bacterial colonies. This paper is organized as follows: Section 2 briefly describes the experimental procedures. A full description of the experimental results follows in section 3. Conclusions and discussions of the results are presented in section 4. Preliminary results of this work have been reported in ref. [19].
2. Experimental procedures Experiments are quite simple, as shown below. Careful attention should be paid to avoiding contamination by miscellaneous bacteria and fungi. First of all, we chose Bacillus subtilis as the sample bacterial species. The reason for the choice is as follows. Bacteria are unicellular organisms, and the species Bacillus subtilis is rod-shaped with about 0.7 ~m diameter and 2 Ixm length, and proliferates through the simplest cell division. This enables us to study most clearly the effect of environmental conditions (nutrient concentration, agar moisture, temperature, and so on) on the colony formation. One more advantage to use this bacterial species is that it is hydrophobic and tends
500
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
to grow two-dimensionally on the surface of agar plates, which enables us to analyze colony patterns much more easily. The bacterial strain was isolated from food and identified as Bacillus subtilis. We used the same strain throughout the present experiments. A solution of 5 g of sodium chloride, 5 g of potassium phosphate dibasic and a certain amount of Bactopeptone (Difco Laboratories, Detroit, USA) as nutrient was made in 1 liter of distilled water, and adjusted at p H 7.1 by adding 6 N hydrochloric acid. The solution was then mixed with 15 g of Bactoagar (Difco). The mixture was heated at 121°C for 15 min to autoclave it and then 20 ml of it was poured into each plastic petri dish of 88 mm diameter. After being kept at room temperature overnight, the agar plates were dried at 50°C for about 40 min. The thickness of the agar plates thus prepared is about 3 mm. The bacterial strain was inoculated on the agar plate surface at the center of the dish. The plates were stored in a humidified box at 35°C. Bacterial colonies incubated and grown on agar plates for a while (typically three weeks) were photographed through transmitted light. The photos were then analyzed digitally by a personal computer through an image scanner to obtain the fractal dimension by means of the box-counting method.
3. Experimental results The colony patterns grow two-dimensionally on the agar plates, since the species used in the present experiments (Bacillus subtilis) is hydrophobic. The effect of nutrient concentration was investigated first. Fig. 1 shows colony patterns incubated three weeks after inoculation at various initial nutrient concentrations. Maximum colony size in the figure (fig. lc) is about 2.5 cm. An increase of the nutrient concentration is found to enhance the colony growth. Particularly, the growth does not take place without nutrient, as seen in fig. la. This means that the local bacterial proliferation (local growth process) at the interface of colonies is governed primarily by the presence of nutrient. It should be noted that already in fig. lc the colony pattern exhibits an outwardly open and randomly branched structure, clearly reminiscent of a two-dimensional D L A pattern. From now on we fix the initial nutrient concentration at I g/l. (Note that this concentration is much less than usual, e.g., 15 g/l.) In fig. 2 a typical example is shown of colony patterns incubated three weeks after inoculation at this nutrient concentration. It really looks like a two-dimensional D L A pattern. In fact, its fractal dimension measured by the box-counting method is about 1.72. Averaged over 14 samples from the same strain which look more or less similar, we obtained a fractal dimension
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
501
b
~i
¸ i
,
Fig. 1. The effect of nutrient concentrations on the colony growth. The photographs were taken three weeks after inoculation. The initial peptone concentrations were (a) 0 g/l, (b) 0.25 g/I and (c) 0.5 g/l (the colony size is about 2.5 cm). Note in particular that no growth occurs without nutrient.
502
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
Fig. 2. A typical example of DLA-like colony patterns incubated at 35°C for three weeks after inoculation on the surface of agar plates containing initially 1 g/l of peptone as nutrient. This pattern has a fractal dimension of D ~ 1.72.
D = 1.73 _+ 0.02. This is in good agreement with that of two-dimensional D L A patterns [8]. As described above, this result does not immediately mean that bacterial colonies are f o r m e d by the D L A mechanism; m o r e evidence is needed. The next problem is whether these colony patterns are really growing through D L A processes or not. In order to examine this, we first tried to observe the existence of the screening effect during the colony growth. This effect is characteristic for the pattern formation in a Laplacian field. As clearly seen in fig. 3, many interior branches were found to stop growing afterwards in spite of their open neighborhood during the incubation period. This evidences the existence of the screening effect of protruding main branches against interior ones in a colony. We next observed the behavior of two neighboring colonies. In fig. 4 the colony pattern is shown incubated three weeks after having inoculated at two points 1 cm apart. As clearly seen in the figure, two neighboring colonies repel each other and never fuse together. This behavior is also another feature of the pattern formation in a Laplacian field. Taking all these experimental results into account, it is now clear that the present bacterial colonies grow through D L A processes. Then, a remaining problem is: What makes the Laplacian field, or what diffuses? In the present case there are two possibilities: (1) Nutrient diffuses in toward the colonies, a n d / o r (2) some waste material discharged by bacteria diffuses out from the colonies. Both cases are equally possible to produce the same D L A patterns.
M. Matsushita and H. Fufikawa
C
Diffusion-limited bacterial colony growth
503
"
Fig. 3. The growth of a DLA-like colony pattern photographed (a) 8, (b) 13 and (c) 19 days after inoculation. Note that many inner branches, typical examples of which are pointed by triangles, are seen to stop growing afterwards, indicating the screening effect.
504
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
Fig. 4. The repulsion of two neighboring colonies. Note that two colonies inoculated initially at two points I cm apart repel each other and never fuse together.
In o r d e r to d e t e r m i n e which o n e is r e a l l y t a k i n g p l a c e , w e p u t t h e n u t r i e n t p e p t o n e at a c o r n e r o f a dish ( e l s e w h e r e n o n u t r i e n t ) , a n d i n o c u l a t e d b a c t e r i a at t h e c e n t e r . A s s e e n in fig. 5, t h e c o l o n y s h o w e d a c l e a r t e n d e n c y to g r o w t o w a r d w h e r e t h e n u t r i e n t was initially p l a c e d . This result i n e v i t a b l y l e a d s us to t a k e t h e first p o s s i b i l i t y a n d c o n c l u d e t h a t t h e n u t r i e n t c o n c e n t r a t i o n field is essential for t h e c o l o n y p a t t e r n to g r o w t h r o u g h t h e D L A p r o c e s s e s .
Fig. 5. Bacterial colony showing a clear tendency to grow toward the nutrient. The nutrient peptone was initially put only at the right corner of the dish in the photo.
M. Matsushita and H. Fujikawa / Diffusion-limited bacterial colony growth
505
4. Discussion
We have inoculated bacteria called Bacillus subtilis on the surface of agar plates which initially contained a small amount of peptone as nutrient, and incubated them for a while. We have found that bacterial colonies grow two-dimensionally on the agar plates and have an outwardly open and randomly branched structure, clearly reminiscent of two-dimensional DLA patterns. We then obtained a fractal dimension of the colony patterns of D = 1.73-+ 0.02, which is very close to that of DLA. We have, in addition, confirmed the existence of the screening effect of protruding main branches against inner ones in a colony, the repulsion between two neighboring colonies and the tendency of colony growth toward the nutrient. All these results clearly indicate that the bacterial colonies grow through the DLA mechanism in a nutrient concentration field. DLA essentially consists of two ingredients [20]: (i) a Laplacian field surrounding a cluster, and (ii) a local growth mechanism at the cluster surface. A diffusion field instead of the Laplacian one due to, e.g., a high concentration of diffusing particles, or the addition of a drift field such as sedimentation gives rise to a morphological change in growing patterns [8]. The modification of the local growth mechanism such as the introduction of anisotropy and surface tension effects is also known to induce a morphological change [8]. Usual bacterial colony patterns do not look like DLA; nevertheless, our present results may be generalized as follows: For the bacterial colony growth in general, the nutrient concentration constitutes the most important field surrounding a colony and the way of bacterial proliferation corresponds to the essential local growth mechanism. Sometimes, the nutrient concentration may be too high, which results in rather compact colony patterns, instead of DLA-like ramified ones. And also, the style of the bacterial proliferation may change from species to species and from genus to genus. Suppose, for instance, that some bacteria tend to proliferate a little bit left-handed. Then the colony may exhibit something like a left-handed spiral galaxy. This kind of colony patterns are actually seen for some bacteria. In other words, although they look different from DLA, it is plausible that in the limit of low nutrient concentrations the growth of bacterial colonies is generally controlled by diffusion-limited processes. Making one more step forward, one may insist that the intrinsic morphology of bacterial colonies can only be obtained in the limit of low nutrient concentrations, since only then the local growth process inherent to a bacterial species may produce some colony pattern inherent to the species. We have also studied the effect of high nutrient concentration and agar plate moisture on the colony morphology [19]. Particularly interesting is the fact that
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M. Matsushita and H. Fufikawa / Diffusion-limited bacterial colony growth
c o l o n y p a t t e r n s t e n d to show the so-called d e n s e - b r a n c h e d m o r p h o l o g y w h e n they grow o n moist agar plates. M o r e extensive e x p e r i m e n t a l i n v e s t i g a t i o n is, h o w e v e r , n e e d e d o n the m o r p h o l o g i c a l c h a n g e in bacterial colonies, a n d we h o p e to r e p o r t o u r results o n this p r o b l e m in the n e a r future.
Acknowledgements T h e a u t h o r s are grateful to Prof. T. M a t s u y a m a for v a l u a b l e suggestions a n d discussions. T h a n k s are also d u e to J. T e r a d a a n d K. M u r a t a for their assistance in the fractal analysis.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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