Background and Experimental Results

There is an obvious case for studying bacteria. For example, bacteria are responsible for a large number of diseases and they are responsible for most of the recycling that takes place. Their use in other areas is clearly going to increase as our understanding of their complex biology becomes clearer. Here we are interested in how a global pattern in bacterial populations can arise from local interactions. Under a variety of experimental conditions numerous strains of bacteria aggregate to form stable (or rather temporarily stable) macroscopic patterns of surprising complexity but with remarkable regularity. It is not easy to explain how these patterns are formed solely by experiment, but they can, however, be explained for the most part, with the use of mathematical models based on the known biology. It is not possible to discuss all the patterns which have been studied experimentally so we concentrate on the collection of diverse patterns observed by Berg and his colleagues (see Budrene and Berg 1991, 1995 and earlier references there) in the bacteria Escherichia coli (E. coli) and Salmonella typhimurium (S. typhimurium).

E. coli and S. typhimurium are common bacteria: for example, E. coli is abundant in the human intestine and S. typhimurium can occur in incompletely cooked poultry and meat. These bacteria are motile and they move by propelling themselves by means of long hairlike flagella (Berg 1983). Berg also mapped the movement of a bacterium over time, and found that the organism's motion approximates a random walk and so the usual Fickian diffusion can be used to describe their random motility; their diffusion coefficient has been measured experimentally.

A key property of many bacteria is that in the presence of certain chemicals they move preferentially towards higher concentration of the chemical, when it is a chemoat-tractant, or towards a lower concentration when it is a repellent. The sensitivity to such gradients often depends on the concentration levels. In the modelling below we shall be concerned with chemotaxis and the sensitivity issue. The basic concept of chemo-taxis (and diffusion) was discussed in Chapter 11, Volume I, Section 11.4 and in more detail in the last chapter. Basically, whether or not a pattern will form depends on the appropriate interplay between the bacterial populations and the chemical kinetics and the competition between diffusive dispersal and chemotactic aggregation.

Budrene and Berg (1991, 1995) carried out a series of experimental studies on the patterns that can be formed by S. typhimurium and E. coli. They showed that a bacterial colony can form interesting and remarkably regular patterns when they feed on, or are exposed to, intermediates of the tricarboxylic acid (TCA) cycle especially succinate and fumarate. They used two experimental methods which resulted in three pattern forming mechanisms. The bacteria are placed in a liquid medium in one procedure, and on a semi-solid substrate (0.24% water agar) in the other. They found one mechanism for pattern formation in the liquid medium, and two in the semi-solid medium. In all of the experiments, the bacteria are known to secrete aspartate, a potent chemoattractant.

Liquid Experiments with E. coli and S. typhimurium

These experiments produce relatively simple patterns which appear quickly, on the order of minutes, and last about half an hour before disappearing permanently. Two types of patterns are observed and are selected according to the initial conditions. The simplest patterns are produced when the liquid medium contains a uniform distribution of bacteria and a small amount of the TCA cycle intermediate. The bacteria collect in aggregates of roughly the same size over the entire surface of the liquid, although the pattern often starts in one general area and spreads from there (Figure 5.1(a)).

In the second type of liquid experiment, the initial density of bacteria is uniform, and the TCA cycle intermediate is added locally to a particular spot, referred to as the 'origin.' Subsequently, the bacteria are seen to form aggregates which occur on a ring centred about the origin, and in a random arrangement inside the ring (Figure 5.1(b)).

Importantly, in these liquid experiments, the patterns are generated on a timescale which is less than the time required for bacterial reproduction and so proliferation does not contribute to the pattern formation process. Also, the bacteria are not chemotactic to any of the chemicals initially placed in the medium, including the stimulant. The experimentalists (H.C. Berg, personal communication 1994) also confirmed that fluid dynamic effects are not responsible for the observed patterns.

Semi-Solid Experiments with E. coli and S. typhimurium

The most interesting patterns are observed in the semi-solid experiments and in particular with E. coli. For these experiments, a high density inoculum of bacteria is placed on a petri dish which contains a uniform distribution of stimulant in the semi-solid medium, namely, 0.24% water agar. Here the stimulant also acts as the main food source for the bacteria, and so the concentrations are much larger than in the liquid experiments. After two or three days, the population of bacteria has gone through 25-40 generations during which time it spreads out from the inoculum, eventually covering the entire surface of the dish with a stationary pattern of high density aggregates separated by regions of near zero cell density. Some typical final patterns are shown in Figures 5.2 and 5.3. The S. typhimurium patterns are concentric rings and are either continuous or spotted while E. coli patterns are more complex, involving a greater degree of positional symmetry between individual aggregates. An enormous variety of patterns has been observed, the most common being sunflower type spirals, radial stripes, radial spots and chevrons. Using time-lapse videography of the experiments reveals the very different kinematics by which S. typhimurium and E. coli form their patterns.

The simple S. typhimurium patterns (Figure 5.2) begin with a very low density bacterial lawn spreading out from the initial inoculum. Some time later, a high density ring of bacteria appears at some radius less than the radius of the lawn and after another

Figure 5.1. Bacterial patterns formed by Salmonella typhimurium in liquid medium. (a) A small amount of TCA was added to a uniform distribution of bacteria. (b) A small amount of TCA was added locally to a uniform distribution of bacteria. (Unpublished results of H. Berg and E.O. Budrene; photographs were kindly provided by courtesy of Dr. Howard Berg and Dr. Elena Budrene and reproduced with their permission.)

Figure 5.1. Bacterial patterns formed by Salmonella typhimurium in liquid medium. (a) A small amount of TCA was added to a uniform distribution of bacteria. (b) A small amount of TCA was added locally to a uniform distribution of bacteria. (Unpublished results of H. Berg and E.O. Budrene; photographs were kindly provided by courtesy of Dr. Howard Berg and Dr. Elena Budrene and reproduced with their permission.)

Bacteria Pattern Formation

Figure 5.2. Typical S. typhimurium patterns obtained in semi-solid medium and visualized by scattered light. Experiments were carried out by Howard Berg and Elena Budrene using the techniques described in Budrene and Berg (1991). About 104 bacteria were inoculated at the centre of the dish containing 10 ml of soft agar in succinate. In (a) the rings remain more or less intact while in (b) they break up as described in the text: the time from inoculation in (a) is 48 hours and in (b), 70 hours. The 1-mm grid on the left of each figure gives an indication of the scale of the patterns. (From Woodward et al. 1995 where more experimental details and results are provided)

Figure 5.2. Typical S. typhimurium patterns obtained in semi-solid medium and visualized by scattered light. Experiments were carried out by Howard Berg and Elena Budrene using the techniques described in Budrene and Berg (1991). About 104 bacteria were inoculated at the centre of the dish containing 10 ml of soft agar in succinate. In (a) the rings remain more or less intact while in (b) they break up as described in the text: the time from inoculation in (a) is 48 hours and in (b), 70 hours. The 1-mm grid on the left of each figure gives an indication of the scale of the patterns. (From Woodward et al. 1995 where more experimental details and results are provided)

Figure 5.3. E. coli patterns obtained in semi-solid medium. Note the highly regular patterns. The light regions represent high density of bacteria. The different patterns (a)-(d) are discussed in the text at several places. (From Budrene and Berg (1991); photographs courtesy of Dr. Howard Berg and reproduced with permission)

Figure 5.3. E. coli patterns obtained in semi-solid medium. Note the highly regular patterns. The light regions represent high density of bacteria. The different patterns (a)-(d) are discussed in the text at several places. (From Budrene and Berg (1991); photographs courtesy of Dr. Howard Berg and reproduced with permission)

time interval, when the lawn has expanded further, a second high density bacterial ring appears at some radius larger than that of the first ring. The rings, once they are formed, are stationary. The rings may remain continuous as in Figure 5.2(a) or break up into a ring of spots as in Figure 5.2(b). The high density aggregates of bacteria in one ring have no obvious positional relation to the aggregates in the two neighbouring rings.

The more dramatic patterns exhibited by E. coli, such as those shown in Figure 5.3, have definite positional relationships between radially and angularly neighbouring aggregates. These relationships seem to be the result of existing aggregates inducing the formation of subsequent ones (Budrene and Berg 1995). Instead of an initial bacterial lawn, a swarm ring of highly active motile bacteria forms and expands outwards from the initial inoculum. The bacterial density in the swarm ring increases until the ring becomes unstable and some percentage of the bacteria are left behind as aggregates. These aggregates remain bright and full of vigorously motile bacteria for a short time, but then dissolve as the bacteria rejoin the swarm ring. Left behind in the aggregate's original location is a clump of bacteria which, for some unknown reason, are non-motile: it is these non-motile bacteria that are the markers of the pattern.

It appears that one, or both, of the speed of the swarm ring and the time at which the dissolution of aggregates occurs, are key elements in the formation of any one pattern. If the dissolution happens quickly, the aggregates appear to be pulled along by the swarm ring, and the non-motile bacteria are left behind as a radial streak as in Figures 5.3(c) and (d). On the other hand if the dissolution happens a little less quickly, the cells from the dissolved aggregate rejoin the swarm ring and induce the formation of aggregates at the rejoining locations and this results in a radial spot pattern. If the dissolution happens even more slowly, the swarm ring becomes unstable before the bacteria from the aggregates have time to rejoin the ring. The ring then tends to form aggregates in between the locations where aggregates already exist which results in a sunflower spiral type of pattern.

Remember that, just as in the liquid experiments, none of the chemicals placed in the petri dish is a chemoattractant. The timescale of these patterns, however, is long enough to accommodate several generations of E. coli, and so proliferation is important here. Consumption of stimulant is also non-negligible, especially in the swarm ring patterns.

Since none of the substrates used are chemoattractants the patterns of E. coli in Figure 5.3 cannot be explained by some external chemoattractants. Chemoattractants, however, play a major role since the bacteria themselves produce a potent chemoattrac-tant, namely, aspartate (Budrene and Berg 1991). Up to this time it had been assumed that the phenomenon of chemotaxis existed in E. coli and S. typhimurium only to guide the bacteria towards a food source. It was only in these experiments of Budrene and Berg (1991, 1995) that evidence was found that the bacteria can produce and secrete a chemoattractant as a signalling mechanism. This is reminiscent of the slime mould Dic-tyostelium discoideum where the cells produce the chemical cyclic AMP as a chemotac-tic aggregative signalling mechanism. In our modelling therefore we focus primarily on the processes of diffusion and chemotaxis towards an endogenously produced chemoat-tractant and how they interact to produce the bacterial patterns.

We should remember that all of these patterns are formed on a two-dimensional domain of the petri dish. With the wide variety of patterns possible it is clear that if the medium were three-dimensional the pattern complexity would be even greater. It would be quite an experimental challenge to photograph them. Here we model only the two-dimensional patterns and show that the models, which reflect the biology, can create the experimentally observed patterns.

What is clear is that chemotaxis phenomena can give rise to complex and varied geometric patterns. How these complex geometries form from interactions between individual bacteria is not easy to determine intuitively from experiments alone.1 In cases such as this, when biological intuition seems unable to provide an adequate explanation, mathematical modelling can play an important, even crucial, role. To understand the patterns, many questions must be answered, often associated with the fine details of the biological assumptions and parameter estimates. For example, are diffusion and

1 It was for this reason that Howard Berg initially got in touch with me. A consequence of the first joint modelling attempts (Woodward et al. 1995) generated informative and interesting experimental and biological questions.

chemotaxis towards an endogenously produced chemoattractant sufficient to explain the formation of these patterns? What is the quantitative role of the chemoattractant stimulant and how quickly must it be produced? What other patterns are possible and what key elements in the experiments should be changed to get them? A review of the biology, the modelling and the numerical schemes used to simulate the model equations is given by Tyson (1996).

Most of the material we consider in detail is based on a series of theoretical studies of these specific patterns (Woodward et al. 1995, Tyson 1996, Murray et al. 1998, Tyson et al. 1999). They proposed mathematical models which closely mimic the known biology. Woodward et al. (1995)—a collaborative work with the experimentalists Drs. Berg and Budrene—considered the less complex patterns formed by S. typhimurium and proposed an explanation for the observed self-organization of the bacteria. Ben-Jacob and his colleagues (Ben-Jacob et al. 1995, 2000; see other references there) have also studied a variety of bacteria both theoretically and experimentally: many of the patterns they obtain are also highly complex and dramatic. The patterns depend, of course, on the parameter values and experimental conditions. In nature, however, bacteria have to deal with a variety of conditions, both hostile and friendly. To accommodate such environmental factors bacteria have developed strategies for dealing with such conditions. These strategies involve cooperative communication and this affects the type of patterns they form. Ben-Jacob (1997; see other references there) has investigated the effect of possible communication processes, such as chemotactic feedback. The consequences of including such cooperativity in the model chemotactic systems is, as would be expected, that the spectrum of pattern complexity is even greater. The analytical and extensive numerical studies of Mimura and his colleagues (see, for example, Mimura and Tsujikawa 1996, Matsushita et al. 1998, 1999, Mimura et al. 2000 and other references there) are particularly important in highlighting some of the complex solution behaviour reaction diffusion chemotaxis systems can exhibit. For example, Mimura et al. (2000) classify the various pattern classes (five of them) and suggest that, with one exception, the morphological diversity can be generated by reaction diffusion models. Mimura and Tsujikawa (1996) considered a diffusion-chemotaxis with population growth and in the situation of small diffusion and chemotaxis they derived an equation for the time evolution of the aggregating pattern. In this chapter we discuss specific bacterial patterns obtained with S. typhimurium and E. coli and, very briefly, those exhibited by Bacillus subtilis which are quite different.

These bacterial patterns are far more elaborate than those observed when chemotac-tic strains grow on media containing nutrients that are attractants (for example, Agladze et al. 1993). They also differ from the travelling waves of aggregating cells of the slime mould Dictyostelium discoideum in that the structures formed by E. coli and S. ty-phimurium, for example, are only temporally stable.

The spatial pattern potential of chemotaxis has been exploited in a variety of different biological contexts. Mathematical models involving chemotaxis (along with reaction diffusion models (Chapters 2 and 3) and mechanochemical models (Chapters 6 and 7) are simply part of the general area of integrodifferential equation models for the development of spatial patterns. The basic Keller-Segel continuum mechanism for pattern formation in the slime mould Dictyostelium discoideum was proposed by Keller and Segel (1970) and was discussed in Chapter 11, Volume I, Section 11.4. A discrete, more biologically based, model (as a consequence of the new biological insights found since then) for the aggregation with appropriate cell signalling is given by Dallon and Othmer (1997). Othmer and Schaap (1998) give an extensive and thorough review of oscillatory cyclic AMP signalling in the development of this slime mould. Since the pioneering work of Keller and Segel (1970, 1971), a considerable amount of modelling effort has been expended on these patterns such as the work on bacteria by Ben-Jacob et al. (1995) who had thresholding behaviour in aspartate production and a cell-secreted waste field in their model. They obtained spatial patterns resembling some of the experimentally observed E. coli patterns. Brenner et al. (1998) performed a one-dimensional analysis of a model mechanism for swarm ring formation of E. coli patterns in a semi-solid medium. They studied the relative importance of the terms in their equations from the point of view of pattern formation and obtained some analytical results: for example, they derived an expression for the number of clumps in a given domain in terms of the model parameters.

Chemotaxis plays an important role in a wide range of practical phenomena such as in wound healing (see Chapter 10), cancer growth (see Chapter 11) and leukocytes moving in response to bacterial inflammation (for example, Lauffenburger and Kennedy 1983 and Alt and Lauffenburger 1987). Until recently, relatively little work had been done where cell populations are not constant; one exception was the travelling wave model of Kennedy and Aris (1980) where the bacteria reproduce and die as well as migrate. It appears that the presence of chemotaxis (or haptotaxis, a similar guidance phenomenon for cells in the mechanical theory of pattern formation discussed in Chapter 6) results in a wider variety of patterns than only reaction and diffusion, for example. Of course, when significant growth occurs during the patterning process the spectrum of patterns is even wider.

It is possibly pertinent to note here that the specific patterns formed by many bacteria depend sensitively on the parameters and on the conditions that obtain in the experiments, including the initial conditions. As such a potential practical application of bacterial patterning is as a quantitative measurement of pollution.

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