Chemotaxis, as we saw in Chapter 11, Volume I, involves the directed movement of organisms up a concentration gradient, and so is like a negative diffusion. However, whereas diffusion of cells depends only on their density gradient, chemotaxis depends on the interaction between the cells, the chemoattractant and the chemoattractant gradient. The general form of the chemotaxis term in the conservation equations is the divergence of the chemotactic flux:

where Jc is the chemotactic flux, x(n, c) is the chemotaxis response function, as yet unknown, with n and c the cell density and chemoattractant concentration respectively. A lot of research has been directed to finding a biologically accurate expression for the chemotaxis function x (n, c). Ford and Lauffenburger (1991) reviewed the main types of functions tried. As in the case of the diffusion term, forms for x(n, c) have been proposed either by working up from a microscopic description of cell behaviour, or by curve fitting to macroscopic results from population experiments. A synthesis of the various approaches suggests that the macroscopic form of Lapidus and Schiller (1976), namely, n x(n, c) =

where k is a parameter is a good one. This seems to give the best results when compared to experimental data, in particular the experiments by Dahlquist et al. (1972) which were designed specifically to pinpoint the functional form of the chemotactic response. Interestingly, the inclusion of all of the receptor level complexity by other researchers did not give any significant improvement over the results of Lapidus and Schiller (1976). The main advantage, a significant one of course, of receptor models is that the parameters can be directly applied to experimentally observable physicochemical properties of the bacteria. It is not necessary, however, to include such detail in a population study. For the modelling and analyses here we are primarily interested in describing the behaviour of the populations of E. coli and S. typhimurium as a whole, so a macroscopically derived chemotaxis coefficient is most appropriate. Based on the above form, we choose (Woodward etal. 1995)

The parameters k1 and k2 can be determined from the experimental results of Dahlquist etal. (1972) and give k2 = 5 x 10-6Mand k1 = 3.9 x 10-9Mcm2s-1.

The proliferation term involves both growth and death of the bacteria. From Budrene and Berg (1995) cells grow at a constant rate that is affected by the availability of succinate. In the semi-solid experiments the stimulant, succinate or fumarate, is the main carbon source (nutrient) for the bacteria whereas in the liquid experiments, nutrient is provided in other forms and is not limiting. We thus assume a proliferation term of the form

( s2 \ cell growth and death = k3n ( k4-t — n I , (5.6)

where the k's are parameters.

Intuitively this form is a reasonable one to take: it looks like logistic growth with a carrying capacity which depends on the availability of nutrient, s. When the bacterial density is below the carrying capacity, the expression (5.6) is positive and the population of bacteria increases. When n is larger than the carrying capacity, the expression is negative and there is a net decrease in population density. Implied in this form is the assumption that the death rate per cell is proportional to n; another possibility, but less plausible perhaps, is simply a constant death rate per cell.

Production and Consumption of Chemoattractant and Stimulant

The model contains one production term (production of chemoattractant), and two consumption terms (uptake of chemoattractant and of stimulant). Due to lack of available data, we have to rely on intuition to decide what are reasonable forms for the production and uptake of chemoattractant.

For the nutrient consumption, we expect that nutrient will disappear from the medium at a rate proportional to that at which cells are appearing. Since the linear birth rate of the cells is taken to be k3k4s2/(k9 + s2) this suggests the following form for the consumption of nutrient by the cells, s2

where the k's are parameters. The consumption form has a sigmoid-like characteristic.

Chemoattractant consumption by the cells could have a similar sigmoidal character. However, the chemical is not necessary for growth and so there is likely very little created during the experiment. So, we simply assume that if a cell comes in contact with an aspartate molecule, it ingests it, which thus suggests a chemoattractant consumption, with parameter k7, of the form chemoattractant consumption = k7nc. (5.8)

The chemoattractant production term has also not been measured in any great detail. We simply know (H.C. Berg, personal communication 1993) that the amount of chemoattractant produced increases with nutrient concentration and probably saturates over time which suggests a saturating function, of which there are many possible forms. To be specific we choose n2

chemoattractant production = k5s-t , (5.9)

where k5 and k6 are other parameters. An alternative nonsaturating possibility which is also plausible is chemoattractant production = k5sn2. (5.10)

In fact both these forms give rise to the required patterns, so further experimentation is needed to distinguish between them, or to come up with some other function. The critical characteristic is the behaviour when n is small since there the derivative of the production function must be positive.

Mathematical Model for Bacterial Pattern Formation in a Semi-Solid Medium

Let us now put these various functional forms into the model word equation system (5.1)-(5.3) which becomes:

where n, c and s are the cell density, the concentration of the chemoattractant and of the stimulant respectively. There are three diffusion coefficients, three initial values (n, c and s at t = 0) and nine parameters k in the model. We have estimates for some of these parameters while others can be estimated with reasonable confidence. There are several, however, which, with our present knowledge of the biology, we simply do not know. We discuss parameter estimates below.

Mathematical Model for Bacterial Pattern Formation in a Liquid Medium

It seems reasonable to assume that the production of chemoattractant and the chemotac-tic response of the cells are governed by the same functions in both the liquid and semisolid experiments. The difference between the two groups of experiments lies more in the timescale and in the role of the stimulant. As mentioned earlier the cells do not have time to proliferate over the time course of the liquid experiments, so there is no growth term in this model. Also, the stimulant is not the main food source for the cells (it is externally supplied) so consumption of the stimulant is negligible. This shows that the liquid experiment model is simply a special case of the semi-solid experiment model. With cell growth, chemoattractant degradation and consumption of stimulant eliminated we are left with the simpler three-equation model d n 2

which has fewer unknown parameters than the semi-solid mode system. The last equation is uncoupled from the other equations. In the case of the simplest liquid experiment, in which the stimulant is uniformly distributed throughout the medium, the third equation can also be dropped.

As mentioned, we have some of the parameter values and can derive estimates for others from the available literature; we also have estimates for some parameter combinations. The product k3k4 is the maximum instantaneous growth rate, which is commonly determined from the generation time, tgen, as ln2

instantaneous growth rate =-.

tgen

For the E. coli experiments, the generation time is of the order of 2 hours, giving an instantaneous growth rate of 0.35/hour. The grouping k3k4/k8 is termed the yield coefficient, and is calculated by experimentalists as weight of bacteria formed weight of substrate consumed

Similarly, the grouping k3k4/k7 is the yield coefficient for the bacteria as a function of chemoattractant (nitrogen source). The parameters k1 and k2 are calculated from measurements of cellular drift velocity and chemotaxis gradients made by Dahlquist et al. (1972).

Parameter |
Value |
Source |

ki |
3.9 x 10-9 M cm2s-1 |
Dahlquist et al. 1972 |

k2 |
5 x 10-6M |
Dahlquist et al. 1972 |

k3 |
1.62 x 10-9 hr ml-1 cell-1 |
Budrene and Berg 1995 |

k4 |
3.5 x 108 cells ml-1 |
Budrene and Berg 1995 |

k9 |
4 x 10-6 M2 |
Budrene and Berg 1995 |

Dn |
2 - 4 x 10-6cm2s-1 |
Berg and Turner 1990; Berg 1983 |

Dc |
8.9 x 10-6cm2s-1 |
Berg 1983 |

Ds |
« 9 x 10-6cm2s-1 |
Berg 1983 |

no |
108 cells ml-1 |
Budrene and Berg 1991 |

so |
1 - 3 x 10-3 M |
Budrene and Berg 1995 |

Budrene and Berg (1995) measured growth rates for their experiments and this let us get reasonably precise determination of the parameters k4, k9 and k3 by curve fitting. They also measured the ring radius as a function of time in the semi-solid experiments. The known parameter estimates are listed in Table 5.1 along with the sources used. We do not have estimates for the other parameters. However, since we shall analyse the equation sytems in their nondimensional form it will suffice to have estimates for certain groupings of the parameters. These are given below in the legends of the figures of the numerical solutions of the equations.

Intuitive Explanation of the Pattern Formation Mechanism

Before analysing any model of a biological problem it is always instructive to try and see intuitively what is going to happen in specific circumstances. Remember that the domain we are interested in is finite, the domain of the experimental petri dish. Consider the full model (5.11)-(5.13). In (5.13) the uptake term is a sink in a diffusion equation and so as time tends to infinity, the nutrient concentration, s, tends to zero. This in turn implies, from (5.11) and (5.12) that eventually cell growth and production of chemoattractant both tend to zero, while consumption of chemoattractant and death of cells continue. So, both the cell density and chemoattractant concentration also tend to zero as time tends to infinity. Thus the only steady state in this model is the one at which (n, c, s) = (0,0, 0) everywhere. But, of course, this is not the situation we are interested in. What it implies, though, is that it is not possible to carry out a typical linear analysis with perturbations about a uniform nonzero steady state. Instead we must look at the dynamic solutions of the equations.

Now consider the model system for the liquid experiments, equations (5.14)-(5.16). The last equation implies that eventually the stimulant will be spatially uniform since it is simply the classical diffusion equation which smooths out all spatial heterogeneities over time. By inspection there is a uniform steady state (n, c, s) = (n0, 0, 0) with no, the initial concentration of cells, being another parameter which can be varied experimentally. From (5.15) the source term is always positive so c will grow unboundedly.

In this case, eventually this concentration will be sufficiently high to significantly reduce the chemotaxis response in (5.14) and in the end simple diffusion is dominant and the solutions become time-independent and spatially homogeneous. So, again, with the liquid experiments, we have to look at the dynamic evolution of the solutions. Here a perturbation of any one of the steady states ((n, c, s) = (n0, 0,0)) results in a continually increasing concentration of chemoattractant. To get anything interesting from the model analyses therefore we have to look for patterns somewhere in that window of time between perturbation of the uniform initial conditions and saturation of the chemotactic response.

It is straightforward to see how the physical diffusion-chemotaxis system for the liquid model (5.14) to (5.16) could give rise to the appearance, and disappearance, of high density aggregates of cells. At t = 0 the cells begin secreting chemoattractant and since the cells are randomly distributed, some areas have a higher concentration of chemoattractant than others. Because of the chemotaxis these groups of higher cell concentration attract neighbouring cells, thereby increasing the local cell density, and decreasing it in the surrounding area. The new cells in the clump also produce chemoat-tractant, increasing the local concentration at a higher rate than it is being increased by the surrounding lower density cell population. In this way, peaks and troughs in cell density and chemoattractant concentration are accentuated. This is not the whole story since diffusion of the cells and the chemicals is also involved and this has a dispersive effect which tries to counter the aggregative chemotactic process and smooth out these peaks and troughs or rather prevent them happening in the first place. It is then the classical situation of local activation and lateral inhibition and which process dominates—aggregation or dispersion—depends on the intimate relation between the various parameters and initial conditions via n0.

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