The Keller-Segel model

I mentioned the Keller-Segel model in a previous post on chemotaxis. In the past I have read, and heard and thought a lot about this model but I had never actually carefully read the 1970 paper where it was introduced. (Keller, E. F. and Segel, L. A., Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399). I have done so now. I also read a paper by Evelyn Fox Keller called ‘Science as a medium for friendship: how the Keller-Segel models came about’ (Bull. Math. Biol. 68, 1033) where she describes the history of the origin of the model. As indicated in the title of the paper the main object which the model was intended to describe was the formation of concentrations of the population density of the cellular slime mould Dictyostelium discoideum under certain circumstances. The title also indicates that the relevance of this model is to the very beginning of this process. Mathematical studies of the model which follow it up to the formation of singularities, and there are many of these in the literature, are probably of little relevance to Dictyostelium. They may, however be of relevance to the population dynamics of other chemotactic organisms, for instance E. coli. References can be found for instance in a paper of Brenner, Levitov and Budrene (Physical mechanisms for chemotactic pattern formation in bacteria, Biophys. J., 74, 1677). In Dictyostelium ‘stickiness’ as Keller and Segel call it (a more dignified sounding name would be ‘adhesion’) comes into play before the cell density gets very high. The mathematical modelling of adhesion in this type of context does not seem to be well developed although some possibilities have been proposed.

Keller and Segel define a parabolic system of four coupled evolution equations. They then simplify this by assuming that some of the variables have already evolved to equilibrium. This results in a system of two equations which is the usual starting point of mathematical papers on the subject. The equations are linearized about a stationary and homogeneous state and a mode analysis is carried out for the resulting linear system. Growing modes indicate instability. The system contains a number of parameters and for certain choices of these parameters instability is found. This is interpreted as the genesis of the concentrations in the population density which are to be modelled. This procedure is very similar to what is done in many analyses of systems in physics. The authors refer to the analysis of the Benard instability. Another source they quote is the work of Turing on pattern formation. The latter work has had a huge influence in mathematical biology. I am reminded of a mechanism in astrophysics, the Jeans instability, which is invoked to explain the formation of galaxies in the early universe.

A feature of the analysis which the authors see as unsatisfactory is that there is no prediction of the spatial scale on which the concentrations occur. They do make some suggestions for overcoming this. In experiments on Dictyostelium aggregation is seen to be accompanied by pulsations. It had been suggested that these are actively controlled by some pacemaker activity of the cells. Keller and Segel raise the idea that pulsations might arise from a system of the type they discuss without the need for additional input. In this context they mention the concept of overstability which as far as I can see just means the occurrence of an eigenvalue of the linearized problem with positive real part and non-zero imaginary part. In the absence of sufficient knowledge of the more recent literature on the subject I do not know whether, or to what extent, the issues raised in this paragraph have been resolved in the meantime.

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