Archive for the ‘chemotaxis’ Category

Collective motion of bacteria and aggregation

March 29, 2010

In a previous post I wrote about chemotaxis, the motion of organisms in response to the concentration of a chemical substance. Here I want to discuss aspects of modelling this phenomenon on a population level. I consider the motion of E. coli in agar with chemotaxis in response to the concentration of aspartate. A presentation of mathematical models for this related to the Keller-Segel model can be found in Betterton and Brenner (Phys. Rev. E64, 61904). The phenomena of interest are the formation and propagation of bands of high concentration of bacteria and formation of localized aggregates. In the latter case this translates in the mathematical models into the occurrence of a singularity in the solution. In these models the phenomena taken into account are motion of the bacteria due to a chemical gradient, production of chemical substances (e.g. aspartate) and consumption of substances (e.g. succinate). The experimental observations and the first attempts to explain them are due to Adler (Science 166, 1588) and Budrene and Berg (Nature 376, 49).

I now want to focus on the situation where a band of bacteria breaks up into localized concentrations which then (in the mathematical models) proceed to singularity formation. Concentrating on a region close to a point of the band it seems reasonable to look at the collapse of an almost uniform cylinder of bacteria. This leads to the problem of formation of singularities in solutions of the Keller-Segel system in three space dimensions which are almost cylindrically symmetric. The case of exact cylindrical symmetry reduces to the Keller-Segel system in two space dimensions. From the point of view of scaling this is a critical problem and the asymptotics of collapse is subtle. (In three space dimensions there are self-similar solutions describing collapse to a point.) This asymptotic behaviour was analysed rigorously by Herrero and Velázquez (Math. Ann. 306, 583). On the basis of numerical work Betterton and Brenner suggested that the solutions of Herrero and Velázquez (HV) were unstable and that they (Betterton and Brenner) had found the generic behaviour. Velázquez (SIAM J. Appl. Math. 62, 1581) showed that the blow-up behaviour which Herrero and he found is stable in the sense of formal expansions. Later sophisticated numerical work by Budd et al. (J. Comp. Phys. 202, 463) led the authors to the conclusion that the HV behaviour is generic. A central idea of the numerical method is to use an adaptive mesh in such a way that the mesh points follow any scaling behaviour present in a solution. This gives a way of avoiding rescaling the mesh according to an expectation of the scaling behaviour and thus perhaps ‘finding the scaling you expect’.

There is a region close to the centre where the density of bacteria can be approximated by the result of applying a time-dependent rescaling to a the corresponding function coming from a static solution of the system. The scaling is the natural scaling of the system and generates a one-parameter family of time-independent solutions starting from any one of them. At least superficially all of this bears a close resemblance to what happens in the formation of singularities in critical wave maps, as described in a previous post. Does this have a deeper meaning?


Dictyostelium aggregation revisited

August 8, 2008

In a previous post on the Keller-Segel model I raised some questions concerning the applicability of the model to the aggregation of Dictyostelium discoideum. A coherent picture of the early stages of this process is presented in a review paper of Ben-Jacob, Cohen and Levine entitled ‘Cooperative self-organization of microorganisms’ (Adv. Phys. 49, 395-554), which I found on the web page of Levine. According to the account given there there is a first phase involving spiral waves and chemotaxis only starts to play a role after that. The tips of the spiral waves define the centres of the aggregation process. See Fig. 31 of the paper. There are also interesting comments on the role of adhesion later in the process of aggregation where streams are formed. All of this is backed up by extensive references. I was interested to see that several of the topics discussed in this blog are mentioned in the paper of Ben-Jacob et. al. (Liesegang rings, chemotaxis, quorum sensing, dynamics of actin polymerization, bacterial motion, Keller-Segel model, Dictyostelium). Thus there may be more unity in my apparently rather random musings than I had realized.

The paper of Ben-Jacob at. al. appears to be a rich repository of ideas about mathematics, physics, biology and their mutual interactions. It also contains many striking pictures of the patterns which microorganisms, in particular bacteria, can produce.

The Keller-Segel model

July 30, 2008

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.


May 25, 2008

Chemotaxis is the process by which cells move in response to gradients in the concentration of a chemical substance. There are many important examples in biology, for example embryonic development, wound healing, homing of white blood cells to a site of infection or the metastasis of cancer cells. Other examples include free-living organisms exemplified by the bacterium Escherichia coli or the cellular slime mould Dictyostelium discoideum. Both of these are favourite model organisms. Chemotaxis in the first is the subject of a book, ‘E. coli in motion‘ by Howard Berg while the second has its own website.

The modelling problems which arise in the study of chemotaxis can be divided into two types. One is to model how an individual cell reacts to a chemical gradient. The other is to model how whole populations of cells react. In the case of the first problem there is a big difference between the case of eukaryotic cells and that of bacteria. This has to do with the difference in size, which makes physics look very different in the two cases and restricts the mechanisms which are conceivable. The fact that eukaryotic cells move by deformation (extension of pseudopodia etc.) make the process much harder to describe theoretically than for a bacterium. Of course they are also more complicated on a biochemical level. When considering models for populations of cells it is particularly interesting to look at the case where the cells produce the chemical themselves. The model for chemotaxis which is probably most popular among mathematicians is the Keller-Segel model. An extensive review of the mathematical literature on this subject has been given by Dirk Horstmann (Jahresbericht der DMV, 105, 104-165; 106, 51-69).

In the case of E. coli, as described in the book of Berg, the mechanism by which the bacterium manages to move in a controlled way involves stochastic elements. There are flagella which move clockwise or anticlockwise and the motion is steered by the probabilities that they rotate in the two possible directions, depending on how many molecules of the substance being detected bind to receptors on the cell surface. Because the cell is so small it is not practical to use spatial differences in concentration to detect the motion. Instead temporal differences are used. In other words the bacterium, instead of asking the question ‘What is the direction of the gradient of the concentration where I am now?’ asks the question ‘how does the concentration change in time if I start moving in this direction’ for a sufficient sample of directions. On a mathematical level it is possible to start from a stochastic model encoding the behaviour of a single cell and derive a continuum model of the motion of a population of cells. For more details see the paper of Horstmann quoted above.

An indication how much (or how little) is understood about the mechanisms of chemotaxis of eukaryotic cells is provided by a talk of Michael Sixt from the Max Planck Institute of Biochemistry I heard in Berlin on 24th January. The following is based on some notes I made at that time. The central theme of the lecture of Sixt was chemotaxis of dendritic cells. These are immune cells which are responsible for collecting samples from tissues and transporting them to the lymph nodes where they are presented to other immune cells such as T-cells. He started by saying that chemotaxis of immune cells needs be fast in order to allow prompt immune responses. The usual explanation says that these cells use adhesion molecules called integrins in order to pull themselves through tissues. It turns out, however, that cells from knockout mice which have no functioning integrins can move as fast as normal cells in vivo. On a surface they cannot. The mechanism of their motion was studied in collagen gels. The motion at the front end took place by actin polymerization. When myosin was deactivated the front of the cells moved as fast as before but the back stayed where it was. The reason for this was that the nucleus, the most inflexible part of the cell gets stuck in the pores of the gel. Changing to a gel with larger pores increased the motility of the myosin-inhibited cells. To see more details other studies were done under agarose. This means that the cells are confined to move between a hard surface and a layer of gel which they cannot penetrate. Neutrophils were able to move faster than some other leukocytes. The reason has to do with their full name – neutrophil polymorphonuclear granulocytes. The irregular form of the nucleus allows it to be pushed more effectively through the pores. This kind of motion could also be very interesting from the point of view of metastasis. A therapy which is based on hitting adhesion molecules would not affect this kind of motion at all. What the adhesion molecules are needed for is extravasation (leaving the blood vessels). Sixt accompanied his talk by striking films taken under the microscope which illustrate the points just described. Recently a paper related to this talk where he was one of the authors appeared (Lammermann et. al., Nature 453, 51-55).