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).