Research A. Rendall

In my research I have recently concentrated on mathematical models which are important for biology and medicine. A particular focus was on immunology.

States of the immune system

Immunology involves interacting systems consisting of cells and the substances they secrete and which they use to communicate with each other. Often these systems are so complex that it is difficult to treat them theoretically in terms of words and pictures alone. One possibility of going further is to use mathematical models. My first project in this area concerned a model of this type which describes different states of the immune system (Th1 or Th2 dominance). As in problems coming from many areas of science the model used is a system of ordinary differential equations depending on parameters. The previous analysis of this model in the literature essentially consisted of some numerical computations for special values of the parameters and initial data. I wanted to understand the global dynamics of this model and if possible prove corresponding theorems. My first paper on this subject was essentially a confirmation of what was already supposed to be true. In a later paper I was able to extend this analysis and discover new phenomena. It had earlier been seen that the system can exhibit two steady states which correspond to Th1 and Th2 dominance. I was able to show that for certain parameter values there are more than two stable steady states, something which was not previously known and which is potentially of therapeutic importance. For the competition between Th1 and Th2 is important for the development of many diseases, both infectious diseases such as tuberculosis and autoimmune diseases such as rheumatoid arthritis and multiple sclerosis. This model is already somewhat outdated since it does not include more recent advances related to the importance of Th17 cells or regulatory T cells. The analysis could, however, presumably be extended to cover such things.

Activation of T cells

The model just discussed describes the behaviour of a population of immune cells. A complementary task is to describe the mechanisms taking place inside an immune cell which determine its behaviour. In this context an important concept is that of signalling pathways. These are sets of coupled chemical reactions which can often be modelled by ordinary differential equations. Many of these pathways propagate information from the surface of the cell (where receptors are affected by substances exterior to the cell) to the nucleus where they influence the behaviour of transcription factors which control how frequently certain genes are read. An example of this is the NFAT pathway which is involved in the activation of T cells. I have written a paper on the dynamics of this pathway as described by ODE. One of the main results is that a large system of ODE which models a component of this pathway has exactly one stationary solution and that all other solutions converge to it.  The proof uses the machinery of chemical reaction network theory. In this paper I also investigated another component of this pathway describing oscillations of the calcium concentration in the cell. I was able to give criteria for the presence or absence of periodic solutions of this system depending on biological parameters. The task of a T cell is to recognize foreign substances and to react accordingly. In this way an immune response is triggered. In this process information is transferred from the surface of the cell to the nucleus by a network of chemical reactions. The NFAT signalling pathway is an important component of this network.

The MAP kinase cascade

One chemical network which is an important component of many signalling pathways and which is, in particular, important in T cell signalling is the MAP kinase cascade. It consists of three layers, each of which consists of repeated phosphorylations of a substrate and the fully phosphorylated from of the substrate in one layer is the catalyst for the phosphorylations in the next layer. Together with Juliette Hell we proved that a single layer with two phosphorylations can lead to the existence of two stable steady states. Moreover, a layer with one phosphorylation above a layer with two can lead to the existence of periodic solutions. These results gave a mathematically rigorous confirmation of conclusions previously obtained by means of simulations and heuristic considerations. They were proved using bifurcation theory and geometric singular perturbation theory.

The Calvin cycle of photosynthesis

Photosynthesis is very important for our life. It is the ultimate source of the food we eat, the oxygen we breath and many fuels (fossil fuels and biofuels). It is important to understand this process better and mathematical models can contribute to doing so. An important part of photosynthesis is carbon fixation, in which carbon dioxide from the atmosphere is used to produce carbohydrates. At the centre of this process there is a system of chemical reactions called the Calvin cycle. There are various mathematical models for this process. Together with Juan Velazquez from the University of Bonn we have investigated the dynamics of a number of these models. One question which is important in this context and which is not yet completely answered is whether these systems can exhibit more than one stable stationary solution. If photosynthesis could work in more than one stationary state then this could be of interest for biotechnology. It might be possible to bring a plant into a state where it produces more biomass than before. We found that some of these systems have solutions where the concentrations of all substances tend to zero at late times and other where the concentrations grow without bound at late times. The presence of solutions of this type could be used as a criterion for considering certain systems as inappropriate for modelling the biological system.

Since the above was written there has been more progress on these questions. Together with Stefan Disselnkötter we were able to show that in some models of the Calvin cycle there are two positive steady states, one of which is stable and the other unstable. I also showed that in a more detailed model of the Calvin cycle due to Pettersson and Ryde-Pettersson there are at least two positive steady states and that in a related model due to Poolman there are at least three.