## Archive for the ‘immunology’ Category

### Itk and T cell signalling

June 18, 2014

I have spent a lot of time thinking about signalling pathways involved in the activation of T cells and ways in which mathematical modelling could help to understand them better. In the recent past I had not found much time to read about the biological background in this area. Last weekend I started doing this again. In this context I remembered that Al Singer told me that Itk was an interesting target for modelling. At that time I knew nothing about Itk and only now have I come back to that, reading a review article by Andreotti et. al. in Cold Spring Harbor Perspectives in Biology, 2010. Before I say more about that I will collect some more general remarks.

The signalling network involved in the activation of T cells is very complex but over time I have become increasingly familiar with it. I want to review now some of the typical features to be found in this and related networks. Phosphorylation and dephosphorylation play a very important role. Phosphate groups can be added to or removed from many proteins, replacing (in animals) the hydroxyl groups in the side chains of the amino acids serine, threonine and tyrosine. The enzymes which add and remove these groups are the kinases and phosphatases, respectively. Often the effect of (de-)phosphorylation is to switch the kinase or phosphatase activity of the protein on or off. This kind of process has been studied from a mathematical point of view relatively frequently, with the MAPK cascade being a popular example. Another phenomenon which is controlled by phosphorylation is the binding of one protein to another, for instance via SH2 domains. An example involved in T cell activation is the binding of ZAP-70 to the $\zeta$-chain associated to the T cell receptor. This binding means that certain proteins are brought into proximity with each other and are more likely to interact. Another type of players are linker or adaptor proteins which seem to have the main (or exclusive?) function of organising proteins spatially. One of these I was aware of is LAT (linker of activated T cells). While reading the Itk paper I came across Slp76, which did not strike me as familiar. Another element of signalling pathways is when one protein cleaves another. This is for instance a widespread mechanism in the complement system.

Now back to Itk (IL2-inducible T cell kinase). It is a kinase and belongs to a family called the Tec kinases. Another member of the family which is more prominent medically is Btk, which is important for the function of B cells. Mutations in Btk cause the immunodeficiency disease X-linked agammaglobulinemia. This is the subject of the first chapter of the fascinating book ‘Case studies in Immunology’ by Geha and Notarangelo. As the name suggests this gene is on the X chromosome and correspondingly the disease mainly affects males. In some work I did I looked at the pathway leading to the transcription factor NFAT. However I only looked at the more downstream part of the pathway. This is related to the fact that in experimental work the more upstream part is often bypassed by the use of ionomycin. This substance causes a calcium influx into the cytosol which triggers the lower part of the pathway. In the natural situation the calcium influx is caused by ${\rm IP}_3$ binding to receptors on the endoplasmic reticulum. The ${\rm IP}_3$ comes from the cleavage of ${\rm PIP}_2$ by ${\rm PLC}\gamma$. This I knew before, but what comes before that? In fact ${\rm PLC}\gamma$ is activated through phosphorylation by Itk and Itk is activated through phosphorylation by Lck, a protein I was very familar with due to some of its other effects in T cell activation.

It seems that in knockout mice which lack Itk T cell development is still possible but the immune system is seriously compromised. Effects can be seen in the differentiation of T-helper cells into the types Th1, Th2 and Th17. The problems are less in the case of Th1 responses because Itk can be replaced by another Tec kinase called Rlk. In the case of Th2 responses this does not work and the secretion of the typical Th2 cytokine IL4 is seriuously affected. The Th17 cells are in an intermediate position, with IL17A being affected but IL17F not. Itk also has important effects during the maturation of T cells. Despite the many roles of Itk there are few cases known where mutations in the corresponding genes leads to medical problems in humans. This kind of mutation is a unique opportunity to learn about the role of various substances in humans, where direct experiments are not possible.

In a 2009 paper of Huck et. al. (J. Exp. Med. 119, 1350) the case of two sisters who suffered from serious problems with immunity is described. In particular they had strong infections with Epstein-Barr virus which could not be overcome despite intensive treatment. They also has an excess of B cells. The older sister died at the age of ten. The younger sister was even more severely affected and stem cell transplantation was attempted when she was six years old. Unfortunately she did not survive that. After extensive investigations it was discovered that both sisters were homozygous for the same mutation in the gene for Itk and that was the source of their problems. Their medical history offers clues to what Itk does in humans. The gene is on chromosome 5 and thus it is natural that its mutations are much more rarely discovered than those of Btk. The mutation must occur in both copies of the gene in order to have a serious effect and this can happen just as easily in females as in males.

### Guillain-Barré syndrome

May 9, 2013

Yesterday I went to a talk by Hans-Peter Hartung about autoimmune diseases of the peripheral nervous system. To start with he gave a summary of similarities and differences between the peripheral and central nervous systems and their relations to the immune system. Of the diseases he later discussed one which played a central role was Guillain-Barré syndrome. In fact he emphasized that this ‘syndrome’ is phenomenologically defined and consists of several diseases with different underlying mechanisms. There is one form which is sporadic in its occurrence and predominant in the western world and another which can take an epidemic form and occurs in China. At a time when medical services in China were very poor this kind of epidemic had very grave consequences. Now, however, I want to return to the ‘classical’ form of Guillain-Barré.

GBS is a disease which is fascinating for the outside observer and no doubt terrifying for the person affected by it. I first learned about it in an account – I do not remember where I read it – of the case of a German doctor. He was on holiday in Tenerife when he fell ill. He recognized the characteristic pattern of symptoms, suspected GBS and got on the first plane home. He wanted to optimize the treatment he got by going to the best medical centre he knew to get treated. The treatment was successful. In GBS the immune system attacks peripheral nerves and this leads to a rapidly progressive paralysis over the course of a few days. In a significant proportion of patients this leads to the control of the muscles responsible for breathing failing and thus to death. For this reason it it is important for the patient to quickly reach a place where the disease will be recognized and they can be put on a ventilator when needed.The disease can then also be treated by plasmapheresis or immunoglobulins. In the talk it was mentioned that in the epidemics in China it was often necessary to put patients on a manual ventilator which was operated their relatives. If this acute phase can be overcome the patient usually recovers rather completely, although some people have lasting damage. It is typical that in a single patient the disease does not recur although there are a small number of cases where there are several relapses and disability accumulates.

It has been suggested that influenza infections, or influenza vaccinations, can lead to an increased risk of developing GBS. This has been an important element of controversies surrounding vaccinations, including those against H1N1. I wrote briefly about this in a previous post. In the talk the speaker mentioned a recent Canadian study indicating a slight risk of GBS due to vaccination against influenza. Nevertheless this risk was still a lot less than that due to actually becoming infected with influenza. There has also been a German study with similar results which, however, has not yet been published. There is another kind of infection which appears to carry a much higher risk, namely that with the bacterium Campylobacter jejuni. I actually mentioned this in my previous post but had completely forgotten about it. In the talk it was pointed out that this infection is quite common while GBS is very rare. So the question arises of why GBS is not more frequent. A possible explanation is that the bacterium is rather variable. The suggested mechanism is molecular mimicry (and it seems that GBS is the first case where molecular mimicry was precisely documented). In other words, certain molecules of the bacterium are similar to molecules belonging to the nervous system. Then it happens that antibodies against the bacterium cause damage to the nerves. Depending on the variant of the bacterium this similarity of the two types of molecules is more or less strong so that the effect is more or less pronounced. There is some idea in this case what exactly the molecules are which show this similarity. They are so-called gangliosides, a type of glycolipids.

This has reminded me of an issue which fascinated me before. Is there a simple explanation of why some autoimmune diseases show repeated relapses while others show a single episode (like typical GBS), a continuous progression or a combination of relapsing and progressive phases at different times? Has anyone collected data on these patterns over a variety of autoimmune diseases?

### Hello Mainz

April 18, 2013

This post is in some sense dual to the earlier one ‘goodbye to Berlin‘. To start with I can confirm that there is no shortage of Carrion Crows (and no Hooded Crows) in Mainz. When I arrived here and was waiting for my landlord to come and let me into my flat I saw some small and intensely green spots of colour in a row of trees in front of me. I knew the source of these – they were what I could see of Ring-Necked Parakeets. I have known for a long time that these birds live wild in England but it was only relatively recently, in the course of my activity looking for jobs, that I realised they were so common in parts of Germany. While in Heidelberg for an interview I observed a big number of them making a lot of noise in a small wood opposite the main railway station. I also saw some of them when I came to Mainz for the interview which eventually led to my present job. In my old institute in Golm I often used to see Red Kites out of my office window. It occurred to me that these might be replaced by Black Kites in Mainz. During my first weekend here I was walking across the campus of the university when I saw a large and unfamiliar bird of prey approaching me. When it came closer I realised that it was a Black Kite. I enjoyed the encounter. Since that I have also seen one from my office window. The Red Kite is a beautiful bird but for some reason I feel closer to its dark relative. It gives me a feeling of the south since the first place I saw these birds many years ago was in the Camargue.

Eva and I have been using Skype to maintain contact. I feel that this big change in our life has not been without benefits for our communication with each other and when I was home last weekend it was a richer experience than many weekends in the last few years. I appreciate the warm welcome I have had from my colleagues here in Mainz and my first days here, while sometimes a bit hectic, have been rewarding. Breaking the routine of years opens up new possibilities. I assured myself that I will not completely have to do without interesting biological talks here by going to a lecture by Alexander Steinkasserer on CD83. This taught me some more about dendritic cells for which this surface molecule is an important marker.

This is the first week of lectures here and yesterday brought the first concrete example of the new direction in my academic interests influencing my teaching, with the start of my seminar on ‘ordinary differential equations in biology and chemistry’. The first talk was on Lotka-Volterra equations. The subjects to be treated by other students in later lectures include ones a lot further from classical topics.

### Talk on mathematical modelling in Karlstad

November 20, 2012

Yesterday I was in Karlstad in Sweden to give a talk on the uses of mathematical modelling in the natural sciences. I was invited to do this by Claes Uggla and I was very happy to have the opportunity to present some of my ideas on this subject. The talk was structured as a series of examples involving applications of different mathematical techniques. Many of these examples have been discussed in some form in this blog during the past few years and indeed a lot of my ideas on the subject were developed in conjunction with the blog posts. The subjects were William Harvey and the circulation of the blood, multidrug therapy for HIV-AIDS, the lizard Uta stansburiana, oscillations near the big bang, Liesegang rings, modelling oscillations in vole populations using a reaction-diffusion system, signal transduction in T cells.

As well as presenting a variety of applications of different types of mathematics I also wanted to explain some mathematical connections between these subjects. One central idea is that structural stability is an issue of key importance in modelling natural phenomena. Most phenomenological models involve parameters or other elements which are not known exactly. Thus to be of interest for applications features of the dynamics of the model should be invariant under arbitrary small perturbations of the system. More precisely, if a model does not possess an invariance of this type but is nevertheless useful this requires some explanation. One possible source of an explanation is the presence of what I call ‘absolute elements’ in the model. For instance, in population dynamics if a population is zero at some time then it will definitely remain zero. This fact is independent of the details of how the population grows when it is non-zero. Similarly a spacetime singularity can define an absolute element in cosmology. When the spacetime metric breaks down this ends the dynamics in a way which is independent of the details of the dynamics of the matter away from the singularity. Thus structural stability can be weakened to the condition of invariance under small perturbations which leave certain submanifolds fixed. This can lead to the appearance of relevant heteroclinic cycles although these are not structurally stable in the absolute sense. It explains the appearance of heteroclinic cycles in the models for lizards and for the big bang in a unified way. In a similar way, restricting the perturbations of a system of chemical reactions to those which leave a particular reaction irreversible can furnish the homoclinic orbit needed to model Liesegang rings.

I have now put a slightly extended version of this talk with references on my web page. On the same day there was a talk by Bernt Wennberg on models for the collective motion of birds and fish, concentrating mainly on kinetic models related to the Boltzmann equation. At the start of his talk he showed some of the well-known pictures of flocks of Starlings over Rome. In the evening I had my own pleasant experience with a flock of birds. A large number of Jackdaws (a couple of hundred) were flying around the central square in Karlstad and calling. For some reason I have become increasingly attached to the Jackdaw over the years. At this point, and without a good excuse, I want to tell a story about Jackdaws from the book ‘King Solomon’s Ring’ by Konrad Lorenz. It is a long time since I read the book and so I hope I do not distort the story too much. At one time Lorenz was living in a small village in Austria where he was regarded by the locals as a bit crazy. One of his interests was the social life of Jackdaws. There were Jackdaws living on the roofs of the houses and he climbed up to get close to them. In order to fit in better with his black subjects he decided to dress in black. The only ‘suitable’ black clothing he could find was a devil’s costume left over from a fancy dress party. No doubt the spectacle of him climbing over the roofs dressed as the devil perfected his reputation with the local inhabitants.

### Conference on systems biology of T cells in Baeza, part 2

October 25, 2012

In the remaining one and a half days of the conference there were another fourteen talks and I will mention some aspects of their contents which attracted my attention. One recurring theme was that the encounter of a T cell receptor (TCR) with the peptide it recognizes bound to an MHC molelcule (pMHC) is often not just the encounter of one TCR with one pMHC but of multiple players. It can be shown by electron microscopy that the TCR tend to cluster on the surface of a T cell even before it has encountered antigen. This is done by attaching gold particles to the TCR so that they show up as black dots on the electron micrograph. It was shown in the talk of Hisse van Santen that a similar thing happens with the pMHC on the surface of antigen presenting cells. Judging from the discussion after the talk it seems that the explanation for this is that the pMHC, which are well known to be produced in the interior of the cell, are exported to the surface in groups. There also seems to be a widely held opinion that signalling through the T cell receptor is absolutely dependent on clustering of TCR. This makes life more complicated than it otherwise might have been. I learned at this conference that experiments on T cell signalling in vitro are often done by using tetramers, i.e. groups of four pMHC which are bound together covalently. In the talk of Wolfgang Schamel described experiments using tetramer binding. He said that this work was linked with some mathematical modelling, done by Thomas Höfer and others, but he did not want to take questions on that. My impression was that the model was an extension of the kinetic proofreading model. It has not yet been published and so I did not yet have an opportunity to look at it. Carmen Molina-París and Balbino Alarcón discussed cooperative effects in T cell receptor binding.

Michal Polonsky showed pictures of individual T cells trapped in small wells in a microfluidic device. When activated they wriggle very vigorously. These are the kind of pictures which could easily make you take a very anthropomorphic view of T cells. The aim of this work is to observe the differentiation, division and death of the cells over long periods (several days). If they were not trapped it would be extremely difficult to follow them under the microscope since they would be liable to run away. A break from the purely scientific talks was provided by a presentation of Dinah Singer about the systems biology programme at the National Cancer Institute in the US, a programme which she runs. Apart from concrete information about funding another aspect of this was the question of what might be learned about the potential for applying systems biology to immunology from existing applications of these ideas to cancer research. Dipankar Nandi talked about a phenomenon I had never heard of before and would never have expected – atrophy of the thymus as a consequence of certain diseases. Finally, I was on more familiar ground with the talk of Isabel Mérida about certain signalling pathways in T cell activation. The substance at the centre of her talk, diacylglycerol kinase, was not familiar to me but the context was. Right at the end of the conference there was a general discussion session planned. This session, which was led by Ed Palmer, ended up being very short. This was due to the (in itself positive) fact that the discussions after (and during) the individual talks had taken up more time than planned. The final discussion was interesting despite its brevity. The basic theme was: if mathematicians are collaborating with immunologists what can each side do to help the other in this process? Interesting points were brought up and we were all sent home with some things to think about.

### Conference on systems biology of T cells in Baeza

October 22, 2012

At the moment I am attending a conference on systems biology of T cells in Baeza. Of the eleven talks today the first nine made no mention of mathematics – there was not a single equation. The tenth, by Zvi Grossmann, did show a couple. Thus the bias today was very much towards experimental immunology. It was interesting for me to be immersed in this atmosphere and I learned a lot of things. There are three things which stick in my mind particularly. The first is the fact, mentioned in the talk of Bruno Kyewski, that antigens mimicking all tissues of the body are presented by medullary epithelial cells in the thymus. This allows future T cells to learn about all self antigens. I asked him afterwards if this includes tissues which are in the immunologically privileged sites, usually poorly accessible to the immune system, like the central nervous system. He confirmed that this is the case. The second is the fact, which came up in the talk of Marisa Torio, that T cell precursors in the thymus have the potential to develop into almost any type of white blood cell. This means that the fate of a cell to become a T cell is in general not decided before it reaches the thymus, the answer to a question I had often asked myself. The third is the description in the talk of Alfred Singer of the way in which it is decided which of the surface molecules CD4 or CD8 a T cell carries. I had already watched a video by Singer on this subject on the NIH web page but one thing I was not aware of was the fact that by binding the protein Lck it is possible for CD4 and CD8 to interfere with T cell signalling. Lck is sequestered and hence is not available for use by the T cell receptor.

Grossmann’s talk was mainly concerned with rather abstract ideas about cell signalling and it was hard for me to get to grips with them. I had the impression that the right mathematical context for these things should be control theory. The last and only really mathematical talk of the day, by Rob de Boer, was a highlight for me and not only for me. At dinner the air was buzzing with conversations on the subject. The talk was on monitoring the dynamics of immune cells by labelling with deuterium and drawing conclusions about their lifetimes. I had heard a talk on a similar subject by de Boer before at a conference in Dresden and I wrote about it briefly in a previous post. I liked that earlier talk but I liked the talk today much more. This was probably less due to the difference in content as to the fact that for whatever reason I now appreciated the significance of this work much better. This is an example where a mathematical model can be used to obtain information about processes in immunology which it is difficult or impossible to obtain in any other way. It is not that the mathematics is complicated, just some explicitly solvable linear ODE. The impressive thing is the direct contact this work makes with real biological questions like ‘how long does a memory T cell live’. Analysing different experiments both using deuterium in human subjects and other more poisonous substances which can only be used in mice originally gave inconsistent answers for lifetimes. With hindsight this arose from the assumption in the models of just one population of cells with a definite death rate. Passing to a model with two classes of cells largely removed the discrepancy. There was another interesting aspect of this lecture and its reception which explains its prevalence at dinner. It has to do with communication between different fields, in this case mathematics and biology. There was a lot of confusion among the audience which was due not to the factual content of the work but to the way the results were described and to the choice of language in describing the results. I should remember for the future that it is not enough to get an interesting result in mathematical biology. It is also necessary to be very careful about formulating it in the right way so as to make its meaning transparent for biologists.

### The NFAT signalling pathway

January 6, 2012

The role of T cells in the immune system is to recognize foreign substances and then take appropriate action. In order for this to happen information must be propagated from the surface of the cell, where the T cell receptor is, to the nucleus in order to initiate DNA transcription. The last step in this process is the binding of a suitable combination of transcription factors to the DNA. NFAT (nuclear factor of activated T cells) is one of these transcription factors. The fact that the associated signalling pathway plays an important role in the activation of T cells explains the name. In fact this substance (or class of substances – there are actually five different ones) are important for signalling in many cells of the immune system. I already mentioned the NFAT signalling pathway, its connection to calcium and a paper on the subject by Salazar and Höfer in a previous post. Now I have written a paper where I look into mathematical aspects of the activation of NFAT by means of dephosphorylation and the role of calcium in this process. Salazar and Höfer introduced a high-dimensional dynamical system and computed stationary solutions in a slightly simplified version of that system. I now proved, using chemical reaction network theory, that for each choice of the many parameters in the system there exists exactly one stationary solution of the full system for each value of the total amount of NFAT in the cell. Every solution with that total amount of NFAT converges to the stationary solution at late times. Furthermore, this solution is well approximated by the explicit solution of the simplified system under a biologically motivated assumption that certain parameters are small enough. The main tool in the proof is the Deficiency Zero Theorem.

The result just mentioned concerns the model for the dephosphorylation process with the stimulation of the cell expressed through fixed choices of the parameters. In reality the stimulation is communicated through the calcium concentration in the cytosol. This means that the parameters in the model for desphosphorylation should be replaced by time-dependent functions which themselves are the result of a dynamical process. The situation is described by Salazar and Höfer with the help of a two-dimensional dynamical system closely related to one introduced by Somogyi and Stucki to describe calcium oscillations in liver cells. In the paper I did some analysis of the model, giving criteria for the stability of the unique stationary solution for given parameter values and the existence of periodic solutions. Hopf bifurcations play a role. The model is closely related to the Brusselator and techniques of proof can be imported from that case. In particular it is important to identify explicit invariant regions for the flow. When a solution of the model for the calcium concentration is such that it tends to a constant at late times then it can be shown that the resulting configuration of the phosphorylation states of NFAT also converges to the situation with constant coefficients previously analysed. When a solution converges to a periodic solution at late times it is not clear what can be said.

### IPEX and CD25

December 20, 2011

In a recent post I wrote about some ideas of Kendall Smith and his role in discovering the cytokine IL-2. On 8th December I heard him give a talk in which he presented various ideas about IL-2, its receptor and Tregs. I discussed some aspects of IL-2 in the last post. When mutant mice are engineered which cannot produce IL-2 they show a strange combination of symptoms which combine immunodeficiency (a reduced capability of the immune system to fight pathogens) and autoimmune disease (an inappropriate reaction of the immune system to host tissues). This probably has to do with the fact that IL-2 is important for the production of both effector T cells and Tregs, which act in opposite directions. Similar phenomena are seen in the disease of humans called IPEX (immune dysregulation, polyendocrinopathy, enteropathy, X-linked syndrome). It is often attributed to a lack of the transcription factor Foxp3 which is of central importance for the function of Tregs. The gene for Foxp3 is on the X chromosome and this explains the way IPEX is inherited and the term X-linked in its name. However, as pointed out by Smith in his talk, one third of patients diagnosed with IPEX have no mutation in the Foxp3 gene. In this context he referred to a paper of Caudy et al. (J. Allergy Clin. Immunol. 119, 482). What is shown in this paper is that there is a different possible cause of IPEX-like symptoms, namely mutations in the gene for CD25, a surface molecule associated to Tregs.

The paper concerns a patient (an eight year old boy) who had suffered a horrific combination of diseases. It was found that he had mutations in both copies of the CD25 gene. The mutation in one copy came from the mother and was a frame shift due to an insertion. In other words, there is a extra base in the DNA which makes the part of the gene after it look like nonsense when it is being transcribed. The mutation in the other copy came from the father and consisted of one base being exchanged. This happed to cause a stop codon so that reading stopped at that point. The combination of these circumstances meant that the boy could not produce CD25 and this was the presumed cause of his disease. His Foxp3 gene was normal. On the other hand other IPEX patients can produce CD25. Thus there appear to be two diseases with related symptoms. The gene coding for CD25 is on chromosome 10, not the X chromosome. This is why two mutations are necessary to produce CD25 deficiency.

What is the connection to IL-2? The IL-2 receptor, which was also discovered by Kendall Smith and his collaborators, consists of three chains called $\alpha$, $\beta$ and $\gamma$. The second and third are always present on the surface of T cells but the first is only present in variable amounts. In fact the $\alpha$ chain of the IL-2 receptor is nothing other than CD25. The $\beta$ and $\gamma$ chains together allow for some IL-2 signalling but strong signalling in response to normal concentrations of IL-2 is only possible with the help of the $\alpha$ chain. In this case it is not only the case that the receptor signals when IL-2 is bound to it. Binding also causes the receptor to be taken into the interior of the cell and destroyed. This process is an important part of the dynamics associated to IL-2. The $\gamma$ chain of the IL-2 receptor also forms part of the receptor for many other cytokines, for instance IL-4. The gene for this receptor is on the X chromosome. When it cannot be produced due to a mutation this leads to a disease called X-linked severe combined immunodeficiency (SCID). In this case the immune system does not function since so much of its signalling system has been disrupted. This is also known as the ‘bubble boy disease’ since children affected by it have to live in a sterile environment.

### Shifting attention towards Tregs

December 3, 2011

Yesterday I heard a talk by Abul Abbas where two of the main themes were regulatory T cells (Tregs) and interleukin 2. Correctly functioning immunity is the result of a balance between effector cells and Tregs and he emphasized that in trying to develop therapies it might be more valuable to concentrate on influencing the regulatory side. He described a mouse model which he has developed for studying autoimmune disease. One criterion in developing this model was that it should concern the target tissue where an antigen is expressed and not the lymphoid tissue. Another is that the target tissue should be easily accessible for doing experiments in vivo. For this reason he chose the skin. In this transgenic model antigen expression can be turned on and off by feeding the mice with doxycyclin. When the antigen is turned on an autoimmune disease results. If it is turned off the mice recover. If it is turned on again the mice get sick again but much less than the first time. This is reminiscent of ordinary immunity which is due to memory effector cells. In this case it seems that there are memory Tregs. This suggests the idea that a possible cause of autoimmune disease in humans could be a lack of memory Tregs.

When IL-2 was first discovered it was known for causing T cells to proliferate and thus strengthening the immune response. More recently it has been found that eliminating IL-2 does not necessarily act in an immunosuppressive way. Apparently it can be replaced by something else in driving the proliferation of effector T cells. On the other hand it also drives the proliferation of Tregs and Abbas argued that this is its most essential function. In that case it cannot be replaced.

The lecturer made a number of interesting comments about themes such as immunology, therapies for immune disorders and cancer, clinical trials etc. I did not note them down and I cannot reproduce them here. Nevertheless I have the impression that a learned of lot of things which I might profit from in the future.

### Calcium oscillations

November 19, 2011

There is evidence to suggest that oscillations in levels of calcium inside and outside cells are used as a signalling mechanism. A variety of mathematical models have been introduced to study this phenomenon. Here I will discuss some aspects of the subject. A more general review can be found in this Scholarpedia article. In the plasma membrane and the endoplasmic reticulum there are pumps which transport calcium ions out of the cytosol. The result is a huge concentration difference between the cytosol on the one hand and the extracellular space and the lumen of the endoplasmic reticulum on the other hand. This can be several orders of magnitude. There are also ion channels in these membranes which, when open, allow the calcium to flow down its gradient. This provides a way to change the calcium concentration in the cytosol very fast and this can cause rapid changes in the behaviour of a cell. In this context it is important that the endoplasmic reticulum has such a high surface area and is so widely distributed in the cell. One type of calcium channels in the ER reacts to the binding of the substance IP${}_3$ (inositol 1,4,5-trisphosphate) to the channel by opening. This effect is also modulated by the calcium concentration in the cytosol. There are calcium channels in the plasma membrane and there is also a certain amount of leakage through both membranes. Transport of calcium in and out of mitochondria can be an important effect. Some combination of these features can lead to oscillations in the calcium concentration in the cytosol. This presents a challenge for mathematical modelling. Ideally a dynamical system consisting of ODEs for the concentrations of various substances would exhibit periodic solutions. Of course a system of this kind must have dimension at least two and several two-dimensional models have been proposed. It could be that several of these models are useful since calcium signalling in different cell types may use different mechanisms. The difficult thing is not to find a model exhibiting oscillations but to find the right model for a particular type of cell. In what follows I consider one type of model. I have chosen this type for two reasons. The first is its simplicity. The second is that it may be relevant to explaining the role of calcium in the activation of T cells.

I consider first a model due to Somogyi and Stucki (J. Biol. Chem. 266, 11068). It is a two-dimensional dynamical system. The two variables are the calcium concentrations in the lumen of the ER and the cytosol, call them $x$ and $y$. The concentration of IP${}_3$ is taken to be constant. The rates of change of $x$ and $y$ are given by $k'y-kx-\alpha f(y)x$ and $kx-k'y+\alpha f(y)x+\gamma-\beta y$. The quantities $k,k',\alpha,\beta,\gamma$ are positive constants while $f$ is a positive function which describes the behaviour of the IP${}_3$ receptor and must be further specified to get a definite model. The inventors of the model remark that setting $k=0$ and $f(y)=\frac{y^2}{a^2+y^2}$ causes this system to reduce to the famous Brusselator, which I have commented on elsewhere. Thus the model can be thought of as a kind of generalized Brusselator and indeed it exhibits similar qualitative behaviour. The choices which are suggested to be appropriate for the cells being studied (in this case hepatocytes) is that $k>0$ and $f$ is given by a Hill function, $f(y)=\frac{y^n}{a^n+y^n}$. Nice features of this system is that it has a unique stationary solution which can be written down explicitly and that it is also possible to get an explicit formula for the characteristic equation of the linearization at that stationary solution. In this way the stability of the stationary solution can be determined, with instability corresponding to the existence of a limit cycle. It is stated that a Hopf bifurcation occurs but there is no discussion of proving this. The general picture seems to be that oscillatory behaviour occurs at intermediate levels of IP${}_3$ stimulation and disappears at levels which are too low or too high. In this paper an alternative version of the model is introduced where in some places $x$ is replaced by $x-y$. This happens when modelling effects driven by the difference of concentrations in the two compartments. Given that $x$ is normally much larger than $y$ it is plausible to replace the difference of concentrations by the concentration in the ER.

The dephosphorylation of the transcription factor NFAT during the activation of T cells has been studied in a paper of Salazar and Höfer (J. Mol. Biol. 327, 31). An important step in the activation process is an influx of calcium caused by release of IP${}_3$. The calcium binds to calmodulin. It also binds to the phosphatase calcineurin which can then be activated by calmodulin. Finally calcineurin removes phosphate groups from NFAT. In this paper a model for calcium dynamics is used which is closely related to the (alternative model) of Somogyi and Stucki. There are three equations but two of them form a closed system which is more or less the Somogyi-Stucki model with a specific choice of receptor activity as a function of the concentration of IP${}_3$. The last equation essentially means that the calcium level is integrated in time to give the concentration of active calcineurin.