Archive for October, 2012

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.

A simple system with two different timescales

October 1, 2012

While struggling with the proof of a result for a model for photosynthesis (which I intend to report on in more detail at a later date) I decided to apply the following principle which I heard about as an undergraduate and is apparently due to Paul Halmos: if there is a mathematical problem you cannot solve then there is also a simpler problem you cannot solve. With this in mind I developed a model problem for what I wanted to do which I still could not solve. In the meantime, with the help of some key input from Juan Velázquez, I can solve the model problem and I will report on that here. It seems that the right conceptual framework for this is that of systems with two different timescales. Consider two functions x(t) and y(t) which satisfy \dot x=x and \dot y=x-y^5. The fact that the power here is exactly five is not so important. This was the power that came up in the problem I was originally interested in and I wanted to rule out any misleading simplifications which might have arisen by replacing it by the power two, for instance. The first equation can be solved explicitly in the form x(t)=\alpha e^t.This reduces the original system to the scalar, but non-autonomous, equation \dot y=\alpha e^t-y^5. The question of interest is whether this equation has solutions which tend to infinity as t\to\infty and if so how many of these are there. A guess at the asymptotics of a solution of this kind which is formally consistent is y=\alpha^{\frac15}e^{\frac15 t}+\ldots. It is possible to take this further by looking for formal power series solutions where y-\alpha^{\frac15}e^{\frac15 t} is a linear combination of integer powers of e^{\frac15 t}. It turns out that there is a solution of this type in the sense of formal power series. In other words, substituting the expression in the equations and comparing coefficients gives a consistent answer. The coefficients are determined uniquely. This means that if there is more than one solution with this type of asymptotics these coefficients cannot distinguish between them and hence cannot be used to parametrize them. In fact there is a one-parameter family of solutions having this type of asymptotics and the difference between any two of these is of order \exp [-Ce^{\frac45 t}] for a constant C. This means that while these solutions decay on a timescale t the difference between them decays on a timescale which is exponentially faster. I am reminded of the term ‘asymptotics beyond all orders’ which I have heard occasionally but I do not know exactly what that means.

How can these results be proved? First introduce a quantity w=1-\frac{y^5}{x} since this can be expected to decay very fast. The evolution equation for y can be rewritten as an evolution equation for w of the form \dot w+5\alpha^{\frac45}e^{\frac45 t}w=q, where the quantity q will eventually be small. This equation can be solved by variation of constants to give an integral equation for w. It contains a parameter \eta_0 which distinguishes the different solutions. The problem of solving the integral equation can be reformulated as a fixed point problem for a mapping \Phi. The key step is to show that, for \alpha sufficiently large and \eta_0 sufficiently small, \Phi maps a suitable set of functions with a certain decay property to itself. The fixed point can then be obtained using the Arzela-Ascoli theorem. The simple system considered here can presumably be treated by simpler methods. My motivation for discussing it here that the technique of proof is of much wider applicability and the conclusions obtained are a model for other problems.


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