Archive for February, 2017

Kestrels and Dirichlet boundary conditions

February 28, 2017

The story I tell in this post is based on what I heard a long time ago in a talk by Jonathan Sherratt. References to the original work by Sherratt and his collaborators are Proc. R. Soc. Lond. B269, 327 (more biological) and SIAM J. Appl. Math. 63, 1520 (more mathematical). There are some things I say in the following which I did not find in these sources and so they are based on my memories of that talk and on things which I wrote down for my own reference at intermediate times. If this has introduced errors they will only concern details and not the basic story. The subject is a topic in population biology and how it relates to certain properties of reaction-diffusion equations.

In the north of England there is an area called the Kielder Forest with a lake in the middle and the region around the lake is inhabited by a population of the field vole Microtus\ agrestis. It is well known that populations of voles undergo large fluctuations in time. What is less known is what the spatial dependence is like. There are two alternative scenarios. In the first the population density of voles oscillates in a way which is uniform in space. In the second it is a travelling wave of the form U(x-ct). In that case the population at a fixed point of space oscillates in time but the phase of the oscillations is different at different spatial points. In general there is relatively little observational data on this type of thing. The voles in the Kielder forest are an exception to this since in that case a dedicated observer collected data which provides information on both the temporal and spatial variation of the population density. This data is the basis for the modelling which I will now describe.

The main predators of the voles are weasels Mustela\ nivalis. It is possible to set up a model where the unknowns are the populations of voles and weasels. Their interaction is modelled in a simple way common in predator-prey models. Their spatial motion is described by a diffusion term. In this way a system of reaction-diffusion equations is obtained. These are parabolic equations and to the time evolution is non-local in space. The unknowns are defined on a region with boundary which is the complement of a lake. Because of this we need not only initial values to determine a solution but also boundary conditions. How should they be chosen? In the area around the lake there live certain birds of prey, kestrels. They hunt voles from the air. In most of the area being considered there is very thick vegetation and the voles can easily hide from the kestrels. Thus the direct influence of the kestrels on the vole population is negligible and the kestrels to not need to be included in the reaction-diffusion system. They do, however, have a striking indirect effect. On the edge of the lake there is a narrow strip with little vegetation and any vole which ventures into that area is in great danger of being caught by a kestrel. This means that the kestrels essentially enforce the vanishing of the population density of voles at the edge of the lake. In other words they impose a homogeneous Dirichlet boundary condition on one of the unknowns at the boundary. Note that this is incompatible with spatially uniform oscillations. On the boundary oscillations are ruled out by the Dirichlet condition. When the PDE are solved numerically what is seen that the shore of the lake generates a train of travelling waves which propagate away from it. This can also be understood theoretically, as explained in the papers quoted above.

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Conference on cancer immunotherapy at EMBL

February 5, 2017

I just came back from a conference on cancer immunotherapy at EMBL in Heidelberg. It was very interesting for me to get an inside view of what is happening in this field and to learn what some of the hot topics are. One of the speakers was Patrick Baeuerle, who talked about a molecular construct which he introduced, called BiTE (bispecific T cell engager). It is the basis of a drug called blinatumomab. This is an antibody construct which binds both CD3 (characteristic of T cells) and CD19 (characteristic of B cells) so that these cells are brought into proximity. In its therapeutic use in treating acute lymphoblastic leukemia, the B cell is a cancer cell. More generally similar constructs could be made so as to bring T cells into proximity with other cancer cells. The idea is that the T cell should kill the cancer cell and in that context it is natural to think of cytotoxic T cells. It was not clear to me how the T cell is activated since the T cell receptor is not engaged. I took the opportunity to ask Baeuerle about this during a coffee break and he told me that proximity alone is enough to activate T cells. This can work not only for CD8 T cells but also for CD4 cells and even regulatory T cells. He presented a picture of a T cell always being ready to produce toxic substances and just needing a signal to actually do it. Under normal circumstances T cells search the surfaces of other cells for antigens and do not linger long close to any one cell unless they find their antigen. If they do stay longer near another cell for some reason then this can be interpreted as a danger sign and the T cell reacts. Baeuerle, who started his career as a biochemist, was CEO of a company called Micromet whose key product was what became blinatumomab. The company was bought by Amgen for more than a billion dollars and Baeuerle went with it to Amgen. When it came on the market it was the most expensive cancer drug ever up to that time. Later Baeuerle moved to a venture capital firm called MPM Capital, which is where he is now. In his previous life as a biochemical researcher Baeuerle did fundamental work on NF\kappaB  with David Baltimore.

In a previous post I mentioned a video by Ira Mellman. At the conference I had the opportunity to hear him live. One thing which became clear to me at the conference is the extent to which, among the checkpoint inhibitor drugs, anti-PD1 is superior to anti-CTLA. It is successful in a much higher proportion of patients. I never thought much about PD1 before. It is a receptor which is present on the surface of T cells after they have been activated and it can be stimulated by the ligand PD1L leading to the T cell being switched off. But how does this switching off process work? The T cell is normally switched on by the engagement of the T cell receptor and a second signal from CD28. In his talk Mellman explained that the switching off due to PD1 is not due to signalling from the T cell receptor being stopped. Instead what happens is that PD1 activates the phosphatase SHP2 which dephosphorylates and thus deactivates CD28. Even a very short deactivation of CD28 is enough to turn off the T cell. In thinking about mathematical models for T cell activation I thought that there might be a link to checkpoint inhibitors. Now it looks like models for T cell activation are not of direct relevance there and that instead it would be necessary to model CD28.

I learned some more things about viruses and cancer. One is that the Epstein-Barr-virus, famous for causing Burkitt’s lymphoma also causes other types of cancers, in particular other types of lymphoma. Another is that viruses are being used in a therapeutic way. I had heard of oncolytic viruses before but I had never really paid attention. In one talk the speaker showed a picture of a young African man who had been cured of Burkitt’s lymphoma by … getting measles. This gave rise to the idea that viruses can sometimes preferentially kill cancer cells and that they can perhaps be engineered to as to do so more often. In particular measles is a candidate. In that case there is an established safe vaccination and the idea is to vaccinate with genetically modified measles virus to fight certain types of cancer.

In going to this conference my main aim was to improve my background in aspects of biology and medicine which could be of indirect use for my mathematical work. In fact, to my surprise, I met one of the authors of a paper on T cell activation which is closely related to mathematical topics I am interested in. This was Philipp Kruger who is in the group of Omer Dushek in Oxford. I talked to him about the question of what is really the mechanism by which signals actually cross the membrane of T cells. One possibility he mentioned was a conformational change in CD3. Another, which I had already come across is that it could have to do with a mechanical effect by which the binding of a certain molecule could bring the cell membranes of two interacting cells together and expel large phosphatases like CD45 from a certain region. In the paper of his I had looked at signalling in T cells is studied with the help of CAR T-cells, which have an artifical analogue of the T cell receptor which may have a much higher affinity than the natural receptor. In his poster he described a new project looking at the effect of using different co-receptors in CAR T-cells (not just CD28). In any case CAR T-cells was a subject which frequently came up at the conference. Something which was in the air was that this therapy may be associated with neurotoxicity in some cases but I did not learn any details.

As far as I can see, the biggest issue with all these techniques is the following. They can be dramatically successful, taking patients from the point of death to long-term survival. On the other hand they only work in a subset of patients (say, 40% at most) and nobody understands what success depends on. I see a great need for a better theoretical understanding. I can understand that when someone has what looks like a good idea in this area they quickly look for a drug company to do a clinical trial with it. These things can save lives.  On the other hand it is important to ask whether investing more time in obtaining a better understanding of underlying mechanisms might not lead to better results in the long run.