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.
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.
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.
August 27, 2012
The multiple futile cycle is a simple type of network of chemical reactions which is often found in biological systems. In a previous post I mentioned it as a component of a slightly more complicated network found in many cells, the MAP kinase cascade. One concrete realization of the multiple futile cycle is a protein which can be phosphorylated at up to sites. All the phosphorylation steps are carried out by one kinase while all dephosphorylation steps are carried out by one phosphatase. Each step is modelled in the Michaelis-Menten way, including an enzyme-substrate complex as one of the species and using mass action kinetics. There results a system of ordinary differential equations with three conservation laws. These represent the conservation of the total amount of the two enzymes and of the substrate protein. In the case , which might be called the simple futile cycle, using the conservation laws to eliminate some of the variables leads to a three-dimensional dynamical system. A basic question is what can be said about the dynamics of solutions of this system.
It has been shown by Angeli and Sontag (Nonlinear Analysis RWA, 9, 128) that in the case every solution converges to a stationary solution and that this stationary solution is unique for given values of the conserved quantities. The proof uses the theory of monotone dynamical systems. The original dynamical system is not monotone and so the first step in their proof is to replace it by another system which is monotone and show that convergence properties of solutions of the second imply convergence properties of solutions of the first. The second step is to prove the convergence of solutions of monotone systems under the additional condition of the existence of a translational symmetry. The paper mentions that this result is dual to a previously known result due to Mierczyński about monotone systems with a conserved quantity. Up to now I thought that the only benefit of knowing that a dynamical system is monotone is the possibility of reducing it to an effective system of one dimension less. This is only interesting if the initial system is of dimension no more than three. What this work has shown me is that knowing that a system is monotone can sometimes be the key to concluding much more. One aspect of the paper of Angeli and Sontag which was a source of confusion for me was a difference in conventions to what I am familiar with from chemical reaction network theory. This seems to be essential for the monotonicity argument and not just a matter of taste. The stoichiometric matrix (or stoichiometry matrix) is defined differently because a reversible reaction is treated as a single reaction rather than as a pair.. I feel a spontaneous preference for the CRNT convention but here it seems that a different one can be a real advantage. In the case of the simple futile cycle an important effect is that the dimension of the kernel of the stoichiometric matrix is three with the CRNT convention and one with the Angeli-Sontag convention.
In another paper (J. Math. Biol. 61, 581) Angeli, De Leenheer and Sontag present a more general theory related to this. Here the hypotheses needed to obtain a suitable monotone system involve the properties of certain graphs constructed from the reaction network. In this theory the notion of persistence of the dynamical system plays an important role. This is the property that a positive solution can never have any limit points on the boundary of the positive region. The case (dual futile cycle) has been considered in a paper of Wang and Sontag (J. Nonlin. Sci. 18, 527). There they are able to show that for certain ranges of the parameters generic solutions converge to stationary solutions. To emphasize the power of the techniques developed in these papers it should be pointed out that they can be applied to systems with arbitarily large numbers of unkowns and parameters and that when they apply they give strong conclusions.
D. Angeli and E. D. Sontag (2008). Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles Nonlinear Analysis: Real World Applications, 9 DOI: 10.1016/j.nonrwa.2006.09.006
D. Angeli, P. De Leenheer and E. D. Sontag (2010). Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. Journal of mathematical biology, 61 (4) PMID: 19949950
August 25, 2012
I have a tendency to use the minimal amount of technology I have to in order to achieve a particular goal. So for instance, having been posting things on this blog for several years, I have made use of hardly any of the technical possibilities available. Among other things I did not assign my posts to categories, just putting them in one long list. I can well understand that not everyone who wants to read about immunology wants to read about general relativity and vice versa. Hence it is useful to have a sorting mechanism which can help to direct people to what they are interested in. Now I have invested the effort to add information on categories to most of the posts. It was easy (though time-consuming) to do and I find that the results are useful. It is helpful for me myself to navigate through the material and it is interesting for me to see at a glance how many posts on which subjects there are. For now on I will systematically assign (most) new posts to a category and the effort to do so should be negligible. This post is an exception since it does not really fit into any category I have.
August 2, 2012
This week I am at a conference on mathematical relativity in Oberwolfach. Today Jared Speck gave a talk on work of his with Igor Rodnianski which answers a question I have been interested in for a long time. The background is that it is expected that the presence of a stiff fluid (a fluid where the speed of sound is equal to the speed of light) leads to a simplification of the dynamics of the solutions of the Einstein equations near spacetime singularities. This leads to the hope that the qualitative properties of homogeneous and isotropic solutions of the Einstein equations coupled to a stiff fluid near their big bang singularities could be stable under small changes of the initial data. This should be independent of making any symmetry assumptions. As discussed in a previous post it has been proved by Fuchsian techniques that there is a large class of solutions consistent with this expectation. The size of the class is judged by the crude method of function counting. In other words there are solutions depending on as many free functions as the general solution. This is a weak indication of the stability of the singularity but it was clear that a much better statement would be that there is an open neighbourhood of the special data in the space of all initial data which has the conjectured singularity structure. This is what has now been proved by Rodnianski and Speck.
This result fits into a more general conceptual framework. Suppose we have an explicit solution of an evolution equation and we would like to investigate the stability of its behaviour in a certain limit . If we expect that solutions with data close to the data for have the same qualitative behaviour then we may try to prove this directly. Call this the forward method. If there is no evidence that this idea is false but it seems difficult to prove it then we can try another method as an interim approach to gain some insight. This is to attempt to construct solutions with the expected type of asymptotics which are as general as possible. I call this the backward method, since it means evolving away from the asymptotic regime of interest. The forward method is preferable to the backward if it can be done. In the case of singularities in Gowdy spacetimes Satyanad Kichenassamy and I applied the backward method and Hans Ringström later used the forward method. It is perhaps worth pointing out that while the forward method is more satisfactory than the backward one both together can sometimes be used to give a better total result than the forward method alone. There are also examples of this in the context of expanding cosmological methods with positive cosmological constant. I applied the backward method while Ringström, Rodnianski and Speck later used the forward method. The result for the stiff fluid with which I started this post also fits into this framework using the forward method. The corresponding result for the backward method was done by Lars Andersson and myself more than ten years ago.
There were two other talks at this conference which can be looked at from the point of view just introduced. One was a talk by Gustav Holzegel on his work with Dafermos and Rodnianski on the existence of asymptotically Schwarzschild solutions. The second was my talk on an apsect of Bianchi models I have discussed in a previous post. Both of these used the backward method.
July 27, 2012
The music did seem to have a positive effect on the synchronization of lectures. Unfortunately it was not always there – for instance it was not there before my talk – and it seems to have been getting less and less. One good thing is that the name tags, as well as showing the usual information have the first name (or nickname) printed in large letters at the top. I find that this can be very useful for recognizing people after only having met them fleetingly.
The plenary talk of Claire Tomlin yesterday was about the HER2 receptor which plays an important role in breast cancer. It is connected to transcription factors in the nucleus by a signalling network containing two main pathways. One of these includes the MAP kinase cascade while another passes through the substance Akt. Excessive activity of this type of signalling can be reduced by a drug called lapatinib, which is a tyrosine kinase inhibitor. There is, however, a problem that this beneficial effect can be neutralized after some time. The speaker described ideas for overcoming this effect based on a study of the signalling network. A result of this analysis is that, counterintuitively, combining the administration of lapatinib with another treatment which increases the concentration of Akt at a different time could lead to a more effective therapy. I did not get the details but this seems like a case where mathematical modelling could actually contribute effectively to cancer treatment by suggesting new strategies. Relations were mentioned to the pattern of hairs on the wings of Drosophila. In her research on biomedical themes she benefits from her background in control engineering and aerodynamics.
The talk of Becca Asquith which I mentioned in the last post was cancelled. Instead there was a lecture by Sandy Anderson who seems to like to cultivate the image of the hard-drinking Scotsman. He started his career in mathematical modelling and then moved a long way towards medical research, now heading a lab at the Moffit Cancer Center in Florida. The subject of his talk was the role of heterogeneity in cancer. He started by giving a view of the importance of cancer (in terms of the number of people it kills) and the trends in the numbers for the different forms. They have mostly been decreasing for many years with the notable exception of lung cancer (for well-known reasons) but the rate of decrease is not very large despite the huge amount of effort, and money, put into cancer research. He said that death in cancer usually does not result from a tumour which stays in its original site but as a result of metastasis. Thus that is the key phenomenon to be understood. This requires an understanding of many different scales and for the talk he concentrated on the cellular scale. He claimed that an important fact that cancer researchers had not taken into account sufficiently until very recently is how heterogeneous tumours are. There is a large variation in the phenotype of the individual cancer cells and the phenotypes are evolving. This evolution is strongly influenced by the environment of the tumour, for instance the structure of the surrounding extracellular matrix. Experiments done on cell cultures may give misleading results since the ‘happy’ cells in the Petri dish with all modern comforts are not under the same pressure as corresponding cells in the body. The more the external pressures are the more the dangerous cells which are going to metastasize dominate over the others. In some cases treatment can accelerate the growth of a tumour. This danger exists if the treatment is given too late. These ideas have arisen by the use of mathematical modelling. These are ‘hybrid models’ which combine discrete and continuous dynamical systems and this is a terms which I have met in several other talks at this conference. One of the conclusions of this research is that it may be a good idea to control cancer cells rather to destroy them. For the attempt to destroy cells may destroy the relatively harmless ones and unleash the dangerous one on their surroundings. Anderson’s talk conveyed the excitement of the application of mathematical modelling in cancer research at this moment and I wonder if some of the young people in the audience might have been recruited.
This afternoon I went to a session on wound healing. There was an introductory lecture by Rebecca Segal and this was helpful for me since I knew very little about the subject. Two of the things I found interesting – I was already primed for this by talking to Angela Reynolds at her poster yesterday – is that immunology (dynamics of neutrophils and macrophages) plays a big role and that ODE models can be useful. Useful means that they can help doctors make decisions how to treat wounds they are confronted with in practise.