July 19, 2014
Last week I was at the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications in Madrid. It was very large, with more than 2700 participants and countless parallel sessions. This kind of situation necessarily generates a somewhat hectic atmosphere and I do not really like going to that type of conference. I have heard the same thing from many other paople. There is nevertheless an advantage, namely the possibility of meeting many people. To do this effectively it is necessary to proceed systematically since it is easy to go for days without seeing a particular person of interest. This aspect was of particular importance for me since I am still at a relatively early stage in the process of entering the field of mathematical biology and I have few contacts there in comparison to my old field of mathematical relativity. In any case, the conference allowed me to meet a lot of interesting people and learn a lot of interesting things.
The first plenary talk, by Charles Fefferman, was on a subject related to a topic I was interested in many years ago. I learned that a lot has happened since I last thought about this. The attempt to model a body of fluid with a free surface leads to considerable mathematical difficulties. When I started working on dynamical models for this kind of situation few people seemed to be interested in proving theorems on the subject. The source of my interest in the subject was the influence of Jürgen Ehlers, who always had a clear vision of what were the important problems. In this way I found myself in the position of a pioneer in a certain research area. Being in that situation has the advantage of not being troubled by strong competition. On the other hand it can also mean that whatever you achieve can be largely ignored and it is not the best way to get wide recognition. Often finishing mathematical research directions gets more credit than starting them. This could no doubt be compensated by suitable advertizing but that was never my strong point. This is a configuration which I have often found myself in and in fact, comparing advantages and disadvantages, I do not feel I need to change it. Coming back to the fluids with free surface, this is now a hot topic and played a prominent role at the conference. When I was working on this the issue of local existence in the case of inviscid fluids was still open. A key step was the work of Sijue Wu on water waves. I learned from the talk of Fefferman that this has been extended in the meantime to global existence for small data. The question which is now the focus of interest is formation of singularities (i.e. breakdown of classical solutions) for large data. Instead of considering the breaking of one wave the idea is to consider two waves which are approaching each other while turning over until they meet. There are already analytical results on parts on this process by Fefferman and collaborators and they plan to extend this to a more global picture by using a computer-assisted proof. Another plenary was by Ingrid Daubechies, who talked about applications of image processing to art history. I must admit that beforehand the theme did not appear very attractive to me but in fact the talk was very entertaining and I am glad I went to hear it.
I gave a talk on my recent work with Juliette Hell on the MAPK cascade in a session organized by Bernold Fiedler and Atsushi Mochizuki. I found the session very interesting and the highlight for me was Mochizuki’s talk on his work with Fiedler. The subject is how much information can be obtained about a network of chemical reactions by observing a few nodes, i.e. by observing a few concentrations. What I find particularly interesting are the direct connections to biological problems. Applied to the gene regulatory network of an ascidian (sea squirt) this theoretical approach suggests that the network known from experimental observations is incomplete and motivates searching for the missing links experimentally. Among the many other talks I heard at the conference, one which I found particularly impressive concerned the analysis of successive MRT pictures of patients with metastases in the lung. The speaker was using numerical simulations with these pictures as input to provide the surgeon with indications which of the many lesions present was likely to develop in a dangerous way and should therefore be removed. One point raised in the talk is that it is not really clear what information about the tissue is really contained in an MRT picture and that this could be an interesting mathematical problem in itself. In fact there was an encouragingly (from my point of view) large number of sessions and other individual talks at the conference on subjects related to mathematical biology.
The conference took place on the campus of the Universidad Autonoma somewhat outside the city. A bonus for me was hearing and seeing my first bee-eater for many years. It was quite far away (flying high) but it gave me real pleasure. I was grateful that the temperatures during the week were very moderate, so that I could enjoy walking through the streets of Madrid in the evening without feeling disturbed by heat or excessive sun.
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 -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 binding to receptors on the endoplasmic reticulum. The comes from the cleavage of by . This I knew before, but what comes before that? In fact 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.
May 14, 2014
In a previous post I wrote about glycolytic oscillations and mentioned a mathematical model for them, the Higgins-Selkov oscillator. Higgins introduced this as a chemical model while Selkov also wrote about some mathematical aspects of modelling this system. When I was preparing my course on dynamical systems I wanted to present an example where the existence of periodic solutions can be concluded by using the existence of a confined region in a two-dimensional system and Poincare-Bendixson theory. An example which is frequently treated in textbooks is the Brusselator and I wanted to do something different. I decided to try the Higgins-Selkov oscillator. Unfortunately I came up against difficulties since that model has unbounded solutions and it is hard to show that there are any bounded solutions except a stationary solution which can be calculated explicitly. For the purposes of the course I went over to considering the Schnakenberg model, a modification of the Higgins-Selkov oscillator where it is not hard to see that all solutions are bounded.
More recently I decided to try to finally find out what happens with the Higgins-Selkov oscillator itself. Reading Selkov’s paper I originally had the impression that he had proved the essential properties of the solutions. This turned out to be mistaken. One obstacle for me was that Selkov cited a theorem from a famous Russian textbook of Andronov et. al. and I did not know what the theorem was. An English translation of the book exists in the university library here but since Selkov only cites a page number I did not know how to find the theorem. I was able to get further when Jan Fuhrmann got hold of a copy of the page in question from the Russian original. This page has an easily identifiable picture on it and this allowed me to identify the corresponding page of the English translation and hence the theorem. I found that, as far as it is applicable to the oscillator problem this was something I could prove myself by a simple centre manifold argument. Thus I realized that the results quoted by Selkov only resolve some of the simpler issues in this problem.
At this stage I decided to follow the direction pointed out by Selkov on my own. The first stage, which can be used to obtain information about solutions which tend to infinity, is to do a Poincare compactification. This leads to a dynamical system on a compact subset of Euclidean space. In general it leads to the creation of new stationary points on the boundary which are not always hyperbolic. In this particular example two new stationary points are created. One of these has a one-dimensional centre manifold and it is relatively easy to determine its qualitative nature. This is what Selkov was quoting the result of Andronov et. al. for. The other new stationary solution is more problematic since the linearization of the system at that point is identically zero. More information can be obtained by transforming to polar coordinates about that point. This creates two new stationary points. One is hyperbolic and hence unproblematic. The linearization about the other is identically zero. Passing to polar coordinates about that point creates three new stationary points. One of them is hyperbolic while the other two have one-dimensional centre manifolds. The process comes to an end. When trying this kind of thing in the past I was haunted by the nightmare that the process would never stop. Is there a theorem which forbids that? In any case, in this example it is possible to proceed in this way and determine the qualitative behaviour near all points of the boundary. The problem is that this does not seem to help with the original issue. I see no way in which, even using all this information, it is possible to rule out that every solution except the stationary solution tends to infinity as tends to infinity.
Given that this appeared to be a dead end I decided to try an alternative strategy in order to at least prove that there are some parameter values for which there exists a stable periodic solution. It is possible to do this by showing that a generic supercritical Hopf bifurcation occurs and I went to the trouble of computing the Lyapunov coefficient needed to prove this. I am not sure how much future there is for the Higgins-Selkov oscillator since there are more modern and more complicated models for glycolysis on the market which have been studied more intensively from a mathematical point of view. More information about this can be found in a paper of Kosiuk and Szmolyan, SIAM J. Appl. Dyn. Sys. 10, 1307.
Finally I want to say something about the concept of feedback, something I find very confusing. Often it is said in the literature that oscillations are related to negative feedback. On the other hand the oscillations in glycolysis are often said to result from positive feedback. How can this be consistent? The most transparent definition of feedback I have seen is the one from a paper of Sontag which I discussed in the context of monotone systems. In that sense the feedback in the Higgins-Selkov oscillator is definitely negative. An increase in the concentration of the substrate leads to an increase in the rate of production of the product. An increase in the concentration of the product leads to an increase of the rate of consumption of the substrate. The combination of a positive and a negative sign gives a negative loop. The other way of talking about this seems to be related to the fact that an increase in the concentration of the product leads to an increase in the reaction rate between substrate and product. This is consistent with what was said before. The difference is what aspects of the system are being regarded as cause and effect, which can lead to a different assignment of the labels positive and negative. The problem as I see it is that feedback is frequently invoked but rarely defined, with the implicit suggestion that the definition should be obvious to anyone with an ounce of understanding. I seem to be lacking that ounce.
April 3, 2014
The MAP kinase cascade, which I mentioned in a previous post, is a biochemical network which has been subject to a lot of theoretical and experimental study. Although a number of results about mathematical models for this network have been proved, many widely accepted results are based on numerical and/or heuristic approaches. Together with Juliette Hell we set out to extend the coverage of rigorous results in this area. Our first results on this can be found in a paper we just posted on q-bio.
The system of equations which is fundamental for this work is that of Huang and Ferrell discussed in my previous post on the subject. I call it the MM-MA system (for Michaelis-Menten via mass action). When this system can be reduced to a smaller system by means of a quasistationary approximation the result will be called the MM system (for Michaelis-Menten) (cf. this post). With a suitable formulation the MM system is a singular limit of the MM-MA system. The MAPK cascade consists of three coupled layers. The first main result of our paper concerns the dual futile cycle, which can be thought of as the second layer of the cascade in isolation (cf. this post). We proved that the MM system for the dual futile cycle exhibits a generic cusp bifurcation and hence that for suitable values of the parameters there exist two different stable stationary solutions (bistability). Using the fact that this system is a singular limit of the system arising from the MM-MA description of the same biological system we then used geometric singular perturbation theory (cf. this post) to conclude that the MM-MA system also shows bistability.
The second main result concerns the system obtained by truncating that of Huang-Ferrell by keeping only the first two layers. It is subtle to find a useful quasistationary approximation for this system and we were put on the right track by a paper of Ventura et. al. (PLoS Comp. Biol. 4(3):e1000041). This allowed us to obtained an MM system which is a limit of the MM-MA system in a way which allows geometric singular perturbation theory to be applied. This leads to the following relative statement: if the MM system for the truncated MAPK cascade has a hyperbolic periodic solution then the same is true for the MM-MA system. To get an absolute statement it remains to prove the existence of periodic solutions of the MM system, which in this case is of dimension three. That there are solutions of this kind is indicated by numerical work of Ventura et. al.
February 11, 2014
This post arises from the fact that there seems to be some constructive interference between various directions I am pursuing at the moment. The first has to do with the course on dynamical systems I just finished giving. This course was intended not only to provide students in Mainz with an extended introduction to the subject but also to broaden my own knowledge. I wrote lecture notes for this in German and having gone to the effort of producing this resource I thought I should translate the notes into English so as to make them more widely available. The English version can be found here. Both versions are on the course web page. The second thing is that I will be organizing a seminar on bifurcation theory next semester and I want it to achieve a wide coverage, even at the risk that its waters, being broad, may be shallow (this is paraphrase of a quote I vaguely remember from Nietzsche). The connections between these two things are that I treated simple bifurcation theory and a little chaos in the course and that going further into the landscape of bifurcations necessarily means that at some point chaos rears its ugly head. The third thing is the fact that it has been suggested that the MAPK cascade, a dynamical system I am very interested in from the point of view of my own research, may exhibit chaotic behaviour, as described in a paper of Zumsande and Gross (J. Theor. Biol. 265, 481). This paper attracted my attention when it appeared on arXiv but it is only now that I understood some of the underlying ideas and, in particular, that the Hopf-Hopf bifurcation plays a central role. This in turn led me to a paper by Stiefs et. al. on chaos in ecological systems (Math. Biosci. Eng. 6, 855). They consider models with predator-prey interaction and a disease of the predators.
A Hopf-Hopf (or double Hopf) bifurcation arises at a stationary point where the linearization has two pairs of non-vanishing purely imaginary eigenvalues. Of course it is necessary to have a system of at least dimension four in order for this to occur. The subset of parameter space where it occurs has codimension two and lies at the intersection of two hypersurfaces on which there are Hopf bifurcations. For this system there is an approximate normal form. In other words the system is topologically equivalent to a system given by simple explicit formulae plus higher order error terms. The dynamics of the model system ignoring error terms can be analysed in detail. For simple bifurcations a system in approximate normal form is topologically equivalent to the model system. For the Hopf-Hopf bifurcation (and for the simpler fold-Hopf bifurcation with one zero and one pair of non-zero purely imaginary eigenvalues) this is no longer the case and the perturbation leads to more complicated dynamics. For instance, a heteroclinic orbit in the model system can break as a result of the perturbation. A lot of information on these things can be found in the book of Kuznetsov. In the paper on ecological systems mentioned above a Hopf-Hopf bifurcation is found using computer calculations and this is described as ‘clear evidence for the existence of chaotic parameter regions’. My understanding of chaos is still too weak to be able to appreciate the precise meaning of this statement.
Using computer calculations Zumsande and Gross find fold-Hopf bifurcations in the MAPK cascade (without explicit feedback) indicating the presence of complex dynamics. If chaos occurs in the ecological system and the MAPK cascade what biological meaning could this have? Ecosystems can often be thought of as spatially localized communities with their own dynamics which are coupled to each other. If the dynamics of the individual communities is of a simple oscillatory type then they may become synchronized and this could lead to global extinctions. If the local dynamics are chaotic this cannot happen so easily and even if a fluctuation which is too big leads to extinctions in one local community, these can be avoided in neighbouring communities, giving the ecosystem a greater global stability. One point of view of chaos in the MAPK cascade is that it is an undesirable effect which might interfere with the signalling function. It might be an undesirable side effect of other desirable features of the system. In reality MAPK cascades are usually embedded in various feedback loops and these might suppress the complex behaviour in the free cascade. Zumsande and Gross investigated this possibility with the conclusion that the feedback loops tend to make things worse rather than better.
November 29, 2013
I have already written two posts about the Michaelis-Menten limit, one not very long ago. I found some old results on this subject and I was on the look-out for some more modern accounts. Now it seems to me that what I need is something called geometric singular perturbation theory which goes back to a paper of Fenichel (J. Diff. Eq. 31, 53). An interesting aspect of this is that it involves using purely geometric statements to help solve analytical problems. If we take the system of two equations given in my last post on this subject, we can reformulate them by introducing a new time coordinate , called the fast time, and adding the parameter as a new variable with zero time derivative. This gives the equations , and , where the prime denotes the derivative with respect to . We are interested in the situation where the equation , which follows from the equations written in terms of the original time coordinate , is equivalent to for a smooth function . The linearization of the system in along the zero set of automatically has at least two zero eigenvalues. For Fenichel’s theorem it should be assumed that it does not have any more zero (or purely imaginary) eigenvalues. Then each point on that manifold has a two-dimensional centre manifold. Fenichel proves that there exists one manifold which is a centre manifold for all of these points. This is sometimes called a slow manifold. (Sometimes the part of it for a fixed value of is given that name.) Its intersection with the plane coincides with the zero set of . The original equations have a singular limit as tends to zero, because multiplies the time derivative in one of the equations. The remarkable thing is that the restriction of the system to the slow manifold is regular. This allows statements to be made that qualitative properties of the dynamics of solutions of the system with are inherited by the system with small but non-zero.
Due to my growing interest in this subject I invited Peter Szmolyan from Vienna,who is a leading expert in this field, to come and give a colloquium here in Mainz, which he did yesterday. One of his main themes was that in many models arising in applications the splitting into the variables and cannot be done globally. Instead it may be necessary to use several splittings to describe different parts of the dynamics of one solution. He discussed two examples in which these ideas are helpful for understanding the dynamics better and establishing the existence of relaxation oscillations. The first is a model of Goldbeter and Lefever (Biophys J. 12, 1302) for glycolysis. It is different from the model I mentioned in a previous post but is also an important part of the chapter of Goldbeter’s book which I discussed there. The model of Goldbeter and Lefever was further studied theoretically by Segel and Goldbeter (J. Math. Biol. 32, 147). On this basis a rigorous analysis of the dynamics including a proof of the existence of relaxation oscillations was given in a recent paper by Szmolyan and Ilona Kosiuk (SIAM J. Appl. Dyn. Sys. 10, 1307). The other main example in the talk was a system of equations due to Goldbeter which is a kind of minimal model for the cell cycle. It is discussed in chapter 9 of Goldbeter’s book.
I have the feeling that GSPT is a body of theory which could be very useful for my future work and so I will do my best to continue to educate myself on the subject.
September 9, 2013
Here I return to a subject which has been mentioned in this blog on several occasions, bifurcation theory. The general set-up is that we have a dynamical system of the form where denotes the unknowns and stands for parameters. A central aim of the theory is to find conditions under which the system is topologically equivalent to a simple model system. In other words it only differs from the model system by an invertible continuous change of parameters. It is natural to start in the case where and . By choosing the coordinates appropriately we can focus on the case and . The subject can be explored by going to successively more complicated cases. If and we have the case that there is no bifurcation. It follows from the implicit function theorem that for parameters close to zero there is exactly one stationary solution close to zero and it has the same character (source or sink) as the stationary solution for . The case where , and is the fold bifurcation, which was discussed in a previous post. In this case these conditions together with the condition imply topological equivalence to a standard system. The case where , , and is the cusp bifurcation. Here topological equivalence to a standard case does not hold in a context with only one parameter. It can be obtained by passing to the case and requiring that a suitable combination of derivatives with respect to and does not vanish. I was able to use this to prove the existence of more than one stable stationary solution in a model for the competition between Th1 and Th2 dominance in the immune system, cf. a previous post on this subject. The system of interest in that case is of dimension four and the fact I could obtain results using a system of dimension one resulted from exploiting symmetry properties. Later I was able to prove the existence of more (four) stable stationary solutions of that system using other methods.
It is possible to talk about fold and cusp bifurcations in higher dimensional systems. This is possible in the case that the linearization of the system at the bifurcation point (which is necessarily singular) has a zero eigenvalue with a corresponding eigenspace of dimension one and no other eigenvalues with real part zero. Then the reduction theorem tells us that near the stationary point the dynamical system is topologically equivalent to the product of a standard saddle with the restriction of the system to a one-dimensional centre manifold. This centre manifold is by definition tangent to the eigenspace of the linearization corresponding to the zero eigenvalue of the linearization at the bifurcation point. It is now rather clear what has to be done in order to analyse this type of situation. It is necessary to determine an approximation of sufficiently high order to the centre manifold and to carry out a qualitative analysis of the dynamics on the centre manifold. In practice this leads to cumbersome calculations and so it is worth thinking carefully about how they can best be organized. A method of doing both calculations together in a way which makes them as simple as possible is described in Section 8.7 of Kuznetsov’s book on bifurcation theory. A number of bifurcations, including the fold and the cusp, are treated in detail there. One way of understanding why the example from immunology I mentioned above was relatively easy to handle is that in that case the centre manifold could be written down explicitly. I did not look at the problem in that way at the time but with hindsight it seems to be an explanation why certain things could be done.
In the case of higher dimensional systems the quantities which should vanish or not in order to get a certain type of bifurcation are replaced by more complicated expressions. In the fold or the cusp is replaced by where and are left and right eigenvectors of the linearization corresponding to the eigenvalue zero. Naively one might hope that for the cusp would be replaced by but unfortunately, as explained in Kuznetsov’s book, this is not the case. There is a extra correction term which involves the second derivatives and which is somewhat inconvenient to calculate. We should be happy that a topological normal form can be obtained at all in these cases. Going more deeply into the landscape of bifurcations reveals cases where this is not possible. An example is the fold-Hopf bifurcation where there is one zero eigenvalue and one pair of non-zero imaginary eigenvalues. There it is possible to get a truncated normal form which is a standard form for the terms of the lowest orders. It is, however, in general the case that adding higher order terms to this gives topologically inequivalent systems. A simple kind of mechanism behind this is the breaking of a heteroclinic orbit. It is also possible that things can happen which are much nore complicated and not completely understood. There is an extended discussion of this in Kuznetsov’s book.