SIAM Conference on the Life Sciences in Charlotte

August 7, 2014

This week I have been attending the SIAM Conference on the Life Sciences in Charlotte. Here I want to mention some highlights from my personal point of view. First I will mention some of the plenary talks. John Rinzel talked about mathematical modelling of certain perceptual phenomena. We are all familiar with the face-vase picture which switches repeatedly between two forms. I had never considered the question of trying to predict how often the picture switches. Rinzel presented models for this and for other related auditory phenomena which he demonstrated in the lecture. I find it remarkable that such apparently subjective phenomena can be brought into such close connection with precise mathematical models. Kristin Swanson talked about her work on modelling the brain cancer known as glioma and its various deadly forms. I had heard her talk on the same theme at the meeting of the Society for Mathematical Biology in Dundee in 2003. Of course there has been a lot of progress since then. This was long before I started this blog but if the blog had existed I would certainly have written about the topic. I will not try to resurrect the old stories from that distant epoch. Instead I will just say that Kristin is heavily involved in using computer simulations to optimize the treatment (surgery, radiotherapy, chemotherapy) of individual patients. One of the main points in her talk this week is that it seems to be possible to divide patients into two broad categories (with nodular or diffuse growth of the tumour) and that this alone may have important implications for therapeutic decisions. Oliver Jensen talked about a multiscale model for predicting plant growth, for instance the way in which a root manages to sense gravity and move downwards. This involves some very sophisticated continuum mechanics which the speaker illustrated by everyday examples in a very effective and sometimes humorous way. The talk was both impressive and entertaining. Norman Mazer talked about the different kinds of cholesterol (LDL, HDL etc.). According to what he said lowering LDL levels is an effective means for avoiding risks of cardiovascular illness but the alternative strategy of raising HDL levels has not been successful. He explained how mathematical modelling can throw light on this phenomenon. My understanding is that the link between high HDL level and lower cardiovascular risks is a correlation and not a sign of a causal influence of HDL level on risk factors. The last talk was by James Collins, a pioneer of synthetic biology. The talk was full of good material, both mathematical and non-mathematical. Maybe I should invest some time into learning about that field.

There was one very interesting subject which was not the subject of a talk at the conference (at least not of one I heard – it was briefly referred to in the talk of Collins mentioned above) but was a subject of conversation. It is a paper called ‘Paradoxical Results in Perturbation-Based Signaling Network Reconstruction’ by Sudhakaran Prabakaran, Jeremy Gunawardena and Eduardo Sontag which appeared in Biophys. J. 106, 2720. It suggests that the ways in which biologists deduce the influence of substances on each other on the basis of experiments are quite problematic. The mathematical content of the paper is rather elementary but its consequences for the way in which theoretical ideas are applied in biology may be considerable. The system studied in the paper is an in vitro reconstruction of part of the MAP kinase cascade and so not so far from some of my research.

Among the parallel sessions those which were most relevant for me were one entitled ‘Algebra in the Life Sciences’ and organized by Elisenda Feliu, Nicolette Meshkat and Carsten Wiuf and one called ‘Developments in the Mathematics of Biochemical Reaction Networks’ organized by Casian Pantea and Maya Mincheva. My talk was in the second of these. These sessions were very valuable for me since they allowed me to meet a considerable number of people working in areas close to my own research interests, including several whose papers were well known to me but whom I had never met. I think that this will bring me to a new level in my work in mathematical biology due to the various interactions which took place. I will not discuss the contents of individual talks here. It is rather the case that what I learned form them will flow into my research effort and hence indirectly influence future posts in this blog. I feel that this conference has gained me entrance into a (for me) new research community which could be the natural habitat for my future research. I am very happy about that. The whole conference was an enjoyable and stimulating experience for me. I noticed no jet lag at all but I must be suffering from a lack of sleep due to the fact that the many things going on here just did not leave me the eight hours of sleep per night I am used to.

 

 

Baruch Blumberg and Hepatitis B

August 6, 2014

This year, at my own suggestion, I got the book ‘Hepatitis B. The hunt for a killer virus.’ by Baruch Blumberg as a birthday present. Blumberg was the central figure in the discovery of the hepatitis B virus and was rewarded for his achievements by a Nobel prize in 1976. The principal content of the book is an account of the story leading up to the discovery. In fact the subtitle is a bit misleading since Blumberg was not hunting for a virus when he started the research which eventually led to it being found. He was interested in polymorphisms, differences in humans (and animals) which lead them to have different susceptibilities to certain diseases. Nowadays this would be done by comparing genes but at that time, before the modern developments in molecular biology, it was necessary to compare proteins. This was done by observing that antibodies in the blood of some individuals reacted with proteins in the blood of others. This is a mild version of what happens when someone gets a transfusion with an incompatible blood group.

Blumberg did a lot of work with blood coming from people living in unusual or extreme conditions. For this he travelled to exotic places such as Suriname, northern Alaska and remote parts of Nigeria. He seems to have had a great appetite for exciting travel and a corresponding dose of courage. He has plenty of adventures to relate. The second protein he found he names the ‘Australia antigen’ since it was common among aborigines. A good source of antibodies was the blood of people who had had many blood transfusions since their immune systems had been confronted with many antigens. In particular they often carried the Australia antigen.

Pursuing the nature of the Australia antigen led  to the realization that it was part of the hepatitis B virus, a virus which causes liver disease and can be spread by blood contact, in particular blood transfusions. The transfusion recipients had become infected with hepatitis B and had produced antibodies to it. Hepatitis B was the first hepatitis virus to be discovered and so why is it labelled ‘B’? In fact people had noticed cases of hepatitis after tranfusions and suspected two viruses, ‘A’ transmitted by contaminated food or water and ‘B’ transmitted by blood contact. There were researchers who had been ‘hunting’ intensively for these viruses and many of them were understandibly not happy when an outsider beat them to it.

For many years Blumberg worked at the Fox Chase Cancer Center in Philadelphia. It was generously funded and the fact that his research had little obvious relation to cancer was not a problem. Once the director of the institute warned that a serious funding cut might be coming. This led Blumberg and colleagues to the idea of developing a vaccine against hepatitis B as a way of making money. Just as Blumberg had not been a virologist when he discovered the virus he was not an expert on vaccines when he developed the vaccine. At that time the need for a vaccine did not seem so urgent since hepatitis B was known as an acute disease which was rarely life-threatening. Later the vaccine acquired a very different significance. There are very many chronic carriers (hundreds of millions worldwide) and a significant proportion of these develop liver cancer after many years. Thus, surprisingly, the hepatitis B vaccine has attained the status of an ‘anti-cancer vaccine’ and has had a huge medical impact.

This book has a very different flavour from the book of Francois Jacob I wrote about in a previous post. Blumberg gives the impression of being a highly cultured person but more than that of an adventurer and man of action. (Along the way he was Master of Balliol College Oxford and director of the NASA Astrobiology Institute.) Jacob also had enough adventures but appears to belong to a more intellectual type, concentrating more on his inner life. In his book Blumberg does not reveal too much which is really personal and always maintains a certain distance to the reader.

 

 

Breaking waves in Madrid

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.

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.

The Higgins-Selkov oscillator

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 t 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.

Proofs of dynamical properties of the MAPK cascade

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.

Lyapunov exponents

March 25, 2014

In the literature on chaos and strange attractors one concept which plays a prominent role is that of Lyapunov exponents. I came across this repeatedly but never understood the definition. I think I have now understood the reason that I had problems. The fact that I have spent so much time with differential geometry in the past years sometimes makes me see the mathematical world through ‘differential geometric spectacles’. I felt that dynamical systems were objects which naturally live on smooth manifolds and that the definition of important concepts should not be dependent on coordinates or the presence of a preferred metric. The usual definitions of Lyapunov exponents appeared to me strongly tied to Euclidean space although I had seen a couple of comments in the literature (without further justification) that the definition was coordinate independent. In what follows I will give a manifestly coordinate independent definition. Incidentally, the definition of Lyapunov exponents is not invariant under changes of the time coordinate and at one time this led to confusion among people studying chaos in cosmological models. The source of the problem was the lack of a preferred time coordinate in general relativity.

Consider a dynamical system defined by a smooth vector field on a manifold M. Let A be a compact subset of the manifold which is invariant under the flow \phi generated by the vector field. The aim here is to define the maximum Lyapunov exponent of a point x_0\in A. The derivative of the flow, J_t=D_x\phi (t,x_0) is a linear mapping from T_{x_0}M to T_{\phi (t,x_0)}M. In the Euclidean space picture J_t is treated as a matrix and this matrix is multiplied by its transpose. What is this transpose in an invariant setting? It could be taken to be the mapping from T^*_{\phi (t,x_0)}M to T^*_{x_0}M naturally associated to J_t by duality. The product of the matrices could be associated with the composition of the linear mappings but unfortunately the domains and ranges do not match. To overcome this I introduce a Riemannian metric g on a neighbourhood of A. It is then necessary to show at the end of the day that the result does not depend on the metric. The key input for this is that since A is compact the restrictions of any two metrics g_1 and g_2 to A are uniformly equivalent. In other words, there exists a positive constant C such that C^{-1}g_1(v,v)\le g_2(v,v)\le Cg_1(v,v) for all tangent vectors v at points of A. Once the metric g has been chosen it can be used to identify the tangent and cotangent spaces with each other at the points x_0 and \phi (t,x_0) and thus to compose J_t and its ‘transpose’ to get a linear mapping B(t) on the vector space T_{x_0}M. This vector space does not depend on t. The eigenvalues \lambda_i (t) of the mapping B(t) are easily shown to be positive. The maximum Lyapunov exponent is the maximum over i of the limes superior for t\to\infty of \frac{1}{t} times the logarithm of \sqrt{\lambda_i (t)}. Note that the ambiguity of a multiplicative constant in the definition of B(t) becomes an ambiguity of an additive constant in the definition of the logarithms and because of the factor \frac{1}{t} this has no effect on the end result.

In general if the maximum Lyapunov exponent at a point x_0 is positive this is regarded as a sign of instability of the solution starting at that point (sensitive dependence on initial conditions) and if the exponent is negative this is regarded as a sign of stability. Unfortunately in general these criteria are not reliable, a fact which is known as the Perron effect. This is connected with the question of reducing the study of the asymptotic behaviour of a non-autonomous linear system of ODE to that of the autonomous systems obtained by freezing the coefficients at fixed times.

The Hopf-Hopf bifurcation and chaos in ecological systems

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.

Interval arithmetic

January 28, 2014

In a recent post I wrote about the relations between mathematics and simulations. In doing so I forgot about one fascinating theme in this area, that of computer-assisted proofs. Here I am thinking about the technique known as interval arithmetic. I got interested in this subject years ago and went as far as to order a book on the subject for the institute library. However I never applied the technique myself. I was reminded of all this since I wanted to say something about the Lorenz system and strange attractors at the end of the course on dynamical systems I am giving at the moment. Then I remembered that there was a well-known result of Warwick Tucker related to the existence of the Lorenz attractor which made use of interval arithmetic. The basic idea of this technique is simple enough. Say I want to calculate the value of a function at a certain point on the computer. A conventional calculation gives an object y which is a rational number together with a certain idea of how accurate this number is. It does not, however, give any rigorous inequality for that number, due to possible errors in the calculation, in particular rounding error. Doing the same calculation by interval arithmetic gives an interval [y_1,y_2]. It constitutes a proof that the desired value of the function is contained in the interval defined by the rational numbers y_1 and y_2. All intermediate steps in the calculation are exact. This kind of approach could be used to give a proof that a certain function has a zero, in other words an existence proof. It could also be used to prove inequalities satisfied by the point where that zero is. The technique can be implemented in solvers for differential equations and thus used to prove results about dynamical systems. For me a computer-assisted proof of this type has the same philosophical satatus as a proof done by hand by a mathematician. There might be a mistake in the computer programme and the input data given to the programme might have been incorrect but this is on the same level as the mistakes a mathematician makes in a manual calculation. This type of proof has a very different status from the result of a numerical simulation done by a probably reliable but not strictly controlled programme.

Why is this technique not more popular? I can think of several reasons. The first is a lack of interest in rigorous proofs among many potential users. The second is that in practise the intervals obtained may be too large to solve the problems of interest. The third is that the calculations may be too slow. If the second or third reason is the main problem then it should be possible to improve the situation by using better algorithms and more computing power.

Monotone dynamical systems

December 29, 2013

In previous posts I have written a little about monotone dynamical systems, a class of systems which in some sense have simpler dynamical properties than general dynamical systems. Unfortunately this subject was always accompanied by some confusion in my mind. This results from the necessity of a certain type of bookkeeping which I was never really able to get straight. Now I think my understanding of the topic has improved and I want to fix this knowledge here. There are two things which have led to this improvement. One is that I read an expository article by Eduardo Sontag which discusses monotone systems in the context of biochemical networks (Systems and Synthetic Biology 1, 59). The other is that I had the chance to talk to David Angeli who patiently answered some of my elementary questions as well as providing other insights. In what follows I only discuss continuous dynamical systems. Information on the corresponding theory in the case of discrete dynamical systems can be found in the paper of Sontag.

Consider a dynamical system \dot x=f(x) defined on an open subset of R^n. The system is called monotone if \frac{\partial f_i}{\partial x_j}\ge 0 for all i\ne j. This is a rather restrictive definition – we will see alternative possibilities later – but I want to start in a simple context. There is a theorem of Müller and Kamke which says that if two solutions x and y of a monotone system satisfy x_i(0)\le y_i(0) for all i then they satisfy x_i(t)\le y_i(t) for all i and all t\ge 0. This can be equivalently expressed as the fact that for each t\ge 0 the time t flow of the dynamical system preserves the partial order defined by the condition that x_i\le y_i for all i. This can be further reexpressed as the condition that y-x belongs to the positive convex cone in R^n defined by the conditions that the values of all Cartesian coordinates are non-negative. This shows the way to more general definitions of monotone flows on vector spaces, possibly infinite dimensional. These definitions may be useful for the study of certain PDE such as reaction-diffusion equations. The starting point is the choice of a suitable cone. This direction will not be followed further here except to consider some other simple cones in R^n.

A monotone system in the sense defined above is also sometimes called cooperative. The name comes from population models where the species are beneficial to each other. Changing the sign in the defining conditions leads to the class of competitive systems. These can be transformed into cooperative systems by changing the direction of time. However for a given choice of time direction the competitive systems need not have the pleasant properties of cooperative systems. Another simple type of coordinate transformation is to reverse the signs of some of the coordinates x_i. When can this be used to transform a given system into a monotone one?. Two necessary conditions are that each partial derivative of a component of f must have a (non-strict) sign which is independent of x and that the derivatives are symmetric under interchange of their indices. What remains is a condition which can be expressed in terms of the so-called species graph. This has one node for each variable x_i and an arrow from node i to node j if \frac{\partial f_j}{\partial x_i} is not identically zero. If the derivative is positive the arrow bears a positive sign and if it is negative a negative sign. Alternatively, the arrows with positive sign have a normal arrowhead while those with negative sign have a blunt end. In this way the system gives rise to a labelled oriented graph. To each (not necessarily oriented) path in the graph we associate a sign which is the product of the signs of the individual edges composing the path. The graph is said to be consistent if signs can be associated to the vertices in such a way that the sign of an edge is always the product of the signs of its endpoints. This is equivalent to the condition that every closed loop in the graph has a positive sign. In other words, every feedback in the system is positive. Given that the other two necessary conditions are satisfied the condition of consistency characterizes those networks which can be transformed by changes of sign of the x_i to a monotone system. A transformation of this type can also be thought of as replacing the positive orthant by another orthant as the cone defining the partial order.

Next I consider some examples. Every one-dimensional system is monotone. In a two-dimensional system we can have the sign patterns (+,+), (-,-) and (+,-). In the first case the system is monotone. In the second case it is not but can be made so by reversing the sign of one of the coordinates. This is the case of a two-dimensional competitive system. In the third case the system cannot be made monotone. A three-dimensional competitive system cannot be made monotone. The species graph contains a negative loop. Higher dimensional competitive systems are no better since their graphs all contain copies of that negative loop.

A general message in Sontag’s paper is that consistent systems tend to be particularly robust to various types of disturbances. Large biochemical networks are in general not consistent in this sense but they are close to being consistent in the sense that removing a few edges from the network make them consistent. This also means that they can be thought of as a few consistent subsystems joined together. Since biological systems need robustness this suggests a topological property which biochemical networks should have compared to random networks. Sontag presents an example where this has been observed in the transcription network of yeast.

A more sophisticated method which can often be used to obtain monotone systems from systems of chemical reactions by a change of variables has been discussed in a previous post. The advantage of this is that together with other conditions it can be use to show that generic solutions, or sometimes even all solutions, of the original system converge to stationary solutions.


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