The Goodwin oscillator

During a talk by Jae Kyoung Kim at last week’s SMB meeting the speaker showed a system of equations and called it ‘a system you all know’. This revealed to me a gap in my knowledge of mathematical biology. The system is the Goodwin oscillator. It is described in Murray’s book on mathematical biology and I am sure I have read the relevant section on some level. This just shows that there a big difference between reading something and understanding its significance and being able to situate in a wider context. Now I have done my homework on this and I will write something about it here. The system, in the form it is given in Murray’s book, is an example of systems of the form . Here the labels on the are supposed to be interpreted modulo . In other words there are equations and is interpreted as . In the Goodwin model itself and the functions are linear for . The function is equal to for constants , and . This function is positive and its derivative is negative. Thus it can be interpreted as representing a negative feedback on the production of . In the context in which it was introduced by Goodwin the quantities , and represent concentrations of mRNA, the enzyme it codes for and the product of a reaction it catalyzes. The substrate of the enzyme is assumed present at a constant level and is not modelled explicitly.

It is known that the system admits a periodic solution if the Hill coefficient is greater than eight and not otherwise. Since this number is considered unrealistically large for the application which inspired the model modifications of this have been considered where periodic solutions can be obtained for lower values of . It is proved by Hastings, Tyson and Webster (J. Diff. Eq. 25, 39) that for the Goodwin system and a larger class of similar models the following is true. The system has a unique steady state and if the linearization of the system at that point has no repeated eigenvalues and at least one eigenvalue with positive real part there exist periodic solutions. This reduces the existence question to the analysis of the linearization. The existence proof relies on the Brouwer fixed point theorem and is similar to a proof I described in a previous post. Although the Goodwin system is three dimensional the method is not restricted to that case. The proof does not give information about the stability of the periodic solutions. In the paper of Hastings et. al. they indicate that an alternative analysis using a Hopf bifurcation can give stability in some cases. However no details of the stability argument are given in that paper.

The Goodwin model was inspired by the fundamental work of Monod and Jacob on gene regulation.  Various things have given me an appetite for learning more about gene regulatory networks and this was increased by some of the talks I heard last week.

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SMB annual meeting in Knoxville, part 2

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.

SMB annual meeting in Knoxville

On Tuesday I will travel to Knoxville for the annual meeting of the Society for Mathematical Biology. On Wednesday I will give a talk there about my work on the NFAT signalling pathway. The programme of the conference is very dense: apart from the times when there are plenary talks there are seven sessions in parallel. My usual tactics at conferences of this type is to choose whole sessions to attend rather than individual talks. Anything else is usually frustrating due to the poor synchronization of the talks in different sessions. Maybe it will be better in this case. It is planned to have music to mark the breaks between talks which will be heard in all the rooms. This could overcome any lack of discipline imposed by the chairs of the individual sessions. Since all the rooms are in one building and, to judge by their numbering, close together it may really be practicable to attend individual talks.

What is the advantage of going to a big conference like this? The primary one is the opportunity of networking with people working in the field. Given that so much of the time is filled up with lectures this will require serious effort. It is good that the list of lectures was available well in advance of the conference. This allows a certain overview of who is taking part. It would have been even better if a full list of participants had also been available in advance. The second most important aspect of the conference is learning new things by actually listening to the talks. Since this is not a subject that I know so well that I can almost predict what the talks will be like just by seeing the titles and authors, there is plenty of opportunity for me here. Making the best of this opportunity will nevertheless require careful planning.

In the schedule there are eleven talks under the heading immunology and in addition a minisymposium on cancer immunology. These are things for me to focus on. There is also one plenary talk (by Becca Asquith) containing the phrase ‘immune response’. There is a session with the title ‘systems biology’ and four talks. My feelings towards this subject are ambivalent. On the one hand the idea – a concentrated theoretical approach to understanding biological systems – seems to me a very good thing. On the other hand I am not convinced by the way this idea has been realized up to now. One problem I see is that the definition of systems biology is rather vague and hence it is difficult to see what the content is. Another is that I have the impression that there is too much dominance of the quantitative over the qualitative (and high throughput over low throughput). My negative impression may just be due to lack of knowledge. In any case, I feel that I want to be enthusiastic about systems biology but I have not yet found the right point of access. A few weeks ago there was a conference on systems biology in Leipzig. I would have liked to attend but was prevented by other commitments. A highlight was a debate between Sydney Brenner and Denis Noble. I was not able to be there and so I was happy when I recently found that a video of it is available on the web. In fact the debate was not marked by strongly conflicting ideas. Both participants stressed that their views were not very far apart. I did not feel that I had a much clearer picture of the subject of the debate afterwards than I did before. Brenner dominated the proceedings. As usual he had a lot of interesting things to say. For instance he talked about a bacterium which adapted to live in I always find it inspiring listening to him and I recently had the opportunity to experience him live in a talk he gave in Berlin with the title ‘Reading the genome’. Through this I came upon a resource where short general articles by Brenner can be found. These one-page texts appeared under the names ‘Loose Ends’ and ‘False Starts’ and were published each month in the journal Current Biology between 1994 and 2000.

Stability of heteroclinic cycles

A heteroclinic orbit is a solution of a dynamical system which converges to one stationary solution in the past and to another stationary solution in the future. A heteroclinic chain is a sequence of heteroclinic orbits where the past limit of each orbit is the future limit of the preceding one. If this sequence is periodic we get what is called a heteroclinic cycle. Given such an object it is of interest to ask about its stability. For an initial datum sufficiently close to the cycle, when does the corresponding solution converge to the cycle at late times? In particular, when is the -limit set of the solution of interest equal to the entire cycle? To obtain information about this question it is useful to consider the linearization of the system about the vertices of the cycle. For a solution of the kind we are looking for, if it exists, will spend most of its time near the stationary points which are the vertices. If during time periods near a vertex it tends to approach the cycle then this is a good sign that the whole solution will approach the cycle. The behaviour of the solution near the stationary solution is determined by the linearization, at least if the stationary solution is hyperbolic.

In a previous post I described a result of Stefan Liebscher and collaborators which provides detailed information on the nature of the initial singularities of some spatially homogeneous spacetimes which are vacuum or where the matter content is described by a perfect fluid with linear equation of state . In that situation the Einstein equations can be reduced to a system of ordinary differential equations, the Wainwright-Hsu system, which treats all Bianchi class A models in a unified way. In particular it includes the type IX models. There is a heteroclinic cycle consisting of three Bianchi type I vacuum solutions. The main theorem of the paper is that there is a codimension one submanifold of initial data for which the -limit set of the corresponding solution is the heteroclinic cycle just described. The qualitative nature of this result is just as in the general discussion above except that the direction of time has been reversed. The system for vacuum solutions is four-dimensional. The vertices of the cycle are embedded in a one-dimensional manifold of stationary solutions and so the linearization must have at least one zero eigenvalue. As a consequence these vertices are not hyperbolic but the problem can be overcome. Of the remaining eigenvalues one is negative and the others are positive. The theorem makes use of the fact that for . In the presence of a fluid there is additional positive eigenvalue . The same idea of proof applies provided . This inequality is equivalent to an inequality for the parameter in the equation of state of the fluid.

An analogue of the vacuum solutions of type IX is given by solutions of type with a magnetic field. The dynamics of these solutions near the singularity was studied a long time ago by Marsha Weaver. In this situation there is a heteroclinic cycle essentially identical to that in the vacuum case. It is then natural to ask whether an analogue of the known theorem in the vacuum case in the paper by Stefan and collaborators holds. Together with Stefan and Blaise Tchapnda we have now written a paper on this subject. It turns out that there is a closely analogous result but that it is a lot harder to prove. The reason is that the eigenvalues of the linearization are in a less favourable configuration. Fortunately a weaker condition on the eigenvalues suffices. Suppose that denotes the eigenvalue at a vertex corresponding to the outgoing orbit in the cycle at that point. Then it suffices to assume that without imposing conditions of the other , provided that another condition on the existence of invariant manifolds is satisfied. The existence of these manifolds is a consequence of the geometric nature of the problem which gives rise to the dynamical system being considered. In this way we get a result on the stability of the heteroclinic cycle in the model with magnetic field. We are also able to remove the undesirable restriction on in the case with fluid. This work gives rise to a number of new questions on possible generalizations of this result. For more information on this I refer to the discussion section of our paper.