## The multiple futile cycle

August 27, 2012

The multiple futile cycle is a simple type of network of chemical reactions which is often found in biological systems. In a previous post I mentioned it as a component of a slightly more complicated network found in many cells, the MAP kinase cascade. One concrete realization of the multiple futile cycle is a protein which can be phosphorylated at up to $n$ sites. All the phosphorylation steps are carried out by one kinase while all dephosphorylation steps are carried out by one phosphatase. Each step is modelled in the Michaelis-Menten way, including an enzyme-substrate complex as one of the species and using mass action kinetics. There results a system of $3n+3$ ordinary differential equations with three conservation laws. These represent the conservation of the total amount of the two enzymes and of the substrate protein. In the case $n=1$, which might be called the simple futile cycle, using the conservation laws to eliminate some of the variables leads to a three-dimensional dynamical system. A basic question is what can be said about the dynamics of solutions of this system.

It has been shown by Angeli and Sontag (Nonlinear Analysis RWA, 9, 128) that in the case $n=1$ every solution converges to a stationary solution and that this stationary solution is unique for given values of the conserved quantities. The proof uses the theory of monotone dynamical systems. The original dynamical system is not monotone and so the first step in their proof is to replace it by another system which is monotone and show that convergence properties of solutions of the second imply convergence properties of solutions of the first. The second step is to prove the convergence of solutions of monotone systems under the additional condition of the existence of a translational symmetry. The paper mentions that this result is dual to a previously known result due to Mierczyński about monotone systems with a conserved quantity. Up to now I thought that the only benefit of knowing that a dynamical system is monotone is the possibility of reducing it to an effective system of one dimension less. This is only interesting if the initial system is of dimension no more than three. What this work has shown me is that knowing that a system is monotone can sometimes be the key to concluding much more. One aspect of the paper of Angeli and Sontag which was a source of confusion for me was a difference in conventions to what I am familiar with from chemical reaction network theory. This seems to be essential for the monotonicity argument and not just a matter of taste. The stoichiometric matrix (or stoichiometry matrix) is defined differently because a reversible reaction is treated as a single reaction rather than as a pair.. I feel a spontaneous preference for the CRNT convention but here it seems that a different one can be a real advantage. In the case of the simple futile cycle an important effect is that the dimension of the kernel of the stoichiometric matrix is three with the CRNT convention and one with the Angeli-Sontag convention.

In another paper (J. Math. Biol. 61, 581) Angeli, De Leenheer and Sontag present a more general theory related to this. Here the hypotheses needed to obtain a suitable monotone system involve the properties of certain graphs constructed from the reaction network. In this theory the notion of persistence of the dynamical system plays an important role. This is the property that a positive solution can never have any $\omega-$ limit points on the boundary of the positive region. The case $n=2$ (dual futile cycle) has been considered in a paper of Wang and Sontag (J. Nonlin. Sci. 18, 527). There they are able to show that for certain ranges of the parameters generic solutions converge to stationary solutions. To emphasize the power of the techniques developed in these papers it should be pointed out that they can be applied to systems with arbitarily large numbers of unkowns and parameters and that when they apply they give strong conclusions.

D. Angeli and E. D. Sontag (2008). Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles Nonlinear Analysis: Real World Applications, 9 DOI: 10.1016/j.nonrwa.2006.09.006

D. Angeli, P. De Leenheer and E. D. Sontag (2010). Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. Journal of mathematical biology, 61 (4) PMID: 19949950

## Organizing posts by categories

August 25, 2012

I have a tendency to use the minimal amount of technology I have to in order to achieve a particular goal. So for instance, having been posting things on this blog for several years, I have made use of hardly any of the technical possibilities available.  Among other things I did not assign my posts to categories, just putting them in one long list. I can well understand that not everyone who wants to read about immunology wants to read about general relativity and vice versa. Hence it is useful to have a sorting mechanism which can help to direct people to what they are interested in. Now I have invested the effort to add information on categories to most of the posts. It was easy (though time-consuming) to do and I find that the results are useful. It is helpful for me myself to navigate through the material and it is interesting for me to see at a glance how many posts on which subjects there are. For now on I will systematically assign (most) new posts to a category and the effort to do so should be negligible. This post is an exception since it does not really fit into any category I have.

## Stable big bang singularities

August 2, 2012

This week I am at a conference on mathematical relativity in Oberwolfach. Today Jared Speck gave a talk on work of his with Igor Rodnianski which answers a question I have been interested in for a long time. The background is that it is expected that the presence of a stiff fluid (a fluid where the speed of sound is equal to the speed of light) leads to a simplification of the dynamics of the solutions of the Einstein equations near spacetime singularities. This leads to the hope that the qualitative properties of homogeneous and isotropic solutions of the Einstein equations coupled to a stiff fluid near their big bang singularities could be stable under small changes of the initial data. This should be independent of making any symmetry assumptions. As discussed in a previous post it has been proved by Fuchsian techniques that there is a large class of solutions consistent with this expectation. The size of the class is judged by the crude method of function counting. In other words there are solutions depending on as many free functions as the general solution. This is a weak indication of the stability of the singularity but it was clear that a much better statement would be that there is an open neighbourhood of the special data in the space of all initial data which has the conjectured singularity structure. This is what has now been proved by Rodnianski and Speck.

This result fits into a more general conceptual framework. Suppose we have an explicit solution $u_0(t)$ of an evolution equation and we would like to investigate the stability of its behaviour in a certain limit $t\to t_*$. If we expect that solutions with data close to the data for $u_0$ have the same qualitative behaviour then we may try to prove this directly. Call this the forward method. If there is no evidence that this idea is false but it seems difficult to prove it then we can try another method as an interim approach to gain some insight. This is to attempt to construct solutions with the expected type of asymptotics which are as general as possible. I call this the backward method, since it means evolving away from the asymptotic regime of interest. The forward method is preferable to the backward if it can be done. In the case of singularities in Gowdy spacetimes Satyanad Kichenassamy and I applied the backward method and Hans Ringström later used the forward method. It is perhaps worth pointing out that while the forward method is more satisfactory than the backward one both together can sometimes be used to give a better total result than the forward method alone. There are also examples of this in the context of expanding cosmological methods with positive cosmological constant. I applied the backward method while Ringström, Rodnianski and Speck later used the forward method. The result for the stiff fluid with which I started this post also fits into this framework using the forward method. The corresponding result for the backward method was done by Lars Andersson and myself more than ten years ago.

There were two other talks at this conference which can be looked at from the point of view just introduced. One was a talk by Gustav Holzegel on his work with Dafermos and Rodnianski on the existence of asymptotically Schwarzschild solutions. The second was my talk on an apsect of Bianchi models I have discussed in a previous post. Both of these used the backward method.

## The Goodwin oscillator

July 30, 2012

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 $\frac{du_i}{dt}=f_i(u_{i-1})-k_iu_i$. Here the labels on the $u_i$ are supposed to be interpreted modulo $n$. In other words there are $n$ equations and $u_0$ is interpreted as $u_n$. In the Goodwin model itself $n=3$ and the functions $f_i$ are linear for $i>1$. The function $f_1$ is equal to $\frac{a}{b+u_n^m}$ for constants $a$, $b$ and $m$. This function is positive and its derivative is negative. Thus it can be interpreted as representing a negative feedback on the production of $u_1$. In the context in which it was introduced by Goodwin the quantities $u_1$, $u_2$ and $u_3$ 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 $m$ 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 $m$. 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.

## SMB annual meeting in Knoxville, part 2

July 27, 2012

The music did seem to have a positive effect on the synchronization of lectures. Unfortunately it was not always there – for instance it was not there before my talk – and it seems to have been getting less and less. One good thing is that the name tags, as well as showing the usual information have the first name (or nickname) printed in large letters at the top. I find that this can be very useful for recognizing people after only having met them fleetingly.

The plenary talk of Claire Tomlin yesterday was about the HER2 receptor which plays an important role in breast cancer. It is connected to transcription factors in the nucleus by a signalling network containing two main pathways. One of these includes the MAP kinase cascade while another passes through the substance Akt. Excessive activity of this type of signalling can be reduced by a drug called lapatinib, which is a tyrosine kinase inhibitor. There is, however, a problem that this beneficial effect can be neutralized after some time. The speaker described ideas for overcoming this effect based on a study of the signalling network. A result of this analysis is that, counterintuitively, combining the administration of lapatinib with another treatment which increases the concentration of Akt at a different time could lead to a more effective therapy. I did not get the details but this seems like a case where mathematical modelling could actually contribute effectively to cancer treatment by suggesting new strategies. Relations were mentioned to the pattern of hairs on the wings of Drosophila. In her research on biomedical themes she benefits from her background in control engineering and aerodynamics.

The talk of Becca Asquith which I mentioned in the last post was cancelled. Instead there was a lecture by Sandy Anderson who seems to like to cultivate the image of the hard-drinking Scotsman. He started his career in mathematical modelling and then moved a long way towards medical research, now heading a lab at the Moffit Cancer Center in Florida. The subject of his talk was the role of heterogeneity in cancer. He started by giving a view of the importance of cancer (in terms of the number of people it kills) and the trends in the numbers for the different forms. They have mostly been decreasing for many years with the notable exception of lung cancer (for well-known reasons) but the rate of decrease is not very large despite the huge amount of effort, and money, put into cancer research. He said that death in cancer usually does not result from a tumour which stays in its original site but as a result of metastasis. Thus that is the key phenomenon to be understood. This requires an understanding of many different scales and for the talk he concentrated on the cellular scale. He claimed that an important fact that cancer researchers had not taken into account sufficiently until very recently is how heterogeneous tumours are. There is a large variation in the phenotype of the individual cancer cells and the phenotypes are evolving. This evolution is strongly influenced by the environment of the tumour, for instance the structure of the surrounding extracellular matrix. Experiments done on cell cultures may give misleading results since the ‘happy’ cells in the Petri dish with all modern comforts are not under the same pressure as corresponding cells in the body. The more the external pressures are the more the dangerous cells which are going to metastasize dominate over the others. In some cases treatment can accelerate the growth of a tumour. This danger exists if the treatment is given too late. These ideas have arisen by the use of mathematical modelling. These are ‘hybrid models’ which combine discrete and continuous dynamical systems and this is a terms which I have met in several other talks at this conference. One of the conclusions of this research is that it may be a good idea to control cancer cells rather to destroy them. For the attempt to destroy cells may destroy the relatively harmless ones and unleash the dangerous one on their surroundings. Anderson’s talk conveyed the excitement of the application of mathematical modelling in cancer research at this moment and I wonder if some of the young people in the audience might have been recruited.

This afternoon I went to a session on wound healing. There was an introductory lecture by Rebecca Segal and this was helpful for me since I knew very little about the subject. Two of the things I found interesting – I was already primed for this by talking to Angela Reynolds at her poster yesterday – is that immunology (dynamics of neutrophils and macrophages) plays a big role and that ODE models can be useful. Useful means that they can help doctors make decisions how to treat wounds they are confronted with in practise.

## SMB annual meeting in Knoxville

July 22, 2012

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 $D_2 O.$ 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

July 12, 2012

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 $\omega$-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 $p=(\gamma-1)\rho$. 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 $\alpha$-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 $-\mu$ is negative and the others $\lambda_1,\lambda_2$ are positive. The theorem makes use of the fact that $\lambda_i>\mu$ for $i=1,2$. In the presence of a fluid there is additional positive eigenvalue $\lambda_3$. The same idea of proof applies provided $\lambda_3>\mu$. This inequality is equivalent to an inequality for the parameter $\gamma$ in the equation of state of the fluid.

An analogue of the vacuum solutions of type IX is given by solutions of type ${\rm VI}{}_0$ 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 $\lambda_1$ denotes the eigenvalue at a vertex corresponding to the outgoing orbit in the cycle at that point. Then it suffices to assume that $\lambda_1>\mu$ without imposing conditions of the other $\lambda_i$, 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 $\gamma$ 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.

## Low throughput biology

April 24, 2012

In modern biology there is a strong tendency to collect huge quantities of data with high throughput techniques. This data is only useful if we have good techniques of analysing it to obtain a better understanding of the biological systems being studied. One approach to doing this is to build mathematical models. An idea which is widespread is that the best models are those which are the closest to reality in the sense that they take account of as many effects as possible and use as many measured quantities as possible. Suppose for definiteness that the model is given by a system of ordinary differential equations. Then this idea translates into using systems with many variables and many parameters. There are several problems which may come up. The first is that some parameters have not been measured at all. The second is that those which have been measured are only known with poor accuracy and different parameters have been measured in different biological systems. A third problem is that even if the equations and parameters were known perfectly we are still faced with the difficult problem of analysing at least some aspects of the qualitative behaviour of solutions of a dynamical system of high dimension. The typical way of getting around this is to put the equations on the computer and calculate the solutions numerically for some initial data. Then we have the problem that we can only do the calculations for a finite number of initial data sets and it is difficult to know how typical the solutions obtained really are. To have a short name for the kind of model just described I will refer to it as a ‘complex model’.

In view of all these difficulties with complex models it makes sense to complement the above strategy by one which goes in a very different direction. The idea for this alternative approach is to build models which are as simple as possible subject to the condition that they include a biological effect of interest. The hope is then that a detailed analysis of the simple model will generate new and useful ideas for explaining biological phenomena or will give a picture of what is going on which may be crude but is nevertheless helpful in practise, perhaps even more helpful than a complex model.

It often happens that in analysing a complex model many of the parameters have to be guessed (perhaps just in an order of magnitude way) or estimated by some numerical technique. It is then justified to ask whether adding more variables and corresponding parameters really means adding information. How can we hope to understand complex models at all? If these were generic dynamical systems with the given number of unknowns and parameters this would be hopeless. Fortunately the dynamical systems arising in biology are far from generic. They have arisen by the action of evolution optimizing certain properties under strong constraints. Given that this is the case it makes sense to try and understand in what ways these systems are special. If key mechanisms can be identified then we can try to isolate them and study them intensively in relatively simple situations. My intention is not to deny the value of high throughput techniques. What I want to promote is the idea that it is bad if the pursuit of those approaches leads to the neglect of others which may be equally valuable. On a theoretical level this means the use of ‘simple models’ in contrast to ‘complex models’. There is a corresponding idea on the experimental side which may be even more necessary. This is to focus on the study of certain simple biological systems as a complement to high throughput techniques. This alternative might be called ‘low throughput biology’. It occurred to me that if I had this idea under this name then it might also have been introduced by others. Searching for the phrase with Google I only found a few references and as far as I could see the phrase was generally associated with a negative connotation. Rather than making an opposition between low throughput and high throughput techniques like David and Goliath it would be better to promote cooperation between the two. I have come across one good example in this in the work of Uri Alon and his collaborators on network motifs. This work is well explained in the lectures of Alon on systems biology which are available on Youtube. The idea is to take a large quantity of data (such as the network of all transcription factors of E. coli) and to use statistical analysis to identify qualitative features of the network which make it different from a random network. These features can then be isolated, analysed and, most importantly, understood in an intuitive way.

## Dynamics of the MAP kinase cascade

April 7, 2012

The MAP kinase cascade is a group of enzymes which can iteratively add phosphate groups to each other. More specifically, when a suitable number of phosphate groups have been added to one enzyme in the cascade it becomes activated and can add a phosphate to the next enzyme in the row. I found this kind of idea of enzymes modifying each other with the main purpose of activating each other fascinating when I first came across it. (The first example I saw was actually the complement cascade which occurs in immunology.) This type of structure is just asking to be modelled mathematically and not surprisingly a lot of work has been done on it. Here I will survey some of what is known.

The MAP kinase cascade is a structure which occurs in many types of cells. It has three layers. The first layer consists of a protein which can be phosphorylated once. The second layer consists of a protein which can be phosphorylated twice by the same enzyme. This enzyme is the phosphorylated form of the protein in the first layer. The third layer also consists of a protein which can be phosphorylated twice by the same enzyme. This enzyme is the doubly phosphorylated form of the protein in the second layer. The protein in the third layer is the one which is called MAP kinase (mitogen activated protein kinase, MAPK). A kinase is an enzyme which phosphorylates something else and so it is not suprising that the protein in the second layer is called a MAP kinase kinase (MAPKK). The protein in the first layer is accordingly called a MAP kinase kinase kinase (MAPKKK). The roles of the players in this scheme can be taken by different enzymes. For concreteness I name those which occur in the case of human T cells. There the proteins in the first, second and third layers are called Raf, MEK and ERK, respectively. The protein which phophorylates Raf, and hence starts the whole cascade, is Ras. It, or rather the corresponding gene ras, is famous as an oncogene. This means that when the gene is not working properly cancer can result. In fact many drugs used in cancer treatment target proteins belonging to the MAP kinase cascade.

A model for the MAP kinase cascade was written down by Huang and Ferrell (PNAS, 93, 10078). They used a description of Michaelis-Menten type where for each basic substance three species are included in the network. These are the substance itself (free substrate), the enzyme and the complex of the two. Of course since in the MAP kinase cascade certain proteins act both as substrate and enzyme in different reactions there is some overlap between these. For clarity this may be called the ‘extended Michaelis-Menten’ description to contrast it with the ‘effective Michaelis-Menten’ description arising from the extended version by a quasi-steady state limiting process. Note that for a given basic reaction network with $m$ species the extended MM description has more than $m$ species but still has mass-action kinetics whereas the effective MM description has $m$ species but kinetics more complicated than mass action. Phosphatases catalysing the reverse reactions are also included in the model. The phosphatase which removes both phosphate groups of ERK is called MKP3.

In the paper the steady states of the model are studied and an input-output relation is computed numerically. The activity of the MAPK is plotted as a function of the concentration of the first enzyme (Ras in the example). A sigmoidal curve is found which corresponds to what is called ultrasensitivity. The dynamical properties of the model are not discussed. In particular it is not discussed whether there might be multistability (more than one stable stationary solution for fixed values of the parameters) or periodic solutions. The authors also did experiments whose results agreed well with the theoretical predictions. The experiments were done with extracts from the oocytes (immature egg cells) of the frog Xenopus laevis.

The possible dynamic behaviour was investigated in later papers. In some of these the effect of adding an additional feedback was considered. This kind of feedback is probably important in real biological systems. It may, for instance, explain why the results of experiments on whole oocytes are different from those done with extracts. Here, for mathematical simplicity, I will restrict to the case without additional feedback, in other words to the original Huang-Ferrell model. Multistability in this type of model was found in a paper of Markevich, Hoek and Kholodenko (J. Cell Biol. 164, 353). They investigate both extended and effective MM dynamics numerically and find bistability for both. In the extended MM model, which is the one I am most interested in here, the phosphorylation is supposed to be distributive. In other the words the kinase is released between the two phosphorylation steps. The alternative to this is called processive phosphorylation. In a paper of Conradi et. al. this result is compared with chemical reaction network theory (CRNT). It is found that while techniques from CRNT yield results agreeing with those of Markevich et. al. for the case where both the kinase and the phosphatase act in a distributive way, if one of these is replaced by a processive mechanism it can be proved using the Deficiency One Algorithm of CRNT that there is no multistationarity. The case with distributive phosphorylation is the special case $n=2$ of what is called a multiple futile cycle with $n$ steps. Wang and Sontag (J. Math Biol. 57, 29) proved upper and lower bounds for the number of steady states in this type of system under certain assumptions on the parameters. In particular this confirms that there can be three steady states (without determining their stability). Going beyond the single layer to the full cascade opens up more possibilities. Numerical evidence has been presented by Qiao et. al. (PLOS Comp. Biol. 9, 2007) that there are periodic solutions. To understand why these should exist it might be best to think of them as relaxation oscillations.

## The Einstein-Boltzmann system

March 13, 2012

The Boltzmann equation provides a description of the dynamics of a large number of particles undergoing collisions, such as the molecules of a gas. The classical Boltzmann equation belongs to Newtonian physics. It has a natural relativistic generalization. The Boltzmann model is adapted to capture the effects of short-range forces acting on short time scales during collisions. The model can be extended to also include the effects of long-range forces generated collectively by the particles. If the forces are gravitational and the description is made fully relativistic then the system of equations obtained is the Einstein-Boltzmann system. In any of the cases mentioned up to now the Boltzmann equation is schematically of the form $Xf=Q(f)$. The term on the left is a transport term giving the rate of change of the function $f$, the density of particles, along a vector field $X$ on phase space. The vector field $X$ is in general determined by the long-range forces. The term on the right is the collision term which, as its name suggests, models the effect of collisions. It is an integral term which is quadratic in $f$. The function $f$ itself is a function of variables $(t,x,p)$ representing time, position and velocity (or momentum).

How is the collision term obtained? It is important to realize that it is in no sense universal – it contains information about the particular interaction between the particles due to collisions. This can be encoded in what is called the scattering kernel. In the classical case it is possible to do the following. Fix a type of interaction between individual particles and solve the corresponding scattering problem. Each specific choice of interaction gives a scattering kernel. Once various scattering kernels have been obtained in this way it is possible to abstract from the form of the kernels obtained to define a wider class. A similar process can be carried out in special relativity although it is more complicated. Any scattering kernel which has been identified as being of interest in special relativity can be taken over directly to general relativity using the principle of equivalence. Concretely this means that if the Boltzmann collision term is expressed in terms of the components of the momenta in an orthonormal frame
then the resulting expression also applies in general relativity.

For a system of evolution equations like the Einstein-Boltzmann system one of the most basic mathematical questions is the local well-posedness of the initial value problem. For the EB system this problem was solved in 1973 by Daniel Bancel and Yvonne Choquet-Bruhat (Commun. Math. Phys. 33, 83) for a certain class of collision terms. The physical interpretation of the unknown $f$ in the Boltzmann equation as a number density means that it should be non-negative. In the context of the initial value problem this means that it should be assumed that $f$ is initially non-negative and that it should then be proved that the corresponding solution is non-negative. In the existence proofs for many cases of the Boltzmann equation the solution is obtained as the limit of a sequence of iterates, each of which are by construction non-negative. The convergence to the limit is strong enough that that the non-negativity of the iterates is inherited by the solution. In the theorem of Bancel and Choquet-Bruhat the solution is also constructed as the limit of a sequence of iterates but no attention is paid to non-negativity. In fact that issue is not mentioned at all in their paper. To prove non-negativity of solutions of the EB system it is enough to prove the corresponding statement for solutions of the Boltzmann equation on a given spacetime background. The latter question has been addressed in papers of Bichteler and Tadmon. On the other hand it is not easy to see how their results relate to those of Bancel and Choquet Bruhat. This question has now been investigated in a paper by Ho Lee and myself . The result is that with extra work the desired posivity result can be obtained under the assumptions of the theorem of Bancel and Choquet-Bruhat. While working on this we obtained some other insights about the EB system. One is that the assumptions of the existence theorem appear to be very restrictive and that treating physically motivated scattering kernels will probably require more refined approaches. In the almost forty years since the local existence theorem there have been very few results on the initial value problem for the EB system (with non-vanishing collision term). We hope that our paper will set the stage for further progress on this subject.