## Modelling the Calvin cycle

March 18, 2013

Some years ago the Max Planck Institute for Molecular Plant Physiology organized a conference on metabolic networks. I decided to see what was going on in the institute next to the one where I work and I went to some of the talks. The one which I found most interesting was by Zoran Nikoloski. His subject was certain models for the Calvin cycle, which is part of photosynthesis. A motivating question was whether photosynthesis can work in two different stable steady states. If that were the case it might be possible to influence the plant to move from one state to another and, in the best case, to increase its production of biomass. This is of interest for biotechnology. Mathematically the question is that of multistationarity, i.e. whether a system of evolution equations admits more than one stationary solution. Beyond this it is of interest whether there can be more than one stable stationary solution. In fact in this context the issue is not that of absolute uniqueness of stationary solutions but of uniqueness within a given stoichiometric compatibility class. This means that the solution is unique when certain conserved quantities are fixed. One thing I found attractive about the presentation was that the speaker was talking about rigorous mathematical results on the dynamics and not just about numerically calculating a few solutions.

If the system is modelled deterministically and diffusion is neglected there results a system of ordinary differential equations for the concentrations of the substances involved as functions of time. It is necessary to choose which substances should be included in the description. In a basic model of the Calvin cycle there are five substances. In the work discussed in the talk of Nikoloski and in a paper he wrote with Sergio Grimbs and others (Biosystems 303, 212) various ODE systems based on this starting point are considered. They differ by the type of kinetics used. They consider mass action kinetics (MA), extended Michaelis-Menten kinetics where the enzymes catalysing the reactions are included explicitly (MM-MA) and effective Michaelis-Menten (MM) obtained from the system MM-MA by a singular limit. The systems MA and MM consist of five equations while the system MM-MA consists of nineteen equations. In the paper of Grimbs et. al. they show among other things that the system MM never admits a stable stationary solution, whatever the reaction constants, while the system MM-MA can exhibit two different stationary solutions.

After the talk I started reading about this subject and I also talked to Nikoloski about it. Later I began doing some research on these systems myself. Some technical difficulties which arose (which I wrote about in a previous post) led me to consult Juan Velázquez and he joined me in this project. Now we have written a paper on models for the Calvin cycle. In a case where there is only one stationary solution and it is unstable it is of interest to consider the final fate of general solutions of the system. For some initial conditions the concentrations of all substances tend to zero at late times. For other data (a whole open set) we were able to show that all concentrations tend to infinity as $t\to\infty$. We called the latter class runaway solutions. These do not seem to be of direct biological relevance but they might be helpful in choosing between alternative models which are more or less appropriate. The proof of the existence of runaway solutions for the MA system is somewhat complicated since this turns out to be a system with two different timescales. The system MM-MA also admits runaway solutions. Although the system is larger than MA the existence proof is simpler and in fact can be carried out in the context of a larger class of systems. Runaway solutions are also found for the system MM.

In the paper of Grimbs et. al. one system is considered which includes the effect of diffusion. Restricting to homogeneous solutions of this system gives a system of ODE called MAdh which is different from the system MA. The difference is that while the concentration of ATP is a dynamical variable in MAdh it is taken to be constant in MA. We showed that the system MAdh has zero, one or two solutions depending on the values of the parameters and that all solutions are bounded. Thus runaway solutions are ruled out. Intuitively this is due to the fact that the supply of energy is bounded but this heuristic argument is far from providing a proof. There are many other models of the Calvin cycle in the literature. In general they consider the reactions between a larger class of substances. It is an interesting task for the future to extend the results obtained up to now to these more general models. This post has been very much concerned with the mathematics of the problem and has not said much about the biology. The reactions making up the Calvin cycle were determined experimentally by Melvin Calvin and I can highly recommend his Nobel lecture as a description of how this was achieved

## Absolute concentration robustness

February 20, 2013

In the past years I have been on the committees for many PhD examinations. A few days ago, for the first time, I was was on the committee for a thesis on a subject belonging to the area of mathematical biology. This was the thesis of Jost Neigenfind and it was concerned with a concept called absolute concentration robustness (ACR).

The concentration of a given substance in cells of a given type varies widely between the individual cells. (Cf. also this previous post). It is of interest to identify mechanisms which can ensure that the steady state concentration of a particular substance is independent of initial data. (This is a way in which the output of a system can be independent of background variation.) In saying this I am assuming implicitly that more general solutions converge to steady states. A more satisfactory formulation can be obtained as follows. In a chemical reaction network there are usually a number of conserved quanitities, say $C_\alpha$. These define affine subspaces of the state space, the stoichiometric compatibility classes. For many systems there is exactly one stationary solution in each stoichiometric compatibility class. The condition of interest here is that the value of one of the concentrations, call it $x_1$, in the steady state solution is independent of the parameters $C_\alpha$. (The other concentrations $x_i,i>1$ will in general depend on the $C_\alpha$.) This property is ACR. I first heard of this in a talk by Uri Alon at the SMB conference in Krakow in the summer of 2011. The basic idea is explained clearly in a paper of Shinar and Feinberg (Science 327, 1389). They present a general theoretical approach but also describe some biological systems where ACR (in a suitable approximate sense) has been observed experimentally. In the terminology of Chemical Reaction Network Theory (CRNT) the examples they discuss have deficiency one. They mention that ACR is impossible in systems of deficiency zero. There is no reason why it should not occur in systems of deficiency greater than one but in those more complicated dynamics make it more difficult to decide whether the property holds or not.

The result of Shinar and Feinberg only covers a class of reaction networks which is probably very restricted. What Neigenfind does in his thesis is to develop more general criteria for ACR and computer algorithms which can check these criteria for given systems. The phenomenon of ACR is interesting since it is a feature which may be more common in reaction systems coming from biology than in generic systems. At least there is a good potential reason why this might be the case.

## Goodbye to Berlin

February 1, 2013

For this post I could not resist the temptation to borrow the title of Christopher Isherwood’s novel although what I am writing about here has very little to do with his book. The connection of the title to the content is that I will soon be leaving Berlin after living here more than fifteen years. I have accepted a professorship at the University of Mainz and I will move there in April. The first time I came to Berlin I landed at Tegel airport and I interpreted the Hooded Crow I saw beside the runway as a good omen. This requires some explanation. In those days the Hooded Crow (Corvus corone cornix) was a subspecies of the Carrion Crow. In the meantime it has been promoted to the rank of a species but that will not concern me here – I am not sure whether I feel I should congratulate it on receiving this honour. It differs from the nominate form (according to the old classification) by having a grey body while Corvus corone corone (Carrion Crow in the narrower sense) is all black. These two forms have the classical property of subspecies that they are allopatric. In other words they occur in more or less disjoint regions. On the boundary between the regions there is little interbreeding. The Orkney Islands where I grew up belong to the land of the Hooded Crow. Most of Great Britain and in fact most of Western Europe belong to the domain of the Carrion Crow. Even Aberdeen, where I studied and did my PhD, belongs to the land of the Carrion Crow. This helps to explain why I associate the Hooded Crow with ‘home’ and the Carrion Crow with ‘foreign parts’. It also has to do with the fact that there was a Hooded Crow which nested regularly in a garden near where I grew up in Orkney. I would climb the tree from time to time to keep an eye on the development of the brood and ring the chicks at the right moment. For these reasons the bird in Tegel seemed to tell me I was coming home. Now I am daring to venture once again (and probably for most of the rest of my life) into the land of the Carrion Crow.

When leaving a place it is natural to think about the good things which you experienced there. What were the best things about Berlin for me? The best thing of all is that Berlin was where I met my wife Eva. Eichwalde, where she lived at that time and where we both live now, has a very special feeling for me which will never go away. (Just for the record, we do not really live in Eichwalde but that is where the nearest train station is, with the result that the platform of the station there has something of the gates of Paradise for me.) Of course I cannot fail to mention the Max Planck Institute for Gravitational Physics which has provided me with a scientific home during all that time. I am grateful to the successive leaders of the mathematical group there, Jürgen Ehlers and Gerhard Huisken, for the working and social environment which they created and maintained. Another important thing about Berlin I will miss is the contact with its excellent research in biology and medicine. I have spent many valuable hours attending the Berlin Life Science Colloquium and I feel very attached to the Paul Ehrlich lecture hall where it usually takes place. The wooden seats are hard but the interest of the lectures was generally more than enough to make me forget that. I will also miss the stimulating atmosphere of the group of Bernold Fiedler at the Free University, which has been a source of a lot of intellectual input and a lot of pleasure.

This is perhaps the moment to say why I am leaving Berlin. Ever since I was a student I have felt a strong allegiance to mathematics. As a child I was concerned with metaphysical questions and later I got interested in physics as the most fundamental part of science. During my undergraduate study I realised that mathematics, and not physics, was the right intellectual environment for me. A key experience for me was that through my study plan I ended up doing two courses on Fourier series, a subject which was new to me, at the same time. One was in physics and one in mathematics. The contrast was like night and day. This may have had something to do with the abilities of the individual lecturers concerned but it was mainly due to essential differences between mathematics and physics. By the end of my studies I had specialized in mathematics and my commitment to that subject has remained constant ever since.

For a long time my strongest connection to mathematics concerned intrinsic aspects of the subject. The significance of applications for me was as a good source of mathematical problems. This has changed over the years and I have become increasingly fascinated by the interplay between mathematics and its applications. At the same time the focus of my interest has moved from mathematics related to fundamental physics to mathematics related to biology and medicine. This change has led to a discrepancy between the research I want to do and the research area of the institute where I work. A Max Planck Institute is by its very nature focussed on a certain restricted spectrum of subjects and this is not compatible with a major change of research direction of somebody working there. This is the reason that I started applying for jobs which fitted the directions of work where my new interests lie. The move to Mainz is the successful endpoint of this process. Moving from a Max Planck Institute to a university will naturally involve spending more time on teaching and less time on research. This does not dismay me. The most important thing is that I will be doing something I believe in. Teaching elementary mathematics and analysis, apart from establishing the basis needed for doing research, is something whose intrinsic value I am convinced of.

## The relativistic Boltzmann equation

January 3, 2013

In a previous post I wrote about the Einstein-Boltzmann system and some recent work on that subject by Ho Lee and myself. One of the things we realized as a consequence of this work is that the known local existence theorem for the Einstein-Boltzmann system requires very restrictive assumptions on the collision kernel. Now we have looked in more detail at other kinds of collision kernel which are closer to what is desirable from the point of view of the physical applications. As a result of this we have written a paper which is concerned with the hard potential type of collision kernel. The subject of the paper is the Boltzmann equation in special relativity or on a homogeneous and isotropic background. This is intended to prepare the ground for similar work on the coupled Einstein-Boltzmann system. The main results are global existence theorems for spatially homogeneous solutions of the Boltzmann equation without any small data restriction. They are analogous to results obtained previously by Norbert Noutchegueme and collaborators for a more restrictive type of collision kernel.

The collision kernel is a function of the relative momentum $g$ and the scattering angle $\theta$. In the case of the classical (i.e. non-relativistic) Boltzmann equation a type of collision kernel which has often been studied is that arising from a power-law interaction between particles. The corresponding collision kernel has a power-law dependence on $g$ and a dependence on $\theta$ which cannot be determined explicitly. It has a non-integrable singularity in $\theta$ at $\theta=0$. It has been observed that properties of solutions of the Boltzmann equation determine two different regimes for the exponent of $g$. The cases between an inverse square and an inverse fifth power force between particles are known as soft potentials. The exponent of $g$ varies from $-4$ to $-1$. Cases with powers of the force more negative than $-5$ are known as hard potentials. As the power in the force ranges from $-5$ to $-\infty$ the exponent of $g$ varies from $-1$ to zero. When the exponent of $g$ is equal to $-1$  there are simplifications in some calculations and this has led to this case being popular among theorists. It is called Maxwell molecules. The limit where the exponent of $g$ tends to zero corresponds to the case of collisions of hard spheres.

Given the importance of the distinction between the soft potential and hard potential cases in the theory of the classical Boltzmann equation it is natural to look for an analogous distinction in the relativistic case. This was done by Dudynski and Ekiel-Jezewska. The analogy seems to be not at all straightforward. This work was carried further by Robert Strain and collaborators, who were able to apply these concepts and obtain a variety of global existence results. In their work the data are not required to be symmetric but are assumed to be close to data for known solutions. Our work is at the opposite extreme with a very strong symmetry assumption (spatially homogeneous) but no smallness requirement. It is modelled on theorems for the classical Boltzmann equation due to Mischler and Wennberg in Annales IHP (Analyse non lineaire) 16, 467. There is an analogue of Maxwell molecules in the relativistic case called Israel molecules but the analogy is not simple. My global conclusion from my experience with this problem is that there are a lot of interesting and challenging open problems around in the study of the relativistic Boltzmann equation and the Einstein-Boltzmann system.

## Talk on mathematical modelling in Karlstad

November 20, 2012

Yesterday I was in Karlstad in Sweden to give a talk on the uses of mathematical modelling in the natural sciences. I was invited to do this by Claes Uggla and I was very happy to have the opportunity to present some of my ideas on this subject. The talk was structured as a series of examples involving applications of different mathematical techniques. Many of these examples have been discussed in some form in this blog during the past few years and indeed a lot of my ideas on the subject were developed in conjunction with the blog posts. The subjects were William Harvey and the circulation of the blood, multidrug therapy for HIV-AIDS, the lizard Uta stansburiana, oscillations near the big bang, Liesegang rings, modelling oscillations in vole populations using a reaction-diffusion system, signal transduction in T cells.

As well as presenting a variety of applications of different types of mathematics I also wanted to explain some mathematical connections between these subjects. One central idea is that structural stability is an issue of key importance in modelling natural phenomena. Most phenomenological models involve parameters or other elements which are not known exactly. Thus to be of interest for applications features of the dynamics of the model should be invariant under arbitrary small perturbations of the system. More precisely, if a model does not possess an invariance of this type but is nevertheless useful this requires some explanation. One possible source of an explanation is the presence of what I call ‘absolute elements’ in the model. For instance, in population dynamics if a population is zero at some time then it will definitely remain zero. This fact is independent of the details of how the population grows when it is non-zero. Similarly a spacetime singularity can define an absolute element in cosmology. When the spacetime metric breaks down this ends the dynamics in a way which is independent of the details of the dynamics of the matter away from the singularity. Thus structural stability can be weakened to the condition of invariance under small perturbations which leave certain submanifolds fixed. This can lead to the appearance of relevant heteroclinic cycles although these are not structurally stable in the absolute sense. It explains the appearance of heteroclinic cycles in the models for lizards and for the big bang in a unified way. In a similar way, restricting the perturbations of a system of chemical reactions to those which leave a particular reaction irreversible can furnish the homoclinic orbit needed to model Liesegang rings.

I have now put a slightly extended version of this talk with references on my web page. On the same day there was a talk by Bernt Wennberg on models for the collective motion of birds and fish, concentrating mainly on kinetic models related to the Boltzmann equation. At the start of his talk he showed some of the well-known pictures of flocks of Starlings over Rome. In the evening I had my own pleasant experience with a flock of birds. A large number of Jackdaws (a couple of hundred) were flying around the central square in Karlstad and calling. For some reason I have become increasingly attached to the Jackdaw over the years. At this point, and without a good excuse, I want to tell a story about Jackdaws from the book ‘King Solomon’s Ring’ by Konrad Lorenz. It is a long time since I read the book and so I hope I do not distort the story too much. At one time Lorenz was living in a small village in Austria where he was regarded by the locals as a bit crazy. One of his interests was the social life of Jackdaws. There were Jackdaws living on the roofs of the houses and he climbed up to get close to them. In order to fit in better with his black subjects he decided to dress in black. The only ‘suitable’ black clothing he could find was a devil’s costume left over from a fancy dress party. No doubt the spectacle of him climbing over the roofs dressed as the devil perfected his reputation with the local inhabitants.

## Conference on systems biology of T cells in Baeza, part 2

October 25, 2012

In the remaining one and a half days of the conference there were another fourteen talks and I will mention some aspects of their contents which attracted my attention. One recurring theme was that the encounter of a T cell receptor (TCR) with the peptide it recognizes bound to an MHC molelcule (pMHC) is often not just the encounter of one TCR with one pMHC but of multiple players. It can be shown by electron microscopy that the TCR tend to cluster on the surface of a T cell even before it has encountered antigen. This is done by attaching gold particles to the TCR so that they show up as black dots on the electron micrograph. It was shown in the talk of Hisse van Santen that a similar thing happens with the pMHC on the surface of antigen presenting cells. Judging from the discussion after the talk it seems that the explanation for this is that the pMHC, which are well known to be produced in the interior of the cell, are exported to the surface in groups. There also seems to be a widely held opinion that signalling through the T cell receptor is absolutely dependent on clustering of TCR. This makes life more complicated than it otherwise might have been. I learned at this conference that experiments on T cell signalling in vitro are often done by using tetramers, i.e. groups of four pMHC which are bound together covalently. In the talk of Wolfgang Schamel described experiments using tetramer binding. He said that this work was linked with some mathematical modelling, done by Thomas Höfer and others, but he did not want to take questions on that. My impression was that the model was an extension of the kinetic proofreading model. It has not yet been published and so I did not yet have an opportunity to look at it. Carmen Molina-París and Balbino Alarcón discussed cooperative effects in T cell receptor binding.

Michal Polonsky showed pictures of individual T cells trapped in small wells in a microfluidic device. When activated they wriggle very vigorously. These are the kind of pictures which could easily make you take a very anthropomorphic view of T cells. The aim of this work is to observe the differentiation, division and death of the cells over long periods (several days). If they were not trapped it would be extremely difficult to follow them under the microscope since they would be liable to run away. A break from the purely scientific talks was provided by a presentation of Dinah Singer about the systems biology programme at the National Cancer Institute in the US, a programme which she runs. Apart from concrete information about funding another aspect of this was the question of what might be learned about the potential for applying systems biology to immunology from existing applications of these ideas to cancer research. Dipankar Nandi talked about a phenomenon I had never heard of before and would never have expected – atrophy of the thymus as a consequence of certain diseases. Finally, I was on more familiar ground with the talk of Isabel Mérida about certain signalling pathways in T cell activation. The substance at the centre of her talk, diacylglycerol kinase, was not familiar to me but the context was. Right at the end of the conference there was a general discussion session planned. This session, which was led by Ed Palmer, ended up being very short. This was due to the (in itself positive) fact that the discussions after (and during) the individual talks had taken up more time than planned. The final discussion was interesting despite its brevity. The basic theme was: if mathematicians are collaborating with immunologists what can each side do to help the other in this process? Interesting points were brought up and we were all sent home with some things to think about.

## Conference on systems biology of T cells in Baeza

October 22, 2012

At the moment I am attending a conference on systems biology of T cells in Baeza. Of the eleven talks today the first nine made no mention of mathematics – there was not a single equation. The tenth, by Zvi Grossmann, did show a couple. Thus the bias today was very much towards experimental immunology. It was interesting for me to be immersed in this atmosphere and I learned a lot of things. There are three things which stick in my mind particularly. The first is the fact, mentioned in the talk of Bruno Kyewski, that antigens mimicking all tissues of the body are presented by medullary epithelial cells in the thymus. This allows future T cells to learn about all self antigens. I asked him afterwards if this includes tissues which are in the immunologically privileged sites, usually poorly accessible to the immune system, like the central nervous system. He confirmed that this is the case. The second is the fact, which came up in the talk of Marisa Torio, that T cell precursors in the thymus have the potential to develop into almost any type of white blood cell. This means that the fate of a cell to become a T cell is in general not decided before it reaches the thymus, the answer to a question I had often asked myself. The third is the description in the talk of Alfred Singer of the way in which it is decided which of the surface molecules CD4 or CD8 a T cell carries. I had already watched a video by Singer on this subject on the NIH web page but one thing I was not aware of was the fact that by binding the protein Lck it is possible for CD4 and CD8 to interfere with T cell signalling. Lck is sequestered and hence is not available for use by the T cell receptor.

Grossmann’s talk was mainly concerned with rather abstract ideas about cell signalling and it was hard for me to get to grips with them. I had the impression that the right mathematical context for these things should be control theory. The last and only really mathematical talk of the day, by Rob de Boer, was a highlight for me and not only for me. At dinner the air was buzzing with conversations on the subject. The talk was on monitoring the dynamics of immune cells by labelling with deuterium and drawing conclusions about their lifetimes. I had heard a talk on a similar subject by de Boer before at a conference in Dresden and I wrote about it briefly in a previous post. I liked that earlier talk but I liked the talk today much more. This was probably less due to the difference in content as to the fact that for whatever reason I now appreciated the significance of this work much better. This is an example where a mathematical model can be used to obtain information about processes in immunology which it is difficult or impossible to obtain in any other way. It is not that the mathematics is complicated, just some explicitly solvable linear ODE. The impressive thing is the direct contact this work makes with real biological questions like ‘how long does a memory T cell live’. Analysing different experiments both using deuterium in human subjects and other more poisonous substances which can only be used in mice originally gave inconsistent answers for lifetimes. With hindsight this arose from the assumption in the models of just one population of cells with a definite death rate. Passing to a model with two classes of cells largely removed the discrepancy. There was another interesting aspect of this lecture and its reception which explains its prevalence at dinner. It has to do with communication between different fields, in this case mathematics and biology. There was a lot of confusion among the audience which was due not to the factual content of the work but to the way the results were described and to the choice of language in describing the results. I should remember for the future that it is not enough to get an interesting result in mathematical biology. It is also necessary to be very careful about formulating it in the right way so as to make its meaning transparent for biologists.

## A simple system with two different timescales

October 1, 2012

While struggling with the proof of a result for a model for photosynthesis (which I intend to report on in more detail at a later date) I decided to apply the following principle which I heard about as an undergraduate and is apparently due to Paul Halmos: if there is a mathematical problem you cannot solve then there is also a simpler problem you cannot solve. With this in mind I developed a model problem for what I wanted to do which I still could not solve. In the meantime, with the help of some key input from Juan Velázquez, I can solve the model problem and I will report on that here. It seems that the right conceptual framework for this is that of systems with two different timescales. Consider two functions $x(t)$ and $y(t)$ which satisfy $\dot x=x$ and $\dot y=x-y^5$. The fact that the power here is exactly five is not so important. This was the power that came up in the problem I was originally interested in and I wanted to rule out any misleading simplifications which might have arisen by replacing it by the power two, for instance. The first equation can be solved explicitly in the form $x(t)=\alpha e^t$.This reduces the original system to the scalar, but non-autonomous, equation $\dot y=\alpha e^t-y^5$. The question of interest is whether this equation has solutions which tend to infinity as $t\to\infty$ and if so how many of these are there. A guess at the asymptotics of a solution of this kind which is formally consistent is $y=\alpha^{\frac15}e^{\frac15 t}+\ldots$. It is possible to take this further by looking for formal power series solutions where $y-\alpha^{\frac15}e^{\frac15 t}$ is a linear combination of integer powers of $e^{\frac15 t}$. It turns out that there is a solution of this type in the sense of formal power series. In other words, substituting the expression in the equations and comparing coefficients gives a consistent answer. The coefficients are determined uniquely. This means that if there is more than one solution with this type of asymptotics these coefficients cannot distinguish between them and hence cannot be used to parametrize them. In fact there is a one-parameter family of solutions having this type of asymptotics and the difference between any two of these is of order $\exp [-Ce^{\frac45 t}]$ for a constant $C$. This means that while these solutions decay on a timescale $t$ the difference between them decays on a timescale which is exponentially faster. I am reminded of the term ‘asymptotics beyond all orders’ which I have heard occasionally but I do not know exactly what that means.

How can these results be proved? First introduce a quantity $w=1-\frac{y^5}{x}$ since this can be expected to decay very fast. The evolution equation for $y$ can be rewritten as an evolution equation for $w$ of the form $\dot w+5\alpha^{\frac45}e^{\frac45 t}w=q$, where the quantity $q$ will eventually be small. This equation can be solved by variation of constants to give an integral equation for $w$. It contains a parameter $\eta_0$ which distinguishes the different solutions. The problem of solving the integral equation can be reformulated as a fixed point problem for a mapping $\Phi$. The key step is to show that, for $\alpha$ sufficiently large and $\eta_0$ sufficiently small, $\Phi$ maps a suitable set of functions with a certain decay property to itself. The fixed point can then be obtained using the Arzela-Ascoli theorem. The simple system considered here can presumably be treated by simpler methods. My motivation for discussing it here that the technique of proof is of much wider applicability and the conclusions obtained are a model for other problems.

## 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.