## Hepatitis C

May 29, 2016

I once previously wrote something about hepatitis C in this blog which was directed to the mathematical modelling aspects. Here I want to write about the disease itself. This has been stimulated by talks I heard at a meeting of the Mainzer Medizinische Gesellschaft. The speakers were Ralf Bartenschlager from Heidelberg and and Stefan Zeuzem from Frankfurt. The first speaker is a molecular biologist who has made important contributions to the understanding of the structure and life cycle of the virus. For this work he got the 2015 Robert Koch prize together with Charles Rice from the Rockefeller University. The second speaker is a clinician.

Hepatitis C is transmitted by blood to blood contact. According to Zeuzem the main cause of the spread of this disease in developed countries is intravenous drug use. Before there was a test for the disease it was also spread via blood transfusions. (At one time the risk of infection with hepatitis due to a blood transfusion was 30%. This was mainly hepatitis B and by the time of discovery of hepatitis C, when the risk from hepatitis B had essentially been eliminated, it had dropped to 5%.) He also mentioned that there is a very high rate of infection in certain parts of Egypt due to the use of unsterilized needles in the treatment of other diseases. Someone asked how the disease could have survived before there were injections. He did not give a definitive answer but he did mention that while heterosexual contacts generally carry little risk of infection with this virus homosexual contacts between men do carry a significant risk. The disease typically becomes chronic and has few if any symptoms for many years. It does have dramatic long-term effects, namely cirrhosis and cancer of the liver. He showed statistics illustrating how public health policies have influenced the spread of the disease in different countries. The development in France has been much more favourable (with less cases) than in Germany, apparently due to a publicity campaign as a result of political motives with no direct relevance to the disease. The development in the UK has been much less favourable than it has even in Germany due an almost complete lack of publicity on the theme for a long time. The estimated number of people infected in Germany is 500000. The global number is estimated as 170 million.

There has been a dramatic improvement in the treatment of hepatitis C in the past couple of years and this was the central theme of the talks. A few years ago the situation was as follows. Drugs (a combination of ribavirin and interferon $\alpha$) could be used to eliminate the virus in a significant percentage of patients, particularly for some of the sub-types of the virus. The treatment lasted about a year and was accompanied by side effects that were so severe that there was a serious risk of patients breaking it off. Now the treatment only lasts a few weeks, it cures at least 95% of the patients and in many situations 99% of them. The side effects of the new treatments are moderate. There is just one problem remaining: the drugs for the best available treatment are sold for extremely high prices. The order of magnitude is 100000 euros for a treatment. Zeuzem explained various aspects of the dynamics which has led to these prices and the circumstances under which they might be reduced in the future. In general this gave a rather depressing picture of the politics of health care relating to the approval and prescription of new drugs.

Let me get back to the scientific aspects of the theme, as explained by Bartenschlager. A obvious question to ask is: if hepatitis C can essentially be cured why does HIV remain essentially incurable despite the huge amount of effort and money spent on trying to find a treatment? The simple answer seems to be that HIV can hide while HCV cannot. Both these viruses have an RNA genome. Since the copying of RNA is relatively imprecise they both have a high mutation rate. This leads to a high potential for the development of drug resistance. This problem has nevertheless been overcome for HCV. Virus particles are continually being destroyed by the immune system and for the population to survive new virus particles must be produced in huge numbers. This is done by the liver cells. This heavy burden kills the liver cells after a while but the liver is capable of regenerating, i.e, replacing these cells. The liver has an impressive capability to survive this attack but every system has its limits and eventually, after twenty or thirty years, the long-term effects already mentioned develop. An essential difference between HIV and HCV is that the RNA of HCV can be directly read by ribosomes to produce viral proteins. By contrast, the RNA of HIV is used as a template to produce DNA by the enzyme reverse transcriptase and this DNA is integrated into the DNA of the cell. This integrated DNA (known as the provirus) may remain inactive, not leading to production of protein. As long as this is the case the virus is invisible to the immune system. This is one way the virus can hide. Moreover the cell can divide producing new cells also containing the provirus. There is also another problem. The main target of HIV are the T-helper cells. However the virus can also infect other cells such as macrophages or dendritic cells and the behaviour of the virus in these other cells is different from that in T-helper cells. It is natural that a treatment should be optimized for what happens in the typical host cell and this may be much less effective in the other cell types. This means that the other cells may serve as a reservoir for the virus in situations where the population is under heavy pressure from the immune system or drug treatment. This is a second sense in which the virus can hide.

Some of the recent drugs used to treat HCV are based on ideas developed for the treatment of HIV. For instance a drug of this kind may inhibit certain of the enzymes required for the reproduction of the virus. There is one highly effective drug in the case of HCV which works in a different way. The hepatitis C virus produces one protein which has no enzymatic activity and it is at first sight hard to see what use this could be for the virus. What it in fact does is to act as a kind of docking station which organizes proteins belonging to the cell into a factory for virus production.

The hepatitis C virus is a valuable example which illustrates the relations between various aspects of medical progress: improvement in scientific understanding, exploitation of that information for drug design, political problems encountered in getting an effective drug to the patients who need it. Despite the negative features which have been mentioned it is the subject of a remarkable success story.

## Flying to Copenhagen without a carpet

May 11, 2016

This semester I have a sabbatical and I am profiting from it by travelling more than I usually do. At the moment I am visiting the group of Carsten Wiuf and Elisenda Feliu at the University of Copenhagen for two weeks. The visit here also gives me the opportunity to discuss with people at the Niels Bohr Institute. Note that the authors of the paper I quoted in the post on NF$\kappa$B were at the NBI when they wrote it and in particular Mogens Jensen is still there now. I gave a talk on some of my work on the Calvin cycle at NBI today. Afterwards I talked to Mogens and one of his collaborators and found out that he is still very active in modelling this system.

I was thinking about my previous visits to Copenhagen and, in particular, that the first one was on a flying carpet. The background to this is that when I was seven years old I wrote a story in school with the title ‘The Magic Carpet’. I do not have the text any more but I know it appeared in the School Magazine that year. In my own version there was also a picture which I will say more about later. But first something about the story, of which I was the hero. I bought the carpet in Peshawar and used it to visit places in the world I was interested in. For some reason I no longer know I had a great wish at that time to visit Copenhagen. Perhaps it was due to coming into contact with stories of Hans Christian Andersen. In any case it is clear that having the chance this was one of the first places I visited using the magic carpet. The picture which I drew showed something closer to home. There I can be seen sitting on the carpet, wearing the blue jersey which was my favourite at that time, while the carpet bent upwards so as to just pass over the tip of the spire of St. Magnus Cathedral in Kirkwall. In the story it was also related that one of the effects of my journey was a newspaper article reporting a case of ‘mass hallucination’. I think my teachers were impressed at my using this phrase at my age. They might have been less impressed if they had known my source for this, which was a Bugs Bunny cartoon.

During my next visit to Copenhagen in 2008 (here I am not counting changing planes there on the way to Stockholm, which I did a few times) I was at a conference at the Niels Bohr Institute in my old research field of mathematical relativity and I gave a talk in that area. Little did I think I would return there years later and talk about something completely different. I remember that there was a photo in the main lecture room where many of the founders of quantum mechanics are sitting in the first row. From my own point of view I am happy that another person who can be seen there is Max Delbrück, a shining example of a switch from physics to biology. My next visit to Copenhagen was for the conference which I wrote about in a previous post. It was at the University. Since that a lot has happened with chemical reaction network theory and with my understanding of it. The lecture course I gave means that some of the points I mentioned in my post at that time are things I have since come to understand in some depth. I look forward to working on projects in that area with people here in the coming days.

## NFκB

May 1, 2016

NF$\kappa$B is a transcription factor, i.e. a protein which can bind to DNA and cause a particular gene to be read more or less often. This means that more or less of a certain protein is produced and this changes the behaviour of the cell. The full name of this transcription factor is ‘nuclear factor, $\kappa$-light chain enhancer of B cells’. The term ‘nuclear factor’ is clear. The substance is a transcription factor and to bind to DNA it has to enter the nucleus. NF$\kappa$B is found in a wide variety of different cells and its association with B cells is purely historical. It was found in the lab of David Baltimore during studies of the way in which B cells are activated. It remains to explain the $\kappa$. B cells produce antibodies each of which consists of two symmetrical halves. Each half consists of a light and a heavy chain. The light chain comes in two variants called $\kappa$ and $\lambda$. The choice which of these a cell uses seems to be fairly random. The work in the Baltimore lab had found out that NF$\kappa$B could skew the ratio. I found a video by Baltimore from 2001 about NF$\kappa$B. This is probably quite out of date by now but it contained one thing which I found interesting. Under certain circumstances it can happen that a constant stimulus causing activation of NF$\kappa$B leads to oscillations in the concentration. In the video the speaker mentions ‘odd oscillations’ and comments ‘but that’s for mathematicians to enjoy themselves’. It seems that he did not believe these oscillations to be biologically important. There are reasons to believe that they might be important and I will try to explain why. At the very least it will allow me to enjoy myself.

Let me explain the usual story about how NF$\kappa$B is activated. There are lots of animated videos on Youtube illustrating this but I prefer a description in words. Normally NF$\kappa$B is found in the cytosol bound to an inhibitor I$\kappa$B. Under certain circumstances a complex of proteins called IKK forms. The last K stands for kinase and IKK phosphorylates I$\kappa$B. This causes I$\kappa$B to be ubiquinated and thus marked for degradation (cf. the discussion of ubiquitin here). When it has been destroyed NF$\kappa$B is liberated, moves to the nucleus and binds to DNA. What are the circumstances mentioned above? There are many alternatives. For instance TNF$\alpha$ binds to its receptor, or something stimulates a toll-like receptor. The details are not important here. What is important is that there are many different signals which can lead to the activation of NF$\kappa$B. What genes does NF$\kappa$B bind to when it is activated? Here again there are many possibilities. Thus there is a kind of bow tie configuration where there are many inputs and many outputs which are connected to a single channel of communication. So how is it possible to arrange that when one input is applied, e.g. TNF$\alpha$ the right genes are switched on while another input activates other genes through the same mediator NF$\kappa$B? One possibility is cross-talk, i.e. that this signalling pathway interacts with others. If this cannot account for all the specificity then the remaining possibility is that information is encoded in the signal passing through NF$\kappa$B itself. For example, one stimulus could produce a constant response while another causes an oscillatory one. Or two stimuli could cause oscillatory responses with different frequencies. Evidently the presence of oscillations in the concentration of NF$\kappa$B presents an opportunity for encoding more information than would otherwise be possible. To what extent this really happens is something where I do not have an overview at the moment. I want to learn more. In any case, oscillations have been observed in the NF$\kappa$B system. The primary thing which has been observed to oscillate is the concentration of NF$\kappa$B in the nucleus. This oscillation is a consequence of the movement of the protein between the cytosol and the nucleus. There are various mathematical models for describing these oscillations. As usual in modelling phenomena in cell biology there are models which are very big and complicated. I find it particularly interesting when some of the observations can be explained by a simple model. This is the case for NF$\kappa$B where a three-dimensional model and an explanation of its relations to the more complicated models can be found in a paper by Krishna, Jensen and Sneppen (PNAS 103, 10840). In the three-dimensional model the unknowns are the concentrations of NF$\kappa$B in the nucleus, I$\kappa$B in the cytoplasm and mRNA coding for I$\kappa$B. The oscillations in normal cells are damped but sustained oscillations can be seen in mutated cells or corresponding models.

What is the function of NF$\kappa$B? The short answer is that it has many. On a broad level of description it plays a central role in the phenomenon of inflammation. In particular it leads to production of the cytokine IL-17 which in turn, among other things, stimulates the production of anti-microbial peptides. When these things are absent it leads to a serious immunodeficiency. In one variant of this there is a mutation in the gene coding for NEMO, which is one of the proteins making up IKK. A complete absence of NEMO is fatal before birth but people with a less severe mutation in the gene do occur. There are symptoms due to things which took place during the development of the embryo and also immunological problems, such as the inability to deal with certain bacteria. The gene for NEMO is on the X chromosome so that this disease is usually limited to boys. More details can be found in the book of Geha and Notarangelo mentioned in  a previous post.

## Stability of steady states in models of the Calvin cycle

April 25, 2016

I have just written a paper with Stefan Disselnkötter on stationary solutions of models for the Calvin cycle and their stability. There we concentrate on the simplest models for this biological system. There were already some analytical results available on the number of positive stationary solutions (let us call them steady states for short), with the result that this number is zero, one or two in various circumstances. We were able to extend these results, in particular showing that in a model of Zhu et. al. there can be two steady states or, in exceptional cases, a continuum of steady states. This is at first sight surprising since those authors stated that there is at most one steady state. However they impose the condition that the steady states should be ‘physiologically feasible’. In fact for their investigations, which are done by means of computer calculations, they assume among other things that certain Michaelis constants which occur as parameters in the system have specific numerical values. This assumption is biologically motivated but at the moment I do not understand how the numbers they give follow from the references they quote. In any case, if these values are assumed our work gives an analytical proof that there is at most one steady state.

While there are quite a lot of results in the literature on the number of steady states in systems of ODE modelling biochemical systems there is much less on the question of the stability of these steady states. It was a central motivation of our work to make some progress in this direction for the specific models of the Calvin cycle and to develop some ideas to approaching this type of question more generally. One key idea is that if it can be shown that there is bifurcation with a one-dimensional centre manifold this can be very helpful in getting information on the stability of steady states which arise in the bifurcation. Given enough information on a sufficient number of derivatives at the bifurcation point this is a standard fact. What is interesting and perhaps less well known is that it may be possible to get conclusions without having such detailed control. One type of situation occurring in our paper is one where a stable solution and a saddle arise. This is roughly the situation of a fold bifurcation but we do not prove that it is generic. Doing so would presumably involve heavy calculations.

The centre manifold calculation only controls one eigenvalue and the other important input in order to see that there is a stable steady state for at least some choice of the parameters is to prove that the remaining eigenvalues have negative real parts. This is done by considering a limiting case where the linearization simplifies and then choosing parameters close to those of the limiting case. The arguments in this paper show how wise it can be to work with the rates of the reactions as long as possible, without using species concentrations. This kind of approach is popular with many people – it has just taken me a long time to get the point.

January 23, 2016

Here I discuss another tool for analysing chemical reaction networks of deficiency greater than one. This is the Advanced Deficiency Algorithm developed by Feinberg and Ellison. It seems that the only direct reference for the mathematical proofs is Ellison’s PhD thesis. There is a later PhD thesis by Haixia Ji in which she introduces an extension of this called the Higher Deficiency Algorithm and where some of the ideas of Ellison are also recapitulated. In my lecture course, which ends next week, I will only have time to discuss the structure of the algorithm and give an extended example without proving much.

The Advanced Deficiency Algorithm has a general structure which is similar to that of the Deficiency One Algorithm. In some cases it can rule out multistationarity. Otherwise it gives rise to several sets of inequalities. If one of these has a solution then there is multistationarity and if none of them does there is no multistationarity. It is not clear to me if this is really an algorithm which is guaranteed to give a diagostic test in all cases. I think that this is probably not the case and that one of the themes of Ji’s thesis is trying to improve on this. An important feature of this algorithm is that the inequalities it produces are in general nonlinear and thus may be much more difficult to analyse than the linear inequalities obtained in the case of the Deficiency One Algorithm.

Now I have come to the end of my survey of deficiency theory for chemical reaction networks. I feel I have learned a lot and now is the time to profit from that by applying these techniques. The obvious next step is to try out the techniques on some of my favourite biological examples. Even if the result is only that I see why the techniques do not give anything interesting in this cases it will be useful to understand why. Of course I hope that I will also find some positive results.

## Elementary flux modes

January 16, 2016

Elementary flux modes are a tool which can be used to take a reaction network of deficiency greater than one and produce subnetworks of deficiency one. If it can be shown that one of these subnetworks admits multistationarity then it can sometimes be concluded that the original network also does so. There are two conditions that need to be checked in order to allow this conclusion. The first is that the network under consideration must satisfy the condition $t=l$. The second is that genericity conditions must be satisfied which consist of requiring the non-vanishing of the determinants of certain matrices. If none of the subnetworks admit multistationarity then no conclusion is obtained for the full network. The core element of the proof is an application of the implicit function theorem. These conclusions are contained in a paper of Conradi et. al. (Proc. Nat. Acad. Sci. USA 104, 19175).

One of the ways of writing the condition for stationary solutions is $Nv(c)=0$, where as usual $N$ is the stoichiometric matrix and $v(c)$ is the vector of reaction rates. Since $v(c)$ is positive this means that we are looking for a positive element of the kernel of $N$. This suggests that it is interesting to look at the cone $K$ which is the intersection of the kernel of $N$ with the non-negative orthant. According to a general theorem of convex analysis $K$ consists of the linear combinations with non-negative coefficients of a finite set of vectors which have a mininum (non-zero) number of non-zero components. In the case of reaction networks these are the elementary flux modes. Recalling that $N=YI_a$ we see that positive vectors in the kernel of the incidence matrix $I_a$ are a special type of elementary flux modes. Those which are not in the kernel of $I_a$ are called stoichiometric generators. Each stoichiometric generator defines a subnetwork where those reaction constants of the full network are set to zero where the corresponding reactions are not in the support of the generator. It is these subnetworks which are the ones mentioned above in the context of multistationarity. The application of the implicit function theorem involves using a linear transformation to introduce adapted coordinates. Roughly speaking the new coordinates are of three types. The first are conserved quantities for the full network. The second are additional conserved quantities for the subnetwork, complementing those of the full network. Finally the third type represents quantities which are dynamical even for the subnetwork.

Here are some simple examples. In the extended Michaelis-Menten description of a single reaction there is just one elementary flux mode (up to multiplication by a positive constant) and it is not a stoichiometric generator. In the case of the simple futile cycle there are three elementary flux modes. Two of these, which are not stoichoimetric generators correspond to the binding and unbinding of one of the enzymes with its substrate. The third is a stoichoimetric generator and the associated subnetwork is obtained by removing the reactions where a substrate-enzyme complex dissociates into its original components. The dual futile cycle has four elementary flux modes of which two are stoichiometric generators. In the latter case we get the (stoichiometric generators of the) two simple futile cycles contained in the network. Of course these are not helpful for proving multistationarity. Another type of example is given by the simple models for the Calvin cycle which I have discussed elsewhere. The MA system considered there has two stoichiometric generators with components $(3,6,6,1,3,0,1)$ and $(3,5,5,1,3,1,0)$. I got these by working backwards from corresponding modes for the MM-MA system found by Grimbs et. al. This is purely on the level of a vague analogy. I wish I had a better understanding of how to get this type of answer more directly. Those authors used those modes to prove bistability for the MM-MA system so that this is an example where this machinery produces real results.

## The German Dr. House

January 14, 2016

The central figure in the American TV series Dr. House is a doctor who is brilliant in the diagnosis of unusual medical conditions but personally very difficult. When I first saw this series I found the character so unpleasant that I did not want to watch the programme. However in the course of time I got drawn in to watching it by the interest of the medical content. While some aspects of this series are quite exaggerated and far from reality the medical parts are very accurate and well researched. As I learned yesterday even details seen there like the numbers on heart monitors accurately reflect the situation being portrayed. I have this information from a lecture I attended yesterday at the Medizinische Gesellschaft Mainz [Mainz Medical Society]. The speaker was Professor Jürgen Schäfer, a man who has become known in the media as the German Dr. House. I am pleased to report that I detected no trace of the social incompetence of Dr. House in Dr. Schäfer.

Jürgen Schäfer is trained as a cardiologist. He and his wife, who is a gastroenterologist, got so interested by the series Dr. House that they would spend time discussing the details of the diagnoses and researching the background after they has seen each programme. Then Schäfer had the idea that he could use Dr. House in his lectures at the University of Marburg. The first obstacle was to know if he could legally make use of this material. After a casual conversation with one of his patients who is a lawyer he contacted the necessary people and signed a suitable contract. At this time his project attracted considerable attention in the media even before it had started. In the lectures he analyses the cases occurring in the series. The students are encouraged to develop their own diagnoses in dialogue with the professor. These lectures are held in the evenings and are very popular with the students. In the evaluations the highest score was obtained for the statement that ‘the lectures are a lot of fun’.

This is only the start of the story. During a consultation in one of the episodes of Dr. House he suddenly makes a deep cut with a scalpel in the body of the patient (one of the melodramatic elements), opens the wound and shows that the flesh inside is black. The diagnosis is cobalt poisoning. After seeing this it occurred to Dr. Schäfer that this diagnosis might also apply to one of his own patients and this turned out to be true. In addition to serious heart problems this patient was becoming blind and deaf. He had had a hip joint replacement with an implant made of a ceramic material. At some point this became damaged and was replaced. In order to try to avoid the implant breaking again the new one was made of metal. The old implant fragmented and left splitters in the body. These had acted like sandpaper on the new joint and at the time of removal it had been reduced to 70% of its original size by this process. As a result large quantities of cobalt was released, resulting in the poisoning. The speaker showed a picture of the operation of another of his patients with a similar problem where the wound could be seen to be filled with a black oily liquid. Together with colleagues Schäfer published an account of this case in The Lancet with the title ‘Cobalt intoxication diagnosed with the help of Dr. House’. Not all his coauthors were happy with this title but Schäfer wanted to acknowledge his debt to the series. At the same time it was a great piece of advertizing for him which lead to a lot of attention in the international media.

Due to his growing fame Schäfer started to get a lot of letters from patients with mysterious illnesses. This was more than he could handle. He informed the administration of the university clinic where he worked that he was going to start sending back letters of this type unopened, since he just did not have the time to cope with them. To his surprise they wanted him to continue with this work and arranged from him to be relieved from other duties. They set up a new institute for him called Zentrum für unerkannte Krankheiten [centre for unrecognized diseases]. This was perhaps particularly surprising since this is a privately funded clinic and the work of this institute costs money rather than making money. The techniques used there include toxicological and genomic analyses.

Here is another example from the lecture. Schäfer’s institute uses large scale DNA analysis to screen for a broad range of parasites in patients with unclear symptoms. In one patient they found DNA of the parasite causing schistosomiasis. This disease is usually got by bathing in infected water in tropical or subtropical areas. The patient tested negatively for the parasite and had never been to a place where this disease occurs. The mystery was cleared up due to the help of a vet of Egyptian origin. He was familiar with schistosomiasis and due to his experience with large animals he was not afraid of analysing very large stool samples. He succeeded in finding eggs of the parasite in the patient’s stool. The diffculty was that the numbers of eggs were very low and that for certain reasons they were difficult to recognise in this case, except by an expert. The patient was treated for schistosomiasis as soon as the genetic results were available but it was very satisfying to have a confirmation by more classical techniques. The mystery of how the patient got infected was solved as follows. As a hobby he kept lots of fish and he imported these from tropical regions. The infection presumably came from the water in his aquarium. We see that in the modern world it is easy to import tropical diseases by express delivery after placing an order in the internet

I do not want to end before mentioning that Schäfer said something nice about how mathematicians can help medical doctors. He had a patient who is a mathematics professor and had the following problem. From time to time he would collapse and was temporarily paralysed although fully conscious. A possible explanation for this would have been an excessively high level of sodium in the body. On the other hand measurements showed that the concentration of sodium in his blood was normal, even after an attack. The patient then did a calculation (just simple arithmetic). On the basis of known data he worked out the amount of sodium and potassium in different types of food and noted a correlation between negative effects of a food on his health and the ratio of the sodium to potassium concentrations. This supported the hypothesis of sodium as a cause and encouraged the doctors to look more deeply into the matter. It turned out that in this type of disease the sodium is concentrated near the cell membrane and cannot be seen in the blood. A genetic analysis revealed that the patient had a mutation in a little-known sodium channel.

I think that this lecture was very entertaining for the audience, including my wife and myself. However this is not just entertainment. With his institute Schäfer is providing essential help for many people in very difficult situations. He has files of over 4000 patients. This kind of work requires a high investment in time and money which is not possible for a usual university clinic, not to mention an ordinary GP. It is nevertheless the case that Schäfer is developing resources which could be used more widely, such as standard protocols for assessing patients of this type. As he emphasized, while by definition a rare disease only effects a small number of patients the collection of all rare diseases together affects a large number of people. If more money was invested in this kind of research it could result in  a net saving for the health system since it would reduce the number of people running from one doctor to another since they do not have a diagnosis.

## The deficiency one algorithm

January 7, 2016

Here I continue the discussion of Chemical Reaction Network Theory with the Deficiency One Algorithm. As its name suggests this is an algorithm for investigating whether a reaction network allows multistationarity or not. It is also associated to a theorem. This theorem says that the satisfaction of certain inequalities associated to a reaction network is equivalent to the existence of a choice of reaction constants for which the corresponding system of equations with mass action kinetics has two distinct positive stationary solutions in a stoichiometric compatibility class. This is true in the context of networks of deficiency one which satisfy some extra conditions. The property of satisfying these, called regularity, seems to be relatively weak. In fact, as in the case of the Deficiency One Theorem, there is a second related result where a pair of zeroes of the right hand side $f$ in a stoichiometric compatibility class is replaced by a vector in the kernel of $Df$ which is tangent to that class.

An example where this theory can be applied is double phosphorylation in the processive case. The paper of Conradi et. al. cited in the last post contains the statement that in this system the Deficiency One Algorithm implies that multistationarity is not possible. For this it refers to the Chemical Reaction Network Toolbox, a software package which implements the calculations of the theorem. In my course I decided for illustrative purposes to carry out these calculations by hand. It turned out not to be very hard. The conclusion is that multistationarity is not possible for this system but the general machinery does not address the question of whether there are any positive stationary solutions. I showed that this is the case by checking by hand that $\omega$-limit points on the boundary of the positive orthant are impossible. The conclusion then follows by a well-known argument based on the Brouwer fixed point theorem. This little bit of practical experience with the Deficiency One Algorithm gave me the impression that it is really a powerful tool. At this point it is interesting to note a cross connection to another subject which I have discussed in this blog. It is a model for the Calvin cycle introduced by Grimbs et. al. These authors noted that the Deficiency One Algorithm can be applied to this system to show that it does not allow multistationarity. They do not present the explicit calculations but I found that they are not difficult to do. In this case the equations for stationary solutions can be solved explicitly so that using this tool is a bit of an overkill. It neverless shows the technique at work in another example.

Regularity consists of three conditions. The first (R1) is a necessary condition that there be any positive solutions at all. If it fails we get a strong conclusion. The second (R2) is the condition $l=t$ familiar from the Deficiency One Theorem. (R3) is a purely graph theoretic condition on the network. A weakly reversible network which satisfies (R3) is reversible. Reading this backwards, if a network is weakly reversible but not reversible then the theorem does not apply. The inequalities in the theorem depend on certain partitions of a certain set of complexes (the reactive complexes) into three subsets. What is important is whether the inequalities hold for all partitions of a network or whether there is at least one partition for which they do not hold. The proof of the theorem uses a special basis of the kernel of $YA_\kappa$ where $\kappa$ is a function on $\cal C$ constructed from the reaction constants. In the context of the theorem this space has dimension $t+1$ and $t$ of the basis vectors come from a special basis of $A_\kappa$ of the type which already comes up in the proof of the Deficiency Zero Theorem.

An obvious restriction on the applicability of the Deficiency One Algorithm is that it is only applicable to networks of deficiency one. What can be done with networks of higher deficiency? One alternative is the Advanced Deficiency Algorithm, which is implemented in the Chemical Reaction Network Toolbox. A complaint about this method which I have seen several times in the literature is that it is not able to treat large systems – apparently the algorithm becomes unmanageable. Another alternative uses the notion of elementary flux modes which is the next topic I will cover in my course. It is a way of producing certain subnetworks of deficiency one from a given network of higher deficiency. The subnetworks satisfy all the conditions needed to apply the Deficiency One Algorithm except perhaps $t=l$.

## The deficiency one theorem

December 2, 2015

Here I continue with the discussion of chemical reaction network theory begun in the previous post. After having presented a proof of the Deficiency Zero Theorem in my course I proceeded to the Deficiency One Theorem following the paper of Feinberg in Arch. Rat. Mech. Anal. 132, 311. If we have a reaction network we can consider each linkage class as a network in its own right and thus we can define its deficiency. If we denote the deficiency of the full network by $\delta$ and the deficiency of the $i$th linkage class by $\delta_i$ then in general $\delta\ge\sum_i \delta_i$. The first hypothesis of the Deficiency One Theorem is that the deficiencies of the linkage classes are no greater than one. The second is that equality holds in the inequality relating the deficiency of the network to those of its linkage classes. The stoichiometric subspace $S$ of the full network is the sum of the corresponding spaces $S_i$ for the linkage classes. The sum is direct precisely when equality holds in the inequality for the deficiencies. The third condition is that there is precisely one terminal strong linkage class in each linkage class ($t=l$). The first conclusion is that if there exists a positive stationary solution there is precisely one stationary solution in each stoichiometric compatibility class. The second is that if the network is weakly reversible there exists a positive stationary solution. Thus two of the conclusions of the Deficiency Zero Theorem have direct analogues in this case. Others do not. There is no statement about the stability of the stationary solutions. Networks which are not weakly reversible but satisfy the three conditions of the Deficiency One Theorem may have positive stationary solutions. In the paper of Feinberg the proof of the Deficiency One Theorem is intertwined with that of another result. It says that the linearization about a stationary solution of the restriction of the system to a stoichiometric compatibility class has trivial kernel. A related result proved in the same paper says that for a weakly reversible network of deficiency zero each stationary solution is a hyperbolic sink within its stoichiometric class. Statements of this type ensure that these stationary solutions possess a certain structural stability.

In the proof of the Deficiency One Theorem the first step (Step 1) is to show that when the assumptions of the theorem are satisfied and there is positive stationary solution $c^*$ then the set of stationary solutions is equal to the set of points for which $\log c-\log c^*$ lies in the orthogonal complement of the stoichiometric subspace. From this the conclusion that there is exactly one stationary solution in each stoichiometric compatibility class follows just as in the proof of the Deficiency Zero Theorem (Step 2). To complete the proof it then suffices to prove the existence of a positive stationary solution in the weakly reversible case (Step 3). In Step 1 the proof is reduced to the case where the network has only one linkage class by regarding the linkage classes of the original network as networks in their own right. In this context the concept of a partition of a network $(\cal S,\cal C,\cal R)$ is introduced. This is a set of subnetworks $({\cal S},{\cal C}^i,{\cal R}^i)$. The set of species is unchanged. The set of reactions $\cal R$ is a disjoint union of the ${\cal R}^i)$ and ${\cal C}^i$ is the set of complexes occurring in the reactions contained in ${\cal R}^i)$. The partition is called direct if the stoichiometric subspace of the full network is the direct sum of those of the subnetworks. The most important example of a partition of a network in the present context is that given by the linkage classes of any network. That it is direct is the second condition of the Deficiency One Theorem. The other part of Step 1 of the proof is to show that the statement holds in the case that there is only one linkage class. The deficiency is then either zero or one. Since the case of deficiency zero is already taken care of by the Deficiency Zero Theorem we can concentrate on the case where $\delta=1$. Then the dimension of ${\rm ker} A_k$ is one and that of ${\rm ker} (YA_k)$ is two. The rest of Step 1 consists of a somewhat intricate algebraic calculation in this two-dimensional space. It remains to discuss Step 3. In this step the partition given by the linkage classes is again used to reduce the problem to the case where there is only one linkage class. The weak reversibility is preserved by this reduction. Again we can assume without loss of generality that $\delta=1$. The subspace $U={\rm im} Y^T+{\rm span}\omega_{\cal C}$ is a hyperplane in $F({\cal C})$. We define $\Gamma$ to be the set of functions of the form $\log a$ with $a$ a positive element of ${\rm ker} (YA_k)$. The desired stationary solution is obtained as a point of the intersection of $U$ with $\Gamma$. To show that this intersection is non-empty it is proved that there are points of $\Gamma$ on both sides of $U$. This is done by a careful examination of the cone of positive elements of ${\rm ker} (YA_k)$.

To my knowledge the number of uses of the Deficiency Zero Theorem and the Deficiency One Theorem in problems coming from applications is small. If anyone reading this has other information I would like to hear it. I will now list the relevant examples I know. The Deficiency Zero Theorem was applied by Sontag to prove asymptotic stability for McKeithan’s kinetic proofreading model of T cell activation. I applied it to the model of Salazar and Höfer for the dynamics of the transcription factor NFAT. Another potential application would be the multiple futile cycle. This network is not weakly reversible. The simple futile cycle has deficiency one and the dual futile cycle deficiency two. The linkage classes have deficiency zero in both cases. Thus while the first and third conditions of the Deficiency One Theorem are satisfied the second is not. Replacing the distributive phosphorylation in the multiple futile cycle by processive phosphorylation may simplify things. This has been discussed by Conradi et. al., IEE Proc. Systems Biol. 152, 243. In the case of two phosphorylation steps the system obtained is of deficiency one but the linkage classes have deficiency zero so that condition (ii) is still not satisfied. It seems that the Deficiency One Algorithm may be more helpful and that will be the next subject I cover in my course.

## Feinberg’s proof of the deficiency zero theorem

November 23, 2015

I discussed the deficiency zero theorem of chemical reaction network theory (CRNT) in a previous post. (Some further comments on this can be found here and here.) This semester I am giving a lecture course on chemical reaction network theory. Lecture notes are growing with the course and can be found in German and English versions on the web page of the course. The English version can also be found here. Apart from introductory material the first main part of the course was a proof of the Deficiency Zero Theorem. There are different related proofs in the literature and I have followed the approach in the classic lecture notes of Feinberg on the subject closely. The proof in those notes is essentially self-contained apart from one major input from a paper of Feinberg and Horn (Arch. Rat. Mech. Anal. 66, 83). In this post I want to give a high-level overview of the proof.

The starting point of CRNT is a reaction network. It can be represented by a directed graph where the nodes are the complexes (left or right hand sides of reactions) and the directed edges correspond to the reactions themselves. The connected components of this graph are called the linkage classes of the network and their number is usually denoted by $l$. If two nodes can be connected by oriented paths in both directions they are said to be strongly equivalent. The corresponding equivalence classes are called strong linkage classes. A strong linkage class is called terminal if there is no directed edge leaving it. The number of terminal strong linkage classes is usually denoted by $t$. From the starting point of the network making the assumption of mass action kinetics allows a system of ODE $\dot c=f(c)$ to be obtained in an algorithmic way. The quantity $c(t)$ is a vector of concentrations as a function of time. Basic mathematical objects involved in the definition of the network are the set $\cal S$ of chemical species, the set $\cal C$ of complexes and the set $\cal R$ of reactions. An important role is also played by the vector spaces of real-valued functions on these finite sets which I will denote by $F({\cal S})$, $F({\cal C})$ and $F({\cal R})$, respectively. Using natural bases they can be identified with $R^m$, $R^n$ and $R^r$. The vector $c(t)$ is an element of $F({\cal S})$. The mapping $f$ from $F({\cal S})$ to itself can be written as a composition of three mappings, two of them linear, $f=YA_k\Psi$. Here $Y$, the complex matrix, is a linear mapping from $F({\cal C})$ to $F({\cal S})$. $A_k$ is a linear mapping from $F({\cal C})$ to itself. The subscript $k$ is there because this matrix is dependent on the reaction constants, which are typically denoted by $k$. It is also possible to write $f$ in the form $Nv$ where $v$ describes the reaction rates and $N$ is the stoichiometric matrix. The image of $N$ is called the stoichiometric subspace and its dimension, the rank of the network, is usually denoted by $s$. The additive cosets of the stoichiometric subspace are called stoichiometric compatibility classes and are clearly invariant under the time evolution. Finally, $\Psi$ is a nonlinear mapping from $F({\cal S})$ to $F({\cal C})$. The mapping $\Psi$ is a generalized polynomial mapping in the sense that its components are products of powers of the components of $c$. This means that $\log\Psi$ depends linearly on the logarithms of the components of $c$. The condition for a stationary solution can be written as $\Psi(c)\in {\rm ker} (YA_k)$. The image of $\Psi$ is got by exponentiating the image of a linear mapping. The matrix of this linear mapping in natural bases is $Y^T$. Thus in looking for stationary solutions we are interested in finding the intersection of the manifold which is the image of $\Psi$ with the kernel of $YA_k$. The simplest way to define the deficiency of the network is to declare it to be $\delta=n-l-s$. A fact which is not evident from this definition is that $\delta$ is always non-negative. In fact $\delta$ is the dimension of the vector space ${\rm ker} Y\cap {\rm span}(\Delta)$ where $\Delta$ is the set of complexes of the network. An alternative concept of deficiency, which can be found in lecture notes of Gunawardena, is the dimension $\delta'$ of the space ${\rm ker} Y\cap{\rm im} A_k$. Since this vector space is a subspace of the other we have the inequality $\delta'\le\delta$. The two spaces are equal precisely when each linkage class contains exactly one terminal strong linkage class. This is, in particular, true for weakly reversible networks. The distinction between the two definitions is often not mentioned since they are equal for most networks usually considered.

If $c^*$ is a stationary solution then $A_k\Psi (c^*)$ belongs to ${\rm ker} Y\cap{\rm im} A_k$. If $\delta'=0$ (and in particular if $\delta=0$) then this means that $A_k\Psi (c^*)=0$. In other words $\Psi (c^*)$ belongs to the kernel of $A_k$. Stationary solutions of this type are called complex balanced. It turns out that if $c^*$ is a complex balanced stationary solution the stationary solutions are precisely those points $c$ for which $\log c-\log c_*$ lies in the orthogonal complement of the stoichiometric subspace. It follows that whenever we have one solution we get a whole manifold of them of dimension $n-s$. It can be shown that each manifold of this type meets each stoichiometric class in precisely one point. This is proved using a variational argument and a little convex analysis.

It is clear from what has been said up to now that it is important to understand the positive elements of the kernel of $A_k$. This kernel has dimension $t$ and a basis each of whose elements is positive on a terminal strong linkage class and zero otherwise. Weak reversibility is equivalent to the condition that the union of the terminal strong linkage classes is the set of all complexes. It can be concluded that when the network is not weakly reversible there exists no positive element of the kernel of $A_k$. Thus for a network which is not weakly reversible and has deficiency zero there exist no positive stationary solutions. This is part of the Deficiency Zero Theorem. Now consider the weakly reversible case. There a key statement of the Deficiency Zero Theorem is that there exists a complex balanced stationary solution $c^*$. Where does this $c^*$ come from? We sum the vectors in the basis of ${\rm ker} A_k$ and due to weak reversibility this gives something which is positive. Then we take the logarithm of the result. When $\delta=0$ this can be represented as a sum of two contributions where one is of the form $Y^T z$. Then $c^*=e^z$. A further part of the deficiency zero theorem is that the stationary solution $c^*$ in the weakly reversible case is asymptotically stable. This is proved using the fact that for a complex balanced stationary solution the function $h(c)=\sum_{s\in\cal S}[c(s)(\log c(s)-\log c^*(s)-1)+c(s^*)]$ is a Lyapunov function which vanishes for $c=c^*$