## Encounter with an aardvark

August 21, 2019

When I was a schoolboy we did not have many books at home. As a result I spent a lot of time reading those which were available to me. One of them was a middle-sized dictionary. It is perhaps not surprising that I attached a special significance to the first word which was defined in that dictionary. At that time it was usual, and I see it as reasonable, that articles did not belong to the list of words which the dictionary was responsible for defining. For this reason ‘a’ was not the first word on the list and instead it was ‘aardvark’. From the dictionary I learned that an aardvark is an animal and roughly what kind of animal it is. I also learned something about its etymology (it was an etymological dictionary) and that it originates from Dutch words meaning ‘earth’ and ‘pig’. Later in life I saw pictures of aardvarks in books and saw them in TV programmes, but without paying special attention to them. The aardvark remained more of an intriguing abstraction for me than an animal.

Yesterday, in Saarbrücken zoo, I walked into a room and saw an aardvark in front of me. Suddenly the abstraction turned into a very concrete animal pacing methodically around its enclosure. I had a certain feeling of unreality. I do not know if aardvarks always walk like that or whether it was just a habit which this individual had acquired by being confined to a limited space. Each time it returned (reappearing after having disappeared into a region not visible to me) the impression of unreality was heightened. I was reminded of the films of dinosaurs which sometimes come on TV, where the computer-reconstructed movements of the animals look very unrealistic to me. Seeing the aardvark I asked myself, ‘if mankind only knew this animal from fossil remains would it ever have been possible to reconstruct the gait I now see before me?’

Another animal I encountered in the Saarbrücken zoo is a species whose existence I did not know of before. This is Pallas’s cat. This is a wild cat with a very unusual and engaging look. The name Pallas has a special meaning for me for the following reason. When I was young and a keen birdwatcher some of the birds which were most exiting for me were rare vagrants from Siberia which had been brought to Europe by unusual weather conditions. A number of these are named after Pallas. I knew almost nothing about the man Pallas. Now I have filled in some background. In particular I learned that he was a German born in Berlin who was sent on expeditions to Siberia by Catherine the Great.

## SMB meeting in Montreal

July 27, 2019

This week I have been attending the SMB meeting in Montreal. There was a minisymposium on reaction networks and I gave a talk there on my work with Elisenda Feliu and Carsten Wiuf on multistability in the multiple futile cycle. There were also other talks related to that system. A direction which was new to me and was discussed in a talk by Elizabeth Gross was using a sophisticated technique from algebraic geometry (the mixed volume) to obtain an upper bound on the number of complex solutions of the equations for steady states for a reaction network (which is then of course also an upper bound for the number of positive real solutions). There were two talks about the dynamics of complex balanced reaction networks with diffusion. I have the impression that there remains a lot to be understood in that area.

At this conference the lecture rooms were usually big enough. An exception was the first session ‘mathematical oncology from bench to bedside’ which was completely overfilled and had to move to a different room. In that session there was a tremendous amount of enthusiasm. There is now a subgroup of the SMB for cancer modelling which seems to be very active with its own web page and blog. I should join that subgroup. Some of the speakers were so full of energy and so extrovert that it was a bit much for me. Nevertheless, it is clear that this is an exciting area and I would like to be part of it. There was also a session of cancer immunotherapy led by Vincent Lemaire from Genentech. He and two others described the mathematical modelling being done in cancer immunotherapy in three major pharmaceutical companies (Genentech, Pfizer and Glaxo-Smith-Kline). These are very big models. Lemaire said that at the moment that there are 2500 clinical trials going on for therapies related to PD-1. A recurring theme in these talks was the difference between mice and men.

This morning there was a talk by Hassan Jamaleddine concerning nanoparticles used to present antigen. These apparently primarily stimulate Tregs more than effector T cells and can thus be used as a therapy for autoimmune diseases. He showed some impressive pictures illustrating clearance of EAE using this technique. A central theme was interference between attempts to use the technique in animals with two autoimmune diseases in different organs, e.g. brain and liver. I was interested by the fact that for what he was doing steady state analysis was insufficient for understanding the biology.

This afternoon, the conference being over, I took to opportunity to visit Paul Francois at McGill, a visit which was well worthwhile.

## The fold-Hopf and Hopf-Hopf bifurcations

June 30, 2019

Bifurcations of a dynamical system $\dot x=f(x,\alpha)$ can be classified according to their codimension. The intuitive idea is that the set of vector fields exhibiting a given type of bifurcation form a submanifold of the space of all vector fields of the given codimension. Well-known examples of bifurcations of codimension one are the cusp bifurcation, which can already occur when the variables $x$ and $\alpha$ are one-dimensional, and the Hopf bifurcation where the variable $x$ must be two-dimensional but the variable $\alpha$ can be one-dimensional. In fact the minimal dimension of the variable $\alpha$ required corresponds to the codimension of the bifurcation. In this post I want to discuss two bifurcations of codimension 2, the fold-Hopf bifurcation, where $x$ must be at least three-dimensional and the Hopf-Hopf bifurcation where $x$ must be at least four-dimensional

The first typical feature of these bifurcations is the configuration of eigenvalues of $D_x f$ at the bifurcation point. For a fold there is an eigenvalue zero. For a Hopf bifurcation there is a complex conjugate pair of non-zero purely imaginary eigenvalues. For a fold-Hopf bifurcation there is a zero eigenvalue and a complex conjugate pair of non-zero purely imaginary eigenvalues. For a Hopf-Hopf bifurcation there are two complex conjugate pairs of non-zero purely imaginary eigenvalues. The basic hope now is that if some genericity conditions are satisfied the system can be locally reduced to a normal form by a transformation of variables. This is true for the fold and Hopf cases but for fold-Hopf and Hopf-Hopf it is no longer true. A weaker goal which can be attained is to reduce the system to an approximate normal form so that the right hand side is the sum of a simple explicit expression and a higher order error term. The genericity assumptions are as follows. For the cusp the steady state should move with non-zero velocity when the parameter is changed and the steady state of the system for the bifurcation value of the parameter should be as non-degenerate as possible. This means that although $f_x=0$ there (which is the bifurcation condition) $f_{xx}\ne 0$. For the Hopf case the eigenvalues which are on the imaginary axis at the bifurcation value should move off the axis with non-zero velocity when the parameter is changed. At the same time the steady state at the bifurcation value should be as non-degenerate as possible. Solutions close to this steady state circle it and a corresponding Poincare mapping can be defined which describes how the distance from the steady state changes when the solution circles it once. Call this $p(x)$. The fact that there is a bifurcation means that $p'(0)=0$ and $p''(0)$ is automatically zero. The non-degeneracy condition is that $p'''(0)\ne 0$.

Now we come to the fold-Hopf bifurcation. One non-degeneracy condition combines conditions from the two simpler bifurcations in a simple way. It says that the position of the steady state and the real part of the eigenvalue move independently as the two parameters are changed. This is condition ZH0.3 in Theorem 8.7 in the book of Kuznetsov. I was confused by the fact that this condition involves a quantity $\gamma (\alpha)$ which is apparently nowhere defined in the book. It does occur on one other page. On that page there is also a $\Gamma (\alpha)$ which is defined and I think that the solution to the problem is that these two are equal. Assuming that that is correct then $\gamma (\alpha)$ is the projection of the position of the steady state onto the kernel of the linearization at the bifurcation point. The remaining non-degeneracy conditions are conditions on the system at the bifurcation value of the parameter. At the moment I do not have an intuition for the meaning of those conditions.

In the case of the Hopf-Hopf bifurcation a genericity assumption which is qualitatively different from those we have seen up to now is a non-resonance condition, condition HH.0 of Kuznetsov. It says that the two imaginary parts of the eigenvalues at the bifurcation point should not exhibit linear relations with integer coefficients. The next condition is that the real parts of the eigenvalues move independently as the two parameters are changed (HH.5 of Kuznetsov). As in the fold-Hopf case the remaining non-degeneracy conditions are conditions on the system at the bifurcation value of the parameter which I do not understand intuitively.

When analysing the Hopf bifurcation it turns out that after doing a suitable transformation of variables and discarding some terms which can be considered small the resulting system is rotationally invariant. In polar coordinates the angular component is constant and the radial component is cubic in the radius. For the fold-Hopf bifurcation it is natural to proceed as follows. We do a linear transformation so that the new axes belong to the eigenspaces of the linearization. Moreover this transformation is chosen so that the restriction of the linearization to the plane correponding to the complex eigenvalues is in standard form. Then the normal form is rotationally invariant and can be expressed in cylindrical polar coordinates. The component in the angular direction depends only on the coordinate $\xi$ along the axis while the other two components depend only on $\xi$ and the radial coordinate $\rho$. Thus the analysis of the phase portrait of the system near the bifurcation point can be reduced to the analysis of a two-dimensional dynamical system on the half-plane $\rho\ge 0$ called the amplitude system. A steady state of the amplitude system with $\rho=0$ corresponds to a steady state of the full system while a steady state of the amplitude system with $\rho>0$ corresponds to a periodic solution of the full system.

The amplitude system contains a parameter $s=\pm 1$ and a parameter $\theta$. The signs of these two quantities are of crucial importance. If $s=1$ and $\theta>0$ then the system can be reduced to normal form. If $s=-1$ and $\theta<0$ then the situation is still relatively simple but adding a small perturbation typically causes a heteroclinic orbit to break. The most difficult case is that where $s\theta<0$ and it is in that case that chaos may occur. A steady state of the amplitude system away from the axis can undergo a Hopf bifurcation and this corresponds to the occurrence of an invariant torus in the full system and is a Neimark-Sacker bifurcation of the limit cycle. This torus can break up for some value of the parameters and this is what leads to chaos.

In the Hopf-Hopf case the normal form involves two angular coordinates where the dynamics are trivial and two radial coordinates $r_1$ and $r_2$. Thus we again obtain a two-dimensional amplitude system, this time defined on a quadrant. Kuznetsov distinguishes between a ‘simple’ and a ‘difficult’ case according to the parameters but at my present level of understanding they both look very difficult. For Hopf-Hopf the truncated normal form is generically never topologically equivalent to the full system. A Neimark-Sacker bifurcation is always present.

As mentioned in a previous post, both bifurcations discussed here have been observed numerically in an ecological model and Hopf-Hopf bifurcations (but not fold-Hopf bifurcations, this was stated incorrectly in the previous post) in a model for the MAPK cascade.

## Light and lighthouses

June 3, 2019

I recently had the idea that I should improve my university web pages. The most important thing was to give a new presentation of my research. At the same time I had the idea that the picture of me on the main page was not very appropriate for attracting people’s attention and I decided to replace it with a different one. Now I have a picture of me in front of the lighthouse ‘Les Éclaireurs’ in the Beagle Channel, taken by my wife. I always felt a special attachment to lighthouses. This was related to the fact that as a child I very much liked the adventure of visiting uninhabited or sparsely inhabited small islands and these islands usually had lighthouses on them. This was in particular true in the case of Auskerry, an island which I visited during several summers to ring birds, especially storm petrels. I wrote some more about this in my very first post on this blog. For me the lighthouse is a symbol of adventure and of things which are far away and not so easy to reach. In this sense it is an appropriate symbol for how I feel about research. There too the goals are far away and hard to reach. In this context I am reminded of a text of Marcel Proust which is quoted by Mikhail Gromov in the preface to his book ‘Metric structures for Riemannian and non-Riemannian spaces’:

‘Même ceux qui furent favorables à ma perception des vérités que je voulais ensuite graver dans le temple, me félicitèrent de les avoir découvertes au microscope, quand je m’étais au contraire servi d’un télescope pour apercevoir des choses, très petites en effet, mais parce qu’elles étaient situées à une grande distance, et qui étaient chacune un monde’

[Even those who were favourable to my perception of the truths which I wanted to engrave in the temple, congratulated me on having discovered them with a microscope, when on the contrary I used a telescope to perceive things, in fact very small, but because they were situated at a great distance, and each of which was a world in itself.]

I feel absolutely in harmony with that text. Returning to lighthouses, I think they are also embedded in my unconscious. Years ago, I was fascinated by lucid dreams. A lucid dream usually includes a key moment, where lucidity begins, i.e. where the dreamer becomes conscious of being in a dream. In one example I experienced this moment was brought about by the fact of simultaneously seeing three lighthouses, those of Copinsay, Auskerry and the Brough of Birsay. Since I knew that in reality it is impossible to see all three at the same time this made it clear to me that I must be dreaming.

The function of a lighthouse is to use light to convey information and to allow people (seafarers) to recognise things which are important for them. Thus a lighthouse is a natural symbol for such concepts as truth, reason, reliability, learning and science. These concepts are of course also associated with the idea of light itself, that which allows us to see things. These are the elements which characterize the phase of history called the enlightenment. Sometimes I fear that we are now entering a phase which is just the opposite of that. Perhaps it could be called the age of obscurity. It is characterized by an increasing amount of lies, deceit, ignorance and superstition. Science continues its progress but sometimes it seems to me like a thin ray among gathering darkness. A future historian might describe the arch leading from the eighteenth to the twenty-first century. I recently watched a video of the Commencement speech of Angela Merkel in Harvard. In a way many of the things she said were commonplaces, nothing new, but listening to her speech and seeing the reactions of the audience it became clear to me that it is important these days to repeat these simple truths. Those of us who have not forgotten them should propagate them. And with some luck, the age of obscurity may yet be averted.

## Book on cancer therapy using immune checkpoints, part 2

April 20, 2019

I now finished reading the book of Graeber I wrote about in the last post. Here are some additional comments. Chapter 7 is about CAR T cells, a topic which I wrote about briefly here. I also mentioned in that post that there is a mathematical model related to this in the literature but I have not got around to studying it. Chapter 8 is a summary of the present state of cancer immunotherapy while the last chapter is mainly concerned with an individual case where PD-1 therapy showed a remarkable success but the patient, while against all odds still alive, is still not cancer-free. It should not be forgotten that the impressive success stories in this field are accompanied by numerous failures and the book also reports at length on what these failures can look like for individual patients.

For me the subject of this book is the most exciting topic in medicine I know at the moment. It is very dynamic with numerous clinical studies taking place. It is suggested in the book that there is a lot of redundancy in this and correspondingly a lot of waste, financial and human. My dream is that progress in this area could be helped by more theoretical input. What do I mean by progress? There are three directions which occur to me. (1) Improving the proportion of patients with a given type of cancer who respond by modifying a therapy or replacing it by a different one. (2) Identifying in advance which patients with a given type of cancer will respond to which therapy, so as to allow rational choices between therapies in individual cases. (3) Identifying new types of cancer which are promising targets for a given therapy. By theoretical input I mean getting a better mechanistic understanding of the ways in which given therapies work and using that to obtain a better understanding of the conditions needed for success. The dream goes further with the hope that this theoretical input could be improved by the formulation and analysis of mathematical models.

What indications are there that this dream can lead to something real? I have already mentioned one mathematical model related to CAR T-cells. I have mentioned a mechanistic model for PD-1 by Mellman and collaborators here. This has been made into a mathematical model in a 2018 article by Arulraj and Barik (PLoS ONE 13(10): e0206232). There is a mathematical model for CTLA-4 by Jansson et al. (J. Immunol. 175, 1575) and it has been extended to model the effects of related immunotherapy in a 2018 paper of Ganesan et al. (BMC Med. Inform. Decis. Mak. 18,37).

I conclude by discussing one topic which is not mentioned in the book. In Mainz (where I live) there is a company called BIONTECH with 850 employees whose business is cancer immunotherapy. The CEO of the company is Ugur Sahin, who is also a professor at the University of Mainz. I have heard a couple of talks by him, which were on a relatively general level. I did not really understand what his speciality is, only that it has something to do with mRNA. I now tried to learn some more about this and I realised that there is a relation to a topic mentioned in the book, that of cold and hot tumours. The most favourable situation for immune checkpoint therapies is where a tumour does in principle generate a strong immune response and has adapted to switch that off. Then the therapy can switch it back on. This is the case of a hot tumour, which exhibits a lot of mutations and where enough of these mutations are visible to the immune system. By contrast for a cold tumour, with no obvious mutations, there is no basis for the therapy to work on. The idea of the type of therapy being developed by Sahin and collaborators is as follows (my preliminary understanding). First analyse DNA and RNA from the tumour of a patient to identify existing mutations. Then try to determine by bioinformatic methods which of these mutations could be presented effectively by the MHC molecules of the patients. This leads to candidate proteins which might stimulate the immune system to attack the tumour cells. Now synthesise mRNA coding for those proteins and use it as a vaccine. The results of the first trials of this technique are reported in a 2017 paper in Nature 547, 222. It has 295 citations in Web of Science which indicates that it has attracted some attention.

## Book on cancer therapy using immune checkpoints

April 19, 2019

In a previous post I wrote about cancer immunotherapy and, in particular, about the relevance of immune checkpoints such as CTLA-4. For the scientific work leading to this therapy Jim Allison and Tasuku Honjo were awarded the Nobel Prize for Medicine in 2018. I am reading a book on this subject, ‘The Breakthrough. Immunotherapy and the Race to Cure Cancer’ by Charles Graeber. I did not feel in harmony with this book due to some notable features which made it far from me. One was the use of words and concepts which are typically American and whose meanings I as a European do not know. Of course I could go out and google them but I do not always feel like it. A similar problem arises from the fact that I belong to a different generation than the author. It is perhaps important to realise that the author is a journalist and not someone with a strong background in biology or medicine. One possible symptom of this is the occurrence of spelling mistakes or unconventional names (e.g. ‘raff’ instead of ‘raf’, ‘Mederex’ instead of ‘Medarex’ for the company which played an essential role in the development of antibodies for cancer immunotherapy, ‘dendrites’ instead of ‘dendritic cells’). As a consequence I think that if a biological statement made in the book looks particularly interesting it is worth trying to verify it independently. For example, the claim in one of the notes to Chapter 5 that penicillin is fatal to mice is false. This is not only of interest as a matter of scientific fact since it has also been used as an (unjustified) argument by protesters against medical experiments in animals. More details can be found here.

Chapter four is concerned with Jim Allison, the discoverer of the first type of cancer immunotherapy using CTLA-4. I find it interesting that in his research Allison was not deriven by the wish to find a cancer therapy. He wanted to understand T cells and their activation. While doing so he discovered CTLA-4, as an important ‘off switch’ for T cells. It seems that from the beginning Allison liked to try certain experiments just to see what would happen. If what he found was more complicated than he expected he found that good. In any case, Allison did an experiment where mice with tumours were given antibodies to CTLA-4. This disables the off switch. The result was that while the tumours continued to grow in the untreated control mice they disappeared in the treated mice. The 100% reponse was so unexpected that Allison immediately repeated the experiment to rule out having made some mistake. The result was the same.

Chapter six comes back to the therapy with PD-L1 with which the book started. The treatments with antibodies against PD-1 and PD-L1 have major advantages compared to those with CTLA-4. The success rate with metastatic melanoma can exceed 50% and the side effects are much less serious. The latter aspect has to do with the fact that in this case the mode of action is less to activate T cells in general than to sustain the activation of cells which are already attacking the tumour. This does not mean that treatments targetting CTLA-4 have been superceded. For certain types of cancer it can be better than those targetting PD-1 or PD-L1 and combinations may be better than either type of therapy alone. For the second class of drugs getting them on the market was also not easy. In the book it is described how this worked in the case of a drug developed by Genentech. It had to be decided whether the company wanted to develop this drug or a more conventional cancer therapy. The first was more risky but promised a more fundamental advance if successful. There was a showdown between the oncologists and the immunologists. After a discussion which lasted several hours the person responsible for the decision said ‘This is enough, we are moving forward’ and chose the risky alternative.

This post has already got quite long and it is time to break it off here. What I have described already covers the basic discussion in the book of the therapies using CTLA-4 and PD-1 or PD-L1. I will leave everthing else for another time.

## Banine’s ‘Jours caucasiens’

April 11, 2019

I have just read the novel ‘Jours caucasiens’ by Banine. This is an autobiographical account of the author’s childhood in Baku. I find it difficult to judge how much of what she writes there is true and how much is a product of her vivid imagination. I do not find that so important. In any case I found it very interesting to read. It is not for readers who are easily shocked. Banine is the pen name of Umm-El-Banine Assadoulaeff. She was born in Baku into a family of oil magnates and multimillionaires. In fact she herself was in principle a multimillionaire for a few days after the death of her grandfather, until her fortune was destroyed when Azerbaijan was invaded by the Soviet Union. In later years she lived in Paris and wrote in French. To my taste she writes very beautifully in French. I first heard of her through the diaries of Ernst Jünger. While he was an officer in the German army occupying Paris during the Second World War he got to know Banine and visited her regularly. It was not entirely unproblematic for her during the occupation when she was visited at her appartment by a German army officer in uniform. She seemed to regard this with humour. The two had a close but platonic relationship.

The society in which Banine grew up was the result of the discovery of oil. Her ancestors had been poor farmers who suddenly became very rich because oil-wells were built on their land. She presents her family as being very uncivilised. They were muslims but had already been strongly affected by western culture. I found an article in the magazine ‘Der Spiegel’ from 1947 where ‘Jours caucasiens’ is described by the words ‘gehören zu den skandalösesten Neuerscheinungen in Paris’ [is one of the most scandalous new publications in Paris]. It also says that her family was very unhappy about the way they were presented in the book and I can well understand that. It seems that she had a low opinion of her family and their friends and the culture they belonged to, although she herself did not seem to mind being part of it. She was attracted by Western culture and Paris was the place of her dreams. As a child she had a German governess. Her mother died when she was very young and after her father had remarried she had a French and an English teacher for those languages. She quickly fell in love with French. On the other hand, she saw having to learn English as a bit of a nuisance. Her impression was that the English had just taken the words from German and French and changed them in a strange way.

After the Russian invasion Banine’s father, who had been a government minister in the short-lived Azerbaijan Republic, was imprisoned. He was released due to the efforts of a man whose motivation for doing so was the desire to marry Banine. She was very much against this. Perhaps the strongest reason was that he had red hair. There was a superstition that red-haired people, who were not very common in that region, had evil supernatural powers. Banine’s grandmother told her a story about an alchemist who discovered the secret of red-haired people. According to him they should be treated in the following way. He cut off their head, boiled it in a pot and put the head on a pedestal. If this was done correctly then the heads would start to speak and make prophecies which were always true. Banine could not help associating her potential husband with this horrible myth. Unfortunately she was under a lot of social pressure and after hesitating a bit agreed to the marriage. Apart from being a sign of gratitude for her father’s release this was also a way of persuading her suitor to use his influence to get a visa for her father to allow him to leave Russia. In the end she accepted this arrangement instead of running away with the man she loved. At this time she was fifteen years old. Her father got the visa and left the country. Later she also got a visa and was able to leave. The last stage of her journey was with the Orient Express from Constantinople to Paris. The book ends as the train is approaching Paris and a new life is starting for her.

## Stability in the multiple futile cycle

April 8, 2019

In a previous post I described the multiple futile cycle, where a protein can be phosphorylated up to $n$ times. About ten years ago Wang and Sontag proved that with a suitable choice of parameters this system has $2k+1$ steady states. Here $k$ denotes the integral part of $n/2$. The question of the stability of these steady states was left open. On an intuitive level it is easy to have the picture that stable steady states and saddle points should alternate. This suggests that there should be $k+1$ stable states and $k$ saddles. On the other hand it is not clear where this intuition comes from and it is very doubtful whether it is reliable in a high-dimensional dynamical system. I have thought about this issue for several years now. I had some ideas but was not able to implement them in practise. Now, together with Elisenda Feliu and Carsten Wiuf, we have written a paper where we prove that indeed there are parameters for which there exist steady states with the stability properties suggested by the intuitive picture.

How can information about the relative stability of steady states be obtained by analytical calculations? For this it is good if the steady states are close together so that their stability can be investigated by local calculations. One way they can be guaranteed to be close together is if they all originate in a single bifurcation as a parameter is varied. This is the first thing we arranged in our proof. The next observation is that the intuition I have been talking about is based on thinking in one dimension. In a one-dimensional dynamical system alternating stability of steady states does happen, provided degenerate situations are avoided. Thus it is helpful if the centre manifold at the bifurcation is one-dimensional. This is the second thing we arranged in our proof. To get the particular kind of alternating stability mentioned above we also need the condition that the flow is contracting towards the centre manifold. I had previously solved the case $n=2$ of this problem with Juliette Hell but we had no success in extending it to larger values of $n$. The calculations became unmanageable. One advantage of the case $n=2$ is that the bifurcation there was a cusp and certain calculations are done in great generality in textbooks. These are based on the presence of a one-dimensional centre manifold but it turns out to be more efficient for our specific problem to make this explicit.

The general structure of the proof is that we first reduce the multiple futile cycle, which has mass action kinetics, to a Michaelis-Menten system which is much smaller. This reduction is well-behaved in the sense of geometric singular perturbation theory (GSPT), since the eigenvalues of a certain matrix are negative. With this in place steady states can be lifted from the Michaelis-Menten system to the full system while preserving their stability properties. The bifurcation arguments mentioned above are then applied to the Michaelis-Menten system.

The end result of the ideas discussed so far is that the original analytical problem is reduced to three algebraic problems. The first is the statement about the eigenvalues required for the application of GSPT. This was obtained for the case $n=2$ in my work with Juliette but we had no idea how to extend it to higher values of $n$. The second is to analyse the eigenvalues of the linearization of the system about the bifurcation point. What we want is that two eigenvalues are zero and that all others have negative real parts. (One zero eigenvalue arises because there is a conservation law while the second corresponds to the one-dimensional centre manifold.) There are many parameters which can be varied when choosing the bifurcation point and a key observation is that this choice can be made in such a way that the linearization is a symmetric matrix, which is very convenient for studying eigenvalues. The third problem is to determine the leading order coefficient which determines the stability of the bifurcation point within the centre manifold.

I started to do parts of the algebra and I would describe it as being like entering a jungle with a machete. I was able to find a direction to proceed and show that some progress could be made but I very soon got stuck. Fortunately my coauthors came and built a reliable road to the final goal.

## The pole-shifting theorem, part 2

March 25, 2019

In the last post I wrote about the pole-shifting theorem but included almost no information about the proof. Now I want to sketch that proof, following the account in the book of Sontag. It is convenient to talk about prescribing the characteristic polynomial rather than prescribing the eigenvalues. The characteristic polynomial contains the information about the set of eigenvalues together with information about their multiplicity. It is helpful to use changes of basis in the state space to simplify the problem. A change of basis leads to a similarity transformation of $A$ and so does not change the characteristic polynomial. It also does not change the rank of $R(A,B)$. Hence the property of controllability is not changed. Which polynomials can be obtained from matrices of the form $A+BF$ also does not change since the change of basis can be used to transform $F$ in an obvious way. Putting these things together shows that when proving the theorem for given matrices it is allowed to pass to new matrices by a change of basis when convenient.

The first step in the proof is to look at the theorem in the case of one control variable ($m=1$). I will use the notation $(A,b)$. In this case the system can be brought into a special form, the controller form, by a change of basis. Then it is elementary to solve for the unique feedback which solves the problem. The next step is to reduce the case of general $m$ to a modified control problem with $m=1$. Let $v$ be any vector with $Bv$ non-zero and $b=Bv$. The idea is to show that there is an $F_1$ such that $(A+BF_1,b)$ is controllable. If this can be done then the result for $m=1$ gives, for a given polynomial $\chi$, a matrix $f$ such that the characteristic polynomial of $(A+BF_1)+bf$ is $\chi$. But $(A+BF_1)+bf=A+B(F_1+vf)$ and so taking $F=F_1+vf$ solves the desired problem.

It remains to find $F_1$. For this purpose a sequence of vectors is constructed as follows. First choose a vector $v$ such that $Bv$ is non-zero and let $x_1=Bv$. Then $x_1$ is a non-zero element of the state space. Next choose $x_2=Ax_1+u_1$, where $u_1$ belongs to the image of $B$, in such a way that $x_1$ and $x_2$ are linearly independent. If this succeeds continue to choose $x_3=Ax_2+u_2$ in a similar way. The idea is to construct a maximal chain of linearly independent vectors $\{x_1,\ldots,x_k\}$ of this type. The claim is now that if $(A,B)$ is controllable $k=n$. Consider the space spanned by the $x_i$. It is of dimension $k$. Since the chain cannot be extended $Ax_k+Bu$ must also belong to this space, for any choice of $u$. In particular $Ax_k$ belongs to the space. Hence the image of $B$ belongs to the space. The definition of the $x_i$ then implies that $Ax_i$ belongs to the space for all $i$ so that the space is invariant under $A$. Putting these facts together shows that the image of $R(A,B)$ is contained in this space. By controllability it must therefore be the whole $n$-dimensional Euclidean space. Next define $F_1x_i=u_i$ for $i=1,2,\ldots,k-1$ and $F_1x_k$ arbitrarily. Then $R(A+BF_1,x_1)=(x_1,\ldots,x_n)$, which completes the proof.

In fact this theorem can be extended to one which describes which polynomials can be assigned when $(A,B)$ is not controllable. They are the polynomials of the form $\chi_1\chi_u$ where $\chi_1$ is an arbitrary monic polynomial of degree $r$ and $\chi_u$ is a polynomial defined by $(A,B)$ called the uncontrollable part of the characteristic polynomial of $A$. What this means is that some poles (the uncontrollable ones) are fixed once and for all and the others can be shifted arbitrarily.

## The pole-shifting theorem

March 22, 2019

Here I discuss a theorem of linear algebra whose interest comes from its applications in control theory.The discussion follows that in the book ‘Mathematical Control Theory’ by Eduardo Sontag. Let $A$ be an $n\times n$ matrix and $B$ an $n\times m$ matrix. We consider the expression $A+BF$ for an $m\times n$ matrix $F$. The game is now to fix $A$ and $B$ and attempt, by means of a suitable choice of $F$, to give the eigenvalues of $A+BF$ specified values. The content of the theorem is that this is always possible, provided a suitable genericity assumption, called controllability, is satisfied. In fact this statement has to be modified slightly. I want to work with real matrices and thus the eigenvalues automatically come in complex conjugate pairs. Thus the correct statement concerns candidates for the set of eigenvalues which satisfy that restriction. Where does the name of the theorem come from? Its source is the fact that eigenvalues of a matrix $M$ can be though of as poles of the function $(\det (M-\lambda I))^{-1}$ or the matrix-valued function $(M-\lambda I)^{-1}$. This is a picture popular in classical control theory. The primary importance of this result for control theory is that the stability of a control system is in many cases determined by a matrix of the form $A+BF$. If we can choose $F$ so that the eigenvalues of $A+BF$ are all real and negative then we have shown how a system can be designed for which the desired state is asymptotically stable. When the state is perturbed it returns to the desired state. It even does so in a monotone manner, i.e. without any overshoot.

What is the genericity condition? It is implest to explain in the case $m=1$. Then $B$ is a column vector and we can consider the vectors $B, AB, A^2B, \ldots$. After at most $n-1$ steps this sequence of vectors becomes constant. Controllability is the condition that the vectors generated in this way span the whole space. This condition can be reformulated as follows. We identify a set of $n$ vectors in $n$ dimensions with the Euclidean space of $n^2$ dimensions. To put it another way we place the vectors side by side as the columns of a matrix. Then the condition of controllability is nothing other than the condition that the rank of the resulting matrix is $n$. The path to the general definition is then simple. The matrices listed before are no longer vectors but we place them side by side to get an $n\times mn$ matrix $R(A,B)$, the reachability or controllability matrix. The condition for controllability is then that this matrix has rank $n$.

There are also other equivalent conditions for controllability, known under the name of the Hautus Lemma. This says that the rank condition (call this condition (i)) is equivalent to the condition (call it condition (ii)) that the rank of the matrix obtained by placing $\lambda I-A$ next to $B$ is $n$ for all complex numbers $\lambda$. It is easily seen that it is equivalent to assume this in the case that $\lambda$ is any eigenvalue of $A$. The proof that (i) imples (ii) is elementary linear algebra. The converse is more complicated and relies on the concept of the Kalman controllability decomposition. The proof of the pole-shifting theorem itself is rather involved and I will not discuss it here.