Monotone systems revisited

December 4, 2019

There are some topics in mathematics and physics which are a lasting source of dissatisfaction for me since I feel that I have not properly understood them despite having made considerable efforts to do so. In the case of physics the reason is often that the physicists who understand the subject are not able to explain it in a way which provides what a mathematician sees as a comprehensible account. In mathematics the problem is a different one. Mathematicians frequently have a tendency (often justified) to discuss things on a level which is as general as possible. This leads to theorems which are loaded down with detail and where the many technical conditions make it difficult to see the wood for the trees. When confronted with such things I sometimes feel exhausted and give up. I prefer an account which builds up ideas step by step from simple beginnings. Here I return to a subject which I have written about more than once in this blog before but where the sense of dissatisfaction remains. I hope to reduce it here.

I start with a system of ordinary differential equations \dot x_i=f_i(x). It should be defined on the n-dimensional Euclidean space or on one of its orthants. (An orthant is the subset of Euclidean space defined by making a choice of the signs of its components. It generalises a quadrant in the two-dimensional case.) The system is said to be cooperative if \frac{\partial f_i}{\partial x_j}>0 for all i\ne j. The name comes from the fact that the equations for the population dynamics of a set of species has this property if each species benefits the others. Suppose we now have two solutions x and \bar x of the system and that x_i(t_0)\le\bar x_i(t_0) for all i at some time time t_0. We may abbreviate this relation by x(t_0)\le\bar x(t_0). Here we see a partial order on Euclidean space defined by the ordering of the components. A theorem of Müller and Kamke says that if the initial data for two solutions of a cooperative system at time t_0 satisfies this relation then x(t)\le\bar x(t) for all t\ge t_0. Another way of saying this is that the time-t flow of the system is preserves the partial order. A system of ODE with this property is called monotone. Thus the Müller-Kamke theorem says that a cooperative system is monotone.

The differential condition for monotonicity can be integrated. If x and \bar x are two points in Euclidean space with x_i=\bar x_i for a certain i and x_j\le\bar x_j for j\ne i then f_i(x)\le f_i(\bar x). To see this we join x to \bar x by a piecewise linear curve where the coordinates other than the ith are increased successively from x_j to \bar x_j. On each segment of this curve the value of f_i does not decrease, as a consequence of the fundamental theorem of calculus. Hence its value at the end of the entire path is at least as big as its value at the beginning. We now want to prove that a certain inequality holds at all times t\ge t_0. In order to do this we would like to consider the first time t_*>t_0 where the inequality fails and get a contradiction. Unfortunately there might be no such time – in principle the condition might fail immediately. To get around this we deform the system for the solution \bar x to \frac{d\bar x_i}{dt}=f_i(\bar x)+\epsilon. If we can prove the result for the deformed system the result for the initial system follows by continuous dependence of the solution on \epsilon. For the deformed system let t_* be the supremum of the times where the desired inequality holds. If the inequality does not hold globally then the system is still defined at t=t_*. For t=t_* we have x_i=\bar x_i for some i and we can assume w.l.o.g. that x_j<\bar x_j for some j since otherwise the two solutions would be equal and the result trivial. The integrated form of the cooperativity condition implies that at t_* the right hand side of the evolution equation for \bar x_i-x_i is positive. On the other hand the fact that it just reached zero coming from positive values implies that the right hand side of the evolution equation is non-positive and we get a contradiction.

A key source of information about monotone dynamical systems is the book of Hal Smith with this title. I have repeatedly looked at this book but always got bogged down quite quickly. Now I realise that for my purposes it would have been much better if I had started with chapter 3. The Müller-Kamke theorem is discussed in section 3.1. The range of application of this theorem can be extended considerably by the following trick, discussed in section 3.5. Suppose that we define y_i=(-1)^{m_i}x_i where each of the m_i are zero or one. This transforms the signs of Df in a certain way and so cooperativity of the system for y corresponds to a certain sign pattern for the entries of Df. A first important condition is that each off-diagonal element of Df(x) should be either non-negative or non-positive. Next, the sign of \frac{\partial f_i}{\partial x_j}\frac{\partial f_j}{\partial x_i} is not changed be the transformation and must thus be non-negative. In the context of population models this can be interpreted as saying that there is no pair of species which are in a predator-prey relationship. Given that these two conditions are satisfied we consider a labelled graph where the nodes are the numbers from 1 to n and there is an edge between two nodes if at least one of the corresponding partial derivatives is non-zero at some point. The edge is then labelled with the sign of this non-zero value. A loop in the graph can be assigned the sign which is the product of those of its edges. It turns out that a system can be transformed to a cooperative system in the way indicated if and only if the graph contains no negative loops. I will call a system of this type ‘cooperative up to sign reversal’. The system can be transformed by a permutation of the variables into one where Df has diagonal blocks with non-negative entries and off-diagonal elements with non-positive entries.

If all elements of Df are required to be non-positive we get the class of competitive systems. It should be noted that being competitive leads to less restrictions on the dynamics of a system (towards the future) than being cooperative. We can define a class of systems which are competitive up to sign reversal. An example of such a system is the basic model of virus dynamics. In that system the unknowns are the populations of uninfected cells x, infected cells y and virus particles v. The transformation y\mapsto -y makes it into a competitive system. In various models of virus dynamics including the immune response the target cells of the virus and the immune cells are in a predator-prey relationship and so these systems can be neither cooperative up to sign or competitive up to sign.

Cedric Villani’s autobiography

November 1, 2019

I have just read Cédric Villani’s autobiographical book ‘Théorème Vivant’. I gave the German translation of the book to Eva as a present. I thought it might give her some more insight into what it is like to be a mathematician and give her some fortitude in putting up with a mathematician as a husband. Since I had not read the book before I decided to read it in parallel. I preferred to read the original and so got myself that. With hindsight I do not think it made so much difference that I read it in French instead of German. I think that the book is useful for giving non-experts a picture of the life of a mathematician (and not just that of a mathematician who is as famous as Villani has become). For this I believe that it is useful that the book contains some pieces of mathematical text which are incomprehensible for the lay person and some raw TeX source-code. I think that they convey information even in the absence of an understanding of the content. On the other hand, this does require a high level of tolerance on the part of the reader. Fortunately Eva was able to show this tolerance and I think she did enjoy the book and learn something more about mathematics and mathematicians.

For me the experience was of course different. The central theme of the book is a proof of Villani and Clement Mouhot of the existence of Landau damping, a phenomenon in plasma physics. I have not tried to enter into the details of that proof but it is a subject which is relatively close to things I used to work on in the past and I was familiar with the concept of Landau damping a long time ago. I even invested quite a lot of time into the related phenomenon of the Jeans instability in astrophysics, unfortunately without significant results. Thus I had some relation to the mathematics. It is also the case that I know many of the people mentioned in the book personally. Sometimes when Villani mentions a person without revealing their name I know who is meant. As far as I remember the first time I met Villani was at a conference in the village of Anogia in Crete in the summer of 2001. At that time he struck me as the number one climber of peaks of technical difficulty in the study of the Boltzmann equation. I do not know if at that time he already dressed in the eccentric way he does today. I do not remember anything like that.

For me the book was pleasant to read and entertaining and I can recommend it to mathematicians and non-mathematicians. If I ask myself what I really learned from the book in the end then I am not sure. One thing it has made me think of is how far I have got away from mainstream mathematics. A key element of the book is that the work described there got Villani a Fields medal, the most prestigious of mathematical prizes. These days the work of most Fields medallists is on things to which I do not have the slightest relation. Villani was the last exception to that rule. Of course this is a result of the general fact that communication between different mathematical specialities is so hard. The Fields medal is awarded at the International Congress of Mathematicians which takes place every four years. That conference used to be very attractive for me but now I have not been to one since that in 2006 in Madrid and I imagine that I will not go to another. That one was marked by the special excitement surrounding Perelman’s refusal of the Fields medal which he was offered for his work on the Poincaré conjecture. Another sign of the change in my orientation is that I am no longer even a member of the American Mathematical Society, probably the most important such society in the world. I will continue to follow my dreams, whatever they may be. Villani is also following his dreams. I knew that he had gone into politics, becoming a member of parliament. I was surprised to learn that he has recently become a candidate for the next election to become mayor of Paris.

Article on mathematical economics

September 28, 2019

I now find myself in the for me a priori surprising situation of being one of the authors of an article in the Journal of Mathematical Economics (85, 17-37). The title is ‘The invariant distribution of wealth and employment status in a small open economy with precautionary savings’. Without having a global overview of the subject I want to explain roughly what is being described by the mathematical model in this article. The idea is that we have a population of individuals who may lose their jobs and find new ones in a way which involves a lot of randomness. It is supposed that when they have employment these individuals save some money in order to be able to support themselves better during periods where they are unemployed. They question is whether these assumptions lead to a long-term stationary distribution of wealth and employment in the population. The starting model is a stochastic one but the consideration of stationary distributions lead to an ODE problem.

So how did I get involved in this project? Klaus Wälde, a professor of economics at the University of Mainz, had a project on this subject with Christian Bayer, a mathematician from the Weierstrass Institute in Berlin. The two of them submitted a paper which contained some partial results on stationary distributions. However they did not have a complete existence proof and the referees insisted that they try to improve on that. Wälde looked for some assistance on this point in the mathematics department in Mainz and ended up talking to me about the matter. The concrete mathematical problem to be solved concerned a certain boundary value problem for a system of ODE with rather singular boundary conditions. At this point the universality of mathematics comes in. It turned out that this problem coming from economics could be solved with the help of a theorem which I proved with Bernd Schmidt in 1991 to treat a problem coming from astrophysics. Along the way I was also able to show that an analyticity assumption which had been made in the formulation of the economics model was unnecessary. I have no intention of going further into the area of mathematical economics – I want to remain focussed on mathematical biology. It is nevertheless a nice feeling to think that with the mathematical training I have I could in principle contribute to almost any field of science.

Science as a literary pursuit

August 24, 2019

I found something in a footnote in the book of Oliver Sacks I mentioned in the previous post which attracted my attention. There is a citation from a letter of Jonathan Miller to Sacks with the idea of a love of science which is purely literary. Sacks suggests that his own love of science was of this type and that that is the reason that he had no success as a laboratory scientist. I feel that my own love for science has a strong literary component, or at least a strong component which is under the control of language. In molecular biology there are many things which have to be named and people have demonstrated a lot of originality in inventing those names. I find the language of molecular biology very attractive in a way which has a considerable independence from the actual meaning of the words. I expect that there are other people for whom this jungle of terminology acts as a barrier to entering a certain subject. In my case it draws me in. In my basic field, mathematics, the terminology and language is also a source of pleasure for me. I find it stimulating that everyday words are often used with a quite different meaning in mathematics. This bane of many starting students is a charm of the subject for me. Personal taste plays a strong role in these things. String theory is another area where there is a considerable need for inventing names. There too a lot of originality has been invested but in that case the result is not at at all to my taste. I emphasize that when I say that I am not talking about the content, but about the form.

The idea of using the same words with different meanings has a systematic development in mathematics in context of topos theory. I learned about this through a lecture of Ioan James which I heard many years ago with the title ‘topology over a base’. What is the idea? For topological spaces X there are many definitions and many statements which can be formulated using them, true or false. Suppose now we have two topological spaces X and B and a suitable continuous mapping from X to B. Given a definition for a topological space X (a topological space is called (A) if it has the property (1)) we may think of a corresponding property for topological spaces over a base. A topological space X over a base B is called (A) if it has property (2). Suppose now that I formulate a true sentence for topological spaces and suppose that each property which is used in the sentence has an analogue for topological spaces over a base. If I now interpret the sentence as relating to topological spaces over a base under what circumstances is it still true? If we have a large supply of statements where the truth of the statement is preserved then this provides a powerful machine for proving new theorems with no extra effort. A similar example which is better known and where it is easier (at least for me) to guess good definitions is where each property is replaced by one including equivariance under the action of a certain group.

Different mathematicians have different channels by which they make contact with their subject. There is an algebraic channel which means starting to calculate, to manipulate symbols, as a route to understanding. There is a geometric channel which means using schematic pictures to aid understanding. There is a combinatoric channel which means arranging the mathematical objects to be studied in a certain way. There is a linguistic channel, where the names of the objects play an important role. There is a logical channel, where formal implications are the centre of the process. There may be many more possibilities. For me the linguistic channel is very important. The intriguing name of a mathematical object can be enough to provide me with a strong motivation to understand what it means. The geometric channel is also very important. In my work schematic pictures which may be purely mental are of key importance for formulating conjectures or carrying out proofs. By contrast the other channels are less accessible to me. The algebraic channel is problematic because I tend to make many mistakes when calculating. I find it difficult enough just to transfer a formula correctly from one piece of paper to another. As a child I was good in mental arithmetic but somehow that and related abilities got lost quite early. The combinatoric channel is one where I have a psychological problem. Sometimes I see myself surrounded by a large number of mathematical objects which should be arranged in a clever way and this leads to a feeling of helplessness. Of course I use the logical channel but that is usually on a relatively concrete level and not the level of building abstract constructs.

Does all this lead to any conclusion? It would make sense for me to think more about my motivations in doing (and teaching) mathematics in one way or another. This might allow me to do better mathematics on the one hand and to have more pleasure in doing so on the other hand.

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?’

As a schoolboy I read that dictionary a lot but I did not read it from beginning to end. For comparison, I am reading the autobiography ‘On the Move: A Life’ of Oliver Sacks and he tells the following story. As a student he won a cash prize and used the money to buy a copy of the Oxford English Dictionary in 12 volumes. He read it from beginning to end. I do feel a certain sympathy for him in this matter but it is an example of the fact that he seems to have been excessive in many things, to an extent which creates a distance between him and me. The book is well written and contains a lot of very good stories and I can recommend it. Nevertheless it is not one of those autobiographies which leads me to identify with the author or to admire them greatly. At this point I have only read about a quarter of the book and so my impression may yet change. I had previously read some of his books with pleasure, ‘Awakenings’, The Man Who Mistook His Wife for a Hat’ and ‘An Anthropologist on Mars’ and it was interesting to learn more about the man behind the books.

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 exciting 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.

Despite this I find the book a very rewarding read due to the stories it tells. It was exciting to read the first chapter which describes the experiences of one of the first patients to experience what seemed like a miracle cure due to treatment with an antibody to PD-L1. I find it fascinating to get an impression of what a person in this type of situation actually lives through. On a personal note, I was happy to see that when the patient met the team of researchers who had developed the treatment one of the people present was Ira Mellman. As I mentioned in a previous post I have been present at a lecture of Mellman. The second chapter describes known cases where an infectious disease can lead to the elimination of a tumour. It describes how, more than a hundred years ago, William Coley tried to turn this observation into a therapy. His success in doing so was very limited and this was unavoidable. The ideas needed to understand what might be going on in such a situation simply did not exist at that time. Without understanding it was impossible to pursue a therapy in a controlled way. I knew something about this story before reading the book but it filled in a lot more background for me. The key figure in the third chapter is Steven Rosenberg. I had not heard his name before. He had an important position at the NIH and pursued research into cancer immunotherapy during a period where there were few returns. One substance which he tried to use therapeutically was IL-2. Here again I was pleased to come across the name of a person who I have heard give a talk, as mentioned in a previous post. This is Kendall Smith, the discoverer of IL-2.

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.

The fifth chapter throws some light on the question why researchers were so sceptical for so long about the idea that the immune system can effectively fight cancer. The central conceptual reason is that in order to interpret the results of certain experiments it is not enough to consider the typical cancer cell. Instead it is necessary to think on a population level and see the tumour as an ecosystem. When cancer cells are attacked in some way and the majority die there will be a few left over which are immune to that particular type of attack. That small population will then expand and the tumour will grow again. The genetic composition of a typical cell in the new tumour will be very different from that of the old one. In the case of an attack by the immune system this gives rise to the concept of ‘cancer immunoediting’. On the road to transforming the experimental results of Allison into a therapy there were further conceptual obstacles along the road. In the phase 2 clinical trial for an antibody against CTLA-4 run by Bristol-Myers-Squibb (which had taken over the company Medarex which had started the development of the drug) the criteria for success were badly chosen. They were based on what might have been good criteria for a chemotherapy but were not good for the new type of therapy. Success was based on the tumours of a certain percentage of patients having shrunk by a certain amount after three months. The problem was that the time scale (which had been chosen to limit the expense) was too short and tumour size was not the right thing to look at. It could be seen that he tumours of certain patients had grown but they reported that they were feeling better. It happened that a patient who was about to die called up months later, after the planned endpoint of the trial and said ‘I’m fine’. What is the explanation for these things? The first aspect is that the immune response required to attack the tumour takes a considerable time to develop and success needs more than three months. The other is that a bigger tumour does not necessarily mean more cancer cells. It can also mean that there are huge numbers of immune cells in the tumour. Imaging the size of the tumour misses that. A similar trial to that of BMS gave similar results and was abandoned. That the same did not happen with the BMS trial was apparently due to someone called Axel Hoos. He persuaded the company to extend the trial and introduced a better endpoint criterion, the proportion of patients who live for a certain time. This led to success and eventually to the approval of the drug ipilimumab. Its rate of success, in the case of metastatic melanoma, is that about 20% of the patients are cured (in the sense that the tumours go away and have not come back until today, the survival curve flattens out at a positive value). The side effects are formidable, due to autoimmune reactions.

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.