Is mathematics being driven out by computers?

In the past two weeks I attended two conferences. The first was the annual meeting of the Deutsche Mathematikervereinigung (DMV, the German mathematical society) in Berlin. The second was the joint annual meeting of the ESMTB (European Society for Mathematical and Theoretical Biology) and the SMB (Society for Mathematical Biology) in Heidelberg. I had the impression that the participation of the SMB was relatively small compared to previous years. (Was this mainly due to the pandemic or due to other problems in international travel?) There were about 500 participants in total who were present in person and about another 100 online. I was disappointed with the plenary talks at both conferences. The only one which I found reasonably good was that of Benoit Perthame. One reason I did not like them was the dominance of topics like machine learning and artificial intelligence. This brings me to the title of this post. I have the impression that mathematics (at least in applied areas) is becoming ever weaker and being replaced by the procedure of developing computer programmes which could be applied (and sometimes are) to the masses of data which our society produces these days. This was very noticeable in these two conferences. I would prefer if we human beings would continue to learn something and not just leave it to the machines. The idea that some day the work of mathematicians might be replaced by computers is an old one. Perhaps it is now happening, but in a different way from that which I would have expected. Computers are replacing humans but not because they are doing everything better. There is no doubt there are some things they can do better but I think there are many things which they cannot. The plenary talks at the DMV conference on topics of this kind were partly critical. There occurred examples of a type I had not encountered before. A computer is presented with a picture of a pig and recognizes it as a pig. Then the picture is changed in a very specific way. The change is quantitatively small and is hardly noticeable to the human eye. The computer identifies the modified picture as an aeroplane. In another similar example the starting picture is easily recognizable as a somewhat irregular seven and is recognized by the computer as such. After modification the computer recognizes it as an eight. This seems to provide a huge potential for mistakes and wonderful opportunities for criminals. I feel that the trend to machine learning and related topics in mathematics is driven by fashion. It reminds me a little of the ‘successes’ of string theory in physics some years ago. Another aspect of the plenary talks at these conferences I did not like was that the speakers seemed to be showing off with how much they had done instead of presenting something simple and fascinating. At the conference in Heidelberg there were three talks by young prizewinners which were shorter than the plenaries. I found that they were on average of better quality and I know that I was not the only one who was of that opinion.

In the end there were not many talks at these conferences I liked much but let me now mention some that I did. Amber Smith gave a talk on the behaviour of the immune system in situations where bacterial infections of the lung arise during influenza. In that talk I really enjoyed how connections were made all the way from simple mathematical models to insights for clinical practise. This is mathematical biology of the kind I love. In a similar vein Stanca Ciupe gave a talk about aspects of COVID-19 beyond those which are common knowledge. In particular she discussed experiments on hamsters which can be used to study the infectiousness of droplets in the air. A talk of Harsh Chhajer gave me a new perspective on the intracellular machinery for virus production used by hepatitis C, which is of relevance to my research. I saw this as something which is special for HCV and what I learned is that it is a feature of many positive strand RNA viruses. I obtained another useful insight on in-host models for virus dynamics from a talk of James Watmough.

Returning to the issue of mathematics and computers another aspect I want to mention is arXiv. For many years I have put copies of all my papers in preprint form on that online archive and I have monitored the parts of it which are relevant for my research interests for papers by other people. When I was working on gravitational physics it was gr-qc and since I have been working on mathematical biology it has been q-bio (quantitative biology) which I saw as the natural place for papers in that area. q-bio stands for ‘quantitative biology’ and I interpreted the word ‘quantitative’ as relating to mathematics. Now the nature of the papers on that archive has changed and it is also dominated by topics strongly related to computers such as machine learning. I no longer feel at home there. (To be fair I should say there are still quite a lot of papers there which are on stochastic topics which are mathematics in the classical sense, just in a part of mathematics which is not my speciality.) In the past I often cross-listed my papers to dynamical systems and maybe I should exchange the roles of these two in future – post to dynamical systems and cross-list to q-bio. If I succeed in moving further towards biology in my research, which I would like to I might consider sending things to bioRxiv instead of arXiv.

In this post I have written a lot which is negative. I feel the danger of falling into the role of a ‘grumpy old man’. Nevertheless I think it is good that I have done so. Talking openly about what you are unsatisfied with is a good starting point for going out and starting in new positive directions.

Advertisement

Another conference on biological oscillators at EMBL in Heidelberg

I recently attended a conference at EMBL in Heidelberg and I very much enjoyed experiencing a live conference for the first time in a couple of years. I heard similar sentiments expressed by many of the other participants at the meeting. This conference at EMBL was a sequel to one which I previously wrote about here. The present event was in hybrid form with many of the speakers remote. There were nevertheless more than a hundred people attending on site. The conference started with a presymposium. This was intended to teach some mathematics to biologists. I attended it since I saw it as an opportunity to learn more about what kind of mathematics is really of interest to biologists. Among the main themes discussed were the relationships between positive feedback and multistability and between negative feedback and oscillations. First there was a one-hour talk by Hanspeter Herzel. Then there was a practical part where we were supposed to play with a computer programme. I had downloaded the necessary programmes (R and RStudio) as recommended but this part of the event was a failure for me. I am simply lacking in basic computer competence. It was not explained to us how to begin using the programme and I was not able to supply this missing information on my own. The first part, the lecture, was more interesting for me. The speaker mentioned a paper which he wrote with others about circadian oscillations in the number of lymphocytes in different tissues (D. Druzd et al., Immunity 46, 120). I had previously wondered about the possible roles of oscillations in immunology but I never thought of that direction. I spoke to Herzel about this in a coffee break. This demonstrates a huge advantage of live versus online conferences. I am sure that the information he and I exchanged over coffee would never have been communicated if the conference had been only online. There is a standard picture in immunology in which antigen is being continuously transported to lymph nodes, where it can activate lymphocytes. A key point of the paper is that this does not happen at a constant rate. Instead the process is highly oscillatory. Lymphocytes reach their highest level in the lymph nodes mainly at the beginning of the active phase  (i.e. the beginning of the dark phase in the mice in which these observations were carried out). This means that the effectiveness of a vaccination or another chemical intervention may depend strongly on the time at which it is administered. Herzel told me about an example where this has been seen in practise in cancer immunotherapy. I decided that I wanted to investigate this more closely. Before I could do that I heard the talk of Francis Levi, which was exactly on this topic. Returning to the paper quoted above, according to Herzel the mathematical content was very elementary, using a linear model. I am happy that simple mathematical models and ideas can lead to useful biological insights. What I do not find so good is that the information on the mathematics presented in the paper is so minimal, even in the supplementary material. There is one aspect of this story which is unclear to me. It is important for the functioning of the immune system that a given T cell visits many lymph nodes in a day. Thus the delays to the entrance or exit from lymph nodes which are supposed to implement the rhythm must act in some kind of averaged sense.

I also had a chance to talk to Levi over coffee and get some additional insights about some aspects of his lecture. He has been working on chronotherapy in oncology for many years. This means the idea that the effectiveness of a cancer therapy can be very dependent on the time of day it is administered. He has applied these ideas in practise but the ideas have not gained wide acceptance in the community of oncologists. There is a chance that this may change soon due to the appearance of two papers on this subject in the prestigious journal ‘The Lancet Oncology’ in November 2021. One of the papers (22, 1648) is by Levi, the other (22, 1777) by Qian et al.

Now let me mention a couple of the other contributions I liked best. On Monday there was a (remote) talk by Albert Goldbeter on the coupling between the cell cycle and the circadian clock. Here, as elsewhere in this conference, entrainment was a central theme. There was a discussion of the role of multiple limit cycles in these models. There was also a (remote) talk by Jim Ferrell. His subject was cataloguing certain aspects of an organism called the mouse lemur. The idea was to have a list of cell types and hormones and to know which cell types produce and are affected by which hormones. There is a preprint on this subject on BioRxiv. One feature of these primates which I found striking is the following. They are much fatter in winter than in summer and this is related to a huge difference in thyroid hormones. If I remember correctly it is a factor of ten. For comparison, in humans thyroid hormones also vary with the time of year but only on the scale of a couple of per cent. In a talk by Susan Golden (live) on the Kai system in cyanobacteria I was able to experience one of the pioneers in that field.

Uğur Şahin, Özlem Türeci and The Vaccine

 

I have just read the book ‘The Vaccine’ by Joe Miller, Uğur Şahin and Özlem Türeci. More precisely, I read the German version which is called ‘Projekt Lightspeed’ but I am assuming that the contents are not too different. The quality of the language in the version I read is high and I conclude from this that it is likely that both the quality of the language in the original and the quality of the translation are high. Miller is a journalist while Şahin and Türeci are the main protagonists of the story told in the book. It is the story of how the husband and wife team of researchers developed the BioNTech vaccine against COVID-19, a story which I found more gripping than fictional thrillers. The geographical centre of the story is Mainz. Şahin and Türeci live there and the headquarters of BioNTech, the company they founded, is also there. In fact when I moved to Mainz in 2013 I lived just a couple of hundred metres from what is now the area occupied by the BioNTech. Since I was interested in biotechnology the building was interesting for me. My first encounter with Şahin was a public lecture he gave about cancer immunotherapy in February 2015 and which I wrote about here. I heard him again in a keynote talk he gave at a conference at EMBL about cancer immunotherapy in February 2017. I was interested to hear his talk but it seems that it did not catch my attention since I did not mention it in the account I wrote of that meeting. One of the last lectures I attended live before the pandemic made such things impossible was at the university medical centre here in Mainz on 13th February 2020. Şahin was the chairman. The speaker was Melanie Brinkmann and the subject the persistence of herpes viruses in the host. I did not detect any trace of the theme COVID-19 in the meeting that day except for the fact that the speaker complained that she was getting asked so many questions on that subject on a daily basis. Later on she attained some public prominence in Germany in the discussion of measures against the pandemic. The book is less about the science of the subject than about the human story involved. I have no doubt that the scientific content is correct but it is not very deep. That is not the main subject of the book.

I now come to the story itself. Şahin and Türeci are both Germans whose parents came to Germany from Turkey. They studied medicine and they met during the practical part of their studies. They were both affected by seeing patients dying of cancer while medicine was helpless to prevent it. They decided they wanted to change the situation and have pursued that goal with remarkable consistency since then. They later came to the University of Mainz. They founded a biotechnology company called Ganymed producing monoclonal antibodies which was eventually sold for several hundred million Euros. They then went on to found BioNTech with the aim of using mRNA technology for cancer immunotherapy. An important role was played by money provided by the Strüngmann brothers. They had become billionaires through their company Hexal which sold generic drugs. They were relatively independent of the usual mechanisms of the financial markets and this was a big advantage for BioNTech. (A side remark: I learned from the book that the capital NT in the middle of the company name stands for ‘new technology’.) In early January 2020 Şahin foresaw the importance of COVID-19 and immediately began a project to apply the mRNA technology of BioNTech to develop a vaccine. The book is the story of many of the obstacles which he and Türeci had to overcome to attain this goal. In the US the vaccine is associated with the name Pfizer and it is important to mention at this point what the role of Pfizer was, namely to provide money and logistics. The main ideas came from Şahin and Türeci. Of course no important scientific development is due to one or two people alone and there are many contributions. In this case a central contribution came from Katalin Karikó.

How does the BioNTech vaccine work? The central idea of an mRNA vaccine is as follows. The aim is to introduce certain proteins into the body which are similar to ones found in the virus. The immune response to these proteins will then also act against the virus. What is actually injected is mRNA and that is then translated into the desired proteins by the cellular machinery. To start with the sequences of relevant proteins must be identified and corresponding mRNA molecules produced in vitro based on a DNA template. The RNA does not only contain the code for the protein but also extra elements which influence the way in which it behaves or is treated within a cell. In addition it is coated with some lipids which protect it from degradation by certain enzymes and help it to enter cells. Karikó played a central role in the development of this lipid technology. After the RNA has been injected it has to get into cells. A good target cell type are the dendritic cells which take up material from their surroundings by macropinocytosis. They then produce proteins based on the RNA template, cut them up into small peptides and display these on their surface. They also move to the lymph nodes. There they can present the antigens to T cells, which get activated. For T cells to get activated a second signal is also necessary and it is fortunate that mRNA can provide such a signal – in the language of vaccines it shows a natural adjuvant activity. In many more popular accounts of the role of the immune system in the vaccination against COVID-19 antibodies are the central subject. In fact according to the book many vaccine developers are somewhat fixated on antibodies and underestimate the role of T cells. There Şahin had to do a lot of convincing. It is nevertheless the case that antibodies are very important in this story and there is one point which I do not understand. Antibodies are produced by B cells and in order to do so they must be activated by the antigen. For this to happen the antigen must be visible outside the cells. So how do proteins produced in dendritic cells get exported so that B cells can see them?

I admire Şahin and Türeci very much. This has two aspects. The first is their amazing achievement in producing the vaccine against COVID-19 in record time. However there is also another aspect which I find very important. It is related to what I have learned about these two people from the book and from other sources. It has to do with a human quality which I find very important and which I believe is not appreciated as it should be in our society. This is humility. In their work Şahin and Türeci have been extremely ambitious but it seems to me that in their private life they have remained humble and this makes them an example to be followed.

 

Dynamics of the activation of Lck

The enzyme Lck (lymphocyte-associated tyrosine kinase) is of central importance in the function of immune cells. I hope that mathematics can contribute to the understanding and improvement of immune checkpoint therapies for cancer. For this reason I have looked at the relevant mathematical models in the literature. In doing this I have realized the importance in this context of obtaining a better understanding of the activation of Lck. I was already familiar with the role of Lck in the activation of T cells. There are two tyrosines in Lck, Y394 and Y505, whose phosphorylation state influences its activity. Roughly speaking, phosphorylation of Y394 increases the activity of Lck while phosphorylation of Y505 decreases it. The fact that these are influences going in opposite directions already indicates complications. In fact the kinase Lck catalyses its own phosphorylation, especially on Y394. This is an example of autophosphorylation in trans, i.e. one molecule of Lck catalyses the phosphorylation of another molcule of Lck. It turns out that autophosphorylation tends to favour complicated dynamics. It is already the case that in a protein with a single phosphorylation site the occurrence of autophosphorylation can lead to bistability. Normally bistability in a chemical reaction network means the existence of more than one stable positive steady state and this is the definition I usually adopt. The definition may be weakened to the existence of more than one non-negative stable steady state. That autophosphorylation can produce bistability in this weaker sense was already observed by Lisman in 1985 (PNAS 82, 3055). He was interested in this as a mechanism of information storage in a biochemical system. In 2006 Fuss et al. (Bioinformatics 22, 158) found bistability in the strict sense in a model for the dynamics of Src kinases. Since Lck is a typical member of the family of Src kinases these results are also of relevance for Lck. In that work the phosphorylation processes are embedded in feedback loops. In fact the bistability is present without the feedback, as observed by Kaimachnikov and Kholodenko (FEBS J. 276, 4102). Finally, it was shown by Doherty et al. (J. Theor. Biol. 370, 27) that bistability (in the strict sense) can occur for a protein with only one phosphorylation site. This is in contrast to more commonly considered phosphorylation systems. These authors have also seen more complicated dynamical behaviour such as periodic solutions.

All these results on the properties of solutions of reaction networks are on the level of simulations. Recently Lisa Kreusser and I set out to investigate these phenomena on the level of rigorous mathematics and we have just put a paper on this subject on the archive. The model developed by Doherty et al. is one-dimensional and therefore relatively easy to analyse. The first thing we do is to give a rigorous proof of bistability for this system together with some information on the region of parameter space where this phenomenon occurs. We also show that it can be lifted to the system from which the one-dimensional system is obtained by timescale separation. The latter system has no periodic solutions. To obtain bistability the effect of the phosphorylation must be to activate the enzyme. It does not occur in the case of inhibition. We show that when an external kinase is included (in the case of Lck there is an external kinase Csk which may be relevant) and we do not restrict to the Michaelis-Menten regime bistability is restored.

We then go on to study the dynamics of the model of Kaimachnikov and Kholodenko, which is three-dimensional. These authors mention that it can be reduced to a two-dimensional model by timescale separation. Unfortunately we did not succeed in finding a rigorous framework for their reduction. Instead we used another related reduction which gives a setting which is well-behaved in the sense of geometric singular perturbation theory (normally hyperbolic) and can therefore be used to lift dynamical features from two to three dimensions in a rather straightforward way. It then remains to analyse the two-dimensonal system. It is easy to deduce bistability from the results already mentioned. We go further and show that there exist periodic and homoclinic solutions. This is done by showing the existence of a generic Bogdanov-Takens bifurcation, a procedure described more generally here and here. This system contains an abundance of parameters and the idea is to fix these so as to get the desired result. First we choose the coordinates of the steady state to be fixed simple rational numbers. Then we fix all but four of the parameters in the system. The four conditions for a BT bifurcation are then used to determine the values of the other four parameters. To get the desired positivity for the computed values the choices must be made carefully. This was done by trial and error. Establishing genericity required a calculation which was complicated but doable by hand. When a generic BT bifurcation has been obtained it follows that there are generic Hopf bifurcations nearby in parameter space and the nature of these (sub- or supercritical) can be determined. It turns out that in our case they are subcritical so that the associated periodic solutions are unstable. Having proved that periodic solutions exist we wanted to see what a solution of this type looks like by simulation. We had difficulties in finding parameter values for which this could be done. (We know the parameter values for the BT point and that those for the periodic solution are nearby but they must be chosen in a rather narrow region which we do not know explicitly.) Eventually we succeeded in doing this. In this context I used the program XPPAUT for the first time in my life and I learned to appreciate it. I see this paper as the beginning rather than the end of a path and I am very curious as to where it will lead.

My first virtual conference (SMB 2020)

At the moment I am attending the annual conference of the Society for Mathematical Biology, which is taking place online. This is my first experience of this kind of format. The conference has many more participants than in any previous year, more than 1700. It takes place in a virtual building which is generated by the program Sococo. I find this environment quite disorienting and a bit stressful. This reaction probably has to do with the facts that I am no longer so young and that I have always tried to avoid social media as much as possible. I am sure that younger generations (and members of older generations with an enthusiasm for new technical developments) have far fewer problems getting used to it. In advance I was a bit worried about setting up the necessary computer requirements to be able to give my talk or even to go to others. In the end it worked out and my talk, given via Zoom, went smoothly. I got some good feedback, I am already convinced that it was worth joining this meeting and I may be less sceptical about joining others of this type in the future. There have been technical hitches. For instance the start of one big talk was delayed by about 20 minutes for a reason of this kind. Nevertheless, many things have gone well. Of course it is much preferable to meet people personally but when that is not possible virtual meetings with old friends are also pleasant.

This meeting has been organized around the subgroups of the society. I am a member of the subgroups for immunology and oncology. In fact my talk got scheduled in the group for mathematical modelling. My greatest allegiance is to the immunology subgroup and so I was happy to see that it was represented so strongly at this conference. It has seven sessions of lectures, made up of 28 talks. If I should choose my favourite from those talks in this section I have heard so far (it is not finished yet) then I pick the talk of Ruy Ribeiro on CD8 cells in HIV infection. I did feel I was missing some necessary background but I nevertheless found the talk useful in introducing me to important ideas which were new to me. The immunology subgroup have managed to get a very prominent speaker for their keynote talk, David Ho. I wrote about his work and its relation to mathematics in one of my first ever posts on this blog, way back in 2008. I am looking forward to hearing his talk later today (which will be on COVID-19). I will now mention some of my other personal highlights from the conference so far. I noticed on the program that Stas Shvartsman was giving a talk. His work has made a positive impression on me in the past and I did have a little e-mail contact with him. On the other hand I never met him personally and I had never heard a talk by him. This was my chance and I went in with high expectations. They were not disappointed. He talked about developmental defects arising from mutations in a single gene. In particular he concentrated on mutations in the Raf-MEK-ERK MAPK cascade. I was familiar with the role of mutations in this cascade in cancer but I had never heard about this other role. Stas described experiments in Drosophila where one base in MEK is mutated. This produces flies with a particular small change in the pattern of the veins in their wings. Interestingly, this does not occur in all flies with the mutation but only in 30% of them (I hope I am remembering the right number). Another talk yesterday which I appreciated was that by Robert Insall, who was talking about aspects of chemotaxis. He started with the phenomenon of the spread of melanoma. Melanoma cells do have a very strong tendency to spread in space and the question is what controls that. Is it chemotaxis along a chemical gradient?He showed pictures of melanoma cells moving fast up a chemical gradient. Then he showed a similar picture showing them moving just as fast without the chemical gradient. So what is going on? This kind of experiment starts with cells concentrated on one side of a region and a spatially homogeneous distribution of a relevant substance. The cells consume this substance, create their own gradient and then undergo chemotaxis along it. The speaker explained how there are many instances of chemotaxis in biology which can only be explained by a self-created gradient and not be a pre-existing one. He also made some interesting remarks about how mathematical modelling can lead to insights in biology which would not be possible with the usual verbal approaches of the biologists.

I also want to mention an interesting conversation I had in the poster session. The poster concerned was that of Daniel Koch. The theme of his work (which has already been published in a paper in J. Theor. Biol.) is that the formation of oligomers of proteins (or their posttranslationally modified variants) can lead to interesting dynamics. At first sight this may sound too simple to be interesting but in fact in mathematics it is often the careful consideration of apparently simple situations which leads to fundamental progress. I imagine that this principle also applies to other disciplines (such as biology) but it is perhaps strongest in mathematics. In any case, I am strongly motivated to study this work carefully. The only question in when it will be, given the many other directions I want to pursue.

Monotone systems revisited

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 . It should be defined on the -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 for all . 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 and of the system and that for all at some time time . We may abbreviate this relation by . 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 satisfies this relation then for all . Another way of saying this is that the time- 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 and are two points in Euclidean space with for a certain and for then . To see this we join to by a piecewise linear curve where the coordinates other than the th are increased successively from to . On each segment of this curve the value of 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 . In order to do this we would like to consider the first time 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 to . If we can prove the result for the deformed system the result for the initial system follows by continuous dependence of the solution on . For the deformed system let 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 . For we have for some and we can assume w.l.o.g. that for some since otherwise the two solutions would be equal and the result trivial. The integrated form of the cooperativity condition implies that at the right hand side of the evolution equation for 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 where each of the are zero or one. This transforms the signs of in a certain way and so cooperativity of the system for corresponds to a certain sign pattern for the entries of . A first important condition is that each off-diagonal element of should be either non-negative or non-positive. Next, the sign of 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 to 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 has diagonal blocks with non-negative entries and off-diagonal elements with non-positive entries.

If all elements of 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 , infected cells and virus particles . The transformation 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.

SMB meeting in Montreal

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.

Book on cancer therapy using immune checkpoints, part 2

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

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.

Herd immunity

I have a long term interest in examples where mathematics has contributed to medicine. Last week I heard a talk at a meeting of the Mainzer Medizinische Gesellschaft about vaccination. The speaker was Markus Knuf, director of the pediatric section of the Helios Clinic in Wiesbaden. In the course of his talk he mentioned the concept of ‘herd immunity’ several times. I was familiar with this concept and I have even mentioned it briefly in some of my lectures and seminars. It never occurred to me that in fact this is an excellent example of a case where medical understanding has benefited from mathematical considerations. Suppose we have a population of individuals who are susceptible to a particular disease. Suppose further that there is an endemic state, i.e. that the disease persists in the population at a constant non-zero level. It is immediately plausible that if we vaccinate a certain proportion of the population against the disease then the proportion of the population suffering from the disease will be lower than it would have been without vaccination. What is less obvious is that if exceeds a certain threshold then the constant level will be zero. This is the phenomenon of herd immunity. The value of depends on how infectious the disease is. A well-known example with a relatively high value is measles, where is about . In other words, if you want to get rid of measles from a population then it is necessary to vaccinate at least 95% of the population. It occurs to me that this idea is very schematic since measles does not occur as a constant rate. Instead it occurs in large waves. This idea is nevertheless one which is useful when making public health decisions. Perhaps a better way of looking at it is to think of the endemic state as a steady state of a dynamical model. The important thing is that this state is asymptotically stable in the dynamic set-up so that it recovers from any perturbation (infected individuals coming in from somewhere else). It just so happens that in the usual mathematical models for this type of phenomenon whenever a positive steady state (i.e. one where all populations are positive) exists it is asymptotically stable. Thus the distinction between the steady state and dynamical pictures is not so important. After I started writing this post I came across another post on the same subject by Mark Chu-Carroll. I am not sad that he beat me to it. The two posts give different approaches to the same subject and I think it is good if this topic gets as much publicity as possible.

Coming back to the talk I mentioned, a valuable aspect of it was that the speaker could report on things from his everyday experience in the clinic. This makes things much more immediate than if someone is talking about the subject on an abstract level. Let me give an example. He showed a video of a small boy with an extremely persistent cough. (He had permission from the child’s parents to use this video for the talk.) The birth was a bit premature but the boy left the hospital two weeks later in good health. A few weeks after that he returned with the cough. It turned out that he had whooping cough which he had caught from an adult (non-vaccinated) relative. The man had had a bad cough but the cause was not realised and it was attributed to side effects of a drug he was taking for a heart problem. The doctors did everything to save the boy’s life but the infection soon proved fatal. It is important to realize that this is not an absolutely exceptional case but a scenario which happens regularly. It brings home what getting vaccinated (or failing to do so) really means. Of course an example like this has no statistical significance but it can nevertheless help to make people think.

Let me say some more about the mathematics of this situation. A common type of model is the SIR model. The dependent variables are , the number of individuals who are susceptible to infection by the disease, , the number of individuals who are infected (or infectious, this model ignores the incubation time) and , the number of individuals who have recovered from the disease and are therefore immune. These three quantities depend on time and satisfy a system of ODE containing a number of parameters. There is a certain combination of these parameters, usually called the basic reproductive rate (or ratio) and denoted by whose value determines the outcome of the dynamics. If the infection dies out – the solution converges to a steady state on the boundary of the state space where . If, on the other hand, there exists a positive steady state, an endemic equilibrium. The stability this of this steady state can be examined by linearizing about it. In fact it is always stable. Interestingly, more is true. When the endemic steady state exists it is globally asymptotically stable. In other words all solutions with positive initial data converge to that steady state at late time. For a proof of this see a paper by Korobeinikov and Wake (Appl. Math. Lett. 15, 955). They use a Lyapunov function to do so. At this point it is appropriate to mention that my understanding of these things has been improved by the hard work of Irena Vogel, who recently wrote her MSc thesis on the subject of Lyapunov functions in population models under my supervision.