Poincaré, chaos and the limits of predictability

March 5, 2017

In the past I was surprised that there seemed to be no biography of Henri Poincaré. I recently noticed that a biography of him had appeared in 2013. The title is ‘Henri Poincaré. A scientific biography’ and the author is Jeremy Gray. At the moment I have read 390 of the 590 pages. I have learned interesting things from the book but in general I found it rather disappointing. One of the reasons is hinted at by the subtitle ‘A scientific biography’. Compared to what I might have hoped for the book concentrates too much on the science and too little on the man. Perhaps Poincaré kept his private life very much to himself and thus it was not possible to discuss these aspects more but if this is so then I would have found it natural that the book should emphasize this point. I have not noticed anything like that. I also found the discussion of the scientific topics of Poincaré’s work too technical in many places. I would have preferred a presentation of the essential ideas and their significance on a higher level. There are other biographies of great mathematicians which made a better impression on me. I am thinking of the biography of Hilbert by Constance Reid and even of the slim volumes (100 pages each) on Gauss and Klein written in East Germany.

On important discovery of Poincaré was chaos. He discovered it in the context of his work on celestial mechanics and indeed that work was closely connected to his founding the subject of dynamical systems as a new way of approaching ordinary differential equations, emphasizing qualitative and geometric properties in contrast to the combination of complex analysis and algebra which had dominated the subject up to that point. The existence of chaos places limits on predictability and it is remarkable that these do not affect our ability to do science more than they do. For instance it is known that there are chaotic phenomena in the motion of objects belonging to the solar system. This nevertheless does not prevent us from computing the trajectories of the planets and those of space probes sent to the other end of the solar system with high accuracy. These space probes do have control systems which can make small corrections but I nevertheless find it remarkable how much can be computed a priori, although the system as a whole includes chaos.

This issue is part of a bigger question. When we try to obtain a scientific understanding of certain phenomena we are forced to neglect many effects. This is in particular true when setting up a mathematical model. If I model something using ODE then I am, in particular, neglecting spatial effects (which would require partial differential equations) and the fact that often the aim is not to model one particular object but a population of similar objects and I neglect the variation between these objects which I do not have under control and for whose description a stochastic model would be necessary. And of course quantum phenomena are very often neglected. Here I will not try to address these wider issues but I will concentrate on the following more specific question. Suppose I have a system of ODE which is a good description of the real-world situation I want to describe. The evolution of solutions of this system is uniquely determined by initial data. There remains the problem of sensitive dependence on initial data. To be able to make a prediction I would like to know that if I make a small change in the initial data the change in some predicted quantity should be small. What ‘small’ means in practice is fixed by the application. A concrete example is the weather forecast whose essential limits are illustrated mathematically by the Lorenz system, which is one of the icons of chaos. Here the effective limit is a quantitative one: we can get a reasonable weather forecast for a couple of days but not more. More importantly, this time limit is not set by our technology (amount of observational data collected, size of the computer used, sophistication of the numerical programs used) but by the system itself. This time limit will not be relaxed at any time in the future. Thus one way of getting around the effects of chaos is just to restrict the questions we ask by limits on the time scales involved.

Another aspect of this question is that even when we are in a regime where a system of ODE is fully chaotic there will be some aspects of its behaviour which will be predictable. This is why is is possible to talk of ‘chaos theory’- I know too little about this subject to say more about it here. One thing I find intriguing is the question of model reduction. Often it is the case that starting from a system of ODE describing something we can reduce it to an effective model with less variables which still includes essential aspects of the behaviour. If the dimension of the reduced model is one or two then chaos is lost. If there was chaos in the original model how can this be? Has there been some kind of effective averaging? Or have we restricted to a regime (subset of phase space) where chaos is absent? Are the questions we tend to study somehow restricted to chaos-free regions? If the systems being modelled are biological is the prevalence of chaos influenced by the fact that biological systems have evolved? I have seen statements to the effect that biological systems are often ‘on the edge of chaos’, whatever that means.

This post contains many questions and few answers. I just felt the need to bring them up.

Kestrels and Dirichlet boundary conditions

February 28, 2017

The story I tell in this post is based on what I heard a long time ago in a talk by Jonathan Sherratt. References to the original work by Sherratt and his collaborators are Proc. R. Soc. Lond. B269, 327 (more biological) and SIAM J. Appl. Math. 63, 1520 (more mathematical). There are some things I say in the following which I did not find in these sources and so they are based on my memories of that talk and on things which I wrote down for my own reference at intermediate times. If this has introduced errors they will only concern details and not the basic story. The subject is a topic in population biology and how it relates to certain properties of reaction-diffusion equations.

In the north of England there is an area called the Kielder Forest with a lake in the middle and the region around the lake is inhabited by a population of the field vole Microtus\ agrestis. It is well known that populations of voles undergo large fluctuations in time. What is less known is what the spatial dependence is like. There are two alternative scenarios. In the first the population density of voles oscillates in a way which is uniform in space. In the second it is a travelling wave of the form U(x-ct). In that case the population at a fixed point of space oscillates in time but the phase of the oscillations is different at different spatial points. In general there is relatively little observational data on this type of thing. The voles in the Kielder forest are an exception to this since in that case a dedicated observer collected data which provides information on both the temporal and spatial variation of the population density. This data is the basis for the modelling which I will now describe.

The main predators of the voles are weasels Mustela\ nivalis. It is possible to set up a model where the unknowns are the populations of voles and weasels. Their interaction is modelled in a simple way common in predator-prey models. Their spatial motion is described by a diffusion term. In this way a system of reaction-diffusion equations is obtained. These are parabolic equations and to the time evolution is non-local in space. The unknowns are defined on a region with boundary which is the complement of a lake. Because of this we need not only initial values to determine a solution but also boundary conditions. How should they be chosen? In the area around the lake there live certain birds of prey, kestrels. They hunt voles from the air. In most of the area being considered there is very thick vegetation and the voles can easily hide from the kestrels. Thus the direct influence of the kestrels on the vole population is negligible and the kestrels to not need to be included in the reaction-diffusion system. They do, however, have a striking indirect effect. On the edge of the lake there is a narrow strip with little vegetation and any vole which ventures into that area is in great danger of being caught by a kestrel. This means that the kestrels essentially enforce the vanishing of the population density of voles at the edge of the lake. In other words they impose a homogeneous Dirichlet boundary condition on one of the unknowns at the boundary. Note that this is incompatible with spatially uniform oscillations. On the boundary oscillations are ruled out by the Dirichlet condition. When the PDE are solved numerically what is seen that the shore of the lake generates a train of travelling waves which propagate away from it. This can also be understood theoretically, as explained in the papers quoted above.

Conference on cancer immunotherapy at EMBL

February 5, 2017

I just came back from a conference on cancer immunotherapy at EMBL in Heidelberg. It was very interesting for me to get an inside view of what is happening in this field and to learn what some of the hot topics are. One of the speakers was Patrick Baeuerle, who talked about a molecular construct which he introduced, called BiTE (bispecific T cell engager). It is the basis of a drug called blinatumomab. This is an antibody construct which binds both CD3 (characteristic of T cells) and CD19 (characteristic of B cells) so that these cells are brought into proximity. In its therapeutic use in treating acute lymphoblastic leukemia, the B cell is a cancer cell. More generally similar constructs could be made so as to bring T cells into proximity with other cancer cells. The idea is that the T cell should kill the cancer cell and in that context it is natural to think of cytotoxic T cells. It was not clear to me how the T cell is activated since the T cell receptor is not engaged. I took the opportunity to ask Baeuerle about this during a coffee break and he told me that proximity alone is enough to activate T cells. This can work not only for CD8 T cells but also for CD4 cells and even regulatory T cells. He presented a picture of a T cell always being ready to produce toxic substances and just needing a signal to actually do it. Under normal circumstances T cells search the surfaces of other cells for antigens and do not linger long close to any one cell unless they find their antigen. If they do stay longer near another cell for some reason then this can be interpreted as a danger sign and the T cell reacts. Baeuerle, who started his career as a biochemist, was CEO of a company called Micromet whose key product was what became blinatumomab. The company was bought by Amgen for more than a billion dollars and Baeuerle went with it to Amgen. When it came on the market it was the most expensive cancer drug ever up to that time. Later Baeuerle moved to a venture capital firm called MPM Capital, which is where he is now. In his previous life as a biochemical researcher Baeuerle did fundamental work on NF\kappaB  with David Baltimore.

In a previous post I mentioned a video by Ira Mellman. At the conference I had the opportunity to hear him live. One thing which became clear to me at the conference is the extent to which, among the checkpoint inhibitor drugs, anti-PD1 is superior to anti-CTLA. It is successful in a much higher proportion of patients. I never thought much about PD1 before. It is a receptor which is present on the surface of T cells after they have been activated and it can be stimulated by the ligand PD1L leading to the T cell being switched off. But how does this switching off process work? The T cell is normally switched on by the engagement of the T cell receptor and a second signal from CD28. In his talk Mellman explained that the switching off due to PD1 is not due to signalling from the T cell receptor being stopped. Instead what happens is that PD1 activates the phosphatase SHP2 which dephosphorylates and thus deactivates CD28. Even a very short deactivation of CD28 is enough to turn off the T cell. In thinking about mathematical models for T cell activation I thought that there might be a link to checkpoint inhibitors. Now it looks like models for T cell activation are not of direct relevance there and that instead it would be necessary to model CD28.

I learned some more things about viruses and cancer. One is that the Epstein-Barr-virus, famous for causing Burkitt’s lymphoma also causes other types of cancers, in particular other types of lymphoma. Another is that viruses are being used in a therapeutic way. I had heard of oncolytic viruses before but I had never really paid attention. In one talk the speaker showed a picture of a young African man who had been cured of Burkitt’s lymphoma by … getting measles. This gave rise to the idea that viruses can sometimes preferentially kill cancer cells and that they can perhaps be engineered to as to do so more often. In particular measles is a candidate. In that case there is an established safe vaccination and the idea is to vaccinate with genetically modified measles virus to fight certain types of cancer.

In going to this conference my main aim was to improve my background in aspects of biology and medicine which could be of indirect use for my mathematical work. In fact, to my surprise, I met one of the authors of a paper on T cell activation which is closely related to mathematical topics I am interested in. This was Philipp Kruger who is in the group of Omer Dushek in Oxford. I talked to him about the question of what is really the mechanism by which signals actually cross the membrane of T cells. One possibility he mentioned was a conformational change in CD3. Another, which I had already come across is that it could have to do with a mechanical effect by which the binding of a certain molecule could bring the cell membranes of two interacting cells together and expel large phosphatases like CD45 from a certain region. In the paper of his I had looked at signalling in T cells is studied with the help of CAR T-cells, which have an artifical analogue of the T cell receptor which may have a much higher affinity than the natural receptor. In his poster he described a new project looking at the effect of using different co-receptors in CAR T-cells (not just CD28). In any case CAR T-cells was a subject which frequently came up at the conference. Something which was in the air was that this therapy may be associated with neurotoxicity in some cases but I did not learn any details.

As far as I can see, the biggest issue with all these techniques is the following. They can be dramatically successful, taking patients from the point of death to long-term survival. On the other hand they only work in a subset of patients (say, 40% at most) and nobody understands what success depends on. I see a great need for a better theoretical understanding. I can understand that when someone has what looks like a good idea in this area they quickly look for a drug company to do a clinical trial with it. These things can save lives.  On the other hand it is important to ask whether investing more time in obtaining a better understanding of underlying mechanisms might not lead to better results in the long run.

Herbal medicine and its dangers

December 18, 2016

I recently heard a talk by Thomas Efferth of the Institute for Pharmacology of the University of Mainz on herbal medicine. There is a common point of view that substances derived from plants are harmless and good while the chemical drugs of standard medicine are evil. The speaker emphasized that plants have good reasons for not being good to those who eat them. They do not have immune systems of the type we do and they cannot run away and so it is natural that they use poisons to defend themselves. Herbal medicines are effective in some cases but they need to be subject to controls as much as do substances obtained by artificial chemical means. In the talk a number of examples of the dangers of ‘natural’ medicines were presented and I will write about some of them here.

The first example is that of Aristolochia. This a large genus of plants, some of which are poisonous. One of these, Aristolochia clematitis, has been extensively used in herbal medicine. It was used extensively in the west in ancient times and is used in traditional Chinese medicine until today. In the talk the story was told of an incident which happened in Belgium. There was a product sold as a means of losing weight which contained a Chinese plant. It sold so well that the manufacturer’s supplies of the plant were running out. When more was ordered a fateful mistake took place. There are two plants which have the same name in China. The one is that which was originally contained in the weight-loss product. The other is the poisonous Aristolochia fangchi and it was the one which was delivered. This led to more than 100 cases of kidney failure in the people using the product. Another way in which plants can be dangerous is as weeds in crop fields. In the Balkans contamination of grain with Aristolochia clematitis led to a kidney disease called Balkan nephropathy, with 35000 recorded cases. The substance, aristolochic acid, which is responsible for the kidney toxicity is also known to be a strong carcinogen. Interestingly, this substance is not poisonous for everyone and its bad effects depend a lot on the variability in liver enzymes among individuals.

A class of substances used by many plants to protect themselves against insects are the pyrrolizidine alkaloids. These substances are hepatotoxic and carcinogenic. They may move through the food chain being found, for instance, in honey. It has been noted that there may be risks associated to the amount of these substances contained in medicinal herbs used both in the West and in China. It was mentioned in the talk that drinking too much of certain types of herbal tea may be damaging to health. The problem is usually not the plants that are the main components of the teas but other plants which may be harvested with them in small quantities. There is at least one exception to this, namely coltsfoot (Tussilago farfara). In one case the death of an infant due to liver disease is believed to be due to the mother drinking this type of tea during pregnancy. After that the sale of coltsfoot was banned in Germany.

There were some remarks in the talk on heavy metals which I found quite suprising. One concerned ayurvedic medicine which has an aura of being gentle and harmless. In fact in many of these substances certain heavy metals are added delibrately (lead, mercury and arsenic). According to Wikipedia more than 80 cases of lead poisoning due to ayurvedic ‘medicines’ have been recorded. Another remark was that there can be significant concentrations of heavy metals in tobacco smoke. The negative health effects of smoking are sufficiently well known but this aspect was new to me.

Another theme in the talk was interactions between herbal medicines and normal drugs. Apparently it is often the case that patients who use herbal remedies are afraid to mention this to their doctors since they think this may spoil the relationship to their practitioner. Then it can happen that a doctor is suprised by the fact that a drug he prescribes is not working as expected. Little does he know that the patient is secretly taking a ‘natural’ drug in parallel. An example is St. John’s wort which is sometimes taken as a remedy for depression. It may work and it has no direct negative effects but it can be problematic because it reduces the effects of other drugs taken at the same time, e.g. the contraceptive pill. It changes the activity of liver enzymes and causes them to eliminate other drugs from the body faster than would normally happen, thus causing an effective reduction of the dose.

We are surrounded by poisonous plants. I was always sceptical of the positive effects of ‘natural’, plant-derived medicines. Now I have realised how seriously the dangers of these substances should be taken.

Honesty is the best policy

December 10, 2016

Yesterday I did the following thought experiment. I imagined a situation where someone asked me two questions, saying that I should answer spontaneously without thinking too long. The first question was ‘Are you happy with your life?’ and the second ‘Are you happy with the society around you?’ My answer to the first question was ‘yes’ and to the second ‘no’. I then started thinking about the cause of the discrepancy between these two answers. I came to the conclusion that it has a lot to do with the concept of ‘honesty’. I believe in and live according to the phrase in the title of this post, ‘honesty is the best policy’ while I feel that in the society around me lies have a huge influence. It is also worth remarking that lies are not the only kind of dishonesty.

If I am honest what is the reason? One important influence is my upbringing. I grew up in a family which was very attached to telling the truth. Here the influence of my mother was particularly strong. What influenced me was not so much what she said on the subject as the example of how she behaved. My mother’s attachment to honesty had a lot to do with her attachment to religion. I did not inherit her religion, becoming an atheist in my teens. I also did not inherit her moral convictions but I did inherit certain patterns of moral behaviour. One thing that stops me telling lies is simply that I find it very unpleasant due to childhood conditioning. Since truth plays a central role in mathematics it is perhaps natural that mathematicians should tend to be truthful, also in everyday life. It might also conversely be the case that among people who go into academia those with a specially strong attachment to truth might tend to go in the direction of mathematics.

Probably the main motivation for lying is the hope to gain some advantage by doing so. This may be short-sighted if despite a short-term profit the net long-term payment is negative. The idea that this is often the case is one motivation for me not to lie. Another is the fact that lies require management in order to profit from them. It is necessary to remember the lies you told so as not to betray yourself and it is also necessary to remember the corresponding true version. A lot of profitable lies require a lot of management and this is stress which I like to avoid. More thoughts in similar directions can be found on this web page
http://www.becomingminimalist.com/why-honesty-is-the-best-policy-for-simplicity/

What types of dishonesty in society bother me? One is political correctness. It means sanctions againt telling the truth, or just speaking plainly rather than in euphemisms, in many situations. There are some cases where there may be good reasons for measures like this but I think that in the majority of cases there are no good reasons. They are based on arbitrary conventions or at best on misguided ideas of well-meaning people. I prefer to speak openly but I often avoid doing so and simply keep quiet in order to avoid problems. I would prefer if this was not so often necessary. A related topic is that of reference letters. When I write references for people applying for academic jobs I generally feel free to tell the truth. For jobs outside academia things are very different. There telling the truth might easily lead to disaster since open criticism is often effectively forbidden. Instead it is necessary to write in code and even then the information which might be helpful for the employer, or for the applicant, might not get through. In other situations it is necessary to be careful when criticising people but it is wrong not to criticise. Constructive criticism can be good. I am happy when I receive constructive criticism although it may be unpleasant at the moment it arrives. I believe that it is also important to make statements in certain situations like ‘in my opinion person X is better than person Y at doing task Z’. This is not a comparison of the value of the two people in general but just in the context of a particular ability.

One other theme I want to mention is marriage, because my marriage is one of the things in my life which contributes most to my happiness. When I use the word ‘marriage’ here I mean it to denote a long-term romantic relationship, not only one which is recognised legally by a piece of paper. I have been married in the latter sense for eight years but the underlying relationship goes back sixteen years. I would not dare to write here about individual marriages (apart from my own) but I will say something about averages. It seems that with time marriages in our society become less and less stable and last for shorter times. I believe in marriage in the old-fashioned sense of ’till death do us part’ and I think it is very unfortunate for many people that they have replaced this by a sequence of shorter-term relationships accompanied by difficult separations. The relation of this topic to that of honesty is as follows. I think that separations often result from the fact that the people involved are pursuing short term gains at the expense of the partner. Then the short-term gains turn into long-term losses. In a talk I heard recently the speaker voiced the opinion that the kind of degradation of marriage (or of love) I have been talking about is due to the fact that many people going into relationships have not had the experience of good marriages in their childhood. I had the advantage of growing up in surroundings where this type of relationship was widespread and I had very direct experience of how it was in the case of my own parents.

Photorespiration and RuBisCO

November 29, 2016

The enzyme at the centre of the fixation of carbon in photosynthesis (which is the ultimate source of our food) is RuBisCO (Ribulose Bisphosphate Carboxylase Oxygenase). It is sometimes called the ‘lazy enzyme’ because it works so slowly. This is unjust since the process is so difficult that RuBisCO is the only enzyme that can do it. So it is like the situation of a strong man who is the only one who can carry very heavy rocks and who is called lazy because it takes him so long to move the rocks. The name already indicates that this enzyme catalyses two different reactions with the same substrate, ribulose bisphosphate. So it might also be claimed that it does not concentrate well. Instead of spending all its time on the useful process of carboxylation it apparently wastes a lot of time on the hobby of oxygenation. Carboxylation produces two molecules of phosphoglycerate (PGA) by combining a molecule of carbon dioxide with one of ribulose bisphosphate. In the oxygenation reaction only one molecule of PGA is produced together with one of phosphoglycolate. At least superficially the latter substance is not only useless but actual causes the cell a lot of trouble getting rid of it. I have not seen a definitive explanation about what purpose this process, photorespiration, might serve. One idea is that it might act as a safety valve. A too large input of energy to the photosynthesis system might damage it and photorespiration can be used to dissipate this energy in a relatively harmless way when necessary. The relative rates of the two reactions are influenced by the concentration of carbon dioxide and for normal atmospheric concentrations the relative rate of photorespiration is quite high.

The discussion up to now has been appropriate for what is called C_3 photosynthesis. The name comes from the fact that the first molecule to be produced is the three-carbon molecule PGA. There is an alternative called C_4 photosynthesis. This also contributes to food production, even to the food I eat. My generic breakfast is cornflakes and the most economically important plant which uses C_4 photosynthesis is maize. In this process carbon dioxide is used to make malate. Malate is not directly useful but is an intermediate. It is transported to an internal compartment (bundle-sheath cells) where it releases carbon dioxide. The result is an increased concentration of carbon dioxide and RuBisCO, which is waiting there, is pushed towards doing more carboxylation at the expense of oxygenation. Since the C_4 process leads to greater crop yields under certain conditions there is a project to genetically engineer rice so as to produce a C_4 variant. There is third type of photosynthesis , CAM photosynthesis for crassulacean acid metabolism. It is used, for instance by pineapples. There are similarities with C_4 with the difference that while in C_4 carbon dioxide is harvested in one place and released in another in CAM it is harvested at one time (during the night) and released at another (during the day).

The importance of dendritic cells

October 30, 2016

I just realized that something I wrote in a previous post does not make logical sense. This was not just due to a gap in my exposition but to a gap in my understanding. I now want to correct it. A good source for the correct story is a video by Ira Mellman of Genentech. I first recall some standard things about antigen presentation. In this process peptides are presented on the surface of cells with MHC molecules which are of two types I and II. MHC Class I molecules are found on essentially all cells and can present proteins coming from viruses infecting the cell concerned. MHC Class II molecules are found only on special cells called professional antigen presenting cells. These are macrophages, T cells and dendritic cells. The champions in antigen presentations are the dendritic cells and those are the ones I will be talking about here. In order for a T cell to be activated it needs two signals. The first comes through the T cell receptor interacting with the peptide-MHC complex on an APC. The second comes from CD28 on the T cell surface interacting with B7.1 and B7.2 on the APC.

Consider now an ordinary cell, not an APC, which is infected with a virus. This could, for instance be an epithelial cell infected with a influenza virus. This cell will present peptides derived from the virus with MHC Class I molecules. These can be recognized by activated {\rm CD8}^+ T cells which can then kill the epithelial cell and put an end to the viral reproduction in that cell. The way I put it in the previous post it looked like the T cell could be activated by the antigen presented on the target cell with the help of CD28 stimulation. The problem is that the cell presenting the antigen in this case is an amateur. It has no B7.1 or B7.2 and so cannot signal through CD28. The real story is more complicated. The fact is that dendritic cells can also present antigen on MHC Class I, including peptides which are external to their own function. A possible mechanism explained in the video of Mellman (I do not know if it is certain whether this is the mechanism, or whether it is the only one) is that a cell infected by a virus is ingested by a dendritic cell by phagocytosis, so that proteins which were outside the dendritic cell are now inside and can be brought into the pathway of MHC Class I presentation. This process is known as cross presentation. Dendritic cells also have tools of the innate immune system, such as toll-like receptors, at their disposal. When they recognise the presence of a virus by these means they upregulate B7.1 and B7.2 and are then in a position to activate {\rm CD8}^+ T cells. Note that in this case the virus will be inside the dendritic cell but not infecting it. There are viruses which use dendritic cells for their own purposes, reproducing there or hitching a lift to the lymph nodes where they can infect their favourite cells. An example is HIV. The main receptor used by this virus to enter the cells is CD4 and this is present not only on T cells but also on dendritic cells. Another interesting side issue is that dendritic cells can not only activate T cells but also influence the differentiation of these cells into various different types. The reason is that the detection instruments of the dendritic cell not only recognise that a pathogen is there but can also classify it to some extent (Mellman talks about a bar code). Based on this information the dendritic cell secretes various cytokines which influence the differentiation process. For instance they can influence whether a T-helper cell becomes of type Th1 or Th2. This is related to work which I did quite a long time ago on an ODE system modelling the interactions of T cells and macrophages. In view of what I just said it ḿight be interesting to study an inhomogeneous version of this system. The idea is to include an external input of cytokines coming from dendritic cells. In fact the unknowns in the system are not the concentrations of cytokines but the populations of cells. Thus it would be appropriate to introduce an inhomogeneous contribution into the terms describing the production of different types of cells.

Modern cancer therapies

October 28, 2016

I find the subject of cancer therapies fascinating. My particular interest is in the possibility of obtaining new insights by modelling and what role mathematics can play in this endeavour. I have heard many talks related to these subjects, both live and online. I was stimulated to write this post by a video of Martin McMahon, then at UCSF. It made me want to systematize some of the knowledge I have obtained from that video (which is already a few years old) and from other sources. First I should fix my terminology. I use the term ‘modern cancer therapies’ to distinguish a certain group of treatments from what I will call ‘classical cancer therapies’. The latter are still of central importance today and the characteristic feature of those I am calling ‘modern’ here is that they have only been developed in the last few years. I start by reviewing the ‘classical therapies’, surgery, radiotherapy and chemotherapy. Surgery can be very successful when it works. The aim is to remove all the cancerous cells. There is a tension between removing too little (so that a few malignant cells could remain and restart the tumour) and too much (which could mean too much damage to healthy tissues). A particularly difficult case is that of the glioma where it is impossible to determine the extent of the tumour by imaging techniques alone. An alternative to this is provided by the work of Kristin Swanson, which I mentioned in a previous post. She has developed techniques of using a mathematical model of the tumour (with reaction-diffusion equations) to predict the extent of the tumour. The results of a simulation, specific to a particular patient, is given to the surgeon to guide his work. In the case of radiotherapy radiation is used to kill cancer cells while trying to avoid killing too many healthy cells. A problematic aspect is that the cells are killed by damaging their DNA and this kind of damage may lead to the development of new cancers. In chemotherapy a chemical substance (poison) is used with the same basic aim as in radiotherapy. The substance is chosen to have the greatest effect on cells which divide frequently. This is the case with cancer cells but unfortunately they are not the only ones. A problem with radiotherapy and chemotherapy is their poor specificity.

Now I come to the ‘modern’ therapies. One class of substances used is that of kinase inhibitors. The underlying idea is as follows. Whether cells divide or not is controlled by a signal transduction network, a complicated set of chemical reactions in the cell. In the course of time mutations can accumulate in a cell and when enough relevant mutations are present the transduction network is disrupted. The cell is instructed to divide under circumstances under which it would normally not do so. The cells dividing in an uncontrolled way constitute cancer. The signals in this type of network are often passed on by phosphorylation, the attachment of phosphate groups to certain proteins. The enzymes which catalyse the process of phosphorylation are called kinases. A typical problem then is that due to a mutation a kinase is active all the time and not just when it should be. A switch which activates the signalling network is stuck in the ‘on’ position. This can in principal be changed by blocking the kinase so that it can no longer send its signals. An early and successful example of this is the kinase inhibitor imatinib which was developed as therapy for chronic myelogenous leukemia (CML). It seems that this drug can even cure CML in many cases, in the sense that after a time (two years) no mutated cells can be detected and the disease does not come back if the treatment is stopped. McMahon talks about this while being understandibly cautious about using the word cure in the context of any type of cancer. One general point about the ‘modern’ therapies is that they do not work for a wide range of cancers or even for the majority of patients with a given type of cancer. It is rather the case that cancer can be divided into more and more subtypes by analysing it with molecular methods and the therapy only works in a very specific class of patients, having a specific mutation. I have said something about another treatment using a kinase, Vemurafenib in a previous post. An unfortunate aspect of the therapies using kinase inhibitors is that while they provide spectacular short-term successes their effects often do not last more than a few months due to the development of resistance. A second mutation can rewire the network and overcome the blockade. (Might mathematical models be used to understand better which types of rewiring are relevant?) The picture of this I had, which now appears to me to be wrong, was that after a while on the drug a new mutation appears which gives the resistance. The picture I got from McMahon’s video was a different one. It seems that the mutations which might lead to resistance are often there before treatment begins. They were in Darwinian competition with other cells without the second mutation which were fitter. The treatment causes the fitness of the cells without the second mutation to decrease sharply. This removes the competition and allows the population of resistant cells to increase.

Another drug mentioned by McMahon is herceptin. This is used to treat breast cancer patients with a mutation in a particular receptor. The drug is an antibody and binds to the receptor. As far as I can see it is not known why the binding of the antibody has a therapeutic effect but there is one idea on this which I find attractive. This is that the antibodies attract immune cells which kill the cell carrying the mutation. This gives me a perfect transition to a discussion of a class of therapies which started to become successful and popular very recently and go under the name of cancer immunotherapy, since they are based on the idea of persuading immune cells to attack cancer cells. I have already discussed one way of doing this, using antibodies to increase the activities of T cells, in a previous post. Rather than saying more about that I want to go on to the topic of genetically modified T cells, which was also mentioned briefly here.

I do not know enough to be able to give a broad review of cellular immunotherapy for cancer treatment and so I will concentrate on making some comments based on a video on this subject by Stephan Grupp. He is talking about the therapy of acute lymphocytic leukemia (ALL). In particular he is concerned with B cell leukemia. The idea is to make artificial T cells which recognise the surface molecule CD19 characteristic of B cells. T cells are taken from the patient and modified to express a chimeric T cell receptor (CAR). The CAR is made of an external part coming from an antibody fused to an internal part including a CD3 \zeta-chain and a costimulatory molecule such as CD28. (Grupp prefers a different costimulatory molecule.) The cells are activated and caused to proliferate in vitro and then injected back into the patient. In many cases they are successful in killing the B cells of the patient and producing a lasting remission. It should be noted that most of the patients are small children and that most cases can be treated very effectively with classical chemotherapy. The children being treated with immunotherapy are the ‘worst cases’. The first patient treated by Grupp with this method was a seven year old girl and the treatment was finally very successful. Nevertheless it did at first almost kill her and this is not the only case. The problem was a cytokine release syndrome with extremely high levels of IL-6. Fortunately this was discovered just in time and she was treated with an antibody to IL-6 which not only existed but was approved for the treatment of children (with other diseases). It very quickly solved the problem. One issue which remains to be mentioned is that when the treatment is successful the T cells are so effective that the patient is left without B cells. Hence as long as the treatment continues immunoglobulin replacement therapy is necessary. Thus the issue arises whether this can be a final treatment or whether it should be seen a preparation for a bone marrow transplant. As a side issue from this story I wonder if modelling could bring some more insight for the IL-6 problem. Grupp uses some network language in talking about it, saying that the problem is a ‘simple feedback loop’. After I had written this I discovered a preprint on BioRxiv doing mathematical modelling of CAR T cell therapy of B-ALL and promising to do more in the future. It is an ODE model where there is no explicit inclusion of IL-6 but rather a generic inflammation variable.

In the beginning was the worm

September 29, 2016

In a previous post I mentioned the book by Andrew Brown whose title I have used here. I came across it in a second hand bookshop in Berkeley when I was spending time at MSRI in 2009. I read it with pleasure then and now I have read it again. It contains the story of how the worm Caenorhabditis elegans became an important model organism. This came about because Sydney Brenner deliberately searched for an organism with favourable properties and promoted it very effectively once he had found it. It is transparent so that it is possible to see what is going on inside it and it is easy to keep in the lab and reproduces fast enough in order to allow genetic research to be done rapidly. The organism sought was supposed to have a suitable sexual system. C. elegans is normally hermaphrodite but does also have males and so it is acceptable from that point of view. One further important fact about C. elegans is that it has a nervous system, albeit a relatively simple one. (More precisely, it has two nervous systems but I have not looked into the details of that issue.) Brenner was looking to understand how genetics determines behaviour and C. elegans gave him an opportunity to make an attack on this problem in two steps. First understand how to get from genes to neurons and then understand how to get from neurons to behaviour. C. elegans has a total of 302 neurons. It has 959 cells in total, not including eggs and sperm. Among the remarkable things known about the worm are the complete developmental history of each of its cells and the wiring diagram of its neurons. There are about 6400 synapses but the exact number, unlike the number of cells or neurons, is dependent on the individual. For orientation note that C. elegans is a eukaryotic organism (in contrast to phages or E. coli) which is multicellular (in contrast to Saccharomyces cerevisiae) and it is an animal (in contrast to Arabidopsis thaliana). Otherwise, among the class of model organisms, it is as simple and fast reproducing as possible. In particular it is simpler than Drosophila, which was traditionally the favourite multicellular model organism of the geneticists.

In this blog I have previously mentioned Sydney Brenner and expressed my admiration for him. I have twice met him personally when he was giving talks in Berlin and I have also watched a number of videos of him which are available on the web and read various texts he has written. In this way I have experienced a little of the magnetism which allowed him to inspire gifted and risk-taking young scientists to work on the worm. Brenner spent 20 years at the Laboratory of Molecular Biology in Cambridge, a large part of it as director of that organization. In the pioneering days of molecular biology the lab was producing Nobel prizes in series. He had to wait until 2002 for his own Nobel prize (for physiology or medicine), shared with John Sulston and Robert Horvitz. In his Nobel speech Brenner said that he felt there was a fourth prizewinner, C. elegans, which, however, did not get a share of the money. My other favourite quote from that speech is his description of the (then) present state of molecular biology, ‘drowning in a sea of data, starving for knowledge’. Since then that problem has only got worse.

Now I will collect some ‘firsts’ associated with C. elegans. It was the first multicellular organism to have its whole genome sequenced, in 1998. This can also be seen as the point of departure for the human genome project. Here the worm people overtook the drosophilists and the Drosophila genome was only finished in 2000. Sulston played a central role in the public project to sequence the human genome and the struggle with the commercial project of Craig Venter. It was only the link between the worm genome project and the human one which allowed enough money to be raised to finish the worm sequence. According to the book Sulston was more interested in the worm project since he wanted to properly finish what he had started. Martin Chalfie, coming from the worm community introduced GFP (green fluorescent protein) into molecular biology. He first expressed it in E. coli and C. elegans. He got a Nobel prize for that in 2008. microRNA (miRNA) was first found in C. elegans. It is the basis of RNA interference (RNAi), also first found in C. elegans. This earned a Nobel prize in 2006. The genetics of the process of apoptosis (programmed cell death) was understood by studying C. elegans. When Sulston was investigated the cell lineage he saw that certain cells had to die as part of the developmental process. Exactly 131 cells die during this process.

To conclude I mention a couple of features of C. elegans going beyond the time covered by the book. I asked myself what we can learn about the immune system from C. elegans. Presumably every living organism needs an immune system to survive in a hostile environment. The adaptive immune system in the form known in humans only exists in vertebrates and hence, in particular, not in the worm. Some related comments can be found here. It seems that C. elegans has no adaptive immune system at all but it does have innate immunity. It has cells called coelomocytes which have at least some resemblance to immune cells. It has six of them in total. Compare this with more than 10^9 immune cells per litre in our blood. C. elegans eats bacteria. These days the human gut flora is a fashionable topic. A couple of weeks ago I heard a talk by Giulia Enders, the author of the book ‘Darm mit Charme’ which sold a million copies in 2014. I had bought and read the book and found it interesting although I was not really enthusiastic about it. Now TV advertising includes products aimed at the gut flora of cats. So what about C. elegans? Does it have an interesting gut flora? The answer seems to be yes. See for instance the 2013 article ‘Worms need microbes too’ in EMBO Mol. Med. 5, 1300.

Models for photosynthesis, part 4

September 19, 2016

In previous posts in this series I introduced some models for the Calvin cycle of photosynthesis and mentioned some open questions concerning them. I have now written a paper where on the one hand I survey a number of mathematical models for the Calvin cycle and the relations between them and on the other hand I am able to provide answers to some of the open questions. One question was that of the definition of the Pettersson model. As I indicated previously this was not clear from the literature. My answer to the question is that this system should be treated as a system of DAE (differential-algebraic equations). In other words it can be written as a set of ODE \dot x=f(x,y) coupled to a set of algebraic equations g(x,y)=0. In general it is not clear that this type of system is locally well-posed. In other words, given a pair (x_0,y_0) with g(x_0,y_0)=0 it is not clear whether there is a solution (x(t),y(t)) of the system, local in time, with x(0)=x_0 and y(0)=y_0. Of course if the partial derivative of g with repect to y is invertible it follows by the implicit function theorem that g(x,y)=0 is locally equivalent to a relation y=h(x) and the original system is equivalent to \dot x=f(x,h(x)). Then local well-posedness is clear. The calculations in the 1988 paper of Pettersson and Ryde-Pettersson indicate that this should be true for the Pettersson model but there are details missing in the paper and I have not (yet) been able to supply these. The conservative strategy is then to stick to the DAE picture. Then we do not have a basis for studying the dynamics but at least we have a well-defined system of equations and it is meaningful to discuss its steady states.

I was able to prove that there are parameter values for which the Pettersson model has at least two distinct positive steady states. In doing this I was helped by an earlier (1987) paper of Pettersson and Ryde-Pettersson. The idea is to shut off the process of storage as starch so as to get a subnetwork. If two steady states can be obtained for this modified system we may be able to get steady states for the original system using the implicit function theorem. There are some more complications but the a key step in the proof is the one just described. So how do we get steady states for the modified system? The idea is to solve many of the equations explicitly so that the problem reduces to a single equation for one unknown, the concentration of DHAP. (When applying the implicit function theorem we have to use a system of two equations for two unknowns.) In the end we are left with a quadratic equation and we can arrange for the coefficients in that equation to have convenient properties by choosing the parameters in the dynamical system suitably. This approach can be put in a wider context using the concept of stoichiometric generators but the proof is not logically dependent on using the theory of those objects.

Having got some information about the Pettersson model we may ask what happens when we go over to the Poolman model. The Poolman model is a system of ODE from the start and so we do not have any conceptual problems in that case. The method of construction of steady states can be adapted rather easily so as to apply to the system of DAE related to the Poolman model (let us call it the reduced Poolman model since it can be expressed as a singular limit of the Poolman model). The result is that there are parameter values for which the reduced Poolman model has at least three steady states. Whether the Poolman model itself can have three steady states is not yet clear since it is not clear whether the transverse eigenvalues (in the sense of GSPT) are all non-zero.

By analogy with known facts the following intuitive picture can be developed. Note, however, that this intuition has not yet been confirmed by proofs. In the picture one of the positive steady states of the Pettersson model is stable and the other unstable. Steady states on the boundary where some concentrations are zero are stable. Under the perturbation from the Pettersson model to the reduced Poolman model an additional stable positive steady state bifurcates from the boundary and joins the other two. This picture may be an oversimplification but I hope that it contains some grain of truth.