## 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.

## A visit to Iceland

September 4, 2016

I just returned from 10 days as a tourist in Iceland. It was not my first time there. Years ago I went on a cruise which included two stops in Iceland, one in Reykjavik and one in Akureyri. In each case there was a short bus tour to see some typical sights – a waterfall, a geysir and a volcanic region with bubbling mud and hot springs. This time I had a chance to see a lot more. I enjoyed the cruise a lot but I had the impression that the majority of the people on the ship were very bored. The main antidote to the boredom offered was lots of food. One day I got up from the dinner table and went on deck. There had been no indication that there might be something interesting to see. When I opened the door I was confronted with a beautifully conical volcano covered with ice, Snaefellsjökull. (I will come back to the interesting issue of Icelandic pronuciation later.) This was one of my strongest impressions from the whole cruise. The volcano is at the end of a long peninsula to the north of Reykjavik. This time I learned that this volcano is the esoteric centre of Iceland and that it was the starting point of Jules Verne’s Journey to the Centre of the Earth. This was not really a point on the programme of the tour this time but it was clearly visible with binoculars from the hotel in Rekjavik where we spent the last two nights. I had one view of it which was monochrome due to the light conditions but where the sharp edges of the crater stood our clearly.

One attraction of Iceland for me was the bird life. It seems that the country has claimed exclusive rights on the puffin. The numerous tourist shops in Reykjavik are called ‘puffin shops’ due to the number of representations of that bird they sell. I also saw puffin offered as one of the constituents of a special menu also including whale and horse meat. We spent one night in Vik and I discovered two stranded fulmars on a grass area not far from the hotel. These birds can only take off from an elevated starting point like a cliff or from water. If one lands on a flat area some distance from the sea then it is doomed unless it gets help. I rescued two of them by carrying them (ten minutes walk including crossing a road with significant traffic) to the sea. Since there had not been a big storm I suppose they had come down during their first flight after leaving the nest. (There were fulmars nesting on inland cliffs on the other side of the hotel from the sea.) Fortunately I still knew how to catch them and pick them up without hurting them or being the victim of their defence mechanism of spitting foul-smelling oil when feeling threatened. I enjoyed seeing a few glaucous gulls in the harbour in Reykjavik on the last day. Probably the last time I saw any was on the cruise I already mentioned. It was also nice to see and hear many whimbrels. The first one already welcomed me at the airport when I arrived.

I felt at home in the natural surroundings in Iceland and after a few days I thought of one explanation. There are very few trees in Iceland and this is just as it is in Orkney where I grew up. The first time I was on the mainland of Scotland when I was four years old I said ‘I don’t like this place – you can’t see anything for trees’. There are many areas in Iceland where there is only sand, rocks, water and ice. I had the impression of seeing what the Earth is really like, without the veil of green which we usually see. I also got an impression of what it is really like to live next to a volcano. I was in a museum close to (and devoted to) Eyafjallajökull, the infamous producer of ash with the complicated name. I learned that one US journalist just called it E15, due to the number of letters. There was a film showing in the museum explaining what people living near the volcano experienced at the time of the last eruption. Coming back to the name, the pronunciation of Icelandic does seems to be a difficult question but also an interesting one. I would like to spend some time understanding it better. There are nice videos on the pronuciation of Eyafjallajökull here and here. The final double l is the really tricky point. There are points of similarity between the Icelandic language and the dialect I grew up with. This is due to the influence of an extinct language called Norn which was spoken on Orkney and Shetland in past centuries and which is related to (Old) Icelandic. For instance the oystercatcher is called tjaldur in Icelandic and chaldro in our dialect.

I also had some culinary experiences. At breakfast in the hotels there was always a bottle of cod liver oil on the table. I remember this liquid from my childhood as a threat used on young children. ‘If you do not behave yourself I will give you a spoonfull of cod liver oil.’ Due to persistent encouragement from Eva I tried a little and found it not as bad as I expected. Our guide also gave us some pieces of Greenland shark to try. He gave  us a warning about the taste and some of the alcholic drink called the black death to wash it down with. It tastes of nothing at first but chewing leads to a strong taste reminiscent of urine. In fact the flesh of the shark is poisonous due to its content of trimethylamine N-oxide. In Iceland it is treated by first burying it for several weeks and then drying it to get rid of the poison. The result is considered a delicacy. The Greenland shark is interesting because of the fact that it was recently discovered that it can live to be four hundred years old, only becoming sexually mature when it is 150. I want to read more about it.

As a final comment on Iceland: the weather was much better than we expected!

## ECMTB 2016 in Nottingham

July 17, 2016

This past week I attended a conference of the ESMTB and the SMB in Nottingham. My accomodation was in a hall of residence on the campus and my plan was to take a tram from the train station. When I arrived it turned out that the trams were not running. I did not find out the exact reason but it seemed that it was a problem which would not be solved quickly. Instead of finding out what bus I should take and where I should take it from I checked out the possibility of walking. As it turned out it was neither unreasonably far nor complicated. Thus, following my vocation as pedestrian, I walked there.

Among the plenary talks at the conference was one by Hisashi Ohtsuki on the evolution of social norms. Although I am a great believer in the application of mathematics to many real world problems I do become a bit sceptical when the area of application goes in the direction of sociology or psychology. Accordingly I went to the talk with rather negative expectations but I was pleasantly surprised. The speaker explained how he has been able to apply evolutionary game theory to obtain insights into the evolution of cooperation in human societies under the influence of indirect reciprocity. This means that instead of the simple direct pattern ‘A helps B and thus motivates B to help A’ we have ‘C sees A helping B and hence decides to help A’ and variations on that pattern. The central idea of the work is to compare many different strategies in the context of a mathematical model and thus obtain ideas about what are the important mechanisms at work. My impression was that this is a case where mathematics has generated helpful ideas in understanding the phenomenon and that there remain a lot of interesting things to be done in that direction. It also made me reflect on my own personal strategies when interacting with other people. Apart from the interesting content the talk was also made more interesting by the speaker’s entertaining accounts of experiments which have been done to compare with the results of the modelling. During the talk the speaker mentioned self-referentially that the fact of his standing in front of us giving the talk was an example of the process of the formation of a reputation being described in the talk. As far as I am concerned he succeeded in creating a positive reputation both for himself and for his field.

Apart from this the other plenary talk which I found most interesting was by Johan van de Koppel. He was talking about pattern formation in ecology and, in particular, about his own work on pattern formation in mussel beds. A talk which I liked much less was that of Adelia Sequeira and it is perhaps interesting to ask why. She was talking about modelling of atherosclerosis. She made the valid point near the beginning of her lecture that while heart disease is a health problem of comparable importance to cancer in the developed world the latter theme was represented much more strongly than the former at the conference. For me cancer is simply much more interesting than heart disease and this point of view is maybe more widespread. What could be the reason? One possibility is that the study of cancer involves many more conceptual aspects than that of heart disease and that this is attractive for mathematicians. Another could be that I am a lot more afraid of being diagnosed with cancer some day than of being diagnosed with heart disease although the latter may be no less probable and not less deadly if it happens. To come back to the talk I found that the material was too abundant and too technical and that many ideas were used without really being introduced. The consequence of these factors was that I lost interest and had difficulty not falling asleep.

In the case of the parallel talks there were seventeen sessions in parallel and I generally decided to go to whole sessions rather than trying to go to individual talks. I will make some remarks about some of the things I heard there. I found the first session I went to, on tumour-immune dynamics, rather disappointing but the last talk in the session, by Shalla Hanson was a notable exception. The subject was CAR T-cells and what mathematical modelling might contribute to improving therapy. I found both the content and the presentation excellent. The presentation packed in a lot of material but rather than being overwhelmed I found myself waiting eagerly for what would come next. During the talk I thought of a couple of questions which I might ask at the end but they were answered in due course during the lecture. It is a quality I admire in a speaker to be able to anticipate the questions which the audience may ask and answer them. I see this less as a matter of understanding the psychology of the audience (which can sometimes be important) and rather of really having got to the heart of the subject being described. There was a session on mathematical pharmacology which I found interesting, in particular the talks of Tom Snowden on systems pharmacology and that of Wilhelm Huisinga on multidrug therapies for HIV. In a session on mathematical and systems immunology Grant Lythe discussed the fascinating question of how to estimate the number of T cell clones in the body and what mathematics can contribute to this beyond just analysing the data statistically. I enjoyed the session on virus dynamics, particularly a talk by Harel Dahari on hepatitis C. In particular he told a story in which he was involved in curing one exceptional HCV patient with a one-off therapy using a substance called silibinin and real-time mathematical modelling.

I myself gave a talk about dinosaurs. Since this is work which is at a relatively early stage I will leave describing more details of it in this blog to a later date.

## An eternal pedestrian

June 13, 2016

I am presently visiting Japan. My host is Atsushi Mochizuki who leads the Theoretical Biology Laboratory at RIKEN in Wako near Tokyo. RIKEN is a research organisation which was founded in 1917 using the Kaiser-Wilhelm-Gesellschaft as a model. Thus it is a kind of Japanese analogue of the Max Planck Society which is the direct descendant of the Kaiser-Wilhelm-Gesellschaft. I had only been in Japan once before and looking at my records I see that that was in August 2005. At that time I attended a conference in Sendai, a place which I had never heard of before I went there. Since then it has become sadly famous in connection with the damage it suffered from the tsunami which also caused the Fukushima nuclear disaster. At least I had even then previously heard of Tohoku University which is located in the city.

Yesterday, sitting by the river in Wako, I was feeling quite meditative. I was in an area where motor vehicles are not permitted. There were not many people around but most of those who were there were on bikes. I started thinking of how this is typical of what I have experienced in many places I have been. On a walk along the Rhine in Mainz or in the surrounding countryside most of the people you see are on bikes. Copenhagen is completely dominated by bikes. In the US cars dominate. For instance when I was in Miami for a conference and was staying at the Biltmore Hotel I had to walk quite a distance to get dinner for an affordable price. In general the only people I met walking on the streets there were other conference participants. When I visited the University of California at Santa Barbara bikes were not the thing on the campus but it was typical to see students with skateboards. Summing up, I have frequently had the experience that as a pedestrian I was an exception. It seems that for normal people just putting one foot in front of the other is not the thing to do. They need some device such as a car, a bike or a skateboard to accompany them. I, on the other hand, am an eternal pedestrian. I like to walk places whenever I can. I walk twenty minutes to work each day and twenty minutes back. I find that a good way of framing the day. When I lived in Berlin there was a long period when I had a one-way travelling time of 90 minutes by train. I am glad to have that behind me. I did not consciously plan being so near to work in Mainz but I am glad it happened. Of course being a pedestrian has its limits – I could not have come to Japan on foot.

My pedestrian nature is not limited to the literal interpretation of the term. I am also an intellectual pedestrian. An example of this is described in my post on low throughput biology. Interestingly this post has got a lot of hits, more than twice as many as any other post on my blog. This is related to the theme of simple and complex models in biology. Through the talks I have given recently in Copenhagen, Berlin and here in Japan and resulting discussions with different people I have become of conscious of how this is a recurring theme in those parts of mathematical biology which I find interesting. The pedestrian may not get as far as others but he often sees more in the places he does reach. He may also get to places that others do not. Travelling fast along the road may cause you to overlook a valuable shortcut. Or you may go a long way in the wrong direction and need a lot of time to come back. Within mathematics one aspect of being a pedestrian is calculating things by hand as far as possible and using computers as a last resort. This reminds me of a story about the physicist John Wheeler who had a picture of a computer on the wall in his office which he called ‘the big computer’. When he wanted to solve a difficult problem he would think about how he would programme it on the computer and when he had done that thoroughly he had understood the problem so well that he no longer needed the computer. Thus the fact that the computer did not exist except on paper was not a disadvantage.

This is the direction I want to (continue to) go. The challenges along the road are to achieve something essential and to make clear to others, who may be sceptical, that I have done so.

## Hepatitis C

May 29, 2016

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

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

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

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

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

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

## Flying to Copenhagen without a carpet

May 11, 2016

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

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

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

## NFκB

May 1, 2016

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

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

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

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

April 25, 2016

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

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

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