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 NFB 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.
May 1, 2016
NFB 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, -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. NFB 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 . 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 and . The choice which of these a cell uses seems to be fairly random. The work in the Baltimore lab had found out that NFB could skew the ratio. I found a video by Baltimore from 2001 about NFB. 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 NFB 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 NFB is activated. There are lots of animated videos on Youtube illustrating this but I prefer a description in words. Normally NFB is found in the cytosol bound to an inhibitor IB. Under certain circumstances a complex of proteins called IKK forms. The last K stands for kinase and IKK phosphorylates IB. This causes IB to be ubiquinated and thus marked for degradation (cf. the discussion of ubiquitin here). When it has been destroyed NFB is liberated, moves to the nucleus and binds to DNA. What are the circumstances mentioned above? There are many alternatives. For instance TNF 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 NFB. What genes does NFB 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 the right genes are switched on while another input activates other genes through the same mediator NFB? 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 NFB 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 NFB 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 NFB system. The primary thing which has been observed to oscillate is the concentration of NFB 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 NFB 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 NFB in the nucleus, IB in the cytoplasm and mRNA coding for IB. The oscillations in normal cells are damped but sustained oscillations can be seen in mutated cells or corresponding models.
What is the function of NFB? 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.
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.
January 23, 2016
Here I discuss another tool for analysing chemical reaction networks of deficiency greater than one. This is the Advanced Deficiency Algorithm developed by Feinberg and Ellison. It seems that the only direct reference for the mathematical proofs is Ellison’s PhD thesis. There is a later PhD thesis by Haixia Ji in which she introduces an extension of this called the Higher Deficiency Algorithm and where some of the ideas of Ellison are also recapitulated. In my lecture course, which ends next week, I will only have time to discuss the structure of the algorithm and give an extended example without proving much.
The Advanced Deficiency Algorithm has a general structure which is similar to that of the Deficiency One Algorithm. In some cases it can rule out multistationarity. Otherwise it gives rise to several sets of inequalities. If one of these has a solution then there is multistationarity and if none of them does there is no multistationarity. It is not clear to me if this is really an algorithm which is guaranteed to give a diagostic test in all cases. I think that this is probably not the case and that one of the themes of Ji’s thesis is trying to improve on this. An important feature of this algorithm is that the inequalities it produces are in general nonlinear and thus may be much more difficult to analyse than the linear inequalities obtained in the case of the Deficiency One Algorithm.
Now I have come to the end of my survey of deficiency theory for chemical reaction networks. I feel I have learned a lot and now is the time to profit from that by applying these techniques. The obvious next step is to try out the techniques on some of my favourite biological examples. Even if the result is only that I see why the techniques do not give anything interesting in this cases it will be useful to understand why. Of course I hope that I will also find some positive results.
January 14, 2016
The central figure in the American TV series Dr. House is a doctor who is brilliant in the diagnosis of unusual medical conditions but personally very difficult. When I first saw this series I found the character so unpleasant that I did not want to watch the programme. However in the course of time I got drawn in to watching it by the interest of the medical content. While some aspects of this series are quite exaggerated and far from reality the medical parts are very accurate and well researched. As I learned yesterday even details seen there like the numbers on heart monitors accurately reflect the situation being portrayed. I have this information from a lecture I attended yesterday at the Medizinische Gesellschaft Mainz [Mainz Medical Society]. The speaker was Professor Jürgen Schäfer, a man who has become known in the media as the German Dr. House. I am pleased to report that I detected no trace of the social incompetence of Dr. House in Dr. Schäfer.
Jürgen Schäfer is trained as a cardiologist. He and his wife, who is a gastroenterologist, got so interested by the series Dr. House that they would spend time discussing the details of the diagnoses and researching the background after they has seen each programme. Then Schäfer had the idea that he could use Dr. House in his lectures at the University of Marburg. The first obstacle was to know if he could legally make use of this material. After a casual conversation with one of his patients who is a lawyer he contacted the necessary people and signed a suitable contract. At this time his project attracted considerable attention in the media even before it had started. In the lectures he analyses the cases occurring in the series. The students are encouraged to develop their own diagnoses in dialogue with the professor. These lectures are held in the evenings and are very popular with the students. In the evaluations the highest score was obtained for the statement that ‘the lectures are a lot of fun’.
This is only the start of the story. During a consultation in one of the episodes of Dr. House he suddenly makes a deep cut with a scalpel in the body of the patient (one of the melodramatic elements), opens the wound and shows that the flesh inside is black. The diagnosis is cobalt poisoning. After seeing this it occurred to Dr. Schäfer that this diagnosis might also apply to one of his own patients and this turned out to be true. In addition to serious heart problems this patient was becoming blind and deaf. He had had a hip joint replacement with an implant made of a ceramic material. At some point this became damaged and was replaced. In order to try to avoid the implant breaking again the new one was made of metal. The old implant fragmented and left splitters in the body. These had acted like sandpaper on the new joint and at the time of removal it had been reduced to 70% of its original size by this process. As a result large quantities of cobalt was released, resulting in the poisoning. The speaker showed a picture of the operation of another of his patients with a similar problem where the wound could be seen to be filled with a black oily liquid. Together with colleagues Schäfer published an account of this case in The Lancet with the title ‘Cobalt intoxication diagnosed with the help of Dr. House’. Not all his coauthors were happy with this title but Schäfer wanted to acknowledge his debt to the series. At the same time it was a great piece of advertizing for him which lead to a lot of attention in the international media.
Due to his growing fame Schäfer started to get a lot of letters from patients with mysterious illnesses. This was more than he could handle. He informed the administration of the university clinic where he worked that he was going to start sending back letters of this type unopened, since he just did not have the time to cope with them. To his surprise they wanted him to continue with this work and arranged from him to be relieved from other duties. They set up a new institute for him called Zentrum für unerkannte Krankheiten [centre for unrecognized diseases]. This was perhaps particularly surprising since this is a privately funded clinic and the work of this institute costs money rather than making money. The techniques used there include toxicological and genomic analyses.
Here is another example from the lecture. Schäfer’s institute uses large scale DNA analysis to screen for a broad range of parasites in patients with unclear symptoms. In one patient they found DNA of the parasite causing schistosomiasis. This disease is usually got by bathing in infected water in tropical or subtropical areas. The patient tested negatively for the parasite and had never been to a place where this disease occurs. The mystery was cleared up due to the help of a vet of Egyptian origin. He was familiar with schistosomiasis and due to his experience with large animals he was not afraid of analysing very large stool samples. He succeeded in finding eggs of the parasite in the patient’s stool. The diffculty was that the numbers of eggs were very low and that for certain reasons they were difficult to recognise in this case, except by an expert. The patient was treated for schistosomiasis as soon as the genetic results were available but it was very satisfying to have a confirmation by more classical techniques. The mystery of how the patient got infected was solved as follows. As a hobby he kept lots of fish and he imported these from tropical regions. The infection presumably came from the water in his aquarium. We see that in the modern world it is easy to import tropical diseases by express delivery after placing an order in the internet
I do not want to end before mentioning that Schäfer said something nice about how mathematicians can help medical doctors. He had a patient who is a mathematics professor and had the following problem. From time to time he would collapse and was temporarily paralysed although fully conscious. A possible explanation for this would have been an excessively high level of sodium in the body. On the other hand measurements showed that the concentration of sodium in his blood was normal, even after an attack. The patient then did a calculation (just simple arithmetic). On the basis of known data he worked out the amount of sodium and potassium in different types of food and noted a correlation between negative effects of a food on his health and the ratio of the sodium to potassium concentrations. This supported the hypothesis of sodium as a cause and encouraged the doctors to look more deeply into the matter. It turned out that in this type of disease the sodium is concentrated near the cell membrane and cannot be seen in the blood. A genetic analysis revealed that the patient had a mutation in a little-known sodium channel.
I think that this lecture was very entertaining for the audience, including my wife and myself. However this is not just entertainment. With his institute Schäfer is providing essential help for many people in very difficult situations. He has files of over 4000 patients. This kind of work requires a high investment in time and money which is not possible for a usual university clinic, not to mention an ordinary GP. It is nevertheless the case that Schäfer is developing resources which could be used more widely, such as standard protocols for assessing patients of this type. As he emphasized, while by definition a rare disease only effects a small number of patients the collection of all rare diseases together affects a large number of people. If more money was invested in this kind of research it could result in a net saving for the health system since it would reduce the number of people running from one doctor to another since they do not have a diagnosis.
November 23, 2015
I discussed the deficiency zero theorem of chemical reaction network theory (CRNT) in a previous post. (Some further comments on this can be found here and here.) This semester I am giving a lecture course on chemical reaction network theory. Lecture notes are growing with the course and can be found in German and English versions on the web page of the course. The English version can also be found here. Apart from introductory material the first main part of the course was a proof of the Deficiency Zero Theorem. There are different related proofs in the literature and I have followed the approach in the classic lecture notes of Feinberg on the subject closely. The proof in those notes is essentially self-contained apart from one major input from a paper of Feinberg and Horn (Arch. Rat. Mech. Anal. 66, 83). In this post I want to give a high-level overview of the proof.
The starting point of CRNT is a reaction network. It can be represented by a directed graph where the nodes are the complexes (left or right hand sides of reactions) and the directed edges correspond to the reactions themselves. The connected components of this graph are called the linkage classes of the network and their number is usually denoted by . If two nodes can be connected by oriented paths in both directions they are said to be strongly equivalent. The corresponding equivalence classes are called strong linkage classes. A strong linkage class is called terminal if there is no directed edge leaving it. The number of terminal strong linkage classes is usually denoted by . From the starting point of the network making the assumption of mass action kinetics allows a system of ODE to be obtained in an algorithmic way. The quantity is a vector of concentrations as a function of time. Basic mathematical objects involved in the definition of the network are the set of chemical species, the set of complexes and the set of reactions. An important role is also played by the vector spaces of real-valued functions on these finite sets which I will denote by , and , respectively. Using natural bases they can be identified with , and . The vector is an element of . The mapping from to itself can be written as a composition of three mappings, two of them linear, . Here , the complex matrix, is a linear mapping from to . is a linear mapping from to itself. The subscript is there because this matrix is dependent on the reaction constants, which are typically denoted by . It is also possible to write in the form where describes the reaction rates and is the stoichiometric matrix. The image of is called the stoichiometric subspace and its dimension, the rank of the network, is usually denoted by . The additive cosets of the stoichiometric subspace are called stoichiometric compatibility classes and are clearly invariant under the time evolution. Finally, is a nonlinear mapping from to . The mapping is a generalized polynomial mapping in the sense that its components are products of powers of the components of . This means that depends linearly on the logarithms of the components of . The condition for a stationary solution can be written as . The image of is got by exponentiating the image of a linear mapping. The matrix of this linear mapping in natural bases is . Thus in looking for stationary solutions we are interested in finding the intersection of the manifold which is the image of with the kernel of . The simplest way to define the deficiency of the network is to declare it to be . A fact which is not evident from this definition is that is always non-negative. In fact is the dimension of the vector space where is the set of complexes of the network. An alternative concept of deficiency, which can be found in lecture notes of Gunawardena, is the dimension of the space . Since this vector space is a subspace of the other we have the inequality . The two spaces are equal precisely when each linkage class contains exactly one terminal strong linkage class. This is, in particular, true for weakly reversible networks. The distinction between the two definitions is often not mentioned since they are equal for most networks usually considered.
If is a stationary solution then belongs to . If (and in particular if ) then this means that . In other words belongs to the kernel of . Stationary solutions of this type are called complex balanced. It turns out that if is a complex balanced stationary solution the stationary solutions are precisely those points for which lies in the orthogonal complement of the stoichiometric subspace. It follows that whenever we have one solution we get a whole manifold of them of dimension . It can be shown that each manifold of this type meets each stoichiometric class in precisely one point. This is proved using a variational argument and a little convex analysis.
It is clear from what has been said up to now that it is important to understand the positive elements of the kernel of . This kernel has dimension and a basis each of whose elements is positive on a terminal strong linkage class and zero otherwise. Weak reversibility is equivalent to the condition that the union of the terminal strong linkage classes is the set of all complexes. It can be concluded that when the network is not weakly reversible there exists no positive element of the kernel of . Thus for a network which is not weakly reversible and has deficiency zero there exist no positive stationary solutions. This is part of the Deficiency Zero Theorem. Now consider the weakly reversible case. There a key statement of the Deficiency Zero Theorem is that there exists a complex balanced stationary solution . Where does this come from? We sum the vectors in the basis of and due to weak reversibility this gives something which is positive. Then we take the logarithm of the result. When this can be represented as a sum of two contributions where one is of the form . Then . A further part of the deficiency zero theorem is that the stationary solution in the weakly reversible case is asymptotically stable. This is proved using the fact that for a complex balanced stationary solution the function is a Lyapunov function which vanishes for
November 17, 2015
EMBL, the European Molecular Biology Laboratory, is an international institution consisting of laboratories at five sites, two in Germany, one in the UK, one in France and one in Italy. I recently attended a meeting on the theme ‘Biological Oscillators’ at the site in Heidelberg. The impressive building is in the form of a double helix. There are two spiral ramps over several stories which are linked by bridges (‘hydrogen bonds’, in German Wasserstoffbrücken). This helix provides an original setting for the poster sessions. The building is reached by ascending steep hills in the area behind the castle. I took the comfortable option of using the bus provided by the institute. This meeting had about 130 participants but I think that the capacity is much greater.
One of the most interesting talks on the first day from my point of view was by Petra Schwille from the Max Planck Institute for Biochemistry. She talked about the Min system which is used by bacteria to determine their plane of division. The idea is that certain proteins (whose identity is explicitly known) oscillate between the ends of the cell and that the plane of division is the nodal surface of the concentration of one of these. The speaker and her collaborators have been able to reconstitute this system in a cell-free context. A key role is played by the binding of the proteins to the cell membrane. Diffusion of bound proteins is much slower than that of proteins in solution and this situation of having two different diffusion constants in a coupled system is similar to the classical scenario known from the Turing instability. It sounds like modelling this system mathematically can be a lot of fun and that there is no lack of people interested in doing so.
There was also a ‘Keynote Lecture’ by Jordi Garcia-Ojalvo which lived up to the promise of its special title. The topic was the growth of a colony of Bacillus subtilis. (The published reference is Nature 523, 550.) In fact, to allow better control, the colony is constrained to be very thin and is contained in a microfluidic system which allows its environment to be manipulated precisely. A key observation is that the colony does not grow at a constant rate. Instead its growth rate is oscillatory. The speaker explained that this can be understood in terms of the competition between the cells near the edge of the colony and those in the centre. The colony is only provided with limited resources (glycerol, glutamate and salts). It may be asked which resource limits the growth rate. It is not the glycerol, which is the primary carbon source. Instead it is the glutamate, which is the primary source of nitrogen. An important intermediate compound in the use of glutamate is ammonium. If cells near the boundary of the colony produced ammonium it would be lost to the surroundings. Instead they use ammonium produced by the interior cells. It is the exterior cells which grow and they can deprive the inner cells of glutamate. This prevents the inner cells producing ammonium which is then lacking for the growth of the outer cells. This establishes a negative feedback loop which can be seen as the source of the oscillations in growth rate. The feasibility of this mechanism was checked using a mathematical model. The advantage of the set-up for the bacteria is that if the colony is exposed to damage from outside it can happen that only the exterior cells die and the interior cells generate a new colony. The talk also included a report on further work (Nature 527, 59) concerning the role of ion channels in biofilms. There are close analogies to the propagation of nerve signals and the processes taking place can be modelled by equations closely related to the Hodgkin-Huxley system.
I will now mention a collection of other topics at the conference which I found particularly interesting. One recurring theme was NFB. This transcription factor is known to exhibit oscillations. A key question is what their function is, if any. One of the pioneers in this area, Mike White, gave a talk at the conference. There were also a number of other people attending working on related topics. I do not want to go any deeper here since I think that this is a theme to which I might later devote a post of its own, if not more than one. I just note two points from White’s talk. One is that this substance is a kind of hub or bow-tie with a huge number of inputs and outputs. Another is that the textbook picture of the basic interactions of NFB is a serious oversimplification. Another transcription factor which came up to a comparable extent during the conference is Hes1, which I had never previously heard of. Jim Ferrell gave a talk about the coordination of mitosis in Xenopus eggs. These are huge cells where communication by means of diffusion would simply not be fast enough. The alternative proposed by Ferrell are trigger waves, which can travel much faster. Carl Johnson talked about mechanisms ensuring the stability of the KaiABC oscillator. He presented videos showing the binding of individual KaiA molecules to KaiC. I was was amazed that these things can be seen directly and are not limited to the cartoons to be found in biology textbooks. Other videos I found similarly impressive were those of Alexander Aulehla showing the development of early mouse embryos (segmentation clock) where it could be seen how waves of known chemical events propagating throught the tissues orchestrate the production of structures in the embryo. These pictures brought the usual type of explanations used in molecular biology to a new level of concreteness in my perception.