Archive for the ‘mathematical biology’ Category

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.

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.

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\kappaB 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\kappaB 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\kappaB 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\kappaB could skew the ratio. I found a video by Baltimore from 2001 about NF\kappaB. 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\kappaB 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\kappaB is activated. There are lots of animated videos on Youtube illustrating this but I prefer a description in words. Normally NF\kappaB is found in the cytosol bound to an inhibitor I\kappaB. Under certain circumstances a complex of proteins called IKK forms. The last K stands for kinase and IKK phosphorylates I\kappaB. This causes I\kappaB to be ubiquinated and thus marked for degradation (cf. the discussion of ubiquitin here). When it has been destroyed NF\kappaB 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\kappaB. What genes does NF\kappaB 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\kappaB? 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\kappaB 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\kappaB 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\kappaB system. The primary thing which has been observed to oscillate is the concentration of NF\kappaB 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\kappaB 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\kappaB in the nucleus, I\kappaB in the cytoplasm and mRNA coding for I\kappaB. 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\kappaB? 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.

The deficiency one algorithm

January 7, 2016

Here I continue the discussion of Chemical Reaction Network Theory with the Deficiency One Algorithm. As its name suggests this is an algorithm for investigating whether a reaction network allows multistationarity or not. It is also associated to a theorem. This theorem says that the satisfaction of certain inequalities associated to a reaction network is equivalent to the existence of a choice of reaction constants for which the corresponding system of equations with mass action kinetics has two distinct positive stationary solutions in a stoichiometric compatibility class. This is true in the context of networks of deficiency one which satisfy some extra conditions. The property of satisfying these, called regularity, seems to be relatively weak. In fact, as in the case of the Deficiency One Theorem, there is a second related result where a pair of zeroes of the right hand side f in a stoichiometric compatibility class is replaced by a vector in the kernel of Df which is tangent to that class.

An example where this theory can be applied is double phosphorylation in the processive case. The paper of Conradi et. al. cited in the last post contains the statement that in this system the Deficiency One Algorithm implies that multistationarity is not possible. For this it refers to the Chemical Reaction Network Toolbox, a software package which implements the calculations of the theorem. In my course I decided for illustrative purposes to carry out these calculations by hand. It turned out not to be very hard. The conclusion is that multistationarity is not possible for this system but the general machinery does not address the question of whether there are any positive stationary solutions. I showed that this is the case by checking by hand that \omega-limit points on the boundary of the positive orthant are impossible. The conclusion then follows by a well-known argument based on the Brouwer fixed point theorem. This little bit of practical experience with the Deficiency One Algorithm gave me the impression that it is really a powerful tool. At this point it is interesting to note a cross connection to another subject which I have discussed in this blog. It is a model for the Calvin cycle introduced by Grimbs et. al. These authors noted that the Deficiency One Algorithm can be applied to this system to show that it does not allow multistationarity. They do not present the explicit calculations but I found that they are not difficult to do. In this case the equations for stationary solutions can be solved explicitly so that using this tool is a bit of an overkill. It neverless shows the technique at work in another example.

Regularity consists of three conditions. The first (R1) is a necessary condition that there be any positive solutions at all. If it fails we get a strong conclusion. The second (R2) is the condition l=t familiar from the Deficiency One Theorem. (R3) is a purely graph theoretic condition on the network. A weakly reversible network which satisfies (R3) is reversible. Reading this backwards, if a network is weakly reversible but not reversible then the theorem does not apply. The inequalities in the theorem depend on certain partitions of a certain set of complexes (the reactive complexes) into three subsets. What is important is whether the inequalities hold for all partitions of a network or whether there is at least one partition for which they do not hold. The proof of the theorem uses a special basis of the kernel of YA_\kappa where \kappa is a function on \cal C constructed from the reaction constants. In the context of the theorem this space has dimension t+1 and t of the basis vectors come from a special basis of A_\kappa of the type which already comes up in the proof of the Deficiency Zero Theorem.

An obvious restriction on the applicability of the Deficiency One Algorithm is that it is only applicable to networks of deficiency one. What can be done with networks of higher deficiency? One alternative is the Advanced Deficiency Algorithm, which is implemented in the Chemical Reaction Network Toolbox. A complaint about this method which I have seen several times in the literature is that it is not able to treat large systems – apparently the algorithm becomes unmanageable. Another alternative uses the notion of elementary flux modes which is the next topic I will cover in my course. It is a way of producing certain subnetworks of deficiency one from a given network of higher deficiency. The subnetworks satisfy all the conditions needed to apply the Deficiency One Algorithm except perhaps t=l.

Feinberg’s proof of the deficiency zero theorem

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 l. 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 t. From the starting point of the network making the assumption of mass action kinetics allows a system of ODE \dot c=f(c) to be obtained in an algorithmic way. The quantity c(t) is a vector of concentrations as a function of time. Basic mathematical objects involved in the definition of the network are the set \cal S of chemical species, the set \cal C of complexes and the set \cal R 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 F({\cal S}), F({\cal C}) and F({\cal R}), respectively. Using natural bases they can be identified with R^m, R^n and R^r. The vector c(t) is an element of F({\cal S}). The mapping f from F({\cal S}) to itself can be written as a composition of three mappings, two of them linear, f=YA_k\Psi. Here Y, the complex matrix, is a linear mapping from F({\cal C}) to F({\cal S}). A_k is a linear mapping from F({\cal C}) to itself. The subscript k is there because this matrix is dependent on the reaction constants, which are typically denoted by k. It is also possible to write f in the form Nv where v describes the reaction rates and N is the stoichiometric matrix. The image of N is called the stoichiometric subspace and its dimension, the rank of the network, is usually denoted by s. The additive cosets of the stoichiometric subspace are called stoichiometric compatibility classes and are clearly invariant under the time evolution. Finally, \Psi is a nonlinear mapping from F({\cal S}) to F({\cal C}). The mapping \Psi is a generalized polynomial mapping in the sense that its components are products of powers of the components of c. This means that \log\Psi depends linearly on the logarithms of the components of c. The condition for a stationary solution can be written as \Psi(c)\in {\rm ker} (YA_k). The image of \Psi is got by exponentiating the image of a linear mapping. The matrix of this linear mapping in natural bases is Y^T. Thus in looking for stationary solutions we are interested in finding the intersection of the manifold which is the image of \Psi with the kernel of YA_k. The simplest way to define the deficiency of the network is to declare it to be \delta=n-l-s. A fact which is not evident from this definition is that \delta is always non-negative. In fact \delta is the dimension of the vector space {\rm ker} Y\cap {\rm span}(\Delta) where \Delta 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 \delta' of the space {\rm ker} Y\cap{\rm im} A_k. Since this vector space is a subspace of the other we have the inequality \delta'\le\delta. 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 c^* is a stationary solution then A_k\Psi (c^*) belongs to {\rm ker} Y\cap{\rm im} A_k. If \delta'=0 (and in particular if \delta=0) then this means that A_k\Psi (c^*)=0. In other words \Psi (c^*) belongs to the kernel of A_k. Stationary solutions of this type are called complex balanced. It turns out that if c^* is a complex balanced stationary solution the stationary solutions are precisely those points c for which \log c-\log c_* 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 n-s. 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 A_k. This kernel has dimension t 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 A_k. 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 c^*. Where does this c^* come from? We sum the vectors in the basis of {\rm ker} A_k and due to weak reversibility this gives something which is positive. Then we take the logarithm of the result. When \delta=0 this can be represented as a sum of two contributions where one is of the form Y^T z. Then c^*=e^z. A further part of the deficiency zero theorem is that the stationary solution c^* in the weakly reversible case is asymptotically stable. This is proved using the fact that for a complex balanced stationary solution the function h(c)=\sum_{s\in\cal S}[c(s)(\log c(s)-\log c^*(s)-1)+c(s^*)] is a Lyapunov function which vanishes for c=c^*

Conference on biological oscillators at EMBL in Heidelberg

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 NF\kappaB. 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 NF\kappaB 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.

Siphons in reaction networks

October 8, 2015

The concept of a siphon is one which I have been more or less aware of for quite a long time. Unfortunately I never had the impression that I had understood it completely. Given the fact that it came up a lot in discussions I was involved in and talks I heard last week I thought that the time had come to make the effort to do so. It is of relevance for demonstrating the property of persistence in reaction networks. This is the property that the \omega-limit points of a positive solution are themselves positive. For a bounded solution this is the same as saying that the infima of all concentrations at late times are positive. The most helpful reference I have found for these topics is a paper of Angeli, de Leenheer and Sontag in a proceedings volume edited by Queinnec et. al.

There are two ways of formulating the definition of a siphon. The first is more algebraic, the second more geometric. In the first the siphon is defined to be a set Z of species with the property that whenever one of the species in Z occurs on the right hand side of a reaction one of the species in Z occurs on the left hand side. Geometrically we replace Z by the set L_Z of points of the non-negative orthant which are common zeroes of the elements of Z, thought of as linear functions on the species space. The defining property of a siphon is that L_Z is invariant under the (forward in time) flow of the dynamical system describing the evolution of the concentrations. Another way of looking at the situation is as follows. Consider a point of L_Z. The right hand side of the evolution equations of one of the concentrations belonging to Z is a sum of positive and negative terms. The negative terms automatically vanish on L_Z and the siphon condition is what is needed to ensure that the positive terms also vanish there. Sometimes minimal siphons are considered. It is important to realize that in this case Z is minimal. Correspondingly L_Z is maximal. The convention is that the empty set is excluded as a choice for Z and correspondingly the whole non-negative orthant as a choice for L_Z. What is allowed is to choose Z to be the whole of the species space which means that L_Z is the origin. Of course whether this choice actually defines a siphon depends on the particular dynamical system being considered.

If x_* is an \omega-limit point of a positive solution but is not itself positive then the set of concentrations which are zero at that point is a siphon. In particular stationary solutions on the boundary are contained in siphons. It is remarked by Shiu and Sturmfels (Bull. Math. Biol. 72, 1448) that for a network with only one linkage class if a siphon contains one stationary solution it consists entirely of stationary solutions. To see this let x_* be a stationary solution in the siphon Z. There must be some complex y belonging to the network which contains an element of Z. If y' is another complex then there is a directed path from y' to y. We can follow this path backwards from y and conclude successively that each complex encountered contains an element of Z. Thus y' contains an element of Z and since y' was arbitrary all complexes have this property. This means that all complexes vanish at x_* so that x_* is a stationary solution.

Siphons can sometimes be used to prove persistence. Suppose that Z is a siphon for a certain network so that the points of Z are potential \omega-limit points of solutions of the ODE system corresponding to this network. Suppose further that A is a conserved quantity for the system which is a linear combination of the coordinates with positive coefficents. For a positive solution the quantity A has a positive constant value along the solution and hence also has the same value at any of its \omega-limit points. It follows that if A vanishes on Z then no \omega-limit point of that solution belongs to Z. If it is possible to find a conserved quantity A of this type for each siphon of a given system (possibly different conserved quantities for different siphons) then persistence is proved. For example this strategy is used in the paper of Angeli et al. to prove persistence for the dual futile cycle. The concept of persistence is an important one when thinking about the general properties of reaction networks. The persistence conjecture says that any weakly reversible reaction network with mass action kinetics is persistent (possibly with the additional assumption that all solutions are bounded). In his talk last week Craciun mentioned that he is working on proving this conjecture. If true it implies the global attractor conjecture. It also implies a statement claimed in a preprint of Deng et. al. (arXiv:1111.2386) that a weakly reversible network has a positive stationary solution in any stoichiometric compatobility class. This result has never been published and there seems to be some doubt as to whether the proof is correct.