## Macronectes giganteus

April 15, 2018

Southern Giant Petrel

This blog is named after the Storm Petrel, Hydrobates pelagicus. It is a small bird, looking superficially like a swallow, and with a wingspan of less than 20 centimetres and a weight of about 30 grams. Looking back to my recent trip to South America, I see that the bird which made the biggest impression on me was a relative of the title species, the Southern Giant Petrel, Macronectes giganteus. It is on quite a different scale, with a wingspan of about two metres and a weight of about 5 kilograms. Thus it approaches the size of one of the smaller species of albatross. In form it looks a bit like a giant version of the Fulmar. The first ones I saw were in the harbour of Ushuaia. I then saw many more in flight during the cruise on the Beagle Channel. Before the trip I was not informed about how to distinguish Macronectes giganteus from the very similar Northern Giant Petrel, Macronectes halli. Fortunately for me, Eva was very active with her camera and took a photograph (see above) of an individual in Ushuaia which shows what I later learned to be a characteristic feature of M. giganteus, namely the fact that the tip of the bill is green. There does exist a light morph which is mainly white but we did not see any of those.

## T cell triggering

April 6, 2018

When reading immunology textbooks I had the feeling that one important point was not explained. The T cell receptor is almost entirely outside the cell and so when it encounters its antigen it cannot transmit this information into the cytosol the way a transmembrane receptor does. But since the activation of the cell involves the phosphorylation of the cytoplasmic tails of proteins associated to the receptor (CD3 and the $\zeta$-chains) the information must get through somehow. So how does this work? This process, which precedes the events relevant to the models for T cell activation I discussed here, is referred to as T cell triggering. I had an idea about how this process could work. If the T cell receptor and the coreceptor CD8 both bind to a peptide-MHC complex they are brought into proximity. As a consequence CD3 and the $\zeta$-chains are then close to CD8. On the other hand the kinase Lck is associated to CD8. Thus Lck is brought into proximity with the proteins of the T cell receptor complex and can phosphorylate them. I had never seen this described in detail in the literature. Now I found a review article by van der Merwe and Dushek (Nature Reviews in Immunology 11, 47) which explains this mechanism (and gives it a name, co-receptor heterodimerization) together with a number of other alternatives. It is mentioned that this mechanism alone does not suffice to explain T cell triggering since there are experiments where T cells lacking CD4 and CD8 were triggered. The authors of this paper do not commit themselves to one mechanism but instead suggest that a combination of mechanisms may be necessary.

I will describe one other mechanism which I find particularly interesting and which I already mentioned briefly in a previous post. It is called kinetic segregation and was proposed by Davis and van der Merwe. One way of imagining the state of a T cell before activation is that Lck is somehow inactive or that the phosphorylation sites relevant to activation are not accessible to it. A different picture is that of a dynamic balance between kinase and phosphatase, between Lck and CD45. Both of these enzymes are active and pushing the system in opposite directions. In an inactivated cell CD45 wins this struggle. When the TCR binds to an antigen on an antigen-presenting cell the membranes of the cells are brought together and there is no longer room for the bulky extracellular domain of CD45. Thus the phosphatase is pushed away from the TCR complex and Lck can take control. This could also represent a plausible mechanism for the function of certain artificial constructs for activating T cells, as discussed briefly here.

This mechanism may be plausible but what direct evidence is there that it really works? Some work which is very interesting in this context is due to James and Vale (Nature 487, 64). The more general basic issue is how to identify which molecules are involved in a particular biochemical process and which are not. The method used by these authors is to introduce selected molecules (including T cell receptors) into a non-immune cell and to see under what circumstances triggering can occur. Different combinations of molecules can be used in different experiments. With these techniques it is shown that the kinetic segregation mechanism can work and more is learned about the details of how it might work.

## The Higgins-Selkov oscillator, part 2

March 29, 2018

Some time ago I wrote about a mathematical model for glycolysis, the Higgins-Selkov oscillator. In fact I now prefer to call it the Selkov oscillator since it is a system of equations written down by Selkov, although it was obtained by modifying a previous model of Higgins. At that time I mentioned some of the difficulties involved in understanding the global properties of solutions of this system. I mentioned that I had checked that this system exhibits a supercritical Hopf bifurcation, thus proving that it has stable periodic solutions for certain values of the parameters. In the hope of getting more insights into this system I decided to make it the theme of a master’s thesis. Pia Brechmann, the student who got this subject, obtained a number of interesting results. After she submitted her thesis she and I decided to carry this a bit further so as to get a picture of the subject which was as comprehensive as possible. We have now written a paper on this subject. I cannot say that we were able to answer all the questions we would have liked to but at least we answered some of them. Here I will mention some of the things which are known and some which are not.

When written in dimensionless form the system contains two parameters $\alpha$ (a positive real number) and $\gamma$ (an integer which is at least two). I already mentioned a result about a Hopf bifurcation and this was originally only proved for $\gamma=2$. In the paper we showed that there is a supercritical Hopf bifurcation for any value of $\gamma$. I also previously mentioned doing a Poincaré compactification of the system and blowing up new stationary solutions which appear in this process and are too degenerate to analyse directly. The discussion of using blow-ups in polar coordinates previously given actually concerned the case $\gamma=2$ and does not seem practical for higher values of $\gamma$. It turns out that the technique of quasihomogeneous directional blow-ups, explained in the book ‘Qualitative theory of planar differential systems’ by Dumortier et al. can be used to treat the general case. This type of blow-up has the advantage that the transformations are given by monomials rather than trigonometric functions and that there is a systematic method for choosing good values of the exponents in the monomials.

We discovered a paper by d’Onofrio (J. Math. Chem. 48, 339) on the Selkov oscillator where he obtains interesting results on the uniqueness and stability of periodic solutions. It was suggested by Selkov on the basis of numerical calculations that for large $\alpha$ there exist solutions with oscillations which grow arbitrarily large at late times (let us call these unbounded oscillations). We were not able to decide on a rigorous level whether such solutions exist or not. They cannot exist when a periodic solution does. When they exist they have to do with a heteroclinic cycle in the compactification. We showed that when a cycle of this kind exists it is asymptotically stable and in that case solutions with unbounded oscillations exist. However we were not able to decide whether a heteroclinic cycle at infinity ever exists for this system. What we did prove is that for all values of the parameters there exist unbounded solutions which are eventually monotone.

We also proved that when the steady state is stable each bounded solution converges to it and that when there exists a periodic solution it is unique and each bounded solution except the steady state converges to that. I find it remarkable that such an apparently harmless two-dimensional dynamical system is so resistent to a complete rigorous analysis.

## Lotka’s system

March 11, 2018

The system of ODE $\dot x=a-bxy$, $\dot y=bxy-cy$ was considered in 1910 by Lotka as a model for oscillatory chemical reactions (J. Phys. Chem. 14, 271). It exhibits damped oscillations but no sustained oscillations, i.e. no periodic solutions. It should not be confused with the famous Lotka-Volterra system for predator-prey interactions which was first written down by Lotka in 1920 (PNAS 6, 410) and which does have periodic solutions for all positive initial data. That the Lotka system has no periodic solutions follows from the fact that $x^{-1}y^{-1}$ is a Dulac function. In other words, if we multiply the vector field defined by the right hand sides of the equations by the positive function $x^{-1}y^{-1}$ the result is a vector field with negative divergence. This change of vector field preserves periodic orbits and it follows from the divergence theorem that the rescaled vector field has no periodic orbits. My attention was drawn to this system by the paper of Selkov on his model for glycolysis (Eur. J. Biochem. 4, 79). In his model there is a parameter $\gamma$ which is assumed greater than one. He remarks that if this parameter is set equal to one the system of Lotka is obtained. Selkov obtains his system as a limit of a two-dimensional system with more complicated non-linearities. If the parameter $\gamma$ is set to one in that system equations are obtained which are related to Higgins’ model of glycolysis. Selkov remarks that this last system admits a Dulac function and hence no sustained oscillations and this is his argument for discarding Higgins’ model and replacing it by his own. The equations are $\dot x=a-b\frac{xy}{1+y+xy}$ and $\dot y=b\frac{xy}{1+y+xy}-cy$. (To obtain the limit already mentioned it is first necessary to do a suitable rescaling of the variables.) In this case the Dulac function is $\frac{1+y+xy}{xy}$.

The fact that the Lotka-Volterra system admits periodic solutions can be proved by exhibiting a conserved quantity. At this point I recall the well-known fact that while conserved quantities and their generalizations, the Lyapunov functions, are very useful when you have them there is no general procedure for finding them. This naturally brings up the question: if I did not know the conserved quantity for the Lotka-Volterra system how could I find it? One method is as follows. First divide the equation for $\dot y$ by that for $\dot x$ to get a non-autonomous equation for $dy/dx$, cheerfully ignoring points where $\dot x=0$. It then turns out that the resulting equation can be solved by the method of separation of variables and that this leads to the desired conserved quantity.

One undesirable feature of the Lotka-Volterra system is that it has a one-parameter family of periodic solutions and must therefore be suspected to be structurally unstable. In addition, if we consider a solution where predators are initially absent the prey population grows exponentially. The latter feature can be eliminated by replacing the linear growth term in the equation for the prey by a logistic one. A similar term corresponding to higher death rates at high population densities can be added in the equation for the predators but the latter modification has no essential effect. This is a Lotka-Volterra model with intraspecific competition. As discussed in the book ‘Evolutionary Games and Population Dynamics’ by Josef Hofbauer and Karl Sigmund, when this model has a positive steady state that state is globally asymptotically stable. The proof uses the fact that the expression which defines the conserved quantity in the usual Lotka-Volterra model defines a Lyapunov function in the case with intraspecific competition. This is an example of the method of obtaining conserved quantities or Lyapunov functions by perturbing those which are already known in special cases.

It follows from Poincaré-Bendixson theory that the steady states in the Lotka model and the Higgins model are globally asymptotically stable. This raises the question whether we could not find Lyapunov functions for those systems. I do not know how. The method used for Lotka-Volterra fails here because the equation for $dy/dx$ is not separable.

## Conference on mechanobiology and cell signalling in Oberwolfach

March 2, 2018

I just attended a conference in Oberwolfach in an area rather far away from my usual interests, although it was about mathematical biology. I did meet some interesting (known and unkown) people and encountered some new ideas. Here I will just discuss two talks which particularly caught my attention.

In a talk of Takashi Hiiragi I learned a number of interesting things about embryology. The specific subject was the mouse embryo but similar things should apply to the human case. On the other hand it is very far away from what happens in Drosophila, for instance. There is a stage where the first eight cells are essentially identical. More precisely there is a lot of random variation in these cells, but no systematic differences. The subsequent divisions of these cells are not temporally correlated. By the time the number of cells has reached thirty-two an important differentiation step has taken place. By that time there are some of the cells which belong to the embryo while the others will be part of the placenta. If the first eight cells are separated then each one is capable of giving rise to a complete mouse. (The speaker did seem to indicate some restriction but did not go into details.) In order to understand the development process better one of these cells is studied in isolation. The cell contains a clock and so it ‘knows’ that it is in the eight cell stage. It then develops into a group of four cells in the same way that the eight cells would normally develop into 32. Differentiation takes place. The key symmetry-breaking step takes place when one end of the cell (at the eight-cell stage) develops an area at one end where actin is absent. This polarization then influences the further motion of the cells. It is interesting that the interaction between the cells in these processes seems to have more to do with mechanical signals then with chemical ones.

There was a talk of Fredric Cohen about cholesterol. His claim was that the concentration of cholesterol as usually measured is not a useful quantity and that the quantity which should be measured is the chemical potential of cholesterol. This has to do with the fact that cholesterol is hardly soluble in water or in hydrophobic liquids. I must say that the term ‘chemical potential’ was something which was always very opaque for me. As a result of this talk I think I am beginning to see the light. The cholesterol in a cell is mainly contained in the cell membrane. However it is not simply dissolved there as single molecules. Instead most of the molecules are interacting with proteins or with other cholesterol molecules. The chemical potential has to do with how many molecules get transferred when the system is connected to a reservoir. Only those molecules which are free are available to be transferred. So the issues seem to be the relationship between the amounts of free and bound molecules and what the real significance of the concentration of free molecules is for understanding a system.

## Energy budget models

February 26, 2018

In a previous post I mentioned a talk I gave on dinosaurs and promised more information at a later date. Now a paper related to this has appeared and I will keep the promise. A basic issue is the way in which dinosaurs regulated their body temperature. The traditional idea was that they were cold-blooded (exotherm) like crocodiles. Later it was suggested that they might have been warm-blooded (endotherm) like birds or mammals. Then it was claimed that this was unrealistic and that they were mesotherm. This means something in between exotherm and endotherm, with a limited control of body temperature. There do exist organisms like this, a notable example being the tuna. I was involved in a project with Jan Werner, Eva Maria Griebeler and Nikos Sfakianakis on this subject. The first published result coming from this effort has now appeared (J. Theor. Biol. 444, 83).

The long-term goal of this project is to understand the evolution of warm-blooded animals in connection with the evolution of birds from dinosaurs. This involves understanding the way in which animals allocate energy to different tasks. How much do they use for generating body heat and how much do they use for other tasks such as maintenance or reproduction? A first step is to find a parametrization of the possible energy allocation strategies. In other words we want to identify suitable variables which could be used to describe the evolutionary process we want to understand. This is the content of the paper which has just appeared. At this point it might be asked how we can find out about the energy consumption of dinosaurs at all. It turns out that there are general relations known between the energy consumption of an animal and its growth rate over its lifetime. Thus the growth curve of a dinosaur gives indirect information about its energy consumption. But how can we get information about the growth curve? This is something I learned in the course of this project. The large bones of dinosaurs exhibit annual growth rings like those known for trees. The rings are of different thicknesses and thus give information on the growth rate in different years.

The paper does not contain a detailed dynamic model of the energy use of an animal over its lifetime. Instead it introduces a set of possible time evolutions depending on a finite number of parameters and then tests (by numerical methods) whether these suffice to reproduce the experimental growth curves of a number of animals with sufficient accuracy. It is also checked that the results obtained are consistent with known facts about the age of sexual maturation of the different species. It turns out that the mathematical model is successful in fitting the experimental constraints. It is found that, as expected, the model predicts that exotherms continue to grow as long as they live while endotherms stop growing at a time comparable to the age of sexual maturity.

## Trip to Ushuaia, Part 2

February 26, 2018

The reason for visiting El Calafate was its proximity to the glacier Perito Moreno. In the garden of our hotel there was a Buff-Necked Ibis, an attractive species which we saw repeatedly during the rest of our trip. Perito Moreno has interesting dynamical properties. I wonder if it has ever been modelled mathematically? Let me describe the process. The glacier comes from a peninsula and its lower end enters a lake, the Lago Argentino. It then proceeds until it has crossed the lake, which is quite narrow at that point. When it has reached the other side it separates one arm of the lake from the main part. The lake has an outflow but none in the smaller separated part. Thus the water level in the separated part rises compared to that in the main part. At the moment the difference in the levels is about thirty meters. This results in flooding of the surrounding land. We visited one farm there where an important part of the grazing land has already been submerged. The process just described leads to the water exterting a strong force on the glacier. This pressure is first released to a limited extent when water starts to flow under the glacier. This flow increases in intensity and produces a kind of arch which is flows under. Pieces of ice break off the arch successively, making it higher and higher. Eventually, about three days after the water has first penetrated the ice the arch is so narrow and high it collapses and then the obstruction has been removed. We are back at the starting point of the process. All this could be seen in a video in the glacier museum. The next breakthrough of the water is expected within the next few months but nobody knows exactly when it will happen.

From El Calafate we travelled overland and crossed the border into Chile.We had hardly crossed the border when we saw our first Andean Condor. This part of Chile has no road connection within the country to the rest of Chile. All necessary goods are imported by ship and the prices are correspondingly high. We were first in Puerto Natales. There it was convenient to observe the local ducks and cormorants along the waterfront. There is a statue of a giant sloth, a creature whose remains were found in a cave in the region. We travelled to the Torres del Paine, a spectacular mountain range. What you can see there is heavily dependent on the weather and we were quite lucky. Only the very top of the largest of the three ‘Cuernos’ (horns) refused to emerge from the clouds as long as we were there. What sounded like thunder turned out to be an avalanche. From Puerto Natales we travelled to Punta Arenas on the Strait of Magellan. There we visited some reconstructions of famous naval vessels. There is the Beagle, with which Darwin travelled, one of the ships of the expedition of Magellan himself and the modified lifeboat with which Shackleton sailed from Elephant Island to South Georgia and thus saved the lives of the members of his expedition. The text of the famous advertisement with which Shackleton recruited men for this expedition is reproduced there. According to Wikipedia the story of this advertisement is apocryphal but the text is so delicious that I cannot resist repoducing it here: ‘Men wanted for hazardous journey. Low wages, bitter cold, long hours of complete darkness. Safe return doubtful. Honour and recognition in event of success’.

From Punta Arenas we flew to Puerto Montt and travelled from there to Puerto Vargas. Here the attraction was the volcano Osorno. This time our luck with the weather seemed to be at an end. Our guide explained in a quite amusing manner how if the weather had been different the wall of cloud which we saw in a certain direction would have been replaced by a view of the beautiful volcano. In fact it turned out that there was a small window before breakfast the next day where the volcano could be seen from the hotel. The last step of the journey was a flight to Santiago. The city did not make a very good impression at first sight. Compared with Buenos Aires all the signs seemed to be reversed. Later we learned that the city is divided very strictly along economic lines. The rich upper part of the city looks quite different to the rest. We learned about the story of Chile as a model system for testing neoliberal theories. Just now the Chileans, who used to be considered as very backward are proud to be doing better (economically) than their neighbours, the Argentinians. We also had the opportunity to learn from our guides about the politics of Allende and Pinochet, in particular that Pinochet still enjoys considerable popularity in Chile.

We flew back from Santiago to Frankfurt via Madrid, our heads full of many images of Argentina, Chile, their people and their natural environment.

## Trip to Ushuaia

February 25, 2018

Eva and I have just come back from a trip to Argentina and Chile. I had been in Argentina twice before, for conferences. The country made a positive impression on me but I did not have time to see very much. At that time I thought I should come back to see more. Now I have finally got around to doing so. The trip could have ended almost before it started since we almost missed our flight. I usually leave plenty of time when travelling to the airport and this was no exception. The ICE which we intended to take to travel from Mainz to Frankfurt airport was cancelled, without a explanation being offered. We then got into the regional train which was the next possibility. It just sat there instead of leaving. Then there was an announcement to say that because of a problem with points it would take a different route and, in particular, would not stop at Frankfurt airport. Our only hope was to travel to the main station in Frankfurt and then back to the airport. The situation seemed chaotic due the problem with the points already mentioned and the fact that there were people on the tracks somewhere. Even the ticket collector on the train did not seem to be able to give us a reasonable suggestion. Eventually we got to the airport and after running throught the airport and jumping a few queues we did get to our flight. We would have preferred a less stressful start to our journey. We flew via Sao Paulo to Buenos Aires and joined the group there. (This was an organized group trip).

The first time I visited Argentina I also flew to Buenos Aires but I saw almost nothing of the city. I just took a taxi from the international airport to the domestic one and my memory was that it drove around the periphery. Looking at how the airports lie this impression was probably mistaken. During my first trip I had a few hours to wait for my continuing flight to Cordoba and I was able to watch the for me exotic gulls, since the airport is close to the water. This time we were warned on arrival about the dangers of the city and how to behave so as to avoid them. We actually had no problems although we did not pay that much attention to security issues. One member of the group was attacked in the middle of the day close to our hotel. A man jumped on his back and tried to steal his watch. I do not think the watch was of particular value but the thief could not know that. Two brave women came and intervened and the thief ran away without the watch. The victim was left with some nasty-looking bruises on his arm but there was no further damage.

Our impression of Buenos Aires was of a beautiful city with a very pleasant atmosphere. It must be said that the days we were there were holidays, so that we experienced the city in a much more relaxed mode than it would be in on a working day. We discovered that there is a very nice nature reserve within easy walking distance of the city. My experience travelling with groups which are not specialised on birds is that if you go anywhere which might be good for birds you see so little that it is frustrating. The present trip was an exception to this rule. In the reserve area near Buenos Aires one of the most prominent species was the Jacana, whch was numerous. Some other prominent sightings were Roseate Spoonbill, Black-Necked Swan (which we later also saw in many other places), Red-Crested Cardinal and a Hummingbird of an undetermined species. From our guides we learned a bit about the complicated subject of Argentinian politics. One story which stuck in my mind and which I reproduce here without further comment is the following. There was a time when many Argentinians were protesting about the meat prices being too high. The Kirchners banned the export of beef. The result was a situation of overproduction which did lead to a decrease in the prices. This led in turn to many producers going out of business or drastically cutting their stocks. The final results were then shortages (beef had to be imported to Argentina from Uruguay!) and that a key national industry had been damaged in a major and probably irreparable way.

After a couple of days in Buenos Aires we flew to Trelew and spent some time exploring the Valdes peninsula, staying in Puerto Madryn. We learned what dry pampa looks like, a brown very dry landscape which in that area forms huge monotone expanses. At that stage I did not find the landscape attractive although other variants we saw in other areas later looked better. We were able to see some of the standard wildlife: sea elephants, sea lions and a huge colony of Magellanic Penguins. The temperature was around 30 degrees (I mean Celcius, not Fahrenheit) and the penguins were suffering a lot from the heat. I was pleasantly suprised to get a close view of an armadillo. The towns in this area was established by settlers from Wales, which explains the curious names. I would be interested to read about the adventures of these pioneers.

After this we flew to the town mentioned in the title of this post, Ushuaia. Before the trip I felt that Ushuaia was more like a mythical place than a real one. But now I have been there. Because of a last-minute change of plane schedule we had less time in Ushuaia than planned. Despite this we were able to take a trip on the Beagle channel with a catamaran in the late afternoon. The weather was excellent. For me this was the highlight of the whole trip. In the town itself we saw Dolphin Gulls and the first Giant Petrels. We visited some seabird islands with breeding colonies of terns and cormorants. We even landed on one island where there were Great Skuas flying around. In one place I saw a couple of Sheathbills on the beach. What was special is that we came to one place where there was a big concentration of fish. There was a corresponding concentration of seabirds, including several Albatrosses. Afterwards one Black-Browed Albatross followed the ship for quite a long time. At 22.00 we caught a flight to El Calafate.

## Conference on quantitative principles in biology at EMBL

November 5, 2017

I just returned from a conference with the title ‘Quantitative Principles in Biology’ at EMBL in Heidelberg. There was quite a lot of ‘philosophical’ discussion there about the meaning attached by different people to the word ‘model’ and the roles of experiment and theory in making scientific progress, in particular in biology. The main emphasis of the conference was on specific scientific results and I will mention a few of them below. In general I noticed a rather strong influence of physics at this conference. In particular, ideas from statistical physics came up repeatedly. I also met several people who had moved their research area to biology from physics, for instance string theory. I was happy that the research field of immunology was very well represented at the conference and that Jeremy Gunawardena said to me that he believes that immunology is one area of biology where mathematics has a lot to contribute to biology.

In a talk of Susanna Manrubia I learned about a class of biological systems I had never heard of before. These are the so-called multipartite viruses. In this type of system a genome is distributed between two or more virus particles. To allow the production of new viruses in a cell it must be infected with all the components. The talk described experiments in which this type of system was made to evolve in the laboratory. They start with foot and mouth virus in a situation where infection takes place with very high frequency. This is followed through many generations. It was found that in this context a bipartite virus could evolve from a normal virus. This did not involve an intermediate situation where (as in influenza) a virus has several segments of genetic material in a single virus particles. When the bipartite virus was propagated in isolation under circumstances where the frequency of infection was much lower it evolved back to the ordinary virus state. There exists a large variety of multipartite viruses in nature. They seem to be most common in plants (where apparently multiple virus infections are very common) but are also found in other organisms, including animals.

I had a poster at the conference on my work on T cell activation with Eduardo Sontag. To my surprise Paul Francois, on whose results this work built, was at the conference and so I had the chance to discuss these things with him. In our work there is a strong focus on the T cell receptor and at the conference there were several other contributions related to the modelling of other receptors. Eugenia Lyaschenko talked about how receptors can sense relative levels of ligand concentration and how endocytosis plays a role in this. Nikolas Schnellbächer had a poster on the theme of how dimerization of receptors can lead to major qualitative changes in their response functions. There are also important differences between homo- and heterodimers. I learned something about the mechanisms which play a role there. Yaron Antebi talked about the dynamical significance of a situation where several related ligands can bind to several related receptors.

Turing instabilities came up repeatedly at the meeting and were getting positive comments. One ‘take-home message’ for me was that the usual story that a Turing instability requires different diffusion constants should be weakened. It is based on the analysis of a system with two components and as soon as there are more than two components no such clear statement can be made. In addition, taking into account cell growth can help to trigger Turing instabilities.

A talk by Pieter Rein ten Wolde deepened my understanding of circadian clocks in cyanobacteria. They have a clock on a post-translational level involving phosphorylations of the KaiABC proteins and also a clock which involves translation. In the talk it was explained how the bacterium needs both of these for its time-keeping. A key point is that the period of the oscillator defining the clock (around 24 hours) can be longer than the period of the cell cycle. Thus this is like a clock which can continue to tick while it is being disassembled into its constituent parts and put together again.

## The matrix-tree theorem

October 29, 2017

When I was an undergraduate graph theory was a subject which seemed to me completely uninteresting and I very consciously chose not to attend a lecture course on the subject. In the meantime I have realized that having become interested in reaction networks I could profit from knowing more about graph theory. One result which plays a significant role is the matrix-tree theorem. It asserts the equality of the number of subgraphs of a certain type of a given graph and a certain determinant. This could be used to calculate numbers of subgraphs. On the other hand, it could be used in the other direction calculate determinants and it is the second point of view which is relevant for reaction networks.

For the first part of the discussion here I follow these notes. If $G$ is an (unoriented) graph then a spanning tree is a connected subgraph which includes all vertices of $G$ and contains no cycles. The graph Laplacian $L$ of $G$ is the matrix for which $L_{ii}$ is equal to the number of edges containing the vertex $v_i$, $L_{ij}=-1$ if $i\ne j$ and there is an edge joining vertex $v_i$ to vertex $v_j$ and $L_{ij}=0$ otherwise. The first version of the matrix tree theorem, due to Kirchhoff, says that the number of spanning trees of $G$ can be calculated in the following way. Choose any vertex $v_j$ of $G$ and remove the $j$th row and $j$th column from $L$ to get a matrix $L_j$. Then compute the determinant of $L_j$. Surprisingly the value of the determinant is independent of $j$. The first version of the theorem can be obtained as a consequence of a second more sophisticated version proved a hundred years later by Tutte. This concerns directed graphs. A vertex $b$ is said to be accessible from $a$ if there is a directed path from $a$ to $b$. A vertex in a directed graph is called a root if every other vertex is accessible from it. A directed tree is a directed subgraph which contains a root and which, when the orientation is forgotten, is a tree (i.e. it is connected and contains no unoriented cycles). There is then an obvious way to define a spanning rooted tree. To formulate the second version of the theorem we introduce the matrix Laplacian $L$ of the directed graph. If $i=j$ the entry $L_{ij}$ is the number of edges with final vertex $i$. $L_{ij}=-1$ if $i\ne j$ and there is a directed edge from $v_i$ to $v_j$. $L_{ij}=0$ if $i\ne j$ and there is no edge connecting $v_i$ and $v_j$. The statement of the theorem is that the number of rooted trees with root $v_j$ is equal to the determinant of $L_j$, a matrix derived from $L$ as before. To see that the first version of the theorem follows the second first associate an oriented graph to an unoriented one by putting in oriented edges joining a pair of vertices in both directions whenever they are joined by an edge in the original graph. Then there is a bijection between trees rooted at $v$ in the unoriented graph and oriented trees rootes at $v$ in the oriented graph. On the other hand the two graphs have the same Laplacian since the number of edges ending at a vertex in the oriented graph is the same as the number of edges having that endpoint in the unoriented graph.

What is the connection of all this with reaction networks? Consider a chemical reaction network with only monomolecular reactions and reaction coefficients all equal to one. Then under mass action kinetics the evolution equations for the concentrations are $\dot x=Lx$, where $L$ is the Laplacian matrix of the network. There is a conserved quantity, which is the sum of the concentrations of all species. A steady state is an element of the kernel of $L$. The next part of the discussion follows a paper of Gunawardena (PLoS ONE, 7(5), e36321). He allows general reaction constants. The notion of the Laplacian is extended correspondingly. If the network is strongly connected (in the terminology of graph theory) or weakly reversible (in the terminology of chemical reaction network theory) the kernel of the Laplacian matrix is one-dimensional. Thus there is precisely one steady state in each stoichiometric compatibility class. It is moreover possible to compute a vector spanning the kernel of $N$ by graph theoretical methods. This also goes back to Tutte. To get the $i$th component first take the product of the reaction constants over a rooted tree and then sum these quantities over the rooted trees with root $v_i$. More generally the dimension of the kernel of $N$ is equal to the number of terminal strong linkage classes. It is also revealed there that the Laplacian corresponds to the matrix $A$ considered by Horn and Jackson. These ideas are also related to the King-Altman theorem of enzyme kinetics. I have the impression that I have as yet only scratched the surface of this subject. I hope I will be able to come back and understand more about it at a later date.