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

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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.

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David Vetter had an older brother who also suffered from SCID and died of infection very young. Thus his parents and their doctors were warned. The brother was given a bone marrow transplant from his sister, who had the necessary tissue compatibility. Unfortunately this did not save him, presumably because he had already been exposed to too many infections by the time it was carried out. The parents decided to have another child, knowing that if it was a boy the chances of another case of SCID were 50%. Their doctors had a hope of being able to save the life of such a child by isolating him and then giving him a bone marrow transplant before he had been exposed to infections. The parents very soon had another child, it was a boy, he had SCID. The child was put into a sterile plastic bubble immediately after birth. Unfortunately it turned out that the planned bone marrow donor, David’s sister, was not a good match for him. It was necessary to wait and hope for an alternative donor. This hope was not fulfilled and David had to stay in the bubble. This had not been planned and it must be asked whether the doctors involved had really thought through what would happen if the optimal variant they had thought of did not work out.

At one point David started making punctures in his bubble as a way of attracting attention. Then it was explained to him what his situation was and why he must not damage the bubble. Later there was a kind of space suit produced for him by NASA which allowed him to move around outside his home. He only used it six times since he was too afraid there could be an accident. His physical health was good but understandably his psychological situation was difficult. New ideas in the practise of bone marrow transplantation indicated that it might be possible to use donors with a lesser degree of compatibility. On this basis David was given a transplant with his sister as the donor. It was not noticed that her bone marrow was infected with Epstein-Barr virus. As a result David got Burkitt’s lymphoma, a type of cancer which can be caused by that virus. (Compare what I wrote about this role of EBV here.) He died a few months after the operation, at the age of 12. Since that time treatment techniques have improved. The patient whose case is described in the book of Geha and Notarangelo had a successful bone marrow transplant (with his mother as donor). Unfortunately his lack of antibodies was not cured but this can be controlled with injections of immunoglobulin once every three weeks.

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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 of species with the property that whenever one of the species in occurs on the right hand side of a reaction one of the species in occurs on the left hand side. Geometrically we replace by the set of points of the non-negative orthant which are common zeroes of the elements of , thought of as linear functions on the species space. The defining property of a siphon is that 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 . The right hand side of the evolution equations of one of the concentrations belonging to is a sum of positive and negative terms. The negative terms automatically vanish on 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 is minimal. Correspondingly is maximal. The convention is that the empty set is excluded as a choice for and correspondingly the whole non-negative orthant as a choice for . What is allowed is to choose to be the whole of the species space which means that is the origin. Of course whether this choice actually defines a siphon depends on the particular dynamical system being considered.

If is an -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 be a stationary solution in the siphon . There must be some complex belonging to the network which contains an element of . If is another complex then there is a directed path from to . We can follow this path backwards from and conclude successively that each complex encountered contains an element of . Thus contains an element of and since was arbitrary all complexes have this property. This means that all complexes vanish at so that is a stationary solution.

Siphons can sometimes be used to prove persistence. Suppose that is a siphon for a certain network so that the points of are potential -limit points of solutions of the ODE system corresponding to this network. Suppose further that is a conserved quantity for the system which is a linear combination of the coordinates with positive coefficents. For a positive solution the quantity has a positive constant value along the solution and hence also has the same value at any of its -limit points. It follows that if vanishes on then no -limit point of that solution belongs to . If it is possible to find a conserved quantity 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.

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On Friday I flew to Chicago in order to attend an AMS sectional meeting. I had been in Chicago once before but that is many years ago now. I do remember being impressed by how much Lake Michigan looks like the sea, I suppose due to the structure of the waves. This impression was even stronger this time since there were strong winds whipping up the waves. Loyola University, the site of the meeting, is right beside the lake and it felt like home for me due to the combination of wind, waves and gulls. The majority of those were Ring-Billed Gulls which made it clear which side of the Atlantic I was on. There were also some Herring Gulls and although they might have been split from those on the other side of the Atlantic by the taxonomists I did not notice any difference. It was the first time I had been at an AMS sectional meeting and my impression was that the parallel sessions were very parallel, in other words in no danger of meeting. Most of the people in our session were people I knew from the conferences I attended in Charlotte and in Copenhagen although I did make a couple of new acquaintances, improving my coverage of the reaction network community.

In a previous post I mentioned Gheorghe Craciun’s ideas about giving the deficiency of a reaction network a geometric interpretation, following a talk of his in Copenhagen. Although I asked him questions about this on that occasion I did not completely understand the idea. Correspondingly my discussion of the point here in my blog was quite incomplete. Now I talked to him again and I believe I have finally got the point. Consider first a network with a single linkage class. The complexes of the network define points in the species space whose coordinates are the stoichiometric coefficients. The reactions define oriented segments joining the educt complex to the product complex of each reaction. The stoichiometric subspace is the vector space spanned by the differences of the complexes. It can also be considered as a translate of the affine subspace spanned by the complexes themselves. This makes it clear that its dimension is at most , where is the number of complexes. The number is the rank of the stoichiometric matrix. The deficiency is . At the same time . If there are several linkage classes then the whole space has dimension at most , where is the number of linkage classes. The deficiency is . If the spaces corresponding to the individual linkage classes have the maximal dimension allowed by the number of complexes in that class and these spaces are linearly independent then the deficiency is zero. Thus we see that the deficiency is the extent to which the complexes fail to be in general position. If the species and the number of complexes have been fixed then deficiency zero is seen to be a generic condition. On the other hand fixing the species and adding more complexes will destroy the deficiency zero condition since then we are in the case so that the possibility of general position is excluded. The advantage of having this geometric picture is that it can often be used to read off the deficiency directly from the network. It might also be used to aid in constructing networks with a desired deficiency.

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To explain the background to this treatment I first recall some facts about T cells. T cells are white blood cells which recognize foreign substances (antigens) in the body. The antigen binds to a molecule called the T cell receptor on the surface of the cell and this gives the T cell an activation signal. Since an inappropriate activation of the immune system could be very harmful there are built-in safety mechanisms. In order to be effective the primary activation signal has to be delivered together with a kind of certificate that action is really necessary. This is a second signal which is given via another surface molecule on the T cell, CD28. The T cell receptor only binds to an antigen when the latter is presented on the surface of another cell (an antigen-presenting cell, APC) in a groove within another molecule, an MHC molecule (major histocompatibility complex). On the surface of the APC there are under appropriate circumstances other molecules called B7.1 and B7.2 which can bind to CD28 and give the second signal. Once this has happened the activated T cell takes appropriate action. What this is depends on the type of T cell involved but for a cytotoxic T cell (one which carries the surface molecule CD8) it means that the T cell kills cells presenting the antigen. If the cell was a virus-infected cell and the antigen is derived from the virus then this is exactly what is desired. Coming back to the safety mechanisms, it is not only important that the T cell is not erroneously switched on. It is also important that when it is switched on in a justified case it should also be switched off after a certain time. Having it switched on for an unlimited time would never be justified. This is where CTLA4 comes in. This protein can bind to B7.1 and B7.2 and in fact does so more strongly than CD28. Thus it can crowd out CD28 and switch off the second signal. By binding to CTLA4 the antibody in ipilimumab stops it from binding to B7.1 and B7.2, thus leaving the activated T cell switched on. In some cases cancer cells present unusual antigens and become a target for T cells. The killing of these cells can be increased by CTLA4 via the mechanism just explained. At this point I should say that it may not be quite clear whether this is really the mechanism of action of CTLA4 in causing tumours to shrink. Alternative possibilities are mentioned in the Wikipedia article on CTLA4.

There are various things which have contributed to my interest in this subject. One is lectures I heard in the series ‘Universität im Rathaus’ [University in the Town Hall] in Mainz last February. The speakers were Matthias Theobald and Ugur Sahin and the theme was personalized cancer medicine. The central theme of what they were talking about is one step beyond what I have just sketched. A weakness of the therapy using antibodies to CTLA4 or the related approach using antibodies to another molecule PD-1 is that they are unspecific. In other words they lead to an increase not only in the activity of the T cells specific to cancer cells but of all T cells which have been activated by some antigen. This means that serious side effects are very likely. An approach which is theoretically better but as yet in a relatively early stage of development is to produce T cells which are specific for antigens belonging to the tumour of a specific patient and for an MHC molecule of that patient capable of presenting that antigen. From the talk I had the impression that doing this requires a lot of input from bioinformatics but I was not able to understand what kind of input it is. I would like to know more about that. Coming back to CTLA4, I have been interested for some time in modelling the activation of T cells and in that context it would be natural to think about also modelling the deactivating effects of CTLA4 or PD-1. I do not know whether this has been tried.

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Consider the MM system for the truncated cascade. The aim is then to find a Hopf bifurcation in a three-dimensional dynamical system with a lot of free parameters. Because of the many parameters is it not difficult to find a large class of stationary solutions. The strategy is then to linearize the right hand side of the equations about these stationary solutions and try show that there are parameter values where a suitable bifurcation takes place. To do this we would like to control the eigenvalues of the linearization, showing that it can happen that at some point one pair of complex conjugate eigenvalues passes through the imaginary axis with non-zero velocity as a parameter is varied, while the remaining eigenvalue has non-zero real part. The behaviour of the eigenvalues can largely be controlled by the trace, the determinant and an additional Hurwitz determinant. It suffices to arrange that there is a point where the trace is negative, the determinant is zero and the Hurwitz quantity passes through zero with non-zero velocity. This we did. A superficially similar situation is obtained by modelling an in vitro model for the MAPK cascade due to Prabakaran, Gunawardena and Sontag mentioned in a previous post in a way strictly analogous to that done in the Huang-Ferrell model. In that case the layers are in the opposite order and a crucial sign is changed. Up to now we have not been able to show the existence of a Hopf bifurcation in that system and our attempts up to now suggest that there may be a real obstruction to doing so. It should be mentioned that the known necessary condition for a stable hyperbolic periodic solution, the existence of a negative feedback loop, is satisfied by this system.

Now I will say some more about the model of Prabakaran et. al. Its purpose is to obtain insights on the issue of network reconstruction. Here is a summary of some things I understood. The in vitro biological system considered in the paper is a kind of simplification of the Raf-MEK-ERK MAPK cascade. By the use of certain mutations a situation is obtained where Raf is constitutively active and where ERK can only be phosphorylated once, instead of twice as in vivo. This comes down to a system containing only the second and third layers of the MAPK cascade with the length of the third layer reduced from three to two phosphorylation states. The second layer is modelled using simple mass action (MA) kinetics with the two phosphorylation steps being treated as one while in the third layer the enzyme concentrations are included explicitly in the dynamics in a standard Michaelis-Menten way (MM-MA). The resulting mathematical model is a system of seven ODE with three conservation laws. In the paper it is shown that for given values of the conserved quantities the system has a unique steady state. This is an application of a theorem of Angeli and Sontag. Note that this is not the same system of equations as the system analogous to that of Huang-Ferrell mentioned above.

The idea now is to vary one of the conserved quantities and monitor the behaviour of two functions and of the unknowns of the system at steady state. It is shown that for one choice of the conserved quantity and change in the same direction while for a different choice of the conserved quantity they change in opposite directions when the conserved quantity is varied. From a mathematical point of view this is not very surprising since there is no obvious reason forbidding behaviour of this kind. The significance of the result is that apparently biologists often use this type of variation in experiments to reach conclusions about causal relationships between the concentrations of different substances (activation and inhibition), which can be represented by certain signed oriented graphs. In this context ‘network reconstruction’ is the process of determining a graph of this type. The main conclusion of the paper, as I understand it, is that doing different experiments can lead to inconsistent results for this graph. Note that there is perfect agreement between the experimental results in the paper and the results obtained from the mathematical model. In a biological system if two experiments give conflicting results it is always possible to offer the explanation that some additional substance which was not included in the model is responsible for the difference. The advantage of the in vitro model is that there are no other substances which could play that role.

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For the Pettersson-MA model we were able to identify a regime in which overload breakdown takes place. This is where the initial concentrations of all sugar phosphates and inorganic phosphate in the chloroplast are sufficiently small. In that case the concentrations of all sugar phosphates tend to zero at late times with two exceptions. The concentrations of xylulose-4-phosphate and sedoheptulose-7-phosphate do not tend to zero. These results are obtained by linearizing the system around a simple stationary solution on the boundary and applying the centre manifold theorem. Another result is that if the reaction constants satisfy a certain inequality a positive solution can have no positive -limit points. In particular, there are no positive stationary solutions in that case. This is proved using a Lyapunov function related to the total number of carbon atoms. In the case of the Poolman-MA model it was shown that the stationary point which was stable in the Pettersson case becomes unstable. Moreover, a quantitative lower bound for concentration of sugar phosphates at late times in obtained.These results fit well with the intuitive picture of what should happen. Some of the results on the Poolman-MA model can be extended to analogous ones for the original Poolman model. On the other hand the task of giving a full rigorous definition of the Pettersson model was postponed for later work. The direction in which this could go has been sketched in a previous post.

There remains a lot to be done. It is possible to define a kind of hybrid model by setting in the Poolman model. It would be desirable to completely clarify the definition of the Pettersson model and then, perhaps, to show that it can be obtained as a well-behaved limiting system of the hybrid system in the sense of geometric singular perturbation theory. This might allow the dynamical properties of solutions of the different systems to be related to each other. The only result on stationary solutions obtained so far is a non-existence theorem. It would be of great interest to have positive results on the existence, multiplicity and stability of stationary solutions. A related question is that of classifying possible -limit points of positive solutions where some of the concentrations are zero. This was done in part in the paper but what was not settled is whether potential -limit points with positive concentrations of the hexose phosphates can actually occur. Finally, there are a lot of other models for the Calvin cycle on the market and it would be interesting to see to what extent they are susceptible to methods similar to those used in our paper.

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There is a standard mathematical model for the MAPK cascade due to Huang and Ferrell. It consists of three layers, each of which is a simple or dual futile cycle. Numerical and heuristic investigations indicate that the Huang-Ferrell model admits periodic solutions for certain values of the parameters. Together with Juliette Hell we set out to find a rigorous proof of this fact. In the beginning we pursued the strategy of showing that there are relaxation oscillations. An important element of this is to prove that the dual futile cycle exhibits bistability, a fact which is interesting in its own right, and we were able to prove this, as has been discussed here. In the end we shifted to a different strategy in order to prove the existence of periodic solutions. The bistability proof used a quasistationary (Michaelis-Menten) reduction of the Huang-Ferrell system. It applied bifurcation theory to the Michaelis-Menten system and geometric singular perturbation theory to lift this result to the original system. To prove the existence of periodic solutions we used a similar strategy. This time we showed the presence of Hopf bifurcations in a Michaelis-Menten system and lifted those. The details are contained in a paper which is close to being finished. In the meantime we wrote a review article on phosphorylation systems. Here I want to mention some of the topics covered there.

The MAPK cascade, which is the central subject of the paper is not isolated in its natural biological context. It is connected with other biochemical reactions which can be thought of as feedback loops, positive and negative. As already mentioned, the cascade itself consists of layers which are futile cycles. The paper first reviews what is known about the dynamics of futile cycles and the stand-alone MAPK cascasde. The focus is on phenomena such as multistability, sustained oscillations and (marginally) chaos and what can be proved about these things rigorously. The techniques which can be used in proofs of this kind are also reviewed. Given the theoretical results on oscillations it is interesting to ask whether these can be observed experimentally. This has been done for the Raf-MEK-ERK cascade by Shankaran et. al. (Mol. Syst. Biol. 5, 332). In that paper it is found that the experimental results do not fit well to the oscillations in the isolated cascade but they can be modelled better when the cascade is embedded in a negative feedback loop. Two other aspects are also built into the models used – the translocation of ERK from the cytosol to the nucleus (which is what is actually measured) and the fact that when ERK and MEK are not fully phosphorylated they can bind to each other. It is also briefly mentioned in our paper that a negative feedback can arise through the interaction of ERK with its substrates, as explained in Liu et. al. (Biophys. J. 101, 2572). For the cascade as treated in the Huang-Ferrell model with feedback added no rigorous results are known yet. (For a somewhat different system there is result on oscillations due to Gedeon and Sontag, J. Diff. Eq. 239, 273, which uses the strategy based on relaxation oscillations.)

In our paper there is also an introduction to two-component systems. A general conclusion of the paper is that phosphorylation systems give rise to a variety of interesting mathematical problems which are waiting to be investigated. It may also be hoped that a better mathematical understanding of this subject can lead to new insights concerning the biological systems being modelled. Biological questions of interest in this context include the following. Are dynamical features of the MAPK cascade such as oscillations desirable for the encoding of information or are they undesirable side effects? To what extent do feedback loops tend to encourage the occurrence of features of this type and to what extent do they tend to suppress them? What are their practical uses, if any? If the function of the system is damaged by mutations how can it be repaired? The last question is of special interest due to the fact that many cancer cells have mutations in the Raf-MEK-ERK cascade and there have already been many attempts to overcome their negative effects using kinase inhibitors, some of them successful. A prominent example is the Raf inhibitor Vemurafenib which has been used to treat metastatic melanoma.

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When I go on a holiday trip somewhere I often like to take a book with me which has some special connection to the place I am going. Often I have little time to actually read the book during the holiday but that does not matter. For Cornwall and, in particular, St. Ives the natural choice was ‘To the Lighthouse’. That novel is set in the Isle of Skye but it is well known that the real-life setting which inspired it (and the lighthouse of the title) was in St. Ives. This lighthouse, Godrevy Lighthouse, cost a little over seven thousand pounds to build, being finished in 1859. In 1892, on one of two visits there, the ten year old Virginia signed the visitors book. The book was sold for over ten thousand pounds in 2011. So in a sense the little girl’s signature ended up being worth more money than the lighthouse she was visiting. Of course, due to inflation, this is not a fair comparison. Looking on my bookshelves at home I was surprised to find that I do not own a copy of ‘To the Lighthouse’. On those shelves I find ‘The Voyage Out’, ‘Jacob’s Room’, ‘Moments of Being’ and ‘Between the Acts’ but neither ‘To the Lighthouse’ nor ‘The Waves’. Perhaps I never owned them and only borrowed them from libraries. I have a fairly clear memory of having borrowed ‘To the Lighthouse’ from the Kirkwall public library. I do not remember why I did so. Perhaps it was just that at that time I was omnivorously consuming almost everything I found in the literature section in that library. Or perhaps it had to do with the fact that lighthouses always had a special attraction for me. An alternative explanation for the fact I do not own the book myself could be that I parted with it when I left behind the majority of the books I owned when I moved from Aberdeen to Munich after finishing my PhD. This was due the practical constraint that I only took as many belongings with me as I could carry: two large suitcases and one large rucksack. I crossed the English Channel on a ferry and I remember how hard it was to carry that luggage up the gangway due to the fact that the tide was high.

I find reading ‘To the Lighthouse’ now a very positive experience. Just a few paragraphs put me in a frame of mind I like. I have the feeling that I am a very different person than what I was the first time I read it but after more than thirty years that is hardly surprising. I also feel that I am reading it in a different way from what I did then. I find it difficult to give an objective account of what it is that I like about the book. Perhaps it is the voice of the author. I feel that if I could have had the chance to talk to her I would certainly have enjoyed it even if she was perhaps not always the easiest of people to deal with. Curiously I have the impression that although I would have found it extremely interesting to meet Proust I am not sure I would have found it pleasant. So why do I think that I may be appreciating aspects of the book now which I did not last time? A concrete example is the passage where Mrs Ramsay is thinking about two things at the same time, the story she is reading to her son and the couple who are late coming home. The possibility of this is explained wonderfully by comparing it to ‘the bass … which now and then ran up unexpectedly into the melody’. I feel, although of course I cannot prove it, that I would not have paid much attention to that passage during my first reading. The differences may also be connected to the fact that I am now married. Often when I am reading a book it is as if my wife was reading it with me, over my shoulder, and this causes me to pay more attention to things which would interest her. A contrasting example is the story about Hume getting stuck in a bog. I am sure I paid attention to that during my first reading and it now conjured up a picture of how I was then, perhaps eighteen years old and still keen on philosophy. After a little thought following the encounter with the story it occurred to me that I knew more of the story about Hume, that he was allegedly forced to say that he believed in God in order to persuade an old woman to pull him out. This extended version is also something I knew in that phase of my life, perhaps through my membership in the Aberdeen University philosophy society. On the other hand this story does come up (at least) two more times in the book and it is a little different from what I remember. What the woman forced him to do was to say the Lord’s Prayer.

I came back from England yesterday and although I did not have much time for reading the book while there I am on page 236 due to the head start I had by reading it before I went on the trip. The day we went to St. Ives started out rainy but the weather cleared up during the morning so that about one o’ clock I was able to see Godrevy lighthouse and look at it through through my binoculars. They also allowed me to enjoy good views of passing gannets and kittiwakes but I think I would have been disappointed if I had made that trip without seeing the lighthouse.

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