I have just read the book ‘The Vaccine’ by Joe Miller, Uğur Şahin and Özlem Türeci. More precisely, I read the German version which is called ‘Projekt Lightspeed’ but I am assuming that the contents are not too different. The quality of the language in the version I read is high and I conclude from this that it is likely that both the quality of the language in the original and the quality of the translation are high. Miller is a journalist while Şahin and Türeci are the main protagonists of the story told in the book. It is the story of how the husband and wife team of researchers developed the BioNTech vaccine against COVID-19, a story which I found more gripping than fictional thrillers. The geographical centre of the story is Mainz. Şahin and Türeci live there and the headquarters of BioNTech, the company they founded, is also there. In fact when I moved to Mainz in 2013 I lived just a couple of hundred metres from what is now the area occupied by the BioNTech. Since I was interested in biotechnology the building was interesting for me. My first encounter with Şahin was a public lecture he gave about cancer immunotherapy in February 2015 and which I wrote about here. I heard him again in a keynote talk he gave at a conference at EMBL about cancer immunotherapy in February 2017. I was interested to hear his talk but it seems that it did not catch my attention since I did not mention it in the account I wrote of that meeting. One of the last lectures I attended live before the pandemic made such things impossible was at the university medical centre here in Mainz on 13th February 2020. Şahin was the chairman. The speaker was Melanie Brinkmann and the subject the persistence of herpes viruses in the host. I did not detect any trace of the theme COVID-19 in the meeting that day except for the fact that the speaker complained that she was getting asked so many questions on that subject on a daily basis. Later on she attained some public prominence in Germany in the discussion of measures against the pandemic. The book is less about the science of the subject than about the human story involved. I have no doubt that the scientific content is correct but it is not very deep. That is not the main subject of the book.

I now come to the story itself. Şahin and Türeci are both Germans whose parents came to Germany from Turkey. They studied medicine and they met during the practical part of their studies. They were both affected by seeing patients dying of cancer while medicine was helpless to prevent it. They decided they wanted to change the situation and have pursued that goal with remarkable consistency since then. They later came to the University of Mainz. They founded a biotechnology company called Ganymed producing monoclonal antibodies which was eventually sold for several hundred million Euros. They then went on to found BioNTech with the aim of using mRNA technology for cancer immunotherapy. An important role was played by money provided by the Strüngmann brothers. They had become billionaires through their company Hexal which sold generic drugs. They were relatively independent of the usual mechanisms of the financial markets and this was a big advantage for BioNTech. (A side remark: I learned from the book that the capital NT in the middle of the company name stands for ‘new technology’.) In early January 2020 Şahin foresaw the importance of COVID-19 and immediately began a project to apply the mRNA technology of BioNTech to develop a vaccine. The book is the story of many of the obstacles which he and Türeci had to overcome to attain this goal. In the US the vaccine is associated with the name Pfizer and it is important to mention at this point what the role of Pfizer was, namely to provide money and logistics. The main ideas came from Şahin and Türeci. Of course no important scientific development is due to one or two people alone and there are many contributions. In this case a central contribution came from Katalin Karikó.

How does the BioNTech vaccine work? The central idea of an mRNA vaccine is as follows. The aim is to introduce certain proteins into the body which are similar to ones found in the virus. The immune response to these proteins will then also act against the virus. What is actually injected is mRNA and that is then translated into the desired proteins by the cellular machinery. To start with the sequences of relevant proteins must be identified and corresponding mRNA molecules produced in vitro based on a DNA template. The RNA does not only contain the code for the protein but also extra elements which influence the way in which it behaves or is treated within a cell. In addition it is coated with some lipids which protect it from degradation by certain enzymes and help it to enter cells. Karikó played a central role in the development of this lipid technology. After the RNA has been injected it has to get into cells. A good target cell type are the dendritic cells which take up material from their surroundings by macropinocytosis. They then produce proteins based on the RNA template, cut them up into small peptides and display these on their surface. They also move to the lymph nodes. There they can present the antigens to T cells, which get activated. For T cells to get activated a second signal is also necessary and it is fortunate that mRNA can provide such a signal – in the language of vaccines it shows a natural adjuvant activity. In many more popular accounts of the role of the immune system in the vaccination against COVID-19 antibodies are the central subject. In fact according to the book many vaccine developers are somewhat fixated on antibodies and underestimate the role of T cells. There Şahin had to do a lot of convincing. It is nevertheless the case that antibodies are very important in this story and there is one point which I do not understand. Antibodies are produced by B cells and in order to do so they must be activated by the antigen. For this to happen the antigen must be visible outside the cells. So how do proteins produced in dendritic cells get exported so that B cells can see them?

I admire Şahin and Türeci very much. This has two aspects. The first is their amazing achievement in producing the vaccine against COVID-19 in record time. However there is also another aspect which I find very important. It is related to what I have learned about these two people from the book and from other sources. It has to do with a human quality which I find very important and which I believe is not appreciated as it should be in our society. This is humility. In their work Şahin and Türeci have been extremely ambitious but it seems to me that in their private life they have remained humble and this makes them an example to be followed.

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The International Congress of Mathematicians takes place every four years and is the most important mathematical conference. In 1900 the conference took place in Paris and David Hilbert gave a talk where he presented a list of 23 mathematical problems which he considered to be of special importance. This list has been a programme for the development of mathematics ever since. The individual problems are famous and are known by their numbers in the original list. Here I will write about the 16th problem. In fact the problem comes in two parts and I will say nothing about one part, which belongs to the domain of algebraic geometry. Instead I will concentrate exclusively on the other, which concerns dynamical systems in the plane. Consider two equations , , where and are polynomial. Roughly speaking, the problem is concerned with the question of how many periodic solutions this system has. The simple example , shows that there can be infinitely many periodic solutions and that the precise question has to be a little different. A periodic solution is called a limit cycle if there is another solution which converges to the image of the first as . The real issue is how many limit cycles the system can have. The first question is whether for a given system the number of limit cycles is always finite. A second is whether an inequality of the form holds, where depends only on the degree of the polynomials. is called the Hilbert number. Linear systems have no limit cycles so that . Until recently it was not known whether was finite. A third question is to obtain an explicit bound for . The first question was answered positively by Écalle (1992) and Ilyashenko (1991), independently.

The subject has a long and troubled history. Already before 1900 Poincaré was interested in this problem and gave a partial solution. In 1923 Dulac claimed to have proved that the answer to the first question was yes. In a paper in 1955 Petrovskii and Landis claimed to have proved that and that for a particular cubic polynomial . Both claims were false. As shown by Ilyashenko in 1982 there was a gap in Dulac’s proof. After 60 years there was almost no progress on this problem. Écalle worked on it intensively for a long time. For this reason he produced few publications and almost lost his job. At this point I pause to give a personal story. Some years ago I was invited to give a talk in Bayrischzell in the context of the elite programme in mathematical physics of the LMU in Munich. The programme of such an event includes two invited talks, one from outside mathematical physics (which in this case was mine) and one in the area of mathematical physics (which in this case was on a topic from string theory). In the second talk the concept of resurgence, which was invented by Écalle in his work on Hilbert’s sixteenth problem, played a central role. I see this as a further proof of the universality of mathematics.

The basic idea of the argument of Dulac was as follows. If there are infinitely many limit cycles then we can expect that they will accumulate somewhere. A point where they accumulate will lie on a periodic solution, a homoclinic orbit or a heteroclinic cycle. Starting at a nearby point and following the solution until it is near to the original point leads to a Poincaré mapping of a transversal. In the case of a periodic limiting solution this mapping is analytic. If there are infinitely many limit cycles the fixed points of the Poincaré mapping accumulate. It follows that this mapping is equal to the identity, contradicting the limit cycle nature of the solutions concerned. Dulac wanted to use a similar argument in the other two cases. Unfortunately in that case the Poincaré mapping is not analytic. What we need is a class of functions which on the one hand is general enough to include the Poincaré mappings in this situation and on the on the other hand cannot have an accumulating set of fixed points without being equal to the identity. This is very difficult and is where Dulac made his mistake.

What about the second question? It has been known since 1980 that . There is an example with three limit cycles which contain a common steady state and another which does not. It is problematic to find (or to portray) these with a computer since three are very small and one very large. More about the history can be found in an excellent article of Iyashenko (Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc. 39, 301) which was the main source for what I have written. In March 2021 a preprint by Pablo Pedregal appeared (http://arxiv.org/pdf/2103.07193.pdf) where he claimed to have answered the second problem. I feel that some caution is necessary in accepting this claim. A first reason is the illustrious history of mistakes in this field. A second is that Pedregal himself produced a related preprint with Llibre in 2014 which seems to have contained a mistake. The new preprint uses techniques which are far away from those usually applied to dynamical systems. On the one hand this gives some plausibility that it might contain a really new idea. On the other hand it makes it relatively difficult for most people coming from dynamical systems (including myself) to check the arguments. Can anyone out there tell me more?

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Yesterday my wife and I got our second vaccination. In the meantime the relevant authority (STIKO) has recommended that those vaccinated once with the product of AstraZeneca should get an mRNA vaccine the second time. The fact that we waited the rather long time suggested to get our second injection meant that the new recommendation had already come out and we were able to get the vaccine of Biontech the second time around. There have not been many studies of the combination vaccination but as far as I have seen those that there are gave very positive results. So we are happy that it turned out this way. This time the arm where I got the injection was sensitive to pressure during the night but this effect was almost gone by this morning. The only other side effect I noticed was an increased production of endorphins. In other words, I was very happy to have reached this point although I know that it takes a couple of weeks before the maximal protection is there.

Every second year there is an event in Mainz devoted to the popularization of science called the Wissenschaftsmarkt. It has been taking place for the last twenty years. Normally it is in the centre of town but due to the pandemic it will be largely digital this year. This year it is on 11th and 12th September and has the title ‘Mensch und Gesundheit’ [rough translation: human beings and their health]. I will contribute a video with the title ‘Gegen COVID-19 mit Mathematik’ [against COVID-19 with mathematics]. The aim of this video is to explain to non-scientists the importance of mathematics in fighting infectious diseases. I talk about what mathematical models can contribute in this domain but also, which is just as important, about what they cannot do. If the public is to trust statements by scientists it is important to take measures against creating false expectations. I do not go into too much detail about COVID-19 itself since at the moment there is too little information available and too much public controversy. Instead I concentrate on an example from long ago where it is easier to see clearly. It also happens to be the example where the basic reproductive number was discovered. This is the work of Ronald Ross on the control of malaria. Ross was the one who demonstrated that malaria is transmitted by mosquito bites and he was rewarded for that discovery with a Nobel Prize in 1902. After that he studied ways of controlling the disease. This was for instance important in the context of the construction of the Panama Canal. There the first attempt failed because so many workers died of infectious diseases, mainly malaria and yellow fever, both transmitted by mosquitos. The question came up, whether killing a certain percentage of mosquitos could lead to a long-term elimination of malaria or whether the disease would simply come back. Ross, a man of many talents, set up a simple mathematical model and used it to show that elimination is possible and was even able to estimate the percentage necessary. This provided him with a powerful argument which he could use against the many people who were sceptical about the idea.

]]>The general idea is as follows. We have a dynamical system depending on parameters. We want to compare it with a normal form which is a standard model dynamical system depending on parameters. Here everything is local in a neighbourhood of , which is a steady state. The number of parameters in the normal form is the codimension. In the end we want to conclude that dynamical features which are present in the normal form are also present in the original system. The normal form for the Hopf bifurcation contains periodic solutions. In fact there are two versions depending on a parameter which can take the value plus or minus one. These are known as supercritical and subcritical. In the supercritical case the periodic solutions are stable, in the subcritical case unstable. The normal form for the Bogdanov-Takens bifurcation also has two cases which may, by analogy with the Hopf case, be called super- and subcritical. The fact we used is that when a Bogdanov-Takens bifurcation exists there is always a Hopf bifurcation close to it. Moreover the super- or subcritical nature of the bifurcation is inherited. What I wanted to investigate is whether the presence of a suitable bifurcation of codimension three could imply the existence of a Bogdanov-Takens bifurcation close to it and whether it might even be the case that both super- and subcritical Bogdanov-Takens bifurcations are obtained. If this were the case then it would indicate an avenue to a statement of the kind that the existence of a generic bifurcation of codimension three of a certain type implies the existence of stable periodic solutions.

The successful comparison of a given system with the normal form relies on certain genericity conditions being satisfied. These are of two types. The first type is a condition on the system defined by setting . It says that the system does not belong to a certain degenerate set defined by the vanishing of particular functions of the derivatives of at . In an abstract way this degenerate set may be thought of a subset of the set of -jets of functions at the origin. This subset is a real algebraic variety which is a union of smooth submanifolds (strata) of different dimensions which fit together in a nice way. The set defining the bifurcation itself is also a variety of this type and the degenerate set consists of all strata of the bifurcation set except that of highest dimension. The family induces a mapping from the parameter space into the set of -jets. The second type of genericity condition says that this mapping should be transverse to the bifurcation set. Because of the genericity condition of the first type the image of lies in the stratum of highest dimension in the bifurcation set, which is a smooth submanifold. The transversality condition says that the sum of the tangent space to this manifold and the image of the linearization at of the mapping defined by is the whole space. Writing about these things gives me a strange feeling since they involve concepts which I used in a very different context in my PhD, which I submitted 34 years ago, and not very often since then.

The following refers to a two-dimensional system. In the Bogdanov-Takens case the bifurcation set is defined by the condition that the linearization about has a double zero eigenvalue. The genericity condition of the first type involves the -jet. The first part BT.0 (cf. this post for the terminology) says that the linearization should not be identically zero. The second part consists of conditions BT.1 and BT.2 involving the second derivatives. The remaining condition BT.3 is the transversality condition (genericity condition of the second type). Now I want to study bifurcations which satisfy the eigenvalue condition for a Bogdanov-Takens bifurcation and the condition BT.0 but which may fail to satisfy BT.1 or BT.2. At one point the literature on the subject appeared to me like a high wall which I did not have the resolve to try to climb. Fortunately Sebastiaan Janssens pointed me to a paper of Kuznetsov (Int. J. Bif. Chaos, 15, 3535) which was a gate through the wall and gave me access to things on the other side which I use in the following discussion. Following Dumortier et al. (Erg. Th. Dyn. Sys. 7, 375) we look at cases where the -jet of the right hand side of the equation for is of the form . It turns out that we can divide the problem into two cases, in each of which one of the coefficients and is zero and the other non-zero. We first concentrate on the case which is that studied in the paper just quoted. There it is referred to as the cusp case. It is the case where BT.2 is satisfied and BT.1 fails. As explained in the book of Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularites and Abelian integrals.) it is useful to further subdivide the case into three subcases, known as the saddle, focus and elliptic cases. These cases give rise to different normal forms (partially conjectural). A bifurcation diagram for the cusp case can be found in the paper of Dumortier et al. Since we are dealing with a codimension 3 bifurcation the bifurcation diagram should be three-dimensional. It turns out, however, that the bifurcation set has a conical structure near the origin, so that its structure is determined by its intersection with any small sphere near the origin. This gives rise to a two-dimensional object which can be well represented in the plane. Note, however, that passing from the sphere to the plane necessarily involves discarding a ‘point at infinity’. It is this intersection which is represented in Fig. 3 of the paper of Dumortier et al. That this is an accurate representation of the bifurcation diagram in this case is the main content of the paper of Dumortier et al.

In the bifurcation diagram in the paper of Dumortier et al. we see that there are two Bogdanov-Takens points and which are joined by a curve of Hopf bifurcations. The points and are stated to be codimension 2 and this indicates that they are non-degenerate Bogdanov-Takens points. There is one exceptional point on the curve of Hopf bifurcations which is stated to be of codimension 2. This indicates that it is a non-degenerte Bautin bifurcation point. This in turn implies that one of the Bogdanov-Takens points is supercritical and the other subcritical. Thus in this case there exist both stable and unstable periodic solutions in each neighbourhood of the bifurcation point. This indicates that the strategy outlined at the beginning of this post can in principle work. Whether it can work in practise in a given system depends on how difficult it is to check the relevant non-degeneracy conditions. I end with a brief comment on the case . The book of Dumortier et al. presents conjectured bifurcation diagrams for all subcases but does not claim complete proofs that they are all correct. In all cases we have two Bogdanov-Takens points joined by a curve of Hopf bifurcations on thich there is precisely one Bautin point. Thus in this sense we have the same configuration as in the case . In a discussion of these matters in the paper of Kuznetsov mentioned above (which is from 2005) it is stated that a complete proof is only available in the saddle case. I do not know if there has been any further progress on this since then.

]]>For simplicity we consider the model in one space dimension. The spatial domain is then a finite closed interval and Neumann boundary conditions are imposed for the concentration of ATP. We prove that for suitable values of the parameters in the model there exist infinitely many smooth inhomogeneous steady states. It turns out that all of these are (nonlinearly) unstable. This is not a special feature of this system but in fact, as pointed out in a paper of Marciniak-Czochra et al. (J. Math. Biol. 74, 583), it is a frequent feature of systems where ODE are coupled to a diffusion equation. This can be proved using a method discussed in a previous post which allows nonlinear instability to be concluded from spectral instability. We prove the spectral instability in our example. There may also exist non-smooth inhomogeneous steady states but we did not enter into that theme in our paper. If stable inhomogeneous steady states cannot be used to explain the experimental observations what alternatives are available which still keep the same model? If experiments only measure time averages then an alternative would be limit sets other than steady states. In this context it would be interesting to know whether the system has spatially homogeneous solutions which are periodic or converge to a strange attractor. A preliminary investigation of this question in the paper did not yield definitive results. With the help of computer calculations we were able to show that it can happen that the linearization of the (spatially homogeneous) system about a steady state has non-real eigenvalues, suggesting the presence of at least damped oscillations. We proved global existence for the full system but we were not able to show whether general solutions are bounded in time, not even in . It can be concluded that there are still many issues to be investigated concerning the long-time behaviour of solutions of this system.

]]>The discussion which follows is based on a paper called ‘Spectral Condition for Instability’ by Jalal Shatah and Walter Strauss. At first sight their proof looks very different from the proof I presented in the last post and here I want to compare them, with particular attention to what happens in the finite-dimensional case. We want to show that the origin is nonlinearly unstable under the assumption that the spectrum of intersects the half of the complex plane where the real part is positive. In the finite-dimensional case this means that has an eigenvalue with positive real part. The spectral mapping theorem relates this to the situation where the spectrum of intersects the exterior of the unit disk. In the finite-dimensional case it means that there is an eigenvalue with modulus greater than one. We now consider the hypotheses of the Theorem of Shatah and Strauss. The first is that the linear operator occurring in the equation generates a strongly continuous semigroup. The second is that the spectrum of meets the exterior of the unit disk. The third is that in a neighbourhood of the origin the nonlinear term can be estimated by a power of the norm greater than one. The proof of nonlinear instability is based on three lemmas. We take to be a point of the spectrum of whose modulus is equal to the spectral radius. In the finite-dimensional case this would be an eigenvalue and we would consider the corresponding eigenvector . In general we need a suitable approximate analogue of this. Lemma 1 provides for this by showing that belongs to the approximate point spectrum of . Lemma 2 then shows that there is a whose norm grows at a rate no larger than and is such that the norm of the difference between and can be controlled. Lemma 3 shows that the growth rate of the norm of is close to . In the finite-dimensional case the proofs of Lemma 1 and Lemma 2 are trivial. In Lemma 3 the lower bound is trivial in the finite-dimensional case. To get the upper bound in that case a change of basis can be used to make the off-diagonal entries in the Jordan normal form as small as desired. This argument is similar to that used to treat in the previous post. In the theorem we choose a ball such that instability corresponds to leaving it. As long as a solution remains in that ball the nonlinearity is under good control. The idea is to show that as long as the norm of the initial condition is sufficiently small the contribution of the nonlinear term on the right hand side of the integral equation will remain small compared to that coming from the linearized equation, which is growing at a known exponential rate. The details are complicated and are of course the essence of the proof. I will not try to explain them here.

]]>I already mentioned that AstraZeneca became unpopular. In connection with the thromboses it was only recommended by the Paul Ehrlich Institute (the official body for such things in Germany) for people over sixty. The argument was that since people over sixty had a higher probability of serious consequences if they got infected the net benefit was positive for them. Doctors were having problems getting rid of their doses. However the situation is very dynamic. Although AstraZeneca is still not recommended for people under 60 (like myself) and it is not administered to them in the official vaccination centres doctors (GP’s and specialists) are allowed to vaccinate people under 60 with that product if they inform them about the risks. They are also allowed to reduce the waiting time between the injections. These political moves have led to many people (particularly young people) wanting to get vaccinated with AstraZeneca. The result is that now instead of being hard to get rid of the AstraZeneca vaccine is scarce. Thus now there is a lot of competition and the whole process is quite chaotic. We were lucky that my wife was offered a vaccination with AstraZeneca at a time when this seemed like asking for a favour rather than offering to do a favour. She asked if I could also be vaccinated by the same doctor and got a positive answer. This was our good fortune.

It was still not clear to us whether we should accept the offer. It is in the nature of the situation that many things concerning the disease and the vaccination are simply not known. On the other hand the amount of information available is huge. All this makes a rational decision difficult. From my point of view the thromboses were so rare that they did not influence my decision at all. The combination of the earlier availability of AstraZeneca for us with the longer waiting time meant that it was not clear which option (taking AstraZeneca now or waiting for Biontech) would lead to being fully immunized sooner. We did not want to shorten the interval for the AstraZeneca vaccination (which our doctor would have accepted) for a reason I will discuss later. A reason for making the choice we did was that this gave the feeling of finally making some progress. In the comparison of the efficiencies of the two products it is important to know that if the criterion considered is the risk of serious illness or hospitalization then according to the official figures AstraZeneca is at least as good as Biontech (both at least 95%). In any case, it was a good feeling to know that we had received one dose of the vaccine. A few hours later I have not noticed any side-effects whatsoever.

Let me return to the question of shortening the time between the injections for AstraZeneca. There is some evidence that this decreases the efficiency of the vaccinations and I have read about two possible mechanisms for this which seem to me at least plausible. The first is connected with the phenomenon of affinity maturation. The first B cells which are activated in response to a certain antigen can undergo somewhat random mutations. This means that the population of B cells becomes rather heterogeneous. The different variants then compete with each other in such a way that those which bind most strongly to the antigen come to dominate the population. In this way the quality of the antibodies is increased. If a second vaccination is given too soon it can interrupt the affinity maturation initiated by the first vaccination before the antibodies have been optimized. The second mechanism is as follows. The immune reponse against the antigen remains active for some time after the vaccination. If second vaccination is given while that is still the case then the antibodies generated in the first vaccination can bind to the vector viruses coming from the second vaccination and prevent them from achieving their intended purpose. These are not established facts but I prefer to have plausible hypotheses than a complete lack of a possible explanation.

]]>Up to this point I have avoided giving precise definitions. So what does nonlinear instability of mean? It means that there is a neighbourhood of such that for each neighbourhood of there is a solution satisfying and for some . In other words, there are solutions which start arbitrarily close to and do not stay in . How can this be proved? One way of doing so is to use a suitable monotone function defined on a neighbourhood of . This function should be continuously differentiable and satisfy the conditions that , for and for . Here is the rate of change of along the solution. Let be sufficiently small so that the closed ball is contained in the domain of definition of . We will take this ball to be the neighbourood in the definition of instability. Let be the maximum of on . Thus in order to show that a solution leaves it is enough to show that exceeds . Consider any solution which starts at a point of other than for . The set where is open and the solution can never enter it for . The intersection of its complement with is compact. Thus has a positive minimum there. As long as the solution does not leave we have . Hence . This implies that if the solution remains in for all then eventually exceeds , a contradiction. This result can be generalized as follows. Let be an open set such that is contained in its closure. Suppose that we have a function which vanishes on the part of the boundary of intersecting and for which on except at . Then is nonlinearly unstable with a proof similar to that just given.

Now it will be shown that if has an eigenvalue with positive real part a function with the desired properties exists. We can choose coordinates so that the steady state is at the origin and that the stable, centre and unstable subspaces at the origin are coordinate subspaces. The solution can be written in the form where these three variables are the projections on the three subspaces. Then is a direct sum of matrices , and , whose eigenvalues have real parts which are positive, zero and negative respectively. It can be arranged by a choice of basis in the centre subspace that the symmetric part of is as small as desired. It can also be shown that because of the eigenvalue properties of there exists a positive definite matrix such that . For the same reason there exists a positive definite matrix such that . Let . Then . The set is defined by the condition . There for a positive constant . On this region , where we profit from the special basis of the centre subspace mentioned earlier. The quadratic term in which does not have a good sign has been absorbed in the quadratic term in which does. This completes the proof of nonlinear instability. As they stand these arguments do not apply to the infinite-dimensional case since compactness has been used freely. A discussion of the infinite-dimensional case will be postponed to a later post.

]]>In the finite-dimensional case we know the solution of the equation with initial datum . It is . It is tempting to keep this formula even when is unbounded, but it must then be supplied with a suitable interpretation. There are general ways of defining nonlinear functions of unbounded linear operators using spectral theory but here I want to pursue another direction, which uses a kind of axiomatic approach to the exponential function . It should have the property that and it should satisfy the semigroup property for all non-negative and . It remains to require some regularity property. One obvious possibility would be to require that is a continuous function from into the space of bounded operators with the topology defined by the operator norm. Unfortunately this is too much. Let us define an operator whenever this limit exists in and to be the linear subspace for which it exists. In this way we get an unbounded operator on a definite domain. The problem with the continuity assumption made above is that it implies that . In other words, if the operator is genuinely unbounded then this definition cannot apply. In particular it cannot apply to our example. It turns out that the right assumption is that for and any . This leads to what is called a strongly continuous one-parameter semigroup. is called the infinitesimal generator of . Its domain is dense and it is a closed operator, which means that its graph is a closed subset (in fact linear subspace) of the product with the topology defined by the product norm. In a case like the example above the problem with continuity is only at . The solution of the heat equation is continuous in in any reasonable topology on any reasonable Banach space for but not for . In fact it is even analytic for , something which is typical for linear parabolic equations.

In this discussion we have said how to start with a semigroup and get its generator but what about the converse? What is a criterion which tells us for a given operator that it is the generator of a semigroup? A fundamental result of this type is the Hille-Yosida theorem. I do not want to go into detail about this and related results here. I will just mention that it has to do with spectral theory. It is possible to define the spectrum of an unbounded operator as a generalization of the eigenvalues of a matrix. The complement of the spectrum is called the resolvent set and the resolvent is , which is a bounded operator. The hypotheses made on the generator of a semigroup concern the position of the resolvent set in the complex plane and estimates on the norm of the resolvent at infinity. In this context the concept of a sectorial operator arises.

My interest in these topics comes from an interest in systems of reaction-diffusion equations of the form . Here is vector-valued, is a diagonal matrix with non-negative elements and the Laplace operator is to be applied to each component of . I have not found it easy to extract the results I would like to have from the literature. Part of the reason for this is that I am interested in examples where not all the diagonal elements of are positive. That situation might be described as a degenerate system of reaction diffusion equations or as a system of reaction-diffusion equations coupled to a system of ODE. In that case a lot of results are not available ‘off the shelf’. Therefore to obtain an understanding it is necessary to penetrate into the underlying theory. One of the best sources I have found is the book ‘Global Solutions of Reaction-Diffusion Systems’ by Franz Rothe.

]]>All these results on the properties of solutions of reaction networks are on the level of simulations. Recently Lisa Kreusser and I set out to investigate these phenomena on the level of rigorous mathematics and we have just put a paper on this subject on the archive. The model developed by Doherty et al. is one-dimensional and therefore relatively easy to analyse. The first thing we do is to give a rigorous proof of bistability for this system together with some information on the region of parameter space where this phenomenon occurs. We also show that it can be lifted to the system from which the one-dimensional system is obtained by timescale separation. The latter system has no periodic solutions. To obtain bistability the effect of the phosphorylation must be to activate the enzyme. It does not occur in the case of inhibition. We show that when an external kinase is included (in the case of Lck there is an external kinase Csk which may be relevant) and we do not restrict to the Michaelis-Menten regime bistability is restored.

We then go on to study the dynamics of the model of Kaimachnikov and Kholodenko, which is three-dimensional. These authors mention that it can be reduced to a two-dimensional model by timescale separation. Unfortunately we did not succeed in finding a rigorous framework for their reduction. Instead we used another related reduction which gives a setting which is well-behaved in the sense of geometric singular perturbation theory (normally hyperbolic) and can therefore be used to lift dynamical features from two to three dimensions in a rather straightforward way. It then remains to analyse the two-dimensonal system. It is easy to deduce bistability from the results already mentioned. We go further and show that there exist periodic and homoclinic solutions. This is done by showing the existence of a generic Bogdanov-Takens bifurcation, a procedure described more generally here and here. This system contains an abundance of parameters and the idea is to fix these so as to get the desired result. First we choose the coordinates of the steady state to be fixed simple rational numbers. Then we fix all but four of the parameters in the system. The four conditions for a BT bifurcation are then used to determine the values of the other four parameters. To get the desired positivity for the computed values the choices must be made carefully. This was done by trial and error. Establishing genericity required a calculation which was complicated but doable by hand. When a generic BT bifurcation has been obtained it follows that there are generic Hopf bifurcations nearby in parameter space and the nature of these (sub- or supercritical) can be determined. It turns out that in our case they are subcritical so that the associated periodic solutions are unstable. Having proved that periodic solutions exist we wanted to see what a solution of this type looks like by simulation. We had difficulties in finding parameter values for which this could be done. (We know the parameter values for the BT point and that those for the periodic solution are nearby but they must be chosen in a rather narrow region which we do not know explicitly.) Eventually we succeeded in doing this. In this context I used the program XPPAUT for the first time in my life and I learned to appreciate it. I see this paper as the beginning rather than the end of a path and I am very curious as to where it will lead.

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