Strongly continuous semigroups

March 19, 2021

An evolution equation is an equation for a function u(t,x) depending on a time variable t and some other variables x which can often be thought of as spatial variables. There is also the case where there are no variables x, which is the one relevant for ordinary differential equations. We can reinterpret the function u as being something of the form u(t)(x), a function of x which depends on t. I am thinking of the case where u(t,x) takes its values in a Euclidean space. Then u(t) should be thought of as taking values in a function space. Different regularity assumptions on the solutions naturally lead to different choices of function space. Suppose, for instance, I consider the one-dimensional heat equation u_t=u_{xx}. Then I could choose the function space to be C^0, the space of continuous functions, C^2, the space of twice continuously differentiable functions or L^p. For some choices of function spaces we are forced to consider weak solutions. It is tempting to consider the evolution equation as an ODE in a function space. This can have advantages but also disadvantages. It gives us intuition which can suggest ideas but the analogues of statements about ODE often do not hold for more general evolution equations, in particular due to loss of compactness. (In what follows the function space considered will always be a Banach space.) We can formally write the equation of the example as u_t=Au for a linear operator A. If we choose the function space to be X=L^p then this operator cannot be globally defined, since A does not map from X to itself. This leads to the consideration of unbounded operators from X to X. This is a topic which requires care with the use of language and the ideas which we associate to certain formulations. An unbounded operator from X to X is a linear mapping from a linear subspace D(A), the domain of A, to X. Just as there may be more than one space X which is of interest for a given evolution equation there may be more than one choice of domain which is of interest even after the space has been chosen. To take account of this an unbounded operator is often written as a pair (A, D(A)). In the example we could for instance choose D(A) to be the space of C^\infty functions or the Sobolev space W^{2,p}.

In the finite-dimensional case we know the solution of the equation u_t=Au with initial datum u_0. It is u(t)=e^{tA}u_0. It is tempting to keep this formula even when A 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 S(t)=e^{tA}. It should have the property that S(0)=I and it should satisfy the semigroup property S(s+t)=S(s)S(t) for all non-negative s and t. It remains to require some regularity property. One obvious possibility would be to require that S is a continuous function from [0,\infty) into the space of bounded operators L(X) with the topology defined by the operator norm. Unfortunately this is too much. Let us define an operator Ax=\lim_{t\to 0}\frac{S(t)x-x}{t} whenever this limit exists in X and D(A) 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 D(A)=X. In other words, if the operator A 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 \|S(t)x-x\|\to 0 for t\to 0 and any x\in D(X). This leads to what is called a strongly continuous one-parameter semigroup. A is called the infinitesimal generator of S(t). 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 X\times X with the topology defined by the product norm. In a case like the example above the problem with continuity is only at t=0. The solution of the heat equation is continuous in t in any reasonable topology on any reasonable Banach space for t>0 but not for t=0. In fact it is even analytic for t>0, something which is typical for linear parabolic equations.

In this discussion we have said how to start with a semigroup S(t) and get its generator (A,D(A)) but what about the converse? What is a criterion which tells us for a given operator (A,D(A)) 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 (A-\lambda I)^{-1}, 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 u_t+D\Delta u=f(u). Here u is vector-valued, D is a diagonal matrix with non-negative elements and the Laplace operator is to be applied to each component of u. 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 D 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.

Dynamics of the activation of Lck

January 26, 2021

The enzyme Lck (lymphocyte-associated tyrosine kinase) is of central importance in the function of immune cells. I hope that mathematics can contribute to the understanding and improvement of immune checkpoint therapies for cancer. For this reason I have looked at the relevant mathematical models in the literature. In doing this I have realized the importance in this context of obtaining a better understanding of the activation of Lck. I was already familiar with the role of Lck in the activation of T cells. There are two tyrosines in Lck, Y394 and Y505, whose phosphorylation state influences its activity. Roughly speaking, phosphorylation of Y394 increases the activity of Lck while phosphorylation of Y505 decreases it. The fact that these are influences going in opposite directions already indicates complications. In fact the kinase Lck catalyses its own phosphorylation, especially on Y394. This is an example of autophosphorylation in trans, i.e. one molecule of Lck catalyses the phosphorylation of another molcule of Lck. It turns out that autophosphorylation tends to favour complicated dynamics. It is already the case that in a protein with a single phosphorylation site the occurrence of autophosphorylation can lead to bistability. Normally bistability in a chemical reaction network means the existence of more than one stable positive steady state and this is the definition I usually adopt. The definition may be weakened to the existence of more than one non-negative stable steady state. That autophosphorylation can produce bistability in this weaker sense was already observed by Lisman in 1985 (PNAS 82, 3055). He was interested in this as a mechanism of information storage in a biochemical system. In 2006 Fuss et al. (Bioinformatics 22, 158) found bistability in the strict sense in a model for the dynamics of Src kinases. Since Lck is a typical member of the family of Src kinases these results are also of relevance for Lck. In that work the phosphorylation processes are embedded in feedback loops. In fact the bistability is present without the feedback, as observed by Kaimachnikov and Kholodenko (FEBS J. 276, 4102). Finally, it was shown by Doherty et al. (J. Theor. Biol. 370, 27) that bistability (in the strict sense) can occur for a protein with only one phosphorylation site. This is in contrast to more commonly considered phosphorylation systems. These authors have also seen more complicated dynamical behaviour such as periodic solutions.

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.

The Bogdanov-Takens bifurcation, part 2

December 30, 2020

This post continues the discussion in the previous post on this subject. It turns out that the (exact) normal form is a system whose dynamics are not easy to analyse. The bifurcation diagram can be found as Figure 8.8 in the book of Kuznetsov. The case presented there is the supercritical one. The diagram can easily be converted into that describing the subcritical case, the most important thing being that the time direction is reversed so that the types of stability are interchanged. The bifurcation point is the origin in the plane of the parameters (\beta_1,\beta_2). The plane can be written as the union of four open regions R_i separated by bifurcation curves. These regions are distinguished by the types of dynamical behaviour which occur for the given values of the parameters. R_1 is simplest, having a trivial dynamics, without steady states or periodic solutions. The boundary of R_1 is a smooth curve T passing through the origin. If the origin is deleted the remainder consists of two connected components T_- and T_+. The curve T_- is the boundary between R_1 and R_2. There a fold bifurcation takes place, producing a saddle point and a sink. The region R_2 is that between T_- and the negative part of \beta_2 axis, which we call H. On that axis a Hopf bifurcation takes place. The sink turns into a source and a stable periodic solution is born. This is a supercritical Hopf bifurcation. For the other sign of the parameter s there is an unstable periodic solution and the Hopf bifurcation is subcritical. This is the reason that we have also adopted the terms sub- and supercritical in the case of a BT bifurcation. (Note that the sign of s can be computed from the parameters a_{ij} and b_{ij}.) On the other side of H we come to R_3. In that region there is a saddle point and the periodic solution already mentioned. It is the unique periodic solution for given values of the parameters. The other boundary of R_3 is a curve C where a non-local bifurcation takes place. On the curve there is an orbit which is homoclinic to the saddle. On the other side of C is R_4 where the homoclinic loop has broken and the qualitative behaviour is similar to that in R_2 except that there is a source instead of a sink. From R_4 we can pass through T_+ and return to R_1 by means of a fold bifurcation. It is perhaps worth mentioning what happens on the upper half of the \beta_2 axis. There is no bifurcation there but the two eigenvalues of the linearization are equal and opposite (a neutral saddle). This type of situation may be dangerous when looking for Hopf bifurcations because it shares certain algebraic properties with a Hopf point without being one. For this reason it could make sense to call it a ‘false Hopf point’, which is my internal name for it.

The proofs of the statements about the homoclinic orbit and the periodic solution are hard and I will just make a few comments on that. There is a subtle rescaling which, in particular, ensures that the two steady states are a fixed distance apart, independent of the parameters. Doing this transformation brings the system into the form of a perturbation of a Hamiltonian system. In the latter case the homoclinic orbit is obtained as a level surface of the Hamiltonian. This perturbation can be analysed by a method due to Pontryagin. Use is made of elliptic functions. These things are sketched in some detail by Kuznetsov in an appendix. He also mentions an alternative method for proving the uniqueness of the periodic orbit due to Dumortier and Rousseau. Concerning the fact that the approximate normal form can be transformed to the exact normal form Kuznetsov only gives a very brief sketch of the proof. Combining what is known about the qualitative behaviour of the solutions in normal form and the fact that any system satisfying the Bogdanov-Takens conditions can be reduced to this normal form provides a method for proving the existence and stability of periodic solutions and the existence of a homoclinic loop in given dynamical systems. The genericity conditions can be easier to check than for a Hopf bifurcation since they are conditions on the second derivatives rather than on the third derivatives. How can it be that it is easier to analyse a more complicated bifurcation than a simpler one? The point is that the hardest work is hidden in the proofs of the theorems about the normal form and does not have to be repeated when analysing a concrete system. The use of a BT bifurcation to help prove the existence of a Hopf bifurcation is connected to the idea of an organizing centre. The idea is to obtain insight into a dynamical system by looking for points with extremely special properties. In a given system a point like this of a given type may not exist but when it exists it may be easier to find than a point with a lower degree of speciality, even if the latter occurs more commonly. For instance we can look for a BT point instead of looking for a Hopf point. This type of strategy may be especially useful in systems where there are many parameters over which there is a lot of control. It gives a way of focussing the search in the parameter space. It is the opposite of the situation where you feel that you are looking for something you expect to be plentiful but do not know where to start.

The Bogdanov-Takens bifurcation

December 29, 2020

Consider a system of ODE of the form \dot x=f(x,\alpha) with parameters \alpha. A steady state is a solution of f(x_0,\alpha_0)=0 and it is a bifurcation point if J=D_x f(x_0,\alpha_0) has at least one eigenvalue on the imaginary axis. A common procedure in bifurcation theory is to start with those cases which are least degenerate. Thus we look at those cases with the fewest eigenvalues on the imaginary axis. If there is only one such eigenvalue, which is then necessarily zero, then I call this a spectral fold point. I use the word ‘spectral’ to indicate that this is a condition which only involves the structure of eigenvalues or eigenvectors. The full definition of a fold point also includes conditions which do not only involve the linearized equation \dot y=Jy. If the only imaginary eigenvalues are a non-zero complex conjugate pair then this is a spectral Hopf point. If the only imaginary eigenvalues are two zero eigenvalues and the kernel is only one-dimensional (so that J has a non-trivial Jordan block for the eigenvalue zero) then we have a spectral Bogdanov-Takens point.

When we have picked a class of bifurcations on the basis of the eigenvalues of J the aim is to show that it can be reduced to a standard form by transformations of the unknowns and the parameters. This is often done in two steps. The first is to reduce it to an approximate normal form, which still contains error terms which are higher order in the Taylor expansion about the bifurcation point than the terms retained. The second is to transform further to eliminate the error terms and reduce the system to normal form. Normal forms for the Bogdanov-Takens bifurcation were derived independently by Bogdanov and Takens in the early 1970’s. In this post I follow the presentation in the book of Kuznetsov, which in turn is based on Bogdanov’s approach. In the notation of Kuznetsov the BT bifurcation is defined by four conditions, denoted by BT.0, BT.1, BT.2 and BT.3. The condition BT.0 is the spectral condition we have already discussed. To be able to have a BT bifurcation the number of unknowns must be at least two. The following discussion concerns the two-dimensional case. This is the essential case since higher dimensions can then be treated by using centre manifold theory to reduce them to two dimensions. The BT bifurcation is codimension two which means that in a suitable sense the set of dynamical systems exhibiting this bifurcation is a subset of codimension two in the set of all dynamical system. Another way of saying this is that in order to find BT bifurcations which persist under small perturbations it is necessary to have at least two parameters. For these reasons we consider the case where there are two variables x and two variables \alpha. The conditions BT.1, BT.2 and BT.3 are formulated in terms of the derivatives of f of order up to two at the bifurcation point. We choose coordinates so that the bifurcation point is at the origin.

By a linear transformation from x to new variables y the system can be put into the form \dot y=J_0y+R(y,\alpha) where J_0 is a Jordan block and the remainder term R is of order two in y and of order one in \alpha. After this a sequence of transformations are carried out leading to new unknowns \eta, new parameters \beta and a new time coordinate. This eventually leads to the equations \dot\eta_1=\eta_2 and \dot\eta_2=\beta_1+\beta_2\eta_1+\eta_1^2+s\eta_1\eta_2+O(|\eta|^3), which is the (approximate) normal form of Bogdanov. Takens introduced a somewhat different normal form. The parameter s is plus or minus one. Because of relations to the Hopf bifurcation I call the case s=-1 supercritical and the case s=1 subcritical. Let us denote the coefficients in the quadratic contribution to R by a_{ij}(\alpha) and b_{ij}(\alpha). The condition BT.1 is that a_{20}(0)+b_{11}(0)\ne 0 and it is required to allow an application of the implicit function theorem. The condition BT.2 is that b_{20}\ne 0 and it is required to allow a change of time coordinate. The final transformation involves a rescaling of both the unknowns and the parameters. The existence of the new parameters \beta as a function of the old parameters \alpha is guaranteed by the implicit function theorem and it turns out that the non-degeneracy condition is equivalent to the condition that the derivative of a certain mapping is invertible at the bifurcation point. If the equation is \dot y=g(y,\alpha) then the mapping is (y,\alpha)\mapsto (g,{\rm tr}D_y g,\det D_y g). This condition is BT.3. This equivalence is the subject of a lemma in the book which is not proved there. As far as I can see proving this requires some heavy calculations and I do not have a conceptual explanation as to why this equivalence holds. Carrying out all these steps leads to the approximate normal form. At this point there is still a lot more to be understood about the BT bifurcation. It remains to understand how to convert the approximate normal form to an exact one and how to analyse the qualitative behaviour of the solutions of the system defined by the normal form. I will leave discussion of these things to a later post.

Paper on hepatitis C

November 13, 2020

Together with Alexis Nangue and other colleagues from Cameroon we have just completed a paper on mathematical modelling of hepatitis C. What is being modelled here is the dynamics of the amount of the virus and infected cells within a host. The model we study is a modification of one proposed by Guedj and Neumann, J. Theor. Biol. 267, 330. One part of it is a three-dimensional model, which is related to the basic model of virus dynamics. It is coupled with a two-dimensional model which gives a simple description of the way in which the virus replicates inside a host cell. This intracellular process is related to the mode of action of the modern drugs used to treat hepatitis C. I gave some information about the disease itself and its treatment in a previous post. In the end the object of study is a system of five ODE with many parameters.

For this system we first proved global existence and boundedness of the solutions, as well as positivity (positive initial data lead to a positive solution). One twist here is a certain lack of regularity of the coefficients of the system. When some of the unknowns become zero the right hand side of the equations is discontinuous. This means that it is necessary to prove that this singular set is not approached during the evolution of a positive solution. The source of the irregularity is the use of something called the standard incidence function instead of simple mass action kinetics. The former type of kinetics has a long history in epidemiology and I do not want to try to explain the background here. In any case, there are arguments which say that mass action kinetics leads to unrealistic results in within-host models of hepatitis and that the standard incidence function is better.

We show that the model has up to two virus-free steady states and determine their stability. The study of positive steady states is more difficult and, at the moment, incomplete. We have proved that there cannot be more than three steady states but we do not know if there is ever more than one. Under increasingly restrictive assumptions on the parameters (restrictions which are unfortunately not all biologically motivated) we show that there is at least one positive steady state, that there is exactly one and that that one is asymptotically stable. Under certain other assumptions we can show that every solution converges to a steady state. This last proof uses the method of Li and Muldowney discussed in a previous post. Learning about this method was one of the (positive) side effects of working on this project. Another was an improvement of my understanding of the concept of the basic reproductive number as discussed here and here. During this project I have learned a lot of new things, mathematical and biological, and I feel that I am now in a stronger position to tackle other projects modelling hepatitis C and other infectious diseases.

This

Lisa Eckhart and her novel Omama

October 2, 2020

Lisa Eckhart is a young Austrian known for a type of comedy which is called Kabarett in German-speaking countries. Characteristic features of this form of entertainment are black humour and political content. In her stage performances she likes to break taboos and flout political correctness. I first happened to see her on TV and I was immediately interested by her performance. Since then I saw her on stage in Mainz, shortly before COVID-19 made going to the theatre impossible for some time. I had also previously seen another Austrian proponent of the same type of performance who I appreciate a lot, Josef Hader, on stage in Mainz. Recently Eckhart published her first novel, ‘Omama’, and I now read it. In her stage performances there are some obstacles to understanding. The first is that her humour relies essentially on ambiguity. The second is that the whole thing goes very fast. The third is that she likes to mix in quite a lot of Austrian dialect. I thought that if there is some similarity between the contents of the novel and that of the performances it might help me to understand more. After all, a novel can be read at the speed the reader desires and it is possible to take time to investigate anything which is unclear. Recently there has been some public controversy in Germany around Eckhart. She has been accused of antisemitism, which I do not believe is justified. In August she was supposed to give a reading from her novel at a literature festival in Hamburg. Her invitation was withdrawn because the organisers were afraid that her appearance there might lead to violent protests by left-wing groups which the police would not be able to control. I find the fact that such a thing can happen a disgrace. What happened to free speech? Later the organisers renewed the invitation in a modified form but this time Eckhart refused, which I can understand. On Wednesday evening I went to read the last few pages of the book but before I got properly started I was called by my wife, who was watching TV. When I came into the living room I understood why she had called me since Lisa Eckhart was on the screen. She was participating in a program on the channel ARTE on the subject of decadence. The word decadence was one which had not gone through my mind for many years but it interested me a long time ago. At that time I read ‘A Rebours’ by Huysmans, ‘Le Soleil des Morts’ by Mauclair and, of course, ‘Les Fleurs du Mal’. Looking back it is hard to imagine how different my world was then, a fact that has less to do with a change in the world than with a change in myself. In any case, when I think about it, it is clear that we are now living in a period of decadence.

Let me finally come to the novel itself. It struck me as a curate’s egg. Parts of it are very good. There are passages where I appreciate the humour and I find the author’s use of language impressive. On a more global level I do not find the text attractive. It is the story of the narrator’s grandmother. (Here is a marginal note for the mathematical reader. Walter Rudin, known for his analysis textbooks, was born in Austria. In¬† a biographical text about him I read that one of his grandmothers was referred to as ‘Omama’.) The expressions are often very crude, with a large dose of excrement and other unpleasant aspects of the human body, and many elements of the story seem to me pointless. There is no single character in the novel who I find attractive. This is in contrast to the novel of Banine which I previously wrote about, where I find the narrator attractive. That novel also contains plenty of crude expressions but there are more than enough positive things to make up for it. I would like to emphasize that just because I find a novel unpleasant to read it does not mean I judge it negatively. A book which I found very unpleasant was ‘Alexis ou le traite du vain combat’ by Marguerite¬† Yourcenar but in that case my conclusion was that it could only be so unpleasant because it was so well written. I do not have the same feeling about Omama. As to the insight which I hoped I might get for Eckhart’s stage performances I have not seen it yet, but maybe I will notice a benefit the next time I experience a stage performance by her.

Determinant of a block triangular matrix

August 26, 2020

I am sure that the fact I am going to discuss here is well known but I do not know a good source and so I decided to prove it for myself and record the answer here. Suppose we have an n\times n matrix M with complex entries m_{ij} and that it is partitioned into four blocks by considering the index ranges 1\le i\le k and k+1\le i\le n for some positive integer k<n. Suppose that the lower left block is zero so that the matrix is block upper triangular. Denote by A, B and C the upper left, upper right and lower right blocks. Then \det M=\det A\det C. This is already interesting in the block diagonal case B=0. To begin the proof of this I use the fact that over the complex numbers diagonalizable matrices with distinct diagonal elements are dense in all matrices. Approximate A and C by diagonalizable matrices with distinct diagonal elements A_n and C_n, respectively. Replacing A and C by A_n and C_n gives an approximating sequence M_n for M. If we can show that \det M_n=\det A_n\det C_n for each n then the desired result follows by continuity. The reason I like this approach is that the density statement may be complicated to prove but it is very easy to remember and can be applied over and over again. The conclusion of this is that it suffices to treat the case where A and C are diagonalizable with distinct eigenvalues. Since the determinant of a matrix of this kind is the product of its eigenvalues it is enough to show that every eigenvalue of A or C is an eigenvalue of M. In general the determinant of a matrix is equal to the determinant of its transpose. Thus the matrix and its transpose have the same eigenvalues. Putting it another way, left eigenvectors define the same set of eigenvalues as right eigenvectors. Let \lambda be an eigenvalue of A and x a corresponding eigenvector. Then [x\ 0]^T is an eigenvector of M corresponding to that same eigenvalue. Hence any eigenvalue of A is an eigenvalue of M. Next let \lambda be an eigenvalue of C and y a corresponding left eigenvector. Then [0\ y] is a left eigenvector of M with eigenvalue \lambda. We see that all eigenvalues of A and C are eigenvalues of M. To see that we get all eigenvalues of M in this way it suffices to do the approximation of A and C in such a way that A_n and C_n have no common eigenvalues for any n. Then we just need to compare the numbers of eigenvalues of the matrices A,C and M. A consequence of this result is that the characteristic polynomial of M is the product of those of A and C. An application which interests me is the following. Suppose we have a partially decoupled system of ODE \dot x=f(x,y), \dot y=g(y), where x and y are vectors. Then the derivative of the right hand side of this system at any point has the block triangular form. This is in particular true for the linearization of the system about a steady state and so the above result can be helpful in stability analyses.

Talk by David Ho on COVID-19

August 22, 2020

On Thursday David Ho gave a keynote lecture at the SMB conference. He talked about work to develop monoclonal antibodies against SARS-CoV-2. He started by apologising, in view of the given audience, that there would be no mathematics in his talk but he did make make clear his continuing belief in the importance of applying mathematics to biology. He has been leading an effort with a precise medical goal – to find effective neutralising antibodies against this new virus. Antibodies were obtained from five patients severely ill with COVID-19. Four of them survived while one later died of the disease. These antibodies were then analysed by biochemical and bioinformatic means to find those which bound best to the spike protein of the virus. In this context I learned some basic things about the virus. The spike, which is used by the virus to enter cells is considered the number one target for antibodies which could be effective in combating the disease. More precisely there are two different subdomains which are possible targets, one more at the tip of the spike (the receptor-binding domain) and another more on the sides (the N-terminal domain), which is a trimer. A number of antibodies were found which bind to the first subdomain or to one of the subunits of the second. Another was found whose binding site is somewhat less local. This whole process was carried out in just few weeks, a remarkable achievement.

The antibodies just mentioned are the therapeutic candidates. The idea is to either produce monoclonal antoibodies with these sequences or possibly versions which are improved so as to be longer-lived. Monoclonal antibodies are known to be extremely expensive when used to treat other diseases, such as cancer. They are also expensive in the present context, but the speaker said that the retail cost depends very much on the quantity produced. In other applications the number of patients is relatively small and the cost correspondingly high. If the antobodies were being used for a very large number of patients the cost would be lower. It would remain problematic for low and middle income countries. It has been discussed that the Gates foundation might make it possible to offer this treatment in poorer countries for fifty dollars a dose. The main advantage of this method compared with that of trying to use antibodies from the serum of patients directly is that it is much more practical to apply on a very large scale. The effectiveness of the antibodies against the disease has been tested in hamsters. Ho made the impression of someone tackling a major problem of humanity head on with some of the best tools available. He said that an article giving an account of the work had appeared in Nature (Potent neutralizing antibodies against multiple epitopes on SARS-CoV-2 spike, Nature 584, 450). In response to one question on one aspect of the treatment he said that the answer was not known but he would just be continuing to a Zoom meeting of researchers leading the attempts to develop therapies which was to discuss exactly that question.

My first virtual conference (SMB 2020)

August 20, 2020

At the moment I am attending the annual conference of the Society for Mathematical Biology, which is taking place online. This is my first experience of this kind of format. The conference has many more participants than in any previous year, more than 1700. It takes place in a virtual building which is generated by the program Sococo. I find this environment quite disorienting and a bit stressful. This reaction probably has to do with the facts that I am no longer so young and that I have always tried to avoid social media as much as possible. I am sure that younger generations (and members of older generations with an enthusiasm for new technical developments) have far fewer problems getting used to it. In advance I was a bit worried about setting up the necessary computer requirements to be able to give my talk or even to go to others. In the end it worked out and my talk, given via Zoom, went smoothly. I got some good feedback, I am already convinced that it was worth joining this meeting and I may be less sceptical about joining others of this type in the future. There have been technical hitches. For instance the start of one big talk was delayed by about 20 minutes for a reason of this kind. Nevertheless, many things have gone well. Of course it is much preferable to meet people personally but when that is not possible virtual meetings with old friends are also pleasant.

This meeting has been organized around the subgroups of the society. I am a member of the subgroups for immunology and oncology. In fact my talk got scheduled in the group for mathematical modelling. My greatest allegiance is to the immunology subgroup and so I was happy to see that it was represented so strongly at this conference. It has seven sessions of lectures, made up of 28 talks. If I should choose my favourite from those talks in this section I have heard so far (it is not finished yet) then I pick the talk of Ruy Ribeiro on CD8 cells in HIV infection. I did feel I was missing some necessary background but I nevertheless found the talk useful in introducing me to important ideas which were new to me. The immunology subgroup have managed to get a very prominent speaker for their keynote talk, David Ho. I wrote about his work and its relation to mathematics in one of my first ever posts on this blog, way back in 2008. I am looking forward to hearing his talk later today (which will be on COVID-19). I will now mention some of my other personal highlights from the conference so far. I noticed on the program that Stas Shvartsman was giving a talk. His work has made a positive impression on me in the past and I did have a little e-mail contact with him. On the other hand I never met him personally and I had never heard a talk by him. This was my chance and I went in with high expectations. They were not disappointed. He talked about developmental defects arising from mutations in a single gene. In particular he concentrated on mutations in the Raf-MEK-ERK MAPK cascade. I was familiar with the role of mutations in this cascade in cancer but I had never heard about this other role. Stas described experiments in Drosophila where one base in MEK is mutated. This produces flies with a particular small change in the pattern of the veins in their wings. Interestingly, this does not occur in all flies with the mutation but only in 30% of them (I hope I am remembering the right number). Another talk yesterday which I appreciated was that by Robert Insall, who was talking about aspects of chemotaxis. He started with the phenomenon of the spread of melanoma. Melanoma cells do have a very strong tendency to spread in space and the question is what controls that. Is it chemotaxis along a chemical gradient?He showed pictures of melanoma cells moving fast up a chemical gradient. Then he showed a similar picture showing them moving just as fast without the chemical gradient. So what is going on? This kind of experiment starts with cells concentrated on one side of a region and a spatially homogeneous distribution of a relevant substance. The cells consume this substance, create their own gradient and then undergo chemotaxis along it. The speaker explained how there are many instances of chemotaxis in biology which can only be explained by a self-created gradient and not be a pre-existing one. He also made some interesting remarks about how mathematical modelling can lead to insights in biology which would not be possible with the usual verbal approaches of the biologists.

I also want to mention an interesting conversation I had in the poster session. The poster concerned was that of Daniel Koch. The theme of his work (which has already been published in a paper in J. Theor. Biol.) is that the formation of oligomers of proteins (or their posttranslationally modified variants) can lead to interesting dynamics. At first sight this may sound too simple to be interesting but in fact in mathematics it is often the careful consideration of apparently simple situations which leads to fundamental progress. I imagine that this principle also applies to other disciplines (such as biology) but it is perhaps strongest in mathematics. In any case, I am strongly motivated to study this work carefully. The only question in when it will be, given the many other directions I want to pursue.

Bifurcation and stability

August 6, 2020

A first step in studying solutions of the dynamical system \dot x=f(x,\lambda) is to look at the set of steady states, the solutions of the equations f(x,\lambda)=0. In the simplest cases these equations can be solved for the unknowns x as a function of the parameters \lambda. A criterion for when this can be done, at least in principle, is given be the implicit function theorem. If f(0,0)=0 then the condition is that A=D_x f(0,0) is invertible. A slightly less favourable situation is that when the original system of n equations for n variables can be reduced to a single equation for a single variable in such a way that when this one equation has been solved the other unknowns can be calculated at steady state. This is related to the case where D_x f(0,0) has rank n-1 and the appropriate analogue of the implicit function theorem is Lyapunov-Schmidt reduction. A brief general description of this technique can be found in a previous post. In the absence of non-zero purely imaginary eigenvalues of A the system has a one dimensional centre manifold and there is a relation between Lyapunov-Schmidt reduction and centre manifold reduction. There are a lot of similarities between these two techniques but also some important differences. In the case of centre manifold theory we obtain the existence of a one-dimensional submanifold, which may be non-unique and may be less regular (in the sense of being smooth or analytic) than the system itself. In the case of Lyapunov-Schmidt reduction we obtain a one-dimensional quotient manifold which is unique and as regular as the system itself. Note that when the latter method is applied some choices must be made but the essential results are independent of those choices.

When it is possible to reduce the equations for steady states to a single equation p(X,\alpha)=0 as discussed above it may still be difficult to determine how many solutions the equation has for fixed \alpha. The function has been denoted by p since in many applications it is a polynomial. The issue of the number of solutions is not my concern here. Instead I assume I already know something about how many solutions exist and I would like to know something about their stability. The question is, under what circumstances a reduction of the existence question to one dimension also leads to a reduction of the stability question to one dimension. Here I discuss a result on this question which was obtained in a paper of Crandall and Rabinowitz with the title ‘Bifurcation, perturbation of eigenvalues and linearized stability’ (Arch. Rat. Mech. Anal. 19, 1083 (1973)). The exposition I will give here is based not on that paper (which I have not read) but on that in the book ‘Singularities and Groups in Bifurcation Theory’ by Golubitsky and Schaeffer.

I will discuss the result in the simplest case I can think of. This is where x is a point in the plane and \alpha is a scalar. I assume that the kernel of A is the x_1-axis and its image the x_2-axis. I assume further that the non-zero eigenvalue of A is -a with a>0. In this situation the equation f_2(x_1,x_2,\alpha)=0 can be written in the form x_2=h(x_1). Substituting this into the equation f_1(x_1,x_2,\alpha)=0 gives an equation of the form g(x_1,\alpha)=0. Here x_1 plays the role of X above. Steady states close to the origin are in one-to-one correspondence with zeroes of g. The main result is that the stability of the steady state at the point X is determined by the sign of the derivative g' of g with respect to X. When that sign is negative the steady state is asymptotically stable and when the sign is positive it is a saddle.