## Archive for May, 2017

### Becoming a German citizen

May 24, 2017

I first moved to Germany in 1987 and I have spent most of the time since then here. The total time I have spent elsewhere since my first arrival in Germany does not add up to more than two years. There is every reason to expect I will spend the rest of my life here. I am married to a German, I have a job here I like and a house. I could have applied for German citizenship a long time ago but I never bothered. Being an EU citizen living in Germany I had almost almost all privileges of a native. The only exception is that I could not vote except in local elections but since I am not a very political person that was not a big issue for me. It was also the case that for a long time I might have moved to another country. For instance I applied for a job in Vienna a few years ago and I might well have taken it if it had been offered to me. Now the chances of my moving are very small and so there is no strong argument left against becoming a German citizen.

What is more important is that there are now arguments in favour of doing so. With the EU showing signs of a possible disintegration the chance that I could lose the privileges I have here as an EU citizen is not so small that it should be neglected. The referendum in which the Scots voted on the possibility of leaving the UK was the concrete motivation for my decision to start the application process. Scotland stayed in the UK but then the Brexit confirmed that I had made the right decision. At the moment there is no problem with keeping British citizenship when obtaining German citizenship and I am doing so. This may change sometime, meaning that I will have to give up my British citizenship to keep the German one, but I see this as of minor importance.

As prerequisites for my application I had to do a number of things. Of course it was necessary to submit a number of documents but I have the feeling that the amount of effort was less than when obtaining the documents needed to get married here. I had to take an examination concerning my knowledge of the German language, spoken and written. It was far below my actual level of German and so from that point of view it was a triviality. It was just a case of investing a bit of time and money. I also had to do a kind of general knowledge test on Germany and on the state where I live. This was also easy in the sense that the questions were not only rather simple for anyone who has lived in the country for some time but they are also taken from a list which can be seen in advance. Again it just meant an investment of time and money. At least I did learn a few facts about Germany which I did not know before. In my case these things were just formalities but I think it does make sense that they exist. It is important to ensure that other applicants with a background quite different from mine have at least a minimal knowledge of the language and the country before they are accepted.

After all these things had been completed and I had submitted everything it took about a year before I heard that the application had been successful. This time is typical here in Mainz – I do not know how it is elsewhere in Germany – and it results from the huge backlog of files. People are queueing up to become German citizens, attracted by the prospect of a strong economy and a stable political system. Yesterday I was invited to an event where the citizenship of the latest group of candidates was bestowed in a ceremony presided over by the mayor. There were about 60 new citizens there from a wide variety of countries. The most frequent nationality by a small margin was Turkish, followed by people from other middle eastern countries such as Iraq and Iran. There were also other people from the EU with the most frequent nationality in that case being British. My general feeling was one of being slightly uneasy that I was engaged in a futile game of changing horses. It is sad that the most civilised countries in the world are so much affected by divisive tendencies instead of uniting to meet the threats confronting them from outside.

### The Routh-Hurwitz criterion

May 7, 2017

I have been aware of the Routh-Hurwitz criterion for stability for a long time and I have applied it in three dimensions in my research and tried to apply it in four. Unfortunately I never felt that I really understood it completely. Here I want to finally clear this up. A source which I found more helpful than other things I have seen is https://www.math24.net/routh-hurwitz-criterion/. One problem I have had is that the Hurwitz matrices, which play a central role in this business, are often written in a form with lots of … and I was never sure that I completely understood the definition. I prefer to have a definite algorithm for constructing these matrices. The background is that we would like to understand the stability of steady states of a system of ODE. Suppose we have a system $\dot x=f(x)$ and a steady state $x_0$, i.e. a solution of $f(x_0)=0$. It is well-known that this steady state is asymptotically stable if all eigenvalues $\lambda$ of the linearization $A=Df(x_0)$ have negative real parts. This property of the eigenvalues is of course a property of the roots of the characteristic equation $\det(A-\lambda I)=a_0\lambda^n+\ldots+a_{n-1}\lambda+a_n=0$. It is always the case here that $a_0=1$ but I prefer to deal with a general polynomial with real coefficients $a_i, 0\le i\le n$ and a criterion for the situation where all its roots have negative real parts. It is tempting to number the coefficients in the opposite direction, so that, for instance, $a_n$ becomes $a_0$ but I will stick to this convention. Note that it is permissible to replace $a_k$ by $a_{n-k}$ in any criterion of this type since if we multiply the polynomial by $\lambda^{-n}$ we get a polynomial in $\lambda^{-1}$ where the order of the coefficients has been reversed. Moreover, if the real part of $\lambda$ is non-zero then it has the same sign as the real part of $\lambda^{-1}$. I find it important to point this out since different authors use different conventions for this. It is convenient to formally extend the definition of the $a_i$ to the integers so that these coefficients are zero for $i<0$ and $i>n$.

For a fixed value of $n$ the Hurwitz matrix is an $n$ by $n$ matrix defined as follows. The $j$th diagonal element is $a_j$, with $1\le j\le n$. Starting from a diagonal element and proceeding to the left along a row the index increases by one in each step. Similarly, proceeding to the right along a row the index decreases by one. In the ranges where the index is negative or greater than $n$ the element $a_n$ can be replaced by zero. The leading principal minors of the Hurwitz matrix, in other words the determinants of the submatrices which are the upper left hand corner of the original matrix, are the Hurwitz determinants $\Delta_k$. The Hurwitz criterion says that the real parts of all roots of the polynomial are negative if and only if $a_0>0$ and $\Delta_k>0$ for all $1\le k\le n$. Note that a necessary condition for all roots to have negative real parts is that all $a_i$ are positive. Now $\Delta_n=a_n\Delta_{n-1}$ and so the last condition can be replaced by $a_n>0$. Note that the form of the $\Delta_k$ does not depend on $n$. For $n=2$ we get the conditions $a_0>0$, $a_1>0$ and $a_2>0$. For $n=3$ we get the conditions $a_0>0$, $a_1>0$, $a_1a_2-a_0a_3>0$ and $a_3>0$. Note that the third condition is invariant under the replacement of $a_j$ by $a_{n-j}$. When $a_0a_3-a_1a_2>0$, $a_0>0$ and $a_3>0$ then the conditions $a_1>0$ and $a_2>0$ are equivalent to each other. In this way the invariance under reversal of the order of the coefficients becomes manifest. For $n=4$ we get the conditions $a_0>0$, $a_1>0$, $a_1a_2-a_0a_3>0$, $a_1a_2a_3-a_1^2a_4-a_0a_3^2>0$ and $a_4>0$.

Next we look at the issue of loss of stability. If $H$ is the region in matrix space where the Routh-Hurwitz criteria are satisfied, what happens on the boundary of $H$? One possibility is that at least one eigenvalue becomes zero. This is equivalent to the condition $a_n=0$. Let us look at the situation where the boundary is approached while $a_n$ remains positive, in other words the determinant of the matrix remains non-zero. Now $a_0=1$ and so one of the quantities $\Delta_k$ with $1\le k\le n-1$ must become zero. In terms of eigenvalues what happens is that a number of complex conjugate pairs reach the imaginary axis away from zero. The generic case is where it is just one pair. An interesting question is whether and how this kind of event can be detected using the $\Delta_k$ alone. The condition for exactly one pair of roots to reach the imaginary axis is that $\Delta_{n-1}=0$ while the $\Delta_k$ remain positive for $k. In a paper of Liu (J. Math. Anal. Appl. 182, 250) it is shown that the condition for a Hopf bifurcation that the derivative of the real part of the eigenvalues with respect to a parameter is non-zero is equivalent to the condition that the derivative of $\Delta_{n-1}$ with respect to the parameter is non-zero. In a paper with Juliette Hell (Math. Biosci. 282, 162), not knowing the paper of Liu, we proved a result of this kind in the case $n=3$.

### Mathematical models for T cell activation

May 2, 2017

The proper functioning of our immune system is heavily dependent on the ability of T cells to identify foreign substances and take appropriate action. For this they need to be able to distinguish the foreign substances (non-self) from those coming from substances belonging to the host (self). In the first case the T cell should be activated, in the second not. The process of activation is very complicated and takes days. On the other hand it seems that an important part of the distinction between self and non-self only takes a few seconds. A T cell must scan the surface of huge numbers of dendritic cells for the presence of the antigen it is specific for and it can only spare very little time for each one. Within that time the cell must register that there is something relevant there and be induced to stay longer, instead of continuing with its search.

A mathematical model for the initial stages of T cell activation (the first few minutes) was formulated and studied by Altan-Bonnet and Germain (PloS Biol. 3(11), e356). They were able to use it successfully to make experimental predictions, which they could then confirm. The predictions were made with the help of numerical simulations. From the point of view of the mathematician a disadvantage of this model is its great complexity. It is a system of more than 250 ordinary differential equations with numerous parameters. It is difficult to even write the definition of the model on paper or to describe it completely in words. It is clear that such a system is difficult to study analytically. Later Francois et. el. (PNAS 110, E888) introduced a radically simplified model for the same biological situation which seemed to show a comparable degree of effectiveness to the original model in fitting the experimental data. In fact the simplicity of the model even led to some new successful experimental predictions. (Altan-Bonnet was among the authors of the second paper.) This is the kind of situation I enjoy, where a relatively simple mathematical model suffices for interesting biological applications.

In the paper of Francois et. al. they not only do simulations but also carry out interesting analytical calculations for their model. On the other hand they do not follow the direction of attempting to use these calculations to formulate and prove mathematical theorems about the solutions of the model. Together with Eduardo Sontag we have now written a paper where we obtain some rigorous results about the solutions of this system. In the original paper the only situation considered is that where the system has a unique steady state and any other solution converges to that steady state at late times. We have proved that there are parameters for which there exist three steady states. A numerical study of these indicates that two of them are stable. A parameter in the system is the number $N$ of phosphorylation sites on the T cell receptor complex which are included in the model. The results just mentioned on steady states were obtained for $N=3$.

An object of key importance is the response function. The variable which measures the degree of activation of the T cell in this model is the concentration $C_N$ of the maximally phosphorylated state of the T cell receptor. The response function describes how $C_N$ depends on the important input variables of the system. These are the concentration $L$ of the ligand and the constant $\nu$ describing the rate at which the ligand unbinds from the T cell receptor. A widespread idea (the lifetime dogma) is that the quantity $\nu^{-1}$, the dissociation time, determines how strongly an antigen signals to a T cell. It might naively be thought that the response should be an increasing function of $L$ (the more antigen present the stronger the stimulation) and a decreasing function of $\nu$ (the longer the binding the stronger the stimulation). However both theoretical and experimental results lead to the conclusion that this is not always the case.

We proved analytically that for certain values of the parameters $C_N$ is a decreasing function of $L$ and an increasing function of $\nu$. Since these rigorous results give rather poor information on the concrete values of the parameters leading to this behaviour and on the global form of the function we complemented this analytical work by simulations. These show how $C_N$ can have a maximum as a function of $\nu$ within this model and that as a function of $L$ it can have the following form in a log-log plot. For $L$ small the graph is a straight line of slope one. As $L$ increases it switches to being a straight line of slope $1-N/2$ and for still larger values it once again becomes a line of slope one, shifted with respect to the original one. Finally the curve levels out as it must do, since the function is bounded. The proofs do not make heavy use of general theorems and are in general based on doing certain estimates by hand.

All of these results were of the general form ‘there exist parameter values for the system such that $X$ happens’. Of course this is just a first step. In the future we would like to understand better to what extent biologically motivated restrictions on the parameters lead to restrictions on the dynamical behaviour.