## Archive for February, 2013

### Absolute concentration robustness

February 20, 2013

In the past years I have been on the committees for many PhD examinations. A few days ago, for the first time, I was was on the committee for a thesis on a subject belonging to the area of mathematical biology. This was the thesis of Jost Neigenfind and it was concerned with a concept called absolute concentration robustness (ACR).

The concentration of a given substance in cells of a given type varies widely between the individual cells. (Cf. also this previous post). It is of interest to identify mechanisms which can ensure that the steady state concentration of a particular substance is independent of initial data. (This is a way in which the output of a system can be independent of background variation.) In saying this I am assuming implicitly that more general solutions converge to steady states. A more satisfactory formulation can be obtained as follows. In a chemical reaction network there are usually a number of conserved quanitities, say $C_\alpha$. These define affine subspaces of the state space, the stoichiometric compatibility classes. For many systems there is exactly one stationary solution in each stoichiometric compatibility class. The condition of interest here is that the value of one of the concentrations, call it $x_1$, in the steady state solution is independent of the parameters $C_\alpha$. (The other concentrations $x_i,i>1$ will in general depend on the $C_\alpha$.) This property is ACR. I first heard of this in a talk by Uri Alon at the SMB conference in Krakow in the summer of 2011. The basic idea is explained clearly in a paper of Shinar and Feinberg (Science 327, 1389). They present a general theoretical approach but also describe some biological systems where ACR (in a suitable approximate sense) has been observed experimentally. In the terminology of Chemical Reaction Network Theory (CRNT) the examples they discuss have deficiency one. They mention that ACR is impossible in systems of deficiency zero. There is no reason why it should not occur in systems of deficiency greater than one but in those more complicated dynamics make it more difficult to decide whether the property holds or not.

The result of Shinar and Feinberg only covers a class of reaction networks which is probably very restricted. What Neigenfind does in his thesis is to develop more general criteria for ACR and computer algorithms which can check these criteria for given systems. The phenomenon of ACR is interesting since it is a feature which may be more common in reaction systems coming from biology than in generic systems. At least there is a good potential reason why this might be the case.

### Goodbye to Berlin

February 1, 2013

For this post I could not resist the temptation to borrow the title of Christopher Isherwood’s novel although what I am writing about here has very little to do with his book. The connection of the title to the content is that I will soon be leaving Berlin after living here more than fifteen years. I have accepted a professorship at the University of Mainz and I will move there in April. The first time I came to Berlin I landed at Tegel airport and I interpreted the Hooded Crow I saw beside the runway as a good omen. This requires some explanation. In those days the Hooded Crow (Corvus corone cornix) was a subspecies of the Carrion Crow. In the meantime it has been promoted to the rank of a species but that will not concern me here – I am not sure whether I feel I should congratulate it on receiving this honour. It differs from the nominate form (according to the old classification) by having a grey body while Corvus corone corone (Carrion Crow in the narrower sense) is all black. These two forms have the classical property of subspecies that they are allopatric. In other words they occur in more or less disjoint regions. On the boundary between the regions there is little interbreeding. The Orkney Islands where I grew up belong to the land of the Hooded Crow. Most of Great Britain and in fact most of Western Europe belong to the domain of the Carrion Crow. Even Aberdeen, where I studied and did my PhD, belongs to the land of the Carrion Crow. This helps to explain why I associate the Hooded Crow with ‘home’ and the Carrion Crow with ‘foreign parts’. It also has to do with the fact that there was a Hooded Crow which nested regularly in a garden near where I grew up in Orkney. I would climb the tree from time to time to keep an eye on the development of the brood and ring the chicks at the right moment. For these reasons the bird in Tegel seemed to tell me I was coming home. Now I am daring to venture once again (and probably for most of the rest of my life) into the land of the Carrion Crow.

When leaving a place it is natural to think about the good things which you experienced there. What were the best things about Berlin for me? The best thing of all is that Berlin was where I met my wife Eva. Eichwalde, where she lived at that time and where we both live now, has a very special feeling for me which will never go away. (Just for the record, we do not really live in Eichwalde but that is where the nearest train station is, with the result that the platform of the station there has something of the gates of Paradise for me.) Of course I cannot fail to mention the Max Planck Institute for Gravitational Physics which has provided me with a scientific home during all that time. I am grateful to the successive leaders of the mathematical group there, Jürgen Ehlers and Gerhard Huisken, for the working and social environment which they created and maintained. Another important thing about Berlin I will miss is the contact with its excellent research in biology and medicine. I have spent many valuable hours attending the Berlin Life Science Colloquium and I feel very attached to the Paul Ehrlich lecture hall where it usually takes place. The wooden seats are hard but the interest of the lectures was generally more than enough to make me forget that. I will also miss the stimulating atmosphere of the group of Bernold Fiedler at the Free University, which has been a source of a lot of intellectual input and a lot of pleasure.

This is perhaps the moment to say why I am leaving Berlin. Ever since I was a student I have felt a strong allegiance to mathematics. As a child I was concerned with metaphysical questions and later I got interested in physics as the most fundamental part of science. During my undergraduate study I realised that mathematics, and not physics, was the right intellectual environment for me. A key experience for me was that through my study plan I ended up doing two courses on Fourier series, a subject which was new to me, at the same time. One was in physics and one in mathematics. The contrast was like night and day. This may have had something to do with the abilities of the individual lecturers concerned but it was mainly due to essential differences between mathematics and physics. By the end of my studies I had specialized in mathematics and my commitment to that subject has remained constant ever since.

For a long time my strongest connection to mathematics concerned intrinsic aspects of the subject. The significance of applications for me was as a good source of mathematical problems. This has changed over the years and I have become increasingly fascinated by the interplay between mathematics and its applications. At the same time the focus of my interest has moved from mathematics related to fundamental physics to mathematics related to biology and medicine. This change has led to a discrepancy between the research I want to do and the research area of the institute where I work. A Max Planck Institute is by its very nature focussed on a certain restricted spectrum of subjects and this is not compatible with a major change of research direction of somebody working there. This is the reason that I started applying for jobs which fitted the directions of work where my new interests lie. The move to Mainz is the successful endpoint of this process. Moving from a Max Planck Institute to a university will naturally involve spending more time on teaching and less time on research. This does not dismay me. The most important thing is that I will be doing something I believe in. Teaching elementary mathematics and analysis, apart from establishing the basis needed for doing research, is something whose intrinsic value I am convinced of.