## Archive for June, 2008

### The Fraunhofer Society

June 27, 2008

In Germany there are several large research organizations, among them the Max Planck Society (which happens to be my employer) and the Fraunhofer Society. Yesterday I went to the inaugural lecture of Ulrich Buller at the University of Potsdam. He is the head of research within the Fraunhofer Society and the theme of his talk was the research strategy of the society. I thought it would be a good opportunity to improve my knowledge of local (and national) scientific politics.

The annual research budgets of Fraunhofer and Max Planck are 1.2 and 1.3 billion euros respectively. The Fraunhofer society has 54 research institutes spread over Germany and over different research fields. The Max Planck Society has 80. (The counting here may not be consistent since it depends on distinguishing between individual institutes and branches of other institutes.) The mission of Max Planck is fundamental research while that of Fraunhofer is applied research. Another big difference between the two organizations is that while the great majority of the funding for Max Planck comes from the government a Fraunhofer Institute has to earn a large part of its money from contracts with industry, after a short honeymoon period. Buller emphasized that a natural consequence of the need to earn money is that a Fraunhofer Institute has to be run in a way which has similarities with the way a private company is run. For this reason, he said, the director of a Fraunhofer Institute must be given a large degree of autonomy in deciding on the direction of research done by his staff. It also means that there is inherent competition between the individual Fraunhofer Institutes which should not be suppressed. At the same time it makes sense to have cooperation between the institutes and this leads to a certain conflict of goals. To try and address this issue the institutes are arranged into groups (Verbünde) with related research fields which try to foster cooperation where it is useful.

One of the most lucrative developments arising from research within the Fraunhofer Society concerns the technology for the MP3 player which was developed at the Fraunhofer Institute for Integrated Circuits in Erlangen. It was clear from the talk that the license fees connected with this are an important and welcome resource for the Society.

A point which was stressed by the speaker was the relatively small percentage of Germans who are qualified to take up jobs in scientific research. (He quoted a figure of 12%.) Here ‘relatively’ means ‘relatively to a number of other countries’. A consequence is that, at least within the national market, Max Planck, Fraunhofer, the universities and other research organizations are competing for a relatively small pool of potential employees. He talked about some of the ways that the Fraunhofer Society is trying to improve the situation from its point of view. He mentioned initiatives to set up collaborations between the different players (Fraunhofer with universities, Fraunhofer with Max Planck etc.) and thus better use the available human resources.

Where is mathematics in all this? It is plausible that there is money to be earned with computer science, but with mathematics? Remarkably there is a Fraunhofer Institute which does mathematics (Fraunhofer Institute for Industrial Mathematics) and it has been extremely successful. The original scepticism of the Society was overcome by a generous contribution from the state of Rheinland-Pfalz (Rhineland-Palatinate). In fact the institute was so successful financially that reponsibility was taken over by the Society earlier than originally planned. This success story is told in an article by the institute’s founding director Helmut Neunzert in Mitteilungen der deutschen Mathematiker-Vereinigung, 4, 262-268. It is impressive that he launched this project when he was already 59. Of course in the end the products sold by the institute do have a lot to do with the use of computers but the real mathematical content is enough to demarcate it from other Fraunhofer Institutes situated squarely in computer science.

### Quorum sensing

June 22, 2008

In a previous post on chemotaxis I briefly discussed the way that bacteria move in response to chemical gradients. This may be described in a rather anthropomorphic way by saying that an individual bacterium has particular goals which it pursues actively. On a similar level of description it turns out that bacteria cooperate with each other within a population. An example of this is the phenomenon known as quorum sensing. It plays a role in the formation of biofilms. These consist of colonies of bacteria surrounded by solid material which they have secreted. Quorum sensing is interesting on a purely theoretical level but it may also have practical consequences. Instead of fighting bacterial infection by attacking individual organisms using antibiotics an alternative might be to interfere with the social life of the bacteria by influencing quorum sensing. Under these circumstances individual bacteria could still survive but the multiplication of their population which leads to disease could be prevented. A useful source of information on quorum sensing is the The quorum sensing site. It should be noted that this is not just a phenomenon which occurs in exceptional cases – it is apparently a widespread phenomenon in bacterial colonies.

In many cases the signalling molecules involved in quorum sensing have been identified. There is also rather detailed information about how the bacteria react to the signal. There has been some discussion as to whether it is not better to talk about diffusion sensing in some cases. This means that the concentration of a molecule produced by a bacterium is sensed but that the presence of other bacteria is not necessary. In this way the organism can assess the volume available for diffusion of the chemical rather than the number of other organisms present. For more information on this see this article. There is already a considerable literature on mathematical models for quorum sensing, using ODE and/or reaction diffusion type models. See for example a paper of Dockery and Keener (Bull. Math. Biol. 63, 91). I do not yet have a good overview of the field.

The first organism in which quorum sensing was observed was Vibrio fischeri. This bacterium is responsible for the bioluminescence of certain marine animals and the production of light is regulated by quorum sensing. When I hear the name Vibrio I automatically think of the sinister member of the genus, Vibrio cholerae. This bacterium, the cause of cholera, has been reported to use quorum sensing but in a negative sense, i.e. the presence of more bacteria can lead to a suppression of their activity. This organism has had a notable career in history and also played a major role in establishing the fame of Robert Koch who identified it as the cause of an epidemic in Alexandria in 1883, in competition with associates of his rival Louis Pasteur.

### Shock waves

June 13, 2008

A partial differential equation is the requirement that partial derivatives up to order $k$ of some unknown function satisfy a certain algebraic relation. An evident prerequisite for this to make sense is that the derivatives of the function up to order $k$ exist. It turns out, however, that this apparently obvious statement is not true. In many cases it is possible to reformulate a PDE so as to make it meaningful to talk about solutions which have less derivatives than appear to be necessary at first sight. This leads to the concept of weak (or generalized) solutions, in contrast to classical solutions which have the $k$ derivatives needed for the straightforward interpretation of the equation.

In hydrodynamics, the study of fluids, weak solutions play an important role due to the phenomenon of shock waves. Solutions of the Euler equations which describe a fluid while neglecting viscosity have the tendency to develop discontinuities in the basic fluid variables (e.g. the density) even when starting from a perfectly smooth configuration. If the solution is to be used for modelling a physical situation beyond the time where a discontinuity appears it is necessary to use weak solutions. It turns out that weak solutions of the Euler equations can be used effectively in hydrodynamics and there are powerful methods for handling them numerically. At the same time it is very difficult to prove rigorous mathematical results about solutions of the Euler equations in the presence of shocks. Most of the theorems available in the literature concern situations which are reduced to an effective problem in one space dimension by means of a symmetry assumption (plane symmetry). Recently a notable exception to this appeared in the form of a book ‘The formation of shocks in 3-dimensional fluids‘ by Demetrios Christodoulou which treats the dynamics of solutions of the Euler equations up to the moment of shock formation without requiring symmetry assumptions. See also the recent review article of Christodoulou for background to this work and a concise history of mathematical developments in hydrodynamics (Bull. Amer. Math. Soc. 44, 581).

If viscosity is included in the description of fluids then the Euler equations are replaced by the Navier-Stokes equations. There is reason to suspect that in this case shock waves are smoothed out and a smooth initial configuration remains smooth in the course of the evolution, for all time. There are simple examples where this can be seen but there is still no global regularity result for Navier-Stokes (and no counterexample). The Clay Foundation has offered a prize of one million dollars for the solution of this problem in either direction. The fact that the prize has not yet been collected is a sign of the difficulty of the problem. For a discussion of this question and its broader mathematical significance I recommend the excellent account of Tao.

What effect does gravity have on the formation of shock waves? It is reasonable to suppose that the answer to this question is ‘almost none’. This intuition applies not only in Newtonian physics but also to the case of a fully relativistic description in the context of general relativity. The gravitational field curves space but in many situations the curvature produced should not affect the qualitative nature of the process of shock formation. A couple of years ago Fredrik Ståhl and I set out to confirm this idea rigorously. The project has been delayed by other things but now we have finished a manuscript and put it on the arXiv (Shock Waves in Plane Symmetric Spacetimes). The result is that there are plane symmetric solutions of the relativistic Euler equations coupled to the Einstein equations of general relativity which lose smoothness in an arbitrarily short time, depending on the initial size of the spatial derivatives of the fluid variables. The fact is used that the mechanism of breakdown bears a sufficiently close resemblance to that in a non-relativistic fluid without gravity. In this one-dimensional setting there are left-moving and right-moving waves in the fluid and gravity leads to a coupling between them. A central idea of the proof of our result is to use changes of variable so as to attain a sufficient amount of decoupling of the degrees of freedom corresponding to the two types of waves.

### Balo’s concentric sclerosis

June 6, 2008

A mathematical model for this rare disease was presented in a paper by Khonsari and Calvez in the journal PLoS ONE, abbreviated as [KC] from now on) and this is my primary source of information for this post. In addition I have used the standard work ‘McAlpine’s Multiple Sclerosis‘ as a reference. The disease was first described by Balo in 1927 and he himself already mentioned a possible connection to Liesegang rings. It is relatively common in the Philippines compared to other parts of the world. The model proposed in [KC] is a system of PDE similar to those used to describe chemotaxis, e.g. the Keller-Segel model. An important difference from reaction diffusion equations is the presence of cross diffusion, i.e. the flux of one diffusive species depends on the gradient of another. The system of [KC] also contains an ODE describing the density of oligodendocytes (the cells responsible for production and maintenance of myelin) which is analogous to the equation describing the formation of a precipitate in certain models of Liesegang rings. Computer simulations using this model show the production of rings for certain values of the parameters involved and not for others. The authors of [KC] compare their model to one based on another mechanism (which has discontinuous coefficients) and argue that their model performs better.

In considering the relation of Balo’s disease to MS an interesting related question is whether MS is one disease or many. A classification into four types has been suggested with different pathological mechanisms. Balo’s disease most closely resembles Type III. If this distinction could be pushed further it might lead to a global increase in understanding. Perhaps different forms require different therapies. Characteristic features of Type III include destruction of oligodendrocytes and signs that the oxygen supply in the area where damage is taking place is not sufficient. Two different general pictures of the formation of the rings of demyelinated tissue have been proposed in the literature. In one of these the demyelination only takes place in certain rings while in the other there is demyelination everywhere followed by remyelination in certainrings. Remyelination, a process by which the myelin sheaths of nerve cells are repaired is known to play an important role in MS. The quality of the new coatings is generally poorer than that of the original ones. Understanding the dynamics of the interplay of demyelination and remyelination is a crucial issue in understanding the progression of MS.

Another disease related to MS (or a special form of it, the question arises here too, it resembles Type II) is neuromyelitis optica (NMO), also known as Devic’s disease. This disease is automimmune in nature and a specific target of autoimmune attacks has been identified, the protein aquaporin 4. This knowledge is useful in distinguishing NMO from typical MS by determining the concentration of antibodies against aquaporin 4 in the blood. For more details see this article in PloS Medicine.