## Archive for November, 2009

### Some vaccinations I have had

November 30, 2009

I recently discovered a blog called WeiterGen (in German) with a lot of valuable information relevant to the public discussion of the influenza vaccine in Germany and elsewhere.There is also information about a notorious skin cream whose claimed medicinal properties have been the subject of some very dubious reporting on German TV recently. In Germany if you have a television set you have to pay the considerable licence fees for the public TV service. A justification often given for this is that it is important that the public have a trustworthy source of objective information. After this skin cream business I will be a lot more sceptical about the information on public TV, expecially things concerned with medical themes.

On a more cheerful note, the blog I just quoted also has an interesting post about the bacterium Mycoplasma pneumoniae as a model organism for use in systems biology.

### Bootstrap arguments

November 22, 2009

A bootstrap argument is an analogue of mathematical induction where the natural numbers are replaced by the non-negative real numbers. This type of argument is a powerful tool for proving long-time existence theorems for evolution equations. For instance, it plays a central role in the proof of the stability of Minkowski space by Christodoulou and Klainerman and the theorem on formation of trapped surfaces by Christodoulou discussed in previous posts. The name comes from a story where someone pulls himself up by his bootstraps, leather attachments to the back of certain boots. This story is often linked to the name of Baron Münchhausen. In another variant he pulls himself out of a bog by his pigtail. This was a person who really lived and was known for telling tall tales. In later years people wrote various books about him and incorporated many other tall tales from various sources. The word ‘booting’ applied to computers is derived from ‘bootstrapping’ in the sense of this story. There are also bootstrap methods used in statistics. They involve analysing new samples drawn from a fixed sample. In some sense this means obtaining more knowledge about a system without any further input of information. It is this aspect of ‘apparently getting something for nothing’ which is typical of the bootstrap. In French the procedure in statistics has been referred to as ‘méthode Cyrano’. Unfortunately is seems that in PDE theory the French have just adopted the English term. I say ‘unfortunately’ because of a fondness for Cyrano de Bergerac. As in the case of Münchhausen there was a real person of this name, this time a writer. However the name is much better known as that of a fictional character, the hero of a play by Edmond Rostand. The non-fictional Cyrano wrote among other things about a trip to the moon. There is also a Münchhausen story where he uses a kind of inverse bootstrap (could there be a PDE analogue here?) to return from the moon. He constructs a rope which he attaches to one of the horns of the moon but it is much too short to reach down to the ground. He climbs to the bottom of the rope, reaches up and cuts off and detaches the part ‘which he does not need any more’ and ties it onto the bottom. He then repeats this process. Returning to Cyrano, he describes seven methods for getting to the moon of which the sixth is the one relevant to the bootstrap. He stands on an iron plate and throws a magnet into the air. The iron plate is attracted by the magnet and starts to rise. Then he rapidly catches the magnet and throws it into the air again. I should point out that Cyrano does not believe in the nonsensical stories he is telling – his aim is a practical one, holding the attention of the Duc de Guiche so as to delay him for a very specific reason.

Now I return to the topic of bootstrap arguments for evolution equations. I have given a discussion of the nature of these arguments in Section 10.3 of my book. Another description can be found in section 1.3 of Terry Tao’s book ‘Nonlinear dispersive equations: local and global analysis‘. A related and more familiar concept is that of the method of continuity. Consider a statement $P(t)$ depending on a parameter $t$ belong to the interval $[0,\infty)$. Let $S$ be the subset consisting of those $t$ for which the statement $P$ is true on the interval $[0,t)$. If it can be shown that $S$ is non-empty, open and closed then it can be concluded that the statement holds for all $t$, by the connectedness of the interval. The special feature of a bootstrap argument is the way in which openness is obtained. Suppose that, starting from $P(t)$, we can prove a string of implications which ends again with $P(t)$. This is nothing other than a circular argument and proves nothing. Suppose, however, that in addition this can be improved so that the statement at the end of the string is slightly stronger than that at the beginning. This improvement is something to work with and is a typical way of proving the openness needed to apply the continuity argument. It is more convenient here to work with the open interval $(0,\infty)$ since we want to look at properties of solutions of an evolution equation defined on the interval $(0,t)$. Let $P(t)$ be the statement that a certain inequality (1) holds on the interval $(0,t)$ and suppose that $P(t)$ implies the statment $Q(t)$ that a stronger inequality (2) holds on the same interval. Things are usually set up so that $Q(t)$ implies by continuity that (2) holds at $t$ and that the the property of being ‘stronger’ then shows that $P(t')$ holds for $t'$ slightly greater than $t$. This shows the openness property. I think the best way to really understand what a bootstrap argument means is to write out a known example explicitly or, even better, to invent a new one to solve a problem which interests you. The key thing is to find the right choice of $P$ and $Q$. What I have described here is only the simplest variant. In the work of Christodoulou mentioned above he uses a continuity argument on two-dimensional sets.

### Watching T cells cross the blood-brain barrier

November 9, 2009

A recent press release by the Max Planck Institute for Neurobiology in Martinsried reports on a paper (‘Effector T cell interactions with meningeal vascular structures in nascent autoimmune CNS lesions’, Nature 462, 94) where detailed information is given on certain aspects of the way that activated T cells cross the blood-brain barrier during the development of the disease EAE in rats. In fact the authors were able to film the behaviour of the cells in living rats over extended time periods. In the best-known type of interaction between white blood cells and blood vessels through which they pass, the cells roll along the wall of the vessels until at some point they stop and exit the vessel by squeezing between the cells forming the wall. In this work the disease is provoked by introducing activated T cells which recognize myelin basic protein. In the case of most blood vessels they roll as just described but in the case of certain blood vessels belonging the BBB they instead crawl along the walls, often against the direction of the blood flow. This type of behaviour has apparently not been seen before in T cells although it is known from some other types of leukocytes. It looks as if the cells are searching for something particular although it is not clear what. Some of them eventually cross the BBB into the central nervous system while others let go and return to the bloodstream.

Once the cells get into the CNS they encounter phagocytic cells which activate them and cause them to produce substances such as interferon $\gamma$ and interleukin 17. This then causes further T cells to be recruited to the CNS, thus leading to the full development of the disease. The identity of these phagocytic cells seems a bit mysterious. They are described in the paper as being intermediate between macrophages and dendritic cells. They are said to be constantly probing the region just outside the vessel walls. What I find particularly interesting about this work is that instead of just obtaining indirect information on what is going on it shows very directly what the cells are doing. The information presented in the paper is much more extensive that what I have just indicated. It has been possible to follow the cells on their way to deeper levels of the brain and to compare these particular T cells to other activated T cells which recognize a different antigen having nothing to do with CNS tissue.

### Influenza vaccines

November 5, 2009

I have recently been reading about influenza vaccines and I am summarizing some of the information I found here. I start with some remarks on the classification of influenza viruses. The first distinction is between influenza A and influenza B viruses. The former are classified further into subtypes HnNm for numbers $m$ and $n$. Well-known examples are H5N1 (which includes the recent ‘bird flu’) and H1N1 (which includes the pandemic of 1918 and the current ‘swine flu’.) Influenza B does not carry a pandemic threat and will not be considered further here. Every year a vaccine is produced for the seasonal flu epidemic (in fact two – one for the southern and one for the northern hemisphere). It is trivalent, being directed against three types of virus. In recent years this has always been of the form H3N2 + H1N1 + B. In particular this is the case for the present vaccine for seasonal flu. It is not expected that this vaccine will be effective against the pandemic H1N1 swine flu. Thus a separate type of vaccine has been developed for that. In the classification H and N stand for haemagglutinin and neuraminidase, two proteins which occur on the surface of the virus and come in different forms in different strains. These are the main molecules of the virus recognized by antibodies. They are involved in the processes by which the virus enters and leaves host cells, respectively.

Next I come to some details concerning the vaccines themselves. I concentrate on those being applied in Germany since this is what would be relevant for me if I got vaccinated myself. I get the impression that there are a lot of unreliable and misleading statements on this subject in the media and so some care is necessary in judging the information available. On the web page of the Paul Ehrlich Institute there is a list of vaccines against seasonal flu approved in Germany in this season. Twenty products are listed. All are classified as inactivated. This means that if manufactured successfully the vaccine cannot lead to any reproduction of the virus. In other words the vaccine uses (parts of) ‘dead’ virus particles. Three of the vaccines are described as ‘virosomal’ which means that they can be administered as a nasal spray. Presumably all the others are administered by injection. Two of them include an adjuvant, a substance which is intended to amplify the immune response. This is one theme which has led to recent controversy in connection with swine flu vaccines and I will return to it later. One vaccine (Optaflu) is said to be produced in cell culture. This is connected to another theme of recent controversy, with discussion in the media about vaccines produced using cancer cells.