## Archive for August, 2009

### Rheumatoid factor

August 30, 2009

We are familiar with the fact that it is possible to do tests for various types of disease. A well-known example are tests for HIV, often called AIDS tests in everyday language. I asked myself the question to what extent there are also reliable tests for autoimmune diseases. If there is no pathogen such as a virus present then one approach to tests is ruled out. But are there others? One relevant example I found is that of rheumatoid arthritis (RA) and tests based on a substance called rheumatoid factor. I will say more about this below but first I wanted to recall one well-known pitfall of tests in general. This shows that the notion of a reliable test is not completely obvious. It is useful to introduce some terminology. The sensitivity of a test is the proportion of people having the disease for which the test gives a positive result. The specificity is the proportion of people not having the disease for which the test gives a negative result. Consider the example of a disease which affects one person in one million and a test where the probability of a positive result in someone who does not have the disease is one in ten thousand. In this case the specificity is 0.9999, which sounds very good. Suppose now, however, one million people are tested. The expected number of people in the sample having the disease is one but the number of people who will test positive is one hundred. In other words a positive test only really says that the probability of the patient having the disease is at least one per cent, a rather weak statement. The problem here is simply that looking for an effect which is very rare can make reliable testing very hard. The quantity which comes out to be one per cent in this example is called the positive predictive value.

After this digression I now come back to rheumatoid factor. Autoimmune diseases can be divided into those where the immune system attacks a particular tissue (e.g. insulin dependent diabetes mellitus where it is the $\beta$ cells of the pancreas which are attacked) and those which are systemic and affect a variety of tissues. RA is of the latter type although its best-known effects concern the joints. The presence of rheumatoid factor is an indication of RA but it may also be present in people suffering from Sjögren’s syndrome or in healthy individuals. Thus, in the above terminology, the specificity of the test is not very high. It also happens that rheumatoid factor is not detectable in people with RA during extended periods. To sum up, this is test for RA but it does not seem to be a very precise one. What kind of substance is rheumatoid factor? I mention at this point, since it is something that used to confuse me, that the words ‘antibody’ and ‘immunoglobulin’ are synonymous. Rheumatoid factor is an immunoglobulin of type M against immunoglobulins of type G. The antigen it binds to is in the Fc part of the antibody. This is a part which is independent of the antigen for which the antibody is specific and only depends on the isotype. The isotype differs between antibodies with different functional properties and is designated by the letter M, D, G, A or E. There is another test for RA with better specificity using anti-citrullinated protein antibodies. Citrulline is a non-classical amino acid. It is not coded for by DNA but can be produced by modification of existing proteins.

What about other autoimmune diseases? Another important systemic autoimmune disease is systemic lupus erythematosus (SLE). Here a common test involves antinuclear antibodies, i.e. antibodies against parts of the cell nucleus. The test has high sensitivity but low specificity. In the case of multiple sclerosis there seems to be no test comparable to those just mentioned. One diagnostic criterion which is used is that of oligoclonal bands. What this means is the occurrence of proteins in the cerebrospinal fluid (actually immunoglobulins of type G) which show up as bands in electrophoresis. It is important that there are a few of those bands (oligo) in the cerebrospinal fluid which are not present in the blood. This is regarded as evidence of activity of the immune system within the central nervous system. The difference to the tests discussed above is that little seems to be known about what substances the antibodies producing the bands are antibodies against.

### How to be unhappy

August 16, 2009

There are many books on the market which give advice as to how to improve your life. I feel attracted to this kind of book, which of course shows something about how satisfied I am with my own life. I particularly like the classic ‘How to stop worrying and start living’ by Dale Carnegie which I have read several times. This is an old book, going back to 1944, but this does not matter. Circumstances change but human nature, which is at the centre of this type of book, does not. I like Carnegie’s down to earth approach. I also like the fact that he makes clear that the main things he has to say are not new – his purpose is to remind us of things which in principle we already know but which are all too often forgotten. I feel that I have really profited from reading that book.

The reason I am writing this post is that I just read a book called ‘Anleitung zum Unglücklichsein’ by Paul Watzlawick, which I came across in the public library. I found this book very amusing and entertaining. The title can be translated as ‘Guide to being unhappy’. I am not aware that it has been translated into English. The list of references includes a book called ‘How to make yourself miserable’ by Dan Greenburg, which may be similar.Returning to the book of Watzlawick, it is not likely to make its readers unhappy. It is full of biting humour. It is a kind of parody of the self-help books I mentioned above but in a sense it can have similar effects. It lays bare certain mechanisms in human thinking and in the communication between individuals. This is not a easy book to read. It is necessary to pay careful attention so as not to occasionally take what is written there literally. Reading the book is a rewarding experience. It underlines the fact that the meaning of the word ‘happiness’ (Glück) is not so obvious as is generally assumed.

### Poisson brackets

August 9, 2009

Poisson brackets are very popular in theoretical physics. In the case of a classical mechanical system described in terms of a Hamiltonian the underlying mathematical structure is a finite-dimensional manifold $M$ and a symplectic structure $\omega$. This object has an inverse $\omega^{-1}$. The Poisson bracket of two functions $f$ and $g$ on $M$ is defined in terms of their exterior derivatives as $\omega^{-1}(df,dg)$. There is no problem here for the mathematician interested in understanding this. What is more difficult is the definition of the Poisson bracket for field theories. Formally this corresponds to allowing the manifold $M$ to be infinite dimensional. Consider for instance a scalar field $\phi$, in one space dimension for maximum simplicity. Thus we are considering functions $\phi (t,\theta)$ and I suppose (simplicity again) that they are periodic in $\theta$. The Lagrangian density is $\frac12 (\phi_t^2-\phi_\theta^2)$. The momentum $\pi$ conjugate to $\phi$ is defined to be the time derivative $\phi_t$. In this case the phase space $M$ is formally the space of functions $(\phi,\pi)$.I will not try to specify what kind of functions – for example we can think of them as being smooth.Given two functionals $F$ and $G$ (i.e. two functions on the infinite dimensional space $M$) the Poisson bracket is defined by the formula $\int_{S^1} \frac{\delta F}{\delta\phi}\frac{\delta G}{\delta\pi}-\frac{\delta F}{\delta\pi}\frac{\delta G}{\delta\phi}$. The derivatives here are functional derivatives. Now, feeling certain that my physicist colleagues will react with amusement or incomprehension, I must admit that I do not understand what a functional derivative is. This formula is a hieroglyph for me.

Let us not be discouraged. The functional derivative looks something like a variational derivative, a concept which is more familiar to me. This basically just means that if I have a functional which is the integral over $S^1$ of a function of $\phi$ and $\pi$ then it is possible to consider variations $(\phi(\lambda),\pi(\lambda))$, intuitively curves in the manifold $M$, and differentiate them with respect to the parameter $\lambda$. If the space $M$ can be defined as a decent infinite-dimensional manifold (e.g. a Banach manifold) then this can be related to constructions of differential geometry on that manifold. The variational derivative just corresponds to the exterior derivative and is a one-form. If we start with smooth functions then this object takes values in the dual space. In other words we encounter distributions. Now in the presence of a volume form many distributions can be identified with functions. The one-form is then integration against that function, which could be called the functional derivative and used in the definition of the Poisson bracket.If the functionals are integrals of expressions depending pointwise on $\phi$ and $\pi$ then this is nothing other than the partial derivative with respect to the same quantity. Notice that with this definition we can reproduce the fact that the momentum $\pi$ is the functional derivative of the Lagrangian with respect to $\phi_t$ If the functional is allowed to depend on the spatial derivative of $\phi$ then a further complication is added since in that case it is necessary to integrate by parts in space to compute the function corresponding to the distribution.If we can interpret the functional derivatives as functions of $\theta$ then the Poisson bracket can be written as $\int_{S^1}\frac{\delta F}{\delta\phi}(\theta)\frac{\delta G}{\delta\pi}(\theta)-\frac{\delta F}{\delta\pi}(\theta)\frac{\delta G}{\delta\phi}(\theta)d\theta$.