## Archive for May, 2010

### Adaptive immunity across the animal kingdom

May 21, 2010

Yesterday I went to a talk by Jim Kaufman from Cambridge which I found very informative. Textbooks on immunology usually concentrate largely on the human immune system and the immune system of the mouse, as the main experimental animal. For a long time it was believed that adaptive immunity was confined to vertebrates (with a few exceptions like lampreys which appeared to lack it). Those animals which have adaptive immunity seem to have the whole package, in the sense that they have antibodies, T cells and MHC molecules. So it looks like that from an evolutionary point of view all these things arose about the same time (on the time scales typical for evolution). In this talk I discovered that very many invertebrates have adaptive immunity. They use different systems from that familiar from vertebrates and often different systems from each other. Insects are included. Drosophila, for instance, has an adaptive immune system based on a single gene which can be spliced in a huge number of different ways. From the references given in the talk it looked to me as if these facts about invertebrates came out in about the last five years.

One of the main themes of the talk was that there is a big divide in adaptive immunity between mammals and other vertebrates. (There is a marsupial which is an exception.) The difference concerns the major histocompatibility complex. The MHC molecules in chickens (to take the principal example discussed in the talk) are more adapted to specific pathogens than is the case in mammals. This might also mean that mammals are particularly susceptible to autoimmune diseases. Apparently not so much is known about autoimmune dieases in non-human mammals or non-mammalian vertebrates. This is understandible from the point of view of the motivation of human beings. Kaufman also pointed out that not very much is known about the frequency of autoimmune diseases in poorer countries.

Another theme of the talk was the evolution of adaptive immunity. I was not able to follow the details very closely. One idea was that T cells appeared relatively early and may have recognized targets directly before there were cells to present them with antigens. Another was that natural killer cells were one of the earliest elements which led to adaptive immunity. In any case I enjoyed the opportunity to obtain an unusually broad view of immunity.

### The principle of symmetric criticality

May 12, 2010

There are many interesting partial differential equations which can be expressed as the Euler-Lagrange equations corresponding to some Lagrangian. Thus they are equivalent to the condition that the action defined by the Lagrangian is stationary under all variations. Sometimes we want to study solutions of the equations which are invariant under some symmetry group. Starting from the original equations, it is possible to calculate the symmetry-reduced equations. This is what I and many others usually do, without worrying about a Lagrangian formulation. Suppose that in some particular case the task of doing a symmetry reduction of the Lagrangian is significantly easier than the corresponding task for the differential equations. Then it is tempting to take the Euler-Lagrange equations corresponding to the symmetry-reduced action and hope that for symmetric solutions they are equivalent to the Euler-Lagrange equations without symmetry. But is this always true? The Euler-Lagrange equations without symmetry are equivalent to stationarity under all variations while the Euler-Lagrange equations for the symmetry-reduced action are equivalent to stationarity under symmetric perturbations. The second property is a priori weaker than the first. This procedure is often implicit in physics papers, where the variational formulation is more at the centre of interest than the equations of motion.

The potential problem just discussed is rarely if ever mentioned in the physics literature. Fortunately this question has been examined a long time ago by Richard Palais in a paper entitled ‘The principle of symmetric criticality’ (Commun. Math. Phys. 69, 19). I have known of the existence of this paper for many years but I never took the trouble to look at it seriously. Now I have finally done so. Palais shows that the principle is true if the group is compact or if the action is by isometries on a Riemannian manifold. Here the manifold is allowed to be an infinite-dimensional Hilbert manifold, so that examples of relevance to field theories in physics are included. The proof in the Riemannian case is conceptually simple and so I will give it here. Suppose that $(M,g)$ is a Riemannian manifold and $f$ a function on $M$. Let a group $G$ act smoothly on $M$ leaving $g$ and $f$ invariant. Let $p$ be a critical point of the restriction of $f$ to the set $F$ of fixed points of the group action. It can be shown that $F$ is a smooth totally geodesic submanifold. (In fact in more generality a key question is whether the fixed point set is a submanifold. If this is not the case even the definition of the principle may be problematic.) The gradient of $f$ at $p$ is orthogonal to $F$. Now consider the geodesic starting at $p$ with initial tangent vector equal to the gradient of $f$. It is evidently invariant under the group action since all the objects entering into its definition are. It follows that this geodesic consists of fixed points of the action of $G$ and so must be tangent to $F$. Hence the gradient of $f$ vanishes.

When does the principle fail? Perhaps the simplest example is given by the action of the real numbers on the plane generated by the vector field $x\frac{\partial}{\partial y}$ and the function $x$. This has no critical points but its restriction to the fixed point set, which is the $y$-axis, has critical points everywhere.