# Hydrobates

A mathematician thinks aloud

## Archive for May, 2011

### Is half of what is in immunology textbooks wrong?

May 31, 2011

Yesterday I heard a talk by Rolf Zinkernagel who won a Nobel prize together with Peter Doherty in 1996 for the discovery of the MHC restriction in the T cell recognition of antigens. What this means is that a T cell which is monitoring the body for harmful substances does not recognise the antigen (the substance itself) alone but a combination of the antigen with a major histocompatibility complex (MHC) molecule. The MHC molecules were originally discovered through their role in tissue transplantation. In order that a transplant not be rejected the MHC molecules of donor and host should be sufficiently similar. The key work which earned the prize was done in Australia and it was interesting to observe that Zinkernagel (who is originally from Switzerland) speaks English with quite a noticeable Australian accent. The talk was clearly meant to be provocative and the speaker did emphasize on more than one occasion that he was exaggerating. My knowledge of the subject is not good enough to allow me to make a comprehensive judgement of his claims but it was clear that he was presenting a picture far from the usual consensus.

One of the statements in the talk is the one which occurs in the title of this post, namely that fifty per cent of what is in immunology textbooks is wrong. This was complemented with the statements that what is difficult is to know which fifty per cent is wrong and that more than fifty per cent is based on fashion and something else which I do not remember. On a more humble note the speaker said that a similar statement might apply to what was in his lecture. This is not going to cause me to lose confidence in immunology textbooks but it is reasonable to adopt the recommendation not to accept what is in the books uncritically.

Zinkernagel said that from an evolutionary point of view our immune system is designed to work well until we are 25 at the most, which was all that was required to guarantee offspring for our ancestors. It is not clear to me that having some more children could not have resulted in a competitive advantage. It may, however, be that before the invention of agriculture it was impossible to feed many children. He emphasized the important role of the period before and shortly after birth in the development of the immune system. In the first months children are protected by antibodies coming from the mother and the claim was that this has a determining influence on the way the immune system works. At the same time he was critical of the standard picture of acquired immunity, where the reponse to a second infection is faster and stronger than that to the first infection and this is the key to immunity. He also suggested that after an individual has been infected with a pathogen such as measles the antigen remains in the body in some form and continues to stimulate the immune system.  This seemed very speculative to me. The lecture contained many comments relating to diverse aspects of immunity (the speaker emphasized the importance of making the distinction between immunity and immunology). One suggestion I found interesting was that by doing intensive research on the relatively benign HIV-2 it might be possible to obtain essential insights for understanding the problem of HIV-1 better.

The lecture hall was very full and this showed that there was a lot of interest in hearing this speaker. On the other hand the questions after the talk were very few and it did not seem to come to a real intellectual engagement between the speaker and the members of the audience asking the questions. Perhaps Zinkernagel has moved so far from the accepted view on many themes in immunology that communication with colleagues has become difficult. Whatever else can be said about this lecture I think it did have the positive feature of stirring up the ideas of the audience and this cannot be a bad thing in science.

Posted in immunology | 4 Comments »

### The quantal theory of immunity

May 24, 2011

I recently read two papers of Kendall Smith on his ‘quantal theory of immunity’ (Cell Research 16, 11 and Medical Immunology 3:3). To start with, it should be noted that the word ‘quantal’ here has nothing to do with quantum theory although Schrödinger’s cat does make a brief cameo appearance in the first paper. The term is used to denote all-or-nothing reactions. In other words as the input to a system is increased continuously there is a threshold where the output changes from a low almost constant level to another much higher almost constant level. The input could be the concentration of a substance surrounding a cell while the output could be the amount of a certain protein produced by the cell.

Smith starts by discussing the ‘clonal selection theory’ of MacFarlane Burnet. In this context it is natural to pose the broad question of the possibility of discovering general theories or laws in biology. These concepts are familiar to us from physics and the question is to what extent they can be adapted to biology. I do not pretend to have a general answer to this. A theory or law in this sense is an idea in science which can be used to explain not just a few phenomena but which gives a pattern which can contribute to explanations in a wide field. An obvious example which comes to mind in biology is natural selection. Burnet’s theory seems to provide an example of this kind in immunology. It says that the ability of the immune system to recognize antigens is localized in certain white blood cells (later identified as the lymphocytes), each of which is specific for one antigen. The number of antigens which can be recognized is enormous and correspondingly there are only very few cells recognizing a given antigen. The large numbers of cells needed to combat a given pathogen are reached by ‘clonal expansion’ – one cell undergoes a population expansion by cell division. The next question is how the immune system manages to cause this proliferation in cases where it is beneficial to the individual and to avoid it in cases where it would be harmful. The quantal theory of immunity is a proposed theory to provide an explanation for this.

Smith’s discussion concentrates on T cells, saying that the case of B cells is similar. The proliferation of T cells is associated to their production of the cytokine IL-2. Thus it is important to understand the causes and effects of the latter. Another important theme which needs to be discussed at this point is the the relation between the behaviour of individual cells and that of cell populations. In an experiment we might measure a property of a population, for instance the total production of IL-2, but it is clear that what is being measured in that case is just an average of what is happening in individual cells. Experiments which focus on the properties of individual cells have revealed that there is a high intrinsic variability in cell populations (and in populations of the organisms whose cells are being studied). It has been suggested that this can provide an evolutionary advantage in some cases.

Let us return to IL-2 of which, as I understand, Smith is more or less the father. The reaction to IL-2 in an individual T cell is quantal. The same is true for the production of IL-2 in response to stimulation of the T cell receptor (together with the co-receptor CD28). The global picture presented in Smith’s papers (seen at low resolution) is the following. The fate of T cells (activation, proliferation, anergy, apoptosis) is determined by the number of T cell receptors stimulated. On the axis representing this number there are several bands representing the different outcomes. Variants of this scheme are presented for T cells in the thymus and in the periphery. Underlying this mechanism there is another similar one where the controlling quantity is the number of occupied IL-2 receptors.

Posted in immunology | 1 Comment »

### Cooperativity and the Hill equation

May 1, 2011

In biology it often happens that a molecule of one substance has several binding sites for another molecule (call it the ligand). This is known as cooperative binding. A classical example is haemoglobin, which has four binding sites for oxygen. If the rate of binding of the ligand at one of the sites is affected (positively or negatively) by the fact that some of the other sites are occupied then this is known as allostery. Suppose for example that an enzyme has two binding sites for a ligand. Using a description of this process of Michaelis-Menten type for the two successive binding events leads to an equation where the right hand side is a quotient of two quadratic polynomials. A derivation of this can be found in Murray’s book ‘Mathematical Biology’. Assuming a reaction where both molecules combine with the enzyme simultaneously gives a different quotient of quadratic polynomials which is particularly simple, namely $\frac {Au^2}{u^2+B}$. This is not likely to be a genuine reaction mechanism but might arise in some way by telescoping two reactions to get a simpler model. It may be interpreted as what is called complete cooperativity: when one molecule of the ligand binds the probability of binding at the other site becomes much higher and so may be supposed to occur essentially immediately.

A generalization of this idea for $n$ substrate molecules is given by the Hill equation $\dot u=f(n)=\frac{Au^n}{u^n+B}$. It was introduced by the physiologist Archibald Hill on a paper on haemoglobin in 1910. (Hill studied mathematics before he turned to physiology. He received the 1922 Nobel Prize for Physiology or Medicine for his work on the production of heat in muscles.) There he first considers what kind of equations might result if several molecules of oxygen bind to one molecule of haemoglobin. He then suggests considering the equation which now bears his name independently of detailed reaction mechanisms. This is a common procedure in modern biology. The Hill equation is used primarily as a phenomenological ansatz. The idea of cooperativity is in the background but the link is not very direct. In particular, it can happen that the Hill equation is used with non-integer values of $n$. An important qualitative feature of the nonlinearity in the Hill equation is that it is sigmoid for $n>1$. In other words the first derivative increases for small values of the argument before decreasing for larger values. This is a feature which may be observed in experimental data. When it is seen it is possible to try to fit it to a description by the Hill equation using a Hill plot. This means fitting the linear relation $\log(\frac{f}{A-f})=n\log u-\log B$. In particular the slope of the graph gives the Hill coefficient $n$.

This equation has no relation to the linear ordinary differential equation called Hill’s equation which is named after the astronomer and mathematician George William Hill.

Posted in dynamical systems, mathematical biology | 1 Comment »
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