Rare event simulation in immunobiology

The ability of the adaptive immune system to discriminate safely between foreign and self molecules is a fundamental ingredient to everyday survival of higher vertebrates (such as ourselves). Unlike innate immune responses, which happen in all kinds of organisms, responses of the adaptive immune system are highly speci c to the target which induced their activation. Until now it is not quite clear how this process of recognition and activation works. The main problem here is to explain and understand a system which is obviously able to recognise one (or a few) type(s) of foreign molecules against an enormous variety of self molecules, although there is no a priori di erence between "self" and "foreign" on the molecular level: There is no such thing as a "self"marker on self molecules to enable an easy discrimination. (This is obvious, because such a marker could easily be forged by foreign intruders). A classi cation of molecules into self and foreign is even unique for every individual. This is most obvious if we think of organ transplantation, where, although it is human, the immune system tries to attack the donor organ because it recognizes it as foreign. A novel approach to this problem of statistical recognition (of one particular foreign signal against a large, uctuating self background) was established by van den Berg, Rand and Burroughs [26] (henceforth referred to as BRB) and further developed by Zint, Baake and den Hollander [28]. In contrast to many existing deterministic models, they formulate an explicit stochastic model. It describes (random) encounters between two crucial types of white blood cells (see Fig. 2): the antigen-presenting cells (APCs), which display a mixture of self and foreign antigens at their surface (a sample of the molecules around in the body), and the T cells, which "scan" the APCs by means of certain receptors and nally "decide" whether or not to react, i.e. to start an immune response. Each T cell is characterised by a speci c type of T cell receptor (TCR), which is displayed in many identical copies on the surface of the particular T cell. A large number (estimated at 10 in [1]) of di erent receptors, and hence di erent T cell types, are present in an individual. However, the number of potential antigen types is still vastly larger (roughly 10; see [16]). Thus, speci c recognition is impossible; this is known as Mason's paradox. The task is further complicated by the fact that every APC displays on the order of thousand(s) of di erent "self" antigen types, in various copy numbers, together with, possibly, one (or a small number of) foreign types. The probability that a T cell reacts to an encounter with a randomly chosen APC has to be very small in order to avoid autoimmune reactions. Some questions may therefore be answered analytically with the help of large deviation theory; others require simulation, but its use has been limited due to the low probabilities involved, at least with the straightforward simulation methods applied so far [26, 28]. Here we present an e cient method of rare event simulation. The article is organized as follows. Sec. 2 recapitulate the biological model; Sec. 3 the simulation method; and Sec. 4 presents some results obtained by applying this method to the T cell model. More details, as well as further results, may be found in [14].