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In-Depth Information
The activation function
f
(
h
i
) determines the intensity of the stimulation of the clone
i
by the idiotypic field
h
i
. These equations are used to study the dynamical regimes:
oscillatory, chaotic, etc. Further details are outside the scope of the present work.
The main task of the immune system is to recognize the antigen by means of cell
receptors. The binding mechanism, whose fine details are mostly unknown, is based
on different physical-chemical effects (short-range noncovalent interactions, hydro-
gen binding, van der Waals interactions, etc.) (Perelson and Weisbuch 1997). The
binding of a receptor with a molecule requires that they complement each other over
a significant portion of their surface. This
generalized shape
is the constellation of
features that determine the binding among molecules (Perelson and Oster 1979).
Under the assumption that the shape can be described by
K
parameters, a point in a
K
-dimensional space (the
shape space
) specifies the generalized shape of the mo-
lecular binding region. Oster and Perelson estimated that in order to be complete the
receptor repertoire should fulfill the following conditions: (1) each receptor should
recognize a set of related epitopes, each of which differs slightly in shape; (2) the
repertoire size should be on the order of 10
16
or larger; (3) at least a subset of the
repertoire should be distributed randomly throughout the shape space (Perelson and
Weisbuch 1997).
Later, Farmer, Packard, and Perelson (1986) introduced the idea of using binary
strings to represent the shape of a receptor. To determine the degree of affinity be-
tween strings it is possible to resort to different string-matching criteria. For instance,
by using a “key-lock” analogy, two binary strings have high affinity if they “com-
plement” each other, that is, when the two strings are lined up every 0 in one corre-
sponds to a 1 in the other and conversely.
The following paragraph is an introduction (by no means complete) to mathe-
matical models of the immune system response and in particular to the class of mod-
els more close to our own approach that will be discussed later in this chapter. At this
point of the discussion we just need to say that we deal with a stochastic and discrete
description of the immune system response at the cellular scale.
8.2.1 An Overview of Discrete Models
A number of discrete models of the immune system working at the cellular scale
have been proposed in the past by using a variety of different techniques and aims.
The first aim in modeling the immune system is to reproduce the (primary and sec-
ondary) response. However, many other aspects of its behavior, like autoimmune
diseases (Weisbuch and Atlan 1988), selection and hypermutation of antibodies
during an immune reaction (Celada and Seiden 1996), autoimmunity and T-
lymphocyte selection in thymus (Morpurgo, Serenthà, Seiden, and Celada 1995), the
immune response to known virus inducing cancer (Melief, Toes, Medema, Van der
Burg, Ossendorp and Offringa 2000), the acquired HIV-related tumors (Carbone and
Gaidano 2001; Varthakavi, Smith, Deng, Sun, and Spearman 2002), etc., have been
studied and modeled. Moreover, it is worth pointing out that the activation of the
immune response against some tumors has been known since the 1950s. Other stud-
ies showed that tumor cells resort to escape mechanisms that prevent the activation
of the immune system (Anachini and Mortarini 1999). Recent extensive reviews on
