Information Technology Reference
In-Depth Information
number of all positions j where x j = y j . XOR ( x , y ) is the vector with 1's in all positions
j where x j
ones ( XOR ( x ,
y )) is the number of those positions where x j = y j and therefore both terms have the
same value. A similar argument can be given for the second case.
Given some i
y j and ones ( XOR ( x , y )) is the number of those 1's, thus n
H , the R/T-function has a maximum exactly for the point y with i
= y , and for all other binary vectors it has some value between 0 and 1. It has a
minimum for the point y which is the complement of i . Thus R/T changes the form of
the lattice of figure 2 in such a way that for a special element i this element becomes
the top element and compl ( i ) the bottom element, cf. figure 3. The function decreases
with growing values of d XOR ( x , y ) from 1 to 0 in an exponential form. This is shown in
figure 7 for n = 10.
1,2
1
0,8
0,6
0,4
0,2
0
0123456789 0
d(x,y)
Fig. 7. The form of the Rogers/Tanimoto function for n = 10
4 Affinity Functions
Shape-spaces are not defined as abstract structures; rather their main purpose is to
describe a special relationship between their elements called affinity . The affinity
between two immune elements certainly depends on their distance in the shape-space.
Several authors, e.g. [2], [3], consider affinity as a constant quantity and are interested
in the total amount of influence of other elements on some immune element x . In [1],
a more detailed concept of affinity is given. According to it, affinity is a time
dependent quantity and in addition depends on the concentration of an element i that
exerts affinity on other elements. I will adopt this approach in the following.
The question is, whether shape is a constant property of immune elements and
therefore also the distances between them, in particular antibodies, or not. The shape
of an antibody can be modified by mutation. However, according to the usual
definitions of mutation, the distances between an element and its mutants are small, so
that the mutants of an element x lie in an
. Therefore
distance will be considered as a constant quantity in the following, i.e. the distance
between y and x and that between y and a mutant of x are taken as equal.
ε
-ball around x with small
ε
 
Search WWH ::




Custom Search