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5.2.1.2
Coutinho and Varela's Idiotypical Network
In Coutinho and Varela's model (Coutinho, 1993; Varela and Coutinho, 1991), both
idiotypes bound to a cell surface and free antibodies are considered. h e concept
of network sensitivity for an idiotype is introduced and defi ned as a function of the
a nity between such idiotype and the network antibodies. Here, m ij denotes the
a nity between two idiotypes of type i and j . h e network sensitivity for the i th
idiotype is denoted by σ i ( t ) and it is thereby defi ned as
N
i
()
t
mf
()
t
(5.5)
ij
j
j
1
f j ( t ) is the amount of free idiotypes of type i . After a maturation process, specifi c
B cells generate free antibodies. h us, a diff erential equation that describes the
change in the concentration of free antibodies is defi ned as
df
()
t
i
(5.6)
kbtmat
()
( ())
t
kft
() ()
t
kft
()
ii
i
2
i
i
3
i
dt
b i ( t ) is the number of idiotypes attached to the surface of B cells.
db t
dt
()
i
(5.7)
kb t pol
()
r
( ())
t
metai
[]
kb t
()
4
i
i
5
i
Mat () is the lymphocyte maturation function. Prol regulates the probability of
proliferation, and meta [ i ] represents the metadynamics that results from adding
resting lymphocytes into the active IN. Both mat () and prol () functions are
considered as bell-shaped.
5.2.1.3
Farmer, Packard, and Perelson's Idiotypical Network
h is IN model considers the microdynamics of the antibodies and antigens inter-
action. In this case, the model keeps track of the proportions of each type of antibodies
among the population. An antibody is considered as a pair of paratope and epitope
( p, e ), which are explicitly represented by binary strings (Farmer et al., 1986).
h e a nity measure takes into account all possible matching by shifting
antibody j. Each shift of antibody i is matched against antibody j . h erefore, the
matching a nity m ij between antibodies i and j is defi ned as
r
d
m
G
e
(
l
k
)
p
( )
l
s
1
(5.8)
ij
i
j
k
1
l
1
 
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