Biomedical Engineering Reference
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independent of
M
the lymphocyte clone size should scale ~
M
3/4
M
1/4
=
M
, where
the first factor comes from the number of service units and the second one from
the number of lymphocytes needed per unit. In (4) we also examine the case in
which lymphocyte movement and antigen growth both depend on the basic
metabolic rate of an organism and also conclude that for that case clone size ~
M
.
2.3. Scaling of the Lymphocyte Repertoire
Next, we ask how many different clones of lymphocytes should be present
in an animal of mass
M
. The number of different clones is called the size of the
immune repertoire and determines how many different antigens the immune
system can recognize.
We assume that antigens mainly enter the body through our intake of food,
liquids, and air. The rate at which a mammal consumes food and air is governed
by its metabolic rate. One can show that the lifetime total metabolic activity of a
mammal scales as ~
M
(4), suggesting that a mammal needs to deal with
cM
an-
tigens during its lifetime, where
c
is some constant.
In order to assess the probability that an immune system with a repertoire of
size
N
can recognize an antigen, Perelson and Oster (5) introduced the idea of
shape-space. In this theory it is assumed each lymphocyte has a receptor that can
recognize antigens in a volume
v
0
of shape space, which has total volume
V
. If
we let F be the probability of the immune system, i.e., all
N
different clones,
failing to recognize an antigen, then
N
F
¬
¬
v
v
-
-
=
1
!
0
exp
N
0
-
-
®
.
[3]
-
-
-
-
V
®
V
The probability of a successful immune response to an infection is then 1 - F,
and the probability,
P
s
, that the organism will successfully repel all
cM
infec-
tions during its lifespan is given by
P
s
= (1 - F)
cM
x exp(-F
cM
).
[4]
This probability should be very near to unity, so we require F
cM
<< 1 or
¬
-
v
0
1
F
=
exp
N
-
<<
.
[5]
-
-
V
®
cM
This in turn implies
