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and
b=1,
leads to:
N
. In the numerical simulations (Fig.3) we obtained
exponents close to 1, instead of 1.38. The difference between the two values is due to
the crude approximation of
f
P
used above (see Fig.3 (
right)
). In the random case, we
obtain:
~
exp(
−
1
38
τ
)
τ
−
2
N
. Here again the exponent is not the same as the one found in [16]
(which was
-2.5
), again due to the approximations used. Nevertheless with this
calculation we were able to understand how two distinctive behaviors can be found and
that the power law behavior in
N
τ
signals the existence of processes with many different
lifetimes. Hence, Fig.3 shows that, even if the system could display a continuum of
different lifetimes, the frustrated system displays a single well defined lifetime.
~
τ
τ
3
10
0
10
-2
10
-4
10
-6
10
-8
1.E+00
N=11
N=51
N=501
N
W
1.E-02
1.5
1.E-04
1.E-06
0
1.E-08
0
125
250
375
500
0
5
10
15
20
25
W
Rank
Fig. 3.
The distribution function
N
τ
(calculated as in Fig.2)
converges quickly to the asymptotic
distribution, when the number of cells in the system varies from
N=11
to
N=501
(Left)
. The
distribution for
N=51
and
N=501
is almost the same, and given by an exponential
)
≅
AN
. This quick convergence shows that the properties of the model do not depend
crucially on the number of cells involved. This shows that the model is robust in the sense that
generalizations to account for spatial effects should not produce different results (provided the
densities are not too low).
(Right)
The distribution of the rank in the IL occupied by a conjugated
cell in the other cell's IL. We used a population with
N=501.
This distribution is directly related
to
f
P
(see equation (4)), which is not uniform as assumed in the calculation of the exponents.
exp(
−
τ
τ
The previous analysis is important to discuss the impact of the introduction of a
new cell into the system. What happens if the frustration is broken? Does the system
break up into a set of long lived interactions (as could happen after introducing a
random cell into the
N=3
system discussed above)?
The recursive (self-similar) structure given by (1) provides a simple answer: for
large
N
, after removing any number of cells from the system, we again obtain a
system in which ILs for the remaining cells have the same structure as the initial ILs.
Hence, if a new randomly generated cell is introduced in the system it can produce a
long lived conjugate and we can view the resulting system as being composed of the
conjugate involving the new cell and the remaining fully frustrated system. This
guarantees that the system remains stable upon introduction of a pathogen.
It should also be remarked that, contrary to the cases where
N=3
or
N=4
, recognition
of the external pathogen should not require an infinitely long-lived binding. Thus, to
define a functional immune system, we invoke
assumption 4
, and determine that a
response will occur for interactions whose lifetimes significantly exceed a typical
lifetime. For instance, in the example of Fig.2, it could be determined that only if a
conjugate lived for 20 units of time, then an effector function would take place.
