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10
0
N
W
10
0
N
W
1000000
1000000
10
-2
10
-2
10000
10000
10
-4
10
-4
100
100
10
-6
10
-6
1
1
0
20
40
60
1
10
100
W
W
Fig. 4.
The distribution
N
τ
for the frustrated system with N=50 cells and 1 pathogenic cell with
a random set of ligands. The distribution function was calculated using the same procedure as
in Fig.3. On the left the distribution function is plotted as in Fig.3, showing that a long tail
appears corresponding to long-lived interactions. On the right the same distribution is plotted in
a double logarithmic scale to highlight the power law behavior emerging for the long-lived
interactions.
In order to be more precise, we now consider some numerical examples. Consider
a population with
N
=51 cells from Fig.3 where one cell has been replaced by a
pathogenic cell, i.e., a cell that presents a foreign peptide. This population can be
simulated constructing the ILs as in (1), for
N=51
cells, but where the presence of the
pathogenic cell (say cell 1) in the others cell's ILs is moved a random number of
positions (up or down). The IL for cell 1 stays the same. In this way we assume that
only the ligands of the pathogenic cell change while the receptors of this cell remain
the same. It is interesting to remark how the distribution N
changes so dramatically
with this single cell substitution (see Fig.4). A power law tail now appears which is
due to the appearance of long-lived interactions. These long-lived interactions involve
the pathogenic cell. In over 100 populations simulated, all the interactions lasting
longer than
τ
=20 iterations steps involved cell 1. This is interesting because it shows
that the system is performing self-nonself recognition with high specificity.
In order to understand how sensitive this discrimination is, we next performed the
same simulation but where the range of changes in the ILs was restricted: the position
of the pathogenic cell in the other cell's ILs was moved only 1 position, up or down.
Typical examples are shown in Fig.5, where it can be seen that there are long lasting
interactions occurring, although in smaller number than in the previous case.
How the system achieves such high sensitivity and specific self-nonself
discrimination can be seen as arising from a
generalized kinetic proofreading
mechanism,
and was first discussed in [16]
.
In the frustrated state interactions have a
probability to terminate given by equation (2), with
q=1-p
. If ILs are changed due to a
change in the rank of the pathogenic cell, then
q
increases for some interactions and
decreases for others. Hence, certain interactions involving the pathogenic cell can
decrease their unbinding probability to
P
*
, while in a first approximation interactions
not involving the pathogenic cell do not change their unbinding probability
P,
given
by (2). Considering the probability that a conjugate remains bound for
τ
τ
time-steps,
we obtain (using (3)):
P
*
/ P
τ
=[(1-P
*
)/(1-P)]
τ−1
P
*
/P .
(5)
τ
