Information Technology Reference
In-Depth Information
Another important point to remark is that if cell D behaved exactly as one of the
other cells already present in the system then the system would not stay in the tolerant
state. This shows that in this model cells are recognized according to how they
function with respect to the whole system. This is a useful property for a protection
system, because it implies that clonal proliferation of an infected cell would not be a
successful strategy for a pathogen. Rather pathogens need to mutate in order to
successfully infect the host. Further, it also shows that a certain level of arbitrariness
exists concerning the definition of the ligands and receptors in the system. What is
required is that cell A senses cell B with maximal avidity, cell B senses with maximal
avidity cell C, and so on. This says nothing about what cell's A receptors are,
allowing them to be somatically defined, as required in an adaptive immune system.
Although the previous solution allows the system to remain in the frustrated state,
it requires that cell D has low avidity relative to all the other cells in the system. This
may not always be achieved in a particular system provided thymic positive selection
has selected reactive cells to span uniformly a complete space of sequences. To see
this more clearly, imagine that a ligand or a receptor are defined through a sequence
of bits and that affinity is proportional to the number of bits in common between the
ligand and the receptor (i.e. through a Hamming distance). Then, provided the set of
receptors in the system is uniformly distributed, it is not possible to define a ligand
that is simultaneously
more
anticorrelated with
all
the receptors in the system. This
remark is important, because it shows that thymic selection may have a double
function which is not only to select reactive cells, but also to provide a uniform
distribution of receptors and ligands. A more detailed analysis of thymic repertoire
selection in the light of the present theory will be discussed in a forthcoming paper.
The previous results are restricted to populations with a small number,
N
, of
elements. Can we generalize these results to arbitrary
N
? For
N
odd it is easy to
establish that there exists a system exhibiting full frustration. Considering that the cell
at position
j
at the interaction list of cell
i
is
L
i
(j),
then the list verifies the requirement:
L
i
(j)= L
u
(N-j)
, where
u= L
i
(j) .
(
1
)
Hence, if cell
i
has on the top position
(j=1)
of its IL, cell
j
, then cell
j
has on the
bottom of its IL cell
i
. This simple rule forces frustration. For a system with an odd
number of elements, it then becomes straightforward to show that such a system never
attains a stable configuration, as there is always at least one unbound element that is
at the top position of the IL of one in the system. Consequently it is always possible to
destabilize at least one pair of bound cells.
The same argument does not apply to systems with
N
even, in which case the
system can converge to a stable configuration. However, due to the complexity of the
cellular interactions, for populations with
N
even the system converges very slowly to
the stable solution. In Fig.2 we see that the number of iterations required grows
exponentially fast with
N.
Hence, although for
N
even
the system has a stable
configuration, the dynamics of the system is governed by the proximity to a
computationally hard problem [45]. Hence, from a biological point of view, the
system behaves as in the
N
odd case. And in fact the duration of cellular contacts
behaves as in the
N
odd case (Fig.2
(left)).
Fig.2
(left)
also shows that interactions' lifetimes decay exponentially. This is not
an obvious result, because Almeida and Vistulo de Abreu [16] obtained a power law
