Biology Reference
In-Depth Information
X
1
X
2
X
n
.
.
X
0
.
X
i-1
X
n-1
.
.
.
X
i
Fig. 4.1
Left
: Representation of the interaction graph of an
n
-switch fully connected with negative
interactions (
in black
) and positive ones (
in red
).
Middle
:
Araucaria
tree growth.
Right
: represen-
tation of the potential
P
in case
n
¼
1:
P
is associated to the 2-switch defined by
μ ¼
1,
c
¼
2,
V
¼
2,
a
i
¼
0.1,
σ ¼
1. The surface of
P
is represented on the state space
E
¼
(X
0
0X
1
)
4.3.2 Example of a Hamiltonian Genetic Regulatory
Network in Immunology
The network controlling the expression of the RAG (Recombination Activating
Gene), responsible of the rearrangements of the
V
(
D
)
J
region of the chromosome 14
in human, giving birth to the T-cell receptors alpha contains strong connected
components (scc): (1) the subnetwork (given in Fig.
4.2
) containing the gene
GATA3 regulating the T helper cell maturation and made of two circuits, one
positive of length 5 and another negative of length 3, tangent on the gene SOCS1
and (2) the subnetwork containing the gene PU.1 and made of two negative circuits
of respective length 6 and 2 (Demongeot and Waku
2012a
; Demongeot and
Demetrius (
submitted
); Demongeot et al.
2011b
). The circuits are Hamiltonian
and the conservative energy is the discrete kinetic energy
E
defined on the circuit
C
by:
X
i
2C
x
i
ð
2
E
ð
C
Þ¼
ð
t
Þ
x
i
ð
t
1
Þ
Þ
=
2
;
where
x
i
(
t
) is the Boolean state of the gene
i
at time
t
(equal to 1 if the gene is
expressing its protein and 0 if it is in silence) and where the transitions on
C
are
either the identity (symbol +) or the negation (symbol
). The number of attractors
of the first scc is equal to 3 and of the second equal to 1 [cf. Demongeot and Waku
(
2012a
), Demongeot and Demetrius (
submitted
), and Demongeot et al. (
2011b
) and
numbers in red in Fig.
4.19
of the Mathematical Annex].
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