Biology Reference
In-Depth Information
Demongeot and Waku
2012a
; Demongeot and Demetrius (
submitted
); van der Pol
and van der Mark
1928
;
http://www.sciencesunivnantesfr_sites_genevieve_tulloue_
4.3.1 Example of a Potential Metabolic Network
in Morphogenesis
Metabolic networks used in plant or animal morphogenesis are fully connected
with all interactions inhibitory, except positive auto-loops representing auto-
catalytic processes. Such systems called n-switches (Demongeot et al.
2000
;
Thellier et al.
2004
; Cinquin and Demongeot
2005
) are driven by the differential
equations ruling the concentrations
x
i
of growth hormones [as auxin for plants
like Araucaria trees—see Fig.
4.1
and Forest and Demongeot (
2006
) and Forest
et al. (
2006
)—or transduction peptides for animals (Jolliot and Prochiantz
2004
)]
in several locations
i
, the value
i
0 being reserved to the location of the first
growth, 1 to the second growth location,
¼
(apex leaves cells for plants and after
first bud cells; medulla primitive cells for animal nervous system and after
bulbar cells) following:
...
Va
i
X
i
c
þ Σ
j¼
0
;
...
;n
a
j
X
j
c
8
i
¼
0
;
...
;
n
;
d
X
i
=
d
t
¼ σ þ
=ð
1
Þμ
i
x
i
;
where
,
V
,
c
, and
a
i
's are, respectively, a constant entry flux, the Hill's maximum
velocity, cooperativity, and affinities, and
σ
μ
i
's the degradation rates of hormones.
The dynamics is ruled by the potential:
X
i
Log
y
i
=
þ Σ
i
a
i
y
i
2
c
þ Σ
i
μ
i
y
i
2
where
y
i
2
P
ð
y
Þ¼σ
2
V
Log
ð
1
Þ=
4
c
=
4
;
¼
X
i
If we replace Hill's kinetics by allosteric Monod-Wyman-Changeux kinetics,
the system remains potential (Demongeot et al.
2007a
,
b
; Glade et al.
2007
).
In the case of plant morphogenesis, the first attractor observed corresponds to the
steady state values
x
0
* verifying
Va
0
x
0
*
c
a
0
x
0
*
c
σ þ
=ð
1
þ
Þμ
0
x
0
*
¼
0and
x
i
*
¼
0,
for
i
0. It corresponds to the growth of the apex, but after a certain time, due to the
negative geotropic plant growth, auxin from apex leaves cells can no more sufficiently
diffuse among the first bud of the plant for inhibiting its cells, then the second steady
state can be expressed, verifying:
>
Va
0
x
1
**
c
a
0
x
1
**
c
σ þ
=ð
þ
Þμ
0
x
1
**
¼
1
0and
x
i
**
1. An example of such successive bud growths is given by
Araucaria tree (Fig.
4.1
, middle).
¼
0, for
i
6¼
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