Biomedical Engineering Reference
In-Depth Information
mentation are expressed transiently, and it is left to a segment polarity network
to maintain the definition of segment boundaries. Segment polarity networks
abound in insect orders, whereas the patterns of stable segmentation are vari-
able. Von Dassow et al. (58,59) suggest that the segment polarity network is a
robust evolutionary module, recruited by different insect species, and provided
with different inputs to produce diverse patterns of segmentation. In order for
this to be the case, parametric variation in reaction coefficients should leave the
patterning ability of the network intact.
In order to model the network, Von Dassow simulated large systems of
coupled first-order differential equations. For example, the rate of transcription
of mRNA
M
i
from gene
E
i
, assuming a concentration of binding transcription
factor
x
i
, a maximum rate of transcription
T
max
, and a rate of decay
dm
i
, is given
by
¯
c
i
x
¡
°
mT
=
dm
,
[6]
¡
°
i
max
c
c
i
kx
+
¢
±
where the parameter
k
determines the value at which the transcription factor
X
i
has half maximum effect on the rate of translation of the gene
E
i
. The subse-
quent translation of
M
i
into a protein
P
i
with a maximum rate of translations
r
max
and a rate of decay
d
p
p
i
is of the form
¯
m
¡
°
p
=
r
d p
.
[7]
i
¡
°
i
max
P
i
mk
+
¢
±
i
These proteins are then free to bind to other proteins forming complexes with
novel transcription activity (e.g., a
p
i
might bind to a
p
j
to induce
x
k
, etc).
Equations of this form assume saturation of enzymes and substrates. As a
consequence, over large variations in parameter values steady-state concentra-
tions of protein products and complexes remain unchanged. Saturation is the
assumption behind the derivation of the familiar Michelis-Menten rate law: the
concentration of substrate is in large excess over the concentration of enzyme
(66). In the limiting case of very high values of the constant
c
, coupled differen-
tial equations can be effectively replaced by Boolean networks. In this case, only
the topology of the network and the initial conditions, not the kinetic constants,
have an influence on steady states. Thus stable variation of segmentation in in-
sect orders might be achieved through variation in initial conditions with disre-
gard for variation in kinetic parameters. Species diversity would derive from
feeding different initial conditions through the same network without regard for
species-specific variation in rate constants. If saturation is not justified this ro-
bust modularity disappears. The empirical validity of saturation in developmen-
tal networks remains to be established.
