Biology Reference
In-Depth Information
BOX 14.1 Mutual Antagonism
The dynamic properties of the molecular regulatory system in
Figure B1
A can be described by a differential equation for
B
a function of B in
Figure B1
B. In
Figure B1
C we plot, as func-
tions of B, the two rate curves:
V
synthesis
¼
[CycB:Cdk1],
dB
dt
¼
k
sb
k
db
B
k
dbc
C
A
B
;
¼
k
sb
;
and
V
degradation
¼ð
k
db
þ
k
dbc
C
A
:
(B1.3)
Þ
B
(B1.1)
The points of
intersection of
these two curves (where
where k
sb
, k
db
and k
dbc
are rate constants. The first subscript,
's' or 'd', refers to the type of reaction, 'synthesis' or 'degra-
dation', the second subscript, 'b' in this case, refers to the
chemical being synthesized or degraded, and the third
subscript, 'c', refers to the enzyme catalyzing the reaction
(whenever relevant). In Eq.
(B1.1)
C
A
is the activity of the
Cdh1:APC complex, which is given by the steady state solu-
tion of the multisite phosphorylation chain in
Figure B1
A.
According to Kapuy et al. [101], C
A
is given by the function
V
synthesis
V
degradation
) are steady state solutions of Eq.
(B1.1)
.
Clearly, the dynamical system may exhibit bistability, depend-
ing on the relative values of its parameters. In
Figure B1
Dwe
indicate how the steady state values of B depend on H, with all
other parameters fixed at their values in
Table B1
.
¼
1
ð
B
.
H
Þ
TABLE B1
Parameter values for the CycB-Cdh1
q
þ
1
model
C
A
¼
B
.
H
1
:
(B1.2)
N
þ
1
ð
Þ
Parameter
Value
Parameter
Value
k
sb
0.1
q
2
where q is the threshold number of phosphate groups above
which Cdh1P
i
is inactive. In Eq.
(B1.2)
, H
k
db
0.1
N
9
¼
activity of the
Cdk-counteracting phosphatase.
A representative set of parameter values for this dynamical
system is given in
Table B1
. For this parameter set we plot C
A
as
k
dbc
1
H
0.25
FIGURE B1
Bistability in a
model of mutual antagonism.
(A) Molecular regulatory network.
'CycB' represents the CycB:Cdk1
heterodimer (a protein kinase) and
'Cdh1' represents the Cdh1:APC
complex (an E3 ubiquitin ligase).
Cdh1 has multiple sites (N)of
phosphorylation by CycB:Cdk1. The
first q states of phosphorylation
(Cdh1, Cdh1-P,
.
, Cdh1-P
q
) are
assumed to be active, and states
Cdh1-P
q
þ
1
,
.
, Cdh1-P
N
inactive. H
is a CDK-counteracting phospha-
tase. (B) Steady-state activity of
Cdh1 as a function CycB-dependent
kinase activity, from Eq.
(B1.2)
. (C)
Rates of synthesis and degradation
of CycB, as functions of CycB-
dependent kinase activity, from Eq.
(B1.3). The intersection points
correspond to two stable steady
states of the dynamical system
(black circles) and one unstable
steady state (white circle). The slope
of line a is k
db
þ
k
dbc
C
T
, and the slope of line b is k
db
. (D) Bifurcation diagram. The steady state values of B, from panel C, are plotted as functions of
H, the activity of the counteracting phosphatase. For 0.184
<
H
<
0.492, the regulatory network has three steady states, two stable (solid lines) and
one unstable (dashed line). The turning points, at H
¼
0.184 and 0.492, are called saddle-node bifurcation points.
(A)
(B)
(C)
(D)
