Biology Reference
In-Depth Information
FIGURE 14.4
A simple bistable switch. (A) Wiring
diagram. The synthesis of protein P is directed by mRNA
M, which is transcribed from a gene controlled by
a promoter (gray box on the double-stranded DNA mole-
cule). The promoter is active when it is bound by dimers (or
tetramers) of P. (B) Rates of synthesis (solid line) and
degradation (dashed line) of P as functions of P/K
P
. Black
circles indicate stable steady states; white circle indicates
an unstable steady state. Refer to Eqs.
(1)
e
(3)
in the text.
(C) Domain of bistability in parameter space. (D) Signal-
response curve. The steady state concentration of P is
plotted as a function of
f$
K
p
for n
¼
4 and
ε
¼
0.1. The
response is bistable for values of
f$
K
p
in the range (0.301,
0.591).
(A)
(B)
Slope =
φ
Kp
(C)
(D)
φ
Kp
φ
Kp
At 'steady state' dM/dt
¼
dP/dt
¼
0. Hence,
gene to turn back on,
f$
K
p
must be reduced below the other
saddle-node bifurcation point at
f$
K
p
¼
0.301. For 0.301
<
k
dm
k
sm
M
k
dm
k
dp
k
sm
k
sp
P
H
ð
P
Þ¼
¼
¼ f
P
(3)
f$
K
p
<
0.591, the reaction network is 'bistable', i.e., it can
persist in one or another of two stable steady states (on or
off). In the bistable zone, the two stable steady states are
separated by an unstable steady state.
The bistable behavior we have illustrated with this
simple model is completely representative of bistability in
more complex networks. The irreversibility of transitions is
related to bifurcation points in a bistable system, and
bistability is a consequence of
k
dm
k
dp
/k
sm
k
sp
is a constant with units nM
-1
. The
steady-state concentrations of mRNA and protein are deter-
mined by solutions of the algebraic equation H(P)
where
f ¼
¼ f
P,
which is a cubic equation for n
¼
2 and a quintic equation for
n
4. From the graphs in
Figure 14.4
B it is easy to see that
this algebraic equation may have one, two or three real
positive roots (i.e., steady-state values of P), depending on
the values of n,
¼
f$
K
p
. In
Figure 14.4
C we indicate how
the number of steady states depends on
and
ε
Positive Feedback
þ
Sufficient Non-linearity
and
f$
K
p
for n
¼
2
ε
þ
Rate-constant Constraints
:
and n
4. In
Figure 14.4
D we show how the steady state
values of P depend on
¼
f$
K
p
for n
¼
ε
¼
0.1.
Figure 14.4
D is known as a one-parameter bifurcation
diagram (see
Table 14.1
for definitions) for the steady-state
solution, P
ss
,ofEqs.
(1)
as a function of the dimensionless
bifurcation parameter
4 and
In our example, the 'positive feedback' is obvious: the
transcription factor P upregulates its own production.
'Sufficient non-linearity' is reflected in the sigmoidal Hill
function with n
1, there can be no bist-
ability.) 'Rate-constant constraints' are evident in
Figure 14.4
C: bistability is exhibited only within a limited
range of rate constant values.
¼
2 or 4. (If n
¼
f$
K
p
. For small values of
f$
K
p
the
gene is being actively transcribed (the gene is 'on') and P
ss
is large. As
f$
K
p
increases, P
ss
steadily decreases until the
gene abruptly turns off at
0.591. At this value of
f$
K
p
the dynamic system is said to undergo a 'saddle-node'
bifurcation (see
Table 14.1
). The upper steady state
(a 'stable node') coalesces with the intermediate steady
state (an unstable 'saddle point') and they both disappear,
leaving only one attractor of the dynamic system, namely
the lower steady state (the alternative stable node). In the
lower steady state gene expression is turned off, and it is not
possible to turn it back on by a small decrease in the value
of
f$
K
p
¼
IRREVERSIBLE TRANSITIONS IN THE
BUDDING YEAST CELL CYCLE
Figure 14.5
shows the molecular interactions underlying
the irreversible transitions of the budding yeast cell cycle.
Before describing these reaction networks in detail, we
must introduce a few simplifications. In budding yeast,
f$
K
p
. The transition is 'irreversible'. In order to coax the
