Molecular Biology of the Cell by Bruce Alberts, Alexander Johnson, Julian Lewis, David Morgan, Martin Raff, Keith Roberts, Peter Walter by by Bruce Alberts, Alexander Johnson, Julian Lewis, David Morg

MATHEMATICAL ANALYSIS OF CELL FUNCTIONS
519
(A)
GENE X
X
GENE Y
Y
(B)
(C)
d[X]
= β X
.
1 [X]
mX –
dt
1 + (K Y [Y]) h Y τ X
Equation set 8–9
d[Y]
= β Y
.
1 [Y]
mY
–
dt
1 + (K X [X]) h X τ Y
[X] st
[Y] st
= β X
. mX . τX
= β Y
. mY . τY
1
1 + (K Y [Y st ]) h Y
1
1 + (K X [X st ]) h X
Equation 8–10
Equation 8–11
Figure 8–81 A graphical nullcline analysis. (A) X inhibits Y and Y inhibits X, resulting
in a positive feedback loop. (B) Equation set 8–9 can be used to determine the rate of
change in the concentrations of proteins X and Y. (C) Equations 8–10 and 8–11 provide
the concentrations of proteins X and Y, respectively, when these concentrations reach
a steady state. (D, E) Blue curves (called nullclines) are plots of [X st ]calculated from
Equation 8–10 over a range of concentrations of [Y st ]. Red curves indicate values of [Y st ]
calculated from Equation 8–11 over a range of concentrations of [X st ]. At an intersection
of the two lines, both [X] and [Y] are at steady state. For plot (D), the binding of both
proteins to their target gene promoters was cooperative (h X and h Y much larger than 1),
resulting in the presence of multiple intersections of the nullclines––suggesting that the
system can assume multiple discrete steady states. In plot (E), the binding of protein
X to the promoter of gene Y was not cooperative (h X close to 1), resulting in only one
nullcline intersection and thus just one likely steady state.
concentration of Y concentration of Y
(D)
(E)
concentration of X
concentration of X
steady state. For example, when there is a low cooperativity of protein X binding
to the promoter of gene Y (that is, a small Hill
MBoC6
coefficient,
n8.611/8.82
h X , in Equation 8–11),
the plot of [Y] is less curved (Figure 8–81E), and it is less likely that there will be
multiple intersections of the two curves.
We emphasized earlier that positive feedback typically generates a bistable system
with two stable steady states. Why does the system modeled in Figure 8–81D
have three? This conundrum can be explained by solving the reaction rate equations
(Equation set 8–9, Figure 8–81B) for various different starting conditions of
[X] and [Y], determining all values of [X] and [Y] as a function of time. Starting
with each set of initial concentrations of [X] and [Y], these calculations produce
a so-called trajectory of points, each indicated by a curved green line on Figure
8–82A. A fascinating pattern emerges: each trajectory moves across the plot and
settles in one of two steady states, but never in the third (middle steady state).
We conclude that the middle steady state is unstable because it cannot “attract”
any trajectories. The system therefore has only two stable steady states. Thus, the
number of stable steady states in a system need not be equal to the total number
of its theoretically possible steady states. In fact, stable steady states are usually
separated by unstable ones, as in our example.
Once this system adopts a fate by settling in one of the two steady states, does
it have the ability to switch to the other state? The numerical solution of Equation
concentration of Y
(A)
concentration of X
concentration of Y
(B)
1
2
concentration of X
Figure 8–82 Analysis of the stability of
a system’s steady states. (A) The dotted
lines are the nullclines for the system shown
in Figure 8–81. Also shown are dynamic
trajectories (green) that show the changes
over time in [X] and [Y], starting at a variety
of different initial concentrations (determined
by solution of Equation set 8–9; see Figure
8–81B). By plotting [X] versus [Y] at each
time point, we find that, although there are
three possible steady states in this system,
the dynamic trajectories converge on only
two of them. The middle steady state is
avoided: it is unstable, being unable to
attract any trajectories. (B) Imagine that
the system is at the upper-left steady state
and experiences a perturbation (black
arrows), such as a random fluctuation in
the production rates of X and/or Y. If the
perturbation is small (arrow 1), the system
will return to the same steady state. On
the other hand, a perturbation that drives
the system beyond the unstable (middle)
steady state (arrow 2) causes it to switch
to the lower-right steady state. The set of
perturbations that a system can withstand
without switching from one steady state
to the other is known as the region of
attraction of that steady state.