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

518 Chapter 8: Analyzing Cells, Molecules, and Systems
Positive Feedback Is Important for Switchlike Responses and
Bistability
We turn now to positive feedback and its very important consequences. First and
foremost, positive feedback can make a system bistable, enabling it to persist in
either of two (or more) alternative steady states. The idea is simple and can be
conveyed by drawing an analogy with a candle, which can exist either in a burning
state or in an unlit state. The burning state is maintained by positive feedback: the
heat generated by burning keeps the flame alight. The unlit state is maintained
by the absence of this feedback signal: so long as sufficient heat has never been
applied, the candle will stay unlit.
For the biological system, as for the candle, bistability has an important corollary:
it means that the system has a memory, such that its present state depends
on its history. If we start with the system in an Off state and gradually rack up the
concentration of the activator protein, there will come a point where autostimulation
becomes self-sustaining (the candle lights), and the system moves rapidly to
an On state. If we now intervene to decrease the level of activator, there will come
a point where the same thing happens in reverse, and the system moves rapidly
back to an Off state. But the transition points for switching on and switching off
are different, and so the current state of the system depends on the route by which
it has been taken in the past—a phenomenon called hysteresis.
A simple case of positive feedback can be seen in a regulatory system in which
a transcription regulator activates (directly or indirectly) its own expression, as in
Figure 8–80A. Positive feedback can also arise in a circuit with many intervening
repressors or activators, so long as the net overall effect of the interactions is activation
(Figure 8–80B and C).
To illustrate how positive feedback can generate stable states, let us focus on a
simple positive feedback loop containing two repressors, X and Y, each of which
inhibits expression of the other (Figure 8–81A). As we saw with Equation set 8–8
(Figure 8–76B) earlier, we can create differential equations describing the rate
of change of [X] and [Y] (Equation set 8–9, Figure 8–81B). We can further modify
these equations to include cooperativity by adding Hill coefficients. As we did
earlier, we can then create equations for calculating the concentrations of [X]
and [Y] when the system reaches a steady state (that is, when (d[X]/dt ) = 0 and
(d[Y]/dt ) = 0; Equations 8–10 and 8–11, Figure 8–81C).
Equations 8–10 and 8–11 can be used to carry out an intriguing mathematical
procedure called a nullcline analysis. These equations define the relationships
between the concentration of X at steady state, [X st ], and the concentration of Y
at steady state, [Y st ], which must be simultaneously satisfied. We can plug in different
values for [Y st ] in Equation 8–10, and calculate the corresponding [X st ] for
each of these values. We can then graph [X st ] as a function of [Y st ]. Next, we repeat
the process by varying [X st ] in Equation 8–11 to graph the resulting [Y st ]. The intersections
of these two graphs determine the theoretically possible steady states of
the system. For systems in which the Hill coefficients h X and h Y are much larger
than 1, the lines in the two graphs intersect at three locations (Figure 8–81D). In
other systems that have the same arrangement of regulators but different parameters,
there might only be one intersection, indicating the presence of only a single
ACTIVATING
INPUT
ACTIVATING
INPUT
ACTIVATING
INPUT
(A)
GENE X
X
(B)
GENE X
X
GENE Y
Y
(C)
GENE X
X
GENE Y
Y
Figure 8–80 Positive feedback of a gene
onto itself through serially connected
interactions. A sequence of activators and
repressors of any length can be connected
to produce a positive feedback loop, as
long as the overall sign is positive. Because
the negative of a negative is positive, not
only circuit (A) and (B) but also circuit (C)
create positive feedback.