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
517
ACTIVATING
INPUT
(A)
GENE X
X
GENE Y
Y
GENE Z
Z
protein concentration
X
Y
Z
(B)
time
ACTIVATING
INPUT
(C)
GENE V
V
GENE W
W
GENE X
X
GENE Y
Y
GENE Z
Z
protein concentration
(D)
time
such that DNA-binding affinity increases at higher concentrations of the transcription
regulator. Cooperativity produces a steeper transcriptional response to
increasing regulator concentration than the response that can be generated by the
binding of a monomeric protein to a single site. A steep transcriptional response
of this sort, when present in conjunction with positive feedback, is an important
ingredient for producing systems with the ability to switch between different discrete
phenotypic states. To begin to understand how this occurs, MBoC6 n8.608/6.79 we need to modify
our equations to include cooperativity.
Cooperative binding events can produce steep S-shaped (or sigmoidal) relationships
between the concentration of regulatory protein and the amount bound
on the DNA (see Figure 15–16). In this case, a number called the Hill coefficient (h)
describes the degree of cooperativity, and we can include this coefficient in our
equations for calculating the bound fraction of promoter (Figure 8–79A). As the
Hill coefficient increases, the dependence of binding on protein concentration
becomes steeper (Figure 8–79B). In principle, the Hill coefficient is similar to the
number of molecules that must come together to generate a reaction. In practice,
however, cooperativity is rarely complete, and the Hill coefficient does not reach
this number.
(A)
bound fraction
(B)
bound fraction =
1
0.5
ACTIVATOR
0.5
0
h=6
h=3
(K A [A]) h for activators, or
for repressors
1 + (K A [A]) h
1 + (K R [R]) h
h=2
concentration of protein A
1/K A
h=1
unbound fraction
1
REPRESSOR
h=6
h=1
h=3
h=2
0
1/K R concentration of protein R
Figure 8–78 Oscillations arising
from delayed negative feedback.
A transcriptional circuit with three
components (A, B) is less likely to oscillate
than a transcriptional circuit with five
components (C, D). The X (light blue),
Y (dark blue), and Z (brown) here represent
transcription regulatory proteins. For the
simulations in (B) and (D), the system was
initiated from random initial conditions for
X, Y, and Z. Oscillations are produced by
a delay induced as the signal propagates
through the loop.
Figure 8–79 How the cooperative
binding of transcription regulatory
proteins affects the fraction of
(K R [R]) h
promoters bound. (A) Cooperativity is
incorporated into our mathematical models
by including a Hill coefficient (h) in the
equations used previously to determine the
fraction of bound promoter (see Figures
8–72E and 8–75A). When h is 1, the
equations shown here become identical to
the equations used previously, and there is
no cooperativity. (B) The left panel depicts a
cooperatively bound transcription activator
and the right panel depicts a cooperatively
bound transcription repressor. Recall
from Figure 8–75B that gene activity
is proportional to bound activator (left
panel) or unbound repressor (right panel).
Note that the plots get steeper as the Hill
coefficient increases.