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
515
bound fraction
(B)
1.0
0.5
0
(A)
bound fraction =
TRANSCRIPTION
ACTIVATOR
K[R]
1 + K[R]
unbound fraction = 1 – bound fraction =
unbound fraction
0.5
0
1/K A concentration of protein A (C) 1/K R
protein production rate = β. m
1
1 + K[R]
1
1
1 + K[R]
TRANSCRIPTION
REPRESSOR
concentration of protein R
Figure 8–75 How promoter occupancy
depends on the binding affinity of a
transcription regulator protein. (A) The
fraction of a binding site that is occupied by
a transcription repressor R is determined
by an equation that is similar to the one
we used for a transcription activator (see
Figure 8–72E), except that in the case of a
repressor we are interested primarily in the
unbound fraction. (B) For a transcription
activator A, half of the promoters are
occupied when [A] = 1/K A . Gene activity is
proportional to this bound fraction. (C) For
a transcription repressor R, gene activity
is proportional to the unbound fraction
of promoters. As indicated, this fraction
is reduced to half of its maximal value
when [R]=1/K R . (D) As in the case of the
transcription activator A (see Figure 8–74),
we can derive equations to assess the
timing of protein X production as a function
of repressor concentrations.
(D)
d[X]
dt
1 [X]
= β. m –
Equation 8–7
1 + K[R] τ X
1
[X st ] = β. m .
1 + K[R]
τ X
Negative Feedback Is a Powerful Strategy in Cell Regulation
Thus far, we have considered simple regulatory systems of just a few components.
In most of the complex regulatory systems that govern cell behaviors, multiple
modules are linked to produce larger circuits that we call network motifs, which
can produce surprisingly complex and biologically useful responses whose properties
become apparent only through mathematical analysis. A particularly common
and important network motif is the negative feedback loop, which can have
dramatically different functions depending on how it is structured.
We take as a first example a network motif consisting of two linked modules
(Figure 8–76A). Here, an input signal initiates the transcription of gene A, which
produces a transcription activator protein A. This activates gene R, which synthesizes
a transcription repressor protein R. Protein R in turn binds to the promoter
of gene A to inhibit its expression. This cyclical organization creates a negative
feedback loop that one can intuitively understand as a mechanism to prevent
proteins from accumulating to high levels. But what can we learn about negative
feedback loops, and their value in biology, by using mathematics to model them?
The negative feedback loop in Figure 8–76A can be modeled using Equation
8–7 (see Figure 8–75D) for the repression of gene A and Equation 8–5 (see Figure
8–74B) for the activation of gene R. Thus, for proteins A and R, we use the set of differential
equations (Equation set 8–8) shown in Figure 8–76B. The two equations
in this set are coupled, which means that they must be solved together to describe
MBoC6 n8.604/8.79
Figure 8–76 A simple negative feedback motif. (A) Gene A negatively
regulates its own expression by activating gene R. The product of gene R is a
transcription repressor that inhibits gene A. (B) Equation set 8–8 can be solved
to determine the dynamics of system components over time. (C) A system with
negative feedback (blue) reaches its steady state faster than a system with
no feedback (red). The plots indicate the levels of protein A, expressed as a
fraction of the steady-state level. The blue line reflects the solution of Equation
set 8–8, which includes negative feedback of gene A by the repressor R. The
red line represents the solution when the rate of synthesis of A was set to a
constant value that is unaffected by the repressor R.
(A)
(B)
fraction steady-state
protein level
(C)
ACTIVATING
INPUT
1
0.5
0
GENE A
A
d[A] β A
. mA
=
–
dt 1 + K R [R]
GENE R
[A]
τ A
d[R]
K
= β R
. A [A] [R]
mR –
dt
1 + K A [A] τ R
Equation set 8–8
R
time