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
513
transcription rate = β
at steady state:
K[A]
1 + K[A]
K[A]
protein production rate = β. m
1 + K[A]
protein degradation rate = [X]
τ X
(A)
d[X]
= protein production rate – protein degradation rate
dt
d[X] K[A] [X]
= β. m – Equation 8–5
dt 1 + K[A] τ X
(B)
(C)
(D)
[X st ] = β. m
K[A]
1 + K[A]
t
[X](t) = [X st ](1 – e – τ ) X
change in [A] in our example changes from about 5 seconds to 2 seconds (see Figure
8–73B). These insights are not accessible from either cartoons or equilibrium
MBoC6 8.620/8.75
equations. This is an unusually simple example; mathematical descriptions such
as differential equations become more indispensible for understanding biological
interactions as the number of interactions increases.
.
τ X
Equation 8–6
fraction steady-state protein level
(E)
1.0
0.5
0
τ 1 τ 2
RESPONSE TIME
DEPENDENCE ON
PROTEIN LIFETIME
time
Figure 8–74 Effect of protein lifetime
on the timing of the response.
(A) Equations for calculation of the rates of
gene X transcription, protein X production,
and protein X degradation, as explained
in the text. (B) Equation 8–5 is an ordinary
differential equation for calculating the
rate of change in protein X in response to
changes in other components. (C) When
the rate of change in protein X is zero
(steady state), its concentration can be
calculated with Equation 8–6, revealing
a direct relationship with protein lifetime
(τ). (D) The solution of Equation 8–5
specifies the concentration of protein X
over time as it approaches its steady-state
concentration. (E) Response time depends
on protein lifetime. As described in the text,
the time that it takes a protein to reach
a new steady state is greater when the
protein is more stable. Here, the blue line
corresponds to a protein with a lifetime that
is 2.5-fold shorter than the lifetime of the
protein in red.
Both Promoter Activity and Protein Degradation Affect the Rate of
Change of Protein Concentration
To understand our gene regulatory system further, we also need to describe the
dynamics of protein X production in response to changes in the amount of transcription
activator protein A. Here again, we use an ordinary differential equation
for the rate of change of protein X concentration—determined by the balance of
the rate of production of protein X through expression of gene X and the protein’s
rate of degradation.
Let us begin with the rate of protein X production, which is determined primarily
by the occupancy of the promoter of gene X by protein A. The binding and
dissociation of a transcription regulator at a promoter generally occurs on a much
faster time scale than transcription initiation, causing many binding and unbinding
events to occur before transcription proceeds. As a result, we can assume that
the binding reaction is at equilibrium on the time scale of transcription, and we
can calculate promoter occupancy by protein A using the equilibrium equation
discussed earlier (Equation 8–3, Figure 8–72E). To determine transcription rate,
we simply multiply the occupied promoter fraction by a transcription rate constant,
β, that represents the binding of RNA polymerase and the subsequent steps
that lead to production of mRNA and protein (Figure 8–74A). If each mRNA molecule
produces, on average, m molecules of protein product, then we can determine
protein production rate by multiplying the transcription rate by m (Figure
8–74A).
Now let us consider the factors that influence protein X degradation and
its dilution due to cell growth. Degradation generally results in an exponential
decline in protein levels, and the average time required for a specific protein to be