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

514 Chapter 8: Analyzing Cells, Molecules, and Systems
degraded is defined as its mean lifetime, τ. In our current example, the rate of degradation
of protein X depends on its mean lifetime τ X , which takes into account
active degradation as well as its dilution as the cell grows. The degradation rate
depends on the concentration of protein X and is calculated by dividing this concentration
by the lifetime (Figure 8–74A).
With equations for rates of production and degradation in hand, we can now
generate a differential equation to determine the rate of change of protein X as a
function of time (Equation 8–5, Figure 8–74B). This equation can be solved by the
numerical methods mentioned earlier. According to the solution of this equation,
when transcription begins, the concentration of protein X rises to a steady-state
level at which the concentration of X is not changing anymore; that is, its rate of
change is zero. When this occurs, rearrangement of Equation 8–5 yields an equation
that can be used to determine the steady-state value of X, [X st ] (Equation 8–6,
Figure 8–74C). An important concept emerges from the mathematics: the steadystate
concentration of a gene product is directly proportional to its lifetime. If lifetime
doubles, protein concentration doubles as well.
The Time Required to Reach Steady State Depends on Protein
Lifetime
We can see from Equation 8–6 (see Figure 8–74C) that when the concentration
of protein A rises, protein X increases to a new steady-state value, [X st ]. But this
cannot happen instantaneously. Instead, X changes dynamically according to the
solution of its differential rate equation (Equation 8–5). The solution of this equation
reveals that the concentration of X over time is related to its steady-state concentration
according to the equation in Figure 8–74D. Once again, mathematics
uncovers a simple but important concept that is not intuitively obvious: following
a sudden increase in [A], [X] rises to a new steady state at an exponential rate that
is inversely related to its lifetime; the faster X is degraded, the less time it takes it to
reach its new steady-state value (Figure 8–74E). The faster response time comes at
a higher metabolic cost, however, since proteins with a rapid response time must
be produced and degraded at a high rate. For proteins that are not rapidly turned
over, the response time is very long, and protein concentration is determined primarily
by the dilution that results from cell growth and division.
Quantitative Methods Are Similar for Transcription Repressors and
Activators
Positive control is not the only mechanism that cells use to regulate the expression
of their genes. As we discussed in Chapter 7, cells also actively shut off genes,
often by employing transcription repressor proteins that bind to specific sites on
target genes, thereby blocking access to RNA polymerase. We can analyze the
function of these repressors by the same quantitative methods described above
for transcription activators. If a repressor protein R binds to the regulatory region
of gene X and represses its transcription, then the fraction of gene binding sites
occupied by the repressor is specified by the same equation we used earlier for
the transcription activator (Figure 8–75A). In this case, however, it is only when
the DNA is free that RNA polymerase can bind to the promoter and transcribe the
gene. Thus, the quantity of interest is the unbound fraction, which can be viewed
as the probability that the site is free, averaged over multiple binding and unbinding
events. When the repressor concentration is zero, the unbound fraction is 1
and the promoter is fully active; when the repressor concentration greatly exceeds
1/K, the unbound fraction approaches zero. Figures 8–75B and C compare these
relationships for a transcription activator and a transcription repressor.
We can create a differential equation that provides the rate of change in protein
X when repressor concentrations change (Equation 8–7, Figure 8–75D). As in
the case of the transcription activator, the steady-state concentration of protein
X increases as its lifetime increases, but it decreases as the concentration of the
transcription repressor increases.