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, [Xst] (Equation 8–6, Figure 8–74C). An important concept emerges from the mathematics: the steady-state 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, [Xst]. 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.
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.
1.0
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