[A:pX] (multiples of initial value) involved. For example, if we double the values of both kon and koff, then Equa tion 8–1 (Figure 8–72C) indicates that the steady-state value of [A:pX] does not change. However, the time it takes to reach 50% of this steady state after a ten-fold 4.5 3.5 2.5 d[A:pX] = rate of complex formation – rate of complex dissociation 1.5 d[A:pX] 0.5 on [A][pX] – koff [A:pX] protein production rate = ˜.m 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 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.
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 1.0 0.5
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.
degraded is defined as its mean lifetime, τ. In our current example, the rate of degradation of protein X depends on its mean lifetime τX, 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).