The most common strategy for solving this problem is to use ordinary differential equations. The equations that describe biochemical reactions have a simple premise: the rate of change in the concentration of any molecular species X (that is, d[X]/dt) is given by the balance of the rate of its appearance with that of its disappearance. For our example, the rate of change in the concentration of the bound promoter complex, [A:pX], is determined by the rates of complex assembly and disassembly. We can incorporate these rates into the differential equation shown in Figure 8–73A (Equation 8–4). When [A] changes, Equation 8–4 can be solved to generate the concentration of [A:pX] as a function of time. Notice that when kon [A][pX] = koff [A:pX], then d[A:pX]/dt = 0 and [A:pX] stops changing. At this point, the system has reached the steady state.
Calculation of all [A:pX] values as a function of time, using Equation 8–4, allows us to determine the rate at which [A:pX] reaches its steady-state value. Because this value is attained asymptotically, it is often most useful to compare the times needed to get to 50, 90, or 99 percent of this new steady state. The simplest way to determine these values is to solve Equation 8–4 with a method called numerical integration, which involves plugging in values for all of the parameters (kon, koff, etc.) and then using a computer to determine the values of [A:pX] over time, starting from given initial concentrations of [A] and [pX]. For kon = 0.5 × 107 sec–1 M–1, koff = 0.5 × 10–1 sec–1 (K = 108 M–1 as above), and [pXT ] = 10–10 M, it takes [A:pX] about 5, 20, and 40 seconds to reach 50, 90, and 99 percent of the new steady-state value following a sudden tenfold change in [A] (Figure 8–73B). Thus, a sudden jump in [A] does not have instantaneous effects, as we might have assumed from looking at the cartoon in Figure 8–72A.
Differential equations therefore allow us to understand the transient dynamics of biochemical reactions. This tool is critical for achieving a deep understanding of cell behavior, in part because it allows us to determine the dependence of the dynamics inside cells on parameters that are specific to the particular molecules
Figure 8–73 Using differential equations to study the dynamics and steady-state behavior of a biological system.
(A) Equation 8–4 is an ordinary differential equation for calculating the rate of change in the formation of bound promoter complex in response to a change in other components. (B) Formation of [A:pX] after a tenfold increase in [A], as determined by solving Equation 8–4. In blue is the solution corresponding to kon = 0.5 × 107 sec–1 M–1 and koff = 0.5 × 10–1 sec–1. In this case, it takes [A:pX ] about 5, 20, and 40 seconds to reach 50, 90, and 99 percent of the new steady-state value. For the red curve, the kon and koff values are doubled, and the system reaches the same steady state more rapidly.