A:pX rate of complex formation = k [A][pX] rate of complex dissociation = koff [A:pX] at steady state: on [A][pX] = koff [A:pX] GENE X GENE Y
Figure 8–71 Diagrams that summarize biochemical relationships. Here, a simple cartoon indicates that gene X represses gene Z (left) whereas gene Y activates gene Z (right).
Figure 8–72 A simple transcriptional interaction. (A) Genes A and X each produce a protein, with the product of gene A serving as a transcription activator to stimulate expression of gene X. As indicated by the green arrow, stimulation depends in part on the binding of protein A to the promoter region of gene X, designated as pX. (B) The binding of protein A to the gene promoter is determined by the concentrations of the two binding partners (denoted as [A] and [pX], in units of mol/liter, or M), the association rate constant kon (in units of sec–1 M–1), and the dissociation rate constant koff (in units of sec–1). (C) At steady state, the rates of association and dissociation are equal, and the concentration of the bound complex is determined by Equation 8–1, in which the two rate constants are combined in the equilibrium constant K. (D) Equation 8–2 can be derived to calculate the steady-state concentration of bound complex at a known total concentration of the promoter [pXT]. (E) Rearrangement of Equation 8–2 yields Equation 8–3, which allows calculation of the fraction of promoter pX that is occupied by protein A.
by the notation in Figure 8–72B. This reaction notation is more informative than the diagrams in our figures, but has its own limitations. Suppose that the concentration of A increases by a factor of ten as a response to an environmental input. If A increases, we intuitively know that A:pX should increase too, but we cannot determine the amount of the increase without additional information. We need to know the affinity of the binding interaction and the concentrations of the components. With this information in hand, we can rigorously derive the answer.
As discussed earlier and in Chapter 3 (see Figure 3–44), we know that the formation of a complex between two binding partners, such as A and pX, depends on a rate constant kon, which describes how many productive collisions occur per unit time per protein at a given concentration of pX. The rate of complex formation equals the product of this rate constant kon and the concentrations of A and pX (see Figure 8–72B). Complex dissociation occurs at a rate koff multiplied by the concentration of the complex. The rate constant koff can differ by orders of magnitude for different DNA sequences because it depends on the strength of the non-covalent bonds formed between A and pX.
We are primarily interested in understanding the amount of bound promoter complex at equilibrium or steady state, where the rate of complex formation equals the rate of complex dissociation. Under these conditions, the concentration of the promoter complex is specified by a simple equation that combines the two rate constants into a single equilibrium constant K = kon/koff (Equation 8–1; Figure 8–72C). K is sometimes called the association constant, Ka. The larger this constant K, the stronger the interaction between A and pX (see Figure 3–44). The reciprocal of K is the dissociation constant, Kd.
To calculate the steady-state concentration of promoter complex using Equation 8–1, we need to account for another complication: both A and pX exist in two forms—free in solution and bound to each other. In most cases, we know the total concentration of pX and not the free or bound concentrations, so we must find a way to use the total concentration in our calculations. To do this, we first specify that the total concentration of pX ([pXT ]) is the sum of the concentrations of free ([pX]) and bound ([A:pX]) forms (Figure 8–72D). This leads to a new equation that allows us to use [pXT ] to calculate the steady-state concentration of the promoter complex ([A:pX]) (Equation 8–2, Figure 8–72D).