Protein A also exists in two forms: free ([A]) and bound to pX ([A:pX]). In a cell, there are typically one or two copies of pX (assuming there is only one gene X per haploid genome) and multiple copies of A. As a result, we can safely assume that from the viewpoint of A, [A:pX] is negligible relative to the total [AT ]. This means that [A] ≈ [AT ], and we can just plug in the values of total [AT ] in Equation 8–2 without incurring appreciable error in the calculation of [A:pX].
Now, we are ready to determine the effects of increasing the concentration of
A. Suppose that K = 108 M–1, which is a typical value for many such interactions. The starting concentration of A is [AT ] = 10–9 M, and [pXT ] = 10–10 M (assuming there is one copy of gene X in a haploid yeast cell, for example, with a volume of around 2 × 10–14 L). Using Equation 8–2, we find that a tenfold increase in the concentration of A causes the amount of promoter complex [A:pX] to increase 5.5fold, from 0.09 × 10–10 M to 0.5 × 10–10 M at steady state. The effects of a tenfold increase in the concentration of A will vary dramatically depending on its starting concentration relative to the equilibrium constant. Only through this mathematical approach can we achieve a thorough understanding of what these effects will be and what impact they will have on the biological response.
To assess the biological impact of a change in transcription activator levels, it is also important in many cases to determine the fraction of the target gene promoter that is bound by the activator, since this number will be directly proportional to the activity of the gene’s promoter. In our case, we can calculate the fraction of the gene X promoter, pX, that has protein A bound to it by rearranging Equation 8–2 (Equation 8–3, Figure 8–72E). This fraction can be viewed as the probability that promoter pX is occupied, averaged over time. It is also equal to the average occupancy across a large population of cells at any instant in time. When there is no protein A present, pX is always free, the bound fraction is zero, and transcription is off. When [A] = 1/K, the promoter pX has a 50% chance of being occupied. When [A] greatly exceeds 1/K, the bound fraction is almost equal to one, meaning that pX is fully occupied and transcription is maximal.
The most important and basic insights for which we, as biologists, depend on mathematics concern the behavior of regulatory systems over time. This is the central theme of dynamics, and it was for the solution of problems in dynamics that the techniques of calculus were developed, by Newton and Leibniz, in the seventeenth century. Briefly, the general problem is this: if we are given the rates of change of a set of variables that characterize the system at any instant, how can we compute its future state? The problem becomes especially interesting, and the predictions often remarkable, when the rates of change themselves depend on the values of the state variables, as in systems with feedback.
Let us return to Equation 8–2 (Figure 8–72D), which tells us that when [A] changes, [A:pX] at steady state will also change to a new concentration that we can calculate with precision. However, [A:pX] does not change instantaneously to this value. If we hope to understand the behavior of this system in detail, we must also ask how long it takes [A:pX] to get to its new steady-state value inside the cell. Equation 8–2 cannot answer this question. We need calculus.