Molecular Biology of the Cell by Bruce Alberts, Alexander Johnson, Julian Lewis, David Morgan, Martin Raff, Keith Roberts, Peter Walter by by Bruce Alberts, Alexander Johnson, Julian Lewis, David Morg

MATHEMATICAL ANALYSIS OF CELL FUNCTIONS
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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:p X 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 p X , depends
on a rate constant k on , which describes how many productive collisions occur per
unit time per protein at a given concentration of p X . The rate of complex formation
equals the product of this rate constant k on and the concentrations of A and
p X (see Figure 8–72B). Complex dissociation occurs at a rate k off multiplied by the
concentration of the complex. The rate constant k off can differ by orders of magnitude
for different DNA sequences because it depends on the strength of the noncovalent
bonds formed between A and p X .
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 = k on /k off (Equation 8–1;
Figure 8–72C). K is sometimes called the association constant, K a . The larger this
constant K, the stronger the interaction between A and p X (see Figure 3–44). The
reciprocal of K is the dissociation constant, K d .
To calculate the steady-state concentration of promoter complex using Equation
8–1, we need to account for another complication: both A and p X exist in two
forms—free in solution and bound to each other. In most cases, we know the total
concentration of p X 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 p X ([p X T ]) is the sum of the concentrations of free
([p X ]) and bound ([A:p X ]) forms (Figure 8–72D). This leads to a new equation that
allows us to use [p X T ] to calculate the steady-state concentration of the promoter
complex ([A:p X ]) (Equation 8–2, Figure 8–72D).
Protein A also exists in two forms: free ([A]) and bound to p X ([A:p X ]). In a cell,
there are typically one or two copies of p X (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:p X ] is negligible relative to the total [A T ]. This means
that [A] ≈ [A T ], and we can just plug in the values of total [A T ] in Equation 8–2
without incurring appreciable error in the calculation of [A:p X ].
Now, we are ready to determine the effects of increasing the concentration of
A. Suppose that K = 10 8 M –1 , which is a typical value for many such interactions.
The starting concentration of A is [A T ] = 10 –9 M, and [p X T ] = 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:p X ] to increase 5.5-
fold, 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, p X , 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 p X 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, p X is always free, the bound fraction is zero, and transcription is