Genes VII

Probably all proteins that have a high affinity for a specific sequence also possess a low affinity for any (random) DNA sequence. A large number of low-affinity sites will compete just as well for a repressor tetramer as a small number of high-affinity sites. There is only one high-affinity site in the E. coli genome: the operator. The remainder of the DNA provides low-affinity binding sites. Every base pair in the genome starts a new low-affinity site. (Just moving one base pair along the genome, out of phase with the operator itself, creates a low-affinity site!) So there are 4.2 10⁶ low-affinity sites.

The large number of low-affinity sites means that, even in the absence of a specific binding site, all or virtually all repressor is bound to DNA; none is free in solution.

We may describe the binding of repressor to DNA by the equilibrium:

in which [Repressor-DNA] is the concentration of repressor bound to DNA, [Free repressor] is the concentration of free repressor, and [DNA] is the concentration of nonspecific binding sites.

What proportion of total repressor protein is free? By rearranging the equation, we see that the distribution of repressor is given by:

Applying the parameters for the lac system, we find that:

• The nonspecific equilibrium binding constant is K_A = 2 10⁶ M^-1.
• The concentration of nonspecific binding sites is 4 10⁶ in a bacterial volume of 10 ^V15 liter, which corresponds to [DNA] = 7 10 ^V3 M (a very high concentration).

Substituting these values gives:

Free : Bound repressor = 10 ^V4.

So all but 0.01% of repressor is bound to (random) DNA. Since there are ~10 molecules of repressor per cell, this is tantamount to saying that there is no free repressor protein. This has an important implication for the interaction of repressor with the operator: it means that we are concerned with the partitioning of the repressor on DNA, in which the single high-affinity site of the operator competes with the large number of low-affinity sites.

In this competition, the absolute values of the association constants for operator and random DNA are not important; what is important is the ratio of K_sp (the constant for binding a specific site) to K_nsp (the constant for binding any random DNA sequence), that is, the specificity.

We can define the parameters that influence the ability of a regulator protein to saturate its target site by comparing the equilibrium equations for specific and nonspecific binding. As might be expected intuitively, the important parameters are:

• The size of the genome dilutes the ability of a protein to bind specific target sites.
• The specificity of the protein counters the effect of the mass of DNA.
• The amount of protein that is required increases with the total amount of DNA in the genome and decreases with the specificity.
• The amount of protein also must be in reasonable excess of the total number of specific target sites, so we expect regulators with many targets to be found in greater quantities than regulators with individual targets.

Figure 10.17 compares the equilibrium constants for lac repressor/operator binding with repressor/general DNA binding. From these constants, we can deduce how repressor is partitioned between the operator and the rest of DNA, and what happens to the repressor when inducer causes it to dissociate from the operator.

Repressor binds ~10⁷ times better to operator DNA than to any random DNA sequence of the same length. So the operator comprises a single high-affinity site that will compete for the repressor 10⁷ better than any low-affinity (random) site. How does this ensure that the repressor can maintain effective control of the operon?

Using the specificity, we can calculate the distribution between random sites and the operator, and can express this in terms of occupancy of the operator. If there are 10 molecules of lac repressor per cell with a specificity for the operator of 10⁷, the operator will be bound by repressor 96% of the time. The role of specificity explains two features of the lac repressor-operator interaction:

• When inducer binds to the repressor, the affinity for the operator is reduced by ~10³-fold. The affinity for general DNA sequences remains unaltered. So the specificity is now only 10⁴, which is insufficient to capture the repressor against competition from the excess of 4.2 10⁶ low-affinity sites. Only 3% of operators would be bound under these conditions.
• Mutations that reduce the affinity of the operator for the repressor by as little as 20-30 have sufficient effect to be constitutive. Within the genome, the mutant operators can be overwhelmed by the preponderance of random sites. The occupancy of the operator is reduced to ~50% if the repressor’s specificity is reduced just 10 .

The consequence of these affinities is that in an uninduced cell, one tetramer of repressor usually is bound to the operator. All or almost all of the remaining tetramers are bound at random to other regions of DNA, as illustrated in Figure 10.18. There are likely to be very few or no repressor tetramers free within the cell.

The addition of inducer abolishes the ability of repressor to bind specifically at the operator. Those repressors bound at the operator are released, and bind to random (low-affinity) sites. So in an induced cell, the repressor tetramers are "stored" on random DNA sites. In a noninduced cell, a tetramer is bound at the operator, while the remaining repressor molecules are bound to nonspecific sites. The effect of induction is therefore to change the distribution of repressor on DNA, rather than to generate free repressor.

When inducer is removed, repressor recovers its ability to bind specifically to the operator, and does so very rapidly. This must involve its movement from a nonspecific "storage" site on DNA. What mechanism is used for this rapid movement? The ability to bind to the operator very rapidly is not consistent with the time that would be required for multiple cycles of dissociation and reassociation with nonspecific sites on DNA. The discrepancy excludes random-hit mechanisms for finding the operator, suggesting that the repressor can move directly from a random site on DNA to the operator. This is the same issue that we encountered previously with the ability of RNA polymerase to find its promoters (see Figure 9.12). The same solution is likely: movement could be accomplished by direct displacement from site to site (as indicated in Figure 10.18). A displacement reaction might be aided by the presence of more binding sites per tetramer (four) than are actually needed to contact DNA at any one time (two).

The parameters involved in finding a high-affinity operator in the face of competition from many low-affinity sites pose a dilemma for repressor. Under conditions of repression, there must be high specificity for the operator. But under conditions of induction, this specificity must be relieved. Suppose, for example, that there were 1000 molecules of repressor per cell. Then only 0.04% of operators would be free under conditions of repression. But upon induction only 40% of operators would become free. We therefore see an inverse correlation between the ability to achieve complete repression and the ability to relieve repression effectively. We assume that the number of repressors synthesized in vivo has been subject to selective forces that balance these demands.

The difference in expression of the lactose operon between its induced and repressed states in vivo is actually 10³ . In other words, even when inducer is absent, there is a basal level of expression of ~0.1% of the induced level. This would be reduced if there were more repressor protein present, increased if there were less. So it could be impossible to establish tight repression if there were fewer repressors than the 10 found per cell; and it might become difficult to induce the operon if there were too many (Lin and Riggs, 1975).

In order to extrapolate in vivo from the affinity of a DNA-protein interaction in vitro, we need to know the effective concentration of DNA in vivo. The "effective concentration" differs from the mass/volume because of several factors. The effective concentration is increased, for example, by molecular crowding, which occurs when polyvalent cations neutralize ~90% of the charges on DNA, and the nucleic acid collapses into condensed structures. The major force that decreases the effective concentration is the inaccessibility of DNA that results from occlusion or sequestration by DNA-binding proteins.

One way to determine the effective concentration is to compare the rate of a reaction in vitro and in vivo that depends on DNA concentration. This has been done using intermolecular recombination between two DNA molecules. To provide a control, the same reaction is followed as an intramolecular recombination, that is, the two recombining sites are presented on the same DNA molecule. We assume that concentration is the same in vivo and in vitro for the intramolecular reaction, and therefore any difference in the ratio of intermolecular/ intramolecular recombination rates can be attributed to a change in the effective concentration in vivo. The results of such a comparison suggest that the effective concentration of DNA is reduced >10-fold in vivo (Hildebrandt et al., 1995).

This could affect the rates of reactions that depend on DNA concentration, including DNA recombination, and protein-DNA binding. It emphasizes the problem encountered by all DNA-binding proteins in finding their targets with sufficient speed, and reinforces the conclusion that diffusion is not adequate (see Figure 9.12).