Protein B has a higher affinity for ligand X; it is half-saturated at a much lower concentration of X than is protein A. Protein A has K_a = 10⁶ m⁻¹; protein B has K_a = 10⁹ m⁻¹.

The K_d increases when allolactose is bound. In other words, the affinity of the Lac repressor for its DNA binding site decreases, leading to its dissociation from the DNA.

The K_d for Lac repressor binding to the operator is likely to increase, as the changes in amino acid sequence are likely to disrupt one or more interactions between the protein and its DNA binding site.

DNA is a polyelectrolyte, and the many negatively charged groups in its backbone are bound by cations, primarily Mg²⁺ ions but also monovalent ions such as K⁺. Binding of a protein to DNA involves displacement of some or all of the ions at the DNA site where binding occurs.

S-5

Nonspecific DNA binding generally involves interactions with the invariant parts of DNA structure—the phosphate and deoxyribose groups in the DNA backbone—and hydrophobic interactions with the nucleotide bases. Binding to a specific DNA sequence requires substantial interaction with groups in the bases that distinguish one nucleotide from another. These features of each base are largely accessible in the major and minor grooves of the DNA.

All of these situations give rise to real or apparent negative cooperativity. Apparent negative cooperativity in ligand binding can be caused by the presence of two or more different types of ligand-binding sites with different affinities for the same ligand on the same or different protein molecules in the same solution. Apparent negative cooperativity can also be observed in heterogeneous protein preparations. There are few well-documented cases of true negative cooperativity.

2.4 × 10⁻⁶ m.

Details: The volume of a cylinder is given by V = πr²h. For a cylinder of r = 0.000050 cm and h = 0.00020 cm, V = 1.57 × 10⁻¹² cc (mL), or 1.57 × 10⁻¹⁵ L. The solution concentration in the cell is 1.20 g/mL, and the protein concentration is 20% of this, or 0.24 g/mL. Thus the cell contains 3.77 × 10⁻¹³ g of protein. If the cell contains 1,000 proteins of the same molecular weight, it contains 3.77 × 10⁻¹⁶ g of a given protein, or 0.24 g per liter of cytosol. Dividing 0.24 g/L by the 100,000 g/L of a 1 m solution gives a concentration of 2.4 × 10⁻⁶ m. This corresponds to just over 2,200 copies of each protein in the cell.

(b), (e), and (g).

We now know that to catalyze a reaction, an enzyme active site must be complementary (in shape and charge) not to the substrate but to the transition state of the reaction that is catalyzed.

The inactivated enzyme will no longer have a measurable k_cat and K_m.

(a) [S] = 1.7 × 10⁻³ m. (b) 0.33V_max, 0.67V_max, and 0.91V_max. (c) The upper (red) curve corresponds to enzyme B ([X] > K_m for this enzyme); the lower (black) curve, enzyme A.

(a) k_cat = 400 s⁻¹. (b) K_m = 10 μm. (c) α = 2, α′ = 3. (d) ANGER is a mixed inhibitor.

(a) [E_t] = 24 ηm. (b) [A] = 4 μm (V₀ is exactly ½V_max, so [A] = K_m). (c) [A] = 40 μm (V₀ is exactly ½V_max, so [A] = 10K_m in the presence of inhibitor).

V_max ≈ 140 mm min⁻¹; K_m ≈ 1 × 10⁻⁵ m.

Movement along the DNA requires ATP hydrolysis. A normal RuvB subunit can still hydrolyze ATP, even if it is adjacent to a mutant subunit in a heterohexameric complex. However, movement along DNA requires cooperation between adjacent subunits, which cannot occur if one of the subunits is mutated.

In principle, ATM and ATR could be enzymes that covalently modify other proteins. In fact, ATM and ATR are the most common type of such proteins: they are protein kinases that add phosphoryl groups to hundreds of cellular protein targets.

It is likely that the helicase is present in one fraction, and another protein or macromolecule needed to activate the helicase is present in the other fraction.