Chapter 5

Protein A has a binding site for ligand \(\mathrm{X}\) with a \(K_{\mathrm{d}}\) of \(3.0 \times 10^{-7}\) ?. Protein \(\mathrm{B}\) has a binding site for ligand \(\mathrm{X}\) with a \(K_{\mathrm{d}}\) of \(4.0 \times 10^{-8} \mathrm{M}\). Calculate the \(K_{\mathrm{a}}\) for each protein. Which protein has a higher affinity for ligand X? Explain your reasoning.

Which of these situations would produce a Hill plot with \(n_{\mathrm{H}}<1.0\) ? Explain your reasoning in each case. a. The protein has multiple subunits, each with a single ligand-binding site. Ligand binding to one site decreases the binding affinity of other sites for the ligand. b. The protein is a single polypeptide with two ligandbinding sites, each having a different affinity for the ligand. c. The protein is a single polypeptide with a single ligand-binding site. As purified, the protein preparation is heterogeneous, containing some protein molecules that are partially denatured and thus have a lower binding affinity for the ligand. d. The protein has multiple subunits, each with a single ligand-binding site. Ligands bind independently to each site, do not affect the binding affinity of other sites, and bind with identical affinities.

The protein calcineurin binds to the protein calmodulin with an association rate of \(8.9 \times 10^{3} \mathrm{M}^{-1} \mathrm{~s}^{-1}\) and an overall dissociation constant, \(K_{\mathrm{d}}\), of 10 \(\mathrm{n} \mathrm{M}\). Calculate the dissociation rate, \(k_{\mathrm{d}}\), including appropriate units.

The \(E\). coli nickel-binding protein binds to its ligand, \(\mathrm{Ni}^{2+}\), with a \(K_{\mathrm{d}}\) of \(100 \mathrm{~nm}\). Calculate the \(\mathrm{Ni}^{2+}\) concentration when the fraction of binding sites occupied by the ligand \((Y)\) is (a) \(0.25\), (b) \(0.6\), (c) \(0.95 .\)

Under appropriate conditions, hemoglobin dissociates into its four subunits. The isolated \(a\) subunit binds oxygen, but the \(\mathrm{O}_{2}\)-saturation curve is hyperbolic rather than sigmoid. In addition, the binding of oxygen to the isolated \(a\) subunit is not affected by the presence of \(\mathrm{H}^{+}, \mathrm{CO}_{2}\), or BPG. What do these observations indicate about the source of the cooperativity in hemoglobin?

Studies of oxygen transport in pregnant mammals show that the \(\mathrm{O}_{2}\) saturation curves of fetal and maternal blood are markedly different when measured under the same conditions. Fetal erythrocytes contain a structural variant of hemoglobin, HbF, consisting of two \(a\) and two \(\gamma\) subunits \(\left(\alpha_{2} \gamma_{2}\right)\), whereas maternal erythrocytes contain \(\mathrm{HbA}\left(\alpha_{2} \beta_{2}\right)\). a. Which hemoglobin has a higher affinity for oxygen under physiologic conditions? b. What is the physiological significance of the different \(\mathrm{O}_{2}\) affinities? When all the BPG is carefully removed from samples of \(\mathrm{HbA}\) and \(\mathrm{HbF}\), the measured \(\mathrm{O}_{2}\)-saturation curves (and consequently the \(\mathrm{O}_{2}\) affinities) are displaced to the left. However, HbA now has a greater affinity for oxygen than does HbF. When BPG is reintroduced, the \(\mathrm{O}_{2}\)-saturation curves return to normal, as shown in the graph. c. What is the effect of BPG on the \(\mathrm{O}_{2}\) affinity of hemoglobin? How can this information be used to explain the different \(\mathrm{O}_{2}\) affinities of fetal and maternal hemoglobin?

A team of biochemists uses genetic engineering to modify the interface region between hemoglobin subunits. The resulting hemoglobin variants exist in solution primarily as \(a \beta\) dimers (few, if any, \(\alpha_{2} \beta_{2}\) tetramers form). Are these variants likely to bind oxygen more weakly or more tightly? Explain your answer.

An antibody with high affinity for its antigen has a \(K_{\mathrm{d}}\) in the low nanomolar range. Assume an antibody binds an antigen with a \(K_{\mathrm{d}}\) of \(5 \times 10^{-8}\) M. Calculate the antigen concentration when \(Y\), the fraction of binding sites occupied by the ligand, is a. \(0.4\), b. \(0.5\), c. \(0.8\), d. \(0.9\).

A monoclonal antibody binds to Gactin but not to F-actin. What does this tell you about the epitope recognized by the antibody?

Some pathogens have developed mechanisms to evade the immune system, making it difficult or impossible to develop effective vaccines against them. a. African sleeping sickness is caused by a protozoan called Trypanosoma brucei, carried by the tsetse fly. The trypanosome surface is dominated by one coat protein, the variable surface glycoprotein (VSG). The trypanosome genome encodes over 1,000 different versions of VSG. All of the cells in an initial infection feature the same VSG coat on their surfaces, and this is readily recognized as foreign by the immune system. However, an individual trypanosome in the broader population will switch and randomly begin expressing a different variant of the VSG coat. All the descendants of that cell will have the new and different protein on their surface. As the population with the second VSG coat increases, an individual cell will then switch to a third VSG protein coat, and so on. b. The human immunodeficiency virus (HIV) has an error-prone system for replicating its genome, effectively introducing mutations at an unusually high rate. Many of the mutations affect the viral protein coat. Describe how each pathogen can survive the immune response of its host.