Chapter 35, Problem 19P

Interpretation Introduction

(a)

Interpretation:

The dissociation constant of protein and antibody A is to be calculated.

Concept introduction:

The protein is made up of different amino acids. The amino acid is the smallest unit of a protein. The protein may be composed of primary, secondary, tertiary and quarternary structure. The antibody is also a type of protein which is derived by plasma cells and it is used by the immune system of the cells.

Answer to Problem 19P

The dissociation constant of a complex between protein and antibody A is ${10}^{-7}\text{M}$.

Explanation of Solution

The dissociation constant is defined as the breaking of larger molecules into its smaller units or it is defined as the ratio of products to the reactants with respect to their stoichiometric number.

The reaction corresponding to the dissociation of PA is given below.

$\text{PA}\to \text{P}+\text{A}$

The dissociation constant of the above equation is shown below.

$K_{\text{d}}=\frac{\left[\text{P}\right]\left[\text{A}\right]}{\left[\text{PA}\right]}$    (1)

The value of dissociation constant is obtained when half of the complex is dissociated as given below.

$K_{\text{d}}=\frac{1}{2}\times$ maximum number of binding sites

$\begin{array}{c}{K}_{\text{d}}=\frac{1}{2}\times 1\\ =0.5\end{array}$

This is the concentration of protein which is obtained and the concentration of the complex which is dissociated.

The concentration of antibody A is obtained from the graph shown below.

Figure 1

From the graph above.

$\begin{array}{c}\log \left[\text{antibody}\right]=-7\\ \left[\text{antibody}\right]={10}^{-7}\text{M}\end{array}$

Substitute values of $\left[\text{P}\right]=0.5\text{M}$, $\left[\text{PA}\right]=0.5\text{M}$ and $\left[\text{A}\right]={10}^{-7}\text{M}$ in equation (1) as shown below.

$\begin{array}{c}{K}_{\text{d}}=\frac{\left[\text{P}\right]\left[\text{A}\right]}{\left[\text{PA}\right]}\\ =\frac{\left[0.5\text{M}\right]\left[{10}^{-7}\text{M}\right]}{\left[0.5\text{M}\right]}\\ ={10}^{-7}\text{M}\end{array}$

Therefore the dissociation constant the complex between protein and antibody A is ${10}^{-7}\text{M}$.

Interpretation Introduction

(b)

Interpretation:

The dissociation constant of protein and antibody B is to be calculated.

Concept introduction:

The protein is made up of different amino acids. The amino acid is the smallest unit of a protein. The protein may be composed of primary, secondary, tertiary and quarternary structure. The antibody is also a type of protein which is derived by plasma cells and it is used by the immune system of the cells.

Answer to Problem 19P

The dissociation constant of a complex between protein and antibody B is ${10}^{-9}\text{M}$.

Explanation of Solution

The dissociation constant is defined as the breaking of larger molecules into its smaller units or it is defined as the ratio of products to the reactants with respect to their stoichiometric number.

Consider a reaction given below.

$\text{PB}\to \text{P}+\text{B}$

The dissociation constant of the above equation is shown below.

$K_{\text{d}}=\frac{\left[\text{P}\right]\left[\text{B}\right]}{\left[\text{PB}\right]}$    (2)

The value of dissociation constant is obtained when half of the complex is dissociated as given below.

$K_{\text{d}}=\frac{1}{2}\times$ maximum number of binding sites

$\begin{array}{c}{K}_{\text{d}}=\frac{1}{2}\times 1\\ =0.5\end{array}$

This is the concentration of protein which is obtained and the concentration of the complex which is dissociated.

The concentration of antibody B is obtained from the graph shown below.

Figure 2

From the graph above, the concentration of the antibody is calculated as follows:

$\begin{array}{c}\log \left[\text{antibody}\right]=-9\\ \left[\text{antibody}\right]={10}^{-9}\text{M}\end{array}$

Substitute values of $\left[\text{P}\right]=0.5\text{M}$, $\left[\text{PB}\right]=0.5\text{M}$ and $\left[\text{B}\right]={10}^{-9}\text{M}$ in equation (1) as shown below.

$\begin{array}{c}{K}_{\text{d}}=\frac{\left[\text{P}\right]\left[\text{B}\right]}{\left[\text{PB}\right]}\\ =\frac{\left[0.5\text{M}\right]\left[{10}^{-9}\text{M}\right]}{\left[0.5\text{M}\right]}\\ ={10}^{-9}\text{M}\end{array}$

Therefore the dissociation constant the complex between protein and antibody B is ${10}^{-9}\text{M}$.

On comparing, it is found that antibody A and B are identical except for one single amino acid. The reason for the generation of antibody B is through gene mutation which permanentlychanges the DNA sequence of a species. It may be hereditary or acquired mutation. Since in this case only one amino acid changes.

Therefore, the mutation is a point mutationthat generated antibody B.