Chapter 7, Problem 14P

Interpretation Introduction

(a)

Interpretation:

An oxygen-binding curve for a hypothetical two subunit hemoglobin with $n=1.8$ and $P_{50}=10\text{ torr}$ should be plotted.

Concept introduction:

Hill equation is represented as follows:

$\log \left(\frac{Y}{1-Y}\right)=n\log \left(p{\text{O}}_2\right)-n\log (P_{50})$

Here, Y- fractional saturation

n − a measure of the degree of cooperativity in ligand binding.

P₅₀ − partial pressure of oxygen at which hemoglobin is half saturated.

The Hill plot of log(Y/1-Y) versus log(P₅₀) should be a linear graph with slope of n.

Answer to Problem 14P

Explanation of Solution

The values of $n=1.8$ and $P_{50}=10\text{ torr}$ are substituted to the Hill equation and the partial pressure of oxygen at different fractional saturation values is calculated. A plot of fractional saturation versus oxygen partial pressure is drawn.

Y	log(Y/(1-Y)	log(Y/(1-Y)-n log (P50)	log (pO2)	pO2
0.1	-0.95424251	0.845757491	0.469865	2.950294
0.2	-0.60205999	1.197940009	0.665522	4.629374
0.3	-0.36797679	1.432023215	0.795568	6.245518
0.4	-0.17609126	1.623908741	0.902172	7.983099
0.5	0	1.8	1	10
0.6	0.176091259	1.976091259	1.097828	12.52646
0.7	0.367976785	2.167976785	1.204432	16.01148
0.8	0.602059991	2.402059991	1.334478	21.60119
0.9	0.954242509	2.754242509	1.530135	33.89493

Interpretation Introduction

(b)

Interpretation:

An oxygen-binding curve for shypothetical two subunit hemoglobin with $n=2,\text{ L = 1000, c = 0}{\text{.01 and K}}_R=1\text{ torr}$ should be plotted using concerted model

Concept introduction:

Concerted model equation is written as follows:

$Y=\frac{\alpha {\left(1+\alpha \right)}^{n-1}+Lc\alpha {\left(1+c\alpha \right)}^{n-1}}{{\left(1+\alpha \right)}^n+L{\left(1+c\alpha \right)}^n}$

Here, Y − fractional saturation

a − ratio between the substrate concentration and the dissociation constant for a ligand binding to a single site in R state.

L − The ratio of the concentrations of the T and R states with no ligands bound

c − the ratio between the dissociation constant for a ligand binding to a single site in R state and that of T state.

n − number of binding sites

Also, the ratio can be calculated as follows:

$\alpha =\frac{p{O}_2}{K_R}$

Here,

pO₂ − partial pressure of oxygen

K_R - the dissociation constant for a ligand binding to a single site in R state.

Answer to Problem 14P

Explanation of Solution

$\begin{array}{l}Y=\frac{\alpha {\left(1+\alpha \right)}^{2-1}+1000\times 0.01\alpha {\left(1+0.01\alpha \right)}^{2-1}}{{\left(1+\alpha \right)}^2+1000{\left(1+0.01\alpha \right)}^2}\\ Y=\frac{\alpha +{\alpha }^2+10\alpha +0.1{\alpha }^2}{1+2\alpha +{\alpha }^2+1000+20\alpha +0.1{\alpha }^2}\\ Y=\frac{11\alpha +1.1{\alpha }^2}{1001+22\alpha +1.1{\alpha }^2}\end{array}$

When $Y=0.1$

$\begin{array}{l}0.1=\frac{11\alpha +1.1{\alpha }^2}{1001+22\alpha +1.1{\alpha }^2}\\ 11\alpha +1.1{\alpha }^2=100.1+2.2\alpha +0.11{\alpha }^2\\ 0.99{\alpha }^2+8.8\alpha -100.1=0\\ \alpha =6.55\end{array}$

$\begin{array}{l}\alpha =\frac{p{O}_2}{K_R}\\ 6.55=\frac{p{O}_2}{1}\\ p{O}_2=6.55\text{ torr}\end{array}$

Likewise, partial pressure of oxygen for several fractional saturation values are calculated and a plot of fractional saturation versus oxygen partial pressure is drawn.

Y	pO2
0.1	6.55
0.3	17.09
0.5	30.15
0.7	53.21
0.9	138.91