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Chapter 32: Problem 9

A typical hormone binds to its receptor with an affinity \(\left(K_{\mathrm{D}}\right)\) of approximately \(1 \times 10^{-9} M\). Consider an in vitro (test-tube) system in which the total hormone concentration is approximately 1 n \(M\) and the total concentration of receptor sites is \(0.1 \mathrm{n} M\). What fraction of the receptor sites is bound with hormone? If the concentration of receptors is decreased to \(0.033 \mathrm{n} M\), what fraction of receptor is bound with the hormone?

Short Answer

Expert verified
When the total concentration of receptor sites is 0.1 nM, the fraction of the receptor sites bound with hormone is 0.5 or 50%. When the concentration is decreased to 0.033 nM, the fraction of the receptor sites bound with hormone increases to approximately 0.75 (or 75%).

Step by step solution

01

Calculate Fraction of Receptor Sites Bound with Hormone for 0.1 nM

Firstly, it's important to convert concentration units to be consistent. The total hormone concentration is 1 nM which equals \(1 \times 10^{-9}M\). The concentration of receptor sites is also given in nM (0.1), which equals \(0.1 \times 10^{-9}M\). Now, substitute these values into the formula to find the fraction of the receptor sites bound: \(\frac{[\text{Hormone}]}{[\text{Hormone}] + K_D} = \frac{1 \times 10^{-9}}{1 \times 10^{-9} + 1 \times 10^{-9}}\)
02

Solve the fraction

Perform the calculation to find the fraction: \(= \frac{1 \times 10^{-9}}{2 \times 10^{-9}} = 0.5\). It means 50% of the receptor sites are bound with hormone.
03

Calculate Fraction of Receptor Sites Bound with Hormone for 0.033 nM

The concentration of receptors is now decreased to 0.033 nM, which equals \(0.033 \times 10^{-9} M\). Substitute into the formula to find the new fraction of the receptor sites bound: \(\frac{[\text{Hormone}]}{[\text{Hormone}] + K_D} = \frac{1 \times 10^{-9}}{1 \times 10^{-9} + 0.33 \times 10^{-9}}\)
04

Solve the fraction

Perform the calculation to find the fraction: \(\frac{1 \times 10^{-9}}{1.33 \times 10^{-9}} = 0.7518796992481203\). Approximately, 75% of the receptor sites are bound with hormone when the receptor concentration is decreased to 0.033 nM.

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