Chapter 19, Problem 10P

(a)

Summary Introduction

To calculate:

The hydrogen ions concentration in external medium and in matrix if the $\text{pH}$ of medium is $7.4$ and $\text{pH}$ of matrix is $7.7$.

Introduction:

Protons are translocated to external medium from mitochondrial matrix. This results in establishment of $\text{pH}$ gradient through out the inner mitochondrial membrane. The driving force for synthesis of ATP (adenosine triphosphate) is provided by the diffusion of protons back into matrix.

Explanation of Solution

Explanation:

$\text{pH}$ of a solution refers to the concentration of hydrogen ions released by it. The formula for calculating $\text{pH}$ is as follows:

$\text{pH}=-\text{log}\left[{\text{H}}^{\text{+}}\right]$

The $\text{pH}$ of the matrix is $7.4$. Substitute the value of $\text{pH}$ into the above formula, and calculate concentration of hydrogen ions. The concentration of ${\text{H}}^{\text{+}}$ in the external medium is as follows:

$\begin{array}{c}-\text{log}\left[{\text{H}}_{\text{external medium}}^{\text{+}}\right]=\text{7}\text{.4}\\ \left[{\text{H}}_{\text{external medium}}^{\text{+}}\right]={\text{10}}^{-\text{7}\text{.4}}\\ \left[{\text{H}}_{\text{external medium}}^{\text{+}}\right]=\text{3}\text{.98}\times {\text{10}}^{-\text{8}}\\ \sim \text{4}\times {\text{10}}^{-\text{8}}\text{ M(Molar)}\end{array}$

The concentration of ${\text{H}}^{\text{+}}$ in the external medium is $\text{4}\times {\text{10}}^{-\text{8}}\text{ M}$. The concentration of ${\text{H}}^{\text{+}}$ in the matrix is as follows:

$\begin{array}{c}-\text{log}\left[{\text{H}}_{\text{matrix}}^{\text{+}}\right]=\text{7}\text{.7}\\ \left[{\text{H}}_{\text{matrix}}^{\text{+}}\right]={\text{10}}^{-\text{7}\text{.7}}\\ \left[{\text{H}}_{\text{matrix}}^{\text{+}}\right]=1.99\times {\text{10}}^{-\text{8}}\\ \sim 2\times {\text{10}}^{-\text{8}}\text{ M}\end{array}$

The concentration of ${\text{H}}^{\text{+}}$ in the matrix is $2\times {\text{10}}^{-\text{8}}\text{ M}$.

Conclusion

Conclusion:

The concentration of ${\text{H}}^{\text{+}}$ in the external medium is $4\times 10^{-8}M$. The concentration of ${\text{H}}^{\text{+}}$ in the matrix is $2\times 10^{-8}M$.

(b)

Summary Introduction

To calculate:

Outside to inside ratio of ${\text{H}}^{\text{+}}$ and energy inherent in the difference in concentration.

Introduction:

The concentration of hydrogen ions reflects $\text{pH}$. The protons are translocated to external medium from matrix. This establishes the proton gradient. As a result of the proton gradient energy is developed which is used to drive the synthesis of ATP.

Explanation of Solution

Explanation:

The concentration of ${\text{H}}^{\text{+}}$ in the external medium is $\text{4}\times {\text{10}}^{-\text{8}}\text{ M}$. The concentration of ${\text{H}}^{\text{+}}$ in the matrix is $2\times {\text{10}}^{-\text{8}}\text{ M}$.

The ratio of the hydrogen ions concentration in the external medium and matrix can be calculated using the following formula:

$\text{Ratio of external }\left[{\text{H}}^{\text{+}}\right]\text{ to internal }\left[{\text{H}}^{\text{+}}\right]=\frac{\text{External }\left[{\text{H}}^{\text{+}}\right]}{\text{Internal }\left[{\text{H}}^{\text{+}}\right]}$

Substitute values of hydrogen ions concentration in the external medium and hydrogen ions concentration in the matrix. The ratio of the concentration of hydrogen ions in the external medium and matrix is as follows:

$\begin{array}{c}\text{Ratio of external }\left[{\text{H}}^{\text{+}}\right]\text{ to internal }\left[{\text{H}}^{\text{+}}\right]=\frac{\text{4}\times {\text{10}}^{-\text{8}}\text{ M}}{2\times {\text{10}}^{-\text{8}}\text{ M}}\\ =2\end{array}$

Free energy inherent in this concentration difference can be calculated using the following formula

$\Delta G=RT\ln \left[\frac{C_2}{C_1}\right]$

Where, $\Delta G$ is Gibb’s free energy

R is gas constant $\left(\text{R}=\text{8}\text{.315 Joule/mole}\cdot \text{Kelvin}\right)$

T is absolute temperature $\text{T}=\text{298 K (Kelvin)}$

Substitute values of C₂ and C₁ in the above equation. The inherent energy of the reaction will be as follows:

$\begin{array}{c}\Delta G=\text{8}\text{.315 J/mol}\cdot \text{K}\times \text{298 Kelvin}\times \ln 2\\ =-1,71\text{7 J/mol}\\ =-1.7\text{ kJ/mol (Kilojoule/mole)}\end{array}$

Negative sign of free energy reflects that the inner transmembrane potential is negative. So energy inherent in the concentration difference is $-1.7\text{ kJ/mol}$.

Conclusion

Conclusion:

Energy inherent in the concentration difference is $-1.7kJ/mol$.

(c)

Summary Introduction

To calculate:

The number of protons in actively respiring mitochondrion of liver given that diameter is $\text{1}\text{.5 μm (micronmeter)}$

Introduction:

The number of protons in respiring mitochondria depends on the volume of the mitochondrion. More the diameter of mitochondrion, more will the protons liberated by it.

Explanation of Solution

Explanation:

The diameter of the mitochondrion is $\text{1}\text{.5 μm}$, which can be used to calculate the volume of mitochondrion. The volume of mitochondrion can be calculated using the formula given below:

$\text{Volume of sphere}=\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}$

Where, π is $3.14$

r is radius of mitochondrion $\text{r}=\raisebox{1ex}{$\text{diameter}$}\!\left/ \!\raisebox{-1ex}{$\text{2}$}\right.$.

Substitute value of r in the formula given above to calculate the volume of mitochondrion. The volume of mitochondrion can be calculated as follows:

$\begin{array}{c}\text{Volume of sphere}=\frac{\text{4}}{\text{3}}\times \text{3}\text{.14}\times {\left(\frac{\text{1}\text{.5}}{\text{2}}\text{ μm}\right)}^3\\ =\frac{\text{4}}{\text{3}}\times \text{3}\text{.14}\times {\left(0.75\times {10}^{-3}\right)}^3\\ =1.77\times {10}^{-6}{\text{ μm}}^3\\ \sim 1.77\times {10}^{-15}\text{ L}\end{array}$

Moles of protons in a respiring mitochondrion can be calculated by the following formula:

$\text{Moles of protons in a respiring mitochondrion}=\left[{\text{H}}^{\text{+}}\right]\times \text{Volume of mitochondrion}$

Substitute value of $\left[{\text{H}}^{\text{+}}\right]$ and volume of mitochondrion in the above formula. The moles of proton can be calculated as follows:

$\begin{array}{c}\text{Moles of protons in a respiring mitochondrion}=2\times {\text{10}}^{-\text{8}}\text{ M}\times 1.77\times {10}^{-15}\text{ L}\\ =\text{3}\text{.54}\times {10}^{-23}\text{ moles}\end{array}$

Number of protons in a respiring mitochondrion can be calculated by using the following formula:

$\begin{array}{c}\text{Number of protons in a respiring mitochondrion}=\text{moles of proton}\times \text{6}\text{.022}\times {\text{10}}^{\text{23}}\\ =\text{3}\text{.54}\times {10}^{-23}\text{ moles}\times \text{6}\text{.022}\times {\text{10}}^{\text{23}}\\ =21.31\\ \sim 21\end{array}$

So, the number of protons in the respiring liver is $21$.

Conclusion

Conclusion:

So, the number of protons in the respiring liver is $21$.

(d)

Summary Introduction

To predict:

Whether the $\text{pH}$ gradient is enough to generate ATP.

Introduction:

The protons are transported from the matrix to external medium. This creates a proton gradient across the inner mitochondrial membrane. The flow of proton from external matrix into matrix derives the synthesis of ATP.

Explanation of Solution

Explanation:

For an actively respiring cell, energy inherent in the concentration difference is $-1.7\text{ kJ/mol}$. For synthesis of ATP from ADP approximately ∆G should be $-\text{30}\text{.5}\text{kJ/mol}$. So the energy is not enough to drive the synthesis of ATP.

Conclusion

Conclusion:

The free energy of the difference in the concentration of hydrogen ions in external medium and matrix is $-1.7\text{ kJ/mol}$ which is not enough to drive the synthesis of ATP.

(e)

Summary Introduction

To suggest:

The way through which necessary energy is provided by ATP synthesis.

Introduction:

The protons are transported from the matrix to external medium. This creates a proton gradient across the inner mitochondrial membrane. The flow of proton from external matrix into matrix derives the synthesis of ATP.

Explanation of Solution

Explanation:

The concentration gradient is not alone enough for driving the synthesis of ATP. So, the sum of energy generated by charge separation and concentration gradient is used to drive the synthesis.

Conclusion

Conclusion:

The overall transmembrane electric potential is developed by concentration gradient and charge separation which together drive the synthesis of ATP.