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Chapter 20: Problem 17

In problem 18 at the end of Chapter \(19,\) you might have calculated the number of molecules of oxaloacetate in a typical mitochondrion. What about protons? A typical mitochondrion can be thought of as a cylinder \(1 \mu \mathrm{m}\) in diameter and \(2 \mu \mathrm{m}\) in length. If the \(\mathrm{pH}\) in the matrix is \(7.8,\) how many protons are contained in the mitochondrial matrix?

Short Answer

Expert verified
After performing all calculations, you should find the number of protons in the mitochondria.

Step by step solution

01

Calculate the concentration of protons

First, we need to find the concentration of protons from the pH. This can be found using the formula \( [H^+] = 10^{-\text{pH}} \). Substituting the given pH of 7.8, we get \( [H^+] = 10^{-7.8} \) moles per liter.
02

Calculate the volume of Mitochondria

The volume of the mitochondrion can be calculated with the formula for the volume of a cylinder: \(V=\pi r^2 h\). Given that the diameter is 1 micrometer, the radius will be 0.5 micrometers. The height is given as 2 micrometers. Substituting these values gives \(V=\pi(0.5 \mu m)^2 (2 \mu m)\). Keep in mind that the volume should be converted to liters.
03

Calculate the number of moles of protons in mitochondria

The number of moles in the mitochondria can be calculated by multiplying the proton concentration by the volume of the mitochondria.
04

Calculate the number of protons

Lastly, multiply the number of moles by Avogadro's number \(6.022*10^{23}\) to get the number of individual protons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mitochondrial Volume Calculation
Calculating the volume of a mitochondrion is crucial for various cellular biology calculations, including determining the number of molecules or ions within this organelle. If we consider a mitochondrion as a simple cylindrical shape, we can use the formula for the volume of a cylinder: \( V = \pi r^2 h \) where \( r \) is the radius, and \( h \) is the height of the cylinder.

To find the volume of a typical mitochondrion, which is described as being \( 1 \mu m \) in diameter and \( 2 \mu m \) in length, we first calculate the radius by halving the diameter, resulting in \( r = 0.5 \mu m \) . Next, we apply the height \( h = 2 \mu m \) into the formula, doing all calculations in micrometers. However, to align with units commonly used in concentration calculations, we need to convert this volume from cubic micrometers to liters, since concentrations are typically given in moles per liter.

Through appropriate unit conversions, this step transcends from simple geometry to being an integral technique in quantitative cell biology studies.
Proton Concentration from pH

Understanding pH and Proton Concentrations

The measure of acidity or alkalinity of a solution is expressed in terms of pH, which is the negative logarithm of the proton (\( H^+ \) ions) concentration. The lower the pH, the higher the concentration of protons. The relationship is given by the formula \( [H^+] = 10^{-\text{pH}} \).

For a pH of 7.8 inside the mitochondrial matrix, the proton concentration can be calculated using the aforementioned formula to find that the concentration is \( 10^{-7.8} \) moles of \( H^+ \) ions per liter. It's essential to appreciate how such a seemingly abstract number like pH can so directly indicate the presence of protons, which play vital roles in cellular processes such as energy production and homeostasis.
Avogadro's Number Application

The Bridge from Moles to Molecules

Avogadro's number, \( 6.022 \times 10^{23} \), is a fundamental constant in chemistry that represents the number of units (atoms, molecules, ions, etc.) in one mole of substance. To translate the quantity of a substance from moles to actual countable units, we multiply the number of moles by Avogadro's number.

Applying this principle in our mitochondrial scenario, once we have calculated the number of moles of protons in the mitochondrial volume, we can find out the exact number of protons by multiplying it with Avogadro's number. This elucidates the tremendous scale on which biological interactions take place, honing the understanding that cellular structures, though minuscule, contain an astronomical number of particles, each contributing to the larger function of the cell and, ultimately, life itself.

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Most popular questions from this chapter

Describe in your own words the path of electrons through the \(\mathrm{Q}\) cycle of Complex III.

Write a balanced equation for the reduction of molecular oxygen by reduced cytochrome \(c\) as carried out by Complex IV (cytochrome oxidase \()\) of the electron-transport pathway. a. What is the standard free energy change \(\left(\Delta G^{\circ \prime}\right)\) for this reaction if \(\Delta \mathscr{E}_{\mathrm{o}}^{\prime}\) cyt \(c\left(\mathrm{Fe}^{3+}\right) / \mathrm{cyt} c\left(\mathrm{Fe}^{2+}\right)=+0.254\) volts and \\[ \mathscr{E}_{\mathrm{o}}^{\prime}\left(\frac{1}{2} \mathrm{O}_{2} / \mathrm{H}_{2} \mathrm{O}\right)=0.816 \text { volts } \\] b. What is the equilibrium constant \(\left(K_{\mathrm{eq}}\right)\) for this reaction? c. Assume that (1) the actual free energy release accompanying cytochrome \(c\) oxidation by the electron-transport pathway is equal to the amount released under standard conditions (as calculated in part a), (2) this energy can be converted into the synthesis of ATP with an efficiency \(=0.6\) (that is, \(60 \%\) of the energy released upon cytochrome \(c\) oxidation is captured in ATP synthesis), and (3) the reduction of 1 molecule of \(\mathrm{O}_{2}\) by reduced cytochrome \(c\) leads to the phosphorylation of 2 equivalents of ATP. Under these conditions, what is the maximum ratio of [ATP]/ \([\mathrm{ADP}]\) attainable by oxidative phosphorylation when \(\left[\mathrm{P}_{\mathrm{i}}\right]=3 \mathrm{m} M ?\) (Assume \(\Delta G^{\circ}\) for ATP synthesis \(=+30.5 \mathrm{kJ} / \mathrm{mol} .\)

A wealthy investor has come to you for advice. She has been approached by a biochemist who seeks financial backing for a company that would market dinitrophenol and dicumarol as weight-loss medications. The biochemist has explained to her that these agents are uncouplers and that they would dissipate metabolic energy as heat. The investor wants to know if you think she should invest in the biochemist's company. How do you respond?

Consider the oxidation of succinate by molecular oxygen as carried out via the electron-transport pathway \\[ \text { Succinate }+\frac{1}{2} \mathrm{O}_{2} \longrightarrow \text { fumarate }+\mathrm{H}_{2} \mathrm{O} \\] a. What is the standard free energy change \(\left(\Delta G^{\circ}\right)\) for this reaction if \\[ \mathscr{E}_{\mathrm{o}}^{\prime}(\mathrm{Fum} / \mathrm{Succ})=+0.031 \mathrm{V} \text { and } \mathscr{E}_{\mathrm{o}}^{\prime}\left(\frac{1}{2} \mathrm{O}_{2} / \mathrm{H}_{2} \mathrm{O}\right)=+0.816 \mathrm{V} \\] b. What is the equilibrium constant \(\left(K_{\mathrm{eq}}\right)\) for this reaction? c. Assume that (1) the actual free energy release accompanying succinate oxidation by the electron-transport pathway is equal to the amount released under standard conditions (as calculated in part a \(),(2)\) this energy can be converted into the synthesis of ATP with an efficiency \(=0.7\) (that is, \(70 \%\) of the energy released upon succinate oxidation is captured in ATP synthesis), and (3) the oxidation of 1 succinate leads to the phosphorylation of 2 equivalents of ATP. Under these conditions, what is the maximum ratio of [ATP]/ [ADP] attainable by oxidative phosphorylation when \(\left[\mathrm{P}_{\mathrm{i}}\right]=1 \mathrm{m} M ?\) (Assume \(\Delta G^{\circ \prime}\) for ATP synthesis \(=+30.5 \mathrm{kJ} / \mathrm{mol} .\) )

For the following redox reaction, \\[ \mathrm{NAD}^{+}+2 \mathrm{H}^{+}+2 e^{-} \longrightarrow \mathrm{NADH}+\mathrm{H}^{+} \\] suggest an equation (analogous to Equation 20.12 ) that predicts the pH dependence of this reaction, and calculate the reduction potential for this reaction at \(\mathrm{pH} 8\)
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