a. What is the standard free energy change \(\left(\Delta G^{\circ}\right)\) for the reduction of coenzyme \(\mathrm{Q}\) by \(\mathrm{NADH}\) as carried out by Complex \(\mathrm{I}\) (NADH-coenzyme Q reductase) of the electron-transport pathway if \(\mathscr{E}_{\mathrm{o}}^{\prime}\left(\mathrm{NAD}^{+} / \mathrm{NADH}\right)=-0.320 \mathrm{V}\) and \(\mathscr{E}_{\mathrm{o}}^{\prime}\left(\mathrm{CoQ} / \mathrm{CoQH}_{2}\right)=\) \(+0.060 \mathrm{V}\) b. What is the equilibrium constant \(\left(K_{\mathrm{eq}}\right)\) for this reaction? c. Assume that (1) the actual free energy release accompanying the NADH- coenzyme Q reductase reaction 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 \(=0.75\) (that is, \(75 \%\) of the energy released upon NADH oxidation is captured in ATP synthesis), and (3) the oxidation of 1 equivalent of NADH by coenzyme Q leads to the phosphorylation of 1 equivalent of ATP. Under these conditions, what is the maximum ratio of [ATP]/ \([\mathrm{ADP}]\) attainable by oxidative phosphorylation when \(\left[\mathrm{P}_{\mathrm{i}}\right]=1 \mathrm{m} M ?\) (Assume \(\Delta G^{\circ \prime}\) for ATP synthesis \(=+30.5 \mathrm{kJ} / \mathrm{mol}\).)

Step by step solution

01

Calculation of Standard Free Energy Change

The standard free energy change \(\Delta G^{\circ}\) is calculated using the formula: \(\Delta G^{\circ}=-nF\Delta E^{\circ}\), where \(n\) is the number of electrons transferred (2 in this case for the reduction of Q to QH2), \(F\) is the Faraday constant (\(F=96485.3 J/V·mol\)), and \(\Delta E^{\circ}\) is the difference in standard reduction potentials between the two half-reactions. Since \(\Delta E^{\circ} = E^{\circ}(oxidation) - E^{\circ}(reduction) = 0.060 V - (-0.320 V) = 0.380 V \), substitute these values into the equation to get \(\Delta G^{\circ}=-2 \times 96485.3 J/V·mol \times 0.380 V = -73268.4 J/mol = -73.3 kJ/mol\)

02

Calculation of Equilibrium Constant

Using the standard free energy change, the equilibrium constant \(K_{eq}\) can be calculated using the Gibbs free energy equation: \(\Delta G^{\circ} = -RT \ln K_{eq}\), where \(R\) is the ideal gas constant (\(R = 8.314 J/K·mol\)) and T is the temperature in Kelvin (Assume room temperature: \(T = 298 K\)). Rearranging for \(K_{eq}\), the equation becomes \(K_{eq} = e^{(\Delta G^{\circ}/(-RT))}\). Substituting \(\Delta G^{\circ} = -73.3 kJ/mol = -73300 J/mol\), \(R = 8.314 J/K·mol\), and \(T = 298 K\) yields \(K_{eq} = e^{(-73300 J/mol / (-8.314 J/K·mol × 298 K))} = e^{29.65} = 9.05 × 10^{12}\)

03

Calculation of [ATP]/[ADP] Ratio

Considering that the efficiency of the reaction is 75%, the actual free energy release will be \(0.75 \times \Delta G^{\circ}\). Additionally, literature value for synthesis of ATP under standard conditions is \(\Delta G^{\circ \prime} = +30.5 kJ/mol = +30500 J/mol\). As per the question conditions, the change in free energy of ATP synthesis reaction is equal to the energy released during the reduction. Hence, \(\Delta G = \Delta G^{\circ \prime} = 0.75 \times \Delta G^{\circ}\). The ratio of [ATP]/[ADP] in equilibrium can be calculated using \(\Delta G = -RT \ln ([ATP]/[ADP])\), which rearranges to \([ATP]/[ADP] = e^{(\Delta G/RT)}\). Substituting the corresponding values gives \([ATP]/[ADP] = e^{((0.75 \times -73300 J/mol) / (8.314 J/K·mol × 298 K))} = e^{-6.99} = 0.00091\)

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Key Concepts

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

Understanding the Role of NADH-coenzyme Q reductase in Energy Production

NADH-coenzyme Q reductase, also known as Complex I, plays a crucial role in cellular respiration within the mitochondria of the cell. It is the first enzyme in the electron-transport pathway, which is a series of reactions that generate energy in the form of ATP, the cellular 'energy currency.' Every time we consume food, our body breaks it down into nutrients, and one of the byproducts is NADH. This molecule contains a high amount of potential energy, which needs to be harnessed carefully.

Complex I facilitates the transfer of electrons from NADH to coenzyme Q (CoQ), and in doing so, it helps convert the stored energy within NADH into a form that can be used to produce ATP. The process involves a series of redox reactions, where NADH is oxidized to NAD+, and CoQ is reduced to CoQH2. The importance of understanding this component of the electron-transport pathway is not just academic; it's essential for grasping how our cells produce energy and for identifying potential therapeutic targets for diseases associated with mitochondrial dysfunction.

The Electron-Transport Pathway: Harnessing Energy

The electron-transport pathway is a complex process that occurs within the mitochondria. It consists of a set of protein complexes and small organic molecules that shuttle electrons derived from food nutrients. These electrons move through the pathway, gradually falling to lower energy levels and releasing energy at each step.

The movement is akin to a ball rolling downhill, and as it rolls, the energy gets transformed, with the help of this pathway, into a gradient of protons across the mitochondrial membrane. This proton gradient is later used by ATP synthase (Complex V) to manufacture ATP during oxidative phosphorylation. Understanding each step and how energy is captured and converted provides students with a clearer picture of how biological systems efficiently use basic chemical processes to sustain life.

Equilibrium Constant Calculation: Predicting Reaction Direction

The equilibrium constant \(K_{eq}\) is a numerical value that indicates the extent to which a reaction will proceed before reaching a state of equilibrium. It's a central concept in chemical thermodynamics, providing a link between the free energy change and the concentrations of reactants and products at equilibrium. When \(K_{eq}\) is very large, as in this case with a value in the order of \(10^{12}\), it implies that the forward reaction is highly favored, and virtually all reactant molecules will be converted to products.

For students learning about equilibrium, understanding that \(K_{eq}\) is derived from the standard free energy change (\(\Delta G^{\circ}\)) under standard conditions, gives them predictive power. They can calculate how the changes in concentrations of reactants and products will affect the direction of a reaction, and for reactions pertaining to biochemistry, they can then infer how these changes might impact cellular processes.

Oxidative Phosphorylation: Synthesizing the Energy Currency ATP

Oxidative phosphorylation is the final stage of the electron-transport pathway, where the energy from electrons transported down enzyme complexes is used to form ATP. ATP synthase, a vital enzyme, uses the energy stored in the form of the proton gradient across the mitochondrial membrane to synthesize ATP from ADP and inorganic phosphate (Pi).

This process is incredibly efficient, converting about 75% of the energy from NADH into ATP, as mentioned in the exercise. By understanding the principles of oxidative phosphorylation, students can appreciate the energy management within a cell, knowing how critical it is for maintaining cellular functions and life itself. When considering the ATP/ADP ratio, they can also explore the regulatory mechanisms that control the production and utilization of ATP in response to cellular demands.