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 \(=+30.5 \mathrm{kJ} / \mathrm{mol} .\) )

Step by step solution

01

Calculation of \(\Delta G^{\circ'}\)

To calculate the standard free energy change \(\Delta G^{\circ'}\) of the reaction, we can use the Nernst equation: \(\Delta G^{\circ'}=-nF\Delta E^{\circ'}\), where \(n\) is the number of electrons transferred (in this case, 2), \(F\) is Faraday's constant (96485.3 C/mol), and \(\Delta E^{\circ'}\) is the difference in standard reduction potentials. \(\Delta E^{\circ'}\) can be calculated as \(\Delta E^{\circ'}=E^{\circ'}(\text{Accepter}) - E^{\circ'}(\text{Donor})\). Given \(E^{\circ'}(\mathrm{Fum} / \mathrm{Succ}) =+0.031 V\) and \(E^{\circ'}(\frac{1}{2}\mathrm{O}_{2} / \mathrm{H}_{2}\mathrm{O}) = +0.816 V\), \(\Delta E^{\circ'} = +0.816 V - (+0.031 V) = +0.785 V.\) Using this in the Nernst equation, we get \(\Delta G^{\circ'} = -2 \times 96485.3 C/mol \times 0.785V = -151.6 kJ/mol\).

02

Calculation of \(K_{eq}\)

The equilibrium constant \(K_{eq}\) can be calculated using the relationship \(\Delta G^{\circ'}= -RT\ln K_{eq}\), where \(R = 8.3145 J/mol/K\) is the gas constant and \(T\) is the temperature in Kelvin. In our case, we will consider room temperature, 298K. So, \(\ln K_{eq} = -\Delta G^{\circ'}/RT = 151600 J/mol / (8.3145 J/mol/K \times 298K)\). Then, \(K_{eq} = e^{(\ln K_{eq})}\).

03

Calculation of max [ATP]/[ADP] ratio

Given that the oxidation of 1 succinate molecule leads to the synthesis of 2 equivalents of ATP, the \(\Delta G^{\circ'}\) for the synthesis of 1 ATP is -151.6 kJ/mol / 2 ATP = -75.8 kJ/mol. Since 70% of the energy is captured in ATP synthesis, the \(\Delta G\) for the synthesis of 1 ATP is -75.8 kJ/mol x 0.70 = -53.1 kJ/mol. Equivalent to +30.5 kJ/mol, the \(\Delta G\) under the specified conditions can be calculated using the equation \(\Delta G = \Delta G^{\circ'} + RT\ln Q\), where \(Q\) is the reaction quotient ([products]/[reactants]). In this case, \(\Delta G = 0\), because the system is in equilibrium. So, \(Q = exp( -\Delta G^{\circ'}/RT) = exp( -(-53.1 kJ/mol/ (8.3145 J/mol/K \times 298 K)))\). Knowing that \(Q = [ATP]/([ADP]*[P_i])\) and \([P_i] = 1 mM\), we get the maximum [ATP]/[ADP] ratio.

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Key Concepts

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

Understanding Standard Free Energy Change

In the realm of biochemistry, the standard free energy change ((DeltaG^{circ})) serves as a pivotal concept, indicating the amount of free energy released or absorbed during a chemical reaction under standard conditions at 1 M concentration, 1 atm pressure, and room temperature (25°C). It's important to recognize that a negative (DeltaG^{circ}) value represents a reaction that is spontaneous, thus releasing energy, which makes it exergonic. Conversely, a positive value would mean the reaction requires energy input and is endergonic.

Using the given redox potentials for the electron donors and acceptors involved in the oxidation of succinate, the calculation of (DeltaG^{circ}) becomes possible through the modified Nernst equation: (DeltaG^{circ}')=-nF(DeltaE^{circ}'), where (DeltaE^{circ}') is the difference in reduction potentials, (F) is the Faraday constant, and (n) denotes the number of moles of electrons exchanged. This fundamental principle not only provides insight into the directionality and spontaneity of reactions but also underpins the energetic feasibility of biochemical pathways, such as the electron-transport chain instrumental in cellular respiration.

Applying the Nernst Equation

The Nernst equation plays a central role in bioenergetics, establishing a relation between the electrical potential of a cell and the concentrations of reactants and products. In the context of the oxidation of succinate, the Nernst equation has been adapted to calculate the standard free energy change for the reaction given the standard reduction potentials of succinate/fumarate and oxygen/water.

The standard reduction potential ((E^{circ}')) indicates a substance's tendency to gain electrons, which is essential for understanding the flow of electrons in biological systems. By subtracting the donor's potential from the acceptor's potential, we obtain the voltage that drives the electron flow in the reaction. This voltage is then applied within the Nernst equation to deduce the (DeltaG^{circ}), thereby embedding this equation as a critical analytical tool for evaluating biochemical reactions and their energy transductions.

Deciphering the Equilibrium Constant

The equilibrium constant ((K_{eq})) is a quantitative measure of a chemical reaction's position at equilibrium, expressed in terms of the concentrations of the products and reactants. A larger (K_{eq}) signifies a greater extent of reaction towards the products, while a smaller constant suggests a reaction favoring the reactants.

In our exercise, the relationship between the equilibrium constant and the standard free energy change is illuminated through the expression (DeltaG^{circ}') = -RT ln(K_{eq}) where (R) is the universal gas constant and (T) is the absolute temperature in Kelvin. Utilizing this relationship permits the calculation of (K_{eq}) from the computed (DeltaG^{circ}'), offering a clear window into the reaction's equilibrium dynamics and the consequential tendencies of the participant species within the biochemical pathway involved.

Evaluating ATP Synthesis Efficiency

ATP synthesis efficiency is a measure of how adeptly the energy from a biochemical process, such as substrate oxidation, is converted into usable chemical energy in the form of ATP. It is an important concept in bioenergetics, as it reflects how living organisms conserve and utilize energy derived from metabolic reactions.

The problem presents a scenario where 70% of the energy from succinate oxidation is successfully harnessed for ATP synthesis, known as the P/O ratio. The efficiency relates directly to the ATP yield from the electron-transport chain and bears implications for the maximum attainable ratio of [ATP]/[ADP], a critical determinant for cellular energetics and metabolic control. Through this lens, efficiency encapsulates both the yield of ATP and the energetic cost of its synthesis, echoing the biological necessity for a balance between energy production and consumption.