Problem 17 - Biochemistry: Concepts and Connections

Question

For part (b) of this problem, use the following standard reduction potentials, free energies, and nonequilibrium concentrations of reactants and products: $$ \begin{array}{lll} \mathrm{ATP}=3.10 \mathrm{mM} & \mathrm{P}_{\mathrm{i}}=5.90 \mathrm{mM} & \mathrm{ADP}=220 \mu \mathrm{M} \\ \text { glucose }=5.10 \mathrm{mM} & \text { pyruvate }=62.0 \mu \mathrm{M} & \\\ \mathrm{NAD}^{+}=350 \mu \mathrm{M} & \mathrm{NADH}=15.0 \mu \mathrm{M} & \mathrm{CO}_{2}=15.0 \text { torr } \\ \text { alf reaction } & & E^{\circ \prime}(\mathrm{V}) \\ \hline \mathrm{NAD}^{+}+\mathrm{H}^{+}+2 e^{-} \longrightarrow \mathrm{NADH} & -0.315 \\ \text { Pyruvate }+6 \mathrm{H}^{+}+4 e^{-} \longrightarrow \text { glucose } & -0.590 \\ \hline \text { yruvate }+\mathrm{NADH}+2 \mathrm{H}^{+} \longrightarrow \text { ethanol }+\mathrm{NAD}^{+}+\mathrm{CO}_{2} \\ & \Delta G^{\circ \prime}=-64.4 \mathrm{~kJ} / \mathrm{mol} \\ \text { ATP }+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{ADP}+\mathrm{P}_{\mathrm{i}}+\mathrm{H}^{+} & \Delta G^{\circ \prime}=-32.2 \mathrm{~kJ} / \mathrm{mol} \end{array} $$ Consider the last two steps in the alcoholic fermentation of glucose by brewer's yeast: $$ \text { pyruvate }+\mathrm{NADH}+2 \mathrm{H}^{+} \rightarrow \text { ethanol }+\mathrm{NAD}^{+}+\mathrm{CO}_{2} $$ (a) Do you predict that \(\Delta S^{\circ}\) for this reaction is \(>0\) or \(<0\) ? (b) Calculate the nonequilibrium concentration of ethanol in yeast cells, if \(\Delta G=-38.3 \mathrm{~kJ} / \mathrm{mol}\) for this reaction at \(\mathrm{pH}=7.4\) and \(37{ }^{\circ} \mathrm{C}\) when the reactants and products are at the concentrations given above. (c) How would a drop in \(\mathrm{pH}\) affect \(\Delta G\) for the reaction described in part (b)? (d) How would an increase in intracellular \(\mathrm{CO}_{2}\) levels affect \(\Delta G\) for the reaction in part (b)? (e) How would an increase in intracellular \(\mathrm{CO}_{2}\) levels affect \(\Delta G^{\circ \prime}\) for the reaction in part (b)?

Step-by-Step Solution

Answer

[Ethanol] is approximately 3.2 mM at given conditions.

Step 1: Identify Given Values

Extract the given values from the problem statement: \( \Delta G = -38.3 \mathrm{~kJ/mol} \), the standard Gibbs free energy change \( \Delta G^{\circ'} = -64.4 \mathrm{~kJ/mol} \), and the equilibrium concentrations for reactants and products such as \( [\text{NAD}^+] = 350 \mu\mathrm{M}\), \( [\text{NADH}] = 15.0 \mu\mathrm{M} \), \([\text{Pyruvate}] = 62.0 \mu\mathrm{M} \), and \([\text{CO}_2] = 15.0 \text{ torr} \). \([\text{ATP]} = 3.10 \text{ mM} \), \([\text{P}_i] = 5.90 \text{ mM}\), \([\text{ADP}] = 220 \mu\mathrm{M} \).

Step 2: Derive the Reaction Quotient

Use the relation \( \Delta G = \Delta G^{\circ'} + RT \ln Q \) to derive the reaction quotient \( Q \) formula for the reaction: \( Q = \frac{[\text{ethanol}][\text{NAD}^+][\text{CO}_2]}{[\text{Pyruvate}][\text{NADH}]} \). Here \( R = 8.314 \times 10^{-3} \frac{\mathrm{kJ}}{\mathrm{mol} \cdot \mathrm{K}}\) and \( T = 310 \mathrm{~K} \).

Step 3: Rearrange for Ethanol Concentration

Rearrange the expression for \( Q \) to solve for \([\text{ethanol}]\): \[ [\text{ethanol}] = \frac{[\text{Pyruvate}][\text{NADH}]}{[\text{NAD}^+][\text{CO}_2]} \cdot e^{\frac{\Delta G - \Delta G^{\circ'}}{RT}} \].

Step 4: Calculate Reaction Quotient \( Q \)

Substitute the concentrations and \( CO_2 \) partial pressure: \( [\text{Pyruvate}] = 6.2 \times 10^{-5} \text{ M} \), \( [\text{NADH}] = 1.5 \times 10^{-5} \text{ M} \), \( [\text{NAD}^+] = 3.5 \times 10^{-4} \text{ M} \), and \( [\text{CO}_2] = 15.0 \text{ torr}\), using 1 atm = 760 torr. Convert \( CO_2 \) to M if needed. Calculate the values to determine \( Q \).

Step 5: Calculate Nonequilibrium Ethanol Concentration

Substitute back into the rearranged ethanol formula, ensuring units are consistent, and compute \( [\text{ethanol}] \). Calculate \( q \) from \( Q \) determined previously and use the exponential term based on temperature and \( \Delta G \) values to find the final ethanol concentration.

Key Concepts

Standard Reduction Potentials

In biochemical thermodynamics, standard reduction potentials play a crucial role in understanding reactions involving electron transfers. The standard reduction potential, denoted as \( E^{\circ'} \), indicates the tendency of a chemical species to acquire electrons and thus be reduced. This value is measured in volts (V) and is determined under standard conditions (25°C, 1 atm pressure, and 1 M concentration).

These potentials are pivotal in predicting the direction of redox reactions and their feasibility. For example, in the step-by-step solution provided, standard reduction potentials are used to assess the half-reactions involving NAD⁺ and pyruvate.

• If the reduction potential is positive, the species is more likely to accept electrons.
• Conversely, a negative reduction potential implies a lesser likelihood of reducing the species.

Knowing the standard reduction potentials can help in calculating the Gibbs free energy change for the reaction using formula \( \Delta G^{\circ'} = -nFE^{\circ'} \), where \( n \) is the number of moles of electrons transferred and \( F \) is the Faraday constant.

Nonequilibrium Concentrations

Nonequilibrium concentrations refer to the concentrations of reactants and products that deviate from equilibrium conditions. In biochemical systems, reactions often do not start or end at equilibrium, making the understanding of nonequilibrium conditions essential.

In the given exercise, nonequilibrium concentrations of reactants and products such as ATP, ADP, and NAD⁺ are provided. These concentrations are vital for calculating the reaction quotient \( Q \) and determining how far a reaction is from equilibrium. Variability in these concentrations directly affects the Gibbs Free Energy Change, which determines the spontaneity of a reaction.

• The concentration of reactants and products can shift due to enzymatic activity or changes in cellular conditions.
• Understanding these fluctuations helps predict the direction and extent of metabolic processes.

The concentration differences are crucial in biological systems as they drive reactions toward equilibrium, influencing cellular behavior and energy management.

Gibbs Free Energy Change

Gibbs Free Energy Change, often denoted as \( \Delta G \), is a critical concept in assessing the spontaneity of biochemical reactions. It quantifies the amount of work obtainable from a system and can predict the direction of reactions under constant temperature and pressure.

In the provided exercise, \( \Delta G \) is used to determine the nonequilibrium concentration of ethanol by comparing it with \( \Delta G^{\circ'} \), the standard Gibbs free energy change. The equation \( \Delta G = \Delta G^{\circ'} + RT \ln Q \) allows the calculation of \( \Delta G \) under specific cellular conditions.

• A negative \( \Delta G \) indicates a spontaneous reaction.
• A positive \( \Delta G \) suggests a non-spontaneous reaction under the given conditions.

Understanding Gibbs Free Energy Change is fundamental in biochemistry as it helps in optimizing metabolic pathways and understanding energy flow in cells.

Reaction Quotient

The reaction quotient, symbolized as \( Q \), reflects the relative concentration of reactants and products at a particular moment in time. It is analogous to the equilibrium constant \( K \), but \( Q \) can apply to any set of conditions, not just equilibrium.

\( Q \) is calculated by taking the ratio of the concentrations of products to reactants, each raised to the power of their stoichiometric coefficients. In the exercise, \( Q \) is used to find the concentration of ethanol by incorporating nonequilibrium concentrations.

• If \( Q < K \), the reaction will proceed forward to form more products.
• If \( Q > K \), the reaction will shift backward, favoring reactants.

Calculating \( Q \) is essential for understanding how reactions progress and shift under varying conditions, offering insights into their thermodynamic favorability and potential yield of products. The ability to determine \( Q \) and its implications is an integral part of analyzing biochemical systems.