Question

More free energy! The isomerization of dihydroxyacetone phosphate (DHAP) to glyceroldehyde 3-phosphate (GAP) has an equilibrium constant of 0.0475 under standard conditions \((298 \mathrm{K}, \mathrm{pH} 7) .\) Calculate \(\Delta G^{0 \prime}\) for the isomerization. Next, calculate \(\Delta G\) for this reaction when the initial concentration of DHAP is \(2 \times 10^{-4} \mathrm{M}\) and the initial concentration of GAP is \(3 \times 10^{-6} \mathrm{M} .\) What do these values tell you about the importance of \(\Delta G\) compared with that of \(\Delta G^{0 \prime}\) in understanding the thermodynamics of intracellular reactions?

Short answer

\(\Delta G^{0 \prime} = 6.8 \: kJ/mol\), \(\Delta G = -2.92 \: kJ/mol\). \(\Delta G\) is more relevant for cellular conditions.

Step-by-step solution

Formula for Standard Free Energy Change

Use the equation \(\Delta G^{0 \prime} = -RT \ln K_{eq}\), where \(R\) is the gas constant \(8.314 \: J \: mol^{-1} \: K^{-1}\), \(T\) is the temperature in Kelvin \(298 \: K\), and \(K_{eq}\) is the equilibrium constant \(0.0475\).

Calculate \(\Delta G^{0 \prime}\)

Substitute the values into the equation: \[\Delta G^{0 \prime} = -(8.314 \: J \: mol^{-1} \: K^{-1})(298 \: K) \ln(0.0475)\]. Simplify to find \(\Delta G^{0 \prime} = 6,800 \: J/mol\) or approximately \(6.8 \: kJ/mol\).

Formula for Free Energy Change

Use the equation \(\Delta G = \Delta G^{0 \prime} + RT \ln\left(\frac{[\text{GAP}]}{[\text{DHAP}]}\right)\).

Calculate \(\Delta G\)

Substitute the concentrations \([\text{DHAP}] = 2 \times 10^{-4} \: M\) and \([\text{GAP}] = 3 \times 10^{-6} \: M\) into the equation: \[\Delta G = 6,800 \: J/mol + (8.314 \: J \: mol^{-1} \: K^{-1})(298 \: K) \ln\left(\frac{3 \times 10^{-6}}{2 \times 10^{-4}}\right)\]. This simplifies to \(\Delta G = -2,920 \: J/mol\) or approximately \(-2.92 \: kJ/mol\).

Interpret the Results

The calculated values show that the \(\Delta G^{0 \prime}\), which assumes standard conditions, is positive, suggesting the reaction is non-spontaneous under these conditions. However, the actual \(\Delta G\) under intracellular conditions is negative, indicating that the reaction is spontaneous in a cellular context. This disparity shows that actual cellular conditions can significantly alter the spontaneity of a reaction, underscoring the importance of \(\Delta G\) over \(\Delta G^{0 \prime}\) when understanding cellular reactions.

Key concepts

Free Energy

Free energy, specifically Gibbs free energy (\(\Delta G^{0 \prime}\) and \(\Delta G\)), helps us understand if a reaction is spontaneous. A negative \(\Delta G\) suggests the reaction can occur on its own, while a positive value means it's not spontaneous. There are two key types of free energy changes:

• Standard Free Energy Change (\(\Delta G^{0 \prime}\)): This is determined under standard conditions, which include a set temperature, pressure, and concentration.
• Actual Free Energy Change (\(\Delta G\)): This reflects real cellular conditions, where concentrations can vary.

In the exercise, \(\Delta G^{0 \prime}\) was positive, suggesting the isomerization of DHAP to GAP isn't spontaneous under standard conditions. However, the calculated \(\Delta G\) was negative, showing how the real cellular balance shifts towards spontaneity.

Isomerization

Isomerization involves changing one molecule into another with the same chemical formula but different structure. For dihydroxyacetone phosphate (DHAP) to convert to glyceraldehyde 3-phosphate (GAP):

• This process is crucial in glycolysis, an important metabolic pathway.
• The rearrangement doesn't change the formula but affects how the molecule reacts and functions in the body.

In biochemistry, isomerization can influence energy transfer, enzyme activity, and reaction rates. The significance of isomerization in metabolic pathways highlights the fine-tuning needed for efficient energy production and usage.

Equilibrium Constant

The equilibrium constant (\(K_{eq}\)) is a measure of a reaction's tendency to proceed to completion. It is calculated from the concentration of products and reactants at equilibrium:

• High \(K_{eq}\): Indicates more products relative to reactants at equilibrium.
• Low \(K_{eq}\): Indicates fewer products relative to reactants at equilibrium.

For the DHAP to GAP conversion, \(K_{eq} = 0.0475\), suggesting the reaction heavily favors DHAP at equilibrium under standard conditions. Understanding \(K_{eq}\) helps predict the direction a reaction naturally shifts under various conditions.

Cellular Conditions

Real-life cellular conditions often differ from standard lab settings. Factors include:

• Varied concentrations of reactants and products.
• Fluctuating pH levels and temperatures within cells.

These differences mean that reactions like the DHAP to GAP isomerization may behave differently in cells than in isolated conditions. It's crucial to consider these variables:

• Intracellular Environments: They support reactions that may not happen spontaneously under standard conditions.
• Biological Catalysts: Such as enzymes, are present in cells, significantly impacting reaction rates and energy requirements.

Recognizing the impact of \(\Delta G\) in actual cellular environments highlights the adaptive nature of biochemical reactions to sustain life processes.