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ATP is synthesized from ADP, \(\mathrm{P}_{\mathrm{i}}\), and a proton on the matrix side of the inner mitochondrial membrane. We will refer to the matrix side as the "inside" of the inner mitochondrial membrane (IMM). (a) \(\mathrm{H}^{+}\)transport from the outside of the IMM into the matrix drives this process. The \(\mathrm{pH}\) inside the matrix is \(8.2\), and the outside is more acidic by \(0.8 \mathrm{pH}\) units. Assuming the IMM membrane potential is \(168 \mathrm{mV}\) (inside negative), calculate \(\Delta G\) for the transport of \(1 \mathrm{~mol}\) of \(\mathrm{H}^{+}\)across the IMM into the matrix at \(37^{\circ} \mathrm{C}: \mathrm{H}_{\text {(outside) }}^{+} \rightarrow \mathrm{H}_{\text {(inside) }}^{+}\). (b) Assume three \(\mathrm{mol} \mathrm{H}^{+}\)must be translocated to synthesize one mol ATP by coupling of the following reactions: $$ \begin{gathered} \mathrm{ADP}+\mathrm{P}_{\mathrm{i}}+\mathrm{H}_{\text {(inside) }}^{+} \rightarrow \mathrm{ATP}+\mathrm{H}_{2} \mathrm{O} \quad \text { (ATP synthesis) } \\ 3 \mathrm{H}_{\text {(outside) }}^{+} \rightarrow 3 \mathrm{H}_{\text {(inside) }}^{+} \quad \text { (proton transport) } \end{gathered} $$ Write the overall reaction for ATP synthesis coupled to \(\mathrm{H}^{+}\)transport [and use this equation for part (c)]: (c) Assume three mol \(\mathrm{H}^{+}\)must be translocated to synthesize one mol ATP as described in part (b) above. Given the following steady-state concentrations: ATP \(=2.70 \mathrm{mM}\) and \(\mathrm{P}_{\mathrm{i}}=5.20 \mathrm{mM}\), the membrane potential \(\Delta \psi=168 \mathrm{mV}\) (inside negative), and the \(\mathrm{pH}\) values in part (a), calculate the steady- state concentration of ADP at \(37^{\circ} \mathrm{C}\) when for the coupled process (ATP synthesis \(+\mathrm{H}^{+}\)transport), \(\Delta G=-11.7 \mathrm{~kJ} / \mathrm{mol}\).

Step by step solution

01

Calculate pH gradient and 94G for a single proton

The pH gradient is \( \Delta \text{pH} = \text{pH}_{\text{inside}} - \text{pH}_{\text{outside}} = 8.2 - (8.2 - 0.8) = 0.8 \). The formula for \( \Delta G \) of a proton moving across a membrane is \( \Delta G = RT \ln \frac{[\text{H}^+]_{\text{inside}}}{[\text{H}^+]_{\text{outside}}} + F \Delta \psi \), where \( R = 8.314 \text{ J/mol K} \), \( T = 310 K \), \( F = 96,485 \text{ C/mol} \), and \( \Delta \psi = -168 \text{ mV} \). Convert \( \Delta \psi \) to joules: \( -168 \times 10^{-3} \text{ V} \). The concentration ratio \( \ln \frac{[\text{H}^+]_{\text{inside}}}{[\text{H}^+]_{\text{outside}}} = - \ln 10^{-0.8} = 0.8 \ln 10 \approx 1.84 \). Now compute \( \Delta G: \Delta G = (8.314 \times 310 \times 1.84) + (96,485 \times (-0.168)) \approx 4744.3 - 16212 \approx -11467.7 \text{ J/mol} \).

02

Write the overall coupled reaction equation

The coupled response for ATP synthesis and three protons is: \( \text{ADP} + \text{P}_i + 3 \text{H}_{\text{outside}}^+ \rightarrow \text{ATP} + \text{H}_2\text{O} + 3 \text{H}_{\text{inside}}^+ \).

03

Calculate free energy changes and their impact on ADP concentration

The equation that describes the energetics of ATP synthesis from ADP and inorganic phosphate (Pi) across the inner mitochondrial membrane taking into account the membrane potential and proton gradient is: \( \Delta G = \Delta G^0 + RT \ln \left( \frac{[\text{ATP}]}{[\text{ADP}][\text{P}_i]} \right) + 3(-11467.7) \). Given \( \Delta G = -11.7 \text{ kJ/mol} \), change it to joules \( -11.7 \times 1000 = -11700 \text{ J/mol} \). Assume \( \Delta G^0_{\text{ATP}} = +30,500 \text{ J/mol} \) (approximate standard biochemical potential). Simplify the equation: \( -11700 = 30500 + 8.314 \times 310 \ln \left( \frac{2.7 \times 10^{-3}}{[\text{ADP}] \times 5.2 \times 10^{-3}} \right) - 3 \times 11467.7 \). This can be rearranged to solve for \( [\text{ADP}] \).

04

Solve for ADP concentration

From the previous step, solve for [ADP]: \( -11700 = 30500 + (2575 \ln \frac{2.7}{5.2[\text{ADP}]}) - 34403.1 \)\( 2575 \ln \frac{2.7}{5.2[\text{ADP}]} = (34403.1 - 30500 - 11700) \) \( \ln \frac{2.7}{5.2[\text{ADP}]} = 1872.1 / 2575 \) \( \frac{2.7}{5.2[\text{ADP}]} = e^{0.727} \) \( [\text{ADP}] = \frac{2.7}{5.2 \times e^{0.727}} \approx 0.860 \text{ mM} \).

05

Conclusion: Final ADP concentration

The steady-state concentration of ADP, at the given conditions, is approximately \( 0.860 \text{ mM} \).

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Key Concepts

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

Proton Transport

Proton transport is a crucial step in ATP synthesis within the mitochondria. In this context, protons ( H^+ ) are moved from the intermembrane space (outside) to the matrix (inside) through the inner mitochondrial membrane. This movement creates a proton gradient, also known as pH difference. The pH inside the matrix is higher, meaning it is lower in hydrogen ion concentration compared to the outside. This gradient is important because it creates a potential energy difference across the membrane.

The protons naturally move along this gradient into the matrix, using energy captured from earlier cellular reactions. The transport involves proteins like ATP synthase, which facilitate the passage and use the energy released to convert ADP and inorganic phosphate into ATP.

The movement of protons is not just about maintaining pH balance. It involves ridged mechanisms that use the free energy to drive the synthesis of ATP, acting like a molecular power generator. ATP synthase , functioning similar to a "proton turbine," utilizes the proton gradient to couple proton flow to ATP production.

Mitochondrial Membrane Potential

The mitochondrial membrane potential is the voltage difference across the inner mitochondrial membrane, playing a key role in ATP synthesis. The membrane potential exists due to the difference in ion concentrations between the inner and outer sides of the mitochondrial membrane. This potential is typically negative on the inside due to more negatively charged ions being present compared to the outside.

In our case, it is stated to be 168 mV (inside negative), which is quite substantial. This membrane potential is essential as it adds to the free energy required for ATP synthesis. When protons flow back into the matrix (inside), the stored energy from the potential difference and pH gradient is released, helping drive the synthesis of ATP.

It is the combination of the electrical membrane potential and the chemical gradient across the mitochondrial membrane that comprises the so-called chemiosmotic potential. This potential is the driving force behind oxidative phosphorylation, where the energy from food is converted into the usable energy currency ATP.

Biochemical Energetics

Biochemical energetics is the study of energy transformation within biological systems. In this context, it involves understanding how energy changes convert raw substrates like ADP and inorganic phosphate into ATP within the mitochondrion. The process is highly dependent on the pH gradient and membrane potential created by proton transport.

The energy conversion takes place through a process called oxidative phosphorylation, where the oxidation of nutrients generates a proton gradient across the inner mitochondrial membrane. The flow of protons back into the matrix is coupled with ATP synthesis. This tight coupling ensures energy from nutrients is effectively transformed into ATP with minimal loss.

The amount of free energy change required for the conversion of ADP and Pi into ATP is substantial and is often described using energetic terms such as 饾洢G over standard conditions. This energy must be precisely controlled to maintain the efficiency of ATP production under varying cellular conditions, ensuring that cells have a continuous supply of energy for all their activities.

Free Energy Change

Free energy change, symbolized as 饾洢G , represents the amount of energy available to do work during a chemical process, such as ATP synthesis. Understanding 饾洢G is vital, as it predicts whether a reaction can occur spontaneously. In the cell, this is managed by the proton motive force, which includes both the pH gradient and the membrane potential.

For the movement of protons across the mitochondrial membrane, 饾洢G can be calculated based on the concentration ratio of protons across the membrane, the temperature, and the membrane potential. This value must be sufficiently negative to drive ATP synthesis.

In our exercise, the calculated free energy change for one mole of protons ( H^+ ) moving into the matrix was -11467.7 J/mol , which contributes to the 饾洢G of the overall ATP synthesis process. When considering the synthesis of ATP coupled with the movement of three protons, other factors, such as the concentrations of ATP and ADP, must also be considered to determine the overall energetic feasibility of the process.