Problem 3 - Biochemistry: Concepts and Connections

Question

Assume a pH gradient of \(4.0\) units across a thylakoid membrane, with the lumen more acidic than the stroma. (a) What is the standard free energy change per mol of protons associated with this gradient at \(25^{\circ} \mathrm{C}\) ? (b) What is the longest wavelength of light that could provide enough energy per photon to pump one proton against this gradient, assuming \(20 \%\) efficiency in photosynthesis and \(T=25^{\circ} \mathrm{C}\) ? (c) What is the standard free energy change per mol \(\mathrm{O}_{2}\) produced? How does this compare to the energy required to drive the synthesis of ATP? (d) The intensity of natural sunlight drops off dramatically beyond \(650 \mathrm{~nm}\). Calculate the maximum number of moles of protons that could be pumped against the gradient by the energy in a mole of photons of \(650 \mathrm{~nm}\) wavelength. Assume \(100 \%\) efficiency and \(T=25{ }^{\circ} \mathrm{C}\) for this calculation.

Step-by-Step Solution

Answer

(a) \(-22.9 \, \text{kJ/mol}\) (b) 1048 nm (c) \(-91.6 \, \text{kJ/mol}\); ATP synthesis needs less energy. (d) \(8.06 \text{ mol protons/mol photons}\) at 650 nm.

Step 1: Calculate the Standard Free Energy Change per Mol of Protons

The standard free energy change (∆G) for a proton gradient can be calculated using the formula: \[ \Delta G = -2.303RT(\text{pH difference}) \]where R is the gas constant \(8.314 \, \text{J/mol·K}\) and T is the temperature in Kelvin. Convert \(25^{\circ} \text{C}\) to Kelvin (298 K). Substituting the values, we get:\[ \Delta G = -2.303 \times 8.314 \, \text{J/mol·K} \times 298 \, \text{K} \times 4.0 = -22879 \, \text{J/mol} \] So, the standard free energy change per mol of protons is \(-22.9 \, \text{kJ/mol}\).

Step 2: Calculate the Longest Wavelength of Light

To calculate the wavelength, we use the energy relation:\[ E = \frac{hc}{\lambda} \]and rearrange it to solve for \(\lambda\):\[ \lambda = \frac{hc}{E} \]where \(h\) is Planck's constant \((6.626 \times 10^{-34} \, \text{J·s})\) and \(c\) is the speed of light \((3 \times 10^8 \, \text{m/s})\). Since the energy required is \(-22.9 \, \text{kJ/mol}\) and efficiency is 20%, only \(-22.9 \, \text{kJ/mol} \times \frac{1}{0.2}\) is required: \[ E_{\text{actual}} = -22.9 \, \text{kJ/mol} \times \frac{1}{0.2} = -114.5 \, \text{kJ/mol} = 1.145 \times 10^5 \, \text{J/mol} \]Convert to energy per photon by dividing by Avogadro's number \((6.022 \times 10^{23} \, \text{mol}^{-1})\):\[ E_{\text{photon}} = \frac{1.145 \times 10^5 \, \text{J/mol}}{6.022 \times 10^{23} \text{mol}^{-1}} \approx 1.897 \times 10^{-19} \, \text{J/photon} \]Substitute back into \(\lambda = \frac{hc}{E_{\text{photon}}}\):\[ \lambda = \frac{6.626 \times 10^{-34} \, \text{J·s} \times 3 \times 10^8 \, \text{m/s}}{1.897 \times 10^{-19} \, \text{J}} \approx 1048 \, \text{nm} \]

Step 3: Calculate the Standard Free Energy Change per Mol O2 Produced

For water splitting in photosynthesis, 4 electrons are needed for 1 \(\text{O}_2\) molecule, equivalent to 4 moles of protons. Thus:\[ \Delta G_{\text{O}_2} = 4 \times -22.9 \, \text{kJ/mol} = -91.6 \, \text{kJ/mol} \]This is the standard free energy change per mol \(\text{O}_2\) produced. The energy required to synthesize ATP from ADP, \(-30.5 \, \text{kJ/mol}\), is roughly a third of this value.

Step 4: Calculate the Maximum Moles of Protons at 650 nm

First, calculate the energy of a photon with wavelength 650 nm:\[ E_{\text{650 nm}} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \, \text{J·s} \times 3 \times 10^8 \, \text{m/s}}{650 \times 10^{-9} \, \text{m}} \approx 3.06 \times 10^{-19} \, \text{J/photon} \]Energy per mole of such photons:\[ E_{\text{mole}} = 3.06 \times 10^{-19} \, \text{J/photon} \times 6.022 \times 10^{23} \text{mol}^{-1} \approx 1.844 \times 10^5 \, \text{J/mol} \]The number of moles of protons that can be pumped against the energy:\[ \text{Moles of protons} = \frac{1.844 \times 10^5 \, \text{J/mol}}{22.9 \times 10^3 \, \text{J/mol}} \approx 8.06 \]

Key Concepts

Thylakoid Membrane

The thylakoid membrane is a critical component of the chloroplast where photosynthesis occurs, specifically within the light-dependent reactions. Think of it as a barrier that houses the necessary components to convert light energy into chemical energy. This membrane is essential because it contains pigments, like chlorophyll, which absorb sunlight.
Located within the chloroplast, the thylakoid membrane creates compartments: the lumen (inside the membrane) and the stroma (outside the membrane). These compartments facilitate different processes during photosynthesis. The membrane itself is designed to optimize the capture of sunlight and manage the movement of ions, which are vital for energy production.
Within the scope of photosynthesis, the thylakoid membrane plays a dual role by both aiding in the absorption of light and contributing to the formation of a proton gradient, crucial for ATP synthesis.

pH Gradient

In photosynthesis, the thylakoid membrane helps establish a pH gradient, which is vital for the energy conversion process. This gradient is formed by the movement of protons (H⁺ ions) from the stroma, where the pH is higher, into the thylakoid lumen, where it becomes more acidic (lower pH).
This difference in pH across the membrane is called the proton motive force. It represents the potential energy stored to perform work, specifically in producing ATP. The steeper the gradient, the greater the potential for energy conversion. A pH difference of 4.0 units means the lumen is significantly more acidic than the stroma, enhancing the proton motive force.
By harnessing this gradient, the energy generated can drive the ATP synthase enzyme to form ATP. Essentially, the pH gradient is nature’s way of storing energy temporarily until it can be converted to a more usable chemical form.

Free Energy Change

Free energy change, \(\Delta G\), is a crucial concept when understanding how substances like protons move and how energy is transferred during photosynthesis. In a proton gradient across a thylakoid membrane, this energy change is calculated using the formula: \[ \Delta G = -2.303 RT(\text{pH difference}) \] where \( R \) is the gas constant and \( T \) is temperature in Kelvin.
This equation calculates the energy required to move protons against the gradient. A negative \(\Delta G\) indicates a release of energy, suggesting the process is spontaneous and favorable. In our example, \( \Delta G = -22.9 \, \text{kJ/mol} \). It implies that energy is available, which can be used to perform work, such as pumping more protons or driving ATP synthesis.
This calculation not only measures the efficiency but also helps compare different energy conversion processes, such as between photosynthesis and ATP synthesis itself.

ATP Synthesis

ATP synthesis is at the heart of energy conversion in photosynthesis, powered by the proton gradient across the thylakoid membrane. The enzyme ATP synthase acts like a turbine driven by the flow of protons back into the stroma from the lumen. This movement through the enzyme facilitates the binding of ADP and inorganic phosphate (Pi) to form ATP.
Each molecule of ATP holds energy that cells can use for various functions. The process of ATP formation is intricately linked to the pH gradient and the inherent energy changes (as calculated using \(\Delta G\)).
The energy efficiency of this process is noteworthy. Ideally, the number of ATP molecules synthesized corresponds to the amount of energy provided by the proton gradient. Often, comparisons are drawn between the free energy change during photosynthesis and the energy storage capacity of ATP, accentuating photosynthesis's role in energy management and conversion.