Question

Rhodopsin is a biological pigment that serves as the primary photoreceptor in vision (Science \(266[1994]: 422)\) The chromophore in rhodopsin is retinal, and the absorption spectrum of this species is centered at roughly 500 nm. Using this information, determine the value of \(q_{E}\) for retinal. Do you expect thermal excitation to result in a significant excitedstate population of retinal?

Short answer

The value of \(q_{E}\) for retinal is approximately \(5.4 \times 10^{-5}\), which is very small. This indicates that thermal excitation does not result in a significant excited state population of retinal at room temperature, as most molecules remain in the ground state.

Step-by-step solution

Calculate the energy of a photon absorbed by retinal

To find the energy of a photon absorbed by retinal, we'll use the formula: \[E_{photon} = \frac{hc}{\lambda}\] Where \(E_{photon}\) is the energy of the photon, \(h\) is the Planck's constant (\(6.63 \times 10^{-34} Js\)), \(c\) is the speed of light (\(3 \times 10^8 ms^{-1}\)), and \(\lambda\) is the wavelength of the absorption spectrum (500 nm). First, we need to convert the wavelength from nm to m: \[\lambda = 500 \times 10^{-9} m\] Now, we can calculate the energy of the photon: \[E_{photon} = \frac{(6.63 \times 10^{-34} Js)(3 \times 10^8 ms^{-1})}{500 \times 10^{-9} m} = 3.98 \times 10^{-19} J\]

Determine the value of \(q_{E}\) for retinal

We'll use the Boltzmann distribution to find the value of \(q_{E}\) for retinal: \[q_E = e^{-\frac{E_{photon}}{k_B T}}\] Where \(E_{photon}\) is the energy of the photon calculated in step 1, \(k_B\) is the Boltzmann constant (\(1.38 \times 10^{-23} J K^{-1}\)), and \(T\) is the temperature (assuming room temperature, 298 K). \[q_E = e^{-\frac{3.98 \times 10^{-19} J}{(1.38 \times 10^{-23} J K^{-1})(298 K)}} = e^{-9.83} \approx 5.4 \times 10^{-5}\]

Assess if thermal excitation results in a significant excited state population of retinal

Since the value of \(q_{E}\) for retinal is very small (\(5.4 \times 10^{-5}\)), we can conclude that thermal excitation does not result in a significant excited state population of retinal. This is because at room temperature, the probability of a molecule being in the excited state due to thermal energy is very low, with most molecules remaining in the ground state.

Key concepts

Rhodopsin

Rhodopsin is a crucial protein in the human eye, acting as a light-sensitive receptor in the rod cells—specialized for low-light vision. Its importance lies in its ability to absorb light and trigger a biological process that allows us to see in dim conditions. This pigment comprises of opsin, a protein, and a retinal chromophore, which is the light-absorbing molecule. When rhodopsin absorbs light, it undergoes a structural change that initiates the visual transduction pathway, converting light into electrical signals interpreted by the brain as visual images.

Rhodopsin's sensitivity to light is finely tuned, with its peak absorption occurring around 500 nm in the blue-green region of the visible spectrum. Without rhodopsin, human night vision would be severely compromised, illustrating this protein's critical role in our visual system.

Retinal Chromophore

The retinal chromophore is an organic molecule bound within rhodopsin and is responsible for absorbing light. This molecule is a form of vitamin A and undergoes a change in its molecular conformation when it absorbs photons, a process called photoisomerization. From a straight-chain (11-cis-retinal) form, it switches to a bent-chain (all-trans-retinal) configuration. This change in shape triggers a series of reactions that ultimately send nerve impulses to the brain.

The efficiency of vision depends not only on the presence of the retinal chromophore but also on its ability to undergo this isomerization quickly and effectively. The regeneration of the 11-cis-retinal after it releases energy is also vital, as this allows rhodopsin to be 'reset' and ready to absorb another photon.

Absorption Spectrum

The absorption spectrum is a graph showing how much light a substance absorbs across different wavelengths. For the retinal chromophore in rhodopsin, it peaks around 500 nm, which corresponds to greenish-blue light. This peak is important because it tells us the most efficient wavelength for activating rhodopsin in vision. The spectrum is determined by the molecular structure of the retinal chromophore and how its electrons are arranged.

Since each photon's energy is inversely proportional to its wavelength, retinal's absorption spectrum informs us that the light it best absorbs carries a significant amount of energy, enough to cause photoisomerization. Understanding this spectrum helps us appreciate why certain light wavelengths are more visually effective under different lighting conditions.

Boltzmann Distribution

The Boltzmann distribution is a statistical distribution that describes the distribution of energy states among particles in a given system. At thermal equilibrium, it predicts that in a group of molecules, there will be a distribution of energy levels with most molecules occupying the lowest energy state, or ground state. The equation incorporates the Boltzmann constant and the temperature of the system to calculate this distribution.

In the context of vision, the distribution suggests that at room temperature, a small fraction of retinal molecules would be thermally excited to a higher energy state. However, from the calculated value of Boltzmann's distribution, we can infer that the probability of thermal excitation to an excited state is negligible, meaning that, practically, light absorption is necessary for retinal to reach an excited state.

Excited State Population

The excited state population for a molecule refers to the number of molecules in an excited energy state at any given moment. It is significantly influenced by the molecule's environment, such as temperature, and the energy of the surrounding photons. In biological systems like the human eye, the population of retinal molecules in the excited state is critical for vision. This population is typically low unless stimulated by light because of the relatively high energy difference between the ground state and the excited state.

The calculation performed using the Boltzmann distribution showed that at room temperature, without light, the excited state population of retinal is small. Consequently, we need photons of adequate energy—like those around 500 nm—to excite retinal and initiate the visual process. This excitation is a precise and efficient way to trigger the complex cascade of events that lead to the perception of light.