Question

Rhodopsin is the pigment in the retina rod cells responsible for vision, and it consists of a protein and the co-factor retinal. Retinal is a \(\pi\) -conjugated molecule which absorbs light in the blue-green region of the visible spectrum, where photon absorption represents the first step in the visual process. Absorption of a photon results in retinal undergoing a transition from the ground-or lowest-energy state of the molecule to the first electronic excited state. Therefore, the wavelength of light absorbed by rhodopsin provides a measure of the ground and excited-state energy gap. a. The absorption spectrum of rhodopsin is centered at roughly 500 . nm. What is the difference in energy between the ground and excited state? b. At a physiological temperature of \(37^{\circ} \mathrm{C}\), what is the probability of rhodopsin populating the first excited state? How susceptible do you think rhodopsin is to thermal population of the excited state?

Short answer

The energy difference between the ground and excited state of rhodopsin is approximately \(3.99 \times 10^{-19} \; \text{J}\). At a physiological temperature of \(37^{\circ}\)C, the probability of rhodopsin populating the first excited state is approximately \(3.62 \times 10^{-32}\). This extremely low probability indicates that rhodopsin is not significantly susceptible to thermal population of the excited state, ensuring that our vision is primarily sensitive to photon absorption rather than thermal fluctuations.

Step-by-step solution

Calculate the energy difference between the ground and excited states

First, we need to convert the wavelength of light absorbed by rhodopsin (500 nm) into energy using the Planck's constant and the speed of light. Using the formula for energy: \[E = \dfrac{hc}{\lambda}\] Where \(E\) is the energy, \(h\) is the Planck's constant (approx. \(6.626 \times 10^{-34} \text{ Js}\)), \(c\) is the speed of light (\(2.998 \times 10^{8} \;\text{m/s}\)), and \(\lambda\) is the wavelength (500 nm, which is equivalent to \(5.00 \times 10^{-7} \;\text{m}\)). Now, plug in the values and calculate the energy difference: \[E = \dfrac{(6.626 \times 10^{-34}\; \text{Js})(2.998 \times 10^{8}\; \text{m/s})}{(5.00 \times 10^{-7} \;\text{m})}\] \[E \approx 3.99 \times 10^{-19}\; \text{J}\]

Calculate the probability of rhodopsin being in the first excited state

Now we need to find the probability of rhodopsin populating the first excited state at 37 °C. We will use the Boltzmann distribution to calculate the probability: \[P = \dfrac{1}{e^{\frac{\Delta E}{k_BT}}}\] Where \(P\) is the probability, \(\Delta E\) is the energy difference between the ground and excited states, \(k_B\) is the Boltzmann constant (\(1.381 \times 10^{-23}\; \text{J/K}\)), and \(T\) is the temperature in Kelvin (37 °C, which is equivalent to 310 K). Now, plug in the values and calculate the probability: \[P = \dfrac{1}{e^{\frac{(3.99 \times 10^{-19}\; \text{J})}{(1.381 \times 10^{-23}\; \text{J/K})(310\; \text{K})}}}\] \[P \approx 3.62 \times 10^{-32}\] So, the probability of rhodopsin being in its first excited state due to thermal fluctuations is approximately \(3.62 \times 10^{-32}\).

Discuss the susceptibility of rhodopsin to thermal population

Judging by the extremely low probability found in Step 2, the rhodopsin molecule is not significantly susceptible to thermal population of the first excited state at physiological temperatures. This is a good thing, as it would ensure that rhodopsin and, in turn, our vision is primarily sensitive to the absorption of photons, rather than influenced by thermal fluctuations.

Key concepts

Rhodopsin

Rhodopsin is an essential light-sensitive pigment found in the photoreceptor cells of the retina, primarily known for its role in vision. It is a combination of a protein called opsin linked to a light-absorbing molecule called retinal. The function of rhodopsin hinges on its unique ability to absorb light, particularly in the blue-green region of the visible spectrum. This absorption is vital as it initiates a sequence of biological responses, ultimately resulting in the perception of light.
When light hits rhodopsin, the retinal molecule within changes configuration via a photochemical reaction. This transformation allows other processes, such as the activation of a signaling cascade, essential for vision, to begin. This phototransduction then enables the brain to convert light stimuli into visual signals.

• Rhodopsin's peak absorption is around 500 nm, catering specifically to low-light conditions.
• Retinal's configuration change is essential for the sensitivity and adaptability of human vision, especially under varying light conditions.

Energy States

In the context of rhodopsin and light absorption, energy states refer to the various levels of energy that the retinal molecule can occupy. These can be broadly categorized into a ground state (lowest energy) and an excited state (higher energy).
The absorption of light is a crucial physical event, leading to the transition of retinal from the ground state to an excited state. This energy transition symbolizes the essential first step in vision, enabling the retinal to perform its function in the rhodopsin molecule.

• The energy difference between these states can be quantified using the wavelength of absorbed light and Planck's law.
• Such transitions are specific to particular wavelengths of light, providing an understanding of the visible spectrum's energy dynamics.

Boltzmann Distribution

The Boltzmann distribution is a statistical tool used to describe the distribution of energy states among molecules at thermal equilibrium. For rhodopsin, it helps us understand the likelihood of molecules occupying higher energy states due to thermal fluctuations.
While the primary role of light absorption in rhodopsin involves moving from a ground state to an excited state, the Boltzmann distribution allows the calculation of how frequently such transitions may occur simply due to thermal energy at body temperature.
For rhodopsin, this probability is extremely low, as calculated using the Boltzmann formula.

• This distribution emphasizes that thermal energy is usually insufficient to excite retinal molecules significantly at body temperature.
• It supports the vision process's dependency on photon absorption rather than thermal excitation, assuring clarity and stability in visual perception.