The first step in the very complex mechanism of vision is the photoinduced cis → trans isomerization — Analytical Chemistry Chemistry Question
Quantum chemistry of vision
The first step in the very complex mechanism of vision is the photoinduced cis → trans isomerization of the chromophore retinal embedded in rhodopsin molecules. Absorption of visible light by cis-retinal causes a change of the configuration of a double bond:
cis-retinal trans-retinal
Show the double bond, which participates in the cis-trans-isomerization. Indicate the reaction coordinate.
Model Answer
Reaction proceeds by rotation of a part of the molecule about the C11–C12 bond:
The rotation angle is the reaction coordinate.
Energies of the reactant and the product were found to be periodic functions of the reaction coordinate x:
Ecis(x) = 1.79(1 – cos(x)),
Etrans(x) = 1.94 + 0.54 cos(x).
Energies are in eV (1 eV = 1.60×10–19 J = 96500 J mol–1), x = 0 corresponds to the reactant, x = π to the product. Draw the energy diagram for this reaction.
Determine the energy change for the reaction and its activation energy in kJ mol–1.
Model Answer
The energy change is the difference between the lowest energies of the trans- and cis-isomers:
Q = Etrans(π) – Ecis(0) = 1.40 – 0 = 1.40 eV = 135 kJ mol–1.
Transition state of reaction is near the region of curve-crossing:
1.79(1 – cos(x)) = 1.94 + 0.54 cos(x),
x = 1.64 = 0.521π = 93.7 °.
Activation energy (reaction barrier) is defined by the energy difference between the transition state and the reagent:
EA = Ecis(1.64) – Ecis(0) = 1.91 eV = 184 kJ mol–1.
This barrier is rather high to be overcome at ambient temperature.
What is the largest wavelength of light that can be absorbed by cis-retinal?
Model Answer
Maximal wavelength is determined by the energy difference between trans- and cis-retinal at x = 0:
ΔE = Etrans(0) – Ecis(0) = 2.48 – 0 = 2.48 eV = 3.97⋅10–19 J.
λ = hc / ΔE = (6.63×10–34 × 3.00×108) / 3.97×10–19 = 5.01×10–7 m = 501 nm.
Let us apply the “particle-in-a-box” model to the electrons present in the conjugated system of cis-retinal. Energy levels of a particle of the mass m locked in an one-dimensional box with the width l are given by:
En = h^2 n^2 / (8 m l^2), n = 1, 2, …
What is the number of electrons in the conjugated system of cis-retinal?
Model Answer
Conjugated electronic system of retinal contains 6 double bonds, that is, 12 π-electrons that occupy 6 lowest energy levels.
Based on your answers on questions 4.4 – 4.5 and using the formula above calculate l. How does this value compare with the structure of retinal molecule?
Model Answer
Absorption of light causes the transition from the highest occupied to the lowest unoccupied level:
ΔE = E7 – E6 = (13 h^2) / (8 m l^2),
where electron mass is m = 9.11×10–31 kg. Hence,
l = √((13 h^2) / (8 m ΔE)) = 6.63×10–34 × √(13 / (8 × 9.11×10–31 × 3.97×10–19)) = 1.41×10–9 m = 1.41 nm.
This value correlates well with the sum of bond lengths in the conjugated system – 6 double bonds and 5 ordinary bonds.