In this problem, we'll discuss some aspects of the most informative sense organ — vision. In Part A, we'll examine geometric optics and vision correction; in Part B, we'll study the properties of human rhodopsins; and in Part C, we'll learn about microbial rhodopsins. 

In your answers please explicitly specify final formula and numerical answer.

Part A. Geometrical Optics and vision correction

Binocular vision and distance measurement

Humans have binocular vision. Having two eyes separated by $s=60$ mm results in each eye forming its own image, slightly different from the image in the other eye. This difference allows us to consciously make quantitative estimates of distances to objects.

Measurement procedure:

1. We put our thumb at arm's length and look with one eye at the object to which we want to measure the distance.
2. We close the first eye, open the second, and look at the distance $\Delta x$ the thumb has moved along the object. For example, we're looking at a car, and the thumb has moved halfway across the car (Figure 1). We know that the characteristic size of the car is $L=4$ m, so $\Delta x=2$ m.
3. Next, we calculate the distance $r$ from the thumb to the object:

$$ r=k \Delta x. $$

A1  ^0.50 Determine the coefficient $k$. The distance between the thumb and the eye is $l=60$ cm.

A2  ^0.20 Determine the distance $r$ to the car from the example above.

Focal length of the crystalline lens

A3  ^0.70 A person observes a tree with a height of $h=3.0$ m at a distance of $d=100$ m. It turns out that the size of the tree's image on the retina is $l=0.6$ mm. Using this data, determine the focal length $F$ of the crystalline lens. The optical system of the eye can be considered to consist of a single thin converging lens (the crystalline lens) and a screen (the retina) on which the image is formed.

Farsightedness and nearsightedness

A4  ^0.40 Choose which lenses (Figure 2) are suitable for glasses to correct farsightedness. Farsightedness is a condition where distant objects are seen clearly but near objects appear blurred. Lenses are made of glass with a refractive index greater than that of air.

A5  ^1.00 A very nearsighted person's eye lacks accommodation, meaning this person's eye can only clearly see objects at a distance of $x = 25$ cm. Determine the focal length of glasses would be needed so that this person could clearly see very distant objects. Here we neglect the distance between the lens of the eye and the lens of the glasses. What lens shape (Figure 2) would be suitable for such glasses?

Refractive surgery

Modern ophthalmology allows for modification of the optical system itself — the cornea. In the procedure the excimer laser precisely "evaporate" tissue layers. A laser (Light Amplification by Stimulated Emission of Radiation) generates coherent radiation by stimulated photon emission in a medium with a population inversion, where the number of particles in the excited state exceeds their number in the ground state. Ophthalmology utilizes gas lasers with the active medium consists of short-lived diatomic molecules known as argon-fluoride excimers, which exist only in the excited state. When they decay, they emit light.

A6  ^0.30 The argon-fluorine excimer laser generates ultraviolet radiation with a wavelength of $\lambda=193$ nm. Calculate the energy of one photon of this radiation.

A7  ^0.40 The "evaporation" of microscopic tissue layers using a controlled laser beam occurs due to several processes, one of which is bond cleavage. Compare the obtained laser photon energy with the energies of chemical bonds in corneal biomolecules. Mark the bonds for which cleavage is possible under the action of such a laser.

$E_{\mathrm{(C-C)}} = 348$ kJ/mol
$E_{\mathrm{(C-N)}} = 305$ kJ/mol
$E_{\mathrm{(N-H)}} = 391$ kJ/mol
$E_{\mathrm{(C-H)}} = 413$ kJ/mol
$E_{\mathrm{(C-O)}} = 360$ kJ/mol
$E_{\mathrm{(O-H)}} = 463$ kJ/mol

Part B. Rhodopsins, color vision and color blindness

Molar mass of rhodopsin

Rhodopsin (visual purple, UniProtID P08100) is a light-sensitive protein responsible for colorless night vision and is located in rods (a type of retinal cell). The functional form of the protein contains not only amino acids but also an additional cofactor molecule: retinal. The form of this protein without the cofactor is called opsin (the apo form).

Entry (amino acid sequence) P08100 from the UniProt database:

$\texttt{>sp|P08100|OPSD_HUMAN Rhodopsin OS=Homo sapiens OX=9606 GN=RHO PE=1 SV=1}$
$\texttt{MNGTEGPNFYVPFSNATGVVRSPFEYPQYYLAEPWQFSMLAAYMFLLIVLGFPINFLTLY}$
$\texttt{VTVQHKKLRTPLNYILLNLAVADLFMVLGGFTSTLYTSLHGYFVFGPTGCNLEGFFATLG}$
$\texttt{GEIALWSLVVLAIERYVVVCKPMSNFRFGENHAIMGVAFTWVMALACAAPPLAGWSRYIP}$
$\texttt{EGLQCSCGIDYYTLKPEVNNESFVIYMFVVHFTIPMIIIFFCYGQLVFTVKEAAAQQQES}$
$\texttt{ATTQKAEKEVTRMVIIMVIAFLICWVPYASVAFYIFTHQGSNFGPIFMTIPAFFAKSAAI}$
$\texttt{YNPVIYIMMNKQFRNCMLTTICCGKNPLGDDEASATVSKTETSQVAPA}$

B1  ^0.70 Calculate the molar mass of the opsin P08100. Round your answer to the nearest whole number. Ignore any post-translational modifications. The molar masses of amino acids and their structures are provided in the answer sheet.

Absorption spectrum

The absorption spectrum of proteins in the UV and visible range (wavelengths $250$–$780$ nm) is determined by the amino acids that make up the protein and additional molecules (cofactors) that may also be part of the protein. Only aromatic amino acids (phenylalanine, tyrosine, and tryptophan), which are found in many proteins, absorb in this range, with an absorption maximum around $280$ nm. Cofactors typically have absorption maxima significantly different from $280$ nm.

B2  ^0.20 The figure shows two absorption spectra (Figure 3). Determine which form of the protein (opsin or rhodopsin) these spectra correspond to.

Rhodopsin extinction coefficient

The extinction coefficient $\varepsilon_{\lambda}^\mathrm{X}$ is a value that characterizes the absorption of light by a molecule $\mathrm{X}$ at a specific wavelength $\lambda$. This value is often used in everyday laboratory practice to quickly and conveniently determine the concentration of molecules in a solution. Absorbance $A_{\lambda}$ is related to the extinction coefficient and concentration as follows:
$$A_{\lambda}=\varepsilon_{\lambda} cl,$$where $c$ is the molar concentration of the substance and $l$ is the optical path length.

It's possible to determine the unknown extinction coefficient for rhodopsin by conducting a chemical reaction that produces a substance with a known extinction coefficient. Adding excess hydroxylamine $\mathrm{NH_2OH}$ and exposing it to light causes the following chemical reaction:

$$\mathrm{Rhodopsin} + \mathrm{NH_2 OH} \rightarrow \mathrm{Opsin} + \mathrm{Retinaloxime}.$$
The extinction coefficient of retinal oxime $\varepsilon_{365}^{\mathrm{RO}}=33600$ M$^{-1}\cdot$cm$^{-1}$. In the laboratory, the absorption spectra of the sample were measured before and after the reaction with excess hydroxylamine (Figure 4). The spectra were measured in the same cuvette.

B3  ^0.50 Determine the concentration of retinal oxime in the solution at the end of the experiment if the optical path was $l=10$ mm.

B4  ^0.50 Determine the extinction coefficient of rhodopsin $\varepsilon_{500}^{Rhodo}$ based on the experimental data.

B5  ^0.40 Determine the number of rhodopsin molecules in the solution if the volume of the solution $V=2$ ml, the optical path $l=10$ mm.

Color vision

Humans possess color vision because the retina contains light-sensitive receptor cells (cones) of three colors. Each type of cone has a maximum sensitivity near one of the three primary colors: blue, green, and red, because it expresses one of three opsin genes: OPN1SW, OPN1MW, and OPN1LW. The sensitivity (response magnitude to light with fixed intensity) of these color receptors is shown in Figure 5. If two different light signals excite these receptors equally, the colors of these light signals will be indistinguishable. For example, a mixture of red and green can be selected so that they are indistinguishable from yellow.

B6  ^0.30 Determine the ratio $r$ of the sensetivities of the long-wavelength $L$ and medium-wavelength $M$ cones to excitation by monochromatic yellow light ($580$ nm).

B7  ^0.80 Determine the intensity ratio $x$ of a mixture of red ($630$ nm) and green ($530$ nm) light so that this light is perceived as yellow from the previous question.

B8  ^1.10 The light now contains a mixture of three colors: red ($650$ nm), green ($547$ nm), and violet ($420$ nm). What intensity ratio ($y$: red/violet, $z$: green/violet) must be taken to produce a color identical to the monochromatic color corresponding to a wavelength of $500$ nm (blue)? Is it possible to obtain the desired monochromatic color by mixing three selected colors?

Genetics of color blindness

Color blindness is a reduced or absent ability to distinguish all or some colors. It is most often an inherited disorder associated with damage to one of the three genes listed above. The gene locations are shown in the table below; the damaged gene variant is recessive.

B9  ^0.50 A father and son have protanopia, but the mother has normal color vision. From whom (mother/father/unable to determine) did the son inherit his protanopia?

B10  ^1.00 A man with deuteranopia and protanopia married a woman with normal vision. They had a son with deuteranopia (without protanopia) and a daughter with protanopia (without deuteranopia). What is the probability of this marriage having a healthy child? What is the probability of having a child with both anomalies?

B11  ^1.10 A man with protanopia and tritanopia married a woman with normal vision. They had a son with tritanopia (without protanopia) and a daughter with protanopia (without tritanopia). What is the probability of having a healthy child from this marriage? What is the probability of having a child with both anomalies?

Part C. Microbial rhodopsins

Microbial rhodopsins are light-sensitive transmembrane proteins. Like animal rhodopsins, microbial rhodopsins contain retinal. The difference is that visual rhodopsins are found in animals, while microbial rhodopsins are found in bacteria, archaea, protozoan eukaryotes, and viruses.

Absorption spectrum 2

To date, more than 10,000 genes encoding various microbial rhodopsins have been identified. While similar in structure and homology, different microbial rhodopsins contain different amino acids, which determine their function and spectral properties. The figure (Figure 6) shows photographs of solutions of different microbial rhodopsins and their possible absorption spectra (in no particular order).

C1  ^0.70 Specify the maximum absorption spectra of the samples (with an accuracy of $\pm 10$ nm).

Proton transfer

Many microbial rhodopsins are proton pumps, meaning they actively pump protons through themselves when exposed to light, thereby transferring a proton from one side of the cell membrane to the other. A simple experiment to determine whether microbial rhodopsins are proton pumps is designed as follows. Microbial rhodopsins are heterologously expressed in the cell membrane of E. coli. A suspension of E. coli cells is placed in a stirred vessel (test tube, volume $V=3$ ml) containing an aqueous NaCl solution. A pH meter electrode is inserted into the vessel (Figure 7). When the vessel is illuminated, the microbial rhodopsins absorb light, transferring a proton across the membrane. The pumping rate of rhodopsins is constant.

The graph (Figure 8) shows the pH value as a function of time after the light is turned on. Before the light was turned on, the system was at equilibrium. Consider that the microbial rhodopsins are located in the inner membrane of E. coli, while the outer membrane allows protons to pass freely.

C2  ^0.30 Determine which direction (outward or inward) the proton is pumped in this experiment? Explain your answer using formulas and schematic diagrams.

C3  ^0.60 Estimate the pumping rate $q$. The pumping rate equals to the number of protons pumped per unit of time by microbial rhodopsins through all the cells of this experimental system.

C4  ^0.50 Estimate the surface area of one E. coli cell using electron microscopy image (Figure 9).

C5  ^0.50 Estimate the pumping rate $q_1$ through one molecule of microbial rhodopsin. The cell concentration in the vessel is $n_{cells}=6.4 \cdot 10^9$ ml$^{-1}$. Electron microscopy analysis of the membrane surface revealed that the concentration of microbial rhodopsin molecules on the membrane surface is $\sigma = 5 \cdot 10^3~\mu m^{-2}$.

Cell membrane permeability

It's clear that with prolonged illumination, the pH doesn't increase infinitely, but rather settles at a new value. This can be explained by the fact that the cell membrane is actually permeable to protons, and proton leakage occurs. The flux $j$ (the number of protons passing through a unit membrane area per unit time) is linearly dependent on the difference in proton concentrations $n_1$ and $n_2$ on either side of the membrane:
$$j=\alpha (n_1-n_2 ),$$here $\alpha$ is the membrane's proton permeability. Here, the concentration is expressed in units per unit volume.

C6  ^0.90 Based on the results of the experiment, determine $\alpha$. Indicate the units of measurement for $\alpha$.