Humans eyes have evolved to perceive light only between approximately 350-700nm, because that form of light is most common to our lifes. Other animals can perceive lights with slightly different fequencies. Can eyes theoretically evolve in a way that they can observe all light as we know it today? If not what would approximately be the maximum amount of frequencies that can be perceived?
Answer
There are some limits on what light can be detected biologically based on physics and chemistry. Although there are animals that can sense more UV or more infrared than humans, they are still subject to these limitations.
How Vision Works
The way most types of vision operate is by using photons to cause a conformational change in an opsin protein by influencing a chemical bond in a sensitive molecule (a chromophore, for example retinal, which changes from 11-cis to all-trans) bound to the opsin. To sense a particular wavelength of light using this mechanism, it is necessary that the energy of the light is sufficient to cause this chemical change in the opsin-chromophore, without being enough energy to destroy the molecule entirely by breaking a bond.
Sensing low-energy (long wavelength) light
Longer wavelength light ('infrared') is also known as heat. Detecting "heat" photons is difficult because they don't have much energy. Therefore, you need a molecule that is very sensitive. It turns out that the opsin-based vision strategies use chromophores that are too stable to sense long wavelength light.
Some snakes have particular sensitivity to infrared, for example the pit viper. However, these animals do not "see" infrared using their eyes: they have another specialized organ called the pit organ. The pit organ uses receptors that are much more like the heat receptors in your skin than the photoreceptors in your eye, but the pit organ uses a pinhole camera effect to organize the incoming light onto a very sensitive heat-sensitive membrane.
Humans of course are able to sense infrared, too, but the same way: through the skin rather than the eyes, and at low resolution. However, if you stand outside in the sun and close your eyes, you can probably still "see" the direction of the sun: one side of your body will feel warmer than the other! In a way, this is like a form of low-resolution "vision" if you want to define that term very broadly. The snake's infrared sensitivity is more like this sensation than eyesight.
When you get to very long wavelengths, for example radio waves, the wavelength is too long and too low-energy to even be relevant at a molecular level: these forms of light will travel right through cells, and individual photons don't carry enough energy to influence molecules in ways that can be sensed by biology.
Sensing high-energy (short wavelength) light
Very short wavelengths of light have the opposite problem: too much energy. Ionizing radiation (from the upper ultraviolet range and shorter) has enough energy to actually free electrons and create ions. This type of radiation damages biological molecules, including DNA, so organisms have a lot of good reasons to avoid these types of radiation. Exposure to short-wavelength radiation can also cause bleaching (permanent destruction) of photosensitive molecules and potentially damaging heating of tissue surrounding pigments, in addition to direct effects on other molecules in the cells.
Insects and other organisms that can see into the UV range see mostly in the near-UV range, >300nm, not too far from our own vision range (roughly corresponding to "UVA"). However, it isn't true that humans can't see these wavelengths due to some lack of sensitivity of our photoreceptors: instead, UV light is (at least partly) blocked by the cornea and lens, presumably to protect the retina from damage!
Humans and other organisms do have some sensitivity to damaging UV wavelengths, just not through vision. Instead, humans and other organisms sense UV based on the damage done within cells, which recruits repair mechanisms and apoptosis to prevent damage from leading to cancer.
Summary:
The chemical processes behind vision determine the range of potential wavelengths that can be sensed. Animals on earth are able to sample from this entire range. Mechanisms to sense wavelengths outside that range would require wholly different approaches, and maybe even a unique biochemistry. The mechanisms would have to be so different that we probably wouldn't refer to those mechanisms as "vision" (for example, even the infrared sensitivity of the pit viper isn't really though of as vision) but we would come up with a new term, such as "radio sensitivity" or "radiation sensitivity" just like we have different names for "hearing" relatively high-frequency pressure waves with the ears but we have "vibration sensitivity" for low-frequency pressure waves felt through the skin.
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