Modeling the response of photoreceptors

Modeling the response of photoreceptors

Our goal now is to develop a way of making quantitative statements about the response of a photoreceptor to a particular light. To do this, we have to formalize the way we talk about light, the way we describe what a photopigment does, and what happens when the two things interact. So far, we’ve been thinking about these things intuitively, but to go further we have to be much more concrete.

Let’s start with formalizing the way we talk about light. We’ve already discussed that there are different kinds of light that differ from one another in terms of their wavelength, which seems to have something to do with color (remember that red and blue light diffracted differently). We also know that light of any wavelength can sometimes be bright and sometimes be faint, this related to the amplitude of light if we’re thinking about light as a wave, and its related to how many light particles (or photons) there are in a stimulus if we’re thinking about light as a particle. Finally, we also know from using our diffraction gratings in Lab #1 that most light sources contain light with different wavelengths. “White” light from a fluorescent tube turns into a rainbow of colors when viewed through a diffraction grating, for example, though red laser light just looks red. To describe a particular light in a useful way, we’ve got to cover all of these bases: We need to describe what wavelengths of light are present in a stimulus, and we should also describe how strong light is at each wavelength. I’m going to argue that this means we should describe light using a list, which we’ll call a spectrum. This word can refer to the kind of list we’re making, but it can also refer to the actual pattern of light and dark stripes you see when you spread out the different wavelengths of light that are present in a mixture of inputs (Figure 1).

 Figure 1 - An emission spectrum of light. The position and color of each stripe shows us what wavelengths of light are present in this stimulus, while the brightness of each stripe tells us how much of that light there is. We want to make a numerical version of this pattern.

Our list will have an entry for each integer wavelength in the range of visible light (let’s say 400nm-700nm), and the magnitude of the number that we put there will tell us the strength of the light at that wavelength. Let’s say a value of zero means that there’s no light at that wavelength, while larger values (say 100, in arbitrary units) means that we have quite a lot of light at that wavelength. In general, a particular light spectrum would look like this:

Now that we’ve got a language for describing lights, we need to think about a language for describing what a photopigment does in response to a light shining on it. To get you thinking about this, I want to remind you of some of the observations you probably made in Lab #3 when you were exposing sunprint paper to different kinds of light. My guess is that you probably noticed the following things:

1)   Exposing the paper to the sun for more time led to a darker mark.

2)   Making sunlight pass through a grey filter led to a fainter mark.

3)   Red and green laser light probably did very little to the paper.

4)   Blue laser light probably led to a very dark mark.

5)   Ultraviolet light (if you could get a good blacklight!) also probably led to a dark mark.

The important thing about these observations is that they indicate that the response of a photopigment depends on both exposure time and wavelength. The influence of exposure time should be easy to understand: Exposing the paper to the sun for more time is also exposing it to more light. By playing with time in Lab #3, you were really playing with amplitude in a way. If you think of light as photons, for example, exposing the paper to the sun for a longer period of time just means that more photons can ping into the paper. The influence of wavelength is more interesting, though: In some cases, it doesn’t matter how much time you spend shining the light on the paper, it seems to never change. That is, you could be shooting tons of red laser light at the sunprint paper for minutes on end, but all those photons don’t lead to any kind of response. This suggests that whatever description we use for a photopigment, it’s going to have to take into account the wavelength of light that’s shining on it.

Figure 2 - Some kinds of light will allow you to make a sun print like this one, but other kinds of light won't change the paper at all. Describing the response of the paper means we have to describe these effects that differ as a function of wavelength AND amplitude.

I want to suggest that it’s probably a good idea to describe photopigments using another list. It will look a lot like the lists that we made to describe lights, but there are some crucial differences in terms of what’s actually in the two lists and what the different entries mean. Like our description of light, I’d like to have an entry for each wavelength that we’re interested in – let’s use the same range we agreed upon for light (400nm-700nm). What I want to do next, however, is try to figure out how well light at each wavelength actually does something to the photopigment. If we’re talking about our sunprint paper, maybe what I want to do is come up with a number of photons that I like (let’s say 1000) and for each wavelength, shoot that many photons at the paper and record what happens. If the paper isn’t any different, I’ll put a zero in the list. If the paper has a really dark mark on it, I’ll write down a ‘1.’  When I’m done, my list might look something like this:

This list (or spectrum) is telling me something different from my other one, and to understand what that is, it’s worth knowing a little bit about real photopigments. The photopigment that’s on your rods is called rhodopsin, and when light shines on it, it absorbs some of the photons that strike it, which causes structural changes in the photopigment molecules (Figure 3). The word I really want you to pay attention there is the word absorbs, because that’s the key to understanding this second list we’ve generated. The photopigment molecules only change shape when they absorb photons and not all photons are easily absorbed by these molecules! Oh sure, some of them are readily soaked up, but others aren’t at all and others are somewhere in the middle. How to make this more concrete? Let’s imagine that we shoot photons at rhodopsin molecules and count how many at each wavelength are actually absorbed. If we know the total number of photons, we can calculate a fraction that tells us what proportion of photons at that wavelength actually “stuck” to the photopigment molecule. That is what the numbers in this second list are, and that’s why they only vary between 0 and 1.

Figure 3 - When rhodopsin absorbs photons, it undergoes structural changes that lead to signaling in the retina. But not all photons of light are absorbed easily!

So now we’ve got ourselves two kinds of spectra: We have a light spectrum to describe lights and we have an absorption spectrum to describe photopigments. Now the big question: How do we use these to describe what a photopigment does in response to a light?

Here’s an assumption I’m going to make to get us started: The response of a photopigment (or photoreceptor with photopigment on it) depends on the total number of photons that it absorbs. You’ll notice that I’m using particle-like language here, but I’ll still need to keep wave properties in mind this whole time. We need both, so I won’t apologize! How do I use my two lists to come up with this number?  The total number of photons absorbed will have to depend both on how many photons there are to be absorbed (in the light) and how good my photopigment is at absorbing photons of that wavelength. For example, let’s say that my light had 80 photons with a wavelength of 410nm. That’s all well and good, but if the photopigment only absorbs 10% of 410nm light, we’ve only got 8 photons that were absorbed at that wavelength (80 * 0.10 = 8). We can work this product out for every wavelength of light that we’re interested in: Multiply the number of photons at that wavelength by the absorption rate, and this will tell us the number of photons that actually “stuck.” Once we’re done, how do we get the grand total? We just add all those numbers up. This means that if we have our two lists, the recipe for calculating what a photoreceptor does in response to a specific light is as follows:

 Or, we can write this more compactly like this:

This is called the dot product of the two lists. We will absolutely see dot products again in other contexts, so get comfortable with this operation! Here, it gives us a single number that describes the strength of a photopigment’s response to a light. This is important because we’re going to use these numbers to examine what you can and can’t do with your different photoreceptors – remember, this is the language that these cells use to describe lights! That light spectrum is fine for describing a light out in the world, but once it gets to your retina, these new numbers are going to be all you have to work with! Next, we’ll talk a bit about the consequences of this transformation, and how we can keep working with these photoreceptor numbers to explain what we see.