 So we spoke about the hydrogen bonds in the helix already in the last lecture, but let's do this using slightly more of our free end denominator now. What is the free energy of forming? Well, you already know that the first hydrogen bond here between residues zero and four that would lock in residues one, two and three. That's one hydrogen bond. If I add a second hydrogen bond, I will lock in three more up to residues two, three and four. So in general, I will have that N residues are stabilized by N minus two H bonds, right? So if we say that what is the delta G for the helix, that's going to mean for N residues, first I have N minus two hydrogen bonds and that's good. That's a let's call that F. That's a good term. I'm gaining energy enthalpy for a hydrogen bond. But to do this, I had to take N residues and lock that into helical shape, right? Locking them into helical shape means losing entropy. So there is going to be T times S here. So it's N residues multiplied by the temperature multiplied by S alpha. That is the entropy difference of moving one residue to a helix. So we have N in two places here. We can write this in a slightly different way. So what if I write this as minus two F H bond, that's constant, plus N multiplied by F H bond minus temperature S alpha. What this gives me is one term here that's constant and one that depends on the number of residues in the helix. And we can write that as delta G in it plus delta G elongation or extension. That's what the book does. So that means that there's one component for the entire helix stability. There is one component that's kind of fixed when we're starting the helix and then a second component that depends on the elongation and that one which is multiplied by the number of residues we put there. Because then we can study this as a function of the residue. The book and a few other places, we can look at this in a few different ways. The probability then of something being in a helix, if I let me draw that up here. The probability of being in a helix, that's e raised to minus delta G in it divided by kT multiplied by e raised to minus delta G elongation. I'm going to need to make that short. So there are kind of two components here. One component that determines the initiation and one that determines the elongation. The second one we're going to need to multiply by n right. And that means that we can kind of consider this term separately. So if we call this something, we call that that is a parameter sigma. And the second one is a parameter s raised to the power of n. So sigma is some sort of equilibrium constant describing how likely it is to initiate a helix. And the s parameter is something that describes for each residue here how likely is it to elongate that into a helix. And then we have to raise that to the number of residues. I'm not going to talk that much about sigma and s. They're just convenient parameters that the book uses, but you might see them in a few places. So in principle, armed with this, we can study the whole equilibrium process when this helix moved from either being a helix or just being an extended coil and see where that takes us if we can say something about these properties. We're going to need something else. It's kind of hard to measure this. So we're going to need to measure how much helix there is in something. And there's a beautiful and very simple method that we can do that with. Remember that I mentioned the property of rotating polarized light. There is a simple method called CD spectroscopy for circular dichroism. And because helices and sheets, their amino acids have slightly different orientation, you're going to have a very characteristic spectrum for alpha helix, second spectrum for beta sheet and the third spectrum for random coil. So if I'm now starting to say change the temperature of the helix, I can observe quickly and cheaply and simply in this equipment roughly how much of my sequence is alpha helical versus how much of set turn into coil. And that's going to allow us to get to these numbers the backward way. But I'll do a little bit more math first. So by specifying the stability in terms of a cost for initiating it, a cost for elongating it per residue and then the n term saying how many residues I have, I can look at three different cases. So if I look at some sort of delta G here as a function of the length, I will have some sort of initiation energy and that's roughly constant. Here we have n. So first I will go up, I will pay my initiation energy delta G in it. That's going to be bad, right? Always bad. If this was positive, helices would form everywhere, would be enough to have one residue helical like that. We know that's not the case. But then there are three possible scenarios. First, assuming that you had a long sequence that were just pro-lens, in this case they would hate to be helix. So I would always pay by extending it. The more residues I try to put in a helix, the worse it would be. So in this case delta G would always be larger than zero. We will never see any trace of a helix. The other possibility is that delta G in it is still positive but then I'm adding a few residues that kind of like to be in a helix but it's not really, not really gigantically positive. So in here, sure, it's better to have a helix of that length than stopping there but the point here is that delta G here is still positive. So this will not spontaneously form a helix. If I force it to go here, it might go here but eventually I would prefer to go back and not have any helix at all. So this is still bad even though we're starting to go downhill. While the third obvious case is of course if I extending this but the extension energy correspond to residues really favoring helix, at this point it's going to be better to be in a helix say of length 20 here than it would be to keep them in the coil and in this case the helix will form. The only question is how good is this and how high is this barrier because the height of this barrier will kind of tell us how long this is going to take in terms of genetics, right? We can study that and it's surprisingly easy for a helix but the helix behaves slightly different to some other things. So I'm going to need to make a bit of an analogy first.