 So this far we've started the helix, we've started the sheet and we've started that special very tight turn. There is a fourth part that I want to mention just for completeness. So we have helix, you have sheet, we have turn, but then we also have this elusive coil. So you might think of the coil as a chain just straight stout. Don't do that even if I occasionally draw them that way because if you take all those fine psi torsions and put them in the straight stout 180 degree states that is just one specific microstate. The probability of finding any specific chain completely straight stout is zero to first approximation. So in principle it's going to be more curled up and randomly curled up. There are some pretty neat ways to estimate roughly how large the size of such a curl would be and I'm going to take you through that because it's a simple and very beautiful derivation that leads to a much cooler physical result. In general if I have a collection of amino acids this is going to be complicated. Each amino acid I can draw as a link and then I start from link one and I go to link n. These can't overlap and that's going to lead to a complication but if I initially ignore that and assume that they can overlap if I go to link n the question is what is the average distance between the first and the last residue going to be here? Well that's not too hard right? I can say that if h as a vector that's my entire chain here that's a sum over all my links sum over all the links I if I had m links and each of these has a distance r i vectors. Amino acids they all look the same so it's going to be a very good approximation target that all these links the first approximation have the same length but then I can calculate the average here or at least the square of the average so the average is going to be the square root of the average of the square. So let's say you want to calculate the average of h squared well that's going to be that sum squared right? That's going to be the sum of i to m of r i j r sorry r i sorry not squared we square the entire sum that means I have two such terms then the second term here I'm now going to have each term here I'm going to multiply by each term in the second one. I can write that as two sums so I have one sum where I have all the squares right r 5 multiplied by r 5 r 17 multiplied by r 17 the ones where the indices are identical plus a second term sorry we're going to need to be an average there average there average there plus a second term here that's a double sum over all the i's not equals to j to m and then I'm going to take the average of that and those two are vectors this simplifies in a very beautiful way that's why I want to take you through it in general if I have one vector and then I take another vector and they are completely unrelated what is the average of their scalar products going to be that entire term will disappear because on average they're going to be zero and that means that this entire average is just really the average of the sum over all elements r i squared but again if all of them have the same length then I can just say that corresponds to m the number of residues multiplied by the average of that square so that and if I'm instead interested in the square root of that so that the square root of the average of h 2 again to get the estimate for the length the point is that's going to be proportional to the square root of the number of residues multiplied by the length of each residue so the point here is that I don't really care about the length of each residue the interesting aspect is what happens when I take n or sorry I said m of these so as this chain grows the average end to end distance will only grow as the square root of m and that's a surprisingly accurate estimate you might argue that this is completely horrible you can't assume that chains can't overlap and that's quite true in principle you will also proteins my amino acids are not completely free to rotate here but remember I wasn't really interested in the specific number or was so that instead of having each amino acid here you could argue that if I move from one amines acid to the second amino acid there's going to be a large correlation with them but by the time I moved say four amino acids apart I'm going to have enough freedom that I can consider them independent so I could think of this as rather the individual residues some sort of persistence or correlation length that is actually four amino acid residue that would change the average length of the links are but it would not change the asymptotic dependence of how this grows as a number of the the number of residues still assuming that they can't overlap is an oversimplification uh paul flory who will say very famous polymer physics went much further into this and show that you can derive even theoretically that if you take this volume exclusion effect into account this value should rather be 0.5 square root it should be roughly 0.6 and today with computers we can show that it's roughly 0.588 with critical exponents effect and again 0.588 it's pretty darn close to 0.5 the take-home message of this derivation is that if you just throw a chain into a coil it's going to be much more coiled up randomly it's going to be packed think about that hydrophobic effect and the so-called molten globular do not think of as a chain as stretched out to one side to the other and this is entirely driven by entropy the chain has a lot of freedom in coil and the reason why it likes to be in coil is because it can use that entropic freedom