 So, let me draw the traditional curves again. We have the energy. We know that the energy goes down right. When we fold, we have the entropy. I'm going to skip the details. The entropy also goes down. And then we want to think about what happens to the free energy. Well, if we have this nucleation condensation model, there is going to be some sort of region here that has formed the core. But that's going to be small. That's going to be very small. And then we have to think about what is how many native contacts do we have in here and how much valuable energy have we fold. So if we look at, say, delta E, that is going to go roughly, first as the volume here. I'm going to need to, I don't care what the proportionality is, but let's say to the very first approximation, it's going to be proportional to the volume in there. So I'll say that it's a constant one multiplied by the volume. But maybe that is too easy an approximation, because it's only the residues inside this volume that have all the beautiful interactions. The volume would be proportional to the number of residues. And for all these plots, I'm thinking of the number of residues that I've folded or density or something. But all the ones on the surface here, they don't have the beautiful contacts yet. So shit, I'm going to need to compensate for that. So I have to subtract some sort of other constant multiplied by the area, the surface of the small folding core. And if the volume is roughly proportional to the number of residues, that means that I'm going to have C1 multiplied by the number of residues minus C2. Well, the area is proportional to the volume raised to powers of two thirds. So this is going to be the number of residues raised to two thirds. Then we do the same for the entropy. In the entropy two, I have, there's going to be another con that let's call it C1 prime. The change here is going to be negative because this is the I'm losing freedom, right? The first approximation, that's proportional to the volume, the residues that are really stuck inside C1 multiplied by volume. But the entropy, well, if I have a small loop sticking out, the part sticking out here, they have also started losing their freedom, not quite as much. But with a bit of hand waving, hopefully you're accepting what I'm saying that with entropy, it's actually worse. Here I also lose from the part on the surface. It's not going to be good to be on the surface. So here I have C2 prime multiplied by area. So what that means is again C1 prime multiplied by N, sorry, minus C1 prime multiplied minus C2 prime multiplied by N raised to the power of two thirds. What that means is that if we look at energy first, first we had something that was proportional to N, but it's not quite that good. Particularly in the beginning, I'm not going to go down quite that quickly. So there will be a deviation here that is proportional to N to two thirds. Entropy, on the other hand, that's going to go down quicker initially. So here there will also be an extra term that is N to the power of two thirds. If we take those and say E minus S or TS, well, you know, these dashed lines, the first approximation, they will cancel, right? So that the remaining parts here, F sharp, is going to be roughly proportional to the number of residues in the folding core raised to the power of two thirds. This looks like a very minor difference. Remember when we talked, we talked about kinetics before, but now I want to say that the time it takes to get across this barrier, the time it took to get across the barrier, do you remember that that was roughly time zero multiplied by the exponent raised to plus delta F plus delta F divided by, let's say, RT. And I don't care about all the constants here. So that's some sort of T zero multiplied by an exponent. Let's call it A multiplied by N raised to two thirds. I don't care what units are in frame. This is what we should compare to Leventhal's paradox. So remember before we said that the folding time would be two raised to the power of N, if N was 100, but it was a very large number. And here I just say the folding time is E, well E is almost two first approximation, raised to something that is proportional to N raised to the power of two thirds. Am I completely crazy? I spent all these hours, you spent a few weeks just to get up with N raised to the power of two thirds instead of N. That's horrible. It's not. You have no idea how fast the exponential function grows with the argument, and that's the trick here. Two raised to the power of 100 is roughly 10 to the power of 30. Two raised to the power of 100 raised to the powers of two thirds itself, that's roughly 3 million. Do you see the difference? 3 million versus 10 to the power of 30. This is orders of orders of orders of orders of orders of magnitude faster that we would have predicted from Leventhal's paradox. And if you actually work out the exact personalities here, we can show that for normal small domains in the ballpark of up to 50 or 100 residues, we would predict folding times ranging from a few seconds up to maybe a minute or a few minutes. It fits perfectly. And that pretty much solves protein folding, or it solves Leventhal's paradox at least. And to sum that up, the proteins we have and the folds we have, the reasons for the size of the folds you observe is that they are that size because they can't be much larger to fold quickly and stably into beautiful stable functional proteins. That's what kind of what Leventhal proposed, right? Second, there will be many proteins that don't fulfill this, and they will not be stable proteins. Evolution will select against it. Sure, there are exceptions, but those exceptions are so rare that first we can ignore them, and the second part to think about is that it could lead to prions if you have two stable states. Why do we talk so much about folding? Well, folding was the problem where we introduced this, but this is equally true for say an ion channel moving between an open and closed state, or say a voltage sensor moving up or down in a membrane. If it's moving slowly in one direction, that means that it's an entropic process where we need to get across the barrier and find the right solution, and in the other direction it might move really fast. Then it's an entropic barrier that we also understand. So any type we're looking at, conformational transitions, all these things that we've learned are equally true, and in many cases we can use similar methods to study them. And that's of course the take-home message for you, that the field today is gradually, I wouldn't say moving away from protein folding, we're still interested in it, but suddenly computer simulations and experiments have become so good at studying, for instance, membrane protein transitions, is that there is an amazing ocean of fascinating problems there that we're working on. So get in touch with me if you want to do a PhD on it.