 When I first introduced the molten globule, I did it from the concept of hydrophobic collapse, which is still true. But now I'm gonna look at this in a slightly different way. What happens when I'm actually bringing things together and what is the influence of the protein actually being a chain? It turns out that this chain literally just means that the protein is a polymer. Just like a piece of plastic, right? But this does not have any of those nice protein properties. It doesn't really have any well-defined function or packaging or anything. It's kind of boring, so let's throw it away. We can study this just with hand-waving and showing you roughly how the energy or entropy should behave. I'll get to that in a second. So if you don't want to follow the duration, just forward a bit. But those of you who are interested, I'm gonna write down a few things. So one case is that I simply have beads that are not connected. And then I want to add a new bead to this system. And I'm just gonna need to invent a few things here. So I will say that that new bead is gonna have the energy epsilon and it has the volume omega. You can use any number you want. And that means that if I have a total number of particles N and a total volume V, some sort of volume in the system, V and N, that means that I can say that the density I have is rho and that's gonna be the number of particles multiplied by the volume of each particle divided by the total volume, right? So that will go from zero to one. When it's one, I'm crowded in here. So what happens when I'm adding one more particle to this? Well, when we're starting the volume available, that's gonna be the total volume divided by N particles. But some of that is gonna be taken already, right? So the part that is taken is the number of particles I've already added multiplied by the volume per particle. So what should write? Once I'm adding this new particle, I'm saying the available volume is roughly equal to the total volume available minus the part I've already occupied. And then I'm dividing per particle. Again, volume available particle. That equals, if I use the expressions above, that's volume divided by N one minus the density, right? Because the volume divided by N there, minus omega divided by N, which is the volume divided by the density. Sorry, volume multiplied by density. And then I can use the volume multiplied by the volume there and dividing by the density. That's gonna be omega divided by rho. So I can simplify this even further. That's omega divided by rho multiplied by one minus rho. And now all the large V's and the large N have disappeared. I only have to care about volume per particle and the density, which is a number going from zero to one, as I've been adding particles. This I can actually plot and see what happens. Why would we like to plot that? Well, there are two things we're interested in here, right? I would like to understand how the free energy is changing as I'm adding that particle. And God, I mean, by now, I hope what equation you should use. You should F equals E minus TS. So we need to look at how the energy is changing and how the entropy is changing, right? But the entropy, that was just really the logarithm of the volume. So there are gonna be two parts here. And I'm gonna run out of space here. So I'm gonna need to erase this. But if we look at the energy first, if I starting from some sort of number, for every particle I'm adding, I get epsilon. So the more particles I'm adding, the lower the energy will be. So some sort of density here. As I'm adding more particles, the density will just go down linearly. Boring, but simple. The entropy, on the other hand, ooh, this is gonna be more complicated. Let's put it as a function of density. And I'm not gonna be, so delta S is gonna be the logarithm of this, but I'm not gonna worry too much. If the density goes towards zero, this is gonna be one over zero. So when the density here is zero, this is gonna be a very large number. And the logarithm of infinity is still infinity. So there's gonna be something up here that's very large. All the way up. As the density goes towards one, well, eventually we'll say zero here. I can't take the logarithm of zero. So I can't really get all the way to one. But I could get so close that when the system is perfectly packed, there's only one way of organizing things. And that means that it's just one way of doing it. And then the entropy would be zero, right? So at some point, the entropy is gonna go to zero and then there will be some sort of shape between them. And I was just guessing here roughly how it looked based on the fact that I know that equation. If we now plot those two in terms of the free energy, F equals E minus TS. It's gonna depend a little bit on what the temperature is. So depending if the temperature goes from low to high, when the temperature is very low, well, when the temperature is low, that means that I don't have a whole lot of influence from the entropy term, the energy will dominate. You can actually show that you will end up with a quay, something that looks roughly that way. And then if I'm increasing the temperature a bit, it might start to look that way. And eventually if I keep increasing the temperature, it's gonna look something like that. So what's we say? Low T to high T. This is beautiful, we love this. This is a phase transition. Do you see here that at high density, this date is gonna be best. And then at some point when we're changing the temperature, it's gonna be better to jump over to the state of low density. And that's actually what happens. For instance, if a gas is undergoing a condensation or something, plain and simple. In a system with some sort of particles that just attract each other, when they start to condense into a bowl or anything that's making them attractive, it will undergo a phase transition. Cool, now we just need to repeat this for the case when they are connected by a chain. But then I need to erase this first, but remember that plot. So if we repeat this for a chain, the first part is actually gonna be easier. Now I have a chain here and at some point we're adding one new monomer to this chain. And we wanna investigate exactly the same thing, what happens here. So the energy is again epsilon. And then we have the volume omega. And we also have N and V, but I'm not gonna use them. I'll save you some time there. The introductory part here is exactly the same as it. The volume available here is really gonna be one minus the density, right? The difference though is that this particle I can't put anywhere. I can only place this in sort of small region that depends on the distance since the previous residue I added. So I can't add it to the entire system. So the available volume here is gonna be roughly some sort of constant omega. Think of omega as the volume available around the previous monomer, multiplied by one minus density. And this just means that if I'm already crowded the entire system I can't really add anything more. And when I'm starting out, the entire volume here is available. You can derive that properly the way I did it a few minutes ago. This looks almost like previous equation, but I don't have one over row here. So if we just, that means that the entropy then, entropy equals K ln volume. And if we don't care too much about K and everything, that's gonna be omega multiplied by one minus row. But omega was just a constant. So if I plot this the same way I did in the previous slide, energy is gonna look exactly the same. Goes down. Entropy, well, when omega, sorry, when row is very close to one, I'm gonna have the same shape there. Then it will reach one specific state. But when omega is zero, this is just gonna be a constant. So I would get, go to some sort of constant value for the entropy there. It's not gonna go to infinity. And what this means is that depending on temperature, you're gonna get something that looks roughly like this or like that, or maybe like that, where that is low and that is high. You see, it looks completely different. And in particular, there is no phase transition here, which is horrible. We don't have a beautiful phase transition anymore. Let me erase this and draw the other plot right next to it so I can compare them. So just to recap, this is when we had a chain. And here's when we had a, should we say a cloud. So in the case of cloud, energy looked exactly the same way. Entropy on the other hand looked like that, went to infinity. And F equals E minus TS had a shape like that. So you actually had a proper phase transition or at the different temperature it might look like that. The reason why this happens is that once, if you're in a cloud, in terms of entropy, the best possible state for all our molecules is to be infinitely for almost infinitely far away from each other. If the volume is limited, I can't go there. But in the world, the further away you are, the more ways we have to organize. If you're in a chain on the other hand, we're bound together. So there is an upper limit and you really can't get further away. And it's really this upper limit that means that we don't have a proper phase transition. I can't get to the point where I'm infinitely far away. So what really happens in the multi-globular is not that you have a phase transition. The phase transitions happens here when molecules first don't interact at all and then they start to interact, condense. In the chain, they are already interacting. They can't get out to infinity. So what happens when I form the multi-globul is really that I'm kind of freezing it in. I wouldn't say freeze, but I'm starting to find some sort of local minimum, but it's continuous. And that's what happens in this hydrophobic effect. It's better rather than being stretched out than have the hydrophobic interactions with water. The hydrophobic effects means that I collapse and at most temperature, that's going to be advantages, but it's not the advantages in the sense of a phase transition. There is no phase transition when we're moving between coil and multi-globular. But for proteins in general, the chain is imperative. If you were not the chain, for most combinations of amino acids, it would be better to just spread the amino acids infinitely far apart. And that would mean that it would be extremely advantageous to be in the completely unfolded state, right? And that would mean that you would need insanely good energies to ever make the folded state. So this chain aspect is kind of important for us or being a polymer, that's important for us to actually be able to fold proteins in realistic time. But the chain, per se, does not make a protein. The chain just means that there is going to be an effect that we condense together, but there is no uniqueness here. There is no magic, there is no phase transition. That will have to come from elsewhere.