 This is both a very fun, potentially difficult, but also enlightening part of the class. It's probably the lecture I like most. We will have some slides with equations today. Don't worry too much. Actually, there are only two somewhat difficult mathematic parts, and they're not difficult in the sense that the equations are difficult. They're difficult because they are abstract. But for both of these, I have made a sequence, a much slower derivation on an iPad recording on the model page, so you can go through them at any pace you want, and if you have questions, don't hesitate to shoot me an email. I can easily make that explanation 10 times longer there. But to avoid overloading you with that, I might deliberately go through them slightly quicker so you don't feel that I'm spending 40 minutes just talking about equations. I'm gonna start with a couple of the last test slides or so we didn't have time to go through yesterday. We spoke a little bit about the hydrophobic effect and I started to hint about the rule of entropy here, at least. I haven't talked so much about the hydrophobic effect and protein folding, I'm gonna come back to that. But I did introduce the Boltzmann distribution and how that relates to energies. So that, you're not quite there yet, but you're partly armed with two parts. We know what proteins look like, the biology. And in addition to that, we started looking at what are the energies involved here. But they're just energies in the sense of physics that you can calculate an energy. The problem there is that the energy scale is infinite. It literally goes from minus infinity to plus infinity. And right now you don't have any gut feelings what energies are important. The problem in biology, in physics you could argue, well it's always good to have lower energies, but the fascinating thing in biology is that we have an involvement of the time scales. So energy levels that are real, energy differences that are really small don't matter. But energy levels that are really big don't matter either. So it's these intermediate energy levels that are interesting because they're gonna determine everything whether things happen or not. And that is really the other side of things, the Boltzmann distribution. Because the Boltzmann distribution is the component that will determine what happens and what does not happen based on what energies we have involved. But before I continue with that, I'm gonna, let's go through some of these study questions at least. If we look at the chemical bonds, why do they form in the first place? Profound silence. Yes, but in fact, my idea is that it's due to electrons, right? Electro favorable configurations of the electrons and that's the explanation of every single molecular interaction we have. And once we have decided that, we pretty much forget the electrons and describe the interactions in simpler ways. So if you, I think we did discuss the few interactions related to bonds between atoms so that would be bonds, angles and torsion so that let's skip number two. If you look at the non-bonded interactions in the protein though, what are they? There are two classes of them. Fund, sorry, Fundavals and? Well, I wouldn't say random, but electrostatic forces. Oh, London, yes, so London dispersion, that is included in, oh, this is real. So the London dispersion forces, that's the dispersive component of Fundavals interactions, it's quite correct actually. But that's things that relate to dipole-dipole interactions, anything that's not charged. And in that case, they're gonna dominate. For anything that is charged, the electrostatics would dominate because it's so much stronger. Now, on the other hand, electrostatics, things, if you look in this room, what interactions would dominate here? It gets complicated on a microscopic level, but if you look at the semi-microscopic level, an important difference, while electrostatics is much stronger, it's typically only on a microscopic level that you have free mobile ions, right? There are a few, that table doesn't have a plus one charge. So while electrostatic forces are large, on macroscopic scale, they tend to average out because you have roughly the same number of positive and negative charges. So that's why in many systems, the Leonard Jones interactions can actually be more important because they're all attractive at long range. So if you look at the strengths of some of these interactions, I kind of already gave it away to you that electrostatics is really strong. Which one is the weakest? Fundavals, yes. With this caveat that they had. And there are also the interactions that we have the most often. I think we also compared electrostatic with dispersion. That's the London forces. But another way of saying this, which ones are the most important for proteins? It's not, sorry? Why? Yes, I would say hydrogen bonds is fine. There's another answer that I think would have been perfectly fine too. Roughly the same order of magnitude as hydrogen bonds. No, so would peptide bonds break in form in a protein typically? So here, I think it's a beautiful example for that point of view, right? Peptide bond is super strong. But because it's so strong, once it's formed, it's no longer relevant. And if it's formed, it's formed, it's formed. It's never gonna break inside a protein. But you have the torsions, right? So the torsions are, they're weaker than the bonds in the sense that the torsions can actually reconfigure. And that will change their energy. And because they're roughly the same ballpark as the hydrogen bonds, these are the intermediate forces that are gonna be governed by the Boltzmann distribution, which is complicated, but also very fun. If you, we can discuss this in terms of an energy landscape. What is an energy landscape? You're right, but well, it's not necessarily limited to proteins, but what are the components, the functions? So it's a three dimensional landscape. What are the X and Y axis? Yes, or the whole point, some sort of coordinates. They can be generalized coordinates, so they can be a torsion angle or something, but the whole point is specific coordinates. So one point in an energy landscape, you typically have a completely frozen in specific conformation of a molecule. And that's what I somewhat introduced yesterday, but we're gonna talk more about it. That's a microstate. It's once when nothing moves, it's uniquely perfectly defined at least in theory. And at that point, we can only talk about energy or potential energy or enthalpy even. But we're having the next question. We started to introduce the concept of entropy. So what is the relation between energy and entropy in these energy landscapes? I deliberately didn't talk about energy. So that's, I think this is a good example of these study questions, right? I try not to have a study question that corresponds exactly to line four or slide 15. So imagine an energy landscape that looks like this. You have a small hole there and then a hole there. Which one is best to be in? Which, where you're gonna have most of, if you have a particle here, where it's most likely to find the particle? Yes, good, you're not asleep. If you have, let's see here. If you have something that looks like this versus like this, where is it more likely to find the particle? Why they have the same energy? Yeah, and that's of course the problem, right? That if you only talk about energy and a specific coordinate, the Boltzmann distribution doesn't really take this into account. Or rather, it does take it into account. But the way the Boltzmann distribution takes this into account that this is state one, this is state two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16. And of course, if you enumerate this, that's all the different state it is. The Boltzmann distribution will take it into account. Because it's not at all more likely to be here. It's just as likely to be in state one, two, three, four. It's just as likely to be in state five as in state one. But you can also be in state two through 15. But then of course, it's human beings, right? That's the way I fool you there. It's nice to think of this is, we can say that this is A and this is B. That's also a state, right? But it's a different kind of state. So state A here just corresponds to a single microscopic state. But state B corresponds to more states. And of course, this is really, who on earth would be so stupid that if you have two causes, why would you call them state and state? Yeah, that's physicists, sorry about that. I can only apologize for a few generations before you've me included. And then we try to keep these apart. And one way, when I wanna be proper about this, I tried to call these microstates. So these should be states that are exact, right? There's no multiplicity or anything in a microstate. And then we want a completely different name for the other state. So let's call that macrostate. So apart from the pronunciation similarity, it's really stupid to call these. But again, I wish this was not the case, but this is what we've decided to call them. And I would be fooling you if I used a different definition. So when we talk about macrostates, that's what we mean these larger states say that we talk about them in the lab or something that internally might have some freedom for the coordinates. We're gonna come back to that a lot today. And then because not just me, the entire world is sloppy. So occasionally we just talk about the state. And then we hope that it's gonna be obvious which type of state we talk about. It usually isn't. So this space, spatial degree of freedom in the amount of volume, that's really what we're getting to with the entropy. And yeah, and I perfectly illustrated this already. So on the next line, I'm sloppy and then I talk about states and microstates. So if I, when I wanna emphasize one of them, I tend to emphasize microstates because a state is what we would have in the lab. So if you just see state as hopefully a macrostate, maybe. I will actually, based on what you know now, how would you explain entropy? Well, you might explain it that way, but that's not the answer I'm looking for. Remember what I say yesterday, entropy is not difficult. Don't fall in the trap of trying to juggle about though it's disorder or anything. It's not complicated. There's a clear definition. Yes. Yeah. It's just logarithm of that number. That's gonna be the correction factor that appears in the Boltzner distribution. It's not more complicated than that. Actually, it will look really beautiful. You're gonna see this will be beautiful in 20 minutes when I go through it. But it's literally a correction factor so that the Boltzner distribution normally realized in using microstates. If we take these correction factors into account, we can use the Boltzner distribution to talk about macrostates. And it's literally to be able to handle that definition. And then we fool you by trying to hand wave and say this is something that you can explain kind of like temperature or feeling or something. It's a purely mathematical construct. But it's a construct that helps you. It's easier to, here it's easier to consider two states than 16. We spoke a little bit about the hydrophobic effect but I'm gonna talk a lot more about that today so I will deliberately skip that. But what can you use the Boltzner distribution? Yeah, well, we're not gonna care about electrons in this class. Electrons don't exist anymore, I just killed them. More generally. So the cool thing with physics in general has to account for all time scales. The neat thing with biology is that we're much more limited, we're looking at life processes. And that's in contrast to physics, I don't have an infinite amount of time. Today we have two hours in my life, I have like 90 years, like 40 years or remaining. And that means that we can certainly say that time scales are interesting. So if you look at the Boltzner distribution, you're comparing things. So you have an exponential raised to something at delta G divided by Kp, right? So that this is something that can either be much larger than one or much smaller than one. If it's much larger than one, then this is gonna be a very small number and the probability will be very low. If it's much smaller than one, the probability will be very high. So the interesting is the relation between free energies and Kt. So what is Kt? Well, it's kind of like an energy scale, right? So what is Kt in this room? Yeah, no, no, Kt is literally just, it's just the energy unit. But concretely, this is a number you need to know when I, there are many things. I hope I'm not gonna have to come by and wake you up at nine, two AM, lots of times. Roughly 2.5 kilojoules per mole at room temperature. So here's the difference between physics, right? Physics considers everything. In biology, biology doesn't happen at 1000 Kelvin. Biology doesn't happen at 0.1 Kelvin. And at room temperature, we need to compare all the energies to this. You could also say roughly 0.6 Kcal. And this might sound completely stupid, but units actually do matter. And there are generations of students who occasionally think that thought that you can kind of exchange the units. No, you can't. It's a factor of 4.18 difference between these. So if at room temperature, if assuming that I have an energy barrier, that's 0.1 kilojoules per mole. Is it an important energy barrier? Exactly, right? So that's kind of if you're biking in the spring now, so there's gravel everywhere. But your bike is not like you're bumping into the gravel. It's just a little bit bumpy, right? You ride over it. So energies that are much lower than Kt, you won't really notice them. You will be able to pass those energy barriers very quickly. So yes, technically they are barriers, but the thermal energy will be able to overcome those barriers. And that's where the thermal barrier is. So the thermal energy, this is roughly the type of energy barriers that will be easy to cross at room temperature. If I have a say an electrostatic energy and it's 300 Kcals per mole, is that an important energy barrier? I like that. You're smart. You're awake. So the obvious thing was that there's a much more important energy barrier. Why is that energy barrier not important? Right, so that's like if you're biking, right? And if you're considering the wall, you're not gonna be able to bike through the wall. So it's not particularly interesting how difficult it is to bike through the wall so that you would just have to go around the wall. But on the other hand, if you have a torsion with like four kilojoules per mole, that's a very interesting energy barrier because you might occasionally be able to overcome it, but occasionally not. And you might start to see that. So what were hydrogen bonds? What energies were hydrogen bonds? So this is the reason why I mentioned both these energies. We are in Sweden and Europe. In principle, we should adhere to the SI system. Principle is the keyword in that sentence. In practice, when it comes to biology, people live in Kcals space. Sorry, that's not my choice. So hydrogen bonds are a few Kcals, two to five or so. It will depend a little bit on the hydrogen bond. But if you wanna make your life easier and avoid, either you have to keep these units in mind all the time or stick to Kcals. Then it's 0.6. So don't make the mistake of, a hydrogen bond is significantly higher than Kt. So they normally, they will not break spontaneously. It is possible to break them, but it's not a free lunch. And we already spoke a little bit about here how volume or the microsates complicate the Boltzmann distribution. What is then the difference between energy and free energy? Exactly, right? Or you could, the other way of saying it is that when you're moving, if we're in microstate land, then we're talking about potential energy or enthalpy if you wanna be strict. Again, if you should be strict, it's always called enthalpy when it's not free energy. Everyone is too lazy to do that. So when we just say energy, we mean enthalpy. If you're looking at the microstates, the Boltzmann distribution, then you should use enthalpy in these terms. And that will describe how likely things are. If you're in macrostate territory, then we're including this volume and how ordered they are or the logarithm of states. So when you're looking at macrostates, you include the entropy. And then the term here you will have is free energy rather than enthalpy. At free energy, there is a reason that, apart from the fact that they would have been good to have a different name for that, the reason for this is what you see down in question 19 here. If, just fine, you might wanna have a look here. You can actually, without using a ton of mathematics, you can show that the free energy will describe the amount of, literally the amount of energy available to perform work in a system. And that's of course why the free energy will determine if there is energy available to perform work, a reaction will happen, right? If you would have to go uphill, the reaction will not happen spontaneously. And that's literally why it's called free energy. It corresponds to work we can do and that's why it will describe what reactions happen. But it is a stupid name. We are, let's say, to just touch briefly on entropy, can you give examples of systems with low versus high entropy? Mm-hmm. Can you give an example of a system with zero entropy? When would you have a perfect crystal? So if a crystal with no impurities in the lab here, would it have zero entropy? At zero Kelvin, yes. At zero Kelvin, any system will have zero entropy. But we can't read zero Kelvin. So it's an ideal state. But at zero Kelvin, there's no motion whatsoever in a system. The system has gone down all the way to down one specific microstate. By definition, there is no entropy anymore. But the cool thing, that applies to any system. Just free, if you freeze anything to zero Kelvin, the entropy is gone. That's actually not something you can prove. That's one of the laws of thermodynamics. We define entropy that way. This is closely related to, how did you actually know? Let's wait for 60 and I'm gonna need to come back to that today. What is the difference between the Helmholtz and Gibbs free energies? In general, both right, because the work has to do with the pressure multiplied by the change in volume. So the whole point, if you're performing work on the environment. And a physicist, that's just a nuisance to a physicist. But chemists occasionally have to take that into account. They're particularly working with gases or something that expands. I think I have a few slides about the hydrogen bonds and the hydrophobic effects today. Otherwise, I will go through that on the blackboard. And I'll come back to the temperature because we have a much more fun definition of that today. I'm gonna speak a little bit more about statistical mechanics. But before that, let me go through these hydrogen bonds, both for proteins and water. So I hinted that the hydrogen bonds could somehow help explain why things collapse. And then I also told you that it's very important. Don't try to guess these things. But write up what is the before state? What is the after state? And then write down what is the free energy, the energy and the entropy before and after? And then look at the differences. So at first sight, this looks really simple that if you have two sides, since you're a donor and acceptor, it would just be a matter of having them form a hydrogen bond and then you've formed a hydrogen bond in the protein. Boom, that's great. We gained energy here corresponding to a hydrogen bond. And if that energy is negative, which is this, otherwise the hydrogen bond would not form. That would mean that there would be a drop in free energy and that would explain how the protein folds. The problem is that proteins don't exist in vacuum. And this goes down to the issue with that you can't use quantum mechanics and ignore the solvent, right? Because those in real solvent, these two groups, they will already be making interactions with waters. So there are two hydrogen bonds formed here. And when this protein is that formed, well, it forms a hydrogen bond here, but then the two water molecules will form a hydrogen bond instead. So that the net difference in enthalpy or energy is zero. There is no change in the number of hydrogen bonds formed. While, so that all the difference here will come down to the difference in entropy, not the difference in enthalpy. And this is really absurd. It's a very strange effect with hydrogen bond. So what is the nature of the hydrogen bond? What type of interaction is it? What is the nature of the hydrogen bond? What type of interaction is the hydrogen bond? Yeah, well, and it's an electrostatic, it's caused by electrostatics, right? Although it's electrons, but it's electrostatic in nature. And it's by, we already spoke that electrostatics are in principle the strongest types of interactions. And yet, absurdly, no, it's the strongest types of interactions that have the largest energies involved. And yet, absurdly enough, the hydrogen bonds will not correspond to any difference in energy. So, and this is actually not a coincidence. These interactions are so strong that the molecules involved will do virtually anything they can to maintain the hydrogen bonds because they would lose so much energy otherwise, right? And because of that, the difference shows up in the entropy instead. I'll come back to this in a second when we talk about the hydrophobic effect. Because if we look at the general molecule, this can be octanol or any small, non-acrus molecule, hydrocarbon in general. You can have that in the yellow part here that would be pure hydrocarbon. And then you can ask how many of these molecules would be solvated in water? It's a very classical experience to do it partition coefficients. And you all know that some molecules such as maybe ethanol, they are perfectly soluble in water, while other molecules are hardly soluble in water at all. And that's what we describe by this so-called hydrophobicity. In the lab, the way you measure this is fairly simple. You just measure, we already know that from the Boltzmann distribution, right? That the probability of something is proportional to the exponential of the difference in free energy divided by KT or RT here if you're a chemist. And you can invert that expression. So that would mean that the delta G, the difference in free energy corresponds to that minus RT and then the logarithm of the probabilities. And these probabilities should really be what is the likelihood of seeing a molecule in water versus the likelihood of seeing the molecule in the liquid? Or instead of likelihood, you can say concentration. So what is the concentration of octanol in water versus octanol in octanol? So these so-called partition coefficients is really fractions that experiment list matter. So pure hydrocarbon or cyclohexane here, I think it is around nine molar, while if you put hexane in water, you would have something like 0.1 millimolar. So virtually all the hexane would prefer to stay in the hexane. There are very few molecules that would go down here. And if you then put that into the equation as room temperature, you get the difference that the solubility, the difference in delta G for putting hexane in water is roughly seven k cal per mole. So note that it's positive. So it's positive means that in principle, it will not happen spontaneously. But a few of the molecules will go there. And that's the difference that in chemistry, we say it in absolute terms, the reaction will not happen. But we're not chemists now. Now we're physicists. The Boltzmann distribution doesn't say that the reaction won't happen. It will say that it's unlikely to happen. So there are only very few molecules we'll see there, but you will see a few of them. Actually, if this was literally zero, we wouldn't be able to take the logarithm of it, right? So then it would just be infinitely high. And you can expand this. We can make this even more complicated that because there is a third phase here, right? On the top, those cyclohexane molecules, they're interacting with each other. So you could also, you could turn this into a triangle. You could have, well, if you have one molecule, one cyclohexane molecule in vacuum, that's not interacting with anything. What happens if they first condense? Remember that we talked about condensed phases the other day. So if they now turn into a liquid, what are the properties of that transition? Or what happens if you take that molecule and put it in water? And if we understand both of those processes, we can also understand the process between those two legs. Why can't we do that? If we knew the two top legs, why do we understand the third leg? Yes, there's state functions, right? So that if it was, if the, basically, if you're starting from one point and if you go the entire lap in this triangle, you must be back at the same state. If you were not back at the same, if the free energy was either positive or negative, I would choose to go in the direction where the free energy was negative. And then that would go as many laps as required and I would create the perpetuum mobile. And that violates the first law of thermodynamics. And here are the problem, there are lots of processes involved here. The only way you can do not try to guess, the only way we can understand this is by looking at, for each of these, what is Delta G, Delta H and Delta S, and then using this seemingly very simple equation. The book goes through this in some details. You can measure these things in the lab. And that's how we, that's actually how we access most of these things. Understanding them on molecular level is super difficult. So if you take a small molecule, what is the charge of cyclohexane? One full cyclohexane molecule. Nothing, it's a sorry, bit of a trick question, Wednesday morning. Meaning that it doesn't have any electrostatic interactions. So you can probably almost guess this, that to first approximation, surprisingly, it's only gonna be Lenard-Jones. And Lenard-Jones interactions, they only depend on how many atoms you're around you. They hardly depend. There is very little difference between Lenard-Jones interactions, whether they are carbons or oxygens. So quite surprisingly, if you're taking one molecule here and moving it from vacuum or gas, either to liquid hydrocarbon or one hydrocarbon in water, the energy difference is virtually the same. It's negative, which is good. Condensed faces like each other, they start to attract. They, the molecules, the molecules start to attract through dispersion. But there is no difference between the liquid or the actual solution, which makes sense, because it's Lenard-Jones interactions. Had it been a charged particle, it wouldn't have been different. But for Lenard-Jones, this makes much more sense when you think about it for a second. On the other hand, if you measure the delta G, the actual cost of the solubility, one hydrocarbon is gonna love to be put in liquid hydrocarbon. So that might be minus four or five kcal. But if you take one hydrocarbon and put it in the water, in this case, it's delta G is plus two. It doesn't like to be put in water at all. And again, that's what you would expect based on your gut feeling about hydrophobicity. If you know, but this is where we like having a very simple equation. The only equation we know is G equals H minus TS. And if we know G, and if we know H, we can solve for TS. So this enables us to tell that TS, the difference in entropy here is roughly minus three while the difference in entropy here is roughly minus 10. So that's for some reason, there must be a drastic reduction in entropy when you're putting one hydrocarbon in solubility. Based on what you know from the equations, fundamental laws of interactions, and the measurements that is not particularly soluble. So what you now know, what happens when you put the hydrocarbon in water, there is a drastic reduction in entropy, but virtually no interaction, sorry, no change in the energy. Why is that? Sorry, I gotta say based on those two, you can then of course solve for all these things between those two, but I'm gonna avoid that for a second. So what happens here, this has been actually been a very long discussion understanding why this is the case and where this entropy drop comes from. That there is an entropy drop that is clear. But what people eventually found out some 30, 40 years ago is that what happens here is that all these water molecules, they participate in hydrogen bonds already. And because the hydrogen bonds are so strong, as I mentioned a few minutes ago, they will do virtually anything to reconfigure and maintain those hydrogen bonds. So what effectively happens is that, and to do that, they will end up with a much more ordered state just around this molecule. They were effective forming a cage-like structure because no water can turn their hydrogen bonds towards the molecule because they can't form that way. So effective you went up with a cage around the water. It's called a clatter rate even. And that of course, that order corresponds, it's a more ordered states and that means that it has lower entropy. So that the number of hydrogen bonds is maintained but the water around the solute will have to order and that explains the cost for solving hydrophobic things. We will come back to what this means, that actually once you form these condensed phases and particularly a liquid phase it's actually a very small cost to then turn this all the way into a crystal or solid or so, which is a little bit related to protein folding but we will come back to that much later in the course. But based on what I showed you on the previous slide, you can now explain the hydrophobic effect. So the hydrophobic effect has to do with these oil drops forming, literally forming drops if you put it in water. So why does this happen? Why is it better if you have two small microscopic pieces of oil, why is it better to have one large drop? Exactly, right? So to first approximation you can say that the amount of bad entropy here, the amount of order you're inducing is roughly proportional to the area and the area of the sphere grows as the radius squared while the volume of the sphere grows as the radius cubed. So that the larger the droplet is, the smaller is the fraction that you have to put on the surface of the sphere. So that's why it's lower free energy to try to separate the oil versus water as much as you can and that's why we form droplets. And I'm already now, I'm not gonna prove this but I'm gonna hint, if this is a protein, what residues do you think are hydrophobic and what residues do you think are hydrophilic? It is actually a real protein and green hydrophilic. So there is something very similar going on in proteins here. The protein two is collapsing almost like the drop in water and turning all its hydrophobic residues towards the inside. It will get more complicated than though because it's not just hydrophobicity that matters for protein but it's by far the most important first step of protein folding. So what we've gone through now is that these energy landscape that we talked about for various small molecules, we can actually consider these energy landscapes for gigantic molecules. And normally in energy landscapes, we tend to focus on the entropy, sorry, the energy, but the entropy somehow describes the width of these energy landscapes. And these two things in combination will determine if you're starting from the top, will we be able to find the bottom states here? So why do we need to find the lowest, the bottom state? It was a certain Danish-sounding name that proposed that. What did Christian Amfinnsen say? Yes, and if we actually, in that case we should probably or rather lowest free energy, right? So in that case, we actually should have an energy landscape that is rather free energy but somehow these low lying states should correspond to the native states. And the point is not so much that we want to find it in terms of our prediction, although that's certainly important in bioinformatics, but at least we want to understand why it happens. But on the way we also have to make sure that there can't be any energy barriers that correspond to hard walls. So we have to find a way to search in this landscape to actually get there. And based on that, you might start to be able to say some things about these structures we looked at. We looked at alpha helices for instance. It's a repetition here. What are the properties of alpha helices and how does this relate to energy entropy and the hydrophobic effect and the Boltzmann distribution? Can we say something about this based on the landscapes? In general, but on the other end, everything is on the outside here, right? All residues are exposed to the outside. But of course, if you have an alpha helix and an interface or something, you can separate it inside from the outside. I mostly think about these hydrogen bonds. So when you turn things into the alpha helix, what will happen to the entropy compared to a free chain? There is a rather drastic reduction in entropy, right? Because we're ordering the system. But the only reason why you would then form an alpha helix is that if all these hydrogen bonds interactions are so advantageous that the gain in hydrogen bonds outweigh the loss of entropy, which it does. Otherwise, we wouldn't see alpha helices. If you compare that with beta strands, so if you look at the individual strand, one of these first, let's pretend the other four don't exist. Would that ever happen spontaneously? If you again, if you think about energy and entropy. So the reason for having two structures and talking about differences, that might give you a clue about what the answer should be. Remember the Ramesh-Hendron diagram, right? The alpha helices is one well-identified area at least in the Ramesh-Hendron diagram. That is true for a beta sheet too. It's also one well-defined part so that rather than allowing the fine psi angles to be anything, we're constraining it to one part of the Ramesh-Hendron diagram. So I would rather argue to first approximation the drop in entropy is just as bad for a beta strand. But if you just look at one strand, we have the challenge that the one strand per se does not necessarily form more favorable hydrogen bonds. You might be forming hydrogen bonds to water and that's the complication I spoke about, but we'll have to come back to that. But the hydrogen, sorry, the beta sheet, and that's the difference between one strand and many forming a sheet. The stability here really comes from the multiple strands making hydrogen bonds with each other. So one single strand will virtually never be stable. You need all of them. They get their stability. It's more of a collective property here. So just to sum this up, we can say if you can talk about the delta G of protein folding, I would argue that is almost entirely hydrophobic effect, at least the initial part. And then towards the end, we're gonna need to do a bit of polishing and packing. This part turns out to be difficult because this will depend on the specific side chains. So the devil is in the detail here. But if you just throw any chain in water, they will have a hydrophobic collapse instantly. Now, of course, there are lots of other interactions. I'm gonna spend a few more minutes before I release you to the break here. We talked a lot about electrostatics. The reason why electrostatics is, sorry, why it's strong is that they have this one over R dependence. But there is also this other part, epsilon, in electrostatics. And I've deliberately ignored the one over four pi here. What is epsilon? Yes, in this case, it would be absolute. But the relative permeability is fine. What is epsilon in water? The relative one. Well, the relative one of vacuum is one, right? So in vacuum, charges see each other. In water is quite the relative, permittivity is quite high. It's around 80. So water is very good at shielding charges. You can have two charges. They can be quite close proximity in water because all the water molecules will orient around them and shield them. So it's gonna be roughly 80 times weaker in water. So if you now take a charge and move that from water into a protein, what is the interior of a protein? That's more like oil, right? Oil roughly has an absolute of maybe two to four or so. I might have said three here in the book. There's actually a typo in this figure. In the first edition of the book, there's a typo in this figure. But if the water is screening the charges, that if you move the charge from water to the inside of a protein, on the inside of the protein, the charges will see each other completely. This could be good if you have two charges of opposite sign. But in general, every charge here will also need to interact with lots of amino acids that are not charged. And that's usually very bad. So it's almost astronomically expensive to put charges on the inside of a protein. So typically, that's part of the side of a protein. We typically don't see charges on the inside of a protein. If there are charged amino acids, they will stick to the surface, the outside. The other thing that can happen where proteins are a bit more complicated that we occasionally call them charged amino acid, but the chemically correct we're saying that they are titratable because you can actually change, depending on the pH, you can get them to either release or take up a proton. So that sometimes these actually prefer to get rid of their charge and become neutral. That is something that costs free energy, but it costs less free energy than putting the charged version on the inside. So if you ever see a charged amino acid in a protein sequence, I would almost eat my left shoe if that is not on the surface, but almost is the keyword. There are some exceptions. We spoke a little bit about polar and charged residues. I will just mention this in the relation to the helix capping. I haven't forgot the break, but I will see one more minute here. Remember that I told that these helices that because all the peptide bonds line up that you create a very large dipole in the entire helix, right? And that means that you have negative, well, you have a small dipole there. Dipoles go from negative to positive charge, negative to positive charge. And this course, if you add up all those small dipoles of the peptide bonds, that corresponds to fairly large dipole that goes here with a, well, the sequence goes in that direction, but that would correspond to on the C-terminus, actually having a fairly large negative charge, and on the N-terminus, the start of the sequence, you would have a fairly large positive charge. And this for the helix, then it actually worked quite nice to have that positive charge in the helix then matched with the negatively charged amino acid here, or vice versa, putting a positively charged amino acid there. So you sometimes see these charged amino acid at the end of helices too. But the inside of the protein would usually be here, and then these would be the parts exposed to solvents. Why do I measure that? Well, this turns out to be super important functionally. This is a, one of the most, I think I mentioned this at the very start of the class. This is a small sensor in your nerve system that is responsible pretty much every single heartbeat. So every time you have a nerve signal that changes the potential across your cell, what happens is that you have, in the red helix here, you have a couple of charged amino acids. And then you see how the clock is ticking there roughly 200 microseconds. This entire helix now, and now you have sort of instant replay here, they will actually move down to a different state. And when this moves down to a different states because the potential over your cell brain changed, that is gonna induce the entire ion channel to opening and then will be a flow of potassium ions through. And that will cause things like a heartbeat, which is kind of nice to have. And then this resets in a millisecond or so, and then it can start again a second later. And of course, if you didn't have those charged amino acid, we would be somewhat screwed here. We wouldn't have heartbeats. So charges are very nice because they can create force in an electric field and the body uses this all the time. So in principle, we think we understand things there, but there are a bunch of things we've glossed over. In particular, we've glossed over all the details about the Boltzmann distribution and I just hand wave it. So after the break here, we are gonna talk about suicides and the physics and other horrible things related to the Boltzmann distribution. So before the break, I hinted that we kind of need to look a bit deeper into statistical mechanics. Hopefully we're not gonna look this deep. There's a great quote. The very first sentence of the book, States of Matter, which is also in the statistical mechanics book by David Goodsley. Statistical mechanics has this strange reputation of being different. It's not a strange. Statistical mechanics is difficult, but I think the most thing is difficult to grasp and that's why it has an unfair reputation of being really hard. The problem with statistical mechanics, I would say, it deals with things we're not used to dealing with rather than having these beautiful, well-defined simple equations for a simple atom or a simple thing sliding on a plank or so you're dealing with systems that contains thousands or millions of particles, actually more than millions, a mole, contains the power of 23 particles, so that it's our way to try to deal with systems that are actually of relevant complexity because there is nothing in this world unless you're doing a super simple proof of concept physical example that only involves one or two particles. So the power of statistical mechanics it actually deals with real things and sorry, reality is difficult. However, in addition, there are lots of branches of physics that deal with reality. The real beauty of statistical mechanics is also it doesn't really assume anything about any system. You're trying to find what are the general principles that are true for any systems if you can't assume anything. And I think that's the really hardest part because you're basically, I'm telling you to, well, discover, do some things, here's a chalk. You only have the backboard, you can't assume anything. So the hardest part at least for me is usually to say, okay, it's not like a matter of what are the boundary conditions, well, it's up to you to define. You have to start by defining a system with the boundary conditions, sorry. So what usually gets difficult is that we're gonna talk about systems like this, a general system with a capital S, it can be any system. But of course, if I haven't assumed anything, the power of this is that if I can show that things must be true without having assumed anything of the system, they're universally true. And particularly if I've only used math to get there. So the power of statistical mechanics, I think it was Einstein who was coined that there's only one branch of science that he's confident will never be completely overhauled. And that's statistical mechanics and thermodynamics because they don't assume anything. Even quantum mechanics is based on observations, right? And then we formulated theories to reflect those observations. It's a great theory and I don't think quantum mechanics will be overhauled. But this has even fewer assumptions. Yes? Yes. It's mostly thermodynamics. But again, even most of the, I would say the laws of thermodynamics, in particular, what is this? They are derived from statistical mechanics. Now of course, there are lots of branches of statistical mechanics. There are things in chemistry or simulations where you start to assume a whole lot more about the systems. But the foundations of statistical mechanics are really truly universal. So I'm not gonna, this is not the proof I expect you to go through, but just to show this, I'm gonna spend two slides here for three, just to at least hand wave that the Boltzmann distributions is really universal. So if you think about any system here, and this could, this system could be the universe. The larger outside system here in the solid box, that's my entire system. And that's the system I wanna understand something about. But to be able to understand something about that system, we're gonna need to look at the smaller part or at least how the energy is distributed in this system because that's what the Boltzmann distribution was about, right? What are the difference way of distributing energy in a system? So if we look at the smaller part in the circle here, that every single possible microstate of this system we can imagine, well, a microstate is a specific given detailed state, right? And in a given microstate, that small part will have a certain energy. So let's look at the state when this small part has the energy epsilon. Whether that is large or small, I don't know, but it's small compared to the rest of the system. And then let's also say that the number of ways you can organize the entire large system, the rest of the system we call the thermostat, it's just a definition. The way we can organize this system so that the small system has the energy epsilon, well, that will depend on the energy. We have no idea what type of, it's a general function. It's not necessarily near. But that's, let's call that M, and that's a function of the energy, in general. And then, and that's the probability of observing the system in that particular state and that is just counting. That will corresponds to the number of microstates where this is true, right? So if this particular, if one particular state corresponds to 12 microstates and another one corresponds to 12 million microstates, it's gonna be one million times more likely to observe it in the second state. Will you follow me there? Good. And then I completely arbitrarily define something else. S equals K logarithm of M. Because M, we're counting, it's larger than one. Well, it's at least one, it's larger than zero. And that means that I'm always allowed to take the logarithm of it. Whether that makes, right now that doesn't make sense, but it's a mathematically allowed operation. So I can use that as a definition. Don't for a second assume that it means anything for now. Now that K, forget about the K, it's a constant. It's just a constant. It's just a constant. We forget, we don't know what Boltzmann is. I have no idea what that is. So this is what we had on the previous slide. We don't know what M is here. It might have a really complicated definition so that what do physicists know when we don't know what a function looks like? We seriously expand it, right? And I'm gonna argue here that, oh, Siri likes Boltzmann. I'm gonna argue that only the first term in this expansion matters. The zeroth term, that's always a constant. So S is in general a function of M here, right? So if you double, does M depend, both S and M depend on the system size. They're not necessarily strictly proportional, but roughly. Energy, on the other hand, is proportional to the system size. And you can actually show that if M has to do with number of states and probabilities, the reason for taking this logarithm is that then you can add it up. So you can actually show that S will be proportional to the number of particles in the system, and so will E. So we have both S and E are proportional to M, the number of particles in the system. And it's in general gonna be very large for the total system. So rather, maybe we should then, instead of doing the series expansion with respect to M, let's do the series expansion with respect to the energy. So the first term is S at the particular energy, at the energy E. The second term, well, that's gonna be the first derivative, so that's D S D E. That's effectively an entropy that is proportional to the number of particles divided by an energy that is proportional to the number of particles. So this term is gonna be roughly independent of the number of particles, right? It's a constant, a constant will matter. The second derivative, then we roughly have units of S that is proportional to the number of particles divided by the square unit of energy, so that we have number of particles divided by square number of particles. What will happen with that term for a very large system? It will go to zero, right? And we were talking about large systems. So this will be true. In the size of an infinite large system, this is gonna hold exactly. And now we completely got rid of the entire complicated dependence. If the system is large enough, this is exact. Which is actually the whole key. Once we've done that, we can just take the first expression and solve for M. So now we just revert it, and that we had took the logarithm before, so then the reverse of that is the exponential. But in addition to that exponential, now we can take that expression for S and insert our series expansion. So an exponential of a sum here, or a difference, so that's that term minus that term. An exponential of a sum that corresponds to the product of two exponentials, right? If you know your exponential laws. So you're gonna have the exponential of that term. Well, we can, let's say that the minus sign belongs to the epsilon. So it's plus that entire term minus that. So that's multiplied by the exponential of that entire term, which is strictly true. And that corresponds to the number of states. And the probability of observing that is again proportional to the number of states. So the probability of serving this distribution, we have here. That's just a boring constant. We don't care about constants. We can replace that with some other constant. So the probability of seeing a system is really proportional to exponential raised to minus the energy in the small part of the system multiplied by a bunch of strange constants. But they're kind of constants, right? Remember, they don't depend on the size of the system. That dsd, it was usually called one over temperature. If people have been a bit smarter when we define temperature, we would have, well, what we call temperature would rather have had at reciprocal scale. This, we would of course, actually, had we defined temperature today, you would have taken that entire expression and said that's what we call temperature or the energy dependence of the system. The problem is that we didn't. We had completely different definitions of temperature long before people started studying statistical mechanics. So the first problem is that we have one over temperature. We can live with that. The other problem is we have completely different units. And to translate between units, we use constants. So kappa is Boltzmann's constant. But had we defined this today, kT would have been the unit of temperature we would have used because it's the natural unit of temperature or one over kT, rather. So we haven't assumed anything and this will hold true for any system if the system is large enough. And this also shows where you're getting the temperature, right? It's not particularly complicated. It's the property of the system surroundings. Again, how many freedoms we have as a function of energy. You could already don't try to understand it. It's just a definition. The complication is that in kindergarten you had a very good idea what this is. That is really a strange constant, but you have absolutely no idea what s is. But from a strictly physical point of view, s is a clear definition that you introduced while p is something strange that appears. So before this class you might have thought that temperature was easy and entropy hard. I've made, you might think that entropy is easy, but unfortunately temperature is hard. So I'm not sure if you're better off. I will again confess that I still don't have a good gut feeling of entropy. You have to, you can plot this in plots and everything, but it's not like temperature. You're not gonna feel spontaneously what it is. But you can certainly plot what is, how does entropy increase as a function of the energy? And then you have different curves here that corresponds to given combinations of entropy and energy. The tangent of this curve is one over temperature. And the blue lines I tend to draw here for a different system, if we draw things where E minus TS is constant here, along such a line, you're gonna have the same free energy. I'll come back to that in a second when we talk about free energy for plots like this. What we can do with this, and I've just probably done it first, is that if you know the Boltzmann distribution, you can determine the probability of being a state I. Actually, you can determine the relative probability. If you want the absolute probability, you're gonna need to normalize this so that you have the sum over all states and the denominator, right? The so-called partition function. The partition function is really simple. If you are having a dice or something with six states, if you're having a protein with a gazillion states, it's kind of complicated because I can't count a gazillion. So this is really simple, beautiful in physics. The only problem for any realistic system, you will never know the complete partition function, which again is this curve, is that if you know the partition function, you will know everything about the system, but there is no way you will know that about any practical system. So the best we can hope to do is approximate partition functions, and an obvious way to approximate partition functions is for instance, computer simulations that you're gonna be doing later in the class. If the other, and if you know these partition functions, you can just sum things up over states. You can calculate energy, you can calculate the entropy, or at least in theory, calculating what is the entropy in a particular state, one sub-state I. Well, in general, that's hard. And there's been a lot of research going on actually in the 20th century about what do we mean by entropy? The book goes through, this is some detail, but to save a little bit of time and avoid having carrying over too many slides on Friday, I'm actually not gonna go through this, but the cool thing is that there's quite a lot of research in information theory, and it turns out the way you define entropy or information contents is actually exactly the same in information theory due to Shannon, and the way we do it in physics. But rather than spending, this is surprisingly a more difficult derivation than the Boltzmann distribution. So instead of wasting 15 minutes on that, I've actually made AAM iPad recording that, where you can follow me going through and marking the equations on the class page if you wanna go through it. But suffice to say, it is possible to determine these things if you, particularly if you like permutation laws, and then you end up with something that has to do with the probability of being in the state multiplied by the logarithm of, well, either the weight or one over the weight, it depends on the minus sign. So if you include this, you can determine all things like free entities and everything directly from the partition function. And that's why physicists are so obsessed with, well, not just physicists, chemists too, like me are obsessed with computer simulations. If we can do computer simulations that at least cover the 95% of phase space that is relevant, we can calculate free entities. You can calculate the entropy. We can calculate all these properties of a system in a computer rather than going in the lab. And we're increasing getting to the point where computers are fast enough to do this. This gets slightly more complicated. There are a bunch of interesting observations one can do here though. In general, you can draw any theoretical line in this entropy versus energy landscape. So you can say that the previous one I showed you had an obvious and would you have the tangent of a temperature with positive? But in theory you could have something where a line where the derivative is negative which would correspond to negative temperature. Can you have negative temperature? We had to have both yes and no. So based on our definitions, that's absolutely nothing to say that that constant has to be larger than zero, right? So based on the definitions, there is nothing that limits it. There is one important word that I deliberately haven't mentioned here but that you usually take for granted. No, but equilibrium. At equilibrium, you can't have negative temperature. But there are plenty of cases where say when gas is expanding through a valve or something, non-equilibrium processes can easily have negative temperature when something is happening. So in general, in physics, the answer is yes. But any stable system you can observe, the answer is typically no. You can also have these strange things. So I'm gonna show in a second that in most of these diagrams that what you have up here on the left is gonna correspond to lower free energy while here you have high free energy. So if you have a curve with this type of shape that would record that any point along this curve you have the highest possible free energy. Nature doesn't seek high free energy. Why would you ever be stable at high free energy? Normally you wouldn't. But that's kind of like balancing on the edge of a knife, right? As long as you're neither pushed to the left nor right, you can balance. But of course the second you have the slightest perturbation you're gonna fall down to the right or the left. So all these things are special cases that can happen, but in practice they will virtually always look like this. That up here you have the green area with even lower free energy than along the blue curve. This tangent defines the temperature and up here you have higher free energy. And again, don't try to understand these plots too much. The system would like to have as much free energy as low free energy as possible, but we can't get further up than it's defined by our temperature in this case. But if you know this, you can start to compare phase transitions and everything by looking a little bit at these curves. So what will happen, not just phase transitions, any type of change in a system, that will change the free energy of the system. So if your instance, you can have a gradual change. Say if you have water at 39 degrees centigrade and you're heating that to 40 degrees centigrade, that will gradually change the properties of a system just a little bit. So that would corresponds to increasing the energy of the water, right? And when we increase the energy of the water, well, the slope of that curve is gonna decrease which corresponds to slightly higher temperature. The entropy will increase too. Nothing dramatic whatsoever. You can plot in slightly different way. You can plot the energy as a function of temperature there and you will have some sort of dependence on temperature. And not entirely surprising that if you just wanna plot the distribution or the probability of seeing this in the first energy and then the second energy as we're gradually heating water from 39 to 40, there's nothing dramatically ever happening. You will gradually have the entire population slowly shift from E1 to E2 and it does so continuously. So that's normal, it makes sense and anything is gonna be a small gradual change in these plots. But all chains don't look like that. Occasionally you have bumps in the road or depressions. So what happens in the case here on the left, you might have an entropy versus energy curve that it has those sort of small bumps there but then it has a depression there and it goes up again. Remember these things that I said that this is gonna be, when the curve is this shape, it's kind of gonna be like balancing on the edge of a knife, right? So that if you are here, it would actually be better to jump down either there or there because those free energies, if you, the second here has lower free energy. So it's bad to be here but you might need to cross this one when you're moving from one state to the other. So here you have some sort of stable region. You have some strange thing happening in between and then you have another stable region. There is nothing special that happens in energy but if you look at this and you, the difference between energies, if you're gradually increasing the energy here, you're gonna have some of the molecules or something in energy one and almost nothing in energy two and as we're heating the system, you're gonna have more and more population in energy two and eventually all of them in energy two but we never have anything in between. They are not happy to be between. This corresponds to a phase transition such as boiling water. So that there's something that changing and that there's some abrupt change in the middle where the things are strange. The span over which this happened, you can actually calculate as the books goes into some detail of doing it but it has to do with KT and the difference between these energy levels. So what is the range? How large is the span over when you're moving from water to water vapor? What is the span over which that phase transition happens? When does water begin to boil? 100 degrees centigrade and when is the boiling process complete at what temperature? 100 degrees, hmm, there's something strange. You're actually wrong. So, but the trick here is that KT, how large is KT in, forget about the moles here, KT at room temperature in kilocalories. KT is an insanely small energy, right? While the difference in energy between a kilo of boiling water and a kilo of liquid water is quite large. So the temperature span here is gonna be the ballpark of 10 to the minus 20 Kelvin which is pretty decent approximate with zero and that's why we say water boils at 100 degrees centigrade but theoretically there is a small span. Now with proteins on the other hand if you're looking at one molecule of a protein you're looking at energies per mole, right? So the energy difference between one protein that is folded versus one protein that is unfolded it's an exceptionally small energy. So suddenly these things are in the same ballpark. So for things on the microscopic level molecules changing phases then you might have a span of 10 degrees. We will come back to these transitions. So everything in protein is not a phase transition but some things are. You can also use the Boltzmann distribution to determine when things happening. I think Burke will go through this in more detail in one of the lecture lectures but this far we mostly looked at the valleys in the Friendi landscape and that's obvious because the Boltzmann distribution typically describes what things happen at equilibrium and at equilibrium you're gonna have all your population in the low free parts of the Friendi landscape. You don't wanna have any population at the peaks, right? But it's not, if we're only looking at the valleys we don't care if there's a brick wall between them that you can ever cross and in practice in life science it matters that you can cross things because any process in biology that would take longer than 100 years in principle can't happen because I would die before it happens. So in biology it's also important to think about kinetics how long does it take to go over these barriers and then we will also need to look at the Friendi how easy it is to go from one state across this barrier and down again. And this is, you can talk about this in terms of how long time takes for transitions or transition rates but Burke will go through that in detail. So when it comes to kinetics and understanding how fast processes happen you have to care about the peaks of the energy landscape. But when it comes to equilibrium distributions we only care about the low levels. So we can actually apply this to secondary structure. We already did this a little bit but if you look at the alpha helices for instance the key thing is there are hydrogen bonds I to I plus four. We, the first hydrogen bond, well the first three residues reform is just gonna be bad because we have to place them in a bad low entropy conformation but we haven't gained any hydrogen bonds. So for the three first residues you're just paying that's really bad. While the fourth residue, the first hydrogen bond we get that is locking three residues in place. The next hydrogen bond also locks three residues in place. So in general that you have a number of residues well N residues are stabilized by N minus two hydrogen bonds. And you can, we can define this any way we want but an easy way of doing it it's saying that the free energy of this transition we have to, if it's a transition it's always a difference. So we have to look at what is the free energy of the alpha helix compared to the coil. So we formed the hydrogen bonds that's good. So for N residues we had N minus two hydrogen bonds N minus two multiplied by the energy of the hydrogen the free energy of the hydrogen bond which is usually gonna be negative. But we also have to put all of those residues in the shape of a helix and that's bad. So that that minus N multiplies T by the delta S or this is actually really the change in entropy when we move it from coil to helix. So whether alpha helix is formed or not will depend on this equation. And since we have N in two terms here it's kind of nice so let's take that first term that does not depend on N minus two multiplied by the energy of one hydrogen bond and then we have a second term where all the N dependences. So this hydrogen bond free energy is negative and it's minus two multiplied by something that's negative so this is gonna be positive term, right? So initially when we're forming an alpha helix there is an initiation energy that's a barrier you're gonna start to pay when you want an alpha helix. So first it's uphill and that's bad. The question is that uphill so bad that it's a brick wall or is uphill, well we can always say it's roughly two hydrogen bonds maybe 10 K Cal. It's a high barrier but one we can't get over at room temperature compared to KT. And then as you were starting to elongate that's where this comes from you can actually show that this term well you can't say what it is but if alpha helix is formed this term will have to be negative, right? Because if it was positive alpha helix would never form. So we already know that in principle it has to be negative. Yes, that's the initiation allegation constant. There are a couple of different ways of treating this with parameters but in the interest of time I'm not gonna go through this I promise there won't be any questions about this. There's more physical way of dealing with all the exponentials. Alpha helixes have a few peculiar properties though that at this point you might think that it's a phase transition but it's actually not. So if you consider different phases such as ice and water normally there would be ice this time of year when I give the class but not this year. Ice and water normally can't coexist. If you have a mixture of ice and water and just leave it to itself either everything will turn to water or everything will turn to ice. And the reason for that is that we have this costly surface tension between the ice and the water area. So you have in real spray in the real world when you have three dimensions the number of residues we have in each phase will be proportional to the volume while the area is proportional to, well the number of, roughly the radius squared if it's a circle or something but and that's gonna correspond to the number of residues raised to parts of two thirds. But that means that for a large system it would be better to get rid of this area then we would get rid of the surface tension and we would still have either all water or all ice. This doesn't really hold for a helix because if you look at that coil up there how many dimensions are there in the coil? You could argue that this segment exists in three dimensions right? But it's the helix each residue here this is really a one-dimensional structure. It's a sequence. So if the question is true we have one more helical residue formed there. That's a one-dimensional process. So if you're adding or removing one helical residue in one dimension the size here doesn't really change. The size of the interface if you count the interface the number of, well the number of interfaces between helix and coil. It doesn't really matter how large this helix is there will always be one place after and one place before it where you're interfacing with the coil. This is a very deep result by LeBlanc-Dauer actually that faces can't coexist in 3D or 2D but in 1D faces can't coexist. So the alpha helix is actually not a strict phase transition but we can still learn a bit of the transitions we use. Oh sorry I actually had this described in more detail here. So the point is not quite protein the reason why I have this is a bit of a memento mori. Protein folding is not just a simple thing this is phase transitions applied to biological matter. They're similar but not the same. In this case we have to understand the balance but it's not strictly a physical phase transition. You can actually, you can show a little bit about the probability of mixing various pieces and helix and coil. A long time ago this course was more physical. I've gradually moved away from that. So in the interest of time I'm not gonna go through this but you can basically calculate what is the probability of initiating a helix to coil transition anywhere in the sequence. What is then the probability of elongating it and then we do a lot of mathematics and then we can actually show that we can estimate that what is the transition midpoint and draw some conclusions about how expensive it is to elongate versus initiate a helix. If you're interested in that let me know and then I will make a screen recording and go through this in detail but it's another one of these that thinks they steal 10 minutes and they don't add a whole lot. But the important thing it was leads to in terms of biological conclusions. That means that we can start, we always pay the initiation energy as I showed you and then it's a matter the elongation energy can be larger than zero then these particular residues will never form a helix. They can be just so slightly smaller than zero. Well, if they're just so slightly smaller than zero what's gonna happen here is that the difference from zero to that point is still positive, right? So this is not an alpha helix. If you already paid the initiation energy it would be better to go here but compared to the start you're worse off. So this helix is not yet long enough to be stable. If you extended that helix, the blue one here it might eventually become stable. This purple one or cyan rather, the cyan helix on the other hand reaches zero free energy here. So at this point it's gonna be better to be in a helix than in the starting conformation and that would spontaneously form an alpha helix. We can't, right now we can, how fast this happens depends on how high this initiation barrier is and in general this is pretty darn fast. It happens in a few nanoseconds or so. You can almost actually in an electron microscope you can pretty much watch alpha helix is growing. So the time, the rate limiting step here has to do with this initiation energy that comes from overcoming the entropy of putting the first few residues in an alpha helix. Once you have the first hydrogen bond it's pretty much downhill. And then the second part we can also calculate if one likes to use this helix coil transition you can calculate how long time these things. But the take home message for the helix that they form very fast. It matters how expensive they are to initiate but it also matters how long, how expensive it is to elongate them. They're relatively low free energy barriers in the ballpark of one or a few k-cals per mole. It will depend what, well, whether it forms a helix or will of course depend on the residues. Some residues like more to be an alpha helix shape. Remember that I told you about one residue in particular that hated to be helix. Which one was that? Proline. Can you understand that better here? So proline had a very peculiar shape, right? So for proline in particular, the entropy drop of restricting it in a helix is gonna be worse. So that is mostly the entropy that we're paying there. Most helixes are roughly 20 to 30 residues long. And that's a bit strange because what I just told you that once you form the helix then it's more frequently downhill at least unless you have something really bad. So wouldn't it be better to have really long helixes? In principle, yes. But the problem is it's also matter of probability. Remember the genetic code and the relative abundance of amino acids. Sooner or later, nature will put a proline there. Or forget proline. Roughly half the amino acids like to be in helical shape. Roughly half of them dislike being helix. So to have a very long helix, if you already have a chain where I've given you 100 residues and they all prefer to be helical, yes, then you would have a very long helix. And there are such proteins. But in general in nature, the likelihood of never having a helix breaking residue, that will become lower and lower and lower the longer the genus, right? So this limitation is mostly due to probability. You can't, you have to rely on the fact that you never get a non-helix liking residue by chance. If we compare that with beta sheets on the other hand, they're quite different. Here the way we start to create the sheet is much more complicated. It's not just a matter of putting two, three residues there, right? We have to form at least two of these strands need to get close to each other. So there's likely a much more complicated initiation part here. On the other hand, once we have started to form the beta sheet, there are gonna be a huge number of hydrogen bonds. So just looking at the structure, you can almost guess that they're gonna be more expensive to form. It will take longer, but eventually there will be a very rapid process once you've formed them. We're gonna look a little bit about that just to give you an idea how we can derive that. And that is the derivation that I will have time to do the last 10 minutes here. Experimentally, it's very fine. A beta sheet formation can take hours or weeks, but sometimes just a millisecond. So it's a huge diversity in times. And the only way to find that out is we need to create a simple model for it. If you want to determine how fast something happens, this model will have to rely on identifying that transition state, the worst part on the way between coil and beta sheet. Why? That has to do with transitions, right? If you need to get from state A to state B, and if there is a barrier, that barrier is bad. The higher the barrier is, the worse it is. So I need to be able to say how fast this could happen. I need to identify the lowest barrier. And the height of that barrier is what determines how fast it's gonna happen. Not the end state, not the beginning state, but how high the barrier is relative to the starting state. So your guess is as good as mine here. I can't prove that this is the lowest barrier. Actually, you can, but that's a lot more work. So I'm gonna hand wave a bit here. So that if you wanna form a beta sheet, there will have to be some sort of minimum length of the sheet. And I will have to start by forming one of these things called the hairpin. And what I'm gonna argue is that the worst state is that you form one entire hairpin here and one extra turn, but not more. That's a little bit of hand waving, but not a huge amount of it. Oh, sorry, there we had a hairpin. The problem is that I'm gonna need to introduce a bunch of definitions. And this one of those things, your guess is as good as mine, but actually they're not because I have some experience. I've done this before and some definitions will be easier to work with. So if we call F beta, that's the free energy of moving one residue inside a single beta hairpin relative to the random coil. That's just the free energy corresponding to the entropy here. And then there's some sort of extra energy or difference if I put it at the very start of the edge, the very start at the very end of a large sheet. We don't even know what the sign there is. So the total free energy at the edge residues is some of those two. And then there will have to be some sort of free energy for the bend, the coil, when I move from one residue to the other. We don't know what the sign of that is either for now. And then I postulate that if sheets can form, we must have the U must be larger than zero and this edge energy must be free energy must be larger than zero. That's strange. I say that these things are bad. Well, but the point is that if it was good to just have these turns, right, you could forget about the beta sheets. The chain would keep just turning all the time. There would be no point in forming the beta sheets. And if this delta F beta was smaller than zero, then it would be better to be on the outside of the sheet than inside the sheet. Same thing there, but then you would just have individual strands, they would never form beta sheets. It would be bad to form the sheets. So since we've observed that we already, we can say something about these two signs. This is a very common reasoning. Look at observations and then we can say something at least about the sign of things. Then there are two scenarios. If the combination of this F beta and delta F beta is smaller than zero, so that it's still, it's still better to be at the edge of a strand than to be in random coil. Then a single, very long beta hairpin would be more stable than a coil. That can happen and only a single turn would then be required for formations. You would literally have one hairpin, both of them would be at the end and you would not have extended sheets. Or you could have that the combination of the two here is larger than zero, so that the individual hairpin is actually not stable, but it's not until you keep adding more of these strands that it becomes stable. And of course the first one, it's easier, so we're gonna focus more on the second part here. That's gonna be a more difficult challenge. Based on that, we can show that for this to happen spontaneously, the formation process, we need one turn and then we need to have two long strands that they need where all residues are exposed to the edge, right, n is the number of residues. And then we need to add one more turn. And for that to happen and for that to be, when that is exactly zero, so that it can happen from this equations, and again in the interest of not keeping you longer than time, we can calculate that there is a certain limit here that if the length of each strand is too short, we're gonna keep paying for those turns. Because I remember, I already said the turns are expensive. Then we're gonna keep paying for the turns, but we don't really gain enough back from the hydrogen bonds and everything and putting things in the beta sheet. So there has to be a certain minimum length to the beta sheet. And based on that length, you can at least estimate roughly what this transition point should be. And I won't take you through all the details of the question, you can do that if you want to, but the key thing is that that will depend on the free energy of putting things in the beta sheet. That might sound obvious, but it's not obvious at all. You remember for the alpha helix? The initiation barrier just depended on two hydrogen bonds. It didn't depend about what residue you had. So it was more or less a constant. It will vary a little bit from helix to helix. But here you have the free energy of this barrier here will depend whether this residue likes to be in helix or not, sorry, in a sheet or not. That can even be, that can be positive or negative. And it can certainly arrange for, say, one K-cal to 10 K-cal. So you have a much larger span. And the book then has to prove that this is, there is no other lowest possible transition state energy, but I'm not gonna care so much about that. But what you can then show, the same way if you use this kinetics, the time it takes to form, that has to do with the time it takes to go over these barriers. Don't worry about the specific time here, that's not important. But the likelihood to go over this barrier, that again had to do with e raised to minus delta G divided by KT. If the delta G of the barrier is very high, it's gonna be a very low probability to go over it. And that means that the rate, how fast you go over it, is gonna be very low. The rate corresponds roughly to one over, it is one over the time. So what that will mean that for beta sheet formation rates, the time it takes to form beta sheet, it contains a bunch of constants here, but the real point, it will depend exponentially on the stabilization energy of the beta sheet. So this can ease, and this is the way it can go from milliseconds to hours. So I'm just gonna show you one slide about that, that's pretty cool actually, that I will come back and repeat this. This explains a bunch of misfolding in nature, in particular things like prions. Are you, do you know what a prion is? This was really popular about 10 years ago when Stanley Proust and I got the Nobel Prize for it. Because it was quite scary. Have you heard about mad cow disease? Bovine's point to form encephalopathy. So that there were a bunch of strange neurological diseases where it appeared so normally most diseases are conveyed by viruses or bacteria, right? And we know, we understand them, you can have some inflammation too, but disease agents are viruses or bacteria and you can destroy them by boiling. And what we then realized that there were certain strange diseases that would somehow be conveyed by protein. Even if you boiled it, you could autoclave it and the disease agent would still survive. It's super scary, diseases that could be spread in ways that we couldn't control at all. And in particular, these permanently spread through animals. And what happened in the UK then that they were feeding cows with the remnants of the slaughter from other cows. You were feeding cows, cow brain. And what then happened is that they suddenly started being very horrible neurological diseases where young people, roughly your age, started becoming like premature Parkinson and everything and they eventually died. And this was traced down to this hamburger meat in the UK that these disease agents were able to spread from the cows to the human. And exactly how these things happen, we don't know that, but what basically happens is that you have proteins that normally have one stable state, but under some conditions, they can very, very, very slowly convert to another state. So it's proteins with two stable states. But what then happens when you have lots of these proteins, the mere presence of these proteins appears to accelerate the entire process. This is a slow process. It would normally take 100 years. So nature over 50 years. And nature solves that by killing you, sorry. It doesn't matter because again, we reproduce in our 30s, right? And that means by the next generation won't be influenced by it until you start eating somebody else's brain. Because this will go straight through the stomach and into your brain. And then you start to accelerate the process and building up these plaques. So we will, I will come back to this a little bit on Friday, but we understand this in reasonable good detail. I understand Prusina got the Nobel Prize for this. Sorry for taking two extra minutes. We are pretty much in sync for Friday. There are, I'm gonna tell a little bit about the differences between helices and sheets. There are a bunch of study questions. If there is one thing that you should do tomorrow, when you partly have a day off, oh, my bad. Understand, EHSFG. You need to know these things like running water and you need to be able to work with them. Look at the surrounding around you, water processes, and think about whether this happens spontaneously. And can you estimate whether things have lower or high entropy? All these other things come out spontaneously, if you understand those five letters. So we don't have particularly difficult equations in this class, but they require some thinking. And then I'm gonna go back and look in the biology and proteins on Friday. Do let me know if you would like me to create some recordings and go through the equations that I skipped over today in more detail. It depends a bit on your interest, but I figured if I'm never gonna ask you to derive them, there is no point in wasting 10 minutes deriving the myself on the backboard. So see you on Friday at 10 a.m.