 I'm not sure whether this is good or bad news. The advantage, after today, the math is going to be easier. But today is likely going to be the hardest math that you have in the entire program. If you, before you start, well, the point, the point, there are two reasons why I go through this math. First, I think it's kind of important to have seen it. And the reason for that is that there are some things that are simply universal in nature. And those are some of the things. The second part is not really complicated math. The math is simple. The hard part is thinking about it. And if there's one thing I wanted to learn that I want you to learn from this course, it's really that it's not dangerous to start defining things, thinking about it. You will likely not in your future career do derivations and statistical mechanics, but you will do derivations. And you will have to sit down and assume things and model things. And I think this is a great training in this way of thinking. But I also want, in particular, the harder equations here, I'm not going to ask you to be able to re-derive them in the exam. Second part is that there are, like, 50 slides or something today. I kind of expect that we won't get through all of them. So whatever I, well, I'm going to spend the time that we need for the early part of statistical mechanics. And I haven't even mentioned anything about alpha helices and beta sheets here in the title for today. If we have time for them, it's great. And otherwise, we'll continue with that tomorrow. So I gave you a small homework task, but I didn't mail it out until yesterday. And the minute after I did it, I realized, wait a second, if I'm pretty much going to finish off statistical mechanics today, and we also have quite a lot of math to cover, it's much better to do that tomorrow. So did you all read that mail? So what you're going to do tomorrow, everything we went through the first week and this, try to explain this to either to me or to your neighbor, but without using all the math. Because it's not really complicated. You don't need all the math to understand it conceptually. But as you would likely figure out, in many cases, it's harder to explain things conceptually than just type what the derivation or definition is. But just to get us restarted, I'm going to bring up these study questions anyway and try to do the same thing we did before the break. And I'm not sure. So we have one new participant here, right? So I'm not sure. Did you go through the lectures last week? Yeah. Yeah. So Bjorn and Ari also told me that to tell you that contact them and they will help you get started with the labs if you need a login or anything. But what I do to mildly force you to start studying and at least thinking about it, I'm not going to go through this. This is up to you. So take a question and start answering it. And the trick is, if you start answering early, you get easy questions. So you stay silent and wait till the end. You get the hard ones. And I also won't grade you on anything you say here. You get a grade based on the exam. So questions, pick one. And then let's start discussing it. 2. 2? Yep. Exactly. And that's when you say the most common helix. That is what that's the right handed alpha helix. Yes. And the point here, if you know that pattern, it also seems very natural to realize there is one pattern that's slightly tighter wound. And then you reduce this pattern by one or it's one that's slightly less tightly wound and then you increase it by one. The names of those other helixes is kind of irrelevant. I was one out of 15. Dysulfide bridge, yeah? Yeah, so the point is a covalent bond in contrast to ionic bonds and everything. And it's all, you write that it's a skeleton that it's strong but why is it so important compared to many other bonds, say the peptide bond or something? Peptide bonds are important too. You already mentioned that it's a bond between two residues but where are those residues located? Yes, we said that. But the point is you have two systems that are located in quite different parts of the sequence, right? So it's not a local bond. And for this reason, they tend to reduce the entropy of the system a lot. It orders the system. So if you have a large structure, you lock in an entire structure or something with a disulfide bond or disulfide bridge. So it's not local. Right? I have to say this because the paradigm bonding pattern is aligned vertically, if you imagine it's in this direction and there was all of them going the same direction. So almost exactly right. So it's due to the hydrogen bonds but not the hydrogen bonds directly, right? So what is the, the reason why I have a large dipole in helix is that you have lots of small dipoles. And those, well indirectly, you could argue it's a hydrogen bond but these are due to the part hydrogen bonding which is really the peptide group. So the peptide groups have this dipole and they all line up perfectly. But of course, they are also the ones forming the hydrogen bonds. And related to that you get something, because of that you get this item in 10 helix capping which is what? Right. And why do you get that? Because it's a positive charge at one end and a negative charge at the other end. Exactly. Well, technically it's not a charge, right? But the dipole is the equivalent of having a charge at one end. So it's at the positive end of the dipole it's really advantageous to have a negative residue and vice versa. And you see these things very clearly in bioinformatics. Other things. Well, yes or no? That depends a little bit when you do think, you know what, I would actually, we're gonna go through this a lot, either today or tomorrow. So we can actually, I realize in hindsight this, we can actually wait with that. One important thing you realize in vacuum, the energy component is really important. But we also have the second you have things in solution the energy pretty much cancels out. So it turns out when it comes to these free energy components there's gonna be something that, there's some sort of barrier initially that we need to get over. But, and depending on how high that barrier area is that's gonna affect the rate by which this can form. And then eventually you get over the barrier there's gonna be something else that stabilizes it. But we're actually gonna see in, it's mostly entropy for both of them actually. But we will, I read in hindsight those were stupid questions to have. We'll go through them later either today or tomorrow. What's the historical way to get secondary structure? So what's the method? Yes, I see this spectroscopy as the bottom end. And I'm not saying that in a negative way. But bottom end is good because it's deep. It's super cheap. Anybody has a seated spectrometer. What's the disadvantage of seated spectroscopy? Yeah, so you only see the average secondary structure content of an entire molecule. You don't get the sequence resolution. And if you would like the sequence resolution you would do what? Yes. And why would you not, why would you do NMR rather than X-ray? No, hydrogen is very relevant for secondary structure. The difference is the two of you, you go and get your sample and get it to the NMR spectrometer at Stockholm University. And it's Monday today by Thursday or so you're gonna have your secondary structure. You start to do X-ray crystallography. By November you might have your protein overexpressed. Then you're gonna need to crystallize it which with a little bit of luck you might have done next summer. Another six months later you will have gotten time on the synchrotron in Grenoble. And just three or four months after that, so that the point is it takes years to get an X-ray structure. If you're lucky, it might not even overexpress. So NMR is cheaper and better. But what would you do in practice? Yes, because in practice computers today get 85% correctness in secondary structure. And the problem with secondary structure isn't exactly defined. Helices and sheets are a bit fuzzy at the end and loops and everything. As always there are exceptions, but overall bioinformatics does the job that's just as good as the best experimental methods. It's not worth trying to do it experimentally. And you already did that in the bioinformatics course, some of you. Why do you gain so much energy from solvating D, E, K and R? So first, what is D, E, K and R? I don't expect you to know these one-letter amino acid alphabet by heart, but long-term if you're gonna work with proteins, it helps. They're charged amino acids. And the main reason then is that if when they are not in water you're gonna pay an enormous penalty for having it charged in a vacuum or equivalently in oil. So for each state you have an energy and an entropy, so that's why it's going in because the difference is that occurs in energy and entropy. Yes, almost. I would even say that an energy landscape, that's a continuous distribution of energy as a function of ideally, in this case I actually didn't say what the landscape is, but the extreme case this would be as a function of the microstates, right? So that each point in the energy landscape would correspond to one microstate. Then there might be some re-and, and again each point has an energy. Some of them are gonna be favorable, others are gonna be unfavorable. And then regions of the energy landscape that we can somehow group into a state that we can observe say a folded state, which is not really just one point, it's a collection of states. The entropy there enters so that how many microstates are there like that? So you can say that entropy is kind of related to exploring the energy landscapes, which part we will explore in practice. Yeah, that's a very good question. How do you think? I really don't know, that's the same question. So the problem is that this is hard of course. Computing an energy landscape with a one residue it's easy, you can just throw it in a computer and calculate it, say a dialanine peptide or something. And we can even, the accuracy you get in a computer there is gonna be less than point one kilojoule per mole. So it's gonna be beautiful. And you're gonna do this week, you're gonna do three more labs on this very simple systems deliberately. But starting from next week, we're gonna move you over and start doing computer labs on real proteins with modern simulation programs. You're actually gonna calculate real energies on real structures and simulate them. So the beautiful thing doing this on one residue, trivial. Doing this on a thousand residues or even a hundred residues, the problem is that your conformational landscape explodes. So doing, if you just have one residue it's gonna be say a hundred by a hundred states it will take, doesn't matter what, you can even do it with quantum mechanics. It's gonna take a couple of hours on a computer. So the problem here is that with a thousand residues is just as simple, but you don't have time. It would take a billion years to calculate on a computer. So you can't, you can't evaluate the complete energy landscape for a system with 32 degrees of freedom. Bad luck. But that doesn't mean that we should give up. I'll come back to that later today because this touches on a very important question in this lecture. 13, difference between states and microstates. Yep. Yes. So that one problem with this, of course, physicists in particular and scientists, yeah, we're not always good at separating between these. So first we took, whenever we're working on a problem I typically call them states because if I'm working on something particular usually you only work with one type of states. So if you should be really careful about this you should call one of them say macro states and that would correspond to something macroscopically an observation you can do in the lab say the temperature or the energy of a system. Now exactly how the systems looks on the inside on the atoms or so you don't care because what you're really, you're interesting in your experimental observables that characterize the state of the system. While the microstates are how all the particles all the components of the system are distributed. And in general there are way many, in general there are billions of microstates that represent one macroscopic state. Exactly. Or you can say that the observables you get in a macrostate is an average of the observables in the microstate or the properties in the microstate. Number eight, tryptophan, what's special with it? Exactly. So and the reason, there are two reasons why that fluorescence is important. Fluorescence in general is of course nice because it's something you can observe experimentally but it had it been alanine that was fluorescent we would have had another problem. Exactly. So you would, every single protein would have fluorescence and you would have fluorescence in 50 different parts. The neat thing is that tryptophan is so rare that most proteins might have one, potentially two tryptophan. And if you mutate out one of them you only have one tryptophan in your protein. And this prorescence then depends on the environment of the tryptophan. So there you can start, you can use this to study properties locally in the protein. And while things like this might sound very detailed and low level by physical properties you should be aware most of the things that you later on are gonna work would say high throughput sequencing or something. How do you think you detect sequences in high throughput sequencing? Not, well it's normally not the rate of tryptophan directly but it's prorescence. So every single way you need to derive data will somehow have to depend on observable, right? And all these observables come back to this. It actually turns out that high throughput sequencing you're also gonna have this problem is how do you, you can't detect every residue because that would be too slow. So you start trying to detect combinations of two, three, four, five residues. But then you also, how do you tell these apart? And then you're not having all these statistics between different states and everything. I don't think you're gonna go through that in that course but anytime you do measurements you're gonna have to worry about states in principle. Let's see. Do anything else that's fun here. Beta sheets twisted and why is the alpha, right handed alpha helix by far the most stable helix? So that's one question. Why is the right handed more stable than the left handed? Well, that's because of the chirality of the amino acids, right? The amino acids we have happen to be more stable with right handed alpha helices. The reason why the alpha helix is the most stable is that it fulfills all these hydrogen bonds and it's relatively relaxed. So all these other two helices that are strained either because it's wound too tightly or not wound hard enough. So the larger helix is actually gonna have vacuum in the middle of it and the 310, the tighter helix is gonna be a bit strained in all the bonds. So it's relaxed but has all the hydrogen bonds. And maybe we should finish off with these things based on the labs. Multiplicity relative to microstates and entropy. Do you remember that? Well, not one microstate that are dependent on the same indicator. So they're effectively is a microstate but they somewhat are an entropy to the system because it's not the same thing having one microstate. It's like having a bigger microstate. Right, so the whole point is that simply if you wanna go to a stage with a given energy there's more than one way to do it suddenly, right? And that's something you have to account for. And I forget what was Bjorn and Dari that explained to one another. I like that explanation. With this, if you are in a given state there are different probabilities to moving to different states, right? And if I am at an energy five you can argue what is the probability of moving to another state with energy say six or four? But if there are suddenly 10 states with one energy but only one other state with the, if there are 10 states with energy six but one state with energy four. In principle, moving up in energy is bad, right? But on the other hand, but it's still a finite probability, it can happen. But the more states like that there are the larger the probability that I will go to one of them. But in the lab you will still just observe that as one macro state with energy six. But this means that this whole multiplicity is something we need to take into account when we decide how probable something is. Even if it's not the best possible states if there are lots of states like that collectively this macro state can still be likely even though each individual micro state is less likely. And that brings us to the last one. Based on that, how would you explain entropy? Exactly, or if you try to explain that without using those advanced definitions? Are there any different states? I think that, yes, I think that was a good way of saying it. And I would just add one thing, experimental observation, right? So that there are more ways to represent the inside of the system that corresponds to one given way of having the outside of the system say the observation. And the point is that right now we're not at all thinking about things like for instance disorder or anything the way that you typically try to introduce entropy as a phenomenological term in chemistry or something. But it's just has to do with, it's basically probabilities. The number of waves we can arrange something. We had a bunch of these things with hydrogen bonds. I won't go through this in detail. And my point here is basically all these processes you do or something we can relate to what are the changes in enthalpy and what are the changes in entropy. And similarly can repeat that inside a protein too and I won't really go through that in detail either. But the difference we have in vacuum all the things look really simple. And in vacuum at least the first approximation unless the molecule completely changes we're likely not gonna change the entropy a whole lot. But in vacuum if you form a bond or something it's almost always a change in enthalpy that you gain energy from the enthalpy from doing something. The caveat is that the second year water or solvent around it we screw up everything in the sense that we're usually just redistributing bonds and everything. And in general again that means that the energy is exactly the same but you're talking about differences in entropy. And that's what's gonna come back to protein folding. So what I'm gonna do today is we're gonna get back a bit to the foundations of statistical mechanics and there will be some math here. But what I want you to think about is are these questions we're gonna try to answer. So first what do we mean by a state and what do we mean by the things that we're still playing around a bit with them trying to cover this in normal words the probability of observing a system in a particular state. What do we mean by that? You might remember that I did this hand-waving derivation of the Boltzmann distribution and everything. Today we're gonna do something pretty cool. We're gonna derive the Boltzmann distribution for a general system that we don't know anything about. And the reason we're bringing that you don't need a whole lot of advanced mathematics to do this. The hard part is that you don't know anything about the system. And this is you for some reason we usually don't do this a whole lot in undergraduate teaching. If you don't know anything about a system it's gonna be up to you to start defining things. What is the energy? What is the number of states? The worst thing that can happen is that we do something wrong and that we might have to go back and change it. I will make a small confession that I know how to do this. I'm usually not gonna screw up. I might make some typos or something that happens all the time. But the point is it is not dangerous to assume something that turns out to be wrong. The worst thing that can happen is that you have to go back and change it. And this is something that you should learn in the long term. Don't be afraid of starting to assume and play around with a system no matter what the system is. What this will help us to answer in addition to the question that how likely is something there's something that I keep bringing up that we haven't really defined. What is stability versus instability? And you probably know this with a gut feeling that stability is somehow the system that's happy where it is. Instability is a system that's gonna move to another state. But that's a pretty unphysical definition. And that in turn will then bring us to some things. What are transitions? What are the barriers? How is that related to free energy? And in particular, this will allow us to start looking at kinetics because there is a fundamental difference here. The Boltzmann distribution, there are two parts of it. The Boltzmann distribution tells you what are the states of things at equilibrium? If a free energy of one state is lower than another free energy, you're gonna be in that state. It's more likely. You will have some population in the other states but it's clear that an individual particle would long term prefer on average to be in the better state. But that doesn't mean you will get there instantly, right? Because there is also a matter of, the question is can I get there? And if there is an infinitely high energy barrier or something, even though in theory it might be better on the other side of the wall, if I can't get through the wall, it doesn't matter. In physics, this is kind of irrelevant because in the long run, things will go to equilibrium. The only problem is that in the long run, we are all dead. So in biology, kinetics matters a whole lot. How fast do things happen? Do they happen fast enough that it's biologically relevant? And in some cases, things happen so fast that it effectively happens instantaneously. And all these things are gonna be super important to understand things later. We will come back to this, our friend the partition function, which relates to your question about when can you determine the entire free end, the landscape or something? And then I'm gonna talk a little bit about phase transitions. And I again, I deliberately haven't said anything about helices and sheets because I don't think we will get there. So if we go back to statistical mechanics, for some reasons that is, no, there are good reasons. Statistical mechanics has its reputation for being complicated, difficult and everything. In many ways it is. It is extremely difficult if you were to try to formulate the equations of every single particle here. And if you, again, if you try to zoom in on a square centimeter or something on this picture and try to describe all the interactions of that particles, and then you multiply that by whatever, 2000 particles or the number you have here, that gets complicated. In theory, you could do this in a computer today that usually works much better and that's in part why we can use statistical mechanics for more complicated systems today. But the point here is, of course, we're not looking at the square centimeter here, right? Because if you look at the square centimeter here, either on the left or the right, there isn't necessarily gonna be a substantial difference. But to you, I think it's very easy for you to tell that there is a difference between the left and the right side here, right? And that has to do with these averages that we're talking about. There is some average order on the right and on average, there's more disorder on the left. So the whole idea with statistical mechanics is that for realistic systems, such as ones we have in biology, but also in physics, water, you have a huge number of particles, not a million, not a billion, but think of a mole or more, 10 to the power of 25 particles. And this means that you can no longer study individual properties. We can talk about an average individual property or something, but as a system, the only thing that really matters to us is what is the average properties of the system, say the density or the energy per particle on the left versus the right. And that also by definitions mean that we cannot study each and every individual state. If you just think, and again, this is a tiny system, right, compared to the ones we're gonna look at later. But if you try to imagine every single possible confirmation of every single possible particle here, it's gonna be an astronomical number. So there's absolutely no way you can study each and every individual microstate. So you can't even observe them. And this is why we somehow need to use statistical mechanics to go from these microstates to the macrostates that we can observe experimentally, such as density or something. And this means that exactly as many of you said before, that we start looking at average properties. And the second you have average properties, this concept of entropy also matters. Because entropy is really the connection between how many microstates will give rise to this macrostate, right? So that's the connection between small and big states. And that suddenly gives us a very, why, when we define entropy, the entropy we typically define as some constants types the logarithm of the microstates. Why is there a logarithm there? You just trusted me when I introduced that before. I said, this is our definition. And of course, you can define, the nice thing with physics is that you can define things almost anywhere you want. The worst thing that can happen is that if you define it in a stupid way, nature will punish you because you end up with very complicated equations. So why did we put a logarithm there? Exactly. I'm not sure whether you did this in the bioinformatics course, that if you imagine the number of states we have here on the left, let's say we can call that x1. And that corresponds to the number of ways we can organize the left side. And then the number of ways we can organize the right side would be x2. The number of ways we can then organize the entire system, that's just probabilities, right? And then we should multiply x1 by x2. And then we keep having these multiplications. So if you have two systems that you do something which we're not used to thinking in terms of multiplying, we wanna, if you double the size of the system, we like to put a two in front of something. And by having a logarithm, we can suddenly call this entropy. And the entropy for the total system is the left side entropy plus the right side entropy. I'm not sure, did you in the bioinformatics course that some of you took, you go through the definition of these alignment scores or something. Okay, but the point, you do exactly the same thing there. An alignment score is really a rate of what is the probability of putting one residue next to another residue. And if you do this for an entire sequence, that's just a gigantic multiplication, very long multiplication. That leads to two problems. First, you're gonna have, start multiplying a lot. Second, these numbers will so small that you will frequently run out of the floating point, the floating point domain of the computers. So there too is makes much more sense, take the logarithm of it and add the numbers instead. But that's a completely arbitrary definition just because we like to make our life simpler. I bet that the person who first did try it without the logarithm first, and then he went back and realized that it's much smarter to do it with the logarithm. The other part of that leads to, so what is temperature then? This is a hard question, I don't expect to do it, but think about it. We don't know anything about, what does temperature really measure in a system? If we think of that in terms of microstates versus microstates. So in a way, temperature describes the interchange between states, right? If you have a high temperature, you have lots of different, then you have more energy in the system and there are more ways that we can arrange things relatively in each other. So high temperature is somehow describing how much freedom we have inside these states related to the energy. And that also means that if there is no temperature, by definition we only have one microstate and then there is no entropy. Mm-hmm. Temperature will not have more entropy than more energy because energy, like, temperature breaks fast, so then... We'll come back to that in a little bit, but the whole way in, it's a good question, but the point is that entropy, energy, and temperature are connected to each other, right? You might remember these definitions. I will come back to this definition. This was deliberately a bit hand-waving, but they're not three-separated. So temperature is something we used to think about well, in the lab or something, but here it's not an independent quantity. Temperature really measures how much the entropy as changes relative to the energy of the system. And we're gonna calculate exactly how much in a second. So this brings me to the mathematics. I've stolen some pictures from the book here. But what I like here is that we don't know anything about the system. We're just gonna consider a system with a capital S. This system can be absolutely anything, and this is the hard part. So what do you do when we don't know anything about something? Well, in principle, we could do what we did before in the course. I could assume that this is a column of gas or something, but the problem, the second I assume something, the derivation is no longer general, right? If I assume something, then the derivation is only valid for the case that I've assumed. So if I want something to be completely generic for any system, I can't assume anything about the system. What we talked about before the Easter break is that physicists usually love to think about this large system or something where the energy is conserved. So we're not really, we're not exchanging any energy with the surrounding world. And so you can think of this, the big square here is either our lab or you could think of it as the universe or something, it doesn't really matter, but it's some sort of large system. You could also think of this as a large, completely isolated solution with a protein in it. And the point is that once we get here, at least the first approximation, there's nothing that comes in or out here. But we're also going to look at a much smaller part. So this is the part we are observing. And this could, you could think of this as your protein. You could think of this as an individual atom that we're looking at or something. So that here's a small system and here's a large system. And the reason in physics, when we say that the energy is conserved or anything, we typically call this large system the thermostat or something. So that the small system can exchange things with this larger system. But the larger system doesn't exchange anything with the surrounding world. And the reason for that is that if the small system then takes something here, that has to be donated by the large system and vice versa. So anything that goes in here is taken from the thermostat. And so this is our world. This is the universe. The large system can be, of course, be divided in tons of different micro states. We can think of each point here as an atom, but again, that's just how we're thinking about it. And if we pick this small system small enough, let's pick just one atom or something so that the second I picked the small system here, this is so small that this kind of corresponds to one microstate only. So if I picked that system and that's perfectly defined and then that one has some small energy epsilon, the rest of the energy must be the total energy minus that epsilon and that is in the thermostat. But of course, the thermostat contains way more than one particle. So the thermostats, if these two particles change place, that's not gonna influence the small system. I can trade place between these two particles, et cetera. So in particular, when the thermostat is very large, there's gonna be a gigantic number of microstates for the thermostat that corresponds to the case when we're observing the energy epsilon in the small system. Do you follow me there? And the hard part here is, of course, that we haven't really assumed anything yet. Now the point here is that if we can, let's just call that something, we can call that M for macrostates, large M. If that is the number of microstates that corresponds to the macrostate when we're observing the small system here, that is kind of a function of the energy. And the probability of observing the case in this one we have exactly epsilon there. Well, the more different ways the thermostat can be arranged, the more likely it is to observe that. That's not an entirely obvious result, but if you think about it for a while, I think you will agree. If every microstate in the thermostat is completely identical, and we are asking you how likely is it to observe this small state in the epsilon state? The more microstates that corresponds to having epsilon there, the more likely it is to observe it that way, right? There's gonna be a proportionality there. And then we just throw in our definition. So the only thing we have defined now, we've said that what is the energy, we've said that the energy is conserved. We're asking how likely is it to have epsilon there. And then we introduce the definition that that's the number of microstates. And then we've already said, or you can say that we define it again, that the entropy of that system, we just say that that's a proportionality factor multiplied by the logarithm of that number of states. The only reason you might think that this is Boltzmann's constant, but it's not. Actually, it is. But this is just a proportionality constant. Because we like to give this S term the units of Jol per Kelvin, energy per temperature. So for now, this could be absolutely anything. So what we've essentially done is that we've divided the system into all these microstates of the large systems. We've divided them into different classes where each class corresponds to this small system being in exactly one state corresponding to epsilon. And this is something that you will likely read up in the book too. I would be astonished if you got this the first time. This is just a completely arbitrary constant. And now we're going to get a little bit more complicated, but this is the most, I think this is the most advanced math we'll do. So entropy, for now this entropy is just a definition. And this is not really, this is not more complicated than the definition. What I say here is just that S equals a constant times the logarithm of the number of microstates. The reason why this looks a bit more complicated is that I've stressed that it's for the thermostat. And I've said that this is a function of how much energy we have in the thermostat. But again, all this stuff is just, I can't even really. All this is saying is S equals K multiplied by the logarithm of the number of microstates. And then you're kind of screwed. How do you get further from that? Try not to look at it. Don't look at your notes. It's much better that you follow this because this is something very typical in physics. So what you do in physics all the time is that you're looking at one state and then you want to understand what happens around the state. Because if you're just looking at one state, you're looking at that state but nothing else. But the second you want to look at a process or something, you typically want to look at the surrounding. So what I've done here is that this state here is kind of capital E. And then we're looking at something small that's changing around it, the epsilon here. And the way you normally do in physics if you're looking at a small variation, you do a series expansion. You remember that from calculus. If you take an arbitrary function, completely arbitrary function, the absolutely easiest approximation of a function at any point, that would be its value, right? If you want to start looking at how a function varies around the value, you would say that you have the f is approximately equal to the value in that point plus delta x times the derivative, right? Delta x or something. So then you're looking at both the value and the derivative. You can add the second derivative, the third derivative and everything. You can go as far as you want. And if you have an infinite number of terms here, you can describe the entire function. And for most physicists, the infinity is usually one, occasionally two here. So that you only use one, two, or occasionally three terms here. So I'm going to do a series expansion. And this is much... Let me throw up the equation first. I think you think that this looks complicated, right? Is this completely obvious to all of you? Good! This is much easier than it looks like. So what I'm saying here really is that a completely arbitrary function here and how that varies around some point, that is just equal to... First, we have the value in that point itself. That's the function value. And this should actually be an approximate equal sign here, not exactly. And then we have... Actually, this should be... It's correct that it's a minus there, but you can think of this plus the delta x, the small variation around that point, multiplied by the derivative, right? And that is the derivative, because this is a function of an energy. But in this case, there is a minus sign before that change, so that minus sign moves in there too. So all I'm saying here is that the way the entropy changes around a value is the value of the entropy in that value. And then the deviation multiplied by the derivative. And all what this means is that this vertical line here means that I take that derivative at a constant value of the energy. So I put in the value of the energy as e here. So a friend of order would then say why do I stop after the first order term here? Why don't I have a second derivative where the third derivative or a fourth or a fifth eleventh derivative? This is going to be pretty bad, because I kept saying at the start of this lecture that this is universal, and then I keep doing this horrible approximations. If this is an approximation, statistical mechanics is not exact. So there's an easy and a hard way to judge that. First, what is the value of epsilon compared to e? Yes, you could even say infinitesimally small, right? Because the whole point, the thermostat is something that is large enough. And large enough could in principle be the universe, while epsilon is very small. However, that is still an approximation. Just because something is small doesn't mean that the next term is zero. That just means that it's small. And you can think of that in a slightly different way. If you think of the system that I showed you, how does the entropy S vary with the number of particles or the size of the system? Yes, so to first approximation, it's proportional to the number of particles, right? So this term is going to be proportional to the number of particles. How does the energy vary with the number of particles? Again, to the first approximation. Yes, so you take something that's proportional to the number of particles divided by something that's proportional to the number of particles. The first approximation, this term is going to be... Yes, it's not proportional to the number of particles. The next term, the second-order term, then, yeah, I would have d2S, dE2. So then you would have something proportional to the number of particles and the square of the number of particles. And for a system that's large enough, that's going to go to zero. So any higher-order term will go to zero as it's not just going to be small. If the system is large enough, it will go to zero. And that's why I actually can have an equal sign here, if the system is large enough. So does that mean we only care about this when the systems are large? Yes. The whole point of statistical mechanics is that's the statistical part of it. Statistical mechanics describe systems that are so large that we need to study average properties rather than individual properties. The cool thing here is that we still haven't really assumed anything about the system, but suddenly we can start saying how things change and what are the proportionality constants here. There's going to be something that's proportional to our small energy here and everything. This is the hard part about physics, but also the beautiful part. You can learn a lot of things by some very mild and simple assumptions. So for instance, the thing here we said about the magnitudes of these terms and everything, that's going to be true for any system. We haven't assumed anything about the system yet. And then you just throw up these things. This I realize is the hard part. Suddenly I think the book 2 suddenly says you're going to solve for m. That's not at all obvious. Why would we solve for m here? Almost. I'll get back one slide. Remember what I said that the probability of observing the system in this state is proportional to m, right? And this, of course, is what we want to get to eventually. How likely is it that this system is in this particular state? And that's why if that is described by m, of course it's m that we want to get. And we already made this definition that the entropy s equals a constant times the logarithm of m. This I think is typically something that if you didn't know what you were going to do, I would do a mistake here. And then you would need to go back and realize, wait a second. What was it really that I was after? If you don't know what you're after, it's very hard to derive it. But if you ping-pong back, I was interested in observing how likely the system, what could that likelihood correspond to? It's m. So let's jump back to here. We now have these expressions that's a function of the entropy. So we can use this equation again backwards and try to get m from it. And again, this is trivial. I'm just taking the exponential of both sides here. So that m equals the exponential of this entropy divided by the constant. Exactly the same thing as we have up there, formulating in a slightly different way. But now we also expanded how this entropy varies around this state, right? So now we can take this definition and put in there and see what happens. Yes, sorry. This is m-term, my bad. I ran out of space there. And if we take that entire expression, so that's a sum or actually a difference in this case, that difference up in the exponential, an exponential of a minus b or an exponential of a plus b is an exponential of a multiplied by an exponential of b, exponential loss. And then the minus sign here will follow us so that this is an exponential of that term multiplied by an exponential of that term including the minus sign. And then I just compress that here to get it to fit on one line. And now we are actually starting to find some really interesting things. How does that first term depend? Remember that we were interested in our small system, the one with dj epsilon, right? How does epsilon come in the first term here? Right, so what would you call the first term? For all intents, it's just a constant, right? We don't care what it is, you could call that c. Because at some point we don't really care about the number of microstates, it's just that the number of microstates is proportional to something. And if you care about proportionality this is just a completely arbitrary constant. So in one way we just approximated away the entire first term here. So that the number of microstates in the thermostat is a constant multiplied by the exponential of minus the energy. And then we have something complicated in here. This derivative of the entropy divided by a constant. Derivative of the entropy with respect to the energy divided by a constant. That is not obvious. But we can actually say something here. The number of microstates is somehow equal to the minus of the energy. So when the energy of the small system changes, it's going to reduce the number of available microstates exponentially, right? And then there are some constants that we haven't really looked into yet. And this is completely arbitrary. It's true for any system. We haven't assumed anything about the system. This is independent of epsilon and that term is actually independent of epsilon 2. So it's only in that small term where epsilon 3 of this small system actually enters. Of course. But that's what we're showing now. We're not there quite yet, but this is the Boltzmann. We just proven that the Boltzmann distribution is true for any system. We haven't even started to talk about Newton's laws or anything. And this is by far the hardest part with thermodynamics and statistical mechanics that if you want to prove that things are universal, by definition you can't assume anything. But the beauty of it is that this does not rely on any observation. This does not rely on the fact that, say, general relativity is true. And so the amazing thing with these things is that it's just based on observations and mathematics. General relativity might, well, you could argue that general relativity might not be true because you can't really combine it with quantum mechanics. These things will never be proven untrue because it's not based on observation. So what the book then gets to and this is what I already hinted, the reason why we're solving for M is that the probability of observing this small part of the system in this state, epsilon, is proportional to the number of microstates of the entire world, the large system, that corresponds to it. So this probability is proportional to the number of microstates in the thermostat still, which in turn is proportional to the beauty with these proportional signs is we can forget about this constant and have an equal sign of the constant. Often exponential where the argument is minus the energy in the system with some large constants in there. And this, of course, is the Boltzmann distribution and these constants we don't really know more about them, so these are going to be important properties of nature. So one of them, this ds, how the entropy changes relative to energy, this is a constant which I, in principle, could call T. The only problem is that that would then be the inverse of what you normally think of as temperature. So that the way we've defined these temperature scales and everything, the Celsius or the Fahrenheit scale were completely phenomenological. And then it turns out that if we now want to define an absolute temperature that behaves the same way you think of temperature, we should invert it. Otherwise we should have a temperature that could go to infinity at zero Kelvin and then it would go to zero at infinity, it would just confuse you. So it's just, you could argue that this is a mistake we did when we first introduced temperature phenomenologically hundreds of years ago. And it's way easier to fix that. So that's why we call it 1 over T, but this is just a proportionality constant. And this other small constant kappa there, that is, of course, equal to your friend Boltzmann's constant. Why do we have the Boltzmann constant there? Is this really just one constant divided by another constant? Exactly. So if we had started from scratch today and derived temperature we would have measured temperature in units of Boltzmann's constant. So the only reason to have Boltzmann's constant is that we're used to thinking of temperature in terms of this arbitrary temperature unit Kelvin or Fahrenheit, well, Kelvin if it's absolute temperature. And to avoid screwing that up we introduced this constant. But this again is really a historical shortcoming that we didn't know of thermodynamics for L. We didn't either you could argue that we didn't know about it when people first started talking about temperature and when Kelvin, for instance, introduced absolute temperature and everything it was simply easier to keep it compatible. So today we would likely have some temperature called X that would be 1 over temperature and measured in units of Boltzmann's constant. And the gas constant is, of course, equivalent to that. The only difference between gas and Boltzmann is whether we measure things per particle or per mole of particles. And in principle, that is the Boltzmann distribution. We just don't call it the Boltzmann distribution yet. The probability of serving a system is proportional to e, the exponential raised to the power of minus the energy multiplied by some proportionality constants which is going to be called divided by kT. So the more you raise the more the energy of our small system goes up, we reduce that number of states exponentially. You can actually show that. The book is a small exercise. I'll show you just for fun. If we look at how likely is it for the system to move to say kT higher energy. kT was rather small. Do you remember how much was kT again? 0.6 watt? Yes. The units are important. So that's a small energy by all intents and purposes. And if we assume what is the likelihood of observing the system is something that has changed there. In this case, we increase the energy of the large system. Well, we know what the definition of those microstates is. That's just the entropy. And then we can take this term and put in the series expansion again. I'm not going to go through the math for you. And that turns out we have this S term and then we have T divided by T and k divided by k. That's 1. So this really corresponds to the logarithm of the number of microstates plus 1. Now, what that means, for every kT you increase the temperature, there are roughly a factor of 3 more microstates. You can go through the math if you want to. It's not difficult. Why do I show you that? Well, this is related to the fact why this kT is such a fundamental constant. So the second you're going up to kT, if you raise a barrier by a factor of 2 kT, it's roughly E squared. So almost a factor of 10. So by the time you're at 4 kT, you reduce the probability by a factor of 100. And this is why it thinks that our significantly larger end kT, the probability goes down extremely quickly. Conversely, when barriers are significantly smaller than kT, the probability that something will happen essentially goes to 1. And this is why the reason why this is such a characteristic energy scale or something at room, not just at room temperature, but for systems in general. Sorry. So again, something in biology, a reaction where the barrier is much lower than kT will virtually always happen. And if a reaction is much higher than kT, it will never happen. And this is no longer just based on hand waving. You've proved that here. The point of this is that we know just a little bit more now. I'm going to go through a couple of more slides and then I think we'll talk about this after the break. We've shown that the Boltzmann distribution is true in general. We have introduced concept of the entropy as a function of the energy. As the energy goes up, there are, well, as the energy varies, the number of microstates goes up by a factor of 3 per kT. And again, the number of microstates that is directly proportional to the entropy, the entropy is the logarithm of the number of microstates. So we can start to think draw a curve where we put that there are a bunch of curves like this in the book. They're not entirely natural, but I think if the entropy by definition is a function of the number of microstates in turn is a function of energy, we can certainly draw a plot where the entropy is a function of the energy, right? Exactly how that curve we look we don't care about right now, but it's some sort of curve that describes different ways where the system can be as a function of different energies. And understanding this conceptually is not entirely easy. This derivative that entered in the exponential that corresponds to one of the temperature, that is literally the slope of the curve. The slope is one over temperature. And here we come into this point that you talked about the size of the system. We can really only define this between these two over a small energy range. If there are so many particles in the system that I can define this derivative, right? Because if there are hardly any particles or anything, these are going to be noisy the number of states. So defining this as continuum properties of the curve is a system to be very, very large. So I can really only define this slope when I have enough particle that I have a smooth curve. If the curve is completely random and everything, there isn't really any slope here. And this means that the whole concept of temperature, I can really only define temperature when you're dealing with things that equilibrium and many particles. I'll come back to that in a second. It is, right? Well, you can certainly define a microscopic temperature, but you can't necessarily measure a temperature. I'll come back to that in a second. Let's see. Yes, I'll come back to that in a second, but just remember that the temperature corresponds to these slopes of the curves, or one over the slopes of the curves. This might seem like more heavy mathematics, but the point here is that I'm going to show you how this is useful. If you now design a completely arbitrary state, I, whatever, you could say that the probability of being in that state, we typically use W there for weight. The weight to be in that state, for instance, is a function of temperature and energy or something. For state I, that would correspond to the exponential of minus the energy of state divided by these constants we already had. And in principle, if I don't have the denominator there, I have a proportional sign here. But proportional signs are bad, because as you remember these slabs, you can't calculate a proportional sign. If you're going to calculate something exactly, you need to know what is the constant, too. And if you talk about probabilities, normally in nature, the probability of all things should sum up so the total probability is one, right? You are in one state. We don't know which state, but if you sum all states, the probability should be one. So they will, by definition, have an association factor here that is the sum over all such states. And that's actually, it's very simple. I just said that. We sum it over all states. So that factor is the sum over all states I, and then I have exactly the same expression. It's just something you add in. So the only reason I add in is rather than saying that the probability of observing something is proportional to, I want to be able to say exactly what is the probability. And then to get rid of that personality sign, I have to divide by the sum. And I think you did this in the labs, but you didn't think of it this way. My point here is that this is one of the deepest results of statistical mechanics, but it's also a very obvious result. All of you could do this. You're just normalizing probabilities over a number of states so that the sum sums to one. So mathematically and, say, statistically, it's completely trivial. The cool thing is that this is exactly what you asked about regarding energy landscapes before. This is the sum over every single possible state of the system. Every single possible microstate or microstate, depending how you see it. Microstate in this case. Do you think this is easy or hard to calculate? It was easy in your lab when you had three states. When you have 10 to the power of 25 states, it's impossible to calculate. So this is the sad and beautiful part. This tells you everything about the world if you have an infinite amount of time and infinite compute capability. If you have a small system, we can calculate this exactly. If you have a small system, on the other hand, what is the point of using statistical mechanics? Then you could just use a computer and calculate all the motions of the particles exactly. The beauty of this we need statistical mechanics when our systems are so large and so complicated that in practice we can't solve this. We can't. And then it's time to give up and close this course and go home right. This is our friend, the partition function. And the cool thing is that let's just assume for a second that we know what the partition function is. That our system is easy enough that we could calculate it with a large supercomputer. We know what the probability is of being in every state. So I can calculate everything. The energy of my system is just to sum over every state what is the probability of being in that state multiplied by the energy of that state. Pure statistics. Sum it up. Super trivial. Entropy. Same way. The total entropy of the system is just to sum over all states what is the probability of being in that state multiplied by the entropy in that state. And I'm actually going to help you a bit because I have a derivation of this. I will take you through that relatively quickly because that's not... What is S i here? What is the entropy of a small state? It's not entirely trivial. The book goes through this in some detail. So if you bear with me for just two, three, five minutes, I will take you through how we can calculate this and it's going to turn out that this is a simple logarithm of the weight of each state. But this is kind of a parenthesis. My point here is that if we know the partition function so that we know these weights you can kind of calculate everything about your system. But to be able to calculate that I can't have something that I don't know what it is. So we're going to need to be able to calculate what S i in each microstatus. The way that both the book and I kind of like to derive this is that so this is the beginning of the parenthesis. If you have a large number of N different systems that can be distributed in smaller systems system 1, 2, 3, etc. all the way up to J. And there are some weights here, weight 1, 2, 3, etc. all the way up to J. The number of systems that you have in each state corresponds to the weight of that multiplied by the total N here. And the sum of all those weights must be 1, 2. Then you have to go into mathematical statistics and start kind of how many different ways are there to sort this balls or dices or something in these different systems. And it turns out that this leads to a long, long, long description where you start looking at probabilities. Sorry, wrong order here. So if you start picking N1 systems for the smallest one that corresponds to well the first one we can pick from capital N the second we can keep from capital N minus 1, etc. And then you need to you need to do this math correctly if you're hung right in your mouth. If you want to go through this in detail, follow the book. I've had some extra we use a sterling formula to get the second expression. And then you just keep doing math, math, math, math. And after a while, you can show that this expression is equal to 1 divided by all these weights to the same weight, multiplied with each other, raised to the power of M. The only thing you're using here is mathematical statistics, really. But we're not, this is not a course of mathematical statistics. I don't really care about it. For a single system the total entropy in the system turns out to be 1 divided by all these weights, rates to itself multiplied with each other. And in that case, the entropy which is the logarithm of the number of such, the logarithm of the number of weights we can distribute that turns out to be the sum over the weight multiplied by the logarithm of 1 over the weight. This is by no means obvious. The point is, the reason why I'm even showing this derivation is just, trust me, this is just a matter of statistics. You don't assume anything about the system, you don't assume anything of the world. But in the end, the entropy of a state is just depends on the weight of the state and the logarithm of the weight of the state. So this really only depends on how probable a state is. What does that mean? Well, if you have, assuming that you have a thousand states and if you have one of those states that is extremely probable then that one is going to be high but it's going to be really low for all the other 999 ones and if W is very high this is a small number and then the logarithm becomes small. So if you have one state where you have almost everything of a low value for this but if you had all your 1000 states had almost exactly the same weight, 1000, then both of the terms are going to be relatively high. So if you have lots of states where it's roughly equally probable to be then this is going to be high but if you have one state where it's much more likely to be it's going to be low. So this also agrees naturally with our concept of disorder or anything. And if you're doing, I'm not sure how far you got into bioinformatics, but it actually turns out that you can define something called the Shannon entropy in informatics and it's actually exactly the same way just the probability of something has nothing to do with physics. You can calculate, you can define this in computer science too exactly the same way. End of parenthesis remember that I said that the energy in a state is just the sum over the weight of each states multiplied by the energy, sorry, the total energy is the sum of the energy in each state multiplied by the weight and in exactly the same way the total entropy in a system is the constant which we can put outside of the sum and then the sum of the definition of the entropy in each sub-state. Sorry? Let's try that in a second but my point here is that we know what, if you know the partition function you know what w i is. We know what this is and we know what that is. If we know the partition function we can calculate these exactly numerically and you might think that I or the book made a typo here I get it. What you really want to know is the free energy why do you want to know the free energy? The free energy, exactly. The free energy determines everything what happens in chemistry and this is how we're getting back to chemistry if a change in free energy leads to a reduction in free energy then that's a process that's going to happen if the free energy goes up that's a process that's not going to happen If I know the free energy of every single point in my energy landscape as you asked about then I know where in the energy landscape I'm going to be then I could predict everything I could instantly predict where a protein is going to be I can predict if a mutation is going to stabilize a protein I can predict whether a protein will fold I can predict literally about any chemical reaction and every single thing you observe in the lab is probably related to their free energies. If I know this this is our holy grail. We know everything about chemistry but we just we know E and we know S so this is the first term here is going to be well you see the W here occurs in both those parts so let's break out the W and then we have an epsilon j minus T and it's just so cumbersome to keep working of the logarithm of 1 over something and if you remember your logarithm laws I can take that that is minus the logarithm of W instead so that minus comes from 1 over W there don't worry about the exact details of the equation the whole point is I've just taken those two expressions and put it in that form the point here is that if you know that you can calculate F exactly for any state the system is in you can instantly say what is the free energy in and again this is an absolute free energy in kilojoules per mole the likelihood of having this particular transition state or anything of your protein or a molecule but what is the point of this derivation if we can't determine that well there is a reason why I bet what people have thought so much with the system because this is complicated and it's also historically it's very much something you do with paper and pen historically you might get up to 10 particles or something at 10 states doing it manually roughly the size of things you did at the lab right anything more complicated than that you won't be able to handle but there is a difference between today and 100 years ago is that we have computers so if you look at this again this is a sum where we have some complicated expression but we also have this weight here so if you have a particular state and the weight is 0.2 is that an important state well yeah kind of it contributes 20% of everything we're seeing right if this weight is 10 to the minus 39 do you think it's very important to have that state so the point is that some states all states are not created equal here some states might be important to consider while other states might not be important to consider so while yes if you do know z exactly you can calculate f exactly but conversely if you know z approximately you can calculate free energies approximately well the whole point this is essentially what you end up doing in a computer simulation right that if we can sample essentially what this is we can sample all the states that have high weights the states that have high weights really corresponds to the ones with low free energy the likelihood that a protein spontaneously would go through a fusion reaction and turn into a nuclear reactor well technically it can't happen it's not probability 0 it's just probability 10 to the minus 5000 or something I think for all intents and purposes the likelihood that protein spontaneously starts to go under fusion but now we start to approximate our partition function but the point is that if I can sample enough of the states that are relevant low free energy I can calculate f compare f between two states and in particular that's what we did very early on in the course if I only care about two states I really only need to sample those two states right to determine how good they are relative to each other the rest of the world doesn't matter so what this means is that the fosters our computers have gotten the better we can calculate properties for systems not just with one particle not just with 10 but gradually going to 1000, 10,000, 100,000 and many of these large computer simulations that we do today is simply that computers are so fast that we can sample systems that have hundreds of thousands of degrees of freedom there is no way we sample them exhaustively but we sample them well enough to do degrees of freedom that are relevant that you actually can start making conclusions and essentially what we're doing we're starting to approximate this partition function this leads to something else that's very important we should take a break here in a couple of minutes you might think in a simulation that you're simulating nature if I take a protein for instance and start to put that in a computer and push the button I can simulate how it moves you probably saw the movies what it was two weeks ago that I showed you these proteins moving that is not what a simulation is that's how a simulation looks it might look absolutely beautiful that you're having velocities and everything but what all simulations do are really that we're sampling things we're sampling as many states as possible and calculating how likely these states are eventually it's going to turn out that it's really advantageous to try to do that by approximating its motions but you can't simulate a motion of a protein you don't know it exactly but we can use it to sample states and this is also the reason why in all these labs where you spend all this time with this dirty toy systems it's much easier to understand this on the simple toy systems and then gradually when we understand those we're going to move back to the real systems the danger with starting with the real systems is that you think that you can do an absolute simulation of nature you can't do that I think this is a great... I know that there was a lot of mathematics here it's a great place to take a break what I'm going to do after the break is that you remember that curve that I drew up how the entropy varies as a function of the energy? we can actually use that to start doing some conclusions about what will happen for different systems and how do systems behave under different circumstances and then we'll see depending on how much time that takes I might have time to start applying this a little bit to alpha helices because many of these things might probably be easier to grasp when you see it applied but there's lots of deep mathematics here so it's definitely worth reflecting a bit about that it's 10... 25 now should we take our 15 minutes or so and meet here at 10 to 11? so the main conclusion from our point if we know simulations in general they approximate the partition function by sampling the system and the beauty here is that only the systems that are relatively likely to happen will be important so if you think of this in this energy landscape those very extreme peaks in red the ones that are extremely unfavorable if you just simulate a system you're never going to visit that point but the whole idea is that nature doesn't really visit it either so it's not really if you determine say the average confirmation of a protein or some average observable that's not really going to be particularly influenced of properties in these extremely unfavorable systems and that's essentially why simulations and everything will work pretty good at approximating the partition function with fast computers what I'm going to talk a little bit about now is plots where we plot this entropy as a function of the energy and this might seem bad but remember what I said in the beginning this allows us to start defining what we mean by these things as instability versus stability of a system so in general the only thing that we've said is that the entropy is a function of the number of microstates and the number of microstates in general is what the energy in a system that's what we showed the first half today and that means that in general the entropy is a function of the energy that means given a particular energy there will always be one value for the entropy and then I also said that in general the slope of this curve is one over the temperature but if you now, for some curves like this one, that slope will be negative so both of these curves will correspond to negative temperature can you have negative temperature? sorry? so this comes down, at the end of the day this comes down to definitions, right? I would actually argue you can have negative temperatures now this is even an absolute temperature you can have negative absolute temperatures but not at equilibrium so what this means that the point is that this blue line would technically be a positive state but it would instantly decide want to move away from it so that it's completely impossible to define this as temperature but if you have something like a gas during the expansion of a gas during a split second or something when the system has not yet reached equilibrium technically you could imagine a system where as the energy goes up the entropy goes down because that's what's happening both in this case, you get more energy and anyway the entropy goes down but that would be that if you are somewhat during a process where things are freezing but for whatever reason the energy goes up a little bit you will virtually never ever be able to observe anything like this because the time server would be infinitesimally small so macroscopically under equilibrium you can never observe a system under conditions like this technically, physically the way you just define a temperature it's formally possible and there are some fun, I used to have a paper on this but it's just physics it's not that important in some cases where people have been able to measure things that correspond to a negative temperature absolute temperature what does this correspond to? well this is a system where as the energy goes up the entropy goes up too so it's kind of more natural but this is a concave region I can definitely define what the derivative is here and this derivative will correspond to a finite positive temperature along this line, I know what the energy here is constant and this line corresponds to E-TS where the temperature is constant along it so this black line here the tangent that would correspond to a given free energy, right? E-TS, that's why it's a line but the point here is that all points further up here in the curve has lower free energy and all points further down there have higher free energy so every single possible state along this blue curve, except my point has lower free energy so that point has the highest possible free energy the system can have that is physically possible but that corresponds to the system balancing on the edge of a knife it's technically it's a local point where the derivative is zero so we're not going to have a force to one or the other direction so in infinitesimally large change either to the left or the right we're going to fall down on one side of the knife's edge so that corresponds to a system that's balancing at the peak of a transition state or something it's not impossible but you're never going to observe it there because it will fall down on one side or the other and conversely a system that is stable always corresponds to a region that's convex here so this is the extreme opposite the blue curve is again the entropy but in this case the black curve here where we are this is the lowest free energy we can have along the blue curve anything that's even lower would be up here I can't reach that on the blue curve and all other points below this black line has higher free energy so this is a system that for this particular energy it has found the absolutely lowest free energy it can be in so in practice all interesting regions that we can observe at equilibrium are going to be convex no I agree don't try to think of what this really means in terms of this system entropy this is just something we have defined my only point with this don't try to understand what this means my only point here is that we can define what's possible along these blue curves don't try to think in terms of what happens here or anything these are pure mathematical definitions the point of this is this slide if we now have a larger curve here and if the energy changes here now we can start to look at multiple points along this curve and then you have different temperatures and in this case temperature this is a higher slope which means that the temperature is lower and if we go up here the temperature goes up and we can draw that in a different way you can so how does the energy vary as a function of temperature and this is way more natural this is how you would think about it so as the temperature goes up here the energy gradually goes up and what this corresponds to is something that can happen smoothly you can be there, you can be there you can be there, you can be there you can be there, there's an any point you pick along this blue curve is going to be a perfectly nice point where you're going to be happy to be if you think of any process or something that changes what does this correspond to something that can change this way smoothly and nicely yes so if you have water at say 10 degrees centigrade you can go all the way up to 99 nothing dramatic happens you can stop at any point you want it's fully reversible and what you're going to see is that there's going to be a distributions in energies or something in your particles and as you go up this distribution will gradually move there is nothing special happening it's kind of boring but it should be boring but the point is that this blue curve can of course look absolutely any way we want there's nothing to say that the blue curve is like this and in particular let's assume that we start to have a blue curve that's starting to have some small depressions that goes up and down a bit so at least the second derivative is varying so here we have one temperature here you have another temperature up here but right in between here there's a concave region remember what I said about the concave region that's this local maximum where we don't really want to be so we can go smoothly all the way up to energy one here and then it's going to be bad we're going to be unstable in this region and then we can be at energy two and then we're happy and stable again and there is some sort of gap between these two that corresponds to a gap in energy but also some sort of you could imagine there is some sort of characteristic temperature here that corresponds to the slope between these two points that's going to be between these two temperatures in principle you can do the same thing here we can say that we can plot how the energy varies as a function of temperature and this curve is going to be roughly the same but now there's going to be some sort of range both in energy and temperature here where we are not stable in this region we don't want to be because that corresponds to the bad part here and this really means that as we're starting to heat the system we have one state here and as we go up we don't move between them so the particles that move away from this state they have to move all the way over to the second state directly you can't be here you don't want to be here this is one side of the knife and that's the other side of the knife and this is true for any system we haven't assumed anything it's just the way entropy things vary so what does this correspond to you think you had a good example water so this would be ice right and then something happens and then you have liquid water so this is a phase transition and you might think that this you can have both the ice at the same time and actually say you can't I'll show you why in a second but there is some friend of 40 would again say that there is something wrong here I keep talking about this temperature range and everything have you ever heard about the temperature range where ice melts so the book goes into some detail you can actually calculate this you don't need anything more than these equations but in particular given what I did before the break I kind of assumed that you wouldn't like me to go the point is that this evaluates to a very simple expression you can approximate this range in temperature is roughly 4 times the Boltzmann constant times this temperature roughly the temperature where we're having the transition happen divided by the difference in energies and then it's just a matter of doing the mathematics here you can say for water sorry for ice melting to water this temperature is roughly what yes, 273 Kelvin we know what the Boltzmann constant is the energy here well that depends on the size of the system but if you're looking at something typical like a kilo of water the phase transition energy is roughly 50 kilo calories not per mole but 50 kilo calories period that's a very large energy compared to Boltzmann constant because you still have Boltzmann constant here so it's technically true there is a range over which water ice melts to water and that's roughly 10 to the minus 24 Kelvin which for all intents and purposes is zero right that macroscopically in the lab you feel like water melts exactly at zero Kelvin so what's the point of going through that if this is so small that in practice this is just one point that it go here and then say boom okay well that's because you're looking at a large macroscopic block that is one entire kilo of ice if this is rather a protein again we can use 273 Kelvin there too or 300 if you want to do it in the lab but the point is that the energy difference for one protein that's the energy difference for one protein molecule and then you may be talking about 50 kilo calories but per mole and that's a difference of 10 to the power of 23 right and suddenly you're talking about an energy range that might be 10 Kelvin or something so for small molecules these are finite regions a protein will have some sort of phase transition that happens over a finite region and I've already spilled a bit of the beans here that protein folding corresponds to phase transitions but they're kind of different from the phase transitions that you would see macroscopically in ice and everything because it's such a small system and in principle I could stop here but this corresponds to what you say is a first order phase transition in physics and if I said first order there must be a second order right so I'm just going to tell you what the difference here a first order phase transition is characterized that there is a fairly abrupt change in energy as a function of the temperature this is a remember this can be a very very narrow region that starts to change significantly and a second order phase transition is almost the same think of that as a second derivative that just means that there is an abrupt change in the slope of the energy as a function of temperature and then there are actually other types of phase transitions too we're not going to go through the higher order phase transitions they're not really relevant for the proteins we're looking at and everything but there are lots of ways to study this in physics if you're interested and that's actually it corresponds to jump in heat capacity so you can measure that too you know what the good news is? this is all the statistical physics you will ever need but with this we can now start to study way more interesting concepts so that when you're looking at thermodynamics thermodynamics is kind of boring in a way because thermodynamics without statistical mechanics that just looks at what happens at equilibrium what happens if you've had an infinitely long time to relax but there is nothing in what I've said here that actually assumes that we always must have equilibrium and everything so we can start to look at how fast reactions happen for instance, transition rates because all this really corresponds to different free energy barriers and how easy it is to move over them or not and the neat thing with that is not whether you could argue that statistical mechanics and just look at the Boltzmann distribution that just tells you if things will happen at some point infinitely far away in time the difference with kinetics we can say when things happen there is a fundamental difference there in the energy landscape so that you can think about I'll come back to that in a slide or two what this means that if we have in general if you have some sort of reaction and this can be any reaction but think of say two small chemical molecules bonding or something you typically say that 0 is the initial state and 1 is the end state what would happen if you just had a downhill slope from 0 to 1 it would instantly go to 1 you couldn't even measure it this would only be limited by the diffusion of the particles and again if two particles are already close to each other there again it would happen instantly you couldn't even measure it so most the reason why most processes or reactions are finite is that it's not just enough to go there but to go there you somehow need to get over some barrier first there are lots of reasons for this barrier it might be that you have super low concentration so that it takes time just to diffuse and find each other that's a typical entropic barrier that you need to seek and sample all parts of the system until the molecules find each other it might be that you need to bind the molecules in sort of strange way or that the molecule have to bind something else that's very unfavorable from an energy point of view that would be an energy barrier so that the whole point there is some sort of barrier free energy and exactly what that is we don't really care but if it doesn't happen instantly there is some sort of barrier here and you can of course imagine a general complicated chemical reaction might go through lots of these barriers we'll come back to that in a second you know everything to derive this yourselves but I'll take you through it anyway what is the likelihood sorry I copied this from the book there is a sharp sign there is an F sharp or something up there what is the likelihood of being at the top of the barrier no but not very likely doesn't kind of well actually that's wrong that depends on the height of the barrier if this is 0.1 kT it might be very likely so in general the number of particles N sharp we have there that's just the Boltzmann distribution we can say that that's roughly the fraction of the particles we have up there compared to the number of particles we have down there that's the Boltzmann distribution e to the minus the difference in free energy and kT right the likelihood of being there compared to being down there and then friend of order you can of course start to say well you should compare the number of particles N sharp to N minus N sharp and everything but in general if this barrier is reasonably high it's going to be a very small fraction of the particles that are up here so it's not the particles that are up here won't really affect the particles in this zero state so the probability of being up here is the original particles multiplied by the Boltzmann distribution and if the delta f is high this is going to be small, if delta f is small this is going to be large and then we need to approximate things a little bit what happens when you're up here to first approximation and which side do you go down to it's 50-50 so in principle if we are up here in a given time the number of particles that falls down should somehow be proportional to the number of particles that are up here and in principle it's a factor of two but we're thinking about orders of magnitude but let's forget about that factor of two let's just assume that once we reach this energy we somehow continue to one I'm well aware that I'm making a factor of two difference error here but it doesn't really matter and let's also assume that if I'm up here the time it takes me to go down that should somehow just be limited by diffusion or something but it's not going to happen instantly it will take say a femtosecond or a picosecond or something so there's going to be some very short time here that corresponds to the process some sort of characteristic time for the process we don't know what that is and if we don't know what something else we'll just call it something tau but if we are at the top the time it takes to go down is tau and of course we can say that our factor two is part of this if it's only a constant right there's two difference we can include that factor of two in that constant but that means that in that time tau the number of particles that moved over was n-sharp divided by n right the fraction of particles that had come up to the top so after the time tau n-sharp divided by n particles have moved over so again approximately the time for all particles to move over well that's going to be n was 100 and n-sharp was 2 that was 2 divided by 2 of 100 particles but that also means that it's going to be 100 divided by 2 such batches right if there are 2 particles in each batch it's going to require 50 batches until all of them are over do you follow me there so that the rough time for all of them to go over is roughly going to be the time for each step and then the number of particles n divided by n-sharp and again we're also here we're kind of ignoring that some particles might go back in the other direction we're just talking about orders of magnitude here and if you're not happy with those orders of magnitude you can put an arbitrary constant as part of tau but what this means we know n divided by n-sharp that's just the opposite of n-sharp divided by n right so that just means remove the minus sign there so this the time it takes for particles to go over here is really the time it takes to go downhill once we are at the top multiplied by an exponential raised to plus the energy barrier divided by kt so what we're saying here is that this time for the reaction to happen is now proportional to an energy barrier with a plus sign the higher that energy barrier is the longer it's going to take you to go over and this increases exponentially and remember what I said before when things happen or not that that's related to kt if this one is significantly higher than kt the time for this to happen is going to grow exponentially and it's going to grow exponentially very quickly and if delta f is much smaller than kt well this time is going to be so small that it would happen all of that will happen instantly essentially so the amazing thing is that now we're moving over from things that are statically distributed until saying that when do things happen now there's of course one horrible thing here we have this tau and I haven't told you what the tau is in general this still depend you could apply this to say cars that are driven between Stockholm and Malmo and in that case one picosecond would be a pretty bad approximation of that time constant but this is just about any system and if you define a system we can define what this time constant is in general with chemical reactions there is more than one way for things to happen you can imagine that you might have a reaction that can happen with or without a catalyst for instance and then there's going to be two times one slow time where it happens without the catalyst and a fast time when it happens with the catalyst which one is going to happen well in principle you should then start what is the probability of going over barrier 1 what is the probability of going over barrier 2 and the point is that of course one barrier is much lower than another all these weights will be dominated by the lowest barrier right so in general if you have multiple pathways the pathway with the lowest barrier is going to dominate the total time it takes what do you think that's actually a very good idea so if you have a system that can happen in three different ways which way it happens one of them is in barrier 1 another 10 and a third 100 this is actually a more important question that you think for a long time we used to think that protein folding only happened one way but if there's one way we've seen for modern computer simulations and these are still too slow for you to do them yourselves but we've seen that in folding at home for instance which are the distributed networks all over the world that protein folding is not a matter of a protein folding one reaction pathway you should rather think of this like a spider's web that there are hundreds of different pathways some pathways are more dominant than others but all of them can happen and depending on how large and complicated the protein for instance is you can have either that one pathway corresponds to 99% or that the most dominant pathway is only 50% it depends on the system the problem though with that is that if you're going to work with that you end up having to do the Boltzmann waiting of all these times and everything and this is just tedious and complicated and everything so when we determine when things happen instead of saying that this transition takes 5 it's much easier to work with the reciprocals and the reciprocals is these transition rates and they're just one over the transition time and the advantage of that that describes how many particles cross per second essentially and if a transition is more likely to happen the transition rate is going to be higher so usually it makes much more sense to talk of things in terms of rates rather than transition times the other nice things with the transition rates is that you can add them so if you have one transition rate at 0.5 another one that's 0.2 and the third one that's 0.1 the transition rate is going to be 5.3 so then you can sum up all these different pathways rather than doing the Boltzmann statistics but they contain exactly the same information whether we talk about times or transition rates sure something amazing regarding what I was like for the show this time there is more you know so that number that's a good question but imagine that N is a mole 6 to the power of 23 would you consider that if you now measure this in the lab which we measure there is where it has the reaction happen right and if one out of 6 times 10 to the power of 23 molecules have moved over I think your answer would be so after this time for this in principle what I guess you might want to say 50% or something here but again we don't worry about those factors too and you can fold those into the because we don't know the exact value of this time anyway right so if you wanted to do this I guess you could do this more accurate and say that you consider that the reaction has happened in half that time because that's when 50% of the molecule have moved over here I would rather say roughly as if all the molecules have moved over the reason why I don't do that is that when 50% of the molecules have moved over we should no longer compare N sharp to N right but we should compare N sharp to N divided by 2 it has just become so much more complicated than in the end the largest area you're going to make is a factor of 2 and in particular in chemistry because of these exponentials we're talking about orders of magnitude factors of 2 rarely matter the other thing that will happen is if you now have multiple of these barriers that are parallel the lowest one will dominate if you have a reaction that needs to go through multiple steps it actually becomes pretty complicated you can do the math and if you have lots of steps that are roughly the same magnitude it gets really complicated but if you have multiple steps let's say 1, 2, 5 and then suddenly 1 that's 25 2 again the highest barrier will dominate there so if you go through multiple steps but one step has a significantly higher barrier that one is going to dominate the waiting time so if they are successive if they happen to have after each other you need to wait for the highest barrier if they are parallel that you can choose one or the other you're going to go over the shortest barrier and if you think about it that probably makes sense right so that's essentially what you need to do we start here and then we apply exactly our normal mathematics to go from here to X however the problem when you are in X one option is to go forward to 1 but another option is to go backward from X to 0 that might not be so obvious here but in particular in this right so here we start with X here the intermediate state here is actually worse so when we are here it's going to be better to go back than to go forward so in this and in general if this is very bad you're going to have a significant back flow of particles and that's why at each intermediate state you will have to account for the fact that you can both go backward and forward but that of course also means that in one state it's much more likely you can assume that it's equilibrium before that transition and there's equilibrium after that transition and then it's only the highest peak that matters and this depends for a small protein for instance it's usually only one barrier that is that they fold all or nothing for a very large protein such as a ribosome or something they have to go through a ton of different steps and that's usually why they take much longer to fold my point for going through this is that this is way more applicable than you might think so the idea is that we're going to try to apply this to secondary structure the last 30 minutes or so and the simplest secondary structure we can imagine is the alpha helix what you occasionally talk about is called helix coil transitions or helix coil equilibrium and that just means that I'm looking at a string of amino acids beat 50 or 100 residues and then you look how much of this string is alpha helix, how much of it is coil that is without structure and how would you measure that? but what method? CD, yes this is the reason why CD's pick to us because suddenly this is more important than you think suddenly you can use this super cheap machine that you can find in any lab and it's so cheap that it can order 10 more if you need it and we can use that to sneak into the atomic detail of entropy and enthalpy for instance of proteins we can look at beta sheets too we'll wait for it actually the random coil we'll probably wait for tomorrow you are experts on the alpha helix now so you already remember that each hydrogen bond goes from residue i to i plus 4 right and this is kind of obvious but so the first residue well so 1, 2, 3 nothing really happens when we form the first the hydrogen bond from residue 0 to 4 this really locks 3 residues into place residue 1, 2 and 3 the first and last are a bit free to move at least half free residue 4 on the other hand when we form the next hydrogen bond here we also look 2, 3 and 4 into place so in general 1 residues are stabilized by n minus 2 hydrogen bonds just the mat so these 3 are stabilized by 1 and these 3 are stabilized by 1 but in principle that's enough that we know how much we stabilize these by in principle we can write down the free energy I bet you wouldn't this is not entirely natural but this is also the free energy of any transition is the free energy of the product starting state right? so going from the free energy difference of going from a coil to an alpha helix that should just be the free energy of the alpha helix minus the free energy of the coil and then let's just assume that we have n residues there is one part here that's energy and we know how many the energy of 1 hydrogen bond we have no idea what that is actually you do but only amateurs would say that an hydrogen bond you're professionals now so you're going to say that the energy bond is just called that FH bond that's the free energy of this hydrogen bond why do we use a lower case F? and in this case because this is the total for the chain and here I'm thinking the energy of an individual hydrogen bond the beauty is you could use X if you want to as long as you define it or Z or you can use the Greek or or something that's the great thing with physics and we know that for n residues we have n minus 2 such hydrogen bonds and right now this seems we haven't really learned anything we've just said that this is the number of hydrogen bonds but we've also taken those residues and forced them to be in a helix because all our residues are in a helix all the n residues and there is some sort of entropy loss because they're no longer free we have to force them to be in a helix we have absolutely no idea what that is now so we'll just use the letter S for it and that S is really the loss of fixating one residue in a helix and if we forget about the lower term there this might seem we don't really well at phase this is not really obvious but there is an n there so we n multiplied by the hydrogen bonds and n multiplied by that term and there is one term here that does not depend on n so that's just for fun right it that way so we have one term here that does not depend on the number of residues and then a second term that depends on the number of residues and we still don't know what these terms are numerically but we can definitely say that that for oh sorry the blue doesn't show very well here this is the cost of starting the helix because whether it's one or five thousand residues that's what you're going to pay while the second cost here this is the one that's proportional to the number of residues we have in the helix can you say and can you guess anything about assuming that you have a small chain that does form alpha helix can you say anything about these terms now you can write this that there is some sort of initiation cost and some sort of the number of residues times the elongation cost and that's why I use those names in it and el happens to be my initial student can you say anything about the terms well can you start saying something about the sign here is total delta f what are the sign of those terms sorry what's negative so what would happen if the initiation was negative you would be on the downhill slope right then it would instantly form alpha helices so I would actually say that's wrong this is kind of an open goal for somebody else to suggest another well it could be zero too I guess but if it's not negative it might be positive so you're going to pay it's not good to form an alpha helix and remember the reason this is what you're going to pay if you wouldn't put one residue or two residues in an alpha helical state it's harder to get those good hydrogen when stabilizing you yet right so initially you're only paying and this is why it's uphill so there is some sort of energy barrier we need to get over this is the energy barrier we need to get over that term on the other hand what sign should that have if we'd like to form alpha helices negative so that as your first we start to pay but as we're growing this eventually this goes down again and if the helix is long enough there's been a lot of energy to form an alpha helix but it will not form instantly because we're not going to get over this barrier instantly there is another way of formulating this which is how people have usually done it in chemistry and it's not as hard as you think exponential and we have our delta f there right just take that delta f and we expand that into initiation and elongation but that delta f is a sum and that's if it's a sum inside an exponential that's a product of two exponentials right and that n we have there times the number of residues well this is the cost of elongating one residue but I'm allowed to take that n and move it out and raise the exponential to that as an exponent it's exponential also you probably don't remember them but it doesn't matter so this entire expression the exponential of the initiation energy we can call sigma and that second term the exponential of the elongation per residue remember the n is outside we can call that s and raise to the power of n and this might sound stupid what's the point of this the probability of something we're being in in alpha helix that's one probability sigma of the initiation multiplied by some sort of elongation raised to the power number of residues we have remember the definition of sigma here I might use that I'll go back and remember remind you about it let's see you know what I might jump a little bit forward here sorry I'll do this first no I will have to go back sorry I was saying my whole point here is that these are going to be very simple parameters that we can actually get from experiments we can measure just by measuring with 3D spectroscopy how likely it is to have a helix how many residues that are in the helical form I can measure what sigma is experimentally and I can measure with s's experimentally and then it's just a matter of if we know those we can solve for the initiation and elongation energy from these equations so we'll go back to that just so shortly how does the helix form then do you think it's a phase transition what type of phase transition and this might be so obvious right we're going through first order phase transitions and it's wrong it's not the phase transition I'll show you why let's compare a helix and this is a long sequence that we have helix this forms in some parts coil in between them and let's compare that to say water and a small piece of ice freezing in the water because that we know is a real phase transition the book goes through this in some detail but I'm going to head wave a little bit about it if you have a small region here that has formed ice the number of particles or water molecules that we have in there is going to be proportional to the volume which is roughly proportional to the cube of the radius right if it's a sphere the area of that sphere on the other hand is going to be roughly proportional to the square of the radius for pr squared so that area is proportional to the number of particles raised to two thirds but the point is that the size of the interface between the ice and the water grows as the number of particles grows if you look at things like size of oil in water or something having this surface area that's usually bad right we don't want to have these areas so what do you think is going to happen here in water this is horrible this is not going to be stable and there is a very deep theorem by Landau and I'm so not going to prove this faces cannot coexist in three dimensions a friend of order would then say but if you go out and look at Stockholm we've just had winter we might still have some water coexisting with ice right so what's wrong with that picture the time constant so the time constant here might be a month rather than five picoseconds but it's not going to be stable even if you keep this at zero degrees you will have either all water or all ice you're not going to have 50-50 water and ice but it might take a very long time to get there so it's a beautiful example of a completely different time constant but a helix this is one dimensional so if you look at this helix again the size of the area between say the water and ice and the helix which is going to be between the helical residues and the coil residues that's one two three four five six seven eight residues in this case right if I add three or four residues here am I changing the area so that the interface regions in a one-dimensional system is constant so in a one-dimensional system as a helix it's perfectly happy to have the two regions coexisting and you can even show this the number of residues that we have put in a helix here that means that the freedom here of the residues that we have not yet put in a helix there's an N minus N and this term will always for a very large number of N here this second term will always dominate so the free energy will actually be negative here so it's very happy to coexist I'm not going to ask you to derive that and this is typically what you see in CD spectroscopy you see helices and coils coexisting and that's what you see in a protein too right in a helix or 100% coil you have a mix of regions and for a small helix you can even calculate what helix length corresponds to the midpoint of the transition where exactly 50% helix 50% probability of being helix and 50% probability of being coil and that means that the probability of elongating it further at that point should be zero if we then assume that this helix can start at any point of the N residues we can finish at any point of the N residues that's N squared but we must start before we finish so that removes half of the option a rough approximation of the number of ways to put a helix inside N large residues is N squared divided by 2 so that is essentially the volume and then we ignore the factor 2 because I don't really care about this constants and then we can show that the entropy of forming this helix corresponds to the logarithm of the number of residues so I can put that into this equation what is the free energy of a helix at the point where I'm perfectly happy midpoint this should suddenly be zero so those two things should be equal and if they are equal I can say that well the number of residues that's just my capital N there right so if I solve for that N and call that N zero I get this expression the exponential of the most divided by 2 kT and remember that I told you to remember the definition of that sigma the initiation parameter that actually corresponds to the square root one over the square root of sigma there's a minus in sigma the point here is not exactly what this definition looks like remember that I told sigma is something we can measure in a cd spectrum if you can measure sigma we measure N so in two minutes in a cd spectrometer we can measure what is the fraction of residues that are helical and suddenly you can use cd spectra to start deriving all these parameters just put in everything like the initiation for a typical alpha helix might be say 4k cal that's a barrier and the elongation might be in the ballpark that we gain 2k cals per mole so the beautiful thing with cd spectrum we haven't really assumed anything about the atomic nature or anything right but you can use simple methods like spectroscopy to peek deep into the free energies in this case of alpha helix formation so this is the barrier we need to get over 4k cal per mole is that a higher or low barrier so what do we compare it to to decide whether it's higher or low sorry? yes so it's like 6-8 kT physicists for this reason they frequently like to think about energies in units of kT so you frequently see physicists plotting and they say it's kT on the y-axis so that's 10 kT which is it's not going to happen instantly but it's small enough that it will happen and then once you're over that by the time you're at 3 residues or so then it starts to be advantageous so you can't form a helix with just 1-2 residues we need to have 3-4 residues before it starts to be stable but then you can also show that this leads to a very large range of energies here when you can actually have a coexistence the helix will form gradually and this is exactly why it's not a phase transition because the first order phase transition was characterized that it has to be abrupt and then you can of course why on earth did I go through all the phase transition if the alpha helix does not go through a first order phase transition but we have another secondary structure right? yes so beta sheets do go through a phase transition and already now you can always remember that alpha helix are super fast and beta sheets are much more complicated they take longer to fold but once they have folded they are very stable and that has to do with those phase transition properties the point of this is that a long time ago people started, they had this great idea that if you used to see this spectroscopy you can take an arbitrary sequence and then you start examining what is the helicity to determine this S and sigma parameters for any amino acid so for alanine S is the elongation is roughly minus 0.4 kcal, means alanine likes to be in an alpha helix glycine well roughly 0.2 which means that it's plus 1 kcal glycine does not like to be in an alpha helix right? so depending on and proline is going to be roughly 3 to 5 kcal per mole which indicates to be in an alpha helix so using this you can actually just adapt these free energies and as a function of the sequence we could say what is the probability of this particular sequence folding in alpha helix don't ever do this this might get to like 30 to 40 percent accuracy and a secondary structure prediction and the reason for this is that this only holds for a single sequence in isolation in water is that you have a mix right? so first you have real sequences you're going to have a real protein where things interact with the surrounding and everything you have all the evolutionary effects of bioinformatics there's going to be orders of magnitude more efficient at doing this but it might be fun to consider that it's not a coincidence that some the reason that some residues like helix this is something we can measure experimentally it's not random it's not purely based on evolution or anything in practice we are better at predicting it with evolution and that brings you to a kind of an important part when you look at protein structures there are two ways you can approach protein structure you can think of this from physics or from evolution and they're actually orthogonal one is not necessarily more right than the other but they're completely fundamental different ways if you look at this from a physics point of view the reason why a protein has a particular fold is because it's the lowest free energy if you look at it from a bioinformatics point of view you don't really use physics but what do you rely on in bioinformatics yes in a way but let's take one step deeper why is one more probably than the other what is the original information you rely on in bioinformatics why are things conserved they exist this a good way but why do things exist there's one word that describes it all why are they useful why do they exist why do you see them evolution and the reason why things are good in evolution is of course why they are stable here but the difference is that you're going to try to do this with paper and pen now nature has had 4.3 billion years of trial and error that's a bit of a head start but the reason why things do evolve is of course why they are stable and good but the way it might sound bad when I say it's cheating but cheating is good when you're allowed to do it by using evolution you allow nature to use 4.3 billion years of information rather than trying to predict it from scratch yourself with a beautiful approach so the point is in alpha helix is that now we can actually look at the rate of formation and we can look at the rate of formation because all these energies we can measure them experimentally you can measure how fast things happen we can measure what is the initiation energy and the initiation we don't have to measure this with some super expensive time resolved experiment right we take the simple experiment we had we calculate what is the initiation energy what is the elongation energy from those I could calculate how high is the first barrier right and when we calculate the first barrier then we can calculate how fast do we expect things to happen and an entire helix forms in something like 100 nanoseconds or so which means that each residue takes a couple of nanoseconds say 5 nanoseconds or so to fault and this turns out to work reasonably well with the things we said before right that 5 nanoseconds at this corresponded to an energy barrier or in the entire helix this was not something that happened instantly there was a bit of a barrier for this to happen but this is still very fast in terms of physical time scales and so in this case I would say that this is the characteristic time scale for extending the helix with one unit so what happens is really that you have depending on what residues you have in your helix you're going to start from zero and there is some sort of barrier you will always have to go over some sort of barrier and then the possibility is that you might either go down a lot you might be almost plus minus zero or you might go up in free energy and all three are possible what do they correspond to different residues right some residues if you have a polyprolene helix it's going to be the red curb polyprolene it's bad to start and it's just gets worse extending it you will never get a helix of polyprolene not an alpha helix that is well if you have lots of residues that love to be in a helix you're going to follow the green curb but in that case there is something bad here if n goes to infinity what happens to the free energy oh yes and is that good or bad it's awesome it's infinitely good can't be too much of a good thing so what's the problem with that so that assuming that you already have a helix with an infinite number of say alanines then it would be awesome what is the probability of having a residue with an infinite number of alanines that I don't remember the but remember what I had the other like what is the relative propensity of the abundance of various amino acids right alanine might be 5-10% so the probability of having an alanine in the first position is 0.1 the probability of having two alanines is 0.1 squared so the likelihood of only having residues that will form alpha helices is of course smaller and smaller and smaller the more residues you want right so in theory you could have this but the probability of never ever having anything that's bad for helix is to go to 0 so at some point there is going to be an average length of these helices and that's the reason why you don't see very short you don't see very short helices in nature because they're not going to be stable enough you don't see very long helices because they're very unlikely to happen it's very very few sequences that could form those so that's why we actually tend to have helices that are in the ballpark of 20 residues or so that's kind of the equilibrium and we might show that tomorrow or not it's not super important you can actually calculate this a little bit more if you want to but I'm taking you through lots of mathematics I'm not going to do that right now you can calculate what is the initiation time and that corresponds exactly to this tau which is the characteristic time multiplied by the exponential of the barrier which is the initiation barrier and then we can just take the sigma parameter and reuse that definition so if we know sigma and we have a guttie rough idea of the rough time scales here maybe 5 ns or so we can calculate how long it takes to initiate a helix but in practice if you have 20 residues or so this helix can start anywhere and the number of residues in a helix we had that on the previous slide that's also related to this sigma so we can actually this leads to a small correction that we can calculate roughly the time that we went to initiate it is roughly proportional to 1 over the square root of this parameter and then the time it takes to extend this to all other residues can also be calculated from the allegation and that turns out to be pretty much the exact same expression but now there are lots of factors too and everything you've discarded my only take home message with this is that an average helix spends roughly half-half time of first being initiated and then forming out to 20 residues or so 20-30 residues we have 10 minutes to noon I'm not going to go through the beta sheets today but I'm just going to sum up the helices for you so the special thing helices form super fast but they do not form instantaneously there is a small barrier within the ballpark of a couple of kcals say 10kt and it takes time to start it but we actually we can't ignore the time it takes to grow the helix it's roughly 50-50 between them but both of them are super fast the beautiful thing is that we can derive all these things from some of the cheapest experiments you can imagine and suddenly we can actually get numbers for all these free energy barriers so now we're no longer and this is the beauty not just forget about the fact that this happens to do with helix formation suddenly we can do some very fundamental theories about how matter and molecules interact what they should interact if you then find a simple way to connect that to an experiment you can use these simple experimental methods to get very detailed information back about the system you're starting free energy differences in particular we're likely going to show this for one of the labs because you might think that this is something simple that people did a hundred years ago I would argue a whole lot of the really cool experiments we and others do today are based on this you find a simple model that's usually based on predicting a free energy relative to two states and then you find a way how likely state A is relative to state B and this might sound abstract but state A and B you might for instance take a protein how likely what determines if a protein is a membrane protein if you can find a way to measure whether a particular helix has inserted in the membrane or not then we have the relative distribution right the second you can measure that we can do the same thing here translate it back to energy differences whether it's stable in membranes or not so that while this might sound and feel like physics almost all chemistry builds on this and that's why you keep seeing this RTL and K everywhere we're inverting the Boltzmann distribution to determine free energies from observations and the characteristic lengths for alpha helices tend to be around 20-30 residues you almost never see anything much shorter because they're not stable enough and it's longer because that would be very unlikely to happen tomorrow I'm going to go through this for beta sheets too but I will deliberately skip through the beta sheets today they are way cooler lots of mathematics there that I think you're going to be pretty happy that we don't have to cover today but the summary here is that what I spend most of the time going through is book chapter 8 Statistical Mechanics read it, read it, read it I do not expect you to be able to hand derive all these equations I need to learn here it's not dangerous to assume things equations are usually your friend rather than your enemy I too remember my second year at university and mechanics there was a course in mechanics too even I found it relatively easy but you know what the mistake I did for every single problem I got I hand-waved and you kind of get it right but hand-waving only goes so far and when things become too complicated something is wrong when you hand-wave and the point is that equations are your friend because you don't have to be smart just follow the equations let the equations do the work for you and when you follow the equations I'm not going to say that you never make mistakes but you make fewer mistakes because you rely on the equations rather than trying to rely on your gut feeling what you think would happen and that's something that you should use whether that has to do with measuring high-tropin sequencing, measuring membrane protein you don't the point is that the equations help you they're difficult the first time you study them but trust me they become your friends I spoke a little bit about helices in chapter 9 but I think this is going to be more interesting tomorrow when we compare helices and sheets and we're actually going to look a little bit into the coil too what determines what the coil is and the homework I'm not going to have a gigantic list of things here so the first part of your homework I mailed you about yesterday tomorrow morning we should talk about this in normal words do you have a gut feeling what all these things mean what do they mean by the fact that what happens with the number of states when the energy goes up or the energy goes down all these letters E, H, S, F, G what do they stand for and mean and when do you use F and when do you use G I already kind of alluded to the stupid constants they're not the stupidest thing and many of them are actually quite natural how does these systems properties change with E or T phase transitions transition rates and barriers are probably more important than the other things here too because these transition rates are not going to be limited to protein structure anything you will ever see is going to be based on a transition rate for instance the some of the newest sequencing techniques that people are developing in the world are no longer based on fluorescence do you know how what is the state of the art or at least in the research when people try to determine sequencing non-pore devices so what you then have is a very thin material long term this might be graphene but today you have other pores today you have a very small pore and then you pull the DNA sequence through this pore it's going to be slow but on a molecular scale this is insanely fast and what you then measure is you measure some sort of property that will vary and that's essentially going to be an interaction or a current or something and if we can then determine how does the current vary when we're pulling this through and that's essentially going to be a transition rate right what is the barrier, does the energy go up or down depending on what you're pulling can we calculate how this changes ideally we should be able to detect whether this is an AG C or T depending on how the energy barrier varies when this goes through I'm not going to cover that in the course but the point is that these simple things with energies they are the basis of everything anything you observe in a lab can be translated into a free energy of some sort it's four minutes earlier that I'm going to leave you today, sorry I'd hope that you would get a little bit more time do you have any questions for me?