 Before I head on, on all this stuff, how did the lab work out yesterday? First, Alex said that he should have reactivated your cards with the one caveat. I think there was one or two persons who for whatever reason was not signed up on the formal course registration yet. That should be fixed later today, but in that case your card might not work until later today, but. And otherwise, lab wise, let's see, you managed to connect to computers, and did Bjorn and Dari instruct you about writing up a small lab report for the lab and everything? Good, you're on top of things, sir. So they are usually very receptive at responding to email to you. The one caveat is that both of them are working with cryo-EM also. And what happens with cryo-EM is that suddenly you get a recording session on the microscope for 24 hours, and then you pretty much try to spend all the time you have in front of that microscope. So suddenly they might just go off the grid for 24 hours, and that's because they're recording, in that case. Although they normally respond to email, I would suggest that we start by doing what we did, not yesterday, but on Wednesday. Caveat, if I'm a bit more confusing than normal, I had to get up at five AM this morning and take a flight up from Malmö. So I'm a bit, not tired, but a bit disordered. So on Wednesday we spoke about electrostatics, van der Waals, hydrogen bonds, all these interactions that start to, well, that form all the interactions of your proteins. Why did we speak about those interactions? Why do you need to know those interactions in the first place? So most of the thing, well, full disclosure, occasionally I will teach you things just because I think they're fun to know. But most of the things we bring up in this course, there is actually a reason. And the reason here is not just that I did my PhD on things like this, that does happen occasionally. But what do all these interactions form? It's much more fundamental than that. One way it doesn't read, well, it's gonna be important for proteins, of course, but we could describe secondary structure without formally knowing all these interactions. So remember all these equations we started to bring up E minus TS and everything. In particular, this equation that you're gonna, you're gonna either gonna love or hate it by the end of this course, but you are gonna know it by heart. All this stuff goes into the E. So the E consists in a very simple molecule, let's say not a protein. Say, whatever, benzene or something. You're gonna have lots of covalent bonds and you're gonna have a bit of electrostatics. And then if this benzene is embedded in water, you're gonna have fundavarsal and adenos interactions, not really hydrogen bonds. And that's it. That's what the E consists of. You can calculate the E. In theory, if you had a powerful enough computer, you could calculate E with quantum chemistry. Just plain and simple. How do you calculate the T? That is one good answer and it's gonna turn out that it is completely wrong. You can measure it. Of course, you all know what temperature is, I hope. And it's gonna turn out that you're all wrong. That is not what temperature is. We can, of course, interpret temperature that way. Remember the things that I said. Some things in this course that might look very complicated like the Boltzmann distribution. They're not really that complicated when you start working with them. And it's gonna turn out that some other things that look very simple and that you think you've known them for 20 years. That's gonna turn out that it's much more complicated than you thought. So, in a way, the temperature is a much more complicated concept than the entropy. And before this course, you all thought entropy was difficult. Entropy is not really that difficult because it's just a definition. We count things with entropy. It's just that if you're gonna count things, you end up having to multiply probabilities and everything, it becomes ugly, difficult. That's just, we wanna be able to add things up and that's just where we introduce the definition. I'll come back to that today. But the point is that the entropy is gonna be fairly difficult to calculate directly in a computer because I can only calculate states that I have visited. And if you look at the small, a single state of a molecule, you can't really calculate all the states it might be able to visit just by looking at one of them. So it turns out that the energy is easy because we can just enumerate it. If we have a computer powerful and not just calculate all the energies, even in a protein, works fine. Entropy, hmm, more complicated. Somewhat related to this stuff you did in the lab yesterday. I don't think that Björn and Dori brought up entropy already yesterday, right? So the point with the lab yesterday was to give you a gut feeling for the Boltzmann distribution and that it's, you can play around with it. The other reason why we did this very simple lab was to also give you a gut feeling for this simple gedankin experiment. You don't, it's much easier to understand equations if you use a simple model for them. But trying to understand these equations and directly working with a protein, you would be too confused by the protein. And there's nothing in these equations that depend on having a protein. Now, of course, we are gonna apply these equations to proteins, but when we apply them to proteins, it's gonna help you that you understood it from a much simpler system. But on Monday, I think it is, you're gonna introduce the entropy in that lab too. So we're gonna continue. Then we spoke about degrees of freedom and there we have the dihedral torsions. So dihedral torsions are complicated. Actually, no, it's not complicated, but they enter in two forms. On the one hand, it's an energy term, right? But on the second, it's a degree of freedom. And it's not an extremely high energy, electrostatics is a much stronger energy term. But the electrostatics is not really a degree of freedom. The positions of the atoms are the degrees of freedoms. And we free, unfortunately, I so wish that there was a clear way, a definition that we call the way that you can move around the bond one thing and that we call the energy variation when we move around the bond another thing. That would have been really good. Unfortunately, that's not the case. We tend to use torsions or dihedrals for both of them completely interchangeably. They're gonna be the important degrees of freedom in proteins. And as we talked about that already, the first lecture at Pipe Psi, we talked a bit about interaction strengths. This I'm gonna come back to today. You need, I don't care about, well, there's gonna be one number that you can't get wrong by a factor of two. But most of the other numbers, it's the orders of magnitude that matters. But again, to have a gut feeling what's important or not, you're gonna need to have a gut feelings about some energy magnitudes. We spoke about energy landscapes, the Boltzmann distributions, and free energy and entropy. So with that, I would suggest that we head on to these study questions and have you answer them or ask more. Should we do what we did on Wednesday and start from the top? Okay. So why do chemical bonds form? One has a orbital that's not full with two electrons. You can get a electron from another atom and then one of them has to be left and the other one stays down. They complete each other's spins. I think that's the way the book defines. Why doesn't it happen to noble gases? Why are they inert? Could you come up? Is it perfectly correct definition? Could you come up with another way of arguing by bonds form? Based on the equation I just had on the whiteboard. So in this case, you're completely right. But if a process happens, just to get to the, if a process happened, it's because what is lower in general? The free energy has to be lower. It's a soup. This is another astronomical stupidity that we use to cause energy and free energy. We should call it something completely different, but I can't change the world. So when any reaction happens, it's because the free energy is lower in the direction that happens. So the free energy, it has to be advantageous free energy wise. In this particular case, these two atoms right next to each other in general, the entropy is not going to change tremendously. And because in that case, we can kind of ignore the entropy term. So in this particular case, it's mainly the energy that's better. And that's also why it works really well to use a quantum chemistry to determine where the bonds form. So then friend of order would say, if these gases are inert, they should never, they should always be gases all the way down to zero Kelvin. So why do eventually even these gases form liquids at low temperature? You can get liquid helium. It's difficult, but you can. It will start making one of the noble gases that I told you. So it has to do with these Lennard-Jones interactions that one atom can induce a slight shift in the electrons in another. And that means that you will start at these very weak so-called dispersions corrections. And that's what we described with these Lennard, well, the attractive part of the Lennard-Jones interactions. We're going to come back to the Lennard-Jones interactions. One of these small curiosities, there's actually not two people, it was the one guy called Lennard-Jones in last name. Related to that, what is the energy of stuff? If in a noble gas where an atoms are infinitely apart, what is the energy? This is a problem in general, because that there is no, just as there is no absolute reference scale for height, you might try to pick the elevation above sea level or something, but in general, there are no absolute scales for anything. So at some point, anytime you talk about interactions, we're going to say, either you can prefer to always work with differences in energies or free energies and everything. That's usually a good idea. Otherwise, if you want to talk about absolute energies, you have to decide where is your zero scale, your zero point. And that's difficult, right? Because the energy in a water molecule is different from the energy in an oxygen molecule with two oxygens or something. So it turns out that the obvious definition that these physicists like, if atoms are infinitely far apart, they don't interact. It would be pretty strange to call that energy 47. But it's completely arbitrary. You need a number. But the point is that you could, of course, say that the energy say, if I hide it in one, that's the normal state of water. That should be zero. But to break it apart, you would need to add energy. And that's how they would mean that you would have a completely arbitrary, if the atoms are not interacting at six. So that from a physical point of view, the obvious zero energies, if atoms are not interacting, there is no interaction energy between them and we call that zero. That leads to something that's a bit strange. The second molecules start to interact and form either liquids or solids or everything. What is the energy gonna be in them? Positive or negative? Because if it was positive, it would be better to split apart, right? Low energy is good. And that means in physics, you frequently talk about these as condensed systems. And condensed just means that they've started to interact. So for any normal condensed system, the energy is gonna be negative. And you will probably see that later on in labs and everything. It looks a bit strange initially, but it's very common to have an energy of minus 300 and that's good. Positive energy on the other hand is bad. And there we are again. So that's when Jones interactions, it's negative, so it's an attraction. So even at all atoms will attract each other at very long distance, even though it's very, very weak. And I don't, I might have a slide about this later, but not today. So we spoke a little bit about how strong interactions are. Yes? Yeah. These questions were just energy. We'll get back to free energy. Ha, ha, ha. If, so roughly, what are the orders of magnitudes? If you, the order of magnitude of electrostatics, how strong could electrostatics be? And various how strong? 10th to the power of 90. Yes, so it technically probably could be 10th to the power of 90 if there's a nuclear device or something, but a few hundred kilojolts or kilocalts doesn't really matter. You can use either kilojolts or kilocalts, but you have to be aware of what you're using and you have to be aware of the difference between those because if you start mixing them, really bad things will happen. And we'll see that later. So what's the strength of the Lenard-Jones interactions? Van der Waals interactions. It's a tiny fraction of that, maybe one kilojol per mole or something, super weak. So which one is most important? Which one is gonna dominate what things do? In many cases, yes, but it's also complicated. So at short range, it's clearly electrostatics that's much more important. But at long range, in particular in a protein, what's gonna happen is that if you compare these interactions, and I know that I will have a slide about this later in the course, don't remember what lecture. What is the sign of electrostatics? Is that attractive or repulsive? So it can be either, right? You can't say. The Lenard-Jones interactions, the long range dispersion corrections, are those attractive or repulsive? So what happens if you have lots of atom interacting and the random signs for all of them? The electrostatics will on average cancel out. It's gonna be very noisy and I'm not saying that it will always cancel out. But if you just add up enough atoms, these weak interactions, while they're very weak, they all have the same sign. So when you add up, eventually, they can actually become quite important because there is no noise, you add them all, they're all attractive. That, sorry, that was a bit of a discussion here. What are the non-bonded interactions in a protein? I think we roughly covered that, but that's again, that's electrostatics and fundamentals interactions. And we also spoke a little bit about how strong they are. And I think we essentially answered six and seven there too. If you don't know about this, look up either in the book or Wikipedia. It is important to know the strength of these. Why is it important? Let's see here. I'll come to that in question 13. If you give me three minutes, I'll explain why it's so important for you to know the magnitudes. So if we go back, this was kind of the homework that I gave you today. What happens with hydrogen bonds when ice melts to water? We spoke about that on Wednesday. A little bit, but the point is you don't break them, right? So you can actually, I think this is an awesome example of how you study things because we can't really determine, it's very difficult to study the structure of liquid water. There was a large research group over at physics here actually led by Anders Nilsson, who's one of the leaders in the world. And they're using basically X-ray, different sort of X-ray scattering techniques to measure things in liquid water. And understanding what actually happens in a liquid is complicated because you can't get a crystal of it. But the neat thing here is that you can do fairly simple measurements like the spectroscopy, measuring how the specific heat, the various things so that you can then use this F equals E minus T as equation to draw some conclusions about it. And one of the things that's all from spectroscopy is that most hydrogen bonds are still there. It's just that they tend to weaken up a bit. I think if you do a simulation or something, at perfect, at zero Kelvin, you would have exactly two hydrogen bonds per water. And the reason that you have two hydrogen bonds that they take part in, sorry, two hydrogels, they take part, both take parts in hydrogen bonds. And then you have these two electron pairs, lone pairs, and they will also take part in hydrogen bonds. So that technically means four, but since each hydrogen bond requires two molecules, so that means that it's four multiplied by 0.5. So it's two hydrogen bonds per water. And that's, as you go to zero Kelvin, that's where we will get. And then as you start warming this up from zero Kelvin, it will become looser and looser and looser, and there'll be actually be very little change until we actually melt things. And then you might jump from say 1.95 to 1.7. And then you will stay at 1.7 until you boil the water. So the hydrogen bonds do not break, but something else happens. So what happens when ice melts to water? Why does ice melt in the first place? So hydrogen bonds go to change from one to the third place. So they will certainly change more. You know what I'm saying? Can you come up with an equation that could help you answer these things? The point of equations, I remember when I was roughly your age, and again I was studying physics, and the problem with physics, I most students think they're pretty smart, so did I. And the problem is it doesn't matter how smart you are, at some point you get to the point you can't hand wave your way through things, because it's complicated. The point of equations is not that they're difficult. Equations help you. You use equations when things become too difficult to handle in your brain, and when you can't really hand wave your way things through. So if you look at the equation again, in one way it's probably pretty stupid to keep striking this out, because I'm gonna write it again, F equals E minus TS. So in principle there should also be a change. But the reason ice melts is that suddenly there is something in this equation that means that it's better to be in the other phase. And as you mentioned, there's a fairly small change in the energy, but it does change, but there's also gonna be a change in entropy, right? So the main thing that happens for water is that we gain entropy by having the waters free. And that means that this term becomes larger, and that means that minus TS becomes lower, and that means that it reduces entropy. So it's important for the waters to get this extra freedom. Because again, breaking the hydrogen bonds, that per se is never good, right? That means you lose energy, and you don't wanna go uphill. So the only reason we do break those hydrogen bonds is that we gain more in the entropy. And that starts to relate to the hydrophobic effect that we hand-waved ourselves through a bit on Wednesday. And I think now you can solve this with this equation instead. So how does this explain the hydrophobic effect? Or if you try to explain the hydrophobic effect using this equation. So we draw things. I will have to strike it out because I need to get through for some more stuff. Here's our hydrophobic solvent, solute. And then we have the waters that now start to form this structure around it, right? And then we have delta F equals delta E minus T, delta S. So what we said on Wednesday is apparently we don't really break a whole lot of hydrogen bonds here. Why don't we just break the hydrogen bonds? Because the solute here is hydrophobic and can't really participate in the hydrogen bonds. It would seem obvious that we should just break the hydrogen bonds. So it comes back to the strength of the interactions, right? Remember, and that's why you need to have this gut feeling. Electrostatic interactions are so expensive that the hydrogen bonds are electrostatic interactions. To breaking those electrostatic interactions and breaking 10 or 20 of them, it's gonna be so expensive that there's no way this could ever be vulnerable. It's gonna precipitate immediately. But then you can never pretty much never, ever get anything non-polar to dissolve in water. And the point is that it's so expensive to lose those hydrogen bonds so that the waters will do anything it can to maintain them. Literally anything. And that anything includes reorganizing itself to forming these shell-like structures around the solute where the number of hydrogen bonds is pretty much exactly constant. So if you look at that equation then, what does that imply for the solvation of molecules, in particular hydrophobic molecules and the hydrophobic effect? It's not entirely trivial, right? So remember what I said, this seemingly simple equation is more complicated than you thought, yes. So the F is free energy, which is we frequently call this energy, but I also should call it enthalpy, minus the temperature multiplied by the entropy. We'll get to the delta G in about 40 minutes. You can think of delta G if you want for now. If an equation is complicated, the first thing I would do, put a delta in front of it, because then you can think about the change. And then you can think, what happens before you have things in water and what happens after you have things in water? How have things changed? And then take one, take it easy. There's not such a long equation. What happens to the first term? The change in energy. How does the energy change when you put this, and again, focus on the hydrogen bonds? How does the energy change? Sorry? Okay, good. So we're halfway there. Nope. And again, you need to learn to think about orders of magnitude. Of course it will change a little bit, right? But it's not gonna be the dominant change. And then we say that roughly zero. What's gonna happen to the entropy? The freedom or the order of this system? And particularly for these waters are in the shell around it. So what was entropy? No, the logarithm of the number of states. So that if there are lots away from, if the molecule can be oriented in lots of different ways, there are lots of states that can adapt. And then the logarithm of it is also gonna be high. But if there are fewer states a molecule can adapt, the entropy is gonna be low. So the end, there's gonna be a fairly significant drop in entropy for this, right? And then sadly, this is where things get complicated. It's not, again, we're not talking about advanced math, but there's a change. There is first, what is the value of the entropy? Second, what is the change of the entropy? And third, what is the sign of this entire term? Because there's also minus sign here. So in this case, delta S would be smaller than zero because there was a drop in entropy. And that means that minus T times delta S would go upright. So that in general for taking something hydrophobic and putting it in water, delta F is gonna be larger than zero, which means that it will cost us free energy. Hydrophobic things don't spontaneously dissolve in water, at least not to a significant extent. And the point is that you can hand where you think, weigh things through this equation, but you need to think about each term. You need to think what happened before, what happened after. And I wasn't kidding with it, I think this is serious. This is a more difficult equation than the Boltzmann distribution. But the good thing is if you take it easy and don't guess, don't try to jump four steps ahead, take one step at a time and make sure that, okay, I'm really certain that delta is zero. Good, then we can forget about that one. T, well, there was no change in temperature, and then we only look at delta S. We're gonna come back to that. So that leads to another question that I might or might not ask at an exam at some point. The hydrophobic effect, is that entropic or entalpic? Entalpy is really energy. So it's an entropic effect. And this is really strange, because it's caused by hydrogen bonds that are caused by electrostatics. So the obvious things would be to say, ah, it's energy, it's entalpy, but it's an entropic effect. And one way you see that is that it's strongly dependent on temperature. So that brings us number 10. What is an energy landscape? But if you hadn't looked at this equation, and if I had said you that there is something and it's caused entirely by hydrogen bonds, is it energy or entropy? Before this class, I would say that everyone, each and every one of you, and including me, would have said, ah, it's obviously energy, so it must be in the E. But the point is not in the E. And the reason, for now you just have to believe me, but it's possible to measure this experimentally, and we'll see that later today. So no, if it's obvious to you after reading this, then you're good, because then you understood the concept. What is the energy landscapes? So the energy landscape is, this is a concept that we like to think, because as we're gonna start to talk about these things, in particular, what are the different delta E's and the different delta S's here? You can't visualize a protein, although with different energies that a protein can have as a function of 30,000 coordinates. So we create something fake, simple, assuming that we just have two degrees of freedom, how does a third variable, such as the energy or the free energy, vary as far as these two degrees of freedom? It's gonna turn out that even in two dimensions, this is a really useful concept. It will help us understand things. So for what can you use the Boltzmann distribution? You used it yesterday. So the Boltzmann distributions will tell you what parts of the energy landscape or what physicists would like to call that phase space. Every single possible confirmation that the molecule, the world can adapt. The physicists are, sorry, we use the Boltzmann distribution and say which one of these are likely and which one of them are not likely. The point is that any of them could be, what do you call it, occupied. But the likelihood that some are occupied is so extremely low that we can almost ignore it, whether the likelihood that others are occupied is so high that we can almost take it for granted. Can you use that to give some concrete examples? I bet you all use this, probably in chemistry, but you're not aware of it. Have you ever been, calculated some sort of energy or something as RT, L, N, K? What you just did there is that you inverted the Boltzmann distribution. Because if you take delta E divided by RT, there should be a minus there, right? E, you put a minus delta E. So the Boltzmann, from the Boltzmann distribution, if you know how likely something is, which is again, that's all these constants tells you. How likely is it to be on the left versus right hand of an equation? If you know that, you can use it to get the energy. And we're gonna see some example of that later on. So the Boltzmann distribution is kind of the, that's the first, the zero-thorner. That tells you everything about physics. It even on Ludwig Boltzmann's gravestone, the equation is written. So number 12, then, in this particular equation, I said R. So why do physicists like to work with K and chemists prefer R? Right, and chemists, if you are a chemist, it's not obvious, you don't wanna count. Counting the number of molecules is absurd because that, well, it's not necessarily absurd. But if a chemist, if you had to count the number of molecules, you would constantly be working with numbers, oh, I have 10 to the power of 38 molecules of this. It's complicated, it's much easier to say that I have 10 moles. So for chemists who need to work with real-life things, you want normal numbers, and then it's very nice to introduce a scaling factor. But for the physicists, the scaling factor is completely arbitrary, and physicists hate arbitrary scaling factors. So you wanna get down to the most fundamental things and count it. The Boltzmann's constant is simply Joel per Kelvin. There's no mole in it. So they differ by 6.02 times 10 to the power of 23. We're gonna, I think we will probably mostly see K, but we might see R a bit too. So what is KT? So rather than asking you how large is it, let's take a step back, what is KT? What is the unit of KT? Energy, yes. So why is KT important? Why could you imagine that KT is important? Think about it. Well, again, we have some sort of probability or densities that's proportional of E to the minus, let's call it Delta F now, divided by KT. So both the denominator and the denominator here have scales of energy, right? So the K, the constant, I'm not sure about you, but Joel per Kelvin, it doesn't tell me anything. Whether something is large or smaller. But energy on the other hand, that we can measure in Joel's or at least Joel's per mole or something. This is the number you need to know. So I'm gonna tell you what it is. It is roughly 0.6 kilo-cals per mole at room temperature, or 2.5 kilo-jolt per mole. You can pick either, but I will kill you unless you remember the units. Because the point that if you screw up the units here, you're a factor four off. So choose whether you like SI units or kilocalories. You can do absolutely anything, but you're gonna need to be consistent in your units. Well, it's an energy. You can measure an energy any way you want. Mole is the reason why you're chemists, most of you. And for chemists, it makes much more sense to measure things. We always measure energy per mole, and for proteins too. So mole is not the unit. Mole is just a scaling factor. As it actually makes sense to, it might sound strange that the, yeah, it's an energy term, and we can always say it per mole. RT is the same. So what does that mean? If a delta F here, if a change in free energy is significantly higher than this, or significantly lower. So what this is, this provides you this kind of the scale on the Boltzmann thermometer, whatever we say that. If a free N, if sorry, if this change here is normally, if you go downhill, if delta F is negative, then it's always good, then things should happen, right? The question is, sometimes we can actually go uphill too. We can cross some energy barriers. And if that energy barrier is significantly lower than these numbers, it's not really a barrier for us, because this is the amount of energy we have due to the thermal noise, the thermal vibrations of atoms. So if a barrier is 0.1 kilojoule per mole, it's like gravel on the road. You're just gonna run over it. You don't notice it. On the other hand, if the energy barrier is, say, 10 k-cals per mole, then we're gonna bump into it. On average, we do not have that much energy. So the reason why you need to know the rough magnitudes of all these energy terms is to be able to compare them to this. This is gonna tell you, well, if there's an energy barrier, you will at some point be able to cross or not. And that will mean that it will turn out that we will occasionally face energy barriers. Say, if you have one of these double bonds, a peptide bond, so rotating around the double bond might be 20 kilocalories per mole, will that happen? So then you should take 20 divided by 0.6, which would be like 30 or 35. So there should e raised to the power of minus 35 is roughly the likelihood that it will happen. We will come back and see whether that's a smaller, large number. Number 14, you can't really solve that yet, but we have to notice that it's important. So how does the volume or the number of states complicate the Boltzmann distribution? So the Boltzmann distribution assumed that we could put a label on each and every state, a molecule or system could be in. But that's not really how it works in practice that you don't go down to the quantum level and look at how each individual bonds. For a protein, for instance, we talk about whether the protein is folded or unfolded. You're not gonna take into account that, well, there might only be one folded states, but there are 500 million states which are the non-folded ones. And that matters because in the lab or macroscopically or whatever you will call it, we tend to group state in a different way than we do on the very lowest level when we look at each other. And here too, people are sloppy, but I think a good definition is called the Boltzmann states for microstates. So the microstates are the ones that really enter in the Boltzmann distribution and they are unique. And if you have 500 microstates with the same energy, there are really 500 different states. And that works fine. In the Boltzmann, if you had 500 of them, you could handle it that way. But that again is typically not how we work in the lab, right? Because then we have the, ah, no, erase it again. Then we have this entropy. And the entropy describes it in the lab that the state with five kilo joule per mole of energy, that consists of more microstates than some other states. So that we somehow, we need to explain that in real life, there are some states that are actually larger than others. I hope this sounds confusing because if it was not confusing, the lab on Monday would not be useful for you. Hopefully it's gonna be much cleaner on Monday. So what is this detailed balance I spoke about? I'll draw a picture, but I won't answer it for you. So they had a state A here and a state B here. That does ring any bell now. Why? And there is one condition that you didn't mention that's important. So that what this basically says that the flow here must be the same as the flow here, an equilibrium. If that was not the case, we would gradually move from A to B or from B to A and then we would not have equilibrium. And so the probability of being in A multiplied by the probability of going from A to B must be the same as the probability of being in B multiplied by the probability of moving from B to A. That looks complicated. But that means P A divided by P B must be P B to A divided by P A to B. The reason why that is cool is that what we have on the left-hand side is just the probability of being in A and the probability of being in B. Can you calculate that? And how would you calculate that? With the Boltzmann distribution. And the point is that's easy. But here on the right-hand side here, we're talking about relations of the flux of how fast processes happen. And I'm not sure about you, but I would not have an easy way to calculate that if you just asked me. So that suddenly there's seemingly this simple Boltzmann distribution, which is simple, also starts to tell us a bit about something, at least how fast things move from left to right relative to how fast things move from right to left here. So you can use these equations to start drawing conclusions about things that are not at all that simple. And we're gonna come back to these type of equations later because how good and how easy it is to move is gonna have to do with how large this energy barrier is versus how large that energy barrier is. So it's gonna be, we will learn a lot about chemistry processes in general and what happened and what does not happen here. So in general, if I did not show you this and ask, seeing those two states, what is the flux from A to B relative to the flux from B to A? That's super complicated, right? Oh, we need to know the details of the chemistry processes. I need to know what the activation energy is and in theory, you probably could calculate it that way. But again, take a step back, let the equations help you. And the point is that the Boltzmann distribution helps us this way. If this was not true, we would not have equilibrium. And because we have equilibrium, we can relate that to the probability of being in each state. And the probability of being in each state, that just had to do with the difference in free energy between them, right? With the Boltzmann distribution. So the Boltzmann distribution, it's one sexy equation and it's not as hard to understand as you might think. E minus TS is more difficult. So then we have this horrible difference between energy and free energy. I will make mistakes. I will, the likelihood that I won't make this mistake in the course and not call free energy energy at some point, it's roughly as, well, it doesn't happen. But what is the difference between free energy and energy? Yes. And I'm in one, again, in hindsight, it's so stupid that we call them both energy, but you have to remember, they are completely different concepts. They just happen to have the same unit. Can you give examples of system with low energy versus high entropy? Oh, sorry, low entropy versus high entropy. Sorry? High entropy. Yes. So ISIS is a system that has extremely low entropy. And as you go to absolute Kelvin, the entropy goes where? To? To zero. As you go to zero, what would the entropy of, as you go to zero Kelvin, what would the entropy become? Zero, why? Yes, but I said that, why can't we call that 47? It would make more, or, obviously, it wouldn't make more sense to have the entropy be zero as room temperature, which is our default state. It has to do with these elevation scales, right? There is, there isn't necessarily any absolute scale. We can say that this is five meter or this is three meter. It doesn't matter because for the physics equations, we'll have to do with the differences. So we have defined this. And it's basically the third law of thermodynamics that as the absolute temperature goes to zero, the entropy goes to zero. There's a completely arbitrary definition that makes a lot of sense, of course, otherwise we wouldn't have made it. But there is, you can't prove that. That's an assumption we make in the definitions. Or assumption, it's a postulate. That's how we define entropy. I think we already spoke about the hydrogen bonds and I think we already spoke about these letters in this equation E minus TS meant. Is there anything else you wanna ask about related to Wednesday or should we push ahead with today's fun topics? I think, let's see, even though it's really late, you got the lecture slides, right? I don't think I have any papers I'm gonna hand out today. So we're gonna talk about these, the F and G's that you mentioned. There are gonna be two types of free energies. And I'm gonna bother you with this for roughly five minutes today and then we will just forget about that. This is not something that you're gonna need to be bothered with the rest of the course, but you have to be aware, somewhere deep back in your brain that there are different types of free energy. We are actually gonna explain what temperature is. It's a little bit of math, but there's not really, most of the math here is not gonna be very complicated, but if there is one recommendation, work through the equations. Actually, the more worried you are about the equations, the more important, take a piece of pen and paper and sit down and work through them. They're not difficult, but they require a bit of thinking. And by thinking and feeling that you understand them, I think that's probably far more efficient than spending two hours reading the book. We're gonna look a little bit more in detail at the hydrophobic effect, to connect it more to E minus TS, we can dig deeper here. We are gonna speak about real protein folding actually, which is much more related to this than you might have thought. And we're gonna talk about the electrostatics and a bit of these charged amino acids again. So, just to talk about this confusion about energy and free energy, one of my colleagues, Vijay Pandey, who's a professor at Stanford, now at Andreessen Horowitz, he mentioned a joke related to this about free energy and everything. So he was sitting at the San Francisco airport at the point of working in the lounge in his computer and then there was this elderly lady sitting right next to him. I said, that looks really advanced. So what are you working on? I said, oh, I'm working on free energy. And I said, oh, that is so important. I think she completely, that the point is everybody makes this mistake. It's real, they have nothing to do with each other. And I so wish that we've had another equation final definition for it, but we don't. The other problem is that life, life in physics is easy. In the extreme case, if I as a physicist had to define something, I always start with the simplest possible case. I simplify by insane amounts. Why should we simplify by insane amounts? It's to be able to focus on the essence of the problem. And as a physicist really the absolutely easiest case, let's assume that we have a completely isolated system. You can't do any, it can't exchange anything with the surrounding. We can't change the volume. We can't change the energy. There is no heat or anything going in or out. There is no work. And in lots of cases, that makes a lot of sense for physics. But for us, it would be fairly boring because if you can't even add or remove heat, absolutely we can't really make anything happen. But if you had a completely isolated system, the only thing that really would matter is the internal state of the system and what the total energy of the system is. And the system would be at equilibrium. But if we wanna make, and as physicists, we like to draw our systems as a box and that whatever is inside of that box, well, that depends. This could be two electrons. You could have one water molecule. You could have say, the atmospheric layers are above Sweden or you could have the universe. So it all depends. But let's assume that we have some well-defined system here that we wanna study. And again, to make our life easier, let's assume that the system itself doesn't really change in shape or anything. The only thing we can add or remove would be heat, energy. In theory, you could of course add energy as work too, but the work becomes a little bit more complicated because it will start to change things like volume. So for now on, let's say there is no work being made. The only thing we can add is energy as heat. So heat can go either go out of the system or into the system. And in that case, the way we're gonna, everything that happens in this system can be described by the free energy equals the, in this case, it actually is the energy of the system minus temperature times the entropy. And this is called, what physicists love to use is called the Helmholtz free energy. This is a simple thing. And for that, we use F. At least if we're good about it, which we're not always. But most of you are not physicists. So in the real world, yes, you can still exchange heat, but in general, you can also change the volume of your system. Why on earth would it change the volume of your system? That sounds really stupid complication. So imagine if you have a test tube and you're salivating a bit of protein in it. Will that change the volume? In general, it will, right? Because the volume in the test tube can change a little bit. So trying to solve a salt in water without allowing the volume of the water to change, it would be insanely difficult. You would need a super complicated equipment to have it, force it to be at constant volume and water has very low compressibility. So I think it would even be impossible. So rather than trying to force a super complicated setup, solving salt, that's probably a bad example. Imagine if you mix water and ethanol, that will change the volume of the system. One liter water plus one liter ethanol, if you mix them, it's gonna be roughly 1.7 liters. So rather than worrying about that and trying to compensate for it with a super complicated setup, we allow the system to do work on the surrounding or the surrounding to do work on the system. It's the same thing, it's just a sign. And you know that work, it's very easy because that has to do with pressure multiplied by volume. So what is the pressure? And then we're doing a change in the volume. That describes how much work we're doing on the system, which can be either positive or negative. So all these three terms have units of energy. And I'm not sure about you, but suddenly this equation becomes complicated. And just as that E included all the bonds, torsions, everything inside the protein, the current pressure multiplied by the current volume, that also describes the state of the system. And that just as the bonds, that's very easy to calculate. It's a state property. So can't we just take those E plus PV and group it into a new letter? We keep introducing, yeah, that's what physicists do. Pick a new letter for it. So this is now not strictly energy anymore. It's not just internal energy. We also have the pressure. And that's why we use this concept of enthalpy. And I so wish I was consistent. So that's why sometimes I call this E enthalpy two because we don't really care about PV here. And at some point you're now gonna start to sign that this is becoming complicated. This is a name, Gibbs free energy. And the way to think about it, think about test tubes, test tube volumes can change. Now, formally you could call that enthalpy two, right? But when we introduced this, I didn't tell you about PV. And that's why it's really, I think you should always call it enthalpy in both cases. If it says H, it's obviously enthalpy. If it says E, it might be a bit strange to call it enthalpy. The reason why I think you should use enthalpy and I try to use enthalpy but I will forget. It's that it makes it very clear that that is enthalpy but that is free energy. Because if you start throwing around energy too sloppily you will make mistakes and start calling F energy. So this looks like it's complicated, right? So you had, I think you had a question there. Oh, was that the G or the F? I'm both gonna make your life simpler here and more complicated. If you have a protein and you solvate a little bit of protein in water. How much protein will you normally solvate in water if you're doing an experiment? A millimolar or something? How much do you think that's gonna change the volume? So what chemists frequently do, in particular in this, yeah, but we can ignore PD. So that whether we call it E or H doesn't really matter. And then we'll call it G. And then you will have that equation but you will call it G or F. And chemists don't care about Helmholtz. So they will just call that D free energy. And that's why things can easily get a bit confusing. And I said, I wish that they were not confusing but I'm not gonna lie to you and pretend that this unique. You will frequently see that G equals E minus TS. And what that probably meant that the person writing down that equation he's really thinking about Gibbs free energy. The main thing is likely chemists who wrote that equation which is fine, we are chemists. But he also thought about PV isn't that important so he's gonna ignore it. So in virtually everything we're gonna do in the course you will frequently see, you will work with delta Gs and we're gonna talk more and more about delta Gs. But since we're gonna ignore PV you will frequently see G equals E minus TS. So that means from now on we can kind of forget Helmholtz. The book might use it in a few cases or be aware that there are two different ones. You can use whichever one you want. So which letter should you use for this? So when it is a free energy, what letter should you use for the free energy? So this goes down to the equation. You can use absolutely any letter you want. You can use theory you could, I could of course say that S equals T minus DF. That would of course need to be right. Well by this I mean enthalpy, by that I mean temperature and by that I mean entropy. If you just define that, that's perfectly fine. Everybody reading your papers likely kind of a bit of trouble understanding your equations because every time they're CG they're gonna assume that it's free energy. So that there isn't really any unique definitions here that you have to use them. You can introduce absolutely anything you want. But to me if I see a delta G in a paper I know by our oh they're talking about a change in free energy. And that's why you should stick to the normal notions. And as chemists delta G is usually the one to use. Physicists would like the delta F. But just make sure that you define what you do and then you're gonna be good. Helmholtz versus Gibbs is a bit related to another concept. We're gonna touch a little bit on that. So I wanna bring it up. Have you heard about intensive versus extensive properties? So intensive are properties that are not really proportional to system size. While extensive properties are. So if you're Swedish intensive are the ones that not change with system size. Or at least not proportional to them. So this, let's start on the right here. That's probably easier. If you take a system and you literally make four copies of it, how many atoms do you have in the system? Or particles? Four times as many. So it's strictly proportional to the size of the system. So that's an extensive unit. It changes proportional to the system. The volume is also four times larger, right? So at the energy again, assuming that we have the same interactions in the system here, the energy is also gonna be four times larger. And the Gibbs free energy then is also gonna be four times larger. Now if we don't allow the volume to change here, if we would force to be in the Helmholtz world, if I just stuffed four times as many particles or at least twice as many particles in the same volume, I would start to change the interactions. I would start to change the pressure in the system. So the real world becomes complicated to work with things like that. And that's why in chemistry we would like to work with Gibbs. If you're twice as much volume, everything is twice as large. It's not the main concept of the course, but I think you will see it. And then we should gonna do some equations. How many of you think you understand temperature? Or at least you did so before this course. All these things we've introduced, remember when I derived the Boltzmann distribution, I just said that it was temperature. And you believe me because you said T. And you all, just as I, when I read a paper, I said delta G and I recognized that free energy. The second you saw T, you assumed that it was temperature and you all know what temperature is because you've seen the thermometer. But what is temperature really? What is 14 Kelvin or 14 degrees centigrade? So you think you know temperature because you've seen something on a measuring device and that somehow correlates with your concept of hot versus cold. That's a pretty darn fussy definition. And it turns out that the temperature in all these equations that pretty much always enter to this KT or something. So one with some sort of proportionality concept, but it's not the universal proportionality constant like Boltzmann's concept that never changed. It's something that depends a bit on the properties of the system. And again, this property can be that the system is hot or the system is cold, which again is a darn fussy definition. And whenever things get too complicated, the point is rely on your equations and let they catch you. So let's look at F equals E minus TS. So if you wanna understand how an equation does, it's very useful to start looking at changes. And then we would start talking about delta Fs and equals delta E's and delta TS. But in particular, if you're a physicist, it's also used a delta in general that can be any change, larger or small. And if you're a physicist, it's very instructive to start thinking about how do things behave around a small point. So assuming that we are somewhere here along the blue curve here, which would be F. Either if we're on the peak here or the trial there or somewhere along the line, how does delta F change, sorry, how does F change if we move a little bit to the left or right? And how is that related to changes in E, T and S? And I'm not sure, it depends if you had different math background. Some of you will have seen this, but this is fairly easy. All of you have done derivatives and this is not more complicated than the derivatives. So if you're looking at first-order derivatives, if you have a function F of X and then we take the derivative, that's one way of writing it right. Another way of writing it says that that's DF DX, right? What does that say? That has to do with the definition of derivatives that you all live through at some point. That is how much, if you look at a very small interval, how much does F change when X changes? And there's actually nothing. You can look at delta, sorry, DF or DX separately. In that case, they're usually called differentials. But all the math is still true. So we're gonna do that. So if we look at F and then we look at the very small DF change around it, which can be either positive or negative, but now it has to be small. And that is gonna correspond to a small change in this entire expression. And it turns out for differentials, exactly the same laws as normal derivative supply. So this would be a small change in DE. And then minus the small change in T multiplied by S. And then we just apply the product rules here. So it's gonna be first T constants multiplied DS and then minus S multiplied DT. So that's always gonna be the change. You have it written down there too. And then when you're looking at this, this isn't suddenly, this is not really gonna help you a whole lot. But that means that we can say that the small change of free energy around something is related to these small changes. And when you don't get further with the general equation, we start to have to assume what is it really that we're interested in? And a good look at equilibrium. At equilibrium, you can start to say about things. So we know that temperature is constant at equilibrium. Otherwise things would be changing. We also need to have a local minimum in F that can actually technically be a maximum too, but. And if that means, well, around this part, the delta F, the DF should not really change by a whole lot. And that means that we can simplify this. Again, we even said that that would mean DF equals DE minus TDS because DT would disappear. And I just said that DF would have to be zero. And then that would mean that DE equals TDS or T equals the derivative of energy with respect to entropy, sorry. This sounds pretty dry to you now. I'll let you melt that for 10 seconds and I'll tell you what it is. Yes. Sorry, which equation are you talking about? That one? Let's see. F, and then we have F. I don't think I'm missing anything. Sorry, my bad. Yes. Yes, there we have F. Thank you. It's another good reason that just looking at equations is not enough. You have to work through them and then you frequently make mistakes. This probably sounds a bit corny and strange. Yes? So here we already know F equals E minus TDS. That's the equation we're starting from, right? And then we're looking in general, whether you're at equilibrium or something, if you're looking at a small variation along this curve, F plus a small difference, there's gonna be a small difference in the free energy. And then we just take the right hand side here and apply the normal product rules for derivatives. And then initially, I deliberately write everything out. It's also gonna do one thing at a time. Of course, I know that the temperature is constant, but don't try to jump forward because in general, temperature might not be constant unless we're at equilibrium. And the second step is that, okay, we are at the equilibrium and then we can start to say that, you know, that term disappears, that F there cancels and that equilibrium, DF should be zero and then we get a much simpler equation. But do one thing at a time. So I frequently do this. I first expand the entire equation and then I try to find which one of these terms are gonna be zero so that I can strike them out. This is the definition of temperature. And this is the formal physical definition of temperature. Yes? So I took this equation that I got and that we have was simply by looking at small changes in E minus TS. So getting to this step was just looking at small differences, applying the product rules. And then I say at equilibrium, there are two things that happen. Well, first, at equilibrium, by definition, we should be at the local minimum of free energy. That is the definition of equilibrium and free energy. And that means that DF must be zero. I also have F both there and there, so I can remove that too, right? And at equilibrium, we also have constant temperature. At the temperature is constant, the change in temperature is zero. So that term also disappears. So we just see that we have zero equals DE minus TDS. And once we are there, then we just solute this again and get back to the derivative here. The reason, there are two reasons for bringing this up. So temperature really explains how much does the energy change in the system as you're allowing it to have more and more microstates. Is that something that you have a good gut feeling for? At least you're not lying and saying, yes. I have no idea what that means. Well, I do have an idea what it means, but it's pretty darned far from trivial. So what we've achieved now in three lessons is that we've pretty much turned your world upside down. That temperature, which was something that you all thought you understood, is actually something pretty complicated. Well, the Boltzmann distribution is something fairly easy. But the cool thing is that we can derive temperature from very simple equations that don't really assume anything of the world. There's nothing here that assumes that temperature is related to velocity of molecules or anything. So this only assumes thermodynamics. And I'm not sure about you, but constantly trying, it's going to turn out that this turns out to be something important. And constantly talking about these derivative slopes and relations between energy and entropy is complicated. And this, in particular, we realize this bears a very close, even identical resemblance to the things that we mean by Holtz versus Gold. It's pretty neat to introduce a concept for it, and then we call that temperature. So this is a strict, unique definition of temperature, the thermodynamic definition of temperature, which is what we use. And this is how you define one Kelvin. It's not defined by the certain velocity of atoms in a particular sample. You define it from energy and entropy. And that's also why it has to be zero at absolute. At the absolute zero point. This slide also looks a bit strange. I'm, well, all of you have seen the exponential distribution. So why in earth do I plot it? Remember in those study questions, I said that the barrier for rotating around two torsions, if that was, say, 20 kilocalts, what was KT again? I will try to ask you this 20 times during the course by the time of the exam. This is pretty much, it's not guaranteed to be on the exam. It's only kind of like 95% certain that it will be on the exam. You have to know what KT is if I wake you up in the middle of the night. So I said, the other thing to remember, I said that the energy for turning around this bond was roughly 20 Kcal per mole. This is why I would strongly, strongly recommend you to know both of these numbers, because if you only know the kilojoule number, what is the first thing you would need to start doing now? You would need to start converting units. But if you know that it's 0.6 Kcal, it's just 20 divided by 0.6. What's that? It's 40, roughly. That's the other point. Don't use your calculators for anything. You need to guess, you need to estimate, then you need to be able to do these things quickly. And if you had to pull out your calculator, it's probably 35 or something to be more accurate, right? It doesn't matter. So that many of these things, you need to have a quick gut feeling. So that the likelihood of these two things happening is E raised to the minus delta, let's call it delta G now, divided by KT. They will say they're turning around, rough, or they're turning around, say, taking a peptide bond, then turning it from sister trans double bond. Let me get back to that in a second. If we now, if you calculate what this quotient is, for instance, 20 divided by 0.6, that is what enters in the exponential function. And that starts to decide how likely things are. So the minus things here would be that it would be one in. So assuming that I had a delta G difference that was exactly equal to one KT, that would be E to the power of one, 2.7. So if you raise the energy in the system by one KT, the likelihood to get across that, if there is a barrier that is one KT, it's roughly one in three that you will get over it. That will happen. It's not always gonna happen, but so that's one kilocalorie, sorry, one KT is roughly the normal scale. You will get over it sometimes and sometimes not. By the time you're at three KT, it's gonna be roughly 10, so that's gonna be one in 10. The three KT is starting to be a fairly high barrier. Do you see what I did just did? I said three KT, I didn't say kilojoules per mole. It turns out it's very convenient to measure energies in units of KT. Because rather than saying that that bond is 20 kilojoules per mole, the first thing that you're gonna say, well, it depends, we have to compare that at a specific temperature, or you can say, well, the barrier I need to get over is roughly 30 KT or 40 KT. And the KT is the normal, that's the unit of temperature you have at the certain, sorry, the unit of energy that we have at a certain temperature. If something is 10 KT, it's roughly 10 times that. And then I know that the argument here is gonna be 10. So rather, and I know why I confused you a bit here, but rather than using kilojoules per mole or kilocalories per mole, when you talk about molecular system, it's very convenient to measure this in units of KT. Because then we don't have to worry about all those moles and everything. So let's start, we go up here. By the time you have, if X equals 10 here, so when something is 10 KT, the likelihood that that's gonna happen is one in 22,000. So what was the energy we spoke about again? That double bond, 20 kilojoules per mole. So how many KT roughly at room temperature? 30 to 40, right? And the temperature is important because at zero, sorry, at zero Kelvin, it would be very different. It would be infinitely high. So that the temperature is relevant here. But if we don't say anything else, we assume room temperature. It's gonna be 40, okay. So the likelihood that this has happened is gonna be somewhere between these two numbers. So for one double bond to turn, you're starting to be in the ballpark will never happen. Those two electrostatic interactions are 300 kilojoules per mole. So that's gonna be in the ballpark of 400. These energy barriers you will not get over. If you doubt that, you can compare this. I think you have it on your slides already. There are a couple of numbers here. By the time you're here, you can pretty much. These are the places I love to be. I will love the bet that I will eat my left shoe if it happens. It's not gonna happen, period. While conversely, of course, if something is significantly smaller than KT, they will happen all the time. You have no, again, the reason for drawing this is that the exponential grows faster than you think. So we're gonna stay very close to one or zero here. By the time you're a little bit below KT, things will happen all the time. And by the time you're a little bit above KT, things will never happen. So that the only energies that are really interesting are the ones that are close to KT. If an energy is significantly above KT, that would be equivalent of me having to go through the wall to get out in the corridor. Technically, I can. It's not an infinite energy. It's just gonna be so astronomically high that it can never happen. And can never happen is the one thing we focus on there, right? Then we're only interested in things that actually can happen. So this, we can start to relate to all these degrees of freedom and the energies we had. So what ballpark were you talking about for a torsion? So you see here, why would it help you to have a gut feeling to know roughly how high a torsion energy is without having to look it up on the internet? What it is that you can't do unless you know how large the energy itself actually is. Because the equations you can probably write down. But why do you need to know roughly how large it is in some unit of energy? To know whether it's gonna happen at room temperature, right? Can you cross this? Can these molecule, for instance, can you rotate the peptide bond? We already said that based on the numbers there, no. They will never have assist trans-isomerization in a peptide bond. But if you look at the normal torsions where there might be a unit of say a few k-cals, the point, it doesn't matter whether it's two or five. It can't be 300 and it's not 0.1. Well, two to five, yeah. If it's five, it's gonna start to be unlikely. This might be one kilo calorie per mole. But then it's gonna depend on the temperature. At three, if it's two or three at low temperature, we might only get up the normal thermal vibrations, which is kT, might only correspond to going up a bit here. In theory, we can go up and across there, but it might be pretty rare. If you now raise the temperature to 900 Kelvin, suddenly the thermal vibrations might be up here. And that means that you will constant across this barrier because this now looks like gravel in the road. Because when the barrier becomes lower than kT, we're not really gonna see it anymore. And conversely, if the barrier is higher than kT, at some point it starts being a brick wall. So there are two parts. You need to know the energies of the interactions to compare them to kT, and that's why you also need to know kT. And the third solution to that is what physicists frequently do. They like to measure energies in units of kT. But you're gonna need to know all three of them. Does that make a little bit of sense at least? Yes. So this has to do with the definitions. So first, if it's negative, we always go downhill, right? But if you look at, again, the easiest thing is to look at the, let's look at two states, A and B, where B is slightly lower in free energy. And then we can look at this different in free energy. We will call that delta G, right? So that the likelihood of being in B, I'm sorry, the likelihood of being in A relative to the likelihood of being in B is gonna be proportional to E raised to minus delta G divided by kT. What happens, let's even say that in the limit where delta G goes to zero, or if it's very small, then there starts to be a very small number here. That's not small, the magnitude is small. It goes close to zero. Anything raised to the power of zero is what? One. So that's roughly one if delta G is zero. So when delta G is very small, they're gonna be just as, it's gonna be roughly the same probability in A as it is in B. Now of course, if you go even further so that delta G starts to become a very large negative number, then it will change and then B will become significantly more probable than A. Did you notice how I just explained this? I went back and let the equation help me. And you can do this too. These are not difficult equations. The exponentials and logarithm is pretty much as advanced math as you're gonna do here, but you need to trust the equation and you need to learn to work with the equations. This is how it looks in practice, I think, let's see. Ah, there. This is a small movie of a peptide. Let's see, it's a four alanine chain that is moving in a simulation. There might even be a water around this, I don't remember, at room temperature. And I think this entire movie probably copped verse 10 or 20 picoseconds or so. Because it's difficult to do experiments at 37 degrees centigrade because you would need to heat the entire lab or you would need to keep the test tube at constant temperature or I would need to, anytime I move the test tube I would need to keep it at 37 degrees all the time. You can do it, but if you wanna do, also in the grand scheme of things, compared to zero Kelvin, whether you are at 298 or roughly, say, 315 or something, it doesn't really matter. It's a very small difference. Proteins are stable at room temperature too. There's actually a very good question that, but it's a pain to do all your experiments at 40 degrees centigrade. Unless you, of course, have it in a test animal or something, but in the lab, room temperature is much easier. So there are a couple of things here, yeah? So that would be, where do these movements come from? Well, in general, right? These are, as you mentioned, these are the thermal vibrations that atoms have a finite energy, which in this case is a whole lot of it's kinetic energy. And the way we describe this simply has to do much. How much energy do they have relative to the amount of freedom they had, or a.k.a. temperature? Trust me, call this temperature, it's so much. So the temperature, we can calculate the kinetic energy. It's the sum of the mass of each atom multiplied by its velocity squared. The reason for showing you, sorry? The kinetic temperature, you've all just done this in per-secondary school. The kinetic energy of any particle is its mass multiplied by its velocity squared, right? And that's what we do here too. It's just that you have many particles here. So for each atom, calculate how much it weighs, square its velocity, and sum it up. The reason for looking at this is the thermal vibrations are larger than you think. This is not a static molecule. It's certainly not a crystal, right? Things move, and they move in real proteins. And now, of course, we're just looking at a very small part of a protein. And for a gigantic protein, this would likely just vibrate or something. But an individual atom or something, they move quite a lot. What are the other things we see here? The bonds here are virtually constant, right? You don't see any large motions in the bonds. And that's why we don't need quantum chemistry. We can just assume that the bond lengths are roughly constant. There are some vibrations in these angles or something, but again, for the entire folding of a protein, do you think these small vibrations of that hydrogen is going to be important? It's fluttering a bit back and forth. But the thing that do happen occasionally is that you occasionally see these transitions. I think, did we see any transition of the torsion? Yeah, I think we see some. Maybe, yeah, that really was one. So that it's these torsion transitions that create the large motions. But in addition to that, the molecule spends most of its time kind of rattling around. And that's also a characteristic of all these distributions. Molecules will spend most of its time rattling around and waiting, having some thermal vibration. And occasionally, they will undergo through larger motions. And if you think about this as an energy diagram, what's going to happen is that, again, we might have something that looks like that. Oh, sorry. I'll try to bring some better pens on Monday. So what happens is that you are in this basin and then you vibrate around here and you go up in energy and then suddenly boom, you head over there. So this would then be a transition of a torsion, right? And when you've done that transition, now we're going to spend some time rattling around here. And then at some point, we might go back here or something. This is, of course, a horrible approximation because I can only draw this in one dimension. If you had two dimensions, I could probably go directly from there to there without passing by that one. And that's why it's nice to at least have a two-dimensional energy landscape. In real, even for this molecule, what could it be? It's like 50 atoms or something. So it would be 150 dimensional landscape. I can't draw that. But to understand it conceptually, it works great even in one dimension. Yep. I don't remember. I think it's rough. A ballpark of picoseconds, 10 picoseconds. If I do this for 10 nanoseconds, it would be, I think I'm in looping this. So you've probably seen it 10 times or something. But this is timescales of picoseconds. At femtoseconds that you would see individual atoms slowly moving. And at nanoseconds, things would start to fold and everything. Yeah. But proteins are written on small timescales. But how long does it take to fold the protein? Milliseconds to seconds. Maybe microseconds so that you will spend a lot of, if you looked at this on a microsecond timescale, you would see gigantic motions everywhere. But then we wouldn't be able to talk about the individual interactions. Let's see. It's 1024. I think this is a great place to take a break. So let's meet here at, say, well, take how we're doing. Let's meet here at quarter to 11 because I have quite a few slides. I might not finish all the slides today, that's fine. So let's meet here at quarter to 11 and then we'll continue to talk about the hydrophobic effect for real. All right. Lots of equations. What I'm gonna try to do now is that we're gonna head back. So now we're gonna do this first jump. Now we're gonna move back to things that we actually have covered and see whether we can understand them better by using these equations. And it was particularly the hydrophobic effect that we've talked a lot about. And this is the hydrophobic effect you're seeing that oil, anything hydrophobic in water is gonna form larger droplets or something to minimize the free energy. And that was also intimately related to these things I were doing. It's the likelihood of spontaneously going from there to there doesn't happen, trust me. I've done that experience on my laptop for years. For some reason, even if I start out here, I always seem to end up with something like there a few weeks later. And it's exactly the same process. So let's think about the concept of a hydrogen bond first. And to make this really easy, let's start a vacuum, which is of course, I know that it's sound realist, but if you have two water molecules and they're first evacue and not interacting, what happens if these should somehow form one hydrogen bond between them in water? And for now, we're just gonna look at these two waters, forget everything else. So we're gonna get one contribution here from the energy. Why do I start talking about energy here? Because if we're gonna need to decide whether this happens or not, we need to rely on our equation. And there are pretty much only two things that matter in our equation. It's E and T S. And in a way, it's even, in a way it's probably the stupid of me to say that E H was smaller than zero. So in the interest of doing things in many steps, I could, there will be an energy related to the formation of the hydrogen bond. For now, let's not care about what it is. There is also gonna be a change in entropy. And for now, we're gonna ignore what it is. But then we don't really have much at all. Then we just have to say, well, there is an E and there is a T S. And the point that we need to learn a little bit more about it. And you can say some things already that is the energy of a hydrogen positive or negative? Well, it says there that it's negative, but why is it negative? So what would happen if the energy of a hydrogen bond was positive? It would be better not to have a hydrogen bond and then we wouldn't see any hydrogen bonds and then we wouldn't talk about them. So by definition, because hydrogen bonds form means that it has to be a negative energy. That's the only thing we know about that. But it's not as bad as you think because you just excluded all positive energies. So it's whatever. And we also know that that has to be a change in energy as we go from that state to that state. Entropy is more complicated, but we can probably hand wave a little bit. So in this case, we have two waters that are completely mobile and flexible. By the time you have formed ice, things are completely locked in and then they can't move at all. So that, and in ice, you would have two hydrogen bonds per water, right? So here, they participate in one. So by the time you've formed one hydrogen bond, you've lost half the entropy of a completely free water. And I'm hand waving. Hand waving is good. But again, I only participate in only, this water only opens half the hydrogen bond and the other water also owns half a hydrogen bond. So it's half a hydrogen bond each for two waters. So in total, I lose roughly the amount of energy corresponding to one, sorry, I lose the amount of entropy corresponding to one completely free rotating water. You agree? So the point is, you see that we can get fairly far just by hand waving a bit. I have no idea what this entropy is. I have no idea whether it's 15 kilocalts per mole or something, but there are some ballpark things. I can also say that the entropy of a rotating water is larger than zero. Why? So it doesn't say delta here. It is an absolute entropy. Entropy can't be negative. So that's just something to keep in mind. If it only says S, it can't be negative. If it says delta S, it can be negative, but S itself can only be positive. And now I'm gonna ask you about something. Let's do this period. I'm gonna, in the interest of time, we might just do this very quickly. Talk to the person right next to you, but we're only gonna spend 60 seconds. Decide whether A or B is true. And now we're talking about the hydrogen formation of two waters going from vacuum and forming one hydrogen bond. So if you didn't solve this yet, I'm gonna give you a little bit of help that I deliberately didn't put in your lecture notes. Think about that one. So I'll break there for your sake, so we don't spend too much time on this particular one. I'll give you, I'll do another one with you. So how many of you think that A is true? How many of you think that B is true? In this particular case, A is true. Can some of you thought, hey, why did you think, all right, let's talk with others. Why did you say that B was true? So if the free energy was positive in this case of going from, is this something good or bad? Will two waters in vacuum if they get close to each other, would they like to form a hydrogen bond or not? So in general, this is probably something that happens simply because we see so many hydrogen bonds. So when you go from here to here, this must be good. And that means that delta F must be what? Negative. And delta F is the change in E minus TS. So we must gain more, that one must be, first it is negative already, and we must gain more than we lose for the entropy for one water. This is a bit of an artificial example, and that's why I did it fairly quickly. In general, this is more complicated because you are saying water or something, it's not just a matter of forming a hydrogen bond, but it's virtually always the case of forming a hydrogen bond, either with one water or with another water. I'm well aware that probably sounds fussy too, so let's make it one notch more concrete. Let's look at forming it in a protein. So we have two cases. Let's look at the top one first, a protein in vacuum. And the problem is proteins are complicated, so let's think about something very simple. You just have a state, and the protein is not really moving here anymore, and let's assume that these two amino acids are always right next to each other, so we don't have to worry about all the torsions are changing shape. But they're so close to each other so that they could form a hydrogen bond, and the only thing we ask, will this hydrogen bond form or not? And in a way that's roughly the same situation you had in the previous slide. So the delta G here is gonna be decided by again, delta E minus T delta S. So what is the change in energy here? Delta E, I just missed it. It's negative, but what is it? What is E H? So it's the energy of one hydrogen bond, right? So that's what we were creating. And you just said that is also negative. So it's say the change in energy should be negative. And again, you could always argue, but what if you start turning the torsion, but again, simplify things. Don't worry about the minor details. So for now, let's assume that there are no motion changes. If there are no changes in motion, or at least no significant changes here, it's also reasonable, I would say, to say that delta S is roughly zero. Why? So this is a bit more of an approximation. It's not strictly true, of course, but if this doesn't really change how flexible the protein is, or anything, it's not really gonna change the number of different microstates the protein can be in. Now, and if these two residues are already next to each other, it will probably lose a little bit of entropy, but it's not gonna have any gigantic change in the entropy for the entire protein. Again, it's not strictly true, but in the extreme case where we simplify a lot, I think I can at least argue. And in that case, the delta F, or the delta G, you see, I already start mixing them up, sorry. So delta G, which is the same as delta F, is EH, because E minus T delta S, and delta S was zero. So that here, the change in free energy is negative, corresponding roughly to the energy of a hydrogen bond. So will this happen or not in the upper part? It's gonna happen. Again, under the general assumption that this doesn't mean a drastic drop in the degree of freedom here or something, so that if the entropy is roughly the same. Sadly, you thought this was easy. The problem is that proteins don't exist in vacuum, and that's why we can't use quantum chemistry. So solvent, what happens instead? Well, we have two hydrogen bonds here, right? Sorry, two, we have the same partners here, but the only problem is they're not gonna dangle around in free air. In the unfolded state, they will have, with probability, according to certainty, they're gonna be forming hydrogen bonds with waters. So there is one two hydrogen bonds there already. And if you're now gonna start having the protein form a hydrogen bond, well, you can certainly form a hydrogen bond between the donor and the acceptor here. The only problem is that then I lose those two hydrogen bonds. So the waters will need to form a hydrogen bond in itself. So this does not change the number of hydrogen bonds. So suddenly delta E is roughly zero. And then the change here is gonna be delta S, which is, again, roughly the loss of entropy of this water. So in real life, when we have things in solution when there are hydrogen bonds in both cases, do you see here that delta F for delta G is suddenly dominated by the entropy instead? And here all bets are kind of off. It depends. Because here also it's gonna suddenly, because this is a balance that's much smaller, and it's very much gonna depend on what are the entropic differences inside the protein. Even if there were some different changes in entropy here, it's not really gonna matter because the change in the hydrogen bond energy is so large. But again, here we don't change the hydrogen bond energy. So it doesn't help us whether it's large. Everything will be dominated by the relative small changes in entropy. So in general, it depends. It depends on how free the protein is before and after, whether it's good or bad. And we gain something by making the waters more free. Sorry, no. The first thing that, maybe the waters are a little bit more free. And then it's gonna depend on the entropy change in the chain. So in reality, it's difficult. It's difficult to prepare, as a predict whether it's gonna happen or not. Do you see the other thing we did? We approached this by calling something as stay. I didn't call them stay day and be here, but that's effectively what we did. And we started to look at the differences in free energy during the process. And you can have the point, dare to hand wave, dare to assume, dare to introduce things. And if that leads to good things, then I would probably go over it once again and say, okay, so I think I understand this, but were my assumptions correct? The second thing I would do, if I were to write something like that, I don't think I'm gonna do that in an exam. Well, actually, maybe it could be a good exam question. If you say, assume that delta S is zero, because I think in this case, if we assume that these are close to each other, there might be some changes in entropy here, but it's likely gonna be much smaller than delta E. You might be wrong, but it's a reasonable, hopefully it's a reasonable assumption. And if it leads to completely arbitrary, strange results at the end, go back and revisit your assumption, wait a second, is that really reasonable? So let's take, and now it starts sound like a course in organic chemistry. I promise you, we'll get to proteins in five minutes. So just as it's difficult to, sometimes difficult to predict the hydrogen bonds, in the case when I move things from vacuum, it turned out that it's fairly easy and I could predict where this thing's gonna happen. It's gonna turn out that the solubility of, sorry, they predicting what proteins and what amino acids are soluble in water and what happens and eventually whether they fold or not, it's very related to the solubility of things like oil in water. And this is neat because we can measure it very accurately in the lab. So if you take some octanol or something and then you measure what fraction of these molecules are gonna end up in the water. This is something that you do in the lab all the time. There we have it. And what you do in the lab is that you call these partition coefficients. And again, you can measure it with spectroscopy or something. What is the fraction of, say, the octanol that ends up in water? And that's what I showed you before the break, right? You've likely all used this, but you used this equation. And the only thing you've done is you take Boltzmann's distribution and then you take the logarithm of both sides. And because you're the enchemist, you would use RT instead of KT. But you're simply applying Boltzmann's distributions. But the good thing here is that X is fairly easy to measure. It's very easy to measure even. While delta G, while delta G itself you couldn't measure, but you can calculate delta G from the X. And this is how we get all the free energy of solubility or similar compounds in water. We calculated from the partition coefficients. So if you take something like, what is that? That's cyclohexane, I think. Cyclohexane in cyclohexane has a concentration of roughly nine molar. Cyclohexane in water will saturate, you're gonna have something like one millimolar because it hates water. It's so hydrophobic. It's given the choice between these two environments, the cyclohexane prefers to stick in the cyclohexane. And if you put these numbers and plug them in the equations on the last slide, you're gonna say that the delta G is roughly seven KKals per mole. We're not quite in each my left two territory, but we're getting there. So that's a fairly high energy. It's definitely not gonna happen spontaneously. Well, or will it? So this is now a difference between what you might historically have learned in chemistry with Boltzmann's distributions. It does happen spontaneously. But it's very rare. You will occasionally see a molecule like that, but there are gonna be very few of the molecules in that state. So a key difference with the Boltzmann distribution and what you might be used to think about free energy before, it's more about probabilities than absolute yes or no. So it's everything will happen, but the probability might at some point be so small that we ignore it. But that's an approximation. So that's definitely not a spontaneous process and it's gonna cost you free energy to solvate hexane in water. And an obvious question then is why does that cost energy? Because that is related to the hydrophobic effect, right? The hydrophobic effect is seeing if you're trying to solute oil and water, it doesn't work. And you know the equation to answer that. So now it's G. Well, at least I'm exercising all the letters. The beauty of this equation is that it also connects. You started to see this, but this will connect the lab. So when we've introduced this, we talked about this on the molecular scale and everything. The cool thing is that this connects things to the lab. So in the lab there are a couple of different things we can do. We can study hexanol in hexanol, so pure hexanol. We can measure how much of this is gonna be solvated in water. You could also try to in theory boil hexanol and evaporate it so that you turn it into hexanol gas. And that would move to just one molecule. The reason why we like gases, again, molecules don't interact. So the difference between gas and liquid, that just has to do the interactions between these different molecules. While that would be the interactions with it in water. And then there are tons of numbers here. Delta G is the easiest one to measure because it's, get it from the partition coefficient and that will determine how much you do see in each state. So Delta G is, oh, and I should have had units here, but it's all K-Cal. So that Delta G we can measure. That Delta G we can measure and we can even measure that too. You don't need to measure all three of them, of course. If you measure two, you can solve for the third. What is Delta H? Well, Delta H is really the amount of enthalpy change, right? And if we assume that there's not, when we move to the gas phase, it's a bit more complicated because then there are differences in pressures and everything. But if at least we move between those two, we can actually calculate how much energy do I have to add to solvate it? And that relates to Delta G too. In this particular case, the Delta H is how much energy do I need to add to boil it? So you can solve for most of these things. The one complicated part is the entropy here. It would be nice to be able to solve for the entropy. Of course, if you know both Delta G and Delta H, you can solve for the entropy, but in some cases that's hard. And again, between any of two of these states, this equation holds, and if you go around the entire lap, it should be, at least if you swap that arrow, if you go around one entire lap, it has to be zero. So how do we calculate Delta S? Or S? Now it sounds like I'm gonna have another slide and talk about derivations of equations. I won't. I'm just gonna spend 20 seconds on this. Just as you did this exercise with Delta F before, you can actually show fairly easy, just as the temperature was the derivative of energy with respect to entropy. You can do it exactly the same way and show that entropy S is the derivative of free energy with respect to temperature. It's exactly the same math. And what this means, just redo the experiment. Do it at 300 Kelvin, 310 Kelvin, 320 Kelvin, and then you're gonna get the slope here related to the temperature. From that slope, I can solve for the entropy. Do that as an exercise. It's not the exercise, I think it's actually, it's even a good exercise for you to do this yourself, rather than me showing it, because it's roughly the same difficulty as the previous one. But the nice thing is that all these plots I talked about, you can measure them in a calorimeter or something, that's very simple lab experiments, or possibly with a spectrometer to get the partition coefficient. And then you can use these very simple experiments to draw deep conclusions about energy, entropy, and free energy of the different systems, and what happens. And that's when you start to see some of these strange things that, at least to me, I think I'm flabbergasted as much we can learn from this. Let's see, ah, no, it's showing on that screen. Delta H there, if you solve it, it turns out that the difference in energy between, in this case, cyclohexane in cyclohexane versus cyclohexane in water, delta H is actually zero. That is not something you can understand. It's, all right, you will be able to understand, but it's so not obvious. Because if you have something that is not soluble, you would explain that it has to do with the energy or something. So all these things I've hand-waved about free energy that, oh, it's mostly entropy because the hydrogen bonds don't move. You just trusted me. I didn't prove that. I just said it was true. The reason why we know it's true is this type of measurements. We know that the actual enthalpy does not change because we can calculate it, or we can calculate it from experiments. And because the enthalpy, we're certainly, the delta G is certainly large and positive, right? And because of that, we can say that, ah, it must be due to the entropy. Not based on hand-waving or arguments that it must be like that way, but based on experiments. Simple experiments show you that a hydrophobic effect is centropic. And again, that's what leads to all these effects that we saw that you need to form larger structures, because if the cost is orienting all these waters, if you move the oil together in larger droplets, the total area here is gonna be smaller, and then there are gonna be fewer waters for which we reduce the entropy here. And it actually turns out that both the entropy and actually both, so how does the entropy or the T delta S term vary with temperature? Well, there's a T there, right? So we'll definitely, the higher the temperature is, the more important that term is gonna be. It actually turns out that the enthalpy also varies quite quickly with S, as with temperatures. As you increase the temperature, these, all the energies and the interactions, they go up and the T delta S term goes up too. So that the net change in free energy, in the case of hydrophobic salvation to determine whether things happen is gonna be the difference between these two terms. And that's the solid line up here with various, it still varies with temperature, but the dependence here is much weaker. And in this case, it's positive. So will this ever happen spontaneously? Pure hydrocarbon in water? Not really, right? And again, plus 8K Kals per mole. How many KT is that? Yeah, maybe 15. I kind of run down a bit to compensate for this 0.6. So e to the power of 15. And you know that that's a pretty large number. So just from that plot, you can start, oh, you're gonna have a very small fraction of that compound in water. If you start to plot this for a bunch of different compounds, and I think I stole this from the book, you can do it for ethane, you can do it for benzene, you can do it for toluene. And somewhere here, you should start to be bored. I didn't sign up for organic chemistry. So why do we have ethane or benzene? Well, the point, all these correspond to amino acid side chains. And that is at least, you can say there could be something on the backbone of the amino acid, but at least the differences between the amino acids will have to correspond to the differences between simple compounds like that. Because it's the only thing that differ if I change an alanine to say a phenylalanine. And that means that all these numbers I showed you, we can calculate with delta G, delta H, T, delta S and everything is to solve with them. And don't worry, you also don't need to know that. But what we see here is that the delta G here is gonna be the larger your hydrophobic compound is, the larger that delta G is. And it's, I will show you that. You can calculate the area of a molecule in a very, it's not entirely easy to implement actually, but the algorithm is easy to do. Just imagine that you have some sort of water ball and roll that over the entire surface and then we calculate how large this dashed red surfaces. And the average radius of a water is roughly 1.4, just based on the volume of water. It's a very simple approximation, but just to get an idea of how large this molecule is. And then we can plot this. So on the y-axis here, I plot the surface, the solvent accessible surface. And on the x-axis, I plot the delta G. And you see that it fits almost perfectly. At least for the, forget about these for a second, because these are actually partly polar, but these small hydrophobic ones, it's exactly proportional to the hydrophobic surface. So the delta G of solvator in them is gonna depend on their hydrophobic surface, which fits very well with this model that it's based on the water forming a clatter-rate structure around the hydrophobic part. What you've done with these other ones is that here we pretty much try to subtract the offset here has this because some of these atoms are not hydrophobic. But forget for that for now. In principle, it's a little bit more complicated if you start looking into proteins or something. So if you start, now we talked about moving from pure cyclohexane or something that, yes, I'll say cyclohexane for now, moving something completely hydrophobic into water. But what if we think about doing the opposite? What if we have something hydrophobic in water? So this would be one oil-like molecule. What's gonna happen with those molecules in water? They will of course move in that direction rather, right? They wanna be together. Once they have moved here, it's actually very little energy that we need to get down to something that's almost a crystal where they're perfectly packed. So that's like it's a 10th or something on the energy. So most of the energy has to do with this hydrophobic effect that you don't wanna have hydrophobic things in water. And then if you were to reduce the temperature by a number of degrees or something, you will get the oil freezing. It's not exactly this property that happened if you put olive oil in the fridge, but it's pretty close. That oil will freeze and form something almost crystal-like at very high temperatures. And that's pretty much what happens in proteins, too. So what's gonna happen for a protein? This is actually a protein, but I've removed everything interesting. What is red and green here? Do you think? And green. So what a protein does is that it turns its hydrophilic parts to the outside and hydrophobic parts to the inside. That is exactly the same thing we saw here. You start there and you go there. Then it's a bit more complicated because you have some hydrophilic amino acids, right? But the hydrophobic parts collapses away from water. Now, just as for the other case, to really form that, you also need to, at some point, get all the details right. So first you have this stretched-out chain that will very quickly collapse. And then at some point, we're gonna form the specific interactions in the secondary structure. And it's the same thing there. It's kind of like a bit of polishing or getting all the details in right. But by far, the first and the strongest effect is gonna be the hydrophobic effect. So just as oil droplets go together into larger oil droplets, a stretched-out protein chain will very quickly collapse into a hydrophobic interior, at least. So that the folding, and now we're just handling it, this is mostly, the problem with things here, this is mostly true right now. I'm gonna modify this a little bit later on in the course. Folding, the first stages of folding is primarily that we need to move all the hydrophobic residues away from water because it's just as bad having, say, a phenylalanine-solid water as it is to have a benzene-solid water. They will go, it's like microseconds, they will turn away from the water and try to face each other. And this is exactly the thing that I've talked to you about for two hours or something now. And rather than doing everything, we're just gonna swap the sign. That's the plot. It's protein folding. So all I did is I swapped the Y-axis here. Apologies that it's a bit hard to read. So in general, for a protein, there's gonna be one contribution from delta H. And there's gonna be, where, again, the lower the temperature is, and there's gonna be one contribution for T delta S. And now we're just talking about the hydrophobic parts, not everything in a protein. And this means that you're gonna end up with some sort of net effect, which is a delta G, which in general is negative now because we just swapped the sign. It will change a bit with temperature and there's gonna be some sort of, if you go too high up in temperature or something, it will eventually unfold. You don't see that, it's gonna be up here somewhere. And again, there's gonna be some sort of intermediate temperature here where the protein is pretty happy to be folded. And you probably see that there's a horrible oversimplification. I haven't worried about detailed hydrogen bonds. I haven't worried about the detailed secondary structure. I haven't worried about the fact that you have charges in proteins. We haven't worried about the fact that you have hydrophilic residues in proteins and everything, but at a third approximation, it's a surprisingly good model. And you can choose to split this any way you want and separate this from the polar or the non-polar residues. We will go through this later on in the course. I'm not gonna body with that. I realize there's quite a lot of new things. So protein folding, I would argue is 90% hydrophobic effect and then 10% packing of fixing up secondary structure. The hydrophobic effect is by far the most important thing of protein folding. But of course, while they, but just for hydrogen bonds, just because the hydrophobic effect is there, it's largely entropic. This is still related to and caused by the interactions in the amino acids that are very much electrostatic, for instance. And we also know that hydrogen bonds are super important. Then we talked about this morning that they are entirely caused by electrostatics, even if it's stopped being entropic. And hopefully you've seen, you know what the equations is to calculate an electrostatic potential. In general, you will see me simplifying things, maybe even oversimplifying things. You have likely seen that the energy between two interacting charges is gonna be one over four pi epsilon zero epsilon R multiplied by charge one, multiplied by charge two divided by the radius, right? And that's formally what we should use if we're strict and correct and using the SI system. How much does four change when you comparing different things? So there's this beautiful story of how they serve many years ago when they cracked, sorry, there's a sideline, when they cracked the encryption of the PlayStation 3, actually. So the Sony engineers had made a mistake. So they had a routine to return a random number. The only point, they didn't have actually have time to implement the routine. So for now they just returned four and then they forgot to fix that routine before they shipped the whole thing. So it returned four, always. Which is, of course, a random number, but it's fairly repetitive. Same thing, pi doesn't really change between folded versus unfolded proteins. Epsilon zero is the permittivity of free space. It doesn't change either. So it's quite, we can include these all the time, but it's more noise. So that there is the alternative, you can have the CGS system which would use completely different units and then you would skip those and just have the units, make sure that it looks multiply, product of the charges divided by epsilon divided by the radius. And I'm not using the CGS system here. I could not care less about the units, but this is my way of simplifying it. I know that the potential will depend on the charges and then it will depend on epsilon and we should divide by the radius and yeah, then there are a bunch of units. I don't care about the units because we're not gonna calculate these in numbers anyway. And you've all done this. If you do this in water, epsilon of water is roughly 80. That's a remarkably high number. Water is really cool at a good at screening things. So when epsilon is high, it means that your effect will be scaled down all the interactions by a factor of 80. And I know hardly any no other compound that does as efficiently as water and is as cheap. On the inside of a protein on the other hand, epsilon is probably something in the order of two to four. And that is because you don't really have water there. It's water that's responsible for the screening. Oil on the other hand is not very good at screening charges. So this will mean you, that means that you can actually calculate how much it costs to move a charge from water to protein and everything. You don't need to know those equations. But my point here is that two charges that are out in water or any charge that's out in water, it will be screened very efficiently by the water. If you take that same charge and move it on the inside of a protein on the other hand, the equations will be not 80 but roughly 25 to 40 times stronger. So when you actually do have charges in oil, those electrostatic interactions are not screened and then they are much stronger than they would be in water. So surprisingly, it's very uncommon to have electrostatics on the inside of proteins but when we do, it's much more important because it's not screened. And we occasionally do have things, these titratable amino acids, right? Because they are charged. Arginine, maybe histidine, certainly lysine aspartic anglotamic acid. Yep. So, epsilon here is just a unit. The epsilon R is a way to describe the medium, that different media end up being, having different efficiency at screening. For now on, let's just say that this, I will show you this, I have slides for it. But for now on, this is just a screening. If you put two charges in water, you get one energy. Sorry, if I put two charges at one on a meter in water I get one energy. If I take this same two charges and put them in oil, their energy is much stronger. So that this is some sort of property of the medium around the charges. And I'll, you'll get to know shortly why it's so much stronger for water. But this is something we can measure in the lab. And if you put two charges in vacuum, it would be one. So ER is the relative one. And that's why I prefer, I don't bother about, I don't hardly even remember all the numbers of the absolute one, it doesn't matter. So what's gonna happen if you now take an arginine and put that on the inside of a protein, the effect of those charges is gonna be immense. And that means that it's gonna cost you a lot of energy to move that to the inside of a protein. Because it might not be paired by hydrogen bonds or anything, it's gonna be so, this is gonna be so expensive that it's horrible. So introducing that charge is much more expensive on the inside of a protein than it would be to do it in water. Because in water the interactions are screen and there are hydrogen bond partners and everything. And this is the main reason why charges basically don't occur on the inside of proteins. There are some exceptions I'll tell you about in a second. So if you, if I show you a model of a protein and there's an arginine right in the middle of the protein, what would you say about my model? It's likely wrong, right? But now let's say that I show you another model where you have an arginine here and a glutamic acid here. Then it likely works, right? Or if I show you a third model where you have an arginine here and a glutamic acid here. You should probably fix the model. So that you can use as lone charges don't occur in proteins. Occasionally you might have pairs of charges. There is even a word for this. You call them, it's almost like a hydrogen bond but it's not quite a hydrogen bond. So you typically call them salt bridges. And the reason that has to do with eye, the connection between ions and salts. Because a free charge, these are effectively, oh, I don't have a protein. But they're effective, these side chains are effectively ions. So the other thing that can of course happen is that when you push this side chain into a protein it can change its titration state. It can either drop or take up a proton to become neutral. Now that's of course not good because in principle they prefer to have their proton here, right? But in many cases it's gonna be even more expensive to introduce this protein into the hydrophobic interior than deprotonate it and then pull the protein. So it's just a matter, are we gonna pay free energy by introducing something even charged on the inside of a hydrophobic medium or are we gonna pay by deprotonating it? There is no way I can win here. And again, that's the reason why occasionally they will occur on the inside. If you have one of these titratable residues on the inside of a protein, it's probably 50 to 75% certainly that is deprotonated. But even then it's gonna be very rare. So if we now compare a couple of things here. We've talked about it. So how strong are various things? One of them I haven't put up yet. What's KT? 0.6 watt. Yes. How strong is the hydrogen bond? It varies a little bit. Let's say 5K Kals or so. That's 10 KT. It's a fairly large number. It doesn't sound large with five, right? But 10 KT, that starts to be barriers you don't spontaneously break hydrogen bonds. So do they have any idea what the typical stability energy at least of a small protein is? Even medium ones, but it turns out that it's not that dependent on the size of the protein. Take a wild guess. Maybe 10, maybe 20 KKals. So you're talking about a couple of hydrogen bonds that stabilize the energy of an entire protein. Well, the energy we take to unfold it because the protein will spawn spontaneously, right? And that's because it's a lower free energy to fold. But the point if I add 10 to 20 kilo calories per mole by heating it, the protein would unfold. So proteins have a very low stabilization compared to any other chemical processes. That sounds awfully stupid. Why on earth would nature do that? Wouldn't it be much smarter to have proteins have a 10 megajole stabilization, 10 megajole per mole stabilization energy? No, they don't have to be flexible. They just have to fold and do their job, right? Well, you could still argue that the flexibility, okay, I'll give you that. Maybe the flexibility is good, but say something else that doesn't have to move. Why can't the protein be super stable? Yes, exactly. Because your body will also need to break them down. And if you a month later would need a million calories of food to be able to afford to break down your protein, you couldn't survive. So this is a delicate balance that proteins must be stable, but they can't be too stable. So I already mentioned a little bit that epsilon is three inside a protein, but I think there's a great comparison to your question what epsilon is in water. So why is epsilon so high in water? That's what we can start by asking ourselves. So if you have, if I introduce a small ion here, the plus here, what's gonna happen with that in water is that all the water molecules around it is gonna turn its negative oxygen to my ion. So effectively all the waters around my ion, they screen the water and they can do that because the water molecules are so small that the entire molecule can reorient the screen. If I move the ion or something, all the water molecules will move with it. You can even show this, you can measure this as a function of frequency. So at low frequency, it's 80. So if I just have a radioactive field, then I'm sort of radio, radio frequency field. And if I try to force the water molecules to move, they will move and they will move very quickly. And then as I increase the frequency, at some point the frequency becomes so high that the water molecule starts to move, but before it's had a chance to move, the field has already changed again. So that at very high frequency fields, epsilon goes down, the water, the entire molecule has to rotate and suddenly it doesn't have time to rotate. And that is the reason why epsilon water is so high. Small molecule, very large dipole in it and it can reorient. In the protein on the other hand, we don't have the water on the inside of the protein, right? Do you have some dipoles inside a protein? Well, we do, the peptide ones, right? Can the peptide ones rotate freely? So the problem is even though you have some dipoles, they can't reorient to screen your charges. You might have another SMM, you might have a aspartic acid or something that pairs up with my charge, but the protein itself is not the flexible medium that can screen charges. And that's why effectively this epsilon, the screening factor is much, much lower. And there's another super corny unit you measure, there's a Farad's per meter, it doesn't matter, forget about it for now. But the point is you can actually measure this inside proteins and it turns out that we, I'm roughly correct at least. And I think that's roughly what I said already. So in vacuum, I don't have this epsilon R. In vacuum, epsilon is, the epsilon R is always zero. So I always measure things relative to vacuum. So in vacuum it would be one and because in vacuum we have no molecules that can rotate around. If you now start having two charges here, if I have a cat ion there and an ion ion there, plus minus, you will see all the waters will help to screen here. And it's gonna screen both that charge and that charge. If you, on large scale, if I have thousands of waters here in the middle, you can start to look at this water as some sort of continuous medium, right? And imagine doing this in the lab. And if I have two ions, two charges placed one centimeter apart, there are gonna be so water molecules between them that it's continuous. And then let's bring them down to one millimeter apart. And one micrometer apart. And one nanometer apart. At some point, you're gonna have these two charges right next to each other with no water between them. What is epsilon? Well, it starts to become hard to define. But epsilon is a macroscopic unit that we use to define in the lab. And at some point in the microscopic scale, this breaks down a bit. You can calculate it with electrostatic interactions. I would guess it might be 10 or so, because you would have a bit of screening from the things around it. But the point is we're not gonna really worry so much exactly what it is, but you have to remember that in general, in a protein, it's low because you have no free molecules that can rotate the screen. And this is not just on the inside of a protein. There's another very important case where this happens. Membrane proteins. That we'll talk more about later. So what surrounds a membrane protein? A membrane. Can you imagine what epsilon is inside a membrane? It's more like oil, right? So this is our first indication that some things will, everything I've said about protein folding here, kind of assumed that we have water around the protein. So it's gonna turn out that some things behave a bit differently from membrane proteins, but we've come back to that in two weeks or so. So I guess you could guess here what it is. It's not gonna be one because the water is around it. The strength here would be insane, but there are some waters around it. It's not gonna be 40 to 80 either because they are close. So I guess on the previous slide, you would be somewhere here. But you also see that it has a tremendous difference in the interaction strain, even whether it's 30 or six. So electrostatics is difficult because the number becomes so large. And this actually happens. You do this all the time. When we saw buffers or ions or anything, water, because then we have to break these charges apart. And instead of having them adjacent to each other, we're gonna add in that a few angstroms apart of water. You can compare that strength with hydrogen bonds. So the hydrogen bond at epsilon 80 is roughly five k cal or so. So it's almost as strong as two full charges, two full ions next to each other. So hydrogen bonds are so strong that they're almost covalent bonds, not quite. But if you look at simulations of proteins, they will stay for milliseconds or something. It's very hard to break hydrogen bonds. I think, yes, oh, I think we're doing good here. I'm gonna be able to let you off, not quite yet, but I'm gonna need another 10 minutes or so. So I'm well aware that there were a bunch of new concepts I introduced here. We came back to the hydrophobicity and spent a lot of time talking about it. Also the hydrophobic effect. And I think it's a good idea. Learn to look at these mirror mirrors that you can either think of moving a hydrophobic compound into water or taking a single hydrophobic compound that is in water and moving it to an entirely hydrophobic environment. The latter is more corresponding to protein folding while the first one would be more corresponding to simply solving something in water. And simply solving something in water is a conceptually easier experiment to do in the lab. But it is the same experiment. It's just a matter of which direction the arrow points in. The only thing, when you're changing the direction of an arrow, the only thing that does is that it adds a minus sign before all your terms and the PNG equations. We've started to talk a little bit more about enthalpy and entropy. So how many times did I mention pressure and volume in all these slides? We're gonna ignore it. We're gonna ignore it completely for now because it's not relevant, not for proteins. Now on the other hand, again, if you're solving ethanol to water 50-50, then it's very important. But the reason why it doesn't matter for proteins is that the protein concentrations are typically in the low milli or micromolars or something, and then the volume effects are negligible. Did you have a question, Orin? Yeah. That is way more complicated. In particular, you start having what is the surface tension in the memory? What is the shape of the memory? Because you're also in a two-dimensional system. In the interest of not confusing you all too much, I love memories. We're gonna talk a lot about memory. Much more than there's a book, but we might wanna wait until after Easter. A little bit, but not much. Because again, that the average concentration of one membrane protein in a membrane is still relatively low. The total concentration of all membrane proteins is quite high. Roughly a third of the mass in membranes are membrane proteins. But for a single membrane protein, then when it's folding, it's a very small fraction of the entire surrounding. We spoke a little bit about free energy of process. I'm well aware. If you haven't seen this before, the first time you see this is confusing. But the one thing I so would like you to do today is it's almost weekend. Spend a couple of hours in this afternoon before you take weekend and try to understand it. Because on Monday, we're first gonna have a second lecture and then I think you're gonna have a lab, a second lab on the entropy parts of the Boltzmann equation of the rest of Monday so that catch some breath and make sure you don't have to sit with this Saturday and Sunday. I will answer emails too if you need some. I've normally I respond to emails within a few hours at least and I will make sure to do it there too. I'm also well aware that how horrible Mondo is. So what I will likely try to do, I saw that this was gonna be down a couple of hours on Monday and I've had it with this, trying to help you to get access is I will try to put all the material we have on a separate site instead. And then I might have to move to Canvas or something next year. But I'll mail it out to you. Still, I only have the emails of you that are on the site and two or three of you have mailed me. If you don't get emails and you had not mailed me, I might not even have your correct emails. So mail me then and I'll make sure to put out some alia so that I can reach you. Remember that green and red plot? It's a bad choice of color because I know people are red, might be red color blind. That is actually called something. It's called a molten globule. And it's gonna be an important concept in protein folding. So I lied just a little bit to you. What happens if I take an amino acid site? I can actually take any amino acid sequence, 100 random amino acids. And if I throw them in a water solution, what's gonna happen is that it's gonna say boom and then a microsecond later they have collapsed into some fluffy bowl. It's not a protein. It's just a random mix of amino acids. They're all gonna be pointing the hydrophobic amino acid to each other on the inside and they're gonna be pointing all the hydrophilic amino acid on the outside. But it's not yet a protein. We don't have all the details. We haven't really formed the details of the secondary structure or anything yet. So this is some sort of intermediate states. And it's, I'm not, well globular, it is globular, it's molten because it's still floppy and flexible and everything. It's kind of like a liquid on the inside. But you will never see under normal conditions. You're never gonna see these pictures we have of a stretched out chain of amino acids. It can't happen. And that's another question. Why won't you see that? I'll have a couple of minutes. So I'll take you through that. So here's my multi globular. Sorry, I will have to do red and green because I'm gonna need to get some more pens here. Hydrophobic on the inside, hydrophilic on the outside. And this is just a mixed blob of amino acids. And then I'll draw another potential orientation. So here's another, here's the same chain. Why won't you see it like this in water? Stretched out. Yes? Yeah, but what's so bad with exposing hydrophobic residues? It is bad. You're right, but I want to drill down deeper. Based on what you've learned with Boltzmann distributions, free energy, entropy and things. Remember my desk? My desktop? How many states like that are there? If you put all, if I require all the torsions to be in trans, because that's, if you really want this to be fully stretched out, they will all have to be trans. There is only one single microstate like that. How many microstates can you imagine if you just throw things in a random bowl? There are going to be quintillions of states like that, right? So entropically, that's a much more favorable thing to do. Even before we even started to talk about hydrophobic versus hydrophilic and surrounding, then of course the entire hydrophobic effect is going to make this even stronger. Well, hydrophobicity will help too. But even if all your residues were hydrophilic, the chains would still not be stretched out because there is only, well, it could be stretched out. But it's, if you have 100 residues, there is only, there's two to the power of 100 combinations and there's only one of them that would be completely stretched out. All the other ones would be more or less collapsed. So it's simply, there are many, and that has to do with the things that for us, this looks like one state and that looks like one state, but they are not the same. They are one state, of course. This is a multi-globular state and this is a completely stretched out state. But if you drill deeper into these states, this corresponds to one microstate where you have all the torsions and the spin. This one internally corresponds to lots of different microstates. There are lots of different ways we can put our amino acids and end up with something that you feel looks roughly like that one. And that has to do with exactly these entropy parts of the Boltzmann distributions. And that's what's gonna be what you test in the lab on Monday afternoon. So then you're gonna start having duplicate states. You have several states with the same energy in some cases. And you will see that, you will see that directly and it's very cool. You will just end up with a, when you start just calculating this probability, it will look like a correction factor. And if you do this, and I'm not gonna spill the beans, because it's gonna be more beautiful than you think. It's gonna be way more beautiful than you think. And I hope, I also hope you can understand entropy. But my point, entropy, trust the equations. Once you trust the equations, entropy is not complicated. Temperature, on the other hand, is super complicated. If you wanna, I deliberately skipped, the book spends a lot of time talking about electrostatics. I don't do it simply because this is not a course on electrical field theory. It's a super fun topic, but you don't really need to dig that deep into it to understand the proteins. So I think it's much more interesting to talk about why folding happens in the best of the biology. But if you wanna, please do try to read chapters five and six in the book. The book typically has one chapter. In most cases, there's one chapter per one hour of lecture. So we typically go through two chapters per day. So if you haven't done this yet, it's important to get started reading the book because we're already six chapters into it. So on Monday morning, I am gonna go through these study questions with you. Talk about differences in the free energies and everything. There is one thing that I did not bring up, but that could be cool to you. We had this definition that free entities somehow corresponds to the amount of energy that is available for work. And you did the bad thing of just trusting me there. You can prove that. And you don't need very advanced mathematics for it. So if you follow that link, there is a webpage that actually goes through and proves that free energy does really correspond to the amount of energy that is available to do work with. And what you will see again, most of these equations, they don't require lots of advanced skill in math, but you need to sit down and think with it. Just because it looks like a trivial equation does not mean that you can solve them in 10 seconds. But I'm not gonna otherwise spend some time with these. I think it's a good way. If you understand all these study questions, you're gonna fly through the exam with flying colors. And that's all I had. So let's finish 15 minutes early.