 Welcome to the second lecture in the biophysics class. I'm going to start today too with a historical background showing you some very old movies. And then we're going to jump into a number of physical concepts that I've divided concept by concept, which hopefully will make it easier both for when you're studying and when you're looking at that later based on the study questions. Let's start with movies. This is an old, very old movie generated by Cyrus Leventhal that was actually displayed on a computer screen, then filmed with 16mm camera and then many, many decades later this was 66. Mike Levent and a few other friends transferred this back to computer and I got these from my friend Mike. This was the first ever computer representations of protein molecules. To you it's obvious that we can look at structures this way. You've probably seen this a ton of times, but in the 1960s it was a miracle that instead of looking in columns of x, y and z coordinates for every single atom you could visualize and try to understand conceptually what a molecule would look like. Myoglobin here was one of the first protein structures ever determined. This is from the LMB in Cambridge and it's a protein that binds oxygen in your muscle tissue and the part that binds oxygen is an iron containing heme group. We'll get back to that later in the class. I'm going to show you a second example also by Cyrus. This is the protein lysosine. It's an enzyme that breaks down bacterial walls and it's present in tears, saliva, mucus, etc. And here too you can see properties of the structure. Later on in the class I'm going to tell you what these helices and beta sheets are. But here too, seeing this, being able to rotate it live and everything was a breakthrough that you couldn't imagine when you were used to just either standing in the lab or considering protein structures as a set of Cartesian coordinates. The person generating this, I just figured that it's fun to see when most of these famous people, when you look at them, you see very senior photos where they're very serious and everything. But the thing when Cyrus did this, he was fairly young. He was also a Mac user. Well, not that type of Mac. Here is the type of Mac Cyrus was using. It's a computer screen and mouse, but note that you don't really have a keyboard. Actually you do have a keyboard. It's the typewriter. But the typewriter is not directly connected to the screen. So you would type, get punch cards or something, let the computer run your program and then you would have this circular screen that you would look at and film it. There is no way you could have direct text output. You might even see a mouse there in the background. And this is what the mouse looked like. Of course, they didn't call it mouse. It was just some sort of weight of a lot. Mac stood for multi-axis computer. I'm not really sure what the multi aspect was. And then for fun, I'll throw in a picture of Cyrus sitting and typing at one of these teleprompters. You're going to hear more about Cyrus and something called Leventhal's paradox later on in the class. But I'll cover that when we get there. The next part is that we're going to be looking at specific elements of protein structure. So when it comes to structure, what we're going to need to decide is what are the atoms in the structure, but just looking at individual atoms is fairly pointless. So we're going to need to consider how and why atoms interact, how strongly they interact and why some atoms prefer to interact with other atoms. You've probably taken this in various forms, either in upper secondary school or at university, but I'm going to introduce this in a slightly different way possibly. This is more chemical than physical way. At the end of the day, the reason why any atoms interact is due to electrons. And electrons is in principle complicated. You should solve the Schrodinger equation. But if you're a chemist, then we need to be able to have an understanding. We tend to use something called orbitals, which is a schematic way to display the way electrons would interact with each other so that we can hand wave about how bonds work. But make no mistake, this is an approximation. And if you haven't heard of orbitals before, you definitely heard of it indirectly in upper secondary school when you talked about electrons in shells. So you might remember you had this first S shell and then three P electrons and then D electrons. And this is largely the same concept. It has to do with various energy levels. In each of these shells, you can have two electrons. And the reason we have two of them is that electrons are paired. And you might also remember that you talk about some sort of spin-up property and some sort of spin-down property. Of course, that has nothing to do with up and down. But you can think of a sine equation, right? A sine is sometimes positive and sometimes negative. But if you now take the sine equation and push it 180 degrees pi, you're now going to have another sine wave. That's also sometimes positive and negative, but it's offset. So when one is positive, the other one is negative. And this is roughly the same concept. The electron maintains its charge. It has to do with density and where the density probability is highest. But that also means for the S orbital, that's easy. That's rotationally spherically symmetric. But for these other orbitals, there is a very clear spatial favorability here. The P orbitals are, think of that as Cartesian axis, X, Y, Z. They are orthogonal. And for the D orbitals, it's even more complicated. So why do I tell this? Well, it turns out that this concept of orbitals is going to tell us when atoms are likely to bind. What happens if two atoms get closer to each other? When will they repel each other versus when will they bind? And it's also going to explain lots of concepts such as hydrogen bombing. What would happen if you had two of these would spin up? So if I erase that spin down for a second and then I put it as spin up. Well, what then would happen is that you would have two electrons that have exactly the same quantum mechanical ground state as they approach each other. And that leads to something called the Pauli exclusion principle. And you can even prove that's not that hard mathematically, that that would lead to an exponentially strong repulsion. So it's going to be very, very bad in terms of energy as they get close. So I'll erase that one. So I don't have something bad written on screen. In practice, it gets slightly more complicated because sometimes if you're a chemist, you can have double bonds and things like that. I'll worry about that when we get there. We might not get there that much in this class. So can we use this to understand binding? Let's try. This is something very simple. It's two hydrogen atoms, I think. And each hydrogen atom has one proton and one electron. And it's not, well, at very high temperatures in the plasma or something, they would be happy that way. But hydrogen is a fairly reactive molecule. So under natural conditions, what's going to happen is that you have these two hydrogens and as they approach each other, what will happen is that if both of these electrons start, instead of going, if you have one hydrogen atom there and then one hydrogen atom there, if this electron and that electron, instead of being separated in the two hydrogen atoms being unpaired, what can happen is that we have our hydrogens and then we bring them really close. That might be a little bit too close. But basically you would have both these electrons encompass both of the hydrogens. What will then happen is that around the left hydrogen atom here, you're effectively going to have both spin up and spin down. And around the right atom, you're also going to have both spin up and spin down because they will pair up that way. So what's now going to happen is that instead of having two incomplete atoms, you're going to have two atoms that each feel that they're complete and this is going to lead to a chemical bond. What complicates things now is that this gets difficult to explain to a teenager or something and then we start inventing all these things. We have covalent bonds, we have ionic bonds. This is what I would call an ionic bond. Sorry, covalent bond, my bad. But in reality in life, this is a sliding scale. It's not going to be one fixed distance when it starts being one bond and it cuts over into being another type of bond. But I'm going to go through a couple of different ways these electrons interact at different distances and see if we can learn a little bit more about that. And I'm not going to go that much deep into the orbitals for now. We'll come back to the orbitals once we start entering the hydrogen bonds. So at close distance then, atoms would form covalent bonds because by pairing up, both electrons would feel like they had a complete first shell would spin up and spin down. If you now take these atoms and pull them further apart, that's not quite going to work that way. But on the other hand, we know that lots of atoms, even such as noble gases that already have two electrons in their, well, that have completely paired shells, they also tend to interact. If you lower the temperature far enough, even helium will eventually become liquid, although that's when it's really cold. So one way or another, these atoms have to interact. If they don't interact, they would prefer to be in gas phase. And what happens with those atoms is that if we draw a nucleus here too, this is an atom, and then we have some sort of completely full electron shell around it. Well, that atom is happy. It doesn't really need anything. So you have lots of plus charges here. I'll just put four pluses, just a symbol. And then we have lots of minus charges out here, minus, minus, minus, minus. That would be a very special shell, but again, just an illustration. The net charge of that system is plus minus zero. No question about it. But what if I take another small molecule here, such as the ones that you looked at the other day, water. And at some intermediate distance here, I have my water oxygen, and then I have two hydrogens. You all remember now that we have these partial charges in this molecule because the oxygen stole the electrons from the hydrogens a bit. And we can draw that rather than drawing plus and minus signs. You could think of that as having a dipole here, right? So we can have a fairly large dipole in the direction from minus to plus. But again, this molecule too is neutral, but it's neutral in the sense that it doesn't have a net charge, but the charges are not equally distributed inside the molecule, and that's what gives it its dipole. Now, if these molecules start to sense each other, what's going to happen is that effectively we have a bit of minus charge here on the oxygen while the plus charges on the hydrogens are further away. We have minus charges here too and plus charges. But what we can then do is that as these molecules get closer, what if we change the representation here on the left a little bit so that we still have this nucleus with plus, plus, plus, plus. But then we shift this electron cloud a little bit away. I'm exaggerating horribly here. What this will now mean is that the negative charges here have moved further away from the water, and this water, again, if I just draw the entire water as a blob here, as minus there and plus there, those minus signs are now going to like that it's plus signs closer here, while we avoid the repulsion and increase the interaction a little bit between them. And another way of doing that, if both of these pens are arrows, you can say that you have one dipole here and you have one dipole here, and they have now lined up a bit. So this is going to create a weak attraction. The net charge here is zero, the net charge there is zero, but they will still attract each other a little bit because the dipole here is inducing an effect in this molecule. But the question wasn't whether a noble gas would interact with water, right? We know that noble gases per se start to interact. Well, what if we had two of these molecules? Let's just think of it. And if you were at absolute zero, nothing would move, right? And then we would likely have this as the best possible state you could imagine. But at finite temperature, atoms oscillate, even electrons oscillate, they oscillate much quicker, that starts happening at like one Kelvin. So at some point in time, if instead of having a water here, I imagine having another similar molecule. Here I have my plus charges and here I have my minus charges. But now it's at high temperature, so at some point this molecule is shaking a little bit. The electrons is going in one way, completely natural as the surface in a glass of water will shake a little bit if I'm carrying it around. But this molecule now has the properties like the water had up here, right? This is effectively a dipole. It's a temporary dipole. It will disappear in a second, but right for now it is a dipole. This second atom will respond to that, because here we now have a net dipole. This one is still plus minus zero, it doesn't have a dipole. But if this molecule now does the same type of shift of the negative charges in that direction, you will effectively under very short time have these dipoles line up again. And again, dipoles lining up. So what we have up here was what we would call an induced dipole. And this in this case is an induced dipole-dipole interaction. This will happen for any atom, even if it's a noble gas, because they have electrons and even if they're not charged, they're going to be very weak, because what will happen a nanosecond later, if actually much less than that, 50 seconds probably, these electrons will have shifted back. And then a little point in time later they might have shifted in the opposite direction. And then this molecule will have to adapt. And what that's going to lead to is that you essentially have an interplay, right? Where these molecules start shifting around, but on average at finite temperature they tend to be aligned. You can calculate the shape of that. And it's a dispersion interaction, and it will go as 1 over r to the power of 6. So it will decay very quickly at large distances, and it's also a very weak interaction, which is the reason that noble gases do not really attract each other by a whole lot. So these are essentially the two extremes. We have hydrogen bonds, very strong electrostatic interactions. And on the other hand, an explanation why things will interact at very, very long distances. There are more interactions, but before we go into all the other interactions, I will start to think about a complication that just happened here. Because this was not supposed to be a class on quantum chemistry, right? But everything we do here is about electrons and quantum. So on the one hand it's great that we can understand interactions because they're due to electronic interactions. The problem with electrons is that we cannot treat electrons classically. There is no way we can treat electrons with simple ball and stick models. So does that mean that we should give up on this class and head over to the quantum mechanics class instead? Well, yes, in many ways we should, right? Because if it is about electronic interactions, I just spent a few slides or boards here justifying that we need quantum chemistry, so of course we need quantum chemistry. You could even argue that I started introducing you to quantum chemistry, even if we didn't use equations for it. But on the other hand we have all these simplified models that I showed you that are literally ball and stick models where we think of an atom as a ball and bonds as sticks. And people keep using them all over the world. So why does that work? Well, you could argue that either that you're a realist or you don't know what you're doing. There are many reasons why people have done this historically. Today we can treat maybe 100 atoms really accurately with quantum chemistry when I was a student with probably six or so. But that is not an excuse, it's not an acceptable excuse because if it's expensive or difficult to do it right, that doesn't mean that it's going to work to do it wrong. So I think we'll have to strike out that argument. The other problem though is that what I have here on the other side, that's not really accurate either. Because what you do in quantum chemistry is that you put things in a computer and then you calculate what is the best possible distribution of electrons. Well, first the best possible one, we are working with time in this class so that you would have to solve the time-dependent relativistic Schrodinger equation. And you can probably do that for like one electron. The other problem is that if you determine the best distribution you're essentially determining what is the structure at zero Kelvin. You need to break it to you but there's not a whole lot of interesting biology happening at zero Kelvin. The other problem when you do this, take any small molecule in life science, that's not going to work. We need water. I spent a large part of last lecture talking about the virtues and importance of water. You can't start doing life science but say that you're going to approximate everything being at zero Kelvin and no water. That is just as stupid as the first item here saying that you're going to do a bad method because the usual method doesn't work. So it turns out that quantum chemistry on the one hand has a very accurate representation of some things, the energy between the atoms that are included, but it has a lousy representation of other things. It doesn't represent motion. It doesn't represent the solvent. And it can be surprisingly difficult for it to get very large systems to work and the time dependence isn't there. The classical models on the other end have other strengths. They are simple so that they're very fast to calculate. That means that we can reach realistic size systems. We can reach systems that involve water. And we can simulate those systems at realistic room temperatures when things actually move so that we reproduce what happens, for instance, if two atoms, not two atoms, but two molecules get close to each other or if a protein is folding. And that's going to be a huge difference that we're going to come back to in this class. So doing things with simple methods looks great as I'm showing you here in this movie. But the point is this simplification has to come from somewhere. I can't just invent things out of thin air. So why does this work? Well, it works because we're cheating. And the cool thing is in real life you're occasionally allowed to cheat. So remember the water molecule. We don't really have that many things to do in water. If we make this simple, you could argue that we have the charge on the oxygen. And no matter what charge I put on the oxygen, I'm going to need to put half that charge of the opposite sign on the hydrogen. So that's really just one number I can tune how the charge is distributed between the oxygens and the hydrogens. To first approximation, I will say that the angle and the bonds here, that I'll take those from quantum chemistry. So I'm not really going to change those. The other part is that we're going to need some sort of parameter describing the repulsion between this. That is, if two atoms get close to each other, how much are they repelling each other? In principle, that is two parameters, one for the oxygens and one for the hydrogens. But you will have to trust me when I say this. If you were to draw and water molecule a real way based on the radius of the atom, it would look completely bonkers. And that's why we don't do it because it would look rough in this way. This would be the oxygen and this would be one hydrogen and this would be the other hydrogen. So you can probably come up with a reasonably easy approximation here. And if we just erase those hydrogens from the Lennard-Jones interactions, the thunderballs, I'll get back to those in a second. So that it's pretty much just one sphere with one radius. That's one parameter and we combine that with the distribution of charts. That is two parameters. I can, in principle, set these two parameters to anything I want. So what should I set it based on? Well, I look at water and there are two very simple things with water. I know what the density of water should be and I know roughly how expensive it should be to say boil water, the heat of vaporization. And I can just change these parameters to make sure that they fit the experimental properties of water. It's a horrible way of cheating. And of course the reason this works is that quantum chemistry would get the same results but from fundamental first principles. Here I'm skipping over all that and just tuning it to fit the experiment. But if that works, then I'm able to reproduce the diffusion coefficient of water as I'm doing in this movie. It works great. And I'm able to do this orders of magnitude cheaper than I could do with quantum chemistry. It's a pretty cool approximation. Who came up with that approximation? Well, there were a number of scientists involved in this. There's a long, long, long story in this field and unfortunately I won't have time to take you through all of it. But it started out in a Schneer-Lifthansa lab in Israel and coming very much from polymer physics. I'm sorry, I might even have said it. The situation in polymer physics is that people started using something called semi-empiric parameterization. That sounds difficult. It's not really the first principle once that would be the quantum chemistry. The fully empiric parameter is that would be if I just tuned everything to fit experiments. And I didn't do that here, right? I took the bond lengths from quantum chemistry. I took the angles from quantum chemistry. At some point you're going to see that we take charges from quantum chemistry. So I kind of use a little bit of quantum chemistry and then I combine that with parameterizing it to fit experiments. And that's why we call it semi-empiric methods. These methods have become immensely popular the last few decades. The first people to come up with this was actually, well, there was Schneer-Lifthansa and Ari Warshall in particular. And based on these methods Ari and a few other colleagues were able to determine very detailed models of large molecules and enzymes where they used quantum chemistry for a small part in the middle and then these semi-empiric models outside of it. And for that work they were rewarded with a Nobel Prize in chemistry in 2013. Almost any high impact paper you read in biology today, if it's a new structure or something, they frequently include a bit of simulation which is kind of fascinating because only 20 years ago this was something very esoteric and theoretical physics. We're not going to start doing simulations quite yet but we will have to look into a little bit about the interactions and how we model interactions in proteins because that's going to be super useful when we start revisiting protein structure next week. So in principle there are many ways atoms interact in molecules but we're going to look primarily at large molecules, proteins, and we're going to try to group this the way we normally do it in our programs. It's not necessarily the only way of doing it but I would argue it's the most common way. The first and simplest interaction if we look at a almost, I was about to say nest but this is a core of a protein, it's a beautiful structure but let's look at just one interaction in the middle here. If these two atoms are moving relative to each other that's going to correspond to a stretching or compression of that bond. That's actually quite expensive to do because remember those electron orbital pairing I told you. If we're stretching that we're going to move the electrons away from its equilibrium state but that will definitely happen a bit at room temperature. There are many ways we can model this and if we're going to do it with quantum chemistry it becomes really complicated. So with quantum chemistry this would be a so-called quantum chemical oscillator and I'm not sure if all of you have studied quantum chemistry or quantum mechanics but in quantum mechanics what that would mean is that you would have a ground state that actually corresponds to a fixed length and if you increase the energy you're going to start to have some of the molecules populate a higher state with also a fixed length that's slightly higher and then etc increasingly higher energies but each length is discrete and that's what we see in these horizontal bars. That would be very complicated to do for us so we're not going to do it. But maybe we can cheat instead. So if you look at that blue curve that blue curve is again very simple approximation. It's called a Morse bond potential and what that describes is roughly at very small distances atoms will repel each other exponentially. That's the Pauli exclusion principle that I already introduced you to and you might know since before. If you start stretching atoms you're also going to have bad things happening, right? That you're taking something away from its equilibrium condition but if you do stretch things far enough at some point the bond will break and bonds can break and when bonds break the atoms will eventually be happy. They won't be quite as happy as when they were together but you can certainly say heat to gas until the point where the two hydrogen atoms and the H2 molecules go away from each other and form a plasma. So the blue potential does describe that but then we think a little bit more about that. Wait a second. If we're going to study proteins around room temperature how frequently is it going to be that we start having things well we're never going to have plasma phase that would be hundreds of thousands of kelvins but how frequent is it that we actually break molecules inside the normal proteins in ourselves? Well it does happen once in a blue moon for very specific chemical reactions but typically when a molecule is stable in an equilibrium state it's neither going to form nor break bonds. And I would argue that accounts for 99% of what we do and if that's 99% we should optimize for that. So I'm going to take the approximation and make an even worse approximation. So I said let's just assume that it's not a quantum harmonica oscillator but a classical harmonica oscillator. So there's just a second-order potential here that the potential of the energy is simply some sort of force constant multiplied by the square of the deviation from the equilibrium bond length. Is that bad? Well in a way it is bad but physics has been bad in that way for centuries. You might know that under another name, Hooke's law. Hooke's law has to do with this the force when you're extending a spring, right? I hate to break it to you, that's wrong too. If you don't know that it's kind of fun because I think every three-year-old kid knows that if you have one of these springs that you're using to walk down a stair or something if you're a three-year-old you can't help trying to see what happens if you tear it apart, right? And at some point you've distorted it so much that it won't go back. That has to do with memory effects and metals and everything. Hooke's law is not too. The reason why we use laws like that in physics is that you can imagine having absolutely any type of function, a potential describing a system. We're going to talk more about potentials in the next lecture but this far we can just say if I have a potential energy here that is some sort of arbitrary shape there is something around the x-axis here, we don't know what it is for now. This might be the deviation from the equilibrium or the length of spring, whatever. The y-axis here is an energy and in physics and life low energy is good. That corresponds to weight being on the floor rather than on the table. So this would be the best possible state here. Now, describing this entire equation is difficult but what if I'm only interested in deviations around this part here? Well, I could start by saying the first approximation here would be the actual value of the function here but it turns out that's not very important because when it comes to energies I can talk about how far I've lifted the weight from the floor but this is my floor or a floor in another building so I can just say that the first approximation that is I'll just adjust my y-axis here to say that this is zero. The second part I would need to account for if I do a series expansion here that would be the derivative but here's the cool thing. What is the first derivative around a local minimum or maximum for that matter? Well, it's zero. It disappears. So I don't have to think of the first derivative. And the next higher part I then need to consider is going to be the second derivative. So if I now take the second derivative and fit that around this local minimum that's what I get exactly a potential like this one and that's why they're so common in physics. So physicists apply this not because it's right but if I don't know anything around a local minimum I can always describe some things with this type of function. How good it is, that depends on how far we go away but for bonds in our case it's going to turn out to be an excellent approximation. There is only one approximation that's even better. We're going to come back to that when we do actual simulations. Remember that quantum mechanical oscillator? It turns out that if we stick to room temperature 99% of the bonds would be in the ground state. So an even simpler way of fixing this would be to give all the bond lengths a fixed distance and say that they should not deviate from that distance. Slightly harder to implement on a computer but much easier to understand in terms of physics. So as I told you about bonds, bonds are strong. We're talking about hundreds of kilocalories per mole. Unfortunately that's something we're going to need to come back to repeat it in this class. How do we count energies? Well, we're going to occasionally going to use kilojoules. We're occasionally going to use kilocalories but it turns out that inside one molecule these more energies would be absurdly small. There would be 10 to the power of minus 25 or something and something tells me that you don't want to use numbers that small because it gets obnoxious. So instead we tend to measure energies in terms of kilojoules per mole. That is, instead of measuring the energy in one bond I measure the energy that would be stored in avogados numbers bonds. And the reason why we've adapted that is that that leads to very nice numbers. You can talk about the energy for bond might be 200 kilocalories per mole or something. 200 is an easy number to work with and I can't forget about all those exponentials. And that is so common that sadly as chemists we occasionally we might occasionally be a bit sloppy and say that the energy is 200 kcal. And when I say 200 kcal you should just take it for granted that I actually mean 200 kcal per mole. There is no way we would have absolute energy that would be 200 kcal inside a molecule. So why do we use kilojoules and kcal? There is an SI standard for this and that would be the kilojou. The problem is that standards are great so people tend to develop their own and if you look at this all over the world and particularly in the US people still tend to use kcal even in parts of Asia. In Europe kilojoules are more common but the sad part both of these number systems are around and that means you need to be aware of both of them. There are very few, I don't think there is anything in this class that's going to depend on the exact 4.184 conversion factor between them but you need to account for this factor of 4. For instance if I ask you about the energy of a hydrogen bond if you make a factor of 4 error that's probably large enough that you might not get that right on the exam. So in summary I would suggest that you look up these numbers. You're going to need to think about kilojoules, kilojoules per mole, kcal and kcal per mole. So try to look up either for bonds if you want to or the other interactions are going to talk about and get the rough idea what are the orders of magnitudes of the various energies you have and see if you can get a gut feeling about which ones of these are high and which ones are low. I will tell you a little bit about it in some of these short video recordings about the different types of interactions. After bonds the second type of interaction we're going to discuss is angles. So I'm using the same type of molecule here but instead of looking at just a relative displacement in the distance between two atoms we're looking at how the angle between three atoms is varying. So if you have atoms i, j and k here we can draw one vector i to j and another vector j to k and then we're looking at how this angle is varying. This is also a fairly strong effect just like the bonds but it's softer than the bonds. And by softer I literally mean that where a bond would not typically change more than one or a few percent under room temperature an angle can change well not ten percent but a few degrees definitely. So if a molecule has to be squeezed in or adapt when we're trying to bind something to it the angles definitely can vary so we need a reasonable model for the angles. So now that you're a skilled physicist and biophysicist what type of model would you pick for the bond angles? This is a function. It's a function that will obviously depend on this angle and you have to tell the truth absolutely no idea what it is. It's a completely random function that with some sort of complicated shape and around this we have some sort of minimum and we would like to model at least the vibrations of a few degrees around this minimum. And since you already listened to the bond slide I hope that you all agree that a reasonably good way of doing that is that we might have a potential that has some sort of force constant multiplied by a different say in the angle squared. We might use a cosine function or something there too it's not crucial but it's the first order approximation of how the energy is varying around a local minimum. It works great for angles and now we can describe both bonds and angles. So we started with an interaction involving two atoms then we had an interaction involving three atoms you can probably guess what's coming next an interaction involving four atoms. So we picked these same four central atoms what if we look at some sort of relative motion that includes all four of them. This is getting a bit trickier to visualize I'm going to try it anyway and I can use my pens here so here's one bond, here's the second bond and here's the third bond so the atoms would be at the end of the pointers here. So if you look at this I can turn this around so I'm not changing any angle here the angle around this one do you see the angle there is fixed and if I instead do the same motion here but if I follow it there do you see that the angle between that, that and that atom that doesn't really change either. So what is changing here if you're looking along the middle bond what is changing is a rotation along the middle bond. There are many names we use for this actually no not many but there are two we occasionally call this a torsion angle and we occasionally call it a dihedral angle and sorry they are used interchangeably in the field. The way we define this is by each triplet of atoms here i, j, k for the first three atoms and then the last three atoms that would be j, k and l each such triplet defines a plane and then we can take the blue plane here and the red plane and there is then an angle between those two planes right that angle is the torsion angle what gets a bit complicated here now is that we're going to need to find a way to describe this and this gets complicated in two ways first if you look at the red and blue planes here there are two ways to describe this do you pick the in this case the small angle or the large angle between them because that's going to be basically in this case one angle phi here is going to be 180 minus the other angle so it matters which one you're choosing the great thing is that we're scientists so that we like to have a standard to define things and in this case we have chosen to define two standards there is a convention that you sometimes call the polymer convention and there is a convention that you call the biochemistry convention what I tell you with this conventions are you're going to skim through that so I'm not going to tell you that's a homework task now for lecture two go out in Wikipedia, look at torsion angles and check what is the biochemistry convention biochemistry and what is the polymer convention you will typically not have to worry because we're normally going to stick to biochemistry but if you start defining and calculating these angles yourself you're going to need to know which one you're picking because if you pick the wrong one things will be pretty garbled and now you might think that you know exactly how you're going to treat this interaction this is a sort of complex interaction that we would like to describe around local equilibrium and we don't know exactly how they work and I think you're going to get this wrong because you would pick the simplest potential is proportional to a force constant multiplied by the square of the displacement of the angle first second order approximation the problem with that is if I take this torsion angle here if I start out here and then I wrote this an entire turn 360 degrees 2 pi well the deviation is now 2 pi but I'm back in exactly the same state as I started and in particular for small molecules small molecules, these barriers are low enough that small molecules can occasionally rotate an entire turn here so for torsion it's not going to work to simply have this very simple second order harmonic description of it so we're going to need something else we're going to need a potential that is periodic and I bet you know about periodic potentials so let's move on to the next slide and see what they look like periodic potentials here is a small animation of a molecule that is very much periodic this small aliphatic hydrocarbon will keep going round and round and round it's a butate and that can actually happen there will certainly be an energy barrier roughly there when the two molecules are superimposed right on top of each other it's going to be better when it's stretched out but it can occur in both those states you actually have two more torsions here because carbon atom 1 and 2 here if that bond rotates that's going to correspond to the three hydrogens rotating and that's an even lower barrier and of course here too the exact form of this potential can be complicated if we calculate it from quantum chemistry but if you want something that is periodic on the unit circle I hope that you would all say trigonometric functions right and that's what we're going to use so the simplest possible trigonometric function would essentially be a sine or cosine in this case I like to have the lowest energy value to be zero rather than having to worry about both positive and negative values so we're just adding a constant here to lift it up so that the baseline here is zero depending on this rotation then I'm going to have an energy here that starts out at a high value then it goes to a low value and then it becomes high again and then it's low again and high again this is an even simpler molecule it's actually an ethane so you just have one bond you're rotating and then we have three hydrogen atoms here and three hydrogen atoms here the reason why that is important is that it's back in a state that's exactly identical to your starting state I'm not sure if you can see the XY axis here maybe not but as I already hinted the reason why this is important is that the energy levels involved here are much much much lower than distortions in bond or torsions so here we might be talking about a handful of kilojoules or k-cals and that's low enough that it's going to happen normally at room temperature which is of course also the reason that we'll realize that we were back in an equivalent periodic state but I started with butane so I need to show you what this would look like in butane so if I remove that one and move to a slightly more realistic potential there are different names trans, cis and gosch that I might not have time to go through here but if you look at that butane molecule again by far the best state here is the one we had straight in the middle here and that would be when the four hydrogen atoms are placed like that sorry for carbon atoms are placed like that the lowest energy by far the worst state we're going to have is if they are placed like that and that's going to correspond to the peak here around zero degrees and that is a hint if you're going to look up biochemistry versus volumetric dimensions now these two other states correspond to things where things are not quite as bad as the cis states so things have rotated roughly a third of a turn away from that one so that means that things are not directly superimposed but there's not quite clashes because it's not quite as good as the fully extended states those are called gosch the history behind that is not important but you see that you have two local minima here the local minima are definitely better than the peaks but they're not as good as the best possible minima between these two called gosch states and the fully trans state you have additional barriers those barriers are not good, they're high but they're not as high as the barrier in the cis state and now things are starting to get pretty complicated for something as simple as butane that I have a very good low state and I have a pretty good low state I have a very bad barrier and I have a somewhat bad barrier the way we represent this is pretty much by using two two cosine functions but they have slightly different periods one that corresponds to a period with 360 degrees and one that corresponds to a third of that you're going to need to have a rough idea roughly what these energies are it's going to turn out that this is super important for proteins and the torsional degrees of freedoms are the ones that are going to describe the entire shape of the chain because once you start rotating around the bond if you have a long chain of connected molecules well, in butane not much happens but if you had 500 other atoms bound at the end of that chain when you rotate this that would mean that the entire rest of the molecule would rotate with it so when we're rotating around bonds in a long chain that tends to change the global confirmation of the entire molecule and now I'm almost getting ahead of myself and talking about protein folding because this is going to be important but that's for later so we're not going to look at an entire protein that's going to use you to the concept of what the torsion bonds mean and why they are important so we want to pick something that is as simple as possible but still has kind of some properties that are protein like and scientists have a toy molecule that we've used for years to study something one of the simplest possible amino acids is alanine we're going to look more in that next week and if you want to take something that is just a tad more complicated than that let's pick two alanines actually this is not even two alanines so I've picked one alanine amino acid in the middle here and then I'm just essentially picking the parts without the side chains so this is a molecule that has in principle four bonds along this long chain here that we could rotate around but I'm going to argue that we do not have four bonds do you remember that I spoke about the amino acids yesterday or last lecture that we had the C alpha and then we had the nitrogen carbon oxygen here then we had a second nitrogen and then we had a second C alpha here and then we had a second carbon there you also had those R groups and then I'm not drawing any hydrogens at all to save a little bit of time this part was this peptide bond that I spoke about in lecture one how it formed and by two amino acids merging together and I already mentioned that this is going to be a very bond with a resonance of electrons all the way from the oxygen to the carbon to the nitrogen to the hydrogen there so there's going to be a net shift that all the electrons have moved down a bit here towards the oxygen and that's effectively going to give this central bond a property as if it was a completely rigid bond this bond will never rotate if we look up here on the right it turns out that two of those bonds involved in this molecules that's going to correspond exactly of this carbon that is not the alpha carbon to nitrogen so we can kind of scratch those out let's forget about both those T-tests for now that means that we just have those other two bonds that we call phi and psi they are by far the most important bonds rotations in amino acids and you're going to need to remember them and you're going to need to know which one they are they're called the torsional angles or the ramachandran angles I'll get back to ramachandran angles in a second and we would somehow like to describe how this molecule is changing as we're moving those so if I start rotating around those two bonds the very first and the very last part of the molecule here is going to rotate and under some conditions the atoms might be clashing a bit so it's bad contact and in other combinations of these two bonds we're likely going to have very good contacts so essentially I have a this is an equation that I'm going to measure potential energy that is a function of two variables and these variables would be the angle phi and the angle psi for now we're not going to worry about the bonds and the torsions and everything I would just assume that those will take the best possible positions and you could of course draw that in MATLAB and a very simple representation would just be a two-dimensional diagram here and in this case blue means good low energy and this is bad high energy so it turns out that you have at least two areas here that are quite good where the molecule is likely to spend time and at least one area because this diagram is periodic so if you go out on the top you're going to re-enter on the bottom here remember the angles are periodic the red area here is going to be a bad part where things are clashing and you do not want to spend time there and it turns out that this was calculated from a simple molecular simulation because if we now take this blue area up here one local minimum and the other local minimum on the other corner there we know what the phi and the psi angle is at the center of each of those minimum and then we can take those conformations and draw them and those are actually the names that we say here in white so we have in the case even of something as simple as the alanine dipeptide it's a single amino acid it just happened to have two of these two of the peptide bonds so that we have both a phi and a psi angle even something as simple as that ends up having two local minima where the structure is fairly happy and you're going to need to trust me for now when I say that you would actually see both of these as room temperature which is a bit strange because one of them will have lower energy than the other so why do we see both? we'll talk more about that next week so I've gone through most interactions now we covered bonds we covered angles we covered torsions and we also covered these ramachandran diagrams that was the two dimensional landscape that showed roughly how the energy would vary as the phi and psi angle chains and that's also why we call those ramachandran torsions occasionally I already talked when I spoke about interactions I already mentioned another type of interaction electrostatics and these so-called van der Waals interactions and that's pretty much the rest take a molecule and just dive in no matter where we are in space you're going to have a ton of atoms either in the protein or in the water or around the protein basically we would never ever have vacuum at room temperature and normal conditions in a test tube there is always an atom right next to you interacting with something and that gets complicated because they're not necessarily a water molecule it's not bound to another water molecule or the protein so we're going to need some sort of way of describing this there are two parts to this I already covered them a little bit one of them has to do with steric interactions that atoms cannot overlap Pauli exclusion principle the other part that is related to that actually it's not really the same process that's these induced dipole-dipole interactions I spoke about the reason why we say that they are related is that if we for a second pretend that all these atoms didn't have charge even if they don't have charge they would still interact so that they can't overlap so you would still have the repulsion and even if they don't have charge such as noble gases they would still have these dipole-dipole interactions at very long distances so there is one repulsive component when they get very close and there is one attractive component at very large distances the second type of normal interaction has to do with electrostatics and that you know since your undergraduate studies a positively charged atom ion will attract a negatively charged ion so in one way the electrostatics is very simple the problem with this if we are now going to start modeling this is that every single if I pick one atom here in the middle that atom might have two or three bonds it might have three or four angles and it might have five torsions but if you pick the atom in the middle how many atoms, other atoms is it interacting with in terms of electrostatics and normal interactions? Hundreds, maybe even thousands so it gets computationally very complicated and expensive to handle this which is a bit difficult let's have a look at how these interactions look like and how strong they are relative to each other so I spoke about normal interactions in terms of electrostatics and van der Waals interactions you already heard in a previous slide that the van der Waals interactions are quite weak, remember they don't condense until very low temperature so can't we just ignore those? well, if I draw lots of atoms here here's one atom and another one fourth, fifth, etc. I'll let you imagine the remaining 5000 ones this one might have a plus sign minus sign, plus, plus, minus minus if you now imagine having 500 atoms here sure, these interactions are strong but most things if I look around me in this room there are pretty much, there are no free charges even inside a battery the positive and negative charges are paired up as ions in the electrolytes so on average, if I look at a few nanometers around an atom, on average there's pretty much exactly the same amount of positive and negative charge so while each individual interaction here is very strong they tend to cancel each other the problem is that they don't cancel each other exactly so there might be hundreds of k-cals per interaction but plus to minus, that's attractive minus sign in potential plus to plus, that's repulsive bad, that's a plus sign in potential so when these cancel out it's going to be very noisy based on the exact positions of all these atoms and at the end of the day that's going to tell us whether it's a good or bad conformation remember, good low energy, bad high energy but if we imagine exactly the same molecules but forget about those charges, so we just look at the repulsion and dispersion part of this again, if we push these atoms very close together they are not going to want to overlap that's exactly the same between all atoms and if you push them close enough imagine a nuclear device or something eventually that repulsion is going to be even stronger than the electrostatics even if they have different charge signs but at very long distance well, I already said that is weak so come on, why can't we just start ignoring that if you look at that one atom and look at 5000 neighbors what's the sign of the interaction with that atom it's attractive that one is also attractive that one is attractive that one is attractive so the attraction or dispersion at long distance has the peculiar effect that all the signs are negative they all attract each other so eventually if I just well, when I add up enough of these the energy will always be negative and that's why you get these effects that if you just reduce the temperature far enough so that you overcome the thermal energy we're going to come back to that later on in the class eventually that small sign will start to dominate and that's why even noble gases condense at some point so we can't ignore this it's much weaker than the electrostatics but where the electrostatics fluctuate with different signs the attraction will always have the same sign and that's why we need to consider both of them so aren't with the knowledge now about the typical interactions we have in atoms and having a little bit of grasp about how atoms move we're now going to revisit the hydrogen bond it's not going to be the last time we revisit the hydrogen bond remember that simple atom oxygen and hydrogen water the miracle of life well let's be proper instead of doing those nasty partial charges let's assume we had an infinite amount of computing power and that we were to calculate the exact electron density around this if we were to do that you're going to need to trust me that we would have higher electron density around the oxygen red here and low electron density around the hydrogens this could only be described with a three-dimensional density function describing where it's high probability of finding electrons and whether it's low probability of finding them and then we'd have to visualize that in some part but since I need something simple what I'm going to take any the electron density that is closer to the oxygen I would just assume that that is placed on the oxygen and that's when we can do the things that we pretend that we have minus say 0.82 is one common water model sorry and then plus 0.41 I almost hit that minus 0.82 that's what I call a partial charge it's really a pure invention I typically need the quantum chemistry to calculate this but this is now a much simpler model where I can apply ball and stick and just use traditional Coulomb interactions for something as simple as water I could in theory parameterize this based on an experimental results and that typically how we do it for water models but if you have something that's part of a drug molecule or something the only way to do this is put this into a computer and run a quantum chemistry program to try to find what is roughly the partial charges for this particular molecule in isolation again this is an approximation there is no question about it but by doing this approximation we're going to be able to handle all these other aspects that I just took you through we're going to be able to handle flexibility in this molecule every torsion here can move we're going to be able to handle what happens when it's interacting with other molecules and what happens if we put it in water so that we pay some but we gain tons of other things and that's roughly increasingly going to be heading we're not going to spend the majority of this class doing computer simulations but this type of sequential modeling is really going to help us understand what happens inside molecules so we already introduced hydrogen bonds a little bit if you just let me erase things here we can start to think what's happening inside a protein all these different interactions we would have in a protein both the electrostatics and fundamentals we have some very strong interactions we have bonds, angles we have some intermediate definition strength and torsions and then we had some very weak ones such as these van der Waals interactions the only way to properly handle these weak non-bonded interactions then I would need to describe this as this 1 over r6 attraction at very long distances, dipole-dipole interactions and then an exponential repulsion at very short distances this is a potential form called the Buckingham potential and this would actually work pretty darn well and this used sometimes in computer simulations the only problem here is that calculating the exponential function can take hundreds of clock cycles on a computer today you can probably do it fast on a graphics card but these models were originally developed in the 60s and 70s and then they most definitely didn't have GPUs is the form there important? this Virginia, that form is super important if you're going to be employed at a national lab and designing nuclear bombs but in a biophysics lab well I haven't checked lately but it's very rare that we have nuclear devices going off in our physics lab biophysics lab that is so I don't really want to I don't worry what's happening if two atoms are literally overlapping each other I just want something very simple and I want something that goes up quickly as I approach unity this is one function here that is literally the dispersion that is 1 over r to the power of 6 I would now like something that and that has a minus sign again it's attractive it's good I would now like some sort of other function with a plus sign but that increases even quicker but if I now take this 1 over r6 and that is stored in a variable if I just square that over r12 and you can do the math here so that the 1 over r12 component here that will increase even faster than 1 over r6 that is just one multiplication which literally takes one clock cycle on any computer today and now I have a plus sign on that one so what that's going to get me is I now have this simplified interaction form where at very short distances this term is going to dominate it's not the exact potential I have down there and at very large distances this term is going to dominate and that is in practice what we all use in computer simulations how good is this approximation well it's not just good it's very very good it's not quite as good as Buckingham but I have to confess that the Buckingham approach too isn't exact either remember when I showed you that atom being surrounded by all other atoms this is really a collective effect particularly the induced I pull measures so that you would actually not just have to calculate pairwise interactions but triple interaction quadruplet interactions pentuplet interactions etc and that's simply completely unrealistic so no matter what functional form we have here if we try to parameterize that from quantum chemistry it's not going to work well but if I have anyway accepted that I'm going to have a simplified form I can now take these parameters and just put one factor on the repulsion and one on dispersion and I try to parameterize this to fit for instance the density and heat of vaporization of small molecules and that means I now get parameters that they're not strictly exact in terms of quantum chemistry but they're very good at describing how atoms either attract each other at long distances or repel each other at least at immediately short distances at some point it will start deviating from the exponential but then we're no longer doing biophysics I don't really care this functional form is called the Lennard-Jones interactions it's actually not a pair of scientists it was called John Lennard-Jones and you will likely see that occasionally we call these van der Waals interactions or we abbreviate VDW personally I would say that van der Waals interactions are all the types of non-monetary interactions including the Buckingham one I just showed you but again there is a bit of nomenclature mismatch there the important thing is that you know what you're talking about these are extremely weak we might be talking about 0.1 k-cal per mole remember that electrostatics and bonds in particular but even electrostatics that tend to be around the same atoms an electrostatic interaction between two atoms that have unit charts and are separated by a few angstrom might be a few hundred k-cals so thousands of times stronger than electrostatics the reason why this was still important was this effect that these all have the same sign while the electrostatics will cancel each other under many conditions so let's look at what the functional form looks like basically if I am going to sum the interactions of all the molecule I would have a double sum over all the pairs of atoms and I would have one term that corresponds to the repulsion and one term that corresponds to the dispersion and once we've done that we can actually translate these parameters and see what types of energies we are talking about in normal molecules you don't need to know that by heart but if one starts doing molecular modeling it can be kind of fun too because roughly knowing these numbers give you a gut feeling for how strongly atoms are interacting so in principle now we have everything to start to calculate energies admittedly in a simplified way but understanding how atoms are interacting in fairly large molecules where bonds can rotate and move let's have a look at that if we bring everything together so if we are to bring all these interactions together this is another beautiful illustration that I got courtesy of Mike Levitt many years ago don't worry too much about the exact shapes in here because again different programs, different scientists tend to use slightly different standards for this but if I want to describe something in a molecule and describe the complete potential there will have to be one term that corresponds to all these harmonic bond vibrations there will have to be one term that corresponds to all the harmonic angle vibrations and those are visualized up there then we are going to need one term describing all these torsion potentials and that's the one in the middle here I can't stress enough how important torsions are and the reason why torsions are important is that they are intermediate you won't know that yet but it's going to turn out that energy variations that are very small in the road we run over them it vibrates a bit but it's not really going to change where we get and energy barriers that are very high they're going to be like brick walls you're not going to get through them so the interesting aspect are the middle the average energy barriers and that's exactly the torsions and again they're illustrated by these rotations around the central bond then you have these Lenard-Jones interactions again this is formulated in a slightly different way where we have a unit of energy and then an equilibrium length there of the bond it's a small mathematical exercise to show that this corresponds to those C6 and C12 numbers I had on the last slide and that would correspond to this repulsion and dispersion part and finally we have an electrostatic energy where this has to do with units whether you're 104 pi epsilon 0 or whether you're using CGS units so forget about the 332 factor there those are the electrostatics hundreds of k-cals this might be 0.1 k-cal torsions a few k-cals and these are hundreds of k-cals too and we're never going to be able to break them so in principle that's everything you need to do to describe what the energy is in one confirmation of a macromolecule but here's the thing the interesting part is we want to compare different confirmations what happens when they move and now things are going to start to be quite different in chemistry because this is much more difficult for quantum chemistry to do than for us so imagine that we take all these things together and sum them up we're going to come back later to how you let a computer do this but if I have a potential if you've studied your undergraduate physics you know that the negative derivative of the potential that's going to correspond to the force on a system again the potential of lifting a weight is the mass times the gravitation factor the height and the force down is the derivative with respect to the height minus the derivative if we know the force on molecules we use Newton's first law if I know the force I can calculate the acceleration force equals mass times acceleration and the acceleration describes how the velocity is changing so if I started from some velocities for instance 0 I can then calculate how the velocities would be a very little while later then I know what the velocity is and the velocity is how the position is changing as a function of time so that means that I can calculate how the positions are varying so this enables me to take small steps and really simulate in a computer how a molecule would move as a function of time the problem is that small is something that should be taken literally you're going to need to do this in orders of femto seconds so this is just one more molecule from the simulation I previously showed you a couple of times moving and I've hidden all his neighbors in this case but the advantage is that computers can do 10th of thousands of these steps per seconds today and we have very large supercomputers that can treat molecules with millions of atoms so this has suddenly become a very useful biochemical tool whereas 20-30 years ago it was something that only theoretical physicists used for very simple systems what can we do with that well let's see if we can revisit some things such as how large molecules move or first maybe the hydrogen bond last lecture I spoke a little bit about hydrogen bonds in terms of partial charges you have the oxygen where you have the negative partial charge because it has stolen some electrons from the hydrogens we talked about orbitals before and the reason why this happens we can describe this with orbitals too all of this is due to the electron density around the atoms but if I were to draw this with orbitals what has actually happened is that the orbitals around the oxygen here they're in tetradural shape so that you kind of have one electron cloud pointing up here and another one pointing up there is ears and then there are two of them as legs what has then happened in a normal water molecule that for two of these orbitals those electrons have paired up with the hydrogens and that's why we have these larger rabbit ears here so for two of those electron pairs there are literally formed pairs and they're formed stable bonds now a water is not reactive so we don't have unpaired electrons here but these pairs are so-called lone pairs they're complete orbitals so it's not going to be super reactive that it's going to start binding to other things but you have a negative partial charge here due to this orbital and another negative partial charge due to this orbital so that means that the entire oxygen here is going to be more negative while the hydrogens up here are more positive and as I mentioned last lecture this is a very strong effect if we then take two of these atoms and bring them close together you're going to get this effect that this water down here is going to have a hydrogen here that is slightly positive and that's going to love to interact last time I said that it was interacting with the oxygen but it's not really interacting with the oxygen it's interacting with those lone electron pairs we're going to need some sort of nomenclature to describe this because we're basically this is a very generous water this water is basically taking his proton the hydrogen there and letting the other water molecule borrow his proton a bit to team up and become a partner to his very lonely electrons there so we call this a donor a hydrogen bond donor and this would be a hydrogen bond acceptor and that is the hydrogen bond there's a you pack standard that we're supposed to draw hydrogen bonds with three dots sorry you pack if you're listening this is a very old illustration that I borrowed this is a strong interaction it's not as strong as a typical bond it's not as strong as a typical full blown electrostatic interaction because remember we have a net zero charge here we have a net zero charge here I'm going to need to tell you what the interact it would be a bit absurd to introduce hydrogen bonds without telling you roughly how strong this is in the case of water you're talking about a few kcals maybe 5kcals around 20 kJ per mole do you see that I made this mistake that I told you about a few times I just said 5kcals bad Eric 5kcals per mole but you're used by physicists now so when I said 5kcals hopefully you assume that it was 5kcals per mole so how do you determine which one is a donor and acceptor now this gets a little bit more complicated because what if you take that water atom assuming that there might be a third water molecule here with two hydrogens this one is also interacting so this hydrogen would now form a hydrogen bond to that oxygen but in this case this is the acceptor and that is the donor so the acceptor and donor is not specific to the molecule in fact as you're going to see later it's very common to have hydrogen bonds inside a single large molecule such as a protein the donor and acceptor criterion is around the specific hydrogen bond so in this hydrogen bond that oxygen is the donor notice it's the oxygen that's the donor not the hydrogen the oxygen donates its hydrogen or proton to this other acceptor that has the lone pair electrons but this in the sense of this particular water molecule that's also participating in this hydrogen bond in this hydrogen bond the same oxygen here acts as a donor and donates this proton to this oxygen as an acceptor so you need to understand the hydrogen bonds you need to understand the donor and acceptor and I should be able to wake you up at 3am in the morning and you need to know what the energy of a hydrogen bond is seriously it's one of the most common questions I ask people in the exam if you flabbergasted that people haven't learned that by heart do yourself a favor look that up and train on that every day there is a reason why it's so important and there is a reason why it's important enough that you need to know it by heart so as we've defined the hydrogen bonds and the donor and acceptor nomenclature we can look a little bit at hydrogen bonds in real molecules I already mentioned water right and in that movie that I showed you I mentioned that water a perfect ice crystal that have exactly two full hydrogen bonds per water that is not as simple as you might think because each hydrogen is participating in one hydrogen bond but the oxygen also has two electron pairs so each water molecule is participating in four hydrogen bonds but it's donor for two of them and it's acceptor for two of them so it's kind of it's participating forming four half hydrogen bonds so the total number of hydrogen bonds is going to be two per water under the ideal scenario now once we start cranking up the temperature and going to say room temperature so when we have liquid water what's going to happen is that the number of hydrogen bonds is going to go just so slightly above two but as long as we're ice it's not going to increase a whole lot and then when we move over to the liquid phase there will be a jump because the atoms will now start to diffuse and move relative to each other but it turns out that almost all the hydrogen bonds are still intact liquid water has an average of 1.7 hydrogen bonds so if we do H2O aqua phase is 1.7 H bonds and if instead they do I'll say X-ray for a crystal that would be roughly 2 H bonds not quite if I were to boil the water then if I would of course go all the way to gas phase I would have zero hydrogen bonds that's not quite true but there will be some of them formed transiently but in principle in gas phase things are so far away from each other that they do not interact we already talked about DNA remember that I mentioned that inside the DNA the recent DNA paired up the way they did because we had this basis purins and purimidines A, G, C and T and these form specific hydrogen bond patterns from a purine to a purimidine and either two hydrogen bonds or three hydrogen bonds under the normal Watson-Crick base pairing that's what gives DNA its specificity if the hydrogen bond was very weak and easier to break the DNA would not maintain its structural integrity and there are a ton of hydrogen bonds in this DNA spiral and that's of course why it's stable and it doesn't deteriorate which we should be fairly happy about because if it deteriorated quickly we would form tumors and have errors in our genetic code but it's not just DNA it's going to turn out that almost all the molecules we work with in this class have hydrogen bonds we're going to talk about protein structure on the very far there you have an alpha helix and in the alpha helix we have tons of hydrogen bonds formed along a staircase it's going to involve those peptide bonds I showed you before the other panel there shows so-called beta sheets we'll introduce the next week and the beta sheets also have a ton of hydrogen bonds so there are two aspects that we can learn already from this lecture remember how I said how the torsions are important because the torsions determine the rotational degrees of freedom that is they determine how a molecule can move on the other hand the strongest interactions that actually form or break are the hydrogen bonds note that form or break is an important modifier there because if I literally tore bonds apart those interactions would be much stronger but those interactions are so stronger that they are brick walls and I won't push my head through a brick wall sorry not even for a lecture but the hydrogen bonds are so strong that they're important to form interactions but they are weak enough that you can actually break them under some conditions for instance if I'm changing temperature or conditions and that's why this is going to be such a miracle that we're going to look at when do hydrogen bonds form can that explain when things are stable or not and the part that allows the molecule to reach those different states will be the torsional degrees of freedom so again you need to know what are the important degrees of freedom in most biomolecules and what are the most important interactions to stabilize degrees of freedoms hydrogen bonds so now we're almost done now we just need to wrap things up and understand how hydrogen bonds describe protein structure and then we can finish this entire class after two lectures I'll deliberately run a little bit ahead of ourselves and imagine if we have a protein and we talked about proteins having these long sequences of amino acids that I described in the first lecture there is some sort of sequence for now I'm going to forget about what it is these green part is the backbone and that's as I already drawn in a couple of cases that is this nitrogen C alpha carbon nitrogen C alpha C nitrogen C alpha C that's the backbone or main chain of the protein structure and then connected to each alpha carbon C alpha we have a side chain which are called R those side chains as we will see later are going to have slightly different properties but some of them are going to like water so called hydrophilic ones other ones are not going to like water hydrophobic or fearing water what I've drawn here is a highly schematic part where the yellow part here corresponds to some sort of side chains that would be hydrophobic and I deliberately I think it might even be Finkelstein's illustration that we've deliberately drawn some water right next to this now waters are not going to like to be right next to something that's hydrophobic why we will have to wait until next lecture or two to show but you can probably guess that already on the previous small snippets here right those water molecules they want to participate in the hydrogen bonds and if you now have something that's a pure carbon here there are no lone pairs there are no hydrogens that molecule is not going to be able to participate in the hydrogen bonds so the waters here right next to the hydrophobic molecules will not be able to participate so what might happen if we now take this protein and throw it inside a cell in a split second what if this protein somehow curls up so it can take these yellow parts and put them together so that the hydrophobic or oil like parts will now be next to each other we already know that's what happens if you throw oil in water right and then all the waters will somehow be out here free to interact with each other in principle that's not entirely wrong that's partly how things actually do work and there are a couple of things here that we're going to need to understand and that's why there will be a few more lectures before we can do this for real first we're going to need to understand what are the different amino acids we have involved in a typical protein and why do they have different properties and what are those properties that's going to come up in two lectures I think we're also going to need to understand what is this concept of state I talk here about some sort of unfolded state and some sort of folded state so first what is a state is that specific XYZ coordinates or something or is it some sort of larger things well to be honest we haven't even defined what a state is this is just random drawings we're going to need to understand a little bit what actually happens with these waters and hydrogen bonds under different conditions when will waters form hydrogen bonds I've already hinted to you actually even explained twice that the reason why water has its properties and in particular such a high boiling point is that those hydrogen those water molecules will do almost anything it takes to maintain their hydrogen bonds so it's not going to be a CCS breaking hydrogen bonds and then you can go to another state so we're going to need to understand the torsional degrees of freedom how these chains will rotate we're going to need to understand what that means this moves over to another state do we get more or less hydrogen bonds and then things might get really complicated because you might have a water here that forms a hydrogen bond with a protein but in this case the same water might be forming a hydrogen bond with another water molecule so all we've done in some cases will be that we have just moved around the hydrogen bond so we still have the same number of hydrogen bonds but they're suddenly formed with different molecules and that is also something that we're going to need to start covering with physics and it turns out that a unifying concept here that's going to come back this has to do with arrangements different ways of arranging molecules and then trying to decide is this arrangement good or bad and this far I've just glossed over that I've just told you and you just believed me when I said a negative energy is good a positive one is bad we're going to need to derive that a bit and you've probably seen this in physics I think you've derided which is quite fun because it's some of the most basic concepts in physics if we're going to do this proper I would be throwing a ton of equations at you caveat I am going to throw a ton of equations at you but if I do that tomorrow I would only have one fifth of the class remaining for lecture 4 which would be a bit of a bummer because I kind of like these things and I won't introduce you to it so we're going to follow Finkelstein here and I'm first going to introduce this with a bit of hand waving in the third lecture without going to too much gory details about physics and then later on when we've had a chance to go back to biology we're going to show this in a more universal way where we don't make as many assumptions but this will arm you with the arms you need to understand models in very generic systems we're going to be able to start drawing conclusions about what processes happen when will Francis a protein fold when will it not fold you will be able to explain the hydrophobic effects and phase transitions and a bunch of things that are borderline pure statistical physics but they are super important and I would argue that long term it's probably the things in my education that I've had most use of the most complicated equations are not necessarily going to be the one that look most difficult the hardest equations are frequently the easiest ones but that is probably all we're going to say about the different conformations today is the final concept that I want to leave you with assuming that for each of these conformations I can calculate what the energy is whether that involves hydrogen bonds and everything I might have a gigantic computer we could do this for the allon 9 dipeptide remember just 2 degrees of freedom I change the phi and the psi torsions the so called ramachandran torsions and then I plot it in a ramachandran diagram and I get the energy as a function of those two torsions in practice when it's the allon 9 dipeptide is such a common molecule we all know what these angles are so we tend to draw that in 2 dimensions but of course we could draw this in 3 dimensions this is also another study on the allon 9 dipeptide and here too red is bad, blue is good it contains pretty much the same information apart from the fact that it's a slightly different study but most of the molecules I've showed you contain way more than 2 degrees of freedom many of the computer simulations we do contain a few millions of degrees of freedom and I'm not sure about you but I find it somewhat difficult to imagine 1 million dimensional spaces so we're going to need to simplify this some way and what we typically do is that we think of some sort of rugged landscape we say that it's high dimensional but it's not really high dimensional this is still just a 2 dimensional landscape it's just that I have lots of minima and maxima here and the reason for those lots of minima and maxima imagine my 1 million atoms there are going to be lots of places where they are very happy and interact closely all the blue parts here and there are likely going to be lots of places where they bump into each other and are not so happy and that would be the green parts of this particular energy landscape and somehow the only thing I have to decide to determine where a protein is is what is the best point in this energy landscape I think or is it that simple because now we also if there was one molecule you could imagine that it's that simple but assuming that this is water and we might have avogados number water molecules in a glass every single water molecule can't be at the same time at once so in particular it might very well be that a particular bond is really good to form but if we would take a very large molecule and stick that so it can't move at all it has to be in the very lowest position here that might be bad for other reasons and for now that will just have to be hand-waving this corresponds closely to a concept that you have touched before entropy you think entropy is going to be difficult if there's one thing I promise you is that after this class you will hopefully not think that entropy is difficult but we're going to need to find tools that describe what do we mean about the distribution in this energy landscape what do we mean by moving in the energy landscape in principle it's bad to be at the peaks here but sometimes you might have to move across a peak to get from a low but not really low as well and to find the really lowest well here in the middle and right now I can't say when will that happen when will it not happen I just hand-waved and claimed to you that well when we have intermediate energies that's good enough but intermediate energy has to be intermediate relative to something else so there has to be an energy scale that we're not aware of yet about for molecular interactions today as always I have prepared a number of study questions for you I will not cover those in detail I won't read them for you but read them here they're also present on the Canvas site for those of you at KTH we organize a seminar when we sit and discuss this before next lecture and some of this might be easier than others there might be a few ones that I haven't covered in detail here we talk about enantiomers in a protein if there is something that you feel is not complete not at all related to what I'm talking about don't worry skip that one it was from the time when I had more information about amino acids in this particular lecture but I will come back to the amino acids next week instead but most of these you should be able to answer with the current knowledge you have if you haven't yet done so start reading chapters 3 and 4 in Finkelstein Finkelstein doesn't go into a whole lot of detail about amino acids either and that's why I've changed this up a little bit there are two very cool papers that I think it could be worth reading on Canvas first one is David Eisenberg that talks a little bit about the way we discovered proteins and secondary structures again that's a bit of wetting the appetite before next lectures and I will also I was able to find the Cyrus Leventhal article in Scientific American from the 1960s when they described this molecular model building by computer it's quite fun those were the days where Scientific American published truly groundbreaking scientific papers not necessarily the type of popular science we think about today not that that's bad with that I'll thank you and see you next lecture