 Good to see you all again. I did a little experiment with two screen recordings on my iPad, where I eventually got the sound working this morning. So they are on the YouTube channel, and there are links from them. But they're mostly going to be the stuff we cover today. When it comes to lecture three, rather than recapping everything there, actually, there are two things I'm going to recap. I'll come back to the alpha helices and beta sheets later today, so I'm not skipping those slides. But if there is one important take home message when it comes to understanding protein folding, you will frequently see this in delta G of, why did we study delta G or delta F? Why is that so important? Based on what you learned yesterday. Zero point on the M. Yeah, sure. Yeah, that's the point of the delta. But the reason why F and G are so important, they describe what will happen and why it will happen. Just studying energy does not get us anywhere. But free energy actually will get us somewhere because it will tell us. So for instance, the reason you can walk on coal when it's burning is that the heat is high, but the energy is not that high. It's simply difficult to interpret things in terms of free energy. Free energy on the other end will always tell us, will a reaction happen or want it? And in particular, when it comes to protein folding, I covered hydrogen bonds and the hydrophobic effect a little bit yesterday. And 90% of protein folding initially is hydrophobic effect. You will have this instant collapse turn all the hydrophobic things to the inside of a protein, keep the water soluble hydrophilic things on the outside. And then somewhere after that, we're going to need to do some sort of packing or polishing, whatever we call it. The devil is in the detail here. It's not entirely easy to get all the side-chain packing right, but to an uneducated eye, that's just going to look like some minor stuff happening at the end. Hydrogen bonds were caused by electrostatics, but it's not the only important electrostatic interaction in protein. In some case, we actually have free charges in protein, not as common as you might think. And you might have studied this already, but so in general, the electrostatic energy of a protein would be a, sorry, not the protein. The electrostatic interaction between any charges would be proportional to the product of the charges. And if we are SI units, there should be 1 over 4 pi epsilon 0 to, right, but then divided by the distance between them. But there are two things that are important. 1 over R here means that they're very long range compared to Lenard-Jones interactions. And the epsilon here is going to have a fairly strong screening effect. So if you look at charges in water where epsilon is in the ballpark of zero, water is amazing. That's because these waters that are small dipoles, they can rotate. And that will mean that they are very efficient at screening charges, so that the entire electrostatic effect will decay fairly rapidly water. The inside of a protein, well, epsilon might be in the ballpark of 3, 4 or so, much more like a pure hydrocarbon. And that effectively is that this screening is roughly a factor 20 to 30 to 40 lower. So that charges see each other at much longer distance inside proteins or any hydrocarbon, for that matter. And if you look at Finkelstein that, say, if you take two proteins and put it at the cost of salivating an ion or something, and you can calculate that this way, it's not an electrostatic, so I'm not going to do it. But an interaction in that a water would be in the ballpark of 1 or 2 kilocalories in electrostatics would maybe be 40 kilocalories in a protein. So that is good in water. It's very easy to have charge interactions in water. But the second you have to, assuming that you have two amino acids that actually are charged. Remember that I said that there are a bunch of charged amino acids. If you have two amino acids with the same charge that are not too far away from each other in a protein, that's actually going to be very bad, because they will see each other at a much longer distance that you would in water. On the other hand, if they have the opposite charge, it's going to be unusually good, because they, too, will attract each other at a very long distance. And that means that when these so-called charged residues, either positive on the left or negative on the right, they are important, and they're frequently very important, not just for folding a protein, but for their function. We call them titratable, and you might remember from upper secondary school, when it comes to acids and bases, you can titrate them and add or remove a proton based on the pH value, right? It's the same thing with these. Depending on the pH value in your surrounding, these can, the positively charged ones can become neutral, and the negatively charged ones can also become neutral, depending on whether you have very low or very high pH. And that's important, for instance, some of the iron channels we study, Lucy are going to talk about this in one day, I hope. Some of them are actually pH-skated. So if you change the pH, the channel will spontaneously open, based purely on electrostatic interactions. And with all this, it actually means that there are clear patterns where, say, polar or charged amino acids occur. Anything that is polar is water-liking, hydrophilic. They tend to occur in loop regions or turns where they are exposed to water, never in the membrane interior. They love to form hydrogen bonds. The charged ones occur in roughly the same places, but even more so in an active site where you're binding a drug or something, because there is usually, the reason to have something charged is usually that there is a very specific interaction. The other thing you might remember that I, a few days ago, I spoke about these helices, right? And when you have a helix, that corresponds to adding up lots of small dipoles so that the entire helix will look like one gigantic dipole. And that means that you can actually stabilize the positive end of the dipole here by having a negative charge to that end or the negative end of the dipole by having a positively charged here, called helix capping. So what could you use this for? Well, there's one thing that you all are using it for every second here. So this is a small channel called KV. And inside your bodies, this is part of what you call a voltage sensor. And that sits in the voltage gated ion channels that are responsible for all your nerve signals. So when my brain decides that I should move my pinky, there goes down a signal here in roughly a tenth of a second. And that's propagated inside these long nerve cells that are maybe half a meter long by a displacement in the sodium versus potassium currents. And when they get one of these small action impulse, it's a change of voltage across the membrane. That causes more channels to open. And then I push this action potential maybe one millimeter further in the nerve cell. And that means I had more channels opening and more channels opening and more channel. And this moves one millimeter at the, well, less than a millimeter, a micrometer at the time. But it goes in a tenth of a second all the way. So it's insanely fast. And these channels, by far the easiest way to make something sensitive to voltage is to add a charge. Because if you take a charge and put it in an electric field, there's going to be a force on the charge. And this is an example of a simulation done almost 10 years ago when David Shaw in New York, where they managed to simulate the so-called gating so that it's a molecular simulation in a computer where you might see some of the side tests that were charges. And then we put an electric field on this and then we're going to see what happens here. So when you add a thing, I think you're going to see this going down in this case because it's closing. Yes. So you see that is gradually, it's actually four charges. And I think you saw two or three of them go down there. So these charges are now on the inside of a protein. And this doesn't look like a holdup for the world. It's a quarter of a millisecond. It's an insanely long and expensive computer simulation. Probably about a factor of 1,000 longer than the ones you're going to run in this course. So the cool thing with this is that it actually enables us to flag individual atoms and see in molecular detail what happens. This is all based on physics that once I have applied a field, it's a lower free energy for this channel to be in the down state. And that's, of course, we would have guessed before but the cool thing by being able to simulate this in a computer, we're kind of proving it, right? That it's only based on physics. There's no magical biology in it. What I'm going to do today is that I'm going to head back. I'm actually going to repeat the duration of the Boltzmann distribution but now I'm going to do this in the general case. And trust me, this is not as complicated as it looks like. Then I'm going to speak a little bit about the partition function which is way more beautiful. And then I'm going to talk a little bit about phase transitions, free energies and also helices and sheets. We'll see how far we get. And the point in that, does this is now more proper statistical mechanics maybe rather than the hand waving statistical physics I used yesterday? The good thing, this is not really difficult. I'm not, the most advanced mathematical concepts I'm going to use the next five, 10 minutes is the exponential function that I hope you've seen before and then a Taylor expansion. That's it. The part that makes this complicated though is that I can't assume anything, right? Because the second I assume something is not general anymore. So the hard part here is juggling with this definition arbitrarily introducing a definition and then learning to work with it. But the math, you could have done this math that up a secondary school. So if I want to know something about a system we're going to somehow need to introduce a system. And a system, no man is an island and neither is a system. If I want to study something small, let's say that one molecule of a protein we looked at that's sitting in a test tube or something and it's surrounded by an ocean of water or other molecules or something. So if I want to study this small part of a system that could be a molecule and I ask the question how likely is it for these molecules to have a particular energy? Well, forget about molecule. For this to be general, I have to say that this is some small arbitrary part of a system and I'm interested in the properties of that small system, right? But then I need to consider what is the rest of the universe? That could be the surrounding in my test tube or literally the rest of the universe if you're studying a galaxy. The total energy is conserved in the universe. And at some point, well, the universe might again be my test tube if that's all I'm interested in. If the total energy in the universe, you know what, let's not call it universe. What physicists like to call this a thermostat. And the only reason we call it a thermostat is that I can exchange heat between these two systems, right? And heat will always flow in the direction of temperature gradient. So it makes sense to call this a thermostat. So if the total energy is conserved and I wanna put this small part in my system, well, the rest of the energy will have to be the thermostat. So the question here is if there are lots of ways to achieve this, there are lots of states where the universe could be divided this way and then it's likely gonna be quite likely for me to observe the system that's way. If there are a few ways this could be achieved, it's less likely for me to observe this particular setup. And again, the abstract part is likely the most difficult for you here. But since we don't know anything, we're gonna need to introduce some definitions. So the definitions we will first introduce is that I say that, well, the probability of seeing something is proportional to the number of such states the system can be in. And that's just this M I talked about yesterday. It's just an arbitrary, the reason why we call it M is of course because it's related to microstates. But it's just a completely arbitrary function. That will depend on the energy. Right now we have no idea how it depends on the energy. You could just, well, you could of course argue that if it's independent of energy, that's also a function of energy. It's just that it's a constant function. It can depend on energy. And you might also have to believe me or accept my hand waving that the likelihood of seeing something in a particular confirmation is proportional to the number of states representing that confirmation. If you wanna ask how likely is for me to win the lottery? That's well, that's proportional to the number of outcome where I would win something on the ticket. And in principle, that doesn't tell you anything. The only thing that tells you that we're somehow counting states here. But if I'm counting, I can say one thing. This is a positive number. That's the only thing I know about it. And now I will again completely arbitrary introduce something that will help us a bit. The worst thing that can happen is that I'm wrong and then 14 hours from now I'm gonna stop the lecture and I'll let you go home and then we can restart but there is a likelihood this might work. If this is a positive number, I'm allowed to take the logarithm of it. And then I put whatever unit constant here and then completely arbitrary other variable. You know that this is gonna be entropy, but pretend you don't. So I can say that I'll introduce S which is a constant multiplied by the logarithm of the number of states. And here's the problem. We still haven't done anything. The teacher has just done a bunch of arbitrary definitions. If you wanna go through this again, I actually made a screen recording of this. This might work for you and then I can spend slightly more time on the equations. So all I really did, this is the same equation as I had on the last slide. I have S is proportional to the logarithm of the number of states. And that doesn't tell you anything. And when you have a complicated function in physics that we find it very difficult to say anything about, what do we do with it? Say hook, slow for describing a spring or something. If we don't know anything particularly about the shape of a function, what do we do? Yes, we love to do series expansions. And normally we stop at the first order. We're gonna do that here too, but for a slightly different reason. For an arbitrary Taylor expansion, I would have F, let's expand this around the value X zero. And then I have a small displacement that I call delta X. Well, I have the value of F at X zero and then plus the displacement multiplied by DF DX taken at the value X zero. That's the first order term. And then we can tell you with delta X squared divided by two faculty, multiplied by D to F DX squared. And then we keep adding terms. Do you agree that far? But that's for a general function. We don't have general functions here. We have something very specific. I am interested in understanding S as a function of E. So F is S and X is E. And remember what I said, we are interested in understanding properties of very large systems. That's the statistical and statistical mechanics. E is proportional to the size of the system. If I have one liter of water under a particular set of conditions, and then I bring in a second liter of water under the same conditions, I have twice as much energy. If you don't believe that, we have a problem. No protests. S, on the other hand, is a bit more complicated. If you're interested in it, see the screen recording. But this has to do the logarithm laws and everything. For now, just trust me, that S will also be proportional to the size of the system. If you don't buy that, what's the screen recording I made for you? And I explain it in detail there. So both S and E are proportional to the size of the system. Twice as large, they are. And that's, of course, why we take the logarithm. It's very nice to be able to add properties. So what happens if I put this into an equation like this? Well, the first order term, if I use this for the entropy, that would be something that's proportional to the entropy, proportional to the system size. Perfectly fine, we need that term. In the second term, let's not care about the displacement, but let's look at the derivative. The derivative is a quotient, right? In the denominator, I have S, that's proportional to the size of the system. And in the denominator, I would then have energies that are also proportional to the size of the system. System divided by system, that's roughly constant, right? So that's a constant term. I will probably have to care about that too. But what happens over here? I have something proportional to the size of the system divided by the square of the size of the system. And as the system turns very large, that term will go to zero. And then in the third order term, it will be proportional to the one over the system size squared, cubed, et cetera. So the higher order up we go, I can ignore all those terms. So by definition, this is not just an approximation. In the limit of large systems, I can ignore all those other terms. And otherwise, it would have been, Hooke's law for springs is an approximation. This is not, this is true for large systems. And 10 to the power of 23 is a fairly large number. So that means that I can take that equation that I have there and just write it with the zero and first order term. So capital E is x zero and minus epsilon is delta x. So this is the value at the point where I'm expanding it, E. And then delta x, which is minus epsilon, multiplied by the derivative taken at that value, right? It's just the first, up to the first order series expansion. The nasty thing is I haven't done anything, right? That this is still, I don't even know what the shape of these equations are. But now we can go backwards. It wasn't so much S that I was interested in, but M. So let's take that equation and solve for M. That's fairly easy. Divide both sides by kappa. And then I take the exponent. So M is the exponent of S divided by kappa. But that we can now take there. We have the S and we have this. So let's insert this Taylor expansion here. So then we get first the constant term here. But all this is now gonna be in the exponent. So I can write this as a separate term and multiply it, right? Addition or subtraction in the exponent because multiplication, if I separate the terms. So there is some term here that is proportional to whatever constant value. The reason to make it, this is just gonna be a constant. I could not care less about what it is. And then a second term, which is this one minus epsilon multiplied by the derivative divided by kappa. So minus epsilon is the energy, the derivative divided by kappa. And this one we're not really gonna care about. And we're actually done. You didn't see it, I think. But so remember what you said the first started. The likelihood of seeing something, the probability of that is proportional to the number of states, m. And m in turn is proportional to, that's why I don't care about that free factor. That's proportional then to this exponential function, the second term on the last slide, minus epsilon, the energy in my small part of the system with some sort of horrible constant here and then a constant kappa. I think I used kappa and k interchange simply, my bad, Merkel, but how do we simplify that further? Well, we can't. We don't know anything about S in general. So here's where we give up. And when we can't simplify it for, I'm not sure about you, but that's a bit cumbersome for me to write all the time. So let's introduce a new letter for that. We might wanna call it F. Actually, F is a very bad idea. It turns out that it's much better to call that one over T. Because otherwise you would have to start to measure temperature in units of one over F. So it just, you see here we're not somehow backwards trying to derive what it is. This just comes out naturally when we wanna decide these probabilities. This strange derivative ends up in the equation. And if I wanna get away from that, I gotta need, this is somehow a property of the environment or the surrounding that is gonna correspond to one over temperature. And that kappa corresponds to Boltzmann's constant. And then we literally have that the probability of observing something is proportional to the exponential function of minus energy divided by kappa and temperature. And we haven't assumed anything about the system. As long as the system is large enough. Then it's a freak of nature, it's of course not the freak, but that this corresponds to absolute temperature. So Kelvin did not know this when he'd write it. These are later results. But I think it's a beautiful way that theoretical physics suddenly converges on the results that we've observed experimentally to. I'm not sure about you, but I've never really had a good feeling for this entropy as a function of energy really is. And it's, I don't think you can necessarily have a great gut feeling for it, but it's still instructive to study it a little bit what it means because it is an important function. So what this basically means that as you are adding energy to a system, how disordered does it get, right? If I'm adding a little bit of energy, how many more states will the system spontaneously partition it? And you can describe this, you can start to draw curves, don't worry, I'm not gonna ask you to necessarily draw curves like these. But in general, the entropy will be a function of the energy. That's what we had on the previous slides. And the question then is that what parts of phase space are allowed? What can happen? And what are the probabilities in different parts? In general, you're gonna have curves like roughly like this, and then the slope of the curve in a particular position is gonna correspond to the temperature, or one over the temperature actually. And any points up here is gonna have lower free energy, but I can't go there because I need to stay along this curve. And any point here is gonna have higher free energy. So I wanna go as far down on this curve I get on the slope. I'll come back to that in a second. If we know that, there are quite a few things that we can measure. We spoke about the Boltzmann factor both now and yesterday, and I kept saying proportional to, that is a bit irritating because you probably don't wanna hear proportional to another 500 times. Sometimes it would be nice to have real probabilities saying not just proportional to, but 14.9%. And that might be what you see in the lab even. By far, if you have three states like you do in the lab, it's very easy, right? If one state is four, another one is five, let's see, that's nine, and the third state is one. The sum is 10. If you divide all of them by 10, it's gonna have 0.4, 0.5, and 0.1, and then the sum will be one. So by far the easiest way to normalize this and get proper weights or probabilities, whatever you wanna call them, is to normalize this by the sum of all states. Easy mathematical statistics. And now, again, the books will leave the change when we use w, that's a weight, but it's basically the same thing as a probability, sorry, that it would be great if people could stick to a standard. I know, sadly, they can't. This capital Z you have in the denominator is the normalization factor that we call the partition function. And this is both the most boring, stupid results we have in the course and the most beautiful one at the same time. It's boring in this, it's just the sum of all the states. It's just the normalization factor. But the conceptual interest in part here is that the hard part, first, it's literally a sum over all the states. And the hand in task you're doing, you have three states, or was it four or five? Hopefully you can count that far. It's not particularly hard. For a protein, how many states is there for a protein? You can't even count them, right? It goes down to 11th house paradox, more than the number of atoms in the universe. That's a somewhat long sum, in particular, we're gonna do it manually. So that the curse in this function is that it's apparently simple if this is a small number. The problem is that these numbers are so astronomically large. But if you know the partition sum, you know everything about your system. And theoretical physicists get something here died when you talked about partition functions and so do I. The reason why this is cool is that if we know the partition functions, sorry, then we literally know all these weights, right? To know those weights, I need to know the denominator too. And if I know that, I can take anything, say the energy. What is the average energy of my protein, not just in one confirmation, but in this test tube? Well, under the conditions I have in the lab, sum up every single state, what is the probability of being a state one, two, three up to a gazillion, and what is the energy of that particular confirmation? Those energies I could at least in theory, calculate by all these bonds, angles, torsions, and everything that we derived a few lectures ago, right? In theory, or if you have a very large computer. So if I know the partition function, I can calculate things from those states. And then I can evaluate the real macroscopic average energy, not just the one confirmation, but the protein in this test tube. And then you can just do the same thing for the entropy. The average entropy of this system is sum over all the states, what is the probability of being a state and what is the entropy in that state? Or can we? Did I tell you how we calculate the entropy? I didn't tell you how we calculate that one, right? So this is a bit of a problem. It's not straightforward how to calculate what is the entropy of a particular sub-state. That is a slightly more cumbersome derivation. It's not hard. And you have to use a bit of mathematical statistics. You need to use, this is basically given a system, how many different ways are they to sort states? And have you all taken mathematical statistics? This comes down to all these exercises where you have red and blue balls and then you start to thinking of permutations and the number of ways to sort them and everything. I'm not gonna spend, if I rush through this in five minutes, you're not gonna learn it. And I can't spend 20 minutes on it. So what I've done, I took 20 minutes and made a screen recording of that. But I'm not gonna go through all those slides in details, but I will just jump to the end results here. It is possible to calculate this and then you end up with something that, it's a simple sum that involves these probabilities. Actually, it's a probability and then the logarithm of one of the probability. The exact shape you don't have to know by heart, but just know that it's probability and the logarithm of probability. There is a screen recording of that when I will take you through it. And I promise I will so not ask you to derive this on the exam and that's why I felt that it's a waste of 20 minutes of precious time with you to try to take you through that. But the point of this is that if you believe that derivation, you can plug that formula into the entropy. And at least if I know the entire summary, if I know every single state and if I know the weight, the likelihood of being in every single state, I couldn't theoretically, what is the entropy say of my protein in this test tube? How disordered is this protein? Is it gonna visit lots of states that is floppy and shaky? Then it's likely pretty disordered. The entropy is gonna be high. If the probability of being in one state is 100%, you can actually do the math here. The logarithm of one is zero, right? So that term is gonna be zero, multiplied by one, that's zero. And then if you're only one state, the entropy will be zero. And you can't be lower than that because it can't be negative. So if there's only one state, it's gonna be zero. If there are more states, the entropy will go up. And there are a few different ways you can, but that means that just by knowing the energy equations, I can solve that equations, by knowing all the different states, the probability of being in them. How do you get those probabilities? Epsilon here, I could get, well, the energy of state 14, I could get by looking at state 14 and calculating it from that, right? How do you get the probability of being in state two in general? You can't, why? What does it depend on? Yeah? Exactly, and that's the problem, right? So that the energy in state two only depends on state two. Because if I have the positions on the atoms, at least if I use a formula that's accurate enough, I can't calculate what the energy in state two is. The likelihood of being in state two, that's the Boltzmann distribution, so that will depend on the energy in all the other states. In the handling task you're doing, that's easy. You have three states, or four. You can calculate the probability of being in each of those states. So if you know all your states, you can calculate the probabilities. And if you can calculate those probabilities, you get the weights or the probabilities. And then both of these equations can be solved. So if you know all the states in a simple system, now we can take E minus TS. Well, let's just plug in both those definitions. It's 30 seconds of math. That minus temperature multiplied by that, it's the same sum, so we can take the entire difference inside that sum. So it depends on the weights and on the energy and then the temperature. That's it. So if I know the energy of every state and I know the weight of the state, I can calculate the free end. Don't worry, you don't not know that by heart. And that's because of my physicists. If I'm gonna ask you anything, I'm gonna ask you to derive something. Knowing things by heart is a useless waste of your brain. What this means, and this is a part that is so accepted deep about the partition function that if you know your partition function, you can calculate not just the free energy, but you can calculate anything about your system exactly. You don't need to do the experiment. You can calculate the heat capacity, the free energy, well, anything. It is defined by the partition function. So that while the partition function is seemingly just a sum, right, the important thing is that to evaluate that sum, I need to know all the individual terms. This is trivial for your computer simulation, but in the real world, this is completely pointless because you can never know all the states. But the beauty of that is that for a long time, this was, again, pure theory, but this has changed completely with computers. Because with computers, we can now, well, I can't do an infinite number of states, but rather than saying historically, physicists, we like to say that infinity is somewhere around five or 10, right? But suddenly we can allow, we can have hundreds of thousands of states. We can let the computer calculate way more complicated things than wherever possible with paper and pen. But even 100,000 is pretty far away from infinity. But the point is that we don't, if you take this protein that what is the likelihood of all my atoms being on top of each other? Technically, that's a valid state. On the other end, you probably know that in practice inside yourselves, I'm not sure about you, but I rarely have nuclear explosions in myself. So that's likely a state we can ignore, right? So that the vast majority of states are gonna be so high energy that the weights are so low that we can in practice ignore them in these sums. And the beauty with computer simulations then is that if I know my partition function approximately that I sample part of the system, I can calculate all these properties and in particular the free energy approximately. But it's not enough to just look at one state. I have to look at all the states and that's why you frequently have to run a small simulation like the one you're doing in the handling task. I realize that might take a while to digest, but this is again why the partition function, it's not that it's complicated, but it's beautiful in what it tells you about the systems. And if we now combine this with look at these, something that are possible versus impossible, I already do some of these curves with entropy related to energy. And even though I don't know anything about the shape yet, it turns out that there are a whole lot of things that are not allowed. In general, these are perfectly fine normal functions, right? But if I add energy, the entropy would go down. But that's a very special setup. In general, if you add energy, you would expect things to be more disordered, right? And if you look at the slopes here, what can we see about them? They are negative, right? So both of them would correspond to negative temperatures. Can you have negative temperatures? A lot of carbon. Do we have any other bits? There's one word I didn't mention here. You can have negative temperatures, but not that equilibrium. So these are non-equilibrium systems. Let's go into it. I'm not gonna cover that in the course, but there's a whole branch of modern statistical mechanics that is non-equilibrium statistical mechanics. For instance, if you have a small valve or something where gas is flowing out, right? While the gas is flowing out of the valve, that's not an equilibrium. And while that happens, you can have negative temperatures. So that corresponds to processes right in the middle of happening. And it's perfectly fine because temperature is just the one over this derivative. And this derivative can be negative. But again, because it's not equilibrium, it doesn't correspond to any system that we can think of or measure. But now and then there is always a paper that they claim to say that they discovered negative temperatures. No, they have not discovered negative temperatures that were discovered by statistical mechanics some 150 years ago. But it's just, you will never see that equilibrium. This is another thing that we, in theory, could happen. We could have a function that would go roughly like this. So this would correspond to, this point on the function is the absolutely highest free energy you can have. They're the free energy is lower and they're the free energy is higher. So if I sit here, this is like balancing on the edge of a knife. We're gonna see that later on in the course when it comes to things in the process of making a transition. Because right at the top of a peak where you're not sure where you're gonna go down left or right, you can be metastable there. So you're not necessarily, you don't perceive a force in either direction, but the second you move epsilon left or right, you're gonna fall down on one of the sides. While by far any normal system that we use to consider, they're gonna look roughly like this. You have a nice smooth curve that goes up and the more energy we add, the derivative will usually go down here. And that means that we have nice stable systems here that you prefer to be at this point which is the lowest possible free energy we have. Any other point along this blue curve would have a higher free energy. Because this entire space has higher free energy than they had at the black line there. So why can I say that the free energy is lower here and the free energy is higher there? That seems completely arbitrary. Why is that particular curve? Why does that correspond to free energy? Remember our friend, E minus TS. And now this unfortunately happens to be plotted in a different way. But if you think of energy as a function of entropy, this line really corresponds to that equation. So if E minus TS is constant, you're gonna be along the black line. So it's not really, it's not anything special. It's just E minus TS. That equation is deeper. Remember I said that that equation is deeper than you thought. So with this mean that you can have some sort of stable state and you can move, if I just change the temperature by adding a removing energy, I will move a little bit along this line but it's gonna be smooth. And if I change the temperature a little bit, I will just move this line a little bit. Nice and stable. And with this, I can now start to see what happens when I change things in a system. And we're not gonna go through hardcore statistical mechanics here but we need to understand a little bit what this would mean for a biosystem. So if I take, the first thing is a gradual change in a system. I literally, let's say that I changed the temperature a little bit. I raised the temperature. I'm starting state one and then I raised the temperature just a little bit. The derivative is lower, the temperature is higher. It's one over the temperature. And that means that I start here and as I'm increasing the temperature, somehow the energy of my system is gonna go up. The exact shape we don't know anything about. I could literally stop at any point between E1 and E2, right? I can stop there, there, there, there, there. Every single point here would be stable. So if I start at E1, as I'm gradually changing this entire distribution of states or energies would gradually move towards higher energies. What would this correspond to in life? To be slightly more concrete. I'm avoiding water, right? If I start at water at 40 degrees and then I add a bit of heat so that that's not boiling, heating. If I start with water at 40 degrees centigrade and then I add the little heat so we're suddenly at 50 degrees centigrade. It's an important change. The system will have different properties. Water at 50 degrees centigrade will have more energy. It will have higher temperature and it will be more disorganized. But there was nothing remarkable that happened anywhere between 40 and 50 degrees. I could have stopped at 41, 42, 43, et cetera. Nothing special. But it describes a smooth transition and the key thing that we are in equilibrium all the time. But life would be pretty boring at equilibrium. So there are some other changes, abrupt changes. Because again, if we, these curves could look, let's assume that the curve that it starts out roughly the way the first curve does but then there's some sort of depression. And then I go up again. So remember that I said that this shape here is an area where we don't really wanna be. This would be balancing on the edge of a knife. But let's just assume that our curve looks like that in some sort of region of space. So I can be stable here, here, here. I can be stable there. Let's not worry about this for now. I can be stable here and I can be stable up here but there is something strange happening here. Now of course I can follow the curves. In theory at least we could imagine being here. And same thing here as the temperature increases the energy would go up. But the point is if I'm sitting here it would be better to fall down there or there. Because remember the line here, right? If I take that point and draw a line here every single point on my line here is gonna be at the higher free energy than the line there or the line there. That's kind of the definition. So I'm at a worse point than being, being right in the middle here is worse than being before it or after it. So there's something bad happening in this interval. And what that would mean is that as I'm gradually adding energy well I start out by having most of the system down here but I never get anything between. So suddenly if I can't be here I start getting population here instead at the higher energy on the other side of the gap. I'm jumping directly to E2. So I can go up to E1 but then I have to take a jump to E2 and eventually if I keep adding energy eventually everything will have moved over to E2 but I do not have anything whatsoever here between. What does this correspond to? Yes. I'm getting the water, right? Where we're at 100 degrees centigrade. And the, so what is the interval? How large is the temperature interval in which water boils? Too bad, statistical mechanics doesn't work. It's obviously flawed, right? You can't describe reality. So there is actually an interval in which water boils. So why don't you see it? I'm not gonna ask you about the boiling temperature of water. Save that for the exam. You can calculate this. And I'm not gonna, again, given that the book, Finkelstein goes into some detail to derive this but this is gonna depend on the energies of the interactions. So if we look at water, this basically gonna be, let's see, I think I have it. Yes, it's a 4KT squared divided by the temperature, the energy interval here. So if you have a very large energy interval here you're gonna have a small delta T. When you're boiling water, you're not boiling a molecule, right? You're boiling maybe a gram of water, a macroscopic quantity. You're boiling an insane number of molecules. And here's where I think chemistry fools you a bit. So what was the energy of a hydrogen bond again? It's not the divergent. There was a point to that question. What was the energy of a hydrogen bond? Yes, and what is the keyword in that answer? Per mole, yes. Let's forget about the mole for a second. Then we should divide that by six times 10 to the power of 23, right? That's a low number. So we really should be talking about the ballpark at 10 to the power of minus 23. Those are the types of energies we're thinking about in a protein. What is the type of energy it takes to say heat water by one degree? If you don't know, there is a trick. You don't have to answer in kilojoules. You can answer in K-cals. Yes, one. That's how one K-cals is defined. Do you see the difference? One K-cal versus 10 to the minus 23. Or kilojoules. You can't even imagine the difference, right? So that in macroscopic quantities, if we're boiling water or a gram of water or something, the energy difference here we're talking about are so large, they're like 10 to the power of 20, at least larger. So that the temperature interval we would have for water here, it is a temperature interval and that temperature interval might be 10 to the minus 20 Kelvin. So between 100 minus 10 to the power of minus 20 and 100.0, I think to first approximation, we can say that it boils at 100 degrees centigrade. You don't have to worry about the interval for macroscopic properties. For a single protein molecule on the other hand, it's 10 to the power of 20 smaller. So when you're looking at these exceptionally small energies on the molecular scale, the energy, well, this gap might actually correspond to having a temperature interval here that might be 10 Kelvin. So protein going through a phase transition or something and some of these are gonna be phase transitions, it's not really gonna happen instantly. There is an entire temperature interval over which we are not really stable. Oh, so I even had that the way to calculate that. What does that mean? Well, part of this is gonna explain stability, but we can also use this to explain what reactions will happen. And I'm gonna do one slide on that before I let you have the break. In the handling task, you look at different energies between different states and then tell you how populated they are, right? But in general, I might, what if I have two states? I have A here and I have B there. Where is it better to be? Yes. How do you get from A to B? That's a problem, right? So if I just drop this point, where is this better to be? It should go straight through the floor, but for some reason it doesn't go straight through the floor. It would be better for it to be another three floors down. But there is a barrier stopping this. So it's not enough to say that it would be better for this point to be one floor down. The energy barrier to get there matters. And this energy barrier is too high. So it's not just A and B that matters. Whatever we have up here also matters. So to get from A to B, you first have to get there. So that the height of this will somehow matter. Will it matter? I'm gonna take you through that after the break. But there is a fundamental difference here that you might wanna think about, physics versus biology. What is the population you would have in A versus B if you're a physicist and you think at equilibrium versus if you're a biologist and think at equilibrium? There are very important differences here. But I'll give you that after the break. So let's meet here at the quarter past. I will get started again, so I make sure to release you on time. And I realized there was a mouthful of equations the first hour here and it's gonna be slightly less equation, the second half. And after today we're gonna head back and look more at proteins if you fit up with equations. So the question I asked you before the break is that there are two parts that are important here. If you're only interested in distributions between A and B. Actually, let's include the hash mark too. At equilibrium, what fraction of the molecules are gonna be at the hash mark? If it's a high barrier to first approximation, right? Because it's not good to be there. It's better to move epsilon to the left or right to move down. So if you're interested in equilibrium to first approximation we can ignore the hash mark. That's not relevant. There are only two states here. There are state A and state B. And if you are, in many ways, if you're a physicist, equilibrium is the most natural and interesting thing to look at, right? And equilibrium, there are only two relevant states in this plot. The only problem is that at equilibrium you are all dead. The universe is still not in equilibrium. And the equilibrium is always relative. I could argue that I'm close enough to equilibrium in my test tube, but at equilibrium things don't happen anymore. And in particular for in biology that's a problem because that's, well, even you, our life spans in the ballpark of 100 years, but there are lots of processes still turn over and everything that might be of so limited time scales that you don't really have time to achieve full equilibrium between different things. So suddenly the transition barrier starts to matter. Just like they do in industry, that's why you have catalyzers. You can't afford to wait forever. And if you care about the speed with which reactions happen, you also need to look at the barriers or the peaks. And here I just drew things in one dimension, but it's really the same thing in a general energy landscape. If you're only interested in equilibrium distributions, you only look at the minimum. But if you're interested in understanding how fast things happen, you need to look at the peaks, or in particular the saddle points to get from one minimum to another. So these peaks matter even though we don't expect, you can't really observe a system at this point, but how high it is will influence how quick the transition has happened. So I think there is a subtask in the handling task where you would tell that you disallow a direct transition between two of them. So you have to go through a higher transition and that will influence how quickly you reach equilibrium. The good thing is that we can at least hand wave ourselves a bit and try to understand how likely it is to reach that barrier, because let's treat that barrier just like it was another state in the Boltzmann distribution. So that's the, and let's forget about those two to the right there. The simple rack, if you are there at the peak, let's call that hash mark. And again, the likelihood of being up there is proportional, we know that. That's the Boltzmann distribution. The exponential functions are raised to the relative energy up there divided by KT, right? And if I, the reason I said delta F here, I don't really care about the absolute thing. So here at the number of molecules I have up here compared to the molecules I have down here, well, the fraction of molecules I have at the transition rate is roughly proportional to the total number of molecules multiplied by this factor. And the reason why I can say, I don't care about a factor of 10, I'm just interested in the orders of magnitude here. So there's gonna be lots of proportional or approximate. If, once I have reached that point, the first approximation we can again say it's probably 50-50 whether you're gonna fall back down there or whether you're gonna continue and reach the other better state, right? If we don't know anything else. And again, just ballpark estimates. So that for all the molecules to go up there, well first, that once a molecule has been there, I'm ignoring half a dozen of factors too here, maybe there is some sort of that this might be proportional to the time it takes for each of them. And then I gotta need it so that, well, the time taken is gonna be proportional to an exponential without the minus sign. So why do I kill the minus sign there? Well, the higher this barrier, it makes sense. The higher the barrier is, the longer it's gonna take, right? So the time it takes for all the molecules to go over, if I, for a second, assume that if the molecules are up there, they will instantly go over so that I have 1% of the molecules there. The time it would take for all of them, if I say I have 100 molecules and in each such batch I have 1% of it, that's gonna be 100, all the molecules divided by that expression, again, roughly. And here I don't account for the fact that as molecules go over, I'm depleting the molecules they had in the original states. That's gonna be another factor, two or three. I don't care about the small factors. So the time it would take for all the molecules to go over is somehow gonna be proportional to one over the time it takes to reach that barrier. And that's why we kill the minus sign. So what this means that the higher this energy is, the longer it's gonna take to go over the barrier. And then there is some sort of arbitrary, this tau is where I have grouped all the sins I just made and all the factors of twos that I should have a really bad conscience about. That's why it works. You don't have to care about the constant. It's just, there is some sort of constant there. And for now we can just say that that's some sort of constant that describes how long it actually takes to move over the entire barrier. So that the time in general, the time it takes to go over a barrier is proportional to the exponential function, but not of minus, but of plus the energy we have to go over divided by kT. That's the time. If you don't like to think, it makes a whole lot of sense to talk about time. If you first have to go over one barrier and then another barrier and then a third barrier and then a fourth barrier, then it's convenient to talk about times because that's like a serial circuit with many resistors. You have to go through all of them. Sometimes there are multiple ways you can go out. If you had to evacuate a room, maybe somebody would jump out the window while somebody else would take that door and the third person would take that door. If there are multiple processes that can go on in parallel, then it actually makes more sense to talk about transition rates that are just really one over the time. With transition rates, you can add if there are multiple independent paths they can take. But to make a long story short, what this does is that it connects free energy, not just to how likely it is to be in A or B, but we can at least approximately use it to determine how fast will a reaction happen? Will it be a slow or fast reaction? And that will of course, in particular for proteins, alpha-heal is a sheets, we can start to at least discuss a little bit how stable are things and will they form slow or fast? So let's try to do that for simple building blocks and light molecules, alpha-healuses first. So a few, two lectures ago I think it was, I showed you the shape of these alpha-healuses and we showed that they're stabilized by hydrogen bonds all over the place, right? And this normal alpha helix is defined by having a hydrogen bond from residue i to i plus four to zero to four, one to five, two to six, et cetera. So that the first hydrogen bonds will kind of lock three more residues in place and the second another three and then another three and another three. So in general, we have N residues that are stabilized by N minus two hydrogen bonds as we're increasing the size of the helix. And the reason why it's N minus two is again that the first hydrogen bond locks three residues in place, so I forced three more residues to be in an alpha helix state but I only got one hydrogen bond from it. So already now you should be able to start hand waving a little bit and that's the reason why I'm using this as an example. What is good and what is bad here? And this is gonna be your task for the second hand in assignment. The reason for that is I wanna teach you, it's not dangerous to work with equations, it's not dangerous to play around and see where you get. And I think this is a great example to teach about what free energy really matters. So if we don't know anything about the process, let's start by making some definitions. So if I wanna see what is the free energy of forming an alpha helix, I need to say what is the free energy in the alpha helix compared to the free energy in the before state that is the coil or the not alpha helical state. And then I need to try to describe what happened. So I will form hydrogen bonds. Hydrogen bond is energy, right? So that's the E. And for end residues, I formed N minus two hydrogen bonds. I have no idea what the energy of the hydrogen bond is. Actually I do, and you do too. But let's just call it FH bond for now to avoid introducing a number. So the energy I'm gaining by putting N residues in here is N minus two multiplied by the energy of the hydrogen bond. And we don't know more about it for now. But for this to happen, I would have to take all those N residues and freeze them in a conformation that corresponds to an alpha helix. So they are no longer free to move. They now have to have the phi and psi angles that correspond to the alpha helical part in the Ramos-Schänder diagram. And that's entropy. So why do we get that term? Why is the entropy important? If this individual residue could choose, would it prefer to be free or would it prefer not to be able to move and have to be locked in the alpha helix? And that's the whole point of it. You want to maximize the entropy so that while we will gain some cost of forming these hydrogen bonds, you're gonna be paying for it dearly here. This is not good. This is the penalty. Otherwise, everything in the world would be hydrogen bonds and alpha helices. And if you just do the math here, that there are parts of the terms here that are proportional to N, right? And then there's one term that is not proportional to N. So you pay roughly two hydrogen bonds in the beginning and then there is some other, sorry, the F here is negative. So minus, minus, will be plus. So you have one term here that is constant and then a term here that depends on the number of residues. So we can describe here that there is some sort of initiation cost that is constant and then an elongation cost for every extra residue we put into an alpha helix. And this is not limited to biophysics. This is what I want you to be able to play around with with equations when it comes to understanding things in free energy. We can, I'm sorry, there are some other ways you can describe this. I'm not in the interest of time I will skip this slide. This used to be important for physicists a long term ago, but nobody uses this way of describing it anymore. Actually, let me go back there. What this will mean is that already now you can say something here. That first term is it gonna be positive or negative? And the second term, will it be positive or negative? That's not trivial. But the way to solve that is to think about what did it mean? If this term was negative, that would mean that it would always be good for any to start forming alpha helix anywhere. The second you have an amino acid, it would instantly want to form an alpha helix. And we kind of know that doesn't happen. Every residue does not form an alpha helix. So we know that we will be paying initially. So that term has to be positive. On the other hand, if this term was also positive, then we would always be paying. It would always be bad to be an alpha helix. So we also know that this term has to be negative. So we will pay initially, but if the helix grows far enough, the second term will outweigh the first one. And at some point, you're gonna have a stable secondary structure element. It's pretty cool to be able to say that without even having seen any of the equations or anything, right? There are different ways you can talk about this. There's a deep result in physics by Lev Landau that's saying that in general, faces can't coexist. If you have, it's a good time of year to have that example. If you look at Vidar Fjadden or something, right now you could argue that there's both ice and water. But if you wait a very long time, it's all gonna be either ice or only water. And the reason for that is that if you only looked at the entropy, mixing ice and water is good. The more mixed things are, the more disordered they are. But there is also surface tension between the parts that are ice and the parts that are water where you can't form hydrogen bonds and everything. And that area is gonna cost you. So that if you separate one large part with ice and one large part with water, you minimize that surface area. And that's not specific to water. That goes for any phase transition in multiple dimensions. Faces can't coexist. But for a helix, it's slightly different. So for a helix is one-dimensional. So the entropy term here is still existing. Mixing helix and coil is always good because of a higher entropy. But the boundary between the helix and the coil, that's just one-dimensional points along this chain. So there is no large surface tension. So an alpha helix can actually mix quite freely between the longer sequence. There's some helix and then there's some coil and helix and coil, helix, coil, coil, coil, helix, coil, helix. So they will in general mix. And that's a slightly boring part here. Alpha helix is not a phase transition. It's smooth. You can unwind that helix by one or two turns and turn it into coil and you can make that helix slightly longer. They can coexist anyway. So this is just like gradually heating water. Helices don't go through a phase transition. So then it was useless for me to go through the phase transition, not quite. There are some ways we can even calculate what is the average length of the helix. And in the interest of time, I'm not gonna go through that because I think going through some real proteins on Friday is more important. But the point is that with these equations, I can essentially estimate at this, sorry, at the midpoint that I no longer have there, we can measure in a fairly simple experiment that when a particular sample, I know that using circular dichroism or something, I can see there's 50% of the sample remaining in the part that would be helical. Remember how I said that you can turn circular polarized light, right? Suffice to say that there are experimental techniques here we can measure how much of this sample is helical. And by using those very simple experiments, we can go the back way and try to determine what is the average length of a helix. But it's not such an important result for the course that it works spending 10 minutes on it. But what all this gets us, if you go through this math and everything or Kasalt-Fingkelstein, we can actually calculate both what the average length of helixes and it should be maybe 10 to 20 residues. And we can also calculate average rates of formation because the rates of formation will correspond to this initialization entity. That's why I took you through the initialization entity. Remember what I said? If we are at state A, and then I need to go through a hash mark there to get state B. So if B is helix and A is coil, this is really the initialization entity. You need to get to the top of the reaction before anything can happen. And of course, this is a biophysics course by teaching this Ralph Helis, but this is universally true for any process in the world you can imagine. If you understand the properties of the transitional states, you understand what process will happen and what process will not happen because it's free energy. And I know that I spoke to some of you during the break, who found free energy a bit complicated. That's true, it is complicated because it's an important concept. And there's this famous radio interview with Albert Einstein, that young school students, some asks him that it's mathematics troubles him. And Einstein said, I can assure you that my problems with mathematics is far worse. And that mathematics isn't easy, that the equations are hard and we work with them because you get valuable things from them. If we understand this initiation energy, and again, then we know that it was roughly the equivalent of two hydrogen bonds, then we can start to estimate roughly how high is this barrier and roughly how long will it take to go over the barrier. And if you do this math for alpha helis, it turns out that the small helix fragment forms in maybe 100 nanoseconds or so, maybe 20 residues. And each residue, it's probably in the ballpark of five nanoseconds. It's exceptionally fast. It's so fast you can, it's even difficult to observe it with a whole lot of experimental methods. You can literally almost see an alpha helix growing in a microscope. Not quite. I'll come back to the Friday where we find alpha helices in your bodies. And the rate limiting step, what is that determines how quick or slow this happens? Not just in an alpha helix, but in general, what determines how fast a reaction happens? The initial energy and the transition. So the difference, the extra energy you have to get to go up to the transition state. And that's a bit long and cumbersome to say, so we, it's the transition state. The transition states determine what reactions happen or not. And this is the observable. You can never observe the system in the real transition state. We have zero population of the transition state on average, but they still determine everything whether things will happen or not. And we can, one can do some math and argue that for a general helix to form, it spends roughly the half of the time waiting to form and the other half extending it. But this is something I had a few years in the course. It's in the interest, I'm not gonna taunt you with that. So to sum up our alpha helices, they form almost instantly. Both initiation and allegation matters. They start to form, but then they grow gradually. It's a relatively low free energy barrier. Maybe one K cal is probably a little bit, maybe two or so, but a few K cals. So is that a higher low barrier? So this is the problem, right? That in life sciences, we have a natural scale of the energies involved. We had the hydrogen bonds that were roughly what? Yeah, two to five K cals. And then this process that might, so that hydrogen bonds is gonna drive this process fairly hard. But what should we compare that to? So it's too high. This is a scary part because you've had this, but I don't think that people force you to think about it. There is a natural energy scale. What is the natural energy scale? In all these equations. What do you compare the energies to? I can't take the exponential function of anything that has units. The exponential function, the argument of that has to be without units. So what do I do with all the energies? In the Boltzmann distribution, the probability of something to happen is proportional to what? The number of? KT, right? So KT always has units of energy. So your energy scale is KT. What is KT? At room temperature. Yes, I should know that. We're in biology. So we're gonna need to talk about K cals per mole or kilojoules per mole. I should translate it to electron bonds. But you would, you can respond in absolutely any units you want. There are two numbers you need to know. And these are numbers just as the hydrogen bonds. You need to know these numbers by heart. Roughly 2.5 kilojoules per mole. Or 0.6 K cal. So the units matter. You can't just say 2.5. Remember what I told you yesterday about biking through ice? If you're gonna decide to bike across an ice barrier, it matters whether it's one centimeter or one meter tall. And if you have no idea about that, you're gonna have a bit of problems in the traffic. So to be able to work in the traffic, you need to have a gut feeling. What ice barriers can you bike across and what ice barriers should you probably not try to bike across, right? The reason why you need to know these numbers by heart is not because I'm gonna ask you at the exam. Actually, it is because I'm gonna ask you at the exam. But the real reason why you need to know numbers like this per art, you need to, when you hear a number, if I say that, oh, the transition barrier for this to happen is 500 mega calories per mole, will that reaction happen at room temperature? No, we can ignore that. That's, of course, one of the likelihood that the sponge will go down to the next door. It's not gonna happen. Completely irrelevant, let's ignore it. On the other hand, if a barrier is 0.01 kilojolts per mole, it's a saying that that's not even a barrier. It's like gravel on the road. We're not gonna see it. We will just pass through it all the time. On the other hand, if a barrier is, say, 5K Cal, it's a factor of 10. If we will likely be able to go over it now and then, we're not gonna go over it all the time. But that corresponds to this, say, 20 centimeter edge of ice. You can't get across it, but you're gonna feel it. And those are actually the most important barriers, because those are barriers that are relevant. It is a barrier. It would take a time to get over it, but we can't get over it. So that any time we talk about energies, these are the energies you need to compare it to. At room temperature. So for most reactions, the obvious way to make a reaction go faster is what? Yeah, because if you heat it up, KT, well, it goes up proportional to the absolute temperature. At 600 Kelvin, these energies are twice what they are at room temperature, and that would change the Boltzmann distribution. So why don't you think that the residues are longer than 20 or 30, the alpha helix are longer than 20 or 30 residues? Wouldn't it be better to keep, if it's good to extend the helix, shouldn't we just keep extending it and extending it and extending it forever? So this is where evolution comes in. Not every single amino acid likes to be in a helix. Polium, for instance, hates to be in a helix, right? We're based on the way that they have its ring and everything, it can't form hydrogen bonds. So sooner or later, you will have an amino acid in the sequence that doesn't like to be in a helix and then it will break it up. The other secondary structure element we looked at were beta sheets. And these are actually much more interesting. This structure, too, is stabilized by what? There are some dashed things on the screen. It's stabilized by hydrogen bonds, right? Lots of hydrogen bonds. Can you imagine what the difference in formation process is here? For the helix, like I said, you could pretty much add one residue at a time. What happens here? So there are likely some fairly strong cooperative effects, right? Because the chain will stick together. If you already have these four chains, if I start to put that chain here, the second I put that residue here, the second residue is pretty much there, right? So what's gonna happen is that once I have some hydrogen bonds, this is almost like a zipper, it goes here, boop, and then you have another eight hydrogen bonds. So it will take a very long time until I have this last fifth strand here, positioned exactly here. But when it happens, suddenly I get all these hydrogen bonds at once. So already now we can start to guess that this might be more of a phase transition thing. So we should pretty much try to ask the same, we know experimentally that they can take hours or weeks, and sometimes just milliseconds. So there is something more complicated than the transition barriers here that are pretty fun. And I think that's gonna be the end of this difficult part of physics here. The only way we can understand this is by looking at this process. I need to understand what is the initiation cost and what does it cost to what I gain from making the sheets larger. That is slightly more complicated, so it will probably take us roughly 10 minutes. But it's not difficult, it's just a little bit of bookkeeping. So if we have a beta sheet, there are many ways in theory that a sheet could form, and I'm just gonna take you through one model. So if you start thinking of a long beta sheet, at some point I'm gonna start to form one hairpin. The reason why this is called hairpin is that it literally looks like a hairpin that you would have in your hair. If you don't even have a hairpin, you just have a bunch of residues lined up after each other, they haven't formed any hydrogen bonds, and at that point we wouldn't even have a beta sheet, so we don't care about that. It can't be, this is the simplest possible beta sheet you can imagine. This will have to have some sort of length, exactly how large that is how to come back to. And at some point you add one more turn, and then we keep adding more residues and more residues, and then we keep extending it that way. So somewhere along this path, there should be the worst possible state. Why do I wanna know the worst possible state? Yeah, remember the transition state that the worst, the highest free energy you have to get to, not necessarily the highest free energy in general, but the highest free, which is basically the lowest path I can take, but I need to take the path. So the worst point along the path I have to take, that's what's gonna determine how slow the process will be. Let's start by looking at this hairpin, and we'll come back to the worst part is in a second. We're gonna need to introduce a number of things, so it's slightly more complicated for the alpha helix, but it's not a whole lot. As always, when we define things, you can define things any way you want. If you're lucky, or have done this before, lucky, you tend to make wiser choices. So to make this slightly easier, if you look at the previous slide here, there are some residues on the inside, and that there are some residues that are on the edge. And just as for the alpha helix, it's good to think about in terms of what is the general free energy for some of our being on the inside, and maybe an extra effect from being on the edge and not having a hydrogen bond there, right? So I'm gonna try to separate those, and instead of saying free energy A and free energy B, I'm gonna say that there is in general just by being in a beta hairpin, every such residue has some sort of free energy F beta. I don't know what it is, right now, I don't even know what the sign is, but there has to be some sort of average such energy. But then if you are at the edge, then I can't form as many hydrogen bonds, and that's likely not good. So there is some sort of extra edge energy, and because I lack imagination, let's just call that delta F beta. Actually, it's not just because of that. I tend to follow the same nomenclature as Finkelstein if you wanna read the book. It makes your life easier. So every residue on the inside in a large sheet would have an energy F beta, while the energy of the rest of you that are facing the water or the surrounding, their total energy would be F beta plus delta F beta. And then at some point I make a turn that is up here, and there could be an extra turn. For every turn I make, let's just say, if I make a U turn, let's use the letter U, there is some sort of energy for it. And right now I don't even know whether the energy is positive or negative. This is something I would strongly advise you. Don't try to be too smart. Don't try to jump ahead. Don't try to say what is the sign of it. Introduce a letter for it. The second you've introduced letters, then you have your stuff that you can work with, and then you can start to think about what they mean. And now I'm gonna say that once we have those definitions, since sheets can form, you must have that the delta F beta must be positive, and you must be positive. Why? And in particular, how do you show things like that? Have you heard the term reducts you add absurdum in mathematics? A useful way of doing mathematical proof is usually assume the opposite, and show that it leads to something that is absurd or unrealistic. So let's assume that it was positive, that it's the U, the turn term, that would be negative. What would that mean? That would mean that it would be good to make turns, right? Forget about forming a beta sheet. My entire sequence would just be making turns. It would be awesome. And we kind of know that proteins don't look that way. They do form helices and sheets. So that's, in general, you cannot say it, but because we know a bit of what proteins look like, we can say that's obviously not realistic. There could be an individual turn that would like to form, but in general, it's not gonna be good to just keep turning because you're not forming hydrogen bonds. So in general, U has to be positive. It's not good. Why can't it be good to be at the edge of the beta sheet? In general, it's okay to do it. Well, if it would be good to be at the edge, right? I would have one hairpin here and I would have one hairpin there and I would have one hairpin there, but it would never be good for those to get together because if they get together, they would lose their edges. So think of this as some kind of surface tension and then suddenly we would try to maximize the surface instead of minimizing it. And same thing here. In general, we know that you tend to form larger beta sheets, so this can't be true in general. So now we know something about other sides. Sorry. There are two scenarios then. If we look at the total energy on the edge, F beta plus the extra edge energy, that can either be smaller than zero or larger than zero. Both these things can be true actually. If the total energy is good, then we wouldn't up in this area. It would be better to have an, instead of forming three different ones, you would just form one infinitely long beta hairpin instead of many short ones. In theory, that could happen, but we also know that from proteins in general, that's pretty rare. So in general, it's not good to just have one long beta hairpin. It would be better to have something larger. And that brings us to the second state here. In general, even for a single beta hairpin, it's not really good to be in the single beta hairpin. The first hairpin we're forming is still pretty bad. So we're still just moving uphill when we're forming the first beta hairpin. And that's also important for me because I wanna understand why it can take so long to form beta sheets. And that I'm more interested in the case where it will take a very long time to form where there's a very long uphill road. So to get further, then I need to understand a bit more. I need to understand how long must this hairpin be? So how many residues do I need to have in it? Well, that too we can understand. If I look at this simple case where the single hairpin is not stable, the free energy of such a hairpin is that I have one turn and then I have two strands where each strand has N residues and they contain both F beta and delta F beta because every single residue is in the hairpin and they face in it. And if I now keep adding one new strand, a third strand, sorry, this is not in time. I think I've realized I wrote the equations in a strange way in your notes here so that if I added one new third strand here, I'm moving from this date and then I'm adding this right. So I'm adding one turn and another N new residues. I already said that the turn per se is bad but at some point it has to be good to add these residues because if it's only bad, it will never become better no matter how many things like that I add. But on the other end, if the entire sheet here is very short so you only have three or four residues, the extra residues I have adding here might not be enough to counterweight what I'm paying for the turn. So there will be some sort of minimum length of this sheet that says how large does it have to be before things start to be good. So if I add one new third strand, there will be costs equivalent to one turn and there will be N residues that are on the inside. And the reason for that, I will still have one strand here on the outside, one strand here on the outside and one strand that is entirely on the inside. So I don't increase the surface but I increase the stuff on the inside. And the smallest, when I exactly balance the turn with that energy, the sum of these two should be zero, right? And to find the smallest number of residues where that's the case, well, move over U to the other side because minus U divided by F and we tend to write U divided by minus F that's gonna be nice in a few slides. So minus U divided by F, we can determine some sort of the minimum length of the strands. I'm gonna use this equation on the next slide. That's the reason why I've had to write. And at this point, we just had to look at this point and guess that if I said that the single hairpin was not stable, so I'm still going uphill. I'm paying energy. This is a single hairpin and then I just added one more turn. The turn will cost me energy so I'm still going uphill, right? And at somewhere here, now I start to add hydrogen bonds here in the third strand but that's roughly what I assume on the previous slide. The point when I start having internal residues, that's when I start to go downhill. So that the worst possible state here is likely that I have one hairpin and one extra turn. Is that exact? No, sure, definitely not. But it's an approximation roughly of what the transition barrier might be. And then I just add up so that I have one turn and then a second turn and then I have two times this smallest number of residues that I needed in the hairpin, multiplied by the energy inside them. And now this N-min thing is for, this is why I don't want this N-min in this equation. That's why I had to derive the equation on the last slide. If I now take the results of N-min there and insert it and do just simplify and strike out some terms, I get to this equation. Not something worth knowing by heart but this depends, the free energy of the transition states depends a little bit on the turn. It depends a bit on the surface energy which is probably related to a hydrogen bond or something. But it also depends on the denominator of the free energy of how likely its residue is to be in a beta sheet. That is very different from the alpha helix. The alpha helix just contains a general entropy term. So what will this mean? I'll jump a bit. Well, in general, the time it will take something, I said that that's proportional to the exponential raised to plus the transition state energy, right? Divided by KT. And the transition state energy, let's forget about KT and everything. I put all that into a constant A. But then this, the free energy of being in a beta sheet ends up in the denominator. But this will vary a lot. Some residues love to be in beta sheet and then this is gonna be a very low energy. Some others hate it to be in a beta sheet but they're gonna be at very high. So you even have an exponential dependence of time depending on whether things are stable to be in a beta sheet or not. And that's why some of them will take a millisecond and others can take 100 years to form beta sheets. I didn't skip those intermediate slides entirely. Finkelstein then goes into, so here too, of course, just as for the alpha helix, what can happen is that you need to go over some sort of transition state that can vary by orders of magnitude here. And until you've reached the transition state, you are just paying and then at some point we're gonna get the payback and then it's better to just extend the beta sheet. But in contrast to alpha helix, you're not talking about microseconds here. This can easily be seconds hours or more. If you would like to do this properly, we actually have to prove that this is the lowest possible transition state energy that I didn't. I just hand waved that this was a reasonable transition state. Why? Why do I need to prove that it's the lowest one? So this is again not limited to biophysics but general to free energy and process in general, all I have done is proved that there is a possible way for a beta sheet to form, right? That has a particular transition state. Let's say that you are very creative. You can come up with a faster way for it to fold. Then they would likely follow your path instead. So most of the molecules will of course follow the fastest path where they have the lowest possible transition state energy. So to really, in principle, I think one would have to go through all the models and prove that there is no other state that can be lower because otherwise this would just be, you can consider this an upper bound to the time. If you could find a faster way, it can happen faster. That would be a catalyst. And here too you can calculate the exact formation times and everything. You're gonna do a little bit of that in the handling tasks. I will leave it for that. So if we sum up beta sheets, they are unstable sheets are extremely slow to form that if they are not residues that really love to be in a beta sheet, they can take hours to weeks in the body and in some cases it can be even slower. There are very high free energy barriers. And this is actually a first order phase transition. It's a real phase transition when you fold these molecules but it's a phase transition on microscopic scale, meaning that it happens over 10 degrees centigrade or something completely different from the phase transitions you would expect between water and ice. So these small building blocks we have, they are similar but also very different. They are similar in the sense that they are stabilized by hydrogen bonds, but for helix the hydrogen bonds are local. And by local we mean that they are to neighbors that are not very far away in your chain, four residues away. It's a gradual growth. You can add one residue at a time but that also makes it very, very fast because again the next turn here, those residues are not very far away. And that gives us a relatively low initiation barrier which means that it folds fast. Beta sheets on the other hand are non-local. So these residues might be one, two, three, four, five, six. Well the next strand here might be 244, 245, 246, 247. And for these two different parts of the chain to find each other, you might have to wait a very, very long time before they spontaneously bump into each other in space. But when they do form, it tends to be all or nothing. It just snaps in place and then you suddenly have 10 hydrogen bonds and they will never let go. And that gives us a very high initiation barrier but it can also be a very low formation. So they are similar but also very different. And after that excursion into theoretical physics, you might think that, oh my God, I set up to do bio physics in this class. This is super important when it comes to real proteins. So maybe you might be a little bit too young now. Have you heard about prions? So some 20 years ago almost, Stan Prusner and a bunch of other people started to realize that there appeared to be infectious agents that were neither viruses or bacteria. And again, if you're not a doctor, you have no idea how scary that is, right? Because in general, you kill both viruses and bacteria by heating it to 200 degrees centigrade or something. We do that all the time in hospitals and everything. But for some reason, there were examples where you could have disease that could spread but it was a virus or bacterium and you couldn't really out-to-clave it or anything. And that would open an entirely new branch of diseases and they think, super scary. And this was first, this was identified in a number of animals. One of them is what's called BSE, bovine spondiform encephalopathy, but also rare diseases in humans called Kreuzfeld-Jakop disease, where you had some sort of strange plaques building up in the brain. And eventually, what Stan Prusner proposed is that these were actually misfolded proteins. Proteins that for some reason end up in two states or end up in a state that is not their native states. And that these proteins, and that this was heavily criticized, but I eventually, Stan Prusner was proven right and he got the Nobel Prize for this some 15 years ago. So these are proteins, this is an example that a protein on the left is the native state in the sense that it's the naturally occurring one that is biologically functioning. But sometimes for reasons that we didn't quite understand, this protein could undergo a transition and start to look like that instead. It's very important to see beta sheets here because beta sheets, these transitions can take a very, very long time to happen. Did they also measure, if you have a beta sheet, it's like to grow, right? So could you, if you have two proteins like that, could you imagine what they would like to do? Merge. If you now have two pairs like that, what would they like to do? Merge. So eventually you might start having these molecules that build up larger structures. And these are likely the plaques that you identify in the brain. There's still a lot of research in Ibiza that, so the idea here is that you have these very gradual slow growth of proteins that end up in some other strange state that is not the biologically active state. They shouldn't be there, but they start to grow whether it's correctly or incorrectly, but it's essentially, it's misfolded. Well, it's not really misfolded, right? Because this is what Amfisa said. Why will the proteins end up in the state? There is one concept we talked about both yesterday and today, that should be, if I ask you why something happens, what is the first thing that you should think of? Why do things happen? Is the free energy in this state lower or higher? Lower. So we need to understand why the free energy there is lower. It goes there because the free energy is lower. So Amfisa pretty much said that it will go to the state that has lowest free energy. But the other thing, but why don't you stay there? That is the biologically active states. That was kind of what Amfisa said that that should be the active state. That should have the lowest free energy. So the way that this likely happens, but now we're venturing a bit further out is that we might have some sort of unfolded state here and then we have a folding and then we have a state here and then we have a very, very high free energy barrier. It goes up to the next floor and then we have something that looks like that. Imagine that this barrier is so high that it would take 50 years to get over it. When this protein folds in your body, it's gonna end up here, right? And if this is a good functioning state, it's gonna work perfectly fine. You will survive. You will even have offspring because you typically have kids before you're 50, right? But eventually, once you turn 50 or 60, some of the proteins start segregating up here. And if you now have these processes where the mere existence of proteins like this is gonna, I have one more slide to go through. The mere existence of these plaques will likely accelerate the formation of even more plaque. So there are two problems here. One of them isn't really a problem that in the medieval days, we used to die when we were 50 or 60 and you certainly had your offspring. There is not really, in an evolutionary perspective, there was not really a whole lot of reason to be healthy in your 50s. It's not gonna give you a better offspring. Today, we tend to live slightly longer than your 50s. It would be nice to be healthy, but there is not a huge evolutionary advantage to it. And that's likely why it hasn't worked out. The other thing that started to happen is that some superstar came up with a great idea that let's take cattle and turn them into cord beef and feed that cord beef brain to other cattle. They're in the UK. And the problem is, remember, these plaques, they're super stable. They're very, very stable. You need to get across that barrier to the other side. So you're not gonna denaturate them. Just because you boil it doesn't mean you destroy it. So now you take your cattle and feed them this plaque to the meat from another brain. This will go straight through their stomach, their intestines, it will go out in their blood up to their brain, where they will start to accelerate things. So now this cattle will have even more of these plaques in their bodies and brains. Then you take these cows and turn them into hamburgers. And that's what leads to this BSC craze some 15 years ago where actually young people started to get in Croix-Fields-Jakob disease because we were eating meat that was infected with prions. And since then we have stopped doing that to cattle. That's all I had for today. There are a bunch of things. I think by far the most important thing. You need to get a gut feeling for free energy. You need to understand this E, S, H, F, G, what they do and what they represent and what they're relevant. And a particular understanding when we talk about energy versus free energy. I did an experiment with screen recordings. I'm more than happy to do three or four more of these. But if there's pointers, I've taken this course before. So there's pointers in meat picking of the things to record. If there are things you don't understand let me know and I will make a separate screen recording of them. There are, in principle, you could start with Hand in Task 2 now but I would suggest you finish the first one first and then we have a bunch of reading instructions for you too. And on Friday, I'm gonna be looking at protein structure biology again. See you on Friday.