 Lecture two. I'm not necessarily going to ask how you did with reading. Actually, I will ask one thing. When it came yesterday, did any of you read the Watson and Crick paper? You should. There are quite a lot of things in the course that I won't necessarily go through and repeat. I will intentionally try not to overload you with material. And there are some in general. If I upload a 20-page paper or even something like the Sanger paper yesterday, I don't expect you to read that in detail. But do try and try hard to read the short stuff I upload because it will help you. There will also be some things you realize in the lecture notes, for instance, that are not covered well by the book. My team is very active in research, so I like telling you about things that appeared last year or even this year, or something for obvious reasons that's not going to be covered in 10-year-old textbooks. In many cases, it might not even be good reading material. Occasionally, I try to provide links where you can read up a bit on this online. But try hard to actually go through the reading material I put there. I will not put 20 pieces for every lecture because then you won't do any of them. But if I put four or five there, go through them because there could be questions about it on the exams. What I like with the Watson and Crick paper, this is a super short paper, but this is an exceptionally British statement towards the end of this paper, very undertoned, understated humor. So as we talked about yesterday, they proposed this double helical structure, right? And they also proposed this pairs of nucleotides. And then they just say one sentence at the end. It has not escaped our notice that this also provides a mechanism for the transfer of genetic information. So this is the entire birth of modern DNA sequencing. So they're not just telling it explicitly. They say that you thought that we hadn't thought of it. We have thought of it. And this is the end. If this had been a paper today, there would have been 40 pages of supporting information, lots of details, proof that this model is correct. I think it's such a beautiful statement because that's the real discovery. And this is, of course, where their model was so much better than Linus Paulings, for instance. One of you asked me yesterday about the pi helix. I looked this up. And unfortunately, the answer is not quite as fun as you, I think. So when Linus, in some of the original protein papers, Linus in particular proposed the alpha helix, which we all know today that it's super famous. He also spent quite a lot of time talking about the gamma helix, which is this 5.1 angstrom structure. The reason why you never hear about it is that it's a parenthesis. It's not important. It's not a structure that actually occurs anywhere. About a year later, during one of the meetings in London, there was another group that proposed the helix that we now know as the 516 helix, the pi helix. And because there were already two helixes called alpha and gamma, well, they can't call it beta helix because we also have the beta sheets, right? So they pretty much came up with another Greek letter and they picked pi. So it's not really more fun than that. The small corollary there, though, is that Linus Pauling was extremely negative to the helix. He said that it can't be true. It's an obviously incorrect helix. And the reason why Pauling thinks it's incorrect, which is actually partly true, is that because of the width of the helix, there's almost a hole in the middle of it and nature a porous vacuum, right? So if these atoms are actually spaced a bit too far apart for them to be happy, so they would all be attracted, but you can't really compress the helix high there because of all the other local structural constraints. But it's yet another example where Pauling actually turned out to be wrong. His gamma helix turned out to be completely relevant while the pi helix actually occurs now and then in a couple of percent of the structures or so. The normal place where you see the pi helix, the 310 helix, you might actually have a small stretch of it in some structures. The pi helix, in contrast, might occur around a proline bulge. It's typically just one turn of a helix that's a bit broader or something. Personally, I would, personally, I would hardly call that a real pi helix, but to each their own, I guess. Today, well, rather than recapping what we did yesterday, I'm going to head straight to these questions I gave you, and I'm not sure where you can read this with the sunlight, but we'll give it a shot anyway. Hey, you went through all these questions yesterday afternoon, right? That's the other thing to think about. I will deliberately not schedule all afternoons for you. I won't schedule all mornings either, particularly when I'm traveling. That's not necessarily the same thing as free time. I expect you to read the books and read through these things for your own sake. Otherwise, you can end up with an insane workload the last 10 days of the course when you suddenly need to read everything. The other thing that I can ask you about, as you might see from the schedule now, last year, the students expressed that they were very happy to have one completely free day per week, if they can. I try to compress everything we do so that, for instance, Thursdays are free. Do you prefer that, or do you prefer me to spread out the events a bit more so that you have an event activity either in the morning or afternoon? Okay, one completely free day. Good. But that, of course, will mean that there will be other days a week when you have teaching in the morning and labs in the afternoon. Dari and Björn will start the labs on Monday too. I think we're fully covered there. I will start with this. Rather than having you pick questions, I would suggest that we just do this in some sort of round robin fashion. So let's start here at the beginning with the first one. How many charts do amino acids do you know? Yes, and then we'll just continue. The point here, this is not an exam, right? I want you to go through this. It's perfectly fine not to answer. If you don't know the answer, guess or tell me what you think. The point is to talk about it. If there are things that are easy, we shouldn't spend time on them. We should spend time on the difficult ones. Which ones are there? Yes, and histidine. You could argue that this is a bit of a trick question. Whether they are charged or not depends on what? BH. Yeah, and the really difficult one here is histidine. The problem with histidine is that formally the pKa value, the value where it changes is 7.4. Unfortunately, that means that you can't predict anything of it. There's also a phenomenon called pKa shifts. So depending on what you're surrounding is, you might either like to release your proton or pick up a proton. So that there is not a single histidine. A single histidine in water will have a pKa value of 7.4. But the second it's in a protein, it's not going to have exactly 7.4. It might be 7.8 or 6.9. Unfortunately for histidines, that means all better. It's by far the worst residue in the world when it comes to predicting anything. Lysine or arginine in contrast? Basically you would be, unless you have experimental evidence in the country, you would be insane to predict that arginine is not charged. But there are exceptions. So let's continue with the next two questions. Acidic. Well, just pick one. What I did not cover yesterday, what do we mean by an acidic amino acid in the first place? Why do we call it an acidic? Because it usually has a carboxyl group, which is of course exactly the same thing you would have in a normal acid, right? So it tends to release a proton and solution. So right next to you then, basic amino acids. And in proteins this typically occurs around one specific atom in these acids. It's always one atom involved. Well, it's a hydrogen that's taken up, right? But the hydrogen is virtually always taken up by a nitrogen. So you have an amine or amide group. So that would be arginine. And arginine is a bit special because the charge is quite spread out. So you have two NH2 groups, one NH. And in lysine you have one NH3 group at the end of it. And then of course we have acetylene tube. Number four. Which amino acid is in carol? Yes, why? Good. This is also something I didn't cover yesterday. But what would happen if we had some D isomer amino acids in a protein? Yes. So of course you could argue that this is partly a completely artificial problem, right? Your bodies don't create the amino acids. So why on earth am I mentioning this? So you could argue that this is just a purely academic example because I'm just telling this because there are some cases where they might exist. Could you imagine using this anyway in biotech? So one problem that's very common, well, a feature that's very common today is that we're moving traditional drugs that you give are small hydrophobic compounds. Think of benzene, right? Slightly more fancy. But they're very small hydrophobic. And the problem with these drugs, as you will see later on in this course, they're pretty boring. They're difficult to get them to do exactly what you want and everything. Imagine if you could create a drug that's an artificial protein. What would happen with such a drug when you eat it? When you take it in a pill in the morning? Yeah, it's digested in your stomach, right? That's what your stomach does to proteins because it recycles the parts. What if such a protein consists of D amino acids? Right, because you could imagine creating a protein that's bio-orthogonal really, right? So that it doesn't interact with the normal enzymes and everything. So it would go straight in your blood. It would behave pretty much like a normal chemical molecule. It could have a very long lifetime. And we occasionally use that in the lab too, not necessarily with the D isomers, but create amino acids that are deliberately not compatible with the normal ones to give them special properties. Sorry, you had a question? No, well, imprints... No, so it... Lies and license statistics. If you look at the very lowest level, one single peptide bond, right? You could have a D alanine isomerized with an L alanine in the sense that they both have the atoms needed to form the bond. But the way this polymerization works in practice, of course, that you have enzymes driving it. So in your body, all the proteins and enzymes in your cell, they would not take up D amino acids and use those D amino acids to polymerize with the L amino acids. So in the very lowest chemistry level, it's correct that they could. In practice, they would not form any stable proteins, and also in practice, your body would never use them. But that's actually a great question. Yes? So it depends. So that these are... this is not a 100% selectivity. So when it comes to the way... the way a protein is digested is really that the peptide bond is stable. It would like to be there. So to break a peptide bond, you have an enzyme that binds the entire complex, the two amino acids, it's a long chain of amino acids, that binds this in a way that you can then break the bond and then release it again. So this requires the entire part here to fit very well in this complex. How well this fits is going to depend on the amino acids. A simple example, again, glycine. Glycine is not carol, right? In the case of glycine, we don't even have this separation. But in general, all this fitting is going to be based on billions of years of evolution, and all this level of evolution has selected for L amino acids. The other part where this is a bit academic, of course, we don't have... we have very few other organisms where there actually are D amino acids, that these amino acids would have to be synthesized entirely artificially. The other thing I mentioned yesterday is that, remember, there are more than 20 amino acids, right? And you can certainly use some of these non-natural amino acids to try to achieve the same effects. This is a huge problem, actually. We'll come back to that later on in the course. But some of the most promising modern pharmaceuticals, are what you call biologicles, that proteins, essentially. Biologicles are just a fancy name for it. The problem for the pharmaceutical industry is that no matter how successful such a drug is, you can never make a blockbuster drug if that requires the patient to go to the doctor and get an injection twice a day. And that's on a number... well, there are a number of reasons for that. Most of us would not accept getting injections unless it's a life-threatening disease, which means, suddenly, that 99% of the diseases are after the question. The other problem in large parts of the world, you might not have the health care and everything to simply sustain this. You might need to keep this protein cold and everything. What every single pharmaceutical company wants is a small pill that you take. So we can make biologicles be administered with pills with a huge amount of money to be made there. Yep. Yes. So there are lots of drugs that are very... it's actually some metabolite that has the real effect. Even when it comes to alcohol, a lot of the poisoning effects are actually the metabolites rather than the ethanol. So I'm not sure whether that relates directly to the deisomers. The deisomers would rather... this is part of a general concept that you would like to get through the digestive tract without breaking down whatever drug you're giving. The digestive tract is a pretty scary place for any type of molecules to be, right? Insanely acidic pH, lots of enzymes that can break down everything. They protect you. So we continue with some of the questions here. That was...let's see. So you had a question before, where were we? What are the levels of structure, organization and protein? What I mentioned yesterday too, so why do we classify structure this way? Rather than saying this way, why do we classify structure in the first place? Yeah. In particular, in addition to function, to understand it, right? Even now this morning we talked about pi helices versus 310 helices versus alpha helices. And I could just refer to, say, pi helix and hopefully understood that that was a slightly wider helix. I could talk about a 310 helix. You understood that that was a slightly more condensed helix. The alternative would be that I would have to stand to talk here about specific hydrogen bonding patterns. In the helix, when the first residue is hydrogen bound to the fourth residue, it becomes so complicated. And this is just on the level of a single helix. So try to understand how large structural features work if you needed to go down to the atoms every time. It would be too complicated. The other reason which I think you touched a bit on the bioinformatics course, and we're going to come back to that later on in the course, is that this is intimately related to evolution. Nature, in bioinformatics we frequently spoke about evolution and domains as in the terms of that's the unit by which evolution happens, right? It's going to turn out that this is intimately related to what can fold. These domains are typically the smallest independent folding units, some sort of building blocks, bricks that proteins well, yourselves tend to reuse in proteins. So describe the relation between sequence structure and function. We're going to spend more time on this too later on in the course. There is a word for this. What's this called? Sequence to structure to function. Central dogmaia. We typically just mentioned in that order, there was one important thing you said here, chicken and egg problem. Normally people would not call this a chicken and egg problem, but I would argue you're right, because there is an arrow back that we virtually never draw. What is that arrow back? Well, so it's not that function never generates structure and structure never generates sequence, but there is a feedback loop here. What's the feedback loop? Well, there's evolution, right? And a particular natural selection. Because a sequence that leads to a bad structure that leads to a bad function will not survive. And conversely, if you have a sequence that leads to a stable structure that is very efficient, this will likely lead to that sequence being emphasized in natural selection. But it's not the direct feedback, it's indirect. So let's, we can continue here at the beginning. Does function induce structure? Well, that's kind of hard. For now it's an open question. Not directly, right? But it's kind of the same thing here. You could argue that natural selection does it. The other thing one could imagine is that what if you have a large protein that somehow binds another protein and the binding of the other protein or the small compound somehow changes the structure. So somehow in general you are right, but I'm just saying that remember what I said yesterday was an exception. So in some cases the function actually can enforce a bit of the structure. But in general you're quite right. The answer is normally no. Yeah. Yeah. Yes. That depends what? That depends on the structure. In some cases yes. But we'll come back to that way. It's like you're six lectures ahead. I love a listeric regulation. We're going to talk lots about it. My favorite research topic. Nine. How are amino acids linked into a protein chain? Yes. And this says a bond that has to do with sharing of electron orbitals. So what type of bond is it? Yes. Well in particular it's a double bond. It's a stiff bond that you normally can't rotate around. This is not, of course, in quantum chemistry. I don't expect you to know any details about electrons and everything. But you should know that the way the peptide bond is set up means that you can't rotate around it normally. And number 10. Methods with which we determine protein structure. So this is cool. Five years ago there is no way I would have expected anybody to say EM. But today I would say EM would like to be my first bet. Another reason for that that we'll come back to later on in the course. Determining a structure with x-ray crystallography takes probably two years. You should purify the protein, overexpress the protein, purify it, and then you should be lucky that you're not scooped. This is a field that kills people's career. You have the most outstanding students spending four years on the project. They're just ready to publish, and then they're scooped by another group that just publish the same structure. With cryoEM, you spend three months on it. It's still not fun being scooped, but you survive being scooped after three months. You don't do it after four years. There is a new method as gradually. We are likely going to see some sort of structure being obtained with neutron scattering. Not yet. I would not say that it's the method of choice, but it could happen in the future. 11. Which ones are the most important degrees of freedom in proteins? Yeah. Why? There are many torsion or dihedral angles in proteins. Which ones are the important ones? And which ones are the ones that majorly change the structure? Yes. The backbone is key here. Every single side chain has torsions, too. But if you rotate the carbon-to-carbon bond around a CH3 group, it's going to cause the hydrogen to rotate a bit. But this chain effect that it's going to have effects later on is what's really going to change it. How do you define a dihedral torsion angle? That might be a bit hard to do just orally here. So I will skip the next one. Cis and trans conformations. So that's... Yes, I would say this is not the R groups, right? This is just the groups, the atoms, around a single bond. Not necessarily amino acids. Not necessarily the R groups. Because the R groups are going to be... There will be three torsions between two R groups. So when it comes to cis or trans, we're just looking at the distribution of the heavy atoms around a single bond. Yes. Cis are there on the same side and trans are on the opposite side. So why are cis conformations almost never observed in peptide bonds? Yeah. So when I... You have no idea how spoiled you are. Because when I started I was a famous professor in London in 1993, I think it was. We did experiments on this. So then you had proteins, models, and then you took a gigantic spreadsheet and divided this in like whatever, 200 small dots. And then you set with an angle and moved this one bit. And then you checked whether they clashed and then you put an X mark in that one. If they didn't clash, you put a circle. That was good. So we spent one entire afternoon to pretty much manually draw a Ramachandran diagram. It was the natural thing to do before you had computers. So what did Anvinsen and Leventhal say? Yes. Even free energy. I'm going to come back to that later. But this is key. What I mentioned yesterday, this is one of those results that all you are going to think it's obvious. This was so not obvious. It's a paramount importance means that protein folding is in principle, at least simple. This obeys... We cannot understand these laws. But what's Leventhal's problem then? Yes. So that... I'm not sure why I told you that explicitly yesterday, but you can even do a back-of-the-envelope calculation. If you have this 10 to the power of 300 conformations, if you could test one of these per, say, nanosecond or something, it would take longer than the age of the universe to fold the protein. So we know that they find the best conformation, but it's impossible for the protein to exhaustively scan all the conformations. And that's going to be a problem. Why is water such a special molecule? Okay. Why? But I would argue that hydrogen is important than NMR because of water. So the special thing with water is that it's an exceptionally polar molecule. There is pretty much no other small molecule that is as polar as water. And this is what is going to create the ability to form very strong hydrogen bonds. You have no idea how strong hydrogen bonds are. They're not quite a normal bond, they're stronger than any normal weak interactions. And this is what, in turn, creates this whole salvation effect that a protein would like to separate itself from water and have a shell around it with water. This is what creates a whole bunch of the effects that we can't solvate, for instance, ions and everything. The entire experiments you tend to do in NMR, for instance, is that you're looking at the hydrogen exchange between a protein and the water and everything. And that, of course, would not work unless you had a clear separation between hydrogens between a protein versus hydrogen in the water. We're going to talk more about water today. So what's the key finding of Watson and Crick and the DNA structure? We talked a little bit about that, but I'll skip that one. Phypsi, omega and chi. So which one is which? Yeah, and you will almost never see that because it's almost always trans, but not always. And chi, yeah. And normally you just talk about chi1, but you can't have a chi2 and chi32, but the further out you are, the smaller the effect of those. So normally we will only talk about phypsi. So phypsi, you need to know immediately what they are. Describing energies that are typical for a few interactions in proteins, well, I'll make it a magnitude of some energies in proteins. So what type of energies are we talking about? So when you say strong, how strong? You can pretty much define this any way you want, right? But it's important to have one example of some sort of reference interaction where you know roughly where they are. And I like this example of two charges that are very close to each other. And then you talk about a couple of 100 kcal per mole. So two unit charges right next to each other, a few 100 kcal. And another example could be a hydrogen bond. That's something that you need to know. Everybody, what's the energy of a hydrogen bond? Five is what? Yes, remember the units. We, I already said yesterday that you spend roughly kcal, I can also want to roughly your body's weight of ATP per day. And these post-translational modifications, it's not really critical to the course, but it's basically when you're adding or removing something on amino acids after the fact that the chain is coming out of the ribosome. There are a bunch of different ways you can classify these amino acids. I will not really go through that in detail, because, and we talk about the Ramachandran diagrams. I will head on to what we're going to do today. I'm going to tell you a little bit about computer history, actually. We're going to think a little bit about models, and I'm going to reintroduce this in with partial charges that we spoke about yesterday. I'm going to dig a whole lot deeper into the energy landscapes, and in particular the Boltzmann function. You're going to do physics today, or at least a bit of math. We're going to start looking into entropy and free energy. These are super important concepts. You're going to hear them like five times in this course. Today we're going to use it relatively easily, but it's going to get more strict later on. Yes, you had a question? Yes. Oh, sorry. I thought I had had those out. Handouts. There they are. And if there was somebody who hadn't signed up, the sign-up sheet is on the very top. This is a movie I got through Mike Leavitt a few years ago. Very old movie. Filmed with a 16mm, or 8mm camera, I think, in Cambridge. This was amazingly cool when it first appeared. So these are static X-ray structures that people put in a computer. They put the coordinates in a computer, and then they calculated the three-dimensional projections so that you can tell the computer to rotate this and kind of try to understand what's happening. I think we had a second one here. That was myoglobin, right? The second one is probably lysosine. Yes, lysosine. Philips was also one of the very early heroes of the laboratory molecular biology. So this is another small, similar structure. To the people working with this, this was heaven. Remember what I showed you yesterday when people are standing and building these gigantic proteins manually on small steel rods? The fact that you could actually use a computer to do this was unheard of. So the people, the person who did this is somebody you have actually heard of. It was a geek. He was using a Mac. Not the type of Mac you're used to though. So this was his Mac. Multi-axis computer. And the geek behind this was actually Cyrus Leventhal. He was one of the first persons to be using computers in this entire field. And you even have something. I'm not sure whether you should call it a mouse, but this was basically the input devices. You didn't have a normal screen. Because computers, you didn't have text on computer screens. So the only screen was kind of like an oscilloscope that could draw these things. And the movies you saw was they put an 8mm camera in front of the screen and filmed it. This is roughly 50 years ago. The very first visualizations. You even have a picture of Cyrus there. And on the Mondo site there I managed to get hold. This was not entirely easy. In 1967 or 8 Cyrus published a paper in Scientific American. In those days Scientific American was one of the most amazing journals you had. Because you had some of the best scientists in the world writing good, popular stories about the very state of the art in research. So there is like a 10 page description how you can use computers to model proteins. It's an amazing read and it's also something I would recommend. Because what's happened over the last 50 years is that today we're doing everything in computers. Not just modeling of proteins, but even determining structures with cryeum and everything. That was a bit of parenthesis. I'm going to come back a little bit to these interactions. I mentioned today, and we spoke a little bit about yesterday, that some of the special, I guess you should say that every single interaction we have is due to electrons. Unless you're looking at sort of nuclear decay when it would be the nuclei, but we're not into nuclear physics. So if everything is based on electrons, why are we not doing quantum chemistry or quantum mechanics? The reason why you form a bond is because of relatively advanced or relatively advanced correlations between the electrons, which you can formulate for instance in orbitals and everything. You can talk about spins with electrons. There is no end to the amount of mathematics here. So that if you really want to understand interactions properly, you have to go drill all the way down to the electrons. The way we typically explain how bonds form and everything is a concept called orbitals, which is also not really part of this course, but that it's isolated electrons very much like to pair up. And when they pair up, you're going to have one electron with the spin. This is just a mathematical concept, but you can say that the spin is pointing up or the spin is pointing down, which is just a way to say that they like to form pairs. But this is a small r. As these atoms or electrons get further and further and further and further away, there's going to be something that happens. So eventually, electrons won't really form bonds anymore. I guess you all know that bonds are in the orders of angstroms, right? So at some point, the electrons are going to prefer to be in different molecules. Now, the way we define energies is that if two particles, any particle, is at an infinite distance away from each other, what is the energy between them? Zero. That's a completely arbitrary definition, but it makes more sense to say that it's zero than 39.3. But you could do that. You could define a physics with any number. It's just physics is all about relative numbers. So we just, if there's one absolute number that we can define, that's same for all systems, let's call it zero. But that means that in most cases, matter interacts when it gets closer. And it interacts favorably. So in almost all energies that we're going to talk about are usually going to be negative. Because if it's zero at infinity and it gets better, low energy is good, it has to be negative. Unless you get to the point where something starts repelling. So that, and electrons actually, there is some sort of intermediate range between infinitely far away when they don't interact and very close so that they form bonds. So even if two atoms don't bind to each other, this water molecule is going to have more plus charges on the hydrogens and more minus charges on the oxygen. Even something like a xenon atom that absolutely has no charge whatsoever. It still has a plus charge inside the nucleus, right? And then some electrons around it. So what's going to happen here is that the water, the dipole in the water molecule will the negative side here is going to push all the electrons a bit away from the nucleus here. Because on average if I'm, sorry, if I'm negative here, I'm going to like that the xenon tried to turn its positive part, that is the nucleus slightly closer to me, while the negatives part slightly further away, right? Now you're talking about a difference about a femtometer or something, it's insanely small. But this will now mean that this xenon atom suddenly has a slight dipole. It's very small. But now there is the xenon that we think of never having interaction is now having a start, not net charge but the net distribution of its charges. This will change all the time but that means that you can even take two xenon atoms, xenon here and xenon there. And because the electrons fluctuate a little bit in one atom that's going to induce fluctuations in the other atom. And now you have two small dipoles. And these dipoles will either like or dislike each other and on average they're going to like each other a little bit. So any type of atom, even if they're not charged at all, are going to have very weak interactions at large distances. And there is a name for this, like London dispersion forces. You're probably more familiar of hearing of these as Lenard Jones or something. But this is the reason why even xenon and neon and helium will eventually condense. So somewhere around very close to zero Kelvin, even noble gases can form solids. These interactions are orders of magnitudes weaker than the charges we spoke about today. Now of course they're formed by electrons but we don't think of these as charge interactions. We don't think of them as electrostatics. But they're going to be important. Why do you think these are important if assuming that they're a factor 100 to 1000 smaller than electrostatic interactions, couldn't we just ignore them? I might have a slide about this later but then I'll skip it in that case. It makes sense, right? If something is a 1000 factor, 1000 smaller any person who's smart should just ignore that. It simplifies life. So this is a problem. What I just said, right, that what is the sign of these? Are these attractive or repulsive? These are not just on average, they're always attractive. At large distances all atoms attract each other and that's why all the noble gases will eventually condense. Electrostatic interactions. Are they attractive or repulsive? So that's the problem. If you now take 1000 interactions and sum them up if it's electrostatics it's going to be noisy. If it's this type of interactions it's now going to be factor 1000 stronger. So eventually as you sum up more and more and more and more and more interactions these features could even start to dominate. So with all these electrons and everything you could argue that we should stop teaching biophysics and we should go back to quantum mechanics. We really need to understand all the gory details here exactly how the electrons are distributed. We're going to need to study another 60 credits of physics and mathematics and we can completely forget about all the biology and life science. In principle, that's correct. You need quantum chemistry to understand life correctly. Quantum mechanics is the correct description. There are a couple of problems with this. Quantum chemistry can only handle roughly 100 atoms. To do this accurately you're going to need to use time dependent theory. You're also going to need to use relativistic theory because electrons move so fast and at somewhere there we can handle one electron almost. So the problem is that this is something that in theory is great but in practice it's completely inadequate. The other problem is that you assume that the nuclei don't move which is kind of bad in biology because everything here is going to be about motions of proteins and everything. So at this point you've completely thrown out the baby with the water. You can have a beautiful descriptions of electrons for assistance where you don't understand anything about the biology or physics. No water. You can't handle water because then that would be way more than 100 atoms and quantum chemistry is not going to help us in this field. You can even argue that this system of two is of course accurately blistered by quantum chemistry. It's just that the way this is a field where we all know that it's ridiculous we don't need quantum chemistry. As you will see in this course when you look at proteins there are a handful of exceptions but in general we don't need quantum chemistry. It's not quantum chemistry that decides when a protein folds. That doesn't mean that it's not correct but we don't need it to understand it. There was a Nobel Prize a couple of years ago that I won't really go through in that much detail. The only reasons are Mike were one of the people who get this which is kind of fun so I worked with him for several years. There are ways to extrapolate, to combine quantum mechanics and simpler scales. The one thing where this actually saves you is when it's enzymes. Because when it comes to breaking or forming bonds for instance in a peptide chain to break that bond you need quantum chemistry. So there are a few cases where this is really cool where you can really understand biology with it and that's why the Nobel Prize in chemistry was awarded to this field in 2013. Oops sorry. Partial charges so before we head on to understand I'm going to try to get on to the point where we're really going to talk about entropy, energy and what we can do with these interactions but to make that useful to you I'm going to need to spend a couple of slides to go through the type of interactions we have in proteins and just so you know what we're talking about. We're going to come back to this later on in the course when you're actually going to start running some simulations and calculations yourself on structures. Partial charges is the first one that I spoke about and that's very much based on motions of electrons too. In a simple molecule such as water the electrons will move slightly towards the water which will mean that the water has a slight negative charge of roughly minus 0.8 and the hydrogens have a charge of roughly 0.4. This in water this effect is huge. If you look at something like benzene you also can have an effect that the carbons will be slightly negatively charged and the hydrogens are slightly positively charged but you see that there is an order of magnitude difference there and you can even take something like a few amino acid or something and calculate the partial charges. This is something you need quantum chemistry for but thank god there are programs that just do it automatically for us. The simple answer is yes, definitely. You could also argue electronegativity, chicken and egg. Electronegativity is a number that we put on an atom to describe how much of this effect we have. So at the very core of it this has to do with the orbitals you have and the electrons and how they relate the number of electrons you have around each atom, some of the properties of the nucleus. You can't translate this to single numbers such as electronegativity, yes. So it's very much related to electronegativity. The other part we spoke a bit about was bond stretching. We typically in a protein or something you can... one thing that you're going to see here and it's important to dare to make assumptions. If you want to try to guess or describe how a bond is changing a bond is super complicated really, right? Lots of electrons. But if you don't know anything else about it try to describe a bond one atom here and one atom here and then you have some sort of spring or potential between them. So that you could say that on average we would like to be at one angstrom if it's carbon to hydrogen. If you're significantly shorter than that you're going to try to push it back to one angstrom and if you're significantly longer than that you're going to try to contract it back to one angstrom. That is what you call an harmonic potential that is an X2 function which is not how it works in practice because eventually if you pull hard enough the bond is going to dissociate and the atoms break, right? So this blue curve would actually be more correct than it's called the Morse potential. We're never going to talk about Morse. To make this really correct you should use a sort of quantum chemistry to say that in quantum mechanics there are discrete levels of energy and this is technically true and if we start there we're never ever going to move beyond bonds in this course. What happens for a normal protein is that your bond is going to stay here. You never have a bond that's eventually you could have a bond in between a carbon and hydrogen that stretches to two angstroms if you have a temperature of roughly one million Kelvin. If you have a temperature of one million Kelvin in your experiment it's not going to be particularly relevant to biophysics anyway, right? You're probably building a nuclear device so that focus on the model we care about in our case we're going to be down here and that's also worrying whether the blue or the green curve is correct here. Yes, you could argue that technically the blue curve is slightly more correct but it's a lot more complicated to work with. So many of the greatest discoveries in physics in general and biophysics in particular have come from people daring to make very simple assumptions. And that's another part what I love with this Watson and Crick paper just look at the model of it. Super, you remove all the atoms. You remove all the details. So think of the concept as a backbone and then pairs of building blocks pairing up. Oh sorry, I should show you some bonds. There are tons of bonds in a protein that you probably understand. I shouldn't be in front of the camera all the time. You can define an angle in a similar way. So in an angle that would be three atoms involved in it and just define the angle between these three atoms it's pretty much the same thing as a bond. These angles can move a bit more because the springs are less stiff but somewhere around 300 Kelvin they might move up to a couple of degrees or something. It's not completely irrelevant but these are not the degrees of freedom we're going to worry about. And as we talked about yesterday you have the torsions that we care about a lot. Since you all know the definition of torsions right now I will move over here instead. So we'll look at a couple of them. Ethane. The average energy here is roughly 3 kilocalories per mole on the peak. For butane this gets significantly more complicated and the reason why butane is more complicated is that depending on orientation along this bond you're either going to have both these two methyl CH3 groups right next to each other. When they are x-ray each other it's going to be really bad roughly 6 kilocalories per mole. The trans state is going to be by far the best but between the trans and cis state there are also going to be these so called gosh states when you just have a small hydrogen staggered right next to the CH3 groups. So in general these torsion potentials or something can get very complicated. You're never going to need to derive them this is also something you do with quantum chemistry other people do it for you. But as a building block it's going to help us to do advanced things. This is an example for the simplest molecule you can imagine in a protein, dialanine, dipeptides. So it's pretty much a single alanine residue. So you have one phi and one psi residue. You need a couple of atoms before and after it so you can actually define them. And then you get an energy diagram like this. So here you have phi on the x-axis psi on the y-axis and the color here corresponds to energies from roughly zero Cal when it's blue up to 2025 when it's red or yellow. The red parts you're never going to visit so in practice you can move between 2, 3, 4 states here. And we also spoke a little bit about Ramos-Chandon diagrams yesterday. The thing to remember with the Ramos-Chandon diagrams is that rather than worrying about specific energies what I showed you in the last slide there you have a specific energy and this is actually correct roughly to within a K-calor. But in most cases you're not interested in understanding the exact levels here. If you actually wanted to do that let a computer do it for you. The point where Ramos-Chandon diagrams are useful is to decide entire areas here that are allowed versus disallowed. So this is likely good. This is likely good. This is horribly bad. It's never going to happen. And that's why both the book and we frequently work with this super simplified Ramos-Chandon diagrams. So just say that it's black if it's disallowed and white if it's allowed. So if you did not have any side chains at all, just the backbone and this is not a real chemical molecule of course. Then you would have there would be one band in the middle here that's disallowed because then the chain would move back onto itself. But apart from that pretty much everything would be allowed. Gray here just means that it's not ideal but it works. So that remember what we've said now a couple of times that the backbone torsion angles are critical because they determine your degrees of freedom. They determine how the protein can move. But what is that they don't determine if you look at this part? Let me phrase that differently. Why do you need the side chains? So why will the side chains help you? Couldn't you form a helix here? You don't get the specificity, right? This structure could form a helix but if you play with the idea you could form a helix here. An alpha helix would be roughly here. The only problem is that it could also form any other structure. So why would this want to stay in an alpha helix? There is no reason why this would stay in an alpha helix. And it doesn't help that one case out of a billion it might be in an alpha helix if it would immediately move away from the alpha helix again. So if you then look at even glycine that is super simple. We still don't have any real side chains. With glycine we've only added the oxygen and the hydrogens to the backbone and we're already starting to rule out a whole lot of part of the structure. And as you move to more complicated ones, alanine is still relatively simple. Lots of gray areas that are kind of okay and virtually all normal amino acids look like this. Compare that to the first one, right? You've excluded 90% So the side chains help us achieve some sort of specificity. The side chains are going to say that only very few structures are allowed and exactly what structures are allowed are going to depend on what side chains you have. And this is where proteins are polymers like normal polymers but they're hetero polymers in that there are different monomers, amino acids in our case in different parts of the structure. But this is also going to mean as I'm sure you know from bioinformatics there's going to be a very specific relationship between what amino acids you have and not just what structure you have but also what function you can achieve. So we spoke a little bit about the electron interactions at long distances and just that you had electrostatics we had bonds, angles, torsions if you're going to describe a protein we're going to need to find a way to describe these electron interactions at long distances. That's very easy we have a some sort of potential that goes as 1 over r to the power of 6 that's a constant but we can determine that constant. How can you could you imagine how you determine that constant? Experiments, but how sorry to be experiment yes but how do we do determine it from an experiment? Yes. So in general there are going to be more constants here but there are lots of this is going to affect lots of things right it's going to affect how hard the atoms are bound to each other that's going to affect like the boiling point of the substance it's going to affect the density of the substance so there are a bunch of relative as long as you have a pure substance say pure O2 or something we can measure experimental properties of that and then decide what should this parameter be for an oxygen atom to do this really accurately if you that's at long distances somewhere if you start pushing atoms closer and closer and closer together they're going to like it more and more and more they start going down the quantum starts overlapping each other that's going to get really bad just as you can't take these two tables and push them into each other at some start you're going to get a super strong repulsion and this is based entirely on quantum chemistry and it's possible to show that that's an exponential for in theory these are pretty difficult to calculate these two are very important to get accurate if you want to build a nuclear bomb if you don't want to build a nuclear bomb it doesn't really matter exactly where the curve is going to be up here so what people in biophysics and most of chemistry actually do rather than having a very complicated exponential you just use a steeper one over the radius to the power of 12 there is a deep history to that calculating an exponential function in computers is relatively expensive and when you're going to simulate this later you're going to calculate this part roughly a billion times per second you don't want something expensive there but we already calculated one over r6 for every distance between two atoms and if you take this number and just multiply it by itself you get one over r12 there is nothing that's really correct about one over r12 but we don't care you just want something that goes up very quickly and it's very steep here to make sure that things don't overlap in practice you're never going to get here and atoms are going to stay down here so the exact shape up here is completely irrelevant and then the way we get these then we have two numbers we're going to need to decide how much repulsion do we have and how much attraction do we have between atoms and the way you normally do this for very simple atoms is that there are two numbers we know for most liquids we know what the heat of vaporization is that is that if you are if you have water at 100 degrees and then you keep adding energy so you evaporate this to form vapor at 100 degrees to first approximation in vapor there are no longer the atoms are infinitely far away from each other right so they're not interacting so this is going to be related to how strongly the atoms interact the heat of vaporization the other part is going to be the density that is well we know in the case of water we have a density of around one that gives you two numbers and you have two parameters to get this is absolutely horrible in a way this should have been derived properly with quantum chemistry but in practice you can cheat by just fitting these numbers to experiments and the person introduced this entire concept with Ari Warshall there's also a reasoning about the Nobel Prize in 2013 this is because if you try to do this from ground up with quantum chemistry there is no way we could treat complicated molecules but by being smart and realize that rather than trying to solve this exactly from physics use the fact that this is super easy to measure in the lab and then you create what you call empirical parameters that this is an average repulsion that we know is going to correspond to the average of what these atoms actually feel it works great yes sorry here oh long story this has to do with spectroscopy and how you measure different wavelengths in spectroscopy I hadn't even seen that in spectroscopy you frequently measure energies in wavelengths because it corresponds to where you have the absorption in this case it's irrelevant I should probably remove that thanks I hadn't even seen in principle those are all the interactions you have and you can describe everything with that but as I alluded to yesterday hydrogen bonds are so special and we're going to come back to them so much that we have a chance to talk a little bit extra about them and we hydrogen bonds are electrostatic interactions but because of their strength and everything we typically treat them as a special electrostatic interaction if you look at a water molecule and in particular the oxygen the way these electrons are normally divided around the oxygen is the electrons are negative and they you could think that the electrons would like to be as far away as possible so this is pretty much going to be a tetrahedral shape where you have the electrons in the corner and when you add two hydrogens there are going to be two hydrogens along two of these orbitals but you're also going to have two unpaired orbitals here but there is some electron density but they don't really form bonds so for a water molecule we're used to drawing the water molecule like that which is completely correct but on the opposite side here there are pointed negative regions that are more negative than the oxygen so it's not just a point on the oxygen that the oxygen itself has lots of negative charge this negative charge is concentrated into regions and that's why anytime you look at water including the movie I showed you yesterday when two atoms are close to each other this water is going to love to turn this negative part towards a hydrogen on another part and that's going to create this connection between them for this to work you're going to need one molecule when you actually have a hydrogen it's not going to be okay with any hydrogen this needs to be a relatively polar hydrogen so you can't take an ethane or methane remember the partial charges I draw you when you had a small hydrocarbon right because the carbon is not really that electronegative so the carbon is not really going to pull the electrons away from the hydrogen it's not going to be very charged but if you have either an oxygen or a nitrogen here or something that pulls the electrons away from the hydrogen the hydrogen will effectively be charged similarly on the other side it's only going to be an atom that actually is relatively electronegative that will have this property of having some sort of unpaired orbitals where you have electrons around it so it's not going to you will never have a hydrogen bond in a pure hydrocarbon but you're going to need nitrogen or nitrogen involved to have these effects so they're only a handful of atoms where it can happen I will come back to this later this is exactly the same property we see in proteins so in proteins you always have an NH group paired with a CO group that's always where we get the hydrogen bonds and that always, virtually always includes the peptide bond there are some other parts that can form hydrogen bonds but in helix what stabilizes the helix are all the peptide groups hydrogen bonds in the peptide groups along the helix so let's try to apply this and see we can understand something this seems a bit abstract but it's not as I mentioned yesterday one of the best ways to study bonds and study energy in bonds is to use spectroscopy because what spectroscopy essentially does is that you can pump energy into a molecule and see does the molecule try to absorb this energy if the molecule did absorb the energy there was some sort of motion in the protein that oscillated with this period and in this case you can use infrared absorption spectroscopy the technique is relevant for us and then you have different wavelengths here in micrometers and then you have different absorption and depending on what liquid we have we're going to get slightly different properties so first you have ice we can have water in a so this is one of these molecules that keeps changing name tetra tetra-methyl chloride no that's not what it's called today I'm too old, never mind it's a methane where you replace all the hydrogens with the chlorides and three is liquid water what does this mean for hydrogen bonds you can learn quite a lot of hydrogen bonds from this just for studying simple things like this or I would also phrase it this way right that when it comes to what you're mainly seeing here is the you're pumping things into the bond between the oxygen and the hydrogen and in ice ice is a relatively uniform measure right so that virtually all these bonds have the same distance and it's rigid same thing with water in the CCl4 it's well the CCl4 is not really going to be involved in this it's just going to be the water so you're going to have individual water molecules that you pump into the JN2 the difference when you actually have this in liquid water is that it gets more complicated and that some of these oxygen hydrogen bonds are inside a single molecule others are going to be interactions between two different molecules you're also going to have the water moving much more so the entire spectrum gets much more spread out but you also see that there is a lot of hydrogen bonds in liquid water just as in ice it's not really that we break the hydrogen bonds that they're not there this is intimately related to a concept that I I only introduced in the very last slide yesterday it's 10 minutes past and let's give me 5 minutes and then I'll give you a break called the hydrophobic effect which I would argue that the hydrophobic effect is the reason why I asked you why water is the most important molecule in biology so the hydrophobic effect is something the first time you see it you're convinced you understand it the more you look about this the more complicated it gets so try to follow me here actually let's take this slow and interrupt me if you don't see where I'm going what the hydrophobic effect is it's really you could argue that this is applied hydrogen bonds what do we know about hydrogen bonds are they strong or weak strong yes so they're strong there's lots of energy in it so if something happens to a molecule or something will the waters try to maintain the hydrogen bonds and if the water had to make some unfavorable say Lennar-Jones interactions to maintain a hydrogen bond what would they do would they give up the Lennar-Jones interactions or would they give up the hydrogen bond they would give up the weak interactions it's more important to keep the hydrogen bonds so waters will do almost anything to maintain hydrogen bonds and as you saw in this movie yesterday perfect ice has on average two hydrogen bonds per water liquid water has 1.7 300 Kelvin temperature difference so that the water do not give up hydrogen bonds until you boil it so what happens if you now take something that's not polar like small carbon the compound doesn't matter this could be an ethane or methane any small molecule that is not at all polar this molecule the grey molecule here will not form a hydrogen bond what happens if you put such a molecule in water yes you're right it will have to make some contact with water but the problem is that these water were already making contact with other waters they were so happy with all their hydrogen bonds and if you're not putting a drop of oil there you're breaking their hydrogen bonds what will they do they will somehow try to rearrange to maintain their hydrogen bonds you can think of this as standing holding hands in a circle around it because if they had such water that had to give up a hydrogen bond would lose 5 kcal per mole so if you now put 2 drops of oil in water what these drops of oil can do the amount of hydrogen bonds that are influenced here are pretty much proportional to the area right? so you have one surface area here and one surface area here but if you take these two hydrophobic parts and put them together the total surface area here is going to be smaller because we gain the area that they're making with each other you have all seen this what happens when you put a drop of oil in water yeah, they don't mix right this is the hydrophobic effect and same thing here if they start making some sort of network they're very rapidly going to create a new network around this so why is the solubility of hydrocarbons in water low sorry, they are they are not polar but that's technically true but then in many cases I would argue that the definition of not polar is that you have low solubility in water which is not quite true you could argue that they don't have dipoles in the molecules but in the context of hydrogen bonds what happens when you put a hydrocarbon in water they can't form hydrogen bonds with water so what happens to the hydrogen bonds that's a very good answer that is completely wrong don't get me wrong it was the one I was looking for just to remember what I said that the hydrogen bonds are strong, right so will the molecules give up their hydrogen bonds they will do anything to avoid giving up those hydrogen bonds anything and as we will come back to after this might be more tomorrow the anything that they will need to do normally water is fairly happy you saw the movie yesterday to maintain all these hydrogen bonds you are going to need to form a very stiff network around this to make sure that there is you are essentially almost form a super thin layer of ice around a hydrophobic solvent to make sure that the hydrogen bonds are maintained no, this is not shielding because had it been an ion it would have been shielded right, if you had a charge the waters would orient if you have a positive ion the waters would orient to turn the negative oxygen towards the ions this is not really shielding there is even a name for this called a clatterate structure or something so that you form a rigid cell around this where at least in the cell they mean I think you can see that here right so that all the yellow lines here are hydrogen bonds so that you see how all these water have oriented and this is actually a simulation so all the water here have oriented around this there is probably an argon atom or something they have oriented around this here to try to maintain all the hydrogen bonds so that they are not broken but what this would mean is that all these water molecules are now stiff, they have almost formed a structure like ice and that is a very well ordered state and that is actually going to be bad so it is going to turn out and that is where we are going to need to understand these concepts of ordering and everything and that is where we are going to need free energy and entropy a bit you do not lose a single hydrogen bond but this ordering is a bit costly because the ordering is bad that is why the solubility is low we are going to come back to that after the break the there are two more slides and then I will give you a break I could take questions too the hydrogen bonds in proteins are kind of similar and they are driven by the hydrophobic effect if we assume that these yellow groups are hydrophobic you can imagine say valine or something something that definitely can form hydrogen bonds in water if you have a stretched out protein they will need to be exposed to water molecules and this is probably not the world's best plot because it looks as if the water molecules love these hydrophobic groups they do not write, this is really bad because these water molecules would need to solvate valine, oil just as a drop of oil will separate from the water could you imagine that it would be great for this protein to fold up put all the hydrophobic groups together this is essentially forming a drop of oil inside the protein and then you put all the waters around it so this is going to be one fairly strong effect of protein folding why proteins fold and why we form stable structures it's all driven by these simple fundamental interactions that it's bad to solvate non-polar groups and we can calculate that it's actually very easy to calculate that if you have a small molecule you can pretty much calculate roughly what the area around the different atoms are what is the rough area of valine what is the rough area of alanine and this works surprisingly well most of the small compounds you can estimate how hydrophobic they are based on how large their non-polar surface area is the book goes through this in a little bit more detail we might come back to that later when we compare amino acids so just to sum it up here before we take a break there are a couple of fundamental interactions we have all these so-called bonded interactions bonds and angles that we're never going to talk about more often today I promise you have the torsion angles the torsion angles is still bonded in the sense that it's an interaction for atoms that stick together and then on top of that you have a couple of non-bonded interactions lena-jones or van der Waals interactions and electrostatics they're much longer distance and there are more atoms involved and then you have these hydrophobic effects which really are electrostatic interactions but the reason we remember that I said that it's good to classify structure because otherwise it gets too complicated if you need to account for all the details you've only seen the beginning of the hydrophobic effects this is going to get complicated and if for every single hydrophobic effect we had to drill down and talk about partial charges on atoms it would get too complicated so that's why we now we're going to start to talk about hydrophobic effects in terms of water and how polar or non-polar water is but this I think is a good place to take a break 20 should we take 15 minutes and actually you can have 20 I don't have that many slides today but a bit of mathematics so let's meet here at 20 minutes to 11 let us get started again I don't you're going to like this I have 20 more slides but it's going to take at least well I will stop by noon but we're not more going to get through the slides because there's going to be some math now as what we will talk about later is that all these interactions will you sum them up right you're going to have different interactions they're going to have different weights but if you know all the small interactions in the protein you could conceive all of them up and say what is the total energy of the protein I'm not saying that it will be easy but it's possible to define now if you have a small molecule say one that can be in cis versus trans or something there is also going to be different confirmations of this molecule this will be better some will be worse and then you could imagine plotting this in a very high dimensional landscape if you just have the phi and psi actually no this is not the phi and psi angle but you could imagine having let's speak this might be better you could imagine having something that's a function of two degrees of freedom one degree of freedom here and one degree of freedom here and then the energy in this case this would be a surface right so that this is good this is almost as good this is bad and that is very bad but how bad is this is this bad that it only happens now and then or is it bad I will eat my left shoe if that ever happens we can probably agree that that but will we ever be here there you will probably definitely be here but how good is this is this going to be so good that everything will be here and there is not a single molecule that's going to be here that's going to be hard to say right now of course this is a super simplified view of this in physics you frequently talk about phase spaces or anything I'm not going to go into details what that is about but if you have a molecule with 100 atoms each of these atoms has an x y and z coordinate so each atom has three degrees of freedom so in principle just with 100 atoms that's like a handful of amino acids you have 300 degrees of freedom before you added water so that's a 300 dimensional landscape that gives you a so for each point in this 300 dimensional landscape you're going to have an energy in practice proteins are going to be rather be like 300,000 dimensional landscapes or something so that you can't draw these landscapes you can have a computer calculate them but it's usually very efficient and educational to think about something that's two dimensional and exactly what these degrees of freedom are we don't really care about they're just a thought model to help us think there are lots of people that have spent lots of time thinking about this in the context of physics not just in energy because that the energy here that's just a z scale how many of these trolls do we have or how many of the peaks and how good are they relative to each other and that's part of a greater concept called statistical mechanics or statistical thermodynamics there are a few very famous people that have started this, this is actually a very fun quote from a book by David Godstein or that you probably can't read this but this is the introduction to the book actually Ludwig Boltzmann has spent much of his life studying statistical mechanics died in 1906 by his own hand Paul Ehrenfest, his student carrying on the work died similarly in 1933 now it is our turn to study statistical mechanics perhaps it will be wise to approach the subject cautiously this is partly true but this is hard many of you the point is I think this is hard too not the next few slides because I've gone through them quite a few times but anything in the world that's worth knowing is hard and this will be some pain if you don't feel some pain going through this hard mathematical parts of the course it's probably because you're not spending enough time on those aspects because as hard as they are these are also going to be some of the things that give us most profound insight about how nature works and I'm well aware that some of you might feel not at all interested in the physics you're going to want to design drugs or something but this is how we design drugs today we use mechanics, we use physics we understand interactions in proteins so one of Boltzmann's greatest discoveries what's something called the Boltzmann distribution have any of you seen this I'm going to try to take it somewhat easy what the Boltzmann distribution basically says is something that I and you I think, took for granted yesterday and earlier today that low energy is good that sounds obvious, the only problem is that it's a complete fake because we define low energy that it's good so it's just a circular reasoning so if we're going to somehow speak about energy I can of course say that low energy is good and that's because it's good to be at low energy so what the Boltzmann distribution does is that it starts to translate energies to probabilities so rather than saying good or bad you say how likely is it for something to happen or for us to observe something that's a more stringent way that saying that it's good or bad you can say that 43% of the molecule is going to be in this state but only 22 in another state then it's roughly twice as good to be in the first as the second one that's a stringent criterion it's not just that it's reasonably good now at the end of the day what's up to you to decide is 0.1% important or is 1% important whether we're talking about hydrogen bonds or binding or something, that's up to you to decide but it's a strict mathematical definition and what the Boltzmann distribution does is that it says that the probability is proportional to the exponential function raised to the power of minus the difference in energy between the two states divided by KT T is the temperature and K is Boltzmann's constant there is an amazing amount of fun physics here forget about this constant for now it's going to be one of the most important things later on but right now it's just something to get the units right the delta energy here means that you're somehow talking about an energy relative to something else we're not going to worry too much about that either right now we don't know the zero point the proportionality here doesn't really matter so much either you could argue that there should be some sort of constant here and we can fit that constant to make sure that the sum of all probabilities should be one right I'm not saying that it's easy to calculate but that's a problem that's possible to solve and problems that we know that it's possible to solve we're not going to worry about let's worry about the ones we don't understand so what the Boltzmann distribution means in practice there's things like this if you're looking at the number of molecules and speed for instance at a very low temperature 100 Kelvin nitrogen for instance, nitrogen gas most molecules are going to have relatively low speed so there's going to be some sort of distribution here there's not going to be a single atom that doesn't have any speed at all because then they don't have energy there are going to be some atoms that have significantly higher speed than average and then there's going to be some sort of distribution here as you increase the temperature we gradually push these molecules to have higher and higher and higher speed and somewhere around 700 Kelvin this starts to get really smeared out and the Boltzmann distribution basically tells that depending on what the difference is the energy is what is the likelihood of observing these states it's a simple molecule here at 700 Kelvin you can calculate what the kinetic energy for that molecule is and then we can determine how likely is it for it to have a specific speed what people normally do actually even in physics is that either you take these things for granted or you just have a teacher throwing out this as a postulate or something and the point is that these things are not very difficult we can derive them and I'm actually going to derive this for you next week we're going to derive this in a super complicated way next week I'm going to show you that this is true for any type of system what is a system? a system is kind of anything right and that's the cool thing it's true for the universe it's true for my computer it's true for anything you look at but how do you prove something but if you're going to prove that something is true for anything you can't make any assumptions right because the second you start to make an assumption it's a special case and this is where it gets complicated super abstract because it has to be abstract if you're going to prove something in the general case and then we're going to need to start to talk about a general system that has a level of energy and a general fluctuation of energy and at that point I'm going to lose all of you so what we're going to do now next week you will follow this but to show you that this holds and that it's not super complicated physics I'm deliberately going to pick a special case and then we show that it's true for a very specific case that of course doesn't mean that it might not at all be true for proteins or anything but let's see this is we can have the lowest part of the slide there too if you have some sort of large test tube or column or something for whatever reason you don't see it should say gas down here it doesn't on the slides so this is a super thin column actually you can imagine that it's let's say that it has an area it doesn't matter what it is and then you have some sort of gas in this one and it has a height we don't know what the height is right now most we have gravity too so molecules will be dragged down here so the further down they are the greater the lower the potential energy of the molecule would be right but of course you also hopefully know the ideal gas law that if you have more molecules down here we're going to increase the pressure so you can't have all the molecules sitting at zero coordinate here there's definitely high pressure so some of the molecules will need to move up a bit even though their energy is higher so at some point we're going to have a distribution here that most molecules will have low energy but a few molecules will have higher energy there was the gas label so we can just say when it comes to deriving things this is also part of the first exercise in teaching you to make some assumptions the instinctive thing is to say that but you have to tell me what the assumptions are the bad thing is that's how we all teach mathematics, that's how we all teach physics and that's how we all usually teach chemistry sorry that's not how it works in the real world when I get a research problem to solve research problems don't come flagged with these are the assumptions you should make that's up to me to make the assumptions and one thing that I think where we frequently fail if we don't teach you to sit down and try to work with it and see where it leads you so this might sound very strange but the way I normally solve problems and hopefully most of you is you start to make some assumptions that sound reasonable and if these reasonable assumptions as long as they lead you forward you continue sometimes you get stuck and then you have to take a step back and try something else but it's not dangerous to make assumptions the worst thing that you can realize is that after a while that the assumptions you made are completely incorrect and you can't make them so we will fail liberally try to make some assumptions so we can put numbers and equations on the things we're talking about and at some point we're going to need to say that I spoke about molecules that were lower or higher up let's introduce a height variable so h here is the height that we are at in this particular gas further and I also mentioned at the top we have low density but high energy and further down we have high density energy so that you can have two opposing forces here gravity pushing things down and pressure pushing some atoms up right and at some point we're going to need to choose a point to study and I would suggest that we study the part that is a specific height so let's think about what happens in this yellow part and an obvious thing is that how many molecules do we have here because that's the number of molecules with a given energy at a given height now that's going to be a function of the height of course or and the height corresponds to potential energy and that's exactly what the Boltzmann function was saying how many molecules do we have or how likely is it to have something as a function of the energy and somewhere we also know at equilibrium these things should be the same we should not have a flow upward down but at equilibrium the upward and downward pressures here should be the same and then we have the ideal gas law if you're a chemist or physicists can follow that side Boltzmann's constant is the same as the ideal gas constant it's just a matter do you count things per particle or do you count things per mole of particles you can do anything you want I think since the book uses Clapeyron's law as a physicist I'm going to do that too in the derivation here but you can change this for r throughout it if you want to so the problem is that we don't know what the volume of this tube is or anything but and of course you could say that the volume is 5 that the radius is 5 centimeters or something but an easier thing is to simply say that you know what the volume doesn't matter because the area is constant so talk about the number of molecules per volume instead of the number of molecules and then we can forget about the volume we also know that the pressure then instead of having the absolute number of atoms we can say that the pressure is the number of volumes per the number of atoms per volume multiplied by kt and we also know that if we change this as a function of the height so that how the pressure changes with height that's just going to depend on how this number of molecules changes as a function of the height so it's just a simple application of the gas law this for we also know that what the potential energy is you hopefully remember that from our secondary school so if we go up just a little bit in this yellow layer dh the weight of the gas pressing down is going to decrease a bit because there are now more molecules under us than over us and the weight of those molecules is the mass times d times n multiplied by the slight height and again we're still calculating this per volume but we also know that it's equilibrium right so if we're now changing the weight of the gas pushing down we also know how the pressure changes when we go up so that if we combine that with the change in pressure and say that these changes must be equal well we just put the equal side and solve a bit there and then we say that the change in the number of particles relative number of particles per volume as a function of the height multiplied by kt is minus mgn from that expression up there do you still follow me so all you've done and this is the normal way you normally find two things that must be the same but express them in different ways that then you can just put an equal side between them and then you try to solve so what we're after here what I'm interested is how n changes as a function of h I still don't have n as a function of h I just have some sort of complicated derivative here but what I would like and I have n here too which complicates things even more so this is a differential equation but I'm somehow we're going to try to work a bit here on how many particles do I have as a function of the height so I can move over the kt there and simply say the change in n per height is equal minus and this is just a constant for now multiplied by n itself and here there are two things you can do either you know this immediately because you've studied physics and mathematics or you start looking into your formulas so here you say that a derivative of something is proportional to a equal to a constant multiplied by that number itself we know that if you take a logarithm of a function and derive that you get one over that n and then the derivative of n inside it and this is so something that's not obvious to you what you would do here you would probably look it up in a formula book this is the sad part here I can't spend an hour going through this even if I had never seen this before I might try things first and then eventually I would oh Jesus wait a second doesn't the logarithm derivative look that way so it's not something that's completely obvious unless you're a mathematician and working with this all the time but since we know that this is true the derivative of a logarithm I can say that you know what what I really had on the last page is that the derivative of the logarithm of the number of particles relative to the height is a constant and then we integrate that and again so that and we get that then we say that the if we integrate the constant right hand side the constant multiplied by a a constant then we're going to say that the logarithm of n is equal to this constant multiplied by the thing we're integrating at h so we're going to add an h there and there should also be some constant but again let's forget about the extra constants for now and then I take the exponential of both sides so the exponential of the logarithm of n is just n and on this side I get the exponential of that entire constant multiplied by h and here's the part rather than worrying about what that constant should be and on the integration I just say that yeah it's proportional there will be some sort of constant up here but we don't care about that for now so what you said is that the number of particles per height is proportional to the exponential raised to but wait a second the mass times the gravity constant times h that's the energy right divided by kt and that's the Boltzmann distribution now of course this was just one specific case and we can't prove this for one million cases because we wouldn't get further on in this course the point here and here's the beauty I didn't assume anything about areas I didn't assume anything about interactions in the gas I didn't assume any details about the ideal gas law I didn't assume anything about the temperature I didn't assume anything about anything right so that the more generally you derive things we know that this can't depend on the specific gas because I didn't make any assumptions about what gas it was so if we could make this even more general it would be even more powerful so if we try to interpret this and now when I say state here if you want to keep this simple it corresponds to height in this gas pipe and the energy really corresponds to the potential energy so that the probability of being at the particular energy state is some sort of constant multiplied by the exponential of minus ea divided by kt where this energy is the energy in the state and as you almost see now do you see that the temperature in Boltzmann's constants they always occur together so you can kind of think of Boltzmann's constants as the unit for temperature right you're always going to see kt as a combination and if you add another state eb well then the probability of being there is proportional to the exponential raised to the minus eb divided by kt the more negative these energies well if this energy is very negative then you're going to have minus minus and that's positive so if you have a very negative energy this term will eventually get very large so the more negative the energy is up here the greater the probability of observing that state and conversely if it's a very high energy the only irritating part here is what is c yeah well it's coming through the integration constant it's not exactly but it's partly the integration constant it's also either we could somewhat try to get this to the integration constant or you should somehow going to enumerate every single possible state of a molecule or they sum up to one neither of those things are particularly attractive because we don't know what an all we know is that an integration constant is a constant we don't know what the constant should be but here's the beautiful thing you almost never care about absolute energies or probably you care about relative things if I want to compare a cis versus a trans state I don't care about what the peptide bond is in I want to know how good is the cis versus the trans state of this particular bond so if we divide these probabilities with each other to say how probable is a relative to b that cancels out the cis so then and now it starts to getting really ugly right now you have quotients of two exponentials do you remember your exponential laws so what happens if you have a quotient of two exponentials that corresponds to the difference the subtraction of the arguments right so what this is really then exponential of minus the difference in energy between a and b and that's even closer and now you see that the second you start to talk about differences the relative probabilities of two states corresponds to exponential functions of differences in energy between them and there is no constant what's there to worry about so the second you know an energy difference between two states you can start to say how likely it is for one of the other trigger already at this point we can start to make some relatively simple conclusions or that you're not going to know this yet but what is KT so temperature is temperature right so what's Boltzmann's constant it's not a coincidence that you don't know that I could probably only name the first three or four digits the point is don't try to remember Boltzmann's constant what did I say does Boltzmann's constant ever occur alone so what is Boltzmann's constant has a Boltzmann's constant always occurs in units of joule per mole per kelvin I'm not sure about you but to me that's not an I know what temperature is I know what pressure is but what is joule per mole per kelvin what's that feel right but Boltzmann's constant multiplied by a temperature has what type of unit energy so we can think of that that energy is going to depend on the temperature so what temperature do we care about we only care about room temperature so the point is that we should know what KT is at room temperature and this is another one just as you need to know what the energy of a hydrogen bond is this is something you need to know period KT and the problem here there are two things two ways you can formulate this depending on whether you're American or European and you're going to need to know both because you're going to collaborate with both of them if you like to formulate things in kilocalories per mole which is still very common it's roughly 0.6 kilocalories per mole at room temperature and if you're a modern educated person using the SCI system and use kilojoules per mole that's going to be 2.5 roughly you need to know this and don't mistake one for the other why is this important what will this help you so what this morning and yesterday we've been talking about a bunch of different energies roughly how large the energy differences are I talked about the energy differences in torsions we talked about energies of a hydrogen bond so what what the Boltzmann constant what KT does this gives you a scale for energies these are the energy difference normal fluctuations in energy at room temperature and that probably sounds fussy to you but let's take this and put it into the Boltzmann distribution at room temperature if state A here has an energy that's roughly 0.6 kilocalories higher than state B you know that state A is going to be less populated than state B because it has a higher energy but this is going to be 1% less populated or a billion times less populated you just put it into this equation so the energy difference here and here's the beauty of it now rather than pulling out your calculator and trying to look up Boltzmann's constants the energy here was roughly 0.6 kcal and KT is 0.6 kcal so that's e to the minus 1 or 1 over e and e is roughly oh you're way too accurate e is 3 now the point is we don't about 2.7a like the third digits come on we're talking about fundamental ideas and models here e is 3 so if something when something changes by this much the likelihood changes by roughly a factor 3 now that's still relevant if you see something one time out of 3 it's still going to happen instead say that something is say 2.5 megajoules per mole if that is an energy difference then the factor here is going to be e to the minus 1000 I would encourage you to do that calculation because your calculator is likely going to say that's not the number because it can calculate numbers that low so when you start going significantly above these numbers you very quickly get to the point what happens and they're multiplied so if you have the first factor 0.6 kcal u go up is a factor 3 the next one is a factor well 3 times 3 right now you're down to 1 tenth so by the time you have 4 of these factors it's going to be 1 in 100 so if you have energy differences that starts being significantly larger than the scale once you're in the ballpark of say 10kt things don't happen anymore if you have an energy difference on the other hand that's 0.1kt well let's try that or 0.01 just to make it easier in that case you have 0.01kt there and kt so that's e raised to the minus 0.01 what's that anytime you get close to 0 anything raised to the power of 0 is 1 so what you're saying if energy differences are smaller than kt the probabilities are roughly the same so that also says that when the energy differences get smaller than kt it doesn't really matter anymore so this is the room temperature absolute scale of energy you can think of this as I'm not sure a road or something that the way the road is when energy differences are significantly higher than this it's like putting a big fence in the road when the energy differences are smaller it's just like a bit of gravel you're not going to notice it's small fluctuations so what does this mean how strong was the hydrogen bond 5W yes you can see that it starts making a difference here when it's a factor of 4 so 5 calories per mole that's almost 10kt and 10kt I just said that's e to the power the likelihood of breaking that in the ballpark of e to the power of 10 e to the power of minus 10 that's a very small number it doesn't sound like a large number but the probability of violating that it gets very low quickly and that's why you also that's why you need to know both kt and the rough order of magnitude of the strengths think about those electrostatic interactions we talked about yesterday how strong were they 300 kcal compare that to that one you will never ever violate that sorry it does not happen that's a point if you end up with something, a model anything you do, an experiment and based on your experiments you're predicting that you should put say two positive charges right next to each other there is something wrong it doesn't matter what is wrong but you should know that that doesn't happen in nature Boltzmann, you don't forbid it but the probability is so small that it will never happen if you find the error there are lots of fun stuff with this Boltzmann also gives something called equilibrium or detailed balance I need to mention this because it's going to come up later on in the course don't worry about this if you don't understand it now what I told you before is that equilibrium things don't happen that's not quite true it's not that equilibrium is not the same thing as zero kelvin at zero kelvin nothing would happen but at equilibrium what you have is that the average properties of the system stay constant but individual atoms can still move so if you have two states A and B, what the Boltzmann distribution tells you is that you have a bunch of molecules in state A and some of those molecules will move over from state A to B you can think of this as what I don't know actually water, a glass of water is great you always have some molecules evaporating from water and you will always have some water molecules condensing from gas so what Boltzmann tells you that the probability, sorry the population in the state multiplied by the probability of moving to the other state if you just have two states they need to be equal and that means that on average we're not going to change the individual molecules will move but the average number of molecules in each state is equivalent and if you now know how many molecules are in state A and how many molecules are in state B you can actually calculate how many molecules move in each direction so suddenly we also have an idea that how frequently is it to move in one direct or the other direct this is going to be important later on in the courses but it's a pretty cool physical concept I just had to mention it how many of you are not confused about this good you're going to spend all of Monday afternoon of this in the lab so the first two physics lab are really going to be about these Boltzmann distributions and you're going to sit down with Björn and Arjen write some very simple programs where you test all these things and measure it and see what happens and there are actually going to be some problems here too because things are not necessarily as easy as you think is it just energy that matters sorry if we needed to evacuate this room which is the best way to evacuate it well it's probably that door I'm not sure whether we yes we probably can open the windows too but if we were 6,000 people in this room well you know student populations go up it would not be the best possible way for all of us to run out through that door so even if it's less ideal to jump out the window it would probably be a good idea for some of us some of the 6,000 try to go out the window so it's not just a matter of the best way to get out but also the number of ways we can get out because everybody can take the best way just as everybody can't have the lowest energy in the gas and another way rather than the simple gas vessel if I don't worry we're not going to redo the calculation for these instantly this is going to be pretty good right because here you can have lots of molecules at low energy but relatively few molecules at high energy and this is going to be the worst one you only have one poor molecule at the best energy but lots of molecules at very high energy but there is nothing I've said this far I just talked about energy the problem here is that the area is now not constant which was my assumption for the first one but somehow and this is the same thing whether you're going to run out through the door of the windows the number of ways you can do things also matters and that's complicated because that's not an energy you don't count doors in energy you don't count the volume here in energy so let's see what we can do there so if we say that we have these two states A and B but rather than saying that there is sorry the circle one is just an example the whole point in the circle one even in the circle you actually only have one point here at the very lowest one then you would have slightly more volume here to tell the truth the circle would probably on average here you would have a larger volume my only point here is I actually have no idea what the distribution would be in the circle but the shape matters and the shape between these two in particular this one would the gas would have lower the average energy of gas would be lower here than it would be there but for any type of system we're just looking at these two states a state A and a state B rather than saying that they're exactly identical let's now say that state A has some sort of volume and if you're a cystist you say that that's volume A and state B has volume B this can be anything and to first approximation let's again if you don't know anything assume simple things to first approximation the number of states is kind of proportional to this volume right that makes sense if there is one way of putting things in state A if you have a volume that's twice as large there well there are probably two ways of putting things there right it's not completely and probably at least so that the probability of being in a state then should both depend on this Boltzmann distribution we had but it should also depend on the relative volume between the two states so even if they had exactly the same energy if the volume was twice as large as state A it should be twice as good to be in A but no sorry it should not be twice as good to be a state A but we would absorb twice we would observe twice as many state A and again to get back to the simple analogy if we had two doors here and one door was twice as wide on average you could get twice as many students out through the door that's twice as wide so we use the thing we had in the previous slide but to say that so the probability of being in state A multiplied by the volume in state A relative to the probability in state B and the volume in state B we're just including the volume there that's the exact thing we had before but now we're adding the volumes in the nominator and denominator too so there you have it there you have the probability there is only one what's the volume we have no idea what the volume is right and this happens all the time in physics if you don't know what something is don't be afraid of assuming if I would say that there was a pressure here just say okay that's pressure A and pressure B for now we have no idea what it is and that's kind of like taking out a mortgage at some point you're going to need to pay it back and unfortunately now we're going to need to pay introducing the volumes here made a lot of sense and I could understand the probabilities the only problem at the end of the day I've now introduced the volume and we don't want that volume so now we're going to need to try to find a way to get rid of this volume it's pretty ugly to have it there important the fact that I can't do anything about it since we have no idea what this volume is can't we somewhat try to group this into the energy any if I have any number V that's just the exponent of the logarithm of V so I can just cheat let's take this volume and instead of having as a factor before this I take the logarithm of I move it up here and say that add the logarithm so this is just cosmetics I haven't done anything here I just hidden it in a separate place and to make to move this even closer to the energy I can of course add kT before it and divide by kT so then I can put it on the same line here this looks way more complicated right but it's not really so the energy we've talked about the energy but the energy is kind of a property of the state this volume is kind of a property of the state too right just as the width is the property of the door so rather than thinking instead of saying energy that I said when we derived this when I talk about a state I can say that's yeah energy minus Boltzmann's constant times the temperature times the logarithm of the volume that doesn't sound that beautiful but if I could make this sound beautiful it would make sense because this is something that's the property of the state but it looks but ugly for now which is a bit sad but you know what let's forget about it for a second we'll see if this worked instead of saying energy we could give this a new name we could call it Susan but if we call this Susan it's not going to make a whole lot of sense nobody would have any idea that this would really describe how likely it is for things to happen how likely is it to go through door A versus door B this would both take the energy of something into account and the size of it the number of states and this is something that people have studied a lot in chemistry for different points of view because this really describes what reactions happen and how likely are they to happen and while this is an energy it's kind of better than an energy because it really describes how much energy is available to do work if this is negative it's going to happen if it's positive it's not going to happen and the concept we use for this is free energy free energy is just a name such as Susan but it's a name that all physicists have agreed on so I would recommend that you call it free energy rather than something else or people will have no idea what you're talking about don't worry too much about what the specific meaning of free is it's a definition so when we talk about this entire part and we say that this is something that also takes the size or the volume of the state into account we call it free energy it's not just energy and in the case of these gas vessels it will also take the volume of the vessel into account at different heights and if you really want to dig deeper into this this free part has to do with how much energy is available to perform work we're going to get back to that now on the one hand I'm super happy here because I have a good great definition that describes when things happen or don't happen I can take the volume into account the only problem is I swept this a bit under the rug by calling it free energy because they all said this horrible KT L and V here and what you do as a physicist when you can't get further then we define things you can't get further here sorry this is somehow going to be depend on how many microscopic configurations you can have in the system and eventually this is going to be proportional to the number of quantum states I don't think you want to do quantum chemistry for the next six lectures so let's just invent the name for this say that Boltzmann's constant times the logarithm of the volume we call that entropy and that's how much disorder there is you've probably all heard of entropy and you've introduced that in terms of disorder how many of you have you heard well not just heard of it how many of you have worked with entropy in any type of classes good how many of you understand entropy that's also what I would have guessed but think about it how many of you understand don't try to understand entropy this is a definition I think the mistake that most people do when they try to teach your entropy they try to they somehow try to have a feeling for what entropy is then entropy is the logarithm of the number of microsecond of the volume of the system for instance the number of quantum states you can never calculate this we're not even going to bother with this volume but that means that it's a well defined property that you can calculate for a system just as you can calculate a temperature or something and the entropy has to do with how many ways you can configure the system the more different ways, the larger the internal volume of a state is the more different ways you can configure a system the larger the entropy and of course you can call that disorder if you want to but at this part it's not hand waving this is as a way a strict definition we leave out the temperature here because the temperature is something that changes the entropy this should just depend on the state somehow and if you do not do a different experiment the different temperature the entropy should still be the same in the state so it makes sense not to include the temperature here and rather than having these horrible logarithms and everything then we can write out the free energy s equals to the energy we've called this energy this far there is no end to the amount of confusion here so if you have energy and free energy you will miss, it takes all the time so for that reason physicists typically call this the enthalpy to stress that this is not the free energy it's just the interaction energy I will I so wish I could tell that I'm going to be strict about this from now on in the course that would be a lie I'm going to say energy when I mean energy I'm going to try to say free energy but if you hear enthalpy it's the E we talk about and then you have minus the temperature multiplied by the entropy and this is typically either you call this F or G it depends whether you're a physicist whether you call it F or G chemists like to call it G, physicists call it F and the first part here describes the interaction for instance the potential energy or the bond energy or the torsions or the electrostatics the second part describes the number of states or the disorder in the system and here's where things start to get interesting and because just before the break what do we talk about and I just realized we don't have the sunlight anymore should we try to turn on the lights again now well now well, we talked about the water right and what was the problem with those hydrogen bonds that there were multiple confirmations and we were the hydrogen bonds and the solution, the salvation of something hydrophobic forced the waters to become more ordered so it's exactly something like that we're talking about here and it's going to turn out that the hydrogen bonds in water energy is more related to these things than that one and here's the beautiful thing after the last two slides now if you just replace every single E in Boltzmann distribution with an F you include the entropy and see how beautiful it is with this definition right the equation looks exactly the same the probability of being in a state A relative to state B is equal to the exponential of minus the difference in free energy between the two states divided by kT and conversely you can of course invert this relation if you know how likely state A is and how likely state B is you can measure that in the lab maybe how much hydrocarbon is sold how much of your say oil is in the solution part of the water versus how much is in the oil phase there you can say that the difference in free energy is the logarithm of those two multiplied by minus kT right so the second you know have an observation of how likely two things are you can translate that and get the free energy from it so we're gradually only going to talk about free energy in the course because this is the cool part free energy that describes what things happen and what things do not happen so you probably still don't really have a good feeling for entropy let me give you an example this is my desktop no it's not my desktop will never look that way how many states does this correspond if you think about the icons here and how they're organized on the desk oh this looks like my desk and unfortunately you now have the handouts here and you would think that this corresponds to many states but it doesn't this is just one state too so is the whole entropy part wrong so why is this just one state this looks super disordered right so the problem is what do we mean by a state a state is really a microscopic state so that this state means that that particular icon has to be in exactly that place and that icon has to be in exactly that place otherwise it's not that state so there is only one specific state that corresponds to exactly this placement of all the icons just as there is only one state that corresponds to exactly that placement of all the icons so this is not as easy as we thought but the key thing is that how many similar states are there to that one there is only a very few states well if you only have a handful of icons there are only so many ways you can place them right so there are relatively few states like this one where you can come up with pretty much an infinite number of states that looks like the second one so that because there are many much more states like this the entropy is going to be higher because what you think about when you ask me about my desk my computer desktop you are not asking me what are the x and y coordinates of the placement of your icons you are asking me how cluttered is your desk right and that is a different question because when you talk about how cluttered or how cluttered or ordered my desk then you are asking a fairly high level question how organized is your desktop that is not the same thing as asking for the x and y coordinates of every single individual pixel file so the point is that there are very few states there are very few macroscopic large scales that corresponds to that one but there are lots of macroscopic large states that corresponds to this one so my kids well they are a bit older now but then when they were younger whenever you go into their room toys there are no special laws of physics that applies to toys or anything so any configuration of the toys is just one configuration but damn it no matter how much you clean up and put the toys in order when you come back the likelihood of finding them in the organized states is very low and that's of course just stupid because each organized state when the toys are completely scattered over the floor that's just one state each such state is not more likely than the ordered state when all the toys are in order but because there are billions and billions of billions of random states that I don't like because the toys are disordered that means that the general state when you say that the likelihood of finding the room in a disordered state is very large but the likelihood of finding the room in an ordered state is very low zero actually so this is also something we haven't really introduced yet but sometimes you need to separate things are you thinking about the large macroscopic system or are you thinking about all the gore details in the system on all the gore details in the system you would eventually have to drill down to quantum mechanics that's usually something you don't want but if you did that you could actually calculate the entropy here if you know how many different ways there are of placing the icons there are only a finite number of pixels on the screen if you calculate every single potential way of putting something you can calculate the entropy exactly occasionally normally you only do that in the information theory in chemistry you would measure it instead but the point is there is nothing secret with entropy entropy is a logarithm of the number of states it's just that you probably don't want to go to quantum chemistry to start calculating that but it's nothing don't think don't try to if you feel that entropy is difficult don't try to get a good feeling for it define it it's the logarithm of the number of microstates or the volume so that leads us to what we can do with this we're going to need to come back to this because it's just as yesterday we did the first easy introduction to biology and amino acids now we also got the first introduction to physics and how we can work with these things and here's where they go hand in hand we needed to define energies what we meant by the biological system so that we can gradually start putting this into these equations but same things now we're not doing equations for equation sake now we gradually want to move back to the biology and see ok so can we now use this to understand the things we went through superficially yesterday on a higher and more formal level and next week we're going to come back and do this more formally too so based on observation and probabilities you can formulate some laws about this in thermodynamics this is not a course in thermodynamics there's pretty much only one law I can expect you to know by heart here but thermodynamics is different from all other science there's not a these are not laws that you prove either mathematically well in most cases or that you measure them but thermodynamics is really based on probability and I think it was Einstein who once said that he was convinced that thermodynamics is the only field of physics and science that we will never ever completely overhaul because it's not based on assumptions or anything it's based on observations things that happen versus do not happen and there are some if we can formulate that in a couple of laws the 0th law here I'm going to skip this actually has to be more how you define temperature it's a very long the first law of thermodynamics has a very long formal definition and there's actually there are many definitions you can have here what does this mean some of you have probably taken this forget about the long fancy words oh sorry the 0th one this is something that people came up with much later because we realized to define temperature if you and I are in equilibrium and you and I are in equilibrium the two of you are also in equilibrium you have to define this is an assumption you have to make so let's forget about that one if I'm going to write them down I had to include that one too what does this one mean energy is not created right energy you can't create or annihilate energy and that you probably all know and that's also important to Boltzmann because if you could create energy from nothing then the whole Boltzmann distribution and everything would fail that energy has to be a property of a state if you change its form you can you can add it and you remove it but it has to go somewhere number 2 here is something that you have it's been fuzzy before and you could argue that it's still a bit fuzzy now that I'm not sure you can read it well in the light but in a natural thermodynamic process the sum of entropies and interaction thermodynamic systems increases and approach a maximum at equilibrium yes you've said here and listen for 2 hours now I'm well aware that this is not natural what does this mean entropy can't decrease total entropy can never ever decrease and this is a problem right well I wouldn't say a problem it's important do you see the content with the first law you say that energy is constant if you look at the sum of the universe the energy we have in the universe is the energy we're going to have in the universe till the end of days it will never change that's not true for entropy so entropy we can certainly say what is the entropy of one state relative to another state or something but the second we start having this sort of process a hydrogen bond forming or something energy moves from one part of the system to another part of the system the sum of entropies is not necessarily constant and if you formulate this in terms of order versus disorder that the universe as a whole always becomes more disordered now you can certainly add order somewhere you can clean a room, you can put things in order but that means that you're creating even more disordered elsewhere this is a losing battle you cannot order the universe the more work you do on something to try to put it in order the more entropy you're adding elsewhere so that the best thing you can do is to do as little as possible it's a great argument against cleaning and at equilibrium this energy has reached a constant value exactly what this constant value is that depends it doesn't change spontaneously but any time you're going through a process the sum of all changes in entropy in the system you're looking at is always larger than zero and then as there's also this corollary as you get close to the absolute zero point the energy approaches the constant minimum value we typically define that as zero but it doesn't have to be that way I'm going to try to use this to explain something to get a little bit more practical here let's look at water a process that at least appears to be simple remember the simulation again yesterday hydrogen bonds in water versus hydrogen bonds in ice you have lots of hydrogen bonds in both states and if you take ice starting from zero kelvin in a computer simulation and then you look at the number of hydrogen bonds as you're increasing temperature this goes up slightly but there is not really anything fundamental you might think that as you're increasing things from zero kelvin that the number of hydrogen bonds would be 2222222222 and then suddenly at zero degree centigrade there is some step function here and you start to lose some hydrogen bonds and you have water that's not how it works the number of hydrogen bonds changes smoothly but there is something strange that happens at zero degree centigrade when you move over from ice to zero from ice to a liquid so what happens is that in ice and here's the problem that you have interactions in an ice lattice so what is the energy here is it positive or negative sorry I think you were right it's just I didn't hear it negative because otherwise it would be infinitely far away from each other what's the energy in water in liquid water if you think about the entropy at zero kelvin you would have a perfect low entropy that's zero so then you can probably also imagine that this is a very well ordered system but let's move away from thinking about entropy as order think of this as volume if all ice if all water molecules here are perfectly in order this goes all the way out to infinity how many states like that are there one what is the logarithm of one zero so that would be zero entropy no entropy how many states roughly like that one are there many so then you're going to have at least a medium entropy at some point you're going to boil the water and then you get very high entropy but the point is that there is no abrupt change either in the energy or in the entropy for each of these states you could imagine having it at any temperature but what happens is that at low if we draw this in e minus ts this is also one of these equations that you will have to learn by heart e minus ts let's see if I can find any way to turn on the lights they seem to have rationalized that part away oh well you can determine what the free energy is of ice if you know what the energy and the entropy is right? similarly you can determine what the energy and the entropy is of liquid water and you can actually again don't think in terms of things can't happen or can't happen but things in terms of probability so if you are at say 100 kelvin the likelihood of observing things in ice compared to observing things in water let's draw that sorry if the free energy equals e minus ts the difference in free energy is the difference in energy minus temperature times the difference in entropy so if you were at 100 kelvin and had to go over to some sort of liquid water what would happen is that the entropy would go up and when the entropy goes up this changes right you're also changing the energy here so then you can start to calculate what is if you were at 100 kelvin what would the change in free energy be would it be good or bad so what happens here at very low temperature at any point here you would certainly have a difference in entropy and you would have a difference in energy but you also have the temperature here right so at very low temperature if the temperature L you can imagine if the temperature is zero this term disappears completely so at zero kelvin what is the only thing that matters energy and the energy is best there so at zero kelvin it doesn't care about entropy at zero kelvin you will always be nice eventually as the temperature goes up here though the second term starts to become more and more and more important right and at some point most of the effect here is going to be based on the entropy so at high temperature you care less about the fact that you might lose some energy but it's very important that you gain entropy by going to the state so what happens in the case of water at zero degree centigrade is that it's the turnover point when this changes sign so suddenly you are in ice you are in ice it's better to be in ice at 100 kelvin it's better to be in ice at 200 kelvin it's better to be in ice at 270 kelvin but at 273 kelvin suddenly these terms change magnitude yes they change size so that it's suddenly better to go over in water but this term has gradually roughly in decreasing slowly and this term has gradually been increasing there is nothing special that happens with either term suddenly the second term is larger than the first one and then it's better to go over in water and you're going to have the same thing when you boil eventually so all these transitions over the processes happens is going to have to do with this balance between entropy and energy it's 11.47 for tomorrow I think you should think about this you can use E-TS to explain the hydrophobic effect but now you're not going to hand wave so tomorrow morning when we go through these things you should explain the hydrophobic effect in terms what happens to the energy between well you have water is interacting with other waters you also have water is interacting with the oil but you also have the entropy of the water and oil so you can use this to explain the hydrophobic effect and whether things are soluble or not I'm deliberately not going to go through that for you because if I go through it you're going to think that it's easy it's not entirely trivial sit down with this this afternoon and try to understand it in terms of free energy the difference is that even this morning when I went through this I hand waved and hand waving is good to understand things but it's also very dangerous because you go wrong when you hand wave you can hand wave for simple systems but the more complicated the systems get sooner or later you go wrong because you think you understand it trust the equations the equations is not a complication they're there to help you so write down all these different states what is, what happened before you put the oil in the water, what happened when the oil is in the water what is the energy before what is the entropy before what is the energy after what is the entropy after and then you can also start thinking what happens when you increase temperature with solubility and if you get stuck this is a great motivation for reading chapter 4 of the book this summer of lectures the most important thing today I think I did was that we started introducing a bit of physics we're going to go through that slightly more complicated next week too but the point is before we do that we're now going to go back to a bit of simple chemistry and biophysics and understanding what this means to proteins we can explain what this is going to mean for an alpha helix beta sheet we're now able to say that not just whether things are good or bad will helix's form under what conditions will proteins form in the first place we have a bunch of study questions for that too, mostly about chemical bonds but also increasingly about Boltzmann distributions and energy landscapes so now it's starting to get a little bit more fun I think that's all I had today we have 10 minutes to spare I'll be happy to take any questions or I can leave you early