 Good morning. In the interest of time, I am going to get started sharp on time here. It's great to see a bunch of brave souls here here at 8.15. As you realized on Friday, I ran out of time for a few slides. I think I've done that before in that lecture, which is no big deal because I have some spare time on Friday when we're going to talk about proteins and the things we're going to bring up today and tomorrow are far more important or at least harder for you to learn on your own so that I'm deliberately going to spend the time necessary for that. What I also experimented with last year and that I might do this year again if you feel you need it, that it's difficult to go through equations or rather it's difficult for you to follow when I go through equations. I've occasionally experimented a bit with doing a screen recording on my iPad while I draw the equations and talking about them. There are a couple of trial lectures for some of the things I'm going to go through in particular tomorrow I think. Let me know if you like them and I might do more. Today we're going to talk about the Boltzmann distribution but before we get to that I'm both going to go through some of the starter questions from last week and I'm going to finish off with the interactions. We didn't have time to go through them. As a Tuesday morning exercise I would suggest we jump straight into some of these questions. I spoke a lot about amino acids when I introduced the molecular structure last week so can you come up with three ways or three properties by which you can classify amino acids. Science polarity and yeah that's fine and pretty much your imagination is the only limitation. Aliphatic versus aromatic is an important one too whether they have this ring hydrocarbons. There is one amino acid that is not chiral. Which one? Yes. Yes sir. Aliphatic versus aromatic. Aliphatic this is fundamental organic chemistry and I realized you're not chemists. So aliphatic carbon chains are just linear long carbon chains. They can certainly have double bonds too but they're traditional while what we call aromatic carbons are typically carbons that involve benzene rings. So aromatic amino acids might be phenylalanine and tyrosine. Their amino acids with the side chains contain one of those rings but it would be perfectly fine to just measure the three you did. I spoke about amino acids and chirality so maybe we should start off with what is chirality. Asymmetric in one way but there are lots of things but they're asymmetric but not chiral. A chair for instance is not a chair has one symmetry axis right here but it's not symmetric if you cut it that way. Exactly. So that's the key. Two molecules who are mirror images of each other but it's not possible to rotate one to the other and that requires one central bond with at least four partners bound and they all have to be different. And that's true for the vast majority of amino acids which give them a bit of peculiar physical properties they will rotate light and a few other things. With one exception there is one amino acid that is not chiral which is glycine and why is it chiral? Exactly. So it doesn't have a side chain right it just has a hydrogen. There is one more highly special amino acid which one? Why is it probably special? So it has this strange ring right the side chain is not just one chain but it's a ring that goes back and binds to the nitrogen in the same amino acid. And what is normally bound to that nitrogen? Normally you have a hydrogen there right and that is the hydrogen that participates in say hydrogen bonds in an alpha helix. So the proline is going to be problematic here it doesn't have that hydrogen so it can not only does it not have the hydrogen it has a bulky ring there. So proline simply it won't fit in most alpha helixes it can't form that hydrogen bond. What would happen if we had some D isomer amino acids in a protein? What is a D isomer? So maybe I can start so what is an isomer in chemistry? So an isomer is example again of mirror molecules right the carality is the center that causes the entire molecules to be isomers. So a D versus L amino acid has to do with this mirror imitus and L is by far the most common in nature. Sorry you can continue not just by far it is the ones that we have in nature. So what would happen if you had a few D amino acids? So we can first let us be simple first we talked about alpha helixes for instance right but the alpha helix is always a given direction and if you now have one amino acid that's a mirror of this it would not be compatible with anything you could not have an alpha helix and then suddenly have a D amino acid in it because the D amino acids would prefer the alpha helix to have the opposite hand of this right so it would break everything completely incompatible with biology well or with the molecular properties. Whether that is good or bad depends it's the first approximation is bad your bodies would never produce the amino acids but occasionally just now and then as you mentioned you might actually want to use this to introduce something that is bio incompatible and the obvious case if you would like to take a drug that you can take orally as a pill but it should not be digested by all the enzymes in your stomach and the reason why that happens that the enzymes have really been trained through evolution to recognize the L amino acids not the D ones and what those amino sorry what those enzymes in your stomach do they are specifically optimized to break something which is a matter of question five how amino acids are linked into a protein hmm why is a peptide bond special sorry right it's a fairly strong and rigid bond you can't rotate around it and it's it's never it's not a bond that would ever split spontaneously it's a very strong covalent bond so what these enzymes do they're gigantic catalyst that causes the amino acid to bind in the right place and this bond cleaves and then you have separate amino acids but once you have formed this chain because in the body we do the opposite right particularly in the ribosome then with the amino acids together based on the gene sequence you have in DNA in particular these triplets the DNA codons at some point we start out with this very long chain that I showed you and in principle that's all we have all we have is our atoms that are bonded in to a very complex large molecule that then packs up in space and we're gonna go through start going through how that happens on Friday but to make your life easier we try to organize this in some sort of symbolic levels to avoid having to focus on every single individual atom all the time what are these levels or what do they represent so why do we why did we end up with these four levels so that's a complete of course a completely arbitrary definition in many ways and I would argue occasionally that's as a physicist that's actually what can be hard with biology everything is not governed by strict laws right some things are just based on observation and this makes sense so the and on each level which you try to identify as commonality so the sequence of course you only have 20 amino acids they are all stitched together all proteins with very few exceptions are linked together in a linear chain with a start and an end it's very rare to have any circles and that means that's an obvious first way to start describing things what is the order in this chain and then as we realized actually even before the first protein structures appeared that there are two very common ways to assemble these either in the spiral forms the helices or up and down while they can be parallel to long extended sheets they can be turns to but two or three different ways and that's reasonable to classify that as a level and that's the secondary structure then and then in principle the only normal structure we're going to talk about is then the third level how do these elements then pack up into a large structure and in principle that's all you need to know but to complicate things further there are a few cases in particular the ribosome itself is a gigantic protein and then you don't this small ball that has curled up that really is the molecule but the ribosome is then a collection of roughly 58 such balls it's a gigantic super molecule and to start describing this it's going to turn out that these have to fold independently that's why we occasionally introduce this tertiary structure but we're not going to talk a whole lot about that in the class we spoke both on Friday and on Wednesday I think it was about the relation between sequence structure and function can you say something about that and that's very much related to 8 to what leads to what yes and this is something I should be able to wake you up at 2 a.m. in the morning and you should instantly say sequence leads to structure leads to function there are not a whole lot of things you need to learn by heart actually that's a great thing with biophysics in one way it's more close to life than physics but it just in contrast to life science we try to avoid learning things too much by heart but knowing some of these mantra will enable you to understand and think better about problems right if somebody else on the why does something have a particular function well that must be explained by the structure and that structure must be explained by the constituent amino acids in the protein but related to that can function ever induce structure so can the structure of the protein ever somehow be dependent on the function so there are a couple of very cool examples I'm going to come back to later in the class or actually no maybe it will be loose you get a chance to tell you about that globular proteins hemoglobin the small molecule that binds oxygen in your blood it turns out that it depends on the species and who you are so a llama has hemoglobin that has a slightly different structure that is much better at binding oxygen than you why but why does the llama need a better ability to bind oxygen in general they live at an altitude of a few thousand meter right so the contents of the oxygen is lower a fetus baby has a different type of fetal hemoglobin that is better at binding oxygen than normal one why because otherwise right that if the mother and the fetus if the fetus would not be able to steal the oxygen from the mother you would not have oxygen being transported from the air to the mother's blood and then over the placenta exchange to the fetus but it was the second you're born this is then a gene that's silenced and then you no longer need it so in principle the answer is no the function itself doesn't induce this but through evolution there are then patterns that will lead to this diversity of structures but they're tiny differences you wouldn't even be able to spot it unless you know what you were looking for and then we started to talk about interactions that are going to continue in a minute here what would you argue are the most important degrees of freedom there are two things are we going to talk about degrees of freedom and we're going to talk more about interactions in a few minutes here so what are the degrees of freedom that are most relevant in the protein because there are a ton of degrees of freedom in a protein right you can have hundred you can have a million atoms multiplied by three that's three million degrees of freedom but not interactions right so what in the back what degrees of freedom in the backbone are the ones that you could describe most of the things with right so there are these rotations around the two bonds that are not the peptide bond the flexible bonds and we just call them phi and psi and lack of imagination and the reason why they are important is that if you rotate a side chain that will only have a local effect if you start rotating along this chain if you rotate in the middle of the protein the entire second half of the protein will rotate so that will have a global change in the confirmation related to that I kind of lied to you and said that that peptide bond is well only kind of I told you that the peptide well you told me that the peptide bond is rigid normally the peptide bonds is always in a trans configuration so that if you look at the entire backbone it would go like the aliphatic chains I drew here straight all the time but and by and that's why in what in chemistry we would call that a trans peptide bond technically it is possible for the peptide bond to be insist but since it can't rotate if you put it insist it's gonna stay insist and that's virtually never observed why the side chains in particular would be close to each other so that unless if you have two glycines it could work but in general any second you actually have a side chain those side chains would then end up being on the same side of the chain and bump into each other the one exception to this is Poland because with proline the entire configuration is different different due to this ring so with proline in an ideal world probably would always be sis while all the other ones were always trans unfortunately we don't live in an ideal world so all the other ones I would normally eat my left shoe if they were not trans not quite that but proline is kind of 50-50 all bets are off with proline it's horrible but you will never be out you can't predict it you wouldn't need a kind of help of a computer or something so related to these degrees of freedoms there were two very important discoveries or a hypothesis by first Christian Ampelsen and then Cyrus Leventhal that I mentioned towards the end of Friday what did they say understanding this will help you understand a ton of topics that we keep repeating in the course almost yes and that's I where I haven't brought this up with you yet but my very reason for being strict the global minimum of free energy do you realize no you don't even I don't realize this is the guy who connected life science to physics that life science is fundamentally even something as complicated proteins all these molecules in our body the way they exist it's not due to some magic factories building the molecule in the specific way you want we just need the blueprint of the molecule and then physics will sort it out awesome result I was so it was so extreme when it was first by that he had of course had to prove this experimentally and everything and the reason you get a Nobel Prize is not just to have the idea but prove that it's actually true at least in general and Cyrus Leventhal yes so an important distinction it's not that Cyrus didn't believe this but Cyrus pointed out that that's funny that's leads an interesting problem that we all agree that they will fold but there are so many degrees of freedom in this so that if you were to search all these degrees of freedom and it would only take like a millisecond or something per degree of freedom it would take you longer than the age of the universe for a protein to fall so there is something fundamental here we don't understand and we still don't understand it we're going to come back to this later on in the class and explain it we spoke a little bit about Ramachandran plots to what was that yes and in particular you can even put dots right if you look at the large protein we could classify all the this particular protein what are the residues what are the fine psi angles and the reason for that is again to try to take a step back that's going to be a recurring pattern we don't want to look at the atoms because if we look at the atoms it's just too much information and rather than worrying about whether the fine psi angle is 59 or 42 I'm gonna let you in a secret you can define what are the fine psi angles that corresponds to helix and the sheet I don't know that it's completely useless knowledge why on earth what I try to know that by heart and I know the rough areas in the Ramachandran diagram that corresponds to helix and sheet and that means I could always if I was ever asked this on exam I could draw this in 30 seconds and then I would do a yeah that must be helix and that must be sheet so most of these complicated concepts are actually there to help you so rather than having to worry about the angles you worry about rough areas and everything pretty much everything that's not helix and sheet we can forget about so we I spoke I ended up by speaking very briefly about the formation differences the properties of helix and sheets we will come back to that one Friday when we talk about the structure but there were some key differences there in particular about locality of interactions and that has to do with that the hydrogen bonds we form in helix is there always hydrogen bonds closed in sequence all hydrogen bonds have to be closed in space right otherwise they won't form but in a helix you're forming a hydrogen bond with a neighbor residue four units away I to I plus four that's also one of those things that you need to know in your sleep I to I plus four so that the residue 14 will form a hydrogen bond with residue 18 etc. I didn't have time to bring up the chemical bonds and interactions so I'm going to get started with that so we have a time to get on to both the Boltzmann distribution today and for all these things don't hesitate if there anything that you don't understand I might not spend long periods during the lectures to talk about it but if there is anything I can always make a small extra recording or something so that you can watch it at your own pace I showed some of these movies and I'm not gonna spend time to go through all of them but the cool part here is that in the 1960s when people started determining x-ray structures everything was rigid we only had exact positions of the atoms and then what a few groups realized that you can actually use computers both to visualize these and possibly even try to improve the structures can we fix things so that we minimize the structures and put the bond and the torsion and the angle in exactly the right position because the experiment is noisy and in conjunction to this they made some of the very first molecular visualizations so these are images I got from a colleague at the L&B and you would probably they didn't exactly do this at their laptops so these were the types of computers they used for it so that typewriter in the rear is how you actually enter the operations in the computer and the only screen you have you don't have a screen that has no text really the only screen you had were these ones where you could visualize it in 3d and you would program them with punch cards so you would spend a week designing your program and then you would hand over the entire stack of punch cards to the secretary and then you would get it back 30 seconds later saying that's a typo on line 47 and then you would have to go back home and you would have to wait another week to get time at 2 a.m. in the morning on the computer so that we have no idea how insanely spoiled you are this is actually a match it's a multi-access computer not quite the one designed by Apple though and Cyrus that we usually see pictures of in a suit and tie that I think he spent most of his time in front of these screens and that's Cyrus in action so he was one of the first physicists slash computer people and for party for historical reason this is very much a field in particular the extra crystallography that has been nominated by physicists because it's hard math but before we go back to physics I'm gonna take you to a slide detouring chemistry so this is not a class on quantum chemistry and in a second we're gonna forget most about quantum mechanics the reason we talk about degrees of freedom a few minutes ago but in addition to the degree the degrees of freedom describes how the molecules can move but in addition to that we also need to think about how will they interact because the interaction it was causes the motions why do they want to be close or far away and at the end of the day this is all determined by the electronic interactions some of you have studied quantum mechanics others haven't for instance we have the Pauli exclusion principles saying that two electrons cannot occupy the same quantum state and that means that there is if you push them too close there is they're gonna repel each other and this whole pattern will also cause that depending on the specific spin of atoms occasionally you can form bonds in particular by pairing up electrons there is a huge amount of theory in there that would easily fill an entire class but chemists like to frequently explain this is slightly easier ways you can use for instance so called orbitals and that's a way of thinking that you probably heard about this SP orbitals in upper secondary school right so chemists have very simple ways of trying to describe when will bonds form and when will they not form and sure at at the basic level there are going to be a ton of electrons involved in bonds on the amino acids but I'm gonna forget about that from now and the reason for that is that those bonds are so strong that they virtually never break even the peptide bond the only case where the peptide bond breaks or is formed is by end times in your stomach or in the ribosome where we build the protein the second the protein is built that bond is there it's stuck so with that we can forget about the bond formation it's not quite that easy though because that we still have to care a little yes I'll come back to that in a second for now we're just talking about interactions we don't worry we're gonna talk about more entropy than you like in this class actually I love entropy so that larger or it becomes slightly complicated because you still have these atoms right and as I mentioned that occasionally we complicate life by trying to organize it so that you have two nuclei of atoms and they can either be close or far away and in nature this is of course a continuum there's nothing magic that happens exactly 1.52 angstrom but as you're moving far away we tend to focus on other things right and that's occasionally we classify things in different ways so you all studied hydrogen bonds and these properties if you have two molecules the properties of one molecule might influence another molecule and in many cases we don't care about this but say if you have a water molecule there you have the electrons to towards the oxygen right and then you have a deficit of electrons in the hydrogens and that means that you're gonna have a small dipole with a plus towards the hydrogens and a minus sign towards the oxygens Xenon doesn't have a dipole but Xenon has both electrons surround in the shell around the nucleus and then a nucleus but when this when the electrons in the Xenon sees this water the electrons might actually decide to move slightly towards my side because that way the electrons will be slightly further away from the oxygen but that will means that the whole Xenon atom will have a small positive partial charge towards the water that's gonna like the oxygen because the oxygen is already negative so by having the electrons just being just so slightly offset where the water here is effectively inducing it's creating a temporary dipole in the Xenon you don't have to limit this to water so if this forget about the water now this is just a fluctuation that you have one see the my Xenon atom here forget about the water it just happened to have the electrons slightly displaced towards my side if that Xenon atom then sees another Xenon atom the other Xenon atom will react to the first Xenon atom say oh you have displaced your electrons then it makes sense for me to display in mine too so that a small temporary fluctuating dipole in one atom can actually induce a dipole in another atom and this happens all the time now a nanosecond later this will have been actually not a ethosecond later this will have been reverted but it means that as the charges fluctuate due to a thermal motion it will be natural these these atom will actually start to attract each other a little bit even though they don't have any formal charge they don't even have a formal dipole and this is the reason why virtually every single substance including helium neon or argon will they won't necessarily stay a gas but eventually they will form a liquid and even a solid and that's due to this very weak dipole-dipole interaction yes no now this flag this fluctuates all the time it's on an ethosecond level right but the whole point they never repel each other unless they get so close so that the electrodes start bumping into each other so at very long distance they're never going to repel each other but any type of fluctuation will on average create some attraction between them it's going to be a tiny attraction but since all the atoms will attract each other the collective part will always be attraction how strong this is depends on how much motion there is if the thermal motion is high you're going to override it but at very low temperature when the molecules don't move as fast and everything this is going to be a more pronounced effect so helium for instance doesn't become a liquid until roughly four degrees Kelvin the cool thing is that you can actually prove this you don't need quantum chemistry for this you can prove that this will lead to an interaction that has one of the power one to them the distance the sixth inverse power of the distance between them and this is a London show this originally and the way you described this you've heard heard about this is called Lenard-Jones interactions they will occur for all the atoms they're weaker than anything else but since they're always attractive at large distance and if you get to very short distance they will eventually repel each other since they can't overlap yep so no that's a good question let's see when it comes to energy at some point all energy is relative right the zero there is no obvious zero level for energy in the world so the way we typically described this that when atoms are infinitely far apart they don't interact at all and that's a reasonable level to say that's zero energy the entire field of physics called condensed matter physics that is the realm when we start in when Adam starts to interact and when Adam starts to interact well under some circumstances they will repel each other but that's not particularly interesting because if they repel each other they will just go away and stop interacting the interesting part is that when Adam starts interacting in favorable ways when they start attracting each other because when they start if they we have first we have ton of water vapor in the air here but those water molecules are so far away that they don't interact but in those water bottles you have for whatever conditions the water atoms like to stick together and that gives the water completely different properties and that happens towards lower temperatures towards higher pressures and everything simply when when the surrounding factors makes it more advantages for the atoms to start interacting so any type of favorable interactions with the molecules start to attract even the water molecules in the air attract here but that attraction is so weak that they haven't condensed and then that these gradually become stronger eventually you will get to a point where we'll condense and if you can as far enough you actually form a liquid the connect the transition between liquid to solid is slightly different I'll show you that in a second price and of course if you wanted to treat this exactly you have to do quantum chemistry this is insanely difficult there's quantum chemistry you would forget about xenon right even for a simple molecule if you're going to be strict here to describe these interactions exactly there is no alternative doing quantum chemistry the good thing with quantum chemistry is we can solve that with paper and pen as long as you don't have more than one electron but then this would be a fairly boring class because there are not a whole lot of biological molecules that only have one electron not any if I think about it the other way of thinking about this is that it would even if you could you could only handle roughly hundred atoms this is insane when I took the classes like this people were proud that we could handle six atoms so hundred atoms is nice but it's certainly no biology maybe one amino acid and then you would not have water around the amino acid did I say quantum chemistry well the problem is that all the quantum mechanics that you teach in a get taught in a physics class here that's also incorrect because to be strictly correct you should have relativistic time decoendant quantum mechanics and now it gets really really really complicated so physics is built on simplifications and we need to simplify things here and the reason the reason simplifications work the proof there is in the eating of the pudding you have to show that your simplifications are reasonable you can't just simplify because you don't have any alternative but if you think the way the chemists do this what you would simplify you would for instance say that all the heavy nuclei they don't move you just look at the electrons that's a horrible approximation so now you're at the point that you're not gonna have water you're not gonna have sorry you're not gonna have any motion whatsoever that means that you're doing everything at zero Kelvin and this class was supposed to be about life science there's absolutely nothing that happens at zero Kelvin so you can be very happy that you got a perfect wave equation for one state which can forget everything about life so the problem is that quantum chemistry the way you normally handle it has other limitations and those hundred atoms you will see your fair share of people who claim that they can do a quantum mechanical calculation of a protein and then there are some things that they ignore such as water you're now gonna start your proteins in back you you're gonna do life science in back you at zero Kelvin good luck with that I know what the result is gonna be there's not a single protein that will have any function whatsoever in back you at zero Kelvin so the point is not necessarily in theory time-dependent risk relativistic quantum mechanics is better but you're gonna throw out the baby with the water due to the other approximations you have to do there and in principle we know that biology works biology has a really cool examples that if I poke myself here I don't die biology has stability properties right it's not really that sensitive to details you can do fairly horrible things with the body and will still survive but even if you start from quantum chemistry you would have other issues you would be extrapolating 15 orders of magnitude and there were a bunch of people who fairly early in the 70s 60s 70s realized that maybe we can do this in another way and those are a reversal in particular came over that let's cheat completely so let's just assume that treat all your atoms like small balls forget entirely about quantum mechanics and then parameterizing so let's say so how much do these balls attract each other you can use Lenard Jones interactions roughly like you learned in undergraduate chemistry have some way of saying that one over our six they should attract each other at very short distances they have to repel there are some charges on the atoms let's pretend that the charge on the atoms doesn't vary that would assume that the electron state close to each nucleus but that might be good enough and then you get enough with a ton of parameters hundreds of parameters just what are all the partial charges in your atoms but the cool thing is that in real life you're occasionally allowed to cheat because we can parameterize we can fit most of those parameters to simpler properties such as liquids so for water we know what the heat of vaporization and the density of water should be and then we tune the parameters to reproduce this this worked so well that they were able to not just minimize proteins they were able to simulate the first enzyme functionality and everything using quantum mechanics for a very small part but literally start to show how atoms are moving and Martin Carplers were among the first people to show that how proteins actually produce move even at hundred Kelvin in an x-ray crystal because hundred Kelvin is not zero Kelvin and they could use this to show why you have this broadening and kind of noise in crystals and this really corresponds to motion in the x-ray crystals again it's a remarkably cool result because they showed us theoretically before we had experimental results and for this they were awarded the Nobel Prize for chemistry in 2013 so if we go into the details there this is just a tiny part of a molecule but you can see that it roughly gets very complicated rather than doing the quantum chemistry way here let's pretend that we are a warshell and how we need a very highly simplified way of describing this you can have some sort of bond there but rather than having the quantum chemical exact correct description of that bond well you can do this in lots of different ways you can introduce the quantum chemistry and if you're now going to have a quantum mechanical oscillator there are certain energy levels that this can adapt and you might remember that from quantum mechanics if you're taking the class but what are a very quickly realized at room temperature you're gonna have 99% of the bond in the ground state and maybe 1% of the bond will be 1% longer yeah that's very important for a protein not so let's just let's just approximate this with that either rigid stick or a small spring that is this length and then forget about yes of course if you simulate this is 5,000 Kelvin this is gonna be bad but it's fairly rare to have 5,000 Kelvin process in the cell so we're gonna be down here so it's a super simple approximation let's just introduce that small spring and rather than having quantum chemistry you get a harmonic function there in theory we can choose to have this exponential to accept that eventually you can stretch a bond but that's virtually never used because bonds typically don't break this works remarkably well it's so remarkable that it's scary we can do exactly the same thing with angles if you have two bonds defined here yeah I can define the angle between these three atoms and this is now a super complicated quantum chemical description with resonances between multiple electrons and everything or we just forget about that and do pretty much exactly the same thing there this angle will only move a couple of degrees it might be 109.5 degrees by default or 120 in this case and it might be 118 or 122 so yeah so maybe not completely unimportant but it's gonna vary so little so that any type of approximation here is gonna be fine and you probably know this as your physicist but why do we use harmonic functions for all these things why use x squared why not something else so this is the problem you've had far too much physics we bombard you with physics so much that you forget about the simple things of physics you might even think that Hooke's law this has to do with Hooke's law when you extend a spring right do you think that Hooke's law is correct the reason for this is that virtually everything in physics is super complicated find a mathematical form to describe that or to make it to be slightly nicer to you let's say that we have some energy here that describes some sort of extension it's not entirely trivial to describe that function but if I start as we're gonna come back to at equilibrium when this system is happy it eventually gonna be at the lowest part here it will find its minimum so by default you're gonna be here right so what do we know here well we can't describe the entire function but we know that we have some sort of function value here the value at x0 then we have a derivative what is the derivative at the minimum 0 so we can forget about the derivative what do we have after the first derivative second derivative so then you have some sort of second derivative at that value and we can now describe how this function varies in a small region around that by multiplying that with the delta x square divided by 2 and that's why all these things are harmonics we have no idea what that second derivative is exactly but it is a number that is a number and that's the K in the equation that's the the spring constant or the harmonic constant so virtually everything we do is that around a minimum we can describe everything as an harmonic and that's why all the bonds and function of their harmonics that's what physicists do if we don't know better the third part though here I'm gonna break a little bit what I just said we spoke about this torsion angles or the dihedral angles the rotation about around phi and psi in principle we can describe this with the same motion but so now I'm gonna need you might see the pattern first I had two atoms for a bond three atoms for an angle and now I'm gonna have four atoms involved for a torsion and now I want to describe what happens when I rotate around this torsion this gets just a tiny bit trickier for two reasons first if you rotate an entire turn 360 degrees or 2 pi for obvious reasons I want to be back at the same value right and if you have a simple molecule will say at ethane or something you're not just gonna have one minimum per term but you might actually have three minimum per turn right so that I can't use a normal harmonic function for these so in most of these cases we end up with say sorry in most of these case I can describe these torsions and now I need to find a mathematical way that is both periodic and that has some sort of repeating pattern and the obvious ways to do that is to use trigonometric functions sine or cosines for a very simple one you might end up with a cosine so this is let's see this is a butane rotating around that bond and if you have butane rotating around that bond you can end up with a function that looks roughly like this now of course it's not gonna look exactly like this because this will depend some electrons and their exact distribution and everything but this turns out to be a really good approximation and really good in this area sec it's gonna be error might be 0.1 k cal or so far lower than the thresholds I'm gonna come back to later today and tomorrow so we're fairly happy with these the energies here are not high the energies here involved here are much lower than the bonds and angles but that's concerned that's actually good because these are energy barriers that we can occasionally go over and that's why again why they are by far the most important degrees of freedom the peptide bond in contrast as a gigantic energy barrier it's so strong that will never rotate spontaneously but that also makes it very boring if it's an energy barrier that's so high you never get over it it's irrelevant and you can make this just as complicated as you want we could in principle take our small if you have a small amino acid here this is the simplest one you can imagine essentially we have a Rammachandran diagram at the phi and psi but instead of now plotting hash marks where the things are allowed or not for each of these for each value of the combination of phi and psi here I can actually plot what the energy here is and then you start end up with the energy landscape that describes that you have a few places with this molecule is happy to be and then lots of high energy regions here in red where it's not strictly impossible but it's very unlikely to be here and this is as complicated as yes just for a single amino acids we're gonna keep making that even worse in a second I will have two more slides and then I'm gonna leave you for the break what all these interactions I brought up this far have in common is that they are bonded so there are things that stick together there are also the what we call non bonded large-scale interactions and that could be charges positive and negative and it can also be these packing effects fundervolso and our Jones interactions that things can bump into each other and everything attracts at long range they're frequently weaker but there's a ton of them they can easily overcome other effects then our Jones interact is the obvious example right there are millions of them but since they're all attractive that's why the water will eventually condense for the charges we end up describing this we can use any charges but instead of having all the electrons and everything here we cheat and I just say that there is a small partial charge on the atoms so for a benzene I would have maybe minus 0.06 on the carbon and plus 0.06 on the hydrogens that's a rough approximation of the wave functions and roughly where the electrons are but around room temperature it's gonna work great you will be able to describe the density of benzene and how the benzene behaves and everything so after the break I have a few more slides in this we're gonna talk a little about the strength of electrostatics and then we're gonna head into the hydrogen bonds so electrostatics is special because it's a very strong interactions it's far orders of magnitude stronger than our Jones interactions and that leads to some unexpected but quite fun effects in particular for water so let's meet here at a quarter past and then I'll continue we spoke about electrostatics right the most special thing about electrostatic is how insanely strong it is and not it's not just strong if you know you're undergraduate physics well you will remember that it electrostatic energy goes as one over the one over r so it's also very long range and just to give a have a hunch about the the energies that might involve if you if you would to have two charges separate by only one angstrom which is admittedly a super short distance you're talking about hundreds of k kals per mole it's an insanely large energy while the bond rotations we spoke about a few slides ago around those torsions right there were a few k calz so that's order of magnitude stronger the other non-bonded interaction where these thunder balls interactions that if I talk about non-bonded action interactions in general we typically call them thunder balls but the nomenclature is a bit fussy here it's chemists not physicists and there are two parts here at very short distances there is repulsion due to the atomic overlap and at very long distances there's this one over r to the power of six interaction that all atoms will attract each other and the best way to you can actually describe that exactly the repulsion by quantum chemistry has to be exponential and the attraction there has to be one over r six unfortunately on most computers the exponential is very difficult to calculate expensive not difficult but it takes a few hundred clock cycles and you're spoiled today but our reward should certainly wasn't spoiled in the 1960s this was this rapidly became a bottleneck but what you then realize unless you're trying to design nuclear weapons you don't you're not you're never going to be in here it doesn't really matter it overlaps all overlap that bad whether it's bad or astronomically bad is really the same so what people fairly quickly came up with that if you have one over r6 you can just square that one clock cycle and get one over r12 which is something that's larger and that's good enough actually it's not good enough at all it's a fairly horrible approximation but you see the problem is worse than that even the equation I showed on the previous slide is incorrect because in quantum chemistry you also need to include triplets of atoms quartets of atoms five body interactions six body interactions all the way up to an infinite number of interactions and that rapidly causes you to just scratch your head and it's just ugly but here's where we can cheat instead if we have an approximate interaction such as this one over r12 and one over r6 which is the Lenard-Jones interaction form well both those C's are constants so we can parameterize them so the cool thing is we tune them to fit experiments we don't derive them bottom up from the fundamental laws of physics but we adjust them to fit experiments and that works remarkably well so by having these parameters we can also stick to pairwise interactions we never look at three atoms at the time and yet we get a functional form that describes reality really well so occasionally cheating is good and these two things together create some very peculiar properties for molecules we already spoke a little bit about hydrogen-bonset the quantum chemical nature of the oxygen will mean that they're effectively four directions where the the free electrons will protrude so along two of these directions those correspond to the bonds we have the hydrogens and then in addition to that you have free or lone electron pairs so there are in addition the water is kind of the each water mole is kind of like a small tetrahedron with two small ears pointing out that we typically don't draw that corresponds to the electrons you see up here there you have and if you're not to have two such waters close to each other that that hydrogen which just has a slight positive partial charge is going to love to interact with this free electron pair here in the other oxygen here and that's really what creates these hydrogen bonds it's not limited to water but this happens anytime you have a large or electronegative atoms such as oxygen or nitrogen that is bound to a hydrogen so then the hydrogen will donate some of its electrons to the oxygen and that oxygen bound to the hydrogen is called donor and then the other oxygen on the other molecule is called an acceptor and then you create a relatively strong bond it's much stronger than a myonic interaction so it's borderline a covalent real chemical bond and this might be a few k-cals or so a water so it's same ballpark as those torsions the hydrogen bond as we already talked about is responsible for a ton of things in addition to water it's really what creates both the structure of DNA RNA the base pairing but also the structure of the beta sheets and the alpha helices so that's a super you could even argue it's the most important non-bond interaction in proteins and in theory this could also explain some things like protein folding so if this long chain if we just have this chain stretched out in water first on the left here what you would then have is that if you have polar or charged residues here or other any side chain that can where the side chain can participate in a hydrogen bond those chain side chains would normally if there are had if there are hydrophilic they would happily participate in hydrogen bonds but if they're hydrophobic you would need to have water packed around them that's not particularly good because those water molecules would not be able to participate in as many hydrogen bonds so just hand waving a bit you can almost it might make sense here to take all those hydrophobic side chains that can't participate in hydrogen bonds and turn them to some sort of inside of the protein while you have all the ones that like to form hydrogen bonds as part of the surface and this is actually one of the first things that happens when you put a protein chain in water that the hydrophobic groups will turn to the inside just like a droplet of oil in water I'm going to come back to that later today but what we've roughly created this far is that if you like equations for any type of small molecule we can formulate fairly simple physical they're super simplified although there are a lot of some Greek letters here but you have bonds angles these torsions there are some impropers here which is a basic way to keep something's planar but we don't have to care about that for now and then you have charges electro statics and you have all these non-button interactions and this is actually enough to describe proteins it's a horrible over simplification compared to quantum chemistry the cool thing here is that I can calculate all the interactions of a protein in a tenth of a millisecond and as you're going to see later this means that we can run very large advanced computer simulations where we actually simulate entire proteins which would be completely impossible if we try to do the quantum chemistry but we will come back to the motions later on if you have a large molecule like that each way you pop the molecule this is fairly boring it's just a single water molecule moving but if you have a protein that you move in different ways for every confirmation every possible orientation of all the atoms you could calculate this V what the potential energy is and then you have a potential energy that in principle is a function of six million degrees of freedom if you have a ribosome and I'm not sure about you but I find it really difficult to think of six million dimensional space so we're going to simplify that slightly and stick to two dimensions so if you have two dimensions we could have an x and a y axis and then as a function of these two which could be Ramachandran sorry it could be the phi and psi torsions right for each value there there would then be a value of the potential energy that would be these surfaces and what we want to now want to understand is that why will be a molecule such as a protein adopt certain confirmations in this landscape and why will it move between them but we don't really have the theoretical tools to do that yet so what we're going to need is a way to say that for an energy landscape in general here what parts are bad and what are good you can probably already now guess that high energy in general is bad while low energy is good but it's not going to be as simple as saying that you will have all the protein here and you will never be here this other blue part here is also kind of good it's just not quite as good as the best part yes for now I'm deliberately gonna leave that open for now let's pretend that we I know that I mentioned free energy earlier today but for now let's pretend that we don't know what free energies we only know about energy so what I defined on the previous slide was a function that calculated potential energy in physics so for a second let's assume that's the only thing we know about strict energy for now those of you are physicists will likely have seen an equation to handle this the Boltzmann distribution right how many of you are familiar with that oh that's me in that case I might take some of this thing slightly fast to save a bit of time but the Boltzmann distribution will determine a lot of things so for instance if you have oxygens as a function of temperature what's the speed distribution is going to be so the Boltzmann distribution will really start to get to the heart of this what things will happen or at least that equilibrium the book goes through this in two stages I'm just gonna do this hand waving version today I might do a small screen recording of this to help you and then tomorrow I'm gonna do this properly the cool thing with doing this the bad thing we're doing with first special cases that I've only proven it for a special case but it might help you a bit to show that we actually can derive this so what the book here Finkelstein does is that if you assume something very simple that you have a gas in a column very narrow column you're gonna have some sort of equilibrium distribution of molecules in this that you're gonna have more molecules at the bottom here where the potential energy is low but if it was only a matter of potential energy all the gas molecules would be at the bottom that can't happen because then the density the pressure here would be too high so some molecules want to be further up but so we need to find a way to balance at each level in a small yellow slice here the number of molecules there is going to be a balance between the gravity and the potential energy pushing it down while the pressure pushing it up because the pressure is higher down there and there are a bunch of definitions one could do here but since we're doing a special case we're not going to worry too much about the detail if you're a chemist you know that PV equals nRT and if you're a physicist you like to say PV equals nKT the only difference between those is that chemists love to calculate in moles why physicists we prefer to count the number of molecules either is fine you just have to be consistent and if you were to do this for a specific case there were a ton of things we would need to introduce the specific volume of the vessel and everything to to avoid that you can just say that instead of large n we use small n so we calculate what are the number of molecules per volume instead of the number of molecules in total and that means that we can also say sorry oops my bad you can also say that what the pressure here is actually is so then the pressure would be nKT we don't have to care about the volume lower case n and if you want to see how this varies if the pressure varies with the height well we can calculate how that varies with the derivative and that's just deriving P with respect to height and that's going to correspond to the difference in the number of molecules with respect to height both K and T are constants here on the other hand you also have the potential energy which at any given volume is the mass in that volume multiplied by the gravitation constant multiplied by the height we're at that's true at any particular level and if you look at a very if you look at a very small piece and then you increase the height by delta H the weight of the gas pressing down in that small area the small yellow thing I showed in the previous slide ups there the difference we have there is going to be that potential energy well mg delta H and then the number of molecules in that and if we have equilibrium here the potential energy acting down and the pressure acting up by definition should be the same so that I'm using that expression from the last slide and then I'm just making those two equal and then I get that equation and with a tiny amount of exercise and Maya might actually do this as a screen recording for you to save some time then we just simplifying this a bit to making it clear we have a derivative with respect to n there and then we have n and then a bunch of constants here that's a differential equation and if you remember your math that actually corresponds to this the derivative of the logarithm and if you do a tiny amount of math you will turn that and that n is proportional to an exponent raised to minus that constant multiplied by the height and this really is the Boltzmann distribution if we say that that is mgh is the energy now this was just a special case but my point is that if you have a special case is not particularly difficult to derive the Boltzmann distribution what I'm going to do tomorrow is that I'm going to show that this is true for any general system without knowing anything whatsoever about the system which is slightly more abstract but it is worth having seen that once but it's a remarkably cool result in physics what Boltzmann really does and Boltzmann is very simple and yet insanely deep the Boltzmann distribution tells you at equilibrium we can calculate the probability of serving two states state a and state b just by comparing the energy and again for now energy is potential energy we don't know anything else and in particular that means that we can compare them so what is the probability of being a state a relative to the probability of being a state b that really has to do with the quotients of those two Boltzmann factors and then the constant in front of them disappears and if you know your exponential laws that means that if you have a quotient here you can actually take the difference between the two terms in the exponent right so the second we're talking about relative probabilities between different states I also end up with a relative difference in energy which is important we no longer have to care about the exact zero level of the energy because we don't know what the zero level of the energy is what this says that I you probably already know is that lower energy states will be more populated and that's going to be the task for you of hand in the first hand in exercise that you should be able to download and work on I still haven't gotten the submission part working the canvas support is working with that something went wrong when we copy the course this year but the first hand is a slightly more complicated this works beautiful in a simple world when we only have states and you look at individual states and comparing their energy but if you have multiple different vessels here which one which shape here would be best energy wise in a way this is not a perfect example because they have different volume but assuming that all these vessels had exactly the same volume which one would be the best where would the atoms have the lowest energy why you are right but so this is the one we started from right this is complicated but so here you hardly have any states with low energy but lots of states with high energy and instinctively if the atoms want to be down here it's good to have a vessel where you have lots of room where they want to be so already instinctively will not just the energy as a function of the height is the same for these two but somehow it matters how much volume you have or how many states you have accessible right so you could argue that in principle everybody would like to be at the front of the concert but you probably want 500 people sitting in the first 20 seats because it's gonna be a bit crowded so some people will accept sitting further back even if you whistle quite as good and this is the where things get slightly more complicated in the real world let's not assume we know anything but let's just start working on this so if you have two states we can well there isn't really anything special here right I can count states it doesn't that just means that some states will have the same energy you're gonna do that in the lab so let's say that we have a state a that have a volume B and the state be sorry state a that has a volume a otherwise you would be fine you can define absolutely anything you want as long as you're fine with the definitions but state a has a volume a I think that sounds better and state B has a volume B B rather than volume you can count it and saying that 50 or 15 or whatever but so let's just say that the number of state here some are proportional to volume then the probability of being in a will be proportional to the volume if all the states of all these states and state energy a have the same energy and there are 500 of those then that's mean that we're roughly going to increase the likelihood by a factor of 500 compared to only having one if all other things were equal so if we now take again the probability of these two ones we should somehow weigh that with the number of states and volume a the number of volume of the number of states for energy a and the number of states in this somehow concept B and this is trivial in a way we haven't done anything I just added a factor in front of each of these so this sounds great but also ugly because I kept introducing constant here now that that's not going to work fine what is the a we have absolutely no idea what the a is let's try a small trick here let's say that if I take something if I have B and if I first take the logarithm of it and then the exponent V is of volumes it's positive so I can always take the logarithm I can instead of V I can write the exponential of the logarithm of V so those V's that on the previous slides were outside of the exponential if it's the exponent of the logarithm well then I can take that logarithm and move it to the exponent here right I haven't done anything this is just mathematical due to I'm just moving things around and then that Katie expression well let's assume that I want same things on the same fraction so then I multiply the logarithm by Katie and divide it by Katie and then I can write it this way and that's the way you're gonna faint because this was supposed to be a simplification and it looks far more horrible than what we started from so this where I should give up and take a step back because in principle it doesn't look that bad there's a ton of constants here but if you get about the constants for a second what I was saying that the probability of being in a relative to the probability being B but now the entire volumes that is still a quotient between two exponential numbers just like the Boltzmann distribution it works like an alternate distribution it's quacks like a Waltzmann distribution everything but you have something ugly complicated here that is not just the energy anymore right it says e minus tk ln volume instead of e so let's just define this we could call them whatever you could call you could come up with any name you wanted it unfortunately we have been really stupid because we didn't come up with any other way to call that free energy there are reasons for calling it the free energy but this has caused generations of students to torment or how do you separate energy from free energy in hindsight we should have called it something completely different but I can't call it something completely different because then you will be completely unable to interact with the rest of the world so we call this free energy because it will have the units of an energy and everything but it's not an energy we should have we should have called it whatever no idea the only problem is that this volume gets complicated it gets really ugly so anything we do we know how to weigh things by a volume the other problem here is that if I now have two things I should start multiplying probabilities and there's to that to be a very beautiful way if I take this entire the logarithm of the volume and say the logarithm of the volume is a new constant I didn't I haven't really defined the volume that exactly so I can certainly take the logarithm of it we know that it's positive and to make it even easier let's build that K the Boltzmann constants let's build that constant into this because if you already have something that's a constant let's throw more constants in it so that we avoid having to see it all the time this is what we call entropy do not for a second try to understand what this means now this is a definition it's merely a definition to make the equations on the previous slide simpler to you it's a mathematical exercise if this works out what if you do this you can instead of having energy the expression we had on the previous slide we call that if and that's not the e minus temperature multiplied by this new concept we had and what this will do is that we'll explain the concept if you have different volumes or different amounts of states available the other really and that's also means that the probability of being in these state a relative to be this will now be something that looks like a Boltzmann distribution but instead of the difference in energy it's the difference in free energy so the free energy takes into account the multiplicity of states and I had two students who made a beautiful lab about this a few years ago and that's your first hand in task you can do this you can design this with a very simple Python program and the second you introduce a degeneracy in the energy level this logarithm will naturally enter this literally counting states on the microscopic level it's not the specific volume or anything this also have a very nice properties if I have one system here a one and another system a two you might have heard of things about order and the number of ways of combining them but probabilities become difficult because you should take the probability of probability one and multiply that with the probability of two right when it has to do with combinations of them the neat thing with taking logarithms is that if you take the logarithms of those you can just add the logarithms sorry yes you add the logarithms as is the entropy is defined as the logarithm of the number of states this becomes a property just like energy the energy in the first state and the energy to to get the total energy just add them up same thing here the number of ways you can combine this or the disorder of whatever you might want to think about the available volume if we use because we use the logarithms of this we end up with a nice problem that has exactly the same features as energy you can add it but this is there's absolutely nothing wrong if you like equations along derivations you're more than welcome to use those expressions I so would not recommend it but the point is that this is not a complicated definition we're using this is something to simplify your lives just don't try to think too hard about it then it's not as natural as energy and it's always going to turn out to be much more natural so that's the first thing that I want to dispel in this course many of you probably have a a hunch that entropy is something really difficult entropy is very easy it's the logarithm of the number of microstates period now it might not be easy with a gut feeling for what that means but it's not the difficult definition at it's completely trivial so if i show you an example here that this is my desktop no the top left is always not my desktop how many states does that correspond to if the way place your icles well no that there are 70 icles right but that is one particular placement of the icles so if you compare how many states there are in each of the cases it actually is that that is just one state there's one particular way of organizing them so how many state does that correspond to yes and that's the first word there is so if you're looking at a snapshot static snapshot it's only one specific state these two are exactly equivalent which does not correspond to your gut feeling about it right this somehow more ordered that's more and the reason for that is that how if you think about how many similar states are there there are very few states that are similar to that one but there are very many states that are similar to that one so where where our minds go wrong here is that you really have two ways of defining states one way of defining state is what is the exact x and y coordinate of every single icon but that is not how you think about the states you think about the states you've taken the 10 000 foot view of this right roughly how ordered are they and roughly what is the organization and we should probably separate these definitions so that you could call it a microscopic state or a micro states that's the x and y coordinates but we prefer to say well if all the icons are neatly lined up that's one or a few states but if all the icons are in a horrible mash there are there are lots of states similar to that so if you just randomly throw them out it's much more likely to turn up in one of those states and that's why that's why you can occasionally interpret entropy in terms of disorder the disorder corresponds to more volumes available and everything but that is just the interpretation of any entropy and i think that's where lots of us go wrong when we somehow try to define entropy as disorder entropy is strictly just the logarithm of the states then it's our minds that tend to associate that with order this can actually explain a lot of things for instance why oil forms droplets in water because it turns out that there's going to be more ways that the waters waters in particular can orient favorably if you put all the oil in one place rather than disperse the oil throughout the water i'm going to come back to that in a second and that's why the reason why we're now going to move over a lot to free energies free energy will be able to tell us what things happen when do phases separate when do have phase transitions when will a hydrogen bond form not just that it can form in theory so free energy is much cooler than energy because free energy starts to interface with chemistry and the real world in a way that the energy never does so in particular free energy will tell us that this virtually this will never happen i'm still waiting for this to happen on my desktop but and if you look at the laws of thermodynamics this really corresponds to the second law here that the entropy of an isolated system not in equilibrium will keep increasing over time and approach a maximum when it reaches equilibrium so you want to spread remember those test those tubes the vessels where you had the gas right that gas wants to spread out over all states and if you're spreading things out over all states the available volume is larger until you've spread it out over all states and that's when you have the max in entropy so that the mere fact that things want to spread out that's what's going to give you the property that the entropy strives to increase if this was not complicated enough first assistant chemist can't ever agree on anything sorry uh so if you're a physicist life is simple you have a well-defined unique system and you only have this interaction i spoke about roughly and yet this system if you're a physicist you always want to isolate the system from the world you might say that you can exchange heat with the rest of the world as we do on the left here and that's what the physicist would call Helmholtz free energy it's plain it's simple you only focus on the essentials the only problem is that it doesn't correspond to reality because in chemistry this is super difficult to achieve in a lab you would need a very special container and what you'd rather have in chemistry you have open test tubes right but if you have a if you take one liter of water and mix that with one liter of ethanol the new volume is not two liters but roughly 1.8 liters so that you're also going to have effects of the pressure and the interaction with the surroundings so in chemistry to be strict you also have to care for the fact that the size of the system can expand or contract and then you need to include this p multiplied by the term the work you're doing on the environment and now i'm going to do something that makes you happy i'm going to exactly ignore that for the rest of this course and the reason this is super in super important difference if you're designing if you're working with gases air airflow or anything because then the the pressure effect can by far be the largest contribution but for proteins and life sizes we typically work with concentrations that in the millimolar or nonimolar right so we have so the concentration of samples we have is so insanely small that this p by v term is going to be a millionth of all the other terms so we will happily ignore it so if you're a chemist we should always work with Gibbs free energy we typically call it delta g but there are so many cases you will just see me saying well we'll talk about g but then i say e minus ts so the whole different difference here the first cases would use e strictly if i should be proper i should use h all the time but we are sloppy everyone is sloppy just as we're sloppy when it comes to the difference between energy and free energy people are sloppy between the difference between f and g because chemists don't care about Helmholtz, chemists will happily use f when they mean Gibbs free energy the only solution is you have to be careful about your definitions by all means you can use z for this if you want to if you properly define that that's what you mean by your free energy in an ideal world we would be super careful but nobody is and even i'm sure that i will screw up in this class too so if it was not enough to confuse well if you think that things were easy now i'm going to make it a bit more difficult this morning you probably thought that entropy was easy but temperature sorry entropy was difficult but temperature was easy i'm going to argue it exactly the opposite way around entropy is a super simple definition that any anybody who knows a logarithm can understand entropy it's just a plain definition temperature on the other hand that is really really difficult so in all those equations i just introduced a t and there was not a single one of you who asked me about what the t is why didn't you ask me about that because it's something you've seen in equations for 25 well 30 years and you just assumed that it was temperature right there was nothing in those equations that i said that it was temperature and in principle this is just a constant in these equations a constant that depends on the conditions where you're performing the experiment and if you just start from this definition we had f equals e minus d s oh my god it's fun so i did yeah those 30 seconds ago the first thing i do is that i dropped i just dropped the pv but i called it f anyway it was not intentional if you look at the very small difference here just an infinitesimally small change df for delta f you can actually use this to extract take the temperature and extract this at the equilibrium if we have an equilibrium and if we argue that this is a property of this environment at equilibrium this should be constant and we also as i hint before that we believe that the free energy has a local minimum at this equilibrium i haven't proven that then a bunch of these terms will disappear because e minus t s should be zero and if you then use that and just solve for the temperature it turns out that the this t constant is really the derivative of the energy with respect to entropy and that is completely obvious to you right it's not to me i have absolutely no idea what that means i have zero gut feeling for that actually that that is an equation i do know but no matter how much i look at that equation it doesn't make sense to me it does not correspond to anything natural it's kind of a slope of as we are increasing the volume how how sensitive is the increase in energy relative to the amount of local volume that's available it's completely non-natural but the really cool thing is that this is a property that we're not defined it's not something that you mentioned at thermometer this is a fundamental property we get from statistical mechanics and the absurd thing is that entropy that any teenager thinks that entropy is really difficult entropy is plain simple temperature is one of the most complicated phenomena in modern physics so it's exactly the opposite of what you're used to having said that you can just this is the thermodynamic definition of temperature it's exact and normally today in the essay this how we define temperature it has nothing to do with the heat and the surrounding that is just how we on a day-to-day basis are used to interpret temperature and armed with this it turns out that we can actually describe quite a few things very well phase transitions we spoke this morning about when things are solid versus when they are a liquid why do things convert between solid and liquid what that will have to correspond to these differences in free energy so at very very low temperature which is the thing we defined on the last slide at special environment conditions you have a very well ordered system so that's going to give you very low energy low energy or potential energy that's good on the other hand it's also a very well ordered system and a very well ordered system corresponds to very low volume rights and that means that the entropy is low so both that term is low and that term is also low on the other end if you have a completed disordered system here the energy is not going to be as good because the interaction is not ideal but on the other hand they're also a much more much larger diversity of states so there are more microscopic states that the system might occur in compare that to all the icons on my desktop right so that the entropy is also going to be larger because the available volume all the different ways we can put these atoms it's much larger so that means either they are low low or they are high high so i can't instantly say which one of this is better which one is better actually i can because the balance here will depend on the constant t here right if t is zero if influence of this s entropy term disappears then it's only a matter of the energy so at a temperature that's sufficiently low you want to be in a well-ordered states and that's eventually why anything will be a solid but as the temperature here increases this term will start to gain an importance and eventually it's more important for the system to have a good low sorry a good high entropy rather than having the lowest possible energy and that's why eventually the molecules at higher temperature it will be more important to have this freedom even if that means paying a bit in terms of having not quite as good energy and aren't with that you can start to solve some yes we will come back to what the phase transition is for now it's a super good questions but that will have to wait until tomorrow that's going to be the next step what actually happens at a phase transition but aren't with this you can already now starts to be banking some of the greatest myths in science you are probably too old for too young for this in 1962 there was an amazing results from the soviet union where Fidyakin and Deryagin published the result where they have claimed that they had discovered poly water so under very special conditions and you push water through very narrow capillaries they argued that they had found a new state of water where water spontaneously started to polymerize so a completely different phase of water it's a liquid phase but that would be more polymer like with a much higher viscosity and everything and this was brand new physics this started to make the reputations at different conferences and everything and the us were terrified that Russia would the soviet union would develop a fully water gap and people even argued that there was some sort of strange hydrogen bond formation pattern between waters that would create some sort of super molecules here they would also argue that the free sea this was very different from all my water you would have a freezing point that was just 240 Kelvin and a boiling point that was much higher here this is of course completely bollocks and based on what you know now you should be able to debunk this I'm well aware that this is not trivial in the interest of time I might not give you five minutes to talk through this but and this is not mentioned in the book that there is a small paper about poly water if you just draw these three phases solid liquid and gas as a function of temperature for any any normal molecule like water at the very lowest temperature we know that the solid should be the best one right the lowest one and similarly at very high temperatures we know that the gas phase should be the best one and that there is some sort of intermediate range where the liquid is best but based just on those two numbers if you have a freezing point of poly water that is down there and then a boiling point that is up here there should be a some sort of pink curve here that is constantly below the green curve right so what would happen to all the water in the world if this was true all the water would prefer to be poly water and I'm not sure about you but I haven't seen a whole lot of poly water around it was completely false actually it was an incorrect result and it was actually due to when you push it through these capillaries you actually get parts of the glass walls and everything to eventually this so you would have silicone dioxide and everything ending up in the water but this was a very it was took about a year and there was gigantic conferences about this and the US were worried that the Russians had discovered something about fundamental physics they didn't understand there is quite a fun novel about this that Kurt Vonnegut wrote called Cat's Cradle where well it's not particularly good novel I guess but part of the picture is that there is a new form of ice called ice nine and the idea with ice nine is that any if ice nine ever comes in touch any water that comes in touch with ice nine will eventually form this new phase and then it leads to some complicating and there there are a ton of different forms of ice lots of different crystal forms at different pressures and temperatures and everything we're not going to go through that in details but then again that has to do with these complications right but the important thing is just armed with us f equals e minus ts it's a at first sight it's a remarkably simple equation it's an exceptionally deep equation that can explain 90 percent of everything you see around you that's it you could even organize it's a bit embarrassing that the yakin and the yakin didn't really sit down five minutes and think about this so i will come back to this several times in this class if there is this is also an equation you need to know if i wake you at two a.m. in the morning look at f equals e minus ts and not just you already know it by heart right but there is a fundamental difference between knowing this equation by heart and understanding what it means and learning to work with it and that's increasingly what i'm going to force you to do learn to work with us and think about what it actually means and as a small exercise we can look at hydrogen bond formation if you have two waters evacuation to water molecules we might want to invest in what happens when you form a hydrogen bond why will a hydrogen bond form well we know what happens that there is a change when we when we gain a hydrogen bond here there is some sort of change in energy and it's no big surprise that that's going to be if the end the energy of something that does interact is smaller than zero it's it's advantageous to have it you will also lose some ability here because there are more ways to orient those molecules but if we now tie these molecules together we're going to lose a bit of freedom here the exact amount of freedom we lose we can calculate each water molecules participates in four hydrogen bonds there there there so that means two hydrogen bonds per molecule on average right because there's a donor and acceptor for each of them and if we assume that if you form both we're in ice we're completely rigid so we have lost half of that so 0.5 by 2 so we've lost entropy corresponding to one freely rotating water and it's less advantageous here so we're already we can say that the delta s here is smaller than zero and now you should be able to answer here what is true for H bond formation is delta e smaller than t delta s or t delta s smaller than delta e why so to how many of this so first do we have any other suggestions that's certainly true how did you try to think about this you have memory like a goldfish what did i tell you 20 seconds ago there is a very simple equation use not the force that's the equation um actually i deliberately cheated a little bit first f equals e minus ts is a bit over simplified the first thing you need to care about anytime there is a process happening there is a before and after right so start by writing down what is the before and what is the after and if it's before and after it's a change then we should think about it's not just f but it's delta f so what is the change in free energy and that corresponds to a change in energy and a change in entropy and you had exactly the right the figure we know that if the hydrogen bond forms if there is a hydrogen bond formation for that to happen what has what does the sign of delta f have to be negative otherwise it would not happen spontaneously the second you've done that it's just you have zero equals delta e minus t delta s and then you just solve for it so don't the point is equations they're a help they help you i so to tell the truth if somebody woke me up in the middle of the night say that delta e minus t delta s i could not say which one is smaller but i could write down this equation and two minutes later i would tell you which one is smaller because i looked at the equation so don't try to hand wave this and jargle and guess what do we always go back to e minus t delta s and think about it the cool thing is that we're not going to particularly complicate the equations but this particular equation we need to know so as i expected i didn't quite have time to finish all the slides today either that's fine but what i'm going to do tomorrow i'm going to start going through a little bit more in detail what this means for things like formation both in proteins and for instance hydrophobic effect when we put oil and water and then i'm going to come back once we've done this hand waving ways of treating free energies then i'm going to come back a little bit and do a slightly more proper physical derivation of the Boltzmann distribution and the reason for that it's not something i expected to know by heart but it's words that have seen it once because it's universal it's true for every single system you don't have to assume anything about the system that's what makes it so amazingly powerful because we haven't assumed anything we know that it's going to remain true forever no matter what other discoveries are made in physics and then armed with that we can take an even deeper tour into more complicated biochemical systems and start looking at what this implies for proteins and that'll finally that's going to turn into protein structure but if there's one thing you should do you should go home and look at this equation and think about what it means for different phases and everything and then we'll talk about phase transition tomorrow