 Good morning everybody. As I told you yesterday, the way I'm going to try to run this is that we're going to start and have you or us discuss things together before I just push ahead and go through another entire lecture. We'll see how well that works. If it doesn't work, I'll just keep lecturing. And but my ideal scenario is that we spend so much time discussing that we don't make it through all the slides and then I will just push those slides to tomorrow's lectures or whatever. I have also updated the schedule so that now your schedule should be complete in the sense that I've also marked all events when we plan to have computer labs. There are actually a total of 11 of them marked. I don't expect that we will have more than 10 but I figured it's so much better for you to know roughly when we plan this and what pace we plan them and then some of those events might disappear. That's probably better than the opposite. Suddenly getting new events. The facility manager for the Crye facility is not here this week. So my plan is that directly off through Easter either Monday, April 4th or Tuesday, April 5th we're also going to go down and do a study visit and have a look not just at the microscopes that we can do anytime but also have a bit of look their images what type of data we can get from these things. And then Dario and Bjorn might tell you a little bit about the data processing they're doing for it too. The final thing when it comes to schedules is that we talked a little bit about yesterday that about research seminars and interest right and one thing I realized we actually have a structured journal club going on every Thursday morning 9 a.m. in this building. Tomorrow we're going to have Laura Oriana from my team talking about epidermal growth factor receptors and models and this is related to cancer models in mice actually. It's pretty fun stuff. She's gone all the way from theory and model building and then she's been able to find likely transition states where these things bind. Is that something you're interested in? So maybe I will at least when these lectures fit I might actually plan for us at 9 a.m. on Thursdays we start by going to this journal club or it's journal club it's a group meeting seminar whatever and then we start the actual lecture at 10 instead. As always the quality of these things I think the quality of the research is always good the quality of a lecture or rather how interesting it is it depends a lot on what you're doing. Some of these can be extremely theoretical you might have a student talking about theory development on refinement methods next week you have somebody talking about super applied stuff but that's the nature of research people do different things. Good I will plan that in the schedule and I've also entered some other things. Again the idea with these seminars they should be voluntary you should go there if you find them interesting I might add even more things but don't feel that basically I'm not going to ask any questions about the research seminars on the exam the examples are going to be on the topic of the course. So what we talked about yesterday as you probably remember that's the advantage of doing things at full pace you don't have a chance to forget. I talked a little bit about experimental not just structure determination but methods in general and I in particular I brought up these two methods x-ray crystallography which has been the this has been the basis of almost everything we know is structure biology but this is gradually giving ground in particular to cryo electron microscopy. There are some other methods I don't even remember whether the book mentions them in particular one called nuclear magnetic resonance NMR. I'm not going to see that NMR is dying that would be wrong but I think it's fair to say that NMR has always had a bit of a special life oh we have a visit here come in and NMR has an advantage of being able to show you how molecules work in solution but it's not really a method that's a resolution vice is competitive with either of these two. We have a visitor today as I mentioned to some of you those six of you are in the PsyLifeLab master program which is not everybody in the course you can have a chance to do internships or whatever yes we call them internships right or summer projects in this building yeah or rather everybody is of course welcome to do these internships but when it comes to the PsyLifeLab students we managed to get this building to pay you a little bit for it so as I told some of you maybe not the newest one NMR is going to spend a couple of minutes at least going through this okay we'll do that later I can compile it thank you so much NMR so sorry for that brief interruption I am actually there are a bunch of other things but since we covered this yesterday I will jump straight ahead to a more fun tutorial these are roughly the study questions I gave you yesterday and I actually I think I removed one because I realized that stuff that we don't really cover until chapter three or four the hydrogen bonds in water sorry about that but then I added I think one or two more so we got all the way to 20 if I just start asking you things absolutely nothing is going to happen here so I have a suggestion you start picking questions and you can so pick a question you would like to answer and answer it and the idea is that if you're quiet and wait you can end up with the really hard questions at the end yes go right ahead yes yes and why exactly the point is has to chirality requires four different groups around something too hard to do this that criterion fails check and then I was really close to start making marks on the screen which I want if you're really smart you might actually have a in theory there are some questions that might have two answers but I think that's extra information for 19 to go no and why not this is actually a hard question that I did not bring out so it's related to something that I'm not sure do we have a separate well it's related to number seven of course the same function yes and it's related to what we call the central dogma of molecular biology which is what sequence leads to structure leads to function but it's a directed graph arrows only go one way now the complication that I did not bring up yesterday the complication that I did not bring up yesterday is that there are there is a phenomenon called convergent evolution and convergent evolution really means that two speed two species or at least two different genes they might not be evolutionary related but they might somehow evolve towards the same functionality and this is complicated because as we brought up in this course in many cases there is really one obvious fold or geometry to achieve a certain functionality right and nature has had four point three billion years of trial and error and if you have four billion years of trial and error and this is a very simple functionality eventually it's likely that both these genes will find the same optimal structure so I would argue that that is the exception that confirms the rule occasionally you can actually have that the function somehow leads to a structure but it's still I would still not say that it it does not induce structure it's not that a specific function means that it has to have a specific function with with a few with a few special exceptions so in general the answer is no you could imagine that if you want a large molecule that should bind a heme group at port of porphyrin then it turns out that it is really nice to have histidines above and below it but that's local structure so locally we might do it but globally globally it never happens much longer answer than I brought up yesterday I think that largely answers seven two yes brownie point sir yes and in particular it would be incompatible there are other things in theory would of course be able to interact right but it would interact in a different way and when you say other proteins other amino acids are the things you're thinking about there that's important because sadly whenever whether we're doing bioinformatics or biophysics we always visualize proteins one-on-one and that's really bad in a way because proteins any protein will interact with at least a hundred or a thousand partners it's it's a soup inside a cell any protein you need to build it somehow in the in the ribosome you need to degrade it that some point the whole point they're not really they're not compatible with all the l isomers that formed every single one of these hundred interaction partners but as you say occasionally that can occasionally it's a really good thing not to be compatible with stuff so that begs another question what if you had a protein that consisted entirely of the isomers yes so the helix would be left-handed is that and and this is this is important right because normally in physics you frequently talk about symmetry and you can even talk about symmetry from particle physics point of view and everything but in general physics is symmetric it doesn't matter whether you you have the same laws of physics whether you're doing something on the left hand of the right hand side right but the problem is that this chirality leads to what you call a symmetry break and that's why for instance an alpha helix is always right-handed right because in principle that's a break of symmetry the mirror image obeys the same physical laws but we can't form you can actually form a left-handed helix but that's not really that's not going to be a mirror even with normal l amino acids you can form a helix going in the other direction but it's going to be a very unfavorable helix that occupies a small part in the ramachandran map so the reason why alpha helix is our right-handed is because we have the isomers we do so why why do we have l isomers not the isomers i don't know nobody knows so at some point maybe maybe nature started to evolve that way by chance yes and of course when nature starts to evolve that way we have to say that way or we're not going to be compatible but that's an unanswered question no because i remember the laws of physics are symmetric right if if we took you your entire body and turned every single amino acid into a d instead of an l amino acid it will still work everything would be everything would be internally consistent and compatible so the laws of physics are the same but these kind of they're two worlds that exist on one one side each of a mountain rich right it's very hard to cross this rich because they're not compatible with each other but the laws of physics are the same almost yes but they're not going to annihilate each other it's just that they won't really talk to each other so there is something else you can i'm not sure whether i'm going to bring this up later in the course so that you don't necessarily have to use d amino acids if you want something that's not that's not digested by the stomach you can use non-natural amino acids so rather than picking one of the normal 20 ones pick something slightly different and then there are a handful of amino acids that are quite common to use to you can for instance there are license chains of different lengths and if these amino acids are different enough they will usually not by be degraded by the normal enzymes in your stomach which is good if you want to have a way to administer a drug okay more questions yep so what do you mean when you say all of them are charged you're actually right and in one way this might be a bad way to pose the question so would you pose this question in a different way i would rather how many amino acids have a net charge so a net charge would mean that the entire molecule has a charge that does not sum up to zero uh so what you're talking about is this property that they're twitter ionic right you have two charges in them but one is positive and one is negative but that means that the net charge is zero and and the problem with that is that that's quite true when the amino acids are isolated but the second you start putting them in a polypeptide right when we form the peptide bonds those net charges disappear so they become polar but so there is no net charge anymore when they're in a protein but otherwise your otherwise your answer was quite right so you mentioned three plus two amino acids bring them up again the names yes that's correct yes and lives in arginine and histidine are what they are positive and let's see there should be some questions somewhere yes we have questions two and three here acidic and basic amino acids they are one of them but are the acidic or basic so that's we have two answers here can we have a third okay they're basic um the reason for that is uh to tell the truth i keep forgetting this too um the way you should think about this is take a paper and pen draw the amino acid so what do we mean when we say that they're basic they well that the point is they behave like a base or versus they behave like an acid right i think if you have an acid and solve it in the water what would that group look like and it's the same for the amino acids uh but i always have to draw these unless it's of course and i have memorized it uh and then we all said two others that you mentioned that the acidic ones that are negatively charged that's a good that's a good way to the other caveat you mentioned is in passing very briefly is it true that they're that they're always charged so when we say that they're charged this is usually at pH seven and if you change the pH enough they will become neutral and there are other amino acids that will become charged and normally lysine and arginine they're at virtually any relevant biological conditions they're always going to have a positive charged and aspartate and glutamate the opposite they're always going to have an negative charge but there is one culprit one really horrible amino acid to deal with when it comes to charges and that's histidine because the problem is that normal those of you have taken care by a chemical inorganic chemistry they might know that you talk about this pka values or something and that's the point where you start to switch the protonation state do you know what this value is for histidine so the problem is right next to seven but that value is if the amino acids exist isolated in solution well that's really interesting if you're a pure inorganic chemistry or something but in life science we talk about these amino acids in proteins and the second this amino acid exists in a protein first you don't have any water around it you might have an hydrophobic environment it's bad to have charges in a hydrophobic environment you might have a positive charge next to it and then you'd up with something called pka shifts meaning that suddenly this point is not at all 6.5 anymore but it might be like five or eight so the problem is when it comes to histidine it's virtually impossible to predict exactly where this protein is going to sit because you have the first you might have you might have two protons you might have one and when you only have one it can sit on either side of the histidine but that doesn't sound like too much of a problem right we can determine that with a structure determination method which one would you pick why so the problem is that the x-ray crystallography won't see the hydrogen and that's that's the only difference between these two states it depends where you have the hydrogen right so why doesn't x-ray crystallography see hydrogens well x-ray crystallography doesn't see any atoms it does sees the electrons and a hydrogen is typically very well a hydrogen typically is not too fond of its electrons so whenever a hydrogen is bonding to something it usually donates it to the electron to the other atom so normally you have hardly have any electrons around the hydrogen and that's why x-ray doesn't see it neutron scattering on the other hand loves to see hydrogens it scatters very well sorry that was not stuff we covered yesterday but under since actually it turns out understanding where how amino acids are titrated and what charges amino acids have that's super important for lots of cases and particularly if for instance if you want to drug that's supposed to pass a cellular membrane or something you don't want it charged that's bad on the other hand if you're having something like a voltage gated ion channel a protein that should somehow be controlled by voltage what do you need them well what happens so what happens with every single heartbeat or something in your cell you have a change in potential so what's going to happen to a ligand if you change the potential if you change the electric field absolutely nothing unless so if you have an electric field a charge in an electric field will be subject to a force so if you change an electric field on a charge that you will pull that quite quite hard actually so the point is at any time I want a component in a protein or something that should move when I change the field we need to have it charged so then we need those charges but then you end up in a bit of a bind here so that on the one if you now have a membrane protein that should change in an electric field you can't have charges because you want it in the membrane but you must have charges to get it to move in an electric field and that's going to be a problem we'll come back to that when we talk about membrane proteins all right do we have more yes we do what are the levels of structure organizations in proteins anybody want to have a go at it yes and what do they represent anybody we call and that's sorry yes you said that secondary sorry you've been said secondary structure my bad and we'll come back to what these domains mean the funny thing is that nowadays we have slightly different definitions of domain in bioinformatics and say physical chemistry in bioinformatics we think of a domain as something that's evolving independently while in the terms of protein structure you typically think of a domain as something that folds independently so remember the well no we have other questions about that so in 99% of cases they're roughly the same but there are a couple of exceptions good we have more questions okay models they always had three strengths of nucleic acid and I think they say that the basis of the inside and the possibly the outside and that they they say that's the salt and not the acid that yeah so those were really important findings and I that's very much related to the key finding so when I said the key finder this leads to something uh so that they propose that there is a specific pairing of the basis that corresponds to the number of hydrogen bonds right so that you have ag c and t and a and t pair and g and c pair but they can't pair any other way and this of course when you have the specific pairing this immediately leads to a proposed copying mechanism for the genetic material you can split them but then when I need to pair up they can there is so for each base there is only one other free base that it can pair with and that's really the funny thing is that they didn't really this was an observation that people had very early on that there appears to be the same amount of a and t and dna there appears to be the same amount and g and c but nobody really drew this conclusion until they saw the structure and I think this is it's important to realize we just saw the polling structure yesterday and it's so easy in hindsight to say that something is stupid while something else is so obvious there were a number of amazingly smart researchers they had the results they saw that it was exactly the same amount of at and gc but nobody thought that this is this is going to change everything in the 20th century about molecular biology so remember that I asked another another question in yesterday's lecture why did it take nine years for Watson and Craig to get the noble prize from the time they published their paper in 53 until they got the price in 62 it's obviously not something you're going to read online so the paper was published in already 1953 this is an important lesson so that again this is not the leak but so that the all the the secrecy has been lifted since more than 50 years ago and there are a couple of professors in the royal academy of sciences that have looked into this and I think they're writing a book about it now even it turned out they were not even nominated until 1960 and that's another one of these results science we usually think of science and hindsight it actually took seven years for anybody to realize that hey this might be a pretty important result and today we think that it's obviously one of the most important results in life science in the 20th century took seven years for anybody to even notice or not notice people of course cited the paper and people in the narrow field felt it was really important but the impact of dna wasn't really obvious then of course 10 years later when we had the genetic code and we started doing molecular biology and everything nowadays is obvious but science does not happen in hindsight and that's I think what you should think of now when it comes to choosing diploma or well so summer projects thesis works PhD studies and everything it's very easy to be seduced by whatever is hottest right now then neither Watson or Crick would have hunted molecular biology it's important to dare to go to the white spots in the map and they are rarely obvious more things so so that's quite right I would just like to defend live in Cyrus a little bit so Cyrus formulated that this is obviously a paradox it's not that he disagreed with Christian but realized there is something here we do not understand because obviously they fold but we can't explain that simple from the laws of physics all right other stuff five psi omega and chi yes and that is by far the least important of those angles you're forgetting one there has to be four atoms in a torsion now you're doing this the really difficult way because when you started I started to think which atom is which don't try to think of those and I which one which one is five that's the bond just be that's the bond just before the c alpha and if I need to know those atom names I will draw it on paper yes and the reason why I just write chi here is that chi one would be the first bond directly after the alpha carbon and then chi two and chi three but the further out you are the less impact you have on the rotation right so the first chi the first chi angle has a fairly major impact and when it comes to predicting side change for instance normally we measure how well we're predicting chi one that's important you occasionally measure how well you're predicting chi two and further out than chi two we don't really care about it and the reason for that at chi three or something is typically just a matter of rotating in an h3 or a c h3 group or something is not really important for structure water we're gonna we can't just skip water because I'm gonna talk more about that today and tomorrow since we talked about these angles how do you define a dahed rule or torsion angle yes yes that's when it comes to the actual angle I think the point is if we forget about the two alternative definitions so what's that's an angle but what type of angle is it exactly so typically this corresponds to rotation around the bond although in principle anytime you have four points you can define a torsion and that's right so which ones are the most important degrees of freedom in a protein why I didn't expect that follow up yes but maybe let's start with the others why are for instance bonds and angles they're obvious degrees of freedom why are they not important or they are important but why are they not relevant for protein structure yeah bonds are hardly at all excited at 300 kelvin and angles might vary a couple of degrees while the torsion degrees of freedoms these energy variations are weak enough that it becomes relevant they can actually change and system trans is what or what the trans is when they're on opposite side two large groups on opposite side of a bond and sis when they're on the same and so what type of energies do we typically talk about in proteins 18 no yes typical energies and that's something we're going to come back to today so that's a very high energy I wouldn't say that that's a typical energy in a protein so what roughly what type of energy is that so that would be an electrostatic energy of two charges separated just by one angstrom typically charges won't get that close to each other so that would be an extreme this that's a type of energy that will never that's a type of bond that would never ever break a covalent bond or something so the typical energies we relate to in protein by far if there's one energy you should remember it's roughly how much is the energy for hydrogen bond and that is related to why water is such an important molecular life five what k cal yes so it's important that the units right and sadly there's both k cal and kilojoules you can use whatever you want but don't mix them up this is a relatively high energy normally it will be formed but it's weak enough that we can occasionally break it and we will come back to later know that the typical stabilization energy for protein is in the order not just one hydrogen bond but maybe 10 or so almost regardless of the size of a protein so they're not the stabilization energies of proteins are fairly low and finally one thing that it just found out what is post-racitational modification since we talked so much about folding I mentioned that briefly yesterday yes and by structure here this usually means that you're modifying an amino acid or something that you might be binding typically it has to be glycosylation or things like that occasionally it can be binding a cofactor hemoglobin for instance need to have the heme group bound but the heme group is it doesn't consist of amino acids you first you fold the protein but while the protein is folded you need to insert this other group sorry so insulin is a very special molecule so what what insulin does is that the chain is actually cut so insulin the funny thing is that many insulin was the first protein ever sequenced by Fred Sanger so insulin actually is cut so that it consists of two chains so the small molecule actually has two parts of it then the funny thing just as I mentioned that it was a fun coincidence that we got the first new structure was hemoglobin and myoglobin that were so closely related I think it was just a coincidence that Fred Sanger started working on insulin it wasn't specifically because of that but yes insulin is special in that way but there are many molecules that have cuts and in particular for higher organisms when genes are separated in introns and exons and I will come back to that we talk about structure that's is there anything else you want to talk about related yesterday or to bring up the push to discuss yes sorry that they predicted the helix by by Matt in well if just in principle I would say it's math and I think it was mostly paper and pen though that they said just as Watson and Crick did what you frequently did since you didn't have computers in those days we were frequently sitting here working with molecular models and then you realize a molecular model is actually a beautiful well before we had computers it's a beautiful way to see what Raman what torsions fine ciphers are allowed even I when I was a student we had a lab in London University in the south of Sweden when we said and did this and see what regions in the Raman Shannon diagram are allowed so what they did is of course they started by seeing what is allowed and then the starting for the regions that are allowed are they some confirmations that are more favorable than others and I would my hunts would be that the alpha helix is likely the structure they found first because really it's a regular structure it's one that's allowed it's makes it very compact and then you can form all these hydrogen bonds now the second you've done this you can of course start to show that this structure you proposed is better than all the other ones you've listed beta sheets is a I would actually argue that beta sheets is a harder structure to predict because beta sheets the individual strand it's simpler but then you need to start to making assumptions about global protein structures how proteins would fold in general just from the point of view of a simple molecule and remember we didn't have protein structures when they did this from the terms of a simple molecule that somehow folds upon itself and alpha helix is a much more reasonable predictable structure the beta strands although the individual strand is simpler it's a more amazing prediction but since yes half math because they of course showed the energetics but also just half paper and pen and sitting with a model model and realize what's possible is that these motifs of like flighting implies that there's something separated by four sure that's not that common so the point with alpha helix is they're so common that Fred you rarely see proteins in alpha helix and remember in 1951 it's not like today when we had whole databases of sequences or anything Fred Tanger's invention was roughly at the same time right so we didn't really have sequences for most proteins we knew that what the average amino acid compositions of them were but we had no idea about what order the amino acids were in and we didn't even realize that well Fred Fred Sanger's result that there is a unique sequence to each protein that is conserved and remember that it's easier to think that people it's just 60 years ago and it's so easy to think that people knew much more than they did that it's a fairly young science good and I will there's one plot that I found on the internet a while ago when it comes to amino acids I kind of like this way and you don't you don't need to know this by heart but the point is there is more than one way to skin a cat and when it comes to amino acids there are lots of way you can divide these proline is all pretty much always unique but then you can say things that are amino acids that are aliphatic that is the one that can do not contain benzene derived groups there are polar ones there are somewhat aromatic groups that phenylalanine tryptophan tyrosine in particular there are ones that are tiny there are ones that are charged negative positive polar non-polar mini hydrophobic you should be aware of a bunch of these classifications and just that you can probably come up with one or two other ones if you want to to for instance their hydrogen bond properties or something and this is of course how nature uses this that you need to find a building block that fits but apart from proline they are typically not unique so in this case if you have a location in your protein where leucine fits very well isolucine is likely also going to fit very well glutamine on the other hand likely not right and if you have a tryptophan and replacing that for a proline that's likely also going to destroy your structure you're probably really aware of this because you have studied this in bioinformatics but again when people first started classifying amino acid they didn't have the sequence databases that we have nowadays the other thing that we talked about yesterday was these ramachandran diagrams but I realized I didn't really tell you I didn't tell you what is what in the ramachandran diagrams and we had these two big regions and these regions corresponds to beta sheets up on the left and then the right-handed alpha helix down here in the middle and then we actually had this left-handed alpha helix over there but you see the relative population that one is so small that you can pretty much forget about it there is always a rule that confirms there's always an exception that confirms the rule in biology but I can't I can't immediately think of any relevant left-handed alpha helix in a structure that I worked with so yet it's a curiosity and I think it's a popular question both in tests and everything to say why are alpha helix is always right handed and that has nothing to do with the ramachandran diagram well it has things to do with the ramachandran diagram but that comes from the chirality it does not come from the fact that it's chance or anything so I think this is no so that first then the points here this is likely just for one protein or something that this is not representative of everything in the database it's actually a very good question I would argue that alpha helix and beta sheet have roughly the same prevalence for globular proteins when it comes to membrane proteins alpha helix are much more common than beta sheets the danger here though is that nowadays we have enough bioinformatics to be able to predict that this is true a permanent danger in biology is that we're always biased by what we've seen this far and in particular for membrane proteins early on it was easier to determine alpha helix structures and if you can only determine alpha helix structure as well suddenly you can end up with lots of structures and they're all alpha helix and then it's very easy to say that by definition all membrane proteins are alpha helix that's not true that's just because we had it was difficult to determine the beta sheet once and now we had some beta sheet once for structure I would still say that it's all but it is very important to be aware of our bias in the structure-determined methods for instance we only see what we can see my plan for today is that we're gonna have a little bit fun I'm going to show you some movies first and then we're going to talk about interactions in proteins so if you felt that there was a torrent of slides yesterday for better and worse we're going to have much fewer slides today I'm going to bring up this comp the concept of saber empiric modeling and part of the interactions is going to be repetition from yesterday and that's quite intentional this leads to some deep questions about how should we model nature in general this quantum mechanics always the answer in life and biology I will well the partial charges are probably part of the interactions I touched upon them yesterday and then we're going to dig a bit deeper into these fundamental properties of proteins and then after the break we're going to have fun because then we're going to start having some I wouldn't say heart physics but there's likely going to be more equations than you've had previously in these master programs so I'm going to start talking about energies energy landscapes and in particular statistical mechanics and the Boltzmann distribution I predict that some of you are going to think that this is a bit difficult in the math and that's why I have fewer slides today we will take the time we need for this if there's one thing I can recommend you if you don't follow me stop me and we'll take it again don't think that you will read this up tonight at home or something it's not going to be easier for you to do this on yourself the good news is that the second part of this slide is going to be a little bit of math and then sometime off through Easter there's going to be one more lecture when actually drive this in the general case and then it's going to be even more math but normally we're going to have less math than we have in these parts the funny thing though is that this will enable us to predict some pretty remarkable things about proteins and actually not just proteins life molecules in general this is some extremely powerful methods from physics and then I'm just going to tempt your appetite a little bit with entropy and free energy we will come back to that tomorrow so if you don't follow me here focus on the Boltzmann distribution that's also what we're going to do in the lab this afternoon so let's see if you can catch what this is this is an old movie so Kendru remember that that was the guy at L&B you got the Nobel Prize for the molecule so how old do you think this movie is first see do you see the molecule in there so that's a prosthetic group that's a molecule that actually binds the oxygen sorry this is a molecule that binds the iron that in turns binds the oxygen so this is if I recall 1966 and the computers we had at that time was this is a short movie there's not quite the type of graphics you have on your phones or something today let's see if I think I should have one more here lies a sign another small molecule so these are not even simulations or anything this are just rendered images where you try to trace out the backbone of these molecules and the reason why people were so amazed with this that the regularity of these and again to you this is likely obvious but in the 1960s people were simply so stunned by the fact that you had these regular but extremely complicated structures in life molecules so this was done by most of this work was done by a computer geek who sat down and worked on a Mac actually that computer geek in question so a Mac at the time was a slightly different type of Mac it's a multi-axis computer so this is the computer to date I'm still not quite sure what this is but I think it's it's not really a mouse but it's something like you use to control the screen here so you don't really have a keyboard so that is the keyboard and how you program the machine and they typically use somewhere you should have a punch card system so the way you would normally program these is that you would have you would use Fortran and then you would write these programs on punch cards sorry you would write the programs and you would translate them to punch card code and then you would put an entire stack of cards in a card reader and the machine would then interpret these holes pretty much you do with chat well pretty much what you do with automated poll stations or something today and turn that into computer code and one of my mentors Mike Levitt you worked not on exactly this type of machine he at one point mentioned that you learned very quickly so when you had your stack of cards which could be like two three hundred cards the first time you fell and dropped these cards you learned that it's very important to draw a diagonal line on the side of the cards you could put them in the right order again because otherwise you had to go back and redo everything and these were the days access to these computers it was not a matter of doing enough and doing a lab as a student Mike mentioned that he used you used to be happy if you get access like the three a.m. in the morning or something and then you could have a couple of hours of time on the computer this is likely one thousandths of the power of your iPhone the cool thing is that that computer geek would Cyrus Leventhal so you think of these people as old gray-haired professors right but Cyrus too was young at one point in time and Cyrus was one of the first people oh you actually see a view one of the molecules there so Cyrus was one of the first people to sit down and work on modeling of proteins by computers not simulations just drawing the structures and there is a famous paper in Scientific American in 1966 which is really hard to get online so a couple of years ago I went down to a library and dug up all the old issues of Scientific American and copied this I have a PDF on the website at least of those of you are logged in or not and I guess since 1966 technically the copyright is probably exposed well it's the copyright has expired so it's exactly 50 years ago so I think I'm fine there it's well worth reading it's like the Scientific Americans is popular science in the 1960s Scientific American was a beautiful paper because popular science today well popular science in the 1960s would probably be considered real science by many today it's a paper written by scientists for scientists but in different areas it's well worth reading so if that was the amusing part I'm going to go through there's one concept that the book brings up and when it comes to bonds it's not going to be the main topic of the course but I want to spend two slides on it anyway because it might help you later and it's related to these things I mentioned about quantum chemistry you're not going to be doing you're hardly going to be doing any quantum mechanics or quantum chemistry in this course actually scratch that you will not be doing any but occasionally it's good to think about what are the limits of classical the way we describe things classically and when do we need quantum chemistry so in general understanding how electrons move is really complicated you can't describe how an electron moves you can only describe probabilities think of these as electron clouds and the only way to really formally correctly determine what happens is that write down the wave equations solve them if there's one electron you can solve it manually otherwise you're going to need a computer and then you're going to need to optimize this to find what is the most advantageous state from an energy point of view meaning the lowest energy and that's going to be the states the two molecules the electrons want to adopt that it becomes really complicated and we're not going to use computers so there are some ways we can actually think about this you're probably some of you might well you should probably be aware of this you can think of electrons as different shells right you have two electrons in the innermost shells and then six more and you can think of this something called balance bond theory that I didn't write down here and the idea that you might remember is that atoms like atoms rather the electrons likes to be in full shells and you can formulate this in the concept called orbitals so the first shell just corresponds to an electron that is completely uniformly divided around the protein technically that's just one shell but then electrons have a property called spin that you don't need to know anything about but think of it as an arrow up or arrow down and since you can have one electron pointing up and one electron pointing down in this shell that gives us two electrons there's actually a very deep concept in quantum mechanics called the powerly exclusion principle that says two identical electrons and it's not even electrons but these fermions particles that have this spin property cannot occupy the same quantum state simultaneously so if they have two in this state one of the knees has spin up and another one needs to spin down if you take two such green balls and try to put some really close together well if both of them have two electrons they're going to repel each other really forcefully and you can show that's an exponential repulsion if you go through all the math in quantum mechanics so that the powerly exclusion principle is really the explanation why all atoms start to repel each other at some point if you push them two are together that's where you get very high energies but if you have an atom say like a hydrogen that just has one electron here it's going to be the s state and have one electron pointing up if you take two such hydrogens and move them together one opportunity for these electrons is to have one electron pointing up and the other one pretty much reverses direction and then you're going to have two atoms but each of these atoms will feel as if it had a full electron shell so you're now going to have both these electrons being happy together one would spin up and one would spin down don't worry i'm not going to ask you about the quantum mechanical part here but the idea here is that under some circumstances the way we pair these electrons if we can make multiple atoms feel as if they have full shells we will form chemical bonds and that's really that's really the main concept behind these so-called balance bond theory that if you get these orbitals overlapping and if they overlap in a good way one up and one down we form a covalent bond this becomes more complicated if we start looking at this that I had in the previous slide is p orbitals one two three because at some point you start having mixing between these orbitals and that's why you have larger larger atoms such as carbon that they can bind up to four atoms so what happens around an alpha carbon for instance when you bind four atoms you actually get one s and the three p orbitals to mix i might have they might say sp3 hybrid hybridization on some slide what i'm really saying by that is that these four orbitals they share the load so that you can bind four other atoms from your carbon so why do i keep bringing this up well the electrons this means that the electrons are responsible for two or three things first they're responsible that saturated orbitals due to this pauli exclusion principle they explain why all atoms repel each other if you push them close enough and if you didn't have an atom if you were to push them even further at some point you can have the protons and neutrons inside the chain repelling each other the protons in particular but the same the very same electrons in this case they also explain why we can form covalent bonds it's pure quantum mechanics and at this point we should start getting really worried because if we're gonna need to understand these interactions we should just change the topic of the course we should spend the next four weeks going through quantum mechanics so what happens if we for a second ignore that go to very large distances yes well in general madams will interact with each other but how do they interact oh sorry they attract each other um well yes and no yeah in principle you're right so first thing is this depends on the charge if you had things like two positive ions right they're going to repel each other that's simple that's electrostatic so we'll hang on with that for a while but if any normal atom that is not charged at large distances to first approximation we should say that they don't interact right because they don't have charges or anything but if you think of a small atom like water remember what i said yesterday that water is a very large negative charge here and positive charge here so this is a dipole i just usually do it at an arrow pointing that way okay now if this is this water molecule is interacting with another atom like xenon charged non-charged the first approximation that shouldn't be anything but the xenon is not a point the xenon consists of a nucleus and b electrons so with this and the nucleus here is quite heavy why the electrons hardly weigh anything so what's gonna happen here our negatively charged oxygen is going to do what to the electrons no what's the charge on the oxygen negative so what will the oxygen try to do so if we take all the electrons on the xenon and push those slightly to you right and while the nucleus and slightly now we're talking about tiny fractions of the radius here right so if we take all these electrons and push them slightly away this is going to be somewhat better we can't push them too far away because these electrons will also be happy to interact with the nucleus and the xenon which is positively charged so then you have a xenon with a positively charged nucleus here and the electrons push slightly to the right but hang on a second that's just what we had in the water so if you have the positive and negative charges no longer centered on each other suddenly we now have a dipole here too so one dipole will induce another dipole in the second atom here and that's occasion what you call dipole to induced dipole interaction the only reason bringing that up first is to get to the second point because it gets more complicated so if we have assuming that this is let's imagine that these are xenos so there that's why there's the same color here at finite temperature things move in particular the electrons the electrons are always more mobile than the nuclear so these electrons at some point they will fluctuate and at some point these electrons will be slightly to the left here and then we spontaneously have a small dipole formed here now this temporary dipole which is tiny and well now I have a dipole and then I have the phenomenon up here right so this temporary dipole will induce another dipole in the second atom that's one way of thinking about it the book does something slightly different you can also if we started by having these confirmations we start to imagine that we had these electrons right on top of everything here what happens if I take all these three sets of electrons and move them slightly to the left as I've done here well there are two parts of it what happens to the interactions with from charge to charge to charge does that change no nothing changes because they're the same relative distance what happens with the interactors from those electrons to those elections to those electrons nothing because they're the same distance but what happens to the interactions between the positive charge negative election positive charge negative electron and positive charge they're slightly better right because suddenly you have the electrons right between two positive charges and that positive charge will be right between two electrons you can think of it any way you want this you can actually derive you don't need quantum mechanics for this this is just small dipoles and purely classical electrostatics and this was proved by london that this is proportional to the sixth inverse power of the distance and it's called dispersion is this important can you imagine anywhere in life where this is relevant no I know I wasn't you can certainly think about it in protease I wasn't thinking about protein something much more fundamental if you think about a noble gas so what do you know about noble gases yes which means yes and that you can formulate now that they don't they don't react or interact with anything right so what do you know at what temperature helium becomes a liquid no well negative 200 Kelvin doesn't exist a couple of Kelvin I think it's usually four Kelvin or so but that's strange why does something become a liquid because the interaction but we just said that it's a noble gas and they don't interact the reasons why even noble gases eventually become liquids are because of dispersion interactions but these are it might tell you something that it happens at four Kelvin these are extremely weak forces they're so weak that they're only really relevant when almost all motion in the system has ceased so that's why noble gases only condense when you're at a couple of Kelvin that seems completely relevant why do I either bother they bring this up yes if you don't have any charges these are the only interactions we had and the other thing that while each of these interactions weak if you have one atom surround if you have one atom interacting with one other atom is not really important not with two but an atom interacts with hundreds or thousands of atoms around it right normally that's in electrostatic step would be relevant because if you have one atom interacting with the first atom that's an attractive force so the second atom might be the opposite sign so let's just guess that's a repulsive force so like the sign of electrostatic interactions fluctuates is sometimes attractive sometimes repulsive what is the sign of these interactions attractive or repulsive they're always attractive right so it's small but if you sum up enough of these no matter how small it is if you sum up enough of them eventually it will become important well with electrostatics you never know it can be the net results of many particles interacting with electrostatic can be either repulsive or attractive the other thing you should be aware of the larger an atom is the larger its electron cloud is and the easier it is to deform the electron cloud and that's why larger atoms tend to have stronger dispersion interactions so that friend of order would just say that sadly we're going to need to change the topic of the course and turn this into a quantum chemistry course the only problem is that there are come and informally that's correct right that we just described a whole bunch of things that you need quantum chemistry to treat really accurately the only problem is that there are compromises in life and that includes quantum chemistry quantum chemistry does something beautifully when I was roughly your age I remember having a teacher speaking that it was so amazing because we could now determine the electron structure of benzene six atoms now there are a lot more than six electrons in that molecule and you probably know what the structure of benzene is so that the fact that a computer could also tell us where the electrons are from a fundamental science point of view it's important but it doesn't really teach us a whole lot about benzene computers are a lot faster today today you can handle roughly 100 atoms and if that doesn't sound like such an amazing advance in like the third well the 25 years since I was your age this is because quantum chemistry scales really bad normally in chemistry if you have 10 times as many of something it would be 10 times costly quantum chemistry typically scales to the fifth or fifth power or so so if you have if you go from 10 one atom to 10 atoms or electrons it would be something like 100 000 times more expensive and that's roughly how much faster computers have got it so the problem I don't know about you but there are not a whole lot of interesting proteins of this size but that's a that's a false argument I can't say that just because I would like to study proteins I'm going to study proteins with sloppy methods because the correct methods I can't handle that's kind of irrelevant right we're not going to teach anything about proteins it's not really valid the other problem is that did I say that the quantum mechanics I briefly described that's completely false because we kind of just assume that we can study the electrons while the atoms don't move to do this correctly you should of course use a time dependent relativistic Schrodinger equation the good news for that is that it's possible to solve that one for one electron and suddenly well the good news we can do really well if you start either a hydrogen atom or a helium plus ion but that's pretty much it computers can do a slightly better job but we're limited to picoseconds of time so that forget about life science it's not even chemistry it's physics but same thing there if that was really important it would not be a valid counter argument the other problem though is that when we study these it's not just a matter of what we want to study in our atoms what you typically do with quantum chemistry if we assume that the cores don't move if we don't have any atoms moving you're studying things that zero kelvin and now we just start to become a bit worried right this is the so-called better method life science at zero kelvin the other part those hundred atoms we're kind of forgetting something we're going to need water so are you seriously going to study life science in vacuum so the problem this is by no means this does by no means mean that quantum chemistry is bad quantum chemistry is a really important method the problem is that the compromises this brings that quantum chemistry is a beautiful way of treating the interactions and these isn't really accurately but there are some pretty hard compromises we would do and people still do in quantum chemistry and that's why we're typically not going to use it now all these arguments at least the first few ones here would be false if we absolutely needed quantum chemistry for that but just because quantum chemistry can describe something or quantum mechanics that doesn't mean that you had to use quantum mechanics for it and in fact as you see on the slide this system too is really well described by quantum mechanics i'm not sure about you but my kids usually don't use quantum mechanics to predict how a football is going to move same thing if i go through this door technically i diffract a little bit because i going through a slit now the slit is not so thin so i don't really it's not really going to influence my life that my atoms start to diffract but technically i do diffract a little bit and this has to do with this complicated concept we need to focus on the most important part of the system and it turns out typically for proteins these large molecules with a few exceptions if you form or break bonds you need quantum chemistry to describe that but remember what we told yesterday bonds don't usually even vibrate a whole lot at 300 kelvin so for all practical purposes if we're going to study proteins a protein at room temperature is more like a football it's important to have the water around it it's important to account for the fact that proteins actually do move the exact quantum chemical interactions of the protein we can usually ignore a bit and i think the conclusion of this is really that quantum mechanics and classical mechanics they really describe things on very different scales of length time resolution and everything this is a funny slide that the noble committee had a couple of years ago you frequently use the cats to describe quantum chemistry based on schrodinger's cat that is duality whether the cat is both alive and dead in the box and this of course isek newton and you can think of this as competing theories and everything but in particular modern science i think we tend to ignore the beauty of isek newton there is no question that quantum mechanics is a more detailed theory in a way what i find beautiful with isek newton isek newton managed to describe the world in a way that five-year-olds can understand it have you ever thought about the concept of a force a force doesn't exist there is no such thing as a force you can think in terms of potential energies or something but a force is really just isek newton's way of describing how things interact and any five-year-old will have a grasp of force if you push something and okay we forget the beauty of simple models and newton's equations and newton's mechanics is one of the most simple and beautiful models there is in history and that's why we're largely going to use it the problem that in some cases in particular enzymes or so you really need quantum quantum mechanics so that there were a couple of groups in israel in particular that started to look into this in the 1960s i'm sorry i think i might have screwed up the order in which i build this slide yes sorry the problem with quantum mechanics is really that you can derive all these parameters and we could sometimes try to derive the charges and everything from quantum mechanics and just extrapolate but extrapolating this would this would be like me trying to point out the location of the george washington monument in dc from this room it's roughly in that direction but if i point here i might as well point to mexico city and that's if you what we're trying to do here if you extrapolate 15 orders of magnitude you're going to be wrong it doesn't matter what theory you use so that's not really going to work and then there was a group by schnerer lifts on at the rehovert in israel small group he had schnerer had one student working from arie or shall who sat down and started to look into this at other alternative ways we can describe this and i really started with quantum mechanics but one of the beautiful things they realized is that you don't need to use quantum mechanics for the entire system in that particular case they looked at enzymes and that the very specific binding site you need quantum mechanics but for the rest of the protein that doesn't form or break bonds you can use something much simpler but the much simpler thing will kill you when you extrapolate so they realized that you can cheat because sometimes in life you're allowed to cheat so one way could of course be through that i try to do the world's most accurate quantum chemistry calculation to determine the parameters of a water molecule so that i will reproduce the specific heat and density of water that would be an amazing calculation right but why on earth should i do that calculation we know what the specific heat and density of water is we can measure it in the lab in 10 minutes so why do you try to use a computer for something we already know so their idea was this really what you call semi-empiric models is that fit these parameters is that and that's what i mean by cheating that we already know the results to adapt the parameters that we will get the density and heat of aberration of these molecules and this was really the beginning of all modern molecular modeling that you could suddenly treat large gigantic systems and i would argue that one of the most important results that this really made it possible to explain how enzymes work that we understood enzymes is mostly about electrostatics and what i mentioned briefly yesterday that an enzyme binds the binds the components we need in the equation so that they can somehow form or break a bond we reduce the energy of that highest barrier we need to get over and then we release the product again and this was really arie in particular who showed in this really early work and they got the noble prize together with martin carplus actually a couple of years ago martin is the one who pioneered simulations and molecular studies of proteins it's very fun mike is my actual my old postdoc advisor it's it's very strange when you realize in the middle of the night that your postdoc advisor got the noble prize i've never thought of mike he's an amazing smart person one of the nicest people but i've never thought him as a noble prize winner and i don't even sure if he himself thinks of himself as a noble prize winner anymore um but this led if we start accepting this issue with some sort of not fake but fitted parameterizations all these complicated things with electron clouds moving and everything we can start to describe that in terms of rather than having electron clouds that are offset we can say that on average you have a slightly larger point charge in this particle and a point charge which is positive on the hydrogen this is of course wrong these are not point like particles these are electron distributions but if we have them as point like particles with some non-unit charge we can treat this with classical methods and in a benzene so water has very high partial charges and something like benzene here you see it has very low partial charges it's a largely largely hydrophobic molecule so that was one part now we can at least we've gotten away from the worst part of all the electrons right we can describe the how charges partial charges allows us to describe the distribution of electrons without using quantum mechanics there is another part called bond stretching um so if we look at a small bond here between two carbons for instance in principle this is really complicated because you should have the the blue potential here is what's called the Morse potential that you can't get very close and at very large distance eventually the atoms are going to release an unbind but that's not even enough because it's also a quantum mechanical oscillator so it's not a classical oscillator along the blue curve but in principle you have these discrete energy levels that are that you can we determine with quantum chemistry where the molecules should be placed but remember what I said that normally on average these things are not excited at room temperature so typically at room temperature we're going to be there and then we can do a plain simple harmonic approximation the green curve here just assume that this is a second order potential sure if we start to having if you start to similar if you start to examine these molecules at a thousand or two thousand Kelvin then you start making errors here but at 300 Kelvin forget about it it's a beautiful approximation in fact the best approximation would likely be to just have a small stiff rod and not allow them to move at all so this is a solved problem we don't need quantum mechanics for bond stretching angle vibrations roughly the same as I mentioned yesterday these can move a little bit a couple of degrees but you don't need quantum mechanics to describe that we can use a classical simple spring function it's not a perfect approximation at 300 Kelvin but pretty darn close so they move a couple of degrees solve problem we don't need to worry about it you're pretty much never going to worry about bonds or angles in a protein and the third part are these torsions or dihedral angles these are considerably more complicated because it's not just a matter of getting a ground state right right okay these are the important degrees of freedom that the energies are so low that they actually do move and we've talked about the definition so the fact that they are easy to alter is good in a way but because they are easy to alter it's not just enough for us to describe them well in the ground state we need to have a reasonably good description what happens when they are in other states and you can describe that in a fairly simple potential just use a sine or cosine function a very tiny molecule like ethane so ch3 ch3 if you have the points where these hydrogens are placed right next to each other are going to be a bit bad and the points where they are facing in different directions they're going to be much better energy-wise the energy for this the peak here is roughly three kilocalories per mole there's a much much much remember the 300 kilocalories per mole we mentioned for electrostatics this is a tiny energy so this is going to be relevant the molecule would be happier there but we will occasionally also see it there let's see i think i can start a movie here yes this is butane this is a much more complicated molecule because you have two large carbons here so here we are in the trans state that's going to be really good and then eventually you're going to go through a bunch of different states until we end up back in the cis state when these two carbons are in the same side and that's going to be pretty bad so here you have a trans state here which is by far the best and then you go through a somewhat bad state you find a somewhat better state again and then you go through the really bad state here so but here too you have energies that are in the bulk of a one to four or five kilocalories per mole so all of them are still fairly low so this was one single torsion in a tiny molecule we have more than a protein so they occasionally if you had the simplest polypeptide we can imagine that would be an alanine that just had some groups so that we have one phi and one psi bond and then we can draw this in a Ramesh and a diagram phi and psi and here we have colored it so red here means really bad energy and blue means really low energy technically this is actually free energy than energy for now you can think of this as energy we will come back to the free energy after the break and here you have one really good state and another really good state and the differences between these to go from one of these states to another you have to go through a peak here that's a couple of kilocalories lower and then you go down in a different well on the other side and these are actually the pictures of both of these reasonably good states this is the best and this one is slightly higher energy but you're never going to be up here we will come back to that picture later this is just an example but this starts connecting to the concepts of energy in general i'm going to spend another five minutes on going quickly going through a couple of Ramesh and then diagrams and then i think this is a really good place to take a break we spoke a little bit about these Ramesh and because remember this is a Ramesh and a diagram but here i have not drawn it as completely forbidden or completely allowed but i've drawn it as areas that are better or worse these diagrams will in general depend on the amino acid so i've stolen a couple of pictures from the Finkelstein book here this would be this is a simplified completely fake Ramesh and a diagram that only consists of carbons and nitrogens so only the backbone and then we would have one region here that would be disallowed because if we well this would basically cause the atoms to collide and large areas that are allowed when we form a glycine so we start having at least the hydrogens there and some oxygens so this is at least a real polypeptide suddenly there are much larger areas here the black ones that are disallowed and a few regions that are okay and this keeps this pattern keeps recurring as we go to larger and larger structures for alanine the completely forbidden areas are roughly the same but these gray areas are pretty bad for alanine it doesn't like to be there and most other amino acids end up looking something like this so this is really good beta sheets and here we now we have our alpha helices and possibly left-handed helix so there are already now remember what we said about Leventhal's paradox and they said that a typical rescue might have each of these torsions might be in 36 different positions and that too might be in 36 different positions there is no way you have 36 squared different conformations here because most of them are being going to be completely forbidden nature will never ever try that so while we have these torsions and proteins are flexible most of these degrees of freedom will end up being forbidden there are not that large regions of space we can visit I will do two more slides and then we're going to take a break the final part these dispersion corrections that we spoke about we need to somehow we need to formulate those and we typically group them so at very close distance we want to somehow say that it should be so if you start pushing atoms really close together things should be exponentially costly and at very long distance we should make sure that we are attractive just as the london forces said the correct way of describing this would actually be to have something that's one over the sixth power of r at large distance that's just a constant and at very short distance we should have something that goes up exponentially the only problem with this is that actually there's not a problem with this this is the beautiful model and to tell the truth is probably the model we should use in the future but when people started this in the 1960s computers were slow and it turns out that it's pretty expensive to calculate an exponential function on your computer well that doesn't matter we need an exponential that's the correct functional form until you start thinking a bit well how frequently is it going to be for you to have your atoms in a protein in here no so there's there are some areas that are really important here if you're friends if you're building a nuclear weapon then it's going to be very important what happens when what's going to happen when you have these pressures start 10 to the power of 10 or something you don't have pressures of a billion bar or something in a protein so with the caveat that I think that this is a better functional form we probably should move to it people quickly decided you know what we can cheat here so can you think of something else rather than have an exponential we would like something that goes up steeply at a very short distance that that's easy to calculate so well the easy if we always already have one over r6 take that number and square it one multiplication and then you have one over r12 and that's in practice what people have been using for 30 years so then you have a potential that looks almost the same it goes up as one over r12 there but it's still attractive at very long distances the reason why I show you this slide is actually this diagram that you also have in the book so the energies we talk about here the energies you're so first the distances here is in the ballpark of two to three angstrom here so it's very so that's when atoms start to the the fund of also lennar jose distance between atoms but the energies you're talking about 0.1 k cal per mole so they're extremely weak interactions but yeah many of them and those are going to be important too i think this is an excellent point for a break and after the break we're going to continue with hydrogen bonds and a little bit of energy landscapes and then we'll start deriving the boltzmann distribution and i have a tons of slides for it the reason i don't think it's going to be that difficult but i pretty much written down the entire derivation so that you have it in the lecture notes too so i i spoke about the lennar jose interactions before the break and the final part of this has to do with hydrogen bonds and proteins that i touched a little upon yesterday hydrogen bonds are typically perfect in ice in ice hydrogen bonds would be perfect so you have the oxygen and the hydrogen covalent bonds but because the oxygen is so electronegative and will pull the electrons to it that will almost create some like half an ion on each hydrogen and that will cause it from these really technically we call it an electrostatic interaction but this one is so strong that it's kind of halfway between a normal non-bonded electrostatic interaction and the covalent bond hydrogen bonds are really important and that brings you to this other question is that what happens if we start from ice when virtually all of these hydrogen bonds are formed and then we start to throw the ice we add heat as you saw in the movie yesterday we somehow break the hydrogen bonds but the question is isn't that you break some hydrogen bonds while others are formed all the time or do you somehow distort all hydrogen bonds a little bit so this is this is actually it actually happens to be the right answer but it's not a trivial answer and it's certain it's something we didn't know until like 20 30 years ago and this is relate this gonna eventually gonna be turned out that this is a phase transition but it's also intimately related to what happens for instance in a protein when a protein folds or something and that's why the book has a plot in it but that's why I'm bringing it up here so this is a very simple model of an IR absorption spectra for the oxygen to hydrogen interaction and basically the wave the wavelength here will give us some information about what type of property this is involved in and we have three types of samples here we have one ice so this is a sample where all the hydrogen bonds are actually formed and then I have a clear peak at a particular frequency here that corresponds to actually having a hydrogen bond the second one is a bit strange so it's water but water is not the solvent here you have a tiny amount of water in a carbon tetrachloride solution so this is water that does not form any hydrogen bonds and then you end up having a resonance at the different a peak at a different frequency so if we had a structure in water where some hydrogen bonds kept being maintained while others were completely broken then you would have a system with two peaks right one peak here and then would go down and you would have one peak here but what happens in practice is that you get this third effect a very smeared out spectrum is that so that all hydrogen bonds have moved a little bit more to the distance when they're not formed very very simple setup and we're going to come back to this when we talk about protein so these the beauty with these simple measurements is literally they are simple you can do it in an equipment that costs a couple of hundred dollars or something and it gives you an absolute answer in this particular case either we would have two peaks by model distribution or one peak we have one peak and that means that all bonds tend to become more loose in water you have no idea about the stuff that's written about hydrogen bonds in water Dave Chandler is an amazing professor in California and he keeps he keeps reinventing this thermodynamics and every time I read a new paper from I'm so astonished water is an extremely complicated liquid and that's but with that we can start to say a little thing about these hydrophobic effect we're going to come back to this later on remember what I said about the hydrogen bonds that they're so strong they're almost a covalent bond it's a lot of energy so waters will do almost anything they can to maintain their hydrogen bonds five kilocalories might not sound like a lot but compare these to the zero point one kilocalories per mole that was an unwanted interactions if you start losing your hydrogen bonds that's a lot of energy in particularly if every single water molecule loses one so what happens if you now put a hydrophobic solvent like Xenon or something in there what these water molecules do they will not just be happy and accept that they suddenly lost half of their hydrogen bonds these molecules will start to reorient so at any cost they will try to find other neighbors to form hydrogen bonds with instead so what you're going to do is that you're going to form a cage like structure around the hydrophobic solute so that these waters pretty much maintain all their hydrogen bonds and this is a bit unexpected right because you would imagine that the reason why the reason why it would expensive to solvate something hydrophobic in water would be that we started to break hydrogen bonds that's not true you keep all those hydrogen bonds but that begs another question so that but in that case why is it so bad to solvate something hydrophobic in water you don't lose any hydrogen bonds from it we'll come back to that tomorrow but that also because because this is so bad you start having these oil water effects that rather than having two small small oil droplets it's better for the oil to form a larger droplet so that you have a smaller interface to water and this also an effect that happens in proteins but i will go through that later so just to sum up all these interactions this is a slide from vikipedia actually and it's kind of nice because it summarized everything we have in proteins you have bond vibrations you have angle vibrations you have torsions you have some lennar jones interactions between any atoms even if there is no direct bond between them you have some electrostatics that we don't show here and then you have some sort of hydrophobic effect that is roughly proportional to the hydrophobic surface area and we'll come back to that how we calculate that tomorrow yep so that's that's a that's quite a way better than you think it's both right and wrong so first the main effect from the torsion angles it's actually not true even if you didn't have anything out here the way electrons and these atom and this atom all right electrons don't like because remember there is a sp3 hybridization here so this atom has electrons in four directions this atom also has electron in four directions even if you had no groups whatsoever bound to this these two atoms would still have a torsion potential that looks roughly like this and that had to do with the way the orbitals are oriented so it's not this effect is not primarily because these two atoms bump into each other or something but it's an effect of the electron distribution along the bond however then it's certainly also an effect that if you rotate their own that bond there's going to be an interaction between that atom and that atom and you can choose how to describe that occasionally we try to include that in the torsion and occasionally we try to include that in the lennar jones interaction it's basically it's up to you to define where you want it just make sure that you're systematic because these are simplified there are approximate descriptions ultimately it's all due to the electrons remember those ramachandan diagrams I showed you we can think of those we only looked at those as two-dimensional plots and different energies but you can think of them drawing them three-dimensionally we have some this is not a strictly a ramachandan diagram but we have a phi in one axis and psi on the other then we can just let the height here be the energy so low means low energy that's good high means high energy that's bad and then you can start to think of this and again even this just two there's just two degrees of freedom will give us something that looks like a landscape right so you're going to have some peaks places where it's bad to be and you're going to have some trolls blue areas here where it's really good to be but now life gets more complicated because if you are in one valley here and want to get over to another valley you will always there is always going to be some ridge you need to cross and depending on how high these ridges are occasionally you will be able to move to one valley from another and occasionally it's going to be too high so you can't get there and you will almost never be on the peaks and this is the so-called concept energy landscape and we can describe this a lot in as much detail as you want they only carry that these are super simple energy landscapes these are two degrees of freedom a typical protein if we forget and again if we do the approximation we used yesterday forget about everything except the torsion angles even for a small protein of say 50 residues you're going to have 100 degrees of freedom so that would be a 101 dimensional plot and I'm not sure about you but I find it really hard to visualize 101 dimensional spaces so these are really good for thinking about things but we should have bear with this really we can only draw these in three dimensions if that's two degrees of freedom but when we talk about energy landscapes we really talk about the ability and possibility of a molecule to explore different regions here and how expensive is it to move from one region to another because what could happen here in principle it's good to be at the low regions and it's bad to be at the high regions but what if I am here that's not the best place this one would be the best place but if this barrier is so high that it takes me a thousand years to get there it doesn't really help right I'm always going to be stuck in what I think is the best of worlds out there because I don't I can't go across the saddle pointer and in many cases that's going to turn out to be the case so we haven't proven this yet but I'm going to wet your the prions we talked about yesterday these proteins that can misfold what I'm going to come back to later on the course is that what likely happens with prions is that the normal native state the biologically normally functioning state is a state like that and we're typically very happy there and normally nothing happens but under some circumstances that we don't quite know about you can cross the barrier to some other states that's even better even lower energy and it's we call that a misfolded state well nature calls it's a better folded state the only problem is that that is bad for you biologically so that's when you get these plaques or something and of course the second that some protein starts to form here that might drop the barrier more and more so you might start having more and more proteins go to the wrong place and for prions this is likely the case so for now you have to take my word for it the only problem here though is that at some point we're going to need to decide how many particles are in red and how many particles are in the different blue places otherwise it's not really going to help us a lot apart from the fact that we can draw some beautiful pictures so this now requires we need to start on what is the probability of being somewhere as a function of what the energy in that state is right and this is complicated because there's pretty much only one way to deal with this and it's called statistical mechanics and it's a pretty tough area in physics the good thing is that we're not going to go through all of physics but there's this beautiful code for you David Goodstein he wrote a book about statistical mechanics in the 1960s and you could well it's sorry it's a bit a Ludwig Boltzmann who spent much of his life studying statistical mechanics died in 1906 by his own hand Paul Ehrenfest the students 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 I can kind of understand the math statistical mechanics is extremely complicated and it's equation after equation after equation and it can be pages of equations when I was a student I remember solving the seven dimensional Gauss integrals by hand with my professor was 34 years old I thought that was a completely natural thing to do but the thing is that it's powerful it's extremely powerful and statistical mechanics and thermodynamics because we don't this is not something that builds on other laws of physics really it's something that builds on observation we see what things happen in nature and I also for this route this is likely the only field of physics that will never ever be overturned because it's based on observation what things do happen in practice the amazing thing is that this also governs everything we see in chemistry biochemistry proteins and to tell the truth most experiments that you work with historically experiments tend to gloss over this fact well we take an average you know those average works great if you work in a test tube and one mole of something but the second you start having single molecule experiments and you're looking at one or two or three molecules you can no longer ignore statistical mechanics because nature is statistical in nature the good thing is that we're not going to go through everything the other good thing is that this was a pain in the days when we had to do everything with paper and pen today we have computers and that makes this is what I would argue why suddenly we can apply this in life science and everything because we let computers do the hard part and this afternoon and tomorrow afternoon you're actually going to do some very simple labs on this with computers which means that you're not going to have to sit down and do the manual labor intensive deduction but there is one thing I'm going to show you which is we're not going to have a whole lot of equations but this is one equation that is important I'm just going to throw this up it's called the Boltzmann distribution and this says the probability of being somewhere in a state and right now I'm not even going to see what the state is you can think of this as the blue point on the map or you can think of the state of an electron the the probability of something happening that is proportional to an exponential function raised to minus the difference in energy or the energy because this probability if it's a difference it should be some difference compared to a reference state divided by kt kt is Boltzmann's constant and t is temperature because we have this is just a constant we're going to ignore that for now what temperature is in the denominator there what this means that if you take something like the speed of atoms in a gas at low temperature you're going to have a narrow distribution where the speed is fairly low but as because it's it's much better to be at low energy states at low temperature but as we're increasing temperature I'm going to get a wider distribution and I will also be able to populate higher energy states in this case higher speed I haven't proven this at all this is just something I've thrown out that this is going to turn out to be the case and in person we can derive this there that might not sound like an amazing thing but it is amazing we can derive this without assuming anything this is both the major pain of physics but also the beauty we connect a system a system that you don't know anything about it just a system with properties normally we say that that properties that we can exchange heat with the rest of the world energy and then you can show that this is universally true if we're going to do that we would spend the next hour or two hours at least and half of you would not show up tomorrow so I'm going to try to avoid that and we might do that later on in the course but just to at least argue that this is reasonable I'm going to do that for a specific case and then it actually turns out that all the properties of that specific case factors out so in the end the result doesn't depend on the side on the specific case at all so I'm going to try to derive this in a fairly simple manner and deriving the Boltzmann distribution ah stop me if you can't follow this so this fairly simple manner is at least these is the example I in the book could imagine so if you have some sort of column or vessel whatever and we fill that with gas without knowing anything you will probably agree with me think of the the earth atmosphere that we're going to have higher density we're going to have more molecules down here right and then we're going to have lower density fewer molecules up here why is that if you think of this being one kilometer higher so yes gravity pulls things down right so it's better to be down but if we had all the molecules at height zero here we would have an almost infinitely high pressure so that pressure causes things to spread over the entire vessel but on average is better to be down so we're going to have more down here but then we're going to have less and less up here and that actually the gravity that is really well described by the potential energy here right the higher up we are the higher the energies and the lower down we are the lower the energies so already here we can see that it's better to be in low energy but we always have something in the high energy so we can describe that energy with the height and we have high density low potential energy downstairs and low density high energy upstairs and as you also said we have two effects here gravity pulls down while pressure counteracts that and pressure would if we had no gravity all the particles would be uniformly distributed in the vessel because there would not be any pressure gradients anymore so let's study one particular tiny region here the yellow one and my question is really how many molecules do we have here remember that's because if we can say how many molecules or at least the density of molecules that really answers the question we were after right how likely is it to be somewhere as a function of the energy of that state how do we determine that without sneaking looking at the sorry so this is the hard part right and that's normally when you do in maths people throw all the information at you in many ways that's good because when you have all the information you start to calculate the exact pressure the exact volume and everything but the forces the beauty and the difficulty it doesn't depend on the exact volume these laws are universal and that is usually when you start out that it's very hard because you don't have any concrete numbers and that you feel that you have nothing to work with you always do have something to work with because we can assume that that volume is v for instance and hopefully in the end it's going to turn out that the answer doesn't depend on v if the answer depends on v something either something went wrong or we used the wrong problem formulation so that one solution to this is to aggressively introduce for instance the height or the volume or whatever we call things but at equilibrium when things don't change here remember that I had these two forces gravity and pressure so when things don't change they have they kind of have to be similarly strong right so that looking at this small volume it has to be the same amount of the gravity must exert the same type of work down as the pressure exerts up if this was different things would change and then I would just wait until it has stabilized and look at it so when things are stable and don't change anymore I should be they should be equal and that also means if I'm standing here at some height h if I go just a tiny bit up dh and that can be infinitesimally small well so what happens here that my pressure will have changed a bit because the pressure is lower higher up right and that change in pressure must correspond exactly to the amount of well the amount of gravity pull on this small volume down so they usually when you have two things in this case gravity and pressure we just need to find when are they going to be equal and then we just try to solve that equation and remove as many things that will be hopefully end up with something simple at the end we're going to try that so the first thing we're going to use is the ideal gas for most of your chemistry right so we're going to start at that side pv equals nrt okay you're going to be happy in a minute if you're not a chemist the pressure times the volume equals the number of particles multiplied the gas constant multiplied by the temperature that is simple if you're a chemist the gas constant is roughly 8.3 per 8.3 the probability chemists love counting in moles physicists hate counting in moles physicists count atoms the only difference here is avogados constant so sorry this is a physics topic so i'm going to change here in physics you say pressure times volume equals this is still an n it's a completely different n than that n of course this is the number of particles instead of the number of moles particles per mole and this k this is means a fundamental constant in nature called Boltzmann's constants and again the difference between k and r is just really avogados number but since we are we're going to try to be physicists here pv equals nkt that just tells how much pressure do we have in a particular volume with the number of particles or atoms in this case and at a particular temperature so you know what now we actually did what you suggested we introduced the volume and i introduced the number of particles but since we don't really know what that volume is rather can't we say we take the volume and divide by that on both sides and then we can have a small n that just means the number of particles per volume and then i've just gotten rid of the volume this is a density if you make the system twice as big we just double the number of particles but i also double the volume the same laws should apply so now i have one one fewer strange pieces that i need to deal with and that means that this equation is now p equals small nkt that was the easy part and well as i said that we're gonna see we would need to see what happens when i move up or down this column right and that's well the only thing that changes when i move up or down that's the height so let's take this small expression and derive it with respect to the position the height so derivative of the pressure with respect to height or p prime we could write to because one dimensional that's well natural constants rarely change with height temperature doesn't change with height so this is really kt multiplied by the derivative of this particle density with respect to height not particularly complicated so now we just know how p varies with height that's one side of the equation the other side is of course the gravity right so let's see how the gravity changes relate to height well gravity we don't need to we're not going to do anything complicated with general relativity or anything so gravity is really described by the potential energy which is the mass multiplied by the gravitation constant multiplied by the height that is how much the potential energy is when we go up the column so if we then go up just a tiny bit dh the weight of the gas pressing down decreases by the mass multiplied by the gravitational constant multiplied by the height and then the number of particles per volume so we know that moving up by little amount that's how much the weight changes but then we also had when we all we also know that the pressure changes by this amount and by definition those differences might be the reason why the pressure changes is because we have fewer particles pushing down right so that if these are not identical i can't gain more in potential energy than what i lose in pressure then we would not have equilibrium so by definition the difference in pressure here dn dh multiplied by kt must be equals to the mass multiplied by the gravitational constant multiplied the number of particles per volume the reason for the minus sign is that the pressure decreases when we go up and then we can divide both of those sides by kt so we simply get the derivative of the number of particles with respect to height is minus a bunch of constant stuff multiplied by the number of particles that's a differential equation first order differential equation which is not very hard to solve thank god uh and i'm not going to expect you to solve differential equations that's not part of the course uh on the other hand i also hate throwing up proofs or anything and then say you're just going to need to trust me what the result is so the way i solve that equation is i have a derivative of n that results in an n there the trick you use to that is say that if you derive a logarithm with respect to something the derivative of the logarithm itself is one over the argument and then we have the chain rule which is going to be the derivative of the argument inside the logarithm so if you just use that trick it turns out that the derivative of the logarithm of the number of particles with respect to the height is just a constant integrating a constant is easy so if we integrate both of those sides with respect to h there we get the logarithm of n and here we just get minus this constant multiplied by h plus a constant right and then we take the exponent of that and that means the constant will actually lead to a constant in the exponent i don't know what that constant is and i don't really care to tell the truth so rather than keeping dragging around this constant i yeah whatever there's going to be a constant and that's why i write proportional to there is some constant and for now i don't care what it is because i don't know what it is so the number of particles is going to be proportional to the exponential because we had a logarithm and took the exponential of both sides raised to minus but wait a second the mass multiply the gravitational constant multiplied by the height here's where i cheat just a little bit i identify well that was the potential energy as the function of height right so rather than having this special case of gravity let's just say i assume that is the energy divided by kt that's cheating a little bit and that's want to air a demonstratum that's what we're supposed to prove the number of particles or the probability of being somewhere is proportional to an exponential raised to some energy relative to a reference state divided by Boltzmann constant multiplied by temperature the book goes through this through but in much less detail the book spends four lines doing it and that's why i have these slides so what is this constant going to be exactly it's hard to say it's something is proportional that when you're going to do a computer lab sorry there is no computer operations they calculate something that's proportional to something when you do this on a computer you're going to need to choose a constant how do you choose the constant so it's easier if you just have four states let's make a simple world here the proportion to being state one the sorry the the probability of being state one is this expression and then it's delta e1 right the probability of state two then we put in energy two the probability for state three is energy three can i need to guess what the probability state four is that's of course energy four right if this is now my very restrictive world i only have four state that i can be in this is my universe this is called phase space every single possible combination of my system four states what is the sum of these probabilities so that you can also say that rather than saying that this constant you can say that this is equal the exponential of this divided by the sum over every single state in this case it would be four states that sum even has a name called the partition function and physicists usually abbreviated by capital z if you only have four states that's pretty easy because you just sum over four states the problem is that in the real world it becomes a bit complicated because you have a couple of not billions not trillions of quadrillions but even you have like 10 to the power of 30 40 50 states you can't if you know the partition function for a system we of course know every single state if you know this you know everything you can predict everything about the system you can predict how it would reactions happen you can predict really in practice we never know that entire sum but if we look at two specific states we can compare them the only reason for mentioning i don't think the book mentions that but in the lab this afternoon because you're going to do things at a computer at some point you're going to need to have the sum over all your states and then you're going to see this capital z this is much more useful than you might think so if we have let's again be limited here we have two states let's say a cis and a trans state of a torsion now you can argue that you know it's much more complicated you have tons of atoms connected to it remember what i said yesterday we should always simplify as much as possible so let's just ignore all the other things for now one bond cis or trans and the probability being in state a we can say that cis that's some constant multiplied by the exponent of that minus that energy divided by kt while state b which should then be trans is the same thing but the energy in the trans state and now i stopped writing a delta energy here and the reason for that in the first case the delta is just proportional to some arbitrary reference date and that can be anything that's like a potential energy you can say that the potential energy of this remote is based on the fact that is one meter above the floor but that's rather arbitrary right why don't i measure it from the street level a wait a second perhaps i should use the sea level there is always no matter what you're doing you're introducing sort of arbitrary reference potential and that's this delta is kind of unnecessary it's up to you to define what your references but if you want to know what is the fraction of how how likely is it to be compared to be in a cis state compared to being in a trans state and there you calculate what is the probability of a divided by the probability of b so that's just a relative distribution of those two and then again that's the quotient of the right-hand sides here and if you know your exponential loss if you take one exponent and divide by another that corresponds to subtracting the arguments up there so this will be an exponent minus delta the difference between state a and b divided by kt so the second you have two states you can compare how likely one state is relative to another directly with the boltzmann distribution it tells you everything that is what you're going to do in the lab this afternoon normally doing this improving this with pen and paper it's tedious a computer a python program can do this you just tell how likely is it to go from one state to another and then eventually you're going to see that you actually hopefully if things go well you're going to retrieve the boltzmann distribution at the end of the labs so the take-home message in lower energy states will always be more populated but how much more populated they are depends on the temperature at the point if you eventually approach zero kelvin everything is going to be in the lowest possible states because this this kind of explodes right divided by zero this would be minus infinity and that would be zero oh sorry that would be plus plus infinity but on the other if the temperature goes up eventually if the temperature is high enough this argument will become zero so at at high enough temperatures relative to your energies energy differences are not going to be important anymore so eventually if you just increase your temperature enough you will be able to go over any barrier and they said you can think of this like an arbitrary energy scale but the ruler on this energy scale the units is kt so kt describes that's the natural energy unit in the universe so what is kt there is a reason why i didn't tell you what the boltzmann constant is it's pointless never guess yes but i don't know what i'm i'm thinking much more concrete than that the number in kilo joules per mole or a kilo calories per mole because we're chemists so that we usually calculate per mole you're starting together you're a bit high you're roughly one order of magnitude too high so kt is so first the wise answer is that nobody thought it was it depends on temperature but at room temperature kt is roughly 0.6 kilo calories per mole or 2.5 kilo joules per mole you can use whatever you want but you need to remember the units but it was 2.5 kilo calories is far too much so what this really means is that if we use kilo calories per mole because that's what i use in the rest of the course any differences that are smaller or substantially smaller than kt that's just going to be like gravel on the road you're not going to feel it we will just all those states are going to be roughly equally probable they're irrelevant so an energy difference of 0.1 kcal it doesn't matter both those states will be equally probable at room temperature at zero kelvin it would be important while something like an hydrogen bond then five kcal that's roughly that's e to the power of minus 10 it starts to be pretty unlikely that we're breaking a hydrogen bond so once you start having energy that are higher or significantly higher than kt those starts to become significant peaks in our energy landscape that it would be hard to get over remember that electrostatic interaction we spoke about 300 kilocalories per mole so there you talk about 300 divided by 0.6 so the probability of crossing that barrier is roughly e to the power of three four hundred you can try to calculate on the computer but you will likely get an error that the computer can't calculate it because the number is too large so the point you will never get over an energy barrier that's 100 kcal or higher and this is this is the beauty that we get that we remember we haven't this there's nothing in here that depends on a protein there's nothing here that depends on water this is a completely universal result for any system that suddenly we have a natural energy scale all energies are not equal difference is much smaller than kt we will just get gloss over we feel like a bump in the road energies much higher than kt are going to be major road bumps that we won't don't want to run into there is another concept that i don't think the book mentions but i'm going to mention this very briefly anyway because point this is not just an equation for corny physicists to realize how and i'm i can say that because i'm a physicist this is not just an equation for physicists to understand that whether one state is more populated than another this actually tells you everything about what the system does to and there's a concept called detail balance that you might think that when a system has i think i mentioned yesterday at equilibrium we're all dead that's not quite true so equilibrium does not necessarily mean that all processes have stopped it's just that when you observe a system there are no net changes in the system but no net changes if you have two states a and b that just means that the flow from a to b is the same as the flow from b to a right so individual atoms particles proteins whatever if you think of a protein what happens in practice protein is unfold all the time but there are other unfold also unfolded proteins that fold all the time so you have this equilibrium between folding and unfolding so that on average you might see that says 90 percent of your proteins are folded and the Boltzmann distribution actually describes this so by definition at equilibrium these fluxes must be the same so the probability of the number of proteins the number of systems that goes from state a to b that is partly the probability of going from state a to b but you can't go from state a unless we were in state a to start with right so it's kind of a condition probability the probability the complete probability of an atom moving from a to b is the probability of first being in a which is either the population or the probability there i kind of use p and n interchangeably here multiplied by the fact okay we were in a but what is then the probability if you are in a of going to be so these are the atoms that will move from a to b and at equilibrium that must by definition be equal with the opposite the ones that already are in b multiplied by the probability if you are in b to go to a and this might look like a completely obvious equation but it's actually not because if you then divide na you know nb on the left side and keep the probabilities on the right side that at equilibrium the population in state a divided by the population in state b so the relative population you can also formulate that as the relative flux between the two states so on the right hand side i'm not talking about how many particles you have in each state in principle i could stand on the top of the ridge and just observe how many particles are going to my left and how many particles are going to my right i know nothing about what is on the left and i know nothing what is on the right but if i just observe this enough i can actually tell what are the relative probabilities of being in these states eventually uh that's pretty much all i'm going to say about the Boltzmann distribution are you confused there's this famous what is it that anybody anybody who isn't confused this is not about the Boltzmann distribution but there is famous good by Einstein i think that actually anybody who isn't confused by quantum mechanics hasn't understood it um it's not trivial the equation here is simple but there is an amazing amount of deep knowledge here because the cool thing this applies to anything it applies to every single particle in your body but we have not derived this specifically for an atom it also applies to footballs you can put the system on any scale you want but that's this abstract nature of things of course what makes it complicated right we don't we can't say what is the system what is the probability it can be any probability just define what your state is and just define what your probability is and you're good that is what you're going to do this afternoon and i think we have a little bit of time here so i will have a chance to go through the rest too the problem is that that's not enough there is some some things that go wrong and there is something we missed here so you're going to start to do the first part at the lab hopefully everything will look really neat and then you're going to go through the second part and realize there are complications since i have 20 minutes or so i will go through this my recommendation is if there is one thing you need to do today focus on the Boltzmann distribution focus on that in the lab the second part i go through now is more to wet your appetite and we're going to come back to this tomorrow remember when i showed you my gas container i just showed you the one on the left but what if it looked like one of those which one of if you had to pick one here and you somehow got scored the more atoms could be at lower energy which one would you pick and assume now they're not equal volume they should be equal volume but why in this case the circle would actually be largest volume so you might very well be right but assume that they all had the total volume the same i would pick the one on i would pick that one because here you have lots of room down here for particles i want to have very low energy but we don't really need a whole lot of room for the ones that are going to have super high energy right but that's just based on a hunch you haven't proven that at all uh so the problem here is that somehow the volume seems to matter and that's the volume that we kind of snuck out and we talked about part average particle density the other problem is what do you mean by volume volume works great if i derive this for a gas but i just said that the whole point was that this was universal for a system so the volume somehow describes how many different ways something can be in in this way how many different ways a particle can be in a container remember when we spoke about the electrons and the spins well that is also a state in a way right an electron has a state so occasionally we can talk about things in some sort of abstract state way where we will still call it volume but this you can think of this how many alternative ways do i have to do something and this could be if i need to enter if i need to exit this room i only have one option to exit it if i had two doors you can imagine that's two possible ways so it's we will call it volume but it's not necessarily volume in the cubic decimeter sense we can think of this as the number of available states too but you know what this is not very hard if we assume this is volume let's just do the math so i'm going to do an advanced postulate the volume of state a is volume a so you see what i'm doing again it's exactly if i don't know something i just define it and call it that and the volume of state b is i didn't write that out it's going to be vb and i would also say to first i will the way i count things here is that the number of states is proportional to volume that's just something i say there is no universal definition of nature what a state is but somehow the the number of good things here is proportional to the volume i can define anything i want these are my equations i think that if you're not in physics that's something that you're usually very afraid to you're you're afraid to make assumptions it's perfectly fine to make assumptions the absolutely worst thing that can happen is that 49 slides from now on i realize i'm in a bind i have not been able to do anything then it's probably time to start going back and revisit some of the assumptions that's not going to happen here but that happens all the time in research i screw up all the time in my assumptions and then somehow the probability should also be proportional to volume right if you have two good well if you have two doors it's twice as easy to exit or something so in one way it's really easy the probability of somehow not just being in state a but being in a volume a that has more than one state compared to volume b then i should just multiply both those expressions to volume as a constant there might be some other constant here too but the other constant cancels out that was easy we just solved it right there is one little problem here what is the little problem i have can you use that in your lab why no there is no proportion it's very straight beautiful equals so what is the volume you're going to use in your lab so this is just a completely arbitrary symbol that i introduced right you have absolutely no idea what that volume is this looks beautiful it's an equation and it's completely unusable in practice and that's the danger that you can introduce you could argue that this is kind of a screw up case you can introduce any number of things i want but at some point i need to get them to cancel out or remove where i'm going to be stuck in something that looks beautiful but it's unusable this is unusable unless i know what the volume is the problem is i don't know what the volume is i have absolutely no idea what the volume is and in principle i'm stuck here in principle the advantage in the church of physics is that what you do when you're stuck yeah yes we'll define something and the energy here is somehow a property of this state right i would like to move this volume up in here so i could somehow group it and say that this is a property of the state let's see if i can do that so i use a trick the volume is just the exponential of the logarithm of the volume exponential and logarithm are the opposites right so that that's that doesn't mean anything and then i can say instead of having the volume as a factor in front of the exponent i get the logarithm of the volume up in the exponent here on both sides and then i can just i would like to divide them both by kt so i say that it's the exponent of minus the energy minus the temperature multiplied by boltzmann's constants multiplied by the logarithm and then divided by kt it looks absolutely horrible i know but you know what the funny thing is that looks like a boltzmann distribution it's exactly the boltzmann distribution but instead of having my beautiful energy i now have an energy minus the temperature multiplied by the boltzmann's constant multiplied by some strange logarithm but somehow so this is kind of a correction factor or something to the energy that we can group together with the energy i still can't solve this because i still have the nasty volume and if i don't know what the volume is i sure don't know what the logarithm of the volume is but this as a physicist if i can't even get further with that definition i can define this i i frequently i always call it energy and remember there were a couple of slides where i said that technically this is a free energy if that is the energy this entire group is what i'm going to call the free energy and we're later going to see that that corresponds to the amount of energy that's available to perform work in a system you don't know that now you're going to have to take my word for it but if you somehow if you have an e can you imagine any other letter we could just call that f next letter in the alphabet that's natural right and then because you don't know in general you don't know what the energy of a system is you will just assume that or assume that you can calculate it this is some other property it's going to have the same units as energy so let's just call that f but that's kind of cheating right because to calculate if i can't calculate the logarithm of the volume i sure can't calculate f just because i changed the symbol so we're going to need to introduce something here this the temperature is kind of nice to have on the outside but Boltzmann's constants multiplied by the logarithm of some volume that's what i'm going to call we need the name for it so i'm going to call it entropy how many of you have stumbled into entropy before and then you try to understand it you try to understand the entropy right that's what all chemists try to do you can't understand entropy this is what physicists do and entropy this is just something that comes out as a result from an equation and rather than have logarithm or this is a physical definition it's going to turn out that this definition is pretty darn useful because it corresponds to something but an entropy just as the energy is a property of a state and again the energy of the remote control as a function of the height over the table if i have states and if i assume i can measure things in principle this is a this s is a property of a state that in theory at least we should be able to measure at least differences should be possible to measure so rather than just having an energy we can define this free energy as being in this is really the expression i had right energy minus temperature multiplied planks constant logarithm of volume but then we replace k l and v with s this might sound really stupid why on earth aren't doing this well the beautiful thing here there are two things that are really beautiful first this now looks exactly like the boltzmann distribution it is the boltzmann distribution by the way but we have a delta f instead of delta e in the argument but the other beautiful part here is that there were kind of two parts in nature we had one part that describes energy in the system the interactions this part s then describes how many states with similar energy are there for instance in this case of volume a if all the particles in volume a had the same property they have the same energy so s here somehow describes how large volume a is that would be s a while s b would describe how large volume b is and rather than volume we could think of this it's not really a volume it's a logarithm of volume you can think of this as the number of available counted microstates or something don't try to understand this from now it's just that somehow magically this s describes the number of available copies of something the neat thing by having t outside because t is not the property of the system t might change the temperature from one experiment to another right so this is a property of the system that is a property of the system the t is a property of an experiment so this is that being a very neat definition because this entropy we can measure the energy we can measure and then we can study things what happens as a function of the temperature here when you those of you have stumbled upon entropy before i probably stumbled upon this definition that it somehow measures disorder in a system right if something is very ordered if something is perfectly ordered that absolute zero kelvin so that is only one single possible confirmation it can be imagine an ice molecule that's so perfect that every single hydrogen bond is perfectly four minutes i know that forget about quantum mechanics and heisenberg uncertainty principle and everything if a perfect ice crystal there is not a single atom you can move in this crystal because you would break a hydrogen bond how many states are that one the logarithm of one is zero so at zero kelvin the entropy of a system is zero there is no disorder and as we're starting to break things there are many if you imagine an ice crystal with 10 well a small ice crystal with 100 molecules how many ways are there where you have one broken hydrogen bond well quite a few right there are even more states we have two broken hydrogen bonds there are even more way of 10 broken hydrogen bonds so the higher you the more disordered your system is the more alternative confirmation there are that have the same energy so normally the more disordered a system is the higher this factor s is going to be and that's why we somehow think of this as disorder but make no mistake it's not that we try to measure what disorder is this is a physical definition that just turns out to be a really good representation of disorder and that's why they don't try to understand it this is a definition it's a Boltzmann's constant multiplied by the logarithm of the states yes oh sorry i should probably have a volume so when i did this is a probability of being in volume a you're quite right i should probably say probability of being in volume a here and probability of being well it's just what i mean here i let a represent more than one state right this is a collection of states and i let b represent something else so i should probably have said volumes here but just that since i got rid of the volume up here it almost felt wrong to keep carrying the v suffix down here but it's the same so a is one state here but that now is not just one a is one state that we can measure but inside a we have lots of smaller states and we typically call those micro states while b is another but we have many smaller states in that okay that sounds confusing you can i think of this ice crystal an ice crystal hundred hundred molecules an average e molecules forms two hydrogen bonds that's one state and we can say what is there's one state when we have 95 percent of the hydrogen bonds formed from a microscopic point of view we can say that's one state that has a particular energy that i can measure now microscopically there are lots of ways to distribute the broken hydrogen bonds right so inside this macroscopic measurable states you have lots of small atomic states and that's really what i talked about these volumes but remember we're going to go through entropy in a lot of detail tomorrow yep in one way it is but the point is that i'm i don't say this is some this is a property of a state that i can measure so it turns out that this free energy but the problem we don't really know what the volume is if you were working with gas particles yes i could measure the volume but if you're working with chemical reactions what are the number of microstates in which an ethane molecule can be beats me i have no idea but if you can measure it turns out that this delta f is a free energy actually measures directly the work we're doing so if you take a molecule and heat it we're adding energy if you're not changing the distribution so this is molecules move over to some other state there is a direct relation between how much energy i've used to heat this molecule and the probability in which they are even though this means that i have lots of multiple states and everything so the beautiful thing i never have to count that volume this is a beautiful way of delta f is something i can see in the lab and then i can actually calculate with delta s i'm sorry what s or delta s is but in a way you're right yes it's just that i've hidden it so that this is now a number that i don't have to think about units in terms of a volume or anything if this sounds complicated i'm going to try to have a couple of simple examples um i used to have an example here that i thought fit really well and that's my kid's room uh and i'll actually i'll bring up and so well in my kid's room um my kids are not so well behaved so that their room tends to look like a bomb has exploded there and then you occasionally clean up all the toys and put them neatly in bookshelves and everything and for some strange reason the next day or the toys are scattered out on the floor again so why if there is now this random force of nature that consists of my kids why don't the toys ever spontaneously go from the floor up in the bookshelf that has to do with entropy right but it's not that easy uh so perhaps we can do some those you might not have kids so let's do something simpler this is a desktop and on my mac you can actually sort things in this groups of folders so that we have multiple icons or something sorted together so you can think of this as a state each of these folders or something is a macrostate so that's something i can measure while all the small apps or anything here that's my macrostates as a lab rat when we're measuring this we don't really care about all the details in there but the next second you're a physicist and then you care deeply about the macrostates and then you try to count the apps inside it so if you think in terms of lab states that how many macroscopic lab states does this correspond to well how many screenshots are there no there's one screenshot there's one picture it's much easier than you think the other alternative is of course this is probably a truer representation of my desktop usually looks like let's ask the same question there how many states does that correspond to so that was the easy question it's the same in terms and now you're thinking like physicists it's one specific distribution of every single app or document on the desktop it does not matter you might as a human being you might think that looks looks neater or something but in terms of state it's one specific distribution of all the components and there could be atoms or apps or toys in my kid's room but as an experimentalist maybe this is the more relevant question how many similar states are there so this is a very well-ordered states so how many similar states are there to that one while how many similar states are there compared to that one yes so there are relatively and trust me i have over i have years of experimental evidence in my kid's rooms here there are very few states that are ordered simply because there are many more states that are random and this is the difference you see that when we first originally defined Boltzmann we only looked at one specific microstate at the time one and one but experimentally you know if you take those two documents and assume that those are the same type of documents if you swap those well it's going to be the same energy in your lab experiments it's not going to change everything you're still going to say that it has the same i don't know what absorbance or something if you take proteins and you say assume that these are proteins and let's say we unfold that molecule but that molecule was unfolded so let's fold that one again you have exactly the same amount of folded protein you're going to get exactly the same spectroscopic signal so experimentally you have many states that give the same signal macroscopically it's one state but microscopically inside it you have many sub states but you have few ordered sub states and many disordered ones and that's what we described with entropy so this one is low s and this one is very high s and we have actually now derived one of the most important things actually because we can't prove this because this is based on our and this is strange why can't i prove this it might feel like as if we proved something but i haven't no we only did that i introduced a bunch of definitions and assumptions and i said that given those assumptions we need to do something the concept of a state we haven't proven what a state is and therefore you can't prove what a state is because this is ultimately a concept in your minds now and this is the problem because this is both the really big problem with thermodynamics but also the beauty this is not based on something else being true it's not that if somebody shows that the law of gravity is slightly different that would change thermodynamics it won't because thermodynamics is based on observations and our concept of things and what we've actually gone through here is the first law of thermodynamics you probably know that we can't create or destroy energy that would be a preparatory mobila the whole concept of entropy is really that the entropy of an isolated system not in equilibrium will increase over time and approach a maximum when it reaches equilibrium so and the total sum of entropy if you're multiple components you can certainly have one part of the system going down in entropy but then another will go up and this considered a disorder in the in the world always increases this is very much related to the arrow of time in modern physics the arrow of time the reason why time goes forward is because of entropy the entropy always increases you can't prove this it's a postulate and then as a consequence at absolute zero the entropy of any system approaches a minimum i said that it was zero but it's actually after you you can always put a constant in front of it if you want because if you can that's like taking your ruler and moving it down you can change the reference level but you know what the beautiful thing is now you can start explaining lots of things with this things have worked a bit hard before but it's going to turn easy we're not going to do this for proteins we will save that for tomorrow's and actually i'm not even going to do proteans or i'm going to do amino acids tomorrow so let's do a phase transition ice if there's one equation sorry if there are two equations the three equations you can look no two equations you should learn today boltzmann distribution is the first one if you have room for another one learn that free energy is energy minus temperature multiplied by entropy this is an equation that's very easy to learn it's very hard to really understand it and even i make mistakes water or h2o can exist either in the form of ice or in the form of liquid water and you all know that around zero degrees centigrade something happens right something happens at hundred degrees centigrade two and virtually all compounds in nature go through these phase transitions so if you look at ice and this is an ideal piece of ice that would really be sero kelvin or close to it but this state is characterized by very very high order but here's the complication you also have a beautiful energy here because every single hydrogen bond is formed right so that the energy here is low the lower the temperature gets the better it is and you also have an extremely low entropy because it's a highly ordered system so that term is low and that term is also low so they're kind of gonna they're gonna have opposite signs here so they will contract each other what can you say from that forget about the right hand side for now the reason you're quiet is that you're smart you can't say anything about it like if i say that the free energy here is 49 will you trust me that's a very bad idea don't run the point you need some sort of scale here right it doesn't mean anything unless you start comparing it to something so this equation is usually pointless until you start looking at differences in free energy differences in free energy always corresponds to either getting work from a system or putting work into a system and it turns out that every single measurement you can do in a lab corresponds to a free energy you're measuring something so the key thing here is to look at the right hand side and the right hand side we have liquid water the energy here is much higher and high energy is bad in physics we don't have condensed systems have low energy when things are advantageous so high energy is bad for a molecule the entropy i say that it's medium you could say that it's high the reason i say medium is that if you turn this to gas it would be even higher right so the problem here what i now did is that okay the term on the left hand side e became higher and the term on the right hand side also became higher so this is still hard to say something about but the third part is really the temperature here because depending on what temperature have we're going to change the balance between energy and s right so if t is zero how important is the entropy if t is zero the entropy is irrelevant right because that term will be gone so at zero degrees you would always try to adopt the state that had the best possible energy and that would be a surprise that's absolutely perfect you would do anything to maintain all the hydrogen bonds no matter how ordered your system became and conversely at infinite temperature you can't don't imagine how you would get there but at the temperature that's high enough the energy becomes irrelevant and everything depends on entropy right so eventually when you're high enough we will do anything to make sure that we can be as disordered as possible no matter how much energy we lose from it and that's eventually what can happen in a gas or something we will lose every single hydrogen bond if necessary because the temperature is so high that is more important to be as disordered as you can be so what happens around zero degree centigrade for instance is that where we are ice at minus five degrees this first term is still just so slightly larger than the second one and then it's better to be ice but the second year at plus one degree centigrade now that term became larger and then it's suddenly better to be liquid water so the entropy really the balance between energy or enthalpy as we're going to call it later and entropy can describe phase transitions we can describe way more complicated things than phase transition with this so i'm going to give you a food for thought the book actually goes through this but i figured you should have some there should be some carrot for reading the book you can explain the hydrophobic effect with this can you imagine how let's leave the podiums out for a second in principle you're right but they would look at the a drop of oil in water or a single xenon added in water xenon doesn't dissolve easily in water either remember that i said this was it's somehow caused by electrostatics right because it's if the solvent didn't have the hydrogen bonds there wouldn't be an issue it's somehow caused by the hydrogen bonds but then we had this perplexing effect but before adding the xenon we had a given number of hydrogen bonds and the second we put the xenon and we have how many hydrogen bonds no roughly the same remember that the water will do anything to maintain their hydrogen bonds so we haven't lost a single hydrogen bond so while it might seem obvious that hydrophobicity is an electrostatic effect it's not the electrostatic energy is identical before and after adding something but the problem is that to be able to maintain all those hydrogen bonds all your waters went through this acrobatics to form this shell structure around your solute is that a more or less well-ordered state no it's a more sorry it's a it's a it's a it's a more ordered state right if you're forming a shell compared to having waters that can move around and move around and anything and now we say that the waters have to have a specific shell structure around one given molecule you're forcing the water to become ordered there is suddenly a fewer we will go through that in detail tomorrow and I will have to say some experimental evidence for it so that the entropy becomes lower when you're putting the xenon in water so that the hydrophobic effect is actually an entropic effect and it's not at all an electrostatic effect although it's caused by electrostatics but it's not an electrostatic effect could you imagine any way to prove this again again and now I'm just standing here and talking and you don't have to believe me can you imagine any way to show this experimentally so what would you predict so there's one problem here say if you start doing experiments with water below zero it's going to be somewhat difficult because it's no longer water it's ice so let's stay above zero so what so what happens here is that as we are increasing the temperature I can start to alter the balance between these two factors right so that as I'm increasing the temperature I can start to look at the solubility of various compounds as a function of temperature and if the solubility of these compounds is the same regardless of temperature then it's an electrostatic effect right there it's an energetic effect at least but if the solubility changes as a function of temperature it's an entropic effect that's a good question there are only two possible answers to it one of them is wrong and the other one is right what do you think right so when you put this more putting a molecule in will increase the entropy right but you also have a minus sign here so that as you're increasing the temperature it becomes easier to solve eight most things in water at least the hydrophobic things like oil or so it's not an extremely strong effect and even at 100 degrees you're not going to have a huge solubility of oil and water but there's definitely there is a small temperature dependence here that's actually an excellent study question I might add that tomorrow too that's pretty much I think I'm not really gonna the key thing here is that the Boltzmann distribution you're gonna be battling with this this afternoon too and I try to get a feeling for what this means it's really it will help you a lot later on and it really explains everything if you then feel that you want to there will be some possibilities for at least to have a quick look in the lab and understand what happens with entropy but focus on understanding the simple part the entropy is just that I want you to have seen it once so that we can come back to it tomorrow discuss it tomorrow morning and tomorrow you're going to have a lab where we actually add the entropy part to it but today you're not going to be worrying too much about entropy in the labs I added a bunch of study questions here too you have them in the slides and I'll try to encode locally in my computers I can hopefully get the lecture up a bit earlier and there is also lecture slides uploaded online so I would suggest that we go through this tomorrow morning the same way as you did this morning if you understand my point if you understand these study questions you're going to pass the course and that's why I tried to occasionally include a couple of them that I haven't talked about in the lectures to read the book and I think that's all I had today do you have any questions