 What I'm going to talk about today is that we're going to start by following up from yesterday first and It's going to be a bit more about free energy not as much equations as yesterday And we're going to continue talking about electrostatics in particular and the hydrophobic effect and the idea here is really today I'm going to sum up what we did all the stuff we did with Boltzmann We're going to talk more about entropy and then this afternoon You're going to do a lab where we started including the entropy into this addition to the Boltzmann distribution Yesterday and tomorrow I'm going to finish off the week and actually talk about protein structure again So what you're going to see that we're going to do frequently in this course is I'm going to re it might seem that I'm rehashing things But I frequently go through things in one two or even three iterations And then we move on the one hand between physics and on the other between chemistry applications So at the very first lecture I ran through proteins fairly quickly and now we spent the time going through the Boltzmann distribution free energy entropy So the idea tomorrow then then we can start going through some of these structures again But then we're actually going to interpret it in terms of free energy in particular So why do we have an alpha helix when an alpha helix is forming? Can we now specifically say in terms of free energy why these things are forming and then after that? We're going to go back to even more physics and equations So but the idea is that you should feel that these are things that we can very directly apply to the proteins And eventually we're even going to see how we can apply them to predict how fast various things What I spoke mostly about yesterday We have a recap here, but I would almost suggest that we're going to move straight to the study questions that we took up yesterday and I Also realized when I record this you hear my voice, but nobody hears your voice So might I might text just say the number of these questions that anybody starts answering Unless you have another idea I would suggest we do it exactly the same way pick a question and then we start answering it or discussing it And then we'll see how far we get Number one Yes, but do chemical bonds always form when two atoms starts having their electrons get close Well, so in some cases atoms repel each other, right? And that's also due to these bonds So when atoms can share electrons in a favorable way in particular filling their electron shells That's when you have these covalence bonds forming, but it's not the case that they always form the bonds and Why are then e.g. noble gases inert? Exactly And that immediately leads to question number two that either you or somebody else can answer if these noble gases are inert Obviously, they can still form liquids or something at low temperatures Now why is that? Yes, so induced dipole to induced dipole interactions and This actually I realized this is something I haven't brought up But it might even seem a bit strange So why and why would we keep talking about these negative energies and everything so in general you're probably all familiar with the ideal gas law Is something right so that when atoms are at least an approximation infinitely apart from each other They don't interact and then then it's kind of obvious if something don't interact It would be very strange to have a positive energy than right So if something don't interact at some point you're gonna need to choose a reference scale for your energy So somewhere you need to put the zero point and putting the zero point where things don't interact seems fairly reasonable, right? So what then happens the second thing start to con you call it condensed because it's condensed matter when things start to condense and form liquids They form liquids because they interact favorably and this means that you're gonna need to have a lower energy And this energy is now going to be lower than zero And that's why we have typically have these interaction positive good interaction negative energies are negative and that I realize How much of a contradiction that sounds positive energies are negative But negative energies are good in physics some other questions Can you even read this? Well, you have it in your hand out those 19 E minus TS, okay? Yes One problem with this that you asked you would like to start to realize now When do we mean energy and when do we mean free energy and In hindsight it's such an astronomically bad decision to call them energy and free energy We like we should likely have called free energy something else like free work or something, but sorry, we can't change 150 or 200 years of physics here I'm gonna hopefully you're not gonna be mixing these up But what you will realize that both I and several other authors we don't always bother with saying this free prefix So eventually we're gonna see even more energy landscapes Technically most of those are actually free energy landscapes, but it gets so tedious to keep adding this prefix So sadly we keep losing the free prefix, but when you have the equations and anything, it's important that you know what you mean right 13 KT forget about forget about K and there is a case Forget about K. Think of the entire combination K multiplied by the temperature at room temperature Sorry Yes, but that's the units. So what is Katie numerically? K cal yes, and here's where you need to remember the units at 0.6 K cal or what in kilojoules Yes, and this is the one thing that you need to know if somebody wakes up at 2 a.m. In the morning because it's gonna be so important not just in this course But when it comes to you and this is not just about physics any type of energy any type of probabilities You're gonna interpret understanding what KT is because this is what this will immediately You will immediately be able to tell people whether things will this happen or will it not happen? I even had a very fun opponent a couple of years ago. We kept asking a student that Well if the transition energy is what was it type 5k calc from all of the will this happen? And the student was of course just I mean he had no idea But yeah, if you have a rough idea with KTS, you just this is gonna be an exponent raised to is at 1 Then it will happen if it's an exponent is raised to minus 500 then it's extremely unlikely that will happen So just having a gut feeling for what KTS me said you can compare energy energy and say is this an energy barrier We will be able to cross or not Yes Because you have the mole kilojoules per mole so that's like K is what the way physicists think here Physicists likes to think in terms of particles. So the normal way you specify the Boltzmann constant is joule per Kelvin But a chemist rarely think about what is the energy for one molecule a chemist would never Virtually never ever so no because you end up with awkward numbers and very low powers, right? So in chemistry we always think in terms of per mole But of course, that's just more less kind of dimensionless It's just a scaling factor and you can equally well specify the Boltzmann's constants and joule per Kelvin per mole It's just that we typically don't do it But what you specify this energy kT you could of course specify this energy per unit or per unit mole and in chemistry We like to think of this per unit mole Yes, I would but but again there is no you just talking about different letters you use for the same thing There's no diet. There's no different in dimension here. It's just a scaling factor Yes And again, so that's just because we chemists tend to use are for this But you can equally well use k as long as you're careful about your units and that's what I kept stressing We said 0.6 kilo joule per mole or 2.5. Sorry 0.6 k cal per mole or 2.5 kilo joule per mole The other thing to be aware of this always enters as a fraction as a quota quotient and the exponent, right So the only thing that's going to be important for you is to make sure that you have the same units in the nominator and the denominator And then they will cancel out So all the energies we talked about bonds and everything that's always k cal per mole or kilo joule per mole And sadly, it's a curse of science if there are two ways of doing it. We for some reason always end up using both Other questions and that this also relates to number 12, right? Physicists usually like to work with Boltzmann's constants while chemists prefer are this is exactly the same fundamental natural constants It's just we use different letters for it You can use you could use z if you wanted to as long as you're really careful specifying what this is what the units are That's gonna cause some royal confusion any time you deal with people because that is also the partition function But as long as you're careful with your definition go right ahead 10. Yep the energy landscapes Right, so this is really a very high that we usually draw these in three dimensions But it's really the energy or actually free energy as a function of all the degrees of freedom in the system and This is sentimentally related to on Vincent and Levent are later on because Ultimately what the molecule needs to do to find the lowest free energy It's gonna need to explore the entire energy landscape and in a real physical system if you have hundred thousand particles Each particle has an x and a y and a z coordinate There are at least three hundred thousand degrees of freedom so even the simp the simp even Thank God for proteins. We can usually ignore things like bonds and everything If you go down and only look at the dihedral torsion So I say that there are two of them per amino acid even a 200 dimensional energy landscape is extremely complicated And it might not sound like the ten or two or two hundred is so different But I have a slide and well, I have a table in two or three slides that we're gonna help you understand that The exponentials are harder to understand than you think Right, that was energy landscape something else 17 examples of systems with lower high entropy Yes, so gas or liquids have always had higher entropy than solids Yep volume argue that's related to your lab last night. So the last year after how did that work? It was it was kind of a new idea that Björn and Ari that I think was good that so normally we would have used equations equations equations So our idea was for you to get a feeling just by looking at these states. It's a bit abstract, of course, but did it work out well? So how what is the problem with this volume exactly? And and I think that was kind of the point in the lab that physicists we keep talking about volume But this is a very abstract volume. So higher entropy that just means that somehow We're opening up additional Additional small building blocks or states where you can put your system and that's literally what I think you did in the last part of the lab you can create additional states where you can be with the same energy and That somehow corresponds to the more of these so-called microstates and Similarly reduced entropy that something is happening to the system that says that you don't have as many microstates that are available to you There are fewer combinations in which you can put the system and I almost think that this is Probably an easier way to think of entropy rather than thinking in terms of this order or disorder Because order and disorder are also so abstract. How do you count disorder? It's easy to count these blocks that you see in different states or something And the funny thing is that not just it's not just easier. It's also more correct Because the way we can the way we define entropy is actually strictly Boltzmann's constants multiplied by the logarithm of the number of microstates and Then you can choose whether you do this per state or per mole of states, right? Let's see here something fun Interactions it's gonna what interactions are strongest and weakest you have some examples of numbers of strength of various interactions Coulomb is extremely strong. Yes, and and of course all these things depend on exactly what distances and charges You have it I don't really care about the details, but it's here to its import It helps a lot to have gut feelings about things because I will rather than spending half an hour calculating it in detail You can straight I'll say this is relevant or it's irrelevant. So a Coulomb interaction might be what strength-wise Yes, and that's probably a very strong one, but the point it can easily be a hundred K cal And if you compare that to say Leonard Jones interaction Yeah, so it can even be 0.1 So that at least two if not three orders of magnitude weaker and and here's the key It doesn't really matter where there's two or three orders of magnitude, right? Because yes, if it starts to matter if it wants to start to be within a factor of 10 or something Then you need to sit down and do the matter and calculate it exactly But if something is 1% or 0.1 percent, it's irrelevant no matter what and this helps you a lot to say that if you Have something that you have both a charge interaction and a dispersion interaction You can pretty much ignore the dispersion part compared to the charge interaction if they're if they're both attractive that is And I think those are also the strongest and weakest interactions if you compare this with torsions in a system What are what's the typical energy for a torsion if you move around between different states by the barriers or the differences between states? Yeah, a couple of K cals and There of course there are certainly exceptions there too that are at some point You might twist the torsion so much so that Adam starts bumping into each other But that's a special case, but the normal things that we would run over is a couple of K cals And that also involve covers comparing electrostatics with dispersion We have the what happens with hydrogen bonds when ice melts to water do some bonds break completely while others are maintained or is it something else? Yes, so this is the key thing This is a gradual transition the opposite of a gradual transition is all or none And you might remember that the day before yesterday I said something about that on the slide Protein stability is different Proteins, they are folded they are folded they are folded until they are no longer folded you don't gradually boil an egg But something happens at roughly a bit over 70 degrees right suddenly the whole egg white denaturates and the egg is boiled Well, it goes from the outside and in but for an individual protein. It's all or none So there's something that appears to be quite different here The other and this of course when you're gradually heating ice, but as a collective You're definitely looking at this macroscopically There's something very fundamental when happen you have ice at minus zero point one degrees and plus zero point one degrees You suddenly have water So there are these whole concept of phase transition is a bit more complicated and we're going to get back to that many times And as you might start see now there is a very clear parallel between protein folding and phase transition This is essentially a phase transition, but it's a more complicated one for what can you use the Boltzmann distribution? Apart from passing this course Which I guess we shouldn't underestimate right now, but it is actually highly useful in practice You're going to use this all the time even if you're in the lab You don't think of this in the lab, but I try if you've ever done lab work. You use the Boltzmann distribution Have you ever when you're calculating relative fractions or anything? Have you seen this RTL and K where K is some sort of K would be a constant describing the relative concentrations or something and they take the logarithm and multiplied by RT to get the Substration you're just inverting that that logarithm is because you're inverting the Boltzmann distribution and then you're multiplying by RT Or KT instead of having it in the dominator. So because the Boltzmann distribution the relative energies of two states Delta e divided by KT that will also tell you what is the relative population of these two states in a lab You typically do this the opposite way, right? It's very easy to measure how many things are in state a and state b and that could be Solubility in various parts of the system or different states of fluorescence when you can measure this then you have the relative Probabilities in the lab you typically do it the other way you have the relative probabilities and you use that to measure the relative energy between these two states Or do you do you measure the relative energies or is it something else we're measuring in the lab? Yes And this is where kind of we kind of had to cheat because I didn't want you to start with entropy in the very first lab unit Virtually every single measurement you're doing in the lab corresponds directly to a free energy Everything in the lab is about free and this and that's why we're going to talk so much about free energy if you understand free energies you understand measurements and Every single measurement you do is pretty much to get to a free energy although in the lab you might not think of that as free energy, but it certainly is one and That brings us to 16. What is the difference between energy and free energy? So that's the mathematical definition of it Which is quite true as I'm gonna have a link to a site today that actually goes through and explain this in detail But the idea that free energy is literally the amount of energy that's available to perform work and that might sound very strange But the whole idea is that if you are in an arbitrary state of a system And if you're at point where there is a gradient in free energy The system will always want to go towards lower lower free energy and when the system moves towards lower free energy Well, it's gonna then the system moves downward and that means that it will produce heat It will give up heat to the surrounding and that is something that becomes available to heat the universe or increase the entropy of the universe And this will go on Until you are at the point that's a local minimum and the free energy and when you are at the local minimum and the free energy we can no longer perform any work and This means that the free energy will naturally describe the direction in which processes go The process will always go in the direction of reduced free energy until it can't move any further You can actually prove this to that the difference is available as free work The book doesn't and it's not really critical to the rest of the course. I'll provide a link for that today Let's see I think those were the most important ones. I have the book doesn't talk about detailed balance But I kind of like it so I'll bring it up anyway. What is detailed balance? Yes, so that the point is that at the equilibrium that doesn't mean that we don't have any processes anymore It's just that the observable the observable quantities are constant But that might marry well me we might have a bunch of proteins that keeps going from the folded to the unfolded state as long as there's the same number of proteins that's goes from unfolded to the folded state and This is essentially a stationary distribution so all the states are the same But that also means that if you just formulate that with the Boltzmann distributions I did yesterday you can either use to predict the Boltzmann distribution We can either look at the relative distribution the pro the probability of being in states the the populations in the states Or we can just measure the frequencies of transitions from a to b and compare that to b to a And it turns out that in modern statistical methods. That's frequently what you do You can basically stand at the edge of the ridge and just look how much is going left and how much is going right Without actually observing what you have in the basins on each side So deep detailed balance means exactly this that at equilibrium if you have two different states It doesn't matter one of them can be thousand times better than the others But at equilibrium the flows in both directions have to be exactly the same and this might sound strange because it's much easier To go from high to low energy than in the opposite direction, right? But the point is that you might have a ten thousand times more molecules in the low energy state So for an individual molecule it's much lower probability to go over but since there are also many more molecules Collectively the flux of the amount of ions per unit of time or something is going to go over It's going to be exactly the same in both and that's the balance part the D good question It's just the detail balance is specifically the flux the flux is balanced. That's probably the detail It's very good question. I could probably look it up. I'm not sure why Yes, if we just said balance nobody would mean what type of balance we spoke about and This is a very specific property of something you call the ensemble that I don't think we will talk about and now we're getting even deeper into statistical Mechanics, which is not really the point of the course One more thing I'm going to bring up Electrostatics with dispersion we already talked about the strength, but there's something important with the sign here that you need to remember so there are two parts to Dispersion is always repulsive and Then we also have the other part, which is No, sorry repulsion is always my bad dispersion is always attractive repulsion is of course always repulsive Dispersion is always attractive, but the point is both parts of these Lana Jones interactions. We know what the sign is It's always the same sign and that means that Although this is Extremely weak if you look at an individual interaction you can virtually always Ignore if they have the same sign at least you can ignore the dispersive part compared to electrostatics But if you start looking at interactions collective ones with a thousand neighbors or something Electrostatics becomes noisy like literally because as you sum things up You have no idea what the sign is going to be at the end That also means that it's a very fragile interaction that you need to treat very accurately if you start making small errors here And then you sum up lots of components with different signs. You have no idea where you're going to end up The danger with the dispersion although it is small. It's very easy to think we can't ignore it in particularly We want to ignore it beyond some sort of cut-off But as these components add up Eventually if you have enough of them they can start to collectively be fairly large exactly because they have the same sign It's good that we call that otherwise. I would have me on tape saying that dispersion is repulsive Which is fun. It's actually a mistake. I made in my thesis. I screwed up the sign of that equations It seems to follow me. Yes Difference between energy and free energy Yes If if the if the energy is the same, yes, if the energy is the same You will want to go to Heinrich. So remember what I said yesterday You frequently talk about isolated system or the universe or something But we also had the first law of thermodynamics the first law of thermodynamics that you can't destroy or create energy So if you have an isolated system by definition that system does not exchange energy with its surroundings and then the energy is what? constant and if the energy is constant Free energy reduces to the entropy So in an isolated system or if you look at the entire universe in the entire universe Entropy will always increase because we can't create or destroy energy But in an individual system, it's certainly it's certainly possible to clean up my desktop or I we can clean my kids room Then we're reducing the entropy there, but that's of course just because we're exchanging work with some other room or in the universe So what we're doing is basically we're swapping Energy and entropy, but it's also that is not an isolated system, right? Suddenly we are exchanging work in heat with everything So every time I clean my kids room or we try to force them to do it every time we do that the entropy in the universe increases The entropy in their room might be reduced, but in the universe it increases I guess something about protein that is folding and filling up Well, that depends you might also get a whole lot better energy in that state, right? Because a protein that is folding is not isolated. It's working in water. You have hydrogens forming and everything It's a really good question, but I would suggest that we come back to it later We're gonna talk a lot about the proteins and free energy So the caveat that these things work if you only exchange work with the surrounding or something and or it's an isolated system It's more complicated because you need to take the water for instance into account What I'm gonna talk about today, and that was great by the way What I'm gonna talk about today is a little bit more about free energy It turns out remember what I said if there are two ways we can do things People will choose both so they turns out that there are two slightly different sorts of free energy One of them is like by physicists and the other ones by chemists and there is a natural reason why There is a small result that we're gonna need later today's I'm just gonna take up a thermodynamic temperature definition It's not a whole lot of math is less than one slide and then we're gonna Revisit the hydrophobic effect that we talked a little bit about that yesterday and there are some somewhat surprising the results there We're gonna talk more about protein folding and electrostatics and titratable amino acids And I think yes the first concept is free energy This is another one of I kind of like these old cartoons that the city utility board kicks the free energy quack out because nothing in life is free I Had a colleague who mentioned that he was sitting in a lounge in the US over and reading a book and it was an old lady sitting right next to her Oh, I see that you're working on free energy. That is really important And I somehow don't think she understood what that book was about And this come against constant answer this the reason being that it's really stupid to mix up energy and free energy I can only apologize in the name of a couple of dozens of generations of scientists We should have used a different name But we didn't so now too you will have to keep mixing up energy and free energy But if you forget about Helmholtz and Gisrael what I usually spoke about we spoke about a system and a system is something abstract in physics Ideally, you could say you have a physicist actually likes to think of a completely isolated system It's simple because nothing will ever ever cross the yellow boundary here This is my miniature universe all the equations energy is constant inside The only problem this is really complicated to work with if you're going to do this in the lab First you need to make sure that pressure volume everything has to be constant here You need to make sure that this is insulated so that it can't exchange any heat with the surrounding Occasionally you do experiments like that, but it's simply it's a pain that you don't really want to work with and it Doesn't correspond because it's so special isolated doesn't correspond to anything real So the first approximate well the first is this is not an approximation actually the first thing we have to deal with is that Systems are usually allowed to exchange heat with the surroundings and you can think of this as it's still isolated The volume here is constant the pressure might if something happens here The pressure might change because again, we will not allow the volume to expand But if a chemical reactions happen inside here that heats up the system this heat will be allowed to escape to the surrounding air And that's simply because it's for a normal system that you might study That's a much more reasonable model for it a protein in general will be able to exchange heat with the water around it or so This is the type of system that we have been working with although I haven't told you And here we have the free energy. There are only really two parts of it We have the energy that might change and if the energy changes that is manifest as exchanging heat with our surrounding and Temperature multiplied by entropy and that is called the Helmholtz free energy This is obviously the free energy if you ask a physicist and that's what the Physicist usually use the letter f for obviously Helmholtz. That means f It's f for free energy There's one complication here though and it's that this is really bad in chemistry How frequently do you do your experiments in perfectly closed volumes? Doesn't happen, right? So in chemistry rather have exactly the same system you can exchange heat But in chemistry the volume of the system can also expand So if you have a small test tube and do a reaction Well, hey if you take water one liter of water and mix that with one liter of ethanol The resulting volume is not going to be two liters is going to be roughly 1.8 or so So the volume can well in that case you actually added to some but something happened here one liter of water one liter of ethanol Something when they make something happen that X in that case made the volume shrink And when the volume shrinks you also have a pressure from the surrounding world Which is usually one atmosphere, right? And if you change a volume in this case If the volume went inwards the rest of the world pushed on us So here we got some work from the rest of the world the pressure times the volume change Conversely if the system were to expand we would do work on the outside world so there in this case There is one extra way we can exchange energy with the outside world by moving heat or we can exchange energy with the outside world by either doing work on the outside world or allowing the outside world to do work on us and This might seem like an unnecessary complication But it's at the end of the day It's much easier to use our equations to describe what you're actually seeing in the lab for instance the test tube Rather than trying to set up a really complicated lab equipment just to be able to mimic the ideal physics scenario You know what I might actually although it's beautiful that we getting spring I might pull down the shade so that you can see the slides and this is bad enough that I need to keep it pushed in So somewhere we need to account for that pressure times volume and what is that? Is it an entropy or an energy like thing? I Mentioned that when we're doing work on the rest of the world or when the rest of the world where it does work on us There is going to be the work change in free energy or and you typically talk about well This work has the effect at the same units as energy and it's going to be the pressure times the volume change and it turns out You actually don't need to know but the energy in the system is going to be the pressure there We had to add a term that it's the pressure times the volume if I Increase the volume in the system that means that there is somehow more energy inside here pushing out and then I increase the free energy So suddenly there's this extra term P So they the free energy here, which if I just use f I'm going to need a new letter here So let's pick G and that's energy and then this pressure effect or volume effect depending on how you see it And then the entropy so now it's a slightly more complicated equation here But in practice that is not an entropy that is again, it's work and work is much more It's it's like an energy right work is energy so To separate this part with this energy like from that part which is entropy like you typically mix the e and the PV into a new letter called H and That H even has a name. We typically call that enthalpy and Suddenly this is actually a pretty good idea because now you no longer have to call it energy and energy in the sense of energy and free energy So you have free energy Enthalpy and entropy So enthalpy are all that the things that we traditionally talk of us energy like right enthalpy is just the property of the system and this is how much energy almost it is and This is entropy part and those together give rise to a free energy and The reason why that is called G is that this is the Gibbs free energy or That is what you typically use it in many cases people who only look at this type of free energy some particular chemists They frequently use F for it Sorry Not my fault So sometimes this is F and sometimes you see F equals H minus TS How do you keep track of all that? It's painful No Because you know exactly what these things are you know that there is a part that is pure energy You know that there's a part that's pressure and you know that's a part that entropy sit down define it decide what you need in this Particular case you can use z x and y if you want to as long as you're very clear what you've defined to what trust yourself The other Good thing is that you get almost gonna get a free lunch here I have slide. This is the reason why having a test tube there. This really corresponds to particular volume changes in experiments How important do you think this term is? It's really important if you're designing nuclear devices For a chemistry lab can pretty much forget about it Again, there are always exceptions if you have if you have a volume change that's large enough for pressure That's large enough. Of course eventually it's gonna be important But in particular in life science the concentration you might have a protein in water It's like it's gonna be one molecule in a thousand or something the resulting volume change on a system for life science processes It's not gonna show up on the scale So in virtually everything you do you can ignore it The reason why and then that case if you can ignore it. Why do I bring it up? Well, it's because you're gonna keep seeing that occasionally people use an e occasionally they use an H For the free energy we're occasionally gonna see G You're occasionally gonna see F and this of course the reason why we had an F if this is G for the Gibbs free energy F is the previous letter in the alphabet or free energy You will see all these equations interchangeably The good thing is that it's really this equation You need to think about whether says F equals E minus TS or G equals H minus TS. It's the same equation For a protein the important part in the first part here is gonna be all these interactions We've described in the protein You don't really care much about the pressure and the second part is gonna be about the entropy But you're gonna have to live with the fact that there are different letters in different equations Yep It can have yes But but again the entropy is included here I'm not saying that this expand when this expansion I'm not saying that when this expansion happens these two terms are constant You might have a in general if you think of a gas right when this expands The molecules and the gas are also an average slightly further away from each other And that's gonna lead to an increase in the energy because they're no longer interacting as hard, right? So I'm not at all saying that these two are constants We still have these two But we also have to account for the fact that we are exchanging work with the rest of the world So you're absolutely right. Nothing says that the other two terms are constants when that happens And then it's enough with a bunch of different things that we're gonna and I will have some slides on that today when we start Studying changes for instance when volume and temperature is constant in principle It's partial difference or partial derivatives and everything, but we don't really speak too much about it There's one thing to be aware of and There is one more good reason why in chemistry it's much nicer to think about the Gibbs free energy If you have a small volume and keep adding particles Actually, you know what let's say that I had this Water or something and if I keep pushing if I keep salvating say proteins in my water The first protein I solvated is one in a one in a billion molecules. It's not gonna have any effect whatsoever The second protein molecule I can add that's also I have two molecules in a billion Now by the point I've by the point that I've added one million protein molecules It's starting to get pretty crowded in here, right? Because the volume is constant the more proteins I push in here the harder It's just gonna be to solvate things or you can think of adding Think of that you could even think of this as a free space without water and I'm just adding gas molecules So as because the volume is constant the more atoms I add the higher the pressure is gonna be and it's gonna be harder and harder And harder to add things so the energy here between my atoms will depend in a pretty complicated way on how many atoms I already have here. It's gonna be higher and higher repulsion between them The Gibbs energy on the other hand as I keep adding atoms they're gonna adapt to whatever ideal volume They would like to have so when I've added a billion atoms the volume of this system might simply have expanded to a billion times larger so the Gibbs Versions of these both the enthalpy and the Gibbs free energy they are proportional to the number of particles the system will just scale up and That makes it very convenient a particularly if you don't know what the volume is in general What is the potential or something per particle? It's very nice to think in terms of these things Have you ever heard about chemical potential? Do you remember what it is? You remember this is a really complicated quantity, right? That was hard to grasp So chemical potential is essentially Gibbs free energy per particle and because the Gibbs free energy is proportional The number of particles right that means that it's a pretty nice number to talk about for particle Many of these things are easier if you just trust the definitions rather than try to understand it We're not gonna talk about chemical potential, but that's the only reason I bring it up So there was one last thing in chapter 4 that I did not cover yesterday and that's just to do with temperature You've been very good. I've had I was a bit nasty in the previous lectures I kept explaining what most of these constants were the gas constant the Boltzmann's constant everything And then there was this capital T that we all just accepted that capital T must mean temperature I call the temperature to for that matter, but We haven't ever really said what temperature is You just think that you know what temperature is based on your surrounding or swimming and or being outside in Stockholm in winter In principle, this is just another constant in all those equations. We went through this came up naturally that there was some sort of proportionality constant And f equals e minus some constant Multipied by the entropy and for now we can just let's just run with that for a second and assume that this is a constant Let's see what type of properties this constant has As I also told you a minute ago the free energy has this property that if you're in a local minimum the free energy won't change So the first derivative by definition here is zero So if we look in this point and then we start to what happens if we move just a little bit So a very small change in free energy, which should be zero here, right? Because the first derivative is zero We can expand that as a small change on the things we have on the right-hand side here So f plus df is first f of course, but then df is first the change in energy Then we have a product here. So we're gonna need to use the product rule for the Differentiation so TS is TDS plus STT and then the minus signs So we have D minus TDS minus STT And at equilibrium if things don't change if this is some sort of property of the system that too should be constant at equilibrium, right? So DT is by definition zero so that term we can start out. So we have that df is equal to DE minus TDS But we are also at a local minimum here. So DF by definition is zero. So DE minus TDS is zero And then we can solve for the T here. So the T is just the derivative of energy with respect to entropy How much does the energy in the system change when you change that number of states you had in your simulation yesterday? You have your system. You're just allowing it to visit more states And there will be some slope on that curve when the energy changes there That is the thermodynamic Definition of temperature pretty cool, right? That's a definition You can certainly derive this the way Celsius or Kelvin everything did it than trying to explain what the temperature scale is But the absolute temperature and everything comes up naturally from thermodynamics, and this is the cool thing that Entropy is a much much much more fundamental property of nature than we usually think when we bring this up in chemistry as disorder or something So the entropy is a more fundamental concept than temperature. Temperature is something we derive from entropy not vice versa Remember this equation or I will refer back to it They only reason bringing up this slide is a because it's curious be because we're gonna use this in a second here It's a how do you mean there or what? How do you mean where the local minimum is coming from? So I did this is just an arbitrary curve. There is some sort of curve that has a local minimum, right? If the free and this is related to what I said about free energy work and equilibrium if we were here Free energy describes the way in which a reaction goes So if you were here, you would not be at equilibrium the free any we would go downhill here and release heat or work to the rest of the World until we are here So when you are at equilibrium, you are by definition at a local minimum of free energy And that's why I said at equilibrium This also means if you're not that equilibrium, you can't really define temperature This might sound strange, but it's trusting trust the definitions Temperature is only defined at equilibrium You I bet you have started the exponential equations But I think this is a pretty fun table to bring up because both you and I and everybody you keep Misunderstanding you keep forgetting how the exponential distribution scales And the reasons here are just taking this dimensionless the point is the argument You have in the exponent is a difference in energy divided by kt And kt it's zero point six k help from all etc But you can also by far these ways to think in energies in terms of kt So if an energy is roughly kt That this kt divided by kt. So e to the minus one is the probability of something happening or One in e to the power of one so one in two point seven If the probability of something happening is one in two point seven that will happen pretty frequently, right? And if something is even smaller it's going to have an even more frequency So things that are in the ballpark of kt or smaller those energies have very easy to get over and those states will be highly populated By the time you reach two point three kt things are going to happen one in ten It's two point three kt it's an energy difference of roughly one k cal per mole or one point five maybe By the time you are at ten kt it's one in twenty two thousand We're not talking about gigantic energy differences here. This is six k cal per mole One in twenty two thousand is going to be in the higher energy By the time you're at fifty kt and again, this is not really an extremely high energy five times ten to the power of twenty and This way now you're getting into the regime of things. I will eat my left shoe if things happen But you see these energies and I think it's and this why it's it's important to be aware of how Insanely fast this number grows There is a there are a couple of fun numbers here That is number of atoms in earth in universe the computational Estimated complexity of chess and since we had the Google AI beat the world championship of go This is the computational complexity of go. It's just e to the power of 392 309 so the problem is the second you are seeing an energy difference of say 185 kt and you have a colleague say yeah, but it might happen now and then So you've got this result and it could happen by chance You're basically trusting a result that just by chance you happen to pick something like this one atom in the universe Like if you see these things and it happened something is wrong. There is no way things will happen down here So that's and it's and if this is where this is just the fun part But the interesting thing is that even though these fairly low energy differences you start to have fairly quick changes There is another way of seeing that we can study these torsional degrees of freedom and this is why torsional degrees of freedoms are so important Here we had differences that corresponded to one or a couple of k cows, right as you mentioned and If something is roughly one k cal that is one divided by 0.6. That's 1.5 or so So normally I've no I don't have any exact y scale, but I say 300 Kelvin when kt is 0.6. I Might be exploring if this is some sort of torsion potential I might just be exploring the lowest part in the torsion potential here If I now increase my temperature to 900 Kelvin then kt would be three times 0.6 k cal Suddenly I changed the cut of here, right because suddenly it's gonna be my energies divided by 1.8 Which is three times higher So the energies I can then explore will go up roughly a factor of three and that so at 900 Kelvin These two peaks that were significant barriers to me at 300 Kelvin at 900 Kelvin I don't really see them anymore suddenly I explore this entire part of the computational landscape or sorry the conformational landscape So that the key thing they're really small energies We don't care about because they're just bumps the really large energies We can never ever go over but this is why the torsions are so important They're just in the right ballpark that they are very sensitive to the temperature We can occasionally go over them, but occasionally not And I think yes, I have a small move with that So this is just an example of a molecule at 300 Kelvin roughly. I it's let's use poly alanine. I think Just an example of how they move so here you certainly see some rotations here that happen pretty frequently This is the this is a simple simulation. We actually have water around it, but I'm not showing that So what you do see is that there are some very fast motions here the angle vibrations they move by 5 or 10 degrees There are some rotations around in particular the seeds three groups or so you will see them rotate That's because they have a very very low rotational barrier. It's easy to go over You have other groups in particular the peptide bond somewhere. Yes, you will never ever see that peptide bond rotate because it's too higher barrier and Most of the other large bonds. I don't think you will see rotating otherwise either So it's just an example in terms of it. It's really important for you to start thinking in terms of energy and what energies will happen or not So that was the energy part of it the other part it has to do with entropy, right? And I said that we're going to talk a little bit about solubility This is oil in water. Why does oil in water look this way? Or rather the fact that it looks that way we can leave aside. Why is this bad? You probably know that the solubility of oil in water is bad, right? So you can we can formulate this with equations too in a second or I can leave that in the book but remember these pictures of desktops and The number of states I showed you yesterday So because of what we need to do with what this forces the water to do and everything What the oil is basically forcing us to go from there to there So the oil rather than having these water all these water molecules that can have lots of degrees of freedom The oil is forcing the water to move to a very much more ordered states at least locally around it and That means that the entropy goes down which will cost us free energy And that's literally why it's bad. So oil is forcing the water to Well, I think what you what I showed you in the previous I were a couple of millimeter large droplets But what this happens around every single oil molecule essentially and the only way to counter some of that is to at least make these Molecules as large as possible because when the molecules are as large as possible You have the smallest area possible, right? And the smaller the area is the fewer waters you have on the surface that will have to orient themselves Areas key were there. I'll come back to that in a second. So that brings us to the next part. Let's look at these hydrogen bonds two waters in vacuum first and Then we can say that these are actually infinitely far apart from each other There are all these jokes that physicists like to do that to a physicist infinity is typically around five or so And that frequently comes down to summing series or something simple approximations that If you're done first term is good second terms help you a bit three term Then you start to have a really good and by the fact by the point you read four or five That's it's enough. You're just talking about principles anyway So to a fit I'm a physicist serious my point to first approximation. These are infinitely far apart from each other But if they were to be in water, they could form a hydrogen bond These just two waters or two individual waters can only form one hydrogen bond between them because The other things are point in other directions. So they would need other neighbors What will happen here if you move from back you to water rather if you think of forming a hydrogen bonds Well, we're certainly gonna gain some energy here and the energy of this bond is good And by good that means that the change in energy from here to here is smaller than zero delta e for the hydrogen bond You will also lose some entropy Because these two molecules are completely free relative to each other, right? And you can think of this I'm gonna argue that you lose an entropy of one freely rotating water because each water will now be tied in A perfect ice crystal you would have two hydrogen bonds per water But this my water here is only having one hydrogen bonds I'm only losing half my freedom, but this water is also losing half its freedom So that's two waters that each lose half their freedom is two multiplied by 0.5, which surprisingly is one So the loss here is the entropy and the change it and should be delta s here is Well the entropy here is larger than it is here Sorry, Delta s should be smaller than zero there if we include the sign my bad. So what happens? For water is the change in energy and this is actually the reason I screwed up the sign I'll tell you So it's the change in energy smaller than T multiplied by Delta s or is the change in entropy Smaller than the change in energy. I'll go back to the previous slide Two waters, I remember this and we have this a couple years. So what happens here? There are two good parts We gain energy going here, but we also lose entropy And the problem is that both those things we kind of act in the same direction on the free energy, right? So that it's good to gain energy, but it's bad that we're going down in a more ordered entropic state And then there are only two options either the gain in energy is better than the change in the entropy part So this goes down more than that term or this term the entropy part goes down more than the energy. I Know this is a bit abstract. It's deliberate. So what do you think don't motivate? Think about this 10 seconds and then I'm gonna ask you to raise the hand where do you believe a or b? There is no credit for getting this right. Actually, it's more fun if a bunch of we get is wrong So who thinks a and who thinks B? So I'm gonna say in this case. So this is which is true when hydrogen bonds form So in general, of course, you could be general thoughts. It could be anything depend in general We didn't say what the temperature is But if we are under conditions when this hydrogen bond wants to form which is true I'm not gonna say talk to the person next to you and try to convince them that you're right And this is why it's much more fun if you have different views Do you want a clue? Because I deliver I deliberately tried to set you up here So and I force you to think what is an entropy and what is an energy you need to compare these What determines what reactions happen? Think about that and see if you can get the answer. So what is what is the sign of Delta F if a hydrogen bond forms? Okay, so this is smaller than zero so Delta E Minus T Delta S must be smaller than zero But then it's just a matter of add T Delta S on both sides. So Delta E Must be smaller than T Delta S Trust the equations Don't try to think don't try and again This was not exactly the world's most complicated chemical reaction or protein folding I brought up here, right? And you immediately got it wrong rather even I got kind of got the sign wrong That you get things wrong when you try to think about things instinctively where do you think I think I can understand entropy you can't Go to your equations decide this equation has to the sign here has to be smaller than zero if Delta E H minus T Delta S H is smaller than zero you just solve that equations it takes two seconds You have the answer in addition to that when you have that answer You're gonna know that you're right rather than have a hunch that I might be right at least 55 percent certain thing So the point is trust the equations don't try to hand wave. So now we're gonna use a more complicated example here If you forget about the small equations I written here first if you're gonna compare Hydrogen bond formation inside proteins both the book and I I I might have been doing this Previously, let's see. I can show you states. I might have been doing this previously No, I don't think I had a slide about this. It sounds instinctively really good If you can form hydrogen bonds, that must be nice, right? But what is the difference in free energy? How much do we make from forming a hydrogen bond and just as I said that solubility water or something can we can think of it as Either as energy or entropy and it turned out to be entropy in the end If you form a hydrogen bond, what does the hydrogen bond depend on we can do exactly the same thing here So this is a small protein amazingly beautiful drawn by yours truly And this is two side chains or anything in a protein really you can imagine than the D here Just means I have a hydrogen bond donor and a hydrogen bond acceptor so like an oxygen and a hydrogen and Then you can do that a vacuum where you form a hydrogen bond directly from the donor to the acceptor Or you can do the same thing in solvent But the complication in the solvent is that or you know what? Let's let's look at the upper part first and then we'll get to the solvent later So the upper part is really really easy you form one hydrogen bond delta The difference in energy is exactly the energy of that hydrogen bond And that is negative right because we're it's good to form a hydrogen bond in my world at least To first approximation I might have drawn this in a beautiful It doesn't really curl up that much But if these are just two side chains and they're already so close to each other at least locally for those sidechains It's not really going to be a gigantic difference in entropy I'm well aware that that's a in particular type donor here. That's probably not strictly true because I would be curling that up But if you bear with me for a second there Let's let's not start to think in terms of protein folding yet, but just one side chance suddenly participating in the hydrogen bond And that means to first approximation The change in free energy is really based on the energy here. We gain a little bit of energy from forming the hydrogen bond Incidentally what I just did now is one of these modeling things And I realized that you might cringe at it, but when he can't assume that they and the difference is entropy is zero Well, you know, I do a first-order approximation This might turn out to be complete crap If it is complete crap, I would eventually notice and there are some ways we can notice because I can measure this experimentally It turns out, but don't be so afraid to assume things You need to be aware that you're making assumptions and you need to put some mark that this is an assumption check later But if you don't know what something is assume and see what see what works out if you make this assumption It's a really fun way and in many cases you get amazingly simple results The problem if you have the same thing in water because proteins don't exist in vacuum in water These donors and acceptors will not just sit out and not have any neighbors If there is a donor on the surface of a protein You can pretty much bet that that's interacting with a hydrogen on a water and an acceptor is going to be interacting with the oxygen on a different water, so donor It depends which way as I forget to think of this You might be right. I might have drawn you think correctly there my bad Search me for all the slides It depends whether you see that the hydrogen is donating it or whether you have the electrons donating it and the hydrogen Bond really has to do with three components, but you're right the standard definition is probably to have the hydrogen as a donor I'll update the slide The point is that you have one hydrogen bond here blue one hydrogen bond there Once you form this hydrogen bond inside the protein, it's blue there and it's blue there So the net difference in the number of hydrogen bonds is a zip not a nil The energy difference here is zero and this is related the same thing when talked about entropy and salvation, right? That it's easy to think it's electrostatics. It means that Electrostatics has to change that's an energy changing the problem is that the difference in energy here too is zero just this is for any type of salvation But what does happen is that remember that I said two waters They each lose the entropy corresponding roughly to half the entropy of a water So there is a change in entropy here So the total difference in free energy here is really due entirely to the entropy part That these two waters are now less free So this is a complicated right to even here and this comes this will come back it turns out electrostatics in complicated molecules it is electrostatics, but it manifests itself in terms of entropy and Pretty much all electrostatics in water as I'm going to see in the after the break is actually manifest in entropy Really strange So we can look in if we go away from this hydrogen bonds for a second and look at salvation This is related what I spoke about before if we actually want to measure this in the lab and forget about the equations in a second We do know that some molecules like water better while other molecules like for instance octanol or cyclohexane some sort of non-polar solvent better It's a real relatively easy at least to measure these two fractions And that's what you call the so-called partition coefficients What fraction of the molecules be moving water versus what fraction of the molecule will be in the non-polar solvents? Virtually every single solubility you measure has been measured as a partition coefficient So why do you measure partition coefficients? Well because that's about counting. It's easy to count Well, it's easy to count molecules you can see them But even things like Absorbance or anything that the second you're measuring concentrations You're essentially counting right and it's something that's easy to see if something is absorbing twice as much the concentration should be twice as high So if we just look at these hydrocarbons and this is the part where you will might have been working with the Boltzner distribution from experiments The relative fraction of something that is in any state is proportional to the exponential raised to the power of minus G or delta G divided by RT and now I use R and I also avoid using the delta here Because no matter what units you have you always have to choose a zero point and the second use choose your zero point That's the same thing as talking about the delta and then we just solve that for G Rather if you tag X1 and X2 two different states We can solve that for the difference between them and that corresponds to that the difference the relative Concentration of these two is proportional to the exponent of minus the difference divided by RT And if we solve for that difference we get the difference in free energy is minus RT the logarithm of the partition coefficients here So what you do anytime you do this in the lab, you're just inverting the Boltzmann distribution These are really easy to measure in the lab and that's how you get free and it is from most lab results Well, for instance when we measuring the fraction to which membrane protein if we want to measure how expensive it is to insert a Membrane protein in the membrane I might talk about that later You actually do this as a function of the sequence I measure how many sequences end up in the water versus how many sequences end up in the membrane and Then we take the logarithm that multiplied by RT And I get what is the free energy of inserting something in a membrane biologically This was to use all the time. There's much more physics than you think. So let's use a simple example here I think this is cyclohexane although it's really dark just from the number of carbons that cyclohexane It's nice. It's just ethane groups. It's a very simple molecule to model It has nice liquid properties at room temperature and it's not poisonous or anything There are lots of other molecules you can use to but cyclohexane is kind of the poster child for non-polar solvents But you can also think of taking one of those cyclohexane molecules and taking it from its neighbors and putting it in water to measure How soluble it is a water or you could even think of taking one cyclohexane molecule then comparing it How expensive it is to have it in vacuum versus how expensive it is to have it in water But the vacuum measurements are usually difficult to do in the lab. So we'll skip those for a second The concentration of cyclohexane in cyclohexane here is roughly nine moles per liter or nine molars And that's simply based on the molecular weight of the molecule But if you take one of these molecules the number of molecules that would actually like to be in water In relation to that number is something like 10 to the minus 4 So base you're never going to see a cyclohexane molecule in water. It's extremely unsolvable And that actually you can calculate what that means and that corresponds to a free energy difference of roughly 6.7 kcals per mole This is not an extremely large number. You're going to see way larger numbers here And that's why I had that slide about understanding the power of the exponential the second You are at a point a couple of kcals you can pretty much forget about things it starts to be relevant and This expanse of it we're going to take a break in a couple of minutes here Let me give me five or six more slides, but we can think of this in a more general way and hydrocarbons And this is the last things I'm going to speak about free energy today I think so we know that the cost of well the difference in free energy Actually, normally I frequently say cost or gain or something. It's a bit dangerous because if you say cost But then suddenly I'm studying the reverse process. It's really easy to pick up the signs I bet that I have a dozen sign difference in my slides and The reason why these sign are for instance when we talked about the entropy of water It's easy to talk about that the entropy the disorder of something then can move Then the disorder of something that can't move and because we all said since entropy is absolute I it's it sounds strange to talk about a negative entropy a change in entropy can be negative But the entropy itself should always be positive and when you start mixing these things up you frequently have sign errors Trust your equations. So rather than talking about gain or cost It's much better say the delta G for this process is Positive 6.7 and then we can interpret that that's bad. It's gonna cost us something. So it is a cost So moving one hydrocarbon from a liquid hydrocarbon phase into an aqueous phase water is bad In this case, it's cyclohexan but it doesn't really matter much which hydrocarbon it is This is not a spontaneous process. It will not happen That also means that it costs free energy to solve a hexane or cyclohexane here in liquid water And that's you're not gonna get a noble price for so the interest in question is why and The why answer here is of course our simple equation here We know what G is the question is this H or is it TS and now I see I started to use 8 here Although the pressure part is irrelevant So there are only Three parts to this really We can look at one molecule in vacuum that does not interact with anything We can put this among a bunch of peer molecules here Or we can think of taking that molecule and moving it directly into water and the difference between those two arrows Is of course the difference moving it from taking one molecule from this phase and moving it to that phase Lots of numbers here. Let's start with something very simple here this one So if you take one cyclohexane molecule and move that to lots of other cyclohexane molecules, I Argue that that change in enthalpy is negative Do you believe me does that sound reasonable and why is that reasonable and all these numbers are in k-cals per mole by the way But it became too plotty if I added the units everywhere Yes, so this space fits non polar dispersion interactions It likes we have advantages dispersion interactions as room temperature Because this is a liquid right and the fact that it is a liquid means that it's good interactions So Delta H here is negative On the other hand there is no question about it that this molecule is freer to move than it is down here So the T Delta S term is also going to be negative here this is a more ordered state and Exactly how large these are in relative relation to each other Of course, we can't say anything about that, but that the H here is negative and T Delta S is negative The difference between this is actually that Delta G is negative here And that's actually something that we can say too because Delta G would describe if something happens spontaneously and you can see in the lab This cyclohexane molecule is a liquid under normal room temperature conditions had it been a gas This would have been better, but it is a liquid So it's actually also reasonable that the Delta G here is negative The surprising part here is that if you take that very same molecule and move it into water The difference in Delta H is exactly the same Why on earth is that? Yeah, but we're talking about physics we're talking about first order approximate I don't care about the second or third digit. I don't even care about the second digit I care about the roughly the first digit first order approximations Yeah, but we're talking about approximations here the key thing what are the electrostatic interactions of this molecule? No, to first approach it well, there are some small partial charges, right? But if these partial charges zero point zero one and these partial charges zero point eight forget There are no electrostatic interactions in that molecule to first approximation So to first approximation all the interactions here will just be dispersions and this is actually more true It probably fits to two number two digits at least if you measure this experimentally So the first approximation it's actually pretty reasonable that Delta H here is roughly the same But we get a gigantic Drop in entropy here that drop in entropy doesn't come from this molecule, of course But that drop in entropy comes from all the waters that now needs to organize around it And if it's really the gigantic drop in entropy here that Cori is the reason why Delta G here is positive And then the difference here This just ends up being the the effect here is just the difference between those two The other thing you might see that I've made it very clear there My arrows only go in one direction because the way the second I specified how my arrows go It's also very easy to say what is Delta H for that process What would happen if I reverse the direction of each arrow here? I would get the opposite side on every single term. This is a point Why do I choose this direction of the arrows? No, because it's completely arbitrary You just need to know what direction of the arrow you're picking and then run with it Don't change the more you change the more you will screw up I screw up all the time too, but pick something stick to it and then trust your equations Don't try the second you try to understand instinctively Well, no, but then the entropy here should be negative The only problem that your arrow was in the other direction and I didn't think about the arrow I thought about the other process think about the arrow think about the term then we'll interpret it later So trust your equations rather than try to make your equations say what you think they should say So the key thing is that the Delta G's at the end describe what actually happens So how do we but I just threw this out. How do we know that this is even correct in the first place? So this is when I'm gonna use this thermodynamic Definition of temperature. This is more or less a repetition of the old slide remember that said that I use this small difference in free energy Delta F and I expanded that into all those different terms and Then I argued that that's made it possible for me to have this thermodynamic definition of temperature Let's use exactly the same type of expansion again in a slightly different way. So here I try to solve for T So if I take this small difference in enthalpy, sorry in an in free energy That will be the differential of the energy minus TDS minus SDT. So the product formula for differential again And at equilibrium. We also had the thing that we had in the last slide right the E minus TDS is zero So the E minus TDS is zero So at constant volume if we allow the temperature to change because during this process the temperature might change The entropy is the derivative of the free energy with respect to temperature This is much easier than you think So if I want to see how much the entropy changes, I just measure my free energy at multiple temperatures Because free literally the entropy is there's a negative sign there too, but it doesn't matter So it's constant pressure. This is really the derivative minus the derivative of the Gibbs free energy with respect to temperature So the way to do here is that Delta G here as I said it is a Delta G But Delta G is a function of temperature So if I just take this in a simple plot and then I see what the derivative is This is how I can solve for Delta S So all these things are actually measured It's not just based on hunches of physics and everything so to first approximation. It's It's actually true to within roughly 1% that the change in enthalpy is the same really strange And that's another one of these examples that we can measure things in the lab and use our equations then to be able to extract small components that individually would be very hard to extract Pure well, it's virtually impossible to measure entropy directly for instance But I can estimate what the entropy is based on a in this case a temperature dependence of the free energy. Oh, sorry This is the difference in specific heat or the heat capacity It doesn't really matter now the book goes through a bunch of equations. Yeah, Alex is a physicist after all You can actually calculate exactly how the difference in heat capacity is that is how much the How much it how much energy How much the relationship between Delta H and 10 the temperature as you're adding heat It doesn't really help our continued discussions and that's why I skipped it Do not need to go through it. It's not a complicated derivation It's roughly the like the one we did for the thermodynamic temperature, but it's He spends I think two pages just playing around with these equations and see a bunch of things You can get from it, but the point is a very basic fundamental property here that you can't calculate And that is really comes back to this hydrophobic effect This is likely a very good place to stop I will have one or two more slides about temperature dependency after the break So I think the first order you understand this now right that the hydrophobic effect is caused by the ordering of all this water in This clatter it like structures around a solute Which surprisingly because the non-bonded interactions here are to first approximation the same That means that there is not an ill difference in enthalpy But there is a gigantic drop in entropy and that's what's going to cost you the free energy And that brings us the question we're gonna find a good answer after the break What happens when I change temperature here? You've got feeling for solubility of things right is that when you increase the temperature the solubility does what? Yes, so what do you think is going to happen here? food for thought over coffee The reason I bring it up is of course, that's not gonna be what happens And it actually makes a whole lot of sense, but we have to trust their equations a little bit It's 10 30 now should we take our 20 minutes or so and meet here at 10 to 11 All right, so we stopped before the break when I talked about the hydrophobic effect And then when I kind of hinted that this is gonna be strange the opposite solubility effect of normal things This is really easy to measure experimentally because if we just measure things as a function of temperature We can measure how much the enthalpy is We can measure how much the temperature Multiplied by Delta S is again just by measuring Delta D as a function of temperature and it turns out that The funny thing here is that both the enthalpy and the entropy term. They're strongly temperature dependent But they're depend temperature dependent in the same way And because they are temperature dependent in the same way and Delta G is the difference between these two Delta G typically has a relatively weak temperature dependence So there are lots of things happening under the hood, but because they cancel out. We don't see such a remarkably strong effect But the thing that does happen the sign of Delta G is positive. It's always positive and when the temperature goes up Delta G becomes more positive So it actually could the higher the temperature is the worse it is to try to solve things a hydrocarbon in water So here it's important to have the sign of our Processes right so in this case we're talking about transferring a pentadicane or cyclohexane, whatever it is from Pentane, I think the book uses from pentane to water And the reason why this keeps going up is that what happens when you increase temperature? So what do you do to the hydrogen bonds as we're increasing the temperature? We break some of them right and that's good or bad Bad what happens to entropy when we increase temperature? Yes, and we see that but here's the key we increased the entropy both in the pentane, right and In the water and in the pentane in water, and we're not really increasing entropy anymore of the pentane when it's in water Rather than the pentane in pentane So it's just that the entropy keeps going up Where we're not really gaining a whole lot of there is no particular reason where we gaining more entropy by having the pentane in water So that's and that's why we just sadly we lose more from the hydrogen bonds than what we gain from the entropy And that's why it's worse and worse and worse until We have some sort of maximum here and that maximum happens when this temperature the entropy part crosses zero The book spends time going through this in detail I won't bother because it's not critical here But it's if you were aware of the curious fact that the maximum here always happens when this term goes through zero I will come back to that thought in a second, and I'm gonna turn it upside down for a reason Yeah So this is this is the entire t-delta s term and of course this means that as I mentioned There is no huge difference in entropy right, but because I'm multiplying this by temperature. I get a curve That's pretty much linear So it's the t there the t is really the temperature dependence in this term But it's the entire term that enters in the free energy remember that I said before that it was kind of curious that The hydrophobic effect was kind of proportional to the surface area and this holds on a microscopic scale too So this is a table that I took from the Finkelstein book. So here you have a bunch of molecules. We have ethane Why on earth are we studying ethane? This is not a course on Organic chemistry. Well, it turns out this this CH3 group If you take half an ethane, that's one CH3 group and that pretty much correspond to the side chain of alanine, right? You can take benzene C686 that pretty much correspond to the side chain of phenylalanine So for every amino acid, there is some sort of molecule that corresponds to the amino acid analog So that is the side chain pretty much without the backbone And toluene also is fairly similar to phenylalanine including the entire the CH2 group there We can calculate these transfers from benzene to water for instance from carbon tetrachloride to water Let's forget about all the details here, but the point there is a delta G in this table It's positive for all of them, but it's more positive for some than others and If you do the math here, it's going to turn out that this delta G is pretty much exactly proportional to the surface area of these molecules and The reason for that is that you can pretty much just trace out the solvent accessible surface area You're typically traces with some sort of probe that has a radius in the bulk of 1.4 angstrom Why do we use 1.4 angstrom? So that's the first approximation of a water molecule a Water isn't spherical But actually it kind of is So you've seen all these small triangles of water right where you have an oxygen and two hydrogens That's how a water looks from an electrostatic point of view But from a non-polar interaction if you start it's kind of a fun exercise You can look up Lenard-Jones parameters for water Both the ones you would use in a simulation and the ones you would use for real The way a water looks in terms of non-polar interactions is that you have a gigantic bowl That's the oxygen and then you have two small Mickey Mouse ears that are the hydrogens These hydrogens are so small that in simulations you frequently even ignore their there So so they're essentially from a Lenard-Jones point of view dispersion and repulsion It's enough to have the oxygen you need the charges on the hydrogens But there because there are so few electrons on the hydrogens. They hardly have any dispersion interactions And that means that it's a surprisingly when you're talking about non-polar interactions It's a surprisingly good Approximation to just have a single ball with a radius of 1.4 angstrom that we roll over the surface and this red surface would then be The solvent accessible surface Which is a pretty good approximation of energy So is this true for all amino acids? Or it's just a couple of amino acids and molecules that I showed you No So what is that these that I once that I showed you had in common they had a hydrophobic part, right? It's more hydrophobic molecules So this is true for if the entire surface is hydrophobic. This works well We can plot this also stolen from the book. It's pretty amazing how well this is glycine alanine valine leucine phenylalanine It's a pretty darn good proportionality here delta G transfer minus sine 2 and then the accessible surface area Then you have all these other molecules that don't really fall on this line But the reason that these molecules have some polar atoms and if you subtract roughly 50 square angstrom per polar atom Again to first approximation how large a polar atom might be then these two would also fall in the same line So the proportionality here the hydrophobic effect is almost exactly proportional to the non-polar surface area of an amino acid This works good enough and high name my colleague at the way they're doing this and this is research product I shouldn't take too many deals because this is not published yet They've gone through a huge number of eaten acids when you're looking in the insertion and efficiencies and how much How much energy I the cost your gains when you insert different peptides in membranes? this holds Beautiful, it's all simple physical chemistry that we can explain membrane property information and other things You can use non-natural amino acids it works for those two But it gets slightly more complicated. Here. We just look that remember this triangle. I showed you that you can I Could take one molecule and think of inserting this molecule either in a pure Hydrocarbon that's consistent just of his peers or inserting it in water So in that direction or moving between these tracks three structure We can take this triangle and move it in a kind of different direction So here we have something that's either gas or in this case probably more of a liquid so these They are condensed these hydro these molecules hydrophobic molecules. They interact, but it's not really it's not a perfect interaction yet What if we introduce another phase rather than just having liquid we can introduce a solid phase to like ice But this is not ice in the terms of water. This would be like frozen hexane most Apart from the noble gases Where there are a couple of them that you would need to get almost except the entire way to absolute zero Even most of these non-polar Compounds will eventually form liquids, but it has to be fairly low temperature. I'm sorry. Not as liquids, but crystals so What I already had in the previous slide that they would take one of these molecules and move it from the hydrocarbon Liquid over to water the difference in enthalpy was pretty much zero But there was a fairly strong effect with entropy and that is the reason why this was costly, right? But we can also take this molecule or rather the entire collection of molecules and let's form a crystal so that it is highly ordered And it actually turns out that does not cost you a whole lot in Entropy we do lose a little bit of entropy We gain a little bit of enthalpy because they're already interaction and these interactions become slightly better But not by a whole lot But that actually means that the delta G for this process is almost exactly zero So that it's fairly easy for these structures to rather form sort of crystal themselves or so So surprising that this appears to be a fairly easy phase transition and This is something that's gonna happen in proteins eventually There is a very delicate balance between on the one hand energy and on the one hand entropy and Depending on temperature, of course, sometimes you will sometimes this will be better and other times this will be better But it's gonna be fairly easy to move between these two That is likely the reason we have protein folding I wouldn't be saying that unless it was true, but we don't know that yet for your point of view This is something we will explore deeper after these to break So this sort of crystal this type of phase transition is actually more similar to the phase transition We're gonna study for proteins The other thing that would happen with proteins here I've oversimplified things and only looked at hydrophobic things Proteins are more complicated than that a protein that only consists of say phenylalanine is not gonna be a very interesting protein So a real protein will consist both of polar and a polar amino acids So what's the protein gonna do a Protein would like to turn its polar amino acids towards the water, right? Why we would like to form some structure like this on the inside of the protein because then we would gain from this But we would also gain from the ones that can form a track so it would look something like this so all the green stuff would be on the outside water liking while the red stuff here should have some sort of internal of the protein and Again, this is a crossover. Simplication is just an example of a protein. We can't pack it perfectly all the time I've taken this from a real protein, but I've erased all the names because we don't really care about the amino acids Here we don't care about the details. It's just the present. It's good to have water soluble things on the outside It's good to take the hydrophobic things on the inside This is not obvious So we think of this in terms of free energy. What does this do? It does two things Delta F. What are the two components? Yes, and what is good here? What do you gain? Yes, you probably you probably the enthalpy is probably slightly lower because these can interact better but remember what I said that There is not a gigantic difference in enthalpy if a hydrocarbon is interacting with itself or water So to first approximation These might be able to form some hydrogen bonds with water that you might gain a little bit of enthalpy on But every hydrogen bond these green residues are forming with water is a hydrogen bond. They're stealing from another water So if we are going to be extreme physicists and say to first approximation here It's all entropy And some of this entropy is bad the entropy that we're sorting things now We're putting all the red stuff on the inside and all the green stuff on the outside That's an entropic term. That is where the entropy becomes goes down. It becomes more ordered. That's gonna be bad for us On the other hand by having these residues that can participate in hydrogen bonds on the outside That's gonna be good because that's allows us to have favorable salvation So it turns out when if this is gonna form it's an entirely entropic process But some terms of the entropy is good and some terms are bad and Then you can probably start to realize it's gonna be a pain to figure out. What is what here? And that's essentially what protein folding is going to be about If you look at three potential difference here, right? That's in this case You can pack hydrophobic things you can certainly have these blue residues interacting with each other if they are hydrophilic That's gonna be good You will even gain some enthalpy there possibly but in this case We're exposing all the hydrophobic residues on the outside and that would be bad You could also imagine things something where we pack all the hydrophobic residues on the inside That would likely be good and we're exposing the hydrophilic on the outside and The other alternative is to somehow try to form a helico structure or something with it And in this case we might have some really favorable interactions between all these parts Now a helix is a very regular structure So we're gonna lose some entropy here and the question is do we gain enough enthalpy to make this favorable anyway? Well, that depends because it's gonna depend what side chance we have here if this means that we end up with lots of side Chains that are hydrophobic pointing out. It's not gonna work So the unifying factor with all these things is gonna be a balance And that's of course why depending on what amino acids you have some of them will form alpha helices Some of them will form beta sheets and it's gonna turn out some of them won't even fall at all Because you don't happen to have a good balance and that one is gonna be happier in the entire extended states There is a result that I'm gonna bring up at the very end of the course But I will kind of wet your appetite by doing this here all the time Is it important that a protein is chain? Of course, I wouldn't bring it up on it. But why is it important that a protein is a chain? But but now you're thinking with your intuition Don't do it and think of equations If you cut up all these and again, this is a gedankin experiment I will ever they are a chain But what if we could cut up all these interactions so that you had a bunch of what is 15 residues or so that were free What would be the best possible way for these residues to organize if they could get infinitely far apart? What would happen to the entropy? There would be no limit to the entropy right that the further apart you are the more space you have available So that the further apart you are the better the free-ended would be and they would go and explore the universe So the point is that if these were not stitched together The completely unfolded state will always be better So by sticking them together nature has forced their way now You can't get infinitely far apart and then you have to explore the other parts would include the folded states So it actually turns out that it's not just a coincidence that these happen to be a chain Protein folding would not be possible unless they were a chain and Don't ask me how nature figured that out So if we think about proteins that Folding is previously we talked about when I took a hydrocarbon and moved one hydrocarbon from a purely hydrocarbon phase or a liquid phase to the water phase Protein folding is pretty much the exact opposite, right? We're taking an amino acid or something. That's pretty much free in water And we're gonna move that Technique it is not to a liquid hydrocarbon phase, but you can think of that as a protein interior or something. That's no longer exposed to water So that's pretty much the exact opposite process So now we should read you ride this and go through and do everything with the opposite sign Or we can just take that plot and flip it upside down So now the scale here goes well forget about the scale here because it's it will still be that up here is unfavorable and down is favorable and If you're good at reading upside down, this is the entropic component and this is the entropic component and this is Delta G So this turns out to this actually remarkably good approximation what happens at protein folding So what you're gonna happen at protein folding is that you have some sort of temperatures ban here Where it's good to fold and if you go to higher temperatures is not gonna fall Then if you go to lower temperatures is actually not gonna fall either. So this is a roughly a second order Sorry a harmonic plot here For proteins to it's usually gonna be the case that the lower the temperature is there is an entropic component here Due to the T and the T Delta S term, but you also have an enthalpy component That is dependent on temperature. So protein folding is pretty much exactly the same thing as these hydrocarbons We're solving in water, but all the arrows point in the other direction And that's why we spend so much time talking about these solubility and everything. It's the same process You can if you want to go into deep details, these are some plots I found online Here too you can either you can think of that hydrocarbons that just solvating water or you can think of folding a protein Same thing here. I've just flipped the plots, but here it becomes a little bit too difficult to read this upside down But you can separate these things in polar interactions non-polar interactions Red one here is the enthalpy and blue is the entropy And the sum of all these is the black bar here So the non-polar interactions for instance the higher the temperature is if we're solving a hydrogen hydrocarbon in water the enthalpy goes up the Entropy part Goes well depending on the temperature We have roughly the same effect Delta Just so I show it's good Polar and non it's probably easier. Let's start with polar and non-polar enthalpy first red here the polar ones as The temperature goes up this because less and less and less favorable because at low temperature This will be able to form stronger hydrogen bonds salt bridges and everything so there it's As we're you know what Go through this yourself. I'm just gonna screw things up. There are too many curves here Give me till tomorrow and I will explain all the details because I'm so I'm so used to talking about this case And that's why just grew up things when I talk about that case go through these thoughts yourself rather than me Picking your mind the important results here is really the black curve. It is almost constant And in particular in the proteins, which is the part I know better So let's focus on this one for proteins. You see exactly this pair that that's very close to zero There is a small regime here where it's actually below zero if the temperature goes up too much it becomes positive So this is if you take an egg and boil it that's bad. It denaturates the proteins will no longer want to form at high temperature But the prediction you're making from all this curves attack if you just keep dropping the temperature a protein will eventually unfold because it's too cold Is that reasonable? It can be reasonable, right? So that's of course some of the results you get from these equations might just be physical things that you can't interpret But do you think it's reasonable? Do you ever see it and again just to explain things in this plot? It makes sense, right for a non-polar interactions the Entropy part shoots up as the temperature up Polar You have the enthalpy part shoot up instead and then for non-polar and the other parts they have a Act in the opposite direction and the net effect is pretty much plus minus zero So if you've never seen something in the lab It's likely reasonable to expect that this is an artifact of the physical equations that there's something that's not right in our model here, right? Or could there be some other reason why you're not seeing this? So in a way, I'm happy that you're focusing your equation that that's really good But this is actually much simpler. So here's room temperature say 40 degrees centigrade in your body's or something 50 60 70 80 or something And then things starts to go bad here in your proteins will unfold Let's go in the other direction here. So we're at 40 30 20 zero minus 10 minus 20 So why don't we see this happening around minus 20 or so? Well, there was one temperature that I went through there. What happens as zero degrees centigrade water freezes and Water has frozen your protein is stuck in an ice cube And when your water is stuck in an ice cube, sorry when your protein is stuck in an ice cube It's not really going to move a whole lot anymore So that technically it will unfold but because it's stuck inside a crystal is not really gonna unfold This is called low temperature unfolding and people have actually been able to see if the last two decades or so So that there are plenty of ways to get for instance water to stay liquid. You can add a lot of salt to it The only problem is that that's a pretty darn good way to denaturate proteins, too So then you're gonna see it unfolding because of the salt not because of the temperature But there are a couple of cases where people have been to be able to create system that With a liquid that in that case not pure water and actually see how the protein unfolds just because of the temperature It's very rare, but the point is that this is completely correct. There's a very narrow span in temperature where proteins are stable Sure, so but of course all these curves will depend on exactly what I mean as this right you have right so that We will shoot this I will move this either to the left or to the right in some cases you might have amino acids The balance here is so small See if you take one of the favorable curves here and move that up just a little bit Suddenly this is going to be positive no matter what the temperature is That's going to lead to a protein that will never fold because it's never negative if it's never negative It will never spontaneously fold You can imagine another protein where you move all these curves down a bit then it's going to be very advantageous to Fold that's likely going to be a very stable and fast-folding protein. So of course the devil is in the detail here This is really sensitive to the amino acid composition now. I'm just talking about general properties But since I said that you can move this to the left and right. Do you think nature ever does that? I think I'm going to talk about that. It's a later lecture. I need to do it through my slides Fix a couple of old typos too while I'm at it What if you're a fish living in the Antarctic We'll be pretty bad if all your proteins unfolded, right? So for an animal living at very cold conditions nature You can definitely see that those proteins have evolved so that it's more advantageous to be stable at lower temperatures And there are even a couple of cases where you develop specific shock proteins called shock proteins That's essentially protect other proteins when it's really cold There are other organisms a bacteria for instance living in the inside of the volcanoes or so And then this has evolved in the opposite direction that those have been evolved to be very very stable at high temperatures So most things like that. They also come with drawbacks If proteins are very stable at some point you're going to need to degrade a protein in your body and That protein was that's kind of like big building the bill if these buildings if they were built like Fort Knox That's really great when you're constructing them or if you run a bank in it But at some point you will need to deconstruct that building and then you're going to pay for it Because it becomes more expensive to deconstruct and it's the same thing with your bodies It's good to have a stable protein when you're going to use it But at some point you need to degrade it and then you're going to need to spend all that energy to degrade the protein So the nature doesn't want your body doesn't want your proteins to be too stable They should be stable enough, but not more and that's why you only see this in organisms that actually need it So the Delta G of protein folding To first approximation, I would say that's 90% hydrophobic effects part of this could be right and the hydrophobic effect in turn is what type of effect Entropy so protein folding is about entropy entropy and entropy if you're going to prioritize it and Then it's a bit of polishing which is pretty much fun of us packing at the end This yes once you've almost folded your protein It's good to get these sites is to really look in you might have been looking at a little bit of that And then there's a protein structure prediction and see if you can form a disulfide bridge Or you can form a bond that's good all other things equal. It's good to have good things But this is not what determines it This might really lock the structure in at the end and we're going to come back to this But if there's one take home message to have from this week is that protein folding is about entropy Which is a really non-obvious result everything we looked at hydrogen bonds these structures If you just look at them, they're full of interactions between oxygen and hydrogens It seems that it's obvious that I spent them lecture in the half talking about all your interactions and energies, right? It's so natural to think that this is about energies, but it's about entropy and That's of course what we spend all the time with the Boltzmann distribution and anything we're gonna spend a lot of time in entropy But that leads us to another thing I'm gonna go back a little bit to interactions here again and electrostatics because the book does it and electrostatics is important So even though it manifests itself as entropy at the end the original interactions are frequently due to electrostatics because these are the strongest ones And this leads us a very old problem that a bunch of very famous scientists have talked thought about the last 50 years or so so we know that In general the potential electrostatic potential between two charges interaction is the product of the charges Divided we're in Europe. So you should you really use 4 pi epsilon R The book I well I like to use 4 pi epsilon R the book occasionally just uses epsilon This is the beauty about physics. We don't care Never say that's just about it's a constant. We would fold into the equations here We're not gonna talk about specific numbers. So we'll worry about that if we need it right now We care about relative strengths and distance dependencies But the problem what is this epsilon? Well, there's actually a typo in the book there that I fixed If you take a small charge and put that In water water has a dielectric coefficients. That is roughly 80 is that lower high It's pretty much the highest dielectric coefficient of any material you can ever imagine. It's insanely high Something like a membrane on the inside or so or in this case a protein all the Finkelstein happens to have water in that box That might have an epsilon around three or so for now you're gonna need to take my word for it What is then? Epsilon of vacuum would be one so this is actually much closer to vacuum than water and Then you can calculate. What is the energy if you're moving this from in this case? Well, what is the energy of having it in the protein compared to having it in water? And then it would just be a matter of solving that equation I'm not gonna bother about numbers here The point is that if you have a charge in vacuum or in a region with very Low dielectric coefficient the electrostatic interactions are gonna be very strong If epsilon was one it would be the same as vacuum and then there is you see you see all charges There is nothing screening you the charges only see each other So what this epsilon does is in the denominator epsilon scales down the strength of the interactions that screens the interactions So water things are like in this case at least a factor of 25 or so weaker and compared to what compared to vacuum Everything is 80 times weaker in water than it's in vacuum So that a charge is pretty two charges the interaction for one charge to another charge of water Suddenly becomes a factor 80 stronger in protein and this can be both good or bad Remember our friends the charged amino acids in water. They are quite happy even though the even if this even if a specific charge Here it does not interact with something. There are a number of hydrogen bonding partners and No matter what the electrostatic interactors between this one and everything else is going to be screened down by a factor of 80 There is always upward interact with If you now take these side chains and put them on the inside of a protein, there are two options Either you find an ideal salt bridge partners. You have a plus and the minus Matching up and then they would probably be happy Or you have an isolated charged amino acid on the inside of a protein How bad is that? We're talking about astronomically bad. We're talking about that regime of the Remember the slide I had about the exponential function of things that we're talking about eats my left shoe bad This does not happen if you have a model where you're predicting that you have an isolated charged amino acid Pointing is raised in the middle of the protein go back to the drawing table And I'm saying this because we have published models like that And not because they're bad. There are exceptions to this which is pretty fun But in general if you see a charge in a membrane or inside of a protein, there is something wrong Nature would not do this So what happens with these titratable amino acids it's important now I didn't call them charged amino acids because what usually happens is that They change the protonation state to either release a proton or take up a proton so that they become Uncharged and that's why I said depending on the pH and this really was called pk shift So if there's something if this amino acid is now suddenly in a surrounding Where there are no partners whatsoever rather than having the water they might actually prefer to take up or release a proton Now mind you that in turn costs energy. That's not free And it's certainly not something that this amino acid would like to do So what rather happens instead of paying this as a penalty in terms of electrostatic interaction? We typically pay this as a penalty that it costs something to titrate this amino acid So that you would still you will still not see you could have a man having a lysine Residubus it instead of having a nitrogen with three high digits at the top you would have an NH2 group That is just polar and not charged. You're not going to see that either in the inside of a protein This is cheaper, but it's This is roughly a factor too cheaper. So you I would I guess I would just have to eat half my left True I would prefer not to do that either so that the difference from the inside of a protein is that All these interactions become so my stores longer 40 K cal maybe instead of 1.5 or so that it would be in water If you take this 40 K cal and compare that to the Boltzmann distribution 40 divided by 0.6 an Exponential to the power of 50 The population you're gonna see in that state and we such a protein would always be unfolded because it will be better For it to expose that to water So there if there are couple and that's actually it might be a good idea to compare a couple of things here So what do you remember what the strength of the hydrogen bond was? five five what? No, five K cal per mole You might think that's a minor error, but it's just a factor of 10 to the power of 23 error There's a famous paper in I Think it's an either astronomy or astrophysics by some Ramachandran not not our Ramachandran in India and There is a formulation point because they're they're looking at particle and galactic physics and there's a correction to this paper that Equation four in this paper has an error with a factor of 10 to the power of 40, but it doesn't change any of the results And that's it's kind of that type of error. It's 10 to the power of 20 24 Katie then and It's a good idea to stick to the same units now or you're just causing yourself a royal amount of pain Yeah, yeah per mole. So five divided let's say six divided by zero point six factor 10 So with the choice of forming a hydrogen bond or not forming a hydrogen bond and that I actually was actually lying to you The other days, that's of course those five as you might have guessed now That's a roughly effective free energy rather than energy So you don't have to bother about the difference between enthalpy and entropy here So it's roughly a factor e to the power of 10 more likely that it will form the hydrogen bond that and that will not form it Hydrogen bonds are strong much stronger than you think So that's the thermal energy so in that case what how much what do you think that the stability of a protein is? because that in the previous slides I draw a couple of these Just figures with Delta G curves right those were pure examples There were absolutely and no y-scales or anything But in principle one could of course measure the stability of a protein even experimentally you don't need to know any physics for that How stable is the typical small protein and these again this varies over more than an order of magnitude, but ballpark and In the grand scheme of things is that roughly what you would expect or it's surprisingly larger surprisingly small or But if you think about these protein structures a long at so an entire protein is pretty much stabilized by the equivalent of 4 hydrogen bonds To me. This is a very small stabilization energy now. You're quite right that it's it's high compared to Katie in a way, right? But but here's the here's the trick of the balance it The better the stabilization energy is the easier. It's likely going to be to fold proteins But the second we need to degrade it. We need to pay back So nature seems to have optimized proteins to be stable enough They're so stable that they can fold and unfold as a function of temperature And if it takes a second then you've had a chance to try a couple of trillions or quadrillion combinations So it doesn't have to be 50 50 distribution between folded and unfolded But it's still I'm I'm kind of amazed that it's only a handful of hydrogen bonds stabilizing an entire protein Now of course as you say it does it will require some energy to break those So what is then epsilon inside the protein? I Said it was roughly three, but I don't really have any justification for that The book goes through in the book goes through this in a lot of details The way you typically do this in physics is that you use continuum electrostatics and you have areas with different epsilon You can create mirror charges and everything forget about that. It's not relevant for the course In hindsight there are going to be some things that we will bring up later that the book doesn't talk about in particular Things like modern simulations, but this is a part of the book that you can completely skip with a good conscience I so promise not to ask you about these reflections That's a really good question So well in one way that's a good answer. It's a wrong answer, but it's wrong But for good reasons and we actually use that occasionally. So why on earth do we use something in simulations? That's bad. I'll get back to that in a second This is a small charge out in water and rather than drawing all my waters here I think a water as a dipole you have a this would be the oxygen and that would be the two hydrogens so these dipoles are going to turn all the negative parts towards the charge and That of course means that these will screen the charge a bit So the main effect here is that you're going to order all the water around it So the main reason why you have this high dielectric coefficient Is really that water has a very strong dipoles and it will reorient the entire water molecule to screen the charge and water molecules are Large charges, so they're strong dipoles and they're small they're easy to rotate and that's why they're really really efficient at screening things here The protein on the other hand Well, there will be some small charges inside this and everything but you will obviously not have the entire protein is rotating So the most of the effect of the protein will somehow come to that that there are water charges Polarizing this on the outside, but it's going to be a much much much weaker effect inside here and You can measure this and the reason why I say this this is roughly three people have measured dielectric coefficients as a function of what type of proteins you have and What you can do more is that you can measure this as a function of the frequency That sounds really stupid why on earth do we measure things as a function of the frequency? Well, at least something's saying what do you have what do you think happens to the epsilon the permittivity as the frequency goes up we can say So what what dialect what dielectric respond what the dielectric coefficient here what this really means We can forget about the charge for a second So if these what next what this water is doing when they feel that there is a positive charge here These water will start to orient themselves To the charge and a very very short time later They will have oriented themselves and then they will screen the charge and because two charges are going to have lots of water between them So they won't really see each other fully They're just going to see mostly water that has oriented themselves from the other charts and that's why we screen charges so efficiently But that takes time So what if I now take this charge and I move this to this side instead? What will this water have to do it will have to rotate by 180 degrees, right? And rather than using charges I can use an electric field here And that's usually how you measure dielectric coefficient We don't keep adding ions, but I take an electric field and then I measure How does the system respond when I have an electric when I put an electric field on it? But it's with electric field is really easy use an alternating current field So now I can take it here and then a second later I swap the sign of the field and a second later I swap the sign Well, a water can rotate quite a lot in one second so the dielectric coefficient is going to be roughly 80 But then you can take the field say with 100 Hertz or a thousand Hertz or a million Hertz and Somewhere here when the field changes too fast. What's going to happen? No, it will go down right because the water won't have time to reorient So it won't be as efficient as screening anymore Yeah, I'm sorry. Yes. Oh, sorry. I my bad. I heard wrong So this will go down from 80 to well 70 60 50 40 etc. And we'll keep going down and Eventually when you have very high frequency It's not going to go all the way to one actually because you still have the electrons and the electrons I don't think you can't it's virtually impossible to have a field that changes so fast that the electrons can't follow the field So that's pretty much infinitely high frequency. You will end up with an epsilon of roughly two So any any component will have that because the electrons can be polarizable along the bonds And this brings us the question of the proteins and everything said When we talk about dielectric coefficients, we use the talking about Continuum mechanics or something that we think of this as a vacuum this way that there is some sort of space and that this Dialectic coefficient is a property of the space in vacuum It would be one if you had some sort of dielectric dielectric material here as in a condenser or something it would have a value that's different from one But the problem is that all our media They are not continue media You can look at this as different scale, but no matter what scale you look at this at is that you have discrete particles here So this epsilon if you're looking at something that's one centimeter from each other that can taste trillions of particles 10 to the power 20 It's very easy to define an experimentally measured epsilon But your point here about epsilon being one is actually better than you think in a way because at some point We're going to be in so small scales that between these two particles. We actually have back you So somewhere here we might even say that if these are all the charges We have in the system. We should just add up all these charges and sum them according to the Coulomb interaction So what's right Well, there isn't a salient thing right or wrong that depends on what scale you're looking at So when you're modeling things and simulating things in the computer We typically use epsilon of one the vacuum value and the reason why that works is that we have these molecules So if you take this and put it in a computer simulation, but you include the water in the simulation You can use an epsilon of one, but when you start simulating this Let's see in a system like that. These waters are of course free to move, right? So on a much larger scale in the entire system, you will actually end up with an epsilon that is about 80 And that's it because the polarizability here is in the molecules. It's not a property of space People have fought for 40 years about this today I would argue that these the simulations models are fairly good consensus that you should just use a value of epsilon one But if you think then think of your protein as a macroscopic medium or continuum medium Epsilon should likely be around the safe. Sorry. I swapped those two epsilon should be three four there No, no, this water 40 to 80 an interaction would be say one five K cal Drop that to 20. It goes up Typical protein then it would be 30 K cal and in vacuum it would be even higher But so think of a protein in the bulk up of epsilon being three to four But you're never really gonna insert that in any equations. This is a very classical result in biophysics This is very much related to the salt solubility in water But we're gonna look at this from a slightly different point of view that Molecule a salt crystal has an insane amount of favorable electrostatic interactions Sodium and chloride sodium chloride Sodium blue chloride green the electrostatic interaction here is amazing Why on earth would a molecule like that ever be solvated in water and what's gonna happen when you're solvated in water? Because salt is very soluble in water. You probably do it every day. You cook pasta or something So we're getting that it's the first part. It's important that these interactions are really favorable But if you take these as individual ions and put them in water Are the interactions gonna be roughly the same more favorable or less favorable the first approximation roughly the same Because these again the electrostatic interactions here are really good and for water It's gonna it's gonna be good, but maybe slightly better One of the biggest effects here though has to do with entropy This is a very bad state of a system Because here everything is perfectly ordered so you can think of having all the water in one part of the system All the ions in one part of the system. That's a highly ordered one So what in practice you do you get individual waters coming and stealing an ion at the time and then you create a system that has higher entropy and The one reason why I brought this up today actually is that you can actually see that Ah My bad now it's going in the wrong direction Hello There we had it there So basically here we have the ions in the crystals You see the water is coming in from your right here and they're stealing one out of meats And then you have a much more or an ion eats And then you have a very disordered system where these ions are scattered to the water But that gives you better entropy and that comes back to the one of the take-home messages today When it comes to all these electrostatics effects in water no matter how strong the interactions are they usually manifest themselves as Entropy the entropy determines everything not the interactions now, of course, that's in a way That's not quite correct because if you didn't have those interactions We would not get the entropic effects, but it's more Entropy than enthalpy if you're ever in doubt between these two go for entropy Famous words through get I bet I can find something from the exam with the result is the opposite But it's not it's not immediately easy to think of something that is more enthalpy than entropy in proteins There are I have two slides. I'm gonna let you up a little bit early today. There are two slides remaining summary Protein folding is largely determined by height of obesity. We went through that a little bit today I hand waved mostly today tomorrow We're gonna go back and repeat some of those things we started the very first lecture but now we're gonna use all the math we went through that you suffered through the last two days and Interpret alpha helices beta sheets, but now we're gonna think of it. What does it do to entropy? What does it do to enthalpy and as you might have guessed it's gonna be a lot of entropy there and not just interactions That hydrophobic hydrophobic and hydrophobic effect in turn is primarily Manifest this entropy Which is a really funny result, but that's that's the way it is funny results are fun We looked a little bit about free and it's a bunch of these processes This is something I did not I showed this but I didn't mention the world You remember this slide with a bunch of green stuff on the outside and a bunch of red stuff on the inside That actually happens in proteins But this is an early state in protein folding that remember that said that 90% of this course is caused by the hydrophobic effect So the second I take a long chain or something and just throw that in water This collapse when all the hydrophobic groups turn to the inside this happened like in a I was about to say a microsecond this probably happens in 10 nanoseconds. It's so Astronomically bad entropy wise to turn hydrophobic things to water So you weak we have time to see it but in a simulation. It's insane how quickly this goes on the inside But this is not a folded protein It's a sort of early intermediate states that you call a molten globular when you have had the hydrophobic Collapse happen, but you haven't really had any of the polishing happen yet And that's polishing is then the last part There's gonna be some entropy there too But at some point we're gonna get from this intermediate state to the really folded state But that comes much later on in the course for now Just if I if I say molten globular you can think of that green and red plot that hydrophobic things on the inside hydrophilic on the outside and Most of the things I've actually virtually everything we've talked about in these three first lectures now covered if you go up through chapter 6 of this book Skip all the part about the electrostatics and mirroring charges and everything. It's not really relevant for what we do It's good to know that their dialectic coefficients of a protein is a bit larger in vacuum But it's much closer to the vacuum value than it is to the water value And then I have a bunch of study questions that you will go through tomorrow There is a link here if you would like to go through this derivation and predict That free energy really core the free part here really corresponds to the amount of energy that is available to perform work You can actually prove a bunch of this stuff if you go through this link It's not a very long derivation or anything, but I somehow felt I don't want to overload you with lots of equations Equation-wise this week there are two things that are important the Boltzmann distribution that you spent the lab on yesterday and Today you're going to come back to the Boltzmann distribution in the lab But today you're going to spend much much more time on entropy now You're going to be dealing with these multiple states see what the multiple states do and this is going to be more similar to Solubility and protein folding than you think actually and at some point in the future We might even have a third lab using this but that's going to be after that we brought out in the proteins I won't go through this in detail now You have them in the handouts probably in type that's too small, but I also have them at the slides There are full resolution slide every night. I upload the full resolution slide copies on the site It's just that that would be 70 pages. I'm not going to print them all and I think that's all I had today and we'll leave 15 minutes early