 Today we're gonna head back to what I spoke about on what was it Wednesday, and we're gonna Speak mostly about proteins hydrogen bonds all these things we started introducing from a biological point of view and the hydrophobic effect in particular But now we're gonna cover all this in terms of well partly electrostatics, but primarily free energy So the idea now is that you're gonna try to make a for slightly more formal physical approach to this rather than just hand wave and Let's see. I've tried to have some things that will intentionally trip you so that you should learn these equations We spoke a little bit about different interactions yesterday degrees of freedoms and proteins, but then I headed on quite rapidly to interactions energy landscapes and we spent the entire second half on Boltzmann distributions free energy and entropy and that's Mostly where we're gonna be today, but I'm today the point is not to derive these anymore I trust that you followed what we did yesterday, but today we're gonna use the Boltzmann distribution in particular to decide what things are likely and what things are not likely and We're then gonna use free energy to try to determine what happens and what doesn't happen So we start with the study questions, you know what should actually try to turn off the lights there after all Does that get too dark for you? Okay Rather than doing this round robin, maybe we should see if all of you if all of you engage you can just pick questions So we can either you can go around and just pick whatever question want to answer or any if you can start to pick Any question and I won't continue until we're answered all 19 of them So let's start with number one Why do chemical bonds form? Yes, and it's even more than chemical bonds right virtually everything you see around you is due to electrons Anything in chemistry is basically electronic interactions The second you start getting into the nucleus its physics and it's basically radioactivity has to do with nucleus But nothing else has to do with nucleus any type of interaction. You can imagine is due to electrons Then there are different ways the electrons might interact We found of us interactions for instance are due to either electron overlap if you're pushing them together or very weak dipole-dipole interactions Electrostatic bonds, sorry covalent bonds are due to electrons be really being shared between two atoms But all are just different aspects of electrons. So why are then noble gases inert? Yes So the book goes into some details to talk about this when bonds form and he makes an argument actually something entirely crazy argument formally You can use the Heisenberg uncertainty principle to show that an electron will actually be more happy Because it is kind of slightly lower energy if it's Delocalized to cover two atoms instead of just covering one atom So if you have two atoms with one electron each instead of having each electron just well Circulate it's not really a circling right because it's an electron cloud But instead of having one electron around each nucleus having both of them share both nuclei Means that they get a slightly lower energy and that's typically why bonds form Now that's major hand-waving because as you say there are also some cases when this will not happen And at the end of the day this is gonna have to do with the pairing of the spins and the orbitals And then we're heading into quantum chemistry, so I'm not gonna go into details there So that leads to question two which you can answer to them So why do even noble gases form liquids at low temperature? Kind of yes, so what is a liquid so that when we talk about matter? The way everybody's know all of us have been introduced to matter is usually through gases You start with the ideal gas law and start to think in terms of atoms interact, right? in real life The most common forms of matter are usually what you call condensed matter and that's matter That is at a much much higher pressure so everything you see around you here is condensed matter except for the air in the room This is condensed matter water in a glass is condensed matter And this has to do with the fact that I spoke about earlier this week when atoms are at infinite distance away from each other They don't interact and we say that their energy is zero To first approximation. That's usually where we well when we have the ideal gas law That's still the case you're in atoms on average do not interact with each other The second Adam start interacting with each other those interactions are attractive Because if they were just repulsive they it would be worse than being infinitely far away from each other And when Adam starts interacting they will at some point move over to these condensed faces when atoms are interacting strongly So the most important part for the for the atoms are then their interactions with their neighbors And the problem is of course a liquid is somehow a condensed face. Well, it is a condensed face So at some point noble gases even noble gases start to interacting closely with other atoms What type of interaction is that? Yep Yeah, so what happens so this has to do with this dipole dipole interactions that I spoke about yesterday Which normally are very weak normally you would ignore them because the temperature around you is so so large that Just a normal thermal vibrations would cancel these weak interactions But at some point as the temperature drops and drops and drops Eventually the thermal vibrations become so small remember that we spoke about Katie, right? yesterday, so as long as these dipole dipole interactions are Weakier than Katie Because Katie at room temperature is relatively large. They're not really gonna matter But as the temperature drops eventually at some point Katie becomes smaller than the energy in these dipole dipole interactions And at that point the dipole dipole interaction is going to be more important than the thermal vibrations And at that point even noble gases start to interact Roughly, where do these where does this happen? Sorry, yes extreme. Well helium is basically extremely close to zero Kellen And that's why helium is so precious that it's a great liquid to use if you're cooling NMR machines Mm-hmm But you know what came remember this remember the thing the fact that you all said and that we agreed that noble gases do not interact with Anything in particular Xenon. I'm gonna come back to this in five ten lectures and it's gonna turn out There are some exceptions to this apparent exceptions at least What are the interactions related to bonds between atoms in a protein? I would say when I say related to bonds typically we talk about bonded interactions So one type of such interactions I'll steal the easy one for you and in the bond itself and then try and bond itself between two items a very simple interaction But there are other similar interactions The bond itself is covalent, right? So typically we typically classify three type at least three different interactions as so-called bonded interactions It's not just the bond itself That we're gonna see later So the bonded interactions are typically bonds angles and torsions Because they all occur between add them if if two add if three atoms do not if there are if there aren't two bonds between these Adams, right? It doesn't make sense to define an angle between them So technically it's not a bond, but it's an interaction. That's of the same type of a bond And what all these interactions really have to do with is how you're distorting the electron Orbitals when you're moving the geometry even the torsion is actually mostly related to how you distort the electron The electron distribution when you're turning around a single bond and that's why you have a very very large potential if you try to rotate around a double bond for instance So when you when you hear the word bonded interactions things bonds angles and torsions But then and then there are some special ones. They sulfides are certainly there too, but we'll come back to those later Non-bonded interactions that are what? Finder vals is one type of them something else Actually, that's a really good question. You could are you? In one way you're right, but hydrophobic interactions We typically classified in even higher level because hydrophobic interactions is going to be an effect of a combination of these interactions But there is something that And hydrophobic interactions are more complicated than we think but you mentioned found a vals interactions The other important non-bonded interaction is what? Lena Jones, I would say Lena Jones is one potential form that you can use to describe fund of all interactions So there's something else that I've been keeping the strongest interactions that are in proteins Well, and a hydrogen bond is due to what type of interactions Yes, electrostatic cool up so both both hydrogen bonds and My brain hurts and hydrophobic interactions The caveat there is that they're slightly higher level interactions that are effect caused by a combination Frequently of both electrostatics bonded interactions Because you're not going to get hydrogen bonds in the water unless you had the oxygen bound to the hydrogen, right? So in a way, this is similar to how you classify protein structure that we're now going to use these lower level physical Interactions if you call them so to classify and describe these more well So we say call them chemical concepts such as hydrophobic interactions and hydrogen bonds So if you then go to five and this is a bit of a trick question with which of these interactions are strong versus weak And I really give some numbers sorry Electrostatic interactions. Yes, and how strong were they 300 for what? For what? Yeah, I have two atoms right next to each other two charges right next to each other because that's important, right? If it's 300 you have to say what corresponds to 300 k cal There is no question that electrostatic interactions are the strongest The weakest Dipole dipole. Well dip a dipole interaction is actually an electrostatic interaction, too But in terms of the the interactions that we described up here, which one of the ones we talked about would you say classifies the weakest? Yes, the dispersion or van der Waals interactions and as you say and this is the problem here because when we talk about electrostatics since everything is ultimately caused by the elections So electrons Technically that's a dipole dipole interaction and the dipole dipole interaction at the lowest level. That's of course electrostatics, too The reason why this is weak and this is beyond the of a hurt you can actually show that a charge interaction has a potential that goes as 1 over r a charge to Dipole interaction is going to be 1 over r squared a dipole dipole interaction is going to be 1 over r cubed but that's if you have net dipoles such as in water and Eventually get all the way down to the London or the dispersion if you have induced a dipole induced dipole Then you're down to 1 over the r6 so that it decays very quickly with distance But the caveat here is what what are the signs of these interactions? Are they attractive or repulsive? Which one then used there to be careful that you can't say a dipole interaction because the dipole interaction If I draw one water molecule here That corresponds to having a dipole there and if there is another water molecule here That's also going to correspond to having a dipole there the interaction sign between those two waters is going to depend on the orientation Of the dipoles you don't want to turn the two oxygens right next to each other So no you can't say that a dipole interaction is attractive you can say that an induced dipole induced dipole interaction is attractive That's a mouthful. So I would suggest say Lena Jones interaction or dispersion in particular So the caveat here is that although dispersions are orders of magnitude weaker They're all attractive in contrast to electrostatics where we don't know anything about the sign So if you sum up enough of them eventually you will get the up you will eventually we can get a net effect. That's actually stronger I Sorry, that's kind of what we had in seven there, too. Let me jump to eight then what happens with hydrogen bonds when ice melts to water Yes So so when we that's a good question when we talk about electrostatics here We're talking about interactions between partial charges on atoms So for instance that you have an oxygen here that's slightly negatively charged in this water you have a hydrogen here That's slightly positively charged. That's going to be a very strong interaction That's the same thing that a permanent charge that just corresponds to an atom an entire molecule that has a net charge, right? There is no fundamental difference The if the entire molecule is neutral you can actually describe the entire molecule with the dipole But there is no strong difference there. So when I when I speak about electrostatics Think of it. It's when we care about charges But whether it's a unit charge or a fractional charge doesn't matter. Oh, well So with electrostatics, of course, you have two atoms of the same time, right? They're gonna repel each other If you have the opposite sign, they're gonna attract So if you have sum up a million interactions like that in a protein in General you can't say whether these parts of the protein will attract to repel each other because it's gonna depend on what atoms You have in exact positions of these atoms how large the charges are It gets noisy and complicated But what you know if you have one million Lennar-Jones interactions unless the atoms are exactly overlapping in general that we've been an attractive force That causes them to attract each other, right because at long distance all the dispersions are attractive Which is the same effect why even helium will condense close to zero Kelvin So number eight was somewhat related to your homework About understanding phase transitions in water Actually, no the phase transition was used a E minus TS So what happens when I smell the water are hydrogen bonds maintained or not? No, I say I would say that the key thing is to word average So what happens is not going to be that some hydrogen bonds are broken completely and all others are maintained because then you would have a block of ice, right? But what happens is that the waters are more mobile so that you will have an average lifetime of each hydrogen bond And in ice that lifetime would be almost infinity because they're all formed all the time What will eventually happen in water is that the hydrogen bond will live for I don't know a couple of Nanoseconds maybe a millisecond. Well, no, it's going to be nanoseconds It probably lives for a nanosecond a couple of nanoseconds And then it breaks for another few nanoseconds and then it reforms with another water and this is how water is liquid, right? So that on average we have roughly 1.7, but it's not the same 1.7 bonds all the time Yes That there is an average lifetime that they form and reform all the time So not that much. So water is an extremely incompressible liquid if you try to take a Take a plastic bottle, right? It's very easy to compress it. The second you have it full of water You can pretty much run a car over it and it's not going to compress. It's extremely hard to compress water So that changing the volume of water To first approximation, it's impossible So that it depends what you mean by the changed area. You can make this experiment actually you can make the experiment Tall narrow glass versus wide low glass, right? It's very easy to change the area because you're changing the height at the same time So the volume of the water stays kind of like that's a general property of a liquid a lick If you're changing a liquid can move right so that there isn't any inherent shape Ice on the other hand has an inherent shape So for ice price you can't even change the area because you couldn't compress ice in one direction and have the ice cube instantly rise So the hydrophobic effect then that you try to introduce already over there. What is what is the hydrophobic effect? And what interactions cause it and what's causing it? Yeah, well, yes the inability of those molecules to form hydrogen bonds. So does that mean that the hydrogen bonds break and that's So what type of and here's where things get complicated when I say interactions up here We only spoke about interactions that's fund evolved or electrostatics or something, right? So what what is You you could imagine scratching out interactions here. What is that causes the hydrophobic effect? Right, so it has to do with this ordering effect, right that we started to talk about as entropy yesterday And the point entropy is not an interaction It's not that you can't calculate what the entropy between two atoms is the entropy has entropy is more of a global effect How free something has to move? So that normally you can't really classify entropy directly as an interaction You can't directly calculate entropy in a simulation. This is where entropy is complicated But still it's the entropy that causes the hydrophobic effect. Yes So entropy when it when entropy goes down to a more ordered state That's less Favorable so that you should always be careful in entropy actually reduces his right word because you always think of how the entropy changes as part of a process so the reason why the reason why the hydrophobic effect happens is that if the Hydrophobic compound would be fully solvated Then every single around every single such solute you would need a small case of water and that would be very unfavorable That would lead to a loss of entropy So instead you tend to Either well if you have oil the oil is going to separate into a big oily blob and that reduces the surface area And that minimizes the entropy loss in that case You don't lose more than necessary. Yep That's actually a very good question. So what do you think? So that's correct because if that Think of this terms in order, right? if it kept breaking all the time that would correspond to the molecule moving just as much as it did before and By definite we said that that doesn't happen the atom is forced to become more ordered So the fact is that you can't estimate exactly my said, but you can estimate the sign That it will because it's more ordered these hydrogen bonds will now have to have a longer lifetime Because there are fewer partners around it. This is also something that's very powerful when you can't estimate anything about something you can probably estimate the sign and Estimating the sign is usually usually worth more than you think because then you can start to make predictions about the direction Whether something will happen whether it's five or five thousand K cal worry about that later, but the sign gets you a very long way So what is an energy landscape then? So this is a Imprisonable and energy is just a function of all the degrees of freedom in the system But this gets mathematically complicated and in particular we will want to understand these things It's very nice to have some sort of simpler view over this Typically we draw energy landscapes in 3d simply because we can and then you have at least two degrees of freedom But I'll draw you an even simpler energy landscape imagine that you have some sort of free energy here, right and Then I have no idea Then you can start making predictions here What happens if you are here and you want to go there? What are the heights of the barrier is here? How good is it to be at different points? And even if you don't and here too just thinking about what does this imply for signs? Is it easier to go over a very low barrier or is it easier to go over a high barrier? We're gonna come back to that later on but it's very instructive to think about these super simple problems Now the problem here is that if you just look at a one-dimensional landscape We might say that I'm standing here and when we have the break I'm gonna head over to the kitchen and there and have coffee Now the straight reaction path for that is for me to go here and go straight through six walls in that direction That's going to be a fairly high energy barrier So if you do that you could say start predicting there is no way the professor can ever get to the coffee machine And So the problem here that was a one-dimensional. This is not what would happen in practice, right? So I would use the two-dimensional feature and go out the door instead and go in the corridor So the problem here is that and that sounds of course that sounds completely stupid But it's not so the point is a two-dimensional landscape you can do things in two dimensions that you can't do in one dimension So at least using at least two dimensions gives us an ability to see things slightly better It's at least possible to here you can if this barrier is too high you can never ever get over it in two dimensions It's at least possible to get around some barriers But you should of course realize that if a protein now has hundred fifty thousand dimensions There are things you can do in hundred fifty thousand dimensions that you don't see in two dimensions But that's it's it's gonna be very instructive to think in terms of energy and a particular free energy landscapes So for what can you then use the Boltzmann distribution and there's even something on the board you could use to think about it So the cool thing the second you have the Boltzmann distribution and you have an energy landscape You can start to say how likely is it to be here or here or here and Not just say that it's slightly better. You could suddenly say that if I give you ten states and what their energies are You can give me the exact distribution of probabilities. How likely is it to be a state one two three and four? so the Boltzmann distribution connects this hand waving whether something is good or bad and to give real probabilities and So we say oh concrete example. Well, I could I guess we could use the gas distribution. We talked about yesterday or you could If you're looking at spectroscopy or something a spectroscopy has to do with an atom normally an atom would be in its ground state And then when you shine light or something on it, you excite the atom to a higher state, right? Now it's going to be very unlikely for the atom to be there in General, but the reason why you're there when you've excited is because you pumped lots of energy into it And when you have all that energy, it's going to be more likely to be in the high energy state and Very quickly later, you're going to lose the energy again because you're falling down to a pre another state And you can even calculate inside atoms you frequently have multiple different energy levels and everything This two are based the Boltzmann distribution So in the Boltzmann distribution when I actually any time you see the Boltzmann distribution you're going to be talking about k for Boltzmann Yes, sorry. So why do you limit that to say near the boiling point? So the point is that it's not limited to the boiling point. You can pick absolutely any temperature you want So this will tell you the probability You can fix a room temperature Some atoms will move from liquid to gas now It's very unlikely there's going to be very small part of them, but they will and the Boltzmann distribution describes this. Yes So this is the cool part. I Yesterday we did this in a slightly hand-waving way and just showed that this was true for the special case of gas, right? But I also hinted that next week we're going to show that this is true for any system in general The cool thing in physics is up to you to define what your system is There is nothing here that assumes that this is an atom or an electron You can imagine a Boltzmann distribution of cars or something So this is just up to you to define something and if you have something that actually has the You need something that has the property of an energy, right? Because the energy is gonna But the second you have something that's a property of the energy you can relate it to probabilities This is the really sexy part for thermodynamics and statistical mechanics that it's not based on a special case It's universally true for systems. Yes What do you mean when you say kt on the Right, but you would like something to be a function of kt. So k, remember Boltzmann's constants is just a constant So if you have kt on the x-axis, you basically just have temperature or energy On the x-axis. So what you're asking to plotting something as a function of temperature So a phase diagram is a measurement So that in some ways the phase diagram is Measured experimental information that contains the same information that you would be right from the Boltzmann distribution Eventually if you had a perfect model now with the Boltzmann distribution Normally, I'm not particularly interested in having a perfect model if you want to have a perfect model You should use a fancy computer or just go into the lab and measure it The important thing with models is usually to go in the opposite way to extract in the pure essence of the problem The simplest possible part try to again try to get away from all the trees So you see the forest So what is the important property of the problem? And that's frequently just a function of one variable or something same thing with the protease, right? 150,000 atom tons of interactions quantum chemistry. These problems you're gonna look and they're super complicated So the only way to understand them is usually try to cut away everything about except one or two important features So why do physicists usually like to work with Boltzmann's constants? Why chemists prefer the gas constant R? Yeah Same thing here as kilojoules versus K cal you can pick absolutely anything you want But you need to know what you're working with So that it's not yet Occasionally I might even say that it doesn't really matter from one point of view it doesn't matter, but you have to be consistent in what you do. Just as this is going to be a major Catastrophe if you start doing equations and sometimes use K cal and sometimes use kilojoules without converting properly between them to make Sure, you're consistent. I will frequently work with Boltzmann's constants. Sorry occupational disease There's nothing wrong with our the gas constant. It's just that in Boltzmann's distributions You almost never use R and that's because we talk about Boltzmann here, but you could just make sure to keep track of your units so the Important question. What is KT? I? Should be able to wake every single one of you up at 3 a.m. In the morning, and this would be like a running water 0.6 watts killer cows And if somebody and if you get into a lab at every single student and the lab is working with kilojoules 2.5 so the point is you need to know both and it's important to have a gut feeling about this because when you're sitting and scanning a table and You see energies This this is the ruler. This is your scale Now if you're a physicist and if you start to look at a particular problem And if you're thinking about distributions, and how likely something is this relates to your question and why I was about the KT It's very nice to frequently plot something in units of KT Because if you start plotting something in units of energy, right whether this happens or not It's going to depend on the temperature But if you plot something in units of KT Every physicist knows once you get about 10 KT or something. It's going to be a fairly high energy regardless of temperature So physicists it can't look very strange to have an energy scale in units of KT But it's really convenient if you don't care about the numbers if you're just after the principles So how does volume or microstates complicate the Boltzmann distribution? Well, I'm asking the questions So you're on the you're heading in the right direction Sorry, I just want I just want to avoid giving you the answer to have this is hard And you so you're both right But this is another one of these complicated things is worth spending a minute on This all comes down to what you mean by state And the problem here is that beauty is in the eye of the boulder Because we we kind of handway we haven't said what the state is right and This has to do with the complicated part that these things are in general true for all systems That sounds great, but that means that you can't completely ignore what we define by a system and You can of course say that a state. I mean, what is the level of water in a glass or something? So when you define a state on a fairly high level or a chemical level or something you can measure in the lab That's one type of doing it another type You could imagine drilling down all the way to the quantum ground state of every single electron in the system And then you would have like several hundred electrons even in an amino acid and Every single configuration of all those electrons is one state and If the two electrons change slightly, it's a different state or if one atom rotates slightly, it's a different state Both of them are true and it depends on what type of system you're working with right? And that's why it very frequently the problem here is that if you make a normal measurement You're not going to be able to spot those differences, but you will spot the different water levels So it makes a lot of sense to define some sort of macro state or macroscopic state the large states the ones we measure and Then there is some under hidden or lower microscopic states and the way a Microscopic state if you drill all the way down to the quantum ground states There is no entropy because every single state is always unique But that's completely impossible to work within practice So what we normally do is that we tend to work with some sort of macroscopic states You can say that the protein is folded or the protein is unfolded But then we're going to need to somehow account for the fact that inside this large measured state There are lots of other small states and that's why we talk about these volumes or number of available states or something And that's what then leads to entropy You're gonna test this in the lab experimentally well experimentally computationally on Monday I think it will make a whole lot more sense after the lab And the beautiful thing is that this is something that comes out natively of the equations So what is then detailed balance? Right, so the problem is now Haha, this is what you're gonna see this on the lab, but if I have one state here that has energy whatever say minus 5 and Then I have one state here that has energy Let's say minus 10 Which one of this is most likely so you it's going to be more likely to find something in minus 10 than minus 5 I think everybody agrees Let me complicate this but let's say that this is minus 9.9 actually no 9.99 Which one is more likely? Minus 10 is still more likely there's no question about it In the interest of time, I'm not gonna draw them, but let's now assume so if you This is any type of process we have here in our lab, whatever I don't have time to imagine what system it is So what you're saying with a system you have a energy levels of minus 9.99 and another energy level of minus 10 If you measure it, it's going to be slightly more likely that you make the measurement 10, right? But let's now assume that we have a billion of these So you have one billion states like that and you have one state like that If you now go into the lab and measure it, what's the most likely outcome? Why? Boltzmann distribution doesn't say that the Boltzmann distribution states should just depend on the energy So the problem here is that some here right Actually the Boltzmann distribution does say that if you include one billion one states in the Boltzmann distribution But if you're just thinking of the two cases energy minus 9.99 versus energy minus 10 Are there two states or are there a billion one states? Well, that depends on whether you account for the fact that there are lots of states inside this one or not, right? And that's where this volume comes in So one way of doing this is of course that I start to draw now and you come back 14 years or so from now When I have drawn a billion of these Another way is that we somehow try to find a correction factor For the multiplicity of the states that there are more states like this And this is the complication because instinctively we just assume that all states are equal and They are but then you're going to need to count every single of those states And the complication here is that we somehow need to account for that in some cases Some states are more likely some states are larger than others in the sense that there's a larger volume inside them There are more microstates that corresponds to the experimentally measured states So the no, sorry now So what is detailed balance then if I draw something like that? So we have this free energy and I'm just gonna Guess the matter. Let's say that we have 90% of the population there and you have 10% of the population there at equilibrium Does that mean that no particles or molecules or whatever it is never move? because Actually, I'll make this very easy for you. I'll say that there's a tiny barrier there. This barrier is so small that it's 0.1 kT So you're gonna cross that barrier all the time right at room temperature, but on the other hand at some point If all these atoms started to flow down here, we would not have been at equilibrium The definition of equilibrium is that some other systems should not really change anymore So if we really are at equilibrium, I should probably do it It probably wouldn't be 10% there were probably more like 0.1, but for the argument sake, let's say that it's 10 So what somehow if you look at this one minute later, you should still have 10% of the molecules there so what detailed balance means is that the flow or flux in molecules in one direction here and The flux on the other hand they those flows must be the same and you can formulate that in saying that the number of particles in state a multiplied by the probability of going from a to b for Each particle then must be the same as the number of particles in state b multiplied by the probability of a going from b to a and Then you can also divide Move around things a little bit so say the fraction of particles in these two states Tells you something of how likely is it to go from b to a versus to go from a to b So you can if you know the population in two states, you can also start to make predictions of how likely this is going to be to Go between states. Yes The right one. Yeah Because it will happen now and then like we don't so first I haven't told you what the energy differences here, right? What if this energy difference is 0.1? It will happen now and then right But on the other hand So I never ever said here that it happens frequently, but there is a difference between zero and infrequently In some cases it matters proteins actually do unfold a small pro a very small protein. I'll come back to this later. You There is a concept called dwell time ratio so that for a very small protein if you folded them They frequently only stay folded like two-thirds of the time They unfold and then they refold and they unfold and refold The way to get around this would be if a protein was infinitely stable if a protein had a 10 megajoule Stability so that once you fold that it was like a brick Would that be good? Why well, there is something else right that some point you will have your body will have to degrade the protein and Imagine if the proteins in your food had a stability of hundred megajoule You could never ever get an engine for food. You couldn't degrade proteins So it turns out and this is actually one of this is we're gonna come back to this later on the course this is the really hard part that The special thing with chemistry and biophysics is that all these energy we're gonna talk about are gonna be fairly low So that you can't just if you're working with bricks you don't have to account for the probability that bricks spontaneously dissolve But with proteins that's gonna be the case We need to take more states into account The difference between energy and free energy we're gonna talk more about that later today so that we can probably skip But can you give examples of systems with low and high entropy? That's related to your homework Sorry ice Yes, so I says low entropy and boiling water high entropy and the entropy is closely related to the hydrogen bonds and the hydrophobic effect We spoke about already right that it's primarily an entropy effects that you have to force them to become ordered and What the letters in F equals E minus T as is we will get to now so I you don't have to answer that one So today we're gonna speak about two types of free energies Helmholtz and Gibbs There is a reason why there are two it's not just that physicists and chemists can't agree Frequently the case of science if there are two ways of defining something you're always gonna find chemistry and physics doing it different ways Oh, sorry. Yes, there are a slide hands out here And it's not the gas constant in one example because the chemist instantly defined this in a way that suits the lab the physics Defines it in a way that suits their theories. It's unfortunately gonna be the same things with free energy There are two slightly different ways of looking at it We're gonna do something pretty cool. I'm gonna define temperature for you And you can do this derivation yourself We're gonna look at the hydrophobic effects more and if I kill E minus TS and now we're finally gonna talk about protein folding and electrostatics And a little bit about titratable amino acids I have no idea where I found this but it's a fun I share shared with you the free and the utility board deals Appropriately with the free energy quack I've used that in a paper once So when it comes to free and this there are as a physicist remember that I told you that it's very good to Define a simple system that drills down to the essence of what we're interested in So you can of course try to do a complicated measure about everything a measurement about everything in this room But if you want to understand how gas works, it's much easier to do that in an isolated system you just have the gas and Try to remove anything that complicates the problem. It's going to make it easier to measure And what physicists frequently do since they work with very simple systems Do you imagine having a system that is somehow? Well, you could imagine having a system that's completely isolated from the environment So it can't exchange mass. It can't exchange it. You can't have any sort of transport. You couldn't even exchange heat Well such a system is a bit boring because you couldn't even measure on it because any measurement is going to be an exchange with the system so the first system where it starts to be a Large system that we're working inside might be isolated But the first system where it starts to be interesting to see where some what happens with the systems as it undergoes through some process Then we're going to need to exchange something with it and this something is usually energy in The sense of heat I can heat the system or I can cool it but everything else here should be constant I should have the same volume. I should have to say well the the same particles inside it I can't move matter in or out change as little as possible and In that case we can say that the free energy of this system depends on two things free The energy of all the interactions in the system We have the temperature which was the third thing and then with the entropy that is how many different ways Can you organize the particles of the states inside that system? That's all I've talked about yesterday and today Physicists like to use the letter f for that and They call that the Helmholtz free energy We're never we're I'm almost always going to ignore these names, but formerly that's the Helmholtz free end Now we're chemists and that's not necessarily how it works in chemistry chemist also has a system in your test tube For instance, and a test tube. Can a test tube exchange heat with this rounding? most definitely yes but what happens if you If you take one liter of ethanol and mix with one liter of water You're not going to get two liters You're gonna get something like 1.8 liters So the problem is in chemistry you might also be changing the volume of the system Very common virtually any process you do changes the volume and When you're changing the volume this actually Dispending on the pressure and everything in the surrounding world This corresponds to either you doing work on the surrounding world pressure work or the surrounding world does work on you The physicists hate this because again as a physicist you like to drill down to the pure essence simple system You ignore that But as a chemistry ignoring a test tube These are all the systems we work with you can't ignore you want something that corresponds as closely to reality in the lab So if you're a chemist you need to account for this work that you can do work And that's going to be the pressure times volume So as a chemists you have to add this term There is an energy of all the interactions in the system We still have the entropy, but you also have this pressure Multiplied by volume so if you're changing something we're also going to need to take into account Does the pressure change or does the volume change? The good thing is that these this term is usually well, I'll get back to the second This gets complicated all the time right so that you can actually imagine Pressure and volume this is also in a way than energy so it makes sense to merge those two and then we call that H So this word enthalpy I introduced yesterday. I kind of lied I told you that we called it enthalpy to avoid confusing it with free energy Enthalpy is the sum of the energy and these volume effects the small work in a volume chemists usually use the letter f for this That can get a little bit complicated right So we should pick a different letter So if you really want to tell these apart We use the letter g and then we call them at the Gibbs free energy The book is a physics of the problem here. It would be great if there was a standard Unfortunately, there isn't you can occasionally see the letter f used here live with it. You might have to define it If you really want to stress these supports at the Gibbs free energy versus the Helmholtz free energy We're chemists in this department. So we're gonna live here To make your life slightly easier though Do you have any idea for a normal system if you have a test tube or something? How large do you think this is for a typical reaction compared to the two other terms? There are of course special cases if you're working at very high pressure if you're really doing an explosion or something Then it can be gigantic, but for a normal kill chemical reaction with liquids in particularly in biochemistry What are the concentrations of typical things in biochemistry? Yeah, none or micro molars or something like maybe millimolar So it's hardly going to be the fraction of molecules that are proteins and test tube is going to be zero to first approximation Which means that the first approximation in biochemistry This one is zero That kind of simplifies your life a bit right because that means that in practice for everything you do these are going to be equivalent And that means that I'm going to be sloppy you will be sloppy too You're gonna just gonna say free energy. Sometimes you're gonna say f sometimes you're gonna say g it doesn't really matter But you should of course be aware that if there are huge differences in pressure it will be important. Oh So the point here is that your test tubes volumes matter. So when we Free free energies. This is actually a state property So what is a state property? I'm well aware that I haven't introduced it But what do you think a state property is? It depends on the state does it depend on something else But temperature is part of a state of a system So the key thing with a state property is it's something that only depends on the state of the system This probably sounds really stupid, right? What else could something depend on rather than the state of the system? so Friction is friction a state property So right if I'm moving something if a well, I have no idea what but if I'm If I have a small physics lab set up and I measure the force it takes to move something if I move it very quickly Over a rough area. I'm gonna need to use lots of energy to move it, right? But if I move it very slowly, I'm gonna need less energy So when it comes any process where you start involving friction or something then it depends on the path you take there How quickly you go or which way you go? It's gonna be easier for me to get to the kitchen through the door than through the wall So everything in the world is not state properties. There are some things that depend on the way you get somewhere This is intimately related to the laws of thermodynamics. I talked about yesterday a state property If you if you start in one state and then I visit a billion other states, and then I'm eventually back at the first date Can I say something about the state properties? They have to be the same a State property only depends on where you are. It can't depend on how you got there This probably sounds I'm well aware that this sounds a bit strange to you imagine that I had imagine that free energy was not a state property and Here is well, no If I'm in the lecture room, I have a free energy or whatever should we say minus 100 arbitrary units Then we have when we get to our kitchen We have a free energy Actually, no going from the lecture room to the kitchen if that costs me 100 But when I go from the kitchen to the lecture room, I can find a better way so that I get 200 back Then you could just keep repeatedly going and you would gain an infinite amount of energy That's a perpetuum of it. It's impossible So that the free energy can only depend on the state so that once you've folded a protein No matter how you folded the protein the difference in free energy between folded and unfolded protein can't depend on how you fold the protein Because if it did any time a free energy depends on the path It would have a perpetuum of it But take the cheap way when you pay and take the expensive way when you get paid you see my point Yes, right so that So the the energy in the excited state That's a state property now depending on how you excite it You might of course waste a whole lot of energy because the laser is shining on other parts of the system or you're exciting in some other ways So that it doesn't mean that there are more or less efficient ways to excite the system But once this small system is excited the energy index in the state can't depend on how you excited So this is related to say that When it comes to state properties, there are a couple of different properties of a system if you have my small system here And if I double the size of it Some things will depend on particles and some things will not depend on particles The other way if say if you have particles here, and I make this system four times larger. So whether you sorry I should have four in both cases in this case. I double the size of the system in this case. I quadruple it, but that's Irrelevant for now, but if you're changing the size of the system Do you also if you change the number of particles in the system? Do you also start changing things like the volume? Now this is complicated a little at least So there are some things that depend on particles the obvious one is the number of particles, right? there are other things that do not depend on the number of particles such as the temperature and Typically we separate these types into two types of properties so called intensive and extensive properties of a system and Extensive properties is something that's Proportional to the size of the system the energy in a system for instance If you just take a system that a given energy and you just copy it hundred times you're gonna have hundred times as much energy But you're still gonna have the same temperature So the temperature is an intensive property But the energy is an extensive property. I have no idea why it keeps showing that menu So the problem if you now take a small system here, that's fixed If you don't account for this volume effects if you're adding more particles, I'm gonna change all the interactions here But since I did not change the volume Suddenly now there's a higher sir. Yes Yeah, never sorry and I know you're saying about something now you're changing the properties of one system, right? When I say if you have one Let's see if you have one test tube in a particular experiment and There is one mole of salt in that one and you have a certain volume and everything you can measure the temperature in that test tube Now you take ten times as much matter, but you also have ten times But you just increase everything a factor of ten That's of course the same thing as just having ten test tubes, right? Then it's not gonna change What you're doing is what I'm Following it what you're doing is this if you have a system if you don't just copy the size of the system We just start to change the properties in the system other things are gonna happen. So in this case if I have two particles and I suddenly double that to four particles But the volume of the system and everything here is now constant Suddenly I will have increased the energy in here. That's the interval. Of course effect temperature The problem with Helmholtz free energy in this case that this is not just Proportional the number of particles might be a bad way of stating it here But proportional to the number of systems or the number of states how many times I've replicated the system might be better Because when the pressure and temperature here changes I don't account for that in the Helmholtz free energy I can't allow the system to change shape Physicists love this because in physics. It's nice not to have to worry about the volume of the system and everything The reason in chemistry why we love this is that in chemistry if you're adding four test tubes You have four times the volumes you Because you allow in the lab those if you have four times as much of everything You're not going to take that and push it into one test tube We just have well, we might have one test tube, but the volume is effectively going to be four times the volume So what that means that we're working with Gibbs and an enthalpy in chemistry This just corresponds to having four times more. Yes, you have more particles, but the free energy per mole Would be the same right? Because you're just studying more of it now the total free energy would be four times larger But if you calculated per mole, it would be the same So that this just illustrates that in chemistry it makes sense to work with Gibbs So from now on you can kind of expect that we always work with Gibbs But we frequently ignore this PV term because it's so small This allows you to do One really cool thing. What is temperature? So before this course when you started if I had asked you if you know if I introduced two concepts to you Yeah, sorry Sorry, maybe it's I Think you're right. I just didn't get it. Maybe it So if you have if you only have one particle doesn't it have any temperature? So it's kind of related to energy, right? But it's you can very easily end up with a circular argument here because energy is related to temperature If I had asked you last week about two concepts because most of you had heard about entropy Temperature and entropy I bet all of you would I said that temperature is easy entropy is difficult It's actually the other way around As we did yesterday entropy is just something you can't define and that comes fairly natural just from the sides of the states Temperature on the other direction pretty difficult if you're going to do it properly from a thermodynamics point of view But with this equation you can actually do it So, you know You have all been used to using derivatives Well, that's not the question. That's a statement and you always done single dimensional derivative say D F DX whatever X and F here is right and You always look at it this way as a combination But what this just means in one dimension it gets more you can't do it in multiple dimensions but in one dimension this really corresponds to quotient of an infinitesimally small change on the y-axis divided by an infinitesimally small change on the x-axis and Mathematically you can actually separate this you can start to thinking of what if I just had an infinitesimally small change on the x-axis You can think it if it if it had a finite extent, we would call it delta x, right? But as the such a small interval goes towards zero This is called a differential and it's perfectly okay. We can work with differentials. It's allowed in physics So if we have our free energy F Equals E minus TS If we now start to think about an arbitrary system such as this curve, this is a free energy as a function of something at equilibrium Free energy is always in a local minimum. Why? Right, but that's just the definition of a local minimum. So that's just a circular definition So why does the F? Why does the derivative here has to be zero at equilibrium? I argued that The free energy at equilibrium is always at a local minimum at any state. We have an equilibrium. That's a local minimum for free energy. Well That's kind of also the definition of a local minimum, right, but you're getting closer So what does free energy mean? I said it was related to the amount of energy available for work, right? So any time a very useful techniques to prove something is to try to what would the opposite imply? If it was not a local minimum so that you had some sort of derivative, what would happen? So let's try that so that you are giving that if we were here and we were here Nothing would happen. We would just stay there So a local minimum literally just means that the derivative of free energy with respect to the variable And this could be a position or something is zero So a free energy describes in the directions in which reactions happen, right? And any time you have a derivative of a free energy with respect to something you're gonna go in the downhill direction That's what the free energy describes So if you're not at the local minimum yet, you're not at equilibrium. Then you're gonna have a chemical reaction Something will happen So eventually of course you will read some sort of local minimum and then you're gonna stop moving Then you are in a stable state and that's the whole definition of equilibrium Equilibrium means that the free energy is locally stable Otherwise So what would happen here is that you would continue downhill until they eventually reach a minimum It's certainly also zero at the local maximum. What would happen there? It is actually a great example Have you ever used a chemical hand warmer? You need these small packs, right? That there's a small metal plaque in it And then you twist the metal plaque and then it becomes warm Kind of so what happens at a minimum technically the derivative is zero So you're not gonna have a force in any direction But you're what happens if if I give you an infinitesimally small push to the left You're gonna go down there, right or if you get an infinitesimally small push to the right you're gonna go down here So this is This is an unstable state. So you have a local maximum You're sitting on the knife's edge and any time you move you're gonna fall down on one side Now so at the local minimum you have a derivative zero every zero derivative is not the local minimum. They're called extreme values so now of course these hand warmers, so this hand warmers actually a liquid That is would normally prefer to be solid at room temperature But when you heat this liquid to roughly 50 degrees it becomes sorry this solid around 50 degrees it becomes liquid and And then then you keep it well you usually do 200 degrees and then it stays liquid because it's This is not perfectly smoothed. It's the whole landscape here is just a little little bit rugged here So what then is what then happens is that short term this one this liquid is stable as a liquid Because there isn't really anything to get it to start freezing But when you're pushing this small metal plaque You're basically giving it just enough of a push to push it over the barrier Because normally there is a little bit of ruggedness in the barrier So it's not going to go over there spontaneously and with this push what now happens with this liquid is that it freezes That sounds really stupid how on earth can something get hot from freezing So well what happens when this freezes that you have less internal energy in a solid than in the liquid So to freeze this now will now have to give up energy to the surrounding So the energy that you give up to the surrounding is What you feel a seat but again local minimum? Yes So there are a couple of things there first when I say that something is a local minimum That is just a local minimum. It's not the globally most stable state of anything the So let's see if we draw that one in some you meant that one, right? So on average remember the gas that we went through yesterday, although this one is better This is in this case. This is the global minimum, and that's just a local minimum But you can't have there's also only a limited number of states here So you can't have every single particle in the same state they would overlap Just as you can't have every single gas molecule at the bottom of the vessel So while it's going to be much more likely to be here maybe 90 percent Some molecules will have to be up here, too And how may and the distribution of these will then depend on the relative energies Sorry the distribution will depend it will depend on the relative energies and the temperature of course My only point here is that the fact that at the minimum That's right at a local minimum Sorry at an equilibrium when molecules don't move anymore when you don't have a chemical process going on You are at the local minimum in free energy Technically, this is actually true for the local maximum too, but that gets complicated So for now, let's assume that it's a local minimum So here the derivative of the free energy is zero and it's something you can prove That's the whole definition of equilibrium So at this point we we always have f equals e minus ts That's always true no matter whether it's during a reaction or at the minimum But if we start looking at what happens when we Move just an infinitesimally small point away from this So if we were at f we're now going to be at f plus this small change the differential df And the way you handle differentials is just that you would handle derivatives so that The small we already have f because we had it here right the small change in df That will now be one small change in e d e And then you should use the derivative product here. So t multiplied by s you're going to have one term t ds and 1s dt And to simplify things a bit we have f on both sides here. So we can strike that one out We also know that if you are now and again, this is a universally true But in our case we're now at equilibrium Because any anything we study at the lab The only point that's not that equilibrium is while the reaction is happening while you're having this hand warmer While the phase transition is happening or something a minute later. You're going to be back at equilibrium So at the equilibrium those are the interesting systems to study and at equilibrium We also know that df equals zero. So we can strike that out too At equilibrium the temperature doesn't change either So we can start to strike out that entire term because dt must be zero if the temperature doesn't change We've removed most of the things here, right? So now we only say that zero d e minus tds must be zero And then you can solve for the temperature So the temperature is really the derivative of energy with respect to entropy That sounds strange. I know but Entropy was just states, right So as your change at the number of states in your system is changing. How is the energy changing in the system? That is the thermodynamic definition of temperature And this is actually why in particular why lord Why lord kelvin has got so famous that it's not just that you took the Celsius scale and transformed it to zero kelvin But that you realize you can define a thermodynamic temperature from very fundamental ways. This is not based on measurements It just happens that this is then related to energy and everything So it's just it's a pretty cool way of defining temperature and this is actually more complicated than entropy With that, I think we should head back a little bit to the Boltzmann distribution Sorry actually the reason why this becomes important is that the temperature also enters in the Boltzmann distribution So you're also comparing an energy to the temperature, right? The delta e divided by kt Now if we were to invent the temperature today We would measure the temperature in units of kt and then you would not have Boltzmann's constant But because there were already lots of other definitions of temperature It makes well, we're going to need to add that small k But for all you can think of the k as a proportionality constant kt is our unit of temperature So what the Boltzmann distribution really does is it measures? What is your energy? And how does that relate to the general trend of how energy changes relative to states? So you're comparing your energy to temperature But then there's going to be a bunch of different combinations with this and the Boltzmann distribution in particular starts adding this exponential As I hinted a little bit about just yesterday that's important to know If something changes by kt and energy changes by kt How much does that influence the probability? A factor Three yes So it's e, right? So it's two kt. It's a factor of 10 And that doesn't sound so bad Until you realize that exponentials today rise very quickly. I'm well aware that this looks stupid, but it's Understanding how insanely quickly exponentials grow Is important in physics and we make this mistake all the time that we don't understand how quickly it is So the small numbers here are fine, right? But by the time you get to x to the power of 100 You can no longer represent this in single precision, which probably rules out many of your cheap calculators at least by the time you're at One Somewhere up here These are around the largest you can get to the power 10 to the power of 300 on a normal computer Otherwise that you're going to need to define a special numeric class or something To give you an idea about how large these numbers are the number of atoms in earth is roughly 10 to the power of 50 I think the next one is the number of atoms in the universe and then you have the complexity of chess and go respectively These are pretty large numbers And somewhere here. I would be a guess is the budget deficit of the united states or something The point here is that it's it's very easy to end up thinking. Yes, it's a small number But it will have will happen now and then right? because we're talking about probabilities and technically This is correct if I put this pen on the floor At some point it might have enough energy to fly up to the table so we can just wait The only problem is that you're going to wait until you're here, right? So at some point here probabilities become so small that it becomes a matter of definition If the probability is so small that it's unlikely to ever happen in the age of the universe Is it even meaningful to think of a probability? And at that point you start to saying that things can't happen But in thermodynamic point it happens now and then it's just Now and then is zero What this means when you start to translating this to say chemistry or something This has to do with all these energy barriers, right? If you can go from one state to another it's going to depend on the barrier between two states Because if you are here and would like to go there You're going to need to cross this energy barrier at least in one dimensions So if something happens or not that will depend on whether you can get across the barrier And here you can of course start to talk about the energies But then we also had this temperature part, right? So at a low temperature say 300 Kelvin If I am here I might I would probably stay in the ground state here because I don't have enough energy to get away from it Technically I now and then I could get up to that energy So you could argue shouldn't I have some population here? Well the problem is to get from here to there. I would need to cross the energy barrier And if you never have enough energy to cross the energy barrier, I will never get there Well, that depends on the energy barrier, right? Remember if this energy barrier is now so high that you start being down here Yeah, of course in this case in this for a small torsion it will happen now then But imagine that this is now a double bond And you start to have an energy here that might be 50 or 100 kT, right? For a double bond will occasionally move over It's just that it's going to be so rare that we start to say that double bonds are rigid. It doesn't happen The difference here is of course that if you just increase the temperature Suddenly the temperature is going to be so high that all these states are quite likely So if you raise the temperature and then cool it down again, this is a fairly efficient way of relaxing a system So all this is going to be probabilities are complicated in that it turns out that both the trials here the minimal matter But the maximum matter too So the minima and I this was something we're going to come back to the local minima the equilibrium that's important for the distribution To know what states once you are at equilibrium if you've had a chance to sample everything in your system What is the population here and what is the population here? The maxima the barriers on the other hand, they're going to be important for what we call kinetics That is how fast can it go from one state to another? Will the reaction be able to happen? Because it doesn't help that in theory there is a good state on the other side if you can't ever get there So that don't don't assume that and this is why I said that's actually that the local maxima you mentioned They're way more important than you think but it's just we're going to come back to that Minima important for distributions equilibrium maxima are important to understand when processes happen in kinetics I will come back to that many times. Don't worry about it for now So let's head back to our real proteins. This is a small movie for protein. I made like forever ago You probably even have some water somewhere, but I don't think I actually we might not have had water in that simulation So this is a small peptide It's alanine alanine alanine s4 alanines And this is moving around pretty happily. I would guess at around 300 kelvin In this context, it's just some sort of random chemical molecule You see a number of things here first you see some sort of very rapid wiggling of atoms, right? And these are the bonds and angles If you had a large protein while these motions sure there are some large motions here, but They're kind of going to average out. They don't matter that much So they vibrate a lot, but for the large properties of the molecule, they're not going to be that important Then we also have Actually, I'm sure there is water here because the reason why this is pointing in a specific direction is that there's some water molecule out there But then you also have these slower motions that you rotate around some of the bonds here now and then They're much slower So can you start to say something about the energy barriers for these motions? yeah So that the energy barrier for the rotations here, right? Is as long as you're staying in the local part here and just vibrating back and forth There are virtually no energy barriers at all. There's nothing that prevents the rotation Now, of course, you can't have the hydrogen change place. That would be a gigantic barrier The torsions on the other hand, they're softer and larger motions But at some point for a torsion to actually rotate in around the entire way It's a physically larger motion, but there are also some larger barrier symbol And before because of these barriers, you will frequently see that it goes back and forth back and forth It keeps sampling and eventually now and then which is also the probability You certainly move over to another state and then you start to sample the other state instead This separation of motions is very characteristic of proteins. Well, any type of complicated biomolecules in general And this also eventually as you're going to see when we start looking at proteins and everything You will be able to move you will be able to treat this as even larger scales a Confirmational transition if a channels opens but moves between a closed channel state and an open channel state or something All these things can be treated with free energy. It's just a matter of the level and degree of free energy we treat at all So let's head back to the oil then and the salvation and the free energy This is what's going to happen microscopically in anything when you put oil in water That the oil will separate into an oily phase and you're going to enact this phase So this is exactly the free energy. We've been working back and forth with Given that it's 1030. I will just let give me I'll do two or three more slides and then we'll have a break So what the oil really does is that it orders my desk So if you just had a dispersion billions of small Oil droplets in water, this is what you would have But what the oil then does is that it moves from this state Until that state that is a much more ordered state Now That is bad It's certainly not good But on the other and the problem is when you were in that state We would have broken a billion hydrogen bonds in the water and that's even worse So it's not that that is a good state, but compared to the alternative. It's the least bad state So instead of having the oil mix completely with water, you have the oil form a separate phase But that's also of course why putting oil in water is going to cost you Oh oil in water then you won't dissolve it at all But because of the entropy loss here that's going to cost you free energy. So it's not something that happens spontaneously Um Given that it's 10 30 and i'm going to go in now and talk concretely about hydrogen bonds both in water and then proteins I would almost suggest so we take a break here. Uh, so let's take 15 minutes and reconvene here at a quarter to 11 And then we're going to talk with hydrogen bonds and hydrophobic effects in terms of e minus ts So last hour before the weekend here So I said that we're going to start looking at hydrogen bonds in water now, but do this Well, let's start with a hydrogen bonding vacuum. I think that's where the book starts with if you have two poor waters here and they are We can completely ignore about everything else. The reason for this is that becomes complicated when we need to start to take everything else into account What happens when these form a hydrogen bond? And how do you work with that? So this is something that I could conceivably ask you on an exam But more important. This is something that you could conceivably face for not just with hydrogen bonds But anytime you need to understand the process What happens in a sequencer and if to base what happens if a base pair breaks or forms in DNA? So the problem with most of these you actually know more about these processes than you think So the first thing to do is to set out what was the state before and what is going to be the end state afterwards What do you know about the beginning state? What do you know about the end state? Can you start to draw some conclusions of what's going to be the change in the process? And then we'll see we can use that to understand it in terms of free energy So the end state is going to be that you have formed a hydrogen bond a single hydrogen bond between two waters And I'm going to argue that The first thing here is that you're going to gain some energy the energy of one hydrogen bond Is there anybody who proposes that or doesn't think that that's a good idea? Good There is also going to be some sort of change in entropy here, right You will lose some entropy here because the two waters up there are completely free But once you get down here, they are now tied up so that What happened here is that of having two molecules that can rotate, you know I just have one molecule that can rotate And exactly how you know what in the interest of time I'm going to skip you can you can argue exactly whether you should say that it's The problem is that there is still some in here and freedom here This molecule can rotate a bit and this molecule can rotate a bit So in principle you have two molecules and each of them lost half their flexibility But effectively you have less freedom and that meant that we lost some lost some entropy here This is pretty much almost Everything we know about this there are some other things we know you know something about the sign of these terms And that's related to what you asked about this morning, right knowing the sign makes a world of difference And then I'm now going to ask you Something so turn to the person right next to you on the next bench and are you which one of this is true? So which of these terms is largest? Well start by talking to each other and try to have an idea and then try to convince the person right next to you That you're right And I'm gonna you're gonna have like one minute for this and then I'm gonna I'm not gonna tell you the answer One of this is true. I'm not going to tell you which one So did you find this easy was it completely obvious to you what this should be? So I'm going to give you one small clue Do you know what one problem here is things get complicated because you have signs And in particular starting to worry about whether something is smaller than larger than something else if there are signs involved Because five is larger than three right, but minus five is smaller than minus three If you try to hand wave about this and you try to guess what it should be you go wrong Not necessarily this time, but sooner or later So think about it this way This is the equation if And this is always true for any change and I gave you the rate that hydrogen bonds do happen So based on that you can just use the equations and solve for this So well, we can do the first part together if a hydrogen bond actually does form what is delta f Negative okay take another 30 second and and go back to that one And decide which one is true now So you're quiet, which I assume means that you got the answer Do you see how the equations help you? I don't remember this It's completely unimportant knowledge What I do if I need to solve that is say well delta f equals delta e minus t delta s That has to be smaller than zero either it was not happens delta e minus t delta s smaller than zero Add t delta s on both sides. So delta e must be smaller than t delta s Equations is not the royal pain that somebody invented for you to have to sit through mats Equations are meant so that you don't have to remember things like that The second you know and that's also why it's so important to understand this equation If you understand this equation They're going to be like 50 formulas in this course that you don't have to know by heart because you can derive them in 10 seconds with pen and paper So they help you so that's super easy then you understand everything about hydrogen bonds Well not so fast Here's the problem This was in vacuum hydrogen bonds biology doesn't happen in vacuum So in vacuum If you now have a hydrogen bond in a protein What happens here is that you have some sort of hydrogen bond forming And that means that you had two parts. We can say an OH group and a well hydroxyl group and then say a lysine or whatever They form a hydrogen bond To first approximation These side tears were not really that free to move or anything right? They're part of a large protein And one single hydrogen bond and again This is technically not true because you of course you constrain the motions of these two parts a little bit But if you start worrying about the small details, you're never going to get anywhere So dare to make assumptions to first approximation These are fairly restricted because they're all bound together So the change in entropy is not going to be that large Approximation But we still have a we will still gain the energy of the hydrogen bond So the change of free energy in this case is going to be the change in energy And so if two atoms are already close to each other, they might form a hydrogen bond in vacuum Makes sense But that's not how it happens in a real protein and solution The problem with the real protein and solutions is that those two side chains they would not be unpaired They would be making hydrogen bonds to watery, right? So that you would already have one hydrogen bond here and you would already have one hydrogen bond there Now, of course, you can get the protein side chains to make a hydrogen bond with each other instead But that just means that these two waters would likely make a hydrogen bond with each other too So suddenly the net change in energy here is zero So what now appeared so simple and beautiful on the last slide that you gain some energy In proteins, you're not going to gain any energy whatsoever when you fold the protein But what does happen now is that you had two water molecules that were roughly bound to the protein, half each These molecules will now be somewhat free to move here in solution, right? They can diffuse away from the protein So in this case, we appear to have gained some entropy Because instead of having my waters locked to the protein, the waters are free to rotate away here And it's roughly the same argument here. Each water gained half a water worth of mobility So two times 0.5 is roughly one a famous worth 2.5 Never mind. You know what I mean So the point here is that the change of free energy in this case Is actually going to be entirely due to entropy not at all energy Do you follow that? So this means that And this is why you could already why in earth do we spend so much talking about oil in a course in biophysics It's going to turn out that when proteins fold This is much more similar to what happened in the hydrophobic effect on oil, right? It's an entirely entropic effect You don't change the number of hydrogen bonds when you fold proteins. The only question is what you're making hydrogen bonds to And the protein here is very much the same thing as folding up as if it was an oil droplet water So here too, sorry you can you can hear too the way to think about this is formulate what state a is Formulate what state b is what are the differences? Can you say anything about? It's going to be impossible to say anything about the absolute values, right? But it's you can probably estimate what the differences are If you find this difficult sit down and work through it It's a remarkably powerful way of deriving simple things out of nothing Because we didn't really assume anything here, right? Some general things about how Some general things about how proteins might look like roughly and the fact that water form hydrogen bonds And suddenly we can say that in a real protein, this is going to have to be an entropic effect This is where I love the physics It's super powerful because it's not just based on a measurement So this again is of course very much related to the hydrophobic effect that if you have a purely hydrophobic compound such as octenol Most of the octenol waters will like oil better a few of them might go into water and this is going to be based on these features But how do we know that? It's true How do we know how many waters are going to be in octa versus water versus octenol and conversely? What the probability for this is Well in general you can measure the concentration, right? If you solvate something in something else you can measure how many molecules are in each state And this is where the lab comes in and the lab. This is super easy to measure so if you have Again for instance octenol you can measure how many water molecules are in the liquid octenol phase And how many are solvated in water just some concentration There is a fancy name for this and it's called Boltzmann inversion Which sounds super complicated, but it's not So normally you know what the energy is and you solve for the probability That's what's how we use Boltzmann distribution yesterday And this just means taking the logarithm of both sides, right? So that the concentrations here are going to be proportional to the probabilities and then we try to solve for the free energy So anytime you can measure something in the lab a concentration you can convert this into a free energy Measuring the energy here is actually much harder, but measuring free energies in the lab is relatively trivial And this is why it's so important for us to both understand and try to derive free energies in terms of states Because then we can start to relate states and properties of states to things you can measure directly in the lab Not just whether it happens or not, but how likely it is to happen So for instance Gunnar von Hein and my colleague here in the department We've studied insertion of membrane proteins for a long time And we want to understand the energetics what happens inside of the membrane protein As we're inside a membrane when you insert a membrane protein How much do we gain from inserting different types of residues in a membrane protein? How should you design a membrane protein? Then the way we do this is simply you create a very simple assay a test setup That we can measure depending on the amino acids I put in a helix How much of the helix inserts is it 1% or 10% or 20% The second you can measure that We get the x and then I can solve for delta g Even though the actual process is super complicated. It involves Well, it involves the gene moving from From the DNA to the air and I into the ribosome and then a translocon in a membrane It's it's insanely complicated system at the time. We didn't even have structures of these But because free energy is a state property It can't matter the way you get there can't matter The likelihood of being in a membrane can only matter whether you are in the membrane or whether you are in water So If we start look at for instance our octanol two concentrations octanol, sorry, not octanol this hexanol No hexane liquid cyclohexane even sorry. I'm tired The concentration of cyclohexane in cyclohexane is roughly nine mole smaller The concentrations equilibrium of cyclohexane in water would be to first approximation nothing, but it's a very small number Because you know this and you can measure it You can calculate that the free energy required of solvating that is roughly seven k cal per mole pretty cool, right? it's not Instinctively, it's not something that should be trivial to measure right now What is the solubility per mole of a compound or something on the lower level in the hydrophobic effect? It's likely 10-minute measurement in the lab and we can derive it This also relates to question. I got the break. I think I might have been confusing you a little bit Just before the break when I spoke about extensive and intensive units So I told you that the free energy is Extensive in the sense that it depends on the system size, right? If I double the system size the free energy was twice as large that's technically correct, but Normally if I measure the free energy just in k cals It would be correct if I have one molecule. I have five k cals if I two molecules. I would have 10 k cals But in practice It's stupid to measure normally. I want to know what is the energy per molecule And the per molecule part is that's the mole here, right? The mole is just a number of molecules if you double the size of the system This is also an extensive unit that also doubles with the size if the size of the system doubles And if you take a quotient of two extensive units, it becomes an intensive unit that does not change with the system size And that's why it's so convenient to calculate free energy in k cals per mole Because if I did this experiment I could say that yes, I gained the 19.7 mega joules That's completely pointless unless, you know, how many hexane molecules we had there But the second we reported in k cals per mole. It doesn't matter how many molecules you had in the experiment So what since the free energy there is positive. I'm not even going to ask you about that I think you will all agree that this is not really a We would say that this is not the spontaneous process right Free energy only goes downhill. You don't go uphill in free energy But that's not strictly true because there was 0.001 k Molecules per liter, right? So there were some cyclic hexane molecules that dissolved in water, but almost none And here's the difference between how we're used to talk about free energies in chemistry Maybe and the Boltzmann distribution In free in chemistry we say a positive free energy change means that it will not happen But in physics and Boltzmann distribution say well, it will happen, but it's not very likely. It's a very low probability But if that probability was zero, we could not invert the Boltzmann distribution and calculate it so It costs free energy to solve a hexane in the water. The next question is why? As you already have the handouts, there's no point in me waiting There is one formula that we're going to use to try to understand that And the good thing is that there are only three letters. Well Technically this h you could have a pv term there too, but we're going to ignore the pressure for now And then there are a couple of different ways we can actually you know what let's draw some different molecules here You can imagine having a molecule up in vacuum here You could have being the cyclohexane in cyclohexane And then you can have the single cyclohexane in water And along these different legs there are Some things we can calculate Virtually all of these calculations we get either from experiments or from solving things That is solving solving the equations so This delta g is the one that we could measure experimentally right and We can actually measure The internal energy in a system. It's not very hard So actually if you're just if you're going to heat the system, how much energy do I have to add to heat the system? That is the total energy that goes into a system And that is really related to all the interactions between the molecules in the system So And it turns out that this polar Sorry, this non-polar molecule all the interactions this molecule is going to be making with his neighbor. What type of interactions are that? sorry Dispair well funda walls in general right it's certainly not going to be electrostatics And based on what you know, is there going to be any significant difference between those interactions depending on whether you're surrounded by water molecules or other cyclohexanes? Because dispersion again to first approximation dispersion correction just depends on the electrons And the density of stuff around you there is electron both in cyclohexane and water And this sounds very counterintuitive, but it's you can actually show it mentioned that's correct So the delta H the actual because there is no significant pressure part. This is just the energy of your interactions It's the same whether you put it in water or whether you put it in Liquid hexane And if delta H is the same in both of those parts You also know that the delta H for this process, which is the salvation part that has to be zero And then you can measure By measuring the amount of heat we need to do to for instance to well if you if you have this if you have the liquid cyclohexane At the boiling point, how much energy do I have to keep adding to boil this? And that's the energy that would be required to start breaking all these interactions, right and all the entropy So that is something you can solve for sorry can measure in the lab And when we know the delta g here, you can actually solve for t delta s2 And same thing here So what ends up happening here? You can separate each of these components in some part that's enthalpy some parts that entropy and then the net effect in the process So here too, we see that when you solve it there's a water. It's almost well It's entirely entropy that's responsible for the hydrophobic effect And if you're really interested in you can look into the details here what happens with the individual interactions of it I'm not gonna those are not so important here And All this you get from a handful of experimental measurements and then using this equation back and forth try to solve solve solve for things The book goes into some more details about this if you want to follow it It's not difficult That's the only actually the difficult part is usually that there are lots of numbers involved, right? You have to keep track. What are the states? Anytime you start to Say that it's a delta h or delta g the direction of the arrow matters Because if you took this arrow and put it in the other direction all these would change sign So that be careful about these spend some time on it spend what do I have in the state a what do I have in state b How are things changing and does it make sense? Just as we defined thermodynamic temperature. I'm not going to go through this But this hey at some point I might ask you to do this you can actually Sorry, just as we define the thermodynamic temperature you can actually see how Temperature itself is also depends on the energy related to the entropy and That actually allows us if we can measure the free energy at different temperatures In the lab you can actually use this to estimate what the entropy in the system is And some of the measurements you had in the previous slides is actually bet by Measuring the free energy for a range of different temperatures and then you use that to solve what the entropy is That sounds really complicated. Can't you just calculate the entropy directly? So that comes down to this problem and entropy is not like an interaction between atom 4 and 97 You can't calculate based on this interaction between the atoms. What is the entropy? Later on you're going to start using some computer simulations and then the problem is you can never ever extract the entropy directly This is also the problem with if you're just doing quantum chemistry simulation quantum chemistry calculations Since atoms don't move in quantum chemistry. You're never going to get the entropy from quantum chemistry So we need the real simulation for this or measuring things in the lab so Let's see here No, I think that's roughly what I already covered right So the the point here is that if I know if I measure the change in free energy here With the different ones I can also count I can solve for everything And then I get both how the free energy changes how the enthalpy changes And how the entropy changes, but it's pretty much the same information. I had two slides ago, so I'll skip that one So that gives us back to the hydrophobic effect I already showed you this slide was yesterday, but now let's see if you can work through this What happens with the free energy in the different parts here when you form this clatter rate E minus ts Anybody so if you look at the water here Once you form this clatter rate structure around the benzene. What happens to the energy? No, I would actually say that because prior sorry prior to this if the prior state was that you just had the Atoms in the water, right the energy is roughly the same You still have your hydrogen bonds But what happens to entropy? It increases rapidly right so the point is you're sacrificing entropy Well, you're gaining a to be able to keep your energy Same thing here. Once you already have two parts solved here What you're doing here by merging them together you decrease the surface area So the surface area here means that you're now gaining some entropy back, right? Again, the number of hydrogen bonds is going to be constant to the first approximation But by now having a smaller surface area around this clatter rate You're gaining some entropy back And that means that all these droplets will move to the same point of water you're going to this it's going to be it's going to In practice oil will not really solve it in water at all, but you can have a separate oily face Do you believe this? Well, sounds like you don't could you imagine any way you could test this in the lab? So here's the part this is not limited to biophysics But all research and chemistry, but you need to come down with an idea something you believe in But at this point it's just a model you can have any number of models you want Remember Linus Pauling sent lots of his crazy models They're receiving a very famous interview with Linus Pauling when he's 60 years old I think it's a long after I think it's after a second Nobel Prize even and there's this us interviewer This is so professor Pauling. How does it come you have so many great ideas? And it has a well David. It starts with having lots of ideas And this is so true You can have any number of crazy ideas you want you should have But at some point you need to find a way to test it If it doesn't agree with experiments, you should likely give it up If it does agree with experiments, you made some progress because you showed that your idea was not just a good idea It's correct So if we believe this we're going to need to find a way to test this Because at this point it's just hand waving by some crazy professor who's arguing that this is the case If you have something in the chemistry lab Or have you ever tried to dissolve sugar in water? Does it dissolve well? No, not well, okay. What happens if you increase the temperature? By boiling the water right the sugar dissolves better So let's just see let's guess what would happen Before we done any measurements. Can you say something about the temperature dependence of these processes? What would you expect to happen? And how should we do that? What should we start with? There is only one equation we care about right now So delta g equals delta e minus t delta s That determines whether things happen and whether they're good or not So that if this one is really negative It's going to be really good And if it gets positive you're suddenly not going to get anything whatsoever anymore that's solvated So there are two parts of this you have an energy part and you have an entropy part and the entropy part is kind of proportional to temperature here So what parts helps you and what parts work against you when you try to solve oil and water? So they when we looked at the energy first A couple of you are shaking your head. What does that mean? Yes, this is one important thing if you think something say it Because the point is that there are no credits for staying silent in science The worst thing that can happen is that you had a noble prize winning discovery But you just sit silent in the seminar and didn't raise your voices a bit sad. You're not going to be in the paper If you think something say it The worst thing there is no there's nothing bad with being wrong The worst thing that can say the worst thing that can happen is that you're wrong. I'm wrong all the time I'm occasionally right to but So that one is kind of completely irrelevant That leaves one part from this the entropy So these ones right if you have a molecule here What are the typical interactions between oil and water? Bonds are not gonna and here's where here's where it's so good for you And why I had to force you to know what interactions do we have in a system? Bond vibrations. I think we can scratch that out angle vibrations We can scratch that out torsions. We can scratch that out electrostatics. We can scratch that out That leaves us funda walls interactions. It's funda walls interactions So that's why that's why it's so great to be able to enumerate the types of interactions We have it has to be funda walls interactions and to first approximation funda walls interactions are due to dispersions Depends roughly on the size of the atoms. There's no fundamental difference between water and some hydrocarbon So what you've said is that the salvation process and the reason why oil is not very soluble in water is because this term is bad So what happens when you're increasing temperature here if you move to 100 degrees centigrade instead? Yeah, so what's what's gonna happen to the solubility of a hydrocarbon in water as you're increasing the temperature? Try that again. Why would okay? Okay. Why would it increase? But this wasn't negative in first place. You said you just sorry Is oil known for its good solubility in water? So so the point here that this was bad at room temperature, right? Forget about this and minus t delta s is bad at room temperature And then you know you make the term significantly higher Is that going to be better or worse? It's going to be far worse So the solubility of oil you have now predicted that that's going to go down as you increase temperature So does it? So when you're boiling pasta water and if you put oil in it does the water oil suddenly dissolve when you boil it? It doesn't And this is you see how this type of solubility is completely the opposite of the solubility of a normal salt or sugar or something And this how you can test it in the lab So now we had an idea that would lead to some things that compared to normal solubility It's a bit strange the opposite effect, but this effect is actually true. We see this effect in the lab So now it's no longer just a good idea that you think that it's entropy responsible all this You've proven that entropy is responsible for it because the lab the experimental results agrees with you And you can measure this very carefully in the lab if you want to That t delta s pretty much goes up nearly with temperature So the entropy itself changes very weakly And then you add the t and that means that the term is roughly proportional to temperature There are some slight changes in delta g mostly because some of these interactions might change a bit with temperature But compared to that one it's almost constant And that means that the cost for solubility a hydrocarbon in water goes up very rapidly with temperature Yes So let's see here Part of this is going to I guess is going to have to do with water effect The other part is going to be that As you're starting to increase the temperature each and every one of these atoms is going to get a slightly higher speed, right? So that the higher speed is going to mean that you have larger kinetic energy The higher speed means that they're going to bump more into each other. That means that you're also going to have some sort of effect that Not only will you have dispersion effects, but you will likely have some energetic effects from the repulsion and everything But the important part of the delta g effect Is roughly where we were expected to be Now i'm well aware this is a bit more complicated than what you just told you Reality is always more complicated what we did we did a super simple model and this super simple model Was in particular that we assume that there is the first approximation. There is really no energy difference whatsoever Okay, that was not strictly true part of this is also the pv term I guess that we've ignored that there is a bit of pressure So that our our approximation was a bit flawed But the point is that we got to the right place anyway, so don't sweat the details dare to make bold approximations Yes, it would exactly but the point is that If our approximation had been completely wrong And that it was really either dominated it that we would have expected a solubility to go up instead So the point is that you can be pretty horribly wrong in your assumptions as long as you don't get the sign completely off So that the signs are more important much more important than you think Do you have a question or The problem I should actually look up what specific system this word that was for because that's an experimental measurement I would say that there they're probably I would guess that there are three effects First you have a small effect from this pv term So when you're increasing something you're actually increasing the volume a bit So you're performing a bit of work on the surrounding Two you're increasing the kinetic energy of the molecules just so slightly And when things start to vibrate a bit more they probably start to bump into each other a bit more Let me look up what specific molecule that that I was for and I can tell you on Monday Well, no the problem is that the delta e First we said delta e here rather than delta h. The point is that when we measure things experimentally here, right? A delta g The first approximation we would say if you just compare the solubility of moving it to water versus moving it to the liquid phase We predicted that this term would dominate the temperature dependence Let me check what the data is for and then I can tell you exactly what was the reason for that delta h If you repeat this for a number of different molecules You can actually see that it's delta g for small hydrophobic compounds Is the first approximation almost exactly proportional to the area? And this relates to what we talked about was yesterday or on Wednesday that The lennar-jones interactions are proportional to the nonpolar area So this also seems to be right if you look at say ethane Which is roughly similar to an alanine side chain Benzene which is roughly similar to phenylalanine or toluene that alcohol of ethane which is roughly similar to phenylalanine This is more important than you think These are called amino acid analogs And they're not just roughly comparable to this that we frequently measure properties of amino acids by studying the corresponding small molecule That is just the side chain and then you've added a hydrogen instead of the alpha carbon And that way we can now measure chemical properties of say phenylalanine side chains So a toluene Well, toluene it's not well nowadays is poisonous I bet we wouldn't be allowed to let you make labs on it, but it's not it's not a dangerous chemical I would say I've done lots of labs on it the great solvent So now you can basically have a bottle of phenylalanine side chains and do experiments on it And you can understand what is the property of phenylalanine side chain compared to Sorry, that should not be p that's be tyrosine that should be tyrosine and that should be phenylalanine It's probably a typo already in the book. So you can compare different side chains in bottles I'll get back to that And for all of these because we can measure it experimentally We can say roughly how much is the solubility in water the transfer free energy How much of that is enthalpy how much is with entropy and you can even calculate the heat capacity, but we won't go into that So if you now want to predict this we can actually calculate an area There are calculating an area of a small molecule is super complicated really Because how should you calculate it if you have a water? Is it important whether the oxygen can reach your atom or is it important if the hydrogen can reach your atom? You can hardly define an area because you also have hold the problem What is all the problem if you have say the electron clouds and everything you can't define a surface on these scales. Can you? Well, no the the honest answer is you can't but you can approximate So what is a water? How large is the water? Well You can calculate the volume the average we know how many water atoms there are in a liter of water All right, so you can calculate the average volume of water and if I recall correctly it's 0.0314 it's pi That's how I remember it 0.0314 cubic nanometers And if I can calculate the volume per water I could say okay So if on average if these water is was a sphere Which is isn't of course, but if I know the volume of the sphere I can calculate the radius So typical water corresponds to a sphere with a radius of roughly 1.4 extra It's not as stupid as you might think because you all you're all used to drawing waters this way Oxygen and a hydrogen and a hydrogen, right? Some of one of you asked was it on monday? I think about the lena-jones radii and everything in practice What a water looks like is you're going to have a large sphere and then you're going to have two small ears that are the hydrogens Because there are so few electrons around the hydrogens. So water is almost spherical in practice If you think about how the electrons work So if you now just take a sphere and imagine rolling the sphere all over your molecule Whether it's a protein or a small hydrophobic compound you can actually calculate what the surface is and that's something You need a computer for today, but it's not that hard This is going to be an approximate surface But the point is that the approximation is roughly the same no matter what molecule you're working on, right? And this is how we can let a computer calculate. What is the surface area of different molecules? And we can do this for protein side chains roughly how large is the accessible surface area for glycine alanine valine leucine phenylalanine And if you now plot this relative to the delta g Of transfer you can see that it's virtually a perfect proportionality The reason why some of these other molecules are further up because they also include some polar groups In this case, it's not just non-polar, but the non-polar ones here. It's exactly proportional to the surface area scarily good This even works when we inserting membrane proteins and membranes by the way It's all non-polar interactions and entropy hydrophobic effect Yes Sorry Which measurement? So the surface area you don't get in the lab the surface area we get in a computer So we give it again the Glycine side chain has doesn't have any sides, right? So there it's no at us whatsoever. It's zero The alanine is just that's corresponds to the area of a methane Roll over that surface let the computer do it. It's actually fairly complicated algorithm to do it internally But you could imagine sitting down and measuring it So that we give the we give the molecules to the computer and then we tell the computer to calculate the surface area of the molecule now Once you have this once you have Solvated something in a some sort of hydrophobic phase Think of a moving to oil The cost of then hardening it and forming a real crystal is relatively low Because you already have exactly the same interactions and everything they're not going to change that much And as you dropping the temperature, there will be a bit of an entropic effect But it's going to be less of an effect than the effect was of solvating this So The big effect is at some point you're going to need to separate the hydrophobic part from the hydrophilic parts But once you have the hydrophobic parts separately for them to somehow solidify or something is a much lower cost That might sound a bit strange but again There is a substantial difference when you have oil in water and try to well just Create an emulsion or something to get it to be solvated versus having the oil separate Once the oil is separate, it's going to be a fairly smooth process if you just drop the temperature for the oil to eventually form some sort of solid So we're going to come back to this for proteins later, but What i'm going to argue is that happens is what happens super quick Is that a protein will take all its hydrophobic parts and turn to the inside of the protein And this is not based on evolution or anything. It's physics any random molecule you put with something like side chains It would have to turn its hydrophobic parts away from the water And then you're essentially forming an oily part on the inside of the protein while you have a hydrophilic part on the outside of the protein Now that's just the initial part where you've separated things from each other And now there's going to be some sort of second process here where we gradually need to freeze the protein to actually form its real structure That's major hand waving right now. We're going to come back to that next week But the real folding of proteins and how they work is partly going to be related to this separate step So what this means for proteins is that if we now go back to the sequence As we said on monday, we have the sequence of amino acid that determined the properties the physiochemical properties Some of them are hydrophobic Gray other ones are polar blue What can now happen is that depending on what specific residues you have they will pair up in different ways And there will be a number of different processes That regulates one of them is that it's going to be good for hydrophobic parts to turn to each other Other is that it's going to be good to form hydrogen bonds in this A third part is that occasionally you might have some very strong salt What do you call a salt bridge if you have a positively charged side chain here and negatively charged side chain there They would like to interact You might have this type of disulfide bridges that you mentioned earlier today. I'm going to come back to them next week so that Depending on the free energy of all these processes and of course what amino acids you have to drive with You're going to form different types of structures in the proteins so the folding is If you now compare the protein folding to what we've talked about about salvation today Because this is a course about proteins not really salivating oil and water What folding does is that you take hydrophobic residues away from water and turn it to the inside of the protein as I showed on the last slide, right That's really the same but opposite part of salvation So if you try to solvate high say hexane, you would take hydrohexane from pure hydrohexane into water Do you see how it's the opposite process? So if you're salivating oil in water, you're trying to take one molecule in oil and put it in water If you have a stretched out protein side chain You have lots of hydrophobic groups pointing out into the water And you would like to take each of these groups and move them to the inside of the protein away from the water So you can have oil back in oil but The nice thing here is that these processes are just the same any type you talk about free energy This is just the direction the arrow is going and it's exactly the same processes And I so that I think we can use a flop like the ones we had but flip it And I so we probably we probably shouldn't try to interpret this plot when it's upside down But if you just flip the x-axis you're going to get something that describes how hydrophobic parts Fold into proteins. You just change the sign of everything And in particular in this case how protein folding Will be influenced by temperature Now this is not the protein so that the exact shape of the curves and everything is irrelevant Where things get complicated though is that there are lots of different things And here's how proteins are different from say plastic or anything else for proteins you have As we spoke about on monday, right? You have these small amino acid large amino acids You have charged amino acids. You have polar amino acids. You have non-polar amino acids And in each of these classes there are a couple of different ones And just as we said With a reasonably good approximation when you increase the temperature The solubility in water of oil What happened then that went down, right? And here's where you're going to need to think hard about keeping track of all the signs So we would now predict that at higher temperature It would be more favorable to move a hydrophobic residue away from water into the protein So that part is the temperature be good If you want to that you can actually separate each of these plots So this would be the free energy of solvates in the polar and non-polar groups. This would be what happens in folding So when you solvate them there is there are some curves for free energy enthalpy and entropy for both for polar and non-polar groups Red here is enthalpy blue is entropy and black would be the free energy So what would happen when you take these different parts and try to solvate them in water is that by default the for the non-polar groups And the polar groups to first approximation. They're roughly the opposite, right? So the net effect here On the free energy when you sum up all this was if the mixture here of the amino acid You would expect that it would be some sort of worst case to be in here But it's really good to be at very low temperatures and it's even better to be at very high temperatures For folding Well, then we're gonna need to flip it down to do it. I just I so don't expect you to follow this. This is just an example In a protein, we would have some non-polar groups. We would have some polar groups For each of these there would be effects on the enthalpy There would be effects on the entropy and there are going to be some net effects on the free energy And what i'm going to argue and this is just hand waving for now is that for a protein You're going to end up with a net delta g which in that case is the Free energy of protein folding really and the free energy of protein folding is going to discern when Is this protein stable? When does the protein want to fold into this particular shape? The native state of an iron channel or whatever and what I'm going to or what I argue now is that This is going to lead to first a very shallow curve See the black curve here is hardly changing at all in particular not compared to all the components But it's not going to be exactly flat So you will have a case where the free energy has some sort of minimum here There's going to be a very Relatively narrow range where the free energy is smaller than zero And if you go to significantly higher temperature, it's bad. And if you go to significantly lower temperature, it's also bad This is very characteristic of proteins that there are small the net free energy differences are small It's very easy to destroy a protein Boil it, denaturate it do whatever you want with it Add a mutation to a protein you can destabilize a protein by some mutations So all these things you saw in bioinformatics course that some mutations are not tolerated well That's because the free energy of folding that protein is suddenly going to be positive So you won't fold it anymore And we're now gradually in the course going to move more over into this and try to study what happens with protein When are proteins stable? What happens if you mutate a protein? Why would the protein not be stable anymore? Or how are we going to change this protein's distributions? so To sum this up if we're going to look at free energy of protein folding, it's to 90 percent hydrophobic effect That's what determines almost everything And that's why we spend so much time talking about oil. It's the same process, but in the other direction there is Roughly 10 percent polishing here that yes, you also need to pack fund of ass interactions and there could be some disulfide bridges and everything But it's primarily caused by the hydrophobic effect But this is of course where the devil is in the detail all this polishing and the specific interactions is what makes different proteins different Let's see. Yes. Oh, we have 20 minutes. I think I will get to 15 slides. Otherwise, we will save those for monday So let's just go back a little bit to the interactions. The book spends some time on electrostatics I'm going to spend just a few slides on it because it's going to be important later on for membrane proteins And we're going to come back to the protein folding many times So I already told you that electrostatics was super important because it's such a strong interaction And the main reason why electrostatics is strong is also that it it only goes as one over r So the products of the charge divided by epsilon and the radius So that means that electrostatics is not just strong. It will also have long range effects So anything you change something it's going to be important But on the other hand The permittivity here the the larger the permittivity here will scale down electrostatics. So what is epsilon here? well, for instance So that's I think the important thing here is that it depends on the surrounding right In water the relative permittivity is roughly 80 times higher than vacuum So in water You're going to have an epsilon of 80 So that's the product of the charges divided by The distance between them and multiplied by 80 in Protein another type or type in the book epsilon might be in the ballpark of two to four A normal hydrocarbon usually has an epsilon around two There's like 40 times stronger And the reason for that is that we will come back to that in a second. There's nothing to screen it Let's see what do you have I think I have so So normally what happens in water is that you can have all the water molecules screen your charge And that's why anything that's charged is relatively soluble in water Inside oil just as oil hates hates to be in water charges hate to be in oil So the second you start to put just as you definitely want all the hydrophobic parts to be inside in the protein That's charge sidechain arginine It's going to be extremely costly for you to take an arginine sidechain away from water And put that inside the protein Because the epsilon is so much lower inside the protein That's of course also hydrophobic effects, but it's hydrophobic effect working the other direction And that's why these three four five charge sidechains are So important because they have these charges and because now they're not just partial charges, but net plus or minus charges. Yes Mainly mainly because the electrostatics is so much stronger effect Remember that we spoke to the strange of interactions, right? Electrostatics is stronger than anything and in this case, it's not just a polar And if it's just a polar molecule, I would say it's roughly the same But in this case it's charged amino acids Polar molecules are effectively dipoles and then goes as one over r to the power of three These are charged and then it's just one over r If you ever see a molecule like this in a membrane protein, the model is likely wrong Likely is the keyword there. I'm going to show you some of these charges in membrane proteins later on in this course And this is where scientists got confused including us But to first approximation you do not expect to see this in a membrane protein Because it would be too expensive to have in the membrane because membranes are oily, right? so If you solve that equation that I had on the previous slide you can actually say that the cost of taking a Charge for instance an ion and put that in water is roughly 40 kilocalories is that Now you know now by now you have the tools to decide what that means So what should you compare 40 kilocalories to? kt and kt is what? 0.6 So 40 divided by 0.6. It's a bit more than 50, right? So this is e to the power of 50 10 to the power of 50 so then roughly the number of atoms on earth Yes, oh, sorry, uh, but it has 40. Yes, you're quite right 40 k k per mole. Otherwise it would be extreme Ha, I've had that slide for I think five years. Nobody has seen it before Good catch This is so expensive that it's eat my left shoe if I ever see it Can't happen You cannot have a charge in oil. It will not happen in nature Already in the 1960s 1968. I think do you know when they did when we determined the first structures of membrane protein? It's 1980s a couple of noble prizes the first ion channel that structure was determined in 1998 when I was a PhD student already in 1968 30 years earlier than that Andrew Parcetian wrote a beautiful paper where he argued about what is required How what do ion channels need to look like? And he came up with this model that an ion channel somehow has to be some sort of barrel shape To create an aqueous environment so that an ion can go through a membrane without being exposed to the hydrophobic environment Because of these reasoning otherwise it would be too expensive. It can't happen. We need to wait to shield it from the bad Polar non polar environment or it would be the energy barriers would be so high that it could never happen I think it was it's a beautiful example of a very simple theoretical reasoning He could come up with all the fundamental models roughly how it would look like and then it took us 30 years to actually see the structures The point though is that in water this costs 1.5 kcal or roughly 2 kT and 2 kT. That's that's perfectly fine And again, that's compared to have a Charges Virtually never occur inside proteins unless there is a second charge right next to it So that means that if you now take a titratable amino acid and force this inside a protein or inside a membrane They will do the same thing as within this very forward They will give up they will try to do anything to avoid this And they will even eventually give up its proton or take up a proton so it becomes neutral instead Now that will cost you other energy you will pay energy there instead But you will not have something charged in a non polar environment so If you compare a couple of things here roughly, what was the energy of a hydrogen bond? 5 kcal. Yes, and you see now how stop I am I start I stop saying per mole You're going to do that too Nobody here would ever dream of saying an absolute because that would be 10 to the minus 20 or something. It gets too complicated So in biology and biophysicists when everybody somebody says kcal, it's per mole Sloppy chemists. What can I say? kT 0.6, so this is roughly 10 kT How stable is the protein the stability energy of a protein? I haven't told you What would you guess? Can it be a megajolt per mole? No, because then we could never ever degrade it and it would it would be too complicated It would cost too much energy for us to degrade food and everything Can it be 0.1 kcal? Why not? Yes Good, so we've started to narrow this in quite a bit It actually turns out it's going to be in the ballpark of 10 maybe 15 or 20 kals per mole Even for a large protein So that the stability of a protein you're talking about maybe two hydrogen bonds three hydrogen bonds They're remarkably unstable Or you might say that it's kind of remarkable how nature they can create this Amazing large molecules and the stability protein are as stable as they have to be but not more Because you don't want to spend unnecessary energy to eventually degrade them But this is what makes them so complicated to understand. These are processes that can go in both directions This is also what makes them very sensitive to mutations, right? Because if you now mutate one residue, so you lose that hydrogen bond You just lost half your stabilization energy in the protein If you now have a second mutation so you lose that other hydrogen bond You will no longer fall that protein And that's why proteins are there are lots of mutations that you can't tolerate In particular if they're general hydrophobic residues or mutate a glutamic acid to an aspartic acid And then there are other Then there are other places if you make pretty much any mutation in that place the protein will instantly unfold We already I think already spoke a little bit what epsilon is inside a protein The book loves this and it goes through lots of details. I I want and you can kind of skip reading this in the book too What happens out in solution is that if you have something like a small ion Here the water these are really waters, but instead of drawing waters you can think of waters as small dipoles And these dipoles if you have a positively charged ion all the dipoles are going to turn their negative side towards the water and screen it And that means that the electrostatic potential decays very rapidly around that water typically there are two or three hydration layers around this and Now on the other hand inside a protein You're going to have a very low epsilon Three or so because there is you see that there are no dipoles inside this protein that can rotate to screen charges, right? And if there are no dipoles that can rotate to screen your charges It's essentially the same as having things in vacuum The reason why this is not exactly a zero is that you still have a phenomenon you have some electrons in here So the electrons will still be able to move to screen your charges There is there are electrons around every single bond There are polar There are sorry there are some partial charges and there are some small groups that will be able to rotate a little bit So it's not strictly vacuum then it would be zero. Sorry, then it would be one But that means that if you now put a protein if you put a if you put a charge out in water The effect of that charge will disappear within a nanometer or something If you put a charge inside a protein you would feel it throughout the entire protein because there is nothing screening you You can actually measure this you can measure the Dialects because it's inside different type of proteins. You can measure how it functions as a function of time and everything. We're not really going to go into detail here But here's the problem that if you ever going to calculate this in a computer, you know Coulomb's law So if I give you two charges and the radius and everything you can definitely calculate this The problem is that what is epsilon? So you put an epsilon equals 80 or epsilon equals three if you ever do this in a computer So here's the problem when we normally introduce this when you think of water when you say that water has epsilon 80 Why do you say water has epsilon 80? That's really an experimental property, right that we know that water is very efficient at screening charges But it's not a molecular property It's we know that we know effectively water screens are very well oil does not screen charges very well So when you drill down all the way to the atomic level What happens at some point if it's one millimeter you can still treat water as a liquid But at some point when you get down to a nanometer or something you're going to need to take a fact Take into account that these are really atoms and molecules. We're working with So the whole way of describing electrostatic in a media with epsilon or something that's a continuum description of it Which will eventually break down Here I think you will agree that 80 might very well work pretty well here But the reason why it's 80. It's because you have interactions here And if you included all these atoms and molecules in the interaction You wouldn't need to say that it's 80 there. You could say that is one So in a computer we always use one because we include all the interactions And why that is important is at some point if you go to Closer and closer and closer distances at some point you would be all the way down here, right? You can't say that you have epsilon 80 here because there is no longer room for any water molecules between your ions So that if you try to treat things with only bulk properties That's going to break down when you get to length scales where you can't treat the medium as bulk anymore And the reason also you can't really define epsilon So the reason why this works well in a simulation in a simulation we allow all these molecules to rotate So the main reason why it's 80 here is that all the water molecules have rotated to turn their positive charges to the negative ion And their negative charges to the positive ion And that effectively creates a system as if you had a much larger shielding effect But in a computer we can handle this really well and you will eventually So the problem is really that we can't We can't define that Yes So first in Coulomb's law There are typically two epsilons you talk about that the epsilon the permittivity of free space vacuum And then we and that is well defined. That's a physical constant So if you have two charges in vacuum and free space, we know exactly what the electrostatic interaction is The problem is that when we work with this in the lab, you rarely work in vacuum So when we handle this in the lab, if we work in water or oil or whatever it is We realize that that law doesn't hold anymore. Somehow the surround it depends on the surrounding And that's why you introduce this relative permittivity to say that oh in this case The shielding is 80 times higher than vacuum that would be water or in oil It might be a factor three higher or so So that's why in the lab if you don't take the molecular details into account We need to start introducing this relative permittivity to somehow describe the shielding My point here is that this shielding is actually a relatively simple phenomena That we can calculate if you include all the atoms Which can of course be a very large computer simulation 30 years ago. We couldn't today. We can't But if you include all the atoms and the molecules and how they are moved and how they shield each other Then we don't have to worry about these relative epsilons If you put this in a computer, I could calculate exactly what the electrostatic interactions with these would be and how this would change because I now have thousands of water molecules around it and This whole apparent difference between an epsilon on the lab scale and on epsilon the microscopic scale disappears Eventually when you're down in vacuum, you're only looking about interaction between charges And if you take all the charges including those in the water into account epsilon is one always one But provided you take them into account Which might not be feasible in mayhem. I think I already spoke a little bit about solubility of Salt in water Remember the thing that we talked about what is the average what is the average electrostatic interaction between two molecules and we talked in the ballpark of a few hundred k cal The second you have water because you have epsilon equals 80 It's like a couple of k cals And this is the caveat with electrostatic electrostatics depends a lot on the environment in particular water And this is another reason why water is one of the important molecules for some reason water is an extremely polar molecule And that means that it's very efficient at shielding charges That is the reason for the hydrophobic effect It's the reason why you have the high solubility of many compounds in water and it's also the reason why proteins can fold Because that protein folding is caused by the hydrophobic effects and the proteins that provides the driving force to protein fold Good we are perfectly on time Just some things up today in the first week We went through many of the things I touched upon on Wednesday But today we tried to use the free energy formula a bit if there's one thing you I want you to do Feel at home with these formula use These things you read in the book take pen and paper and sit down with it and try to understand it yourself Rather than just reading straight in the book And can you understand some of these very simple processes just by separating what the Energy and entropy terms are and as you also see me doing already after two days. I didn't last very long this year I I never ever bothered about separating between energy and enthalpy You see that right? It's kind of pathetic because I told you where's the you should You're going to start ignoring this too in two days because in practice The important thing for us is to separate what is energy or enthalpy versus what is entropy The specific things with the pressures and everything that we don't really care about it in this context um Free energy of processes is something we're going to come back to and this is will eventually be even more important protein So thinking in terms of this Boltzmann distributions and energies is something that's going to help you a lot I Wet your appetites a little bit by arguing that protein folding Really corresponds to some sort of separation that the hydrophobic parts have to be turned to the inside of a protein There is a word for that something called a molten globule So this is not really folded protein But it's also a pre folded state when you have done The oily parts are on the inside and the hydrophobic hydrophilic parts are on the outside We're going to come back to this many times next week But this is just the first connection that we're not doing basic physical chemistry here We're going to use this to understand proteins And surprisingly this is really cool. I haven't emphasized this way What is by far the strongest effect? The strongest interaction we know is electrostatics, right? And hydrogen bonds That's kind of an electrostatic effect. You agree with that too, right? So the strongest effect we have is hydrogen bonds and hydrogen bonds is an electrostatic effect Hydrogen bonds, it's what's responsible for the hydrophobic effect You agree with that too, right? So the hydrophobic effect It's not a done an electrostatic effect. It's not being an entropic effect There have been generations of scientists that are going wrong with that. So electrostatics Electrostatics in water and the hydrophobic effect to determine whether things are solvated Absurdly enough at the end of the day, it ends up not being electrostatics at all But this manifests itself through entropy Why? Last question and I will stop So this has to do with what I said on Wednesday Because these interactions are so strong Water will do anything required to maintain them So we will not lose the electrostatic interactions, but you're going to maintain the electrostatic interactions But while maintaining the electrostatic interactions, you're going to be paying and you're paying with entropy And that's why the hydrophobic effect ends up being an entropic effect So this course is increasingly going to be more about entropy than interactions. Entropy is super important Most of the things I told today are chapters five and six of this book I'm not doing this a double pace, by the way That this book is meant that each chapter is supposed to be like a 40 45 minute lecture So that's why we typically go through two lectures per lecture here I have a bunch of study questions that we will go through on Monday to spend the afternoon here Well, no, you're going to have fun this afternoon spend some time during the weekend Sunday evening or so to go through this And then we'll talk about them on Monday morning and on Monday afternoon. You're also going to have labs I Hope that I have an email with all the locations of the classrooms in the course now I will post that on the course webpage Good. Enjoy the weekend