 Good afternoon. I would suggest we get started I'll do this in a right. Let's see And I know actually all my slides are in English. I'll do this in English no matter what I always get so confused when I talk Swedish group meetings or anything I Hope you had a nice weekend. I got a chance to Browse through at least some of the study questions and start reading if you haven't done it yet This is a last reminder to get started and reading not just the lecture slides, but also the book What I'm gonna do today is head back more into physics that I would suspect that most of you are a bit more familiar with than the chemistry We're gonna head come back to the chemistry a bit but not so much the low-level chemistry about molecules but talk more about proteins later in this week and on Friday, I'm gonna go through a bunch of different globular and Fibrous proteins and the next Monday I'm likely gonna have Lucy give a talk about membrane proteins It's a bit of a shame because it's our research and I love talking about that But I have another engagement at Stockholm University now on Friday we spoke a lot about amino acids at protein building blocks and went through a bunch of their chemistry The fact that they are chiral that is super important We spoke a little bit about the degrees of freedom. We have in proteins and the reason for doing this is that Compared to the molecules you're so used to say from physics or so a Protein is so complicated that you can't try to understand it By approaching 1500 or worse 15,000 or 150,000 atoms at once You need to simplify it and the way we simplify this is that on each level We try to focus on the key degrees of freedom the key interactions and then somehow simplify this into higher-level building blocks And when it came to the amino acids We relatively early drew the conclusion that the most important degree of freedom for them are these rotations around bonds Not because they're the highest energy and we'll come back today in A slightly different manner and argue why these are important and in particular I picked out these two torsions the Phi and Psi torsions and The reason I picked them is that they are soft enough that I can rotate around them in contrast to the peptide wand Which is of course that having the peptide bond is super important But once you form the peptide bond it tends to be ridded so it doesn't change the Functions of proteins are they were determined by these high-tech characteristics We spoke a little bit about two important results by Amphinsen and Leventhal And that might very well be a good way of repeating things. What did Amphinsen say and what did Leventhal say? classical exam question Close but no cigar Very close, but there is one key word that missed before energy That you actually we haven't been through but it's a free empty free empty And as you will see later today, there is might be a world of difference between those two So what did Leventhal say? Yes Yes So this is the problem right that Amphinsen has shown a beautiful experiment that a protein doesn't need the cell You can refold the proteins but as under the right conditions a protein like any other stupid molecule will refold Spontaneously in a test tube so it has to be governed entirely by the laws of physics But on the other hand we know just looking at these five psi torsions Which is an extremely simplified way of looking at it But this is where you see the beauty of simplicity right if we just ignore everything else and just Dare to make this order of magnitude Approximations and just assuming that there can be two three states per residue by the time you have hundred residues It's such an astronomically large number of states that there is no way nature can test all of these within a millisecond and find the right states So there is something here. We don't understand One thing that is going to help solve that a bit has to do with this Hierarchy of structures and we introduced the secondary structure elements in particular the alpha helix and the beta sheet that I will come back to today and The reason why it made sense to introduce those writers that there were these regions in this ramachandran diagram Describing the fine psi torsions and there were a few large islands that were common to see and these very much Corresponded to the alpha helix and the beta sheet and because they are so common And you will see that today that once you start forming say an alpha helix That tends to grow. It's very rare Well, you will never have two residue in a helix and then one residue in a sheet and then one residue in a helix So that you tend to have regions in a protein that form a larger building block and this hierarchy of structure will partly help explain Leventhal's paradox We also came to the conclusion that again in the interest of simplicity It makes a lot of sense to describe things with classical interactions not because We can't handle a well part of this is because we can't handle the quantum chemistry But part of it is that virtually almost everything we're going to explain in biology does not need quantum chemistry We rarely break or form bonds on the other hand some of the things that To be able to treat quantum chemistry with computers you need to introduce a bunch of approximations such as atoms not moving zero Kelvin You can't have any solvent and at that point those solutions would be completely horrible from a biological point of view So in biology it actually it's not that we are not doing the right thing It's actually quantum chemistry while it might seem More accurate the solutions that we tend to introduce when we do physics actually turn out to be horrible from biological point of view And then I very briefly touched upon hydrogen bonds, but I'm gonna come back and spend a lot more time on that today So based on all these things that we went through last week You could even formulate some sort of interaction description of a molecule here Let's see if we can even get it moving a bit We have a bunch of bonds that you can describe if we don't have any better form for it. I'll just use an harmonic We have a number of angles This particular small water molecule is actually completely rigid, but you could imagine letting the angel here very We had these torsions or dihedral's where bonds are rotating forget about this term for now It's not very important. Do you have electrostatics charges interacting and we have these Lenard Jones interactions that Describe first that atoms can't overlap, which is really the Pauli exclusion principle and At long distances these induced dipoles will mean that all atoms interact and at some point even noble gases will condense There is a name for this and this was this was very much the brainchild of Ari Warshall and Schneerlitz on in the 1960s well So they coined the term force field for this which are really horrible simplified Approximations, but the point is that they are there are lots of constants here and the constants are something we can parametrize from experiments And that's the beauty in all this we can cheat because I know what the density of water is So I can take that model on the left and parametrize it to make sure that I roughly reproduce the properties of liquid water And that suddenly means that in contrast to quantum chemistry. I can simulate water I can simulate freezing of water. I can simulate how water is boiling I can try to calculate how much it will cost to solve it hydrocarbon in water I can calculate the diffusion of water all these things that are actually super important if I actually want to look at water What I can't describe though is that if you want to form some HCO plus science or it's minus science So I can't explain the processes of the hydrogens Well, the hydrogen leaving the H2O plus to form an H3O plus and know it's minus So there are limitations with everything, but in biology. This is gonna life sciences This is usually way more efficient than quantum chemistry So today I'm gonna continue this I'm gonna talk a little bit more about hydrogen bonds and then We are not really actually I'm not gonna gonna talk so much about ramachandran plots I'm fairly quickly gonna head into energy landscapes And then the Boltzmann distribution that I know you're a bit familiar with or you would better be But we're gonna process from a slightly different point here In this course, it's important that you get understand the Boltzmann distribution and get a gut feeling for what it actually will mean for biology and that turns out to be a bit different in physics and At the end if we have time we're gonna speak about electrostatics and charges in proteins biological material But hydrogen bonds first this was the few slides I didn't have time to cover on Friday Given all these interactions we have hydrogen bonds are kind of easy, but also kind of complicated So hydrogen bonds on the lowest level if you want to understand this on the quantum level It's pretty complicated and that has to do with the electron clouds around the oxygen and a few other atoms in particular nitrogen Where you actually have a tetrahedral shape of these orbitals And when these oxygen participates in bond it's will form two bonds with the hydrogens But you're also gonna so that makes it happen protos up there But in addition to those two bonds you're also gonna have two free pair of no two pairs of Electrons here that are pointing in the other directions like small ears or something and Because the oxygen has stolen a bit of the electrons from the hydrogens that it means that the hydrogens will have a not Just small but actually pretty large roughly 0.04. No, sorry 0.4 Positive charts while the oxygen is fairly negatively charged in particular out at this years So if we now take two such water molecules and put them close to each other That hydrogen which is positively charged is gonna love to interact with that negatively charged Electron orbital on another water so you form this hydrogen bond that we dash but in practice the hydrogen bonds They're really strong. They're not as strong as a real bond that but they're far stronger than the normal electrostatic interaction And then there are a ton of details about the angles you can form and everything is not just an electrostatic interaction It should be plus minus 30 degrees from there to there to there. Let's not go into those details But there's one thing that we need to know how strong are they? Roughly 4 to 5 kilo calories per mole or say 15 to 20 kilo joules per mole That is different in life sciences biology biophysics versus traditional physics There will be some numbers you have to learn by heart and the reason why you need to learn them by heart is that Physics kind of involves all scales of length time and energy, right? But in life sciences, there are some characteristic scales. These interactions are gonna be important And that means that if any other if another energy is significantly larger than this It might help to break a hydrogen bond, but if it's significantly lower, it won't so say a few kilo calories per mole 4 to 5 These as we're gonna see later are super important in forming alpha helices You have tons of hydrogen bonds formed here in the helix and that explains why this is a fairly stable secondary structure element, right? But once you form them, you don't want to disrupt them because you would lose a lot of energy and Similarly in beta sheets there are a ton of hydrogen bonds between the different sheets here and there too That's why it's a stable secondary structure element And on an even higher level it also explains this entire DNA spiral the double helix All the bases between them they're kept together by hydrogen bonds Which means that once you go through this entire replication or translation We're gonna need some proteins with enough energy to actually break this apart, which is not entirely trivial That's both good and bad So would it be what would happen if you had a very low interaction energy here if it was easy to tear DNA apart Would it be good? So first DNA would not be a stable right and it would likely you would have more reading It would have more errors in DNA and everything and it would hurt your genetic material So but on the other end then why don't we make it real bonds? Why don't we make it 10 times higher 50 kilo calories per mole surely that would be even better, right? That would even less DNA damage Exactly right because suddenly you would need a diet that would be 10,000 kilo calories per day Because the energy for your cells to work we need to replicate the DNAs your cells also need to tear it apart And for some reason virtually all processes in life Has optimized this again 4.3 billion years of trial and error nature tends to optimize processes so that they cost as much energy as required But not more So that this tends to be optimized to optimize well accepting some errors in DNA because we have other mechanisms that can control and fix errors But not so much errors that we end up throwing too many proteins away because there were too many errors and exactly Why how it has been optimized without though actually we know a bit, but that's a separate story And you might already now start to guess that this is gonna have something to do with protein folding too that whenever a protein chain stretched out there will be some residues that Well depending on whether hydrophobic or hydrophilic They will form hydrogen bonds either with water or they will have to organize around the hydrophobic groups And then when we fold the proteins some things will happen here. We're gonna we're gonna disrupt some hydrogen bonds We might form some new hydrogen bonds And we might simply have a better split between hydrophobic and hydrophilic groups, but we'll come back to that in a few minutes But this so-called hydrophobic effect is gonna be one important interaction So if we just hand wave her a bit, there are a few interactions. We might have bond lengths We might there will also be some angles that could be important We have some linear Jones interactions and also charges that I haven't written here You have some torsion angles and then in some sort of hand-waving fashion that we're not quite sure about yet There is some sort of hydrophobic effect oil oily stuff hydrocarbons. They should not be in water They want to turn away from water, but exactly why you don't know yet And in general if we now take this molecule and this is a still a tiny molecule just an amino acid and if we place This in lots of different conformations Well depending on what confirmation it has it's gonna have an energy that's better or worse Remember that force field plot I showed you that you can given the position of all atoms you can calculate that energy Well, in fact, you're gonna let a computer do it But for every confirmation, there is an energy we can't put on the molecule whether you can calculate it or not And at least any practice again, this is what is it 20 atoms that would mean a 60 dimensional function pretty complicated We're not even gonna try to visualize that but it's important it's very useful as some sort of mind tool to think of a multi-dimensional thing and Because we're humans. It's hard to think in more than three dimensions And that usually means that we drew they draw things as a function of two variables So you have the function value on the z axis that is again a function of some sort of arbitrary x and y variables here I don't care what those variables are in principle. There should be 60 axis is here, but that's too much But in general depending on how the move we molecule there will be some things that are higher energy and there will be some things that are lower energy and to move between these you will have to cross barriers and The obvious question then is what is good versus bad here? And since you're a physicist I can you will probably agree with right away that having low energy is good Having high energy is bad. The question is how bad is that and how good is that is that good enough? And is that too bad? Will if we if we are here, will it be too costly to cross that barrier to go down there? And the short answer that is of course it depends But we want to be slightly more quantitative than it depends And that's where it comes in that there are natural energy scales in biophysics. It's not just energy energy, right? So what was that characteristic energy that we were thinking of? That's one example there. Sorry Yes, but how many well four to five And it actually you are you partly right because in a large protein You could imagine maybe having ten hydrogen bonds or so that actually sounds a little bit It's not a whole lot more than that But you're talking about single to possible that did double-digit kcals We're never talking about mega joules, and we're never talking about millijoules And that means that there is a natural Z axis here, right? There will be I Biked here today And given that how I see this outside when you're back in the suddenly you're the entire bike is shaking because there Is small pieces of ice on the road on there. I don't really care, right because the bike would just go over it I don't care about it and then well there might be a barrier of ten centimeters of ice I can still get over that but I have to go a bit slower on the other hand the big two meter barrier Or so I'm not even going to try to go over it and that's different than in physics that there are characteristic scales I don't care about the one centimeter the one millimeter. I won't even feel Ten centimeters starts to matter and one meter is impossible Which is again compared to all the possible lengths of scales in the universe is a very narrow range there that determines whether something is possible or not Being that you are physicists you probably all know what's going to determine the distributions between these and that's the Boltzmann distribution Have all of you worked with the Boltzmann distribution have all of you taken a step back course Statistical mechanics One brave soul, okay, but that's great Because then actually there might be a point in me driving this. Are you familiar with the Boltzmann distribution? This is a seemingly very simple equation, but I would argue it's one of the deepest in physics Why is this deep? Well The reason why this is deep there is famous saying that and I'm not sure whether how much of a rumor it is that when Albert Einstein was One asked is there one branch of physics that's never ever going to be overturned and then his point was right That's been my statistical mechanics Or even thermodynamics and the point is that this is not based on observation things like quarks or anything That's a model based on the observations. We have this far, right? But what makes many of these things complicated, they're not really based on observations They're based on general reasoning about arbitrary systems We don't necessarily have to make assumptions and because we don't make assumptions There are no assumptions here that are based on present observations And of course because it's not based on observations. It's not really going to be overturned by new observations But that also what makes it complicated And I there is a famous there's a famous quote in the in a book Statistical mechanics written by David Goodstein They talk about how Boltzmann died for his own hand and then Aaron fest one of his students also committed suicide I think the student of Aaron fest also committed suicide and say now it's out to interpose statistical mechanics Perhaps advice to do it with a bit of caution As that make has this reputation for being super super difficult, and it's sure if you're gonna be a professor of statistical mechanics It is difficult But just understanding and getting this gut feeling isn't really that difficult, but you have to let it take time So what this ultimately Determines is that as a function of an energy in some sort of arbitrary state You can think of putting this molecule in a particular confirmation But this is really any molecule system anything Depending on what what the energy is how likely is it to observe the molecule in that state which we call density or row and That describes for instance of a gas You might have heard that you probably might have heard of the maximum Boltzmann distribution at low temperature The average velocity of gas molecules is fairly low, but you have a distribution of them And as the temperature goes up the average velocity of the gas molecules becomes higher and higher and higher It's essentially proportional to mass times the law the average velocity squared multiplied by the mass divided by two And that goes for most things that if you're boiling water, of course the you're adding energy The water becomes hotter and at some point it's going to become so hot that it becomes water vapor instead of liquid water It's possible to derive this for a completely general system and we're gonna do that tomorrow If I were to do that now half at least in the times when I had this class and half of the class are biologists They would die if I started with that so we're gonna do what the book on protein physics actually does and start by deriving this for a special system Simply because to make this a little bit more accessible to you It's gonna turn out at the end of the day all the properties of this special system will disappear and the special system that the book shows that that I will follow is that Think of they have a sort of pillar full of gas And we all you can probably all say that the pillar could even be the earth's ecosystem The density of gas is higher the further down you are saying the atmosphere well very far up in the atmosphere You're gonna have lower density And the reason for that is that well if all the gas The higher up your heart the higher the potential energy is right So all other things equal You would like to be lower down and you can test that yes If you just drop anything it won't slow potential energy But on the other hand we can't have all the gas molecules down here because all the gas molecules were down here Well, the energy here would be close to this or the density here would be close to infinite than that would be horribly bad So some of the gas molecules will have to be further up But it makes sense to have more of them down here and then so sort of gradient going up And since we're anyway simplifying there's no point. Well unless you love the spherical polar coordinates and everything There's no point in doing this for the earth, right? Let's just assume that we have some sort of very narrow vessel that just goes up and there is some sort of variable here that determines the height and As a function of height we have some sort of number of gas molecules or density of gas molecules And that's what we would like to determine So as a function of the height how many gas molecules do we have or density and the two opposing forces as I drawn there already is that We have Sorry, that's a bit we have gravity pushing down all the motor molecules So all the air gas molecules would like to go down on the other hand We have the pressure acting in the other direction And these are equations that you've gone through in undergraduate physics and that's the reason why it's much easier to do the simple system So then we're gonna have a couple of equations here They're not so I'm gonna take the reason why I don't draw this on the I might draw something on the blackboard But it's it's not particularly difficult and I think I've added all the intermediate steps here We have the gas law and all the chemists know this by heart. It's PV equals NRT, but your physicists So let's replace that R with a K PV equals NKT What is the difference between these two? Yes So chemists are used to working in the lab and in practice and this is the same thing with natural scale your physicists you It's obvious you count the molecules But a chemist would never dream of adding six times ten to the six times ten to the twenty molecules They want to add 0.05 moles of the molecule. It makes much more sense when you're sitting in the lab. So for the chemists, it's easier I Even said that those hydrogen bonds do you remember with a 4.5 kilo Joules per mole 4.5 kilo Joules for one bond that would be an insanely high energy But rather saying oh six four point five times ten to the minus 20th Joule That's difficult to work with the chemist everything in chemistry. We like to calculate per mole and you're gonna do it, too Pick any here, but I'm not going to do it for both. So let's just stick to KT and that makes sense We're gonna see the results so in principle the Clapeyro's law that I hope you know the gas law PV equals NKT We have the pressure and we have a volume. We have a total number of Molecules at least in this small field. We're looking at the small yellow band on the previous slide and then Boltzmann's constant K and the temperature T And in principle, I could introduce an arm I could introduce say what the volume is I could say what the area of the vessel is and if we love to do bookkeeping That's not gonna hurt us But it also makes sense, you know what let's see we can factor out some things so there are fewer things We have to care about and one of the things you can do on the left-hand side there if we divide both sides by the volume I get that well just pee on the left-hand side and then the number of atoms per volume They multiply by KT That is much nicer because let's just say that let's calculate per volume And then I don't have to care about the area of the vessel or anything So that then we introduce the lower case n and the lower case n is just the number of water molecules per volume So then we simplify Clapeyro's law a bit to say P equals NKT again Not really that is not a lack of generalization So then we know how the pressure varies with the number of particles and remember the number of particles is a function of the height H and the pressure is also function of the height in principle those should be functions, but I'm lazy when I wrote these equations and Then if I want to see how things are changing as a function of the height It makes us to see what the derivatives there are well if it's only P and N that varies the derivative of P with respect to the height That's going to be the derivative of N with respect to the height and K and here just constants So they just Remain constants. Do you follow me that far? I haven't really done a whole lot there So this describes How the pressure components acting the upwards pressure the other part is the downwards pressure that comes from gravity and I'm gonna hand wave just a little bit here that in that small yellow band We had we have a certain mass and That mass is a certain energy What is pressure? Well pressure is force per area But per area we can forget about because we calculated everything per area per volume so that area part disappears But I'm gonna need to calculate how the energy changes because forces the derivative of energy, right? So that the total energy of in that small the potential energy in that small band is going to be the mass of the molecules multiplied by the gravitational constant multiplied by The height at where we are And if we want to And that's reach molecule and if I now want to look at how many such molecules do I have in that yellow band? Well, that would be the D8 the width of that yellow band, right? so the mass Multiplied by the gravitational constant Multiplied by the small D8 said that's how we're changing and then the number of molecules per volume That's the total change per volume when we go up But then I want to calculate how much this is changing because that's right. That was the energy. That's the change in energy So the derivative would then be How that is changing relative to the end, right? I'll come back to that in a second. So again pushing Up on the previous slide. We had the change in pressure as we go upwards dp dh Equals the derivative of the number of particles with respect to the height multiplied by kt. That was acting up In the other direction now, we have to be careful We already had this mgn term pushing down But that is pushing down and here I am if I'm going to put this equal I Will have to swap the sign here, right? Because that's one of the acting in different directions So that if I'm moving up the number of Yes, if I'm moving up, it's easier to think of this way if mgn is positive Mass is positive g is positive and the number of particles is always positive If that is positive that minus sign would make the right-hand side negative when I go up And we also know with our gut feeling right as we go up We expect the number of particles to decrease so this derivative should be negative, too It's not entirely easy to follow otherwise, but the difference is that if we're looking at the Gravitational part we're thinking of how there's this pushing down And if I want to put that equal to the force pushing up, they are equal, but they're equal in magnitude They're not equal in directions to actually put them equal I will have to swap the sign of one of them to say that they're equal And again in this case by far the easiest thing is to try to reason and realize it Do I have the right sign? If I go up a bit, how should that term change this sign and how should that term be a sign? This is not as difficult as it seems so that we now have the derivative of the number of particles with respect to the height multiplied by this constant equals minus mgn and Then I just move things around a bit here The kt I can divide on both sides. I have the derivative of n equals some sort of constant multiplied by n itself And remember n was a function n is a function of h Do you follow me that far? And you are physicists you should probably be able to guess what form n of h has So if I have a function and then I take the derivative and I get the same function, but a constant in front of it Yes, it should be exponential And you are it would be perfect actually what I would have said I would recognize it Okay, it must be an exponential and that must be the constant up in the exponential if you don't know that another way and again This is one of the things that in if you're a mathematician This might be obvious, but the way to derive this if you're not the physicist And you haven't worked all your life with exponentials is to remember that the law that if you derive a logarithm you get If you derive a logarithm of a function you get one divided by the function multiplied by the derivative of the function And that means that we can take this The derivative of the log that's exactly what I had on the last slide, right? If I take n here and divide n by both sides I have one divided by n multiplied the derivative is equal to a constant So that would correspond to that and that's what I had here, right? So that's the derivative of the logarithm of n is equal to that constant that I integrate that So the logarithm of n is that constant multiplied by n And then I take the exponential of both sides so that we say that n is Proportional to an exponential function And then minus mgh divided by kt. Why is it just proportional and not equal to an exponential function? So what happens here in this step, right? When we integrate the left-hand side there and the right-hand side I should add a constant in the integration But then when I take the exponential That constant there if you remember your exponential laws that will mean that the constant will show up in front of the exponential there Will be an arbitrary constant. I have to confess that I am incredibly sloppy with this and In constant the mathematicians in life. So yeah, whatever. It's just equal and the proportion to is roughly the same thing You might even see me saying so that n equals an exponential and then we implicitly think well Yeah, but you're gonna normalize it at some point and I could cheat and say that well technically I didn't say what they what the units of n is so I can have that constant in my units But right now we have no idea how we're going to normalize it. So let's forget about that constant for now So what they just say that in this case? We're saying that the number of particles at the height in this vessel is Proportional to the exponential function and then we look at this and it just a mgh just turns out to be equal to the energy, right? Sorry, this should not even be if I have that delta energy here This should be the energy relative to the zero point So the number of particles with the density is homo proportional to the exponential of minus an energy divided by Katie But completely arbitrary, I'm sorry completely arbitrary special case I haven't proven this in general, but this is uniquely valid and it has nothing to do with the system as you're gonna see tomorrow That you might have seen before But the important part for our course is to think about what this means So if we look at two states and again think of a molecule if you want to but this could be any state a system could be in So the probability of being in an arbitrary state a probability or density to be with energy a that's a constant multiplied by the exponential to minus the energy in the state a divided by Katie And it's exactly the same thing for the B states the probability of being in the B state is the same constant the same Exponential and everything but now it's the energy be there This is what you're going to be playing around with in your first hand in task And even though you've known the Boltzmann distribution the point of this hand in task is to give you a gut feeling for this what this will really mean in a real system and how large again depending on what the energy scales are What do we mean by a low versus high probability? And in particular that this constant is fairly irritating. I have no idea what this constant is But here's where physics or at least math comes to the rescue It's very rare that I just want to know what is the probability of being in an arbitrary state In most cases, it's enough. How likely is state a compared to state B? I always want to look at differences and The second if I want to look at differences I can take the quote out these two right and the beautiful thing then I can just strike out the seas they disappear and That might initially look like a horrible complicated the quote of exponentials But again, if you know your exponential laws, that means that you can take the difference Sorry, this can be expressed as one exponential and then you just put the difference between those two energies and the exponent there is that So that the probability the relative probability of these two states is an exponential minus the difference in energy divided by kt That in particular it means it doesn't matter where our zero point was whether the zero point of this height in this cylinder was at Sea level or starting from this floor or from the basement. It doesn't matter Because I just look at the relative difference That is going to rescue us all the time in proteins to be I don't care what the zero level is I'm just looking at relative things The other thing we generally say that the lower the energy is in a state the more populated it will be and Again, if you know your mathematics and this might sound stupid, but you have no idea how fast the exponential function grows Do you know that the exponential raised to the power of 50? Is that a large number? How large? More than the number of atoms in the universe That the exponent I like I think the exponential to the power of 20 is the number of possible moves in go so that It's insane completely insane and that also means that the second and energy starts going up And you might think that it's it just doubled just with up a tiny bit You will very quickly get to the point where you have almost no density whatsoever there So almost everything will be in the lowest level state in theory You might have a little bit a higher level up, but there is a very strong driving force to go to the lowest level state And that goes back to Anfisa's observation, right? And that's why well this far we have it just that energy which means that the next slide is a beautiful illustration There is one complication here that I was lying to you Assuming and I realize and I said this is a really bad slide because these four shapes should have had the same volume Sorry about that If we assume that they have the same volume the first approximation And if we did the matter which one remember what I said, it's good. It's good to be low level, right? Actually, you can't compare those two so we focus on those two which one is worse than which one is best If you had to put gas in them, which one the same amount of gas in those two triangles, which one would have the lowest energy? Why Does everyone agree and in some time I'm gonna have a peer challenge later on so this case you're quite right, but So exactly where in the equation did that enter? So that the problem here is that I only looked at energy right In the end of potential energy here, how is the potential energy there different from the potential energy there? Yeah, so the problem is not captured in our equation So there is something we're missing and The part we are missing is that I kind of assumed that it was equal right so that the number of atoms you could have Into it we can have more here. I think we all agree. How do we measure more? We can think in terms of volume and if we have And here it's not ideal either because I would this is infinitesimally small Assuming that this was ten times wider here. We would intrinsically think we can have ten times more here than there, right? So somehow we need to weigh this by the amount of volume And that's fairly easy So let's just say that state a has a particular volume VA and state B has a particular volume VB That's another thing that I know you might have done this in physics before but for the biologists students I keep going through this Don't be afraid of assuming I Sadly, I think it's It comes back to this way I think the way we teach math and physics is completely stupid because we we only ask you to solve problems that we know the answer And we're going to instruct you so that you know when you start solving the problem that you have the right answer, right? That's how you do that exams, you know that this is the differential equation either, you know exactly how to solve it Or you have no idea This carrot part is a research is like the second one when I again if I know how to solve a problem is not research When I need to solve a problem I start with paper and pen and I sit and write and sometimes it works And if I work the week well if I worked an afternoon on this problem, and I don't get anywhere Then I probably try to take a step back. Let's look in the literature. Is there a smarter way to approach this? And it's the same thing here. Don't don't be afraid of making assumptions The app what is the absolute worst thing that can happen is that we don't get further And then we're gonna need to deal with that problem then so there's something I don't know what it is just introduce it as a constant and will either will be able to factor it out later or we'll deal it with it when that happens So that the number of states if I have two states where I have a certain energy There are two places where I can put balls or molecules or something so I think you all agree that Making this proportional to the volume makes sense, right? So that the proportional of being a volume a or state a and the proportional to be in volume B Well, I should multiply these probabilities with the volumes I take exactly the same terms I had in the last slide, but I now put the volumes in front of them Say volume a is now much much much larger even though the energies are the same if volume a is much larger It's better to be an a more likely to be an a 10 times larger. It's 10 times like here The only problem is that there are now a one two three four constants here. Yeah, that's fun not But let's Let's yeah, sorry No, so that that's a good question said that the energy is a function of age But at this function rather than assuming that the area of the volume was same everywhere, right? We now allow the volume and I know here I'm not necessarily talking about the age just but for any state you can be in there is an energy in the state and That we determined by the energy, but there is also somehow a size of this state So if I look at the previous slide the energy there is the same as the energy there, but you have more volume here So if you have the same energy But in one if I have the same energy here But I have one state there and I have the same in a year, but there are three states here We're gonna see three times more particles there than that, right? But I this I'm not sure about you but having these volumes and everything complicates things So we can use a very simple trick and here to The proof is in the eating of the pudding you're allowed to introduce absolutely anything that's mathematically valid and then we'll see how far we get So I can just say that the volume well I could write that the exponential of the logarithm of the volume. That's just a stupid expansion But if I now do that That would mean that the logarithm of the volume well then I have two exponentials right an exponential multiplied by another exponential So the logarithm of the volume would then appear inside the exponential So I could write this complicated expression here I say that the logarithm of the volume a there plus the logarithm of the volume be there And then I can even multiply them by kt and this again This might look completely horrible, but the point is that if you go back two sides This looks exactly like the Boltzmann distribution But instead of the energy is there so energy minus Kt logarithm of volume So it looks exactly like the Boltzmann distribution, but it doesn't just have the energy it has a small volume component there, too So it says e minus tk l and v That has looked beautiful and simple to you In principle, we're gonna have a break back. I hate two more minutes to finish this thought How do we simplify that because there's many things but not beautiful Can you simplify it? So at some point no, we don't know really know what the volume is, right? So I can't say anything about the wall because again it now forget about that gas system We're gonna argue things that are universally valid and there are no universal truths about volume Well, they have to be positive, which is good so I can take the logarithm of it, but that's all I can say So what you do if a first is if you can't simplify things further we define So I'm not sure about you, but I don't like So this is where we introduce on the let's just take the horrible k l and v things and introduce on the word let's call it s Have if you whether you've touched upon entropy or not before it doesn't matter Entropy is not something you can deeply understand or have a gut feeling for it This is just a constant in our equations and then we introduce a name for it So this somehow describes the property whether there are many ways of putting molecules It's of course. It's kind of related to volume But if you don't like to think in terms of cubic centimeters or something This is something. It's just a letter that somehow describes the system It describes that there is lots of freedom in the large bowl But very little freedom in the pointy ball and I don't introduce temperature. Why don't I introduce temperature? Because the temperature might change right and if the temperature changes the shapes of those vessels don't change So it's nice. I don't want something that this should only depend on the vessel or the system while the temperature depends on well the temperature So that horrible expression that I had on the last slide now In each of these is a natural e minus t and then k l and v just says s So it says e minus ts there and e minus ts there. So that's or seemingly a horrible expression just when I had the Boltzmann distribution when I did not care about the volume I just said that the difference or something was proportional to the exponential minus delta e the difference in energy divided by kt And if I just introduce this topic concept instead e minus ts I get exactly the same thing It looks exactly like a Boltzmann distribution But this now also accounts for this that different states might have different volumes or different things So sort of internal property of the state. I'm going to come back to that after the break. We're so not done with entropy But the point with entropy is that trust your maths and physics here. Don't try to have a gut feeling for this This is easy, right? This is something we have chosen to define We're going to see after the break that it makes sense, but that's just a bonus point. I Stole two minutes here. So let's meet here seven minutes past and we'll continue. All right Let's jump back into the deep side of the energy and entropy point. I Know that some of you told me during the break that you take it course is so statistical physics and everything We're going to come back to that more both today and tomorrow But the point is right now that there are two causes there are energy And I think that all of you have a gut feeling with what energy is that when energy is high There is lots of things and I don't start moving fast and what we somehow need to Check now is whether can we have some sort of gut feeling for this entropy thing? The main reason for introducing entropy is really that it makes our life simpler rather than having to worry about Volumes and everything there is something in that we can say that it's a property of the system or At least the property of the state the system is in right now And if we introduce this this makes our equations very simple and when our equations are simple We're happy because we can't handle them and that means that instead of just saying comparing the energies of two systems I can again this F the free energy I can compare the free energy between two systems and it's a simple world that we can just have the wall-spin distribution and we're happy This my friends is something that you need to know I should be able to wake you up at 3 a.m. In the morning and you should say f equals e minus ts If you don't know that by the exam, I will literally kill you And I'm not sure where to go that but pretty much the rest of the course is going to be about this equation This is way deeper than you think If you think that the exponential error, this is a fairly easy equation as simple This is a deep profound equation that can actually be quite difficult to understand the implications of so I'm gonna Use this morning. I used to use a picture of my kids rooms here, but I think they they graphed it So this is a completely anonymous professors desktop You can probably guess which is mine at work at least Rather than think about gas molecules this too is a system and you can think about states here So how many states do these things correspond to it's a bit of a difficult question So we can cheat a bit here, and I think the point here is that this is just really What do we mean by states? So one simple way of saying state well a particular Distribution of pixels on my screen. That's a state if I have exactly the same color of every single pixel on my screen It's obviously the same state, but if one pixel has changed color. It's a different state That would be the lowest minimal But on the other end if I take if I take one of the icos up there and move with one pixel to the left You wouldn't even notice you would likely steal it the same or the desktop, right? So that there are kind of two ways we can define state one on the lowest possible level if a single atom has moved and Then maybe on a slightly him a similar level that how many similar states are there and occasionally I'm gonna call the top one Microstates with I and this one macro states with an A The point here is that there is just one state up there The second point that might be less obvious is that there is just one state down there, too Because that is one particular distribution of pixels on my screen Or one particular Distribution and position and velocity of atoms in a molecule a microstate It doesn't teach us anything And that goes on if we're only looking at energy a state as a state as a state It doesn't matter whether there are the states that look like it or not But the point is that there are relatively few ordered states that looks like roughly like that one But that there are a ton of states that looks roughly like that one So somewhere if we forget about the individual pixels if you want to talk about this crazy professors How ordered is desktop is this? Macroscopic the large-scale states Well, that is one state While we want to want to say we might want to group all the states whether I close all over the desktop as one sort of macroscopic way of describing disorder and This is where this gut feeling of entropy as disorder comes in. There's lots of volume here There are lots of different ways we can distribute things There are lots of ways of organizing things that correspond to this Ln volume being high That is a high entropy or disorder if you want to think about it and there's a gut feeling While if there is only one way of organizing things well by definition that would correspond to this narrow piece of the Vessel, but there was only one way of having states very low number of different states low volume low Microstates, whatever you will call it low entropy order But the point is that don't try to think too much about that Entropy is something we define. What is entropy? Yeah, the logarithm of the number of microstates And you are physicists to the micro you can choose Do you stay at the classical limit and you want to define things by the positions and velocities of your atoms? Or would you like to go quantum? You're more than welcome to go quantum, but not in my course There it is possible to define the number of microstates and properly define the entropy is the logarithm of that multiply with some sort of constant and And the point with that is that then we might choose to interpret this as disorder But it's much easier to start from the plane definition And that actually turns out that it helps us to understand some other things too Why is it difficult to solvate oil and water? This is this hydrophobic effect we spoke about I don't expect you that this would be an obvious answer right now, but in that case I would argue that the reason why it's difficult to solve in oil and water is that the energy is going to be the same You don't lose energy But something happens with the entropy here is very unfavorable for the second term That the entropy is this is a much more ordered state, but let's go through that and drive it with the help of that equation What is the likelihood that this will happen spontaneously? I keep using my computer, but for some reason I never go spontaneously from there to there Because there are way many more states even if the likelihood of going from one state to another is always the same Because there are so many more states like that. So if you just randomly end up in a state It's likely going to be one of those not that one And this corresponds to one of the laws of thermodynamics actually that the entropy of an isolated system will either go up or be the same Let's see. Yes, I think I have a couple of slides about that. Let's see. Maybe not Let's define it this way. I Have slides about this later, but then I can skip them later So what's going to happen in water? There will be some slides on this By default all the waters they live happily and have lots of hydrogen bonds formed with each other So you have a network full of hydrogen bonds and there are lots of good interactions here And waters are also fairly free to diffuse. You might have seen that in the video I showed you last week, right and Now I put a big blob of oil in this water Does oil participate in hydrogen bonds? No, so what will happen here? Let's do this. I have my top of oil and then I have lots of water here You just broke all the hydrogen bonds How much energy was each hydrogen bond? Yeah, well 40 fibers so that it can be I think between 2 and 6 I would say that that's a lot of energy Like multiply by 5 or 10. That's a shit load of energy. That's astronomically bad That's eat my left shoe bad That's not going to happen. There is no way you could lose that many hydrogen bonds in the system That system could not even exist in that shape That energy is so much that the poor waters They will do virtually anything in their power to maintain their hydrogen bonds because it's so much energy in them So what these waters will do they're not going to stand for this They will because they're also free to move, right? So the what the waters will do and I'm bad at drawing here, but they will reorganize around This molecule into kind of a network So that you can still have hydrogen bonds and sorry, this is not perfectly drawn here So they will do anything they can to form a mesh around the oil so they can maintain their hydrogen bonds And they will maintain their hydrogen bonds. You can even measure that So the net change in hydrogen bonds of this process is plus minus zero Because it would be so dramatically bad to lose them and Then we're in a bit of a strange situation because it we didn't lose any energy the energy and this oil thing is exactly the same thing It was when we started But what has happened? So remember the movie I saw when I said that the waters were fair quite free to diffuse, right? They could write as what does that correspond to in terms of volume or the number of states they can be in They have a lot of volume available lots of states available. What has happened here? Yes, so that we don't know how much more but it's radically lower They're radically less free So the volume is going to be much lower and then the logarithm of the volume is also much lower So the entropy is much lower here If you just looked at the original Boltzmann distribution that would not help us one yota Because the volume or free energy doesn't matter, but if we take our small equation here, that's the free energy Equals give it some space between the letters here because I'm adding gonna add something in front of them The free energy equals E minus T as well. That might not tell you a whole lot Usually, it's difficult to think in absolute terms. It's much better to think of changes So I think the delta in free energy is the delta in energy minus Temperature multiplied by the delta the change in entropy So in this process the change in energy was roughly plus minus zero But the change in entropy well the entropy decreased And then you have minus there and the minus sign in front So that means that the free energy went up and we don't go uphill So this is a bad process that would the free energy would go up. It's not the spontaneous process So we could not describe that process just by looking at the energy But when we include the entropy or think of this as a terms of free energy, we can start to explain things like that We'll come back to that And that is very much related to this entropy of an isolated system Not in equilibrium, but just keep increasing over time and a pros and maximum when it reaches equilibrium The problem when we define things defining things is good The only problem with defined things that different people have a tendency to define things in different ways And when they define things in different ways scientists can't even agree on letters It's amazing that there are these the difficult things like introducing the SI system that we can agree on But what letter we're going to use for what property we can't agree on So when you are physicists we frequently talk about the free energy just the way I wrote energy minus the entropy part and then we use F if You're gonna do this in a proper way This is called the Helmholtz free energy and the Helmholtz free energy means that you have some sort of system enclosed in yellow here Where we're only exchanging heat with the environment You can't change the atoms can't go in and out and most importantly you can't do work. You can't change the volume The reason for that is that if you are actually changing the volume if you can change the size of the system We should also count for this work multiplied by pressure times volume Because even if I have the same temperature I might exert work on the environment mechanical work And that is something I have to take into account In general, this is super important in physics This is called the Gibbs free energy and it's probably really the proper free energy. It includes more In chemistry, this is the one we should always almost always use why Well the product chemical systems are usually open to the rest of the world If you have things in a test tube the volume here can change a bit, right? And then you're doing work against the air pressure In particular room temperature the normal chemical reactions we have we're going to have Zero point zero we're in a millimolar or something with proteins. So In principle, we should use g we should call delta g In practice, I'm just going to use that one because the pressure times volume doesn't matter in biophysics But if you're a proper physicist if you're doing say designing nuclear weapons or something where the pressure is high Then you need Gibbs free energy I'm not gonna I'm gonna continue the physics a bit because this leads to a bunch of really fun stuff So in the previous slide, did you think that entropy was a complicated feature? So there was only one thing that's more complicated than entropy in those equations And the final thing that is the one thing that normally cared about What on earth is this? This is some sort of feature that will change depending on the Environment properties of the system on the system It's not really a property of the volume, but it's also not the property of the energy So why didn't any of you react? Yeah, but I didn't say there was temperature. I just used the letter t And then you all happily assume that is temperature It is of course temperature, right? But the point is that just because you've used something every day The reason why you're confused by entropy is that it's something you're not using every day And then you get an entropy is simple Logorithm of the microstates this on the island is a super complicated feature We can derive what that is So we Use start this equation f equals e minus ts. You see that it's going to come back And then we look you're we can look at a small change here delta f for I like differentials because it's the mathematically proper way of doing it But you can use delta f if you've referred So if you look at the small change around this and we're also going to assume that we are close to local minimum Well, the differential here You can use the normal derivative laws here so that if f equals e minus ts If I add a small change df here That would on the right hand side correspond first we have e minus ts That's the f and then plus d e And then ts and dt multiplied by s. So just the product laws of derivatives So that and then I can strike out f on the left hand side and I can't strike out f on the right hand side, right? So I have df equals d e minus ts minus stt That is always true But if I now Say that we are close to a local minimum in the free energy Which is some sort of which is the equilibrium Then by definition the delta f should be if I do a very small move left to right the delta f should the first approximation be zero and that means that delta f which is zero equals d e Minus ts and again at the equilibrium the temperature should be constant. So dt equals zero So that means that d e minus ts is zero And if you just solve there that means t is the eds you're actually you're allowed I'm not sure whether you calculated the differentials. It's one of those things that your secondary school teacher would kill you for it, but This is actually All the differentials that are quite okay. You can separate them So temperature is really just the derivative that how much is the energy of your system changing When the disorder increases Do you ever got feeling for that Suddenly that definition of entropy was pretty easy, right entropy is just the logarithm of the microstates And it's somewhere here. I started to get confused. What on earth is this derivative? So i'm sorry to break it to that your entire life You thought the temperature was easy and entropy was difficult. It's the opposite entropy is trivial Temperature on the other end is a super complicated concept So that's the other why on earth do we have boltzmann's constant in all these equations So what's the units of boltzmann's constant? I see this is probably the way when you're because you're so used to the equations you haven't thought about what they mean It's joule per kelvin So they somehow translate between temperature the units temperature and the units of energy If you were to if you had been the first one if we had come up with this today We would likely measure temperature in somehow units of energy And then put boltzmann's constant to one It would have made all our equations so much simpler But the strange thing and this is very much a coincidence That this concept that you're getting from some of these deep equations and remember Tomorrow i'm going to show that these equations are universal for any system They don't you don't have to assume anything about physics. So this strange derivative that we get out here Corresponds perfectly one to one to the concept you're used to measuring at a thermometer. Yes Uh at equilibrium the properties of a system are not changing If the temperature keeps increasing, we're not at equilibrium Because it's something on the system is changing. So that's Well, once we once we are at the equilibrium, um, let's see how should I justify that? Let me get back to that tomorrow. I like it babe because I agree. I'm gonna be a bit shorter time It's actually a good question because once we are at the equilibrium the entropy no longer increases So from that point of view you are kind of correct But to make sure that I can finish the slides, let me think of a better way of explaining that until tomorrow We're going to use this the reason we're introducing this equation that they will help us Let's pick something complicated phase transitions water ice melting into water If we look at the ice you can probably agree that there are lots of hydrogen bonds formed in ice that gives you a low nice energy Is the entropy here low? Yes, it's not just low. It's low low low At zero kelvin. This is a super ordered system in principle. There is only one way you can organize the molecules that way Liquid water on the other hand the energy is high Is it good or bad to have a high energy? All other things equal. It's bad to have high energy. So we would not form water. So something So that and the entropy or the disorder is higher Which is somehow good So what happens from water else? Well We have our equation again This equation will explain everything at very low temperature The t term is small and that means that this entire term is not going to be so important at zero kelvin It disappears completely, right? So at very low temperature the energy term is by far the most important So that we want to optimize for low energy meaning you want to be ice As the temperature goes up eventually this term is going to dominate It's much more important to make sure that that total term is low And if this term should be small, there is a minus sign ahead of it That means that the s should be high, right? So at very high temperature It's much more important to be in the state that gives you high entropy And that means that some sort of threshold is going to be better to move over into a different phase And that explains everything about phase transitions And that brings us to another fun concept. Um, I'm going to hand out a slide here Paper, poly water, have you heard of it? So this was a Fun story in the 1960s that came out of Russia and Fidyakin and Deryagin So what you know now For any type of state Phase stresses, let's just look at this as a function of temperature. We might have gas water vapor We might have Aqueous water, liquid, and we might have solid water that is ice. Each of this has some sort of free energy And at each of these temperature at very high temperature We know that the gas should be lowest and best Then there's some sort of intermediate phase where liquid is best and at very low temperature the solid is best So the lowest line here is the best one So what happened with Fidyakin and Deryagin in 1962? They reported that they could under some conditions see spontaneous water condensation in capillaries under room temperature And their argument was that they had discovered a new phase of water called poly water That was a polymerization of water that you would pretty much have a network of hydrogen bonds It's not quite ice, but it's polymer like And it would have a freezing point in the ballpark of 240 to 213 kelvin and the boiling point just above 500 kelvin Completely brand new phase Based on what I just have told you And this was as you will see from this paper that the US was so scared They were they were literally arguing that Russia was developing a poly water gap And the next bureau of standard technologies tried to repeat this and everything There's something fundamentally wrong with this and this is the reason you have not heard about it And you should be able to debunk this now Using that diagram up there and our friend f equals e minus ts Talk to the person to the left or right of you and see if you can come up I'll give you one minute because we're a bit short of time So I'll give you one last clue here that the only condition you would observe this Would be some states where this water would have the lowest free entity So that if this was true, this would mean that between the freezing point here and the boiling point There should be another curve here Where poly water should have whatever black or whatever line we would have Right, otherwise, we would never observe it Because if the if the black line for poly water was higher than the liquid and the ice part here By definition, we would never be good to be in that case and it would be better to be normal water So if there was I'm not saying this theoretically It's not impossible that there could of course be a black curve here That's between these two points that would go from there To there, but what would that mean? Would we ever observe normal water? All the water in the world would prefer to be poly water, right? It's impossible And one of the person to observe this it was Richard Feynman, of course I realize that it's completely it doesn't matter It doesn't matter the conditions you did the experiment under it can't be true It's impossible by definition And this is the power because we're you don't need to know anything about the experiment for this If you're arguing that this is a naturally occurring phase of water that would be stable That all what then it would be more stable than normal water And then we would not have normal water So that's again the power of this extremely simple equation And this has actually happened in the literature too. I'm not sure where you've got Kurt Vonnegut cat's cradle So that there is in this book there is a story about a new form of ice There are eight Chris different types of ice form and he He argues that there was the story of the book is basically that there's a new type of ice called ice nine That would freeze already at 150 degrees Fahrenheit, which is roughly 50 degrees centigrade And the point is that basically everything that this comes in touch with We then turn into ice nine and this would spread But it's the same thing there if this was to all the water in the world would have turned into ice nine, right? That's because once you've turned into ice nine you will never go back Which I think probably was kind of the point of the book Let's try to do something slightly more concrete and look at hydrogen bonds So we have two water molecules in vacuum They are poor and don't have any hydrogen bonds and then they find each other in water and make one hydrogen bond Let's look at that very simple process You will gain and you will lose some things here You will gain the energy of one hydrogen bond Let's call that e h And delta e h well that if it's a gain here that the what we're gaining there Let's let's just say that the change in energy is delta e h and then by definition that should be smaller than zero Because it's good, right? We're gaining something But we're also losing something We're losing Well, we have we had we used to have two freely rotating waters And now we're constraining one degree of freedom between them so that you kind of We kind of losing half the freedom for each of two waters So let's just call that delta s h for one water And now things get complicated because if delta s is the change of the process It's becoming more ordered and that means that delta s h must be negative to right Don't assume here. You need to think about what will it mean the direction you're going more ordered less ordered higher energy Less energy. What are the signs? It doesn't get a whole lot simpler than that. So now I'm going to do the pure challenge again and Which one of those two is true is the loss in energy lower Than the change in entropy or is the change in entropy more important than the change in energy for these two Let's do one minute together And I hear somebody just sighing which is perfectly fine. I too would sigh if I tried to handle it this way and guess and hand wait So there is an equation. I wonder which one we might be able to use Think of this equation instead And now I put the delta s here because again think in terms of what has to change We also saw in this time we already hand waived what the sign of that term must be We already hand waived about what the sign of the delta s says was b And you know that hydrogen bonds do form spontaneously in water So that This has to be negative. Otherwise, it would not force spontaneously So try to use that now and see if it works better So should we go through it? Is that the delta e h was that positive or negative? Negative because we're definitely gaining one in Japan. So that first say is going to help us The del forget about the minus sign here. Yeah, the delta s h Was that positive or negative? Negative because we're becoming more ordered But then there is a this negative and then we have a minus sign in front of it. So minus minus is plus So this component is positive So that component can't be the one helping you to form the hydrogen bond The reason the hydrogen bond forms must be because that term is more negative than that term is positive And for waters to water moving to vacuum is probably this is roughly half a kilo calorie per mole and that's roughly five So this is a factor of 10 difference Do it piece by piece Put up this equation Don't skip the part where you have to think about the sign And what is reasonable and then as much you say I think but the point the equation is to help It's not a burden. Don't try to hand wave You can't you can't hand wave your way through and guess these things you need to write down the equations Think what the different terms mean and then it becomes easy. Yes No, so remember that the free and that's a good point The world will strive towards minimizing free energy. There is actually a study question that on a link You can prove that the free energy corresponds to the amount of energy available to do work For now and that would take me 10 minutes to derive So I'm not going to do it But any process that can happen will go in the direction where the free energy Is reduced So if the free energy for a process is positive, it's not going to happen spontaneously You will go from high to low free energy Just as if I drop something I will go from high to low energy So the free energy essentially describes what processes happen or not And that's why it's so important in chemistry If the free energy of folding a protein is negative, it's going to fold If the free energy of folding a particular protein would be positive, it would never fold So free energy pretty much connects physics with reality The reason why I love to think in terms of change is because it forces you to say what which is the state before Which is the state after? Because again, if I were to say that if I would say how much does it cost to break a hydrogen bond Then you need to reverse everything right and then so what was before what was after And in that case the energy would be positive and the delta s would also be positive You can think of this inside a protein too Because remember that I said that most amino acids will happily many amino acids will happily form hydrogen bonds So you can if you do this in vacuum This will be exactly the same thing applies that the change here corresponds roughly to the energy But when I'm doing this for a real protein insolvent what happens is that Well before I folded the protein, I would be making hydrogen bonds to water And once I have folded the protein, I will be making hydrogen bonds in the protein But just as for the oil and water thing Those two waters will form the hydrogen bonds. You see that the number of hydrogen bonds is really constant here And that's the complication in reality That the number of high there are hardly any processes where the number of hydrogen bonds change But what happens is that this is going to cause differences in entropy while the difference in entropy Sorry in energy is virtually zero And that's why I had to introduce interview most of the folding most of the complicated parts here is going to correspond to entropy Same thing with the large protein whenever large protein that is not folded that there are gazillion states that can be in Once it has folded Anfins observation is one unique state very low entropy So this is going to be more about entropy than it The other reason why we love free energy is that it's we can directly connect it to experiments So if you take say an Octanol or a hex cyclohexane some so something that likes to be an oil phase And then we take how much of this would like to be a water The the aqueous concentration or the solubility it could be a salt The way we measure this I can just calculate how much of it is solvated versus what is the concentration when I have it in pure form reasonably straightforward experiments to do But we already know that The equation the Boltzmann distribution in particular so that the concentration in a state or the probability of being in a state Is proportional to the exponential of minus Energy or in this case free energy divided by k t or if it's a canist r t, right But take the logarithm of both sides of this equation and then we can solve for delta g So delta g is equals minus r t And then the logarithm of two different observations. So this say what is The concentration of alcohol Octanol in the aqueous phase versus the concentration in the pure alcohol phase Both those numbers or the relative difference between I can measure in the experiment If I can measure in the experiment I just solve for it and then I get the delta g difference from it And that's usually how we get delta g's we just measure it experimentally And that enables us to do a couple of fun things here We can for instance a concentration of cyclohexane. It's roughly 10 moles per liter of molar While if we put it in water, it's 1 to the minus 4 10 to the minus 4 And you can show that that corresponds to free energy difference of roughly 7 k kals per mole But there are many different states. You can have one molecule in gas phase You can have many in the solvent phase and you can have one molecule solved in water And between all of these you can measure. What is the free energy? What is the change in energy or enthalpy? H is just e plus this Pressure-times volume term that we're not going to care about You can imagine that it said either and what is the change in entropy All these things can either be measured in the lab or I can solve for it using that equation Delta g equals h minus ts Yes Yes Yes, I am It's an I should have a bit of bad conscience for this has to do with typically the equations We're applying in biophysics the concentrations are so low That is virtually nothing I do that will change the volume of pressure in the system That means that I know that by definition the pressure is going to be one millionth of the other things So may I call pa as a physicist? I should probably have my physicist card revoked, but I don't care So yes, we will ignore pressure throughout this course. You should have a little bit of bad conscience for it, but I won't nail you for it The But it's not entirely trivial how to get to that right just as I derived the thermodynamic temperature a few slides ago I can actually see how the entropy changes in this system And I do that in a roughly similar fashion in the interest of time I'm not going to go through it in detail, but If you do a bit of equations that if you measure the delta g for the same system the same process But do this at multiple different temperatures just looking at this equation To first approximation. This is going to be roughly the same, right? So that now changes. How does this change with temperature? The proportionality factor to temperature is the entropy, right? So if I just change how is this equation changing with temperature, I can get the entropy from there So I can get the total change in free energy Gibbs or Helmholtz by just measuring the concentration I can get the entropy by changing how it varies with temperature And then I can just solve for the energy or enthalpy So then I can get all three components And this actually works The book has a couple of examples for cyclohexane and everything in particular Finkelstein. This is not at all well described in the large book And what you see here too is that Whether you're putting something in a pure liquid phase full of hydrocarbon or an aqua solution The delta H is roughly the same. I gain lots of interactions But the big difference is entropy It's really bad entropy wise to put it in water, but it's not so bad to put in put it with other hydrocarbons And the point here now is no the point there It's not just hand waving the things that I hand wave that argue about this clath rate formation the networks around an oil drop That was hand waving The reason why it's two is that we can show that it leads to the right experimental observations There is absolutely no difference in energy This molecule has just as much energy In this case as it had there it's not changing the number of hydrogen bonds If this had been a matter of breaking hundreds of hydrogen bonds, I would have seen a gigantic difference in energy We don't it's all entropy And this leads to a bit of a other fun paradox that the hydrogen bond What did I say about the hydrogen bond? What was it mainly caused by? The reason the hydrogen bond it was electrostatics, right? but It's not really energy So it's it's an the hydrogen bond itself is caused by electrostatics that gives us these peculiar properties But at the end of the day the net effect of that is entropy not energy So the hydrophobic effect is an entropic effect Not an energy effect You can see that in another way The temperature dependence Normally if a process depends on energy And you if let's say that it'd be bad energy that I had broken a number of hydrogen bonds What would happen if I just added energy? With the process if I heated it up would I increase the solubility? But what's if it's entropy if that term is the bad one what happens when I increase the temperature? Does that term become less or more important? More important, right? So that the higher the temperature is what happens with the solubility of oil and water Yes, it gets worse. It's the opposite And you all know this when you've been cooking pasta if you put a bit of oil in the water It doesn't dissolve just because you boil the water, right? If anything it becomes worse And that's also and that's the absurd thing that although this is an effect caused by electrostatics The individual hydrogen bond the hydrophobic effect is an entropic effect. Not an energetic effect I've been tormenting you a bit much and but the point of this is this is super important for proteins because proteins are going to be Hydrophobic on the inside while hydrophilic on the outside So I hate to break it to you, but we're neither going to cooking pasta or work with oil here But proteins are fairly oily So lots of the side chains of proteins are say tryptophanes or they're basically a aromatic ring Isolucine, leucine there are lots of amino acid side chains that are just hydrophobic And just as if you put a benzene ring in water in isolation, this would form a cathode structure this network around that And just as for oil what this means if you had two of these Well, then I would need one structure around the left one and one around the right one And what will happen is that you will minimize this area by bringing them together Because that area the the volume is the same here, but the total area around them is smaller And that is exactly this hydrophobic effect. You're seeing that if you pour a lot of oil into water the phases will not mix They will separate That's going to happen for protein too So if we take a small protein, I'm sorry about the coloring in the sorry if anybody's red green color blind And the red are the hydrophobic residue. This is a real protein The green are the water loving hydrophilic residues So you do see how the protein instantly and this will happen in a split second after you put it in water All the water loving residues are going to be on the surface and all the water hating residues will be buried on their side It's exactly the same effect as when you're solvating oil in water And we're almost out of time here, so I'm going to need to move a couple of slides to tomorrow too But what this free energy is going to be about Is given a particular sequence of amino acids Will it somehow be random or is it going to turn into some sort of maybe helix? Or it's going to turn into some sort of beta sheets Well, that will depend on the residues right because depending you have in the patterns of the residues Some of these structures are going to be more or less favorable And in this case blue means water loving loving to be on the outside and gray means water hating love to be on the inside Which one would you guess is most favorable? Yes, because it can't turn all the water hating ones to the inside So they don't have to face the water while the water outside is all the water loving residues But that's just hand waving for now. We're going to come back to that So and this is the last slide I'll do before I let you go What we can summarize today with is pretty much there is some sort of this landscapes We spoke about and we will keep using them the energy is by far the most important part of it We want to have low energy But the entropy also matters so far out here when we start we might think of this as a large volume There are lots of different states. We can put our protein or whatever molecule we have in it And as the energy here goes down You will also have less and less and less and less freedom, right? That is bad Having less freedom means that the entropy goes down and we're going to lose free energy But there's going to be some sort of balance here that The energy must go down more than it costs me to lose the entropy And what we're going to spend both tomorrow and a bit further down the course talking about is really this balance balancing energy Which is usually good when it goes down while entropy where it usually hurts us when we fold things And then we're going to apply this to a bunch of different proteins and secondary structures You should have the hand in task for today That you can start working on and I would actually encourage you to work on that now because it will help you understand these concepts And in particular, it's going to help you when I go understand two things slightly more mathematically tomorrow We have the study questions and there should be some reading instructions both for Lorel and Finkelstein I would strongly suggest you follow Finkelstein. He described in a far easier way than Nordland or I will do my utmost to make sure that I put up this recording by 6 p.m Today so that you can watch the lecture a second time because it's not entirely easy to follow Do go through it a second or third time if you need to And with that see you tomorrow