 So the alchemical transformations we looked at they're awesome to calculate free energies cheaply The only problem is it did so along a leg where the end result was an unphysical state So I can't compare that to anything in experiment So let's go back a second and think about what we were after we were after the state when I have a protein in water And you can imagine that protein and being in the other room and then I had a ligand in water They're not seeing each other. They're not interacting But then we go along a path here where it says delta G bind And the other state my target state is then going to be the protein that has the ligand bound and they're all in water That is an expensive and difficult calculation to do an expensive Not just expensive in terms of dollars But because it's expensive and difficult to do it's going to be noisy. It's going to be a bad result and it will take a long time What I did on the previous slide is that I took this ligand and just disappeared it So these are physical states. This is going to be non physical states. So if I have a protein, but there's dummy In water. Well, the dummy doesn't really exist. It's not interacting with anything But I've taken internal charges and removed them and everything and that corresponds to some internal change in free energy that I've no idea how to define but As long as I can calculate that there will be some sort of free energy Let's try to do the same thing here The protein in water is the same And that's in the other room. You can think of it as an imaginary one And then I take my ligand and turn that into a dummy in water So again The lower states here are non physical I can do that and then I can also check what are the free engines going to be along these paths Let's just give them some names delta g1 Delta g2 you can probably guess what this is going to be Delta g3 The only reason for this to make sense is if these three paths end up being easier or simpler than doing the expensive path And that's going to be the case Because remember you already saw some of these, right? This one was easy. We said that in the previous slide This one looks really complicated, but it's not as complicated as you might think. Remember that I said Technically I I'm thinking of the protein in water when I'm doing these changes But if I'm only interested In the difference here the ligand or the dummy that they're not interacting with the protein at all So if I want to go from that state to that state I can think literally think of the protein and water being in the other room So this just means that I have a small ligand in water and I gradually disappear it the same way we did there That's if that's easy. This is even easier I don't have to worry at all about the protein So that leaves delta g2 that might turn out to be difficult. In fact, it's not difficult This dummy the only reason I even say dummy there is that there might be some internal state that I'm not accounting for So here I have a protein in water Here I have a protein in water. You might say there's a dummy in water But remember the dummy is not interacting with anything So and there is a dummy here and there is a dummy here And that means I have a dummy That's not interacting with anything and then I have my protein in a lot of water So these two states are going to be the same And if they are the same I can say that that equals zero And if that equals zero I can use these properties of a free energy cycle And let's start up here This is I never try to remember these by heart But I just use the definitions of my delta g's and go in the directions of the arrows Or I go against the arrows and add a minus sign So I'm going to start up here and then I go against delta g bind. So there's minus delta g bind And I'm going against delta g1 so minus delta g1 Plus delta g2 But that's going to be zero Plus delta g3 Then I'm back where I started so the net sum there must be zero Delta g2 was zero so I can strike that one out and then I solved for delta g bind Putting that on the other side And that means that delta g bind here is just the difference between two free energies. So it's delta g3 minus delta g1 This is a so-called free energy cycle And you should know this concept Occasionally we call them a thermodynamic cycle too What this means is that I get my expensive complicated delta g here as a difference between two other delta g's Occasionally we even call that delta delta g values. It's not so common if I try to calculate a solubility But here I turn the ligand into a dummy What if instead of a dummy I have a whole series of compounds? I might want to calculate the binding energy for methanol ethanol propanol butanol Well, maybe I start from butanol and then I try to turn that into propanol And then I take propanol and turn that into ethanol and ethanol into methanol That way I will be able to get the difference in the free energy between say propanol and butanol And in many cases that's good enough because I'm not sure it's nice to have the absolute binding free energy But in most cases I have an entire series of compounds And I'm most interested which one of these binds best and which binds worst and what is the trend in them And in that case since I'm getting these numbers It's a difference that in turn is a difference, right? Then it's very common that I call those values for delta delta g values And that's frequently what we get out of the so-called free energy cycle This is a remarkably useful result and that means that at least for small molecules We can actually use molecular dynamic simulations to calculate binding free energies accounting both for enthalpy And entropy and if you're lucky you're going to get an accuracy here That's like half a k cal per mole or something which is roughly the same as you had in a modern experiment And today I would say computers are getting to the point where they can do this at least equally fast as an experiment Wait 10 years computers will be doing this at least 10 times faster than an experiment And it's already significantly cheaper to do this in a computer than in an experiment And that's we're going to come back to when we talk about drug design later