 This is FKBP, it's FK501 binding protein. Can you guess the name of the small molecule that's bound? That's right, that's FK501. It's just a boilerplate simple protein that several groups have used to develop methods to calculate free energies. Ideally I would like to calculate the binding energy of that molecule but pulling it away and calculating how much I gain or lose when I take it all the way out to water that's complicated for the reasons that I shared with you a minute ago. But what if I instead could calculate another process? What if I could just make this molecule disappear into a ghost by removing all the interactions? That's not quite a physical process but if you bear with me for a second that if we just assume that happens it's going to lead to some nice things. So I can do that, I can easily define two potential functions. VA when the molecule is present and then VB when I've turned all the interactions to zero. So VB would mean that molecule is gone or it's pretty much, it's a ghost instead and in VA molecule is present. Now in reality a molecule would not become a ghost but bear with me for a second here. So I can't do this instantly because if I did this instantly I would create a vacuum gap in the system and it would lead to tons of bad things. But what if I instead define a new potential here that is a function of a parameter lambda and then I just interpolate between these two systems 1 minus lambda times VA plus lambda times VB. What that will mean is that when lambda equals 0 I have VA and when lambda equals 1 I have VB and then I interpolate between them. So this now means that I can simulate the system at many different points maybe pick 10 points between 0 and 1 here and gradually forces molecule to disappear. How is that going to be useful? Well I won't have time to prove that to you but there is a deep result telling us that I can actually get the difference in free energy between these two states by integrating the derivative of V with respect to lambda over lambda. There are two parts here. I don't calculate this numerically. This derivative of the linear interpolation is minus VA plus VB C dependence on lambda. So I can calculate this analytically and then this is an average over the ensemble, right? That's why I need a simulation. And the deep result is that when I actually average this I no longer get the potential where that came from but actually the free energy. I don't expect you to understand that. It's for now on you should just believe it. But what that means is I'm still having an unphysical process here. I'm turning something into a ghost but along that unphysical process this will allow me to fairly easily calculate an actual free energy along a reaction coordinate. In this case the coordinate of gradually disappearing the FK501 molecule or any other molecule for that matter. What does that have to do with reality? Well, not much. But there is an important property of all these values. Potentials and free energies are state variables. It's only the state that matters, not the way we got there. A brick has a potential energy. It doesn't matter where it was yesterday. It's only the state that matters. Otherwise we would violate the first law of thermodynamics. And in particular that means that if I have a state A here that is a real state and I would like to move from to B with also a real state but for some reason there is a brick wall between them in reality. It would be a very high energy barrier. As long as I can find some sort of imaginary physical way to go from A to B all these dotted states on the way they can be fake. The only thing that matters is that the starting state is real and the ending state is real. If I have found a path between them I can use simulations and cheat and take a path where the path itself is actually impossible to calculate the free energy difference between them. And we're going to use that to take paths between states that appear strange but allow us to calculate the actual binding free energy.