 these different interactions we would have in a protein, both the electrostatics and van der Waals. We have some very strong interactions, we have bonds, angles, we have some intermediate definitional strength and torsions, and then we had some very weak ones such as these van der Waals interactions. The only way to properly handle these weak non-bonded interactions then I would need to describe this as this 1 over r6 attraction at very long distances, dipole-dipole interactions, and then an exponential repulsion at very short distances. This is a potential form called the Buckingham potential and this would actually work pretty darn well and it is used sometimes in computer simulations. The only problem here is that calculating the exponential function can take hundreds of clock cycles on a computer. Today you can probably do it fast on a graphics card but these models were originally developed in the 60s and 70s and then they most definitely didn't have GPUs. Is the form there important? Well yes Virginia, that form is super important if you're going to be employed at a national lab and designing nuclear bombs. But in a biophysics lab? Well I haven't checked lately but it's very rare that we have nuclear devices going off in our physics lab, biophysics lab that is. So I don't really want to, I don't worry what's happening if two atoms are literally overlapping each other. I just want something very simple and I want something that goes up quickly as I approaching unity. So if I have this one function here that is literally the dispersion that is 1 over r to the power of 6, I would now like something that and that has a minus sign. Again it's attractive, it's good. I would now like some sort of other function with a plus sign but that increases even quicker. But if I now take this 1 over r6 and that is stored in a variable, if I just square that then I get 1 over r12 and you can do the math here so that the 1 over r12 component here that will increase even faster than 1 over r6. That is just one multiplication which literally takes one clock cycle on any computer today and now I have a plus sign on that one. So what that's going to get me is I now have this simplified interaction form where at very short distances this term is going to dominate. It's not the exact potential I have down there and at very large distances this term is going to dominate and that is in practice what we all use in computer simulations. How good is this approximation? Well it's not just good, it's very very good. It's not quite as good as Buckingham but I have to confess that the Buckingham approach too isn't exact either. Remember when I showed you that atom being surrounded by all other atoms. This is really a collective effect, particularly the induced dipole measures so that you would actually not just have to calculate pairwise interactions but triple interaction, quadruplet interactions, pent-toplet interactions et cetera and that's simply completely unrealistic. So no matter what functional form we have here, if we try to parameterize that from quantum chemistry, it's not going to work well. But if I have any way except that I'm going to have a simplified form, I can now take these parameters and just put one factor on the repulsion and one on dispersion and then I try to parameterize this to fit for instance the density and heat of vaporization of small molecules and that means I now get parameters that they're not strictly exact in terms of quantum chemistry but they're very good at describing how atoms either attract each other at long distances or repel each other at least at intermediately short distances. At some point it will start deviating from the exponential but then we're no longer doing biophysics, I don't really care. This functional form is called the Lennard-Jones interactions. It's actually not a pair of scientists, it was called John Lennard-Jones and you will likely see that. Occasionally we call these van der Waals interactions or we abbreviate VDW. Personally I would say that van der Waals interactions are all the types of normal interactions including the Buckingham one I just showed you but again there's a bit of nomenclature mismatch there. The important thing is that you know what you're talking about. These are extremely weak. We might be talking about 0.1 k cal per mole. Remember that electrostatics and bonds in particular but even electrostatics that tend to be around the same atoms. An electrostatic interaction between two atoms that have unit charge and are separated by a few angstrom might be a few hundred k cal. So thousands of times stronger than electrostatics. The reason why this was still important was this effect that these all have the same sign while the electrostatics will cancel each other under many conditions. So let's look at what the functional form looks like. Basically if I am going to sum the interactions in all a molecule I would have a double sum over all the pairs of atoms and I would have one term that corresponds to the repulsion and one term that corresponds to the dispersion. And once we've done that we can actually translate these parameters and see what types of energies we're talking about in normal molecules. You don't need to know that by heart but if one starts doing molecular modeling it can be kind of fun too because roughly knowing these numbers give you a gut feeling for how strongly atoms are interacting. So in principle now we have everything to start to calculate energies admittedly in a simplified way but understanding how atoms are interacting in fairly large molecules where bonds can rotate and move. Let's have a look at that if we bring everything together.