 So I've rewritten the three curves I had in the previous concept and now we're going to think about what happens when the protein actually folds because it's not quite as easy that any hetero-polymer withdrawing water is going to fold this way. If we think about these entropy and energy plots that we looked at in phase transitions they're going to be very instructive here. So in general for any chain if it's just that fulfills these conditions you're going to get a curve where the entropy changes as a function of energy. Remember those plots that they could tell us about the state distribution and they're going to go down to some sort of roughly fixed value. And this is going to correspond to some sort of energy density here that I have many many many many many many states here it's essentially a continuum and at some point I'm start having a fewer and fewer and fewer states. But here remember that these derivatives was one over a temperature I'm going to call that the critical temperature. What's happens here is that for any chain at some point when this becomes steeper and steeper and steeper that corresponds to a lower temperature when we reach this temperature we're kind of stuck here and you're going to be stuck at the lowest energy level there which is fine but that is not folding that is just that we're the viscosity is becoming so high so at some point I can't move anymore and then we stop moving. That is not true for all sequences because some sequences of amino acids will instead look roughly this way that we will have s as a function of e where it will go down but it will essentially continue down a bit. You probably don't see that part there but what I mean there is that the energy levels here there are still many many many of them and continues but then there are a few very low lying ones. What does that correspond to? Well imagine that my fingers on one hand here is thicker than the other I will be able to get close but I won't really be able to pack them well right? So what I mean by these low lying levels number one two and the best one here might be that might not be number one that number two and the best one might be this so this is kind of unique right? I can't be almost there either I'm here or I have to jump all the way up to the next level so here there is also some sort of one over Tc here which is the temperature when I gradually go down but there is also something you can think of some sort of melting temperature that you can jump all the way to the lowest energy and it's going to turn out that this is what really defines the protein. Forget about the temperatures there if you're not interested in the physics part that there should be one or in some cases two low lying states that are significantly lower than the other ones that creates a uniqueness. Remember when we're making a jump we're moving across a free energy barrier that creates a well defined state and this is really what creates the phase transition. So the phase transition here is from the molten globule to the native state or from the native states to the molten globule. We have a clear change there and we can't really be we can't be stable on the barrier itself and it turns out that this means that we can reason about some things you talked about before like the prions. So what is the the probability of having one state here? We can compare that. If we compare this to this characteristic temperatures these characteristic temperatures might be in the ballpark of 350 Kelvin or so but maybe we should order magnitude estimates are interesting right? Let's just say that the delta here the difference in this energy level is roughly 20 times higher than the kt values or the kt see here the smooth parts. So that would mean that the probability of a protein being able to jump across that barrier again this is not strictly Boltzmann because I'm talking about different sequences here but if this is e to the minus 20 we're talking about something in the ballpark of 10 to the minus 8 so it's again it's going to be 1 in 10 or 100 million sequences that will have these properties where they will form nice stable proteins and just as we hinted when we talked about prions prions are essentially proteins where we just happen to have two of these that are really good and to first approximation the likelihood of that happening is going to be the square so say 10 to the minus 15 or so it's going to be very rare but it will happen now again in nature. So the take-home message here is that proteins are unique they are far more unique than you think they're polymers and they're hetero polymers and the magic happens because of the beautiful side chain packing.