 To understand how this long chain forms let's limit ourselves to just looking at the phi and the psi bonds. Actually I'll make things even easier. Remember that from a Shandlen diagram that there were just two or three regions that was happy. So let's take a this amino acid and say it can kind of be in an A state or a B state and I'll forget everything. This is trivial right? It's two states here multiplied by two states here, multiplied by two states here, multiplied by two there, two to the power of eight or something that's piece of cake. The problem is that you have more than eight amino acids in a protein and this comes back to that what I mentioned that the exponential function grows faster than you think. In this case it's a power function but it's similar. So if I have two states per residue and then I have n residues, well that means if I have 100 residues remember those were among the smallest proteins I had in that table. That's going to be two to the power of 100 different combinations and that's roughly 10 to the power of 30 which is around the largest number you can represent in a single position. Let's assume that nature is super fast at testing things out here that I can test one new conformation per picoseconds. There is no way it's that fast but let's just assume that. If I test one per 10 to the minus 12 seconds that would mean that this would take 10 to the power of 18 seconds to test every single conformation. Well that's not much, just seconds right? It's 30 billion years. The world is only 4.3 billion years old. So there is no there is no theoretical way for this chain to test every single possible conformation and yet it does. The person who first realized this this paradox is Osiris Leventhal who did those movies I showed you in lecture two and it's even called Leventhal's paradox. So what Leventhal's paradox says is that we know that this change finds the thermodynamic minimum. Christian Anfess improved that and yet there is absolutely no way for the chain to test every single possible conformation. So how does it even find that conformation? We don't know. One way or another it does but it can't be brute force searching. Brian Robson did a fun cartoon of this many years ago. The great protein making this jump that we have no idea about. And for almost 20 or 30 years this had remained a gigantic challenge. So we do know that the proteins do take the form but how does the process happen? This is going to be super important if we want to engineer proteins and there's been remarkable progress in this I would say the last 10 years. This recently as 2020 we're now finally able to predict protein structures accurately. This is not going to be a trivial problem to solve. It will depend on free energy. It will depend on things happening in a certain sequence and of course at the end we are not necessarily going to do brute force searching. But to understand this hierarchy we will have to go to the next step in studying the so-called secondary structure elements of proteins.