 So what we got for the beta sheet formation was that the barrier along this formation path in the transition state Looked this way, and in particular. We had the stability and the denominator What this will lead to is that we can you can insert that exactly the same way And so that the time is going to be proportional to some sort of constant tau Raised to and multiplied by an exponential raised to plus delta f. So then we get that entire person So the time of formation is going to be roughly Time constant tau raised to e plus f. So that will be minus two u Delta f beta divided by f beta That we can just write as some sort of time constant tau I'll introduce a constant for most of these things. So it's going to be an exponential raised to minus a Divided by f beta where a won't really depend on the amino acids, but we have the f beta here If I now We can draw the same plots as we did for alpha helix is here here We don't have the separate initiation cost, but this is for an entire sheet and this is a key difference compared to the alpha helix The alpha helix is kind of seeded at one place and then grows But for the beta sheet we need the entire beta sheet forming and that's going to give us Exactly these properties that we had for the phase transitions. In fact beta sheets are a phase transition It's an all-or-nothing transition We can't do the helix coil mixing that we did for the helix and it's also going to mean that you don't have this Temperature span over 5-10 degrees where they gradually form Depending on what the values of these f beta are you can either have things that pretty much never ever form beta sheets We can have things that will form beta sheets fairly quickly or we can have things that Well in principle, it's better to be longer than short But they're not yet so stable that it will be better to have a sheet than a coil But what will this one mean? Well Remember how fast the exponential function grows, right? And here we don't have a constant and you can also remember that solubility part And I also mentioned that some residues prefer to be in a helix versus some prefer to be in a coil and some prefer to be in a sheet if This were to vary between say minus 5 kT to plus kT or 10 kT You will have a very huge difference in span here and in fact It's an exponential, right? So if we just draw this some sort of stability here You might you might want to add a minus sign It depends on whether we say stable in beta sheet or stable in solvent and then we say what the time here would be If this is not very stable in a beta sheet This is going to be exponentially expensive and then if it's very stable in a beta sheet that can form very fast again, it comes directly out of this equation and This leads to the peculiar properties of beta sheets that some of them will form virtually instantly Remember what I said a single hairpin can form as fast as an alpha helix But if you have very large beta sheet that in particular contains some residues that they We can put them in a beta sheet, but they're not thrilled about being in a beta sheet It's just slightly better than being in an alpha helix This is going to be very close to zero and with this gets very close to zero You're gonna have an exponential that just explodes and then we're gonna have a time that just explodes So how long can those times be? Well for a single hairpin it could be just as fast as the helix microseconds But for some really unstable things here. We're talking about seconds minutes hours or maybe years Which led to something very interesting that we didn't know one through roughly 30 years ago