 We'll be really in two parts. You heard from Andrea Parmigiani that Vladimir was a person with very broad interests, which ranged over everything, and particular soft matter physics. So the first part of my talk, I'm going to try to do a little bit of intellectual archaeology. I'm going to try to find a strange which contributed to the theory you just heard from Sergei, the Loreman Rochelle theory, from the viewpoint of soft matter physics. I'm a soft metaphysicist. So maybe it's irrelevant, maybe not. It's just the way how I look at it and how I like to think about his theory. It will involve a little bit of mathematics, but do not be afraid, OK? We'll just take it easy. In the second part, I'm going to try to think about ways how we can enrich the theory, because theories are built collectively. We put stones together, and they put up a big stone, and I'm going to try to put a little stone on top of the big stone without further. So my first contact with soft matter physics was in the form of liquid crystals, and it turns out that liquid crystals have a lot to do with this Loreman Rochelle theory for viruses. So we're going to take a big intellectual detour, but we're going to get back to viruses, I promise you. So we're going to start slower with pneumatic liquid crystals. These are the kind of things you have in your watches, which from which you can get your displays. And they have been looked at since 1888, and they've been long known. And I'm going to look at orientational phase transition. You're familiar with positional ordering, like freezing, but here it has to do with elongated molecules, which are in the shape of these cigars. In an isotropic phase, they are randomly at all orientations, and in the pneumatic phase, they have particular orientations, which causes birefringence scattering of light, and all of these nice display properties. It responds to electrical fields. That's why you can change the output. So from the viewpoint of phase transitions, you, for what's called a thermal tropic liquid crystal, as a function of temperature, you go from an isotropic phase, where these molecules point in all possible direction to this pneumatic phase. And by the way, that comes from a Greek word, which means thread. So if you look at these, you'll find nice thread-like defects. So the order parameter is 0 at high temperatures, and then there's this critical temperature where it makes a big jump. Now, not a big jump, a small jump, then it increases and saturates. So you call it the first order phase transition, associated with a discontinuous jump. Now, when this was very natural that you think of the ordering, you want to characterize the ordering with an order parameter, and you do that by just saying, well, it's the direction of these molecules double headed because it doesn't matter which direction is which. But apart from the direction, there's also a magnitude, an amplitude to what degree are they ordered. And there I was informed that you should think of it in terms of the second Legendre polynomial. Where does that come from? Second Legendre polynomial. I felt I'm a Dutch farmer that they were just trying to pull the wool over my eyes. So anyway, here you have it. Ordering is defined by this function, second Legendre polynomial. This is a 3x squared minus 1 over 2, and x here is cos theta. So the recipe, they tell you, is, well, you have to take all of these molecules, find the angle of the molecules with the ordering direction. This is the z. This is this pneumatic director. And average the cosine squared of theta. If it's isotropic, then the cosine squared of theta is 1 third and s equals 0. That's the isotropic phase. And if it's aligned along the z direction, you get 3 minus 1 is 2 divided by 2 is 1. So the order parameter goes to 1. Well, I can think of a lot of functions for which it should be true. It seems so arbitrary. I had to learn about Legendre polynomials in classes about electrostatics. And I despised them. I didn't see any reason why I should, where this has come from. There's a very nice or somewhat exotic type of pneumatic liquid crystals where you get ordering not just in one direction, but in two directions. You can think of these as molecules in the shape of books. They first align the long axis. And they can still freely rotate. But then when they order in that direction, you pick out the second one. So this is this pneumatic director as before. But now you pick another direction perpendicular to one. So biaxial, they're two axes. So they're two order parameters. The first one is the same as before, s. And we have now a new one. We call t from transverse. And that describes to what extent these molecules are ordered by having the large faces pointing in this direction or any other direction in this plane. OK, now let's see what you're told there. What you're told there by this person, Fraser, is the order parameter for s is the same as before. But now for the transverse one, you're told you have to go to the associated Legendre polynomials. Oh, OK, all right. What is that? That is 3 times 1 minus x squared p22. OK, all right. So you can do that. And yeah, you find that if you have ordering along the z direction, then it all works out. It will, this term particular, this is the complex i. This tells you. This is sort of like an x, y model. And it tells you in what direction it orders. OK, it's nothing to do with viruses, right? So this is pretty boring for you. OK, we'll get there. All right, associated Legendre polynomial, where does this come from? Well, there's a good reason. It's not all this show-off, mathematical show-off. Suppose we have a solution that molecules point in what direction. So you might start to think about viruses, densities in different directions. So we can then measure, as an experimentalist, the density of molecules along a certain spherical angle omega, and then do that for all omegas. And I get a density of molecules along a particular orientation in space. And now there is real serious mathematics, which says that, well, if you have a function which depends on the direction on the surface of a sphere, a solid angle, then you can expand this in spherical harmonics. And if you either are familiar with it, because you know about electrostatics and quantum mechanics, and you don't like them because they're messy, or you haven't seen them, and you still don't like them. So for us, just think of it as an analog of Fourier transformation, just waves. But instead of plane waves along a surface, these are like waves, sorry, along a plane. These are waves on the surface of a sphere. And you get exactly the same thing. So this is Fourier transformation of the surface of a sphere. This is a rigorous result. We call this a basis set. And so we have two indices, L and M. L can be anything from 0 to infinity. And M, called the magnetic quantum number, runs from minus L to plus L. I'll get to this part in a moment. These are simply well-known functions. And there you have our friend, the associated Legendre polynomial. That's where it comes from, from these spherical harmonics. So this is a theorem. Any function rho of omega, sufficiently smooth, under some conditions, can be expanded. And these are the expansion coefficients, just like Fourier coefficients. And that's just the order parameter, very simple. So if you have orientational ordering, then that is fully defined by this set of numbers, Q, L, M. So Q tells you what is the weight of the spherical harmonic L, M, in this expansion. And that is the order parameter. That's why these Legendre polynomials. So there's a good reason. It's not just show-off. It's a sensible choice to define your expansion in terms of these Legendre polynomials. So now we understand why they appeared in uniaxial and biaxial pneumatics. But the moment you do this, you realize that you've opened Pandora's box. An infinite number of diseases can pop out. Because nothing is going to stop you. It so happens that biaxial pneumatics correspond to L equals 2, M equals 0. Biaxial pneumatics, L equals 2. And then M equals 0, plus or minus 2. That's the non-zero terms in here. But the sky is the limit. You can go to L equals 456. Nothing stops you. So physicists, particularly in the early 80s, they climbed up this ladder to larger L. And they found that this was very popular in Israel. To look at it, there's a complex series of liquid crystals called cholesteric and blue faces, which are chiral, by the way, right? They are chiral. And they have plus or minus 1 there. So this plus or minus 1 may have something to do with chirality. OK, why should we stop at L equals 2? We press on where no one went before to L equals 4. They haven't been found in nature. But at least we can do them on the computer. They're called cubanic liquid crystals. If you make computer simulation of cubes, then before these cubes freeze, you can cook up a face with orientational order, but not positional order. This was done by Nelson and Toner. So we're slowly working towards Lorman and Rochelle. Now, this Nelson here is David Nelson. He was my post-doc mentor and very influential guy in soft matter physics. You won't miss him if you see him. He went on to L equals 6. And he shifted the paradigm, as we would say. Well, what did he do? He went away from liquid crystals, and you already heard from Sergei. It has something to do with quasi crystals. Where does that come from? Well, let's suppose that we are in a liquid close to freezing, maybe in a glassy state or in a supercool system. So particles begin to cluster. Can the FCC locally face-centered cubic or hexagonal close packed or icosahedral? Oh, icosahedral. So we imagine that we have a cluster of atoms, and I can play the same game. I can measure a density profile of this sort, rho of omega, to look at how maybe these clusters can kind of line up without positional ordering. They may have the same direction without positional ordering. And then I can do the same Fourier decomposition with server surface sphere. And I can say, well, should it be FCC, HCP, or icosahedral? And the question is, what is the best choice? Now, that question has come out earlier in the work of Alexander and McTague. And you do this with Lando theory. That's where Lando theory is. Lando theory is very good in telling you how symmetries appear. So this is left Lando in one of his younger years. I suppose he is in his 20s here, sure. OK, 50s. All right. OK, he looks young to me. So you've heard about Lando theory, and I'm going to tell you about Lando theory in the sense of quantum mechanics, a few incomprehensible rules which just seem to work. We're not going to try to justify it. Here is what left Lando tells you if you have to find out the appearance of symmetry, like, for instance, viruses. OK, rule number one. You do this with free energy, and you have to write this free energy as a sum of scalar invariance of these QLM. What does that mean? It's actually very simple. It just means that you need to write down terms which don't change if you rotate your coordinate system. You can take any coordinate system, and it shouldn't depend on how you rotate it. Makes sense, right? OK, so he says, take these scalar invariance and build them up term by term. That's how the free energy should look like. S03, that's the group of rotations. Don't just these coordinates rotations. Rule number two. That's Sergei already mentioned that. Use one irreducible representation. OK, just means pick one L. Of course, which L is a question we saw already. You can just walk down the letter. And rule number three, minimize the expression you get with respect to QLM. That's the rules. They've proven to be very useful for finding out about symmetries. So a lot can be said about these three rules, but these are the ones we're going to apply. So if you do that, then this is what you get. This looks horrible, but we'll walk you through it. This is the quadratic invariance. So here, there's no linear invariant, but the quadratic one looks like sum m equals minus L plus L. L is fixed, rule number two. And use sum of the squares. That's an invariant under rotation. Called the quadratic invariance always looks like that. It's very simple. Things get a little hairier if you go to the cubic invariant. Here's how it looks like. Use sum over all m1, m2, and m3. And as Sergei already mentioned, m1 percent, 2 percent, 3 is 0. These are now three terms, q, m1, l, and 1, l, and 2. They're boring. These mysterious things are called Wigner 3J symbols. Now either you have had physics or chemistry. Paul, do you know about Wigner 3J symbols? Chemistry. So chemists have seen them physicists. And if you haven't, don't worry. We'll see in a moment where they come from. And finally, you get quadratic invariance. Good. These terms are W and U. They're phenomological parameters. They have to come out of some more microscopic description. But Landau informs us that close to a continuous or near-continuous transition, the free energy, when you change symmetry, must have this form. No matter how complex the system, it must be this one. Yeah, well, it's a little bit simpler way to see it by saying, well, we want to write down a free energy as a function of its density on the surface of a sphere. And well, if you do a simple Taylor expansion, there's no linear term. Because the average density is fixed by the number of particles. Then you get rho squared, rho cubed, rho to the fourth. And if you stick in this Fourier decomposition, out pops the quadratic invariant here, expansion rho, the cubic invariant here. Now you can see where this comes from because you get three spherical harmonics that you have to integrate together over the surface of a sphere. And that's exactly what this weakness 3J symbol is. And I wrote to the fourth, which is awful, so I won't show it. So here, we have the free energy we've done step number one, step number two, and we only now need to minimize with step number three. So if you do that for L equals six, remember? That's what Nelson did. It turns out the icosahedral symmetry has the lowest free energy. Hooray! We begin to understand maybe where icosahedral symmetry comes from. This is great. Here's how it looks like. You have a combination of spherical harmonics. Notice how few there are. Just one, two, three. It's only five and minus five. This is m equals zero. This is just a constant, so you can as well forget about it. I think this should be six zero. I'm sorry. This is not a constant. So this is very simple. The only thing which is left is the amplitude, the pre-factor. So L equals six is called an icosahedral spherical harmonic. It's just a linear combination of spherical harmonics which transforms as an icosahedron. And the only thing left is the pre-factor. If you stick this into the free energy, this is what you get, a simple polynomial expansion, quadratic cubic quartic. If you plot it, it looks like this. And if the minimum is at zero, then the isotropic phase is stable. And if you vary r, let's say you make this r a little bit smaller, so the curvature becomes less than this minimum becomes non-zero. And you get a nice first-order phase transition at, well, r not necessarily zero, but somewhere near zero. Here you go from an isotropic phase, boom, first-order transition, just like for pneumatics. And there you get the icosahedral phase. Very simple. It's a first-order transition. And it's a first-order transition because like Sergei already said, there's a cubic term here. So for L equals six, you have a cubic term. It's non-zero. It's a first-order transition. So it means there's nucleation and growth and hysteresis and all the wonderful things we associate with first-order phase transitions. So this is very simple. And what's important, Nelson and company checked for the stability, meaning that you already heard from Sergei, fluctuations are important. After you find an extremum, you need to check that it's stable if you push it a little bit away from the extremum. If you push it away, it should go back. So it's a minimum. The whole possible direction in phase space, you can check it and it's stable. So that looks like a wonderful starting point to think about viruses, don't you think? But keep in mind that this is a model for a global phase transition, orientation-ordering. Viruses are very small. So it's a little bit hairy. And so now we come to this paper by Norman and Rochelle. So now we've gone over this intellectual background and here's a picture of the capsid canine parvovirus. It's the smallest, parvo means small. So this is a small virus. And Norman and Rochelle noticed that the density of these kind of viruses can be very nicely explained in terms of an expansion spherical harmonics. That's by itself, you can always do an expansion spherical harmonics because it's a complete set, so you can do it. But they noticed that this particular virus, which is a T equals one virus, can be very nicely fitted by just sticking to L equals 15. And then you need some sort of linear combination. Here you have the order parameters and here you get the spherical harmonics. So the L equals 15 icosahedral spherical harmonic, that's just like we saw for L equals six, is just a particular combination. You see the maxima here, one, two, three, four, five. One, two, three, four, five. Minimum in between, ta, ta, ta, ta. Well, you don't get all the details right, but that's okay. That might be higher order terms. So it's not bad. And I encourage you to read the paper to see how nice these simple expansions seem to explain the density of viruses. It's very interesting. Okay, so first of all, let's go a little bit back what we mean that icosahedral spherical harmonic. So a little bit more intellectual history. And this is Mr. Felix Klein. This is a different Klein you heard before. He lived in the 1850s. And he's the one who invented them. He was a number theorist. And the statement is that for certain L, not all L, there is a unique linear combination of spherical harmonics. You call it Y sub H. I don't know where H comes from. That transforms as a scalar under the icosahedral symmetry group or as a pseudo scalar. I'll tell you in a moment what I mean with scalar and pseudo scalar. So if you take even L, like electrical six by Nelson and the crazy crystals and super cooling, then it's even under inversion. If you go for R to minus R, then you get basically the same density. So that's electrical six. This is electrical 10. And they become progressively more complex. Then for all L you have 15, which we just discussed, 21, which Sergei discussed, 25, 27, these pseudo scalar densities because they're all under inversion. And as Sergei already mentioned, there is something interesting about it. You flip chirality. If you invert this structure, you get a different chirality. If this is left-handed, this becomes right-handed. You can kind of see it by looking at the way these triangles pointing here. The blue triangles are pointing one way and here the blue triangles are pointing in another way. So you create chiral isomers, all right? And the next thing they said is you should only use the even ones. You shouldn't, excuse me, you should only use the odd ones. Capsa densities cannot be represented by the even Ls. Why not? Well, they're made of amino acids. Proteins are made of amino acids. Amino acids have a chirality. You cannot just flip the density. The left and right-handed amino acids are different. So if you take this capsid protein, which appropriately from the Russian origin is called the sputnik, if you take this sputnik protein, then it's very different if you take left-handed and right-handed. So they concluded since capsid densities are definitely not invariant under inversion, you should only use the odd spherical harmonics. And since you're only allowed to use the odd spherical harmonics, the smallest capsid should correspond to the smallest odd icosahedral spherical harmonics, that is L equals 15. And there we have the T equals one viruses by the power of pure reason. I can guess the density, or rather they guess the density of small viruses. It's beautiful, right? Beautiful idea. So powerful virus T equals one should have a density proportional to the icosahedral spherical harmonic L equals 15. The only thing unknown is the amplitude just as before for the quasicrystals. And if you stick it into the free energy, you get a particularly simple expression. You get a quadratic part and you get a quartic part. There's no cubic part because it's made of odd L, since it's odd under inversion if you integrate over the surface of the sphere for every plus R, you get minus on the opposite side. So it cancels. So the odd terms cancel and that means you have a second order phase transition. If you now plot this amplitude as a function of R15, when it changes sign, the amplitude nicely rises from zero. Again, Sergei mentioned that already. So it's a second order phase transition and it has an interesting feature. If you lower R15 for positive, you have positive part here and then when R15 is zero but then changes sign, the positive curvature here becomes negative. So you have two minima. These two minima correspond to left and right-handed chirality. Okay. The position of the minima just follows from R15 and you. And we call this spontaneous chiral symmetry breaking. So the theory by Lorman and Rochelle has spontaneous chiral symmetry breaking coinciding with the assembly of the virus and a continuous very specific transition. Oops. I want to tell you something briefly why this is actually a very attractive picture because viral assembly shouldn't have a large barrier because you can't go over it. You rather have a continuous transition. So it's actually a very nice idea to propose that this is a continuous phase transition. So that's my understanding of the background of the Lorman-Rochelle theory. So the next part is going to see, well, questions. First is, is it a second-order transition? Keep in mind, we're only looking here at this collective assembly pathway. You heard from Mike Hagen. This is the kind of situation where we imagine a kind of disordered aggregate transitioning to an ordered system. So you can see there's a kind of orientational ordering transition from this aggregate to an ordered shell. And there are experiments. You heard about them from Bill Gelbart in UCLA. They did this for CCMV and here you see the RNA outlined. The blue here is the RNA. As you, I like the word to use titrate in because physicists have no idea what titrate in means. So here, what you do is you titrate in proteins. And when I talk to physicists, they have no, but they don't dare to say, what's titrate in? So I love to say it. You titrate in the proteins. That means you just raise the concentration. So as you do that and change the acidity, you can go from a disordered to an ordered structure. So that's the kind of transition. And another speaker at this conference, Guillaume Trissay, use fluorescent thermal shift assay and he finds it's a first-order transition. It's a little worrisome. The second thing is this spontaneous chiral symmetry breaking because the uniform state, before it orders, already is chiral, right? It's made of capsid proteins which are chiral. So the uniform state and the ordered state both are chiral, the same chirality. So you can't really have spontaneous chiral symmetry breaking if the components have a chirality. So it's like you have doing a magnetic phase transition where you have an ordering field. It's a bit of a problem. But the biggest problem for us had to do with this thermodynamic stability. So let's remember this free energy, London rule number two, which you have to minimize. Well, if you check, you'll find that this is an extremum and you still need to check that if you put little fluctuations around it, it stays at the extremum. So we took this solution of Lore-Mann and Rochelle and put a small fluctuation around it. So we said, let's perturb the order parameter in an infinitesimal way, in an arbitrary way, but we keep L equals 15 and see what happens with the free energy. Well, because this is an extremum, you don't get linear terms, but you get quadratic terms like a harmonic oscillator. And this is how it looks, no surprise. Sum over all m, m prime. And here is this delta Q. It's like a displacement away from equilibrium. And these are the coefficients, so it's like a matrix, 31 by 31. And you can show, basically by diagonalizing this, that this matrix must have positive eigenvalues. That's what thermodynamic stability means. So when we did it, and that is not easy, it was a massive numerical calculation, we computed the eigenvalues, means diagonalizing a 31 by 31 matrix, we found seven negative eigenvalues. It's a saddle point. So the Eichler-Sahid, L equals 15 state. Unlike L equals 6, it's not stable. So what is stable? What is the lowest energy state? Well, we found that in the L equals 15, it's d5. This is done by minimization using Mathematica. So we cannot prove it. We just did the best job we can by trying to search in this huge 31-dimensional space. And this is what we came up with, d5. That means that it should have one 5-fold symmetry axis. You see it here. And 5 2-fold axis, they're a little odd, because if you take this 2-fold axis, you have also to flip the density. OK, this doesn't look like a virus at all. And the horrible thing was, all odd icosahedral harmonics are unstable. Everyone we checked, at least. So you can make a stupid joke that since we have just learned that viruses should be odd icosahedral spherical harmonics, and since they're all unstable, there are no viruses with icosahedral symmetry. We have proven by rigorous Lando theory that we shouldn't get the common cold. That's not a funny joke, right? But OK, I'll just give it. Sorry? OK, Paul, you're welcome. The Dutch are known for many things, but not for being particularly funny. OK. The only stable ones we found was 6, 10, 12, and 18. They're thermodynamically stable. Allical 16 is unstable. And we found that in the mathematical literature applied math, this is actually an active field. They studied these even L icosahedral spherical harmonics, and they discovered that instability starts with allical 16. Mr. Matthews, who turned out one of our referees and says, why didn't you cite my paper? OK, good. So there is instability. So maybe the allical six states, and even L, we should have another look, and that's exactly I'm going to echo the point with Sergei made. Lando theory doesn't exactly tell you what you're going to put up this structure. And if you take, like, the picorna virus, you have 12 identical pentagons, so 15 cap proteins. They are stable in solution. So assembly is not from proteins. Assembly starts from these collectives. They have these pentamers floating in solution. So Lando theory is never going to give you the density of this thing. I mean, where would it come from? And you can only hope that it tells you the density of, let's say, the center of mass of these things. So what's wrong with saying reinterpreting this and saying, no, no, these are not single proteins. This represents the density of capsimers. So you might have a perfectly achiral icosahedron, and nothing stops you from putting, from decorating this with chiral units. It's like you have a cubic Bravais lattice, and then you put molecular units in, which may be chiral. It may change the symmetry. You have to check, but it doesn't have to. So there is no intrinsic violation in starting from an achiral mathematical lattice on which you put these guys. So we said maybe we can interpret rho for omega as the density of capsimers centers. And for l equals 10, this is CCMV. It also works. This would be the pentamers, and this would be the hexamers. By the way, for CCMV, they barely show chirality. You would have to look very hard to see that they are. So we would say this is l equals 10, and maybe it's not a bad idea to have a first sort of transition there. It's still suspicious, because if this thing begins to fluctuate and wobble, you would guess that it begins to communicate some of this thermal fluctuation. They're going to pick up on the chirality. Is it possible you completely hide this chirality? Intuitively, I would say I'm on the side of Lohrmann and Rochelle. The chirality ought to show up, and this doesn't have any. So something is not quite right. So we suggested two small extensions of the Lohrmann-Rochelle theory. The first one is, since we're already in Russia, we stay there, and we go from not just London, but London-Brasovsky. So Brasovsky was a student, and from London, I assume. No. OK. Then I don't know. OK. Brasovsky extended London theory. Is that correct? Good. And he applied it to all sorts of problems in liquid crystal, soft matter physics, and what he said was, well, remember, this was this integral of the density. He said, you can do a much nicer job if you don't just assume that this depends on the density, but also allow it to depend on the gradient of the density. So where does this strange term come from, and what is K0? This is the key difference. This is old-fashioned London theory, and this is London-Brasovsky theory. By the way, I'm not saying that liquid crystals are necessarily represented by this. This is just a very simplest way to include a gradient of rho. K0 here is a wave number. It's 2 pi over a characteristic length, a, which is the protein size, or the capsimish size, is the microscopic length scale of the system, 2 pi over a. And you can understand why Brasovsky included that term, because imagine that you want to have a density wave. Rho of r is e to the ikr. Along some direction, this is like a director, and K0 is the wave factor. So you'd like to have a theory with waves, like density waves. In fact, that's the original paper of Lorema and Rochelle was about density waves. So let's put these density waves in by hand. They put them in by hand. Here we say we write in a free energy, which produces density waves. And in fact, if you work with not plus squared plus K0 squared on this wave, you just bring down two factors of ik0. That is minus K0 squared, plus K0 squared, e of 0. So you see immediately that this theory tells you the density is going to consist of waves. And these terms, these nonlinear terms, are going to tell you how they interact. So this is the standard way you look at crystallization. So if you think of the formation of a viral shell as the crystallization of a spherical surface, then this is in fact a natural theory. I'll try to convince you there's an advantage. At first, you might say, what's wrong with just keeping the simple form? Nothing. So here's an example of how long the Rossovsky theory works. Let's take three density waves along the direction of an equilateral triangle. You put them in. You sum over them. This is called a star. And lo and behold, you get a hexagonal lattice. You can take other combinations as well. In this particular case, you get a first order transition again, because the cubic term is nonzero. Because k1 plus k2 plus k3 is zero. So these two terms do not cancel if you take the cubic term. And this theory has been extensively applied in all sorts of problem in liquid crystals. And it's very rich. You can now look at the competition between different symmetries. Far beyond simple Landau theory, you can get micellar faces, hexagonal faces, cubic faces. It's a very rich theory. And we're going to do this on a spherical surface. So we take this term, our old friend, and I'll stuff it in the Landau-Rossovsky theory. And what you get looks very similar to what we used to have. There are really only two differences. You now sum over all L. You no longer pick L. It's not up to you. It's not the, I think, electrical six. Where does it come from? The value for L itself has to come out of minimization of the free energy. So now L is no longer fixed. It's by itself a parameter. So you sum over all L. For the rest, it looks the same, right? This is the quadratic invariant. This is the cubic invariant. That's the quartic invariant. Except we have L1, L2, L3. L is no longer fixed. So the big payoff is that you now get an expression for the stability for different L. This R sub L here is our old R, the one which you got from the quadratic term. These two terms come from the London-Brasovsky one. L times L plus 1 minus K0R squared. This is this wave number, K0, 2 pi O for the protein size. And R is the radius of the shell. And with this, you can now predict the L value of the capsid, which actually is a, it's a parameter. It's like a variational parameter. Here's how it works. You plot this term for different L. They're all parabolas as a function of K0R. No, they're not parabolas. They are more complex. But they all have a minimum when K0R is close to L squared. So this is the L equals 15. This is R 15. This is R 16. This is R 17. Now suppose you reduce R. The transition is exactly like Sergei said. It's where you first hit 0. The point where instability starts is the parabola which goes to 0 first. So in this interval, it's this parabola which goes unstable first at this point. But if you increase K0R in this interval when K0R is bigger than 16, it's the second parabola. So there's an infinite sequence of parabolas. And for given K0R, you can predict which L is going to appear. RL is a minimum and K0R is L. So the big advantage of Lambda-Brasowski is that you can look at the competition between different L. Step two is the correlative. So French colleagues will recognize Pierre-Jules de Genne who thought a lot about how chirality should be included in a Lando theory. And he said that you start with the A chiral theory with scalars. And then in order to include chirality, like for instance in cholesterically with crystals, you have to add the lowest order pseudo scaler. Now for cholesterics, it's a very simple term. You write it down immediately. Alas, it's not so simple if you have to construct your pseudo scaler from the normal to the surface and the gradient of the density. This is the lowest order pseudo scaler. I can't help it. It's fourth order. So it is still within the realm of fourth order expansion. It has this form. You can see it's a pseudo scaler because this is an axial vector dotted into basically a vector. And so it's a pseudo scaler. Those of you who come from soft matter physics may recognize this as related to the pseudo scaler of hell free and prose when they looked at chiral surfaces. That's the way it is. I'm not going to justify it. I'm just going to tell you that's what it came out. It was proven by a brilliant postdoc of mine, Sanjay Dharmavaram. I couldn't do it. I tried. And but he came up with a solution. So now we try again with this, let's call it boosted Lorman-Rogel theory. We just put in these little mathematical extensions. OK, we still get L equals 6 and 10 to be stable minima and nothing new. It's a first order transition for even L because we have an cubic invariant. But there is a small change. When you look at the thermal fluctuations around, that is just what Sergey mentioned. If you look at the thermal fluctuations, they are chiral. So even though the density, the ground state, is a chiral, now because of this thermal fluctuations, they begin to talk to each other. Chirality appears. So both the isotropic state and these structures are now chiral. With that, there's no spontaneous chiral symmetry breaking, which I would argue shouldn't appear. It's gone. And we can conclude that at least Dix, these four icosahedral spherical harmonics are possible representations for viral assembly. They are kosher. That still leaves the odd. We've seen that the odd ones must be physical because the Parvo virus worked beautifully with L equals 15. And L equals 15 remains unstable when I stick to 1L. If I stick to 1L, to 1 parabola, I can't save L equals 15. But I can change K0R and I can mix in a little bit of the neighborhood. Now I know why I like London-Brasovsky theorem because it allows me to look at the competition between different L. So what we did is let's say we're looking at a transition point in this region where these two sectors compete with each other. And then we ask, could there be a stable icosahedral order parameter composed of two irreducible representations, L equals 15 and L equals 16? This is a study in 64 dimensions. I have finally beaten my colleagues in string theory. They're always both about 50 dimensions and you have 60 dimensions. Do that. OK. 64 dimension in theory. However, and this is all done by my friends, my colleagues. I didn't do it. We did an extensive numerical study of the Landau theory. For L equals 15, it's non-icosahedral d5 and other stuff. For L equals 16, it's non-icosahedral, other stuff. But magically, remarkably, in this intervening region where they talk to each other, somehow these two unstable states interfere constructively and stabilize the icosahedral state, which now appear as mixtures of different L. If you make K0R run in this direction, less than 16, it's more K0R. It's more like the L equals 15. If it's going to here, it's more like L equals 16. So this is a chiral structure, mostly L equals 15. And this is a chiral structure, mostly L equals 16. This one is 16 maxima. Maybe that one, this is 72 maxima. So the only thing which happens if you go from here to there is that this maxima points up. So we have rescued the L equals 15 states at the cost of complexity of having to mix in, if you want, Sergei calls it fluctuations. You have to just mix in some of the other Ls, and then you restore it. No spontaneous chiral symmetry breaking? That's nice. That's when trouble starts. You remember Lavlando? He is not happy. There is some sort of, you can hear the vibrations coming up that we violated his second rule. Only one irreducible representation. This is not just some sort of parasitic order parameter you drag along. We need really both irreducible representation to get an icosahedral state. Lando's ire inspired one of our referees to get really upset. How dare we? How dare we? We were violating this basic rule. We were able, in the end, to convince him that this is what comes out. We can't help it. There's very good reasons for this rule. It works nearly always, but apparently not always. So we found here a new region for study of Lando theory, where this rule is no longer applicable. Quickly two examples, then I'll be done. If you stick to the left-hand side, you get the Parvo virus, which is still very nice. It's a little density. Maybe you can even think of that as explaining that. I don't want to push it. If you go to the other direction, when you see these peaks appear, looks a lot like the polyomavirus, a 72 maxima. OK. Is there any confirmation of this idea? Well, you can do numerical simulations of 72 particles on a spherical surface, maybe with Lenard-Jones interaction, just as a check, to see if anything I said makes sense. No chirality, so we don't have this chiral pseudo-scaler. Paul, do you recognize this? Yeah, you do it. This was done by Roya and Paul. It's Stefan Paquet here. Oh, there we are. Three people in our audience did this. They said, well, what is the minimum energy state? And they got a zoo of states, nearly the same energy. D5. Did you remember D5? That's what we got. D3, icosahedral tetrahedral D2. Woo. The icosahedral state is just very stable. If you change the perimeter of a very small interval, so the icosahedral state is very fragile. It competes with other structures with completely different symmetry. So this is the reference. Has it been published, Paul? Yes, OK. So you see my scholarship is failing here. Why was icosahedral symmetry selected for viruses? It's not just free energy, because the icosahedral state is just a little sliver of stability for this T-pull-zeffen structure. So this tells us that icosahedral states are quite delicate. Roya and Bill and myself showed that you can further stabilize them if you allow spheres of two sizes, smaller and bigger. And if you just tune the ratio right, you can stay. But there must be a reason why icosahedral symmetry was selected. So in conclusion, Lorman-Rochall theory is, with some extension, is a very powerful mathematical tool for the study of capsid assembly. We found two modes. First, for mixed L, it's a continuous transition or weekly first order. For pure L, for certain limited even L, you have strongly first order transition. So you can get two completely different modes of assembly. Of course, this is the more interesting one. Viral assembly is a wonderful laboratory for the testing and maybe violation of lambda theory. Big experimental problems. How do we test first order, second order fluctuations? We are lacking the tools because these structures are so small compared to the wavelength of light. Finally, I want to thank my group, particularly Sanjay Dharmavaram, who did all of this hard analytical work finding this pseudo-scaler, Joseph Rathnick, who invented or developed codes for doing these searches. Amit Singh, Josh Kelly, Bill Kluge. Many may remember him. He was my dear friend and colleague. He died last year. And I also want to thank Bill Galbart and Chuck and Rhys for providing us with these data on assembly and with that. I'm sorry I went a little bit over time, but that's it. Sorry to keep you from your lunch. Thank you.