 I hope you will be able to organize more in the next years. This work has been done, so this presentation will be a lot different from the presentation you have had this week. It's more a mathematical presentation, and it's not really on viruses, but it's on self-assembly nanoparticles. I hope you will not be too bored, and some nice questions will pop up. This work has been done in collaboration with Raiden Tuarok at the University of York, and with a structural biologist, Peter Burkart, at the University of Connecticut. And the idea that we want to explore in this talk is if we can classify some specific class of self-assembly nanoparticles, namely the one created in Peter's Burkart lab, and classify them based on geometry. Unluckily, for this kind of self-assembly nanoparticle, they don't obey to the Casper and Kluge classification, because they assemble from a polypeptide engineered in such a fashion that it forms pentamers and trimmers, but many more pentamers than the one allowed by the Casper and Kluge classification. So we have to come up with something different, and the idea is to try to mimic a classification that already exists, the classification that has been given by Fowler and Manopoulos for fullerines. While self-assembling polypeptide nanoparticles, well, as the number of proteins in the data banks is increasing, there are new possibilities to design synthetic nanoparticles, and they are used for medical applications. They can be used for drug delivery, gene therapy, and in particular, the one I'm focusing on is used for vaccination. The idea is not new in the field. The first one in thinking about doing synthetic self-assembly polypeptide nanoparticles where Padilla Kovl and Yeats in 2001, they tried or they engineered a fusion protein that is shown here on the cartoon on the left, that had two domains, one domain in green with the ability to form dimers and the second domain in red that had the ability to form trimmers, and then they linked these two regions. And they were able to form liars of these polypeptides, but they were also able to form closed cages of these polypeptides with a chosen symmetry. And in particular, here is a computer model of this raedral nanoparticle formed by 12 of these fusion proteins with an edge of about 15 nanometers. But it took them 10 years to be able, at the end, to really crystallize the nanoparticle, and they did it in 2012. So the specific system I wanted to focus on is a system that has been developed by Peter Burkart at the University of Connecticut, and it assembles from a three-dimensional building block, and this building block is a polypeptide that has two domains. If you look at this cartoon, it has one domain in blue, shown in blue here, that it has the ability to assemble with other two blue domains. So it forms a trimer. And then it is linked to a second domain, shown in green here, that has the ability to assemble with other four domains of the same kind, and it forms pentamers. And they are able to form a self-assembly nanoparticle, so the nanoparticle self-assemble. And here is a computer model of a nanoparticle made by 60 fusion proteins. In this case, the nanoparticle has been reconstructed with icosahedral symmetry, and in this case, the Casper and Clue classification would work because you just have 12 pentamers in this configuration. But for the other nanoparticles that they are trying to construct, then the classification would not work. So this reconstruction has been done by Burkart's lab. And what are they using this nanoparticle for? They want to use it as a repetitive antigen display system. They further extend the polypeptide that they have by surface protein of the pathogen. So they put, or at the trimeric end, or at pentameric end, or on both ends, some fragments of the surface protein of the pathogen, and they further extend this polypeptide. And they let the nanoparticle assemble. And at the end, you have a form nanoparticle where on the outer surface you have the antigen that are displayed. And they are displayed in several copies. And the other thing is they are displayed or in trimmers or in pentamers, or for some of the vaccines that they are trying, they are displayed in both cases, so in trimmers and in pentamers. And they are trying to do this for several diseases, for SARS, for influenza, and they have a trial that is starting for malaria for the test trying in a short. Why do they want to try to use this new kind of vaccination? Well, because the cost of production are lower, because they have told me that the particle is stable at room temperature, so they don't have the problem of maintaining the code of chain. And also that for people that are afraid by injection, this is the good news. It will not be given by injection, but you will breathe it through the nose. So when they contacted us, the information that they gave us where that the nanoparticle was given had these properties to assemble in trimmers and pentamers, so the polypeptide had the ability to assemble in trimmers and pentamers. And they were also able to tell us that there were some from the scanning transmission microscopy, there were some mass peak corresponding to particles that were made around 230 to 310 and 360 chains. So they were puzzled and they asked, and they were not able to reconstruct these nanoparticles with the usual Casper and Clu classification, they were puzzled, and they asked us if there was a mathematical way to reconstruct the geometry of the nanoparticles. So what I want to point out is that I will just focus on the connectivity of this polypeptide. I will see this polypeptide as if there would be a string and I will explain you which is the mathematical object that I will use, and I will just look at the connectivity between the different proteins. So, but as I said, and luckily the Casper and Clu classification doesn't work. So the question that we want to answer is what are the symmetric structure that can occur as a result of self-assembly? And symmetric is in brackets because the classification that we give and that I will show here is for symmetric nanoparticles up to 360 polypeptides. But the procedure that we have constructed will also work with the nanoparticles that don't have any symmetry. So the procedure has four different steps. Step number one, we determine the possible topology of the particles. By topology, I mean the connectivity between the different proteins. Step two, we determine if there is a symmetry which constrain does the symmetry impose on the nanoparticle. And the step three and step four are connected one with the other. We base our construction on fullerene. So we look for a fullerene that we call somehow equivalent and we will say what's equivalent to the nanoparticles, fullerene equivalent to the nanoparticle. And then when we know which fullerene we want to use, we get back to the nanoparticle. So let's get started. This is the final slide, not of the presentation, but of the work. I show you the final results so you know where I want to get. The idea is that I want to find a mathematical object that represent my nanoparticle. And I represent my nanoparticle by what's called in mathematics a spherical graph. So it's an object that has some vertices. This sphere represented in white represent vertices that we say of degree five. There are five edges coming in. And other vertices of degree three that are the spheres are here shown in black. And this corresponds at the position of the trimmers and of the pentamers. And then we connect this vertices by edges and the edges represent the polypeptile. We don't take into account here this edge between the two different domains. We're just thinking as a first approximation because we just want to look at the connectivity as a straight edge. So as I said, we want to model our nanoparticle by a graph. And here shown in a cartoon, you see that you have vertices of degree three in red and vertices of degree five in blue. What is important to see is that you don't have any edges between vertices of degree five and degree five or edges of degree three and degree three. So the edges just connects degree five with degree three. And this comes from the fact that we are modeling the polypeptile as the connection between the trimmers and the pentamers. And this is what in mathematics we call a bipartite graph. So we have two set of vertices and these are just connected one with the other. So for a graphonosphere or in general a graph, a relation like this that it's called Euler formula holds and it relates the number of vertices in the graph with the number of edges in the graph and the number of faces. And then we are in a specific system and the specific system assembles with these polypeptides that form trimmers and pentamers. So we have a further and another condition and the other condition is that the number capital N of the fusion protein they are into this nanoparticle must be a multiple of 15 because we want all the pentamers and all the trimmers formed in the nanoparticle. So from this condition and this condition we obtain that we can express the number of faces by this formula where M is dependent on the number of fusion protein that we have in our nanoparticle. Then there is some nice theorem in mathematics. We don't look at the theorem, we just know that it holds for our nanoparticles and what this theorem tells us is that there is a relation between the number of edges and the number of vertices. And if we translate this relation for our nanoparticle so in the number of edges and the number of trimmers and pentamers that we have in our nanoparticle we come out with this relation here. So we have a necessary condition and the condition is that M, the small m that we have found before that defines the number of faces in the nanoparticle and that is dependent on the number of polypeptides that we have in the nanoparticle has to be greater or equal than four. And in the case that it's exactly equal to four then you find again the nanoparticles that I've shown before at the beginning for which we had the computer model with 60 polypeptides. So at this point we have modeled our nanoparticle by a graph that has vertices of degree three and vertices of degree five. We can also look at the faces of our graph. And the faces of our graph for as the, because since the graph is a bipartite graph will be all with an even number of edges and this depends on the fact that the vertices of degree five degree three and degree five have to alternate around the boundaries of the faces. So we made a choice and the choice that we made was to choose just rhombic and hexagonal faces but in principle you can also choose bigger faces octagonal or the cagonal faces. And you can write down a relation, an equation about the number of edges in your tiles. In this case you will have four edges for a rhomb, six edges for a hexagon and the number, this total number of edges will be equal of two times the number of polypeptides or edges in the nanoparticle because each edge is shared between two adjacent faces. And from this relation and the one that, and the question that we had before on the faces we can express the number of rhombic faces in dependence of m, the small n that we had in the first slide. So we know that the number of rhombs and the number of hexamons it fully determined when we know how many polypeptides are in our nanoparticle. In this case if we get back to the first example, the case in which m is equal to four, then the nanoparticle graph associated to the nanoparticle is a graph of what is called a rhombic triacontadron. And it's a polyadron that has a icosadral symmetry and has just rhombic faces. In fact for m equal to four there are no hexagonal faces. So this would be the nanoparticle graph, the mathematical object we will look at when we are trying to classify the nanoparticles. In the specific case we had before. Here is just a table that shows that you can compute how many rhombic and how many hexagonal faces you have and they have made special choices and the special choices are motivated by the next slide where I make some choices about symmetry. So at this point what we have done, we have associated to our nanoparticle mathematical object as spherical graph that has vertices of degree three and degree five and we have computed how many depending on the number of polypeptide in the nanoparticle we have computed how many hexagonal and how many rhombic faces we have into the graph. Now we want to look at symmetry and the reason we want to look at symmetry is that if we work with symmetry we are allowed just to work in the fundamental domain and this makes things much easier. But as I said before we were also able to work without symmetry. We will focus here just on tetrahedral, octahedral and icosahedral symmetry but we also have classified particles with lower symmetry. So we have reconstructed particles with lower symmetry. We call capital P the number of pentagonal cluster or vertices of degree five in the fundamental domain and we call capital T the number of triangular cluster or vertices of degree three in the fundamental domain. So as an example I look at the tetrahedral symmetry and here you have a cartoon of a grid of a tetrahedral where I have highlighted the fundamental domain and you can see that there are special position at the vertices, there are two threefold sides at the vertices of at the corners of the faces of the tetrahedron and at the center of the faces of the tetrahedron. So you can put into relation the number, you can express the number of polypeptides that you have in your nanoparticle via the number of pentamers in the fundamental domain or via the number of trimmers in the fundamental domain. So let's do it for the pentamers. So how many pentamers you will have in the nanoparticle with capital N polypeptides will have a number P of pentamers sitting in the fundamental domain. So capital P is the number of pentamers sitting in the fundamental domain. You have 12 copies of the fundamental domain that tasselate the tetrahedron. So you multiply by 12, and then you multiply by five, because you have five proteins in the pentamer. You can do the same thing for the trimmers. So trimmers in the fundamental domain are capital T. You multiply by 12, because 12 is the number of copies of the fundamental domain. And you multiply by three, because three is the number of proteins in the trimer. But then you have to take care also of these special positions here. And you have four copies of these special positions, and four copies of this special position. So if there is a trimer sitting in one of these positions, then alpha or beta will be equal to one. Otherwise, if there are no trimmers sitting in this position, alpha and beta are equal to zero. And here are some of the possible chain numbers up of 360. These are all the chain numbers for which we have particles that can have tetrahedral symmetry. And for this, we have computed where the trimmers sit if they stay in the fundamental domain, or if they sit in some of the special positions. And you can do the same thing for the icosahedral symmetry. The only difference is that you have to keep in mind that now you can put a pentamer in this special position. And you compute it in a similar way, and you find a similar solution for the icosahedral particle and also for the octahedral particle. The reasoning is always the same. The only thing you have to keep in mind here is that we have made a choice. We have chosen just to work with rhombic and hexagonal faces. Therefore, we will not have in our model any solution with the octahedral symmetry. Because in this special position, you cannot put any trimer, and you cannot put any pentamer because it's a fourfold position. But you cannot put also any of the tiles that we have chosen. Because the tiles that we have chosen, rhombs and hexagons, are decorated by trimmers and pentamers along the boundary. And so they will not fit with the fourfold rotations there. So at this point, what we have, we have modeled our nanoparticle by a graph with vertices of degree 3, with vertices of degree 5. We know, depending on the number of polypeptides that we have in the nanoparticle, how many rhombic faces we have and how many hexagonal faces we have in the nanoparticle. And if there is any symmetry in the nanoparticle, we also know where the trimmers will sit. They will sit in the fundamental domain or in special positions. Now we want to classify all of these particles. And in order to do this, we refer to fullerins. So fullerins are carbon molecules, Cn. And they can be represented by three regular spherical graphs. And here is an example, for example, for C68. They have hexagonal faces and they have pentagonal faces. And the number of pentagonal faces is constrained by Euler formula. These pentagonal faces can be isolated, or as in this case, they can be adjacent. There is a classification that has been given by Fowler and Manopoulos. And this is based on a construction in the case for the Ecosadral fullerins, due to Goldberg and Coxeter. And for low symmetry fullerins on a construction given by Fowler and Cremonesteer. And the thing that you have to keep in mind is that for each Cn, it's not that you just have this molecule, but you can have many more isomers. So I'm just showing one. But then when you're looking at a procedure, there can be many more isomers. So in principle, you can reconstruct many more and classify many more nanoparticles. So this is the core of the procedure. What we have in figure A, a portion of the graph of the nanoparticle. In black, you have the vertices of degree 3. In white, you have the vertices of degree 5. And then you have rhombic and hexagonal faces. So we go from A to B. And going from A to B, we add in each hexagonal face a vertex of degree 3. And we connect this vertex of degree 3 to the vertices of degree 5. So in figure B, we have a graph that is still related to the nanoparticle graph. But it has vertices of degree 3 in black and red. And it has vertices of degree 5 and 6 in white. Then we take each of these rhombic faces. And for each of these rhombic faces, we connect the vertices of degree 3 by a dashed line that will become an edge of a new graph. After we have connected all the vertices of degree 3 through the rhombs, we cancel all the edges with the continuous line. So all the black edges. And we arrive from C to D, where we are left with this graph with the dashed lines. And the vertices, all vertices of degree 3 ignore for the moment these white points, they are not part of the graph. They are just there for reference. And this corresponds to a fullerine. So starting from a nanoparticle, we are able to get to a fullerine in a unique way. But the problem is to get back. So what we have done here, going from A to D, is what we have called the vertex addition pool. We have added some vertices. Now we want to get back. So suppose you have a graph that is made by hexagons, has hexagonal and pentagonal faces, and has just vertices of degree 3. And it's a graph corresponding to a fullerine. Now we want to get back. And in order to do this, you have to color just one vertex for each hexagon. And each hexagon has to have one vertex colored. And the choice will not be unique. So for each isomer that you will have here, it could be that you have no solution. So there is no coloring. Or it could be that you have more than one solution. And when you have colored your vertices, then you add the central points. And then you connect these central points, the white points, with all the vertices of degree 3. And you get back. You cancel the dashed lines. You return to a nanoparticle graph. And then you cancel the other vertices, the colored vertices. And you are left with a nanoparticle graph. So this is the idea under the procedure. So suppose you have a nanoparticle with capital N polypeptides. And you want to know if you can associate to this specific fullerine. So which fullerine Cn can we associate to it? Well, if you start from A, doing exactly the addition rule that we have explained in the slide before. Here we have a number of vertices of degree 3 that is equal to 5m. And we also know from the slides before how many hexagonal faces we have in our nanoparticle. And we have m minus 4 nanoparticle, m minus 4 hexagonal faces. So if we add up these two quantities, we know that in the portion of the graph shown in B, you will have 6m minus 4 vertices of degree 3. And this vertices will corresponds to the vertices that belong to the fullerine. So the fullerine we are looking for is a fullerine that is equal to C6m minus 4. Keep in mind that you have a fullerine 6m minus 4, but you can have many isomers from this fullerine. So what do we have at this point? We have a procedure that allows us to go from the nanoparticles to associate the fullerine. And we can get back, but not in a unique way. What we want to do is to classify the nanoparticles. So we go through the classification for the fullerines and then try to apply this to the classification for the nanoparticles. So in the case of the fullerine with icosahedral symmetry, a construction like the one for Caspar and Kluge works, you can parametrize your fullerine by two parameters and you can classify your fullerines by T numbers. In this case, you have T equal to 7. And the total number of atoms that you have in your fullerine will be 20 times T. In the case, and this was the thing that was nice for us, is in the case of lower symmetry, there is a classification given by Fowler-Cremon and Steer. And here I show the construction just for the tetrahedral fullerines. And what you do is that the fullerine is given by four parameters. So if you start from O and you move in the direction E1 by a step E, I, so you move one in this direction and then you move by a step J equal to 2 in this direction. You are arriving P. You draw OP, the edge OP. And this is the edge of an equilateral triangle, what's called a big equilateral triangle. And you will have four in the final net for the tetrahedral particle. Then you start again from O. You move of a step 0 H equal to 0 in the direction E1. And then you move of a step 2 in the direction E2 in this direction. So 0, 2, you're arriving Q. You have an edge from 0 to Q. And you draw here an equilateral triangle, a small equilateral triangle. And you have four copies of this triangle in your fullerine. And what you get by product, you get this scalene triangle and you get 12 copies of this. So your final net of the fullerine will be cut out from this form that is made by four big triangles, four small triangles, and 12 scalene triangles. And you can find on this representation also the axis the rotation axis. So rotation three-fold axis sit in the center of the equilateral triangle. And the two-fold axis sit in the middle of the edge QP. This is the representation for C68. So now we try to put all the information that we have just in one slide and look what we can come up with. So suppose you look for a tetrahedral nanoparticle that has 180 polypeptides. Then since it has 108 polypeptides, small m will be equal to 12. And you know that you will have to take the fullerine equivalent that is C with 6m minus 4, while m is equal to 12. So you will have to take the fullerine C68. And you go and cast in the Fowler and Manopoulos classification. And you look for all the isomers that have C68. 10 minutes. OK. And you look for all the isomers that have C68, that corresponds to C68. And you look for all the ones that have all the tetrahedral symmetry or a higher symmetry. In this case, I've chosen the one that has this parameterization. You can draw the fullerine. And then you get back to step one, where you were computing the number of roms and number of hexagons in your nanoparticle. And you go back to step two, where you were computing where the trimmers and where the pentamers were sitting in the fundamental domain. And if some of them were sitting in a special position. What I've not said here is that the fundamental domain is between the red dots and the middle points there. So the two-fold axis is there and the center. And the two-fold axis here and there. So this orange region corresponds to three times to the fundamental domain. So it's enough to decorate the fundamental domain. And then you can apply the symmetry group to reproduce the whole particle. So now we have to place the coloring, where you already have the solution here. But suppose you don't have the red dots. And you have to understand where to put the red dots. You have some hints, some necessary condition from step one and from step two. You know that you have to place eight hexagons in the final particle. And you know that you don't have any trimmers sitting in the special position. You don't have any trimmers on the symmetry axis. So you know that the points that you are going to color are these special points that are sitting in this case on the symmetry axis. Then you do the construction that we have done before. So you connect the vertices, all the central points with the vertices of degree three, and you cancel the edges that you are not interested. And you arrive at this representation, this error representation corresponding to the nanoparticle graph, where the red points are the points that have been colored in the nanoparticle but that have been colored in the fullerine, but that are not part of the nanoparticle graph. And here is the classification going from fullerines to nanoparticle just for the tetradural nanoparticles. So we are taking all the possible nanoparticles corresponding to this number of proteins because these are the ones that allow tetradural symmetry. We have computed the corresponding fullerines. We have looked at which symmetry were in the fullerines. We have looked for the corresponding parameterization and after doing the coloring, we have seen that for some of them, the symmetry is conserved and for some of them, it's not possible to do the coloring. Whether for others, it's possible to have more than one coloring. And in some cases, the symmetry for example here is even lowered. And this is the corresponding atlas just for the three copies of the fundamental domain for all the particles that we have taken into account just for the tetradural symmetry. We have done this also for lower symmetries. And when you have this, you can also reconstruct the full three-dimensional graph. So you start from the three-dimensional fullerine graph and you arrive to the three-dimensional nanoparticle graph and then you give this to the structural biologists and they get crazy, but I don't know how they do it. Somehow they fit the proteins inside and because they can move, the two vertices are two different layers and they reconstruct the nanoparticles. And here are some of the nanoparticles that have been reconstructed in this representation. Just, oh no, nothing appears. Well, it doesn't matter. And just the other thing that I wanted to say and then I finish. As we have seen before, antigens are attached to the particle surface. I don't know if it doesn't show. But what is important from an article from Dinsen and Vogelstein is the density of the antigen of the surface and the distance of the antigen of the surface. So having a model of the nanoparticle could help to understand which nanoparticle could be better with respect to others. And here we have computed the distance between the trimmers in the nanoparticles and the pentamers in the nanoparticle, but just in the case that the nanoparticle are represented by a graph on the sphere. So it's an idealized model, still an idealized model. It's not really the final conformation that the nanoparticle we have. So what we have done using tools from graph theory, we have developed a predictive tool to construct all possible configuration of the building blocks in the polypeptidic nanoparticles with a given coordination number three and five in this case. And the symmetric configuration can be classified by a generalization of the quasi equivalence that covers arbitrary quasi unfold position and arbitrary non-incosadral symmetry. And we have also given a procedure that holds when the particle is not symmetric. Obviously this has to be done numerically, so still a lot of work to do. And I thank my collaborators, Dr. Eidman-Tvarok at the University of York, Peter Burk at the University of Connecticut, and Ringo and Mueller at the Udza and Truma in Basel. And thank you, I'm happy to take questions. Thank you.