 So thank you. Can you hear me? Yes? So first I would like to thank the organizers for setting up such a nice meeting. So my name is Martin Castell-Nouveau. I'm from France in Lyon and today I would like to present you some recent works that we performed in our group over the last two years and so they all deal about viral self-assembly and I will have two parts. One part will be more specialized on a single object, namely following the self-assembly of a single capsule and try to understand some specific features giving rise to specific shape of viruses. And in the second part I will try to speak about some collective effects, so how a pool of proteins will self-assemble and what is the dynamics associated to this. So as you after two days of workshop you you all know this kind of structures for icosahedral viruses so they are icosahedral because they have some set of different symmetries. You can observe for example in this in these reconstructions here that they range from different sizes from 20 nanometers to more than 60 nanometers and an idealized view of these structures will be this one which is an assembly of let's say triangles where you can identify very easily some hexamers where are six proteins back together and you can also identify some pentramers where you have only five proteins at specific locations. So I don't know if somebody explained it but roughly the t-numbers that are used to classify the icosahedral viruses are measuring basically the distance between two pentramers on a particular lattice okay the lattice of the hexagonal path but not all viruses have this shape some viruses have more irregular shaped and especially elongated shapes this is the case for example for some retro viruses like HIV here with this famous conical core shape RSV which is another retrovirus which has this funny coffin-like shape with something that looks flat here and kind of apex at the other side. This one is a bacteriophage but it looks like a icosahedron that has been stretched so it's just like elongated cylinder and this one among our cure virus is called a lemon-shaped virus which has this very weird shape and so basically if we try to understand what is the difference between the shaped in these viruses we are led to the following conclusions so indeed in all these structures either regular or irregular you can identify hexamers and pentramers and from topological rule if you have a surface that you want to close you must have at least 12 or maximum 12 net pentramers in the structure so for the regular one you would say that the the typical feature is that the pentramers have an equal distance between each other's right why here they are kind of face separated here for example you you have seven pentramers here you have five pentramers this is the same here and this one well this is not a virus this comes from carbon phases this is a double nano cone where you have three on top three at the bottom and six at the waist but you have a total of 12 pentramers but these pentramers are irregularly placed so the question I'm asking today is can we identify what is the property at the level of a single protein that give rise to this particular shape how a protein will know that it will end up into a icosahedron or into a elongated structure to do that we are trying to to understand the elasticity of these objects I'm making the link between the elasticity and the capsid shape and I have some specific discussion about HIV so this problem has been already attacked by numerical simulations in particular in some groups for example in group of boy as Andy and the simplest way to address this these questions of shape is to have this minimal model where in which in fact the building blocks are triangles that are deformable those triangles wants to share some edge to have some adhesion and you must be able to compute some elastic energy for your system and you try to see for example if the the minimization of elastic energy is able to to reproduce the shape you observe so this is not what I'm going to do today and I will try to to to provide some analytical arguments in order to understand the shape that you observe in such simulations but so basically with this model the elasticity will be composed of two parts basically you will have an in-plane stress in in-plane stretching or compression of the protein in their own plane and you will have an out-of-plane curvature as well which tells you about the bending the local bending of the surface so in the in the in the discrete model the bending arise with a potential a kind of quadratic potential around a preferred angle and this is an important parameter we introduce some preferred angle assuming that in fact the protein is by by itself if itself assembled with us some other proteins you want to have some local curvature okay but at the scale of two proteins this this is an angle between the two proteins and the other potential is just the stretching of the bones the stretching of the triangle edges and this just reflects the fact that maybe upon the assembly you will have some compression or stretching okay so given that you have some spontaneous curvature to your surface at the at the beginning of the self-assembly you expect to have some kind of isotropy growth so you will have a kind of sphere covered by triangles but if you do that with just triangles and preserving the the isotropy then you you you see that in fact the triangles that sit on the edge but starts to be compressed okay well I'm going to explain in a minute why but this is exactly the same case if you take a sphere and try to cover the sphere with a flat sheet if you want to to cover this the the sphere you will have to include some folds or wrinkles in the structure okay and this this this tells you that you have a too much too large amount of material to cover the sphere in fact okay so we can estimate we can estimate roughly the the cost due to this this this growth along the sphere if you take a disk for example and you try to put this disk on the top of the sphere following the sphere then the the rim here we have a reduced value and you can evaluate for the whole structure you can evaluate the cost of confining this disk onto the surface and you see that this is a strong scaling low because it goes as R to the power of 6 so it's a large value and the radius of the sphere comes with R to power of 4 so it means that at the very beginning of the the the growth there is a huge mechanical stress that is rising up due to the spontaneous erosion so you have to find ways to reduce this this stress this mechanical stress well you can find you can find some strategies to reduce this stress by looking at the equations describing the internal stress or the in-plane stress so this equation this is one of the equation one of the focal form of common equation it is related to the so the stress tensor which tells you the state of stress inside the thin shell you introduce some specific function here which measure which measure the stress and this this classical equation is written here this is one of the two focal form of common equations giving the the equilibrium solutions for the shape of plates for example and it has two source terms okay and these are very important in fact so this green source term is the defect density by defects we mean topological defects but if we if we try to understand this in terms of our discrete model it means that we have some panchambers so the topological defects from continue theory are just disclinations or panchambers inside the structure and the second term is the Gaussian curvature for the Gaussian curvature it's just the product of the two principle curvature of a surface that you can define at any point and you see that well if there were no such terms in fact the stress will be minimal okay so in a sense you by just writing this question you identify two strategies in order to relax the stress the first one is that well if you have Gaussian curvature then you have to induce defects in order to compensate so to have a minimal state of stress that's one strategy and that that will be the strategy called the defect inclusion or panchamber inclusion through the assembly again the other strategy is to have no defects and no Gaussian curvature but still you want to have some curvature but you can have such a curvature by taking a cylinder or cone for example we have vanishing Gaussian curvature and you want to end up with some cylinders cylinder that has been seen in simulations in a group of Royazandi for example simulations very similar to this the model I described before can we estimate the cost of defect inclusion and to see how is it better than just having this isotropic growth so in fact including a defects or panchamber in the solution amounts to perform such an operation here you have a network of hexagons and you will cut out a wedge of 60 degrees and then you try to glue the thing you try to glue the both parts if you do that if you stay flat well you expect to have to pay a strong stretching price right but if you go to 3d in fact very easily you are able to to form a cone in fact and so the different cost here so here if you stay flat you don't have the bending energy but you have the stretching which goes like the the area of the the disc you are considering now if you perform buckling and have a cone then you see that you just have a logarithmic growth which is much more affordable in terms of mechanical stress so pay attention this is a zero spontaneous curvature calculation that you can make for non-zero spontaneous curvature but it's more complicated so with one defect indeed we can reduce the state of stress but if you keep on going the growth you expect to have to maybe include more defects and we will come up to this point in a minute if you include more defects or more panchambers in fact you see that you will curve more and more your structure okay more precisely for example here you have one defect two three four five and why from a geometric argument is it favorable to have multiple defects in order to to to have a less a less amount of stress well recall that in fact by going from the disc to the sphere the rim was reduced the rim land was reduced but if you just include one defect you are performing such as thing you are reducing the rim land because you removed this part okay so you can compute the number the the length that you remove from the system in order to cover up the sphere okay by adding by adding more defects and so in fact this land for reduction is easier in order to follow the curvature of the sphere so for the other for the other possibility of a strategy to reduce the mechanical stress well just have to pay bending energy because here don't have any stretching if I take a flat sheet of paper I can roll it without any tearing stretching or compressing and you see that it goes like the the area of the other surface and you have also the radius of spontaneous curvature that that comes up so more or less what we have is that if we start from this if we start from this structure with isotropic growth on only hexagons if you are at high spontaneous curvature you would expect to have at first one pantamers but then you have a cascade of pantamers in your end up with an icosahedron and if you are on the other side that for the weak spontaneous curvature then you expect to have a cylinder that will growth indefinitely and you don't have to include any defects so that this is a problem for a virus because it needs to be closed we'll see we'll comment about that in a minute so more precisely for example you can compute for example in the high spontaneous curvature scheme you can compute for example the energy without any defects this is this blue curve and with the 12 defects this is this red curve and in between you see that there is a cascade that the more you add defects the better it is in order to follow the structure okay and you reach eventually icosahedral symmetry on the other side if you have a small curvature and you compare for example this is the energy with the 12 defects you compare with cylinders you see that depending on the radius of the cylinders sometimes you have a smaller energy by going to the cylinders instead of having the defects and so you observe these uncapped cylinders and in between and we we identify some typical scaling for the typical radius of spontaneous curvature you expect to have intermediate shapes that are strongly dependent on the way you self-assemble so indeed with this continuous calculation you you you are able to explain why in the simulations of Roya for example for small radius of small radius of spontaneous curvature you expect to have this icosahedron and on the other side you have the cylinders so I'm I'm just trying to comment now on the special case of HIV so some experiments have been performed quite a long time ago into the self-assembly of just ca which is the part which is the protein that makes up the conical core in HIV and depending on the presence of the genome or not cylinders were quite often observed right now if you look at cones inside particles that are produced inside cells well you make an interesting correlation as well most of the time you observe some cones but sometimes you observe some cylinders but the genome is outside here a cylinder here the genome is outside okay so this raised the possibility that in fact in order to answer the question how would you close your surface if you have an infinite cylinders maybe this is the interaction with the genome that induces the defects so we started to to make some simulations and this is an ongoing work but indeed if you take a surface that wants to grow separately without any interactions with something else if it wants to grow just like a cylinder if you perform the growth in the presence of an attractive sphere then you are able to induce the presence of punch embers so you change the way the surface wants to grow and so we started to to have kind of face diagram but this is very rough basically with the addition energy and the ratio between the spheres and the spontaneous curvature and you expect to have some phases where you don't have any addition some phases where you have complete covering and in some phases you observe some some cones for example so this this will end my first part about the single objects and the main conclusion is that at the level of a single protein this is really the spontaneous curvature so at the very local scale this is the spontaneous curvature that will determine the shape you will end up if you would have rather icosahedron or or more elongated shape and importantly it seems that for the case of HIV but maybe for other viruses as well the interaction of the growing protein surface with something else it could be the genome or it could be the outer membrane for HIV for example is susceptible to give rise to punch embers in order to close the structure so now I move to the to the second part we are going to look at many intermediates now not just one particle and we are going to to focus to focus specifically on one type of self-assembly that is found for HIV again which is what we call the open self-assembly so it means that in fact you have a self-assembly scheme where you have some proteins that wants to minimize their overall energy by associating but there is a source of monomer and there is an outflux of complete particle okay so this is not a closed self-assembly we would be just this case right but here you have an entry and an exit in your system the fact that you have an entry and an exit will all you allow you to reach eventually some stationary state okay so we we try to to to understand the dynamics of such a self-assembly scheme and so the idea to do that is to write down the kinetic equations and so basically we have a pool of monomer which is fed externally by constant flux we have the self-assembly and the pool of monomer couples to all these reactions because you we consider a monomer addition model and the last step is irreversible because you you you lose the the complete particle so the interesting quantity to consider in order to understand what is going on is the local flux which tells you if locally you are going forward or backward and you have also the detail balance and the interesting so there is another also an interesting phenomena is that you expect to have some line tension in your system because you grow your system there there is a border between the particle that is growing and the rest of the membrane for example you expect to have some barrier energetic barrier and this energetic barrier in fact is the net effect of this energetic barrier is coupled to the monomer populations this is this graph here this is the height of the energy barrier this is the number the size of the intermediate two compute and you see that depending on the monomer population you are able to modulate the height and the width of the of the of this barrier and this will have some strong consequences on the on the assembly on the dynamics so if you solve numerically the equations I just showed you will end up with a profile like this so these are different profiles so this is the size distribution as function for different time and you see that in fact you have some assembly waves with with some some parts of intermediates which are relatively well localized okay so you have a propagation of kind of front and in fact sorry it's well it's hidden more or less in the in the effective in the effective parameters the bending energy will will come up in this in this epsilon this is the net energy you gain by forming the by forming the intermediate and if you have bending energy for the membrane you lower this net energy this is an energy gain okay so if you if you bend you you expect to to loss to lose something so the gain will be small we can discuss this afterwards sure so the interesting point is that well the monomers that you have in the solution they have they are used by two two two processes the first process is to nucleate an intermediate right we call this nucleation and there is an elongation which is which amounts to grow the intermediate you just nucleated and the monomers how have to share between classes that will nucleate or elongate and here for example if you compute along the time so these are rescaled units but if you compute on the time you compare for example the elongation current here which is roughly constant but there's a slight drop here to the nucleation current you see that nucleation current is strongly localized so you nucleate a whole bunch of intermediates but then you stop nucleating and then the things travels and that's why you observe this assembly weight here for example if you look at the outcome this is the flux outside so you expect to have as an outcome as if you measure the number of particles that emitted by by the this cell for example you expect to have some can what what we call viral burst it means that for some time you have a strong production of full particles but then you have a lower production then a stronger production then a lower production before reaching eventually some stationary state where you you have this kind of production so we don't have really experimental evidence as precise as the these predictions but if you look for example at cells that are producing producing that are infected and producing HIV for example you see sometimes you see a real like this where you seem to have all the particles in the same state here you don't have particles with few few proteins you see mainly particles with many proteins that are almost closed okay so that that goes to that that fits well at least qualitatively with this kind of prediction but we expect to have more more exponential data so this is really related to the to the presence of of of the barrier here in fact this barrier depends the the the height of the barrier depends on the monomer population and so as the monomer are are divided between monomers that will nucleate and will grow eventually you see that here the elongation is almost constant but this is the nucleation which is highly localized meaning that here for example if I look at an intermediate time let's say 400 which is the violet here you see that well you have something which is localized because you you stopped nucleating you just you use almost all the monomers to propagate because you have a high barrier you cannot create anymore and these these are on the other side of the barrier and they will propagate and you have also an interesting comparison if you want to compare the what what I define as the open self-assembly with a one-shot self-assembly so what is a one-shot self-assembly so suppose you perform the open self-assembly for a certain time and remember you we have an influx of monomer so if you count the total number of monomers you fed in your system you could try the following experiments take this number and plug all the monomers at the same time zero and then you stop putting monomers and you see that the dynamics are very different in fact so this is again the color hate gives you the size distribution you have the time you have the size here and you see that in fact if you all monomers are probably in one shot then you have a large number of nuclei but you have no more monomers to propagate them so you have to disassemble or wait for disassembled or to recycle in order to propagate so if you make a quantitative comparison between the two kind of self-assembly so let's say for example that we are going to compare the two self-assembly at time ten thousand okay so with a flux one at time ten thousand it means that I can plug at the very beginning ten thousand effective monomers because these are reduced units and so if you compare the black curve and the red curve you see that in the experiments where you provided all the monomers at once you have at the given time you have a much less efficient production if you provide the monomers by constant flux but small flux then you have you are able to go faster than if you provide all the monomers together that that's an interesting prediction and again the reason is that you have to recycle for the one-shot self-assembly you have to recycle because you created so many nuclei that you don't have any more monomer to propagate them so you have to disassemble some nuclei in order to make some of them progress and this makes you the process this process which has been called in literature kind of kinetic trap is indeed smaller so I arrived to the to the conclusion to this part and just in order to summarize so for this open self assembly problem the presence of an energetic barrier is very important because it gives you this particular dynamics in the form of assembly waves and if you look at just the out the the output of virus production you expect to have some burst okay not continuous production but some burst before reaching eventually some stationary state and the interesting prediction as well might be that well if you want to perform an experiment of self assembly by using the standard physical chemist way you put all the monomers together maybe if you try to to put some flux of monomers you might enhance the the the speed of the reaction and so with this I would like to thank you for your attention and yes I would like to thank some collaborators in particular thank you