 Welcome back for the last talk of the day, which will be given by Mike Hagan from Brandeis, assembly budding, and then encapsulation by Polyhedra Schell's Mike. OK, thanks, Christian. And thanks to all the organizers. I'll join everyone else in thanking them both for bringing us to this wonderful place and bringing such a wonderful group of people together. And so what I want to talk about is motivated by something I think many of us are interested in, which is in how do proteins that start out in disordered positions like this get into this well-organized shell? And in looking at one of these beautiful structures, it's easy to give the proteins all the credit for driving this process. But as I think we've heard many times over the past couple of days, if you have a substrate like, say, RNA, then you have to ask what role is the substrate playing in the assembly process? So what I'd like to do today is look at two different substrates with different architectures than the RNA and ask how are they affecting the assembly process and how does the pathways and the outcomes depend on the properties of the substrate? So the first substrate we're going to look at is when the proteins are assembling on a fluid membrane, like a lipid bilayer, in the context of assembly and budding during the egress of an enveloped virus. Then the second one we're actually going to step outside of the virus world and into the bacterial world and look at a system in which icosahedral shells assemble around a cargo consisting of many unconnected particles and those particles are encapsulated in the shell. Okay, so to start with the assembly on a fluid membrane, as I said, this is in the context of enveloped viruses which need to acquire an outer membrane envelope by budding through a membrane in their host cell. And this includes many of the nastier human pathogens. The particular family of viruses I'm going to look at is the alpha virus family and this is motivated through many discussions with Tuli. And so here I'm showing a schematic of the alpha virus or this is Cymbus virus and it has two layers. On the inside you have RNA that's surrounded by capsid protein and this is called the nucleocapsid core and then you have an outer layer which consists of the membrane which is shown in green and then these transmembrane glycoproteins which span the membrane and interact with the nucleocapsid on the inside, okay? The nucleocapsid may or may not have icosahedral symmetry but the outer layer is comprised of 80 of these glycoprotein trimmers that are arranged with T equals four icosahedral symmetry. So in general the enveloped viruses are thought to undergo or acquire their membrane by two different pathways. In one pathway the proteins, the nucleocapsid is actually assembling on the membrane and driving deformation of the membrane. In the other pathway, the nucleocapsid first assembles into cytoplasm, it then targets the membrane, recruits glycoproteins and drives budding. And alpha viruses are interesting in this context because the traditional view is that this is the pathway by which they undergo budding that the nucleocapsid first assembles in the cytoplasm then undergoes budding. And so the view is that this is very much driven by the nucleocapsid but why this is interesting and a little bit controversial is because there's been several works that show that you actually don't need the nucleocapsid in order to get budding. So there are several works in which the capsid protein is either impaired or completely knocked out and they still see budding of particles and in fact in this recent work they actually saw budding of infectious particles. So this raises a couple of questions. What is the relative importance of the glycoproteins and the nucleocapsid in driving this process and more basically what do you even need this nucleocapsid for? If you can get infectious particles without it why does the virus bother with this? So what if anything is having the nucleocapsid contributing to the budding process? So those are the questions that we wanna look at. We're gonna use rather simple computational models to try to answer those questions basically by comparing the assembly in the presence and absence of a nucleocapsid. So we're gonna have to use a simplified model if we want to look at the time and length scales relevant to this process. And so we're gonna try to make a model that is as simplified as we can make it while still retaining the essential physics and geometry of the problem. So it's believed that these glycoprotein trimmers are actually the basic unit that undergoes the assembly process because they're heavily intertwined within a trimer. And so in our model the elemental unit is going to be a model trimer. And to model the trimer we've basically stuck these three conical particles together and these are gonna move as a rigid body. So they roughly have the shape of the real trimer. Now we have interactions between the interior beads of these conical particles. And because of the cone angle when these come together they're gonna have a preferred curvature. And Guillermo, my postdoc who did this work has set the cone angle so that if you let these assemble in the absence of a membrane they'll very reliably produce capsids or shells in which you have 80 of these cones. And it's at least roughly has icosahedral symmetry. So if we want to have the nucleocapsid present again we're gonna model it in a way that's as simple as possible. It's been shown that the nucleocapsid doesn't necessarily have icosahedral symmetry or it's not required to have it. So we're gonna model it just as a rigid isotropic sphere and it has short range interactions with the base of the cones modeling the fact that the cytoplasmic domain has short range interactions with the nucleocapsid. How can you get 80 combining? So you have 80 of the trimmers. So it's 240 of the cones. So there's one other feature of the model. It's been shown in a number of systems that the viral proteins typically when they're free in solution are in confirmations that are actually incompatible with assembling into the capsid and that they have to interconvert into a relatively unfavorable confirmation to be active for assembly. And we've shown recently that this having this feature can significantly enhance the robustness of assembly to variations in the parameters. And we found that we needed to include this feature in order to have productive assembly at the very high concentrations of these glycoproteins that you see in the cell membrane during an infection. And so we've included that in our model. Okay, we need one other ingredient, which is a membrane. And again, we need a model for a membrane that we can simulate on rather long time scales. And so we're using this model from Cook and Marcus DiCerno which you have three beads per lipids. So one head group and then two hydrophobic tail beads. And there's interactions between these that you can readily tune to capture the properties of different biological bilayers. So then our glycoproteins are gonna span this membrane and the hope is that they're gonna undergo dynamics to make something like this. We're gonna simulate that dynamics using Brownian dynamics and to answer Dennis's question before he asks it. We also did other sets of simulations where we include all of the long range hydrodynamic interactions and we see qualitatively the same behavior. So what I'm gonna show you now is an example of a typical simulation when we have no nucleocapsid present. And again, the way we're gonna draw these is the active subunits are in this purple color and inactive subunits are in this brown color. So what you're seeing here is the initial stages of the assembly process. They're proceeding quite rapidly. By the way, here is the same thing as here. Here you're just seeing a cutaway view. What you'll see is that at first we see assembly proceeds very rapidly due to the curvature. It's deforming the membrane, but right about now it's starting to slow way down. And that happens just as we get to about two thirds of the way complete when we develop a significant negative curvature at the neck. And here we skipped over that process because we got tired of waiting and eventually we undergo a thermal fluctuation where a scission is gonna happen and then the capsid floats away. But so I think the most interesting feature of this assembly process was the dramatic slowdown that occurred after we got to about two thirds complete when we had that negative curvature. And I'll have more to say about that. Here I'm gonna share the same thing but with the nucleocapsid present. And what you can see is that there's actually very little visual difference in this process. Once again, we see that assembly is fast at first. We developed this inward curvature. Notice that the proteins tend to be absent from that region with the inward curvature. And that is, in fact, why you have such a dramatic slowdown at that point. Yes? We've made it so that they're commensurate. You could try other, but we focused on the case where they're commensurate. So what I want you to see in this is that overall the process is very much the same with and without the nucleocapsid. In both cases, you have this slowdown that occurs when you have the inward curvature. I'll show you in a little bit though that there are some differences when you have the nucleocapsid present. First I wanna focus on the case where it's not present. What I'm showing here is the typical outcome at the end of rather long simulations as a function on this axis of the strength of interactions between the glycoproteins. What you see is when the interactions are very weak, not surprisingly, nothing's able to nucleate and so you get no assembly. If we make the interactions moderately strong, so these blue points here, then you see complete shells as I showed you in the movie. If you make them overly strong, then you get kind of sort of complete but messy shells called the holy shells. And this is not unexpected when you do empty capsid assembly. If you make the interactions too aggressive, you'll tend to get poor assembly like this. So one interesting case is this narrow region at relatively weak interactions. What we find here is that the assembly actually stalls and we have what we think is an equilibrium state where we have this partial assembly and the proteins are never able to make it over this barrier. And I should say the reason why this barrier occurs is that it's actually unfavorable for the proteins to sit within this inwardly curved region and we tried many different protein geometries and we find that this is very generic. Yes? I'll show you what happens with a harder to bend membrane. So we think this could be possibly of some interest to the case of HIV because, so we're not including in these simulations any host cell machinery to drive scission. And if you think about a system that's sitting for a long time in this state and it recruits scission machinery, in that case, you would expect that you would be left with a hole on the bottom of the capsid and you do indeed see that. Yes? So we actually looked at that in previous simulations where we had a much more simplified capsid and we did find that if you have heterogeneities present that the line tension can help drive the assembly process in one of the ways in which it helps drive the assembly process is it actually changes the shape of this neck so it makes it more shallow. So I'm quite sure that if we go and put a heterogeneity into these simulations we also would see this probably being alleviated less metastable so they do play an important role the heterogeneities but just to simplify, I have fewer parameters we focused on a homogeneous membrane in these simulations. Okay, so one of the things that surprised us when we looked at these complete shells is that the size is not at all uniform in particular here I'm showing as I vary the strength of the interaction strength that the mean size of the shells that assemble and you see that it varies over a pretty significant range and they're always larger than the preferred size without the membrane which was the 80 trimmers. We see that as the interactions get stronger we're approaching 80 trimmers but we never actually get there and eventually it starts making these holy capsids. This is what's surprising to us because normally when you make the interaction stronger you expect to see your assemblies getting less similar to what you think is the ground state and here they're tend to get more similar and if you compare that to the case with the nucleocapsid you see that in that case the butted shells tend to be much closer to the preferred size. So what's happening here? We think that we can actually explain this with a very simple equilibrium model. These simulations of course are at finite time so they're not necessarily reaching equilibrium but we can still get insight from an equilibrium model. So what we've done in this model is we've asked what is the elastic free energy as a function of the bud size taking into account the penalties for bending the membrane away from its preferred curvature which is to be in a flat state having no curvature and that's pretty well captured by the health rich free energy which just says that the cost, the elastic energy cost is quadratic in the curvature and there's a coefficient which is the membrane bending modulus. We add to that deviations of the shell from its preferred curvature because if it's making these large shells then it's at something other than its preferred curvature and again to leading order that cost is gonna be quadratic in the difference of the curvature. Now if you consider a state that just before it undergoes scission, remember we're not interested in the scission process, you can integrate over the whole bud and you get this expression. This is the contribution from the membrane which is actually independent of the size of the bud but then you have this term which again comes from the deviations from the intrinsic preferred curvature. So here N zero is the 80 trimmers. So looking at this expression, you might think that this, you'll see that this free energy is minimized by having N equal to N zero but it turns out what's relevant is what is the equilibrium distribution of bud sizes in a system with a finite concentration of subunits so you can derive that with a simple density functional theory and you arrive at this kind of very familiar law of mass action here. This is the concentration of buds with different sizes N. It's proportional to a Boltzmann factor where this is the free energy of a bud with N subunits but then that's modulated by this term out front which is the equilibrium monomer density raised to the power N and this accounts for the translational entropy cost of putting the subunits in. If you optimize this, you'll find that the optimal shell size is actually that which minimizes the free energy per subunit and because of that you end up with an expression that says the optimal shell size is going to be the intrinsic preferred shell size but now increased by a factor which depends on the ratio of the membrane and shell bending moduli. And this is easy to understand physically. There's a competition between the membrane which would prefer to be flat and so wants the buds to be bigger and the glycoprotein shell which wants to stay closer to its preferred size and depending on the ratio of these you see as the membrane gets stronger the shell size is going to get larger. If you take this expression and you compare it to the simulation results you see that it actually fits them pretty well without having to introduce any parameters. And the key point here is that as you increase this glycoprotein interaction strength you're also increasing the shell bending modulus because that depends on the strength of the interactions as I tried to deviate away from the preferred curvature. We recalculated that increase is a linear in the interaction strength. And then so why would the nucleocapsid stay so much closer to the preferred size? Basically because the nucleocapsid is a rigid sphere so it effectively increases this shell bending modulus so that this whole term goes closer to one. And so this is just one parameter value so to get to Bill's question we ran a number of simulations over a range of different membrane and shell bending moduli so all of these different color symbols are for different values of the membrane and shell bending moduli. What we're showing here is the number of subunits in the shell as a function of this dimensionless ratio and you see that all the data claps on to each other and they pretty well follow this theoretical line up until we get to a threshold value of this ratio at which point suddenly they fall away from the line. And the reason why this fall away occurs is that at this point the membrane bending modulus is simply too large and the shell is not actually able to complete itself and you end up with these large holes at the bottom and in fact these points up here correspond to this metastable state I showed you earlier where it's never able to get past this point. Okay so I think this is an interesting prediction and it shows interesting consequence of having nucleocapsid present which is that you avoid all of this polymorphism in the assembly product. So I think this is a prediction that we hope to have tested in Tully's lab. At the moment there is experiments that are consistent with it but I can't say they really test it. So most relevant are these experiments from Rhyskian where they were looking at alpha virus and as I said they had buds that occurred in the absence of nucleocapsid. What they found when they analyzed those buds is that they are in fact tend to be larger up to twice as large as the case with the nucleocapsid present and they're much more polydispersed. So here are shown micrographs with the nucleocapsid present and they're pretty uniform. Here you have some example buds that occur in the absence of the nucleocapsid and they're larger and more polydispersed. Now there are some caveats with this because we don't know the bending modulus of the shell in these experiments and we don't know the preferred curvature either and there are some other systems where they don't seem to see this. So I think this is at this point an interesting prediction that we would like to see tested. So with that I'm gonna move on to the second topic. How much time do I have left? 13 minutes, excellent. So the second topic is now we're moving to this different substrate architecture where we have many unconnected particles that are going to be encapsulated. This is relevant of course in these artificial systems where groups are re-engineering viruses to assemble around some say enzymes in their components so that you can have a customizable nanoreactor. It turns out bacteria thought of doing this a long time before humans did and it's in the form of something called bacterial micro-compartments. So what these are is internal compartment within bacteria and their function is to sequester a particular metabolic process within the cell. And I'll show you how that works in a second but these are found in about 20% of bacteria and there's many different metabolic processes that are facilitated by these structures. What I'm showing here is an example of one well-studied bacterial micro-compartment which is found in cyanobacteria. So these are the so-called blue-green algae. It's a picture of them here. They're responsible for these toxic algae blooms. They're also responsible for a significant fraction of carbon fixation that happens on the planet. So they're pretty important. And what you're seeing here is an example of a micro-compartmentist as black blob that you see here. And this micro-compartment is known as the carboxysome. That's because its function is actually to facilitate carbon fixation by the cyanobacteria. And these are popular enough that if you look on Wikipedia you can see examples of purified shells. And from these you can get a sense of what the architecture of this shell is. It's roughly icosahedral, which you can see from the facets. And here is a schematic of its structure. And so the shell is comprised of hexameric and pentameric protein oligomers. Presumably the hexamers are tiling the facets and then you have 12 of the pentameric units at the vertices. And then inside you have a dense complex of enzymes. And in the case of the carboxysome you have the enzyme rubisco and then other components that are needed to convert the products of photosynthesis into sugars. And the reason why the carboxysome is needed is that rubisco is this very ancient enzyme and it's very ineffective at the ambient concentrations of CO2 in a bacteria. It simply is unable to drive the Calvin cycle. It needs to be brought to very high concentrations along with CO2 inside of the shell and then it can do its job. So this has been known for a long time but until recently I wasn't known how these things actually assembled to arrive at this structure. So recently there are experiments done in a couple of groups I'm showing here, experiments from Cheryl Kerfeld's group. And what's shown here is in red is showing the outline of cyanobacterial cells and they've labeled the rubisco in green. And so after they induced the carboxysome genes the first thing that happens is that all of the rubisco in the cell undergoes phase separation forming a single green clump, such as you see here. Then over the course of time you'll see these clumps will separate into multiple clumps. And so what's happening during this separation process you can get more insight if you look at a higher resolution. And so in these micrographs what's shown is if you look at these two clumps one of them is amorphous as it is at the beginning of the experiment whereas the other is faceted presumably having the structure of the carboxysome. And so the interpretation is that the shell proteins have adsorbed on this initial complex of rubisco which they call the procarboxysome and then assembly has taken place such that the shell essentially carved off a piece of the procarboxysome ending up with it on the interior. So the question then is what are the conditions under which this assembly and brutting process can actually proceed? What are the mechanisms by which the shell will undergo closure and what does the assembly product distribution depend on? What are the key factors? So those are the kinds of questions that we can ask again with relatively simple simulations. Okay, and so we're gonna use a model where again it's thought and there's evidence from AFM experiments that the basic assembly unit in this system is these oligomers the pentameric and the hexameric oligomers and so those are gonna be the basic units in our model. And I'm gonna start with a model in which these pentamers and hexamers interact with each other by an adjustable interaction strength but in the initial model we've made these interactions specific enough that this is the only complete shell that can assemble. In a little bit we're gonna relax that approximation. Okay, and then there's a, the potential is such that there's a preferred angle that these face-face angles that these interact with. Okay, we need also a model for the cargo. This is the actual structure of the Rubisco hollow enzyme for the simulations I'm gonna show you remodeling as a spherically symmetric particle that has short range attractions modeled by a Leonard Jones potential. We did also use a more complicated model that represented the eight-fold structure of this and we get similar results. Okay, it turns out in the real system there's an auxiliary carboxyzone protein that's driving interactions between the cargoes and also between the cargos and the shells. In both cases we're gonna represent that interaction by a direct pair interaction between the cargo or between the cargo and the shell subunits. And so in all we have three different interactions that we have to worry about, the shell shell interactions, the shell cargo interactions and the cargo cargo interactions. And the question is as we vary these interactions what kind of assembly behaviors do we see? I'm gonna here show you an example of a typical simulation and this was runout parameters where you see rapid formation of this globule of the cargo and then the shell proteins have started to adsorb onto it. And what you'll see is that at first adsorb onto it in a disordered manner, very reminiscent of the way the capsid assembly occurs in the Gelbart-Nobler experiments. But now they're starting to realize that they can interact with each other. The shell is gonna start to assemble. And if you wait a while you'll end up in this case with two shells that are separated by this kind of neck of cargo particles. What needs to happen at this point is shell proteins need to get in there, realize that they can interact with the open line and they eventually do and they'll cut the neck and now we've had complete budding of our complete functional carboxysome, five minutes, okay? So we can ask how this depends on the perimeter values in interest of time. I'll just show you one example. So here we're varying the strength of the shell cargo interaction. Here we're varying the shell-shell interaction strength. And there's a regime of modern interaction strengths where you get good assembly. If the interactions are too weak then you either get no assembly at all or it gets stalled at this globule state. The interactions are too strong you tend to get malformed assemblies. So one interesting feature is that this is all at a fixed cargo-cargo interaction strength. If you reduce the cargo-cargo interaction strength such that you don't get this initial phase separation of the cargo anymore, then you'll see assembly and cargo coalescence happening at the same time so you have a different assembly pathway. And this appears to be a pathway that another type of carboxysomes use. So we see here that the cargo-cargo interaction strength is the key parameter that determines the two different pathways that are seen in two different carboxysome systems. So what I want to do in the last three minutes is ask what happens if we allow the shell to form other shell sizes and how does the size of the product depend on, for instance if you have the cargo. And this is motivated by some experiments that showed in the normal situation in a cyanobacteria they have rather large polydispersed shells that form about 200 nanometers in size but if they assemble them in such a way that they can't assemble around cargo they're small and monodispersed around 20 nanometers in size. So an inference from this is that the presence of the cargo is driving the shell proteins which have this rather small preferred size to form much larger shells. And so we're curious, could our model show something like that happening? So to skip over some details we've made it now less specific so it can form any size shell but we're gonna use parameters where in the absence of cargo it still reliably forms these small T equals three shells. So here is what happens if we now give it cargo. What you're gonna see is that it's actually forming much larger shells. In this case they're a little bit larger. You see that this is maybe closer to T equals four in size. If we make the shell cargo strength even stronger what you're gonna see is that it essentially just completely wraps the globule and so we get a much larger shell. And the reason why in this case the cargo is making the shell larger is primarily because in the carboxyzoam system it's only the hexamers that interact with the cargo and the pentamers only come in once there's enough hexamers there so they can interact with neighboring hexamers. So as a consequence of that as you make this shell interaction, shell cargo interaction stronger it takes longer and longer to get those 12 pentamers in there and so you get a larger shell. The cargo-cargo interaction in this case is strong enough that it will phase separate. If it's assembling by the other pathway where they're assembling at the same time then you don't see large shells like that. So this is just quantifying what I said there. On this axis we're varying the shell cargo affinity and you see that as it goes up the mean caps and size goes up until you get to a point where this is too strong and what happens is these drop down now because you get nucleation of shells happening throughout the system and now they're starting to assemble by this other pathway and then we don't see large shells forming anymore. So I think I'm out of time. So let me just try to give this some context. How this fits kind of into cell biology. So it's becoming very apparent that liquid-liquid phase separation plays an important role in organizing cell interiors. There's a big question how the size of these phase separated domains is controlled. Here we see one mechanism where protein shell assembly can actually control the size of a domain but we see also that in turn that the presence of that phase separated cargo is affecting the size of the protein shell. So there's an interesting feedback between the cargo and the shell assembly. And so with that, we have some theory for this also but I will have to skip over that. And so let me just thank Guillermo Lazaro the postdoc who did the alpha virus work. The carboxyzones were started by Jason Perimeter and then have been taken over by my grad student, Fari Moherani. And thanks for your attention.