 I also want to start by thanking the organizers for giving this great opportunity to be here. And of course, my talk wouldn't be here if I did not have these fantastic collaborators. We started this work with my former postdoc, Vlad Belly. And today, I'm going to focus on most recent work with Bo Peng and Hamid Shosie. OK, so I don't have to show the next slide for this audience, but nevertheless, maybe there are a couple of people in here for whom I want to emphasize the fact that when you take a capsid, inside we have the genome, which is negatively charged due to the phosphate groups. And the n-capsulating capsid is multiple copies of the same similar kind of protein. And they usually carry positive charges. And that's the reason why we have this title of electrostatics. And of course, broadly speaking, there are two kinds of shapes. One is the icosahedral shape. Like this is a tobacco, mosaic virus, satellite tobacco, mosaic virus. And we had this character already today, earlier today. This is TMB. This is more helical. And then we can have a composite with the phage. And the genome could be single-sanded RNA, single-sanded DNA, double-sanded RNA, double-sanded DNA. And today, I'm going to focus my attention on these icosahedral viruses with single-sanded RNA. And that's the place where electrostatics plays a role. And this beast is an entirely different universality class where electrostatics is not that important to my mind. So where do I start? I get into trouble right away. So this is a typical protein, which is constituting the capsid. And somebody who plays with paper and pencils, this is a total, total intimidating mess. And I see the alpha helices, the beta sheets. Of course, I took a lot of courses in biochemistry and biophysics. But nevertheless, this is very intimidating. There is a hairpin in here. And I look at another character. It's extremely scary to me. Maybe I want to do something. Maybe not to deal with this at all. The second thing is the partner. If you take RNA and you take any of these genomes and this RNA structure, we'll see some discussions about it more in the rest of the conference. Each one is a highly, highly branched structure. And it's a negatively charged. It's a highly branched structure. And of course, all these attributes are contributing the size and shape of this RNA and how that's going to interact with the capsid. This is my problem. How am I going to deal with this? And then in one afternoon, we had a lucky break. Just putting our feet up on the chair, just in front of a blackboard, we were looking at it. We looked at several viruses, RNA viruses. And the amount of RNA packaged was different. The number of vertices could be the same. You can have a variety of them. And what we observed was, generically speaking, the typical motive of the building block, this every protein molecule had this building block. There is a hydrophobic domain. And there is a tail, which carried dominantly a positive charge, which is protruding into the lumen of the virus. And of course, this virus is floating around water. You know, you have a solution with salty conditions. And the outside is also charged. And there is also another domain in here, which is also charged, which is going to be exposed to the exterior world. See, this was a typical motive. Then we thought we'll just do a paper and pencil calculation in about 30 minutes. That was the goal. And you are not going to believe me when I say it's only 30 minutes calculation, but people, experts in here, know how this is done. Let's imagine, just for fun, let's imagine the problem is like this. I have a spherical vessel. And then I have a charge of bristles protruding into the room. And then I have my RNA, which is a flexible polyelectrolyte, which is going to be RNA, single cell RNA is flexible. That's going to be inside this lumen. Let me just do this. And how would I do this? I want to start with a Hamiltonian. So yes, I don't have an internet here. That's OK. But I don't see it on my screen. How do I get rid of that? Let me see. I don't see it in here. Sorry? Yeah, I'm not that well-coordinated with my mouse. But I can tell you, right? Please. All right, so basically, right? So that is the cationic protein brush. See, it is not there. No, go back. Please go back. Yeah, so this is very unique. It's OK. No problem. You can still read it, right? It's not important. You can't read it anyway. So here, the blue is RNA. So it has all the confirmations of this RNA. Single cell RNA is in the blue color. And the bristles are the protein tails. And then the hydrophobic domain is somewhere there. And the red color corresponds to all the confirmations of these protein tails. And then everything else, the electrostatics. There is interaction between blue and blue. There is interaction between red and red. Interaction between blue and red. And there is lots of salt in here. There is also interaction from the salt. So the electrostatics coming from RNA, capsule proteins, and the salt, right? And then what do you do? That's the partition sum. You calculate the free energy, minimize the free energy, and get to the optimal confirmation. This is something you can do analytically. In spite of all the complicated equations I'm showing you, it isn't doable analytically. And the answer is, as some of you already know, is the following. It makes very simple predictions. Prediction number one is that if I look at the, this is a very key equation that I must wait for this gentleman. Yeah, I hope you'll give me extra two minutes. Oh, yes, sure. Not at all. No problem. Because the first equation is a very key equation, which comes out in a very simple-minded way. But you have to do the work. You have to do the analytical work. If you do it, the answer is the amount of DNA, single cell and RNA, which is a packaged number of repeat units that is packaged into this virus is simply proportional to total positive charge carried by all these bruises. That's a simple prediction. This is not a novel idea. If you take any atom in the periodic table, we believe in electroneutrality. When you put one extra electron, that's not as stable as the neutral case. No problem, don't worry. See, this is my life. Don't look at it. So that's a prediction. So this total length of RNA that is packaged is proportional to total positive charge carried by all these tails. The second prediction is the density profile of this polynucleotide away from the wall will have this depletion zone. It'll kick up like this, where reaches the maximum come off. I did not say anything about the sequence. I did not say anything about packaging signal in here. Because in my Hamiltonian, it's very difficult to put in the sequence to do it in the way, in the style with which we do analytical calculations. Then I have to resort to computer and then I have to do all the things which we have done separately later on. So these are the predictions. And these predictions have turned out to be good. Surprisingly, given these approximations, simple calculations, these are the data. These are not simulations. These are mother nature's data over billions of years for all kinds of viruses. They fall on this very nice muscle curve, exactly the way we predicted. Namely, lambda is proportional to total charge. And similarly, if you look at the density profile, we heard references to Jack Johnson, writing from his laboratory. They changed the sequences and the virus are similarly the same. And the whole process takes place independent of the sequence. And they said that genetic content is different. And the diversity of RNA transcripts, but they all package following electron neutrality. Now, so what? This has been a good suggestion, because now you can package a DNA with anything that is possibly charged. It doesn't have to be a capsid protein. You can, for gene therapy, you would take DNA, package it, or put cargo inside the capsid proteins. It doesn't have to be an RNA. It can be anything that is negatively charged. As long as it follows the electron neutrality rule, you can package as much as you want. And philosophically, the message is that the viruses assembled in a way that the total charges have to be balanced and meant. It probably in the early stages of evolution, it was a co-evolution. It's not only central dogma. There is only RNA dictating everything, but also they were going back and forth. And that's what the implication is. Now let's look at the kinetics of assembly. Now the situation is slightly different. So let's just go to the basic silicon-free argument. Suppose I have a placket of some size linear states, or two-dimensional placket. And then you can compose the free energy of formation of this nucleus. And that will have a nucleation barrier. And that is a line tension. That is a bulk gain in free energy. And then based upon this, you can think about nucleation time going like into the one over super saturation or super cooling, depending upon which variable you use. And then once that is bound, it will grow by various entropic and energetic encounters of the various individual components at the perimeter. And then it will grow. And then eventually it will close. And then based upon conventional arguments of the classical nucleation theory, one could make a prediction that the growth rate would also be e to the minus one over t minus t. Similar formulas, but the pre-factors would be different. This is, again, a very simple argument one could make. And then you do simulations. What we did was we did Langevin Dynamics simulations. I must make it very clear here. We use the fluctuation dissipation theorem. And the friction coefficient of an entity is appropriate to that fluctuation dissipation theorem. So we do Langevin Dynamics properly, in view of the common that we heard before. When you do this, what we showed was there is a very narrow range of a temperature and a protein concentration when this assembly will take place. And we were absolutely very grateful to this Spanish team. We do not know who they are. But they did a fantastic site, the mutagenesis work. They did a lot of experiments. And then they came up with the minimum structure that you need in order for this virus to take place. So we piggybacked on their work. And the reference is given in our work. And we could show how this happens. We could follow the kinetics, how the kinetics is taking place. To make a long story short, in here, it also involves a nucleation barrier. So in the absence of the RNA, capsidic proteins alone could assemble if the conditions are right. There is a barrier in here. That barrier is substantially reduced in the presence of RNA. And also, the amount that the barrier is reduced depends upon the RNA sequence. The RNA sequence contributes the kinetics of the assembly, but not to the final structure. That's basically the summary. And sometimes when the virus particle without the genome may not be able to assemble, but the RNA would allow it to assemble. And therefore, in fact, what happens is this analysis of these data tells us that we could use the classical nucleation growth theory for understanding this assembly of these viruses. And in fact, in order to check this, I'm going to be very quick in here, the prediction. This was a classical argument. The time for nucleation will go like this. And that's the law that we saw. And similarly, the growth rate is predicted to be that way. And that's exactly what we saw. Whether we look at the nucleation rate or the growth rate, that the melting temperature or dissolution temperature is the same, and it has to be. So this verifies theoretically that nucleation growth mechanism is operating for the growth kinetics of a virus. Yes, Robin? Right. Not in our case. We can discuss this. Not in our case. We calculated the nucleation rate. Not in our case. It's a matter of definition of how you calculate your line tension and also bulk energy. It depends upon that, right? Right. We can pick. Robin, we can talk about this later. Can we talk about this later? Because I'm just showing simulation data and the analysis. They fit perfectly fine. We can cobble about the magnitude. But it's not at all. I'll show you. It's not at all 100 kT. So we are talking about 6 kT. And I'll come back to this energy later on. Incidentally, the ATP hydrolysis, for some of the people who are not familiar, ATP hydrolysis is 20 kT. And I'm talking about 6 kT, 6 k to the 10 kT. So in these simulations, here, oh, sorry. Here, we took MEM, minus-minut virus, I think it's called. Now, that's the one they had. External data, we wanted to piggyback on them. We took that virus. We take that as a typical example where there was data. So that's what we analyzed. But the point that I'm trying to make is a global point. It is not a particular example that I want to talk about. I'm talking about a global point. That is, nucleation growth mechanism is a good platform to think about the kinetics of viruses. So let me show you this. This is from our friends. I think we saw this graph already. Yes, of course, experimentally, the very beautiful fluorescence correlation spectroscopic experiments clearly show that it was a really two-stage mechanism. And you can go back and look at all the arguments, and look at the data. They match extremely well with the simulation data. And also, this is the most recent work. So what do I want to conclude? This is the conclusion for Robin. The conclusion is, I'm not done yet. This is only the first part. In general, electrostatics dominates single-stranded RNA virus assembly, not double-stranded DNA. It controls single-stranded RNA virus assembly. And the spirit of my argument is captured in the model of the very first book on virology. This is by Luria, a 1953 book. I must read it loud, particularly for younger people here. There is an intrinsic simplicity of nature, and the ultimate contribution of science resides in the discovery of unifying, simplifying generalizations rather than in the description of isolated situations, Robin, in the visualization of simple overall patterns. What do I want to share with you next? I want to know what the fate of this virus is. I made this virus. This is what happens. Now, how does it get demolished? We had a beautiful lecture before. I want to see. I want to understand how to break up this virus. So there are lots of versions. Let me just simply go in here. There are two ways it can happen. One is, it completely dissolves, uncomplexation between the polyanion and polycateon. The other one is, it slithers through. Lots of holes are on the surface of this virus. It slithers, it undergoes a translocation and goes out. And of course, they'll recombine and then do proliferation of the viruses, et cetera, et cetera. But that's the question that I want to understand. And how do I want to understand? This is something that is like a child's game. Some of you might laugh at me for being so simple-minded. And I went about three ways. The first way is, let me take this virus using this electrostatic principle. I have some amount of RNA in here. Now let me arbitrarily consider. Suppose I increase the amount of RNA, length of the RNA, arbitrarily. Let me say that it's going to be an excess. That excess is given by n minus q. q is a total positive charge on the protein. And n is a total negative charge on the RNA. And this v is the volume of my capsid of the virus. So theta is my order parameter. So what I want to know is, what is the free energy of this virus as a function of theta? How does that happen? When is it going to be non-stable? How much price I have to pay to package more and more of my RNA into the virus? The second thing that I want to do is, OK, so let me take a complex of a partially charged polymer and negatively charged polymer. And they are complex. And I want to break it up. How do I break it up? I bring an invader. I bring a competitor. And I see whether that is going to pry this blue out of this red. And I make a partner with the red. And what is the force involved? How do I think about it? And what are the energetics involved? What are the appropriate conditions? The other strategy is to look at a translocation to the exterior. Because I already told you, the virus outside of the virus is also polar, because it has to be water friendly. So inside it is going to be capsid here. And then I want to put it out. And how does that happen? What are the energetics involved? In order to decompose. So now the question is, I want you to ponder over the following question. Suppose I have a virus like this. And I have my brissel. That's my starting point. And then I put my DNA, my RNA, sorry. It's not ready enough. My RNA. And I'll make it a little longer, make it a little charged. But then there are also, as we talked about, there are lots of proteins in here. Lots of enzymes in here. Then put some enzymes here, some enzymes here. And of course, they're soft. Plus, minus, plus, minus. And outside, plus, minus, plus, minus. Let's take general. Because I can always put the concentration of these added proteins to be 0 in my general problem. That's the easy limit. So let's think about this. How would we do this? So there are conformational fluctuations of the tails, the RNA, and also these proteins that I'm adding. But most importantly, it turns out, is the small ions, which are mobile, they create osmotic pressure. The translational entropy of these small ions is dominating much more than the conformational contributions to the free energy. It turns out, if you do the calculations. But now I have to make sure that I keep the, down in equilibrium. Namely, the chemical potential of the salt inside is same as the chemical potential outside. I have to make that condition. Otherwise, I'm not doing it right. Let's do this calculation. Something remarkable happens. If you do the calculation, what I find is a following. This is the free energy density as a function of theta that I access the genome that I'm putting in. The global free energy minimum corresponds to that number. That's about 0.2. That value of theta is being 0.2, 0.21, something like this. But then, this is for a particular salt concentration. Now I add the salt concentration. I increase the salt concentration. When I increase the salt concentration, that minimum turns out to be a little bit lower. But still, that's a global free energy minimum. Then when I go to about 0.25 molar monovillant salt, then that minimum goes away. It undergoes a phase transition. It is not a stable global free energy minimum anymore. This line is no genome at all. So it becomes unstable. The system becomes unstable. So when I add more and more salt, the virus doesn't like to have the genome inside. A very simple calculation, but it is kind of indicative of the stability of the system. Then you can calculate the susceptibility, et cetera, et cetera. So that's the conclusion number one. Now let me go to the second case, namely, uncomplexation. I think it's unstable. It becomes unstable. Unstable means that the encapsulated RNA inside the virus is unstable state. I'd love to see that. I would love to see those data, please. Already I was worried that 0.25 molar is very high. But my biologist friend told me that that's OK. There is lots of changes in the inside of the cell. But 0.6 molar, that's very interesting. Please wait for a couple of more slides, which may be related to a comment that you are making. But this is inside the virus. I'm talking about this is inside the virus. Let me go to the second question. Suppose I have a complex. And I want to uncomplex this. So let's think about this for a second. We all know that when you have two charges, we know the force between them. That is coulomb. That is 1 over r squared. And we know that in an electrolyte solution, the force between two charges is given by Debye-Hockel approximately, or Passan Boltzmann. It's going to look like this. It turns out what would be the force between two topologically coordinated charged entities? One is a poly cation, poly anion. None of those, obviously, cannot be, because the topological correlation contributes. And then we compute them. We calculate them. Analytically, we can calculate from the free energy. And also compute them to verify our approximations of reason number one. Then I make this complex. Then what I do is, as I'm repeating myself, bring an invader. And that invader is going to kick this blue out and then make the new pair. And I want to know what the substitution time is. Because that substitution time is critical for me to understand how I'm going to break up my virus inside itself, if this mechanism were to be operated. So let me show you one movie here. Here is a, first of all, this is a complexation. So in this complexation, the counterions are absorbing on the backbone of the chain here, green counterions. And then here is white counterions there. And when they are mating, the counterions are being released. And I used to think the electrostatic attraction is a dominant one in dictating the process. It turns out the release of counterions is actually dominating in the complexation mechanism. Because they are getting a lot of entropy. The release counterions have experienced a lot of translational entropy. And that is creating the complex. Now I have made this complex. Then what I do is, I break it. Break it means I bring an invader. And the blue and red is a complex. And now green is an invader. Please look at this. Lots of counterions are on the green. And that's a pair. And the pair is struggling to remain as a pair. Because this is a bigger, deliberately brought a bigger one playing around. And here also the counterions are condensing there. It turns out, eventually the breaker will take place. It turns out the dominant force responsible for the substitution is counterion release only in that domain where the attack is taking place. However much charge you bring, it doesn't matter at all. Only at a place where the new chain is attacking, how much counterions that are released, that's the one that is contributing. So based upon the simulations, we looked at the time needed for substitution. And of course, when the ratio of the new chain, length of the new chain versus the previous chain is 1, it's infinity. The substitution will not take place at all. It turns out. And then eventually it goes down. This is 30, 30 length, this is 60, 60 length. And when the ratio is about 50% larger, 1.5, of the length of the new chain to the earlier chain, then beyond that, there is no gain in bringing a longer chain. There's no gain at all. So all that you have to do, if you want to break up this complex, you need to bring a chain which is about 50% longer. Doing more than that is not necessary. Now what is the driving force for this? Again, as I was repeating myself, this is entropy, release of counter ions. That's why electrostatic is not playing that important role in this substitution process, release of counter ions. Then we thought, instead of this, why don't you just challenge this complex with a salt? Because we thought it will loosen up. That's related to your earlier question. Suppose, then what we did was, here is the simulation. There are lots of salt ions from moving around. And if you're challenging it with a 0.5 molar sodium chloride type salt, then it breaks apart. Not 0.25 molar that came with the analytical calculation. With this simulation, 0.5 was good enough for short chains. We are talking about only 30 monomer chains. For that, 0.4, if you did 0.4 molar, it would not break apart. We have to go to very high salt here for this mechanism. So now let me go to the third one. How do I bring the polymer from the interior to the exterior? So here is the problem. Here I go back to theory. So what we do is, let me start with the inside. There's a charge density of the polymer, which is q. Charge density on the surface with the sigma. Salt concentration inside and outside is the same. And r, because I want to make sure the dynamic remains all OK. And r is the radius of this vesicular sphere. And then I want to take this to the outside. I want to calculate the free energy difference. And the way we do, as the experts in the audience know, is write it as a time-dependent short-engine equation. Vp is all the potentials coming from the polymer, polymer, polymer, polymer in here. And ves is between the polymer and the surface. And then for this particular problem, ves and the potential from the surface is written as a diva-huckel. And depending upon whether the exterior problem or interior problem, it's all standard textbook material. And v0 is an important quantity here. Various parameters come into one combined quantity. Sigma is a charge density on the surface. Q is a charge density on the polymer. This is a persistent length of the polymer. Kappa is the inverse diva length, depends upon concentration. And lb is the beer of length. For water, it's about 0.7 nanometers. So that is the setup of the problem. Then what we do is you convert this after massaging these various variables, convert it into this time-independent short-engine equation, and rx, the potential, the effective potential is like this. And then here, when you solve for the eigenvalue problem, then you can calculate the entropic contribution to the free energy. And then this is the entropic contribution to free energy. But the key variable here is this combined parameter. This is my interaction parameter, sigma, q diva length. And then there's a chain swelling, which is, you know, because when you change the salt, the polymer chain will change its confirmation. And that's the beer of length. So let me show the results. There are lots of puzzling results I just want to share as a challenge. We are not to fade it out in terms of analytical work, but the numerical simulations are very good. So the potential inside and outside, of course, has to be continuous. That's the way it is. And then the effective potential has a turning point. People who work with the quantum mechanics will know this. This is really a painful exercise. There's an int inside, outside, inside. There is region 1, region 2. We use a WKB. We're technically challenging. That's what we had to do. We did that. So inside what we found was this parameter 24 times pi times sigma critical charge density q, beer of length, has a function of inverse divide length in units of the persistent length, monomer length. There is a universal law that we get. If this quantity is higher than this curve, if it's above this curve, we will have complexation. And remarkably, that condition depends upon kappa as a 5 half power law. In the outside, asymptotically, the exponent turns to 11 over 5. Again, above these curves for different sizes of these viruses, we will have adsorption. And below that, there will not be any adsorption. Then there is one more remarkable thing that comes out of these calculations, which is if you consider the RNA inside, flexible polyelectric inside, there is a spontaneous selection of a virus size. This came out as a big surprise to us. And that spontaneous size, kappa times that radius of this virus depends upon the interaction parameter would be as a power law, 2 third power law, roughly 2 third power law. There must be an elegant analytical derivation, which I don't have yet. And we can trace the origin of this behavior, why there is a spontaneous selection and solicitor, but there is a spontaneous selection. And now let's look at taking the polymer from inside to outside. Again, we were very surprised. There could be a kubble there. The change in free energy to take the RNA from inside to outside for identical conditions is only about few kT. And of course, that depends upon solid concentration, but it is really modest driving force for expulsion. This is fantastic news. When you think about it philosophically, this is really fantastic. That's why it happens so easily. But this is what this calculation tells me. And it's really very small. That doesn't mean that there is no barrier between the initial and final state. In order for it to go through, it has to go through an entropic barrier. This is another life of me where I have been thinking about a polymer translocation. Christian knows that. I have been doing this. So the way the RNA is going to go through this capsid wall from interior to exterior, there are lots of free energy landscapes. You can write focal point description, da, da, da. And that's pretty much under control in my domain. And there is a barrier. That barrier is about 6 kT. It's not humongous. That's about 6 kT for these kind of polymers. So with these arguments, then let me conclude by saying that. So regarding the fluctuations in the genome size, Don and Iquibrim tells me that there is a phase transition in genome size. There is beyond certain amount of genome. According to this argument, it is not stable. An uncomplexation by competitor, substitution driven by counter and release. And the threshold length for effective substitution is really finite, very small. And I believe this uncomplexation is probably taking place in the cellular environment by salt gradient, not by bringing a competitor. I thought there was going to be some competitor which is going to break it, but I do not know. But I think that seems to be possible. And finally, in terms of translocation in exterior, the criteria for adsorption on the interior and exterior can be very easily derived. Only a few kT as a driving force for expulsion. I think I'm pretty much done. Once more, I love this quote from Luria. I must say that again. And I'm very grateful to my students. Thank you very much. OK. I'm yours. Now kill me.