 Do you hear me? Is this microphone working? Yes. Very good. So is everybody else? I want to send the organizers for giving me the opportunity to attend this interesting meeting. I have to apologize, I do not really belong to the circle of experts in viruses and I'm not sure organizers, although they did in general a very good job, they may have made one mistake of inviting me and I apologize in advance if you find my presentation boring or uninteresting. What I will tell you is a bit of a story about physics of branched polymers which does have something to do with viruses but I will be mostly concentrated on branched polymers themselves will touch on viruses a little bit. But I will start with this picture which shows the story of my first work about viruses. I was thinking about how double helical DNA could be ejected from the bacteriophage. Some of the people here know this work and in fact Ries Garmin, he has a very interesting poster over here where his work with Manoharan actually opens the new ground in terms of experimental observation of this phenomenon. So what the phenomenon is you have a spring and you have DNA which is driven outside by the spring which is described by this little very trivial equation here. The only trick of course there are two tricks one is to find what is the friction coefficient psi and another one what is the free energy. So you first have to solve within this framework you first have to solve equilibrium problem how much free energy they are given the amount of DNA inside and we attempted to solve this problem and this was actually the first it was supposed to be the first publication in the series but and what we did is we derived this type of structure which is called spool and now it's considered to be either trivial or wrong and so on. But the interesting point is that Anonio's referee said about this first part that what we constructed was a perpetual motion machine. I am still puzzled by how one can construct a perpetual motion machine by just calculating the free energy of the equilibrium structure. So the reason why I am recalling this story is because I thought maybe this Anonio's referee is now sitting in the room. Yes so I thought maybe this person could explain to me privately what he or she actually meant. But really most of my talk I will be talking about this branched polymers and I have here two pictures from two papers one is drawn by Angela Rosa and another by either Robin Brunschma or his student Joshua Kelly. I do not know who did that. Kelly. Okay so this is simpler. This is more artful. But the idea is this is still the same and you know this idea better than I do and I will be assuming that this approximation which is obviously an imperfect approximation I will assume that it works and I will consider something of this sort. Okay so what do people know about these branched polymers. It started in 1949 by Ziemann and Stockmeyer and what they showed is that ideal branched polymer which means branched polymer which has no absolute volume, different pieces so we don't see each other, go through each other freely and its branched pattern is completely random. This object here is the size, the radius which is proportional to its molecular weight or number of monomers to the power one quarter. This is famous result by Ziemann and Stockmeyer which is basic for the field. So the interpretation of that result is the following. Imagine that you have this branched structure and this is how I abstract. I want to think about branched structure. I don't want to think about real RNA. This is my abstract view and what I want to do is I want to emphasize one particular, I call it chemical diameter. Other people call it ladder distance when they think about DNA. I will keep calling it diameter. Something that a typical connection between one and the other and I can choose any one basically random one. And then the statement is because this polymer is random as I have been shown, showed to us if this branching pattern is random then the length of this object, chemical length of this thick line is proportional to square one square root and then this line folds in space such that its size in space is another square root. You have square root of square root and this is end to the one quarter. This is how Ziemann and Stockmeyer theory works. Now important distinction which I want to emphasize and which I want you to keep in mind all the time at least those of you who want to listen to me is if you have this branched object whose branching pattern is repeated here you can swell it, expand, you have to expand it because it's too compact to live in the real three dimensional space. So you have to expand it and you can do it in two ways. Either you can stretch every of this connecting little chain like this while branching pattern is still the same. Chemical structure or in RNA terminology the secondary structure of RNA is the same, it's not affected. Or you can do it differently. You can rearrange if it's possible. You can rearrange branching pattern to make it for instance like this comb and then you can compensate for some excessive stretching of some of these red chains. So there are two modes. One corresponds to quenched or fixed branching pattern and another corresponds to a nil branching pattern. So this is the main distinction and I will follow it. Actually what happened is it is the information particularly useful for our chairman. Many years ago my friends Sasha Gutin, Eugene Shakhnovich and myself we had two months of sort of free time and in that time we performed four papers on, we wrote four papers on branched polymers. So far two of them are a little bit known to people and two of them are completely unknown. So Robin there are two more which you can learn. Okay and very recently Angela Rosa, I favorize and Michael Rubinstein and myself we wrote a review on this annealed system. So how does it work? How do we describe the simplest approach to describe the behavior of these two types of branched polymers? So first of all we describe them in terms of three relationships. First of all how overall radius or generation radius grows with the molecular weight and this is characterized by the power nu. Second of all how this chemical diameter grows with L. And Avi Benchotovs that for this virus like carnage this thing goes like into the two sorts so there is two sorts. And then there is another relation between oral size and this path length. If you assume that this is ideal polymer this quantity will be one half. But generally there is this relation. Now according to Florie we think that free energy of this object consists essentially of entropy and energy. This is minus T s entropy and this is energy. And energy is not particularly remarkable and uninteresting. It's just energy of some clump of material at the given density. But entropy is very dependent on what you can do. You can only stretch your bones or you can also rearrange your branching pattern. If you can only stretch then the first naive expectation which turns out to be right but for non-trivial reasons would be that it goes more or less like you take one of these chemical diameters and you stretch it and this is the distance between its ends and this is the length of this one of these chemical diameters. But this is very non-trivial because this formula looks like you stretch only one diameter and everything else is in order and does not participate in space. So in fact this formula is right for subtle reason which I will not even explain and which I will illustrate using this picture on the next slide. But if you have an annealed polymer which can rearrange its branching pattern then you have an additional entropy which is here and which we can, that one we can understand. You have additional possibility to rearrange. So this is independent of R so therefore it is independent of how the object is placed in space. So about this formula what is L? I told you it is one of the chemical diameters but which one? It turns out that Kramer's theorem, and here is Henry Kramer's very interesting command, Kramer's theorem. The most known part of Kramer's theorem is about computing J.H. Radius, every J.H. Radius of this branched object. But I want to do more. I want to do probability distribution because the exponential of this entropy is the probability distribution of J.H. Radius. So I want to do more and I want to find probability distribution and it turns out that probability distribution is controlled by maximum eigenvalue of certain matrix which I will explain in a second while classical Kramer's theorem tells about average which is controlled by trace by the sum of diagonal elements. So generally this Kramer's matrix is constructed the following way. You take matrix elements K, M which corresponds to bond K and bond M and on the three bond K and bond M they separate the three to three parts. The part on the left of bond K on the right of bond M and in the middle. It's like a sandwich. So you take K of K the number of monomers in this red region times M of M which is the number of monomers in blue region. You multiply and you are almost home. I thought you are home but you are not because I thought about it. I forgot about the fact that in the world there are some negative numbers except in addition to positive numbers and George Kelly showed that in fact sometimes matrix elements are with plus and sometimes with minus and he has perfect algorithm which shows when it's plus and with this minus then this L turns out to be maximum eigenvalue of this Kramer's matrix. So I invite you to realize that there is simple computational algorithm and for those who want to use this it's an easy computational trick. Now I have to go back one second. Okay so if you look at the annealed situation for annealed polymer already this formula predicts something interesting independently of what is R independently of how your polymer swells in space. If you minimize this over L you find optimal chemical structure optimal branching pattern for given degree of swelling which means for the given value of nu. For given value of nu you find value of rho and this is a prediction which you can test and you see all the data available in the literature go very well with this prediction so this is a reasonable prediction. That means annealed object do follow this separate optimization of their chemical structure with respect to their branching pattern and there are more details about various indices which also are in good agreement but let me skip that. So based on this Flory theory what we did in the recent paper which was in a special issue dedicated to somebody in the room we use this theory to describe how much work you have to do in order to compress given RMA in a way that you into the closed volume and you know better than I do that randomly reshuffle RMA or have different branching pattern and therefore we expect that there is different value of this L different values of this chemical diameter or more specifically eigen value of Craners matrix the work needed to be done will be different. So the idea is that in the representation of free energy the interaction part is pretty much the same independent of branching pattern. It depends only on the amount of material on the number of monomers it does not depend very much on the type of branching but the elastic part the entropic part is very sensitive to the type of branching pattern. So this part you can interpolate using Flory Hagen theory or Wanderwald theory but the result is insensitive to it. So our answer in terms of this work of compression our answer is basically this one the difference between randomly reshuffled RMA and real RMA of this particular virus I have no idea what this virus is I have no knowledge of that but there is a difference between randomly reshuffled RMA whose work of compression is given by this line and below it there is a green line. In fact we have two couples two pairs of these lines one for Wanderwald's approximation another for Flory Hagen's approximation and this seems quite uninteresting but the difference is predicted quite reliably. The difference between these two types of RMA is given by this. So that's what we predict for this work of compression and the difference of work of compression and in the end of the day there is this simplified formula which tells you that the difference of work of compression between molecules I and J let's say randomly reshuffled and the real viral one is controlled by the difference of this leading Hagen values of Kramers matrix. I'm not sure Kramers would have liked to discover that his matrix is somehow related to this viruses but it is related. So the question now is can we do better? This is Flory theory this is based on the trick of assuming that there is elastic part and interaction part and it is known in physics that this is sort of a magic there is some magic consolation of errors we never know whether it will work for Flory theory or not. So Flory theory is a nice thing but it's capricious. So can we do more reliably or can we develop a more sophisticated tool to do such calculations? So since the main assumption which I did not emphasize the main assumption of this work of compression calculation was that secondary structure of RMA does not change at all when you compress. So let us now in reality we know from the simulations like Vienna and others that there are many secondary structures maybe hundreds or thousands I forgot within fraction of KT of the ground state. So maybe let's go to the opposite limit and say that there are many different secondary structures and that is annealed limit. That is nothing else but annealed limit. So let's go to annealed limit and let us consider what happens in the annealed limit and our goal now is to develop a good honest sophisticated physics theory. So how would we do it? As a reminder there is a well developed machinery theoretical machinery to study the confinement of flexible polymer. This actually turns out to be based on Schrodinger equation. There is mapping between this problem of confinement of polymer chain and the problem of confinement of quantum particle. And that will show to you on the simplest possible example how this analogy works. So the first example I know that many people are not particularly knowledgeable about quantum mechanics but one thing that people typically remember about quantum mechanics is uncertainty principle. Uncertainty principle tells you that uncertainty of coordinate times uncertainty of momentum is plant constant. So if you want to confine your particle within distance r that means you have to as you go to smaller smaller r you have to give it more and more momentum because r times momentum should be still plant constant. So plant constant divided by r will be the amount of momentum it has. Plant constant divided by r squared will be kinetic energy. The more you confine your particle so poor particle confined in tight prison cell runs very fast back and forth and has a lot of kinetic energy. Similarly poor polymer chain confined in tight compartment it goes back and forth like crazy and has much entropy. It turns out that this mapping entropy is the same as kinetic energy maps to one another it's not the same but it maps to one another as a result the confinement of linear polymer chain always go the confinement price always goes like one over r square. This is nothing but uncertainty principle. You can do it mathematically by doing this Schrodinger equation. Now interesting remark for experts how about this picture when you confine instead of confining a flexible polymer you confine let's say warm like chain or double helical DNA. In this case Schrodinger equation description is not working but it turns out that in the early days of quantum mechanics there was a man named Oscar Klein he developed a version of quantum mechanics which is exactly this and you can solve hydrogen atom in Klein's quantum mechanics and this is your DNA in a composite which I see. Yes it is the same. Yes yes it is the same Klein's Klein Gordon equation. Yes. The confinement Yes. It's unfavorable. It's unfavorable. It increases free energy which means it decreases entropy. It's unfavorable. It's very unfavorable. It's very unfavorable historical circumstance that entropy is defined with minus sign but yeah it's unfavorable. Okay so now in order to develop this theory you need to know something of the Jean's legacy. He studied this annealed branch polymer but he did not consider placing it in space. We'll just partition some of fluctuating secondary structure but you can yeah there is the order of my slides is slightly illogical I apologize I must have messed up these two slides but anyway we will manage. So speaking about confinement work we were talking about confinement so what confinement work now let's consider confinement not of linear chain for which from uncertainty principle I derived for you that confinement costs 1 over r squared. So what we're going to do is we're going to divide it. Now for branch polymer it will cost 1 over r to the power of 4 and it follows from these three lines of logic. There is a radius without confinement. Remember the human stockmire into the one quarter. Now confinement entropy must be some power of this ratio r over rg what else could be there. There could be nothing else unitless. So some power x and how do we find this x. We say that confinement entropy must be proportional to n must be extensive entropy is always extensive so you obtain this formula. So it's not uncertainty principle it is something else. It is one or well there is minus sign. Minus sign is missing and also I didn't say that x is positive. The way it is written x would be negative. x here should be minus 1 minus 4. So there is minus 4. x is minus 4. I think it's ok. It is true but in this case density is not the right thing to follow. I can explain it in the following way. If you have a large branch structure and you want to confine it in this place. That means you have to take this part of your branch structure. It should be compacted to go there in this orange size domain. You have to take this one. You have to confine this one too. You have to confine every piece and this is why it is extensive because it has no exclusive volume so you confine every piece independently of every other piece. You are right that my statement that entropy is always extensive is a cheap one. I wanted to bypass it but the essence of the argument is very rigorous. It is indeed linear. Ok, thank you. So following now Degen we developed a little more sophisticated version of Degen's theory in which we follow spatial dependence of everything. This equilibrium is very complex. The equations are the same equations which Degen derived except we have now spatial dependence. So we consider how this object goes into confinement. Accordingly, inside the potential well we have some circumstances. Some values of this potential energy is little phi. And this phi 1 is potential energy for the chain ends. Phi 3 is potential energy for branch points. And this is perhaps low potential energy inside the confinement volume. Very high potential energy outside. Something like this. And this p star is the eigenvalue so we have this equation. So now if you look at this equation, equation for psi is the distribution of the end points of arbitrary chemical diameter. Remember chemical diameter, secret line. This is the distribution of end points. This chemical diameter is linear polymer. It's not a big surprise that this is Schrodinger equation. It is the same Schrodinger equation which I had three slides before. This Schrodinger equation for psi is the same thing here. Same Schrodinger equation for linear polymer which is now chemical diameter of a branch structure. Except this potential energy for branch points is now replaced by effective potential energy which involves the idea that every monomer of this chemical diameter actually is also pulled or pushed or somehow affected by the whole long branch that Emmerder is from. So this is how this mathematics works. And accordingly there is this function which describes how branchy it is. And this function is the solution of this non-linear equation. For a long time I was much puzzled how these complex looking equations could possibly overcome the fact that this is Schrodinger equation. Schrodinger equation obeys uncertainty principle. Uncertainty principle gives you 1 over r squared. And this should give you 1 over r to the power 4. That was a big puzzle. And I know for some people mathematical puzzles sounds artificial but for us theorists it is a subject that does not allow us to sleep quietly at night. So it was capital G where is capital G. Oh, this one. This one is my artful notation for 3. This is how I write 3 in Russian. This is space dimension bill. This is 6, a square over 6. This comes from space dimension. Okay, so we publish this paper in a journal which I doubt anyone in this audience ever publishes in. It's called low temperature physics. And the reason why we publish it there is because my teacher, Ilya Lipschitz turns 100 this year and we were invited to contribute to this issue of the journal. So what we did is we considered these very equations. We honestly solved them and we discovered the desired 1 over r to the 4. And there is no reason why we published them. The reason how it works is the following. We first of all considered this free polymer away from the potential well. Then we dip it into the potential well but we assume that potential well is very extended in space. And then we slowly hugged this polymer into smaller and smaller volume potential well. And that required this work little omega and little omega comes in these equations in the form of eigenvalues. And it turns out that this effective field which enters in Schrodinger equation through mathematical manipulations, this effective field is not flat bottom potential well. Beautiful surprise is that this field is a harmonic oscillator field. It becomes parabolic because branch points which sit in the middle of this potential well, these branch points are repelled strongly from the wall. Because from each branch point the whiskers of the branches come out and these whiskers touch the boundaries and say no, no, no, don't go there. It's very unfavorable. Please don't approach the walls. And this repulsion keeps branch points in the middle. And that repulsion generates this effective field big five which generates parabolic potential. And that solves the problem. Okay, so here are our numerical data which confirm this analytical result and so on. Now I think in the interest of time I will skip this story which is the analogy between branch RNAs found in some type of viruses. And territories which are found in chroma, chroma, chroma DNA, I will skip that. And I will acknowledge that most of what I told you was done with Robin and actually on Robin's push because he's a real virus expert. He keeps pushing me and I'm not an expert at all. So Josh Kelly is his students, Rylef, And Geli Rote and Michael Rubinstein, we collaborated a lot on these annealed polymers. Guten and Schachnovich were involved many years ago in this branch polymer work. Roya recently involved me in some simulating discussion. There are some pictures here. And I want to thank you for attention, but before I finish I want to spend one to spend one minute going completely out of the box and saying the following. I was sitting here for several days listening to talks. And since I'm completely out of the field, I thought maybe I can allow myself to make a stupid speculation, which may be useful or may be completely useless. But I could not resist thinking when I was listening to you that the following. In general, when we look at life around us, we realize that although life forms are infinitely diverse, it is still believable somehow that possible possibilities of life forms are even more diverse. That evolution could have developed many more things which didn't find necessary to develop. On the other hand, looking at part of the biosphere, which is viruses, looking through the lens of your presentations, I saw that diversity of viruses given the limited number of components and limited size and so on, it seems that possibly the diversity of viruses approaches what is possible, what is allowed by the laws of physics. And is it not possible? I just want to ask you maybe this question is idiotic. And OK, so disregard it. But I want to ask you, is it not possible to maybe challenging but is it not possible to start thinking about the following question. Based on laws of physics and knowing that there are three components, proteins, either RNA or DNA in limited amount, and maybe lipid membrane. In some cases, maybe there is some sort of ergodic description of what you can do, what are in principle possible combination of these ingredients. That would be an interesting thing to answer if anyone could. At least I would be interested to know the answer. Thank you.