 even by Henry Orland, he couldn't come unfortunately, but we were fortunate enough that Angelo Rosa from Pista and Mario Nicodemini University of Maple Foods agreed to start picking him. And so they will give us a course on polymer teaching without thinking. So, yeah, thank you. Uh-oh, good job. Yeah, it works, can you hear me? Yes, okay, so welcome everyone. So, sorry, just do the sheet with the present. Okay, maybe you'll give it to me later. So, as Matteo said, my name is Angelo Rosa. I am a researcher in CISA, you will see CISA, so this Friday is a picture of Piazza Unità d'Italia, you already know probably, few years ago. And so, I just give you a short introduction of our group, I mean, the group who might belong to in CISA is the group on molecular and statistical biophysics, there are, so the main focus of this group is to study molecular, let's say, biomolecular systems, and we study, basically, all the realm of molecular, of biomolecules, like proteins, DNA, RNA and chromosomes, so you will see more about chromosomes during this class. So, why introduce the group is because if you, I mean, if any one of you is interested, there will be soon the admission to our PhD in physics and chemistry or biological systems, this is the official name, where we apply tools of statistical mechanics or computational physics to study those molecules that I mentioned to you before. So, the deadline to apply for the entrance examination is quite tight, unfortunately, the next week is 12 of March. In general, it works, there is an exam, so the exam is divided in a written part, and then if you pass the written part, there is the oral part, so if you want more details you can just look at the web page. If you feel you are interested, but you think maybe you are not in time for this time, there will be an accession this September if we don't cover all the places that we make available now. So, that depends if we don't fill all the places, because we don't have enough time, I mean, the people will not be admitted because they fail the exam, so please, I mean, if you are interested, please have a look, and Giovanni Bussi will tell you more about the research topics on Friday for the people who will go to CISA. So, this was just to tell you, especially because, I mean, the deadline for applying is very close, so that's why I mentioned to you now. So, that's all about self-promotion, let's say. So, and we'll start about these lectures. So, as already Matteo told you, these lectures are about polymer physics of chromosome folding. So, this is a relatively, say, a new subject of research, not polymer physics, polymer physics is very old, but the application of polymer physics to chromosome is quite, is relatively new, and I will tell you now why. So, we'll present the first with a few slides, just I will tell you about some very soft biological introductions. I mean, I understand that for, I mean, can be difficult for physicists, I'm a physicist as well, and I don't, I know very little about biology, so it's very complicated. But I will tell you about the phenomenology, so you, in order that you can understand what I will tell you later. Then, since the lectures are about polymer physics of chromosome folding, and I'm pretty sure probably none of you knows about polymer physics because it's a subject that is normally not taught in normal courses at university. I will tell you about generic polymer physics, I mean, generic modeling at the blackboard. So, I will tell you about theory. So, the only background that is required is some very basic statistical mechanics. I mean, nothing more profound than that. And then there will be more focused lectures next week, three lectures, I think, by Professor Mario Nicodemi from the University of Naples. And Mario will tell you about more, I mean, new experiments on chromosomes to analyze the chromosome structure, the nuclear structure. And he will tell you also about modeling. But in order to understand those modeling, that modeling, you need the polymer physics. So, that's the purpose of my lectures. And this is a bunch of references. So, the first two references are about polymer physics. So, the first one is a very classical one. It's written, it was written by leaf sheets, I mean, the famous leaf sheet. Alexander Grossberg and Sergei, yeah, sorry, Alexey Koklov. It was published, as you can see, almost 40 years ago. So, the subject is, as you can see, I mean, it's not really new. And it's about statistical mechanics of polymers with excluded volume interactions. So, you will find also something about field theory. So, for the ones on view that are more skilled or more interested in things like topology and soft matter or renormalization group, this is a very good reference. It's very broad and very interesting. I mean, still nowadays it's very, very interesting and very profound. I really like it. Then there is these, more very recent, it was published last December. References on pedagogical introduction to polymer conformation is more similar to what I'm going to tell you now. It's a brilliant review. I mean, I really suggest it because it covers a lot of topics. You'll find a lot of references. It does not cover polymer dynamics, only polymer conformation. So, polymers at equilibrium, but different kind of polymers. So, linear polymers being polymers, polymer with different architectures. And it's very interesting. And it's also mainly devoted to physicists. So, it's very, I mean, it's about physical modeling. Then, the rest of these references is about applications of polymer physics to chromosomes. So, this is a very small selection. Now, I mean, you can find many more. So, this one, again, by Shura Grossberg. I mean, the same Grossberg of here, but four tiers, a bit four tiers later. He, I mean, he explained to you, I mean, the effect of topology for the description of chromosomes. I mean, why the effect of topology between different polymer chains is important to understand how chromosomes we have in the nuclei of the cell is more, I mean, it's a review more for physicists than biologists, I would say, because it's quite, it's a bit technical, but it's very, very interesting. But it also talks about experimental results. So, it's a very nice reading. So, this review is by myself and a researcher in Paris, Christophe Zimmer. This is very pedagogical. Basically, there is no math there, but it's more, in fact, it was more for biologists and it's about to explain the different polymer models that were proposed in the literatures up to 2014. I would say that it's already a bit of data instance because it's a very fast growing field. So, there are new models that appear in their, I mean, very last years. And then the rest of those reviews is also about generic polymer modeling for chromosomes. There is a very nice review by Mario, Nicodemi, which is probably, which is more related to his work. So, as you can see, I mean, it's a very nice subject and there is a, I mean, a super new review. It was just published one month ago in this journal, Traffic by Helmut Schisser. So, I think, I mean, if you want a copy of these slides, you can send to me an email or I think if these lectures are registered, I think they are recorded. So, maybe, I mean, if you are interested in reading these papers. So, now, so why the problem of, I mean, why is the problem is timely? I mean, why one should think to apply polymer physics to understand chromosomes? So, the fact that now there are a lot of experimental technique that are applied to investigate the spatial conformations of chromosomes inside the nuclei of the cell. So, you probably know that we, I mean, as humans, we belong to some particular realm of, let's say, of living beings. So, we are eukaryotes. Eukaryotes means that our cells have a nucleus, almost all our cells have a nucleus, and inside the nucleus there are chromosomes. And inside the chromosomes are made by DNA plus some proteins, but basically are made by DNA. And the chromosomes is, the single chromosomes is the single structure in which organizes the genome. So, the old genome, all our DNA, let's say, is organized in different chromosomes. And chromosomes in the cells can be studied now by different techniques. So, the first technique is, this one is called the fluorescence in situ hybridization. It's a very interesting technique, what people do, they can really put fluorescent probes to tag specific portion of the genome. So, you can, in order to see them, to see at the microscope. And this is a picture of how a single nucleus, this is a chicken cell, I mean, it's a chicken nucleus and chicken form a specific cell line. And this is how chromosome appears. So, each chromosome is, can be seen, can be, let's say, seen according to different colors. So, different colors correspond to different chromosomes. This is a nucleus, so it's a very powerful technique. And another technique that is now, that was, let's say, that was proposed only a few years ago, is called the chromosome conformation capture. That's why it's called 3C, CCC, CCC. And the chromosome conformation capture technique allows you to measure how frequent chromosomes interact, I mean, they are close to each other inside the nucleus of the cells. So, they give you quantitative information on the, not only on the special positioning of the chromosome, but also how frequently you can find, I don't know, chromosome one to chromosome 13. And this is important because biologists know now that accord, I mean, depending on how chromosomes are localized inside the nucleus of the cell, they can perform, or they cannot perform, or they perform in not the right way their functions inside the cells. And the third technique, so these two techniques actually do not allow to study chromosomes in vivo, let's say, to study how chromosomes move, for instance, while this third technique is called, sorry, you can study chromosomes dynamics by using green fluorescent protein, maybe you ever did, that was awarded the Nobel Prize a few years ago. So, the idea is that you insert a specific protein, which I think was derived from some jellyfish inside, so you really, you cut some portion of the genome, you insert these proteins, and then these proteins this protein is, I mean, when it's excited by some light becomes green, and then you can follow how these things move inside the cell. But since it's attached to a chromosome, what you can see is the dynamics of the chromosome. And you can imagine that by, you can, for instance, measure the mean square displacement in time of these locus on the chromosome, and apply genetic technique like, I mean, one, for instance, Langevin equation to model, also to model this kind of displacement. And so, all these techniques are quantitative, so they give you numbers. And since, in the end, DNA and then chromosomes are polymers, so it's interesting to see, okay, can I use what I know from polymer physics to understand what I observe from experiments? And the answer is, yes, in the sense that polymers and physics can tell you a lot about chromosome inside the cells. I will tell you a bit more about it later. First, you have to know polymer physics first to appreciate the results. But just let me conclude with a bit more of phenomenology. So this is a picture, of course, but, I mean, what I'm going to tell you about is a, a genetic can be called like the problems of chromosome folding. So namely, how chromosomes are organized inside the nucleus of the cell. The nucleus is a very important organelle inside all our cells where chromosomes are contained. So they are physically separated from the rest of the cell. Probably it was, this was due to evolution in order to protect our genomes, and also the genomes of all the other eukaryotes from being, let's say, dispersed in the cell. Because it's protected somehow. So actually this is a very interesting system for a polymer physicist because it happens. So you think that each DNA, I mean, each single filament of DNA, which belongs to a single chromosome is one centimeter in length. While the dimension of the nucleus is only 10 microns. So you can imagine that this poses a very important problem. How chromosomes are packed inside such a tight volume? I mean, it poses a very important question. I mean, how they are packed in such a way that they are also efficiently packed in order to, that the information which is contained, I mean, inside the genome can be accessed by all the cellular machinery. And as you can see, I mean, that will, I mean, I will show you that these, also generates a lot of interesting physics. That I will tell you a bit more probably Thursday after I will tell you about generic polymer physics. So this is the last slide for biologists, from the biological point of view. Yes, yes. So the percentage is not much. It's about 10, 20% because of course I mean, chromosomes and DNA is very long, but it's also very thin. So the only percentage of chromosomes, which I mean by chromosome, the DNA plus the proteins because it's not only naked DNA. There are also some proteins, actually many proteins. It's about 20%. The rest of the cell is filled with other bodies. Sorry, the nucleus is filled by other bodies, also the cell, I mean, the rest of the cell is filled by bodies, but the nucleus is filled by other bodies and so it's pretty crowded, I would say. And this in general, I mean, actually Eukaryotes all look more or less, they all look the same. We are not so different from fruit fly or even east. I mean, the Saccharomyces cerevisia, which is the budding yeast for making bread or beer, it's also a eukaryote and it's actually one of the most studied organisms and it's quite close to us, I mean, for many things. I mean, in spite of its simplicity. Yes, and then this is last slide for biology, let's say. So this is a bit of a genetic phenomenology. So the cell cycle, first of all, almost any cell undergoes a cycle. So, namely, during the cycle, the DNA of which cell is replicated is doubled in order to form two new cells. So this is pretty common knowledge, I think you know that. So how, in chromosome two, I mean, they do not stay, I mean, they are very dynamic during this cell cycle. Cell cycle can be divided into parts. One part, one, let's say broadly speaking, one is mitosis and the other part is interface. During mitosis, chromosome take these, don't have a point, okay, they take these shape that you are probably familiar with. It's like two elongated roads. But during the interface, it's which is the part of the cell that takes most time, I mean, basically normal cells that applicable in to 24 hours roughly and 23 hours of these 24 hours are taken by interface. And during interface, chromosome do not take that road like shape, but they are more dispersed, let's say inside the nucleus. And form a particular organization which is called territorial organization in sense that each chromosome is dispersed but does not occupy all the nucleus. It occupies a small region of the nucleus which is called the territory. And that discovery was done about 30 years ago. It was a very, a really fundamental discovery in biology. And I mean, the kind of polymorphism that we'll tell you about will be, I mean, it will be in order to understand this peculiar organization of chromosomes inside the cells. So this territorial organization is common. I mean, it's basically any organism as it. So it's a very generic feature. And so it's quite important that you retain it. And the picture here just to conclude that is about how chromosomes are formed. So chromosomes are formed, as I said, by DNA plus some proteins. And this compound of DNA and proteins is called the chromatin. And the chromatin is the main structural unit of the chromosome. So broadly speaking, you have DNA. You have proteins which are called the instance and these stones are electrically charged, positively charged. Since DNA is negatively charged, then DNA wraps around this instance. Okay? And these stones are very important because they contribute to regulate how the DNA, I mean, the genome works. Because when, I mean, when it's because of this interaction, the DNA is tightly packed, it cannot be accessed by what are called, for instance, transcription factors. Transcription factors are some proteins that contribute to transcribe the DNA. When they are, it's not so tightly packed, then instead it can be accessed. So in this structure, this kind of, let's say, conformation is very important to understand. I mean, it's very important to understand how the genome works. But for time being, I mean, you can more or less forget about it in sense that you have, you can just retain that chromosomes are, can be considered like long filaments and basically polymers. So I can close these. I hope it closes, yes. And so we can just start my introduction to polymers. So, I mean, first of all, what are polymers? So polymers are long molecules. And where you have basically one single unit, one single unit is called a monomer. And these unit is attached one after the other. So a simple example, I already mentioned to you of a polymer is DNA. So you have the four bases which are attached one after the other. And the process, I mean, the process that starts, the contributes to the formation of polymers called polymerization. And basically, it is a process where one monomer attaches one after the other. It's a process that can go on basically forever in sense that, because I mean the kind of interaction. So if you have a single monomer, let's say, and you have some, which is functional, so they can be, they can attach to another monomer, then can go this way, and then so on and so forth. So the DNA is a specific sample of linear polymer. Since that, if you stretch from one end to the other, it's like a strain. I mean, it's a linear strain. But there are other polymers which are not linear. They are, for instance, they are called branched polymers. So branched polymers are polymers which are made of units which can be polyfunctional. So for instance, you can imagine that you attach one monomer here, and then there are two links, no? That favor more the presence of branches. And so for instance, can be, can go this way. And also they can go on and on forever. And then there is also polymers which are circular. So namely like ring, they are called ring polymers. For instance, in the DNA of prokaryotes, so not eukaryotes, we are eukaryotes, but prokaryotes, for instance, bacteria, like E. coli, is formed by single, it's a single, so the genome of E. coli is a single DNA, is, yeah, it's just one formed by one single filament of DNA which is circular. So it's a ring polymer. So as you can see, there are also some application in biology. So how, so given this kind of very broad introduction, I mean how physicists have applied their, let's say, their tools to understand the polymer. So basically polymer physics can be divided in two parts. It's like about studying the conformations of polymer at equilibrium. In general, polymers in some solvent and also studying the dynamics of polymers. So for this lecture, I will tell you about a bit also, more bits about dynamics, but I mean to start, I will tell you about polymer conformation at equilibrium. So one is to imagine like, let's say, polymers like at equilibrium at some temperature T, so it's like a real statistical mechanics model. So in order, so what, I mean, what characterizes polymer? I mean, it's very few things. And one thing is the flexibility. So imagine just from the abstract point of view that you have a structure like this, one after the other. So what it happens is that these bonds here are not completely rigid. So they are subject to thermal fluctuation. So in general, let's say the deviation from, so that has some kind of, can you see all me from? So if you have, this is a sort of direction for this bond. So then you have the next bond here. What it happens that at equilibrium, the next bond will not have exactly the same direction as the one before, no? But it will be a bit, on average, will be a bit, let's say a slightly different direction from the previous one, and so on and so forth, okay? So in the end, I mean, if you have many monomers and in general for a polymer, you have many monomers because in general, if you call the number of monomers N, what you want, I mean, in general in polymer physics, you study the limit of N, which is much larger than one. So in the end, for very long polymers, you will have a situation where the initial direction of your polymer will be completely forgotten. So it will be completely correlated from the original direction. So that means that, I mean, at the first approximation, the most simple model for a polymer is a model where you have monomers, okay, which I number like I from zero to N, so I have N plus one monomers, linked one after the other by rigid bonds, okay? And so it's zero, one, two, three, four, N minus one, and N. So I imagine that, so as a first approximation, I imagine that the direction of this bond is completely correlated from the direction of this bond. So they are free to rotate around this inch. And so for the rest. And I call the length of this bond B. In general, in the literature, you can find B on, or you can find also another notation, but I'll stay simply on this notation B. And this is the length of the bond, which is fixed. I take it as fixed. Then I will a bit relax this constraint. And then what I ask is how, so first of all, I want to ask about conformation. So of course, I mean, this is kind of, this is one single conformation, but as I told you, polymers are free to move in the environment. So in the end, what I'm interested in is the typical size of the polymer as a function of the number of monomers. I mean, why these? Because, I mean, it can be shown that the typical size of my polymer now I will be more precise, but in general, the typical size of the polymer grows as a power law of the number of monomers. So there is a scaling behavior. And this is generic for any polymer model. No, if you stretch, so stretch conformation is one particular conformation. I'm taking all possible conformation. So this is an average, if you want, okay, maybe put any average, over all possible conformation. So this is a rigid bond, they are free to rotate. So I'm taking an average of all possible conformation in space for this ensemble. They are all, and I also assume, okay, so this is the generic model. I also assume now that there are no volume interaction between different monomers, namely any monomer can overlap with any other monomer, which is of course not physical, but I mean, in some cases, it's a good approximation. And that's it. And for the rest, these bonds are rigid and they are free to rotate. So in this bond here, it's free to rotate around this inch. So it takes an all spatial conformation. It's clear, it's very simple. And in general, sorry, I removed this average here because I will be more precise on the definition of the observable. In general, the typical side of the polymer grows as a power law of these components. And the first, let's say, task we are going to face is to determine for this specific model, I mean, what is new? Okay, in general, this problem is very simple. I mean, you can see that it's a bit simple for the following reason that imagine that, okay, I take the first bond, sorry, the first monomer here has fixed in space because, I mean, of course, it doesn't matter where the absolute position of the polymer in space. And then of course, I mean, if I say that this thing, I mean, this bond can stay here, then the next one can be everywhere and so on and so forth. This is very similar to what you have for an overall model for a run or walk, right? I mean, run or walk, not in time as the typical run or walk, but a run or walk in space, okay? So imagine that you have something that sits here, then these things can take any other possible direction and so on and so forth. Then this is like a kind of a model for a run or walk. But we'll derive, I mean, now things in a rigorous way. So there are two quantities that typically people study for this model. So in general, to study, let's say the overall size, one quantity is the so-called average square and the 20 distance. So namely, you have, so this is the vector that points to monomer zero and this is the vector that points to monomer n. And this is some frame, the origin of the frame. So then what I'm going to compute is this quantity. And this is defined as, so this I call the r zero. So this is r one and this is r n, okay? So that's the vector that points from the origin of the frame to the bit. So this is r n minus r zero square. Average. So you can ask why I took square and then I averaged. I could take just r n minus r zero average. Of course that will give you zero because I mean any direction is possible. So by symmetry that gives you zero. So I need some quantity that, I mean it's positive. It's positive definite. It's definite positive. So this is the standard quantity that people takes in the account. And so now this quantity is very simple to calculate actually because now I introduce vectors. I mean I introduce the bond vectors t like t one, t two, etc. And these vectors are defined like t i is equal by definition to r i plus one, sorry r i minus r i minus one for r i going from zero to n minus one. I have of course n bonds vectors and n plus one position. So this is very advantageous if I introduce this quantity because now, oh finish the space. So because now this quantity r square ee this can be simply defined like. So this is, so actually r n minus r zero. This is just the sum for i that goes from one to n of t i. You see that because this is just the difference between the last vector and the first vector position. And then this is just the sum of all bond vectors. Okay, sorry. Yes, I should take off the project. Yes, good question, I want to do that off. Yes, maybe it's a bad, okay, let me, no, I love some this way from one to one. Yes, and then so that goes like sum. So this is exact. So if I introduce this thing here, so this is the sum i from one to n of t i square. And now I can, so this is just equal to sum i and j from one to n t i scalar product t j. And I can put this sum outside. I write it here because otherwise probably you can't see. And so this becomes like sum i and j from one to n t i scalar t j. And now you just average. So now you can see that what you need to do is to just distinguish between the terms where i is equal to j and when i is different from j. So this is just the sum i from one to n of t square. Plus twice the sum where i is from one to n minus one and j from i plus one and then t i scalar t j average. But now you can see that it's very easy to do because all t i have the same length by construction. As I told you that they are only equivalent. And so this is just b square times n plus the cross, plus the average of the cross correlation, let's say between bond i and bond j. But since they are all independent, this is just zero. Okay, so in the end, plus zero. And so in the end, this quantity is just equal to b square n. And if you compare with this formula here, you see that my exponent nu, my relative here, nu is equal to one half. So which is, as I said, if you are familiar with random walk, this is just like a random walk. Okay, in space. Is it clear? You have a question? Yeah. They are independent. So we are limited distance. How would you attach if you consider that you are independent, so you have potential or something like that? I will tell you a bit more later about how to do that. In general, this thing is non-zero. So there are different possibilities. Either, okay, for this model, which is, okay, this model is called, I didn't mention that, it's called the freely jointed, or freely rotating, it depends, sometimes from the book, freely jointed chain, FJC. And this is very, let's say, the most simple model for a polymer chain. So the assumption of this model is that this term goes to zero. And this is, of course, the simplest assumption you can take. But then you can have another assumption where this correlation is non-zero. And probably I will tell you a bit more tomorrow. I don't think I have the time today. And if it's non-zero, then you have developed correlation. But then you need to see how, I mean, how this correlation behave as a function of the distance of the bond vectors along the chain. So I can tell you that if these correlations are, let's say are, for instance, exponentially damped, so namely they are very short range, then you have still the law of the random walk on very long scale. So in general, you can renormalize everything if you want. And then you get something which is similar to a random walk. So it's again a random walk. But if these correlations are long range, then you need to see, I mean, then you really need to see how, I mean, there can be correction to this linear term. And in general, I can determine the correction are, and then if you introduce this correct, so then the corrections to the linear term are larger. Actually, not correction, but this correlation, they give a more important time than the linear one. Other question? So if there are no questions, so this is the first result that you need to retain so that we put it, no, no, this is exact. No, it's very simple. It's like a random walk. With a fixed jump. Okay. No, there is no approximation here is exact. No, okay, it can be a mean field if I make some, I mean, if I introduce a model with some of the known volume interaction and then. But the new is not, yeah. So I see what you mean. So if you want, yes, this is equivalent like a mean field. Yeah, approximation, yes, for a polynomial. Yeah, okay, I see. Well, we'll discuss deviation from this model. So I presume this is very simple, so I can be fast on it. So as I said, so this is the first result. So for a model, let's say in general, this is always true. So for a model, for a polymer model with very long polymer without extruded volume interaction, the typical size goes like B and to the one-other. And I let you notice one thing which is very important that this result does not depend on the dimension. Okay, I'd listen to me, but talk to me a bit too much, sorry. I mean, if you have a question, ask, sorry. No, there was, it's a bit difficult to me to keep the attention. So this result does not depend on dimension. So I'm, because in the end of what I did, I never used the fact that you are in 3D or 2D or 1D, whatever, so this, that's all everywhere. So we will see that when we introduce volume interaction, then there will be an effect which depends on dimension, on the exponent, no. So the fact now that I can use, so actually I have a law which is completely related to the law, the random walk, allows me to also to discuss not only the average, so this is basically the average size, the average size of my polymer chains. But what if I want to discuss not only the average properties but also, I mean the complete, I mean, say the properties which also deviate a little bit from the average one, then I need, I mean, I can ask, what is not only the average value of the polymer, but what is the average distribution, sorry, what is the distribution of the, for the values of these, for instance, of the end-to-end distance? And this is simple to derive because if you, I mean, you can now, since I told you that this is very similar, this is basically the same math of a random walk, then one can ask what is the typical distribution of the sides of my polymer? And the typical distribution of the side of my poly, if this is basically the same mathematics of the random walk, it would be like a Gaussian. You know about this distribution of the end, say of the distances that a random walk can perform is a Gaussian, right? So that means the following, that the distribution of my end-to-end distance is so namely for the end-to-end distance of a polymer being R, this is this quantity, so it's two pi NB square B over two, the exponential of minus the R square of two NB square. Yes. What if we substitute R? So this is the probability that the end-to-end distance takes a value R, so distribution on probability distribution, this is given by this formula, which substitute. E, E, E, E, E. E, E, E, E, E. In the end-to-end distance, sorry. It's the same as before, okay? In the end-to-end distance. Yeah, this is, if I have, if I ask what is the probability distribution that my polymer has an end-to-end distance of length R, so it's in this direction, this is the distribution, okay? And this is, can be derived exactly, now I'll tell you how you can derive it. Just I'll give you the first passages. Otherwise I don't have the time to go to give you more details on the rest, but the shape is like the one of a Gaussian, because this is exactly the same, because this is linked to the similarity between the polymer chains with no exclusive volume, without the exclusive volume interaction, and the random walk, so it's a Gaussian. Yes, yes, this one? No, no, it's not the space, it's from the model, from the model, okay, you can find also, no, I said that it provides, so you have this model, so the model is the following. You have a collection of monomers linked one after the other, so it's the most simple model that you can have for a polymer, for a linear polymer, so you have a monomer, one after the other, right? And then I have no correlations between consecutive direction, right? So that means that the typical size of my object grows as the square root of its component, of the total number of components, which is the same as a random walk. So then, okay, I forgot to mention this, if, so these results first does not depend on the dimension of my system. So, namely, this is, this depends, this is the same that you are in three dimension, four dimension, infinite dimension, yeah. Any possible dimension, okay. So then I ask, what is the, not all, so this is the average value, the average size of my polymer. So then I ask, because I need this expression later, so what is the typical distribution of these sizes, of the size of my polymer? And the typical distribution is a Gaussian, and sorry, I forgot, D is the dimension of my space. So the distribution, let's say, has a dependence on the dimension, on this, on this pre-factor here. And you can see that this distribution is in line with these results, because if you calculate R square on this distribution, that gives you NB square, okay? Because the variance here is NB square. Okay, so this is the typical distribution of the sizes of an idea, let's say, is called an ideal polymer, ideal because I have no extruded volume interaction. Okay, so it's very simple. Okay, so now, because I want to say a bit more on, I mean, what, so now, how you, for instance, so this can be an exercise you can do yourself, but I will tell you how you can derive this formula because it's quite instructive, and you can derive it in that way. So you, we can stay in 3D, but in the other, I mean, in different dimension is the same. So in order to derive it, so you have to calculate the average value of RN minus R0 minus R, so this is the average value of this delta function. So this delta function blocks the end-to-end distance to stay equal to R, okay? And then you have to calculate the average value of this delta function on all possible. So given, let's say, all possible conformations of my polymer, okay? So this average is done on, by using the following weight. So this is equal to the integral delta RN minus R0 minus R. So then I have to define, let's say, the weight for my chains, but that you know because as I told you, so the only constraint for the chain is that the bonds are fixed and that the bonds are free to rotate. So then the physical, let's say the Boltzmann weight for my chain is, thank you, this is not at all mistakes, is given by the product from I to one to N of VT4 pi V square delta function TI minus B. So this is the TI. So basically, since, so it's very simple not to describe. So you have the teach bond is the independent from the other and each bond has a fixed length. So this is just the delta function. So this is a product of delta function where each delta function, I mean, it imposes that the length of the bond is fixed. And then the integral is the integral over, I mean, over all these of the DTI, so where the TI is the direction of the bond. Is it clear the shape of this weight? No. So you have, so each bond is independent from the other. I mean, as any possible direction, right? So if, and then each bond has a fixed length. So you have to imagine like it's a road in space that is free to rotate and they are all independent. So this is just the product of all these things, right? And you have to put, so four pi B square is to make these weight, it is just the normalization of these statistical weight. And then I have to do the product of all possible directions. Clear or not? Yes, no? So you have, okay, let's say the, so each bond leaves on a sphere, right? Oh, sorry, on the surface of the sphere because it's fixed, right? So you have your chain. So each bond is the direction of each bond. First of all, each bond has a fixed length, right? And then the direction of each bond is independent of the rest, right? So then I can write these as a product of all possible bonds. You know, that's the weight of, statistical weight on my chain, right? Maybe you can tell me what's not clear. Yes, this is in 3D, yeah, I told you, you know, I was doing the calculation D equal three, yes. Sorry, it's again. No, but this one is D, the DTE, the integral. And this is a delta function that imposes that the bond has the length B, each bond has a length B. So then, is it more, yes? Oh, if you want, yes. See, see, it's more complicated if you think in this term. I mean, this is just, I mean, if you have, suppose that you don't have this, so you don't have the delta function. So this is just a collection of n roads, independent roads, and then you, and then it gives you just, if you make the integral, this is just one, okay? So, it's very simple, it's nothing. So then, in order to, if you want, in order to, so then now I'm fixing the end-to-end distance to be equal to R, okay? So then, in order to calculate to derive the end-to-end distribution, so to make this integral, what you do, you, first of all, you resort to the Fourier representation for this delta function. So you write this thing like dk over two pi cube, the e to the ik, rn minus r0 minus r, okay? So this is the 3D representation of delta function in terms of, I mean, the Fourier representation for the delta function, okay? e to the i, yes. You know this, you know, yeah, yeah, sorry. i, i, yeah, i, yeah, yeah, yeah, exponential of i, but I think it's clear, no? Well, hope so. Okay. This one? Yes, it's the, yeah. Okay. But it's similar to what? This, I don't understand. I mean, I don't write very well, I don't know. So this is the Fourier, you know that, no? The Fourier representation of the delta function, right? So then, what you do, you insert this representation here, okay? So let me, because the blackboard is not enormous, so I introduce it here, sorry. So I want to show you how you can derive, so this is vk to pi cube e to the i k dot r minus r0 minus i, okay? And then there is this thing here. So then, what I do, I switch the two integrals. So I move this outside and this inside. So this is equal to vk over to pi cube. This is e to the i, of course, k, times rn minus r0 minus r times this thing. Pi i equal to n dt i divided by two, sorry, four pi b square delta t i minus b, okay? And now it's, I forgot anything, no. Yes, and then you have this integral that is made on this term here, okay? So now what you do, you write this thing, rn minus r0, like the sum from i to n of t i. Moving because this is just by construction, right? And so that means the following, that this thing is the integral in dk, that you can put outside the term with e to the minus i k r because it does not depend on this term. And this is the integral pi i one n dt i divided by four pi b square delta t i minus b. And you can see that e to the, what's called it, i k dot t i, okay? Because this is a sum, sorry, this is the exponential of i k times the sum of these. And so this, so each t i is independent from the other, so I can put it, I can reframe like this. And this is nothing that is a very simple integral because since each t i is independent from the others, this is just the integral on each t for pi b square delta t minus b e to the i k t. And this to the power of n because each t is independent from the others, okay? And so this integral is very easy to do. Actually, it's exact. You can do it your way, you can do it your way, you can do it your way, you can do it your way, and you can do it yourself. I just give you the solution. So this integral here, finish the space, it's the sign of k b divided by k b, okay? You can see it from very far away. So if you do this integral, that gives you this sign over, sorry, it's a sign, yes, sign of k b divided by k b. So then, so up to now, this formula is exact, so I've made no approximation, okay? So now what you need to do is a very simple approximation. Because you can observe the following thing that this term here, it's always smaller than one, okay? And since it's always smaller than one, and you have a term which is smaller than one to a power law of n, and then, as I told you, it's typically very large because polymers in general have very big n. So then you can just, you can think that the terms that contribute to this integral are the ones where k b is very close to zero. Okay, so this is like one minus k b over a square over six. So this is the Taylor expansion for this term. And then you, so you insert this term here. So this is like one minus k b square over six. This is an approximation. And then you make this term, you make, let's see, yet another approximation because you say that this term one minus k b square over six to the n is like, sorry, I don't know how to write it here. So this is just like the, approximately, the exponential of minus n k b square over six. So you can see it has the same Taylor expansion of this thing. And now it's, you are done because now what you have to do is just the integral, let's say, it's like the Fourier transformation of an exponential, sorry, of a Gaussian, which is also a Gaussian, of course. You know that. No, it's this one. It's approximately, sorry. You have difficulty understanding the notation, sorry. So it's approximately the exponential of this thing. And then what you need to do is just the integral. So this Fourier integral of 3D of the exponential, which is also, of the Gaussian, which is also a Gaussian. And that gives you the formula for the Gaussian function that gives you the n-to-n distribution function of an idea of polymer. And you can do the same thing in any dimension you want. Again, I have not, sorry, I did it in three dimension, but in any dimension, it's doable. It's clear, this is the way I did it. So there is a, so as I told you before doing this expansion, I, the formula is exact. After doing this expansion, the formula is no longer exact, of course. And you can see why it's no longer exact. Sorry? Yes. Yes. No, say, you mean before I did this? Yes, and because the dimensionality's here, okay? So this is, well, it's here, and here too, because here you have this thing, otherwise you would have a different weight here if you are in a different dimension. You are not, for instance, in two dimension, you will be like two pi b, you know? This is four pi b squared is in three dimension, right? So, so this derivation holds for three dimension, okay? So I will probably, okay, tomorrow, because I have some notes, I will give you this, okay, I will send you the page for the notes. I will show you, but the calculation in the end is very simple. So now that you have this, you can put this formula here and you do this integral, I mean, you can try yourself. You do this integral back and you get precisely the formula for the end-to-end distribution function of a Gaussian chain in three dimension. I mean, the formula that I showed you two years before. And so you can see, I mean, where, you can see why the formula is approximated. Okay, let's take just some space here. So what is the approximate, I mean, apparently the calculation, okay, of course there is the approximation, but where it's more evident is approximation. So the approximation is more evident here. So I write it again, E, E equal to R. So the formula, so I write it in three dimension, no? My Gaussian. So where is the approximation? The approximation, I mean, can be seen here because this formula tells you that I can stretch my polymer to any extent and still the distribution of my distance is non-zero, okay? But we started from a model where the bonds are rigid. So I mean, if I stretch it, I mean, enough, then of course, I mean, I cannot reach any possible distance for the end-to-end. When it's over, I mean, completely stretched, it will be like the end-to-end distance will be N times B, okay? But the maximum distance for this formula is not limited. So let's say at full extension NB, the probability for this end-to-end distance will be exponentially small, but will be non-zero. And this is a consequence of this approximation here. So the distribution is, let's say, approximately Gaussian, but it's not exactly Gaussian. But you can verify, for instance, you can simulate, I mean, if you know how to, for instance, you can simulate just a random walk on a lattice, for instance. You can verify that the Gaussian approximation, it's a very good approximation, provided that the number of components of your polymer is large enough. So this approximation is very useful and we will keep it from now on, okay? And another, so you can try to do this calculation, and another important properties of these, that has some connection actually to chromosomes, because I mentioned to you that there exist now these experiments where people are able to measure contacts between, let's say, parts of the chromosome on the cells, is that the probability, so this is called the ring closure probability, the probability that the end-to-end distance is equal to zero. This is, of course, this is simple, you would just put R zero here. This is just three over two pi, and this square, three over two. So which means that the probability of, it's called the ring closure probability of cyclization probability goes like n to the minus three over two. And this is a specific feature of the Gaussian model. If you want to generalize to the dimension, this goes like n to the minus d over two in the dimension. And we will see that if you introduce, so this is a very specific property of a Gaussian polymer because if we introduce a excluded volume interaction in our, let's say, if we consider a more realistic model for a polymer, this is no longer true, okay? And this is a property that actually is measured in chromosomes, and people see that they do not belong, they do not behave as Gaussian polymer, actually, they are quite different. And so by, sorry, so just by looking at these exponents in chromosomes, they can, people can develop different kind of modeling. So it's quite an important feature, yes, please. It's to the ring closure probability. So you ask, what is the probability that the two ends clump close? You can close the polymer. Is it like the same as return probability in a random walk language? No, no, but this is not the average radius. I agree with you, I agree with you, it's not zero. This would be just the radius of a ring polymer, of an ideal ring polymer with no excluded volume interaction, which is actually, it's not very different from a linear polymer. It has the same power low behavior, like it's n to the one over two. It's with a different pre-factor, but the scaling dependent is the same. But this is not the average side of a ring polymer. This is the probability that a linear polymer, the two ends of a linear polymer goes together as a function of the number of components. And it's exactly the same as the return probability, yes. And in the dimension, I mean, it's very easy to generalize, it just be over two. And so now, I'm sorry, I don't think I have time, but what time is it now, sorry? Okay, I'll set it down. I start at 2.30, right? Okay, so now we want to have a more realistic, so this is a very, I mean, this is the starting, really starting point of a polymer. Oh, okay, first of all, one thing that I let you notice, which can be very useful before going to show you how you can generalize by introducing exclusive volume effect is the following. So actually, in spite of its simplicity I mean, these properties, it's very, I mean, it tells you a lot. So suppose that, okay, first of all, it teaches also how to coarse grain polymers in this way, I mean, in this way. Suppose that, again, you have your freely jointed chain. And that you take, so you said, okay, I'm not interested in local scales. I want a model which is called coarse grain. These are typical things that people in polymer phases do. So what I imagine that I consider, let's say a super monomer, which let's say I divide my polymer into, I mean, I into subsection. And this subsection has some size which is schematized by this, let's say, super monomer. And this one after the other and so on and so forth. Okay, now this formula, so suppose that the number of monomers here is sufficiently large. So this formula tells me how I can coarse grain the model because basically the average size of these, let's say super monomer, it's also described according to this distribution. Okay, because any subsection of this model, it's also, it's like an independent model from the rest because as I said, the junction here are independent, okay? So that means that suppose that I call as small the number of monomer that are contained inside this super monomer, then the size, the distribution of the size of this super monomer, which I call the small r, let's say, it's given precisely by this quantity, okay? And then the model that I can consider here, let's say with this more coarse-grained, we need a more coarse-grained approach. It's a model where I have a collection of monomer linked by elastic springs. You see that because it's like an elastic potential roughly. Yeah, so up to here is clear? Yes, okay, so I repeat this. So as I said, suppose that you consider, let's say that you divide your polymer into a collection of n subchains where each subchain has n small n monomers. So then I have, if you want, I have, I don't know how to call it m, I call it m, the number of, let's say, super monomers, this is equal to capital N divided by small n. And so then what I want is a description in terms of m of these spheres, let's say. So then I say since, I mean, since each part, each of these subchains is also, I mean, of course, it's a chain in itself. So it's also subjected to thermal fluctuation. Then the typical size of each one of these subchains has to go like this way, no? Because it's just any sub, because they are all independent. So each sub-portion has to follow the same law. Yes, no, has to be much smaller, I would say. No, has to be, so let's say N, capital N has to be much larger than one. N has to be much larger than one. And if you want, this is not really necessary, but you can take capital N much larger than N as well, so. We are all, we always work in this limit. I mean, where polymers are very long, I mean, we don't want a finite size effect. And so then, since the, I mean, each one of these monomers is, let's say, you can imagine to now that you get rid of, let's say, the small scale details and you center, you take the center of your super monomers and you connect to the next center and so on and so forth. So then, this bond is fluctuating according to this law. The center of the circle, yeah, roughly. You can consider the center of mass, for instance, yes. What did I change, sorry? No, no, I said this is the size of one of these super monomers. Okay, so this is a way, and because this allows me to introduce what actually is used more in polymer physics, which is the Gaussian chain. So Gaussian chain is basically a model where you have, it's similar to the Frigian chain, but now you don't have anymore a collection of rigid bonds, but you have a collection of springs because these are like, let's say it's like a model with springs, yes. Yes, simply because you can center this thing here, right? So then the monomers, which are comprised between these two centers, between these two centers, which are, I mean, the center of these monomers comprised by n small n monomers. I mean, as fluctuation, they have to be the same fluctuation as this one. So just to give you a flavor, I mean how how you can proceed to coarse grained and to have a more generalized model than the Frigian chain. Because the Frigian chain is okay, but there's this constraint of a very fixed bond, so you cannot really work with it. So it's in general, for in polymer phases, people work not with rigid bonds, but with the springs. So in general, you have a collection of consecutive monomers connected by entropic springs, if you want. Which came, which come precisely from the Gaussian distribution of the end to end distance, which is approximately Gaussian in the case of Frigian chain, but is for a Gaussian chain is exact actually, okay, yes. This one? Yes. Okay, let me see, let me see, yes. It's, if you want, it's the end to end distance between two centers, yes, it's small r, yeah. And so now, and because I want to, so I went a bit fast here because I want to introduce a more realistic model. So suppose, so again, the first feature, the most important feature for, basically, for an ideal polymer is that the end to end distance, the typical size grows as the square root of n, and as I told you, you can define another observable, which is the, which I would call the ring closure probability. I can call it, I can call it like PC of n, that goes like n to the minus three over two in three dimension. So this is the, this is the probability that a polymer make a ring, or if you want in the dimension, the over two, okay. So now, what it happens if we have, yes. No, this is the probability, no, this is what is the probability that given the free ends of a polymer chain, I make, I can make a ring, okay. So if you want, this introduces another exponent, which is a bit less traditional in polymer physics, but is another, let's say, scaling exponent. It tells you that it's a, it's power low, the, which is, as some of you observed is just the return probability. But it's, as I said, it's very important because it's a relationship with chromosomes. And I think Mario next week will tell you a lot about that. So suppose that now we have a, so, but our, I mean, as I said, this model is, so this is in the dimension. So this model is okay, I mean, as far as we can neglect extruded volume interaction. But of course, this is, first of all, this is not granted. And second, I mean, we have extruded volume interaction in polymer. So that means that you have a polymer where two monomers, when, because of the thermal fluctuation, when they go close in space, they do not like each other, simply because they cannot stay on top of each other. So they repel each other. So how, I mean, if, I mean, how this low and also this one changes because of the presence of the extruded volume interaction. So this is a very important problem. And actually this problem cannot be solved exactly. And so one has to do some approximation. So you have your polymer, right? So I did, I mean, maybe I wasn't explicit, but actually I made a very important approximation that you have, let's say, two monomers that can occupy any position in the space, which is of course not realistic, no? Because monomers repel each other, right? For instance, if you have, I don't know, any kind of polymer, DNA, whatever. So if you have a long enough filament of DNA inside the nucleus, for instance, you cannot find that the same, I don't know, base can stay on top of another base. So there are effective extruded volume repulsion, no? Between different monomers on a whole new polymer chain. And that has very deep influence on the sides of the polymer in general. So the, so what I want to do is to calculate how extruded volume interaction influence the side, the typical sides of the polymer. Again, I let you notice that this low does not depend on the space dimension, because if, but now we will see that if we introduce a extruded volume interaction, actually now we will have an exponent, which depends indeed on the dimension of the space where the polymer lives. And so they actually, or unfortunately, there is no exact treatment of extruded volume interaction for polymer. So one has to do a simple, one has to do some simplification. And the more general, not more general actually, the more simple, but for sure the more simple theory that allows you to calculate exponent for the polymer is the so-called the Flory theory that was formulated by Flory, I think in the 50s, probably, yes. So the Flory theory is very simple and the idea is the following. You define an effective free energy of your system and of course the free energy will be given in terms of an energetic term that takes into account extruded volume interaction between the polymer, between, yes, the monomers of your polymer and an entropy, okay, right, and this is the temperature. So now my, so this is the general idea. So then this is an effective free energy. So this effective free energy will be a function of the typical size of my polymer. So the energetic term and so the entropy. So then by minimizing this function, I will get the typical size of a polymer as a function of the number of monomers. Ah, yes, that will be also, of course, write it again. So that will be a function of n and r. And this will be a function of n and r minus T s of n and r, okay. And now the task is how I can, I don't know, calculate, estimate, whatever, I mean you can, these two quantities. So I need an estimate for e and an estimate for s. Okay, is it clear, the problem? And then by minimizing this effective free energy, I will get, let's say, possibly, I will get the answer to my problem. So this is the task. And how Flory did, I mean, was in a very clever and, in the end, in a very, very simple way. So, of course, I mean, we will, first we will play with the most simple of this problem. So namely, when we have a monomer, I mean, we have a nomopolymer. So I mean, a homopolymer, I mean, a polymer where all monomers are the same. And where the two monomer, I mean, two monomers, anytime two monomers come close into each other, they just repent, they don't, they don't like. So it's a kind of situation, which, I mean, polymer language is called in, when a polymer is embedded in a good solvent. So a polymer, of course, is not free. I mean, it's always in some solvent. So some solvent means that you have molecules around. So meaning in good solvents, what does it mean in good solvents? In good solvents means that the polymer likes more, the monomers of the polymer likes more to interact with the solvent than with itself. And that basically is the same as introducing an effective repulsion between the monomers of the polymer. Right? Good solvent, good solvent, yes. The opposite is in bad solvent, meaning that the polymer does not like to match the solvent. It's like, what it says, polymer is, when you have a hydrophobic polymer in water, for instance, then to compact, right, to shrink, yes. But we treat first, I mean, we can do also a flow theory for, in the case of bad solvent, but first we do it in a good solvent, which is the simplest case, actually. So how we determine these two terms? Okay, first of all, we do the entropy because it's simpler. And the entropy, actually, it's simpler for the following reason, because Flory, so okay, yes, Flory made a very simple assumption. So he said, okay, I neglect correlations in my polymer chain, so of course, I mean, because I introduce exclusive volume effect, there are strong correlation between monomers, but I said, okay, I neglect this correlation, and I assume, but this is a very strong hypothesis, and I suppose that one has to verify a posteriori, right? So I assume that my entropy is just described by the, so it's, if you want, it would be the logarithm of some quantity, and let's say of the number of states of the possible conformation of my polymer, and the number of states are approximated by using precisely the, let's say, the end-to-end distribution function because that's a measure of the typical conformation of the size of your polymer that I just derived before, namely by the Gaussian, okay? So and if you take the logarithm of the Gaussian, so then you have that your entropy will be given by three, okay, let's do it in the dimension VR squared divided by NB, oh, I think there is a two, yes, two NB squared, and this quantity, so you can just take it, we can get rid of the pre-factor, and so it just took, so this is the flow rate, so that's the, because, okay, you have to do something, I mean, a bit slower, so you take, so suppose that you ask what is the number of states who have end-to-end distribution R, right, right? So then that thing will be, so if you have, okay, let me write this, so that will be like exponential of minus, let's say VR squared, because I want to do it in the dimension, LB squared, right, then, you know, if I want the number of states, so I can multiply this by four pi R squared VR, right, because, yeah, okay, by definition, oh, delta R, if you want to keep it finite, means some small R, right, but then, right, so then I can take it as a measure of, as an estimate of the entropy for the number of, yeah, so if I take the logarithm of this quantity, that's a measure of the entropy for the states, let's say, that have end-to-end distance R, now you can say, okay, I have also this term, but this term, it's basically grows, it actually goes like the logarithm of something, which is a power law of R, but it's always smaller than this one, sorry, I can see, and then you can keep it just this one, it will not influence the rest of the stuff, it's a small, maybe it's a, I would say it's a, I mean, it's a smaller term compared to this one, okay, so that's the first approximation that Flory did, but if you have the states, no, it's plus, yeah, let me see, no, it's a lot of the states, no, otherwise, no, no, it's, right, okay, so then, yes, next term, which is a bit more difficult, but it's still simple, simple, so we need to determine, I mean, we need to, not determine, we need to make a guess for the energy term related to the fact that there are all your monomers inside the space occupied by your polymer chains and they interact, I mean, let's say by pairwise interaction, so in order to derive this term to give you a physical flavor, I mean, how to derive it in a simple way, I consider the following thing, let's, yes, because this is a kind of, no, I see your question, so this is, this is true, I mean, you can always, yeah, it's always defined up to a constant, so it's, you can add any constant here, yeah, so it's, yeah, I mean, of course, it's defined, it's a free answer, it's defined up to a constant, so your friend is right, so it's, okay, let me first, because otherwise, I don't know if I have enough time, so how we can estimate this quantity, so this quantity can be estimated, I mean, I give you a general derivation and then we make an approximation. So, I consider a very simple situation where I have pairwise interaction between monograms and we'll show you that this is perfectly acceptable in the case of the Flory theory for a polymer in a good solvent, so in general, these, let's say the repulsion is, can be written like this way, so one half the sum over n from zero to capital N and m from zero to capital N of something Rn minus Rn, okay, so this is a collection overall possible interaction and then assume that they are pairwise, they are two-body interaction, okay. So now, this can be written also in this way like sum and n, sum and zero and n of the following quantity, Vr minus r prime, let me write this way, delta r minus rm delta minus rm and this is in the r in the r prime, okay. So I just rewrote this by using Flutter. I just wrote it by, I mean, it's just a rewriting, I use just a delta function, nothing special. And now this function, okay, can be written, sorry, this function, this formula can be written like one half the integral of Vr minus Vr prime, okay. V is a generic expression for the potential, I mean, it's a generic, it's just a generic expression from two-body interaction and a polymer. I don't give it any specific shape, you'll be just repulsive because in the end I don't know how it's done, I mean, it's very complicated but I assume that only, I mean, I assume that it can be, I mean, to put approximation, we describe it as a potential which is a short range, zero, so namely zero at large distances and let's say infinity if you want at short distance and the short distance is the typical size of a monomer. Okay, this is a good approximation but I keep it general, so this is a short-range expression for my potential, is it like, sorry? No, it's not exponential, it's really hardcore. Exponential for small distances, ah, yes, yes. But I keep it just general, I mean, I don't want to be too specific. And so then, what I do, I move this thing out and I write, so this will be like, sorry, the r, the r prime, yes, awesome. The r, the r prime and then this will be like the sum n from zero to n of delta r minus r, sorry, r and this rn, so this m equal to zero delta r minus rm, the sum n, zero to n delta r prime minus rm, wait, yes. This is n and this is n, sorry. This is m and this is n, the previous one, this is n, m and this is n, is it clear? No, I mean, because I want to arrive to this formula. So this is just the distribution, I mean the density distribution of a position r and this is the density distribution of a position r prime of my monomers, let's say, right? Is it, yeah, it's just, I mean, rewriting. So this thing, sorry, I lost this vr, sorry, the r, vr prime, vr minus vr prime. So actually, this can be just rewritten, I mean, it's blackboard, because it's a bit, so this thing can be rewritten, sorry if you can't see from too far. So I write it here, so this thing can be written as one over half, the r, vr prime, vr minus vr prime, rho r, rho r prime. Sorry if I did this thing horrible, but I don't know how to write it. And this quantity, so this thing here, if you want, it's just a double integral on the density of, I mean, it's a double integral on the density distribution of my monomers, okay? And so now, suppose that this, as I told you, these potential is short-ranged, so basically you can approximate this like, I mean, for r equal r prime if you want a delta function. So this is like one over half, v zero, where v zero is, or if you want to be zero. So it's the potential in zero, suppose that you can keep it finite, I mean you can do some, it's better this notation, v zero, so namely some constant, but I mean, when you keep this, I mean short-range, and this is, it survives just one integral for rho r square, so this is just an approximation, okay? And but this is a good way that in order to get an estimate for our energy, for the following reason, because now we can write the energy in this way, yes. So we can, let's say, get inspired by this shape because this can be written like some constant, v zero, the density of my polymer, by the density, I mean the density of the monomers actually, by density of the monomers I mean the density of the monomers in the space occupied by the polymers, so that will be like n r d square, because I have a square here, multiplied by assuming that I take a sort of a mean field approximation, where this thing is, let's say, uniform, multiplied by the volume occupied by the polymer, because it's exactly the same expression as here. And as I always do, I neglect pre-factor. I just keep this v zero, which has the dimension of a volume, and if you are familiar with, I don't know, the theory of gas or two-body, this is proportional actually to the second virial coefficient, okay, so this is equal, if I do the calculation, it's v zero, n square divided by r over d, r to the power d, and these are actually the two terms of the Flory derivation, because then I am done, so I have an estimate, as you can see, it's a very simple term, which is proportional to, it's a two-body interaction term, plus the n entropy, or minus n entropy, sure. So the two terms that I can put it here now, and derive, I mean, and if I, now what I will do, I will just do a simple derivative with respect to r, and then I will get, and I will put that to zero, and then I will get the typical size of my polymer as a function of the number of monomers, okay, so this is equal to v zero, n square divided by r over d, plus r square n v square, yeah, yeah, it's the term here, but now we can see it's very simple, yes? Yes, yeah, yeah, we did two approximation, no, absolutely, I agree with you. So the first approximation is this one, and the second approximation is this one, and by some magic, these two approximations exactly cancel, these two approximations, I mean, we have an exact term that magically cancel each other, and now that I will do the minimization, I'll show you that the results that comes out is very good, so it's close to what would be the exact, well, let's say more or less exact values, because now it's very easy. You can just, yes? I'll just finish it easier then, because otherwise, because I'm very tired, because I have to tell you also something about chromosomes, sorry, otherwise I wouldn't understand. And then I just do the derivation, it's, I mean, a derivative, I think you can do it, it's very simple. So I do the derivative of f and r with respect to r, and tomorrow I will comment on these results. So if you do that, you get g zero, n squared to, sorry, minus d, r d plus one, plus t to r divided by n d squared, okay? Yes? Sorry, what? Because I wanted the minimum of the free energy. Yeah, yeah. Ah, sorry, I thought it was the obvious, sorry. Okay, it's a tritid of, yeah, exactly. So if I do that, then I get, okay, again I get, sorry, I will have r d plus, r d plus two, goes like n to the three, and I just, okay, to go faster I just negate the pre-fact, but I'm just interested in the exponent, which means that r goes like n three d plus two, okay? Which means that my exponent nu is equal to three d plus two. So now you can appreciate the fact that the exponent I get is depending on the dimension. So it's completely different from what I have before. And you can see, you can specialize for the different dimension. So for instance, for d equal one, nu is equal to one, which according to you makes sense or not? I would say yes, no, because it is a polynomial with a screw volume interaction is completely stretched. So it's exact. It's what you would expect. In the end, d equal one is trivial, the problem, but it's good that it gives the right exponent. So then you have d equal two, which gives you an exponent which is three over four, which, okay, this is not tremendously obvious, but it's also exact. And that was derived by, let me think, I think it was Duplantier who derived the exact result for a two, let's say, for a polynomial in two dimension with a screw volume interaction. But this is the result, I mean, to derive exact is quite complicated. So it's really technical, but it's good that, I mean, in some sense, with this simple theory, you get the exact result as well. Then you have in d equal three, and this is really exact, I mean, it's really three over four. Then in d equal three, you have three over five. And this is not exact in sense that what is thought to be the true value is, so this is three over five is zero six. That what is thought to be the true value is 0.588, and it can be derived by using renormalization group. But I mean, it's quite remarkable that this approximate result is quite close. And then in d equal four, nu is equal one over two. So, and tomorrow, okay, we'll be more specific on that, but that means that in d equal four, you get, again, the Gaussian behavior. And now just the last question, d larger than four, what do you expect? Okay, it's of course smaller than one alpha. Is it nu, if you will. So, that's not possible too, but you can simulate it. I am. Yeah, yes. And hopefully, it's greater than four for the theorem. For d equal one four, well, I will see it tomorrow more specific because of course now it's late. But in d equal four, mean, for the larger than four, means that exclusive volume interactions are irrelevant. So, simply, if they exist, but they don't as well as the polymer. Okay, that's related, if you want, to the encounter probability. Basically, the polymer never go back. It doesn't, they don't feel it, it does feel itself. But tomorrow, I will show you that how to derive, d equal four is the critical dimension for the theory. And tomorrow, I will show you how you can, let's say, prove it a bit more rigorously. But also, already, this is a piece of proof, if you want. So, sorry if I went too fast, but otherwise, I can't tell you anything about chromosome. So, yes.