 Okay, so, so we start this is about the second part for related to polymer models for a chromosome. So I had the cold this weekend and the source of some fever so I don't feel very well. So, forgive me if I will not be too precise. So in the first part of the, I mean, the things that I described to you last, I mean, last lesson, concerned the, I mean some models that were proposed in the, let's say in the course of the years. And that we somehow we named like say looped model. So, namely, at the basis of the description of chromosomes by using polymers. There's the fact that you have polymers where there are, there is the probabilistic formation of some loops. Okay. The presence of those loops is necessary to describe the experimental data. If you have questions about that. Okay. If not, we move to the second part. So the, I mean, this division in models is not really sharp. It's more like to make an order on all the models that were proposing the same in the first of the years. And also because there are different physical feature that at the basis of this model. So, the next topic is related to the role of topological constraints in the, in the, in the folding on chromosomes. So in the, in chromosome conformations inside the nuclei of the cell. So what I mean by topological constraints. So topological concern is, I mean, something very simple. So suppose that, I mean, related to this is topological concern for polymers and for chromosomes. So suppose that you have polymers in solution. These things, they say, just move the kind of way we saw last week. And then it happens that this, the, there are these fibers. I mean, the polymer fiber, the chroma, you know, the chroma fiber that come close to each other. And then they bump into each other. And they, because of this bumping, I mean, what it happens that they cannot, I mean, it happens, of course, I mean, you can expect that they cannot cross each other. This is a, let's say, an important ingredient in polymer solution, the fact that you cannot cross the two fibers which are closed in space. And then these, this constraint, I mean, of not crossing imposes certain rules of what you can expect for polymers and for chromosomes as well. Okay, so the idea, I mean, let's say the topological constraints and the importance of topological constraints in polymers, let's say, as have been pointed out since the very early days of polymer physics, the idea that they are also important from chromosomes instead is much more recent. To my knowledge, the first work that pointed out on the importance of topological constraints for polymer, sorry for chromosomes, was this, I mean, was this work by Shura Grossberg and his collaborator, where they proposed, I mean, this original idea of the fractal globule. So what is a fractal globule? A fractal globule is, if you want, globules or globules or globules. So, namely, it's a very simple idea. It's a sort of, it's a polymer with a space filling. Namely, you have, so the polymer lives in a volume. All the sites of these volumes are occupied. So it's in jargon is what is called an Hamiltonian walk. So basically it's a space filling curve. So all sites are occupied and it's not a self crossing. So it's self avoiding. So it's not only self avoiding, but it's also space filling, okay. And part of the ingredient is that what is close in space, sorry, what is close in sequence is also close in space. So, namely, you have to think in this way. So suppose that you have a lattice. It's a much, then the crampal globule is done in this way. So, namely, you have portions of the chain, which are, let's say, close in the sequence. They are also close in space. And they form, let's say crampals of increasing scale. So it's a fractal object. Okay, this curve has, let's say, has one important feature that does not form knots. Okay, this is in 2D. There are never knots, but suppose that you have it in 3D, of course, we are talking about, I mean, 3D, I mean, we are talking about chromosomes. So we are in 3D. They never form knots by construction. And suppose that you pull from one of these ends. You can, let's say, open it very, very easily because there are no, basically there are no, since there are no knots that, let's say, that are, that can be detrimental to the swelling of this chain. Then you can open this curve, I mean, very, very easily. And this structure was proposed in the, let's see, in the, yeah, end of 9, end of 8, sorry, in order to model chromosome. But why was proposed, I mean, this kind of model for modeling chromosome. Well, the idea of this guy was, okay, if, I mean, chromosomes are very long polymer, right? And if you, so this is a long polymers, if you have knots and you have, let's say, topological constraints between close by chain, they can be detrimental to how chromosome work inside the nuclei of the cell. So, yeah, so this kind of structure can be, let's say, much more efficient to ensure the proper functioning of the chromosomes in vivo, yes. Well, it's a bit by construction. No, Hamiltonian walk, well, there are two things. So this is an Hamiltonian walk, okay? Hamiltonian walk by construction are space filling. It's the definition of an Hamiltonian walk. But on top of that, this construction does not have knots. So because you have to imagine it's a bit, okay, let's do it, this is on a lattice, let's do it in the continuum. So that will be something this way. And you have a sort of first crumple. Then you have a close by crumple. Okay, and you have this way. And then, so, and that was how it was imagine in, let's say, in your original work, okay? So, of course, in 2D, you never have knots. In 3D, you can, even for a Hamiltonian walk, you can have knots, no? Yeah, yeah, for a Hamiltonian walk. This is a sort of restricted class of Hamiltonian walk, if you want, okay? So now the question is, can, I mean, does this thing has any relevance? I mean, how about topological constraints in chromosome and if they can be measured? So that's not easy, of course, okay? And it's a bit indirect measurement. But actually, in, yes, this is a tree because it's space filling, okay? So in 2D, it will be one over one half. In 2D, it will be one half. So the fractal dimension of this object is one over D. Sorry, it's D, okay? So if you want, this is a polymer, so it grows like always, like N to the nu, okay? Yeah? In this case, nu is equal to one over D. D is the dimension of space. And the fractal dimension of this object is Df is equal to one over nu. So it's D. The fractal dimension is defined as the number of components that are contained inside a region of size R, no? So then N is proportional to R to one over nu. And then the fractal dimension is one over nu, okay? And this is a real factor. So this exponent is, it holds at each scale, I mean, until you reach, of course, the other side of the object. But, I mean, yeah. Okay. So let's see. Yes. So this, I mean, up to this work, let's say that was more like a theoretical speculation than real. I mean, there were no, that was published in the physical journal in the, in the latest, there were no real data that supported this idea at that time. So, but then it came out this work in the, later, I mean, in 2009. Actually, this work was the original work of high about IC. I mean, this technique that I mentioned to you a few days, a few days ago, last week. So in this, in this work, the authors pointed out that in order to explain their data. So they needed to use the crumpled global to need to explain the experimental data. And, you see, that was achieved much, like the confirmation of this model appeared like 20 years, 20 years later. So what they did, I have to go close here. So they, they did a simulation of, they did like Monte Carlo simulation of a polymer model. And they did, and they simulated the two, two different ensembles. So one ensemble was the compatible with an equilibrium global and equilibrium global is, is very different from the fact that global equilibrium global, equilibrium global is basically a global where you allow local change relaxation. In particular, you allow for basically chain crossing and nothing. And so the confirmation, it appears like this. So if you, let's say, if you call or like rainbow, the polymer chain, you can see that all portion of the chain are completely mixed. While if you simulate the crumb, a fractal globule, you obtain a structure. I mean, if you still rainbow color, the, the, the, your, your, your polymer fiber, you can see that different colors. Stay together. I mean, they stay specially close to each other. And they are, let's say, like separated in, in space. Or if you want, they are, let's say, they, they belong to different regions of, of, of your global. What's important, I mean, because you mentioned the fractal dimension, I mean, there is a very specific, specific point to address, is that the equilibrium globule is very different from a fractal, from a fractal globule. Because the equilibrium globule has a fractal dimension of two, because locally it's, it's like a random walk. I mean, on small scales it's like a random walk. Okay. While a fractal globule has always a dimension of three, I mean, in all, I mean, in all, in all small, small scales. And then they compare the, the, so why they, they, they simulate the fractal globule, they resource it to this model, because they were, as I said, they were experimental data. And the experimental data are the following. So what they measure are contacts. As I said, they use, so that was the original paper they would wear, they were introducing high C. So then they measure contact frequencies between these three. How cancel it? Well, it's a fractal of dimension three. It's a space. Oh, okay. It's a special factor. So it has dimension three. It can be, you know, why not? No, not necessary. I mean, it can be, but that's a special case. So that's, that's a real factor. So it always look the same to, I mean, it's a space. Yeah. It scales scaling value. Okay. Maybe the dimension is no, no, no, no, no, the dimension is the inverse of no, because it's defined like. So you need, so you define a region of linear sites are. Right. And you ask how many, I don't know how many mass, how much, how much, how much mass I have inside, right? So, and then that's end to the, like, the F, the power there, and then the F is one overload. Okay. But why, as I was saying, why, why they, let's say they resorted to this model. It was because they had this experimental data. So what they measured there, if you, if you remember about this high C technique, so they measure contact frequencies between inside the genome of human cell. And, and in particular, they were able to, I mean, after they measure all the, let's say all possible contacts between all possible chromosomes inside, inside the cells. They were able to measure what's called the contact frequency between, let's say between those selected spots on the, on chromosomes as a function of the genomics distance between these spots. So basically, you have your chromosome here, you can measure how frequently two sites interact as a function of the genomic distance L between, between them. And they average between all chromosomes. So they did really a sort of ensemble measurement. Okay. So this, if you want, it's the same as the return probability. So this is for a random walk. If you had a random walk, that would be like one over L to the three over two in 3D. But what they found from the experiments is the, is the following. So this is the curve you get from the experiment. And they found a region, this one between, which is about between one mega, one million base pair and 10 million base pair whose slope is not, let's say, is not 1.5. But it's about one. Okay. So it's something very different. And so then I said, they made the DC hypothesis that maybe it's not, let's say it's not the equilibrium global. This is a very good model. This is a good model for the chromosome. But there's to be something like the fractal global, because if you measure the, so then what they did, they simulated the fractal global, and then they measure the corresponding contact frequency between the simulated model. And they got a slope which is about minus one. I mean, it decreases the contact frequency as one. Providing in some sense the first experimental evidence that chromosome can indeed be reasonably well described by this kind of model. Okay, so that was the first experiment, let's say the first experimental. Sorry, of this kind of model. Yes. So the different color. Yeah, it's a kind of way to show different regions. So you have your polymer fiber. Right. And you start coloring it like rainbow coloring. So you, you start from, so you see, I don't know if you see, so you have your, so from the last. So that's a linear polymer. So that's the first monomer and the last monomer. So then you start like red. I don't know, green. And you'll do this way. And the reason to do that, I mean, it's just nothing so special, but the reason to that is the following. I think it's very evident that regions which are closed in sequence for the factor global are also close in space. So you see, so they, they stay together somehow while that's not the case for the equilibrium global. So okay, maybe here it's not super evident because I think the polymer they simulate is not enormously long. You would see much better result of course for very long for much longer polymer. I mean, it's kind of, it's kind of evident. Okay. And so why they simulated, so how they simulated this is, well, it's a bit technical, but if you are interested, maybe you can ask to me, to me later. They use the, anyway, they use Monte Carlo simulations of a polymer model. But I don't want to go too much into this. I want to talk about different stuff. So the role of topological constraint seems to be very important. As I said for, because in the end it seems to be important seems to all actually well described the, the, the situ, I mean the kind of physics or biology that you have for chromosomes. As I said, that was done for, that's, that's for human chromosomes. Of course, I would expect the same happening, the same for other kind of mammoths. And in fact, the people they measure this, they use the high C for other mammoths and they found very, very similar results. I mean, for these contact frequencies and so on and so forth. And I think, I mean, the next few days, Mario Nicodemi will tell you, I mean, more, more about that. So now, actually the role of topological constraints to describe the, let's say, the physics of a chromosome is important, not only for eukaryotes, like, like us, but it's also important in prokaryotes. Because I mentioned to you this because it's, I think it's a very interesting work. It was done, yeah, 2006. And so this model was introduced to by Jun and Mulder to describe, to describe, let's say, the separation of the two different newly formed chain of DNA after the application. I mean, in E. coli, in Escherichia coli. So, namely, in a bacterium. So, the E. coli, as I said, I mean, maybe I mentioned to you last week, contains only one single chromosome of one million base pair, roughly, made of only of DNA, so it's much simpler than eukaryotes. But it's, I mean, it's a, I mean, it duplicates very fast, so it's a, but it's a, let's say, it's one of the most studied, probably, organisms in biology. And what it happens, I mean, one critical question for it is how, I mean, what are the physical mechanisms that, I mean, that leads to the, that lead to the formation of the two, two new cells. And the way these people model this, so basically what you have is the following. You have one, one single chromosome. So then it starts to duplicate and the application, it happened, it's like, so basically you have one structure of this. I think that is called, you have a spot which is called the replication fork. And here the new, so this is the single filament of DNA. And here you have two new filaments of DNA, this and this one. So it opens like this. So it's applicates in this way. Okay. Very simple. So then a certain point they will stay linked for, for a very short time at some point and then they, they separate. And they go to the two separate parts of the cell. And then the cell can replicate and then that if you, so you have two new, two new cells of obi-coli. So in this paper, these guys have actually simulated the process of the application. So they used, again, they used some Monte Carlo algorithm. So they, they, they had one single polyp. So they really started to simulate the process of the application here. So they doubled, let's say, the content if you want of, of the, of polymer, the polymer content in the, inside the, the simulation. And in that show they came with a very beautiful results because they show that only, so they didn't have here any specific cellular means, I mean active mechanism or like it was just pure polymer. So if you have one polymer, you start with the second polymer, start introducing more and more polymer just to, to simulate the application. And what they find, they found at the very, at the very end is that the two chains separated, let's say, spontaneously. Okay. So that was a very nice result because it showed that basically the separation of the chromosomes follows from the fact that you have repulsion between two close by chains. Repulsion and that's very important. The fact that E. coli is as a, so the, the, the, the, the cells of E. coli as a, is elongated the sort of tube. Okay. It's a cylinder almost. And, and they show that these, the cell, having a cell conformation of this way helps in separating the two chains. Okay. And they show this by, by Monte Carlo simulation and they show that also that, so the, so they had also some mapping to the time. And they show that actually this mechanism is compatible with what you observe experimentally. So this is just to show you another, in this case also there are topological constraints between these chains because these guys actually they never, so they really simulated this process. And if you think a little bit, this, in this, during this process, you also never have the formation of, of not again. So in a sense, the, the role of topological constraints here is a bit similar. So somehow it's similar to what you have also for the, for the cramp. So it's not exactly the same, but it's a amount similar. And it shows to you the importance of this ingredient for the description of the same for the physics of chromosomes inside the nuclei of the cells. And it's okay. This is E. coli. So I'm not an expert of E. coli. So I want to tell you more about that. But it's important that you know that people have analyzed or so. And so they have applied polymers physics also to this kind of to this kind of organism, which are somehow are simpler. Of course, then then then you call it. Okay. Question. So I think this one to mention to you. So these, okay, this was my work, but I think this is maybe I mentioned to you later. What I want to, to, to say to you is the following. So, so far, you, okay, there is a, there is a general question here. So what I told you so far concerned models, where we have neglected the secret, let's say the heterogeneity of the sequence. What I mean is the following DNA chromosomes are not homogeneous polymer, right? I mean, you have the sequence of DNA, you have these stones, which are not, which are not regularly placed inside the chromosome. So it's a very heterogeneous polymer. So it seems a bit strange. I mean that you neglect such kind of ingredient in your modeling. Okay. So the features, which appear general, which, like, for instance, the contact frequency, the, the, the, the slope, the, the, I mean, it's compatible with one as I told you is general. So was found also for, for other mammals. So it seems a common feature, but there are other features were not, I mean, which are not so general. So they are specific of the organism. So in order to model it, you need to introduce models where you have sequence heterogeneity. So you have information of related to the specific sequence of the chromosomes. So now this is a very, so if you do that, I mean, I have to say that the task becomes, I mean, the task of simulating chromosomes becomes immediately very demanding. Okay. Because, okay, as I told you, first of all, chromosomes are very long polymer in general. So if you also introduce the sequence heterogeneity, so then you introduce, I mean, I mean, first of all, you have to establish how you want to introduce it. And to relate it to what kind of problem, which kind of problem you want to address. But introducing it immediately makes the problem much more demanding from the computational point of view. So this kind of technique, so this kind of modeling can be done only for or simple organisms or so far, I mean, at least it was done to the best of my knowledge or to all chromosomes, to model single chromosome, for instance, for human or for drosophila because drosophila melanogaster is like the fruit fly. And there's a genome which is big but not so big. So it's a bit, it's an intermediate genome. So it's genome, the genome of drosophila is I think about, I don't remember, I think it's about 20 million base pair. So it's not as big as the human one, but it's still quite big, I mean, to simulate. So this work, the one that I want to mention to you was done and proposed a few years ago in the group of Christophe Zimmer at the Institut Pasteur in France. So what they simulated here, they proposed them actually a model, a full, let's say a whole chromosome model for East. Saccharomyces cerevisia. Why this? Because East is, as I told you, is in Eucalyptus, but it's a very simple one. So the genome is very small. And so now you can simulate it, I mean, you can simulate all the new nucleus of saccharomyces cerevisia. What they did is the following. So they introduced a model for chromosomes here. So this is their model confined inside the spherical region, which models the nucleus. And they also introduced a specific feature of saccharomyces cerevisia, which is the presence of nucleolus. The nucleolus is a region within the nucleus, which, so the physical conformations of, I mean, the physical formation of chromosomes in saccharomyces cerevisia are the following. So all chromosomes are attached here to a region which is called the spindle pole body, spindle pole body. And here there is a region which occupies a large portion of the nucleus. I think 25% of the nucleus, more or less, this one. And this region is the nucleolus. And it faces, I mean, it's on the opposite side with respect to the spindle pole body. Okay, so now, so this imposes a stronger constraint, I mean, on how, I mean, on the regions, I mean, on the, let's say, on the regions, which can be occupied by chromosomes, not because basically I'm out of these regions. I mean, they do not penetrate. The nucleolus is formed, actually it is formed by, the nucleolus is formed, it's also formed by DNA, by some sort of chromosome. But they belong, so it is formed by only, it's formed by some regions which are at the end of some chromosomes. And so this, if you want, imposes, the nucleolus is this one, is this sort of sausage-like region, this one. And so it's made by some portion of DNA which is more compact, more, let's say, which has a thicker size compared to the rest of the DNA inside the cells. So they impose this constraint inside their model. And so then they did, they simulated this model. They used here, they used molecular dynamics simulation. And what they found actually is quite remarkable because, again, also, apart from these effects which you want are still extruded volume effects, so in the end they are not biologically related. They are just the kind of, the kind of ingredients that you put in a normal, if you want, polymer model. They compared their results to the experiment. So they basically, they mapped their region to real biological region. And they found that the spatial positioning, this is the comparison. So this is predicted and this is measured. They found that the kind of prediction of the model works rather well when compared to what they measure in experiments for some region. So these are some specific regions on the chromosome. This is the spindle pole body. So these maps are called, how do they call them? I think they're called like chromosome charts, I think, something like that. Basically it's a two-dimensional chart, it's like a map. And the different columns are related to the density of the spot that can be detected. So this is the spindle pole body. As you can see, the spindle pole body, of course, moves. This is CEN4, it's a specific region on chromosome, yes, on chromosome 4. These are the telomeres, the ending of chromosomes. And this is ribosomal DNA. The ribosomal DNA is this one. It's the one that is associated to the nucleolus. So you can see that the model, I mean, is simple, because it's only physics-based, is able to reproduce experimental data quite well. And they also compare the contact frequencies between loci, and then measure, and predict it from high-C experiments, because there are also high-C experiments for yeast, and they found that the agreement is quite good. So this is just to give you a flavor of what you can do for modeling. They are like a self-avoiding chain. No, they have no space-filling. No, they are not, because the density is about 10%, 15%, In general, actually, in general, the density of DNA, well, density of chromatin inside the unicorn cell is not high, it's about 10%, something like that, okay. And this is the same here, for, in general, this is a number which is more or less the same for any eukaryotes, the only, actually I have to say, the only ingredients that, so there are two main ingredients here. The, with exception to this part, which is the, the part which is related to the nucleolus, they are the chromosomes, they are different colors, but the chromosomes are always described by the same model, it's a self-avoiding polymer model. Then one end is attached to the spindle pole body, and the other, and the other end is free, with exception of some of these ends which form the nucleolus, okay, which is modeled as a, as a sort of thicker polymer, because this is what people know from, from the biology, okay. And apart from that, then it's, let's say, it's basically, it's basically a polymer phase, in the sense that then you have, you, you, you, you, you, let's say, you, you measure what, because then they, they did it, so they apply the molecular dynamic simulation to this model, and then it's, and then that's it, so in the sense that you don't have any specific interaction, you, you have some specificity in the sequence, but you don't have any specific interaction, just as through the volume, okay. Exactly, in fact, the, in fact the saccharomyces cerevisia, okay, there are some more, there are the possible, many possibilities for that, but it's a sort of an exception. So saccharomyces cerevisia has a slope which is more compatible with 1.5 that, with one, and this is probably related to the fact that chromosomes of saccharomyces cerevisia are very short, they're not very, they're not long, very short in the sense that the average, so the genome is very small, but in the average saccharomyces cerevisia chromosome is about 500,000 base pair compared to the average human chromosome which is 100 million base pair is, is much short, and this is probably a possible reason that, that gives you, why it gives you this slope. I mean people are really discussing this kind of slope a lot, probably tomorrow Mario will tell you also more about it, but you, you always also to be a bit careful in sense that, so this is, this, I mean from the polymer perspective, measuring a slope, it gives you just one single exponent, okay, but give, I mean one single exponent does not identify a model, because you need more exponents possibly to identify a reliable polymer model, because you don't, you don't need, you need, so basically the, the, the, the, the, the, the decrease, I mean the exponent with which the contact frequencies decrease is, is one, but it's not, they are not, and this probably is not related to the exponent do, so you should in principle have a technique which measures also the exponent two in order to be more quantitative on the polymer model, at least to, to, to decide to choose that I want this and not this, but, but unfortunately by high C you cannot measure this exponent, okay, so of course I mean the, the fractal glob is very attractive, because as I told you is, it gives a, it's a model where close portion of the genome are also, so close, which, I mean, portion of the genome which are close in secret are also close in space, and you can really imagine to pull one strand after the other, and that, that, that's possible, so it's, it's, so their topological constraints are much less detrimental than for a model where you have everything mixed together, okay. So more recently, this is a paper by, a work by Daniel Jost, Cedric Vayan in, from the, from the Colonial Marsupial Union, so what they did here, so they sort of introduced the model, so as I told you, so as I was telling before, so if you introduce a sequence of interogeneity, I mean the model becomes quite complex, so what they introduced here was a model to explain some data for Drosophila melagonogaster, so the model is the following, so you have a polymer model, and then it's always a, I mean, polymer, it's a sort of mid and spring polymer, so it's the standard polymer description, but now the, the, the exception is the following, that you have five, if I remember correctly, okay, it's not important, maybe four, but what's important is the following, that you design monomers, so it's a sort of copolymer, so you have four different kind of monomers, okay, so you can imagine to color each one of these monomers along the chain, in order to describe different states of the chroma, now I mentioned to you which this kind of differences are, and so, and then you have a preferential interaction between regions of the same kind, so namely blue and blue region, like each other more than blue and green region, and blue and yellow region and blue and black region, okay, so that was their idea, so this kind of model is not new from the polymer physics point of view, and since that, that was introduced, I mean, many, many years ago, in order to describe the physics of, as I said, the physics of copolymers, and what these models, so if you simulate these kind of monomers, so, you end up, so let's say, that depends a bit on the temperature, okay, but in general, what you see here in this kind of model is what's called microface separation, so namely regions which are, which, I mean, regions which like between themselves, like the blue and blue region, tend to form portion which are segregated with respect to the yellow one, which are also segregated, I mean, between themselves and the rest of the polymer, so this phenomenon is called microface separation because, I mean, the reason for calling it microface is just jargon, but because this space separation happens within the same, within the polymer, okay, you would have macroface separation if, for instance, you have two polymers or a solution of polymers with two or more species of poly, of monomers, just to mention you why this called microface separation, so now, why they introduced this kind of modeling was because they wanted to describe DNA of some chromosomes of the endosomial, which, let's say, people found, I mean, people found basically that they could be described by essentially four epigenetic states, so you know what's epigenetic, so epigenetics is a sort of new frontier of DNA, so, so we, genetic is, no, but you have the sequence of DNA, you think that everything is related to the sequence, which is transmitted to the sounds, blah, blah, blah, so that's, okay, that's true, of course, but there is also something which is called epigenetic, so epigenetic is the kind of, let's say, modifications of DNA, I mean, our DNA, which are not, which cannot be transmitted to the offspring, but which essentially remain within the same order, and this is due to the fact that DNA goes into some transformation during his life, so, for instance, and this is because DNA interacts with some specific proteins during the course of our life, so this epigenetic mechanism actually affects continuously the life of all living beings, but they cannot be transmitted to the sons, simply because you are not going to modify the sequence, you are just going to modify the chemistry of the DNA, and this can be done, for instance, by some process which are called, for instance, methylation, acetylation, so this process, you are going just to modify the chemistry of the single base by modifying, for instance, the interaction with these stones, and of course they cannot be transmitted because you are not changing the sequence, but you are changing how the DNA works, and, of course, these, so, modification by epigenesis is, okay, this, I mean, happens continuously, but it's not always positive because also, for instance, some cancers, for instance, are due to epigenetic modifications of our DNA, and also DNA of mammals or other species in general, okay, so it can be dangerous, they are not like mutations because, again, we are not changing the sequence, they are just changing, if you want, epigenesis, they are just changing the chemistry of DNA, and so now people, I mean, at least for Drosophila, a group of experimentalists has observed that the epigenet, so they associated, basically, two chromosomes in Drosophila, they looked at the chromosome in Drosophila, and they saw that the chromosome can be roughly divided into four epigenetic states, so basically the DNA has some, sorry, they identified some chemist, some chemical state, and they can be, so basically they made some clustering, if you want, and they saw that there are only four relevant states, one they called yellow, active, green, which is associated to the presence of some proteins along the sequence of DNA, which interact with DNA, then blue, which is a polycomb, is a species, is a kind of, is also a protein, which is essential for the function of Drosophila, and then black, which is repressive chromatin, so chromatin is going to repress, I mean, the expression of DNA, so, and the idea of this model is that all these regions, so it's an hypothesis, of course, all these regions prefer to interact between themselves, but not among, I mean, not across themselves, so this is their result, they, okay, they measure different things, they saw, as I said, the microface separation, so, namely, for instance, the black, and in this case, it's orange, okay, the black and the yellow region stay physically, they stay physically separated, they form a, they form a microface, and they were also able to compare this model to experimental results, so, let's see, yes, so, this is the experimental results, and this is their model, of course, they were, so, they had to know, I mean, so, they knew from the biology, where, I mean, how to position their, on their model, I mean, their different monomers, and then, I mean, after they did it, again, they did a simulation based on Monte Carlo, I think, they were able to compare with experimental data, and they found a remarkable good agreement with experimental data, so now they had a model that, at least, I mean, for this specific case, is going to work, to describe very well, let's say, again, it's still a bit the physics of the chromosome, because it's always a polymer model, but with sequence heterogeneity, so, yeah, these diagrams, so, these, this one, you mean, so, this diagram, no, no, so, let me see, so, these diagrams describe, so, the contact frequencies between the different regions, okay, so, the reason why, for instance, they appear like a chess pattern, these, I think, well, now I don't remember what to say, but these appears, I think this is because at the beginning, the conformations are very ordered, this is for the, probably for the initial conformations, then they become disordered, for instance, okay, so, let's start from this, which is similar, so, this is for the random coil, so, random coil is just a random polymer, so, you can see, you have the diagonal is very, is right, means that you have all, all, your interaction are confined to the diagonal, and then that's it, this one, yes, so, this is at the experiment, yes, this is the experiment, yeah, you have clustered, yes, so, this is specific feature of contact frequencies, in general, and this is for drosophila, and this is the experiment, so, what people, I'm not sure if this is the experiment, I think it's possible that this is the experiment, this is the simulation, but, okay, the message is that they look similar, and, so, okay, let's go maybe first to experiment, so, this is what people are observing experiments, so, you have, okay, your main diagonal, this is, these are the way of observing many diagonal because things which are closed in space, which also interact much more, and then you have, like, sort of squares, and these squares are region where you have preferential interaction between, let's say, chromosome fragments, and, of course, these things are not trivial because then, because this thing is reproducible, so, that means that it's really, is really telling you something about the biology of the chromosome, so, then, they use this model in order to see if they could reproduce this result, experimental result, and they, I mean, they could, they did it remarkably, remarkably well, so, this, here, is instead because, okay, in their model, actually, you have a sort of, so, you have many possible states, that depends a bit on the temperature, so, you have the coil, then you have multistability, namely, you have things which are still, so, you see, these things are still very, so, they start to, basically, to collapse because they have the same color, but they are not, they are not really forming anything so compact, instead, here, you have something which resembles, I mean, which is much more compact than this thing, so, and, so, what they observe there is that you have the participation of all these possible states, then, according to the temperatures that you are investigating, you have some states which are, which have higher probability than the others or not, but, in general, all these states participate to, to, let's say, to, to, to, to, to the face diagram of your, of your model, okay? My question, you want to do a small break? Well, okay, we do have two, two, three, three, four minutes break here. Okay, so, the models that I have, I have, I mean, I have presented so far, I mean, can be, I mean, broadly speaking, can be categorized as a sort of direct modeling, so, what I mean is the following, you have your polymer models with some kind of hypothesis, I mean, on, about your model, for instance, let's say, the, the one for yeast with the presence of a, a sausage-like region, and then you use, you, I mean, you implement this model, I mean, in some sort of simulation routine, and then you get the results. So, this is kind of what I would call direct modeling. I mean, you have your polymer model with some hypothesis, and you get the results, and then you see, you compare to experimental data. Maybe if it compares good, then you have done good from the very beginning, otherwise, you have to refine your model somehow. Otherwise, there is another approach which has become very popular recently, especially in connection with this contact map. It is the following. You use, I mean, a bit similar, I don't know if you're familiar with that, but a bit similarly with the idea in proteins, people have used NMR, especially NMR data for proteins to reconstruct the 3D conformation of, of proteins. So, they use the NMR data as a spatial constraint, and then they use the, they use it, they use this data to reconstruct how the protein looks like in space. So, this is a similar idea that you have here, I mean, that you have also for chromosomes. You use the contact between, that you can measure by high C in chromosomes, and you do an inverse modeling, namely use these as constraints, and you reconstruct the chromosome conformations inside the cells by using this data as inputs. Of course, I mean, the two, so this is not, I mean, the two approaches are not unrelated in the sense that you can use some, I mean, a specific polymer model with spatial constraints in order to reconstruct the chromosomes. But there are also other approaches which are a bit different. So, that's why I was mentioning to you these. So, the first, the first attempt, I mean, in this respect was done in 2002, where, when people proposed, let's say, the first, the very first, first version of high C at the time was called chromosome conformational capture 3C. So, what people did there, they used the, the spatial constraint, sorry, they used the data that, from 3C has a spatial constraint for a, for a, for a model, which was not a polymer model, actually. It was a, a sort of, so it was something very simple in the end. So, you have a different, so this is the matrix of constraints. This matrix appears like, so it tells you which is close to which for a system of coordinates, you, let's say, you divided this by squares, in squares, and then you associate, so, and then you derive, so you say that D, for instance, one is close to, I don't know, three. And then you use what is called distance geometry, which is a sort of a routine to reconstruct objects in space, starting from some constraints. And then you get this shape. This thing was very simple, because in the end, here, you don't have too many constraints, because this was done for East. So, again, it was very short. So, what they, so, in the end, they were able to reconstruct the shape for this specific chromosome, which I think was a chromosome 3. Yeah, chromosome 3 for East, which appear like this. So, in particular, they found that these, the two telomeres, so this part and this part here, were found relatively close in space, which is also something which is known from the biology. So, in this sense, they were able to find back results which was known from biology. So, now, with the new high-see technique, things become more complicated. And so, one has to reconstruct, so, first of all, one has more data, but having more data means that you need to satisfy more constraints. So, the thing is, so, computationally, this task becomes very demanding. So, this paper was published a few years ago. It was about the complete reconstruction by using high-see data of East genome. And, I mean, the reconstruction, they get, so, because of the, I mean, from this reconstruction, they found many results that were already known in the literature, but proving that reconstruction can be an efficient tool to get back chromosome, I mean, to get back the shape of chromosome and the shape of the genome by using high-see data. So, in this case, they used, I think they used a sort of polymer model. So, which I mean, what I mean is the fact that you have all your chromosomes, and each chromosome is a polymer, and then you implement the constraints there. But that's not the only approach. There is other kind of approaches, for instance, this one, where they used a tool, which is not based on polymer. So, namely, you have just beads, let's say, in space, and then you implement the constraints on these beads, but it's always by using high-see measurement. But these beads are not, let's say, even if they are the same chromosomes, they are not linearly connected. So, this, to me, looks quite a strong assumption, especially because then in the end, you want to, I mean, you want to retrieve that monomers which belong to the same chromosome, they have to stay close in sequence. So, apparently, I mean, if you read this work, they were able to find that, I mean, at the end, but they did not put that as an input. So, basically, they did not use polymers. They just used disconnected beads. And then they implemented the routine in order to reconstruct some specific locus on the human chromosome. So, why they used these, so these things which, I mean, maybe you can look a bit strange, you know, why don't you use a polymer model and then implement in a polymer model spatial constraints. And the reason is why, actually, they used that tool that was, that was basically developed for proteins in order to reconstruct the 3D shape of proteins by using NMR data. So, it was kind of, let's say, improvement of this numerical tool applied to chromosomes. In that case, there were no, let's say, there were no implementation of the fact that different monomers belong to the same chain, so they have to stay linearly connected. And that's the reason why they did not use that. So, probably, I have an impression that this kind of tool can be used only for very short fragment on the, which, where they implemented it on the chromosomes. But for longer, I mean, if you have, let's say, genome-wide measurement and you wanted to satisfy those constraints, I think you need to use a model where monomers, which, I mean, which belong to the same chain, don't have to stay linearly connected. Otherwise, I think you never, I mean, it would be very difficult to, I mean, I think I have an impression to satisfy these constraints. Sorry? You can use, sorry, what? Yeah, maybe. No, but that was already done a few, few years ago. Yeah, sure. Actually, there, yeah, I didn't put that here, but there was a paper published, I think, last year. No, okay, they did not use it for chromosome reconstruction. They used machine learning to apply to high C data to sort of derive a sort of effective force field. A force field between monomers in a polymer model that, that become then compatible with the data in order to simulate chromosomes. And they found that, I don't remember, they found that, so you have a sort of, you have, I don't know, 10, I'm saying some number, but I think it's about like 10 monomers, I mean, 10 different kind of monomers with different interaction potential. So it's a bit similar to the kind of model I mentioned to you before, this copolymer model. But that was derived directly on the data by using, on the high C data and they used machine learning to learn this force field. Okay, but it's a different story. So they did not use for chromosome reconstruction. Actually it's a bit more kind of a polymer model. Yes, and this is, so yeah, this is a similar work. Actually they used a very similar tool where they reconstructed, but at much, using much more coarse grain, so their monomer, one single, I mean, one single monomer, it's, I mean, it's much larger because they wanted to try the construction of human chromosomes here. So you need to do, to be more coarse grain and it was, that will never work. And already like this is very demanding. And so this is just to give you an idea what people can do, can do now. So to conclude, I want to just to mention to you very, very last thing, I mean, how also polymer dynamics and namely the Rouse model as, I mean, can be relevantly used, used to model inside chromosome dynamics, which so far we have not discussed. So this is all about, let's say, chromosome conformation. So in space, why, but chromosome dynamics is important as well. So this is, okay, this is a bit of, no, blah, blah, blah. So you remember, okay, but the only thing important, you remember that, so the Rouse model gives you this behavior, which is sub diffusive. This is in general for any polymer model, even if you apply the Rouse model, but to a self avoiding polymer, you still get always sub diffusion. So this is because of how polymer relaxes from small scales to larger scale. So now there are many, many experiments on polymer dynamics. I just want to highlight this one was done on Easter. It's a very recent one. It was quite remarkable because they can follow chromo, let's say, so what they did, they tagged a specific, okay, but different loci, sorry, on in saccharomyces cerebitia for different chromosomes. So these are the loci they tagged. And then they follow the dynamics of this tag in time. And then so they were able, so you can really see them with a camera. So they were able to reconstruct the XYZ coordinates. Okay, so from the coordinates in time, they measure the mean square displacement of these loci in time. So you can really measure the dynamics. And this is their results, this one. And, okay, apart from the noise, I mean, of course, you have always had some noise. They found that the dynamics appears, I mean, remarkably were described on average by the Rauss model. So this is precisely the same quantity that I mentioned to you a few days ago. So the mean square displacement of a single monomer, and they found that the exponent of the mean square displacement is about 1.5, which is very close to the expectation of the Rauss model. So you don't, that means also that probably you don't need something too sophisticated to describe the dynamics of the system in the sense. For human things, it could be probably more complicated. There are different, I mean, there are other results about the time to mention this. But it's very interesting that by using a simple model, you can have, I mean, you can reproduce the data. In the end, the Rauss model is very simple. I also want to let you know, I mean, maybe notice also that the curves say, I mean, for different, for different loci seems to stay on top of each other. So that means that there is some kind of, I mean, you can use just one single model to describe all these data. So it's very, yes. This one, this is the mean square displacement, so it's over time. And this is, it goes like t to the 0, t to the 1.5. Okay, so it's sub-diffusive. Well, all the time up to where they were able to measure it. It means up to about 500, 5,000, sorry, yeah, 500 seconds, which and scales here up to 0, 4 micrometer square, which is 0, 6 micrometer, which is quite long. I mean, which is, I mean, so in, as a scale is not, it's quite remarkable because the total, so the, let's say, the total and the linear size of a nucleus of this is about 2 microns, so the diameter, 2 micrometer, okay. So, I mean, the average size of a nuclear, the nucleus of this. So, we said that these things is following dynamics up to scales of 0, 6, about micrometers, which means that, well, it's not the average, I mean, it's, yeah, it's 1, 4 from this, no? So it's quite long. So then you ask what, why they didn't follow, they didn't went further. Well, the problem that we, this kind of experiments is that there is a phenomenon which happens at late times, which is called photo bleaching. Photo bleaching is a kind of, so basically you start to see everything white, I mean, everything appears white, while this way called photo bleaching. And that's a kind of problem which is perfectly known in this kind of technique. And some, as sooner or later that appears and then you cannot visualize things anymore. So then that's why they stopped here. Probably, I mean, with improvement of technique, they can go farther. Of course, I mean, if, but that's quite, actually, it's quite remarkable as a measure, especially because they were able to, yeah, for almost for order of management, a bit more than for order of management, which is quite an achievement. Okay. And then I would just to, well, okay, you know, I want to just to conclude to tell you which, because probably it would be a bit useful for what Mario wants to tell you tomorrow. So which kind of tools do you use for describing chromosome? Well, basically, since these objects are quite complicated, you almost, you have always to resort to numerical simulations. Okay. So there are, so there are two kinds of molecular simulations, yeah, numerical methods which are Monte Carlo or molecular dynamics or both. I mean, you can do even both. So depending what you want to do, I mean, you can use one or the other. Which, I mean, who of you are familiar with Monte Carlo or molecular dynamics? All. Okay, good. So I want to tell you, so then I don't go into, because I explained this to biologists, to what Monte Carlo is. So of course, I mean, but you are not familiar with Monte Carlo for polymers. No, okay. So Monte Carlo, so just a simple example. So Monte Carlo for polymer, I mean, of course it depends on the problem again, but to give, just to give you an idea and to explain things in an easier way. So I just think, I'm just using a polymer on the lattice, because in the end if you have enough, I mean, polymers are long enough, it's not important if you're on the lattice or off lattice. And you can always move from lattice to off lattice. So the most, I mean, the simplest example are of course the random walk and the self avoiding walk, because they are also the two simplest polymer models. So how you do, okay, of course for a random walk you don't really need simulation, because you can do everything exactly, yeah, let's do it nonetheless. So how you do random walk on a lattice, this is a 2D task, it's very simple, no? You start from, I mean, a node of the lattice which you call the origin, and you move randomly in one of the possible directions. Then you move on the other, the other, the other, the other. So you use, for doing that you use some random number generator, of course. And that's it. What's the main feature of the random walk is that even if you fall on a site which is occupied, you don't care, I mean, it's okay, because it's a random walk. So this is called simple sampling, because you are generating all conformations independent from one another. It's a very simple algorithm, I mean, it's really two lines of algorithm. And for another walk is okay, no? Because it's very simple to write, and that's fine. For a self-avoiding walk, of course, you can still use the same kind of algorithm, which is, again, it's called simple sampling. And you can do the following. So you start from, again, from the origin, then you move this way, this way, this way, this way. But then it happened that if you do this thing here, so if it happens that your next move, your next step, let's say, fall onto a site which is already occupied, then what happens? I mean, in order to generate the right statistics, is that you have to reject completely this walk, all the walk, not only the last move, all the walk you have to reject, in order to generate the statistics correctly. Okay? So this is a plot, just to show you that things work well. This is a plot for the end-to-end square distance of a self-avoiding walk, of a random walk. Sorry, a random walk and a self-avoiding walk. Random walk is in red, self-avoiding walk is in green. And then you can see that the end-to-end square distance grows as it's expected to be. Actually, the random walk, the self-avoiding walk is a bit down. I probably, because there are no, I didn't use too much statistics. And the problem is that I used simple sampling for generating self-avoiding walks. Now, simple sampling becomes rapidly inefficient to generate self-avoiding walks in this way. I mean, already for a self-avoiding walk of about 30 steps, basically you never, I mean, almost never generate new walks when you do your simulation. This is because you have a lot of crossing in your walk. So that means that the simple sampling is not a very efficient tool to generate self-avoiding walk. Fortunately, there exists, I didn't mention that to you. Ah, yes. So fortunately, there exists much more efficient method. Sorry, I didn't have the time to talk about that. But if you are interested, I really suggest to you to look at, to this review by Sokal, Alan Sokal, which is called Monte Carlo Methods for Self-Avoiding Walk. And this is only for data, I think, it's almost into, I mean, every data is for 2D. You can use the same algorithm for 3D or other the walk. It's a very good review. It explains very rigorously all the alternative methods that were devised to produce in an efficient way in self-avoiding walk. And I think it's very important because basically self-avoiding walk, it's the basis for any realistic polymer simulation. Yes. Yeah, but then you need to use a different, I mean, there is an algorithm which is called PERM, Pruden-Rich-Rosenblut Method. What's that? And this method actually works precisely a bit in this way. So any time you fall into a site which is already occupied, you just reject the last one and then you select another one until you go into the direction, I mean, where there is no occupation. But in order to use this method, you need to re-weight your walk. And then you need to use different weights for it. This method was proposed by Rosenblut and Rosenblut. It's a paper published on Journal of Chemical Physics. I think it was in the 50s. So it's a, oh, maybe later, no, probably it's 50. And there was a sort of method to use, to not discard completely all walk. But then you need to introduce some weights and then you have to re-weight your walk at the very end in order to get the right statistics because otherwise you don't get the right statistics. You can try, you mean, but you don't really get the right statistics. Otherwise you need to resort to method which, two methods that use real, I mean, real Monte Carlo, you devise some moves which can be more or less efficient in order to simulate the self-awareness walk. And that, that review is really brilliant. I mean, there is a lot of information. And also for people who are more mathematically skilled, it proves a lot of theorem saying that this, I don't know, this, this scheme is ergodic. So it proves the ergodicity of some moves. So it's, so if you, if you like this kind of things, I think it's a nice, it's a nice reading. So otherwise, there is another possibility which is molecular dynamics, which is not, which is not like more efficient or less efficient, depends on which kind of problem we want to address. In general, molecular dynamics is useful because you have access somehow also to the dynamics, of course, no, of the problem. Why Monte Carlo not real? I mean, you can also do some dynamics with Monte Carlo, but it's much less efficient and less obvious. Well, with molecular dynamics, you can do it immediately. But in many, many cases it is slower to achieve equilibrium with molecular dynamics. Not the less, I mean, it's a, it's a very used technique. And molecular dynamics is very simple. From my point of view, it's much simpler than Monte Carlo in the sense that in the end, it's just the numerical integration of the Newton's equation. So you have your system of particles and particles. You have the acceleration, and then times the mass, then you equate the acceleration times mass to the forces. And the forces, of course, can be very complicated, but are always forces. And then you need to do the integration numerical. Of course, I mean, there is a bit of, let's say, tricky tricky parts because, of course, numerical integration can be, I mean, it's not always trivial. There are different techniques that were devised. This technique is called so basically the idea is that you do Taylor expansion on the right where there are the acceleration. And then you use, let's say, the forward time and reverse the back time. You sum those together and then you arrive to see such kind of equations. So this is a bit of the speed. In general, it can be a bit more complicated, but not much more complicated than that. This is the verlet algorithm. There are other algorithms which are not more efficient, but which are more, let's say, which are in principle more rigorous, if you want. I'm not going to mention that. This is just to tell you that, in general, this is not so complicated. And then if you have your forces, in general, forces are derived from the harmonic potential, then you need to do derivation of the potential for the forces. For instance, if you want to simulate a Gaussian polymer, what you usually do, you write the equation for the, let's say, for the harmonic potential between the beads. So then what you need to implement in the molecular analysis, not the potential, but it's the forces. So you need to do the derivative of the harmonic potential there. And then that's it. So just to give you some, I mean, how molecular dynamics would work. So this is actually simulation of the paper by Christophe Zimmer, the one that I mentioned to you before. So this is a simulation using molecular dynamics of, ok, it's very short, but a little bit longer, but of the east genome. So the sausage-like part, this one, is the part close to the nucleus, and the other part is chromosome. So this is a very, I mean, a small selection of the trajectory, because otherwise, I mean, the moving becomes quite heavy. But as you can see, I mean, these things, I mean, moves somehow. So this is for two different angles, I mean, of vision, let's say. What? To do this, because you have access to dynamics. Ah, well, it depends when you stop. But it depends. In general, ok, this is a, this is not a matter which is coarse-graining of your model. But in general, well, I would say with current models they go, sometimes scale goes from a millisecond to tens of seconds in real time. So you're simulating real, something like that. But it depends on the resolution of your model, of course. If you are, for instance, if you are not, so this is eukaryotes. But if you want to simulate, for instance, the single chromosome of E. coli, there are people that are doing that, using, for example, E. coli, then you go to smaller scales because you use a finer model. Ok, because then you can really simulate the single DNA base pair. Because it's not, I mean, it's long but not super long. It's one million base pair. It's not impossible to simulate. While for let's say for eukaryotes even for the simple yeast it's already quite complicated. And for, I mean, for higher eukaryotes it's impossible. So this is just to give you a flavor. This is my simulation, actually. This is to simulate the human chromosome. One single human chromosome. And so one single human chromosome, as I told you, this is 100 megabase pair. So I need to do some coarse graining in order to simulate it. So which means that I have thousands of monomers actually I have about 30,000 monomers per one single chromosome. And each monomer is about, so counts about 3,000 base pair. So it's quite coarse grained. But on larger scale I mean, you imagine that coarse graining is not so it means that you can describe only scales from 3,000 base pair up to your scales of your chromosome. But coarse graining in some sense is you hope then you have to, of course, to compare to the experimental result, but you hope that large scale are not affected on the level of description that you have a smaller scale. This is a bit the principle of coarse graining. So coarse graining is basically it's like integration of the smaller degrees of freedom assuming that if you integrate out the smaller degree of freedom you have to just do some normalization but you don't do the physics at large scales. That's a bit the principle. But you have to do that. I mean if you want to describe larger structure it's I mean otherwise it's impossible. So I think it's about time. So with this I've finished. So I don't know if you have questions. So I send the slides to Erika to the secretariat. I don't know if you have questions. She has it. So I don't know if you have questions otherwise