 Ok, primer vse, da vši malo zelo? Ovo je nekaj zelo, da je, da je, da je... ...zaprej vse? Ok, vse je zelo. Vse da tamo zelo, da smo počeli, da, ...zato, da je pravda, ...pozirušnjenja o tezno teori, ...po polymer, ...po polymer, ...po polymer, ...po polymer, ...en vse ino za, ...ko se dopreluje vse, ...en polymer. Tko vsem, da je neko... ...bej pa vse, da sem... Tako, imamo zelo svoje energije, kaj je odstavljeno z dva kontribucija, jedno z energijem, ko je odstavljeno z interakciju tubovi, minus zentropi, ok. In smo zelo, da energija kontribucija je, da je odstavljena z interakciju tubovi, kako je terga, da je entropija, je pripravila je, je bilo, je bilo, je bilo, gašan, logratem do n-20 gašan, n-20 function do gašan polimer. Kajte si to površati, minimizujte, in se si, da je, da je, in do no. We found that no is equal to 3d plus 2, and we, which means that the equal 1 is no equal 1, d equal 2, no equal 3 over 4, d equal 3, no equal 3 over, 3 over 5, and d equal 4, no equal 1 alpha, and for d larger than 4, it's expected to be no equal 1 alpha. So one can say that for d larger or equal to 4, no has to be 1 alpha, OK? And so we discussed this a little bit, and we said, we noticed that the prediction is remarkably, in spite of the simplicity, is remarkably accurate because it's exact in d equal 1, where the exponents are not trivial, let's say, is, I told you a little bit to trust me on this, this is exact in d equal 2, and is exact for the larger or equal to 4. Well, for d equal 3, it's not exact, but it's a good approximation because the exact value is about 0, 59 instead of 0, 6, which is this number here. So one now, OK, first two things. We noticed that the critical dimension here is d equal 4, namely for d larger or equal to 4, the conformation of the polymer is Gaussian again, namely the extruded volume effect do not count too much, do not count, I mean they are not relevant, and in order to see that in a very simple way, one can just look back to this expression, to the two body interactions, right? And in order to see why these are irrelevant, one has to do in the following way, one has to insert here, in this formula, the expression for the unperturbed radius, OK? So, namely, I insert this here, and that goes like kB t, d0 n square divided by n to the d over 2, OK, b to the d, which is v0 b over d, n 2 minus d over 2, or v0 bd, n if 2 is minus d minus 4 divided by 2. So what you notice is the following, that if d is smaller than 4, this contribution goes to infinity, for n goes to infinity, for n going to infinity, right? So that means that two body contributions are important and are relevant, so they cannot be neglected, which is the kind of information contained here. While for d larger than 4, this thing goes to 0, which means that two body interaction can be neglected, OK? So very simple. For d equal to 4, you can imagine there are logarithmic correction as a tiny crossover, let's say. So one can say, OK, but here, in this expression, I mean, we, so one can say, OK, two body interactions are not important in d equal to 4, but what if I consider 3, 4, 5 body interactions? Because this, I have been neglected here, I just counted two body interaction. So those kind of contribution count for d larger than 4, and one can do, one can actually can do the same exercise. So approximately three body interaction, so if I write, yeah, I write it like 3, for instance, here, KVT. So that will be like V3, there will be some coefficient, right? Question? V3, and then there could be, I mean, the contribution to the three body interaction can be calculated exactly the same manner. So it will be like nRd to the power 3, so it's like rho cube, the density to the cube, times Rd. So it can be calculated exactly the same way. And this is V3 n cube divided by R2d. So now to, yeah. So sorry? Ah, just to distinguish by this, I say it's like three body interaction, if you want this is E2. Just a notation, just to distinguish, okay. Yes. Ah, yeah, more or less, yeah. This is basically, these coefficients are the virial, they fall from the virial expansion. Okay, so this is the second virial coefficient, which, okay, should be like V2, if you want. That's the third virial coefficient so on and so forth, proportional to. So if I insert here, now again R, let's say the ideal behavior, what I get is like something like that. V3 n3 R to the d, sorry, n to the d, right? And then this is equal to V3 n to the minus d minus 3. So as you can see, actually these contributions are even less relevant because they become relevant only below d equal, sorry, they become relevant only for d smaller than 3, okay. But they are still, so for d equal 4, actually they do not contribute at all, they contribute even less than those. And so on and so forth. So adding new terms to the virial expansion does not add anything. So if you want to have, sorry. Yeah, exactly, you can neglect it. It's just for a polymer in a good solvent, you can take only the two body interaction, that's it. Okay. If you want, it's the only fixed point. If you do some kind of renormalization group, it's the only fixed point. For d less than 4, for d larger than 4, it's the Gaussian fixed point. So everything is ideal. Of course, I mean, for this particular condition, what I mean is that you have this kind of behavior for a self-abiding polymer in, I mean, we were only to body interaction count. So now you can also play a little bit. So now that we have verified that three body terms do not count too much, actually you can play a bit. So with this term here. So in general, you can a bit generalize the, the flority, or in so, because it's, it becomes very accurate in can be even for other conditions. So it does not, it's not only accurate for the particular case of, let's say, of the self-abiding polymer. So for a polymer in good solvent condition, but this becomes accurate also when you have, okay, this will be probably more clear later, but it becomes also accurate when you have, when practically there is a partial screening of two body interaction. What I mean is the following. When you have a polymer in a, so I mean, this kind of model, I mean, this kind of, let's say, theoretical framework that I told you now, it holds only for a single polymer chain in a solvent, but the situation can be more complicated. For instance, in the case of, let's say, chromosomes, which is, let's say, the next topic of these lectures, you don't have only one single chain. We have more chains, I mean, more, more chromosomes, so it's a, so it's a many chains system, which means that the situation that describes the physics of chromosomes is not the one of a single polymer chain, but it's one of multiple polymer chains that are all together in some, let's say, restricted space, and they interact together, so it's more like the situation of a so-called polymer solution. Okay. So by polymer solution, I mean the following. And suppose that you start from, if you have a container of some volume, and you start adding polymers, and you start from very dilute condition. By very dilute condition, I mean the following, that you have your polymer chains here, polymer, no, let's change here, and these polymer chains, so the typical size of your polymer chains is, that we know, this is b to n nu, right? And what it happened, that suppose that you take an enormous volume in such a way that, let's say, that the chain are, I mean, the typical size of this chain is much smaller than the volume, which is, let's say, the typical volume available for one single chain. Then each chain belongs, I mean, behaves like isolated, no? So the physics that describe this single chain is like the one for a single polymer chain. But suppose that you start to compact the system, so you make it smaller, I mean, the volume, I mean, you start to make it smaller, smaller, smaller. Then what it happened, that these polymers becomes, I mean, they start to become to overlap, actually, okay, like this. And no, no, no, it's, oh, maybe I did, I did a smaller, I should do a smaller volume. So I suppose that you take a smaller volume, you start to compress the system. Okay, okay, let's take a smaller. Then what it happens, no, in the chains are always there, then they start to get packed, right, in like this. And what it happens, eventually, they do not longer behave like single polymer chain, because they start to interact with each other, right? And then what it happened, that because of the kind of interaction between each other, and because in the end the polymers are soft object, they start to interpenetrate, okay? And so by interpenetration, eventually, it means that if, so I move from here, so this is called a dilute solution, dilute solution. Then when they start to interpenetrate, it's called a semi, well, yeah, it depends on the language, but in general it's called a semi dilute solution. And then eventually I go to maximum compaction, if you want, even smaller, when it, let's say, in the volume span by each chain, there are other chains. So everything is interpenetrating, everything is compact. And that's called concentrated solution. It depends a bit by the language, but concentrated solution. Or if you want something, you can call it a melt. So in the extreme case where all the solvent has been removed by the solution, and you have only polymers, you have a melt. So namely the only molecules that you have inside your system are molecules coming from the polymers, okay? So these particular conditions, oh, actually these conditions, and these conditions, is the one that is more, I mean, this one that you can, you have to use, and you will see it later, to describe, to start modeling chromosomes inside the nuclei of the cell. This is the kind of condition, because you have many chains inside the restricted amount of volume, which is the nucleus. And in order to model it, you have to use the physics of semi-dilute solution. And what has to do with fluorine? I mean, because fluorine can tell you something about how this chain would behave. And that's the following. So suppose that, so we use a very extreme situation. We use the situation where we are in a melt, yes, semi-dilute. Yes, we will see it a bit later. I mean, why it's this, I mean, just to start telling, I mean, just to anticipate that briefly now. I mean, why these conditions are interesting? Because they are described, I mean, you can imagine that a bit, but they are described by different, I mean, you will find different physics than one of dilute solution. It's more complicated. You can expect that, because you will have a lot of interaction because of the presence of many chains here, right? And actually, the condition where, the concentrated solution condition is a bit simpler to describe the semi-dilute one. In spite of the fact that you have, I mean, you can expect that you have much more interactions here. And the reason is the following. And that was the first realization of this thing was done by, actually by Flori again. And the fact that in concentrated solution, so when you have only the polymers, almost only polymers, you have a partial, if not complete screening of tuboid interaction within the single chain. And so what I mean is the following. It's like tuboid interactions. Count much less than the situation when the single chain is isolated. And this looks like a bit of a surprising result. But the explanation actually is quite intuitive. And again, it's due to Flori. And the reason why it's like this is the following. So in a concentrated solution, or let's say briefly in a melt, the concentration, so suppose that you take, so let's say in general, the concentration of your monomer, because everything is packed, so you have a real compact system. The overall concentration of your monomer, c, I call it c over r, is roughly constant. See, r is the position of, it's some position in your system. The concentration of your system is roughly constant. But you remember, the interaction of, to say, the tuboid interaction, is proportional to c r square, to the concentration square. But as I told you, this is roughly constant. So this is actually a constant square. So this is for tuboid interaction. So that means that the force between two monomers, which is just, let's say proportional to the derivative of u with respect to r, just if you want the gradient, is just zero, is negligible. Okay, so it's a bit of an intuitive argument. So that means that tuboid interactions do not count in a melt. Because of, for one single reason, because the volume occupied by one single chain is actually penetrated by the presence of the other chains. And the other chains remove the effect of the tuboid interaction within one single chain. So it's like they are effectively screened. This is a genuine, many-body interaction. So because of it, so that requires a bit of a proof, but I mean, you can somehow, you have to believe me now. So the flow return here does not go, so I call it e2, just to say that it's tuboid interaction, is no longer of this kind, but is v2 over, so it has a screening, so it has a one over n term, times n squared over times rd. And so this is effectively like equal to v2 n over rd. So that's, so, okay, this is for single chains, while this is for a melt. So as you can see, contribution to tuboid interaction in a melt is much more depressed, because it has a one over n term. And that comes from the fact that you have, basically a lot of interaction are suppressed because of the presence of the other chains inside the volume of one single chain. That actually, it's not immediate to derive, it comes, well, it comes from, so basically you have to study the system as a, it's called, as a mixture of different polymers, okay. And then, so it's like, so actually you start from a system in like of a and b particles, where things can, so you study the entropy of mixing of the system, and you, but you treat the system, so in the end you treat the system as, like any part is equal to the rest. And then that gives you a second virial coefficient, which is one over n, the single virial coefficient of one single monomer, let's say. And it's not immediate to derive, it's a, we can, maybe I can show it tomorrow. So for time being, let's take it this way. I wanted just to show you, actually I wanted more to point out the fact that you can still use the fluorite or even for, for this polymer solution that somehow they apply, they are more, this is the correct description, if you want to use it for chromosome. And, but it's not, no, it's not obvious, it's not like something, something trivial. And actually that, so because of it, actually, well, because of it, actually you can see immediately one thing. So if I put, so suppose that I, again, I put the impersturbed radius here, so I just do the same exercises before. So I have that r goes like n, so it's b, n over one alpha. So what, sorry, what I have here, I substitute this thing here, and I get that e2 over kbt goes like v2, n, b to the d, again, n d over 2, and that goes like v2 over bd, n one minus d over 2. Or n minus d minus 2 over 2. And that means, so it's set consistent, which is to some extent, that means that for d equal 3 now, this term is no longer important. So that means that, and that was a consequence of the results of Flory, that in d equal 3 now, and in a melt, polymer chains are ideal. So that means d equal 3 melt, r goes like b, n one over 2. And that's actually, which is the correct result. So in a melt, linear chains are ideal. OK, not a completely ideal, because there are, but they, the sides behaves like a Gaussian chain. So you have a different exponent than the, compared to the situation where the single chain is isolated. And that's the consequence of the fact that you have the screening of a student volume effect. So it's a, this is quite important. I would suggest that you, it's an important result in polymer phase, so you should, I mean, depends what you want to do in the future, but it's an important result to take into account. This thing, so what I did is, so this is, so maybe tomorrow I can provide you, I don't know, maybe tomorrow, but let's say, for timing, just believe me that this means that this, this, sorry, the two-body interaction are screened because of the presence of the many chains, OK? You know, this thing, right? So why it's v2 over n, and why v2 to square root of n, this is not obvious, this, I mean, requires a bit of more formalism, but, OK. So then, that means that the second, I mean, let's say, the contribution to the energy is v2n to r over d. So then I can repeat, because I want to know what is the, which is the critical dimension for this, let's say, new flow theory, OK? So I substitute the unperturbed radius here, I mean, as the same way I did before, and then I get this formula, and which means that for d equal, so this, this two-body interaction count whenever d is smaller than 2, yeah, is smaller than 2, for d larger than 2, in particular for d equal 3, the two-body interaction do not count for polymer in a melt, OK, which means that the chain, a single, a linear chain in a melt, so when you have many other chains within which interact with it, is a Gaussian polymer, OK. So as two-body interactions are, because two-body interactions are screened, and so the behavior of a single chain in dilute condition or in a melt is completely different, and that is something one has to take into account when, for instance, describes, I mean, one will see for the physics of chromosome, is a relevant condition, because we have many chains packed together basically. So that's why, and I wanted to tell it this thing now, because in a sense it's a good, it's timely because you can use, you can still use the flow theory. So you can see the flow theory is very powerful, actually predicts, even in spite of its simplicity, predicts a lot of things. Any question? OK, so I want to stay a bit more on the flow theory. So we go, so now, OK, for this, we go back to the case, because I mean, I want to also stress about criticisms on the, I want to tell you about criticism for the theory. I mean, where are, I mean, which, what are is the weaknesses of the theory. So we remain on single chains, because that's a good example, actually, where things do not work, I mean, which things do not work well. And that's the following. So we said that this case nu is equal to 3 over 2, right? So first of all, why the theory, which is important to stress, why the theory seems to work so well, because in the end we have neglected a lot. First of all, we have for sure made a very bad answer about the entropy, because as you can imagine, the entropy of a self-avoiding polymer is much less than the entropy of just a Gaussian polymer. No, you have actually, for a Gaussian polymer, you have much more configuration that you can achieve compared to a self-avoiding polymer, because basically the conformation of a self-avoiding polymer are done in this way. You take all the conform, possible conformation of a Gaussian polymer and you remove conformations, which overlap. But as you can imagine, the conformations which are overlapping are much more than the ones which are not overlapping. So this one, it's a serious hypothesis. So why things work well, and the explanation you typically find in the textbook is the following, that for possible reasons, you have either overestimated the entropy, which is what I'm saying now, but you have also overestimated the energy term. So you have some sort of two kind of overestimation that compensate each other by some magic somehow, by some series of fortunate events, and then you get an exponent, which is almost right. But nonetheless, the things do not work perfectly, because one assumption of the Florida theory is the fact that the end-to-end distribution function, Pr, is basically Gaussian, because this is the basis of the theory. So it's R squared and B squared. Yes, it's basically a Gaussian. That was basically the foundation of the theory. Actually, this thing is wrong. I mean, it's really wrong. So what I mean by wrong, I mean that suppose that you are simulating a self-avojding walk, and you can measure the end-to-end distribution function, you will never find this formula. So you can spot that the Gaussian approximation is completely wrong. And actually, so yes, your question. No, it's wrong also on a short scale. Yes, and now I was, why it's wrong? I mean, what is the correct expression? The question? Yeah, just interrupting. No? Yes, for a self-avojding poem. So this expression is wrong. It's the one that I've used, of course, Floria's used. Of course, it's not me that invented this thing, but it's wrong. And the correct one is, which I have to remember now by heart, so it's something p over r. Capital thing I just know. So let's say the correct one, p over r, is something that has to go like r to the theta times the exponential of minus k, so it is a bit r to b n to the nu, 1 over 1 minus nu. So I say, OK, that's a bit, looks a bit ugly, but that's the correct expression. And that's a function, it's called red and r. That provides a very good description of the end-to-end distribution function for a self-avojding poem. And why to come to your question? Well, this function, first of all, OK, first of all, this function is a sort of generalization of the Gaussian function, of the Gaussian function, for the following reason. So this nu is precisely the same exponent nu. So in particular, if you have theta equal zero and nu equal one-half, OK, you get exactly back the expression for the Gaussian, which is, OK, which is the correct, why I tell you now. While, so this is for a Gaussian, OK, I know theta is another exponent. Then I will tell you what is that. Because it tells us also why the flow rate, where is wrong the flow rate theory. While, for a Gaussian or idea of polymer, OK, let's say idea of polymer. While for a self-avojding polymer, we have theta larger than zero, and ooh, I mean larger than one-half, let's say, for a self-avojding, so, well, actually that should be written more precisely this way, be n to the nu. All right, I mean, the dependence is the same, but just to be annoyed, why it's important that factor, I will tell you now. So, to answer your question, why, you know, just to answer your question, why this, I mean, this is the correct format, even intuitively, because actually, whenever, actually this term, actually is depressing the fact that you can make the two ends close together, because that's impossible, actually, because we have extruded volume effects, even between the two ends, right? So then this exponent theta has to be larger than zero, OK, first. And second, your, so that term here, basically come from the fact that extruded volume effect in some sense overstreč the chain compared to the, let's say, to the case where you don't have those extruded volume effect. And so, that's why this function has this some sort of funny behavior. That was proposed for the first time by these two guys. Yeah. It's not, no, it's not totally empirical. So, sorry. So, this is actually for r equals zero that has to be so, because that means that you have, basically, the probability to close the chain has to be zero. For larger r, OK, of course this, you can neglect this term because it's just for a large r. Then for larger r, that comes from the fact that, as I said, so it's because you are so, let's say, you have the extruded volume effect tend to overstretch the chain. So, if you take this to contribution and to account, this is, actually, that this formula is not completely XR, but it provides a very, very good description of the data. In particular, you mean take this to go, to use that in flow? Or, what do you mean? Ah, you mean, yeah, exactly. So, what do you mean that I use this here? Or no? Or go back? Or you mean the reverse? Yeah. No, I mean, you can do that, but you get worst. You get the worst result. That's because you are, actually, the reason for that is because you are correcting only this term, but you are not correcting this. So, this, that's remain uncorrect. Of course, you can correct this, but then you have to put some kind of effective correction because you cannot predict that priori. And then of course, you can do that, but then it's a fit. I mean, then you don't, you need something that you need to fix and you don't have it. But my point here, I mean, more than that, is that this formula. So, this is correct. I mean, if you have, for instance, a simulation of a safe avoiding walk, you can use this formula to, for instance, to fit to the data and find theta. Okay, no, you have, and you can find theta. Okay, you don't have no, but you can find also no from the day, from your simulation, of course. But then, what I mean, is that you can find either no and theta. And actually theta is, is very, it's quite different from zero. So, there was one question. You, yeah, sorry. I will just. K is a constant and actually, to find it, you fix it by imposing that this function has norm one. So, it's a constant. So, if you integrate this, you impose that this is normal one, this k is given in some terms of, in terms of some gamma function, but it's a constant. Yes. Well, the dimension enter in no and theta, because that, those depends on the dimension. But the shape is the, let's say, no, the shape also will change. Depends on dimension, because depends on theta in no. But the dimension is hidden here. Okay. And, yes. Well, that's the kind of explanation that is given in test book. To be honest, it's not never been completely clear to me, but it has to be something like that, because for sure, this thing, for sure, this has been overestimated, right? Because we have much less conformation for a, for a, for a self avoiding chain with respect to a Gaussian chain. And so then this has to be, was underestimated as well, in order to compensate for the right, for the, for the, for the reason why we get in a very good exponent. But in the end, that's the only exponent we get because we have no way by using the flow theory to get the theta exponent, which actually is quite important as well. Because that gives you the, let's say, that is correlated to the formation of contacts between the ends of a, of a self avoiding polymers. This, this theta. And, just to conclude this, actually theta is, it's, actually is related to the entropy of of the self avoiding work, in which way. So, so it's, it's connected in, in this way, actually is, so if you, so for a, in general for a polymer, this is quite, quite general for a polymer, the number, so let's say the, the partition function of a, of a polymer, typically goes like mu to the n, by some n to the, some exponent gamma minus one. Okay. So this is, the now, the total, let's say the total number of work for some polymer model. This is quite general. That are composed by n monomers. Okay. So that, that's how they grow. So there is a, let's say an exponential part and a power law part. Okay. So, this in general is quite understood if you do it on a lattice. So, suppose we go a bit step back. So suppose that we are analyzing random works. So, super simple. On a, on a cubic lattice. So for a random work on a cubic lattice, d dimension, random work in d dimension. So this function goes like 2d, 2d n. Okay. Or if, and there's no time here, if you want gamma is equal to zero. Okay. And why this is so, because anytime I'm here, I have 2d possible direction where I can go. So that's the, that's actually, this is exact is no, is not in an approximation. This is for the random work. So as you can see that, this formula works well for a random work. For a self avoiding work, always on d dimension. This is, I mean, it is as this kind of former. Okay. And this move here, it's a constant, which depends on, which depends on the lattice. It's not universal. It depends on the lattice you are using to model the system. So if you are on a cubic lattice, it's different that you are on anicom. The depends on the still exponential, but depends on the kind of lattice you are using. While this is actually the interesting dependence, because this exponent does not depend on the lattice you are using. It's universal. So you can use the more convenient lattice if you want. In order, it depends on your code. So that's it, seen quite well in for a self avoiding work. And now, this exponent here, actually gamma is related to theta, because theta is equal to gamma minus one divided by nu. So this exponent theta is actually connected to the entropy of your polymer, which is, if you want this bit obvious, because that's related to the contacts you can form in your chains. And if you want the contacts, it's the only thing that distinguish a self avoiding polymer from a random walk. Because the contacts are forbidden in, in a, let's say, yeah, forbidden in point contact. You cannot overlap two monomers by constructing a self avoiding polymer. So it's quite natural that theta is related to the entropy of your work. Okay. Gamma, as far as I know, so in a generic expression from gamma does not exist, but I can give you the value, for instance, in 3D. Well, here it looks like it's almost exact. So for instance, in 3D, just to show you that this thing is not negligible. So gamma, gamma, of course for a self avoiding walk in 3D, it's equal to, well, here the value is 1 dot 160. Okay. It's approximated like seven over six. So it's larger than one, of course. Which means that my exponent theta, it's one over six divided by, and I have to use nu in 3D, which I can use, and I can use the flow approximation. No, it's three over five. And so this is equal to five over 18, which is about, oh, I tried this slide. It's zero to seven five. So as you can see, it's not negligible at all. In fact, if you put, I mean, if you plot, if you have, I mean, your simulation for self avoiding polymer, and you plot this data, you will see, and you plot this, you will see that your probability distribution function will goes to zero this way. Or are going to zero, sorry, this way. And then it increases. Okay. So it's an important effect. It cannot really be, cannot really be negative. Okay, yes. This one, think, but to be honest, now don't remember. And, but I don't think it's, it's entirely, but yeah, I think you should, yeah, let me see, I think you should start from here. To be honest, I have not really reviewed how to derive this, but it's, let me see, maybe it's here. I think you're probably, okay, so this, as I said, you derive it from, let's say, it's because self avoiding polymers are overstretched with respect to standard Gaussian polymer. And, but then this, so this exponent theta is basically related to the, yeah, when you, it's when you go to, to, to small distances, and I think, if you go to small distances, you can, so you can relate, so, so you can write the, ee, yeah, I think you, you relate the, ee, yeah, you write the number of conformations that have small distances, through these, and then I think it is the path, and that's path, and then you get the relation between these two kind of response. So you can see, they are not independent, but theta and nu are independent. So I mean, they are two, so once you have the two, you can derive gamma, or if you have gamma, you can find theta, but you have, these two exponents are, are not, are completely independent, so they are, they are both necessary, and in particular, the theta exponent, that's, that's why it's profile, the, the flow theory, it's, it's where the flow theory is profoundly wrong, it's wrong that, it can, it doesn't predict, it doesn't have a prediction for the exponent theta, or if you want the exponent theta, it's zero for the, for by construction, for the flow theory, and that's it's more severe defect, but okay, in spite of that, it works, apart from that, it works quite, quite well, and I want just to conclude with, with this, maybe then we do just a small break with the following, so suppose, because this is something that we have said yesterday, so suppose, suppose that I have, sorry, actually I, you remember yesterday, we had, so for a, for a Gaussian chain, so let me, yeah ideal polynomial, ideal polynomial, so you remember that I derived the, the looping probability, you know, the fact that we have two ends, and I want to know what is the probability that for two points on the, on the, on the chain, I mean at the ends of the chain, what is the probability that this thing close, and I told you that this, so the, I don't remember which notation I've used, yes, but the, the closing probability, p that r equal zero, for a Gaussian chain that goes like one over, n, b times d over two, this is in the dimension, right, so in the, as someone of you observe yesterday, this is just related to the return probability for a random walk, that's a, that's the day, the dependence for a, for a, for a random walk, yes, but how that changes for a step avoiding walk, so also as a polymer, so if you, then that's quite easy to derive now, because we have everything, it would just use this function, and then you have that p for r, well, here we cannot use really zero, because we know that for r equal to zero, that just, I mean basically the probability is zero by construction, because two more of this cannot overlap, but you can always say that I took a, let's say a microscopic distance, let's say the size of, roughly the size of my monomers in order to derive my closure probability, and that will go like nb, it will be like nu d plus theta, okay, if I use, and you can see why, because nu theta comes from here, right, while the terms d come simply from the normalization of my or my function, and if you want, this is a perfect general, you can see immediately, this is a generalization of this, because if I use here, theta equal zero, and nu equal one half, which is the prediction of a foregatian polymer, I just recover this formula. So this is quite good that you take into your mind, because this property, I mean the contact, probably some people can actually measure now in chromosome, so now we will, I mean when I will tell you a bit more on chromosome, we can, we will actually compare these two functions to real biological data, and we will, there are some conclusions about it, if they are adequate, no, not adequate, I mean we will see. Actually, if you put numbers here, you, for instance, in d equal three, the solar we can just compare, so what you get here, so suppose that I have d equal three, so that goes like one over nB, one over five, sorry, one, yeah, three over two, which is one dot five, while if you put d equal three here, let's put numbers, so I have nB, so I have, for nu is three over five, d is equal three again, and theta, I said it's forgot, is five over 18, horrible. Five over 18, so now I have two bit, so it's one nB, so it's three over five, plus five divided by 18, so I simplify this, so this is one over nB, so it's 59 divided by 30, so it's about one, so that's about, so it's 59, it's not 60, but it's very close, so it's one over nB to the square, roughly, I mean it's a bit less, so as you can see, and as you can expect, this law here is much steeper, not much, it's steeper than this one, which is just nB one dot five, so this property will be very useful when we analyze, I mean we discuss experiments on chromosome, because we will compare to those laws, because it's exactly what people can measure in chromosome, this is a close, let's say contact probability, and so that's very useful to compare to this kind of theoretical prediction, okay, what time is it now? Do you want to do s-break? Yeah. Half past three. Can we make five minutes break? Do I take some water, too? Theta, yes. Yes, so in fact, maybe I can, sorry. Ah, it's just a constant. It depends on the, actually it's not universal, it depends on the lattice you have. So, no, that describes, we can talk later on, or tomorrow, I mean, I'm here, so. So, yeah, I think it's useful if I tell you this thing that I have in mind, but I need a bit of more time, so I will be more specific tomorrow, because otherwise I will be, I have to rush a bit, sorry. So this is something, actually I have not told you when we mentioned, so we go a little step back to the, let's say, to the ideal polymers, because this is also, somehow it's useful. So we know that, so actually the kind of, I mean, the system that I've mentioned so far, it's for, I mean, the kind of, all the theory that I've done so far is for, let's say, I call it free polymers, free polymers, I mean, polymers who are not in confinement. So, either if it is a Gaussian chain or a self-avoiding walk, and in general, even a polymer in solution, it's, actually these properties are for a polymer in bulk. Even when I did the picture, when I took a polymer switch, I started to compress, the system is always meant to be, so I just repeated the drawing a bit larger. Even in this case, so for a met, I mean, the typical side, so this is the container, but the typical sides of my polymer chain, I said that, I said that R goes like B and one-half for a polymer, for a linear chain in a melt. So, but still, this quantity is much smaller than, if I call it L-box, than the typical size of, let's say, of my container. So, if you want a confinement effect can be neglected. So, it's real, like, it's like in a bulk. So, why I said that? I said that because, actually, this condition is not obvious, that is relevant, for instance, for, again, for chromosome, because we know that we have a nucleus, okay, and we have our polymers here, I mean, our chromosome, which are more or less swollen. But now, this is in real, I mean, we need to wonder, if the confinement is important, namely, if the equilibrium size of the chain is much smaller, comparable, or, so, to say, so, if the chain, if the size of the chain, when it's free, is much smaller, comparable, or much larger than the nucleus, than the size of the nucleus. So, just to give you an example of a, I mean, of the biological systems we are going to discuss. And that's very important, of course, because in the first case, I mean, if the typical size of my, let's say, of my chromosome, when it's free, is much smaller than the size of the nucleus, then, I mean, this is like a real solution, if you want. So, the nucleus does not impact, at least does not impact too much on the size of the polymer. If it's comparable, starts to have some impact, and if the free size, so, the size of the chromosome, let's say, when it's free, I mean, without the confinement, much larger than the size, when it, I mean, is much larger than the size of the nucleus, then, I have to expect that the confinement effects are important, OK. So, we are going to discuss a little bit, what it happens to polymers, so, I go back again to, to generic features of polymers, when they are in confinement, OK. It's different from the melt. Well, the melt, yeah, it's a good question, thanks. But, of course, the situation is not completely similar. OK, I will, sorry, who asked that, because I was, ah, yeah. So, I will be more specific to tomorrow, because today I don't have the time. But the situation is the following. So, in a melt, rigorously, I have, as I said, I have still many chains. And as I said, ah, so, you expect, because of the Flory theorem, that the, let's say, the size of my chain at equilibrium has to grow, as I said, as a Gaussian polymer. But then, so, this, I mean, if you calculate, if you put here the typical side of a monomer, times the number of my monomers, then I get a number and then I have to compare to the size of the container. If, this can be much smaller than the size of the container. I have many chains, right? So then, in this case, the confinement is practically negligible. Maybe I have some confinement effects at the border of my container, but in the end, but inside, that doesn't matter, no? Any chain does not feel the presence of the confinement, right? While, I mean, this is not always the case. So, yeah? I guess. But confinement, so, they are both confined, but in confinement, I mean that, the, let's say, the size of the polymer when it's free, let's say, so, in free space, it's much larger, or larger, or, let's say, of the size of the box. And in that case, the things change a lot, actually. And as I said, I mean, tomorrow will be on specific, but one thing that you can immediately see, so, actually, there are some equations that are exact, and tomorrow, I will tell you more about that. But let's take the most simple system, which is, again, the Gaussian chain, right? Because we know everything, and it's, there are no extruded volume effect, and it's simple. So, for the Gaussian chain, we know, no? What is R? As to go, like, B and to the one-half. OK. And then, suppos, I want, I confine this thing to a sphere of some diameter, so I have, and I have my chain. Then, this thing has to be smaller, or, if you want, in order to get, to stay hamperturb from the condition to be free, has to be smaller or larger. Of, let's say, of the diameter of the sphere. So, then, it's a bit of a matter of definition, because I'm neglecting, let's say, I'm neglecting pre-factors. But if I call this diameter D, then this thing has to be, in order to be hamperturb, to remain hamperturb, has to be smaller, let's say, of the diameter of the sphere. Which means that polymers, remain hamperturb, if n is smaller, or comparable to d to the b square. Right? So, this is the condition of the polymer. So, polymers, so there is, let's say, crossover length for the polymers. And for this length, if I took a polymer, which is much smaller than this length, so I can call it, and see, just to make like the crossover. If polymers are smaller than, they are shorter than this length, then they stay in, I mean, the container does not play in a particular role. But if they are larger, of course they, I mean, it's a serious ingredient, it's a serious actor of the game, right? And that's very simple, if you want to describe, I mean at least for a Gaussian chain, how things have to go. Well, one can be more rigorous, but in the end it's, I mean, what you should expect is very simple. So, if you have, so actually it's very useful, if you, so you have your, okay, let's do it, yeah, let's do it here. So we have our sphere. I just take a sphere because it's similar, but of course, I mean, you can take a cube, I mean, that doesn't change too much. So I have my chain here, and then, and then goes like this. And so the physical, I mean, the way you can describe the system is the following. So then, now it's important, so you take any, if you take pairs of loci, oh loci loci, pairs of monomer, sorry, on the chain, then you should expect the following. If you calculate the spatial distance between them, and I call it this, think n, no, this is the, sorry. This is a single polymer folded, is a Gaussian polymer, so it's an ideal polymer, folded inside the sphere. So there are no extra polymer, so I can put any, I mean, any length can stay there, because there are no, so it's a bit unrealistic situation. But okay, for time being, let's do this, because it's, that can be done, I'll show you tomorrow, it can be done exactly. I mean, it can be done everything, it can be done exactly. Or for time being, just let me, let me just be qualitative. So if I, so I define, so I took two pairs of monomers on my chain, and then I calculate the distance between these two monomers, the average distance, let's say, between these two monomers as a function of n, and of course, I mean, you can see what would be the result. I mean, that has to be like n, b, n to the one-half, okay. When, then I can use this result, when n is smaller or equal to d over b square, right? And then, of course it's simple, what can do this polymer, can only have a distance, which is the order, of the diameter of my container, right? And it should be like this. Sorry? Yeah, it has to be of this order. Actually, you can derive the precise relation, but it's the same order. It's the maximum length that it can get, no? Because it has to explore, and on average, has to stay on this order of magnitude. No, R is the average distance between two pairs of monomers on the chain, and I'm asking how this quantity grows in terms of the linear distances on the polymer, right? So for a free polymer, this is always like this, because it's a Gaussian polymer, but here is no longer, the polymer is no longer free, because it's in a container, and then this thing, because of this relationship, has to be like a Gaussian relationship up to this contour length, and then has to be order of D for larger length, because simply the polymer is confined. So any, let's say, it's like having any two random monomers, any two random points, if you want inside the sphere, and then the average distance between them has to be over the order of diameter. That's it. But so then, this is quite important, because you can see that there is a, actually a crossover length that tells you when confinement effect becomes relevant. Okay. And tomorrow, because again, I need to be, I mean, I want to be a bit more specific on this. You can derive the, and actually for Gaul, only for ideal polymers, you can derive an exact formalism. Actually, I think it's quite useful. We'll tell you about tomorrow, and we will also discuss a little bit about confined polymer, because even by using extruding volume effects, because they are quite useful, and also people can do experiment on that, apart from the fact that, of course, they are related to chromosome, it's not the, I mean, they have more general application than just chromosome. And, but we can do it tomorrow, because I need a bit more time. So if you have questions, maybe I can ask where we are. Yeah. This is a decolidation assumption, right? Yes. So overall thing doesn't matter the smaller things. Yes. Is it because of the Gaussian nature? It's because of the Gaussian nature, yes. So because when you go here, right, basically it's like having, it's reflecting, it's like having a mirror. And then, because this direction is the correlated. Actually, things become more complicated. I didn't talk about it and yeah, I'll see if I have the time, but things become more complicated. If you, so this is a, so Gaussian polymer, or, I mean, this, or, also Frigendi chain, whatever, they are example of flexible polymers, right? But suppose I have a, what is called a warm like chain, which is a semi flexible polymer, then I have another, yet another lamb scale, which is the persistence lamp, or the cool lamp, whatever. And then, this is, this also, also play a role, because basically the semi flexible polymer is characterized by a lamb scale. It's like, if you want, it's like a, yeah, it's like a rigid rod, it's like a spaghetti. So below some scale, my polymer is very stiff, above some scale becomes to be, because of thermal fluctuation becomes more and more flexible. And of course, you can imagine, if you have this such kind of polymer, and my confinement, I mean, D is comparable, or smaller, than these persistence length. And then the conformation of the object is completely different. It's no longer described by this. But of course, I mean, it becomes more complicated. But since it's, but I don't know if I have the time to do that. I will probably not, since it's not really relevant for, for the, let's say, for the chromosome, for the biological part that I have to tell you, I mean, in the last lectures, and we will not talk about that. But it's relevant if you have, for instance, DNA in very strong confinement. So that will be important. But I mean, it will not be important for what I'm going to tell you in the rest room. We will not talk about that probably. Yes. No, it's much smaller. Yeah. For what? For DNA, LP, DNA, it's equal to 150 base pair, which means it's about 50 nanometer. Yes. 50 nanometer? No, sorry, the opposite. Is, no, no, it's right. It's 50 nanometer, which is 150 base pair. You know, it's correct. Yes. No, no, it's correct. It's correct. And this is for double-stranded DNA in, let's say, in normal conditions, in normal salt. Because the persistence length change, change, I think not twice, but change can change considerably according to the to the salt conditions. Because DNA is a charge. But those, the experiments where people have measured this were in, it's a physiological circuit with, I think they use the magnesium ions, Mg2 plus. And that should be more or less, I want to say something, not correct, but it's, I mean, like saying, like saying physiological condition. And in this condition, they found these values for, this is the accepted value for DNA. And that's of course, it's not small. It's, you can see it's 150 base pair. That means that you really, I mean, it depends by what are you doing. But since you can prepare, actually, for instance, the people can study DNA on a slab, or in channels, or nanopores, whatever. And in that case, and they are very nanopores, so they are very small. They are much smaller than these, like the linear sides. They are much smaller than these. So then, this physics, I mean, this physics does not apply there. But for chromosomes, this is not relevant because, I mean, even for bacteria, bacteria have, the DNA of bacteria is one mega base pair, which is, this is, has to be, it's basically DNA. So, which means that it's about 1,000, not even more, no? Is, no, cimandamila is 5,000. Larger than these. And so it's doesn't, doesn't matter. I mean, the confinement is not tight. So, while it matters, for instance, for viruses. So DNA, viruses with DNA, obviously there are also viruses with RNA, but viruses for RNA, sorry, with the DNA, the viral capsid is actually is comparable, or smaller than the 15 nanometer. So, the DNA is basically make some kind of order inside the viral capsid. Then for viruses we have to take into account confinement and the semi flexibility is not simple, but we are not going to talking about vile. So, which is a totally fascinating subject. It's not because I don't like it. It's, it's, I mean, it's a different physics. Okay. For, I mean, for chromosome doesn't matter. I mean, it's not really relevant. I mean, so we can use this physics. Okay. So it will continue tomorrow. If you have a question, just so you can just ask.