 Wedi announced.. Yeah, I think we can start, OK. So, welcome back, so I hope you liked the lecture by, some of you are asking about the exam, I don't know, I mean, so, because Angie hasn't sent me yet the question, so between today and tomorrow. zato sem sem se zelo prišel. Mateo, tezvom, da sem tudi, da sem tudi tudi tudi, da sem tudi tudi, da sem tudi tudi tudi v komputeru. Tako, tudi tudi tudi lektur, da so vrstko, ne, včasno nisi neselj, včasno jaz so pričo. Zato sem izgledal nekaj tahto, bo, da je tato, srednih poslednjih, srednih poslednjih. In aplikacije z RNA sekondariške stručje, sem sem še težko bilo zelo nekaj, zato sem zelo však v zelo polimerji, ker nekaj naprej, srečno, nekaj je zelo na taj statistik, v polimerji, sem je, nekaj, sketeči, sketeči koncept. Przvine je vse vzena. Izgleda je genetih, zato se je zelo zelo zelo izgleda. Zelo vse početno. Zato je malo materijali in zato In vseč... Prejemo, to neče vseč. Prejemo, da so. Tako. I vseč, je to prejbe načinite lekture, ne, ne, ne, je to pdf. Sledi od in otroj skul in, ki se začali v Boulder, kološtri, v 2012. I je to prejbe načinite, Oni so počkali od Šura, Prof. Šura Grozberg in Mikol Rubisenov, da bi se počkali dve veliki autoritivi v teoretičnih, počkali v teoretičnih, počkali v teoretičnih in softnih materičnih. Tako, imam vse materijanje, ker je več, več bolj organizovali. ima pristimi reju. Vse možeš izmahite z micromolekulstv, v kaj mi je odrežil, m.e. 2017. A vse on kreja esklao materijale. Tkaj su brems, pravko 50 prijoč, ima se na vse načinve koncept, ni naprej pristcos, tako zbah ju prezentuje materijale, akas li vse koncept. If you are interested in the details, how calculations are done, you can just download the references in there. But otherwise, it provides a very, very nice overview, also on some new topics. So it's very well organized. Otherwise, there are books, but let's say all the essential materials contents here and here so you can just download these two and if you want to know more. What is a polymer? Just a bit of semantics. So polymers comes from Greek, means many parts. And the reason why it means many parts is because these are chain molecules, which consist of basically from 100 to this is one billion monomeric units, so single molecules that are linked one after the other by some chemical reaction. And the chemical reaction that produces polymers is the process of polymerization, which is a chain reaction. So basically you have a single molecule, so in this case is styrene, which is this molecule here, that polymerizes into a chain, which is polystyrene, by forming a covalent bond between these side chains. And so these reactions in principle, if conditions allow, can go on forever, in principle, let's say. And this is, let's say, it's a big thing in sense that these molecules are huge. I mean, are amongst the largest molecules you can find in nature. They become, I mean, on the molecular level, they become almost, so basically they pass from being, let's say, here you have quantum mechanics, if you want, I mean, in sense that this is real molecule from something which becomes so big, so large, that becomes macroscopic, so you can see that. I mean, just with a microscope, nothing too fancy. And because of these, as I will try to tell you, so because of these properties, that molecules become huge, I mean, there is a lot of properties that you can describe just by using classical mechanics. So now the same kind of reaction is also, well, this is the same kind of reaction here, and you have many of these examples, I mean, you don't have only polystyrene, polystyrene, you have polystyrene, polystyrene, this is basically the common plastic, I mean, you can find for instance plastic bags and the other. So this kind of reaction does not only produce such a kind of molecules which is basically a linear molecule, because you can alter, I mean, you can go from one to the other, so on and so forth. So this is a linear molecule. But you can have much more. First of all, you can have this kind of architecture, so you can have a branched architecture, namely where the growth of another chain can move from a monomer and produce, I mean, due to the presence of multifunctional monomers, it can be more complicated in the sense that you don't have only a single monomer, but you can have many, I mean, the single species, let's say, but you can have many. And in such a way you can form a so-called copolymer, or sometimes it's also called hetero polymer, it's a bit of a choice in choosing words, but let's say it's the same thing. So for instance, copolymer is, you have seen, RNA is a copolymer, we have at least, I mean, we have four kind of different monomers that are joined one after the other and they form, I mean, the RNA is the same, or proteins, such kind of things. So now polymer exists in different states or different, maybe, dilution. So for instance, if you have a polymer, which is, I mean, in a solution, which is very dilute, basically you have a solvent copolymer, and this is very dilute, I mean, in the worst and very dilute condition, this is called a dilute solution, it's a sort of, you can consider it as like a sort of gas of a gas of polymers. Otherwise, if you start removing the solvent, or if you increase the concentration of the polymer, you have a so-called semi dilute solution and depending on the quantity of the polymer, if you have basically only polymers, you have a so-called melt, which I mean, can live in different conditions, it can be a more liquid melt or a more solid like melt, I mean, that really depends on the preparation conditions. So polymers basically are everywhere and they exist also, as I said, in solid state, in a sort of solid state condition and this is, for instance, the natural rubber, for instance, the one of, let's say, natural rubber, the one you can extract from the trees, or also the vulcanized rubber that you have that you use for building tires. It's the same material, it's just that the chemical condition is different. Then you have polymer glasses, for instance, the one that you use for contact lenses, they are sort of polymer glasses, or semi-crystalline polymers that are almost solid. There are more orders somehow. So they really are everywhere. So scales. So this is just an example. So this is something that maybe you want to remember, just to understand a bit the physics of these objects. So KBT at room temperatures. Let's say, it can be expressed in a very natural units if you deal with polymer, which is, this is a very easy number to remember, it's 4 piconeutron per nanometer. So this is a force, and this is a distance, so the two make an energy. So at room temperatures, KBT is about 4 piconeutron per nanometer. This is a very convenient unit for molecules, because basically the average bonded distance in a polymer is on the order of nanometer. And so this is a way to remember instantly these scales. So now, what is the amount of energy of breaking covalent bonds is about 10,000 K. So it's very, very huge. So this means that basically covalent bonds is stable at room temperature. And this is the reason why you can form such macromolecules. So this may be a number that was already told you by Ange. So this means that bonds, if you want, are not in terminal equilibria. They are forever, unless you really pull it a lot. And you have other feature that I will tell you now in a moment that a polymer can be bent, and this bending is actually on the order of KBT, or so a bit more, but occasionally on the order of KBT. And this means that polymers are easily bendable, I mean are more or less easily bendable at room temperature. And this is one feature that probably in the end you have already seen with RNA, you can form such a stache, so as to bend. So now, the typical monomer size is in the order of let's say, angstrom, so from 50 nanometers to angstrom. And as I said, the real polymerization is really variable. I mean, it really depends on the species. It can go from 10 to 10 to the 9. So it's enormous. For instance, DNA in the cells are in disorder. Our DNA, it consists of billion of base pair of DNA. So it's real in this order. I mean, are the monks the largest biopolymer, actually you have in nature. So this means that the contour length of a polymer, which means that you have your chain and you stretch from one end to the other, and you measure how it is long, can go from the order of nanometers to meters. For instance, again, the DNA inside our cells it occupies, I mean, it's the length, in each nucleus of each cell is around one meter. Ok, so it's a pretty impressive length. If you do that, especially because the diameter of the cell is only about 10 microns, ok. Yes. It's not in equilibrium because room temperature is 4 piconutron per nanometer so you convert it is much less than this number. Ok, so this means that the bond is there forever, that doesn't break. No, it's not in thermal equilibrium because the energy of breaking a covalent bond is much higher than the room temperature in this sense. Ok, while bending, as I'll show you in a moment for a polymer it's a clear feature and is on the order of KBT. So you can easily bend it, but you cannot break it. You can break it, but you have to use some chemical reagent. It's about yeah, maybe this is a bit smaller but it was just to give you an idea. It's on the sides of between... No, no, it can be... I mean, so if you want this on the order of let's say on the order of atomic dimension if you want but overall, I mean if you have a relatively complex polymer like for instance DNA, the typical monomer size is on the order of a few nanometers, two, three nanometers. Ok, so it's a bit variable but it can be on the order of for instance if you have let's say a classical yeah, already this it's on the order I think one nanometer about the four polystyrene. Ok, but I mean it's a bit variable. Ok. Can be, can be this is just these are just order of magnitude, ok. But but let's say in principle can be as I said the reaction that this polymerization reaction can go on forever provided you give that you feed your polymer I mean that you construct your polymer in the right condition. Then at some point it cannot really go on forever because basically then what it happens ok, naturally sometimes it forms some precipitate so you in the real solution so you cannot do that anymore but let's say in principle can go on forever that actually I don't have slides about that but that was one of the major discoveries in chemistry it was done in 20s because ok, that was done by I don't remember now it was a German guy now I don't remember the name. That was a I mean it turned no prize for this cover and this cover was precisely the understanding that these macromolecules may exist, ok. Because at the beginning of the 20th century I mean there were of course polymers already existing in nature, ok. But people thought that the properties of the material they observed was due to some basically that molecules were just aggregated ok, just, you know like a salt or things like that they did not realize that molecules were not aggregated were really covalently because they didn't know the covalent bond but they were really covalently bonded after that they created macromolecules and this is a big difference in the 1930s or so in was a German now I don't remember the name I will tell you later, ok. Was there some hand? No, yeah. Yes. So yes, this is a good question that really depends in a plastic so these they are not only chains but they are also they are actually more similar to networks ok, I am not sure how exactly they are prepared but let's say you have your chain and they form a structure like this and then between the different chains there are cross links so they use some in general they use a sulfur bridge and they make reaction between the different points these bridges that keeps this thing together ok, but it can have I mean sooner or later maybe this is something that you have observed this bond can break because they are not the same nature of the covalent bond and so the plastic basically degrades it may take a lot of time of course we know that plastic unfortunately is polluting a lot but it can break and of similar reaction it also makes the difference between natural rubber and the let's say vulcanized rubber the one that is used for winter tires that's why they are so elastic because they create this sulfur bridge that eventually when they degrade you have to change your your tires ok, but before degrading you have to change your tires before maybe that also is another story ok yeah, so for instance this is the kind of slide that can well, this is just an example how polymers can assist in different in different conditions and even you may have, ok, but I mean just to give you an example ok, we don't have to to re-insist on that too much I mean depending on the dilution condition the same polymers pieces can give you let's say give rise to different materials I mean it's the same let's say if you want, it's the same material by different dilution condition and you can have different application ok, but this is just to give you an idea no need to insist on that too much so, as I said polymers can so they are not really interesting I mean there doesn't only in simple artificial polymer ok, or natural polymer but relatively simple but exist biopolymers and of course I mean this is the main topic of these lectures lectures is that there exist proteins DNA RNA DNA sorry, it exists also in a very complex I mean in a complex molecules when you have DNA plus proteins you form what is called the chromatin which is the main constituent of DNA the main constituent of our genome is organized in especially in eukaryotes so eukaryotes are the organisms like us that whose cell contains a nucleus ok, so and we don't have only DNA but we have a more complex polymer which is called a chromatin ok, and this is not the main topic of this lecture but let's say it's just to give you a few examples and of course the main object here is RNA no, RNA that forms this structure this branch structure and this is also a biopolymer ok so, actually understanding a bit the physics the statistical physics of polymer will also pass to understand hopefully a bit of the physics of these objects ok, of course these are much more complex because it's heteropolymers but maybe we can understand something about that, so and yes and so this is a bit of the same scale but related to biopolymers so they exist as I said in very different scales and as you can see DNA is a very nice example of a big macromolekius because the degree of polymerization goes up to 10 to the 9, 10 to the 10 that depends a bit on the organisms but let's say this is the real strength of DNA it's important that it is so long in order to to be robust, I mean to maintain the genetic information and RNA is smaller maybe you have seen that from angel lectures and we will I will try to tell you a bit more maybe a bit more tomorrow about this and then you have proteins, lipids, polysaccharides not the sector of the angel lecture so as I said, I mean so how do we understand the statistical physics I mean why we can do some sort of statistical physics of polymers so now the important point of polymers is that they I mean let's say the bond, the covalent bond that connects one monomer to the other is not it's fixed of course however there are situations like the ones that you have here that basically you may have a sort of a bit of a torsion between one covalent bond and the next one and actually this is because you have some let's say these atoms lives in 3D so there is some sort of a bit of steric hindrance between these monomers ok and one can think of that steric hindrance by saying that let's say the orientation between one covalent bond and the next one basically it has all it can be different however there are three preferential states two are equivalent and one sorry two are equivalent in terms of energy different orientation in terms of the angle between the two ok and one has another minimum so this state by the steric hindrance one is called two gauche they are equivalent so they can be they just matter of the two orientation and one is when they are more aligned and this is the most stable one and which is called trans so now you have to imagine that this orientation ok this different orientation is randomly distributed so this means that locally the polymer is straight it's rigid but because we have this small deviation from let's say the same bond it starts to have on the longer scale these bonds start to correlate from one chosen initial position ok and this means that if you have many many monomers inside the chain the total number of conformation ok you may have for just a single chain is enormous it goes basically exponential ok in this sense in this respect one cannot say that so you have sort of internal flexibility of the chain intrinsic flexibility so this means that we cannot speak of an atom that exists in some specific state but we have to speak of many many possible conformation and then the properties of polymer I mean to talk about the properties of the polymer we do to talk of properties on average and from some specific ensemble so this means that we can use some tools from statistical physics in order to understand let's say the basics physics of polymers so these ideas let's say in from many many remarkable people for in let's say starting from the thirties by Kuhn, Flori, Huggins, Stockmeyer James, Gut, Zim and Raus in probably the most remarkable person here was was PJ Flori who wrote at least two couple of books which are let's say very influential in polymers and he also got the Nobel Prize for chemistry for basically for his work on polymer science and starting from the sixties the same idea where sort of reviewed and put in a more broad perspective because people like Sam Edwards, Pierre Gilles de Gen and Edé Cloiseau basically they use the concept of statistical physics actually quantum field theory in order to understand the physical properties of polymers and in particular polymer solution because they show basically that the same ideas that used from quantum field theory that can be exported especially these like renormalization group they can be exported to the physics of polymers and actually these based on this idea Pierre Gilles de Gen got the Nobel Prize in physics precisely for having developed this connection between polymers and let's say in quantum field theory so let's say here we have at least two Nobel Prize for I mean given to polymer scientists ok and of course I mean the others which was also remarkable scientists were not Nobel Prize but I mean they developed a lot of ideas that we still use nowadays and the idea is totally finding on this concept that let's say describing polymer conformation is a statistical physics problem so so the idea this is the fact that polymer can bend ok so now provided they are let's say long enough that the number of monomers is sufficient ok so now of course I mean polymers let's say different polymers bend differently so this is not particularly profound you have already seen an example that so if the very first lecture you mentioned that you have I mean let's say the molecular level RNA and DNA are not so different in the sense that you have four basis for one and four basis for the other however because DNA forms this double helix is much more rigid than RNA so in fact there is a quantity that I am going to define later and actually this is an example of DNA so ok forget this elastic road model let's say so you have this is a model for a short fragment of DNA so because you have these double helix these filaments are very rigid on the local scale in fact you need to have a very very very long filament of DNA in order to see appreciably the molecule can bend ok ehm so this is a bit of let's say of a general concept so now I think I will use a bit of a black ball so now what is the so ok so now this is a bit of sorry I don't know if you have a question up to this point ok so yes about this one no here correlations means the following that well this is especially related to polymer solution correlation means that if you have a polymer ok let's think like a string ok and this is another this is another polymer chain ok so ehm let's say the kind of the interaction you have here so imagine that this is a monom in fact I have a slide about that later so this kind of interaction ehm are let's say are ehm typically short range in space ok so basically tomorrow we have to stay really close ehm to fill each other but because you have a linear chain here and another linear chain here the molecules become strongly correlated ok because the interaction ehm basically if you want this kind of information ehm because the polymer is linearly connected or linear in this case it's just a linear polymer ehm this linear correlation makes that ehm you start to develop some long range correlation ehm along the profile along the linear profile of the chain so in this sense you have a correlation ok and this is only due to the fact that ehm you have a macromolegius there is this chain connectivity between the different monomers you don't have such a kind of long range correlation if you have just a gas of monomers ok in a gas of monomers you just I mean if you have exclusive volume you can use I don't know the theory of van der Waals or things like that you don't have such a kind of description in polymers just because you have a linear connectivity so this is the main the main defense actually not even I mean we don't need to already on a single chain ok this is something that I'll try to tell you in a moment ok this is a very important thing ehm in fact this change everything so it changes the story from ehm well from the physics of ideal chains where you start describing the physics neglecting the this kind of interactions ok and then the description is relatively easy from non-ideal chains namely where you have this kind of interaction so basically the two cluster molecules they they become to two different universality classes as was shown by some edwards ok I think it was the first person that showed that ok ehm ok so and in fact this is my was my next slide so how we ehm so I will try to be I mean I hope the slides are clear otherwise we do that on the blackboard so how I mean so the first thing ok how we tackle the problem of let's say a statistical physics description of polymers ok so there are many different models so by ideal ideal chains I mean that is at the very beginning so we have a certain amount of flexibility in a chain ok so on a very first approximation we can model a polymer chains like a collection of consecutive bonds ok so we have of some intrinsic ehm bond distance that I call B ok and that's it ok so this is the simplest model I can have the length of the bond is fixed because as I said this bond basically that cannot be broken so let's take it as free as fixed in space and then ehm I mean all the freedom of moving it around ok so this model is called the freely jointed chain model ok and I mean it's very simple ok because basically you don't have anything I mean you just chain connectivity ok and now because let's say as I said so the polymer can assume all sort of conformation because you can bend each bond I mean its connection here how much you want and you can reorient the polymer in many different shapes ok ehm so you are interested in basically the average properties of polymers because I mean of course I mean you will look different from another realization from the same ensemble so you need to find a description on average so the typical quantity that is measured in this in this case is the so called ah maybe I can use ok I use a different is the so called mean I mean it's the end to end distance so basically you have so if this is the coordinate of monomer one sorry ok and this is coordinate of monomer N assuming that I have N monomers ok you may want to measure this quantity ok and then average over all possible conformation ok so you we make an ensemble on a choice that we average over all possible chain conformation ok however we immediately see a problem with this quantity because if we do that this quantity just averages to zero right because you have all sort of possible orientation in space so then we do not use this quantity but use another quantity which is this quantity square end average end average ok and this is the quantity that appears here ok now you may understand that ah this is nothing less that a random walk in space ok because my chain so let's say this bond start from here and goes here the next one can have any possible orientation with respect to this one and so on and so forth for the others so this means that the mean square end to end distance has to respect has to grow linearly with the total number of monomer N ok and so this is a simple result of course but it's a very I mean it's the first important result that basically the result number zero that we have in polymer physics because it tells you that provided that chain is ideal namely we neglect any possible interaction between the monomer which is of course a very big simplification but it's the let's say the first model then the typical chain size can be described like the square root of this quantity grows with the square root of the number of monomer so this is low number zero in polymer physics then starting from this we may define another quantity and this quantity is called the kuhn length of the chain basically the kuhn length of the chain is defined as the mean square end to end distance ok divided by the total contour length of the chain so the contour length of the chain is imagine that you have polymer chain you stretch it from one end to the other and you measure how long it is ok and this quantity is of course in this case would be N the total number of bonds inside the chain times B which is the bond length ok so now because NB square divided by NB ok this is equal to B in this case ok and this is the kuhn length so this is the definition ok in this case the result is very simple ok but there is another situation where this result is not that simple and that basically it let's say it is considered another model which is a bit more a complication with respect to this so imagine that now you don't have your the angles between let's say this bond and this bond like completely free but the angle between this one let's say you have the following model because it increases the complication ok so imagine that this angle here ok while you have only freedom of this rotation here on this angle which I called phi ok so now this is another model which is somehow more similar I mean it is a better suited if you want to describe this kind of situation where basically the angle between one bond and the next one are not completely some hindrance so now this model is called the freely rotating chain ok this one freely rotating chain rotating chain and now for the same let's say for the same model you can compute always exactly maybe you can try it as an exercise I mean it's not complicated the same quantity here, the mean square so if you compute that and if theta is this angle here which is fixed you get these results so this in the limit of very long chain when n goes to, let's say it becomes very very large ok there are corrections here but not very important so now you can perform the same exercise here you can define again you can apply again this definition to derive an expression for the Kuhl length ok and the Kuhl length is, I mean if you do it if you do the math because then let's say the contour length is still nb and you get this quantity here so now you can see that because theta is well sorry cos n theta is always smaller than 1 so you can see that in this case the contour length is longer than the bond size and this has to be so because let's say you have introduced a sort of correlation between the bonds ok so now the interesting thing of these these are just very simple ideas but the interesting thing of these two models, let's say is that the mean square end to end distance always depends on the number of monomer n in the same way they are always linear in the number of monomer n and this is a very important result because it tells you that somehow these two models on larger scale they basically have the same physics locally they are different but on larger scale nothing really changes and in fact these allows me to well I can consider another quantity actually before I forget which is actually more useful if you don't have a linear polymer but if you have for instance a branch polymer you cannot really define an end to end distance because the polymer has many ends so a better quantity is the so called the average square generation radius so the average square generation radius is so you measure the center of mass of the chain of your molecule of your polymer and you compute the mean square distance from the center of mass and you average it over all possible and you get another measure basically of the extension of your polymer chain in space it's different from the mean square end to end distance but it's more general another quantity for instance that you can measure is so called contact probability between the two ends of the polymer but this is not very important here I don't want to to point too much the important point is that we can define actually so forget this exponent here and define an exponent now as I said for an ideal polymer the mean square end to end distance and you can see you can prove that also the mean square generation radius they both grow with the same power law of n and this allows me to define an exponent which in polymer phases define as the exponent nu so now this exponent nu is very important because it identifies and we will see that in a moment the so called universality class of the polymer model but say of the physics of the polymer that you are considering but what do I mean by that so for an ideal for ideal polymers my exponent nu we have measured it actually we have computed it is one half but I mean this is because I am actually so is the next slide this is because is one half because basically the polymer provided is long enough can be understood as a random walk in space yes it's a variance if you want it's the variance of the quantity distance of the monomer from the center of mass so you have to imagine a polymer like cloud of points and this actually is a quantity that you can define for any collection of data points so you have to imagine the polymer nu as a collection of points linear connected for instance you measure the center of mass which here I don't know will be more or less here so this is the position of the center of mass and you compute the distance one by one you square it and you compute the average and again all the possible conformation and that's it then you do the math so the important point is that because these two quantities they are two different quantities they have very different physical meaning square radiation radius is more fundamental instance that you can define it even for a circular polymer or for a branch polymer for any collection while the end-to-end distance is defined only for a linear chain because you have only two turns so these are more general the important point here namely for a linear chain in this case the two quantities on the polymer side the square root of those the average polymer side they increase with the same power law of N so this can be seen it's pretty simple yes no it's you have both here there is a type to fix it so it's the average no no it's the same thing because you first do an average over let's say the fixed conformation and then you average over all possible conformation the both this average here is an ensemble average you see so maybe I can it's important so maybe I can you see you have two if you only have two averages here you have an average the single conformation ok and then you make a second average over all possible conformation ok this looks a bit more complicated but in the end of course it's a bit more complicated let's say but in the end it's not so complicated it's a bit probably less obvious but it really describe it's interesting in one sense that if you think to it like a cloud it gives you the average distance from the center of mass of the cloud but the important point is that both obey these scaling relationship which is pretty general which again allows me to define this exponent nu and this exponent nu actually describe identifies many the universality class what I mean by that here it's written but it's not completely exact in the sense that it identifies the right physics of the problem that allows to describe them in terms of some specific polymer model ok the reason why I mention this contact probability is because this has become more and more popular nowadays related especially to some specific experiments where apply physics to DNA experiments inside the cells because this contact probability this is mainly for a linear polymer it tells you basically gives the propensity for the two ends to come into special proximity and this is also scales as a power law of n but of course now it decays with n because if you have longer longer polymer of course the chance that they come into contact becomes smaller polymer length and these two exponents they both identify the specific universality class of the polymer but let's say for RNA it's more important this one something that you can measure easily ok so now what are these exponents if we can tell one thing for ideal chains for ideal chains ideal chain 2 monomer here 2 monomer here can basically come into close proximity as much as they want they don't have volume they are like invisible to each other the only thing that matters for an ideal chain is the connectivity on the chain and that's it in this case we have sort of random orientation of the bonds it's basically the free agent in chain model or this slightly more complicated model free rotating chain but provided that the chains are long enough basically the exponent nu does not change so basically these two models they are locally different but they fall into the same universality class so this is the key concept in polymer physics in particular this exponent nu is one half because basically the polymer can be assimilated to a random walk in space and in fact you have the same kind of exponent of a random walk I mean when you describe a random walk in time you have that the mean square end to end distance from the original position grows like the square root of time this is like it grows like the square root of n it's the same physics if you want and because it's the same physics you can you can prove this is not to cover maybe you can try that actually you don't have only average quantity here you have a whole distribution for the end to end distance which you can compute exactly and because I mean let's say you have let's say the mapping to a random walk then you can immediately see that the end to end distribution function namely the probability that your chain the density, let's say the probability density that your chain has an end to end distance is a Gaussian because let's say the distribution function the density distribution function of a let's say of a particle that diffuses this Gaussian then this is the same thing it has to be Gaussian as well with a variance that is equal to nb square yes ah ok this enters into another story but because I'm not talking about this subject which is very interesting I have slides here but not for this lecture so there are experiments in this, ok just go back because you ask I think it's interesting so there are so it's more related to chromatin ok so chromatin inside the cells it's a mess in sense that you have you have to think that this is the nucleus of the cell for instance in our cells and you have a lot of DNA here ok, you have so the linear sites of a nucleus is about the microns or amino average that depends on the cell the linear length of the total amount of DNA which is more like organizing chromatin but it's not important, let's say so the linear length of DNA is on the order of meters ok, so all DNA in one single cell of your body is one meter long which is, so you can see that you have five order of minus of difference so this is really a big tangle inside the cell so nowadays there are some clever people that have developed a technique which is called chromatin conformation capture that what it measures so suppose that you imagine that your chromatin actually it is a polymer so what it measures it measures the frequency of contacts between chromatin fragments inside the cells ok and now they have seen that they made a lot of statistics about that so we are in this domain here now and what they measure they have measured precisely this quantity here so how frequent DNA or chromatin if you prefer interact inside the bodies of the nuclei of the cell they have measured this quantity for chromatin and actually they measure this exponent and this exponent turns out so if you are curious that is not this one that you can have just because it is a random walk but it is less steep it is actually closer to one so it is n to the minus one not n to the minus three over two which means that most likely the correct physics of the chromatin is another story but let's say it's not the one of a random walk but it's completely different so this quantity can be measured for RNA so to be honest I'm not sure but I don't think I mean probably it's not very interesting I mean people at least have not asked that question but the important point is that basically all this physics is done I mean it's let's say it's known if if chains are idea so now what does it happen if chains are known idea and so let me oh wow ok I'm not talking too much I'm talking too much but not all I wanted to say ok anyway let's go a bit fast so this is this is what it happens if two monomers actually here it's cannot share the same position in space this means that let's say real polymers have extruded volume this means that if you have a monomer here and a monomer here connected by a chain they cannot occupy the same position in space just because they are solid molecules if you want ok so now this introduces something that your friend has asked before introduces strong correlation along the chain now this correlation arrives not because this extruded volume but the extruded volume remains really a local effect in the sense that the two monomers have to really be brought into close spatial proximity to fill each other so you can think that this is not very important I mean it's a bit unconsequential no because I mean if they have to stay close I mean but the possibility they come close is not I mean it's rare if you want so this should not change the physics but a lot because because you have chain connectivity this introduces correlations long range correlation because let's say the short range correlation in space introduces long range correlation because of this connectivity and this changes completely the physics of polymers because this long range correlation now introduces a very complicated ingredient that has never I mean to nowadays been solved exactly of course I mean I've been solved in many accurate manners up to the precision you want but not exactly and this enters into domain of the so called self avoiding polymers I mean the word I think is self explanatory it means that polymers repel each other I mean locally so now this changes the physics completely because you can show somehow that in three dimension we have a new exponent which is new which as I say is a value which is approximately 3 over 5 0.6 and there is a reason for that and mention later and also the distribution function is no longer gaussian but is well it's called of the red and the closed type and this has something that I will try to tell you a bit more tomorrow with two different exponents so this exponent t which is the exponent of let's say of the exponential there is still an exponential part the exponential part is no longer the argument of the exponential part is no longer parabola but it's different from the parabola it's an exponent which is about 2.45 in d equal 3 and also the exponent this exponent gamma is no longer let's say is not 3 over 2 but it's something larger is about 2.6 which actually is much larger than one of the gaussian chain which means that contacts decay is steeper which is not so difficult to understand because I mean you have exclusive volume so you have less contact less favorable so these results are very well accepted nowadays and there has been proven in experiments also by computer simulation because you can do computer simulation on this object now the important point here and this is something I want to highlight that this exponent that I summarize here are in d equal 3 because now the exponent nu whatever theta t they also depend on dimension and this is a very important point for ideal chain this exponent here do not for ideal chain the exponent nu or the exponent gamma here but let's say the exponent nu does not depend on the dimension so you have a random walk so you have a random walk in space as the exponent nu is the same if you have 1d, 2d, 3d, any d while the exponent here the result here are for 3 dimensional self avoiding polymers this is a very important point sorry this is sorry what I want to say I skipped so well this is a bit of this is something that is a bit related to if you are familiar with the physics of let's say non-ideal gases and let's say the effective interaction of between two monomers let's say inside the polymer can be understood in terms of this myer function actually on the integral of this function basically this function so u is the effective potential that you have between these two monomers that depends on the physical distance between these two molecules and this typically let's say if you take the exponential let's say of this potential divided by kbt it has this shape so basically the propensity that two monomers are close in space is 0 because we have let's say you cannot have physical overlap and then it goes to let's say to this quantity minus 1 and gives you this myer function I don't know if you have ever seen that this typically appears when you do the virial expansion for a non-ideal gas for a real gas basically the integral of this quantity is proportional to the so-called exclusive volume between between the two monomers and and well depending on the value of this function you can have, this is just a bit of knowledge I mean you can have in this review a bit better explanation on that I mean that will take me more days in order to explain this better but let's say that for a self avoiding polymer the one that we are interested in we are in this domain between these two domains well the exclusive volume is sort of proportional to the typical volume occupied by a single monomer so let's say that surrounds the polymer it's called athermal and this goes in very high temperature or in good solvent, namely where you have a part of attraction at very large scale, but typically what it dominates, the physics of the polymer is the repassion on local scale so we can skip this so we are really in this regime so I don't know how much time sorry let me see what ah ah, well, I'm not said too much so we should finish at what time, 12, 30? I don't remember ok ok, so ok, I think I can tell you what I want to say so again this is the physics of reaction where we have exclusion between ah bitmonomer where they occupy the same position in space so two monomers do not like each other so they have to repel each other so the first person who understood correctly actually the physics of this object was PJ Flory as I said that was he got the Nobel prize in chemistry for his work on polymers and basically PJ Flory understood almost everything actually on how to describe this object in approximate way but he understood almost everything he invented he formulated a very simple theory ok in order to estimate precisely the exponent nu for non-ideal chain ok and his theory consists of two parts so the Flory theory is very popular actually and it works extremely well although it contains a lot of approximations ok, so basically the Flory theory is entirely built on the formulation of a sort of effective free energy for the single polymer chain where you have two terms so like any free energy we have an enthalpy term and the enthalpy term is due to the excluded volume interaction in a chain and the entropy term is basically related to the fact that the polymer molecule can occupy many positions in space so these two terms so now the overall idea of Flory was to let's say ok, let's try to estimate these two terms the first term of the entropy is relatively easy so he said let's say anansas for it he said ok, let's let's forget for the time being that my polymer let's say two monomers do not like each other they want to repel and let's use for the entropy of the polymer chain the same entropy of the ideal chain so let's let's use the same formula for it that's one step the first, the other step was to have an estimate of the enthalpy contribution ok and he said ok, this is related basically to the two body interaction because two monomers let's say one monomer interacts with the other ones and you have all the collection of different monomers and so this is the terms contained here times the volume occupied by the chain so basically this term here is a sort of let's say self density of let's say of the n monomers which occupy a region of sides r, of linear sides r and if you want this is a density square times the volume occupied by the polymer chain and he got such an expression for the free energy of a chain which is again this is in 3D which is relatively simple and now the only thing that you have to do you have to take the probability with respect to r which is let's say the variable you want and make it equal to 0 and if you do this simple operation you get an exponent ok, 0.6 for the exponent nu that actually is not exact so you get an expression which is 3 over 5, it's not exact but it's pretty close to exact and pretty close to the exact one and the exact one I mean is the one that you can compute in this particular certificate means or you can just get from numerical simulation actually the true value ok, forget this part here the true value is a value that you can compute up to any precision you want by using for instance a procedure which is coarse-grained placer normalization group that was done by many number of people and this value here is 0.588 I mean just by this simple calculation you almost get everything yes yes so this is a simple point to understand so this entalpic term ok is let me ok so is let's say let me do this proportional tool is given by the following approximation so because this is is related to as I said to the second virial coefficient so it's a two-body interaction in nature so this is proportional to rho which is density let's say the density of the monomers within the volume occupied by the polymer square because it's two-body interaction so you have one monomer interacted with the other one let's say suppose that this is the total volume span by my polymer and this is the typical size of this domain let's say span by the polymer then rho is forget prefactor let's say is n divided by r cube ok so this is my rho you do to the square and then so this is a sort of density of the enthalpy contribution times ok times r cube which is the volume occupied by the polymer so that's the estimation given by flow can be slightly more sophisticated but the essential physics is there if you want to understand better this term this is a bit like if you are familiar with the physics of real gas it's when you do the virial expansion of the of the pressure so we know that for an ideal gas actually it's pretty much the same thing so for an ideal gas we know that for an ideal gas we have pv equal n kbt which can be also written like p equal rho kbt rho is the density for a non-ideal gas you have correction here 1 plus 1 plus rho ok 1 plus coefficient let's say rho coefficient 2 rho plus coefficient 3 rho square blah blah blah and this is the virial expansion so now this thing here second virial second virial term is the same thing that you have here you are treating that as a gas of monomers what you are neglecting here in this approximation is the chain connectivity but the approximation is pretty good chain connectivity basically easy introduced what is the the chain connectivity I mean the fact that you still have a polymer and the chain connectivity it contains in the entropic term as you can see the approximation is quite rough nothing that this has to work because this is this is an approximation but if you bring these two things together it gives you a pretty pretty good result ok notice here I did it in 3D but you can do the same result in any D and actually that would be simply like n r over D and this becomes r to the D in fact the exercise can be repeated in any dimension and you will get that in the D dimension the exponent is new now it depends on D and is equal to 3 over D plus 2 well if you go into this if you are more curious I mean we can talk much more later but let's say so the result here is a sort of miracle in the sense that it works perfectly well almost well in any dimension you can do simulation for instance polymers in any dimension and you will see that this exponent is very well I mean this formula simple formula here it matches very well the result ok I mean to be honest I think nobody has really yet understood why these things work so well ok ok well this is a bit of a sorry it's a bit of a repetition of what I said here let's see again so that yes that this the end-to-end distribution function but this ah ok this is an important point so you can immediately see why I mean aware if you want the flow theory is wrong I mean yeah it's wrong in some sense because as I said I'm assuming at least flow re-assumed sorry no where is it yes flow re-assumed in in computing the entropy that the end-to-end distribution function let's say basically is the same you see that it's r squared over n so this is basically the logarithm of the gaussian ok it's still I mean flow re-assumed that the entropy of a gaussian sorry non-ideal chain is the same of the entropy of a gaussian chain this of course is a very strong approximation I mean it works in the end but I mean to a very good approximation but it's a bit mysterious so now if we go back to what I just said before to you the distribution function of the end-to-end distance for a non-ideal polymer is non gaussian it's strongly non gaussian but this is something that you can measure actually this is the topic of the main topic shown here so this is the end-to-end distribution function in this format is not important for a non-ideal chain in d equal to and in d equal three dimensions so this is basically a self-avoiding chain on a plane and this is a self-avoiding chain in 3d so apart from the shape is different of course but the main point here is that for a very short distance this quantity has to go to zero just because the two ends cannot sorry the two monomers cannot come into a spatial proximity so if you do the same plot for a gaussian it would go to a finite value for x equal to zero when the distance is equal to zero just because gaussian is a phantom chain so they can come close to each other so in the fluid theory you do not use such a shape I mean you use just a gaussian so in spite of the fact that this is a very strong assumption the theory still works but it's clearly wrong because if you measure this quantity it's strongly non gaussian I mean you can really see the thing here so these are the numerical results the points and the line is let's say this sort of red and yellow or shape so the function describes very well the numerical condition the numerical experiment so now you can maybe object but why we don't use directly a shape of this of this kind instead of using a gaussian for the estimation of the entropy you can do that but if you do that you do worse than this so that's why I said this is a sort of miracle and probably people have not really understood this theory works so well but it works and actually we see that it works also in other more complicated concepts so this is I think one of my last slides so this is an important point it actually has also a bit of to do with what I want to tell you about RNA so at this point probably tomorrow that polymers are intrinsically fractal objects namely they are scaling variant this means that a part of the polymer is like the old so if you measure let's say for instance the end-to-end distance of a sub chain of made of small n monomers the critical exponent you get provided that let's say your sub chain is long enough so where you can see the fractality is the same as the same exponent no it's the same between the two and and of course also for the distribution function you can measure the end-to-end distribution function let's say make the proper scaling to the corresponding end-to-end distance the different curves they will overlap so this is a if you want that this comes from the fact that let's say you have this universal properties related to the this exponent nu and this means that you can for instance ask what is the corresponding fractal dimension and you get the corresponding fractal dimension you have to a bit invert the problem so you can ask how many monomers are contained in a typical size of special size of length r and so you have to invert basically this formula here so from r as a function of n you have to go from n as a function of r and then the fractal dimension of a polymer chains with critical n nu is just one over nu ok, so this is basic physics of this object and so this is about dynamics but this is not really you I mean it's interesting but at this point probably I don't think I have time to tell you in detail and well I think this is a good point to stop because by the way we are almost done done no, no break so tomorrow I will so I hope this will be the necessary introduction to let you understand about, because we talk about a bit of branch polymers we will see why they are much more complicated the linear polymer of course and what they have to do with with RNA and how they can be used to understand these properties at least large scale physical properties of RNA so I don't know if you have question on this first part yes so your is a question or is a no ok, so yes the way you formulated the question seems that you know the answer that's why I ask no simple and clear explanation of these well, here there is no actually there is no real fractality inserted in the model ok, so the idea of flow was basically I don't think he knew about fractals because it was before the word fractal was invented this theory was formulated I think in the 50s ok or maybe late 40s now I think the 50s, early 50s that was that well the idea was curious to understand I mean because he had a measurement on real polymer of this exponent no and they seemed to deviate from one half of a Gaussian chain of an ideal chain so he wanted to understand if this exponent can be let's say can be derived from simple physical means ok, and he said let's try this I assume that because my polymer is very dilute so then only two-body interaction let's say really monomer-monomer-per interaction should be important and then I estimate my enthalpy contribution like this, like let's say rho squared times r to the cube let's say the time of the volume of the polymer and then I do not do sofisticate to my to the estimation of the entropic contribution I just use the same entropic contribution of the ideal chain, ok, because if you want this is just the logarithm of the exponential it's a you know I have so the let's say this has to be proportional to the total number of conformation we tend to end distance r, so this has to be proportional to the exponential of minus r squared, let's say and then I take the logarithm and I get this, ok and then I match the two if I match the two I get this formula but I mean there is no obvious reason why this thing should really work because you have a lot of correlations between the monomers because of the presence of the chain connectivity actually there is a lot of work in the literature and of course I don't have the time to explain this where people have tried to provide a physical intuition why this thing works but it's not entirely clear so some papers I've seen, they started from let's say a well-justified model for polymers with monomer-monomer interaction and they make a sort of mean field approach to it so it's more rigorous, ok if you do that you get the same result as first so this is a sort of mean field theory, ok, but if you try to go beyond mean field you do worst so it's kind of strange I mean if you use the same theory and then you go in field you make a computer correction in this sense it's a bit mysterious ok the more robust approach where actually it's not based on mean field is by using renormalization group ok, this is a bit it's more complicated but well if you are interested in that in these reviews there is a mention to the right literature but the idea is the following it's not so complicated so you have your polymer and this is actually it's more based on fractality and the idea is the following it's summarized by this question here so you have your polymer chain what you need to do in order to compute these let's say in a systematic way this exponent what you do is the following so if you are familiar with renormalization group let's say you do a bit the same thing you apply coarse graining namely you say ok I sort of assume that let's say a certain bunch let's say g of my monomers they form a sort of super monomer you can call it in this way and you compute the typical size in terms of the more microscopic variables at the same time you compute so in this gives you a sort of typical size of your let's say super monomers then self consistently you compute also how let's say how the basically the exclusive volume between these two super monomers let's say renormalized in the same way and you now have two equations that describe you how the typical size of this super monomers depends on the microscopic parameter and the typical the typical exclusive volume size between these two super monomers belongs in terms of the microscopic parameter of the same microscopic parameters and you repeat the procedure so if you have done the calculation correctly this procedure will not depend anymore on the microscopic scale like when you do let's say in this model when you want to compute the critical exponent they will not depend on the microscopic details and they go to a fixed point and the fixed point is different from the Gaussian one and you get systematic correction to this exponent so this is the rigorous procedure in many different ways in and to me the most let's say intuitive one is in real space like when you do cost gaining of this model in real space but for polymer is more intuitive because you are really using polymer quantities and if you do that you get the correction is the calculation a bit calm but they are not too complicated at least for a linear chain then if you have more complicated architecture it becomes more common for linear chain it is not so complicated and this is a renormalization you can compute the correction and you get if you take a sufficient let's say if you include correction and correction you get this exponent here but you can see it is not very different I mean if you are ok so somehow this describes the physics not too bad ok so I'll see you tomorrow