 No. So, do, do I have your attention, yes? So for the some, I mean, because you ask. So for the exam tomorrow, so the, let's say the, the weight is structure you have, I mean you will have a, I mean a list of questions, I ta je zelo najbolj, da jaz sem učujem, skupajte do vso nekaj dobro. Tako. Tako, je zelo, da je... Masih, da jaz sem učujem, da je... Zelo, da je, da jaz sem učujem, da jaz sem učujem, zelo, da jaz sem učujem, zelo. Čekaj, jaz sem učujem, da jaz sem učujem, zelo. in v komputeru, ker je veliko spes, in je bilo vsega, tako da je ne v druži. Zelo konceptu. Super. Užitejte svojo svet. Znamo, da je bilo na zelo v zelo v materijali, zelo, da se prišli, da je bilo vsega. Zelo ne bilo komplikatni. Poj Edgarine? Ne znam. Tega v picočku na taj poten. If you can bring materia, I would say yes. I think it's one hour and half in total time, so it's not very long. I don't know, you have other? Who asks? Sorry. I don't know you have other materia. Nismo ne zidim, da je etan natruh, ve. Pomin imam prebst. Zanimamo sprem. Odavimo je na to. Danes vstupijo nekaj, da je zemljen na wizijo relčen. Zanimamo, da mi ve quasi ne potrabilo vzaj. tudi tudi, ki je še vidjamo. Zelo, da teb Nem Boom tudi je ocu ljubek vse se tem, da se isto priče tudi, bodo sem neče dobljela tudi vse neče dobljela, pa izred sem čudovati volumen. Tudi teb po nekakov, zelo sem razljuval. zelo zelo, da je zelo, da je tudi zelo. Zelo je početno izgleda, da je tudi početno izgleda. Izgleda početno izgleda. To je sreda energija. To je, da je... moja energija je k. Zelo da imam tudi treba, in zelo antropi. Če bi se naredil. Zelo je zelo, da je tudi sreda energija. Kb je boljceman, ko je zelo zelo. Zelo, da sem bomo nedeteta, da se včešte. Tako, to, da bomo počutili, tako, to je izgleda, če je počutila, tako, je vsega monomer densitva vsega volima vsega volima vsega volima vsega volima vsega volima. Przesta, sem izgleda, vzgleda, kako sem ga počutila. Nekoj, dar je zelo, je to, vsega volima, vsega volima, vsega volima. Avške, vsega volima vsega, vsega, ampak pa je, da sem še pravidna, da zelo prit, je to, mjela, je to, ko je pravidna tukaj, zelo je to, ki počešel, da se je Between tudi 2 molekul ki drugi ga. Zelo to prit, to je pravidna tukaj, squadnja, imamo spet frač, ko je pravidna. Prefaktor in negrativam prefaktor. Znamenje, prefaktor numerikala, ker sem bil Raz, zelo sem bil všim da. Masne, tko ne jaz tudi izgleda v istimitvi kvaliti. Tudi tudi izgleda, če je raz, zelo je zelo, n-r, je densit, nr monomer tudi z volumem vzela všim trvene. Tukaj tukaj je kruč, ker so kako imelem vama trih. Tukaj, vse je isti vse vzgubila v v 2. n squared r to z d. Pon reviewsa le pwesi. A kar tukaj povrano, tukaj, Seveda je... Zelo je izgleda prejam, da je naredena kB vs. logaritm kaj je tukaj ložite svih srbi hrti. Ok... So situacije sem, pa naredena tukaj ložite kaj... In pak, kako delilo izgleda, ase da, Flori bilo vsego od vsega vsega, je bo nekaj, da je bilo šega gaušanished Baj začeli. Ko se je vse, je to načal, ampak zelo. To z analizim, začeli je to doživ Kožane. Na seglucki se z別nimi. Spodg nekaj. Na 10 monomer. Jse, zrej, ovo je dobro. So vsega tega. In taj je zelo prefakto. Znam, da je, da se... da je tudi konečno propečne, ker smo v vsočenju se, da je tudi nr. poslupovana konformacija v polimeru n po 10 monomersi, je tudi pravda eksponentialna. Tudi je tudi ne prefakto, ne jaz je prefakto. Zato je to ne 한국u v finačnji komputč. Zato smo zelo vzpečnje, tudi na logicalne različnosti. Fay statistical reasons. And then too add here to the fact of function of factor which gives the total which is proportional to the number of conformation with this sis r. And this is exponential of minus r square over nB square, times a factor something like that where is this typical monomer sis, Ta bovanje je vrstivaj. If I do all this math and I took the logarithm, in the end, I have the following expression, which is the one written there, which is this quantity here, so it is V2 n square r to the d. And this minus log of this quantity, ki ima plus r squared n b squared, plus konstant, ok, but constant, which do not depend on r. Ok, and this is the quantity that is written there. And I have to do this minimization, and basically I have to take the derivative of this quantity with respect to r, and just making it zero. Ok, so then I do the f over r, no, and then I impose it equal to zero, and from this I get that r has to be like bn to the nu, ok, where this nu turns out to be 3 divided by d plus 2. Ok, ok, so one thing that I have not told you yesterday is that from here you can see that for d equal 4, ok, because now this quantity depends on the dimension, so it's already, as I said, it's already non-trivial result, because actually it has to be so, no, because excluded volume interactions, of course, are more important in lower dimension, simply the polymer has less space if you want, right, so it feels more the effect of excluded volume. And they decrease, ok, as d becomes larger and larger. However, this formula stops in the validity of some dimension, and the dimension is d equal 4, because if you insert here d equal 4, you get one-half, and one-half is the exponent of, let's say, for an ideal polymer, ok. This means that this, let's say, for, basically the correct result is the form that nu is equal to 3d plus 2 for d smaller or equal 4, and nu is equal one-half for d larger than 4. So, and this 4, ok, is called the critical dimension of associated to the excluded volume interactions. So, that means that for d larger than 4 you still have excluded volume interaction, but they are not important, ok. The polymer remains, is described by this exponent, by the, let's say, the Gaussian exponent, ok. So, this is something quite important, ok. Ok, so why, so now let's go to, so I tell you that, because let's go now to, so this is not important, so let's go now to, yes, yes, yes. So, this is estimating this, well, the argument can be made slightly more, actually more rigorous, but this is just, let's say, the more hand-waving way of deriving it, ok. Yeah, because of that, I mean, the more rigorous thing would be required to much time, and I don't have such time. Ok, so now, yes. Sorry? Hawaii. Ok, so good question. So, you can, well, there is a, again, and also an hand-waving explanation. Look at this formula here, ok. So, you have that, this quantity here is always large, I mean, you can check it easily. It's always larger than one alpha, provided d is smaller than, is smaller, let's say, or equal. I mean, for d equal 4 is exactly one alpha. No, it's 3 over 6, so one alpha. For d equal smaller than 4, it's always larger than one alpha. This means that correctly, that's what the volume interaction makes the polymer more swollen with respect to ideal conditions, ok. For d larger than 4, this becomes smaller than one alpha, which would make, let's say, exclusive volume interaction, making the polymer less swollen than the ideal condition, which is a bit absurdity, ok. So, this means that for d larger than 4, this is the right exponent. Just basically exclusive volume interaction just stops to be important, ok, in this respect. If you want to see it more, slightly more rigorous, lean, you can look at this quantity here, ok. Because if you look at, let's say, to the estimation of this exclusive volume interaction, because if I look at at this quantity here, ok, maybe green is not the best choice. Red. So, if you look at this quantity here, the way to compute these, I mean, to estimate this critical dimension, so basically replace r, ok, replace r by let's say by its ideal size, ok. So, make the substitution here, ok. So, you would get something like v2 n squared times, ok, b to the d, which is a prefactor, n d over 2, ok. So, then this thing becomes v2 d to the d, ok, fine. And then n 2 minus d over 2, ok. So, then you see that this can be also I mean, this is, of course, the same as n. So, minus d minus 4 divided by 2, ok. So, now, for d large smaller than 4 this term here and for n large, which is, let's say, the limit of very large volume, this quantity here goes to infinity, ok. Which means that this term cannot be neglected when I mean, compared to ideal conditions, let's say, when you have a nice volume. Viseversa, for d larger than 4, this term goes to 0. So, which means that it's not important, ok. And it's, ok, so this is another way of seeing the same thing, if you want. Actually, it's typically the more solid argument how you see such a thing, ok. And in fact, d equal 4 is called the critical dimension. I mean, critical dimension with respect to let's say, to the two-body of the volume interaction, ok. I mean, this is now, I mean, it's well understood, I mean, this has to be so, ok. So, now, just I mean, I repeated this argument a bit here, because it's a bit necessary for the following. So, can I move further? You have had the question? Ok. So, now, what I'm going to say now is a bit of an application of let's say, polymer theory, let's say, but now for branched polymer applied to RNA secondary structure, ok. This is a bit of a work we did actually with Tangier. I mean, it's more broad in the sense that this theory of, let's say, of branched polymer was already applied to these to RNA molecules, especially to RNA secondary structure. I mean, not basically, you have seen with Tangier that, let's say, RNA molecules of course, they live in 3D, so it's important not only the secondary structure, but it's especially important in tertiary structure when basically these molecules really fold in 3D space. I'm not talking about that, I'm talking about this part here. And now one can sort of model or understand this secondary structure by using the physics of branched polymer. Ok. So, now, I mean, there are some other applications here, but they are not important in this level. So, what are branched polymer? Ok, so far we have talked about, so these are maybe needed here, so just branched polymer are, let's say, a more complicated class of polymers, because they are not linear, but they can be branched. And this happens so, let's say, the typical view of polymers is that you have monomer units followed by other monomer units and they keep growing, and they form linear chains. Ok. However, it can happen that you may have monomers not with, let's say, doubly functional monomers, but each monomer can be 3 functional. Ok, that means that two monomers can be attached here. Ok. Something like that. And so on and so forth. So, you can have here and it keeps growing, but now this monomer can be 2 functional and they increase. Ok. And you form, let's say, a branched structure. Ok. So now, why this is relevant for RNA, because you see, no, if you have this doubly, let's say, the RNA falls into, I mean, like this, then let's say on large scale you can, I mean, imagine that you forget these, let's say, these double folding here, if you want. And you adjust focusing on, let's say, the backbone of this structure. So the backbone can be thought as a branched structure. Ok. So this is the idea. And the idea will be like, ok, let's for the time being, forget about these, let's say, the base pairing and just focus on the backbone and try to see if we can understand if we can predict or say something about the structure of the backbone in terms of what we know about the physics of branch polymer. That's a bit of the idea. I mean, if there is anything interesting that we can do. So now the physics of branch polymer is, of course, I mean, you may imagine that it is quite more complicated, actually, than linear polymers. And in fact you can introduce, I mean, a larger set of, let's say, scaling exponents. So now, for linear polymer, we introduce this exponent nu that describes how, let's say, the total polymer size increases with the total number of monomers. So this exponent is still, let's say, this definition, is still perfectly valid also for branch polymer, because we have a number of monomers and we can always ask what is the typical generation radius as a function of the number of monomers. Now you see why it's important to define the generation radius because the end-to-end distance now is defined because we have many ends, actually. Then you have another quantity. OK, another quantity is, and this is actually something that is interesting for what I'm going to say, is suppose that let's say, suppose that you have, let's say, you take two monomers along the chain, and you ask what is the average path, I mean, on the backbone between these two monomers. OK, so now this quantity and you average over all possible realization of your polymer chain. So now this quantity is also scales, let's say, with total number of monomers with some specific exponent, which is called rho, OK, it's defined in this way. And you have two more exponents. So another exponent, so this is clear, so what I'm doing, let's say, I'm picking to, let's say, for instance a monomer here and another monomer here and I'm computing, let's say, this path here in terms of the number of bonds between these two monomers. I let you notice one thing, this path is unique, OK, because I'm not considering, so I don't have cycles in my molecules. OK, so then if I pick two monomers, the path that connects them is unique. The minimum path, I'm going from here to here to here to here, OK, I have no other choices. So I'm doing this for any pair of monomers inside the molecules, OK. And then I'm averaging over all possible conformation, because the bonds can be rearranged everywhere. And so this is the quantity I'm looking at, and I mean I can define this exponent here, because it has to scale with total number of monomers of the chain. Then I have another exponent here, which is defined as the following. So now suppose that I have my molecules and I cut at random one bond, for instance this one. Then I'm considering how many monomers are on the left part and how many monomers are on the right part, OK, and I, I mean just measure them. And I average over all possible cuts in the chain, OK. And so this is a sort of this is called the average size as a function of, so the average the mean branch size, sorry. And this is also as a scaling property. This is also scaled as the total number of monomers or bonds inside the chain as an exponent, which is called which is defined epsilon, OK. So now these two guys Fandres, Munch and Madras proved that whatever it is, let's say the molecule you have and what is the ensemble is epsilon is equal to rho, OK. If you are in this, I mean this is a very, very nice work. It's a bit mathematical work. So if you are, if you like mathematical details, they prove this by using statistical physics reasoning. So it's not a simple argument, OK. For the time, I mean just accept it. I mean it's a very impressive work that you did here. And then you have another exponent, OK. This is less interesting for what I was saying for your personal culture. So suppose that again you take any two monomers on the chain, you compute the path between them and you measure the spatial distance, OK. So this quantity scales with the path length as L to the two new paths. So you see why basically I introduce an exponent to new, but it's related to the path on the chain. It's like this is an independent polymer and you are measuring, let's say, another exponent. And let's say now, by scaling argument it's easy to see that new path is not independent by rho, but it's just the ratio between the two. It has to be so, OK, just by scaling argument. But this is not important from God's angle to say, so I mean just keep it, but I mean yes, this one. No, actually you take two points on the chain, two monomers, for instance these two and you do it for any possible path and you measure how many steps I have to walk on the molecule to go from here to here. OK, that's easy, that's it. The next you mean this one? Second and the third. Ah, the third is done. Suppose that I cut this one bond. You count how many monomers are left here and how many are left here. OK, that separates in two independent molecules. OK, and you have so, and then you have some size and some size, and you do all this do the same operation for all bonds and you average the possible conformation of the polymer. OK, you get this quantity. Subbranches, the typical size of the subbranches. OK. To be honest, in the calculation I always took the smallest of the two to avoid finite size effect. I mean, you can take just one because I repeated the procedure for all possible polymer conformation because you have to imagine that the ensemble I'm talking about is where let's say this branch it's a randomly branching point arranged everywhere. So you have a high number or possible it's even higher than the ones of a linear chain. I don't have only, let's say, the exponentially high contribution from the fact that the chain can assume all possible spatial conformation but they have only another entropy if you want associated to the fact that the branches can be arranged everywhere. So in this creates let's say the other like yeah, this creates a higher ensemble if you want. Is it OK? So this is the physical meaning of this spawn. So now, OK, for this, so now this can be also understood in terms of the flow theory. I have to do some let's say some additional ingredients. I mean, the contributors of this work were let's say Isankson and Lubenski and yeah, this work here especially I mean, if you look at this it's a bit simpler work. It's based on the use of the flow theory. OK. And so let's look again at this quantity here. So now, the flow theory again for now for branch polymer randomly branch polymer OK, so I mean it contains again the same terms. OK, so always an enthalpy contribution and entropy. OK, so now this contribution here OK, I'll explain this terms that are on the blackboard, on the slide. So this terms here is the same. OK, so doesn't doesn't change. Approximation, of course. But doesn't change with respect to one that I have defined and I have introduced for linear change. So this is still let's say v2 n squared r to the d. Really the same. I mean, why doesn't change? I mean, I can make the same approximation if you want, I mean conceptually. Because to compute this term I have, let's say I have not really used any information connected in a chain. I mean, OK. So it's just like it's just like, let's say, collection of particles with a screw volume interaction. And that's it. So the fact that is, I mean, linear chain or branch polymer doesn't really matter for estimating this term. So this is why it remains the same. OK. Well, the trickest part is this one which is the entropic part. OK. Because as I said now the entropy is more complicated. And this is the main basically contribution of this work here. Which is, by the way, it's not a long paper. I think it's four, five pages. But I mean, it's pretty clear. So in my opinion. So if you want to understand a bit, you can look at this paper here. So and now it is estimated. So let's, OK, sorry, I can't set the molecule, but let's draw it again. Basically, the scheme that I have there. OK. So now this term, if you want, it actually contains two contribution according to these guys there. OK. So I write let's say S over so I write D like S1 divided by KB plus S2 divided by KB. OK. So these two contribution are highlighted here. This one and this one. OK. So the reason why I call it this way is for the following reasons. So this first part here let's say S1 is sort of reminds of the same entropy contribution that is let's say that comes from a linear chain. OK. For a linear chain, you remember, I had that this is R squared divided by N saluda by NB square. OK. Here in this case, because I have a branch polymer let's say so in this term here comes from the fact that I have, let's say, a linear strand of N monomers. OK. Here my effective linear strand is not made by a monomer but it's made by on average, let's say, by L over B monomers, because let's say my linear strand is the one that is effectively playing a role is the typical length of a single path on the chain. OK. So then I divide by L over B because this is the number of monomer belonging to an effective path. OK. And this is why I have this term here. Which is basically this one. This is the first one. OK. So in these are more or less the same physical, of course I mean I have many possible paths but let's say the idea of, let's say, the hypothesis of these people is that that tends eventually only in some prefactor because again here I'm completely neglecting the prefactor. And the second term here, this one, which is the less obvious, let's say, is related to the entropy of branching. OK. Because as I said, I mean the branches can be, I mean all these branches can be arranged everywhere again. So if you want this is a bit of a term which is similar to this, I mean in let's say in the power law form but is of this kind. OK. And so what is the physical meaning of this? So suppose that so now I have to work a sort of in the space of branch of, let's say, of the branches nest. So where now the typical size is not played by R but is played by the length of the path. OK. And then I have to divide by NB squared to the usual term which is R squared over NB squared but where R now it's L, it's not R because this is an entropy associated to how I can rewire my branches on all possible molecules. OK. I don't know if I've been clear. I mean it's the same, it's a sort of Gaussian term if you want, but where now the typical size is L so it's the average path R. OK. So then I have two contribution here which is this one which one is the elastic contribution and the other is the entropy of let's say the entropy associated to the all possible size in real space plus the entropy associated to all possible sizes but in because of the rewiring of the branches in the space of all possible connections. And now I can in fact if I for the time being I completely neglect these well, OK let's do it story. So now I have these so in the overall I have the following let's say I basically now these these free energy which is now it can be written like this. OK. Cancel this so now I have V2 N squared over R to the D plus this one which is R squared LB this term here plus this one so now this term here is more complicated in sense that because now I have two variable so it's a sort of function of two variables the free energy I don't have only to minimize with respect to R I have also to minimize with respect to L because it's also let's say there are two variables one is the spatial sides of the polymer and the other one is the typical path on the branched on the branched molecules. OK. So as you can see it's I mean this is simple, I mean it's not complicated but let's say these it's more sophisticated like of course I mean because the molecule itself is more sophisticated. OK. So the interesting one thing that you can OK. I mean we can see another feature of these OK. OK, first of all we can see quite of a simplification in this formula that the two terms here oh well sorry the terms here the square volume interaction as I said does not depend on L. OK. So you can let's say here you have to perform the two following minimization you have to do first the derivative of F with respect to L and imposing that equal to 0 and then the derivative of F with respect to R and then imposing it equal to 0 in these two conditions. OK. But now the first one is the one that you can do first because this basically at this term does not enter. OK. Because does not depend on L. So you minimize this first and then you minimize the second one. Yes. Yes. No this is OK maybe drawing is not ideal but this means that this defines what is this R of L. Actually you don't have to worry because I'm not going to take I mean this quantity is not applicable I mean it cannot be measured in RNA. OK. So it is rated in RNA but if you want to know about this so this quantity here it measures how to say this so take to let's say a pair of monomers along the branched polymer you compute the typical special distance is a function of the linear path on these molecules OK. And this is this quantity. OK. Is a function of path length is not as a function of the total number of monomers. OK. So that's the thing. While here R is the the typical monomers, the typical molecule size is a function of total number of monomers to separate quantity. OK. I mean I'm not saying that they are not related actually are related by this scaling relation. That's why this new path exponent which is defined for the paths is the ratio between where row. OK. I mean I understand maybe it's not obvious but this let's say but this is a bit critical I mean if we want to apply this to RNA molecules. OK. For let's say if I'm assuming that the typical secondary structure of RNA molecules I mean we look like this then in the secondary structure let's say analysis what I can measure I can effectively measure the length of the paths OK. And this quantity here which is the ones that I'm going to talk about so this one and this one. OK. But I don't have access to neither to this one because that will involve a measurement of the molecule in space and if I'm measuring the secondary structure I don't have access to that. OK. I need I need some spatial information but which I don't have. OK. So when I'm talking about yes. So now basically you have to do this exercise. And so I think I have it in the next slide. Yes. So let me see. Yes. So the if you do if you minimize this one the result is so it gives you this one. So basically you can try yourself. OK. If you do this. OK. It gives that R like L over B cube times N. OK. So this is anisimatomy. It gives you just this. OK. So now of course I mean R because now I'm minimizing with respect to L. OK. And of course R from this one will contain the terms which still depend on L and the terms which depends on N. OK. This one. So in order then to minimize to this you put relationship back here. Now you have that F will depend only on L and N. You minimize it and you get a relationship between L and N. OK. And this will give you from this will give L as a function of N which defines my exponent which I call it rho. OK. And and then once you have that you basically reinsert this one here and you have a relationship between actually rho and N. OK. Which is the relationship which appears here. OK. So now to well this is a bit of a so I mean so now you have sort of some prediction. OK. For for the exponent N for the exponent sorry we have a relationship between the exponent rho and the exponent N. OK. So you can compare to to the available let's say numerical results in the literature and this is a bit of a comparison but I mean this is a bit technical so I think we can we can skip this. So what is more what is more important also that you can also move further of this of this flow theory in compute it's the corresponding distribution function for this quantities. OK. And this all obey some let's say some scaling relationship. OK. And let's see maybe this is a bit so let's let's look at this. OK. Let's maybe let's go to RNA and hopefully that will become a bit clearer. OK. So this is the maybe some kind of secondary structure that I guess you have already discussed a bit with OK. So RNA we know that RNA because of the base of the base pairing. OK. It takes this peculiar form. OK. So you have that because of this base pair formation you have what is called the secondary structure. OK. And this is basically I mean it's so this is so this is a pairing. OK. But it's not I mean it's a pairing along the sequence and though and you have so this is for a very short RNA fragment but you have a longer one you can form many of this base pairing and you can have a larger molecule that will resemble like a branch structure. OK. Then this structure will fold presumably in 3D space and you have the so called the tertiary structure but here I mean what I'm going to talk about here it's a bit this secondary structure not the tertiary one. OK. Because this is a bit the goal of this of this lecture. OK. So now this is something that is related to a larger molecules where you have you have talked with Ange about multi loop so this is a structure where you have basically you have this multi loop of degree 5 means that you have this structure which is not completely base pair but you have many arms that protrude from it and this is called the multi loop of degree 5 because you have 1, 2, 3, 4, 5 let's say arms protruding from this quantity it's one of these bigger monster let's say from bigger RNA molecules that produce such kind of branched branching structures so this is a specific example from a long non-coding RNA which is just a specific RNA molecules which is 600 nucleotide long that form this kind of structure OK. I don't know actually why it's called Braveheart but that's that's the name for this molecule. So now this kind of I mean I guess Ange has told you that so this kind of structures are the ones that are predicted based on some let's say the empirical force field so basically we have some programs that try so you have your many RNA sequence and what you would like to know is how many possible faults these long RNA may produce so there are let's say there are empirical programs produced in the literature that can be fed by your favorite they can be fed by your favorite RNA molecules and they will produce a certain number of these typical faults OK. Secondary structure so they will give you a hint I mean an idea of what is the secondary structure associated with that that molecule I mean we know that these secondary structures are not unique in the sense that you have many typical I mean a lot, I mean a very huge number just because the pairing between the different nucleotides is not unique OK. I think that was a stress many times already so because it's not unique you can actually produce a large number of these molecules OK. And the idea is that OK because you can produce a large number of these molecules this can be mapped into a corresponding high number of branch molecules and it I mean and maybe by comparing to this kind of theory we learn something from let's say from the typical statistics of these of these molecules. So now this is not completely new and it's quite known in the literature in the sense that there were a I don't know if I have no sorry so this is a collection of let's say of precisely of these kind of results so people in the literature have measured the following. So suppose so you have many many molecules of RNA molecules and what you do you are using precisely this program, these tools to predict secondary structure and you do so once you have let's say your produced molecules from from these programs what you do you can analyze the output of course in this program we have introduced a quantity which is called which was called the maximum ledder distance. Basically this quantity measures the typical let's say the largest path on these let's say on the resulting branched molecule as a function of the total size of the molecules. If you want people have measured this not the typical L but let's say the average largest L but in the end it's the same thing because by scaling argument this I mean the the maximum ledder distance should always obey to the same scale in relationship. As a function of the total number of base pair of total number of bases in RNA molecules namely as a function of total of n if you want where n here now is not n but it's n then nucleotides so now you have to just imagine that instead of a but it means from the it's the same thing. So people have measured this quantity so for for RNA molecules and they found here you have I guess you have a picture from you have an indication for these quantities I mean why this is so interesting I mean I've been so excited about that because you can do two experiments here one experiment is I measured this quantity for specific RNA in particular they measure that for RNA viruses so many RNA viruses here one is the tobacco but you can do it for many of them and you can compare the same quantity for random RNA molecules and random I mean that you construct artificially they don't exist in nature artificial RNA molecules made for instance which homogeneous composition of bases like 25% of A, T, C and U and then you but they are not like you say that the total number of nucleotides is some number so 25% is A, 25% is T, 25% is T, 25% is U and you just make a random sequence out of it and you use these the same programs to measure the faults and interesting results they got is that the not random RNA namely the RNA of viruses have a more compact path distance then do one let's say for random RNA so this is so the results for viruses are this one so this is let's say the maximum letter distance is this one so the results for random RNA is here this one so it's systematically higher the results for random RNA which means that problem of course not surprisingly that viruses I mean living organs have an RNA which is cannot be considered as random but has to be I mean it's different ok and this also means that presumably evolution as far as of course I mean this should be expected but this is the first let's say one of the first quantitative experiments that have shown that there is I mean there is an amount of non randomness that can be quantified for real for real bios and the quantity that people have measured is precisely this one ok is it clear so you can I don't know why I didn't put the references ah yes ah yes these people here so a lot of work was done was done in the group of Bill Gelbart in the United States plus the collaborators of course and I think to my knowledge the first work that has put forward these that have used this sort of relationship is this one predicting the sizes of a large RNA molecules published on PNAS as you can see in 2008 now so now it's 16 years ago ok so this is another plot basically showing the same thing probably this is with less points this is more clear so here you have maximum ladder distance if you want this quantity here as a function of the sequence length ok so what these people have found have found the following relationship have found that if you measure the maximum ladder distance for random RNA the curve you found is the one that is shown here these are the points for random RNA and they found the dependence as a function of the total number of nucleotides of about two-thirds 0,67 that was let's say it's a fit so this relationship seems very robust basically and these are the points that you have here as you can see it's for different families so these kind of things so the tobacco is here the tobacco virus is well it's one of the largest one ok so basically how do they just a few details on how do they get these points they so let's say how do they get the random RNA and then they compare to real RNA they had the same so they did two things actually I said here so they consider a homogeneous composition of nucleotides namely 25%, 25%, 25%, 25% it fixed n sorry it fixed the number of nucleotides or they've also taken let's say the non-homogeneous composition of each RNA virus and they have reshafled it so basically they said ok the composition not the one, let's say not the one that you have been linear in real sequences the amount of nucleotide is the same but it's let's say you randomize the position along the sequence so this is another way of doing randomness if you want and if you do this so if you compare this to operation you basically get the same exponent here however for the real and non-random one let's say taken from nature and which let's say with the distribution that you have you get this data as you can see this maximum letter distance is comparably lower with respect to the number one and and this has to be I mean according to the interpretation of course but I think it's really I mean something that you can believe in that is sort of is a sort of let's say in information saying that evolution has pushed strongly for having real non-random sequence not only non-random in let's say in terms of the choice of the right amount of of base pair composition of base sorry of basis composition but also in terms of how let's say the molecules are exactly allocated along your sequence because then you have to form this base pair and this base pair has to be functional of course because this virus is I mean sort of so this is for instance this is something that you can also see a bit qualitatively so this is a phage so a phage is a virus that infects bacteria ok is a single strand RNA virus it contains about 4,000 nucleotides so this is the sort of conformation you get for its corresponding secondary structure and this is the same let's say secondary structure you have for a similarly composed RNA but random I mean randomize the one of the way that I mentioned here the same size more or less is 4,000 nucleotides is 4,200 otherwise it's pretty much close you see that the non random one has a much larger than this one you can also appreciate that by just by by inspection and this is a bit the message contained here so what these gentlemen also did and this is an hypothesis they said ok now I mean again this is a a ladder distance is a sort of measure of the size of your molecules but it's not a measure of the size in space because let's say this kind of cause, I mean of this kind of structure you get for let's say from these programs they only tells you how much frequently two bases interact, they form base pairing but they don't tell you I mean the extension of molecules in space because they don't know anything about that ok so this is the important thing however what these people have done so in this is an hypothesis is the following ok now I have my maximum ladder distance so then assuming that the maximum ladder distance is let's say is a measure of the average path on the molecules then I can assume that ok then the average path falls randomly in my in space and then so the associated generation radius, so now this is the generation radius of the molecules in space has to be just has to just grow like the square root of let's say the this ladder distance because I mean if this is randomly folding in space then this has to be like if you want just a random polymer to be like the square root of this quantity and this seems very interesting conclusion for the following reason because if you take the square root of this number which is very close to 2 third you get that how the typical RNA molecule has to fold in space has to be like 1 over basically n to the 1 cube, 1 over 3 and if you have such a kind of folding means that my polymer the RNA molecules has to be basically space filling has to be compact you see this you see this because if you compute let's say the typical monomer density within the space occupied by the volume so this has to be like n again like this divided by the typical size this one sorry the typical size r cube and because r I'm predicting that has to be like 1 over 3 this one then so you have r to the 3 and 1 third is n, so n over n just cancels and this means that this is order of 1 so basically the typical density of let's say of my RNA molecules inside its volume is constant and the polymer is space filling so it's just occupying all sort of possible monomer inside the space and this is so this is an hypothesis of course it's a prediction more than a hypothesis because of this work but it's an interesting one because it tells you that it's an interesting one for two reasons the first reason is that because the molecule is compact then can fit inside can be efficiently if you want inside its own volume and the second in my opinion more interesting conclusion is that by measuring let's say something which doesn't know anything about let's say the special folding of the molecule you actually are giving a prediction about how the molecule would fold so now the interesting conclusion of this work is that why this is important why the density has to be constant is important also in other respect because you have to think how the virus is and in RNA viruses I don't know if I mentioned to you but they are the RNA is containing some in a structure which is on the capsid of the virus so now the capsule of the virus is a very tight environment so if RNA attains the maximum compaction then it means that basically it has done efficiently this work let's say the RNA can really stay inside its own volume and this is again this is a conclusion that you can only deduce from the properties of these which again doesn't know anything about the special size so it's somehow itself consistent with the physical expectation and otherwise it's the only, yes because you are only applying let's say the physics of these paths, okay you don't know anything about the special size I don't know if well in the end we have a bit repeated this analysis but maybe sorry, so it was confession ah, okay sorry I don't know what sorry, let me just check with the time no, because I mean maybe we are entering too much details later so it's probably because this it's the thing that I wanted to tell you so now maybe it's good if you have some question this point, yes ah, this is simple because so it's not so suppose so now, okay, what I can measure I hope this is clear it's only by this the outcome of my numerical experiments I can measure this maximum ladder distance which is basically the longest path on this branch 3 this is something that I mean it's not complicated to measure I mean you just count so this is what you get these are just an example but you can have many so this is for a again this is for a real virus it's a virus that attacks bacteria and this is for the similar size RNA I think also we do more or less the same composition but just randomize I mean that I just shuffle the position on the sequence of the different nucleotides, okay and I get this thing here that you can see that it has a larger distance okay sorry, it has a larger maximum ladder distance okay now, suppose that as you can see, I mean from here if this makes more intuitive you can consider this one and you measure this quantity here or this quantity here as a function of certain number of nucleotides and you get this scaling relationship that basically the maximum ladder distance as a function of the number of nucleotides scales like n to about two-third in sequence is random because I mean there is a complication when you cannot do really I mean this is the work we did with Ange but it is a bit too detailed so I don't think it is really work, we don't have time in if I had more time I could explain you but let's say if you have only this kind of measure it is difficult to get any scaling maybe you can have an idea but it is difficult to get any scaling for specific molecules because basically you have only so to get a scaling you need many n for a single molecule n is fixed so you cannot do any scale scaling out of it although it can be done but it is a bit more complicated so I don't have the time to talk about that but the interesting conclusion is that but you can do that for certainly for random molecules in the sense that you have the same it is a random molecule you can create it artificially you can do that if you want you can play such a kind of game or you can do what these people have also done here and also I think Andrzej has done this kind of work you can have your specific RNA sequence because RNA sequence in viruses they span in enormous range so you can have small RNA small in terms of nucleotide or very large RNA in terms of very large number of nucleotide so you have such a kind of data on top of that you can have the same sequence composition and you just randomize it you can move nucleotide around and you can create your own ensemble of folds and you can this is also a sort of randomness then and then you can repeat the game and you can compute this maximally as a function of number of nucleotides you observe this power law after that so now we have this power law now and this is a bit some kind of discussion that did in this work I think it was already in this first two work you can say that ok, now I have my maximum the distance as a function of n now assuming that this ladder is a sort of let's say now this ladder has to fold in space but how can fold in space in kaj fold just randomly it's like a random walk then I have no other possibilities that the generation rates of the polymer in space of the average RNA molecules in space has to be just because it's just like a random polymer has to be the square root of the maximum ladder distance no, exactly like if you have a random polymer you have the square root of the number of monomes if you have a linear chain and then if I took the square root of the maximum ladder distance it has to be like n to the one third you see, it's just the square root of this is it clear? well, because all the random points follow on this curve also if you take these points here jamil, which are non-random and you randomize that in the way I said that you just reshuffle the sequences they all fall more or less here if you want to think no, I don't think there is this data here but yeah, they are all fall on this actually there is it's not completely true there is a small difference but okay, yeah, okay I will not complete this up, they don't fall this curve refers to an homogeneous composition of nucleotide if you don't have a non-homogeneous one but it's still random in the sense that you randomize along the sequence this curve I think can become a bit lower but the scaling is the same so this exponent is still one third so this is the important thing actually it's more than so this is a log-log plot maybe it's not evident but it's a log-log plot spacing is not the same so it's a log-log plot so if you are doing the same curves but for different folds and the composition not homogeneous what you can see that you have another set of points but the curves I think it should become a bit lower but the slope is the same which is the important thing the fact that so basically this means that the pre-factor is different and that can maybe be related to the composition here but the scaling remains the same which is of course the important information of course the pre-factor is also interesting but it's probably related to but if you want the pre-factor is sort of renormalization of the effective bond let's say the effective bond here but if you don't touch the scaling behavior then it's fine so this is the important thing that enters by the way in all polymer theories it's the scaling, it's not the pre-factor yeah, they are error bars this of course depends on the statistic of your sample here I'm not sure because it's the report from this paper I'm not sure if they are error bars or standard deviation but they should be error bars I think at least I just don't remember I'm sure it's written in the paper because why do they think they are too large ok, no ok, but consider that well, producing these folds the statistic is not really enormous anyway I think when you produce these folds I don't remember exactly the right statistic but typically it's not like you have an ensemble an exponentially large ensemble when even for random RNA I think the amount of folds you have is on the order of if you are lucky with this kind of program calculation is about maybe 1000 so 1000 means that you have an error you know about what, 30 or one third of the value so it's possible that the error bars are quite I mean they are not negligible they are computed by some calculation they are quite because what this code does ok, the Ange has mentioned to you the presence of pseudonauts these folds what they do they put all possible they spare they have an energy score and they minimize this energy score but of course they produce sequences so you can give I think I have ah yes so this is how it's very empirical there's nothing really holy in that so this is called Vienna RNA it's called Vienna I think because it was developed in Vienna and you have a different version of it it's one of the it's very used in the work with Ange we use this but if I remember correctly there are others anyway so you have an energy score that you need to minimize inserting that your favorite RNA sequence and so basically the minimization of this energy score so you have a term which is just an offset which has to be there for some technical reasons then you have a term which favors or penalizes I mean depend on the amount of these term branches I mean like this or a term that favors or penalizes the number of unper nucleotides and that produces these more or less I mean a more or less large number of these of these holes and that's it but out of it you don't get only one structure one possible structure you get many so then you make a ranking and well this is a bit rule of thumb you may decide how many of those you want to retain because the energy you get they are all very close to each other but in the end I think in the best case scenario you get 1,000 so I have to say as you can see now let's say the first version of this Vienna René was published so now this is the newest version but the first version of this Vienna René was published in 1981 so not really let's say wow and there you go and there are now new versions but all the time they have I mean in these new releases they have changed a bit these weights here so the way these terms were established were based on let's say on the best agreement with experiment I think it was from calorimetric measurements but otherwise it's very empirical the important thing of let's say a very important assumption of these kind of models is they completely neglect the present they don't take them into account the presence of pseudo nodes because as I guess they make the things much more complicated and much more if you introduce pseudo nodes in the calculation basically the problem starts to be you can do that only for very short RNA molecules if you are going to large RNA molecules like the ones analyzed here the problem becomes impossible to solve the number of conformations you have to estimate becomes exponentially large so let's do that I can give the slides but this is a bit too much detail no, what was fun what we found is that basically so these terms I didn't put the precise values here but basically from it changed in the course of the years so in the first version this branching term was higher compared to this amperonuclotide no, sorry, it was higher no, sorry, it was in the sense that branches were penalized more so now in the newer version branches are less penalized compared to amperonuclotide but it becomes a bit it's a bit of an empirical work but still, if you want to get understanding about how your RNA molecules should work you use this kind of tools there are others but they also produce different results but it's kind of it's basically the first thing you are using and there is a server for it in case you want to play you can just use it, it's called V Vienna again it's called Vienna RNA and well, there is a lot of stuff we have done but this is just the acknowledgement of the different people which I think deserve some acknowledgement and otherwise I don't know I think we can stop here because I think it's a bad time I don't know if you have questions or questions about this okay so I guess I'll see you tomorrow for this so as I said it's just a basic question so don't be too worried what time is it, 11? 11, right? okay I'll come a bit earlier that's it