 The task for this morning is Mattia Bertu Bertini, who presented us this work on omelsovering polymers with unconstrained topology. So, Mattia, go ahead. As for you, 25 minutes, things like that. Hello, everybody. Today, I will talk about the work that I've been doing during this year. It's part of my PhD program in biophysics here in CISA, under the supervision of Angelo. The title of my presentation is a computational model for melds of polymerings with unconstrained topology. So as the title suggests, I will deal with melds of polymerings. That means solution of identity of polymerings. And with unconstrained topology, I mean that I will deal with polymers that can be knotted or unknotted and is there topology can change over time. And regarding a couple of rings, they can be either a link or a link with different kinds of complexity. So the field of study of rings with non-trivial topology, let's say, can be traced back to 40 years ago. It started from Pierre-Gilles Dejeun when he envisioned the possibility to synthesizing polymer materials from topological interlinked rings. From that, he coined the name of Olympic gels because such structure would resemble the Olympic symbol. And from that moment on, a lot of people, both from experimentalists, and they started to characterize and synthesize these kind of materials. And also, simulation and models have been started to be studied and developed. What is interesting is that such structure exists in nature and they can be found, for example, in the mitochondrial DNA of a tripanosomal parasite. This structure is called the kinetoplasm DNA and is made up by thousands of mini-circular DNA maintained through the action of a family of enzyme called topoisomerase. These enzymes act as sketch in the right picture, so the enzyme cuts a DNA strand, make pass another strand of the DNA, and then it moves back the cut strand. In this way, they can cutenate the DNAs. And if the DNA strands belong to the same, for example, to the same DNA, they can also form knots. And inspired by these kind of systems, an experiment has been done three years ago in Stemford in a laboratory led by Spakowitz. And in which they were able, they showed that they were able to synthesize the Olympic hydrogels starting from a dense solution of uncatenated DNA rings through the action of topoisomerase. They showed that the topological state of this Olympic hydrogel was dynamical in the sense that decatenation or decatenation were happening over time through the action of topotoo. And so the network was not quenched, let's say, with quenched topology, but also what they showed that by inactivating the topoisomerase, what they got was a proper network frozen and with interesting features. They performed the logical measurements. And what they saw was that the system with active topoisomerase resulted to be codified by the action of this enzyme respect to the uncatenated case in the sense that the relaxation of this system was happening at a time scale that was faster respect to the case of uncatenated rings. This is because the topological constraints due to the chain uncrossability were resolved through the action of topotoo and the relaxation happened faster. So the system was more fluid, let's say, respect to the system without topotoo. And inactivating topotoo instead, what they got was a network that was behaving like a solid. So this kind of system are interesting because their properties, let's say, can be tuned by tuning the action of an enzyme. So for this reason, we got interested in this kind of systems. And we wanted to build a model, a computational model, that is able to characterize better this kind of systems. So we started looking at a really simple computational model that is based on Monte Carlo Dynamics or Lattice. The Lattice site can be either empty in this model occupied by one monomer so that they implement excluded volume interaction or maximum to occupied by two consecutive monomers. In this way, like in this Lattice site, for example, there are two monomers belonging to the site. On the same Lattice site, there is this store length. And this store length implements a kind of elasticity of the polymer. But it's important to stress that excluded volume interaction are implemented in this kind of system. And the dynamics is really simple. It's local so that you can rely on the dynamical properties. You can study the dynamical properties of the system. And it is based just on randomly picking a monomer from a random polymer, attempt a move towards a nearest neighbor site. And these moves are accepted if the chain connectivity is preserved so that polymer has to be still connected all the monomers of the polymer. And the site is either empty because of the excluded volume interaction or occupied by one of its nearest neighbor on the polymer backbone. So in this way, the store length can diffuse along the backbone of the polymer, for example. The problem, let's say, the problem of this computational model is that it preserves the topology of the system. And so we had to add a new Monte Carlo move. We had to come up with a new scheme that was taking into account the change in the topology due to the action, for example, of the topoisomerase. We wanted to find a still a local move because we wanted to study the dynamical properties of the systems. And what we thought about was that maybe swapping two monomers, two non-consecutive monomers, which are nearest neighbor on the lattice, could do this job. Like this is just a sketch picture. So you have two nearest neighbor monomer, let's say on the lattice, on this lattice, strange lattice. You swap the monomers. And then from a catenated structure, you would get an uncatenated one. So we looked at all possible, at all possible conformation on the lattice that we were studying that was, by the way, FCC lattice, such that we looked at all possible conformation, such that the swap of the monomers preserved the chain connectivity on the lattice. And what we found was that there were 24 conformation whose monomers center can be swapped and do not cause a change in the topology. And I mean this kind of conformation. I refer to conformation because we looked at three plates of monomers. And we were swapping the central monomers. And we looked if the chain connectivity was preserved and what happened to the topology. In this case, this is one of the 24. You start from this. You swap the central monomers, so the green and the yellow. You get this. And if they are within two polymers, you start from this, and you get this. And of course, the topology, the two rings were uncatenated and they remain uncatenated. No change in the topology can happen within this swapping move. Instead, we found 12 conformation whose monomers centers can be swapped and do cause a change in the topology. And this is one of the 12. So you start from this. You get this. And within two polymers, let's say. So you start from this configuration. For example, the red polymer is on top of the blue one. They are uncatenated. But by swapping the green with the yellow, as you see, the red passed through the blue one. And so they are uncatenated. So the system has changed the topology. From two unlinked rings, we got one catenated structure made up by two rings. These examples are referring to two rings. But also this can happen within the same polymer. So if you swap two monomers belonging to the same polymer, what you cause is not a catenation, but you form knots as the ones that I was showing at the beginning. And so once we got these 12 conformation, we got our swapping move that is basically we run over all the couples on our system. And when we find two monomers, two conformation, that are within the 12 that we found, we swap the central model. So once we establish this scheme, the only parameter, let's say, of this model is the rate of action of this move. So how many times or how far in time we chose to use this Monte Carlo move, the fact that happens at a finite rate, we wanted to do that because topoisomeries, for example, work at a finite rate. So it is important, at least for us, that this move had a finite rate of action. For the simulation that we, for the results that I will show you in a moment, we chose the inverse rate, so the timescale of action of this move to be 10 to the 4 Monte Carlo step, to be in the same regime that was seen in the experiment of Spakowitz. And once we established this model, what we, okay, before the results, just the last simulation details, we have studied the three systems that they were the one in play in the experiment of Spakowitz. So we studied a system of a tight density of untangled rings or uncatenated rings that they were the starting point of the experiment of Spakowitz. Then we switched on the Monte Carlo, the new Monte Carlo move, the swapping move that was mimicking the action of topo two. And we studied the system of rings with strength crossing. Then we also studied the system where the topo two was inactive, let's say, basically we took some conformation of equilibrium from the system with the strength crossing. And then we run the standard Monte Carlo dynamics without the swapping move so that the topology was quenched over the entire simulation. We studied bulk solution at tight density 1.23 because this density was already simulated for the untangled case for uncatenated rings. So to have a benchmark at least for that system, we studied a system made up by M polymarines, each of them formed by N-pids and monomers. And we simulated system such that the product of the two. So basically we kept fixed every time the number of monomers in total that we simulated that was around 100,000. And we arranged between polymers made up by 40 beads to up to 640 beads. So now I will show you the results of these simulations for the pre-different systems. So we start with the static properties. We looked at the new exponent that is the universal exponent that relates the degree of polymerization of the polymer. So the number of beads that build the polymer to the M squared ratio radius. So to the sides of the polymer, this exponent for the untangled case or so for the system of uncatenated rings is known to have a crossover from the, this value 0.588 that is the self-avoidable result to new equal one-third passing through this regime of 0.4. As you can see, that is the orange system, the orange core being on the left. As you see, we recovered the new equal 0.4. So we are in this regime for our simulation. We didn't get the one-third because to get that exponent, you have to simulate really, really long polymers. What is interesting is that instead for the system with the dynamical topology, so the system with strength crossing and the system with quench topology, what, as you see, the sides of these polymers are basically are bigger compared to the untangled case because the topo two makes well the rings. So these mechanisms of strength crossing make the rings to become bigger, establishing the discatenated structure. And also the exponent changes, so not just the value of the mean squared ratio radius, but also the exponent change. And we found that it follows the new equal one-thirds that is the exponent for ideal rings. But what we wanted to see and to check if these rings actually belong to the same universality class of the ideal rings, because ideal rings have the exponent of new equal one-thirds. And as you see on the right picture, we looked at the statistics of the mean squared ratio radius plotted in a kind of universality distribution. And as you see, for large x, that is the square root of the mean squared ratio radius divided by its average value. So for, let's say, large scale, the system with strength crossing overlap to the system of ideal rings. So for large scale, we don't see differences between ideal rings and the rings with strength crossing. Instead for small scale, so for small x, we see that there is a deviation between the rings with strength crossing and the ideal ones. And especially the probability for this rings with strength crossing is lower respect to the ideal case, because of course the rings with strength crossing as they have still the excluded volume interaction. And so they cannot be so compressed as the ideal case. So the system are different and they do not belong to the same classes, universality class, let's see. Then we looked at other static properties to properties that basically characterize the topology of the system. We looked at the statistics of the knots. Knots can be detected by topological invariance. And in this case, we used the Alexander polynomials that we computed through the unopened package called the kymonot that was developed in ERN-C signed the group of Mikaeletti. And we computed the spectrum of the knots and its probability distribution as a function of the sites of the polymer rings. This KT is the degree of complexity of the knot, the simplest knot has complexity equal to three, that is the triple knot and as it goes higher, the knot become more and more complex. What we also studied was that is in the inset of the plot is the probability of knotting. So to find in the solution ring that is knotted as a function of the sites of the rings, we fitted these points, let's say, with a power law and we found that the exponent was really near to two that for us made sense because the mechanism that creates knots is a kind of two-body mechanism because it takes two monomers that have to be swapped to create a knot. And so for this reason, it makes sense that the probability goes to the N square that is the density square of the monomers inside the polymer, let's say. And okay, here it's a conformation in which I highlighted the knotted part within the ring. This is a knot with complexity three, so the simplest one, the triple knot as I've said before. And but we found, as you see, for the biggest rings that we simulated also, knots with degree equal to 11. So really, really complex knots can be formed within this framework. Then we looked at the statistics of linking numbers, so another topological invariance that this time measure how many times to close the cork, so the rings that I was simulating were, if they were catenated and if yes, how many times they were going one for each other. This is computed by this Gauss linking number that makes, that gives us integer numbers, zero if they are uncatenated, one if there is just a simple catenation, and so on as the complexity of the linking becomes more goes higher, let's say. We computed again the probability distribution of this Gauss linking number. We found that the probability distribution of G was decaying exponentially from G greater than one, let's say, and the decay length of the Gauss linking number, also in this case, of course, depends on N, the size of the rings. Since we were also interested, as I showed at the beginning, in the system of the experiment, a network was formed, we looked also at the properties of the network that was arising from the catenation of the different rings. So we mapped the system of rings onto a network in which the nodes of the network represents the rings, and the two nodes were connected if the two rings were interlinked one with each other. Then we looked at which scale if there was and if there is, let's say, transition to a fully connected network. And as you can see, there is a transition, a sharp transition towards a fully connected network. So what I'm plotting is that on the left is the mean ring fraction, the mean ring fraction belonging to the largest connected component of the network divided by the number of chains. So if this number is close to one, means that all the rings belongs to the same network structure. This transition happens for the rings with length equal to 160 with a linking degree that is the vertex degree of the network, the average vertex degree of the network that is between two and three. So it means that when the rings are connected on average to other free rings, then there is this transition towards a fully connected network. And as you see in a moment, this transition to a fully connected network will influence also the dynamics of the system with quench topology. So the system in which the rings are connected and the action of strength crossing is no more present as you will see just in the next slide. So with this, I conclude the static properties. So we studied the new exponent and the topological properties of these structures. Then we looked at the dynamical properties. Okay, here there are just an example of the conformation, couple of rings. And in this case on the left, there is two rings that are linked with Gaussian number equal to one. As you can see here, there is the linkage part. Instead for G equal to two already, it's quite complex and you cannot detect at least I didn't detect really where they were passing to each other. So it's really mandatory let's say to compute this integral because you cannot do by eye the computation of the linking number. And okay, then we switch to dynamical properties. We looked mainly at the square displacement of the center of masses, this G free. And in these plots, I'm plotting N. So the number of monomers belonging to the ring times the G free for the three different system that I was simulating. And for the system with untangled rings, we found the properties that were already found in the paper that I was referring at the beginning. So there is a universal, let's say, universal time in which the G free N times G free do not depend on N because let's say the topological interaction are not important instead of large times N times G free depends on N because the diffusion coefficients for system for concentrated solution of rings, the diffusion coefficient depends as one over N to some power that is greater than one. And so as you see, there is the N times G free goes down as N goes higher because of these dependence on the diffusion coefficient. That is not seen in the case of rings with strength crossing as you see at large times, all the curves collapse onto each other basically. There is a really slightly variation between N because the acts of strength crossing make screen basically the topological constraint due to the chain uncrossability. And so the diffusion coefficient takes again the dependence as one over N. Instead, what is interesting is that for the permanently catenated rings, as you see for the biggest sizes that we simulated at large times N times G free saturates. That means that these rings are not diffusing over space anymore. These belong to this giant network structure. This giant network structure is stacked and these rings are stacked as well. And they just diffuse over a tiny space surrounded by all the other catenated rings. And so they are basically frozen also over space. And here I made just a comparison between the three curves for the three different systems. Just to highlight the differences between the untangled rings and the rings with strength crossing. As you see, the N squared displacement takes, there is a deviation between the untangled and the ones with strength crossing at a time scale that is the one of the rate of action of strength crossing. So just to highlight the fact that the system of rings with strength crossing goes faster because of the strength crossings. Then, okay, just to finish, we also did another experiment, computational experiment. We wanted to see what happens to the dynamics by varying the rate of action of the strength crossing move. And we varied it. And what I'm plotting here in this plot is the ratio between the diffusion coefficients of the system with strength crossings divided by the diffusion coefficient of the uncatenated rings, the untangled rings, as a function of the ratio of the time scale of action of strength crossing divided by the diffusion time. So the relaxation time of the untangled system. And as you see, there is a crossover from a region for when the, this value of the rate of the time scales is smaller than one. So when the time scale of action of the strength crossing are faster respect to the relaxation of times of the untangled system, the strength crossing indeed accelerate the dynamics of the system as I showed you in the last picture. But if we take, let's say, two slow mechanism of strength crossing, it can happen that actually the system is slowed down by the action of strength crossing. This is due to the fact that long-lived the catenated structure are formed by this mechanism. These rings become to refuse at this time scale at tau D, but then they find the presence of other rings that are catenated and so they have to diffuse over space with other rings, with other linked rings. And of course in this case, they go slower respect if they were alone. So for this reason, what we find is that not always this mechanism can accelerate the dynamics and can make the system relax faster respect to the untangled case as it was seen in the case of the experimental Spakowitz that was working up in the regime for which this ratio is smaller than one. And with this I conclude. So I presented the Monte Carlo scheme that is able to change the topology dynamically over time and to study the dynamics of the system. We characterize the properties of the different system that we simulated both the statics, the topological properties, the dynamical properties as a function of the size of the rings. And finally what we tried to see and we saw is that the strength crossing mechanism may or may not accelerate the dynamics. In some case, depending on the time scale of relaxation of the untangled system, it may slow down the system. And in the near future, what we would like to do is to perform a systematic study of this system barring the density to look if there are scaling properties that characterize the topological properties of the system also to come up with some scaling laws for the network structure that would be interesting. We would like to implement and to see what happens if we add banding rigidity to the rings because as you may know, the DNA as a polymer is a stiff polymer as a finite banding rigidity. And so we would like to take into account also this feature in our simulation. And it would be interesting also to simulate explicitly the diffusion of top isomerase because in this case, basically we model the system as if the top of two was always present everywhere in the system. So for a regime in which the top of two is really rich within the simulation but it may be that there is a low concentration of this enzyme and then diffusion is important. And so it could be interesting to study this mechanism. And if you want to know more, there is a preprint of this work with Angelo. And if you want, you can ask of course question now or at the other town and thanks for your attention. Thank you very much, Mattia. Maybe we have time for one quick question. Otherwise, Mattia said also for the other speakers for today there is the session on Gator dot town. Okay, so there are no questions. So I remind you that the session on Gator dot town starts at 2 p.m.