 Ok, da smo povrdujati, da smo se povrdujati o vrštivnih vrštivnih in elektroferetnih patenji. Tako, da so tudi izvajte, da so smo vrštivnih in začali, da smo vrštivnih, kako so vse organizi, kaj je Kristian Mikkelejti in da so vlasti vrštivnih. So smo da v Davida Mikkelejti, pa je dnev vrštivno z Andrei Stasiak in Luka Tobijana. And the title here hopefully you get an idea of. We used this title, so we had a selection between titles and we said, yes, we want to use this title because we want the reader to say I want to understand what the hell is topological friction after reading this paper. So let's see whether we can get there. So one thing that will be important for the talk is that DNA is highly compact v svoj biologičnih vseh. To je nekaj, ki se všem izgleda in izgleda v nekaj vseh izgleda Edimbroscijnji festival, tako da počnem s kvizami, ki na pop-kvizami. Vseh, ki ne boš, nekaj se prišli v mene. To je, da se počnem, kako je kompatne DNA. Zato, ki smo počnem, kaj je DNA v svoj bolj, in kaj smo počnem, kaj se počnem, kako je zelo, kaj se počnem, kaj se počnem v Edimbro, in se da se koristina, ovo bo v glasglu, Na njer vse izgleda je, Na vse, ne zvom dissolve, dobro lahko všivno, tako da so tko si plut, njega zelo, s njih nekdo, to je pošlišnji, updated iz vse pošlišnji, da se zelo v taj pluto in s lega vsah, vse zelo prist, od SOEV neko tega pisila z 10 mikronov, kaj sem od 103 kg of the 13 cell, it reaches about that much. At all level, so this I think is quite an impressive number and if we now look more scientifically at each different organism, you would see that this is a micrograph of a DNA inside a bacteriophage which will be important for the first part of the talk. Vseh, kakje je vrstav na obrstav, zelo je 15 nm in DNA je 10 mikron. Kaj je zelo, da vrstav na obrstav in zelo se, da je ina, zelo je tko zelo. In in in v Nuhmanj, tako bila spremno prej, DNA je zelo vzelo. Takaj, čas je veči inoč, zelo se početnjo vstav, nekaj ne se vrstav prizat, neko za to, da je oto početnjo vzelo. So, I will start. So the first part of the talk will be on the physics of DNA inside bacteriophages, so physics of DNA within these viruses. And the second part of the talk will be trying to understand with a simple model how experimentalists can tell which not is which. So thinking about bacteriophages as I said to you before, this is the capsid, so this is the container, you can think of a sphere where you stuff in the DNA of the virus, which has to infect the bacteria. So, there has been a number of experimental studies of this system, because it's perhaps the simplest living organism that you can think of. And the DNA inside, so there has been a few key papers about ten years ago, which shows that at least in the outer layers, so close to the capsid wall, the DNA is arranged in a spool-like fashion. At least, so these are the reconstruction that the experimentalists need. So this will have some importance of what we'll say after. So this is the bacteriophage, it's about an icosahedral, so it's made by proteins. This is the DNA inside. These are many of these viruses. So this is for images. You can also study the DNA notedness. And it will come as no surprise that since the DNA is so compactified, it will be highly noted. So for instance, this is P4 virus. So if this is P4 virus, sorry, this would have about 10,000 base pairs of DNA in. And people have measured how much the DNA there is noted. And you can see that in the wild type, so this is the wild type, 47%, so it's about half of the DNA molecules that were looked at experimentally by arsuaga and co-workers, yeah, are noted. And if you have some mutants, so the trick that the P4 has in order to infect the cell is that it will keep one of the ends of the DNA rooted, so you've got a sphere, you've got an opening, and one end of the DNA is rooted, so it's easy for the virus to attack the bacterium. So this is the wild type. And if you have a Taylor's mutant, so this routine is lost, so both DNA ends are inside the capsid, and as a result, you have many more knots. I should point out here that in order to define a knot for a physical string, which is not closed, that is open, you can, for instance, keep it fixed as it two ends. So, for instance, if I have my extension cord, they are often noted, so this one is noted, for instance, you've got two knots, so this is not a closed string, but it's clear enough that it's noted, because if I keep the two ends fixed, I will not be able to undo it. If they can rotate, and if they are free to move, I cannot undo the knot, but it still makes sense to think that if I have two ends, I can close it and then I can ask myself how many times when I close it I will have a knot. So in this case, we can give a meaning to the notion of an open knot. So this is what happens if we have a bit untidy and we store our headphones cord in the pocket, for instance. When you take it out, it will look a little bit like that, especially if you have two or three, and good luck if you want to see which kind of knot it is or if it is a knot or knot, that's a monster. It's very difficult to characterize. So there are a couple of things that I want to tell you about knot theory, which are important for the first part of the talk, which I should just review. So these are the simplest knots in Tate's table. So this is the unknot. This is a trifoil. This is a full one. So these are the two knots that I have here. We are just looking at the number of crossings. So for instance, this is a full one, because it's in its minimal projection. So this is this one. It's got one, two, three, four crossing. And this is a knot, which appears quite often if you have extension calls, indeed. And then you can go on in the 5-1 and the 5-2. So if you have five crossings, you have two types of topologically inequivalent curves, so you cannot convert one into the other. So that's why they give the name 5-1 and 5-2. So all this I want to also remind you of something that will be important for us is that we have two important classes of knots. One is the torus knot, which are these red curves here. This is 3-1. This is the pentafoil. The next one up would be the 7-1. And there are many other knots, which are torus and which are more complicated than this. And the blue guys are twist knots. So these are knots, which come up very often in an extension cord. And we'll discuss them a little bit more just now. So torus knots are knots that you can draw on the surface of a torus. So that's how you can draw the trifoil, for instance, here on the surface of a torus. Or if you want, you can do that on a donut as the editor convinced us to do in this physics world paper that we had. So these are the torus knots, the 3-1, the 5-1, 7-1, and so on, and so forth. So this would be important for the rest of the talk. So if I look, so this is my extension cable in Edinburgh. This is longer than the one I have here. Yet you see, so this is got, for instance, if we just review what we've been saying. This is 1, 2, 3, 4 crossing. This is a 4-1, so it's a twist knot. And this is 1, 2, 3, 4, 5. So this is a 5. It's actually a 5-2. So all these knots appear quite a lot because, so here, for instance, I got a 3-1 and a 4-1. And the reason they appear so often and the reason they are called twist knots is that you can easily do them by mistake if you make a plectonym, so you twist up a string, and then just by mistake you take one end and you put the hoop here, you create a knot, and this will be a twist knot. An important thing about this knot is that they have a knot in number 1. So if I just undo one crossing, I have an unknot. So this is different from the torus knots. So the unknot in number is the number of crossings that you have to undo in order to go from a knot to an unknot. So an important property of the twist knot is that you have unknot in number 1. So if you undo this crossing, for instance, you will undo the knot. That's why this is a twist knot. So one question that was always puzzling for us is how experimentalists can use their techniques to be able to tell so accurately one knot from another one. So you will know that they do by gel electrophoresis, but I think still that it's quite remarkable. So what is done here is that these are a series of twist knots, and these are the knots that are in a phage, in a bacterial phage. So this is again from the paper by Arsuaga and Muriel and co-workers, where they characterized by gel electrophoresis the complexity of the knots inside the phage. So what they did basically, they ripped the capsid open, so this is the virus, remember, they took the knot or the piece of DNA which was there, ligated it, and then ran it through a gel, similar to the one that Matthew was showing to you before in this first part of the talk. And then different knots migrate at different speed. I will say much more about this after. So what is remarkable is that you can tell quite a few of the simple knot up to ten cross in a list just by looking at how fast they move. So if it moves as fast as a trefoil, it will be a trefoil. So you don't need to look at it. If you do a calibration. So this is quite impressive, I think, that such a simple technique, just telling you how quickly you go through a gel can tell you about the knot type. It can tell you also about other topological things, but the knot will be most important for us today. And if you look here, and if you calibrate your gel electrophoresis experiment correctly, we see that you find the knot. So there's this spot here. This spot corresponds to the trefoil, and then you have a very faint spot which corresponds to the full one. So in these bacteriophages under high confinement the twist knot, which appears very often with the extension cable, appears very rarely in the DNA inside the capsid. And we'll be interested to understand if you can understand the mechanism why this twist knot is not present. The five knots, there are two kind of knots. One is the five ones, the pentafoil. This is the torus knot. And the other one is the one which I showed to you with the extension cable. That's a five-two, that's a twist knot. So again, so this is more difficult to discriminate, but you can do that with gel electrophoresis. So the pentafoil here it is, and here you have a missing spot for the five-two. So what is happening is that we seem to have more torus knot than twist knot. And actually you can do more careful experiment. So this was a groundbreaking experiment a few years ago, about ten years ago, but many more have been done and they all confirmed this remarkable thing that you don't have many twist knots within a bacteriophage. So within a bacteriophage the DNA is no different than in solution, but in solution you find a lot of these four one knots, not in the bacteriophage. So the confinement has to have something to do with this peculiar finding. Okay, so roundabout at this time we got quite interested, and I'll tell you, so I'll be more interested in the ejection, but I need to tell you a little bit about the knot spectrum because otherwise it would be difficult to follow what I say after. So we wanted to understand, we wanted to understand whether we could design a simulation to understand the knot spectrum so the kind of knots that are formed within a bacteriophage. So the first go that we had was very, very simple. This was done with Monte Carlo kinetics, with a kinetic Monte Carlo model, which only consider the fact that the DNA, the bits of DNA have a thickness, so if you are hydrated, the DNA has above the thickness of 2.5 nanometer, compare it to the 15 nanometer that is the sphere that it is confined in. So we have two lens scale there, and we got another lens scale which tells us how stiff the DNA is. This is the persistence length, which is the length of which thermal fluctuation will buckle your fiber, and which is again 15 nanometer. So it's quite remarkable that the persistence length is very similar to the size of the container that you are in normally, and indeed this is of some relevance. And so we did the first simulation only as thickness and persistence length. So we put a piece of DNA with the right stiffness inside the sphere and wanted to see what happens. And this is what happens. So you have a messy structure which is resembling a bit of a very messy spool. That's not, yeah, some kind of spools would look like that, but it's very, very messy, so it's a horrible structure that will be highly entangled. And the deep reason, well, the physics reason why this appears is indeed the fluctuation of the DNA which you can think of as persistence length are about the size of the container. So if the container was very small the stiffness would be, the DNA would look very stiff, so it would order more. Here the DNA looks relatively floppy at that scale, so that's why you get a disorder structure. And if you run this simulation which we did for realistic confinement you can put or not electrostatic, it doesn't matter much. What will matter is something that we will see in a few minutes. You cannot even, we were able to identify essentially none of the knots which we generated. So these are almost certainly noted, but we don't know what they are. They might be the unknot. It's very unlikely. We don't know what any of these knot is because they are not identification routines which are available. They struggle if you go to a very high number of crossings. So they can do very well until you have 16 crossings, perhaps a bit more if you simplify, but not if you have these monsters. So the bottom line from here is that if you take a simple model where you only have a semi-flexibility and thickness, so you have a very honest model of DNA which works pretty well in solution. It works very well there. If you put it in a sphere, it doesn't do a great job in the sense that clearly the experiment shows that we can see some simple knots because we have a very nice spot for the unknot, nice spot for the trifle, and there is nothing here which is so simple. And also it does not resemble the spool-like ordering that we saw in the micrographs. So people were finding pictures like this in six years or so ago. And one theory which was out there is that the micrographs were actually overdoing the kind of symmetry so they were making the DNA look like more order than it was. So people thought about this for a while. And we actually thought about this for a while. So we then tried to put in a simple interaction with the idea that maybe the simulation is not right. Maybe we are missing something. So the idea here is that DNA strands are chiral, and if you want to write a simple model which does that in solution you might get away without worrying about that, but not in confinement. Because if you have strands, so if you have chiral objects that we use to demonstrate this point when you lock them at very close packing they will lock at an angle. Which depends on the groove details of the fusilli in this case and it will do that in DNA as well because you have grooves there and you have a helicity. It turns out that this is actually something which is well known in DNA physics. People have done experiments and shown that the molecules can form a liquid crystalline solution which is exactly what this is. So when you tweak the concentration you can undergo a transition from anisotropic to anematic or a cholestetic phase. So in practice if we remember what Slobodan was saying today if you have chiral objects you expect a cholestetic but the pitch is so large that within the confine of the capsid is very similar to anematic so the angle is about a few degrees. So here comes this interaction, let's see what happens. There you go. So this is a movie from Christian that we had in the first paper on this. And you see that if we have p4 like DNA with this interaction where the parameter were tuned on the basis of this liquid crystal DNA experiment you form nice pools there very different physics from a very simple interaction. And these cases which will look exactly the same so I will go on from now. So this interaction would not be very important in confinement because it only kicks in when the strands are very close together so it will not be very important in solution so in an unconfined situation but it will be very important in confinement when you force the DNA strands to come close together in this angle. So these are the configuration that we get. Again they are very order, there is a very high level of liquid crystallinity and there are some have in defects as we call them like this. DNA is a liquid crystal that doesn't tell the head from the tail. And this is what we would expect for a liquid crystal in polymer. So now the surprising thing perhaps is that with this kind of interaction the situation changes a lot. So this is surprising perhaps up to a certain extent but nevertheless it's quite pleasing that if you look at the so this is a system which, so this is a collection of a couple of hundred simulations where we repeated for each simulation the packaging process so we had a sphere and we fed the DNA in this is in principle a non-equilibrium process so we didn't want to worry about the glassy autocollabation time in this case because every time we had a single different simulation and we just recorded how many times in this case we found a trifle how many times we find a full one and so on and so forth. So this is a thermodynamic system is only that it will get stuck in a metastable state so you have to do it many, many times if you want to sample correctly the distribution. So the results are interesting because you see that you can now, first of all you can characterize the knots because before we couldn't do it without the interaction we have a lot of trifle a lot of pentafoil so we have some 7-1 so relatively large amount of knots that are simple that we know and we know that they are appearing in the gel electrophoresis or the phage as well. The 4-1 is present but it's very, very suppressed the 5-2 is another twist knot it's very, very suppressed so at least within the simulation we can understand that we can understand how this bias comes through so this is through DNA-DNA interaction which you would not have considered in a simple course-grain model for DNA. So our view is that they come from a line in interaction which gives some liquid crystal in order and if you want you can understand this by thinking that if you with liquid crystal interaction can stabilize somehow a spool like the spool that sailors use to put away your harpoon so these pools are very ordered and they resemble a bit a torus so if you can't form a knot there it will be more likely to be a torus knot so this is a very simple mind explanation why this torus knots appear so now so the bottom line from here is that we can just by tweaking the potential a little bit so just by putting some more detail that we would have thought at the beginning simple model to understand the knot spectrum so which knots form within a phage so now the question that we had at the beginning of the project is given that knot's form and we know that more than 90% of the mutant p4 is knotted in order now for the bacteriophage to kill the bacterium it has to make so that the DNA will go inside the bacterium so it has to eject properly and if you have a knot like the ones that I had before you could imagine that the knot would form a hindrance so the DNA would not be able to eject properly so what happens if we simulate ejection so these are some selected simulations here so this is a knotted chain out from the capsid so you should imagine that here you got a sphere and you got an opening here so I got p4 DNA and then I let it out through this opening so the entropy would be much larger outside the virus of course so the DNA wants to get out but the question is how it does that and you naively might think that the knot could form a blockage so that if you had a knotted structure it would create some kind of topological friction just to come back to the title and stop the ejection ok so what we are seeing now is another so the knot has come out so this is another complicated torus knot it's a 9-1 so this has got a knot in number 4 so it's quite complicated and still it comes out no problem you see no difference and yes another very complicated configuration is one of the few that we couldn't classify and it goes away no problem again and one thing that I hope you will guess in the end is that if I ask you what kind of knot it is we don't know but I pretty sure bet that this is going to be a torus knot so the reason why all these knots get out so quickly or do not have many problems getting out we at least thought is linked to the idea of knot localization so if you have if you are tying shoelaces or if you are pulling on a knot it will localize so you see the knot will become very small so when you pull on a knot it will localize that's how we tie our shoelaces on the other hand when we compress a knot it will delocalize no sorry I said at the beginning this is knot we have done simulation with Debye-Huckel interactions and these are actually we put Debye-Huckel interaction and they don't make much of a difference because it's so screened in physiological conditions so the shoelaces will help the physics so it will give you a force to push out but it won't change much the topological friction so the knot when you compress are delocalized so you cannot look at a small region say close to the aperture of the capsule and say is it not it or not so there is no local structure that can create a blockage so that's why the DNA gets out whether it is a torus knot so remember that these simulations were all done with a spool so we're all done with a Karel interaction so it's quite remarkable that we've done a few years after we got again at this we try to study more systematically the ejection simulation so you see at that time the ejection is very slow because it's only entropy driven so these are very slow simulations at the time so we were like quite a lot of statistics so that's why we didn't do them straight away so the red guy here so this is the percentage of ejected beads so it goes from 0 to 100 there is again p4 DNA as a function of time in millisecond in kinetic Monte Carlo we can map to a realistic time scale by using stock slot and diffusion and a few more things so I just here give you the time in millisecond it doesn't matter so much though so I would like you to notice the scale so these are the interactions with the Karel bias on so you see that the DNA gets nicely out it's almost linear actually the way it does that whereas with without colostatic interaction it still gets out sometimes three times 50 or so that I'm plotting here it has gone out but it takes a long time it's much slower so this is an example of topological friction because if you were doing a simple theory with a thermodynamic theory it would be very difficult to guess that the DNA so the DNA concentration is the same, the charge will be the same semi-flexibility is the same the only thing which changes is that there are some small interaction in this case liquid crystalline which has favored a conformation with respect to another so in this case perhaps I should speak about conformational friction which has much less friction than the mass as you would perhaps expect and this is something that is difficult to get with a simple theory so this is how you can see it microscopically so this pool is like the harpoon so if you have a harpoon that's why the sailor put it so nicely if you want to put it out it's easy maybe there is a toss not but it still gets out quite easily if this was a mess it would be much more much slower to come out the bottleneck which is created here is purely topological and is due to the conformation so the entangled nature of this structure makes it difficult to make it eject and in particular you see that you need a lot of rearrangement so this suggests to me that you could suggest an experiment whereby in interaction as to make the current interaction smaller or larger you should be able to see differences in the ejection dynamics for instance ok so there is another thing which I think is quite cute is that I showed you only a part of the trajectories first so look at the red ones so now we are considering only the trajectory with the chiral bias so these are the one which work but not all of them start immediately so this is another prediction we have done and that as far as I know is not quite been tested experimentally perhaps because the time scales according to our simulation are pretty small anyway so there is a stochastic element in this ejection so some of the DNA don't eject immediately and that's due to the fact that they are in a dormant state so if you have a look here look at how the spool is oriented and look at this guy here so you see that this spool so the opening is here so this spool has to do a rotation and has to present itself nicely so that the ejection can start so before that happens it will just stay put so it will just stay put and then when it starts it go very fast ok so there is just a two state biphasic dynamic response in the progress of the ejection so if we look deeper so this is so this is quite a simple plot but there are too many things so this is the knot type and this is the probability of the knots and this is the percentage of knots which has been ejected ok so from this plot we can see two things one if you look at this is a small probability so a small percentage of ejected this should say percentage of ejected DNA sorry you can see that you have an unknot you have a 3-1, 5-1 is essentially the same distribution that we looked at before when the DNA was all in so when the DNA starts to get out it's perhaps nice to see and expected but you see that you don't populate the twist knot so you stay in the torus knot so even when you get out you always have more torus knot on top of this you see that the knot becomes simpler as it the DNA ejects as again you would expect but it's nice to see it in the simulation and quantitatively so another thing that you can see you can follow so for instance that we follow here one of the ejection simulation when there was no lag ok and we can look at the percentage of ejected DNA so this should be in milliseconds it goes from 0 to 7 and you can find here what so this is the percentage of ejected DNA so you see that it's not quite linear it's got some small pauses but it goes all the way so at the beginning this guy was one of the complicated knots like the last one that I showed to you that was ejected very happily so it's got more than 30 crossings in the minimal projection but if you wait longer a little bit you see that it will simplify it will have 20 and then it will go on this progression of torus knot from the 9 one to the 7 one to the 5 one to the 3 one then the un knot this is very typical of what happens if you think about the un knot in number the twist knots have an un knot in number of 1 so they can be undone with only one single uncrossing so the torus knots have more uncrossing numbers so basically what this says is that ejection proceeds in a stepwise way whether you undo one crossing at a time it's like you had a big spool and you are slipping away your tether little by little so that you can undo one crossing at a time so you got a large loop here and you are just teasing out an end so that you make the spool simpler and simpler no there are pauses there are not so many pauses but there are pauses and they resemble so there are experiments and people do see pauses and there are several explanations yes the pausing becomes yeah it's yes you have more pauses as you go on I should review that curve but we did have that we did look at that frequency of pauses so some pauses are extremely long so they just get stuck so very few cases of this so this DNA with carial bias get stuck and in the few cases so this is very episodic but in the few cases that we have seen this for instance is one of the few remarkable example of twist knots so the twist knot is slow to get out and if you look at how so this is one lucky case where we could actually find how this evolved and this is a 5-2 it becomes an unknown knot and then it becomes another quite complicated knot that get stuck then so again it's very different from the nice linear progression that you see with a torus knot so the topological friction is much higher, it gets stuck whereas the torus knot so the reason that we think that the knots can be infective so the phages with the knots can be infective is that they form this nice kind of torus knots that can be undone one crossing at a time so they have a low topological friction in the last 15 minutes or so I will discuss some results that we got quite excited last year and we are also working on some generalization right now on the simulation of the gel ecrophoresis experiment so I will go back to that this is trying to understand how experimentalists can say so confidently which knot is which ok so this is one of the other another slide from the talk with the wit so this is how the agarose gel that experimentalist use looks like so it's a bit like Swiss cheese you can think about it like that or a network with holes and you can ask yourself which knot will move faster in this gel will it be the trifoil, the unknot or the most complex knot or there will be no difference at all you want to guess? D, yes, that's right so the more complex knots will move faster because they are smaller and they can squeeze more easily in the gel or at least that's what you would think right so this argument works beautifully for sedimentation so even if you have knots going out in a disco gel you can see this beautifully and the reason in that case is that very similar that we have a competition between gravity or something else electric force and the friction so the the force is equal to the friction times the velocity and the stock's drug or the friction becomes smaller if you have a smaller object so a smaller object will feel less friction so it will go faster, that's it that's not the whole story though and that's why we were interested in this so a lot of biologists and biophysicists use two-dimensional gel electrophoresis so the idea is that you can have a little bit better separation so this particular case you have a first direction where the where the field is weak where the knots move according to their complexity so the more complex knots will move faster and then you have another direction so the knot will be moving very slowly there and you've got another direction in which you put a stronger field and in general in the stronger field direction or in general you said in some cases at least you have no monotonic mobility so this is telling you that the unnot is moving faster than say the twist knot the 4-1 ok so as a result if you combine the two you will have an electrophoretic arc and this is indeed the one that was used in phage knots to discriminate and to separate them so if you only do one direction you can't see them all so clearly so we wanted to understand why and in principle this has implication for other things that's what we're interested in now so if you have supercoiling for instance here or if you have no sorry so these are dimers supercoiling is not shown but if you could have denatured loops so this is a DNA which is denatured at several different degrees you could have a DNA which is more or less supercoiled as we have heard today so it's got more ride if it is a loop it will not get rid of the supercoiling or you could have a linked DNA so gel electrophoresis will show you how to separate all of these and it will do that by using so there's been a nice work by Weber et al so this is Paolo de Los Rios and Giovanni Dietrich so there is a surprise there was not too much simulation work on this interesting problem I think and what Giovanni and others did is they found out that it's important when you want to understand the mobility of knots in a gel to understand to account properly for the topological interaction between the gel which is modeled pretty much in the same way as Matthew showed to us this morning and the polymer so this kind of they're not quite you can think of them as threading in a way but it's a little different so these kind of interactions are going to affect the dynamics importantly however in no situation we find some non-monotonic behavior so you could either have that the drift speed goes down with complexity or goes up with complexity but you could never find something non-monotonic so our contribution was to consider something slightly different so in the work by Weber they had a regular gel exactly the same as Matthew showed to us today in the first part of his talk we consider that another gel you could think that another thing might be important so that the gel might not be regular but it might be cut here and there so you could have dangling ends which model agarose filaments which here are completely rigid but we check that even if you make them flexible the results don't change much and this could affect things a bit in principle because they could provide a way so in this case rather than threading so the gel could pierce through a loop and perhaps change its mobility and I will try to convince you that this is probably what happens so in this case p is the probability with which we broke one of the regular sides in the cube so if p is zero we would have a gel made by nice little cubes and if p is different than zero we have dangling ends so I will try to show the results so I will try to go through the results in the remaining five minutes or so so this is a drift speed as a function of the average crossing number this is the N naught the trifoil for one so the not complexity goes higher here and there is no surprise that you go a nice linear result so this is what we have would happen a weak field would you produce the nice linear behavior that you find in experiment however if you run the simulation at a moderate field you find that you have a hint of this no monotonicity so we were quite excited by that and wanted to understand at least in our simulation why this comes about so here the N naught is not the slowest but it's faster than the full one so the full one is the slowest of all in this particular case if in this case in the simulation as I showed there is a bit of irregularity because this simulation takes quite a lot of averaging to do but there is a clear signature that we find in anotrophoretic arc which is very similar to the one that people find routinely in experiment where the zero so in this case the full one is the slowest but you can change this hierarchy by tuning parameters anecdotically we could see that when we did this simulation there were cases in which indeed the danglin ends or the gel were piercing or threading let's use the loops so one important clue to understand what the reason is for the no monotonicity and that it is indeed linked to the danglin end is that we can redo the same simulation at p equal to zero regular gel no danglin ends so this is the case with danglin ends you see that the speed is no monotonic with average crossing number or not complexity in this case we could never like in the web simulation we could never find a single case in which we had no monotonicity always monotonic so there can be some discontinuity but it's always monotonic so again in this particular case we could never find a tight gel and the more complicated nodes always go faster so if you look at single no trajectories this is the position as a function of time danglin ends you see that they get stuck several of these guys get stuck without danglin ends they don't get stuck if you look at the radius of generation that is the size it can oscillate so the DNA has to move and the contract that is goes through the gel but you don't see any signature in the velocity curve and here on the other hand again you get stuck when you get stuck your radius generation increases so this is again all due to this thread in or piercing and we wanted to understand this a bit more quantitatively so we can compute from the molecular dynamics in this case simulation that we did they are very similar to the one that Matthew described today actually so this is the number of hitting events so we can ask how often is that a loop will be impaled on one of the danglin ends and this is the number of these events so the unknot is a bit fatter so it gets impaled a bit more often you see but there is a competition so the unknot would go slower with this because it becomes entangled more often but this entanglement time as a function of average crossing number has the opposite behavior so if you have a simple knot even if it gets entangled it will be able to undo itself quite quickly whereas a more complicated knot will take much longer time and you have to consider the competition between the probability of hitting and the disentanglement time and this is what gives you the non-monotonic behavior in the velocity as a function of average crossing number indeed here we made a simple theory so this is our data and this is a theory where we don't have fitting parameters we have matched the fitting parameters to the ones that we have found in the simulation and we have done a very simple problem in which we have a bias random walk which can hit an impalement and get stuck and can disentangle with a time taken from the distribution probability of times that we measured in simulations so we can fit rather well the results here and this is a predictive data so these are not more ecodynamic simulations in this model we can change the parameter here with the radiation of the unknot divided by the mesh size of the gel so in going from the black curve to the purple curve you are essentially making the gel smaller smaller if you want the polymer larger and larger what matters is the ratio between the length scales so you see that the non-monotonicity can be controlled there physically so if you are stuck you want to the unknot the unknot is not stuck permanently so it will be able to disengage yeah it disentangles from the matrix in the sense that you have a dangling end which is penetrating and then you get out you can maybe call it something yeah so let me go through briefly the main conclusions in the first part I showed you that you can do simulation with a coarse grain model where you take into account the fusilli-like interaction between DNA and which allow you to understand that the knot spectrum should enhance torus knots as a function as opposed to twist knots by simulating ejection you can also see that these torus knots get out more quickly and more easily than twist knot and I hope you like this I would like to describe them as topological friction so the topological friction of these knots is lower than that of twist knot and that's why these phages can still be infective because their knots can get out very quickly and then I showed you some results about how to understand from a theoretical point of view the experiments of gel electrophoresis in particular the non-monotonic behavior which is observed at intermediate strength where the velocity of a knot first goes down and then goes up again so which goes against the normal idea that we have that Stokes law would make faster more complex knots and as I showed to you we have a strong hint or a strong clue that this is linked to the fact that the gel can get entangled temporarily with the polymer thank you very much