 So I would like to thank the organizers, Christian Michelletti and Enzo Orlandini. Enzo I never told you but I have a small, just two minute story to tell you. So Enzo Orlandini was a postdoc in our lab for many years and we wrote a paper together. So it was Garel Orlan Orlandini. And one day I received a phone call from a good friend of mine that some of you may know, Bill Eaton. And he told me, what is this paper? Is it a joke? Okay, so it was not a joke. So anyway, thanks for inviting me. So I will talk about... My talk is a little bit related to the talk we heard just before by Maria, except that it's a physical point of view from some problems which are very related, which are the problem of nothing and nothing in polymers, but I'm pretty sure it can be generalized to the kind of problem that you have been talking about. So in the first part of my talk, I will talk about nothing and nothing of polymers using Langevin bridges, which is a technique that I will explain. And then in the second part, if I have time, I never know if I am enough or too much, I will talk about nuts in DNA and in RNA, sorry. Nuts in DNA, you know everything about it, but nuts in RNA is a different subject. And if I don't have time, and even if I have time, I will speak very briefly about it. I strongly recommend you to go to see the poster by Marco Di Stefano, who is a co-author of this work. So about DNAs, we know that there is a lot of coiling and supercoiling, undercoiling in DNA. And there are many nuts in DNA, and you have seen these pictures hundreds of time, but I cannot resist to show it. So these nuts are present, sometimes they are functional, but sometimes these supercoiling or undercoiling or nuts have to be disentangled. And for that matter, the cell will produce an enzyme, which is topoisomerase one or two, which can relax the torsion or a nut DNA. So these are, okay, this is for people who don't know what are nuts. I guess in this audience, they are known. And topo one, so there are two types of topoisomerase, topo one, which doesn't use ATP, which gets only one piece at a time, and then relaxes the torsion around one strand, and then reconnects, and topo two, which uses ATP, and cuts both strands and reconnects. So the questions that have been touched upon before and that I will try to answer or give a method to answer is how does DNA knot and a knot, what are the set of moves to go from one knotted structure to another, and the approach that I will describe is based on dynamics. So it's really a physical approach. It's not a mathematical approach where you look for minimal set of moves which will disentangle. It's really based on Langevin dynamics, which is what people believe to be the diffusive dynamics occurring for polymers. And the question is how does a polymer go from one initial given configuration to a final conformation? So you have one initial conformation, which is, let's say, a certain knot, a final conformation, which is another kind of knot, and you ask what are the stochastic paths which are going from one to another? It's really the paths which are generated by a Langevin equation, but you want to generate this path unbiased. The only thing you want to do, in general, if you start with an initial conformation, you will sample everything, and of course the target that you're looking for is of zero measure, so it's very improbable cases. So the question is can you bias your dynamics, your Langevin dynamics, in a way that you will converge always to the correct final state, but without biasing the statistics? In other words, you generate the real path. And to do that, the standard, well, not the standard method, but the method is Langevin bridges. And to illustrate this method, I will illustrate it on a single particle in one dimension just because the equations are simple to put up, and then I will generalize it for a polymer. So, and this work was published first in 2011, and then with Satya Majumdar on several cases. So the question is the following. So this method originally was not developed to study nothing of DNA or RNA, but rather to study barrier crossing in proteins. And of course, when you have barrier crossing, it's events which are exponentially rare because you have to cross a barrier, so you have this commerce time which is exponentially large, and the time fluctuations of the system, the system fluctuates between this minimum, then suddenly goes there, and etc. And these events of crossing barrier are exponentially rare, and it's very difficult to observe numerically when you do protein simulations. So the method was developed in order to try, and the interesting physics of course is in this region, in the transition region, because this is just harmonic oscillations in the bottom of the two wells. So the method was developed then, and what I will show you is an adaptation of this method to the case of polymers. So the starting point is the overdamped Langevin Dynamics. So you take a Brownian Langevin Dynamics. So it's a particle who is moving in a potential u of x, and gamma is a friction coefficient, and you have a Gaussian white noise which is uncorrelated, so the expectation value of this noise is zero, and the correlation function of the noise is a delta function. So of course, this is what mathematicians call Eto process. I write it in the physics way. In mathematics, they would write dx equals f dt plus dbt. But I understand it better like this, and the friction coefficient is related to the diffusion coefficient by the so-called Einstein relation. So usually what one does is one would... So of course, in the case of a polymer, as we will see, you discretize your polymer, or you consider a set of beads, and you don't have only one coordinate, but you have n coordinates which correspond to the n different monomers of the system. And then you discretize the time. So there are many ways to discretize the time. There is this Eto versus Tratanovich. It doesn't matter. I will not dig into the details, but you can solve this equation. And in fact, from this equation, you can write a path integral representation. And one way that people have been using, and which was mentioned this morning, to do the sampling between the two points is the so-called transition path sampling. So one way to visualize it is on the path integral representation of the Langevin equation. This Langevin equation, you can write... By using the equation, you can write what is the probability of a given path, x1, x2, xn in time. And the probability of having a given path, x1, x2, xn is just the product of the probabilities of the noise eta. And therefore, you can write the probability to start at xi at time 0 and n at xf at time tf as an exponential with this kind of action. And from this action, it's exactly what was mentioned this morning. So there are many ways to do transition path sampling. Essentially, what you do is you start with the initial trajectory with fixed endpoints xi, xf. And then you move all the x's in the middle by a certain delta x. And then this will... If you move one of the x's, this will imply a certain variation of your exponent here. And then you accept or reject this deformation with a metropolis kind of algorithm. So the difficulties of this method is that the sampling space is huge because you have all the discretization in time, plus if you have more than one degree of freedom, you have many, many degrees of freedom. And as a result, if you have barriers in your system, it depends very much on the initial trajectory. So instead of that, I used something which I call launch van bridges, which was in fact introduced by a mathematician, Dob, who is a famous American probabilist. So the idea is the following. If you consider all paths of a single particle, which starts at x0 times 0 and which ends at xf times tf. So I consider all the paths which are joining the extremities like this. And now I ask among all these paths what is the probability to find a particle at point x at time t? So the conditional probability for such a path to go through x at time t is given by this curly p, which is just the product that the particle starts at x0 times 0 and goes at x at time t, times the product that the particle at x times t will end up at xf times tf. That's just the Markovian property of the probability for this path and properly normalized, of course. Now this p of x and t, so which is just the forward probability, this is the backward probability, this one satisfies the Fokker-Planck equation because it's a stochastic motion, launch van, so this one satisfies the Fokker-Planck equation. This one satisfies the adjoint Fokker-Planck equation which is the reverse Fokker-Planck equation. And so p satisfies the Fokker-Planck equation which is here. So d dx and the current is just dp dx plus beta du dx times p. Beta is the inverse temperature, of course. And the adjoint or reverse Fokker-Planck is dq dt equals minus d dx squared plus db du dx, dq dx, which you can obtain by just taking the adjoint of the first equation. Now if you remember, the conditional probability is this curly p, which is q times p. So now you write d curly p by dt and you use the multiplication rule for derivatives and you get this exact equation and this exact equation, so it's dp dx plus d dx beta u minus 2 log q times p. So this exact is just a trivial manipulation and this exact equation shows you that the curly p, the conditional probability satisfies a Fokker-Planck equation absolutely identical to this one, except that there is an additional potential here in beta u, which is this 2 log q, where q again is the probability to go from xt to xftf. So that's the only difference and this term is of course the term which is conditioning your particle to end up at xf at time tf. So now if you remember that the Fokker-Planck equation as written like this is derived from a Langevin equation in the potential u, then this equation where the potential is beta u minus 2 log q is therefore derived from a Langevin equation where the potential is beta u minus 2 log q. This is the new potential. And q, so in other words, you have this Langevin equation. It's a conditioned equation, conditioned, I don't know how to call it. If you start a particle, so q is p of xf tf xt, if your particle starts at x0 at time 0 and you solve, you iterate this equation, it will end up at xf at time tf with the exact probability without any bias. It's just like if you erase all the path which don't go to the target, but you have exactly the right statistics in your path. Okay, so, and I wrote Giorginov because when you, okay, when you talk to mathematicians and they tell you, oh, but this is absolutely trivial, it's a consequence of Giorginov theorem. Okay, so I'm sure some people know what is Giorginov theorem. Okay, anyway, this equation is exact. The equation is Markovian, but it depends through this function q on all the future of the trajectory, but it's a function of x and t, so it's a Markovian equation, and it doesn't bias the statistics of the trajectory. So some example, some trivial examples before going to polymers. So if you take a single particle, a Brownian particle, and you ask what are the trajectories of a particle to start at a given point x0 at time 0 and xf at time tf, the function q is just the Green's function of the free particle, so you solve it exactly, and if you plug it into the equation, the condition-Langevin equation for Brownian motion is just this equation. So you see that this equation, which is exact, you can solve it very trivially, when t goes to tf, so this is like a spring, a time-dependent spring, and the strength of the springs diverges when t goes to tf, so the particle is driven to x equals xf, whatever you do. And you can solve it, of course, trivially, and it takes no time. This is a sample of 1,000 trajectories starting at minus 1 and ending at plus 1, and it takes a fraction of seconds to generate all these. Another example is the so-called Brownian excursions. So Brownian excursions, if it's the same, you go from, so you have a plane, and you're not your trajectories, so it's just to illustrate the method you have. You start here and you look at path, which will end here, and which are not allowed to go into the lower half plane. So the Green's function, then you get by mirror image, by subtraction of mirror image, and this is what you have here, and so the equation that you get is here, and when you take, if the final point is on the wall, you get this equation. I mean, you can play all kinds of games, and again, you can get trajectories, which are extremely easy to generate and to obtain. Final example that I want to show, because it's the most relevant to, to polymers is the case of Einstein-Ulenberg process, which is the case when you have the potential, is a quadratic harmonic potential, then the bridge equation, you can show that it takes this form with a cos k over gamma, where k is the rigidity of the, of the harmonic potential, like this. And an interesting fact is that this equation doesn't depend on the sign of the, of k, which means that the conditional path in a potential is the same as the conditional path in the opposite one. And this brings to the apparent paradox, but it's not, it's a fact of life, that for instance the time for a particle to go from here to here is exactly the same as for the particle to go from here to here. Okay, so now, and for low dimensional systems you can calculate things exactly by looking at the eigenstates of the Fokker-Plank equation and you can get things exactly. So I will continue now and I go back to polymers now. Now assume that we are in excess of topoisomerase. So the chains can be modeled as Gaussian-Phantom chains. So the chains, I assume that the chains can cross each other. There is no resistance because you have, whenever they get in contact, some topoisomerase will be around and will allow the crossing of the chains. So the model I will take for the Phantom chain is a Gaussian chain. So I take a discrete model, but it can be a continuous model and then it would be some integral dR, dS, square, etc. But for the drawings and everything, I use discrete chains. So this is just the elastic bonds which are making up the chains. And then we know that DNA has some bending rigidity. And so I use a model of chain which is a semi-flexible chain where for the bending rigidity I take the square of the square of the second derivative of R. Of course, this is not exact, but it's a way to introduce in the system the bending rigidity. So this coefficient K is related to the bending rigidity and this is the Kuhn length. And I look for a circular chain which means that Rn plus 1 equals Rn and R0 equals Rn. And then the standard dynamics that you expect for such a chain is the so-called Langevin. So in that case it's called Langevin-Raus dynamics which I write here. dRn dt equals d. So then it's the d minus d beta H by dRn. So it's the force which is the gradient of the derivative of beta H with respect to N. So it is this term. This term comes from the elastic energy. This term comes from the bending rigidity plus random noise. So this will produce the diffusion of the system but of course if you start from a given noted configuration there is no chance that you will ever reach any interesting configuration that you're interested in at any finite time. So then, so we assume that we have a certain configuration Rn0. So Rn now, N is a set of the N monomers. So we have a certain configuration at time 0. The final chain is Rnf at time tf. And then the, so as I said in the one-dimensional case the bridge equation is just the standard Langevin equation where you add this term and this term, this additional conditioning force is just 2d times the derivative of flow q where q is this probability that given the particle at Rn, the configuration Rn at time t it will be in the final state Rnf which is the state that your target state at time tf. And of course what I forgot to say is that this can be easily generalized if you add an external force and for linear chains which are not closed. Just for this example I do it without external, so adding an external force you would just have an additional fn and linear means that you don't have periodic boundary conditions. So everything can be calculated exactly. This function q in the case of a Gaussian chain for the model that I proposed here this can be exactly calculated and then you can solve everything in Fourier space so you go to Fourier components, rho p the Hamiltonian takes this form with 1 minus cosine omega p and this is the bending, this is the elasticity this is the bending rigidity term and the equation, the bridge equation in Fourier components for the chain reads this form and this is very similar to this Olnstein-Ulenbeck process I was telling you about before and this is the Lange van Britsch equation which is exact for the specific model I am talking about and this omega which appears here is the sum of these two terms so of course this one goes like omega squared this one goes like omega to the 4 and that's the elasticity term goes like, elasticity term goes like omega squared and this one goes like omega 4 so the idea is that you solve now this equation in Fourier space so with these Fourier modes, rho tilde n and then if you want to see what's happening in real space you just Fourier transform back to real space and of course you don't have to do it at every time step so in fact the most time consuming when you solve this equation it's extremely simple to solve you can solve it up to sizes 400 I think easily and probably even more if you use a sophisticated computer the only time consuming part is when you do Fourier transform inverse to real space and I'm sure that you can make it fast by using FFT which I don't know really how to do but anyway so there is one difficulty which I didn't mention which is that the final configuration so in the form I did there is a one to one correspondence between the monomer of the original configuration so if this is one, two, three let's say n there is a one to one, I use the one to one correspondence between so like this because I give the initial configuration the final configuration but in fact if you look at the nothing or unnotting what you should allow is for a circular permutation of the indices you can map the final state can be any circular permutation of the final configuration that you give because because there is no specific assignment it's just the nothing that is interesting it's not really the conformation of the chain it's the fact that the final chain will be noted or not so it can be done because the final state can be anything you want the final conformation is therefore a superposition of all circular permutation and one can again write an exact bridge equation and solve them numerically so I show you so of course this is so I believe there is not really very well defined path in the nothing or unnotting of chains because it's a purely diffusive model there is no real energy involved except of course the bending rigidity that you don't want to mess too much about and so what you're supposed to do in this game is you're supposed to start with many initial conditions and for each initial condition you're supposed to do many runs with different noise histories so I'll show you so this is the initial configuration of the chain it's a 4-1 knot for instance this is a 0 and not a chain and let's see if it works so this is the kind of dynamics that you get when you solve this equation okay I'm sorry going a little bit down so you can choose any time you want you can choose the initial final configuration you can choose the bending rigidity the coon length the time etc this was a chain of 240 monomer etc and as I said if you want to do statistics on the possibilities and on the knotting and knotting you should do simulations with many initial conditions and with many noise realizations so let me show you some results so for instance this is initial state 5-1 knot and you look at the knotting of a 5-1 knot so in most cases in many cases at least the 5-1 goes through so this is the most interesting graph so this is the 5-1 and this is the time as a function of time so this is the knot state of the chain so for instance to go from 5-1 to 0-1 first the system goes to 5-2 then to 3-1 then to 0 and the interesting thing is that the system could go directly from 5-1 to 3-1 to 0-1 but it doesn't do it it goes through the 5-2 in many cases so some other interesting cases are 5-1 to 0-1 these are cases where the knotting goes directly through 3-1 without going through 3-2 and you see that the incidence of the persistence length so this is persistence length 50 persistence length 150 the incidence is not very important so this is the number of crossing this is the right yeah sorry well yeah the factor of almost the factor of 50% or something like that yes yes indeed another interesting okay let me continue this is 5-2 to 0-1 and 0-1 to 5-2 so 5-2 to a knot so you go from 5-2 3-1 a knot and if you do the reverse stochastically you have a knot 3-1 5-2 this is the RMSD to the initial condition or to the final condition and you see that you have some kind of stochastic reversibility when you go from 5-2 to a knot or a knot to 5-2 the interesting this is ongoing work with Christian it's not yet published almost published hopefully the interesting things are that 5-1 sometimes goes through 5-2 usually it goes through 3-1 but it could go directly to 0-1 but it doesn't and the lifetime of 5-1 is a few times longer than that of 5-2 and this is a little bit contrary to the fact that the incidence of 5-2 is double that of 5-1 so which means that the incidence of knot is not necessarily related to their lifetime is it 2? oh it's 5-2 yes 5-2 yes ok sorry yes yes of course yes I'm sorry ok I will not this how much time do I have ok 10 minutes so ok maybe so maybe I will stop with this and I will skip what to do when there is self interaction or self I will go I will conclude this part and I just want to say a few words about RNA knots so to conclude this part it's possible to study within this simple model the exact transition path between various knotted conformations so these paths are paths which you generate the path to entangle or disentangle in presence of topoisomerase it's possible to study several entangled paths and see how they separate how they entangle or disentangle of course the model for semi-flexible chain is a bit primitive model in the sense that the bond length is not fixed but it's fixed on the average and the question of what happens you can ask the question for open chains how to unknot them in absence of topoisomerase or with topoisomerase which is not 100% efficient and therefore this is the case when you deal with really self avoiding walks okay so this is ongoing work and for this part I will stop here and now I'll go to the more general to another problem that we have been studying with Christian and Marco di Stefano which is the problem of the existence of knots in RNA so I've been working quite a lot on RNA in recent years and so we know that there is a lot of knots in DNA we saw some examples before these are all the knots in the electrophoretic bands that are obtained we had also done a study with Christian about noted proteins and we found that about about 2% of all the PDB proteins are noted so in polymeric theory the probability of a knot decays exponentially with the size of the chain there is a theorem which shows that exactly mathematically in physics it was inferred since a long time that the probability of a knot decays exponentially with size so as a result in double stranded DNA there are very frequent very many knots 2% of all the PDB proteins typically have knots and the most complex knot is a sixth knot and then the question the natural question is what about RNA are there knots in RNA or not? not NOT of course so what we did is we extracted from the PDB all the RNA chains hybridized not hybridized and if you take all the fragments these correspond to 7000 about 7000 distinct RNA fragments each chain is circularized using the minimally invasive scheme and they compute the Alexander polynomial docker code in order to detect knots present in the RNA so this was work as I said for all questions I will ask my lawyer so ok Christian why do you does this satisfy you? yes I have my knot expert ok so the result is the following there are only 3 knotded structure among the 7000 structures which are obtained I will show them a little bit in more detail so there is a 16 crossing prime knot in this RNA which is comprised of 3,170 bases there is a figure of 8 knot in this in this RNA there are a figure of 8 and 3 trifold knots in this RNA all the knots which were found in RNA were found in cryoem structures I have nothing against cryoem it's a great ok it's a great method so this is for instance this protein 2gyA0 this is the trifold knot that was obtained now it turns out that this knot for this protein there is an X-ray there is an X-ray resolution and this X-ray resolution doesn't show this knot so this is the place where the knotting takes place and in the X-ray there is no knot so first example this one again this is in this case this is one of the knots so in this cryoem these two structures are obtained by cryoem so in this one there is a knot but in the corresponding X-ray structure no knot and finally the third one the third one there is no X-ray and therefore we cannot be very conclusive so the one thing that I want to emphasize is that all these structures are obtained from cryoem there is probably an error in the structure of 3JYX5 when I write there is probably it's because the authors themselves question the resolution in the region where there is the knot they are not very sure about that 2JYA0 so these two RNAs may have a genuine knot but again it could be an artifact of cryoem and as we saw there are very few very close homologues of these structures which have no knots so the conclusion the present conclusions is that knots are very rare in RNA and possibly nonexistent however some knots were designed in the late 90 or yes mid-1995 1996 by Simon and his group and the idea is that it's quite easy to make a twist knot because essentially when you have a pairing of bases in RNA it makes a helix so if you have a loop at the end if you can thread one of the ends here through the loop you have a nice twist knot and these people produced synthetic RNA which had this property and they even showed that there is one topoisomerase which can reduce the knot and un-knot this knot so we have also come out with some ideas about how to I'm almost done I'll be done before how to design RNA knots and one of the simplest way to do is to take a so-called if you construct a H pseudo knot so the H pseudo knot in RNA is a structure which is like this yes so this is an H pseudo knot so what you have here is a helix and you have a helical part like this and then in this part you have also this is also helical RNA helix takes 10.7 bases to make one turn so if this is long enough if this helix is long enough there is a possibility to thread because it's turning it's making one turn or more there is a possibility to thread this through this and make a genuine knot so whether you make a knot or not the representation, the secondary structure representation doesn't see the existence of a knot it's completely standard like this but if you look in the PDB file there is a certain number of RNAs which have both helices long enough so that by doing some simple engineering on the second helix it would be possible probably to thread it through and make knots ok so conclusion for the RNA the question is are there any RNA knots in RNA and the solution probably not and to end I'll show this picture which was done by Francesca Michaletti Christian's daughter so this is not an RNA because it has a knot and it's inspired by a painting by Magrid which is called this is not a pipe and thank you very much for your