 Really a pleasure to be back in Trieste so I was talking to Sophie in the lunch break and we realized that this has already been like eight years and and this was actually the first time the two of us met and Also, I'm the first time I probably met at least some of you and Ever since it's been quite an interesting journey. So this feels almost like a family reunion to me So what I like really a lot about this community is that this is really a truly Interdisciplinary community and this is just a list of people I was fortunate to publish with in the last couple of years So there are people from chemistry biologists most of them are physicists though and no mathematician to this point So but this may change also in the future To give you a small outline I will present be presenting to you in the next 40 minutes or so a couple of projects and The recurring topic of these projects is the application of cost-grant models So I'll start off with talking about Not in polymer melts. So if you take out a single chain What is the probability of being noted and what are the consequences for? Theoretical modeling of these polymers then we'll switch gears a little bit and I talk a little bit about cost-grant modeling of DNA and How likely it is that not occurring DNA? Then we'll switch gears once again and look how I'll talk a little bit about dynamics So how nuts can pass through each other and finally I'll also talk a little bit about nuts in protein So this will be more or less a general introduction so we'll hear more about that probably from Sophie tomorrow and I'll finish off also with Introducing to you a cost-grant model with which we can sort of have at least some idea why nuts are so rare in proteins So let me start with something Very simple namely just remind you about nuts in random walks And I have to apologize to all mathematicians at this point So when I'm talking about nuts, I'll talk about nuts in open chains So that means that we typically have to apply a closure in most cases What we'll do is just we'll extend this From the center of mass through the end points and make a big loop and whatever is captured in there Is analyzed by the use of an Alexander polynomial So random knots are actually the first models which have also been studied by computer simulation Here in this paper by Wolo Gotski et al. in some Russian journal in 1974 and they already realized that knots are Very Prevalence so random walks tend to be highly knotted and if you think about it This actually makes sense because it's very easy to set up a random walk in in three-dimensional space And when you just put one beat after the other with a fixed distance You create lots of local entanglements, which you can also see here in this reduced representation of this random walk over here so lots of local entanglements and also lots of knots and Polymer physicists like this model quite a bit because this is actually one of the few models where you can actually do some Hard analytical calculations with whenever you have something like excluded volume The arguments in polymer physics get kind of hand-waving and and this is the only model Essentially a variance of this ideal chain that you can do like real calculations with So if we go to the slightly more sophisticated model and we add Excluded volume interactions. So now the our beats have some volume things change quite dramatically So we go from a highly knotted state in the random walk to an essentially unknotted state So for instance if you have a chain of 1,000 beats the knotting probability depending on which closure you use is about 1% and The reason for this is that all these knots on the local scale Once you introduce volume are sort of driven off and you you you what you obtain is essentially an unknotted chain To finish the discussion about single chains what you can also investigate are chains in bad solvent conditions So for the computer simulators here So what you do in your implicit solvent model is essentially turned on the temperature and and then the attractive interaction Become prevalent and what you get there is essentially you can test with this polymer over here So if you exert thermal fluctuations to this polymer and then at some point You find the ends and you pull and then this ends up typically knotted And this is exactly what also happens on the molecular level and it doesn't really make a difference If you put it in a cage like in the in the case you can see over here, or if you just let it collapse Both configurations are essentially highly knotted All right after this short introduction So let me switch gears a little bit and talk about polymemels. So your polymer melt is essentially a Big bowl of very long spaghetti which you can you know shake around a little bit and you end up with a configuration Which looks a little bit like this. So it's it's very much intermingled and Each chain has a different color here. So you can see that it essentially goes back and forth as you would expect Just to remind you a little bit about the theoretical aspects of this thing So in polymer physics and also in chemistry These chains or chains within such a melt are often modeled as random walks Yeah, and let me just provide you with some hand-waving argument why this is the case. So this is an argument with This goes back to an idea of flurry and the way I presented can be found in the book of Dijen the scaling concepts In polymer physics. So if this is your your chain or your melt over here, you can paint one of your Molecules in one color. So let's paint it red. So if you're spaghetti, you you know You choose one spaghetti, which has a different color and then you look at the concentration profiles so this is the center of mass of this single chain and Within in the vicinity of the center of mass of this chain you'll get an increased concentration of that chain and On the same time you get a dip in the concentration of the surrounding chains Yeah, and you have to add it at these two contributions up You essentially get something like a homogeneous contribution and what you can do now is you Determine the forces Occur from inter chain interactions and these forces are essentially repulsive so the From here you are sort of pushed downhill whereas attractive forces arise from this energy minimum over here and the the argument goes that overall there are no effective forces acting on this morning mass and So there are something like no interaction and therefore the chain should behave like a random walk so Just to briefly remind you how you actually map such chains onto random walks so you say essentially we we take the mean squared end-to-end distance and They should be the same of the random walk and your chain in the melt and they should also have the same contour links That means the maximum extension of your chain should be the same and when you do that after some some some algebra You see that the effective random walk is somewhat smaller than your chain and in our case where we have a flexible Chain, it's roughly half the size of the chain So now we have everything all the ingredients which we need to test this theory if random walk is actually a good representation And then we look at nothing probabilities and then something interesting happens So this is actually a collaboration with a group of Hendrik Meyer's in Strasbourg. So these are people who have been for some time sort of arguing that you know Random walk might not be an ideal representation So this is the probability of observing the knot in such a melt So the density is about point seven and the chain length is up to one thousand So these are large-scale molecular dynamic simulations which we've analyzed and you see here that the probability is Slightly below ten percent. Yeah, so one out of ten Chains contains a knot However, if you look at yes Yeah, so these are very long simulations Yeah, that's a very good question So what you can do is you can actually look at individual chains and and see If the knot is there all the time or if it disappears and you can come up with some time of correlation time And and these things they really run for years. Yeah, so this takes a huge computational effort So it's it's not not something you do on your laptop All right, and and when we compare that with the corresponding random walk what we see essentially here is That It's highly knotted. Yeah, and there's a large discrepancy between what you observe for your chain in the melt and The random walk which is actually a first indication that they do not reproduce the structure of the chain Yeah, and then this also doesn't change if we change the density of the melt Of course in the limit of infinite dilution we obtain the self-awarding walk Which we have actually here on this axis more or less and but if you increase the density a little bit You actually go up to do something like slightly higher than ten percent But but it doesn't make it really a difference because you you never go to eighty percent up here Yes Okay The knottedness essentially Yeah, so we will see later on I will give some other examples that Analyzing knottedness actually is a very fine gauge for the overall structure of a chain Yeah, and and this we essentially use to to to say if this is a good representation or not Yeah, these are all linear polymers of course yeah all right and Something similar can actually be observed when we look at the sizes of knots So again the sizes predicted by the effective random walk model are much smaller than the sizes observed In the chain and by size I mean that if you have a knot here and you start cutting off from both end then this is essentially whatever remains Defines the size of the knot and as as all of you will know that The knot spectrum of course is also different because if you have something with a knotting probability of 80 percent You will not only observe trefoil knots, but many many very complicated knots Whereas in the polymer in melt with a knotting probability of 10 percent you only observe or mostly observe over here and When I put the transparencies together I actually realized that this is also an indication of not localization in these belts So this exponent is smaller than one so they are weakly localized in the melt All right, so we were thinking hard about how to save the day and come up with some other ideal chain models to To take these effects into account However, we didn't really succeed. So I will just present you one of these failed attempts Because I think it's a very interesting model and I was not aware of it actually before talking to Nathan Clispey At another conference in the beginning of this year. So this is actually published in the conference proceeding from from back then So this model I'm talking about now is called the finite memory walker Yeah, so this is essentially a mixture between self-avoiding walk and random walk So up to a certain interaction distance. So if you have a beat here you Regarded as a as a self-avoiding walk and beyond that. It's just a random walk Yeah, so in the vicinity. It's a self-avoiding walk and for larger scales. It's it's a random walk and if you Change this interaction lengths. So you go from nearest neighbor to, you know, 30 beats on each side The knotting probability of course changes from the random walk and Approaches a probability that you would get for self-avoiding walk in the limit of interaction range equal to change Okay, so this already looks kind of okay, but you already see here that it's difficult to really get down to 10% so even for inter-engine range to 30 it didn't really work out and What's When we finally realized that this is not really working is when we looked at the probability for finding a trefanoid of a particular size Yeah, so this is over here. So here you get the Distribution or the probability distribution essentially for the random walk here you get it for the self-avoiding walk this is our chain in the melt and What you observe here. So for instance for a finite memory walker of size 30 You follow the distribution of the self-avoiding walk until interaction range 30 and then it jumps essentially and Approaches the distribution of the random walk So this is again an indication that up to the interaction range It behaves like a self-avoiding walk, but beyond that it behaves like random walk and in no case We can ever you know obtain the distribution which we get for the melt and we also tried other things like you know Reducing the excluded volume of all chains and then all all these kind of models didn't really work so let me just briefly mention what actually did work and These are tether chains. So if you remember from the first the second transparency So I was showing you this this this case where we have a self-avoiding walk Which when we put it in bad solvent condition it collapses to a globular state and the tether point is actually Exactly the transition point and again when you go to polymer theory class People will tell you that at the transition the energetic attraction and the entropic propulsion essentially cancel out each other And again this should be treated as a random walk So we tested that and what we did is we changed the temperature in our model or the solvent quality If you like until the end-to-end distance was the same as in the melt and When we now look at the nothing probabilities we see that They are very similar to the polymers in the well So the difference is about one to one to two percent and you have to keep in mind that the model now is slightly different because The the polymers in the melt are essentially a thermal whereas now we of course we need some attraction to Enforce the transition. All right, so this already looks quite promising So and when we look now at the distribution again, so we are looking at the probability of finding a travel knot of a particular size Now it all matches up and we essentially repute use with our tether polymer The the distribution of a polymer in the melt. So that essentially means two things first of all theta polymers are not random walks either, but if you want Okay, so maybe I should stay that careful But if you want a single-chain model that reproduces the structures of chains in the melt Actually a tether polymer does a very good job. Yeah, so the nothing probabilities are very very similar and And and this is sort of our our best guess for for describing polymers in in belts Yeah, of course. Yeah. Yeah. Yeah, okay Yeah, so of course you there's similar arguments for polymers in the melt So there are the people from Schrasburg who say that they are logarithmic corrections to that and and the same applies essentially to the theta polymers Yeah, so that's that's completely correct All right, so let me switch gears a little bit and talk about related topic namely ring melts and I only mentioned that briefly Because it it's created quite some attention also from people not in the nothing community Namely for the description of chromosome Territories so here there's a nice picture from 2005 and this is essentially shows you the distribution of chromosomes In the nucleus and you can see that they are quite localized so a particular chromosome occupies a certain amount of space and the reason for that is even though they are like open chains That this is probably a highly non equilibrium state. Yeah, and the equilibrium time is just very large I've read papers which calculated it to be something like 300 years or something like that so you are essentially looking at a non equilibrium confirmation and so there were Lots of discussions where people would come up with a model from polymer physics Which sort of reproduces a similar behavior and these are so-called ring melts. So now we have unnoted Melt of unnoted drinks and you can see already over here in comparison to the open chain melt that the chains tend to be much more localized and this is how people draw the analogy between the two and We also did some simulations using GPU with the who in the molecular dynamics program and the results were actually published in the conference proceeding from Conference or from organized by Tetsuo the gushy in 2011 and you can see here that the scaling Actually approaches one third or if you have the squared radius of duration two-thirds And this is actually an indication that these chains in the melt the ring chains in the melt They behave like globular structures and when you look at them You see that they are almost like a like a real globe you and then just for the fun of it We also put knots in there. So again, this has no connection to real biology, but just we did just did it nevertheless and What we observe there is essentially That for very small chains So if you have a chain of size 100 and you make a melt of these chains they're not essentially occupies the whole chain and it reduces the The size of the chain Considerably so in this region we see a big difference between let's say free one-knotted chains and five one-knotted chains as Compared to the unknotted rings over here. However, when you increase the chain size To a couple of thousands They sort of approach each other and they also go to the one-third scaling limit over here Just like the rings and this is again an indication at least that Knots tend to be localized and if you just make the chain long enough the presence of the rock doesn't really make a big difference anymore All right so this is what I wanted to say about melts and now we'll go back to single chains and cause how cause-grained models Can be helpful in this context? So this is just an introductory slide I copied from from ox for nano-poor sequences. You can find many of those like in the internet. So this is the How nano-poor sequencing is supposed to work? So this is a new emerging technology and people are very excited about it To replace so-called next-generation sequencing devices and the idea Is you have a port so this can be a biological poor or synthetic poor and you? Thread a single DNA so this is double-strand DNA over here and here it's separated into a single strand You thread that through here here's a little piece which sort of holds the chain in place All the time you apply a voltage across the membrane so that ions can flow through the poor and depending on what basis are Located at the bottleneck of this poor you get a different electrical signal and then this can be correlated to the basis So to my knowledge So this thing is still sort of under development, so it's not ready yet to really replace next-generation sequencing devices But if they sort of get rid of these problems This is very interesting because these devices can be made very cheaply so you can buy one of those things from from this company for $1,000 and So this could really revolutionize and bring this thing to your doctor's office or even you know in your pocket eventually if if it works at some point I should say and one of the advantages of this Nanopore sequencing is actually that you can Thread very long chains through this device Yeah, so currently only about 5,000 to 10,000 base pairs and later on you will see that you know this might also be connected to knots at some point But the division is clearly to extend that to much larger chains and eventually sequence hundreds of thousands of base pair in in a single run All right, so how do we model DNA? So so this is kind of the DNA which we all have in mind and I think this is from Wikipedia actually So this is DNA in all its atomistic glory Of course if we want to determine nothing probabilities for half a million base pairs I mean this is not the model one should choose so we again take a physicist approach and sort of zoom out a Little bit and then DNA starts looking like that Yeah, and this is something we know how to handle with coarse-grained models so this is essentially your your semi flexible polymer and There are certain components which you have to take into account Maybe the most important is the stiffness Then one should also take account exclude volume interactions So again, I don't want to go into details here but if you just treat it as an idle chain or a warm like chain you will not get the correct results and Finally one can think about electrostatics, but it turns out that these electrostatics are actually screened So we don't need to take them into account explicitly But what we do is we sort of incorporate them in some kind of effective thickness and this is pretty much along the line of a paper from Wologotsky in the early 90s Which also tried to to come up with such a model Based on nodding probabilities from 5,000 to 10,000 base pairs However in their model they made an additional assumption namely that the persistence length of DNA is 50 nanometer and in contrast to that we didn't do that so we just took their experimental data and of course fitting becomes much more complicated by then and Come up with parameters for the stiffness and the effective thickness of our beats and as a consistency check So this is also more for the people interested in polymer physics. We actually determined The persistence length of this chain and it turns out that it's exactly 50 nanometers So again this is an indication that if you get the nodding spectrum right and you have the Good core screen model then this will also give you other structural properties of your chain for instance the persistence length So let's look at the probabilities So if you go beyond 200,000 base pairs with this cost range model already more than 50% of your chains are noted Yeah, and it's actually interesting from an historic perspective because this first model by first paper by Wologotsky in 74 They already made some predictions about nothing in DNA But of course this was based solely on a random walk and they tend to over predict the probability of DNA So they are like, you know, like somewhere over here Maybe if you do it with a pure random walk model, but nevertheless, I think it's actually quite amazing all right then most of the knots are of course travel knots and recently Actually, just a couple of weeks ago. There was a paper published by the case Decker group Ken blazer and and sure cross back was also involved in that which gave us new data on long DNA chains using solid state nanopause and Of course our work at that point was completely independent from them So we didn't use the data to fit anything But we were using the data from the early 90s and you can see that they actually match quite nicely So this is experimental values for one molar KCL However, it's not a perfect match because what we are looking at is a physiological soil concentration So for physiological concentration, we probably expect it to be somewhere over here Yeah, but nevertheless, I think it's it's quite amazing that independently we came to the same conclusions using experiment and computer simulations We can also now determine sort of the typical size of such a knot By looking at the distribution and we define the travel diameter something like two times the radius of duration and the most likely size is around 200 nanometer and that doesn't really change if you increase the size of your DNA and After all, this is a topic which really needs to be addressed in my opinion At least if one wants to go beyond let's say the five or ten thousand base pairs which are currently investigated with these devices and So what will happen actually? So here's actually Sort of a plot which gives you some ideas about scales. So this is DNA of 16,000 base pairs This is the typical size of your knot over here. These are typical sizes of nanopores. So 5 nanometer 10 nanometer 20 nanometer and you see already that this is kind of a floppy object Yeah, so it will not probably clock your nanopore sequencing device Or get stuck or something like that But nevertheless it might influence the ionic current and this is probably the more serious problem And there are actually some some some very nice papers on the dynamics by one of the organizers the group of Christian Micheletti which sort of addresses the dynamics of how how you thread such a chain through a hole and So our paper was also just published last week actually after some journey Odyssey I should go and then one of the remarks of the referees was that we should not only look at Trefoil knots and figure of eight knots, but also at the complete knotting spectrum and There we actually made a quite an interesting discovery Namely that the knots which are more complicated than trefoil knots are not other prime knots But they are composite knots of trefoil knots So up to 500,000 base pairs you essentially just observe Trefoils two trefoils three trefoils and and very few other knots actually so that's interesting so for instance here we have the figure of eight knot and Compare that with the probability of observing two trefoil knots or even three trefoil knots Disco over here. Yeah, so in some cases it's even more likely to observe three trefoil knots than observing one figure of eight And it's it's a highly non-trivial thing actually because one could argue Okay, maybe this is just the product probability, but that doesn't add up either so currently we we don't have a really good Theoretical model of how these things actually come into being so this also gives me the the chance to talk a little bit about the dynamics of these composite knots and this is a project which is Emerged as kind of a fun project and Goes back actually quite a few years when I was still a postdoc with Meran Carter and and at some point He was suggesting such a project as sort of An interesting thing to look at and but it took like in the end like ten more years to really get it done so what we have here are two plates and we have a polymer Confined between these two blades and you have like two knots So one over here and one over here, and then you just do your molecule and that's an emic simulation I also have a video of that with music, but but I think 90% of the people have already seen that more than five times I decided to just make the pictorial thing this time So what you see is essentially that these knots can actually go back and through each other back and forth Yeah, and this is actually something which is sort of counter-intuitive to begin with But if you think about it, it's actually not that difficult to understand But before I come up with a solution, so let me just show you how we analyze that So this is essentially the distance between the locations of the two knots So if you have a positive number that means that the green not is over here and the red not is over here If the distance is zero we are in this intermediate state where they are intertangled and if we are negative Distance it means that they have changed positions and then the trajectory looks a little bit like that So it goes back and forth and spend some time in this intermediate region So what you can do is you can just project that Onto this axis and you get a probability distribution and as good physicists We know that probability distributions are always related to free energies So if you take the negative logarithm of this thing you get something like a topological free energy And you can see that depending on how far the walls are apart This is actually The intermediate state becomes more likely or the separated state becomes more likely of course if you push the walls very close The knots would like to stay in the intermediate state where the two are intertangled and if you you know pull them really apart then they would like to be separated and So one of the questions is okay what actually happens in real DNA or real polymers and when you don't have these these walls and so we did Monte Carlo simulations and Analyzed configurations which just had these two specific knots in there And then we did essentially the same analysis and what turns out if you go to very long chains that Each state is more or less equally likely So this over here is because you reach the end of the chains of one infinite chain You would expect this to be flat, but in the middle You still get us a very very small dip Yeah, and this is again the intermediate state, but if you look at the scale over here This is only about one KT. So what we observe here is that The combined state in which the two knots are intertangled is slightly preferred to the state where the two separated and Of course, this is a purely entropic effect. So we have something like an entropic interaction or entropic attraction of knots But as the effect is very small So Still take that with a grain of salt. Oh This is just how we define it. So if you have a knot over here and a knot over here and they change positions Then then this is negative. Yeah, so but of course this distribution should be symmetric All right, so let's switch gears once again and come to the last topic which is knots and proteins and This is sort of just a summary maybe about what Has been done in the past couple of years from from a theoretical point of view maybe So people including us have looked at knots in the protein data bank and this has already been mentioned Before that knots tend to be very rare. So this is just sort of a semi-complete list of knots which occur there. So currently we have 12 knotted proteins and Most of them are simple travel knots to figure figure of eight knots one five two and one six-fold knots which was already mentioned before and You know depending on how you compute these numbers. So you end up. So I think you were suggesting 2% So I would probably go a little bit even further down So maybe to 1% so these things are actually very rare Yeah, and then of course a couple of questions emerge And I think in the last 10 years or when we lost at least compared to when we last met here in Priest I think some of the issues have made some progress and we are now in a more comfortable position actually to address some of these questions so For instance, why are these knots so rare? Okay, if you think about your your globular polymer at least from from a polymer physics point of view You know you would think that you know all proteins should be knotted If you ask a biologist you say no proteins will be knotted, but that's a different story Then how do these things actually fold? Yeah, and again They have been very beautiful experiments by by Sophie Jackson and and and very interesting simulations and from from Joanna and other people who really give us I feel a better understanding of what's going on there in these days and So I'll talk a little bit about that also later on so, but let me start with sort of the newest addition to the protein knot family this is actually a very interesting before because before we only had enzymes and DNA binding proteins actually Which contain knots and for the first time we've observed a knotted membrane protein Yeah, and and this is what it looks like. So these are two two structures from different sub-families of This transmembrane protein and again like questions like how do these things actually get into the membrane? How do they fold all these things arise? once again, we actually had with a bachelor's student did a small literature research and Recently a couple of weeks ago came up with such a phylogenetic tree, which was done by somebody else These guys over here Pippman and Hershey in a journal called rice So this is a very So to be frank, I didn't know that such a journal exists and it's not something I read on a weekly basis as a physicist I'd say but nevertheless they they made a phylogenetic tree of this family and One of the structures which has been resolved in the PDB is located down here actually and the other one is located over here And so they go back to a common origin and that actually is an indication again that Potentially the whole super family actually contains that knot and that is actually a very old structure For the experts. This is kind of interesting because the sequence identity between these two structures is only 10% That's very uncommon typically structural homologues. They have at least like 30% sequence identity all right, so this is actually something I'd like to Talk a little bit about so the regular thing is essentially when you find a knot in one protein You will also find it in instructional homologues So as an example, I've picked this nice five-fold knot here, which is present in human And there's also a homolog which is present in yeast and even though you know at least in Germany that there's a nice connection between yeast and human the idea is Probably just like in the case before that it goes back to a common ancestor and that means that the Structure is potentially hundreds of millions and maybe billions of years old so the knot has been there for quite some time and An interesting side remark which I tend to make at this point is that this five-two knot has a structural homolog Which is abundant in everybody's brain? Okay, so this is again a story people have heard probably like five or six times at least So if you look very closely, it's located right over here so you can also make kind of a back-of-anvil of calculation and When you add up the numbers You realize that ten to the nineteen so that everybody has ten to the nineteen to ten to the twenty of these knotted molecules in your brain and this was sort of an Heureka moment for me but There are also lots of funding opportunities as pointed out by David touching in the last Get together in Warsaw. So this protein is related to Parkinson's disease and certain types of cancers So so maybe that's finally hoped to to get some funding for that as well All right, so then the question arises If these knots are always preserved and the answer is no, there's one counter example Which we found in in 2006 and the protein is named transcarb amylase and what you see over here Is the not a tomologue at least not a part and the unnot a tomologue which is located over here So this is missing this arc over here essentially, so you have to really look carefully to to see that this thing is actually not and this is not and So how this came into being was maybe just by the deletion of the corresponding sequence in the gene And this was actually when I saw that for the first time a perfect candidate for creating an unnoted protein from another one But as you will see soon that this is not how it actually played out However, there's also a very nice bioinformatics study by Raffaello and Christian So if you want to learn about the origin of this particular knot You should have a look at that as well So the difference actually is not that exciting. So it just catalyzes like different reaction so this carboxyl group over here moves over here and another version and Therefore we have some more space over here for an additional acetyl group. So In the end, it's nothing really exciting So this is nothing, you know, they have been of course lots of speculation and people are still speculating what knots are really good for in that case It's probably not that exciting the current speculations are that it Maybe improves resistance against degradation or thermal fluctuations and these things and there's actually some some evidence by by simulation at least Which support this plane? All right, so let's move on This is again, I think another milestone in the field which which came into being after the last year's conference And this is the first creation of an artificial knot in the group of tortiers So what they did is they took a homo dimeric structure and connected the end of one protein with the beginning of the other one And by doing that they created an artificial truffle knot and they actually pose this not a structure and it's also what probably happened in the beginning when these Structures came into being in the first place So when you look at the structure many of them actually still have remains of these dimeric structures in them so that from a from a protein engineering perspective I find very exciting because you know, maybe at some point this will be just Within the toolbox of you know, how you design proteins and not just structure but also topology will eventually play The last thing on the second to last thing I would like to address is The issue of folding and this is actually a simulation. I did about 10 years ago So this is again, we're coarse-grained models play a role So you have your basic polymer models So you have some extra volume spring potentials angle potentials torsion potentials and in addition you have something like a native potential which essentially gives a negative energy Contributions to amino acids which are close to each other in the native state Yeah, so this way you enforce that the protein will actually fold into the correct confirmation and Again, I quickly realized that there are experts in this which are much better in this than Myself and and one expert is also here Okay John is so she's the one who's was really Has has progressed this field considerably with these kind of simulations And so I was fortunate to publish this paper together with her Where we looked at the folding of this six-fold knot and this looks already quite complicated So give let me give you my poor man's version of it So again, you see this dimeric structure over here the red part and the blue part and those are connected By this wide arc if you want and if you just flip that over The knot essentially disentangles so even though you may have a very complicated knot to begin with in the end It may not be so difficult to fold But again, you know, these are coarse-grained simulations and They are just here essentially to give you a feeling for these kind of things and of course This needs to be confirmed with with corresponding experiments All right, so the last topic in sort of to close the circle Once again, so I will talk about a coarse-grained toy model, which is very popular among physicists namely the so-called HP model H stands for hydrophobic and P for polar. So this is essentially a lattice protein, which has the The property that in low-energy states or native like states It forms a hydrophobic core just like in real proteins. Yeah, and this this model is goes back to candle in 1985 and has been popular amongst physicists that he's ever since and so with this model I Wanted to to address the question. Why not are so rare in proteins? So first I thought actually the formation of this hydrophobic core over there may have something to do with the rarity of the knot But in the end this didn't out turn out to be true so what we did here we simulated 100 sequences the size of each so we 50 percent H 50 percent P and Chains of size 500 and determine knotting probabilities of native like states so these are very low energy states of that model and Of course, I need to point out that this model has Significant degeneracy so these ground states are not unique. So which actually enables us to determine these knotting probabilities But and let me just remind you that these are very difficult simulations Yeah, so again, I tried it on my own and realized that this is too difficult for me And I had to team up with Thomas we used to sort of the world leading expert for these type of models And even with him and very optimized code and using all the tricks of the trade We still needed like five million CP where I was to to get this graph So so don't try this at home If you have to so So if you look at the mean knotting probability and compare that with a homo polymer of similar density You essentially end up with the same More or less the same knotting probabilities So it doesn't really make a difference if you have a hydro polymer or homo polymer However, you see that there's some variance here So you have sequences which are more prone to be knotted and sequences which are less prone to be knotted. So this Let us to the idea that we can sort of intelligently design sequences Which contain very few knots or many knots actually yeah and Just to put this into perspective. So this will give nature a route to evolve to a universe which is is actually I'm not so okay. I've already used up all of my time. So let me just Skip through that very quickly. So the I'm not at states in this protein lettuce universe They are more like the sandwich like structures and here you can also construct states which go back and forth I'd be happy to talk more about that in the coffee break and this already brings me to the end of my talk and To sort of if I have to put Everything into context again The only thing I'd like to say is that you know the use of these car screen models can be very useful For addressing specific issues not only in polymer science But also in this in the context of molecular biology and this is actually something I believe we as physicists can contribute to this community And with this I'd like to close and thank you for your attention