 possible for the patterns that we see into data. And after analyzing some of the statistical features of the contact probabilities that are emerging from high C and gamma experiments, I moved on to discuss some very, very simple and schematic model of what could be some of the mechanisms underlying chromosome folding. And remember, we delve into the strings and binders model. The concept behind this is very, very basic. The idea is that long a sequence, a DNA sequence, you have binding sites engraved in the figure for the binders of the strings and binders model. And so by diffusion, the binders can bridge those binding sites. And they can loop the polymer and fold it. And the key message is that now independently, this is the lesson from polymer physics, independently on the little fine details of the model, or the specific model you consider. I mean, the interaction potential or anything like that. This type of self-interacting polymers have a phase diagram, which looks like one summarized here. And on the x-axis, you have the concentration of the binders. And on the y-axis, you have the binding affinity. How strong is the binding energy of a binder to its cognate binding site? And interestingly, say, a broad class of interacting polymers have the coil-to-globul transition. So you have a transition point, which is this theta line here. And you move from the phase where the polymer is in a coil state, so randomly open and folded, to a state where it is instead collapsed into a globular lump. And the two are, say, two distinct universality classes. And so the idea is that a filament of DNA or different portions along such a filament could be neither one or the other of those fundamental thermodynamic states. Of course, this is far from trivial assumptions. But what I tried to show you yesterday is that, for instance, average contact probabilities can be decently well-described by this kind of assumption. So literally starting from the textbooks of the 80s, you can compute the contact probabilities in the two states. You consider a mixture. And you can explain over pre-orders of magnitude, really not much fitting parameters. Just one, the composition of the mixture, the counterprobability, the average counterprobability is shown from the data. What I also tried to delve into is to describe another type of data, which comes not from high sea, but from fish, let's say microscopy. With microscopy, if you label different DNA sites, you can access the physical distance between those two sites. And you can measure that. And you can compute distributions and moments. In particular, I tried to take your attention on the so-called moment ratio, which you see the ratio of the average of the fourth power of the distance of the two sites of interest divided by the square of the average squared distance. This is an important quote in polymer physics because this is dimensionless. So either your model described this in a good way, or you have no fine-tuning parameters of whatever. And what I showed you beneath is a collection of experimental data. I understood that I have to explain this again, so let me go through it again. What I'm showing you here is a number of experimental data points taken from that paper. And the different colors correspond to different chromosomes, different loci on those chromosomes, different cell types. And on the y-axis, you have the experimental recorded moment ratio. And on the x-axis, the data is ordered in such a way that you see they're ordered according to the genomic distance of the two sites considered. So for instance, that point corresponds to two sites which are at roughly 1 10 of a megabase apart. That point is at 10 megabase apart. And what is surprising, this is what I tried to convey yesterday, but I understood I wasn't good enough. What is interesting is that, you see, there is an accumulation point from above at around three-halves. Whatever the distance, the genomic distance, whatever the cell type, whatever the locus, whatever the chromosome, roughly they all come to 1.5 from above. Why? And what I'm showing you here on the right instead is what is the prediction of equilibrium polymer physics. If you take the two thermodynamic phases I told you about, the coil and the globular thermodynamic phase of an interacting polymer, you know that in the bulk of the thermodynamic phase, the distribution of fluctuations is Gaussian. And so the moment ratio must be exactly three-halves. So what I'm showing you here, the x-axis is different. The y-axis is as before the moment ratio. The x-axis here is an axis that allows us to move across the theta point, the transition point. So at low concentration, you remember our polymer is in the open state. And in fact, you get three-halves for the moment ratio. When you cross the transition and you move into the contact state, the globular phase, again, you see you get three-halves because once again, the system is Gaussian. It is well known instead that around the transition, at the critical point, fluctuations are no longer the captions. And the moment ratio shoots up. And interestingly, the scale is very similar to the scale that you see in the experiments. And so this is telling us that, well, we understand first, why there is an accumulation point at three-halves. Second, why it is from above and not from below. Third, that many, many experiments maybe have been conducted really in size which are in either the open or the closed phase, because they are close to three-halves. There are other loci, other regions, which have scattered out of the three-halves accumulation point. However, they well reside within the range predicted by those toy models, which could imply that those sites are moving across the transition from open to contact or the other way around. So summarizing the impression we have when we look at average contact data, average distance data, that we can make sense of them all by simple concepts of basic polymer physics. The next step I wanted to attack is patterns. Because models such as the Fractal Globally Model or the Self-Formatting Work Model of polymers cannot explain tads and so on. If you look at their contact matrices in a self-formatting work, it's uniform. It is only an average genomic distance effect. And instead we want to understand what's the origin of the patterns we see in the data, because that is where the action is. And so yesterday I closed my lecture trying to give you a sense of how within this context of the strings and binders model or any other self-interacting polymer model, you can make sense of the pattern. And I consider a variant of the toy model I showed you before, where now you have two colors. And it says that along the polymer chain, still a toy model. Along the polymer chain, though, there are two types of binding sites, red and green. The red binding sites are bridged by red binders, and the green binding sites by the green binders. And for such a system, really it's easy to guess how it falls from what we just discussed, because you immediately see that it falls like that and all the possible combination in the different tumor-like phases. So you form a red and a green lump. And they are not technical. They are not independent, because there is still a portion of the polymer, which holds them together. But they are almost independent. And so if you map in this toy model the content matrix, it has two blocks, similar to what is seen in the experiment, the tats. And you must now start guessing, how can I produce more complex patterns? And I, again, gave you an example of how you can obtain high-order domains. And in the evolution of the toy model I discussed, I consider the variant now, where not only you have the two red and green binding sites, but now you had a blue set of binding sites, which go across the two subsets, the red and the green. Again, here it's easy to understand what is happening, though maybe it's less trivial than before. Because interestingly, it turns out that the red and the green first come together, each on its own. And then the blue binding sites bring together the two previously independent domains. And you form a high-order structure, which looks like that in the toy model. No, no, no, this is equilibrium. The reason you keep the red and the green compact, and then the blue bridge together, is just a matter of, say, genomic proximity. It is much easier for the red set to come together first, because they are localized and the green the same. And then, only then, you start bridging the blue. But this is the equilibrium, so it's not a matter of time. But nevertheless, yes, there can be lots of complication in this kind of things. So in this toy model, you see that you start producing interactions across the two blocks which previously were virtually independent. And once again, this is to try to visualize how you can obtain high-order patterns, such as the metadata that we discussed. However, this was a sort of basic introduction to give you a sense of how patterns can emerge in models such as the single-minders or in self-interacting polymer models. However, this can be pushed to match real data to good accuracy. And this is what I want now to discuss. So what you see on the top here is now well- familiar with that high seed data, so contact probabilities. And this is a region of roughly 5, 6 mega around an important gene, FA4, a gene which is involved in embryo development. And I will show you its link to human diseases, congenital diseases. And what you see up here is the patterns of contacts across that region. You see, for instance, that there is a TAD or a meta TAD which encompasses FA4. This is the enhancer of FA4. This is the regulator of FA4. And instead here, there is sort of loser structure. In here, little TADs or whatever, sub-TADs, the definition of TADs and sub-TADs as I stress, this is very, very loose. Here, instead, you see what the strings and minders model does. In overall, you see, the agreement is comparatively good considering the simplicity of the ingredients that we have in this type of model. So not only the average behavior is reproduced, but also the structure of the patterns, the sub-TADs, and so on, whatever they are. The correlation overall is 95%. And this type of quality of explanation of experimental data is found genome-wide with this type of model. So we can explain the way in which the genome fold, the entire genome fold, and 95% of accuracy with this type of methods. And you see here, since now we have the polymer model of that, we can reconstruct the 3D structure, which correspond to that. And notice, this is not one structure. You have to imagine this in a statistical mechanics sense. So the polymer folds in its thermodynamic phase, and then you say it's moving. There's not one state, there's an entropy associated. So different, many different conformations. If you, this is PDF, but I also have movies that I can show you. But we have to imagine that the polymer is breathing, is moving in time. So the contacts that you see there are sort of average over time. At a given time point, you may only see some of them. And you may start guessing what is a tador, a meta-tador, whatever are those IO structures. For instance, look at here. The color scheme here is the one corresponding to that strip. And so you see this little sub-tador whatever is associated to that color, dark red, then pink. Maroon, and so on. And you see how it folds in a complex conformation. It folds bands on itself. And that's what heuristically would have called, we would have called the meta-tard. And you may see that here that are, for instance, structures which correspond to tads or something, they are because you see these two regions, which may be responsible for the two sub-tads, are also folding together, and so on. Of course, there is a price to pay for that, because the modal is no longer as simple as the toy modals I showed you before, where you have distinct sectors for colors, and you have a few colors here to explain the data, 12 main colors have to be introduced. So the modals predict that there are at least 12 main different colors, and overall 24. So the modal is predicting that there are, what are the magnitude, a dozen factors, particles which mediate the interaction. And then I will tell you how one can go and look for them exactly as we do in high energy physics. We predict that there is a particle which has an opposite charge with respect to electrons, and go and look for it. So what I would like to do now is to give you a sense of what those binding factors, at least within the modal, which of course can be wrong, because what I'm showing you now is a fit. I will tell you later on how we tested this. But by now, this could be all wrong. This is just a fit. Anyway, I want to try to give you a sense of what those binding sites are. So I told you that the modal, this is a cartoon of that region as roughly 12 or so colors. And what you see here is a distribution of the different types of binding sites along the DNA sequence. So you see, for instance, there is a bunch of green binding sites here at the beginning. And they may be responsible for that little subcut there. And then there is another bunch immediately after, which is distinct type, and so on. So an explanation of what paths are is emerging. If there are regions where there is a dominant type of binding sites, you see, of course, a specific pattern there of interaction. But that's it. And now we understand from fundamental principles how that arises. Notice also that there are binding domains, such as the swan word, the pink, which are spread along the region. This is producing long-range contacts. And so the interactions between the paths, if you want. And so the arising of metatars and high-holder structures. Once you have the modal predictions about the binding sites, an obvious thing to do is to try to see whether they correlate with some known factors or some specific type of sequence along the genome. You can ask, what are the typical binding sites that correlate with yellow? Or is CTC half, you remember that protein, which was so important, is CTC half correlated with some of those binding sites? I mean, look at where yellow is. Is CTC half located where yellow is? And so on. And of course, you can run that type of analysis. And the result is summarized here. So what you see listed here is a number of chromatin factors. I'm not entering into many details, but if you want, I may be to. Some of those, you have seen them. CTC half, you recognize it. CTC half, pull two is polymerase. And what you have here, instead, is the different types of colors, so green, red, and so on. And what this has shown us is that a single color predicted by the modal, in general, is not a single protein, a single molecule, but is a combination of. So just a few protein factors. Just a few molecular factors can produce a combinatorially large number of different types of binding sites. And in this way, with few chemical ingredients, you can produce the complexity of factors that we are discussing. Just for CTC half, let me give you a sense of what I'm trying to say. You see, CTC half is not correlated with only one color, but it correlates with this, with the neighboring one, but that one, and so on. And CTC half is also associated with other factors, and in different combinations, according to different colors. And by the way, what we found is that there is a subgroup of colors for this locus, locus in biogeomins, for this region, roughly 30% of the colors, which do not correlate with known factors. But they are important to explain the architecture of the locus. And so this is the potential discovery of new particles, new factors, which must yet to be discovered, which are responsible for the formation of the architecture. So a very quick explanation of how we guess the modal. And this is machine learning. And I'm not entering into details. You have a lot of that. And it is the following. Starting from good old physics, you guess at each term of the standard modal one by one. Now with advancement of computers and so on, starting from experimental data, we can have a machine learning approach to try to guess which are the colors where they are located, what's the minimal modal that you need to introduce, explain the data, and so on. And so we developed Prisma, which is a machine learning algorithm to do that. And it's very easy. I just want to give you the flavor of what we do. Suppose you know that your system has a given contact matrix. You can ask, what's the minimal optimal modal, which, based only on the strings and binders, physics, explains the data. And so this is what we do. And in this way, we predict the colors, the minimal number of colors, so binding domains along the polymer, which explain those data. But machine learning here is only complicated name to say, let's try to make a fit with the power of computers we have today. But that's it. Of course, once we have the modal, the Hamiltonian of the system, we start asking all sorts of questions, including, for instance, what's the effect of a mutation? And this is what I now want to discuss. We thought that a stringent test of this type of models would be to make predictions on the effects of mutation. Because up to now, this is a fit. I showed you we can explain the data with very good accuracy in all words and so on. But how can we test that as we do in theoretical physics? Well, if you think about, or at least we thought about that, a very stringent test is predicting the effects of mutations. The idea is the following. Suppose you know your model for your wild type logs, wild type in biology means the normal genome. So you know what's the polymer, which is described as a normal case for your genomic region of interest. The idea is that you can implement a mutation in that. So for instance, you have a deletion in the example shown. You take out that portion, that segment. And then out of only theoretical physics, you predict how the system refolds. What would be the contact matrix? And I call this a stringent test because there are no whatsoever fitting parameters. Take your polymer model and just compute polymer physics. So you compute the contact matrix and you compare it with experiments. And we did that, not just for a deletion. We did that for deletions, duplications, inversions, so all sort of mutations. To see whether we can explain, predict their effects and compare, test the model against, experiments in cells bearing exactly those mutations. And so this is the way we tested the model. And additionally, of course, the excitement is that you can do that for mutations, which are linked to human diseases. To see if out of theoretical physics, we can predict whether a child will have a malformed limb or cell. And this is what I want to show you now. So this is the data you've seen before. This is the FA4 region. The FA4 gene, you remember the TAD or beta TAD, you see there and so on. And this is the model and this is the experiment I showed you before. Now we consider the del B plus deletion. This is a deletion, you should see cats, that portion of the region, so it takes out FA4. And this is what the model predicts. When you take the wild type, the original model, and you take that out, the model predicts that the system refolds like that. So contacts are rearranged in the way you see. And Stefan Mundl is in Berlin, run independent experiments, blind test of the model, in cells carrying exactly the same mutation. And that mutation was chosen because this is a, we have a patient with brachydactyl. Because that mutation is known to produce brachydactyl. That is to say, as you see, a malformation of your lens. And this is the experimental data. I think this is interesting because you see that only the model predicts overall the contact patterns that you see in the data. But what I find exciting is that the model predicts new contacts in the terminology of biologists at topic contacts. So regions of contacts which are absent in the original system. And so you would have no way to predict, starting from only experimental data. And they are confirmed in the experiments. And those contacts are responsible for the disease. I try to explain you why. You see, when you take out that section, that segment, the new contacts imply that there is that region in dark blue which starts interacting with that other region. And the first is the region where the enhancer of FA4 is sitting. So the enhancer of FA4, rather than interacting with FA4 in the mutated system, starts interacting with the next gene, PAX3. So the enhancer has a wrong target. This is what physics predicts and is confirmed experimental. There's a wrong target and activates the wrong gene. And this is what Stefan showed. PAX3 is upregulated and leads to the disease. So by the way, you see here the three destructures, how they refold and so on. But I don't want to delve into many details. And of course, we repeated that for a number of cases. This is Delby I showed you, but this is a duplication. So the duplication means that that region has been duplicated. In that duplication, which gives syndacty another lame malformation, there is a patient with that. I stress this. This is not an idea of work. This is hard life work. So in this case, the modal predicts that, you see there are ectopic contacts here. In this case, the second enhancer of FA4 that you add by the duplication activates Win6, that other gene, which is upregulated in the mutated cells and brings and gives the malformation, et cetera, et cetera, et cetera. Mutation. What do you mean? Oh, no, that's a more complex story. We are going one step at a time. By now, we can tell. Look, if you have that mutation, a given set of genes will be affected. And if you know what those genes do, then you can think that they may have a phenotype. So you may have a disease or something. By now, we can only tell out of physics whether a given genetic mutation will impact the regulation of some gene. OK, so to wrap up, I think this type of results are exciting for at least two reasons. From the point of view of the theoretical physicist, this is very exciting. With simple counts of polymer physics of the 80s, we can predict how our genome falls, how our gene are regulated, which an answer, count at which gene. You see, we are really delving into the fundamental of how life works. From the other reason of excitement is that, as much as with pencil and paper, we could predict the existence of the Higgs boson. As theoretical physicists, now we can tell a kid if he's going to have a disease. And we can do that not just for the few mutations which impact one gene, but we can do that for every single mutation, including those in the non-coding part of our genome. And so you see the promise here is that we can really change medicine, genetic diagnostic, which today is really a discipline in its infancy. This is bringing a quantitative level to genetics. In the five time, I will tell you how this can open huge implications in business. Of course, these type of results are not restricted to the region I showed you. We have data genome-wide. I want to go fast. This is another region, another disease-associated region. This is a new man-associated region. And again, you can redo all what I explained to you, but I don't want to spend too much time on that. And another type of implications you can extract from physics is to drive not just the pair-wise contact matrix, but you can establish what are contacts overall. So you can see if there are triplets, if there are cases where a given gene is controlled not by one regulator, but by two, by three, four. And what we discovered is that, in fact, there are lots of combinatorial regulation of genes. So for instance, there are lots of many body contacts. They are exponentially more abundant than what you would expect in a randomly-folded polymer. And we made predictions on that. We expect those three regulators come together to that gene. In the technology I showed you, our technology, GUM, was the first capable at assessing multiple contact because I see, at least in the original version, can only measure pair-wise contact. With GUM, instead, you can, if you remember that, that's the technology where you extract slices and you count who's in. Since that's all statistics, you can also extract what are triplets, four droplets of interactions. In GUM, we indeed showed that there are multiple contacts. Regulatory processes are very complex. They can involve highly sophisticated regulatory landscapes for the genes. And now, for the sake of, let me skip that. But the key message is that we predict also multiple regulatory contacts. And they can be confirmed experimentally once again. This is an example of the model predictions in entire chromosomes. This is entire chromosome 11. This is real data. This is the model. And again, correlation 95% and so on. Let me skip this. Before the break, what I would like to discuss is that now is I told you that with the model, you predict that there are particles, new particles, which mediate interactions. Of course, we wanted to go and find them. And I want to show you an example where we did that. And again, very close analogy to high energy physics, if you want, though this is low energy chemistry. I would call it this way. So to do that, we considered a simple locus, a simple region, a region where there are few colors. And so it's easy to go and test for the particles which mediate interactions. In the region, we considered as the Huxby locus. This is the name the biologists give to it. Huxby is a cluster of genes which are important in development. And what I'm showing you is that region, it's roughly 1 mega around that cluster, in mouse embryonic stem cells. So where you can do all sorts of experiments without ethical or with minor ethical implications and so on. And you see, this region is marked basically by two colors, red and green. It has in the toy model I showed you. And so starting from the colors predicted by the model, you can produce the content matrices theoretically. You can say, I expect that, say, SNF 8 interacts with SNX 11. SNX 11 is here. SNF 8 is here. And there is roughly 1 megabase. So you see, with the model, you can extract non-trivial implications, non-trivial predictions. There are two genes which are 1 mega apart. And you predict the very common content. And you also predict that it must be a factor, a green factor, which enables the content. And Annapombo in Berlin brought the experiments to test this. And not only Shiran, Highsea, but also cryo fish. So an advanced version of fish microscopy, which has a very high resolution. And to cut short the long story, what she found is that the model set of context found in the model is indeed found experimentally. But Shiran, tribal collocalization experiments. So by fluorescence, not only she marked the two genes of interest, say SNF 8 and SNX 11. So those two are marked. But she also marked the proteins which we guessed could be involved in the contact. And you see here, green is chosen because the genes are active. And so the guess was poll 2 must be involved. And in fact, by tribal collocalization experiment, she found that poll 2 is involved. So that is one of the components of the molecular factor, which holds the two genes together. The red genes are different class. Red genes are now named Poise genes. So genes where the genes is almost active in the sense that there is another factor which prevents poll 2 from transcribing from a full transcription. And that's linked to PRC2, another chemical factor. And what Anna showed is that PRC2 collocalized with red gene. So that's one of the factors involved in bringing together the red genes in this toy model. So this is an example where we know the folding, used that to guess what could be the particles, the molecules which mediate the interactions. And we found at least some of the particles which mediate the interaction. And this type of models can be used to test a number of different hypotheses. For instance, as I showed you, CTCF is involved in genome folding. And so one could use the models to test whether CTCF is responsible for the folding of the locus. One way to do so is to create a model of the region where you think that it's CTCF responsible for the context. And so you map the CTCF binding sites, which are the orange one. And from that model, you extract out of polymer physics what would be the expected content matrix. And you can do that also in other scenarios. You can say, no, it's PULT2 only, which produce context. And so you can map where PULT2 binds. PULT2 is the polymerase. And you can see what is the appearance of the content matrix in that scenario. And this is instead a scenario which I showed you before to cut short a long story. And of course, in this scenario, the one I discussed, you can have also different sub-scenarios. I tried to, I dealt with that to give you the sense of how far you can go with this type of prediction. The scenario shown here, which is the one I discussed before, is where red and green fold in the same cell at the same time. In this variant, your cell population is a mixture in a fraction of cells. The green come together, but the red are not. And in the other fraction of cells, the red come together, but the green not. And so you can distinguish the two out of polymer physics, because you have different factors in the context. And so when they run the experiments, they allow us to dissect which of those different scenarios are the correct one. And so the conclusion we came to, which gave the cover to Nature Genetics, is that it is homotypic interaction, simultaneous homotypic interaction. That is to say, you see, the pattern found experimentally correspond to that situation, where in the same cell, the red come together and the green come together at the same time. And instead, in this case, CTCF doesn't work. CTCF doesn't explain folding, which doesn't imply that, as I will show you later, that CTCF is not important in folding, in a number of other cases. OK, so it's roughly 10 o'clock. So I would stop for 15 minutes. We can meet again at quarter past and try to wrap up. OK, guys, so shall we start again? OK, so if more or less we are all back, so let's start again. Let's try to wrap up. Come on. OK, in the last few minutes of my lectures, what I want to try to discuss with you is another type of mechanism, which we think could be underlying chromosome folding. I discussed this scenario, which is a classical scenario of biology, where you have molecules, which thermodynamically bridge, cognate binding sites on a chromosome. And there are plenty of examples now about that. However, there is another classical scenario in biology, whereby there is a negative mechanism of loop extrusion. The idea is that you have another type of molecules, which bind to DNA and start extruding a loop. And the candidate for that has been envisioned in the combination of CTCF and cohesin. So this is what I want to discuss now. The idea is the following. I showed you how CTCF is an important molecule involved in chromosome folding. What these people envisage is a scenario, which I would describe as a non-equilibrium scenario, not based on thermodynamics, as the one I discussed before. In this picture, you have that. Let me try to draw a picture. We have DNA. And along with the sequence, the molecules binds, which is an active matter. So the molecules binds and start actively extruding a loop of DNA. So that at a later time becomes that, and at a later time becomes that, and so on. The idea is that you spend energy. So while in the previous scenario, in the thermodynamic scenario, if you think about it, the cell has no need of burning energy, because the energy required is absorbed by the thermal butt. So free, no problem. In this scenario, instead, there is a molecular motor which extrudes DNA actively. And the idea is that that motor stops when it finds CTCF binding sites. Interestingly, oppositely direct CTCF binding site. You still have not developed the ability to loop through this next step in this course. So you saw that. This is a molecule which binds an active matter. The active matter starts extruding a DNA loop, forcing this out. So you spend energy, ATP. And step by step, the loop becomes larger and larger. And in the orders area, the motor stops when it encounters oppositely directed CTCF binding sites. So this is a CTCF binding site, and this is another. When the two are brought in, the molecular machinery, it stops. And so this has been named the loop extrusion model. And the molecular factors involved have been envisaged to be CTCF, as I told you. The two oppositely directed CTCF sites are here shown as the green and the red. And the yellow molecule is the extruder. So the object I mentioned before. And the extruder has been envisaged to be cohesion, for instance. Cohesion is another factor known to be involved in chromium organization. And in this model, in the loop extrusion model, if you add an attraction force between all the other sites in the chain, so my color scheme, the pink, attract each other by some force, whose nature is not clear. But anyway, if you combine those ingredients, you see that you can explain, and comparatively well, folding at low psi, where CTCF is playing a major role. And I show you here one example. This is experimental data. This is an experiment showing where CTCF peaks are. The color scheme is the one I mentioned before. So the two directions of those binding sites are highlighted by the different colors, red and green. And this is the result of that model. You see, this is an alternative. This is a different scenario. It's not thermodynamics. It's off the equilibrium. There is a motor pulling out DNA. It works neatly in this case. And in a sense, it's simple. You only need CTCF and that motor. I've shown you that this cannot work in other cases, as the one I've discussed just before. But at least here it is working well. And the variant of this, though, is called slip link model. It's a variant. The variant was introduced by Davide Marelluzzo and us. In this variant, you have a similar mechanism, but there is no need for active extrusion. So the idea is that your extruding factor binds here, and DNA starts moving randomly back and forth into that molecule. So you don't need to have an active extrusion mechanism. And so you do not burn energy. And the random sliding stops when one of the two CTCF sites enters into the machinery. So this is a variant of the one I showed before, which is based on random diffusion. Technically speaking, though, this is not yet an equilibrium model, because this is a diffusion process with two absorbing boundaries, the CTCF sites. So if you had not the two CTCF sites which stops the process, that would be a thermal process, an equilibrium process. But since you have the two boundaries, this is not equilibrium. In the sense that distribution is not the one you expect out of thermodynamics, because the diffusion stops at the two absorbing boundaries and still CTCF sites. So technically, this is also not equilibrium. Nevertheless, the slip link also works comparatively well. And so what I wanted to do to try to wrap up this is to compare the different cases and give you an overview of what we think in the literature now as the situation. So by the way, what I'm showing you here is how the strings and binders model performs on the same locus. And you see that, again, more or less vapor forked similarly. So the advantage of the strings and binders is that this is a very simple thing, based on well-established concepts of polymer physics, equilibrium, no need of energy. And additionally, you don't make assumptions of what could be the factor. You derive the factors, which could be responsible for folding. But this is not a unique scenario. There are scenarios where you have both equilibrium processes, in particular in the loop extrusion, region loop extrusion. You also have inactive mechanism, and active mechanism is known to occur in the way in which cells work. And to cut short a longer discussion, my personal impression is that we are now starting scratching at the surface of a world of complexity in the sense that it's very likely that real chromosomes are folded by different mechanisms. Some are those I illustrated to you. Thermodynamics, off-equilibrium, active. But maybe there is more to discover about that. I told you that with the strings and binders, where we do not make assumptions on the factors, we derive the factors, we see that, say, 30% of the factors do not match with anything we know. So definitely there is an entire world of mechanisms, which we have yet to discover. Though I hope I conveyed the message that is with comparatively simple ingredients and concepts of theoretical physics, of statistical mechanics, we can start understanding the mechanism of folding and then regulation of our genome. And it is maybe those of you who will come to this field that is left open to delve further into that and to highlight the, I think, broad complexity of mechanism and factors which are involved into that. And the promises, of course, are formidable because we can understand how life works and we can really make huge advancements in medicine. So I want to wrap up and maybe open the discussion if you have any comments or if you have any questions. And what I try to discuss with you is that there are new technologies, as the gamma, which is the one we developed, which are allowing, which are revolutionizing this field because we have qualitative data, as we have in physics. So it's not just an observation. Say, I see the chromosome fold. No, I can tell exactly how they fold experimentally. And this is opening, is paving the way to people like us. And I tried to give you a sense of how chromosome folds started from the data without having yet introduced the models. And the data themselves are showing complexity of factors, which have been heuristically named tards, major tards, and so on. But then, by using physics, we can understand those factors and the origin of those factors. What tards are, why they are so complicated sometimes, not just a neat triangle, and why there are major tards, so high-holder interactions, and what are the factors responsible for that. And with polymer physics, we can explain the data with very high accuracy. And interestingly, we can make predictions. Testable predictions, which are not only exciting from an intellectual point of view, as I said, but have significant medical implications. And right me, in case you are interested in this, we have my groups in Naples and in Berlin. We have openings for postdoc and PhD. We like very good candidates. So if you like this, and you have a good city, right me, and let's see how it goes. So I thank you very much for your attention. And if you have any questions, they're in common, and want to open a discussion, antical issues, whatever. Just go ahead. Thank you.