In this talk, Dr. Tina Subic presents a differentiable loop extrusion model (DLM) that integrates biophysical principles of cohesin-driven loop extrusion with sequence‑dependent parameters to generate and fit Hi‑C contact maps. She begins by outlining the biological background of loop extrusion, cohesin legs, and CTCF control points, and explains why traditional 3D molecular dynamics simulations and black‑box machine‑learning approaches are insufficient for mechanistic insight. The DLM reformulates stochastic loop extrusion as a deterministic 1‑D model with four key sequence‑dependent parameters: right/left leg slowdown (0–1) and decay rates, plus a chromatin interaction “stickiness” parameter S (‑1 to 1) that captures non‑cohesin contacts and repulsion. Toy examples demonstrate how adding control points reproduces TAD‑like features (arms, bright corners) and how the stickiness term further shapes contact patterns. The model is discretized (resolution parameter h) and fitted to experimental Hi‑C data using simple projected gradient descent with an L2 loss, achieving good qualitative agreement. From the fitted parameters, Subic extracts genome‑wide profiles of slowdown, decay, and stickiness, and correlates them with genomic annotations: CTCF and RAD21 align with cohesin‑dependent parameters, while active transcription markers (e.g., RNA‑Pol II, enhancer marks) correlate with both slowdown (indicating transcription‑mediated slowing of cohesin) and stickiness (enhancer‑promoter “sticky” interactions). These findings support a bidirectional relationship where transcription influences 3D organization and vice versa. The talk concludes with future directions—using the learned parameters to predict chromatin organization from sequence, extending the model to capture compartmentalization, inter‑chromosomal interactions, and varying resolutions. The audience asks about inter‑chromosomal modeling, compartmentalization, and resolution handling, to which Subic acknowledges the current model’s limitations and outlines possible extensions.
Questions this video answers
No, the current DLM only models intrachromosomal contacts because cohesin is not known to link different chromosomes. The chromatin interaction term could be expanded, but the model would need substantial extensions.
The stickiness term S is currently a coarse proxy for many mechanisms, including A/B compartments. Future work could decompose S into separate terms to explicitly model compartmental effects.
Yes. The model is discretized with a step size parameter h, allowing analysis at different resolutions. Different h values can be chosen to match the resolution of the Hi‑C data.
Chapters
All right. Hi, everybody. I'm Tina Shubitz. I'm a postdoc in the group of Maria Chikina at the University of Pittsburgh. And I'm really excited to be here today and present you a differentiable loop extrusion model. So gene expression is a very complex process. It involves multiple protein and RNA complexes. And especially in eukaryotes, it also involves enhancers, which are areas of the genome located quite far away from the promoter. And that basically means that in order to regulate the gene expression, they actually have to come in physical proximity of the promoter in order to act on it. So it's then important to understand which parts of the chromatin come in close proximity to understand actually how gene expression is regulated in eukaryotes. So the technological advancements of the past decade allow us now to get genome-wide contact maps where we can inspect how frequently we can find locus I and locus J. And we then get this kind of contact maps where in this particular case, brighter color means that those two areas are found more frequently together. So, but you can see that this is not random. So that means that there's a mechanism bringing the two parts of a chromosome together. And it's hypothesized that this is done by the so-called loop extrusion process. A loop extrusion involves a cohesin, which is actually a dimer of two chromatin cohesin legs, which are protein motors. And what they do is they walk in opposite directions. And when they do that, they actually start extruding a chromatin loop in the middle. But there's also control points located at the chromatin, which are hypothesized to be CTCF sites. And the— what they do is like once cohesin arrives at the control points, the dynamics of the cohesin changes. For example, it can be slowed down, but it's also more likely to detach those control points because of proteins that are located at those areas. So now, and of course, then you can imagine that if the area that is located between two convergent control points is more likely to be found together, and we refer to those areas as topologically associated related domains or TADs. So now let's go back to that contact map that I showed you before. We can see actually those loop conformations on the actual contact map. So in the area close to the diagonal, we observe a really high contact frequency, and that's because this is within one TAD and the cohesin has not yet encountered the control points. Now, if we— now what happens if once the cohesin encounters one of the control points? Well, that area here is going to be then— other areas of the chromatin will be more likely contact with that area because cohesin is staying longer at that point. So we start observing those arms in the area. But the other leg, meanwhile, can keep on extruding the chromatin. So it keeps on traveling until it also encounters a control point. And this you can see on the contact map by this extremely bright point there. But the chromatin can also come in close contact in means other than through cohesin and the loop extrusion. And we can see those areas in such off-diagonal squares. And chromatin mitoses just interact with each other because, for example, it might be in open or closed conformation. So now that we have contact maps, we can acquire them experimentally, and we have a basic hypothesis. How do we actually test additional biological— what other additional biological mechanisms might be affecting these contact maps? And importantly, what is the role of the sequence in those contact maps? Well, we can use computational approaches to study either of those two areas. So the two primary ways we can investigate these contact maps computationally is with 3D molecular dynamics simulations or with machine learning models. 3D molecular dynamics simulations basically explicitly test biological mechanisms such as loop extrusion, or as has been shown recently, also, for example, how loop extrusion interacts with the transcription process. Now, machine learning models, on the other hand, they predict contact maps from the sequence and other context-providing experimental inputs such as ChIP-seq or ATAC-seq. And they do a fantastic job at predicting the contact maps from sequence. And there's been plenty of models, machine learning models designed in the past years. Now, the way they do that, though, is they map one-dimensional sequence into two-dimensional contact map through a much, much higher dimensional structure that is really hard to interpret. So we cannot really map how biological mechanisms might be affecting the contact map formation. So can we— so now if I abstract those approaches, molecular— 3D molecular dynamics simulations stays by law and study biological mechanisms through the simulation in order— and we come to the contact maps, whereas machine learning models take sequence in the context through a black box to arrive at the contact map. Now, Kuni actually merged the three, all the three together, all three together to arrive at the final contact map. And why would you want to do that? Well, we would want to, for example, study how biological mechanism and sequence interact. So what we'd like to propose is to go through a differentiable biophysics-based model so that we can arrive at physical parameters which we can then easily map to a sequence. This, because it's biophysics-based, it would incorporate biological mechanism, but because it's differentiable, it can be easily plugged into optimization and other machine learning approaches. So for the rest of the talk, I will present you a differentiable loop extrusion model, or DLM. So I will first describe how we designed the model, and then we will just look at the toy model where we will map parameters that we design into a contact map. And then after, we will fit our model to the data, and that way we will find, acquire parameters from the contact map or from the experimental data. So let's first design the model. So the way, for example, molecular dynamic simulation treats the loop extrusion, we have a cohesin that is then walking down a 3-dimensional polymer, polymer, but we will want to, and those models are usually stochastic. What we will do is we will convert that into a 1-dimensional model and we'll keep it deterministic. And what we will also do, we will split the right leg and the left leg and assume that if the right leg and the left leg start from the same position, they just belong to the same cohesion. So now let's just focus on one leg. So what are the parameters that we need to have? Well, the cohesin is walking, so it walks with a certain velocity v. And it also can detach from the chromatin, which I'll refer to as the decay rate. But of course, there's also control points at the chromatin. So that basically means that those parameters are sequence dependent. So that means that the— once the cohesin encounters a control point, Of course, the rates of the detachment or the decay rate will change. It will increase. So it's, as I already mentioned, sequence-dependent. And velocity will slow down because it might get blocked at those control points. But rather than modeling the change in velocity, I will actually look at another parameter, which I refer to as a slowdown. And I will be observing how the cohesion is slowing down depending on the sequence. So I will be looking at the parameter that has a value between 0 and 1. Okay, so now we have the sketch of the model, and once we have a sketch, we can, we can design the equations and study how the concentration of the right leg and the concentration of the leg, left leg, change over time and across the space. And both those, both two equations just have two, two terms. One is describing the walking and the other one is describing the decay. Now, if we solve that for the steady state, we arrive to the concentration of the cohesin that is bound to the locus I and the locus J by simply summing the concentration of the right leg and of the leg— left leg. And we can do that because we are assuming that those two legs started at the same position. And now if we do that for all starting positions on the chromatin, we can arrive at the contact maps. Now let's look at that equation here again. We can see that these two equations involve parameters that depend on the sequence, namely the slowdown of the right and the left leg and the decay of the right and the left leg. So these four parameters should be the ones that determine cohesion-dependent features of the contact map. So to convince you, let's look at the toy model. We'll first look at, uh, what happens when there's no control points. Then we will add one control point, and then we will have two converging control points, which should result in a TAD. And then we will also just look at what happens if we add one more control point. Okay, so what we will be looking at is we will look at the parameters, the slowdown, the, and the decay of the left and of the right leg. So if there's no control point, there should be no change in those parameters.. So there should be no slowdown or no change in the decay rate in either of the, in either of the legs. And what we simply see is a, is a decay function. So that checks out. What happens now when we add one control point? So in that case, we will be looking at the bottom plot, uh, at the, uh, which we'll be looking at the control point for the left leg. And in that case, there should, uh, we will implement it as a slowdown. At that point and also as an increase in the decay. And what you can see is we're getting those arms that you have observed in the experimental contact maps. So now if we add one more convergent control point, so the one for the right leg, now you see that we also observe all the features of the tap. We see both of the arms and we also see, hope you can see it on the screen, but there's like a brighter point at the end where both legs of the cohesin are stalled, so the cohesin spends more time at that particular location. So, and of course, we can now study the interplay of control points or CTCF sites, and you can start observing phenomena such as nested TADs. So I hope I convinced you that with this, with this model, we can already recapitulate quite a large number of features of the contact maps. However, Chromatin loci are also— basically, chromatin is a chemically active polymer. So if we have an enhancer and promoter, they might also stick together longer once they come into close proximity. So the contact frequency between i and j of a locus i and j should not depend only on cohesin. Yes, it depends on the cohesin because that's something that brings them together. But then the two contacts might also linger together for example, like if they belong to the same compartment, A versus B, or for example, if there's like proteins holding them together. And we can imagine that cohesin might not be the only way to bring those contacts together because chromatin is also constantly diffusing within the nucleus. So we also can assume that this chromatin interaction term might be independent of the cohesin concentration. And what we haven't mentioned so far is also that we also observe repelling between different parts of the chromatin because You can think of parts of chromatin as like oil and water. They can also repel each other. So we will assign the value of this parameter to be between -1 and 1. So now let's look at how the toy model would look like if we add this chromatin interaction parameter. So we can now look at a bit larger area with several different control points. And then let's say that this parameter is varying across the chromatin region, changing the sign, for example, when it encounters a control point. Now you can see that this toy model is starting to look quite a lot like the contact maps from the experimental data that I've showed you at the beginning. So what would happen if I then fit the model to the data? Can we discover novel control points in the genome that would organize the chromatin that way? So let's try to fit that model. And what I've used is I've used a simple projected in descent and a very simple loss function, just the pixel-wise L2 error. And as you can see, we can actually fit our model quite well to the experimental data. And we are— so we can see qualitatively we are able to recapitulate quite a large area, uh, quite a large area of the contact map. So now that we fitted our model, how do the physical parameters look like? Well, we of course We see that in the areas where there's like more, where there's more TADs or there's more dynamics, the parameters are also like way more varying. And the area within this one larger TAD, of course, there's not much happening here. But this should then depend on the sequence, right? As I already mentioned, the control points are hypothesized to be CTCF sites. So those areas correspond to the CTCF sites. So in the vertical lines, I'm now showing you the CTCF sites that are supposed to be blocking either right or the left leg. And what I want you to appreciate that for the most part, they actually do correlate with the changes in the parameters, but not always. There's like areas where, there's areas where, for example, here, there is a change in the parameters, but there's no CTCF site. So there must be something else also happening in the genome that might be affecting the loop extrusion model. So what we can do now that we have physical parameters from the fitted model, we can do a correlation study. And we have done the correlation study for the whole chromosome 10 for different markers. And so now let's look at some of the markers. First, let's do some reality check. And we see that the parts of the loop extrusion machinery, meaning CTCF and RAD21, they are actually correlating with cohesin dependent parameters, which is the slowdown. So great, sanity check. But there's like air— but this marker here, for example, is strongly correlated with the slowdown, and this marker is a marker for active gene transcription. So basically that tells us that transcription slows down the cohesin. And this is an exciting result because this year a paper came out where they've done the study where they've done 3D molecular dynamics simulations where they've studied how polymerase might be interacting with the, with the cohesin, and I've done experimental knockout of the polymerase, and they came to the same conclusion. So this is exciting. And at the same time, I already mentioned at the beginning, the enhancers might stay at promoter areas for a prolonged manner. So let's also look at the parameter S. I refer to this chromatin interaction as S because I see those areas as sticky. So let's— and you can see actually that markers that that mark enhancers, they also are sticky. So this basically tells us that transcription itself is also involved in the regulation of the spatial organisation of the chromatin, which then again affects the transcription. So it's a backward and forward process, which I find fascinating. And this is where we are at the moment in this project. We're really in the early stage. But so far, I hope that I convinced you that we can actually discover the chromatin organization points, or at least confirm the previously discovered ones, and that we've confirmed that both loop extrusion machinery and transcription affect how the genome is organized in 3D. And we've done that only with 5 vector parameters, which all have physical meaning. So in the future, we hope that actually we would be able to use our model to use sequence and the experimental context to also predict the chromatin organisation points. And with that, I would just like to thank you for your attention. And I'd also like to thank my group, especially Maria, for trusting me with this exciting project. So thank you. Thanks so much. We have time for 2 questions. Probably I'll go by order.
Yeah, thank you. Great talk. Yeah, I wonder whether your modeling has been applied to interchromosomal context rather than just intrachromosomal context. And do you think you will need additional models to model that?
Oh, yeah, absolutely. So right now, this would only include— because obviously cohesion, at least not to my knowledge, cannot really link to. So at least that part. But for example, like the— this chromatin interaction term might be expanded to a larger area. But of course, the model would definitely have to be expanded. Yeah, this is a very simple model.
Very neat talk. My question is closely related to that. Have you thought about incorporating compartmentalization?
Oh, yeah. So that's a phenomenal question. So right now, this chromatin interaction term is kind of thought thought of as, in a way, a projection of all the potential mechanisms that might be acting on the chromatin, meaning like all the, as already tiny little areas such as some cancer promoter areas, but also like the A and B compartments. But of course, it would be extremely exciting to then decompose this chromatin interaction into several terms, and then also study how specific parts or specific areas and fit this extended model to the data and learn a bit more detailed picture of interactions. Yeah, thank you for the question.
Hi, very nice talk. So Hi-C matrices have this concept of resolution, right? And depending on the resolution, you'll see different kinds of structures. So is, are you also adhering to that? And like, is there, are you using resolution in your model?
Yes, I do. I didn't show it. So this model, let me just go— wait, but there you go. So I haven't— oh, actually, let me show you. Let me go to the supplementary slides. There we go. So once you— so in order to arrive at the steady state, I've discretized the model, and in this discretization process, I've actually included this parameter h, which I haven't shown because we were only looking at one resolution. But indeed, you can actually— we can account for different resolutions. In with this particular model. Yeah. Thank you so much. Thank you.
You know where the— which one? This one? Yeah. Is it— do you know if it's okay in Google Slides or do you want to try and get it into PowerPoint? If you haven't checked it in Google Slides, I think it's safer in Google Slides. It was made
