# Entropic Competition between Supercoiled and Torsionally Relaxed Chromatin Fibers Drives Loop Extrusion through Pseudo-Topologically Bound Cohesin

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

_{LJ}= 10 nm containing 400 bp of DNA wrapped around two nucleosomes [27] and ~70 bps of linker DNA. The total size of our beaded chain was 150 beads, which represents a smaller loop of 60 kbp. The beads were bound by a strong harmonic potential, and a fully repulsive interaction potential in order to model excluded volume was installed. Furthermore, we imposed a bending stiffness K

_{b}= 5, giving the fiber persistence length of 50 nm [28]. A classical beaded chain model does not have torsional stiffness. In order to include the torsional stiffness into the model, we included additional virtual beads that did not possess excluded volume interaction and exhibited only hydrodynamic drag γ with surrounding media. These additional beads are attached periaxially with respect to the main axis of the chromatin chain and their set to γ = γ

_{R}= 1. Subsequently, these periaxial beads are interlocked by the torsional potential that creates an energy penalty for torsional deformation. Parameters and detailed procedures for how we build polymer models with torsional stiffness can be found in our earlier work [29,30,31].

_{c}) = γ

_{c}of the virtual periaxial beads and adjacent real bead was increased for a particular bead found in the inner cross-section of the ring, defined as x

_{c}, along the simulated fiber. In the simulations, we investigated the effect of increased drag γ

_{c}on the levels of accumulated supercoiling and speed of loop extrusion, while we used values of γ

_{c}to be between 2 and 200 times larger than the drag of real beads γ

_{R}= 1. In order to load the ring on the fiber, we used a modified approach described by Bonato et al. [33]. The ring was initially positioned around a loading bead (x

_{c}= 0) in a folded handcuff conformation with two rings of 7 beads wrapped around the fiber. In the next step, the bending stiffness of cohesin was reinstalled, which caused the opening of the folded handcuff configuration. In addition to these initial steps, the bonds between beads that created the joint section of the handcuffs configuration were removed. At the same time, excluded volume interaction between cohesin ring and loading bead was increased to 3 σ

_{b}(while maintaining excluded volume between beads of the fiber at 1 σ

_{b}), in order to prevent the pseudo-topologically threaded ring from sliding off from the chain. The increased size of the excluded volume may represent an increased body of RNA polymerase+TOP1 motor, which starts introducing supercoiling behind the ring just after its loading on the fiber. The summary of the bead setup with parameters and model equations of the interactions employed in our coarse-grained model is provided in Table S1.

_{c}= γ

_{R}= 1, i.e., where there is no excessive friction between cohesin and the fiber, and no accumulation of supercoiling was observed. We found such conditions when our motor turned once per 36,000 integrations. Subsequently, we performed very long simulations and no visual writhing or evident pushing of cohesin ring was observed over a very long simulation run. Our integration step was set to ∆τ = 0.0025 time units, and one full rotation took 90 time units. Next, we determined the relation between simulated time units and the physical time by the approach used by Di Stefano et al. such that the time unit corresponds to Stokes’ time τ = 6πησ

^{3}/k

_{B}T = 4.5 µs*η, where η is the viscosity of surrounding media [36]. In order to obtain 10 rotations per second, we considered that the viscosity of surrounding media is 240 times that of pure water. This value is reasonable, as the values of viscosity in the presence of molecular crowders in living cells were reported experimentally to be as high as 220,000 times that of pure water [37]. The longest simulations with γ

_{c}= 2 γ

_{R}took three weeks to simulate, and the shortest time, with runs of γ

_{c}= 200 γ

_{R}, took about two days. Calibration by using increased viscosity helped to save computational time and made performing the simulations of the loop extrusion mediated by friction 240 times faster.

## 3. Results and Discussion

#### 3.1. The Rate of Loop Extrusion Is Controlled by the Friction Imposed by Cohesin

_{c}= 2. In the case of very high friction between cohesin and the fiber, the movement is even more straightforward, with much less fluctuation of the instantaneous values of loop size (Figure 1b). The extrusion also becomes very symmetric. The time of extrusion of the whole loop in the case of the highest γ

_{c}in the simulation was 15 times faster than the lowest γ

_{c}= 2. In the case of employing low gammas, the extrusion becomes more stochastic and asymmetric. The simulations also indicate that the dependence of the loop extrusion rate on the friction saturates when the increase of γ

_{c}from 2 to 20 speeds up the extrusion 4 times, but a further increase of γ

_{c}by a factor of 10 to γ

_{c}= 200 speeds extrusion only 2 times. This indicates that there exists an optimum value of friction mediating the loop extrusion, after which the loop extrusion rate reaches its maximum, and later, at much higher frictions, the extrusion would probably decrease or even stop. In our simulations, however, we do not explore frictions that high as this would require decreasing integration steps and would make the simulations unfeasible in terms of the computational time.

_{c}of the cohesin on the chromatin fiber for each frame of the simulation. The sum of the calculated writhe and twist gives the value of linking number ∆Lk according to Fuller’s theorem [41]. This can be used to calculate the density of supercoiling as σ = (Lk − Lk

_{0}) / Lk

_{0}= ∆Lk/Lk

_{0}, where Lk

_{0}is the linking number of the relaxed state, and here, it will correspond to the number of turns of relaxed DNA Lk

_{0}= ~40 turns per bead [42]. During extrusion of the loop, the supercoiling inside the loop is relaxed by the influx of relaxed portions of the fiber into the loop, but also by escaping of supercoiling through the ring by axial rotation of fibers. During the simulations, from several tens to hundreds of rotations of the motor are performed. This is consistent with the size of the loop, which is 60 kbp; hence, the extrusion of the whole fiber should take time in order of 10s of seconds. The total number of rotations of polymerase introducing supercoiling at a rate of 10 rotations per second should be also in the order of hundreds. The calculated linking numbers indicate that in the case of large gammas, about 87.5% of rotations are relaxed. In the case of simulations with low γ

_{c}s, the loop loses above 99% of rotations both due to axial rotations as well as by dissolving the supercoiling in the un-supercoiled portions of chromatin flowing through the cohesin ring into the loop. The fluctuations of the linking number in terms of its amplitude originate mostly from the fluctuation of the twist. The overall fluctuations are more intense in the case of the systems with lower settings of γ

_{c}. In the case of lower friction, the accumulation of supercoiling has a strong non-equilibrium character. On the other hand, the accumulation of supercoiling in the case of larger friction strongly limits the effusion of supercoiling by axial rotations, while the gradient of supercoiling between the motor and position of cohesin quickly re-establishes, yielding the linear dependence of ∆Lk with time.

#### 3.2. Mathematic Model of Transcriptionally Driven Loop Extrusion

_{R}imposed on the periaxial beads in the simulations along the fiber is the same, and set to γ

_{R}= 1 for all x except the position of cohesin x

_{c}. The value is increased at the position of cohesin x

_{c}, such that D

_{c}= ${D}_{\sigma}\left({x}_{\mathrm{c}}\right)$ = k

_{B}T/γ

_{c}= ${D}_{\sigma}$γ

_{R}/γ

_{c}with ε = k

_{B}T = 1. Initially, at the time τ = 0, the fiber is torsionally relaxed $\sigma \left(x,\tau \right)=0$. For times τ > 0, the supercoiling is continuously introduced at the position of x = 0, which is mathematically treated as a setting of the boundary condition ${\left(\partial \sigma /\partial t\right)}_{x=0}={\dot{\sigma}}_{\mathrm{P}}+{D}_{\sigma}\left({\partial}^{2}\sigma /\partial {x}^{2}\right)$, where the speed of production is given as ${\dot{\sigma}}_{\mathrm{P}}=$10 rotations per second. The equation is solved within the region x ϵ <0, x

_{c}>, which represents the size of the loop. Because the movement of cohesin is one of the most prominent features of our proposed model, this needs to be included in the solution. Hence, we need to describe the movement of cohesin and solve the diffusion of supercoiling as a problem with moving boundary set as ${\left(\partial \sigma /\partial t\right)}_{x={x}_{c}}={D}_{\sigma}\left(\partial \sigma /\partial x\right)-{D}_{\sigma}\left({x}_{c}\right)\left(\partial \sigma /\partial x\right)$ [45]. The movement of cohesin can be described by the equation for displacement from the overdamped Langevin equation, where we neglect the noisy part of the equation.

_{c}reflects the dissipation of the energy during the movement. It should be noted that the value of friction between the fiber and the rings is not the result of simulations but is introduced as a parameter setting into the model. Simulating the friction starting from first principles would require the employment of a fully atomistic model, and it would encounter several issues starting from the computational complexity of the problem [46]. The atomistic computer simulations of chromatin fiber with a comparable size to ours were undertaken recently, while the system of 83 kbp consisted of one billion atoms [47]. However, such simulations were only possible thanks to the employment of 100 thousand CPUs, while the time scales of seconds or minutes of real time were far beyond the reach of the atomistic simulations, as we could not take advantage of coarse-grained time units. If one was able to undertake the simulations of this kind of magnitude though, the friction coefficient could be explored as a measured property dependent also on cohesin ring mass, its hydrodynamic drag, etc., which would be then a part of Equation (2). Instead, in our simulations, we explore the behavior of supercoiling and loop extrusion rate for a wide range of frictions, which is a parameter setting. Hence, the rings themselves could be even fixed, and we would still obtain the same simulation for a given setting of friction. The wide range of frictions employed is plausible based on the experimental work of Stigler et al. [40], which suggested that friction can be very high. Physically, the friction arises from a combination of inter-surface adhesion, surface roughness and deformation. Based on the results reported by Stigler et al. [40], the molecular surface of chromatin appears to be very rugged for a diffusing cohesin due to the presence of individual nucleosomes, nucleosome arrays and other protein obstacles and machinery. In our model, we suppose a uniform distribution of the obstacles that are represented by a given averaged setting of the friction parameter γ

_{c}. In principle, however, the beaded model also provides the possibility for implementing a site-specific friction within the resolution given by the particular level of coarse-graining employed in the beaded model. This would be interesting, e.g., to investigate the effects of local modulation of friction due to chromatin folding and the presence of protein/DNA regulatory hubs such as those identified at specific sites on micro-C contact maps [48]. At the same time, we consider the right term of Equation (2) to be the change of internal energy associated with the drop in energy of supercoiling as new relaxed portions of fiber infuse into the loop once the cohesin advances in movement. In our mathematical model, we assume that the temporary drop of energy of supercoiling induced by cohesin movement propels the cohesin movement and loop extrusion. The energy of supercoiling can be expressed as u = K.∆Lk

^{2}, where the constant K represents energy of fiber’s force interactions such as bond stretching, angle bending and torsional flexibility [42].

_{c}. The fitted dependencies are shown in Figure 1 and Figure 2, together with the values of loop size and linking number obtained by coarse-grained simulations. The values of model parameters are obtained per two fibers going through one ring. As our set of equations treats the problem in 1 dimension with one motor on one side and cohesin boundary on the other side, the coefficients obtained should be therefore halved in order to properly grasp the fact that effusion occurs through two fibers embraced by a single ring. As underlined in the paper by Bonato et al., using a similar approach to mathematical modeling in order to describe the movement of cohesin ring along the fiber, this aspect of the mathematical treatment does not qualitatively affect the results [33].

_{Tw}= 2.0 and D

_{Wr}= 0.1, giving the value of diffusivity of supercoiling as a weighted average with contributions of each based on whether the system accommodates a stable plectonemic or straight conformation. In our simulations, we also see the fluctuations of ∆Lk affected in greater part by the fluctuations of twist, implying that the dynamics of twist is much faster. Additionally, we note that the diffusivity of supercoiling in real units obtained from transforming σ

^{2}/τ will be much larger than the diffusivity of whole plectonemes observed by Dekker et al. to be D = 0.1 kbp2/s [50]. However, the diffusivity of supercoiling along the fiber agrees well with expected rotational mobility of DNA that was experimentally observed to be in order of thousands of rotations per second [51]. As observed from the simulations and supported also by mathematical modeling, in the case of large gammas, the equilibrium state of the gradient of supercoiling along the fiber is relatively quickly re-established after each movement of cohesin, as the high friction with cohesin limits losses of supercoiling (Video S2). As result, the value of the average supercoiling of the loop increases rather linearly over time. On the other hand, in the case of low gammas, the gradient of supercoiling along the fiber is more time-dependent as the semipermeable boundary formed by cohesin allows escaping of supercoiling by axial rotations to a much greater extent (Video S3). As result, the average value of ∆Lk starts to increase more intensely in later stages of the loop extrusion. This is consistent with the expectation that in an infinitely long loop, the average density of supercoiling would approach a ratio of turns introduced by polymerase versus the average loop extrusion rate. Additionally, we would like to stress, the purpose of the mathematical model presented in this section is not to provide a full quantitative treatment for describing precisely the loop extrusion in vivo by two parameters; rather, we aimed to provide an additional support in terms of qualitative agreement between the simulated data and fits from the mathematical model that would support the proposed mechanism of the transcriptionally driven loop extrusion mediated by friction between cohesin rings and chromatin fibers.

#### 3.3. Towards Entropically Driven Loop Extrusion, Osmotic Pressure and Other Models

_{sim}= 3.4 × 10

^{3}. The halved value per arm is 1.7 × 10

^{3}, which is similar to the value of the energy constant given by Vologodskii as K = 1100, and it is referred to as a dimensionless constant [52]. On the other hand, it was pointed out that the dimension of the constant is probably per bp [42]. In our case, we use arbitrary units in terms of unitless beads on both sides of the equation; hence, a change of the units would not change the constant obtained from the fit. The difference of the fitted value of the energy constant is most likely given due to the internal settings of the force field in our model and can be improved in future works. In order to decouple contributions of other forces acting on the extrusion, one would need to go into a more detailed description of the energy term. For example, Bonato et al. have shown that chromatin stiffness and compaction play a crucial role in enhancing diffusive loop extrusion [33]. In their model and subsequent mathematical description, they proposed that the change of the loop’s internal energy follows the equation $u\left({x}_{\mathrm{c}}\right)/{k}_{\mathrm{B}}T=8{l}_{p}/{x}_{\mathrm{c}}^{2}+c\mathrm{log}\left({x}_{\mathrm{c}}\right)$, where the first term on the right side of the equation represents energy penalty due to the chromatin stiffness that would drop as the loop size grows, AND the second term represents the entropic cost of looping. When these terms are added to our energy term given by the energy of supercoiling u = K.∆Lk

^{2}in (Equation (2)), the fitted value of K drops to 1550 (c = 0.03); however, the quality of the fit remains the same (Figure S1). In general, adding more terms into the functional decreases degrees of freedom and keeps the quality of fits the same if not improved. Additionally, one may think of other energy contributions acting in favor of the loop extrusion, such as opening the folded conformation of the cohesin rings in the early stages of the simulations. On one hand, incorporating the energy functional used by the diffusive model into our model provides a perspective for merging both models that both exhibit enhancement of the loop extrusion by entropic mechanism. On the other hand, providing a rigorous description of the energy term and decoupling energy contributions arising from bending, stretching, unfolding the cohesin, motor activity that acts as a constant speed, i.e., infinite energy motor, etc. is beyond the scope of the present paper. Moreover, the levels of coarse-graining in our work and in Bonato et al.’s work are currently different, which makes a direct comparison of the results complicated. Thus, we instead intend to show that the proposed mechanism is still able to drive loop extrusion at varying levels of friction, accumulated supercoiling and non-topologically bound cohesins, with proof-of-concept simulations supported by a mathematical description. As the energy term ∂u/∂x is a definition of chemical potential μ, the loop extrusion driven by the change of the chemical potential can be considered as an entropic process. The difference of the chemical potentials on the interface determined by the moving cohesin, μ(σ, p + Π) on supercoiled side and μ(σ = 0) on relaxed side, corresponds to the definition of osmotic pressure. Hence, one may think of an analogy of the osmotic process when the difference of concentrations on the interface drives the solvent into the concentrated phase. In our case, the solvent would be represented by relaxed portions of the fiber that “dissolve” the accumulated supercoiling within the extruded loop. The similar influx of chromatin fiber works in the mechanism of loop extrusion by an osmotic ratchet [53], where the fiber “dissolves” the concentration of cohesin rings loaded on the fiber.

#### 3.4. Biological Contexts and Implications

_{c}= 2, 5, 20 and 200 that were the main scenarios discussed in this paper. The loop extrusion rates are shown as a function of a product of the imposed friction. As the figure shows, the loop extrusion rate still increases with increasing friction as the energy of accumulated supercoiling increases, but it follows a logarithmic relation, showing the saturating effect on the loop extrusion rate. Biologically, the speeds should be such that the entire human genome could be extruded within minutes [14]. We may see that biologically relevant rates consistent with experimental measurements between 0.5–2 kbps [14] are achieved mainly for lower settings of γ

_{c}≤ 20.

_{c}= 2 to −10 for the largest friction γ

_{c}= 200 employed. These values, however, represent the total value along the chain, while in reality, the supercoiling creates a gradient from the position from transcription site (TSS) x = 0 towards the site of release of supercoiling at the position of cohesin, x = x

_{c}. The site-specific linking number ∆Lk calculated at a given bead can be considered as the bead’s local density of the supercoiling σ. In this way, for γ

_{c}= 200, the values measured and modeled at the position of the bead representing transcription site in our simulations reach σ(x = 0) = −12%. At the same time, the supercoiling drops towards the position of cohesin, where the value is maintained by the semipermeable boundary at σ(x

_{c}) = −5%. The evolution of ∆Lk along the chain and simulation time obtained from the mathematical model is given in Videos S2 and S3. The profile of ∆Lk obtained as average over the simulation run is given in Figure S2. The total value of Lk is the integral under the curve. In the case of low friction, the values range from σ = −7% at the TSS to σ(x

_{c}) = −0.2%, where the supercoiling drops more dramatically along the fiber, at the position of cohesin. The values of local supercoiling at the transcription site are a bit lower than predicted by the stochastic model of supercoiling-dependent transcription by Brackley et al. [44]. However, the values are in good agreement with the local DNA supercoiling determined for genes with low, medium and high expression around the position of TSS in terms of crosslinks (CLs) by psoralen photobinding in vivo by Kouzine et al. [55]. The faster decay of supercoiling in the simulations with lower γ

_{c}s is consistent with the profiles measured by Kouzine et al.

**Figure 3.**The higher friction between cohesin ring and fiber induces higher levels of supercoiling and extrudes loops at higher rates. (

**a**) Loop extrusion rate for different settings of γ

_{c}imposing friction between cohesin ring and chromatin fiber, resulting in different levels of accumulated supercoiling. The loop extrusion is shown as a function of imposed friction between cohesin and fibers. The insets show snapshots from the end of the simulations at the lowest and highest settings of the friction. (

**b**) The contact maps are calculated during the loop extrusion. The contact maps were calculated along the trajectories as the average over 10 simulation runs. The maps show the loss of ant-diagonal when the loop extrusion is more stochastic, i.e., random, when lower settings of friction between cohesin ring and chromatin fiber allow the fiber to sample larger conformational space. The data acquisition for the contact maps calculation stopped once the ring slipped away from the chromatin fiber. Biologically, the rings should stay stuck to CTCF’s proteins at borders of the domain [56], which would emphasize the loss of the anti-diagonal in simulations with lower friction.

## 4. Conclusions

## Supplementary Materials

_{c}= 200; Video S2: Mathematical modeling of loop extrusion at high levels of supercoiling accumulated at high friction, γ

_{c}= 200; Video S3: Mathematical modeling of loop extrusion at low levels of supercoiling accumulated at low friction γ

_{c}= 2; Table S1: Model setup of bead interactions for chromatin fiber and cohesin; Figure S1: Fits of the system of mathematical Equations (1) and (2) over the simulated linking number and loop sizes when incorporating the energy terms for stiffness; Figure S2: Profile of supercoiling along the fiber obtained as average over the whole simulation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The simulations of loop extrusion. (

**a**) The loop sizes are shown as a dependence of time for four settings of friction between cohesin ring and chromatin fiber γ

_{c}= 2, 5, 20 and 200. Graphs indicate that the rate of loop extrusion increases, with larger friction induced at the position of the ring. The red lines show the fits obtained by numerical solving of the system of differential equations proposed to describe the process of entropically driven extrusion (Equations (1) and (2)). (

**b**) The progress of loop extrusion for individual arms of the loop in terms of their sizes is calculated as a difference of the position of the ring on the arm and the position of the motor. The graph shows that the extrusion becomes more stochastic for lower settings of γ

_{c}s, making the extrusion also more asymmetric.

**Figure 2.**The simulations of supercoiling. (

**a**) The graphs show accumulations of linking numbers in terms of White’s formula ∆Lk = ∆Tw + ∆Wr. The accumulation of the linking number is shown as dependence of time for four settings of friction between cohesin ring and chromatin fiber γ

_{c}= 2, 5, 20 and 200. (

**b**) Density of supercoiling obtained as σ = ∆Lk/ℓ. The red lines are the fits obtained from solving the differential system of equations describing the accumulation of supercoiling with a moving permeable boundary defined by the position of cohesin.

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**MDPI and ACS Style**

Rusková, R.; Račko, D.
Entropic Competition between Supercoiled and Torsionally Relaxed Chromatin Fibers Drives Loop Extrusion through Pseudo-Topologically Bound Cohesin. *Biology* **2021**, *10*, 130.
https://doi.org/10.3390/biology10020130

**AMA Style**

Rusková R, Račko D.
Entropic Competition between Supercoiled and Torsionally Relaxed Chromatin Fibers Drives Loop Extrusion through Pseudo-Topologically Bound Cohesin. *Biology*. 2021; 10(2):130.
https://doi.org/10.3390/biology10020130

**Chicago/Turabian Style**

Rusková, Renáta, and Dušan Račko.
2021. "Entropic Competition between Supercoiled and Torsionally Relaxed Chromatin Fibers Drives Loop Extrusion through Pseudo-Topologically Bound Cohesin" *Biology* 10, no. 2: 130.
https://doi.org/10.3390/biology10020130