Chromatin Loops as Allosteric Modulators of Enhancer-Promoter Interactions

The classic model of eukaryotic gene expression requires direct spatial contact between a distal enhancer and a proximal promoter. Recent Chromosome Conformation Capture (3C) studies show that enhancers and promoters are embedded in a complex network of looping interactions. Here we use a polymer model of chromatin fiber to investigate whether, and to what extent, looping interactions between elements in the vicinity of an enhancer-promoter pair can influence their contact frequency. Our equilibrium polymer simulations show that a chromatin loop, formed by elements flanking either an enhancer or a promoter, suppresses enhancer-promoter interactions, working as an insulator. A loop formed by elements located in the region between an enhancer and a promoter, on the contrary, facilitates their interactions. We find that different mechanisms underlie insulation and facilitation; insulation occurs due to steric exclusion by the loop, and is a global effect, while facilitation occurs due to an effective shortening of the enhancer-promoter genomic distance, and is a local effect. Consistently, we find that these effects manifest quite differently for in silico 3C and microscopy. Our results show that looping interactions that do not directly involve an enhancer-promoter pair can nevertheless significantly modulate their interactions. This phenomenon is analogous to allosteric regulation in proteins, where a conformational change triggered by binding of a regulatory molecule to one site affects the state of another site.


Introduction
Distal enhancer elements in higher eukaryotes are essential for regulating gene expression [1][2][3][4]. In conjunction with transcription factor binding and nucleosome modifications, the classic model of enhancer function requires the direct spatial contact between enhancers and their target promoters ( Figure 1A) [1][2][3][4]. Recent studies have started to reveal the complexity of the enhancer-promoter (E-P) interaction network, where each enhancer can influence multiple promoters, and each promoter may be influenced by multiple enhancers [5][6][7][8]. In addition, gene expression and E-P interactions occur within higher-order three-dimensional chromatin organization, which is characterized by an intricate network of interactions at multiple scales. For example, below 1Mb, chromatin is organized into continuous 500-900kb regions of enriched contact frequency called topologically associated domains (TADs) [9,10]. TADs were found to be about 90% cell-type independent (2763/3000 conserved boundaries between two assayed cell types [9]). Within TADs, additional cell-type specific looping interactions are formed [6,11,12]. These observations raise an important question; namely, how can E-P contacts be affected by looping interactions between other regulatory elements in their genomic neighborhood?
Two models for how proximal looping interactions can modulate E-P contacts have been proposed: the decoy model and the topological model (experiments [13][14][15], reviewed in [16][17][18][19]). The decoy model suggests that insulating elements directly interact with the enhancer, sequestering it from the promoter, and thereby directly hinder E-P interactions. The topological model proposes that two regulatory elements in the vicinity of the enhancer and promoter can interact with each other to form a chromatin loop; this, in turn, affects E-P contacts.
Evidence supporting the topological model includes experiments in multiple organisms ( Figure  1B-D) [20][21][22][23]. For example, Kyrchanova et al. [21] recently observed that a single Drosophila gypsy element placed between an enhancer and a promoter did not change their interactions; however, introducing two gypsy elements changed E-P interactions depending on gypsy position and orientation. The authors explain these observations by gypsy-gypsy looping interactions. We note that while this gypsy element consisted of twelve repeated copies of the Su(Hw) binding site, elements with fewer sites are sufficient for insulation [24,25]. The regulatory effects of loops may also be relevant at larger genomic distances; in mice, a regulatory element with multiple larger (25kb and 55kb) loops was suggested to control multiple E-P contacts at the H19 locus [22]. Analogously, loops between insulating elements were suggested to modulate the activity of silencing elements [23].
It remains unclear whether, and to what extent, the looping interactions between other regulatory elements can mediate E-P contacts. When these looping interactions do not directly involve the E-P pair, their effect is reminiscent of allosteric regulation in proteins [26,27], where binding of a regulator molecule to one site changes the state of another site or the whole protein. A classic example is the binding of allolactose to the lac repressor at the regulatory domain. While allosteric interactions in proteins are mediated by the protein structure, we propose that interactions between genomic sites could be mediated by local changes in the conformational ensemble of the chromatin fiber.
Polymer simulations provide an ideal testing ground to investigate the allosteric effects of a loop on E-P contacts; many loci can be probed simultaneously at high resolution, and more complicated looping arrangements can be systematically characterized. Previously, Mukhopadhyay et al. [28] used polymer simulations to demonstrate that the topological model of insulation applies to an unconfined system of two fused chromatin rings; namely, two loci within the same ring interact more frequently than loci in different rings. We extend this line of inquiry by asking whether forming loops may affect interactions at scales exceeding the loop size, e.g. interactions of a loop with the rest of the chromosome or between loci in the vicinity of the loop.
Here we use polymer models to study how 15-60kb chromatin loops can influence E-P contacts. We note that 3C-based studies have only begun to provide unbiased data at sufficient resolution to build polymer models of a particular locus [29], and the fine structure of the chromatin fiber in vivo remains largely unknown [30,31]. Thus, for generality, we model chromatin as a long homogeneous flexible fiber with only a few additional looping interactions between specific elements, as described below. Synthesizing results from the literature, we primarily focused our simulation analysis on two important arrangements of the loop-forming elements relative to an E-P pair: (1) an enhancer is flanked by loop-forming elements, while a promoter is beyond the loop ( Figure 1E); and (2) both loop-forming elements are located in the genomic region between an enhancer and a promoter ( Figure 1F).
We find that loops can significantly insulate or facilitate the frequency of E-P interactions, depending on the loop location relative to the E-P pair. We consider a variety of situations and parameters, including: E-P genomic distance, stiffness of the chromatin fiber, size of the loop, topological constraints on the chromatin fiber (i.e. topoisomerase II activity), chromatin density, the number of looping elements, and excluded volume interactions. We find that different mechanisms underlie insulation and facilitation; insulation occurs due to steric exclusion by the loop, while facilitation occurs due to an effective shortening of the E-P genomic distance. We additionally consider how insulation and facilitation would be observed in microscopy studies and find substantial differences from how they would manifest in 3C-based studies. Taken together, our results suggest that due to its polymer nature, chromatin allows for interactions to be mediated in an allosteric manner, i.e. formation of a contact between two sites can insulate or facilitate interactions between other loci in the vicinity.

Model and analysis of simulations
Using equilibrium simulations of a confined polymer chain, we study how chromatin loops affect E-P contact frequency in their vicinity. We model chromatin as a semi-flexible polymer fiber with excluded volume; the fiber consists of 15nm diameter monomers, each representing three nucleosomes or 500bp, with a persistence length of 3 monomers (Figure 1G, Methods) [32]. Unless otherwise noted, we allow occasional chromatin fiber crossing by setting a finite energy cost (using a truncated repulsive potential) for two monomers to occupy the same volume, which accounts for topoisomerase II (topo-II) activity (see Methods). Thus, two regions of the chain can spontaneously cross through each other with a probability controlled by the energy penalty of co-occupancy. To account for the dense arrangement of chromatin within the nucleus, we confine the chromatin fiber to impose a 2% volume density. We later vary volume density from 1% to 20% (see below), which is consistent with current estimates of chromatin volume density in the interphase nuclei of higher eukaryotes [33]. Since the flexibility of the chromatin fiber in vivo is incompletely characterized, we varied flexibility in our simulations and found quantitatively similar results (see below).
For each set of conditions and looping interactions, we performed Langevin dynamics simulations using OpenMM [34] (see Methods and Video S1) and sampled conformations from the resulting equilibrium ensemble; these conformations were subsequently analyzed to compute contact frequencies (see below, and Methods). To investigate the effects of a chromatin loop on a larger region of chromatin, we model a loop by forming an irreversible bond between a pair of monomers and allowing the whole polymer to equilibrate (Figure 1E, 1F, see Methods). We considered loops of sizes L=15kb, 30kb, and 60kb, and a 2.5kb loop with a more flexible chromatin fiber (see Discussion), in a proportionally sized genomic region of length 33*L, i.e. 1Mb for a 30kb loop ( Figure S1). Our polymer model contains no additional sequence-specific details, and thus generally addresses how E-P interactions are altered in the vicinity of a loop. The model remains agnostic to the chromatin organization at larger genomic scales, assuming that the simulated region is contained within a single TAD [35].
For each set of parameter values and loops, we generate an equilibrium ensemble of conformations and compute the contact frequency between loci (monomers) in this ensemble (Figure 2A, Table S1 for parameter values). We display pairwise contact frequencies using heatmaps (Figure 2B), as typical for Hi-C and 5C experiments. Our simulated heatmaps are characterized by two features: (i) a decay of contact frequency as a function of increasing genomic distance, and (ii) an off-diagonal interaction between the loop bases. The first feature follows from the polymer connectivity of the simulated chromatin fiber. The second feature alters the typical decline in the contact frequency and is of primary interest in this study.
For a given position of the enhancer and the promoter, we can compute the contact frequency ratio as the contact frequency in the model with a loop, divided by the contact frequency for an otherwise equivalent model without a loop. Contact frequency ratios below 1 indicate insulation, whereas ratios above 1 correspond to facilitation. Unless noted otherwise, we report contact frequency ratios for a 30kb loop and a 50kb E-P genomic distance. We note that each simulation contains information regarding every possible position of the enhancer and the promoter. From this, we can compute contact frequency ratios as a function of E-P distance and location. Below we examine how the loop length and the E-P spacing affect observed phenomena.

Chromatin loops can insulate or facilitate enhancer-promoter interactions
We used the simulated heatmaps of pairwise contact frequency to investigate the two important arrangements of the E-P pair and the loop from the literature (Figure 1).
The first arrangement involves a chromatin loop formed by elements flanking an enhancer, such that the enhancer is located within the chromatin loop and the promoter is located outside of the loop ( Figure 2C). Since the enhancer and promoter are equivalent in our polymer model, this scenario also describes a promoter flanked by a pair of loop-forming elements and an enhancer located outside of the loop. Simulations show that for 50kb E-P spacing, formation of such a 30kb loop leads to a ~35% reduction in E-P contacts, serving as an insulator (contact frequency ratio of 0.64, Figure 2D). Below we refer to this arrangement as insulation.
The second arrangement constitutes a chromatin loop located in the genomic region between the enhancer and promoter, i.e. both loop-forming elements are located between the enhancer and promoter ( Figure 2C). Formation of such a loop facilitates E-P interactions by increasing their contact frequency by more than 4-fold (contact frequency ratio of 4.15, Figure 2D).
Next we examined how E-P spacing affects the magnitude of loop-induced insulation or facilitation. Interestingly, the two effects behave differently; while facilitation diminishes with E-P genomic distance, insulation appears to be independent of distance ( Figure 3A). These results reveal an important difference between loop-induced facilitation and insulation: facilitation is a local phenomenon, and insulation is a global effect.
To better understand insulation, we varied the position of the enhancer within the loop. We found that insulation is weaker when the enhancer is placed in the middle of the loop (0.75 contact frequency ratio), and strengthens as the enhancer approaches the base of the loop (0.49 contact frequency ratio, Figure 3B). We note that an extreme case of topological-model insulation is in fact similar to decoy-model insulation, which occurs when the enhancer is placed at the base of the loop. In this scenario, we observe stronger insulation because the enhancer is permanently interacting with the other loop base, sterically hindering interactions between the enhancer and all other loci. This can be seen as dark stripes at the positions of the loop base monomers on the heatmap; the profile of interactions of the loop base with the rest of the fiber is detailed in Figure S2. Below we identify and discuss mechanisms underlying insulation and facilitation.

Chromatin fiber flexibility, topological constraints, and overall density do not underlie insulation or facilitation
To test the generality of insulation and facilitation, we varied several biologically relevant and physical characteristics of our model, many of which have not been fully characterized in vivo.
First, we investigated the importance of chromatin fiber flexibility by simulating chromatin fibers with different persistence lengths. We found that fiber flexibility does not significantly affect insulation or facilitation ( Figure S3). This is consistent with the fact that both phenomena are observed at distances much larger than the persistence length, and thus in our simulations do not emerge solely due to fiber stiffness. As such, cartoons with rigid, stiff loops should in many cases be understood as schematics [36]; renderings of three-dimensional chromatin loops from our models are shown in Figure 1E, 1F, and 2A. We also note that for simulations with larger and smaller loop sizes, the main qualitative features of the heatmap remain the same ( Figure S1).
Next, we studied the effect of topological constraints, as they have been suggested to play an important role in chromosome organization [33,37,38]. To investigate this, we performed simulations both with and without allowing two regions of the chromatin fiber to cross, which may respectively correspond to cells with active and inactive topo-II. We found that insulation and facilitation are observed irrespective of the topological constraints ( Figure S4A). We note that the terms topological model, topologically-associated domains (TADs), and topological constraints all refer to distinct, and likely unrelated, concepts. In particular, our results demonstrate that the topological model of insulation is independent from topological constraints on the chromatin fiber. Additionally, topological constraints are distinct from other topological effects such as supercoiling of the chromatin fiber [39]; supercoiling can lead to significant insulation and facilitation [39], but may be more relevant for bacterial chromosome organization [40].
Third, we assessed the influence of chromatin density on insulation and facilitation. In particular, active and inactive chromatin environments are known to have respectively lower and higher densities [33,41]. We performed simulations at densities ranging from low (1%) to high (20%) volume density ( Figure S4B). We found that while both insulation and facilitation remain qualitatively present at all densities, they are roughly twice as strong at 1% vs. 20% density. This finding indicates that low density, as found in active chromatin, is important for the magnitude of both possible regulatory effects.
Together, these variations in our model suggest that insulation and facilitation exist across a range of biologically relevant parameter values. However, they indicate that insulation and facilitation do not mechanistically follow from the fiber stiffness, topological constraints, or overall density.

Fundamental properties of polymers underlie insulation and facilitation
To understand the mechanisms of insulation and facilitation, we performed simulations of a phantom polymer chain, which lacks excluded volume interactions ( Figure S5). Remarkably, elimination of excluded volume interactions completely abolishes the insulation effect. In contrast, the degree of facilitation remains largely unaffected by the elimination of excluded volume (reduced from 4.15 to 3.20). We note that phantom chain simulations do not adequately describe chromosomes, but nevertheless can provide useful insights into polymer behavior; here, they demonstrate how steric exclusion by a loop can give rise to insulation.
The loss of insulation in simulations without excluded volume interactions led us to investigate the spatial relationship between the loop and the rest of the polymer fiber. We found that the spatial density of monomers from other regions of the fiber is depleted near the loop, i.e. the loop sterically excludes interactions with the rest of the polymer (Figure 4A). Interestingly, regions immediately outside the loop are also sterically excluded by the loop; we find 20-50% insulation for regions up to 6kb away from the loop (Figure S6A).
Facilitation does not depend on excluded volume interactions, but depends on E-P distance. Therefore we considered how facilitation might arise from an effectively shortened E-P distance imposed by the intervening loop; in particular, we compared contact frequency ratios for the facilitation arrangement and for a simulation without a loop but at a 30kb smaller genomic distance (i.e. shortened by the loop size). Indeed, we see that these are in almost complete agreement, demonstrating that facilitation results from the effectively shortened genomic distance ( Figure 4B).
To get further insight into the mechanisms of insulation and facilitation, we performed in silico fluorescence in situ hybridization (FISH) by calculating the distribution of E-P spatial distances across many conformations (Figure 4C). To consistently compare insulation and facilitation, we considered them at an E-P distance of 90kb, where both effects have approximately the same fold change (contact frequency ratios of 0.75 and 1.3). For insulation, we observe only a small shift in the overall distribution of E-P spatial distances (mean increased by 3%). This confirms that insulation occurs not because the E-P pair is much further away on average, but due to steric exclusion of the promoter by the loop engulfing the enhancer. For facilitation, however, the distribution of E-P spatial distances shifted more strongly (mean decreased by 9%). These results highlight that differences in contact frequency are not always proportionally reflected in differences in mean spatial distances. Moreover, our results show that both effects could be hard to detect by microscopy, but facilitation would be more evident than insulation.
Together, these results provide evidence for the mechanisms underlying insulation and facilitation. For insulation, regions within the loop are sterically excluded from making contacts with the rest of the polymer fiber. For facilitation, the E-P pair has an effectively shorter genomic distance.

Intra-loop interactions and two-loop models
The analyses above focused on understanding how a single loop affects E-P contact frequency at genomic distances exceeding the loop size. For E-P genomic distances less than the loop size, both elements can be positioned within the loop. In this case, we found that interactions are facilitated, consistent with previous results [28]. However, the degree of facilitation depends on the relative position of the elements in the loop (Figure S6B). Placing the enhancer at one loop base and the promoter at the other can greatly facilitate their interaction frequency. On the contrary, with the enhancer at the loop base and the promoter in the enhancer-proximal portion of the loop, the facilitation effect may disappear, likely due to the superposition of intra-loop facilitation with the insulating properties of the loop bases ( Figure S2B).
Many enhancers, promoters, and loop-forming elements can be present in a given genomic region, opening the possibility for more complicated scenarios of insulation and facilitation. Towards this end, we performed simulations where two consecutive loops were formed. We observed qualitatively similar insulation and facilitation in the two-loop case, for the two arrangements similar to those we initially focused on ( Figure S7). Within this two-loop element, the average contact frequency between loci within one loop is higher than the average contact frequency between loci from different loops. In this sense, the two loops are insulated from each other as well as from the rest of the fiber. This is consistent with simulations of an isolated system of two fused rings [28]. Moreover, these results show that the concept of steric exclusion, which underlies insulation for a single loop, applies to the two-loop case as well. In particular, each loop in the two-loop model sterically excludes the other, as well as the rest of the chromatin fiber.

Discussion
Using a polymer model of chromatin, we found that a single loop in the vicinity of an E-P pair can either insulate or facilitate their interactions. These effects have a considerable magnitude, with about 2-fold insulation and 3-5 fold facilitation of E-P contact frequency, which is comparable to generally observed changes in gene expression [42].
Collectively, experiments have observed that different local arrangements of regulatory elements can lead to complex patterns of gene expression. For example, one insulating element between an E-P pair can decrease gene expression, yet two elements between the same E-P pair do not [15,16,18,19]; it was hypothesized that the two elements cancel each other out by forming a loop. Our model shows that this loop would in fact facilitate E-P interactions. Additionally, our model predicts that if the second element were placed outside the E-P pair, the resulting loop would indeed insulate E-P interactions. Note that since the exact quantitative relationship between E-P contact frequency and gene expression or phenotype remains largely unknown, we focus on qualitative comparisons between our model and these experimental studies.
The indirect modulation of E-P contacts by chromatin loops is often referred to as the topological model [16,17,19], a term used rather vaguely. Studies that consider the topological model often assume a particular mechanism whereby the loop alters E-P interactions. Specific mechanisms include: sliding along DNA [43], lamina attachments [19,44], and inter-nucleosome interactions [17]. Our simulations show that the formation of the loop itself can insulate or facilitate E-P interactions, due to the polymer nature of chromatin, independent of specific molecular mechanisms.
These effects can be best understood in terms of allosteric regulation. In particular, interactions that are responsible for the formation of the loop do not necessarily directly prevent or form E-P contacts. Instead, they steer the conformational ensemble of the chromatin fiber toward or away from conformations where an enhancer and a promoter are in contact. This mechanism of action is analogous to classical allosteric regulation in proteins [26], and particularly to disordered proteins, where binding of an allosteric substrate changes the protein conformation, which in turn alters the structure of a distant active site [45]. We note that the concept of allostery has also been useful for understanding other systems, including nucleosome-mediated transcription factor binding cooperativity [46,47].
The polymer mechanisms underlying insulation and facilitation arise due to two different effects. Facilitation results from the effectively shortened E-P genomic distance due to the loop and is likely robust to the molecular details of the chromatin fiber. Insulation arises due to excluded volume interactions and steric exclusion by the loop, and thus depends on the chromatinchromatin affinity. In polymer physics terms, insulation would require good solvent conditions, which are likely satisfied in active decondensed chromatin. We further note that altering chromatin-chromatin affinity may allow further modulation of insulation strength, for example through chromatin modifications that change the net charge of nucleosome tails.
In specific biological systems, the detailed structure and flexibility of the chromatin fiber may become relevant. When the chromatin loop size approaches several persistence lengths, the loop could become very rigid. Consequentially, its effects may depend on the molecular details of the loop-forming elements, including their orientation as observed in a recent study [21]. However, the insulation and facilitation we observe may still manifest with similar strength at smaller genomic distances for more flexible or loosely packed chromatin fiber; changing these parameters causes small loops to behave similarly to larger loops ( Figure S1D). We note that many processes may locally increase chromatin flexibility, either uniformly or through the formation of kinks [38], including the loss/unwrapping of nucleosomes [48]. Finally, the fine details of loop formation may be very important when the E-P pair is within the loop or near the loop bases, as we observed large variations in facilitation from subtle differences in E-P position in these cases.
Reconciling views of chromosome organization from 3C-based and microscopy studies remains an important challenge [49]. In our simulations, we found that changes in contact frequency are not always accompanied by equal changes in the mean spatial distance between two loci. In particular, insulation changes the distribution of spatial distances at small values, while having little effect on the mean. Changes in the spatial distribution at small distances could be difficult to detect experimentally and would require many cells to be assayed. Our results also suggest that integrating 3C-based and microscopy data can provide mechanistic insights.
Another important aspect of in vivo networks of local looping interactions is that they may be both dynamic over the course of the cell cycle [32] and different between cells [50]. Our results for insulation and facilitation by fixed loops, where the bases of the loop are always connected, remain relevant for dynamic loops while they are present. Roughly speaking, the effect on insulation or facilitation for a given loop is proportional to its frequency of occurrence in a cell.
Given the complexity of the local looping network, it is likely that there are multiple dynamic loops in the vicinity of the enhancer and promoter. While we studied the permanent single and double loop systems, our results provide intuition even to these more complicated systems. For instance, the global nature of insulation implies it can hinder interactions between enhancers and any number of promoters. Conversely, facilitation is local and thus specific to the regions that directly flank the loop. Together, our results highlight the complex and non-local grammar of regulatory elements surrounding enhancers and promoters. In conjunction with emerging biological data, future simulations will provide additional insight into the consequences of chromatin's polymer nature for allosteric modulation E-P interactions.

Polymer model
Model overview: We modeled chromatin as a fiber of monomers connected by harmonic bonds. Unless noted, each spherical monomer had a diameter of 15 nm and represented 500bp, or approximately three nucleosomes. A permanent loop was formed by connecting two monomers with a harmonic bond of the same strength as the bonds between all consecutive monomers. This permanently brings the loop bases into contact. Two such loops were formed in the two-loop simulations. A three-point interaction force was used to impose a bending energy and account for the rigidity of the fiber. To model volume interactions, monomers interacted via a shifted Lennard-Jones potential, which is a computationally efficient purely-repulsive potential. Unless noted otherwise, the Lennard-Jones potential was truncated at U=3kT as specified below to allow occasional fiber crossing. Simulated polymers were confined to a sphere at a given density and initialized from an unentangled polymer conformation.
Polymer models were simulated with OpenMM, a high-performance GPU-assisted molecular dynamics software (https://simtk.org/home/openmm). We used an in-house openmm-polymer library to efficiently set up polymer simulations with OpenMM, and to analyze simulation results. openmm-polymer is publicly available on the Bitbucket online repository: http://bitbucket.org/mirnylab/openmm-polymer. Scripts used to perform simulations, build contact maps, and calculate insulation/facilitation are available in the "examples" folder of the openmm-polymer library; those scripts can be modified to incorporate any arrangement of loops and calculate facilitation or insulation for any parameter values.
Simulations were characterized by 4 parameters: loop size, number of loops, fiber stiffness, and system density. Total polymer length was always chosen to be approximately 33 * loop size. The initial conformation for all simulations was an unentangled polymer ring. Simulations for a phantom chain were performed by switching off inter-monomer Lennard-Jones interactions. Choice of parameters for various models is summarized in Table S1.

Forces and Langevin Dynamics simulations
Adjacent monomers were connected by harmonic bonds with a potential U = 25*(r -1) 2 (here and below, energy is in units of kT). The stiffness of the fiber was modeled by a three point interaction term, with the potential U = k*(1-cos(α)), where α is an angle between neighboring bonds, and k is a parameter controlling stiffness. A value of k=3 was used for most simulations; k=2 and k=4 were used for simulations with lower and higher stiffness; k=2 was used for simulations with the smallest (10 monomer) loop.
Neighboring monomers interacted via a shifted Lennard-Jones (LJ) repulsive potential U = 4 * (1/r 12 -1/r 6 ) + 1, for r<2 1/6 ; U=0 for r > 2 1/6 (for details see [32]). To account for the activity of type-II topoisomerase, we allowed fiber crossing by truncating the shifted LJ potential at an energy of E cutoff = 3 kT. For energy U more than 0.5 * E cutoff , the LJ potential was modified as: U softened = 0.5 * E cutoff * (1 + tanh(2*U/E cutoff -1)). To avoid numerical instabilities, the interaction radius r was truncated at r = 0.3 via: r truncated = (r 10 + 0.3 10 ) 1/10 , which introduced a negligible shift in the final softened potential. We note that our simulations were performed at thermodynamic equilibrium, and thus the rate at which the fibers were allowed to pass through each other does not influence the equilibrium properties of the system; the only relevant factor is whether the system is allowed to change its topological state or is "locked" to the unknotted topological state. We explored both scenarios in our simulations. Spherical confinement was realized as a potential linearly increasing at a rate of k=5kT/mon when the radius was larger than the confinement radius. We simulated our model using Langevin Dynamics, performing 80,000 blocks of 3000 time steps (240,000,000 time steps total). For the fiber lengths considered here, polymer simulations reached equilibrium in less than 1000 blocks; this was confirmed by observing that monomer displacement saturates at about 500 blocks. Polymer conformations starting with block 1000 were used for our analysis. We note that this study focuses on equilibrium aspects of chromatin loops and that simulated time is not specifically matched to the time-scale of E-P interactions in vivo; chromatin loop dynamics are beyond the scope of this study.
An Andersen thermostat was used to keep the kinetic energy of the system from diverging. The time step was then chosen to ensure conservation of kinetic energy and lack of fiber crossing with the non-truncated Lennard-Jones potential. The absence of fiber crossing in this case was confirmed by the conservation of Alexander's polynomial for a 50000-monomer ring simulated at a high density of 0.85 for 1,000,000,000 time steps.

Initialization and starting conformations
Since our simulations were performed at thermodynamic equilibrium, the starting conformation does not affect properties of the resulting heatmap; for simulations with fixed topology (i.e. no fiber crossing), only the topological state of the starting conformation is relevant. For simulations with or without fiber crossings, we initialized our simulations from an un-entangled polymer state created as described below. We started with a 4-monomer ring on a cubic lattice. We then chose one bond at random, and tried to extend the polymer at this location by two monomers, by making a bond into a kink. To do this, we considered another bond, obtained by shifting this bond by one in a random direction perpendicular to the bond (choosing one out of 4 possible directions). If both locations of the shifted bond were free, the polymer was extended to incorporate this bond. For example, if a chosen bond was going in +z direction: … -> (0,0,0) -> (0,0,1) -> … , and we attempted to grow it in the -y direction (chosen randomly out of +x, -x, +y, -y), we would check positions (0,-1,0) and (0,-1,1). If both of them were free, the polymer sequence would be changed to … -> (0,0,0) -> (0,-1,0) -> (0,-1,1) -> (0,0,1) -> … . If at least one of the positions of the shifted bond was occupied, selection of the random bond was repeated. The process was repeated until the polymer grew to the desired length. Since no polymer fibers can pass between the old bond and a kink, this process preserves the original topology and creates an un-entangled polymer.

Calculating and analyzing heatmaps
To obtain heatmaps, we first found all contacts within each polymer conformation. A contact was defined as two monomers being at a distance less than 2 monomer diameters. Contacts for all pairs of monomers were then put on a heatmap (i.e. a 2000-monomer polymer produced a 2000x2000 heatmap). When calculating contact frequency ratios for insulation and facilitation, averaging was performed over small regions of the heatmap to reduce sampling noise. Unless noted, we report the average value for insulation over a region of the heatmap, by averaging over monomers in the promoter-proximal third of the loop and over a +/-3 monomer E-P separation. The range of insulation values for different positions in the promoter-proximal half of the loop is shown in Figure 3B. For facilitation, we average over a region of the heatmap defined by a +/-3 monomer E-P distance and a +/-((E-P genomic distance -loop length) / 6) monomer offset from a symmetric placement of an E-P pair around the loop bases (e.g. +/-6 monomers in Figure 2B). For the case when the E-P pair was within the loop, no averaging was used, since this occurs at short E-P distances where many contacts occur. To calculate the contact frequency ratio, we used simulations without a loop to calculate the expected frequency; for all parameter values, two simulations without a loop and ten simulations with a loop were performed, with a newly generated starting conformation for each simulation.

Calculating spatial density around a loop
To calculate normalized spatial density around the loop, we analyzed the model with default parameters (i.e. as in Figure 2), where a loop connects monomers 970 and 1030 of a 2000monomer ring.
1. For each conformation, we found the center of mass (COM) of the 60 loop monomers. We then defined "distal" monomers as monomers 0-950 and 1050-2000 (outside of the loop, plus 20 monomers, or 10kb, away from the loop base). We then counted how many distal monomers were at each spatial distance from the loop COM, averaging over all conformations and using bins starting at 0 with a step size of .5 monomers.
2. We then account for the fact that at larger spatial distances, a greater portion of the spatial shell exceeds the confining boundary. To this end, we took the position of the loop center of mass in a given conformation, and performed step 1 for a COM from a conformation in a different run (i.e., for conformation X in run Y, we took COM of the loop from the conformation X in the run (Y+1) modulus 10).
3. We then divided spatial densities from 1 by spatial densities from 2 to obtain normalized spatial densities.
To create a no-loop control, we repeated steps 1-3 for the monomers exactly opposite from the center of the loop, exactly repeating the same procedure (i.e. assuming 60-monomer loop began at monomer 1970, recalling that the polymer is closed into a length-2000 ring).

Simulated FISH distributions
To calculate simulated FISH distributions, we considered an E-P distance at which the magnitude of insulation and facilitation are comparable (90kb) and analyzed the model with default parameters. We then iterated over conformations and calculated the spatial distance between the E-P pair for both arrangements as well as for a control arrangement without a loop (i.e. monomers exactly opposite from the center of the loop, as for the spatial density calculation). The spatial distances were binned starting at 0 with a step size of 0.5 monomers.

Figure 1. Enhancer-promoter pairs in the context of other interactions. Experimental
Studies, (A) Illustration of an enhancer (in yellow) spatially interacting with a promoter (blue) along a chromatin fiber. This coloring convention continues throughout the paper. (B) A recent study in Drosophila suggested a 7kb chromatin loop formed between Su(Hw) insulators (orange) could decrease E-P interactions (red "X") [20]. (C) Conversely, a 3kb chromatin loop in the region between enhancer and promoter was proposed to increase E-P interactions. (D) Five arrangements for proposed looping interactions from three studies, left to right, [21], [22], and [23]. (left) a single Drosophila gypsy element between an enhancer and a promoter did not change their interactions (top), however an additional gypsy element upstream of the enhancer decreased E-P interactions (bottom) [21]. (center) at the mouse H19 locus, a regulatory element with multiple larger loops (55kb and 25kb) was suggested to control multiple E-P contacts; the enhancer can regulate the promoter before the loop, but cannot regulate the promoter within the loop [22]. (right) chromatin loops may also modulate spatial interactions between silencing elements (e.g. PRE, black triangles) and their target promoters [23]. The promoter within the loop is not silenced (top), whereas the promoter beyond the loop is silenced (bottom).