How subtle changes in 3D structure can create large changes in transcription

Animal genomes are organized into topologically associated domains (TADs), which exhibit more intra-domain than inter-domain contact. However, the absolute difference in contact is usually no more than twofold, even though disruptions to TAD boundaries can change gene expression by 8-10 fold. Existing models fail to explain this superlinear transcriptional response to changes in genomic contact. Here, we propose a futile cycle model where an enzyme stimulated by association with its products can exhibit bistability and hysteresis, allowing a small increase in enhancer-promoter contact to produce a large change in expression without obvious correlation between E-P contact and promoter activity. Through mathematical analysis and stochastic simulation, we show that this system can create an illusion of enhancer-promoter specificity and explain the importance of weak TAD boundaries. It also offers a mechanism to reconcile recent global cohesin loop disruption and TAD boundary deletion experiments. We discuss the model in the context of these recent controversial experiments. Together, these analyses advance our interpretation and understanding of cis-regulatory contacts in controlling gene expression, and suggest new experimental directions.

We wondered whether these apparent contradictions can be reasonably reconciled with known 64 biochemical mechanisms. To answer these questions we examined simple biophysical models of 65 proximity-dependent E-P communication grounded in known physical laws (chemical master 66 equation). We identify mechanisms whose relevance for the spatial organization of the genome 67 has not previously been considered, which we show provide a simple reconciliation of all three 68 apparent contradictions. We discuss the implications for this revised view of contact-mediated cis-69 regulation of gene expression for interpreting experimental results. 70 71

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A simple model of hypersensitivity to changes in contact frequency 73 To better understand the connection between E-P contact frequency and transcriptional behavior, 74 we started with a quantitative analysis of recently published super-resolution microscopy data 75 (Mateo et al., 2019). These experiments used Optical Reconstruction of Chromatin Architecture 76 to observe 3D path of chromatin through two neighboring regulatory domains. The upstream 77 domain contained the gene Ubx and its enhancers, and the downstream region contained the 78 gene abd-A and its enhancers (Fig. 1A). In wildtype embryos, interactions between the two 79 domains, including enhancers and promoters, were less frequent than within the domains, 80 producing two clearly distinct TADs in population average data (Fig 1A). Deletion of a few kilobase 81 elements at the border of these TADs resulted in a clear increase in the interaction between the 82 domains, though only by a factor of 1-2.5 fold across the population (Fig 1A). While the structures 83 are easily distinguished at the population level, the structure of single cells are more variable, with 84 25% of individual wildtype cells showing some cross-TAD border mixing between the Ubx 85 enhancers and the abd-A promoter, compared to 50% of mutant cells (Supp. Fig 1A). The 86 change in gene expression, as assayed by smFISH, is hypersensitive to this moderate change in 87 contact frequency, with an average 5-fold increase in mRNA counts of abd-A (Fig. 1B). Similar 88 changes are also seen for Abd-B (Supp. Fig. 2). From these single cell-analyses, we concluded 89 that the quantitative disconnect between weak TAD borders (see also Supp. Fig 3) and the large 90 transcriptional effects of border disruption (see also Supp. Fig. 4) is not an artefact of Hi-C or of 91 population averaging. Instead, small differences in the frequency of contact appear sufficient to 92 drive large changes in expression, requiring us to reexamine the text-book picture of E-P 93 regulation (Alberts et al., 2018).  107  108  Inspired by the hypersensitive cell-signaling pathways dependent on futile-cycle competition  109 between phosphatases and kinases, we considered a model of enhancer-mediated transcription 110 ( Fig. 1C) in which the transcription rate of the promoter is governed by the amount of transcription 111 promoting 'tag' that accumulates there. For simplicity, we assume transcription is linearly 112 proportional to tag amount. The tag could be an accumulating condensate of transcription factor, 113 or a histone modification, such as acetylation. The promoter has an intrinsic rate of tag addition 114 (r 0 ) and removal (g), and the enhancer regulates this competition by contributing to tag addition 115 when it interacts (e) with the promoter (Fig. 1C). Finally, we assume the enzyme that adds the 116 tag can also physically interact with this output product, increasing its effective addition rate from 117 r 0 to r 1 , r 2 , …, r n-max where n is the number of tag molecules currently associated, and r n > r n-1 . 118 This could equivalently be achieved by stabilizing the binding of enzyme by associating with the 119 tag, increasing its local concentration instead of catalytic rate. The probabilities of product 120 association and dissociation are proportional to the amount of existing product, with respective 121 rate constants k n and τ n . The importance of this product interaction, the constraints on the 122 parameter values, and range of known molecular factors which may execute these behaviors, are 123 discussed below. 124 125 This model readily exhibited a hypersensitive regime in which small fold changes in E-P contact 126 lead to large fold changes in transcription (Fig. 1D), around the experimentally observed range 127 (Fig 1A, 1B). This hypersensitive behavior suggests that the relatively weak effects of most TAD 128 boundaries (Supp. Futile-cycle promoters can create an illusion of enhancer-promoter specificity 132 We next explored how differences in promoter properties affect enhancer-promoter 133 responsiveness. We considered two promoters which both experience a two-fold increase in 134 enhancer-promoter interaction at the start of the simulation ( Fig. 2A), for example, due to removal 135 of a neighboring strong TAD boundary. Both promoters start in a low transcription state, but the 136 second one has a slight (two-fold) higher intrinsic tag removal rate, g ( Fig. 2A). Promoter 1 137 exhibited a large increase in transcription, yet Promoter 2 remains almost unaffected by this 138 perturbation, exhibiting only a small shift (Fig. 2B). Because of its higher intrinsic affinity for the 139 tag removal machinery, Promoter 2 is still in the sub-linear response regime (Fig. 1D). Further 140 increase in enhancer promoter interaction would be sufficient to drive it into a highly transcribing 141 state like Promoter 1. This behavior is interesting, as many experimentally reported perturbations 142 involve bystander genes that respond little to gains in enhancer activity interpreted as evidence of a "lock-and-key" mechanism for gene regulation (van Arensbergen et  145 al., 2014). In that model, enhancer-promoter specificity is achieved by different distinct pairs of 146 biochemical locks and keys, rather than by spatial organization (Fig. 2C). These simulations 147 illustrate that the apparent specificity may be an illusion resulting from different tipping points in 148 the type of promoters considered here.

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Hysteretic in futile-cycle promoters creates memory at multiple timescales, a potential 160 explanation why global TAD disruption causes little transcriptional change 161 We next examined the population level transcription dynamics of simulated cells exposed to a 162 mild change in E-P contact, like the 2-fold or less expected from TAD border disruption due to 163 border deletion or cohesin removal ( Fig. 3A & B). We focused this analysis on promoters in the 164 hypersensitive regime, since promoters in the sub-linear regimes exhibit limited sensitivity to mild 165 changes in E-P-contact. Simulated cells with a promoter started in an active state which 166 experienced a two-fold reduction in contact frequency showed little change in expression by the 167 end of the early equilibration period (Fig. 3C). However, by the late time point, the most cells 168 exhibited dramatically decreased expression (Fig. 3C). This response was much slower than 169 observed for inactivation of an enhancer, where most cells dropped to low expression during the 170 early period (Fig. 3C). Thus, the transcription output of these promoters is robust to short 171 structural perturbations, even though it remains rapidly responsive to chemical signalling 172 changes, such as deactivation of an enhancer. The hysteretic behavior of the model was further examined by comparing the response to a 195 change in contact frequency as a function of initial state. Promoters that start at low 196 tag/transcription levels follow different response curves from those that start high (Fig. 4A). This 197 results from feedback where promoters which have a high level of the tag recruit more of the tag 198 addition enzyme and thus require less interaction from the enhancer to counteract tag removal. 199 Consequently, these promoters generally stayed highly transcribing even at EP contact levels 200 that are insufficient to overcome tag removal from promoters that started with few tags. As 201 illustrated in the simulations above, the degree difference between the curves depends on time 202 (Fig. 4B). Promoters that experienced a very low rate of E-P contact rapidly converged to a 203 common distribution in which most of the population is silent, independent of the starting state. 204 Those that experienced a high rate of contact rapidly converged to a distribution which is mostly 205 actively transcribing. Promoters with intermediate contact rates exhibited much longer memory of 206 the starting state (Fig. 4B).

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Transcription and enhancer-promoter contact may be largely uncorrelated in single cells 219 Having examined the temporal dynamics at the population scale, we turned the dynamic 220 behaviors in single cells. We asked whether a promoter driven by contact-dependent enhancer 221 activity and futile cycles, as described above, would be expected to show strong or weak 222 correlation between E-P looping and transcription at the single cell level. This parallels 223 experimental measurements we and others recently made ( Fig. 5A & B). From our simulated 224 cells, we computed the odds ratios for observing transcription in a window given observation of 225 contact in that window (Fig 5C). Across the different regimes of E-P contact frequency, the Odds 226 Ratios are statistically greater than 1, but only by a small margin of 0-30%, paralleling our recent 227 empirical observation (Mateo et al., 2019). Since the frequency of interaction only tips the scales 228 in the existing futile cycles of promoter modification, rather than acting as a triggering event for 229 transcription, a strong correlation is not expected. The Odds Ratio still remains different from 1 230 however, as promoters which are experiencing a high frequency of EP contact are more likely to 231 transcribe, increasing the probability the two events co-occur. 232 233 To further study the relation between E-P contact and transcription, we examined contact 234 frequency in the stochastic model relative to the timing of transcription events. Previously, the 235 lack of change in E-P distance across any time scale relative to detected transcription bursts was 236 interpreted to refute a contact-dependent model of gene enhancer-mediated regulation 237 (Alexander et al., 2019) (Fig. 5D). For a moderate number of simulated bursts (~500), we also 238 find no detectable increase in proximity at any timescales relative to that of the burst (Fig. 5E). 239 This is surprising at first, since by construction transcription is dependent on contact. Because 240 the transition also depends on promoter-intrinsic tag addition and removal events, which also 241 occur stochastically, many E-P contact events don't result in transcription and many that do still 242 exhibit variable timing, so that with small numbers of total events the dependence is washed out 243 in the stochastic noise. Simulations involving a similar number of total cells as experiments also 244 lacked detectable correlation between the fraction of time spent transcribing and average E-P 245 distance or contact rate for individual cells, same as observed in the experimental data (Supp Fig  246  5 A,B,D,E). Substantially increasing the number of simulated cells uncovered weak but 247 statistically significant correlations in each of these cases (Fig. 5E, and Supp. Fig. 5 C,F), similar 248 to the observations from the Drosophila data (Fig 5A). Thus, the Futile Cycle transcription model 249 shows that enhancers could regulate genes like Sox2 or the Bithorax genes in a purely contact-250 dependent manner of communication, even without strong correlation between contact events 251 and transcription arising at the single cell level.

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Minimal parameter requirements for hypersensitivity 268 Thus far we have examined the behavior of a simple, discrete chemical model of promoter 269 regulation (Fig. 1C) in a regime in which the model exhibits hypersensitivity and hysteresis, using 270 stochastic simulations. However, not all parameter values for tag addition rates and removal rates 271 will support these behaviors. We used a theoretical systems analysis approach to determine 272 general requirements on the parameter values needed for these dynamic properties to arise. 273 From a simple mass-action analysis of the chemical system, we derive a closed-form expression 274 for the rate of change in tag concentration (d[a]/dt) as a function of the current tag concentration 275 ([a]) (Fig. 6A) (see Methods), also called a phase portrait (Strogatz, 2018). For an appropriate 276 region of parameter space (discussed below), d[a]/dt has up to three unchanging values, or critical 277 points, where the curve crosses zero (Fig. 6D). can be seen the second zero is unstable, and the third is stable. This bistability produces 281 hysteresis in the system: promoters which start with low levels of expression (or tag) slowly 282 increase [a] as the E-P frequency is raised to the point where the left and center critical point 283 merge (a saddle-node bifurcation), which leads to a sharp discontinuous jump in the stable 284 expression levels to the high state (Fig. 6B). 285 286 Fig 6  287 288  We find two simple prerequisites for the Futile Cycle to exhibit bistable behavior. First, there must 315 be at least three degrees of stimulation (n max ≥ 2) for the enzyme's tag addition rate as a function 316 of tag amount to be a sigmoidal curve, enabling three critical points (Fig. 6E) (see Supplementary 317 Information for a derivation). Second, it requires increasing stimulation of the successive states 318 (i.e. r n > r n-1 ), creating positive feedback in tag addition. If tag-associated states have less activity 319 than non-associated states the production rate will not be an increasing function of total tag 320 amount, preventing the existence of multiple stable states (Fig. 6F) Our simulations indicate such conclusions are premature. A futile cycle promoter exhibits not just 357 hypersensitive response to contact interactions, but also substantial regimes of sublinear or 358 insensitive response. Promoter specific differences in the intrinsic addition or removal of tags can 359 render one promoter unresponsive to an enhancer that activates a second promoter, even though 360 the two experience similar absolute interaction frequency. Indeed, the unresponsive one may 361 even experience a higher interaction frequency (e.g. because it is physically closer). It is thus 362 premature to interpret such results as evidence of biochemical specificity, or to dismiss a role for 363 chromatin structure. This is notable, since transgene assays indicate most regulatory elements 364 will promiscuously activate most promoters if placed close enough ( removing TADs and so-called "loops" or corner points from contact maps genome wide (Rao et 384 al., 2017). Surprisingly, this structural perturbation resulted in little transcriptional changes, hardly 385 any gene changing nascent transcription levels by more than a factor of two. This apparently 386 contradicted results in earlier and concurrent reports, in which disruption of single TAD boundaries 387 resulted in substantial change in expression to the local genes, leading to recent debate in the 388 community. While it is important to note that the experiments were conducted in different cellular 389 backgrounds, our modeling work demonstrates that a simple explanation is plausible. Two fold or 390 smaller differences in contact frequency in the model show significant hysteresis, and will retain 391 their initial transcription rate for many hours after such perturbation, but this memory is lost on 392 longer time scales. As transcription was assayed hours after the contact changes in the degron 393 experiment, little change was detected, whereas in experiments which genetically deleted border 394 elements, transcription was assayed days and cell generations later, at which point larger effects 395 were detected for some promoters. 396 397 Notably 3 These data have supported speculation that enhancers must be able to influence promoter activity 432 from a distance, e. The model analyzed here clearly demonstrates that it is possible for transcription to be regulated 436 by enhancer-promoter contact and yet show little temporal correlation, even in single-cell time-437 lapse data. This is because the E-P contact only changes the promoter state, toggling between a 438 condition in which transcription events are likely, and one in which they are less likely. This 439 explanation can be directly tested with larger datasets-with sufficient observations, a weak 440 correlation should be detectable (as seen in the fixed cell data), since cells experiencing a higher 441 frequency of contact are more likely to have a high level of tag and thus more likely to transcribe, 442 even though it is not a one-to-one dependency. 443 444

Repression also supports hypersensitivity and hysteresis 445
While for simplicity the model assumed that the tags accumulated at the promoter are activating, 446 such that transcription is proportional to tag level, it is equally plausible that futile cycles of 447 repressive tags at the promoter account for hypersensitivity of some promoters. In this case we 448 model transcription as inversely proportional to the concentration of the promoter tag. The distal 449 regulatory element which adds more of the tag should be termed a silencer rather than an 450 enhancer. In response to increased contact with the silencer, the promoter transitions through a 451 weakly-sensitive ON regime, a hypersensitive bistable regime, to a weakly-sensitive OFF regime 452 (Supp. Fig. 3). Alternatively, a model in which the enhancer removes the repressive tag from the 453 promoter supports all three regimes (Supp. Fig. 3). Notably, the Futile Cycle model as discussed here doesn't require the "tag" be a histone 480 modification or even an enzyme product. Any molecular group which can accumulate at the 481 promoter and exhibit a higher likelihood of molecular addition to promoters which already contain 482 the molecule will do. Micro-condensates of transcription factors are another interesting candidate. 483 Individual molecules fusing and re-dissolving out of the condensate provide the necessary 484 features of a futile cycle. Larger condensates are more likely to retain molecules and grow larger, 485 since they offer more weak multivalent interactions, a process called Ostwald Ripening (Marqusee 486 and Ross, 1983), and are more likely to fuse with other clusters, providing the necessary positive 487 feedback. Many of the transcription factors associated with gene regulation have recently been 488 shown to exhibit these properties, forming visible condensates of many molecules and exhibiting 489 rapid recovery after photo-bleaching indicative of futile cycling into and out of the cluster (Strom 490 and Narlikar, 2020). As we develop improved tools to measure the protein occupancy of individual 494 promoters in single cells along with chromatin structure, it will be possible to test the dynamic 495 interplay between contact and protein / modification accumulation more directly. 496 497 Enhancer cooperativity/super-enhancer hubs cannot explain hypersensitivity. 498 It has been suggested cooperative, hub-like interactions among multiple enhancers targeting the 499 same promoter may explain the hypersensitivity of transcription to enhancer-contact frequency. 500 This proposal draws its intuition from the superficial similarity between the hypersensitivity of 501 transcription to contact and the hypersensitivity to transcription factor (TF) binding that can arise 502 through cooperative interactions among TFs. The idea is that while the individual E-P contacts 503 may be weak and transient, much like individual TF-DNA interactions, if they stabilize one 504 another, the system can show hypersensitive transitions: A perturbation removing a weak TAD 505 boundary could subtly increase the contact frequency for all enhancers in a super-enhancer 506 cluster one side of the boundary with a promoter on the other side --much like a subtle increase 507 in TF concentration can the binding probability for each TF with its target site. If this subtle 508 increase in interaction is sufficient for a second enhancer to join the promoter before the first one 509 dissociates, the cooperative interactions between them make the connection last even longer and 510 so the third joins, further stabilizing the complex and so forth. Thus a small initial change has led 511 to a dramatic change in total contact duration, which in this view explains the strong transcriptional 512 changes without invoking promoter properties. However, available data in which contact 513 frequency was measured before and after structural perturbation fail to support this model --514 perturbations of initially weak boundaries result largely in weak changes in absolute contact 515 frequency, not the strong changes predicted by formation or collapse of a cooperative hub (e.g. 516 Fig 1b, Supp. Figs 3-4). 517 518 Conclusion 519 We have described and analyzed a simple model of enhancer-contact-dependent promoter 520 activity. In this view, the promoter is not only a middleman in the regulation of transcription. It does 521 not only relay the activity state of the enhancer into transcriptional activity of its target gene, nor 522 function as a simple amplifier of enhancer activity to set the levels of expression. Instead, through 523 accumulation of local tags, the promoter is able to integrate its history of interaction with one or 524 more regulatory elements and respond in striking nonlinear ways to changes in these signals. By 525 'accumulation of local tags' we postulate nothing more than that the promoter is epigenetically 526 regulated (in the broadest sense of 'epigenetic). In addition to its core sequence, the accumulation 527 of other factors at the promoter, be they histone modifications or transcription factors, provides 528 the necessary mechanism for the dynamical system to exhibit memory. 529 530 This view of the promoter as an epigenetic regulator and integrator of signals is not new, and 531 does not posit the existence of any unknown molecular mechanisms. However, a minimal model 532 that captures the key features of such promoters we have shown can exhibit a much more 533 complex dependency on chromatin structure than previously acknowledged. These behaviors 534 offer a simple explanation to controversies of major importance to chromatin structure and 535 transcription community. These include (1) why TAD boundaries can be weak and why insulators 536 only mildly affect contact frequency and yet be major regulators of transcription (2) why many 537 genes thought to be responsive to distal enhancers are insensitive to some structural changes 538 ( In brief, these simulations model the chemical master equation in explicit time and discrete time 550 steps, with discrete transition probabilities for the addition and removal of promoter tags. For 551 simplicity, we equate promoter modification and transcription, reducing the total parameter space 552 without loss of generality. 553 554 Continuum limit calculations 555 The ordinary differential equation (ODE) model was derived from the discrete chemical master 556 equation using mass-action kinetics as described below. The stimulated enzyme example 557 presented below is only one implementation of the futile cycle model, but used here because it's 558 the simplest and most intuitive system that recapitulates the desired quantitative behavior. 559 560 Model parameter definitions Consider a promoter-bound enzyme that attaches molecular tags 561 to a promoter at some basal rate 0 . This enzyme can also associate with promoter-bound tags 562 up to some number , which changes the tagging rate to . The enzyme-acetyl association 563 rates are dependent on the current association state (e.g. cooperativity), while dissociation is 564 assumed to be constant (for now, but can be generalized later). "Free" tags not associated with 565 the enzyme but still promoter-bound are removed from the promoter with a rate constant . 566 567 Reaction and rate equations 568 For illustrative purposes, we will start with the = 2 case (later, we'll show this is sufficient to 569 capture nonlinear behavior of interest). Then the enzyme has three "association states" ( = 0, Several variations of this metric exist, but we define it as follows: first the contact-frequency map 619 is normalized relative to linear genomic distance by dividing each off-diagonal row by the sum of 620 the row. This improves the interpretability of the measure, as the inter-domain triangle contains 621 more interactions further from the diagonal than the intra-domain ones (Supp. Fig 3). We then 622 scan the whole genome, computing the average, normalized interaction-frequency within each of 623 the 3 triangular regions shown in Supp.