Action at a distance in transcriptional regulation
AAction at a distance in transcriptional regulation
William Bialek, a,b
Thomas Gregor, a,c and Gaˇsper Tkaˇcik d a Joseph Henry Laboratories of Physics, and Lewis–Sigler Institute forIntegrative Genomics, Princeton University, Princeton NJ 08544 USA b Initiative for the Theoretical Sciences, The Graduate Center,City University of New York, 365 Fifth Ave, New York NY 10016 c Department of Developmental and Stem Cell Biology UMR3738, Institut Pasteur, 75015 Paris, France d Institute of Science and Technology Austria, Am Campus 1, A-3400 Klosterneuburg, Austria (Dated: December 18, 2019)There is increasing evidence that protein binding to specific sites along DNA can activate thereading out of genetic information without coming into direct physical contact with the gene. Therealso is evidence that these distant but interacting sites are embedded in a liquid droplet of proteinswhich condenses out of the surrounding solution. We argue that droplet–mediated interactions canaccount for crucial features of gene regulation only if the droplet is poised at a non–generic point inits phase diagram. We explore a minimal model that embodies this idea, show that this model hasa natural mechanism for self–tuning, and suggest direct experimental tests.
In multicellular organisms, the transcription of genesinto messenger RNA is controlled by the binding of tran-scription factor proteins to “enhancer” sites that can beseparated from the gene by tens of thousands of basepairs along the DNA sequence [1–6]. Close approach ofenhancers to their target promoters has been inferredfrom cross–linking experiments [7], and there is directevidence that the action of the enhancer requires physi-cal proximity to the promoter site where transcription isinitiated [8]. But proximity is not contact: the most re-cent measurements indicate that the enhancers and theirtarget promoters remain separated by 150 −
350 nm evenduring active transcription [8–12].How is the apparent action at a distance possible? In-teractions between the enhancer and promoter could betransmitted along the length of the DNA molecule, but itseems more plausible that this interaction is transmittedacross the shorter three dimensional distance [13]. Re-cent observations indicate that there is a medium for thistransmission, a condensed droplet of the protein “medi-ator” which surrounds the promoter [14]; these dropletsalso contain high concentrations of RNA polymerase [15],are associated with foci of active transcription [16], canform in vitro [17], and contain other co–activating fac-tors [18]. We propose that the droplet acts as a largerscale version of an allosteric protein [19–21]: in the sameway that the protein structure allows binding of smallmolecules at one site to influence binding or enzymaticactivity at a distant site, the droplet would allow bindingof transcription factors (TFs) at an enhancer site to in-fluence activity at the distant promoter site (Fig 1). Wewill argue that this is possible only if the droplet is ata non–generic point in its phase diagram, and that thecollective interactions among the enhancer sites can drivethe system toward such points.Two facts will be crucial to our discussion. First, tothe extent that the mediator droplets are similar to otherexamples of intracellular phase separation [22], they will be liquid–like [23], and thus in general will not transmitstructural changes across hundreds of nanometers. Sec-ond, gene expression can be controlled in a quantitative,graded fashion in response to changing concentrations oftranscription factors [24–27].If we did not have the constraint of graded responses,we could imagine that binding of TFs to enhancer sitestriggers droplet condensation, and that this is the es-sential mechanism of regulation, as proposed for “super–enhancers” [18, 28]. But this is an all–or–none mecha-nism, and it is difficult to harness the triggering of phaseseparation to generate a quantitatively graded responseto changes in TF concentration. The existence of dropletsby itself does not solve the problem.
FIG. 1: The DNA strand (black) surrounds a condenseddroplet (orange). Promoter site is marked by an arrow (red),enhancer sites as blocks (green) with transcription factors(magenta and blue) both bound to these sites and freely dif-fusing. a r X i v : . [ q - b i o . S C ] D ec Even if transcription requires droplet condensation,there are pathways for regulation once the droplet hasformed. In eukaryotes transcription involves a very largenumber of different proteins, and it is plausible that manyof these components condense into the droplet. Withmultiple components the phase diagram is more com-plicated than just two phases [29, 30], so droplets cancondense and still have additional degrees of freedom re-lated to the addition or expulsion of different molecularspecies. Let us summarize these variables by an orderparameter φ ( (cid:126)r ), which can vary with position (cid:126)r insidethe droplet. These are the degrees of freedom that can propagate interactions through the droplet.The simplest model envisions a set of K identical bind-ing sites for a single class of transcription factors (at po-sitions (cid:126)r i ), plus one promoter site (at (cid:126)r a ), all embeddedin a droplet. These binding sites typically will be ar-rayed across multiple enhancers, all of which can con-tribute to regulating transcription. The relevant vari-ables are σ i = ± A = { , } for the inactive and active states of thepromoter. All of these variables couple to the order pa-rameter, and it is important that these couplings are spa-tially local. The free energy is then F [ φ ( (cid:126)r ); { σ i } , A ] = F [ φ ( (cid:126)r )] + E A − k B T c/c ) K (cid:88) i=1 σ i + K (cid:88) i=1 g i σ i φ ( (cid:126)r i ) + g a Aφ ( (cid:126)r a ) , (1)where g i is the interaction between the order parameterand binding of TFs to the enhancers, g a is the interactionbetween the order parameter and the active vs inactivestate of the promoter, E is the free energy differencebetween the two states of the promoter in the absence ofTFs, c is the concentration of these factors, and c is the“bare” binding constant of the TF to the enhancer sites.Let’s assume that, as in conventional models of al-lostery, the transmission of information can be describedas an equilibrium thermodynamic effect [31–33]. Hence,we define an effective free energy by integrating out thefluctuations of the order parameter, e − F eff ( { σ i } , A ) /k B T = (cid:90) D φ exp (cid:18) − F [ φ ( (cid:126)x ); { σ i } , A ] k B T (cid:19) , (2)where D φ is the measure for integration over φ ( (cid:126)r ). F eff is composed of independent and interacting parts, F eff = F + F int , and to leading order in the couplings we have F = E A − k B T c/c ) K (cid:88) i=1 σ i (3) F int = − K (cid:88) i , j=1 g i g j k B T C ( r ij ) σ i σ j − K (cid:88) i=1 g a g i k B T C ( r i a ) σ i A, (4)where we choose coordinates so that the average orderparameter is zero in the Boltzmann distribution definedby F , and C ( r ) is the correlation function of the orderparameter fluctuations in this distribution, (cid:104) φ ( (cid:126)r i ) φ ( (cid:126)r j ) (cid:105) ≡ C ( r ij ) , (5)with r ij = | (cid:126)r i − (cid:126)r j | . The question of whether the droplettransmits information from the enhancer to the promoter becomes the question of whether fluctuations in the orderparameter are correlated over these long distances [34].In liquids at generic parameter values, density fluctu-ations have a short correlation length ξ , so that C ( r ) (cid:39) e − r/ξ , with ξ on the nanometer scale, and thus thesemodes cannot support action at a distance. There can beadditional degrees of freedom associated with the orienta-tional ordering of molecules in the droplet, or with con-formational changes of these molecules, but again withgeneric parameters we expect to find small ξ . The al-ternative is that the parameters describing the dropletare at a non–generic point in the phase diagram, wherethe correlation length can become long, and this is the“critical droplet” scenario we explore here.Close to a first–order phase transition the free energy F has two nearly degenerate minima [34]. In a suffi-ciently small droplet, fluctuations in the order parameterare dominated by flickering between these minima, andthere is an effective surface tension that keeps the en-tire droplet in one minimum, so that C ( r ) becomes onlyweakly dependent on distance [35]. Close to a second–order phase transition the correlation length diverges,and C ( r ) decays very slowly, as power of distance. Ei-ther of these scenarios seems to require some tuning ofthe droplet parameters, to which we return below.If all the transcription factor binding sites couple tothe droplet in the same way, then we should have all the g i = g . We can capture the essential predictions of thismodel if all the distances r ij and r a are roughly equal toa typical R , in which case we can simplify to F int = − J K (cid:88) i , j=1 σ i σ j − J a K (cid:88) i=1 σ i A, (6)with two parameters J = ( g /k B T ) C ( R ) and J a = -1 c/c m ean a c t i v i t y critical dropletMWC FIG. 2: Mean promoter activity as function of the TF con-centration. Results from Eqs (3) and (6) with K = 8 sites, J = 0 . k B T , J a = k B T , and E = k B T ln(100), comparedwith the corresponding MWC model ( J a = 0 . k B T , J = 0).Note that interaction energies are on the order of k B T or less,but the droplet generates a very steep response to changingTF concentrations. ( gg a /k B T ) C ( R ). To gain intuition, we note that if J = 0then Eq (6) becomes identical to the Monod–Wyman–Changeux (MWC) model for allosteric proteins [20, 32],with the A = 1 / g i , which describes in-teraction of the TF with the surrounding droplet, andis determined by the face of the protein opposite fromthe DNA binding domain. A generalization is to imaginethat we have K a binding sites for activators at concen-tration c a and K r sites for repressors at concentration c r . Then at low repressor concentrations there is coop-erative activation, but at higher repressor concentrationthe system approximates a switch that depends on theratio of powers of the concentrations, c K a a /c K r r .A second generalization is to have multiple nearbybinding sites within one enhancer interact more strongly,perhaps through additional degrees of freedom, and thenlet the emergent states of multiple enhancers couple tothe droplet. The system could then approximate logi-cal operations corresponding to combinations of ANDswithin each enhancer and ORs among enhancers.An important implication of this model is that tran-scription is regulated not by the binding of transcriptionfactors to individual binding sites, but rather by an in-tegrated signal from multiple binding sites that are dis- tributed around the droplet of size R . If we think ofthe transcriptional output as a “measurement” of the TFconcentration, then the accuracy of this measurement islimited by the random arrival of molecules [36–38]; thesmallest concentration differences δc to which a systemcan respond reliably is given by δcc (cid:39) √ D(cid:96)cτ . (7)where c is the background concentration, D is the diffu-sion constant, τ is the time over which the system canaverage, and (cid:96) is the linear size of the sensitive element.If the response is driven by a single binding site, then (cid:96) is the size of that site, but if the system integratesover many binding sites, then (cid:96) can approach the lineardimensions of the entire array of sites, in our case thesize of the droplet, which is ∼ × larger than individ-ual binding sites. From Eq (7), responses which wouldrequire hours of integration at a single site thus becomereliable in minutes. Transcription factor concentrationsare so low that this difference can be crucial [37, 39].Poising a condensed droplet near a critical point seemsto require fine tuning of its parameters. Cells can exertexquisite control over protein and nucleic acid concentra-tions [39, 40], but matching the concentrations of crucialmolecules to their critical values still seems difficult. Inour case, however, there is a thermodynamic driving forcethat pushes the system toward conditions where correla-tion lengths are long. To estimate this effect, let’s assumethat the droplet has a critical point when one of its com-ponents is at concentration x . The chemical potential ofthe surrounding solution holds the concentration close tosome mean concentration ¯ x , and variations ∆ x aroundthis mean cost a free energy F (cid:39) (¯ nk B T / x/ ¯ x ) ,where ¯ n (cid:39) R ¯ x is the mean number of molecules in thedroplet. But at a concentration x the correlation lengthwill be ξ = a | x / ( x − x ) | ν [34], where a is a micro-scopic length scale. Away from criticality, the gain infree energy from interaction among K binding sites is F (cid:39) − ( J / K ( K − e − R/ξ , and in total we have2 F ¯ nk B T (cid:39) (∆ x/ ¯ x ) − A exp (cid:20) − Ra (cid:12)(cid:12)(cid:12)(cid:12) x ∆ x + ∆ x (cid:12)(cid:12)(cid:12)(cid:12) ν (cid:21) , (8)with ∆ x = ¯ x − x and A = ( J /k B T ) K ( K − / ¯ n . Thedominant component of the droplet is present at only¯ n ∼
100 [14], so with K ∼
10 binding sites it is easy tohave A ∼
1; to be conservative we consider A = 0 . R ∼
150 nm and the molecularscale a ∼ R/a = 5. Assuming that the chemical potential alonesets ¯ x = 1 . x , we see that the possibility of mediatinginteraction among binding sites creates a sharp minimumof the free energy at the critical point, sufficient to pullthe system very close to criticality (Fig 3).Taking this thermodynamic driving force seriously, wenote that when transcription is active, enhancer bind- concentration x/x -10-505101520 f r ee ene r g y F / k B T FIG. 3: Free energy as a function of concentration in thedroplet, from Eq (8), with ¯ n = 100, ∆ x = x / A = 0 . R/a = 5, and ν = 1 /
2. Note the weak minimum at the x = x + ∆ x , set by the chemical potential, which is dominatedby the minimum at criticality, x = x . ing sites with states that are “aligned” to this activa-tion have a free energy that is lower by J a ∝ C ( R ),and since C ( R ) decreases with r this generates a smallforce pulling the enhancer toward the promoter. In con-trast, enhancers in states that are not contributing toactivation of transcription have a free energy that ishigher by J a and a force pushing enhancer and promoterapart. These small forces are balanced by a stiffness,which also determines the thermal fluctuations in theenhancer–promoter distance. The result is that alignedvs anti–aligned enhancers should be at different meandistances from the promoter site, and this displacementis ∆ R/R ∼ ( J a /k B T )( δR/R ) , where δR is the stan-dard deviation of the distance R , and we assume that d ln C ( R ) /d ln R ∼
1. These displacements should be di-rectly observable, for example by measuring the positionsof different enhancers for the pair-rule genes in the earlyfly embryo [41]. More generally this suggests that single–molecule observations of enhancer motions could be con-nected, quantitatively, to the energetics of cooperativetranscriptional activation.To summarize, a large number of transcription fac-tor binding sites, embedded in a droplet that surroundsthe promoter, will generate cooperative regulation if thedroplet is poised near special points or lines in its phasediagram where correlation lengths become long. In thisscenario the droplet functions much like an allosteric pro-tein, but this is possible only because of the proximity tocriticality. This is similar to the long–ranged interactionsbetween proteins that we expect to see in a membrane[42] if lipid compositions are close to a critical point, asobserved [43–45]; it has also been suggested that chro- matin itself is close to a sol/gel phase boundary [46].There is a much wider range of ideas about how critical-ity could play a role in biological function [47, 48], butwhat is special in our example is that we have identified,as an intrinsic part of the functional behavior, a mecha-nism that drives the system toward its critical point, andperhaps this is more general. Consequences of this ther-modynamic driving force should be directly observable inthe physical positions of enhancer and promoter sites.We thank L Barinov, SA Kivelson, and MS Levinefor helpful discussions. This work was supported inpart by the US National Science Foundation, throughthe Center for the Physics of Biological Function (PHY–1734030) and Grant PHY–1607612; by National Insti-tutes of Health Grants P50GM071508, R01GM077599,and R01GM097275; and by Austrian Science Fund grantFWF P28844. [1] W De Laat and D Duboule, Topology of mammalian de-velopmental enhancers and their regulatory landscapes.
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