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Facilitateddiffusionofproteinsonchromatin O. Beńichou,1 C. Chevalier,1 B.Meyer,1 and R. Voituriez1 1LaboratoiredePhysiqueTheóriquedelaMatie`reCondenseéCNRS-UPMC,4PlaceJussieu,75255ParisCedex (Dated:January27,2011) Wepresentatheoreticalmodeloffacilitateddiffusionofproteinsinthecellnucleus.Thismodel,whichtakes intoaccountthesuccessivebindingandunbindingeventsofproteinstoDNA,reliesonafractaldescriptionof thechromatinwhichhasbeenrecentlyevidencedexperimentally.Facilitateddiffusionisshownquantitativelyto befavorableforafastlocalizationofatargetlocusbyatranscriptionfactor,andeventoenabletheminimization ofthesearchtimebytuningtheaffinityofthetranscriptionfactorwithDNA.Thisstudyshowstherobustness ofthefacilitateddiffusionmechanism,invokedsofaronlyforlinearconformationsofDNA. 1 1 0 PACSnumbers: 2 n Thenowwellestablishedtheoryoffacilitateddiffusionex- modelwhichquantitativelyaddressesthesetwoquestions. a plains how DNA-binding proteins can in principle find their J target sites on DNA efficiently. This model describes search 6 trajectoriesasalternatingphasesoffreediffusioninthebulk 2 cytoplasmand1-dimensionaldiffusionalongtheDNAstrand, called sliding, which is made possible by sequence indepen- ] h dent interactions of proteins with DNA. Since the seminal c work[1],suchpathwayshavebeenevidencedexperimentally e both in vivo [2] and in vitro [3–5] thanks to single molecule m technics, and theoretical aspects have been refined [6–9], in - t particular highlighting that such strategies can minimize the a search time for a target site by a proper tuning of the pro- st tein/DNAinteraction[10–13]. FIG.1:Facilitateddiffusiononchromatin:Achromatinbindingpro- . tein (blue) searches for a target locus (yellow) on chromatin (red). t All these theoretical approaches rely on a schematic de- a Thebinding(resp.unbinding)rateisdenotedbyλ (resp.λ ). scriptionofDNAasa1-dimensionallinearchainalongwhich 2 1 m aproteincandiffuse,surroundedbyanhomogeneousmedium d- in which the protein performs regular diffusion. More re- At the theoretical level, modelling facilitated diffusion in n cently,crowdingeffectshavebeenincorporatedinthesemod- thecellnucleusraisestwoproblems: (i)first,totakeintoac- o els both for the sliding [14] and the bulk cytoplasmic phases count the switching dynamics of the protein between a state c [4], leading to more realistic descriptions of gene regulation boundtothechromatinandafreelydiffusingstateinthenu- [ kineticsinprokaryots. cleoplasm, and (ii) second to model the diffusion phase of a 2 However, such models are clearly inapplicable to eukary- proteinboundtoacomplexstructuresuchaschromatin.Point v ots, in which the DNA is packed in the cell nucleus [15] (i)hasbeenstudiedinthecontextofintermittentsearchstrate- 8 and forms a complex structure called chromatin which is far gies[21],andgeneralmethodstocalculatemeansearchtimes 5 fromasimple1-dimensionalchain. Evenifafullbottom-up forintermittenttrajectorieshavebeendeveloped.Ontheother 7 description of the in vivo DNA organization remains out of hand, the full distribution of the first-passage time (FPT) for 4 reach, theoretical ideas [16], and now growing experimental diffusion in fractals has recently been obtained in [22] and . 6 evidencesindicatethatthechromatinhasahierarchizedarchi- enables to tackle point (ii) under the assumption, backed by 0 tecturewhichdisplaysfractalpropertiesatleastoverthe100 experiments[17–19],thatthechromatinisfractal. 0 nm – 10 µm range. Indeed, textural image analysis, neutron In this letter, we gather and extend these new tools to de- 1 : scattering[17], rheologytechnics[18]andmorerecentlythe velop a theoretical model of facilitated diffusion in the cell v Hi-Cmethod[19]revealedindependentlyafractalstructureof nucleus. More precisely, we calculate analytically the mean Xi the chromatin characterized by a fractal dimension df which search time for a target for a protein which alternates diffu- wasfoundintherange2.2−3. Despitethiscomplexstructure sionphasesonthechromatinwhichisassumedtohaveafrac- r a ofthechromatin,theswitchingdynamicsofproteinsbetween talstructure,andfreediffusionphasesinthenucleoplasm(see aDNAboundstateandafreelydiffusingstate,whichcharac- Fig. 1). Underthesehypothesis, weshowquantitativelythat terizesfacilitateddiffusioninprokaryots, seemstobealsoat facilitateddiffusionineukaryotscansignificantlyspeedupthe workinthenucleus,asevidencedontheexamplesofhistons, searchprocess,andthatitenablestominimizethesearchtime high-mobility group proteins, and more generally chromatin bytuningtheaffinityoftheproteinwithDNA.Theseresults binding proteins [20]. This naturally raises the questions of arequalitativelysimilartothecaseofprokaryots,andsuggest determiningwhethertheclassicalfacilitateddiffusionmecha- that facilitated diffusion is a robust mechanism. At the theo- nismcanbeefficientalsointhecomplexnuclearenvironment, reticallevel, thisstudyyieldsasaby-productthecalculation andwhetheritcanbeusedtoregulateandoptimizegeneex- ofthedistributionoftheFPTaveragedoverthestartingpoint pression in eukaryots. This letter presents a first theoretical foraparticlediffusinginafractalstructure,whichremaineda 2 challengeinthefield[23–26]. Alternatively,onecanmakeuseoftherenewalequation[29] Search time distribution for simple diffusion in a fractal whichreadsinLaplacespace Pˆ (s) = Wˆ (s)/Wˆ (s). Us- TS TS TT medium. Wefirstconsideraproteinwhichremainsinanad- ing the symmetry relation W (t)Wstat =W (t)Wstat, the aver- ji i ij j sorbedstateanddiffusesonthechromatin,whichismodelled age over S can be taken and yields an exact expression of by a discretized domain D of volume N, and, following ex- theGFPTdistribution: Φˆ (s)=Wstat/(sWˆ (s)).Takingnext T T TT perimentalobservations[17–19],ischaracterizedbyafractal the large volume limit, the propagator Wˆ (s) can be evalu- TT dimensiondf andatypicalsizeR∝ N1/df. Wecalculatehere atedusingtheO’ShaughnessyandProcacciaformalism[30], the distribution of the search time, defined as the FPT at the whichleadsto target,averagedoverthestartingpositionoftheprotein. This (cid:16)√ (cid:17) firsttechnicalstepisnecessarytoaddresstheproblemoffacil- (cid:32) 4 (cid:33)ν Γ(1+ν) Iν A(cid:104)τT(cid:105)s aitnatiemdpdoirfftaunstiothnedoirsectuicsasledquienstthioenn.eTxhtepparroatgerianpohf;pboessiitdieosn,rit(ti)s, ΦˆT(s)= A(cid:104)τT(cid:105)s Γ(1−ν) × I−ν(cid:16)√A(cid:104)τT(cid:105)s(cid:17), (2) isassumedtoperformasymmetricnearestneighborrandom whereν=d /d ,A=2(1−ν2)/ν,andI (andlaterJ )denote walkonthechromatinwithaconstanthoppingrate(setto1). f w ν ν Besselfunctions.Introducingtherescaledvariableθ=t/(cid:104)τ (cid:105), To account for the complex organization of DNA-DNA con- T onefinallyobtainsbyinverse-Laplacetransforming(2): tactpointsthechromatincannotbedescribedasalinearchain: itiseffectivelybranchedeveniftheDNAislinear,yieldinga Q (θ) = exp(−θ) (d <d ) connectivitypotentiallylargerthan2whichtakesintoaccount T w f intersegmentaltransfer. Theresultingdynamicsischaracter- 22νν Γ(1+ν) (cid:88)∞ J (α ) = α1−2ν ν k oizfedthebymtehaenwsqalukardeimdiesnpsliaocnemdewndtewfiinthedtitmhreo:ug(cid:68)rh2(tth)e(cid:69)∝scat2l/idnwg, (1−ν2)Γ(1−ν) k=0 k J1−ν(αk) asunmdethdetonuacclteaarsmreeflmecbtrianngewwahllisc.hAbosuwnedsptrhoececehdrotmoashtionwis, tahse- × exp−2(1α2k−νν2)θ (dw >df), search time distribution is independent of these microscopic (3) details of the chromatin conformation, and is governed only by its larger scale properties which are characterized by df wheretheαk’saretherealzerosofJ−ν. andd . w WedenotebyW (t)thepropagator,i.e. theprobabilitythat ji theprotein,startingatsiteiatt=0,isatsite jattimet,and generation 6 generation 4 write Wstat for the stationary probability at site j. We will 1 theory j n make use of the pseudo-Green function of the walk defined o bainygteeHrnejesirtei=cdf(cid:82)iutn=∞n0tc(htWieojngi(ltof)b(−ta)lWwFiPsjltalTt)bd(etG,dFaenPndTo)ttehadetbLayagpiflˆva(escn)e.ttaWrragenesatfroseritmheeToref, FPT distributi0.1 a Sierpinski gasket N = 263 embedding cubes which is the FPT at site T averaged over the starting site S G N = 163 embedding cubes with weight Wstat. We thus define the probability density Φ ed 1 theory of the GFPT bSy ΦT(t) = (cid:80)Nj=1WsjtatPTj(t), where PTj is thTe rescal probabilitydensityoftheFPTatT startingfromagivensite j. b Critical percolation cluster 0.1 Generalexpressionshavebeenderivedforthefirstmoment 0 0.5 rescaled tim1e θ 1.5 2 of both the FPT [27] and the GFPT [28] (see also [23–26] forspecificexamples),andmorerecently,thehigherFPTmo- FIG.2: DistributionoftheGFPTfordiffusioninfractalmediumin mentshavebeendeterminedinthelarge-volumelimitN (cid:29)1 thecompactcase. Numericalsimulationsfordifferentsystemsizes [22]. In the case of non-compact exploration (d < d ), it N (symbols)areplottedagainstthetheoreticalpredictionofEq. 3 w f (plainlines). aSierpinskygasket(targetattheapex)andbcritical reads bond percolation cluster embedded in a 3D cubic lattice (average (cid:68)τn (cid:69)=n!(cid:104)τ (cid:105)nHTT −HTS, (1) takenoverrandomtargetsandclusterrealizations). TS T H TT where(cid:104)τ (cid:105) = H /Wstat isthemeanGFPT[28]. Usingnext This very general result, confirmed by numerical simula- that H WTstat = HTTWstTat, deduced from detailed balance, and tionsonvariousfractalsets(seefig.2and[31])showsthatthe ji i ij j (cid:68) (cid:69) GFPT distribution takes a universal form indexed by d and averaging over S, we obtain τn = n!(cid:104)τ (cid:105)n, from which it f T T dw only for any fractal structure, independently of its micro- can be deduced immediately that the GFPT distribution is a scopic details. In particular, it showsthe applicabilityof our simpleexponentialofmean(cid:104)τ (cid:105). T approachtochromatinundertheassumptionthatitisfractalat In the compact case (dw > df) ho(cid:68)weve(cid:69)r, it can be shown leastatsufficientlylargescale. Asillustrativeexamples,sim- thatduetoastrongerdependenceof τn onS [22,27],the ulations were performed on structures such as critical perco- TS averageoverS mustbetakenbeforethelargeN limit,which lationclusters,whichcapturethelargescalepropertiesofnu- makestheasymptoticformof[22]unusableforthispurpose. clearDNAorganization:theyare(i)fractalscharacterizedby 3 well defined dw and df, (ii) naturally embedded in euclidean Intermittent wona ltkh eG SMieFrpPinTs kvia graiastkioetn o wf g. ern. .t o5 λ(316 (6d sisitseos)c wiaittiho tna rfgreetq a.)t tahned a pλe2 x(rebind freq.) spaceand(iii)disordered.Thesefractalscanthereforebeseen 1.6 as minimal models of chromatin beyond the linear chain de- T1.4 P scription. F G Searchtimedistributionforfacilitateddiffusioninafractal an 1.2 e medium. Following the classical picture of facilitated diffu- d m 1 sion, we now consider that the protein can desorb from the ale cphlarsommabteinfowreitrhebraintediλn1g,taondthtehecnhrformeealtyindi(ffseueseFiing.th1e).nIuncltehois- resc00..68 λλλ222 === 000...00000115 firstapproach,weadoptameanfieldtreatmentofthephases 0.4 λ2 = 0.05 offreediffusion:weassumethatthedurationofsuchaphase λ2 = 0.1 is exponentially distributed with mean τ2 = 1/λ2, and fur- 0.2 λλ22 == 01.5 ther suppose that the protein rebinds at a position which is 0 0.1 1 10 100 1000 uniformlydistributedonthechromatin. Wedetermineinthis A τT λ1 paragraphthemeantimenecessaryfortheproteinstartingat a random position on the chromatin to reach a target locus FIG. 3: Mean search time for facilitated diffusion on a Sierpinski on the chromatin for the first time. We denote by (cid:104)T (cid:105) this gasketofgeneration5(N =366)withatargetattheapex.Numerical T mean GFPT for facilitated diffusion, and by F (t) the GFPT simulations(symbols)andtheoreticalprediction(Eq. 7,plainlines) T ofthemeanGFPT,rescaledbyitsvalueforλ = 0,isplottedasa probability density. Using tools developed in the context of √ 1 functionofx = A(cid:104)τ (cid:105)λ forvariousλ . intermittent search strategies [11, 21] , it can be shown that 0 T 1 2 the Laplace transform Fˆ is expressed in terms of the distri- T butionΦˆ oftheGFPTforsimplediffusiononthechromatin, whichisTgivenbyEq.(2): 1.6 λλ22==00..000011 NN==6115469 T1.4 λ2=0.005 N=1776  −1 FP λ2=0.05 N=620 FˆT(s)=ΦˆT(λ1+s)1− (cid:16)11−+ΦˆsT(cid:17)((cid:16)λ11++ss)(cid:17) . (4) mean G1.21 λλλ222===001.. 01 N5 = N 1N=6=922894813 λ1 λ2 aled 0.8 Assuming that diffusion on the chromatin is compact (as in resc0.6 mostexamplesoffractalsembeddedin3Dspace[32])anus- ingexpression(2)forΦˆ finallyyieldsanexplicitexpression 0.4 T oftheLaplacetransformeddistributionoftheGFPT: 0.2 0 FˆT(s)=(s+λ1)(s+λ2)Iν(xs) 0.1 1 10A τT λ1100 1000 (cid:34) Γ(1−ν) (cid:35)−1 (5) × x2νs(s+λ +λ )I (x )+λ λ I (x ) FIG. 4: Mean search time for facilitated diffusion on critical bond 4νΓ(1+ν) s 1 2 −ν s 1 2 ν s percolationclustersembeddedina3Dcubiclatticewitharandomly √ locatedtarget.Numericalsimulations(symbols)andtheoreticalpre- where xs ≡ A(cid:104)τT(cid:105)(s+λ1). Onecancheckthattheclassical diction(Eq. 7,plainlines)ofthemeanGFP√T,rescaledbyitsvalue resultoffacilitateddiffusionona1-dimensionalDNA[11,21] forλ =0,isplottedasafunctionofx = A(cid:104)τ (cid:105)λ forvariousλ 1 0 T 1 2 isrecoveredford = 1andd = 2. ThemeanGFPT(cid:104)T (cid:105)is andsystemsizeN. f w T thenreadilyobtainedbywriting (cid:104)T (cid:105)=−(cid:32)∂FˆT(cid:33) =(cid:32) 1 + 1 (cid:33)1−ΦˆT(λ1), (6) Beforecommentingonthisresult,notethatalimitdistribution T ∂s s=0 λ1 λ2 ΦˆT(λ1) GtimTeofΘth=etG/(cid:104)FTPT(cid:105)c.anDbeenoobtitnaginUed=bys(cid:104)cTon(cid:105)sidetrhiengLtahpelarceescvaalreid- T ∞ T ∞ whichfinallyyieldsthecentralresultofthispaper: able associated to Θ, and writing Gˆ (U)= Fˆ (U/(cid:104)T (cid:105) ), T T T ∞ we find using (5) in the large volume limit Gˆ (U) ∼ (1 + (cid:104)TT(cid:105)=(cid:32)λ11 + λ12(cid:33)4ΓνΓ(1(1−+νν))x02νIνI(−xν0(x)0) −1, (7) GUT)−(1Θ,)w∼hiecxhp(y−ieΘld)s. immediatelythesimpleexpTonentialform Optimalsearchtime. Wenowcommentonpreviousresults with x0 ≡ xs=0. This exact expression of the mean search and focus on the minimization of the mean search time (7) timeforfacilitateddiffusioninafractalmedium,validatedby for facilitated diffusion. First note that (cid:104)T (cid:105) ∝ N in the T ∞ numericalsimulationsonvariousexamplesoffractalswhich largevolumelimit, whichshowsimmediatelythatfacilitated mimic the large scale properties of chromatin (see Figs 3, 4 diffusion if faster than diffusion alone, which would yield a and[31])canbesimplifiedasfollowsinthelargeN limit: search time scaling as Ndw/df [28]. Actually, the search time can be minimized as a function of the desorption rate λ , as (cid:32) 1 1 (cid:33) x2νΓ(1−ν) soonastheadsorptionrateλ islargeenough. Quantitativ1ely, (cid:104)T (cid:105)∼(cid:104)T (cid:105) ≡ + 0 . (8) 2 T T ∞ λ1 λ2 4νΓ(1+ν) thefunction(cid:104)TT(cid:105)exhibitsaminimumvalueforsomeλ1 =λm1in 4 ifthevalueofthederivativeof(cid:104)T (cid:105)withrespecttoλ atλ = site even in the case of eukaryots, under the assumption that T 1 1 0isnegative. Thissetsthefollowingconditiononλ : the chromatin has a fractal organization, which seems veri- 2 fied experimentally. We add that this approach is indepen- 4ν(4−ν2) dentofthemicroscopicstructureoftheDNAandcouldalso λ (cid:62)λmin = , (9) 2 2 (cid:104)τ (cid:105)(5+2ν)(1−ν)2 be valid to some extent in the case of prokaryots, where the T DNA, even if less densely packed than in eukaryots seems whichisinpracticesatisfiedforalargeenoughchromatinvol- tohavearathercompactorganizationsignificantlydeparting ume. Underthiscondition,adirectcalculationshowsthatthe fromalinearchain. Usingtypicalexperimentalvaluesd (cid:39)3 w optimal value λm1in can be expanded in the large volume limit and df (cid:39) 2.5, one finds that the search time is minimized as: for λmin/λ (cid:39) 1/5. This suggests that the adsorption time 1 2 λmin 1−ν Γ2(ν)sin(πν)(cid:32) 2ν2 (cid:33)ν shouldbesignificantlylargerthanthetimeoffreediffusionto 1 (cid:39) − . (10) minimizethesearch, incontrastwiththeclassicalprediction λ ν π (cid:104)τ (cid:105)λ (1+ν)(1−ν)2 2 T 2 [10, 11]. 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