A Symmetry Breaking Model for X Chromosome Inactivation
aa r X i v : . [ q - b i o . S C ] J a n A Symmetry Breaking Model for X Chromosome Inactivation
Mario Nicodemi a and Antonella Prisco ba Dip. di Scienze Fisiche, Univ. di Napoli ‘Federico II’, INFN, Via Cintia, 80126 Napoli, Italy b CNR Inst. Genet. and Biophys. ‘B. Traverso’, Via P. Castellino 111, 80131 Napoli, Italy
In mammals, dosage compensation of X linked genes in female cells is achieved by inactivationof one of their two X chromosomes which is randomly chosen. The earliest steps in X-inactivation(XCI), namely the mechanism whereby cells count their X chromosomes and choose between twoequivalent X, remain mysterious. Starting from the recent discovery of X chromosome colocalizationat the onset of X-inactivation, we propose a Statistical Mechanics model of XCI, which is investi-gated by computer simulations and checked against experimental data. Our model describes howa ‘blocking factor’ complex is self-assembled and why only one is formed out of many diffusiblemolecules, resulting in a spontaneous symmetry breaking (SB) in the binding to two identical chro-mosomes. These results are used to derive a scenario of biological implications describing all currentexperimental evidences, e.g., the importance of colocalization.
X chromosome inactivation (XCI) is the phenomenonin female mammal early embryo cells by which one oftheir two X chromosomes, randomly chosen, is tran-scriptionally silenced, and epigenetically inherited in de-scendants, to equalize the dosage of X genes productswith respect to males [1, 2, 3]. Crucial aspects of thischromosome-wide stochastic regulatory mechanism, nec-essary to survival, still elude comprehension despite be-ing the focus of substantial interest for their importantscientific and medical implication (see [1, 2, 3] and Ref.stherein). Starting from the important discovery of Xcolocalization during XCI establishment [4, 5], in thispaper we propose a Statistical Mechanics model of theearly steps of XCI.Actually, XCI is a multistep process involving [1, 2, 3]:“counting” the number of the X chromosomes of thecell, “choice” of the inactive X, its silencing and main-tenance. Silencing and maintenance start being under-stood: the former is induced by the action of the
Xist gene transcript, and maintenance of the inactive stateis a paradigm of epigenetic inheritance [3, 6]. Countingand choice are, instead, in many respects still mysteri-ous, though it is well established they are controlled byyet unknown sites located within a 1 Mb region on theX, the X-chromosome-inactivation center (
Xic ), contain-ing several genes and regulators [1, 2], such as the
Xist gene. We also know that cells having a normal numberof autosomes (non sex chromosomes) and extra copies ofthe X chromosome have only one active X, irrespectiveof the number of X’s [1, 2, 7].This biological scenario suggests [1, 2, 3] that “con-trolling factors” for counting and choice derive from au-tosomes and interact with cis-acting regulatory sequenceson the X chromosomes, whose position within the
Xic isstill unknown. Current models postulate the existenceof a “blocking factor” (BF) [1, 2, 3], a complex made ofX and autosomal factors, binding to the
Xic of just onechromosome per diploid cell preventing its inactivation,as the second unprotected
Xic in a female cell is inacti- vated by default. Multiple factors models were proposedas well [8, 9]. These models do not account, though, forthe discovery that colocalization, i.e., a physical proxim-ity, occurs between X chromosomes at the onset of XCI,specifically in the
Xic region, a phenomenon shown to benecessary for XCI [4, 5]. To comprehend the role of
Xic colocalization, a description of the system is demandedthat considers the influence of the spatial configurationon the interaction of the
Xic with the BF.The molecular nature of the blocking factor is itselfstill unknown: it must be a unique entity to performits function, though, several considerations (e.g., degra-dation, over-production problems) exclude the possibil-ity that it is a single protein or RNA molecule. On theother hand, if a diffusible controlling factor is producedin several copies that can statistically reach the target,asymmetric binding to two equivalent chromosomes mustbe explained. While the BF can be envisaged as a uniquecomplex formed by autosomally derived molecules, whyonly one is formed is not understood.We introduce a Statistical Mechanics schematic modelof the diffusible “controlling factors” theory of X inacti-vation and we explain how a supermolecular complex canbe self-assembled and why only one is formed out of manymolecules, resulting in a spontaneous symmetry breaking(SB) in the binding to two identical chromosomal targets.We use, then, our new insights on the “blocking factor”to derive biologically relevant implications and depict acomprehensive scenario of experimental evidences thathighlights the implications of
Xic colocalization with re-spect to the kinetics of XCI.
Our model -
In our model, for simplicity, we includejust the essential components of the process we are inter-ested in (see Fig.1): the two relevant proximal portionsof, say, the
Xic , where the diffusible controlling factorsare assumed to bind, and a portion of space surround-ing them. Such a volume includes an initially randomdistribution of molecular controlling factors originatedby autosomes. These factors are represented by diffus-ing particles having an affinity for their target regions onthe two X chromosomes, as well as a reciprocal affinityamong themselves, as they can form a complex.In our schematic description, the two
Xic segments areparallel, at a given distance L in some units d (of theorder of the unknown molecular factors size), in a volume(a cubic lattice with spacing d ) of linear sizes L x =2 L , L y = L and L z = L around them (see Fig.1). Thediffusing factors randomly move from one to a nearestneighbor vertex on the lattice. On each vertex no morethan one particle can be present at a given time [10].Each particle interacts with its nearest neighbors viaan effective energy E . Below, we mainly discuss thecase where E is of the order of a ‘weak’ hydrogen bondenergy, say 6 kJ/mole, which at room temperature cor-responds to E = 2 . kT [11] (the “random walk” modelis recovered if E = 0). The probability of a particleto move to a neighboring empty site is proportional tothe Arrhenius factor r exp( − ∆ E/kT ), where ∆ E is theenergy change in the move, k the Boltzmann constantand T the temperature [11, 12]. The prefactor r is thereaction kinetic rate (setting the time scale here), de-pending on the nature of the molecular factors and ofthe surrounding viscous fluid (for example, r = 30 sec − is a typical value of biochemical kinetics). Finally, sincethe Xic chromosome segments have an affinity for parti-cles, each lattice site belonging to the chromosomes has abinding energy E X (equal for the two X’s) with particles;for simplicity we take E X = 2 . kT too.The idea we illustrate below is that the molecularfactors interaction, E , induces cluster formation (seeFig.1): when a freely diffusing particle collides with acluster of other particles, it tends to “stick” to them,which produce cluster growth. We show that if E isabove a given threshold, E ∗ (of the order of ‘weak’ hy-drogen bonds), a phase transition occurs and the manyclusters eventually coalesce in a single major “complex”.Interestingly, the time, τ , to form the complex rapidlygrows with the X segments distance, L , explaining theimportant role of X colocalization. Computer Simulations -
We studied by Monte Carlosimulations [13] the dynamics and the final state attainedby the system. The size of our lattice is L = 16 d (wechecked our results for L as large as 128 d ), with peri-odic boundary conditions. Averages are over at least 256runs and time is given in units of Monte Carlo latticesweeps [13]. The fraction of particles per lattice site inthe examples below (see Fig.1) is c = 25 · − .Pictures of the system state at given time slices dur-ing a typical evolution are shown in Fig.1, which com-pares the simple “random walk” model ( E = 0) withthe present model ( E /kT = 2 . E = 2 . kT ,clusters of particles form, which end up in a single big cluster covering only one of the ‘chromosomes’. This phe-nomenon, similar to nucleation where DNA acts as a seed[14], illustrates how the formation of a single “complex”and the spontaneous breaking of the binding symmetrybetween the two X chromosomes can occur. Notice thatif the chromosome affinity tends to zero, E X →
0, a sin-gle cluster of particles is still eventually formed, but itsbinding to the X’s is unlikely.
FIG. 1: We show the evolution of our particle system, aroundtwo equally binding “chromosomes”, when the effective par-ticle interaction energy is E = 2 . kT ( left ) and E = 0( right ), starting from the same initial random configurationat t = 0. When E = 0, a “random walk” diffusion is found.The evolution is drastically different for E = 2 . kT , wheredroplets of particles are formed, ending up in a single clus-ter covering just one of the two equivalent chromosomes and,thus, breaking their binding symmetry. Fig.2 shows the evolution, during the same kind of runof Fig.1, of the average concentration around the chro-mosome on the left, ρ l , and on the right, ρ r ( ρ l = N l /V l where N l is the number of particles in a cylinder, ofradius R = 2 . d and volume V l = πR L z , centeredaround the left chromosome; analogously ρ r = N r /V r ).In the “random walk” case, E = 0, the naive expecta-tion from the symmetry between the two chromosomesthat ρ l ( t ) = ρ r ( t ) is indeed verified at all times, t ; ρ l ( t )and ρ r ( t ) slightly grow up to a value comparable to theinitial random one. In the E /kT = 2 . ρ l and ρ r start from the same initial value, c , but at some pointone of the two has a crisis as particles are all taken in theregion of the other X whose local concentration gets oneorder of magnitude larger than at the beginning.The assembling of a single factor is attained when abalance is achieved between the entropy reduction andenergy gain in the process: for a given concentration ofthe particles, c , only when the interaction energy, E , isabove the phase transition line value E ∗ ( c ) (broken linein the phase diagram of the right inset in Fig.2), a singlemajor complex is formed and, thus, the original bindingsymmetry between the two chromosomes is broken. -4 -3 -2 -1 0 1 2 log(t[h]) ( t )
10 20 30 L / c E . / k T BRKN X-SYMX-SYM. FIG. 2: The average concentration around the chromosomeon the left, ρ l , and on the right, ρ r , of Fig.1 are shown as afunction of time in a logarithmic scale. Circles refer to thecase E = 2 . kT and squares to E = 0 (filled symbols for ρ l , empty symbols for ρ r ). If E = 0, the symmetry betweenthe two chromosomes is preserved during the evolution and ρ l ( t ) = ρ r ( t ). When E /kT = 2 .
4, the symmetry is broken:after a transient, ρ l grows one order of magnitude larger thanat the starting point and ρ r → Right inset
The systemphase diagram, derived by simulations, is shown in the inter-action energy-concentration plane, ( E , c ), where the brokencurve fits the phase transition line. Left inset
We plot thetime, τ , to assemble the BF as a function of the distance L between the two X segments in a box of given size L x = l , L y = l / L z = l / l = 64 d , E X = E = 2 . kT and c = 25 · − ). Specifically, we show the ratio τ ( L ) /τ ,where τ is the reference value for L = 16 d , the distanceconsidered in the rest of the paper. The fit is: τ = a + bL . Discussion -
The SB mechanism we illustrated canexplain why only one X chromosome per cell is active:it is the only one to recruit a sufficient number of con-trolling factors. The time, τ , to complete the assem-bling of the blocking factor complex (BF) can be de-fined from the dynamics of the system “order parame-ter”, m ( t ) = | ρ l ( t ) − ρ r ( t ) | / ( ρ l + ρ r ), which at long timesis approximately exponential: m ( t ) ∝ exp( − t/τ ). AlikeBrownian processes, τ increases as the square of the Xsegment distance, L : τ ∼ L (see left inset of Fig.2).This suggests that only when the X’s colocalize the BF can be assembled in a time short enough to be useful onbiological time scales.Differences in the affinities of the X, E X , induced forinstance by deletions of binding sites on one of them, re-sult in a decisive breaking of the symmetry in particlebinding: the X with less affinity remaining “naked” and,thus, unprotected from inactivation. This explains biasedXCI in embryonic tissues resulting from allelic differencesin Xic sequences, (e.g., in the
Xce locus [15] or other re-gions [1, 2, 3]). XCI in female cells and lack of XCI inmale cells could derive from a similar mechanism. Withrespect to XCI in polyploid cells, in our simulations thehigher is c the larger the probability to have, at interme-diate stages, two clusters, bound to two X, large enoughto act as BF’s. This could describe the stochastic natureof the ‘X chromosome/autosome ratio effect’ [1, 2, 3].The SB model also rationalizes recent important dele-tion experiments across the Xic region, known to affectchoice and counting (see [8, 9, 16, 17, 18, 19, 20] andRef.s therein). We summarize here, in particular, thephenotype of three deletions, which were instrumental indefining the role of the region 3’ to
Xist in counting andchoice, namely ∆65kb[16],
Tsix ∆ CpG [9] and
Xite ∆ L [19].The ∆65kb deletion removes 65kb of DNA in the Xic region relevant to the chromosome activation [16].∆65kb causes non-random inactivation of the deleted Xin heterozygous XX cells[16], and X inactivation in XYcells[18]. The explanation from BF models is that the∆65kb deletion removes the binding sites for the block-ing factor and the complex cannot bind the X any more.Interestingly, the X chromosome bearing the deletion isnot active, not even in male cells.The behavior of male cells is, however, drastically dif-ferent in the case of shorter deletions. The analysis oftwo smaller non-overlapping deletions within the abovementioned ∆65kb sequence, namely the
Tsix ∆ CpG dele-tion, removing the
Tsix promoter, and the
Xite ∆ L mu-tation, removing Xite , added further important informa-tion. In heterozygous XX cells, the
Tsix ∆ CpG deletioncauses non-random inactivation of the deleted X, whereasin XY cells the
Tsix ∆ CpG deleted X remains active[9].The
Xite ∆ L phenotype is analogous [19]. Finally, in ho-mozygous Tsix ∆ CpG
XX mutants the choice of the activeX is still random[21], but, importantly, in a fraction ofcells both X’s are inactivated (“chaotic counting” [8]).As usual single BF models cannot explain these re-sults [8], the simulations with the SB version we pro-pose (summarized in Fig.3) allow a fresh look at the
Tsix ∆ CpG and
Xite ∆ L data. In our model, the block-ing factor is a cluster of transacting factors which canbind several sites on a chromosome at the same time.The Tsix ∆ CpG deletion reduces the total blocking fac-tor/chromosome affinity. In our model, the differencein the affinity, E X , of the wild type and of the deletedchromosome explains, as described before, why choice isskewed in the heterozygous XX cells (see Fig.3A,B). Atvariance with the results from the longer ∆65kb deletion,however, in the case of the smaller Tsix ∆ CpG and
Xite ∆ L deletions, the mutated X remains active in XY cells. Thisis easily understood within the SB model: if binding sitesare found in both the regions deleted by Tsix ∆ CpG and
Xite ∆ L , then each mutation will reduce the affinity ofthe chromosome for the blocking factor, though neitherdeletion will fully erase the overall affinity. Thus, inXY cells the blocking factor can still bind the deletedX chromosome (see Fig.3C,D), since there is no compet-ing wild type X. Random choice in homozygous XX isexplained as before (see Fig.3E). “Chaotic counting” [8]in homozygous deleted mutants derives from an analo-gous mechanism: deletions significantly reduces the totalX-chromosome/particles affinity (see Fig.3F,G) and, in afraction of cells, the blocking factor doesn’t bind at all. XX ia ∆ XX ia s ∆ A) L B) X ∆ i D) YX s ∆ a C) Y L XX i ∆ XX ia ss ∆∆ ss ∆ iiiXX LL ∆∆ iXY G) F)E) H)
Autos.Trans. BF FIG. 3: A pictorial summary of SB model results. Panels
A,B) consider heterozygous XX cells with either a compara-tively long deletion in the
Xic region (X L ∆ ), say ∆65kb[16],or a shorter deletion (X s ∆ ), say Tsix ∆ CpG [9]. The mutatedX, having a reduced overall affinity for the blocking factor(BF), looses on average the competition for it ( skewed inac-tivation ). C,D)
In XY cells, the effects of X s ∆ and X L ∆ canbe quite different: X L ∆ being unable to bind the BF. E,F)
In homozygous XX cells X s ∆ , with a reduced affinity, suc-ceeds in binding the BF only in a fraction of cases ( “chaoticcounting” [8]), whereas G) X L ∆ is unable to bind BF. H) InXY cells, transgenic autosomes with long enough
Xic inser-tions can bind the BF and the X is inactivated in a fractionof cases.
Transgene insertions into autosomes have also been an-alyzed [15, 22, 23]. When long
Xic transgenes are intro-duced, in multiple copies [15], into autosomes of male ES cells, inactivation of the single X occurs in a fraction ofthe cells [22, 23]. In our view the mutated autosomes canbind the BF and compete with the X for it, leaving thereal X chromosome prone to inactivation (see Fig.3H).The simple version of the SB model here discussed con-siders only a single kind of BF complex, though the modelcould easily accommodate more than one (in a Potts-likevariant), as recently proposed in [8]. It does not eitherimply that only one kind of soluble factors is involved.Summarizing, the SB regulatory mechanism we pro-pose describes the self-assembling of the blocking factorfrom diffusible DNA binding molecules and explains whyonly one is formed, i.e., the binding symmetry of thetwo equivalent X’s is broken. The emerging picture ofits properties helps in delineating a scenario of biologi-cal implications reconciling within a single framework theexisting experimental evidences (e.g., X colocalization).In our model “counting” and “choice” are no longer dis-tinct phenomena: they are regulated by the SB stochasticmechanism where time is an important parameter. Moregenerally, the simplicity and robustness of the SB mech-anism, illustrated here for XCI, suggest it can be widelyused in random monoallelic expression processes [24]. [1] Avner, P., and Heard, E. (2001)
Nat. Rev. Genet. , 59.[2] Lucchesi, J.C., Kelly, W.G. and Panning, B. (2005) Annu. Rev. Genet. , 615.[3] Chow, J.C., Yen, Z., Ziesche, S.M., Brown, C.J. (2005) Annu. Rev. Genomics and Hum. Genet. , 69-92.[4] Na Xu, Tsai, C.-L., Lee, J.T. (2006) Science , 1149.[5] Bacher, C.P., et al . (2006)
Nat. Cell Biol. Letters , 293.[6] de Napoles M., et al. (2004) Dev. Cell. , 663-76.[7] Gartler,S.M., Riggs,A.D. (1983) Ann. Rev. Gen. , 155.[8] Lee, J.T. (2005) Science , 768-71.[9] Lee, J.T., Lu, N. (1999)
Cell. , 47-57.[10] This version is a generalization of the Ising model [12].[11] Watson, J.D., et al. “Molecular Biology of the Gene” ,Benjamin Cummings (2003).[12] Stanley, H.E. “Introduction to Phase Transitions andCritical Phenomena” , Clarendon Press (1971).[13] Binder, K. (1997) Rep. Prog. Phys. , 487.[14] Cacciuto, A., et al. (2004) Nature , 404.[15] Heard, E., et al. (1999)
Mol. Cell. Biol. , 3156-3166.[16] Clerc, P., and Avner, P. (1998) Nat. Genet. , 249-53.[17] Clerc, P., Avner, P. (2003) Sem. Cell. Dev. Biol. , 85.[18] Morey, C., et al. (2004) EMBO J. , 594-604.[19] Ogawa, Y., Lee, J.T. (2003) Mol Cell. , 731-43.[20] Morey, C., et al. (2001) Hum. Mol. Genet. , 1403-11.[21] Lee, J.T. (2002) Nat. Genet. , 195-200.[22] Lee, J.T., Strauss, W.M., Dausman, J.A., Jaenisch, R.(1996) Cell. , 83-94.[23] Herzing, L.B., et al. (1997) Nature , 272-5.[24] Singh, N., et al. (2003)
Nat. Genet.33