A thermodynamic switch for chromosome colocalization
aa r X i v : . [ q - b i o . S C ] S e p A thermodynamic switch for chromosomecolocalization
Mario Nicodemi a,b (*), Barbara Panning c ,and Antonella Prisco da Department of Physics and Complexity Science, Universityof Warwick, UK, and b INFN Napoli, Italy c Department of Biochemistry and Biophysics,University of California San Francisco, California, USA d CNR Inst. Genet. and Biophys. ‘Buzzati Traverso’,Via P. Castellino 111, Napoli, ItalyNovember 13, 2018
A general model for the early recognition and colocalization of homologousDNA sequences is proposed. We show, on a thermodynamic ground, how thedistance between two homologous DNA sequences is spontaneously regulated bythe concentration and affinity of diffusible mediators binding them, which actas a switch between two phases corresponding to independence or colocalizationof pairing regions.Chromosome recognition and pairing is a general feature of nuclear organi-zation. In particular, these phenomena have a prominent role (and are compar-atively better studied) in meiosis, the specialized cell division necessary for theproduction of haploid gametes from diploid nuclei. During the prophase of thefirst meiotic division, homologous chromosomes identify each other and pair viaa still mysterious long-distance reciprocal recognition process [1, 2, 3].Many hypotheses exist on the mechanisms underlying the early stages ofcoalignment of homologs along their length (see ref.s in [1, 2, 3]). A long-standing idea is that pairing may occur via unstable interactions, such as adirect physical contact between DNA duplexes (the “kissing model”, see, e.g.,[4]). Pairing initially based on non permanent interactions has the importantadvantage of preventing ectopic association between non-homologous chromo-somes, and avoid topologically unacceptable entanglements, leaving space toadjustments [4]. Several mechanisms could contribute to the outcome of theprocess, e.g., costrained motion of chromosome in territories, bouquet forma-tion at telomeres, tethering to the nuclear envelope. While chromosome fullalignment includes several stages, the early physical contact and colocalizationcould be driven by specific chromosomal regions bridged by molecular media-tors. In this complex scenario, though, the crucial question on the mechanicalorigin of early recognition and pairing remains unexplained.Here we explore the thermodynamic properties of a recognition/pairing mech-anism based on weak, biochemically unstable interactions between specific DNA1equences and molecular mediators binding them. We show that randomly dif-fusing molecules can produce a long-distance interaction mechanism whereby ho-mologous sequences spontaneously recognize and become tethered to each other.This colocalization mechanism is tunable by two “thermodynamic switches”,namely the concentration of molecular mediator and their affinity for their bind-ing sites. When threshold values in the concentration, or affinity, of mediatorsare exceeded, homologous sequences are joined together, else they move inde-pendently.
Model:
Our model includes (see Fig.1) two homologue segments involvedin mutual recognition and pairing, described as a self-avoiding bead chains, awell established model of polymer physics [5], and a concentration, c , of Brow-nian molecular factors having a chemical affinity, E X , for them. We investi-gate the thermodynamics properties of the system by Monte Carlo (MC) com-puter simulations [6]. For computational purposes, chromosomal segments andmolecules are placed in a volume consisting of a cubic lattice with spacing d (our space unit, of the order of the molecular factors length) and linear sizes L x = 2 L , L y = L and L z = L (see Fig.1). In each simulation, the ‘beads’ ofthe chromosomal segments start from a straight, vertical line configuration, ata distance L from each other, and molecular mediators from a random initialdistribution. Diffusing molecules randomly move from one to a nearest neighborvertex on the lattice. On each vertex no more than one particle can be presentat a given time. The chromosomal segments diffuse as well on such a latticeperforming a Brownian motion under the constraint that two proximal ‘beads’on the string must be within a distance √ d from each other (i.e., on next ornearest next neighboring sites on the lattice). For the sake of simplicity, we dis-regard here the rest of the chromosomes and DNA segment ends are costrainedto move tethered to the bottom and top plane of the system volume (Fig.1).When neighboring a chromosomal chain, molecules interact with it via a bind-ing energy E X . Below, we mainly discuss the case where E X is of the orderof a “weak” hydrogen bond-like energy, say 3 kJ/mole, which at room temper-ature corresponds to E X = 1 . kT [7]. In our simulations, at each time unit(corresponding to a MC lattice sweep) the probability of a particle to move to aneighboring empty site is proportional to the Arrhenius factor r exp( − ∆ E/kT ),where ∆ E is the energy barrier in the move, k the Boltzmann constant and T the temperature [7, 8]. The factor r is the reaction kinetic rate, depending onthe nature of the molecular factors and of the surrounding viscous fluid, andsets the time scale. We employ r = 30 sec − , a typical value in biochemicalkinetics. Averages are over up to 2048 runs from different initial configurations. Results:
First we show how the interaction of chromosomes with molecularmediators drives colocalization. To this aim, we calculated the thermodynamicequilibrium value of the average square distance (relative to the system linearsize L ) between the two chromosomal segments: d = 1 N N X z =1 h r ( z ) i L (1)where N is the number of beads in each string (here N = L ) and h r ( z ) i is the average (over MC simulations) of the square distance of the beads at2height’ z . The average value of d is maximal when the two ‘chromosomes’float independently and decreases if parts of the polymers become colocalized,approaching zero when a perfect alignment is attained.The equilibrium distance, d , depends on the concentration, c , of mediators.At low concentration (see Fig.2, e.g., c < c ) d has a value of the order ofthe system size (around 40% of L ), corresponding to the expected averagedistance of two independent strings undergoing Brownian motion in a box ofsize L ; a typical configuration for c = 0 .
3% being shown in Fig.1 panel A).Indeed, the physical basis for the independence of chromosomes exposed to alow concentration of mediating molecules is intuitive: pairing can occur whenbridges are formed by molecules attached to couples of binding sites. A singlebridging event, however, can be statistically quite unlikely since ‘weak’ bonds arebiochemically unstable and to form a bridge a diffusing molecule must first find(and bind) a site on one chromosome and then together they have to successfullyencounter the second one.Fig.2 shows, however, that when c is higher than a threshold value, c tr (for E X = 1 . kT , c tr ≃ . d collapses to zero: this is the sign that thetwo ‘chromosomes’ have colocalized; a typical picture of the system state, for c = 2 . c is high enough chancesincrease to form multiple bridges and, as they reinforce each other, configura-tions where molecules hold together the two polymers become stabilized. Thethreshold concentration value, c tr , corresponds to the point where such a pos-itive mechanisms becomes winning, and can be approximately defined by theinflection point of the curve d ( c ). Alike phase transitions in finite-size systems[6, 8] (see below), around c tr there is a crossover region which can be located,for instance, between the concentrations c and c (see Fig.2) defined by thecriterion that d is close within 5% to the random or zero plateau value (for E X = 1 . kT , c ≃ .
3% and c ≃ d , we plot thesquared fluctuations of the distance (i.e., its statistical variance), ∆ d ( c ), as afunction of the concentration of mediators. For c < c , both d ( c ) and ∆ d ( c )have the non zero value found for non interacting Brownian strings in the in-dependent diffusion regime (∆ d ∼ d ( c ) = 0 for c > c inthe tight colocalization regime. Interestingly, in the crossover region, d ( c ) issmaller than in the purely random regime, although it has marked fluctuations(∆ d ( c ) can be even larger than d ( c )). This situation is illustrated by a pic-ture of a typical configuration, for c = 0 . c isabove a critical value, c tr , i.e., in the ‘colocalization phase’. Conversely, when c is below c tr , d ( c ) has the same value found for two non interacting Brown-ian strings. This is the ‘random phase’, where chromosomes are independent.The concentration of mediators acts as a switch between the two phases, whilearound the critical threshold chromosomes undergo transient interactions.A similar effect is found when, for a given (high enough) concentration, c , thechemical affinity, E X , of binding sites is changed (see Fig.2 lower panel): when E X is smaller than a threshold value, E tr , the two polymers float independentlyone from the other. Around E tr a crossover region is found, and as soon as3 X gets larger than E tr , an effective attraction between polymers is establishedand they are spontaneously colocalized. Another potential layer of regulationof the system is the number of binding sites for molecular mediators. In fact, areduction in the number of binding sites produces the same effect of a reductionin the affinity of mediators, that is, chromosomes become unable to find andbind each other.The pairing mechanisms illustrated above has a thermodynamics origin. Itis a ‘phase transition’ [8] occurring when entropy loss due to polymer colocal-ization is compensated by particle energy gain as they bind both polymers,the lower E X the higher the concentration, c , required. Actually, the transi-tion is found in a broad region of the ( E X , c ) plane, as shown in Fig.3 wherethe system phase diagram is plotted in a range of typical biochemical valuesof “weak” binding energies E X . For very low values of E X the colocalizationcan be, instead, impossible. The overall properties of such a phase diagram(independent v.s. colocalized chromosomes) are robust to changes in the modeldetails, though the precise location of the different phases can be affected [8].Summarizing, when soluble mediators bind a specific recognition sequence onhomologous chromosomes, recognition and colocalization of homologs can oc-cur, as a result of a robust and general thermodynamic phenomenon, namelya phase transition occurring in the system. The higher the affinity of media-tors for chromosomal binding sites, the lower is the threshold concentration ofmediators that promotes colocalization (see Fig.3). Discussion:
We described a general colocalization mechanism, groundedon thermodynamics, whereby specific regions of a pair of chromosomes can spon-taneously recognize each other and align. Physical juxtaposition is mediated bysequence-specific molecular factors that bind DNA via weak, non permanent,biochemical interactions. When the concentration/affinity of molecular media-tors is above a critical threshold an effective attraction between their bindingregions is generated, leading to a close alignment; else chromosomes float awayfrom each other by Brownian motion. In the threshold crossover region, pairingsites undergo transient interactions: the average distance is shorter than in thepurely random regime, but marked fluctuations are observed.In our simulations, the two homologous pairing regions are described aspolymers diffusing with their ends tethered to the upper and lower planes of thesystem box. This recalls telomeres tethering to the nuclear envelope observedat meiosis. While it is not a prerequisite for the switch mechanism, on the otherhand, it can enhance the switch effects [1, 2, 3]. Releasing such a constraintdoesn’t change the general results, but pairing regions would collapse in a moredisordered geometry. The overall properties of the phase diagram (independentvs. colocalized chromosomes) are robust to changes in the model details [9]. Amodel including many a pair of chromosomes has longer equilibration times, asexpected in a crowded environment, yet, its phase diagram is unchanged. Thescenario is also unaltered in the case of mediators that interact with each otherand aggregate.An implication of this model is that a cell can regulate the initiation ofhomologous chromosome interaction by up-regulating the concentration of me-diators or their affinity for DNA sites (e.g., through changes in the chromatinor by a chemical modification of the mediator). This switch has general androbust roots in a thermodynamics phase transition [8], irrespective of ultimate4olecular and biochemical basis. In real cells, specific short chromosomal re-gions (“pairing centers”) could mediate the early steps of homolog recognition,and act as a seed and reference point to a subsequent stable long scale chro-mosomal pairing, which could involve additional mechanisms. A speculation isthat the threshold effect can be exploited to ensure a precise control of pairingformation/release, while the presence of a crossover region in concentration toreduce undesired entanglements. The initial binding molecules could, in turn,help the sequences in recruiting complexes later used to other purposes (e.g., inpairing stabilization, synapsis, recombination).In the present model individual mediators do not need to be strongly bind-ing to glue homologous chromosomes together, and any molecules with abovethreshold affinity can induce attraction. Specificity of colocalization amongmany chromosome pairs could be, indeed, obtained by sets of molecules bind-ing, with higher affinities, specific homologous sequences. While the molecularmediators considered here are supposed to have more than one “DNA bind-ing domain”, proteins that can bind a single DNA site, but are able to makeprotein-protein interactions, could also mediate co-localization. As a pair oflinked proteins is, in fact, a single molecular mediator the thermodynamics pic-ture is unchanged. Finally, direct DNA duplex interactions [4] could replace, orhelp, binding molecules. A duplex kissing site would correspond in our modelto a binding site with a molecular mediator already attached, so the overallbehavior should be similar.Experimental discoveries on meiotic pairing have accomplished huge pro-gresses, but the mechanisms for homologue early coalignment are still unclear[1, 2, 3]. In
C. elegans , for instance, homologs proper pairing is primarily regu-lated by special telomeric regions, known as “pairing centers” (PCs) [10, 11, 12].Homologous PCs interact, during early prophase, with HIM/ZIM Zn-finger pro-teins which are necessary to mediate pairing [13, 14]. Specific sites and proteinsare also involved in meiotic pairing of
Drosophila . In male, on the X and Ychromosomes, a 240bp repeated sequence in the intergenic spacer of rDNA actsas a pairing center, and autosomes pair, as well, by the interaction of a num-ber of sites (see ref.s in [2, 3]). A similar behavior is observed in
Drosophila female [15, 16]. In
Drosophila males, special proteins, SNM and MNM, havebeen also discovered which bind X-Y and autosomal pairing sites at prophaseI, and are required for pairing [17]. The question is open whether the presentmodel applies to such an experimental scenario. In a picture where pairing ismediated by unstable interactions, thermodynamics dictates, anyway, a preciseframework showing that minimal “ingredients”, such as soluble DNA bindingmolecules and homologous arrays of binding sites, can in fact be sufficient forpairing if the balance of mediator concentration and DNA affinity is appropriate.Our thermodynamic switch theory is prone to experimental tests (e.g., theexistence of threshold effects in mediator concentration, c ). It can be exploited,as well, for a quantitative understanding of the effects on pairing, e.g., of dele-tions (which can be modeled here by reducing the binding site number within L ), or of chemical modifications of binding sequences (modeled by changes in E X ), and to guide the search for candidates for chromosomal sites and inter-action mediators. Finally, the general message of the model may be applicableto various cellular processes that involve the spatial reorganization of DNA innuclear space (e.g., organization of chromosomal loci and territories, justappo-sition of DNA sequences in transcriptional regulation, somatic pairing, pairing5f X chromosomes at the onset of X inactivation [1, 2, 3, 18, 19, 20, 21, 22, 23]).We thank N. Kleckner and A. Storlazzi for very helpful discussions andcritical reading of the manuscript. References [1] Zickler, D. & Kleckner N. (1998) The Leptotene-Zygotene transition ofmeiosis. Annu. Rev. Genet. 32, 619.[2] Gerton J.L., Scott Hawley R. (2005) Homologous chromosome interactionsin meiosis: diversity amidst conservation. Nature Rev. Gen. 6, 477.[3] Zickler D. (2006) From early homologue recognition to synaptonemal com-plex formation. Chromosoma 115: 158174[4] Kleckner N. & Weiner B.M. (1993) Potential Advantages of Unstable Inter-actions for Pairing of Chromosomes in Meiotic, Somatic, and PremeioticCells.
Cold Spring Harbor Symposia on Quantitative Biology
Vol. LVIII,553.[5] M. Doi and S.F. Edwards, “The Theory of Polymer Dyn amics” , ClarendonPress. (1984).[6] Binder K. (1997) Applications of Monte Carlo methods to statisticalphysics. Rep. Prog. Phys. 60, 487.[7] Watson J.D., Baker T.A., Bell S.P., Gann A., Levine M., Losick R. (2003) “Molecular Biology of the Gene” , Benjamin Cummings.[8] Stanley H.E. (1971) “Introduction to Phase Transitions and Critical Phe-nomena” , Clarendon Press.[9] Nicodemi M., Prisco A., in preparation.[10] McKim, K.S., Howell, A.M., and Rose, A.M. (1988). The effects of translo-cations on recombination frequency in Caenorhabditis elegans. Genetics120, 9871001.[11] Villeneuve, A.M. (1994). A cis-acting locus that promotes crossing overbetween X chromosomes in Caenorhabditis elegans. Genetics 136, 887902.[12] MacQueen A.J., Phillips C.M., Bhalla N., Villeneuve A.M., Dernburg A.F.(2005) Chromosome sites play dual roles to establish homologous synapsisduring meiosis in
C.Elegans . Cell 123, 1037-1050.[13] Phillips CM, Wong C, Bhalla N, Carlton PM, Weiser P, Meneely PM,Dernburg AF, (2005) HIM-8 binds to the X chromosome pairing centerand mediates chromosome-specific meiotic synapsis. Cell 123:10511063[14] Phillips C.M., Dernburg A.F. (2006) A family of Zinc-Finger proteins isrequired for Chromosome-specific pairing and synapsis during meiosis in
C. Elegans . Devel. Cell 11, 817. 615] Hawley RS, Irick H, Zitron AE, Haddox DA, Lohe A, New C, WhitleyMD, Arbel T, Jang J, McKim K et al (1992) There are two mechanisms ofachiasmate segregation in Drosophila females, one of which requires hete-rochromatic homology. Dev Genet 13:440467[16] Dernburg AF, Sedat JW, Hawley RS (1996) Direct evidence of a role forheterochromatin in meiotic chromosome segregation. Cell 86:135146[17] Thomas S.E., Soltani-Bejnood M., Roth P., Dorn R., Logsdon J.M.Jr., andMcKee B. (2005) Identification of two proteins required for conjunction andregular segregation of achiasmate homologs in
Drosophila male meiosis. Cell123, 555.[18] Na Xu, Tsai C.-L., Lee J.T. (2006) Transient Homologous ChromosomePairing Marks the Onset of X Inactivation. Science 311, 1149.[19] Bacher C.P., Guggiari M., Brors B., Augui S., Clerc P., Avner P., Eils R.,Heard E. (2006) Transient colocalization of X-inactivation centres accom-panies the initiation of X inactivation. Nat. Cell Biol. Letters 8, 293.[20] Nicodemi M., Prisco A. (2007) A Symmetry Breaking Model for X Chro-mosome Inactivation. Phys. Rev. Lett. 98, 108104.[21] Nicodemi M., Prisco A. (2007) Self-assembly and DNA binding of the block-ing factor in X Chromosome Inactivation, PLoS Comp. Bio. 3, 2135-2142.[22] Lanctˆot C., Cheutin T., Cremer M., Cavalli G., Cremer T. (2007) Dynamicgenome architecture in the nuclear space: regulation of gene expression inthree dimensions. Nat. Rev. Gen. 8, 104.[23] Misteli T (2007) Beyond the sequence: cellular organization of genomefunction. Cell 128, 787-800. 7 =0.3% A) c=0.9% B) c=2.5% C) Figure 1: Pictures of typical configurations, from computer simulations, of themodel system at thermodynamic equilibrium, in the two described phases dis-cussed in Fig.2 (panel A , independent motion; panel C , colocalization) and theirintermediate crossover region (panel B ), for the shown values of the concentra-tion of molecular mediators, c (here E X = 1 . kT ).8 .1 0.2 0.5 1.0 2.0 5.0 log(c[%]) d , d [ % ] dd C1 Ctr
C 2 Ex/kT d , d [ % ] dd E1 E tr
E 2 Figure 2:
Top panel
The equilibrium chromosome average square distance, d , is shown as a function of the concentration of binding molecules, c (here themolecule/chrom. affinity is E X = 1 . kT ): for c < c tr ≃ . d approachesvalues as big as the system size and chromosomes are randomly and indepen-dently diffusing (horizontal dotted lines give the values found for pure randomwalks); for c > c tr , d rapidly decays to zero, showing that they have colocal-ized. Around c tr there is a crossover regime, approx. between c and c , wherechromosomes tend to align since d is smaller than in the region where theymove independently, but its fluctuations, ∆ d , are of the order of d ; here chro-mosomes are only transiently colocalizing. Bottom panel
A similar behaviouris found when d is plotted as a function of the chemical affinity, E X , shownhere for c = 0 . .2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Ex/kT l og ( c [ % ]) INDEPENDENTCOLOCALIZED