in the complex mechanisms of the functional regulation of genome , the spatial distribution of dna loci plays a key role .
actually , the genome nuclear architecture deeply affects genes transcription status and can determine the success or   the failure of several cellular processes [ 18 ] .
a vital , nontrivial genome arrangement is the result of a number of interactions between distant dna loci which can pair and/or associate with nuclear scaffolding elements ( e.g. , membrane , matrix , etc . ) .
a wealth of examples can be cited : from the homologous chromosomes recognition and pairing , necessary , for example , during meiosis [ 911 ] , to the clustering of genes around specific nuclear structures , such as the 
longstanding unresolved issues are the mechanisms whereby distant dna loci are able to  cross - talk  in order to achieve the right space configuration , and how the cell is able to organize in space and time such events .   here , we set in a statistical mechanics framework the most recent discoveries about the molecular bases of such  dna cross - talk  phenomena . as a result , we show how it is possible to build a general , thermodynamic - based model , which answers the above questions .
we refer to a well - established biological picture , whereby a dna locus needs the presence of soluble binding molecules to target another dna segment .
starting from these experimental evidences , we analyzed a physics model describing the interaction between a couple of polymers and a concentration c of binding molecules in a lattice .
we showed that these two ingredients  say , a set of soluble molecules able to bind a number of dna sites are sufficient to enable a 
dna cross - talk ,  which eventually results in the colocalization of the dna loci . by means of extensive monte carlo computer simulations
we dissected in fine details how the dna cross - talk is influenced by changes of some parameters , such as the binding molecules concentration , the dna binding sites number , and their relative affinity .
we observed a threshold effect : the dna loci colocalization is possible provided that the above mentioned parameters pass specific threshold values . only in this case
the dna loci stop their brownian diffusion and , as a result of a thermodynamic phase transition , pair off .  
while this rationalizes the experimentally proved importance for dna cross - talks of the molecular factors concentration and of the length of the relative dna attaching regions ( see   and below ) , it suggests some possible regulatory mechanisms by means of well - described cellular strategies , like the upregulation of a dna binding molecules or the modification of chromatin structure .
moreover , our analysis of the dynamics of dna colocalization , which is currently under investigation in several wet experiments , provides some interesting , testable results . in the following , before introducing our model in more details , we briefly report on some of the many experiments , which point out the special role played by precise dna segments and specific dna binding molecules for the  dna cross - talks  .
protein - mediated chromatin loopsa variety of examples are known of chromatin loops , with fundamental regulatory roles , which are the result of molecule - mediated interactions between different dna loci .
a naive t cell can differentiate into a th1 or a th2 cell depending on which locus among the ifng on chromosome 10 or the th2 on chromosome 11 becomes activated [ 14 , 15 ] .
the cell maturation is accompanied by a change in the chromatin architecture within the above mentioned loci . in particular , on th2 cell activation
, the chromatin at th2 locus adopts a trascriptionally active structure consisting of a series of small loops .
these loops are realized thanks to satb1 protein , which is rapidly expressed upon the cell maturation and binds 9 sites within th2 locus .
another example is the occurrence of two chromatin loops at kit gene , which are achieved through the contacts between different dna loci mediated by gata proteins .
the loop able to activate kit is produced by the binding of gata-2 protein to the loci - situated + 5 kb and   114 kb with respect to the kit promoter .
instead , gata-1 intervenes to mediate the interaction between the loci + 58 kb and + 5 kb in order to form the loop which downregulates kit .
so , the control of the active / downregulated status of kit gene , essential for erythropoiesis ( the production of red blood cells ) , is thought to be accomplished through the regulation of the relative expression level of gata-1 and gata-2 proteins .
a variety of examples are known of chromatin loops , with fundamental regulatory roles , which are the result of molecule - mediated interactions between different dna loci .
a naive t cell can differentiate into a th1 or a th2 cell depending on which locus among the ifng on chromosome 10 or the th2 on chromosome 11 becomes activated [ 14 , 15 ] .
the cell maturation is accompanied by a change in the chromatin architecture within the above mentioned loci . in particular , on th2 cell activation ,
the chromatin at th2 locus adopts a trascriptionally active structure consisting of a series of small loops .
these loops are realized thanks to satb1 protein , which is rapidly expressed upon the cell maturation and binds 9 sites within th2 locus .
another example is the occurrence of two chromatin loops at kit gene , which are achieved through the contacts between different dna loci mediated by gata proteins .
the loop able to activate kit is produced by the binding of gata-2 protein to the loci - situated + 5 kb and   114 kb with respect to the kit promoter .
instead , gata-1 intervenes to mediate the interaction between the loci + 58 kb and + 5 kb in order to form the loop which downregulates kit .
so , the control of the active / downregulated status of kit gene , essential for erythropoiesis ( the production of red blood cells ) , is thought to be accomplished through the regulation of the relative expression level of gata-1 and gata-2 proteins . 
					
x chromosomes dynamic arrangement during x - chromosomes inactivationan even more striking event , which entails the organization of two whole chromosomes , happens during x - chromosomes inactivation ( xci).xci is one of the most mysterious aspects in current mammalian x biology .
it is the phenomenon occurring in the female mammalian cells , which leads to the silencing of one of its x chromosomes randomly chosen , to equalize the dosage of x products with respect to males , having only one x [ 1720 ] .
the x chromosomes  cross - talk  and their spatial arrangement is fundamental for the success of xci.at the beginning of process , the xci regulatory sequences on the x 's couple , called xic ( x - inactivation center ) , have to be juxtaposed [ 21 , 22 ] .
the interaction allowing this colocalization is mediated by a protein - rna bridge which includes ctcf protein , able to bind a cluster of about 50 sites in the tsix / xite sequence within xic .
the random choice of the x to be inactivated follows and , then , while the active x moves to the nuclear membrane , the inactive x associates with the nucleolus , to maintain its silenced status .
an even more striking event , which entails the organization of two whole chromosomes , happens during x - chromosomes inactivation ( xci).xci is one of the most mysterious aspects in current mammalian x biology . it is the phenomenon occurring in the female mammalian cells , which leads to the silencing of one of its x chromosomes randomly chosen , to equalize the dosage of x products with respect to males , having only one x [ 1720 ] .
the x chromosomes  cross - talk  and their spatial arrangement is fundamental for the success of xci.at the beginning of process , the xci regulatory sequences on the x 's couple , called xic ( x - inactivation center ) , have to be juxtaposed [ 21 , 22 ] . the interaction allowing this colocalization
is mediated by a protein - rna bridge which includes ctcf protein , able to bind a cluster of about 50 sites in the tsix / xite sequence within xic .
the random choice of the x to be inactivated follows and , then , while the active x moves to the nuclear membrane , the inactive x associates with the nucleolus , to maintain its silenced status .
on the basis of the experimental evidences , we built a schematic model for dna cross - talk , including a couple of dna segments endowed with a number of binding sites ( bss ) for a set of molecular factors ( mfs ) ( see figure 1 ) .   for sake of simplicity , dna segments as well as mfs move in a lattice whose spacing is d0 , of the order of magnitude of the bs length ( see section 4 ) . the lattice dimensions are lx = 2l , ly = l and lz = l in units of d0 .  
they are formed by l nonoverlapping beads , each included in a single lattice site , which randomly diffuse under the constraint that two proximal beads must be on next or nearest next neighboring sites ( nonbreaking constraint ) . in each dna locus , a number n0 of beads are bss .  
mfs are represented by brownianly diffusing beads , and , to a first approximation , only a single mf at a time can be present in a lattice site .
when an mf and a polymer bs are in neighboring sites , they interact via an effective energy e ; mfs can bind a couple of bss from both dna segments at the same time ( in resemblance of several known mediating proteins , e.g. , ctcf ; see figure 1 ) .
the energy function  of the system is the following : 
 ( 1)=er,rnb(r)nm(r ) , 
							 where nb(r)=0,1 is the occupation variable of bss ; it is 1 if the lattice site at position r is occupied by a bs ; otherwise it is 0 ( a site can not be occupied by more than one kind of particle ) .
the sum runs on all nearest neighboring lattice sites r and r. we studied equilibrium and dynamic features of this system by means of monte carlo ( mc ) computer simulations .  
our schematic model does not include all the complex dna regulatory phenomena taking place within the cell nucleus .
however , the cross - talking mechanism we envisaged here is general , since it grounds on thermodynamic bases ( see next sections ) , and for this reason it is not affected by the complexity of the model .
thus , while the approximations we use do not have any influence on cross - talking efficiency , they allow us to explore a wide range of parameters combinations , without the drawback of computational unfeasibility .
as for the parameter values , we chose weak biochemical energy values for e ( ~0/5kt ) , typical for dna - protein interactions [ 2528 ] . the mf concentration , c , ( expressed as the ratio between the mfs number and the lattice sites number ) , spanned the range 0.01/1% , which corresponds to typical nuclear protein concentration ( see section 4 ) .
the binding sites number n0 is 24 ( i.e. , of the order of magnitude of the known case of ctcf binding sites in tsix / xite ) ; however , we also varied it to verify the bss deletion effects ( see figure 4 ) .
the lattice we considered has l = 32 and periodic boundary conditions ( see section 4 for further details ) .
time is given in units of monte carlo lattice sweeps ; details about the conversion from mc time to real time are given in the section 4 .
all the following results are averaged over up to 500 runs . when an mf concentration c is present within the lattice and the polymers have   n0   bss each , an effective attraction is induced between the couple of polymers , whose intensity depends on the value of the mf - bs interaction energy e.   this is shown by the normalized mean square distance between the polymers , defined as 

( 2)d2(t)=1n0z=1n0r2(z , t)l2 , 
							 that is , the square distance between two beads at the same height z , r(z , t ) , averaged over the n0 bss and over different realizations ( as the symbols    indicate ) .
l is used for normalization . in figure 2 the plot of d(t ) with c = 0.2% and n0 = 24 for two values of the energy e is illustrated . at t
= 0 the mfs were randomly placed within the lattice sites , and the polymers were in their straight vertical configuration , at a distance l from each other ( so d(0 ) = 100% ) .
it is evident that the increase of e from 1.4 kt to 2.2 kt determines very different equilibrium states for the system : while at lower energy , d saturates at the value expected from a couple of independent random walk polymers ( d ~ 40% ) ; when e = 2.2 kt , d eventually goes down to   ~0 , thus revealing that the effective attraction is now high enough to make dna loci pair off .  

d(t ) shows an initial linear behaviour , determined by the random diffusion of the polymers when they are far from each other , and a final exponential saturation to a plateau which is dependent , as we saw , on the values of e. a good fit function , which includes both the above mentioned time regimes , is the following ( superimposed fits in figure 2(a ) ) : 
 ( 3)d2(t)=d2()+[d2(0)d2()+at1+bt]et/ , 
							 where d( ) is the final equilibrium value , while a , b ,   and  are fit parameters which depend on e , c ,   and n0 .   to get more insight into the dynamics of this  cross - talk mechanism  and into the key role of the molecular factors , we monitored the time behaviour of the average fraction f of the mfs attached to a single dna locus .
the role of mfs as mediators of the dna loci interaction clearly emerges in figure 2(b ) , where the plot of f(t ) , for the same values of the parameters used above , is shown together with d(t ) fit functions .
for both the energies e , f(t ) starts from the same value , due to the initial random distribution of the mfs in the lattice sites .
when e = 1.4 kt , f(t ) simply saturates in a time scale which is about 4 orders of magnitude smaller than that of d(t ) ; actually , mfs have a size much smaller than polymers , so their dynamics is much faster . if the energy is increased to e = 2.2 kt and the dna loci pair off ( d saturates at   ~0 ) , a more interesting time evolution is found .
in fact , now , two time regimes are easily distinguished : during the first   ~0.5 minute , f(t ) has a behaviour similar to that in the previous case , except for the higher plateau due to the higher mf - bs affinity .
this first time regime corresponds to the initial mf binding to the dna loci which , still out of the action range of the mf - induced effective potential , independently diffuse .
yet , just when d begins to exponentially fall down to   ~0 , f(t ) rises to a second plateau : as soon as dna loci , during their random diffusion , get closer , some mfs start to stably bind both of them , and , as a result , f(t ) increases .
this is evidenced in figure 2(c ) , showing that quite all the bound mfs actually form  bridges  between the dna loci after   ~1 minute .
keep together the two dna loci , so that pairing is finally produced ( see figure 1 ) .
the f(t ) analysis , while allows an immmediate distinction between two dynamical regimes , makes the mf mediating role evident : we saw that colocalization takes place as a result of mf bridges realization , which , however , occurs at thermodynamic equilibrium only for certain values of the parameters . to better understand this issue
, we now carry out an analytic , though approximate calculation for the pairing probability p.   for sake of simplicity , we considered the two dna segments close to each other in their straight configuration . if a random distribution of a concentration c of mfs is presumed , a number cn0 of them will be present , on average , between the n0 bss of the dna loci . for each of these mfs two states
the   energy equal to   2e corresponds to the former case , while the energy is 0 in the latter .
thus , according to the canonical distribution , p = exp(2e / kt)/(1 + exp(2e / kt ) ) and q = 1/(1 + exp(2e / kt ) ) are , respectively , the probability that a single mf bridge is formed or not .  
we can define the pairing probability p as the probability that at least one mf bridge is built between the dna loci . since cn0 is the total number of the possible mf bridges , the following formula holds : 
 ( 4)p=1(q)cn0=11[1+exp  ( 2e / kt)]cn0 
							 according to which p increases with ( c , e , n0 ) .
while this calculation , in its simplicity , helps understanding how the mfs presence induces an effective attraction at thermodynamic equilibrium which is stronger with higher ( c , e , n0 ) , in the following section we seek for the exact thermodynamics of the system by means of mc simulations .
what emerged from mc simulations is that , even if the intensity of the effective potential always increases with ( c , e , n0 ) , the system reacts to a change in one of these parameters only if a threshold value is crossed : below this threshold , dna loci are independent , above they colocalize , as a result of a thermodynamic phase transition [ 13 , 31 ] .
in fact , as we know from thermodynamics law , the equilibrium state of a system must correspond to a free energy minimum : so , the dna loci form a thermodynamic stable couple only when the energy gain coming from the mf  bridges  is able to compensate the entropy loss which results from colocalization .
however , in our finite size system , a narrow intermediate regime , where dna loci couples are unstable , is found as well ; the existence of such a regime is indeed typical for phase transitions in finite system . 

figure 3 shows the thermodynamic equilibrium value of the mean square distance d figure 3(a ) and of the fraction of colocalized dna loci , p ( figure 3(b ) ) , as function of the interaction energy e ( here we fixed c = 0.2% and n0 = 24 , as before ) . while for e  e * = 1.75 the random values for d and p were measured ( d ~ 40% and p ~ 0% , orange horizontal lines ) , as soon as e  e * , d suddenly falls down to 0% ,    and , correspondingly , p increases to 100% ( blue horizontal lines ) . in the crossover region around e * ( defined by the criterion p(e * ) = 50% ) , intermediate values for p and d are found , since dna loci couples are continuously formed and disrupted .  
as we explain at the beginning of this section , in the thermodynamic limit ( i.e. , with infinitely large system ) , e * would mark the transition point between two phases ( a  brownian phase  below it , a  colocalization phase  above ) , and a power - law behaviour would be found around e * , for the order parameters d(e ) and p(e ) . we fitted the quantitative data from the simulations , and we observed that power - law functions are also compatible with p(e ) and d(e ) for e  e * in our finite - sized system ( see dashed gray lines in figures 3(a ) and 3(b ) ) : 
 ( 5)d2(e)=drand2(ee0ed2),p(e)=(ee0ep) , 
							 where drand  40% , e0 = 1.7 kt ,   ed = 1.31 kt , ep = 0.16 kt ,   and  = 0.43 . since
, in a cellular process , the time needed to accomplish each task is also a very important factor , we studied the behaviour of  ( see formula ( 3 ) ) , which represents the characteristic time scale to reach the thermodynamic equilibrium ( and colocalization , above threshold values of ( c , e , n0 ) ) as function of e ( figure 3(c ) ) .
an increasing behaviour is observed due to the slower dna loci diffusion when the mf - bs bonds get stronger .
the  jump  at   e * signals the phase transition occurrence . for fitting we used power - law functions ( superimposed gray lines ) , since this is the expected behaviour for the system relaxation time near a transition point , in the thermodynamic limit .
however , in our finite - sized case , we found that exponential functions can fit the data as well .
we also varied all the three parameters ( c , e , n0 ) , in order to find the 3d phase diagram which is plotted in figure 4 .
the yellow circles mark the transition points between the  brownian  and the  colocalization  phase . it is shown that , as predicted by the approximate calculation of p ( see ( 4 ) )
, the intensity of the effective attraction depends as well on mf concentration c and bs number n0 ; hence , dna loci colocalization is also triggered when threshold values in c and n0 are passed .
a power - law surface fits the data well ( see ): we found cen0  cost with  = 4 and  = 1.1 .
we can conclude that colocalization is possible only if a triplet of parameters ( c , e , n0 ) which is above this surface is given .
we presented a very general mechanism which can account for different phenomena involving spatial chromatin organization .
the mechanism we envisaged can produce the association between two dna loci , without the intervention of any molecular motor able to actively move them ( e.g. , actin / myosin system ) , since the energy needed is provided by the surrounding thermal bath .
the only required ingredients are those emerged in several experiments ( see section 1 ) : say , the presence of diffusive molecules ( molecular factors , mfs ) which are able to bind specific sites ( binding sites , bss ) within the dna loci ( figure 1 ) .
we carried out a detailed dissection of a schematic but quantitative model , both in its thermodynamic and dynamical aspects , by means of extensive monte carlo computer simulations .
we found that the dna loci  cross - talk  and the consequent colocalization are enabled only when the mf concentration c , the bs number n0 ,   and the mf - bs interaction energy e are above specific threshold values , satisfying the relation cen0 = cost (  = 4 and  = 1.1 ) .
one important consequence of this result is that a cell can trigger or release pairing , for example , simply by upregulating / downregulating the mf production ( i.e. , by tuning c ) or by means of chromatin modification ( i.e. , by changing e ) .
moreover , while these threshold effects can explain the results of several experiments ( e.g. , the importance of mediator concentration , see section 1 and [ 13 , 31 ] ) , they are prone to be quantitatively tested , for example , by means of bs deletions . actually , for the first time , we introduced a model which describes such phenomena in a quantitative way : we are able to predict the likelihood of pairing and its probability distribution , once the controlling parameters ( c , e , n0 ) are given .
understanding the pairing dynamics is interesting , especially because several experiments are currently investigating it .
importantly , we measured the time scale  needed to reach full colocalization , as function of the system parameters .
we showed that (e ) is an increasing function ; analogously , it can be shown that  always increases with c and n0 as well , for the same physical reasons ( see above and [ 13 , 33 ] ) .
this results in a nontrivial , counterintuitive prediction : if one ( or more ) of the control parameters ( c , e , n0 ) is reduced ( e.g. , n0 could be reduced by dna deletions , e by chemical manipulations ) at a level still above the threshold for pairing , pairing not only would occur , but it would be speeded up .
we think that the general mechanism we envisaged here , with its thermodynamic robust roots , can apply to lots of cellular processes involving dna loci spatial organization ( e.g. , meiosis , xci , chromosome territories , etc . ) and can mark very relevant progress in their comprehension .
for mc computer simulations we used a lattice with l = 32 , and so with dimensions lx = 64 , ly = 32 ,   and lz = 32 , in units of d0 , the lattice spacing constant . in this lattice , we represented dna segments as directed polymers .
these choices allowed faster simulations , though they do not affect the results we discussed : if such constraints are released , the free energy minimization , above specific thresholds in ( c , e , n0 ) , would determine the dna loci colocalization as well . by referring to the known case of ctcf binding in tsix / xite region on x chromosome [ 22 , 29 ] , we assume that the order of magnitude of d0 corresponds to the linear dimension of 20 bp ( the length of a ctcf binding site ) .
however , we also checked that our results remain essentially unchanged in larger lattices ( l as large as 128 ) and with longer dna loci ( up to 128 beads ) . in the implementation of mc stochastic dynamics
, we considered the probability for a particle to move to an empty next neighboring site , proportional to the arrhenius factor exp(/kt ) ,  being the energy change due to the move , k the boltzmann constant and t the room temperature [ 30 , 32 ] . to measure p , the fraction of colocalized dna loci
colocalized  two dna segments at a distance less than the 10% of the lattice linear size l. a rough calculation can be made to estimate the molecular concentration in real nuclei , corresponding to the mf volume concentrations c. since in our model the number of molecules per unit volume is c / d0 , the molar concentration is  = c/(d0a ) , where a is the avogadro number .
provided that d0  10 nm ( see above ) , we obtain that the typical nuclear protein concentration ,  ~ 1 mol / l , corresponds to c ~ 0.1% .
recent experiments have proved that human dna loci , while at long times , show a constrained motion , brownianly diffuse at short times .
the order of magnitude of their typical short time diffusion constant is d = 1 m / h . for sake of simplicity , we took the diffusion constant of a free polymer ( i.e. , with e = 0 ) in our lattice equal to d ; this gives the conversion factor from mc unit time to real time ( 1 monte carlo lattice sweep  30 microseconds ) .
 the recognition and pairing of specific dna loci , though crucial for a plenty of important cellular processes , are produced by still mysterious physical mechanisms . 
 we propose the first quantitative 
 model from statistical mechanics , able to clarify the interaction allowing such  dna cross - talk  
 events . 
 soluble molecules , which bind some dna recognition sequences , produce an effective attraction 
 between distant dna loci ; if their affinity , their concentration , and the relative dna binding 
 sites number exceed given thresholds , dna colocalization occurs as a result of a thermodynamic 
 phase transition . in this paper , after a concise report on some of the most recent experimental results , 
 we introduce our model and carry out a detailed  in silico  analysis of it , by means of monte 
 carlo simulations . 
 our studies , while rationalize several experimental observations , result in very 
 interesting and testable predictions . 
1. Introduction
2. Results and Discussion
3. Conclusions
4. Materials and Methods