System Dynamics.

<p>The average fraction of paired chromosomes, <i>p</i>, is plotted as a function of time, <i>t</i>, for three values of the concentration of molecular factors, <i>c</i> (here <i>E</i> = 1.2 <i>kT</i>), belonging to three different regimes: Brownian <i>c</i> = 0.3%; crossover <i>c</i> = 2.5%; Pairing <i>c</i> = 5%. After an initial diffusive behavior, chromosomes attain their equilibrium pairing state exponentially in time (superimposed fit: <i>p</i>(<i>t</i>)∝[1−exp(−<i>t</i>/<i>τ</i>)]). Inset: The average time scale, <i>τ</i>, to attain the equilibrium pairing state is plotted as function of <i>c</i> (for <i>E</i> = 1.2 <i>kT</i>). <i>τ</i> increases with <i>c</i> because the higher <i>c</i>, the higher is the average number of molecules bound to DNA and, consequently, proportionally lower the <i>Xic</i> diffusion constant. The superimposed fit is a linear function.</p

Binding sites deletions.

<p>The figure shows the pairing fraction, <i>p</i>, in heterozygous deletions, as a function of the remaining fraction, <i>f</i>, of original binding sites. In the ‘Wild Type’case (<i>f</i> = 1) the system is chosen to be in the ‘Pairing phase’(here <i>c</i> = 5%, <i>E</i> = 1.2 <i>kT</i>) and the equilibrium value of the fraction of paired chromosomes is <i>p</i> = 100%. The pairing fraction, <i>p</i>, has a non linear behavior as function of <i>f</i>, with a crossover region around <i>f</i>∼50%. Short deletions, preserving a large fraction of BSs, say, <i>f</i>>70%, have tiny effects on the pairing fraction, while deletions with <i>f</i><30% erase pairing. Inset: The average time, <i>τ</i>, to approach the equilibrium pairing state is plotted as function of <i>f</i>. When <i>f</i> is reduced, <i>τ</i> is shorter, since less MFs are bound to <i>Xic's</i> which, in turn, have an higher effective diffusion constant.</p

Typical equilibrium configurations.

<p>Pictures of typical configurations of our model system at thermodynamic equilibrium (here <i>E</i> = 1.2 <i>kT</i>). (A) Polymers conformation for a value of the concentration of molecular factors (MFs) <i>c</i> = 0.3% (Brownian phase, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi-1000244-g004" target="_blank">Figure 4</a>), (B) for <i>c</i> = 2.5% (crossover region), (C) for <i>c</i> = 5% (Pairing phase). The polymers, representing <i>Xic</i> segments responsible for pairing, are formed by a set of linked beads (not visible because of magnification); green beads are the binding sites (BSs) interacting with the floating molecular factors (MFs, yellow beads). The BSs form a cluster of <i>n</i>₀ = 24 sites, which is of the order of magnitude of the clustered CTCF binding sites found in the <i>Tsix/Xite</i> region (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi-1000244-g001" target="_blank">Figure 1</a>). MFs can bind more than a single BS at the same time, as much as CTCF molecules which have multiple DNA binding domains.</p

Diagram of the <i>Xic</i> region involved in X chromosome pairing.

<p>The location of <i>Xpr </i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi.1000244-Augui1" target="_blank">[6]</a> and <i>Tsix/Xite </i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi.1000244-Xu1" target="_blank">[4]</a>,<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi.1000244-Bacher1" target="_blank">[5]</a>, the regions involved in pairing at the onset of X-Chromosome Inactivation (XCI), is mapped within the X-Inactivation center (<i>Xic</i>). The red line with arrows highlights the area where <i>Xpr</i> has been localized <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi.1000244-Augui1" target="_blank">[6]</a>. The enlargement of the <i>Tsix/Xite</i> region reports the discovered binding sites for CTCF <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi.1000244-Xu2" target="_blank">[7]</a>.</p

Value of Probabilities (from Figure 5) for the BF To Bind the “Left” or “Right” X, <i>p_l</i> (Filled Circles) and <i>p_r</i> (Empty Circles), and To Be Unbound, <i>p_u</i> (Squares), Are Plotted After 20 Hours (i.e., After the Transient Regime from the Initial Configuration) as a Function of the Relative Skewing Energy between the “Chromosomes,” Δ<i>E</i>

<div><p>Here <i>E_x</i> = 4<i>kT</i>, <i>E</i>₀ = 2.4 <i>kT</i>, and <i>c</i> = 0.025.</p><p><b>The inset</b> shows <i>p_u</i> as a function of <i>E_x</i> when Δ<i>E</i> = 0 (here <i>p_l</i> = <i>p_r</i> = (1 − <i>p_u</i>) / 2). Interestingly, <i>p_u</i> is nonzero also for comparatively high values of <i>E_x</i>.</p></div

In Our Lattice Model Particles Interact with Those on Nearest Neighbor Vertexes, via an Effective Energy <i>E</i>₀

<p>In the left picture, no energy barrier has to be crossed for the particle to move to its left neighbor (Δ<i>E</i> = 0). In the central picture, the particle breaks one bond and the barrier is Δ<i>E</i> = <i>E</i>₀; in the right picture Δ<i>E</i> = 2<i>E</i>₀, since the moving particle has two neighbors.</p

Equilibrium state as function of <i>c</i>.

<p>The equilibrium value of the fraction of paired chromosomes, <i>p</i>, is plotted as function of the concentration, <i>c</i>, of binding molecular factors, for a given value of their affinity, <i>E</i> (here, <i>E</i> = 1.2 <i>kT</i>). When the concentration is below a threshold value <i>c</i>^*≃2.3%, no stable pairing is observed (<i>p</i>∼0) and the chromosomes randomly float away from each other (‘Brownian phase’). Above threshold, <i>p</i> saturates to 100%, as a phase transition occurs (to the ‘Pairing phase’) and chromosomes spontaneously colocalize, their driving force being an effective attraction of thermodynamics origin.</p

Phase diagram.

<p>The diagram shows the thermodynamic equilibrium state of the system in the (<i>E</i>, <i>c</i>) plane, for a range of typical biochemical binding energies, <i>E</i>, and concentrations, <i>c</i>. Circles mark the line <i>c</i>^*(<i>E</i>) delimiting the transition from the Brownian phase, where chromosomes diffuse independently, to the Pairing phase, where chromosomes are juxtaposed (the superimposed fit is a power law).</p

Probability Plot of the Blocking Factor Complex to Bind, at Time <i>t</i>, the “Left” and “Right” Chromosome, <i>p_l</i>(<i>t</i>) (Filled Circles), and <i>p_r</i>(<i>t</i>) (Empty Circles), and the Probability To Be Unbound, <i>p_u</i>(<i>t</i>) (Squares)

<p>The case shown here is for <i>E_x</i> = 4<i>kT</i>, Δ<i>E</i> = 0, <i>E</i>₀ = 2.4 <i>kT</i>, and <i>c</i> = 0.025.</p

Distance and collision times distribution.

<p>(A) The distribution of the normalized distance, <i>ND</i> (0<<i>ND</i>≤1), between the two X chromosomes is plotted at two time frames (in the phase where pairing occurs, here <i>c</i> = 5%, <i>E</i> = 1 <i>kT</i>). The initial distribution corresponds to randomly located chromosome positions (<i>t</i> = 0 h); while colocalization progresses a peak in becomes visible and saturates at 48 h. In the inset the corresponding experimental data (from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000244#pcbi.1000244-Xu1" target="_blank">[4]</a>) are reported. (B) The cumulative frequency distribution of ‘paired chromosomes’(i.e., having <i>ND</i><0.1), under the same conditions of (A), is shown. (C) Probability distribution of the time <i>t_collision</i> required by a chromosome to encounter for the first time the other (i.e., to be located within a normalized distance, <i>ND</i>, less than 0.1 from it) with the same values for <i>E</i> and <i>c</i> used in (A) and (B). An exponential behaviour is found (superimposed fit).</p