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

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

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

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

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

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

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

Additional file 2 of Mosaic autosomal aneuploidies are detectable from single-cell RNAseq data

Analysis report. A.html file that details all the analysis included herein, including additional figures referred to in the manuscript as well as some further analyses. (HTML 8315 kb