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MN and AP conceived and designed the experiments, analyzed the data, and wrote the paper. MN performed the experiments and contributed reagents/materials/analysis tools.

The authors have declared that no competing interests exist.

X chromosome inactivation (XCI) is the phenomenon occurring in female mammals whereby dosage compensation of X-linked genes is obtained by transcriptional silencing of one of their two X chromosomes, randomly chosen during early embryo development. The earliest steps of random X-inactivation, involving counting of the X chromosomes and choice of the active and inactive X, are still not understood. To explain “counting and choice,” the longstanding hypothesis is that a molecular complex, a “blocking factor” (BF), exists. The BF is present in a single copy and can randomly bind to just one X per cell which is protected from inactivation, as the second X is inactivated by default. In such a picture, the missing crucial step is to explain how the molecular complex is self-assembled, why only one is formed, and how it binds only one X. We answer these questions within the framework of a schematic Statistical Physics model, investigated by Monte Carlo computer simulations. We show that a single complex is assembled as a result of a thermodynamic process relying on a phase transition occurring in the system which spontaneously breaks the symmetry between the X's. We discuss, then, the BF interaction with X chromosomes. The thermodynamics of the mechanism that directs the two chromosomes to opposite fates could be, thus, clarified. The insights on the self-assembling and X binding properties of the BF are used to derive a quantitative scenario of biological implications describing current experimental evidences on “counting and choice.”

In diploid cells, most genes are expressed from both alleles; the most notable exception to this rule is X-linked genes in female mammals. During embryo development, one X chromosome, randomly selected, is transcriptionally silenced in female cells, so that the levels of X-derived transcripts are equalized in XX females and XY males [

The complete phenomenon leading to X inactivation involves several steps: counting of X chromosomes in the cell, choice of the inactive X, initiation and spreading of silencing on the designated inactive X, and maintenance of the inactive status through subsequent cell divisions [

On the X, the DNA segment controlling silencing is the X-chromosome–inactivation center (

The basic observations listed above ground the hypothesis that “controlling factors” for counting and choice derive from autosomes and interact with cis-acting regulatory sequences on the X chromosomes. Some models explain counting and random choice in XCI [

The nature of the BF (and its binding site on the X) is still unclear: it might be a unique nuclear component, such as an attachment site on the membrane, though it is mostly assumed [

We study here a Statistical Mechanics quantitative version of the “BF” theory of X inactivation to answer such a question in the light of the above recent experiments. Within the framework of such a model, called the “symmetry breaking” (SB) model [

In our model we consider the relevant proximal portions of the two

A random initial configuration of molecular factors, deriving from autosomes, surrounds two parallel and equally binding X chromosome segments relevant to XCI. Molecules have a reciprocal affinity, _{0}, and can bind each other. As times goes on, particles form clusters (A), and if _{0} is larger than a given threshold value ^{*} (a value of the order of a weak hydrogen bond, see text) clusters coalesce into a single major complex attached to only one of the chromosomes (B).

In our schematic description, we consider a simple geometric configuration where the two X segments are parallel, at a given distance _{0} (of the order of the molecule size), in a volume of linear sizes _{x}_{y}_{z} = ^{3} vertexes, with spacing _{0}. The diffusing factors randomly move from one to a nearest neighbor vertex on such a lattice. On each vertex no more than one particle can be present at a given time (see

In the left picture, no energy barrier has to be crossed for the particle to move to its left neighbor (Δ_{0}; in the right picture Δ_{0}, since the moving particle has two neighbors.

From a Statistical Mechanics point of view, as each molecule interacts with those on its lattice nearest neighbors with an energy, _{0}, the system is characterized by its total energy, i.e., by the following Hamiltonian:
_{0} is the effective interaction energy, the sum is over all nearest neighbors pairs _{i}_{0} is in the range of hydrogen bond energies [_{X}_{X}_{X}_{0}. The present SB model [

We investigated by Monte Carlo (MC) computer simulations [_{0}exp(−Δ_{f}_{i}_{f}_{i}_{0} + _{X}_{0}, the bare reaction kinetic rate, related to the ultimate biochemical nature of the molecular factors and of the surrounding viscous fluid. We use _{0} = 30 s^{−1}, a typical value in biochemical kinetics, which sets the time unit here. The “random walker” model is recovered when _{0} = 0, i.e., in absence of interaction.

We recorded the probability distribution _{0} = 0), after a short transient, two small stable peaks are formed in correspondence with the location of the two chromosomes (x_{l},y_{l}) = (L/2,L/2) and (x_{r},y_{r}) = (3L/2,L/2). In the _{0} = 6 kJ/mole case, at long times it is apparent that one of the early two peaks is going to dominate by far the other (if _{0} = 6 kJ/mole, the interaction between particles leads to the formation of a single major “complex” [

In contrast to the “random walk” case _{0} = 0 (right), if _{0} = 6 kJ/mole (left), particles accumulate after a transient around a single “chromosome” as the region around the other one is depleted.

We stress that only when a precise balance between entropy reduction and energy gain is achieved in the cluster assembling process is a single complex formed, i.e., the symmetry between the X chromosomes is broken. At a given concentration of the particles, _{0}, is above a critical threshold value E^{*}(_{0},c), the region where the symmetry is broken extends broadly [

(A) System “order parameter,” _{l}_{r}_{l}_{r}_{l}_{r}_{0} = 6 kJ/mole and squares to _{0} = 0. If _{0} = 0, the symmetry between the two chromosomes is preserved during the evolution and _{0} = 6 kJ/mole, the symmetry is broken: after a transient,

(B,C) The asymptotic value_{0} are plotted as a function of particle interaction energy, _{0} >(normalized by the thermal energy scale _{0} ∼ 1.6 kT signals the transition to the SB phase (see text).

Now we turn to the dynamics of the complex formation. _{l}_{r}_{l}_{r}_{l}_{r}_{l}_{l}_{l}_{l}_{l}_{l}_{l}_{l}_{0} and volume _{l}^{2}_{z}, centered around the left chromosome; analogously, _{r}_{r}_{r}_{0} = 6 kJ/mole, _{∞} − (_{∞} − _{0})exp(−t / τ_{0}), where _{0} and _{∞} are its initial and final values, and τ_{0} the characteristic time scale of the assembling process. The equilibrium value of _{∞}, and of τ_{0} depends on the three model parameters (_{X}_{0},_{∞}, as a function of _{0} is shown in ^{*} is apparent: both _{∞} and τ_{0} have a drastic change of behaviour at ^{*}. In particular, _{∞}(_{0}) is very close to zero (i.e., no major complex is formed, as particles are evenly distributed is space) for _{0} < ^{*}, while it becomes definitely larger than zero for _{0} > ^{*} (i.e., a single major complex is formed and attached to one X). The behavior of τ_{0} (see _{0}, and has a jump at ^{*}. We also found that the lower the affinity of the chromosomes, _{X}

In the phase where the single complex is formed, τ_{0} can be interpreted as the waiting time to have the majority of particles around only one of the X's, i.e., as the characteristic time scale of XCI. The process which results in the aggregation of a single complex has an early stage where molecules tend to bind “sticky” chromosomes (see _{0}, to complete the aggregation of the single complex increases with the square of the X segments distance (as expected in Brownian-like diffusion processes). This suggests that only when the X's colocalize can the complex be assembled in a time short enough to be useful on the cell time scales. For instance, if the average distance between X's without colocalization is a factor of 10 larger, the assembling time is increased by two orders of magnitude, which is far longer than the cell cycle itself.

Summarizing, in the SB model, the molecular factor interaction induces formation of clusters; if _{0} is above a given threshold, ^{*}, a thermodynamics phase transition occurs and clusters eventually coalesce in a single major “complex” [

We now turn to the role in our model of _{X}_{X}_{Xleft}_{Xright}_{Xleft}_{X}_{Xright}_{Xright}_{X}

The binding of the BF results in a relative abundance of molecular factors around the “left” or “right” X, much larger than expected by a random fluctuation. So, in the phase where the BF is formed, after measuring the average concentration of molecules around the “left”, _{l}_{r}_{l}_{r}_{l}_{r}_{l}_{r}_{u}_{l}_{r}_{u}_{Xleft}_{Xright}

We first discuss the case where Δ_{l}_{r}_{u}_{X}_{0} = 2.4 ^{−2}). Initially, we have _{u}_{0}) = 1 and _{l}_{r}_{u}_{l}_{r}_{l}_{r}_{l}_{r}_{u}_{u}

The case shown here is for _{x}_{0} = 2.4

The long time values of _{u}_{l}_{r}_{X}_{u}_{l}_{r}_{u}_{X}_{u}_{X}_{X}_{X}_{X}_{u}_{u}_{X}

Here _{x}_{0} = 2.4

_{u}_{x}_{l}_{r}_{u}_{u}_{x}

We now discuss the case where a non zero gap, Δ_{u}_{l}_{r}_{X}_{r}_{u}_{Xleft}_{Xright}_{Xleft}_{Xrigh}_{u}_{l}

The SB model describes the self-assembling of a single controlling complex, biologically interpreted as the “BF” in XCI, that attaches to one of the two X segments, designating the active X, as it is the only one to be protected by a dense enough coating of molecules [

We showed that even a small difference in the values of the affinities, _{X}

We now discuss important experiments on “counting” and “choice” which consider deletions on X chromosomes or transgenic insertions into autosomes. In our perspective, a deletion in a segment, including binding sites for the molecular components of the BF, results in the reduction of the chemical affinity of such a sequence for the BF; analogously, the insertion of a similar segment into an autosome results in the possibility of the autosome to bind the BF. So, with reference to the data reported in the previous section, we can interpret, on a quantitative ground, deletion and insertion experiments. We first consider an important deletion which was instrumental in defining the role of the region 3′ to ^{ΔCpG} deletion [

Δ65 kb causes nonrandom inactivation of the deleted X in heterozygous XX cells [^{ΔCp} [

The X chromosome bearing the Δ65 kb deletion is not active in (XY) male cells as well. Importantly, other shorter deletions nested into the Δ65 kb have been described that cause ectopic X inactivation in male cells. The Δ

The effect of these short deletions, unexplained by usual BF models, can be simply understood within the SB model framework, which can explain, as well, their “probabilistic” character, where only a fraction of cells initiate ectopic XCI. As the Y chromosome doesn't bind the BF, in our schematic model it can be described as an “X” with Δ_{X}_{u}_{X}_{X}^{ΔCpG} deleted X remains active [

The analysis of reinsertions into the Δ65 kb deletion, in heterozygous female cells, provides striking evidence that choice can be dissected away from its likely downstream effector mechanism [

In homozygous ^{ΔCpG} XX mutants (i.e., female cells with both X's mutated), the choice of the active X is still random [_{X}

Transgene insertions into autosomes have also been analyzed [

The BF model, focusing on the mechanisms designating the active X chromosome, could represent only some aspects of “counting and choice” in XCI. For instance, some experimental results suggest that cis-acting regulatory sequences in the

The chain of events that follow the binding of the BF has not been determined yet and multiple layers of regulation may contribute to choice. A possible hypothesis is that the BF upregulates

Finally, the SB mechanism shows that there is a typical “time scale” for the protection of the active X to occur, as the supermolecular BF takes some time to bind and grow on a randomly designated chromosome. At intermediate time points, factors accumulate on both chromosomes (see _{0}, to form the final complex rapidly grows with the X segment distance,

Summarizing, in this study we devised a physical mechanism (illustrated via a schematic SB model) for the self-assembling of a single supermolecular complex which spontaneously breaks the binding symmetry of two equivalent targets. This embodies a new stochastic regulatory mechanism resulting from collective behavior at a molecular level.

In the SB model scenario, we explained by quantitative simulations how an X-inactivation theory based on a supermolecular controlling complex can physically work, independently from its ultimate biochemical details. It ascribes the features of random X inactivation to the mechanism of assembly and binding of the BF. In the present view, the BF is a cluster of transacting factors which can bind many a site on a chromosome at the same time and coat, in particular, a region regulating (directly or indirectly)

A comprehensive scenario emerges from our SB schematic model, explaining the variety of experiments using deletions or transgenic insertions to investigate “counting and choice” in XCI. Further evidence supporting our picture is found in recent papers [

The SB regulatory mechanism, discussed here for XCI, relies on a switch that has a thermodynamics origin, a phase transition occurring in the system [

In our simulations, particles start from a random initial configuration in the space (see _{0} up to _{0} in order to check that our results are robust to size changes. We use periodic boundary conditions, and the averages shown below are over up to 1,024 runs from different initial configurations. Monte Carlo step unit is a lattice sweep [

blocking factor

symmetry breaking

X chromosome inactivation

X-chromosome–inactivation center

X inactive-specific transcript