Thermodynamic pathways to genome spatial organization in the cell nucleus

Abstract

The architecture of the eukaryotic genome is characterized by a high degree of spatial organization. Chromosomes occupy preferred territories correlated to their state of activity and, yet, displace their genes to interact with remote sites in complex patterns requiring the orchestration of a huge number of DNA loci and molecular regulators. Far from random, this organization serves crucial functional purposes, but its governing principles remain elusive. By computer simulations of a statistical mechanics model, we show how architectural patterns spontaneously arise from the physical interaction between soluble binding molecules and chromosomes via collective thermodynamics mechanisms. Chromosomes colocalize, loops and territories form, and find their relative positions as stable thermodynamic states. These are selected by thermodynamic switches, which are regulated by concentrations/affinity of soluble mediators and by number/location of their attachment sites along chromosomes. Our thermodynamic switch model of nuclear architecture, thus, explains on quantitative grounds how well-known cell strategies of upregulation of DNA binding proteins or modification of chromatin structure can dynamically shape the organization of the nucleus.

Figure 1

Model. The figure shows two representative snapshots from our three-dimensional computer simulations. In the left panel, a self-avoiding walk (SAW) polymer is shown, as it floats randomly within the assigned volume without forming stable loops. In the right panel the volume also contains a concentration c = 0.04% of Brownian molecules (yellow) having an affinity E_X = 4 kT for a fraction f = 1/3 of the polymers beads (shown in a darker shade). As molecules can bind more than one polymer site, loops can be formed. However, they are stable, and confine the polymer in a closed territory (as in the case shown here), only if c is above a threshold value (see Fig. 2). The SAW chains shown here comprise n = 64 beads.

Figure 2

Thermodynamic switches for intrachromosomal contacts and loop formation. The equilibrium average gyration radius, R_g², of the model polymer pictured in Fig. 1, depends on the affinity, E_X, of its binding sites for a set of molecular factors, on the concentration, c, of those factors, and on the fraction, f, of polymer beads which can bind molecules. R_g represents the radius of a sphere enclosing the polymer: it has a maximum (R_g² = 1 in our normalization) when folding is random and a minimum when the polymer loops on itself in a lump (the horizontal red line is the radius of a compact sphere formed by the polymer). In the left panel, R_g² is shown as a function of E_X, for a given value of c and f (here c = 0.04%, f = 1/3). For E_X below a threshold value, E_tr ≃ 3kT, R_g² is ∼1 and the polymer is on average open. For E_X > E_tr, R_g² collapses, as the polymer forms a looped territory. In the central panel, R_g² is shown as a function of c, for a given E_X and f (here E_X = 4 kT, f = 1/3). In addition, in this case a threshold effect is observed (c_tr ≃ 0.01%), although a broader crossover region exists where the level of folding can be tuned. The right panel shows the sharp threshold of R_g² as a function of f (f_tr ≃ 0.1, here c = 0.04%, E_X = 4 kT), illustrating that only in presence of multiple sites (i.e., above f_tr) the polymer can be folded in loops. In all the above cases, loops are thermodynamically stable only above the threshold values, as a consequence a phase transition occurring in the system. By tuning affinities/concentrations, the cell can act, thus, on a thermodynamic switch to form and release loops and territories.

Figure 3

Scaling properties. (Lower panel) The average gyration radius, R_g, relative to polymer model considered into the left panel of Fig. 2, is plotted as a function of the polymer chain length n. The picture shows the ratio R_g²(n)/R_g²(64) (since n = 64 is the reference case dealt with in the rest of the article) for n = 16, 32, 64, 128. In the phase where the polymer is open, i.e., for E_X = 1kT < E_tr (see left panel of Fig. 2), the average gyration radius, R_g (solid circles), scales with n as a power law R_g ∼ n^ν with an exponent ν ∼ 0.6 (52,53) (superimposed fit, dashed line). In the looped phase, i.e., for E_X = 4 kT > E_tr, R_g (open circles) scales as n^1/3 (superimposed fit, long dashed line), showing that the polymer is lumped in a compact conformation. (Upper panel) The threshold energy, E_tr, relative to the left panel of Fig. 2, is a function of the polymer chain length n. Here we plot the ratio E_tr(n)/E_tr(64) (where E_tr(64) ≃ 3kT). The superimposed fit is $E_{\text{tr}}\left(n\right)=E_{\text{tr}}^{\infty }+A/n^B$, where E_tr(n) is the threshold energy for a chain of size n, $E_{\text{tr}}^{\infty }\sim 0.96E_{\text{tr}}\left(64\right)$, the extrapolated value for an infinitely long system, A ∼ 0.47E_tr(64), and B ∼ 0.5 a fitting coefficient and exponent.

Figure 4

Phase diagram. The state of the polymer/chromosome (see Fig. 1) at thermodynamic equilibrium is summarized by this phase diagram in a range of values of weak biochemical affinities, E_X, and concentration, c, of its binding molecules (here f = 1/3). When E_X and c are below the transition line, E_tr(c) (open circles), the polymer is open (as sketched in the inset) and no stable loops can be formed. Above threshold, instead, the system enters the region where the polymer is folded and looped on itself.

Figure 5

Two snapshots are shown from computer simulations of our two polymer model. In the left panel, the polymers float independently within the assigned volume. In the right panel, the volume also includes a concentration, c = 0.3%, of molecules (yellow particles), which can bind simultaneously each polymer once at any of their specific loci (darker sites, here in a fraction f = 1/2 with affinity E_X = 4 kT). When c is above a threshold value (see Fig. 6), as in the case shown, thermodynamically stable bridges can be formed between the polymers, which spontaneously tend to pair parts of, or all of, their chains.

Figure 6

Thermodynamic switches for interchromosomal contacts. The equilibrium average distance, d², of the two polymer model pictured in Fig. 5, is a function of the affinity, E_X, of their binding sites for diffusing molecules; of the concentration, c, of molecules; and of the fraction, f, of polymer binding sites. In the left panel, d² is plotted as a function of E_X (here c = 0.3%, f = 1/2). When E_X is smaller than a threshold, E_tr ∼ 3.5kT, d² is maximal (d² = 1 in our normalization) and the polymers float independently one from the other. For E_X > E_tr, d² drastically decreases, as the polymers are spontaneously colocalized. In the central panel, d² is shown as a function of c (here E_X = 4 kT, f = 1/2) and a threshold appears as well (c_tr ∼ 0.07%), surrounded by a crossover region. In the right panel, the sharp threshold of d² as a function of f is shown (f_tr ∼ 0.4, here E_X = 4 kT, c = 0.3%): only multiple binding sites, above f_tr, can achieve polymer colocalization. The mechanism driving polymer colocalization is an effective reciprocal attraction of thermodynamic origin, related to a phase transition: below threshold, molecules bridging by chance the polymers do not succeed in holding them in place; above threshold, bridges are thermodynamically stabilized. Molecular mediators act, then, as a thermodynamic switch to spontaneous formation and release of polymer stable contacts.

Figure 7

Territorial maps. The relative positions of three polymers can be regulated by the concentration of specific molecular factors. (Inset) A configuration is shown from our computer simulations of a three polymer model. A specific molecular factor can bind polymers 1 (pink) and 2 (blue), while a different factor binds polymers 2 and 3 (orange). Both molecular factors have here a concentration c = 0.13% (E_X = 4 kT, f = 1/2), but they are not shown for clarity. (Main panel) The average distance between polymers 1 and 2, $d_{12}^2$ (squares), decreases as a function of c (the distance between 2 and 3 equals $d_{12}^2$, and is not shown). As an indirect effect of the attraction within pairs 1-2 and 2-3, the distance between 1 and 3, $d_{13}^2$ (diamond), decreases as well, remaining, however, above $d_{12}^2$. The three polymers, thus, tend to form a triangle with two short equal edges (corresponding to d₁₂ and d₂₃) and a longer edge (i.e., d₁₃). In general, by tuning c, E_X, and f, a variety of configurational patterns can be spontaneously attained. Notably, since polymers 1 and 3 compete for bridging the sites of polymer 2, they physically interfere and $d_{12}^2$ is larger than in the case of an isolated couple (yellow lower line, from Fig. 6). A proper spatial organization of chromosomes in territories and within territories could also help in minimizing physical interference and entanglement.

Figure 8

Schematic illustration of thermodynamic switches and their effects at different levels of system organization. (Top panel) The assembly of intrachromosomal contacts and loops is thermodynamically possible only when the concentration/affinity of binders (circles) exceeds precise threshold values. At that point, previously randomly and independently diffusing molecules and chromosomes spontaneously generate an organized pattern, in a process reversible by downregulation of the switch. Specific conformations can be attained by site specificity of a set of molecular mediators. (Bottom panel) Similar threshold and self-organization mechanisms act for establishing contact between remote loci and, at a higher scale, relative positions of territories. A variety of patterns, encoded in the location of a number of binding sites along chromosomes, can be precisely selected via thermodynamics effects by a combinatorial use of a set of molecular mediators (rectangles).