Interplay of chromatin organization and mechanics of the cell nucleus

Abstract

The nucleus of eukaryotic cells is constantly subjected to different kinds of mechanical stimuli, which can impact the organization of chromatin and, subsequently, the expression of genetic information. Experiments from different groups showed that nuclear deformation can lead to transient or permanent condensation or decondensation of chromatin and the mechanical activation of genes, thus altering the transcription of proteins. Changes in chromatin organization, in turn, change the mechanical properties of the nucleus, possibly leading to an auxetic behavior. Here, we model the mechanics of the nucleus as a chemically active polymer gel in which the chromatin can exist in two states: a self-attractive state representing the heterochromatin and a repulsive state representing euchromatin. The model predicts reversible or irreversible changes in chromatin condensation levels upon external deformations of the nucleus. We find an auxetic response for a broad range of parameters under small and large deformations. These results agree with experimental observations and highlight the key role of chromatin organization in the mechanical response of the nucleus.

Figure 1

Schematic representation of the cell nucleus as a compressible polymer gel. The more compacted state of chromatin, heterochromatin, is depicted in green, while the mostly extended configuration of chromatin, euchromatin, is depicted in purple. The fraction of heterochromatin in the deformed configuration might be different from that in the reference state due to the coupling between chemical reactions and deformations. In (i)–(v), we depict schematically the different contributions to the chromatin free energy density, given by Eq. 2. (i) Schematic representation of the maximum packing of chromatin $n_{\mathrm{m}}$; (ii) and (iii) represent the attractive and repulsive forces between the different states of the chromatin; (iv) represents the entropy due to the microstates associated with a given macroscopic heterochromatin fraction ϕ; and (v) represents the effective difference of chemical potential between euchromatin and heterochromatin. Note that the nuclear membrane and nucleolus are not included in the model. Their inclusion in the picture is intended to enhance comprehension of the nucleus’s structural intricacies.

Figure 2

Heterochromatin fraction, ϕ, as a function of the volumetric deformation, quantified by J. Compression or expansion of the nucleus changes the heterochromatin fraction. (a) For $\Delta {\mu }^{\ast }=-2$, the fraction of heterochromatin changes continuously with J. (b) For $\Delta {\mu }^{\ast }=-5$, two stable heterochromatin states (solid lines) can coexist with an unstable state (dashed line). The region of multistability is marked as a shaded region, and the insets depict the fraction of heterochromatin for each stable curve. The fraction of heterochromatin can jump discontinuously between the upper and lower stable branches as J is changed.

Figure 3

The shaded gray area represents the parameter space for which the heterochromatin fraction, ϕ, displays multistability.

Figure 4

(a and b) Dimensionless free energy density $f^{\ast }\left(J,\varphi \left(J\right)\right)$ as a function of J. The cases of (a) $\Delta {\mu }^{\ast }=-2$ and (b) $\Delta {\mu }^{\ast }=-4$. The equilibrium points $f^{\ast }\left(J_{\text{eq}}\right)$ are depicted as red points. The nuclear volume and heterochromatin fraction at the equilibrium points are shown schematically as an inset. (c) Map of the mechanical behavior of the nucleus for small deformations. The different regions divide the parameter space in which the nucleus has a single equilibrium volume from regions with two stable volumes. Those regions are then further classified based on their value of the Poisson’s ratio, ν. The region where the nucleus behaves as an auxetic solid, $\nu <0$, is delimited by the black curve. (d) The same as (c) but for the case of fast deformations for which the heterochromatin fraction remains frozen, $\varphi =\varphi \left(J_{eq}\right)$.

Figure 5

Behavior of the cell nucleus under large uniaxial deformation in the case of (a–c) $\Delta {\mu }^{\ast }=-1$ and (d and e) $\Delta {\mu }^{\ast }=-2$. The curves represent the transverse stretch, ${\lambda }_{\perp }$, as a function of the uniaxial stretch, ${\lambda }_{\parallel }$, which is imposed externally. The equilibrium points are represented as red dots. Solid lines represent regions where the behavior is auxetic. On the top, we represent schematically the deformed configuration (shaded) superimposed to the closest equilibrium configuration (in white). The black dashed line represents a uniaxial deformation of a representative incompressible solid.