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.

Significance

For over a century, it has been known that mechanical cues influence the shape and volume of cells. The same stimuli influence the chromatin organization inside the cell nucleus. This interplay has important pathological consequences, possibly leading to the reprogramming of healthy cells. Ascertaining the origin of these phenomena is a long-standing question in biological physics. We employ a polymer physics approach to unravel how mechanics and chromatin organization are coupled inside the cell nucleus. Our results explain several experimental observations and enable the study of gene activation driven by deformations in biologically relevant situations.

Introduction

The nucleus is arguably the most important organelle of eukaryotic cells. It contains the genetic information and regulates the sections of DNA that are being actively transcribed into proteins and those that are silent. To fit inside the nucleus, the DNA is wrapped around histone proteins to form the nucleosomes. The resulting structure is called chromatin, and its further self-organization is an intense subject of research (1,2,3,4,5,6,7). It can be found in a more condensed and genetically silent conformation, heterochromatin, or in a less condensed and genetically active conformation, euchromatin, which may undergo microphase separation (8). A specific portion of the genome can be silenced by turning euchromatin into heterochromatin through enzymatic modification of the histone proteins, such as histone deacetylation or histone methylation, or as a result of the deformation of the nucleus. During the interphase, the chromatin permeates most of the nucleus, which can be considered as a semi-dilute polymer solution (1).

Cells in the human body are subject to a variety of mechanical stimuli acting at multiple scales (9). At the single-cell level, mechanical signals guide cell fate and differentiation in stem cells (10) and migration strategies in cancer cells (11). The stiffness of the surrounding matrix alters the force-generating capacity of cancer cells (12), which affects their metastatic potential (13). Collective processes such as wound healing, tumor generation, and tissue homeostasis are related to the physical microenvironment and the mechanical forces applied by individual cells (14). In addition, extracellular mechanical stimuli are subsequently transmitted to the nucleus via the cytoskeleton, leading to nuclear deformation (15).

Deformation of the nucleus modulates nuclear organization, chromatin structure, and, in turn, gene expression (16,17). This can occur when the deformation of the nucleus stretches the chromatin fiber, thereby changing its conformation. The stretching unpacks parts of the heterochromatin, activating the transcription of the genetic information they contain (18,19). Besides direct stretching of the chromatin fiber, experiments have shown that the mechanical environment and physical stimuli are key regulators of gene transcription and expression in various physiological contexts. When cultured on rigid or soft substrates, fibroblasts show different levels of chromatin compaction and extensive changes in chromatin structure and accessibility (20). Nuclear deformations due to confinement (21), substrate stretching (22), fluid flow (23), and migration (24) have been associated with significant changes in chromatin conformation and subsequent changes in gene expression. These findings suggest that mechanical signals of different types may regulate the expression of genes through different pathways (16,17,25). Interestingly, the condensation or decondensation of chromatin, in turn, changes the mechanical properties of the nucleus, leading to a bi-directional feedback loop between nuclear mechanics and chromatin conformation (15).

Investigation of nuclear mechanics at the individual cell level (26,27,28,29) and the tissue level (30) suggest that the nucleus is a soft solid that is significantly stiffer than the cytoplasm. Atomic force microscopy experiments showed that the nucleus displays a power-law spectrum of relaxation times (31), which is similar to the behavior of a critical gel (32). Experiments using custom-made tweezers and isolated nuclei demonstrated that chromatin is the main mechanical element of the nucleus at small and medium strains, while the nuclear envelope is responsible for significant strain hardening at larger strains (33). The authors also found that the stiffness of the nuclei depends on the chromatin condensation level. These results are supported by more recent experiments using microrheology of fluorescent histones, which show that condensation of chromatin induced by cell differentiation slows down the movements of the chromatin and stiffens the nucleus (34). At short length scales, the interphase chromatin behaves similarly to a viscoelastic liquid (34,35,36), with intriguing nonequilibrium effects due to active chromatin remodeling (37,38,39,40,41,42). Finally, loss of heterochromatin and a consequent decrease of chromatin compaction can significantly change the Poisson’s ratio of the nucleus, possibly leading to an auxetic behavior (43,44).

The present work is motivated by two experimental evidences: 1) the mechanical behavior of the cell nucleus depends strongly on the chromatin compaction and can be auxetic in the case of highly decondensed chromatin (43,44) and 2) deformation of the nucleus can trigger condensation of the chromatin, possibly increasing the biochemical markers that correspond to condensed chromatin (20,21,22). These findings suggest that the interplay of biochemical reactions and externally imposed deformations is key to understanding the mechanical response of the nucleus (15). To unravel the coupling between these phenomena, we develop a mechanical model of the cell nucleus that includes biochemical reactions that condense and decondense chromatin.

Materials and methods

We consider the nucleus as a compressible polymer gel (45), which, in the reference state, is spherical (Fig. 1). In this work, we focus on the mechanical role of chromatin in the cell nucleus. To do so, we neglect the nuclear envelope and other nuclear components and only consider chromatin in our model. The position of a point in the reference configuration, ${\mathrm{\Omega }}_0$, is denoted by the vector $\mathit{X}$. We define the displacement field $\mathit{u}\left(\mathit{X}\right)$ that specifies the displacement of a point from its reference position $\mathit{X}$ to its new position $\mathit{x}=\mathit{X}+\mathit{u}\left(\mathit{X}\right)$. The deformation gradient is given by $\mathit{F}=\mathit{I}+\partial \mathit{u}/\partial \mathit{X}$. The reference configuration therefore represents the state in which there are no deformations, $\mathit{F}=\mathit{I}$.

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.

The chromatin represents the polymeric component of the gel, and it can be in two states: euchromatin or heterochromatin. To model the dynamic coexistence of heterochromatin and euchromatin phases of chromatin, we follow previous works on single heteropolymer chains with fluctuating monomer identities (46,47,48,49), which we extend to the case of a polymer gel representing the cell nucleus. We denote with n the chromatin number density in the reference state, and we denote ϕ as the local heterochromatin fraction. The local fraction of euchromatin is $1-\varphi$. Motivated by experimental observations (50,51,52), we assume that the heterochromatin-heterochromatin interactions are attractive while heterochromatin-euchromatin and euchromatin-euchromatin interactions are repulsive.

In what follows, we study the behavior of the nucleus using an approach based on equilibrium thermodynamics, where nonequilibrium effects are introduced as effective contributions to the free energy. We define the free energy density as the sum of a purely elastic contribution and the free energy due to the chromatin mixing as $f=f_{el}+f_{ch}$. We assume a neo-Hookean model for the elastic contribution,

$$f_{el}=\frac{G}{2}\left(\text{Tr}\left(\mathit{C}\right)-2\ln J-3\right)\text{,}$$ (1)

where G denotes the shear modulus, $\mathit{C}$ denotes the right Green-Cauchy deformation tensor $\mathit{C}={\mathit{F}}^T\mathit{F}$, and J is the determinant of the deformation tensor, $J=\det \left(\mathit{F}\right)$, representing the volume of the nucleus compared to that in the reference configuration. Here, we neglect the dependence of the shear modulus on the heterochromatin fraction (34) and use a constant value for G. The free energy density due to the chromatin mixing is given by a Flory-Huggins contribution

$$\frac{f_{ch}}{k_{B}T}=J\left(n_{\mathrm{m}}-\frac{n}{J}\right)\log \left(1-\frac{n}{n_{m}J}\right)-\frac{\chi n^2{\varphi }^2}{J{k}_{B}T}+\frac{{\chi }_{\mathrm{R}}{n}^2}{J{k}_{B}T}\left(1-\varphi \right)+n\left(\varphi \log \varphi +\left(1-\varphi \right)\log \left(1-\varphi \right)\right)-\frac{n\varphi \Delta \mu }{k_{B}T}\text{,}$$ (2)

where $k_B$ is the Boltzmann constant and T is the temperature. A schematic depiction of each term appearing in Eq. 2 is shown in Fig. 1, i–v. The first term on the right-hand side of Eq. 2 represents the free energy density due to the entropy of mixing all the molecules suspended in the nucleoplasm. Its derivative with respect to J, which is proportional to the osmotic pressure, goes to infinity as the local chromatin fraction, given by $n/J$, approaches the maximum packing $n_{\mathrm{m}}$ (27,53). As a result, the local concentration of chromatin can never exceed its maximum packing value $n_{\mathrm{m}}$. The second and third terms on the right-hand side of Eq. 2 follow the classic contribution of the Flory-Huggins theory (54). The second term represents the enthalpic contribution due to the attractive interactions between the segments of chromatin that are in the heterochromatin state. The third term represents the free energy density due to the repulsive interactions between the euchromatin and the heterochromatin and between euchromatin and itself. The fourth term on the right-hand side of Eq. 2 represents the entropy due to all possible microscopic configurations corresponding to a given macroscopic fraction of heterochromatin, ϕ. This term penalizes the deviation of the heterochromatin fraction from $\varphi =0.5$. Intuitively, the number of possible microscopic arrangements of chromatin into heterochromatin or euchromatin is at a maximum when $\varphi =0.5$. Instead, only one arrangement is possible when either $\varphi =0$ or $\varphi =1$ because the chromatin is either fully heterochromatin or fully euchromatin. Finally, the last term on the right-hand side of Eq. 2 represents the contribution due to an effective chemical potential difference, $\Delta \mu$, between the euchromatin and heterochromatin. For $\Delta \mu <0$, formation of euchromatin is favorable, and the opposite occurs for $\Delta \mu >0$. We emphasize that $\Delta \mu$ does not represent the thermodynamic chemical potential difference. We use it to model the nonequilibrium biochemical reactions that control the levels of euchromatin and heterochromatin (55). Here, we consider $\Delta \mu$ constant, but there is evidence that cells can modulate the balance between condensed and decondensed chromatin through the influx or outflux of transcription factors through the nuclear envelope (25,56). These nonequilibrium effects could be included by using an effective chemical potential difference that depends on the volume of the nucleus or that changes in time. The chromatin contribution to the free energy density, given by Eq. 2, displays some similarity to the form proposed by Dormidontova et al. (47) for a single heteropolymer chain with fluctuating monomer identities. The main difference between these two approaches is the difference between the grand canonical ensemble, which is employed in the present work and controlled by chemical potential $\Delta \mu$, and the canonical ensemble, which is used by Dormidontova et al. (47) and controlled by the number of monomers.

To reduce the number of independent parameters, we make f dimensionless using $n_m{k}_{B}T$. The dimensionless free energy density is denoted as $f^{\ast }=f_{el}^{\ast }+f_{ch}^{\ast }$. We find five dimensionless numbers: 1) the dimensionless shear modulus given by $G^{\ast }=G/\left(n_m{k}_{B}T\right)$; 2) and 3) the dimensionless interaction energy, ${\chi }^{\ast }=\chi n_{\mathrm{m}}/k_{B}T$ and ${\chi }_{\mathrm{R}}^{\ast }={\chi }_{\mathrm{R}}{n}_{\mathrm{m}}/k_{B}T$, 4) the chromatin density in the reference state normalized by its maximum density $n^{\ast }=n/n_m$; and 5) the dimensionless effective chemical potential difference $\Delta {\mu }^{\ast }=\Delta \mu /k_{B}T$. The shear modulus, G, of the nucleus is reported to be on the order of $G\approx 100-200\text{Pa}$ (29). An upper value for the maximum packing of chromatin, $n_m$, is given by the close packing density of nucleosomes considered as cylinders of radius $10\text{nm}$ and height $4\text{nm}$, as $n_m\approx 7.2\times {10}^5\mu {\mathrm{m}}^{-3}$. The true maximum density of chromatin is likely larger than our estimate because we ignored the volume of the DNA that links the nucleosomes and the volume already occupied by proteins and macromolecules in the nucleoplasm. Assuming that the repulsive interactions are mainly due to steric repulsion between nucleosomes (57), we can use standard polymer thermodynamics to estimate ${\chi }_{\mathrm{R}}\approx k_{B}T/n_{\mathrm{m}}$ (58). Finally, we expect the effective potential $\Delta \mu$ to be on the order of a few $k_{B}T$. By using the estimates of the physical parameters described above, we fix $G^{\ast }=0.5$ and ${\chi }_{\mathrm{R}}^{\ast }=1$. The dimensionless value of the chromatin density, $n^{\ast }$, in eukaryotic cells could lie between 0.01 and 0.3 (1). Here, we consider $n^{\ast }=0.2$ (59), and we vary the two remaining dimensionless numbers ${\chi }^{\ast }$ and $\Delta {\mu }^{\ast }$.

We use the dimensionless free energy density to analyze the behavior of the nucleus and its response to external deformations. Note that in our model, the deformation of the nucleus and the fraction of heterochromatin, ϕ, are unknown and must be obtained by minimizing the free energy. We anticipate that purely shear deformations that do not change the local volume of an element, $J=1$, only change the elastic part of the free energy that is independent of ϕ and, therefore, do not introduce changes to the fraction of heterochromatin.

Results and discussion

While the model for the mechanical behavior of the nucleus is formulated for spatially variable quantities, here, we study its predictions obtained assuming a homogeneous distribution of chromatin and uniform nuclear deformations: change of volume and uniaxial.

Chromatin condensation changes under controlled volumetric deformation

Here, we look at the changes in heterochromatin fraction, $\varphi \left(J\right)$, as the nuclear volume shrinks or expands. To do so, we need to look for the local maxima and minima of the free energy with respect to ϕ for a fixed value of J. These points satisfy $\partial f^{\ast }/\partial \varphi =0$, which yields

$$2\arctan \left(1-2\varphi \right)+\Delta {\mu }^{\ast }+\frac{{\chi }_R^{\ast }}{J}+\frac{2{\chi }^{\ast }\varphi }{J}=0\text{.}$$ (3)

The solution to Eq. 3 yields the heterochromatin fraction ϕ when the nucleus is constrained to the volume J. The heterochromatin fraction is coupled to the deformation of the nucleus, and it grows as the volume of the nucleus is reduced (Fig. 2, a and b) because nuclear compression increases the chromatin concentration and forming more heterochromatin reduces the total free energy due to their attractive interactions. In the case of $\Delta {\mu }^{\ast }=-2$, the heterochromatin fraction ϕ grows continuously as the volume ratio J is reduced (Fig. 2 a). This implies that a nuclear compression-expansion cycle leads to reversible heterochromatin formation. The chromatin condenses forming heterochromatin during compression and then decondenses it once the nucleus is expanded back to its initial volume. Instead, in the case of $\Delta {\mu }^{\ast }=-5$, different fractions of heterochromatin can coexist for a given J (Fig. 2 b). We find either one stable solution or the coexistence of two stable solutions with an unstable solution. This case represents a nucleus whose chromatin in the euchromatin state is highly favored. The behavior just described implies that the chromatin condensation level, ϕ, can jump from one branch to another if the volume fraction, J, is increased or decreased beyond some threshold values, effectively introducing a hysteresis cycle shown as a shaded area.

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.

We find analytical results for the parameter range where single or multiple solutions to Eq. 3 exist. The critical values of the parameters for which two solutions exist discriminate regions where one or three solutions exist. These points are those for which the left-hand and right-hand sides of Eq. 3 are tangent, which yields the condition $2{\chi }^{\ast }\left(\varphi -{\varphi }^2\right)/J+1=0$. The two sides of Eq. 3 are tangent only if ${\chi }^{\ast }/J>2$, and the two tangent points are given by $\varphi =1/2\left(1\pm \sqrt{{\chi }^{\ast }/J-2J/{\chi }^{\ast }}\right)$. By plugging the expression of the two tangent points into Eq. 3, we obtain two curves that discriminate the regions where multiple heterochromatin fractions coexist from regions where there is a single solution. The multistability region of the parameter space is shown as the gray shaded area in Fig. 2 b. Since, ${\chi }_{\mathrm{R}}^{\ast }$ and J are positive numbers, multistability is possible only if $\Delta {\mu }^{\ast }<-2$. Nuclei of cells that preferentially contain euchromatin, $\Delta {\mu }^{\ast }\ll 0$, could display a hysteresis cycle under volumetric compression and subsequent expansion. Fig. 3 helps rationalize the results presented in Fig. 2, a and b. In the case of $\Delta {\mu }^{\ast }=-2$ and ${\chi }_{\mathrm{R}}^{\ast }=1$, there are no values of ${\chi }^{\ast }$ and J for which multiple stable heterochromatin fractions coexist, which agrees with the curves plotted in Fig. 2 a. Conversely, by changing the value of J in the case of $\Delta {\mu }^{\ast }=-5$ and ${\chi }_{\mathrm{R}}^{\ast }=1$, the heterochromatin fraction, ϕ, first enters and then exits the region of multistability (Fig. 3). This agrees with the results reported in Fig. 2 b showing multistability at intermediate values of J.

Figure 3.

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

Equilibrium volume of the nucleus

The reference configuration of the nucleus, $J=1$, minimizes the elastic part of the free energy $f_{el}$, but in general, it does not minimize the total free energy, which is the sum of the elastic part and the contribution due to the chromatin, $f_{ch}$. Indeed, the chromatin introduces an osmotic pressure, which, at equilibrium, must be balanced by the elastic deformation of the nucleus. This leads to a swollen, $J>1\text{,}$ or shrunken, $J<1$, nucleus compared to its reference configuration. Note that there may be more than one configuration that minimizes the total free energy, possibly leading to the coexistence of multiple equilibrium configurations. In the absence of external loads, the free energy density is characterized by two variables only, J and ϕ, $f^{\ast }=f^{\ast }\left(J,\varphi \right)$. To find all possible equilibrium configurations, we need to find the minima of the free energy with respect to these two variables.

We analyze the free energy profile along a line $\varphi \left(J\right)$ that satisfies $\partial f^{\ast }/\partial \varphi =0$. Local minima and maxima of the free energy must be located along this line. To do so, we proceed as in the previous subsection, and we first compute the extrema, $\varphi \left(J\right)$, which satisfies $\partial f^{\ast }/\partial \varphi =0$. We then plug these values into the expression of the total free energy $f^{\ast }$, which is then given by $f^{\ast }\left(J,\varphi \left(J\right)\right)$ and thus depends on J only.

For $\Delta {\mu }^{\ast }=-2$ and weak attractive interactions, ${\chi }^{\ast }=0.25$, there is one minimum of the free energy for $J_{\text{eq}}\approx 1.4$ (Fig. 4 a). The heterochromatin fraction corresponding to this minimum is ${\varphi }_{\text{eq}}=\varphi \left(J_{\text{eq}}\right)\approx 0.2$. This equilibrium point represents a moderately swollen nucleus with mostly decondensed chromatin. In the case of stronger attractive interactions ${\chi }^{\ast }=1$ and $\Delta {\mu }^{\ast }=-2$, another minima appears at $J_{\text{eq}}\approx n^{\ast }$. The new equilibrium point represents a highly compacted nucleus with condensed chromatin ${\varphi }_{\text{eq}}\approx 1$. These swollen and collapsed configurations can coexist, and the nucleus can jump between them if the external pressure is changed. If the attractive interactions are increased further, ${\chi }^{\ast }=2$, the nucleus displays again a single equilibrium point at $J_{\text{eq}}\approx n^{\ast }$, representing a compacted nucleus.

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, $\varphi =\varphi \left(J_{eq}\right)$.

The situation becomes more complicated for $\Delta \mu \ast =-4$ (Fig. 4 b). In the case of ${\chi }^{\ast }=0.25$, the behavior is qualitatively similar to that shown in Fig. 4 a. However, for ${\chi }^{\ast }=1$ and ${\chi }^{\ast }=3$, the free energy is not a single-valued function of J. We identify three different branches, two stable and one unstable. In the case of ${\chi }^{\ast }=3$, one stable branch extends from $J>1$ to $J\approx 0.6$. This branch has a local minimum for $J_{\text{eq}}\approx 1.4$, which corresponds to a swollen nucleus with mostly heterochromatin. Next, there is an unstable branch that exists for intermediate values of J, approximately between $J\approx 0.6$ and $J\approx 0.8$. While this branch displays a local minimum with respect to J, this point is not a local minimum of the function $f^{\ast }\left(J,\varphi \right)$ but is a saddle point instead. The second derivative of $f^{\ast }$ with respect to ϕ is negative along this branch. Finally, the last branch extends from $J\approx n^{\ast }$ to $J\approx 0.8$ and has a minimum in $J_{\text{eq}}\approx n^{\ast }$, which represents a compacted nucleus with a highly condensed chromatin.

Elastic behavior and Poisson’s ratio for small deformations

Motivated by experiments connecting the chromatin condensation to the mechanical behavior of the nucleus, we use the model to compute the dimensionless bulk modulus, $K^{\ast }$, at the equilibrium points, $J=J_{eq}$, as

$$K^{\ast }=J\frac{{\partial }^2{f}^{\ast }}{\partial J^2}{|}_{J=J_{eq}}\text{.}$$ (4)

From linear elasticity theory, we can compute the Poisson ratio as $\nu =\left(3K^{\ast }-2G^{\ast }\right)/\left(2G^{\ast }+6K^{\ast }\right)$. From the condition $\nu <0$, it follows that the nucleus response to small deformations is auxetic if $G^{\ast }>3/2K^{\ast }$. We repeat the free energy analysis performed in the previous section for different interaction strengths, ${\chi }^{\ast }$, and chemical potential differences, $\Delta {\mu }^{\ast }$. For each couple of these dimensionless numbers, we compute the free energy and find the stable equilibrium points. Then, for each equilibrium point, we evaluate the bulk modulus and the Poisson ratio.

The linear elastic behavior of the nucleus (Fig. 4 c) is divided into four regions: a region with one equilibrium point with a positive Poisson’s ratio, a region with two equilibrium points with a positive Poisson’s ratio, a region with a single auxetic equilibrium point, and a region with two equilibrium points, one of which has a negative Poisson’s ratio. We can link the nuclear mechanical behavior predicted in Fig. 4 c with recent experimental findings, which found that the Poisson’s ratio of the cell nucleus becomes negative upon decondensation of chromatin (43). Decondensing chromatin in our model corresponds to either reducing the magnitude of ${\chi }^{\ast }$ or reducing $\Delta {\mu }^{\ast }$. Reducing the magnitude of ${\chi }^{\ast }$ weakens the attractive interactions between heterochromatin states and thus favors the decondensation of chromatin and the formation of euchromatin. Likewise, changing the effective chemical potential difference $\Delta {\mu }^{\ast }$ toward more negative values moves the equilibrium balance between euchromatin and heterochromatin and promotes the formation of euchromatin. If we take a nucleus that belongs to the region on the top right of Fig. 4 c where $\nu >0$ and decondense its chromatin by reducing the magnitude of ${\chi }^{\ast }$ or reducing $\Delta {\mu }^{\ast }$, it can enter a region where $\nu <0$.

In the analysis above, we considered slow deformations for which the fraction of heterochromatin changes quasi-statically with the imposed deformations to minimize the total free energy. Yamamoto and Schiessel (60) argue that the rate of deformation of the nucleus should be compared with the characteristic time required by biochemical reactions that assemble and disassemble nucleosomes, which can be on the order of minutes. They argue that some experiments are performed under the fast deformation regime and showed that under fast deformations, the nucleus always behaves as an incompressible solid. We show that this is not the case in our model. We model fast deformations around an equilibrium point as deformations for which the fraction of heterochromatin remains frozen, $\varphi ={\varphi }_{\text{eq}}$. We compute the apparent bulk modulus that would be measured under a fast deformation as

$$K_{\text{fast}}^{\ast }=J\frac{{\partial }^2{f}^{\ast }}{\partial J^2}{|}_{J=J_{eq},\varphi ={\varphi }_{\text{eq}}}\text{.}$$ (5)

Under fast deformations (Fig. 4 d), the region where the Poisson’s ratio is negative is smaller than the case of slow deformations (Fig. 4 c). Nevertheless, there is still a large parameter space for which $\nu <0$. Similarly to the case of slow deformations, by reducing the attractive interactions between heterochromatin, ${\chi }^{\ast }$, the linear elastic behavior of the nucleus can transition from a nonauxetic to an auxetic behavior. This finding suggests that decondensing chromatin may lead to an auxetic response regardless of the deformation rate. Choosing values of $n^{\ast }$ or $G^{\ast }$ smaller than those used in Fig. 4 does not qualitatively change the results (Fig. S1).

Uniaxial deformation

In this section, we extend our analysis beyond the linear elastic regime and investigate how the nucleus responds to large uniaxial deformations. In this case, the nucleus displays two principal stretches that are given by ${\lambda }_{\parallel }$, and ${\lambda }_{\perp }$, parallel and perpendicular to the axis of deformation, respectively. The determinant of the deformation gradient, J, is given by $J={\lambda }_{\parallel }{\lambda }_{\perp }^2$, and the trace of the right-Cauchy tensor, $\text{Tr}\left(\mathit{C}\right)$, is given by $\text{Tr}\left(\mathit{C}\right)={\lambda }_{\parallel }^2+2{\lambda }_{\perp }^2$. We fix the stretch imposed, ${\lambda }_{\parallel }$, and we compute the stretch along the transversal direction, ${\lambda }_{\perp }$, and the fraction of heterochromatin, ϕ, by minimizing the free energy. In the absence of deformations, the nucleus is at equilibrium, the stretches are given by ${\lambda }_{\parallel }={\lambda }_{\perp }=J_{\text{eq}}^{\frac{1}{3}}$, and the fraction of heterochromatin is given by $\varphi \left(J_{\text{eq}}\right)$. Note that there can be more than one equilibrium point (see Fig. 4, a–c).

In Fig. 5, a–c, we plot the transversal stretch, ${\lambda }_{\perp }$, as a function of the externally imposed uniaxial stretch, ${\lambda }_{\parallel }$, for $\Delta {\mu }^{\ast }=-1$ and different attraction energies, ${\chi }^{\ast }$. The combination of $\Delta {\mu }^{\ast }$ and ${\chi }^{\ast }$ chosen here corresponds to points both inside and outside the region of auxetic behavior in the linear elastic regime (Fig. 4 c). In the case of auxetic nuclei, the transversal stretch, ${\lambda }_{\perp }$, decreases/increases when the uniaxial stretch, ${\lambda }_{\parallel }$, decreases/increases. Fig. 5, a–c, shows that the nucleus behavior near the equilibrium points follows that predicted by the linear elastic analysis (Fig. 4 c). The curves for ${\chi }^{\ast }=0.4$ and ${\chi }^{\ast }=0.5$ (Fig. 5 a) display a single equilibrium point $J_{\text{eq}}\approx 1$, around which the nucleus behavior is auxetic. Interestingly, the auxetic behavior extends beyond the linear deformation regime. When the nucleus behaves as an auxetic material (solid line), the slope of the curve increases as ${\lambda }_{\parallel }$ is decreased from its equilibrium value (Fig. 5 a). This implies that the auxetic behavior of the nucleus becomes more pronounced for large deformations, in agreement with experimental findings (43). The auxetic behavior persists until a critical value of the uniaxial compressive stretch is reached, beyond which the nucleus behaves as an incompressible solid. In the case of an incompressible solid, volume conservation implies that ${\lambda }_{\perp }\propto {\lambda }_{\parallel }^{-\frac{1}{2}}$.

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.

The curves for ${\chi }^{\ast }=0.65$ and ${\chi }^{\ast }=0.7$ consider the cases of nuclei with two equilibrium points (Fig. 5 b). These nuclei are auxetic near their swollen equilibrium point, $J_{\text{eq}}\approx 1$, and they behave almost as incompressible solids for small deformations around the shrunk state, $J_{\text{eq}}\approx n^{\ast }$. The transversal stretch of a shrunken nucleus differs depending on whether the nucleus is compressed or extended. Under large-amplitude compression, the transversal stretch of the shrunk nucleus increases to keep the volume constant. However, under uniaxial extension, the transversal stretch of the shrunken nucleus first decreases and then increases. This implies that a shrunken nucleus becomes auxetic for large extensional deformations. This finding is consistent with recent experiments (44). Finally, the curve for ${\chi }^{\ast }=1$ corresponds to the case of a shrunken nucleus with $J_{\text{eq}}\approx n^{\ast }$ that behaves essentially as an incompressible solid (Fig. 5 c).

In Fig. 5, d and e, we consider the case of $\Delta {\mu }^{\ast }=-2$. The curves for ${\chi }^{\ast }=0.4$ and ${\chi }^{\ast }=0.5$ display a single equilibrium point corresponding to a swollen nucleus (Fig. 5 d). In these cases, for small deformations, the nucleus displays a positive Poisson’s ratio, in agreement with the linear elastic predictions (Fig. 4 c). Nevertheless, as the nucleus is compressed further, the transversal stretch first increases and then decreases, marking a transition to an auxetic behavior. A nucleus can display positive Poisson’s ratios for small deformations but behave as an auxetic solid at large deformations. As a consequence, the region where the nucleus behaves as an auxetic solid can be larger than that predicted by the linear elastic analysis (Fig. 4 c). Finally, the curves at ${\chi }^{\ast }=1$ (Fig. 5 d) show that there can be two stable coexisting branches. If the nucleus is in the swollen equilibrium, under sufficient uniaxial compression, the transversal stretch can jump to the lower branch. While on the lower branch, the nucleus effectively behaves as an incompressible solid.

Conclusion

Here, we have considered the nucleus as a compressible chromatin polymer gel. Following previous works on heteropolymers with fluctuating monomer identities (46,47,48,49), we assumed that the polymer gel can be in a compact state, heterochromatin, and a less compact state, euchromatin. These two states experience different attractive or repulsive interactions and can be converted to each other through a generic chemical reaction, which models, in an effective way, the different strategies employed by the cell to change the condensation levels of the chromatin. This is different from the previous approaches, which considered a specific mechanism of nucleosome assembly and disassembly (60) or a simplified model (59).

To reduce the number of parameters, we made several simplifications. First, we assumed that the mechanical behavior of the nucleus follows a neo-Hookean constitutive model, which is valid for moderate strains up to 20%. Second, we neglected the viscoelastic behavior of the nucleus and investigated the long-time response that minimizes the free energy. The viscoelastic nature of the nucleus plays a role in the transient response to mechanical stimuli. Third, we considered the mechanical environment and spatial constraints as the only factors able to alter the level of chromatin condensation. We did not include biochemical factors in the model, such as yes-associated protein signaling, although their effect on the chromatin organization could be easily included in the model. Fourth, the model does not include the nuclear envelope, which has been shown to have a strong signature on the mechanical response (61) and is responsible for the stiffening of the nucleus at large strains (33). In addition, the nuclear envelope acts as a potential mechano-sensing mechanism that may indirectly influence the level of chromatin condensation. Finally, we neglected active stresses that drive coherent flows inside the nucleus (37,39,42). These effects are not captured in our model but could be included by considering a power input driven by molecular motors and chemical reactions using the Onsager variational principle (62).

Despite its simplicity, the model makes interesting predictions by changing two parameters only. The fraction of heterochromatin is coupled to the deformation of the nucleus and can change under an imposed compression or expansion. The formation of heterochromatin under cyclic compression and expansion of the nucleus can be reversible or irreversible depending on the parameters and the amplitude of the deformation. In the absence of externally imposed deformations, the nucleus displays a single equilibrium volume or coexistence between two equilibrium volumes. The transition between a simply stable and bistable situation depends on the condensation levels of the chromatin and suggests the existence of critical points around which fluctuations could have long time and spatial correlations (63). Such bistable behavior appears to differ from the intramolecular phase transition predicted for heteropolymers with fluctuating monomer identities (47) because the ensembles considered in these works are different. For small nuclear deformations, we find that the Poisson’s ratio can be negative for a broad range of parameters. Our model predicts that the nucleus can behave as an auxetic solid even if the fraction of heterochromatin remains frozen (60). Under large uniaxial deformations, the model predicts that a swollen nucleus that is not auxetic for small deformations can become auxetic for large deformations. Similarly, a nucleus in the shrunken state, which is essentially incompressible for small deformations, can become auxetic under large uniaxial extensional deformations.

These predictions agree with different experimental observations. Experiments using fibroblasts squeezed between two glass plates and uniaxial extension of a cell monolayer showed the formation of heterochromatin, which was reversible upon the removal of the deformation (21,22). Conversely, recent experiments showed that cells migrating through very narrow constriction, which severely squeezes their nuclei, display irreversible formation of heterochromatin (24). These findings are consistent with our results (Fig. 2, a and b), which show that the formation or loss of heterochromatin can be reversible or irreversible, depending on the magnitude of the externally imposed deformation and the initial chromatin condensation state. The Poisson’s ratio of the nuclei of stem cells changes from positive in their native state to negative in their transition state (43), decreasing as the applied deformation increases. The change of the mechanical behavior of the nucleus between the two types of stem cells depends on the fraction of heterochromatin. Decreasing the fraction of heterochromatin promotes the transition to an auxetic behavior (43,44). These observations are compatible with the predictions of the model (Fig. 4 c). The effect of decondensing the chromatin in the experiments can be realized in the model by either decreasing ${\chi }^{\ast }$ or decreasing $\Delta {\mu }^{\ast }$. By doing so, the linear elastic behavior of the nucleus can change from nonauxetic to auxetic (Fig. 4 c). Our model predicts that the Poisson’s ratio can decrease as the uniaxial compressive deformation is increased (Fig. 5 a), which also agrees with experimental findings (43,44). Experiments have shown that nuclei can behave as an incompressible solid for small deformations but that they can behave as an auxetic solid for larger deformations (44). This feature is also predicted by our model (Fig. 5 d). While good qualitative agreement between the model and the experiments is obtained by changing two parameters only, we note that all of the above-mentioned experiments were carried out using nuclei with an intact nuclear envelope. It is possible that the nuclear envelope, which was neglected in our model, plays a relevant role in the experimental observations because it is linked to the chromatin through the linker of nucleoskeleton and cytoskeleton (LINC) complex. Assessing the importance of the nuclear envelope in the phenomenology described in this work is part of our future endeavors.

In summary, our results demonstrate that the coupling between chromatin conformation changes and deformations is a key factor in determining the mechanical behavior of the cell nucleus. While the distribution of chromatin and the deformations experienced by real nuclei are often inhomogeneous, we restricted our study to spatially uniform deformations and assumed a homogeneous distribution of chromatin. In a spatially extended model, as the nucleus is deformed, different domains of euchromatin will form and disappear at different positions. Therefore, our approach should be considered as, effectively, an average over the entire nucleus. Nevertheless, the model can be readily implemented into finite element libraries and opens the possibility of investigating the interplay of inhomogeneous chromatin conformation changes and deformations during confined cell migration, indentation experiments, and micropipette aspiration.

Author contributions

M.D.C. developed the model and performed the numerical calculations. M.D.C. and M.J.G.-B. wrote the paper, prepared the figures, analyzed the results, and obtained funding.

Editor: Alexandra Zidovska.