mESC culture.

WT mESCs (Bruce 4 C57BL/6J, male, EMD Millipore, Billerica, MA) and later derivatives were cultured under 2i conditions (StemSure D-MEM [Wako Pure Chemicals, Osaka, Japan], 15% of fetal bovine serum, 0.1 mM β-mercaptoethanol, 1 × MEM nonessential amino acids [Wako Pure Chemicals], a 2 mM L-alanyl-L-glutamine solution [Wako Pure Chemicals], 1000 U/mL LIF [Wako Pure Chemicals], 20 mg/mL gentamicin [Wako Pure Chemicals], 3 mM CHIR99021, and 1 mM PD0325901) in a 0.1% gelatin-coated dish.

Lbr/LamA DKO cell lines.

To disrupt Lama and Lbr genes simultaneously, we applied a CRISPR-Cas9 system [15,16]. In particular, we used Cas9 nickase that is reported to have fewer off-target effects [17]. To increase the gene disruption rate and simplify detection of the deletion mutation, we designed two sets of sgRNAs (S1A Fig). Next, 5 × 10⁴ WT mESCs were seeded in a gelatin-coated 24-well plate and cultured for 12 h. The cells were transfected with 90 ng each of pKLV-EF-LMNA-1L, -1R, -2L, -2R, and pKLV-EF-LBR-1L, -1R, -2L and -2R and 300 ng of pBSKDB-CBh-Cas9n-pA using Lipofectamine 2000 (Life Technologies, Gaithersburg, MD). After 4 h, the medium containing Lipofectamine was replaced with a fresh regular medium. At 12 h post-transfection, the cells were treated with puromycin (2 μg/mL) for 36 h. Then, the medium was replaced with a fresh medium without puromycin and stood incubating for 48 h. After that, the cells were trypsinized and seeded in a gelatin-coated 10-cm culture dish. Seven days later, we picked 48 colonies and briefly performed PCR to detect gene disruption using specific primers (Lmna-F-gPCR and Lmna-F-gPCR for Lmna; Lbr-F-gPCR and Lbr-R-gPC for Lbr, S2 Table). Next, some of the candidate clones were subjected to immunofluorescence assays using anti-LBR, anti-LamA/C, and anti-lamin B1 antibodies (Fig 1B and S1B Fig). We obtained four clones (DKO-4, -33, -36, and -48) without LamA/C and LBR expression (S1B Fig). To examine the nucleotide sequence changes in these cells, we first performed southern blots using Lmna, Lbr, and Amp probes as previously described [47]. Probes for Lmna and Lbr were prepared by means of universal M13 primers, the PCR DIG Probe Synthesis Kit (Roche Diagnostics, Mannheim, Germany) and pBSK-Lmna-probe and pBSK-Lbr-probe, respectively (S2 Table). For LamA and Amp detection, genomic DNA was digested with SpeI. For Lbr detection, genomic DNA was digested with BmtI and BstEII (S1C Fig). Furthermore, we determined genomic DNA sequence around CRISPR target sites of LamA and Lbr in clones DKO-4 and -36, in which both genes were expected to be inactivated (S1D and S1E Fig). In this study, we mainly used clone DKO-4 as representative of the DKO cell lines unless otherwise indicated. We confirmed that the number of CC clusters steadily decreases during differentiation process in DKO-36 cells. The average number of CC clusters in DKO-36 mESCs, neural stem cells (NSCs) and neurons at 16 days post-differentiation were 20.3 ± 6.69, 7.96 ± 2.87 and 4.49 ± 3.00, respectively. DKO-4 mESCs expressed pluripotency markers, Nanog and Oct4 (S2 Fig), but not NSC and neuron markers (S3 Fig and S4 Fig).

Differentiation into postmitotic neurons.

NSC differentiation was induced using a protocol adapted from Conti et al. [48]. Briefly, mESCs were trypsinized and resuspended in the N2B27 medium [1:1 mix of DMEM/F12 (Thermo Fisher Scientific) supplemented with N2 (Life Technologies) and Neurobasal medium (Thermo Fisher Scientific) supplemented with B27 (Life Technologies)]. Next, 2 × 10⁵ cells were seeded in gelatin-coated 6-cm plates. At 5 days post-differentiation, the cells were resuspended in the N2 expansion medium [49] and plated in uncoated T-75 flasks to allow for neurosphere outgrowth. Cell aggregates were allowed to settle under the force of gravity in a 15-mL tube for 10 min. The cells were seeded in 6 mL of a fresh NS expansion medium in poly-ornithine/laminin-coated 6-cm plates. Following outgrowth of cells (3–7 days), the entire population was trypsinized and seeded as single cells in poly-ornithine/laminin-coated 6-cm plates in the expansion medium. After several passages, the cultures were uniformly positive for an NSC marker, Nestin (S2 Fig to S5 Fig).

For neuronal differentiation, NSCs were trypsinized, and 5 × 10³ cells were seeded in each well of a poly-ornithine/laminin-coated 8μ-slide (ibidi, Martinsried, Germany) in the N2B27 medium supplemented with FGF-2 (5 ng/mL) (neural induction medium). Half of the medium was replaced every 2–3 days to maintain conditioning of the medium. After the cells spent 6 days under these conditions, we replaced the media with a 50:50 mixture of DMEM/F12 and Neurobasal medium (Thermo Fisher Scientific, Waltham, MA) supplemented with 0.25× N2 plus 1× B27 and without EGF or FGF. At 9 days postdifferentiation, the cultures were positive for a neuron marker, βTub III (S2 Fig to S5 Fig).

Mice.

The use of animals in the present study was in accordance with the Guidelines for Animal Experiments of Nagoya University. All animal experiments were conducted with the approval of the Animal Research Committee in Nagoya University. C57BL/6J mice were purchased from CLEA (Shizuoka, Japan), and maintained under a cycle of 12-hour light and 12-hour dark with free access to food and water. For ex vivo imaging, 15-days-old C57BL/6J mice were used.

Model development.

We extended the model from a previous study [12]. The key point of our modeling is to describe the dynamics of subnuclear compartments by linking them to domains using phase-field functions (Fig 2A). We first designed each domain by means of energy functions defined by the nucleus, ϕ₀(x,t), chromosome territories, ϕ_m(x,t) (1≤m≤N), heterochromatin, ψ(x,t) and chromocenters (CCs), ψ₀(x,t), where x∈Ω in Rⁿ and t>0. N represents the total number of chromosomes in the nucleus, and we typically chose N = 8 for representative simulations.

We first defined the basal free energy functions using Ginzburg-Landau free energy [50,51] for the nucleus, chromosome territories, heterochromatin and CCs according to the following equation: where are gradient coefficients.

Second, we formed subnuclear compartments using a restriction condition and by defining volumes. The restriction conditions are given as follows:

(S₁) each chromosome should be restricted in the nuclear domain [52],

(S₂) no heterochromatin domains should escape from the given chromosome domain [52],

(S₃) the chromosome domains tend to be separated from each other and constitute separate territories [19],

(S₄) no CC should escape from the given heterochromatin domain and chromosome domain [52].

The conditions, S₁–S₄ are described as where β₀, β_ψ, β_ϕ and are positive constants and indicate the intensity of domain territories. h(ϕ_m) is given for , which is technically used to induce energetic asymmetry between ϕ = 0 and ϕ = 1 to drive the interface while keeping ϕ = 0 and ϕ = 1 local minima. E₀ and E₁ are fundamental formulations for setting up nuclear and subnuclear compartments.

Next, we define chromosome and heterochromatin volumes according to the following equation: where

V₀(t) is the volume of nucleus and , and are the target volumes of chromosomes, heterochromatin, and CCs, respectively, which is determined and calculated by the size of cell nucleus at time t divided by each number. That is, the target volumes are temporally changed depending on the nuclear volume changed temporally. We define and . In the case of fixed nucleus with a constant volume V₀, we set , . The first term of E₂ ensures the condition that the nucleus is fully occupied by chromosomes.

As an additional condition that is unnecessary to regenerate inverted nuclear architecture but is required to produce conventional nuclear architecture, we can describe the role of LBR and/or LamA/C between heterochromatin and the nuclear envelope [2] in the model by the affinity between nuclear function, ϕ₀, and heterochromatin domain function, ψ, as follows: where γ is a constant that determines the strength of the affinity. Note that γ > 0 reflects the presence of LBR and/or Lam A/C on the interior of the nuclear envelope and their tethering of heterochromatin to the nuclear periphery. In contrast, γ = 0 represents the absence of LBR and Lam A/C expression and means that heterochromatin and the nuclear envelope do not interact with detectable affinity.

Now, the total energy of subnuclear compartments is expressed as: (1)

Then, we take the functional derivatives of Eq (1) with respect to ϕ_m (1≤m≤N), ψ and ψ₀, which drives the system to evolve with time, satisfying: where μ is a positive constant that represents the mobility of each phase-field. Calculating the above equations yields the reaction-diffusion model as follows: (2) where

Note that to generate conventional nuclear architecture as the initial conditions, we solved the system (2) with >0, but γ = 0 is always assumed in the simulations for inverted nuclear architecture because we do not need the condition for affinity between heterochromatin and the nuclear envelope in the nuclear architecture reorganization process.

Now, we nondimensionalize the model (2) for time scale T and spatial scale L incorporating and into the model (2). Let us define the mobility constant, μ, as: and set:

We then obtain the following system by removing the tilde: (3)

In the model, we assumed that the nucleus changes its size and shape independently of chromatin states, so that the phase-field ϕ₀ is given to be an independent function, describing the nucleus unrelated to the other phase-field functions. That is, we directly control the nuclear deformation using the following equation: where is a deformation function, which controls movement of the interface of the nuclear phase-field. For example, to make a circular nucleus from an elliptical one in 2D that is initially given, we define it as (4) where x₀ is the center coordinate of the nucleus, r(t) is the radius of the circle, and dist(x,r) = |x−x₀|−r(t). A₀ is the positive value controlling the speed of deformation, so that we can make the difference of dynamic degree in deformation by adjusting A₀. By defining r(t) as a decreasing function, we can generate the decrease in nuclear size.

Fusion of heterochromatin regions and CCs.

In the above modeling, we implicitly assumed the fact that heterochromatin regions and CCs in different chromosomes are fused when they are in contact with each other [6,9,10]. We described all heterochromatin domains and CCs in a nucleus using a single phase-field function. Such description leads to the fusion when the interfaces of two separate domains are close, as shown in S9B and S9C Fig.

Parameter validation.

Because we captured subnuclear compartments as domains, the key element determining the dynamics of heterochromatin or CC clustering is the interface of phase-field function. The interface region, {x|0<ϕ(x)<1}, can be interpreted as a confinement region of chromatin movement as shown in S9D Fig. That is, the thickness of the interface, denoted by γ, can be interpreted as the radius of the confinement region, which was found to have a value less than 4 μm [11].

In fact, we can explicitly calculate the thickness of the interface [53] as follows: where λ is a given constant defining the interface region, such that λ≤ϕ≤1−λ. Although it is very difficult to estimate all parameters of the model from experimental data, we validated our representative parameters by using the scale of the confinement region (δ) and the spatial scale of nuclear size.

For the simulations, Ω is given by L_x×L_y, where L_x and L_y are horizontal and perpendicular lengths, respectively, in the square shown in S9A Fig. We estimated the spatial scale directly from the images of our experiments and from a previous report [6], where the size of the nucleus in two dimensions is approximated to 4 μm to 8 μm for the short diameter (the y-axis diameter), , and 6 μm to 12 μm for the long diameter (the x -axis diameter), . We simulated the nondimensionalized system for L_x×L_y = 1.8×1.8 square nucleus and L_ld×L_sd = 1.6×1.0 elliptic nucleus by the scaling by means of . That is, the dimensional length of the short diameter and long diameter of the elliptic nucleus are expressed as . Therefore, the nuclear size is chosen in the feasible range with respect to the real-world data. We also chose ε_ϕ such that δ is in the feasible range. The detailed values of ε_ϕ in S3 Table satisfy and for λ = 0.15.

On the other hand, we estimated time scale T using the time scale of nuclear deformation. To validate the time scale that we chose for the simulation, we compared qualitative dynamics and temporal data that were obtained from a previous report [6] and in our experiments of model cell system. Note that the scale of interface thickness (i.e., the confinement length scale of subnuclear compartment movement) is independent from the time scale of the model dynamics because we defined the mobility of phase-fields as μ = T⁻¹ in system. That is, the time scale of nuclear architecture reorganization process is determined by the time scale of nuclear deformation, T. We also ran simulations with a nondimensionalized system (3) so that the results are consistent without dependence on the specific nuclear size. The detailed representative parameters are given in S3 Table.

The main results in the paper was presented for N = 8, but the effect of dynamic deformation on CC clustering is not dependent on the number of N. The example result for the case of N = 12 is shown in S9E Fig.

The measure of nuclear deformation.

In numerical simulations, we obtained dynamic deformation of the nucleus, with fixing of the center of the nucleus, and we created several shapes randomly. That is, the deformation magnitude of all shapes can be quantified via measurement how much the boundary of a nucleus differs from the boundary of a circle of the same area (Fig 2H). We calculated the deformation degree by

The case of low deformation degree (Fig 2G) was formulated with choosing smaller value of A₀ that chosen in the case of high deformation degree (Fig 2E). The average of total deformation degree () of Fig 2G was about 30% smaller than that of Fig 2E.

The effect of nuclear size on nuclear architecture reorganization.

During the nuclear architecture reorganization in a rod photoreceptor cell, we found that this process is accompanied by a 40% reduction in nuclear volume (by 20% of the area in 2D) and convergence of its shape from an ellipsoidal to spherical one during the differentiation [6]. To test whether the effect of the nuclear size decrease critically affects the clustering of CCs, we conducted simulations with a 20% decrease in the nuclear area. In many simulations, we observed that the nuclear size decrease does not have such a critical effect on clustering of CCs (S10 Fig). We could not find that the decrease in nuclear size promotes CC clustering advantageously without sufficient deformation of the nucleus, indicating that the distance between CCs can be decreased because of a decrease in the nucleus size but its effect is negligible, and the dynamic nuclear deformation is a sufficient condition for the nuclear architecture reorganization.

Nuclear deformation definition.

The nucleus of mouse rod photoreceptor cell has been described to be deformed from an ellipsoidal to spherical shape over time scales of several weeks during differentiation. Here, we refer this deformation as “quasi-static” nuclear deformation. We defined the time scale of nuclear deformation, T_d, by the time for moves of the nuclear envelop 1 μm. Then, the speed of nuclear deformation (S_d) can be estimated as . We also defined the limited time for the nuclear architecture reorganization as T_p, which is reported to be from several dozen days to several months in a rod photoreceptor cell [6], and we set T_p = 41 days in the representative parameter set. On the other hand, it was found that specific genomic regions and subnuclear compartments are confined and immobile over distances greater than 0.4 μm for more than 1 hour scale [11]. That is, the movement speed of a subnuclear compartment (S_c) can be estimated as S_c≤0.4 (μmh⁻¹).

Now, by comparing the time scale or speed of nuclear deformation with the speed of the subnuclear compartment, we classified nuclear deformation as follows:

• Static state nucleus when S_d = 0,
• Quasi-static nuclear deformation when S_d≪S_c and T_d≈T_p,
• Dynamic nuclear deformation when S_d≫S_c and T_d≪T_p.

In the simulation, we typically chose S_d≈0.088 (μmh⁻¹) with T_d = T_p days for quasi-static deformation, and S_d≈2.4 (μmh⁻¹) with T_d = 25 min for dynamic deformation.