Nuclear Mechanics Model

Nuclear Mechanics Model

Overview

The nuclear mechanics model describes how cell-generated traction is transmitted to the nucleus and converted into nuclear deformation, YAP/TAZ activity, and mechanogenomic outputs.

In the mechanogenomic virtual cell, this module receives the traction predicted by the motor-clutch model and computes the effective nuclear stress:

$$\sigma_{nuc}(E)$$

This nuclear stress drives changes in:

• projected nuclear area;
• time-dependent nuclear deformation;
• lamin A/C-dependent mechanical gating;
• YAP/TAZ nuclear activity;
• mechanosensitive gene activation.

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Biological interpretation

The nucleus is treated as a central mechanosensor.

Forces generated at focal adhesions are transmitted through:

• the actin cytoskeleton;
• the actin cap;
• the LINC complex;
• the nuclear envelope;
• the nuclear lamina.

The mechanical response of the nucleus depends strongly on the nuclear lamina, especially lamin A/C.

In this model, lamin A/C is represented by the dimensionless parameter:

$$\ell$$

Higher ell represents a stiffer nucleus.
Lower ell represents a softer and more deformable nucleus.

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Model input

The input of the nuclear mechanics module is the traction generated by the motor-clutch model:

$$T(E)$$

where E is the substrate or tissue stiffness.

The motor-clutch model converts substrate stiffness into cellular traction:

$$E \rightarrow T(E)$$

The nuclear mechanics model then converts cellular traction into nuclear stress:

$$T(E) \rightarrow \sigma_{nuc}(E)$$

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Force transmission to the nucleus

Not all traction generated at the substrate reaches the nucleus. Part of the force is dissipated through the substrate, adhesions, and cytoskeleton.

The effective nuclear stress is modeled as:

$$\sigma_{nuc}(E) = T(E) \frac{\kappa}{\kappa + k_c}$$

where:

Symbol	Meaning
sigma_nuc	Effective mechanical stress transmitted to the nucleus
T(E)	Cellular traction generated by the motor-clutch module
kappa	Effective substrate stiffness
k_c	Clutch stiffness

The transmission factor is:

$$\frac{\kappa}{\kappa + k_c}$$

This term represents the fraction of force effectively transmitted through the mechanical system toward the nucleus.

Biologically, this corresponds to force propagation from focal adhesions to actin structures and then to the nuclear envelope through LINC-associated coupling.

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Lamin A/C as a nuclear mechanical gate

Lamin A/C controls the mechanical resistance of the nucleus.

The model represents relative lamin A/C level as:

$$\ell$$

Lamin A/C affects nuclear mechanics in two main ways:

1. It increases the stress required for nuclear deformation.
2. It modulates the mechanical gating of YAP/TAZ nuclear activity.

The half-saturation stress for nuclear deformation is:

$$s_{1/2} = s_0 \ell$$

where:

Symbol	Meaning
s_1/2	Stress required to reach half-maximal nuclear deformation
s_0	Basal stress scale
ell	Relative lamin A/C level

A higher lamin A/C level shifts the nuclear deformation response toward higher stress.

In biological terms:

• low lamin A/C makes the nucleus easier to deform;
• high lamin A/C makes the nucleus more resistant to deformation;
• lamin A/C acts as a mechanical gate between cytoskeletal force and nuclear response.

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Steady-state nuclear area

The main morphological output of the model is the steady-state projected nuclear area:

$$A_{ss}(E)$$

The model uses a saturating stress-response function:

$$A_{ss}(E) = A_{min} + (A_{max}-A_{min}) \frac{\sigma_{nuc}(E)} {\sigma_{nuc}(E)+s_{1/2}}$$

Substituting the lamin-dependent half-saturation stress:

$$s_{1/2} = s_0 \ell$$

gives:

$$A_{ss}(E) = A_{min} + (A_{max}-A_{min}) \frac{\sigma_{nuc}(E)} {\sigma_{nuc}(E)+s_0\ell}$$

where:

Symbol	Meaning
A_ss(E)	Steady-state projected nuclear area
A_min	Minimum or basal projected nuclear area
A_max	Maximum projected nuclear area under high stress
sigma_nuc	Effective nuclear stress
s_0 ell	Lamin-dependent resistance to deformation

Interpretation:

• At low nuclear stress, A_ss approaches A_min.
• At high nuclear stress, A_ss approaches A_max.
• A stiffer lamina requires higher stress to reach the same projected nuclear area.

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Time-dependent nuclear deformation

Nuclear deformation is not assumed to be instantaneous.

The model assumes first-order relaxation toward the stiffness-dependent steady state:

$$\tau \frac{dA}{dt} = A_{ss}(E) - A(t)$$

The analytical solution is:

$$A(t) = A_{ss}(E) + \left[A_0 - A_{ss}(E)\right]e^{-t/\tau}$$

where:

Symbol	Meaning
A(t)	Nuclear area at time t
A_0	Initial nuclear area
A_ss(E)	Steady-state nuclear area determined by stiffness
tau	Characteristic time scale of nuclear adaptation

This equation allows the model to capture progressive nuclear spreading over time on stiff substrates.

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YAP/TAZ mechanical gating

The model also predicts nuclear YAP/TAZ activation.

First, it defines a mechanical gating function:

$$u(\sigma) = \frac{1} {1+ \exp\left[ -\frac{\sigma-\sigma^*/\ell}{w} \right]}$$

where:

Symbol	Meaning
u(sigma)	Mechanical activation gate
sigma*	Basal activation threshold
ell	Relative lamin A/C level
w	Width of the activation transition

This gate represents the idea that sufficient nuclear stress is required to promote nuclear-envelope tension and YAP/TAZ nuclear entry.

The YAP nuclear-to-cytoplasmic ratio is modeled as:

$$YAP_{N/C} = 1 + (R_{max}-1) u(\sigma) \ell \frac{\sigma} {\sigma+\sigma_s}$$

where:

Symbol	Meaning
YAP_N/C	Nuclear-to-cytoplasmic YAP ratio
R_max	Maximum possible YAP activation
u(sigma)	Mechanical gating function
ell	Relative lamin A/C level
sigma_s	Stress scale for signal saturation

This module converts nuclear stress into a mechanotranscriptional signal.

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Population-level nuclear area model

The nuclear-area distribution is modeled as a mixture of two populations:

$$P(A|E,t) = \phi \mathcal{N}(A;\mu_b,\varsigma_b) + (1-\phi) \mathcal{N}(A;\mu_m(E,t),\varsigma_m)$$

where:

Symbol	Meaning
phi	Fraction of the basal population
mu_b	Mean nuclear area of the basal population
varsigma_b	Dispersion of the basal population
mu_m(E,t)	Mean nuclear area of the mechanosensitive population
varsigma_m	Dispersion of the mechanosensitive population

The basal population is assumed to be relatively insensitive to stiffness.

The mechanosensitive population follows the nuclear mechanics model and changes with stiffness and time.

This mixture explains why population averages can mask mechanical responses.

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Contact inhibition and effective clutch number

At longer culture times, cell-cell contacts increase as cultures become more confluent.

This can shift mechanical coupling away from cell-substrate adhesions and toward cell-cell adhesions.

The model represents this as a reduction in the effective number of substrate clutches:

$$n_c^{eff}(t) = n_c^0 [1-\beta c(t)]$$

where:

Symbol	Meaning
n_c_eff	Effective number of substrate clutches
n_c_0	Initial number of substrate clutches
beta	Strength of contact inhibition
c(t)	Confluence level

Confluence is modeled as:

$$c(t) = 1-e^{-t/t_c}$$

This introduces an integrin-to-cadherin mechanical switch into the model.

Biologically, this represents the progressive engagement of cell-cell adhesion as cultures become denser.

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Mechanogenomic output

The nuclear mechanics module predicts that increasing stiffness activates a mechanosensitive nuclear program.

Representative gene modules include:

Module	Example genes
YAP/TAZ-TEAD signaling	YAP1, WWTR1, TEAD2, TEAD4, CCN2
Nuclear envelope and lamina	LMNA, LMNB2, NUP93, TPR, TMPO
Adhesion and cytoskeleton	VCL, ILK, SRC, MYH9, MYL9, FLNA, CFL1
Matrix remodeling and fibrosis	LOX, COL1A1, COL1A2, VIM, ACTA2

The response is expected to be nonlinear or threshold-like for some genes, especially during transitions to high-stiffness fibrosis-like environments.

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Connection to hepatic fibrosis

Hepatic fibrosis can be interpreted as a progressive increase in tissue stiffness.

Approximate stiffness values used in the model are:

Fibrosis stage	Approximate stiffness
F0	~4 kPa
F1	~7 kPa
F2	~9.5 kPa
F3	~13 kPa
F4	~23-26 kPa

The nuclear mechanics model uses this stiffness axis to predict how nuclear stress, nuclear area, YAP/TAZ activity, and mechanosensitive gene expression change during fibrosis progression.

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Role in the mechanogenomic virtual cell

The nuclear mechanics model converts cellular traction into nuclear and transcriptional outputs:

$$T(E) \rightarrow \sigma_{nuc}(E) \rightarrow A(t),\;YAP/TAZ,\;genes$$

It is the mechanosensitive core of the virtual cell.

Together with the motor-clutch model, it provides a minimal physical-computational framework linking substrate stiffness to nuclear mechanotransduction and fibrosis-associated mechanogenomic trajectories.

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Summary

Component	Role
sigma_nuc(E)	Effective mechanical stress reaching the nucleus
ell	Relative lamin A/C level controlling nuclear stiffness
A_ss(E)	Steady-state projected nuclear area
A(t)	Time-dependent nuclear area
YAP_N/C	Nuclear-to-cytoplasmic YAP activation
P(A|E,t)	Population-level nuclear-area distribution
n_c_eff(t)	Effective clutch number under contact inhibition

The nuclear mechanics model is therefore the bridge between physical force sensing and mechanogenomic regulation.