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Mathematical Modelling of Mechanotransduction via RhoA Signalling Pathways

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Pith's one-line read A two-way RhoA–FAK feedback loop can explain why cells deform similarly on soft and stiff surfaces.

desk verdict Solid two-way coupling model, but the homeostasis plateau is computed in a large-deformation regime the linear elasticity assumption cannot support. read the letter →

arxiv 2501.04407 v2 pith:PS7RVG26 submitted 2025-01-08 math.NA cs.NAphysics.bio-phq-bio.CB

Sofie Verhees , Chandrasekhar Venkataraman , Mariya Ptashnyk This is my paper · ORCID
classification math.NAcs.NAphysics.bio-phq-bio.CB MSC 65M6092C1035K5774B05
keywords mechanotransductionRhoAsignallingfocaladhesionkinasemechanicalhomeostasisreaction-diffusionequationsbulk-surfacefiniteelementslinearelasticitysubstratestiffness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that mechanical homeostasis—cells deforming by roughly the same amount whether they sit on a soft or a stiff surface—can emerge from a minimal mechanistic model rather than from a tuned control system. The model couples the RhoA signalling pathway to cell elasticity in two directions: mechanical stress activates focal adhesion kinase (FAK), and active FAK increases the cell's Young's modulus. In simulations, this two-way loop makes the magnitude of cell deformation and cell volume change nearly flat across substrate stiffnesses from 0.1 kPa to 7 GPa, while a model with constant cell stiffness shows no such plateau. A sympathetic reader would take the paper to be claiming that the feedback loop, not any single reaction, is the mechanism behind stiffness homeostasis.

What carries the argument

The central object is a two-way feedback loop. Stress activates FAK through the term $C_1\,\mathrm{tr}(\sigma)_+\,\phi_d$, and the cell's Young's modulus increases with active FAK according to the power law $E_c(\phi_a)=k_7(1+(k_8\phi_a)^p)$, where active FAK is used as a proxy for F-actin. This loop is embedded in a bulk-surface reaction-diffusion system for inactive and active FAK in the cytoplasm, active RhoA on the membrane, and a linear elasticity problem whose boundary traction depends on active RhoA. The numerical machinery is a bulk-surface finite element method with semi-implicit time stepping; the power law is the bridge that converts a biochemical signal into a mechanical property, and removing that bridge removes the homeostasis.

What would settle it

Measure, in a single cell type, both active FAK concentration and deformation on substrates spanning 0.1 kPa to 7 GPa. If deformation follows substrate stiffness monotonically instead of plateauing, or if cell stiffness is unchanged when active FAK is elevated, the claimed two-way mechanism is falsified. Pharmacological FAK inhibition should also destroy the plateau if the mechanism is causal.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that bidirectional coupling between signalling and cell mechanics is sufficient to reproduce mechanical homeostasis. The authors extend a spatial RhoA signalling model to include linear elastic cell deformation, with the cell's Young's modulus given by $E_c(\phi_a)=k_7(1+(k_8\phi_a)^p)$ and with FAK activation driven by the positive part of the stress trace, $C_1\,\mathrm{tr}(\sigma)_+\,\phi_d$. When both couplings are present, steady-state cell deformation and $\mathrm{div}(u)$ become nearly constant over a wide range of ECM stiffnesses, while active FAK and active RhoA still rise in a threshold-like Hill-function manner. This reproduces the qualitative experimental observations of mechanical homeostasis and of threshold-like stiffness responses. The plateau does not occur when the cell is assigned constant stiffness or when stress feedback onto signalling is switched off, so the paper concludes that the bidirectional coupling itself is the load-bearing mechanism.

Load-bearing premise

The homeostasis prediction rests on the assumed law that cell stiffness rises with active FAK according to $E_c(\phi_a)=k_7(1+(k_8\phi_a)^p)$, with active FAK standing in for F-actin; if real cells stiffen differently, or if FAK is not a good proxy for F-actin, the plateau could disappear.

Editorial extensions

If this is right

  • If the central claim is correct, models that treat cell stiffness as a fixed constant will miss the homeostasis plateau; stiffness must be allowed to respond to signalling.
  • The reduced FAK–RhoA model reproduces the threshold-like dependence of signalling on substrate stiffness seen in experiments, so the threshold response is not an artefact of the full pathway.
  • Cell shape becomes a control variable: axisymmetric and lamellipodium-shaped cells show different concentrations and different threshold responses, implying that geometry is part of the mechanosensing circuit.
  • A linear viscoelastic version of the model produces the same qualitative behaviour, so the homeostasis result does not depend on the purely elastic constitutive law.
  • The framework is designed to be extended to viscoelastic or poroelastic mechanics, nuclear deformation, and spatially varying substrate stiffness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same feedback loop would predict that a cell pre-exposed to a stiff substrate retains a higher Young's modulus and therefore deforms less on a later soft substrate, offering a concrete mechanism for mechanical memory that could be tested by sequential-culture experiments.
  • Beyond the paper, pharmacological inhibition of FAK or RhoA should erase the plateau: deformation should then scale with substrate stiffness as it does in the constant-stiffness case, providing a direct causal test of the model.
  • Beyond the paper, the fitted parameters $k_8$ and $p$ could be estimated from combined live imaging of active FAK and atomic force microscopy stiffness maps, turning the fitted constitutive law into a measurable prediction.
  • Beyond the paper, because the plateau appears only when both couplings are active, one-way stiffness-sensing models may misestimate how cells respond to matrix gradients, especially in slowly varying environments where the bidirectional term dominates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript proposes a reduced bulk-surface reaction-diffusion model for inactive and active FAK and active RhoA, coupled to linear elasticity of the cell. Substrate stiffness activates FAK on the membrane, the cell Young's modulus is made to depend on active FAK concentration through Eq. (2), and the boundary traction depends on active RhoA through Eq. (4). The model is simulated for 2D and 3D stimuli, for axisymmetric and lamellipodium-shaped cells, and the results are used to claim a threshold-like response to substrate stiffness and a robustness of cell deformation to stiffness changes, interpreted as mechanical homeostasis. The paper also presents a bulk-surface finite element method and verifies its convergence on a manufactured solution.

Significance. If the central claim holds, the paper provides a minimal mechanistic explanation of mechanical homeostasis: bidirectional coupling between RhoA signalling and cell mechanics can make cell deformation insensitive to ECM stiffness. The paper has clear strengths: the reduced model is shown to reproduce the qualitative results of the full model of [18]; the numerical scheme is benchmarked on a manufactured solution with the expected orders of convergence; and a parameter sensitivity analysis is provided. However, the main emergent predictions are tied to the assumed power-law stiffness law and to the linear small-strain approximation, so the significance of the homeostasis claim depends on additional validation that the manuscript does not currently provide.

major comments (2)
  1. [Mathematical model for mechanotransduction, Eq. (3); Results, Figs. 1 and 4] The model is formulated under the explicit assumption of small deformations, leading to the linear elastic equation (3). Yet the results report maximum displacement magnitudes of 7 µm for the 2xD stimulus (Fig. 1) and 7.5 µm for the 3D stimulus (Fig. 4) on a cell with |Y|=1193 µm^3, i.e. an equivalent spherical diameter of roughly 13 µm and radius of roughly 6.5 µm. Displacements comparable to the cell radius imply O(1) strains, far outside the small-strain regime in which Eq. (3) is valid. The central homeostasis claim is read off from div(u) and |u| in exactly this regime (Figs. 3, 5, and 7). The Discussion admits that the linear elastic assumption 'is limiting as it assumes small deformations', and the viscoelastic check in S1 Appendix A.7 retains linear kinematics. A finite-strain computation, or a restriction of the homeostasis claim to genuinely small strains, is needed before the plateau can be regarded as a robust prediction of the model rather than a possible artifact of the linearization.
  2. [Eq. (2), Table 1(b), and S1 Appendix A.6] The homeostatic plateau is largely a consequence of the assumed constitutive law E_c(phi_a)=k7(1+(k8 phi_a)^p), with k7, k8, and p fitted to results for phi_a in [18] and F-actin in [38]. The threshold-like dependence on substrate stiffness is similarly inherited from the Hill-type terms and parameters taken from [18]. Because these fitted inputs are used to generate the very curves that are then presented as emergent predictions, the claim that homeostasis 'emerges' from the bidirectional coupling is only as strong as the empirical support for Eq. (2). The sensitivity analysis in S1 Appendix A.6 varies the parameters of Eq. (2) by ±10% and ±20%, but it does not test the form of the law itself. The authors should either test alternative functional forms of E_c(phi_a), or present independent data that constrain Eq. (2), before the homeostasis result is presented as a model prediction rather than a restatement of the fitted stiffness law.
minor comments (4)
  1. [Methods, first paragraph] There is a typo: 'mechanostranduction' should be 'mechanotransduction'.
  2. [Results, Fig. 8 caption] The caption contains a duplicated phrase: 'simulation results showing rho_a, phi_d, phi_a and |u| simulation results showing phi_a and rho_a for the model...' should be cleaned up.
  3. [Introduction, last paragraph] 'mechanisms that underlay the mechanical homeostasis' should be 'mechanisms that underlie mechanical homeostasis'.
  4. [References] Reference [43] is cited before [42] in the text; the citation order in the bibliography could be adjusted if the journal requires strict numerical ordering.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: fitted inputs are external and the homeostasis plateau is not imposed by construction.

full rationale

The derivation chain is self-contained relative to the claims that matter. The biochemical submodel is explicitly a reduction of the Scott et al. model (Eq. 1), with the Michaelis-Menten E/(C+E) and Hill ((gamma*phi_a)^n+1) terms inherited from that external source; the paper frames the resulting threshold-like dependence as reproduction and validation against [18,44], not as a new prediction derived from mechanics. The abstract's phrase "novel emergent features such as ... threshold-like response" overstates novelty because that threshold is structurally present in Eq. (1), but this is a presentation issue, not a circular derivation of the paper's central homeostasis claim. The parameters k7, k8, and p in the cell-stiffness law Eq. (2) are fitted to external F-actin/FAK data ([18,38]), while the homeostasis plateau is not fitted to homeostasis data: it appears only when the bidirectional feedback sigma -> phi_a -> E_c is active, and it is compared qualitatively with [28]. Self-citations ([14,60]) occur only for standard numerical and multiscale tools and are not load-bearing. The paper explicitly flags the linear-elastic small-deformation limitation in the Discussion; that is an accuracy risk, not circularity. No equation is shown to be equivalent by construction to another, and no fitted parameter is renamed as a prediction of the central emergent quantity. The stated but deferred existence proof is a completeness gap, not a circular step.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim depends on several parameters fitted to prior simulation data and on phenomenological coupling laws. No new physical entities, such as particles, forces, or dimensions, are introduced beyond the mathematical projection P, which is a construction rather than a physical entity.

free parameters (6)
  • C1 = 0.1 (kPa s)^-1; range 0-2 explored
    Coupling strength of cell stress to FAK activation; introduced in this paper, not fitted to specific data, but explored over a range.
  • k6 = 0.1 s^-1
    Traction coefficient in boundary condition (4); fitted to yield a deformation magnitude of 0-10 um as in [18].
  • k7 = 0.2 kPa
    Baseline Young's modulus in Eq (2); fitted to results for active FAK in [18].
  • k8 = 2.4245 dm^3/umol
    FAK sensitivity of cell stiffness in Eq (2); fitted to results for active FAK in [18].
  • p = 2.6
    Exponent in the stiffness law (2); fitted to results for active FAK and F-actin in [18] and [38].
  • nu_c = 0.3
    Poisson ratio of the cell; estimated from the literature range 0.17-0.66 [42].
assumptions (6)
  • domain assumption Linear elasticity with small deformations (Eq. 3)
    The cell is treated as a linear elastic material; small deformation justifies solving signalling on the reference configuration. Stated in the Methods section.
  • ad hoc to paper Cell Young's modulus is a power-law function of active FAK (Eq. 2)
    Proposed based on [18] and [38]; k7, k8, and p are fitted to simulation results. This law is load-bearing for the homeostasis claim.
  • domain assumption Substrate stiffness activates FAK through a Michaelis-Menten term k3 E/(C+E) (Eq. 1)
    Inherited from the reference model [18]; this functional form generates the threshold-like response.
  • domain assumption RhoA activation uses a Hill-type term k5((gamma phi_a)^n + 1)(M_rho/|Y| - rho_a/n_r) (Eq. 1)
    Inherited from [18]; assumes conservation of total RhoA mass.
  • ad hoc to paper FAK is activated by the positive part of the trace of the stress tensor, C1 tr(sigma)+ (Eq. 6)
    Phenomenological proxy for stress-induced FAK activation; no direct molecular derivation is given.
  • ad hoc to paper Boundary traction depends on active RhoA as k6 P(rho_a nu) (Eq. 4)
    Models RhoA-driven contractility; P projects out rigid motions.

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Cite this review

Pith. "Pith review of Mathematical Modelling of Mechanotransduction via RhoA Signalling Pathways." pith.science (2026). https://pith.science/paper/PS7RVG26

@misc{pith2026250104407,
  author       = {Pith},
  title        = {Pith review of: Mathematical Modelling of Mechanotransduction via RhoA Signalling Pathways},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PS7RVG26}},
  note         = {Machine review of arXiv:2501.04407}
}
read the original abstract

We derive and simulate a mathematical model for mechanotransduction related to the Rho GTPase signalling pathway. The model addresses the bidirectional coupling between signalling processes and cell mechanics. A numerical method based on bulk-surface finite elements is proposed for the approximation of the coupled system of nonlinear reaction-diffusion equations, defined inside the cell and on the cell membrane, and the equations of elasticity. Our simulation results illustrate novel emergent features such as the strong dependence of the dynamics on cell shape, a threshold-like response to changes in substrate stiffness, and the fact that coupling mechanics and signalling can lead to the robustness of cell deformation to larger changes in substrate stiffness, ensuring mechanical homeostasis in agreement with experiments.

Figures

Figures reproduced from arXiv: 2501.04407 by the authors.

Figure 1
Figure 1. Numerical simulation results showing ρa, ϕd, ϕa and |u| for the model in Eqs (3)-(6) for the axisymmetric shape and in the case of 2xD stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec = 0.6 kPa (ϕa ̸→ Ec); (B) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec = f(ϕa) (ϕa → Ec); (C) C1 = 0.1 (kPa s)−1 (σ → ϕa) and Ec = 0.6 kPa (ϕa ̸→ Ec); (D) C1 = 0.1 (kPa s)−1 (… view at source ↗
Figure 2
Figure 2. Numerical simulation results showing ρa, ϕd, ϕa and |u| for the model in Eqs (3)-(6) for the lamellipodium shape and in the case of 2xD stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec = 0.6 kPa (ϕa ̸→ Ec); (B) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec = f(ϕa) (ϕa → Ec); (C) C1 = 0.1 (kPa s)−1 (σ → ϕa) and Ec = 0.6 kPa (ϕa ̸→ Ec); (D) C1 = 0.1 (kPa s)−1 … view at source ↗
Figure 3
Figure 3. Simulation results showing the mean, 1 |Ω| R Ω · dx, min and max values of f(ϕa), div(u), ϕa and ρa as functions of substrate stiffness E. We consider different couplings with four different values for C1 and two different shapes at T = 100 s by which time the results are at a steady state. All other parameter values as in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Numerical simulation results showing ρa, ϕd, ϕa and |u| for the model in Eqs (3)-(6) for the axisymmetric shape and in the case of the 3D stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec = 0.6 kPa (…
Figure 5
Figure 5. Figure 5: Simulation results showing the mean, 1 |Ω| R Ω · dx, min and max values of f(ϕa), div(u), ϕa and ρa as functions of substrate stiffness E. We consider different couplings with four different values for C1 and two different shapes at T = 100 s by which time the results …
Figure 6
Figure 6. Figure 6: Numerical simulation results showing ρa, ϕd, ϕa and |u| for the model in Eqs (3), (4), and (6) for the axisymmetric shape and in the case of the 3D stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec =…
Figure 7
Figure 7. Figure 7: Simulation results showing the mean, 1 |Ω| R Ω · dx, min and max values of f(ϕa), div(u), ϕa and ρa as functions of substrate stiffness E, in the case of the model in Eqs (3), (4) and (6) and 3D stimulus. We consider different couplings, four different values for C1, a…
Figure 8
Figure 8. Figure 8: Numerical simulation results showing ρa, ϕd, ϕa and |u| simulation results showing ϕa and ρa for the model in Eqs (3), (4), and (6) for the axisymmetric shape and in the case of 2xD stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C…
Figure 9
Figure 9. Figure 9: Numerical simulation results showing ρa, ϕd, ϕa and |u| for the model in Eqs (3)-(6) for the lamellipodium shape and in the case of the 3D stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec = 0.6 kPa …
Figure 10
Figure 10. Figure 10: Numerical simulation results showing ρa, ϕd, ϕa and |u| for the model in Eqs (3), (4), and (6) for the lamellipodium shape and in the case of the 3D stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec…
Figure 11
Figure 11. Figure 11: Numerical simulation results showing ρa, ϕd, ϕa and |u| for the model in Eqs (3), (4), and (6) for the lamellipodium shape cells and 2xD stimulus at a steady state at T = 100 s. Four different scenarios are considered: (A) C1 = 0 (kPa s)−1 (σ ̸→ ϕa) and Ec = 0.6 kPa (…
Figure 12
Figure 12. Figure 12: Simulation results showing the mean, 1 |Ω| R Ω · dx, min and max values of f(ϕa), div(u), ϕa and ρa as functions of substrate stiffness E, in the case of the model in Eqs (3), (4) and (6) and 2xD stimulus. We consider different couplings, four different values for C1,…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.