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arxiv: 1906.08962 · v1 · pith:IYNBOY3Dnew · submitted 2019-06-21 · 🧬 q-bio.CB · cond-mat.soft· math.DS· physics.bio-ph

Cell Shape and Durotaxis Follow from Mechanical Cell-Substrate Reciprocity and Focal Adhesion Dynamics: A Unifying Mathematical Model

Elisabeth G. Rens , Roeland M.H. Merks This is my paper

Pith reviewed 2026-05-25 18:38 UTC · model grok-4.3

classification 🧬 q-bio.CB cond-mat.softmath.DSphysics.bio-ph
keywords focal adhesionsdurotaxiscell shapesubstrate stiffnesscellular Potts modelfinite element modelcell contractilitymechanosensing
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The pith

A multiscale model shows focal adhesion growth under cell contractility explains stiffness-dependent cell shapes and durotaxis.

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

The paper introduces a computational model in which cells pull on substrates through focal adhesions that grow and stabilize in response to the forces they experience. On stiffer substrates these adhesions reach higher forces more quickly within any given time window, producing larger adhesions that resist detachment and permit greater cell spreading and elongation. The same rules also cause cells to migrate toward stiffer regions because adhesions there remain more stable. The model couples a cellular Potts representation of the cell with finite-element mechanics of the substrate and ordinary differential equations for each focal adhesion. If the four listed assumptions hold, the observed range of cell morphologies and durotactic behavior follows directly from mechanical reciprocity between cell and substrate.

Core claim

The mechanosensitive behavior of single-focal adhesions, cell contractility and substrate adhesivity together suffice to explain the observed stiffness-dependent behavior of cells. Cells apply forces onto the substrate through FAs; the FAs grow and stabilize due to these forces; within a given time-interval, the force that the FAs experience is lower on soft substrates than on stiffer substrates due to the time it takes to reach mechanical equilibrium; and smaller FAs are pulled from the substrate more easily than larger FAs. Together these assumptions provide a unifying model for cell spreading, cell elongation and durotaxis in response to substrate mechanics.

What carries the argument

Multiscale model that couples the cellular Potts model for cell shape and motility, a finite-element model for substrate deformation, and differential equations describing growth and detachment of each focal adhesion under the four stated mechanical assumptions.

If this is right

  • Cell area increases monotonically with substrate stiffness because larger focal adhesions form and persist on stiffer material.
  • Cells adopt elongated shapes on substrates of intermediate stiffness where force balance favors directional extension before full spreading occurs.
  • Cells migrate up stiffness gradients because focal adhesions disassemble more readily on the softer side.
  • The same rules reproduce rounded shapes on soft substrates, spindle-like shapes on medium-stiffness substrates, and flat spread shapes on rigid substrates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The model implies that measured differences in focal-adhesion lifetime across stiffness gradients should be sufficient to predict durotactic speed without invoking additional intracellular signaling pathways.
  • Varying cell contractility in the model should shift the stiffness range at which cells transition from rounded to elongated shapes, offering a testable prediction for experiments that pharmacologically modulate myosin activity.
  • Extending the substrate model to include spatial gradients in adhesivity rather than stiffness alone could show whether the same focal-adhesion rules also produce haptotaxis.

Load-bearing premise

Within any given time interval the force experienced by focal adhesions is lower on soft substrates than on stiffer substrates because mechanical equilibrium takes longer to reach on the soft material.

What would settle it

Direct measurement showing that focal adhesions on soft substrates reach the same force magnitude as on stiff substrates within the same short time window, or experiments in which altering the FA growth rate parameter fails to change cell shape or durotaxis as predicted.

Figures

Figures reproduced from arXiv: 1906.08962 by Elisabeth G. Rens, Roeland M.H. Merks.

Figure 1
Figure 1. Figure 1: Flowchart of the multiscale CPM. (A) CPM calculates cell shapes in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Model M1 predictions: Cell area increases with increasing substrate [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Model M2 predictions: Cells elongate on substrates of intermediate [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Range of stiffness on which cells elongate depends on myosin motor [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Durotaxis as a result of integrin catch-bond dynamics. (A) Ten [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

Many animal cells change their shape depending on the stiffness of the substrate on which they are cultured: they assume small, rounded shapes in soft ECMs, they elongate within stiffer ECMs, and flatten out on hard substrates. Cells tend to prefer stiffer parts of the substrate, a phenomenon known as durotaxis. Such mechanosensitive responses to ECM mechanics are key to understanding the regulation of biological tissues by mechanical cues, as it occurs, e.g., during angiogenesis and the alignment of cells in muscles and tendons. Although it is well established that the mechanical cell-ECM interactions are mediated by focal adhesions, the mechanosensitive molecular complexes linking the cytoskeleton to the substrate, it is poorly understood how the stiffness-dependent kinetics of the focal adhesions eventually produce the observed interdependence of substrate stiffness and cell shape and cell behavior. Here we show that the mechanosensitive behavior of single-focal adhesions, cell contractility and substrate adhesivity together suffice to explain the observed stiffness-dependent behavior of cells. We introduce a multiscale computational model that is based upon the following assumptions: (1) cells apply forces onto the substrate through FAs; (2) the FAs grow and stabilize due to these forces; (3) within a given time-interval, the force that the FAs experience is lower on soft substrates than on stiffer substrates due to the time it takes to reach mechanical equilibrium; and (4) smaller FAs are pulled from the substrate more easily than larger FAs. Our model combines the cellular Potts model for the cells with a finite-element model for the substrate, and describes each FA using differential equations. Together these assumptions provide a unifying model for cell spreading, cell elongation and durotaxis in response to substrate mechanics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a multiscale model coupling the cellular Potts model for cell morphology, a finite-element description of the deformable substrate, and ODEs governing focal adhesion (FA) assembly, growth, and detachment. It claims that four explicit assumptions—(1) cells apply contractile forces via FAs, (2) these forces promote FA growth and stabilization, (3) within each discrete time interval the net force transmitted through an FA is lower on soft substrates because mechanical equilibrium is not reached, and (4) smaller FAs detach more readily—together suffice to reproduce stiffness-dependent cell spreading, elongation, and durotaxis.

Significance. If the numerical implementation demonstrates that assumption (3) emerges from the coupled CPM-FE time scales rather than from parameter tuning, the work would supply a concrete, falsifiable link between single-FA force kinetics and emergent cell-level mechanosensing. The explicit separation of cell contractility, substrate adhesivity, and FA dynamics is a methodological strength that could be tested against independent FA lifetime measurements.

major comments (2)
  1. [Abstract] Abstract, assumption (3): the claim that force experienced by FAs is lower on soft substrates within a fixed time interval rests on the FE solver not reaching equilibrium on the FA-assembly time scale. The manuscript must demonstrate (e.g., via supplementary time-step convergence tests) that the chosen FE time step and effective modulus produce measurably slower FA growth on compliant gels; otherwise the stiffness dependence is inserted by construction rather than derived from the premises.
  2. [Results] The central unification requires that the four assumptions generate the observed behaviors without additional stiffness-dependent rules. Because assumptions (1), (2) and (4) are standard while (3) supplies the explicit stiffness link, the results section should report the FA force histories and growth rates on substrates spanning at least two orders of magnitude in modulus, together with the corresponding cell aspect ratios and durotactic indices, to allow direct assessment of whether the time-to-equilibrium effect is sufficient.
minor comments (1)
  1. [Methods] The abstract states that the model “describes each FA using differential equations” but does not list the ODEs or the numerical coupling scheme between the CPM, FE mesh, and FA variables; these should be supplied in the methods with explicit time-stepping order.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive critique. The comments highlight the need to explicitly verify that the stiffness dependence arises from the coupled time scales rather than from parameter choices. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract, assumption (3): the claim that force experienced by FAs is lower on soft substrates within a fixed time interval rests on the FE solver not reaching equilibrium on the FA-assembly time scale. The manuscript must demonstrate (e.g., via supplementary time-step convergence tests) that the chosen FE time step and effective modulus produce measurably slower FA growth on compliant gels; otherwise the stiffness dependence is inserted by construction rather than derived from the premises.

    Authors: We agree that explicit verification is required. In the revised version we will add a supplementary section containing time-step convergence tests for the FE solver across substrate moduli spanning at least two orders of magnitude. These tests will quantify the time required to reach mechanical equilibrium and show that, within the FA-assembly time scale used in the model, the net force transmitted through each FA remains lower on compliant substrates, thereby confirming that assumption (3) emerges from the separation of time scales rather than from ad-hoc tuning. revision: yes

  2. Referee: [Results] The central unification requires that the four assumptions generate the observed behaviors without additional stiffness-dependent rules. Because assumptions (1), (2) and (4) are standard while (3) supplies the explicit stiffness link, the results section should report the FA force histories and growth rates on substrates spanning at least two orders of magnitude in modulus, together with the corresponding cell aspect ratios and durotactic indices, to allow direct assessment of whether the time-to-equilibrium effect is sufficient.

    Authors: We will expand the Results section to include the requested quantitative data. New figures will display representative FA force histories and instantaneous growth rates for substrates ranging from 0.1 kPa to 100 kPa, together with the resulting cell aspect ratios and durotactic indices obtained under identical parameter sets. These plots will demonstrate that the four assumptions alone are sufficient to produce the stiffness-dependent phenotypes without additional rules. revision: yes

Circularity Check

0 steps flagged

No circularity: behaviors emerge from explicit mechanistic assumptions without reduction to inputs by construction.

full rationale

The paper states four assumptions upfront as modeling premises and implements them in a coupled CPM-FE simulation to produce cell spreading, elongation, and durotaxis. No equation or step equates a derived quantity to a fitted input or prior self-citation; the stiffness dependence is injected via assumption (3) as an explicit premise rather than recovered tautologically. The derivation chain therefore runs from stated premises to simulated outcomes in the standard direction for a mechanistic model and remains self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central claim rests directly on four explicit assumptions about focal adhesion mechanics that are treated as domain knowledge rather than derived; the model adds no new entities but introduces several free parameters in the differential equations and Potts rules.

free parameters (2)
  • FA growth and stabilization rates
    Parameters in the differential equations that control how focal adhesions enlarge and stabilize under force are modeling choices that must be set to produce the target behaviors.
  • FA detachment threshold
    The size or force threshold at which smaller focal adhesions detach is a free parameter in the model.
axioms (4)
  • domain assumption cells apply forces onto the substrate through FAs
    Listed as assumption (1)
  • domain assumption the FAs grow and stabilize due to these forces
    Listed as assumption (2)
  • domain assumption within a given time-interval, the force that the FAs experience is lower on soft substrates than on stiffer substrates due to the time it takes to reach mechanical equilibrium
    Listed as assumption (3); this is the load-bearing premise that encodes stiffness dependence
  • domain assumption smaller FAs are pulled from the substrate more easily than larger FAs
    Listed as assumption (4)

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Reference graph

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