How cells pull · MatriXense Learn
MatriXense Learn · Lesson 2 of 2

How cells
pull.

Traction force in 2D and 3D: from beads to forces, and why collagen is so hard.

01Why forces matter02Measuring in 2D, then 3D03The collagen problem

Why we care

Cells pull to feel, to move, and to build.

Feel

Reach out, grab, pull. The pull-back tells a cell how stiff its world is.

Decide

Stem cells on soft, medium and stiff gels lean toward nerve, muscle and bone fates.

Close wounds

Myofibroblasts contract tissue like a drawstring.

Fibrosis

More pull, stiffer tissue, even more pull. A vicious loop.

Invasion

Cancer cells pull to move, and carve tracks to follow.

A little physics

Force, stress, strain, stiffness.

Stretching a block of gel
Force F

Newtons. Cells pull with nanonewtons.

σ = F / A   stress
ε = ΔL / L₀   strain
E = σ / ε   stiffness
Plastic is ~10⁶ × stiffer

than most soft tissue. You can't stretch a wall.

History

Measuring the invisible.

1980Wrinkles

Cells wrinkle thin silicone rubber.

Harris, Wild & Stopak · Science
1999Beads in a gel

Bead displacements → forces. TFM is born.

Dembo & Wang · Biophys J
2002Fourier trick

Fast 2D reconstruction (FTTC).

Butler et al. · Am J Physiol Cell Physiol
2010Into 3D

Cells inside synthetic, linear gels.

Legant et al. · Nature Methods
2016Collagen

Nonlinear, fibrous networks.

Steinwachs et al. · Nature Methods

2D traction force microscopy

Beads under a cell, two pictures.

Cell on a soft gel with beads
1 · Soft gel + beadsPolyacrylamide of known stiffness, beads just below the surface.
2 · Stressed, then relaxedImage with the cell, detach it (e.g. trypsin), image the relaxed gel.
3 · Displacement fieldAn arrow for each patch of gel. Usually below 1 µm.u(x) = xstressed − xrelaxed

From displacements to forces

Forward is easy. Backward is hard.

Forward problem
T̂ = argmin ‖ u − G T ‖² + λ ‖ T ‖²fit the data, but penalise wild forces · λ = regularisation strength

The 3D challenge

Collagen breaks every assumption.

1Fibrous

Pores of several µm. Force travels along fibres.

2Nonlinear

Stiffens as it stretches. No single E.

3Remodelled

Cells bundle, align and cut fibres. Not all deformation springs back.

4Viscoelastic

It flows a little. The answer depends on time.

5Blurry in z

Axial resolution ~2–3× worse, chasing sub-µm motion.

6Huge data

Hours of 3D stacks, with bleaching and drift.

3D traction force

Every bead around the cell moves inward.

Cell in a bead-filled gel · drag to turn

How fast the field fades

Illustrative curves. Fibres carry force further than a linear gel.

Make or break

Reference, drift, noise.

Relaxing the cell
ReferenceRelax the cell at the end (cytochalasin D, or detergent). That's zero force.
DriftShared motion of beads far from the cell, measured and subtracted.
Noise floorLocalisation jitter of still beads. Here ~24 nm. Below it: not force.

The pipeline

From beads to forces.

Beads at two timepointsreference later
1 · BeadsEvery bead found in 3D at two timepoints.xref, x(t)
2 · Drift and matchingGlobal drift removed, each bead matched to itself.x(t) − d(t) ↔ xref
3 · Displacement fieldInterpolated in 3D. Grey is below the noise floor.u(x) = x(t) − d(t) − xref
4 · Strain → stress → energyStrain from the gradient of u, stress from the gel's stiffness.ε = ½(∇u + ∇uᵀ) · W = ½∫σ:ε dV

Example · one fibroblast

How hard is it working?

Strain energy density0 min
0.63pJ

Strain energy one fibroblast puts into collagen in 2 hours.

24nm

Noise floor of bead localisation, so gentle cells still count.

The material model

A straight line, or a curve?

Stress vs strain (illustrative)
Linear gel

Polyacrylamide: double the stretch, double the stress.

Collagen

Soft, then strain-stiffening as fibres align. Softer under compression as fibres buckle.

Use a nonlinear model, or stay at small strain and say so.

Many cells

Neighbours pull on the same beads.

Two cells, overlapping fields
47–232%

Force error fitting each cell on its own.

2–17%

Force error fitting all cells together.

Remodelling

Force leaves a footprint.

Two fibroblasts · collagen0 min
Fibre directionaligned to cell–cell axis

Densification by distance from the cell

Fibre alignment (order parameter S)

Densification at 0–25 µm matches a measured fibroblast (+5.1 %). Other values are a simulated example.

Recap

A gentle pull, on the whole world.

2Dbeads + regularised inverse
3D collagenfibres, nonlinear, drift, noise
Strain energyhow hard it works
Remodellingwhat it leaves behind

It isn't just sitting there. It's pulling.

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