How cells move · MatriXense Learn
MatriXense Learn · Lesson 1 of 2

How cells
move.

Cell migration in 2D and 3D: trajectories, mean squared displacement, and migration patterns.

01Why it matters02From movie to numbers03Reading the MSD

Why we care

Biology is full of cells going somewhere.

Development

Neural crest cells cross the embryo to build your face, nerves and skin pigment.

Repair

Fibroblasts crawl into a wound and start rebuilding the tissue.

Defence

T cells scan lymph nodes at around 10 µm per minute, looking for trouble.

Disease

Most cancer deaths come from metastasis: cells that leave, travel and settle elsewhere.

2D vs 3D

A skater on ice, or a hiker in a forest.

On a dish · 2D
The ice rink.One flat, rigid surface. Easy to image, easy to repeat.
In tissue · 3D
The forest.Collagen fibres on every side, gaps often smaller than the nucleus.

Migration in 2D

Protrude, adhere, contract, release.

A crawling cell · top viewCrawling
1 · ProtrudeActin grows at the front and pushes out a thin sheet, the lamellipodium.
2 · AdhereIntegrins grab the surface and cluster into focal adhesions: the feet.
3 · ContractMyosin pulls on actin. The body is hauled forward. This is traction force.
4 · ReleaseAdhesions at the rear let go, and the tail follows.

Migration in 3D

In 3D, the matrix sets the rules.

Mesenchymal
Grab, pull, cut.Spindle-shaped, adhesive, uses MMPs to make a path. Slow but strong. Fibroblasts, many cancer cells.
Amoeboid
Squeeze through.Round and soft, fast, no cutting. The stiff nucleus limits how small a gap it can pass.

Step 1 · detection

No stain. Just light.

Brightfield · z = 40 µm0 cells in focus
Why not a dye?

Dyes and strong light can stress cells, and stressed cells move differently.

The contrast flip

Out of focus one way: bright core. The other way: dark core. In focus: almost invisible.

→ x, y and z, label-free

Step 2 · linking

Connect the dots, get a trajectory.

Detections in each frame
Linking

Match each cell to itself in the next frame. Hard cases: crossing, dividing, leaving the field.

A trajectory
r(t) = ( x(t), y(t), z(t) )
t (min)xyz (µm)
0112.487.141.0
10118.990.642.2
20123.097.841.7
…………

Measure 1 · speed

Speed depends on how often you look.

The true path
v = | r(t+Δt) − r(t) | / Δt
Every 1 min

–

Every 30 min

–

Same cell. Report Δt, and compare like with like.

Measure 2 · directness

Equally fast. Not equally far.

Two cells · same path length L
D = d / L

d: start-to-end distance. L: total path length. 1 = straight, near 0 = turning.

Watch out

D falls the longer you track. Compare tracks of equal duration.

Measure 3 · MSD

Mean squared displacement, step by step.

One trajectory
MSD(τ) = ⟨ | r(t+τ) − r(t) |² ⟩averaged over all t, and over cells

Measure 3 · MSD

The slope tells you the kind of walk.

log MSD vs log τ
α = 1 · random walk
MSD = 4Dτ (2D)
α = 2 · directed
MSD = v²τ²

α < 1 · confined caged or anchored

Persistent random walk
MSD = 2S²P [ τ − P(1 − e−τ/P) ]

S: speed. P: persistence time.

Measure 4 · direction

Who is following what?

Chemotaxis · attractant →

Chemotaxis · durotaxis · contact guidance

FMI = Δxgradient / L

0 = no preference, > 0 = toward the signal.

Putting it together

Reading migration like handwriting.

Randomα ≈ 1 · low D
Persistentα 2 → 1 · long runs
Directedhigh FMI
Confinedα < 1 · MSD plateaus
Collectiveneighbours aligned
Stop-and-golook at the distribution

2D vs 3D analysis

3D is not 2D plus one.

3D track and its flat shadow
MSD3D = 6Dτ
MSDxy = 4Dτ

A flat view keeps two thirds of a random walk, and can miss vertical motion entirely.

But z is blurrier

Precision along the optical axis is lower. Measure in 3D, check z, and state what you report.

2D vs 3D migration

Does 2D predict 3D?

Same cells, dish vs gel
Often, no

Fast on a dish is not fast in a gel. Fibrillar 3D migration looks more like 1D migration on lines.

Doyle et al. · J Cell Biol 2009

Why

No floor · nucleus vs pore size · ~10⁶× softer · fibre guidance. 3D steps are anisotropic and heavy-tailed.

Wu et al. · PNAS 2014

MSD: a first test

Same Δt, per dimension. Compare α, S, P. Then check step distributions, direction memory and anisotropy.

Example · simulated data

All of it, in one gel.

400 × 400 × 200 µm · 12 h0 h
PersistentRandomDrag to turn

Mean squared displacement

Speed and persistence in-plane, where precision is best. Direction and depth in full 3D.

Before you analyse

Five classic traps.

1Short tracks

Long-lag MSD rests on few pairs. Trust lags up to ~¼ of track length.

2Leaving the field

Fast cells exit first, so long tracks favour slow cells.

3Drift

A sliding gel makes every cell look directed. Subtract shared motion.

4Divisions

Cells round up and stop. Split the track at division.

5What is n?

Gels, donors or experiments are your units. Not the thousand cells.

Recap

One list of positions. A lot of biology.

Speedhow active
Directness · Phow decisive
MSD αwhat kind of walk
FMIwhat it follows

Not wiggling dots. Decisions.

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