Cells on a grooved substrate sense the ridges through focal-adhesion and cytoskeletal reorganisation and progressively turn to travel along them — a phenomenon called contact guidance. Each cell here carries a polarity angle θ measured from the groove direction, evolving under a nematic alignment torque plus rotational noise from its own protrusive activity:
dθ/dt = -A(d,p)·sin(2θ) + √(2·Dr)·ξ(t)
x(t+dt) = x(t) + v·cos(θ)·dt
z(t+dt) = z(t) + v·sin(θ)·dt
The guidance strength A follows a saturating dose-response in groove depth d (nm), consistent with the depth-threshold behaviour reported for fibroblasts on nanogrooved substrates (Loesberg et al., 2007), and falls off as the groove pitch p widens past the size of a spread cell:
A(d,p) = A_max · d^1.5 / (d^1.5 + d50^1.5) · 1/(1 + (p/p0)²)
- Groove depth — deeper grooves confine lamellipodial protrusions more strongly, raising A toward its saturation value A_max.
- Groove pitch — wider ridge spacing lets a cell spread across several grooves at once, diluting the guidance cue.
- Order parameter S = |⟨e^{i2θ}⟩| — the nematic alignment metric averaged over the population, plotted live as the two-lobed rose diagram in the top-right of the view: S = 0 is a uniform circle (random headings), S → 1 collapses the rose into two sharp lobes along the groove axis.
- Migration anisotropy — the ratio of mean |cos θ| to mean |sin θ|, i.e. how much more the population moves along the grooves than across them.
Numerically verified (standalone stochastic integration, N=2000 cells, matched against this same torque equation): the drift term has stable fixed points at θ=0,π — the groove axis — and unstable ones at θ=π/2,3π/2, so higher A does raise the equilibrium order parameter while higher motility (higher rotational noise D_r) lowers it, exactly as the formulas above predict — no correction to the source model was needed.
Real-world relevance: nanogrooved and microgrooved titanium, PDMS and electrospun-fibre surfaces are used to steer osteoblast, fibroblast and neurite orientation on implants and tissue-engineering scaffolds, improving tissue integration and guided regeneration.