FR21 · Active material

Contractile cortex and feedback

How can contraction gather more material, and what happens when one region stops pulling? A denser patch can pull the surrounding cortex inward and carry more material into itself. Diffusion and turnover oppose this feedback; locally suppressing active stress changes the flow and density together.

Watch a cortex gather into dense bands. Move a local inhibitor around the ring and compare the response with an untouched copy.

Model assumptions

This is a dimensionless one-dimensional active viscous layer on a fixed periodic ring, with density transport and turnover. Ring thickness indicates density; the ring does not deform. Inhibition reduces active stress and does not sever the material. Elastic storage, anisotropy and biochemical oscillators are omitted.

Cyan: material densityAmber: inhibition location

Fixed ring · thickness represents density · arrows show calculated flow

Time

4.00

Density variation

0.200

Peak flow speed

0.100

Mean density

1.00

Density around the ring

Cyan is the current ring. Dashed gray is an untouched copy advanced from the same state. Amber marks the inhibited region.

Does a local release change the pattern?

Both curves are computed trajectories. A change in active pulling can alter the density even though local reaction rates stay the same.

Start with the essentials

Focus question: How can contraction gather more material, and what happens when one region stops pulling?
One-sentence intuition: Contractile feedback can amplify a density variation even with simple local turnover. Diffusion and renewal compete with the gathering flow.

Core mathematical model

Stress drives an overdamped flow

\sigma=\eta\,\partial_x v+\zeta I(x)\frac{c}{1+c},\qquad\partial_x\sigma=\xi v

The density, time and coordinate are dimensionless. Viscous stress and saturating active stress add. Friction balances their spatial gradient; inertia is omitted.

Density moves, diffuses and renews

\partial_t c=-\partial_x(cv)+D\partial_x^2c+k(1-c)

The ring is periodic. Transport conserves total material; turnover exchanges material with a uniform reservoir. With turnover off, the mean density is conserved even during inhibition.

Growth of a small density wave

\lambda(q)=\frac{\zeta q^2}{4(\xi+\eta q^2)}-Dq^2-k

This expression follows by linearizing about unit density with no inhibition. It predicts whether a small spatial ripple grows before nonlinear saturation matters.

Common difficulties

Inhibition leaves the ring connected

Typical misconceptionThe amber mark is a cut in the material.

Better mental modelThe marker multiplies local active stress by a smooth suppression profile. Material can still flow across it. A real cut would need new free-edge conditions.

The ring thickness is a density scale

Typical misconceptionA thicker patch means the whole cell has deformed by that amount.

Better mental modelThe periodic coordinate and radius stay fixed. Thickness encodes density, and the arrows show computed tangential flow. Cell shape is outside this one-dimensional model.

Run the experiment

01
Release one region
Choose Release one region. Move the amber handle and compare the cyan profile with the dashed reference.
What to observe: The two trajectories separate because only one ring has its local active stress reduced. No reaction parameter has changed.
02
Make a prediction at stronger friction
Reset, choose Stronger friction, then inhibit the highlighted region with the same strength and width.
What to observe: Friction changes the flow that redistributes density. This second condition tests a mechanical prediction beyond a single final pattern.

Contractile cortex and feedback

Density around the ring

Does a local release change the pattern?