How contraction draws more material together

Follow material around a cortex ring

The cell cortex is a thin layer beneath the membrane, containing actin filaments and molecular motors. Motors consume chemical energy and generate tension. We extract one periodic direction and follow its density and tangential flow; a ring makes the periodic connection explicit.

Coordinate $x$ increases around the ring, with $0\le x<24$ . Relative density is $c$ and positive $v$ points along increasing $x$ . All quantities are dimensionless. Positive total stress denotes tension, and $\xi>0$ measures drag against the environment.

Reading: M. Nishikawa

Stress gradients drive the flow

Neglect inertia and balance the stress gradient against drag. Viscous stress is $\eta\partial_xv$ . Active tension takes the saturating form $\zeta c/(1+c)$ , which stays bounded at high density.

Uniform tension does not drive a flow. A slightly denser region has a higher active tension, producing gradients toward its peak. Convergent flow can bring in more material and increase tension again: mechanical positive feedback.

This is an active viscous material model. It contains neither elastic displacement nor stored elastic energy. The source paper includes biochemical regulation; the single density and simple turnover here isolate one part of the feedback.

\sigma=\eta\partial_xv+\zeta I(x)\frac{c}{1+c},\qquad \partial_x\sigma=\xi v,\qquad \xi v=\eta\partial_x^2v+\partial_x\!\left[\zeta I(x)\frac{c}{1+c}\right]

The active coefficient is positive. Inhibition defaults to unity. Velocity is solved from periodic force balance.

Reading: M. Nishikawa

Density changes through flux and turnover

Material flux along the positive direction is $cv-D\partial_xc$ . The first term follows the flow and the second diffuses down concentration gradients. Density changes through incoming minus outgoing flux, plus exchange with a reservoir.

Turnover $k(1-c)$ supplies a region below reference density and removes material above it. With turnover off, the total amount around the periodic ring is conserved. With turnover on, the total approaches its reference value.

\partial_tc=D\partial_x^2c-\partial_x(cv)+k(1-c),\qquad \frac{d}{dt}\int_0^{24}c\,dx=k\left(24-\int_0^{24}c\,dx\right)

Periodic transport terms cancel on integration. This global relation independently checks the density update.

Calculate the contractile growth rate

Add a small wave around $c=1,v=0,I=1$ . The slope of the saturating function there is $1/4$ . At wave number $q$ , viscosity and drag divide the velocity response by $\xi+\eta q^2$ .

Substituting the velocity into the density equation gives three contributions: diffusion smooths, turnover restores reference density, and active contraction amplifies disturbances. Spatial instability occurs when the active contribution wins. Unequal diffusion of two species is not required for this mechanism.

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

This continuous linear result applies near the homogeneous state before local inhibition. The numerical stencil and flux discretization also affect discrete growth rates.

Reduce active stress in a small region

The intervention multiplies active stress by $I(x)=1-d\exp[-s(x,x_0)^2/(2\ell^2)]$ . Here $s$ is shortest periodic distance to $x_0$ , $d$ is inhibition depth and $\ell$ is width. Defaults $d=0.9$ and $\ell=0.8$ leave one tenth of the original active stress at the center.

The ring remains connected and retains viscous coupling. Severing material with a laser would change its mechanical boundaries and needs another model. Comparing inhibited and control evolutions separates intervention response from natural evolution.

Inhibiting a density peak and inhibiting a trough need not produce the same response. The next chapter retains known initial spatial variation, adds observation noise and tests predictions in a condition withheld from fitting.

Worked example

Switch contraction on and off at one scale

Take $\xi=1,\eta=0.5,D=0.08,k=0.1,\zeta=3$ and continuous wave number $q=1$ . The active term is $3/[4(1+0.5)]=0.5$ , diffusion contributes $0.08$ and turnover contributes $0.1$ , giving $\lambda=0.32>0$ .

Turning contraction off gives growth rate $-0.18$ , so the disturbance decays. Actual wave numbers on the ring are discrete. For quantitative comparison on the finite ring, use $q=2\pi n/24$ with an integer mode number.

Try the reasoning

Connect a density peak to convergent flow

Locate regions where the default flow converges toward a density peak.
Compare weak and strong contraction. Predict which supports growing disturbances.
Drag the inhibition region and compare the inhibited and control records. Explain the sequence from altered stress through velocity response to density redistribution.

Exercises and explanations

Does uniform nonzero active tension require a nonzero velocity?
Read the explanation
No. A stress gradient drives flow in this model. Uniform density, uniform tension and zero velocity satisfy the periodic force balance.
With $k=0$ but active contraction retained, how should integrated density change?
Read the explanation
It remains constant. Flow and diffusion redistribute material within the periodic ring without net escape, even while strong local peaks form.
After inhibition a point becomes less dense. How can you distinguish transport away from local breakdown?
Read the explanation
Compare flux divergence with the turnover term and inspect total amount. The flux and turnover calculation separates these contributions.

Course sources and calculation scope

The derivations, diagrams and numerical examples are authored for this course. Source locations below distinguish adopted equations, teaching extensions and the one exact published example recalculated here. Interactive fields and noisy observations are model-generated.

A. M. Turing, The Chemical Basis of Morphogenesis (1952), Philosophical Transactions B 237, 37–72
Read §4, pp. 42–43, for the two-cell numerical example and §6 for mode analysis. The reproduced result is early linear growth in §4. No source figure is copied; the Schnakenberg experiments use a different model.
J. Schnakenberg, Simple Chemical Reaction Systems with Limit Cycle Behaviour (1979), Journal of Theoretical Biology 81, 389–400
Original reaction-model reference. The dimensionless irreversible teaching form is specified here; its equilibrium, Jacobian and numerical examples are derived in the course. Parameters are not biological measurements.
D. Barkley, A Model for Fast Computer Simulation of Waves in Excitable Media (1991), Physica D 49, 61–70
Read the two-variable equations in §2 and the discussion of applicability. The experiment solves those continuous equations with its own discretization and teaching initial conditions; source images and rotation frequencies are not reproduced.
E. J. Crampin, E. A. Gaffney and P. K. Maini, Reaction and Diffusion on Growing Domains (1999), Bulletin of Mathematical Biology 61, 1093–1120
Read §3, equations 9–20, on conservation, transport and growth. The paper mainly studies one-dimensional growth. The two-dimensional uniformly expanding square is a separately derived teaching extension, not a reproduction of its mode-doubling figures.
M. Nishikawa, S. R. Naganathan, F. Jülicher and S. W. Grill, Controlling Contractile Instabilities in the Actomyosin Cortex (2017), eLife 6, e19595
Read Appendix equations 4–9 on density flux, active viscous stress, saturation and force balance. This teaching reduction uses one density, linear turnover and local stress inhibition. It does not fit embryo data or reproduce Figure 2. The source is CC BY.
A. L. Krause, M. A. Ellis and R. A. Van Gorder, Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds (online 2018; issue 2019)
Read equations 4–6 for moving-domain conservation, 14–15 for material coordinates and 44–45 for Schnakenberg reactions and Jacobian. The periodic square and parameters are teaching choices; ellipse geometries and Figures 1–4 are not reproduced.
L. Giomi, Geometry and Topology of Turbulence in Active Nematics, arXiv:1409.1555v2 (2015); Physical Review X 5, 031003
Frontier branch reading on two-dimensional incompressible active nematics, vortex statistics and defects. Those orientation and fluid equations belong to FR22; their results are outside the one-dimensional scalar cortex model.