Test a pattern mechanism with an intervention

Specify the competing explanations first

The active candidate is the preceding cortex model with one unknown, contractility $\zeta$ . The passive candidate sets active stress to zero while retaining the same diffusion and turnover. Both receive the complete initial density and all other parameters. The intervention changes only active stress.

The passive candidate then predicts identical inhibited and control evolutions, hence zero paired response. The active candidate changes flow and predicts a response varying in position and time. An intervention that also damages material or changes turnover would require those effects in the candidates.

At position $x_j$ and time $t_r$ , subtract control density from inhibited density. Both begin with the same field, preventing an initially different peak from being counted as an intervention effect.

y_{jr}=c_{\mathrm{inhibited}}(x_j,t_r)-c_{\mathrm{control}}(x_j,t_r)+\epsilon_{jr}

Noise is added directly to the paired difference. Separately noisy measurements followed by subtraction would require a different difference-noise variance.

State spatial variation and noise separately

Each friction condition starts with the default cosine density plus a sine disturbance of amplitude $0.04$ and evolves for four time units under its own friction. Each condition has a distinct saved initial field shared by all candidates in that condition. It is a known initial condition, separate from material parameters.

Each condition samples 16 equally spaced sites at times $0.25,0.5,0.75,1$ , giving 64 observations. Errors $\epsilon_{jr}$ are independent, zero-mean Gaussian values with standard deviation $\sigma$ . There is no process noise or unknown spatial friction.

Use only friction $\xi=1$ to choose the least-squares fit from $\zeta=1,1.25,\ldots,5$ . Freeze that parameter and predict friction $\xi=2$ . Observations in this held-out condition do not influence parameter selection.

Known initial structure, unknown spatial material heterogeneity and random noise are distinct. This exercise handles the first and observation noise. Unknown heterogeneity needs its own spatial scale, amplitude and estimation procedure.

Observe prediction error as noise increases

The table uses the first saved noise seed, 510210, rounded to three significant figures. Root-mean-square error squares each prediction error, averages all sites and times, then takes the square root. Its unit remains dimensionless density difference. The generating contractility is three.

At low noise the active fit recovers the generating parameter and predicts the held-out condition better. As noise rises the estimate shifts and the errors draw closer. The question becomes how much the record can discriminate.

Eight fixed seeds are saved per noise level. The active candidate has lower held-out error in all eight at standard deviations 0.02 and 0.08, but only seven at 0.3. These finite examples illustrate variability; they do not establish a general success probability.

\operatorname{RMSE}=\sqrt{\frac1{64}\sum_{j,r}\bigl(y_{jr}-\widehat y_{jr}\bigr)^2}

Compute held-out error after freezing the fitted parameter. Further tuning on these observations removes their held-out status.

Same-model synthetic observations, seed 510210, held-out friction two. Raw observations, initial fields and predictions are available for recalculation.
Noise $\sigma$	Fitted $\widehat\zeta$	Active error	Passive error
$0.02$	$3$	$0.0215$	$0.0741$
$0.08$	$2.75$	$0.0870$	$0.119$
$0.3$	$1.75$	$0.328$	$0.343$

Before transferring the exercise to tissue

The data generator belongs to the active family used for fitting, giving that model-form assumption an inherent advantage. This same-model exercise tests sampling, fitting and held-out evaluation, and illustrates specified noise effects. It is not independent biological validation.

With tissue, first test intervention specificity, initial conditions, boundaries and imaging error. Candidate models must share the same observation definition. A response can reject one passive candidate while remaining compatible with several active regulatory networks.

This suggests useful frontier measurements: concentration and flow recorded together with intervention. Density locates material, velocity helps distinguish transport from reaction, and a held-out condition asks whether the explanation extends beyond the fitted image.

Reading: M. Nishikawa

Worked example

How strongly does the smaller error speak?

For the first record at noise $0.08$ , calibration gives $\widehat\zeta=2.75$ . Held-out RMSE is approximately $0.0870$ for the active candidate and $0.119$ for the passive one. The active candidate predicts better in this specified synthetic comparison with a known initial field and ideal intervention.

The difference does not identify a molecular pathway or attach biological units to the fitted value. If stronger noise reverses the ranking, examine whether the record contains enough information, then consider additional times, repetitions or a more discriminating intervention.

Download the authored records

Synthetic intervention observations and predictions (JSON)

Try the reasoning

Reserve a condition before tuning

Fix an initial field and inhibition position, then record inhibited-minus-control density.
Choose contractility using one friction condition only, then freeze it.
After changing the friction preset, reapply the fitted contractility. Use the downloadable record and accompanying script for exact held-out replay. Separate parameter error, observation noise and model-form error.

Exercises and explanations

Why can repeatedly tuning on a held-out record invalidate the reported prediction error?
Read the explanation
Parameter selection has used its information. Reserve a new condition or record for final evaluation and report the selection process.
Does a zero prediction here mean that every passive physical mechanism has zero intervention response?
Read the explanation
No. Zero follows from the assumed intervention affecting only active stress. If diffusion, connectivity or turnover also change, passive candidates can respond and need to be reformulated.
What can still be checked independently in a same-model exercise?
Read the explanation
Conservation relations, passive limits, analytic linear spectra and another numerical solver can test implementation. Biological model validation requires independent observations, interventions and prediction conditions.

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.