On Growth and Form, and Function: Reusable Regulatory Handles Control Phenotypic Variation
Benedikt Hartl, Milton L. Montero, Marcello Barylli, Sebastian Risi, Michael Levin
cs.NE, cs.LG
2026-09-24
Rank-one LoRA on neural cellular automata: 2.6% of weights parametrically control phenotype scaling and transfer zero-shot across all tested scaffold-sharing morphologies.
In 1917, D'Arcy Thompson drew a famous set of grids: lay Cartesian coordinates over one fish's anatomy, deform the grid, and you often land on the body plan of a related species. That shape relations can be described by simple geometric transforms has been settled for a century. The missing half is causal: which change in the underlying developmental dynamics grows one body plan into the other? The paper calls this Thompson's century-old inverse problem.
You cannot run that experiment in living tissue at will, so the authors work with neural cellular automata (NCAs) as a minimal model: every cell in a grid holds a 32-dimensional state vector and runs the same two-layer network that reads its neighbors and updates itself; from a single seed cell the collective grows a target image, here an emoji, and the shared network stands in for a gene-regulatory network. The question becomes whether coherent morphological transformations can be encoded as low-dimensional directions in that network's weight space. The team spans Michael Levin's group at Tufts' Allen Discovery Center and Sebastian Risi at the IT University of Copenhagen (also Sakana AI).
Three stages.
First, a shared scaffold. 120 Noto emoji targets are trained jointly: the scaffold weights W0 and one rank-16 LoRA adapter per target are optimized together, with a one-hot activation selecting which adapter grows. After training the adapters are discarded; W0 corresponds to no phenotype and serves purely as the common reference frame. Ten parallel workers merge their scaffolds into a shared checkpoint every 100 iterations with 0.9:0.1 weighting.
Second, learned scale directions. Two rank-one LoRA adapters, one horizontal and one vertical, are attached to a fish-growing NCA (W0 plus the fish adapter) and trained jointly over 25 prescribed scalings (base 37x37, the full grid of sizes {25,31,37,43,49}). The coefficient is logarithmic, beta(s) = beta0 log(s/s0), so stretch and shrink are symmetric in weight space and the base size maps to zero modulation. Final weights are W0 + target adapter + betax Deltax + betay Deltay; rank one means 320 parameters, 2.6% of the 12,416 in the network.
Third, a census of weight space. With the scaffold frozen, one rank-16 LoRA is fitted for each of roughly 25,000 emoji-vendor pairs (nine vendors), then PCA is run over the full weight space to see which phenotypic features the components track.
| Item | Result |
| Scale control | two rank-one adapters (320 params, 2.6%) cover all 25 trained scalings |
| Out-of-distribution | trained on 5 sizes, still effective across the continuous [15,57] grid (LPIPS, 16 stochastic rollouts per point) |
| Cross-phenotype | the same Deltax/Deltay transfer zero-shot to a lizard, a fire extinguisher and other structurally different targets, internal features preserved |
| Weight-space structure | PC0 vs horizontal extent r=-0.71, PC1 vs vertical r=-0.73, cross terms only 0.07/0.08 |
| Degeneracy | learned vs PCA scale directions: cosine similarity about 0.1 |
Zero-shot transfer is the strongest result. The scale directions learned on the fish, applied unchanged to NCAs growing a lizard or a fire extinguisher, rescale them while internal detail and proportions survive, and anisotropic scaling produces an effective shear on oriented shapes. The adapters encode system-level geometric control, not memorized features of the fish.
The census produced its own findings. The first two principal components align with horizontal and vertical extent, and translating any phenotype's weights along them compresses or stretches its developmental outcome. The mean weight difference between OpenMoji-style targets and the rest acts as a style direction that adds the vendor's typical black outline. The centroid difference between symmetrically split targets and the rest acts as a fission direction: above roughly beta = 2 it splits any phenotype along the vertical axis, and the reverse direction fuses split ones.
The degeneracy is itself informative: learned scale directions and PCA-derived scale directions are nearly orthogonal (cosine about 0.1) while controlling the same transformations, meaning multiple regulatory paths lead to the same macroscopic outcome within one substrate, which the authors connect to poly-computing.
LoRA here works as an instrument, not a fine-tuning shortcut. Bolted onto a self-organizing dynamical system, it exposes an interpretable semantic geometry in parameter space: size, style and fission are directions you can walk along. For weight-space interpretability researchers this is a compact case study where the "condition" of a conditional generative process is a low-rank perturbation of the generative rule itself, not a token at the input.
For Levin's line of work on bioelectricity and regenerative medicine the implication is larger: Pattee's 1973 anticipation of system-level control handles gets a computational instance, where macroscopic outcomes shift without instructing every cell individually. The stated long-term goal is an "anatomical compiler" that generates developmental programs on demand, the way image models generate pictures. Honestly, today this is a proof of concept on 2D emoji with no physics.
The authors' own list: an abstract 2D grid without physical constraints; weights that cannot be mapped to specific genes or morphogens; entangled directions, since the scale PCs also shift brightness and the style direction also shrinks size; and fine-grained reconstruction of individual dynamics needs thousands of components, so the geometry is only semantic at the level of coarse features.
Two further caveats. Universality is scoped to one scaffold: zero-shot transfer requires every NCA to share the jointly trained W0, and the paper itself argues that independently trained NCAs are related by gauge symmetries that make their weights incomparable, so "universal directions" cover less than the phrase suggests. And the regeneration study in the appendix is a negative result: transferring the weight difference between regenerative and non-regenerative versions onto other phenotypes produced largely dysfunctional dynamics, so the regenerative-medicine pitch does not yet work inside the model itself.
There is also no quantitative comparison against full fine-tuning or existing conditioning schemes such as goal-guided NCAs; the LPIPS heatmaps are qualitative evidence. Disclosure: the work was partly funded by Astonishing Labs, co-founded by Levin.