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On Growth and Form, and Function: Reusable Regulatory Handles Control Phenotypic Variation

Benedikt Hartl, Milton L. Montero, Marcello Barylli, Sebastian Risi, Michael Levin

arXiv:2609.29755v1cs.NEcs.LG

TL;DR

The paper asks how simple morphological transformations map onto developmental regulatory dynamics. It uses LoRA-adapted neural cellular automata to test for compact, reusable regulatory directions, finding that low-rank adaptations control scaling and other phenotypic variations. These results provide a computational realization of Thompson-style transformations in an NCA model.

  • Problem

    The paper asks how coherent transformations between related anatomies correspond to changes in developmental regulation, an inverse problem not resolved by shape alone.

    Source: §1 · p. 2

  • Method

    The authors apply low-rank adaptation to a pretrained neural cellular automaton, modeling how compact changes to its shared regulatory network redirect collective growth.

    Source: §1 · pp. 2, 4

  • Results

    Rank-one adaptations implement horizontal and vertical scaling, whose combinations generalize beyond training and transfer across phenotypes; approximately 25,000 shared-scaffold adapters reveal directions associated with extent, style, and symmetrical fission.

    Source: §1 · p. 5

  • Takeaways & Limitations

    The results computationally realize Thompson-style grid transformations as controllable low-dimensional directions in an NCA’s regulatory weight space.

    Source: §1 · pp. 2, 5

    Initial efforts to map non-regenerative to regenerative NCA solutions did not yield a generalizable hyper-direction, leaving transfer of regenerative capabilities for future work.

    Source: §3.3 · p. 11

Abstract

from arXiv · show

How phenotypic transformations are implemented by changes in underlying regulatory dynamics remains a central question in developmental biology. Inspired by D'Arcy Thompson's 1917 "On Growth and Form", we ask whether coherent large-scale transformations of morphology can be encoded as low-dimensional modulations of a self-organizing developmental system. We use neural cellular automata (NCAs) as bio-inspired models of distributed development, in which a shared local regulatory network grows target morphologies from a single cell. We apply low-rank adaptation (LoRA) to pretrained NCAs, representing each adapted developmental program as a low-rank modulation of a fixed regulatory scaffold. Horizontal and vertical scaling of a fully grown 2D emoji phenotype can each be implemented by rank-one adaptations. Their linear combinations parametrically control phenotype size, generalize beyond the training distribution, and compose with target-specific adapters. Strikingly, adaptations learned for one phenotype transfer zero-shot across structurally and semantically diverse phenotypes sharing the same reference scaffold, while largely preserving internal features. This suggests reusable system-level hyper-directions of scale rather than morphology-specific transformations. From approximately 25,000 independently trained phenotype-specific NCA adapters with a shared scaffold, we further identify latent low-dimensional directions that functionally control phenotypic variation including scaling, style, and symmetrical fission. Together, our results provide a computational realization of D'Arcy Thompson's remarkable grid transformations in a 2D NCA---a minimal cybernetic tissue in which variations of fully grown emoji phenotypes can be encoded, combined, and controlled through low-dimensional directions in regulatory weight space.

1 Introduction

The paper asks how simple transformations of organismal form map onto changes in developmental regulation, extending D’Arcy Thompson’s geometric insight toward a mechanistic account. It tests whether compact regulatory shifts in neural cellular automata (NCAs) can control coherent phenotypic variation.

1 Introduction

  • Thompson’s grid transformations relate biological shapes geometrically, but leave open which developmental processes produce the differences and where or when they must change.The paper frames this as an inverse problem: finding the developmental-dynamics modulation corresponding to a simple transformation between anatomies.Source: §1 · p. 2
  • NCAs model cells as agents whose shared regulatory rules let them grow and maintain a target morphology through repeated local interactions.The paper uses this framework to examine how parameter changes in cellular regulation may redirect collective development.Source: §1 · p. 4
  • The study applies LoRA to a pretrained NCA scaffold and reports that rank-one adaptations control horizontal and vertical scaling, with combinations generalizing beyond training and transferring across target phenotypes.A shared-scaffold collection of approximately 25,000 phenotype-specific adapters also reveals directions associated with phenotypic extent, style, and symmetrical fission.Source: §1 · p. 5
  • The authors propose that low-dimensional regulatory directions can change macroscopic features without prescribing detailed cellular trajectories, offering a computational account of reusable control handles.They suggest analogous compact regulatory changes could provide a route for evolution to access coherent transformations of form and function.Source: §1 · p. 5

2 Methods

The methods use neural cellular automata to model collective growth from local cell updates, then adapt their shared regulatory network with low-rank parameter changes.

2.1 Neural Cellular Automata

  • In an NCA, cells on a grid use local neighborhood information and a shared trainable ANN update rule to produce collective self-organization.The ANN replaces the fixed update function of a conventional cellular automaton; the described architecture uses a 3 × 3 Moore neighborhood and identity and Sobel perception filters.Source: §2.1 · p. 5
  • The shared cellular policy is optimized so the NCA grows toward a predefined target pattern from an initial seed within a developmental time frame.The visible output is the RGBA portion of the complete grid state.Source: §2.1 · p. 6

2.2 Low-Rank Adaptation of Neural Cellular Automata

  • LoRA adapts a frozen reference NCA by adding trainable low-rank factors to its weights, refining baseline behavior toward a new target.This replaces adjustment of all layer weights with optimization of the low-rank factors, reducing the number of trainable parameters.Source: §2.2 · p. 6
  • The method applies corresponding LoRAs to hidden and output layers, while keeping the perception block fixed in the implementation described.The passage notes that trainable convolutional or attention-based perception kernels could also be modified with LoRAs.Source: §2.2 · p. 6

3 Results

The results show that compact LoRA changes to a shared NCA scaffold can control scale and transfer across phenotypes, while weight-space analyses reveal additional directions for scale, style, and fission.

3.1 LoRA-NCA: Finetuning Pretrained Neural Cellular Automata

  • NCAs model development as repeated local updates by a shared regulatory ANN, and the study asks whether compact parameter changes can redirect this collective growth toward transformed phenotypes.A pretrained NCA supplies a reference regulatory program for testing persistent, parametrically controlled transformations and transfer across morphologies.Source: §3.1 · p. 7, §3.2 · p. 8

3.2 Learned Regulatory Hyper-Directions control Phenotypic Scale-Variation

  • All 25 tested horizontal–vertical scaling transformations were implemented with rank-one adaptations that also generalize to unseen sizes.The paper reports that rank one comprises 320 parameters, or 2.6% of the NCA’s full ANN parameter count.Source: §3.1 · p. 7, §3.2 · p. 8

3.3 LoRA finds Universal Mechanistic Hyper-Directions of Phenotypic Variation, in Emoji-World

  • Target-specific rank-16 adapters can be composed with rank-one scale directions to apply continuous scaling while retaining a target-specific modulation.The combined weights add the target-specific modulation and the scale-dependent adapters to the shared scaffold.Source: §3.3 · p. 9
  • The same scale adapters transfer zero-shot to tested phenotypes sharing a scaffold, including out-of-distribution sizes, while largely preserving internal features.Anisotropic scaling can also produce effective shear transformations for oriented phenotypes.Source: §3.3 · pp. 10–11
  • Initial mappings between non-regenerative and regenerative NCAs did not yield a generalizable hyper-direction, leaving regenerative transfer unresolved.The authors present regenerative control as a potential application for future work, not an established result.Source: §3.3 · p. 11

3.4 Phenotypic Variation is encoded in Latent Regulatory Geometry

  • Across approximately 25,000 adapters, PC0 and PC1 correlate with horizontal and vertical extent (r = −0.71 and r = −0.73) and modulate growth along those axes.These empirical scale directions have only approximately 0.1 cosine similarity to the learned scale directions, and the PCs also affect brightness.Source: §3.4 · pp. 11, 13
  • The OpenMoji direction adds blackish outlines while preserving baseline features, whereas the fission direction splits targets vertically and its inverse can fuse them, sometimes losing details.Robust splitting occurs above a target-specific threshold of β ≳ 2.Source: §3.4 · p. 13

4 Discussion

Rank-one LoRA adaptations control horizontal and vertical scaling, combine beyond training conditions, and transfer across phenotypes sharing a scaffold. Analysis of roughly 25,000 adapters also reveals functional directions for scale, style, and fission, alongside limits on their specificity and biological interpretation.

4 Discussion

  • Rank-one adaptations control horizontal and vertical scaling, combine to generalize beyond training conditions, and transfer across scaffold-sharing phenotypes while largely preserving internal features.The two scaling adaptations amount to approximately 2.6% of the NCA’s total parameters.Source: §4 · p. 13
  • Across approximately 25,000 phenotype-specific adapters, principal components correlate with horizontal and vertical extent, while subset differences yield functional directions for style and symmetrical fission.Adding these directions to phenotype-specific adaptations changes NCA dynamics to compress or stretch outcomes, transfer style, or split and fuse morphology.Source: §4 · p. 14
  • NCA weights cannot yet be assigned directly to particular genes, morphogens, physiological signals, or developmental events, and equivalent biological control handles remain unknown.The authors identify physically and biologically grounded models as a possible route to finding such handles.Source: §4 · p. 14
  • Learned and PCA-derived scale directions control the same phenotypic extents despite having at most about 10% cosine similarity, indicating degenerate morphogenetic control.Distinct weight-space directions can modulate the same macroscopic feature.Source: §4 · p. 14
  • Scale, style, and fission handles provide multiple regulatory paths to phenotypic variation; their transfer and recombination are proposed as a computational analogue of facilitated variation.The discussion suggests that reuse could expand accessible variation through recombination rather than only through new point mutations.Source: §4 · pp. 14–15
  • Whether reproductive machinery uses stress-guided, heritable regulatory shifts or actively selects and combines functional handles remains an open question.The proposed stress-guided process is conditional on regulatory change being coupled to inheritance.Source: §4 · p. 15
  • Leading weight-space directions are not fully feature-specific, and detailed reconstruction requires thousands of components; their biophysical meaning remains elusive in an abstract 2D grid.Future work is proposed on more structured representations and more biophysically grounded models.Source: §4 · p. 15
  • The authors propose a path toward inferring local rule changes for desired anatomical transformations and a future conditional anatomical compiler.They suggest that biophysically grounded extensions may help identify interventions for regeneration and developmental repair.Source: §4 · p. 15

Conflict of Interest

The authors disclose partial funding from Astonishing Labs and M.L.’s financial and invention-related ties to the company.

Conflict of Interest

  • Tufts research was partially funded under a Sponsored Research Agreement with Astonishing Labs; M.L. is a co-founder and shareholder, and the company has certain rights to associated inventions.Source: Conflict of Interest · p. 16

A.1 NCA Parameters

The NCA uses 32 state channels and local neighborhood features processed by pointwise convolutional layers to generate residual updates. Training evolves pools of developmental states against normalized, padded emoji targets.

A.1 NCA Parameters

  • Each cell carries 32 state channels, with RGBA and alpha-based viability; identity and Sobel filters produce 96 neighborhood features processed by two pointwise convolutional layers.A ReLU activation and zero-initialized output kernel generate residual state updates, while cell-wise dropout implements asynchronous updates.Source: §A.1 · p. 24
  • Targets are resized to 37 × 37 pixels, padded onto a 67 × 67 lattice, and grown from one living seed using pools of 1,024 states evolved for 128 steps.Training evaluates mean-squared reconstruction loss at a randomly selected step from 64 through 127.Source: §A.1 · p. 24

A.2 Shared-Baseline Training

A shared scaffold is trained from multiple emoji targets, then frozen as the reference for independently fitted target adapters. A separate procedure learns paired rank-one adaptations for prescribed horizontal and vertical scaling transformations.

A.2 Shared-Baseline Training

  • The shared scaffold was jointly optimized with rank-16 adapters for 120 Noto Emoji targets across ten workers, then retained after removing the target-specific adapters.Workers periodically merged their scaffold checkpoints into a shared checkpoint.Source: §A.2 · p. 24

A.3 Frozen-Baseline Finetuning

  • With the scaffold frozen, approximately 25,000 rank-16 LoRA factors were independently fitted for available emoji–vendor pairs.Targets came from nine emoji artwork sources, and each factor was saved at its lowest observed training loss.Source: §A.3 · p. 24

A.4 Imposed Mappings of Phenotypic-Transformation and Regulatory Modulation

  • Two rank-one LoRA factors—one horizontal and one vertical—were jointly trained across 25 prescribed transformations while the baseline regulatory scaffold remained fixed.The same adapter pair was learned across the training transformations rather than fitting a separate adapter to each target.Source: §A.4 · p. 25

A.5 Regeneration Experiments

LoRA did not make a non-regenerative scaffold significantly more regenerative, while phenotype-specific regenerative training improved regeneration for its target but did not transfer successfully to another phenotype.

A.5 Regeneration Experiments

  • LoRA training with a regenerative loss did not make a non-regenerative scaffold significantly more regenerative.The attempt used adversarial phenotypic ablations during and after development.Source: §A.5 · p. 25
  • A phenotype-specific regenerative adapter improved regeneration for its target, but its translation to another phenotype produced a largely dysfunctional NCA.The adapter had very low cosine similarity to the non-regenerative adapter for the same phenotype.Source: §A.5 · p. 25

B.1 Deep Features as a Perceptual Metric

The authors use a perceptual loss to compare transformed targets and NCA outcomes in a semantically rich latent space, alongside mean-square error.

B.1 Deep Features as a Perceptual Metric

  • Perceptual loss compares encoded target features with developmental outcomes in a semantically rich latent space.The feature encoder is AlexNet.Source: §B.1 · p. 25
  • MSE mostly tracks transformation size and offers limited information about pixel-wise or semantic quality.Source: §B.1 · p. 25
  • The authors report average and standard deviation of perceptual loss, together with average MSE, for out-of-distribution scaling and cross-phenotype experiments.Source: §B.1 · p. 25

B.2 Principal Component Analysis

PCA comparisons show that full weight-space representations expose more semantic structure than raw LoRA factors, while faithful reconstruction of NCA dynamics still needs many components. The tested scale adaptations also transfer across phenotypes sharing a scaffold.

B.2 Principal Component Analysis

  • All tested target phenotypes scaled through the same hyper-directions, including beyond training conditions, supporting universal scale directions for NCAs with a shared scaffold.Training used the blue fish target; the perceptual-loss comparison indicates transfer across the tested phenotypes.Source: §B.2 · p. 26, Figure 6 · p. 26
  • Full weight-space representations show more semantic structure than raw LoRA factors, while SVD-canonicalized LoRAs also show considerably more structure than raw factors.The LoRA decomposition is non-unique, so the analysis fixes its gauge using SVD of the effective adaptation.Source: §B.2 · pp. 25, 27
  • High-fidelity reconstruction of NCA dynamics requires thousands of principal components, although full weight-space representations reveal phenotypic details with a few hundred.Few leading PCs capture rough morphology, while fine details often require thousands, especially for LoRA or canonicalized LoRA parameters.Source: §B.2 · p. 27

B.3 Twin-Score

The twin-score detects two-component emoji phenotypes by combining component size balance with shape similarity, including mirror symmetry.

B.3 Twin-Score

  • The twin-score is intended to detect phenotypes with two similarly sized and shaped components.Foreground fragments below 5% of total area are discarded, and samples not containing exactly two remaining components receive a score of zero.Source: §B.3 · p. 27
  • Component shapes are compared after normalization to 32 × 32 masks, using intersection-over-union and allowing a left–right reflection.Source: §B.3 · p. 27
  • The final score multiplies size balance by shape similarity, with higher values indicating similarly sized components of similar or mirror-symmetric shape.Source: §B.3 · p. 27

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