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

arXiv: https://arxiv.org/abs/2609.29755 · HTML: https://mlsanity.com/paper/2609.29755/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 98 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [From Entropy to Epiplexity: Rethinking Information for Computationally Bounded Intelligence](https://mlsanity.com/paper/2601.03220/) (2026)
- [Questioning Representational Optimism in Deep Learning: The Fractured Entangled Representation Hypothesis](https://mlsanity.com/paper/2505.11581/) (2025)
- [HyperNCA: Growing Developmental Networks with Neural Cellular Automata](https://mlsanity.com/paper/2204.11674/) (2022)
- [There's Plenty of Room Right Here: Biological Systems as Evolved, Overloaded, Multi-scale Machines](https://mlsanity.com/paper/2212.10675/) (2022)
- [ARC-NCA: Towards Developmental Solutions to the Abstraction and Reasoning Corpus](https://mlsanity.com/paper/2505.08778/) (2025)
- [Goal-Guided Neural Cellular Automata: Learning to Control Self-Organising Systems](https://mlsanity.com/paper/2205.06806/) (2022)
- [Neural cellular automata: applications to biology and beyond classical AI](https://mlsanity.com/paper/2509.11131/) (2025)
- [CAX: Cellular Automata Accelerated in JAX](https://mlsanity.com/paper/2410.02651/) (2024)
- [Neural Cellular Automata Manifold](https://mlsanity.com/paper/2006.12155/) (2020)
- [EngramNCA: a Neural Cellular Automaton Model of Memory Transfer](https://mlsanity.com/paper/2504.11855/) (2025)
- [Heuristically Adaptive Diffusion-Model Evolutionary Strategy](https://mlsanity.com/paper/2411.13420/) (2024)
- [Engineering morphogenesis of cell clusters with differentiable programming](https://mlsanity.com/paper/2407.06295/) (2024)
- [Meta-Learning an Evolvable Developmental Encoding](https://mlsanity.com/paper/2406.09020/) (2024)
- [Neural Particle Automata: Learning Self-Organizing Particle Dynamics](https://mlsanity.com/paper/2601.16096/) (2026)
- [Learning Developmental Scaffoldings to Guide Self-Organisation](https://mlsanity.com/paper/2605.14998/) (2026)
- [Intelligence from Learnable Novelty](https://mlsanity.com/paper/2607.18433/) (2026)
- [Remapping and navigation of an embedding space via error minimization: a fundamental organizational principle of cognition in natural and artificial systems](https://mlsanity.com/paper/2601.14096/) (2026)
- [BraiNCA: brain-inspired neural cellular automata and applications to morphogenesis and motor control](https://mlsanity.com/paper/2604.01932/) (2026)

## Abstract (arXiv)

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.