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

Inspired by D'Arcy Thompson's "On Growth and Form," this study demonstrates that coherent large-scale morphological transformations in neural cellular automata can be encoded and controlled through reusable, low-dimensional regulatory adaptations, enabling zero-shot transfer of scaling and other phenotypic variations across diverse target morphologies.

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

Published 2026-09-25✓ Author reviewed ⓘ

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For over a century, biologists have grappled with a fundamental question: how does a single fertilized egg know how to build a complex animal, and how do small changes in the instructions lead to the vast diversity of shapes we see in nature? This field, known as evolutionary developmental biology, seeks to understand the link between the microscopic rules cells follow and the macroscopic forms they create. A key idea in this quest is that the body is not built like a static blueprint, but rather emerges from a dynamic process where cells constantly communicate, sense their neighbors, and adjust their behavior to reach a target shape. If we could understand the "knobs" and "switches" that control this process, we might be able to explain how evolution reshapes organisms or even how to guide healing in damaged tissues. The challenge has been finding a way to describe these complex, distributed changes without getting lost in the trillions of individual molecular interactions.

In a new study, researchers have created a simplified digital model to test whether large-scale changes in body shape can be controlled by adjusting just a few high-level settings. They used a system called a neural cellular automaton, which is essentially a grid of artificial cells. Each cell in this grid is a tiny computer that looks at its immediate neighbors and decides what to do next based on a shared set of rules, much like a gene-regulatory network in a real organism. By training these artificial cells to grow into specific shapes, such as emojis of fish or lizards, the researchers established a baseline where the cells reliably build the target form. The core question was whether they could take this trained system and make it grow a different size or shape by making only tiny, coordinated adjustments to the rules the cells follow, rather than rewriting the entire rulebook for every new shape.

The team discovered that they could indeed control the size and shape of these digital organisms using a technique that allows for minimal changes to the underlying rules. Instead of retraining the entire system for every new size, they applied small, low-rank adjustments to the shared rules. Think of it as turning a few specific dials on a complex machine rather than rebuilding the machine from scratch. They found that a single, simple adjustment to the rules was enough to stretch the digital organism horizontally, while another simple adjustment stretched it vertically. Remarkably, these adjustments were not just memorizing how to make one specific fish bigger; they represented a general principle of scaling. When the researchers applied these same rule adjustments to digital organisms that looked completely different—such as a lizard or a fire extinguisher—the new shapes grew correctly in size and proportion, preserving their unique internal features while changing their overall dimensions. This suggests that the system had learned a universal way to control scale that works across many different forms.

To see if these findings were just a lucky accident with one type of shape, the researchers trained thousands of these digital organisms on a vast library of about 25,000 different emoji targets. They then analyzed the mathematical space where all the rule adjustments lived. In this space, they found clear, organized directions that corresponded to specific biological traits. Moving in one direction made the organisms wider, while moving in another made them taller. They even found a direction that changed the artistic style of the shape, making it look like a specific cartoon style with a thick black outline, and another direction that could split a single shape into two symmetrical halves, like a twin birth. These directions were not random; they were consistent and reusable. The researchers showed that by combining these different adjustments, they could create complex changes, such as stretching a shape while also changing its style, all without losing the core identity of the organism.

The study suggests that the complex process of building a body might be governed by a few powerful, high-level controls that can be tuned to produce a wide variety of outcomes. While this work was done in a computer simulation and does not yet prove that real biological cells use exactly these same mechanisms, it offers a compelling proof of concept. It demonstrates that coherent, large-scale changes in form can be encoded as simple, low-dimensional modulations of a shared regulatory system. This provides a new way to think about how evolution might reshape organisms: perhaps by tweaking a few key parameters in the developmental rules, rather than inventing entirely new instructions for every new shape. The findings also open the door to future possibilities where scientists might one day identify similar "handles" in real biological systems to guide regeneration or correct developmental errors, turning the century-old puzzle of how growth and form are related into a solvable engineering problem.

Technical Summary: LoRA-NCA – Low-Rank Regulatory Control of Phenotypic Variation

Problem Statement
The paper addresses a central inverse problem in developmental biology and evolutionary theory: if two anatomical forms are related by a simple geometric transformation (such as the grid deformations proposed by D'Arcy Thompson in On Growth and Form), what is the corresponding modulation of the underlying developmental dynamics that physically generates them? While Thompson demonstrated that morphological variations can be mapped via coordinate transformations, the causal developmental mechanisms—specifically how microscopic regulatory changes (e.g., in gene regulatory networks) translate into macroscopic, coherent shape changes—remain largely unidentified. The authors seek to determine whether coherent phenotypic variations can be encoded as low-dimensional modulations of a self-organizing developmental system, effectively finding "reusable regulatory handles" to control growth and form without micromanaging individual cellular trajectories.

Methodology
The authors utilize Neural Cellular Automata (NCAs) as minimal, bio-inspired in silico models of distributed development. In this framework:

System Architecture: A grid of artificial cells (agents) updates its numerical state based on a shared Artificial Neural Network (ANN) that acts as a regulatory machinery (analogous to gene regulatory networks). Cells perceive their local neighborhood (Moore neighborhood) and update their states to collectively grow a target morphology from a single seed.
Low-Rank Adaptation (LoRA): Instead of retraining the entire network for every new target or transformation, the authors apply LoRA to a pretrained NCA. They freeze the baseline regulatory parameters ( $\theta_0$ ) and introduce trainable low-rank matrices ( $A$ and $B$ ) to modulate the weights ( $W_T = W_0 + \Delta W$ ).
Experimental Design:
1. Scaling Experiments: The authors trained LoRA adapters to implement horizontal and vertical scaling transformations on a baseline "fish" emoji. They tested whether rank-one adaptations could parametrically control size and generalize to out-of-distribution scaling factors.
2. Zero-Shot Transfer: They investigated whether scaling adapters learned for one phenotype could be applied to other, structurally distinct phenotypes (e.g., lizards, fire extinguishers) that share the same frozen regulatory scaffold ( $W_0$ ).
3. Large-Scale Empirical Discovery: The authors trained a dataset of approximately 25,000 phenotype-specific LoRA-NCAs (rank $r=16$ ) on diverse emoji targets using a single shared scaffold. They performed Principal Component Analysis (PCA) on the resulting weight space ( $\Delta W$ ) to empirically discover latent "hyper-directions" corresponding to specific phenotypic features (size, style, symmetry).

Key Results

Efficient Control of Scale: Rank-one adaptations (comprising only ~2.6% of the total parameters) were sufficient to robustly implement horizontal and vertical scaling transformations. These adaptations generalized to scaling factors outside the training distribution.
Universal Hyper-Directions: Scaling adapters learned for a single phenotype transferred zero-shot to diverse, unrelated phenotypes sharing the same scaffold. This indicates that the adapters represent system-level geometric controls (universal hyper-directions of scale) rather than memorized features of a specific morphology.
Semantic Weight Space Geometry: Analysis of the 25,000 LoRA-NCAs revealed a semantically structured regulatory weight space:
- Principal Components: The first two principal components (PC0, PC1) strongly correlated with horizontal and vertical phenotypic extent, respectively.
- Style and Fission: Mean differences between subsets of the dataset yielded functional directions for "style transfer" (e.g., converting outputs to an "OpenMoji" style with black outlines) and "symmetrical fission" (splitting a phenotype into two symmetric halves).
Degeneracy and Orthogonality: The empirically derived PCA directions for scaling were only weakly aligned (cosine similarity $\approx 0.1$ ) with the explicitly learned rank-one scaling directions. This suggests a degeneracy in morphogenetic control, where distinct directions in parameter space can modulate the same macroscopic feature, supporting the concept of poly-computing in biological substrates.

Significance and Claims
The paper claims to provide a computational realization of D'Arcy Thompson's grid transformations, demonstrating that coherent anatomical changes can be controlled by interventions along low-dimensional directions in regulatory weight space.

Mechanistic Interpretation: The results suggest that biological form is a robust yet plastic outcome of self-regulated dynamics, where system-level outcomes can be redirected through compact changes in regulatory rules without micromanaging detailed cellular events. This aligns with Howard Pattee's hypothesis that effective control in hierarchical systems arises through a selective loss of microscopic detail.
Evolvability and Facilitated Variation: The existence of reusable, transferable hyper-directions offers a computational analogue for "facilitated variation." It suggests that evolution could access coherent phenotypic changes by recombining existing regulatory handles rather than requiring novel mutations for every trait.
Bioengineering Implications: The work proposes a systems-biology strategy where desired anatomical outcomes are addressed through distributed, tailored regulatory modulations. While the current model is abstract (2D grid without physical constraints), the authors posit that identifying analogous handles in biophysically grounded models could eventually enable the "inverse problem" of inferring local rule modulations for regeneration, developmental repair, or cancer treatment.

Limitations and Modesty
The authors explicitly note that the current NCAs are abstract models; their weights cannot yet be directly mapped to specific genes, morphogens, or physiological signals. The extracted hyper-directions are not fully feature-specific (e.g., scaling directions also affect brightness), and the physical meaning of the weight space remains partly elusive. The paper frames these results as a proof-of-concept for a "minimal cybernetic tissue," acknowledging that future work is required to embed these regulatory handles in more biophysically grounded models to achieve true predictive digital twins of biological systems.

Technical Summary: LoRA-NCA – Low-Rank Regulatory Control of Phenotypic Variation

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