Developmental Dimensionality

Why ferns are fractal: A mathematical theory of GRN constraint manifolds and morphological possibility

Overview

This repository presents a mathematical framework explaining why ancient plant lineages (ferns, lycophytes, liverworts) exhibit recursive, fractal-like geometry while angiosperms display extraordinary morphological diversity.

The core insight: A gene regulatory network (GRN) is not an algorithm computing morphology—it is a low-dimensional constraint manifold through which high-dimensional molecular dynamics must flow. A narrow channel cannot transmit complex morphological signals, regardless of the underlying dynamics.

The central equation bounds morphological complexity by channel dimension:

$$K(M) \leq O(D \log \frac{1}{\epsilon})$$

where $D$ is the developmental dimensionality (measured by participation ratio) and $K(M)$ is the Kolmogorov complexity of the morphology.

Key Results

Empirical finding: Duckweed (Spirodela polyrhiza), despite being an angiosperm, has developmental dimensionality $D_{\text{dev}} = 1.99$—matching ancient liverworts and 3.7× lower than Arabidopsis ($D_{\text{dev}} = 7.30$).

This confirms the central prediction: morphological complexity, not phylogenetic age, determines developmental dimensionality.

Species	Clade	D_dev	Morphology
Marchantia polymorpha	Liverwort	1.97	Thalloid
Spirodela polyrhiza	Angiosperm	1.99	Minimal frond
Selaginella moellendorffii	Lycophyte	2.02	Fractal branching
Equisetum hyemale	Horsetail	4.55	Segmented stem
Physcomitrium patens	Moss	5.95	Simple gametophyte
Arabidopsis thaliana	Angiosperm	7.30	Complex rosette

The Story in Figures

1. The GRN as a Constraint Funnel

High-dimensional molecular dynamics funneled to a low-dimensional developmental manifold.

2. Limited Decisions at Each Growth Step

Fern cells have ~3 independent choices; angiosperm cells have ~8.

3. Same Channel at Every Scale → Fractal Geometry

The same GRN constrains growth at frond, pinna, and pinnule levels.

4. Trajectory Convergence

Narrow channels produce stereotyped outcomes; wide channels enable diversity.

Mathematical Framework

The paper proves:

1. Participation Ratio Bounds (Theorem 3.1): $1 \leq \text{PR}(\mathbf{C}) \leq r$

2. GRN Cost Scaling (Theorem 4.1): $C_{\text{total}}(D) = \Theta(D^{2\gamma})$

3. Survival Threshold (Theorem 5.1): $L^* = -\tau \ln(1 - C/\mu)$

4. Dimensionality-Geometry Correspondence (Theorem 7.1): Low-D channels are constrained to self-similar morphologies.

Repository Structure

├── paper/
│   ├── developmental_dimensionality.tex    # Main manuscript
│   ├── developmental_dimensionality.pdf    # Compiled PDF
│   └── references.bib                      # Bibliography
├── code/
│   ├── compute_geometric_plants_dimensionality.py  # D_dev computation
│   ├── create_manifold_story.py                    # Generate figures
│   ├── create_grn_metabolic_figures.py             # Metabolic visualizations
│   └── create_lsystem_geometry.py                  # L-system figures
├── figures/                                # Publication figures
├── data/                                   # Expression data (GEO)
└── archive/                                # Old drafts

Running the Code

# Compute developmental dimensionality
python code/compute_geometric_plants_dimensionality.py

# Generate manifold story figures
python code/create_manifold_story.py

# Generate all figures
python code/create_grn_metabolic_figures.py
python code/create_lsystem_geometry.py

Requirements: numpy, pandas, matplotlib, scipy

Data Sources

• Arabidopsis: AtGenExpress GSE5630 (60 developmental stages)
• Moss: GEO GSE142053 (20 samples)
• Geometric plants: GEO GSE270814 (Marchantia, Selaginella, Equisetum)
• Duckweed: GEO GSE235730 (13 samples, 6 developmental stages)

Citation

@article{todd2026developmental,
  title={Developmental Dimensionality and Morphological Geometry:
         A Mathematical Framework for Plant Evo-Devo},
  author={Todd, Ian},
  journal={In preparation},
  year={2026}
}

Related Work

This paper is part of a research program on dimensional constraints in biological systems:

• Minimal Embedding Dimension — Information geometry bounds
• Oscillatory Scaling Limits — Coherence time constraints

License

MIT License. See LICENSE for details.

Author

Ian Todd Sydney Medical School, University of Sydney itod2305@uni.sydney.edu.au ORCID: 0009-0002-6994-0917