🔬 condensed matter

Control of morphology and topology in a lattice model of branching morphogenesis

This paper presents a lattice model of morphogen-controlled branching morphogenesis that integrates non-equilibrium physics and developmental biology to investigate growth patterns across diffusion-limited and stochastic surface growth regimes, while demonstrating that local topological control enables steady-state branched clusters exhibiting power-law scaling with consistent fractal exponents across different lattice geometries.

Original authors: Christian Hanauer, Frank Jülicher, Efe Ilker

Published 2026-07-28

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CC BY 4.0

Original authors: Christian Hanauer, Frank Jülicher, Efe Ilker

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 or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of physics and the rules of life are having a very intense, very productive argument. On one side, you have non-equilibrium physics, the study of how things change, move, and grow when they aren't sitting still—think of how a drop of ink spreads in water or how a snowflake forms. On the other side is developmental biology, the study of how a single cell becomes a complex, living creature with lungs, blood vessels, and nerves. The big question connecting them is: How do living things build their intricate, branching shapes? Why do our lungs look like trees, and why do our blood vessels look like rivers? Scientists know that tiny chemical messengers, called morphogens, act like traffic controllers, telling cells when to grow and where to go. But the big mystery is how these chemicals manage to create shapes that are not just messy blobs, but organized, branching networks that can even fix themselves if they get damaged. Understanding this is like finding the secret blueprint for how nature builds its most complex machines.

Now, enter a team of researchers who decided to build a digital playground to solve this puzzle. They created a computer model—a grid of tiny squares, like a giant checkerboard—where they simulated cells growing and shrinking based on the instructions of these chemical messengers. Think of it as a video game where the "cells" are pixels that can turn on or off, but instead of a player controlling them, the pixels react to a glowing, invisible fog of chemicals that they produce themselves. The researchers wanted to see if they could make these digital cells grow into shapes that looked like real biological branches, and even more impressively, could they make them shrink back down without falling apart?

The results were fascinating. When the researchers let the cells grow freely, they could tune the "fog" to create two very different types of shapes. If the chemical fog spread out very far, the cells grew into wild, spiky, fractal trees that looked exactly like the classic "Diffusion-Limited Aggregation" (DLA) patterns seen in physics—think of lightning striking or frost forming on a window. But if the fog stayed very close to the cells, they grew into tight, round, compact blobs, similar to how a colony of bacteria might spread on a petri dish. This confirmed that the range of the chemical signal is the dial that controls whether nature builds a spiky tree or a round bush.

However, the real magic happened when the researchers added a special rule: the "topology constraint." In the real world, if a branch of a tree breaks off, the tree doesn't fall apart into two separate trees; it stays as one connected object. In their simulation, the researchers programmed the cells to follow this rule. They found that when they allowed the cluster to grow and then shrink (a process called degrowth) while keeping it connected, something extraordinary occurred. The cluster didn't just bounce back and forth between its grown and shrunk states; it settled into a steady, fluctuating state. It was like a living thing that had found its perfect balance, constantly adding and removing cells but always maintaining its tree-like shape.

In this steady state, the clusters exhibited a unique "personality" that was different from both the wild, spiky trees and the round, compact blobs. When the researchers measured how big these steady-state clusters were compared to how many cells they contained, they found a specific mathematical pattern. The size of the cluster grew according to a power law with an exponent of about 0.68 on a square grid and 0.67 on a hexagonal grid. This number is special because it sits right at the theoretical limit for how extended a branching structure can get in two dimensions. It suggests that by simply adding and removing cells while keeping the structure connected, the system naturally finds the most efficient, stretched-out shape possible.

The paper also showed that this process is not reversible in a simple way. If you grow a cluster and then shrink it, the path it takes to shrink is different from the path it took to grow. It's like walking up a mountain on a steep, winding trail and then coming down a different, smoother path; you end up at the same spot, but the journey leaves a different mark. This "history dependence" means the shape of the cluster remembers how it got there.

Ultimately, this study suggests that the secret to building complex, resilient biological networks might not be a rigid, pre-planned blueprint, but rather a simple, local rule: "Grow if the chemical signal says so, shrink if it says stop, but never let the whole thing fall apart." By combining the chaotic energy of growth with the strict discipline of staying connected, nature (or at least this computer model) can create structures that are both flexible and strong, capable of adapting to damage while maintaining their essential form. The researchers propose that this simple interplay between chemical signals and topological rules could be the key to understanding how our own bodies build and maintain their vital, branching networks.

Technical Summary: Control of Morphology and Topology in a Lattice Model of Branching Morphogenesis

Problem Statement
Branched biological networks, such as vasculature, lungs, and neurons, exhibit specific geometries and topologies that are critical for their function. While physical models like Diffusion-Limited Aggregation (DLA) successfully describe fractal growth in non-equilibrium systems, they fail to capture key biological features: regulation by signaling molecules (morphogens), the ability to undergo degrowth (reversal of growth), and the maintenance of steady-state homeostasis. Furthermore, biological networks often require specific topological constraints (e.g., maintaining a single connected component without loops or holes) that standard stochastic growth models do not inherently enforce. This paper addresses the open problems of how network topology is controlled, how growth and degrowth are regulated, and how steady branched patterns are maintained.

Methodology
The authors propose a stochastic lattice model that couples the occupation dynamics of cells with the reaction-diffusion dynamics of morphogens.

Lattice Representation: A multicellular cluster is represented on a square (or hexagonal) lattice of size $L \times L$ . Sites are either occupied ( $n_r=1$ ) or empty ( $n_r=0$ ).
Morphogen Dynamics: The morphogen concentration $c_r$ follows a reaction-diffusion equation where production and degradation rates depend on whether a site is inside or outside the cluster. The system is solved in the quasistatic limit, assuming the morphogen field relaxes faster than the cluster changes.
Growth and Degrowth Rules: The transition rates for a site to become occupied ( $\nu^+_r$ $ν_{r}^{+}$ ) or empty ( $\nu^-_r$ $ν_{r}^{-}$ ) are governed by local morphogen gradients.
- Growth occurs at empty sites adjacent to occupied ones, driven by the morphogen gradient pointing outward.
- Degrowth occurs at occupied sites adjacent to empty ones, driven by the gradient pointing inward.
Topological Constraints: To control topology, the authors introduce a local operator $f_r$ based on the Euler characteristic ( $\chi = C - H$ , where $C$ is connected components and $H$ is holes). Transitions are permitted only if they preserve $\chi$ (specifically maintaining a tree-like structure with $C=1, H=0$ ). This prevents the formation of holes or the separation of the cluster into disconnected components.
Simulation: The system is simulated using Kinetic Monte Carlo methods, exploring regimes from pure growth to simultaneous growth and degrowth (steady state).

Key Contributions and Results

Morphogen-Controlled Morphology:
The model demonstrates that the length scale of the morphogen gradient ( $\lambda_{out}$ ) dictates the cluster morphology.
- DLA Limit: In the limit of no degradation and no external source ( $\lambda_{out} \gg L$ ), the model reproduces highly branched, fractal structures characteristic of DLA.
- Eden Limit: In the limit of strong degradation ( $\lambda_{out} \ll L$ ), the model yields compact, almost circular clusters similar to the Eden model.
- Intermediate Regime: A crossover between these two limits is observed, governed by the diffusion-degradation length.
Scaling Behavior during Growth:
The radius of gyration ( $R_g$ ) scales with cluster size ( $N$ ) as $R_g \propto N^\beta$ .
- In the DLA limit, $\beta \approx 0.57 \pm 0.01$ .
- In the Eden limit, $\beta \approx 0.48 \pm 0.01$ .
- The imposition of topological constraints (preserving a tree-like structure) has a negligible effect on these scaling exponents during the pure growth phase, consistent with previous findings for variable topology.
Role of Topology in Growth and Degrowth:
The study highlights a distinct difference between growth with and without topological constraints during degrowth.
- Without Constraints: Degrowth leads to the pinching off of branches, creating disconnected components and altering the Euler characteristic.
- With Constraints: The cluster maintains a single connected, tree-like structure. Branches elongate rather than pinch off, resulting in a larger radius of gyration for small cluster sizes compared to the unconstrained case. This reveals a history dependence and irreversibility in the growth-degrowth cycle.
Steady-State Dynamics and Novel Scaling:
By introducing simultaneous growth and degrowth (logistic growth rate balanced by a constant degrowth rate), the system reaches a fluctuating steady state.
- New Scaling Exponent: Steady-state clusters with topological constraints exhibit a scaling exponent $\beta = 0.68 \pm 0.01$ on a square lattice and $\beta = 0.67 \pm 0.01$ on a hexagonal lattice.
- Significance of Exponent: This value is significantly higher than that of standard DLA growth ( $\approx 0.59$ ) and approaches the theoretical ceiling of $2/3$ for maximal extension in 2D DLA processes. The authors note that this saturation is achieved with relatively small cluster sizes ( $10^1 - 10^3$ particles), suggesting that the steady-state topology-preserving dynamics drives the cluster to explore maximally extended configurations, rather than requiring symmetry breaking or lattice anisotropy.

Significance and Claims
The paper claims to present a minimal model that bridges concepts from non-equilibrium physics and developmental biology. Its primary significance lies in:

Generalizing the classical DLA scenario to include morphogen-controlled growth, degrowth, and homeostasis.
Demonstrating that local topological rules can maintain global structural integrity (tree-like topology) while allowing for dynamic fluctuation in size.
Identifying a distinct class of steady-state fractal clusters with scaling properties ( $\beta \approx 2/3$ ) that differ from pure growth processes, suggesting that the interplay of growth and degrowth under topological constraints is a key mechanism for achieving specific biological network architectures.

The authors conclude that while the model captures generic features of branching morphogenesis, the specific biochemical mechanisms that could implement the required topological operators in real biological systems remain an open question for future investigation.

Technical Summary: Control of Morphology and Topology in a Lattice Model of Branching Morphogenesis

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