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Quantitative BiologyEnglishPublished

Heterogeneous Operator Assemblages Yield Hexagonal Self‑Reproducing Cells in a Multiphase Model

October 31, 2026arXiv: 2610.40524v1

A new theoretical framework combines differential heterogenesis with autopoiesis to explain how living systems can develop historicity and maintain themselves. The framework uses heterogeneous operator assemblages that obey two closure constraints: a fixed–point condition that keeps the interface stable, and a reproduction condition that preserves the operator composition across divisions. In a concrete example, the Gray–Scott reaction–diffusion equation is used outside a free surface while an anisotropic, gradient‑dependent diffusion operator is used inside. Numerical simulations show that this heterogenetic assembly produces hexagonal cells that retain their shape after each division and fill the domain until the substrate is exhausted. In contrast, a homogeneous Gray–Scott model with identical operators on both sides of the interface generates round cells that lose their shape when multiplied.

Table 1 shows a three‑column scheme. The first column lists features provided by differential heterogenesis. The second column lists the two structural constraints required for living systems. The third column lists the outcomes that result when the constraints are applied: self‑maintaining assemblies and inherited reproduction.
Already given by differential heterogenesisConstraint added hereWhat the constraint yields
Spatial assemblage of heterogeneous operators composed over a domainFixed-point condition 𝒜⁡(ϕ)=ϕ, hence 𝒜⁡(Γ)=Γthe assemblage persists as a solution of its own production, and the form it generates is self-maintaining rather than transient
Temporal assemblage of operators superposed and modulated in timeReproduction condition C𝒜τ​(𝒜)=𝒜the assemblage is reproduced with variation, since the whole stratification of operators is transmitted, so the form generated is inherited rather than re-invented
Table 1. Summary of how the differential heterogenesis framework integrates with autopoiesis to build a theory linking historicity and self‑maintenance.Sarti, Alessandro, Citti, Giovanna · arXiv:2610.40524v1 · Table 1: · CC BY 4.0
Figure 6 illustrates a simulation starting from a single mother cell. The interior region has anisotropic diffusion whereas the exterior follows Gray–Scott kinetics. As the cell reproduces, it remains hexagonal, while in a separate panel a homogeneous model shows round cells losing shape upon division. (1/6)Figure 6 illustrates a simulation starting from a single mother cell. The interior region has anisotropic diffusion whereas the exterior follows Gray–Scott kinetics. As the cell reproduces, it remains hexagonal, while in a separate panel a homogeneous model shows round cells losing shape upon division. (2/6)Figure 6 illustrates a simulation starting from a single mother cell. The interior region has anisotropic diffusion whereas the exterior follows Gray–Scott kinetics. As the cell reproduces, it remains hexagonal, while in a separate panel a homogeneous model shows round cells losing shape upon division. (3/6)Figure 6 illustrates a simulation starting from a single mother cell. The interior region has anisotropic diffusion whereas the exterior follows Gray–Scott kinetics. As the cell reproduces, it remains hexagonal, while in a separate panel a homogeneous model shows round cells losing shape upon division. (4/6)Figure 6 illustrates a simulation starting from a single mother cell. The interior region has anisotropic diffusion whereas the exterior follows Gray–Scott kinetics. As the cell reproduces, it remains hexagonal, while in a separate panel a homogeneous model shows round cells losing shape upon division. (5/6)Figure 6 illustrates a simulation starting from a single mother cell. The interior region has anisotropic diffusion whereas the exterior follows Gray–Scott kinetics. As the cell reproduces, it remains hexagonal, while in a separate panel a homogeneous model shows round cells losing shape upon division. (6/6)
Figure 6. Heterogeneous assembly producing hexagonal cells. The anisotropic diffusion operator inside the level set shapes the cells into hexagons, a contrast to the round cells seen when the operator is homogeneous.Sarti, Alessandro, Citti, Giovanna · arXiv:2610.40524v1 · Figure 6: · CC BY 4.0

The model’s construction relies on a free‑boundary multiphase system. The invariant interface γ satisfies γ\mathcal{A}_{ au}(ϕ)=ϕ, which follows from a Schauder‑type argument showing that a nonlinear, non‑local operator maps a closed bounded convex set into a compact subset of itself. The numerical evidence, however, is the only way to confirm the two closure conditions for this heterogeneous system; analytical existence and stability proofs are not yet available.

These results illustrate how changing the local operator across a moving interface can break rotational symmetry and impose a preferred geometry on the emerging structure, while still allowing for self‑maintaining, self‑reproducing dynamics governed by historical constraints.

Source paper

A heterogenetic theory of living systems

Quantitative Biology
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Heterogeneous Operator Assemblages Yield Hexagonal Self‑Reproducing Cells in a Multiphase Model | arXiv News