Hierarchical p-Adic Framework for Gene Regulatory Networks: Theory and Stability Analysis

Abstract

Gene regulatory networks exhibit hierarchical organization across scales; capturing this structure mathematically requires a metric that distinguishes regulatory influence at each level. We show that the ultrametric of the $p$-adic integers $\mathbb{Z}_p$ -- whose self-similar nested-ball structure is a natural fractal encoding of multi-scale organization -- provides such a framework. Embedding the $N$-gene state space into $\mathbb{Z}_p$ and working over the complete, algebraically closed field $\mathbb{C}_p$, we prove the existence of rational functions that interpret the discrete dynamics and construct hierarchical approximations at each resolution level. These constructions yield a stability measure $μ$ -- aggregating how the dynamics contracts or expands across resolution levels -- and a ball-level classification of fixed points -- contracting, expanding, or isometric -- extending the attracting/repelling/indifferent trichotomy of non-Archimedean dynamics from points to balls. A key result is that $μ$ and the classification, although their definition and dynamical meaning require the analytical tools of $\mathbb{C}_p$, are fully determined by the discrete data. Minimizing $μ$ over all $N!$ gene orderings defines an optimal regulatory hierarchy; for the Arabidopsis thaliana floral development network ($N=13$, $p=2$), a $μ$-minimizing ordering places known master regulators -- UFO, EMF1, LFY, TFL1 -- in the leading positions and recovers the accepted developmental hierarchy without biological input beyond the transition map.

Figures (6)

• Figure 1: Fractal tree of $\mathbb{Z}_5$ at two resolutions. Each node branches into $p{=}5$ children, one per residue class; the pattern at each node reproduces the whole at a smaller scale. Points sharing a long common root-to-leaf path agree in many leading $p$-adic digits and are $p$-adically close (Remark \ref{['rem:hierarchical_partition']}). The branching continues identically at every scale; passing from (a) to (b) simply reveals finer detail within the same self-similar structure.
• Figure 2: Ultrametric property: two intersecting balls in $\mathbb{C}_p$ share a point $z$; recentering at $z$ shows one ball is contained in the other. The isosceles property of ultrametric triangles is illustrated inline in the text.
• Figure 3: Hierarchical partition of $\mathbb{Z}_p$ for $p=5$. The ball $\mathbb{Z}_p = B_1(0)$ is the disjoint union of five balls ${B}_{1/5}(m)$, $m=0,\ldots,4$ (radius $1/5$, level 1). Each is subdivided into five balls of radius $1/25$ (level 2); inside the ball $m=0$ (i.e. ${B}_{1/25}(0)$ at level 2), one of those is further subdivided into five balls of radius $1/125$ (level 3). The same pattern repeats at every scale.
• Figure 4: Tree representation of $\mathbb{Z}_p$. Each node at depth $n$ corresponds to a ball $B_{1/p^n}(m)$; its $p$ children are the sub-balls at level $n{+}1$. Edge labels indicate the $n$-th digit $a_n \in \{0,\ldots,p{-}1\}$; a root-to-leaf path encodes the $p$-adic expansion of a configuration. (a) $\mathbb{Z}_2$, levels $n{=}0$--$3$; the highlighted path traces $m=5=(1,0,1)_2$. (b) $\mathbb{Z}_5$, levels $n{=}0$--$1$, with one branch expanded to $n{=}2$; the remaining branches subdivide identically. Compare with the spatial (nested-ball) view in Figure \ref{['fig:padic_balls_hierarchy']}.
• Figure 5: A. thaliana floral development network ($N=13$ genes). Node colors: red = top-ranked genes in $\pi^*$ (UFO, EMF1, LFY, TFL1), dark teal = organ identity genes (AP1, AP2, AG, PI, AP3), light teal = other regulatory genes (FT, FUL, WUS, SEP). Edge styles: solid green arrows = activation, dashed red = repression, solid gray = regulation with context-dependent effect. Network topology and interaction types from mendoza1998dynamicsespinosa2004gene.

Theorems & Definitions (51)

• Proposition 2.1: Corollary 2.3.4 in Gouvea2020p
• proof
• Proposition 2.2: Proposition 2.4 in benedetto2019dynamics
• proof
• Remark 2.3: Hierarchical Partition and Nested Balls
• Lemma 2.4: Ball-truncation equivalence
• proof
• Proposition 2.5: Proposition 3.25 in benedetto2019dynamics
• Remark 2.6: Terminology
• Proposition 2.7: Theorem 4.18 in benedetto2019dynamics