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Recursive Self-Composite Approach Towards Structural Understanding of Boolean Network

Jongrae Kim, Woojeong Lee, Kwang-Hyun Cho

TL;DR

The paper addresses understanding the structural dynamics of Boolean networks beyond attractor analysis by focusing on the logic update rules themselves. It introduces a recursive self-composite framework where $x(p)= f[x(p-1)]$, enabling convergence to a kernel logic with preperiod $p^*$ and period $\ell^*$, such that $f^{p^*+r}(x)=f^{p^*+r+\ell^*}(x)$. The approach defines a kernel set $K$ of states and proves the existence of a kernel logic across all networks; it also introduces a leaping and filling strategy to accelerate convergence and avoid exponential growth of rule strings. Demonstrations on a simple network, a 37-state T-cell receptor network, and a 90-state cancer signaling network illustrate how the method reveals hidden algebraic structure and reduces effective state space, enabling insights into rare-event logics and potential control targets.

Abstract

Boolean networks have been widely used in many areas of science and engineering to represent various dynamical behaviour. In systems biology, they became useful tools to study the dynamical characteristics of large-scale biomolecular networks and there have been a number of studies to develop efficient ways of finding steady states or cycles of Boolean network models. On the other hand, there has been little attention to analyzing the dynamic properties of the network structure itself. Here, we present a systematic way to study such properties by introducing a recursive self-composite of the logic update rules. Of note, we found that all Boolean update rules actually have repeated logic structures underneath. This repeated nature of Boolean networks reveals interesting algebraic properties embedded in the networks. We found that each converged logic leads to the same states, called kernel states. As a result, the longest-length period of states cycle turns out to be equal to the number of converged logics in the logic cycle. Based on this, we propose a leaping and filling algorithm to avoid any possible large string explosions during the self-composition procedures. Finally, we demonstrate how the proposed approach can be used to reveal interesting hidden properties using Boolean network examples of a simple network with a long feedback structure, a T-cell receptor network and a cancer network.

Recursive Self-Composite Approach Towards Structural Understanding of Boolean Network

TL;DR

The paper addresses understanding the structural dynamics of Boolean networks beyond attractor analysis by focusing on the logic update rules themselves. It introduces a recursive self-composite framework where , enabling convergence to a kernel logic with preperiod and period , such that . The approach defines a kernel set of states and proves the existence of a kernel logic across all networks; it also introduces a leaping and filling strategy to accelerate convergence and avoid exponential growth of rule strings. Demonstrations on a simple network, a 37-state T-cell receptor network, and a 90-state cancer signaling network illustrate how the method reveals hidden algebraic structure and reduces effective state space, enabling insights into rare-event logics and potential control targets.

Abstract

Boolean networks have been widely used in many areas of science and engineering to represent various dynamical behaviour. In systems biology, they became useful tools to study the dynamical characteristics of large-scale biomolecular networks and there have been a number of studies to develop efficient ways of finding steady states or cycles of Boolean network models. On the other hand, there has been little attention to analyzing the dynamic properties of the network structure itself. Here, we present a systematic way to study such properties by introducing a recursive self-composite of the logic update rules. Of note, we found that all Boolean update rules actually have repeated logic structures underneath. This repeated nature of Boolean networks reveals interesting algebraic properties embedded in the networks. We found that each converged logic leads to the same states, called kernel states. As a result, the longest-length period of states cycle turns out to be equal to the number of converged logics in the logic cycle. Based on this, we propose a leaping and filling algorithm to avoid any possible large string explosions during the self-composition procedures. Finally, we demonstrate how the proposed approach can be used to reveal interesting hidden properties using Boolean network examples of a simple network with a long feedback structure, a T-cell receptor network and a cancer network.
Paper Structure (6 sections, 5 theorems, 32 equations, 3 figures)

This paper contains 6 sections, 5 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Recursive Self-Composite Boolean network
  3. Motivations
  4. Main Results
  5. Examples
  6. Conclusions & Future Works

Key Result

Theorem 1

All recursive self-composition of the synchronous Boolean network given by eq:boolean_network_compact converges to a steady-state logic or a cyclic logic, i.e, where $p^*$ between 0 and $2^n$ is the minimum number of recursions when the logic starts repeating itself, $r$ is a non-negative integer and $\ell^*$ between 1 and $2^n$ is the period of logic cycles.

Figures (3)

  • Figure 1: The interaction graph and state transition map of the Boolean network, \ref{['eq:x123_simple_example_01']}, are shown. There are three steady-states, $\{ s = (000), (011), (111) \}$ and one cycle with the period 2, $\{ s= (010) \leftrightarrow (001) \}$, where $s=(x_1 x_2 x_3)$.
  • Figure 2: The Boolean logic in \ref{['eq:x123_simple_example_01']} switches between two update rules.
  • Figure 3: The Boolean logic in \ref{['eq:x1234567_simple_example']} cycles between six update rules.

Theorems & Definitions (13)

  • Definition 1: Recursive Self-Composite
  • Theorem 1: Convergence of Recursive Self-Composition
  • Remark 1
  • Theorem 2: Longest Cycle Upper Bound
  • Definition 2: Kernel States Set
  • Theorem 3: Kernel States Set of Converged Logic
  • Theorem 4: Longest Length Cycle
  • Definition 3: Kernel Logic
  • Theorem 5: Existence of Kernel Logic
  • Remark 2: Leaping & Filling
  • ...and 3 more