Phenotype control and elimination of variables in Boolean networks

TL;DR

This work analyzes how reducing Boolean networks by eliminating variables impacts asymptotic dynamics and phenotype-control strategies. It introduces a mediator-based structural condition that preserves minimal trap spaces under reduction and compares three control paradigms—attractor-based, trap-space-based, and value-propagation control—across update schemes SD, AD, and GD. The authors provide positive results for control by value propagation and, under mediator removal, for control of minimal trap spaces, but reveal extensive counterexamples where projection or reduction breaks existing control strategies or yields new ones in the reduced network. They show that reduction is more predictable for propagation-compatible control and trap-space preservation, while attractor control is more fragile under elimination, and they outline practical implications for model reduction in biological Boolean networks. Overall, the paper offers a nuanced roadmap for leveraging reduction in phenotype control, with concrete theorems and counterexamples guiding when and how reduced models can inform controls on the original system.

Abstract

We investigate how elimination of variables can affect the asymptotic dynamics and phenotype control of Boolean networks. In particular, we look at the impact on minimal trap spaces, and identify a structural condition that guarantees their preservation. We examine the possible effects of variable elimination under three of the most popular approaches to control (attractor-based control, value propagation and control of minimal trap spaces), and under different update schemes (synchronous, asynchronous, generalized asynchronous). We provide some insights on the application of reduction, and an ample inventory of examples and counterexamples.

Figures (10)

• Figure 1: (a) Interaction graph and state transition graphs of a Boolean network in 3 components. States in boxes are representative states w.r.t. the component $n = 3$, which is not autoregulated. (b) Interaction graph and state transition graphs of the Boolean network obtained from the network in (a) by elimination of component 3. $\mathrm{AD}(\rho(f))$, $\mathrm{SD}(\rho(f))$ and $\mathrm{GD}(\rho(f))$ coincide. The transitions $110 \to 010$ and $110 \to 011$ are lost in the reduction.
• Figure 2: Schematics summarizing the idea behind elimination of a non-autoregulated component (component $n$ in the figure). (a) Effect on the update functions: all instances of $x_n$ are substituted with the update function $f_n$ of $n$. (b) Effect on the interaction graph: paths of length two that are mediated by $n$ become direct interactions or can disappear with the reduction. (c) Effect on the asynchronous dynamics: $\sigma(x)$ is the representative state of $(x, x_n)$. Only transitions that start from a representative state are guaranteed to be preserved.
• Figure 3: Example illustrating that, if $S_n \neq \star$, then $C(\rho(f),S_{[n-1]})$ and $\rho(C(f,S))$ can differ.
• Figure 4: $S = \mathbb{B}^3$ is an attractor-control strategy for $P=0{\star}{\star}$ for an asynchronous dynamics (case (a)), for a synchronous dynamics (case (b)). On the other hand, $S$ is not an MTS-control strategy. (c): $S = \mathbb{B}^3$ is an MTS-control strategy for $P=0{\star}{\star}$, since the unique minimal trap space is the fixed point $000$. $S$ is not an attractor-control strategy in any of the three dynamics, given the existence of the attractor $\{001, 010, 011, 100, 101, 110\}$.
• Figure 5: Relationship between the three different approaches to phenotype control studied in this paper. The black double-lined arrows indicate total inclusion of phenotype control: any control by value propagation is an attractor-control and MTS-control strategy. Gray double-lined arrows with a slash indicate that the relationship is not always true. A reference to a counterexample is provided.

Theorems & Definitions (50)

• Example 2.1
• Example 2.2
• Example 2.3
• Remark 2.4
• Example 2.5
• Proposition 2.6
• proof
• Definition 2.7
• Definition 2.8
• Example 2.9