Reduction for asynchronous Boolean networks: elimination of negatively autoregulated components

TL;DR

This work extends the standard variable-elimination approach for asynchronous Boolean networks to include vertices with optional negative autoregulation, carefully analyzing how reductions affect the attractor landscape and interaction structure. It formalizes the generalized reduction using maps $\mathcal{R}^a$ and $\mathcal{S}^a$, yielding a reduced network $\tilde{f}$ that preserves certain transitions and, under additional conditions, cyclic attractors; it also demonstrates that naive mediator-node removals can alter asymptotic behavior. The authors establish sufficient structural conditions on the interaction graph to guarantee attractor preservation, demonstrate that a chain of mediator variables can change the number of attractors, and provide an alternate proof that the number of attractors is bounded by $2^{|I|}$ where $|I|$ is the size of a minimum positive feedback vertex set. Overall, the results broaden the practical applicability of reduction for large networks while clarifying limitations and enabling bounds on long-term dynamics. The work has potential implications for scalable analysis of gene regulatory networks and related discrete dynamical systems.

Abstract

To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph.

Figures (4)

• Figure 1: Commutative diagrams that illustrate the definition of the reduction method described in naldi2009reductionnaldi2011dynamicallyveliz2011reduction.
• Figure 2: Illustration of the reduction method described in naldi2009reduction. Representative states are shown in boxes. When the second variable is eliminated, transitions starting from representative states are preserved. The asynchronous dynamics on $\mathbb{B}^3$ on the left reduces to the asynchronous dynamics on $\mathbb{B}^2$ on the right.
• Figure 3: Illustration of the effect of elimination of one variable ($v$) on asynchronous state transition graphs in case of no loops in $G(f)(x)$ at $v$ (left) and a negative loop in $G(f)(x)$ at $v$ (right). In the first case, only transitions that start at the representative state $\mathcal{R}(x)$ are preserved. In the second case, transitions out of both $\mathcal{R}^0(x)$ and $\mathcal{R}^1(x)$ are preserved.
• Figure 4: Asynchronous state transition graphs for the map $f(x_u,x_v,x_w)=(\bar{x}_u,x_u,(x_u\wedge x_w)\vee(\bar{x}_v\wedge x_w)\vee(x_u\wedge \bar{x}_v))$ (left) and the one obtained from $f$ by eliminating $v$ (right). The state transition graphs have one cyclic attractor and two cyclic attractors respectively.

Theorems & Definitions (32)

• Example 2.1
• Proposition 2.2
• Theorem 2.3
• Remark 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof