Abstract
Effective control of biological systems can often be achieved through the control of a surprisingly small number of distinct variables. We bring clarity to such results using the formalism of Boolean dynamical networks, analyzing the effectiveness of external control in selecting a desired final state when that state is among the original attractors of the dynamics. Analyzing 49 existing biological network models, we find strong numerical evidence that the average number of nodes that must be forced scales logarithmically with the number of original attractors. This suggests that biological networks may be typically easy to control even when the number of interacting components is large. We provide a theoretical explanation of the scaling by separating controlling nodes into three types: those that act as inputs, those that distinguish among attractors, and any remaining nodes. We further identify characteristics of dynamics that can invalidate this scaling, and speculate about how this relates more broadly to non-biological systems.
Highlights
Effective control of biological systems can often be achieved through the control of a surprisingly small number of distinct variables
Readers familiar with Boolean network dynamics can proceed to the section
In the context of theoretical biology, their relevance was first highlighted by Kauffman[23], who identified cell types with the attractors of network dynamics, allowing for their mathematical study
Summary
Our brief discussions about fixed-point and cyclic attractors reveal that—once the input nodes are pinned, and in the absence of additional structure in the dynamics that significantly undermine the approximate randomness of the transition matrix (for an example of this see the section on Random Networks)—we can expect our iterative pinning procedure to converge quickly. These exceptions can be understood by considering the effect of a strong bias toward activation on the set of fixed-point attractors: those fixed points with few or no activated nodes are expected to have small basins because most perturbations will lead toward states with more excited nodes. When the bias toward activation is removed in the “balanced” networks, these exceptions disappear and CK sizes are again characterized by
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