On the Number of Control Nodes in Boolean Networks With Degree Constraints

Area of Science:

• Computational Biology
• Network Science
• Systems Biology

Background:

• Boolean networks (BNs) are widely used to model complex biological systems.
• Controlling BNs is essential for understanding and manipulating system behavior.
• The minimum control node set problem addresses the minimal intervention needed to reach a target state.

Purpose of the Study:

• To derive nontrivial lower and upper bounds for the minimum control node set in Boolean networks with degree constraints.
• To analyze these bounds for four specific types of Boolean networks: k-k-XOR-BNs, simple k-k-AND-BNs, k-k-AND-BNs with negation, and k-k-NC-BNs.
• To investigate the applicability of these findings to networks utilizing AND and OR functions.

Main Methods:

• Combinatorial analysis of four types of Boolean networks with specified indegree and outdegree (k).
• Division of network nodes into three disjoint sets.
• Extension of the time required to reach a target state.
• Utilization of necessary conditions for network controllability.

Main Results:

• Derivation of four bounds: general lower bound, best-case upper bound, worst-case lower bound, and general upper bound for the minimum control node set size.
• Discovery of meaningful results and phenomena related to network control.
• Demonstration that results for AND functions are also applicable to OR functions.

Conclusions:

• The study provides a comprehensive analysis of the minimum control node set problem for degree-constrained Boolean networks.
• The derived bounds offer valuable insights into the controllability of these networks.
• The findings have implications for understanding and controlling complex biological systems modeled by Boolean networks.