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1. The Dynamics of Canalizing Boolean Networks

   https://doi.org/10.1155/2020/3687961  

   Paul, Elijah; Pogudin, Gleb; Qin, William; Laubenbacher, Reinhard (January 2020, Complexity)

   Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. In this paper, we investigate the dynamics of a random Boolean network with such properties using analytical methods and simulations. From our simulations, we observe that Boolean networks with higher canalizing depth have generally fewer attractors, the attractors are smaller, and the basins are larger, with implications for the stability and robustness of the models. These properties are relevant to many biological applications. Moreover, our results show that, from the standpoint of the attractor structure, high canalizing depth, compared to relatively small positive canalizing depth, has a very modest impact on dynamics. Motivated by these observations, we conduct mathematical study of the attractor structure of a random Boolean network of canalizing depth one (i.e., the smallest positive depth). For every positive integer ℓ , we give an explicit formula for the limit of the expected number of attractors of length ℓ in an n -state random Boolean network as n goes to infinity. 

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2. Edgetic perturbations to eliminate fixed-point attractors in Boolean regulatory networks

   https://doi.org/10.1063/1.5083060  

   Campbell, Colin; Albert, Réka (February 2019, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   The dynamics of complex biological networks may be modeled in a Boolean framework, where the state of each system component is either abundant (ON) or scarce/absent (OFF), and each component's dynamic trajectory is determined by a logical update rule involving the state(s) of its regulator(s). It is possible to encode the update rules in the topology of the so-called expanded graph, analysis of which reveals the long-term behavior, or attractors, of the network. Here, we develop an algorithm to perturb the expanded graph (or, equivalently, the logical update rules) to eliminate stable motifs: subgraphs that cause a subset of components to stabilize to one state. Depending on the topology of the expanded graph, these perturbations lead to the modification or loss of the corresponding attractor. While most perturbations of biological regulatory networks in the literature involve the knockout (fixing to OFF) or constitutive activation (fixing to ON) of one or more nodes, we here consider edgetic perturbations, where a node's update rule is modified such that one or more of its regulators is viewed as ON or OFF regardless of its actual state. We apply the methodology to two biological networks. In a network representing T-LGL leukemia, we identify edgetic perturbations that eliminate the cancerous attractor, leaving only the healthy attractor representing cell death. In a network representing drought-induced closure of plant stomata, we identify edgetic perturbations that modify the single attractor such that stomata, instead of being fixed in the closed state, oscillates between the open and closed states. 

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3. pystablemotifs: Python library for attractor identification and control in Boolean networks

   https://doi.org/10.1093/bioinformatics/btab825  

   Rozum, Jordan C.; Deritei, Dávid; Park, Kyu Hyong; Gómez Tejeda Zañudo, Jorge; Albert, Réka; Cowen, ed., Lenore (December 2021, Bioinformatics)

   Abstract Summarypystablemotifs is a Python 3 library for analyzing Boolean networks. Its non-heuristic and exhaustive attractor identification algorithm was previously presented in Rozum et al. (2021). Here, we illustrate its performance improvements over similar methods and discuss how it uses outputs of the attractor identification process to drive a system to one of its attractors from any initial state. We implement six attractor control algorithms, five of which are new in this work. By design, these algorithms can return different control strategies, allowing for synergistic use. We also give a brief overview of the other tools implemented in pystablemotifs. Availability and implementationThe source code is on GitHub at https://github.com/jcrozum/pystablemotifs/. Supplementary informationSupplementary data are available at Bioinformatics online. 

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4. Network Restructuring Control for Conic Invariance with Application to Neural Networks

   https://doi.org/10.1109/CDC.2018.8618729  

   Singh, Matthew F.; Ching, ShiNung (December 2018, Conference on Decision and Control)

   Recent advances in the study of artificial and biological neural networks support the power of dynamic representations-computing with information stored as nontrivial limit-sets rather than fixed-point attractors. Understanding and manipulating these computations in nonlinear networks requires a theory of control for abstract objective functions. Towards this end, we consider two properties of limit-sets: their topological dimension and orientation (covariance) in phase space and combine these abstract properties into a single well-defined objective: conic control-invariant sets in the derivative space (i.e., the vector field). Real-world applications, such as neural-medicine, constrain which control laws are feasible with less-invasive controllers being preferable. To this end, we derive a feedback control-law for conic invariance which corresponds to constrained restructuring of the network connections as might occur with pharmacological intervention (as opposed to a physically separate control unit). We demonstrate the ease and efficacy of the technique in controlling the orientation and dimension of limit sets in high-dimensional neural networks. 

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5. Transcriptional Regulatory Network Topology with Applications to Bio-inspired Networking: A Survey

   https://doi.org/10.1145/3468266  

   Roy, Satyaki; Ghosh, Preetam; Ghosh, Nirnay; Das, Sajal K (November 2022, ACM Computing Surveys)

   The advent of the edge computing network paradigm places the computational and storage resources away from the data centers and closer to the edge of the network largely comprising the heterogeneous IoT devices collecting huge volumes of data. This paradigm has led to considerable improvement in network latency and bandwidth usage over the traditional cloud-centric paradigm. However, the next generation networks continue to be stymied by their inability to achieve adaptive, energy-efficient, timely data transfer in a dynamic and failure-prone environment—the very optimization challenges that are dealt with by biological networks as a consequence of millions of years of evolution. The transcriptional regulatory network (TRN) is a biological network whose innate topological robustness is a function of its underlying graph topology. In this article, we survey these properties of TRN and the metrics derived therefrom that lend themselves to the design of smart networking protocols and architectures. We then review a body of literature on bio-inspired networking solutions that leverage the stated properties of TRN. Finally, we present a vision for specific aspects of TRNs that may inspire future research directions in the fields of large-scale social and communication networks. 

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