The Influence of Canalization on the Robustness of Boolean Networks

AbstractTime- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by k-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to c-sensitivity and provides formulas for the activities and c-sensitivity of general k-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

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On the Lyapunov Exponent of Monotone Boolean Networks †

Mathematics ◽

10.3390/math8061035 ◽

2020 ◽

Vol 8 (6) ◽

pp. 1035

Author(s):

Ilya Shmulevich

Keyword(s):

Lyapunov Exponent ◽

Dynamical Systems ◽

Boolean Functions ◽

Boolean Network ◽

Discrete Dynamical Systems ◽

Boolean Networks ◽

Asymptotic Formulas ◽

Small Perturbations ◽

Monotone Boolean Functions ◽

Almost All

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

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Deriving a Boolean dynamics to reveal macrophage activation with in vitro temporal cytokine expression profiles

BMC Bioinformatics ◽

10.1186/s12859-019-3304-5 ◽

2019 ◽

Vol 20 (1) ◽

Cited By ~ 1

Author(s):

Ricardo Ramirez ◽

Allen Michael Herrera ◽

Joshua Ramirez ◽

Chunjiang Qian ◽

David W. Melton ◽

...

Keyword(s):

Boolean Network ◽

Macrophage Activation ◽

Expression Profiles ◽

Network Models ◽

Cytokine Expression ◽

Boolean Networks ◽

Experimental Results ◽

Activated Macrophages ◽

Activation Patterns

Abstract Background Macrophages show versatile functions in innate immunity, infectious diseases, and progression of cancers and cardiovascular diseases. These versatile functions of macrophages are conducted by different macrophage phenotypes classified as classically activated macrophages and alternatively activated macrophages due to different stimuli in the complex in vivo cytokine environment. Dissecting the regulation of macrophage activations will have a significant impact on disease progression and therapeutic strategy. Mathematical modeling of macrophage activation can improve the understanding of this biological process through quantitative analysis and provide guidance to facilitate future experimental design. However, few results have been reported for a complete model of macrophage activation patterns. Results We globally searched and reviewed literature for macrophage activation from PubMed databases and screened the published experimental results. Temporal in vitro macrophage cytokine expression profiles from published results were selected to establish Boolean network models for macrophage activation patterns in response to three different stimuli. A combination of modeling methods including clustering, binarization, linear programming (LP), Boolean function determination, and semi-tensor product was applied to establish Boolean networks to quantify three macrophage activation patterns. The structure of the networks was confirmed based on protein-protein-interaction databases, pathway databases, and published experimental results. Computational predictions of the network evolution were compared against real experimental results to validate the effectiveness of the Boolean network models. Conclusion Three macrophage activation core evolution maps were established based on the Boolean networks using Matlab. Cytokine signatures of macrophage activation patterns were identified, providing a possible determination of macrophage activations using extracellular cytokine measurements.

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Random Networks with Quantum Boolean Functions

Mathematics ◽

10.3390/math9080792 ◽

2021 ◽

Vol 9 (8) ◽

pp. 792 ◽

Cited By ~ 1

Author(s):

Mario Franco ◽

Octavio Zapata ◽

David A. Rosenblueth ◽

Carlos Gershenson

Keyword(s):

Boolean Functions ◽

Boolean Network ◽

Complex Dynamics ◽

Boolean Networks ◽

Stable Complex ◽

Complex Dynamic ◽

Random Boolean Networks ◽

Quantum Simulator ◽

Novel Model ◽

Unstable Dynamics

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

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The Dynamics of Canalizing Boolean Networks

Complexity ◽

10.1155/2020/3687961 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14

Author(s):

Elijah Paul ◽

Gleb Pogudin ◽

William Qin ◽

Reinhard Laubenbacher

Keyword(s):

Gene Regulatory Networks ◽

Regulatory Networks ◽

Boolean Network ◽

Boolean Networks ◽

Molecular Networks ◽

Biological Applications ◽

Expected Number ◽

Modeling Framework ◽

Gene Regulatory ◽

The Stability

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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On the Number of Driver Nodes for Controlling a Boolean Network to Attractors

10.1101/395442 ◽

2018 ◽

Author(s):

Wenpin Hou ◽

Peiying Ruan ◽

Wai-Ki Ching ◽

Tatsuya Akutsu

Keyword(s):

Biological Networks ◽

Boolean Network ◽

Network Models ◽

Boolean Networks ◽

Reasonable Assumption ◽

Control Problems ◽

Expected Number ◽

Linear Complex ◽

Simulation Results ◽

Control Complex

AbstractIt is known that many driver nodes are required to control complex biological networks. Previous studies imply that O(N) driver nodes are required in both linear complex network and Boolean network models with N nodes if an arbitrary state is specified as the target. In this paper, we mathematically prove under a reasonable assumption that the expected number of driver nodes is only O(log2N + log2M) for controlling Boolean networks if the targets are restricted to attractors, where M is the number of attractors. Since it is expected that M is not very large in many practical networks, this is a significant improvement. This result is based on discovery of novel relationships between control problems on Boolean networks and the coupon collector’s problem, a well-known concept in combinatorics. We also provide lower bounds of the number of driver nodes as well as simulation results using artificial and realistic network data, which support our theoretical findings.

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ROBUSTNESS OF TRANSCRIPTIONAL REGULATION IN YEAST-LIKE MODEL BOOLEAN NETWORKS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127410026228 ◽

2010 ◽

Vol 20 (03) ◽

pp. 929-935 ◽

Cited By ~ 1

Author(s):

MURAT TUĞRUL ◽

ALKAN KABAKÇIOĞLU

Keyword(s):

Transcriptional Regulation ◽

Boolean Functions ◽

Boolean Network ◽

Regulation Of Gene Expression ◽

Boolean Networks ◽

Transcription Level ◽

Yeast Saccharomyces Cerevisiae ◽

Boolean Network Model ◽

Global Topology ◽

Model Networks

We investigate the dynamical properties of the transcriptional regulation of gene expression in the yeast Saccharomyces Cerevisiae within the framework of a synchronously and deterministically updated Boolean network model. With a dynamically determinant subnetwork, we explore the robustness of transcriptional regulation as a function of the type of Boolean functions used in the model that mimic the influence of regulating agents on the transcription level of a gene. We compare the results obtained for the actual yeast network with those from two different model networks, one with similar in-degree distribution as the yeast and otherwise random, and another due to Balcan et al., where the global topology of the yeast network is reproduced faithfully. We, surprisingly, find that the first set of model networks better reproduce the results found with the actual yeast network, even though the Balcan et al. model networks are structurally more similar to that of yeast.

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Construction Method of Probabilistic Boolean Networks Based on Imperfect Information

Algorithms ◽

10.3390/a12120268 ◽

2019 ◽

Vol 12 (12) ◽

pp. 268

Author(s):

Katsuaki Umiji ◽

Koichi Kobayashi ◽

Yuh Yamashita

Keyword(s):

Gene Regulatory Networks ◽

Boolean Functions ◽

Regulatory Networks ◽

Imperfect Information ◽

Boolean Network ◽

Construction Method ◽

Boolean Networks ◽

Probabilistic Boolean Network ◽

Gene Regulatory ◽

Selection Probabilities

A probabilistic Boolean network (PBN) is well known as one of the mathematical models of gene regulatory networks. In a Boolean network, expression of a gene is approximated by a binary value, and its time evolution is expressed by Boolean functions. In a PBN, a Boolean function is probabilistically chosen from candidates of Boolean functions. One of the authors has proposed a method to construct a PBN from imperfect information. However, there is a weakness that the number of candidates of Boolean functions may be redundant. In this paper, this construction method is improved to efficiently utilize given information. To derive Boolean functions and those selection probabilities, the linear programming problem is solved. Here, we introduce the objective function to reduce the number of candidates. The proposed method is demonstrated by a numerical example.

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A Multilayer Structure Facilitates the Production of Antifragile Systems in Boolean Network Models

Complexity ◽

10.1155/2019/2783217 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Hyobin Kim ◽

Omar K. Pineda ◽

Carlos Gershenson

Keyword(s):

Complex Systems ◽

Real World ◽

Multilayer Structure ◽

Boolean Network ◽

Single Layer ◽

Network Models ◽

Boolean Networks ◽

Multilayer Structures ◽

Multilayer Networks ◽

Engineered Systems

Antifragility is a property from which systems are able to resist stress and furthermore benefit from it. Even though antifragile dynamics is found in various real-world complex systems where multiple subsystems interact with each other, the attribute has not been quantitatively explored yet in those complex systems which can be regarded as multilayer networks. Here we study how the multilayer structure affects the antifragility of the whole system. By comparing single-layer and multilayer Boolean networks based on our recently proposed antifragility measure, we found that the multilayer structure facilitated the production of antifragile systems. Our measure and findings will be useful for various applications such as exploring properties of biological systems with multilayer structures and creating more antifragile engineered systems.

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Graphene-based 3D XNOR-VRRAM with ternary precision for neuromorphic computing

npj 2D Materials and Applications ◽

10.1038/s41699-021-00236-x ◽

2021 ◽

Vol 5 (1) ◽

Author(s):

Batyrbek Alimkhanuly ◽

Joon Sohn ◽

Ik-Joon Chang ◽

Seunghyun Lee

Keyword(s):

Neural Network ◽

Energy Consumption ◽

Recognition Accuracy ◽

Material Selection ◽

Weighted Sum ◽

Device Design ◽

Key Factors ◽

Neuromorphic Computing ◽

Device Scaling ◽

The Impact

AbstractRecent studies on neural network quantization have demonstrated a beneficial compromise between accuracy, computation rate, and architecture size. Implementing a 3D Vertical RRAM (VRRAM) array accompanied by device scaling may further improve such networks’ density and energy consumption. Individual device design, optimized interconnects, and careful material selection are key factors determining the overall computation performance. In this work, the impact of replacing conventional devices with microfabricated, graphene-based VRRAM is investigated for circuit and algorithmic levels. By exploiting a sub-nm thin 2D material, the VRRAM array demonstrates an improved read/write margins and read inaccuracy level for the weighted-sum procedure. Moreover, energy consumption is significantly reduced in array programming operations. Finally, an XNOR logic-inspired architecture designed to integrate 1-bit ternary precision synaptic weights into graphene-based VRRAM is introduced. Simulations on VRRAM with metal and graphene word-planes demonstrate 83.5 and 94.1% recognition accuracy, respectively, denoting the importance of material innovation in neuromorphic computing.

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Controllability of Boolean networks with intermittent disturbances

Modern Physics Letters B ◽

10.1142/s0217984920503091 ◽

2020 ◽

Vol 34 (28) ◽

pp. 2050309

Author(s):

Tao You ◽

Hailun Zhang ◽

Mingyu Yang ◽

Xiao Wang ◽

Yangming Guo

Keyword(s):

Gene Expression ◽

Boolean Network ◽

Boolean Networks ◽

Genetic Mutations ◽

Intermittent Control ◽

Mathematical Tool ◽

Pulse Control ◽

Expression Of Genes ◽

Complicated Process ◽

Living Being

In biological systems, gene expression is an important subject. In order to clarify the specific process of gene expression, mathematical tools are needed to simulate the process. The Boolean network (BN) is a good mathematical tool. In this paper, we study a Boolean network with intermittent perturbations. This is of great significance for studying genetic mutations in bioengineering. The expression of genes in the internal system of a living being is a very complicated process, and it is clear that the process is trans-ageal for humans. Through the intermittent control and pulse control of the BN, we can obtain the trajectory of gene expression better and faster, which will provide a very important theoretical basis for our next research.

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