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R (2006) Methods of robustness analysis for boolean models of gene control networks
 IET Systems Biology
"... As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network ..."
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Cited by 47 (17 self)
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As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuoustime character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model’s gene pattern prediction as a function of the timescales of the various processes. 1
Controllability of Boolean control networks via the PerronFrobenius theory
 AUTOMATICA
, 2012
"... Boolean control networks (BCNs) are recently attracting considerable interest as computational models for genetic and cellular networks. Addressing controltheoretic problems in BCNs may lead to a better understanding of the intrinsic control in biological systems, as well as to developing suitable ..."
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Cited by 13 (3 self)
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Boolean control networks (BCNs) are recently attracting considerable interest as computational models for genetic and cellular networks. Addressing controltheoretic problems in BCNs may lead to a better understanding of the intrinsic control in biological systems, as well as to developing suitable protocols for manipulating biological systems using exogenous inputs. We introduce two definitions for controllability of a BCN, and show that a necessary and sufficient condition for each form of controllability is that a certain nonnegative matrix is irreducible or primitive, respectively. Our analysis is based on a result that may be of independent interest, namely, a simple algebraic formula for the number of different control sequences that steer a BCN between given initial and final states in a given number of time steps, while avoiding a set of forbidden states.
2008b) Quantitative analysis of robustness and fragility in biological networks based on feedback dynamics
 Bioinformatics
"... Motivation: It has been widely reported that biological networks are robust against perturbations such as mutations. On the contrary, it has also been known that biological networks are often fragile against unexpected mutations. There is a growing interest in these intriguing observations and the u ..."
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Cited by 11 (2 self)
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Motivation: It has been widely reported that biological networks are robust against perturbations such as mutations. On the contrary, it has also been known that biological networks are often fragile against unexpected mutations. There is a growing interest in these intriguing observations and the underlying design principle that causes such robust but fragile characteristics of biological networks. For relatively small networks, a feedback loop has been considered as an important motif for realizing the robustness. It is still however not clear how a number of coupled feedback loops actually affect the robustness of large complex biological networks. In particular, the relationship between fragility and feedback loops has not yet been investigated till now. Results: Through extensive computational experiments, we found that networks with a larger number of positive feedback loops and a smaller number of negative feedback loops are likely to be more robust against perturbations. Moreover, we found that the nodes of a robust network subject to perturbations are mostly involved with a smaller number of feedback loops compared with the other nodes not usually subject to perturbations. This topological characteristic eventually makes the robust network fragile against unexpected mutations at the nodes not previously exposed to perturbations. Contact:
A Pontryagin Maximum Principle for Multi–Input Boolean Control Networks
"... A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. Boolean networks have been studied extensively as models for simple artificial neural networks. Recently, Boolean networks gained considerable interest as models for biological syst ..."
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Cited by 9 (3 self)
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A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. Boolean networks have been studied extensively as models for simple artificial neural networks. Recently, Boolean networks gained considerable interest as models for biological systems composed of elements that can be in one of two possible states. Examples include genetic regulation networks, where the ON (OFF) state corresponds to the transcribed (quiescent) state of a gene, and cellular networks where the two possible logic states may represent the open/closed state of an ion channel, basal/high activity of an enzyme, two possible conformational states of a protein, etc. Daizhan Cheng developed an algebraic statespace representation for Boolean control networks using the semi–tensor product of matrices. This representation proved quite useful for studying Boolean control networks in a controltheoretic framework. Using this representation, we consider a Mayertype optimal control problem for Boolean control networks. Our main result is a necessary condition for optimality. This provides a parallel of Pontryagin’s maximum principle to Boolean control networks.
Boolean Dynamics of Biological Networks with Multiple Coupled Feedback Loops
, 2007
"... Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little ..."
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Cited by 8 (5 self)
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Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little attention paid to the dynamical role of a feedback loop, which is a circular chain of interactions between Boolean variables. We note that such feedback loops are ubiquitously found in various biological systems as multiple coupled structures and they are often the primary cause of complex dynamics. In this article, we investigate the relationship between the multiple coupled feedback loops and the dynamics of Boolean networks. We show that networks have a larger proportion of basins corresponding to fixedpoint attractors as they have more coupled positive feedback loops, and a larger proportion of basins for limitcycle attractors as they have more coupled negative feedback loops.
Observability of Boolean Networks: A GraphTheoretic Approach
, 2013
"... Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables and outputs. BNs are recently attracting considerable interest as computational models for genetic and cellular networks. We consider the observability of BNs, that is, the possibility of uniquely determining the ..."
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Cited by 7 (3 self)
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Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables and outputs. BNs are recently attracting considerable interest as computational models for genetic and cellular networks. We consider the observability of BNs, that is, the possibility of uniquely determining the initial state given a time sequence of outputs. Our main result is that determining whether a BN is observable is NPhard. This holds for both synchronous and asynchronous BNs. Thus, unless P=NP, there does not exist an algorithm with polynomial time complexity that solves the observability problem. We also give two simple algorithms, with exponential complexity, that solve this problem. Our results are based on combining the algebraic representation of BNs derived by D. Cheng with a graphtheoretic approach. Some of the theoretical results are applied to study the observability of a BN model of the mammalian cell cycle.
The Rise of the Regulatory
 State,” J. Econ. Lit
, 2003
"... approaches to modelling developmental gene ..."
MinimumTime Control of Boolean Networks
, 2012
"... Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables. BNs are recently attracting considerable interest as computational models for biological systems and, in particular, as models of gene regulating networks. Boolean control networks (BCNs) are Boolean networks wi ..."
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Cited by 5 (3 self)
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Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables. BNs are recently attracting considerable interest as computational models for biological systems and, in particular, as models of gene regulating networks. Boolean control networks (BCNs) are Boolean networks with Boolean inputs. We consider the problem of steering a BCN from a given state to a desired state in minimal time. Using the algebraic statespace representation (ASSR) of BCNs we derive several necessary conditions, stated in the form of maximum principles (MPs), for a control to be timeoptimal. In the ASSR every state and input vector is a canonical vector. Using this special structure yields an explicit statefeedback formula for all timeoptimal controls. To demonstrate the theoretical results, we develop a BCN model for the genetic switch controlling the lambda phage development upon infection of a bacteria. Our results suggest that this biological switch is designed in a way that guarantees minimal time response to important environmental signals.
A model of sequential branching in hierarchical cell fate determination
 IEEE Transactions on Automatic Control, Special Issue on Systems Biology
, 2008
"... ..."
Topologybased cancer classification and related pathway mining using microarray data
 Nucleic Acids Res
, 2006
"... Cancer classification is the critical basis for patienttailored therapy, while pathway analysis is a promising method to discover the underlying molecular mechanisms related to cancer development by using microarray data. However, linking the molecular classification and pathway analysis with gene ..."
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Cited by 4 (0 self)
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Cancer classification is the critical basis for patienttailored therapy, while pathway analysis is a promising method to discover the underlying molecular mechanisms related to cancer development by using microarray data. However, linking the molecular classification and pathway analysis with gene network approach has not been discussed yet. In this study, we developed a novel framework based on cancer classspecific gene networks for classification and pathway analysis. This framework involves a novel gene network construction, named ordering network, which exhibits the powerlaw nodedegree distribution as seen in correlation networks. The results obtained from five public cancer datasets showed that the gene networks with ordering relationship are better than those with correlation relationship in terms of accuracy and stability of the classification performance. Furthermore, we integrated the ordering networks, classification information and pathway database to develop the topologybased pathway analysis for identifying cancer classspecific pathways, which might be essential in the biological significance of cancer. Our results suggest that the topologybased classification technology can precisely distinguish cancer subclasses and the topologybased pathway analysis can characterize the correspondent biochemical pathways even if there are subtle, but consistent, changes in gene expression, which may provide new insights into the underlying molecular mechanisms of tumorigenesis.