Hidden Order of Boolean Networks

It is a common belief that the order of a Boolean network is mainly determined by its attractors, including fixed points and cycles. Using the semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of the Boolean networks, this article reveals that in addition to this explicit order, there is a certain implicit or hidden order, which is determined by the fixed points and limit cycles of their dual networks. The structure and certain properties of dual networks are investigated. Instead of a trajectory, which describes the evolution of a state, the hidden order provides a global horizon to describe the evolution of the overall network. We conjecture that the order of networks is mainly determined by the dual attractors via their corresponding hidden orders. Then these results about the Boolean networks are further extended to the k -valued case.

On the number of driver nodes for controlling a Boolean network when the targets are restricted to attractors

It 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 order to cope with this intrinsic difficulty, we consider a special case of the control problem in which the targets are restricted to attractors. For this special case, we mathematically prove under the uniform distribution of states in basins that the expected number of driver nodes is only O(log₂N+log₂M) for controlling Boolean networks, where M is the number of attractors. Since it is expected that M is not very large in many practical networks, the new model requires a much smaller number of driver nodes. 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.

Observability of Boolean Control Networks Using Parallel Extension and Set Reachability

This brief reviews various definitions of observability for Boolean control networks (BCNs) and proposes a new one: output-feedback observability. This new definition applies to all BCNs whose initial states can be identified from the history of output measurements. A technique called parallel extension is then proposed to facilitate observability analysis. Furthermore, a technique called state transition graph reconstruction is proposed for analyzing the set reachability of BCNs, based on which new criteria for observability, single-input sequence observability, and arbitrary-input observability, are obtained. Using the proposed techniques, this brief proves that the problem of output-feedback observability can be recast as that of stabilizing a logic dynamical system with output feedback. Then, a necessary and sufficient condition for static output feedback observability is proposed. The relationships between the different definitions of observability are discussed, and the main results are illustrated with examples.

Identifying a Probabilistic Boolean Threshold Network From Samples

This paper studies the problem of exactly identifying the structure of a probabilistic Boolean network (PBN) from a given set of samples, where PBNs are probabilistic extensions of Boolean networks. Cheng et al. studied the problem while focusing on PBNs consisting of pairs of AND/OR functions. This paper considers PBNs consisting of Boolean threshold functions while focusing on those threshold functions that have unit coefficients. The treatment of Boolean threshold functions, and triplets and -tuplets of such functions, necessitates a deepening of the theoretical analyses. It is shown that wide classes of PBNs with such threshold functions can be exactly identified from samples under reasonable constraints, which include: 1) PBNs in which any number of threshold functions can be assigned provided that all have the same number of input variables and 2) PBNs consisting of pairs of threshold functions with different numbers of input variables. It is also shown that the problem of deciding the equivalence of two Boolean threshold functions is solvable in pseudopolynomial time but remains co-NP complete.

Partition-Based Solutions of Static Logical Networks With Applications

Given a static logical network, partition-based solutions are investigated. Easily verifiable necessary and sufficient conditions are obtained, and the corresponding formulas are presented to provide all types of the partition-based solutions. Then, the results are extended to mix-valued logical networks. Finally, two applications are presented: 1) an implicit function (IF) theorem of logical equations, which provides necessary and sufficient condition for the existence of IF and 2) converting the difference-algebraic network into a standard difference network.

Identification of Boolean Networks Using Premined Network Topology Information

This brief aims to reduce the data requirement for the identification of Boolean networks (BNs) by using the premined network topology information. First, a matching table is created and used for sifting the true from the false dependences among the nodes in the BNs. Then, a dynamic extension to matching table is developed to enable the dynamic locating of matching pairs to start as soon as possible. Next, based on the pseudocommutative property of the semitensor product, a position-transform mining is carried out to further improve data utilization. Combining the above, the topology of the BNs can be premined for the subsequent identification. Examples are given to illustrate the efficiency of reducing the data requirement. Some excellent features, such as the online and parallel processing ability, are also demonstrated.

Systemic modeling myeloma-osteoclast interactions under normoxic/hypoxic condition using a novel computational approach

Interaction of myeloma cells with osteoclasts (OC) can enhance tumor cell expansion through activation of complex signaling transduction networks. Both cells reside in the bone marrow, a hypoxic niche. How OC-myeloma interaction in a hypoxic environment affects myeloma cell growth and their response to drug treatment is poorly understood. In this study, we i) cultured myeloma cells in the presence/absence of OCs under normoxia and hypoxia conditions and did protein profiling analysis using reverse phase protein array; ii) computationally developed an Integer Linear Programming approach to infer OC-mediated myeloma cell-specific signaling pathways under normoxic and hypoxic conditions. Our modeling analysis indicated that in the presence OCs, (1) cell growth-associated signaling pathways, PI3K/AKT and MEK/ERK, were activated and apoptotic regulatory proteins, BAX and BIM, down-regulated under normoxic condition; (2) β1 Integrin/FAK signaling pathway was activated in myeloma cells under hypoxic condition. Simulation of drug treatment effects by perturbing the inferred cell-specific pathways showed that targeting myeloma cells with the combination of PI3K and integrin inhibitors potentially (1) inhibited cell proliferation by reducing the expression/activation of NF-κB, S6, c-Myc, and c-Jun under normoxic condition; (2) blocked myeloma cell migration and invasion by reducing the expression of FAK and PKC under hypoxic condition.

Synchronization design of Boolean networks via the semi-tensor product method

We provide a general approach for the design of a response Boolean network (BN) to achieve complete synchronization with a given drive BN. The approach is based on the algebraic representation of BNs in terms of the semi-tensor product of matrices. Instead of designing the logical dynamic equations of a response BN directly, we first construct its algebraic representation and then convert the algebraic representation back to the logical form. The results are applied to a three-neuron network in order to illustrate the effectiveness of the proposed approach.

Dynamics of influenza virus and human host interactions during infection and replication cycle

The replication and life cycle of the influenza virus is governed by an intricate network of intracellular regulatory events during infection, including interactions with an even more complex system of biochemical interactions of the host cell. Computational modeling and systems biology have been successfully employed to further the understanding of various biological systems, however, computational studies of the complexity of intracellular interactions during influenza infection is lacking. In this work, we present the first large-scale dynamical model of the infection and replication cycle of influenza, as well as some of its interactions with the host's signaling machinery. Specifically, we focus on and visualize the dynamics of the internalization and endocytosis of the virus, replication and translation of its genomic components, as well as the assembly of progeny virions. Simulations and analyses of the models dynamics qualitatively reproduced numerous biological phenomena discovered in the laboratory. Finally, comparisons of the dynamics of existing and proposed drugs, our results suggest that a drug targeting PB1:PA would be more efficient than existing Amantadin/Rimantaine or Zanamivir/Oseltamivir.

Binary higher order neural networks for realizing Boolean functions

In order to more efficiently realize Boolean functions by using neural networks, we propose a binary product-unit neural network (BPUNN) and a binary π-ς neural network (BPSNN). The network weights can be determined by one-step training. It is shown that the addition " σ," the multiplication " π," and two kinds of special weighting operations in BPUNN and BPSNN can implement the logical operators " ∨," " ∧," and " ¬" on Boolean algebra 〈Z(2),∨,∧,¬,0,1〉 (Z(2)={0,1}), respectively. The proposed two neural networks enjoy the following advantages over the existing networks: 1) for a complete truth table of N variables with both truth and false assignments, the corresponding Boolean function can be realized by accordingly choosing a BPUNN or a BPSNN such that at most 2(N-1) hidden nodes are needed, while O(2(N)), precisely 2(N) or at most 2(N), hidden nodes are needed by existing networks; 2) a new network BPUPS based on a collaboration of BPUNN and BPSNN can be defined to deal with incomplete truth tables, while the existing networks can only deal with complete truth tables; and 3) the values of the weights are all simply -1 or 1, while the weights of all the existing networks are real numbers. Supporting numerical experiments are provided as well. Finally, we present the risk bounds of BPUNN, BPSNN, and BPUPS, and then analyze their probably approximately correct learnability.