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**1 - 6**of**6**### Steady state analysis of Boolean molecular network models via model reduction and computational algebra

, 2014

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### An Efficient Steady-State Analysis Method for Large Boolean Networks with High Maximum Node Connectivity

, 2015

"... Boolean networks have been widely used to model biological processes lacking detailed kinetic information. Despite their simplicity, Boolean network dynamics can still capture some important features of biological systems such as stable cell phenotypes represented by steady states. For small models, ..."

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Boolean networks have been widely used to model biological processes lacking detailed kinetic information. Despite their simplicity, Boolean network dynamics can still capture some important features of biological systems such as stable cell phenotypes represented by steady states. For small models, steady states can be determined through exhaustive enumeration of all state transitions. As the number of nodes increases, however, the state space grows exponentially thus making it difficult to find steady states. Over the last several decades, many studies have addressed how to handle such a state space explosion. Recently, increasing attention has been paid to a satisfiability solving algorithm due to its potential scalability to handle large networks. Meanwhile, there still lies a problem in the case of large models with high maximum node connectivity where the satisfiability solving algorithm is known to be computationally intractable. To address the problem, this paper presents a new partitioning-based method that breaks down a given network into smaller subnetworks. Steady states of each subnetworks are identified by independently applying the satisfiability solving algorithm. Then, they are combined to construct the steady states of the overall network. To efficiently apply the satisfiability solving algorithm to each subnet-work, it is crucial to find the best partition of the network. In this paper, we propose a method that divides each subnetwork to be smallest in size and lowest in maximum node connectivity. This minimizes the total cost of finding all steady states in entire subnetworks. The pro-posed algorithm is compared with others for steady states identification through a number of simulations on both published small models and randomly generated large models with differing maximum node connectivities. The simulation results show that our method can scale up to several hundreds of nodes even for Boolean networks with high maximum node connectivity. The algorithm is implemented and available at

### Improving BDD-based Attractor Detection for Synchronous Boolean Networks

"... ABSTRACT Boolean networks are an important formalism for modelling biological systems and have attracted much attention in recent years. An important direction in Boolean networks is to exhaustively find attractors, which represent steady states when a biological network evolves for a long term. In ..."

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ABSTRACT Boolean networks are an important formalism for modelling biological systems and have attracted much attention in recent years. An important direction in Boolean networks is to exhaustively find attractors, which represent steady states when a biological network evolves for a long term. In this paper, we propose a new approach to improve the efficiency of BDD-based attractor detection. Our approach includes a monolithic algorithm for small networks, an enumerative strategy to deal with large networks, and two heuristics on ordering BDD variables. We demonstrate the performance of our approach on a number of examples, and compare it with one existing technique in the literature.

### Synthesising Executable Gene Regulatory Networks from Single-cell Gene Expression Data

"... Abstract. Recent experimental advances in biology allow researchers to obtain gene expression profiles at single-cell resolution over hundreds, or even thousands of cells at once. These single-cell measurements provide snapshots of the states of the cells that make up a tissue, instead of the popula ..."

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Abstract. Recent experimental advances in biology allow researchers to obtain gene expression profiles at single-cell resolution over hundreds, or even thousands of cells at once. These single-cell measurements provide snapshots of the states of the cells that make up a tissue, instead of the population-level averages provided by conventional high-throughput ex-periments. This new data therefore provides an exciting opportunity for computational modelling. In this paper we introduce the idea of viewing single-cell gene expression profiles as states of an asynchronous Boolean network, and frame model inference as the problem of reconstructing a Boolean network from its state space. We then give a scalable algo-rithm to solve this synthesis problem. We apply our technique to both simulated and real data. We first apply our technique to data simulated from a well established model of common myeloid progenitor differentia-tion. We show that our technique is able to recover the original Boolean network rules. We then apply our technique to a large dataset taken dur-ing embryonic development containing thousands of cell measurements. Our technique synthesises matching Boolean networks, and analysis of these models yields new predictions about blood development which our experimental collaborators were able to verify. 1

### A Parallel Attractor Finding Algorithm Based on Boolean Satisfiability for Genetic Regulatory Networks

"... In biological systems, the dynamic analysis method has gained increasing attention in the past decade. The Boolean network is the most common model of a genetic regulatory network. The interactions of activation and inhibition in the genetic regulatory network are modeled as a set of functions of th ..."

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In biological systems, the dynamic analysis method has gained increasing attention in the past decade. The Boolean network is the most common model of a genetic regulatory network. The interactions of activation and inhibition in the genetic regulatory network are modeled as a set of functions of the Boolean network, while the state transitions in the Boolean network reflect the dynamic property of a genetic regulatory network. A difficult problem for state transition analysis is the finding of attractors. In this paper, we modeled the genetic regulatory network as a Boolean network and proposed a solving algorithm to tackle the attractor finding problem. In the proposed algorithm, we partitioned the Boolean network into several blocks consisting of the strongly connected components according to their gradients, and defined the connection between blocks as decision node. Based on the solutions calculated on the decision nodes and using a satisfiability solving algorithm, we identified the attractors in the state transition graph of each block. The proposed algorithm is benchmarked on a variety of genetic regulatory networks. Compared with existing algorithms, it achieved similar performance on small test cases, and outperformed it on larger and more complex ones, which happens to be the trend of the modern genetic regulatory network. Furthermore, while the existing satisfiability-based algorithms cannot be parallelized due to their inherent algorithm design, the proposed algorithm exhibits a good scalability on parallel

### Characterization of Reachable Attractors Using Petri Net Unfoldings

"... Abstract. Attractors of network dynamics represent the long-term be-haviours of the modelled system. Their characterization is therefore cru-cial for understanding the response and differentiation capabilities of a dynamical system. In the scope of qualitative models of interaction net-works, the co ..."

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Abstract. Attractors of network dynamics represent the long-term be-haviours of the modelled system. Their characterization is therefore cru-cial for understanding the response and differentiation capabilities of a dynamical system. In the scope of qualitative models of interaction net-works, the computation of attractors reachable from a given state of the network faces combinatorial issues due to the state space explosion. In this paper, we present a new algorithm that exploits the concurrency between transitions of parallel acting components in order to reduce the search space. The algorithm relies on Petri net unfoldings that can be used to compute a compact representation of the dynamics. We illustrate the applicability of the algorithm with Petri net models of cell signalling and regulation networks, Boolean and multi-valued. The proposed ap-proach aims at being complementary to existing methods for deriving the attractors of Boolean models, while being generic since it applies to any safe Petri net.