We provide the first classification of different types of RandomBoolean Networks (RBNs). We study the differences of RBNs depending on the degree of synchronicity and determinism of their updating scheme. For doing so, we first define three new types of RBNs. We note some similarities and differences between different types of RBNs with the aid of a public software laboratory we developed. Particularly, we find that the point attractors are independent of the updating scheme, and that RBNs are more different depending on their determinism or non-determinism rather than depending on their synchronicity or asynchronicity. We also show a way of mapping non-synchronous deterministic RBNs into synchronous RBNs. Our results are important for justifying the use of specific types of RBNs for modelling natural phenomena.

Abstract. Living cells are extremely well-organized autonomous systems, consisting of discrete interacting components. Key to understanding and modeling their behavior is modeling their system organization. Four distinct chemical toolkits (classes of macromolecules) have been characterized, each combinatorial in nature. Each toolkit consists of a small number of simple components that are assembled (polymerized) into complex structures that interact in rich ways. Each toolkit abstracts away from chemistry; it embodies an abstract machine with its own instruction set and its own peculiar interaction model. These interaction models are highly effective, but are not ones commonly used in computing: proteins stick together, genes have fixed output, membranes carry activity on their surfaces. Biologists have invented a number of notations attempting to describe these abstract machines and the processes they implement. Moving up from molecular biology, systems biology aims to understand how these interaction models work, separately and together. 1

Motivation: Intervention in a gene regulatory network is used to help it avoid undesirable states, such as those associated with a disease. Several types of intervention have been studied in the framework of a probabilistic Boolean network (PBN), which is essentially a finite collection of Boolean networks in which at any discrete time point the gene state vector transitions according to the rules of one of the constituent networks. For an instantaneously random PBN, the governing Boolean network is randomly chosen at each time point. For a context-sensitive PBN, the governing Boolean network remains fixed for an interval of time until a binary random variable determines a switch. The theory of automatic control has been previously applied to find optimal strategies for manipulating external (control) variables that affect the transition probabilities of an instantaneously random PBN to desirably affect its dynamic evolution over a finite time horizon. This paper extends the methods of external control to context-sensitive PBNs.

Probabilistic Boolean Networks (PBNs) comprise a graphical model based on uncertain rule-based dependencies between nodes and have been proposed as a model for genetic regulatory networks. As with any algebraic strucicf theckxx--zkfjx#[xk of important mappings between PBNs isckT--#G for both theory andapplic-kfjj This paper treats the ckxjH[[kfjj of mappings to alter PBNstruc-#V while at the same time maintaining cintaining with the original probability strucilit It ctkx[[jH projecHkfj onto sub-networks, adjuncwork of new nodes, resolution reducuti mappings formed by merging nodes, and morphological mappings on the graph structure of the PBN. It places PBNs in the framework of many-sorted algebras and in that context defines homomorphisms between PBNs.

Motivation: We have hypothesized that the construction of transcriptional regulatory networks using a method that optimizes connectivity would lead to regulation consistent with biological expectations. A key expectation is that the hypothetical networks should produce a few, very strong attractors, highly similar to the original observations, mimicking biological state stability and determinism. Another central expectation is that, since it is expected that the biological control is distributed and mutually reinforcing, interpretation of the observations should lead to a very small number of connection schemes.

Probabilistic Boolean networks (PBNs) have recently been introduced as a promising class of models of genetic regulatory networks. The dynamic behaviour of PBNs can be analysed in the context of Markov chains. A key goal is the determination of the steady-state (long-run) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the long-term influence of a gene on another gene or determine the long-term joint probabilistic behaviour of a few selected genes. Because matrix-based methods quickly become prohibitive for large sizes of networks, we propose the use of Monte Carlo methods. However, the rate of convergence to the stationary distribution becomes a central issue. We discuss several approaches for determining the number of iterations necessary to achieve convergence of the Markov chain corresponding to a PBN. Using a recently introduced method based on the theory of two-state Markov chains, we illustrate the approach on a sub-network designed from human glioma gene expression data and determine the joint steadystate probabilities for several groups of genes. Copyright # 2003 John Wiley & Sons, Ltd.

this article, we give an overview of GSP and describe how pattern recognition and network analysis are central to diagnosis and therapy for genetic diseases

Abstract. In this paper we study homomorphisms of Probabilistic Regulatory Gene Networks(PRN) introduced in [2]. The model PRN is a natural generalization of the Probabilistic Boolean Networks (PBN), introduced by I. Shmulevich, E. Dougherty, and W. Zhang in [14], that has been using to describe genetic networks and has therapeutic applications, see [15]. In this paper, our main objectives are to apply the concept of homomorphism and ǫ-homomorphism of probabilistic regulatory networks to the dynamic of the networks. The meaning of ǫ is that these homomorphic networks have similar distributions and the distance between the distributions is upper bounded by ǫ.Additionally, we prove that the class of PRN together with the homomorphisms form a category with products and coproducts. Projections are special homomorphisms, and they always induce invariant subnetworks that contain all the cycles and steady states in the network. Here, it is proved that the ǫ-homomorphism for 0 < ǫ < 1 produce simultaneous Markov Chains in both networks, that permit to introduce the concept of ǫ-isomorphism of Markov Chains, and similar networks.

It is acknowledged that the presence of positive or negative circuits in regulatory networks such as genetic networks is linked to the emergence of significant dynamical properties such as multistability (involved in differentiation) and periodic oscillations (homeostasis). Rules proposed by the biologist R. Thomas assert that these circuits are necessary for such dynamical properties. These rules have been studied by several authors. Their obvious interest is that they relate the rather simple information contained in the structure of the network (signed circuits) to its much more complex dynamical behaviour. We prove in this article a non-trivial converse of these rules, namely that certain positive or negative circuits in a regulatory graph are actually sufficient for the observation of a restricted form of the corresponding dynamical property, differentiation or homeostasis. More precisely, the crucial property that we require is that the circuit be globally elementary. We then apply these results to the vertebrate immune system, and show that the 2 elementary functional positive circuits of the model indeed behave as modules which combine to explain the presence of the 3 stable states corresponding to the Th0, Th1 and Th2 cells.

This paper considers key issues in the emerging field of genomic signal processing and its relationship to functional genomics. It focuses on some of the biological mechanisms driving the development of genomic signal processing, in addition to their manifestation in gene-expression-based classification and genetic network modeling. Certain problems are inherent. For instance, small-sample error estimation, variable selection, and model complexity are important issues for both phenotype classification and expression prediction used in network inference. A long-term goal is to develop intervention strategies to drive network behavior, which is briefly discussed. It is hoped that this nontechnical paper demonstrates that the field of signal processing has the potential to impact and help drive genomics research