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Recent Development and Biomedical Applications of Probabilistic Trairatphisan et al. Cell Communication and Signaling 2013, 11:46 http://www.biosignaling.com/content/11/1/46 REVIEW Open Access Recent development and biomedical applications of probabilistic Boolean networks Panuwat Trairatphisan1*, Andrzej Mizera2,JunPang2, Alexandru Adrian Tantar2,4, Jochen Schneider3,5 andThomasSauter1 Abstract Probabilistic Boolean network (PBN) modelling is a semi-quantitative approach widely used for the study of the topology and dynamic aspects of biological systems. The combined use of rule-based representation and probability makes PBN appealing for large-scale modelling of biological networks where degrees of uncertainty need to be considered. A considerable expansion of our knowledge in the field of theoretical research on PBN can be observed over the past few years, with a focus on network inference, network intervention and control. With respect to areas of applications, PBN is mainly used for the study of gene regulatory networks though with an increasing emergence in signal transduction, metabolic, and also physiological networks. At the same time, a number of computational tools, facilitating the modelling and analysis of PBNs, are continuously developed. A concise yet comprehensive review of the state-of-the-art on PBN modelling is offered in this article, including a comparative discussion on PBN versus similar models with respect to concepts and biomedical applications. Due to their many advantages, we consider PBN to stand as a suitable modelling framework for the description and analysis of complex biological systems, ranging from molecular to physiological levels. Keywords: Probabilistic Boolean networks, Probabilistic graphical models, Qualitative modelling, Systems biology Background fail nevertheless to offer a quantitative determination of A large number of formal representation types that exist the system’s dynamics due to their inherent qualitative in Systems Biology are used to construct distinctive math- nature. ematical models, each with their own strengths and Probabilistic Boolean networks (PBNs) were introduced weaknesses. On one hand, deciphering the complexity in 2002 by Shmulevich et al. as an extension of the of biological systems by quantitative methods, such as Boolean Network concept and as an alternative for mod- ordinary differential equation (ODE) based mathemat- elling gene regulatory networks [3]. PBNs combine the ical models, yields detailed representations with high rule-based modelling of a BN, as introduced by Kauff- predictive power. Such an approach is however often man [4-7], with uncertainty principles, e.g., as described hampered by the low availability and/or identifiability by a Markov chain [8]. In terms of applications, anal- of kinetic parameters and experimental data [1]. These ogously to the case of traditional BNs, the qualitative limitations often result in the generation of relatively nature of state and time in a PBN framework allows small quantitative network models. On the other hand, for modelling of large-scale networks. The integrated qualitative modelling frameworks such as the Boolean stochastic properties of PBNs additionally enable semi- Networks (BNs), allow for describing large biological net- quantitative properties to be extracted. Existing analytic works while still preserving important properties of the methods on PBNs allow for gaining a better under- systems [2]. The models pertaining to this latter class standing of how biological systems behave, and offer in addition the means to compare to traditional BNs. *Correspondence: [email protected] Examples are the calculation of influences which rep- 1 Life Sciences Research Unit,University of Luxembourg, Luxembourg resent the quantitative strength of interaction between Full list of author information is available at the end of the article certain genes [3], or the determination of steady-state © 2013 Trairatphisan et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Trairatphisan et al. Cell Communication and Signaling 2013, 11:46 Page 2 of 25 http://www.biosignaling.com/content/11/1/46 distributions to quantitatively predict the activity of cer- trol of PBNs’, ending with the relationship between PBNs tain genes in steady state [8]. and other probabilistic graphical models in Section ‘Rela- It has been shown in the past years that the use of tionship between PBNs and other probabilistic graphical PBNs in the biological field is not limited to the molecu- models’. Later, in Section ‘PBN applications in biological lar level, but also can potentially be linked to applications and biomedical studies’ we present a broad summary of in clinic. To name a few, Tay et al. constructed a PBN PBN applications as a representation of biological net- to demonstrate the interplay between dengue virus and works followed by a discussion on the future applications different cytokines which mediate the course of disease of PBN in Systems Biology and Systems Biomedicine. A in dengue haemorrhagic fever (DHF) [9]. Ma et al. pro- short conclusion is given in Section ‘Conclusion’. cessed functional Magnetic Resonance Imaging (fMRI) signals to infer a brain connectivity network comparing Introduction to probabilistic Boolean networks between Parkinson’s disease patients and healthy subjects and their dynamics [10]. Even though the research efforts on PBNs in this Boolean networks direction are just sprouting, the results from such PBN A Boolean Network (BN) G(V, F), as originally introduced studies can provide a first clue on a disease’s etiology and by Kauffman [4-7], is defined as a set of binary-valued progression. As PBNs are highly flexible for data integra- variables (nodes) V ={x1, x2, ..., xn} and a vector of tion and as there exist a number of computational tools Boolean functions f = (f1, ..., fn).Ateachupdating for PBN analysis, PBN is a suitable modelling approach epoch, referred to as time point t (t = 0, 1, 2, ...), the to integrate information and derive knowledge from omic state of the network is defined by the vector x(t) = scale data which should in turn facilitate a physician’s (x1(t), x2(t), ..., xn(t)),wherexi(t) is the value of variable decision-making process in clinic. xi at time t, i.e., xi(t) ∈{0, 1} (i = 1, 2, ..., n). For each { } For the past decade, PBNs were the object of extensive variable xi there exists a predictor set xi1 , xi2 , ..., xik(i) studies, both theoretical and applied. Among theoretical and a Boolean predictor function (or simply predictor) fi topics, there are steady-state distribution, e.g., [11-13], being the i-th element of f that determines the value of xi network construction and inference, e.g., [14-16], net- atthenexttimepoint,i.e., work intervention and control, e.g., [17-19]. Several minor topics were investigated as well, including reachability + = analysis [20] or sensitivity analysis [21]. Other studies xi(t 1) fi(xi1 (t), xi2 (t), ..., xik(i) (t)),(1) dealt with PBNs in biological systems at multi-level such as gene regulatory networks [22-24], signal transduction networks [25], metabolic networks [26], and also physi- where 1 ≤ i1 < i2 < ··· < ik(i) ≤ n.Since ological networks [9,10] which could potentially link to the predictor functions of f are time-homogenous, the medicine as previously mentioned. In parallel, a number notation can be simplified by writing fi(xi1 , xi2 , ..., xik(i) ). of computational tools which facilitate the modelling and Without loss of generality, k(i) can be defined to be analysis of PBNs are also continuously developed [27-29]. a constant equal to n for all i by introducing ficti- Given the continuous development in this area due to tious variables in each function: the variable xi is ficti- the broad on-going range of research on PBNs, we offer tious for a function f if f (x1, ..., xi−1,0,xi+1, ..., xn) = a state-of-the-art overview on this modelling framework. f (x1, ..., xi−1,1,xi+1, ..., xn) for all possible values of A comparison of PBN to other graphical probabilistic x1, ..., xi−1, xi+1, ..., xn. A variable that is not fictitious is modelling approaches is also enclosed, specifically with referred to as essential.Thek(i) elements of the predictor { } respect to Bayesian networks. Last but not least, a view set xi1 , xi2 , ..., xik(i) are referred to as the essential pre- of the theoretical and applied research on PBNs as mod- dictors of variable xi.Thevectorf of predictor functions els for the study of multi-level biomedical networks is constitutes the network transition function (or simply the included. network function). The network function f determines the In order to provide a coherent overview of the recent time evolution of the states of the Boolean network, i.e., advances on PBN, we start with several theoretical x(t + 1) = f (x(t)). Thus, the BN’s dynamics is determin- aspects, organised as follows: an introduction to PBNs and istic. The only potential uncertainty is in the selection of associated dynamics are given in Section ‘Introduction to the initial starting state of the network. probabilistic Boolean networks and their dynamics’, the Given an initial state, within a finite number of steps, construction and inference of PBNs as models for gene the BN will transition into a fixed state or a set of states regulatory networks are presented in Section ‘Construc- through which it will repeatedly cycle forever. In the first tion and inference of PBNs as models of gene regulatory case, each such fixed state is called a singleton attractor, networks’,structural intervention and external control are whereas in the second case, the set of states is referred to discussed in Section ‘Structural intervention and con- as a cyclic attractor.Anattractor is either a singleton or Trairatphisan et al. Cell Communication and Signaling 2013, 11:46 Page 3 of 25 http://www.biosignaling.com/content/11/1/46 a cyclic attractor.
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