Distributional observability of probabilistic Boolean networks

doi:10.1016/j.sysconle.2021.105001

Systems & Control Letters

Volume 156, October 2021, 105001

https://doi.org/10.1016/j.sysconle.2021.105001 Get rights and content

Abstract

A probabilistic Boolean network (PBN) is a discrete network composed of a family of Boolean networks together with a set of probabilities governing the selection of a Boolean network at each time step. We introduce in this paper a novel observability problem for PBNs. Specifically, we assume that the values of the initial state of the PBN are not known with certainty, but can be described by probability distributions. We ask ourselves under which conditions it is possible to uniquely determine the probability distribution of initial states when giving only knowledge of the evolution of the probability distributions of network outputs. We propose a complete answer to this problem using a linear algebra approach. Several examples, both artificial and real-world, are given and illustrate the viability of the proposed theoretical results.

Introduction

Mathematical modeling of biological systems is an attractive way for systematically studying the functions of biological processes and their relationship with the environment. One of the frequently used frameworks for modeling biological systems is that of Boolean models. After assembling individual components and interactions involved in a system, Boolean network (BN) models can properly capture the qualitative temporal behavior of the system without the need for much kinetic details [1]. Typically, a BN is presented in the form of a (deterministic) nonlinear system, while it was shown in [2] that such nonlinear dynamics can always be mapped invertably to a standard discrete-time linear dynamics using an algebraic state representation approach. The main idea behind is that a BN with $n$ variables has $2^{n}$ possible states, and if such states are represented by canonical vectors of length $2^{n}$ , then the one-step transitions between states can be equivalently specified by a $2^{n} \times 2^{n}$ matrix. This formal simplicity makes it comparatively easy to formulate and solve control theoretic problems for BNs, and thereby has stimulated a significant increase in the number of studies in this area [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. For some recent work on the analysis and control of BNs based on other approaches, see, e.g., [13], [14], [15].

A probabilistic Boolean network (PBN) is a stochastic extension of the classical (deterministic) BN. It may be viewed as a collection of BNs endowed with a probability structure determining the likelihood with which a constituent network is active. PBNs retain the appealing properties of BNs such as requiring few kinetic parameters, but are more robust in the face of uncertainty [16]. The algebraic state representation has also proved a nice framework for addressing control-related problems in PBNs [17], [18], [19], [20], [21], [22], [23].

Observability is one of the main topics in control theory. The classical definition of observability, roughly speaking, says that a system is observable if any two distinct initial states can be distinguished by observing the output of the given system [24]. Observability and related problems for classical BNs have been extensively studied by many authors; see, e.g., [25], [26], [27], [28], and more recently [29], [30], [31], [32], [33], [34], and references therein. Compared with the many studies focusing on BNs, the investigation of the observability of PBNs is currently at an early stage. The authors of [35] considered a type of observability for PBNs, which requires that for any two distinct initial states there is a nonzero probability that different output sequences are produced so that the two states become distinguishable. Such an observability notion was revisited in a recent study [36] and was referred to as finite-time observability in probability. Two slightly different notions of observability were also explored in that study. The first one is termed finite-time observability with probability one, and requires that for any two distinct initial states the corresponding two outputs on a finite time window are distinct with probability one. The second is termed asymptotical observability in distribution and it corresponds to the case where for any pair of distinct initial states, the probability that the outputs are distinct tends to one as the length of the observation window grows to infinity. The authors of [37] suggested an alternative characterization of the property of observability with probability one, and proposed set-theoretic algorithms for determining all possible initial (or final) states compatible with a given output sequence. A necessary and sufficient condition for a multiplex PBN (i.e., a set of PBNs linked through interconnected layers) to be finite-time observable with probability one was given in [38].

We consider in this paper a novel observability problem for PBNs. The problem is to distinguish between the probability distributions of initial states in a PBN from knowledge of the probability distributions of outputs over time. In many existing research works on PBNs, both theoretical and practical, the initial state of the PBN is assumed to be randomly distributed according to a certain probability distribution [39], [40], [41], [42]. This is not only because such an assumption is in better agreement with the stochastic nature of the model, but also due to the fact that there are many real situations in which the initial state of the model cannot be known exactly. We therefore consider initial probability distributions rather than deterministic initial states for starting a PBN (see Remark 1 below for more comments on the motivation). The fact that the initial state is a random variable and that the updating rule of the network is randomly chosen at each time step leads to the fact that the output of a PBN at any given time is also a random variable. Since every stochastic process induces a family of finite dimensional distributions that determines the stochastic process uniquely (see, e.g., [43]), it is then quite reasonable to think that two initial state distributions of a PBN are indistinguishable statistically if the corresponding output processes have the same finite dimensional distributions. Observability based on such indistinguishability is then a direct extension of the classical definition when dealing with PBNs, and is to some extent more stochastic in nature compared to the aforementioned three existing observability notions. Clearly, this type of distributional observability is stronger than the property of finite-time observability in probability, but neither implies nor is implied by the other two observability notions. The main question addressed in this paper is under which conditions a PBN is initial-distribution observable. We provide a complete answer to this question. As in previous studies [35], [36], [37], [38], the theoretical framework developed in this paper is based on the powerful algebraic representation approach.

The remainder of this paper is organized as follows. Section 2 contains the basic notation and briefly reviews PBNs. The main contributions, establishing conditions for distributional observability, are presented in Section 3. Several examples, both contrived and biological, are given in Section 4. A summary of the paper is provided in the final section.

Section snippets

Preliminaries

Let $δ_{k}^{i}$ denote the $i$ th $k \times 1$ canonical basis vector, and $Δ_{k}$ the set consisting of the canonical vectors $δ_{k}^{1}, \dots, δ_{k}^{k}$ . The set of all $k \times r$ matrices whose columns are canonical vectors is denoted by $L^{k \times r}$ . Let $A$ and $B$ be $k_{1} \times r_{1}$ and $k_{2} \times r_{2}$ matrices, respectively, and let $l$ be the least common multiple of $r_{1}$ and $k_{2}$ . The (left) semitensor product [44] of $A$ and $B$ is a $(k_{1} l / r_{1}) \times (r_{2} l / k_{2})$ matrix, denoted by $A ⋉ B$ , and defined by $A ⋉ B = (A \otimes I_{l / r_{1}}) (B \otimes I_{l / k_{2}})$ , where $\otimes$ is the Kronecker product of matrices and $I_{l / r_{1}}$ and $I_{l / k}$

Main results

Let us consider a PBN with output, described by the algebraic representation² $x (t + 1) = F_{θ (t)} ⋉ u (t) ⋉ x (t), y (t) = H x (t),$ $x \in Δ_{N}, u \in Δ_{M}, y \in Δ_{Q} .$ As assumed above, ${θ (t)}$ is a Bernoulli process with outcome space ${1, \dots, S}$ and probability $p_{i}$ for the occurrence of outcome $i$ , $F_{i} \in L^{N \times N M}$ for each $1 \leq i \leq S$ , and $H \in L^{Q \times N}$ . We denote by $y |_{μ, u}$ the output stochastic process generated by (6) corresponding to the initial state distribution

Examples

This section presents some examples that illustrate the use of our results.

Example 1

Consider a PBN as in (6), with $N = 4$ , $M = Q = 2$ , $F_{1} = [δ_{4}^{1} δ_{4}^{4} δ_{4}^{4} δ_{4}^{3} δ_{4}^{3} δ_{4}^{4} δ_{4}^{4} δ_{4}^{3}]$ , $F_{2} = [δ_{4}^{1} δ_{4}^{2} δ_{4}^{2} δ_{4}^{4} δ_{4}^{3} δ_{4}^{4} δ_{4}^{4} δ_{4}^{4}]$ , $H = [δ_{2}^{1} δ_{2}^{2} δ_{2}^{1} δ_{2}^{2}]$ , and the selection probabilities given by $p_{1} = 0.6$ and $p_{2} = 0.4$ . The matrices $P_{u}$ ’s and $J_{z}$ ’s are $P_{δ_{2}^{1}} = [\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0.4 & 0.4 & 0 \\ 0 & 0 & 0 & 0.6 \\ 0 & 0.6 & 0.6 & 0.4 \end{bmatrix}], P_{δ_{2}^{2}} = [\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0.6 \\ 0 & 1 & 1 & 0.4 \end{bmatrix}],$ $J_{δ_{2}^{1}} = diag (1, 0, 1, 0), J_{δ_{2}^{2}} = diag (0, 1, 0, 1),$ and the subspace $V_{0}$ is $V_{0} = im H^{⊤} = span \{{[1, 0, 1, 0]}^{⊤}, {[0, 1, 0, 1]}^{⊤}\}$ . Direct calculation shows that $P_{δ_{2}^{1}}^{⊤} V_{0} = P_{δ_{2}^{2}}^{⊤} V_{0} = span {[1,$

Summary

We have addressed in this paper the problem about observability of a PBN. We assume that the initial states of the PBN are not explicitly known but are instead described by probability distributions. The problem is about whether one is able to distinguish between different initial state distributions of the network using only the probability distributions of outputs over time. We have provided a complete solution to this problem, by giving a necessary and sufficient condition for a PBN to be

CRediT authorship contribution statement

Rui Li: Conceptualization, Methodology, Writing – original draft, Writing – review & editing. Qi Zhang: Conceptualization, Writing – original draft, Writing – review & editing. Jianlei Zhang: Methodology, Writing – original draft, Writing – review & editing. Tianguang Chu: Conceptualization, Methodology, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors gratefully acknowledge constructive and insightful feedback received from the anonymous reviewers and the Associate Editor.

This research was supported by the National Natural Science Foundation of China under Grants 61503375, 61673027, 62073174, and 62073175, the Humanity and Social Science Youth Foundation of Ministry of Education of China under Grant 20YJCZH228, and the Excellent Young Scholars Funding Project in UIBE under Grant 19YQ10.

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