Sampling-rate-dependent probabilistic Boolean networks

External control of a genetic regulatory network is used for the purpose of avoiding undesirable states, such as those associated with a disease. To date, intervention has mainly focused on the external control of probabilistic Boolean networks via the associated discrete-time discrete-space Markov processes. Implementation of an intervention policy derived for probabilistic Boolean networks requires nearly continuous observation of the underlying biological system since precise application requires the observation of all transitions. In medical applications, as in many engineering problems, the process is sampled at discrete time intervals and a decision to intervene or not must be made at each sample point. In this work, sampling-rate-dependent probabilistic Boolean network is proposed as an extension of probabilistic Boolean network. The proposed framework is capable of capturing the sampling rate of the underlying system.     

Intervention in context-sensitive probabilistic Boolean networks

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. RESULTS: This paper treats intervention via external control variables in context-sensitive PBNs by extending the results for instantaneously random PBNs in several directions. First, and most importantly, whereas an instantaneously random PBN yields a Markov chain whose state space is composed of gene vectors, each state of the Markov chain corresponding to a context-sensitive PBN is composed of a pair, the current gene vector occupied by the network and the current constituent Boolean network. Second, the analysis is applied to PBNs with perturbation, meaning that random gene perturbation is permitted at each instant with some probability. Third, the (mathematical) influence of genes within the network is used to choose the particular gene with which to intervene. Lastly, PBNs are designed from data using a recently proposed inference procedure that takes steady-state considerations into account. The results are applied to a context-sensitive PBN derived from gene-expression data collected in a study of metastatic melanoma, the intent being to devise a control strategy that reduces the WNT5A gene's action in affecting biological regulation, since the available data suggest that disruption of this influence could reduce the chance of a melanoma metastasizing.     

On the long-run sensitivity of probabilistic Boolean networks

Boolean networks and, more generally, probabilistic Boolean networks, as one class of gene regulatory networks, model biological processes with the network dynamics determined by the logic-rule regulatory functions in conjunction with probabilistic parameters involved in network transitions. While there has been significant research on applying different control policies to alter network dynamics as future gene therapeutic intervention, we have seen less work on understanding the sensitivity of network dynamics with respect to perturbations to networks, including regulatory rules and the involved parameters, which is particularly critical for the design of intervention strategies. This paper studies this less investigated issue of network sensitivity in the long run. As the underlying model of probabilistic Boolean networks is a finite Markov chain, we define the network sensitivity based on the steady-state distributions of probabilistic Boolean networks and call it long-run sensitivity. The steady-state distribution reflects the long-run behavior of the network and it can give insight into the dynamics or momentum existing in a system. The change of steady-state distribution caused by possible perturbations is the key measure for intervention. This newly defined long-run sensitivity can provide insight on both network inference and intervention. We show the results for probabilistic Boolean networks generated from random Boolean networks and the results from two real biological networks illustrate preliminary applications of sensitivity in intervention for practical problems.     

The complex fluctuations of probabilistic Boolean networks

Probabilistic Boolean networks (PBNs) are extensions of Boolean networks (BNs), and both have been widely used to model biological systems. In this paper, we study the long-range correlations of PBNs based on their corresponding Markov chains. PBN states are quantified by the deviation of their steady-state distributions. The results demonstrate that, compared with BNs, PBNs can exhibit these dynamics over a wider and higher noise range. In addition, the constituent BNs significantly impact the generation of 1/f dynamics of PBNs, and PBNs with homogeneous steady-state distributions tend to sustain the 1/f dynamics over a wider noise range.     

Gene perturbation and intervention in probabilistic Boolean networks

MOTIVATION: A major objective of gene regulatory network modeling, in addition to gaining a deeper understanding of genetic regulation and control, is the development of computational tools for the identification and discovery of potential targets for therapeutic intervention in diseases such as cancer. We consider the general question of the potential effect of individual genes on the global dynamical network behavior, both from the view of random gene perturbation as well as intervention in order to elicit desired network behavior. RESULTS: Using a recently introduced class of models, called Probabilistic Boolean Networks (PBNs), this paper develops a model for random gene perturbations and derives an explicit formula for the transition probabilities in the new PBN model. This result provides a building block for performing simulations and deriving other results concerning network dynamics. An example is provided to show how the gene perturbation model can be used to compute long-term influences of genes on other genes. Following this, the problem of intervention is addressed via the development of several computational tools based on first-passage times in Markov chains. The consequence is a methodology for finding the best gene with which to intervene in order to most likely achieve desirable network behavior. The ideas are illustrated with several examples in which the goal is to induce the network to transition into a desired state, or set of states. The corresponding issue of avoiding undesirable states is also addressed. Finally, the paper turns to the important problem of assessing the effect of gene perturbations on long-run network behavior. A bound on the steady-state probabilities is derived in terms of the perturbation probability. The result demonstrates that states of the network that are more 'easily reachable' from other states are more stable in the presence of gene perturbations. Consequently, these are hypothesized to correspond to cellular functional states. AVAILABILITY: A library of functions written in MATLAB for simulating PBNs, constructing state-transition matrices, computing steady-state distributions, computing influences, modeling random gene perturbations, and finding optimal intervention targets, as described in this paper, is available on request from is@ieee.org.     

Computation of steady-state probability distributions in stochastic models of cellular networks

Cellular processes are "noisy". In each cell, concentrations of molecules are subject to random fluctuations due to the small numbers of these molecules and to environmental perturbations. While noise varies with time, it is often measured at steady state, for example by flow cytometry. When interrogating aspects of a cellular network by such steady-state measurements of network components, a key need is to develop efficient methods to simulate and compute these distributions. We describe innovations in stochastic modeling coupled with approaches to this computational challenge: first, an approach to modeling intrinsic noise via solution of the chemical master equation, and second, a convolution technique to account for contributions of extrinsic noise. We show how these techniques can be combined in a streamlined procedure for evaluation of different sources of variability in a biochemical network. Evaluation and illustrations are given in analysis of two well-characterized synthetic gene circuits, as well as a signaling network underlying the mammalian cell cycle entry.     

An extension of the steady-state approximation of the kinetics of enzyme-containing systems

An extension of the steady-state approximation of multi-enzyme systems is obtained, which can also be applied when enzyme concentrations are of the same order of magnitude as substrate concentrations. This extension contains as a first order approximation the extension given by Vergonet and Berendsen. The mathematical procedure however is different and avoids difficulties inherent in their theory.     

Stochastic Boolean networks: An efficient approach to modeling gene regulatory networks

ABSTRACT: BACKGROUND: Various computational models have been of interest due to their use in the modelling of gene regulatory networks (GRNs). As a logical model, probabilistic Boolean networks (PBNs) consider molecular and genetic noise, so the study of PBNs provides significant insights into the understanding of the dynamics of GRNs. This will ultimately lead to advances in developing therapeutic methods that intervene in the process of disease development and progression. The applications of PBNs, however, are hindered by the complexities involved in the computation of the state transition matrix and the steady-state distribution of a PBN. For a PBN with n genes and N Boolean networks, the complexity to compute the state transition matrix is O(nN22n) or O(nN2n) for a sparse matrix. RESULTS: This paper presents a novel implementation of PBNs based on the notions of stochastic logic and stochastic computation. This stochastic implementation of a PBN is referred to as a stochastic Boolean network (SBN). An SBN provides an accurate and efficient simulation of a PBN without and with random gene perturbation. The state transition matrix is computed in an SBN with a complexity of O(nL2n), where L is a factor related to the stochastic sequence length. Since the minimum sequence length required for obtaining an evaluation accuracy approximately increases in a polynomial order with the number of genes, n, and the number of Boolean networks, N, usually increases exponentially with n, L is typically smaller than N, especially in a network with a large number of genes. Hence, the computational complexity of an SBN is primarily limited by the number of genes, but not directly by the total possible number of Boolean networks. Furthermore, a time-frame expanded SBN enables an efficient analysis of the steady-state distribution of a PBN. These findings are supported by the simulation results of a simplified p53 network, several randomly generated networks and a network inferred from a T cell immune response dataset. An SBN can also implement the function of an asynchronous PBN and is potentially useful in a hybrid approach in combination with a continuous or single-molecule level stochastic model. CONCLUSIONS: Stochastic Boolean networks (SBNs) are proposed as an efficient approach to modelling gene regulatory networks (GRNs). The SBN approach is able to recover biologically-proven regulatory behaviours, such as the oscillatory dynamics of the p53-Mdm2 network and the dynamic attractors in a T cell immune response network. The proposed approach can further predict the network dynamics when the genes are under perturbation, thus providing biologically meaningful insights for a better understanding of the dynamics of GRNs. The algorithms and methods described in this paper have been implemented in Matlab packages, which are attached as Additional files.     

概率布尔型基因调节网络的分解方法研究

基于Iyla Shumlevich等提出的概率布尔型网络(PBN)模型,计算网络中任意两基因之间量化了的相互影响,给出相应的有向带权图模型。通过对模型的标准化分析,找出关键节点,以各关键节点为中心,对网络划分,通过计算子网络间交互信息,确定各子网络边界,以达到对网络的最优分解。     

An automatic method for deriving steady-state rate equations.

A method is described for systematically deriving steady-state rate equations. It is based on the schematic method of King & Altman [J. Phys. Chem. (1956) 60, 1375-1378], but is expressed in purely algebraic terms. It is suitable for implementation as a computer program, and a program has been written in FORTRAN IV and deposited as Supplementary Publication SUP 50078 (12 pages) at the British Library (Lending Division), Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem. J. (1977) 161, 1-2.     

Boolean networks using the chi-square test for inferring large-scale gene regulatory networks

Background  

Boolean network (BN) modeling is a commonly used method for constructing gene regulatory networks from time series microarray data. However, its major drawback is that its computation time is very high or often impractical to construct large-scale gene networks. We propose a variable selection method that are not only reduces BN computation times significantly but also obtains optimal network constructions by using chi-square statistics for testing the independence in contingency tables.     

Guiding the self-organization of random Boolean networks

Random Boolean networks (RBNs) are models of genetic regulatory networks. It is useful to describe RBNs as self-organizing systems to study how changes in the nodes and connections affect the global network dynamics. This article reviews eight different methods for guiding the self-organization of RBNs. In particular, the article is focused on guiding RBNs toward the critical dynamical regime, which is near the phase transition between the ordered and dynamical phases. The properties and advantages of the critical regime for life, computation, adaptability, evolvability, and robustness are reviewed. The guidance methods of RBNs can be used for engineering systems with the features of the critical regime, as well as for studying how natural selection evolved living systems, which are also critical.     

An approximation method for the solution of diffusion and related problems

An approximation method is introduced which enables a number of diffusion-type problems to be solved in an approximate but simple manner. Many cases require only the solution of a simple first-order differential equation. The method is applied to a number of cases in which the exact solutions are available. A comparison shows that the method is quite satisfactory in these cases. The method is applied to diffusion problems with rate of consumption proportional to concentration or to the square of the concentration. In the latter case, the result obtained is essentially the same as that found by H. G. Landau (1950) after elaborate calculations.     

Kinetic model for bifunctional multisubstrate enzymes. steady-state approximation

The steady-state approximation of the generalized model of a bifunctional multisubstrate enzyme is considered. Cases when the reactions catalyzed by the bifunctional enzyme are independent of one another and when one reaction influences the other are considered. Shunting of the kinetic mechanisms and activation of one reaction by the second reaction catalyzed by the bifunctional enzyme are analyzed. The kinetic equations are derived and qualitatively analyzed. It is shown in all cases discussed that the functional relationships between the kinetic parameters observed and the concentrations of the substrates participating in the reactions can be unambiguously defined on the basis of the concept of relationship of the intermediate enzyme forms in the enzymatic process.     

A probabilistic approach to compact steady-state kinetic equations for enzymic reactions

We describe a method for deriving kinetic equations based on the simplification of a complex graphical scheme of steady-state enzymic reactions to one that is comprised of an unbranched pathway. It entails compressing unbranched multi-step sequences into one step, and fusing some graph nodes into a single node. The final form of the equations is compact and well structured, and it simplifies the choice of independent kinetic parameters. The approach is illustrated by an analysis of representative two- and three-substrate reactions.     

Boolean analysis of cell regulation networks

A comparison is made between the predictions of the Boolean and continuous analysis of a regulation model when the formation of two mediators interacting by cross-inhibition is stimulated by one or two specific signals. For such a system, the Boolean analysis reproduces the characteristics of behaviour previously predicted by continuous analysis (multiple stable states of opposite type, discontinuous transition, and associated hysteresis phenomenon). The qualitative agreement between the two methods allows a qualitative but rigorous treatment of regulation systems in which the Boolean analysis is applicable. From a general schematic representation of interaction in bidirectional control systems, we analyse by the Boolean method a large range of possible systems of increasing complexities which could theoretically apply. Previously unforeseen consequences of some systems are described. After that, we give a logical analysis of a well-known system (negative loop grafted with additional external controls) and discuss the application of such a system to explain certain oscillatory phenomena in the cell, showing the disrupting role of an additional control on the expected behaviour. Thus, when the analysis of a model including a negative loop does not indicate the possibility of experimentally suggested oscillations, we propose other simple logical structures which can predict this behaviour. Finally, we show a logical analysis of an opposite type of example of cell regulation where the biochemical observations can be accounted for simply by a negative loop grafted with one input variable.     

Properties of Boolean networks and methods for their tests

One of the major challenges in complex systems biology is that of providing a general theoretical framework to describe the phenomena involved in cell differentiation, i.e., the process whereby stem cells, which can develop into different types, become progressively more specialized. The aim of this study is to briefly review a dynamical model of cell differentiation which is able to cover a broad spectrum of experimentally observed phenomena and to present some novel results.     

Quasi-steady-state approximation for chemical reaction networks

 The parameter embedding leading to the quasi-steady-state approximation of Heinrich [9] is investigated within the theory of invariant manifolds of Fenichel [4] in order to clarify the essential assumptions needed for this reduction to a low dimensional system. In particular, the concept of pool-variables can be avoided in this generalized approach. Moreover, the dominating influence of the slow subnetwork over the complementary fast subnetwork is interpreted geometrically and in chemical terms and this can be seen as an “enslaving” of the fast subsystem by the slow subsystem. Finally, the results are applied to a system of slime mould communication [6, 7, 13] and to a maltose transport system [2, 3]. Received: 16 June 1997