Identification of control targets in Boolean molecular network models via computational algebra

Background

Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered is that of Boolean networks. The potential control targets can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system.

Results

This paper presents a method for the identification of potential intervention targets in Boolean molecular network models using algebraic techniques. The approach exploits an algebraic representation of Boolean networks to encode the control candidates in the network wiring diagram as the solutions of a system of polynomials equations, and then uses computational algebra techniques to find such controllers. The control methods in this paper are validated through the identification of combinatorial interventions in the signaling pathways of previously reported control targets in two well studied systems, a p53-mdm2 network and a blood T cell lymphocyte granular leukemia survival signaling network. Supplementary data is available online and our code in Macaulay2 and Matlab are available via http://www.ms.uky.edu/~dmu228/ControlAlg.

Conclusions

This paper presents a novel method for the identification of intervention targets in Boolean network models. The results in this paper show that the proposed methods are useful and efficient for moderately large networks.

     

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.     

Intervention in a family of Boolean networks

MOTIVATION: Intervention in a gene regulatory network is used to 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 a collection of Boolean networks in which the gene state vector transitions according to the rules of one of the constituent networks and where network choice is governed by a selection distribution. The theory of automatic control has been applied to find optimal strategies for manipulating external control variables that affect the transition probabilities to desirably affect dynamic evolution over a finite time horizon. In this paper we treat a case in which we lack the governing probability structure for Boolean network selection, so we simply have a family of Boolean networks, but where these networks possess a common attractor structure. This corresponds to the situation in which network construction is treated as an ill-posed inverse problem in which there are many Boolean networks created from the data under the constraint that they all possess attractor structures matching the data states, which are assumed to arise from sampling the steady state of the real biological network. RESULTS: Given a family of Boolean networks possessing a common attractor structure composed of singleton attractors, a control algorithm is derived by minimizing a composite finite-horizon cost function that is a weighted average over all the individual networks, the idea being that we desire a control policy that on average suits the networks because these are viewed as equivalent relative to the data. The weighting for each network at any time point is taken to be proportional to the instantaneous estimated probability of that network being the underlying network governing the state transition. The results are applied to a family of Boolean networks 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. AVAILABILITY: The software is available on request. SUPPLEMENTARY INFORMATION: The supplementary Information is available at http://ee.tamu.edu/~edward/tree     

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.     

Inference of Boolean Networks Using Sensitivity Regularization

The inference of genetic regulatory networks from global measurements of gene expressions is an important problem in computational biology. Recent studies suggest that such dynamical molecular systems are poised at a critical phase transition between an ordered and a disordered phase, affording the ability to balance stability and adaptability while coordinating complex macroscopic behavior. We investigate whether incorporating this dynamical system-wide property as an assumption in the inference process is beneficial in terms of reducing the inference error of the designed network. Using Boolean networks, for which there are well-defined notions of ordered, critical, and chaotic dynamical regimes as well as well-studied inference procedures, we analyze the expected inference error relative to deviations in the networks'' dynamical regimes from the assumption of criticality. We demonstrate that taking criticality into account via a penalty term in the inference procedure improves the accuracy of prediction both in terms of state transitions and network wiring, particularly for small sample sizes.     

Uncovering operational interactions in genetic networks using asynchronous Boolean dynamics

Biological networks of large dimensions, with their diagram of interactions, are often well represented by a Boolean model with a family of logical rules. The state space of a Boolean model is finite, and its asynchronous dynamics are fully described by a transition graph in the state space. In this context, a model reduction method will be developed for identifying the active or operational interactions responsible for a given dynamic behaviour. The first step in this procedure is the decomposition of the asynchronous transition graph into its strongly connected components, to obtain a “reduced” and hierarchically organized graph of transitions. The second step consists of the identification of a partial graph of interactions and a sub-family of logical rules that remain operational in a given region of the state space. This model reduction method and its usefulness are illustrated by an application to a model of programmed cell death. The method identifies two mechanisms used by the cell to respond to death-receptor stimulation and decide between the survival and apoptotic pathways.     

Logical and symbolic analysis of robust biological dynamics

Logical models provide insight about key control elements of biological networks. Based solely on the logical structure, we can determine state transition diagrams that give the allowed possible transitions in a coarse grained phase space. Attracting pathways and stable nodes in the state transition diagram correspond to robust attractors that would be found in several different types of dynamical systems that have the same logical structure. Attracting nodes in the state transition diagram correspond to stable steady states. Furthermore, the sequence of logical states appearing in biological networks with robust attracting pathways would be expected to appear also in Boolean networks, asynchronous switching networks, and differential equations having the same underlying structure. This provides a basis for investigating naturally occurring and synthetic systems, both to predict the dynamics if the structure is known, and to determine the structure if the transitions are known.     

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 impact of function perturbations in Boolean networks

MOTIVATION: A network is said to be robust relative to a certain network characteristic if a small change in network structure does not significantly affect the characteristic. From the perspective of network stability, robustness is desirable; however, from the perspective of intervention to exert influence on network behavior, it is undesirable. For Boolean networks, there are two fundamental types of robustness. One type pertains to perturbing the state of the network and the other to perturbing the rule-based structure. RESULTS: This article explores the impact of function perturbations in Boolean networks from two aspects: (1) analysis: predict the impact on network state transitions and attractors via analytical approaches or identify a perturbation by observing its consequences; (2) synthesis: preserve or modify the network characteristics, especially attractors, by introducing a judicious change to the functions. The results are applied to achieve intervention that structurally alters the network to achieve a more favorable steady-state distribution and to identify the function perturbation that has led to altered observed behavior. The intervention procedure is applied to a WNT5A network to reduce the risk of metastasis in melanoma, and the identification procedure is applied to a Drosophila melanogaster segmentation polarity gene network to identify regulatory function perturbation.     

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.     

Intervention in gene regulatory networks via greedy control policies based on long-run behavior

Background  

A salient purpose for studying gene regulatory networks is to derive intervention strategies, the goals being to identify potential drug targets and design gene-based therapeutic intervention. Optimal stochastic control based on the transition probability matrix of the underlying Markov chain has been studied extensively for probabilistic Boolean networks. Optimization is based on minimization of a cost function and a key goal of control is to reduce the steady-state probability mass of undesirable network states. Owing to computational complexity, it is difficult to apply optimal control for large networks.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

Template-based intervention in Boolean network models of biological systems

Motivation

A grand challenge in the modeling of biological systems is the identification of key variables which can act as targets for intervention. Boolean networks are among the simplest of models, yet they have been shown to adequately model many of the complex dynamics of biological systems. In our recent work, we utilized a logic minimization approach to identify quality single variable targets for intervention from the state space of a Boolean network. However, as the number of variables in a network increases, the more likely it is that a successful intervention strategy will require multiple variables. Thus, for larger networks, such an approach is required in order to identify more complex intervention strategies while working within the limited view of the network’s state space. Specifically, we address three primary challenges for the large network arena: the first challenge is how to consider many subsets of variables, the second is to design clear methods and measures to identify the best targets for intervention in a systematic way, and the third is to work with an intractable state space through sampling.

Results

We introduce a multiple variable intervention target called a template and show through simulation studies of random networks that these templates are able to identify top intervention targets in increasingly large Boolean networks. We first show that, when other methods show drastic loss in performance, template methods show no significant performance loss between fully explored and partially sampled Boolean state spaces. We also show that, when other methods show a complete inability to produce viable intervention targets in sampled Boolean state spaces, template methods maintain significantly consistent success rates even as state space sizes increase exponentially with larger networks. Finally, we show the utility of the template approach on a real-world Boolean network modeling T-LGL leukemia.

Conclusions

Overall, these results demonstrate how template-based approaches now effectively take over for our previous single variable approaches and produce quality intervention targets in larger networks requiring sampled state spaces.

     

Algorithms for identifying Boolean networks and related biological networks based on matrix multiplication and fingerprint function.

Due to the recent progress of the DNA microarray technology, a large number of gene expression profile data are being produced. How to analyze gene expression data is an important topic in computational molecular biology. Several studies have been done using the Boolean network as a model of a genetic network. This paper proposes efficient algorithms for identifying Boolean networks of bounded indegree and related biological networks, where identification of a Boolean network can be formalized as a problem of identifying many Boolean functions simultaneously. For the identification of a Boolean network, an O(mnD+1) time naive algorithm and a simple O (mnD) time algorithm are known, where n denotes the number of nodes, m denotes the number of examples, and D denotes the maximum in degree. This paper presents an improved O(momega-2nD + mnD+omega-3) time Monte-Carlo type randomized algorithm, where omega is the exponent of matrix multiplication (currently, omega < 2.376). The algorithm is obtained by combining fast matrix multiplication with the randomized fingerprint function for string matching. Although the algorithm and its analysis are simple, the result is nontrivial and the technique can be applied to several related problems.     

Identification of bifurcation transitions in biological regulatory networks using Answer-Set Programming

Background

Numerous cellular differentiation processes can be captured using discrete qualitative models of biological regulatory networks. These models describe the temporal evolution of the state of the network subject to different competing transitions, potentially leading the system to different attractors. This paper focusses on the formal identification of states and transitions that are crucial for preserving or pre-empting the reachability of a given behaviour.

Methods

In the context of non-deterministic automata networks, we propose a static identification of so-called bifurcations, i.e., transitions after which a given goal is no longer reachable. Such transitions are naturally good candidates for controlling the occurrence of the goal, notably by modulating their propensity. Our method combines Answer-Set Programming with static analysis of reachability properties to provide an under-approximation of all the existing bifurcations.

Results

We illustrate our discrete bifurcation analysis on several models of biological systems, for which we identify transitions which impact the reachability of given long-term behaviour. In particular, we apply our implementation on a regulatory network among hundreds of biological species, supporting the scalability of our approach.

Conclusions

Our method allows a formal and scalable identification of transitions which are responsible for the lost of capability to reach a given state. It can be applied to any asynchronous automata networks, which encompass Boolean and multi-valued models. An implementation is provided as part of the Pint software, available at http://loicpauleve.name/pint.

     

A framework to find the logic backbone of a biological network

Background

Cellular behaviors are governed by interaction networks among biomolecules, for example gene regulatory and signal transduction networks. An often used dynamic modeling framework for these networks, Boolean modeling, can obtain their attractors (which correspond to cell types and behaviors) and their trajectories from an initial state (e.g. a resting state) to the attractors, for example in response to an external signal. The existing methods however do not elucidate the causal relationships between distant nodes in the network.

Results

In this work, we propose a simple logic framework, based on categorizing causal relationships as sufficient or necessary, as a complement to Boolean networks. We identify and explore the properties of complex subnetworks that are distillable into a single logic relationship. We also identify cyclic subnetworks that ensure the stabilization of the state of participating nodes regardless of the rest of the network. We identify the logic backbone of biomolecular networks, consisting of external signals, self-sustaining cyclic subnetworks (stable motifs), and output nodes. Furthermore, we use the logic framework to identify crucial nodes whose override can drive the system from one steady state to another. We apply these techniques to two biological networks: the epithelial-to-mesenchymal transition network corresponding to a developmental process exploited in tumor invasion, and the network of abscisic acid induced stomatal closure in plants. We find interesting subnetworks with logical implications in these networks. Using these subgraphs and motifs, we efficiently reduce both networks to succinct backbone structures.

Conclusions

The logic representation identifies the causal relationships between distant nodes and subnetworks. This knowledge can form the basis of network control or used in the reverse engineering of networks.

     

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 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 computing architectures.     

生物分子网络弹性研究进展

弹性是生物分子网络重要且基础的属性之一,一方面弹性赋予生物分子网络抵抗内部噪声与环境干扰并维持其自身基本功能的能力,另一方面,弹性为网络状态的恢复制造了阻力。生物分子网络弹性研究试图回答如下3个问题：a. 生物分子网络弹性的产生机理是什么？b. 弹性影响下生物分子网络的状态如何发生转移？c. 如何预测生物网络状态转换临界点,以防止系统向不理想的状态演化？因此,研究生物分子网络弹性有助于理解生物系统内部运作机理,同时对诸如疾病发生临界点预测、生物系统状态逆转等临床应用具有重要的指导意义。鉴于此,本文主要针对以上生物分子网络弹性领域的3个热点研究问题,在研究方法和生物学应用上进行了系统地综述,并对未来生物分子网络弹性的研究方向进行了展望。