Reverse Engineering Boolean Networks: From Bernoulli Mixture Models to Rule Based Systems

A Boolean network is a graphical model for representing and analyzing the behavior of gene regulatory networks (GRN). In this context, the accurate and efficient reconstruction of a Boolean network is essential for understanding the gene regulation mechanism and the complex relations that exist therein. In this paper we introduce an elegant and efficient algorithm for the reverse engineering of Boolean networks from a time series of multivariate binary data corresponding to gene expression data. We call our method ReBMM, i.e., reverse engineering based on Bernoulli mixture models. The time complexity of most of the existing reverse engineering techniques is quite high and depends upon the indegree of a node in the network. Due to the high complexity of these methods, they can only be applied to sparsely connected networks of small sizes. ReBMM has a time complexity factor, which is independent of the indegree of a node and is quadratic in the number of nodes in the network, a big improvement over other techniques and yet there is little or no compromise in accuracy. We have tested ReBMM on a number of artificial datasets along with simulated data derived from a plant signaling network. We also used this method to reconstruct a network from real experimental observations of microarray data of the yeast cell cycle. Our method provides a natural framework for generating rules from a probabilistic model. It is simple, intuitive and illustrates excellent empirical results.     

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.

     

Mean-field Boolean network model of a signal transduction network

In this paper we provide a mean-field Boolean network model for a signal transduction network of a generic fibroblast cell. The network consists of several main signaling pathways, including the receptor tyrosine kinase, the G-protein coupled receptor, and the Integrin signaling pathway. The network consists of 130 nodes, each representing a signaling molecule (mainly proteins). Nodes are governed by Boolean dynamics including canalizing functions as well as totalistic Boolean functions that depend only on the overall fraction of active nodes. We categorize the Boolean functions into several different classes. Using a mean-field approach we generate a mathematical formula for the probability of a node becoming active at any time step. The model is shown to be a good match for the actual network. This is done by iterating both the actual network and the model and comparing the results numerically. Using the Boolean model it is shown that the system is stable under a variety of parameter combinations. It is also shown that this model is suitable for assessing the dynamics of the network under protein mutations. Analytical results support the numerical observations that in the long-run at most half of the nodes of the network are active.     

Kolmogorov complexity of epithelial pattern formation: The role of regulatory network configuration

The tissues of multicellular organisms are made of differentiated cells arranged in organized patterns. This organization emerges during development from the coupling of dynamic intra- and intercellular regulatory networks. This work applies the methods of information theory to understand how regulatory network structure both within and between cells relates to the complexity of spatial patterns that emerge as a consequence of network operation. A computational study was performed in which undifferentiated cells were arranged in a two dimensional lattice, with gene expression in each cell regulated by identical intracellular randomly generated Boolean networks. Cell–cell contact signalling between embryonic cells is modeled as coupling among intracellular networks so that gene expression in one cell can influence the expression of genes in adjacent cells. In this system, the initially identical cells differentiate and form patterns of different cell types. The complexity of network structure, temporal dynamics and spatial organization is quantified through the Kolmogorov-based measures of normalized compression distance and set complexity. Results over sets of random networks that operate in the ordered, critical and chaotic domains demonstrate that: (1) ordered and critical networks tend to create the most information-rich patterns; (2) signalling configurations in which cell-to-cell communication is non-directional mostly produce simple patterns irrespective of the internal network domain; and (3) directional signalling configurations, similar to those that function in planar cell polarity, produce the most complex patterns, but only when the intracellular networks function in non-chaotic domains.     

Reprogramming cell fates: reconciling rarity with robustness

The stunning possibility of “reprogramming” differentiated somatic cells to express a pluripotent stem cell phenotype (iPS, induced pluripotent stem cell) and the “ground state” character of pluripotency reveal fundamental features of cell fate regulation that lie beyond existing paradigms. The rarity of reprogramming events appears to contradict the robustness with which the unfathomably complex phenotype of stem cells can reliably be generated. This apparent paradox, however, is naturally explained by the rugged “epigenetic landscape” with valleys representing “preprogrammed” attractor states that emerge from the dynamical constraints of the gene regulatory network. This article provides a pedagogical primer to the fundamental principles of gene regulatory networks as integrated dynamic systems and reviews recent insights in gene expression noise and fate determination, thereby offering a formal framework that may help us to understand why cell fate reprogramming events are inherently rare and yet so robust.     

Membrane Potential Controls Adipogenic and Osteogenic Differentiation of Mesenchymal Stem Cells

Background

Control of stem cell behavior is a crucial aspect of developmental biology and regenerative medicine. While the functional role of electrophysiology in stem cell biology is poorly understood, it has become clear that endogenous ion flows represent a powerful set of signals by means of which cell proliferation, differentiation, and migration can be controlled in regeneration and embryonic morphogenesis.

Methodology/Principal Findings

We examined the membrane potential (V_mem) changes exhibited by human mesenchymal stem cells (hMSCs) undergoing adipogenic (AD) and osteogenic (OS) differentiation, and uncovered a characteristic hyperpolarization of differentiated cells versus undifferentiated cells. Reversal of the progressive polarization via pharmacological modulation of transmembrane potential revealed that depolarization of hMSCs prevents differentiation. In contrast, treatment with hyperpolarizing reagents upregulated osteogenic markers.

Conclusions/Significance

Taken together, these data suggest that the endogenous hyperpolarization is a functional determinant of hMSC differentiation and is a tractable control point for modulating stem cell function.     

Common origins of blood and blood vessels in adults?

After embryonic development, the vast majority of cells are differentiated and all organs are in place. Growth of the organism then ensues and continues until adulthood, whereupon cell division largely ceases. In some tissues, notably the bone marrow, skin, and gut, cell proliferation continues throughout life to replace cells lost by attrition. This regeneration is fueled by rare, long-lived, and largely quiescent stem cells that give rise to committed progenitors, which in turn generate large numbers of fully differentiated cells. Mounting evidence suggests that such cells can significantly contribute to tissue repair and regeneration in adults and may therefore prove beneficial for autologous cell and gene therapies. This review focuses on the potential of adult stem cells to give rise to hematopoietic and vascular cells. We discuss evidence that a highly purified population of adult stem cells, termed SP cells, serves as a hematopoietic progenitor and can contribute to vascular regeneration after injury. We also discuss the potential relationship of these cells to the embryonic hemangioblast.     

Molecular network control through boolean canalization

Boolean networks are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Recently, dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming. This work studies the role of canalization in the control of Boolean molecular networks. It provides a method for identifying the potential edges to control in the wiring diagram of a network for avoiding undesirable state transitions. The method is based on identifying appropriate input-output combinations on undesirable transitions that can be modified using the edges in the wiring diagram of the network. Moreover, a method for estimating the number of changed transitions in the state space of the system as a result of an edge deletion in the wiring diagram is presented. The control methods of this paper were applied to a mutated cell-cycle model and to a p53-mdm2 model to identify potential control targets.     

The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model

Methods for modeling cellular regulatory networks as diverse as differential equations and Boolean networks co-exist, however, without much closer correspondence to each other. With the example system of the fission yeast cell cycle control network, we here discuss these two approaches with respect to each other. We find that a Boolean network model can be formulated as a specific coarse-grained limit of the more detailed differential equations model for this system. This demonstrates the mathematical foundation on which Boolean networks can be applied to biological regulatory networks in a controlled way.     

Boolink: a graphical interface for open access Boolean network simulations and use in guard cell CO2 signaling

Signaling networks are at the heart of almost all biological processes. Most of these networks contain large number of components, and often either the connections between these components are not known or the rate equations that govern the dynamics of soluble signaling components are not quantified. This uncertainty in network topology and parameters can make it challenging to formulate detailed mathematical models. Boolean networks, in which all components are either on or off, have emerged as viable alternatives to detailed mathematical models that contain rate constants and other parameters. Therefore, open-source platforms of Boolean models for community use are desirable. Here, we present Boolink, a freely available graphical user interface that allows users to easily construct and analyze existing Boolean networks. Boolink can be applied to any Boolean network. We demonstrate its application using a previously published network for abscisic acid (ABA)-driven stomatal closure in Arabidopsis spp. (Arabidopsis thaliana). We also show how Boolink can be used to generate testable predictions by extending the network to include CO₂ regulation of stomatal movements. Predictions of the model were experimentally tested, and the model was iteratively modified based on experiments showing that ABA effectively closes Arabidopsis stomata at near-zero CO₂ concentrations (1.5-ppm CO₂). Thus, Boolink enables public generation and the use of existing Boolean models, including the prior developed ABA signaling model with added CO₂ signaling components.

An open-source, graphical interface for the simulation of Boolean networks is presented, applied to an abscisic acid signaling network in guard cells, and extended to include input from CO₂.     

smedinx-11 is a planarian stem cell gap junction gene required for regeneration and homeostasis

The largely unknown mechanisms that regulate adult stem cells probably involve signals from neighboring differentiated cells. Gap junction channels providing direct cell-cell communication via small molecules are a crucial component of morphogenesis and normal physiology. However, no specific gap junction protein has yet been functionally linked to adult/somatic stem cell behavior in vivo or to organ regeneration. We report the identification and characterization of smedinx-11--an innexin gap junction channel gene expressed in the adult stem cells (neoblasts) of the planarian Schmidtea mediterranea. smedinx-11 RNAi treatment inhibits regeneration and abrogates neoblast maintenance. Moreover, smedinx-11 expression is enriched in an irradiation-sensitive subpopulation (;X2') and is required for proper expression of other stem cell-specific markers. Analyses of the smedinx-11 downregulation phenotype revealed a striking anterior-posterior neoblast gradient. Our data demonstrate a novel role for gap junction proteins and suggest gap junction-mediated signaling as a new and tractable control point for adult, somatic stem cell regulation.     

Hub-centered gene network reconstruction using automatic relevance determination

Network inference deals with the reconstruction of biological networks from experimental data. A variety of different reverse engineering techniques are available; they differ in the underlying assumptions and mathematical models used. One common problem for all approaches stems from the complexity of the task, due to the combinatorial explosion of different network topologies for increasing network size. To handle this problem, constraints are frequently used, for example on the node degree, number of edges, or constraints on regulation functions between network components. We propose to exploit topological considerations in the inference of gene regulatory networks. Such systems are often controlled by a small number of hub genes, while most other genes have only limited influence on the network's dynamic. We model gene regulation using a Bayesian network with discrete, Boolean nodes. A hierarchical prior is employed to identify hub genes. The first layer of the prior is used to regularize weights on edges emanating from one specific node. A second prior on hyperparameters controls the magnitude of the former regularization for different nodes. The net effect is that central nodes tend to form in reconstructed networks. Network reconstruction is then performed by maximization of or sampling from the posterior distribution. We evaluate our approach on simulated and real experimental data, indicating that we can reconstruct main regulatory interactions from the data. We furthermore compare our approach to other state-of-the art methods, showing superior performance in identifying hubs. Using a large publicly available dataset of over 800 cell cycle regulated genes, we are able to identify several main hub genes. Our method may thus provide a valuable tool to identify interesting candidate genes for further study. Furthermore, the approach presented may stimulate further developments in regularization methods for network reconstruction from data.     

On the dynamics of random Boolean networks subject to noise: Attractors, ergodic sets and cell types

The asymptotic dynamics of random Boolean networks subject to random fluctuations is investigated. Under the influence of noise, the system can escape from the attractors of the deterministic model, and a thorough study of these transitions is presented. We show that the dynamics is more properly described by sets of attractors rather than single ones. We generalize here a previous notion of ergodic sets, and we show that the Threshold Ergodic Sets so defined are robust with respect to noise and, at the same time, that they do not suffer from a major drawback of ergodic sets. The system jumps from one attractor to another of the same Threshold Ergodic Set under the influence of noise, never leaving it. By interpreting random Boolean networks as models of genetic regulatory networks, we also propose to associate cell types to Threshold Ergodic Sets rather than to deterministic attractors or to ergodic sets, as it had been previously suggested. We also propose to associate cell differentiation to the process whereby a Threshold Ergodic Set composed by several attractors gives rise to another one composed by a smaller number of attractors. We show that this approach accounts for several interesting experimental facts about cell differentiation, including the possibility to obtain an induced pluripotent stem cell from a fully differentiated one by overexpressing some of its genes.