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1.  Flower Development as an Interplay between Dynamical Physical Fields and Genetic Networks 
PLoS ONE  2010;5(10):e13523.
In this paper we propose a model to describe the mechanisms by which undifferentiated cells attain gene configurations underlying cell fate determination during morphogenesis. Despite the complicated mechanisms that surely intervene in this process, it is clear that the fundamental fact is that cells obtain spatial and temporal information that bias their destiny. Our main hypothesis assumes that there is at least one macroscopic field that breaks the symmetry of space at a given time. This field provides the information required for the process of cell differentiation to occur by being dynamically coupled to a signal transduction mechanism that, in turn, acts directly upon the gene regulatory network (GRN) underlying cell-fate decisions within cells. We illustrate and test our proposal with a GRN model grounded on experimental data for cell fate specification during organ formation in early Arabidopsis thaliana flower development. We show that our model is able to recover the multigene configurations characteristic of sepal, petal, stamen and carpel primordial cells arranged in concentric rings, in a similar pattern to that observed during actual floral organ determination. Such pattern is robust to alterations of the model parameters and simulated failures predict altered spatio-temporal patterns that mimic those described for several mutants. Furthermore, simulated alterations in the physical fields predict a pattern equivalent to that found in Lacandonia schismatica, the only flowering species with central stamens surrounded by carpels.
PMCID: PMC2965087  PMID: 21048956
2.  Stochastic Boolean networks: An efficient approach to modeling gene regulatory networks 
BMC Systems Biology  2012;6:113.
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.
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 efficiency 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.
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.
PMCID: PMC3532238  PMID: 22929591
3.  Floral patterning defects induced by Arabidopsis APETALA2 and microRNA172 expression in Nicotiana benthamiana 
Plant molecular biology  2006;61(4-5):781-793.
Floral patterning and morphogenesis are controlled by many transcription factors including floral homeotic proteins, by which floral organ identity is determined. Recent studies have uncovered widespread regulation of transcription factors by microRNAs (miRNAs), ~21-nucleotide non-coding RNAs that regulate protein-coding RNAs through transcript cleavage and/or translational inhibition. The regulation of the floral homeotic gene APETALA2 (AP2) by miR172 is crucial for normal Arabidopsis flower development and is likely to be conserved across plant species. Here we probe the activity of the AP2/miR172 regulatory circuit in a heterologous Solanaceae species, Nicotiana benthamiana. We generated transgenic N. benthamiana lines expressing Arabidopsis wild type AP2 (35S∷AP2), miR172-resistant AP2 mutant (35S∷AP2m3) and MIR172a-1 (35S∷MIR172) under the control of the cauliflower mosaic virus 35S promoter. 35S∷AP2m3 plants accumulated high levels of AP2 mRNA and protein and exhibited floral patterning defects that included proliferation of numerous petals, stamens and carpels indicating loss of floral determinacy. On the other hand, nearly all 35S∷AP2 plants accumulated barely detectable levels of AP2 mRNA or protein and were essentially non-phenotypic. Overall, the data indicated that expression of the wild type Arabidopsis AP2 transgene was repressed at the mRNA level by an endogenous N. benthamiana miR172 homologue that could be detected using Arabidopsis miR172 probe. Interestingly, 35S∷MIR172 plants had sepal-to-petal transformations and/or more sepals and petals, suggesting interference with N. benthamiana normal floral homeotic gene function in perianth organs. Our studies uncover the potential utility of the Arabidopsis AP2/miR172 system as a tool for manipulation of floral architecture and flowering time in non-model plants.
PMCID: PMC3574581  PMID: 16897492
APETALA2; Arabidopsis; microRNA; miR172; Nicotiana benthamiana
4.  A Tutorial on Analysis and Simulation of Boolean Gene Regulatory Network Models 
Current Genomics  2009;10(7):511-525.
Driven by the desire to understand genomic functions through the interactions among genes and gene products, the research in gene regulatory networks has become a heated area in genomic signal processing. Among the most studied mathematical models are Boolean networks and probabilistic Boolean networks, which are rule-based dynamic systems. This tutorial provides an introduction to the essential concepts of these two Boolean models, and presents the up-to-date analysis and simulation methods developed for them. In the Analysis section, we will show that Boolean models are Markov chains, based on which we present a Markovian steady-state analysis on attractors, and also reveal the relationship between probabilistic Boolean networks and dynamic Bayesian networks (another popular genetic network model), again via Markov analysis; we dedicate the last subsection to structural analysis, which opens a door to other topics such as network control. The Simulation section will start from the basic tasks of creating state transition diagrams and finding attractors, proceed to the simulation of network dynamics and obtaining the steady-state distributions, and finally come to an algorithm of generating artificial Boolean networks with prescribed attractors. The contents are arranged in a roughly logical order, such that the Markov chain analysis lays the basis for the most part of Analysis section, and also prepares the readers to the topics in Simulation section.
PMCID: PMC2808677  PMID: 20436877
5.  An Efficient Algorithm for Computing Attractors of Synchronous And Asynchronous Boolean Networks 
PLoS ONE  2013;8(4):e60593.
Biological networks, such as genetic regulatory networks, often contain positive and negative feedback loops that settle down to dynamically stable patterns. Identifying these patterns, the so-called attractors, can provide important insights for biologists to understand the molecular mechanisms underlying many coordinated cellular processes such as cellular division, differentiation, and homeostasis. Both synchronous and asynchronous Boolean networks have been used to simulate genetic regulatory networks and identify their attractors. The common methods of computing attractors are that start with a randomly selected initial state and finish with exhaustive search of the state space of a network. However, the time complexity of these methods grows exponentially with respect to the number and length of attractors. Here, we build two algorithms to achieve the computation of attractors in synchronous and asynchronous Boolean networks. For the synchronous scenario, combing with iterative methods and reduced order binary decision diagrams (ROBDD), we propose an improved algorithm to compute attractors. For another algorithm, the attractors of synchronous Boolean networks are utilized in asynchronous Boolean translation functions to derive attractors of asynchronous scenario. The proposed algorithms are implemented in a procedure called geneFAtt. Compared to existing tools such as genYsis, geneFAtt is significantly faster in computing attractors for empirical experimental systems.
The software package is available at
PMCID: PMC3621871  PMID: 23585840
6.  An Evaluation of Methods for Inferring Boolean Networks from Time-Series Data 
PLoS ONE  2013;8(6):e66031.
Regulatory networks play a central role in cellular behavior and decision making. Learning these regulatory networks is a major task in biology, and devising computational methods and mathematical models for this task is a major endeavor in bioinformatics. Boolean networks have been used extensively for modeling regulatory networks. In this model, the state of each gene can be either ‘on’ or ‘off’ and that next-state of a gene is updated, synchronously or asynchronously, according to a Boolean rule that is applied to the current-state of the entire system. Inferring a Boolean network from a set of experimental data entails two main steps: first, the experimental time-series data are discretized into Boolean trajectories, and then, a Boolean network is learned from these Boolean trajectories. In this paper, we consider three methods for data discretization, including a new one we propose, and three methods for learning Boolean networks, and study the performance of all possible nine combinations on four regulatory systems of varying dynamics complexities. We find that employing the right combination of methods for data discretization and network learning results in Boolean networks that capture the dynamics well and provide predictive power. Our findings are in contrast to a recent survey that placed Boolean networks on the low end of the “faithfulness to biological reality” and “ability to model dynamics” spectra. Further, contrary to the common argument in favor of Boolean networks, we find that a relatively large number of time points in the time-series data is required to learn good Boolean networks for certain data sets. Last but not least, while methods have been proposed for inferring Boolean networks, as discussed above, missing still are publicly available implementations thereof. Here, we make our implementation of the methods available publicly in open source at
PMCID: PMC3689729  PMID: 23805196
7.  Within and between Whorls: Comparative Transcriptional Profiling of Aquilegia and Arabidopsis 
PLoS ONE  2010;5(3):e9735.
The genus Aquilegia is an emerging model system in plant evolutionary biology predominantly because of its wide variation in floral traits and associated floral ecology. The anatomy of the Aquilegia flower is also very distinct. There are two whorls of petaloid organs, the outer whorl of sepals and the second whorl of petals that form nectar spurs, as well as a recently evolved fifth whorl of staminodia inserted between stamens and carpels.
Methodology/Principal Findings
We designed an oligonucleotide microarray based on EST sequences from a mixed tissue, normalized cDNA library of an A. formosa x A. pubescens F2 population representing 17,246 unigenes. We then used this array to analyze floral gene expression in late pre-anthesis stage floral organs from a natural A. formosa population. In particular, we tested for gene expression patterns specific to each floral whorl and to combinations of whorls that correspond to traditional and modified ABC model groupings. Similar analyses were performed on gene expression data of Arabidopsis thaliana whorls previously obtained using the Ath1 gene chips (data available through The Arabidopsis Information Resource).
Our comparative gene expression analyses suggest that 1) petaloid sepals and petals of A. formosa share gene expression patterns more than either have organ-specific patterns, 2) petals of A. formosa and A. thaliana may be independently derived, 3) staminodia express B and C genes similar to stamens but the staminodium genetic program has also converged on aspects of the carpel program and 4) staminodia have unique up-regulation of regulatory genes and genes that have been implicated with defense against microbial infection and herbivory. Our study also highlights the value of comparative gene expression profiling and the Aquilegia microarray in particular for the study of floral evolution and ecology.
PMCID: PMC2843724  PMID: 20352114
8.  Target Genes of the MADS Transcription Factor SEPALLATA3: Integration of Developmental and Hormonal Pathways in the Arabidopsis Flower 
PLoS Biology  2009;7(4):e1000090.
The molecular mechanisms by which floral homeotic genes act as major developmental switches to specify the identity of floral organs are still largely unknown. Floral homeotic genes encode transcription factors of the MADS-box family, which are supposed to assemble in a combinatorial fashion into organ-specific multimeric protein complexes. Major mediators of protein interactions are MADS-domain proteins of the SEPALLATA subfamily, which play a crucial role in the development of all types of floral organs. In order to characterize the roles of the SEPALLATA3 transcription factor complexes at the molecular level, we analyzed genome-wide the direct targets of SEPALLATA3. We used chromatin immunoprecipitation followed by ultrahigh-throughput sequencing or hybridization to whole-genome tiling arrays to obtain genome-wide DNA-binding patterns of SEPALLATA3. The results demonstrate that SEPALLATA3 binds to thousands of sites in the genome. Most potential target sites that were strongly bound in wild-type inflorescences are also bound in the floral homeotic agamous mutant, which displays only the perianth organs, sepals, and petals. Characterization of the target genes shows that SEPALLATA3 integrates and modulates different growth-related and hormonal pathways in a combinatorial fashion with other MADS-box proteins and possibly with non-MADS transcription factors. In particular, the results suggest multiple links between SEPALLATA3 and auxin signaling pathways. Our gene expression analyses link the genomic binding site data with the phenotype of plants expressing a dominant repressor version of SEPALLATA3, suggesting that it modulates auxin response to facilitate floral organ outgrowth and morphogenesis. Furthermore, the binding of the SEPALLATA3 protein to cis-regulatory elements of other MADS-box genes and expression analyses reveal that this protein is a key component in the regulatory transcriptional network underlying the formation of floral organs.
Author Summary
Most regulatory genes encode transcription factors, which modulate gene expression by binding to regulatory sequences of their target genes. In plants in particular, which genes are directly controlled by these transcription factors, and the molecular mechanisms of target gene recognition in vivo, are still largely unexplored. One of the best-understood developmental processes in plants is flower development. In different combinations, transcription factors of the MADS-box family control the identities of the different types of floral organs: sepals, petals, stamens, and carpels. Here, we present the first genome-wide analysis of binding sites of a MADS-box transcription factor in plants. We show that the MADS-domain protein SEPALLATA3 (SEP3) binds to the regulatory regions of thousands of potential target genes, many of which are also transcription factors. We provide insight into mechanisms of DNA recognition by SEP3, and suggest roles for other transcription factor families in SEP3 target gene regulation. In addition to effects on genes involved in floral organ identity, our data suggest that SEP3 binds to, and modulates, the transcription of target genes involved in hormonal signaling pathways.
The key floral regulator SEPALLATA3 binds to the promoters of a large number of potential direct target genes to integrate different growth-related and hormonal pathways in flower development.
PMCID: PMC2671559  PMID: 19385720
9.  Flower Development 
Flowers are the most complex structures of plants. Studies of Arabidopsis thaliana, which has typical eudicot flowers, have been fundamental in advancing the structural and molecular understanding of flower development. The main processes and stages of Arabidopsis flower development are summarized to provide a framework in which to interpret the detailed molecular genetic studies of genes assigned functions during flower development and is extended to recent genomics studies uncovering the key regulatory modules involved. Computational models have been used to study the concerted action and dynamics of the gene regulatory module that underlies patterning of the Arabidopsis inflorescence meristem and specification of the primordial cell types during early stages of flower development. This includes the gene combinations that specify sepal, petal, stamen and carpel identity, and genes that interact with them. As a dynamic gene regulatory network this module has been shown to converge to stable multigenic profiles that depend upon the overall network topology and are thus robust, which can explain the canalization of flower organ determination and the overall conservation of the basic flower plan among eudicots. Comparative and evolutionary approaches derived from Arabidopsis studies pave the way to studying the molecular basis of diverse floral morphologies.
PMCID: PMC3244948  PMID: 22303253
10.  A Dynamical Systems Hypothesis of Schizophrenia 
PLoS Computational Biology  2007;3(11):e228.
We propose a top-down approach to the symptoms of schizophrenia based on a statistical dynamical framework. We show that a reduced depth in the basins of attraction of cortical attractor states destabilizes the activity at the network level due to the constant statistical fluctuations caused by the stochastic spiking of neurons. In integrate-and-fire network simulations, a decrease in the NMDA receptor conductances, which reduces the depth of the attractor basins, decreases the stability of short-term memory states and increases distractibility. The cognitive symptoms of schizophrenia such as distractibility, working memory deficits, or poor attention could be caused by this instability of attractor states in prefrontal cortical networks. Lower firing rates are also produced, and in the orbitofrontal and anterior cingulate cortex could account for the negative symptoms, including a reduction of emotions. Decreasing the GABA as well as the NMDA conductances produces not only switches between the attractor states, but also jumps from spontaneous activity into one of the attractors. We relate this to the positive symptoms of schizophrenia, including delusions, paranoia, and hallucinations, which may arise because the basins of attraction are shallow and there is instability in temporal lobe semantic memory networks, leading thoughts to move too freely round the attractor energy landscape.
Author Summary
One of the hallmarks of schizophrenia is the complexity and heterogeneity of the illness. We propose that part of the reason for the inconsistent symptoms may be a reduced signal-to-noise ratio and increased statistical fluctuations in different cortical brain networks. The novelty of the approach described here is that instead of basing our hypothesis purely on biological mechanisms, we develop a top-down approach based on the different types of symptoms and relate them to instabilities in attractor neural networks. Schizophrenia is characterized by cognitive, negative, and positive symptoms. We propose which characteristic effects in a dynamical system could cause these symptoms, and investigate our hypothesis in a computational model. We implement an integrate-and-fire network model and focus on the alterations of synaptic channels activated via NMDA and GABA receptors. We found that a decrease in the NMDA receptor conductance could contribute to both the cognitive and negative symptoms by reducing the neuronal firing rates and the stability of the attractor states. A reduction of both NMDA and GABA conductance causes an instability of the attractor states related to the positive symptoms. Overall, we provide a framework to discuss schizophrenia in a dynamical system framework.
PMCID: PMC2065887  PMID: 17997599
11.  Noise in a Small Genetic Circuit that Undergoes Bifurcation 
Complexity  2005;11(1):45-51.
Based on the consideration of Boolean dynamics, it has been hypothesized that cell types may correspond to alternative attractors of a gene regulatory network. Recent stochastic Boolean network analysis, however, raised the important question concerning the stability of such attractors. In this paper a detailed numerical analysis is performed within the framework of Langevin dynamics. While the present results confirm that the noise is indeed an important dynamical element, the cell type as represented by attractors can still be a viable hypothesis. It is found that the stability of an attractor depends on the strength of noise related to the distance of the system to the bifurcation point and it can be exponentially stable depending on biological parameters.
PMCID: PMC1456069  PMID: 16670776
cell types; attractors; genetic networks; stability; robustness; stochastic processes; Langevin dynamics
12.  Boolean Network Model Predicts Knockout Mutant Phenotypes of Fission Yeast 
PLoS ONE  2013;8(9):e71786.
Boolean networks (or: networks of switches) are extremely simple mathematical models of biochemical signaling networks. Under certain circumstances, Boolean networks, despite their simplicity, are capable of predicting dynamical activation patterns of gene regulatory networks in living cells. For example, the temporal sequence of cell cycle activation patterns in yeasts S. pombe and S. cerevisiae are faithfully reproduced by Boolean network models. An interesting question is whether this simple model class could also predict a more complex cellular phenomenology as, for example, the cell cycle dynamics under various knockout mutants instead of the wild type dynamics, only. Here we show that a Boolean network model for the cell cycle control network of yeast S. pombe correctly predicts viability of a large number of known mutants. So far this had been left to the more detailed differential equation models of the biochemical kinetics of the yeast cell cycle network and was commonly thought to be out of reach for models as simplistic as Boolean networks. The new results support our vision that Boolean networks may complement other mathematical models in systems biology to a larger extent than expected so far, and may fill a gap where simplicity of the model and a preference for an overall dynamical blueprint of cellular regulation, instead of biochemical details, are in the focus.
PMCID: PMC3777975  PMID: 24069138
13.  Gene perturbation and intervention in context-sensitive stochastic Boolean networks 
BMC Systems Biology  2014;8:60.
In a gene regulatory network (GRN), gene expressions are affected by noise, and stochastic fluctuations exist in the interactions among genes. These stochastic interactions are context dependent, thus it becomes important to consider noise in a context-sensitive manner in a network model. As a logical model, context-sensitive probabilistic Boolean networks (CSPBNs) account for molecular and genetic noise in the temporal context of gene functions. In a CSPBN with n genes and k contexts, however, a computational complexity of O(nk222n ) (or O(nk2 n )) is required for an accurate (or approximate) computation of the state transition matrix (STM) of the size (2 n ∙ k) × (2 n ∙ k) (or 2 n × 2 n ). The evaluation of a steady state distribution (SSD) is more challenging. Recently, stochastic Boolean networks (SBNs) have been proposed as an efficient implementation of an instantaneous PBN.
The notion of stochastic Boolean networks (SBNs) is extended for the general model of PBNs, i.e., CSPBNs. This yields a novel structure of context-sensitive SBNs (CSSBNs) for modeling the stochasticity in a GRN. A CSSBN enables an efficient simulation of a CSPBN with a complexity of O(nLk2 n ) for computing the state transition matrix, where L is a factor related to the required sequence length in CSSBN for achieving a desired accuracy. A time-frame expanded CSSBN can further efficiently simulate the stationary behavior of a CSPBN and allow for a tunable tradeoff between accuracy and efficiency. The CSSBN approach is more efficient than an analytical method and more accurate than an approximate analysis.
Context-sensitive stochastic Boolean networks (CSSBNs) are proposed as an efficient approach to modeling the effects of gene perturbation and intervention in gene regulatory networks. A CSSBN analysis provides biologically meaningful insights into the oscillatory dynamics of the p53-Mdm2 network in a context-switching environment. It is shown that random gene perturbation has a greater effect on the final distribution of the steady state of a network compared to context switching activities. The CSSBN approach can further predict the steady state distribution of a glioma network under gene intervention. Ultimately, this will help drug discovery and develop effective drug intervention strategies.
PMCID: PMC4062525  PMID: 24886608
Gene regulatory networks; Boolean networks; Stochastic Boolean networks; Context dependent; Gene perturbation; Intervention; Context switch; Steady state distribution; p53 network; glioma network
14.  Functional recapitulation of transitions in sexual systems by homeosis during the evolution of dioecy in Thalictrum 
Sexual systems are highly variable in flowering plants and an important contributor to floral diversity. The ranunculid genus Thalictrum is especially well-suited to study evolutionary transitions in sexual systems. Homeotic transformation of sexual organs (stamens and carpels) is a plausible mechanism for the transition from hermaphroditic to unisexual flowers in this lineage because flowers of dioecious species develop unisexually from inception. The single-copy gene PISTILLATA (PI) constitutes a likely candidate for rapid switches between stamen and carpel identity. Here, we first characterized the expression pattern of all B class genes in the dioecious species T. dioicum. As expected, all B class orthologs are expressed in stamens from the earliest stages. Certain AP3 lineages were also expressed late in sepal development. We then tested whether orthologs of PI could potentially control sexual system transitions in Thalictrum, by knocking-down their expression in T. dioicum and the hermaphroditic species T. thalictroides. In T. dioicum, we found that ThdPI-1/2 silencing caused stamen primordia to develop into carpels, resulting in male to female flower conversions. In T. thalictroides, we found that ThtPI silencing caused stamen primordia to develop into supernumerary carpels, resulting in hermaphroditic to female flower conversions. These phenotypes illustrate the ability for homeotic mutations to bring about sudden and potentially adaptive changes by altering the function of a single gene. We propose a two-step evolutionary model where transitions from hermaphroditic to unisexual plants in Thalictrum result from two independent mutations at a B class gene locus. Our PI knockdown experiments in T. thalictroides recapitulate the second step in this model: the evolution of female plants as a result of a loss-of-function mutation in a B class gene.
PMCID: PMC3842162  PMID: 24348491
B class genes; PISTILLATA; VIGS; RNAi; ranunculid; ABC model; sex determination; MADS box genes
15.  The regulatory network of E. coli metabolism as a Boolean dynamical system exhibits both homeostasis and flexibility of response 
BMC Systems Biology  2008;2:21.
Elucidating the architecture and dynamics of large scale genetic regulatory networks of cells is an important goal in systems biology. We study the system level dynamical properties of the genetic network of Escherichia coli that regulates its metabolism, and show how its design leads to biologically useful cellular properties. Our study uses the database (Covert et al., Nature 2004) containing 583 genes and 96 external metabolites which describes not only the network connections but also the Boolean rule at each gene node that controls the switching on or off of the gene as a function of its inputs.
We have studied how the attractors of the Boolean dynamical system constructed from this database depend on the initial condition of the genes and on various environmental conditions corresponding to buffered minimal media. We find that the system exhibits homeostasis in that its attractors, that turn out to be fixed points or low period cycles, are highly insensitive to initial conditions or perturbations of gene configurations for any given fixed environment. At the same time the attractors show a wide variation when external media are varied implying that the system mounts a highly flexible response to changed environmental conditions. The regulatory dynamics acts to enhance the cellular growth rate under changed media.
Our study shows that the reconstructed genetic network regulating metabolism in E. coli is hierarchical, modular, and largely acyclic, with environmental variables controlling the root of the hierarchy. This architecture makes the cell highly robust to perturbations of gene configurations as well as highly responsive to environmental changes. The twin properties of homeostasis and response flexibility are achieved by this dynamical system even though it is not close to the edge of chaos.
PMCID: PMC2322946  PMID: 18312613
16.  Comparative Transcriptional Profiling Provides Insights into the Evolution and Development of the Zygomorphic Flower of Vicia sativa (Papilionoideae) 
PLoS ONE  2013;8(2):e57338.
Vicia sativa (the common vetch) possesses a predominant zygomorphic flower and belongs to the subfamily Papilionoideae, which is related to Arabidopsis thaliana in the eurosid II clade of the core eudicots. Each vetch flower consists of 21 concentrically arranged organs: the outermost five sepals, then five petals and ten stamens, and a single carpel in the center.
Methodology/Principal Findings
We explored the floral transcriptome to examine a genome-scale genetic model of the zygomorphic flower of vetch. mRNA was obtained from an equal mixture of six floral organs, leaves and roots. De novo assembly of the vetch transcriptome using Illumina paired-end technology produced 71,553 unigenes with an average length of 511 bp. We then compared the expression changes in the 71,553 unigenes in the eight independent organs through RNA-Seq Quantification analysis. We predominantly analyzed gene expression patterns specific to each floral organ and combinations of floral organs that corresponded to the traditional ABC model domains. Comparative analyses were performed in the floral transcriptomes of vetch and Arabidopsis, and genomes of vetch and Medicago truncatula.
Our comparative analysis of vetch and Arabidopsis showed that the vetch flowers conform to a strict ABC model. We analyzed the evolution and expression of the TCP gene family in vetch at a whole-genome level, and several unigenes specific to three different vetch petals, which might offer some clues toward elucidating the molecular mechanisms underlying floral zygomorphy. Our results provide the first insights into the genome-scale molecular regulatory network that controls the evolution and development of the zygomorphic flower in Papilionoideae.
PMCID: PMC3578871  PMID: 23437373
17.  Relationships between probabilistic Boolean networks and dynamic Bayesian networks as models of gene regulatory networks 
Signal processing  2006;86(4):814-834.
A significant amount of attention has recently been focused on modeling of gene regulatory networks. Two frequently used large-scale modeling frameworks are Bayesian networks (BNs) and Boolean networks, the latter one being a special case of its recent stochastic extension, probabilistic Boolean networks (PBNs). PBN is a promising model class that generalizes the standard rule-based interactions of Boolean networks into the stochastic setting. Dynamic Bayesian networks (DBNs) is a general and versatile model class that is able to represent complex temporal stochastic processes and has also been proposed as a model for gene regulatory systems. In this paper, we concentrate on these two model classes and demonstrate that PBNs and a certain subclass of DBNs can represent the same joint probability distribution over their common variables. The major benefit of introducing the relationships between the models is that it opens up the possibility of applying the standard tools of DBNs to PBNs and vice versa. Hence, the standard learning tools of DBNs can be applied in the context of PBNs, and the inference methods give a natural way of handling the missing values in PBNs which are often present in gene expression measurements. Conversely, the tools for controlling the stationary behavior of the networks, tools for projecting networks onto sub-networks, and efficient learning schemes can be used for DBNs. In other words, the introduced relationships between the models extend the collection of analysis tools for both model classes.
PMCID: PMC1847796  PMID: 17415411
Gene regulatory networks; Probabilistic Boolean networks; Dynamic Bayesian networks
18.  Robustness in Regulatory Interaction Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic and Genetic 
Regulatory interaction networks are often studied on their dynamical side (existence of attractors, study of their stability). We focus here also on their robustness, that is their ability to offer the same spatiotemporal patterns and to resist to external perturbations such as losses of nodes or edges in the networks interactions architecture, changes in their environmental boundary conditions as well as changes in the update schedule (or updating mode) of the states of their elements (e.g., if these elements are genes, their synchronous coexpression mode versus their sequential expression). We define the generic notions of boundary, core, and critical vertex or edge of the underlying interaction graph of the regulatory network, whose disappearance causes dramatic changes in the number and nature of attractors (e.g., passage from a bistable behaviour to a unique periodic regime) or in the range of their basins of stability. The dynamic transition of states will be presented in the framework of threshold Boolean automata rules. A panorama of applications at different levels will be given: brain and plant morphogenesis, bulbar cardio-respiratory regulation, glycolytic/oxidative metabolic coupling, and eventually cell cycle and feather morphogenesis genetic control.
PMCID: PMC2790118  PMID: 20057955
robustness in regulatory interaction networks; attractors; interaction graph boundary; interaction graph core; critical node; critical edge; updating mode; microRNAs
19.  Network Class Superposition Analyses 
PLoS ONE  2013;8(4):e59046.
Networks are often used to understand a whole system by modeling the interactions among its pieces. Examples include biomolecules in a cell interacting to provide some primary function, or species in an environment forming a stable community. However, these interactions are often unknown; instead, the pieces' dynamic states are known, and network structure must be inferred. Because observed function may be explained by many different networks (e.g., for the yeast cell cycle process [1]), considering dynamics beyond this primary function means picking a single network or suitable sample: measuring over all networks exhibiting the primary function is computationally infeasible. We circumvent that obstacle by calculating the network class ensemble. We represent the ensemble by a stochastic matrix , which is a transition-by-transition superposition of the system dynamics for each member of the class. We present concrete results for derived from Boolean time series dynamics on networks obeying the Strong Inhibition rule, by applying to several traditional questions about network dynamics. We show that the distribution of the number of point attractors can be accurately estimated with . We show how to generate Derrida plots based on . We show that -based Shannon entropy outperforms other methods at selecting experiments to further narrow the network structure. We also outline an experimental test of predictions based on . We motivate all of these results in terms of a popular molecular biology Boolean network model for the yeast cell cycle, but the methods and analyses we introduce are general. We conclude with open questions for , for example, application to other models, computational considerations when scaling up to larger systems, and other potential analyses.
PMCID: PMC3614996  PMID: 23565141
20.  Boolean Network Model for Cancer Pathways: Predicting Carcinogenesis and Targeted Therapy Outcomes 
PLoS ONE  2013;8(7):e69008.
A Boolean dynamical system integrating the main signaling pathways involved in cancer is constructed based on the currently known protein-protein interaction network. This system exhibits stationary protein activation patterns – attractors – dependent on the cell's microenvironment. These dynamical attractors were determined through simulations and their stabilities against mutations were tested. In a higher hierarchical level, it was possible to group the network attractors into distinct cell phenotypes and determine driver mutations that promote phenotypic transitions. We find that driver nodes are not necessarily central in the network topology, but at least they are direct regulators of central components towards which converge or through which crosstalk distinct cancer signaling pathways. The predicted drivers are in agreement with those pointed out by diverse census of cancer genes recently performed for several human cancers. Furthermore, our results demonstrate that cell phenotypes can evolve towards full malignancy through distinct sequences of accumulated mutations. In particular, the network model supports routes of carcinogenesis known for some tumor types. Finally, the Boolean network model is employed to evaluate the outcome of molecularly targeted cancer therapies. The major find is that monotherapies were additive in their effects and that the association of targeted drugs is necessary for cancer eradication.
PMCID: PMC3724878  PMID: 23922675
21.  A Hybrid Model of Mammalian Cell Cycle Regulation 
PLoS Computational Biology  2011;7(2):e1001077.
The timing of DNA synthesis, mitosis and cell division is regulated by a complex network of biochemical reactions that control the activities of a family of cyclin-dependent kinases. The temporal dynamics of this reaction network is typically modeled by nonlinear differential equations describing the rates of the component reactions. This approach provides exquisite details about molecular regulatory processes but is hampered by the need to estimate realistic values for the many kinetic constants that determine the reaction rates. It is difficult to estimate these kinetic constants from available experimental data. To avoid this problem, modelers often resort to ‘qualitative’ modeling strategies, such as Boolean switching networks, but these models describe only the coarsest features of cell cycle regulation. In this paper we describe a hybrid approach that combines the best features of continuous differential equations and discrete Boolean networks. Cyclin abundances are tracked by piecewise linear differential equations for cyclin synthesis and degradation. Cyclin synthesis is regulated by transcription factors whose activities are represented by discrete variables (0 or 1) and likewise for the activities of the ubiquitin-ligating enzyme complexes that govern cyclin degradation. The discrete variables change according to a predetermined sequence, with the times between transitions determined in part by cyclin accumulation and degradation and as well by exponentially distributed random variables. The model is evaluated in terms of flow cytometry measurements of cyclin proteins in asynchronous populations of human cell lines. The few kinetic constants in the model are easily estimated from the experimental data. Using this hybrid approach, modelers can quickly create quantitatively accurate, computational models of protein regulatory networks in cells.
Author Summary
The physiological behaviors of cells (growth and division, differentiation, movement, death, etc.) are controlled by complex networks of interacting genes and proteins, and a fundamental goal of computational cell biology is to develop dynamical models of these regulatory networks that are realistic, accurate and predictive. Historically, these models have divided along two basic lines: deterministic or stochastic, and continuous or discrete; with scattered efforts to develop hybrid approaches that bridge these divides. Using the cell cycle control system in eukaryotes as an example, we propose a hybrid approach that combines a continuous representation of slowly changing protein concentrations with a discrete representation of components that switch rapidly between ‘on’ and ‘off’ states, and that combines the deterministic causality of network interactions with the stochastic uncertainty of random events. The hybrid approach can be easily tailored to the available knowledge of control systems, and it provides both qualitative and quantitative results that can be compared to experimental data to test the accuracy and predictive power of the model.
PMCID: PMC3037389  PMID: 21347318
22.  An extended gene protein/products boolean network model including post-transcriptional regulation 
Networks Biology allows the study of complex interactions between biological systems using formal, well structured, and computationally friendly models. Several different network models can be created, depending on the type of interactions that need to be investigated. Gene Regulatory Networks (GRN) are an effective model commonly used to study the complex regulatory mechanisms of a cell. Unfortunately, given their intrinsic complexity and non discrete nature, the computational study of realistic-sized complex GRNs requires some abstractions. Boolean Networks (BNs), for example, are a reliable model that can be used to represent networks where the possible state of a node is a boolean value (0 or 1). Despite this strong simplification, BNs have been used to study both structural and dynamic properties of real as well as randomly generated GRNs.
In this paper we show how it is possible to include the post-transcriptional regulation mechanism (a key process mediated by small non-coding RNA molecules like the miRNAs) into the BN model of a GRN. The enhanced BN model is implemented in a software toolkit (EBNT) that allows to analyze boolean GRNs from both a structural and a dynamic point of view. The open-source toolkit is compatible with available visualization tools like Cytoscape and allows to run detailed analysis of the network topology as well as of its attractors, trajectories, and state-space. In the paper, a small GRN built around the mTOR gene is used to demonstrate the main capabilities of the toolkit.
The extended model proposed in this paper opens new opportunities in the study of gene regulation. Several of the successful researches done with the support of BN to understand high-level characteristics of regulatory networks, can now be improved to better understand the role of post-transcriptional regulation for example as a network-wide noise-reduction or stabilization mechanisms.
PMCID: PMC4108923  PMID: 25080304
23.  Identifying dynamical modules from genetic regulatory systems: applications to the segment polarity network 
BMC Bioinformatics  2007;8:413.
It is widely accepted that genetic regulatory systems are 'modular', in that the whole system is made up of smaller 'subsystems' corresponding to specific biological functions. Most attempts to identify modules in genetic regulatory systems have relied on the topology of the underlying network. However, it is the temporal activity (dynamics) of genes and proteins that corresponds to biological functions, and hence it is dynamics that we focus on here for identifying subsystems.
Using Boolean network models as an exemplar, we present a new technique to identify subsystems, based on their dynamical properties. The main part of the method depends only on the stable dynamics (attractors) of the system, thus requiring no prior knowledge of the underlying network. However, knowledge of the logical relationships between the network components can be used to describe how each subsystem is regulated. To demonstrate its applicability to genetic regulatory systems, we apply the method to a model of the Drosophila segment polarity network, providing a detailed breakdown of the system.
We have designed a technique for decomposing any set of discrete-state, discrete-time attractors into subsystems. Having a suitable mathematical model also allows us to describe how each subsystem is regulated and how robust each subsystem is against perturbations. However, since the subsystems are found directly from the attractors, a mathematical model or underlying network topology is not necessarily required to identify them, potentially allowing the method to be applied directly to experimental expression data.
PMCID: PMC2233651  PMID: 17961242
24.  Cell-type specific analysis of translating RNAs in developing flowers reveals new levels of control 
Combining translating ribosome affinity purification with RNA-seq for cell-specific profiling of translating RNAs in developing flowers.Cell type comparisons of cell type-specific hormone responses, promoter motifs, coexpressed cognate binding factor candidates, and splicing isoforms.Widespread post-transcriptional regulation at both the intron splicing and translational stages.A new class of noncoding RNAs associated with polysomes.
What constitutes a differentiated cell type? How much do cell types differ in their transcription of genes? The development and functions of tissues rely on constant interactions among distinct and nonequivalent cell types. Answering these questions will require quantitative information on transcriptomes, proteomes, protein–protein interactions, protein–nucleic acid interactions, and metabolomes at cellular resolution. The systems approaches emerging in biology promise to explain properties of biological systems based on genome-wide measurements of expression, interaction, regulation, and metabolism. To facilitate a systems approach, it is essential first to capture such components in a global manner, ideally at cellular resolution.
Recently, microarray analysis of transcriptomes has been extended to a cellular level of resolution by using laser microdissection or fluorescence-activated sorting (for review, see Nelson et al, 2008). These methods have been limited by stresses associated with cellular separation and isolation procedures, and biases associated with mandatory RNA amplification steps. A newly developed method, translating ribosome affinity purification (TRAP; Zanetti et al, 2005; Heiman et al, 2008; Mustroph et al, 2009), circumvents these problems by epitopetagging a ribosomal protein in specific cellular domains to selectively purify polysomes. We combined TRAP with deep sequencing, which we term TRAP-seq, to provide cell-level spatiotemporal maps for Arabidopsis early floral development at single-base resolution.
Flower development in Arabidopsis has been studied extensively and is one of the best understood aspects of plant development (for review, see Krizek and Fletcher, 2005). Genetic analysis of homeotic mutants established the ABC model, in which three classes of regulatory genes, A, B and C, work in a combinatorial manner to confer organ identities of four whorls (Coen and Meyerowitz, 1991). Each class of regulatory gene is expressed in a specific and evolutionarily conserved domain, and the action of the class A, B and C genes is necessary for specification of organ identity (Figure 1A).
Using TRAP-seq, we purified cell-specific translating mRNA populations, which we and others call the translatome, from the A, B and C domains of early developing flowers, in which floral patterning and the specification of floral organs is established. To achieve temporal specificity, we used a floral induction system to facilitate collection of early stage flowers (Wellmer et al, 2006). The combination of TRAP-seq with domain-specific promoters and this floral induction system enabled fine spatiotemporal isolation of translating mRNA in specific cellular domains, and at specific developmental stages.
Multiple lines of evidence confirmed the specificity of this approach, including detecting the expression in expected domains but not in other domains for well-studied flower marker genes and known physiological functions (Figures 1B–D and 2A–C). Furthermore, we provide numerous examples from flower development in which a spatiotemporal map of rigorously comparable cell-specific translatomes makes possible new views of the properties of cell domains not evident in data obtained from whole organs or tissues, including patterns of transcription and cis-regulation, new physiological differences among cell domains and between flower stages, putative hormone-active centers, and splicing events specific for flower domains (Figure 2A–D). Such findings may provide new targets for reverse genetics studies and may aid in the formulation and validation of interaction and pathway networks.
Beside cellular heterogeneity, the transcriptome is regulated at several steps through the life of mRNA molecules, which are not directly available through traditional transcriptome profiling of total mRNA abundance. By comparing the translatome and transcriptome, we integratively profiled two key posttranscriptional control points, intron splicing and translation state. From our translatome-wide profiling, we (i) confirmed that both posttranscriptional regulation control points were used by a large portion of the transcriptome; (ii) identified a number of cis-acting features within the coding or noncoding sequences that correlate with splicing or translation state; and (iii) revealed correlation between each regulation mechanism and gene function. Our transcriptome-wide surveys have highlighted target genes transcripts of which are probably under extensive posttranscriptional regulation during flower development.
Finally, we reported the finding of a large number of polysome-associated ncRNAs. About one-third of all annotated ncRNA in the Arabidopsis genome were observed co-purified with polysomes. Coding capacity analysis confirmed that most of them are real ncRNA without conserved ORFs. The group of polysome-associated ncRNA reported in this study is a potential new addition to the expanding riboregulator catalog; they could have roles in translational regulation during early flower development.
Determining both the expression levels of mRNA and the regulation of its translation is important in understanding specialized cell functions. In this study, we describe both the expression profiles of cells within spatiotemporal domains of the Arabidopsis thaliana flower and the post-transcriptional regulation of these mRNAs, at nucleotide resolution. We express a tagged ribosomal protein under the promoters of three master regulators of flower development. By precipitating tagged polysomes, we isolated cell type-specific mRNAs that are probably translating, and quantified those mRNAs through deep sequencing. Cell type comparisons identified known cell-specific transcripts and uncovered many new ones, from which we inferred cell type-specific hormone responses, promoter motifs and coexpressed cognate binding factor candidates, and splicing isoforms. By comparing translating mRNAs with steady-state overall transcripts, we found evidence for widespread post-transcriptional regulation at both the intron splicing and translational stages. Sequence analyses identified structural features associated with each step. Finally, we identified a new class of noncoding RNAs associated with polysomes. Findings from our profiling lead to new hypotheses in the understanding of flower development.
PMCID: PMC2990639  PMID: 20924354
Arabidopsis; flower; intron; transcriptome; translation
25.  Multistable switches and their role in cellular differentiation networks 
BMC Bioinformatics  2014;15(Suppl 7):S7.
Cellular differentiation during development is controlled by gene regulatory networks (GRNs). This complex process is always subject to gene expression noise. There is evidence suggesting that commonly seen patterns in GRNs, referred to as biological multistable switches, play an important role in creating the structure of lineage trees by providing stability to cell types.
To explore this question a new methodology is developed and applied to study (a) the multistable switch-containing GRN for hematopoiesis and (b) a large set of random boolean networks (RBNs) in which multistable switches were embedded systematically. In this work, each network attractor is taken to represent a distinct cell type. The GRNs were seeded with one or two identical copies of each multistable switch and the effect of these additions on two key aspects of network dynamics was assessed. These properties are the barrier to movement between pairs of attractors (separation) and the degree to which one direction of movement between attractor pairs is favored over another (directionality). Both of these properties are instrumental in shaping the structure of lineage trees. We found that adding one multistable switch of any type had a modest effect on increasing the proportion of well-separated attractor pairs. Adding two identical switches of any type had a much stronger effect in increasing the proportion of well-separated attractors. Similarly, there was an increase in the frequency of directional transitions between attractor pairs when two identical multistable switches were added to GRNs. This effect on directionality was not observed when only one multistable switch was added.
This work provides evidence that the occurrence of multistable switches in networks that control cellular differentiation contributes to the structure of lineage trees and to the stabilization of cell types.
PMCID: PMC4110729  PMID: 25078021

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