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1.  Decomposition of Gene Expression State Space Trajectories 
PLoS Computational Biology  2009;5(12):e1000626.
Representing and analyzing complex networks remains a roadblock to creating dynamic network models of biological processes and pathways. The study of cell fate transitions can reveal much about the transcriptional regulatory programs that underlie these phenotypic changes and give rise to the coordinated patterns in expression changes that we observe. The application of gene expression state space trajectories to capture cell fate transitions at the genome-wide level is one approach currently used in the literature. In this paper, we analyze the gene expression dataset of Huang et al. (2005) which follows the differentiation of promyelocytes into neutrophil-like cells in the presence of inducers dimethyl sulfoxide and all-trans retinoic acid. Huang et al. (2005) build on the work of Kauffman (2004) who raised the attractor hypothesis, stating that cells exist in an expression landscape and their expression trajectories converge towards attractive sites in this landscape. We propose an alternative interpretation that explains this convergent behavior by recognizing that there are two types of processes participating in these cell fate transitions—core processes that include the specific differentiation pathways of promyelocytes to neutrophils, and transient processes that capture those pathways and responses specific to the inducer. Using functional enrichment analyses, specific biological examples and an analysis of the trajectories and their core and transient components we provide a validation of our hypothesis using the Huang et al. (2005) dataset.
Author Summary
Understanding how cells differentiate from one state to another is a fundamental problem in biology with implications for better understanding evolution, the development of complex organisms from a single fertilized egg, and the etiology of human disease. One way to view these processes is to examine cells as “complex adaptive systems” where the state of all genes in a cell (more than 20,000 genes) determines that cell's “state” at a given point in time. In this view, differentiating cells move along a path in “state space” from one stable “attractor” to another. In a 2005 paper, Sui Huang and colleagues presented an experimental model in which they claimed to have evidence for such attractors and for the transitions between them. The problem with this approach is that although it is intuitively appealing, it lacks predictive power. Reanalyzing Huang's data, we demonstrate that there is an alternative interpretation that still allows for a state space description but which has greater ability to make testable predictions. Specifically, we show that these abstract state space trajectories can be mapped onto more well-known pathways and represented as a “core” differentiation pathway and “transient” processes that capture the effects of the treatments that initiate differentiation.
PMCID: PMC2791157  PMID: 20041215
2.  Collective Dynamics of Specific Gene Ensembles Crucial for Neutrophil Differentiation: The Existence of Genome Vehicles Revealed 
PLoS ONE  2010;5(8):e12116.
Cell fate decision remarkably generates specific cell differentiation path among the multiple possibilities that can arise through the complex interplay of high-dimensional genome activities. The coordinated action of thousands of genes to switch cell fate decision has indicated the existence of stable attractors guiding the process. However, origins of the intracellular mechanisms that create “cellular attractor” still remain unknown. Here, we examined the collective behavior of genome-wide expressions for neutrophil differentiation through two different stimuli, dimethyl sulfoxide (DMSO) and all-trans-retinoic acid (atRA). To overcome the difficulties of dealing with single gene expression noises, we grouped genes into ensembles and analyzed their expression dynamics in correlation space defined by Pearson correlation and mutual information. The standard deviation of correlation distributions of gene ensembles reduces when the ensemble size is increased following the inverse square root law, for both ensembles chosen randomly from whole genome and ranked according to expression variances across time. Choosing the ensemble size of 200 genes, we show the two probability distributions of correlations of randomly selected genes for atRA and DMSO responses overlapped after 48 hours, defining the neutrophil attractor. Next, tracking the ranked ensembles' trajectories, we noticed that only certain, not all, fall into the attractor in a fractal-like manner. The removal of these genome elements from the whole genomes, for both atRA and DMSO responses, destroys the attractor providing evidence for the existence of specific genome elements (named “genome vehicle”) responsible for the neutrophil attractor. Notably, within the genome vehicles, genes with low or moderate expression changes, which are often considered noisy and insignificant, are essential components for the creation of the neutrophil attractor. Further investigations along with our findings might provide a comprehensive mechanistic view of cell fate decision.
PMCID: PMC2920325  PMID: 20725638
3.  Floral Morphogenesis: Stochastic Explorations of a Gene Network Epigenetic Landscape 
PLoS ONE  2008;3(11):e3626.
In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5–10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.
PMCID: PMC2572848  PMID: 18978941
4.  Asymmetry in Erythroid-Myeloid Differentiation Switch and the Role of Timing in a Binary Cell-Fate Decision 
GATA1-PU.1 genetic switch is a paradigmatic genetic switch that governs the differentiation of progenitor cells into two different fates, erythroid and myeloid fates. In terms of dynamical model representation of these fates or lineages corresponds to stable attractor and choosing between the attractors. Small asymmetries and stochasticity intrinsically present in all genetic switches lead to the effect of delayed bifurcation which will change the differentiation result according to the timing of the process and affect the proportion of erythroid versus myeloid cells. We consider the differentiation bifurcation scenario in which there is a symmetry-breaking in the bifurcation diagrams as a result of asymmetry in external signaling. We show that the decision between two alternative cell fates in this structurally symmetric decision circuit can be biased depending on the speed at which the system is forced to go through the decision point. The parameter sweeping speed can also reduce the effect of asymmetry and produce symmetric choice between attractors, or convert the favorable attractor. This conversion may have important contributions to the immune system when the bias is in favor of the attractor which gives rise to non-immune cells.
PMCID: PMC3851994  PMID: 24367366
GATA1-PU.1 switch; differentiation; immune cells; pluripotent cells
5.  An incoherent regulatory network architecture that orchestrates B cell diversification in response to antigen signaling 
B cell receptor signaling controls the expression of IRF-4, a transcription factor required for B cell differentiation. This study shows that IRF-4 regulates divergent B cell fates via a ‘kinetic-control' mechanism that determines the duration of a transient developmental state.
The intensity of signaling through the B cell receptor controls the level of expression of IRF-4, a transcription factor required for B cell differentiation. The rate of IRF-4 production dictates the extent of antibody gene diversification that B cells undergo upon antigen encounter before differentiating into antibody-secreting plasma cells.Computational modeling and experimental analyses substantiate a model, whereby IRF-4 regulates B cell fate trajectories via a ‘kinetic-control' mechanism.Kinetic control is a process by which B cells pass through an obligate state of variable duration that sets the degree of cellular diversification prior to their terminal differentiation.An incoherent regulatory network architecture, within which IRF-4 is embedded, is the basis for realization of kinetic control.
The generation of a diverse set of pathogen-specific antibodies, with differing affinities and effector functions, by B lymphocytes is essential for efficient protection from many microorganisms. Antibody gene diversification in B cells is mediated by two molecular processes termed class-switch recombination and somatic hypermutation (CSR/SHM) (F1A). The former enables the generation of antibodies with the same antigen-binding specificity, but different effector domains, whereas the latter results in a repertoire of antibodies with a range of affinities for a given antigen containing the same effector domain. CSR/SHM occurs in antigen-activated B cells before their terminal differentiation into plasma cells. The transcription factor IRF-4 is required for CSR/SHM as well as plasma-cell differentiation, with its highest levels of expression being necessary for the latter. IRF-4 acts in the context of a network of regulators that include Blimp-1, Pax5, Bach2 and Bcl-6 (F1B). Despite extensive characterization of these individual factors, how the network responds to sensing of antigen by the B cell antigen receptor (BCR, antibody molecule expressed on cell surface) to regulate the extent of antibody gene diversification and plasma-cell differentiation remains to be addressed.
To address this issue, we assemble a computational model. The model reveals two contrasting scenarios that can underlie B cell fate dynamics. In one case, the initial rate of IRF-4 production controls a binary cell fate choice that involves either going to the CSR/SHM state or to the plasma-cell state; the time spent in the CSR state is relatively insensitive to the initial rate of IRF-4 production (herein called ‘basic bistability'). In the other case, IRF-4 drives all cells through a transient CSR/SHM state, but the initial rate of IRF-4 production sets its duration (‘kinetic control'). Both scenarios predict that increasing the initial rate of IRF-4 production favors the generation of plasma cells at the expense of CSR/SHM, but they differ fundamentally with respect to the underlying gene expression patterns.
To distinguish between these two scenarios experimentally, we utilize two different genetic models. The first involves the B1-8i transgenic mouse whose B cells express a rearranged V187.2 VDJ Ig heavy chain gene segment that is specific for the hapten nitrophenol (NP). The second is a newly developed mouse model that allows exogenous control of IRF-4 expression in naive primary B cells using a tet-inducible allele. Using these models, we show that (i) BCR signal strength sets the initial rate of IRF-4 accumulation and (ii) the concentration of IRF-4 is sensed by an incoherent gene regulatory network architecture to regulate the extent of CSR/SHM prior to plasma-cell differentiation. Our results are consistent with the ‘kinetic-control model' in which the levels of BCR-induced IRF-4 expression control the duration of an obligate CSR/SHM state that enables B cell diversification before terminal differentiation into plasma cells. Evidence for the transient CSR/SHM state is corroborated by both patterns of gene expression and the presence of AID-dependent mutations in individual non-switched plasmablasts.
Our results provide a molecular framework for understanding how B cells balance the competing demands for Ig CSR and SHM with the secretion of antibodies during humoral immune responses. The key feature of the network architecture that allows IRF-4 to coordinate the two competing states of gene expression in a temporal manner is that it simultaneously but asymmetrically activates both sides of a bistable mutual repression circuit. Because the two effects of the primary regulator antagonize each other, we describe the circuit as being based on an ‘incoherent' regulatory motif. Other incoherent regulatory motifs in varied biological systems are also associated with the acquisition of transient cell states, and we consider how the kinetic-control mechanism proposed by us could more generally serve to translate developmental cues into elaborate morphogenetic patterns.
The B-lymphocyte lineage is a leading system for analyzing gene regulatory networks (GRNs) that orchestrate distinct cell fate transitions. Upon antigen recognition, B cells can diversify their immunoglobulin (Ig) repertoire via somatic hypermutation (SHM) and/or class switch DNA recombination (CSR) before differentiating into antibody-secreting plasma cells. We construct a mathematical model for a GRN underlying this developmental dynamic. The intensity of signaling through the Ig receptor is shown to control the bimodal expression of a pivotal transcription factor, IRF-4, which dictates B cell fate outcomes. Computational modeling coupled with experimental analysis supports a model of ‘kinetic control', in which B cell developmental trajectories pass through an obligate transient state of variable duration that promotes diversification of the antibody repertoire by SHM/CSR in direct response to antigens. More generally, this network motif could be used to translate a morphogen gradient into developmental inductive events of varying time, thereby enabling the specification of distinct cell fates.
PMCID: PMC3130558  PMID: 21613984
BCR signal strength; bistability; gene regulatory network; ghost of a fixed point; Irf4
6.  Quasi-potential landscape in complex multi-stable systems 
The developmental dynamics of multicellular organisms is a process that takes place in a multi-stable system in which each attractor state represents a cell type, and attractor transitions correspond to cell differentiation paths. This new understanding has revived the idea of a quasi-potential landscape, first proposed by Waddington as a metaphor. To describe development, one is interested in the ‘relative stabilities’ of N attractors (N > 2). Existing theories of state transition between local minima on some potential landscape deal with the exit part in the transition between two attractors in pair-attractor systems but do not offer the notion of a global potential function that relates more than two attractors to each other. Several ad hoc methods have been used in systems biology to compute a landscape in non-gradient systems, such as gene regulatory networks. Here we present an overview of currently available methods, discuss their limitations and propose a new decomposition of vector fields that permits the computation of a quasi-potential function that is equivalent to the Freidlin–Wentzell potential but is not limited to two attractors. Several examples of decomposition are given, and the significance of such a quasi-potential function is discussed.
PMCID: PMC3481575  PMID: 22933187
multi-stable dynamical system; non-equilibrium dynamics; quasi-potential; state transition; epigenetic landscape; Freidlin–Wentzell theory
7.  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
8.  Noise-Driven Stem Cell and Progenitor Population Dynamics 
PLoS ONE  2008;3(8):e2922.
The balance between maintenance of the stem cell state and terminal differentiation is influenced by the cellular environment. The switching between these states has long been understood as a transition between attractor states of a molecular network. Herein, stochastic fluctuations are either suppressed or can trigger the transition, but they do not actually determine the attractor states.
Methodology/Principal Findings
We present a novel mathematical concept in which stem cell and progenitor population dynamics are described as a probabilistic process that arises from cell proliferation and small fluctuations in the state of differentiation. These state fluctuations reflect random transitions between different activation patterns of the underlying regulatory network. Importantly, the associated noise amplitudes are state-dependent and set by the environment. Their variability determines the attractor states, and thus actually governs population dynamics. This model quantitatively reproduces the observed dynamics of differentiation and dedifferentiation in promyelocytic precursor cells.
Consequently, state-specific noise modulation by external signals can be instrumental in controlling stem cell and progenitor population dynamics. We propose follow-up experiments for quantifying the imprinting influence of the environment on cellular noise regulation.
PMCID: PMC2488392  PMID: 18698344
9.  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
10.  Neural Network Mechanisms Underlying Stimulus Driven Variability Reduction 
PLoS Computational Biology  2012;8(3):e1002395.
It is well established that the variability of the neural activity across trials, as measured by the Fano factor, is elevated. This fact poses limits on information encoding by the neural activity. However, a series of recent neurophysiological experiments have changed this traditional view. Single cell recordings across a variety of species, brain areas, brain states and stimulus conditions demonstrate a remarkable reduction of the neural variability when an external stimulation is applied and when attention is allocated towards a stimulus within a neuron's receptive field, suggesting an enhancement of information encoding. Using an heterogeneously connected neural network model whose dynamics exhibits multiple attractors, we demonstrate here how this variability reduction can arise from a network effect. In the spontaneous state, we show that the high degree of neural variability is mainly due to fluctuation-driven excursions from attractor to attractor. This occurs when, in the parameter space, the network working point is around the bifurcation allowing multistable attractors. The application of an external excitatory drive by stimulation or attention stabilizes one specific attractor, eliminating in this way the transitions between the different attractors and resulting in a net decrease in neural variability over trials. Importantly, non-responsive neurons also exhibit a reduction of variability. Finally, this reduced variability is found to arise from an increased regularity of the neural spike trains. In conclusion, these results suggest that the variability reduction under stimulation and attention is a property of neural circuits.
Author Summary
To understand how neurons encode information, neuroscientists record their firing activity while the animal executes a given task for many trials. Surprisingly, it has been found that the neural response is highly variable, which a priori limits the encoding of information by these neurons. However, recent experiments have shown that this variability is reduced when the animal receives a stimulus or attends to a particular one, suggesting an enhancement of information encoding. It is known that a cause of neural variability resides in the fact that individual neurons receive an input which fluctuates around their firing threshold. We demonstrate here that all the experimental results can naturally arise from the dynamics of a neural network. Using a realistic model, we show that the neural variability during spontaneous activity is particularly high because input noise induces large fluctuations between multiple –but unstable- network states. With stimulation or attention, one particular network state is stabilized and fluctuations decrease, leading to a neural variability reduction. In conclusion, our results suggest that the observed variability reduction is a property of the neural circuits of the brain.
PMCID: PMC3315452  PMID: 22479168
11.  Representational Switching by Dynamical Reorganization of Attractor Structure in a Network Model of the Prefrontal Cortex 
PLoS Computational Biology  2011;7(11):e1002266.
The prefrontal cortex (PFC) plays a crucial role in flexible cognitive behavior by representing task relevant information with its working memory. The working memory with sustained neural activity is described as a neural dynamical system composed of multiple attractors, each attractor of which corresponds to an active state of a cell assembly, representing a fragment of information. Recent studies have revealed that the PFC not only represents multiple sets of information but also switches multiple representations and transforms a set of information to another set depending on a given task context. This representational switching between different sets of information is possibly generated endogenously by flexible network dynamics but details of underlying mechanisms are unclear. Here we propose a dynamically reorganizable attractor network model based on certain internal changes in synaptic connectivity, or short-term plasticity. We construct a network model based on a spiking neuron model with dynamical synapses, which can qualitatively reproduce experimentally demonstrated representational switching in the PFC when a monkey was performing a goal-oriented action-planning task. The model holds multiple sets of information that are required for action planning before and after representational switching by reconfiguration of functional cell assemblies. Furthermore, we analyzed population dynamics of this model with a mean field model and show that the changes in cell assemblies' configuration correspond to those in attractor structure that can be viewed as a bifurcation process of the dynamical system. This dynamical reorganization of a neural network could be a key to uncovering the mechanism of flexible information processing in the PFC.
Author Summary
The prefrontal cortex plays a highly flexible role in various cognitive tasks e.g., decision making and action planning. Neurons in the prefrontal cortex exhibit flexible representation or selectivity for task relevant information and are involved in working memory with sustained activity, which can be modeled as attractor dynamics. Moreover, recent experiments revealed that prefrontal neurons not only represent parametric or discrete sets of information but also switch the representation and transform a set of information to another set in order to match the context of the required task. However, underlying mechanisms of this flexible representational switching are unknown. Here we propose a dynamically reorganizable attractor network model in which short-term modulation of the synaptic connections reconfigures the structure of neural attractors by assembly and disassembly of a network of cells to produce flexible attractor dynamics. On the basis of computer simulation as well as theoretical analysis, we showed that this model reproduced experimentally demonstrated representational switching, and that switching on certain characteristic axes defining neural dynamics well describes the essence of the representational switching. This model has the potential to provide unique insights about the flexible information representations and processing in the cortical network.
PMCID: PMC3213170  PMID: 22102803
12.  Predicting Pancreas Cell Fate Decisions and Reprogramming with a Hierarchical Multi-Attractor Model 
PLoS ONE  2011;6(3):e14752.
Cell fate reprogramming, such as the generation of insulin-producing β cells from other pancreas cells, can be achieved by external modulation of key transcription factors. However, the known gene regulatory interactions that form a complex network with multiple feedback loops make it increasingly difficult to design the cell reprogramming scheme because the linear regulatory pathways as schemes of causal influences upon cell lineages are inadequate for predicting the effect of transcriptional perturbation. However, sufficient information on regulatory networks is usually not available for detailed formal models. Here we demonstrate that by using the qualitatively described regulatory interactions as the basis for a coarse-grained dynamical ODE (ordinary differential equation) based model, it is possible to recapitulate the observed attractors of the exocrine and β, δ, α endocrine cells and to predict which gene perturbation can result in desired lineage reprogramming. Our model indicates that the constraints imposed by the incompletely elucidated regulatory network architecture suffice to build a predictive model for making informed decisions in choosing the set of transcription factors that need to be modulated for fate reprogramming.
PMCID: PMC3056652  PMID: 21423725
13.  Hematopoietic differentiation: a coordinated dynamical process towards attractor stable states 
BMC Systems Biology  2010;4:85.
The differentiation process, proceeding from stem cells towards the different committed cell types, can be considered as a trajectory towards an attractor of a dynamical process. This view, taking into consideration the transcriptome and miRNome dynamics considered as a whole, instead of looking at few 'master genes' driving the system, offers a novel perspective on this phenomenon. We investigated the 'differentiation trajectories' of the hematopoietic system considering a genome-wide scenario.
We developed serum-free liquid suspension unilineage cultures of cord blood (CB) CD34+ hematopoietic progenitor cells through erythroid (E), megakaryocytic (MK), granulocytic (G) and monocytic (Mo) pathways. These cultures recapitulate physiological hematopoiesis, allowing the analysis of almost pure unilineage precursors starting from initial differentiation of HPCs until terminal maturation. By analyzing the expression profile of protein coding genes and microRNAs in unilineage CB E, MK, G and Mo cultures, at sequential stages of differentiation and maturation, we observed a coordinated, fully interconnected and scalable character of cell population behaviour in both transcriptome and miRNome spaces reminiscent of an attractor-like dynamics. MiRNome and transcriptome space differed for a still not terminally committed behaviour of microRNAs.
Consistent with their roles, the transcriptome system can be considered as the state space of a cell population, while the continuously evolving miRNA space corresponds to the tuning system necessary to reach the attractor. The behaviour of miRNA machinery could be of great relevance not only for the promise of reversing the differentiated state but even for tumor biology.
PMCID: PMC2904736  PMID: 20553595
14.  Place Cell Rate Remapping by CA3 Recurrent Collaterals 
PLoS Computational Biology  2014;10(6):e1003648.
Episodic-like memory is thought to be supported by attractor dynamics in the hippocampus. A possible neural substrate for this memory mechanism is rate remapping, in which the spatial map of place cells encodes contextual information through firing rate variability. To test whether memories are stored as multimodal attractors in populations of place cells, recent experiments morphed one familiar context into another while observing the responses of CA3 cell ensembles. Average population activity in CA3 was reported to transition gradually rather than abruptly from one familiar context to the next, suggesting a lack of attractive forces associated with the two stored representations. On the other hand, individual CA3 cells showed a mix of gradual and abrupt transitions at different points along the morph sequence, and some displayed hysteresis which is a signature of attractor dynamics. To understand whether these seemingly conflicting results are commensurate with attractor network theory, we developed a neural network model of the CA3 with attractors for both position and discrete contexts. We found that for memories stored in overlapping neural ensembles within a single spatial map, position-dependent context attractors made transitions at different points along the morph sequence. Smooth transition curves arose from averaging across the population, while a heterogeneous set of responses was observed on the single unit level. In contrast, orthogonal memories led to abrupt and coherent transitions on both population and single unit levels as experimentally observed when remapping between two independent spatial maps. Strong recurrent feedback entailed a hysteretic effect on the network which diminished with the amount of overlap in the stored memories. These results suggest that context-dependent memory can be supported by overlapping local attractors within a spatial map of CA3 place cells. Similar mechanisms for context-dependent memory may also be found in other regions of the cerebral cortex.
Author Summary
The activity of ‘place cells’ in hippocampal area CA3 systematically changes as a function of the animal's position in an arena as well as contextual variables like the color or shape of enclosing walls. Large changes to the local environment, e.g. moving the animal to a different room, can induce a complete reorganization of place-cell firing locations. Such ‘global remapping’ reveals that memory for different environments is encoded as separate spatial maps. Smaller changes to features within an environment can induce a modulation of place cell firing rates without affecting their firing locations. This kind of ‘rate remapping’ is still poorly understood. In this paper we describe a computational model in which discrete memories for contextual features were stored locally within a spatial map of place cells. This network structure supports retrieval of both positional and contextual information from an arbitrary cue, as required by an episodic memory structure. The activity of the network qualitatively matches empirical data from rate remapping experiments, both on the population level and the level of single place cells. The results support the idea that CA3 rate remapping reflects content-addressable memories stored as multimodal attractor states in the hippocampus.
PMCID: PMC4046921  PMID: 24902003
15.  Boolean genetic network model for the control of C. elegans early embryonic cell cycles 
BioMedical Engineering OnLine  2013;12(Suppl 1):S1.
In Caenorhabditis elegans early embryo, cell cycles only have two phases: DNA synthesis and mitosis, which are different from the typical 4-phase cell cycle. Modeling this cell-cycle process into network can fill up the gap in C. elegans cell-cycle study and provide a thorough understanding on the cell-cycle regulations and progressions at the network level.
In this paper, C. elegans early embryonic cell-cycle network has been constructed based on the knowledge of key regulators and their interactions from literature studies. A discrete dynamical Boolean model has been applied in computer simulations to study dynamical properties of this network. The cell-cycle network is compared with random networks and tested under several perturbations to analyze its robustness. To investigate whether our proposed network could explain biological experiment results, we have also compared the network simulation results with gene knock down experiment data.
With the Boolean model, this study showed that the cell-cycle network was stable with a set of attractors (fixed points). A biological pathway was observed in the simulation, which corresponded to a whole cell-cycle progression. The C. elegans network was significantly robust when compared with random networks of the same size because there were less attractors and larger basins than random networks. Moreover, the network was also robust under perturbations with no significant change of the basin size. In addition, the smaller number of attractors and the shorter biological pathway from gene knock down network simulation interpreted the shorter cell-cycle lengths in mutant from the RNAi gene knock down experiment data. Hence, we demonstrated that the results in network simulation could be verified by the RNAi gene knock down experiment data.
A C. elegans early embryonic cell cycles network was constructed and its properties were analyzed and compared with those of random networks. Computer simulation results provided biologically meaningful interpretations of RNAi gene knock down experiment data.
PMCID: PMC4029147  PMID: 24564942
16.  Detecting cellular reprogramming determinants by differential stability analysis of gene regulatory networks 
BMC Systems Biology  2013;7:140.
Cellular differentiation and reprogramming are processes that are carefully orchestrated by the activation and repression of specific sets of genes. An increasing amount of experimental results show that despite the large number of genes participating in transcriptional programs of cellular phenotypes, only few key genes, which are coined here as reprogramming determinants, are required to be directly perturbed in order to induce cellular reprogramming. However, identification of reprogramming determinants still remains a combinatorial problem, and the state-of-art methods addressing this issue rests on exhaustive experimentation or prior knowledge to narrow down the list of candidates.
Here we present a computational method, without any preliminary selection of candidate genes, to identify reduced subsets of genes, which when perturbed can induce transitions between cellular phenotypes. The method relies on the expression profiles of two stable cellular phenotypes along with a topological analysis stability elements in the gene regulatory network that are necessary to cause this multi-stability. Since stable cellular phenotypes can be considered as attractors of gene regulatory networks, cell fate and cellular reprogramming involves transition between these attractors, and therefore current method searches for combinations of genes that are able to destabilize a specific initial attractor and stabilize the final one in response to the appropriate perturbations.
The method presented here represents a useful framework to assist researchers in the field of cellular reprogramming to design experimental strategies with potential applications in the regenerative medicine and disease modelling.
PMCID: PMC3878265  PMID: 24350678
Cellular reprogramming; Transdifferentiation; Dedifferentiation; Stability; Attractor; Positive circuit; Reprogramming determinants
17.  Model Checking to Assess T-Helper Cell Plasticity 
Computational modeling constitutes a crucial step toward the functional understanding of complex cellular networks. In particular, logical modeling has proven suitable for the dynamical analysis of large signaling and transcriptional regulatory networks. In this context, signaling input components are generally meant to convey external stimuli, or environmental cues. In response to such external signals, cells acquire specific gene expression patterns modeled in terms of attractors (e.g., stable states). The capacity for cells to alter or reprogram their differentiated states upon changes in environmental conditions is referred to as cell plasticity. In this article, we present a multivalued logical framework along with computational methods recently developed to efficiently analyze large models. We mainly focus on a symbolic model checking approach to investigate switches between attractors subsequent to changes of input conditions. As a case study, we consider the cellular network regulating the differentiation of T-helper (Th) cells, which orchestrate many physiological and pathological immune responses. To account for novel cellular subtypes, we present an extended version of a published model of Th cell differentiation. We then use symbolic model checking to analyze reachability properties between Th subtypes upon changes of environmental cues. This allows for the construction of a synthetic view of Th cell plasticity in terms of a graph connecting subtypes with arcs labeled by input conditions. Finally, we explore novel strategies enabling specific Th cell polarizing or reprograming events.
PMCID: PMC4309205
logical modeling; signaling networks; T-helper lymphocyte; cell differentiation; cell plasticity; model checking
18.  Systems biology of stem cell fate and cellular reprogramming 
Stem cell differentiation and the maintenance of self-renewal are intrinsically complex processes requiring the coordinated dynamic expression of hundreds of genes and proteins in precise response to external signalling cues. Numerous recent reports have used both experimental and computational techniques to dissect this complexity. These reports suggest that the control of cell fate has both deterministic and stochastic elements: complex underlying regulatory networks define stable molecular ‘attractor’ states towards which individual cells are drawn over time, whereas stochastic fluctuations in gene and protein expression levels drive transitions between coexisting attractors, ensuring robustness at the population level.
PMCID: PMC2928569  PMID: 19738627
19.  Epigenetic Landscapes Explain Partially Reprogrammed Cells and Identify Key Reprogramming Genes 
PLoS Computational Biology  2014;10(8):e1003734.
A common metaphor for describing development is a rugged “epigenetic landscape” where cell fates are represented as attracting valleys resulting from a complex regulatory network. Here, we introduce a framework for explicitly constructing epigenetic landscapes that combines genomic data with techniques from spin-glass physics. Each cell fate is a dynamic attractor, yet cells can change fate in response to external signals. Our model suggests that partially reprogrammed cells are a natural consequence of high-dimensional landscapes, and predicts that partially reprogrammed cells should be hybrids that co-express genes from multiple cell fates. We verify this prediction by reanalyzing existing datasets. Our model reproduces known reprogramming protocols and identifies candidate transcription factors for reprogramming to novel cell fates, suggesting epigenetic landscapes are a powerful paradigm for understanding cellular identity.
Author Summary
Traditionally, standard development has been viewed as a one-way process; an organism starts as a single cell (embryonic stem cell, ESC) that divides into a multitude of mature cell types (skin cells, heart, liver, etc). But, in 2006 Takahashi and Yamanaka revolutionized this view by stochastically converting skin cells into cell types resembling ESC (called induced pluripotent stem cells, iPSC). Following this groundbreaking experiment, other reprogramming protocols have been found, so now scientists can switch between a variety of cell types such as ESC, skin, liver, neurons, and heart. This has already revolutionized the understanding of biology and could change the future of medicine. A common metaphor for development is Waddington's landscape, in which an ESC is like a ball rolling down a hill which eventually ends in a valley (mature cell type). In this paper, we make this analogy more precise by developing a mathematical model of cellular development. Using data on real cell types, we can provide insight into existing reprogramming protocols and potentially predict new reprogramming protocols.
PMCID: PMC4133049  PMID: 25122086
20.  Changes of Mind in an Attractor Network of Decision-Making 
PLoS Computational Biology  2011;7(6):e1002086.
Attractor networks successfully account for psychophysical and neurophysiological data in various decision-making tasks. Especially their ability to model persistent activity, a property of many neurons involved in decision-making, distinguishes them from other approaches. Stable decision attractors are, however, counterintuitive to changes of mind. Here we demonstrate that a biophysically-realistic attractor network with spiking neurons, in its itinerant transients towards the choice attractors, can replicate changes of mind observed recently during a two-alternative random-dot motion (RDM) task. Based on the assumption that the brain continues to evaluate available evidence after the initiation of a decision, the network predicts neural activity during changes of mind and accurately simulates reaction times, performance and percentage of changes dependent on difficulty. Moreover, the model suggests a low decision threshold and high incoming activity that drives the brain region involved in the decision-making process into a dynamical regime close to a bifurcation, which up to now lacked evidence for physiological relevance. Thereby, we further affirmed the general conformance of attractor networks with higher level neural processes and offer experimental predictions to distinguish nonlinear attractor from linear diffusion models.
Author Summary
A recent psychophysical experiment showed that participants do adjust their decisions (change their mind) based on further evidence, which was processed only after the first decision was made. The established notion of (perceptual) decision-making as a decision variable evolving in time until a termination criterion is reached does not incorporate these changes of mind. In the biophysically-realistic attractor model, the mean firing rates of neural populations encoding the decision alternatives act as the decision variable. In line with neurophysiological evidence from decision-related neurons in the lateral intraparietal cortex, a decision is made if a fixed firing rate threshold is crossed. We propose here that a change of mind is induced if this decision threshold is crossed a second time, namely by the neural population encoding the initially losing alternative, which thus overtakes the population that first crossed the decision threshold. Interestingly, we found this more likely to happen the further the system is pushed towards a regime where decision-making is no longer unambiguous, but both neural populations can fire at elevated rates. This, besides, corresponds to higher incoming activity and thus faster and less accurate decisions and suggests that the brain operates over the whole range of inputs enabling decision-making.
PMCID: PMC3121686  PMID: 21731482
21.  Cancer attractors: A systems view of tumors from a gene network dynamics and developmental perspective 
Cell lineage commitment and differentiation are governed by a complex gene regulatory network. Disruption of these processes by inappropriate regulatory signals and by mutational rewiring of the network can lead to tumorigenesis. Cancer cells often exhibit immature or embryonic traits and dysregulated developmental genes can act as oncogenes. However, the prevailing paradigm of somatic evolution and multi-step tumorigenesis, while useful in many instances, offers no logically coherent reason for why oncogenesis recapitulates ontogenesis. The formal concept of “cancer attractors”, derived from an integrative, complex systems approach to gene regulatory network may provide a natural explanation. Here we present the theory of attractors in gene network dynamics and review the concept of cell types as attractors. We argue that cancer cells are trapped in abnormal attractors and discuss this concept in the light of recent ideas in cancer biology, including cancer genomics and cancer stem cells, as well as the implications for differentiation therapy.
PMCID: PMC2754594  PMID: 19595782
22.  Combining Network Modeling and Gene Expression Microarray Analysis to Explore the Dynamics of Th1 and Th2 Cell Regulation 
PLoS Computational Biology  2010;6(12):e1001032.
Two T helper (Th) cell subsets, namely Th1 and Th2 cells, play an important role in inflammatory diseases. The two subsets are thought to counter-regulate each other, and alterations in their balance result in different diseases. This paradigm has been challenged by recent clinical and experimental data. Because of the large number of genes involved in regulating Th1 and Th2 cells, assessment of this paradigm by modeling or experiments is difficult. Novel algorithms based on formal methods now permit the analysis of large gene regulatory networks. By combining these algorithms with in silico knockouts and gene expression microarray data from human T cells, we examined if the results were compatible with a counter-regulatory role of Th1 and Th2 cells. We constructed a directed network model of genes regulating Th1 and Th2 cells through text mining and manual curation. We identified four attractors in the network, three of which included genes that corresponded to Th0, Th1 and Th2 cells. The fourth attractor contained a mixture of Th1 and Th2 genes. We found that neither in silico knockouts of the Th1 and Th2 attractor genes nor gene expression microarray data from patients with immunological disorders and healthy subjects supported a counter-regulatory role of Th1 and Th2 cells. By combining network modeling with transcriptomic data analysis and in silico knockouts, we have devised a practical way to help unravel complex regulatory network topology and to increase our understanding of how network actions may differ in health and disease.
Author Summary
Different T helper (Th) cell subsets have an important role in regulating the immune response in inflammatory diseases. Th1 and Th2 cells are thought to counter-regulate each other, and alterations in their balance result in different diseases.This paradigm has been challenged by recent clinical and experimental data. Because of the large number of genes involved in regulating Th1 and Th2 cells, assessment of this paradigm by experiments or modelling is difficult. In this study, we combined novel algorithms for network analysis, in silico knockouts, and gene expression microarrays to examine if Th1 and Th2 cells had counter-regulatory roles. We constructed a directed network model of genes that regulated Th1 and Th2 cells through text mining and manual curation. We identified four cycles in the gene expression dynamics, three of which expressed genes that corresponded to Th0 (Th1/Th2 precursor), Th1 and Th2 cells. The fourth cycle contained the expression of a mixture of Th1 and Th2 genes. We found that neither in silico knockouts of the Th1 and Th2 attractor genes nor gene expression microarray data from patients and healthy subjects supported a counter-regulatory role of Th1 and Th2 cells.
PMCID: PMC3002992  PMID: 21187905
23.  Dynamical Modeling of the Cell Cycle and Cell Fate Emergence in Caulobacter crescentus 
PLoS ONE  2014;9(11):e111116.
The division of Caulobacter crescentus, a model organism for studying cell cycle and differentiation in bacteria, generates two cell types: swarmer and stalked. To complete its cycle, C. crescentus must first differentiate from the swarmer to the stalked phenotype. An important regulator involved in this process is CtrA, which operates in a gene regulatory network and coordinates many of the interactions associated to the generation of cellular asymmetry. Gaining insight into how such a differentiation phenomenon arises and how network components interact to bring about cellular behavior and function demands mathematical models and simulations. In this work, we present a dynamical model based on a generalization of the Boolean abstraction of gene expression for a minimal network controlling the cell cycle and asymmetric cell division in C. crescentus. This network was constructed from data obtained from an exhaustive search in the literature. The results of the simulations based on our model show a cyclic attractor whose configurations can be made to correspond with the current knowledge of the activity of the regulators participating in the gene network during the cell cycle. Additionally, we found two point attractors that can be interpreted in terms of the network configurations directing the two cell types. The entire network is shown to be operating close to the critical regime, which means that it is robust enough to perturbations on dynamics of the network, but adaptable to environmental changes.
PMCID: PMC4219702  PMID: 25369202
24.  A Systems Approach Reveals Regulatory Circuitry for Arabidopsis Trichome Initiation by the GL3 and GL1 Selectors 
PLoS Genetics  2009;5(2):e1000396.
Position-dependent cell fate determination and pattern formation are unique aspects of the development of plant structures. The establishment of single-celled leaf hairs (trichomes) from pluripotent epidermal (protodermal) cells in Arabidopsis provides a powerful system to determine the gene regulatory networks involved in cell fate determination. To obtain a holistic view of the regulatory events associated with the differentiation of Arabidopsis epidermal cells into trichomes, we combined expression and genome-wide location analyses (ChIP-chip) on the trichome developmental selectors GLABRA3 (GL3) and GLABRA1 (GL1), encoding basic helix-loop-helix (bHLH) and MYB transcription factors, respectively. Meta-analysis was used to integrate genome-wide expression results contrasting wild type and gl3 or gl1 mutants with changes in gene expression over time using inducible versions of GL3 and GL1. This resulted in the identification of a minimal set of genes associated with the differentiation of epidermal cells into trichomes. ChIP-chip experiments, complemented by the targeted examination of factors known to participate in trichome initiation or patterning, identified about 20 novel GL3/GL1 direct targets. In addition to genes involved in the control of gene expression, such as the transcription factors SCL8 and MYC1, we identified SIM (SIAMESE), encoding a cyclin-dependent kinase inhibitor, and RBR1 (RETINOBLASTOMA RELATED1), corresponding to a negative regulator of the cell cycle transcription factor E2F, as GL3/GL1 immediate targets, directly implicating these trichome regulators in the control of the endocycle. The expression of many of the identified GL3/GL1 direct targets was specific to very early stages of trichome initiation, suggesting that they participate in some of the earliest known processes associated with protodermal cell differentiation. By combining this knowledge with the analysis of genes associated with trichome formation, our results reveal the architecture of the top tiers of the hierarchical structure of the regulatory network involved in epidermal cell differentiation and trichome formation.
Author Summary
The establishment of single-celled leaf hairs (trichomes) from pluripotent epidermal (protodermal) cells provides a powerful system to determine the gene regulatory networks involved in plant cell fate determination. Two transcription factors—GL1 and GL3—have been associated with the initiation of trichome formation; yet only a handful of GL1-GL3–regulated genes have previously been characterized. In this study, we combined expression analyses performed in a number of different genotypes to identify a minimal set of about 500 genes associated with trichome formation. We also used ChIP-chip to identify a set of about 20 genes that are immediate targets of GL3 and GL1. Many more genes are targeted by GL1 or by GL3, likely in cooperation with other bHLH of MYB partners, but not by both GL1 and GL3. As predicted for genes involved in the initiation of epidermal cell fate determination, several of the GL3/GL1 direct targets are expressed early during trichome formation, including the transcription factors MYC1 (bHLH), SCL8 (GRAS), and genes involved in the regulation of the endocycle (SIM and RBR1). Co-expression analyses permitted us to identify sets of target genes likely downstream of the GL3/GL1 regulated transcription factors, providing the first steps towards building the regulatory network associated with the differentiation of protodermal cells into trichomes.
PMCID: PMC2642726  PMID: 19247443
25.  Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation 
Cellular differentiation, reprogramming and transdifferentiation are determined by underlying gene regulatory networks. Non-adiabatic regulation via slow binding/unbinding to the gene can be important in these cell fate decision-making processes. Based on a stem cell core gene network, we uncovered the stem cell developmental landscape. As the binding/unbinding speed decreases, the landscape topography changes from bistable attractors of stem and differentiated states to more attractors of stem and other different cell states as well as substates. Non-adiabaticity leads to more differentiated cell types and provides a natural explanation for the heterogeneity observed in the experiments. We quantified Waddington landscapes with two possible cell fate decision mechanisms by changing the regulation strength or regulation timescale (non-adiabaticity). Transition rates correlate with landscape topography through barrier heights between different states and quantitatively determine global stability. We found the optimal speeds of these cell fate decision-making processes. We quantified biological paths and predict that differentiation and reprogramming go through an intermediate state (IM1), whereas transdifferentiation goes through another intermediate state (IM2). Some predictions are confirmed by recent experimental studies.
PMCID: PMC3808556  PMID: 24132204
potential landscape; dynamical path; differentiation and reprogramming; non-adiabatic

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