Why a simple model of genetic regulatory networks describes the distribution of avalanches in gene expression data

Evolution of hierarchy and irreversibility in theoretical cell differentiation model

The process of cell differentiation in multicellular organisms is characterized by hierarchy and irreversibility in many cases. However, the conditions and selection pressures that give rise to these characteristics remain poorly understood. By using a mathematical model, here we show that the network of differentiation potency (differentiation diagram) becomes necessarily hierarchical and irreversible by increasing the number of terminally differentiated states under certain conditions. The mechanisms generating these characteristics are clarified using geometry in the cell state space. The results demonstrate that the hierarchical organization and irreversibility can manifest independently of direct selection pressures associated with these characteristics, instead they appear to evolve as byproducts of selective forces favoring a diversity of differentiated cell types. The study also provides a new perspective on the structure of gene regulatory networks that produce hierarchical and irreversible differentiation diagrams. These results indicate some constraints on cell differentiation, which are expected to provide a starting point for theoretical discussion of the implicit limits and directions of evolution in multicellular organisms.

Potential long consequences from internal and external ecology: loss of gut microbiota antifragility in children from an industrialized population compared with an indigenous rural lifestyle

Human health is strongly mediated by the gut microbiota ecosystem, which, in turn, depends not only on its state but also on its dynamics and how it responds to perturbations. Healthy microbiota ecosystems tend to be in criticality and antifragile dynamics corresponding to a maximum complexity configuration, which may be assessed with information and network theory analysis. Under this complex system perspective, we used a new analysis of published data to show that a children's population with an industrialized urban lifestyle from Mexico City exhibits informational and network characteristics similar to parasitized children from a rural indigenous population in the remote mountainous region of Guerrero, México. We propose then, that in this critical age for gut microbiota maturation, the industrialized urban lifestyle could be thought of as an external perturbation to the gut microbiota ecosystem, and we show that it produces a similar loss in criticality/antifragility as the one observed by internal perturbation due to parasitosis by the helminth A. lumbricoides. Finally, several general complexity-based guidelines to prevent or restore gut ecosystem antifragility are discussed.

Emergent Criticality in Coupled Boolean Networks

Early embryonic development involves forming all specialized cells from a fluid-like mass of identical stem cells. The differentiation process consists of a series of symmetry-breaking events, starting from a high-symmetry state (stem cells) to a low-symmetry state (specialized cells). This scenario closely resembles phase transitions in statistical mechanics. To theoretically study this hypothesis, we model embryonic stem cell (ESC) populations through a coupled Boolean network (BN) model. The interaction is applied using a multilayer Ising model that considers paracrine and autocrine signaling, along with external interventions. It is demonstrated that cell-to-cell variability can be interpreted as a mixture of steady-state probability distributions. Simulations have revealed that such models can undergo a series of first- and second-order phase transitions as a function of the system parameters that describe gene expression noise and interaction strengths. These phase transitions result in spontaneous symmetry-breaking events that generate new types of cells characterized by various steady-state distributions. Coupled BNs have also been shown to self-organize in states that allow spontaneous cell differentiation.

On the Criticality of Adaptive Boolean Network Robots

Systems poised at a dynamical critical regime, between order and disorder, have been shown capable of exhibiting complex dynamics that balance robustness to external perturbations and rich repertoires of responses to inputs. This property has been exploited in artificial network classifiers, and preliminary results have also been attained in the context of robots controlled by Boolean networks. In this work, we investigate the role of dynamical criticality in robots undergoing online adaptation, i.e., robots that adapt some of their internal parameters to improve a performance metric over time during their activity. We study the behavior of robots controlled by random Boolean networks, which are either adapted in their coupling with robot sensors and actuators or in their structure or both. We observe that robots controlled by critical random Boolean networks have higher average and maximum performance than that of robots controlled by ordered and disordered nets. Notably, in general, adaptation by change of couplings produces robots with slightly higher performance than those adapted by changing their structure. Moreover, we observe that when adapted in their structure, ordered networks tend to move to the critical dynamical regime. These results provide further support to the conjecture that critical regimes favor adaptation and indicate the advantage of calibrating robot control systems at dynamical critical states.

Cancer and Chaos and the Complex Network Model of a Multicellular Organism

In the search of theoretical models describing cancer, one of promising directions is chaos. It is connected to ideas of "genome chaos" and "life on the edge of chaos", but they profoundly differ in the meaning of the term "chaos". To build any coherent models, notions used by both ideas should be firstly brought closer. The hypothesis "life on the edge of chaos" using deterministic chaos has been radically deepened developed in recent years by the discovery of half-chaos. This new view requires a deeper interpretation within the range of the cell and the organism. It has impacts on understanding "chaos" in the term "genome chaos". This study intends to present such an interpretation on the basis of which such searches will be easier and closer to intuition. We interpret genome chaos as deterministic chaos in a large module of half-chaotic network modeling the cell. We observed such chaotic modules in simulations of evolution controlled by weaker variant of natural selection. We also discuss differences between free and somatic cells in modeling their disturbance using half-chaotic networks.

Boolean modelling as a logic-based dynamic approach in systems medicine

Molecular mechanisms of health and disease are often represented as systems biology diagrams, and the coverage of such representation constantly increases. These static diagrams can be transformed into dynamic models, allowing for in silico simulations and predictions. Boolean modelling is an approach based on an abstract representation of the system. It emphasises the qualitative modelling of biological systems in which each biomolecule can take two possible values: zero for absent or inactive, one for present or active. Because of this approximation, Boolean modelling is applicable to large diagrams, allowing to capture their dynamic properties. We review Boolean models of disease mechanisms and compare a range of methods and tools used for analysis processes. We explain the methodology of Boolean analysis focusing on its application in disease modelling. Finally, we discuss its practical application in analysing signal transduction and gene regulatory pathways in health and disease.

Attractor-Specific and Common Expression Values in Random Boolean Network Models (with a Preliminary Look at Single-Cell Data)

Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the "common sea" (CS) as the set of nodes that take the same value in all the attractors of a given network realization, and the "specific part" (SP) as the set of all the other nodes, and we study their properties in different ensembles, generated with different parameter values. Both the CS and of the SP can be composed of one or more weakly connected components, which are emergent intermediate-level structures. We show that the study of these sets provides very important information about the behavior of the model. The distribution of distances between attractors is also examined. Moreover, we show how the notion of a "common sea" of genes can be used to analyze data from single-cell experiments.

Semi-Adaptive Evolution with Spontaneous Modularity of Half-Chaotic Randomly Growing Autonomous and Open Networks

Up until now, studies of Kauffman network stability have focused on the conditions resulting from the structure of the network. Negative feedbacks have been modeled as ice (nodes that do not change their state) in an ordered phase but this blocks the possibility of breaking out of the range of correct operation. This first, very simplified approximation leads to some incorrect conclusions, e.g., that life is on the edge of chaos. We develop a second approximation, which discovers half-chaos and shows its properties. In previous works, half-chaos has been confirmed in autonomous networks, but only using node function disturbance, which does not change the network structure. Now we examine half-chaos during network growth by adding and removing nodes as a disturbance in autonomous and open networks. In such evolutions controlled by a ‘small change’ of functioning after disturbance, the half-chaos is kept but spontaneous modularity emerges and blurs the picture. Half-chaos is a state to be expected in most of the real systems studied, therefore the determinants of the variability that maintains the half-chaos are particularly important in the application of complex network knowledge.

Effective connectivity determines the critical dynamics of biochemical networks

Living systems comprise interacting biochemical components in very large networks. Given their high connectivity, biochemical dynamics are surprisingly not chaotic but quite robust to perturbations-a feature C.H. Waddington named canalization. Because organisms are also flexible enough to evolve, they arguably operate in a critical dynamical regime between order and chaos. The established theory of criticality is based on networks of interacting automata where Boolean truth values model presence/absence of biochemical molecules. The dynamical regime is predicted using network connectivity and node bias (to be on/off) as tuning parameters. Revising this to account for canalization leads to a significant improvement in dynamical regime prediction. The revision is based on effective connectivity, a measure of dynamical redundancy that buffers automata response to some inputs. In both random and experimentally validated systems biology networks, reducing effective connectivity makes living systems operate in stable or critical regimes even though the structure of their biochemical interaction networks predicts them to be chaotic. This suggests that dynamical redundancy may be naturally selected to maintain living systems near critical dynamics, providing both robustness and evolvability. By identifying how dynamics propagates preferably via effective pathways, our approach helps to identify precise ways to design and control network models of biochemical regulation and signalling.

Engineering self-organized criticality in living cells

Complex dynamical fluctuations, from intracellular noise, brain dynamics or computer traffic display bursting dynamics consistent with a critical state between order and disorder. Living close to the critical point has adaptive advantages and it has been conjectured that evolution could select these critical states. Is this the case of living cells? A system can poise itself close to the critical point by means of the so-called self-organized criticality (SOC). In this paper we present an engineered gene network displaying SOC behaviour. This is achieved by exploiting the saturation of the proteolytic degradation machinery in E. coli cells by means of a negative feedback loop that reduces congestion. Our critical motif is built from a two-gene circuit, where SOC can be successfully implemented. The potential implications for both cellular dynamics and behaviour are discussed.

Dynamical properties and path dependence in a gene-network model of cell differentiation

In this work, we explore the properties of a control mechanism exerted on random Boolean networks that takes inspiration from the methylation mechanisms in cell differentiation and consists in progressively freezing (i.e. clamping to 0) some nodes of the network. We study the main dynamical properties of this mechanism both theoretically and in simulation. In particular, we show that when applied to random Boolean networks, it makes it possible to attain dynamics and path dependence typical of biological cells undergoing differentiation.

Statistical physics of liquid brains

Liquid neural networks (or 'liquid brains') are a widespread class of cognitive living networks characterized by a common feature: the agents (ants or immune cells, for example) move in space. Thus, no fixed, long-term agent-agent connections are maintained, in contrast with standard neural systems. How is this class of systems capable of displaying cognitive abilities, from learning to decision-making? In this paper, the collective dynamics, memory and learning properties of liquid brains is explored under the perspective of statistical physics. Using a comparative approach, we review the generic properties of three large classes of systems, namely: standard neural networks (solid brains), ant colonies and the immune system. It is shown that, despite their intrinsic physical differences, these systems share key properties with standard neural systems in terms of formal descriptions, but strongly depart in other ways. On one hand, the attractors found in liquid brains are not always based on connection weights but instead on population abundances. However, some liquid systems use fluctuations in ways similar to those found in cortical networks, suggesting a relevant role for criticality as a way of rapidly reacting to external signals. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

Concepts in Boolean network modeling: What do they all mean?

Boolean network models are one of the simplest models to study complex dynamic behavior in biological systems. They can be applied to unravel the mechanisms regulating the properties of the system or to identify promising intervention targets. Since its introduction by Stuart Kauffman in 1969 for describing gene regulatory networks, various biologically based networks and tools for their analysis were developed. Here, we summarize and explain the concepts for Boolean network modeling. We also present application examples and guidelines to work with and analyze Boolean network models.

Evolving Always-Critical Networks

Living beings share several common features at the molecular level, but there are very few large-scale “operating principles” which hold for all (or almost all) organisms. However, biology is subject to a deluge of data, and as such, general concepts such as this would be extremely valuable. One interesting candidate is the “criticality” principle, which claims that biological evolution favors those dynamical regimes that are intermediaries between ordered and disordered states (i.e., “at the edge of chaos”). The reasons why this should be the case and experimental evidence are briefly discussed, observing that gene regulatory networks are indeed often found on, or close to, the critical boundaries. Therefore, assuming that criticality provides an edge, it is important to ascertain whether systems that are critical can further evolve while remaining critical. In order to explore the possibility of achieving such “always-critical” evolution, we resort to simulated evolution, by suitably modifying a genetic algorithm in such a way that the newly-generated individuals are constrained to be critical. It is then shown that these modified genetic algorithms can actually develop critical gene regulatory networks with two interesting (and quite different) features of biological significance, involving, in one case, the average gene activation values and, in the other case, the response to perturbations. These two cases suggest that it is often possible to evolve networks with interesting properties without losing the advantages of criticality. The evolved networks also show some interesting features which are discussed.

Disturbance in human gut microbiota networks by parasites and its implications in the incidence of depression

If you think you are in control of your behavior, think again. Evidence suggests that behavioral modifications, as development and persistence of depression, maybe the consequence of a complex network of communication between macro and micro-organisms capable of modifying the physiological axis of the host. Some parasites cause significant nutritional deficiencies for the host and impair the effectiveness of cognitive processes such as memory, teaching or non-verbal intelligence. Bacterial communities mediate the establishment of parasites and vice versa but this complexity approach remains little explored. We study the gut microbiota-parasite interactions using novel techniques of network analysis using data of individuals from two indigenous communities in Guerrero, Mexico. Our results suggest that Ascaris lumbricoides induce a gut microbiota perturbation affecting its network properties and also subnetworks of key species related to depression, translating in a loss of emergence. Studying these network properties changes is particularly important because recent research has shown that human health is characterized by a dynamic trade-off between emergence and self-organization, called criticality. Emergence allows the systems to generate novel information meanwhile self-organization is related to the system's order and structure. In this way, the loss of emergence means a depart from criticality and ultimately loss of health.

Ensembles, dynamics, and cell types: Revisiting the statistical mechanics perspective on cellular regulation

Genetic regulatory networks control ontogeny. For fifty years Boolean networks have served as models of such systems, ranging from ensembles of random Boolean networks as models for generic properties of gene regulation to working dynamical models of a growing number of sub-networks of real cells. At the same time, their statistical mechanics has been thoroughly studied. Here we recapitulate their original motivation in the context of current theoretical and empirical research. We discuss ensembles of random Boolean networks whose dynamical attractors model cell types. A sub-ensemble is the critical ensemble. There is now strong evidence that genetic regulatory networks are dynamically critical, and that evolution is exploring the critical sub-ensemble. The generic properties of this sub-ensemble predict essential features of cell differentiation. In particular, the number of attractors in such networks scales as the DNA content raised to the 0.63 power. Data on the number of cell types as a function of the DNA content per cell shows a scaling relationship of 0.88. Thus, the theory correctly predicts a power law relationship between the number of cell types and the DNA contents per cell, and a comparable slope. We discuss these new scaling values and show prospects for new research lines for Boolean networks as a base model for systems biology.

Modeling the Role of the Microbiome in Evolution

There is undeniable evidence showing that bacteria have strongly influenced the evolution and biological functions of multicellular organisms. It has been hypothesized that many host-microbial interactions have emerged so as to increase the adaptive fitness of the holobiont (the host plus its microbiota). Although this association has been corroborated for many specific cases, general mechanisms explaining the role of the microbiota in the evolution of the host are yet to be understood. Here we present an evolutionary model in which a network representing the host adapts in order to perform a predefined function. During its adaptation, the host network (HN) can interact with other networks representing its microbiota. We show that this interaction greatly accelerates and improves the adaptability of the HN without decreasing the adaptation of the microbial networks. Furthermore, the adaptation of the HN to perform several functions is possible only when it interacts with many different bacterial networks in a specialized way (each bacterial network participating in the adaptation of one function). Disrupting these interactions often leads to non-adaptive states, reminiscent of dysbiosis, where none of the networks the holobiont consists of can perform their respective functions. By considering the holobiont as a unit of selection and focusing on the adaptation of the host to predefined but arbitrary functions, our model predicts the need for specialized diversity in the microbiota. This structural and dynamical complexity in the holobiont facilitates its adaptation, whereas a homogeneous (non-specialized) microbiota is inconsequential or even detrimental to the holobiont's evolution. To our knowledge, this is the first model in which symbiotic interactions, diversity, specialization and dysbiosis in an ecosystem emerge as a result of coevolution. It also helps us understand the emergence of complex organisms, as they adapt more easily to perform multiple tasks than non-complex ones.

Criticality Distinguishes the Ensemble of Biological Regulatory Networks

The hypothesis that many living systems should exhibit near-critical behavior is well motivated theoretically, and an increasing number of cases have been demonstrated empirically. However, a systematic analysis across biological networks, which would enable identification of the network properties that drive criticality, has not yet been realized. Here, we provide a first comprehensive survey of criticality across a diverse sample of biological networks, leveraging a publicly available database of 67 Boolean models of regulatory circuits. We find all 67 networks to be near critical. By comparing to ensembles of random networks with similar topological and logical properties, we show that criticality in biological networks is not predictable solely from macroscale properties such as mean degree ⟨K⟩ and mean bias in the logic functions ⟨p⟩, as previously emphasized in theories of random Boolean networks. Instead, the ensemble of real biological circuits is jointly constrained by the local causal structure and logic of each node. In this way, biological regulatory networks are more distinguished from random networks by their criticality than by other macroscale network properties such as degree distribution, edge density, or fraction of activating conditions.

How Criticality of Gene Regulatory Networks Affects the Resulting Morphogenesis under Genetic Perturbations

Whereas the relationship between criticality of gene regulatory networks (GRNs) and dynamics of GRNs at a single-cell level has been vigorously studied, the relationship between the criticality of GRNs and system properties at a higher level has not been fully explored. Here we aim at revealing a potential role of criticality of GRNs in morphogenesis, which is hard to uncover through the single-cell-level studies, especially from an evolutionary viewpoint. Our model simulated the growth of a cell population from a single seed cell. All the cells were assumed to have identical intracellular GRNs. We induced genetic perturbations to the GRN of the seed cell by adding, deleting, or switching a regulatory link between a pair of genes. From numerical simulations, we found that the criticality of GRNs facilitated the formation of nontrivial morphologies when the GRNs were critical in the presence of the evolutionary perturbations. Moreover, the criticality of GRNs produced topologically homogeneous cell clusters by adjusting the spatial arrangements of cells, which led to the formation of nontrivial morphogenetic patterns. Our findings correspond to an epigenetic viewpoint that heterogeneous and complex features emerge from homogeneous and less complex components through the interactions among them. Thus, our results imply that highly structured tissues or organs in morphogenesis of multicellular organisms might stem from the criticality of GRNs.

Stability and structural properties of gene regulation networks with coregulation rules

Coregulation of the expression of groups of genes has been extensively demonstrated empirically in bacterial and eukaryotic systems. Such coregulation can arise through the use of shared regulatory motifs, which allow the coordinated expression of modules (and module groups) of functionally related genes across the genome. Coregulation can also arise through the physical association of multi-gene complexes through chromosomal looping, which are then transcribed together. We present a general formalism for modeling coregulation rules in the framework of Random Boolean Networks (RBN), and develop specific models for transcription factor networks with modular structure (including module groups, and multi-input modules (MIM) with autoregulation) and multi-gene complexes (including hierarchical differentiation between multi-gene complex members). We develop a mean-field approach to analyse the dynamical stability of large networks incorporating coregulation, and show that autoregulated MIM and hierarchical gene-complex models can achieve greater stability than networks without coregulation whose rules have matching activation frequency. We provide further analysis of the stability of small networks of both kinds through simulations. We also characterize several general properties of the transients and attractors in the hierarchical coregulation model, and show using simulations that the steady-state distribution factorizes hierarchically as a Bayesian network in a Markov Jump Process analogue of the RBN model.

CABeRNET: a Cytoscape app for augmented Boolean models of gene regulatory NETworks

BACKGROUND

Dynamical models of gene regulatory networks (GRNs) are highly effective in describing complex biological phenomena and processes, such as cell differentiation and cancer development. Yet, the topological and functional characterization of real GRNs is often still partial and an exhaustive picture of their functioning is missing.

RESULTS

We here introduce CABERNET, a Cytoscape app for the generation, simulation and analysis of Boolean models of GRNs, specifically focused on their augmentation when a only partial topological and functional characterization of the network is available. By generating large ensembles of networks in which user-defined entities and relations are added to the original core, CABERNET allows to formulate hypotheses on the missing portions of real networks, as well to investigate their generic properties, in the spirit of complexity science.

CONCLUSIONS

CABERNET offers a series of innovative simulation and modeling functions and tools, including (but not being limited to) the dynamical characterization of the gene activation patterns ruling cell types and differentiation fates, and sophisticated robustness assessments, as in the case of gene knockouts. The integration within the widely used Cytoscape framework for the visualization and analysis of biological networks, makes CABERNET a new essential instrument for both the bioinformatician and the computational biologist, as well as a computational support for the experimentalist. An example application concerning the analysis of an augmented T-helper cell GRN is provided.

CoGNaC: A Chaste Plugin for the Multiscale Simulation of Gene Regulatory Networks Driving the Spatial Dynamics of Tissues and Cancer

We introduce a Chaste plugin for the generation and the simulation of Gene Regulatory Networks (GRNs) in multiscale models of multicellular systems. Chaste is a widely used and versatile computational framework for the multiscale modeling and simulation of multicellular biological systems. The plugin, named CoGNaC (Chaste and Gene Networks for Cancer), allows the linking of the regulatory dynamics to key properties of the cell cycle and of the differentiation process in populations of cells, which can subsequently be modeled using different spatial modeling scenarios. The approach of CoGNaC focuses on the emergent dynamical behavior of gene networks, in terms of gene activation patterns characterizing the different cellular phenotypes of real cells and, especially, on the overall robustness to perturbations and biological noise. The integration of this approach within Chaste’s modular simulation framework provides a powerful tool to model multicellular systems, possibly allowing for the formulation of novel hypotheses on gene regulation, cell differentiation, and, in particular, cancer emergence and development. In order to demonstrate the usefulness of CoGNaC over a range of modeling paradigms, two example applications are presented. The first of these concerns the characterization of the gene activation patterns of human T-helper cells. The second example is a multiscale simulation of a simplified intestinal crypt, in which, given certain conditions, tumor cells can emerge and colonize the tissue.

Investigating the relation between stochastic differentiation, homeostasis and clonal expansion in intestinal crypts via multiscale modeling

Colorectal tumors originate and develop within intestinal crypts. Even though some of the essential phenomena that characterize crypt structure and dynamics have been effectively described in the past, the relation between the differentiation process and the overall crypt homeostasis is still only partially understood. We here investigate this relation and other important biological phenomena by introducing a novel multiscale model that combines a morphological description of the crypt with a gene regulation model: the emergent dynamical behavior of the underlying gene regulatory network drives cell growth and differentiation processes, linking the two distinct spatio-temporal levels. The model relies on a few a priori assumptions, yet accounting for several key processes related to crypt functioning, such as: dynamic gene activation patterns, stochastic differentiation, signaling pathways ruling cell adhesion properties, cell displacement, cell growth, mitosis, apoptosis and the presence of biological noise. We show that this modeling approach captures the major dynamical phenomena that characterize the regular physiology of crypts, such as cell sorting, coordinate migration, dynamic turnover, stem cell niche correct positioning and clonal expansion. All in all, the model suggests that the process of stochastic differentiation might be sufficient to drive the crypt to homeostasis, under certain crypt configurations. Besides, our approach allows to make precise quantitative inferences that, when possible, were matched to the current biological knowledge and it permits to investigate the role of gene-level perturbations, with reference to cancer development. We also remark the theoretical framework is general and may be applied to different tissues, organs or organisms.

An extended gene protein/products Boolean network model including post-transcriptional regulation

Background

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.

Results

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.

Conclusions

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.

On the dynamical properties of a model of cell differentiation

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

Criticality is an emergent property of genetic networks that exhibit evolvability

Accumulating experimental evidence suggests that the gene regulatory networks of living organisms operate in the critical phase, namely, at the transition between ordered and chaotic dynamics. Such critical dynamics of the network permits the coexistence of robustness and flexibility which are necessary to ensure homeostatic stability (of a given phenotype) while allowing for switching between multiple phenotypes (network states) as occurs in development and in response to environmental change. However, the mechanisms through which genetic networks evolve such critical behavior have remained elusive. Here we present an evolutionary model in which criticality naturally emerges from the need to balance between the two essential components of evolvability: phenotype conservation and phenotype innovation under mutations. We simulated the Darwinian evolution of random Boolean networks that mutate gene regulatory interactions and grow by gene duplication. The mutating networks were subjected to selection for networks that both (i) preserve all the already acquired phenotypes (dynamical attractor states) and (ii) generate new ones. Our results show that this interplay between extending the phenotypic landscape (innovation) while conserving the existing phenotypes (conservation) suffices to cause the evolution of all the networks in a population towards criticality. Furthermore, the networks produced by this evolutionary process exhibit structures with hubs (global regulators) similar to the observed topology of real gene regulatory networks. Thus, dynamical criticality and certain elementary topological properties of gene regulatory networks can emerge as a byproduct of the evolvability of the phenotypic landscape.

Dynamically critical systems are those which operate at the border of a phase transition between two behavioral regimes often present in complex systems: order and disorder. Critical systems exhibit remarkable properties such as fast information processing, collective response to perturbations or the ability to integrate a wide range of external stimuli without saturation. Recent evidence indicates that the genetic networks of living cells are dynamically critical. This has far reaching consequences, for it is at criticality that living organisms can tolerate a wide range of external fluctuations without changing the functionality of their phenotypes. Therefore, it is necessary to know how genetic criticality emerged through evolution. Here we show that dynamical criticality naturally emerges from the delicate balance between two fundamental forces of natural selection that make organisms evolve: (i) the existing phenotypes must be resilient to random mutations, and (ii) new phenotypes must emerge for the organisms to adapt to new environmental challenges. The joint effect of these two forces, which are essential for evolvability, is sufficient in our computational models to generate populations of genetic networks operating at criticality. Thus, natural selection acting as a tinkerer of evolvable systems naturally generates critical dynamics.

Cell-cell interaction and diversity of emergent behaviours

Despite myriads of possible gene expression profiles, cells tend to be found in a confined number of expression patterns. The dynamics of Boolean models of gene regulatory networks has proven to be a likely candidate for the description of such self-organisation phenomena. Because cells do not live in isolation, but they constantly shape their functions to adapt to signals from other cells, this raises the question of whether the cooperation among cells entails an expansion or a reduction of their possible steady states. Multi random Boolean networks are introduced here as a model for interaction among cells that might be suitable for the investigation of some generic properties regarding the influence of communication on the diversity of cell behaviours. In spite of its simplicity, the model exhibits a non-obvious phenomenon according to which a moderate exchange of products among adjacent cells fosters the variety of their possible behaviours, which on the other hand are more similar to one another. On the contrary, a more invasive coupling would lead cells towards homogeneity.

A dynamical model of genetic networks for cell differentiation

A mathematical model is proposed which is able to describe the most important features of cell differentiation, without requiring specific detailed assumptions concerning the interactions which drive the phenomenon. On the contrary, cell differentiation is described here as an emergent property of a generic model of the underlying gene regulatory network, and it can therefore be applied to a variety of different organisms. The model points to a peculiar role of cellular noise in differentiation and leads to non trivial predictions which could be subject to experimental testing. Moreover, a single model proves able to describe several different phenomena observed in various differentiation processes.

Evolvability and Speed of Evolutionary Algorithms in Light of Recent Developments in Biology

Biological and artificial evolutionary systems exhibit varying degrees of evolvability and different rates of evolution. Such quantities can be affected by various factors. Here, we review some evolutionary mechanisms and discuss new developments in biology that can potentially improve evolvability or accelerate evolution in artificial systems. Biological notions are discussed to the degree they correspond to notions in Evolutionary Computation. We hope that the findings put forward here can be used to design computational models of evolution that produce significant gains in evolvability and evolutionary speed.

Salivary gland branching morphogenesis: a quantitative systems analysis of the Eda/Edar/NFkappaB paradigm

BACKGROUND

Ectodysplasin-A appears to be a critical component of branching morphogenesis. Mutations in mouse Eda or human EDA are associated with absent or hypoplastic sweat glands, sebaceous glands, lacrimal glands, salivary glands (SMGs), mammary glands and/or nipples, and mucous glands of the bronchial, esophageal and colonic mucosa. In this study, we utilized EdaTa (Tabby) mutant mice to investigate how a marked reduction in functional Eda propagates with time through a defined genetic subcircuit and to test the proposition that canonical NFkappaB signaling is sufficient to account for the differential expression of developmentally regulated genes in the context of Eda polymorphism.

RESULTS

The quantitative systems analyses do not support the stated hypothesis. For most NFkappaB-regulated genes, the observed time course of gene expression is nearly unchanged in Tabby (EdaTa) as compared to wildtype mice, as is NFkappaB itself. Importantly, a subset of genes is dramatically differentially expressed in Tabby (Edar, Fgf8, Shh, Egf, Tgfa, Egfr), strongly suggesting the existence of an alternative Eda-mediated transcriptional pathway pivotal for SMG ontogeny. Experimental and in silico investigations have identified C/EBPalpha as a promising candidate.

CONCLUSION

In Tabby SMGs, upregulation of the Egf/Tgfalpha/Egfr pathway appears to mitigate the potentially severe abnormal phenotype predicted by the downregulation of Fgf8 and Shh. Others have suggested that the buffering of the phenotypic outcome that is coincident with variant Eda signaling could be a common mechanism that permits viable and diverse phenotypes, normal and abnormal. Our results support this proposition. Further, if branching epithelia use variations of a canonical developmental program, our results are likely applicable to understanding the phenotypes of other branching organs affected by Eda (EDA) mutation.

A publish-subscribe model of genetic networks

We present a simple model of genetic regulatory networks in which regulatory connections among genes are mediated by a limited number of signaling molecules. Each gene in our model produces (publishes) a single gene product, which regulates the expression of other genes by binding to regulatory regions that correspond (subscribe) to that product. We explore the consequences of this publish-subscribe model of regulation for the properties of single networks and for the evolution of populations of networks. Degree distributions of randomly constructed networks, particularly multimodal in-degree distributions, which depend on the length of the regulatory sequences and the number of possible gene products, differed from simpler Boolean NK models. In simulated evolution of populations of networks, single mutations in regulatory or coding regions resulted in multiple changes in regulatory connections among genes, or alternatively in neutral change that had no effect on phenotype. This resulted in remarkable evolvability in both number and length of attractors, leading to evolved networks far beyond the expectation of these measures based on random distributions. Surprisingly, this rapid evolution was not accompanied by changes in degree distribution; degree distribution in the evolved networks was not substantially different from that of randomly generated networks. The publish-subscribe model also allows exogenous gene products to create an environment, which may be noisy or stable, in which dynamic behavior occurs. In simulations, networks were able to evolve moderate levels of both mutational and environmental robustness.

Critical dynamics in genetic regulatory networks: examples from four kingdoms

The coordinated expression of the different genes in an organism is essential to sustain functionality under the random external perturbations to which the organism might be subjected. To cope with such external variability, the global dynamics of the genetic network must possess two central properties. (a) It must be robust enough as to guarantee stability under a broad range of external conditions, and (b) it must be flexible enough to recognize and integrate specific external signals that may help the organism to change and adapt to different environments. This compromise between robustness and adaptability has been observed in dynamical systems operating at the brink of a phase transition between order and chaos. Such systems are termed critical. Thus, criticality, a precise, measurable, and well characterized property of dynamical systems, makes it possible for robustness and adaptability to coexist in living organisms. In this work we investigate the dynamical properties of the gene transcription networks reported for S. cerevisiae, E. coli, and B. subtilis, as well as the network of segment polarity genes of D. melanogaster, and the network of flower development of A. thaliana. We use hundreds of microarray experiments to infer the nature of the regulatory interactions among genes, and implement these data into the Boolean models of the genetic networks. Our results show that, to the best of the current experimental data available, the five networks under study indeed operate close to criticality. The generality of this result suggests that criticality at the genetic level might constitute a fundamental evolutionary mechanism that generates the great diversity of dynamically robust living forms that we observe around us.

Dynamic transcriptomic response to acute hypertension in the nucleus tractus solitarius

Baroreceptor afferents project to the cardiovascular region of the nucleus tractus solitarius (cvNTS), and their cvNTS target neurons may play a role in governing the sensitivity and operating range of the arterial baroreceptor reflex (baroreflexes). Recent studies have shown differential gene and protein expression in the cvNTS in response to changed arterial pressure. However, the extent of these responses is unknown. Therefore, we collected differential global gene expression data in a time series following acute hypertension in awake, freely moving rats. To acquire statistically significant results and place them in functional context, we overcame several quality control requirements and developed novel analytical approaches. The physiologically new findings from the study are that acute hypertension causes very extensive, time-varying gene regulatory changes, many involving neuronal function-specific genes and systems of genes. We use standard genomic analysis methods to manage the large data sets and to develop results such as heat maps to examine patterns and clusters in the gene regulation. We used the Gene Ontology categories to provide functional context. To place our findings in the context of the relevant literature, we developed two graphical representations of the networks implicated, linking receptors and channels to signaling pathways. The results point to the multivariate complexity of the response and implicate a group of receptors as candidates for mediating nucleus tractus solitarius baroreflex function in hypertension by identifying concurrent upregulation of receptor genes. We were able to make transcription factor binding predictions and record dysregulation of heart rate correlated with the transcriptional response.