Robustness, flexibility, and the role of lateral inhibition in the neurogenic network

The central regulatory circuit in the gene network controlling the morphogenesis of Drosophila mechanoreceptors: an in silico analysis

Identification of the mechanisms underlying the genetic control of spatial structure formation is among the relevant tasks of developmental biology. Both experimental and theoretical approaches and methods are used for this purpose, including gene network methodology, as well as mathematical and computer modeling. Reconstruction and analysis of the gene networks that provide the formation of traits allow us to integrate the existing experimental data and to identify the key links and intra-network connections that ensure the function of networks. Mathematical and computer modeling is used to obtain the dynamic characteristics of the studied systems and to predict their state and behavior. An example of the spatial morphological structure is the Drosophila bristle pattern with a strictly defined arrangement of its components - mechanoreceptors (external sensory organs) - on the head and body. The mechanoreceptor develops from a single sensory organ parental cell (SOPC), which is isolated from the ectoderm cells of the imaginal disk. It is distinguished from its surroundings by the highest content of proneural proteins (ASC), the products of the achaete-scute proneural gene complex (AS-C). The SOPC status is determined by the gene network we previously reconstructed and the AS-C is the key component of this network. AS-C activity is controlled by its subnetwork - the central regulatory circuit (CRC) comprising seven genes: AS-C, hairy, senseless (sens), charlatan (chn), scratch (scrt), phyllopod (phyl), and extramacrochaete (emc), as well as their respective proteins. In addition, the CRC includes the accessory proteins Daughterless (DA), Groucho (GRO), Ubiquitin (UB), and Seven-in-absentia (SINA). The paper describes the results of computer modeling of different CRC operation modes. As is shown, a cell is determined as an SOPC when the ASC content increases approximately 2.5-fold relative to the level in the surrounding cells. The hierarchy of the effects of mutations in the CRC genes on the dynamics of ASC protein accumulation is clarified. AS-C as the main CRC component is the most significant. The mutations that decrease the ASC content by more than 40 % lead to the prohibition of SOPC segregation.

Notch receptor/ligand diversity: contribution to colorectal cancer stem cell heterogeneity

Cancer cell heterogeneity is a key contributor to therapeutic failure and post-treatment recurrence. Targeting cell subpopulations responsible for chemoresistance and recurrence seems to be an attractive approach to improve treatment outcome in cancer patients. However, this remains challenging due to the complexity and incomplete characterization of tumor cell subpopulations. The heterogeneity of cells exhibiting stemness-related features, such as self-renewal and chemoresistance, fuels this complexity. Notch signaling is a known regulator of cancer stem cell (CSC) features in colorectal cancer (CRC), though the effects of its heterogenous signaling on CRC cell stemness are only just emerging. In this review, we discuss how Notch ligand-receptor specificity contributes to regulating stemness, self-renewal, chemoresistance and cancer stem cells heterogeneity in CRC.

Localization of Noise in Biochemical Networks

Noise, or uncertainty in biochemical networks, has become an important aspect of many biological problems. Noise can arise and propagate from external factors and probabilistic chemical reactions occurring in small cellular compartments. For species survival, it is important to regulate such uncertainties in executing vital cell functions. Regulated noise can improve adaptability, whereas uncontrolled noise can cause diseases. Simulation can provide a detailed analysis of uncertainties, but parameters such as rate constants and initial conditions are usually unknown. A general understanding of noise dynamics from the perspective of network structure is highly desirable. In this study, we extended the previously developed law of localization for characterizing noise in terms of (co)variances and developed noise localization theory. With linear noise approximation, we can expand a biochemical network into an extended set of differential equations representing a fictitious network for pseudo-components consisting of variances and covariances, together with chemical species. Through localization analysis, perturbation responses at the steady state of pseudo-components can be summarized into a sensitivity matrix that only requires knowledge of network topology. Our work allows identification of buffering structures at the level of species, variances, and covariances and can provide insights into noise flow under non-steady-state conditions in the form of a pseudo-chemical reaction. We tested noise localization in various systems, and here we discuss its implications and potential applications. Results show that this theory is potentially applicable in discriminating models, scanning network topologies with interesting noise behavior, and designing and perturbing networks with the desired response.

Genetic and Molecular Interactions between HΔCT, a Novel Allele of the Notch Antagonist Hairless, and the Histone Chaperone Asf1 in Drosophila melanogaster

Cellular differentiation relies on the highly conserved Notch signaling pathway. Notch activity induces gene expression changes that are highly sensitive to chromatin landscape. We address Notch gene regulation using Drosophila as a model, focusing on the genetic and molecular interactions between the Notch antagonist Hairless and the histone chaperone Asf1. Earlier work implied that Asf1 promotes the silencing of Notch target genes via Hairless (H). Here, we generate a novel HΔCT allele by genome engineering. Phenotypically, HΔCT behaves as a Hairless gain of function allele in several developmental contexts, indicating that the conserved CT domain of H has an attenuator role under native biological contexts. Using several independent methods to assay protein-protein interactions, we define the sequences of the CT domain that are involved in Hairless-Asf1 binding. Based on previous models, where Asf1 promotes Notch repression via Hairless, a loss of Asf1 binding should reduce Hairless repressive activity. However, tissue-specific Asf1 overexpression phenotypes are increased, not rescued, in the HΔCT background. Counterintuitively, Hairless protein binding mitigates the repressive activity of Asf1 in the context of eye development. These findings highlight the complex connections of Notch repressors and chromatin modulators during Notch target-gene regulation and open the avenue for further investigations.

Computational analysis of GAL pathway pinpoints mechanisms underlying natural variation

Quantitative traits are measurable phenotypes that show continuous variation over a wide phenotypic range. Enormous effort has recently been put into determining the genetic influences on a variety of quantitative traits with mixed success. We identified a quantitative trait in a tractable model system, the GAL pathway in yeast, which controls the uptake and metabolism of the sugar galactose. GAL pathway activation depends both on galactose concentration and on the concentrations of competing, preferred sugars such as glucose. Natural yeast isolates show substantial variation in the behavior of the pathway. All studied yeast strains exhibit bimodal responses relative to external galactose concentration, i.e. a set of galactose concentrations existed at which both GAL-induced and GAL-repressed subpopulations were observed. However, these concentrations differed in different strains. We built a mechanistic model of the GAL pathway and identified parameters that are plausible candidates for capturing the phenotypic features of a set of strains including standard lab strains, natural variants, and mutants. In silico perturbation of these parameters identified variation in the intracellular galactose sensor, Gal3p, the negative feedback node within the GAL regulatory network, Gal80p, and the hexose transporters, HXT, as the main sources of the bimodal range variation. We were able to switch the phenotype of individual yeast strains in silico by tuning parameters related to these three elements. Determining the basis for these behavioral differences may give insight into how the GAL pathway processes information, and into the evolution of nutrient metabolism preferences in different strains. More generally, our method of identifying the key parameters that explain phenotypic variation in this system should be generally applicable to other quantitative traits.

Microbes adopt elaborate strategies for the preferred uptake and use of nutrients to cope with complex and fluctuating environments. As a result, yeast strains originating from different ecological niches show significant variation in the way they induce genes in the galactose metabolism (GAL) pathway in response to nutrient signals. To identify the mechanistic sources of this variation, we built a mathematical model to simulate the dynamics of the galactose metabolic regulation network, and studied how parameters with different biological implications contributed to the natural variation. We found that variations in the behavior of the galactose sensor Gal3p, the negative feedback node Gal80p, and the hexose transporters HXT were critical elements in the GAL pathway response. Tuning single parameters in silico was sufficient to achieve phenotype switching between different yeast strains. Our computational approach should be generally useful to help pinpoint the genetic and molecular bases of natural variation in other systems.

Landau theory for cellular patterns driven by lateral inhibition interaction

The phenomenology of Landau theory with spatial coupling through diffusion has been widely used in the study of phase transitions and patterning. Here we follow this theory and apply it to study theoretically and numerically continuous and discontinuous transitions to periodic spatial cellular patterns driven by lateral inhibition coupling. As opposed to diffusion, lateral inhibition coupling drives differences between adjacent cells. We analyze the appearance of errors in these patterns (disordered metastable states) and propose mechanisms to prevent them. These mechanisms are based on a temporal-dependent lateral inhibition coupling strength, which can be mediated, among others, by gradients of diffusing molecules. The simplicity and generality of the framework used herein is expected to facilitate future analyses of additional phenomena taking place through lateral inhibition interactions in more complex scenarios.

Computational study of parameter sensitivity in DevR regulated gene expression

The DevRS two-component system plays a pivotal role in signal transmission and downstream gene regulation in Mycobacterium tuberculosis. Under the hypoxic condition, phosphorylated DevR interacts with multiple binding sites at the promoter region of the target genes. In the present work, we carried out a detailed computational analysis to figure out the sensitivity of the kinetic parameters. The set of kinetic parameters takes care of the interaction among phosphorylated DevR and the binding sites, transcription and translation processes. We employ the method of stochastic optimization to quantitate the relevant kinetic parameter set necessary for DevR regulated gene expression. Measures of different correlation coefficients provide the relative ordering of kinetic parameters involved in gene regulation. Results obtained from correlation coefficients are further corroborated by sensitivity amplification.

microRNA-14 as an efficient suppressor to switch off ecdysone production after ecdysis in insects

The precise increase and decrease of hormone ecdysone are critical for accurate development in insects. Most previous works focus on transcriptional activation of ecdysone production; however, little is known about the mechanism of switching off ecdysone biosynthesis after ecdysis. Here, we showed that the precursor microRNA-14 (pre-miR-14) encodes two mature miRNAs in silkworm; both of these two mature miRNAs regulate various genes in the ecdysone-signalling pathway. Bmo-miR-14-5p targets on nine genes whereas Bmo-miR-14-3p targets on two genes in the same pathway. These two mature miRNAs increased immediately after the ecdysis, efficiently suppressing the 20-hydroxyecdysone (20E) biosynthesis, the upstream regulation, and the downstream response genes. Knocking down either of two mature miRNAs or both of them delays moult development, impairing development synchrony in antagomir-treated groups. In addition, overexpressing Bmo-miR-14-5p but not Bmo-miR-14-3p significantly affected the 20E titer and increased the moulting time variation, suggesting that Bmo-miR-14-5p, though it is less abundant, has more potent effects in development regulation than Bmo-miR-14-3p. In summary, we present evidence that a pre-miRNA encodes two mature miRNAs targeting on the same pathway, which significantly improves miRNA regulation efficiencies to programmatically switch off ecdysone biosynthesis.

RACIPE: a computational tool for modeling gene regulatory circuits using randomization

Background

One of the major challenges in traditional mathematical modeling of gene regulatory circuits is the insufficient knowledge of kinetic parameters. These parameters are often inferred from existing experimental data and/or educated guesses, which can be time-consuming and error-prone, especially for large networks.

Results

We present a user-friendly computational tool for the community to use our newly developed method named random circuit perturbation (RACIPE), to explore the robust dynamical features of gene regulatory circuits without the requirement of detailed kinetic parameters. Taking the network topology as the only input, RACIPE generates an ensemble of circuit models with distinct randomized parameters and uniquely identifies robust dynamical properties by statistical analysis. Here, we discuss the implementation of the software and the statistical analysis methods of RACIPE-generated data to identify robust gene expression patterns and the functions of genes and regulatory links. Finally, we apply the tool on coupled toggle-switch circuits and a published circuit of B-lymphopoiesis.

Conclusions

We expect our new computational tool to contribute to a more comprehensive and unbiased understanding of mechanisms underlying gene regulatory networks. RACIPE is a free open source software distributed under (Apache 2.0) license and can be downloaded from GitHub (https://github.com/simonhb1990/RACIPE-1.0).

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0594-6) contains supplementary material, which is available to authorized users.

Interrogating the topological robustness of gene regulatory circuits by randomization

One of the most important roles of cells is performing their cellular tasks properly for survival. Cells usually achieve robust functionality, for example, cell-fate decision-making and signal transduction, through multiple layers of regulation involving many genes. Despite the combinatorial complexity of gene regulation, its quantitative behavior has been typically studied on the basis of experimentally verified core gene regulatory circuitry, composed of a small set of important elements. It is still unclear how such a core circuit operates in the presence of many other regulatory molecules and in a crowded and noisy cellular environment. Here we report a new computational method, named random circuit perturbation (RACIPE), for interrogating the robust dynamical behavior of a gene regulatory circuit even without accurate measurements of circuit kinetic parameters. RACIPE generates an ensemble of random kinetic models corresponding to a fixed circuit topology, and utilizes statistical tools to identify generic properties of the circuit. By applying RACIPE to simple toggle-switch-like motifs, we observed that the stable states of all models converge to experimentally observed gene state clusters even when the parameters are strongly perturbed. RACIPE was further applied to a proposed 22-gene network of the Epithelial-to-Mesenchymal Transition (EMT), from which we identified four experimentally observed gene states, including the states that are associated with two different types of hybrid Epithelial/Mesenchymal phenotypes. Our results suggest that dynamics of a gene circuit is mainly determined by its topology, not by detailed circuit parameters. Our work provides a theoretical foundation for circuit-based systems biology modeling. We anticipate RACIPE to be a powerful tool to predict and decode circuit design principles in an unbiased manner, and to quantitatively evaluate the robustness and heterogeneity of gene expression.

Cells are able to robustly carry out their essential biological functions, possibly because of multiple layers of tight regulation via complex, yet well-designed, gene regulatory networks involving a substantial number of genes. State-of-the-art genomics technology has enabled the mapping of these large gene networks, yet it remains a tremendous challenge to elucidate their design principles and the regulatory mechanisms underlying their biological functions such as signal processing and decision-making. One of the key barriers is the absence of accurate kinetics for the regulatory interactions, especially from in vivo experiments. To this end, we have developed a new computational modeling method, Random Circuit Perturbation (RACIPE), to explore the dynamic behaviors of gene regulatory circuits without the requirement of detailed kinetic parameters. RACIPE takes a network topology as the input, and generates an unbiased ensemble of models with varying kinetic parameters. Each model is subjected to simulation, followed by statistical analysis for the ensemble. We tested RACIPE on several gene circuits, and found that the predicted gene expression patterns from all of the models converge to experimentally observed gene state clusters. We expect RACIPE to be a powerful method to identify the role of network topology in determining network operating principles.

Robustness, flexibility, and sensitivity in a multifunctional motor control model

Motor systems must adapt to perturbations and changing conditions both within and outside the body. We refer to the ability of a system to maintain performance despite perturbations as "robustness," and the ability of a system to deploy alternative strategies that improve fitness as "flexibility." Different classes of pattern-generating circuits yield dynamics with differential sensitivities to perturbations and parameter variation. Depending on the task and the type of perturbation, high sensitivity can either facilitate or hinder robustness and flexibility. Here we explore the role of multiple coexisting oscillatory modes and sensory feedback in allowing multiphasic motor pattern generation to be both robust and flexible. As a concrete example, we focus on a nominal neuromechanical model of triphasic motor patterns in the feeding apparatus of the marine mollusk Aplysia californica. We find that the model can operate within two distinct oscillatory modes and that the system exhibits bistability between the two. In the "heteroclinic mode," higher sensitivity makes the system more robust to changing mechanical loads, but less robust to internal parameter variations. In the "limit cycle mode," lower sensitivity makes the system more robust to changes in internal parameter values, but less robust to changes in mechanical load. Finally, we show that overall performance on a variable feeding task is improved when the system can flexibly transition between oscillatory modes in response to the changing demands of the task. Thus, our results suggest that the interplay of sensory feedback and multiple oscillatory modes can allow motor systems to be both robust and flexible in a variable environment.

Convergent occurrence of the developmental hourglass in plant and animal embryogenesis?

BACKGROUND

The remarkable similarity of animal embryos at particular stages of development led to the proposal of a developmental hourglass. In this model, early events in development are less conserved across species but lead to a highly conserved 'phylotypic period'. Beyond this stage, the model suggests that development once again becomes less conserved, leading to the diversity of forms. Recent comparative studies of gene expression in animal groups have provided strong support for the hourglass model. How and why might such an hourglass pattern be generated? More importantly, how might early acting events in development evolve while still maintaining a later conserved stage?

SCOPE

The discovery that an hourglass pattern may also exist in the embryogenesis of plants provides comparative data that may help us explain this phenomenon. Whether the developmental hourglass occurs in plants, and what this means for our understanding of embryogenesis in plants and animals is discussed. Models by which conserved early-acting genes might change their functional role in the evolution of gene networks, how networks buffer these changes, and how that might constrain, or confer diversity, of the body plan are also discused.

CONCLUSIONS

Evidence of a morphological and molecular hourglass in plant and animal embryogenesis suggests convergent evolution. This convergence is likely due to developmental constraints imposed upon embryogenesis by the need to produce a viable embryo with an established body plan, controlled by the architecture of the underlying gene regulatory networks. As the body plan is largely laid down during the middle phases of embryo development in plants and animals, then it is perhaps not surprising this stage represents the narrow waist of the hourglass where the gene regulatory networks are the oldest and most robust and integrated, limiting species diversity and constraining morphological space.

A model study of the morphogenesis of D. melanogaster mechanoreceptors: the central regulatory circuit

Macrochaetes (large bristles) are sensor organs of the Drosophila peripheral nervous system with a function of mechanoreceptors. An adult mechanoreceptor comprises four specialized cells: shaft (trichogen), socket (tormogen), neuron, and glial cell (thecogen). All these cells originate from a single cell, the so-called sensor organ precursor (SOP) cell. Separation of the SOP cell from the encompassing cells of the imaginal disc initiates a multistage process of sensory organ development. A characteristic feature of the SOP cell is the highest amount of the proneural proteins AS-C as compared with the encompassing ectodermal cells. The accumulation of proneural proteins and maintenance of their amount in the SOP cell at a necessary level is provided by the gene network with the achaete-scute gene complex (AS-C) as its key component. The activity of this complex is controlled by the central regulatory circuit (CRC). The CRC comprises the genes hairy, senseless (sens), charlatan (chn), scratch (scrt), daughterless (da), extramacrochaete (emc), and groucho (gro), coding for the transcription factors involved in the system of direct links and feedbacks and implementation of activation-repression relationships between the CRC components. The gene phyllopod (phyl), involved in degradation of the AS-C proteins, is also associated with the CRC functioning. In this paper, we propose a mathematical model for the CRC functioning as a regulator of the amount of proneural AS-C proteins in the SOP cell taking into account their degradation. The modeling has demonstrated that a change in the amount of proneural proteins in the SOP cell is stepwise rather than strictly monotonic. This prediction can be tested experimentally.

Cellular metabolic network analysis: discovering important reactions in Treponema pallidum

T. pallidum, the syphilis-causing pathogen, performs very differently in metabolism compared with other bacterial pathogens. The desire for safe and effective vaccine of syphilis requests identification of important steps in T. pallidum's metabolism. Here, we apply Flux Balance Analysis to represent the reactions quantitatively. Thus, it is possible to cluster all reactions in T. pallidum. By calculating minimal cut sets and analyzing topological structure for the metabolic network of T. pallidum, critical reactions are identified. As a comparison, we also apply the analytical approaches to the metabolic network of H. pylori to find coregulated drug targets and unique drug targets for different microorganisms. Based on the clustering results, all reactions are further classified into various roles. Therefore, the general picture of their metabolic network is obtained and two types of reactions, both of which are involved in nucleic acid metabolism, are found to be essential for T. pallidum. It is also discovered that both hubs of reactions and the isolated reactions in purine and pyrimidine metabolisms play important roles in T. pallidum. These reactions could be potential drug targets for treating syphilis.

Evolution of early development in dipterans: reverse-engineering the gap gene network in the moth midge Clogmia albipunctata (Psychodidae)

Understanding the developmental and evolutionary dynamics of regulatory networks is essential if we are to explain the non-random distribution of phenotypes among the diversity of organismic forms. Here, we present a comparative analysis of one of the best understood developmental gene regulatory networks today: the gap gene network involved in early patterning of insect embryos. We use gene circuit models, which are fitted to quantitative spatio-temporal gene expression data for the four trunk gap genes hunchback (hb), Krüppel (Kr), giant (gt), and knirps (kni)/knirps-like (knl) in the moth midge Clogmia albipunctata, and compare them to equivalent reverse-engineered circuits from our reference species, the vinegar fly Drosophila melanogaster. In contrast to the single network structure we find for D. melanogaster, our models predict four alternative networks for C. albipunctata. These networks share a core structure, which includes the central regulatory feedback between hb and knl. Other interactions are only partially determined, as they differ between our four network structures. Nevertheless, our models make testable predictions and enable us to gain specific insights into gap gene regulation in C. albipunctata. They suggest a less central role for Kr in C. albipunctata than in D. melanogaster, and show that the mechanisms causing an anterior shift of gap domains over time are largely conserved between the two species, although shift dynamics differ. The set of C. albipunctata gene circuit models presented here will be used as the starting point for data-constrained in silico evolutionary simulations to study patterning transitions in the early development of dipteran species.

Competition in notch signaling with cis enriches cell fate decisions

Notch signaling is involved in cell fate choices during the embryonic development of Metazoa. Commonly, Notch signaling arises from the binding of the Notch receptor to its ligands in adjacent cells driving cell-to-cell communication. Yet, cell-autonomous control of Notch signaling through both ligand-dependent and ligand-independent mechanisms is known to occur as well. Examples include Notch signaling arising in the absence of ligand binding, and cis-inhibition of Notch signaling by titration of the Notch receptor upon binding to its ligands within a single cell. Increasing experimental evidences support that the binding of the Notch receptor with its ligands within a cell (cis-interactions) can also trigger a cell-autonomous Notch signal (cis-signaling), whose potential effects on cell fate decisions and patterning remain poorly understood. To address this question, herein we mathematically and computationally investigate the cell states arising from the combination of cis-signaling with additional Notch signaling sources, which are either cell-autonomous or involve cell-to-cell communication. Our study shows that cis-signaling can switch from driving cis-activation to effectively perform cis-inhibition and identifies under which conditions this switch occurs. This switch relies on the competition between Notch signaling sources, which share the same receptor but differ in their signaling efficiency. We propose that the role of cis-interactions and their signaling on fine-grained patterning and cell fate decisions is dependent on whether they drive cis-inhibition or cis-activation, which could be controlled during development. Specifically, cis-inhibition and not cis-activation facilitates patterning and enriches it by modulating the ratio of cells in the high-ligand expression state, by enabling additional periodic patterns like stripes and by allowing localized patterning highly sensitive to the precursor state and cell-autonomous bistability. Our study exemplifies the complexity of regulations when multiple signaling sources share the same receptor and provides the tools for their characterization.

A computational model of the effect of gene misexpression on the development of cortical areas

Brain function depends on the specialisation of brain areas. In the murine cerebral cortex, the development of these areas depends on the coordinated expression of several genes in precise spatial patterns in the telencephalon during embryogenesis. Manipulating the expression of these genes during development alters the positions and sizes of cortical areas in the adult. Qualitative data also show that these genes regulate each other's expression during development so that they form a regulatory network with many feedback loops. However, it is currently unknown which regulatory interactions are critical to generating the correct expression patterns to lead to normal cortical development. Here, we formalise the relationships inferred from genetic manipulations into computational models. We simulate many different networks potentially consistent with the experimental data and show that a surprising diversity of networks produce similar results. This demonstrates that existing data cannot uniquely specify the network. We conclude by suggesting experiments necessary to constrain the model and help identify and understand the true structure of this regulatory network.

Modeling Notch signaling: a practical tutorial

Theoretical and computational approaches for understanding different aspects of Notch signaling and Notch dependent patterning are gaining popularity in recent years. These in silico methodologies can provide dynamic insights that are often not intuitive and may help guide experiments aimed at elucidating these processes. This chapter is an introductory tutorial intended to allow someone with basic mathematical and computational knowledge to explore new mathematical models of Notch-mediated processes and perform numerical simulations of these models. In particular, we explain how to define and simulate models of lateral inhibition patterning processes. We provide a Matlab code for simulating various lateral inhibition models in a simple and intuitive manner, and show how to present the results from the computational models. This code can be used as a starting point for exploring more specific models that include additional aspects of the Notch pathway and its regulation.

Regulation of neuronal differentiation at the neurogenic wavefront

Signaling mediated by the Delta/Notch system controls the process of lateral inhibition, known to regulate neurogenesis in metazoans. Lateral inhibition takes place in equivalence groups formed by cells having equal capacity to differentiate, and it results in the singling out of precursors, which subsequently become neurons. During normal development, areas of active neurogenesis spread through non-neurogenic regions in response to specific morphogens, giving rise to neurogenic wavefronts. Close contact of these wavefronts with non-neurogenic cells is expected to affect lateral inhibition. Therefore, a mechanism should exist in these regions to prevent disturbances of the lateral inhibitory process. Focusing on the developing chick retina, we show that Dll1 is widely expressed by non-neurogenic precursors located at the periphery of this tissue, a region lacking Notch1, lFng, and differentiation-related gene expression. We investigated the role of this Dll1 expression through mathematical modeling. Our analysis predicts that the absence of Dll1 ahead of the neurogenic wavefront results in reduced robustness of the lateral inhibition process, often linked to enhanced neurogenesis and the presence of morphological alterations of the wavefront itself. These predictions are consistent with previous observations in the retina of mice in which Dll1 is conditionally mutated. The predictive capacity of our mathematical model was confirmed further by mimicking published results on the perturbation of morphogenetic furrow progression in the eye imaginal disc of Drosophila. Altogether, we propose that Notch-independent Delta expression ahead of the neurogenic wavefront is required to avoid perturbations in lateral inhibition and wavefront progression, thus optimizing the neurogenic process.

Tension and Robustness in Multitasking Cellular Networks

Cellular networks multitask by exhibiting distinct, context-dependent dynamics. However, network states (parameters) that generate a particular dynamic are often sub-optimal for others, defining a source of “tension” between them. Though multitasking is pervasive, it is not clear where tension arises, what consequences it has, and how it is resolved. We developed a generic computational framework to examine the source and consequences of tension between pairs of dynamics exhibited by the well-studied RB-E2F switch regulating cell cycle entry. We found that tension arose from task-dependent shifts in parameters associated with network modules. Although parameter sets common to distinct dynamics did exist, tension reduced both their accessibility and resilience to perturbation, indicating a trade-off between “one-size-fits-all” solutions and robustness. With high tension, robustness can be preserved by dynamic shifting of modules, enabling the network to toggle between tasks, and by increasing network complexity, in this case by gene duplication. We propose that tension is a general constraint on the architecture and operation of multitasking biological networks. To this end, our work provides a framework to quantify the extent of tension between any network dynamics and how it affects network robustness. Such analysis would suggest new ways to interfere with network elements to elucidate the design principles of cellular networks.

Multitasking pervades our daily lives. For example, the technological devices that we increasingly rely upon are now engineered with such multifunctionality or “integration” in mind. Similarly, cellular networks also multitask in that they generate multiple, distinct dynamics according to their operating context. Here we show that differences in parameter spaces that underlie different dynamics thus cause a “tension”, which ultimately constrains network operation. In particular, our analysis reveals that tension negatively impacts robustness by reducing accessibility of parameters able to accomplish two tasks and reduces their ability to withstand perturbations. The presence of tension and its negative impact on network robustness represents a fundamental, generic constraint on the operation of different multitasking networks.

Global analysis of dynamical decision-making models through local computation around the hidden saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity.

Intracellular trafficking in Drosophila visual system development: a basis for pattern formation through simple mechanisms

Intracellular trafficking underlies cellular functions ranging from membrane remodeling to receptor activation. During multicellular organ development, these basic cell biological functions are required as both passive machinery and active signaling regulators. Exocytosis, endocytosis, and recycling of several key signaling receptors have long been known to actively regulate morphogenesis and pattern formation during Drosophila eye development. Hence, intracellular membrane trafficking not only sets the cell biological stage for receptor-mediated signaling but also actively controls signaling through spatiotemporally regulated receptor localization. In contrast to eye development, the role of intracellular trafficking for the establishment of the eye-to-brain connectivity map has only recently received more attention. It is still poorly understood how guidance receptors are spatiotemporally regulated to serve as meaningful synapse formation signals. Yet, the Drosophila visual system provides some of the most striking examples for the regulatory role of intracellular trafficking during multicellular organ development. In this review we will first highlight the experimental and conceptual advances that motivate the study of intracellular trafficking during Drosophila visual system development. We will then illuminate the development of the eye, the eye-to-brain connectivity map and the optic lobe from the perspective of cell biological dynamics. Finally, we provide a conceptual framework that seeks to explain how the interplay of simple genetically encoded intracellular trafficking events governs the seemingly complex cellular behaviors, which in turn determine the developmental product.

Different cell fates from cell-cell interactions: core architectures of two-cell bistable networks

The acquisition of different fates by cells that are initially in the same state is central to development. Here, we investigate the possible structures of bistable genetic networks that can allow two identical cells to acquire different fates through cell-cell interactions. Cell-autonomous bistable networks have been previously sampled using an evolutionary algorithm. We extend this evolutionary procedure to take into account interactions between cells. We obtain a variety of simple bistable networks that we classify into major subtypes. Some have long been proposed in the context of lateral inhibition through the Notch-Delta pathway, some have been more recently considered and others appear to be new and based on mechanisms not previously considered. The results highlight the role of posttranscriptional interactions and particularly of protein complexation and sequestration, which can replace cooperativity in transcriptional interactions. Some bistable networks are entirely based on posttranscriptional interactions and the simplest of these is found to lead, upon a single parameter change, to oscillations in the two cells with opposite phases. We provide qualitative explanations as well as mathematical analyses of the dynamical behaviors of various created networks. The results should help to identify and understand genetic structures implicated in cell-cell interactions and differentiation.

Robust selection of sensory organ precursors by the Notch-Delta pathway

The patterning of multicellular organisms is robust to environmental, genetic, or stochastic fluctuations. Mathematical modeling is instrumental in identifying mechanisms supporting this robustness. The principle of lateral inhibition, whereby a differentiating cell inhibits its neighbors from adopting the same fate, is frequently used for selecting a single cell out of a cluster of equipotent cells. For example, Sensory Organ Precursors (SOP) in the fruit-fly Drosophila implement lateral inhibition by activating the Notch-Delta pathway. We discuss parameters affecting the rate of errors in this process, and the mechanism (inhibitory cis interaction between Notch and Delta) predicted to reduce this error.

Robustness portraits of diverse biological networks conserved despite order-of-magnitude parameter uncertainty

MOTIVATION

Biological networks are robust to a wide variety of internal and external perturbations, yet fragile or sensitive to a small minority of perturbations. Due to this rare sensitivity of networks to certain perturbations, it is unclear how precisely biochemical parameters must be experimentally measured in order to accurately predict network function.

RESULTS

Here, we examined a model of cardiac β-adrenergic signaling and found that its robustness portrait, a global measure of steady-state network function, was well conserved even when all parameters were rounded to their nearest 1-2 orders of magnitude. In contrast, β-adrenergic network kinetics were more sensitive to parameter precision. This analysis was then extended to 10 additional networks, including Escherichia coli chemotaxis, stem cell differentiation and cytokine signaling, of which nine exhibited conserved robustness portraits despite the order-of-magnitude approximation of their biochemical parameters. Thus, both fragile and robust aspects of diverse biological networks are largely shaped by network topology and can be predicted despite order-of-magnitude uncertainty in biochemical parameters. These findings suggest an iterative strategy where order-of-magnitude models are used to prioritize experiments toward the fragile network elements that require precise measurements, efficiently driving model revision.

CONTACT

jsaucerman@virginia.edu

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

From Notch signaling to fine-grained patterning: Modeling meets experiments

Notch signaling is the canonical signaling pathway between neighboring cells. It plays an important role in fine-grained patterning processes such as the formation of checkerboard-like differentiation patterns and sharp boundaries between developing tissues. While detailed information about many of the genes and proteins involved have been identified, we still lack a quantitative mechanistic understanding of these processes. Here we discuss several recent studies that provide novel insights into Notch-dependent patterning by combining mathematical models with quantitative experimental results. Such approaches allow identification of mechanisms and design principles controlling how patterns are generated in a reproducible and robust manner.

A dynamical model of ommatidial crystal formation

The crystalline photoreceptor lattice in the Drosophila eye is a paradigm for pattern formation during development. During eye development, activation of proneural genes at a moving front adds new columns to a regular lattice of R8 photoreceptors. We present a mathematical model of the governing activator-inhibitor system, which indicates that the dynamics of positive induction play a central role in the selection of certain cells as R8s. The "switch and template" patterning mechanism we observe is mathematically very different from the well-known Turing instability. Unlike a standard lateral inhibition model, our picture implies that R8s are defined before the appearance of the complete group of proneural cells. The model reproduces the full time course of proneural gene expression and accounts for specific features of the refinement of proneural groups that had resisted explanation. It moreover predicts that perturbing the normal template can lead to eyes containing stripes of R8 cells. We observed these stripes experimentally after manipulation of the Notch and scabrous genes. Our results suggest an alternative to the generally assumed mode of operation for lateral inhibition during development; more generally, they hint at a broader role for bistable switches in the initial establishment of patterns as well as in their maintenance.

Mechanism and significance of cis-inhibition in Notch signalling

Notch receptors in a given cell are activated by cell surface ligands in neighbouring cells but can also be inhibited by the ligands present within the same cell. This process is known as cis-inhibition of Notch. Additionally, reciprocal cis-inhibition of the ligands by Notch has also been observed, albeit to a limited extent. Here, we review the mechanisms, functional relevance and potential implications of these cis-inhibitory interactions for Notch-mediated fate decisions.

Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth

Turgor generates the stress that leads to the expansion of plant cell walls during cellular growth. This has been formalized by the Lockhart equation, which can be derived from the physical laws of the deformation of viscoelastic materials. However, the experimental evidence for such a direct correlation between growth rate and turgor is inconclusive. This has led to challenges of the Lockhart model. We model the oscillatory growth of pollen tubes to investigate this relationship. We couple the Lockhart equation to the dynamical equations for the change in material properties. We find that the correct implementation of the Lockhart equation within a feedback loop leading to low amplitude oscillatory growth predicts that in this system changes in the global turgor do not influence the average growth rate in a linear manner, consistent with experimental observations. An analytic analysis of our model demonstrates in which regime the average growth rate becomes uncorrelated from the turgor pressure.

Quantitative variation in autocrine signaling and pathway crosstalk in the Caenorhabditis vulval network

BACKGROUND

Biological networks experience quantitative change in response to environmental and evolutionary variation. Computational modeling allows exploration of network parameter space corresponding to such variations. The intercellular signaling network underlying Caenorhabditis vulval development specifies three fates in a row of six precursor cells, yielding a quasi-invariant 3°3°2°1°2°3° cell fate pattern. Two seemingly conflicting verbal models of vulval precursor cell fate specification have been proposed: sequential induction by the EGF-MAP kinase and Notch pathways, or morphogen-based induction by the former.

RESULTS

To study the mechanistic and evolutionary system properties of this network, we combine experimental studies with computational modeling, using a model that keeps the network architecture constant but varies parameters. We first show that the Delta autocrine loop can play an essential role in 2° fate specification. With this autocrine loop, the same network topology can be quantitatively tuned to use in the six-cell-row morphogen-based or sequential patterning mechanisms, which may act singly, cooperatively, or redundantly. Moreover, different quantitative tunings of this same network can explain vulval patterning observed experimentally in C. elegans, C. briggsae, C. remanei, and C. brenneri. We experimentally validate model predictions, such as interspecific differences in isolated vulval precursor cell behavior and in spatial regulation of Notch activity.

CONCLUSIONS

Our study illustrates how quantitative variation in the same network comprises developmental patterning modes that were previously considered qualitatively distinct and also accounts for evolution among closely related species.

Evolutionary computation for the design of a stochastic switch for synthetic genetic circuits

Biological systems are inherently stochastic, a fact which is often ignored when simulating genetic circuits. Synthetic biology aims to design genetic circuits de novo, and cannot therefore afford to ignore the effects of stochastic behavior. Since computational design tools will be essential for large-scale synthetic biology, it is important to develop an understanding of the role of stochasticity in molecular biology, and incorporate this understanding into computational tools for genetic circuit design. We report upon an investigation into the combination of evolutionary algorithms and stochastic simulation for genetic circuit design, to design regulatory systems based on the Bacillus subtilis sin operon.

Pattern, growth, and control

Systems biology seeks not only to discover the machinery of life but to understand how such machinery is used for control, i.e., for regulation that achieves or maintains a desired, useful end. This sort of goal-directed, engineering-centered approach also has deep historical roots in developmental biology. Not surprisingly, developmental biology is currently enjoying an influx of ideas and methods from systems biology. This Review highlights current efforts to elucidate design principles underlying the engineering objectives of robustness, precision, and scaling as they relate to the developmental control of growth and pattern formation. Examples from vertebrate and invertebrate development are used to illustrate general lessons, including the value of integral feedback in achieving set-point control; the usefulness of self-organizing behavior; the importance of recognizing and appropriately handling noise; and the absence of "free lunch." By illuminating such principles, systems biology is helping to create a functional framework within which to make sense of the mechanistic complexity of organismal development.

The emergence of modularity in biological systems

In this review, we discuss modularity and hierarchy in biological systems. We review examples from protein structure, genetics, and biological networks of modular partitioning of the geometry of biological space. We review theories to explain modular organization of biology, with a focus on explaining how biology may spontaneously organize to a structured form. That is, we seek to explain how biology nucleated from among the many possibilities in chemistry. The emergence of modular organization of biological structure will be described as a symmetry-breaking phase transition, with modularity as the order parameter. Experimental support for this description will be reviewed. Examples will be presented from pathogen structure, metabolic networks, gene networks, and protein-protein interaction networks. Additional examples will be presented from ecological food networks, developmental pathways, physiology, and social networks.

Neutral evolution of robustness in Drosophila microRNA precursors

Mutational robustness describes the extent to which a phenotype remains unchanged in the face of mutations. Theory predicts that the strength of direct selection for mutational robustness is at most the magnitude of the rate of deleterious mutation. As far as nucleic acid sequences are concerned, only long sequences in organisms with high deleterious mutation rates and large population sizes are expected to evolve mutational robustness. Surprisingly, recent studies have concluded that molecules that meet none of these conditions--the microRNA precursors (pre-miRNAs) of multicellular eukaryotes--show signs of selection for mutational and/or environmental robustness. To resolve the apparent disagreement between theory and these studies, we have reconstructed the evolutionary history of Drosophila pre-miRNAs and compared the robustness of each sequence to that of its reconstructed ancestor. In addition, we "replayed the tape" of pre-miRNA evolution via simulation under different evolutionary assumptions and compared these alternative histories with the actual one. We found that Drosophila pre-miRNAs have evolved under strong purifying selection against changes in secondary structure. Contrary to earlier claims, there is no evidence that these RNAs have been shaped by either direct or congruent selection for any kind of robustness. Instead, the high robustness of Drosophila pre-miRNAs appears to be mostly intrinsic and likely a consequence of selection for functional structures.

Epidermal patterning in Arabidopsis: models make a difference

The leaf and root epidermis in Arabidopsis provide ideal systems in which to explore the mechanisms that underlie the patterned assignment of cell fates during development. Extensive experimental studies have uncovered a complex interlocked feedback network that operates within the epidermis to coordinate the choice between hair and nonhair fates. A number of recent studies using mathematical models have begun to study this network, highlighting new mechanisms that have subsequently been confirmed in model-directed experiments. These studies illustrate the potential of integrated modeling and experimentation to shed new light on developmental processes. Moreover, these models enable systems-level comparative analyses that may help understand the origin and role of properties, such as robustness and redundancy in developmental systems and, concomitantly, the evolution of development itself.

Integrative multicellular biological modeling: a case study of 3D epidermal development using GPU algorithms

BACKGROUND

Simulation of sophisticated biological models requires considerable computational power. These models typically integrate together numerous biological phenomena such as spatially-explicit heterogeneous cells, cell-cell interactions, cell-environment interactions and intracellular gene networks. The recent advent of programming for graphical processing units (GPU) opens up the possibility of developing more integrative, detailed and predictive biological models while at the same time decreasing the computational cost to simulate those models.

RESULTS

We construct a 3D model of epidermal development and provide a set of GPU algorithms that executes significantly faster than sequential central processing unit (CPU) code. We provide a parallel implementation of the subcellular element method for individual cells residing in a lattice-free spatial environment. Each cell in our epidermal model includes an internal gene network, which integrates cellular interaction of Notch signaling together with environmental interaction of basement membrane adhesion, to specify cellular state and behaviors such as growth and division. We take a pedagogical approach to describing how modeling methods are efficiently implemented on the GPU including memory layout of data structures and functional decomposition. We discuss various programmatic issues and provide a set of design guidelines for GPU programming that are instructive to avoid common pitfalls as well as to extract performance from the GPU architecture.

CONCLUSIONS

We demonstrate that GPU algorithms represent a significant technological advance for the simulation of complex biological models. We further demonstrate with our epidermal model that the integration of multiple complex modeling methods for heterogeneous multicellular biological processes is both feasible and computationally tractable using this new technology. We hope that the provided algorithms and source code will be a starting point for modelers to develop their own GPU implementations, and encourage others to implement their modeling methods on the GPU and to make that code available to the wider community.

Systems biology of embryogenesis

The development of a complete organism from a single cell involves extraordinarily complex orchestration of biological processes that vary intricately across space and time. Systems biology seeks to describe how all elements of a biological system interact in order to understand, model and ultimately predict aspects of emergent biological processes. Embryogenesis represents an extraordinary opportunity (and challenge) for the application of systems biology. Systems approaches have already been used successfully to study various aspects of development, from complex intracellular networks to four-dimensional models of organogenesis. Going forward, great advancements and discoveries can be expected from systems approaches applied to embryogenesis and developmental biology.

Specialization can drive the evolution of modularity

Organismal development and many cell biological processes are organized in a modular fashion, where regulatory molecules form groups with many interactions within a group and few interactions between groups. Thus, the activity of elements within a module depends little on elements outside of it. Modularity facilitates the production of heritable variation and of evolutionary innovations. There is no consensus on how modularity might evolve, especially for modules in development. We show that modularity can increase in gene regulatory networks as a byproduct of specialization in gene activity. Such specialization occurs after gene regulatory networks are selected to produce new gene activity patterns that appear in a specific body structure or under a specific environmental condition. Modules that arise after specialization in gene activity comprise genes that show concerted changes in gene activities. This and other observations suggest that modularity evolves because it decreases interference between different groups of genes. Our work can explain the appearance and maintenance of modularity through a mechanism that is not contingent on environmental change. We also show how modularity can facilitate co-option, the utilization of existing gene activity to build new gene activity patterns, a frequent feature of evolutionary innovations.

Propagation of kinetic uncertainties through a canonical topology of the TLR4 signaling network in different regions of biochemical reaction space

Background

Signal transduction networks represent the information processing systems that dictate which dynamical regimes of biochemical activity can be accessible to a cell under certain circumstances. One of the major concerns in molecular systems biology is centered on the elucidation of the robustness properties and information processing capabilities of signal transduction networks. Achieving this goal requires the establishment of causal relations between the design principle of biochemical reaction systems and their emergent dynamical behaviors.

Methods

In this study, efforts were focused in the construction of a relatively well informed, deterministic, non-linear dynamic model, accounting for reaction mechanisms grounded on standard mass action and Hill saturation kinetics, of the canonical reaction topology underlying Toll-like receptor 4 (TLR4)-mediated signaling events. This signaling mechanism has been shown to be deployed in macrophages during a relatively short time window in response to lypopolysaccharyde (LPS) stimulation, which leads to a rapidly mounted innate immune response. An extensive computational exploration of the biochemical reaction space inhabited by this signal transduction network was performed via local and global perturbation strategies. Importantly, a broad spectrum of biologically plausible dynamical regimes accessible to the network in widely scattered regions of parameter space was reconstructed computationally. Additionally, experimentally reported transcriptional readouts of target pro-inflammatory genes, which are actively modulated by the network in response to LPS stimulation, were also simulated. This was done with the main goal of carrying out an unbiased statistical assessment of the intrinsic robustness properties of this canonical reaction topology.

Results

Our simulation results provide convincing numerical evidence supporting the idea that a canonical reaction mechanism of the TLR4 signaling network is capable of performing information processing in a robust manner, a functional property that is independent of the signaling task required to be executed. Nevertheless, it was found that the robust performance of the network is not solely determined by its design principle (topology), but this may be heavily dependent on the network's current position in biochemical reaction space. Ultimately, our results enabled us the identification of key rate limiting steps which most effectively control the performance of the system under diverse dynamical regimes.

Conclusions

Overall, our in silico study suggests that biologically relevant and non-intuitive aspects on the general behavior of a complex biomolecular network can be elucidated only when taking into account a wide spectrum of dynamical regimes attainable by the system. Most importantly, this strategy provides the means for a suitable assessment of the inherent variational constraints imposed by the structure of the system when systematically probing its parameter space.