Dynamic Maternal Gradients Control Timing and Shift-Rates for Drosophila Gap Gene Expression

Approximated gene expression trajectories for gene regulatory network inference on cell tracks

The study of pattern formation has benefited from our ability to reverse-engineer gene regulatory network (GRN) structure from spatiotemporal quantitative gene expression data. Traditional approaches have focused on systems where the timescales of pattern formation and morphogenesis can be separated. Unfortunately, this is not the case in most animal patterning systems, where pattern formation and morphogenesis are co-occurring and tightly linked. To elucidate patterning mechanisms in such systems we need to adapt our GRN inference methodologies to include cell movements. In this work, we fill this gap by integrating quantitative data from live and fixed embryos to approximate gene expression trajectories (AGETs) in single cells and use these to reverse-engineer GRNs. This framework generates candidate GRNs that recapitulate pattern at the tissue level, gene expression dynamics at the single cell level, recover known genetic interactions and recapitulate experimental perturbations while incorporating cell movements explicitly for the first time.

Combinatorial interpretation of BMP and WNT controls the decision between primitive streak and extraembryonic fates

BMP signaling is essential for mammalian gastrulation, as it initiates a cascade of signals that control self-organized patterning. As development is highly dynamic, it is crucial to understand how time-dependent combinatorial signaling affects cellular differentiation. Here, we show that BMP signaling duration is a crucial control parameter that determines cell fates upon the exit from pluripotency through its interplay with the induced secondary signal WNT. BMP signaling directly converts cells from pluripotent to extraembryonic fates while simultaneously upregulating Wnt signaling, which promotes primitive streak and mesodermal specification. Using live-cell imaging of signaling and cell fate reporters together with a simple mathematical model, we show that this circuit produces a temporal morphogen effect where, once BMP signal duration is above a threshold for differentiation, intermediate and long pulses of BMP signaling produce specification of mesoderm and extraembryonic fates, respectively. Our results provide a systems-level picture of how these signaling pathways control the landscape of early human development.

Automatic design of gene regulatory mechanisms for spatial pattern formation

Gene regulatory mechanisms (GRMs) control the formation of spatial and temporal expression patterns that can serve as regulatory signals for the development of complex shapes. Synthetic developmental biology aims to engineer such genetic circuits for understanding and producing desired multicellular spatial patterns. However, designing synthetic GRMs for complex, multi-dimensional spatial patterns is a current challenge due to the nonlinear interactions and feedback loops in genetic circuits. Here we present a methodology to automatically design GRMs that can produce any given two-dimensional spatial pattern. The proposed approach uses two orthogonal morphogen gradients acting as positional information signals in a multicellular tissue area or culture, which constitutes a continuous field of engineered cells implementing the same designed GRM. To efficiently design both the circuit network and the interaction mechanisms-including the number of genes necessary for the formation of the target spatial pattern-we developed an automated algorithm based on high-performance evolutionary computation. The tolerance of the algorithm can be configured to design GRMs that are either simple to produce approximate patterns or complex to produce precise patterns. We demonstrate the approach by automatically designing GRMs that can produce a diverse set of synthetic spatial expression patterns by interpreting just two orthogonal morphogen gradients. The proposed framework offers a versatile approach to systematically design and discover complex genetic circuits producing spatial patterns.

Automatic design of gene regulatory mechanisms for spatial pattern formation

Synthetic developmental biology aims to engineer gene regulatory mechanisms (GRMs) for understanding and producing desired multicellular patterns and shapes. However, designing GRMs for spatial patterns is a current challenge due to the nonlinear interactions and feedback loops in genetic circuits. Here we present a methodology to automatically design GRMs that can produce any given spatial pattern. The proposed approach uses two orthogonal morphogen gradients acting as positional information signals in a multicellular tissue area or culture, which constitutes a continuous field of engineered cells implementing the same designed GRM. To efficiently design both the circuit network and the interaction mechanisms-including the number of genes necessary for the formation of the target pattern-we developed an automated algorithm based on high-performance evolutionary computation. The tolerance of the algorithm can be configured to design GRMs that are either simple to produce approximate patterns or complex to produce precise patterns. We demonstrate the approach by automatically designing GRMs that can produce a diverse set of synthetic spatial expression patterns by interpreting just two orthogonal morphogen gradients. The proposed framework offers a versatile approach to systematically design and discover pattern-producing genetic circuits.

A bistable autoregulatory module in the developing embryo commits cells to binary expression fates

Bistable autoactivation has been proposed as a mechanism for cells to adopt binary fates during embryonic development. However, it is unclear whether the autoactivating modules found within developmental gene regulatory networks are bistable, unless their parameters are quantitatively determined. Here, we combine in vivo live imaging with mathematical modeling to dissect the binary cell fate dynamics of the fruit fly pair-rule gene fushi tarazu (ftz), which is regulated by two known enhancers: the early (non-autoregulating) element and the autoregulatory element. Live imaging of transcription and protein concentration in the blastoderm revealed that binary Ftz fates are achieved as Ftz expression rapidly transitions from being dictated by the early element to the autoregulatory element. Moreover, we discovered that Ftz concentration alone is insufficient to activate the autoregulatory element, and that this element only becomes responsive to Ftz at a prescribed developmental time. Based on these observations, we developed a dynamical systems model and quantitated its kinetic parameters directly from experimental measurements. Our model demonstrated that the ftz autoregulatory module is indeed bistable and that the early element transiently establishes the content of the binary cell fate decision to which the autoregulatory module then commits. Further in silico analysis revealed that the autoregulatory element locks the Ftz fate quickly, within 35 min of exposure to the transient signal of the early element. Overall, our work confirms the widely held hypothesis that autoregulation can establish developmental fates through bistability and, most importantly, provides a framework for the quantitative dissection of cellular decision-making.

Speed and navigation control of thymocyte development by the fetal T-cell gene regulatory network

T-cell differentiation is a tightly regulated developmental program governed by interactions between transcription factors (TFs) and chromatin landscapes and affected by signals received from the thymic stroma. This process is marked by a series of checkpoints: T-lineage commitment, T-cell receptor (TCR)β selection, and positive and negative selection. Dynamically changing combinations of TFs drive differentiation along the T-lineage trajectory, through mechanisms that have been most extensively dissected in adult mouse T-lineage cells. However, fetal T-cell development differs from adult in ways that suggest that these TF mechanisms are not fully deterministic. The first wave of fetal T-cell differentiation occurs during a unique developmental window during thymic morphogenesis, shows more rapid kinetics of differentiation with fewer rounds of cell division, and gives rise to unique populations of innate lymphoid cells (ILCs) and invariant γδT cells that are not generated in the adult thymus. As the characteristic kinetics and progeny biases are cell-intrinsic properties of thymic progenitors, the differences could be based on distinct TF network circuitry within the progenitors themselves. Here, we review recent single-cell transcriptome data that illuminate the TF networks involved in T-cell differentiation in the fetal and adult mouse thymus.

Stability of Pattern Formation in Systems with Dynamic Source Regions

We explain the principles of gene expression pattern stabilization in systems of interacting, diffusible morphogens, with dynamically established source regions. Using a reaction-diffusion model with a step-function production term, we identify the phase transition between low-precision indeterminate patterning and the phase in which a traveling, well-defined contact zone between two domains is formed. Our model analytically explains single- and two-gene domain dynamics and provides pattern stability conditions for all possible two-gene regulatory network motifs.

Using phenotypic plasticity to understand the structure and evolution of the genotype-phenotype map

Deciphering the genotype-phenotype map necessitates relating variation at the genetic level to variation at the phenotypic level. This endeavour is inherently limited by the availability of standing genetic variation, the rate of spontaneous mutation to novo genetic variants, and possible biases associated with induced mutagenesis. An interesting alternative is to instead rely on the environment as a source of variation. Many phenotypic traits change plastically in response to the environment, and these changes are generally underlain by changes in gene expression. Relating gene expression plasticity to the phenotypic plasticity of more integrated organismal traits thus provides useful information about which genes influence the development and expression of which traits, even in the absence of genetic variation. We here appraise the prospects and limits of such an environment-for-gene substitution for investigating the genotype-phenotype map. We review models of gene regulatory networks, and discuss the different ways in which they can incorporate the environment to mechanistically model phenotypic plasticity and its evolution. We suggest that substantial progress can be made in deciphering this genotype-environment-phenotype map, by connecting theory on gene regulatory network to empirical patterns of gene co-expression, and by more explicitly relating gene expression to the expression and development of phenotypes, both theoretically and empirically.

Towards a physical understanding of developmental patterning

The temporal coordination of events at cellular and tissue scales is essential for the proper development of organisms, and involves cell-intrinsic processes that can be coupled by local cellular signalling and instructed by global signalling, thereby creating spatial patterns of cellular states that change over time. The timing and structure of these patterns determine how an organism develops. Traditional developmental genetic methods have revealed the complex molecular circuits regulating these processes but are limited in their ability to predict and understand the emergent spatio-temporal dynamics. Increasingly, approaches from physics are now being used to help capture the dynamics of the system by providing simplified, generic descriptions. Combined with advances in imaging and computational power, such approaches aim to provide insight into timing and patterning in developing systems.

The early Drosophila embryo as a model system for quantitative biology

With the rise of new tools, from controlled genetic manipulations and optogenetics to improved microscopy, it is now possible to make clear, quantitative and reproducible measurements of biological processes. The humble fruit fly Drosophila melanogaster, with its ease of genetic manipulation combined with excellent imaging accessibility, has become a major model system for performing quantitative in vivo measurements. Such measurements are driving a new wave of interest from physicists and engineers, who are developing a range of testable dynamic models of active systems to understand fundamental biological processes. The reproducibility of the early Drosophila embryo has been crucial for understanding how biological systems are robust to unavoidable noise during development. Insights from quantitative in vivo experiments in the Drosophila embryo are having an impact on our understanding of critical biological processes, such as how cells make decisions and how complex tissue shape emerges. Here, to highlight the power of using Drosophila embryogenesis for quantitative biology, I focus on three main areas: (1) formation and robustness of morphogen gradients; (2) how gene regulatory networks ensure precise boundary formation; and (3) how mechanical interactions drive packing and tissue folding. I further discuss how such data has driven advances in modelling.

Homology of process: developmental dynamics in comparative biology

Comparative biology builds up systematic knowledge of the diversity of life, across evolutionary lineages and levels of organization, starting with evidence from a sparse sample of model organisms. In developmental biology, a key obstacle to the growth of comparative approaches is that the concept of homology is not very well defined for levels of organization that are intermediate between individual genes and morphological characters. In this paper, we investigate what it means for ontogenetic processes to be homologous, focusing specifically on the examples of insect segmentation and vertebrate somitogenesis. These processes can be homologous without homology of the underlying genes or gene networks, since the latter can diverge over evolutionary time, while the dynamics of the process remain the same. Ontogenetic processes like these therefore constitute a dissociable level and distinctive unit of comparison requiring their own specific criteria of homology. In addition, such processes are typically complex and nonlinear, such that their rigorous description and comparison requires not only observation and experimentation, but also dynamical modelling. We propose six criteria of process homology, combining recognized indicators (sameness of parts, morphological outcome and topological position) with novel ones derived from dynamical systems modelling (sameness of dynamical properties, dynamical complexity and evidence for transitional forms). We show how these criteria apply to animal segmentation and other ontogenetic processes. We conclude by situating our proposed dynamical framework for homology of process in relation to similar research programmes, such as process structuralism and developmental approaches to morphological homology.

Dynamical modules in metabolism, cell and developmental biology

Modularity is an essential feature of any adaptive complex system. Phenotypic traits are modules in the sense that they have a distinguishable structure or function, which can vary (quasi-)independently from its context. Since all phenotypic traits are the product of some underlying regulatory dynamics, the generative processes that constitute the genotype-phenotype map must also be functionally modular. Traditionally, modular processes have been identified as structural modules in regulatory networks. However, structure only constrains, but does not determine, the dynamics of a process. Here, we propose an alternative approach that decomposes the behaviour of a complex regulatory system into elementary activity-functions. Modular activities can occur in networks that show no structural modularity, making dynamical modularity more widely applicable than structural decomposition. Furthermore, the behaviour of a regulatory system closely mirrors its functional contribution to the outcome of a process, which makes dynamical modularity particularly suited for functional decomposition. We illustrate our approach with numerous examples from the study of metabolism, cellular processes, as well as development and pattern formation. We argue that dynamical modules provide a shared conceptual foundation for developmental and evolutionary biology, and serve as the foundation for a new account of process homology, which is presented in a separate contribution by DiFrisco and Jaeger to this focus issue.

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.

Inference of dynamic spatial GRN models with multi-GPU evolutionary computation

Reverse engineering mechanistic gene regulatory network (GRN) models with a specific dynamic spatial behavior is an inverse problem without analytical solutions in general. Instead, heuristic machine learning algorithms have been proposed to infer the structure and parameters of a system of equations able to recapitulate a given gene expression pattern. However, these algorithms are computationally intensive as they need to simulate millions of candidate models, which limits their applicability and requires high computational resources. Graphics processing unit (GPU) computing is an affordable alternative for accelerating large-scale scientific computation, yet no method is currently available to exploit GPU technology for the reverse engineering of mechanistic GRNs from spatial phenotypes. Here we present an efficient methodology to parallelize evolutionary algorithms using GPU computing for the inference of mechanistic GRNs that can develop a given gene expression pattern in a multicellular tissue area or cell culture. The proposed approach is based on multi-CPU threads running the lightweight crossover, mutation and selection operators and launching GPU kernels asynchronously. Kernels can run in parallel in a single or multiple GPUs and each kernel simulates and scores the error of a model using the thread parallelism of the GPU. We tested this methodology for the inference of spatiotemporal mechanistic gene regulatory networks (GRNs)-including topology and parameters-that can develop a given 2D gene expression pattern. The results show a 700-fold speedup with respect to a single CPU implementation. This approach can streamline the extraction of knowledge from biological and medical datasets and accelerate the automatic design of GRNs for synthetic biology applications.

Precision of tissue patterning is controlled by dynamical properties of gene regulatory networks

During development, gene regulatory networks allocate cell fates by partitioning tissues into spatially organised domains of gene expression. How the sharp boundaries that delineate these gene expression patterns arise, despite the stochasticity associated with gene regulation, is poorly understood. We show, in the vertebrate neural tube, using perturbations of coding and regulatory regions, that the structure of the regulatory network contributes to boundary precision. This is achieved, not by reducing noise in individual genes, but by the configuration of the network modulating the ability of stochastic fluctuations to initiate gene expression changes. We use a computational screen to identify network properties that influence boundary precision, revealing two dynamical mechanisms by which small gene circuits attenuate the effect of noise in order to increase patterning precision. These results highlight design principles of gene regulatory networks that produce precise patterns of gene expression.

Quantifying Temperature Compensation of Bicoid Gradients with a Fast T-Tunable Microfluidic Device

As a reaction-diffusion system strongly affected by temperature, early fly embryos surprisingly show highly reproducible and accurate developmental patterns during embryogenesis under temperature perturbations. To reveal the underlying temperature compensation mechanism, it is important to overcome the challenge in quantitative imaging on fly embryos under temperature perturbations. Inspired by microfluidics generating temperature steps on fly embryos, here we design a microfluidic device capable of ensuring the normal development of multiple fly embryos as well as achieving real-time temperature control and fast temperature switches for quantitative live imaging with a home-built two-photon microscope. We apply this system to quantify the temperature compensation of the morphogen Bicoid (Bcd) gradient in fly embryos. The length constant of the exponential Bcd gradient reaches the maximum at 25°C within the measured temperatures of 18-29°C and gradually adapts to the corresponding value at new temperatures upon a fast temperature switch. The relaxation time of such an adaptation becomes longer if the temperature is switched in a later developmental stage. This age-dependent temperature compensation could be explained if the traditional synthesis-diffusion-degradation model is extended to incorporate the dynamic change of the parameters controlling the formation of Bcd gradients.

Gene regulatory networks in Drosophila early embryonic development as a model for the study of the temporal identity of neuroblasts

Genes belonging to the "gap" and "gap-like" family constitute the best-studied gene regulatory networks (GRNs) in Drosophila embryogenesis. Gap genes are a core of two subnetworks controlling embryonic segmentation: (hunchback, hb; Krüppel, Kr; giant, gt; and knirps, kni) and (hb; Kr; pou-domain, pdm; and, probably, castor, cas). Of particular interest is that (hb, Kr, pdm, cas) also specifies the temporal identity of stem cells, neuroblasts, in Drosophila neurogenesis. This GRN controls the sequential differentiation of neuroblasts during the asymmetric cell division. In the last decades, modeling of the patterning of gene ensemble (hb, Kr, gt, kni) in segmentation was in the center of attention. We show that our previously published and extensively studied model at a certain level of external factors is able to reproduce temporal patterns of (hb, Kr, pdm, cas) in neurogenesis with minor evolutionary explicable modifications. This result testifies in favor of a hypothesis that the similarity of two gene ensembles active in segmentation and neurogenesis is a result of co-option of the network architecture in evolution from the common ancestral form. By means of the model dynamical analysis, it is shown that the establishment of the robust patterns in both systems could be explained in terms of the action of attractors in the gap gene dynamical system. We formulate the common principles underlying the robustness of both GRNs in segmentation and neurogenesis due to the similar functional organization of the gene ensembles as having the same evolutionary origin.

The dynamic transmission of positional information in stau- mutants during Drosophila embryogenesis

It has been suggested that Staufen (Stau) is key in controlling the variability of the posterior boundary of the Hb anterior domain (xHb). However, the mechanism that underlies this control is elusive. Here, we quantified the dynamic 3D expression of segmentation genes in Drosophila embryos. With improved control of measurement errors, we show that the xHb of stau- mutants reproducibly moves posteriorly by 10% of the embryo length (EL) to the wild type (WT) position in the nuclear cycle (nc) 14, and that its variability over short time windows is comparable to that of the WT. Moreover, for stau- mutants, the upstream Bicoid (Bcd) gradients show equivalent relative intensity noise to that of the WT in nc12-nc14, and the downstream Even-skipped (Eve) and cephalic furrow (CF) show the same positional errors as these factors in WT. Our results indicate that threshold-dependent activation and self-organized filtering are not mutually exclusive and could both be implemented in early Drosophila embryogenesis.

Dynamic positional information: Patterning mechanism versus precision in gradient-driven systems

There is much talk about information in biology. In developmental biology, this takes the form of "positional information," especially in the context of morphogen-based pattern formation. Unfortunately, the concept of "information" is rarely defined in any precise manner. Here, we provide two alternative interpretations of "positional information," and examine the complementary meanings and uses of each concept. Positional information defined as Shannon information helps us understand decoding and error propagation in patterning systems. General relativistic positional information, in contrast, provides a metric to assess the output of pattern-forming mechanisms. Both interpretations provide powerful conceptual tools that do not compete, but are best used in combination to gain a proper mechanistic understanding of robust patterning.

Lighting up the central dogma for predictive developmental biology

Although the last 30years have witnessed the mapping of the wiring diagrams of the gene regulatory networks that dictate cell fate and animal body plans, specific understanding building on such network diagrams that shows how DNA regulatory regions control gene expression lags far behind. These networks have yet to yield the predictive power necessary to, for example, calculate how the concentration dynamics of input transcription factors and DNA regulatory sequence prescribes output patterns of gene expression that, in turn, determine body plans themselves. Here, we argue that reaching a predictive understanding of developmental decision-making calls for an interplay between theory and experiment aimed at revealing how the regulation of the processes of the central dogma dictate network connections and how network topology guides cells toward their ultimate developmental fate. To make this possible, it is crucial to break free from the snapshot-based understanding of embryonic development facilitated by fixed-tissue approaches and embrace new technologies that capture the dynamics of developmental decision-making at the single cell level, in living embryos.

Dynamic control of the T-cell specification gene regulatory network

Specification of multipotent blood precursor cells in postnatal mice to become committed T-cell precursors involves a gene regulatory network of several interacting but functionally distinct modules. Many links of this network have been defined by perturbation tests and by functional genomics. However, using the network model to predict real-life kinetics of the commitment process is still difficult, partly due to the tenacity of repressive chromatin states, and to the ability of transcription factors to affect each other's binding site choices through competitive recruitment to alternative sites ("coregulator theft"). To predict kinetics, future models will need to incorporate mechanistic information about chromatin state change dynamics and more sophisticated understanding of the proteomics and cooperative DNA site choices of transcription factor complexes.

Inherency of Form and Function in Animal Development and Evolution

I discuss recent work on the origins of morphology and cell-type diversification in Metazoa – collectively the animals – and propose a scenario for how these two properties became integrated, with the help of a third set of processes, cellular pattern formation, into the developmental programs seen in present-day metazoans. Inherent propensities to generate familiar forms and cell types, in essence a parts kit for the animals, are exhibited by present-day organisms and were likely more prominent in primitive ones. The structural motifs of animal bodies and organs, e.g., multilayered, hollow, elongated and segmented tissues, internal and external appendages, branched tubes, and modular endoskeletons, can be accounted for by the properties of mesoscale masses of metazoan cells. These material properties, in turn, resulted from the recruitment of “generic” physical forces and mechanisms – adhesion, contraction, polarity, chemical oscillation, diffusion – by toolkit molecules that were partly conserved from unicellular holozoan antecedents and partly novel, distributed in the different metazoan phyla in a fashion correlated with morphological complexity. The specialized functions of the terminally differentiated cell types in animals, e.g., contraction, excitability, barrier function, detoxification, excretion, were already present in ancestral unicellular organisms. These functions were implemented in metazoan differentiation in some cases using the same transcription factors as in single-celled ancestors, although controlled by regulatory mechanisms that were hybrids between earlier-evolved processes and regulatory innovations, such as enhancers. Cellular pattern formation, mediated by released morphogens interacting with biochemically responsive and excitable tissues, drew on inherent self-organizing processes in proto-metazoans to transform clusters of holozoan cells into animal embryos and organs.

Modularity, criticality, and evolvability of a developmental gene regulatory network

The existence of discrete phenotypic traits suggests that the complex regulatory processes which produce them are functionally modular. These processes are usually represented by networks. Only modular networks can be partitioned into intelligible subcircuits able to evolve relatively independently. Traditionally, functional modularity is approximated by detection of modularity in network structure. However, the correlation between structure and function is loose. Many regulatory networks exhibit modular behaviour without structural modularity. Here we partition an experimentally tractable regulatory network-the gap gene system of dipteran insects-using an alternative approach. We show that this system, although not structurally modular, is composed of dynamical modules driving different aspects of whole-network behaviour. All these subcircuits share the same regulatory structure, but differ in components and sensitivity to regulatory interactions. Some subcircuits are in a state of criticality, while others are not, which explains the observed differential evolvability of the various expression features in the system.

Oscillations in well-mixed, deterministic feedback systems: Beyond ring oscillators

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I present a way of breaking down regulatory networks to find Hopf bifurcations. This helps find optimal conditions for oscillations in dynamical systems models of these networks.

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In a model of negative auto-regulation of a gene by its dimeric protein, it is optimal for the monomer to degrade faster than the mRNA and the mRNA to degrade faster than the dimer.

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Adding a weak positive feedback loop to a repressilator increases the probability of oscillations.

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The optimal degradation rate of species in the sub-loop is higher than that of species outside it.

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The opposite is true for a negative feedback sub-loop or a very strong positive feedback sub-loop.

A ring oscillator is a system in which one species regulates the next, which regulates the next and so on until the last species regulates the first. In addition, the number of the regulations which are negative, and so result in a reduction in the regulated species, is odd, making the overall feedback in the loop negative. In ring oscillators, the probability of oscillations is maximised if the degradation rates of the species are equal. When there is more than one loop in the regulatory network, the dynamics can be more complicated. Here, a systematic way of organising the characteristic equation of ODE models of regulatory networks is provided. This facilitates the identification of Hopf bifurcations. It is shown that the probability of oscillations in non-ring systems is maximised for unequal degradation rates. For example, when there is a ring and a second ring employing a subset of the genes in the first ring, then the probability of oscillations is maximised when the species in the sub-ring degrade more slowly than those outside, for a negative feedback subring. When the sub-ring forms a positive feedback loop, the optimal degradation rates are larger for the species in the sub-ring, provided the positive feedback is not too strong. By contrast, optimal degradation rates are smaller for the species in the sub-ring, when the positive feedback is very strong. Adding a positive feedback loop to a repressilator increases the probability of oscillations, provided the positive feedback is not too strong, whereas adding a negative feedback loop decreases the probability of oscillations. The work is illustrated with numerical simulations of example systems: an autoregulatory gene model in which transcription is downregulated by the protein dimer and three-species and four-species gene regulatory network examples.

Modeling Edar expression reveals the hidden dynamics of tooth signaling center patterning

When patterns are set during embryogenesis, it is expected that they are straightly established rather than subsequently modified. The patterning of the three mouse molars is, however, far from straight, likely as a result of mouse evolutionary history. The first-formed tooth signaling centers, called MS and R2, disappear before driving tooth formation and are thought to be vestiges of the premolars found in mouse ancestors. Moreover, the mature signaling center of the first molar (M1) is formed from the fusion of two signaling centers (R2 and early M1). Here, we report that broad activation of Edar expression precedes its spatial restriction to tooth signaling centers. This reveals a hidden two-step patterning process for tooth signaling centers, which was modeled with a single activator-inhibitor pair subject to reaction-diffusion (RD). The study of Edar expression also unveiled successive phases of signaling center formation, erasing, recovering, and fusion. Our model, in which R2 signaling center is not intrinsically defective but erased by the broad activation preceding M1 signaling center formation, predicted the surprising rescue of R2 in Edar mutant mice, where activation is reduced. The importance of this R2-M1 interaction was confirmed by ex vivo cultures showing that R2 is capable of forming a tooth. Finally, by introducing chemotaxis as a secondary process to RD, we recapitulated in silico different conditions in which R2 and M1 centers fuse or not. In conclusion, pattern formation in the mouse molar field relies on basic mechanisms whose dynamics produce embryonic patterns that are plastic objects rather than fixed end points.

Encounters across networks: Windows into principles of genomic regulation

Gene regulatory networks account for the ability of the genome to program development in complex multi-cellular organisms. Such networks are based on principles of gene regulation by combinations of transcription factors that bind to specific cis-regulatory DNA sites to activate transcription. These cis-regulatory regions mediate logic processing at each network node, enabling progressive increases in organismal complexity with development. Gene regulatory network explanations of development have been shown to account for patterning and cell type diversification in fly and sea urchin embryonic systems, where networks are characterized by fast coupling between transcriptional inputs and changes in target gene transcription rates, and crucial cis-regulatory elements are concentrated relatively close to the protein coding sequences of the target genes, thus facilitating their identification. Stem cell-based development in post-embryonic mammalian systems also depends on gene networks, but differs from the fly and sea urchin systems. First, the number of regulatory elements per gene and the distances between regulatory elements and the genes they control are considerably larger, forcing searches via genome-wide transcription factor binding surveys rather than functional assays. Second, the intrinsic timing of network state transitions can be slowed considerably by the need to undo stem-cell chromatin configurations, which presumably add stability to stem-cell states but retard responses to transcription factor changes during differentiation. The dispersed, partially redundant cis-regulatory systems controlling gene expression and the slow state transition kinetics in these systems already reveal new insights and opportunities to extend understanding of the repertoire of gene networks and regulatory system logic.

A re-inducible gap gene cascade patterns the anterior-posterior axis of insects in a threshold-free fashion

Gap genes mediate the division of the anterior-posterior axis of insects into different fates through regulating downstream hox genes. Decades of tinkering the segmentation gene network of Drosophila melanogaster led to the conclusion that gap genes are regulated (at least initially) through a threshold-based mechanism, guided by both anteriorly- and posteriorly-localized morphogen gradients. In this paper, we show that the response of the gap gene network in the beetle Tribolium castaneum upon perturbation is consistent with a threshold-free ‘Speed Regulation’ mechanism, in which the speed of a genetic cascade of gap genes is regulated by a posterior morphogen gradient. We show this by re-inducing the leading gap gene (namely, hunchback) resulting in the re-induction of the gap gene cascade at arbitrary points in time. This demonstrates that the gap gene network is self-regulatory and is primarily under the control of a posterior regulator in Tribolium and possibly other short/intermediate-germ insects.

A damped oscillator imposes temporal order on posterior gap gene expression in Drosophila

Insects determine their body segments in two different ways. Short-germband insects, such as the flour beetle Tribolium castaneum, use a molecular clock to establish segments sequentially. In contrast, long-germband insects, such as the vinegar fly Drosophila melanogaster, determine all segments simultaneously through a hierarchical cascade of gene regulation. Gap genes constitute the first layer of the Drosophila segmentation gene hierarchy, downstream of maternal gradients such as that of Caudal (Cad). We use data-driven mathematical modelling and phase space analysis to show that shifting gap domains in the posterior half of the Drosophila embryo are an emergent property of a robust damped oscillator mechanism, suggesting that the regulatory dynamics underlying long- and short-germband segmentation are much more similar than previously thought. In Tribolium, Cad has been proposed to modulate the frequency of the segmentation oscillator. Surprisingly, our simulations and experiments show that the shift rate of posterior gap domains is independent of maternal Cad levels in Drosophila. Our results suggest a novel evolutionary scenario for the short- to long-germband transition and help explain why this transition occurred convergently multiple times during the radiation of the holometabolan insects.

Different insect species exhibit one of two distinct modes of determining their body segments (known as segmentation) during development: they either use a molecular oscillator to position segments sequentially, or they generate segments simultaneously through a hierarchical gene-regulatory cascade. The sequential mode is ancestral, while the simultaneous mode has been derived from it independently several times during evolution. In this paper, we present evidence suggesting that simultaneous segmentation also involves an oscillator in the posterior end of the embryo of the vinegar fly, Drosophila melanogaster. This surprising result indicates that both modes of segment determination are much more similar than previously thought. Such similarity provides an important step towards our understanding of the frequent evolutionary transitions observed between sequential and simultaneous segmentation.

Speed regulation of genetic cascades allows for evolvability in the body plan specification of insects

During the anterior-posterior fate specification of insects, anterior fates arise in a nonelongating tissue (called the "blastoderm"), and posterior fates arise in an elongating tissue (called the "germband"). However, insects differ widely in the extent to which anterior-posterior fates are specified in the blastoderm versus the germband. Here we present a model in which patterning in both the blastoderm and germband of the beetle Tribolium castaneum is based on the same flexible mechanism: a gradient that modulates the speed of a genetic cascade of gap genes, resulting in the induction of sequential kinematic waves of gap gene expression. The mechanism is flexible and capable of patterning both elongating and nonelongating tissues, and hence converting blastodermal to germband fates and vice versa. Using RNAi perturbations, we found that blastodermal fates could be shifted to the germband, and germband fates could be generated in a blastoderm-like morphology. We also suggest a molecular mechanism underlying our model, in which gradient levels regulate the switch between two enhancers: One enhancer is responsible for sequential gene activation, and the other is responsible for freezing temporal rhythms into spatial patterns. This model is consistent with findings in Drosophila melanogaster, where gap genes were found to be regulated by two nonredundant "shadow" enhancers.

Decoding temporal interpretation of the morphogen Bicoid in the early Drosophila embryo

Morphogen gradients provide essential spatial information during development. Not only the local concentration but also duration of morphogen exposure is critical for correct cell fate decisions. Yet, how and when cells temporally integrate signals from a morphogen remains unclear. Here, we use optogenetic manipulation to switch off Bicoid-dependent transcription in the early Drosophila embryo with high temporal resolution, allowing time-specific and reversible manipulation of morphogen signalling. We find that Bicoid transcriptional activity is dispensable for embryonic viability in the first hour after fertilization, but persistently required throughout the rest of the blastoderm stage. Short interruptions of Bicoid activity alter the most anterior cell fate decisions, while prolonged inactivation expands patterning defects from anterior to posterior. Such anterior susceptibility correlates with high reliance of anterior gap gene expression on Bicoid. Therefore, cell fates exposed to higher Bicoid concentration require input for longer duration, demonstrating a previously unknown aspect of Bicoid decoding.

DOI:http://dx.doi.org/10.7554/eLife.26258.001

A spectrum of modularity in multi-functional gene circuits

A major challenge in systems biology is to understand the relationship between a circuit's structure and its function, but how is this relationship affected if the circuit must perform multiple distinct functions within the same organism? In particular, to what extent do multi‐functional circuits contain modules which reflect the different functions? Here, we computationally survey a range of bi‐functional circuits which show no simple structural modularity: They can switch between two qualitatively distinct functions, while both functions depend on all genes of the circuit. Our analysis reveals two distinct classes: hybrid circuits which overlay two simpler mono‐functional sub‐circuits within their circuitry, and emergent circuits, which do not. In this second class, the bi‐functionality emerges from more complex designs which are not fully decomposable into distinct modules and are consequently less intuitive to predict or understand. These non‐intuitive emergent circuits are just as robust as their hybrid counterparts, and we therefore suggest that the common bias toward studying modular systems may hinder our understanding of real biological circuits.