Analysis of gap gene regulation in a 3D organism-scale model of the Drosophila melanogaster embryo

Dynamic Modeling of Transcriptional Gene Regulatory Networks

Diverse cellular phenotypes are determined by groups of transcription factors (TFs) and other regulators that influence each others' gene expression, forming transcriptional gene regulatory networks (GRNs). In many biological contexts, especially in development and associated diseases, the expression of the genes in GRNs is not static but evolves in time. Modeling the dynamics of GRN state is an important approach for understanding diverse cellular phenomena such as cell-fate specification, pluripotency and cell-fate reprogramming, oncogenesis, and tissue regeneration. In this protocol, we describe how to model GRNs using a data-driven dynamic modeling methodology, gene circuits. Gene circuits do not require knowledge of the GRN topology and connectivity but instead learn them from training data, making them very general and applicable to diverse biological contexts. We utilize the MATLAB-based gene circuit modeling software Fast Inference of Gene Regulation (FIGR) for training the model on quantitative gene expression data and simulating the GRN. We describe all the steps in the modeling life cycle, from formulating the model, training the model using FIGR, simulating the GRN, to analyzing and interpreting the model output. This protocol highlights these steps with the example of a dynamical model of the gap gene GRN involved in Drosophila segmentation and includes example MATLAB statements for each step.

Embryonic geometry underlies phenotypic variation in decanalized conditions

During development, many mutations cause increased variation in phenotypic outcomes, a phenomenon termed decanalization. Phenotypic discordance is often observed in the absence of genetic and environmental variations, but the mechanisms underlying such inter-individual phenotypic discordance remain elusive. Here, using the anterior-posterior (AP) patterning of the Drosophila embryo, we identified embryonic geometry as a key factor predetermining patterning outcomes under decanalizing mutations. With the wild-type AP patterning network, we found that AP patterning is robust to variations in embryonic geometry; segmentation gene expression remains reproducible even when the embryo aspect ratio is artificially reduced by more than twofold. In contrast, embryonic geometry is highly predictive of individual patterning defects under decanalized conditions of either increased bicoid (bcd) dosage or bcd knockout. We showed that the phenotypic discordance can be traced back to variations in the gap gene expression, which is rendered sensitive to the geometry of the embryo under mutations.

Evaluation of BMP-mediated patterning in a 3D mathematical model of the zebrafish blastula embryo

Bone Morphogenetic Proteins (BMPs) play an important role in dorsal-ventral (DV) patterning of the early zebrafish embryo. BMP signaling is regulated by a network of extracellular and intracellular factors that impact the range and signaling of BMP ligands. Recent advances in understanding the mechanism of pattern formation support a source-sink mechanism, however it is not clear how the source-sink mechanism shapes patterns in 3D, nor how sensitive the pattern is to biophysical rates and boundary conditions along both the anteroposterior (AP) and DV axes of the embryo. We propose a new three-dimensional growing Partial Differential Equation (PDE)-based model to simulate the BMP patterning process during the blastula stage. This model provides a starting point to elucidate how different mechanisms and components work together in 3D to create and maintain the BMP gradient in the embryo. We also show how the 3D model fits the BMP signaling gradient data at multiple time points along both axes. Furthermore, sensitivity analysis of the model suggests that the spatiotemporal patterns of Chordin and BMP ligand gene expression are dominant drivers of shape in 3D and more work is needed to quantify the spatiotemporal profiles of gene and protein expression to further refine the models.

Classification-Based Inference of Dynamical Models of Gene Regulatory Networks

Cell-fate decisions during development are controlled by densely interconnected gene regulatory networks (GRNs) consisting of many genes. Inferring and predictively modeling these GRNs is crucial for understanding development and other physiological processes. Gene circuits, coupled differential equations that represent gene product synthesis with a switch-like function, provide a biologically realistic framework for modeling the time evolution of gene expression. However, their use has been limited to smaller networks due to the computational expense of inferring model parameters from gene expression data using global non-linear optimization. Here we show that the switch-like nature of gene regulation can be exploited to break the gene circuit inference problem into two simpler optimization problems that are amenable to computationally efficient supervised learning techniques. We present FIGR (Fast Inference of Gene Regulation), a novel classification-based inference approach to determining gene circuit parameters. We demonstrate FIGR’s effectiveness on synthetic data generated from random gene circuits of up to 50 genes as well as experimental data from the gap gene system of Drosophila melanogaster, a benchmark for inferring dynamical GRN models. FIGR is faster than global non-linear optimization by a factor of 600 and its computational complexity scales much better with GRN size. On a practical level, FIGR can accurately infer the biologically realistic gap gene network in under a minute on desktop-class hardware instead of requiring hours of parallel computing. We anticipate that FIGR would enable the inference of much larger biologically realistic GRNs than was possible before.

Emergence of form in embryogenesis

The development of form in an embryo is the result of a series of topological and informational symmetry breakings. We introduce the vector-reaction-diffusion-drift (VRDD) system where the limit cycle of spatial dynamics is morphogen concentrations with Dirac delta-type distributions. This is fundamentally different from the Turing reaction-diffusion system, as VRDD generates system-wide broken symmetry. We developed 'fundamental forms' from spherical blastula with a single organizing axis (rotational symmetry), double axis (mirror symmetry) and triple axis (no symmetry operator in three dimensions). We then introduced dynamics for cell differentiation, where genetic regulatory states are modelled as a finite-state machine (FSM). The state switching of an FSM is based on local morphogen concentrations as epigenetic information that changes dynamically. We grow complicated forms hierarchically in spatial subdomains using the FSM model coupled with the VRDD system. Using our integrated simulation model with four layers (topological, physical, chemical and regulatory), we generated life-like forms such as hydra. Genotype-phenotype mapping was investigated with continuous and jump mutations. Our study can have applications in morphogenetic engineering, soft robotics and biomimetic design.

A gene expression atlas of a bicoid-depleted Drosophila embryo reveals early canalization of cell fate

In developing embryos, gene regulatory networks drive cells towards discrete terminal fates, a process called canalization. We studied the behavior of the anterior-posterior segmentation network in Drosophila melanogaster embryos by depleting a key maternal input, bicoid (bcd), and measuring gene expression patterns of the network at cellular resolution. This method results in a gene expression atlas containing the levels of mRNA or protein expression of 13 core patterning genes over six time points for every cell of the blastoderm embryo. This is the first cellular resolution dataset of a genetically perturbed Drosophila embryo that captures all cells in 3D. We describe the technical developments required to build this atlas and how the method can be employed and extended by others. We also analyze this novel dataset to characterize the degree and timing of cell fate canalization in the segmentation network. We find that in two layers of this gene regulatory network, following depletion of bcd, individual cells rapidly canalize towards normal cell fates. This result supports the hypothesis that the segmentation network directly canalizes cell fate, rather than an alternative hypothesis whereby cells are initially mis-specified and later eliminated by apoptosis. Our gene expression atlas provides a high resolution picture of a classic perturbation and will enable further computational modeling of canalization and gene regulation in this transcriptional network.

Making models match measurements: model optimization for morphogen patterning networks

Mathematical modeling of developmental signaling networks has played an increasingly important role in the identification of regulatory mechanisms by providing a sandbox for hypothesis testing and experiment design. Whether these models consist of an equation with a few parameters or dozens of equations with hundreds of parameters, a prerequisite to model-based discovery is to bring simulated behavior into agreement with observed data via parameter estimation. These parameters provide insight into the system (e.g., enzymatic rate constants describe enzyme properties). Depending on the nature of the model fit desired - from qualitative (relative spatial positions of phosphorylation) to quantitative (exact agreement of spatial position and concentration of gene products) - different measures of data-model mismatch are used to estimate different parameter values, which contain different levels of usable information and/or uncertainty. To facilitate the adoption of modeling as a tool for discovery alongside other tools such as genetics, immunostaining, and biochemistry, careful consideration needs to be given to how well a model fits the available data, what the optimized parameter values mean in a biological context, and how the uncertainty in model parameters and predictions plays into experiment design. The core discussion herein pertains to the quantification of model-to-data agreement, which constitutes the first measure of a model's performance and future utility to the problem at hand. Integration of this experimental data and the appropriate choice of objective measures of data-model agreement will continue to drive modeling forward as a tool that contributes to experimental discovery. The Drosophila melanogaster gap gene system, in which model parameters are optimized against in situ immunofluorescence intensities, demonstrates the importance of error quantification, which is applicable to a wide array of developmental modeling studies.

Mechanisms of scaling in pattern formation

Many organisms and their constituent tissues and organs vary substantially in size but differ little in morphology; they appear to be scaled versions of a common template or pattern. Such scaling involves adjusting the intrinsic scale of spatial patterns of gene expression that are set up during development to the size of the system. Identifying the mechanisms that regulate scaling of patterns at the tissue, organ and organism level during development is a longstanding challenge in biology, but recent molecular-level data and mathematical modeling have shed light on scaling mechanisms in several systems, including Drosophila and Xenopus. Here, we investigate the underlying principles needed for understanding the mechanisms that can produce scale invariance in spatial pattern formation and discuss examples of systems that scale during development.

The importance of geometry in mathematical models of developing systems

Understanding the interaction between the spatial variation of extracellular signals and the interpretation of such signals in embryonic development is difficult without a mathematical model, but the inherent limitations of a model can have a profound impact on its utility. A central issue is the level of abstraction needed, and here we focus on the role of geometry in models and how the choice of the spatial dimension can influence the conclusions reached. A widely studied system in which the proper choice of geometry is critical is embryonic development of Drosophila melanogaster, and we discuss recent work in which 3D embryo-scale modeling is used to identify key modes of transport, analyze gap gene expression, and test BMP-mediated positive feedback mechanisms.

The synthesis-diffusion-degradation model explains Bicoid gradient formation in unfertilized eggs

Precise formation of morphogen gradients is essential to the establishment of reproducible pattern in development. Mechanisms proposed for obtaining the requisite precision range from simple models with few parameters to more complex models involving many regulated quantities. The synthesis-diffusion-degradation (SDD) model is a relatively simple model explaining the formation of the Bicoid gradient in Drosophila melanogaster, in which the steady-state characteristic length of the gradient is determined solely by the rates of diffusion and degradation of the morphogen. In this work, we test the SDD model in unfertilized D. melanogaster eggs, which contain a single female pronucleus and lack the nuclear division cycles and other zygotic regulatory processes seen in fertilized eggs. Using two-photon live imaging as well as a novel method for quantitative imaging based on decorrelation of photoswitching waveforms, we find that the Bicoid gradient is longer and shallower in unfertilized eggs as compared to the gradient at the same time points in fertilized eggs. Using a means of measuring the Bicoid lifetime by conjugation to a photoconvertible fluorophore, we find that the lifetime is correspondingly longer in unfertilized eggs, providing qualitative and quantitative agreement with the predictions of the SDD model.

BMP signaling in wing development: A critical perspective on quantitative image analysis

Bone Morphogenetic Proteins (BMPs) are critical for pattern formation in many animals. In numerous tissues, BMPs become distributed in spatially non-uniform profiles. The gradients of signaling activity can be detected by a number of biological assays involving fluorescence microscopy. Quantitative analyses of BMP gradients are powerful tools to investigate the regulation of BMP signaling pathways during development. These approaches rely heavily on images as spatial representations of BMP activity levels, using them to infer signaling distributions that inform on regulatory mechanisms. In this perspective, we discuss current imaging assays and normalization methods used to quantify BMP activity profiles with a focus on the Drosophila wing primordium. We find that normalization tends to lower the number of samples required to establish statistical significance between profiles in controls and experiments, but the increased resolvability comes with a cost. Each normalization strategy makes implicit assumptions about the biology that impacts our interpretation of the data. We examine the tradeoffs for normalizing versus not normalizing, and discuss their impacts on experimental design and the interpretation of resultant data.