Quantitative Models of Developmental Pattern Formation

Deterministic genetic barcoding for multiplexed behavioral and single-cell transcriptomic studies

Advances in single-cell sequencing technologies have provided novel insights into the dynamics of gene expression and cellular heterogeneity within tissues and have enabled the construction of transcriptomic cell atlases. However, linking anatomical information to transcriptomic data and positively identifying the cell types that correspond to gene expression clusters in single-cell sequencing data sets remains a challenge. We describe a straightforward genetic barcoding approach that takes advantage of the powerful genetic tools in Drosophila to allow in vivo tagging of defined cell populations. This method, called Targeted Genetically-Encoded Multiplexing (TaG-EM), involves inserting a DNA barcode just upstream of the polyadenylation site in a Gal4-inducible UAS-GFP construct so that the barcode sequence can be read out during single-cell sequencing, labeling a cell population of interest. By creating many such independently barcoded fly strains, TaG-EM enables positive identification of cell types in cell atlas projects, identification of multiplet droplets, and barcoding of experimental timepoints, conditions, and replicates. Furthermore, we demonstrate that TaG-EM barcodes can be read out using next-generation sequencing to facilitate population-scale behavioral measurements. Thus, TaG-EM has the potential to enable large-scale behavioral screens in addition to improving the ability to multiplex and reliably annotate single-cell transcriptomic experiments.

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.

Microfluidic Tissue Engineering and Bio-Actuation

Bio-hybrid technologies aim to replicate the unique capabilities of biological systems that could surpass advanced artificial technologies. Soft bio-hybrid robots consist of synthetic and living materials and have the potential to self-assemble, regenerate, work autonomously, and interact safely with other species and the environment. Cells require a sufficient exchange of nutrients and gases, which is guaranteed by convection and diffusive transport through liquid media. The functional development and long-term survival of biological tissues in vitro can be improved by dynamic flow culture, but only microfluidic flow control can develop tissue with fine structuring and regulation at the microscale. Full control of tissue growth at the microscale will eventually lead to functional macroscale constructs, which are needed as the biological component of soft bio-hybrid technologies. This review summarizes recent progress in microfluidic techniques to engineer biological tissues, focusing on the use of muscle cells for robotic bio-actuation. Moreover, the instances in which bio-actuation technologies greatly benefit from fusion with microfluidics are highlighted, which include: the microfabrication of matrices, biomimicry of cell microenvironments, tissue maturation, perfusion, and vascularization.

To be more precise: the role of intracellular trafficking in development and pattern formation

Living cells interpret a variety of signals in different contexts to elucidate functional responses. While the understanding of signalling molecules, their respective receptors and response at the gene transcription level have been relatively well-explored, how exactly does a single cell interpret a plethora of time-varying signals? Furthermore, how their subsequent responses at the single cell level manifest in the larger context of a developing tissue is unknown. At the same time, the biophysics and chemistry of how receptors are trafficked through the complex dynamic transport network between the plasma membrane-endosome-lysosome-Golgi-endoplasmic reticulum are much more well-studied. How the intracellular organisation of the cell and inter-organellar contacts aid in orchestrating trafficking, as well as signal interpretation and modulation by the cells are beginning to be uncovered. In this review, we highlight the significant developments that have strived to integrate endosomal trafficking, signal interpretation in the context of developmental biology and relevant open questions with a few chosen examples. Furthermore, we will discuss the imaging technologies that have been developed in the recent past that have the potential to tremendously accelerate knowledge gain in this direction while shedding light on some of the many challenges.

Patterns of Polar Domains in a Spatiotemporal Model of Interacting Polarities

Polarity fields are known to exhibit long distance patterns, in both physical and biological systems. The mechanisms behind such patterns are poorly understood. Here, we describe the dynamics of polarity fields using an original physical model that generalizes classical spin models on a lattice by incorporating effective transport of polarity molecules between neighboring sites. We account for an external field and for ferromagnetic interactions between sites and prescribe the time evolution of the system using two distinct dissipative classes for nonconserved and conserved variables representing polarity orientation and magnitude, respectively. We observe two main types of steady-state configurations-disordered configurations and patterns of highly polar spots surrounded by regions with low polarity-and we characterize patterns and transitions between configurations. Our results may provide alternative pattern-generating mechanisms for materials endowed with polarity fields.

Tools to reverse-engineer multicellular systems: case studies using the fruit fly

Reverse-engineering how complex multicellular systems develop and function is a grand challenge for systems bioengineers. This challenge has motivated the creation of a suite of bioengineering tools to develop increasingly quantitative descriptions of multicellular systems. Here, we survey a selection of these tools including microfluidic devices, imaging and computer vision techniques. We provide a selected overview of the emerging cross-talk between engineering methods and quantitative investigations within developmental biology. In particular, the review highlights selected recent examples from the Drosophila system, an excellent platform for understanding the interplay between genetics and biophysics. In sum, the integrative approaches that combine multiple advances in these fields are increasingly necessary to enable a deeper understanding of how to analyze both natural and synthetic multicellular systems.

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.

Improving Parameter Inference from FRAP Data: an Analysis Motivated by Pattern Formation in the Drosophila Wing Disc

Fluorescence recovery after photobleaching (FRAP) is used to obtain quantitative information about molecular diffusion and binding kinetics at both cell and tissue levels of organization. FRAP models have been proposed to estimate the diffusion coefficients and binding kinetic parameters of species for a variety of biological systems and experimental settings. However, it is not clear what the connection among the diverse parameter estimates from different models of the same system is, whether the assumptions made in the model are appropriate, and what the qualities of the estimates are. Here we propose a new approach to investigate the discrepancies between parameters estimated from different models. We use a theoretical model to simulate the dynamics of a FRAP experiment and generate the data that are used in various recovery models to estimate the corresponding parameters. By postulating a recovery model identical to the theoretical model, we first establish that the appropriate choice of observation time can significantly improve the quality of estimates, especially when the diffusion and binding kinetics are not well balanced, in a sense made precise later. Secondly, we find that changing the balance between diffusion and binding kinetics by changing the size of the bleaching region, which gives rise to different FRAP curves, provides a priori knowledge of diffusion and binding kinetics, which is important for model formulation. We also show that the use of the spatial information in FRAP provides better parameter estimation. By varying the recovery model from a fixed theoretical model, we show that a simplified recovery model can adequately describe the FRAP process in some circumstances and establish the relationship between parameters in the theoretical model and those in the recovery model. We then analyze an example in which the data are generated with a model of intermediate complexity and the parameters are estimated using models of greater or less complexity, and show how sensitivity analysis can be used to improve FRAP model formulation. Lastly, we show how sophisticated global sensitivity analysis can be used to detect over-fitting when using a model that is too complex.

Build to understand: synthetic approaches to biology

In this review we discuss how synthetic biology facilitates the task of investigating genetic circuits that are observed in naturally occurring biological systems. Specifically, we give examples where experimentation with synthetic gene circuits has been used to understand four fundamental mechanisms intrinsic to development and disease: multistability, stochastic gene expression, oscillations, and cell-cell communication. Within each area, we also discuss how mathematical modeling has been employed as an essential tool to guide the design of novel gene circuits and as a theoretical basis for exploring circuit topologies exhibiting robust behaviors in the presence of noise.

Regulation of ERK-MAPK signaling in human epidermis

Background

The skin is largely comprised of keratinocytes within the interfollicular epidermis. Over approximately two weeks these cells differentiate and traverse the thickness of the skin. The stage of differentiation is therefore reflected in the positions of cells within the tissue, providing a convenient axis along which to study the signaling events that occur in situ during keratinocyte terminal differentiation, over this extended two-week timescale. The canonical ERK-MAPK signaling cascade (Raf-1, MEK-1/2 and ERK-1/2) has been implicated in controlling diverse cellular behaviors, including proliferation and differentiation. While the molecular interactions involved in signal transduction through this cascade have been well characterized in cell culture experiments, our understanding of how this sequence of events unfolds to determine cell fate within a homeostatic tissue environment has not been fully characterized.

Methods

We measured the abundance of total and phosphorylated ERK-MAPK signaling proteins within interfollicular keratinocytes in transverse cross-sections of human epidermis using immunofluorescence microscopy. To investigate these data we developed a mathematical model of the signaling cascade using a normalized-Hill differential equation formalism.

Results

These data show coordinated variation in the abundance of phosphorylated ERK-MAPK components across the epidermis. Statistical analysis of these data shows that associations between phosphorylated ERK-MAPK components which correspond to canonical molecular interactions are dependent upon spatial position within the epidermis. The model demonstrates that the spatial profile of activation for ERK-MAPK signaling components across the epidermis may be maintained in a cell-autonomous fashion by an underlying spatial gradient in calcium signaling.

Conclusions

Our data demonstrate an extended phospho-protein profile of ERK-MAPK signaling cascade components across the epidermis in situ, and statistical associations in these data indicate canonical ERK-MAPK interactions underlie this spatial profile of ERK-MAPK activation. Using mathematical modelling we have demonstrated that spatially varying calcium signaling components across the epidermis may be sufficient to maintain the spatial profile of ERK-MAPK signaling cascade components in a cell-autonomous manner. These findings may have significant implications for the wide range of cancer drugs which therapeutically target ERK-MAPK signaling components.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0187-6) contains supplementary material, which is available to authorized users.

Modeling Scalable Pattern Generation in DNA Reaction Networks

We have developed a theoretical framework for developing patterns in multiple dimensions using controllable diffusion and designed reactions implemented in DNA. This includes so-called strand displacement reactions in which one single-stranded DNA hybridizes to a hemi-duplex DNA and displaces another single-stranded DNA, reversibly or irreversibly. These reactions can be designed to proceed with designed rate and molecular specificity. By also controlling diffusion by partial complementarity to a stationary, cross-linked DNA, we can generate predictable patterns. We demonstrate this with several simulations showing deterministic, predictable shapes in space.

Systems biology of cancer: a challenging expedition for clinical and quantitative biologists

A systems-biology approach to complex disease (such as cancer) is now complementing traditional experience-based approaches, which have typically been invasive and expensive. The rapid progress in biomedical knowledge is enabling the targeting of disease with therapies that are precise, proactive, preventive, and personalized. In this paper, we summarize and classify models of systems biology and model checking tools, which have been used to great success in computational biology and related fields. We demonstrate how these models and tools have been used to study some of the twelve biochemical pathways implicated in but not unique to pancreatic cancer, and conclude that the resulting mechanistic models will need to be further enhanced by various abstraction techniques to interpret phenomenological models of cancer progression.

A computational approach towards a gene regulatory network for the developing Nematostella vectensis gut

Background

The starlet sea anemone Nematostella vectensis is a diploblastic cnidarian that expresses a set of conserved genes for gut formation during its early development. During the last decade, the spatial distribution of many of these genes has been visualized with RNA hybridization or protein immunolocalization techniques. However, due to N. vectensis' curved and changing morphology, quantification of these spatial data is problematic. A method is developed for two-dimensional gene expression quantification, which enables a numerical analysis and dynamic modeling of these spatial patterns.

Methods/Result

In this work, first standardized gene expression profiles are generated from publicly available N. vectensis embryo images that display mRNA and/or protein distributions. Then, genes expressed during gut formation are clustered based on their expression profiles, and further grouped based on temporal appearance of their gene products in embryonic development. Representative expression profiles are manually selected from these clusters, and used as input for a simulation-based optimization scheme. This scheme iteratively fits simulated profiles to the selected profiles, leading to an optimized estimation of the model parameters. Finally, a preliminary gene regulatory network is derived from the optimized model parameters.

Outlook

While the focus of this study is N. vectensis, the approach outlined here is suitable for inferring gene regulatory networks in the embryonic development of any animal, thus allowing to comparatively study gene regulation of gut formation in silico across various species.

Spatiotemporal control of gene expression using microfluidics

Accurate spatiotemporal regulation of genetic expression and cell microenvironment are both essential to epithelial morphogenesis during development, wound healing and cancer. In vivo, this is achieved through the interplay between intrinsic cellular properties and extrinsic signals. Amongst these, morphogen gradients induce specific concentration- and time-dependent gene expression changes that influence a target cell's fate. As systems biology attempts to understand the complex mechanisms underlying morphogenesis, the lack of experimental setup to recapitulate morphogen-induced patterning in vitro has become limiting. For this reason, we developed a versatile microfluidic-based platform to control the spatiotemporal delivery of chemical gradients to tissues grown in Petri dishes. Using this setup combined with a synthetic inducible gene expression system, we were able to restrict a target gene's expression within a confluent epithelium to bands of cells as narrow as four cell diameters with a one cell diameter accuracy. Applied to the targeted delivery of growth factor gradients to a confluent epithelium, this method further enabled the localized induction of epithelial-mesenchymal transitions and associated morphogenetic changes. Our approach paves the way for replicating in vitro the morphogen gradients observed in vivo to determine the relative contributions of known intrinsic and extrinsic factors in differential tissue patterning, during development and cancer. It could also be readily used to spatiotemporally control cell differentiation in ES/iPS cell cultures for re-engineering of complex tissues. Finally, the reversibility of the microfluidic chip assembly allows for pre- and post-treatment sample manipulations and extends the range of patternable samples to animal explants.

A discrete model of Drosophila eggshell patterning reveals cell-autonomous and juxtacrine effects

The Drosophila eggshell constitutes a remarkable system for the study of epithelial patterning, both experimentally and through computational modeling. Dorsal eggshell appendages arise from specific regions in the anterior follicular epithelium that covers the oocyte: two groups of cells expressing broad (roof cells) bordered by rhomboid expressing cells (floor cells). Despite the large number of genes known to participate in defining these domains and the important modeling efforts put into this developmental system, key patterning events still lack a proper mechanistic understanding and/or genetic basis, and the literature appears to conflict on some crucial points. We tackle these issues with an original, discrete framework that considers single-cell models that are integrated to construct epithelial models. We first build a phenomenological model that reproduces wild type follicular epithelial patterns, confirming EGF and BMP signaling input as sufficient to establish the major features of this patterning system within the anterior domain. Importantly, this simple model predicts an instructive juxtacrine signal linking the roof and floor domains. To explore this prediction, we define a mechanistic model that integrates the combined effects of cellular genetic networks, cell communication and network adjustment through developmental events. Moreover, we focus on the anterior competence region, and postulate that early BMP signaling participates with early EGF signaling in its specification. This model accurately simulates wild type pattern formation and is able to reproduce, with unprecedented level of precision and completeness, various published gain-of-function and loss-of-function experiments, including perturbations of the BMP pathway previously seen as conflicting results. The result is a coherent model built upon rules that may be generalized to other epithelia and developmental systems.

Quantification of variability in trichome patterns

While pattern formation is studied in various areas of biology, little is known about the noise leading to variations between individual realizations of the pattern. One prominent example for de novo pattern formation in plants is the patterning of trichomes on Arabidopsis leaves, which involves genetic regulation and cell-to-cell communication. These processes are potentially variable due to, e.g., the abundance of cell components or environmental conditions. To elevate the understanding of regulatory processes underlying the pattern formation it is crucial to quantitatively analyze the variability in naturally occurring patterns. Here, we review recent approaches toward characterization of noise on trichome initiation. We present methods for the quantification of spatial patterns, which are the basis for data-driven mathematical modeling and enable the analysis of noise from different sources. Besides the insight gained on trichome formation, the examination of observed trichome patterns also shows that highly regulated biological processes can be substantially affected by variability.

Cellular resolution models for even skipped regulation in the entire Drosophila embryo

Transcriptional control ensures genes are expressed in the right amounts at the correct times and locations. Understanding quantitatively how regulatory systems convert input signals to appropriate outputs remains a challenge. For the first time, we successfully model even skipped (eve) stripes 2 and 3+7 across the entire fly embryo at cellular resolution. A straightforward statistical relationship explains how transcription factor (TF) concentrations define eve's complex spatial expression, without the need for pairwise interactions or cross-regulatory dynamics. Simulating thousands of TF combinations, we recover known regulators and suggest new candidates. Finally, we accurately predict the intricate effects of perturbations including TF mutations and misexpression. Our approach imposes minimal assumptions about regulatory function; instead we infer underlying mechanisms from models that best fit the data, like the lack of TF-specific thresholds and the positional value of homotypic interactions. Our study provides a general and quantitative method for elucidating the regulation of diverse biological systems. DOI:http://dx.doi.org/10.7554/eLife.00522.001.

Modeling spatiotemporal dynamics of bacterial populations

Quantitative modeling of spatiotemporal dynamics of cells facilitates understanding and engineering of biological systems. Using a synthetic bacterial ecosystem as a workbench, we present the approach to mathematically simulate the spatiotemporal population dynamics of the ecosystem. A description of ecosystem's genetic construction and model development is firstly given. Parameter estimation and computational approach for the derived partial differential equations (PDEs) are then given. Spatiotemporal pattern formation is computed by numerically solving the PDE model. Biodiversity of the ecosystem and its impacts by cellular seeding distance and motility are computed according to the cell distribution patterns.

Evolution of BMP signaling in Drosophila oogenesis: a receptor-based mechanism

The bone morphogenetic protein (BMP) signaling pathway is a conserved regulator of cellular and developmental processes in animals. The mechanisms underlying BMP signaling activation differ among tissues and mostly reflect changes in the expression of pathway components. BMP signaling is one of the major pathways responsible for the patterning of the Drosophila eggshell, a complex structure derived from a layer of follicle cells (FCs) surrounding the developing oocyte. Activation of BMP signaling in the FCs is dynamic. Initially, signaling is along the anterior-posterior (A/P) axis; later, signaling acquires dorsal-ventral (D/V) polarity. These dynamics are regulated by changes in the expression pattern of the type I BMP receptor thickveins (tkv). We recently found that signaling dynamics and TKV patterning are highly correlated in the FCs of multiple Drosophila species. In addition, we showed that signaling patterns are spatially different among species. Here, we use a mathematical model to simulate the dynamics and differences of BMP signaling in numerous species. This model predicts that qualitative and quantitative changes in receptor expression can lead to differences in the spatial pattern of BMP signaling. We tested these predications experimentally in three different Drosophila species and through genetic perturbations of BMP signaling in D. melanogaster. On the basis of our results, we concluded that changes in tkv patterning can account for the experimentally observed differences in the patterns of BMP signaling in multiple Drosophila species.

Influence of cell-to-cell variability on spatial pattern formation

Many spatial patterns in biology arise through differentiation of selected cells within a tissue, which is regulated by a genetic network. This is specified by its structure, parameterisation and the noise on its components and reactions. The latter, in particular, is not well examined because it is rather difficult to trace. The authors use suitable local mathematical measures based on the Voronoi diagram of experimentally determined positions of epidermal plant hairs (trichomes) to examine the variability or noise in pattern formation. Although trichome initiation is a highly regulated process, the authors show that the experimentally observed trichome pattern is substantially disturbed by cell-to-cell variations. Using computer simulations, they find that the rates concerning the availability of the protein complex that triggers trichome formation plays a significant role in noise-induced variations of the pattern. The focus on the effects of cell noise yields further insights into pattern formation of trichomes. The authors expect that similar strategies can contribute to the understanding of other differentiation processes by elucidating the role of naturally occurring fluctuations in the concentration of cellular components or their properties.

Computational approaches to developmental patterning

Computational approaches are breaking new ground in understanding how embryos form. Here, we discuss recent studies that couple precise measurements in the embryo with appropriately matched modeling and computational methods to investigate classic embryonic patterning strategies. We include signaling gradients, activator-inhibitor systems, and coupled oscillators, as well as emerging paradigms such as tissue deformation. Parallel progress in theory and experiment will play an increasingly central role in deciphering developmental patterning.

On the evolution of morphogenetic models: mechano-chemical interactions and an integrated view of cell differentiation, growth, pattern formation and morphogenesis

In the 1950s, embryology was conceptualized as four relatively independent problems: cell differentiation, growth, pattern formation and morphogenesis. The mechanisms underlying the first three traditionally have been viewed as being chemical in nature, whereas those underlying morphogenesis have usually been discussed in terms of mechanics. Often, morphogenesis and its mechanical processes have been regarded as subordinate to chemical ones. However, a growing body of evidence indicates that the biomechanics of cells and tissues affect in striking ways those phenomena often thought of as mainly under the control of cell-cell signalling. This accumulation of data has led to a revival of the mechano-transduction concept in particular, and of complexity in general, causing us now to consider whether we should retain the traditional conceptualization of development. The researchers' semantic preferences for the terms 'patterning', 'pattern formation' or 'morphogenesis' can be used to describe three main 'schools of thought' which emerged in the late 1970s. In the 'molecular school', the term patterning is deeply tied to the positional information concept. In the 'chemical school', the term 'pattern formation' regularly implies reaction-diffusion models. In the 'mechanical school', the term 'morphogenesis' is more frequently used in relation to mechanical instabilities. Major differences among these three schools pertain to the concept of self-organization, and models can be classified as morphostatic or morphodynamic. Various examples illustrate the distorted picture that arises from the distinction among differentiation, growth, pattern formation and morphogenesis, based on the idea that the underlying mechanisms are respectively chemical or mechanical. Emerging quantitative approaches integrate the concepts and methods of complex sciences and emphasize the interplay between hierarchical levels of organization via mechano-chemical interactions. They draw upon recent improvements in mathematical and numerical morphogenetic models and upon considerable progress in collecting new quantitative data. This review highlights a variety of such models, which exhibit important advances, such as hybrid, stochastic and multiscale simulations.

Stem cell niche dynamics: from homeostasis to carcinogenesis

The stem cell microenvironment is involved in regulating the fate of the stem cell with respect to self-renewal, quiescence, and differentiation. Mathematical models are helpful in understanding how key pathways regulate the dynamics of stem cell maintenance and homeostasis. This tight regulation and maintenance of stem cell number is thought to break down during carcinogenesis. As a result, the stem cell niche has become a novel target of cancer therapeutics. Developing a quantitative understanding of the regulatory pathways that guide stem cell behavior will be vital to understanding how these systems change under conditions of stress, inflammation, and cancer initiation. Predictions from mathematical modeling can be used as a clinical tool to guide therapy design. We present a survey of mathematical models used to study stem cell population dynamics and stem cell niche regulation, both in the hematopoietic system and other tissues. Highlighting the quantitative aspects of stem cell biology, we describe compelling questions that can be addressed with modeling. Finally, we discuss experimental systems, most notably Drosophila, that can best be used to validate mathematical predictions.

Modeling morphogen gradient formation from arbitrary realistically shaped sources

Much of the analytical modeling of morphogen profiles is based on simplistic scenarios, where the source is abstracted to be point-like and fixed in time, and where only the steady state solution of the morphogen gradient in one dimension is considered. Here we develop a general formalism allowing to model diffusive gradient formation from an arbitrary source. This mathematical framework, based on the Green's function method, applies to various diffusion problems. In this paper, we illustrate our theory with the explicit example of the Bicoid gradient establishment in Drosophila embryos. The gradient formation arises by protein translation from a mRNA distribution followed by morphogen diffusion with linear degradation. We investigate quantitatively the influence of spatial extension and time evolution of the source on the morphogen profile. For different biologically meaningful cases, we obtain explicit analytical expressions for both the steady state and time-dependent 1D problems. We show that extended sources, whether of finite size or normally distributed, give rise to more realistic gradients compared to a single point-source at the origin. Furthermore, the steady state solutions are fully compatible with a decreasing exponential behavior of the profile. We also consider the case of a dynamic source (e.g. bicoid mRNA diffusion) for which a protein profile similar to the ones obtained from static sources can be achieved.

Secreted, receptor-associated bone morphogenetic protein regulators reduce stochastic noise intrinsic to many extracellular morphogen distributions

Morphogens are secreted molecules that specify cell-fate organization in developing tissues. Patterns of gene expression or signalling immediately downstream of many morphogens such as the bone morphogenetic protein (BMP) decapentaplegic (Dpp) are highly reproducible and robust to perturbations. This contrasts starkly with our expectation of a noisy interpretation that would arise out of the experimentally determined low concentration (approximately picomolar) range of Dpp activity, tight receptor binding and very slow kinetic rates. To investigate mechanisms by which the intrinsic noise can be attenuated in Dpp signalling, we focus on a class of secreted proteins that bind to Dpp in the extracellular environment and play an active role in regulating Dpp/receptor interactions. We developed a stochastic model of Dpp signalling in Drosophila melanogaster and used the model to quantify the extent that stochastic fluctuations would lead to errors in spatial patterning and extended the model to investigate how a surface-associated BMP-binding protein (SBP) such as Crossveinless-2 (Cv-2) may buffer out signalling noise. In the presence of SBPs, fluctuations in the level of ligand-bound receptor can be reduced by more than twofold depending on parameter values for the intermediate transition states. Regulation of receptor-ligand interactions by SBPs may also increase the frequency of stochastic fluctuations providing a separation of timescales between high-frequency receptor equilibration and slower morphogen patterning. High-frequency noise generated by SBP regulation is easily attenuated by the intracellular network creating a system that imitates the performance of a simple low-pass filter common in audio and communication applications. Together, these data indicate that one of the benefits of receptor-ligand regulation by secreted non-receptors may be greater reliability of morphogen patterning mechanisms.

Growing up is stressful: biophysical laws of morphogenesis

Would it not be nice to understand the rules that govern how a small and round zygote reforms itself into a full blown three-dimensional and structured organism? The past decades have provided us with a wealth of knowledge about molecular mechanisms, intracellular behaviors, and tissue organization. However, we still do not know how to systematically integrate molecular mechanisms into descriptions that operate at larger scales involving higher-order structures such as the actomyosin cell cortex or an entire tissue. For development, it is the biophysical laws by which these structures deform, move, and restructure that are essential for morphogenetic rearrangements at developmental length- and time-scales. Recent years have seen the advent of systematic approaches for identifying these laws and ways to determine associated physical behaviors. Here I attempt to paint an intuitive picture of the mechanical concepts that are important for large-scale developmental rearrangements, and I briefly review the technique of laser ablation for measuring associated physical quantities and testing physical models.

Self-similar dynamics of morphogen gradients

Morphogen gradients are concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues and play a fundamental role in various aspects of embryonic development. We discovered a family of self-similar solutions in a canonical class of nonlinear reaction-diffusion models describing the formation of morphogen gradients. These solutions are realized in the limit of infinitely high production rate at the tissue boundary and are given by the product of the steady state concentration profile and a function of the diffusion similarity variable. We solved the boundary value problem for the similarity profile numerically and analyzed the implications of the discovered self-similarity on the dynamics of morphogenetic patterning.

Computational design of synthetic regulatory networks from a genetic library to characterize the designability of dynamical behaviors

The engineering of synthetic gene networks has mostly relied on the assembly of few characterized regulatory elements using rational design principles. It is of outmost importance to analyze the scalability and limits of such a design workflow. To analyze the design capabilities of libraries of regulatory elements, we have developed the first automated design approach that combines such elements to search the genotype space associated to a given phenotypic behavior. Herein, we calculated the designability of dynamical functions obtained from circuits assembled with a given genetic library. By designing circuits working as amplitude filters, pulse counters and oscillators, we could infer new mechanisms for such behaviors. We also highlighted the hierarchical design and the optimization of the interface between devices. We dissected the functional diversity of a constrained library and we found that even such libraries can provide a rich variety of behaviors. We also found that intrinsic noise slightly reduces the designability of digital circuits, but it increases the designability of oscillators. Finally, we analyzed the robust design as a strategy to counteract the evolvability and noise in gene expression of the engineered circuits within a cellular background, obtaining mechanisms for robustness through non-linear negative feedback loops.

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.

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.

Cell-sorting at the A/P boundary in the Drosophila wing primordium: a computational model to consolidate observed non-local effects of Hh signaling

Non-intermingling, adjacent populations of cells define compartment boundaries; such boundaries are often essential for the positioning and the maintenance of tissue-organizers during growth. In the developing wing primordium of Drosophila melanogaster, signaling by the secreted protein Hedgehog (Hh) is required for compartment boundary maintenance. However, the precise mechanism of Hh input remains poorly understood. Here, we combine experimental observations of perturbed Hh signaling with computer simulations of cellular behavior, and connect physical properties of cells to their Hh signaling status. We find that experimental disruption of Hh signaling has observable effects on cell sorting surprisingly far from the compartment boundary, which is in contrast to a previous model that confines Hh influence to the compartment boundary itself. We have recapitulated our experimental observations by simulations of Hh diffusion and transduction coupled to mechanical tension along cell-to-cell contact surfaces. Intriguingly, the best results were obtained under the assumption that Hh signaling cannot alter the overall tension force of the cell, but will merely re-distribute it locally inside the cell, relative to the signaling status of neighboring cells. Our results suggest a scenario in which homotypic interactions of a putative Hh target molecule at the cell surface are converted into a mechanical force. Such a scenario could explain why the mechanical output of Hh signaling appears to be confined to the compartment boundary, despite the longer range of the Hh molecule itself. Our study is the first to couple a cellular vertex model describing mechanical properties of cells in a growing tissue, to an explicit model of an entire signaling pathway, including a freely diffusible component. We discuss potential applications and challenges of such an approach.

In developing animal tissues, cells can often re-arrange locally and mix relatively freely. However, in some stereotypic and crucially important instances during body development, cells will strictly not intermingle, and instead form sharp boundaries along which they will sort out from each other. This mechanism helps organisms to establish signaling centers and to maintain distinct cellular identities. Often, cells at such boundaries will remain in close physical contact and are morphologically alike. Thus, the boundary itself can be difficult to observe unless the expression status of specific marker genes is monitored experimentally. How are these ‘compartment boundaries’ established? Here we devise a computational model that aims to describe one such boundary in a well-studied animal tissue: the developing wing primordium of Drosophila melanogaster. We model the production, diffusion and local sensing of an essential signaling molecule, the Hedgehog protein. We reveal one possible mechanism by which Hedgehog sensing can influence the mechanical properties of cells, and compare the simulated outcome to observations in experimentally perturbed, actual wing discs. Our relatively simple model suffices to establish a straight and stable compartment boundary.

Structural discrimination of robustness in transcriptional feedforward loops for pattern formation

Signaling pathways are interconnected to regulatory circuits for sensing the environment and expressing the appropriate genetic profile. In particular, gradients of diffusing molecules (morphogens) determine cell fate at a given position, dictating development and spatial organization. The feedforward loop (FFL) circuit is among the simplest genetic architectures able to generate one-stripe patterns by operating as an amplitude detection device, where high output levels are achieved at intermediate input ones. Here, using a heuristic optimization-based approach, we dissected the design space containing all possible topologies and parameter values of the FFL circuits. We explored the ability of being sensitive or adaptive to variations in the critical morphogen level where cell fate is switched. We found four different solutions for precision, corresponding to the four incoherent architectures, but remarkably only one mode for adaptiveness, the incoherent type 4 (I4-FFL). We further carried out a theoretical study to unveil the design principle for such structural discrimination, finding that the synergistic action and cooperative binding on the downstream promoter are instrumental to achieve absolute adaptive responses. Subsequently, we analyzed the robustness of these optimal circuits against perturbations in the kinetic parameters and molecular noise, which has allowed us to depict a scenario where adaptiveness, parameter sensitivity and noise tolerance are different, correlated facets of the robustness of the I4-FFL circuit. Strikingly, we showed a strong correlation between the input (environment-related) and the intrinsic (mutation-related) susceptibilities. Finally, we discussed the evolution of incoherent regulations in terms of multifunctionality and robustness.

The effects of weak genetic perturbations on the transcriptome of the wing imaginal disc and its association with wing shape in Drosophila melanogaster

A major objective of genomics is to elucidate the mapping between genotypic and phenotypic space as a step toward understanding how small changes in gene function can lead to elaborate phenotypic changes. One approach that has been utilized is to examine overall patterns of covariation between phenotypic variables of interest, such as morphology, physiology, and behavior, and underlying aspects of gene activity, in particular transcript abundance on a genome-wide scale. Numerous studies have demonstrated that such patterns of covariation occur, although these are often between samples with large numbers of unknown genetic differences (different strains or even species) or perturbations of large effect (sexual dimorphism or strong loss-of-function mutations) that may represent physiological changes outside of the normal experiences of the organism. We used weak mutational perturbations in genes affecting wing development in Drosophila melanogaster that influence wing shape relative to a co-isogenic wild type. We profiled transcription of 1150 genes expressed during wing development in 27 heterozygous mutants, as well as their co-isogenic wild type and one additional wild-type strain. Despite finding clear evidence of expression differences between mutants and wild type, transcriptional profiles did not covary strongly with shape, suggesting that information from transcriptional profiling may not generally be predictive of final phenotype. We discuss these results in the light of possible attractor states of gene expression and how this would affect interpretation of covariation between transcriptional profiles and other phenotypes.

Switch and template pattern formation in a discrete reaction-diffusion system inspired by the Drosophila eye

We examine a spatially discrete reaction-diffusion model based on the interactions that create a periodic pattern in the Drosophila eye imaginal disc. This model is known to be capable of generating a regular hexagonal pattern of gene expression behind a moving front, as observed in the fly system. In order to better understand the novel "switch and template" mechanism behind this pattern formation, we present here a detailed study of the model's behavior in one dimension, using a combination of analytic methods and numerical searches of parameter space. We find that patterns are created robustly, provided that there is an appropriate separation of timescales and that self-activation is sufficiently strong, and we derive expressions in this limit for the front speed and the pattern wavelength. Moving fronts in pattern-forming systems near an initial linear instability generically select a unique pattern, but our model operates in a strongly nonlinear regime where the final pattern depends on the initial conditions as well as on parameter values. Our work highlights the important role that cellularization and cell-autonomous feedback can play in biological pattern formation.

The measure of success: constraints, objectives, and tradeoffs in morphogen-mediated patterning

A large, diverse, and growing number of strategies have been proposed to explain how morphogen gradients achieve robustness and precision. We argue that, to be useful, the evaluation of such strategies must take into account the constraints imposed by competing objectives and performance tradeoffs. This point is illustrated through a mathematical and computational analysis of the strategy of self-enhanced morphogen clearance. The results suggest that the usefulness of this strategy comes less from its ability to increase robustness to morphogen source fluctuations per se, than from its ability to overcome specific kinds of noise, and to increase the fraction of a morphogen gradient within which robust threshold positions may be established. This work also provides new insights into the longstanding question of why morphogen gradients show a maximum range in vivo.

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.

Robust dynamical pattern formation from a multifunctional minimal genetic circuit

BACKGROUND

A practical problem during the analysis of natural networks is their complexity, thus the use of synthetic circuits would allow to unveil the natural mechanisms of operation. Autocatalytic gene regulatory networks play an important role in shaping the development of multicellular organisms, whereas oscillatory circuits are used to control gene expression under variable environments such as the light-dark cycle.

RESULTS

We propose a new mechanism to generate developmental patterns and oscillations using a minimal number of genes. For this, we design a synthetic gene circuit with an antagonistic self-regulation to study the spatio-temporal control of protein expression. Here, we show that our minimal system can behave as a biological clock or memory, and it exhibites an inherent robustness due to a quorum sensing mechanism. We analyze this property by accounting for molecular noise in an heterogeneous population. We also show how the period of the oscillations is tunable by environmental signals, and we study the bifurcations of the system by constructing different phase diagrams.

CONCLUSIONS

As this minimal circuit is based on a single transcriptional unit, it provides a new mechanism based on post-translational interactions to generate targeted spatio-temporal behavior.

Organism-scale modeling of early Drosophila patterning via bone morphogenetic proteins

Advances in image acquisition and informatics technology have led to organism-scale spatiotemporal atlases of gene expression and protein distributions. To maximize the utility of this information for the study of developmental processes, a new generation of mathematical models is needed for discovery and hypothesis testing. Here, we develop a data-driven, geometrically accurate model of early Drosophila embryonic bone morphogenetic protein (BMP)-mediated patterning. We tested nine different mechanisms for signal transduction with feedback, eight combinations of geometry and gene expression prepatterns, and two scale-invariance mechanisms for their ability to reproduce proper BMP signaling output in wild-type and mutant embryos. We found that a model based on positive feedback of a secreted BMP-binding protein, coupled with the experimentally measured embryo geometry, provides the best agreement with population mean image data. Our results demonstrate that using bioimages to build and optimize a three-dimensional model provides significant insights into mechanisms that guide tissue patterning.

The extracellular regulation of bone morphogenetic protein signaling

In many cases, the level, positioning and timing of signaling through the bone morphogenetic protein (BMP) pathway are regulated by molecules that bind BMP ligands in the extracellular space. Whereas many BMP-binding proteins inhibit signaling by sequestering BMPs from their receptors, other BMP-binding proteins cause remarkably context-specific gains or losses in signaling. Here, we review recent findings and hypotheses on the complex mechanisms that lead to these effects, with data from developing systems, biochemical analyses and mathematical modeling.

Inferring Drosophila gap gene regulatory network: a parameter sensitivity and perturbation analysis

BACKGROUND

Inverse modelling of gene regulatory networks (GRNs) capable of simulating continuous spatio-temporal biological processes requires accurate data and a good description of the system. If quantitative relations between genes cannot be extracted from direct measurements, an efficient method to estimate the unknown parameters is mandatory. A model that has been proposed to simulate spatio-temporal gene expression patterns is the connectionist model. This method describes the quantitative dynamics of a regulatory network in space. The model parameters are estimated by means of model-fitting algorithms. The gene interactions are identified without making any prior assumptions concerning the network connectivity. As a result, the inverse modelling might lead to multiple circuits showing the same quantitative behaviour and it is not possible to identify one optimal circuit. Consequently, it is important to address the quality of the circuits in terms of model robustness.

RESULTS

Here we investigate the sensitivity and robustness of circuits obtained from reverse engineering a model capable of simulating measured gene expression patterns. As a case study we use the early gap gene segmentation mechanism in Drosophila melanogaster. We consider the limitations of the connectionist model used to describe GRN Inferred from spatio-temporal gene expression. We address the problem of circuit discrimination, where the selection criterion within the optimization technique is based of the least square minimization on the error between data and simulated results.

CONCLUSION

Parameter sensitivity analysis allows one to discriminate between circuits having significant parameter and qualitative differences but exhibiting the same quantitative pattern. Furthermore, we show that using a stochastic model derived from a deterministic solution, one can introduce fluctuations within the model to analyze the circuits' robustness. Ultimately, we show that there is a close relation between circuit sensitivity and robustness to fluctuation, and that circuit robustness is rather modular than global. The current study shows that reverse engineering of GRNs should not only focus on estimating parameters by minimizing the difference between observation and simulation but also on other model properties. Our study suggests that multi-objective optimization based on robustness and sensitivity analysis has to be considered.

Morphogen gradient formation

How morphogen gradients are formed in target tissues is a key question for understanding the mechanisms of morphological patterning. Here, we review different mechanisms of morphogen gradient formation from theoretical and experimental points of view. First, a simple, comprehensive overview of the underlying biophysical principles of several mechanisms of gradient formation is provided. We then discuss the advantages and limitations of different experimental approaches to gradient formation analysis.

Particle tracking model of electrophoretic morphogen movement reveals stochastic dynamics of embryonic gradient

Some developmental events rely on an electrophoretic force to produce morphogenetic gradients. To quantitatively explore the dynamics of this process, we constructed a stochastic model of an early phase of left-right patterning: serotonin movement through the gap junction-coupled blastomeres of the Xenopus embryo. Particle-tracking simulations showed that a left-right gradient is formed rapidly, quickly reaching a final stable level. The voltage difference was critical for producing a morphogen gradient of the right steepness; gap junctional connectivity and morphogen mass determined the timing of the gradient. Endogenous electrophoresis drives approximately 50% of the particles across more than one cell width, and approximately 20% can travel across half the embryo. The stochastic behavior of the resulting gradients exhibited unexpected complexity among blastomeres' morphogen content, and showed how spatiotemporal variability within individual cells resulted in robust and consistent gradients across the embryonic left-right axis. Analysis of the distribution profile of gradient gain values made quantitative predictions about the conditions that result in the observed background level of laterality defects in unperturbed frog embryos. This work provides a general model that can be used to quantitatively analyze the unexpectedly complex dynamics of morphogens in a wide variety of systems.

Stable, precise, and reproducible patterning of bicoid and hunchback molecules in the early Drosophila embryo

Precise patterning of morphogen molecules and their accurate reading out are of key importance in embryonic development. Recent experiments have visualized distributions of proteins in developing embryos and shown that the gradient of concentration of Bicoid morphogen in Drosophila embryos is established rapidly after fertilization and remains stable through syncytial mitoses. This stable Bicoid gradient is read out in a precise way to distribute Hunchback with small fluctuations in each embryo and in a reproducible way, with small embryo-to-embryo fluctuation. The mechanisms of such stable, precise, and reproducible patterning through noisy cellular processes, however, still remain mysterious. To address these issues, here we develop the one- and three-dimensional stochastic models of the early Drosophila embryo. The simulated results show that the fluctuation in expression of the hunchback gene is dominated by the random arrival of Bicoid at the hunchback enhancer. Slow diffusion of Hunchback protein, however, averages out this intense fluctuation, leading to the precise patterning of distribution of Hunchback without loss of sharpness of the boundary of its distribution. The coordinated rates of diffusion and transport of input Bicoid and output Hunchback play decisive roles in suppressing fluctuations arising from the dynamical structure change in embryos and those arising from the random diffusion of molecules, and give rise to the stable, precise, and reproducible patterning of Bicoid and Hunchback distributions.

Systems biology of the self-regulating morphogenetic gradient of the Xenopus gastrula

The morphogenetic field concept was proposed by experimental embryologists to account for the self-regulative behavior of embryos. Such fields have remained an abstract concept until the recent identification of their molecular components using a combination of genetics, biochemistry, and theoretical modeling. One of the best studied models of a morphogenetic field is the Dorsal-Ventral (D-V) patterning of the early frog embryo. This patterning system is regulated by the bone morphogenetic protein (BMP) signaling pathway and an intricate network of secreted protein antagonists. This biochemical pathway of interacting proteins functions in the extracellular space to generate a D-V gradient of BMP signaling, which is maintained during extensive morphogenetic movements of cell layers during gastrulation. The D-V field is divided into a dorsal and a ventral center, in regions of low and high BMP signaling respectively, under opposite transcriptional control by BMPs. The robustness of the patterning is assured at two different levels. First, in the extracellular space by secreted BMP antagonists that generate a directional flow of BMP ligands to the ventral side. The flow is driven by the regulated proteolysis of the Chordin inhibitor and by the presence of a molecular sink on the ventral side that concentrates BMP signals. The tolloid metalloproteinases and the Chordin-binding protein Crossveinless-2 (CV2) are key components of this ventral sink. Second, by transcriptional feedback at the cellular level: The dorsal and ventral signaling centers adjust their size and level of BMP signaling by transcriptional feedback. This allows cells on one side of a gastrula containing about 10,000 cells to communicate with cells in the opposite pole of the embryo.

Accuracy of positional information provided by multiple morphogen gradients with correlated noise

Normal development of multicellular organisms requires cells to respond properly according to their positions. Positional information is often provided to cells as concentrations of diffusive chemicals called morphogens with spatial gradients. However, the spatial profiles of their concentrations include various kinds of noises, making positional information unreliable. In many developmental systems, multiple morphogen gradients are adopted to specify the spatial position along a single axis, presumably to achieve a sufficiently high precision of information on the location of each cell. In this paper, we ask how the precision of positional information depends on the number of morphogens. We derive a formula for the limit of precision when each cell adopts the maximum-likelihood estimation of the "true" position from noisy inputs. The precision increases with the number of morphogens and interestingly it also depends on the correlation of noises. The positional specification can be made more precisely if their gradients are of the opposite (same) direction when noises of the two morphogens are positively (negatively) correlated. The formula also tells us a minimum number of morphogens needed to achieve a given precision of positional information. We illustrate the theory by analyzing experimental data for the gradients of two diffusive chemicals, Bicoid and Caudal, in the early development of Drosophila embryo. The analysis suggests that combined information provided by the two chemicals is able to give accurate positional information in the middle part of the embryo, where the embryo segmentation occurs in later stages, much more than near both ends.