Whole-embryo modeling of early segmentation in Drosophila identifies robust and fragile expression domains

Body plan evolvability: The role of variability in gene regulatory networks

Recent computational modeling of early fruit fly (Drosophila) development has characterized the degree to which gene regulation networks can be robust to natural variability. In the first few hours of development, broad spatial gradients of maternally derived transcription factors activate embryonic gap genes. These gap patterns determine the subsequent segmented insect body plan through pair-rule gene expression. Gap genes are expressed with greater spatial precision than the maternal patterns. Computational modeling of the gap-gap regulatory interactions provides a mechanistic understanding for this robustness to maternal variability in wild-type (WT) patterning. A long-standing question in evolutionary biology has been how a system which is robust, such as the developmental program creating any particular species' body plan, is also evolvable, i.e. how can a system evolve or speciate, if the WT form is strongly buffered and protected? In the present work, we use the WT model to explore the breakdown of such Waddington-type 'canalization'. What levels of variability will push the system out of the WT form; are there particular pathways in the gene regulatory mechanism which are more susceptible to losing the WT form; and when robustness is lost, what types of forms are most likely to occur (i.e. what forms lie near the WT)? Manipulating maternal effects in several different pathways, we find a common gap 'peak-to-step' pattern transition in the loss of WT. We discuss these results in terms of the evolvability of insect segmentation, and in terms of experimental perturbations and mutations which could test the model predictions. We conclude by discussing the prospects for using continuum models of pattern dynamics to investigate a wider range of evo-devo problems.

Demixing in Binary Mixtures with Differential Diffusivity at High Density

Spontaneous phase separation, or demixing, is important in biological phenomena such as cell sorting. In particle-based models, an open question is whether differences in diffusivity can drive such demixing. While differential-diffusivity-induced phase separation occurs in mixtures with a packing fraction up to 0.7 [S. N. Weber et al. Binary mixtures of particles with different diffusivities demix, Phys. Rev. Lett. 116, 058301 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.058301], here we investigate whether demixing persists at even higher densities relevant for cells. For particle packing fractions between 0.7 and 1.0 the system demixes, but at packing fractions above unity the system remains mixed, exposing re-entrant behavior in the phase diagram that occurs when phase separation can no longer drive a change in entropy production at high densities. We also find that a confluent Voronoi model for tissues does not phase separate, consistent with particle-based simulations.

Functional analysis of the Drosophila eve locus in response to non-canonical combinations of gap gene expression levels

Transcription factor combinations play a key role in shaping cellular identity. However, the precise relationship between specific combinations and downstream effects remains elusive. Here, we investigate this relationship within the context of the Drosophila eve locus, which is controlled by gap genes. We measure spatiotemporal levels of four gap genes in heterozygous and homozygous gap mutant embryos and correlate them with the striped eve activity pattern. Although changes in gap gene expression extend beyond the manipulated gene, the spatial patterns of Eve expression closely mirror canonical activation levels in wild type. Interestingly, some combinations deviate from the wild-type repertoire but still drive eve activation. Although in homozygous mutants some Eve stripes exhibit partial penetrance, stripes consistently emerge at reproducible positions, even with varying gap gene levels. Our findings suggest a robust molecular canalization of cell fates in gap mutants and provide insights into the regulatory constraints governing multi-enhancer gene loci.

Shaping the scaling characteristics of gap gene expression patterns in Drosophila

How patterns are formed to scale with tissue size remains an unresolved problem. Here we investigate embryonic patterns of gap gene expression along the anterior-posterior (AP) axis in Drosophila. We use embryos that greatly differ in length and, importantly, possess distinct length-scaling characteristics of the Bicoid (Bcd) gradient. We systematically analyze the dynamic movements of gap gene expression boundaries in relation to both embryo length and Bcd input as a function of time. We document the process through which such dynamic movements drive both an emergence of a global scaling landscape and evolution of boundary-specific scaling characteristics. We show that, despite initial differences in pattern scaling characteristics that mimic those of Bcd in the anterior, such characteristics of final patterns converge. Our study thus partitions the contributions of Bcd input and regulatory dynamics inherent to the AP patterning network in shaping embryonic pattern's scaling characteristics.

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.

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.

Analyzing the stability of gene expression using a simple reaction-diffusion model in an early Drosophila embryo

In all complex organisms, the precise levels and timing of gene expression controls vital biological processes. In higher eukaryotes, including the fruit fly Drosophila melanogaster, the complex molecular control of transcription (the synthesis of RNA from DNA) and translation (the synthesis of proteins from RNA) events driving this gene expression are not fully understood. In particular, for Drosophila melanogaster, there is a plethora of experimental data, including quantitative measurements of both RNA and protein concentrations, but the precise mechanisms that control the dynamics of gene expression during early development and the processes which lead to steady-state levels of certain proteins remain elusive. This study analyzes a current mathematical modeling approach in an attempt to better understand the long-term behavior of gene regulation. The model is a modified reaction-diffusion equation which has been previously employed in predicting gene expression levels and studying the relative contributions of transcription and translation events to protein abundance [10,11,24]. Here, we use Matrix Algebra and Analysis techniques to study the stability of the gene expression system and analyze equilibria, using very general assumptions regarding the parameter values incorporated into the model. We prove that, given realistic biological parameter values, the system will result in a unique, stable equilibrium solution. Additionally, we give an example of this long-term behavior using the model alongside actual experimental data obtained from Drosophila embryos.

Quantitative Comparison of the Anterior-Posterior Patterning System in the Embryos of Five Drosophila Species

Complex spatiotemporal gene expression patterns direct the development of the fertilized egg into an adult animal. Comparisons across species show that, in spite of changes in the underlying regulatory DNA sequence, developmental programs can be maintained across millions of years of evolution. Reciprocally, changes in gene expression can be used to generate morphological novelty. Distinguishing between changes in regulatory DNA that lead to changes in gene expression and those that do not is therefore a central goal of evolutionary developmental biology. Quantitative, spatially-resolved measurements of developmental gene expression patterns play a crucial role in this goal, enabling the detection of subtle phenotypic differences between species and the development of computations models that link the sequence of regulatory DNA to expression patterns. Here we report the generation of two atlases of cellular resolution gene expression measurements for the primary anterior-posterior patterning genes in Drosophila simulans and Drosophila virilis By combining these data sets with existing atlases for three other Drosophila species, we detect subtle differences in the gene expression patterns and dynamics driving the highly conserved axis patterning system and delineate inter-species differences in the embryonic morphology. These data sets will be a resource for future modeling studies of the evolution of developmental gene regulatory networks.

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

Causality analysis detects the regulatory role of maternal effect genes in the early Drosophila embryo

In developmental studies, inferring regulatory interactions of segmentation genetic network play a vital role in unveiling the mechanism of pattern formation. As such, there exists an opportune demand for theoretical developments and new mathematical models which can result in a more accurate illustration of this genetic network. Accordingly, this paper seeks to extract the meaningful regulatory role of the maternal effect genes using a variety of causality detection techniques and to explore whether these methods can suggest a new analytical view to the gene regulatory networks. We evaluate the use of three different powerful and widely-used models representing time and frequency domain Granger causality and convergent cross mapping technique with the results being thoroughly evaluated for statistical significance. Our findings show that the regulatory role of maternal effect genes is detectable in different time classes and thereby the method is applicable to infer the possible regulatory interactions present among the other genes of this network.

The hidden simplicity of biology

Life is so remarkable, and so unlike any other physical system, that it is tempting to attribute special factors to it. Physics is founded on the assumption that universal laws and principles underlie all natural phenomena, but is it far from clear that there are 'laws of life' with serious descriptive or predictive power analogous to the laws of physics. Nor is there (yet) a 'theoretical biology' in the same sense as theoretical physics. Part of the obstacle in developing a universal theory of biological organization concerns the daunting complexity of living organisms. However, many attempts have been made to glimpse simplicity lurking within this complexity, and to capture this simplicity mathematically. In this paper we review a promising new line of inquiry to bring coherence and order to the realm of biology by focusing on 'information' as a unifying concept.

Temporal and spatial dynamics of scaling-specific features of a gene regulatory network in Drosophila

A widely appreciated aspect of developmental robustness is pattern formation in proportion to size. But how such scaling features emerge dynamically remains poorly understood. Here we generate a data set of the expression profiles of six gap genes in Drosophila melanogaster embryos that differ significantly in size. Expression patterns exhibit size-dependent dynamics both spatially and temporally. We uncover a dynamic emergence of under-scaling in the posterior, accompanied by reduced expression levels of gap genes near the middle of large embryos. Simulation results show that a size-dependent Bicoid gradient input can lead to reduced Krüppel expression that can have long-range and dynamic effects on gap gene expression in the posterior. Thus, for emergence of scaled patterns, the entire embryo may be viewed as a single unified dynamic system where maternally derived size-dependent information interpreted locally can be propagated in space and time as governed by the dynamics of a gene regulatory network.

How pattern formation is regulated relative to the size of an organism is unclear. Here, Wu et al. take data from gap gene expression in flies of different sizes together with simulations, identifying how scaling emerges dynamically and that local patterning influences global gene regulatory networks.

Quantitative analysis and modeling probe polarity establishment in C. elegans embryos

Cell polarity underlies many aspects of metazoan development and homeostasis, and relies notably on a set of PAR proteins located at the cell cortex. How these proteins interact in space and time remains incompletely understood. We performed a quantitative assessment of polarity establishment in one-cell stage Caenorhabditis elegans embryos by combining time-lapse microscopy and image analysis. We used our extensive data set to challenge and further specify an extant mathematical model. Using likelihood-based calibration, we uncovered that cooperativity is required for both anterior and posterior PAR complexes. Moreover, we analyzed the dependence of polarity establishment on changes in size or temperature. The observed robustness of PAR domain dimensions in embryos of different sizes is in agreement with a model incorporating fixed protein concentrations and variations in embryo surface/volume ratio. In addition, we quantified the dynamics of polarity establishment over most of the viable temperatures range of C. elegans. Modeling of these data suggests that diffusion of PAR proteins is the process most affected by temperature changes, although cortical flows appear unaffected. Overall, our quantitative analytical framework provides insights into the dynamics of polarity establishment in a developing system.

Tissue patterning and cellular mechanics

In development, cells organize into biological tissues through cell growth, migration, and differentiation. Globally, this process is dictated by a genetically encoded program in which secreted morphogens and cell-cell interactions prompt the adoption of unique cell fates. Yet, at its lowest level, development is achieved through the modification of cell-cell adhesion and actomyosin-based contractility, which set the level of tension within cells and dictate how they pack together into tissues. The regulation of tension within individual cells and across large groups of cells is a major driving force of tissue organization and the basis of all cell shape change and cell movement in development.

Global sensitivity analysis of a dynamic model for gene expression in Drosophila embryos

It is well known that gene regulation is a tightly controlled process in early organismal development. However, the roles of key processes involved in this regulation, such as transcription and translation, are less well understood, and mathematical modeling approaches in this field are still in their infancy. In recent studies, biologists have taken precise measurements of protein and mRNA abundance to determine the relative contributions of key factors involved in regulating protein levels in mammalian cells. We now approach this question from a mathematical modeling perspective. In this study, we use a simple dynamic mathematical model that incorporates terms representing transcription, translation, mRNA and protein decay, and diffusion in an early Drosophila embryo. We perform global sensitivity analyses on this model using various different initial conditions and spatial and temporal outputs. Our results indicate that transcription and translation are often the key parameters to determine protein abundance. This observation is in close agreement with the experimental results from mammalian cells for various initial conditions at particular time points, suggesting that a simple dynamic model can capture the qualitative behavior of a gene. Additionally, we find that parameter sensitivites are temporally dynamic, illustrating the importance of conducting a thorough global sensitivity analysis across multiple time points when analyzing mathematical models of gene regulation.

What does it take to evolve an enhancer? A simulation-based study of factors influencing the emergence of combinatorial regulation

There is widespread interest today in understanding enhancers, which are regulatory elements typically harboring several transcription factor binding sites and mediating the combinatorial effect of transcription factors on gene expression. The evolution of enhancers poses interesting unanswered questions, for example, the evolutionary time taken for a typical enhancer to emerge or the factors shaping its evolution. Existing approaches to cis-regulatory evolution have often ignored the combinatorial nature and varied biochemical mechanisms of gene regulation encoded in enhancers. We report on our investigation of enhancer evolution through the use of PEBCRES, a framework for evolutionary simulation of enhancers that employs a mechanistic and well-supported sequence-to-expression model to assign fitness to the evolving enhancer genotype. We estimated the time necessary to evolve, from genomic background, enhancers capable of driving complex gene expression patterns similar to those involved in early development in Drosophila. We found the time-to-evolve to range between 0.5 and 10 Myr, and to vary greatly with the target expression pattern, complexity of the real enhancer known to encode that pattern, and the strength of input from specific transcription factors. To our knowledge, this is the first estimate of waiting times for realistic enhancers to evolve. The in silico evolved enhancers had, with a few interesting exceptions, site compositions similar to those seen in real enhancers for the same patterns. Our simulations also revealed that certain features of an enhancer might evolve not due to their biological function but as aids to the evolutionary process itself.

Mid-embryo patterning and precision in Drosophila segmentation: Krüppel dual regulation of hunchback

In early development, genes are expressed in spatial patterns which later define cellular identities and tissue locations. The mechanisms of such pattern formation have been studied extensively in early Drosophila (fruit fly) embryos. The gap gene hunchback (hb) is one of the earliest genes to be expressed in anterior-posterior (AP) body segmentation. As a transcriptional regulator for a number of downstream genes, the spatial precision of hb expression can have significant effects in the development of the body plan. To investigate the factors contributing to hb precision, we used fine spatial and temporal resolution data to develop a quantitative model for the regulation of hb expression in the mid-embryo. In particular, modelling hb pattern refinement in mid nuclear cleavage cycle 14 (NC14) reveals some of the regulatory contributions of simultaneously-expressed gap genes. Matching the model to recent data from wild-type (WT) embryos and mutants of the gap gene Krüppel (Kr) indicates that a mid-embryo Hb concentration peak important in thoracic development (at parasegment 4, PS4) is regulated in a dual manner by Kr, with low Kr concentration activating hb and high Kr concentration repressing hb. The processes of gene expression (transcription, translation, transport) are intrinsically random. We used stochastic simulations to characterize the noise generated in hb expression. We find that Kr regulation can limit the positional variability of the Hb mid-embryo border. This has been recently corroborated in experimental comparisons of WT and Kr- mutant embryos. Further, Kr regulation can decrease uncertainty in mid-embryo hb expression (i.e. contribute to a smooth Hb boundary) and decrease between-copy transcriptional variability within nuclei. Since many tissue boundaries are first established by interactions between neighbouring gene expression domains, these properties of Hb-Kr dynamics to diminish the effects of intrinsic expression noise may represent a general mechanism contributing to robustness in early development.

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.

Evolutionary Design of Gene Networks: Forced Evolution by Genomic Parasites

The co-evolution of species with their genomic parasites (transposons) is thought to be one of the primary ways of rewiring gene regulatory networks (GRNs). We develop a framework for conducting evolutionary computations (EC) using the transposon mechanism. We find that the selective pressure of transposons can speed evolutionary searches for solutions and lead to outgrowth of GRNs (through co-option of new genes to acquire insensitivity to the attacking transposons). We test the approach by finding GRNs which can solve a fundamental problem in developmental biology: how GRNs in early embryo development can robustly read maternal signaling gradients, despite continued attacks on the genome by transposons. We observed co-evolutionary oscillations in the abundance of particular GRNs and their transposons, reminiscent of predator-prey or host-parasite dynamics.

Shadow enhancers enable Hunchback bifunctionality in the Drosophila embryo

Hunchback (Hb) is a bifunctional transcription factor that activates and represses distinct enhancers. Here, we investigate the hypothesis that Hb can activate and repress the same enhancer. Computational models predicted that Hb bifunctionally regulates the even-skipped (eve) stripe 3+7 enhancer (eve3+7) in Drosophila blastoderm embryos. We measured and modeled eve expression at cellular resolution under multiple genetic perturbations and found that the eve3+7 enhancer could not explain endogenous eve stripe 7 behavior. Instead, we found that eve stripe 7 is controlled by two enhancers: the canonical eve3+7 and a sequence encompassing the minimal eve stripe 2 enhancer (eve2+7). Hb bifunctionally regulates eve stripe 7, but it executes these two activities on different pieces of regulatory DNA--it activates the eve2+7 enhancer and represses the eve3+7 enhancer. These two "shadow enhancers" use different regulatory logic to create the same pattern.

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.

The embryo as a laboratory: quantifying transcription in Drosophila

Transcriptional regulation of gene expression is fundamental to most cellular processes, including determination of cellular fates. Quantitative studies of transcription in cultured cells have led to significant advances in identifying mechanisms underlying transcriptional control. Recent progress allowed implementation of these same quantitative methods in multicellular organisms to ask how transcriptional regulation unfolds both in vivo and at the single molecule level in the context of embryonic development. Here we review some of these advances in early Drosophila development, which bring the embryo on par with its single celled counterparts. In particular, we discuss progress in methods to measure mRNA and protein distributions in fixed and living embryos, and we highlight some initial applications that lead to fundamental new insights about molecular transcription processes. We end with an outlook on how to further exploit the unique advantages that come with investigating transcriptional control in the multicellular context of development.

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.

Global parameter estimation for thermodynamic models of transcriptional regulation

Deciphering the mechanisms involved in gene regulation holds the key to understanding the control of central biological processes, including human disease, population variation, and the evolution of morphological innovations. New experimental techniques including whole genome sequencing and transcriptome analysis have enabled comprehensive modeling approaches to study gene regulation. In many cases, it is useful to be able to assign biological significance to the inferred model parameters, but such interpretation should take into account features that affect these parameters, including model construction and sensitivity, the type of fitness calculation, and the effectiveness of parameter estimation. This last point is often neglected, as estimation methods are often selected for historical reasons or for computational ease. Here, we compare the performance of two parameter estimation techniques broadly representative of local and global approaches, namely, a quasi-Newton/Nelder-Mead simplex (QN/NMS) method and a covariance matrix adaptation-evolutionary strategy (CMA-ES) method. The estimation methods were applied to a set of thermodynamic models of gene transcription applied to regulatory elements active in the Drosophila embryo. Measuring overall fit, the global CMA-ES method performed significantly better than the local QN/NMS method on high quality data sets, but this difference was negligible on lower quality data sets with increased noise or on data sets simplified by stringent thresholding. Our results suggest that the choice of parameter estimation technique for evaluation of gene expression models depends both on quality of data, the nature of the models [again, remains to be established] and the aims of the modeling effort.

Using evolutionary computations to understand the design and evolution of gene and cell regulatory networks

This paper surveys modeling approaches for studying the evolution of gene regulatory networks (GRNs). Modeling of the design or 'wiring' of GRNs has become increasingly common in developmental and medical biology, as a means of quantifying gene-gene interactions, the response to perturbations, and the overall dynamic motifs of networks. Drawing from developments in GRN 'design' modeling, a number of groups are now using simulations to study how GRNs evolve, both for comparative genomics and to uncover general principles of evolutionary processes. Such work can generally be termed evolution in silico. Complementary to these biologically-focused approaches, a now well-established field of computer science is Evolutionary Computations (ECs), in which highly efficient optimization techniques are inspired from evolutionary principles. In surveying biological simulation approaches, we discuss the considerations that must be taken with respect to: (a) the precision and completeness of the data (e.g. are the simulations for very close matches to anatomical data, or are they for more general exploration of evolutionary principles); (b) the level of detail to model (we proceed from 'coarse-grained' evolution of simple gene-gene interactions to 'fine-grained' evolution at the DNA sequence level); (c) to what degree is it important to include the genome's cellular context; and (d) the efficiency of computation. With respect to the latter, we argue that developments in computer science EC offer the means to perform more complete simulation searches, and will lead to more comprehensive biological predictions.

A primer on thermodynamic-based models for deciphering transcriptional regulatory logic

A rigorous analysis of transcriptional regulation at the DNA level is crucial to the understanding of many biological systems. Mathematical modeling has offered researchers a new approach to understanding this central process. In particular, thermodynamic-based modeling represents the most biophysically informed approach aimed at connecting DNA level regulatory sequences to the expression of specific genes. The goal of this review is to give biologists a thorough description of the steps involved in building, analyzing, and implementing a thermodynamic-based model of transcriptional regulation. The data requirements for this modeling approach are described, the derivation for a specific regulatory region is shown, and the challenges and future directions for the quantitative modeling of gene regulation are discussed.

TWO-LAYER MATHEMATICAL MODELING OF GENE EXPRESSION: INCORPORATING DNA-LEVEL INFORMATION AND SYSTEM DYNAMICS

High-throughput genome sequencing and transcriptome analysis have provided researchers with a quantitative basis for detailed modeling of gene expression using a wide variety of mathematical models. Two of the most commonly employed approaches used to model eukaryotic gene regulation are systems of differential equations, which describe time-dependent interactions of gene networks, and thermodynamic equilibrium approaches that can explore DNA-level transcriptional regulation. To combine the strengths of these approaches, we have constructed a new two-layer mathematical model that provides a dynamical description of gene regulatory systems, using detailed DNA-based information, as well as spatial and temporal transcription factor concentration data. We also developed a semi-implicit numerical algorithm for solving the model equations and demonstrate here the efficiency of this algorithm through stability and convergence analyses. To test the model, we used it together with the semi-implicit algorithm to simulate a Drosophila gene regulatory circuit that drives development in the dorsal-ventral axis of the blastoderm-stage embryo, involving three genes. For model validation, we have done both mathematical and statistical comparisons between the experimental data and the model's simulated data. Where protein and cis-regulatory information is available, our two-layer model provides a method for recapitulating and predicting dynamic aspects of eukaryotic transcriptional systems that will greatly improve our understanding of gene regulation at a global level.

Comparative approaches to the study of physiology: Drosophila as a physiological tool

Numerous studies have detailed the extensive conservation of developmental signaling pathways between the model system, Drosophila melanogaster, and mammalian models, but researchers have also profited from the unique and highly tractable genetic tools available in this system to address critical questions in physiology. In this review, we have described contributions that Drosophila researchers have made to mathematical dynamics of pattern formation, cardiac pathologies, the way in which pain circuits are integrated to elicit responses from sensation, as well as the ways in which gene expression can modulate diverse behaviors and shed light on human cognitive disorders. The broad and diverse array of contributions from Drosophila underscore its translational relevance to modeling human disease.

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.

Using graph-based assessments within socratic tutorials to reveal and refine students' analytical thinking about molecular networks

Biological systems, from the molecular to the ecological, involve dynamic interaction networks. To examine student thinking about networks we used graphical responses, since they are easier to evaluate for implied, but unarticulated assumptions. Senior college level molecular biology students were presented with simple molecular level scenarios; surprisingly, most students failed to articulate the basic assumptions needed to generate reasonable graphical representations; their graphs often contradicted their explicit assumptions. We then developed a tiered Socratic tutorial based on leading questions designed to provoke metacognitive reflection. The activity is characterized by leading questions (prompts) designed to provoke meta-cognitive reflection. When applied in a group or individual setting, there was clear improvement in targeted areas. Our results highlight the promise of using graphical responses and Socratic prompts in a tutorial context as both a formative assessment for students and an informative feedback system for instructors, in part because graphical responses are relatively easy to evaluate for implied, but unarticulated assumptions.

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

The axial bodyplan of Drosophila melanogaster is determined during a process called morphogenesis. Shortly after fertilization, maternal bicoid mRNA is translated into Bicoid (Bcd). This protein establishes a spatially graded morphogen distribution along the anterior-posterior (AP) axis of the embryo. Bcd initiates AP axis determination by triggering expression of gap genes that subsequently regulate each other's expression to form a precisely controlled spatial distribution of gene products. Reaction-diffusion models of gap gene expression on a 1D domain have previously been used to infer complex genetic regulatory network (GRN) interactions by optimizing model parameters with respect to 1D gap gene expression data. Here we construct a finite element reaction-diffusion model with a realistic 3D geometry fit to full 3D gap gene expression data. Though gap gene products exhibit dorsal-ventral asymmetries, we discover that previously inferred gap GRNs yield qualitatively correct AP distributions on the 3D domain only when DV-symmetric initial conditions are employed. Model patterning loses qualitative agreement with experimental data when we incorporate a realistic DV-asymmetric distribution of Bcd. Further, we find that geometry alone is insufficient to account for DV-asymmetries in the final gap gene distribution. Additional GRN optimization confirms that the 3D model remains sensitive to GRN parameter perturbations. Finally, we find that incorporation of 3D data in simulation and optimization does not constrain the search space or improve optimization results.