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Throughout development, complex networks of cell signaling pathways drive cellular decision-making across different tissues and contexts. The transforming growth factor β (TGF-β) pathways, including the BMP/Smad pathway, play crucial roles in determining cellular responses. However, as the Smad pathway is used reiteratively throughout the life cycle of all animals, its systems-level behavior varies from one context to another, despite the pathway connectivity remaining nearly constant. For instance, some cellular systems require a rapid response, while others require high noise filtering. In this paper, we examine how the BMP-Smad pathway balances trade-offs among three such systems-level behaviors, or "Performance Objectives (POs)": response speed, noise amplification, and the sensitivity of pathway output to receptor input. Using a Smad pathway model fit to human cell data, we show that varying non-conserved parameters (NCPs) such as protein concentrations, the Smad pathway can be tuned to emphasize any of the three POs and that the concentration of nuclear phosphatase has the greatest effect on tuning the POs. However, due to competition among the POs, the pathway cannot simultaneously optimize all three, but at best must balance trade-offs among the POs. We applied the multi-objective optimization concept of the Pareto Front, a widely used concept in economics to identify optimal trade-offs among various requirements. We show that the BMP pathway efficiently balances competing POs across species and is largely Pareto optimal. Our findings reveal that varying the concentration of NCPs allows the Smad signaling pathway to generate a diverse range of POs. This insight identifies how signaling pathways can be optimally tuned for each context., (© 2024. The Author(s).)

Positional information encoded in signaling molecules is essential for early patterning in the prosensory domain of the developing cochlea. The sensory epithelium, the organ of Corti, contains an exquisite repeating pattern of hair cells and supporting cells. This requires precision in the morphogen signals that set the initial radial compartment boundaries, but this has not been investigated. To measure gradient formation and morphogenetic precision in developing cochlea, we developed a quantitative image analysis procedure measuring SOX2 and pSMAD1/5/9 profiles in mouse embryos at embryonic day (E)12.5, E13.5, and E14.5. Intriguingly, we found that the pSMAD1/5/9 profile forms a linear gradient up to the medial ~ 75% of the PSD from the pSMAD1/5/9 peak in the lateral edge during E12.5 and E13.5. This is a surprising activity readout for a diffusive BMP4 ligand secreted from a tightly constrained lateral region since morphogens typically form exponential or power-law gradient shapes. This is meaningful for gradient interpretation because while linear profiles offer the theoretically highest information content and distributed precision for patterning, a linear morphogen gradient has not yet been observed. Furthermore, this is unique to the cochlear epithelium as the pSMAD1/5/9 gradient is exponential in the surrounding mesenchyme. In addition to the information-optimized linear profile, we found that while pSMAD1/5/9 is stable during this timeframe, an accompanying gradient of SOX2 shifts dynamically. Last, through joint decoding maps of pSMAD1/5/9 and SOX2, we see that there is a high-fidelity mapping between signaling activity and position in the regions that will become Kölliker's organ and the organ of Corti. Mapping is ambiguous in the prosensory domain precursory to the outer sulcus. Altogether, this research provides new insights into the precision of early morphogenetic patterning cues in the radial cochlea prosensory domain., (© 2023. The Author(s).)

Transforming growth factor-β1, -β2, and -β3 (TGF-β1, -β2, and -β3) are secreted signaling ligands that play essential roles in tissue development, tissue maintenance, immune response, and wound healing. TGF-β ligands form homodimers and signal by assembling a heterotetrameric receptor complex comprised of two type I receptor (TβRI):type II receptor (TβRII) pairs. TGF-β1 and TGF-β3 ligands signal with high potency due to their high affinity for TβRII, which engenders high-affinity binding of TβRI through a composite TGF-β:TβRII binding interface. However, TGF-β2 binds TβRII 200-500 more weakly than TGF-β1 and TGF-β3 and signals with lower potency compared with these ligands. Remarkably, the presence of an additional membrane-bound coreceptor, known as betaglycan, increases TGF-β2 signaling potency to levels similar to TGF-β1 and -β3. The mediating effect of betaglycan occurs even though it is displaced from and not present in the heterotetrameric receptor complex through which TGF-β2 signals. Published biophysics studies have experimentally established the kinetic rates of the individual ligand-receptor and receptor-receptor interactions that initiate heterotetrameric receptor complex assembly and signaling in the TGF-β system; however, current experimental approaches are not able to directly measure kinetic rates for the intermediate and latter steps of assembly. To characterize these steps in the TGF-β system and determine the mechanism of betaglycan in the potentiation of TGF-β2 signaling, we developed deterministic computational models with different modes of betaglycan binding and varying cooperativity between receptor subtypes. The models identified conditions for selective enhancement of TGF-β2 signaling. The models provide support for additional receptor binding cooperativity that has been hypothesized but not evaluated in the literature. The models further showed that betaglycan binding to the TGF-β2 ligand through two domains provides an effective mechanism for transfer to the signaling receptors that has been tuned to efficiently promote assembly of the TGF-β2(TβRII) 2 (TβRI) 2 signaling complex., Competing Interests: Declaration of interests The authors declare no competing interests., (Copyright © 2023 Biophysical Society. All rights reserved.)

In response to the growing computational intensity of the healthcare industry, biomedical engineering (BME) undergraduate education is placing increased emphasis on computation. The presence of substantial gender disparities in many computationally intensive disciplines suggests that the adoption of computational instruction approaches that lack intentionality may exacerbate gender disparities. Educational research suggests that the development of an engineering and computational identity is one factor that can support students' decisions to enter and persist in an engineering major. Discipline-based identity research is used as a lens to understand retention and persistence of students in engineering. Our specific purpose is to apply discipline-based identity research to define and explore the computational identities of undergraduate engineering students who engage in computational environments. This work will inform future studies regarding retention and persistence of students who engage in computational courses. Twenty-eight undergraduate engineering students (20 women, 8 men) from three engineering majors (biomedical engineering, agricultural engineering, and biological engineering) participated in semi-structured interviews. The students discussed their experiences in a computationally-intensive thermodynamics course offered jointly by the Biomedical Engineering and Agricultural & Biological Engineering departments. The transcribed interviews were analyzed through thematic coding. The gender stereotypes associated with computer programming also come part and parcel with computer programming, possibly threatening a student's sense of belonging in engineering. The majority of the participants reported that their computational identity was "in the making." Students' responses also suggested that their engineering identity and their computational identity were in congruence, while some incongruence is found between their engineering identity and a creative identity as well as between computational identity and perceived feminine norms. Responses also indicate that students associate specific skills with having a computational identity. This study's findings present an emergent thematic definition of a computational person constructed from student perceptions and experiences. Instructors can support students' nascent computational identities through intentional mitigation of the gender stereotypes and biases, and by framing assignments to focus on developing specific skills associated with the computational modeling processes., (© The Author(s) 2022.)

We propose a PDE-constrained shape registration algorithm that captures the deformation and growth of biological tissue from imaging data. Shape registration is the process of evaluating optimum alignment between pairs of geometries through a spatial transformation function. We start from our previously reported work, which uses 3D tensor product B-spline basis functions to interpolate 3D space. Here, the movement of the B-spline control points, composed with an implicit function describing the shape of the tissue, yields the total deformation gradient field. The deformation gradient is then split into growth and elastic contributions. The growth tensor captures addition of mass, i.e. growth, and evolves according to a constitutive equation which is usually a function of the elastic deformation. Stress is generated in the material due to the elastic component of the deformation alone. The result of the registration is obtained by minimizing a total energy functional which includes: a distance measure reflecting similarity between the shapes, and the total elastic energy accounting for the growth of the tissue. We apply the proposed shape registration framework to study zebrafish embryo epiboly process and tissue expansion during skin reconstruction surgery. We anticipate that our PDE-constrained shape registration method will improve our understanding of biological and medical problems in which tissues undergo extreme deformations over time.

Numerous stages of organismal development rely on the cellular interpretation of gradients of secreted morphogens including members of the Bone Morphogenetic Protein (BMP) family through transmembrane receptors. Early gradients of BMPs drive dorsal/ventral patterning throughout the animal kingdom in both vertebrates and invertebrates. Growing evidence in Drosophila, zebrafish, murine and other systems suggests that BMP ligand heterodimers are the primary BMP signaling ligand, even in systems in which mixtures of BMP homodimers and heterodimers are present. Signaling by heterodimers occurs through a hetero-tetrameric receptor complex comprising of two distinct type one BMP receptors and two type II receptors. To understand the system dynamics and determine whether kinetic assembly of heterodimer-heterotetramer BMP complexes is favored, as compared to other plausible BMP ligand-receptor configurations, we developed a kinetic model for BMP tetramer formation based on current measurements for binding rates and affinities. We find that contrary to a common hypothesis, heterodimer-heterotetramer formation is not kinetically favored over the formation of homodimer-tetramer complexes under physiological conditions of receptor and ligand concentrations and therefore other mechanisms, potentially including differential kinase activities of the formed heterotetramer complexes, must be the cause of heterodimer-heterotetramer signaling primacy. Further, although BMP complex assembly favors homodimer and homomeric complex formation over a wide range of parameters, ignoring these signals and instead relying on the heterodimer improves the range of morphogen interpretation in a broad set of conditions, suggesting a performance advantage for heterodimer signaling in patterning multiple cell types in a gradient., Competing Interests: The authors have declared that no competing interests exist.

Identification of individual cells in tissues, organs, and in various developing systems is a well-studied problem because it is an essential part of objectively analyzing quantitative images in numerous biological contexts. We developed a size-dependent wavelet-based segmentation method that provides robust segmentation without any preprocessing, filtering or fine-tuning steps, and is robust to the signal-to-noise ratio. The wavelet-based method achieves robust segmentation results with respect to True Positive rate, Precision, and segmentation accuracy compared with other commonly used methods. We applied the segmentation program to zebrafish embryonic development IN TOTO for nuclei segmentation, image registration, and nuclei shape analysis. These new approaches to segmentation provide a means to carry out quantitative patterning analysis with single-cell precision throughout three dimensional tissues and embryos and they have a high tolerance for non-uniform and noisy image data sets.

Pattern formation by bone morphogenetic proteins (BMPs) demonstrates remarkable plasticity and utility in several contexts, such as early embryonic development, tissue patterning and the maintenance of stem cell niches. BMPs pattern tissues over many temporal and spatial scales: BMP gradients as short as 1-2 cell diameters maintain the stem cell niche of the Drosophila germarium over a 24-h cycle, and BMP gradients of several hundred microns establish dorsal-ventral tissue specification in Drosophila , zebrafish and Xenopus embryos in timescales between 30 min and several hours. The mechanisms that shape BMP signaling gradients are also incredibly diverse. Although ligand diffusion plays a dominant role in forming the gradient, a cast of diffusible and non-diffusible regulators modulate gradient formation and confer robustness, including scale invariance and adaptability to perturbations in gene expression and growth. In this Review, we document the diverse ways that BMP gradients are formed and refined, and we identify the core principles that they share to achieve reliable performance., Competing Interests: Competing interestsThe authors declare no competing or financial interests., (© 2021. Published by The Company of Biologists Ltd.)

Purpose of Review: Partial differential equation (PDE) mathematical models of biological systems and the simulation approaches used to solve them are widely used to test hypotheses and infer regulatory interactions based on optimization of the PDE model against the observed data. In this review, we discuss the ability of powerful machine learning methods to accelerate the parametric screening of biophysical informed- PDE systems., Recent Findings: A major shortcoming in more broad adaptation of PDE-based models is the high computational complexity required to solve and optimize the models and it requires many simulations to traverse the very high-dimensional parameter spaces during model calibration and inference tasks. For instance, when scaling up to tens of millions of simulations for optimization and sensitivity analysis of the PDE models, compute times quickly extend from months to years for sufficient coverage to solve the problems. For many systems, this brute-force approach is simply not feasible. Recently, neural network metamodels have been shown to be an efficient way to accelerate PDE model calibration and here we look at the benefits and limitations in extending the PDE acceleration methods to improve optimization and sensitivity analysis., Summary: We use an example simulation to quantitatively and qualitatively show how neural network metamodels can be accurate and fast and demonstrate their potential for optimization of complex spatiotemporal problems in biology. We expect these approaches will be broadly applied to speed up scientific research and discovery in biology and other systems that can be described by complex PDE systems., Competing Interests: Conflicts of Interest No potential conflicts of interest relevant to this article were reported.

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

Neutrophil migration is essential for inflammatory responses to kill pathogens; however, excessive neutrophilic inflammation also leads to tissue injury and adverse effects. To discover novel therapeutic targets that modulate neutrophil migration, we performed a neutrophil-specific microRNA (miRNA) overexpression screen in zebrafish and identified 8 miRNAs as potent suppressors of neutrophil migration. Among those, miR-199 decreases neutrophil chemotaxis in zebrafish and human neutrophil-like cells. Intriguingly, in terminally differentiated neutrophils, miR-199 alters the cell cycle-related pathways and directly suppresses cyclin-dependent kinase 2 (Cdk2), whose known activity is restricted to cell cycle progression and cell differentiation. Inhibiting Cdk2, but not DNA replication, disrupts cell polarity and chemotaxis of zebrafish neutrophils without inducing cell death. Human neutrophil-like cells deficient in CDK2 fail to polarize and display altered signaling downstream of the formyl peptide receptor. Chemotaxis of primary human neutrophils is also reduced upon CDK2 inhibition. Furthermore, miR-199 overexpression or CDK2 inhibition significantly improves the outcome of lethal systemic inflammation challenges in zebrafish. Our results therefore reveal previously unknown functions of miR-199 and CDK2 in regulating neutrophil migration and provide directions in alleviating systemic inflammation., Competing Interests: The authors declare no conflict of interest., (Copyright © 2019 the Author(s). Published by PNAS.)

Cell migration plays an important role in physiology and pathophysiology. It was observed in the experiments that cells, such as fibroblast, leukocytes, and cancer cells, exhibit a wide variety of migratory behaviors, such as persistent random walk, contact inhibition of locomotion, and ordered behaviors. To identify biophysical mechanisms for these cellular behaviors, we developed a rigorous computational model of cell migration on a two-dimensional non-deformable substrate. Cells in the model undergo motion driven by mechanical interactions between cellular protrusions and the substrate via the balance of tensile forces. Properties of dynamic formation of lamellipodia induced the persistent random walk behavior of a migrating cell. When multiple cells are included in the simulation, the model recapitulated the contact inhibition of locomotion between cells at low density without any phenomenological assumptions or momentum transfer. Instead, the model showed that contact inhibition of locomotion can emerge via indirect interactions between the cells through their interactions with the underlying substrate. At high density, contact inhibition of locomotion between numerous cells gave rise to confined motions or ordered behaviors, depending on cell density and how likely lamellipodia turn over due to contact with other cells. Results in our study suggest that various collective migratory behaviors may emerge without more restrictive assumptions or direct cell-to-cell biomechanical interactions.

In both vertebrates and invertebrates, spatial patterning along the Dorsal-ventral (DV) embryonic axis depends on a morphogen gradient of Bone Morphogenetic Protein signaling. Scale invariance of DV patterning by BMPs has been found in both vertebrates and invertebrates, however the mechanisms that regulate gradient scaling remain controversial. To obtain quantitative data that can be used to address core questions of scaling, we introduce a method to tune the size of zebrafish embryos by reducing varying amounts of vegetal yolk. We quantified the BMP signaling gradient in wild-type and perturbed embryos and found that the system scales for reductions in cross-sectional perimeter of up to 30%. Furthermore, we found that the degree of scaling for intraspecies scaling within zebrafish is greater than that between Danioninae species.

Protein post-translational modifications (PTMs) serve to give proteins new cellular functions and can influence spatial distribution and enzymatic activity, greatly enriching the complexity of the proteome. Lipidation is a PTM that regulates protein stability, function, and subcellular localization. To complement advances in proteomic identification of lipidated proteins, we have developed a method to image the spatial distribution of proteins that have been co- and post-translationally modified via the addition of myristic acid (Myr) to the N terminus. In this work, we use a Myr analog, 12-azidododecanoic acid (12-ADA), to facilitate fluorescent detection of myristoylated proteins in vitro and in vivo. The azide moiety of 12-ADA does not react to natural biological chemistries, but is selectively reactive with alkyne functionalized fluorescent dyes. We find that the spatial distribution of myristoylated proteins varies dramatically between undifferentiated and differentiated muscle cells in vitro. Further, we demonstrate that our methodology can visualize the distribution of myristoylated proteins in zebrafish muscle in vivo. Selective protein labeling with noncanonical fatty acids, such as 12-ADA, can be used to determine the biological function of myristoylation and other lipid-based PTMs and can be extended to study deregulated protein lipidation in disease states., (Copyright © 2017 by the American Society for Biochemistry and Molecular Biology, Inc.)

Advanced molecular probing techniques such as single molecule fluorescence in situ hybridization (smFISH) or RNAscope can be used to assess the quantity and spatial location of mRNA transcripts within cells. Quantifying mRNA expression in large image sets usually involves automated counting of fluorescent spots. Though conventional spot counting algorithms may suffice, they often lack high-throughput capacity and accuracy in cases of crowded signal or excessive noise. Automatic identification of cells and processing of many images is still a challenge. We have developed a method to perform automatic cell boundary identification while providing quantitative data about mRNA transcript levels across many images. Comparisons of mRNA transcript levels identified by the method highly correlate to qPCR measurements of mRNA expression in Drosophila genotypes with different levels of Rhodopsin 1 transcript. We also introduce a graphical user interface to facilitate analysis of large data sets. We expect these methods to translate to model systems where automated image processing can be harnessed to obtain single-cell data. The described method: •Provides relative intensity measurements that scale directly with the number of labeled transcript probes within individual cells.•Allows quantitative assessment of single molecule data from images with crowded signal and moderate signal to noise ratios.

Background: Acute myelogenous leukemia (AML) progresses uniquely in each patient. However, patients are typically treated with the same types of chemotherapy, despite biological differences that lead to differential responses to treatment., Results: Here we present a multi-lineage multi-compartment model of the hematopoietic system that captures patient-to-patient variation in both the concentration and rates of change of hematopoietic cell populations. By constraining the model against clinical hematopoietic cell recovery data derived from patients who have received induction chemotherapy, we identified trends for parameters that must be met by the model; for example, the mitosis rates and the probability of self-renewal of progenitor cells are inversely related. Within the data-consistent models, we found 22,796 parameter sets that meet chemotherapy response criteria. Simulations of these parameter sets display diverse dynamics in the cell populations. To identify large trends in these model outputs, we clustered the simulated cell population dynamics using k-means clustering and identified thirteen 'representative patient' dynamics. In each of these patient clusters, we simulated AML and found that clusters with the greatest mitotic capacity experience clinical cancer outcomes more likely to lead to shorter survival times. Conversely, other parameters, including lower death rates or mobilization rates, did not correlate with survival times., Conclusions: Using the multi-lineage model of hematopoiesis, we have identified several key features that determine leukocyte homeostasis, including self-renewal probabilities and mitosis rates, but not mobilization rates. Other influential parameters that regulate AML model behavior are responses to cytokines/growth factors produced in peripheral blood that target the probability of self-renewal of neutrophil progenitors. Finally, our model predicts that the mitosis rate of cancer is the most predictive parameter for survival time, followed closely by parameters that affect the self-renewal of cancer stem cells; most current therapies target mitosis rate, but based on our results, we propose that additional therapeutic targeting of self-renewal of cancer stem cells will lead to even higher survival rates.

Dicot leaves are composed of a heterogeneous mosaic of jigsaw puzzle piece-shaped pavement cells that vary greatly in size and the complexity of their shape. Given the importance of the epidermis and this particular cell type for leaf expansion, there is a strong need to understand how pavement cells morph from a simple polyhedral shape into highly lobed and interdigitated cells. At present, it is still unclear how and when the patterns of lobing are initiated in pavement cells, and one major technological bottleneck to addressing the problem is the lack of a robust and objective methodology to identify and track lobing events during the transition from simple cell geometry to lobed cells. We developed a convex hull-based algorithm termed LobeFinder to identify lobes, quantify geometric properties, and create a useful graphical output of cell coordinates for further analysis. The algorithm was validated against manually curated images of pavement cells of widely varying sizes and shapes. The ability to objectively count and detect new lobe initiation events provides an improved quantitative framework to analyze mutant phenotypes, detect symmetry-breaking events in time-lapse image data, and quantify the time-dependent correlation between cell shape change and intracellular factors that may play a role in the morphogenesis process., (© 2016 American Society of Plant Biologists. All Rights Reserved.)

In a Wall Street Journal article published on April 5, 2013, E. O. Wilson attempted to make the case that biologists do not really need to learn any mathematics-whenever they run into difficulty with numerical issues, they can find a technician (aka mathematician) to help them out of their difficulty. He formalizes this in Wilsons Principle No. 1: "It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations." This reflects a complete misunderstanding of the role of mathematics in all sciences throughout history. To Wilson, mathematics is mere number crunching, but as Galileo said long ago, "The laws of Nature are written in the language of mathematics[Formula: see text] the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word." Mathematics has moved beyond the geometry-based model of Galileo's time, and in a rebuttal to Wilson, E. Frenkel has pointed out the role of mathematics in synthesizing the general principles in science (Both point and counter-point are available in Wilson and Frenkel in Notices Am Math Soc 60(7):837-838, 2013). We will take this a step further and show how mathematics has been used to make new and experimentally verified discoveries in developmental biology and how mathematics is essential for understanding a problem that has puzzled experimentalists for decades-that of how organisms can scale in size. Mathematical analysis alone cannot "solve" these problems since the validation lies at the molecular level, but conversely, a growing number of questions in biology cannot be solved without mathematical analysis and modeling. Herein, we discuss a few examples of the productive intercourse between mathematics and biology.

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., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Discovery in developmental biology is often driven by intuition that relies on the integration of multiple types of data such as fluorescent images, phenotypes, and the outcomes of biochemical assays. Mathematical modeling helps elucidate the biological mechanisms at play as the networks become increasingly large and complex. However, the available data is frequently under-utilized due to incompatibility with quantitative model tuning techniques. This is the case for stem cell regulation mechanisms explored in the Drosophila germarium through fluorescent immunohistochemistry. To enable better integration of biological data with modeling in this and similar situations, we have developed a general parameter estimation process to quantitatively optimize models with qualitative data. The process employs a modified version of the Optimal Scaling method from social and behavioral sciences, and multi-objective optimization to evaluate the trade-off between fitting different datasets (e.g. wild type vs. mutant). Using only published imaging data in the germarium, we first evaluated support for a published intracellular regulatory network by considering alternative connections of the same regulatory players. Simply screening networks against wild type data identified hundreds of feasible alternatives. Of these, five parsimonious variants were found and compared by multi-objective analysis including mutant data and dynamic constraints. With these data, the current model is supported over the alternatives, but support for a biochemically observed feedback element is weak (i.e. these data do not measure the feedback effect well). When also comparing new hypothetical models, the available data do not discriminate. To begin addressing the limitations in data, we performed a model-based experiment design and provide recommendations for experiments to refine model parameters and discriminate increasingly complex hypotheses.

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

Mathematical modeling of transcription factor and signaling networks is widely used to understand if and how a mechanism works, and to infer regulatory interactions that produce a model consistent with the observed data. Both of these approaches to modeling are informed by experimental data, however, much of the data available or even acquirable are not quantitative. Data that is not strictly quantitative cannot be used by classical, quantitative, model-based analyses that measure a difference between the measured observation and the model prediction for that observation. To bridge the model-to-data gap, a variety of techniques have been developed to measure model "fitness" and provide numerical values that can subsequently be used in model optimization or model inference studies. Here, we discuss a selection of traditional and novel techniques to transform data of varied quality and enable quantitative comparison with mathematical models. This review is intended to both inform the use of these model analysis methods, focused on parameter estimation, and to help guide the choice of method to use for a given study based on the type of data available. Applying techniques such as normalization or optimal scaling may significantly improve the utility of current biological data in model-based study and allow greater integration between disparate types of data., (Copyright © 2013 Elsevier Inc. All rights reserved.)

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., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

Bone Morphogenetic Proteins (BMPs) are critical for pattern formation in many animals. In numerous tissues, BMPs become distributed in spatially non-uniform profiles. The gradients of signaling activity can be detected by a number of biological assays involving fluorescence microscopy. Quantitative analyses of BMP gradients are powerful tools to investigate the regulation of BMP signaling pathways during development. These approaches rely heavily on images as spatial representations of BMP activity levels, using them to infer signaling distributions that inform on regulatory mechanisms. In this perspective, we discuss current imaging assays and normalization methods used to quantify BMP activity profiles with a focus on the Drosophila wing primordium. We find that normalization tends to lower the number of samples required to establish statistical significance between profiles in controls and experiments, but the increased resolvability comes with a cost. Each normalization strategy makes implicit assumptions about the biology that impacts our interpretation of the data. We examine the tradeoffs for normalizing versus not normalizing, and discuss their impacts on experimental design and the interpretation of resultant data., (Copyright © 2012 Federation of European Biochemical Societies. Published by Elsevier B.V. All rights reserved.)

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.

Lipid droplets are complex organelles that exhibit highly dynamic behavior in early Drosophila embryo development. Imaging lipid droplet motion provides a robust platform for the investigation of shuttling by kinesin and dynein motors, but methods for imaging are either destructive or deficient in resolution and penetration to study large populations of droplets in an individual embryo. Here we report real-time imaging and quantification of droplet motion in live embryos using a recently developed technique termed "femtosecond-stimulated Raman loss" microscopy. We captured long-duration time-lapse images of the developing embryo, tracked single droplet motion within large populations of droplets, and measured the velocity and turning frequency of each particle at different apical-to-basal depths and stages of development. To determine whether the quantities for speed and turning rate measured for individual droplets are sufficient to predict the population distributions of droplet density, we simulated droplet motion using a velocity-jump model. This model yielded droplet density distributions that agreed well with experimental observations without any model optimization or unknown parameter estimation, demonstrating the sufficiency of a velocity-jump process for droplet trafficking dynamics in blastoderm embryos., (Copyright © 2012 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

Bone morphogenetic protein (BMP) signaling controls development and maintenance of many tissues. Genetic and quantitative approaches in Drosophila reveal that ligand isoforms show distinct function in wing development. Spatiotemporal control of BMP patterning depends on a network of extracellular proteins Pent, Ltl and Dally that regulate BMP signaling strength and morphogen range. BMP-mediated feedback regulation of Pent, Ltl, and Dally expression provides a system where cells actively respond to, and modify, the extracellular morphogen landscape to form a gradient that exhibits remarkable properties, including proportional scaling of BMP patterning with tissue size and the modulation of uniform tissue growth. This system provides valuable insights into mechanisms that mitigate the influence of variability to regulate cell-cell interactions and maintain organ function., (Copyright © 2011 Elsevier Ltd. All rights reserved.)

The Steady State (SS) probability distribution is an important quantity needed to characterize the steady state behavior of many stochastic biochemical networks. In this paper, we propose an efficient and accurate approach to calculating an approximate SS probability distribution from solution of the Chemical Master Equation (CME) under the assumption of the existence of a unique deterministic SS of the system. To find the approximate solution to the CME, a truncated state-space representation is used to reduce the state-space of the system and translate it to a finite dimension. The subsequent ill-posed eigenvalue problem of a linear system for the finite state-space can be converted to a well-posed system of linear equations and solved. The proposed strategy yields efficient and accurate estimation of noise in stochastic biochemical systems. To demonstrate the approach, we applied the method to characterize the noise behavior of a set of biochemical networks of ligand-receptor interactions for Bone Morphogenetic Protein (BMP) signaling. We found that recruitment of type II receptors during the receptor oligomerization by itself doesn't not tend to lower noise in receptor signaling, but regulation by a secreted co-factor may provide a substantial improvement in signaling relative to noise. The steady state probability approximation method shortened the time necessary to calculate the probability distributions compared to earlier approaches, such as Gillespie's Stochastic Simulation Algorithm (SSA) while maintaining high accuracy.

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.

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., (Copyright (c) 2010 Elsevier Inc. All rights reserved.)

During the development of some tissues, fields of multipotent cells differentiate into distinct cell types in response to the local concentration of a signalling factor called a morphogen. Typically, individual organisms within a population differ in size, but their body plans appear to be scaled versions of a common template. Similarly, closely related species may differ by three or more orders of magnitude in size, yet common structures between species scale to have similar proportions. In standard reaction-diffusion equations, the morphogen range has a length scale that depends on a balance between kinetic and transport processes and not on the length or size of the field of cells being patterned. However, as shown here for a class of morphogen-patterning systems, a number of conditions lead to scale invariance of the morphogen distribution at equilibrium and during the transient approach to equilibrium. Equilibrium scale invariance requires conservation of the total binding site number and total input flux. Dynamic scale invariance additionally requires sufficient binding to slow the diffusion of ligand. The equations derived herein can be extended to the study of other perturbations to gain further insight into the processes regulating the robustness and scaling of morphogen-mediated pattern formation.

In many developing systems, the fate of a cell is determined by its position in a time-independent spatial distribution of a morphogen. However, during dorsal-ventral patterning in the Drosophila embryo, an initial low-level signal refines to a narrow, high-intensity band. This refinement suggests that cells respond to the local transient morphogen distribution that results from interactions between bone morphogenetic proteins (BMPs), their receptors, the BMP-binding proteins Sog and Tsg, the metalloprotease Tld, and a putative, positively regulated component that locally enhances surface binding of BMPs within the region of high signaling. We develop a computational model for dorsal surface patterning and show that, when positive feedback of a cell surface BMP-binding protein is incorporated, bistability in the kinetic interactions transduces the transient BMP distribution into a switch-like spatial distribution of the BMP-bound receptor. We also show that the inclusion of positive feedback leads to the observed contraction of signaling, because cells near the dorsal midline outcompete adjacent lateral cells for limited amounts of BMP. In the model, cells interpret the morphogen distribution by differentiating according to the history of their exposure rather than to a threshold concentration in a static spatial gradient of the morphogen.

Pharmacokinetic models for ethanol metabolism have contributed to the understanding of ethanol clearance in human beings. However, these models fail to account for ethanol's toxic metabolite, acetaldehyde. Acetaldehyde accumulation leads to signs and symptoms, such as cardiac arrhythmias, nausea, anxiety, and facial flushing. Nevertheless, it is difficult to determine the levels of acetaldehyde in the blood or other tissues because of artifactual formation and other technical issues. Therefore, we have constructed a promising physiologically based pharmacokinetic (PBPK) model, which is an excellent match for existing ethanol and acetaldehyde concentration-time data. The model consists of five compartments that exchange material: stomach, gastrointestinal tract, liver, central fluid, and muscle. All compartments except the liver are modeled as stirred reactors. The liver is modeled as a tubular flow reactor. We derived average enzymatic rate laws for alcohol dehydrogenase (ADH) and acetaldehyde dehydrogenase (ALDH), determined kinetic parameters from the literature, and found best-fit parameters by minimizing the squared error between our profiles and the experimental data. The model's transient output correlates strongly with the experimentally observed results for healthy individuals and for those with reduced ALDH activity caused by a genetic deficiency of the primary acetaldehyde-metabolizing enzyme ALDH2. Furthermore, the model shows that the reverse reaction of acetaldehyde back into ethanol is essential and keeps acetaldehyde levels approximately 10-fold lower than if the reaction were irreversible.