Diversity and robustness of bone morphogenetic protein pattern formation.

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.

Computational modeling of TGF-ß2:TßRI:TßRII receptor complex assembly as mediated by the TGF-ß coreceptor betaglycan.

Transforming growth factor-ß1, -ß2, and -ß3 (TGF-ß1, -ß2, and -ß3) are secreted signaling ligands that play essential roles in tissue development, tissue maintenance, immune response, and wound healing. TGF-ß ligands form homodimers and signal by assembling a heterotetrameric receptor complex comprised of two type I receptor (TßRI):type II receptor (TßRII) pairs. TGF-ß1 and TGF-ß3 ligands signal with high potency due to their high affinity for TßRII, which engenders high-affinity binding of TßRI through a composite TGF-ß:TßRII binding interface. However, TGF-ß2 binds TßRII 200-500 more weakly than TGF-ß1 and TGF-ß3 and signals with lower potency compared with these ligands. Remarkably, the presence of an additional membrane-bound coreceptor, known as betaglycan, increases TGF-ß2 signaling potency to levels similar to TGF-ß1 and -ß3. The mediating effect of betaglycan occurs even though it is displaced from and not present in the heterotetrameric receptor complex through which TGF-ß2 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-ß 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-ß system and determine the mechanism of betaglycan in the potentiation of TGF-ß2 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-ß2 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-ß2 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-ß2(TßRII)2(TßRI)2 signaling complex.

Mitofusin 2 regulates neutrophil adhesive migration and the actin cytoskeleton.

Neutrophils rely on glycolysis for energy production. How mitochondria regulate neutrophil function is not fully understood. Here, we report that mitochondrial outer membrane protein Mitofusin 2 (MFN2) regulates neutrophil homeostasis and chemotaxis in vivoMfn2-deficient neutrophils are released from the hematopoietic tissue, trapped in the vasculature in zebrafish embryos, and not capable of chemotaxis. Consistent with this, human neutrophil-like cells that are deficient for MFN2 fail to arrest on activated endothelium under sheer stress or perform chemotaxis on 2D surfaces. Deletion of MFN2 results in a significant reduction of neutrophil infiltration to the inflamed peritoneal cavity in mice. Mechanistically, MFN2-deficient neutrophil-like cells display disrupted mitochondria-ER interaction, heightened intracellular Ca2+ levels and elevated Rac activation after chemokine stimulation. Restoring a mitochondria-ER tether rescues the abnormal Ca2+ levels, Rac hyperactivation and chemotaxis defect resulting from MFN2 depletion. Finally, inhibition of Rac activation restores chemotaxis in MFN2-deficient neutrophils. Taken together, we have identified that MFN2 regulates neutrophil migration via maintaining the mitochondria-ER interaction to suppress Rac activation, and uncovered a previously unrecognized role of MFN2 in regulating cell migration and the actin cytoskeleton.This article has an associated First Person interview with the first authors of the paper.

Heterodimer-heterotetramer formation mediates enhanced sensor activity in a biophysical model for BMP signaling.

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.

Phenotypical microRNA screen reveals a noncanonical role of CDK2 in regulating neutrophil migration.

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.

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

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

Interplay Between the Persistent Random Walk and the Contact Inhibition of Locomotion Leads to Collective Cell Behaviors.

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.

Fluorescent imaging of protein myristoylation during cellular differentiation and development.

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.

LobeFinder: A Convex Hull-Based Method for Quantitative Boundary Analyses of Lobed Plant Cells.

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.

Mechanisms of scaling in pattern formation.

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

Model-based analysis for qualitative data: an application in Drosophila germline stem cell regulation.

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.

The role of mathematical models in understanding pattern formation in developmental biology.

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.

Quantitative model analysis with diverse biological data: applications in developmental pattern formation.

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.

Optimal Performance Objectives in the Highly Conserved Bone Morphogenetic Protein Signaling Pathway.

Throughout development, complex networks of cell signaling pathways drive cellular decision-making across different tissues and contexts. The transforming growth factor ß (TGF-ß) pathways, including the BMP/Smad pathway, play crucial roles in these 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.

Optimal performance objectives in the highly conserved bone morphogenetic protein signaling pathway.

Throughout development, complex networks of cell signaling pathways drive cellular decision-making across different tissues and contexts. The transforming growth factor ß (TGF-ß) 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.

"I Think I Am Getting There" Understanding the Computational Identity of Engineering Students Participating in a Computationally Intensive Thermodynamics Course.

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.

Early radial positional information in the cochlea is optimized by a precise linear BMP gradient and enhanced by SOX2.

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.

Atypical peripheral actin band formation via overactivation of RhoA and nonmuscle myosin II in mitofusin 2-deficient cells.

Cell spreading and migration play central roles in many physiological and pathophysiological processes. We have previously shown that MFN2 regulates the migration of human neutrophil-like cells via suppressing Rac activation. Here, we show that in mouse embryonic fibroblasts, MFN2 suppresses RhoA activation and supports cell polarization. After initial spreading, the wild-type cells polarize and migrate, whereas the Mfn2-/- cells maintain a circular shape. Increased cytosolic Ca2+ resulting from the loss of Mfn2 is directly responsible for this phenotype, which can be rescued by expressing an artificial tether to bring mitochondria and endoplasmic reticulum to close vicinity. Elevated cytosolic Ca2+ activates Ca2+/calmodulin-dependent protein kinase II, RhoA, and myosin light-chain kinase, causing an overactivation of nonmuscle myosin II, leading to a formation of a prominent F-actin ring at the cell periphery and increased cell contractility. The peripheral actin band alters cell physics and is dependent on substrate rigidity. Our results provide a novel molecular basis to understand how MFN2 regulates distinct signaling pathways in different cells and tissue environments, which is instrumental in understanding and treating MFN2-related diseases.

Label-free imaging of lipid-droplet intracellular motion in early Drosophila embryos using femtosecond-stimulated Raman loss microscopy.

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.

Efficient calculation of steady state probability distribution for stochastic biochemical reaction network.

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.