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# Organism-scale modeling of early Drosophila patterning via Bone Morphogenetic Proteins

^{*}Agricultural and Biological Engineering, Weldon School of Biomedical Engineering, and Bindley Bioscience Center. 225 S. University St., Purdue University, West Lafayette, IN 47907, USA ;

^{*}corresponding authors

## Abstract

Advances in image acquisition and informatics technology have led to organism-scale spatio-temporal 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 3D model provides significant new insights into mechanisms that guide tissue patterning.

## Introduction

In many systems, spatially-patterned cellular differentiation is regulated by signaling molecules called morphogens, which initiate spatio-temporal patterns of gene expression in a concentration-dependant manner [Turing, 1952, Wolpert, 1969, Driever and Nusslein-Volhard, 1988]. In early Drosophila embryos, a morphogen composed of a heterodimer of Decapentaplegic (Dpp) and Screw (Scw), two members of the bone morphogenetic protein (BMP) family (Figure 1 A,B), directs patterning of the dorsal ectoderm [Shimmi et al., 2005, O’Connor et al., 2006]. Unlike classical morphogen systems that rely on the slow spreading of a molecule from a localized source to establish a gradient, BMPs in the early Drosophila embryo are secreted from a broad region making up the dorsal-most 40% of the embryo circumference. Subsequently they are dynamically concentrated into a narrow region centered about the dorsal midline that makes up only 10% of the embryo circumference (Figure 1 A,B) [Sutherland et al., 2003, Wang and Ferguson, 2005, O’Connor et al., 2006].

**...**

A number of extracellular regulators contribute to the dynamics and localization of BMP signaling (Figure 1 B,C). Laterally-secreted Short gastrulation (Sog) and dorsally-secreted Twisted gastrulation (Tsg) diffuse from their regions of expression and form a heterodimer inhibitor (Sog/Tsg) that binds to Dpp-Scw, preventing it from binding to receptors. The cell matrix may mediate the formation of this complex, since it has recently been shown that collagen can bind both BMPs and Sog, thereby facilitating their association (Figure 1 D) [Wang et al., 2008]. Since Dpp-Scw is secreted throughout the dorso-lateral region, this leads to a gradient of inhibitor-bound Dpp-Scw that is high laterally and low at the dorsal midline, and an opposing gradient of free Dpp-Scw that is high at the dorsal midline. The dorsally-secreted metalloprotease Tolloid (Tld) processes Sog only when Sog is bound to BMP ligands, and the degradation of Sog by Tld further enhances both the gradient of inhibitor-bound Dpp-Scw and of free Dpp-Scw. Thus, extracellular Dpp-Scw is redistributed by a combination of binding to inhibitor, processing of this complex, and diffusion.

Simultaneously, receptors and other surface-localized binding proteins compete with Sog to bind the available Dpp-Scw. Dpp-Scw activates signaling by binding to and recruiting the type I receptors, Thickveins (Tkv) and Saxophone (Sax), into a high-order complex containing two subunits of the type II receptor Punt [Shimmi et al., 2005]. The receptor complex phosphorylates Mad (pMad), a member of the Smad family of signal transducers, and phosphorlyated Mad binds to the co-Smad Medea forming a complex that then accumulates in the nucleus, where it regulates gene expression in a concentration-dependent manner [Schmierer et al., 2008].

While complex formation and transport favor a net movement of ligand towards the dorsal midline of the embryo, positive feedback in response to pMad signaling is needed to further concentrate the surface localized Dpp/Scw at the dorsal midline [Wang and Ferguson, 2005]. A loss of extracellular BMP regulators or positive feedback impedes attenuation of pMad laterally and the accumulation of pMad signaling at the dorsal midline. While feedback, extracellular transport, and signal transduction each provide a specific mode of Dpp/Scw signal regulation, it is the dynamic interaction of these regulatory mechanisms that patterns the dorsal surface of Drosophila embryos. Not only does the mechanism work under optimal laboratory conditions, but dorsal surface patterning appears to be remarkably resilient to non-ideal conditions such as temperature fluctuations, reductions in the level of regulatory factors such as Tsg, ectopic gene expression, and other perturbations [Eldar et al., 2002, Umulis et al., 2006]. These issues illustrate the complexity of the problem and suggest that we can no longer rely solely on genetic and biochemical data to fully explain this rather simple patterning problem.

To address a number of unanswered questions about Dpp-Scw mediated patterning and to take full advantage of the available data on Drosophila development, we developed a methodology that seamlessly integrates biological information in the form of pre-patterns, geometry, mechanisms and training data into an organism-scale model of the blastoderm embryo that is based on a reaction-diffusion description of patterning. The mathematical model is simulated using the widely-available computational frameworks Comsol and Matlab (Figure 2 and Figure S1 of the Supporting Online Materials (SOM)), which makes extensive use of the model and methodology feasible. We developed an image analysis protocol to obtain model training and initial condition data and to calculate population statistics for patterns of pMad signaling in *wt* and mutant *D. melanogaster*. We discovered that both the mean and variability of pMad signaling along the DV axis depends on AP position and the specific choice of threshold. Using mutations previously considered robust, we were able to detect differences between mutant and wild-type pMad signaling patterns, which provided an information-rich data set for model training and testing the contributions of diverse positive feedback mechanisms and of proteins that concentrate BMPs at the cell surface. Unexpectedly, we found that geometry also has a large impact on the predicted patterns of BMP-bound receptors, whereas prepatterned expression of receptors and other modulators of signaling did not greatly effect model-data correspondence. We found that if the embryo geometry is perturbed slightly in the model, then including the prepattern information greatly enhanced the model’s ability to fit the observed pMad patterns, which suggests that the prepatterns may mitigate the effects of slightly misshaped embryos. We further identified conditions in the model that improve scale-invariance of patterning and tested the model predictions by staining for pMad in different species of Drosophila. These studies demonstrate that building a model based on image data and training the 3D model against multi-dimensional expression data provides insights into the properties of several important developmental principles including positive feedback, biological robustness, and scale invariance.

## Results and Discussion

### Reproducibility of pMad signaling in late cycle 14 embryos

To begin our study, we first developed an image acquisition, normalization, and analysis strategy to obtain training and pre-pattern data for the model. To minimize errors in mounting and to normalize embryo geometries within a sample, individual embryo images are numerically rotated and re-projected into an orientation with dorsal side up (see SOM Fig. S2-S3). To normalize intensity measurements, we extended the method previously used for analyzing Bicoid intensity distributions [Gregor et al. 2007b] (see SOM Fig. S2-S3). pMad signaling distributions between embryos within a population are highly reproducible, as shown in Figure 3 A. After analysis within a population, we compared pMad patterns between *D. melanogaster wt* and other perturbed populations, including embryos with homozygous and heterozygous mutations in BMP signaling components, hypomorphic alleles of pathway members, ectopic expression of pathway genes, and embryos from different species of Drosophila (mean distributions for *wt* and 3 heterozygous mutant populations are shown in Figure 3 B). First, for *wt* embryos, we measured the width and variability of pMad level-sets for different choices of the concentration thresholds (T). Embryo-to-embryo variability in the mean pMad width (μ) for a given T-value is quantified by the coefficient of variation (CV = σ/μ), a dimensionless quantity that relates the standard deviation σ to the mean along the entire AP axis (Figure 3 C and S4-S6). From T ≈ 0.15 to T ≈ 0.4, the CV for width is less than ~ 0.2 which corresponds to ~ +/ − 2 nuclei in the cross-section. The variability of the CV along the AP axis is minimized at T ≈ 0.2 – 0.25 (Figure 3 C) and increases as the threshold is increased. Thus the embryo-embryo variability is minimized at relatively low thresholds of ≈ 20% of the maximum observed pMad signaling amplitude, which defines the range of thresholds in which we expect to have the greatest ability to detect differences between populations of *wt* and mutant embryos.

### Variability of pMad distributions

Whether or not the pMad distributions in dorsal blastoderm cells are insensitive to changes in gene copy number is unresolved since different groups arrive at different conclusions [Mizutani et al., 2005,Wang and Ferguson, 2005,Eldar et al., 2002,Shimmi et al., 2005,Umulis et al., 2008]. To quantify the “robustness” or insensitivity of the patterns, we measured the difference in width of normalized pMad between *wt* and perturbed cases at threshold positions up to 40% of the maximum normalized *wt* level. Prior analysis that compared the mean width of pMad signal intensity at different cross-sections chosen along the AP axes did not take into account embryo-embryo variability of the population [Mizutani et al., 2005,Wang and Ferguson, 2005,Eldar et al., 2002,Shimmi et al., 2005]. Furthermore, earlier studies did not consider the potential differences in pattern variability between different positions along the AP axis. At 50% embryo length (*x/L _{x}* =0.5), the relative levels of mean pMad signaling vary greatly in amplitude near the DV midline (

*y/L*= 0), but it is not clear if there are measurable differences between the width of the pMad profile for different level sets between T = 0.2 and 0.4 (Figure 3 D). Furthermore, Figure 3 (C) shows that the variability of the width in pMad varies along the AP axis and depends on the specific T-level used for the comparison of the patterns. Given that earlier approaches to investigating the question of pattern robustness were based on one or two cross-sections at a single threshold level where there can be substantial variability, it is easy to see that different conclusions could be drawn from equivalent data sets, thus leading to debate in regards to the “robustness” of patterning.

_{x}Before an appropriate measure of robustness for pMad signaling patterns between different genotypes can be developed, one must first consider the natural variability that exists within each population. As a minimal criterion, we propose that if one cannot reject the null hypothesis in a students t-test for a comparison of widths at a threshold level between two sample populations, then the patterning mechanism is by default “robust.” However, measuring a difference in the width of pMad does not preclude robustness, since a definition of robustness must follow, not precede, the measurement of a statistically significant difference between populations.

To compare pMad distributions between different populations of embryos that each have a deficiency in the level of at least one BMP regulator (Figure 3 D), we used the student’s t-test of significance on the width of pMad level-sets for specific threshold concentrations of 20% and 40% of the maximum scaled concentration (Figure 3 D-F and Figure S4). With an α value of 0.05 as the significance criterion, we found large segments along the AP axis where the null hypothesis (equal means) was rejected. This suggests that there are statistically-significant, measurable differences in the width of pMad staining for specific threshold levels, particularly for the *sog* and *scw* heterozygous mutants. Surprisingly, the ability to reject the null hypothesis depends strongly on the specific threshold of pMad. For low threshold values (T=0.2), the width of pMad in *sog*^{+/−} embryos is ≈ 2 – 4 nuclei (cells) wider than *wt*, but for higher threshold values (T=0.4), the width of pMad is virtually indistinguishable from *wt* for the majority of the AP length (Compare Figures 3 E and F
*sog*^{+/−} line). In *D. melanogaster*, heterozygous mutants are viable under standard laboratory nutrient and temperature conditions of 18–25°C, suggesting that the measurable changes in pMad signaling do not significantly disrupt development. In the context of dorsal surface patterning by BMPs, the image analysis data demonstrates that there are measurable differences between the pMad distributions of *wt* and heterozygous mutant embryos, which provide related, but unique data sets that can be used for model testing and optimization.

Following the normalization of pMad between populations of *wt* and mutant embryos, we extended the methodology to quantify distributions of gene expression of upstream components with the principal goal of incorporating quantitative prepattern information directly into a mathematical model of BMP regulation. Previous work has demonstrated approximate boundaries of gene expression for *dpp*, *tsg*, *sog*, *tld*, and *tkv* based on alkaline phosphatase staining, but a recent report on the *bcd* mRNA distribution demonstrated that detecting mRNA expression by alkaline phosphatase is much less quantitative than fluorescent *in situ* hybridization (FISH) [Spirov et al., 2009]. Quantitative gene expression patterns for *tkv* [Fowlkes et al., 2008], *sog* and *tld* were obtained, but we were unable to quantify the expression of *tsg* and *dpp* by FISH and boundaries were approximated from alkaline phospotase stained images (Figure 4 A).

**...**

### Comparison of BMP-mediated feedback mechanisms

There are three general issues that can be addressed using a 3D model that cannot be addressed by 1D models. The molecular mechanisms of BMP patterning have been inferred, in part, from the pMad distribution in embryos in which DV patterning genes are ectopically expressed along the AP axis, either using the Gal4-UAS system or by injection of mRNA. While pMad dynamics suggest a role for positive feedback, additional evidence for positive feedback is based on non-uniform AP distributions of pMad, pMad signaling losses, and the distribution of surface-bound Dpp in embryos injected with activated *tkv* mRNA on the dorsal surface [Wang and Ferguson, 2005]. Thus any model that postulates the molecular nature of the positive feedback must be able to reproduce the observed anterior-posterior pMad signaling profile and the pattern of surface -bound Dpp. [Wang and Ferguson, 2005]. Secondly, earlier models were developed and tested against 1D data, which greatly reduced the information density used for model optimization and hypothesis testing. Building a mathematical model based on quantitative biological images and training the model against multidimensional image data introduces additional constraints without increasing the number of unknowns. This should increase our ability to compare alternative, plausible mechanisms that could provide new insight into the regulation of BMP signaling. Finally, by using a 3D model one can determine the sensitivity of pMad signaling and/or other patterns of activity to perturbations in geometry, size, and gene expression pre-patterns.

To measure the relative contributions of 1) embryo geometry, 2) upstream gene expression, and 3) the specific choice of positive feedback mechanism, we developed several different versions of the embryonic patterning model. To measure the role of geometry on patterning, we constructed two different model geometries of the embryo: 1) an embryo in the shape of a symmetrical prolate ellipsoid; and 2) a reconstruction of an experimentally-determined non-symmetric embryo derived from the VirtualEmbryo [Fowlkes et al., 2008]. To avoid excessive computational time we separated the analysis of positive feedback from the analysis of gene expression and embryo shape. First we investigated alternative positive feedback mechanisms using only the realistic embryo shape with constant *tkv* and *tld*, and once a positive feedback mechanism was selected, we used it for all further studies to determine dependence of the model results on embryo shape and upstream gene expression patterns. For each different geometry and gene expression scenario, we allowed the parameters for the “winning” positive feedback mechanism to be re-optimized to remove any bias towards a specific geometry and/or gene expression pattern.

To investigate the role of feedback, we developed 9 versions of the model that differ in the specific molecular mechanism of feedback (Figure 1 C; Equations and parameters are provided in the supplemental experimental methods). The models for the test were selected based upon the idea of “mechanism” conservation. While the core signal transduction pathway for BMP signaling is widely conserved, the mechanisms of extracellular regulation vary greatly depending on context and many secreted molecules such as Crossveinless-2 (Cv-2) and Sax play diverse roles in different environments [Bangi et al. 2006, Serpe et al., 2008]. Thus, while Drosophila DV patterning does not rely on HSPGs, Sizzled-like molecules, or ADMP, other molecules like Cv-2, Viking, and Sax may play novel mechanistic roles. The feedback mechanisms we selected for the screen are functionally equivalent to those involved in other BMP signaling contexts [Ambrosio et al., 2008,Kelley et al., 2009,Fujise et al., 2003,Wang and Ferguson, 2005,Muraoka et al., 2006]. The nine models tested were 1) no feedback, 2) positive feedback of receptor levels, 3) inhibited endocytosis, 4) enhanced endocytosis, 5) inhibited Tld processing, 6) enhanced Tld processing, 7) positive feedback of an surface-bound BMP-binding protein (SBP), 8) decreased release of ligand from ligand-receptor complexes, and 9) enhanced formation of type I-type II- Dpp/Scw complexes (Figure 1 C).

The “core” reaction-transport equations, which contain no feedback, for dorsal surface patterning are:

Equations (1)-(7) make up the “no-feedback” model for BMP patterning and specific equations for each positive feedback mechanism are provided in section 3 of the SOM. Here **x** is position in Cartesian coordinates, *D _{i}* is the diffusion coefficient for species

*i*,

*k*,

_{j}*k*are the forward and reverse kinetic rate constants for reaction

_{−j}*j*,

*δ*is the endocytosis rate, and

_{e}*(x) is the production rate as a function of position for species*

_{i}*i*. In addition to equations (1)-(7), we specify the initial and boundary conditions as determined from image analysis and embryo geometry. The model was solved using the finite element method.

Each model was trained by optimizing the spatial distribution of BMP-bound receptors (*BR*) predicted by the model against the population mean pMad distribution in *wt* embryos. To measure the best model-fit to the experimental data, we calculated the root-mean-squared-deviation (RMSD) version of the coefficient of variation (CV (RMSD)= RMSD/μ) [Fomekong-Nanfack et al., 2007]. The RMSD is a measure of the distance between the normalized model results for the BMP-receptor (*BR*) distribution and the corresponding pMad distribution. Here, μ is the average normalized concentration for each image. To remove non-embryo regions from the calculation of CV/RMSD, each image was masked by the x-y orthogonal projection of the mean embryo shape. Typical model results, residuals for the difference between the orthogonal projection of the model against the mean pMad levels, and model dynamics are shown in Figure 4 B-F.

Once a base-case for each feedback mechanism was determined by optimizing the positive feedback module against *wt* pMad data, each model was optimized against 8 additional mutant data sets by varying a single parameter that corresponds to the respective experimental perturbation. For instance, once a parent parameter set was found for the model that incorporates positive feedback of receptors, the model was then optimized against *bcd>dpp* embryos by changing a single parameter that corresponds to the strength of the *bicoid* driver and expression of BMP ligand from the anterior end of the embryo. The mutant data sets used for training correspond to the following genotypes: scw^{+/−}, sog^{+/−}, tsg^{+/−}, bcd>dpp, bcd>2Xdpp, bcd>sog, bcd>dpp,sog, and *tsg ^{−}; bcd>dpp,scw*. To compare alternate feedback mechanisms we used two measures: comparison of median and upper/lower quartiles for RMSD/μ values (Figure 4 B), and comparison of the mean using a t-test of significance with α = 0.05. The box plots in Figure 4 B suggest that the positive feedback modules that 1) inhibit endocytosis, 2) increase Tld processing rate, 3) produce an SBP, 4) slow the release of ligand from receptors, and 5) enhance formation of active type I/II receptor-ligand complexes have a smaller median CV(RMSD) value than the model with no feedback. However, only the SBP model has a statistically lower mean (p=0.044) when compared to the model with no feedback. If the significance criterion is relaxed slightly to α = 0.1, receptor cooperativity (p=0.082) emerges as a good candidate mechanism, whereas the next best feedback model has a p-value of 0.28. A statistically-significant difference means that, on average for a given criterion (e.g. α = 0.05), but not for all the training sets, the model with positive feedback of an SBP produced lower RMSD/μ values when comparing the model and mean population data for each genotype. In some positive feedback cases (e.g. +rec), the best model fit for a number of the mutant cases was worse than the fit achieved with no feedback, because each mutant case is constrained by the parent set of parameters determined for

*wt*embryos.

The SBP positive feedback model achieves the best overall fit by increasing the binding site density dynamically in response to prior pMad signaling. Here, positive feedback produces a non-signaling complex that is endocytosed. The increase in binding site density over time dynamically shortens the length scale for BMP transport by increasing the probability of binding to a surface localized molecule and being endocytosed.

### Contributions of geometry and prepatterns to predicted distributions of BMP-receptor complexes

To determine the relative contribution of upstream gene expression and embryo geometry on pMad signaling, we extended the SBP positive feedback model to include the embryonic pre-pattern for the relative gene expression of the secreted factors Sog, Tld, and Tkv (Figure 4 A) [Fowlkes et al., 2008]. For each measured mRNA distribution, we tested a number of different scenarios to calculate the benefit of including expression pattern data as opposed to approximating the distributions by a constant. The expression patterns of *dpp* and *tsg* were approximated from alkaline phosphatase images available in the literature and online (Berkeley Drosophila Genome Project) [Mason and Marsh, 1998, Shimmi et al., 2005]. To incorporate the imaging data directly into the model, each mean expression pattern is fit to a 2D Fourier sine series and re-projected onto the 3D embryo geometry (Figure 4 A and Fig. S3). Boundaries from Alkaline phosphatase images were approximated by measuring the extents of gene expression for a single embryo.

The sensitivity of the model to the geometrical approximation and upstream gene expression is shown in Figure 4 C. For each test, we used the *sog* mRNA distribution as determined by image analysis, and calculated four different cases for both the real geometry and the ellipsoidal model. Case 1 is uniform Tld and uniform receptors, Case 2 is uniform Tld, non-uniform receptors, Case 3 is non-uniform Tld, uniform receptors, and Case 4 is non-uniform Tld, non-uniform receptors. Surprisingly, the additional biological data did not lead to a substantial reduction in the CV(RMSD) values, but in the ellipsoidal model, the addition of the non-uniform distributions of Tkv and Tld reduced the CV(RMSD) values by ≈ 25%. When both non-uniform Tld and Tkv distributions are used in Case 4 of the ellipsoidal model, the CV(RMSD) is nearly equivalent to Case 4 in the model with the real embryo geometry. These data suggest that the non-uniform distributions may provide a more reproducible pMad pattern and buffer the system against individual variability in embryo geometry. Additionally, the model not only captures the non-uniform AP distribution of pMad, but also captures the sharpening and contraction of *BR* that corresponds well with the observed pMad dynamics (Figure 4 E,F) [Umulis et al., 2006].

### Potential roles of type-IV collagen in BMP-mediated dorsal surface patterning

While feedback of a “non-receptor” mechanism is a prediction of the modeling analysis, it has recently been shown that ubiquitous type-IV collagen binds Dpp-Scw and may act as an exchange factor that partially regulates receptor-ligand complex formation (Figure 1 D) [Wang et al., 2008]. Thus an equivalent positive feedback mechanism to the SBP model could locally modify the binding affinity between Dpp-Scw and collagen thereby increasing the binding site density. It has also been suggested that type-IV collagen regulates assembly of the Sog/Dpp-Scw complex, but it is not clear how this impacts the model of BMP patterning [Wang et al., 2008]. Type-IV collagen is ubiquitously expressed in the blastoderm embryo and binds both Dpp-Scw and Sog. Since the role of secreted binding proteins in the regulation of BMP-receptor interactions has been analyzed elsewhere [Serpe et al., 2008], we focus on mechanisms of collagen-mediated formation of Sog/Dpp-Scw complexes and transport. We developed a modified version of equations (1)-(7) to account for the different order of complex formation suggested by Wang et al. [Wang et al., 2008]. Rather than Sog binding to the co-factor Tsg to form the complex Sog/Tsg, we assumed that Sog and Dpp-Scw bind to collagen, and then bind to each other while attached to collagen. In this scenario the Sog/Dpp-Scw complex remains bound to the collagen matrix until Tsg binds to Sog/Dpp-Scw to form Sog/Tsg/Dpp-Scw, which is then released from the collagen matrix and thereafter does not rebind (Equations given in supplemental experimental procedures). To keep the model as simple as possible, we made the following assumptions: 1) the binding and release of Dpp-Scw and Sog to collagen is rapid relative to other processes such as production and diffusion and can be treated as having equilibrated [Umulis, 2009], 2) the total amount of collagen does not change in time, and 3) the binding of Sog and Dpp-Scw to collagen is weak (Biacore binding assays in [Wang et al., 2008] support this assumption: for Dpp + Viking binding *K _{D}* = 746nM). Other balances between the rates of transport to and reaction on an immobile surface are analyzed elsewhere [Othmer, 1976].

To investigate the impact of reduced diffusion rates, we coordinately lowered the diffusion of both Sog and Dpp-Scw and compared the model predictions to the mean distribution of pMad for *wt* embryos. A decrease in the diffusivity of Sog and Dpp-Scw first decreases the RMSD/μ value to a minimum and then increases it again as the effective diffusivity is decreased further (Figure 5 A). To investigate whether changing the order of Sog/Tsg-Dpp-Scw complex formation affects the model fit to the data, we measured the RMSD/μ values for two models that differ in the order of inhibitor-ligand complex formation. We observed a moderate increase in the RMSD/μ value in the model in which Sog binds to Dpp-Scw before binding to Tsg, but both models require rapid formation of the ligand inhibitor complex (Figure 5 B). Earlier 1D models also require either rapid formation of the complex, tight binding or both and have dissociation constants of 0nM (irreversible binding) [Eldar et al., 2002], 0.01nM (binding between Sog/Tsg and ligand) [Mizutani et al., 2005], and 0.03nM (binding between Sog/Tsg and ligand) [Umulis et al., 2006]. Independent of the order of complex formation, when a lower kinetic constant for binding between Sog/Tsg and Dpp-Scw (or for binding between Sog and Dpp-Scw) is used in the 3D model, the distribution of *BR* does not agree well with pMad distributions in *wt* embryos (Figure 5 B).

**...**

The reported kinetics from Biacore analysis for binding between the Sog homologue Chordin (perfused) and the Dpp homologue BMP-2 (immobilized) gives *k _{off}* =3.4 · 10

^{−3}sec,

*k*=0.28 · 10

_{on}^{−3}nM

^{−1}sec, and a dissociation constant of

*K*=

_{D}*k*/

_{off}*k*≈ 12nM [Rentzsch et al., 2006]. As shown in Figure 5 B, if the measured forward rate of

_{on}*k*=0.28 · 10

_{on}^{−3}nM

^{−1}sec

^{−1}is used in the model, results do not correspond well with observed pMad distributions (Figure 5 B).

Much has been done toward estimating effective binding rates of molecules in solution and binding to receptors on a surface from solution in terms of the rates of the component steps – diffusion to the target/receptor and the intrinsic binding once a complex is formed [Lauffenburger and Linderman, 1993], and there are estimates for similar processes on membranes, where the diffusion coefficients are usually three or more orders of magnitude less [Lauffenburger and Linderman, 1993, Kholodenko et al., 2000]. The measured forward rate constant for Chordin binding to BMP-2 (kon =0.28 · 10^{−3} nM^{−1} sec^{−1}) depends on two factors: the diffusion/transport-limited rate (*k _{diff}*) and the intrinsic reaction rate constant (

*k*) as shown by equation 8:

_{r}
The diffusional contribution *k _{diff}* can be approximated by 4πDs, where D is the sum of the diffusion coefficients of the components (≈O(10)μsec

^{−1}) and s ≈ 5 – 10nm (5 is used) is the encounter radius [Lauffenburger and Linderman, 1993]. This leads to the estimate

*k*≈ 0.76nM

_{diff}^{−1}sec. For BMP-2 + Chordin binding the measured effective rate constant is

*k*=0.28 · 10

_{on}^{−3}nM

^{−1}sec

^{−1}[Rentzsch et al., 2006], and using

*k*and

_{diff}*k*in (8) to solve for

_{on}*k*gives

_{r}*k*≈

_{r}*k*=0.28 · 10

_{on}^{−3}nM

^{−1}sec. This means that in solution, the binding between Chordin and BMP-2 is reaction-limited, and only a small fraction of collisions lead to complex formation. However as shown in Figure 5 B, the model requires forward binding rates that approach the diffusion limit and not the reaction limit for binding reactions in solution. If the model requires much tighter and faster binding then the

*in vitro*-determined kinetics for Chordin + BMP-2, how can the model predictions be reconciled with the binding data? Clearly other factors must be involved, but identifying them is a complex issue that requires a detailed model of all the processes involved, and to date this has not been done. Here we simply suggest how a protein scaffold might influence the formation of multi-protein complexes, and suggest a scenario in which the rate of formation of Sog/Dpp-Scw may be enhanced by the presence of collagen.

Three basic issues related to transport and reaction kinetics in the presence of collagen can be summarized as follows.

- How important are crowding effects in solution, on scaffolds and on membranes? In particular, how does one correctly describe the transport of molecules along a scaffold or membrane – Is Fick’s law appropriate or does molecular crowding and volume exclusion [Grasberger et al., 1986] require a more detailed constitutive relation? A recent prediction concerning crowding is that for a fixed number of reacting particles there is an optimal density of obstacles in solution that minimizes the time needed for a random walker to find a target [Schmidt et al., 2009]. If the density is too low the particles cannot find each other, whereas if it is too high diffusion limits impede the rate of formation.
- If localized collagen leads to local increases in concentration of both Sog and Dpp-Scw, can this lead to a sufficient increase in the rate of complex formation on collagen?
- How do the free energy changes of bimolecular interactions change on a surface, as compared with solution? Can one predict with sufficient accuracy the relation between the kinetic constants for reactions on the surface knowing kinetic constants measured in well-mixed solutions? This involves steric factors, such as better alignment in the surface, that will usually change from solution to surface, as well as differences in internal energy of the molecules in the two environments.

While crowding effects, free energy changes, and local increases in concentration can affect dimerization and complex formation once attached to collagen, little data is available regarding the first and third factors, and we focus solely on the second: how collagen may increase formation rates by concentrating Sog and Dpp-Scw.

Once bound to collagen, the diffusion of Sog and Dpp-Scw molecules would be greatly reduced and the rate of formation of Sog/Dpp-Scw complexes may well be transport limited. To determine how collagen modifies the rate of formation of complexes, we investigated two possibilities: (i) collagen is tightly localized to the inner membrane of the PV space and (ii), collagen extends a distance *h* into the PV space which has overall height *L _{z}* (SOM section 4). If collagen is localized tightly near the surface of the cells that line the PV space, we can approximate the collagen matrix as a thin sheet (2D) and use the diffusion limit

*k*≈

^{2D}*2πD/ln(b/s)*where

*b*is one half the mean distance between molecule on the surface,

*D*is the diffusion coefficient for molecules attached to collagen on the surface, and

*s*is the encounter radius [Lauffenburger and Linderman, 1993]. (For the derivation of the equations relating to collagen-mediated complex formation, see SOM section 4). A typical value for diffusion limited binding reactions in 2D is k+ ≈ 3 *10

^{−1}μ

^{2}sec

^{−1}[Lauffenburger and Linderman, 1993].

In the absence of Tsg, the change in the number of Sog/Dpp-Scw complexes (*N _{SBtot}*) in time is the sum of the rates of binding and release of Sog (S) + Dpp-Scw (B) in solution and the rates of binding and release of collagen-bound Sog (

*SC*) and bound Dpp-Scw (

_{n}*BC*) where

_{n}*C*is collagen. For simplicity we only discuss the result for a 2D collagen matrix layer of area A at the base of the PV space (volume =ALz) and we assume that the concentration is spatially uniform; then the governing equation is

_{n}
The areal concentrations (with units of [mol./area]) of collagen-bound BMP (*BCn*) and collagen-bound Sog (*SCn*) depend on the total density *C _{nT}* of collagen present and the binding kinetics between Collagen+Sog and Collagen + Dpp-Scw. If the dissociation constant for Collagen + Sog is

*K*and for Collagen + Dpp-Scw is

_{S}*K*, and the surface binding reactions are fast relative to other processes, we can determine an effective formation rate for the 2D reactions. Dividing equation 9 by the volume (

_{B}*L*), defining

_{z}A**C**=

_{nT}*C*and

_{nT}/L_{z}**BC**=

_{n}SC_{n}*BC*as the volumetric concentrations by averaging the surface densities over the local volume, and simplifying gives:

_{n}SC_{n}/L_{z}
Here *k _{+}L_{z}* is a second-order rate constant for the surface reaction, with units [(mol./L

^{3}time)

^{−1}], and the ratio $\Omega \equiv {\mathbf{C}}_{\mathbf{nT}}^{2}\u2215\left({K}_{B}{K}_{S}\right)$ is dimensionless. For a relatively slow 2D diffusion-limited surface reaction,

*k*≈ 3 × 10

_{+}^{−1}μ

^{2}sec

^{−1}[Lauffenburger and Linderman, 1993]. If the collagen binding site density is of order 30/μ

^{2}, if

*K*and

_{B}*K*are of order 100 nM, and

_{S}*L*is taken as the height of the PV space (0.5μ) then Ω ~ 1 and this leads to the estimate

_{z}*k*≈ 0.1nM

_{+}L_{z}^{−1}sec

^{−1}. The collagen mediated rate of

*k*≈ 0.1nM

_{+}L_{z}^{−1}sec

^{−1}is in the range needed for proper pattern formation in the model (Figure 5 B). This rate is ≈ 320X greater than the binding between free Sog and Dpp-Scw in solution, but of course it is predicated on the estimates used. If both binding constants are 1μM (recall

*K*for Dpp + Viking is ≈ 746nM) the estimated rate is only 3.2 times that in solution. The density of binding sites may be much higher as well, and Ω could still be in the range necessary to provide a substantial boost in the kinetics of Dpp-Scw + Sog complex formation. If

_{B}*k*≈

_{−}*k*and

_{off}*(k*Ω

_{+}L_{z}*)/k*>> 1 then the effective dissociation constant for the total concentration of complexes (

_{on}*SB*) is

_{tot}*K*≈

_{D}*k*Ω

_{off}/(k_{+}L_{z}*)*≈ 0.03nM, which is exactly in the range necessary for peak formation by the shuttling mechanism with freely diffusible Sog and Dpp-Scw. Similar relationships can also be derived by assuming two different volume fractions in the PV space, but in those cases as well the conclusions are contingent on estimates of unknown rates. If the above estimates hold, the increase in the apparent kinetic rate constant in going from solution to the collagen matrix can be very large and far exceeds the potential reduction in kinetic constants by slowing the diffusion of binding partners once attached to collagen.

However, one problem remains: the Sog/Dpp-Scw complex is still attached to the collagen matrix, which would hinder transport and potentially the reaction with other regulators. A possible resolution of this problem was also presented recently in the context of Drosophila [Wang et al., 2008]. Specifically, the results indicate that the co-factor Tsg facilitates the release of the Sog/Dpp-Scw complex from the type-IV collagen during Drosophila embryonic patterning [Wang et al., 2008]. After release, the Sog/Tsg/Dpp-Scw complex could then participate in other processes such as Tld-mediated cleavage of Sog, which would release the Dpp-Scw ligand for signaling. If Tsg facilitates the release relatively rapidly, the results presented in Wang et al. [Wang et al., 2008] imply an interesting caveat that should be considered when thinking about mechanisms of BMP regulation. The effective amount of Sog/Dpp-Scw in solution (the mobile phase) is controlled by processes on the surface that occur very rapidly and favor the formation of the complex. The limiting step for the formation of the complex would be binding to the type-IV collagen or other binding site (perhaps Cv-2).

### Model results for ectopic gene expression of BMP regulators

With the full 3D model, we can calculate the *BR* distributions for various Gal4-UAS experiments and determine the relative strength of the driver-responder combinations and make predictions to further elucidate BMP regulation. We calculated the *BR* distribution for over 15 Bicoid-Gal4-UAS experiments for which pMad distributions were available, and while the model captures many aspects of the experimental images, it does not produce qualitatively similar results for every combination (Figure 6 A-V; A’-V’).

*BR*calculated using the computational model. See SOM section 3 and Figure S8 for additional discussion of the

**...**

In particular, the model captures the expected distributions in a number of ectopic gene expression experiments, including the rescue of *tsg-;bcd>tsg*, the residual pMad signaling in *bcd>tld,dpp,sog* and *bcd>sog,dpp* embryos. Intriguingly, the model results do not correspond well with the pMad distributions in Figure 6D, E, and R. In each of these cases, the predicted *BR* levels at the anterior end are higher than the observed pMad distribution. All three examples correspond with increased expression of BMP ligands near the anterior end of the embryo. It is remarkable that pMad signaling is so low in a region where there should be a large amount of BMP ligands. In Figure 6 R the model predicts high levels of *BR* near the dorsal midline, whereas the pMad distribution “splits” into two lateral stripes (Additional modeling results are given in Fig. S5). The signaling gaps observed in Figure 6 B,D,E, and R are reminiscent of the pMad signaling gaps observed in embryos with ectopic addition of activated *tkv ^{act}* mRNA [Wang and Ferguson, 2005] (Figure 6 S, S’). To determine if the model could capture the signaling gap observed in the

*tkv*mRNA experiments, we simulated the ectopic addition of a fixed amount of mRNA at the anterior end and measured two quantities: the expected level of pMad signaling, which is the sum of Tkv

^{act}^{act}and

*BR*, and the total distribution of surface-associated BMP. Remarkably, the model captures the observed features in both the expected pMad distribution and the level of surface localized BMP (6 S, S’) including the signaling shadow between the Tkv

^{act}pool in the posterior and

*BR*in the anterior (S) and the anterior-facing wavefront of surface localized BMP (S’) (Figure S5). According to the model, the additional binding of Dpp-Scw to receptors provides a sink that dynamically lowers the range of Dpp-Scw. The limited distribution of Dpp-Scw effects the distribution of Sog/Tsg/Dpp-Scw complexes, which ultimately changes the directionality of the Dpp-Scw flux. The shadows result from the dynamic imbalance of Dpp-Scw binding and Sog/Tsg-mediated transport.

### Scale-invariance of BMP-mediated signaling

The data presented above clearly indicates that the specific embryo geometry has a large impact on the ability of the model to predict levels of *BR* that are in quantitative agreement with the observed pMad distributions. Geometry predominately impacts the distances travelled by BMPs and thus the range of ligands and inhibitors will vary along the AP axis. One would expect that a patterning process dependent on domain geometry would also depend on the overall size or scale of the domain. In general, reaction-diffusion equations similar to those used for BMP patterning are highly dependent on the length-scale of the system and do not lead to the preservation of proportion observed in numerous biological contexts [Othmer and Pate, 1980].

A recent theoretical study of BMP patterning of *Xenopus laevis* embryos suggests that a single-ligand shuttling mechanism does not lead to scale-invariance. In *Xenopus*, scale-invariance might actually be mediated by the combined action of two types of ligands: the BMPs and ADMP. However, ADMP is absent in *Drosophila*, which suggests one of two possibilities: 1) BMP signaling between different sized embryos is not scale-invariant, or 2) alternative mechanisms to the proposed mechanism for *Xenopus* BMP scaling lead to *Drosophila* BMP patterning scale-invariance. To investigate the question of scaling, we calculated the distribution of *BR* in the 3D model by increasing the size of the embryo while keeping all other quantities constant. The “shuttling only” mechanism with fixed production rates (concentration/time) and fixed receptor concentrations, does not lead to scale-invariance of the *BR* distribution. While the *BR* distribution is largely unchanged in smaller embryos (Figure 7 A) that correspond roughly with the size expected for *D. busckii* [Gregor et al., 2005], the pattern of *BR* in embryos ≈ 575μ in length (Figure 7 B) is qualitatively different than the pattern predicted in *D. melanogaster* (≈ 400μ in length). In simulations where the embryo is slightly larger than expected for *D.virilis* or ≈ 750μ in length, the pattern of *BR* splits at ≈ 25% embryo length and leads to two parallel stripes that reconverge at the posterior pole (Figure 7). Before measuring pMad signaling in different size embryos, we tested how an alternative scale-invariance hypothesis might impact the model predicted results.

*BR*levels for

*D. virilis*and

*D. busckii*and predicted distribution for embryos 750μ in length.(A-C) are the expected

*BR*levels when conservation conditions are imposed for the total amounts

**...**

Neglecting positive feedback for a moment, equations (1)-(7) show that the distribution of *BR* complexes depends on a number of linear and non-linear reactions that occur in the perivitelline space. Specifically, the level of *BR* depends on the level of free BMP ligands, which in turn depends on four kinetic processes: 1) production of ligand, 2) binding and release to the Sog/Tsg complex, 3) receptor binding and release, and 4) release from the inhibitor complex by the destruction of Sog by Tld. If we first consider equations 1-7 with zero production of Sog, the equation for Tsg becomes decoupled from the equations for BMP ligands and receptors. In the zero Sog scenario, the level of *BR* depends only on interactions between ligand and receptors. This leads to the simplified set of equations for BMP-receptor dynamics (equations 11-12) that are similar to those proposed for dynamic scale-invariance of AP patterning of *Drosophila* embryos by Bcd [Umulis, 2009,Gregor et al., 2007a,Umulis et al., 2008]:

By replacing the ^{2} by ${L}^{-2}{\stackrel{\u2012}{\nabla}}^{2}$ where ${L}^{-2}{\stackrel{\u2012}{\nabla}}^{2}$ is the dimensionless Laplacian, assuming binding equilibration in the linear range, and adding equations 11 and 12 makes the requirements for scale-invariance of the simplified system more transparent:

Previously, we have suggested for linear morphogen patterning systems that scale-invariance occurs automatically if the following conditions are met [Umulis, 2009,Umulis et al., 2008]:

- Patterning occurs along a surface (Dimensional analysis suggests that the PV space can be approximated as a 2D surface wrapped around a 3D core).
- Proportions are constant between individuals that differ in size, which means that for a constant PV space thickness, the area (A) scales in proportion to the characteristic length squared (i.e. A σL
^{2}). This is true within a species, and approximately true between different species. - The total number of binding sites, decay rates and other processes that may naturally scale with size are constant. This is true, for instance, if receptor levels in the PV space scale with nuclear density.
- Constant total morphogen production in molecules/time (Hypothesis).
- Morphogen gradient interpretation by total binding site occupancy (molecules) and not the total concentration (molecules/area) of occupied binding sites (Hypothesis. This has been demonstrated for receptor occupancy and Activin signaling but this has not yet been tested for BMPs in
*Drosophila*[Dyson and Gurdon, 1998]).

Constancy of total receptor number and total production rates (molecules/time) leads to *R _{tot}* =

*N*and ${\varphi}_{B}^{\ast}\left(\underset{\u2012}{\xi}\right)={\Phi}_{B}^{\ast}\left(\underset{\u2012}{\xi}\right)\u2215{L}^{2}$. Imposing these conditions on equation 13, and defining

_{tot}/(σL^{2})*Λ = N*leads to the equation (15):

_{tot}/(σK_{m})In the “zero Sog” case given by equation 15, the level of free BMP ligand is automatically scale-invariant at steady-state and may also be scale-invariant dynamically during the approach to steady-state depending on the value of Λ. With Sog secretion the situation is much more complicated, primarily due to the nonlinear binding steps that occur between Sog, Dpp-Scw, and Tsg in the PV space. However, the conservation may lead to some scale-invariance, even if it is not “perfect” scale-invariance, as can be shown for linear systems. To test the scale-invariance of the full patterning model, we hypothesized that the secretion of zygotic Dpp-Scw, Sog, Tsg, and the total level of Tld, receptors, and feedback rates in the PV space scale with the number of nuclei as outlined above. The model results deviate less from the *D. melanogaster* predictions, but the patterns are not perfectly scale-invariant (Figure 7 A’-C’). The shape of the *BR* distribution changes in different-sized embryos, but the difference between predicted patterns in embryos becomes much more pronounced as the size is increased from the 400μ length used for the *D. melanogaster wt* base case. The difference in the patterns between the shuttling-only model and the model that includes constancy of secretion rates is also much more pronounced as the simulated embryo size becomes larger (compare Figure 7 B to B’ and C to C’). Since there are large qualitative differences between the predictions made by the two mechanisms, and it was not clear *a priori* that dorsal surface patterning is actually scale-invariant, we stained for pMad signaling in *D. busckii* (≈ 350μ) and *D. virilis* (≈ 575 + μ) (Figure 7 D-G). We found that the ratio of mean width to embryo length (*w/Lx*) for pMad (T=0.2) along the anterior-posterior axis is scale-invariant, but it is not an example of a “perfectly” scale-invariant system. Instead, while *w/L _{x}* is constant between species, the shape of the distribution deviates from

*D. melanogaster*at different positions along the AP axis. Intriguingly, pMad staining in

*D. virilis*is more consistent with a mechanism employing conservation conditions than without the conditions (Figure 7 B, B’ and E). In the model, predictions for embryos 575μ and 750μ in length,

*BR*levels begin to diminish at the poles, the profile bows out near the AP midline, and the maximum amplitude shifts from the position at approximately 1/3 embryo length to approximately 1/2 embryo length. This is consistent with the mean pMad observed in

*D. virilis*, and is also consistent with our preliminary data for pMad staining in

*Musca domestica*embryos.

### The relationship between embryo geometry and dorsal surface patterning

Lastly, we were interested in how changes in embryo proportion impacts the model predicted distributions of *BR*. To test this, we scaled the model geometry for the embryo along the x-axis only, keeping the y and z-scales constant. Furthermore, we tested both the shuttling only and shuttling w/ conservation condition models and show the results of the latter here (Figure 8). For the simulations, we assumed that the mRNA prepatterns were scaled consistently with the geometric scalings. The predicted patterns change appreciably as the geometry of the embryo is changed. In “short” embryos, *BR* levels bow out and accumulate at the poles where Sog would be lowest and Tld highest. In 0.75X embryos, *BR* forms 3 local maxima at ≈25%, 33%, and 75% embryo length and as embryos are stretched in the x-direction beyond 1.0X, pMad signaling is lost at ≈25% and 75% embryo length.

### Future directions

The model results for mechanisms of scale-invariance, collagen, and the relationship between shape and pattern can be experimentally tested to support or refute the “predictions” made in this paper. First, to test the relationship between nuclear density, binding site density (i.e. Tkv and Collagen) and pMad patterning, one could stain for pMad signaling in embryos where the local density of nuclei is disrupted as was done to demonstrate the relationship between nuclei and dpERK patterning [Coppey et al., 2008]. To investigate the dependence of dorsal surface patterning on shape one could quantitatively measure pMad signaling in other species of *Drosophila* such as *D. pseudoobscura, D. erecta* or other *Diptera* including *Lucilla sericata* which both have embryo shapes that differ significantly from *D. melanogaster* [Gregor et al. 2005, Lott et al. 2007].

There are a number of ways to extend the current model to address new and emerging questions of pattern formation. First, one could couple the computational modeling framework with the SpatioTemporal atlases of gene expression to test alternative mechanisms of AP patterning and crosstalk between the AP and DV signals [Fowlkes et al., 2008]. Numerous genes are not strictly AP or DV spanning and their expression is the result of dynamically integrating signals that originate along orthogonal axes. 3D spatiotemporal modeling of Drosophila provides a robust platform to investigate the mechanisms of signal integration.

Another extension of this work is to develop a coupled volumetric/surface patterning embryonic model to understand the balance of processes for Bicoid transport along the AP axis [Bergmann et al., 2007,Gregor et al., 2007a, Umulis et al., 2008, Spirov et al., 2009]. Earlier we simulated the Bicoid transport problem at the embryo periphery for the ectopic expression simulations, but during the last 4 nuclear division cycles there is exchange of Bicoid between nuclei that line the periphery and the embryo core that conspires to produce a quasi-steady-state of nuclear Bicoid concentration [Gregor et al., 2007a, Coppey et al., 2007]. Furthermore, with the full 3D model, we are not limited to transport by diffusion, and one could extend AP patterning studies to include cytoplasmic movement and directed transport [Gregor et al., 2007a,Foe and Alberts, 1983, Hecht et al., 2009].

## ACKNOWLEDGMENTS

The authors gratefully acknowledge the Rosen Center for Advanced Computing at Purdue University, the Minnesota Supercomputing Institute, NIH Grant GM29123 to H.G.O. and the reviewers’ comments.

## Footnotes

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## Contributor Information

David M. Umulis, ^{*}Agricultural and Biological Engineering, Weldon School of Biomedical Engineering, and Bindley Bioscience Center. 225 S. University St., Purdue University, West Lafayette, IN 47907, USA .

Osamu Shimmi, ^{†}Institute of Biotechnology, University of Helsinki, 00014 Helsinki, Finland .

Michael B. O’Connor, ^{‡}Howard Hughes Medical Institute Investigator, and Department of Genetics, Cell Biology, and Development, Minneapolis, MN 55455, USA ; Email: ude.nmu@ronnocom.

Hans G. Othmer, ^{§}School of Mathematics and Digital Technology Center, Minneapolis, MN 55455, USA ; Email: ude.nmu.htam@remhto.

## References

- Ambrosio AL, Taelman VF, Lee HX, Metzinger CA, Coffinier C, De Robertis EM. Crossveinless-2 Is a BMP feedback inhibitor that binds Chordin/BMP to regulate Xenopus embryonic patterning. Dev. Cell. 2008;15:248–260. [PMC free article] [PubMed]
- Bergmann S, Sandler O, Sberro H, Shnider S, Schejter E, Shilo BZ, Barkai N. Pre-steady-state decoding of the Bicoid morphogen gradient. PLoS Biol. 2007;5:e46. [PMC free article] [PubMed]
- Coppey M, Berezhkovskii A, Kim Y, Boettiger A, Shvartsman S. Modeling the bicoid gradient: diffusion and reversible nuclear trapping of a stable protein. Dev. Biol. 2007;312:623–630. [PMC free article] [PubMed]
- Coppey M, Boettiger A, Berezhkovskii A, Shvartsman S. Nuclear trapping shapes the terminal gradient in the Drosophila embryo. Curr. Biol. 2008;18:915–919. [PMC free article] [PubMed]
- Driever W, Nusslein-Volhard C. The bicoid protein determines position in the Drosophila embryo in a concentration-dependent manner. Cell. 1988;54(1):95–104. [PubMed]
- Dyson S, Gurdon JB. The interpretation of position in a morphogen gradient as revealed by occupancy of activin receptors. Cell. 1998;93:557–68. [PubMed]
- Eldar A, Dorfman R, Weiss D, Ashe H, Shilo BZ, Barkai N. Robustness of the BMP morphogen gradient in Drosophila embryonic patterning. Nature. 2002;419(6904):304–8. [PubMed]
- Foe V, Alberts B. Studies of nuclear and cytoplasmic behaviour during the five mitotic cycles that precede gastrulation in Drosophila embryogenesis. J. Cell. Sci. 1983;61:31–70. [PubMed]
- Fomekong-Nanfack Y, Kaandorp J, Blom J. Efficient parameter estimation for spatio-temporal models of pattern formation: case study of Drosophila melanogaster. Bioinformatics. 2007;23:3356–3363. [PubMed]
- Fowlkes C, Hendriks C, Kernen S, Weber G, Rbel O, Huang M, Chatoor S, DePace A, Simirenko L, Henriquez C, Beaton A, Weiszmann R, Celniker S, Hamann B, Knowles D, Biggin M, Eisen M, Malik J. A quantitative spatiotemporal atlas of gene expression in the Drosophila blastoderm. Cell. 2008;133:364–374. [PubMed]
- Fujise M, Takeo S, Kamimura K, Matsuo T, Aigaki T, Izumi S, Nakato H. Dally regulates dpp morphogen gradient formation in the drosophila wing. Development. 2003;130(8):1515–22. [PubMed]
- Grasberger B, Minton AP, DeLisi C, Metzger H. Interaction between proteins localized in membranes. Proc. Natl. Acad. Sci. U.S.A. 1986;83:6258–6262. [PMC free article] [PubMed]
- Gregor T, Bialek W, de Ruyter van Steveninck RR, Tank DW, Wieschaus EF. Diffusion and scaling during early embryonic pattern formation. Proc Natl Acad Sci U S A. 2005;102(51):18403–18407. [PMC free article] [PubMed]
- Gregor T, Wieschaus EF, McGregor AP, Bialek W, Tank DW. Stability and nuclear dynamics of the bicoid morphogen gradient. Cell. 2007a;130(1):141–152. [PMC free article] [PubMed]
- Gregor T, Tank DW, Wieschaus EF, Bialek W. Probing the limits to positional information. Cell. 2007b;130(1):153–164. [PMC free article] [PubMed]
- Hecht I, Rappel WJ, Levine H. Determining the scale of the Bicoid morphogen gradient. Proc. Natl. Acad. Sci. U.S.A. 2009;106:1710–1715. [PMC free article] [PubMed]
- Jaeger J, Surkova S, Blagov M, Janssens H, Kosman D, Kozlov K, Myasnikova E. Manu, Vanario-Alonso C, Samsonova M, Sharp D, Reinitz J. Dynamic control of positional information in the early Drosophila embryo. Nature. 2004;430(430):368–71. [PubMed]
- Kelley R, Ren R, Pi X, Wu Y, Moreno I, Willis M, Moser M, Ross M, Podkowa M, Attisano L, Patterson C. A concentration-dependent endocytic trap and sink mechanism converts Bmper from an activator to an inhibitor of Bmp signaling. J. Cell Biol. 2009;184:597–609. [PMC free article] [PubMed]
- Kholodenko BN, Hoek JB, Westerhoff HV. Why cytoplasmic signalling proteins should be recruited to cell membranes. Trends Cell Biol. 2000;10:173–178. [PubMed]
- Lander AD. Morpheus unbound: reimagining the morphogen gradient. Cell. 2007;128(2):245–256. [PubMed]
- Lauffenburger D, Linderman J. Receptors: Models for Binding, Trafficking, and Signaling. Oxford University Press; New York: 1993.
- Lecuyer E, Yoshida H, Parthasarathy N, Alm C, Babak T, Cerovina T, Hughes T, Tomancak P, Krause H. Global analysis of mRNA localization reveals a prominent role in organizing cellular architecture and function. Cell. 2007;131:174–187. [PubMed]
- Lott S, Kreitman M, Palsson A, Alekseeva E, Ludwig MZ. Canalization of segmentation and it evolution in Drosophila. Proc Natl Acad Sci U S A. 2007;104:10926–31. [PMC free article] [PubMed]
- Surkova A. Manu, Spirov A, VV G, H J, AR K, O R, CE V-A, DH S, M S, J R. Canalization of gene expression and domain shifts in the Drosophila blastoderm by dynamical attractors. PLoS. Comput. Biol. 2009;5(3):e1000303. [PMC free article] [PubMed]
- Mason E, Marsh J. Changes in the pattern of twisted gastrulation gene expression among Drosophila species. J. Mol. Evol. 1998;46:180–187. [PubMed]
- Mizutani CM, Nie Q, Wan FY, Zhang YT, Vilmos P, Sousa-Neves R, Bier E, Marsh JL, Lander AD. Formation of the BMP activity gradient in the Drosophila embryo. Dev. Cell. 2005;8(6):915–24. [PMC free article] [PubMed]
- Muraoka O, Shimizu T, Yabe T, Nojima H, Bae YK, Hashimoto H, Hibi M. Sizzled controls dorso-ventral polarity by repressing cleavage of the Chordin protein. Nat. Cell Biol. 2006;8:329–338. [PubMed]
- O’Connor MB, Umulis DM, Othmer HG, Blair SS. Shaping BMP morphogen gradients in the Drosophila embryo and pupal wing. Development. 2006;133:183–93. [PubMed]
- Othmer HG, Pate E. Scale-invariance in reaction-diffusion models of spatial pattern formation. Proc Natl Acad Sci U S A. 1980;77(7):4180–4184. [PMC free article] [PubMed]
- Reeves G, Muratov C, Schpbach T, Shvartsman S. Quantitative models of developmental pattern formation. Dev. Cell. 2006;11:289–300. [PubMed]
- Rentzsch F, Zhang J, Kramer C, Sebald W, Hammerschmidt M. Crossveinless 2 is an essential positive feedback regulator of Bmp signaling during zebrafish gastrulation. Development. 2006;133:801–11. [PubMed]
- Schmidt JD, Kamber E, Kondev J. Lattice model of diffusion-limited bimolecular chemical reactions in confined environments. Phys. Rev. Letts. 2009;102 (21 pages = 218302) [PubMed]
- Schmierer B, Tournier AL, Bates PA, Hill CS. Mathematical modeling identifies Smad nucleocytoplasmic shuttling as a dynamic signal-interpreting system. Proc. Natl. Acad. Sci. U.S.A. 2008;105:6608–6613. [PMC free article] [PubMed]
- Serpe M, Umulis D, Ralston A, Chen J, Olson D, Avanesov A, Othmer H, O’Connor M, Blair S. The BMP-binding protein Crossveinless 2 is a short-range, concentration-dependent, biphasic modulator of BMP signaling in Drosophila. Dev. Cell. 2008;14:940–953. [PMC free article] [PubMed]
- Shimmi O, Umulis D, Othmer H, O’Connor M. Facilitated transport of a Dpp/Scw heterodimer by Sog/Tsg leads to robust patterning of the Drosophila blastoderm embryo. Cell. 2005;120(6):873–86. [PubMed]
- Spirov A, Fahmy K, Schneider M, Frei E, Noll M, Baumgartner S. Formation of the bicoid morphogen gradient: an mRNA gradient dictates the protein gradient. Development. 2009;136:605–614. [PMC free article] [PubMed]
- Sutherland DJ, Li M, Liu X-Q, Stefancsik R, Raftery LA. Stepwise formation of a SMAD activity gradient during dorsal-ventral patterning of the Drosophila embryo. Development. 2003;130(23):5705–5716. [PubMed]
- Turing A. The chemical basis of morphogenesis. Phil. Trans. R. Soc. London. 1952;237:37–72.
- Umulis D. Analysis of dynamic morphogen scale-invariance. J. Roy. Soc. Interface. 2009 [PMC free article] [PubMed]
- Umulis D, O’Connor M, Othmer H. Robustness of embryonic spatial patterning in Drosophila melanogaster. Curr. Top. Dev. Biol. 2008;81:65–111. [PubMed]
- Umulis DM, Serpe M, O’Connor MB, Othmer HG. Robust, bistable patterning of the dorsal surface of the Drosophila embryo. Proc Natl Acad Sci U S A. 2006;103(31):11613–11618. [PMC free article] [PubMed]
- Wang X, Harris R, Bayston L, Ashe H. Type IV collagens regulate BMP signalling in Drosophila. Nature. 2008;455:72–77. [PubMed]
- Wang YC, Ferguson EL. Spatial bistability of Dpp-receptor interactions during Drosophila dorsal-ventral patterning. Nature. 2005;434(7030):229–34. [PubMed]
- Wolpert L. Positional information and the spatial pattern of cellular differentiation. J Theor Biol. 1969;25(1):1–47. [PubMed]

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- Robustness of embryonic spatial patterning in Drosophila melanogaster.[Curr Top Dev Biol. 2008]
*Umulis D, O'Connor MB, Othmer HG.**Curr Top Dev Biol. 2008; 81:65-111.* - The Decapentaplegic morphogen gradient: from pattern formation to growth regulation.[Nat Rev Genet. 2007]
*Affolter M, Basler K.**Nat Rev Genet. 2007 Sep; 8(9):663-74.* - The importance of geometry in mathematical models of developing systems.[Curr Opin Genet Dev. 2012]
*Umulis DM, Othmer HG.**Curr Opin Genet Dev. 2012 Dec; 22(6):547-52. Epub 2012 Oct 27.* - Type IV collagens regulate BMP signalling in Drosophila.[Nature. 2008]
*Wang X, Harris RE, Bayston LJ, Ashe HL.**Nature. 2008 Sep 4; 455(7209):72-7.* - A positive role for Short gastrulation in modulating BMP signaling during dorsoventral patterning in the Drosophila embryo.[Development. 2001]
*Decotto E, Ferguson EL.**Development. 2001 Oct; 128(19):3831-41.*

- A microfluidic array for large-scale ordering and orientation of embryos[Nature methods. 2011]
*Chung K, Kim Y, Kanodia JS, Gong E, Shvartsman SY, Lu H.**Nature methods. 2011 Feb; 8(2)171-176* - Image analysis and empirical modeling of gene and protein expression[Methods (San Diego, Calif.). 2013]
*Trisnadi N, Altinok A, Stathopoulos A, Reeves GT.**Methods (San Diego, Calif.). 2013 Jul 15; 62(1)10.1016/j.ymeth.2012.09.016* - The importance of geometry in mathematical models of developing systems[Current opinion in genetics & development. ...]
*Umulis DM, Othmer HG.**Current opinion in genetics & development. 2012 Dec; 22(6)547-552* - Sizzled-Tolloid interactions maintain foregut progenitors by regulating Fibronectin-dependent BMP signaling[Developmental cell. 2012]
*Kenny AP, Rankin SA, Allbee AW, Prewitt AR, Zhang Z, Tabangin ME, Shifley ET, Louza MP, Zorn AM.**Developmental cell. 2012 Aug 14; 23(2)292-304* - Multistep molecular mechanism for Bone morphogenetic protein extracellular transport in the Drosophila embryo[Proceedings of the National Academy of Scie...]
*Sawala A, Sutcliffe C, Ashe HL.**Proceedings of the National Academy of Sciences of the United States of America. 2012 Jul 10; 109(28)11222-11227*

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- Organism-scale modeling of early Drosophila patterning via Bone Morphogenetic Pr...Organism-scale modeling of early Drosophila patterning via Bone Morphogenetic ProteinsHoward Hughes Medical Institute Author Manuscripts. Feb 16, 2010; 18(2)260PMC

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