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

Transcription

1 Organism-scale modeling of early Drosophila patterning via Bone Morphogenetic Proteins David M. Umulis, Osamu Shimmi, Michael B. O Connor, and Hans G. Othmer October 20, 2008 Abstract Mathematical models of embryonic development are formulated to illuminate how the spatio-temporal expression of genes that presages the adult body plan of an organism is controlled, but many have limited utility because they oversimplify crucial aspects such as the geometry, the molecular mechanisms, and other components in the system being modeled. To circumvent these limitations we developed a datadriven, 3D, organism-scale model of bone morphogenetic protein (BMP)-mediated embryonic patterning in Drosophila. We tested 7 different receptor/feedback mechanisms and 8 different geometry/gene expression scenarios for their ability to reproduce the mean distributions of pmad signaling in both wild-type and more than twenty different mutant embryos. We found that positive feedback of a secreted BMP binding protein, coupled with the measured embryo geometry, provides the best agreement between model and experiment. The inclusion of all important factors in a 3D model represents a significant step forward in the systems biology of development. D. Umulis, Present address: Agricultural and Biological Engineering, 225 S. University St., Purdue University, West Lafayette, IN 47907, USA O. Shimmi, Institute of Biotechnology, University of Helsinki, Helsinki, Finland M. O Connor, Howard Hughes Medical Institute Investigator, and Department of Genetics, Cell Biology, and Development, Minneapolis, MN 55455, USA moconnor@umn.edu H. Othmer, School of Mathematics and Digital Technology Center, Minneapolis, MN 55455, USA othmer@math.umn.edu

2 1 Introduction In many cases, patterned cellular differentiation during development relies upon the spatiotemporal activity of molecules called morphogens [4, 20, 25]. Morphogens are molecules that generate form or cell fate in a concentration dependant manner and are often secreted factors that diffuse through an extracelluar environment to form concentration gradients. In early Drosophila embryos, patterning of the dorsal ectoderm is directed by by a heterodimer of Decapentaplegic (Dpp) and Screw (Scw) two members of the bone morphogenetic protein (BMP) family (Figure 1A,B) [16,18]. This system is unique in that unlike classical morphogen systems that rely on the slow spreading of a molecule from a localized source to generate a shallow gradient, BMPs in the early Drosophila embryo are instead first secreted from broad region making up the dorsal-most 40% of the embryo circumference and then they are dynamically concentrated into a narrow region centered about the dorsal midline that makes up only 10% of the embryo circumference [16, 19, 22]. The Dpp/Scw heterodimer activates signaling by binding to and recruiting two type I receptor Thickveins (Tkv) and Saxophone (Sax) into a high order complex containing two subunits of the type II receptor Punt [18]. The receptor complex phosphorylates Mad, a member of the Smad family of transcription factors. Phosphorlyated Mad binds to the co-smad Medea and the complex accumulates in the nucleus, where it regulates gene expression in a concentration dependent manner [17]. A number of extracellular regulators contribute to the dynamics and localization of pmad signaling. 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 (Figure 1 B). The dorsally secreted metalloprotease Tolloid (Tld) processes Sog specifically when Sog is bound to BMP ligands. The successive binding of Dpp/Scw to Sog/Tsg and degradation of Sog by Tld establishes a concentration gradient of Dpp/Scw/Sog/Tsg with a maximum laterally and a minimum at the dorsal midline. Thus the flux of Dpp/Scw/Sog/Tsg is from high concentration laterally to low concentration dorsally, and the BMP-ligands are transported by the inhibitor and deposited in a narrow stripe of cells at the dorsal midline. An amazing feature of Drosophila development is that it is surprisingly resilient to a wide range of 2

3 perturbations, including increases/decrease in gene copy number, ectopic gene expression, embryo size and temperature [5, 12, 15, 21, 22, 24]. Recently several mechanistic mathematical models based solely on the extracellular components have been developed to test the plausibility of the shuttling mechanism and identify properties that confer the experimentally observed robustness [5, 15]. While partially successful, the models failed to generate the sharp narrowing in the BMP distribution that is seen in wildtype embryos [16]. Additional experimental evidence has demonstrated that positive feedback in response to BMP signaling is required for proper dynamics and localization of Dpp and pmad [24]. While the target of BMP feedback remains elusive, more recent mathematical analysis that incorporates positive feedback into the model is better able to capture the dynamic contraction in BMP distribution with time and the resilience of the system to perturbations in the levels of many of its components [22]. 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. 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 shape of Dpp-bound to the surface of the embryo [24]. While the existing mathematical models provide a glimpse into the mechanisms of BMP regulation during embryo development, they are limited by the restriction to 1D geometry, symmetry assumptions, and simplified initial/boundary conditions. Furthermore, the 3D spatial distribution of signaling and gene expression contains a wealth of information that can be used to infer the molecular mechanisms of in vivo regulation [8, 13]. Even in cases where the geometry appears symmetrical and amenable to analysis along a plane or line that dissects the organism, no studies have been carried out to test whether this reduction is appropriate. For instance, the Drosophila embryo is frequently modeled as a line for both AP and DV patterning, but the embryo is not symmetric and the cross-sectional circumference is different at each point along the AP 3

4 axis, gene expression is non-uniform along all axes, and as we show here for BMP patterning, the shape of pmad signaling is non-uniform along both the AP and DV axes. To overcome prior limitations in modeling and analysis, determine mechanisms of resilience, and analyze the relationship between upstream gene expression and downstream signaling, we developed a data-driven, organism-scale model of the Drosophila embryo. 2 Image analysis To develop an organism-scale mathematical model of the Drosophila embryo, we need five components: 1) the geometry, 2) a mechanistic hypothesis, 3) suitable initial and boundary conditions, 4) observations for comparison, and 5) a measure of model success (A diagram for the 3D model development workflow is provided in supplemental Figure S1 in the Supporting Online Materials (SOM)). The first step in developing the model was to obtain training and comparison data for model testing by staining for pmad in populations of mid-late cycle 14 embryos. To minimize the staining and mounting error, we applied an automated geometric rotation and scaling process to each image (SOM Figure S2, S3). Following this, the 2D orthogonal projection of fluorescence intensity on the surface of the embryo is rescaled to a normalized concentration such that the χ 2 distribution of individuals about the population mean is minimized (for details see SOM section 2 and Figures S4) [10]. With mounting, image capture, staining, and other sources of experimental error minimized, the remaining variability is closer to the true biological variability of the population. Data points for normalized pmad concentration for 15 D. melanogaster wt embryos at x/lx = 0.5 (Lx is embryo length) are shown in Figure 2 A, along with the population mean and error bars. In addition, we determined the x-y projection of the mean pmad distribution for a large number of genotypes, including scw +/, tsg +/, sog +/, D. virilis, and D. busckii embryos (Figure 2 B). Surprisingly, there is a consistent, non-uniform AP distribution of both normalized pmad concentration and width in wt and mutant D. melanogaster embryos (Figure 2 B), and patterns that diverge slightly from D. melanogaster in D. virilis and D. busckii. In D. melanogaster, pmad intensity peaks at 1/3 embryo length and contains other local minima and maxima of the concentration and width of the pmad stripe. Since BMP signaling activates or suppresses genes in a concentration-dependent manner, we measured 4

5 the width and variability of pmad level-sets for different choices of the concentration thresholds (T). To determine the variability of pmad width along the AP axis, the mean width (µ) +/ the standard deviation (σ) was calculated for each embryo population (SOM Figure S5, S6). Interestingly, there are specific regions along the AP axis that exhibit higher embryo-to-embryo variability of pmad in wt and mutant alleles, as well as in the different Drosophila species tested. Embryo-to-embryo variability in the mean pmad width for a given T-value is quantified by the coefficient of variation (CV = σ/µ), a dimensionless quantity that relates the variability to the mean along the entire AP axis (Figure 2 (C)). 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 CV along the AP axis is minimized at T (Figure 2 C) and increases as the threshold is increased. This means that AP-spanning dorsally-localized genes would have the least embryo-embryo variability if they respond to thresholds that correspond to 25% of the maximum observed pmad signaling. We speculate that this defines the boundary for high level BMP targets such as race, but additional experiments are needed to map the input-output response. Having found that there is relatively low embryo-embryo variability in pmad within a population, we extended our analysis to detect differences in pmad signaling between different populations that each have a deficiency in the level of at least one BMP regulator (Figure 2 D). For the comparisons, 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 2 D-F and Figure S5). 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 differences in the width of pmad for specific threshold levels, and in particular 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 2 E and F sog +/ line). Whether or not the pmad distributions on the dorsal surface are robust with respect to changes in gene copy number is unresolved, since different groups arrive at different conclusions [5, 15, 18, 21, 22, 24], but the analysis presented here may explain some of this apparent 5

6 ambiguity. We propose that if one cannot reject the null hypothesis, then the patterning mechanism is robust. Additionally, measuring a difference does not preclude robustness, since a definition of robustness must follow and not precede a statistical difference. In this context development proceeds normally, which suggests that pmad signaling is robust even though the patterns are different between wt and heterozygous mutant embryos. This means that differences in BMP signaling may be corrected by later events, or that the system has some capacity for embryo-embryo variability, and differences manifest themselves into as yet, unobservable differences between individuals within a population. We speculate that BMP patterning is an example of the latter and additional experiments that map surface-bound Dpp levels to pmad levels and the expression of downstream targets will provide information as to whether robustness is gained by extracellular or intracellular regulation of BMP signaling. To measure the sensitivity of BMP patterning to changes in size, we compared the pmad distributions between D. melanogaster, D. virilis, and D. busckii. On average, the scaled pmad width (w/lx) is constant across species, but not across individuals within a species (Figure 2 G) as has also been shown for Bicoid [9]. While the AP averaged pmad width is constant across species, the same is not true point-wise along the AP axis. For both D. busckii and D. virilis, large segments along the AP axis are significantly wider or narrower than D. melanogaster at all thresholds tested (T=0.2 shown in Figure 2 H). The most notable difference occurs between D. virilis and D. melanogaster embryos with the D. virilis embryos bowing out near the AP midline. While the shapes of pmad signaling are similar, they are not perfectly scale-invariant, and since BMP signaling has a profound impact on the development of morphology, this may be one of the earliest developmental stages whereby small changes in gene expression and signaling influence morphological evolution [14]. 3 Organism-scale modeling Our next objective was to develop an organism-scale model of the Drosophila embryo by directly incorporating our experimental results for pmad signaling and gene expression, the 3D geometry [8], and different mechanisms. The bioimage based model represents a seamless integration of upstream gene expression data, geometry, mechanism, and downstream pmad training data into a computational tool for in silico hypothesis 6

7 testing. To measure the relative contributions of 1) embryos geometry, 2) upstream gene expression, and 3) specific choice of positive feedback mechanism, we developed numerous different versions of the embryonic model. To measure the role of geometry from 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 [8]. Next, we developed 7 versions of the model that differ in the specific molecular mechanism of positive feedback (Figure S7 and SOM section 3). The seven 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, and 7) positive feedback of an autonomous BMP binding protein. The reaction-transport equations for dorsal surface patterning have the form: c = φ + R (c, x, p) (1) t φ = D c. Here, c is the vector of concentration, R is the vector of reaction rates, x is position in Cartesian coordinates, D is the diffusion coefficient, and p is the vector of kinetic parameters (for a full set of equations see the SOM section 3). In addition to equation 1, we specify the initial and boundary conditions as determined from the image analysis and embryo geometry. The model was solved using the finite element method (see SOM section 3.1). Each model was trained by optimizing the spatial distribution of BMP-bound receptors (BR) from 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 (RM SD) = RM SD/µ) [7] between the normalized model results for the BMP-receptor (BR) distribution and the corresponding pmad distribution in a region of each image masked by the x-y orthogonal projection of the embryo shape used for the modeling (Figure 3 B) and plotted the residuals (Figure 3 D) (SOM section 3.2). Once optimized to wt, each model was optimized against 8 additional mutant data sets by optimizing 7

8 a single parameter that corresponds to the respective experimental perturbation. For instance, once a parent parameter set was found for positive feedback of receptors, the model was 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 images 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. Lastly, to measure the relative contribution of upstream gene expression on pmad signaling, we incorporated the embryonic pre-pattern for the relative gene expression of the secreted factors Sog, Tld, and Tkv (see SOM Fig. S8) [8]. 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) [14, 18]. 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 3 A and Fig. S8). Boundaries from Alkaline phospatase images were approximated by measuring the extents of gene expression for a single embryo. Example solutions for the distribution of BMP-Receptor (BR) at 60 minutes for different embryo geometries, constant upstream gene expression, and different positive feedback modules are shown in Figure 3 B and C. The positive feedback modules that 1) inhibit endocytosis, 2) increase Tld processing rate, or 3) produce a membrane associated binding protein have a smaller median CV(RMSD) value than the model with no feedback, however only the binding protein model has a statistically lower mean when compared to the model with no feedback. In the other positive feedback cases, the best model fit for a number of the mutant cases was worse than the fit achieved with no feedback since each mutant case is based on the parent set of parameters determined for wt embryos. The sensitivity of the model to the geometrical approximation and upstream gene expression is shown in Figure 3 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 8

9 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. 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 3 E,F) [22]. While the reason for the strong dependence of the embryo geometry on the ability to reproduce the expected pmad distributions is unclear, these data suggest that the non-uniform distributions may provide a more reproducible pmad pattern in lieu of the specific embryo shape and buffer the system against individual variability in embryo geometry. Another possibility is that the shape of Drosophila embryos evolved to minimize the effect of changes in upstream gene expression on critical patterning events, such as BMP patterning. For example, the tsg gene shows large differences in gene expression in embryos between different species of Drosophila [14] that may lead to quantitative differences in pmad signaling. 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 delineate BMP regulation. In this paper 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 4 A-V; A -V ). 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 4 D, 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 4 R, the model predicts high levels of BR near the dorsal midline, whereas the pmad distribution splits into two lateral stripes (Additional modeling results and predictions 9

10 are given in Fig. S9-S10). The signaling gaps observed in Figure 4 B,D,E, and R is reminiscent of the pmad signaling gaps observed in embryos with ectopic addition of activated tkv act mrna [24] (Figure 4 S, S ). To determine if the model could capture the signaling gap observed in the tkv act 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 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 (4 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 ) (For additional explanation see SOM Figure S11). According to the model, the additional binding of Dpp/Scw to receptors feeds back into the distribution of Dpp/Scw/Sog/Tsg complexes and changes the directionality of the Dpp/Scw flux. Intriguingly, the extracellular BMP shuttling mechanism with feedback does not predict BR distributions that correspond well with our observations of pmad in D. busckii and D. virilis embryos (Figure 4 T,U). This is surprising since it has recently been suggested that the homologous BMP shuttling mechanism in Xenopus automatically confers scale-invariance if a number of conditions are met [1]. One possibility for this difference is that the model of BMP patterning in Xenopus embryos is based on a 1D section of a 3D system that greatly simplifies a rather complex geometry, and it remains to be seen if the proposed BMP shuttling mechanism leads to scale-invariance in a more geometrically-accurate model of the Xenopus embryo. An alternative scaling strategy that is more universal than a shuttling mechanism arises naturally under two conditions that can be applied to more complicated geometries: conservation of receptor binding sites, and conservation of the total flux of molecule secretion [11, 21]. When the conservation conditions are included, the calculated levels of BR correspond well with our observations of pmad staining. It should be noted also, that in very large embryos (> 750µ) (Figure 4 V,V ), the model predicts that pmad signaling will first disappear from the poles, while remaining higher near the AP midline. Indeed, this leads to a canoe-shaped distribution of pmad as observed in the mean D. virilis embryo, and is consistent with our preliminary data of pmad staining in Musca domestica embryos. 10

11 4 Conclusions Development of the BMP patterning mechanism into a data-driven, organism-scale model of the Drosophila blastoderm embryo led to a number of new findings and biological insights. First, we found that pmad signaling is non-uniform along the AP axis in wt and mutant embryos, and that the shape of pmad signaling is different between different species of Drosophila. Additionally, we measured statistically-significant differences in pmad signaling between populations previously found to be robust and found an optimal threshold level that minimizes embryo-embryo and intra-embryonic variability of pmad signaling. By independently optimizing 7 different patterning models against the pmad training data, we found only one positive feedback mechanism that statistically enhanced the model fit versus the base model. We measured the relative contributions of additional biological data for geometry and the distribution of upstream patterning components to improve the model, and found that the greatest single improvement is gained by using the real embryo geometry as opposed to an ellipsoidal approximation. Further analysis and testing of the model reproduces most known pmad signaling patterns obtained by ectopic expression of DV genes along the AP axis, mrna injection, heterozygous mutants, predicts the rescue by anterior expression of tsg in tsg mutant embryos, and provides insight into the physics behind the pmad signaling gaps observed in tkv act injection experiments and tld hypomorphic alleles. Lastly, we used the model to test two different hypotheses of embryonic scale-invariance: BMP shuttling and conservation of binding sites. In summary, the 3D model allows for a much richer tool for hypothesis testing when compared to simpler 1D models. There are a number of ways to improve and extend the 3D model to address additional questions. One extension of the model could be to include the new data that demonstrates a role for type IV collagen during embryonic patterning [23]. While the collagen provides a scaffold for the formation of extracellular inhibitor complexes, we speculate that including these processes may improve the model s ability to capture the observed pmad distributions in the ectopic gene expression experiments not currently captured by the 3D model. Recently, the spatiotemporal gene-expression profiles in Drosophila have been quantitatively mapped to a 3D virtual reconstruction of the blastoderm embryo. By mapping over 95 genes at 6 time cohorts, most of the known regulatory interactions for AP patterning were obtained and hundreds of new interactions were 11

12 predicted. By coupling information-rich data sets with 3D mechanistic models of development, additional interactions can be discovered, and competing hypothetical mechanisms can be compared and judged based on their ability to reproduce existing patterns of gene expression. One 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 [2,11,21]. 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 [3,11]. 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 [6, 11]. The coupling of statistical image analysis methods, direct incorporation of data from multiple sources for gene expression and geometry, dynamic simulation of BMP patterning in an organism-scale model, and optimization of multi-dimensional modeling results with the experimental data provides novel insights and represents a major step forward in systems biology. 5 ACKNOWLEDGMENTS The authors gratefully acknowledge the the Rosen Center for Advanced Computing at Purdue University, the Minnesota Supercomputing Institute, NIH Grant GM29123 to H.G.O. and the reviewers comments. M.B.O. is an Investigator with the Howard Hughes Medical Institute. References [1] D. Ben-Zvi, B. Shilo, A. Fainsod, and N. Barkai. Scaling of the BMP activation gradient in Xenopus embryos. Nature, 453: , Jun [2] S. Bergmann, O. Sandler, H. Sberro, S. Shnider, E. Schejter, B. Shilo, and N. Barkai. Pre-steady-state decoding of the Bicoid morphogen gradient. PLoS Biol., 5:e46, Feb

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16 Figure 1: (A) 3D geometry of the Drosohila blastoderm embryo and computational mesh used for models. (B) Schematic of BMP DV patterning network (cross-section view). Note the? on the diagram for the positive feedback, which is still unknown. 16

17 Figure 2: pmad variability and population statistics 17

18 Figure 2: (A) Normalized wt pmad level vs position y/lx at the cross-section x/lx = 0.5. Lx is the mean length of the AP axis. Red dots are the normalized concentration, the solid blue line is the mean pmad in the cross-section and the error bars correspond to +/ σ. (B) Composite pmad distribution for (beginning at top left) D. melanogaster wt (15), scw +/ (12), sog +/ (9), tsg +/ (18), D. busckii (14), and D. virilis (18). (C) Mean coefficient of variation for pmad width averaged along the AP axis for wt as a function of threshold (CV = σ/w: standard deviation / width). Error bars (standard deviation of CV) provide a measure of CV variability along the AP axis. (D) Mean pmad distribution along a cross-section at x/lx = 0.5. (E-F,H) Mean width of pmad varies along the AP axis. Thin lines represent the average width for the respective embryos (see legend in upper-left). Colored data points are superimposed over the lines for the mean width in regions where the null hypothesis (mean widths between the mutant and wt are equal) is rejected with α = Threshold values (E) T = 0.2, (F) T = 0.4. (G) Average pmad width along AP axis is constant between species. (H) Scale-invariance does not occur at all positions along the AP axis since the shape of the pmad distribution is different between species. 18

19 Figure 3: (A) Left: distribution of gene expression and/or source of information: zygotic tkv levels were obtained from Charless Fowlkes (personal communication) and the gene expression atlas [8]. The Tld distribution and Sog secretion distribution were determined by fluorescent in situ hybridization of tld and sog mrna [13]. Parentheses indicate sample size. (A) Right: prepattern distributions as they appear in the 3D model. The Bicoid distribution was determined solving a simple reaction diffusion equation in the periphery of the embryo [21]. (B-C) Box-and-whisker plots of minimum of CV(RMSD) (RM SD/µ) found in the full 3D model for different feedback mechanisms (B), contributions of Tkv and Tld non-uniform distributions, and (C) embryo shape. Upper and lower limits of the box represent the upper and lower quartile, and the red line is the median CV(RMSD) value for the series of embryos tested. In (B) the cases compared are: no positive feedback, positive feedback of receptors, negative and positive feedback on endocytosis rates, positive and negative feedback of Tld processing, and positive feedback of a BMP-binding non-receptor [22,24] in (B) all models were solved with uniform levels of Tkv and Tld. (C) Case 1: uniform Tkv and Tld, Case 2: uniform Tld, non-uniform Tkv, Case 3: uniform Tkv, non-uniform Tld, and Case 4: non-uniform Tld, non-uniform Tkv for the real embryo geometry (first four) and ellipse approximation of embryo geometry. (D) typical residual result for the difference between the model result and the mean pmad distribution. (E) BMP-bound receptor levels begin broad and low and contract in time to produce the non-uniform distribution of occupied receptors. (F) BR levels at x/lx =

20 Figure 4: 3D model results 20

21 Figure 4: (A-S) Mean pmad staining for ectopic gene expression experiments and/or mutant alleles. (A - S ) Levels of BR calculated using the computational model. (T-V) Calculated BR levels for D. virilis and D. busckii and predicted distribution for embryos 750µ in length.(t-v) are the expected BR levels when conservation conditions are imposed for the total amounts of protein production, and/or protein levels. (T - V ) Same as (T-V) except without conservation on the total amount of production/proteins in the system (i.e. the concentrations are constant). 21

22 Organism-scale modeling of early Drosophila patterning via Bone Morphogenetic Proteins: Supporting Online Materials (SOM) David M. Umulis, Osamu Shimmi, Michael B. O Connor, and Hans G. Othmer October 20, Workflow for 3D model development Figure 1 shows the organization image acquisition, analysis, model development and model testing. 2 Image Processing and Analysis To determine if the mean width of pmad signaling is different between Drosophila melanogaster wt, heterozygous sog, tsg, scw, and other Drosophila species, one must first determine the embryo to embryo variability for each genotype. For wt, and sog, tsg, scw embryo staining, a number of experimental conditions were carefully controlled to limit the variability caused by measurement and experimental error. For other experiments that took place after the initial characterization of wt and heterozygous mutants, we used an internal wt control. Ideally, one would measure the actual levels of Dpp-GFP, as done in Drosophila wing D. Umulis, Present address: Agricultural and Biological Engineering, 225 S. University St., Purdue University, West Lafayette, IN 47907, USA O. Shimmi, Institute of Biotechnology, University of Helsinki, Helsinki, Finland M. O Connor, Howard Hughes Medical Institute Investigator, and Department of Genetics, Cell Biology, and Development, Minneapolis, MN 55455, USA moconnor@umn.edu H. Othmer, School of Mathematics and Digital Technology Center, Minneapolis, MN 55455, USA othmer@math.umn.edu

23 Figure S 1: Workflow for model development of data-driven organism-scale model. 2

24 imaginal discs [4], however the levels of Dpp-GFP are below the levels of detection during the syncitial stages of embryo development, requiring antibody amplificaiton. An alternative approach employed herein by our group and by others is to measure the downstream pmad signaling by antibody staining, ensuring identical experimental conditions for labeling and fixed exposure settings for image acquisition [1, 5]. Embryos in late nuclear division cycle 14 are selected based on the onset of cellularization and the initiation of cephalic furrow formation. Microscope settings are determined for a single embryo and kept constant for the remaining embryos within each genotype and between genotypes. For the large-scale analysis of pmad signaling between different Drosophila species and mutant populations, a number of image processing steps must be carried out to remove the geometric and staining variability introduced by the observer. 2.1 Acquisition and data alignment First, a z-series of 15 slices of pmad fluorescent intensity from each embryo are combined into a single extended focus image using Zeiss Axiovision 4.5 software. Each image is converted to integer matrices where each element in the matrix represents the pixel position of the image, and the integer of each element varies from based on the intensity of the image at that point (0 is black and 255 is white). The matrices are converted to format double that spans from 0 to 1 instead of 0 to 255 and a binary image is computed for a threshold value of 0.2 to find the region of high pmad intensity. Each binary pmad distribution is then fit to an ellipse, and the angle between the horizontal plane and the principal axis of the ellipse is determined. The original grayscale matrix is rotated by the angle of the ellipse above horizontal to re-align the embryos on the x-axis. All embryos are re-positioned and re-scaled to create new images where all embryos are consistently aligned, centered, and scaled. Each 2D pmad profile from the rotated image is then mapped to a 3D surface, rotated in the y-z direction to correct for tilt and then remapped into a new 2D projection of the embryo (Figure 2A). Example pmad distributions after scaling and rotation are shown in Figure 3. 3

25 Figure S 2: (A) Each grayscale image rotated in 3D cartesian coordinates in two steps: first the embryos are rotated in x,y coordinates and then mapped to an ellipse for y-z rotation. Additionally, embryos are re-scaled in the x and y-directions to compensate for natural variability in size. (B). Equations for image normalization. 4

26 Figure S 3: This image shows 15 wt embryos immediately after rotation and scaling, and immediately before normalization. 5

27 2.2 Fluorescence normalization To remove embryo-embryo staining variability, we extended the method developed by Gregor et al. to two dimensions [3]. As previously described, we assume that the observed fluorescence intensity of each image is proportional to the concentration plus some background noise by the relationship I n (x, y) = A n c n (x, y)+b n [3]. In summary, we choose values of A n and B n for each embryo so that the difference between each individual concentration and the population mean concentration is minimized. To calculate the differences, we measure the integrated mean-sqaured error between the measured intensity I n and the approximate intensity, which is equal to A n c(x, y) + B n. The sum of individual tests is a χ 2 distribution, and the optimal choices of A n and B n leads to a minimum χ 2 value. To find the values of A n and B n that minimize the χ 2 value, we use the built in Matlab non-linear solver fminsearch. As done previously [3], we select B n so that the mean of each profile is zero [3]. Since the system is underdetermined, we find a linear family of A n values for each population and therefore are free to choose an appropriate scaling for the mean pmad distribution. We set the maximum of the mean pmad concentration for Drosophila melanogaster wt embryos equal to one and the minimum of the mean concentration equal to zero. Since we are comparing pmad distributions between different populations of embryos that may have different maximum pmad signaling levels, it does not make sense to normalize the maximum of each mean distribution to 1 as we did for the wt case. Instead, we use the ratio of the brightest 5% of pixels from the raw intensity data for the mutant population against the brightest 5% of pixesl for the wt population. The maximum concentration for the mean of each mutant population is then set to the ratio of pixels and the minimum for each mutant population is set to zero. While this method introduces some potential for systematic error, we minimized this by using embryos stained on the same day that were stained with the same antibody stock solution, and imaged using identical microscopy settings. To correct for differences in staining between different days, we used an internal wt control and normalized the mean wt pmad distribution from the control images against the original data set. The scalings determined by normalizing wt between different samples are then imposed on the remaining images in the new experiment and the data can then be compared directly with the original data set. Specifically, we used this methodology for the tld hypomorphic allele tld 7M89. the data for the ectopic gene expression experiments was normalized to 1 due to smaller 6

28 Figure S 4: Example cross-section distribution of pmad before and after minimization of the χ 2 norm. (A) Before and (B) after normalization. population sizes and/or the wt background. 2.3 Haplosufficiency of BMP patterning To determine the embryo to embryo variability of embryos within each genotype, the width of pmad staining is computed for different choices in the threshold value. To produce the composite images in the main text, we used an intensity threshold of 0.2. Following the image processing, the mean width for the population of embryos is determined for each genotype. To determine the variability of patterning, the standard deviation of the mean width is computed for the population at each x-position along the AP axis of individual embryos (Figure S6 A-F). The typical embryo to embryo standard deviation spans approximately +/ 2 nuclei diameters near the anterior end of the embryo, and expands to nearly +/ 3 nuclei near the posterior end of the embryo for all genotypes (Figure S6A-F). We did not measure a difference in the magnitude of embryo-embryo variability in the different populations of embryos (Figure S6 G,H). 7

29 Figure S 5: pmad width depends on AP position. 8

30 Figure S 5: The difference in pmad width depends upon the threshold and position along the AP axis. The mean pmad width is shown at (A) x/lx = 0.25, (B) x/lx = 0.5, and (C) x/lx = Note that the width of pmad in sog +/ is relatively wider in (C) than in (A). 3 3D modeling equations and methods The equations that govern BMP patterning for the 3D model are based on our current knowledge of the patterning pathway, and our hypotheses regarding the nature of the embryonic positive feedback. All species except those localized to the surface are assumed to be freely diffusible. As mentioned in the main text, the specific mechanism of feedback is not known. The positive feedback models tested are shown in Figure S7. The kinetic parameters and associated parameter values are listed in Table 1. The elliptical approximation of the embryo measures 400µ long and 180µ in diameter at the AP midline. The real geometry was determined by interpolation of the nuclei point cloud data in the VirtualEmbryo [2]. A series of slices along the AP axis were extracted from the VirtualEmbryo data and re-assembled into a 3D mesh in Matlab. The program we developed to build the reconstructed 3D embryo can be applied to any 2D data set, as long as the dimensions of each 2D slice and the distance between each slice is known. This is a common format for z-section data obtained using confocal microscopy. Since BMP patterning takes place in a thin, fluid-filled gap at the periphery of the embryo, the domain is approximated as a 2D sheet wrapped around a three dimensional object. The production of Sog, Tsg, and the BMP ligands are limited to specific regions along the surface of the blastoderm embryo as discussed in the main text. To incorporate upstream patterns of gene expression for sog, tld, and tkv directly into the 3D model, we fit a Fourier sine series to the mean FISH images using linear regression. Once the Fourier coefficients were found, we incorporated the Fourier approximation in the finite-element framework and projected the 2D distributions onto the 3D embryo geometry. Note: When fiting data to a Fourier sine seris, there are small regions that have negative values. Since these are nonphysical values, we set every small negative number to zero in the FEM model. Also, we calculated the approximate curves using a high order multi-polynomial, which provided a good fit (Figure S8 B), however they were superseded by the fourier sine series which has a number of advantages including fewer parameters and easier implementation. 9

31 10 Figure S 6: pmad variability.

32 Figure S6: (A-F) Mean +/- standard deviation width of pmad for T=0.25. (A) wt, (B) scw +/, (C) sog +/, (D) tsg +/, (E) D. busckii, (F) D. virilis. The embryo-embryo variability as measured by the standard deviation (G) and the coefficient of variation (H) in pmad patterning is constant between the mutants and species tested. Figure S 7: While the target of BMP-mediated feedback is not known, there are numerous plausible mechanisms of positive feedback. (A) Positive feedback of a receptor. (B) Negative feedback on receptor endocytosis. (C) Positive feedback on receptor endocytosis. (D) Negative feedback on Tld processing. (E) Positive feedback on Tld processing. (F) Positive feedback of a surface-bound binding protein. 11

33 Figure S8: (A) Mean FISH distributions (or distributions from the VirtualEmbryo) are incorporated directly into the 3D model. The tld distribution is shown here. (B) Example overlay of an approximate solution (polynomial from earlier testing shown here). (C) Functional form of sine series used for fitting. 12

34 Name Description Value Units Inhibitor Binding k 2 Binding Sog/Tsg 6.0 * 10 1 nm 1 min 1 k 2 Reverse 3.6 * 10 0 min 1 k 3 Binding I to B 11.4 * 10 0 nm 1 min 1 k 3 Reverse 3.6 * 10 1 min 1 Receptor Binding k 5 Binding B to R 4.8 * 10 2 nm 1 min 1 k 5 Reverse 8.0 * 10 0 min 1 Diffusion D S Diffusion, Sog 45 µ 2 sec 1 D T Diffusion, Tsg 60 µ 2 sec 1 D B Diffusion, BMP 65.7 µ 2 sec 1 D IB Diffusion, I/BMP 38 µ 2 sec 1 D I Diffusion I 40.5 µ 2 sec 1 Production, etc. φ S Sog secretion 1.36 µm*min 1 φ T Tsg secretion 48 nm*min 1 φ B BMP dimer secretion 1 nm*min 1 Tld Tld levels 10 nm Degradation δ T sg Tsg Degrade 5.0 * 10 2 min 1 δ E Internalization rate 3.0 * 10 2 min 1 Table 1: Parameter values for kinetic processes held constant during all wt simulations. 3.1 Model implementation The equations describing dorsal surface patterning in Drosophila are nonlinear, coupled, partial differential equations. Furthermore, we wanted to solve the equations on a non-trivial domain to study the spatial effects of patterning in three dimensions. To solve these equations, it is most convenient to use the Finite Element Method (FEM), which has a number of advantages over other commonly used methods such as the finite difference method. One advantage is that boundary conditions can be specified exactly in the weak formulation eliminating some round off error common to other methods. Specifically, we use Galerkin s Finite Element Method (GFEM). The GFEM is a special case of a more general method known as the method of weighted residuals. Similar to approximating a model fit to data points using the method of least-squares, the method of weighted residuals minimizes the error between the unknown exact solution and 13

35 an approximation to that solution. Suppose a solution to the reaction-diffusion partial differential equation is required. c t = 2 c + f (c) n c = 0 on Ω c (t = 0) = c 0 (x) (1) The function f(c) is non-linear. The first step required for solving the pde using the FEM is to break up the spatial domain into a number of subsections known as elements. This process, called meshing, or triangulation, divides the large and sometimes complex geometry into a large number of smaller simpler regions for calculation. Next, an approximate solution is selected that is composed of a sum of basis or test functions. N c (x, t) ĉ (x, t) = c i (t) Φ i (x) (2) i=1 Here we use quadratic basis functions for increased accuracy. When the approximation is substituted into the original equation, it no longer equals zero but instead it equals some error (residual). c t 2 c f (c) = 0 ĉ t 2 ĉ f (ĉ) =0 ĉ t 2 ĉ f (ĉ) = r (ĉ) We can approximate the exact solution by choosing test function coefficients c i (t) that make the residual equal to zero at a number of points equal to the number of unknown coefficients for the basis functions in the approximation. A generally superior method for solving for the constants is known as the method of weighted residuals. We cannot require the approximate solution to satisfy the partial differential equation at every single point. Instead, we can determine the values c i (t) for the approximation that satisfy a set of N conditions established by the method of weighted residuals. Since the residual form of the partial differential equation is equal to zero, we can require that the equation multiplied by a weighting function and integrated 14

36 over the domain also be equal to zero. R i Ω [ ] c w i t 2 c f (c) da = 0 (3) In this way, we can use the same number of weighting functions that we have unknown constants, construct a set of N equations with N unknowns and solve them using a linear or non-linear solver. The most common method used by engineers is to use the basis function for the approximation to the solution as the weighting function for the weighted residual integral. This is known as Galerkin s method. w i = Φ i (x), R i Ω [ ] c Φ i (x) t 2 c f (c) da = 0 (4) Before substituting in the approximation for c, it is convenient to convert the residual into what is known as the weak form. The residual terms containing 2 nd order derivatives and higher are integrated by parts. Recalling that (Φ i c) = Φ i 2 c + Φ i c, we obtain: Applying the divergence theorem leads to: R i = [ ] c (Φ i c) + Φ i c + Φ i t Φ if (c) da = 0 Ω [ (Φ i c)] da = [ Φ i c n]ds Ω Ω c n = 0 on Ω R i = [ ] c Φ i c + Φ i Ω t Φ if (c) da = 0 In this case, the weak form is completely equivalent to the original strong form of the residual. For example, here is the weak form for the equation for Sog (with a symmetry condition across the y=0, z>0 15

37 plane): S n = 0 on Ω (5) [ ] S R i = Φ i (D S S) + Φ i t Φ i (φ S k 2 S T + k 2 I δ S S) da = 0 (6) Ω Then, the approximation for c can be substituted into the weak form of the residual. c (x, t) ĉ (x, t) = N c j (t) Φ j (x) c t ĉ t = N R i = [ N c j Φ i Φ j + N Ω j=1 j=1 j=1 j=1 dc j dt Φ j (x) dc j dt Φ jφ i Φ i f (ĉ) ] da = 0 While not immediately obvious, this gives N ordinary differential equations in time to solve for c j. [ ] N dc j Φ j Φ i da = [ ] N c j Φ i Φ j da + [Φ i f (ĉ)] da Ω j=1 dt Ω j=1 Ω M dc dt = Ac + g (c), where M ij = Φ i Φ j da, A ij = Φ i Φ j da, g i = Φ i f (ĉ) da Ω Ω Ω Finally the values for c i (t) are computed by solving the set of non-linear odes. To solve the systems of equations, we used the weak formulation package of Comsol Multiphysics 3.4. We solved the models and performed the optimization on a number of 64-bit platforms including a custom built Windows XP x64 machine, a custom personal supercomputer running Linux x64, and on the Steele supercomputer (Rosen Center for Advanced Computing; Purdue University). For parameter optimization, we used a non-linear steepest descent method in Matlab. Since we were concerned that we may be finding local solutions, we also used an adaptive simulated annealing (ASA) method for D. melanogaster wt embryos and compared the solution residual and optimal parameters against those found using the gradient method and found that ASA did find better solutions, however the residual was within 5% of the gradient methods. 16

38 3.2 Equations and parameters for positive feedback modules No feedback B t = D B 2 B + φ B (x) k 3 I B + k 3 IB + λt ld IB k 5 B R + k 5 BR (7) S t = D S 2 S + φ S (x) k 2 S T + k 2 I (8) T t = D T 2 T + φ T (x) k 2 S T + k 2 I + λt ld IB δ T T (9) I t = D I 2 I + k 2 S T k 2 I k 3 I B + k 3 IB (10) IB = D IB 2 IB + k 3 I B k 3 IB λt ld IB t (11) BR = k 5 B R k 5 BR δ E BR dt (12) R tot = R + BR (13) Name Description Value Units R tot Total Receptor Level µm λ T ld Tld process rate 1.07 * 10 1 min 1 Table 2: Optimized receptor and positive feedback parameters for: No feedback 17

39 3.2.2 Positive feedback of receptor B t = D B 2 B + φ B (x) k 3 I B + k 3 IB + λt ld IB k 5 B R + k 5 BR (14) S t = D S 2 S + φ S (x) k 2 S T + k 2 I (15) T t = D T 2 T + φ T (x) k 2 S T + k 2 I + λt ld IB δ T T (16) I t = D I 2 I + k 2 S T k 2 I k 3 I B + k 3 IB (17) IB = D IB 2 IB + k 3 I B k 3 IB λt ld IB t (18) BR = k 5 B R k 5 BR δ E BR (19) dt ) R tot = (R basal + ΛBRν Kh ν + (20) BRν R = R tot BR (21) Name Description Value Units R tot Total Receptor Level µm λ T ld Tld process rate 1.10* 10 1 min 1 Λ Positive feedback amplitude µm*min 1 K H Hill parameter 18.1 nm ν Cooperativity/Gene Param. 2 (fixed) Dimensionless Table 3: Optimized receptor and positive feedback parameters for: positive feedback of receptor 18

40 3.2.3 Negative regulation of endocytosis B t = D B 2 B + φ B (x) k 3 I B + k 3 IB + λt ld IB k 5 B R + k 5 BR (22) S t = D S 2 S + φ S (x) k 2 S T + k 2 I (23) T t = D T 2 T + φ T (x) k 2 S T + k 2 I + λt ld IB δ T T (24) I t = D I 2 I + k 2 S T k 2 I k 3 I B + k 3 IB (25) IB = D IB 2 IB + k 3 I B k 3 IB λt ld IB (26) t ( BR BR ν ) = k 5 B R k 5 BR δ E Λ 1 dt Kh ν + BR (27) BRν R tot = R basal ( ) (28) δ E Λ 1 BRν Kh ν+brν R = R tot BR (29) Name Description Value Units R tot Total Receptor Level 1.65 µm λ T ld Tld process rate 1.09 * 10 1 min 1 Λ Positive feedback amplitude Dimensionless K H Hill parameter nm ν Cooperativity/Gene Param. 2 (fixed) Dimensionless Table 4: Optimized receptor and positive feedback parameters for: negative feedback of endocytosis rate 19

41 3.2.4 Positive regulation of endocytosis B t = D B 2 B + φ B (x) k 3 I B + k 3 IB + λt ld IB k 5 B R + k 5 BR (30) S t = D S 2 S + φ S (x) k 2 S T + k 2 I (31) T t = D T 2 T + φ T (x) k 2 S T + k 2 I + λt ld IB δ T T (32) I t = D I 2 I + k 2 S T k 2 I k 3 I B + k 3 IB (33) IB = D IB 2 IB + k 3 I B k 3 IB λt ld IB (34) t ) BR = k 5 B R k 5 BR δ E (1 + ΛBRν dt Kh ν + BR (35) BRν R tot = R basal δ E ( 1 + ΛBRν K ν h +BRν ) (36) R = R tot BR (37) Name Description Value Units R tot Total Receptor Level 2.25 µm λ T ld Tld process rate 1.04 * 10 1 min 1 Λ Positive feedback amplitude 0.10 Dimensionless K H Hill parameter nm ν Cooperativity/Gene Param. 2 (fixed) Dimensionless Table 5: Optimized receptor and positive feedback parameters for: Positive regulation of endocytosis 20

42 3.2.5 Negative regulation of Tld ( B t = D B 2 B + φ B (x) k 3 I B + k 3 IB + λt ld Λ 1 BR ν Kh ν + BRν ) IB k 5 B R + k 5 BR (38) S t = D S 2 S + φ S (x) k 2 S T + k 2 I (39) ( T t = D T 2 BR ν ) T + φ T (x) k 2 S T + k 2 I + λt ld Λ 1 Kh ν + IB δ T T (40) BRν I t = D I 2 I + k 2 S T k 2 I k 3 I B + k 3 IB (41) ( = D IB 2 BR ν ) IB + k 3 I B k 3 IB λt ldλ 1 IB (42) IB t BR dt K ν h + BRν = k 5 B R k 5 BR δ E BR (43) R tot = R + BR (44) Name Description Value Units R tot Total Receptor Level 2.37 µm λ T ld Tld process rate 2.49 * 10 1 min 1 Λ Positive feedback amplitude Dimensionless K H Hill parameter nm ν Cooperativity/Gene Param. 2 (fixed) Dimensionless Table 6: Optimized receptor and positive feedback parameters for: negative feedback on Tld processing rate 21

43 3.2.6 Positive regulation of Tld ) B t = D B 2 B + φ B (x) k 3 I B + k 3 IB + λt ld (1 + ΛBRν Kh ν + IB BRν k 5 B R + k 5 BR (45) S t = D S 2 S + φ S (x) k 2 S T + k 2 I (46) ) T t = D T 2 T + φ T (x) k 2 S T + k 2 I + λt ld (1 + ΛBRν Kh ν + IB δ T T (47) BRν I t = D I 2 I + k 2 S T k 2 I k 3 I B + k 3 IB (48) ) = D IB 2 IB + k 3 I B k 3 IB λt ld (1 + ΛBRν Kh ν + IB (49) BRν IB t BR dt = k 5 B R k 5 BR δ E BR (50) R tot = R + BR (51) Name Description Value Units R tot Total Receptor Level 3.00 µm λ T ld Tld process rate 1.22 * 10 1 min 1 Λ Positive feedback amplitude 0.56 Dimensionless K H Hill parameter nm ν Cooperativity/Gene Param. 2 (fixed) Dimensionless Table 7: Optimized receptor and positive feedback parameters for: positive feedback of Tld processing 22

44 3.2.7 Positive feedback of a secreted BMP binding protein like type IV collagen or Cv-2 B t = D B 2 B + φ B (x) k 3 I B + k 3 IB + λt ld IB k 4 B C + k 4 BC k 5 B R + k 5 BR (52) S t = D S 2 S + φ S (x) k 2 S T + k 2 I (53) T t = D T 2 T + φ T (x) k 2 S T + k 2 I + λt ld IB δ T T (54) I t = D I 2 I + k 2 S T k 2 I k 3 I B + k 3 IB (55) IB = D IB 2 IB + k 3 I B k 3 IB λt ld IB (56) t C t = ΛBRν K ν h + BRν k 4B C + k 4 BC k 7 BR C + k 7 BCR δ E C (57) BC = k 4 B C k 4 BC k 6 BC R + k 6 BCR δ E BC t (58) BCR = k 6 BC R] + k 7 BR C] k 6 BCR k 7 BCR δ E BCR t (59) BR = k 5 B R k 5 BR + k 7 BCR k 7 BR C δ E BR dt (60) R tot = R + BR + BCR (61) 3.3 Comparison of upstream gene expression contribution We extended the modeling to measure the relative enhancement gained in the model by including additional layers of biological information. Using the positive feedback of an SBP model, we carried out four different cases in both the elliptical geometry and the actual embryo shape. The first case explored in each geometry was uniform Tld and uniform Tkv and the two parameters that correspond to the levels of each were simultaneously optimized against the wt pmad distribution. For case 2, three parameters are optimized: zygotic (non-uniform) Tkv contribution, basal Tkv levels, and the constant for Tld level. For case 4, three parameters were optimized: constant receptor level, non-uniform Tld contribution, the x-scaling for the Tld distribution (to account for the dynamic contraction of Tld away from the AP poles during division cycle 14), and the basal Tld levels. For case 4, five parameters were optimized corresponding to the non-uniform 23

45 Name Description Value Units R tot Total Receptor Level nm λ T ld Tld process rate * 10 1 min 1 Λ Positive feedback amplitude nm*min 1 K H Hill parameter nm ν Cooperativity/Gene Param. 2 (fixed) Dimensionless Additional parameters that were NOT optimized k 4 Binding B to C 2.0 * 10 0 nm 1 min 1 k 4 Reverse 4.0 * 10 0 min 1 k 6 Binding BC to R 1.0 * 10 0 nm 1 min 1 k 6 Reverse 2.0 * 10 1 min 1 k 7 Binding BR to C 2.5 * 10 1 nm 1 min 1 k 7 Reverse 2.0 * 10 1 min 1 Table 8: Optimized receptor and positive feedback parameters for: positive feedback of a surface-associated binding protein such (Kinetics for forward and reverse binding were NOT optimized. Instead, they were approximated from published values for Cv-2 [6 8]) Tkv and non-uniform Tld distributions. The non-uniform zygotic production for Sog, Tld, and Tkv was determined using FISH as described earlier. To account for the maternal component of Tkv, we allowed the model to optimize both the zygotic component and a constant maternal level. For Sog, the images showed a clear distribution of sog mrna along the lateral section with some non-uniformity along the AP axis. The high level sog mrna staining decayed rapidly dorsally to background staining, therefore we do not consider an additional constant contribution for the Sog production. The images shown here for tld mrna staining show the non-uniform AP distribution. Embryos mounted laterally (90-degrees from current orientation) show higher levels of background staining throughout the dorsal surface. To account for the low-level uniform tld expression, we optimized the tld distribution in the model by allowing for a uniform and non-uniform contribution to Tld production. The equations for the Sog production and the non-uniform distributions of Tkv and Tld are summarized by the 24

46 following equations: φ S (y, z) = Λ f(y, z) (62) λ T ld(x, y) = Tld conc λ (g(x, y) + Tld frac ) (63) R tot (x, y) = R conc (h(x, y) + Tkv frac ) (64) Here, f(y, z), g(x, y), and h(x, y) are the Fourier sine series fits of the images. The non-uniform Sog distribution is used for each of the cases of non-uniform Tkv and Tld levels. The quantity Tld conc λ is treated as a single parameter by assuming a constant Tld concentration scale of 10nM. The optimized parameters for Cases 1-4 in the main text are given in Tables 9 and 10. Case Tld conc λ R conc Tld frac Tkv frac conditions g = 0; h = g = h = Table 9: Table of parameters for optimal fit with elliptical geometry. Tld conc λ has units of min 1, R conc has units of nm, and Tld frac, Tkv frac, g, and h are dimensionless. Case Tld conc λ R conc Tld frac Tkv frac conditions g = 0; h = g = h = Table 10: Table of parameters for optimal fit with non-uniform geometry. Tld conc λ has units of min 1, R c onc has units of nm, and Tld frac, Tkv frac, g, and h are dimensionless. Since we observed a dynamic evolution of Tld from nuclear division cycle 13 to 14 where Tld begins closer to the poles in cycle 13 embryos and migrates towards the AP midline in cycle 14 embryos, we relaxed the x-scale for the Tld distribution in the optimization. The Tld values marked with an correspond with simulation results that are not very sensitive to the non-uniform Tld distribution. In the cases here, the x-scaling factor became > 1 which suggests that the optimal Tld distribution is flatter than our observations 25

47 in cycle 14 embryos. The Tld distribution has the smallest effect on the residuals for the model fit versus the observations of pmad signaling in wt embryos. To further investigate whether the Tld contribution is significant, we examined the tld distribution in mutant embryos that either over-express tld or contain a hypomorphic allele of tld that has a reduced catalytic activity. Earlier reports suggested that reducing the amount of Tld leads to wt-like distributions of pmad signaling [1, 8]. However, pmad signaling is sensitive to reductions in the activity of Tld. To demonstrate this, we stained for pmad in tld 7M89 hypomorphic alleles. While we were unable to distinguish a difference between wt and tld 7M89 heterozygous embryos, homozygous tld 7M89 embryos show a loss of pmad near the AP midline and a loss of zen expression (Figure S9). Since tld expressions is non-uniform along the AP axis, it is natural to consider that the distribution of Tld protein may play a role in the loss of pmad signaling. Reducing Tld processing rates in the model for the real embryo geometry did not lead to the observed loss of pmad signaling along the AP axis regardless of the specific rate of Tld processing. Since the RMSD values are only weakly sensitive to the non-uniform Tld distribution in the model, we relaxed the constraints imposed on the Tld distribution to find conditions that can reproduce the observed pmad distribution in the tld 7M89 embryos. We found good agreement between the model results and the qualitative shapes of pmad signaling, however the model shows a greater loss of Tld near the AP midline and higher amplitude peaks before normalization. The loss of pmad signaling corresponds well with zen gene expression in tld 7M89 embryos (Figure S9). 26

48 Figure S 9: Mean pmad width for Tld hypomorphic allele (red) vs. D. melanogaster wt at (A) T=0.2 and (B) T=0.3. The overall intensity of pmad staining for the hypomorphic allele is lower than for wt embryos which leads to a more pronounced dip in width and intensity at x/lx = 0.45 and a gap in pmad signaling. (C) pmad signaling in tld 7M89 embryos. (D) Computational results. (E) zen gene expression in tld 7M89 mutant embryos. 27

49 3.4 Additional computational results In addition to the expression patterns for the ectopic gene expression experiments, we used the model to predict the distribution of BR in a series of cases that are we do not yet have experimental data. With the model, we can also carry out experiments that are too technically difficult to carry out at the moment such as micro-injection of sirna for BMP regulators (Figure 10). Figure S10: Additional predictions and the distribution of BMP regulators.]distribution of (A) Sog, (B) Tsg, and (C) BMP-receptor (BR) in wt embryos. (D-E) BR in early (D) and late (E) cycle 14 in eve2>dpp;dpp embryos. (F) BR in dpp +/, (G) tld +/, (H) eve2>sog, and (I) anterior RNAi against sog. (J-K) Rescue of sog by twt>sog. (J) Sog distribution, (K) BR distribution. (L) Simulated removal of Sog secretion from hypothetical RNAi injection against sog along the right side of the embryo. 28

50 3.5 Explanation of pmad signaling loss The tld hypomorphic allele is not the only observation of pmad loss along the AP axis. Another notable example of pmad loss occurs in embryos injected with activated tkv mrna [9]. Immediately adjacent to the region of injection and high pmad signaling, there is a shadow, or loss of pmad signaling. The positive feedback leads to dorsal surface accumulation of Dpp/Scw and the high levels of positive feedback in the region of tkv act, which also leads to the depletion of Dpp/Scw and pmad signaling in the region immediately to the anterior of the injection. To understand what leads to the local depletion of signal, we used the model to observed the flux of Dpp/Scw/Sog/Tsg complexes in wt, tld hyp, and tkv act embryos. In wt embryos the flux of molecules is predominantly pointed towards to DV midline at all positional along the AP axis 11. There are local maxima and minima of the flux that lead to the non-uniform shape of pmad along the AP axis. In the tld hyp, and tkv act cases, the directionality of the Dpp/Scw/Sog/Tsg flux changes so that the inhibitor actually moves Dpp/Scw away from specific regions along the AP axis adjacent to areas that either have weak Tld activity, or increased Dpp/Scw binding. Similar types of patterns are also expected in embryos with ectopic expression of sog by eve2-gal4. It is interesting to note that the strength of positive feedback (presumably downstream of extracellular patterning) has dramatic effects on upstream processes that position the ligand for signaling. 4 ACKNOWLEDGMENTS References [1] A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B. Z. Shilo, and N. Barkai. Robustness of the BMP morphogen gradient in Drosophila embryonic patterning. Nature, 419(6904):304 8, [2] C. Fowlkes, C. Hendriks, S. Kernen, G. Weber, O. Rbel, M. Huang, S. Chatoor, A. DePace, L. Simirenko, C. Henriquez, A. Beaton, R. Weiszmann, S. Celniker, B. Hamann, D. Knowles, M. Biggin, M. Eisen, and J. Malik. A quantitative spatiotemporal atlas of gene expression in the Drosophila blastoderm. Cell, 133: , Apr

51 Figure S 11: Dpp/Scw/Sog/Tsg flux. 30