PDF《Scaling of the BMP Morphogen》

0 (ventral pole). At x = L (dorsal pole), Chordin is produced with a constant ?ux DChd?[Chd] = ηChd, whereas Admp is produced with a ?ux that depends on the signaling level S(x, t): DLig?[Admp] = αAdmp(S). The ?ux of the other three components is zero. Initial Conditions: Initial values of [Chd], [Admp] and the complexes are set to zero, while [Bmp] =

2 * 10?1 ?M, uniformly across the embryo. Geometry: We consider a 1-dimensional geometry with x =

0 at the ventral pole, and x = L at the dorsal pole. The actual geometry of the embryo, and how it corresponds to this 1-dimensional geometry is explained in section 7.

3 1.2 Notes about the Assumptions of the Screen Constant levels of Bmp: Bmp in our model corresponds to the three ligands BMP2, BMP4 and BMP7. For simplicity, we assumed in the screen that the total level of Bmp is ?xed, ignoring the feedback-mediated zygotic activation of bmp4. As such, the model corresponds most closely to embryos depleted of bmp4 and bmp7, but do express BMP2 which is not activated by high BMP signaling [1]. As was shown by Reversade et al [2], such embryos established a normal DV pattern in full or dorsal-halved embryos. Zygotic regulation of bmp4, as well as its degradation are included in the numerical solutions of speci?c systems. They do not alter the results (see section

2 below). Core vs. Full Network: The presented model includes only what appears to be the conserved core of the patterning network. This is necessary to allow for a tractable screen. In section

6 we show that an extended model, which includes additional components, establishes a similar activation pro?le. 1.3 Screen Parameters and Execution Constants: In all networks we considered [Xlr] = 10?2 ?M, αAdmp(S) = 10?3 T4 Admp S(L,t)4+T4 Admp ?M sec?1 , TAdmp = 10?4 ?M and embryo length L = 1000?M. All other nine parameters were modi?ed systematically around the following log-mean values: ? DLig = DChd = DComp = 1?m2 sec?1 ? kAdmp = kBmp = 0.1?M?1 sec?1 ? λAdmp Chd = λBmp Chd = λChd = 0.1?M?1 sec?1 ? ηChd = 101.5 ?M?m sec?1 Note that there is little data regarding the in vivo rate constants and di?u- sion coe?cients. We have assigned similar values to equivalent parameters (e.g. all di?usion coe?cients are equal) to allow for maximum exploration of rate-constants ratios. Indeed, qualitatively-di?erent behaviors of the model depend on the ratios of parameters (see also analytical analysis below, sec- tion 4). We created a nine dimensional grid centered at the mean values described. Each parameter was sampled at three points: mean value and two additional points, spanning two orders of magnitude. For the Chordin ?ux ηChd, we

4 sampled

4 points covering

3 orders of magnitude. Each point on this grid corresponds to a particular realization of the model, and we considered all possible combinations of parameters on this grid. 1.4 Screen Results: Identifying a Consistent Network For each network (point on the nine-dimensional grid), we solved numeri- cally for the steady-state activation pro?le, using a home-modi?ed version of MATLAB'

s PDE solver. This was done twice: ?rst for the full embryo, and then for the half embryo, assigning L = 500?m. A network was scored consistent if its steady-state activation pro?le was bi- ologically valid, scaled with embryo size, and was robust to perturbations in parameter values. Speci?cally, the activation pro?le had to comply with the following requirements: 1. Biologically valid: (a) DV polarity: peaked at the ventral 25% of the embryo and reached a minimum at the dorsal pole. (b) Steep: exhibited a dynamic range of at least two orders of mag- nitude. This ensured the ability to de?ne several thresholds for gene expression. (c) Continuous increase: rose by less than two orders of magnitude over the dorsal-half. This condition ?ltered out step function like pro?les in which the entire embryo except the dorsal pole had a high activation level. 2. Scaling with embryo size: (a) Biologically valid in both full and half-sized embryos. (b) Scaling: The relative positions (scaled by embryo length) cor- responding to the activation thresholds S = 10?2 ?M and S = 10?1 ?M shifted by less than 20% between full and half embryos. 3. Robust: All parameters were doubled and halved to create a nine di- mensional cube around the grid point examined. A set was considered robust if at least 50% of the vertices in this cube displayed: (a) Biologically valid gradient. (b) Scaling with embryo size.