PDF《Scaling of the BMP Morphogen》

Scaling of the BMP Morphogen Activation Gradient in Xenopus embryos Supplementary Information Danny Ben-Zvi*, Ben-Zion Shilo*, Abraham Fainsod? and Naama Barkai*§ Departments of Molecular Genetics* and Physics of Complex Systems§ Weizmann Institute of Science Rehovot, Israel Department of Cellular Biochemistry and Human Genetics?, Faculty of Medicine, Hebrew University, Jerusalem, Israel Contents

1 Numerical Screen

3 1.

1 Screen Equations

3 1.2 Notes about the Assumptions of the Screen

4 1.3 Screen Parameters and Execution

4 1.4 Screen Results: Identifying a Consistent Network

5 1.5 Screen Results: Parameter Ratios

6 1.6 Shuttling Coe?cient

6 2 Numerical Solutions of Speci?c Cases

7 2.1 General parameters

7 2.2 Speci?c Parameters

8 3 Steady-State of the Shuttling Model: Analytical Treatment

12 3.1 Steady-State of the shuttling Model: Analytical Solution . . .

12 3.2 Necessary Conditions for Scaling and Robustness

15 3.3 Scaling Limits of the System

15 3.4 Analysis of Shuttling with a Single Ligand

16 3.5 Di?erential Signaling by Admp and Bmp

17 4 Analytic Treatment of the General Model

18 4.1 An E?ective Transport Measure

18 4.2 Analytic Treatment of the General Model

18 4.3 Low Transport: TLig eff >

1 20

5 Mutual Regulation of chordin, admp and bmp4 Expression

24 6 Dimensionless Equations

27 7 An Extended Model of the Dorso-Ventral Patterning Network

28 8 Geometry of the Model

34 1 Figure S-1: The BMP patterning network. (a-d): Patterning network in Drosophila (a,c) and Xenopus (b,d): The expression domains of the components comprising the BMP patterning network and the resulting BMP activation gra- dient are shown in a-b. The interactions between components are shown in c-d. (e): A schematic representation of the inhibition-based mechanism. Patterning is governed by an inhibition gradient established over a ?eld of an activator. The activator may be uniformly distributed. Note that positive feedback of BMP ex- pression may result in graded BMP expression as well, but this is not required for the generation of the gradient itself and does not constitute a main aspect of the patterning mechanism. d, l and v stand for Dorsal, Lateral and Ventral regions of the embryo. Inhibitor in gray, activator in black and the total activator (free and in complex) in dashed black. (f): A schematic representation of the shuttling-based mechanism. Patterning relies on physical translocation of the BMP ligands to the ventral region, leading to an activation gradient that arises primarily from the graded distribution of the BMP ligands themselves. E?ective shuttling requires the binding of ligand to the inhibitor to facilitate its di?usion, and the release of the ligand by cleavage of the complex. Notations as in 1e.

2 1 Numerical Screen In this section we describe the numerical screen. We de?ne the reaction- di?usion equations used, the assumptions undertaken, explain how the screen was executed and discuss the analysis of its results. 1.1 Screen Equations As described in the main text, our core model consists of two BMP ligands: [Bmp] and [Admp], and a single BMP inhibitor, Chordin ([Chd]). Chordin binds both Bmp and Admp, with the respective on-rates kBmp and kAdmp creating the complexes [ChdBmp] and [ChdAdmp] respectively. Chordin is cleaved by the protease Xlr with the cleavage rates λChd, λBmp Chd and λAdmp Chd , corresponding to its free or ligand-bound form. All components are allowed to di?use with the respective di?usion coe?cients DLig, DChd, DComp. De- noting the concentration of a protein M by [M], the dynamics is given by the following set of reaction-di?usion equations: ?[Chd] ?t = DChd?2 [Chd] ? [Chd] kAdmp[Admp] + kBmp[Bmp] + λChd[Xlr] (1a) ?[Admp] ?t = DLig?2 [Admp] ? kAdmp[Chd][Admp] + λAdmp Chd [Xlr][ChdAdmp] (1b) ?[Bmp] ?t = DLig?2 [Bmp] ? kBmp[Chd][Bmp] + λBmp Chd [Xlr][ChdBmp] (1c) ?[ChdAdmp] ?t = DComp?2 [ChdAdmp] + kAdmp[Chd][Admp] ? λAdmp Chd [Xlr][ChdAdmp] (1d) ?[ChdBmp] ?t = DComp?2 [ChdBmp] + kBmp[Chd][Bmp] ? λBmp Chd [Xlr][ChdBmp] (1e) The biologically relevant output was considered to be the signaling level S(x): S(x, t) = [Admp](x, t, ) + [Bmp](x, t) (2) Boundary Conditions: All ?uxes vanish at x =