The Interaction of Surface Geometry with Morphogens

✍ Scribed by FREDERICK W. CUMMINGS

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 286 KB
• Volume: 212
• Category: Article
• ISSN: 0022-5193
• DOI: 10.1006/jtbi.2001.2377

✦ Synopsis

Expressions are given for the Gauss and Mean curvatures of a surface of thickness h. The two curvatures, (K and H), which are given at each point of the middle surface, are adequate to describe the surface. The sheet thickness varies with position in the middle surface bisecting the apical and basal surfaces. The de"nitions of K and H are in terms of radii of curvature, but such radii are not appropriate variables for determining how morphogens in the surface may couple to the geometry. More suitable expressions are developed here. Two important geometrical constraints must be satis"ed, namely the famous Gauss}Bonnet theorem, and an inequality stemming from the de"nition of the two curvatures. It is argued that these constraints are of great usefulness in determining the form of the coupling of morphogens to the geometry. In particular, when two key morphogens su$ce to determine surface geometry, explicit expressions are suggested to determine both Gauss (K) and Mean (H) curvatures as functions of invariant morphogen densities.