This is a EPSRC funded project that is concerned with the modelling of morphogenesis and morphological evolution of simple organisms such as hydroids and jellyfish. The objective is to explore the morphological themes shared by these organisms based on the current understanding on cell signalling and pattern formation and generate autonomous morphological system that couples spatial patterning and shape change.

The model is suggested by Cummings. He focuses on A the morphological evolution of an epithelial cell layer that is arisen from a feedback mechanism. The feedback mechanism essentially couples a diffusible chemical, called morphogen, that forms a pattern on the epithelium according to its local geometry, specified by the metric tensor. This pattern then alters the geometry through specifying the local curvatures. This leads to a new geometry, i.e. the epithelium deforms. This new geometry in turn modifies the morphogen pattern and the process continues. A brief description of the model is presented in postscript format (1Mb).

Axisymmetric Simulations

A number of morphogenetical sequences of axisymmetric geometries have been simulated that mimick different stages of biological evolution, such as gastrulation and evagination. They are produced by specifying the curvatures of the desired geometries and the evolutions begin from a sphere. The simulations are presented in animations. Each contains the evolution of a 3D axisymmetric geometry, the cross-section of the geometry and the corresponding Gaussian curvature, metric and morphogen solutions.

Gastrulation 3D (3.4Mb qt). Cross-section (1.5Mb qt). Solutions (2.5Mb qt).

Evagination 3D (2.3Mb qt). Cross-section (1.4Mb qt). Solutions (2.2Mb qt).

"Cup" formation 3D (12.2Mb qt). Cross-section (1.7Mb qt). Solutions (2.5Mb qt).

Reproduction 3D (6.6Mb qt). Cross-section (1.1Mb qt). Solutions (2.1Mb qt). B

Symmetry Breaking

In order to produce non-axisymmetric 'true' 3D geometries such as those containing one or more tentacles, solutions to both the metric tensor and the Euclidean geometry are non-trivial. Symmetry breaking involves the reconstruction of the coordinate intrinsic to the creature's surface geometry, i.e. the body-fitted curvilinear coordinate and the mapping of this coordinate to conventional Euclidean space such as the Cartesian coordinate. The intrinsic coordinate is characterised by its orthogonal, isothermal properties and is constructed using mesh generation technique . A brief introduction to the one dimensional models and the extension to non-axisymmetric 'true' 3D models are shown in the following pdf file (0.5Mb). The artificial growth of three cnidarians: polyp, medusa and hydranth are simulated. The animated sequences that are shown in quicktime and mpeg formats. For all cases, the geometries initiate from a unit sphere to their corresponding final shapes.

Polyp quicktime (0.28Mb qt), mpeg (0.75Mb mpg).

Medusa quicktime (0.28Mb qt), mpeg (0.80Mb mpg).

Hydranth quicktime (0.25Mb qt), mpeg (0.80Mb mpg).

The incorporation of morphogen to the model and the investigation to the feedback from morphogen to Gaussian curvature are currently underway.

Presentations

A working group meeting was held in the Santa Fe Institute in 29-31 July 2000 at which the computational simulations of axisymmetric biological form (see below) was presented with a description of the model. The report of the meeting and related preprints may be found in the url. The computational simulations were also presented in the form of posters in the following two conferences:

1. International Conference on Mathematics in Biology, Annual Meeting of the Society for Mathematical Biology, Salt Lake City, Utah, US 3-5 August 2000 (Poster: slc.eps (1.0Mb) ).

2. British Societies for Cell and Developmental Biology, Joint Spring Meeting: Cell and Tissue Morphogenesis Brighton, UK 3-6 April 2001 (Poster: ssx.eps (0.8Mb) ).

Seminars on the work were presented at

3. Mathematical Biology ,meeting at the University of Leeds 14/3/01

4. Scientific Computing Institute University of Utah April 2001

5. Santa Fe Institute Lunchtime Seminar Series 30th July 2001 A PDF file containing the material presented is: sfitalk.pdf

Publications

The following publications have been submitted or are in draft form being revised for submission

1. A Computational Model for Organism Growth based on Surface Mesh Generation. C.H. Leung and Martin Berzins. (Submitted to Journal of Computational Physics February 2002)

B PDF file containing this: submitted version.

2. An Investigation of the Morphological Evolution and the Morphospace based on Cummings Model (working draft only) C.H. Leung B.C. Goodwin D.C.Hogg and M. Berzins.

PDF file containing this: draft paper .

3. From Genetic Networks to Morphogenesis and Back Again N.Monk and B. Goodwin Draft in preparation.

2. An Implementation of the Cummings Method fro Embryology involving Feeback between Morphogenetic Field and Epithelial Shape (working draft only) C.H. Leung and M. Berzins.

PDF file containing this: draft paper . Last Modified: 20th July 2002