Technical Papers and Presentations

In his seminal work, “A Chemical Basis for Morphogenesis,” Alan Turing proposed a mechanism for pattern formation in nature. Turing patterns occur in a broad range of natural systems, from desert vegetation to animal markings. Turing mechanisms have also been implicated in processes including cellular differentiation and limb development. The chlorine dioxide-iodine-malonic acid (CDIMA) reaction produces Turing patterns in a chemical system. Its photosensitivity makes it ideal to study growing Turing patterns, as the patterns are suppressed via bright light and allowed to develop in shaded regions. The shaded region is increased in size over time at varying growth rates. Previous work has demonstrated that varying the growth rate of the pattern selects for different morphologies. In a previous study, we used COMSOL Multiphysics^® simulation software to integrate the Lengyel-Epstein reaction-diffusion model in conjunction with CDIMA experiments to examine an experimentally accessible but idealized model of growth.

Recently, we have extended our use of COMSOL^® to replicate biological growth modes more closely and efficiently. The Lengyel-Epstein model is implemented with the Coefficient Form PDE physics, and the domain size is increased with a Parametric Sweep, using the Time-Dependent solver at each step. Several novel growth modes are examined. First, the mesh for the domain can be configured to strictly add elements to the outside as it expands. This is similar to our previous work, but uses one domain instead of two (illuminated and dark) domains. Second, the mesh can be configured to allow a fixed number of elements, which each expand as the system grows, akin to cellular elongation common in plant growth. Third, by illuminating the domain with intensity varying with an s-shaped function, the pattern can be suppressed more on the exterior than the interior, causing the pattern to grow from the inside. Finally, the mesh can be configured to expand elementwise until a critical size, then split into different elements. This reflects cellular division and should provide insight into the development of patterns in biological growth. COMSOL® makes these alternative growth methods accessible and allows us to closely model what is observed in nature. This study will test previously observed trends for more realistic models of growth and may reveal new morphological trends.

As this study consists of four growth modes each with several parameters, often 200+ parametric combinations for each growth type are required. The COMSOL® Application Builder is used to produce easy-to-use applications to quickly simulate the various types of growth in a wide range of parameters.

COMSOL has proved to be an invaluable tool for the study of Turing pattern growth due to the ability to finely manipulate the mesh and the fast computing speed, made even more efficient using the Application Builder. Recent simulations have revealed some regions in parameter space select for novel types of growth, such as parallel stripes across the domain, which were not observed in our previous experiments. These studies are ongoing and will provide us with a more complete understanding of how Turing patterns develop in biological systems.