Reaction Diffusion Systems

Reaction diffusion systems are evolving patterns which result from various types of reactions and product diffusion within a dynamic medium. Biological reaction diffusion systems within the submembrane cytoplasm have been suggested by Conrad and Liberman (1982) as a mechanism of biological information representation. In their model, reaction diffusion patterns of the energy rich nucleotide, cyclic AMP, which are regulated by the membrane are the texture of cytoplasmic information. Propagation and interaction of chemical, nonlinear waves lead to pattern formation in a number of chemical and biological media (Winfree and Strogatz, 1984). In the well studied "Belousov-Zhabotinsky reaction," spiral chemical reaction waves propagate at uniform speed and interact with other waves to produce complex patterns. Waves radiate from spiral centers at a rate of a few millimeters per minute as the spirals turn in about one minute. Several chemical reactions with suitable diffusion rates and visible color changes of reaction products show these characteristic patterns, as do cultured amoeba cells responding to pulses of cyclic AMP (Figure 1.9). Similar phenomena have also been reported in retinal and cortical nerve nets and in heart muscle. Smaller scale reaction diffusion patterns are accordingly faster.

Figure 1.9: Self organizing spatial and temporal patterns described by the chemical reaction-diffusion system known as the Belousov-Zhabotinsky reaction and emulated by biological systems. With permission from Arthur Winfree.

Winfree and Strogatz (1984) have studied the 3 dimensional behavior of reaction diffusion systems. They find that reaction diffusion waves commonly appear as involute spirals or scrolls radiating from tiny rotating activity patterns called "organizing centers." The scrolls emanate from their central organizing axis which typically forms a closed ring or toroidal vortex. The origin of the waves is defined as a phase singularity whose immediate neighborhood is a rotating pattern of chemical activities, the pivot of the rotating spiral wave from which it radiates. The ostensibly flat spiral is actually a cross section of a three dimensional wave shaped like a scroll which emerges from a filament of singularity in 3 dimensions (Figure 1.10).

Figure 1.10: Three dimensional computer simulation of a reaction-diffusion system. A filamentous organizing center emanates "scroll ring" patterns. With permission from Arthur Winfree.

Cytoplasmic microtubules and centrioles are organizing centers which could behave like the singularities described by Winfree and Strogatz. Dynamic activities of the cytoskeleton may release diffusing waves of calcium ions which can alter the nature of surrounding cytoplasm by sol-gel transformations (Chapter 5). Coding by microtubule associated proteins (MAPs) and other factors could result in reaction-diffusion patterns specific to the dynamic state of the organizing center. Such patterns could suffice as short term memory in cells ranging from simple protozoa to human brain neurons. Another type of interactive, 3 dimensional pattern with interesting properties is the hologram.

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