Pattern formation

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Pattern formation in a computational model of dendrite growth.

The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.

In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. Pattern formation is controlled by genes. The role of genes in pattern formation is well seen in the anterior-posterior patterning of embryos from the model organism Drosophila melanogaster (a fruit fly).

Examples

Further information: Patterns in nature

Examples of pattern formation can be found in Biology, Chemistry, Physics and Mathematics, and can readily be simulated with Computer graphics, as described in turn below.

Biology

Further information: Regional specification and Morphogenetic field

Animal markings, segmentation of animals, phyllotaxis, neuronal activation patterns like tonotopy, and predator-prey equations' trajectories are all examples of how natural patterns are formed.

In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an embryo assume complex forms and functions. The process of embryogenesis involves coordinated cell fate control. Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a morphogen gradient, followed by short distance cell-to-cell communication through cell signaling pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues. This conceptual model was first described as the French flag model in the 1960s.

Anterior-posterior axis patterning in Drosophila

One of the best understood examples of pattern formation is the patterning along the future head to tail (antero-posterior) axis of the fruit fly Drosophila melanogaster. The development of this fly is particularly well studied, and it is representative of a major class of animals, the insects. Other multicellular organisms sometimes use similar mechanisms for axis formation, although signal transfer between the earliest cells of many developing organisms is often more important than in Drosophila.

See Drosophila embryogenesis

Growth of Colonies

Bacterial colonies show a large variety of beautiful patterns formed during colony growth. The resulting shapes depend on the growth conditions. In particular, stresses (hardness of the culture medium, lack of nutrients, etc.) enhance the complexity of the resulting patterns.

Other organisms such as slime moulds display remarkable patterns caused by the dynamics of chemical signalling.

Vegetation patterns

Main article: patterned vegetation

Tiger bush is a vegetation pattern that forms in arid conditions.

Vegetation patterns such as tiger bush and fir waves form for different reasons. Tiger bush consists of stripes of bushes on arid slopes in countries such as Niger where plant growth is limited by rainfall. Each roughly horizontal stripe of vegetation absorbs rainwater from the bare zone immediately above it. In contrast, fir waves occur in forests on mountain slopes after wind disturbance, during regeneration. When trees fall, the trees that they had sheltered become exposed and are in turn more likely to be damaged, so gaps tend to expand downwind. Meanwhile, on the windward side, young trees grow, protected by the wind shadow of the remaining tall trees.

Chemistry

Further information: reaction–diffusion system and Turing Patterns

• Belousov-Zhabotinsky reaction
• Liesegang rings

Physics

Bénard cells, Laser, cloud formations in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads. Dendrites in solidification, liquid crystals. Solitons.

Mathematics

Sphere packings and coverings. Mathematics underlies the other pattern formation mechanisms listed.

Computer graphics

Pattern resembling a Reaction-diffusion model, produced using sharpen and blur.

Some types of automata have been used to generate organic-looking textures for more realistic shading of 3d objects.

A popular photoshop plugin, KPT 6, included a filter called 'KPT reaction'. Reaction produced reaction-diffusion style patterns based on the supplied seed image.

A similar effect to the 'KPT reaction' can be achieved with convolution functions in digital image processing, with a little patience, by repeatedly sharpening and blurring an image in a graphics editor. If other filters are used, such as emboss or edge detection, different types of effects can be achieved.

Computers are often used to simulate the biological, physical or chemical processes that lead to pattern formation, and they can display the results in a realistic way. Calculations using models like Reaction-diffusion or MClone are based on the actual mathematical equations designed by the scientists to model the studied phenomena.

See also: Cellular automaton

Analysis

Further information: Gradient Pattern Analysis

The analysis of pattern-forming systems often consists of finding a Partial differential equation model of the system (the Swift-Hohenberg equation is one such model) of the form

$\frac{\partial u}{\partial t} = F(u,t)$

where F is generically a nonlinear differential operator, and postulating solutions of the form

$u(\mathbf{x},t) = \sum_j z_j(t) e^{i\mathbf{k}_j\cdot\mathbf{x}} + z_j(t)^* e^{-i\mathbf{k}_j\cdot\mathbf{x}}$

where the $z_j$ are complex amplitudes associated to different modes in the solution and the $\mathbf{k}_j$ are the wave-vectors associated to a lattice, e.g. a square or hexagonal lattice in two dimensions. There is in general no rigorous justification for this restriction to a lattice.

Symmetry considerations can now be taken into account, and evolution equations obtained for the complex amplitudes governing the solution. This reduction puts the problem into the form of a system of first-order Ordinary differential equation, which can be analysed using standard methods (see dynamical systems). The same formalism can also be used to analyse bifurcations in pattern-forming systems, for example to analyse the formation of convection rolls in a Rayleigh-Bénard experiment as the temperature is increased.

Such analysis predicts many of the quantitative features of such experiments - for example, the ODE reduction predicts hysteresis in convection experiments as patterns of rolls and hexagons compete for stability. The same hysteresis has been observed experimentally.

Bibliography

• Ball, Philip (2009). Nature's Patterns: a tapestry in three parts. 1:Shapes. 2:Flow. 3:Branches. Oxford. ISBN 978-0199604869. 

External links

• SpiralZoom.com, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
• '15-line Matlab code', A simple 15-line Matlab program to simulate 2D pattern formation for reaction-diffusion model.