Everything about Pattern Formation totally explained
The science of
pattern formation deals with the visible, (
statistically)
orderly outcomes of
self-organisation and the common principles behind similar
patterns.
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 best understood in the anterior-posterior patterning of embryos from the model organism
Drosophila melanogaster (fruit fly).
Examples
Biology
Animal markings, segmentation of animals,
phyllotaxis, neuronal activation patterns like
tonotopy,
Predator-prey equations' trajectories.
In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in
a developing tissue assume complex forms and functions by 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.
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
Drosophila is particularly well studied, and it's representative of a major class of animals, the insects or
insecta. Other multicellular organisms sometimes use similar mechanisms for axis formation, although the relative importance of signal transfer between the earliest cells of many developing organisms is greater than in the example described here.
See
Drosophila embryogenesis
Growth of Bacterial Colonies
Bacterial colonies show a large variety of beautiful patterns formed during colony growth. Experiments show that the resulting shapes depend on the growth conditions. In particular stresses (hardness of the culture medium, lack of nutrients, etc) seem to enhance the complexity of the resulting patterns.
See
Bacterial patterns
Chemistry
see
reaction-diffusion systems and
Turing Patterns
Physics
Bénard cells,
Laser,
cloud formations in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads.
Computer graphics
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 'kpt reaction' can be achieved, with a little patience, by repeatedly sharpening and then blurring an image in many graphics applications. If other filters are used, such as
emboss or
edge detection, different types of effects can be achieved.
Analysis
The analysis of pattern-forming systems often consists of finding a
PDE model of the system (the
Swift-Hohenberg equation is one such model) of the form
» are the wave-vectors associated to a
lattice, for example 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
ODEs, 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.
Further Information
Get more info on 'Pattern Formation'.
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