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[[File:Self-organizing-Mechanism-for-Development-of-Space-filling-Neuronal-Dendrites-pcbi.0030212.sv003.ogv|thumb|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 fate determination|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|Patterns in nature}}
Examples of pattern formation can be found in Biology, Chemistry, Physics and Mathematics,<ref>Ball, 2009.</ref> and can readily be simulated with Computer graphics, as described in turn below.

=== Biology ===
{{further|Regional specification|Morphogenetic field}}

[[Animal markings]], segmentation of animals, [[phyllotaxis]],<ref>Ball, 2009. Shapes, pp. 231–252.</ref> neuronal activation patterns like [[tonotopy]], and [[Lotka–Volterra equation|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.<ref>Ball, 2009. Shapes, pp. 261–290.</ref> The process of [[embryogenesis]] involves coordinated [[cell fate determination|cell fate]] control.<ref name=Lai>{{cite journal|author=Eric C. Lai|title=Notch signaling: control of cell communication and cell fate|url=http://dev.biologists.org/content/131/5/965.full.pdf|doi=10.1242/dev.01074|pmid=14973298|volume=131|issue=5|date=March 2004|pages=965–73}}</ref><ref name=Tyler>{{cite journal|title=Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition|authors=Melinda J. Tyler, David A. Cameron|journal=Vision Research|volume=47|issue=4|pages=501–511|year=2007|doi=10.1016/j.visres.2006.08.025}}</ref><ref name=Meinhard>{{cite web|title=Biological pattern formation: How cell talk with each other to achieve reproducible pattern formation|author=Hans Meinhard|url=http://www.biologie.uni-hamburg.de/b-online/e28_1/pattern.htm|date=2001-10-26|title=Biological pattern formation}}</ref> 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 [[insect]]s. 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 [[Bacterial patterns|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.<ref>Ball, 2009. Branches, pp. 52–59.</ref>

Other organisms such as [[slime mould]]s display remarkable patterns caused by the dynamics of chemical signalling.<ref>Ball, 2009. Shapes, pp. 149–151.</ref>

==== Vegetation patterns ====
{{Main|patterned vegetation}}
[[File:Tiger Bush Niger Corona 1965-12-31.jpg|thumb|[[Tiger bush]] is a [[patterned vegetation|vegetation pattern]] that forms in arid conditions.]]

[[patterned vegetation|Vegetation patterns]] such as [[tiger bush]]<ref name=TigerBush>{{cite book | title=Banded vegetation patterning in arid and semiarid environments | publisher=Springer-Verlag | author=Tongway, D.J., Valentin, C. & Seghieri, J. | year=2001 | location=New York|isbn=978-1461265597}}</ref> and [[fir wave]]s<ref name=FirWave>{{cite web | url=http://tiee.esa.org/vol/v1/figure_sets/disturb/disturb_back4.html | title=Fir Waves: Regeneration in New England Conifer Forests | publisher=TIEE | date=22 February 2004 | accessdate=26 May 2012 | author=D'Avanzo, C.}}</ref> 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.<ref name=TigerBush/> 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.<ref name=FirWave/>

=== Chemistry ===
{{expand section|date=March 2013}}
{{further|reaction–diffusion system|Turing Patterns}}

* [[Belousov–Zhabotinsky reaction|Belousov-Zhabotinsky reaction]]
* [[Liesegang rings]]

=== Physics ===
{{expand section|date=March 2013}}
[[Bénard cell]]s, [[Laser]], [[cloud|cloud formation]]s in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads. [[dendrite (crystal)|Dendrites]] in [[freezing|solidification]], [[liquid crystal]]s. [[Soliton]]s.

=== Mathematics ===
{{expand section|date=March 2013}}
[[Sphere packing]]s and coverings. Mathematics underlies the other pattern formation mechanisms listed.

=== Computer graphics ===

[[File:Homebrew reaction diffusion example 512iter.jpg|thumb|right|Pattern resembling a [[Reaction-diffusion]] model, produced using sharpen and blur.]] Some types of [[Finite state machine|automata]] have been used to generate organic-looking [[Texture (computer graphics)|textures]] for more realistic [[Shader|shading]] of [[3D modeling|3d objects]].<ref>[http://www.cc.gatech.edu/~turk/reaction_diffusion/reaction_diffusion.html Reaction-Diffusion<!-- Bot generated title -->]</ref><ref>{{cite web|title=Reaction-Diffusion Textures|author=Andrew Witkin,Michael Kassy|url=http://www.cs.cmu.edu/~aw/pdf/texture.pdf|doi=10.1145/122718.122750|journal=Proceedings of the 18th annual conference on Computer graphics and interactive techniques|year=1991|pages=299–308}}</ref>

A popular photoshop plugin, [[Kai's Power Tools|KPT 6]], included a filter called 'KPT reaction'. Reaction produced [[Reaction–diffusion system|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 [[Unsharp masking|sharpening]] and [[Box blur|blurring]] an image in a graphics editor. If other filters are used, such as [[Image embossing|emboss]] or [[edge detection]], different types of effects can be achieved.

Computers are often used to [[Computer simulation|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 ==
<!--Is this relevant? It's entirely uncited...-->
{{further|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|Swift-Hohenberg equation]] is one such model) of the form

: <math>\frac{\partial u}{\partial t} = F(u,t)</math>

where ''F'' is generically a [[nonlinear system|nonlinear]] [[differential operator]], and postulating solutions of the form

: <math> 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}}</math>

where the <math>z_j</math> are complex amplitudes associated to different modes in the solution and the <math>\mathbf{k}_j</math> are the wave-vectors associated to a [[lattice (group)|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 system]]s). The same formalism can also be used to analyse [[bifurcation theory|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.{{Citation needed|date=March 2013}}

== References ==

{{reflist|33em}}

== Bibliography ==

* {{cite book|authorlink=Philip Ball|author=Ball, Philip|title=Nature's Patterns: a tapestry in three parts. 1:Shapes. 2:Flow. 3:Branches|publisher=Oxford|year=2009|isbn=978-0199604869}}

== External links ==
* [http://spiralzoom.com/Science/patternformation/patternformation.html ''SpiralZoom.com''], an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
* [http://books.google.co.uk/books?id=T_i0o_55MNwC&printsec=frontcover&dq=introduction+to+computational+mathematics#v=onepage&q=pattern%20formation&f=false, '15-line Matlab code'], A simple 15-line Matlab program to simulate 2D pattern formation for reaction-diffusion model.

{{Genarch}}

{{DEFAULTSORT:Pattern Formation}}
[[Category:Developmental biology]]