Pattern formation: Difference between revisions

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{{Short description|Study of how patterns form by self-organization in nature}}

The science of '''pattern formation''' deals with the visible, ([[statistical similarity|statistically]]) orderly outcomes of [[self-organisation]] and the common principles behind similar [[pattern]]s.

{{Complex systems}}

[[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]]s in space and time. PatternThe role of genes in pattern formation is controlledan byaspect genes.of [[morphogenesis]], Thethe rolecreation of diverse [[anatomy|anatomies]] from similar genes, now being explored in patternthe formationscience isof [[evolutionary developmental biology]] or evo-devo. The mechanisms involved are bestwell understoodseen in the anterior-posterior patterning of embryos[[embryo]]s from the [[model organism]] ''[[Drosophila melanogaster]]'' (a fruit fly), one of the first organisms to have its morphogenesis studied, and in the [[eyespot (mimicry)|eyespots]] of butterflies, whose development is a variant of the standard (fruit fly) mechanism.

==Patterns in nature==

==Examples==

{{Further|Patterns in nature}}

==Biology==

Examples of pattern formation can be found in biology, physics, and other scientific fields,<ref>Ball, 2009.</ref> and can readily be simulated with computer graphics, as described in turn below.

:''See also: [[Regional specification]], [[Morphogenetic field]]''

[[Animal markings]], segmentation of animals, [[phyllotaxis]], neuronal activation patterns like [[tonotopy]], [[Lotka–Volterra equation|predator-prey equations]]' trajectories.

===Biology===

In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in

{{Further|Evolutionary developmental biology|Morphogenetic field}}

a developing tissue assume complex forms and functions by coordinated [[cell fate determination|cell fate]] control<ref>http://cistron.ca/1_pattern_formation.shtml Essay: Pattern formation in fruit fly wings</ref><ref>http://www.biologie.uni-hamburg.de/b-online/e28_1/pattern.htm Essay: 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 [[French flag model]] in the 1960s.

Biological patterns such as [[animal markings]], the segmentation of animals, and [[phyllotaxis]] are formed in different ways.<ref>Ball, 2009. ''Shapes'', pp. 231–252.</ref>

===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 is 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.

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> [[Embryogenesis]], such as [[Drosophila embryogenesis|of the fruit fly ''Drosophila'']], involves coordinated [[cell fate determination|control of cell fates]].<ref name=Lai>{{cite journal |author=Eric C. Lai |title=Notch signaling: control of cell communication and cell fate |doi=10.1242/dev.01074 |pmid=14973298 |volume=131 |issue=5 |date=March 2004 |pages=965–73 |journal=Development|doi-access= |s2cid=6930563 }}</ref><ref name=Tyler>{{cite journal |title=Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition |author=Melinda J. Tyler |author2=David A. Cameron |journal=Vision Research |volume=47 |issue=4 |pages=501–511 |year=2007 |doi=10.1016/j.visres.2006.08.025 |pmid=17034830|s2cid=15998615|doi-access=free }}</ref><ref name=Meinhard>{{cite web|title=Biological pattern formation: How cell[s] talk with each other to achieve reproducible pattern formation |author=Hans Meinhard |agency= Max-Planck-Institut für Entwicklungsbiologie, Tübingen, Germany |url=http://www1.biologie.uni-hamburg.de/b-online/e28_1/pattern.htm |date=2001-10-26 }}</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|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 of positional information cues. This conceptual model was first described as the [[French flag model]] in the 1960s.<ref>{{cite journal |doi=10.1016/S0022-5193(69)80016-0 |author=Wolpert L |title=Positional information and the spatial pattern of cellular differentiation |journal=J. Theor. Biol. |volume=25 |issue=1 |pages=1–47 |date=October 1969 |pmid=4390734 |bibcode=1969JThBi..25....1W }}</ref><ref>{{cite book |author=Wolpert, Lewis |title=Principles of development |publisher=Oxford University Press |___location=Oxford [Oxfordshire] |year=2007 |isbn=978-0-19-927536-6 |edition=3rd |display-authors=etal}}</ref> More generally, the morphology of organisms is patterned by the mechanisms of [[evolutionary developmental biology]], such as [[heterochrony|changing the timing]] and positioning of specific developmental events in the embryo.<ref>{{cite journal |last1=Hall |first1=B. K. |title=Evo-Devo: evolutionary developmental mechanisms |journal=International Journal of Developmental Biology |date=2003 |volume=47 |issue=7–8 |pages=491–495 |pmid=14756324}}</ref>

:''See [[Drosophila embryogenesis]]''

Possible mechanisms of pattern formation in biological systems include the classical [[reaction–diffusion]] model proposed by [[Alan Turing]]<ref>S. Kondo, T. Miura, "Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation", Science 24 Sep 2010: Vol. 329, Issue 5999, pp. 1616-1620 DOI: 10.1126/science.1179047</ref> and the more-recently-found [[elastic instability]] mechanism which is thought to be responsible for the fold patterns on the [[cerebral cortex]] of higher animals, among other things.<ref name="Mercker">{{cite journal |last1=Mercker |first1=M |last2=Brinkmann |first2=F |last3=Marciniak-Czochra |first3=A |last4=Richter |first4=T |title=Beyond Turing: mechanochemical pattern formation in biological tissues. |journal=Biology Direct |date=4 May 2016 |volume=11 |page=22 |doi=10.1186/s13062-016-0124-7 |pmid=27145826|pmc=4857296 |doi-access=free }}</ref><ref>Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632</ref>

===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.

====Growth of colonies====

:''See [[Bacterial patterns]]''

Bacterial colonies show a [[bacterial patterns|large variety of 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 signaling.<ref>Ball, 2009. ''Shapes'', pp. 149–151.</ref> Cellular embodiment (elongation and adhesion) can also have an impact on the developing patterns.<ref>{{Cite journal|last1=Duran-Nebreda|first1=Salva|last2=Pla|first2=Jordi|last3=Vidiella|first3=Blai|last4=Piñero|first4=Jordi|last5=Conde-Pueyo|first5=Nuria|last6=Solé|first6=Ricard|date=2021-01-15|title=Synthetic Lateral Inhibition in Periodic Pattern Forming Microbial Colonies|journal=ACS Synthetic Biology|volume=10|issue=2|language=en|pages=277–285|doi=10.1021/acssynbio.0c00318|pmid=33449631|pmc=8486170|issn=2161-5063}}</ref>

====Vegetation patterns====

==Chemistry==

{{Main|patterned vegetation}}

see [[reaction-diffusion]] systems and [[Alan_Turing#Pattern_formation_and_mathematical_biology|Turing Patterns]]

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

*[[Belousov-Zhabotinsky reaction]]

*[[Liesegang rings]]

[[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=978-1-4612-6559-7}}</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 | access-date=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/> In flat terrains, additional pattern morphologies appear besides stripes – hexagonal gap patterns and hexagonal spot patterns. Pattern formation in this case is driven by positive feedback loops between local vegetation growth and water transport towards the growth ___location.<ref>{{cite journal |author=Meron, E |title=Vegetation pattern formation: the mechanisms behind the forms |journal=Physics Today |volume=72 |issue=11 | pages=30–36 |year=2019 |doi=10.1063/PT.3.4340|bibcode=2019PhT....72k..30M |s2cid=209478350 }}</ref><ref>{{cite journal |author=Meron, E |title=From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study|journal=Annual Review of Condensed Matter Physics |volume=9 | pages=79–103 |year=2018 |doi=10.1146/annurev-conmatphys-033117-053959|bibcode=2018ARCMP...9...79M|doi-access=free}}</ref>

==Mathematics==

[[Sphere packing]]s and coverings.

==Physics=Chemistry===

{{expand section|date=March 2013}}

[[Bénard cells]], [[Laser]], [[cloud formation]]s in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads. [[dendrite (crystal)|Dendrites]] in [[solidification]], [[liquid crystals]], the structure of [[foams]] <ref> [http://engweb.swan.ac.uk/~gabbriellir/javaview/p42a_comparison.html Geometrical models for foams]</ref>.

{{Further|reaction–diffusion system|Turing patterns}}Pattern formation has been well-studied in chemistry and chemical engineering, including both temperature and concentration patterns.<ref name=":0">{{Cite journal|last1=Gupta|first1=Ankur|last2=Chakraborty|first2=Saikat|date=January 2009|title=Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors|journal=Chemical Engineering Journal|volume=145|issue=3|pages=399–411|doi=10.1016/j.cej.2008.08.025|issn=1385-8947}}</ref> The [[Brusselator]] model developed by [[Ilya Prigogine]] and collaborators is one such example that exhibits [[Turing instability]].<ref>{{Citation|last1=Prigogine|first1=I.|title=Self-Organisation in Nonequilibrium Systems: Towards a Dynamics of Complexity|date=1985|work=Bifurcation Analysis: Principles, Applications and Synthesis|pages=3–12|editor-last=Hazewinkel|editor-first=M.|publisher=Springer Netherlands|doi=10.1007/978-94-009-6239-2_1|isbn=978-94-009-6239-2|last2=Nicolis|first2=G.|editor2-last=Jurkovich|editor2-first=R.|editor3-last=Paelinck|editor3-first=J. H. P.}}</ref> Pattern formation in chemical systems often involves [[Chemical oscillator|oscillatory chemical kinetics]] or [[Autocatalysis|autocatalytic reactions]]<ref name=":1">{{Cite journal|last1=Gupta|first1=Ankur|last2=Chakraborty|first2=Saikat|date=2008-01-19|title=Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions|journal=Chemical Product and Process Modeling|volume=3|issue=2|doi=10.2202/1934-2659.1135|s2cid=95837792|issn=1934-2659}}</ref> such as the [[Belousov–Zhabotinsky reaction|Belousov–Zhabotinsky]] or [[Briggs–Rauscher reaction|Briggs–Rauscher reactions]]. In industrial applications such as chemical reactors, pattern formation can lead to temperature hot spots, which can reduce the yield or create hazardous safety problems such as a [[thermal runaway]].<ref>{{Cite journal|last1=Marwaha|first1=Bharat|last2=Sundarram|first2=Sandhya|last3=Luss|first3=Dan|date=September 2004|title=Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors†|journal=The Journal of Physical Chemistry B|volume=108|issue=38|pages=14470–14476|doi=10.1021/jp049803p|issn=1520-6106}}</ref><ref name=":0" /> The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying [[Reaction–diffusion system|reaction-diffusion system]].<ref name=":0" /><ref name=":1" />

Similarly as in chemical systems, patterns can develop in a weakly ionized plasma of a positive column of a glow discharge. In such cases, creation and annihilation of charged particles due to collisions of atoms corresponds to reactions in chemical systems. Corresponding processes are essentially non-linear and lead in a discharge tube to formation of striations with regular or random character.<ref>{{cite journal|title=Nonlinear Properties of High Amplitude Ionization Waves|author1=Igor Grabec |journal=The Physics of Fluids|volume=17|year=1974|issue=10 |pages=1834–1840|doi=10.1063/1.1694626 |bibcode=1974PhFl...17.1834G |doi-access=free}}</ref><ref>{{cite journal|title=Desynchronization of striations in the development of ionization turbulence|author1=Igor Grabec |author2=Simon Mandelj |journal=Physics Letters A|volume=287|year=2001|issue=1–2 |pages=105–110|doi=10.1016/S0375-9601(01)00406-6 |bibcode=2001PhLA..287..105G }}</ref>

Other chemical patterns include [[Liesegang rings]].

==Computer graphics==

[[Image:Homebrew reaction diffusion example 512iter.jpg|thumb|right|320px|Reaction diffusion-like pattern produced using sharpen and blur.]] Some types of [[automata]] have been used to generate organic-looking [[Texture (computer graphics)|textures]] for more realistic [[Shaders|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>http://www.cs.cmu.edu/~aw/pdf/texture.pdf</ref>.

===Physics===

A popular photoshop plugin, [[Kai's Power Tools|KPT 6]], included a filter called 'KPT reaction'. Reaction produced [[Reaction diffusion|reaction-diffusion]] style patterns based on the supplied seed image.

{{expand section|date=March 2013}}

When a planar body of fluid under the influence of gravity is heated from below, [[Rayleigh–Bénard convection|Rayleigh-Bénard convection]] can form organized cells in hexagons or other shapes. These patterns form on the [[Granule (solar physics)|surface of the Sun]] and in the [[Earth's mantle|mantle of the Earth]] as well as during more pedestrian processes. The interaction between rotation, gravity, and convection can cause planetary atmospheres to form patterns, as is seen in [[Saturn's hexagon]] and the [[Great Red Spot]] and stripes of [[Jupiter]]. The same processes cause ordered [[List of cloud types|cloud formations]] on Earth, such as [[Cirrus radiatus|stripes]] and [[Altostratus undulatus cloud|rolls]].

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 [[Embossing#Embossing in image processing|emboss]] or [[edge detection]], different types of effects can be achieved.

In the 1980s, [[Lugiato–Lefever equation|Lugiato and Lefever]] developed a model of light propagation in an optical cavity that results in pattern formation by the exploitation of nonlinear effects.

In addition, computers are often used to [[Computer simulation|simulate]] the biological, physical or chemical processes -described above- that lead to pattern formation, and they are then able to display the results in a realistic way (applications of virtual reality for Science). Calculations are based on the actual mathematical equations designed by the scientists to model the studied phenomena.

[[Precipitation (chemistry)|Precipitating]] and [[Freezing|solidifying]] materials can crystallize into intricate patterns, such as those seen in [[snowflake]]s and [[Dendrite (crystal)|dendritic crystals]].

==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

===Mathematics===

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

{{expand section|date=March 2013}}

[[Sphere packing]]s and coverings. Mathematics underlies the other pattern formation mechanisms listed.

{{Further|Gradient pattern analysis}}

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

===Computer graphics===

:<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>

[[File:Homebrew reaction diffusion example 512iter.jpg|thumb|right|Pattern resembling a [[reaction–diffusion]] model, produced using sharpen and blur]]

{{Further|Cellular automaton}}

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.

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>Greg Turk, [http://www.cc.gatech.edu/~turk/reaction_diffusion/reaction_diffusion.html Reaction–Diffusion]</ref><ref>{{cite book|author1=Andrew Witkin |author2=Michael Kassy |title=Proceedings of the 18th annual conference on Computer graphics and interactive techniques - SIGGRAPH '91 |chapter=Reaction-diffusion textures |chapter-url=https://www.cs.cmu.edu/~aw/pdf/texture.pdf|doi=10.1145/122718.122750|year=1991|pages=299–308|isbn=0-89791-436-8 |s2cid=207162368 }}</ref>

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 [[ODE]]s, 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.

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.

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.

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.

==See also==

*[[Morphogenesis]]

*[[Reaction-diffusion]]

*[[Regional specification]]

*[[embryogenesis]]

*[[embryo]]

*[[model organism]]

*[[Drosophila embryogenesis]]

*[[Tagmosis]]

*[[Projective Geometry]]

*[[Gradient Pattern Analysis]]

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.

{{Genarch}}

==References==

{{Reflist|30em}}

==Bibliography==

* {{cite book|author-link=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=978-0-19-960486-9}}

==External links==

* [https://web.archive.org/web/20190514060955/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. (archived 14 May 2019)

* [httphttps://wwwbooks.texrdgoogle.com/books?id=T_i0o_55MNwC&q=pattern+formation%2C 'TexRD15-line Matlab code'], aA freesimple software15-line Matlab program to experimentsimulate 2D pattern formation with different kinds offor reaction-diffusion models (simulations in 2 dimensions; basic interface with text parameters)model.

*[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.

{{Patterns in nature}}

==References==

{{reflistGenarch}}

{{Complex systems topics}}

{{Authority control}}

{{DEFAULTSORT:Pattern Formation}}

[[Category:Pattern formation| ]]

[[Category:Developmental biology]]

[[Category:Articles containing video clips]]

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