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{{short description|Set of genes all related by point mutations that have equivalent function or fitness}}
{{distinguish|Neural network|Network-neutral data center}}
A '''neutral network''' is a set of [[gene]]s all related by [[point mutation]]s that have equivalent function or [[fitness (biology)|fitness]].<ref name="van Nimwegen 9716–20">{{cite journal|last=van Nimwegen|first=E|author2=Crutchfield, JP |author3=Huynen, M |title=Neutral evolution of mutational robustness.|journal=Proceedings of the National Academy of Sciences of the United States of America|date=Aug 17, 1999|volume=96|issue=17|pages=9716–20|pmid=10449760|doi=10.1073/pnas.96.17.9716|pmc=22276|arxiv=adap-org/9903006|bibcode=1999PNAS...96.9716V|doi-access=free}}</ref> Each node represents a gene sequence and each line represents the mutation connecting two sequences. Neutral networks can be thought of as high, flat plateaus in a [[fitness landscape]]. During [[Neutral theory of molecular evolution|neutral evolution]], genes can randomly move through neutral networks and traverse regions of [[sequence space (evolution)|sequence space]] which may have consequences for [[robustness (evolution)|robustness]] and [[evolvability]].

==Genetic and molecular causes==
{{see also|Neutral mutation|Robustness (evolution)}}
Neutral networks exist in [[fitness landscape]]s since proteins are [[robustness (evolution)|robust]] to mutations. This leads to extended networks of genes of equivalent function, linked by [[neutral mutation]]s.<ref>{{cite journal|last=Taverna|first=DM|author2=Goldstein, RA |title=Why are proteins so robust to site mutations?|journal=Journal of Molecular Biology|date=Jan 18, 2002|volume=315|issue=3|pages=479–84|pmid=11786027|doi=10.1006/jmbi.2001.5226}}</ref><ref>{{cite journal|last=Tokuriki|first=N|author2=Tawfik, DS |title=Stability effects of mutations and protein evolvability.|journal=Current Opinion in Structural Biology|date=Oct 2009|volume=19|issue=5|pages=596–604|pmid=19765975|doi=10.1016/j.sbi.2009.08.003}}</ref> Proteins are resistant to mutations because many sequences can fold into highly similar [[protein fold|structural folds]].<ref>{{cite journal|last=Meyerguz|first=L|author2=Kleinberg, J |author3=Elber, R |title=The network of sequence flow between protein structures.|journal=Proceedings of the National Academy of Sciences of the United States of America|date=Jul 10, 2007|volume=104|issue=28|pages=11627–32|pmid=17596339|doi=10.1073/pnas.0701393104|pmc=1913895|bibcode=2007PNAS..10411627M|doi-access=free}}</ref> A protein adopts a limited ensemble of native conformations because those conformers have lower energy than unfolded and mis-folded states (ΔΔG of folding).<ref>{{cite journal|last=Karplus|first=M|title=Behind the folding funnel diagram.|journal=Nature Chemical Biology|date=Jun 17, 2011|volume=7|issue=7|pages=401–4|pmid=21685880|doi=10.1038/nchembio.565}}</ref><ref>{{cite journal|last=Tokuriki|first=N|author2=Stricher, F |author3=Schymkowitz, J |author4=Serrano, L |author5= Tawfik, DS |title=The stability effects of protein mutations appear to be universally distributed.|journal=Journal of Molecular Biology|date=Jun 22, 2007|volume=369|issue=5|pages=1318–32|pmid=17482644 |doi=10.1016/j.jmb.2007.03.069|s2cid=24638570}}</ref> This is achieved by a distributed, internal network of cooperative interactions ([[hydrophobic]], [[polarity (chemistry)|polar]] and [[covalent bond|covalent]]).<ref>{{cite journal|last=Shakhnovich|first=BE|author2=Deeds, E |author3=Delisi, C |author4= Shakhnovich, E |title=Protein structure and evolutionary history determine sequence space topology.|journal=Genome Research|date=Mar 2005|volume=15|issue=3|pages=385–92|pmid=15741509 |doi=10.1101/gr.3133605 |pmc=551565|arxiv=q-bio/0404040}}</ref> Protein structural robustness results from few single mutations being sufficiently disruptive to compromise function. Proteins have also evolved to avoid [[protein aggregation|aggregation]]<ref>{{cite journal|last=Monsellier|first=E|author2=Chiti, F |title=Prevention of amyloid-like aggregation as a driving force of protein evolution.|journal=EMBO Reports|date=Aug 2007|volume=8|issue=8|pages=737–42|pmid=17668004|doi=10.1038/sj.embor.7401034|pmc=1978086}}</ref> as partially folded proteins can combine to form large, repeating, insoluble [[Amyloid#CITEREF2002|protein fibrils]] and masses.<ref>{{cite journal|last=Fink|first=AL|title=Protein aggregation: folding aggregates, inclusion bodies and amyloid.|journal=Folding & Design|year=1998|volume=3|issue=1|pages=R9-23|pmid=9502314|doi=10.1016/s1359-0278(98)00002-9|doi-access=free}}</ref> There is evidence that proteins show negative design features to reduce the exposure of aggregation-prone [[beta-sheet]] motifs in their structures.<ref>{{cite journal|last=Richardson|first=JS|author2=Richardson, DC |title=Natural beta-sheet proteins use negative design to avoid edge-to-edge aggregation.|journal=Proceedings of the National Academy of Sciences of the United States of America|date=Mar 5, 2002|volume=99|issue=5|pages=2754–9|pmid=11880627|doi=10.1073/pnas.052706099|pmc=122420|bibcode=2002PNAS...99.2754R|doi-access=free}}</ref>
Additionally, there is some evidence that the [[genetic code]] itself may be optimised such that most point mutations lead to similar amino acids ([[neutral mutation|conservative]]).<ref>{{cite journal|last=Müller|first=MM |author2=Allison, JR |author3=Hongdilokkul, N |author4=Gaillon, L |author5=Kast, P |author6=van Gunsteren, WF |author7=Marlière, P |author8=Hilvert, D |title=Directed evolution of a model primordial enzyme provides insights into the development of the genetic code.|journal=PLOS Genetics|year=2013|volume=9|issue=1|article-number=e1003187|pmid=23300488|doi=10.1371/journal.pgen.1003187|pmc=3536711 |doi-access=free }}</ref><ref>{{cite journal|last=Firnberg|first=E|author2=Ostermeier, M |title=The genetic code constrains yet facilitates Darwinian evolution.|journal=Nucleic Acids Research|date=Aug 2013|volume=41|issue=15|pages=7420–8|pmid=23754851|doi=10.1093/nar/gkt536|pmc=3753648}}</ref> Together these factors create a [[distribution of fitness effects]] of mutations that contains a high proportion of neutral and nearly-neutral mutations.<ref name="Hietpas 7896–901">{{cite journal|last=Hietpas|first=RT|author2=Jensen, JD |author3=Bolon, DN |title=Experimental illumination of a fitness landscape.|journal=Proceedings of the National Academy of Sciences of the United States of America|date=May 10, 2011|volume=108|issue=19|pages=7896–901|pmid=21464309|doi=10.1073/pnas.1016024108|pmc=3093508|bibcode=2011PNAS..108.7896H|doi-access=free}}</ref>

==Evolution==
Neutral networks are a subset of the sequences in [[sequence space (evolution)|sequence space]] that have equivalent function, and so form a wide, flat [[plateau]] in a [[fitness landscape]]. [[Neutral theory of molecular evolution|Neutral evolution]] can therefore be visualised as a population diffusing from one set of sequence nodes, through the neutral network, to another cluster of sequence nodes. Since the majority of evolution is thought to be neutral,<ref name=Kimura83>Kimura, Motoo. (1983). ''The neutral theory of molecular evolution.'' Cambridge</ref><ref>{{cite journal | last1 = Kimura | first1 = M. | year = 1968 | title = Evolutionary Rate at the Molecular Level | journal = Nature | volume = 217 | issue = 5129| pages = 624–6 | doi=10.1038/217624a0 | pmid=5637732| bibcode = 1968Natur.217..624K| s2cid = 4161261 }}</ref> a large proportion of gene change is the movement though expansive neutral networks.

===Robustness===
{{see also|Robustness (evolution)}}
[[File:Neutral network.png|thumb|400px|Each circle represents a functional gene variant and lines represents point mutations between them. Light grid-regions have low [[fitness (biology)|fitness]], dark regions have high fitness. ('''a''') White circles have few neutral neighbours, black circles have many. Light grid-regions contain no circles because those sequences have low fitness. ('''b''') Within a neutral network, the population is predicted to evolve towards the centre and away from 'fitness cliffs' (dark arrows).]]

The more neutral neighbours a sequence has, the more [[robustness (evolution)|robust to mutations]] it is since mutations are more likely to simply neutrally convert it into an equally functional sequence.<ref name="van Nimwegen 9716–20"/> Indeed, if there are large differences between the number of neutral neighbours of different sequences within a neutral network, the population is predicted to evolve towards these robust sequences. This is sometimes called circum-neutrality and represents the movement of populations away from cliffs in the [[fitness landscape]].<ref>{{cite journal|last=Proulx|first=SR|author2=Adler, FR |title=The standard of neutrality: still flapping in the breeze?|journal=Journal of Evolutionary Biology|date=Jul 2010|volume=23|issue=7|pages=1339–50|pmid=20492093|doi=10.1111/j.1420-9101.2010.02006.x|s2cid=7774510|doi-access=free}}</ref> An interplay between evolutionary forces such as; mutation rate, recombination rate, population dynamics and the structure of the neutral landscape will effect the characteristics of the robustness and the expected distribution of genotypes with in the neutral network. Especially the ratio between mutation and recombination rate have been shown to effect localization of genotypes with in the neutral network, either towards high [[Eigenvector centrality|eigencentrality]] or attractors of specific recombination dynamics.<ref>{{Cite journal |last=Sella |first=Yehonatan |last2=Bergman |first2=Aviv |date=2025-06-03 |title=Robustness revisited: On the neutral evolution of centrality and localization |url=https://www.pnas.org/doi/10.1073/pnas.2421006122 |journal=Proceedings of the National Academy of Sciences |volume=122 |issue=22 |article-number=e2421006122 |doi=10.1073/pnas.2421006122 |pmc=12146759 |pmid=40440062}}</ref>

In addition to in silico models,<ref name="van Nimwegen">{{cite journal |author1=van Nimwegen E. |author2=Crutchfield J. P. |author3=Huynen M. | title = Neutral evolution of mutational robustness | journal = PNAS | year = 1999| volume = 96 | issue = 17 | pages = 9716–9720 | doi=10.1073/pnas.96.17.9716 | pmid=10449760 | pmc=22276| bibcode=1999PNAS...96.9716V |doi-access=free |arxiv=adap-org/9903006 }}</ref> these processes are beginning to be confirmed by [[experimental evolution]] of [[cytochrome P450]]s<ref name="Bloom 29">{{cite journal|last=Bloom|first=JD|author2=Lu, Z |author3=Chen, D |author4=Raval, A |author5=Venturelli, OS |author6= Arnold, FH |title=Evolution favors protein mutational robustness in sufficiently large populations.|journal=BMC Biology|date=Jul 17, 2007|volume=5|page=29|pmid=17640347 |doi=10.1186/1741-7007-5-29 |pmc=1995189|arxiv=0704.1885 |doi-access=free }}</ref> and [[B-lactamase]].<ref name="Bershtein 2008 1029–1044">{{cite journal|last=Bershtein|first=Shimon|author2=Goldin, Korina |author3=Tawfik, Dan S. |title=Intense Neutral Drifts Yield Robust and Evolvable Consensus Proteins|journal=Journal of Molecular Biology|date=June 2008|volume=379|issue=5|pages=1029–1044|doi=10.1016/j.jmb.2008.04.024|pmid=18495157}}</ref>

===Evolvability===
{{see also|Evolvability}}
Interest in the interplay between [[genetic drift]] and selection has been around since the 1930s when the shifting-balance theory proposed that in some situations, genetic drift could facilitate later adaptive evolution.<ref>{{cite journal|last=Wright|first=Sewel|title=The roles of mutation, inbreeding, crossbreeding and selection in evolution|journal=Proceedings of the Sixth International Congress of Genetics|year=1932|pages=356–366}}</ref> Although the specifics of the theory were largely discredited,<ref>{{cite journal|last=Coyne|first=JA|author2=Barton NH |author3=Turelli M |title=Perspective: a critique of Sewall Wright's shifting balance theory of evolution|journal=Evolution|year=1997|volume=51|issue=3|pages=643–671|doi=10.2307/2411143|pmid=28568586|jstor=2411143}}</ref> it drew attention to the possibility that drift could generate cryptic variation that, though neutral to current function, may affect selection for new functions ([[evolvability]]).<ref>{{cite journal|last=Davies|first=E. K.|title=High Frequency of Cryptic Deleterious Mutations in Caenorhabditis elegans|journal=Science|date=10 September 1999|volume=285|issue=5434|pages=1748–1751|doi=10.1126/science.285.5434.1748|pmid=10481013}}</ref>

By definition, all genes in a neutral network have equivalent function, however some may exhibit [[promiscuous activities]] which could serve as starting points for adaptive evolution towards new functions.<ref>{{cite journal|last=Masel|first=J|title=Cryptic genetic variation is enriched for potential adaptations.|journal=Genetics|date=Mar 2006|volume=172|issue=3|pages=1985–91|pmid=16387877|doi=10.1534/genetics.105.051649|pmc=1456269}}</ref><ref>{{cite journal|last=Hayden|first=EJ|author2=Ferrada, E |author3=Wagner, A |title=Cryptic genetic variation promotes rapid evolutionary adaptation in an RNA enzyme.|journal=Nature|date=Jun 2, 2011|volume=474|issue=7349|pages=92–5|pmid=21637259|doi=10.1038/nature10083|s2cid=4390213|url=https://www.zora.uzh.ch/id/eprint/59818/1/Hayden_Ferrada_Wagner_cryptic_Nature_revised.pdf}}</ref> In terms of [[sequence space (evolution)|sequence space]], current theories predict that if the neutral networks for two different activities overlap, a neutrally evolving population may diffuse to regions of the neutral network of the first activity that allow it to access the second.<ref>{{cite journal|last=Bornberg-Bauer|first=E|author2=Huylmans, AK |author3=Sikosek, T |title=How do new proteins arise?|journal=Current Opinion in Structural Biology|date=Jun 2010|volume=20|issue=3|pages=390–6|pmid=20347587|doi=10.1016/j.sbi.2010.02.005}}</ref> This would only be the case when the distance between activities is smaller than the distance that a neutrally evolving population can cover. The degree of interpenetration of the two networks will determine how common cryptic variation for the promiscuous activity is in sequence space.<ref>{{cite book|last=Wagner|first=Andreas|title=The origins of evolutionary innovations: a theory of transformative change in living systems|publisher=Oxford University Press|location=Oxford [etc.]|isbn=978-0-19-969259-0|date=2011-07-14}}</ref>

== Mathematical Framework ==

The fact that neutral mutations were probably widespread was proposed by Freese and Yoshida in 1965.<ref>Freese, E. and Yoshida, A. (1965). The role of mutations in evolution. In V Bryson, and H J Vogel, eds. Evolving Genes and Proteins, pp. 341-55. Academic, New York.</ref> [[Motoo Kimura]] later crystallized a theory of neutral evolution in 1968<ref>{{cite journal | last1 = Kimura | first1 = M | year = 1968 | title = Evolutionary Rate at the Molecular Level | journal = Nature | volume = 217 | issue = 5129| pages = 624–6 | doi = 10.1038/217624a0 | pmid = 5637732 | bibcode = 1968Natur.217..624K | s2cid = 4161261 }}</ref> with King and Jukes independently proposing a similar theory (1969).<ref>{{cite journal | last1 = King | first1 = JL | last2 = Jukes | first2 = TH | year = 1969 | title = Non-Darwinian Evolution | journal = Science | volume = 164 | issue = 3881| pages = 788–97 | doi = 10.1126/science.164.3881.788 | pmid = 5767777 | bibcode = 1969Sci...164..788L }}</ref> Kimura computed the rate of nucleotide substitutions in a population (i.e. the average time for one base pair replacement to occur within a genome) and found it to be ~1.8 years. Such a high rate would not be tolerated by any mammalian population according to [[John Scott Haldane|Haldane]]'s formula. He thus concluded that, in mammals, neutral (or nearly neutral) nucleotide substitution mutations of [[DNA]] must dominate. He computed that such mutations were occurring at the rate of roughly 0-5 per year per gamete.
[[File:SimpleGenotypePhenotypeMap.jpg|thumb|A simple genotype–phenotype map.]]
In later years, a new paradigm emerged, that placed [[RNA]] as a precursor molecule to [[DNA]]. A primordial molecule principle was put forth as early as 1968 by [[Francis Crick|Crick]],<ref>{{cite journal | last1 = Crick | first1 = FH | year = 1968 | title = The origin of the genetic code | journal = Journal of Molecular Biology | volume = 38 | issue = 3| pages = 367–79 | doi = 10.1016/0022-2836(68)90392-6 | pmid = 4887876 }}</ref> and lead to what is now known as [[RNA world|The RNA World Hypothesis]].<ref>{{cite journal | last1 = Robertson | first1 = MP | last2 = Joyce | first2 = GF | year = 2012 | title = The origins of the RNA world | journal = Cold Spring Harbor Perspectives in Biology | volume = 4| issue = 5| article-number = a003608| doi = 10.1101/cshperspect.a003608 | pmid = 20739415 | pmc = 3331698 }}</ref> [[DNA]] is found, predominantly, as fully [[base pair]]ed double helices, while biological [[RNA]] is single stranded and often exhibits complex base-pairing interactions. These are due to its increased ability to form [[hydrogen bond]]s, a fact which stems from the existence of the extra [[Hydroxy group|hydroxyl]] group in the [[ribose]] sugar.

In the 1970s, Stein and [[Michael Waterman|M. Waterman]] laid the groundwork for the combinatorics of [[RNA]] secondary structures.<ref>{{cite journal | last1 = Stein | first1 = P.R. | last2 = Waterman | first2 = M.S. | year = 1978 | title = On some new sequences generalizing the Catalan and Motzkin numbers | journal = Discrete Math. | volume = 26 | issue = 3| pages = 261–272 | doi = 10.1016/0012-365x(79)90033-5 | doi-access = free }}</ref> [[Michael Waterman|Waterman]] gave the first graph theoretic description of [[RNA]] secondary structures and their associated properties, and used them to produce an efficient [[Principle of minimum energy|minimum free energy]] (MFE) folding algorithm.<ref>M.S. Waterman. Secondary structure of single - stranded nucleic acids. Adv. Math. I (suppl.), 1:167–212, 1978.</ref> An [[RNA]] secondary structure can be viewed as a diagram over N labeled vertices with its Watson-Crick [[base pair]]s represented as non-crossing arcs in the upper half plane. Therefore, a [[Nucleic acid secondary structure|secondary structure]] is a scaffold having many sequences compatible with its implied base pairing constraints. Later, [[Smith–Waterman algorithm|Smith and Waterman]] developed an algorithm that performed local sequence alignment.<ref name=":1" /> Another prediction algorithm for [[RNA]] secondary structure was given by [[Ruth Nussinov|Nussinov]]<ref name=":2">{{cite journal | last1 = Nussiniv | display-authors = etal | year = 1978 | title = Algorithms for Loop Matchings | journal = SIAM Journal on Applied Mathematics | volume = 35 | issue = 1 | pages = 68–82 | doi = 10.1137/0135006 |jstor=2101031}}</ref> Nussinov's algorithm described the folding problem over a two letter alphabet as a planar graph optimization problem, where the quantity to be maximized is the number of matchings in the sequence string.

Come the year 1980, Howell et al. computed a generating function of all foldings of a sequence<ref>{{cite journal | last1 = Howell | first1 = J.A. | last2 = Smith | first2 = T.F. | last3 = Waterman | first3 = M.S. | year = 1980 | title = Computation of generating functions for biological molecules | journal = SIAM J. Appl. Math. | volume = 39 | page = 119133 | doi = 10.1137/0139010 }}</ref> while [[David Sankoff|D. Sankoff]] (1985) described algorithms for alignment of finite sequences, the prediction of RNA secondary structures (folding), and the reconstruction of proto-sequences on a phylo-genetic tree.<ref>{{cite journal |last1=Sankoff |first1=David |title=Simultaneous Solution of the RNA Folding, Alignment and Protosequence Problems |journal=SIAM Journal on Applied Mathematics |date=October 1985 |volume=45 |issue=5 |pages=810–825 |doi=10.1137/0145048}}</ref> Later, [[Michael Waterman|Waterman]] and [[Temple F. Smith|Temple]] (1986) produced a [[Polynomial-time|polynomial time]] [[dynamic programming]] (DP) algorithm for predicting general [[RNA]] secondary structure.<ref>{{cite journal | last1 = Waterman | first1 = M.S. | last2 = Smith | first2 = T.F. | year = 1986 | title = Rapid dynamic programming algorithms for RNA secondary structure | journal = Adv. Appl. Math. | volume = 7 | issue = 4| pages = 455–464 | doi = 10.1016/0196-8858(86)90025-4 | doi-access = free }}</ref> while in the year 1990, John McCaskill presented a polynomial time DP algorithm for computing the full equilibrium partition function of an RNA secondary structure.<ref>{{cite journal | last1 = McCaskill | first1 = John | year = 1990| title = The Equilibrium Partition Function and Base Pair Binding Probabilities for RNA Secondary Structure | journal = Biopolymers | volume = 29 | issue = 6–7| pages = 1105–19 | doi = 10.1002/bip.360290621 | pmid = 1695107 | hdl = 11858/00-001M-0000-0013-0DE3-9 | s2cid = 12629688 | hdl-access = free }}</ref> This changed the dominant calculation of RNA folding from a mapping of sequence to a particular 3D structure, to a mapping of sequence to a whole weighted ensemble of structures, which smooths RNA fitness, which depends on sequence via folding, facilitating more nearly neutral nets.

M. Zuker, implemented algorithms for computation of MFE [[RNA]] secondary structures<ref>{{cite journal | last1 = Zuker | first1 = Michael | last2 = Stiegler | first2 = Patrick | year = 1981 | title = Optimal Computer Folding of Large RNA Sequences Using Thermodynamics | journal = Nucleic Acids Research | volume = 9| issue = 1| pages = 133–148| doi = 10.1093/nar/9.1.133 | pmc = 326673 | pmid = 6163133 }}</ref> based on the work of [[Ruth Nussinov|Nussinov]] et al.,<ref name=":2" /> [[Smith–Waterman algorithm|Smith and Waterman]]<ref name=":1">{{cite journal | last1 = Smith | first1 = Temple F. | last2 = Waterman | first2 = Michael S. | year = 1981 | title = Identification of common molecular subsequences| journal = [[Journal of Molecular Biology]] | volume = 147 | issue = 1| pages = 195–197 | doi = 10.1016/0022-2836(81)90087-5 | pmid = 7265238 }}</ref> and Studnicka, et al.<ref>{{Cite journal|last1=Studnicka|first1=Gary M.|last2=Rahn|first2=Georgia M.|last3=Cummings|first3=Ian W.|last4=Salser|first4=Winston A.|date=1978-09-01|title=Computer method for predicting the secondary structure of single-stranded RNA|journal=Nucleic Acids Research|volume=5|issue=9|pages=3365–3388|doi=10.1093/nar/5.9.3365|pmid=100768|pmc=342256|issn=0305-1048}}</ref> Later L. Hofacker (et al., 1994),<ref>{{cite journal | last1 = Hofacker | first1 = I.L. | last2 = Fontana | first2 = W. | last3 = Stadler | first3 = P.F. | display-authors = etal | year = 1994 | title = Fast folding and comparison of RNA secondary structures | journal = Monatsh Chem | volume = 125 | issue = 2| page = 167 | doi = 10.1007/BF00818163 | s2cid = 19344304 }}</ref> presented The [[ViennaRNA Package|Vienna RNA package]], a software package that integrated MFE folding and the computation of the partition function as well as base pairing probabilities.

[[Peter Schuster (theoretical chemist)|Peter Schuster]] and W. Fontana (1994) shifted the focus towards sequence to structure maps ([[Genotype-phenotype distinction|genotype–phenotype]]) . They used an inverse folding algorithm, to produce computational evidence that [[RNA]] sequences sharing the same structure are distributed randomly in [[Sequence space (evolution)|sequence space]]. They observed that common structures can be reached from a random sequence by just a few mutations. These two facts lead them to conclude that the sequence space seemed to be [[Percolation theory|percolated]] by neutral networks of nearest neighbor mutants that fold to the same structure.<ref name=":0">{{Cite journal|last1=Schuster|first1=Peter|last2=Fontana|first2=Walter|last3=Stadler|first3=Peter F.|last4=Hofacker|first4=Ivo L.|date=1994-03-22|title=From Sequences to Shapes and Back: A Case Study in RNA Secondary Structures|journal=Proceedings of the Royal Society of London B: Biological Sciences|language=en|volume=255|issue=1344|pages=279–284|doi=10.1098/rspb.1994.0040|issn=0962-8452|pmid=7517565|bibcode=1994RSPSB.255..279S|s2cid=12021473}}</ref>

In 1997, C. Reidys Stadler and [[Peter Schuster (theoretical chemist)|Schuster]] laid the mathematical foundations for the study and modelling of neutral networks of [[RNA]] secondary structures. Using a [[Random graph|random graph model]] they proved the existence of a threshold value for connectivity of random sub-graphs in a configuration space, parametrized by λ, the fraction of neutral neighbors. They showed that the networks are connected and [[Percolation theory|percolate]] sequence space if the fraction of neutral nearest neighbors exceeds λ*, a threshold value. Below this threshold the networks are partitioned into a largest [[giant component]] and several smaller ones. Key results of this analysis where concerned with threshold functions for density and connectivity for neutral networks as well as [[Peter Schuster (theoretical chemist)|Schuster]]'s shape space conjecture.<ref name=":0" /><ref>{{Cite web|url=http://www.tbi.univie.ac.at/papersold/papers/Abstracts/reidys_diss.pdf|title=Neutral networks of RNA Secondary Structures}}</ref><ref>{{Cite journal|last1=Hofacker|first1=Ivo L.|last2=Schuster|first2=Peter|last3=Stadler|first3=Peter F.|title=Combinatorics of RNA secondary structures|journal=Discrete Applied Mathematics|language=en|volume=88|issue=1–3|pages=207–237|doi=10.1016/s0166-218x(98)00073-0|year=1998|doi-access=}}</ref>

==See also==
*[[Neutral theory of molecular evolution]]
*[[RNA world]]
*[[Nucleic acid secondary structure]]

==References==
{{reflist|35em}}
{{genarch}}

[[Category:Evolutionary biology]]
[[Category:Genetics]]

Neutral network (evolution) - История изменений

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