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among the population and can confer advantages such as helping to subvert the ability of a host's immune system to recognise it in a subsequent infection. The more mutations the virus makes during replication, the more likely it is to avoid recognition by the immune system and the more diverse its
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for an explanation of the selective advantages of this). However, mutations are not, as a general rule, beneficial, and if it accumulates too many harmful mutations, it may lose some of its biological features which have evolved to its advantage, including its ability to reproduce at all.
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will allow). Drugs have been created to increase the mutation rate of the viruses in order to push them over the critical boundary so that they lose self-identity. However, given the criticism of the basic assumption of the mathematical model, this approach is problematic.
875:{\displaystyle {\begin{matrix}{\frac {\partial z}{\partial t}}&=&{\frac {{\dot {x}}y-x{\dot {y}}}{y^{2}}}\\&&\\&=&{\frac {a(1-Q)xy-x(aQx+by)}{y^{2}}}\\&&\\&=&a(1-Q)z-(aQz^{2}+bz)\\&&\\&=&z(a(1-Q)-aQz-b)\\\end{matrix}}}
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To avoid error catastrophe, the amount of information lost through mutation must be less than the amount gained through natural selection. This fact can be used to arrive at essentially the same equations as the more common differential presentation.
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554:
37:
The term is most widely used to refer to mutation accumulation to the point of inviability of the organism or virus, where it cannot produce enough viable offspring to maintain a population. This use of Eigen's term was adopted by
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be the probability of a member of the second group returning to the first (via an unlikely and very specific mutation). The equations governing the development of the populations are:
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refers to the cumulative loss of genetic information in a lineage of organisms due to high mutation rates. The mutation rate above which error catastrophe occurs is called the
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450:
91:
Consider a virus which has a genetic identity modeled by a string of ones and zeros (e.g. 11010001011101....). Suppose that the string has fixed length
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1791:
Pariente, N; Sierra, S; Airaksinen, A (2005). "Action of mutagenic agents and antiviral inhibitors on foot-and-mouth disease virus".
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159:
241:
At this point, we make a mathematical idealisation: we pick the fittest strain (the one with the greatest reproduction rate
82:
how many mutations can occur during each replication before the population of viruses begins to lose the ability to survive?
1612:"Mutagenesis by human immunodeficiency virus reverse transcriptase: incorporation of O6-methyldeoxyguanosine triphosphate"
1916:
900:
1529:
1504:: in general, large genomes are required for accurate replication (high replication rates are achieved by the help of
60:
Error catastrophe is predicted in certain mathematical models of evolution and has also been observed empirically.
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270:); and we then group the remaining strains into a single group. Let the concentrations of the two groups be
57:
would amplify the errors until the cell was inviable. This theory has not received empirical support.
1611:
95:
and that during replication the virus copies each digit one by one, making a mistake with probability
1715:
1568:
1559:
Eigen M (October 1971). "Selforganization of matter and the evolution of biological macromolecules".
1512:
to persist. Which comes first and how does it happen? An illustration of the difficulty involved is
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or less. This corresponds to half the offspring surviving; namely the half with the correct genome.
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426:{\displaystyle {\begin{cases}{\dot {x}}=&a(1-Q)x+bRy\\{\dot {y}}=&aQx+b(1-R)y\\\end{cases}}}
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billion (10) base units long. This means that the replication mechanism for human DNA must be
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549:{\displaystyle {\begin{cases}{\dot {x}}=&a(1-Q)x\\{\dot {y}}=&aQx+by\\\end{cases}}}
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in a theory for cellular aging, in which errors in the translation of proteins involved in
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is the negative log of probability. Therefore, a genome can only survive unchanged when
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under what parameter values does the original population persist (continue to exist)?
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which replicate close to the error threshold have a genome size of order 10 (10000)
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1704:"The maintenance of the accuracy of protein synthesis and its relevance to ageing"
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Due to the mutations resulting from erroneous replication, there exist up to
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1658:"Perspective-Lethal Mutagenesis of HIV by Mutagenic Ribonucleoside Analogs"
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and colleagues to describe the strategy of lethal mutagenesis to cure
1616:
Mutation
Research/Fundamental and Molecular Mechanisms of Mutagenesis
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operate very close to the critical mutation rate (i.e. the largest
1088:{\displaystyle z(\infty )>0\iff a(1-Q)-b>0\iff (1-Q)>b/a.}
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This result is more popularly expressed in terms of the ratio of
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The population persists if and only if the steady state value of
1302:{\displaystyle L\ln {(1-q)}\approx -Lq>\ln {(1-s)}\approx -s}
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43:
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Taking a logarithm on both sides and approximating for small
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The information lost can be quantified as the genome length
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There was an earlier use of the term introduced in 1963 by
1198:{\displaystyle z(\infty )>0\iff (1-Q)=(1-q)^{L}>1-s}
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is 0.99 - a very small string length in terms of genes.
231:{\displaystyle {\dot {x}}_{j}=\sum _{i}a_{i}Q_{ij}x_{i}}
67:) during replication. The resulting mutations increase
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1610:
Hizi, A; Kamath-Loeb, AS; Rose, KD; Loeb, LA (1997).
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is a genome with one bit which always mutates. Since
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248:) and assume that it is unique (i.e. that the chosen
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distinct strains derived from the parent virus. Let
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be the probability of a virus in the first group (
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960:{\displaystyle z(\infty )={\frac {a(1-Q)-b}{aQ}}}
436:We are particularly interested in the case where
63:Like every organism, viruses "make mistakes" (or
16:Loss of genetic information due to mutation rates
1366:more accurate than for the RNA of RNA viruses.
970:(which is deduced by setting the derivative of
30:in his mathematical evolutionary theory of the
8:
286:) mutating to a member of the second group (
135:denote the probability of a virus of strain
1841:"Examining The Theory of Error Catastrophe"
985:is strictly positive. i.e. if and only if:
890:achieves a steady concentration over time,
46:by using mutagenic ribonucleoside analogs.
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1897:Examining the theory of error catastrophe
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72:population will be (see the article on
1370:Information-theory based presentation
7:
1662:AIDS Research and Human Retroviruses
113:denote the concentration of strain
99:independently of all other digits.
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1382:times the replication error rate
124:denote the rate at which strain
1386:. The probability of survival,
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1805:10.1016/j.virusres.2004.11.008
1656:Loeb, LA; Mullins, JI (2000).
1430:{\displaystyle Lq<-\ln {S}}
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1768:Evolutionary Biology of Aging
1628:10.1016/S0027-5107(96)00217-5
1110:, then the condition becomes
977:So the important question is
26:. Both terms were coined by
1857:10.1128/JVI.80.1.20-26.2006
1312:reducing the condition to:
1933:
1708:Proc. Natl. Acad. Sci. USA
1106:of individual digits: set
1702:Orgel, Leslie E. (1963).
1839:Summers; Litwin (2006).
1500:mystery for biologists,
1496:The result introduces a
894:settles down to satisfy
274:with reproduction rates
87:Basic mathematical model
1774:Oxford University Press
1674:10.1089/088922200309539
1561:Die Naturwissenschaften
1337:{\displaystyle Lq<s}
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444:and instead consider:
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1476:Some viruses such as
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128:reproduces; and let
1917:Population genetics
1845:Journal of Virology
1720:1963PNAS...49..517O
1573:1971NW.....58..465E
1540:Mutational meltdown
1516:can only be 100 if
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1102:and the error rate
139:mutating to strain
55:protein translation
1581:10.1007/BF00623322
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83:
78:
75:
70:
66:
61:
58:
56:
52:
47:
45:
41:
40:Lawrence Loeb
35:
33:
29:
28:Manfred Eigen
25:
21:
1851:(1): 20–26.
1848:
1844:
1834:
1826:
1821:
1796:
1792:
1786:
1767:
1756:
1711:
1707:
1697:
1685:. Retrieved
1665:
1661:
1651:
1639:. Retrieved
1622:(1): 41–50.
1619:
1615:
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1564:
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1439:
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291:
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249:
242:
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153:is given by
147:
145:
140:
136:
129:
125:
118:
114:
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103:
101:
96:
92:
90:
81:
79:
74:biodiversity
69:biodiversity
62:
59:
51:Leslie Orgel
48:
36:
32:quasispecies
19:
18:
1776:. pp.
1482:hepatitis C
1392:information
1348:RNA viruses
1108:b/a = (1-s)
1906:Categories
1668:(1): 1–3.
1546:References
1352:base pairs
290:) and let
255:satisfies
1912:Pathology
1793:Virus Res
1687:3 October
1641:3 October
1420:
1414:−
1358:is about
1354:. Human
1294:−
1291:≈
1281:−
1271:
1256:−
1253:≈
1243:−
1233:
1216:one gets
1190:−
1168:−
1150:−
1140:⟺
1127:∞
1060:−
1050:⟺
1037:−
1028:−
1015:⟺
1002:∞
941:−
932:−
911:∞
886:Assuming
860:−
848:−
839:−
775:−
763:−
696:−
681:−
635:˙
623:−
614:˙
589:∂
581:∂
511:˙
489:−
469:˙
405:−
373:˙
339:−
319:˙
187:∑
171:˙
1875:16352527
1829:, p. 140
1813:15649564
1764:(1991).
1748:13940312
1682:10628810
1597:38296619
1524:See also
1498:Catch-22
563:we have
266:for all
1866:1317512
1778:147–152
1716:Bibcode
1636:9067414
1589:4942363
1569:Bibcode
1506:enzymes
1466:
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561:z = x/y
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1739:299893
1736:
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276:a>b
261:> a
117:; let
65:mutate
1593:S2CID
1488:that
1478:polio
1446:q = 1
1442:L = 1
272:x , y
268:i ≠ j
1871:PMID
1809:PMID
1744:PMID
1689:2021
1678:PMID
1643:2021
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1861:PMC
1853:doi
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1724:doi
1670:doi
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1577:doi
1480:or
1360:3.3
1356:DNA
1100:a:b
44:HIV
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