1. Requirements for information processing 2. The chemistry of Darwinian evolution 3. RNA sequences and structures 4. Consequences of neutrality 5. Evolutionary optimization of RNA structure
RNA folding determination of RNA function molecular recognition catalysis binding to : ground state transition state aptamers ribozymes The paradigm of structural biology
5' - end N 1 O CH 2 O GCGGAU UUA GCUC AGUUGGGA GAGC CCAGA G CUGAAGA UCUGG AGGUC CUGUG UUCGAUC CACAG A AUUCGC ACCA 5'-end 3’-end N A U G C k = , , , OH O N 2 O P O CH 2 O Na � O O OH N 3 O P O CH 2 O Na � O Definition of RNA structure O OH N 4 O P O CH 2 O Na � O O OH 3' - end O P O Na � O
N = 4 n N S < 3 n Criterion: Minimum free energy (mfe) Rules: _ ( _ ) _ � { AU , CG , GC , GU , UA , UG } A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs
What is neutrality ? Selective neutrality = = several genotypes having the same fitness. Structural neutrality = = several genotypes forming molecules with the same structure.
Reference for postulation and in silico verification of neutral networks
many genotypes � one phenotype
One-error neighborhood GUUAAUCAG GUAAAUCAG GUGAAUCAG GCCAAUCAG GUCUAUCAG GGCAAUCAG GUCGAUCAG GACAAUCAG GUCCAUCAG CUCAAUCAG GUCAUUCAG UUCAAUCAG G A C U G A C U G GUCAAUCAG AUCAAUCAG GUCACUCAG GUCAAUCAC GUCAAACAG GUCAAUCAU G U C A A GUCAAUCAA G C A G GUCAACCAG G U GUCAAUAAG C A G A G U U GUCAAUCUG U C C G A C C U A G A C U A A C The surrounding of U A G U U G A G GUCAAUCAG in sequence space G A G
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
GGCUAUCGUA U GUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUU A GACG GGCUAUCGUACGUUUAC U CAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACG C UUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGC C AUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGU G UACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUA A CGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCC U GGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCA C UGGACG G G A U GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGG U CCCAGGCAUUGGACG C U GGCUA G CGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG G A GGCUAUCGUACGUUUACCC G AAAGUCUACGUUGGACCCAGGCAUUGGACG C G CC C A GG GGCUAUCGUACGUUUACCCAAAAG C CUACGUUGGACCCAGGCAUUGGACG G C U UGGA A U C UACG U G U C A G U AAG UC U A U C C C AA One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
Number Mean Value Variance Std.Dev. Total Hamming Distance: 150000 11.647973 23.140715 4.810480 Nonzero Hamming Distance: 99875 16.949991 30.757651 5.545958 Degree of Neutrality: 50125 0.334167 0.006961 0.083434 Number of Structures: 1000 52.31 85.30 9.24 1 (((((.((((..(((......)))..)))).))).))............. 50125 0.334167 2 ..(((.((((..(((......)))..)))).)))................ 2856 0.019040 3 ((((((((((..(((......)))..)))))))).))............. 2799 0.018660 4 (((((.((((..((((....))))..)))).))).))............. 2417 0.016113 5 (((((.((((.((((......)))).)))).))).))............. 2265 0.015100 6 (((((.(((((.(((......))).))))).))).))............. 2233 0.014887 7 (((((..(((..(((......)))..)))..))).))............. 1442 0.009613 8 (((((.((((..((........))..)))).))).))............. 1081 0.007207 9 ((((..((((..(((......)))..))))..)).))............. 1025 0.006833 10 (((((.((((..(((......)))..)))).))))).............. 1003 0.006687 11 .((((.((((..(((......)))..)))).))))............... 963 0.006420 12 (((((.(((...(((......)))...))).))).))............. 860 0.005733 13 (((((.((((..(((......)))..)))).)).)))............. 800 0.005333 14 (((((.((((...((......))...)))).))).))............. 548 0.003653 15 (((((.((((................)))).))).))............. 362 0.002413 16 ((.((.((((..(((......)))..)))).))..))............. 337 0.002247 G G A U 17 (.(((.((((..(((......)))..)))).))).).............. 241 0.001607 C U 18 (((((.(((((((((......))))))))).))).))............. 231 0.001540 G A 19 ((((..((((..(((......)))..))))...))))............. 225 0.001500 C G CC C A GG 20 ((....((((..(((......)))..)))).....))............. 202 0.001347 G C U UGGA A U C UACG U G U C A G U AAG UC U A U Shadow – Surrounding of an RNA structure in shape space: C AUGC alphabet, chain length n=50 C C AA
1. Requirements for information processing 2. The chemistry of Darwinian evolution 3. RNA sequences and structures 4. Consequences of neutrality 5. Evolutionary optimization of RNA structure
Charles Darwin. The Origin of Species . Sixth edition. John Murray. London: 1872
Motoo Kimuras population genetics of neutral evolution. Evolutionary rate at the molecular level. Nature 217 : 624-626, 1955. The Neutral Theory of Molecular Evolution . Cambridge University Press. Cambridge, UK, 1983.
The average time of replacement of a dominant genotype in a population is the reciprocal mutation rate, 1/ � , and therefore independent of population size. Is the Kimura scenario correct for frequent mutations?
d H = 1 = = lim ( ) ( ) 0 . 5 x p x p → 0 1 2 p d H = 2 = lim ( ) x p a → 0 1 p = − lim ( ) 1 x p a → 0 2 p d H ≥ 3 random fixation in the sense of Motoo Kimura Pairs of genotypes in neutral replication networks
for comparison: � = 0, � = 1.1, d = 0 Neutral network: Individual sequences n = 10, � = 1.1, d = 1.0
Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance d H (X i, ,X j ) = 1.
Neutral network: Individual sequences n = 10, � = 1.1, d = 1.0
Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance d H (X i, ,X j ) = 2.
N = 7 Computation of sequences in the core of a neutral network
N = 7 Neutral networks with increasing � : � = 0.10, s = 229
N = 24 Neutral networks with increasing � : � = 0.15, s = 229
N = 70 Neutral networks with increasing � : � = 0.20, s = 229
Extension of the notion of structure
Extension of the notion of structure
GGCCCCUUUGGGGGCCAGACCCCUAAAGGGGUC ((((((((((((((.....)))))))))))))) -26.30 ((((((....)))))).((((((....)))))) -25.30 .(((((((((((((.....))))))))))))). -24.80 (((((((((((((.......))))))))))))) -24.50 ((((((....)))))).(((((......))))) -23.40 (((((......))))).((((((....)))))) -23.30 ..((((((((((((.....)))))))))))).. -23.10 (((((((((((((......)))).))))))))) -23.00 .((((((((((((.......)))))))))))). -23.00 (((((((.((((((.....)))))).))))))) -22.80 ((((((((.(((((.....))))).)))))))) -22.70 ((((((....))))))..(((((....))))). -22.70 ((((((.(((((((.....))))))).)))))) -22.20 (((((((((.((((.....)))).))))))))) -22.10 mfe-weight: 0.7196 (.((((((((((((.....)))))))))))).) -21.90 .(((((((((((((.....)))))))))))).) -21.90 ((((((....))))))...((((....)))).. -21.60 (((((((..(((((.....)))))..))))))) -21.50 .((((((((((((......)))).)))))))). -21.50 (((((......))))).(((((......))))) -21.40 Suboptimal structures and partition function .((((((.((((((.....)))))).)))))). -21.30 ..(((((((((((.......))))))))))).. -21.30 of a small RNA molecule: n = 33
GGCUAUCGUACGUUUAC C CAAAAGUCUACGUUGGACCCAGGCA U UGGACG (((((.((((..(((......)))..)))).))).))............. -7.30 ..........((((((.((....((((.....))))...))...)))))) -6.70 ..........((((((.((....(((((...)))))...))...)))))) -6.60 ..(((.((((..(((......)))..)))).)))..((((...))))... -6.10 (((((.((((..(((......)))..)))).))).))..(........). -6.00 (((((.((((..((........))..)))).))).))............. -6.00 .(((.((..((((..((......))..))))..))....)))........ -6.00 GGCUAUCGUACGUUUAC A CAAAAGUCUACGUUGGACCCAGGCAUUGGACG (((((.((((..(((......)))..)))).))).))............. -7.30 .(((.((..((((..((......))..))))..))....)))........ -6.50 .(((.....((((..((......))..))))((....)))))........ -6.30 ..(((.((((..(((......)))..)))).)))..((((...))))... -6.10 (((((.((((..(((......)))..)))).))).))..(........). -6.00 (((((.((((..((........))..)))).))).))............. -6.00 .(((...((((((..((......))..))))...))...)))........ -6.00 GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCA A UGGACG (((((.((((..(((......)))..)))).))).))............. -7.30 ..(((.((((..(((......)))..)))).)))..(((.....)))... -7.20 ..........((((((.((....((((.....))))...))...)))))) -6.70 ..........((((((.((....(((((...)))))...))...)))))) -6.60 (((((.((((..(((......)))..)))).))).))((.....)).... -6.50 (.(((.((((..(((......)))..)))).))).)(((.....)))... -6.30 .((((.((((..(((......)))..)))).))).)(((.....)))... -6.30 .....(((.((((..((......))..)))))))..(((.....)))... -6.30 (.(((.((((..(((......)))..)))).)))..(((.....))).). -6.10 .....((..((((..((......))..))))..)).(((.....)))... -6.10 ......(((.((((...((....((((.....))))...)).)))).))) -6.10 (((((.((((..(((......)))..)))).))).))..(........). -6.00 (((((.((((..((........))..)))).))).))............. -6.00 .(((.((..((((..((......))..))))..))....)))........ -6.00 ......(((.((((...((....(((((...)))))...)).)))).))) -6.00
Extension of the notion of structure
Extension of the notion of structure
R 1D 2D GGGUGGAAC CACGAG GUUC CACGAG GAAC CACGAG GUUCCUCCC G 3 13 23 33 44 R 1D 2D 23 13 33 C G C G C G A A A A G/ A A C G C C G G G C G C G C A U A U U A U A A U A U G C G C G C G C G C G C A A U A /G A U G C 13 3 G C G CCC 44 1D 2D C G 33 GG 23 R 5' 3’ A A C G C G -1 -28.6 kcal·mol A U A U -1 -28.2 kcal·mol G C G C U U G C 3 G C An RNA switch G C 44 5' 3’ JN1LH -1 -28.6 kcal·mol J.H.A. Nagel, C. Flamm, I.L. Hofacker, K. Franke, -1 -31.8 kcal·mol M.H. de Smit, P. Schuster, and C.W.A. Pleij. Structural parameters affecting the kinetic competition of RNA hairpin formation. Nucleic Acids Res. 34 :3568-3576 , 2006 .
A ribozyme switch E.A.Schultes, D.B.Bartel, Science 289 (2000), 448-452
Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase ( A ) and a natural cleavage ribozyme of hepatitis- � -virus ( B )
The sequence at the intersection : An RNA molecules which is 88 nucleotides long and can form both structures
Two neutral walks through sequence space with conservation of structure and catalytic activity
RNA 9 :1456-1463, 2003 Evidence for neutral networks and shape space covering
Evidence for neutral networks and intersection of apatamer functions
Neutrality in molecular structures and its role in evolution : • Neutrality is an essential feature in biopolymer structures at the resolution that is relevant for function. • Neutrality manifests itself in the search for minimum free energy structures. • Diversity in function despite neutrality in structures results from differences in suboptimal conformations and folding kinetics. • Neutrality is indispensible for optimization and adaptation.
1. Requirements for information processing 2. The chemistry of Darwinian evolution 3. RNA sequences and structures 4. Consequences of neutrality 5. Evolutionary optimization of RNA structure
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
Evolution of RNA molecules as a Markow process and its analysis by means of the relay series
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