2. Some Mathematical Challenges from Life Sciences Part III Peter Schuster, Universität Wien Peter F.Stadler, Universität Leipzig Günter Wagner, Yale University, New Haven, CT Angela Stevens, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig and Ivo L. Hofacker, Universität Wien Oberwolfach, GE, 16.-21.11.2003

3. 1.Mathematics and the life sciences in the 21 st century 2.Selection dynamics 3.RNA evolution in silico and optimization of structure and properties

4. Definition of RNA structure

5. Sequence space of binary sequences of chain lenght n=5

6. Population and population support in sequence space: The master sequence

7. Population and population support in sequence space: The quasi-species

8. The increase in RNA production rate during a serial transfer experiment

9. Mapping from sequence space into structure space and into function

10. Folding of RNA sequences into secondary structures of minimal free energy,  G 0 300 GGCGCGCCCGGCGCC GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA UGGUUACGCGUUGGGGUAACGAAGAUUCCGAGAGGAGUUUAGUGACUAGAGG

11. The Hamming distance between structures in parentheses notation forms a metric in structure space

12. Replication rate constant: f k =  / [  +  d S (k) ]  d S (k) = d H (S k,S  ) Evaluation of RNA secondary structures yields replication rate constants

13. The flowreactor as a device for studies of evolution in vitro and in silico Replication rate constant: f k =  / [  +  d S (k) ]  d S (k) = d H (S k,S  ) Selection constraint: # RNA molecules is controlled by the flow

14. The molecular quasispecies in sequence space

15. Evolutionary dynamics including molecular phenotypes

16. In silico optimization in the flow reactor: Trajectory (biologists‘ view)

17. In silico optimization in the flow reactor: Trajectory (physicists‘ view)

18. Movies of optimization trajectories over the AUGC and the GC alphabet AUGCGC

19. Statistics of the lengths of trajectories from initial structure to target ( AUGC -sequences)

20. Endconformation of optimization

21. Reconstruction of the last step 43  44

22. Reconstruction of last-but-one step 42  43 (  44)

23. Reconstruction of step 41  42 (  43  44)

24. Reconstruction of step 40  41 (  42  43  44)

25. Reconstruction of the relay series

26. Change in RNA sequences during the final five relay steps 39  44 Transition inducing point mutationsNeutral point mutations

27. In silico optimization in the flow reactor: Trajectory and relay steps

28. Birth-and-death process with immigration

29. Calculation of transition probabilities by means of a birth-and-death process with immigration

30. Neutral genotype evolution during phenotypic stasis Neutral point mutationsTransition inducing point mutations 28 neutral point mutations during a long quasi-stationary epoch

31. A random sequence of minor or continuous transitions in the relay series

34. Probability of occurrence of different structures in the mutational neighborhood of tRNA phe Rare neighbors Main transitions Frequent neighbors Minor transitions

35. In silico optimization in the flow reactor: Main transitions

36. 00 09 31 44 Three important steps in the formation of the tRNA clover leaf from a randomly chosen initial structure corresponding to three main transitions.

37. Statistics of the numbers of transitions from initial structure to target ( AUGC -sequences)

38. Statistics of trajectories and relay series (mean values of log-normal distributions)

39. Transition probabilities determining the presence of phenotype S k (j) in the population

41. The number of main transitions or evolutionary innovations is constant.

42. Neutral genotype evolution during phenotypic stasis Neutral point mutationsTransition inducing point mutations 28 neutral point mutations during a long quasi-stationary epoch

43. Variation in genotype space during optimization of phenotypes Mean Hamming distance within the population and drift velocity of the population center in sequence space.

44. Spread of population in sequence space during a quasistationary epoch: t = 150

45. Spread of population in sequence space during a quasistationary epoch: t = 170

46. Spread of population in sequence space during a quasistationary epoch: t = 200

47. Spread of population in sequence space during a quasistationary epoch: t = 350

48. Spread of population in sequence space during a quasistationary epoch: t = 500

49. Spread of population in sequence space during a quasistationary epoch: t = 650

50. Spread of population in sequence space during a quasistationary epoch: t = 820

51. Spread of population in sequence space during a quasistationary epoch: t = 825

52. Spread of population in sequence space during a quasistationary epoch: t = 830

53. Spread of population in sequence space during a quasistationary epoch: t = 835

54. Spread of population in sequence space during a quasistationary epoch: t = 840

55. Spread of population in sequence space during a quasistationary epoch: t = 845

56. Spread of population in sequence space during a quasistationary epoch: t = 850

57. Spread of population in sequence space during a quasistationary epoch: t = 855

59. Mapping from sequence space into structure space and into function

61. The pre-image of the structure S k in sequence space is the neutral network G k

62. Neutral networks are sets of sequences forming the same structure. G k is the pre-image of the structure S k in sequence space: G k =  -1 (S k )  {I j |  (I j ) = S k } The set is converted into a graph by connecting all sequences of Hamming distance one. Neutral networks of small RNA molecules can be computed by exhaustive folding of complete sequence spaces, i.e. all RNA sequences of a given chain length. This number, N=4 n, becomes very large with increasing length, and is prohibitive for numerical computations. Neutral networks can be modelled by random graphs in sequence space. In this approach, nodes are inserted randomly into sequence space until the size of the pre-image, i.e. the number of neutral sequences, matches the neutral network to be studied.

63. Mean degree of neutrality and connectivity of neutral networks GC,AU GUC,AUG AUGC

64. A connected neutral network

65. A multi-component neutral network

66. Degree of neutrality of cloverleaf RNA secondary structures over different alphabets

67. Reference for postulation and in silico verification of neutral networks

72. The compatible set C k of a structure S k consists of all sequences which form S k as its minimum free energy structure (the neutral network G k ) or one of its suboptimal structures.

73. The intersection of two compatible sets is always non empty: C 0  C 1  

74. Reference for the definition of the intersection and the proof of the intersection theorem

75. A sequence at the intersection of two neutral networks is compatible with both structures

76. Barrier tree for two long living structures

79. A ribozyme switch E.A.Schultes, D.B.Bartel, Science 289 (2000), 448-452

80. Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase (A) and a natural cleavage ribozyme of hepatitis-  -virus (B)

81. The sequence at the intersection: An RNA molecules which is 88 nucleotides long and can form both structures

82. Two neutral walks through sequence space with conservation of structure and catalytic activity

83. Examples of smooth landscapes on Earth Massif Central Mount Fuji

84. Examples of rugged landscapes on Earth Dolomites Bryce Canyon

85. Evolutionary optimization in absence of neutral paths in sequence space

86. Evolutionary optimization including neutral paths in sequence space

87. Example of a landscape on Earth with ‘neutral’ ridges and plateaus Grand Canyon

88. Neutral ridges and plateus