Mathematical Challenges in Life Sciences Part III

2. Some Mathematical Challenges from Life SciencesPart IIIPeter Schuster, Universität WienPeter F.Stadler, Universität LeipzigGünter Wagner, Yale University, New Haven, CTAngela Stevens, Max-Planck-Institut für Mathematik in den Naturwissenschaften, LeipzigandIvo L. Hofacker, Universität WienOberwolfach, GE, 16.-21.11.2003

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

4. Definitionof 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. GGCGCGCCCGGCGCC GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA UGGUUACGCGUUGGGGUAACGAAGAUUCCGAGAGGAGUUUAGUGACUAGAGG Folding of RNA sequences into secondary structures of minimal free energy, G0300

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

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

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

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. AUGC GC Movies of optimization trajectories over the AUGCand the GCalphabet

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

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

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

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

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

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

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 Sk(j) in the population

41. The number of main transitions or evolutionary innovationsis constant.

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

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