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Mathematical Challenges in Life Sciences Part III

Explore the intersection of mathematics and life sciences in the 21st century, focusing on selection dynamics and RNA evolution. Discover the population support in sequence space, mapping from sequence to structure, and in silico optimization.

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Mathematical Challenges in Life Sciences Part III

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  1. 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

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

  3. Definitionof RNA structure

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

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

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

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

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

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

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

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

  12. 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

  13. The molecular quasispecies in sequence space

  14. Evolutionary dynamics including molecular phenotypes

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

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

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

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

  19. Endconformation of optimization

  20. Reconstruction of the last step 43  44

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

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

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

  24. Reconstruction of the relay series

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

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

  27. Birth-and-death process with immigration

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

  29. 28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations Neutral point mutations Neutral genotype evolution during phenotypic stasis

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

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

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

  35. 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.

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

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

  38. Transition probabilities determining the presence of phenotype Sk(j) in the population

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

  40. 28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations Neutral point mutations Neutral genotype evolution during phenotypic stasis

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

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

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

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

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

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

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

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

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