Mathematische Probleme aus den Life-Sciences

Transcription

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2 Mathematische Probleme aus den Life-Sciences Peter Schuster Institut für Theoretische Chemie und Molekulare Strukturbiologie der Universität Wien Vortragsreihe Mathematik im Betrieb Dornbirn,

3 Web-Page for further information:

4 1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie

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6 Fully sequenced genomes Organisms 751 projects 153 complete (16 A, 118 B, 19 E) (Eukarya examples: mosquito (pest, malaria), sea squirt, mouse, yeast, homo sapiens, arabidopsis, fly, worm, ) 598 ongoing (23 A, 332 B, 243 E) (Eukarya examples: chimpanzee, turkey, chicken, ape, corn, potato, rice, banana, tomato, cotton, coffee, soybean, pig, rat, cat, sheep, horse, kangaroo, dog, cow, bee, salmon, fugu, frog, ) Other structures with genetic information Source: NCBI 68 phages 1328 viruses 35 viroids 472 organelles (423 mitochondria, 32 plastids, 14 plasmids, 3 nucleomorphs) Source: Integrated Genomics, Inc. August 12 th, 2003

7 1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie

8 Wolfgang Wieser. Die Erfindung der Individualität oder die zwei Gesichter der Evolution. Spektrum Akademischer Verlag, Heidelberg A.C.Wilson. The Molecular Basis of Evolution. Scientific American, Oct.1985,

9 Genomics and proteomics Large scale data processing, sequence comparison... Evolutionary biology Optimization through variation and selection, relation between genotype, phenotype, and function,... Mathematics in 21st Century's Life Sciences Developmental biology Gene regulation networks, signal propagation, pattern formation, robustness... Neurobiology Neural networks, collective properties, nonlinear dynamics, signalling,... Cell biology Regulation of cell cycle, metabolic networks, reaction kinetics, homeostasis,...

10 Genomics and proteomics Large scale data processing, sequence comparison... E. coli: Length of the Genome Nucleotides Number of Cell Types 1 Number of Genes Man: Length of the Genome Nucleotides Number of Cell Types 200 Number of Genes

11 Number of genes in the human genome The number of genes in the human genome is still only a very rough estimate

12 Elimination of introns through splicing genomic DNA mrna AAA The gene is a stretch of DNA which after transcription and processing gives rise to a mrna

13 Sex determination in Drosophila through alternative splicing The process of protein synthesis and its regulation is now understood but the notion of the gene as a stretch of DNA has become obscure. The gene is essentially associated with the sequence of unmodified amino acids in a protein, and it is determined by the nucleotide sequence as well as the dynamics of the the process eventually leading to the m-rna that is translated.

14 The same section of the microarray is shown in three independent hybridizations. Marked spots refer to: (1) protein disulfide isomerase related protein P5, (2) IL-8 precursor, (3) EST AA057170, and (4) vascular endothelial growth factor Gene expression DNA microarray representing 8613 human genes used to study transcription in the response of human fibroblasts to serum V.R.Iyer et al., Science 283: 83-87, 1999

15 Developmental biology Gene regulation networks, signal propagation, pattern formation, robustness... Three-dimensional structure of the complex between the regulatory protein cro-repressor and the binding site on -phage B-DNA

16 Cascades, A B C..., and networks of genetic control Turing pattern resulting from reactiondiffusion equation? Intercelluar communication creating positional information Development of the fruit fly drosophila melanogaster: Genetics, experiment, and imago

17 Linear chain Network Processing of information in cascades and networks

18 Albert-László Barabási, Linked The New Science of Networks. Perseus Publ., Cambridge, MA, 2002

19 Distributed network Small world network Albert-László Barabási, Linked The New Science of Networks. Perseus Publ., Cambridge, MA, 2002

20 Albert-László Barabási, Linked The New Science of Networks Perseus Publ., Cambridge, MA, 2002

21 Formation of a scale-free network through evolutionary point by point expansion: Step 000

22 Formation of a scale-free network through evolutionary point by point expansion: Step 001

23 Formation of a scale-free network through evolutionary point by point expansion: Step 002

24 Formation of a scale-free network through evolutionary point by point expansion: Step 003

25 Formation of a scale-free network through evolutionary point by point expansion: Step 004

26 Formation of a scale-free network through evolutionary point by point expansion: Step 005

27 Formation of a scale-free network through evolutionary point by point expansion: Step 006

28 Formation of a scale-free network through evolutionary point by point expansion: Step 007

29 Formation of a scale-free network through evolutionary point by point expansion: Step 008

30 Formation of a scale-free network through evolutionary point by point expansion: Step 009

31 Formation of a scale-free network through evolutionary point by point expansion: Step 010

32 Formation of a scale-free network through evolutionary point by point expansion: Step 011

33 Formation of a scale-free network through evolutionary point by point expansion: Step 012

34 Formation of a scale-free network through evolutionary point by point expansion: Step 024

35 links # nodes Analysis of nodes and links in a step by step evolved network

36 Cell biology Regulation of cell cycle, metabolic networks, reaction kinetics, homeostasis,... The bacterial cell as an example for the simplest form of autonomous life The human body: cells = eukaryotic cells bacterial (prokaryotic) cells; 200 eukaryotic cell types

37 A B C D E F G H I J K L 1 Biochemical Pathways The reaction network of cellular metabolism published by Boehringer-Ingelheim.

38 The citrate, tricarboxylic acid or Krebs cycle (enlarged from previous slide)

39 Kinetic differential equations d x i = f ( x, x,, x n ; k, k,, k m ) ; i 1, 2, n d t, 1 2 K 1 2 K = K Reaction diffusion equations x i 2 = Di xi + f ( x, x,, xn; k, k,, km ); i 1,2, n t, 1 2 K 1 2 K = K Parameter set k j ( T, p, ph, I, K ; x1, x2, K, xn ); j= 1,2, K, m Concentration Solution curves: x i xi ( t); i = 1, 2,..., n General conditions: T, p, ph, I,... Initial conditions: x i ( 0); i = 1, 2, K, n Time t Boundary conditions: boundary... s normal unit vector... u Dirichlet, Neumann, x s i, = f ( r, t); i = 1, 2, K n x r i s r = uˆ xi = f (, t) ; i= 1,2, K, n u The forward-problem of chemical reaction kinetics

40 Kinetic differential equations d x i = f ( x, x,, x n ; k, k,, k m ) ; i 1, 2, n d t, 1 2 K 1 2 K = K x t Reaction diffusion equations i = i i (, 1 2 n 1 2 m 2 D x + f x, x, K, x ; k, k, K, k ); i = 1,2, K n General conditions: T, p, ph, I,... k Parameter set T, p, p H, I, K ; x, x, K, x ) ; j = 1, 2, j ( 1 2 n K, m Initial conditions: x i ( 0); i = 1, 2, K, n Boundary conditions: boundary... s normal unit vector... u Dirichlet, Neumann, x s r i, = f ( r, t); i = 1,2, K n x r i s r = uˆ xi = f (, t) ; i= 1,2, K, n u Data from measurements xi ( tk); i= 1, 2,..., n; k= 1, 2,..., N x i The inverse-problem of chemical reaction kinetics Concentration Time t

41 Neurobiology Neural networks, collective properties, nonlinear dynamics, signalling,... A single neuron signaling to a muscle fiber

42 Neurobiology Neural networks, collective properties, nonlinear dynamics, signalling,... dv d t = C 1 3 I g Na m h ( V VNa) g K n 4 M ( V V K ) g l ( V V l ) dm = α (1 m) dt dh dt dn dt m β m = α (1 h) h β h = α (1 n) n β n m h n Hogdkin-Huxley OD equations

43 The human brain neurons connected by to synapses

44 Evolutionary biology Optimization through variation and selection, relation between genotype, phenotype, and function,... RNA molecules Bacteria Higher multicelluar organisms Generation time generations 10 6 generations 10 7 generations 10 sec 27.8 h = 1.16 d d 3.17 a 1 min 6.94 d 1.90 a a 20 min d a 380 a 10 h a a a 10 d 274 a a a 20 a a a a Time scales of evolutionary change

45 1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie

46 Element of class 1: The RNA molecule

47 Stock Solution Reaction Mixture 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 N ( t) N ± N The flowreactor as a device for studies of evolution in vitro and in silico

48 Replication rate constant: f k = / [ + d S (k) ] d S (k) = d H (S k,s ) f 6 f 7 f5 f 0 f f 4 f 1 f 2 f 3 Evaluation of RNA secondary structures yields replication rate constants

49 3'-End 5'-End Randomly chosen initial structure Phenylalanyl-tRNA as target structure

50 50 S Average structure distance to target d Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Trajectory (physicists view)

51 36 Relay steps Number of relay step 38 Evolutionary trajectory Time Average structure distance to target d S Endconformation of optimization

52 36 Relay steps Number of relay step 38 Evolutionary trajectory Time Average structure distance to target d S Reconstruction of the last step 43 44

53 36 Relay steps Number of relay step Average structure distance to target d S 42 Evolutionary trajectory Reconstruction of last-but-one step ( 44) Time

54 36 Relay steps Number of relay step Average structure distance to target d S Evolutionary trajectory Reconstruction of step ( 43 44) Time

55 36 Relay steps Number of relay step Average structure distance to target d S Evolutionary trajectory Reconstruction of step ( ) Time

56 Average structure distance to target d S 10 Relay steps Number of relay step Evolutionary trajectory Time Evolutionary process Reconstruction Reconstruction of the relay series

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

58 50 Relay steps S Average structure distance to target d Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Trajectory and relay steps

59 Number of relay step 28 neutral point mutations during a long quasi-stationary epoch Average structure distance to target d S Uninterrupted presence Evolutionary trajectory Time (arbitrary units) Transition inducing point mutations Neutral point mutations Neutral genotype evolution during phenotypic stasis

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

75 Element of class 2: The ant worker

76 Ant colony Random foraging Food source Foraging behavior of ant colonies

77 Ant colony Food source detected Food source Foraging behavior of ant colonies

78 Ant colony Pheromone trail laid down Food source Foraging behavior of ant colonies

79 Ant colony Pheromone controlled trail Food source Foraging behavior of ant colonies

80 RNA model Foraging behavior of ant colonies Element RNA molecule Individual worker ant Mechanism relating elements Mutation in quasi-species Genetics of kinship Search process Optimization of RNA structure Recruiting of food Search space Sequence space Three-dimensional space Random step Mutation Element of ant walk Self-enhancing process Replication Secretion of pheromone Interaction between elements Mean replication rate Mean pheromone concentration Goal of the search Target structure Food source Temporary memory RNA sequences in population Pheromone trail Learning entity Population of molecules Ant colony Learning at population or colony level by trial and error Two examples: (i) RNA model and (ii) ant colony

81 1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie

82 O 5' - end CH 2 O N 1 5'-end GCGGAUUUAGCUCAGUUGGGAGAGCGCCAGACUGAAGAUCUGGAGGUCCUGUGUUCGAUCCACAGAAUUCGCACCA 3 -end O OH N k = A, U, G, C O P O CH 2 O N 2 Na O O OH O P O CH 2 O N 3 3'-end Na O 5 -end Definition of RNA structure O O P OH O CH 2 O N 4 70 Na O O OH O Na P O O 3' - end

83 James D. Watson and Francis H.C. Crick Nobel prize fifty years double helix Stacking of base pairs in nucleic acid double helices (B-DNA)

84 C G C C Å 54.4 U = A C C Å 56.2 Watson-Crick type base pairs

85 O N N O N H H N O N G=U N H H Deviation from Watson-Crick geometry O N N H O N U=G O H N H N N N Deviation from Watson-Crick geometry H Wobble base pairs

86 Sequence 5'-End GCGGAUUUAGCUCAGDDGGGAGAGCMCCAGACUGAAYAUCUGGAGMUCCUGUGTPCGAUCCACAGAAUUCGCACCA 3'-End 3'-End 5'-End 70 Secondary structure Symbolic notation 5'-End 3'-End A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs

87 Minimal hairpin loop size: n lp 3 Minimal stack length: n st 2 Recursion formula for the number of acceptable RNA secondary structures

88 Computed numbers of minimum free energy structures over different nucleotide alphabets P. Schuster, Molecular insights into evolution of phenotypes. In: J. Crutchfield & P.Schuster, Evolutionary Dynamics. Oxford University Press, New York 2003, pp

89 RNA sequence RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Biophysical chemistry: thermodynamics and kinetics Empirical parameters Inverse folding of RNA: Biotechnology, design of biomolecules with predefined structures and functions RNA structure Sequence, structure, and function

90 5 -end 3 -end A C 0 Free energy G S 5 (h) S 6 (h) S 7 (h) S 4 (h) S 8 (h) (h) S 3 (h) S (h) 1 S 2 (h) S 9 Suboptimal conformations C G C U U A A U U C G G G C A A U A A U C A C C U G A G G C U G U U G G G A C A C G U G G C U A G G S 0 (h) Minimum of free energy The minimum free energy structures on a discrete space of conformations

91 hairpin loop hairpin loop hairpin loop stack free end joint stack stack stack free end stack free end bulge internal loop hairpin loop stack hairpin loop multiloop hairpin loop stack stack stack Elements of RNA secondary structures as used in free energy calculations free end free end

92 5 -end 3 -end free energy of stacking < 0 gij, kl + h( nl ) + b ( nb ) + i ( ni G0 = ) stacks of base pairs hairpin loops bulges A C C G C U U A A U U C G G G C A A U A A U C A C C U G A G G C U G U U G G G A C A C G U G G C U internal loops L A G G Folding of RNA sequences into secondary structures of minimal free energy, G 0 300

93 Maximum matching An example of a dynamic programming computation of the maximum number of base pairs Back tracking yields the structure(s). [i,k-1] [ k+1,j ] i i+1 i+2 k j-1 j j+1 X i,k-1 X k+1,j X { X, max (( X + 1 X ρ )} i, j+ 1 = max i, j i k j 1 i, k 1 + k+ 1, j) k, j+ 1 Minimum free energy computations are based on empirical energies GGCGCGCCCGGCGCC RNAStudio.lnk GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA UGGUUACGCGUUGGGGUAACGAAGAUUCCGAGAGGAGUUUAGUGACUAGAGG

94 Maximum matching j i G G C G C G C C C G G C G C C An example of a dynamic programming computation of the maximum number of base pairs Back tracking yields the structure(s). [i,k-1] i i+1 i+2 k X X i,k-1 [ k+1,j ] j-1 j X k+1,j j+1 { X, max (( X + 1 X ρ )} i, j+ 1 = max i, j i k j 1 i, k 1 + k+ 1, j) k, j+ 1 1 G * * G * * C * * G * * C * * G * * C * * C * * C * * G * * G * * C * * G * * 1 14 C * * 15 C * Minimum free energy computations are based on empirical energies GGCGCGCCCGGCGCC GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA RNAStudio.lnk UGGUUACGCGUUGGGGUAACGAAGAUUCCGAGAGGAGUUUAGUGACUAGAGG

95 1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie

96 A single neuron signaling to a muscle fiber

97 A B Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

98 Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

99 Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

100 dv d t = C 1 3 I g Na m h ( V VNa) g K n 4 M ( V V K ) g l ( V V ) l dm = α (1 m) dt dh dt dn dt m β m = α (1 h) h β h = α (1 n) n β n m h n Hogdkin-Huxley OD equations Hhsim.lnk Simulation of space independent Hodgkin-Huxley equations: Voltage clamp and constant current

101 r L V V g V V n g V V h m g t V C x V R l l K K Na Na π 2 ) ( ) ( ) ( = m m t m m β m α = ) (1 h h t h h β h α = ) (1 n n t n n β n α = ) (1 Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = V( ) with = x + t Hodgkin-Huxley equations describing pulse propagation along nerve fibers

102 d V 2 R d ξ dv θ + dξ [ ] 3 g m h( V V ) + g n ( V V ) + g ( V V ) 2π r L 1 2 = C 4 M Na Na K K l l θ d m dξ = α m (1 m) β m m Hodgkin-Huxley PDEquations θ d h d ξ = α h (1 h) β h h Travelling pulse solution: V(x,t) = V( ) with = x + t θ d n d ξ = α n (1 n) β n n Hodgkin-Huxley equations describing pulse propagation along nerve fibers

103 100 V [mv] [cm] T = 18.5 C; θ = cm / sec

104 T = 18.5 C; θ = cm / sec

105 T = 18.5 C; θ = cm / sec

106 40 30 V [mv] [cm] T = 18.5 C; θ = cm / sec

107 T = 18.5 C; θ = cm/sec

108 T = 18.5 C; θ = cm/sec

109 Propagating wave solutions of the Hodgkin-Huxley equations

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111 Acknowledgement of support Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No , 10578, 11065, 13093, 13887, and Universität Wien Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European Commission: Project No. EU Siemens AG, Austria Österreichische Akademie der Wissenschaften Österreichische Akademie der Wissenschaften The Santa Fe Institute and the Universität Wien The software for producing RNA movies was developed by Robert Giegerich and coworkers at the Universität Bielefeld

112 Coworkers Walter Fontana, Santa Fe Institute, NM Universität Wien Christian Reidys, Christian Forst, Los Alamos National Laboratory, NM Peter Stadler, Bärbel Stadler, Universität Leipzig, GE Ivo L.Hofacker, Christoph Flamm, Universität Wien, AT Andreas Wernitznig, Michael Kospach, Universität Wien, AT Ulrike Langhammer, Ulrike Mückstein, Stefanie Widder Jan Cupal, Kurt Grünberger, Andreas Svrček-Seiler, Stefan Wuchty Ulrike Göbel, Institut für Molekulare Biotechnologie, Jena, GE Walter Grüner, Stefan Kopp, Jaqueline Weber

113 Web-Page for further information:

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