Different kinds of robustness in genetic and metabolic networks

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2 Different kinds of robustness in genetic and metabolic networks Peter Schuster Institut für Theoretische hemie und Molekulare Strukturbiologie der Universität Wien Seminar lecture Linz,

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

5 enomics and proteomics Large scale data processing, sequence comparison... E. coli: Length of the enome Nucleotides Number of ell Types 1 Number of enes Man: Length of the enome Nucleotides Number of ell Types 200 Number of enes

6 Fully sequenced genomes Organisms 751 projects 153 complete (16, 118 B, 19 E) (Eukarya examples: mosquito (pest, malaria), sea squirt, mouse, yeast, homo sapiens, arabidopsis, fly, worm, ) 598 ongoing (23, 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: NBI 68 phages 1328 viruses 35 viroids 472 organelles (423 mitochondria, 32 plastids, 14 plasmids, 3 nucleomorphs) Source: Integrated enomics, Inc. ugust 12 th, 2003

7 Wolfgang Wieser. Die Erfindung der Individualität oder die zwei esichter der Evolution. Spektrum kademischer Verlag, Heidelberg Wilson. The Molecular Basis of Evolution. Scientific merican, Oct.1985,

8 Replication: DN 2 DN + Transcription: DN RN Nucleotides mino cids Lipids arbohydrates Small Molecules Metabolism Food Waste Ribosom mrn Protein enetic ode mrn Translation: mrn Protein The gene is a stretch of DN which after transcription gives rise to a mrn

9 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 , and (4) vascular endothelial growth factor ene expression DN 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

10 Elimination of introns through splicing genomic DN mrn The gene is a stretch of DN which after transcription and processing gives rise to a mrn

11 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 DN 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-rn that is translated.

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

13 Developmental biology ene 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-DN

14 Development of the fruit fly drosophila melanogaster: enetics, experiment, and imago

15 Linear chain Network Processing of information in cascades and networks

16 lbert-lászló Barabási, Linked The New Science of Networks. Perseus Publ., ambridge, M, 2002

17 Distributed network Small world network lbert-lászló Barabási, Linked The New Science of Networks. Perseus Publ., ambridge, M, 2002

18 lbert-lászló Barabási, Linked The New Science of Networks Perseus Publ., ambridge, M, 2002

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

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

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

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

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

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

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

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

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

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

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

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

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

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

33 links # nodes nalysis of nodes and links in a step by step evolved network

34 Structures in Directed Networks lbert-lászló Barabási, Linked The New Science of Networks. Perseus Publ., ambridge, M, 2002

35 ell 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, and 200 eukaryotic cell types

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

37 The citric acid or Krebs cycle (enlarged from previous slide).

38 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 oncentration Solution curves: x i xi ( t); i = 1, 2,..., n eneral 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

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 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 eneral 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 oncentration Time t

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

41 The human brain neurons connected by to synapses

42 Evolutionary biology Optimization through variation and selection, relation between genotype, phenotype, and function,... RN molecules Bacteria Higher multicelluar organisms eneration 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

43 O 5' - end H 2 O N 1 5'-end UUUUUUUUUUUUUUUUU 3 -end O OH N k =, U,, O P O H 2 O N 2 Na O O OH O P O H 2 O N 3 3'-end Na O 5 -end RN O O P OH O H 2 O N 4 70 Na O O OH O Na P O O 3' - end Definition of RN structure

44 James D. Watson, 1928-, and Francis rick, 1916-, Nobel Prize fifty years double helix The three-dimensional structure of a short double helical stack of B-DN

45 Sequence 5'-End UUUUDDMUYUUMUUUTPUUU 3'-End 3'-End 5'-End 70 Secondary structure

46 5' 3' Plus Strand Synthesis 5' 3' Plus Strand 3' Synthesis 5' 3' Plus Strand Minus Strand 3' 5' 5' Plus Strand 5' Minus Strand omplex Dissociation + 3' 3' omplementary replication as the simplest copying mechanism of RN omplementarity is determined by Watson-rick base pairs: and =U

47 () + I 1 f 1 I 1 + I 1 () + I 2 f 2 I 2 + I 2 dx / dt = f x - x i i i i Φ = x (f i - Φ i ) Φ =Σ j f j x j ; Σ x = 1 ; i,j =1,2,...,n j j () + I i f i I i + I i [I i] = x i 0 ; i =1,2,...,n ; [] = a = constant f m = max { f j ; j=1,2,...,n} () + I m f m I m + I m x m (t) 1 for t () + I n f n I n + I n Reproduction of organisms or replication of molecules as the basis of selection

48 Selection equation: [I i ] = x i 0, f i > 0 n n ( f φ ), i = 1,2, L, n; x = 1; φ = f x f dx i = xi i i i j j j dt = = 1 = 1 Mean fitness or dilution flux, φ (t), is a non-decreasing function of time, n dφ = dt i= 1 f i dx dt i = f 2 ( ) 2 f = var{ f } 0 Solutions are obtained by integrating factor transformation ( 0) exp( f t) xi i xi () t = ; i = 1,2, L, n n x ( ) ( f jt) j = j 0 exp 1

49 s = ( f 2 -f 1 ) / f 1 ; f 2 > f 1 ; x 1 (0) = 1-1/N ; x 2 (0) = 1/N 1 Fraction of advantageous variant s = 0.1 s = 0.02 s = Time [enerations] Selection of advantageous mutants in populations of N = individuals

50 5' 3' Plus Strand 5' 3' U UU Plus Strand Insertion 3' 5' 3' Minus Strand U U Plus Strand 5' 3' Deletion Point Mutation Mutations in nucleic acids represent the mechanism of variation of genotypes.

51 Theory of molecular evolution M.Eigen, Self-organization of matter and the evolution of biological macromolecules. Naturwissenschaften 58 (1971), J. Thompson, J.L. McBride, On Eigen's theory of the self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21 (1974), B.L. Jones, R.H. Enns, S.S. Rangnekar, On the theory of selection of coupled macromolecular systems. Bull.Math.Biol. 38 (1976), M.Eigen, P.Schuster, The hypercycle. principle of natural self-organization. Part : Emergence of the hypercycle. Naturwissenschaften 58 (1977), M.Eigen, P.Schuster, The hypercycle. principle of natural self-organization. Part B: The abstract hypercycle. Naturwissenschaften 65 (1978), 7-41 M.Eigen, P.Schuster, The hypercycle. principle of natural self-organization. Part : The realistic hypercycle. Naturwissenschaften 65 (1978), J. Swetina, P. Schuster, Self-replication with errors - model for polynucleotide replication. Biophys.hem. 16 (1982), J.S. Mcaskill, localization threshold for macromolecular quasispecies from continuously distributed replication rates. J.hem.Phys. 80 (1984), M.Eigen, J.Mcaskill, P.Schuster, The molecular quasispecies. dv.hem.phys. 75 (1989), Reidys,.Forst, P.Schuster, Replication and mutation on neutral networks. Bull.Math.Biol. 63 (2001), 57-94

52 I 1 + I j f j Q j1 I 2 + I j Σ dx / dt = f x - x i j jq ji j i Φ Φ = Σ f x j j i ; Σ x = 1 ; j j Σ Q i ij = 1 f j Q j2 f j Q ji I i + I j [I i ] = x i 0 ; i =1,2,...,n ; [] = a = constant () + I j f j Q jj I j + I j Q = (1-) p ij l-d(i,j) p d(i,j) p... Error rate per digit f j Q jn I n + I j l... hain length of the polynucleotide d(i,j)... Hamming distance between I i and I j hemical kinetics of replication and mutation as parallel reactions

53 ... U... d =2 d =1 H... U UU... H d =1 H... UU... ity-block distance in sequence space 2D Sketch of sequence space Single point mutations as moves in sequence space

54 Mutant class Binary sequences are encoded by their decimal equivalents: = 0 and = 1, for example, "0" =, "14" =, "29" =, etc Sequence space of binary sequences of chain lenght n=5

55 TTTTTTTTTTTTTTT... TTTTTTTTTTTTTT... Hamming distance d (I,I ) = H (i) (ii) (iii) d (I,I) = 0 H 1 1 d (I,I) = d (I,I) H 1 2 H 2 1 d (I,I) d (I,I) + d (I,I) H 1 3 H 1 2 H 2 3 The Hamming distance between sequences induces a metric in sequence space

56 Mutation-selection equation: [I i ] = x i 0, f i > 0, Q ij 0 dx i n n n = f Q x x i n x f x j j ji j i φ, = 1,2, L, ; i i = 1; φ = j j j dt = = 1 = 1 = 1 f Solutions are obtained after integrating factor transformation by means of an eigenvalue problem x i () t = n 1 k n j= 1 ( 0) exp( λkt) c ( 0) exp( λ t) l = 0 ik ck ; i = 1,2, L, n; c (0) = n 1 k l k= 0 jk k k n i= 1 h ki x i (0) W 1 { f Q ; i, j= 1,2, L, n} ; L = { l ; i, j= 1,2, L, n} ; L = H = { h ; i, j= 1,2, L, n} i ij ij ij { λ ; k = 0,1,, n 1} 1 L W L = Λ = k L

57 Quasispecies Uniform distribution Error rate p = 1-q Quasispecies as a function of the replication accuracy q

58 oncentration Master sequence Mutant cloud Sequence space The molecular quasispecies in sequence space

59 e 1 x 1 l 0 e 1 e 3 x 3 x 2 e 2 e 2 e 3 l 2 l 1 { } 3 The quasispecies on the concentration simplex S 3 = x 0, i = 1, 2,3; x = 1 i i= 1 i

60 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 RN secondary structures yields replication rate constants

61 Hamming distance d (S,S ) = H (i) (ii) (iii) d (S,S) = 0 H 1 1 d (S,S ) = d (S,S ) H 1 2 H 2 1 d (S,S) d (S,S) + d (S,S) H 1 3 H 1 2 H 2 3 The Hamming distance between structures in parentheses notation forms a metric in structure space

62 Stock Solution Reaction Mixture Replication rate constant: f k = / [ + d S (k) ] d S (k) = d H (S k,s ) Selection constraint: # RN molecules is controlled by the flow N ( t) N ± N The flowreactor as a device for studies of evolution in vitro and in silico

63 3'-End 5'-End Randomly chosen initial structure Phenylalanyl-tRN as target structure

64 oncentration Master sequence Mutant cloud Off-the-cloud mutations Sequence space The molecular quasispecies in sequence space

65 enotype-phenotype Mapping I { S { = ( I { ) S { f = ƒ( S ) { { Evaluation of the Phenotype Mutation Q {j f 2 I 2 f 1 I1 I n I 1 f n I 2 f 2 f 1 I n+1 f n+1 f { Q f 3 I 3 Q f 3 I 3 f 4 I 4 I { f { I 4 I 5 f 4 f 5 f 5 I 5 Evolutionary dynamics including molecular phenotypes

66 50 S verage distance from initial structure 50 - d Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Trajectory (biologists view)

67 50 S verage structure distance to target d Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Trajectory (physicists view)

68 36 Relay steps Number of relay step 38 Evolutionary trajectory Time verage structure distance to target d S Endconformation of optimization

69 36 Relay steps Number of relay step 38 Evolutionary trajectory Time verage structure distance to target d S Reconstruction of the last step 43 44

70 36 Relay steps Number of relay step verage structure distance to target d S 42 Evolutionary trajectory Reconstruction of last-but-one step ( 44) Time

71 36 Relay steps Number of relay step verage structure distance to target d S Evolutionary trajectory Reconstruction of step ( 43 44) Time

72 36 Relay steps Number of relay step verage structure distance to target d S Evolutionary trajectory Reconstruction of step ( ) Time

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

74 Transition inducing point mutations Neutral point mutations hange in RN sequences during the final five relay steps 39 44

75 50 Relay steps S verage structure distance to target d Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Trajectory and relay steps

76 Number of relay step 28 neutral point mutations during a long quasi-stationary epoch verage 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

77 50 Relay steps Main transitions verage structure distance to target d S Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Main transitions

78 Three important steps in the formation of the trn clover leaf from a randomly chosen initial structure corresponding to three main transitions.

79 U Movies of optimization trajectories over the U and the alphabet

80 Frequency Runtime of trajectories Statistics of the lengths of trajectories from initial structure to target (U-sequences)

81 lphabet Runtime Transitions Main transitions No. of runs U U Statistics of trajectories and relay series (mean values of log-normal distributions)

82 Minimum free energy criterion Inverse folding of RN secondary structures The idea of inverse folding algorithm is to search for sequences that form a given RN secondary structure under the minimum free energy criterion.

83 Structure

84 3 -end 5 -end U U U U U Structure ompatible sequence

85 U U U U U U U U U U ompatible sequence Structure 5 -end 3 -end

86 U U U U U U U U U U ompatible sequence Structure 5 -end 3 -end Base pairs: U, U, U, U Single nucleotides: U,,,

87 U U U U U U U U U U Structure Incompatible sequence 5 -end 3 -end

88 Initial trial sequences Stop sequence of an unsuccessful trial Target sequence Target structure S k Intermediate compatible sequences pproach to the target structure S k in the inverse folding algorithm

89 Minimum free energy criterion 1st 2nd 3rd trial 4th 5th Inverse folding of RN secondary structures The inverse folding algorithm searches for sequences that form a given RN secondary structure under the minimum free energy criterion.

90 Theory of genotype phenotype mapping P. Schuster, W.Fontana, P.F.Stadler, I.L.Hofacker, From sequences to shapes and back: case study in RN secondary structures. Proc.Roy.Soc.London B 255 (1994), W.rüner, R.iegerich, D.Strothmann,.Reidys, I.L.Hofacker, P.Schuster, nalysis of RN sequence structure maps by exhaustive enumeration. I. Neutral networks. Mh.hem. 127 (1996), W.rüner, R.iegerich, D.Strothmann,.Reidys, I.L.Hofacker, P.Schuster, nalysis of RN sequence structure maps by exhaustive enumeration. II. Structure of neutral networks and shape space covering. Mh.hem. 127 (1996), M.Reidys, P.F.Stadler, P.Schuster, eneric properties of combinatory maps. Bull.Math.Biol. 59 (1997), I.L.Hofacker, P. Schuster, P.F.Stadler, ombinatorics of RN secondary structures. Discr.ppl.Math. 89 (1998), M.Reidys, P.F.Stadler, ombinatory landscapes. SIM Review 44 (2002), 3-54

91 S k = ψ( I. ) f k = fs ( k ) Sequence space Structure space Real numbers Mapping from sequence space into structure space and into function

92 S k = ψ( I. ) f k = fs ( k ) Sequence space Structure space Real numbers

93 S k = ψ( I. ) f k = fs ( k ) Sequence space Structure space Real numbers The pre-image of the structure S k in sequence space is the neutral network k

94 Neutral networks are sets of sequences forming the same structure. k is the pre-image of the structure S k in sequence space: k = -1 (S k ) π { j (I j ) = S k } The set is converted into a graph by connecting all sequences of Hamming distance one. Neutral networks of small RN molecules can be computed by exhaustive folding of complete sequence spaces, i.e. all RN 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.

95 -1 k k j j k = ( S) U I ( I) = S λ j = 12 / 27 = 0.444, λ k = (k) j k onnectivity threshold: / κ- λcr = 1 - κ -1 ( 1) λ λ lphabet size : U = 4 > λ k cr.... < λ k cr.... network k is connected network k is not connected cr ,U U,U U Mean degree of neutrality and connectivity of neutral networks

96 connected neutral network

97 multi-component neutral network iant omponent

98 3'-End 3'-End 3'-End 3'-End 5'-End 5'-End 5'-End 5'-End lphabet Degree of neutrality U U U U Degree of neutrality of cloverleaf RN secondary structures over different alphabets

99 Reference for postulation and in silico verification of neutral networks

100 Structure S k Neutral Network k k k ompatible Set k The compatible set k of a structure S k consists of all sequences which form S k as its minimum free energy structure (the neutral network k ) or one of its suboptimal structures.

101 Structure S 0 Structure S 1 The intersection of two compatible sets is always non empty: 0 1

102 Reference for the definition of the intersection and the proof of the intersection theorem

103 U U U U U 3 -end Minimum free energy conformation S 0 Suboptimal conformation S 1 U U U U U U U U U U sequence at the intersection of two neutral networks is compatible with both structures

104 S 1 S 0 Barrier tree for two long living structures basin '1' long living metastable structure basin '0' minimum free energy structure

105 Kinetics of RN refolding between a long living metastable conformation and the minmum free energy structure

106 cknowledgement of support Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No , 10578, 11065, , and Universität Wien Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European ommission: Project No. EU Siemens, ustria The Santa Fe Institute and the Universität Wien The software for producing RN movies was developed by Robert iegerich and coworkers at the Universität Bielefeld

107 oworkers Walter Fontana, Santa Fe Institute, NM Universität Wien hristian Reidys, hristian Forst, Los lamos National Laboratory, NM Peter Stadler, Bärbel Stadler, Universität Leipzig, E Ivo L.Hofacker, hristoph Flamm, Universität Wien, T ndreas Wernitznig, Michael Kospach, Universität Wien, T Ulrike Langhammer, Ulrike Mückstein, Stefanie Widder Jan upal, Kurt rünberger, ndreas Svrček-Seiler, Stefan Wuchty Ulrike öbel, Institut für Molekulare Biotechnologie, Jena, E Walter rüner, Stefan Kopp, Jaqueline Weber

108 Web-Page for further information:

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