Von der Thermodynamik zu Selbstorganisation, Evolution und Information

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2 Von der Thermodynamik zu Selbstorganisation, Evolution und Information Peter Schuster Institut für Theoretische hemie und Molekulare Strukturbiologie der Universität Wien Kolloquium des Physikalischen Instituts Heidelberg,

3 Equilibrium thermodynamics is based on two major statements: 1. The energy of the universe is a constant (first law). 2. The entropy of the universe never decreases (second law). arnot, Mayer, Joule, Helmholtz, lausius, D.Jou, J.asas-Vázquez,.Lebon, Extended Irreversible Thermodynamics, 1996

4 Stock Solution Reaction Mixture p T T ds env ds i d e S ds 0 ds 0 ds 0 Isolated system U = const., V = const., ds 0 losed system T = const., p = const., d = du-pdv- TdS 0 Open system ds = + ds 0 ds env ds = d S+ d S ds i i 0 e Entropy changes in different thermodynamic systems

5 S max 2 ( ds) < 0 U,V,equil Entropy pproach towards equilibrium Enlarged scale Fluctuations around equilibrium Time Thermodynamics of closed systems: Second law Entropy is a non-decreasing function S S max Entropy and fluctuations at equilibrium

6 Stock Solution [a] = a 0 Reaction Mixture [a],[b] B B B B B B B B B B B B Flow rate r = R -1 Reactions in the continuously stirred tank reactor (STR)

7 oncentration a [a 0 ] Flow rate r [t -1 ] B Reversible first order reaction in the flow reactor

8 oncentration a [a 0 ] Flow rate r [t ] = 0 = = = + B 2 B B utocatalytic second order and uncatalyzed reaction in the flow reactor

9 oncentration a [a 0 ] Flow rate r [t -1 ] = 0 = = = = +2 B 3B B utocatalytic third order and uncatalyzed reaction in the flow reactor

10 utocatalytic third order reactions Direct, + 2 X 3 X, or hidden in the reaction mechanism (Belousow-Zhabotinskii reaction). Multiple steady states Oscillations in homogeneous solution Deterministic chaos Turing patterns Spatiotemporal patterns (spirals) Deterministic chaos in space and time Pattern formation in autocatalytic third order reactions.nicolis, I.Prigogine. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations. John Wiley, New York 1977

11 utocatalytic second order reactions Direct, + I 2 I, or hidden in the reaction mechanism hemical self-enhancement ombustion and chemistry of flames Selection of laser modes Selection of molecular species competing for common sources utocatalytic second order reactions are the basis of selection processes. The autocatalytic step is formally equivalent to replication or reproduction.

12 Stock Solution [] = a 0 Reaction Mixture: ; I, k=1,2,... k k 1 + I 2 I 1 1 d 1 k 2 + I 2 I 2 2 d 2 k 3 + I 2 I 3 3 d 3 k 4 + I 2 I 4 4 d 4 k 5 + I 2 I 5 5 d 5 Replication in the flow reactor P.Schuster & K.Sigmund, Dynamics of evolutionary optimization, Ber.Bunsenges.Phys.hem. 89: (1985)

13 oncentration of stock solution a 0 + I + I I 1 + I 2 I I 2 I I 2 I I 2 I I 2 I 5 5 k > k > k > k > k Flow rate r = R -1 Selection in the flow reactor: Reversible replication reactions

14 oncentration of stock solution a 0 + I 1 + I 2 I I 2 I I 2 I I 2 I I 2 I 5 5 k > k > k > k > k Flow rate r = R -1 Selection in the flow reactor: Irreversible replication reactions

15 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

16 () + 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

17 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

18 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

19 Evolution of Populations: Ronald Fisher s conjecture Mean fitness is a non-decreasing function f = k x k (t) f k / k x k (t) f max Thermodynamics of closed systems: Entropy is a non-decreasing function Second law S S max

20 The origins of changes in genotypes or mutations are either replication errors or damage of nucleic acid molecules.

21 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.

22 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

23 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

24 TTTTTTTTTTTTTTT... TTTTTTTTTTTTTT... 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 induces a metric in sequence space

25 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

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

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

28 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

29 Decrease in mean fitness due to quasispecies formation The increase in RN production rate during a serial transfer experiment

30 Ronald Fisher s conjecture of optimization of mean fitness in populations does not always hold for replication-mutation systems: In general evolutionary dynamics the mean fitness of populations may also decrease monotonously or even go through a maximum or minimum. Optimization of fitness is, nevertheless, fulfilled in most cases, and it can be understood as a useful heuristic.

31 Learning and origin of information on the level of populations: The change in populations as a consequence of adaptation to the environment creates information (on the environmental conditions). The population learns through mutation and selection. Information is conserved through replication that reproduces mutants and error-free copies in the same way.

32 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

33 Sequence 5'-End 3'-End UUUUDDMUY UUMUUUTPUUU 3'-End 5'-End 70 Secondary structure Tertiary structure '-End 3'-End Symbolic notation The RN secondary structure is a listing of, U, and U base pairs. It is understood in contrast to the full 3Dor tertiary structure at the resolution of atomic coordinates. RN secondary structures are biologically relevant. They are, for example, conserved in evolution.

34 Minimum free energy criterion 1st 2nd 3rd trial 4th 5th Inverse folding UUUUUUUUUUUU UUUUUUUUUUUUUUUUU UUUUUUUUUUUUU UUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUU The inverse folding algorithm searches for sequences that form a given RN secondary structure under the minimum free energy criterion.

35 riterion of Minimum Free Energy UUUUUUUUUUUU UUUUUUUUUUUUUUUUU UUUUUUUUUUUUU UUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUU Sequence Space Shape Space

36 S k = ψ( I. ) f k = fs ( k ) Sequence space Phenotype space Non-negative numbers Mapping from sequence space into phenotype space and into fitness values

37 S k = ψ( I. ) f k = fs ( k ) Sequence space Phenotype space Non-negative numbers

38 S k = ψ( I. ) f k = fs ( k ) Sequence space Phenotype space Non-negative numbers The pre-image of the structure S k in sequence space is the neutral network k

39 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.

40 Step 00 Sketch of sequence space Random graph approach to neutral networks

41 Step 01 Sketch of sequence space Random graph approach to neutral networks

42 Step 02 Sketch of sequence space Random graph approach to neutral networks

43 Step 03 Sketch of sequence space Random graph approach to neutral networks

44 Step 04 Sketch of sequence space Random graph approach to neutral networks

45 Step 05 Sketch of sequence space Random graph approach to neutral networks

46 Step 10 Sketch of sequence space Random graph approach to neutral networks

47 Step 15 Sketch of sequence space Random graph approach to neutral networks

48 Step 25 Sketch of sequence space Random graph approach to neutral networks

49 Step 50 Sketch of sequence space Random graph approach to neutral networks

50 Step 75 Sketch of sequence space Random graph approach to neutral networks

51 Step 100 Sketch of sequence space Random graph approach to neutral networks

52 -1 k k j j k = ( S) U I ( I) = S λ j = 12 / 27, 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 Mean degree of neutrality and connectivity of neutral networks

53 multi-component neutral network iant omponent

54 connected neutral network

55 U U U U U U U U U U U U U U U U U U U U ompatible Incompatible 5 -end 5 -end 3 -end 3 -end ompatibility of sequences with structures sequence is compatible with its minimum free energy structure and all its suboptimal structures.

56 k k Neutral network 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 (neutral network k ) or one of its suboptimal structures.

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

58 1 : : The intersection of two compatible sets is always non empty: 1 2

59 Optimization of RN molecules in silico W.Fontana, P.Schuster, computer model of evolutionary optimization. Biophysical hemistry 26 (1987), W.Fontana, W.Schnabl, P.Schuster, Physical aspects of evolutionary optimization and adaptation. Phys.Rev. 40 (1989), M..Huynen, W.Fontana, P.F.Stadler, Smoothness within ruggedness. The role of neutrality in adaptation. Proc.Natl.cad.Sci.US 93 (1996), W.Fontana, P.Schuster, ontinuity in evolution. On the nature of transitions. Science 280 (1998), W.Fontana, P.Schuster, Shaping space. The possible and the attainable in RN genotypephenotype mapping. J.Theor.Biol. 194 (1998), B.M.R. Stadler, P.F. Stadler,.P. Wagner, W. Fontana, The topology of the possible: Formal spaces underlying patterns of evolutionary change. J.Theor.Biol. 213 (2001),

60 3'-End 5'-End Randomly chosen initial structure Phenylalanyl-tRN as target structure

61 Stock Solution Reaction Mixture Fitness function: f k = / [ + d S (k) ] d S (k) = d s (I k,i ) The flowreactor as a device for studies of evolution in vitro and in silico

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

63 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

64 50 S verage distance from initial structure 50 - d Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Trajectory (biologists view)

65 50 S verage structure distance to target d Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor: Trajectory (physicists view)

66 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

67 Shift Roll-Over α α a Flip a α a b Double Flip a α b β β Main or discontinuous transitions: Structural innovations, occur rarely on single point mutations losing of onstrained Stacks Multiloop

68 50 Relay steps Main transitions verage structure distance to target d S Uninterrupted presence Evolutionary trajectory Time (arbitrary units) In silico optimization in the flow reactor

69 Shortening of Stacks Elongation of Stacks Multiloop Minor or continuous transitions: Occur frequently on single point mutations Opening of onstrained Stacks

70 Statistics of evolutionary trajectories Population size N Number of replications < n > rep Number of transitions < n > tr Number of main transitions < n > dtr The number of main transitions or evolutionary innovations is constant.

71 Stable trn clover leaf structures built from binary, -only, sequences exist. The corresponding sequences are readily found through inverse folding. Optimization by mutation and selection in the flow reactor has so far always been unsuccessful. 5'-End 3'-End The neutral network of the trn clover leaf in sequence space is not connected, whereas to the corresponding neutral network in U sequence space is very close to the critical connectivity threshold, cr. Here, both inverse folding and optimization in the flow reactor are successful The success of optimization depends on the connectivity of neutral networks.

72 ...Variations neither useful not injurious would not be affected by natural selection, and would be left either a fluctuating element, as perhaps we see in certain polymorphic species, or would ultimately become fixed, owing to the nature of the organism and the nature of the conditions.... harles Darwin, Origin of species (1859)

73 daptive Periods End of Walk Fitness Random Drift Periods Start of Walk enotype Space Evolution in genotype space sketched as a non-descending walk in a fitness landscape

74 Evolutionary design of RN molecules D.B.Bartel, J.W.Szostak, In vitro selection of RN molecules that bind specific ligands. Nature 346 (1990), Tuerk, L.old, SELEX - Systematic evolution of ligands by exponential enrichment: RN ligands to bacteriophage T4 DN polymerase. Science 249 (1990), D.P.Bartel, J.W.Szostak, Isolation of new ribozymes from a large pool of random sequences. Science 261 (1993), R.D.Jenison, S..ill,.Pardi, B.Poliski, High-resolution molecular discrimination by RN. Science 263 (1994), Y. Wang, R.R.Rando, Specific binding of aminoglycoside antibiotics to RN. hemistry & Biology 2 (1995), Jiang,. K. Suri, R. Fiala, D. J. Patel, Saccharide-RN recognition in an aminoglycoside antibiotic-rn aptamer complex. hemistry & Biology 4 (1997), 35-50

75 mplification Diversification Selection ycle enetic Diversity Selection Selection cycle used in applied molecular evolution to design molecules with predefined properties no Desired Properties??? yes

76 Retention of binders Elution of binders hromatographic column The SELEX technique for the evolutionary design of aptamers

77 Secondary structures of aptamers binding theophyllin, caffeine, and related compounds

78 additional methyl group Dissociation constants and specificity of theophylline, caffeine, and related derivatives of uric acid for binding to a discriminating aptamer TT8-4

79 Schematic drawing of the aptamer binding site for the theophylline molecule

80 ptamer binding to aminoglycosid antibiotics: Structure of ligands Y. Wang, R.R.Rando, Specific binding of aminoglycoside antibiotics to RN. hemistry & Biology 2 (1995),

81 tobramycin U U U U U U U U U U U U RN aptamer Formation of secondary structure of the tobramycin binding RN aptamer L. Jiang,. K. Suri, R. Fiala, D. J. Patel, Saccharide-RN recognition in an aminoglycoside antibiotic-rn aptamer complex. hemistry & Biology 4:35-50 (1997)

82 The three-dimensional structure of the tobramycin aptamer complex L. Jiang,. K. Suri, R. Fiala, D. J. Patel, hemistry & Biology 4:35-50 (1997)

83 ribozyme switch E..Schultes, D.B.Bartel, One sequence, two ribozymes: Implication for the emergence of new ribozyme folds. Science 289 (2000),

84 Two ribozymes of chain lengths n = 88 nucleotides: n artificial ligase () and a natural cleavage ribozyme of hepatitis- -virus (B)

85 The sequence at the intersection: n RN molecules which is 88 nucleotides long and can form both structures

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

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

88 Sequence of mutants from the intersection to both reference ribozymes

89 Reference for postulation and in silico verification of neutral networks

90 oworkers Walter Fontana, Santa Fe Institute, NM hristian Reidys, hristian Forst, Los lamos National Laboratory, NM Peter Stadler, Universität Leipzig, E Ivo L.Hofacker, hristoph Flamm, Universität Wien, T Bärbel Stadler, ndreas Wernitznig, Universität Wien, T Michael Kospach, 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