The Advantage of Using Mathematics in Biology

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2 The Advantage of Using Mathematics in Biology Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Erwin Schrödinger-Institut Wien,

3 Web-Page for further information:

4 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

5 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

6 The Fibonacci numbers Leonardo da Pisa Fibonacci Filius Bonacci ~1180 ~1240

7 The Fibonacci numbers generation # pairs Bodo Werner, Universität Hamburg, 2006

8 The Fibonacci numbers Johannes Kepler ( )

9 Space filling squares The Fibonacci spirals

10 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

11 Gregor Mendel ( ) Gregor Mendel s experiments on plant genetics Versuche über Pflanzen-Hybriden. Verhandlungen des naturforschenden Vereines in Brünn 4: 3 47, Über einige aus künstlicher Befruchtung gewonnenen Hieracium-Bastarde. Verhandlungen des naturforschenden Vereines in Brünn 8: 26 31, 1870.

12 Gregor Mendel s experiments on plant genetics

13 Gregor Mendel s experiments on plant genetics

14 Molecular explanation of Mendel s expriments recombination

15 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

16 alleles: A 1, A 2,..., A n frequencies: x i = [A i ] ; genotype: A i A k Fitness values: a ik = f(a i A k ), a ik = a ki Ronald Fisher ( ) Ronald Fisher s selection equation dφ dt = 2 ( 2 2 < a > < a > ) = 2 var{} a 0

17 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

18 A. M. Turing. The chemical basis of morphogenesis. Phil.Trans.Roy.Soc. London B 327:37, 1952 Alan Turing ( ) Spontaneous pattern formation in reaction diffusion equations

19 Boris Belousov and Anatol Zhabotinskii Boris Belousov Vincent Castets, Jacques Boissonade, Etiennette Dulos and Patrick DeKepper, Phys.Rev. Letters 64:2953, 1990 Experimental verification of Turing patterns in chemical reactions

20 Turing patterns on animal skins and shells James D. Murray Hans Meinhardt Alfred Gierer

21 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

22 A single neuron signaling to a muscle fiber

23 Alan Hodgkin A. L. Hodgkin and A. F. Huxley. A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. Journal of Physiology 117: , 1952 Andrew Huxley The Hodgkin-Huxley equation

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

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

26 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 A single neuron signaling to a muscle fiber

27 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

28 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

29 Temperature dependence of the Hodgkin-Huxley equations

30 Systematic investigation of pulse behavior

31 100 V [mv] [cm] T = 18.5 C; θ = cm / sec

32 T = 18.5 C; θ = cm / sec

33 T = 18.5 C; θ = cm / sec

34 40 30 V [mv] [cm] T = 18.5 C; θ = cm / sec

35 Propagating wave solutions of the Hodgkin-Huxley equations

36 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

37 Chemical kinetics of molecular evolution M. Eigen, P. Schuster, `The Hypercycle, Springer-Verlag, Berlin 1979

38 Stock solution: activated monomers, ATP, CTP, GTP, UTP (TTP); a replicase, an enzyme that performs complemantary replication; buffer solution Flow rate: r = R -1 The population size N, the number of polynucleotide molecules, is controlled by the flow r N ( t) N ± N The flowreactor is a device for studies of evolution in vitro and in silico.

39 Chemical kinetics of replication and mutation as parallel reactions

40 Manfred Eigen s replication-mutation equation

41 Mutation-selection equation: [I i ] = x i 0, f i > 0, Q ij 0 dx i n n n = Q f x x i n x f x j ij j 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

42 constant level sets of Selection of quasispecies with f 1 = 1.9, f 2 = 2.0, f 3 = 2.1, and p = 0.01 parametric plot on S 3

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44 Formation of a quasispecies in sequence space

45 Formation of a quasispecies in sequence space

46 Formation of a quasispecies in sequence space

47 Formation of a quasispecies in sequence space

48 Uniform distribution in sequence space

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

50 Chain length and error threshold p n p n p n p n p Q n σ σ σ σ σ ln : constant ln : constant ln ) ln(1 1 ) (1 max max = K K sequence master superiority of ) (1 length chain rate error accuracy replication ) (1 K K K K = = m j j m m n f x f σ n p p Q

51 Quasispecies Driving virus populations through threshold The error threshold in replication

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53 1. Fibonacci Rabbits, plants, and the golden ratio 2. Mendel Colors, genes, and inheritance 3. Fisher Synthesis of genetics and Darwinian evolution 4. Turing The origin of patterns 5. Hodgkin and Huxley Neurons and PDEs 6. Error thresholds and antiviral strategies 7. Neutral networks How evolution works

54 RNA sequence: GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Biophysical chemistry: thermodynamics and kinetics Empirical parameters RNA structure of minimal free energy Sequence, structure, and design

55 The Vienna RNA-Package: A library of routines for folding, inverse folding, sequence and structure alignment, kinetic folding, cofolding,

56 RNA sequence: GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Iterative determination of a sequence for the given secondary structure Inverse Folding Algorithm Inverse folding of RNA: Biotechnology, design of biomolecules with predefined structures and functions RNA structure of minimal free energy Sequence, structure, and design

57 The inverse folding algorithm searches for sequences that form a given RNA secondary structure under the minimum free energy criterion.

58 Sequence space and structure space

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60 Space of genotypes: I = { I, I, I, I,..., I } ; Hamming metric N Space of phenotypes: S = { S, S, S, S,..., S } ; metric (not required) M N M ( I) = j S k -1 G k = ( S ) U I ( I) = S k j j k A mapping and its inversion

61 Degree of neutrality of neutral networks and the connectivity threshold

62 A multi-component neutral network formed by a rare structure: O < Ocr

63 A connected neutral network formed by a common structure: O > Ocr

64 Stochastic simulation of evolution of RNA molecules

65 Randomly chosen initial structure Phenylalanyl-tRNA as target structure

66 In silico optimization in the flow reactor: Evolutionary Trajectory

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69 A sketch of optimization on neutral networks

70 Application of molecular evolution to problems in biotechnology

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

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

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

74 A ribozyme switch E.A.Schultes, D.B.Bartel, Science 289 (2000),

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

76 The sequence at the intersection: An RNA molecules which is 88 nucleotides long and can form both structures

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

78 Web-Page for further information:

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