The Advantage of Using Mathematics in Biology Peter Schuster - PowerPoint PPT Presentation

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, 15.04.2008

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

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

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

Leonardo da Pisa „Fibonacci“ – Filius Bonacci ~1180 – ~1240 The Fibonacci numbers

1 2 3 4 5 6 generation 1 1 2 3 5 8 # pairs Bodo Werner, Universität Hamburg, 2006 The Fibonacci numbers

Johannes Kepler (1571-1630) The Fibonacci numbers

The Fibonacci spirals Space filling squares

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

Gregor Mendel (1882-1884) Gregor Mendel‘s experiments on plant genetics Versuche über Pflanzen-Hybriden. Verhandlungen des naturforschenden Vereines in Brünn 4 : 3–47, 1866. Über einige aus künstlicher Befruchtung gewonnenen Hieracium-Bastarde. Verhandlungen des naturforschenden Vereines in Brünn 8 : 26–31, 1870.

Gregor Mendel‘s experiments on plant genetics

Gregor Mendel‘s experiments on plant genetics

Molecular explanation of Mendel‘s expriments – recombination

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

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 (1890-1962) Φ ( ) d { } = < > − < > = ≥ 2 2 a a a 2 2 var 0 dt Ronald Fisher‘s selection equation

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

A. M. Turing. The chemical basis of morphogenesis. Phil.Trans.Roy.Soc. London B 327 :37, 1952 Alan Turing (1912-1954) Spontaneous pattern formation in reaction diffusion equations

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

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

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

A single neuron signaling to a muscle fiber

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 : 500-544, 1952 Andrew Huxley The Hodgkin-Huxley equation

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

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

d V 1 = − − − − − − 3 4 I g m h V V g n V V g V V ( ) ( ) ( ) Na Na K K l l d t C M dm = α − − β m m Hogdkin-Huxley OD equations ( 1 ) m m dt dh = α − − β h h ( 1 ) h h dt dn = α − − β n n ( 1 ) n n dt A single neuron signaling to a muscle fiber

∂ ∂ 2 V V 1 = + − + − + − π 3 4 C g m h V V g n V V g V V r L ( ) ( ) ( ) 2 Na Na K K l l ∂ ∂ 2 R x t ∂ m = α − − β m m ( 1 ) Hodgkin-Huxley PDEquations m m ∂ t ∂ h = α − − β h h ( 1 ) Travelling pulse solution: V ( x,t ) = V ( � ) with h h ∂ t � = x + � t ∂ n = α − − β n n ( 1 ) n n ∂ t Hodgkin-Huxley equations describing pulse propagation along nerve fibers

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

Temperature dependence of the Hodgkin-Huxley equations

Systematic investigation of pulse behavior

6 T = 18.5 C; θ = 1873.33 cm / sec 5 � [cm] 4 3 2 1 100 50 0 -50 ] V m [ V

T = 18.5 C; θ = 1873.3324514717698 cm / sec

T = 18.5 C; θ = 1873.3324514717697 cm / sec

18 16 T = 18.5 C; θ = 544.070 cm / sec 14 � [cm] 12 10 8 6 40 30 20 10 0 -10 ] V m [ V

Propagating wave solutions of the Hodgkin-Huxley equations

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

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

Stock solution : activated monomers, ATP, CTP, GTP, UTP (TTP); a replicase, an enzyme that performs complemantary replication; buffer solution r = � R -1 Flow rate : 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 .

Chemical kinetics of replication and mutation as parallel reactions

Manfred Eigen‘s replication-mutation equation

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

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

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Uniform distribution in sequence space

Quasispecies Uniform distribution 0.00 0.05 0.10 Error rate p = 1-q Quasispecies as a function of the replication accuracy q

Chain length and error threshold ⋅ σ = − ⋅ σ ≥ ⇒ ⋅ − ≥ − σ n Q p n p ( 1 ) 1 ln ( 1 ) ln σ ln ≈ n p constant : K max n σ ln ≈ p n constant : K max p = − n Q p ( 1 ) K replicatio n accuracy p error rate K n chain length K f = σ m superiorit y of master sequence K ∑ ≠ − x f ( 1 ) m j j m