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
The Fibonacci numbers Leonardo da Pisa Fibonacci Filius Bonacci ~1180 ~1240
The Fibonacci numbers generation 1 2 3 4 5 6 # pairs 1 1 2 3 5 8 Bodo Werner, Universität Hamburg, 2006
The Fibonacci numbers Johannes Kepler (1571-1630)
Space filling squares The Fibonacci spirals
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) Ronald Fisher s selection equation dφ dt = 2 ( 2 2 < a > < a > ) = 2 var{} a 0
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
Turing patterns on animal skins and shells James D. Murray Hans Meinhardt Alfred Gierer
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
A B 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.
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
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 ) ( ) ( ) ( 1 4 3 2 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
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
Temperature dependence of the Hodgkin-Huxley equations
Systematic investigation of pulse behavior
100 V [mv] 50 0-50 1 2 3 4 5 6 [cm] T = 18.5 C; θ = 1873.33 cm / sec
T = 18.5 C; θ = 1873.3324514717698 cm / sec
T = 18.5 C; θ = 1873.3324514717697 cm / sec
40 30 V [mv] 20 10 0-10 6 8 10 12 14 16 18 [cm] T = 18.5 C; θ = 544.070 cm / sec
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 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.
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 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
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 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
Quasispecies Driving virus populations through threshold The error threshold in replication
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
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
The Vienna RNA-Package: A library of routines for folding, inverse folding, sequence and structure alignment, kinetic folding, cofolding,
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
The inverse folding algorithm searches for sequences that form a given RNA secondary structure under the minimum free energy criterion.
Sequence space and structure space
Space of genotypes: I = { I, I, I, I,..., I } ; Hamming metric 1 2 3 4 N Space of phenotypes: S = { S, S, S, S,..., S } ; metric (not required) 1 2 3 4 M N M ( I) = j S k -1 G k = ( S ) U I ( I) = S k j j k A mapping and its inversion
Degree of neutrality of neutral networks and the connectivity threshold
A multi-component neutral network formed by a rare structure: O < Ocr
A connected neutral network formed by a common structure: O > Ocr
Stochastic simulation of evolution of RNA molecules
Randomly chosen initial structure Phenylalanyl-tRNA as target structure
In silico optimization in the flow reactor: Evolutionary Trajectory
A sketch of optimization on neutral networks
Application of molecular evolution to problems in biotechnology
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.
Structure S 0 Structure S 1 The intersection of two compatible sets is always non empty: C 0 C 1
Reference for the definition of the intersection and the proof of the intersection theorem
A ribozyme switch E.A.Schultes, D.B.Bartel, Science 289 (2000), 448-452
Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase (A) and a natural cleavage ribozyme of hepatitis- -virus (B)
The sequence at the intersection: An RNA molecules which is 88 nucleotides long and can form both structures
Two neutral walks through sequence space with conservation of structure and catalytic activity
Web-Page for further information: http://www.tbi.univie.ac.at/~pks