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

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

The Fibonacci numbers

Johannes Kepler (1571-1630)

The Fibonacci numbers

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

• n 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

Ronald Fisher (1890-1962)

alleles: A1, A2, ..... , An frequencies: xi = [Ai] ; genotype: Ai·Ak Fitness values: aik = f(Ai·Ak), aik = aki

( )

{ }

var 2 2

2 2

≥ = > < − > < = Φ a a a dt d

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

Alan Turing (1912-1954)

• A. M. Turing. The chemical basis of morphogenesis.

Phil.Trans.Roy.Soc. London B 327:37, 1952 Spontaneous pattern formation in reaction diffusion equations

Boris Belousov Vincent Castets, Jacques Boissonade, Etiennette Dulos and Patrick DeKepper, Phys.Rev. Letters 64:2953, 1990 Boris Belousov and Anatol Zhabotinskii

Experimental verification of Turing patterns in chemical reactions

James D. Murray Hans Meinhardt Alfred Gierer

Turing patterns

• n 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 Andrew Huxley

• A. L. Hodgkin and A. F. Huxley. A Quantitative Description
• f Membrane Current and its Application to Conduction and

Excitation in Nerve. Journal of Physiology 117: 500-544, 1952 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.

) ( ) ( ) ( 1

4 3 l l K K Na Na M

V V g V V n g V V h m g I C t d V d − − − − − − =

m m dt dm

m m

β α − − = ) 1 ( h h dt dh

h h

β α − − = ) 1 ( n n dt dn

n n

β α − − = ) 1 (

Hogdkin-Huxley OD equations

A single neuron signaling to a muscle fiber

L r 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 m

β α − − = ∂ ∂ ) 1 ( h h t h

h h

β α − − = ∂ ∂ ) 1 ( n n t n

n n

β α − − = ∂ ∂ ) 1 (

Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = V() with = x + t

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = V() with = x + t

[ ]

L r V V g V V n g V V h m g d V d C d V d R

l l K K Na Na M

π ξ θ ξ 2 ) ( ) ( ) ( 1

4 3 2 2

− + − + − + =

m m d m d

m m

β α ξ θ − − = ) 1 ( h h d h d

h h

β α ξ θ − − = ) 1 ( n n d n d

n n

β α ξ θ − − = ) 1 (

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

Temperature dependence of the Hodgkin-Huxley equations

Systematic investigation of pulse behavior

50

• 50

100 1 2 3 4 5 6 [cm] V [ m V ]

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

• 10

10 20 30 40 V [ m V ] 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 N t 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: [Ii] = xi 0, fi > 0, Qij 0 Solutions are obtained after integrating factor transformation by means

• f an eigenvalue problem

f x f x n i x x f Q dt dx

n j j j n i i i j j n j ij i

= = Φ = = Φ − =

∑ ∑ ∑

= = = 1 1 1

; 1 ; , , 2 , 1 , L

( ) ( ) ( ) ( ) ( )

) ( ) ( ; , , 2 , 1 ; exp exp

1 1 1 1

∑ ∑ ∑ ∑

= = − = − =

= = ⋅ ⋅ ⋅ ⋅ =

n i i ki k n j k k n k jk k k n k ik i

x h c n i t c t c t x L l l λ λ

{ } { } { }

n j i h H L n j i L n j i Q f W

ij ij ij i

, , 2 , 1 , ; ; , , 2 , 1 , ; ; , , 2 , 1 , ;

1

L L l L = = = = = = ÷

−

{ }

1 , , 1 , ;

1

− = = Λ = ⋅ ⋅

−

n k L W L

k

L λ

constant level sets of Selection of quasispecies with f1 = 1.9, f2 = 2.0, f3 = 2.1, and p = 0.01 parametric plot on S3

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

Error rate p = 1-q

0.00 0.05 0.10

Quasispecies Uniform distribution

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 ) 1 ( ln 1 ) 1 (

max max

≈ ≈ − ≥ − ⋅ ⇒ ≥ ⋅ − = ⋅ K K

sequence master

• f

y superiorit ) 1 ( length chain rate error accuracy n replicatio ) 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

Empirical parameters Biophysical chemistry: thermodynamics and kinetics RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function

Sequence, structure, and design

RNA structure

• f minimal free

energy

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 Inverse Folding Algorithm Iterative determination

• f a sequence for the

given secondary structure

RNA structure

• f minimal free

energy

Inverse folding of RNA: Biotechnology, design of biomolecules with predefined structures and functions

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

A mapping and its inversion

• Gk =

( ) | ( ) =

• 1

U

• S

I S

k j j k

I

( ) = I S

j k Space of genotypes: = { I

S I I I I I S S S S S

1 2 3 4 N 1 2 3 4 M

, , , , ... , } ; Hamming metric Space of phenotypes: , , , , ... , } ; metric (not required) N M = {

Degree of neutrality of neutral networks and the connectivity threshold

A multi-component neutral network formed by a rare structure: < cr

A connected neutral network formed by a common structure: > cr

Stochastic simulation of evolution

• f 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

Gk Neutral Network

Structure S

k

Gk C

• k

Compatible Set Ck

The compatible set Ck of a structure Sk consists of all sequences which form Sk as its minimum free energy structure (the neutral network Gk) or one of its suboptimal structures.

Structure S Structure S

1

The intersection of two compatible sets is always non empty: C0 C1

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