The mathematics of Darwins theory of evolution 1859 and 150 years - - PowerPoint PPT Presentation

The mathematics of Darwin‘s theory of evolution 1859 and 150 years later Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Österreich und The Santa Fe Institute, Santa Fe, New Mexico, USA

„The Mathematics of Darwin Legacy“ Centro Internacional de Mathemática, Lisbon, 23.-24.11.2009

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

"La Filosophia è scritta in questo grandissimo libro, que continuamente ci stà aperto innanzi à gli occhi (io dico l’universo) ma non si può intendere se prima non s’impara à intender la lingua, e conoscer i caratteri, nei quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi. & altre figure Geometriche ...", „Philosophy [science] is written in this grand book, the universe ... . It is written in the language of mathematics, and ist characters are triangles, circles and other geometric figures; …. „ Galileo Galilei. 1632. Il Saggiatore. Edition Nationale, Bd.6, Florenz 1896, p.232. Galileo Galilei, 1564 - 1642

"La Filosophia è scritta in questo grandissimo libro, que continuamente ci stà aperto innanzi à gli occhi (io dico l’universo) ma non si può intendere se prima non s’impara à intender la lingua, e conoscer i caratteri, nei quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi. & altre figure Geometriche ...", „Philosophy [science] is written in this grand book, the universe ... . It is written in the language of mathematics, and ist characters are triangles, circles and other geometric figures; …. „ Galileo Galilei. 1632. Il Saggiatore. Edition Nationale, Vol.6, Florenz 1896, p.232. Galileo Galilei, 1564 - 1642

If Charles Darwin would have written the „Origin“ in mathematical language, how would he have done it? What did we learn about evolution from in vitro experiments? Quantitative systems biology – A challenge for biologists, chemists, physicists, and mathematicians !

If Charles Darwin would have written the „Origin“ in mathematical language, how would he have done it? What did we learn about evolution from in vitro experiments? Quantitative systems biology – A challenge for biologists, chemists, physicists, and mathematicians !

1 , ;

1 1 1

= = + =

− +

F F F F F

n n n

Leonardo da Pisa „Fibonacci“ ~1180 – ~1240 Thomas Robert Malthus 1766 – 1834

1, 2 , 4 , 8 ,16 , 32 , 64, 128 , ... geometric progression exponential growth

The history of exponential growth

Leonhard Euler, 1717 - 1783

n n

n x x ) 1 ( lim ) ( exp + ≡

∞ →

Exponential function and exponential growth

Pierre-François Verhulst, 1804-1849

( )

t r

e x C x C x t x C x x r dt dx

−

− + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ) ( ) ( ) ( ) ( , 1

The logistic equation, 1828

( )

Φ r x x C Φ x r x r C x x r x C x x r x − = = ≡ − = ⇒ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = dt d : 1 , ) t ( dt d 1 dt d

Darwin

[ ]

( ) ( )

∑ ∑ ∑

= = =

= − = − = = = =

n i i i j j n i i i j j j n i i i i n

x f Φ Φ f x x f f x x C x x

1 1 1 2 1

; dt d 1 ; X : X , , X , X K

( )

{ }

var 2 2 dt d

2 2

≥ = > < − > < = f f f Φ

Generalization of the logistic equation to n variables yields selection

Three necessary conditions for Darwinian evolution are: 1. Multiplication, 2. Variation, and 3. Selection. Darwin discovered the principle of natural selection from empirical observations in nature.

Gregor Mendel (1822-1884)

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

Ronald Fisher (1890-1962)

Darwin Mendel alleles: A1, A2, ..... , An frequencies: xi = [Ai] ; genotypes: Ai·Aj fitness values: aij = f (Ai·Aj), aij = aji

( )

∑ ∑ ∑ ∑ ∑

= = = = =

= = = − = − =

n j j j i n j n i ji i n i ji j j j i n i ji j

x x x a Φ n j Φ x a x x Φ x x a x

1 1 1 1 1

1 und (t) mit , , 2 , 1 , t d d K

( )

{ }

var 2 2 dt d

2 2

≥ = > < − > < = a a a Φ Ronald Fisher‘s selection equation: The genetical theory of natural selection. Oxford, UK, Clarendon Press, 1930.

If Charles Darwin would have written the „Origin“ in mathematical language, how would he have done it? What did we learn about evolution from in vitro experiments? Quantitative systems biology – A challenge for biologists, chemists, physicists, and mathematicians !

Generation time (optimal) Population size (maximal) Mutation per replication event Bacteria 20 min 1010 1/400 – 1/300 Viruses variable 1012 1 RNA molecules 1 –10 sec 1015 tunable

The world of in vitro evolution experiments

Richard Lenski, 1956 -

Bacterial evolution under controlled conditions: A twenty years experiment. Richard Lenski, University of Michigan, East Lansing

Bacterial evolution under controlled conditions: A twenty years experiment. Richard Lenski, University of Michigan, East Lansing

1 year

Epochal evolution of bacteria in serial transfer experiments under constant conditions

• S. F. Elena, V. S. Cooper, R. E. Lenski. Punctuated evolution caused by selection of rare beneficial mutants.

Science 272 (1996), 1802-1804

1 year

Epochal evolution of bacteria in serial transfer experiments under constant conditions

• S. F. Elena, V. S. Cooper, R. E. Lenski. Punctuated evolution caused by selection of rare beneficial mutants.

Science 272 (1996), 1802-1804

Variation of genotypes in a bacterial serial transfer experiment

• D. Papadopoulos, D. Schneider, J. Meier-Eiss, W. Arber, R. E. Lenski, M. Blot. Genomic evolution during a

10,000-generation experiment with bacteria. Proc.Natl.Acad.Sci.USA 96 (1999), 3807-3812

Innovation by mutation in long time evolution of Escherichia coli in constant environment Z.D. Blount, C.Z. Borland, R.E. Lenski. 2008. Proc.Natl.Acad.Sci.USA 105:7899-7906

Innovation by mutation in long time evolution of Escherichia coli in constant environment

Z.D. Blount, C.Z. Borland, R.E.

• Lenski. 2008.

Proc.Natl.Acad.Sci.USA 105:7899-7906

Three necessary conditions for Darwinian evolution are: 1. Multiplication, 2. Variation, and 3. Selection.

Charles Darwin, 1809-1882

All three conditions are fulfilled not only by cellular organisms but also by nucleic acid molecules – DNA or RNA – in suitable cell-free experimental assays:

Darwinian evolution in the test tube

Taq = thermus aquaticus

Accuracy of replication: Q = q1 · q2 · q3 · … · qn

The logics of DNA replication

RNA replication by Q-replicase

• C. Weissmann, The making of a phage.

FEBS Letters 40 (1974), S10-S18

1 1 2 2 2 1

and x f dt dx x f dt dx = =

2 1 2 1 2 1 2 1 2 1 2 1

, , , , f f f f x f x = − = + = = = ξ ξ η ξ ξ ζ ξ ξ

ft ft

e t e t ) ( ) ( ) ( ) ( ζ ζ η η = =

−

Complementary replication as the simplest molecular mechanism of reproduction

Christof K. Biebricher, 1941-2009

Kinetics of RNA replication

C.K. Biebricher, M. Eigen, W.C. Gardiner, Jr. Biochemistry 22:2544-2559, 1983

Evolution in the test tube: G.F. Joyce, Angew.Chem.Int.Ed. 46 (2007), 6420-6436

RNA sample Stock solution: Q RNA-replicase, ATP, CTP, GTP and UTP, buffer

• Time

1 2 3 4 5 6 69 70 Application of serial transfer technique to evolution of RNA in the test tube

Decrease in mean fitness due to quasispecies formation

The increase in RNA production rate during a serial transfer experiment

Manfred Eigen 1927 -

∑ ∑ ∑

= = =

= = − =

n i i n i i i j i n i ji j

x x f Φ n j Φ x x W x

1 1 1

, , 2 , 1 ; dt d K

Mutation and (correct) replication as parallel chemical reactions

• M. Eigen. 1971. Naturwissenschaften 58:465,
• M. Eigen & P. Schuster.1977. Naturwissenschaften 64:541, 65:7 und 65:341

∑ ∑ ∑ ∑

= = = =

= = − = − =

n i i n i i i j i i n i ji j i n i ji j

x x f Φ n j Φ x x f Q Φ x x W x

1 1 1 1

, , 2 , 1 ; dt d K

Factorization of the value matrix W separates mutation and fitness effects.

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

= = = = − =

∑ ∑ ∑

= = = 1 1 1

; 1 ; , , 2 , 1 , φ φ 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

Phenomenon Optimization of fitness Unique selection outcome Selection yes yes Recombination and selection Independent genes yes no Recombination and selection Interacting genes no no Mutation and selection no yes

The Darwinian mechanism of variation and selection is a very powerful optimization heuristic.

The Darwinian mechanism and optimization of fitness

Chain length and error threshold

n p n p n p 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 length chain rate error accuracy n replicatio ) 1 ( K K K K

∑ ≠

= − =

m j j m n

f f σ n p p Q

Quasispecies

Driving virus populations through threshold

The error threshold in replication: No mutational backflow approximation

W = G

• F

0 , 0 largest eigenvalue and eigenvector

diagonalization of matrix W „ complicated but not complex “ fitness landscape mutation matrix „ complex “ ( complex )

sequence

• structure

„ complex “

mutation selection

Complexity in molecular evolution

The single peak fitness landscapes corresponding to a mean field approximation

Error rate p = 1-q

0.00 0.05 0.10

Quasispecies Uniform distribution

Stationary population or quasispecies as a function of the mutation or error rate p

Fitness landscapes and the search for error thresholds

Error threshold on single-peak and hyperbolic landscapes

Error threshold on single-peak, linear, and step-linear landscapes

Fitness landscapes showing error thresholds

Error threshold on a single peak fitness landscape with n = 50 and = 10

Error threshold: Individual sequences n = 10, = 2 and d = 0, 1.0, 1.85

Motoo Kimuras population genetics of neutral evolution. Evolutionary rate at the molecular level. Nature 217: 624-626, 1955. The Neutral Theory of Molecular Evolution. Cambridge University Press. Cambridge, UK, 1983.

Motoo Kimura

Is the Kimura scenario correct for frequent mutations?

5 . ) ( ) ( lim

2 1

= =

→

p x p x

p

dH = 1

a p x a p x

p p

− = =

→ →

1 ) ( lim ) ( lim

2 1

dH = 2 dH ≥3

1 ) ( lim , ) ( lim

• r

) ( lim , 1 ) ( lim

2 1 2 1

= = = =

→ → → →

p x p x p x p x

p p p p

Random fixation in the sense of Motoo Kimura Pairs of neutral sequences in replication networks

• P. Schuster, J. Swetina. 1988. Bull. Math. Biol. 50:635-650

A fitness landscape including neutrality

Neutral network: Individual sequences n = 10, = 1.1, d = 1.0

Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance dH(Xi,,Xj) = 1.

Neutral network: Individual sequences n = 10, = 1.1, d = 1.0

Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance dH(Xi,,Xj) = 2.

N = 7 Neutral networks with increasing : = 0.10, s = 229

Adjacency matrix

many genotypes

• ne phenotype

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

Evolution of RNA molecules as a Markow process

Replication in the flow reactor as a stochastic process with two absorbing barriers

10 12 14 16 18 20 22 Population size 0.2 0.4 0.6 0.8 1 P r

• b

a b i l i t y t

• r

e a c h t h e t a r g e t s t r u c t u r e

AUGC GC

Probability of a single trajectory to reach the target structure

Computer simulation using Gillespie‘s algorithm: Replication rate constant: fk = / [ + dS

(k)]

dS

(k) = dH(Sk,S)

Selection constraint: Population size, N = # RNA molecules, is controlled by the flow Mutation rate: p = 0.001 / site replication N N t N ± ≈ ) ( The flowreactor as a device for studies

• f evolution in vitro and in silico

Evolution in silico

• W. Fontana, P. Schuster,

Science 280 (1998), 1451-1455

Phenylalanyl-tRNA as target structure Structure of randomly chosen initial sequence

In silico optimization in the flow reactor: Evolutionary Trajectory

28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations change the molecular structure Neutral point mutations leave the molecular structure unchanged

Neutral genotype evolution during phenotypic stasis

Evolutionary trajectory Spreading of the population

• n neutral networks

Drift of the population center in sequence space

A sketch of optimization on neutral networks

If Charles Darwin would have written the „Origin“ in mathematical language, how would he have done it? What did we learn about evolution from in vitro experiments? Quantitative systems biology – A challenge for biologists, chemists, physicists, and mathematicians !

Systems biology or quantitative biology is the chemistry of whole cells and organisms Challenges for theorists and mathematicians: 1. Very large numbers of variables and parameters in ODE modeling 2. Stochastic effects because of very low particle numbers 3. Complex nonlinear reaction networks 4. Complex spatial structures in specific aggregates and compartments

Three-dimensional structure of the complex between the regulatory protein cro-repressor and the binding site on -phage B-DNA

1 2 3 4 5 6 7 8 9 10 11 12 Regulatory protein or RNA Enzyme Metabolite Regulatory gene Structural gene

A model genome with 12 genes

Sketch of a genetic and metabolic network

A B C D E F G H I J K L 1

Biochemical Pathways

2 3 4 5 6 7 8 9 10

The reaction network of cellular metabolism published by Boehringer-Mannheim.

The bacterial cell as an example for the simplest form of autonomous life Escherichia coli genome: 4 million nucleotides 4460 genes The structure of the bacterium Escherichia coli

Evolution does not design with the eyes of an engineer, evolution works like a tinkerer.

François Jacob. The Possible and the Actual. Pantheon Books, New York, 1982, and Evolutionary tinkering. Science 196 (1977), 1161-1166.

• D. Duboule, A.S. Wilkins. 1998.

The evolution of ‚bricolage‘. Trends in Genetics 14:54-59.

The difficulty to define the notion of „gene”. Helen Pearson, Nature 441: 399-401, 2006

ENCODE Project Consortium. Identification and analysis of functional elements in 1% of the human genome by the ENCODE pilot project. Nature 447:799-816, 2007

ENCODE stands for ENCyclopedia Of DNA Elements.

Coworkers

Karl Sigmund, Universität Wien, AT Peter Stadler, Bärbel M. Stadler, Universität Leipzig, GE Paul E. Phillipson, University of Colorado at Boulder, CO Heinz Engl, Philipp Kügler, James Lu, Stefan Müller, RICAM Linz, AT Jord Nagel, Kees Pleij, Universiteit Leiden, NL Walter Fontana, Harvard Medical School, Martin Nowak, Harvard University, MA Christian Reidys, Nankai University, Tien Tsin, China Christian Forst, Los Alamos National Laboratory, NM Thomas Wiehe, Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE Ivo L.Hofacker, Christoph Flamm, Andreas Svrček-Seiler, Universität Wien, AT Kurt Grünberger, Michael Kospach , Andreas Wernitznig, Stefanie Widder, Stefan Wuchty, Jan Cupal, Stefan Bernhart, Lukas Endler, Ulrike Langhammer, Rainer Machne, Ulrike Mückstein, Erich Bornberg-Bauer, Universität Wien, AT

Universität Wien

Acknowledgement of support

Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No. 09942, 10578, 11065, 13093 13887, and 14898 Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05 Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European Commission: Contracts No. 98-0189, 12835 (NEST) Austrian Genome Research Program – GEN-AU: Bioinformatics Network (BIN) Österreichische Akademie der Wissenschaften Siemens AG, Austria Universität Wien and the Santa Fe Institute

Universität Wien

Thank you for your attention!

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