the mathematics of biological evolutionpks/presentation/innsbruck-13.pdf · 2013-09-25 · the...

The Mathematics of Biological Evolution

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austria and

The Santa Fe Institute, Santa Fe, New Mexico, USA

ÖMG-DMV Congress 2013 Minisymposium: Mathematics of the Planet Earth

Innsbruck, 23.– 24.09.2013

Web-Page for further information:

http://www.tbi.univie.ac.at/~pks

1. Prologue

2. Mathematics of Darwin‘s natural selection

3. Mendel, Fisher and population genetics

4. Mutations and selection

5. What means neutrality in evolution?

6. Evolution in simple systems

7. Some origins of complexity in biology

A + X 2 X …… asexual reproduction

A + X + Y X + Y + Z {X,Y} …… sexual reproduction

Two modes of reproduction in biology

viruses, bacteria, some higher organisms (eukaryotes)

most higher organisms (eukaryotes) …… obligatory with mammals

Leonardo da Pisa „Fibonacci“

~1180 – ~1240

1,0 10

11

==

+= −+

FFFFF nnn

The history of exponential growth

1, 2 , 4 , 8 ,16 , 32 , 64, 128 , ...

geometric progression


Thomas Robert Malthus 1766 – 1834

Leonhard Euler, 1717 - 1783

nn n

xx )1(lim)(exp +≡ ∞→

exponential function

A + X 2 X …… asexual reproduction

growthlexponentia)(exp)0()( tfxtxxfdtdx

=⇒=

[ ] fitnessconst 00 fakaxakdtdx

=⇒==⇒= A

[A] = a ; [X] = x

Reproduction and exponential growth

Comparison of curves for unlimited growth

1and1)0(tonormalized0

====tdt

dxxxkdtdx α

α

1. Prologue







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.

No attempt has been made to cast the principle into theorems.

Pierre-François Verhulst, 1804-1849

The logistic equation, 1828

( ) tfeyCyCyty

Cyyf

dtdy

−−+=

−=

)0()0()0()(,1

C …… carrying capacity of the ecosystem

Was known 30 years before the

‘Origin of Species’

Generalization of the logistic equation to n variables yields selection

( )ΦfxxCΦxf

xfCxxfx

Cxxfx

−==≡

−=⇒

−=

dtd:1,)t(

dtd1

dtd

Darwin

Generalization of the logistic equation to n variables yields selection

[ ]

( ) ( ) ∑∑

∑

==

=

=−=−=

====

n

i iijjn

i iijjj

iin

i iiin

xfΦΦfxxffxx

ffCxx

11

121

;dt

d

)X(;1;X:X,,X,X

{ } 0vardtd 22 ≥=><−><= fffΦ

A + Xi 2 Xi ; i = 1, 2 , … , n

Evolutionary optimization

{ }

0lim1lim

,max

)(lim

,

1

max

==

=

=Φ=Φ

≠∞→

∞→

∞→

mjjt

mt

nm

mt

xx

fff

ft

fftft jjj −=Φ−= )()(δφ

f1 = 0.99, f2 = 1.00, f3 = 1.01

fxfxfxf

kxfdt

dxkk

k

=++=Φ

=Φ−=

332211

3,2,1;)(

simplex S3 is an invariant set e3 …… “corner equilibrium”

Phase diagram of Darwinian selection

f1 = 1 , f2 = 2 , f3 = 3 , f4 = 7

Before the development of molecular biology mutation was treated as a “deus ex machina”

1. Prologue







Recombination in Mendelian genetics

Gregor Mendel 1822 - 1884

Mendelian genetics

The 1:3 rule

The results of the individual experiments Gregor Mendel did with the garden pea pisum sativum.

Darwin

Ronald Fisher‘s selection equation: The genetical theory of natural selection. Oxford, UK, Clarendon Press, 1930.

Ronald Fisher (1890-1962)

( ) { } 0var22dtd 22 ≥=><−><= aaaΦ

Mendel

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

( )

∑∑ ∑

∑∑

== =

==

==

=−=−=

n

j jjin

j

n

i ji

in

i jijjjin

i jij

xxxaΦ

njΦxaxxΦxxax

11 1

11

1und(t)mit

,,2,1,td

d

Fitness in Fisher’s selection equation

jifa

xaxxaxa

ij XX

22222112

2111 2)(

fitnessmean

=++=Φ x

1. Prologue







organism mutation rate per genome

reproduction event

RNA virus 1 replication retroviruses 0.1 replication bacteria 0.003 replication eukaryotes 0.003 cell division eukaryotes 0.01 – 0.1 sexual reproduction

John W. Drake, Brian Charlesworth, Deborah Charlesworth and James F. Crow. 1998. Rates of spontaneous mutation. Genetics 148:1667-1686.

Hermann J. Muller 1890 - 1967

Thomas H. Morgan 1866 - 1945

The logic of DNA (or RNA) replication and mutation

Manfred Eigen 1927 -

∑∑

∑

==

=

==⋅=

=−=

n

i iin

i iijiji

jin

i jij

xfΦxfQW

njΦxxWx

11

1

,1,

,,2,1;dt

d

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

Mutation-selection equation: [Ii] = xi 0, fi > 0, Qij 0

Solutions are obtained after integrating factor transformation by means

of an eigenvalue problem

fxfxnixxQfdtdx n

j jjn

i iijn

j jiji ==Φ==Φ−= ∑∑∑ === 111

;1;,,2,1,

( ) ( ) ( )( ) ( )

)0()0(;,,2,1;exp0

exp01

1

1

0

1

0 ∑∑ ∑∑

=

=

−

=

−

= ==⋅⋅

⋅⋅=

n

i ikikn

j kkn

k jk

kkn

k iki xhcni

tcb

tcbtx

λ

λ

{ } { } { }njihHBnjibBnjiQfW ijijiji ,,1,;;,,1,;;,,1,; 1 ======÷ −

{ }1,,1,0;1 −==Λ=⋅⋅− nkBWB k λ

1,,1,0;)( −=Φ−= nkdt

dkk

k λbb

( ) ( )( )

)0()0(;,,2,1;0

0lim1 00

1 00

00 ∑∑ =

=

∞→ ====n

i iin

j j

iiit xhcni

cbcbxtx

The quasispecies is the long-time solution of the mutation-selection equation.

Definition of quasispecies

3210:theoremFrobeniusPerron λλλλ ≥≥>−

=⇔

0

20

10

00

nb

bb

bλ

Phase diagram of the mutation-selection system

)( 000 Φ−= λbb

dtd

Selection Selection-recombination Selection-mutation

Method of solution

integrating factor qualitative analysis integrating factor, eigenvalue problem

Linearity yes no yes

Optimization of yes yes no

Unique optimum yes no no optimum

Invariance of Sn yes yes no

Uniqueness of solution

yes no yes

Selection of fittest fittest/coexistence quasispecies

Comparison of mathematical models of evolution

1. Prologue







Motoo Kimura’s 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, 1924 - 1994

Motoo Kimura

mean time of fixation: 4 Ne … effective population size

mean time of replacement: -1 … reciprocal mutation rate

mk

mk

k

kk

,,1;

,,1;2

=→

=→+

BXXXA

λ

λ

( ))()()(

)(,,)(,)()(

21

21 2211,,,

tPtPtP

ntntntPtP

m

m

nnn

mmnnn

⋅⋅⋅=

=====

XXX

neutrality: all probabilities are equivalent and the densities identical

( ) ( )ntPtPtPntPntPndt

dPnnnn

n ==−++−= +− )()(;)(2)()1()()1( 11 Xλ

knn

k

nn

n tt

kn

knknn

tttP

−

−

−−+

+

= ∑=

+

22

220

),(min

0

0 111

)(00

λλ

λλ

BXXXA

→

→+λ

λ 2

( ) ( )ntPtPtPntPntPndt

dPnnnn

n ==−++−= +− )()(;)(2)()1()()1( 11 Xλ

knn

k

nn

n tt

kn

knknn

tttP

−

−

−−+

+

= ∑=

+

22

220

),(min

0

0 111

)(00

λλ

λλ

( ) ( )0

1)(,2)(,)( 00

20

n

tttPtntntE

+

===λλλσ XX

1)(lim 0 =∞→ tPt

( )m

kk tttPtHtPtH

+

==<=λ

λ1

)()(,)( 0,,0,00 T

m

jmk

jk tt

jm

tH)1(

)()(0 λ

λ+

=

−

=∑

Sequential extinction times Tk

Sequential extinction times Tk

λλ

λ

mmE

kkmE k

≈−

=

⋅−

=

1)(

1)(

1T

T

Sewall Wrights fitness landscape as metaphor for Darwinian evolution

Sewall Wright. 1932. The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: D.F.Jones, ed. Int. Proceedings of the Sixth International Congress on Genetics. Vol.1, 356-366. Ithaca, NY.

Sewall Wright, 1889 - 1988

Evolution as a global phenomenon in genotype space

many genotypes one phenotype

Complexity in molecular evolution

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

1. Prologue







Selma Gago, Santiago F. Elena, Ricardo Flores, Rafael Sanjuán. 2009. Extremely high mutation rate of a hammerhead viroid. Science 323:1308.

Mutation rate and genome size

Replicating molecules

RNA replication by Q-replicase

C. Weissmann, The making of a phage. FEBS Letters 40 (1974), S10-S18

Charles Weissmann 1931-

Kinetics of RNA replication

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

Christof K. Biebricher, 1941-2009

Paul E. Phillipson, Peter Schuster. 2009. Modeling by nonlinear differential equations. Dissipative and conservative processes. World Scientific Publishing, Hackensack, NJ.

Paul E. Phillipson, Peter Schuster. 2009. Modeling by nonlinear differential equations. Dissipative and conservative processes. World Scientific Publishing, Hackensack, NJ.

replicase e(t)

plus strand x+(t) minus strand x-(t)

total RNA concentration

xtot(t) = x+(t) + x-(t)

complemetary replication

Application of molecular evolution to problems in biotechnology

Viruses

The error threshold in replication

quasispecies

driving virus populations through threshold

leth

al m

utag

enes

is

Application of quasispecies theory to the fight against viruses

Esteban Domingo 1943 -

Molecular evolution of viruses

1. Prologue







The bacterial cell as an example for the simplest form of autonomous life

Escherichia coli genome: 4 million nucleotides

4460 genes

The spatial 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.

François Jacob, 1920-2013

Sketch of a genetic and metabolic network

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

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

ENCyclopedia Of DNA Elements

Coworkers

Walter Fontana, Harvard Medical School, MA

Josef Hofbauer, Universität Wien, AT

Martin Nowak, Harvard University, MA

Christian Reidys, University of Southern Denmark, Odense, DK

Karl Sigmund, Universität Wien, AT

Peter Stadler, Bärbel M. Stadler, Universität Leipzig, DE

Thomas Wiehe, Universität Köln, DE

Ivo L.Hofacker, Christoph Flamm, Universität Wien, AT

Paul Phillipson, University of Boulder at Colorado, CO

Christian Forst, University of Texas, Southwestern Medical School, TX

Erich Bornberg-Bauer, Universität Münster, DE

Universität Wien

Thank you for your attention!

Web-Page for further information:

http://www.tbi.univie.ac.at/~pks

the mathematics of biological evolutionpks/presentation/innsbruck-13.pdf · 2013-09-25 · the...

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