Modeling Evolutionary Processes: Evolution from the Viewpoint of a Physicist Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Steps in Evolution: Perspectives from Physics, Biochemistry and Cell Biology – 150 Years after Darwin Bremen, 28.06.– 05.07.2009
Web-Page for further information: http://www.tbi.univie.ac.at/~pks
What is information ? • Information is (only) what is understood. • Information is (only) what creates information. Carl Friedrich von Weizsäcker, 1912-2007, German physicist and philosopher. Information in biology • Understanding of information is interpreted as decoding, • maintenance of information requires reproduction, and • creation of information occurs through adaptation to the environment by means of a Darwinian mechanism of variation and selection.
1. Darwin‘s two pathbreaking ideas 2. Dynamics of Darwinian evolution 3. RNA evolution in the test tube 4. Stochasticity in evolution 5. Evolutionary optimization of RNA structure
1. Darwin‘s two pathbreaking ideas 2. Dynamics of Darwinian evolution 3. RNA evolution in the test tube 4. Stochasticity in evolution 5. Evolutionary optimization of RNA structure
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.
− f f = = 2 1 0 . 1 s f 1 Two variants with a mean progeny of ten or eleven descendants
= = = ( 0 ) 9999 , ( 0 ) 1 ; 0 . 1 , 0 . 02 , 0 . 01 N N s 1 2 Selection of advantageous mutants in populations of N = 10 000 individuals
Charles Darwin drew a tree of life and suggested that all life on Earth descended form one common ancestor time Charles Darwin, The Origin of Species , 6th edition. Everyman‘s Library, Vol.811, Dent London, pp.121-122.
Modern phylogenetic tree: Lynn Margulis, Karlene V. Schwartz. Five Kingdoms . An Illustrated Guide to the Phyla of Life on Earth . W.H. Freeman, San Francisco, 1982.
Duplication of genetic information Deoxyribonucleic acid – DNA The carrier of digitally encoded information
Time Reconstruction of phylogenies through comparison of molecular sequence data
1. Darwin‘s two pathbreaking ideas 2. Dynamics of Darwinian evolution 3. RNA sequences and structures 4. Stochasticity in evolution 5. Evolutionary optimization of RNA structure
Reproduction of organisms or replication of molecules as the basis of selection
Selection equation : [I i ] = x i � 0 , f i > 0 ( ) dx ∑ ∑ = − φ = n = φ = n = , 1 , 2 , , ; 1 ; i L x f i n x f x f i i = i = j j 1 1 i j dt Mean fitness or dilution flux , φ (t), is a non-decreasing function of time , ( ) φ = ∑ n dx d { } 2 = − = ≥ 2 i var 0 f f f f i dt dt = 1 i Solutions are obtained by integrating factor transformation ( ) ( ) ⋅ 0 exp ( ) x f t = = i i ; 1 , 2 , L , x t i n ( ) ( ) ∑ i n ⋅ 0 exp x f t = j j 1 j
Chemical kinetics of replication and mutation as parallel reactions
Mutation-selection equation : [I i ] = x i � 0, f i > 0, Q ij � 0 dx ∑ ∑ ∑ = n − φ = n = φ = n = , 1 , 2 , , ; 1 ; i L f Q x x i n x f x f = j ji j i = i = j j 1 1 1 j i j dt Solutions are obtained after integrating factor transformation by means of an eigenvalue problem ( ) ( ) ∑ − 1 n ⋅ ⋅ λ l 0 exp c t ( ) ∑ n = = = = ik k k 0 ; 1 , 2 , , ; ( 0 ) ( 0 ) k L x t i n c h x ( ) ( ) ∑ ∑ − i 1 k = ki i n n ⋅ ⋅ λ 1 i 0 exp l c t = = jk k k 1 0 j k { } { } { } ÷ = = = − = = = 1 ; , 1 , 2 , L , ; l ; , 1 , 2 , L , ; ; , 1 , 2 , L , W f Q i j n L i j n L H h i j n i ij ij ij { } − ⋅ ⋅ = Λ = λ = − 1 ; 0 , 1 , L , 1 L W L k n k
Perron-Frobenius theorem applied to the value matrix W λ W is primitive: (i) is real and strictly positive 0 (ii) λ > λ ≠ for all 0 k 0 k λ (iii) is associated with strictly positive eigenvectors 0 (iv) is a simple root of the characteristic equation of W λ 0 (v-vi) etc. W is irreducible: (i), (iii), (iv), etc. as above λ ≥ λ ≠ (ii) for all 0 k 0 k
p = 0 Formation of a quasispecies in sequence space
p = 0.25 p cr Formation of a quasispecies in sequence space
p = 0.50 p cr Formation of a quasispecies in sequence space
p = 0.75 p cr Formation of a quasispecies in sequence space
p � p cr Uniform distribution in sequence space
Quasispecies Driving virus populations through threshold The error threshold in replication
Molecular evolution of viruses
A fitness landscape showing an error threshold
= = = = Single peak fitness landscape: and 1 K f f f f f 0 1 2 N f σ = 0 ∑ = − N Quasispecies as a function of the mutation rate p ( 1 ) x f x 0 i i 1 i f 0 = � = 10 = κ master sequence ; n K I N 0
Fitness landscapes showing error thresholds
Error threshold: Individual sequences n = 10, � = 2 and d = 0, 1.0, 1.85
1. Darwin‘s two pathbreaking ideas 2. Dynamics of Darwinian evolution 3. RNA evolution in the test tube 4. Stochasticity in evolution 5. Evolutionary optimization of RNA structure
Three necessary conditions for Darwinian evolution are: 1. Multiplication, 2. Variation , and 3. Selection. Variation through mutation and recombination operates on the genotype whereas the phenotype is the target of selection . One important property of the Darwinian scenario is that variations in the form of mutations or recombination events occur uncorrelated with their effects on the selection process . All conditions can be fulfilled not only by cellular organisms but also by nucleic acid molecules in suitable cell-free experimental assays.
RNA sample Time 0 1 2 3 4 5 6 69 70 � Stock solution: Q RNA-replicase, ATP, CTP, GTP and UTP, buffer D.R.Mills, R.L.Peterson, S.Spiegelman, An extracellular Darwinian experiment with a self-duplicating nucleic acid molecule . Proc.Natl.Acad.Sci.USA 58 (1967), 217-224 Application of serial transfer to RNA evolution in the test tube
Reproduction of the original figure of the β serial transfer experiment with Q RNA D.R.Mills, R,L,Peterson, S.Spiegelman, An extracellular Darwinian experiment with a self-duplicating nucleic acid . Proc.Natl.Acad.Sci.USA molecule (1967), 217-224 58
Cross-catalysis of two RNA enzymes leads to self-sustained replication Tracey A. Lincoln, Gerald F. Joyce, Science 323 , 1229-1232, 2009
Amplification : 1.5 µ 10 10 Exponential growth levels off when the reservoir is exhausted (l.h.s.). RNA production in serial transfer experiments (r.h.s.) Tracey A. Lincoln, Gerald F. Joyce, Science 323 , 1229-1232, 2009
RNA evolution of recombinant replicators Tracey A. Lincoln, Gerald F. Joyce, Science 323 , 1229-1232, 2009
Application of molecular evolution to problems in biotechnology
5' - end N 1 O CH 2 O GCGGAU UUA GCUC AGUUGGGA GAGC CCAGA G CUGAAGA UCUGG AGGUC CUGUG UUCGAUC CACAG A AUUCGC ACCA 5'-end 3’-end N A U G C k = , , , OH O N 2 O P O CH 2 O Na � O O OH N 3 O P O CH 2 O Na � O Definition of RNA structure O OH N 4 O P O CH 2 O Na � O O OH 3' - end O P O Na � O
N = 4 n N S < 3 n Criterion: Minimum free energy (mfe) Rules: _ ( _ ) _ � { AU , CG , GC , GU , UA , UG } A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs
RNA sequence: GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA Iterative determination of a sequence for the Inverse folding of RNA : given secondary RNA folding : structure Biotechnology, Structural biology, design of biomolecules spectroscopy of Inverse Folding with predefined biomolecules, Algorithm structures and functions understanding molecular function 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.
many genotypes � one phenotype
One-error neighborhood GUUAAUCAG GUAAAUCAG GUGAAUCAG GCCAAUCAG GUCUAUCAG GGCAAUCAG GUCGAUCAG GACAAUCAG GUCCAUCAG CUCAAUCAG GUCAUUCAG UUCAAUCAG G A C U G A C U G GUCAAUCAG AUCAAUCAG GUCACUCAG GUCAAUCAC GUCAAACAG GUCAAUCAU G U C A A GUCAAUCAA G C A G GUCAACCAG G U GUCAAUAAG C A G A G U U GUCAAUCUG U C C G A C C U A G A C U A A C The surrounding of U A G U U G A G GUCAAUCAG in sequence space G A G
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
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