Complexity in Evolutionary Processes Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA 7th Vienna Central European Seminar on Particle Physics and Quantum Field Theory Vienna, 26.– 28.11.2010
Web-Page for further information: http://www.tbi.univie.ac.at/~pks
1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution
1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution
= + = = F F F F F ; 0 , 1 + − n 1 n n 1 0 1 Thomas Robert Malthus Leonardo da Pisa 1766 – 1834 „Fibonacci“ ~1180 – ~1240 1, 2 , 4 , 8 ,16 , 32 , 64, 128 , ... geometric progression exponential growth n ⎛ + ⎞ 1 1 5 ⎜ ⎟ ≈ f ⎜ ⎟ n 2 5 ⎝ ⎠ The history of exponential growth
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.
⎛ − ⎞ dx x x C ( 0 ) = = ⎜ ⎟ r x x t 1 , ( ) ( ) + − − r t dt ⎝ C ⎠ x C x e ( 0 ) ( 0 ) Pierre-François Verhulst, 1804-1849 The logistic equation, 1828
− f f = = s 2 1 0 . 1 f 1 Two variants with a mean progeny of ten or eleven descendants
Numbers N 1 (n) and N 2 (n) N 1 (0) = 9999 , N 2 (0) = 1 ; s = 0.1 , 0.02 , 0.01 Selection of advantageous mutants in populations of N = 10 000 individuals
⎛ − ⎞ x x x x d d = ⇒ = − ⎜ ⎟ r x r x r x 1 ⎝ C ⎠ C dt dt x d ( ) ≡ = = − r x Φ C x r Φ ( t ) , 1 : dt [ ] = ∑ = = n x x C X , X , K , X : X ; 1 = n i i i i 1 2 1 x ( ) d ( ) j = − ∑ = − = ∑ n n x f f x x f Φ Φ f x ; = = i i j j 1 i i j j 1 i i dt Darwin ( ) Φ d { } = < > − < > = ≥ 2 2 f f f 2 2 var 0 dt Generalization of the logistic equation to n variables yields selection
1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution
Taq = thermus aquaticus Accuracy of replication: Q = q 1 · q 2 · q 3 · … · q n The logics of DNA replication
Point mutation
x d ∑ n j = − = W x x Φ j n ; 1 , 2 , , K ji i j = i 1 dt ∑ ∑ n n = Φ f x x i i i = = i i 1 1 Manfred Eigen 1927 - 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
x d ∑ ∑ n n j = − = − = W x x Φ Q f x x Φ j n ; 1 , 2 , , K ji i j ji i i j = = i i dt 1 1 ∑ ∑ n n = Φ f x x i i i = = i 1 i 1 Factorization of the value matrix W separates mutation and fitness effects.
integrating factor transformation eigenvalue problem Solution of the mutation-selection equation
1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution
Quasispecies Uniform distribution Stationary population or quasispecies as a function of the mutation or error rate p 0.00 0.05 0.10 Error rate p = 1-q
The no-mutational backflow or zeroth order approximation
quasispecies The error threshold in replication and mutation
1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution
single peak landscape „Rugged“ fitness landscapes
Error threshold on the single peak landscape
linear and multiplicative landscape Smooth fitness landscapes
The linear fitness landscape shows no error threshold
Make things as simple as possible, but not simpler ! Albert Einstein Albert Einstein‘s razor, precise refence is unknown.
Sewall Wright. 1931. Evolution in Mendelian populations. Genetics 16:97-159. -- --. 1932. The roles of mutation, inbreeding, crossbreeding, and selection in evolution. In: D.F.Jones, ed. Proceedings of the Sixth International Congress on Genetics, Vol.I. Brooklyn Botanical Garden. Ithaca, NY, pp. 356-366. -- --. 1988. Surfaces of selective value revisited. The American Naturalist 131:115-131.
Build-up principle of binary sequence spaces
single peak landscape „realistic“ landscape Rugged fitness landscapes over individual binary sequences with n = 10
Error threshold: Individual sequences n = 10, � = 2, s = 491 and d = 0, 1.0, 1.875
d = 0.100 d = 0.200 n = 10, f 0 = 1.1, f n = 1.0, s = 919 Case I : Strong Quasispecies
d = 0.100 d = 0.195 n = 10, f 0 = 1.1, f n = 1.0, s = 637 Case III : Multiple transitions
d = 0.199 d = 0.200 n = 10, f 0 = 1.1, f n = 1.0, s = 637 Case III : Multiple transitions
Paul E. Phillipson, Peter Schuster. (2009) Modeling by nonlinear differential equations. Dissipative and conservative processes. World Scientific, Singapore, pp.9-60.
� 1 , � 1 � largest eigenvalue and eigenvector diagonalization of matrix W „ complicated but not complex “ � = G W F mutation matrix fitness landscape „ complex “ ( complex ) sequence structure � „ complex “ mutation selection Complexity in molecular evolution
1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution
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
The inverse folding algorithm searches for sequences that form a given RNA secondary structure under the minimum free energy criterion.
What is neutrality ? Selective neutrality = = several genotypes having the same fitness. Structural neutrality = = several genotypes forming molecules with the same structure.
I I I I I I Space of genotypes: = { , , , , ... , } ; Hamming metric 1 2 3 4 N S S S S S S Space of phenotypes: = { , , , , ... , } ; metric (not required) 1 2 3 4 M �� N M � ( ) = I S j k U � � -1 � � S I S G k = ( ) | ( ) = I k j j k � A mapping and its inversion
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 C U G C A C C U G A 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
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
One error neighborhood – Surrounding of an RNA molecule of chain length n=50 in sequence and shape space
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