Evolution and Thermodynamics Useful and Misleading Analogies Peter Schuster From Thermodynamics to Dynamical Systems Emmerich Wilhelm‘s 60th Birthday Universität Wien, 17.05.2002
Equilibrium thermodynamics is based on two major statements: 1. The energy of the universe is a constant (first law). 2. The entropy of the universe never decreases (second law). Carnot, Mayer, Joule, Helmholtz, Clausius, …… D.Jou, J.Casas-Vázquez, G.Lebon, Extended Irreversible Thermodynamics , 1996
Stock Solution Reaction Mixture p d e S T dS env T d S i � � � dS 0 dS 0 dS 0 Isolated system Closed system Open system U = const., V = const., T = const., p = const., � dS = dS env + dS 0 0 � � dS dG dU pdV TdS = - - 0 dS d S d S = + i e � d S 0 i Entropy changes in different thermodynamic systems
2 ( d S ) < 0 U,V,equil S max Enlarged scale Approach towards Fluctuations around Entropy equilibrium equilibrium Time Entropy and fluctuations at equilibrium
Thermodynamics of closed systems: Entropy is a non-decreasing function S(t) → S max Second law Evolution of Populations: Mean fitness is a non-decreasing function f(t) = � k x k (t) f k / � k x k (t) � f max Ronald Fisher‘s conjecture
Stock Solution [a] = a0 Reaction Mixture [a],[b] A B B A A A A A B A B A A B A A A � B - 1 Flow rate r = R A A A B A B A B B B B A A Reactions in the continuously stirred tank reactor (CSTR)
1.2 -1 Flow rate r [t ] Concentration a [a ] 0 4.0 6.0 8.0 2.0 10.0 1.0 0.8 0.6 A B Reversible first order reaction in the flow reactor
1.2 -1 Flow rate r [t ] Concentration a [a ] 0 0.50 0.75 1.00 1.25 0.25 1.0 0.8 A + B 2 B 0.6 Autocatalytic second order reaction in the flow reactor
1.2 -1 Flow rate r [t ] Concentration a [a ] 0 0.50 0.75 1.00 1.25 0.25 1.0 � = 0 � = 0.001 � = 0.1 0.8 A + B 2 B 0.6 A B � � = Autocatalytic second order and uncatalyzed reaction in the flow reactor
1.2 -1 Flow rate r [t ] Concentration a [a ] 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 1.0 0.8 A +2 B 3B 0.6 Autocatalytic third order reaction in the flow reactor
1.2 -1 Flow rate r [t ] Concentration a [a ] 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 1.0 � = 0 � = 0.001 � = 0.0025 0.8 � = 0.007 A +2 B 3B 0.6 � � = A B Autocatalytic third order and uncatalyzed reaction in the flow reactor
Autocatalytic third order reactions Multiple steady states � Oscillations in homogeneous solution Direct, A + 2 X 3 X , or hidden in the reaction mechanism Deterministic chaos (Belousow-Zhabotinskii reaction). Turing patterns Spatiotemporal patterns (spirals) Deterministic chaos in space and time Pattern formation in autocatalytic third order reactions G.Nicolis, I.Prigogine. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations . John Wiley, New York 1977
Autocatalytic second order reactions Chemical self-enhancement Combustion and chemistry � Direct, A + I 2 I , or hidden in of flames the reaction mechanism Selection of laser modes Selection of molecular or organismic species competing for common sources Autocatalytic second order reaction as basis for selection processes. The autocatalytic step is formally equivalent to replication or reproduction.
Stock Solution [A] = a 0 Reaction Mixture: A; I , k=1,2,... k k 1 A + I 2 I 1 1 d 1 k 2 A + I 2 I 2 2 d 2 k 3 A + I 2 I 3 3 d 3 k 4 A + I 2 I 4 4 d 4 k 5 A + I 2 I 5 5 d 5 Replication in the flow reactor P.Schuster & K.Sigmund, Dynamics of evolutionary optimization, Ber.Bunsenges.Phys.Chem. 89 : 668-682 (1985)
Concentration of stock solution a0 A + I 2 I 1 1 2 A + I + I A + I 2 I 2 2 1 A + I 1 A + I 2 I 3 3 A + I 2 I 4 4 A + I 2 I 5 5 A k > k > k > k > k 1 2 3 4 5 Flow rate r = � R-1 Selection in the flow reactor: Reversible replication reactions
Concentration of stock solution a0 A + I 2 I 1 1 A + I 2 I 2 2 A + I 1 A + I 2 I 3 3 A + I 2 I 4 4 A + I 2 I 5 5 A k > k > k > k > k 1 2 3 4 5 Flow rate r = � R-1 Selection in the flow reactor: Irreversible replication reactions
(A) + I 1 I 1 I 1 + k 1 Σ (A) + I 2 I 2 Φ I 2 + dx / dt = k x - x j i i i j k 2 Φ = Σ ; Σ = 1 k x x i i i i i [A] = a = constant I j I j (A) + (A) + I j I j + + k = max {k ; j=1,2,...,n} k j k j m j � � � x (t) 1 for t m I m (A) + (A) + I m I m + k m s = (k m+1 -k m )/k m I n (A) + (A) + I n I n + + k n Selection of the „fittest“ or fastest replicating species
1 Fraction of advantageous variant 0.8 0.6 s = 0.1 s = 0.02 0.4 0.2 s = 0.01 0 0 200 600 1000 400 800 Time [Generations] Selection of advantageous mutants in populations of N = 10 000 individuals
A A A A A G G C C G G G U U U G C U C C U C G U G C C -3’ 5’- = adenylate A 27 16 � 4 = 1.801 10 possible different sequences = uridylate U = cytidylate C Combinatorial diversity of sequences: N = 4 � = guanylate G Combinatorial diversity of heteropolymers illustrated by means of an RNA aptamer that binds to the antibiotic tobramycin
5' 3' Plus Strand G C C C G Synthesis 5' 3' Plus Strand G C C C G C G 3' Synthesis 5' 3' Plus Strand G C C C G Minus Strand C G G G C 5' 3' Complex Dissociation 3' 5' Plus Strand G C C C G Complementary replication as + the simplest copying 5' 3' mechanism of RNA Minus Strand C G G G C
5' 3' Plus Strand G C C C G 5' 3' GAA UCCCG AA GAA UCCCGUCCCG AA Plus Strand G C C C G Insertion C 3' G 5' 3' Minus Strand G G C G G C GAAUCCA GAAUCC CGA A 3' 5' Deletion Plus Strand G C C C G C Point Mutation Mutations represent the mechanism of variation in nucleic acids
Σ Φ I 1 dx / dt = k Q ji x - x I j + j i i i j k j Q 1j Σ i Φ = Σ ; Σ = 1 ; k x x Q ij = 1 I j I 2 + i i i i i n-d(i,j) d(i,j) Q = (1-p) p ij k j Q 2j p .......... Error rate per digit d(i,j) .... Hamming distance I j (A) + I j I j + k j Q jj between I and I i j k j Q nj I j I n + Chemical kinetics of replication and mutation
Master sequence Mutant cloud n o i t a r t n e c n o C Sequence space The molecular quasispecies in sequence space
Master sequence Mutant cloud “Off-the-cloud” n mutations o i t a r t n e c n o C Sequence space The molecular quasispecies and mutations producing new variants
Ronald Fisher‘s conjecture does not hold in general for replication-mutation systems : In general evolutionary dynamics the mean fitness of populations may also decrease monotonously or even go through a maximum or minimum. It does also not hold in general for recombination of many alleles and general multi-locus systems in population genetics. Optimization of fitness is, nevertheless, fulfilled in most cases, and can be understood as a useful heuristic.
Optimization of RNA molecules in silico W.Fontana, P.Schuster, A computer model of evolutionary optimization . Biophysical Chemistry 26 (1987), 123-147 W.Fontana, W.Schnabl, P.Schuster, Physical aspects of evolutionary optimization and adaptation . Phys.Rev.A 40 (1989), 3301-3321 M.A.Huynen, W.Fontana, P.F.Stadler, Smoothness within ruggedness. The role of neutrality in adaptation . Proc.Natl.Acad.Sci.USA 93 (1996), 397-401 W.Fontana, P.Schuster, Continuity in evolution. On the nature of transitions . Science 280 (1998), 1451-1455 W.Fontana, P.Schuster, Shaping space. The possible and the attainable in RNA genotype- phenotype mapping . J.Theor.Biol. 194 (1998), 491-515
Three-dimensional structure of phenylalanyl-transfer-RNA
5'-End 3'-End Sequence GCGGAU UUA GCUC AGDDGGGA GAGC M CCAGA CUGAAYA UCUGG AGMUC CUGUG TPCGAUC CACAG A AUUCGC ACCA 3'-End 5'-End 70 60 Secondary Structure 10 50 20 30 40 Symbolic Notation 5'-End 3'-End Definition and formation of the secondary structure of phenylalanyl-tRNA
Genotype-Phenotype Mapping Evaluation of the � = � S � ( ) I � S � Phenotype I � ƒ f = ( S ) � � f � Q � f 1 j f 1 Mutation I 1 f 2 f n+1 I 1 I n+1 I 2 f n f 2 I n I 2 f 3 I 3 Q Q I 3 f 3 I � I 4 f 4 f � I 5 I 4 I 5 f 4 f 5 f 5 Evolutionary dynamics including molecular phenotypes
Criterion of Minimum Free Energy UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG Sequence Space Shape Space
ψ Sk = ( ) I. fk = ( f Sk ) Non-negative Sequence space Phenotype space numbers Mapping from sequence space into phenotype space and into fitness values
Stock Solution Reaction Mixture The flowreactor as a device for studies of evolution in vitro and in silico
Recommend
More recommend
