Some Mathematical Challenges from Molecular Biology Part I Peter Schuster Institut für Theoretische Chemie und Molekulare Strukturbiologie der Universität Wien Mathematisches Kolloquium Zürich, 11.11.2003
Web-Page for further information: http://www.tbi.univie.ac.at/~pks
Prolog – Mathematics and the life sciences in the 21 st century 1. 2. Replication kinetics of RNA molecules and evolution 3. RNA evolution in silico 4. Sequence-structure maps, neutral networks, and intersections 5. Reference to experimental data 6. Summary
Prolog – Mathematics and the life sciences in the 21 st century 1. 2. Replication kinetics of RNA molecules and evolution 3. RNA evolution in silico 4. Sequence-structure maps, neutral networks, and intersections 5. Reference to experimental data 6. Summary
Genomics and proteomics Large scale data processing, sequence comparison ... Evolutionary biology Developmental biology Mathematics in Optimization through variation and Gene regulation networks, selection, relation between genotype, signal propagation, pattern 21st Century's phenotype, and function, ... formation, robustness ... Life Sciences Neurobiology Cell biology Neural networks, collective Regulation of cell cycle, properties, nonlinear metabolic networks, reaction dynamics, signalling, ... kinetics, homeostasis, ...
Genomics and proteomics Large scale data processing, sequence comparison ... 4×10 6 Nucleotides E. coli : Length of the Genome Number of Cell Types 1 Number of Genes 4 000 3×10 9 Nucleotides Man : Length of the Genome Number of Cell Types 200 Number of Genes 30 000 - 100 000
Fully sequenced genomes Fully sequenced genomes • Organisms 751 751 projects 153 153 complete (16 A, 118 B, 19 E) ( Eukarya examples: mosquito (pest, malaria), sea squirt, mouse, yeast, homo sapiens, arabidopsis, fly, worm, …) 598 598 ongoing (23 A, 332 B, 243 E) ( Eukarya examples: chimpanzee, turkey, chicken, ape, corn, potato, rice, banana, tomato, cotton, coffee, soybean, pig, rat, cat, sheep, horse, kangaroo, dog, cow, bee, salmon, fugu, frog, …) • Other structures with genetic information 68 68 phages 1328 1328 viruses 35 35 viroids 472 472 organelles (423 mitochondria, 32 plastids, 14 plasmids, 3 nucleomorphs) Source: NCBI Source: Integrated Genomics, Inc. August 12 th , 2003
The same section of the microarray is shown in three independent hybridizations. Marked spots refer to: (1) protein disulfide isomerase related protein P5, (2) IL-8 precursor, (3) EST AA057170, and (4) vascular endothelial growth factor Gene expression DNA microarray representing 8613 human genes used to study transcription in the response of human fibroblasts to serum V.R.Iyer et al ., Science 283 : 83-87, 1999
Wolfgang Wieser. Die Erfindung der Individualität oder die zwei Gesichter der Evolution. Spektrum Akademischer Verlag, Heidelberg 1998. A.C.Wilson. The Molecular Basis of Evolution. Scientific American, Oct.1985, 164-173.
Developmental biology Gene regulation networks, signal propagation, pattern formation, robustness ... Three-dimensional structure of the complex between the regulatory protein cro-repressor and the binding site on � -phage B-DNA
Development of the fruit fly drosophila melanogaster : Genetics, experiment, and imago
Cell biology Regulation of cell cycle, metabolic networks, reaction kinetics, homeostasis, ... The bacterial cell as an example for the simplest form of autonomous life The human body: 10 14 cells, 10 13 eukaryotic cells and 10 13 bacterial (prokaryotic) cells, � 9 � and � 200 eukaryotic cell types
A B C D E F G H I J K L Biochemical Pathways 1 2 3 4 5 6 7 8 9 10 The reaction network of cellular metabolism published by Boehringer-Ingelheim.
The citric acid or Krebs cycle (enlarged from previous slide).
Kinetic differential equations d x = = i f ( x , x , , x ; k , k , , k ) ; i 1 , 2 , , n K K K 1 2 n 1 2 m d t Reaction diffusion equations ∂ x = ∇ 2 + = Solution curves: x t ( ); = 1, 2, ... , i n i D x f ( x , x , , x ; k , k , , k ) ; i 1 , 2 , , n K K K i i i 1 2 n 1 2 m ∂ t x i Parameter set Concentration = k ( T , p , p H , I , ; x , x , , x ) ; j 1 , 2 , , m K K K j 1 2 n General conditions: , , pH , , ... T p I t Time = Initial conditions: x i ( 0 ) ; i 1 , 2 , , n K � Boundary conditions: boundary ... s � normal unit vector ... u x s = = f ( r , t ) ; i 1 , 2 , , n Dirichlet , K i ∂ x r r Neumann , = ⋅ ∇ s = = i ˆ u x f ( r , t ) ; i 1 , 2 , , n K ∂ i u The forward-problem of chemical reaction kinetics
Kinetic differential equations d x = = i f ( x , x , , x ; k , k , , k ) ; i 1 , 2 , , n K K K 1 2 n 1 2 m d t Reaction diffusion equations ∂ x = ∇ + = i D 2 x f ( x , x , , x ; k , k , , k ) ; i 1 , 2 , , n K K K ∂ i i 1 2 n 1 2 m t General conditions: , , pH , , ... T p I = Initial conditions: x i ( 0 ) ; i 1 , 2 , , n K Parameter set � = Boundary conditions: boundary ... s k ( T , p , p H , I , ; x , x , , x ) ; j 1 , 2 , , m K K K � j 1 2 n normal unit vector ... u r x s = = Dirichlet , f ( r , t ) ; i 1 , 2 , , n K i ∂ x r r Neumann , = ⋅ ∇ s = = i u ˆ x f ( r , t ) ; i 1 , 2 , , n K i ∂ u Data from measurements x t ( ); = 1, 2, ... , ; = 1, 2, ... , i n k N i k x i Concentration The inverse-problem of chemical reaction kinetics t Time
Neurobiology Neural networks, collective properties, nonlinear dynamics, signalling, ... A single neuron signaling to a muscle fiber
The human brain 10 11 neurons connected by � 10 13 to 10 14 synapses
Evolutionary biology Optimization through variation and selection, relation between genotype, phenotype, and function, ... 10 6 generations 10 7 generations Generation time 10 000 generations RNA molecules 10 sec 27.8 h = 1.16 d 115.7 d 3.17 a 1 min 6.94 d 1.90 a 19.01 a Bacteria 20 min 138.9 d 38.03 a 380 a 10 h 11.40 a 1 140 a 11 408 a Higher multicelluar 10 d 274 a 27 380 a 273 800 a 2 × 10 7 a 2 × 10 8 a organisms 20 a 20 000 a Time scales of evolutionary change
Prolog – Mathematics and the life sciences in the 21 st century 1. 2. Replication kinetics of RNA molecules and evolution 3. RNA evolution in silico 4. Sequence-structure maps, neutral networks, and intersections 5. Reference to experimental data 6. Summary
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'-e nd 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 � 3'-end O RNA O OH 5’-end N 4 O P O CH 2 O Na � 70 O O OH 60 3' - end O P O 10 Na � O 50 20 30 40 Definition of RNA structure
James D. Watson, 1928- , and Francis Crick, 1916- , Nobel Prize 1962 1953 – 2003 fifty years double helix The three-dimensional structure of a short double helical stack of B-DNA
5'-End 3'-End Sequence GCGGAUUUAGCUCAGDDGGGAGAGCMCCAGACUGAAYAUCUGGAGMUCCUGUGTPCGAUCCACAGAAUUCGCACCA 3'-End 5'-End 70 60 Secondary structure 10 50 20 30 40
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 Complementary replication as the 3' 5' simplest copying mechanism of RNA Plus Strand G C C C G Complementarity is determined by Watson-Crick base pairs: + 5' 3' G � C and A = U Minus Strand C G G G C
f 1 (A) + I 1 I 1 I 1 + f 2 (A) + I 2 I 2 I 2 + Φ = ( Φ ) dx / dt = x - x f x f i - i i i i i Φ = Σ ; Σ = 1 ; i,j f x x =1,2,...,n j j j j j i � i =1,2,...,n ; [I ] = x 0 ; i f i I i [A] = a = constant (A) + (A) + I i + + I i fm = max { ; j=1,2,...,n} fj � � � xm(t) 1 for t f m I m (A) + (A) + I m I m + f n I n (A) + (A) + I n I n + + Reproduction of organisms or replication of molecules as the basis of selection
Selection equation : [I i ] = x i � 0 , f i > 0 ( ) dx ∑ ∑ n n = − φ = = φ = = i x f , i 1 , 2 , , n ; x 1 ; f x f L i i i j j = = dt i 1 j 1 Mean fitness or dilution flux, φ (t), is a non-decreasing function of time, ( ) φ n dx d = ∑ { } 2 = − = ≥ i 2 f f f var f 0 i dt dt = i 1 Solutions are obtained by integrating factor transformation ( ) ( ) ⋅ x 0 exp f t ( ) = = x t i i ; i 1 , 2 , , n L ( ) ( ) i ∑ = n ⋅ x 0 exp f t j j j 1
s = ( f 2 - f 1 ) / f 1 ; f 2 > f 1 ; x 1 (0) = 1 - 1/N ; x 2 (0) = 1/N 1 Fraction of advantageous variant 0.8 0.6 s = 0.1 s = 0.02 0.4 0.2 s = 0.01 0 0 200 600 800 1000 400 Time [Generations] Selection of advantageous mutants in populations of N = 10 000 individuals
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