Kinetic differential equations d x = = i f ( x , x , K , x ; k , k , K , k ) ; i 1 , 2 , K , n 1 2 n 1 2 m d t Reaction diffusion equations ∂ x = ∇ 2 + = i D 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 � = k ( T , p , p H , I , ; x , x , , x ) ; j 1 , 2 , , m Boundary conditions: boundary ... s 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 , i = ˆ ⋅ ∇ s = = u 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
Neurobiology Neural networks, collective properties, nonlinear dynamics, signalling, ... d V 1 = − − − − − − 3 4 I g m h ( V V ) g n ( V V ) g ( V V ) Na Na K K l l d t C M dm = α − − β Hogdkin-Huxley OD equations ( 1 m ) m m m dt dh = α − − β ( 1 h ) h h h dt dn = α − − β ( 1 n ) n n n dt
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
1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie
Element of class 1: The RNA molecule
Stock Solution Reaction Mixture Replication rate constant: f k = � / [ � + � d S (k) ] � (k) = d H (S k ,S � d S ) Selection constraint: # RNA molecules is controlled by the flow ≈ ± N ( t ) N N The flowreactor as a device for studies of evolution in vitro and in silico
Replication rate constant: f k = � / [ � + � d S (k) ] � (k) = d H (S k ,S � d S ) f 6 f 7 f 5 f 0 f � f 4 f 3 f 1 f 2 Evaluation of RNA secondary structures yields replication rate constants
3'-End 5'-End 70 60 10 50 20 40 30 Randomly chosen Phenylalanyl-tRNA as initial structure target structure
50 S d � 40 t e g r a t o t e 30 c n a t s i d e r u 20 t c u r t s e g a r 10 e v A Evolutionary trajectory 0 0 250 500 750 1000 1250 Time (arbitrary units) In silico optimization in the flow reactor: Trajectory ( physicists‘ view )
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 44 Endconformation of optimization
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 43 44 Reconstruction of the last step 43 � 44
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 42 43 44 Reconstruction of last-but-one step 42 � 43 ( � 44)
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 41 42 43 44 Reconstruction of step 41 � 42 ( � 43 � 44)
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 40 41 42 43 44 Reconstruction of step 40 � 41 ( � 42 � 43 � 44)
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time Evolutionary process 39 40 41 42 43 44 Reconstruction Reconstruction of the relay series
Transition inducing point mutations Neutral point mutations Change in RNA sequences during the final five relay steps 39 � 44
50 Relay steps S d � 40 t e g r a t o t e 30 c n a t s i d e r u 20 t c u r t s e g a r 10 e v A Evolutionary trajectory 0 0 250 500 750 1000 1250 Time (arbitrary units) In silico optimization in the flow reactor: Trajectory and relay steps
Average structure distance Uninterrupted presence Number of relay step 08 to target dS 10 � 12 28 neutral point mutations during 20 14 a long quasi-stationary epoch Evolutionary trajectory 10 0 250 500 Time (arbitrary units) Transition inducing point mutations Neutral point mutations Neutral genotype evolution during phenotypic stasis
Variation in genotype space during optimization of phenotypes Mean Hamming distance within the population and drift velocity of the population center in sequence space.
Spread of population in sequence space during a quasistationary epoch: t = 150
Spread of population in sequence space during a quasistationary epoch: t = 170
Spread of population in sequence space during a quasistationary epoch: t = 200
Spread of population in sequence space during a quasistationary epoch: t = 350
Spread of population in sequence space during a quasistationary epoch: t = 500
Spread of population in sequence space during a quasistationary epoch: t = 650
Spread of population in sequence space during a quasistationary epoch: t = 820
Spread of population in sequence space during a quasistationary epoch: t = 825
Spread of population in sequence space during a quasistationary epoch: t = 830
Spread of population in sequence space during a quasistationary epoch: t = 835
Spread of population in sequence space during a quasistationary epoch: t = 840
Spread of population in sequence space during a quasistationary epoch: t = 845
Spread of population in sequence space during a quasistationary epoch: t = 850
Spread of population in sequence space during a quasistationary epoch: t = 855
Element of class 2: The ant worker
Ant colony Random foraging Food source Foraging behavior of ant colonies
Ant colony Food source detected Food source Foraging behavior of ant colonies
Ant colony Pheromone trail laid down Food source Foraging behavior of ant colonies
Ant colony Pheromone controlled trail Food source Foraging behavior of ant colonies
RNA model Foraging behavior of ant colonies Element RNA molecule Individual worker ant Mechanism relating elements Mutation in quasi-species Genetics of kinship Search process Optimization of RNA structure Recruiting of food Search space Sequence space Three-dimensional space Random step Mutation Element of ant walk Self-enhancing process Replication Secretion of pheromone Interaction between elements Mean replication rate Mean pheromone concentration Goal of the search Target structure Food source Temporary memory RNA sequences in population Pheromone trail ‘Learning’ entity Population of molecules Ant colony Learning at population or colony level by trial and error Two examples: (i) RNA model and (ii) ant colony
1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie
5' - end N 1 O CH 2 O 5'-e nd GCGGAU UUA GCUC AGUUGGGA GAGC CCAGA G CUGAAGA UCUGG AGGUC CUGUG UUCGAUC CACAG A AUUCGC ACCA 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 O OH Definition of RNA structure 5’-end N 4 O O CH 2 P O Na � 70 O O OH 60 3' - end O P O 10 Na � O 50 20 30 40
James D. Watson and Francis H.C. Crick Nobel prize 1962 1953 – 2003 fifty years double helix Stacking of base pairs in nucleic acid double helices (B-DNA)
4 6 5 4 7 6 6 8 5 � C G 3 1 1 4 9 2 C ’ 2 1 C ’ 3 1 2 2 55.7 � 54.4 � 10.72 Å 4 6 5 4 7 6 6 8 5 3 U = A 1 1 9 4 2 C ’ 2 1 C ’ 3 1 2 57.4 � 56.2 � 10.44 Å Watson-Crick type base pairs
O N H O G=U N N O H N N Deviation from H N Watson-Crick geometry H O N U=G H N O N O H N N Deviation from N H Watson-Crick geometry N H Wobble base pairs
5'-End 3'-End Sequence GCGGAUUUAGCUCAGDDGGGAGAGCMCCAGACUGAAYAUCUGGAGMUCCUGUGTPCGAUCCACAGAAUUCGCACCA 3'-End 5'-End 70 60 Secondary structure 10 50 20 40 30 � Symbolic notation 5'-End 3'-End A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs
Minimal hairpin loop size: n lp � 3 Minimal stack length: n st � 2 Recursion formula for the number of acceptable RNA secondary structures
Computed numbers of minimum free energy structures over different nucleotide alphabets P. Schuster, Molecular insights into evolution of phenotypes . In: J. Crutchfield & P.Schuster, Evolutionary Dynamics. Oxford University Press, New York 2003, pp.163-215.
RNA sequence Biophysical chemistry: thermodynamics and kinetics Inverse folding of RNA : RNA folding : Biotechnology, Structural biology, design of biomolecules spectroscopy of with predefined biomolecules, structures and functions Empirical parameters understanding molecular function RNA structure Sequence, structure, and function
5’-end 3’-end A C C (h) S 5 (h) U S 3 G C (h) U A S 4 A U (h) U G S 1 (h) S 2 C G (h) S 8 � 0 G C (h) Free energy G (h) S 9 S 7 A U A A A C C (h) U S 6 A U G G C C Suboptimal conformations A G G U U U G G G A C C A U G A G G G C U G (h) S 0 Minimum of free energy The minimum free energy structures on a discrete space of conformations
hairpin loop hairpin hairpin loop loop stack free stack stack joint end stack bulge free end free end stack internal loop stack hairpin loop hairpin loop multiloop hairpin loop stack stack stack Elements of RNA secondary structures free free as used in free energy calculations end end
5’-end 3’-end A C C U G C U A A U U G C G G C A U A A A C C U A U G G free energy of stacking < 0 C C A G G U U U G G G A C C A U G A G G G C U G ∑ ∑ ∑ ∑ ∆ 300 = + + + + G g h ( n ) b ( n ) i ( n ) L 0 ij , kl l b i stacks of hairpin bulges internal base pairs loops loops Folding of RNA sequences into secondary structures of minimal free energy, � G 0 300
Maximum matching An example of a dynamic programming computation of the maximum number of base pairs Back tracking yields the structure(s). [i,k-1] [ k+1,j ] i i+1 i+2 k j-1 j j+1 X i,k-1 X k+1,j { ( ) } = + + ρ X max X , max ( X 1 X ) + ≤ ≤ − − + + i , j 1 i , j i k j 1 i , k 1 k 1 , j k , j 1 Minimum free energy computations are based on empirical energies GGCGCGCCCGGCGCC GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA RNAStudio.lnk UGGUUACGCGUUGGGGUAACGAAGAUUCCGAGAGGAGUUUAGUGACUAGAGG
Maximum matching j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 i G G C G C G C C C G G C G C C 1 G * * 1 1 1 1 2 3 3 3 4 4 5 6 6 An example of a dynamic programming computation 2 G * * 0 1 1 2 2 2 3 3 4 4 5 6 of the maximum number of base pairs 3 C * * 0 1 1 1 2 3 3 3 4 5 5 4 G * * 0 1 1 2 2 2 3 4 5 5 Back tracking yields the structure(s). 5 C * * 0 1 1 2 2 3 4 4 4 6 G * * 1 1 1 2 3 3 3 4 7 C * * 0 1 2 2 2 2 3 8 C * * 1 1 1 2 2 2 [i,k-1] [ k+1,j ] 9 C * * 1 1 2 2 2 10 G * * 1 1 1 2 11 G * * 0 1 1 12 C * * 0 1 i i+1 i+2 k j-1 j j+1 13 G * * 1 14 C * * X i,k-1 X k+1,j 15 C * { ( ) } = + + ρ X max X , max ( X 1 X ) + ≤ ≤ − − + + i , j 1 i , j i k j 1 i , k 1 k 1 , j k , j 1 Minimum free energy computations are based on empirical energies GGCGCGCCCGGCGCC GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA RNAStudio.lnk UGGUUACGCGUUGGGGUAACGAAGAUUCCGAGAGGAGUUUAGUGACUAGAGG
1. Komplexität in der Biologie 2. Evolutionäre Optimierung und Lernen im Ensemble 3. Strukturbildung von Biomolekülen als kombinatorisches Problem 4. Modellbildung in der Neurobiologie
A single neuron signaling to a muscle fiber
B A Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.
d V 1 = − − − − − − 3 4 I g m h ( V V ) g n ( V V ) g ( V V ) Na Na K K l l d t C M dm = α − − β ( 1 m ) m m m dt dh = α − − β ( 1 h ) h h h dt dn = α − − β ( 1 n ) n n n dt Hogdkin-Huxley OD equations Hhsim.lnk Simulation of space independent Hodgkin-Huxley equations: Voltage clamp and constant current
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