• • • • • • • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 011
• • • • • • • • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 012
• • • • • • • • • • • • • • • • • • • • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 024
• • 2 2 2 • 2 • • • 2 3 • 3 • • 3 3 links # nodes • • 2 14 • 2 14 2 • 10 3 6 • • 5 5 2 2 • 2 10 1 • 5 12 1 • 12 • • 14 1 3 • 2 3 • • • • 2 2 2 2 Analysis of nodes and links in a step by step evolved network
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 = = = f x k x x x k k k ( ; ) ; ( , , ) ; ( , , ) K K 1 n 1 m d t Reaction diffusion equations ∂ x = ∇ + D 2 x f x k ( ; ) Solution curves : ( ) x t ∂ t x i (t) Concentration Parameter set = k ( T , p , p H , I , ) ; j 1 , 2 , , m K K j General conditions : T , p , pH , I , ... t x ( 0 ) Initial conditions : Time Boundary conditions : � boundary ... S , normal unit vector ... u x S = Dirichlet : g ( r , t ) ∂ x S = = ⋅ ∇ ˆ Neumann : u x g ( r , t ) ∂ u The forward problem of chemical reaction kinetics (Level I)
Kinetic differential equations d x = = = f ( x ; k ) ; x ( x , , x ) ; k ( k , , k ) K K 1 n 1 m d t Reaction diffusion equations ∂ x 2 = ∇ + Genome: Sequence I G D x f ( x ; k ) Solution curves : x t ( ) ∂ t x i (t) Concentration Parameter set = ( G I ; , , , , ) ; 1 , 2 , , k j T p p H I j m K K General conditions : T , p , pH , I , ... t x ( 0 ) Initial conditions : Time Boundary conditions : boundary ... S , normal unit vector � ... u x S = Dirichlet : g ( r , t ) ∂ x S = = ⋅ ∇ ˆ u x g ( r , t ) Neumann : ∂ u The forward problem of cellular reaction kinetics (Level I)
Kinetic differential equations d x = = = f ( x ; k ) ; x ( x , , x ) ; k ( k , , k ) K K 1 n 1 m d t Reaction diffusion equations ∂ x = ∇ + 2 ( ; ) D x f x k ∂ t General conditions : T , p , pH , I , ... x ( 0 ) Initial conditions : Genome: Sequence I G Boundary conditions : boundary ... S , normal unit vector � ... u Parameter set x S = = Dirichlet : g ( r , t ) k j ( G I ; T , p , p H , I , ) ; j 1 , 2 , , m K K ∂ x S = = ⋅ ∇ Neumann : ˆ u x g ( r , t ) ∂ u Data from measurements x (t ); = 1, 2, ... , j N j x i (t ) j Concentration The inverse problem of cellular t reaction kinetics (Level I) Time
The forward problem of bifurcation analysis in cellular dynamics (Level II)
The inverse problem of bifurcation analysis in cellular dynamics (Level II)
A model genome with 12 genes 1 2 3 4 5 6 7 8 9 10 11 12 Regulatory protein or RNA Regulatory gene Enzyme Structural gene Metabolite Sketch of a genetic and metabolic network
A + B � X Stoichiometric equations 2 X � Y Sequences Y + X � D Vienna RNA Package SBML – systems biology markup language d a d b Structures and kinetic = = − k a b 1 d t d t parameters Kinetic differential equations d x = − − 2 k a b k x k x y 1 2 3 d t d y = 2 − k x k x y ODE Integration by means of CVODE 2 3 d t d d = k x y 3 d t Solution curves x i (t) Concentration t Time The elements of the simulation tool MiniCellSim SBML : Bioinformatics 19 :524-531, 2003; CVODE : Computers in Physics 10 :138-143, 1996
ATGCCTTATACGGCAGTCAGGTGCACCATT...GGC genotype DNA string TACGGAATATGCCGTCAGTCCACGTGGTAA...CCG genotype-p h e not y p mapping e RNA m RNA genetic regulation network RNA and protein structures Protein enzymes and small Metabolism metabolic reaction network molecules transport system cell membrane Recycling of molecules environment nutrition waste The regulatory logic of MiniCellSym
The model regulatory gene in MiniCellSim
The model structural gene in MiniCellSim
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
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.
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 A single neuron signaling to a muscle fiber
Gating functions of the Hodgkin-Huxley equations
Temperature dependence of the Hodgkin-Huxley equations
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
∂ ∂ 2 1 V V = + − + − + − π 3 4 C [ g m h ( V V ) g n ( V V ) g ( V V ) ] 2 r L Na Na K K l l ∂ ∂ 2 R x t ∂ m = − − α ( 1 m ) β m m m ∂ t ∂ h = − − α ( 1 h ) β h Hodgkin-Huxley partial differential equations (PDE) h h ∂ t ∂ n = − − α ( 1 n ) β n n n ∂ t Hodgkin-Huxley equations describing pulse propagation along nerve fibers
∂ ∂ 2 1 V V = + − + − + − π 3 4 C θ [ g m h ( V V ) g n ( V V ) g ( V V ) ] 2 r L M Na Na K K l l ∂ ξ ∂ ξ 2 R ∂ m = − − θ α ( 1 m ) β m m m ∂ ξ Hodgkin-Huxley ordinary differential equations ∂ h (ODE) = − − θ α ( 1 h ) β h h h ∂ ξ ∂ n Travelling pulse solution: V ( x,t ) = V ( � ) with = − − θ α ( 1 n ) β n n n ∂ ξ � = x + � t Hodgkin-Huxley equations describing pulse propagation along nerve fibers
100 50 ] V m [ V 0 -50 1 2 3 4 5 6 � [cm] T = 18.5 C; θ = 1873.33 cm / sec
T = 18.5 C; θ = 1873.3324514717698 cm / sec
T = 18.5 C; θ = 1873.3324514717697 cm / sec
40 30 20 ] V m [ 10 V 0 -10 6 8 10 12 14 16 18 � [cm] T = 18.5 C; θ = 544.070 cm / sec
T = 18.5 C; θ = 554.070286919319 cm/sec
T = 18.5 C; θ = 554.070286919320 cm/sec
Propagating wave solutions of the Hodgkin-Huxley equations
∂ ∂ 2 1 V V = + + − − + − + − π 3 4 C θ [ g m ( h n n ) ( V V ) g n ( V V ) g ( V V ) ] 2 r L M Na 0 0 Na K K l l ∂ ξ ∂ ξ 2 R ∂ m = − − θ α ( 1 m ) β m Hodgkin-Huxley ordinary differential equations m m ∂ ξ (ODE) ∂ n = − − θ α ( 1 n ) β n n n ∂ ξ Travelling pulse solution: V ( x,t ) = V ( � ) with � = x + � t V = + β = − ≈ − α α ; 0 . 125 exp ( ) 0 . 125 ( 1 V ) V n 0 n 80 80 E Na An approximation to the Hodgkin-Huxley equations
Propagating wave solutions of approximations to the Hodgkin-Huxley equations
Evolutionary biology Optimization through variation and selection, relation between genotype, phenotype, and function, ... Selection and Genetic drift in Genetic drift in Generation time adaptation small populations large populations 10 6 generations 10 7 generations 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 Multicelluar organisms 10 d 274 a 27 380 a 273 800 a 2 × 10 7 a 2 × 10 8 a 20 a 200 000 a Time scales of evolutionary change
Genotype = Genome Mutation GGCUAUCGUACGUUUACCCAAAAAGUCUACGUUGGACCCAGGCAUUGGAC.......G Fitness in reproduction: Unfolding of the genotype: Number of genotypes in RNA structure formation the next generation Phenotype Selection Evolution of phenotypes
I 1 I j + Σ Φ dx / dt = f Q ji x - x f j Q j1 i j j j i I j I 2 + Σ i Φ = Σ ; Σ = 1 ; f x x Q ij = 1 j j i j j � i =1,2,...,n ; f j Q j2 [Ii] = xi 0 ; I i I j + [A] = a = constant f j Q ji l -d(i,j) d(i,j) I j (A) + I j Q = (1- ) p p + I j ij f j Q jj p .......... Error rate per digit l ........... Chain length of the f j Q jn polynucleotide I j d(i,j) .... Hamming distance I n + between Ii and Ij 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 n n = − φ = = φ = = i f Q x x , i 1 , 2 , , n ; x 1 ; f x f L j ji j i i j j = = = dt j 1 i 1 j 1 Solutions are obtained after integrating factor transformation by means of an eigenvalue problem ( ) ( ) ∑ − n 1 ⋅ ⋅ λ c 0 exp t l ( ) ∑ n ik k k = = = = x t k 0 ; i 1 , 2 , , n ; c ( 0 ) h x ( 0 ) L ( ) ( ) ∑ ∑ i − k ki i n n 1 = i 1 ⋅ ⋅ λ c 0 exp t l jk k k = = j 1 k 0 { } { } { } ÷ = = = − = = = 1 W f Q ; i , j 1 , 2 , , n ; L ; i , j 1 , 2 , , n ; L H h ; i , j 1 , 2 , , n L l L L i ij ij ij { } − ⋅ ⋅ = Λ = λ = − 1 L W L ; k 0 , 1 , , n 1 L k
Formation of a quasispecies in sequence space
Formation of a quasispecies in sequence space
Formation of a quasispecies in sequence space
Formation of a quasispecies in sequence space
Uniform distribution in sequence space
Quasispecies Uniform distribution 0.00 0.05 0.10 Error rate p = 1-q Quasispecies as a function of the replication accuracy q
Chain length and error threshold ⋅ σ = − ⋅ σ ≥ ⇒ ⋅ − ≥ − σ n Q ( 1 p ) 1 n ln ( 1 p ) ln σ ln ≈ p constant : n K max p σ ln ≈ n constant : p K max n = − n Q ( 1 p ) replicatio n accuracy K p error rate K n chain length K f = σ m superiorit y of master sequence K ∑ ≠ f j j m
Reaction Mixture Stock Solution Replication rate constant: f k = � / [ � + � d S (k) ] � d S (k) = d H (S k ,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
Randomly chosen Phenylalanyl-tRNA as initial structure target structure
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