Computational and Mathematical Challenges from Molecular Life - PowerPoint PPT Presentation

• • • • Formation of a scale-free network through evolutionary point by point expansion: Step 003

• • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 004

• • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 005

• • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 006

• • • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 007

• • • • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 008

• • • • • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 009

• • • • • • • • • • • Formation of a scale-free network through evolutionary point by point expansion: Step 010

• • • • • • • • • • • • 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).

Stefan Bornholdt. Less is more in modeling large genetic networks. Science 310 , 449-450 (2005)

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

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

The model regulatory gene in MiniCellSim

The model structural gene in MiniCellSim

Evolutionary time: 0000 Number of genes : 12 06 structural + 06 regulatory Number of interactions : 15 04 inhibitory + 10 activating + + 1 self-activating A genabolic network formed from a genotype of n = 200 nucleotides

Evolutionary time scale [generations]: 0000 initial network Evolutionary time : 0000 , initial network 20 TF00 TF01 TF02 TF03 15 SP04 TF05 SP06 SP07 10 SP08 SP09 TF10 SP11 5 Stationary state 0 100 1000 10000 1e+05 Intracellular time Intracellular time scale

Evolution of a genabolic network : Initial genome: random sequence of length n = 200 , AUGC alphabet Gene length: n = 25 Simulation with mutation rate: p = 0.01 Evolutionary time unit >> intracellular time unit

Number of genes : total / structural genes regulatory genes

Evolution of a genabolic network : Initial genome: random sequence of length n = 200 , AUGC alphabet Gene length: n = 25 Simulation with mutation rate: p = 0.01 Evolutionary time unit >> intracellular time unit Recorded events: (i) Loss of a gene through corruption of the start signal “ TA ” (analogue of the “ TATA Box”), (ii) creation of a gene, (iii) change in the edges through mutation-induced changes in the affinities of translation products to the binding sites, and change in the class of genes (tf � sp). (iv)

Statistics of one thousand generations Total number of genes: 11.67 � 2.69 5.97 � 2.22 Regulatory genes: 5.70 � 2.17 Structural genes:

Neurobiology Neural networks, nonlinear dynamics, collective properties, 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.

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 d V d V = + 3 − + 4 − + − π 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 d d d m = − − θ α ( 1 m ) β m m m ξ d Hodgkin-Huxley ordinary differential equations d h (ODE) = − − θ α ( 1 h ) β h h h ξ d d n Travelling pulse solution: V ( x,t ) = V ( � ) with = − − θ α ( 1 n ) β n n n ξ d � = 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 d V d 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 d d d m = − − θ α ( 1 m ) β m Hodgkin-Huxley ordinary differential equations m m ξ d (ODE) d n = − − θ α ( 1 n ) β n n n ξ Travelling pulse solution: V ( x,t ) = V ( � ) with d � = 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

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

Replication rate constant: f k = � / [ � + � d S (k) ] � d S (k) = d H (S k ,S � ) f 6 f 7 f 5 f 0 f 4 f � f 3 f 1 f 2 Evaluation of RNA secondary structures yields replication rate constants

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

Randomly chosen Phenylalanyl-tRNA as initial structure target structure

In silico optimization in the flow reactor: Evolutionary Trajectory

28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations Neutral point mutations leave the change the molecular structure molecular structure unchanged Neutral genotype evolution during phenotypic stasis