Ideas on the Classification of Gene Regulatory Dynamics Peter - PowerPoint PPT Presentation

Ideas on the Classification of Gene Regulatory Dynamics Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Symposium in Honor of René Thomas Bruxelles, 30.– 31.05.2008

Happy Birthday René Greetings from the Eastern Alps Hochgall 3436 m

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

1. Forward and inverse problems in biology 2. Regulation kinetics and bifurcation analysis 3. Reverse engineering of dynamical systems

1. Forward and inverse problems in biology 2. Regulation kinetics and bifurcation analysis 3. Reverse engineering of dynamical systems

Kinetic differential equations d x = = = ( ; ) ; ( , , ) ; ( , , ) K K f x k x x x k k k 1 1 n m d t Reaction diffusion equations ∂ x = ∇ + 2 ( ; ) Solution curves : ( ) D x f x k x t ∂ t x i (t) Concentration Parameter set = ( T , p , p H , I , ) ; j 1 , 2 , , m K K k j General conditions : T , p , pH , I , ... t ( 0 ) Initial conditions : x Time Boundary conditions : � boundary ... S , normal unit vector ... u x S = Dirichlet : ( , ) g r t ∂ S = x = ⋅ ∇ ˆ Neumann : ( , ) u x g r t ∂ u The forward problem of chemical reaction kinetics (Level I)

Kinetic differential equations d x = = = ( ; ) ; ( , , ) ; ( , , ) K K f x k x x x k k k 1 1 n m d t Reaction diffusion equations ∂ x 2 = ∇ + Genome: Sequence I G ( ; ) Solution curves : ( ) D x f x k x t ∂ t x i (t) Concentration Parameter set = ( G I ; , , , , ) ; 1 , 2 , , K K k j T p p H I j m General conditions : T , p , pH , I , ... t ( 0 ) Initial conditions : x Time Boundary conditions : boundary ... S , normal unit vector � ... u x S = Dirichlet : ( , ) g r t ∂ S = x = ⋅ ∇ ˆ ( , ) Neumann : u x g r t ∂ u The forward problem of biochemical reaction kinetics (Level I)

Kinetic differential equations d x = = = ( ; ) ; ( , , ) ; ( , , ) K K f x k x x x k k k 1 1 n m d t Reaction diffusion equations ∂ x = ∇ + 2 ( ; ) D x f x k ∂ t General conditions : T , p , pH , I , ... ( 0 ) Initial conditions : x Genome: Sequence I G Boundary conditions : boundary ... S , normal unit vector � ... u Parameter set x S = = Dirichlet : ( , ) ( G I ; , , , , ) ; 1 , 2 , , g r t K K k j T p p H I j m ∂ S = x = ⋅ ∇ Neumann : ˆ ( , ) u x g r t ∂ u Data from measurements (t ); = 1, 2, ... , x j N j x i (t ) j Concentration The inverse problem of biochemical t Time reaction kinetics (Level I)

The forward problem of bifurcation analysis (Level II)

The inverse problem of bifurcation analysis (Level II)

1. Forward and inverse problems in biology 2. Regulation kinetics and bifurcation analysis 3. Reverse engineering of dynamical systems

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

States of gene regulation in a bacterial expression control system – Jacob - Monod model

States of gene regulation in a bacterial expression control system – Jacob - Monod model

States of gene regulation in a bacterial expression control system – Jacob - Monod model

synthesis degradation Cross-regulation of two genes

∂ F > 0 ∂ n p p = Activation : ( ) j F p + n i j K p j K = Repression : ( ) F p + n i j K p j ∂ = , 1 , 2 F i j < 0 ∂ p Gene regulatory binding functions

= = = dq [ G ] [ G ] const . g = Q − Q 1 ( ) 1 2 0 k F p d q 1 1 2 1 1 = = [ Q ] , [ Q ] , dt q q 1 1 2 2 = = [ P ] , [ P ] p p dq = − 1 1 2 2 Q Q 2 ( ) k F p d q 2 2 1 2 2 dt n p = Activation : ( ) j F p dp + i j n = − P P K p 1 k q d p j 1 1 2 1 dt K = Repression : ( ) F p + n i j dp K p = − P P 2 j k q d p 2 2 2 2 = , 1 , 2 dt i j − ϑ ϑ = = ϑ Stationary points : ( ( )) 0 , ( ) p F F p p F p 1 1 1 2 2 1 2 2 2 1 Q P Q P k k k k ϑ = ϑ = 1 1 , 2 2 1 2 Q P Q P d d d d 1 1 2 2 Qualitative analysis of cross-regulation of two genes: Stationary points

∂ ∂ ⎛ ⎞ F F ⎜ 1 1 ⎟ Q Q k k − 1 1 ∂ ∂ 0 Q d ⎜ ⎟ p p 1 1 2 ⎧ ⎫ ∂ ∂ ⎜ ⎟ ∂ ⎪ ⎪ − & 0 F F Q x d 2 2 Q Q = = = A ⎨ ⎬ 2 k k i ⎜ ⎟ a 2 2 ∂ ∂ ∂ ⎪ ij ⎪ p p x ⎩ ⎭ ⎜ ⎟ 1 2 j − 0 0 P P ⎜ ⎟ k d 1 1 ⎜ ⎟ − 0 0 P P ⎝ ⎠ k d 2 2 ∂ ∂ F F = = Cross regulation : 1 2 0 ∂ ∂ p p 1 2 ∂ F − − ε 0 0 1 Q Q d k 1 1 ∂ p 2 ∂ F Q Q − − ε 2 0 Q Q 0 ε = = A - I D K d k 2 2 ∂ p P P 1 D K − − ε 0 0 P P k d 1 1 − − ε 0 0 P P k d 2 2 Qualitative analysis of cross-regulation of two genes: Jacobian matrix

Q Q ⋅ = ⋅ D K = ⋅ − ⋅ and hence Q P P Q Q P Q P D K K D D D K K P P K D M. Marcus. Two determinant condensation formulas. Linear Multilinear Algebra . 22 :95-102, 1987. I. Kovacs, D.S. Silver, S.G. Williams. Determinants of commuting-block matices. Am.Math.Mon . 106 :950-952, 1999. ( )( ) ∂ F − − ε − − ε − 1 Q P Q P d d k k 1 1 1 1 ∂ p ⋅ − ⋅ = = Q P Q P 2 ∂ ( )( ) D D K K F − − − ε − − ε 2 Q P Q P k k d d 2 2 2 2 ∂ p 1 ( )( )( )( ) ∂ ∂ F F = − − ε − − ε − − ε − − ε − = 1 2 0 Q P Q P Q Q P P d d d d k k k k 1 1 2 2 1 2 1 2 ∂ ∂ p p 2 1 + Q + Q + P + P + = ( ε ) ( ε ) ( ε ) ( ε ) 0 d d d d D 1 2 1 2 ∂ ∂ F F = − Q Q P P 1 2 D k k k k 1 2 1 2 ∂ ∂ p p 2 1

+ + + + + = Q Q P P ( ε ) ( ε ) ( ε ) ( ε ) 0 d d d d D 1 2 1 2 ∂ ∂ F F Eigenvalues of the Jacobian of the = − Q Q P P 1 2 D k k k k 1 2 1 2 ∂ ∂ cross-regulatory two gene system x x 2 1

= − Q Q P P D d d d d OneD 1 2 1 2 + + + + + + Q Q Q P Q P Q P Q P P P ( )( )( )( )( )( ) d d d d d d d d d d d d = 1 2 1 1 1 2 2 1 2 2 1 2 D Hopf + + + Q Q P P 2 ( ) d d d d 1 2 1 2

s > 0.5: bistability both genes on or both genes off Regulatory dynamics at D � 0 , act.-act., n=2

s > 1.29: stable limit cycle gene activity oscillating Regulatory dynamics at D � 0 , act.-rep., n=3

Regulatory dynamics at D > D Hopf , act.-repr., n=3

s > 0.794: bistability gene 1 on and gene 2 off or gene 1 off and gene 2 on Regulatory dynamics at D � 0 , rep.-rep., n=2

Hill coefficient: n Act.-Act. Act.-Rep. Rep.-Rep. 1 S , E S S 2 E , B(E,P) S S , B(P 1 ,P 2 ) 3 E , B(E,P) S , O S , B(P 1 ,P 2 ) 4 E , B(E,P) S , O S , B(P 1 ,P 2 ) E ...... „extinction“, both genes off S ...... „stable fixed point“ with both genes (partially) active O ..... „oscillations“, stable limit cycle B ...... „bistability“ P 1 ..... gene 1 on and gene 2 off P 2 ..... gene 1 off and gene 2 on P ...... both genes active

n p = Activation : ( ) j F p + n i j K p j K = Repression : ( ) F p + n i j K p j m p = j Intermedia te : ( ) F p κ + κ + κ + + i j 2 n K p p p 1 2 3 j j j = ≤ ≤ − , 1 , 2 ; 1 1 i j m n

3.67 > s > 2.02: bistability both genes on or both genes off s > 3.67: bistability and stable limit cycle both genes off or gene activity oscillating Regulatory dynamics, int.-act., m=2, n=4

17.96 > s > 3.883: bistability gene 1 on and gene 2 off or gene 1 off and gene 2 on s > 17.96: bistability and stable limit cycle gene 1 on and gene 2 off or gene activity oscillating Regulatory dynamics, rep.-int., m=2, n=4

( )( ) ∂ F − − ε − − ε − 0 1 Q P P Q d d k k 1 1 1 1 ∂ p 3 ∂ ( )( ) F ⋅ − ⋅ = − − − ε − − ε Q P Q P 2 0 P Q Q P k k d d 2 2 2 2 ∂ d d k k p 1 ∂ ( )( ) F − − − ε − − ε 0 3 P Q Q P k k d d 3 3 3 3 ∂ p 2 ∂ ∂ ∂ F F F = − 1 2 3 Q Q Q P P P D k k k k k k 1 2 3 1 2 3 ∂ ∂ ∂ p p p 3 1 2 Upscaling to more genes: n = 3

An example analyzed and simulated by MiniCellSim The repressilator : M.B. Ellowitz, S. Leibler. A synthetic oscillatory network of transcriptional regulators. Nature 403 :335-338, 2002