The Continuous -Calculus An Algebra for Biochemical Modelling - PowerPoint PPT Presentation

Introduction The Calculus Example Conclusions The Continuous π -Calculus An Algebra for Biochemical Modelling Marek Kwiatkowski School of Informatics University of Edinburgh 14 Oct 2008, CMSB joint work with Ian Stark M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Outline Introduction: ODEs and Process Algebras 1 The Continuous π -Calculus 2 Example: the KaiABC circadian clock 3 Future work and conclusions 4 M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Ordinary Differential Equations k 1 k 2 S + E P + E C k − 1 k 3 ∅ P M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Ordinary Differential Equations k 1 k 2 d [ S ] S + E P + E C = − k 1 [ S ][ E ] + k − 1 [ C ] k − 1 dt d [ E ] = − k 1 [ S ][ E ] + k − 1 [ C ] + k 2 [ C ] dt d [ C ] = k 1 [ S ][ E ] − k − 1 [ C ] − k 2 [ C ] dt k 3 d [ P ] = k 2 [ C ] − k 3 [ P ] ∅ P dt M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Ordinary Differential Equations k 1 k 2 d [ S ] S + E P + E C = − k 1 [ S ][ E ] + k − 1 [ C ] k − 1 dt d [ E ] = − k 1 [ S ][ E ] + k − 1 [ C ] + k 2 [ C ] dt d [ C ] = k 1 [ S ][ E ] − k − 1 [ C ] − k 2 [ C ] dt k 3 d [ P ] = k 2 [ C ] − k 3 [ P ] ∅ P dt M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Process Algebras k 1 k 2 S + E P + E C k − 1 k 3 ∅ P M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Process Algebras k 1 k 2 △ S + E P + E C = a ( x , y ) . ( x . S + y . P ) S k − 1 △ = ( ν u )( ν r ) a ( u , r ) . ( u . E + r . E ) E △ = τ. 0 P k 3 S | ... | S | E | ... | E ∅ P M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Process Algebras k 1 k 2 △ S + E P + E C = a ( x , y ) . ( x . S + y . P ) S k − 1 △ = ( ν u )( ν r ) a ( u , r ) . ( u . E + r . E ) E △ = τ. 0 P k 3 S | ... | S | E | ... | E ∅ P M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions ODEs vs PAs ODEs: PAs: M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions ODEs vs PAs ODEs: PAs: continuous discrete M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions ODEs vs PAs ODEs: PAs: continuous discrete deterministic non-deterministic/stochastic M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions ODEs vs PAs ODEs: PAs: continuous discrete deterministic non-deterministic/stochastic monolithic modular (compositional) M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions ODEs vs PAs ODEs: PAs: continuous discrete deterministic non-deterministic/stochastic monolithic modular (compositional) specify dynamics specify interactions M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions ODEs vs PAs ODEs: PAs: continuous discrete deterministic non-deterministic/stochastic monolithic modular (compositional) specify dynamics specify interactions very popular relatively unknown M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Syntax: species and processes Species: A , B :: = 0 | π 1 . A 1 + · · · + π n . A n D ( � a ) | A | B | ( ν M ) A M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Syntax: species and processes Species: Processes: A , B :: = 0 | π 1 . A 1 + · · · + π n . A n P , Q :: = c · A | P � Q c ∈ R ≥ 0 D ( � a ) | A | B | ( ν M ) A (thus P is an element of R S ) M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Syntax: affinity networks Names represent protein interaction sites. b c k 1 k 2 k 4 k 5 k 3 f e a d An affinity network gives their interaction structure. M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Semantics dP dt : immediate behaviour element of R S equivalent to an ODE system M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Semantics dP dt : immediate behaviour ∂ P : interaction potential element of R S×C×N element of R S equivalent to a transition system equivalent to an ODE system M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Semantics dP dt : immediate behaviour ∂ P : interaction potential element of R S×C×N element of R S equivalent to a transition system equivalent to an ODE system △ ∂ ( P � Q ) = ∂ P + ∂ Q d ( P � Q ) dP + dQ △ = + ∂ P � ∂ Q dt dt dt M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Semantics dP dt : immediate behaviour ∂ P : interaction potential element of R S×C×N element of R S equivalent to a transition system equivalent to an ODE system △ ∂ ( P � Q ) = ∂ P + ∂ Q d ( P � Q ) dP + dQ △ = + ∂ P � ∂ Q dt dt dt △ 1 A → F � 1 B = Aff ( x , y )( � F · G � − � A � − � B � ) x y → G M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Example: a simple chemical reaction network k 1 k 2 S + E P + E C k − 1 k 3 ∅ P M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Example: a simple chemical reaction network k 1 k 2 △ S + E P + E C = a ( x , y ) . ( x . S + y . P ) S k − 1 △ = ( ν M ) b � u , r � . act . E E △ = τ @ k 3 . 0 P k 3 c E · E � c S · S ∅ P M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Example: a simple chemical reaction network k 1 k 2 △ S + E P + E C = a ( x , y ) . ( x . S + y . P ) S k − 1 △ = ( ν M ) b � u , r � . act . E E △ = τ @ k 3 . 0 P k 3 c E · E � c S · S ∅ P u r a k − 1 k 2 k 1 act b M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions Example: a simple chemical reaction network k 1 k 2 △ S + E P + E C = a ( x , y ) . ( x . S + y . P ) S k − 1 △ = ( ν M ) b � u , r � . act . E E △ = τ @ k 3 . 0 P k 3 c E · E � c S · S ∅ P u r a k − 1 k 2 k 1 act b M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions The KaiABC circadian clock of Synechococcus elongatus k ps k ps k ps C 0 C 1 C 6 · · · b 0 f 6 ˜ ˜ ˜ C 0 C 1 C 6 · · · ˜ ˜ ˜ k dps k dps k dps M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions The model △ ( ν M i )( τ @ k ps . C i +1 + τ @ f i . ˜ C i = C i + τ @ k dps . C i − 1 + a i � act i � . ( u i . C i + r i . C i +1 )) ˜ △ τ @˜ k ps . ˜ C i +1 + τ @ b i . C i + τ @˜ k dps . ˜ C i − 1 + b i . b ′ . B ˜ C i = C i B ˜ △ τ @˜ k ps . B ˜ . (˜ C i | B | B )+ τ @˜ k dps . B ˜ a ′ . AB ˜ C i +1 + τ @ k Bb C i = C i − 1 +˜ a i . ˜ C i i AB ˜ △ τ @˜ k ps . AB ˜ C i +1 + τ @˜ k Ab . ( B ˜ C i | A | A )+ τ @˜ k dps . AB ˜ C i = C i − 1 i △ A = a ( x ) . x . A +˜ a . 0 △ B = b . 0 △ P = c A · A � c B · B � c C · C 0 · · · · · · · · · a 0 a 6 ˜ ˜ a 0 a 6 b 0 b 6 k Af k Bf ˜ 6 6 k Af k Bf ˜ k Af k Af 0 0 0 0 k vf k vf a ′ b ′ ˜ a ˜ b a M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions The model: no autonomous phosphorylation △ ( ν M i )( τ @ k ps . C i +1 + τ @ f i . ˜ C i = C i + τ @ k dps . C i − 1 + a i � act i � . ( u i . C i + r i . C i +1 )) ˜ △ τ @˜ k ps . ˜ C i +1 + τ @ b i . C i + τ @˜ k dps . ˜ C i − 1 + b i . b ′ . B ˜ C i = C i B ˜ △ τ @˜ k ps . B ˜ . (˜ C i | B | B )+ τ @˜ k dps . B ˜ a ′ . AB ˜ C i +1 + τ @ k Bb C i = C i − 1 +˜ a i . ˜ C i i AB ˜ △ τ @˜ k ps . AB ˜ C i +1 + τ @˜ k Ab . ( B ˜ C i | A | A )+ τ @˜ k dps . AB ˜ C i = C i − 1 i △ A = a ( x ) . x . A +˜ a . 0 △ B = b . 0 △ P = c A · A � c B · B � c C · C 0 · · · · · · · · · a 0 a 6 ˜ ˜ a 0 a 6 b 0 b 6 k Af k Bf ˜ 6 6 k Af k Bf ˜ k Af k Af 0 0 0 0 k vf k vf a ′ b ′ ˜ a ˜ b a M. Kwiatkowski, I. Stark The Continuous π -Calculus

Introduction The Calculus Example Conclusions The model: no autonomous phosphorylation M. Kwiatkowski, I. Stark The Continuous π -Calculus