Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Membrane Systems Combining Variable Molecular Structures with Discretised Reaction Kinetics From a Toy to a Tool in Systems Biology Thomas Hinze Friedrich-Schiller University Jena Department of Bioinformatics at School of Biology and Pharmacy thomas.hinze@uni-jena.de November 25, 2009 Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Outline CSMs with Incomplete Protein Activation Information: KaiABC Oscillator 1. Motivation 2. Cell signalling modules (CSM) 3. P system framework Π CSM 4. The KaiABC Oscillator: A circadian clock 5. Case Study Π KaiABC 6. Network reconstruction by artificial evolution # 7. The SBMLevolver: a two-level evolutionary algorithm time 8. Evolved networks: a selection 1 0 0 1 1 0 0 0 1 1 0 0 1 0 9. Ongoing study: control system-based 1 0 1 1 0 1 0 1 0 0 1 0 0 1 specification of circadian oscillators Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Combinatorial Explosion of Protein Activation States • Tumor suppressor protein p53: 27 phosphorylation sites • Up to 2 27 = 134 , 217 , 728 distinguishable activation states • Each state: individual constituent of reaction network www.rcsb.org/pdb −> education corner 1tsr 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Cell Signalling Module Characteristics • Intracellular reaction network acting as functional unit • Composed of proteins carrying phosphorylation sites • Interactions between individual activation states Facts • Dynamical behaviour essential to understand function • Often partially unknown • Reconstruction as challenging task in systems biology • Reverse engineering by integrative approach Idea to manage complexity: Capturing each protein by a specific string-object instead of separate species per activation state Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Specification of String-Objects Assuming two alphabets: V (for protein names), V ′ (for protein ∈ V ∪ V ′ properties); w.l.o.g # , ¬ , ∗ / Syntax for string-objects by regular set S = V + · { # } · (( V ′ ) + ∪ {¬} · ( V ′ ) + ∪ {∗} ) � ∗ � Protein properties • x : property x present (e.g. specific phosphate attached) • ¬ x : property x absent (e.g. specific phosphate removed) • ∗ : placeholder for arbitrary property setting Examples • prot1 # p # ∗ # ¬ p (subsumes activation states of prot1) • KaiC # ¬ KaiA # KaiB # 4 (prot. complex, 4 ligands attached) = ⇒ Application of reaction rules requires string matching Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Π CSM : System Components Let � S � be the set of all multisets over S . Π CSM = ( V , V ′ , R 1 , . . . , R r , f 1 , . . . , f r , A , C , ∆ τ ) with R i ∈ � S � × � S � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . is a reaction rule composed of two finite multisets f i : � S � − → N . . . . . . . . . . . . . . . . . . . .is a function corresponding to discrete kinetics of reaction R i A ∈ � S � . . . . . . . . . . . . . . . . . . . is a multiset of axioms representing the initial molecular configuration C ∈ R + spatial capacity of the module (vessel or compartment) ∆ τ ∈ R + . . . . . . . . . . . . . . . . . . . . . . . . . . . time discretisation interval Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Π CSM : Matching Let S be a string-object syntax. Two string-objects match iff there is at least one common wild card-free representation: Match S × S ⊆ Match { ( p # p 1 # p 2 . . . # p m , s # s 1 # s 2 . . . # s m ) | ( p = s ) ∧ � = m ∈ N ∀ j ∈ { 1 , . . . , m } : [( p j = s j ) ∨ ( p j = ∗ ) ∨ ( s j = ∗ ) ∨ (( p j = ¬ q ) ∧ ( s j � = q )) ∨ (( s j = ¬ q ) ∧ ( p j � = q ))] } • Match is a symmetric relation • Requires minimal similarity between string-objects with incomplete information • Uncertainty interpreted as arbitrary replacement by available properties Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Π CSM : Matching Let S be a string-object syntax. Two string-objects match iff there is at least one common wild card-free representation: Example V = {C, E} V’ = {D, T, p} C#D#−p C#T#−p C#D#p C#D#* C#T#p C#*#p C#*#* E#*#* C#D#p C#D#−p C#T#p C#T#−p C#*#p C#D#* C#*#* E#*#* Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Π CSM : Dynamical System Behaviour (I) • Successive progression of configuration L t ∈ � S � over A A B 2 A + B ➜ C A time t ∈ N starting from axioms A D B C B A D • ∆ τ : span between t and t + 1 • Conflict handling by prioritisation A A 2 A + B ➜ C ? 2 A + D ➜ E of reaction rules D B B A L 0 = L 0 , 0 = A � L t , 0 ⊖ Reactants t , 1 ⊎ Products t , 1 if Reactants t , 1 ⊆ L t , 0 L t , 1 = L t , 0 otherwise . . . � L t , r − 1 ⊖ Reactants t , r ⊎ Products t , r if Reactants t , r ⊆ L t , r − 1 L t + 1 = L t , r = L t , r − 1 otherwise Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Π CSM : Dynamical System Behaviour (II) Estimation of multisets Reactants t , j and Products t , j at time t concerning reaction R j = ( A j , B j ) ∈ � S � × � S � denoted A j ( a 1 ) a 1 + . . . + A j ( a p ) a p − → B j ( b 1 ) b 1 + . . . + B j ( b q ) b q includes • Matching between string-objects in L t and those in A j • Consideration of stoichiometry captured by multisets A j , B j • Evaluation of kinetic law expressed by scalar function f j Reactants t , j = { ( e 1 , ∞ ) , . . . , ( e p , ∞ ) } ∩ L t , j − 1 � � � � . . . f j · e 1 ∈ Match ( a 1 ) e p ∈ Match ( a p ) ( e 1 , A j ( a 1 )) , . . . , ( e p , A j ( a p )) � � Products t , j = { ( e 1 , ∞ ) , . . . , ( e p , ∞ ) } ∩ L t , j − 1 � � � � . . . · f j e 1 ∈ Match ( a 1 ) e p ∈ Match ( a p ) ( b 1 , B j ( b 1 )) , . . . , ( b q , B j ( b q )) � � Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Π CSM : Discrete Reaction Kinetics Scalar function f j provides number of turns for application of reaction rule R j . Rate constant: k j = ˆ k j · C · ∆ τ (Euler). f ( L t ( α )) | Match ( A j ) ∩{ ( α, ∞ ) }| f j ( L t ) = k j � ˆ ∀ α ∈ Match ( A j ) ∩ Match ( L t ) : ( R j =( A j , B j )) f ([ Z ]) Selected kinetic laws ˆ Kinetics Activation Repression v reac. rate f ([ Z ]) = [ Z ] ˆ Mass-Action [Z] − reactant conc. (no saturation) v v reac. rate reac. rate [ Z ] [ Z ] ˆ f ([ Z ]) = ˆ f ([ Z ]) = “ ” Michaelis-Menten 1 − Θ+[ Z ] Θ+[ Z ] [Z] [Z] reactant conc. reactant conc. (saturation) Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
Motivation CSM Π CSM KaiABC Oscillator Π KaiABC Artificial Evolution SBMLevolver Evolved Networks CMC11 Circadian Clocks Characteristics • Self-sustained biochemical oscillators • Period of approx. 24 hours persisting under constant environmental conditions (e.g. constant darkness) • Temperature compensation within physiological range • Capability of entrainment by external stimuli (e.g. light/dark or temperature cycles) • Reaction system with at least one feedback loop High scientific impact because . . . • Circadian clock as a potential universal property of life • Self-sustainability and high precision of bio-oscillators • Chronobiological control systems for manifold processes • Several independent evolutionary origins assumed Membrane Systems + Variable Molecular Structures + Discretised Reaction Kinetics Thomas Hinze
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