Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Membrane Systems in Algebraic Biology: From a Toy to a Tool Thomas Hinze Friedrich Schiller University Jena School of Biology and Pharmacy Department of Bioinformatics thomas.hinze@uni-jena.de January 29, 2009 Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Membrane Systems: Inspired by Cells and Tissues www.zum.de = ⇒ Capturing specialties of intra- and intercellular processes Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks (I) Nested Compartments Delimited by Permeable Membranes = ⇒ Spatial regions wherein chemical reactions can occur Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks (II) Dynamics in Compartmental Cell Structure www.reactome.org endocytosis merge exocytosis www.cancer.gov separation division www.wikipedia.org creation dissolution gemmation = ⇒ Plasticity initiated by dedicated reaction networks Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks (III) Complex Polymeric Biomolecules in Low Concentrations receptors external signal ligands hormones, factors, ... endocrine (dist.) enzyme−linked paracrine (near) autocrine (same cell) ion−channel G−protein−linked GDP GTP activation cascade cell membrane cell response phospholipid bilayer ATP phosphorylation activation by protein kinases ADP signal transduction, gene expression transformation, amplification via pathways cytosol nucleus inner membrane genomic dna = ⇒ Reaction pathways forming specific biomolecules Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Outline Membrane Systems in Algebraic Biology: From a Toy to a Tool 1. A Primer: Reaction systems composed of discrete entities 2. Membrane systems: Some introductory examples 3. Features and varieties: Classification and properties 4. Π CSN : A modelling framework for cell signalling networks # 5. Bio-Applications: Appetizers for systems biologists time 1 0 0 1 1 0 0 6. Membrane systems as computing devices 0 1 1 0 0 1 0 1 0 1 1 0 1 0 7. The P page: An online repository and more 1 0 0 1 0 0 1 8. Quo vadis: Concluding remarks and outlook Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Multiset: Molecular Configuration within a Membrane Example A { ( A , 3 ) , ( B , 2 ) , ( C , 0 ) , ( D , 1 ) } A L = D B supp ( L ) { A , B , D } = B A card ( L ) = 6 contents of a toy membrane Definitions Multiset: Let F be a set. A multiset over F is a mapping F : F − → N ∪ {∞} that specifies the multiplicity of each element a ∈ F . Support: Let F : F − → N ∪ {∞} be a multiset. A set S ⊆ F is called supp ( F ) iff S = { s ∈ F | F ( s ) > 0 } . Cardinality: card ( F ) := � F ( a ) a ∈ F Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Term Rewriting: Employ a Reaction by Set Operations A A B 2 A + B ➜ C A D B D C B A Example { ( A , 3 ) , ( B , 2 ) , ( C , 0 ) , ( D , 1 ) } ⊖ { ( A , 2 ) , ( B , 1 ) } ⊎ { ( C , 1 ) } = { ( A , 1 ) , ( B , 1 ) , ( C , 1 ) , ( D , 1 ) } Multiset operations Difference: F ⊖ G := { ( a , max ( F ( a ) − G ( a ) , 0 )) | a ∈ F \ G } Sum: F ⊎ G := { ( a , F ( a ) + G ( a )) a ∈ F ∪ G } Union: F ∪ G := { ( a , max ( F ( a ) , G ( a ))) | a ∈ F ∪ G } Intersection: F ∩ G := { ( a , min ( F ( a ) , G ( a ))) | a ∈ F ∩ G } Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Coping with Conflicts: Introducing Process Control A A B 2 A + B ➜ C A D B C B A D A 2 A + B ➜ C A ? 2 A + D ➜ E D B B A Possible strategies to decide among satisfied reactions Nondeterminism: Maximal parallel enumeration of all potential scenarios Prioritisation of reactions: Determinisation by a predefined order for applicability of reactions Stochasticity: Randomly select a satisfied reaction Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Evolution over Time Time-discrete iteration scheme • Starting from an initial molecular configuration L 0 • Iterative term rewriting for transition(s) L t → L t + 1 . Each iteration corresponds to a discrete period in time ∆ τ • Within each iteration turn, applicable reactions are figured out and subsequentially employed once or several times (e.g. kinetic function f : L → N in concert with discretised kinetic laws) • Obtaining a derivation tree that lists sequences of molecular configurations as nodes Considered aspects • Suitability for small amounts of reacting particles (e.g. cell signalling) • Compliance with mass conservance for undersatisfied reaction scenarios Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks First Example: A Single Membrane System Π PR = ( V , T , [ 1 ] 1 , L 0 , R ) V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . system alphabet T ⊆ V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . terminal alphabet [ 1 ] 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . compartmental structure L 0 ⊂ V × ( N ∪ {∞} ) . . . . . . . multiset for initial configuration R = { r 1 , . . . , r k } . . . . . . . . . . . . . . . . . . . . . . . set of reaction rules Each reaction rule r i consists of two multisets (reactants E i , products P i ) such that r i = ( { ( A 1 , a 1 ) , . . . , ( A h , a h ) } , { ( B 1 , b 1 ) , . . . , ( B v , b v ) } ) . We write in chemical denotation: r i : a 1 A 1 + . . . + a h A h − → b 1 B 1 + . . . + b v B v ⇒ Index i specifies priority of r i : r 1 > r 2 > . . . > r k . = Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks First Example: A Single Membrane System Π PR = ( V , T , [ 1 ] 1 , L 0 , R ) V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . system alphabet T ⊆ V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . terminal alphabet [ 1 ] 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . compartmental structure L 0 ⊂ V × ( N ∪ {∞} ) . . . . . . . multiset for initial configuration R = { r 1 , . . . , r k } . . . . . . . . . . . . . . . . . . . . . . . set of reaction rules Each reaction rule r i consists of two multisets (reactants E i , products P i ) such that r i = ( { ( A 1 , a 1 ) , . . . , ( A h , a h ) } , { ( B 1 , b 1 ) , . . . , ( B v , b v ) } ) . We write in chemical denotation: r i : a 1 A 1 + . . . + a h A h − → b 1 B 1 + . . . + b v B v ⇒ Index i specifies priority of r i : r 1 > r 2 > . . . > r k . = Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Dynamical Behaviour of Π PR Iteration scheme for configuration update incrementing discrete time points t ∈ N ( a , α a , k ) | ∀ a ∈ V � L t + 1 = ∧ α a , 0 = card ( L t ∩ { ( a , ∞ ) } ) ∧ β a , i = card ( E i ∩ { ( a , ∞ ) } ) ∧ γ a , i = card ( P i ∩ { ( a , ∞ ) } ) � α a , i − 1 + f i · γ a , i − f i · β a , i iff ∀ a ∈ V : α a , i − 1 ≥ f i · β a , i ∧ α a , i = α a , i − 1 else ∧ i ∈ { 1 , . . . , k } � System output (distinction empty/nonempty configuration): supp t = 0 ( L t ∩ { ( w , ∞ ) | w ∈ T } ) ⊆ T �� ∞ � Membrane Systems in Algebraic Biology Thomas Hinze
Motivation Prerequisites Membrane Systems Bio-Applications Computing Devices Concluding Remarks Simulation Study of a Concrete Toy System Π PR ( { Z 0 , Z 1 , Z 2 , Z 3 , Z 4 , N , Y , 0 , 1 , Φ , C } , { Φ } , [ 1 ] 1 , L 0 , { r 1 , . . . , r 12 } ) Π PR = { ( Z 0 , 2 ) , ( 0 , 1 ) , ( 1 , 1 ) , ( C , 1000 ) } L 0 = r 1 : Z 3 + 1 + C − → Y + Φ + 1 r 5 : Z 0 + 1 + C − → Z 1 + 1 + Z 0 r 9 : Z 2 + 1 + C − → Z 3 + 1 r 2 : Z 4 + 0 + C − → Y + Φ + 0 r 6 : Z 0 + 0 + C − → Z 2 + 0 + Z 0 r 10 : Z 2 + 0 + C − → N + 0 r 3 : Y + 1 + C − → Y + Φ + 1 r 7 : Z 1 + 1 + C − → N + 1 r 11 : Z 3 + 0 + C − → N + 0 r 4 : Y + 0 + C − → Y + Φ + 0 r 8 : Z 1 + 0 + C − → Z 4 + 0 r 12 : Z 4 + 1 + C − → N + 1 1000 C number of particles Φ 800 Objektanzahl 600 400 200 N Y 0 0 0.05 0.1 0.15 0.2 time course Zeitskala Dynamical simulation was carried out using MatLab ( ∆ τ = 10 − 4 ). Example comes from transformation of a finite automaton. Membrane Systems in Algebraic Biology Thomas Hinze
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