A bigraph-based framework for protein and cell interactions Giorgio Bacci Davide Grohmann Marino Miculan Department of Mathematics and Computer Science University of Udine, Italy MeCBIC 2009 5th September 2009, Bologna 1 / 29
Abstract Machines of Systems Biology (Cardelli 08) gene regulatory networks, stochastic π -calculus, regulation Hybrid Systems, . . . Genes directs protein embedding, membrane construction h c s u κ -calculus, n m i e w signals and events c β Binders, t o o o r h n Brane Calculus, π -calculus, p fi / n n BioAmbients, s Bio-PEPA, e e e h k s CLS+, . . . LCLS, . . . a w r m e / g e r u e l a h t w o r P Q A B s implements fusion/fission Proteins Membranes holds receptors/reactions signal processing, confinements, metabolism regulation storage, transport 2 / 29
Abstract Machines of Systems Biology (Cardelli 08) gene regulatory networks, stochastic π -calculus, regulation Hybrid Systems, . . . Genes The tower of directs protein embedding, informatic models membrane construction h (Milner 09) c s u κ -calculus, n m i e w signals and events c β Binders, t o o o r h n Brane Calculus, π -calculus, p fi / n n BioAmbients, s Bio-PEPA, e e e h k s CLS+, . . . LCLS, . . . a w r m e / g e r u e l a h t w o r P Q A B s implements fusion/fission Proteins Membranes holds receptors/reactions signal processing, confinements, metabolism regulation storage, transport 2 / 29
Abstract Machines of Systems Biology (Cardelli 08) gene regulatory networks, stochastic π -calculus, regulation Hybrid Systems, . . . Genes directs protein embedding, In this talk: bigraphs membrane construction h c s u κ -calculus, n m as a formal framework theory for i e w signals and events c β Binders, t o o o r integrating and comparing models h n Brane Calculus, π -calculus, p fi / n n BioAmbients, s Bio-PEPA, e e e h k s CLS+, . . . LCLS, . . . a w r m e / g e r u e l a h t w o r P Q A B s implements fusion/fission Proteins Membranes holds receptors/reactions signal processing, confinements, metabolism regulation storage, transport 2 / 29
Abstract Machines of Systems Biology (Cardelli 08) gene regulatory networks, stochastic π -calculus, regulation Hybrid Systems, . . . Genes directs protein embedding, In this talk: bigraphs membrane construction h c s u κ -calculus, n m as a formal framework theory for i e w signals and events c β Binders, t o o o r integrating and comparing models h n Brane Calculus, π -calculus, p fi / n n BioAmbients, s Bio-PEPA, e e e h k s CLS+, . . . LCLS, . . . a w r m e / g e we focus on these levels r u e l a h t w o r P Q A B s implements fusion/fission Proteins Membranes holds receptors/reactions signal processing, confinements, metabolism regulation storage, transport 2 / 29
Interactions we want to model Let take as example the vesicle formation process: 3 / 29
Interactions we want to model Let take as example the vesicle formation process: protein interactions „ complexations « de-complexations 3 / 29
Interactions we want to model Let take as example the vesicle formation process: protein membrane interactions reconfigurations „ complexations « ` fissions and fusions ´ de-complexations 3 / 29
Interactions we want to model Let take as example the vesicle formation process: protein-membrane protein membrane interactions interactions reconfigurations „ complexations 0 1 protein configurations « that trigger a membrane ` fissions and fusions ´ @ A de-complexations reconfiguration 3 / 29
Talk outline 0. Introduction to Bigraphs 1. Biological Bigraphs and Bio β framework + syntax + well-formedness + semantics 2. Example: vesicle formation 3. Formal comparison results 4 / 29
A (very short) introduction to Bigraphs (Milner 01) bigraph y 0 y 1 G : � m, X � →� n, Y � 0 1 v 2 2 v 0 0 v 1 v 3 1 place graph link graph x 0 x 1 G P : m → n G L : X → Y roots ... ...outer names 0 1 y 0 y 1 v 0 v 2 v 2 v 3 v 0 v 3 v 1 v 1 sites ... ...inner names 0 1 2 x 0 x 1 5 / 29
. . . bigraphs continued (basic notation) outer name root (region) y 0 y 1 y 2 0 1 K control port K M v 0 e 1 1 v 2 0 node v 1 edge e 0 x 0 x 1 site inner name place = root or node or site link = edge or outer name point = port or inner name 6 / 29
. . . bigraphs continued (definition) . . . we take advantage of the variant of (Bundgaard-Sassone 06) where edges have type. Signature: �K , ar , E� Bigraphs: G P = ( V , ctrl , prnt ): m → n (place graph) G L = ( V , E , ctrl , edge , link ): X → Y (link graph) G = ( V , E , ctrl , edge , prnt , link ): � m , X � → � n , Y � (bigraph) = ( G P , G L ) 7 / 29
Why using bigraphical theory Using bigraphs is convenient for many reasons: + connectivity together with locality + lots of successful encodings (CCS, π -calculus, Ambient Calculus, Petri nets, . . . ) + local reaction rules + construction of compositional bisimilarities for observational equivalences + general tools (see BPL project) 8 / 29
Talk outline 0. Introduction to Bigraphs 1. Biological Bigraphs and Bio β framework + syntax + well-formedness + semantics 2. Example: vesicle formation 3. Formal comparison results 9 / 29
Proteins and bonds in bigraphs: intuition Protein signature: �P , ar , { v , h }� Sites can be visible, hidden, or free, determining the protein interface status hidden x free x bond visible hidden GTP GDP ν y . ( G (1 y + ¯ 3 + 4 x + 5) | GTP (1 y )) ν y . ( G (1 y + ¯ 3 + 4 x + ¯ 2 + ¯ 2 + ¯ 5) | GDP (1 y )) (*) Edge types could be extended to capture phosphorilated states (and more) 10 / 29
Bio β syntax and bigraphical meaning Systems P , Q ::= ⋄ | A p ( ρ ) | � S � P �� | P ∗ Q | ν n . P p n � P | f n � � S � P �� (pinch and fuse) Membranes S , T ::= 0 | A ap ( ρ ) | S ⋆ T p ⊥ n � S | f ⊥ (co-pinch and co-fuse) n � S � P �� S P membrane contents Ra (1 + 2 x ) ∗ � Ma (1 x ) ⋆ Mb (1 y ) � Rb (1 + 2 y ) ∗ C (1) �� 11 / 29
Well-formedness conditions The syntax is too general : many syntactically correct terms do not have a clear biological meaning. Definition (Well-formedness) Graph-likeness: free names occurs at most twice + only binary bonds Impermebility: protein bonds cannot cross the double layer Action pairing: actions and co-actions have to be well paired Action prefix: no occurrences of action terms within an action prefix ?? hyper edges � = bonds impermeability violated! 12 / 29
Well-formedness conditions The syntax is too general : many syntactically correct terms do not have a clear biological meaning. Definition (Well-formedness) Graph-likeness: free names occurs at most twice + only binary bonds Impermebility: protein bonds cannot cross the double layer Action pairing: actions and co-actions have to be well paired Action prefix: no occurrences of action terms within an action prefix Well-formedness is ensured by a type system 12 / 29
Type system Γ 1 ; Γ 2 ⊢ K : τ (Judgement) free names of K free actions occurring once occurring in K a Bio β term . . . occurring twice (system/membrane) (empty) ǫ ∈ { 0 , ⋄} A ∈ P ∀ x ∈ fn ( ρ ) . | ρ, x | ≤ 2 { x ∈ fn ( ρ ) | | ρ, x | = 1 } ; { x ∈ fn ( ρ ) | | ρ, x | = 2 } ⊢ A ( ρ ) : ∅ (prot) ∅ ; ∅ ⊢ ǫ : ∅ (action) t ∈ { p , p ⊥ , f } Γ 1 ; Γ 2 ⊢ P : τ x / ∈ Γ 1 τ ↾ { x } = ∅ Γ 1 ; Γ 2 ⊢ K : ∅ act ( K ) = ∅ ( ν -prot) Γ 1 , x ; Γ 2 ⊢ t x � K : { t x } Γ 1 ; Γ 2 \ { x } ⊢ ν x . P : τ Γ 1 ; Γ 2 , x ⊢ P : τ ∪ { t x , t ⊥ { t x , t ⊥ t ∈ { p , f } x } x } ∩ τ = ∅ (co-f) ( ν -action) x ; ∅ ⊢ f ⊥ x : { f ⊥ Γ 1 ; Γ 2 ⊢ ν x . P : τ x } op ∈ {∗ , ⋆ } Γ 1 , Γ; Γ 2 ⊢ K : τ ∆ 1 , Γ; ∆ 2 ⊢ L : σ Γ 1 , Γ; Γ 2 ⊢ S : τ Γ; ∆ 2 ⊢ P : σ ( τ ↾ Γ ) ⊥ = σ ↾ Γ ( τ ↾ Γ ) ⊥ = σ ↾ Γ (Γ 1 ∪ Γ 2 ) ∩ (∆ 1 ∪ ∆ 2 ) � = ∅ (Γ 1 ∪ Γ 2 ) ∩ ∆ 2 � = ∅ (par) (cell) Γ 1 , ∆ 1 ; Γ 2 , ∆ 2 , Γ ⊢ K op L : τ ∪ σ Γ 1 ; Γ 2 , ∆ 2 , Γ ⊢ � S � P �� : τ ∪ σ 13 / 29
Properties of the type system Proposition (Unicity of type) Let K a Bio β term. If Γ 1 ; Γ 2 ⊢ K : τ and ∆ 1 ; ∆ 2 ⊢ K : σ , then Γ 1 = ∆ 1 , Γ 2 = ∆ 2 and τ = σ Theorem (Well-formedness) A Bio β system P is well-formed if and only if Γ 1 ; Γ 2 ⊢ P : τ . . . later subject reduction 14 / 29
Semantics: Bio β reactive system A Bio β reactive system (Π , → ) is parametrized over two reaction rule specifications: + Protein reactions: similar to chemical reaction rules, but with (essential) spatial informations + Mobility configurations: protein configurations that trigger membrane re-modeling Reactions for Membrane transport are fixed � indeed, biological membrane modifications � are very limited: only pinching and fuse 15 / 29
Membrane transport: pinch T T p ⊥ p n pinch-in n − � P S Q S P Q p n � P ∗ � p ⊥ n � S ⋆ T � Q �� → � T � � S � P �� ∗ Q �� T T p ⊥ p n pinch-out n − S P � Q Q S P � p ⊥ n � S ⋆ T � p n � P ∗ Q �� → � S � P �� ∗ � T � Q �� 16 / 29
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