A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti Maggiolo-Schettini A Calculus of Looping Sequences for Milazzo Troina Modelling Microbiological Systems Introduction The Calculus of Looping Sequences Terms Roberto Barbuti Andrea Maggiolo–Schettini Structural Congruence Dinamics Paolo Milazzo Angelo Troina An Application Sporulation Virus Replication Dipartimento di Informatica, Universit` a di Pisa, Italy Verification Conclusions Ruciane-Nida – September 29, 2005 1/17
Introduction A Calculus of Looping Sequences for Modelling Microbiological Formal models for systems of interactive components can be Systems easily used or adapted for the modelling of biological Barbuti Maggiolo-Schettini phenomena Milazzo Troina ◮ Examples: Petri Nets, π –calculus, Mobile Ambients Introduction The Calculus of The modelling of biological systems allows: Looping Sequences Terms Structural Congruence 1. the development of simulators Dinamics 2. the verification of properties An Application Sporulation Virus Replication 3. (hopefully) the prediction of unknown behaviours Verification Conclusions In this work: 1. we introduce a calculus for microbiological systems 2. we use the calculus for modeling an example of interaction among bacteria and bacteriophage viruses 2/17
The Calculus of Looping Sequences (CLS) A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti We assume a set E of elementary constituents. A Term T Maggiolo-Schettini Milazzo of CLS is given by the following grammar: Troina � � � � L � � Introduction T ::= a T · T T ⌋ T T | T T � � � � The Calculus of Looping Sequences where a is a generic element of E Terms Structural Congruence Dinamics An Application Sporulation Virus Replication The operators are: Verification T · T : Sequencing Conclusions � � L : Looping (if T is a sequence, it can rotate) T T 1 ⌋ T 2 : Containment (if T 1 is a looping, it contains T 2 ) T 1 | T 2 : Parallel composition (groups separated terms) 3/17
Example of Terms A Calculus of Looping Sequences for Modelling Microbiological Systems c e c e Barbuti h Maggiolo-Schettini Milazzo Troina d d a c Introduction g f The Calculus of g f a b a b Looping Sequences Terms b Structural Congruence Dinamics (a) (b) (c) An Application Sporulation Virus Replication Verification Conclusions � � L ( a ) a · b · c � L · a · b �� � L ( b ) c · d · e � L ⌋ h ) · a · b � L ⌋ ( f · g | f · g ) � � ( c ) ( c · d · e 4/17
Structural Congruence A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti The Structural Congruence ≡ is the least congruence Maggiolo-Schettini Milazzo relation on terms satisfying associativity of | and · , Troina right–associativity of ⌋ and the following axioms: Introduction The Calculus of A 1 . ( T 1 | T ) · T 2 ≡ ( T 1 · T 2 ) | T ≡ T 1 · ( T 2 | T ) Looping Sequences A 2 . ( T 1 | T 2 ) ⌋ T ≡ ( T 1 ⌋ T ) | T 2 Terms Structural Congruence � L ≡ � L | T 1 � � Dinamics A 3 . T | T 1 T An Application A 4 . T | T 1 | T 2 ≡ T | T 2 | T 1 Sporulation Virus Replication A 5 . ( T 1 ⌋ T 2 ) ⌋ T 3 ≡ T 1 ⌋ ( T 2 | T 3 ) Verification � L ≡ � � � L Conclusions A 6 . T 1 · T 2 T 2 · T 1 A 7 a . a ⌋ T ≡ a | T A 7 b . ( T 1 · T 2 ) ⌋ T ≡ ( T 1 · T 2 ) | T 5/17
Dinamics of the Calculus (1) A Calculus of Looping Sequences for Modelling Microbiological Systems Let V be a set of term variables ( X , Y , Z , . . . ). Let T be the Barbuti Maggiolo-Schettini set of ground terms and T V be the set of term which may Milazzo Troina contain variables. Introduction ◮ An istantiation is a function σ : V → T . Let Σ be the The Calculus of set of all the possible istantiations Looping Sequences Terms ◮ T σ denotes the term obtained by replacing any variable Structural Congruence Dinamics X with σ ( X ) in T An Application Sporulation Virus Replication Verification A Rewrite Rule is a triple ( T , T ′ , Σ ′ ) where: Conclusions ◮ T , T ′ ∈ T V ◮ variables in T ′ are a subset of those in T ◮ Σ ′ are istantiations which can be applied to T and T ′ 6/17
Dinamics of the Calculus (2) A Calculus of Looping Sequences for Modelling Microbiological Systems A rule ( T , T ′ , Σ ′ ) can be applied to all terms T σ s.t. σ ∈ Σ ′ Barbuti Maggiolo-Schettini Milazzo Example: ( b · X · b , c · X · c , Σ ′ ) Troina where Σ ′ = { σ ∈ Σ | occ ( a , σ ( X )) = 0 } Introduction ◮ can be applied to b · c · b (producing c · c · c ) The Calculus of Looping Sequences ◮ cannot be applied to b · a · b Terms Structural Congruence Dinamics An Application Formally, given a set of rules R , evolution of terms is Sporulation Virus Replication described by the transition system given by the least relation Verification Conclusions → satisfying ( T , T ′ , Σ ′ ) ∈ R σ ∈ Σ ′ T σ → T ′ σ and closed under structural congruence and all the operators 7/17
Application to a Microbiological System A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti Maggiolo-Schettini We describe bacteria and bacteriophage viruses as terms of Milazzo Troina the calculus Introduction The Calculus of Reproduction of bacteria (sporulation) and of the viruses Looping Sequences (replication) are described as sets of rewrite rules Terms Structural Congruence Dinamics An Application → T ′ [ C ] Alternative notation for rewrite rules: T − Sporulation Virus Replication ◮ C is condition on instantiations Verification Conclusions ◮ Σ ′ is the set of istantiations which satisfy C ◮ Example: b · X · b − → c · X · c [ occ ( a , σ ( X )) = 0] 8/17
Sporulation (Reproduction of Bacteria) (1) A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti Maggiolo-Schettini Milazzo Troina Introduction DNA The Calculus of Looping Sequences The bacterium Step 1: Duplication Step 2: Prespore Terms Structural Congruence Dinamics An Application Sporulation Virus Replication Verification Conclusions Step 4: Release Step 3: Coat 9/17
Sporulation (2) A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti Maggiolo-Schettini Milazzo Troina � L ⌋ DNA b � BACTERIUM ::= m · . . . · m Introduction � �� � n The Calculus of � L ⌋ DNA b � Looping Sequences PRESPORE ::= m · . . . · m Terms � �� � Structural Congruence n Dinamics 2 � L ⌋ PRESPORE An Application � SPORE 1 ::= c · . . . · c Sporulation � �� � Virus Replication n Verification 2 Conclusions � L ⌋ PRESPORE � SPORE 2 ::= d · . . . · d � �� � n 2 10/17
Rules for Sporulation (1) A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti ´ L ⌋ ( DNA b | X ) ` S 1 . m · . . . · m − → Maggiolo-Schettini | {z } Milazzo n Troina ´ L ⌋ ( DNA b | DNA b | X ) ` m · . . . · m [ occ ( DNA b , X ) = 0] Introduction | {z } n The Calculus of Looping Sequences Terms ´ L ⌋ ( DNA b | DNA b | X ) ` Structural Congruence S 2 . m · . . . · m − → Dinamics | {z } n An Application ´ L ⌋ ( DNA b | PRESPORE | X ) ` Sporulation m · . . . · m Virus Replication | {z } Verification n Conclusions ´ L ⌋ ( X | PRESPORE | Y ) ` S 3 . m · . . . · m − → | {z } n ´ L ⌋ ( X | SPORE 1 | Y ) ` m · . . . · m | {z } n 11/17
Rules for Sporulation (2) A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti Maggiolo-Schettini Milazzo ´ L ⌋ ( X | SPORE 1 | Y ) Troina ` S 4 . m · . . . · m − → | {z } Introduction n ´ L ⌋ ( X | Y ) ` The Calculus of SPORE 1 · m · . . . · m Looping Sequences | {z } Terms n Structural Congruence Dinamics ´ L ⌋ X ´ L ⌋ X ) | SPORE 2 ` ` An Application S 5 . SPORE 1 · m · . . . · m − → ( m · . . . · m Sporulation | {z } | {z } Virus Replication n n Verification Conclusions ´ L ⌋ DNA b ` S 6 . SPORE 2 − → d · . . . · d | m · . . . · m | {z } | {z } n n 2 12/17
Virus Replication (1) A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti Maggiolo-Schettini Milazzo Troina Introduction DNA The Calculus of Looping Sequences The bacteriophage Step 1: Adsorption Step 2: Penetration Terms Structural Congruence Dinamics An Application Sporulation Virus Replication Verification Conclusions Step 3: Replication Step 4: Maturation Step 5: Release � L ⌋ DNA v � VIRUS ::= v · . . . · v � �� � k 13/17
Rules for Virus Replication (1) A Calculus of Looping Sequences for Modelling Microbiological Systems Barbuti Maggiolo-Schettini Milazzo Troina Introduction ´ L ⌋ X ´ L ⌋ X ` ` V 1 . VIRUS | m · . . . · m − → VIRUS · m · . . . · m The Calculus of | {z } | {z } Looping Sequences n n Terms Structural Congruence ´ L ⌋ X Dinamics ` V 2 . VIRUS · m · . . . · m − → An Application | {z } Sporulation n ´ L ⌋ ( X | DNA v ) | v · . . . · v Virus Replication ` m · . . . · m Verification | {z } | {z } Conclusions n k 14/17
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