The Calculus of Looping Sequences Roberto Barbuti, Giulio Caravagna, - PowerPoint PPT Presentation

The Calculus of Looping Sequences Roberto Barbuti, Giulio Caravagna, Andrea MaggioloSchettini, Paolo Milazzo, Giovanni Pardini Dipartimento di Informatica, Universit` a di Pisa, Italy Bertinoro – June 7, 2008 Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 1 / 59

Outline of the talk Introduction 1 Cells are complex interactive systems The Calculus of Looping Sequences (CLS) 2 Definition of CLS The EGF pathway and the lac operon in CLS CLS variants 3 Stochastic CLS LCLS Spatial CLS Future Work and References 4 Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 2 / 59

Cells: complex systems of interactive components Main actors: ◮ membranes ◮ proteins ◮ DNA/RNA ◮ ions, macromolecules,. . . Interaction networks: ◮ metabolic pathways ◮ signaling pathways ◮ gene regulatory networks Computer Science can provide biologists with formalisms for the description of interactive systems and tools for their analysis. Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 3 / 59

Outline of the talk Introduction 1 Cells are complex interactive systems The Calculus of Looping Sequences (CLS) 2 Definition of CLS The EGF pathway and the lac operon in CLS CLS variants 3 Stochastic CLS LCLS Spatial CLS Future Work and References 4 Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 4 / 59

The Calculus of Looping Sequences (CLS) We assume an alphabet E . Terms T and Sequences S of CLS are given by the following grammar: � � � L ⌋ T � � � T ::= S S T | T � � � � S ::= ǫ a S · S where a is a generic element of E , and ǫ is the empty sequence. The operators are: S · S : Sequencing � � L S : Looping ( S is closed and it can rotate) T 1 ⌋ T 2 : Containment ( T 1 contains T 2 ) T | T : Parallel composition (juxtaposition) � � L ⌋ T . Actually, looping and containment form a single binary operator S Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 5 / 59

Examples of Terms � � L ⌋ ǫ ( i ) a · b · c � � L ⌋ � � L ⌋ ǫ ( ii ) a · b · c d · e � � L ⌋ ( f · g | � � L ⌋ ǫ ) ( iii ) a · b · c d · e Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 6 / 59

Structural Congruence The Structural Congruence relations ≡ S and ≡ T are the least congruence relations on sequences and on terms, respectively, satisfying the following rules: S 1 · ( S 2 · S 3 ) ≡ S ( S 1 · S 2 ) · S 3 S · ǫ ≡ S ǫ · S ≡ S S T 1 | T 2 ≡ T T 2 | T 1 T 1 | ( T 2 | T 3 ) ≡ T ( T 1 | T 2 ) | T 3 � � L ⌋ ǫ ≡ T ǫ � � L ⌋ T ≡ T � � L ⌋ T T | ǫ ≡ T T ǫ S 1 · S 2 S 2 · S 1 We write ≡ for ≡ T . Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 7 / 59

CLS Patterns Let us consider variables of three kinds: term variables ( X , Y , Z , . . . ) sequence variables ( � x , � y , � z , . . . ) element variables ( x , y , z , . . . ) Patterns P and Sequence Patterns SP of CLS extend CLS terms and sequences with variables: � � � L ⌋ P � � � � � P ::= SP SP P | P X � � � � � � � � SP ::= ǫ a SP · SP x � x where a is a generic element of E , ǫ is the empty sequence, and x , � x and X are generic element, sequence and term variables The structural congruence relation ≡ extends trivially to patterns Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 8 / 59

Rewrite Rules P σ denotes the term obtained by replacing any variable in T with the corresponding term, sequence or element. Σ is the set of all possible instantiations σ A Rewrite Rule is a pair ( P , P ′ ), denoted P �→ P ′ , where: P , P ′ are patterns variables in P ′ are a subset of those in P A rule P �→ P ′ can be applied to all terms P σ . Example: a · x · a �→ b · x · b can be applied to a · c · a (producing b · c · b ) cannot be applied to a · c · c · a Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 9 / 59

Formal Semantics Given a set of rewrite rules R , evolution of terms is described by the transition system given by the least relation → satisfying P �→ P ′ ∈ R P σ �≡ ǫ P σ → P ′ σ T → T ′ T → T ′ � � L ⌋ T → � � L ⌋ T ′ T | T ′′ → T ′ | T ′′ S S and closed under structural congruence ≡ . Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 10 / 59

CLS modeling examples: the EGF pathway (1) Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 11 / 59

CLS modeling examples: the EGF pathway (2) phosphorylations EGF EGFR SHC nucleus CELL MEMBRANE Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 12 / 59

CLS modeling examples: the EGF pathway (3) First steps of the EGF signaling pathway up to the binding of the signal-receptor dimer to the SHC protein The EGFR,EGF and SHC proteins are modeled as the alphabet symbols EGFR , EGF and SHC , respectively The cell is modeled as a looping sequence (representing its external membrane): � � L ⌋ ( SHC | SHC ) EGF | EGF | EGFR · EGFR · EGFR · EGFR Rewrite rules modeling the first steps of the pathway: � � L ⌋ X �→ � � L ⌋ X EGF | EGFR · � x CMPLX · � x (R1) � � L ⌋ X �→ � � L ⌋ X CMPLX · � x · CMPLX · � y DIM · � x · � y (R2) � � L ⌋ X �→ � � L ⌋ X DIM · � x DIMp · � x (R3) � � L ⌋ ( SHC | X ) �→ � � L ⌋ X DIMp · � x DIMpSHC · � x (R4) Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 13 / 59

CLS modeling examples: the EGFR pathway (4) A possible evolution of the system: � � L ⌋ ( SHC | SHC ) EGF | EGF | EGFR · EGFR · EGFR · EGFR � � L ⌋ ( SHC | SHC ) ( R 1) − − − → EGF | EGFR · CMPLX · EGFR · EGFR � � L ⌋ ( SHC | SHC ) ( R 1) − − − → EGFR · CMPLX · EGFR · CMPLX � � L ⌋ ( SHC | SHC ) ( R 2) − − − → EGFR · DIM · EGFR � � L ⌋ ( SHC | SHC ) ( R 3) − − − → EGFR · DIMp · EGFR � � L ⌋ SHC ( R 4) − − − → EGFR · DIMpSHC · EGFR Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 14 / 59

CLS modeling examples: the lac operon (1) Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 15 / 59

CLS modeling examples: the lac operon (2) � � L ⌋ ( lacI · lacP · lacO · lacZ · lacY · lacA | polym ) Ecoli ::= m Rules for DNA transcription/translation: x �→ lacI ′ · � lacI · � x | repr (R1) polym | � x · lacP · � y �→ � x · PP · � y (R2) � x · PP · lacO · � y �→ � x · lacP · PO · � y (R3) x · PO · lacZ · � � y �→ � x · lacO · PZ · � y (R4) � x · PZ · lacY · � y �→ � x · lacZ · PY · � y | betagal (R5) x · PY · lacA �→ � � x · lacY · PA | perm (R6) � x · PA �→ � x · lacA | transac | polym (R7) Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 16 / 59

CLS modeling examples: the lac operon (3) � � L ⌋ ( lacI · lacP · lacO · lacZ · lacY · lacA | polym ) Ecoli ::= m Rules to describe the binding of the lac Repressor to gene o, and what happens when lactose is present in the environment of the bacterium: repr | � x · lacO · � y �→ � x · RO · � y (R8) � x · RO · � y �→ repr | � x · lacO · � y (R9) repr | LACT �→ RLACT (R10) RLACT �→ repr | LACT (R11) �� � L ⌋ ( perm | X ) �→ � � L ⌋ X x perm · � x (R12) � � L ⌋ X �→ � � L ⌋ ( LACT | X ) LACT | perm · � x perm · � x (R13) betagal | LACT �→ betagal | GLU | GAL (R14) Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 17 / 59

Some variants of CLS Full–CLS ◮ The looping operator can be applied to any term � � � L ⌋ c � L ⌋ d are allowed ◮ Terms such as a | b CLS+ ◮ More realistic representation of the fluid nature of membranes: the looping operator can be applied to parallel compositions of sequences ◮ Can be encoded into CLS Stochastic CLS ◮ The application of a rule consumes a stochastic quantity of time LCLS (CLS with Links) ◮ Description of protein–protein interactions at the domain level Spatial CLS ◮ Description of physical size and position of entities Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 18 / 59

Outline of the talk Introduction 1 Cells are complex interactive systems The Calculus of Looping Sequences (CLS) 2 Definition of CLS The EGF pathway and the lac operon in CLS CLS variants 3 Stochastic CLS LCLS Spatial CLS Future Work and References 4 Roberto Barbuti (Universit` a di Pisa) The Calculus of Looping Sequences Bertinoro – June 7, 2008 19 / 59