Brane Calculi
Introduction
A biological cellular membrane is an closed surface that can perform
various molecular functions.
Membranes are not just containers: they are coordinators and active
sites of major activity.
“For a cell to function properly, each of its numerous
proteins must be localized to the correct cellular membrane
or aqueous compartment.”
Proteins may be embedded in membranes, and can act on both sides
of the membrane simultaneously.
Introduction
Cells contain a huge number of membranes...
...and they are not static!
Introduction
For example, it is possible for a membrane to gradually create a bubble
that then detaches,
or for a bubble to merge with a membrane,
but it is not possible for a bubble to “jump across” a membrane (only
small molecules can do that).
Introduction
New bubbles can appear autonomously (usually emtpy) or dissolve.
Finally, a membrane can engulf a bubble (this is called phagocytosis)
and a bubble can be exected (exocytosis)
Brane Calculi
Brane Calculi are a family of process calculi for describing membrane
interactions.
I
I
I
I
PEP Calculus
MBD Calculus
Other calculi including molecules
...
The idea is to have terms describing the membrane structures, and to
have a process on each membrane describing what the membrane
could do
Processes are based on actions
Two membranes can interact (e.g. merge) if they have two
corresponding actions on their processes.
The Syntax of Brane Calculi
Systems P, Q, . . . describe the membrane structures
is the empty membrane
◦ is the juxtaposition (parallel composition) of membranes
! represents moltitudes of (equivalent) membranes
σ(|P|) denotes a single membrane, whose associated process (brane) is
σ and which contains system P
The Syntax of Brane Calculi
Branes σ, τ, . . . describe a process of a membrane
0 is the empty brane (does nothing)
| is the parallel composition of branes
! is process replication (describes recursion)
a.σ is sequential composition of actions (action prefixing)
The Structural Congruence Relation
The PEP Calculus
Different calculi of the Brane Calculi family can be obtained by considering
different sets of actions.
The first calculus we consider has Phagocytosis, Exocytosis and
Pinocytosis as possible actions
It is called PEP Calculus
Phagocytosis: a membrane engulfs another one
Exocytosis: a membrane expels another one
Pinocytosis: a membrane engulfs nothing (it creates a bubble inside itself)
In the first two there are two membranes which interact, while in the last
one there is one only membrane performing the action
The PEP Calculus: Phagocytosis
Phagocytosis:
n is the
⊥
n (σ) is
action executed by the membrane which is engulfed
the corresponding action (co–action) executed by the
membrane which engulfs the other one
n is the “communication channel”, the two membrane can perform
the two actions if they use the same n
σ is the process that will be associated with the bubble surrounding
the engulfed membrane
The PEP Calculus: Phagocytosis
Phagocytosis, semantics:
n .σ|σ0 (|P|)
◦
⊥
n (ρ).τ |τ0 (|Q|)
→ τ |τ0 (|ρ(|σ|σ0 (|P|)|) ◦ Q|)
The PEP Calculus: Exocytosis
Exocytosis:
n is the action executed by the membrane which is ejected
⊥
n is the corresponding action (co–action) executed by the
membrane which ejects the other one
n is the “communication channel”, the two membrane can perform
the two actions if they use the same n
The PEP Calculus: Exocytosis
Exocytosis, semantics:
⊥
n .τ |τ0 (|
n .σ|σ0 (|P|)
◦ Q|) → P ◦ σ|σ0 |τ |τ0 (|Q|)
The PEP Calculus: Pinocytosis
Pinocytosis:
}(σ) is the action executed by the membrane which is create a bubble
σ is the process that will be associated with the new bubble
Semantics:
}(ρ).σ|σ0 (|P|) → σ|σ0 (|ρ(| |) ◦ P|)
The PEP Calculus
Actions of the PEP Calculus are called bitonal actions because they
preserve bitonality of membrane contents
assume that nested membranes are colored by alternating two colors,
after phagocytosis, exocytosis and pinocytosis, colors are still
alternating
The PEP Calculus is Turing Complete!
The PEP Calculus: Semantics
Summing up, the semantics of the PEP Calculus is the least transition
relation satisfying the following axioms
(phago)
(exo)
n .σ|σ0 (|P|)
⊥
n .τ |τ0 (|
(pino)
◦
⊥
n (ρ).τ |τ0 (|Q|)
n .σ|σ0 (|P|)
→ τ |τ0 (|ρ(|σ|σ0 (|P|)|) ◦ Q|)
◦ Q|) → P ◦ σ|σ0 |τ |τ0 (|Q|)
} (ρ).σ|σ0 (|P|) → σ|σ0 (|ρ(| |) ◦ P|)
and closed wrt ◦ P , σ(| |) and ≡
The MBD Calculus
The MBD Calculus
Actions of the MBD Calculus can be translated into sequences of actions
of the PEP Calculus
Example: the mate action:
Example: Virus Infection
By using Brane Calculi we can describe a virus infection process.
Some steps of the process (e.g. vRNA replication) require actions we have
not seen.
Translation of Brane Calculi into CLS
Brane Calculi can be translated into CLS
this shows the expressiveness of CLS
a simulator based on CLS can be used to simulate Brane Calculi
We define a bisimulation relation for Brane Calculi
bisimilarity is preserved by the translation
We consider only the PEP Calculus, but the translation of other Brane
Calculi is similar.
The Calculus of Looping Sequences (CLS)
We assume an alphabet E. Terms T and Sequences S of CLS are given
by the following grammar:
S LcT T |T
S ::= a S · S
T
::= S
where a is a generic element of E, and is the empty sequence.
The operators are:
S · S : Sequencing
L
: Looping (S is closed and it can rotate)
S
T1 c T2 : Containment (T1 contains T2 )
T |T : Parallel composition (juxtaposition)
Actually, looping and containment form a single binary operator S
L
c T.
Structural Congruence
The Structural Congruence relations ≡S and ≡T are the least
congruence relations on sequences and on terms, respectively, satisfying
the following rules:
S1 · (S2 · S3 ) ≡S (S1 · S2 ) · S3
S · ≡S · S ≡S S
T1 | T2 ≡T T2 | T1
T1 | (T2 | T3 ) ≡T (T1 | T2 ) | T3
L
L
L
T | ≡T T
c ≡T S1 · S2 c T ≡T S2 · S1 c T
We write ≡ for ≡T .
Dinamics of the Calculus (1)
Let TV be the set of terms which may contain variables of three kinds:
term variables (X , Y , Z , . . .)
sequence variables (e
x , ye, e
z , . . .)
element variables (x, y , z, . . .)
T σ denotes the term obtained by replacing any variable in T with the
corresponding term, sequence or element.
A Rewrite Rule is a pair (T , T 0 ), denoted T 7→ T 0 , where:
T , T 0 ∈ TV
variables in T 0 are a subset of those in T
A rule T 7→ T 0 can be applied to all terms T σ.
Example: a · x · a 7→ b · x · b
can be applied to a · c · a (producing b · c · b)
cannot be applied to a · c · c · a
The Encoding Function
We translate systems of the PEP calculus into CLS terms.
We define an encoding function {[ ]} that takes a PEP system and
results in a pair of a CLS sequence and a set of alphabet symbols.
{[ ]} : PEP → T × P(E)
Operators and actions of the encoded system are translated into elements
of the sequence.
as regards systems: , ◦ , ! and (| |) are translated into
0,circ,bangS and brane, respectively
as regards branes: 0, | , and ! are translated into 0,par and bangB,
respectively
Phagocytosis and exocytosis actions are translated into sequences of
⊥
two elements, namely n , ⊥
n , n and
n are translated into
⊥
⊥
· n,
· n, · n and
· n, respectively
pinocytosis } is translated into }
0, circ, bangS, brane, par , bangB, ,
⊥
, , }, n ∈ E
The Encoding Function
We translate systems of the PEP calculus into CLS terms.
We define an encoding function {[ ]} that takes a PEP system and
results in a pair of a CLS sequence and a set of alphabet symbols.
{[ ]} : PEP → T × P(E)
The encodings of the operands and of the action parameters follow in the
sequence the encodings of the corresponding operators and actions,
respectively, and are delimited by symbols acting as separators.
the set of symbols returned by the encoding contains all these
separators.
The alphabet symbol act is used in the result of the encoding as a
program counter:
during the evolution of the term it preceeds every element which
encodes a currently active action.
Example: consider the simple PEP system ◦ {[ ◦ ]} = ( act · circ · a · 0 · a · 0 ,
{a}
)
The Encoding Function
Definition (Encoding) The translation of a system P of the PEP
calculus into CLS is the term T ∈ T such that, for some (finite) E ⊂ E, it
holds {[P]} = (T , E ), where {[·]} : PEP → T × P(E) is given by the
following recursive definition:
{[]} = act · 0, ∅
{[P1 ◦ P2 ]} = act · circ · a · P10 {/act } · a · P20 {/act }, {a} ∪ E1 ∪ E2
where {[Pi ]} = (Pi0 , Ei ), E1 ∩ E2 = ∅ and a ∈ E \ (E1 ∪ E2 )
{[!P]} = act · bangS · P 0 {/act }, E
where {[P]} = (P 0 , E )
{[σ(|P|)]} = act · brane · a · σ 0 {/act } · a · P 0 {/act }, {a} ∪ EP ∪ Eσ
where {[P]} = (P 0 , EP ), [[σ]] = (σ 0 , Eσ ),
a ∈ E \ (EP ∩ Eσ ) and EP ∩ Eσ = ∅
The Encoding Function
where [[·]] : Branes → T × P(E) is given by the following recursive
definition:
[[0]] = act · 0, ∅
[[σ1 |σ2 ]] = act · par · a · σ10 {/act } · a · σ20 {/act } · a, E1 ∪ E2 ∪ {a}
where [[σi ]] = (σi0 , Ei ), E1 ∩ E2 = ∅ and a ∈ E \ (E1 ∪ E2 )
[[!σ]] = act · bangB · a · σ 0 {/act } · a, E ∪ {a}
where [[σ]] = (σ 0 , E ) and a ∈ E \ E
[[
n .σ]]
= act ·
· n · a · σ 0 {/act } · a, E ∪ {a}
where [[σ]] = (σ 0 , E ) and a ∈ E \ E
[[
⊥
n (ρ).σ]]
= act ·
⊥
· n · a · ρ0 {/act } · a · σ 0 {/act } · a, Eρ ∪ Eσ ∪ {a}
where [[ρ]] = (ρ0 , Eρ ), [[σ]] = (σ 0 , Eσ ) and a ∈ E \ (Eρ ∪ Eσ )
The Encoding Function
[[
n .σ]]
= act ·
· n · a · σ 0 {/act } · a, E ∪ {a}
where [[σ]] = (σ 0 , E ) and a ∈ E \ E
[[
⊥
n .σ]]
= act ·
⊥
· n · a · σ 0 {/act } · a, E ∪ {a}
where [[σ]] = (σ 0 , E ) and a ∈ E \ E
[[}(ρ).σ]] = act · } · a · ρ0 {/act } · a · σ 0 {/act } · a, Eρ ∪ Eσ ∪ {a}
where [[ρ]] = (ρ0 , Eρ ), [[σ]] = (σ 0 , Eσ ) and a ∈ E \ (Eρ ∪ Eσ )
Example of Translation
Let us consider the PEP system !P where P =
n (|
|) ◦
⊥
n (0)(|
|).
According to the semantics of the PEP calculus the system may evolve as
follows:
!P ≡ !P ◦
n (|
|) ◦
⊥
n (0)(|
|) → !P ◦ 0(|0(|0(| |)|) ◦ |) ≡ !P
By applying the encoding to the system we obtain the following CLS term:
act ·bangS ·circ ·e ·brane ·b· ·n·a·0·a·b·0·e ·brane ·d ·
How it evolves? We need rewrite rules!
⊥
·n·c ·0·c ·0·c ·d ·0
Rewrite Rules for the Translation
We give a set of rewrite rules to be applied to terms obtained by the
translation of PEP systems:
the set of the rewrite rules is the same for any PEP systems
and it is inspired by the semantics of the PEP Calculus
Rewrite Rules for the Translation
e
act · par · x · ye · x · e
z ·x ·w
L
c X 7→
e
act · ye · act · e
z ·w
L
c X (par)
act · circ · x · ye · x · e
z 7→ act · ye | act · e
z
act · brane · x · ye · x · e
z 7→
e | act · 0 7→ x · w
e
x ·w
act · ye
e
x ·w
L
L
(circ)
c act · e
z
c act · 0 7→
(brane)
e
x ·w
L
act · 0
7→ act · 0
act · bangS · 0 7→ act · 0
e
act · 0 · x · w
L
c X 7→
e
x ·w
L
e
act · bangB · 0 · w
L
c X 7→
e
act · 0 · w
cX
L
L
(sc1,2)
(sc3,4)
(sc5)
cX
(sc6)
Rewrite Rules for the Translation
act ·
L
L
e c X | act · · xn · x 0 · ye0 · x 0 · e
z0 c Y
· xn · x · ye · x · e
z ·x ·w
L
L
L
e c (X | act · ye c act · ye0 · e
7
→
act · e
z ·w
z0 c Y )
(phago)
⊥
act ·
⊥
L
· xn · x · ye · x · e
z c (X | act ·
· xn · x 0 · ye0 · x 0 · e
z0
L
7
→
Y | act · ye · e
z · act · ye0 · e
z0 c X
c Y)
(exo)
L
L
c (X | act · ye )
(pino)
act · bangS · e
x 7→ act · bangS · e
x | act · e
x
(bangS)
e
act · } · x · ye · x · e
z ·x ·w
L
L
c X 7→
e
act · e
z ·w
e
act · bangB · x · ye · x · w
7→
L
cX
L
e cX
act · bangB · x · ye · x · act · ye · w
(bangB)
Example of Translation
Let’s come back to the example: !P where P =
n (|
|) ◦
⊥
n (0)(|
|).
According to the semantics of the PEP calculus the system may evolve as
follows:
!P ≡ !P ◦
n (|
|) ◦
⊥
n (0)(|
|) → !P ◦ 0(|0(|0(| |)|) ◦ |) ≡ !P
By applying the encoding to the system we obtain the following CLS term:
act ·bangS ·circ ·e ·brane ·b· ·n·a·0·a·b·0·e ·brane ·d ·
which may evolve as follows (let us call it T )....
⊥
·n·c ·0·c ·0·c ·d ·0
Example of Translation
T → T | act · circ · e · brane · b ·
· brane · d ·
⊥
·n·a·0·a·b·0·e
·n·c ·0·c ·0·c ·d ·0
→ T | act · brane · b ·
⊥
(bangS)
·n·a·0·a·b·0
·n·c ·0·c ·0·c ·d
(circ)
L
→∗ T | act · · n · a · 0 · a c act · 0
L
act · ⊥ · n · c · 0 · c · 0 · c c act · 0
2 × (brane)
L
L
L
→ T | act · 0 c (act · 0| act · 0 c act · 0 c act · 0) (phago)
| act · brane · d ·
→∗ T
Correctness of the Translation
The translation of the PEP Calculus is correct:
it is sound and complete
Soundness: for each step performed by a PEP system P there exists a
corresponding sequence of steps performed by its translation
Technical detail: we need a normal form hT i for a term T
Theorem (Soundness) Given a system P of the PEP calculus:
P → P 0 =⇒ ∃T .∃P 00 . s.t. {[P]} →∗ T , hT i ≡ h{[P 00 ]}i and P 00 ≡ P 0 .
Correctness of the Translation
The translation of the PEP Calculus is correct:
it is sound and complete
Completenss: the translation of a PEP system P does not perform
executions which does not correspond to an execution of P
Technical detail: we need again a normal form hT i for a term T
Theorem (Completeness) Given a system P of the PEP calculus:
{[P]} →∗ T =⇒ ∃P 0 s.t. hT i ≡ h{[P 0 ]}i and either P ≡ P 0 or P →∗ P 0 .
Towards Bisimulations for Brane Calculi
In order to define a notion of bisimulation for the PEP calculus, we
introduce a labeled semantics.
denotes the silent (internal) action.
We assume a set L = {( n , σ(|R|)), ( ⊥
n , σ(|R|)), (
N , R ∈ PEP, σ ∈ Branes} of transition labels.
n , σ(|R|)),
|n∈
` denotes a generic label.
A transition with a label in ` represents the potential action a system can
perform when inserted in a suitable context.
n ,σ(|R|)
P −−−−−→ P 0 , means that P has a component σ(|R|) that may enter
another membrane
⊥
,σ(|R|)
n
Q −−−
−−−→ Q 0 , means that Q can engulf another system σ(|R|)
When P and Q are composed by ◦, they can evolve together, with a
silent action, to a new system in which σ(|R|) is inside Q 0
A Labeled Semantics for the PEP Calculus
Definition (Labeled Semantics) The labeled semantics of the PEP
calculus is given by the labeled transition system generated by the
following inference rules:
n .σ|σ0 (|P|)
n ,σ|σ0 (|P|)
−−−−−−−→ (Ph1)
⊥
⊥
n (ρ).τ |τ0 (|Q|)
,σ(|R|)
n
−−−
−−−→ τ |τ0 (|ρ(|σ(|R|)|) ◦ Q|)
(Ph2)
⊥
n ,σ(|R|)
,σ(|R|)
n
P −−−−−−→ P 0
Q −−−
−−−→ Q 0
0
0
P ◦Q →
− P ◦Q
To be continued....
(Ph3)
A Labeled Semantics for the PEP Calculus
Definition (Labeled Semantics) more rules......
n .σ|σ0 (|P|)
n ,σ|σ0 (|P|)
−−−−−−−→ (E1)
n ,σ(|R|)
P −−−−−−→ P 0
⊥
n .τ |τ0 (|P
◦ Q|) →
− R ◦ σ|τ |τ0 (|Q ◦ P 0 |)
}(ρ).σ|σ0 (|P|) →
− σ|σ0 (|P ◦ ρ(| |)|)
P→
− P0
`
Q→
− Q0
`
P ◦Q →
− P ◦ Q0
P ◦Q →
− P0 ◦ Q
(E2)
(Pi1)
`
P→
− P0
σ(|P|) →
− σ(|P 0 |)
`
(Par1,Par2)
(Br1)
Strong Bisimulation for the PEP Calculus
Definition (PEP Strong Bisimulation) A binary relation κ on PEP
terms is a strong bisimulation if, given P and Q such that PκQ, the
following conditions hold:
P→
− P 0 =⇒ ∃Q 0 such that Q →
− Q 0 and P 0 κQ 0
Q→
− Q 0 =⇒ ∃P 0 such that P →
− P 0 and Q 0 κP 0
a,R
a,R 0
a,R
a,R 0
P −−→ P 0 =⇒ ∃Q 0 such that Q −−→ Q 0 , RκR 0 and P 0 κQ 0
Q −−→ Q 0 =⇒ ∃P 0 such that P −−→ P 0 , RκR 0 and Q 0 κP 0
The strong bisimilarity l is the largest of such relations.
PEP Bisimilarity and Translation into CLS
We have seen that the PEP Calculus can be translated into CLS.
In CLS we have bisimulations.
The translations of two bisimilar PEP systems are bisimilar CLS terms?
YES!
Theorem (Full Abstraction) Given two systems P, Q of the PEP
calculus, the following holds:
P l Q ⇐⇒ {[P]} ≈ {[Q]} .