Brane Calculi
Presented by Jesús F. Almansa
[email protected]
Mobile Calculi Course
BRICS, University of Aarhus
December 2004
Brane Calculi – p. 1/1
Motivation
Biological Systems
Imprecise descriptions
Complex...
Need to be formalized.
Brane Calculi – p. 2/1
Motivation
Biological Systems
Imprecise descriptions
Complex...
Need to be formalized.
In particular, membranes have their own dynamics.
Brane Calculi – p. 2/1
Motivation
Biological Systems
Imprecise descriptions
Complex...
Need to be formalized.
In particular, membranes have their own dynamics.
Motivation
Design
Brane Calculi – p. 2/1
Motivation
Biological Systems
Imprecise descriptions
Complex...
Need to be formalized.
In particular, membranes have their own dynamics.
Motivation
Design
Previous work:
P-System: dismatch with reality
BioSpy: calculate with molecules
BioAmbients: calculate with molecules, add membranes
Brane Calculi: calculate on membranes
Brane Calculi – p. 2/1
The Papers
Bitonal Membrane Systems,
Interactions of Biological Membranes
Luca Cardelli
Brane Calculi,
Interactions of Biological Membranes
Luca Cardelli
Brane Calculi – p. 3/1
Membrane Systems
Finite set of simple, closed and smooth curves
Brane Calculi – p. 4/1
Membrane Systems
Finite set of simple, closed and smooth curves
Brane Calculi – p. 4/1
Membrane Systems
Finite set of simple, closed and smooth curves
Alternated Orientation: Bitonality
Brane Calculi – p. 4/1
Membrane Systems
Finite set of simple, closed and smooth curves
Alternated Orientation: Bitonality
Reactions: (Instantaneous) transformations bitonality-preserving
“locally”
Brane Calculi – p. 4/1
Membrane Systems
Finite set of simple, closed and smooth curves
Alternated Orientation: Bitonality
Reactions: (Instantaneous) transformations bitonality-preserving
“locally”
Brane Calculi – p. 4/1
Membrane Systems
Finite set of simple, closed and smooth curves
Alternated Orientation: Bitonality
Reactions: (Instantaneous) transformations bitonality-preserving
“locally”
Some bio-reactions are atonal, but abs-atonality is mostly
unrealistic. Hence, ruled-out.
Brane Calculi – p. 4/1
Membrane Reactions
Brane Calculi – p. 5/1
Membrane Reactions
Mito
Brane Calculi – p. 5/1
Membrane Reactions
Mito
Mate
Brane Calculi – p. 5/1
Membrane Reactions
Mito
Mate
Endo
Brane Calculi – p. 5/1
Membrane Reactions
Mito
Mate
Endo
Exo
Brane Calculi – p. 5/1
{Endo,Exo} is complete
Brane Calculi – p. 6/1
{Endo,Exo} is complete
Moreover, Endo is splitted:
Brane Calculi – p. 6/1
The Leap to Abstraction
Endo is not spontaneous, but regulated by membranes (i.e. its
embedded proteins)
Brane Calculi – p. 7/1
The Leap to Abstraction
Endo is not spontaneous, but regulated by membranes (i.e. its
embedded proteins)
A Formalization:
Actions “on” membranes, not “inside”.
Brane Calculi – p. 7/1
The Leap to Abstraction
Endo is not spontaneous, but regulated by membranes (i.e. its
embedded proteins)
A Formalization:
Actions “on” membranes, not “inside”.
Action/co-action interaction style.
Brane Calculi – p. 7/1
The Leap to Abstraction
Endo is not spontaneous, but regulated by membranes (i.e. its
embedded proteins)
A Formalization:
Actions “on” membranes, not “inside”.
Action/co-action interaction style.
A calculus of membrane reactions.
Brane Calculi – p. 7/1
Syntax
Systems P, Q ::=
Branes
σ, τ ::=
Actions a, b
::=
| P ◦ Q | !P | σLP M
0 | σ|τ | !σ| a.σ
Brane Calculi – p. 8/1
Syntax
Systems P, Q ::=
Branes
σ, τ ::=
Actions a, b
::=
τ |σLP M
| P ◦ Q | !P | σLP M
0 | σ|τ | !σ| a.σ
τ
P
σ
Brane with σ, τ and contents P
Brane Calculi – p. 8/1
Congruence ≡, Reactions →
(P, ◦, ) comutative monoid
(σ, |, 0) comutative monoid
the usual...
Brane Calculi – p. 9/1
Congruence ≡, Reactions →
(P, ◦, ) comutative monoid
(σ, |, 0) comutative monoid
the usual...
P →Q
P ◦R→Q◦R
P →Q
σLP M → σLQM
P ≡ P0
P 0 → Q0
P →Q
Q0 ≡ Q
Brane Calculi – p. 9/1
Congruence ≡, Reactions →
(P, ◦, ) comutative monoid
(σ, |, 0) comutative monoid
the usual...
P →Q
P ◦R→Q◦R
P →Q
σLP M → σLQM
P ≡ P0
P 0 → Q0
P →Q
plus the effect of actions
Q0 ≡ Q
Brane Calculi – p. 9/1
Actions
Actions a, b
::= Bn | ⊥nB (σ) | Cn | C⊥n | (σ)
Brane Calculi – p. 10/1
Actions
Actions a, b
::= Bn | ⊥nB (σ) | Cn | C⊥n | (σ)
Phago:
Bn .σ|σ0 LP M ◦ ⊥nB (ρ).τ |τ0 LQM → τ |τ0 LρLσ|σ0 LP MM ◦ QM
Brane Calculi – p. 10/1
Actions
Actions a, b
::= Bn | ⊥nB (σ) | Cn | C⊥n | (σ)
Phago:
Bn .σ|σ0 LP M ◦ ⊥nB (ρ).τ |τ0 LQM → τ |τ0 LρLσ|σ0 LP MM ◦ QM
Exo:
C⊥n .τ |τ0 LCn .σ|σ0 LP M ◦ QM → P ◦ σ|σ0 |τ |τ0 LQM
Brane Calculi – p. 10/1
Actions
Actions a, b
::= Bn | ⊥nB (σ) | Cn | C⊥n | (σ)
Phago:
Bn .σ|σ0 LP M ◦ ⊥nB (ρ).τ |τ0 LQM → τ |τ0 LρLσ|σ0 LP MM ◦ QM
Exo:
C⊥n .τ |τ0 LCn .σ|σ0 LP M ◦ QM → P ◦ σ|σ0 |τ |τ0 LQM
Pino:
(ρ).σ|σ0 LP M → σ|σ0 LρLM ◦ P M
Brane Calculi – p. 10/1
Actions Depicted
Brane Calculi – p. 11/1
Example: Mate
Proposition:
σ0 |mate n .σLP M ◦ τ0 |mate ⊥n .τ LQM →∗ σ0 |σ|τ0 |τ LP ◦ QM
Brane Calculi – p. 12/1
Example: Mate
def
mate n = Bn . Cn0 .σ
def
mate ⊥
n =
⊥
⊥
n B (Cn0
. Cn00). C⊥
n00 .τ
Proposition:
σ0 |mate n .σLP M ◦ τ0 |mate ⊥n .τ LQM →∗ σ0 |σ|τ0 |τ LP ◦ QM
Brane Calculi – p. 12/1
Example: Mate
def
mate n = Bn . Cn0 .σ
def
mate ⊥
n =
⊥
⊥
n B (Cn0
. Cn00). C⊥
n00 .τ
Proposition:
σ0 |mate n .σLP M ◦ τ0 |mate ⊥n .τ LQM →∗ σ0 |σ|τ0 |τ LP ◦ QM
Homework: Drip (Mito with 0), Bud (Mito with 1)
Brane Calculi – p. 12/1
Example: Viral Reproduction
Brane Calculi – p. 13/1
Example: Viral Reproduction
Almost... molecules are needed
Brane Calculi – p. 13/1
Nice, but...
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Purely combinatorial
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Purely combinatorial
communication could be added...
a, b ::=
...o2o n | o2o ⊥n (m) | s2s n | s2s ⊥n (m) | p2c n | p2c ⊥n (m)
assuming τ {l ← m}
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Purely combinatorial
communication could be added...
a, b ::=
...o2o n | o2o ⊥n (m) | s2s n | s2s ⊥n (m) | p2c n | p2c ⊥n (m)
assuming τ {l ← m}
...and name restriction...
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Purely combinatorial
communication could be added...
a, b ::=
...o2o n | o2o ⊥n (m) | s2s n | s2s ⊥n (m) | p2c n | p2c ⊥n (m)
assuming τ {l ← m}
...and name restriction...
...and choice...
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Purely combinatorial
communication could be added...
a, b ::=
...o2o n | o2o ⊥n (m) | s2s n | s2s ⊥n (m) | p2c n | p2c ⊥n (m)
assuming τ {l ← m}
...and name restriction...
...and choice...
...and all π?
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Purely combinatorial
communication could be added...
a, b ::=
...o2o n | o2o ⊥n (m) | s2s n | s2s ⊥n (m) | p2c n | p2c ⊥n (m)
assuming τ {l ← m}
...and name restriction...
...and choice...
...and all π?
No equivalence
Brane Calculi – p. 14/1
Nice, but... what kind of calculus is this?
Purely combinatorial
communication could be added...
a, b ::=
...o2o n | o2o ⊥n (m) | s2s n | s2s ⊥n (m) | p2c n | p2c ⊥n (m)
assuming τ {l ← m}
...and name restriction...
...and choice...
...and all π?
No equivalence
Biologically meaningful?
Brane Calculi – p. 14/1
Comparative Exercise: Security Applications
Ambients in the air...
Brane Calculi – p. 15/1
Comparative Exercise: Security Applications
Ambients in the air...Pure and Safe
Brane Calculi – p. 15/1
Comparative Exercise: Security Applications
Ambients in the air...Pure and Safe
P
Cap
::=
::=
(ν n)P | 0 | P ◦ Q | !P | n[P ] |Cap.P
~ n | out n | out
~ n | open n |open
in n | in
~ n|
Brane Calculi – p. 15/1
Comparative Exercise: Security Applications
Ambients in the air...Pure and Safe
P
Cap
::=
::=
(ν n)P | 0 | P ◦ Q | !P | n[P ] |Cap.P
~ n | out n | out
~ n | open n |open
in n | in
~ n|
to be explored...
Brane Calculi – p. 15/1