2004-06-01 12:35:58 1
Brane Calculi
Interactions of Biological Membranes
Luca Cardelli
Microsoft Research
Abstract. We introduce a family of process calculi with dynamic nested membranes. In contrast to
related calculi, including some developed for biological applications, active entities here are
tightly coupled to membranes, and can perform interactions on both sides of a membrane. That is,
computation happens on the membrane, not inside of it.
1 Introduction
A biological cellular membrane is an oriented closed surface that can perform various
molecular functions. Membranes are not just containers: they are coordinators and active sites
of major activity1
. Large functional molecules (proteins) are embedded in membranes, with
consistent orientation, and can act on both sides of the membrane simultaneously. The
consistent orientation of such proteins induces an orientation on the membrane. Freely
floating molecules interact with membrane proteins, and can be sensed, manipulated, and
pushed across by active molecular channels. Membranes come in different kinds,
distinguished mostly by the proteins embedded in them, and typically consume energy to
perform their functions.
One of the most remarkable properties of biological membranes is that they form a twodimensional
fluid (a lipid bilayer) embedded in a three-dimensional fluid (water). That is,
both the structural components and the embedded proteins freely diffuse on the twodimensional
plane of the membrane (unless they are held together by specific mechanisms).
Moreover, membranes float in water, which may contain other molecules that freely diffuse in
that three-dimensional fluid. Membrane themselves are impermeable to most substances, such
as water and protons, so that they partition the three-dimensional fluid.
Many membranes are highly dynamic: they constantly shift, merge, break apart, and are
replenished. But the transformations that they can support are rather limited, partially because
orientation must be preserved, and partially because membrane transformations need to be
fairly continuous. For example, it is possible for a membrane to gradually buckle and create a
bubble that then detaches, or for such a bubble to merge back with a membrane, but it is not
possible for a bubble to “jump across” a membrane (only small molecules can do that).
The fluid-within-fluid structure inspires the basic organization of our Brane Calculi2
,
which is characterized by two commutative monoids, each representing a kind of fluid. The
specific transformations that we have selected are further inspired by (some of) the biological
1
“For a cell to function properly, each of its numerous proteins must be localized to the correct cellular
membrane or aqueous compartment.” [9] p.675.
2
“Brane” is a common abbreviation for “membrane” in physics.
2004-06-01 12:35:582
constraints. However, within the general structure of Brane Calculi there is scope for refining
or ignoring such constraints.
One of the constraints one may adopt is the preservation of orientation (e.g., membranes
of different orientation should not merge). A related constraint is bitonality, which requires
nested membranes to have opposite orientations, so that the orientations can be coded by
coloring systems in two tones, as in Figure 1, where P and Q represent arbitrary subsystems.
Preservation of bitonality means that reactions must preserve the even/odd parity with which
components are nested inside membranes: note that P and Q remain on the same color
background in each reaction. This means, in particular, that in a sequence of bitonal reactions
there is never any actual mixing of fluids from inside and outside any given membrane,
although external fluids can be brought inside if safely wrapped in another membrane.
Bitonality is common in cellular-scale living systems. Although not universal, it inspires a
collection of basic reactions that are biologically implementable, and that are different from
those of calculi that are not biologically inspired.3
Figure 1 Examples of Bitonal Reactions
The reactions illustrated in Figure 1 can be formalized and studied on their own [2].
However, in this paper we use them only as informal guides for more detailed calculi, where
the reasons “why” those reactions happen are made more apparent.
2 Basic Framework
2.1 Syntax and Reactions
The basic structure of Brane Calculi consists of two commutative monoids with replication:
we use for composition of systems, with unit , and | for composition of membranes, with
unit 0. Replication (!) is used to model the notion of a “multitude” of components of the same
kind, which is in fact a standard situation in biology. Quantitative refinements are possible
[12] and certainly desirable.
Systems consist of nested membranes, and membranes consists of collection of actions.
Actions are left unspecified at the moment, and are detailed in the following sections. The
familiar notion of structural congruence of processes [11] is applied to systems and
membranes, characterizing their fluidity properties. Reactions happen only at the level of
systems, and are caused only by actions on membranes.
3
The framework in which Brane Calculi are formalized originates in the study of calculi for mobile agents [3].
In that context, sandboxing an applet on its arrival at a site is, in fact, a bitonal operation: it maintains the
separation between safe regions (of internal origin) and unsafe regions (of external origin). We are not aware of
proposals to use sandboxing as a basic operations in that context; here, it corresponds to phagocytosis.
2004-06-01 12:35:58 3
Syntax
Systems P,Q ::= ¦ P Q ¦ !P ¦ σ P nests of membranes
Branes σ,τ ::= 0 ¦ σ|τ ¦ !σ ¦ a.σ combinations of actions
Actions a,b ::= … (detailed later)
Figure 2 Brane Graphical Notation
We abbreviate a.0 as a, and 0 P as P , and σ as σ .
Structural Congruence
P Q Q P
P (Q R) (P Q) R
P P
!
!(P Q) !P !Q
!!P !P
!P P !P
0
P Q P R Q R
P Q !P !Q
P Q ∧ σ τ σ P τ Q
σ|τ τ|σ
σ|(τ|ρ) (σ|τ)|ρ
σ|0 σ
!0 0
!(σ|τ) !σ|!τ
!!σ !σ
!σ σ|!σ
σ τ σ|ρ τ|ρ
σ τ !σ !τ
σ τ a.σ a.τ
Basic Reactions
P Q P R Q R
P Q σ P σ Q
P P’ ∧ P’ Q’ ∧ Q’ Q P Q
We write * for the reflexive and transitive closure of .
Within this framework, our Basic Brane Calculus is the one gradually introduced in
Sections 3 and 4. Possible extensions are discussed in Section 5. Orthogonally, one could add
restriction operators to both systems and membranes, in the style of π-calculus [11], with
extrusion rules such as ((νn)σ) P (νn)(σ P ) if n fn(P). The bound names n would be the
ones used in the following sections to identify pairs of related actions and co-actions.
3 Bitonal Interactions
Bitonal interactions [2] are inspired by endocytosis/exocytosis (the second reversible reaction
in Figure 1). Endocytosis is the process of incorporating external material into a cell by
σ
A membrane σ
with actions σ and contents
2004-06-01 12:35:584
“engulfing” it with the cell membrane (without breaking the membrane or letting the material
cross it). Exocytosis is the reverse process.
3.1 Definitions
Endocytosis, thus described, is an uncontrollable process that can engulf an arbitrary amount
of material. We are interested in more controllable interactions, therefore we specialize
endocytosis into two basic operations: phagocytosis, engulfing just one external membrane,
and pinocytosis, engulfing zero external membranes. In addition we have exocytosis, which is
itself sufficiently controllable. Each action usually comes with a co-action that it is intended
to interact with, indicated by the symbol (pinocytosis does not have a co-action).
Bitonal Actions
Actions a ::= … ¦ n ¦ n(σ) ¦ n ¦ n ¦ (σ) phago , exo , pino
Precedence: a.σ|τ stands for (a.σ)|τ, and !σ|τ stands for (!σ)|τ. The subscripted names n are
used to pair-up related actions an co-actions; we omit them when there is no ambiguity. Cophago
is indexed by a membrane σ; this σ becomes the new membrane that engulfs the
outside material: conceptually it is related to a piece of the old membrane. Exo causes
irreversible mixing of membranes: since membranes are fluids, there is in general no way to
untangle two membranes once they have merged. Incidentally, this implies that merging is
often not a desirable operation.
Bitonal Reactions
Phago n.σ|σ0 P n(ρ).τ|τ0 Q τ|τ0 ρ σ|σ0 P Q
Exo n.τ|τ0 n.σ|σ0 P Q P σ|σ0|τ|τ0 Q
Pino (ρ).σ|σ0 P σ|σ0 ρ P
One can see that the parity of nesting of P and Q is preserved in all these reactions, hence they
preserve the bitonal coloring of those subsystems.
Figure 3 Phago, Exo, Pino (shaded for emphasis)
(ρ).τ
n(ρ).τ
σ
τ
n.σ
σ0 τ0 ρ σ0
σ
σ0
σ
τ
τ0
τ0
σ0
n.σ
σ0
n.σ
n.τ
σ0
σ τ
τ0
σ ρ
2004-06-01 12:35:58 5
3.2 Derived Bitonal Interactions
The Mito reaction, as illustrated in Figure 1 is another uncontrollable process that can split a
membrane at an arbitrary place. To make it more controllable, we specialize it into two basic
operations: budding, splitting off one internal membrane, and dripping, splitting off zero
internal membranes. In addition we have mating (a.k.a. merging or fusion), the obvious
merging of membranes, which is itself sufficiently controllable.
These three bitonal operations, mating, budding, and dripping, can be derived from the
previous three. The derivations are not meant to be biologically significant: they are just a test
of expressive power. In practice one would want to consider these as primitives at the same
level as Phago, Exo, and Pino, since they all have direct implementations in cellular
mechanisms.
Figure 4 Mate, Bud, Drip (shaded for emphasis)
Mate causes irreversible membrane mixing, as in Exo. In Bud, the fresh membrane ρ that
surrounds the bud is a parameter of the co-action, similarly to the situation with Phago. Drip
is similar to Pino, but towards the outside.
The encodings of Mate, Bud, and Drip follow the single basic idea that Mito/Mate in
Figure 1 can be encoded with a sequence of three Endo/Exo operations.
Figure 5 Mito/Mate by 3 Endo/Exo (basic technique)
mate n.τ
τ0
maten.σ
σ0
τ0
σ τ
σ0
τ0
σ0
budn.σ
σ0
budn.σ
bud n(ρ).τ τ
τ0
σ0
dripn(ρ).σ σ
σ0
ρ
ρ
σ0
σ
σ0
σ
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Mate
maten.σ n. n’.σ
mate n.τ n( n’. n”). n”.τ
maten.σ|σ0 P mate n.τ|τ0 Q * σ|σ0|τ|τ0 P Q
maten.σ|σ0 P mate n.τ|τ0 Q =
n. n’.σ|σ0 P n( n’. n”). n”.τ|τ0 Q Phago n
n”.τ|τ0 n’. n” n’.σ|σ0 P Q Exo n’
n”.τ|τ0 n”|σ|σ0 P Q Exo n”
σ|σ0|τ|τ0 P Q
Bud
budn.σ n.σ
bud n(ρ).τ ( n(ρ). n’). n’.τ
bud n(ρ).τ|τ0 budn.σ|σ0 P Q * ρ σ|σ0 P τ|τ0 Q
bud n(ρ).τ|τ0 budn.σ|σ0 P Q =
( n(ρ). n’). n’.τ|τ0 n.σ|σ0 P Q Pino
n’.τ|τ0 n(ρ). n’ n.σ|σ0 P Q Phago n
n’.τ|τ0 n’ ρ σ|σ0 P Q Exo n’
ρ σ|σ0 P τ|τ0 Q
Drip
dripn(ρ).σ ( (ρ). n)). n.σ
dripn(ρ).σ|σ0 P * ρ σ|σ0 P
dripn(ρ).σ|σ0 P =
( (ρ). n)). n.σ|σ0 P Pino
n.σ|σ0 (ρ). n P Pino
n.σ|σ0 n ρ P Exo n
ρ σ|σ0 P
3.3 Example: Viral Infection, Part 1
Certain kinds of viral infection mechanisms represent an ideal example of bitonality in action.
A virus is too big to just cross a cellular membrane. It can either punch its DNA or RNA
through the membrane, essentially performing a Mate, or it can enter by utilizing standard
cellular endocytosis pathways, as shown in Figure 6.
The Semliki Forest virus consists of a capsid containing the viral RNA (the
nucleocapsid). The nucleocapsid is surrounded by a membrane that is similar to the cellular
membrane (in fact, it is obtained from it “on the way out”). This membrane is however
enriched with a special protein that plays a crucial trick on the cellular machinery, as we shall
see shortly. The virus is brought into the cell by phagocytosis, thus wrapped by an additional
2004-06-01 12:35:58 7
membrane layer; this is part of a standard transport pathway into the cell. As part of that
pathway, an endosome compartment merges with the wrapped-up virus. At this point, usually,
the endosome causes some reaction to happen in the material brought into the cell. In this
case, though, the virus uses its special membrane protein to trigger an exocytosis step that
deposits the naked nucleocapsid into the cytosol. The careful separation of internal and
external substances that the cell usually maintains has now been subverted. The nucleocapsid
is in direct contact with the inner workings of the cell, and can begin doing damage. First, the
nucleocapsid disassembles itself, depositing the viral RNA into the cytosol. This vRNA then
follows three distinct paths. First it is replicated (either by cellular proteins, or by proteins that
came with the capsid), to provide the vRNA for more copies of the virus. The vRNA is also
translated into proteins, again by standard cellular machinery. Some proteins are synthesized
in the cytosol, and form the building blocks of the capsid: these self-assemble and incorporate
a copy of the vRNA to form a nucleocapsid. The virus envelope protein is instead synthesized
in the Endoplasmic Reticulum, and through various steps (through the Golgi apparatus) ends
up lining transport vesicles that merge with the cellular membrane, along another standard
transport pathway. Finally, the newly assembled nucleocapsid makes contact with sections of
the cellular membrane that are now lined with the viral envelope protein, and buds out to
recreate the initial virus structure outside the cell.
Figure 6 Viral Infection and Reproduction ([1] p.279)
The initial and final stages of the virus lifecycle can be coded up as follows.
virus . nucap
nucap !bud|X vRNA
cell membrane cytosol
membrane ! (mate)|!
!
"
#
$ #% &
"
!
'
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cytosol endosome Z
endosome !mate |!
viral-envelope bud ( . )
envelope-vesicle .viral-envelope
In the first phase (infection, Figure 7), the nucleocapsid (i.e., the capsid with the viral RNA
inside, abbreviated “nucap”) places itself in the cytosol:
virus cell * membrane nucap cytosol
We next assume that, by interaction with the available cellular machinery in the cytosol, the
nucap causes the production of some number of copies n and m of envelope-vesicles and
nucaps, leaving some modified cytosol’. (In section 4.6 we detail the mechanisms involved,
including the unspecified cytosol’, X, Z, Z’.)
nucap cytosol * nucapn
envelope-vesiclem
cytosol’
In the final phase (reproduction, Figure 8), the virus reassembles itself outside the cell:
membrane nucap envelope-vesicle Z’ * membrane Z’ virus
Figure 7 Viral Infection
Figure 8 Viral Reproduction
The level of abstraction in this code has been chosen to be as close as possible to the one
in the picture. This is important, because we rarely understand the finest details of biological
. nucap ! (mate)|! !mate |! Z Phago
! (mate)|! mate nucap !mate |! Z Mate
! (mate)|! !mate |! nucap Z Exo
! (mate)|! !mate |! nucap Z
endosome
virus cell
membrane
endosome
endosome’
endosome
vesiclemembrane
membrane
membrane
! (mate)|! .bud ( . ) !bud|X vRNA Z’ Exo
! (mate)|! |bud ( . ) !bud|X vRNA Z’ Bud
! (mate)|! Z’ . nucap
nucap
infected cell
envelope-vesicle
virus
nucapenvelope
infected cell
membrane
2004-06-01 12:35:58 9
processes, and even if we did, we still would not want to model every molecule individually.
The reality of virus infection is of course much more complex, and the modeling could be
correspondingly refined. But one has to be able to choose an appropriate level of abstraction:
Brane Calculi aim to provide such a level of abstraction for dynamic membrane
transformations.
4 Molecules
We have not discussed free-floating molecules so far, to emphasize membrane interactions.
Still, a primary function of membranes and of their embedded proteins is to shuttle molecules
across, and it is important to include this ability in our models. In this section we discuss only
small molecules, the ones that can easily cross or be transported across membranes. See
sections 4.7 and 5.4 for a discussion of large molecules.
Membranes may let certain small molecules through by simple diffusion. Usually,
however, they shuttle specific molecules through molecular channels that are implemented by
sophisticated membrane-bound proteins (represented by our actions). Membranes are also a
favorite mooring point of catalysts that cause free-floating molecules to interact with each
other without crossing the membrane (e.g. in processes as basic as protein synthesis).
Moreover, free-floating molecules can act as communication tokens between different
membranes. A simplifying assumption for now is that small molecules do not change, do not
have internal structure, and do not interact among themselves. All interactions between small
molecules are mediated by membranes.
4.1 Definitions
Membranes can bind molecules on either sides of their surface, and can release molecules on
either sides of their surface. Usually, coordinated bindings and releases happen completely or
not at all, as in the antiporter in Figure 10. Because of this, we integrate in a single new action
the ability to bind and release multiple molecules simultaneously.
Molecules and Molecular Actions
Systems P,Q ::= … ¦ m systems extended with molecules m M
p,q ::= m1 … mk multisets of molecules
Actions a,b ::= … ¦ p1(p2) q1(q2) bind&release of molecules
B&R p1 p1(p2) q1(q2).σ|σ0 p2 P q1 σ|σ0 q2 P
A set of molecules M is added to the syntax of systems. A bind&release action is added to the
set of actions. This action (Figure 9) binds, in general, a multiset of molecules outside the
membrane (p1) and a multiset of molecules inside the membrane (p2); if that is possible, it
instantly releases a multiset of molecules outside the membrane (q1) and a multiset of
molecules inside the membrane (q2). (Conservation of mass or energy is not enforced, and
must be designed in.)
2004-06-01 12:35:5810
Figure 9 Bind and Release
Obvious special cases are the separate binding and release on a single side; we omit ( ):
p1( ) bind outside q1( ) release outside
(p2) bind inside (q2) release inside
4.2 Example: Chemical Reactions
A chemical reaction between molecules can be represented as a catalyst: an always empty
membrane that enables a reaction via an appropriate bind-outside&release-outside action.
Therefore, an explicit catalyst has to be present for a certain reaction to happen. This may be a
bit artificial for simple chemistry, but most biological reactions are actively controlled or
enhanced by catalysts.
p q ! p( ) q( ) Chemical reaction
p q p q q p Reversible reaction
For example, the reaction forming a peptide bond between two amino acids (with residues R1
and R2
) can be written:
R1
R2
PeptideBonding R1
-COOH H2N-R2
R1
-CO-HN-R2
H2O
4.3 Example: Compartment Conditions
We can use an appropriate bind-inside&release-inside action to model chemical reactions that
are specific to a given compartment; we call these conditions of the compartment. For
example, certain chemical reactions happen only at a certain acidity, which is a compartmentwide
property. An appropriate condition on the membrane of a compartment can represent
acidity, and the evolution of conditions and compartments can represent changes of acidity.
For example, the merging of a vesicle carrying some reagents with an endosome having a
certain acidity condition, can cause the reagents to react after the merge because they find
themselves in a compartment with the right acidity condition.
p q ! (p) (q) Condition causing p to change into q
p q p q q p Reversible condition
p q|σ P Compartment-wide condition affecting P
p q|σ p p q|σ q A condition-driven reaction
4.4 Example: Molecular Pumps and Channels
A plant vacuole is a specialized membrane that stores nutrients, e.g. salt. The breakdown of
ATP on the external surface of the vacuole, via a proton pump, is used to charge the interior
((
σ
p1(p2) q1(q2).β
)# σ
β
*)#+ *+
2004-06-01 12:35:58 11
of the vacuole with protons (H+
). In general, several other specialized pumps and channels
can be powered by such a charge. In a plant vacuole, a passive (but selective) ion channel can
let chlorine ions (Cl–
) in, attracted by the excess electric charge of H+
. Transporting sodium
ions (Na+
) inside is more difficult, because those are naturally repelled by the excess charge
of H+
. A proton antiporter, however, can swap an Na+
outside with an H+
inside.
Figure 10 Molecular Channels
Each pump and channel is represented by a replicated bind&release action. These actions
are then assembled as the membrane of an initially empty vacuole.
Plant Vacuole
ProtonPump ! ATP( ) ADP Pi(H+
H+
)
IonChannel ! Cl–
(H+
) (H+
Cl–
)
ProtonAntiporter ! Na+
(H+
) H+
(Na+
)
PlantVacuole ProtonPump | IonChannel | ProtonAntiporter
This is of course a qualitative representation of the process. Attaching reaction rates to
the actions, as in Stochastic π-calculus [12] should yield quantitative modeling. Accurately
modeling this situation should be quite interesting, because the reaction rates depend on the
concentrations on both sides of the membrane.
4.5 Examples: Molecularly-Triggered Membrane Interactions
Molecular interactions can trigger membrane interactions, simply by sequencing the two
kinds of actions on a membrane. In the following example, membrane A produces a molecule
that stimulates membrane B to eat A:
Eat Me
A n( ). P
,-
,-
,"
–
Proton
Pump
Ion
Channel
" –
,-- ,- -
,-
Proton
Antiporter
2004-06-01 12:35:5812
B n( ) . (ρ) Q
A B P n n( ) . (ρ) Q
P (ρ) Q
ρ P Q
Pinocytosis, in reality, may incorporate molecular nutrients into the cell. Our basic
pinocytosis operation does not do that, but it can be used as follows to recognize and
incorporate external nutrients. Here n is a nutrient molecule, and C is a cell that recognizes it,
transports it, and stores it in an internal vesicle.
Seek and Store
seekn !n( ) . ( (n).matestore)
store !mate store
C seekn store
n C seekn (n).matestore store
seekn matestore n store
seekn store n
4.6 Example: Viral Infection, Part 2
We can now complete the central part of the virus reproduction cycle, as shown in Figure 11.
In section 3.3, we still had to provide a mechanism for the following reaction:
nucap cytosol * nucapn
envelope-vesiclem
cytosol’
This can be obtained by the following definitions.
Figure 11 Nucleocapsid Replication (detail of Figure 6)
Nucleocapsid structure
nucap capsid vRNA
capsid !bud | disasm
disasm disasm-trigger(vRNA) vRNA( )
"
!
2004-06-01 12:35:58 13
a) vRNA replication (Figure 11 middle-right)
vRNA-repl vRNA vRNA vRNA
b) Capsomer translation and nucleocapsid assembly (Figure 11 top-right)
capsomer-tran !vRNA( ) vRNA( ).drip(capsomers)
capsomers vRNA( ) (vRNA).capsid
c) Virus envelope protein translation and transport (Figure 11 bottom-right)
ER !vRNA( ) vRNA( ). drip( .viral-envelope) Nucleus
Cytosol contents
cytosol endosome !disasm-trigger
vRNA-repl capsomer-tran ER
A nucap particle is defined as a capsid containing vRNA (we do not model any other
content of the capsid, for simplicity). The capsid surface is capable of either budding from the
cell (as in section 3.3), or of disassembling the nucap by pushing the vRNA outside the capsid
in response to some trigger molecule found in the cytosol (we do not model the fate of the
disassembled capsid). There are then three paths that the newly freed vRNA follows:
(a) vRNA is replicated by the standard cellular machinery found in the cytosol:
vRNA-repl vRNA vRNA-repl vRNA vRNA
(b) The cellular machinery (modeled here by a fictitious empty membrane “capsomer-tran”
with an active surface) translates vRNA into capsomer proteins that self-assemble (by
dripping) into an entity that inserts vRNA from the cytosol into an empty capsid, hence
producing a nucap:
capsomer-tran vRNA * capsomer-tran nucap
(c) The E.R. translates vRNA into viral-envelope proteins that are collected (by dripping) into
envelope-vesicles that are ready to merge ( ) with the cellular membrane as shown in section
3.3:
ER vRNA * ER vRNA envelope-vesicle
Finally, the cytosol is defined as containing all the ingredients needed for this process.
The whole reaction then works as follows. By the disassembly of the nucap, we first
obtain (where !bud is the capsid residue):
nucap cytosol * cytosol vRNA !bud
Then, the vRNA gets replicated (a), and the cytosol can interact to assemble nucaps (b) and
produce envelope vesicles (c), obtaining any number of copies n,m,p of the respective
components, and some residue:
nucap cytosol * nucapn
envelope-vesiclem
cytosol vRNAp
!bud
4.7 Protein Complexes
The handling of protein complexes requires more sophistication in the structure of molecules.
See for example the κ-calculus [5], for an expressive notation for molecular complexes that
2004-06-01 12:35:5814
includes state parameters and binding constructs, and that can realistically model protein
interaction networks. Our bind&release mechanism and the rewrites of κ-calculus should
mutually generalize; we think this is a promising direction for combining complexation with
membrane operations. Here we just describe a simple extension of our framework, by adding
complex formation, m1:m2, between simple molecules:
Molecular Complexes
Systems P,Q ::= … ¦ c systems extended with complexes c
Complexes c,d ::= m ¦ c:d basic molecules m M, or complexation
p,q ::= c1 … ck multisets of complexes
Actions a,b ::= … ¦ p1(p2) q1(q2) bind&release of complexes
B&R p1 p1(p2) q1(q2).σ|σ0 p2 P q1 σ|σ0 q2 P
Then, we can use the bind&release operator to express, e.g. complexation on the inside
surface of a membrane:
(m1 m2) (m1:m2)
Protein synthesis in the E.R. has the following structure: membrane bound ribosomes
take amino acids (bound to tRNA) from one side of the membrane, and produce complexes
(polypeptides) on the other side of the membrane. Hence, decomplexation, membranecrossing,
and complexation are combined in a single process. A completely satisfactory
description of this process, though, probably requires either restriction [5], to model the
identity of the polypeptide being assembled, or some further notions of complexation with
membrane-bound proteins.
5 Extensions
In this section we discuss possible extensions that fit well into the Brane Calculi framework.
5.1 Communication
Although much can be done with purely combinatorial operators, as in the Basic Brane
Calculus considered so far, it is possible to add communication operations in the style of CCS
or BioAmbients, assuming a substitution τ{p←m} of name m for name p in τ.
On-Membrane Communication (CCS style)
Actions a,b ::= … ¦ p2pn(m) ¦ p2p n(m)
peer to peer p2pn(m).σ | p2p n(p).τ | ρ P σ | τ{p←m} | ρ P
Cross-Membrane Communication (BioAmbients style)
Actions a,b ::= … ¦ s2sn(m) ¦ s2s n(m) ¦ p2cn(m) ¦ p2c n(m) ¦ c2pn(m) ¦ c2p n(m)
sibling to sibling s2sn(m).σ|σ0 P s2s n(p).τ|τ0 Q σ|σ0 P τ{p←m}|τ0 Q
parent to child p2cn(m).σ|σ0 p2c n(p).τ|τ0 Q P σ|σ0 τ{p←m}|τ0 Q P
2004-06-01 12:35:58 15
child to parent c2p n(p).τ|τ0 c2pn(m).σ|σ0 Q P τ{p←m}|τ0 σ|σ0 Q P
5.2 Choice
A choice operation can be added to membranes:
Choice
Branes σ,τ ::= … ¦ σ+τ
Its main impact is that all reaction rules must then consider more complex normal forms for
membranes, of the form (a.σ+σ1)|σ0 P instead of a.σ|σ0 P . There may be ways to hide this
complexity behind appropriate notation, particularly in absence of binding operators.
A good use for choice is to express a shuffle operator a.b.σ+b.a.σ, which is natural when
considering individual proteins triggered by two independent binding sites. On the other hand,
common forms of choice can be embedded directly in the notation for molecules [5]. Choice
at the system level, instead of the membrane level, does not seem very realistic.
Exercise: define (without using choice) a pair of isolation actions isln, isl n, such that:
isl n.σ|σ0 P isln.τ|τ0 Q * σ τ|τ0 0 σ0 P Q
that is, isl n.σ, when triggered by its co-action, isolates σ as the only residual, and makes
the rest of its membrane and its contents inaccessible. Then, use a pair of isolation actions in
parallel to implement a limited form of choice.
5.3 Atonal Transport
Although we have emphasized bitonal operators, there are situations in which simple in-out
transport operators, as in BioAmbients [14], may be preferable. One example is when
representing a protein with multiple interaction domains as a (fictitious) membrane (see [14]
for a detailed discussion). When a protein is represented that way, protein transport in/out of a
(real) membrane takes the form of atonal operations (ones that do not preserve bitonality).
Atonal situations may also arise at higher levels of organization, as when a cell enters the
bloodstream through a vessel wall.
The following transport operations are similar to the ones in BioAmbients:
Actions a,b ::= … ¦ inn ¦ in n ¦ outn ¦ out n
In inn.σ|σ0 P in n.τ|τ0 Q τ|τ0 σ|σ0 P Q
Out out n.τ|τ0 outn.σ|σ0 P Q σ|σ0 P τ|τ0 Q
Alternatively, one can think of adding a single atonal primitive to Phago/Exo/Pino in order to
encode In/Out. A simple solution is:
wrap(σ).τ|τ0 P σ τ|τ0 P
so that wrap + exo = out, and wrap + phago + exo = in.
2004-06-01 12:35:5816
Figure 12 Atonal Reactions
It is conceivable that a simple type system may keep the bitonal and atonal parts of a
system separate. It is also conceivable that empty membranes (representing molecules) may
harmlessly assume a double tonality, violating bitonality only in a weak sense. This could be
achieved by restricting In/Out to the empty membrane case:
SmallIn inn.σ|σ0 in n.τ|τ0 P τ|τ0 σ|σ0 P
SmallOut out n.τ|τ0 outn.σ|σ0 P σ|σ0 τ|τ0 P
This way, although really changes tone in reactions, the systems is consistently bitonal both
before and after reactions. Again, a minimal atonal extension could consist of:
SmallWrap wrap(σ).τ|τ0 σ τ|τ0
Exercise: show that it is possible to represent small molecules m as empty membranes
molm , for an appropriate definition of molm, in such a way that an operation similar to
bind&release of Section 4.1 is definable. Hints: choice is useful; limit the exercise to
sequential bind&release of individual molecules, rather than atomic bind&release of multiple
molecules.
5.4 Free-Floating Proteins as Membranes
Free-floating proteins are large molecules with complex dynamic behavior and multiple
independent domains of interaction: they can interact with membranes and with each other,
and can act as catalysts for smaller molecules. In section 4.7 we have discussed how to model
protein complexes directly. It may also seem reasonable to model such large molecules as
“small membranes”, that is, as membranes σ with multiple surface actions but (normally)
empty contents.
In this view, a free floating protein inside a membrane is just a membrane inside a larger
membrane. This idea and the issues it raises are discussed in [14]. (A different proposal is to
assume multi-domain molecules as primitive [4][5].)
One problem with representing molecules as membranes, in general, is that molecules
can “squeeze through” membranes or through their channels, while membranes cannot.
Situations where large molecules cross membranes are however, limited, and can sometimes
be modeled by other mechanisms. One common case is when proteins and RNA cross the
nuclear membrane through its pores. The nuclear membrane is a double membrane, so
wrap(σ).τ
..
σ0 τ0
inn.σ in n.τ
//
τ0σ0
σ
σ0
σ
τ
τ0σ0
outn.σ
σ0
outn.σ
out n.τ
σ0
τ0
σ τ
00
τ0
τ
τ0
τ
τ0
σ
2004-06-01 12:35:58 17
crossing it can be modeled bitonally by Phago and Exo through the lumen. (This is slightly
artificial, but an accurate geometrical modeling of the nuclear double membrane and its
toroidal pores would in any case require a 3D calculus.)
The problem of complex formation and breaking ([14], Section 3.2) also has a bitonal
solution. Assuming proteins are represented as empty membranes σ , τ with all their
domains on σ and τ, then complexation is simply merging of two such membranes, σ|τ
(modulo some interaction). Breakup can be achieved by Pino, to recreate internally the protein
fragments, ρ σ1 τ1 followed by Bud to separate them, ρ1 σ2 ρ2 τ2 , and finally
by two Exo, σ τ .
Enzyme interactions ([14], Section 3.3) also have a bitonal solution for enzymes reacting
with proteins (as opposed to small molecules). Two proteins σ ,τ can bind to an enzyme
ρ by Phago, ρ ρ1 σ ρ2 τ , followed by Mate to bring them in contact,
ρ ρ3 σ τ , followed by their interaction, e.g. again Mate, ρ ρ3 σ|τ , followed by
Exo to release the catalyzed product, σ|τ ρ . However, the production of enzymes has
to be modeled as the production of membranes, not of molecules, and this might be awkward.
5.5 Bitonal Brane Calculi
While the operations of the Basic Brane Calculus are bitonal in nature (i.e. they preserve the
nesting parity of subsystems, with the exception of molecules in bind&release), the calculus
framework does not build-in bitonality.
A proper Bitonal Brane Calculus would, instead, adopt a syntax of alternating colored
brackets σ1 σ2 σ3 σ2 … , with an assumption that the tone-dual of a reaction is also a
reaction. (This could also be achieved by type distinction, instead of syntactic distinctions.)
All the figures resulting from such a calculus could be consistently shaded in two alternating
tones, and atonal operations like In, Out, or Wrap could not be directly supported because
they would violate the alternation.
Exercise: show that a bitonal calculus (with Phago+Exo+Pino and alternating brackets)
can emulate the atonal calculus (with Phago+Exo+Pino+Wrap). Hint: double walling.
6 Encoding Brane Calculi
Are Brane Calculi really novel, of can they be easily encoded in other calculi? The obvious
comparison is with the closely related BioAmbients Calculus. Let us consider the simplest
possible idea for a translation P† into BioAmbients, namely “in brane” actions (Figure 13):
σ P † [σ† | P†]
Where the membrane σ is converted into a process inside a membrane […], at the same level
as the translation of P. Consider now the induced translation of Exo:
Exo n.τ|τ0 n.σ|σ0 P Q P σ|σ0|τ|τ0 Q
Exo† [ .τ† | τ0† | [ .σ† | σ0† | P†] | Q†] P† | [σ† | σ0† | τ† | τ0† | Q†]
Where we would have to devise an appropriate definition for .τ† and .σ† so that Exo†
had the prescribed behavior. A problem, though, is already apparent. The Exo rule separates P
from σ|σ0 on the r.h.s., and it can do so because the separation between σ|σ0 and P is built into
the term n.σ|σ0 P on the l.h.s.. In Exo†, though, the process .σ† | σ0† | P† on the l.h.s. is a
featureless composition; how does the rule “know” to split off P† precisely at that position?
2004-06-01 12:35:5818
To avoid this loss of structure, it is necessary to put more structure in the translation:
σ P † [[σ†] | P†] or:
σ P † [σ† | [P†]] “Ball bearing” encoding
This requires more complicated encodings of operations, which need to cross multiple level of
brackets and therefore have atomicity problems.
Figure 13 Exo Encodings
Our suspicion is that an encoding of Brane Calculi in Ambients-like calculi may be
possible, but it is not easy and almost certainly not practically usable.
7 Conclusions
How are “bio”-calculi different from other process calculi? Both in Brane Calculi and in
BioAmbients, (and in BioSPI [13], before that), we have used standard concepts and
techniques developed for calculi of concurrency and mobility. We believe that Brane Calculi
are beginning to confront some of the pragmatic issues discovered with BioAmbients, by
emulating more closely biological processes, in the same way that BioAmbients removed the
need for some artificial encodings in BioSPI.
The issue of choosing “realistic” primitives is a tricky one. At one extreme, only the
precise mechanisms that have an existing biological implementation are realistic, and those
usually have extremely sophisticated and still only partially understood molecular-level
implementations. However, even without understanding the molecular details, it is possible to
distinguish operations that work via dedicated molecular machinery from those that do not. At
the other extreme, biological systems have general constraints and invariants that determine
which operations are at least in principle realistic (and which are not). Membrane orientation
is one such invariant: it is actively maintained by living cells by consistently orienting
proteins on the membrane surface. Bitonality is another invariant, at least in some regimes of
operation; it derives from certain transformations of oriented membranes that produce deeper
nestings: the basic bitonal structure of a cell and its organs is due to such transformations that
happened during evolution ([1] p. 556). These biological invariants suggest a different set of
“potentially realistic” basic operations for concurrent calculi than ones that had been
considered before.
Another basic aspect of biological membranes is their nature as a two-dimensional fluid
embedded in a three-dimensional fluid; this is in fact more fundamental than any orientability
or bitonality considerations. This means that there are at least two commutative monoids
involved, and not the single one usually seen in process calculi. The formalization of these
Original
“on brane”
actions
“In brane”
encoding
attempt
“Ball bearing”
encoding
2004-06-01 12:35:58 19
two monoids adds complexity, but supports the notion of computation on the membrane, that
is, of computation that is directly aware of conditions on both sides of the membrane. Trying
to emulate this fluid-in-fluid structure by other encodings is awkward (see Section 6),
although the issue has been valiantly confronted in BioAmbients.
References
[1] B.Alberts, D.Bray, J.Lewis, M.Raff, K.Roberts, J.D.Watson. Molecular Biology of the Cell. Third
Edition, Garland.
[2] L.Cardelli. Bitonal Membrane Systems – Interactions of Biological Membranes. To appear.
[3] L.Cardelli and A.D.Gordon. Mobile Ambients. Theoretical Computer Science, Special Issue on
Coordination, D. Le Métayer Editor. Vol 240/1, June 2000. pp 177-213.
[4] V.Danos, M.Chiaverini. A Core Modeling Language for the Working Molecular Biologist. 2002.
[5] V.Danos and C.Laneve. Formal Molecular Biology. Theoretical Computer Science, to Appear.
[6] L.H.Hartwell, J.J.Hopfield , S.Leibler , A.W.Murray: From molecular to modular cell biology. Nature.
1999 Dec 2;402(6761 Suppl):C47-52.
[7] H. Kitano: A graphical notation for biochemical networks. BIOSILICO 1:169-176, 2003.
[8] K.W. Kohn: Molecular Interaction Map of the Mammalian Cell Cycle Control and DNA Repair
Systems. Molecular Biology of the Cell, 10(8):2703-34, Aug 1999.
[9] H.Lodish, A.Berk, S.L.Zipursky, P.Matsudaira, D.Baltimore, J.Darnell. Molecular Cell Biology. Fourth
Edition, Freeman.
[10] H.H.McAdams, A.Arkin : It's a noisy business! Genetic regulation at the nanomolar scale. Trends
Genet. 1999 Feb;15(2):65-9.
[11] R.Milner. Communicating and Mobile Systems: The ππππ-Calculus. Cambridge University Press, 1999.
[12] C.Priami. The Stochastic pi-calculus. The Computer Journal 38: 578-589, 1995.
[13] C.Priami, A.Regev, E.Shapiro, and W.Silverman. Application of a stochastic name-passing calculus to
representation and simulation of molecular processes. Information Processing Letters, 80:25-31, 2001.
[14] A.Regev, E.M.Panina, W.Silverman, L.Cardelli, E.Shapiro. BioAmbients: An Abstraction for
Biological Compartments. Theoretical Computer Science, to Appear.
[15] Systems Biology Markup Language. <www.sbml.org>