Technical Report CoSBi 07/2006
Modelling Cellular Processes using
Membrane Systems with Peripheral and
Integral Proteins
Matteo Cavaliere
The Microsoft Research – University of Trento Centre for Computational and Systems
Biology
Sean Sedwards
The Microsoft Research – University of Trento Centre for Computational and Systems
Biology
This is a preliminary version of a paper that will appear in
Proceedings of the International Conference on Computational Methods in Systems
Biology, CMSB06, Lecture Notes in Bioinformatics, 4210:108–126, 2006.
Abstract
Membrane systems were introduced as models of computation inspired by the
structure and functioning of biological cells. Recently, membrane systems have
also been shown to be suitable to model cellular processes. We introduce a new
model called Membrane Systems with Peripheral and Integral Proteins. The model
has compartments enclosed by membranes, floating objects, objects associated to
the internal and external surfaces of the membranes and also objects integral to
the membranes. The floating objects can be processed within the compartments
and can interact with the objects associated to the membranes. The model can
be used to represent cellular processes that involve compartments, surface and
integral membrane proteins, transport and processing of chemical substances. As
examples we model a circadian clock and the G-protein cycle in yeast saccharomyces
cerevisiae and present a quantitative analysis using an implemented simulator.
1 Introduction
Membrane systems are models of computation inspired by the structure and the function
of biological cells. The model was introduced in 1998 by Gh. P˘aun and since then many
results have been obtained, mostly concerning computational power. A short introduc-
tory guide to the field can be found in [12], while an updated bibliography is available via
the web-page [18]. Recently (see, e.g., [10]), membrane systems have been successfully
applied to systems biology and several models have been proposed for simulating biolog-
ical processes (e.g., see the monograph dedicated to membrane computing applications
[5]).
By the original definition, membrane systems are composed of an hierarchical nest-
ing of membranes that enclose regions (the cellular structure), in which free-floating
objects (molecules) exist. Each region can have associated rules, called evolution rules,
for evolving the free-floating objects and modelling the biochemical reactions present in
cell regions. Rules also exist for moving objects across membranes, called symport and
antiport rules, modelling cellular transport. Recently, inspired by brane calculus [3], a
model of a membrane system, having free-floating objects and objects attached to the
membranes, was introduced in [2]. The attached objects model the proteins that are
embedded in lipid bilayer cell membranes. In [2], however, objects are associated to an
indivisible membrane which has no concept of inner or outer surface, while in [4] ob-
jects (peripheral proteins) are attached to either side of a membrane. In reality, many
biological processes are driven and controlled by the presence of specific proteins on the
appropriate side of and integral to the membrane: there is a constant interaction between
floating chemicals and embedded proteins and between peripheral and integral proteins
(see, e.g., [1]). Receptor-mediated processes, such as endocytosis (illustrated in Figure
1) and signalling, are crucial to cell function and by definition are critically dependent
on the presence of peripheral and integral membrane proteins.
One model of the cell is that of compartments and sub-compartments in constant
communication, with molecules being passed from donor compartments to target com-
partments by interaction with membrane proteins. Once transported to the correct
1
Figure 1: Endocytosis of LDL (Essential Cell Biology, 2/e,
c
2004 Garland Science)
compartment, the substances are then processed by means of local biochemical reactions.
Motivated by these ideas we extend the model presented in [4], introducing a model
having peripheral as well as integral proteins.
In each region of the system there are floating objects (the floating chemicals) and,
in addition, objects can be associated to each side of a membrane or integral to the
membrane (the peripheral and integral membrane proteins). Moreover, the system can
perform the following operations: (i) the floating objects can be processed/changed inside
the regions of the system (emulating biochemical rules) and (ii) the floating and attached
objects can be processed/changed when they interact (modelling the interactions of the
floating molecules with membrane proteins).
The proposed model can be used to represent cellular processes that involve floating
molecules, surface and integral membrane proteins, transport of molecules across mem-
branes and processing of molecules inside the compartments. As examples, we model a
circadian clock and the G-protein cycle in saccharomyces cerevisiae, where the possibil-
ity to use, in an explicit way, compartments, membrane proteins and transport rules is
very useful. A quantitative analysis of the models is also presented, performed using an
extended version of the simulator presented in [4] (downladable at [19]). The simulator
employs a stochastic algorithm and uses intuitive syntax based on chemical equations
(described in appendix B).
2 Formal Language Preliminaries
Membrane systems are based on formal language theory and multiset rewriting. We now
briefly recall the basic theoretical notions used in this paper. For more details the reader
can consult standard books, such as [8], [15], [6] and handbook [14].
Given the set A we denote by |A| its cardinality and by ∅ the empty set. We denote
by N and by R the set of natural and real numbers, respectively.
As usual, an alphabet V is a finite set of symbols. By V
∗
we denote the set of all
strings over V . By V
+
we denote the set of all strings over V excluding the empty
string. The empty string is denoted by λ. The length of a string v is denoted by |v|. The
concatenation of two strings u, v ∈ V
∗
is written uv.
The number of occurrences of the symbol a in the string w is denoted by |w|
a
.
2
A multiset is a set where each element may have a multiplicity. Formally, a multiset
over a set V is a map M : V → N, where M(a) denotes the multiplicity of the symbol
a ∈ V in the multiset M.
For multisets M and M
0
over V , we say that M is included in M
0
if M(a) ≤ M
0
(a)
for all a ∈ V . Every multiset includes the empty multiset, defined as M where M(a) = 0
for all a ∈ V .
The sum of multisets M and M
0
over V is written as the multiset (M + M
0
), defined
by (M + M
0
)(a) = M(a) + M
0
(a) for all a ∈ V . The difference between M and M
0
is
written as (M −M
0
) and defined by (M − M
0
)(a) = max{0, M(a)−M
0
(a)} for all a ∈ V .
We also say that (M + M
0
) is obtained by adding M to M
0
(or viceversa) while (M −M
0
)
is obtained by removing M
0
from M. For example, given the multisets M = {a, b, b, b}
and M
0
= {b, b}, we can say that M
0
is included in M, that (M + M
0
) = {a, b, b, b, b, b}
and that (M − M
0
) = {a, b}.
If the set V is finite, e.g. V = {a
1
, . . . , a
n
}, then the multiset M can be explicitly
described as {(a
1
, M(a
1
)), (a
2
, M(a
2
)), . . . , (a
n
, M(a
n
))}. The support of a multiset M is
defined as the set supp(M) = {a ∈ V | M(a) > 0}. A multiset is empty (hence finite)
when its support is empty (also finite).
A compact notation can be used for finite multisets: if M = {(a
1
, M(a
1
)), (a
2
, M(a
2
)),
. . . , (a
n
, M(a
n
))} is a multiset of finite support, then the string w = a
M(a
1
)
1
a
M(a
2
)
2
. . . a
M(a
n
)
n
(and all its permutations) precisely identify the symbols in M and their multiplicities.
Hence, given a string w ∈ V
∗
, we can say that it identifies a finite multiset over V , writ-
ten as M(w), where M(w) = {a ∈ V | (a, |w|
a
)}. For instance, the string bab represents
the multiset M(w) = {(a, 1), (b, 2)}, that is the multiset {a, b, b}. The empty multiset is
represented by the empty string λ.
3 Operations with Peripheral and Integral Proteins
Let V denote a finite alphabet of objects and Lab a finite set of labels.
As is usual in the membrane systems field, a membrane is represented by a pair
of square brackets, [ ]. A membrane structure is an hierarchical nesting of membranes
enclosed by a main membrane called the root membrane. To each membrane is associated
a label that is written as a superscript of the membrane, e.g. [ ]
1
. If a membrane has the
label i we call it membrane i.
A membrane structure is essentially that of a tree, where the nodes are the membranes
and the arcs represent the containment relation. In this paper we avoid a formal mapping
in the interest of the intuitiveness of the description, however, being a tree, a membrane
structure can be represented by a string of matching square brackets, e.g., [ [ [ ]
2
]
1
[ ]
3
]
0
.
To each membrane there are associated three multisets, u, v and x over V , denoted
by [ ]
u|v|x
. We say that the membrane is marked by u, v and x; x is called the external
marking, u the internal marking and v the integral marking of the membrane. In general,
we refer to them as markings of the membrane.
The internal, external and integral markings of a membrane model the proteins at-
tached to the internal surface, attached to the external surface and integral to the mem-
3
brane, respectively.
In a membrane structure, the region between membrane i and any enclosed mem-
branes is called region i. To each region is associated a multiset of objects w called the
free objects of the region. The free objects are written between the brackets enclosing
the regions, e.g., [ aa [ bb ]
1
]
0
.
The free objects of a membrane model the floating chemicals within the regions of a
cell.
We denote by int(i), ext(i) and itgl(i) the internal, external and integral markings
of membrane i, respectively. By free(i) we denote the free objects of region i. For
any membrane i, distinct from a root membrane, we denote by out(i) the label of the
membrane enclosing membrane i.
For example, the string
[ ab [ cc ]
2
a| |
[ abb ]
1
bba|ab|c
]
0
represents a membrane structure, where to each membrane are associated markings and
to each region are associated free objects. Membrane 1 is internally marked by bba (i.e.,
int(1) = bba), has integral marking ab (i.e., itgl(1) = ab) and is externally marked by c
(i.e., ext(1) = c). To region 1 are associated the free objects abb (i.e., free(1) = abb). To
region 0 are associated the free objects ab. Finally, out(1) = out(2) = 0. Membrane 0 is
the root membrane. The string can also be depicted diagrammatically, as in Figure 2.
Figure 2: Graphical representation of [ ab [ cc ]
2
a| |
[ abb ]
1
bba|ab|c
]
0
When a marking is omitted it is intended that the membrane is marked by the empty
string λ, i.e., the empty multiset. For instance, in [ ab ]
u|v|
the external marking is
missing, while in the case of [ ab ]
|v|x
the internal marking is missing.
3.1 Operations
We introduce rules that describe bidirectional interactions of floating objects with the
membrane markings which we call membrane rules. These rules are motivated by the
behaviour of cell membrane proteins (e.g., see [1]) and therefore permit a level of abstrac-
tion based on the behaviour of real molecules. We denote the rules as attach
in
, attach
out
,
de − attach
in
and de − attach
out
, defined:
4
![Figure 5: Reaction scheme and simulation results of noise-resistant circadian clock of [16]](/figures/figure-5-reaction-scheme-and-simulation-results-of-noise-1u07a0jy.webp)
![Figure 4: evol rule [ a → b ]ib|c|. Free objects can be rewritten inside the region and the rewriting can depend on the integral and internal markings of the enclosing membrane.](/figures/figure-4-evol-rule-a-b-ib-c-free-objects-can-be-rewritten-f7tts876.webp)
