=Paper=
{{Paper
|id=None
|storemode=property
|title=Multi-cell Modelling Using Coloured Petri Nets Applied to Planar Cell Polarity
|pdfUrl=https://ceur-ws.org/Vol-724/paper10.pdf
|volume=Vol-724
}}
==Multi-cell Modelling Using Coloured Petri Nets Applied to Planar Cell Polarity==
https://ceur-ws.org/Vol-724/paper10.pdf
Proceedings of the 2nd International Workshop on Biological Processes & Petri Nets (BioPPN2011)
online: http://ceur-ws.org/Vol-724 pp.135-150
Multi-cell Modelling Using Coloured Petri Nets
Applied to Planar Cell Polarity
Qian Gao1 , Fei Liu3 , David Tree2 , and David Gilbert1
1
School of Information Systems, Computing and Maths, Brunel University, UK
http://www.brunel.ac.uk/siscm
2
School of Health Science and Social Care, Brunel University, UK
http://www.brunel.ac.uk/about/acad/health/healthsub/bioscience
{qian.gao,david.tree,david.gilbert}@brunel.ac.uk
3
Computer Science Department, Brandenburg University of Technology Cottbus,
Germany
http://www-dssz.informatik.tu-cottbus.de/DSSZ
liu@Informatik.tu-Cottbus.de
Abstract. Modelling across multiple scales is a current challenge in Sys-
tems Biology. In this paper we present an approach to model at different
spatial scales, applied to a tissue comprising multiple hexagonally packed
cells in a honeycomb formation in order to describe the phenomenon of
Planar Cell Polarity (PCP).
PCP occurs in the epithelia of many animals and can lead to the align-
ment of hairs and bristles. Here, we present an approach to model this
phenomenon by applying coloured Petri Nets (CPN). The aim is to dis-
cover the basic principles of implementing CPN to model a multi-cellular
system with a hierarchical structure while keeping the model mathemat-
ically tractable. We describe a method to represent a spatially defined
multi-scale biological system in an abstract form as a CPN model, in
which all reactions within a cell are categorised into two main types,
each cell is sub-divided into seven logical compartments and adjacent
cells are coupled via the formation of intercellular complexes. This work
illustrates the issues that need to be considered when modelling a multi-
cellular system using CPNs. Moreover, we illustrate different levels of
abstraction that can be used in order to simplify such a complex system
and perform sophisticated high level analysis. Some preliminary analysis
results from animation and stochastic simulation are included in this pa-
per to demonstrate what kinds of sequential analysis can be performed
over a CPN model.
Keywords: coloured Petri nets, modelling, planar cell polarity
1 Introduction
With the rapid growth of data being generated in the biological field, it has be-
come necessary to organise the data into coherent models that describe system
behaviour, which are subsequently used for simulation, analysis or prediction.
135
2 Qian Gao, Fei Liu, David Tree, and David Gilbert
Modelling plays a crucial role in facilitating the understanding of complex bio-
logical mechanisms from an holistic viewpoint.
A large variety of modelling approaches have already been applied to model
a wide array of biological systems (see [1] for a review). Among them, Petri
nets are particularly suitable for representing and modelling the concurrent,
asynchronous and dynamic behaviour of biological systems. Since Reddy et al.
[2] introduced the application of qualitative Petri nets to modelling of metabolic
pathways, a large variety of applications of Petri nets (e.g. stochastic Petri nets,
timed Petri nets, continuous Petri nets, and hybrid Petri nets, etc.) have been
developed for modelling and simulating different types of biological systems [3],
[4].
Modelling across multiple scales is a current challenge in Systems Biology.
There is the need to model at different spatial scales to describe, for example,
intracellular locality in compartments, organelles and vacuoles, as well as inter-
cellular locality in terms of intercellular communication by complex formation
across cell gaps, and by cytokines (intercellular messengers), and higher levels of
organisation into tissues and organs composed of many cells. However, standard
Petri nets do not readily scale to meet these challenges, and current attempts to
simulate biological systems by standard Petri nets have been mainly restricted so
far to relatively small models. Standard Petri nets tend to grow quickly for mod-
elling complex systems, which makes it more difficult to manage and understand
the nets, thus increasing the risk of modelling errors. Two known modelling con-
cepts improving this situation are hierarchy and colour. Hierarchical structuring
has been discussed a lot, e.g. in [5], while the colour has gained little attention
so far.
While there is a lot of reported work on the application of different classes
of standard Petri nets to a variety of biochemical networks, see [4] for a recent
review, there are only a few which take advantage of the additional power and
ease of modelling offered by coloured Petri nets (CPN). To our knowledge, the
existing applications of CPN in systems biology can only be seen in [6], [7], [8], [9],
[10], [11], [12], [13]. Moreover, these existing studies usually resort to Design/CPN
[14] or its successor CPN Tools [15] in order to model and analyse biological
systems. However neither tool was specifically designed with the requirements of
Systems Biology in mind. Thus they are not suitable in many aspects, e.g. they
do not directly support stochastic or continuous modelling, nor the simulative
analysis of the models by stochastic or deterministic simulation.
Building upon the lessons learnt so far, we extend our software tool Snoopy
[16], [17] by specific functionalities and features to support editing, simulating
and analysing of biological models based on coloured qualitative, stochastic and
continuous Petri nets. By doing so, we not only provide compact and readable
representations of complex biological systems, but also do not lose the analysis
capabilities of standard Petri nets, which can still be supported by automatic
unfolding. Moreover, another attractive advantage of CPN for a biological mod-
eller is that they provide the possibility to easily increase the size of a model
consisting of many similar subnets just by adding new colours.
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Multi-cell modelling Using CPN 3
Modelling in Biology tends to emphasise molecular details. Yet in biological
networks that involve more than a few components the typical situation is that
many details are unknown, and it is imperative to devise an approach that can
be insightful and predictive even in the absence of complete knowledge. Our
strategy was based on building an abstract model of PCP which attempts to
identify the key biological aspects (e.g. formation of intercellular complexes),
and then constructing a more detailed but simple model which parameterises
the many unknowns.
In this paper, our aim is to use CPN to describe an intercellular and intra-
cellular signalling model that replicates the phenomenon of PCP in Drosophila
wing. The epithelial cells in this organ are hexagonally packed in a 2-dimensional
honeycomb lattice. The model incorporates an abstract description of informa-
tion flow within the cell, and a representation of inter-cellular communication
through the formation of protein complexes, so that local (transmembrane) sig-
nalling produces a global effect over the entire organ. It should be noted that
approach presented in this paper is applied to an abstract model of PCP in order
to illustrate the application of CPN to PCP signalling. Specifically, we focus on
the way to include the honeycomb structure and logical compartments into the
construction of our multi-cellular model.
This paper is structured as follows: in Subsection 1.1 we introduced the bio-
logical background of planar cell polarity, followed by Subsection 1.2 briefly de-
scribing coloured Petri Nets and Section 2 describing our model and approaches,
followed by the conclusion.
1.1 Planar Cell Polarity
Planar cell polarity (PCP) refers to the orientation of cells within the plane
of the epithelium, orthogonal to the apical-basal polarity of the cells. This po-
larisation is required for many developmental events in both vertebrates and
non-vertebrates. Defects in PCP in vertebrates underlie developmental abnor-
malities in multiple tissues including the neural tube, the kidney and the inner
ear (reviewed in [18]). The signalling mechanisms underlying PCP have been
studied most extensively in the epithelia of the fruit fly Drosophila melanogaster
including the wing, the abdomen, the eye, and the bristles of the thorax. Genetic
studies in the wing and eye in the 1990s led to the proposal of a PCP signalling
pathway involving the PCP proteins Frizzled (Fz), Dishevelled (Dsh) and Prickle
(Pk) (reviewed in [19]). In the late 1990s and 2000 further genetic analysis, in-
cluding the discovery of more PCP proteins, e.g. Flamingo (Fmi) and Van-Gogh
(Vang), and data on the sub-cellular localisation of these proteins in normal and
mutant situations, have led to the formulation of more complex models of PCP
signalling. In this paper we apply CPN to the models formulated in the fly wing
as a means to gain insight into mechanism of PCP.
The adult Drosophila wing comprises about 300,000 hexagonal cells each of
which contain a single hair which points in an invariant distal direction, see
Figure 1. This hair comprises actin bundles and is extruded from the membrane
at the distal edge of the cell during pupal development at the conclusion of
137
4 Qian Gao, Fei Liu, David Tree, and David Gilbert
PCP signalling. Preceding this ultimate manifestation of PCP, PCP signalling
occurs such that the proteins adopt an asymmetric localisation within each cell.
At the initiation of PCP signalling Fmi, Fz, Dsh, Vang and Pk are all present
symmetrically at the cell membrane. At the conclusion of PCP signalling Fmi
is found at both the proximal and distal cell membrane, Fz and Dsh are found
exclusively at the distal cell membrane and Vang and Pk are found exclusively
at the proximal cell membrane. Through the interpretation of various genetic
experiments a consensus view of the signalling events has been formulated that
centres on the communication between these proteins at cell boundaries. The
distally localised Fmi, Fz and Dsh recruit Fmi, Vang and Pk to the proximal
cell boundary and vice versa. Since the localisation of the distal and proximal
proteins appear to be mutually exclusive a completely polarised arrangement
of protein localisation results. The PCP proteins are thus thought to mediate
the cell-cell communication that comprises PCP signalling and that they are
involved in establishing the molecular asymmetry within and between cells which
is subsequently transformed into the polarisation of the wing hairs (reviewed
in [20]). The result is a polarisation of individual cells and local alignment of
polarity between neighbouring cells.
(a) (b)
Fig. 1. Drosophila: (a) Whole wing; (b) Schematic of hexagonal cells with hairs
Systems biology and mathematical modelling have been applied to PCP sig-
nalling by Amonlirdviman et al. [21] (extended in [22]) and Le Garrec et al. [23]
(applied to the Drosophila eye in [24]). Both models centre around the idea of
amplification of polarity via asymmetric complex formation of the core proteins.
Both models rely on numerical simulations in two dimensions for fields of hexag-
onal or approximately hexagonal cells. Therefore, they tend to be rather complex
and do not lend themselves to mathematical analysis very easily. Furthermore,
because of the lack of appropriate biological data, the feedback mechanisms in
these models are mainly based on assumptions.
138
Multi-cell modelling Using CPN 5
In this paper, we apply CPN to PCP signalling in a generic setting that
encompasses a broad class of specific models, ranging from a single cell model
to a multi-cellular model. To this end, we have developed an abstract model for
the generation of PCP to investigate the signalling by implementing animation
analysis and stochastic simulation analysis.
1.2 Coloured Petri Nets
Coloured Petri nets (CPN) [25], [26] are a discrete event modelling formalism
combining the strengths of Petri nets with the expressive power of program-
ming languages. Petri nets provide the graphical notation and constructions for
modelling systems with concurrency, communication and synchronisation. The
programming languages offer the constructions for the definition of data types,
then used for creating compact models. This is the greatest advantage of CPN.
CPN consist, as do standard Petri nets, of places, transitions and arcs. In
systems biology, places also represent species (chemical compounds) while transi-
tions represent any kind of chemical reactions. Each place gets assigned a colour
set and contains distinguishable coloured tokens. A distribution of coloured to-
kens on all places together constitutes a marking of a CPN. Each transition may
have a guard, which is a Boolean expression over defined variables. The guard
must be evaluated to true for the enabling of the transition if it is present. Each
arc gets assigned an expression, which is a multiset type of the colour set of the
connected place.
The variables associated with a transition consist of the variables in the guard
of the transition and in the expressions of arcs connected to the transition. Be-
fore the expressions are evaluated, the variables must get values assigned with
suitable data types, which is called binding [26]. A binding of a transition cor-
responds to a transition instance in the unfolded net. Enabling and firing of a
transition instance are based on the evaluation of its guard and arc expressions.
If the guard is evaluated to true and the preplaces have sufficient tokens, the
transition instance is enabled and may fire. When a transition instance fires, it
removes coloured tokens from its preplaces and adds tokens to its postplaces,
i.e. it changes the current marking to a new reachable one. The colours of the
tokens that are removed from preplaces and added to postplaces are decided by
arc expressions. The set of markings reachable from the initial marking consti-
tutes the state space of a given net. These reachable markings and instances of
transitions between them constitute the reachability graph of the net.
Thus CPN has the ability to tackle the challenges arising from modelling
biological systems beyond one spatial scale, for example, repetition of compo-
nents, which is the need to describe multiple cells each of which has a similar
definition; and organisation of components, which refers to how cells are organ-
ised into regular or irregular patterns over spatial networks in one, two or three
dimensions.
In the following, we give the formal definition of CPN and briefly describe
the tool for modelling CPN.
139
6 Qian Gao, Fei Liu, David Tree, and David Gilbert
Definition In CPN, there are different types of expressions, arc expressions,
guards and expressions for defining initial markings. An expression is built up
from variables, constants, and operation symbols. It is not only associated with
a particular colour set, but also written in terms of a predefined syntax. In the
following, we denote by EXP a set of expressions that comply with a predefined
syntax. The formal definition for coloured Petri nets is as follows [25], [26].
Definition 1 P
(coloured Petri net). A coloured Petri net is a tuple
N =< P, T, F, , C, g, f, m0 >, where:
– P is a finite, non-empty set of places.
– T is a finite, non-empty set of transitions.
– F
P is a finite set of directed arcs.
– is a finite,
P non-empty set of colour sets.
– C : P → Pis a colour function that assigns to each place p ∈ P a colour
set C(p) ∈ .
– g : T → EXP is a guard function that assigns to each transition t ∈ T a
guard expression of the Boolean type.
– f : F → EXP is an arc function that assigns to each arc a ∈ F an arc
expression of a multiset type C(p)M S , where p is the place connected to the
arc a.
– m0 : P → EXP is an initialisation function that assigns to each place p ∈ P
an initialisation expression of a multiset type C(p)M S .
If we consider special arcs, e.g. read arcs or inhibitor arcs, we can get coloured
qualitative (extended) Petri nets. If the transitions are associated with random
(or deterministic) firing rates, we will get coloured stochastic (or continuous)
Petri nets [17].
Modelling tool In Snoopy, we have implemented functionalities for edit-
ing, and animating/simulating coloured qualitative Petri nets (QP N C ), coloured
stochastic Petri nets (SP N C ) and coloured continuous Petri nets (CP N C ) [17],
[27]. In our implementation, QP N C is a coloured extension of extended qual-
itative place/transition net (extended by different types of arcs, e.g. inhibitor
arc, read arc, reset arc and equal arc [28]), SP N C is a coloured extension of
biochemically interpreted stochastic Petri nets introduced in [28] and [29], and
CP N C is a coloured extension of continuous Petri nets introduced in [28]. In
this paper, the drawing, animation and simulation of coloured Petri net models
for PCP are all done by Snoopy.
2 Modelling approach to apply CPN to PCP
We build an abstract model of PCP which only contains the key biological
aspects and other relevant information which are essential for the construction of
our models. Our study is obviously incomplete, as it does not explicitly identify
all relevant genes and molecules, but it provides a useful framework permitting
the future undertaking of further research to fulfil the understanding of PCP.
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Multi-cell modelling Using CPN 7
2.1 Abstract model of PCP
In this paper we model the dynamics of the regulatory protein network which
controls PCP at two stages of refinement regarding the details of localisation
and communication. In the first stage we represent the cell by a highly abstract
model, encoded as a (non-coloured) Petri net. The second stage model is more
refined and is encoded by a coloured Petri net. Both models describe the cytosol
as well as the proximal and distal regions of the cell.
We assume that production of key signalling proteins occurs only in the cy-
tosol and these are degraded constitutively throughout the cell. However, the
proteins are distributed asymmetrically within the cell due to an internal trans-
port network. Drosophila wing cells are approximately hexagonal and form a
regular honeycomb lattice. The core machinery which controls PCP signalling
is uniform across the Drosophila wing. Our model is an abstract description of
PCP which includes only the key structure and biological aspects of PCP in
order to establish the colour sets principles for each cell, and each compartment
within a cell. Therefore, our abstract model is an extremely simplified version
of PCP to begin with which only includes essential components and structure
and eliminates the duplication of molecular species (places) at the distal and
proximal edges of a cell. For example, Fz, Dsh, Pk and Vang exist at both edges
of a cell but asymmetrically distribute at a particular edge of the cell, Fz and
Dsh at the distal edge while Pk and Vang at the proximal edge. However, they
occur only at the particular side of a cell in our abstract model in order to obtain
a minimal simplified model which still satisfies the essential need to process the
signalling. Thus, in our model, the polarity will be arisen by this asymmetrical
distribution of proteins at the distal and proximal edges of each cell together
with the intercellular communication. However the power of coloured Petri nets
facilitates the construction of a large scale model of PCP in the wing, based on
a pattern describing a single cell communicating with its neighbours.
2.2 Simple Petri net model for a single cell
We categorise all reactions involved in each cell into two main types: (1) pro-
duction and transport of proteins; (2) transformation of proteins (reactions that
process the signal). These describe the key biological aspects of PCP and also
satisfy our requirement for a simple pattern model which can be used to establish
CPN colours for the modelling problem. We firstly sub-divide each cell into four
spatial regions: (1) the extracellular space (labelled as communication), where
the intercellular complexes form, to the (2) proximal edge (left-hand side of each
cell) in order to process intercellular signal between two adjacent neighbouring
cells, (3) transport, and (4) distal (right-hand side of each cell). As a result, one
cell contains five places (molecular species, A, B, C, D, E), three transitions
(reactions, e.g. r1) and four spatial regions (e.g. proximal in the blue text box),
see Figure 2 for details. In the model, places D lef t and E lef t indicate that
these two molecular species are from the left-hand side neighbouring cell(s).
141
8 Qian Gao, Fei Liu, David Tree, and David Gilbert
D_left B r1 A r3 D
r4
E_left C
communication proximal transport distal
Fig. 2. Petri net model for a single cell: (a) Four spatial regions of a cell: they are
labelled as communication, proximal, transport and distal; (b) Places (circle) and tran-
sitions (square): they are an abstract representation of the reactions involved in PCP.
Production and transport of proteins is represented by places and transitions labelled
in green, transformation of proteins is labelled in other colours. It should be noted that
the labelled colours here do not provide any information about CPN coloursets , they
are only used for demonstration purpose.
2.3 CPN model for pipeline of simple cells
Since PCP exhibits a high replication in terms of reactions and structure, we
can simply use the single cell model from the first step as the pattern for the
construction of our model of a pipeline of linked simple cells. Thus we create
a model which is capable of folding any number of adjacent neighbouring cells
using CPN in which a different colour is assigned for each individual cell.
We use the single cell model to start with and then assign a constant N to
generate N adjacent cells. A simple colour set named CS with N colours is
created to assign to each place, and a variable x with the type CS is assigned for
each arc except the one from place D to transition r4, which gets the expression
[x > 1] − x, read as ”if x is greater than 1, then it will return the predecessor
of x”, and meaning that the N cells are linked in a linear pipeline rather than a
circuit, see Figure 3 for details.
[x>1]−x
CS CS
x x
C CS
E r4
1‘all() D
x
CS CS
x x
x 3 x
B r1 A r3
Fig. 3. CPN model for cells linked in a pipeline. The declarations are as follows:
colourset CS = int with 1 − N , variable x: CS. The arc expression [x > 1] − x indicates
that the first cell is not linked to the last, meaning cells are linked in a linear pipeline.
142
Multi-cell modelling Using CPN 9
2.4 Refined Petri net model for a single cell
Our refined model exploits the power of CPN to describe repeated structures,
and is inspired by Noe et al. [30] who proposed the idea of compartmental ki-
netic modelling. As described in Section 1.1, the transmission of signalling mainly
occurs at the distal and proximal edges of each cell, whereas, the cytosol only in-
volves proteins production and transport. Thus, we need a central compartments
to represent the cytosol and several compartments for the distal and proximal
edge in each cell. Moreover, we does not consider the communication between
the current cell and its north and south neighbouring cells in our model. As a re-
sult, we sub-divide each biological cell into seven virtual compartments (labelled
as number 1, 2, ...7 in blue in Figure 4), three compartments each for the prox-
imal and distal membrane edges, and one compartment for the cytosol, whilst
each compartment involves all reactions in the single cell model. Here we aim
to establish the framework of applying CPN to PCP signalling – this division
of compartments is an initial approach, which will then be further developed in
a more sophisticated manner if required. Because Drosophila wing cells form a
regular honeycomb lattice there is the need to impose a hierarchical structure
over the model, which we express as a regular hexagonal array of cells, each of
which comprises seven virtual compartments, see Figure 4.
Fig. 4. Compartmentalised Drosophila wing epithelial cell in the context of a frag-
ment of the wing tissue: (1) The coordinate in each cell represents the locality of its
corresponding cell in the honeycomb lattice; (2) Each virtual compartment in a cell is
labelled by number 1 to 7, illustrated by cell (3, 2).
Next we re-construct a Petri net model for a single cell by considering the
seven virtual compartments (Figure 5). In this model, each place or transition
belongs to a specific compartment, e.g. places D and E are located in three
compartments 2, 3, 4 (labelled as vc2, vc3, vc4 in Figure 5) .
143
10 Qian Gao, Fei Liu, David Tree, and David Gilbert
D_3_NW
B_5 r1_5 r3_2 D_2
vc5 vc2
r4_5
E_3_NW
C_5
D_4_NW
r4_6
vc3
E_4_NW
vc6 B_6 r1_6 A r3_3 D_3
D_2_SW
r4_6
E_2_SW
C_6
vc4
D_3_SW
B_7 r1_7 r3_4 D_4
vc7
r4_7
E_3_SW
C_7
vc1
Fig. 5. Refined Petri net model for a single cell with seven compartments (labelled vc1,
vc2,..., vc7). Each place or transition belongs to a specific compartment, indicated by
a number given as a suffix in place or transition names. N W and SW denote two left
neighbours of the current cell.
2.5 CPN model for honeycomb lattice of refined cells
We now describe the construction of a CPN model for PCP with compartment
division, following the procedure below.
First, we code cells of PCP as colours of a colour set, i.e. representing the
locality of each cell using colours. We have chosen to model a 12-cell fragment
of the wing tissue, see Figure 4, as this will give us an adequate size over which
to explore the behaviour of our model. From the figure we can see that it is
easy to use two-dimensional coordinates e.g. (x, y) to represent the cells in the
rectangular honeycomb lattice, which can be defined by the compound colour set
product in Snoopy. For this, we define two simple colour sets Row and Column,
denoting the row and column of the lattice respectively, based on which we define
a product colour set CS1 to represent the coordinates of cells.
Second, we code the virtual compartments as colours. We do not represent
them as numbers 1,2,...,7 without considering their localisation within the cell,
but use a matrix for compartments, i.e. by using a pair of coordinates (a, b) to
denote the location of each compartment in the matrix so that we can clearly
144
Multi-cell modelling Using CPN 11
distinguish between them. It should be noted that the middle compartment
(cytosol) is represented as three rectangles in the matrix in order to conform
with the overall matrix structure, whereas, only one colour set is used for this
compartment rather than three colour sets.
Third, we create variables that are used in the guard of transitions and in
the expression of arcs connected to transitions. In six virtual compartments
for the proximal and distal edges, each arc has been assigned an expression
which includes two pairs of coordinate (x, y, a, b), meaning that the arc links the
associated place to a particular transition in (a, b) compartment of (x, y) cell. In
the middle virtual compartments, the arc expression changes to (x, y, a + 1, 2)
and (x, y, a − 1, 2) which indicates the arcs which associate place A to transition
r1 in proximal (left-hand) compartments are denoted as a + 1, while, those link
A to the transition r3 in distal (right-hand) compartments are denoted as a − 1.
Next, we represent the neighbourhood between neighbouring cells. For this,
we define two neighbour functions, N W and SW , denoting two left neighbours
of the current cell.
Finally, we generate a CPN model for PCP, illustrated in Figure 6. See Table
1 for all declarations.
CSdistal
D
NW(x,y,a,b,r4) ++ r4 (x,y,a−1,2)
(x,y,a,b) (x,y,a,b) (x,y,a+1,2) (x,y,a,b)
SW(x,y,a,b,r4)
30
B A D
NW(x,y,a,b,r4) ++ r1 r3
SW(x,y,a,b,r4) CSproximal CSmiddle CSdistal
1‘all()
(x,y,a,b)
E
C
CSdistal CSproximal
Fig. 6. CPN model describing cells with seven compartments in a 2-D matrix.
Based on what has been obtained from the above models, we will in the future
be able to build a more sophisticated model of PCP which includes all detailed
reactions according to our current understanding of the biological system. This
will facilitate our ability to better understand mechanism of PCP signalling and
provide reliable predictions to help guide the design of biological experiments
which can help to fill in gaps in our knowledge of the system.
3 Analysis
CPNs enjoy a large variety of analysis techniques, ranging from informal ani-
mation or simulation to formal structural analysis or state space analysis. As
the models constructed in this paper are still very abstract, we only use anima-
tion and stochastic simulation. The analysis reported here was performed on the
model which comprises multiple cells and a matrix representing compartments
within each cell.
145
12 Qian Gao, Fei Liu, David Tree, and David Gilbert
Table 1. Declarations for the coloured Petri net model in Figure 6.
Declaration
colourset Row = int with 1 − M ;
colourset Column = int with 1 − N ;
colourset ComR = int with 1 − R;
colourset ComC = int with 1 − C;
colourset CSr4 = enum with c5, c6 1, c6 1, c7;
colourset CS1 = product with Row × Column;
colourset CS2 = CS1 with x%2 = 1&y%2 = 0|x%2 = 0&y%2 = 1;
colourset CS = product with Row × Column × ComR × ComC;
colourset CS4 = CS3 with x%2 = 1&y%2 = 0|x%2 = 0&y%2 = 1;
colourset CSdistal = CS4 with b = 3;
colourset CSproximal = CS4 with b = 1;
colourset CSmiddle = CS4 with b = 2;
variable x : Row;
variable y : Column;
variable a : ComR;
variable b : ComC;
variable r4 : CSr4;
constant M = int with 5;
constant N = int with 5;
constant C = int with 3;
constant R = int with 3;
function CSproximal N W (Row x,Column y,ComR a,ComC b);
function CSproximal SW (Row x,Column y,ComR a,ComC b);
3.1 Animation analysis
We first performed animation analysis (i.e. at the level of the token game) over
our CPN model. Our expectation is that protein diffusion is fast. The relevant
time-scale in this context is the typical time for diffusion of a membrane protein
from one side of a cell to the opposite side which is of the order of 10 minutes. In
comparison, the asymmetric pattern of protein localisation arises on a time scale
of several hours. This is exactly what our model has shown when we manipulate
automatic animation in Snoopy. Thus, it illustrates the reliability of applying
CPN to model PCP.
3.2 Stochastic simulation analysis
We use the Gillespie stochastic simulation algorithm (SSA) [31] for the CPN
model of PCP in Figure 6. Some of the results that have been produced over an
interval of 180 per run are illustrated in Figure 8. The current understanding of
the biological system is that the production of the hair is related to the concen-
tration of several species, including actin which is believed to be responsible for
the formation of the hair itself. We wish to validate our model by demonstrat-
ing that at an abstract level actin is concentrated at the most distal part of a
146
Multi-cell modelling Using CPN 13
cell, designated as the future site for prehair formation. In our model, place E
represents actin and other key proteins, distributed at the distal edge of the cell
(virtual compartments vc2, vc3 and vc4) during signalling. Figure 7 shows how
E changes over time in the three distal compartments (vc2, vc3 and vc4) of cell
(3, 4) and Figure 8 displays the final concentration of the coloured place E in
virtual compartments vc2, vc3 and vc4 for each of the 12 cells (refer to Figure 4
for mappings). The results clearly show that the major accumulation of actin
occurs in virtual compartment vc3 for each of the cells except for cells (2,5) and
(4,5) which do not have distal neighbours and thus lack inter-cellular communi-
cation in that direction. The accumulation of actin in vc3 corresponds exactly to
the location of the prehair formation at the most distal vertex of each cell, see
Figure 1, and we find that it is highest in vc3 for cells (3,2) and (3,4) which have
the maximum number of neighbouring cells (6 each) in the honeycomb lattice.
Fig. 7. Stochastic simulation result: time course plots for the value of E within cell
(3, 4) in the three virtual compartments vc2 (3,4,1,3), vc3 (3,4,2,3) and vc4 (3,4,3,3).
Refer to Figure 4 for mappings.
Fig. 8. Stochastic simulation result: final values for E in virtual compartments vc2,
vc3 and vc4 for each of the 12 cells which are labelled by their identification tuple,
refer to Figure 4 for mappings.
147
14 Qian Gao, Fei Liu, David Tree, and David Gilbert
4 Conclusion
In this paper, we have presented our current work applying CPN techniques to
construct a PCP model in order to explore the mechanisms that drive PCP.
Our aim has been to provide a proof of principle for the use of CPN to model
a multi-cellular system with a hierarchical structure while keeping the model
mathematically tractable.
The model we have developed has allowed us to generate behaviours as a
first step to explaining the complex behaviours observed in the biological system
and to explore the implications of variations in the model. Our analysis confirms
that the behaviour of the model correctly shows the major accumulation of actin
occurring in the most distal part of the cell, corresponding to the location of the
prehair formation in wing cells of Drosophila.
However, the ability of the current model to make predictions and provide an
accurate picture of PCP signalling is limited by its lack of biological detail. In on-
going work we are refining this abstract model into a more detailed model, which
includes exploring alternative ways in which to model the cellular machinery of
PCP signalling. With this refined model we will be able not only to perform
simulations of PCP signalling in wild-type cells but also on patches of mutant
cells in a wild-type background. Our long term goal is to facilitate a better
understanding of the mechanisms that drive PCP, and to make predictions about
the behaviour of the system when it is perturbed by the mutation of specific
genetic components.
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