ion 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 paper to demonstrate what kinds of sequential analysis can be performed over a CPN model.
With the rapid growth of data being generated in the biological ¯eld, it has become necessary to organise the data into coherent models that describe system behaviour, which are subsequently used for simulation, analysis or prediction. Modelling plays a crucial role in facilitating the understanding of complex biological mechanisms from an holistic viewpoint.
A large variety of modelling approaches have already been applied to model
a wide array of biological systems (see [
Modelling across multiple scales is a current challenge in Systems Biology.
There is the need to model at di®erent spatial scales to describe, for example,
intracellular locality in compartments, organelles and vacuoles, as well as
intercellular 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
modelling complex systems, which makes it more di±cult to manage and understand
the nets, thus increasing the risk of modelling errors. Two known modelling
concepts improving this situation are hierarchy and colour. Hierarchical structuring
has been discussed a lot, e.g. in [
While there is a lot of reported work on the application of di®erent classes
of standard Petri nets to a variety of biochemical networks, see [
Building upon the lessons learnt so far, we extend our software tool Snoopy
[
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 intracellular 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 information °ow within the cell, and a representation of inter-cellular communication through the formation of protein complexes, so that local (transmembrane) signalling produces a global e®ect 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. Speci¯cally, 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 biological background of planar cell polarity, followed by Subsection 1.2 brie°y describing coloured Petri Nets and Section 2 describing our model and approaches, followed by the conclusion. 1.1
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
polarisation is required for many developmental events in both vertebrates and
non-vertebrates. Defects in PCP in vertebrates underlie developmental
abnormalities in multiple tissues including the neural tube, the kidney and the inner
ear (reviewed in [
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
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 [
(a)
(b)
Systems biology and mathematical modelling have been applied to PCP
signalling by Amonlirdviman et al. [
In this paper, we apply CPN to PCP signalling in a generic setting that encompasses a broad class of speci¯c 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 (CPN) [
CPN consist, as do standard Petri nets, of places, transitions and arcs. In systems biology, places also represent species (chemical compounds) while transitions represent any kind of chemical reactions. Each place gets assigned a colour set and contains distinguishable coloured tokens. A distribution of coloured tokens on all places together constitutes a marking of a CPN. Each transition may have a guard, which is a Boolean expression over de¯ned 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.
Before the expressions are evaluated, the variables must get values assigned with
suitable data types, which is called binding [
Thus CPN has the ability to tackle the challenges arising from modelling biological systems beyond one spatial scale, for example, repetition of components, which is the need to describe multiple cells each of which has a similar de¯nition; and organisation of components, which refers to how cells are organised into regular or irregular patterns over spatial networks in one, two or three dimensions.
In the following, we give the formal de¯nition of CPN and brie°y describe the tool for modelling CPN.
De¯nition In CPN, there are di®erent types of expressions, arc expressions,
guards and expressions for de¯ning 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 prede¯ned syntax. In the
following, we denote by EXP a set of expressions that comply with a prede¯ned
syntax. The formal de¯nition for coloured Petri nets is as follows [
N =< P; T; F; P; C; g; f; m0 >, where: { P is a ¯nite, non-empty set of places. { T is a ¯nite, non-empty set of transitions. { F is a ¯nite set of directed arcs. { P is a ¯nite, non-empty set of colour sets. { C : P ! P is a colour function that assigns to each place p 2 P a colour set C(p) 2 P. { g : T ! EXP is a guard function that assigns to each transition t 2 T a guard expression of the Boolean type. { f : F ! EXP is an arc function that assigns to each arc a 2 F an arc expression of a multiset type C(p)MS , where p is the place connected to the arc a. { m0 : P ! EXP is an initialisation function that assigns to each place p 2 P an initialisation expression of a multiset type C(p)MS .
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) ¯ring rates, we will get coloured stochastic (or continuous)
Petri nets [
Modelling tool In Snoopy, we have implemented functionalities for
editing, 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 ) [
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 ful¯l the understanding of PCP. In this paper we model the dynamics of the regulatory protein network which controls PCP at two stages of re¯nement regarding the details of localisation and communication. In the ¯rst stage we represent the cell by a highly abstract model, encoded as a (non-coloured) Petri net. The second stage model is more re¯ned 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 cytosol and these are degraded constitutively throughout the cell. However, the proteins are distributed asymmetrically within the cell due to an internal transport 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 simpli¯ed 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 simpli¯ed model which still satis¯es 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
We categorise all reactions involved in each cell into two main types: (1) production 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 ¯rstly 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 ¯ve 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).
D_left E_left r4
B C r1
A r3
D communication proximal transport distal
Since PCP exhibits a high replication in terms of reactions and structure, we can simply use the single cell model from the ¯rst 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 di®erent 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.
CS E x r4 x x
[x>1]−x CS CS
B
C x r1 1‘all()
CS x 3 x
A
CS x D r3
Our re¯ned model exploits the power of CPN to describe repeated structures,
and is inspired by Noe et al. [
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 speci¯c compartment, e.g. places D and E are located in three compartments 2, 3, 4 (labelled as vc2, vc3, vc4 in Figure 5) .
D_2 r4_5 r4_6 r4_6 r4_7
C_5 C_6 C_7 B_6 r1_6
A r3_3
D_3 B_7 r1_7 r3_4
D_4 vc1 vc2 vc3 vc4
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 ¯gure 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 de¯ned by the compound colour set product in Snoopy. For this, we de¯ne two simple colour sets Row and Column, denoting the row and column of the lattice respectively, based on which we de¯ne 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 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 de¯ne 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.
D E CSdistal NW(x,y,a,b,r4) ++ SW(x,y,a,b,r4) NW(x,y,a,b,r4) ++ SW(x,y,a,b,r4) CSdistal r4 (x,y,a,b) (x,y,a,b) (x,y,a+1,2) (x,y,a−1,2)
(x,y,a,b)
B CSproximal
r1 (x,y,a,b)
C CSproximal 30 A CSmiddle 1‘all() r3
D CSdistal
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 ¯ll in gaps in our knowledge of the system. 3
CPNs enjoy a large variety of analysis techniques, ranging from informal animation or simulation to formal structural analysis or state space analysis. As the models constructed in this paper are still very abstract, we only use animation and stochastic simulation. The analysis reported here was performed on the model which comprises multiple cells and a matrix representing compartments within each cell. We ¯rst performed animation analysis (i.e. at the level of the token game) over our CPN model. Our expectation is that protein di®usion is fast. The relevant time-scale in this context is the typical time for di®usion 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
We use the Gillespie stochastic simulation algorithm (SSA) [
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 ¯rst step to explaining the complex behaviours observed in the biological system and to explore the implications of variations in the model. Our analysis con¯rms 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 ongoing work we are re¯ning 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 re¯ned 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 speci¯c genetic components.