=Paper=
{{Paper
|id=Vol-1373/paper3
|storemode=property
|title=A Colored Petri net approach for spatial Biomodel Engineering based
on the modular model composition framework Biomodelkit
|pdfUrl=https://ceur-ws.org/Vol-1373/paper3.pdf
|volume=Vol-1373
|dblpUrl=https://dblp.org/rec/conf/apn/BlatkeR15
}}
==A Colored Petri net approach for spatial Biomodel Engineering based
on the modular model composition framework Biomodelkit==
https://ceur-ws.org/Vol-1373/paper3.pdf
A coloured Petri net approach for spatial
Biomodel Engineering based on the modular
model composition framework Biomodelkit
Mary Ann Blätke1ú and Christian Rohr2ú
1
Otto-von-Guericke University Magdeburg,
Chair of Regulatory Biology
Universitätsplatz 2, D-39106 Magdeburg, Germany
mary-ann.blaetke@ovgu.de
http://www.regulationsbiologie.de/
2
Brandenburg University of Technology Cottbus-Senftenberg,
Chair of Data Structures and Software Dependability,
Postbox 10 13 44, D-03013 Cottbus, Germany,
christian.rohr@b-tu.de,
http://www-dssz.informatik.tu-cottbus.de
Abstract. Systems and synthetic biology require multiscale biomodel
engineering approaches to integrate diverse spatial and temporal scales
in order to understand and describe the various interactions in biological
systems. Our BioModelKit framework for modular biomodel engineering
allows to compose multiscale models from a set of modules, each describ-
ing an individual molecular component in the form of a Petri net. In this
framework, we do now propose a feature for spatial modelling of molec-
ular biosystems. Our spatial modelling methodology allows to represent
the local positioning and movement of individual molecular components
represented as modules. In the spatial model, interactions between com-
ponents are restricted by their local positions. In this context, we use
coloured Petri nets to scale the modular composed spatial model, such
that each molecular component can exist in an arbitrary number of in-
stances. Thus, a modular composed spatial model can be mapped to the
cellular arrangement and different cell geometries.
Keywords: Modular Model Composition, Spatial Modelling, Multiscale
Biomodel Engineering, Coloured Petri nets
1 Introduction
Systems biology aims at describing and understanding complex biological pro-
cesses on a systems level. Therefore, systems biology employs modelling and
simulation as indispensable tools to describe, predict and understand biological
systems in an integrative and quantitative context. Besides complex interactions,
ú
Corresponding authors
M Heiner, AK Wagler (Eds.): BioPPN 2015, a satellite event of PETRI NETS 2015,
CEUR Workshop Proceedings Vol. 1373, 2015.
38 MA Blätke, C Rohr
models do also need to integrate diverse temporal and spatial scales spanning
the biological systems. Multiscale biomodel engineering goes beyond standard
modelling approaches in systems biology and addresses physical problems as im-
portant features at multiple scales in time and space [1]. Current challenges and
methodologies used so far in multiscale biomodel engineering have been reviewed
in [2] and [3].
Here, we focus on the spatial aspects in multiscale biomodel engineering,
which have been mostly neglected in the description of intracellular processes
until now [1]. In particular, we demonstrate, based on the BioModelKit frame-
work for modular biomodel engineering [4], how to extend plain models of intra-
cellular processes to spatial models without their reimplementation. The models
in our case are composed from modules, where each module describes the func-
tionality of a certain molecular component in the form of a Petri net. The use
of coloured Petri nets in our approach allows to represent different numbers of
module instances for each component. To our knowledge, the methodology for
spatial modelling in the context of modular model composition, which we suggest
in this paper, is unique.
As modelling tool, we chose Snoopy [5], because it supports low-level and
coloured Petri net network classes, as well as the concept of logical (fusion)
nodes and hierarchical modelling.
In the next section, we will shortly describe the BioModelKit framework and
summarize the use of coloured Petri nets for multiscale modelling. Afterwards,
in Section 3, we introduce our spatial modelling methodology as a new feature
of the BioModelKit framework. Section 4 applies the introduced methodology
for spatial modelling to a simple example of a molecular interaction between two
proteins represented as modules. In the last section, we give a short summary
and outlook.
2 Previous Work
2.1 BioModelKit Framework for Modular Model Composition
The BioModelKit framework (BMK framework) is a tool for modular biomodel
engineering [4], see Fig. 1. The main motivation behind BMK framework was
to develop a modelling environment, where modules are specifically designed for
the purpose of model composition. The modularisation approach used in BMK
framework was inspired by the natural composition of biomolecular systems,
where molecular components (genes, mRNAs and proteins) are the main build-
ing blocks. Thus, each molecular component is represented as a self-contained
module, describing the underlying functionality using the formal language of
Petri nets. Interface networks, which are part of each module describe the inter-
actions with other molecular components and are used to automatically couple
respective modules [4].
Since, the functionality of genes, mRNAs and proteins is diverse, different
module types have been defined in BMK framework, as well as allelic influence
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 39
Model-based
Biomolecular Systems
Predictions
...
Molecular Experimental Boolean SBML
Mechanisms Data Networks Models
Forward Reverse
Transformation Transformation
Engineering Engineering
High-Throughput
Analysis
Modules
Gene mRNA Protein Protein Causal Allelic
Modules Modules Modules Degrdation Influence Influence
Modules Modules Modules
Model Composition
Upload
Wildtype/
Add Modules to Collection
+ Algorithmic Mutation Alternative
Biomodelkit Set of
Database Modules Models
+ Space-Attributes Spatial
Models
Fig. 1: Overview of the BioModelKit Framework for Modular Model Composi-
tion.
modules and causal influence modules to capture also correlations with missing
mechanistic descriptions [6,7]. Modules can be generated by forward and reverse
engineering approaches or by transforming boolean models or models provided
in the systems biology markup language (SBML) into modules [8,9], see Fig. 1.
The web-interface of BMK framework (www.biomodelkit.com [4]) includes a
feature to submit modules and to create a model annotation file in the BMK
markup language (BMKml, unpublished work). The submitted module and its
annotation have to be curated by an administrator before publicly releasing them
by storing their content in a relational MySQL database (BMKdb). Another
feature of the BMK framework is a model composition algorithm, which allows
to automatically compose comprehensive models from a set of chosen modules.
In addition, the composed model can also be modified by applying algorithms
mimicking single/double gene knock-outs or structural mutations of the included
molecular components (unpublished work).
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
40 MA Blätke, C Rohr
2.2 Coloured Petri Nets
We use Petri nets (PN ) as modelling paradigm, which gives us a complete
formalised and standardised framework, as well as an intuitive way of modelling
concurrent behaviour.
In systems biology, as well as in other fields, it’s quite common that parts of
larger models have similar structures. In such a case simplifying the model via
reusing that part, instead of having redundant structures is demanded. Coloured
Petri nets (PN C ) are a modelling paradigm that fits well in such a case.
We are using Snoopy [10] as modelling and simulation tool, thus we describe
PN C how they are defined there.
coloursets:
enum species := red, green, blue
product complex := species, species
variables:
species x
species y
species 2‘red++
1‘green species 2‘red
x x
A
A
complex
(x,y) complex
(x,y)
1‘(green,blue)
y [x<>y] AB [x<>y] AB
y
species species
2‘green++
2‘green
B 1‘blue B
(a) Before Firing (b) After Firing
Fig. 2: Example PN C of abstract complex formation.
We use the coloured Petri net in Fig. 2 as an example. It represents an ab-
stract complex formation of two species of different kind into one complex. The
model contains two coloursets, first a simple colourset named species of type
enum, including the colours red, green and blue. Second a product colourset
named complex, its colours are 2-tuples of the species colourset. The net consists
of three places A, B and AB and one transition. The colourset species is asso-
ciated with the places A and B and the place AB has colourset complex. The
variables x and y are used in the arc inscriptions and the transition guard. The
transition guard x <> y determines that only tokens of different colour are valid
bindings for the variables x and y. The arc inscriptions x and y on the incoming
We summarize the following net classes together under the term Petri net (PN ):
Qualitative Petri net (QPN ), eXtended Petri net (X PN ), Continuous Petri net
(CPN ), Stochastic Petri net (SPN ) and Hybrid Petri net (HPN ). The same goes
for the coloured Petri nets (PN C ).
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 41
arcs of the transition define its precondition, i.e. there have to be at least one
token on place A and one token on place B and they have to be of different
colour due to the guard. The arc inscription (x, y) on the outgoing arc of the
transition defines the production of one complex token. In Fig. 2a the place A
has two tokens of colour red and one token of colour green and the place B has
two tokens of colour green and one token of colour blue. This gives the following
bindings for the variables x and y: (red, green), (red, blue), (green, blue). We se-
lected the binding (green, blue) and let the transition fire. One green token from
place A and one blue token from place B are consumed and one (green, blue)
token is produced on place AB, see Fig. 2b.
Much more extensive descriptions how to use coloured Petri nets in systems
biology are given in [11,12,13]. Besides the animation of the coloured Petri net,
it is possible to unfold every PN C into an uncoloured PN [11]. So it is possible
to apply any analysis and simulation technique available for uncoloured Petri
nets on coloured Petri nets too.
Up to this, modelling biochemical systems using coloured Petri nets did not
incorporate spatial aspects or movement in space. But this can be included in
the model as shown by Gilbert et al. [14]. Therefore the space is discretised into
a grid of one, two or three dimensions and a position in the grid is represented
by a single place. This works fine if there is no need to distinguish between the
entities on each position. One can model the diffusion of substances using this
approach quite well, as presented in [14].
This can be extended to more complex reaction-diffusion systems, as shown
in [15]. More examples of using coloured Petri nets for modelling of biological
systems including spatial aspects are [16,17].
All models above have in common that they model space by discretisation
into a grid and having one subnet (ranging from a single place to a complex
network) per grid position. This is handy, if the entities moving around have no
internal behaviour or state and there is no need to distinguish them. But if that
is the case, the internal network has to move around as well and this leads to
some issues on modelling and simulation. Parvu et al. [17] used this approach
for a model of phase variation in bacterial colony growth. The bacteria have
two different states, i.e. two places A and B representing the two states and
two transitions for changing the state are needed. In order to let the bacteria
move around, the whole subnet is needed in every grid position. Incorporating
this in the coloured model is straightforward, but the size of the unfolded model
increases drastically. This has an impact on the analysis and simulation of the
model and may lead to inconvenient run times.
While the approaches of representing space via discretisation into grid-places
fit well in the shown cases, it is not practical in our use case, because we have
complex subnets moving around. We present our approach of incorporating space
by adding coordinate places in the following section.
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
42 MA Blätke, C Rohr
3 Spatial Modelling Methodology for Modular Composed
Models
Before we start with the formal description of the spatial transformation al-
gorithm, we have to introduce some general definitions which apply to our
modular modelling approach. A module Mi is defined by a quintuple Mi =
(Pi , Ti , fi , vi , m0,i ) according to the general definition of quantitative Petri nets.
Each module Mi consists of nci + 1 components, the main component C0 and
nci interacting components. Thus, each module Mi represents a set of compo-
nents Ci = {C0 , C1 , . . . , Cnci }. The mapping of a place pij œ Pi of a module
pi
Mi to a set of components Ci j ™ Ci is given by the relation g : Pi æ Ci ,
pi
such that g(pij ) = Ci j . A place pij with |g(pij )| > 1 represents a complex
pi
of |g(pij )| components. The set Ki = {Ci j | |g(pij )| > 1} contains all com-
plexes among the components in Ci . A transition tij œ Ti of a module Mi with
|g(•tij ) fl g(tij •)| > 1 represents an interaction with at least two different involved
components. The total set of all interacting transitions in a Module Mi is given
by TiIA = {’tij œ Ti : |g(•tij ) fl g(tij •)| > 1}.
A set of modules defines a modular composed model M = {M1 , . . . , Mn },
where n is the number of modules. Consequently, the modular composed model
can also be defined as M = (P M , T M , f M , v M , mM 0 ) according to the general
definition of quantitative Petri nets with the following relations:
t
– P M = t Pi , where Mi œ M - total finite and non-empty set of places.
– T M = t Ti , where Mi œ M - total finite and non-empty set of transitions.
– f M = fi , where Mi œ M - total set of directed arcs, weighted by a non-
negative t integer value.
– v M :t Ti æ H, where Mi œ M - total set of firing rates.
– mM 0 Pi æ N0 , ÷pikÕ , pjkÕÕ with pikÕ œ Pi , pjkÕÕ œ Pj , where
pkÕ = pjkÕÕ , {pikÕ : pjkÕÕ } æ pM
i
k œP
M
, mM
0 (pk ) = max(m0,i (pkÕ ), m0,j (pkÕÕ ))
M i j
In addition to the definitions above, the following relations can be derived:
t
– CM = t Ci ,twhere Mi œ M - total set of components
– g M : Pi æ tCi , where Mi œ M - total set of place component relations
– TIA
M
™ TtM = TiIA , where Mi œ M - total set of all interacting transitions
– K M = Ki , where Mi œ M - total set of complexes.
Spatial Transformation Algorithm For the spatial transformation of the
flat modular composed model the following procedure needs to be executed.
Step 1: Explicit Encoding of Local Positions. The position of each component
Ci œ C M is explicitly encoded by d places pC
1 , . . . pd (termed coordinate places),
i Ci
which can be interpreted as coordinates, where d, d Ø 1, defines the number of
axes (e.g. 1D, 2D or 3D grid). The marking m(pC j ) of a place pj defines the
i Ci
current coordinate value, which must be restricted by a lower mL (pC j ) and
i
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 43
upper bound mU (pC j ) to represent the boundaries of the encoded grid, such
i
that, mL/U (pj ) > 0 and mL (pC
Ci
j ) < mU (pj ).
i Ci
Step 2: Local Restriction of Interactions. To restrict the executability of each
transition t œ TIAM
, the firing rate h(t) must be multiplied by a boolean relation
b(t) describing a defined neighbourhood relation: hIA (t) = b(t) ú h(t), t œ TIA M
.
If the neighbourhood relation claims that the distance between components in-
volved in the interaction represented by a transition t œ TIA M
must be zero, b(t)
has to I
be defined as follows:
q|g(•t)fig(t•)|≠1 q|g(•tfig(t•)| qd Cj 2
1, k=1 (m(pk ) ≠ m(pk )) = 0
Ci
b(t) = qi=1 qj=i+1
q Cj 2
0, k=1 (m(pk ) ≠ m(pk )) ”= 0
|g(•t)fig(t•)|≠1 |g(•t)fig(t•)| d Ci
i=1 j=i+1
In addition, read edges, connecting each transition t œ TIA M
and the coordinate
places of the respective components have to be added, such that
fReadEdge (p1≠d , t) = 1.
g(•t)fig(t•)
Step 3: Explicit Encoding of Local Position Changes. To encode the position
changes for a component Ci œ C M two different scenarios have to be considered
dependent on the state of interaction:
1. Local position change of individual components:
For each component Ci œ CM and each coordinate place pC Ci Ci
j œ {p1 , . . . , pd }
i
two transitions tj,L and tj,U are needed to incrementally decrease or increase
Ci Ci
the amount of tokens. The transition tC j,L subtracts tokens from the coordi-
i
nate place pj till m(pj ) = mL (pj ). Therefore, the following edges have
Ci Ci Ci
to be introduced f M (pC j , tj,L ) = 1 and fReadEdge (pj , tj,L ) = mL (pj ) + 1.
i Ci M Ci Ci Ci
The transition tj,U adds tokens to the coordinate place pj till m(pC
Ci Ci
j ) =
i
mU (pCj ). Therefore, the following edges have to be introduced
i
f (tj,U , pC
M Ci
j ) = 1 and fInhibitorEdge (pj , tj,U ) = mU (pj ). To ensure that
i M Ci Ci Ci
the position of the component Ci can only be changed if it does not in-
teract with another component Cj , i ”= j, additional inhibitory edges for
each transition tCj,L/U have to be introduced: fInhibitorEdge (PIS , tj,L/U ) = 1,
i Ci Ci
where PIS Ci
™ P M and PIS Ci
= {’p œ P M : Ci œ g(p) · |g(p)| > 1}.
2. Local position change of complexes:
The local position of components forming a complex ki œ K M have to be
updated consistently during the local position change. For each complex
ki œ K M and each dimension j, 1 Æ j Æ d, two transitions tkj,L i
and tkj,U
i
are
needed to incrementally decrease or increase the amount of t tokens. The tran-
sition tkj,L
i
removes tokens from the set of coordinate places Ch œki pC j till at
h
least for one component Ch œ C the condition m(pj ) = mL (pj ) is ful-
M Ch Ch
filled. Therefore, for each component Ch œ ki the following edges have to be
introduced f M (pC j , tj,L ) = 1 and fReadEdge (pj , tj,L ) = mL (pj ) + 1. The
h ki M C h ki Ch
t
transition tj,U adds tokens to the set of coordinate places Ch œki pC
ki
j
h
till at
least for one component Ch œ C the condition m(pj ) = mU (pj ) is ful-
M Ch Ch
filled. Therefore, for each component Ch œ ki the following edges have to be
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
44 MA Blätke, C Rohr
introduced f M (tkj,U
i
j ) = 1 and fInhibitorEdge (pj , tj,U ) = mU (pj ). To
, pC h M Ch ki Ch
ensure that the position of the complex ki can only be changed if it is actually
formed additional read edges for each transition tkj,L/U
i
have to be introduced:
fReadEdge (PIS , tj,L/U )
ki ki
= 1, where ki
PIS ™ PM and
ki
PIS = {’p œ P M | ki = g(p)}. Furthermore, it must be excluded for each
component Ch œ ki that interacts with other components using a different
binding site. Therefore, all co-existing interactions have to be determined
ki
Kcoex = {’kj œ K M : ki fl kj ”= ? | kj ”= ki }. All places representing a
complex kj œ Kcoex
ki
have to be added to each transition tkj,L/U
i
using an
inhibitory edge: fInhibitoryEdge (PCOEX
ki
, tkj,L/U
i
) = 1, where PCOEX
ki
= {’p œ
P : g(p) = k, k œ Kcoex }.
M ki
To allow the movement of co-existing complexes which use different interac-
tion sites simultaneously, the above described procedure has to be applied
to all possible combination of co-existing complexes, compare Section 4.
For simplicity reasons the firing rate of each transition tkj,L/U
i
and tC h
j,L/U
is given by Fick’s laws of diffusion [18]. Furthermore, we assume equidistant
subvolumes with the width and hight h = 1 and set all diffusion coefficient to
one. Please note, it is straightforward to define the diffusion coefficients more
precisely based on experimental results.
Step 4: Encoding of Component Instances by Applying Coloured Petri nets. A
colourset ‡C
simple
i
with 1 ≠ qCi colours needs to be specified for each component
Ci œ C , where qCi œ N defines the number of instances for a component Ci .
M
The total set of simple coloursets is given by À simple = {‡Csimple
1
simple
, . . . , ‡|C M | }.
For each ‡C simple
i
œ À simple a variable aCi needs to be specified. All edges f (p, t)
and f (t, p) of the flat model M for which it is true, that a component Ci œ
CM , where Ci œ g(p) and |g(p)| = 1 are extended to the multiset expression
aCi ‘f (p, t), or respectively aCi ‘f (t, p). The total set of simple coloursets À simple
is mapped to a subset of places according to the relation S simple : À simple æ
P M , such that S simple (‡C simple
i
) = {p œ P M | Ci œ g(p) · |g(p)| = 1}.
Each complex ki œ K , where ki represents a subset of components, such
M
that ki ™ C r , is represented
M
by a compound colourset of type product
compound
‡ ki = Cj œki ‡C simple
j
. The total set of compound coloursets is given by
À compound = {‡kcompound
1
compound
, . . . , ‡|K M| }. The total set of compound coloursets
À compound is mapped to remaining subset of places according to the relation
S compound : À compound æ P M, such that
S (‡ki
compound compound
) = {p œ P | ki = g(p)}. All edges f (p, t) and f (t, p) of
M
the flat model M for which its is true, that a t complex ki œ K M , where ki = g(p)
are extended to the multiset expression Cj œki aCj ‘f (p, t), or respectively
t
Cj œki a ‘f (p, t). The marking and firing rates are kept constant over all place
Cj
and transition instances, such that marking of each place p œ P M is represented
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 45
by all()‘m0 (p) and the firing rates of each transition t œ T M all()‘h(t), where
all() is a function that extracts all instances of a coloured node.
4 Example
For demonstration purposes we introduce in Fig. 3 a running example of a mod-
ular composed model, which consists of two modules, a module for Protein A
and a module for Protein B. The module of Protein A describes the complex for-
mation and cleavage between the two ligand binding domains of Protein A and
Protein B (P roteinA_LBD, P roteinB_LBD). The formation of the complex be-
tween Protein A and Protein B (P roteinA_LBD__P roteinB_LBD) is the trig-
ger for the phosphorylation of a tyrosine residue at Protein B (P roteinB_T Y R,
P roteinB_T Y Rp). To phosphorylate Protein B, the catalytic domain of Protein
A needs to be in an active state (P roteinA_CD_active). The catalytic domain
of Protein A can switch between being active or inactive (P roteinA_CD_active,
P roteinA_CD_inactive). In the module of Protein B, the subnet describing the
interaction between Protein A and Protein B is redundant. Redundant subnets
are called interface networks (indicated by logical (fusion) places and transitions
shaded in grey), and are used to automatically couple modules. An additional
subnet in the module of Protein B explains the complex formation and cleavage
between the phosphorylated Tyrosine of Protein B (P roteinB_T Y Rp) and a SH2
domain of Protein C (P roteinC_SH2). The Module of Protein C is not given
in this example. Since this paper is not dealing with kinetic aspects, we assume
mass action kinetics and set all kinetic coefficient to one. It is straightforward
to replace this assumption with more detailed kinetic descriptions.
Module of Protein A Module of Protein B
ProteinA LBD ProteinA LBD
ProteinB LBD ProteinB LBD
AB t1 AB t2 AB t1 AB t2
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
BC t1
AB t3 AB t3
ProteinB TYRp ProteinB TYR ProteinB TYRp ProteinC SH2 ProteinC SH2 ProteinB TYR
ProteinB TYRp
ProteinA CD active ProteinA CD active
BC t2
A t2 A t1
ProteinA CD inactive
Fig. 3: Running example with two protein modules.
The module of Protein A (M1 ) contains two components
C1 = { P roteinA, P roteinB } and the module of Protein B (M2 ) contains three
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
46 MA Blätke, C Rohr
components C2 = {P roteinA, P roteinB, P roteinC }. For the composed Model
M = {M1 , M2 }, we get the following mapping of places to the components:
– g M ({P roteinA_LBD, P roteinA_CD_active, P roteinA_CD_inactive})
=P roteinA
– g M ({P roteinB_LBD, P roteinB_T Y R, P roteinB_T Y Rp}) =P roteinB
– g M ({P roteinB_SH2}) =P roteinC
– g M ({P roteinA_LBD__P roteinB_LBD}) = {P roteinA, P roteinB }
– g M ({P roteinB_T Y Rp__P roteinC_SH2}) = {P roteinB, P roteinC }
Furthermore places P roteinA_LBD__P roteinB_LBD and
P roteinB_T Y Rp__P roteinC_SH2 represent two complexes
k1 = {P roteinA, P roteinB } and k2 = {P roteinB, P roteinC }. The set of inter-
acting transitions is given by TIA
M
= {AB_r1, AB_r2, AB_r3, BC_r1, BC_r2}.
For the spatial model we assume a two dimensional grid (d = 2) of the size
5 ◊ 5 for each component given by the constants xDimA = xDimB = xDimC = 5
and yDimA = yDimB = yDimC = 5.
Step 1 of the spatial transformation algorithm introduces two coordinate
places representing the x- and y-coordinate of each component, e.g. for com-
ponent P roteinA we add two places P roteinA_X and P roteinA_Y , see Fig. 4.
We chose the marking of the places representing the local position according to
the following assumption: component P roteinA is initially positioned at (1,1),
component P roteinB at (3,3) and component P roteinC at (4,4).
XY-Position of Protein A XY-Position of Protein B XY-Position of Protein C
4 4
ProteinA X ProteinA Y ProteinB X ProteinB Y ProteinC X ProteinC Y
Fig. 4: Encoding of the local positions for each component in the composed
modular model.
Step 2 of the spatial transformation algorithm restricts the interaction of
components dependent on their local position. The restriction applies only to
transitions in the set TIAM
. We assume that components can only interact, if
their local positions are identical, meaning the distance between them must be
zero. Therefore, the firing rates of transitions AB_t1, AB_t2 and AB_t3 must
be multiplied
I with the boolean expression bAB (t):
1, distAB = 0
bAB (t) = with
0, distAB ”= 0
distAB = (m(P roteinA_X)≠m(P roteinB_X))2 +(m(P roteinA_Y )≠m(P roteinB_Y ))2
The places representing the local positions of component P roteinA and compo-
nent P roteinB have to be connected by read edges to the transitions AB_t1,
AB_t2 and AB_t3, compare Fig. 5. Accordingly, the firing rates of transitions
BC_t1 and BC_t2 must be multiplied with the boolean expression bBC (t):
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 47
I
1, distBC = 0
bAB (t) = with
0, distBC ”= 0
distBC = (m(P roteinB_X)≠m(P roteinC_X))2 +(m(P roteinB_Y )≠m(P roteinC_Y ))2 .
The places representing the local positions of component P roteinB and com-
ponent P roteinC have to be connected by read edges to the transitions BC_t1
and BC_t2, compare Fig. 5.
Restricted Interaction - Only if ProteinA (X,Y) = ProteinB (X,Y)
ProteinA X ProteinB X ProteinA X ProteinB X ProteinA X ProteinB X
AB t1 AB t2 AB t3
ProteinA Y ProteinB Y ProteinA Y ProteinB Y ProteinA Y ProteinB Y
Restricted Interaction - Only if ProteinB (X,Y) = ProteinC (X,Y)
ProteinB X 4 ProteinC X ProteinB X 4 ProteinC X
BC t1 BC t2
ProteinB Y 4 ProteinC Y ProteinB Y 4 ProteinC Y
Fig. 5: Restriction of interactions depending on the local position of the involved
components.
Step 3 of the spatial transformation algorithm encodes the local position
change in respect to the interaction state of the components. In our example we
assume only movement along the horizontal and vertical axes. For each axis
two transitions are needed to either increase or decrease the marking value
of the respective coordinate place of a component with respect to the grid
size. A component can only move as a single entity if it is not interacting at
the same time with other components. Therefore we have to check if the cor-
responding places representing the interaction states (complexes) are empty,
e.g. in case of component P roteinB it can only move as single entity if places
P roteinA_LBD__P roteinB_LBD and P roteinB_T Y Rp__P roteinC_SH2
are empty, see Figure 6(A). To move components forming a complex or which
build co-existing complexes, the coordinates of all involved components have
to be updated simultaneously, see Figure 6(B). Form the definitions above,
we know that component P roteinB can form a complex with P roteinA and
P roteinC . Thus, these two complexes can co-exist because component P roteinB
uses different interaction sites. To move the complex of component P roteinA
and component P roteinB we need to check whether the corresponding place
P roteinA_LBD__P roteinB_LBD is marked and if the place
P roteinB_T Y Rp__P roteinC_SH2 is empty. In contrast, to move the complex
of component P roteinB and component P roteinC we need to check whether the
place P roteinB_T Y Rp__P roteinC_SH2 is marked and if place
P roteinA_LBD__P roteinB_LBD is empty. And to move the co-existing com-
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
48 MA Blätke, C Rohr
plex of component P roteinA, component P roteinB and component P roteinC , we
need to check if both places P roteinA_LBD__P roteinB_LBD and
P roteinB_T Y Rp__P roteinC_SH2 are marked.
Step 4 of the spatial transformation algorithm has to be applied to represent
more than one instance of each component, compare Fig. 7 and 8. In our example
the number of instances for each component is three, which we represent by the
constants numA = numB = numC = 3. For each component Ci œ CM we define
a simple colourset:
– int csProteinA := 1 - numA,
– int csProteinB := 1 - numB,
– int csProteinC := 1 - numC
where colourset csP roteinA is mapped to the places with the relation
g(p) =P roteinA, colourset csP roteinB is mapped to the places with the rela-
tion g(p) =P roteinB and colourset csP roteinC is mapped to the places which
fulfil relation g(p) =P roteinC . The coordinate places of each component have to
be bound to the respective colourset as well. Furthermore, we need to define a
variable for each simple colourset:
– csProteinA A
– csProteinB B
– csProteinC C
All in-going and out-going arcs of places bound to one of the simple coloursets de-
fined above have to carry the respective variable as arc expression. Since, we have
two binary complexes k1 = {P roteinA, P roteinB } and k2 = {P roteinB, P roteinC },
we need to define a compound colourset for each as product of the respective
simple coloursets:
– product csProteinA_ProteinB := csProteinA, csProteinB
– product csProteinB_ProteinC := csProteinB, csProteinC
where colourset csP roteinA_P roteinB is mapped to the places with the relation
g(p) = {P roteinA, P roteinB } and colourset csP roteinB_P roteinC is mapped to
the places with the relation g(p) = {P roteinB, P roteinC }. All in-going and out-
going arcs of places bound to one of the compound coloursets defined above have
to carry a 2-tuple of respective variables as arc expression.
Fig. 9 presents one exemplifying stochastic simulation run of the final spatial
model of Fig. 7 and 8. In Fig. 9(A), we depict the movement of all instances of
components P roteinA, P roteinB and P roteinC on separate two-dimensional grids
of the size 5◊5. During the simulation time three complexes between instances of
component P roteinA and component P roteinB could be obtained (P roteinA_1 +
P roteinB_1,P roteinA_1+P roteinB_2,P roteinA_3+P roteinB_3), as well as two
complexes between instances of component P roteinB and component P roteinC
(P roteinB_1 + P roteinC_1,P roteinB_3 + P roteinC_3). The simulation result
also shows that the distance between the corresponding instances of compo-
nents forming a complex is zero, which means that they can only move has one
entity. Subfigure (1) and (2) of Fig. 9(B) shows that the instance P roteinB1 is
interacting with instance P roteinA1 and instance P roteinC1 at the same time
near the end of the simulation run, thus the two complex states co-exist.
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 49
(A)
Movement of Protein A Movement of Protein B Movement of Protein C
ProteinC X
ProteinA X ProteinB X
XL A XR A XL B XR B XL A 4 XR A
2 xDimA 2 xDimB 2 xDimC
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD ProteinB TYRp ProteinC SH2
ProteinB TYRp ProteinC SH2
ProteinC Y
ProteinA Y ProteinB Y
YD A YU A YL B YD A 4 YU A
YD B
2 yDimA 2 yDimB 2 yDimC
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
ProteinB TYRp ProteinC SH2
ProteinB TYRp ProteinC SH2
(B)
Movement of complex ProteinA ProteinB Movement of complex ProteinB ProteinC Movement of complex ProteinA ProteinB ProteinC
2 xDimA
ProteinA X
ProteinB X ProteinB X ProteinB X
2 xDimB xDimB xDimB
2
ProteinA X ProteinC X ProteinC X
XL AB XR AB XL BC 4 XR BC XL ABC 4 XR ABC
2 xDimA 2 xDimC 2 xDimC
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2
2 yDimA
ProteinA Y
ProteinB Y ProteinB Y ProteinB Y
2 yDimB yDimB yDimB
2
ProteinA Y ProteinC Y ProteinC Y
YL AB YD AB YD BC 4 YU BC YD ABC 4 YU ABC
2 yDimA 2 yDimC 2 yDimC
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2
Fig. 6: Local position change dependent on the state of interaction. (A) compo-
nents are not in complex, (B) components are in complex with each other.
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
50 MA Blätke, C Rohr
Module of Protein A Module of Protein B
csProteinA
csProteinA 1‘all()
ProteinA LBD 1‘all() ProteinA LBD
A A A A
csProteinB ProteinB LBD 1‘all() csProteinB ProteinB LBD 1‘all()
AB t1 AB t2 AB t1 AB t2
B B B B
(A,B) (A,B) (A,B)
(A,B)
csProteinA ProteinB csProteinA ProteinB
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
BC t1
(B,C) B
AB t3 (A,B) C csProteinC AB t3 (A,B) csProteinB
csProteinB csProteinB csProteinB ProteinC
ProteinB TYRp ProteinB TYR 1‘all() ProteinB TYRp ProteinC SH2 ProteinC SH2 ProteinB TYR 1‘all()
B B 1‘all() csProteinBB B
A C B ProteinB TYRp A
csProteinA (B,C) csProteinA
ProteinA CD active ProteinA CD active
A BC t2
A
A t2 A t1
A A
csProteinA
ProteinA CD inactive 1‘all()
XY-Position of Protein A XY-Position of Protein B XY-Position of Protein C
9 9 12 12
csProteinA csProteinA csProteinB csProteinB csProteinC csProteinC
ProteinA X ProteinA Y ProteinB X ProteinB Y ProteinC X ProteinC Y
1‘all() 1‘all() 3‘all() 3‘all() 4‘all() 4‘all()
Restricted Interaction - Only if ProteinA (X,Y) = ProteinB (X,Y)
csProteinA csProteinB csProteinA csProteinB csProteinA csProteinB
ProteinA X 9 ProteinB X ProteinA X 9 ProteinB X ProteinA X 9 ProteinB X
1‘all() AB t1 3‘all() 1‘all() AB t2 3‘all() 1‘all() AB t3 3‘all()
A B A B A B
A csProteinA A csProteinA A csProteinB
csProteinA B csProteinB B csProteinB B
ProteinA Y 9 ProteinB Y ProteinA Y 9 ProteinB Y ProteinA Y 9 ProteinB Y
1‘all() 3‘all() 1‘all() 3‘all() 1‘all() 3‘all()
Restricted Interaction - Only if ProteinB (X,Y) = ProteinC (X,Y)
csProteinB csProteinC csProteinB csProteinC
ProteinB X 9 12 ProteinC X ProteinB X 9 12 ProteinC X
3‘all() BC t1 4‘all() BC t2 4‘all()
B C 3‘all() B C
B C csProteinB B C
csProteinB csProteinC csProteinC
ProteinB Y 9 12 ProteinC Y ProteinB Y 9 12 ProteinC Y
3‘all() 4‘all() 3‘all() 4‘all()
– simple coloursets: – constants:
• int ProteinA := 1 - numA • int numA := 3
• int ProteinB := 1 - numB • int numB := 3
• int ProteinB := 1 - numC • int numC := 3
– compound coloursets: • int xDimA := 5
• product ProteinA_ProteinB := Pro- • int yDimA := 5
teinA, ProteinB • int xDimB := 5
• product ProteinB_ProteinC := Pro- • int yDimB := 5
teinB, ProteinC • int xDimC := 5
– variables: • int yDimC := 5
• ProteinA A
• ProteinB B
Fig. 7: Part 1: Instantiation of modules using coloured Petri nets.
5 Conclusions & Outlook
We presented a new approach for incorporating spatial aspects into modular
composed models. A new approach was necessary, because existing techniques
(see Section 2.2) are not suitable for model composition. In particular, we demon-
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 51
Movement of Protein A Movement of Protein B Movement of Protein C
csProteinB csProteinC
csProteinA
ProteinC X
ProteinA X ProteinB X
XL A A A XL B B B XL C C C
XR A 9 XR B 12 XR C
1‘all() 3‘all() 4‘all()
2‘A xDimA‘A 2‘B xDimB‘B 2‘C xDimC‘C
(A,all()) (A,all()) (all(),C) (all(),C)
(all(),B) (all(),B)
csProteinA ProteinB csProteinB ProteinC
csProteinA ProteinB
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD ProteinB TYRp ProteinC SH2
(B,all()) (B,all())
csProteinB ProteinC
ProteinB TYRp ProteinC SH2
csProteinA csProteinC
csProteinB
ProteinA Y ProteinB Y ProteinC Y
A A B B C C
YD A YU A YD B 9 YD C 12 YU C
YU B
1‘all() 3‘all() 4‘all()
2‘A yDimA‘A 2‘B yDimB‘B 2‘C yDimC‘C
(A,all()) (A,all()) (all(),C) (all(),C)
(all(),B) (all(),B)
csProteinA ProteinB
csProteinA ProteinB csProteinB ProteinC
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
ProteinB TYRp ProteinC SH2
(B,all()) (B,all())
csProteinB ProteinC
ProteinB TYRp ProteinC SH2
Movement of complex ProteinAProteinB Movement of complex ProteinBProteinC Movement of complex ProteinAProteinB ProteinC
csProteinA
2‘A ProteinA X xDimA‘A
1‘all()
csProteinB csProteinB csProteinB
ProteinB X ProteinB X ProteinB X 3‘all()
A
A
9 B 9 B 9 B
B B B
2‘B 3‘all() 2‘B 3‘all()
csProteinA xDimB‘B csProteinC xDimB‘B 2‘B csProteinC xDimB‘B
ProteinA X ProteinC X ProteinC X
XL AB A A XL BC C C XR BC XL ABC C C
XR AB 12 12 XR ABC
1‘all() 4‘all() 4‘all()
2‘A xDimA‘A 2‘C xDimC‘C 2‘C xDimC‘C
(A,B) (A,B) (all(),B) (all(),B) (A,B) (A,B)
csProteinA ProteinB csProteinA ProteinB csProteinA ProteinB
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
(B,all()) (B,all()) (B,C) (B,C) (B,C) (B,C)
csProteinB ProteinC csProteinB ProteinC csProteinB ProteinC
ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2
csProteinA
2‘A ProteinA Y yDimA‘A
1‘all()
csProteinB csProteinB csProteinB A
ProteinB Y ProteinB Y A ProteinB Y
9 9 9
B B 2‘B B B B B
2‘B 3‘all() yDimB‘B 3‘all() yDimB‘B 2‘B 3‘all() yDimB‘B
csProteinA csProteinC csProteinC
ProteinA Y ProteinC Y ProteinC Y
YD AB A A YD BC C C C C
YU AB 12 YU BC YD ABC 12 YU ABC
1‘all() 4‘all() 4‘all()
2‘A yDimA‘A 2‘C yDimC‘C 2‘C yDimC‘C
(A,B) (A,B) (all(),B) (all(),B) (A,B) (A,B)
csProteinA ProteinB csProteinA ProteinB csProteinA ProteinB
ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD ProteinA LBD ProteinB LBD
(B,all()) (B,all()) (B,C) (B,C) (B,C) (B,C)
csProteinB ProteinC csProteinB ProteinC csProteinB ProteinC
ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2 ProteinB TYRp ProteinC SH2
Fig. 8: Part 2: Instantiation of modules using coloured Petri nets.
strated based on the BioModelKit framework for modular biomodel engineer-
ing [4], how to extend plain models of intracellular processes to spatial models
without their reimplementation. The models in our case are composed from mod-
ules, where each module describes the functionality of a certain molecular com-
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
52 MA Blätke, C Rohr
(A)
(1) ProteinA (2) ProteinB (3) ProteinC
Instanz1 Instanz1 Instanz1
5 Instanz2 5 Instanz2 5 Instanz2
Instanz3 Instanz3 Instanz3
4 4 4
3 3 3
y
y
y
2 2 2
1 1 1
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
x x x
(B)
(1) Complex of ProteinA_1 and ProteinB_1 (2) Complex of ProteinA_1 and ProteinB_2
1 4 1 4
Distance
Distance
Marking
Marking
0.5 2 0.5 2
0 0 0 0
0 1 2 3 4 5 0 1 2 3 4 5
Time Time
(3) Complex of ProteinA_3 and ProteinB_3
1 4
Distance
Marking
0.5 2
0 0
0 1 2 3 4 5
Time
(4) Complex of ProteinB_1 and ProteinC_1 (5) Complex of ProteinB_3 and ProteinC_3
1 4 1 4
Distance
Distance
Marking
Marking
0.5 2 0.5 2
0 0 0 0
0 1 2 3 4 5 0 1 2 3 4 5
Time Time
Fig. 9: Stochastic Simulation of the Spatial Model.
ponent in the form of a Petri net. To transform a flat modular composed model
into a spatial model the following steps have to be performed: (1) components
have to be equipped with individual spatial attributes to represent their localisa-
tion and movement in the biomolecular system, (2) components are only allowed
to interact if they fulfil certain neighbourhood conditions, (3) the movement of
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Spatial modelling based on modular modelling 53
components depends on their state of interaction, e.g. interacting components
forming a complex can only move as one entity. The use of coloured Petri nets
in our approach allows us to represent individual numbers of module instances
for each component.
In our approach we use d different places per module to hold the spatial
informations. The value of d is usually 1, 2 or 3 for one-, two- or three-dimensional
space. The position of a module is characterized by the number of tokens on these
places, e.g. ProteinA_X = 3 and ProteinA_Y = 2 is position (3,2) in two-
dimensional space. Furthermore, we add transitions to each module to enable
movement and interaction of modules. The structure of the non-spatial modules
remains the same, while converting it into a spatial module. So it is possible to
revert it back again easily.
The use of places holding spatial information does not restrict our approach
to discrete space, but allows us to model continuous space as well by using
continuous places. This is not possible using the grid-places approach presented
in Section 2.2.
The whole process does not depend on the module and can be applied easily
to any module of the BMKdb. Thus it fits quite well in the BMK framework
presented in Section 2.1. The spatial transformation algorithm will be a new
feature in the next release of the BMK online tool.
Further investigation is needed in terms of simulation. Adding space surely
increases the computational complexity and the question is, how can we challenge
this.
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