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 describing an individual molecular component in the form of a Petri net. In this framework, we do now propose a feature for spatial modelling of molecular 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 components 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 instances. Thus, a modular composed spatial model can be mapped to the cellular arrangement and dierent cell geometries.
Systems biology aims at describing and understanding complex biological
processes 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
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
important features at multiple scales in time and space [
Here, we focus on the spatial aspects in multiscale biomodel engineering,
which have been mostly neglected in the description of intracellular processes
until now [
As modelling tool, we chose Snoopy [
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 2.1
BioModelKit Framework for Modular Model Composition
The BioModelKit framework (BMK framework) is a tool for modular biomodel
engineering [
Since, the functionality of genes, mRNAs and proteins is diverse, dierent module types have been defined in BMK framework, as well as allelic influence
Forward Engineering
Reverse Engineering
Transformation Transformation <sbml> ... </sbml>
mRNA
Upload
Add Modules to Col ection
Set of
High-Throughput
Analysis Model Composition + Algorithmic Mutation + Space-Attributes
modules and causal influence modules to capture also correlations with missing
mechanistic descriptions [
The web-interface of BMK framework (www.biomodelkit.com [
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 [
A 2‘red++ 1‘green x y
[x<>y] 2‘green++ B 1‘blue (a) Before Firing (x,y) complex
AB species
2‘red species
A B x y 2‘green
(x,y) complex [x<>y]
AB (b) After Firing
We use the coloured Petri net in Fig. 2 as an example. It represents an abstract complex formation of two species of dierent 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 associated 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 dierent 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). 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 dierent 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 selected 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 [
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. [
This can be extended to more complex reaction-diusion systems, as shown
in [
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. [
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. Spatial Modelling Methodology for Modular Composed Models Before we start with the formal description of the spatial transformation algorithm, 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 components Ci = {C0, C1, . . . , Cnci }. The mapping of a place pij oe Pi of a module Mi to a set of compopnients Cipij ™ Ci is given by the relation g : Pi ae Ci, such that g(pij ) = Ci j . A place pij with |g(pij )| > 1 represents a complex of |g(pij )| components. The set Ki = {Cipij | |g(pij )| > 1} contains all complexes among the components in Ci. A transition tij oe Ti of a module Mi with |g(•tij ) fl g(tij •)| > 1 represents an interaction with at least two dierent involved components. The total set of all interacting transitions in a Module Mi is given by TiIA = {’t ij oe 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, vM, m0M) according to the general definition of quantitative Petri nets with the following relations: – P M = t Pi, where Mi oe M – T M = t Ti, where Mi oe M – f M = t fi, where Mi oe M
negative integer value. – vM : t Ti ae H, where Mi oe M - total set of firing rates. – m0M t Pi ae N0, ÷p ikÕ , pjkÕÕ with pikÕ oe Pi, pjkÕÕ oe pikÕ = pjkÕÕ , {pikÕ : pjkÕÕ } ae p kM oe P M, m0M(pkM) = max(m0,i(pikÕ ), m0,j (pjkÕÕ )) - total finite and non-empty set of places. - total finite and non-empty set of transitions. - total set of directed arcs, weighted by a non
Pj , where In addition to the definitions above, the following relations can be derived: – CM = t Ci , where Mi oe M - total set of components – gM : t Pi ae t Ci, where Mi oe M - total set of place component relations – TIMA ™ T M = t TiIA, where Mi oe M - total set of all interacting transitions – KM = t Ki, where Mi oe 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 oe CM is explicitly encoded by d places p1Ci , . . . pdCi (termed coordinate places), awxheisch(ec.agn. 1bDe,in2tDeroprre3tDed garsidc)o.oTrdhienamteasr,kwinhgerme (dp,jCdi)Ø of a place pCi defines the 1, defines the number of j current coordinate value, which must be restricted by a lower mL(pjCi ) and upper bound mU (pjCi ) to represent the boundaries of the encoded grid, such that, mL/U (pjCi ) > 0 and mL(pjCi ) < mU (pjCi ).
Step 2: Local Restriction of Interactions. To restrict the executability of each transition t oe TIMA , 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 oe TIMA . If the neighbourhood relation claims that the distance between components involved in the interaction represented by a transition t oe TIMA must be zero, b(t) has to be defined as follows:
I1, q|ig=(1•t)fig (t•)|≠1
q|ig=(1•t)fig (t•)|≠1 b(t) = 0, qqj||jgg==((••ii++ttfi)g11fig (t(•t•)|)|qqdkdk==11((mm((ppkCkCii))≠ ≠ mm((ppkCkCjj))))22 =”=00 In addition, read edges, connecting each transition t oe TIMA and the coordinate places of the respective components have to be added, such that fReadEdge(p1gd(•t)fig (t•), t) = 1.
≠ Step 3: Explicit Encoding of Local Position Changes. To encode the position changes for a component Ci oe CM two dierent scenarios have to be considered dependent on the state of interaction: 1. Local position change of individual components:
For each component Ci oe CM and each coordinate place pjCi oe { p1Ci , . . . , pdCi } two transitions tjC,Li and tjC,Ui are needed to incrementally decrease or increase the amount of tokens. The transition tjC,Li subtracts tokens from the coordinate place pjCi till m(pjCi ) = mL(pjCi ). Therefore, the following edges have to be introduced f M(pjCi , tjC,Li) = 1 and fRMeadEdge(pjCi , tjC,Li) = mL(pjCi ) + 1. mU (pjCi ).
The transition tjC,Ui adds tokens to the coordinate place pjCi till m(pjCi ) =
Therefore, the following edges have to be introduced f M(tjC,Ui , pjCi ) = 1 and fIMnhibitorEdge(pjCi , tjC,Ui ) = mU (pjCi ). To ensure that the position of the component Ci can only be changed if it does not interact with another component Cj , i ”= j, additional inhibitory edges for each transition tjC,Li/U have to be introduced: fInhibitorEdge(PICSi , tj,L/U ) = 1, Ci where PICSi ™ P M and PICSi = {’ p oe P M : Ci oe g(p) · | g(p)| > 1}. 2. Local position change of complexes:
The local position of components forming a complex ki oe KM have to be updated consistently during the local position change. For each complex ki oe KM and each dimension j, 1 Æ j Æ d, two transitions tjk,iL and tjk,iU are needed to incrementally decrease or increase the amount of tokens. The transition tjk,iL removes tokens from the set of coordinate places tChoek i pjCh till at least for one component Ch oe CM the condition m(pjCh ) = mL(pjCh ) is fulfilled. Therefore, for each component Ch oe ki the following edges have to be introduced f M(pjCh , tjk,iL) = 1 and fRMeadEdge(pjCh , tjk,iL) = mL(pjCh ) + 1. The transition tjk,iU adds tokens to the set of coordinate places tChoek i pjCh till at least for one component Ch oe CM the condition m(pjCh ) = mU (pjCh ) is fulfilled. Therefore, for each component Ch oe ki the following edges have to be introduced f M(tjk,iU , pjCh ) = 1 and fIMnhibitorEdge(pjCh , tjk,iU ) = mU (pjCh ). To ensure that the position of the complex ki can only be changed if it is actually formed additional read edges for each transition tjk,iL/U have to be introduced: =
PIkSi 1,
where fReadEdge(PIkSi , tjk,iL/U ) ™ PIkSi = {’ p oe P M | ki = g(p)}. Furthermore, it must be excluded for each component Ch oe ki that interacts with other components using a dierent binding site. Therefore, all co-existing interactions have to be determined Kckoiex = {’ kj oe KM : ki fl kj ”= ? | kj ”= ki}. All places representing a complex kj oe Kckoiex have to be added to each transition tjk,iL/U using an inhibitory edge: fInhibitoryEdge(PCkiOEX , tjk,iL/U ) = 1, where P ki COEX = {’ p oe P M : g(p) = k, k oe Kckoiex}.
To allow the movement of co-existing complexes which use dierent interaction sites simultaneously, the above described procedure has to be applied to all possible combination of co-existing complexes, compare Section 4. P M
and j,L/U and tCh
For simplicity reasons the firing rate of each transition tki j,L/U
is given by Fick’s laws of diusion [
Step 4: Encoding of Component Instances by Applying Coloured Petri nets. A colourset ‡ Csiimple with 1 ≠ qCi colours needs to be specified for each component Ci oe CM, where qCi oe N defines the number of instances for a component Ci. The total set of simple coloursets is given by À simple = {‡ Csi1mple, . . . , ‡ simple . |CM| } For each ‡ Csiimple oe 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 oe CM, where Ci oe 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 Ssimple : À simple ae P M, such that Ssimple(‡ Csiimple) = {p oe P M | Ci oe g(p) · | g(p)| = 1}.
Each complex ki oe KM, where ki represents a subset of components, such ‡thkcaoimtpkouin d™ = r ‡ Csijmple. The total set of compound coloursets is given by CM, is represented by a compound colourset of type product
Cjoek i À compound = {‡ kco1mpound, . . . , ‡ compound}. The total set of compound coloursets |KM| À compound is mapped to remaining subset of places according to the relation Scompound : À compound ae P M, such that Scompound(‡ kcoimpound) = {p oe P M | ki = g(p)}. All edges f (p, t) and f (t, p) of the flat model M for which its is true, that a complex ki oe KM, where ki = g(p) are extended to the multiset expression tCjoek i aCj ‘f (p, t), or respectively tCjoek i aCj ‘f (p, t). The marking and firing rates are kept constant over all place and transition instances, such that marking of each place p oe P M is represented by all()‘m0(p) and the firing rates of each transition t oe T M all()‘h(t), where all() is a function that extracts all instances of a coloured node. 4
For demonstration purposes we introduce in Fig. 3 a running example of a modular 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 formation 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 between Protein A and Protein B (P roteinA_LBD__P roteinB_LBD) is the trigger 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 coecient to one. It is straightforward to replace this assumption with more detailed kinetic descriptions.
Module of Protein A
BC t1 BC t2
Module of Protein B
ProteinA LBD
ProteinB LBD AB t1
AB t2
AB t3 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: – gM({P roteinA_LBD, P roteinA_CD_active, P roteinA_CD_inactive}) =P roteinA – gM({P roteinB_LBD, P roteinB_T Y R, P roteinB_T Y Rp}) =P roteinB – gM({P roteinB_SH2}) =P roteinC – gM({P roteinA_LBD__P roteinB_LBD}) = {P roteinA, P roteinB} – gM({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 interacting transitions is given by TIMA = {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 component 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
with
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 TIMA . 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 with the boolean expression bAB (t): 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 component 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): I1, distBC = 0
with bAB(t) =
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 component 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 ProteinA Y
AB t1
ProteinB X
ProteinA X ProteinB Y
ProteinA Y
AB t2
AB t3 ProteinB X
ProteinA X ProteinB Y
ProteinA Y
ProteinB X ProteinB Y ProteinB X ProteinB Y
Restricted Interaction - Only if ProteinB (X,Y) =ProteinC (X,Y) BC t1 4 ProteinC X
ProteinB X
BC t2 4 ProteinC Y
ProteinB Y 4 ProteinC X 4 ProteinC Y
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 corresponding 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 dierent 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 complex 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 oe 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 relation 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 defined 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 outgoing 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 components 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.
ProteinA X xDimA yDimA Movement of Protein C
ProteinC X 4
xDimC csProteinB ProteinB TYRp
AB t3 (A,B) csProteinB ProteinB TYR 1‘al () 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 demonMovement of complex ProteinAProteinB
Movement of complex ProteinBProteinC
Movement of complex ProteinAProteinB ProteinC XL B YD B 2‘B XL BC
2‘B YD BC
BioModelKit framework for modular biomodel
engineering [
2 Time 3 (3) Complex of ProteinA_3 and ProteinB_3 2 Time 3 2 Time 3 4 4 4 5 4 3 y 2 1 4
ce 2 tsan i D ce 2 tsan i D ce 2 tsan i
D 50 4 50 4 50 1 4 g n i rk0.5 a M 00 1 g n i rk0.5 a M 00 1 1 (1) ProteinA (2) ProteinB (3) ProteinC Instanz1 Instanz2 Instanz3
Instanz1 Instanz2 Instanz3 1 2 4
5 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 localisation and movement in the biomolecular system, (2) components are only allowed to interact if they fulfil certain neighbourhood conditions, (3) the movement of 5 4 3 y 2 1 2 Time 3
4 2 Time 3 4
50 Instanz1 Instanz2 Instanz3 5 4 50
ce 2 tsan i
D 4
ce 2 tsan i D 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 dierent 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 twodimensional 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.