We present a new Petri net simulation environment to enable the processing of experimental data to gain usable new insights about biological systems. Therefore, a powerful mathematical modeling concept - xHPNbio (extended Hybrid Petri Nets for biological applications) - has been defined which is properly adapted to the demands of biological processes. This specification is used for the PNlib (Petri Net library), realized by means of the object-oriented modeling language Modelica, which can be easily integrated for simulation processing in any other network modeling tool, as described in the last part of this paper. There, we briefly describe VANESA, a user-friendly biological network-modeling tool that uses the PNlib and the xHPNbio formalism for the simulation of biological networks.
Modern computer techniques and large memory capacities make it possible to
produce an enormous amount of molecular data stored in huge databases. This data is
indispensable for the scientific progress but does not necessarily lead to insight about
the functionality of biological systems. To improve the understanding of molecular
mechanisms, modern techniques focus on network analysis. The question which is
posed here is, what model formalism is appropriate and which simulator? Numerous
model formalisms have been proposed for modeling and simulation biological
systems (see e.g. [
In the recent years, Petri nets with their various extensions are becoming
increasingly popular. They have been proven to be as universal graphical modeling concept
for representing biological systems in nearly all degrees of abstraction. They support
the qualitative modeling approach as well as the quantitative one. Furthermore, the
biological processes can be modeled discretely as well as continuously and, in
addition, discrete and continuous processes can also be combined within a Petri net model
to so-called hybrid Petri nets first introduced by David and Alla (e.g. [
Despite several works and publications with Petri net approaches, there is a serious problem relating to the lacking unity of concepts, notations, and terminologies. Therefore, to show the research community the power of Petri nets, we have analyzed the demands for carefully modeling biological systems and specially developed a Petri net formalism which is called xHPNbio (extended Hybrid Petri Net for biological applications).
This Petri net concept is the specification of the new simulator based on the
objectoriented modeling language Modelica. A user-friendly graphical model reconstruction
in addition to a well-prepared visualization of simulation results is achieved by
connection the new Petri net simulator to VANESA, a powerful and easy-to-use
biological modeling tool [
Reddy et al. proposed the application of Petri net formalism (introduced by Carl
Adam Petri in 1962) for biological network modeling in order to represent and analyze
metabolic pathways in a qualitative manner [
Moreover, Hofestädt and Thelen expanded the approach of Reddy by introducing
functional Petri nets to enable quantitative modeling of biochemical networks [
Due to the fact that a random behavior of molecular reactions at low concentrations
has been observed in many experiments, Goss and Peccoud introduced stochastic
Petri nets [
A reasonable way for modeling concentrations of biological compounds is by
places containing real token numbers instead of integers and transitions which fire as a
continuous flow specified by an assigned speed. The transformation from the discrete
to the continuous Petri net concept was first introduced by David and Alla in 1987
and they replaced the term token by mark because tokens relate mostly to integer
quantities [
Furthermore, Alla and David and proposed combing the discrete and the
continuous Petri net concept to so-called hybrid Petri nets [
Despite these mentioned works and publications, there is a serious problem regarding the lacking unity of concepts, notations, and terminologies. The definition of Petri nets is not standardized; every author has his/her own definitions which are partly not precise enough, not common, or contradictory. Hence, to show the research community the power of Petri nets, they have to be defined precisely together with the corresponding processes, which are essential for the simulation. This has been done in this paper; based on the mentioned Petri net concepts, formalism has been developed which is able to represent nearly all kinds of biological processes. It is called xHPNbio (extended Hybrid Petri Nets for biological applications).
Two common tools are already available for modeling biological processes with the Petri net formalism. The first one is the commercial tool Cell Illustrator and the second one is the freely available tool Snoopy.
The Cell Illustrator is a commercial, widely-used tool available as a Java Web Start
application that enables to draw, model, elucidate, and simulate complex biological
processes and systems based on extended hybrid functional Petri nets [
Snoopy is a freely available unifying Petri net framework to investigate
biomolecular networks [
Hence, these problems led to the development of a new Petri net simulation environment specified by the established xHPNbio formalism. The xHPNbio elements are modeled object-oriented which allows an easy way to maintain, extend, and modify them. Furthermore, the hybrid simulation is performed by an appropriate Modelicatool. With this several solver settings can be adapted in order to achieve reliable simulation results. Moreover, the xHPNbio formalism is already integrated in VANESA, an easy-to-use biological modeling tool. Using VANESA scientists are able to reconstruct and simulate biological pathways either by drag-and-drop or by loading networks from databases and transforming in the appropriate xHPNbio formalism in one software application. 3
The xHPNbio formalism comprises three different processes, called transitions: discrete, stochastic, and continuous, two different states, called places: discrete and continuous, and four different arcs: normal, inhibition, test, and read arc.
Discrete places contain a non-negative integer quantity, called tokens or marks while continuous places contain a non-negative real quantity, called marks. These marks initiate transitions to fire according to specific conditions. These firings lead mostly to changes of the marks in the connected places.
Discrete transitions are provided with delays and firing conditions and fire first
when the associated delay is passed and the conditions are fulfilled. These fixed
delays can be replaced by exponentially distributed random values, then, the
corresponding transition is called stochastic transition. Thereby, the characteristic
parameter of the exponential distribution can depend functionally on the markings of
several places (cp. [
Both – discrete and stochastic transitions - fire by removing the arc weight from all input places and adding the arc weight to all output places. On the contrary, the firing of continuous transitions takes places as a continuous flow determined by the firing speed which can depend functionally on markings and/or time. Places and transitions are connected by “normal” arcs which are weighted by non-negative integer and real numbers, respectively. But also functions can be written at the arcs depending on the current markings of the places and/or time.
Places can also be connected to transitions by test, inhibition, and read arcs. Then their markings do not change during the firing process. In the case of test and inhibitor arcs, the markings are only read to influence the time of firing while read arcs only indicate the usage of the marking in the transition, e.g. for firing conditions or speed functions. If a place is connected to a transition by a test arc, the marking of the place must be greater than the arc weight to enable firing. If a place is connected to a transition by an inhibitor arc, the marking of the place must be less than the arc weight to enable firing. In both cases the markings of the places are not changed by firing. The same place can be connected to the same transition by a test and, in addition, by a normal arc as well as by an inhibitor and normal arc. These arcs are called double arcs.
It is important to mention that a discrete transition always fires in a discrete manner by removing and adding marks after a delay is passed regardless of whether a discrete or a continuous place is connected to it. However, a continuous transition always fires in a continuous flow so that a discrete place can only be connected to continuous transitions if it is input as well as output of the transition with arcs of the same weight. In this way, the continuous transition can only be influenced by the discrete place but the discrete marking cannot be changed by continuous firing. Hence, the conversion from discrete to continuous markings and vice versa is always performed by discrete transitions connected to continuous places.
A formal definition of an xHPNbio is given below. Therefore, at first the xHPNformalism is introduced which is then expanded to xHPNbio by providing the Petri net elements with a biological meaning.
, , Definition 1. The tuple ( is a xHPN if ─ = ─ ─ ─
1, 2, … , is a finite set of discrete places, = 1, 2, … , is a finite set of continuous places, = { 1, 2, … , } is a finite set of discrete transitions, = { 1, 2, … , } is a finite set of stochastic transitions, , ,
The involved places may change their markings only in a discrete manner. Continuous
changes of involved places are not allowed because then the putative firing times have to be
recalculated the whole time as the continuous change takes place.
─ = { 1, 2, … , } is a finite set of continuous transitions,
─ , , , TS, and are pairwise disjoint,
─ ⊆ ( × ∪ × ∪ × ∪ × ∪ ×
set of normal arcs from places to transitions, where →
place to transition ,
─ ⊆ ( × ∪ × ∪ × ∪ × ∪ ×
of normal arcs from transitions to places, where →
transition to place ,
∪ × ) is a
denotes the arc from
∪ × ) is set
denotes the arc from
, ∈
then → ∈ if and
only if → ∈
and
→ = → ,
─ : { → ℕ0, → ℝ≥0} are the minimum capacities of the places,
─ : { → ℕ0, → ℝ≥0} are the maximum capacities of the places,
─ ℯ: ( ∪ ) → { , } are the resolution types of the places for
type-1conflicts either priority or probability resolution,
─ : ( ∪ ) → ℕ: ℯ( ) = ∧ ∈ , ( ∪ ) → [
Hazard functions Maximum
speeds xHPNbio Biological meaning
metabolites, enzymes, substances, substrates, products, signals, genes, proteins, cells, complexes, activators, inhibitors, repressors, RNA biochemical reactions, metabolic reactions, interactions, regulatory reactions, signal transduction reactions, chemical reactions, binding, phosphorylation
molecules, concentrations, cells
activation mechanisms
catalysis
stoichiometric coefficients, yield coefficients
biological knowledge
stochastic kinetics
kinetics effects/laws
metabolic networks, signal transduction networks, regulatory networks, chemical networks, cell cycle, cell communication, diseases, population dynamics, flux networks, cultivation processes
This xHPN formalism has been transformed to the modeling language Modelica (see section 4) to enable graphical modeling, hybrid simulation, and animation. The execution of a hybrid simulation requires the definition for activating and firing transitions as well as the resolution of possible conflicts.
A discrete/stochastic transition in an xHPN is active if the markings of all input places do not fall below the minimum capacities when the arc weights are removed, and the maximum capacities of all output places ( ) may not be exceeded when the arc weights are added. Additionally, the input places connected by test arcs must have more marks than the arc weights and the places connected by inhibitor arcs must have less marks than the arc weights; read arcs do not influence the activation of a transition.
However, the activation process of continuous transitions requires a differentiation between connected continuous and discrete places. A continuous transition is active if all continuous input places
have either a marking greater than their minimum capacities or they are fed by at least one input transition, i.e. the input speed is not zero. Additionally, all continuous output places have either a marking less than their maximum capacities or they are emptied by at least one output transition, i.e. the output speed is not zero. The connected discrete places have to fulfill the same conditions as mentioned above for activating a discrete transition. In addition, the markings of input places connected by test arcs have to be greater than the arc weights and markings of places connected by inhibitor arcs have to be less than the arc weights.
Definition 3. The tuple ( is an xHPN. A discrete/stochastic transition ∈ ( ∪
) is active if and only if , , , , and ⎧ ⎪ and and and the condition must be fulfilled.
A continuous transition ∈ ∀ ∈ :
( ) > ( ) ∨ ( ( ) = ( ) ∧ > 0) ⎪ ( ) > ⎩ ( ) < → ℐ ( ) ∶ ( ) + → ≤ ( )
is active if and only if ∶ ( ) < ( ) ∨ ( ( ) = ( ) ∧ > 0). → ∈ → ∈ ∀ ∈ ∶ ( ) > and ∀ ∈ and the condition must be fulfilled.
An active transition has to be enabled by all input and output places to become firable. Thereby, enabled discrete transitions wait until the assigned delay is elapsed and stochastic transitions fire first when the putative firing time is reached. However, continuous transitions fire immediately when they are enabled.
Several conflicts can occur when the places have to enable their connected active transitions. Possibly, a discrete place or a continuous place connected to discrete transitions has not enough marks to enable all output transitions simultaneously or cannot receive marks from all active input transitions due to the maximum capacity. Then a conflict arises that has to be resolved (type-1-conflict). This can be either done by providing the transitions with priorities or probabilities. In the first case, a deterministic process decides which place enables which transitions and in the second case the enabling is performed at random; thereby transitions assigned with a high probability are chosen preferentially.
Another conflict can occur between a continuous place and two or more continuous
transitions when the input speed is not sufficient to fire all output transitions with the
instantaneous speed (see equation (1)) (type-2-output-conflict) or when the output
speed is not sufficient to fire all input transitions with the speed of equation (1)
(type2-input-conflict). This conflict is solved by sharing the speeds proportional to the
assigned maximum speeds (see [
If a conflict occurs between a place and continuous as well as discrete/stochastic transitions, the discrete/stochastic transitions take always priority over the continuous transitions (type-3-conflict).
A last conflict can occur when a discrete place has not enough marks to enable all connected continuous transitions (type-4-conflict). This is solved by prioritization of the involved transitions.
The firing is then performed in the following way. Discrete transitions fire by removing as much marks as the arc weights from all input places and by adding as many marks as the arc weights to all output places. However, the firing process of continuous transitions take place as a continuous flow with a maximum speed assigned to every transition. The recalculation of a discrete marking is described by an algebraic equation while a continuous marking is recalculated by a differential equation describing the flow of the continuous firing and an algebraic equation representing the firings of discrete transitions. Definition 4. The tuple ( , , , , , , , , ℐ, ℛ, , , , ℯ, , , ℎ, , , is an xHPN. The firing process of an active continuous transition ∈ by a negative mark change of all continuous input places which is expressed by the is described differential equation and a positive mark change of all continuous output places which is expressed by the differential equation ( ) where is the instantaneous speed of transition and calculated by the following = min
min ∈ ( → ) ⋅
, 1 → ∈ ( ) 1 (1) min ∈ ( ) input places where
is the set of continuous input places of with ( ) = ( ) ,
is the set of continuous output places of
with ( ) = ( ) ,
is the set of all continuous firing input transitions,
is the set of all continuous firing output transitions. If the transition is involved in a
type-2-conflict, this speed has to be adapted appropriately (see [
An active discrete transition ∈
waits time units before it fires and a stochastic transition ∈
fires when the putative firing time is reached which is calculated based on the hazard function ℎ. Both fire by removing the arc weight from all ′( ) = ( ) − → and by adding the arc weight to all output places ′( ) = ( ) + → The marking of a discrete place ∈
can be recalculated by the following algebra ′( ) = ( ) + −
→ , ∈ ( ) ∈ ( ) whereby sitions and transitions.
( ) ⊆ ( ∪ ) is the set of all discrete/stochastic firing input tran∪ ) is the set of all discrete/stochastic firing output
The continuous mark change of a continuous place ∈ aid of the following differential is performed with the ( ) and, in addition, by the following algebraic equation for the discrete mark change caused by firing connected discrete transitions ( ) =
The xHPN definition and the corresponding definitions for activation, enabling, and
firing mentioned above have been implemented by means of the object-oriented
modeling language Modelica to enable graphical hierarchical modeling, hybrid simulation,
and animation [
The Modelica models are described on the textual level by discrete, algebraic, and differential equations and by schematics on the graphical level. A schematic consists of connected components which are defined by other components or on the lowest level by equations in the Modelica syntax. The components have connectors which describe the interaction between them. By drawing a line from one component to another, a connection is established to enable interactions. In this manner a model is constructed. Several components can be structured in libraries, called packages, which provides hierarchical modeling. Moreover, the wrapping technique enables the representation of sub-models consisting of several connected components by a specific adapted icon in order to simplify the modeling process. Then, the sub-models can be used several times in the same or in different models and, in addition, it offers an easy-to-use-model at the top level with an intuitive and familiar adapted view.
For graphical modeling, simulation, and animation an appropriated environment is needed. Several commercial and open- source tools are available. A full list can be found on the Modelica homepage (www.modelica.org).
Each of the xHPN components - transitions, places, and arcs - is modeled by an
own Modelica model which are organized and structured in a Modelica package,
called PNlib (Petri Net library). All components are defined by discrete
(eventbased), algebraic, and differential equations (cp. [
The main process in the place model is the recalculation of the marking after firing a connected transition. In the case of the discrete place model, this is realized by the discrete equation when fire pre(reStart) then t = if fire then pre(t)+firingSumIn–firingSumOut else reStartTokens; end when; whereby pre(t) accesses the marking t immediately before the transitions fire. To this amount, the arc weight sum of all firing input transitions is added and the arc weight sum of all firing output transitions is subtracted from it. Additionally, the tokens are reset to reStartTokens when the user-defined condition reStart becomes true; this could be a global condition.
The marking of continuous places can change continuously as well as discretely. This is implemented by the following construct der(t) = conMarkChange; when discreteFire then
reinit(t, t+discreteMarkChange); end when; when reStart then
reinit(t,reStartMarks); end when; whereby the der-operator access the derivative of the marking t according to time. The continuous mark change is performed by a differential equation while the discrete mark change is performed by the reinit-operator within a discrete equation. This operator causes a re-initialization of the continuous marking every time when a connected discrete transition fires. Additionally, the marking is re-initialized by reStartMarks when the condition reStart becomes true.
The main process of the transition is to check if it can fire. When it is possible,
discrete and stochastic transitions wait as long as the (random) delay is passed while the
continuous transitions fires continuously with a speed calculated in the transition
model. Via connector variables, the places report the transitions their markings and
the transitions report when they fire. The conflicts which could arise in xHPN models
have to be resolved by the mentioned methods to have successful and reliable
simulation (see [
In this section we briefly demonstrate the PNlib possibilities in the software application VANESA (www.vanesa.sf.net). VANESA is a powerful and easy-to-use network modeling tool to members of the laboratory, which combines different fields of studies such as life science, database consulting, modeling and visualization for a semi-automatic and lab-validated reconstruction of biological networks. The application helps to model experimental results that can be expanded with database information to perform modern biological network analysis. Based on project experimental data and the integrated databases in VANESA, users are able to explore and reconstruct any biological system, which can be further transformed, simulated, and analyzed in the language of Petri nets.
Therefore, VANESA uses the PNlib and xHPNbio formalism for simulation processing. One feature of VANESA is the possibility to automatically transform any kind of network into the language of an xHPNbio. Thus, users are able to model and simulate dynamic systems within one tool. Petri net simulations are performed in the background, not visible to users. Simulation results are visualized in plots and animated within the active window (see Fig. 1). 6
A powerful Petri net simulation environment has been developed to enable the processing of experimental data to usable new insights about biological systems. The mathematical modeling concept xHPNbio serves as specification for the presented simulation environment. The xHPNbio elements are modeled object-oriented by discrete, algebraic, and differential equations in the Modelica language. This allows an easy way to maintain, extend, and modify them. The hybrid simulation is performed by an appropriate Modelica tool which usually comprises several possibilities to adapt the solver settings in order to achieve reliable simulation results.
The mathematical modeling concept xHPNbio, was specially developed based on the demands of biological processes, and is so powerful but also so universal and generic that it is an ideal all-round-tool for modeling and simulation of nearly all kinds of processes such as business processes, production processes, logistic processes, work flows, traffic flows, data flows, multi-processor systems, communication protocols, and functional principals.
The PNlib will be freely available on the Modelica homepage (www.modelica.org) soon. However, in this paper we have already presented a software application, called VANESA, which uses the PNlib for simulation processes of biological networks.
In addition, a future goal is to provide an open source Petri net simulation tool for
up till now the PNlib works only with the commercial Modelica tool Dymola. This
demands a further development of the open source Modelica tool OpenModelica to
get the PNlib to work with it because some Modelica features are not yet supported.
The University of Applied Sciences Bielefeld is already closely involved in the
further development of the OpenModelica tool [