Quantitative Modeling of Biochemical Networks

R. Hofestädt1* and S. Thelen2

1University of Magdeburg, Department of Computer Science
2University of Bonn, Department of Computer Science
* corresponding author:
Otto-von-Guericke-Universität Magdeburg, Institut für Technische und Betriebliche Informationssysteme, AG / Bioinformatik und Medizinische Informatik, Universitätsplatz 2, D-39106 Magdeburg, Germany


Today different database systems for molecular structures (genes and proteins) and metabolic pathways are available. All these systems are characterized by the static data representation. For progress in biotechnology, the dynamic representation of this data is important. The metabolism can be characterized as a complex biochemical network. Different models for the quantitative simulation of biochemical networks are discussed, but no useful formalization is available. This paper shows that the theory of Petrinets is useful for the quantitative modeling of biochemical networks.

Keywords: Bioinformatics, Biochemical Networks, Modeling and Simulation, Petrinets


Methods of biotechnology allow the analysis of biochemical reactions and the isolation, sequencing, analysis, and synthesis of genes and proteins [1]. This opens a wide area of applications and will produce various changes in science and human behavior. However, in medicine new drugs could be designed by using these data methods of biotechnology and biocomputing [2]. To make this technology more useful, enormous efforts are necessary. The dream of drug design and gene therapy can become reality, if interdisciplinary efforts are successful. Therefore, the phenomena of gene regulation and the modeling of biochemical reactions has to be analyzed [3]. The new research area, that tries to solve these problems, is called metabolic engineering [4,5]. The goal of this research field is to develop and implement tools in practice and theory which will carry out the analysis and synthesis of metabolic engineering. In the case of theoretical parts, bioinformatics has already developed different tools to accomplish this version. However, database systems for genes (EMBL, GENBANK) and proteins (PIR, SWISSPROT) are available via Internet. Moreover, the Boehringer company is collecting the data of all analyzed biochemical reactions. This data is presented by the Boehringer pathway chart [6]. Since the beginning of 1997 the electronic representation of the biochemical pathways is available via internet [7]. The main gap is still the dynamic representation of these molecular data. Different models and simulation shells are developed, but this gap still exists [8].

Biochemical pathways can be interpreted as complex graphs. Each node represents a metabolite and each edge represents a biochemical reaction which is catalyzed by specific molecular structures. Therefore, the application of the theory of Petrinets seems to be useful. The Petrinet application of biochemical reactions was introduced by Reddy et al. 1993 [9]. In this paper it is shown that Petrinets can easily simulate qualitative biochemical reactions. The problem of this presentation [9] is that gene regulation processes cannot be simulated. It is not possible to simulate the kinetic effect. The first gap could be closed, showing that different classes of conditions can be interpreted as genes, proteines, or enzymes [10]. Using this formalization, cell communication and gene regulation can easily be simulated. Moreover, the simulation of kinetic effects and the feedback control of biochemical reactions is very important. In this paper we present the extension of our formalization [10], which allows the quantitative modeling of regulatory biochemical networks.


Metabolic engineering is the improvement of cellular activities by manipulating the enzymatic transport and the regulatory functions of the cell with the use of DNA recombination technology [4]. The opportunity to introduce heterologous genes and regulatory elements distinguishes metabolic engineering from traditional genetic approaches to improve the strain. Metabolic engineering includes manipulation of protein processing pathways as well as of pathways involving smaller metabolites. At present, metabolic engineering is more a collection of examples than a codified science. The main features of metabolic engineering can be subdivided into two parts: the theoretical and the practical part. The synthesis and creating of new products or new reactants and the synthesis of hybrid metabolic networks belong to the practical part. In this presentation, the theoretical part of metabolic engineering will be discussed. Biochemical data has to be stored by using integrative database systems. Moreover, specific analysis algorithms have to be implemented. The main task is to develop and implement interactive simulation environments, which allow the quantitative discussion of metabolic processes. Therefore, integrative simulation environments must be implemented, which allow the simulation of gene regulation, biosynthesis, and cell communication.

The recombination of phenotypes and features in organisms can be carried out by using methods of DNA recombination. In the first step specific genes, which represent the desirable phenotype (for example body length), have to be isolated and integrated into a specific genome (e.g. Ti-plasmid). Gene transfer into the organism will be realized by infection of the vector molecule. This recombinant process can produce the corresponding phenotype of this gene. A well known example is the so-called 'super mouse' [11], which contains the growth gene of the rat. By expressing this gene, the body length of the 'super mouse' is double the length of a mouse. A popular research field is to identify genetic defects, which will produce metabolic diseases. To repair such defects by using methods of biotechnology is the main task of human genetics. The first step is to identify and modify defect genes. The main problem is the regulation of genetic activity. This is the reason that cellular control mechanisms are analyzed in the field of molecular genetics and biotechnology. Metabolism represents a highly connected system of biochemical reactions, gene regulation mechanisms, and cell communication processes. Therefore, the main task is to develop new models and simulation shells which will allow to modify complex metabolic processes.


Biochemical Networks

The metabolism is based on biochemical reactions. To understand the behavior of biochemical networks, modeling and simulation are important [3]. In the case of genes, enzymes, and biochemical reactions, database systems are available, which represent the analyzed molecular data. Several models have been developed, but the main gap in this area of modeling and simulation is the development of an integrative model and simulation shell [12], which allows the dynamic representation of biochemical networks. The meaning of "integrative" is that this model enables discussion of biosynthetic processes, gene regulation processes, and cell communication processes. Therefore, integrative models allow the discussion of regulatory metabolic networks.

The genetic information (DNA) controls metabolism indirectly. The protein synthesis of structural genes produces specific enzymes which catalyze biochemical reactions. The transcription of these genes has to be regulated by enzymatic mechanisms. The fundamental model of gene regulation is based on the model of Jacob and Monod [13] for the synthesis of the Lactose operon. The primary unit of the gene regulation is the operon, which consists of the promoter, the operator, the gene(s), and the terminator sequence. The RNA polymerase identifies the promoter sequence of the operon and carries out the transcription process. The affinity of the promoter/RNA polymerase complex is defined by specific DNA signal structures, which are called the Pribnow box and the -35 box of the promoter (prokaryotes). Homeotic genes, transposons, enhancers and silencers demonstrate that gene regulation is a complex process. The metabolic control of a cell is defined by biochemical reactions, which change substrates into products (SP). This can be done spontaneously or catalyzed by specific enzymes (S-EP). Most of the biochemical reactions are 2-way processes, which are catalyzed by enzymes (SEP). Therefore, concentration rates are important. In some cases specific molecules, which are called inhibitors (I), are able to reduce the flux. However, the flux of biosynthetic processes is controlled by enzyme affinity, enzyme concentration, and reaction rate (p). These parameters can be modified by proteins and enzymes, which are called influence proteins.

Figure 1: Abstract model of biochemical reactions

In the case of 2-way biochemical reactions, the enzyme will catalyze biochemical reactions from the higher to the lower level of the concentration rates. Moreover, kinetic effects are important [14]. Most biochemical reactions follow the Michaelis-Menten kinetic scheme, which is characterized by the following equation:

V = -dS/dt = Vmax * S / S + Km

where S is the substrate concentration at the given rate of reaction, V the maximum rate of hydrolysis and Km the Michaelis constant. V and Km are two constants that characterize the interactions of the enzyme with its substrate. Enzymes can be controlled by modifying the affinity, efficiency, and specificity of the enzyme. However, genes and their regulation mechanisms, biosynthesis and their catalytic, cell communication processes and liveliness of all these components are called elementary metabolic processes, which define the behavior of the metabolism. All these processes build metabolic networks, which are interconnected with elementary metabolic processes influencing each other in a well defined way.

Related Works

The simulation of metabolic processes is based on specific models, which can be subdivided into the classes of abstract, discrete, and analytical models. The abstract models are based on automata and logical models, which permit the global discussion of fundamental aspects. The goal of analytical models is the exact quantitative simulation, where the analysis of kinetic features of enzymes is important. The paper of Waser et al. [15] presents a computer simulation of phosphofructokinase. This enzyme is part of the glycolysis pathway. Waser et al. model all kinetic features of the metabolic reaction by computer simulation. This computer program is based on chemical reaction rules, which are described by differential equations. Franco and Canelas simulate the purine metabolism by differential equations, where each reaction is described by the relevant substance and the catalytic enzyme using the Michaelis-constant of each enzyme [16]. Discrete models are based on state transition diagrams. Simple models of this class are based on simple production units, which can be combined. Overbeek presented an amino acid production system, a black-box with an input-set and an output-set describing a specific production unit [17]. The graphical model of Kohen and Letzkus [18], which allows the discussion of metabolic regulation processes, is representative for the class of graph theoretical approaches. They expand the graph theory by specific functions which allow the modeling of dynamic processes. In this case the approach of Petrinets is a new method. Reddy et al. [9] presented the first application of Petrinets in molecular biology. This formalism is able to model metabolic pathways. The highest abstraction level of this model class is represented by expert systems [19] and object oriented systems [20]. Expert systems and object oriented systems are developed by higher programming languages (Lisp, C++) and allow the modeling of metabolic processes by facts/classes (proteins and enzymes) and rules/classes (chemical reactions). The grammatical formalization is able to model complex metabolic networks [21].


Petrinets, a graph-oriented formalism, allow the modeling and analysis of systems, which comprise properties such as concurrency and synchronization. A Petrinet consists of transitions and places, which are connected by arcs. In the graphical representation, places are drawn as circles, transitions are drawn as thin bars or as rectangles, and arcs are drawn as arrows. The places and transitions are labeled with their names. Places may contain tokens, which are drawn as dots. The vector representing the number of tokens in each place is the state of the Petrinet and is referred to as marking. The marking can be changed by the firing of the transitions, which is determined by arcs. The arcs can further be divided into input and output arcs. Generally, arcs may have multiplicities greater than one. In the following only single arcs are assumed. A transition is said to be enabled, if all places connected with input arcs contain tokens. An enabled transition may fire by removing a token from each place connected with an input arc and adding a token to each place connected with an output arc. The transitions can be divided into immediate transitions, firing without delay, and timed transitions, firing after a certain delay. A Petrinet is a bipartite directed graph, which can be represented graphically. The places contain indistinguishable tokens, which can be fired by the transitions. The vector representing the number of tokens in each place is called the marking of the net. The reachability graph contains all markings reachable from an initial marking. Formal definitions of elementary Petrinets and theoretical results about their structural properties can be found in [22]. Several extensions of elementary Petrinets were proposed for more compact models and a higher level of abstraction. In colored Petrinets tokens can be distinguished, and in predicate-transition nets tokens are expressions, which are manipulated by the firing of transitions. Initially, Petrinets model only qualitative aspects of a system. In order to include quantitative aspects, every place can hold a well defined number of tokens. The capacity of the place defines the maximum of tokens which can be held by this place. However, the definition of the firing rule has to be extended. The input and output arc will be labeled by integer values. In the case of the input arc, this value states that the transition can fire, if each input place represents equal or more tokens than the input arc will specify. The firing process of a transition will delete tokens and will produce tokens into the output places. The number of deleting and producing tokens will be specified by the arrow weight.

More formally we will define some basic terms.

A net N = (P,T,F) is defined as
- P and T are finite disjunct sets and
- F is a subset of the set (P x T) (T x P).

A place-transition net consists of
- places (represented as circles);
- transitions (represented as boxes);
- arrows from places to transitions and from transitions to places;
- a capacity indication for every place;
- a weight for every arrow (represented as a number);
- an initial marking, defining the initial number of tokens for every place.

In a place-transition net
- a marking is indicated by the number of tokens in every place;
- a place p is in the pre-set (or post-set) of a transition t, if there is an arrow from p to t (or an arrow from t to p);
- a transition t is activated, if
   1. for every place p from the pre-set of t the weight of the arrow from p to t is not greater than the number of tokens indicated at p;
   2. for every place p in the post-set of t the number of tokens at p increased by the weight of the arrow from t to p is not greater than the capacity of p;
- an activated transition t will occur in the number of tokens at every place p is decreased by g, if g is the arrow weight of (p t) and in that the number of tokens at every place p' is increased by g', if g' is the arrow weight of (t p').


Self-modified Petrinets

The main feature of metabolic processes is that the concentration of metabolites will influence the reaction activity of biochemical processes. Therefore, the actual concentration of any metabolite is an important component of the model. This can be done by the extension of the place-transition net including the self-modified component, which was at first defined by Valk [23]. The main feature of this formalization is that each identifier of any place can be used as a parameter of any arrow weight formula.

Example 1: The concentration of the enzyme is important for the biocatalytic process. Using self-modification networks the biocatalytic reaction of lactose into glucose and galactose can be described as follows: each unit lactose will produce one unit of galactose and one unit of glucose. The reaction will only be activated if the enzyme ß-galactosidase is available. Therefore, the concentration of ß-galactosidase will be used. The arrow weight from the place ß-galactosidase to the transition reaction will be ß-galactosidase. This set will be used and produced.

Figure 2: A self-modified Petrinet for the biocatalytic reaction of lactose

Based on these ideas we give a formal definition of self-modified Petrinets and self-modified Petrinets with capacity.

Definition: N = (P,T,F,Vs,m0) is called self-modified Petrinet, iff
- (P,T,F) is a net,
- Vs : P x PN x T N with PN := PN,
- m0 start configuration.

Definition: N = (P,T,F,Ku,Ko,Vs,m0) is called self-modified Petrinet with capacity, iff
- (P,T,F,Vs,m0) is a self-modified net,
- Ku: P N is the minimal capacity of each place,
- Ko: P N is the maximal capacity of each place,
- mo: P N a start mark with Ku(p) < mo(p) < Ko(p).

Petrinets with Functions

The self-modified Petrinet allows the modeling of biochemical processes using actual concentrations. Moreover, it makes sense to model this biocatalytic reaction using functions, which allow each transition to simulate kinetic effects. The calculation of the dynamic biocatalytic process can be realized by using functions for specifying the arrow weight. Moreover, complex relations and conditions can be combined which will activate transitions. Functional Petrinets are specific predicate/transition networks [24], which represent abstract networks. Regarding predicate/transition networks, two kinds of modification will be described. Tokens of the same place can be of different types and the arrow weight will be described by using a specific description language. In the case of biochemical modeling, we only need the second feature of predicate/transition networks, because the possibility of using functions allows the mapping of natural numbers, where identifiers of different places within the net can be used as variables.

Definition: N = (P,T,F,VF,m0) is called functional net, iff
- (P,T,F) is a net,
- VF(f) { g(x1,..,xn} | g: PN x ... x PN N }, n N,
- m0 a start configuration of N.

Example 2: The biocatalytical reaction of example 1 will be extended using the functional description. In the following a linear dependence is suggested. The start configuration m0 = (100,20,0,0) means: 100 units of lactose and 20 units of the enzyme ß-galactosidase. The linear factor is n = 2.

Figure 3: A self-modified Petrinet modeling the lactose biocatalytic process using functions.

Simulation with Petrinets

The formalization of biochemical reactions by the Petrinet model as described in this paper allows the simulation of biochemical networks. At the beginning of the modeling process we have to identify the metabolites (places), the biochemical reactions (transitions), and their relations, which will define the structure of the model (arcs and the arrow weights). Moreover, we have to define the start configuration (tokens into places). Based on this configuration transitions will be enabled, and the firing process will produce new configurations of the Petrinet. However, based on this formalization all possible new configurations can be calculated using the matrix formalization of this method. Regarding example 2 the corresponding matrix and the vector of the start configuration will be:

Lactose			- (n * ß-Galactosidase)
ß-Galactosidase		- (n * ß-Galactosidase) + (n * ß-Galactosidase)
Galactose		+ (n * ß-Galactosidase)
Glucose			+ (n * ß-Galactosidase)

start configuration: (100,20,0,0).

If m0 is the vector of the start configuration, C the actual reaction matrix, and x the vector, that represents the firing transition, the new configuration can be calculated using the following equation:

m':= m0 + Cx.

After each generation the actual vector x has to be calculated. The simulation of a Petrinet can be a sequential or a parallel process. Regarding the sequential simulation only one transition can be fired. However, regarding parallel simulations activated or enabled transitions can be fired simultaneously.

Definition: Let N = (P,T,F,Ku,Ko,V,m0) be a Petrinet. For each t T the mapping t+ and t- is defined as:
  Vk(t,p) if p tF and (tp) represent an arrow weight using a variable
  t+(p) := { Vs(p,q,t) if p tF and the arrow weight of (tp) represent q
  VF(f)(x1,..,xn) if p tF and the arrow (tp) represent a function
  0 otherwise

  Vk(t,p) if p tF and (tp) represent an arrow weight using a variable
  t-(p) := { Vs(p,q,t) if p tF and the arrow weight of (tp) represent q
  VF(f)(x1,..,xn) if p tF and the arrow (tp) represent a function
  0 otherwise

t(p) := t+(p) - t-(p).

The parallelism effect is a dynamic feature and not a structural component of the Petrinet.

Definition: Let N = (P,T,F,Ku,Ko,V,m0) be a Petrinet, U T a transition set and m the actual configuration of P. The set U is called simultaneous application set regarding m, iff

U- := tU t- and U+:= tU t+.

Features of Petrinets

A fundamental feature is reachability. Regarding a Petrinet and a start configuration, we have to answer the question, as to whether a specific configuration can be produced. Mayr et al. showed that the reachability question is solvable for each Petrinet [22]. However, the complexity of this algorithm allows no practical solution. We know that we need exponential growth in time to solve this problem [22].

The limitation of Petrinets is another important feature. A Petrinet which represents places with a restricted account of tokens is called a limited Petrinet. A Petrinet using capacities is limited by definition. An algorithm for the examination of the limitation of Petrinets was presented. However, using self-modified Petrinets the detection of limitation is an unsolvable problem [23].

The capacity value is important for the detection of bottlenecks. The definition of capacities permits fixing an interval for each metabolite, which represents the normal scope of this concentration. Moreover, the detection of bottlenecks can be reduced to the reachability problem. Using small Petrinets the reachability graph can be constructed in practice, which permits calculating all bottlenecks.

Biochemical networks represent a set of biochemical reactions which are highly connected. To analyze metabolic pathways, all activated biochemical reactions are of importance. However, death and liveliness of transitions and configurations must be considered. A transition is called death, if it can never be enabled. Otherwise the transition is called liveliness. The detection of death and liveliness depends on the reachability problem.


Description of the Model

The formalization of Reddy et al. [9] does not permit modeling the kinetic effects of biochemical reactions. However, our extension allows a flexible modeling process. Therefore, we have to consider the following aspects:
- actual arrow weight depends on the actual configuration,
- inhibitor metabolites reduce the concentration of the metabolites,
- a transition can also be activated without inhibitors and activators,
- for the detection of bottlenecks and critical configurations we define concentration borders
for every place which will be tested after the firing of each transition.

The extension of our model will be demonstrated in figure 4.

Figure 4: Extension of the model of Reddy et al. [9].

Activators and inhibitors will be used regarding the actual configuration. The arrow weight is no longer a constant description. In our example F and H design the concentration of the activator and the inhibitor. F and H are elements of the arrow weight description. Therefore, the Petrinet represents self-modified arrows. The reaction t produces the product RP using three units of S1 and four units of S2. These values are multiplied with a factor, that represents the actual concentration of activators, inhibitors, and the actual configuration. For every place capacity values are defined. These values signal concentration intervals, which describe the correct flux of the metabolism.

Biocatalytic Reaction

For the simulation of the flux of biochemical reactions our model has to be able to represent the concentrations of the simulated metabolites. Using our formalization of self-modified Petrinets these features are available. The identifier of any place, which represents the enzyme, can be used as a parameter inside any definition of an arrow weight. Therefore, our model is able to simulate biocatalytic reactions using the theory of Petrinets. The transitions are biochemical reactions, which will not consume the enzyme, and the places are metabolites of these reactions. The formal description of a simple biocatalytic reaction can be given:
P = { Substance, Enzyme, Product }
T = { Reaction }
F = { (Substance, Reaction), (Enzyme, Reaction), (Reaction, Enzyme), (Reaction, Product) }

Regarding this process, Reaction is the catalysed transition of one unit of the Substance into two units of the Product. The arrow weight can be defined:
V(Substance, Reaction) = 1
V(Enzyme, Reaction) = Enzyme
V(Reaction, Reaction) = Enzyme
V(Reaction, Product) = 2

The flexible simulation will use functions as:
V(Substance, Reaction) = 1 * Enzyme
V(Reaction, Product) = 2 * Enzyme

Figure 5: Petrinet representation of the biocatalytic reaction using self-modified arcs.

Gene Regulation

With regard to gene regulation processes, we have to distinguish positive and negative control. Both regulation processes can be either inducible or repressible. Regarding the positive control, we know that the activator will initiate the transcription process. Regarding the inducible positive control, an effector element (inducible enzyme) will enable the transcription process. However, the effector element will activate the activator element. Both elements will build a protein complex which enables the transcription process. The formal description of the positive control is:
  P = { activator inactive, effector, activator active, protein }
  T = { conformation, transcription }
  F = { (activator inactive, conformation), (effector, conformation),
(conformation, activator active), (activator active, transcription),
(transcription, protein) }

With regard to repressive positive control, the effector element will repress the transcription process. The catalytic element is inducible for the transcription process until the effector element will appear. The effector element will inactivate the activator.
  P = { activator active, effector, activator inactive, protein }
  T = { conformation, transcription }
  F = { (activator active, conformation), (effector, conformation),
(conformation, activator inactive), (activator active, transcription),
(transcription, protein) }

The negative control will be controlled by a repressor element which will repress the transcription process. Regarding the inducible negative control, an active repressor will suppress the transcription process. The enzyme complex of the active repressor and the operator will prevent the transcription process. The effector element inactivates the repressor that will enable the transcription process.
  P = { repressor active, repressor/operon, inductor, repressor inactive, active operon, protein }
  T = { connection, docking/RNA polymerase, transcription }
  F = { (repressor active, connection), (connection, repressor/operon),
(repressor/operon, docking/RNA polymerase), (inductor, docking/RNA polymerase),
(docking/RNA polymerase, repressor inactive), (docking/RNA polymerase, active operon),
(active operon, transcription), (transcription, protein) }

The last type of gene regulation processes discussed here is the repressible negative control. The main feature of this control mechanism is that the appearance of the inductor will suppress the transcription process. Without the inductor the operon is active. The repressor is inactive and will be activated by the inductor element. The activated repressor will deactivate the operon, the transcription process will be blocked up.
  P = { repressor inactive, inductor, repressor active, repressor/active operon, active operon, protein}
  T = { conformation, docking repressor, transcription}
  F = { (inductor, conformation), (repressor inactive, conformation),
(conformation, repressor active), (repressor active, docking repressor),
(docking repressor, repressor/active operon), (active operon, transcription),
(transcription, protein) }

Cell Communication

Regarding cell communication processes, which are based on exocytose and endocytose or cellular gaps, the formalization by using Petrinets is simple. Uptake of metabolites by a cell can be formalized using a transition without incoming arrows. On the other hand, if substances leave the cell, we use transitions without outgoing arrows. Regarding specific receptor activities new transitions have to be added.

Wall Chart Representation

All analyzed biochemical reactions are collected by the Boehringer company [6]. Moreover, the KEGG information system represents the static representation of this biochemical data [7]. Based on that data our model allows the dynamic representation of biochemical pathways. In this chapter we will discuss the representation of the glycolysis using our Petrinet model. This biochemical network is a subset of the Boehringer pathway chart. The glycolysis is an important biochemical process which allows the metabolic production of energy. Most of the biochemical reactions of glycolysis are biochemical reactions, which are controlled by positive (metabolites) and negative components (ADP, Insulin). However, the Petrinet modeling of biochemical processes makes regulation components visible. In our Petrinet representation the inhibitor process of P-enol-pyruvate can be shown directly. Glycolysis is a complex example which consists of eight reactions (transitions); different metabolites are connected. Our Petrinet representation describes the biochemical process in direction of the glycolysis. The effects of the enzymes are shown by bold arcs. The positive and negative influence of substances will be shown, using bi-directional interrupt arcs.

Figure 6: Petrinet modeling of the glycolysis pathway.


An important task of Molecular Bioinformatics is to develop information systems for the simulation of biochemical networks. Therefore, models have to be defined which are able to simulate biochemical networks based on the static data representation [7]. A lot of different models are presented [2, 8], but we are still looking for a useful formalization which will solve this task. Petrinets belong to the class of discrete models, which also allow quantitative analysis. Quantitative and qualitative simulations are important in order to understand the molecular behavior of biochemical reactions. Moreover, kinetic effects can be studied directly using this method.

The first Petrinet approach for the simulation of metabolic pathways was presented by Reddy et al. 1993 [9]. This approach is based on the condition event net and discusses qualitative aspects. Moreover, positive and negative components are not included, and the dynamic behavior of biochemical reactions is not represented. Using our approach, the modeling of metabolic networks is possible. This formalization can be used, for example, for the dynamic representation of the Boehringer pathway chart [6], which differs between two domains: the genetic pathways and the domain of metabolic reactions. The advantage of our approach is:
- the graphical representation is a model which corresponds to biochemical reactions,
- the components of our model are substances (places) and reactions (transitions),
- the relations between substances will be characterized by directed arcs.

Our qualitative model permits a biochemical reaction to consume and produce concentrations. This biochemical behavior is similar to the pre- and post-conditions of Petrinets, and our definition of self-modification permits the representation of kinetic effects. Moreover, consuming substances depend on the actual concentration of substances, and the kinetic behavior can be discussed in detail using functions as specific arrow weights. Quantitative changes can also be modeled by the modification of the structure of the Petrinet which allows the discussion of the influence of specific substances. Moreover, the rates of the places can be modified. By means of modifying the actual arrows, new reactions can be defined.

Our formalization is a parallel and discrete model which allows the quantitative simulation of metabolic processes. In the research field of biotechnology and molecular medicine the quantitative simulation of metabolic pathways is important. The detection of genetic defects, which will modify metabolism (metabolic defects), is one approach. The detection of metabolic defects is important, because these effects cause metabolic diseases. Therefore, regarding the corresponding Petrinet the detection of metabolic bottlenecks is necessary. The capacity component of our formalization allows the detection of metabolic bottlenecks. Moreover, we can identifiy and discuss the causal reason for this effect.


This work was supported by the Ministry of Science and Art of the Government of Rheinland-Pfalz.


  1. von Heijne, G. (1987). Sequence Analysis in Molecular Biology. Academic Press, San Diego.

  2. Hofestädt, R., Lengauer, T., Löffler, M. and Schomburg, D. (1997). Bioinformatics. LNCS 1278, Springer-Verlag, Heidelberg.

  3. Hofestädt, R., Collado-Vides, J., Löffler, M. and Mavrovouniotis, M. (1996). Modelling and Simulation of Metabolic Pathways, Gene Regulation and Cell Differentiation. BioEssays 18, 333-335.

  4. Bailey, J. (1991). Toward a Science of Metabolic Engineering. Science 252, 1668-1674

  5. Mavrovouniotis, M., Stephanopoulos, G. and Stephanopoulos, G. (1990). Computer-Aided Synthesis of Biochemical Pathways. Biotechnol. Bioeng. 36, 1119-1131.

  6. Michal, G. (1993). Biochemical Pathways. Boehringer Mannheim, Penzberg.

  7. Kanehisa M., and Goto, S. (1997). A Systematic Analysis of Gene Functions by the Metabolic Pathway database. In: Suhai, S. (ed.), Theoretical and Computational Methods in Genome Research, Plenum Press, New York, pp. 41-56

  8. Collado-Vides, J., Hofestädt R., Löffler M. and Mavrovouniotis, M. (1996). Modeling and Simulation of Gene and Cell Regulation. Dagstuhl-Seminar-Report 130.

  9. Reddy, V. N., Mavrovouniotis, M. L. and Liebman, M. N. (1993). Petri Net Representation in Metabolic Pathways. In: Hunter, L. et al. (eds.). Proceedings First International Conference on Intelligent Systems for Molecular Biology, AAAI Press, Menlo Park, pp. 328-336.

  10. Hofestädt, R. (1994). A Petri Net Application of Metabolic Processes. Journal of System Analysis, Modelling and Simulation 16, 113-122.

  11. Gardner, E., Simmons, E. and Snustad, D. (1991). Principles of Genetics. John Wiley and Sons, New York

  12. Hofestädt, R. and Meineke, F. (1995). Interactive Modelling and Simulation of Biochemical Networks. Comput. Biol. Med. 25, 321-334.

  13. Jacob, F., and Monod, J. (1961). Genetic regulatory mechanisms in the synthesis of proteins. J. Mol. Biol. 3, 318-356.

  14. Höfer, T. and Heinrich, R. (1993). A Second-order Approach to Metabolic Control Analysis. J. Theor. Biol. 164, 85-102.

  15. Waser, M., Garfinkel, L., Kohn, C. and Garfinkel, K. (1983). Computer Modeling of Muscle Phosphofructokinase Kinetics. J. Theor. Biol. 103, 295-312.

  16. Franco, R. and Canela, E. (1984). Computer simulation of purine metabolism. Eur. J. Biochem. 144, 305-315.

  17. Selkov, E., Basmanova. S., Gaasterland, T., Goryanin, I., Gretchkin, Y., Maltsev, N., Nenashev, V., Overbeek, R., Panyushkina, E., Pronevitch, L., Selkov jr., E. and Yunus, I. (1996). The Metabolic Pathway Collection from EMP: the Enzymes and Metabolic Pathways Database. Nucleic Acids Research 24, 26-28.

  18. Kohn, M. and Letzkus, W. (1982). A Graph-theoretical Analysis of Metabolic Regulation. J. Theor. Biol. 100, 293-304.

  19. Brutlag, D., Galper, D. and Millis, D. (1991). Knowledge-based simulation of DNA metabolism: prediction of enzyme action. Comput. Appl. Biosci. 7, 9-19.

  20. Stoffers, H. et al. (1992). METASIM: object-oriented modeling of cell regulation. Comput. Appl. Biosci. 8, 443-449.

  21. Collado-Vides, J. (1991). A Syntactic Representation of Units of Genetic Information - A Syntax of Units of Genetic Information. J. Theor. Biol. 148, 401-429.

  22. Baumgarten, B. (1992). Petri Netze. BI Verlag, Mannheim.

  23. Valk, R. (1978). Self-Modifying Nets: A Natural Extension of Petrinets. LNCS 62, 464-476

  24. Reisig, W.(1986). Petri Netze. Springer-Verlag, Heidelberg.