=Paper=
{{Paper
|id=None
|storemode=property
|title=Model Transformation of Metabolic Networks using a Petri Net Based Framework
|pdfUrl=https://ceur-ws.org/Vol-827/8_DanielMachado_article.pdf
|volume=Vol-827
|dblpUrl=https://dblp.org/rec/conf/acsd/MachadoCRRTF10
}}
==Model Transformation of Metabolic Networks using a Petri Net Based Framework==
https://ceur-ws.org/Vol-827/8_DanielMachado_article.pdf
Model transformation of metabolic networks
using a Petri net based framework
Daniel Machado1 , Rafael S. Costa1 , Miguel Rocha2 , Isabel Rocha1 , Bruce
Tidor3 , and Eugénio C. Ferreira1
1
IBB-Institute for Biotechnology and Bioengineering/Centre of Biological
Engineering, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal
{dmachado,rafacosta,irocha,ecferreira}@deb.uminho.pt
2
Department of Informatics/CCTC, University of Minho, Campus de Gualtar,
4710-057 Braga, Portugal
mrocha@di.uminho.pt
3
Department of Biological Engineering/Computer Science and Artificial Intelligence
Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
tidor@mit.edu
Abstract. The different modeling approaches in Systems Biology create
models with different levels of detail. The transformation techniques in
Petri net theory can provide a solid framework for zooming between these
different levels of abstraction and refinement. This work presents a Petri
net based approach to Metabolic Engineering that implements model re-
duction methods to reduce the complexity of large-scale metabolic net-
works. These methods can be complemented with kinetics inference to
build dynamic models with a smaller number of parameters. The central
carbon metabolism model of E. coli is used as a test-case to illustrate
the application of these concepts. Model transformation is a promising
mechanism to facilitate pathway analysis and dynamic modeling at the
genome-scale level.
1 Introduction
Systems Biology provides a new perspective in the study of living systems and
embraces the complexity emerging of interactions among all biological compo-
nents. Combining theory and experiments, scientists build models to explain and
predict the behavior of the systems under study. Metabolic Engineering is one
of the fields where this perspective has proven useful through the optimization
of metabolic processes for industrial applications [28, 2].
Modeling in Systems Biology is an iterative process as the life-cycle of a
model is comprised of successive refinements using experimental data. Different
approaches, such as top-down, bottom-up or middle-out [18] are used depending
on the purpose of the model and the type of data available for its construction.
In Metabolic Engineering there are macroscopic kinetic models that consider
the cell as a black-box converting substrates into biomass and products, which
are typically used for bioprocess control. On the other hand, there are reaction-
network-level models, either medium-scale dynamic models with detailed kinetic
Recent Advances in Petri Nets and Concurrency, S. Donatelli, J. Kleijn, R.J. Machado, J.M. Fernandes
(eds.), CEUR Workshop Proceedings, ISSN 1613-0073, Jan/2012, pp. 103–117.
104 Petri Nets & Concurrency Machado et al.
information derived from literature and experimental data [3], or genome-scale
stoichiometric reconstructions derived from genome annotation complemented
with literature review [5].
Although the ultimate goal of Systems Biology is a complete understanding
of the cell as a whole, not only it is extremely difficult to collect all the kinetic
information necessary to build a fully detailed whole-cell model due to the lack
of experimental data and model identifiability concerns, but also the computa-
tional cost of simulating the dynamics of a system with such detail would be
tremendous. Therefore, there is a need to fit the level of detail of a model to
the specific problem at hand. For instance, Metabolic Pathway Analysis (MPA)
has been useful in the analysis of metabolism as a way to determine, classify
and optimize the possible pathways throughout a metabolic network. However,
due to the combinatorial explosion of pathways with increasing number of re-
actions, it is still infeasible to apply these methods in genome-scale metabolic
reconstructions without decomposing the network into connected modules [23,
24]. This zooming in and out between different levels of abstraction and connect-
ing parts with different levels of detail is a feature where formal methods and
particularly Petri nets may play an important role. Concepts such as subnet-
work abstraction, transition refinement or node fusion, among others, have been
explored in Petri net theory [8] and may provide the theoretical background for
method development.
In previous work, we reviewed different modeling formalisms used in Systems
Biology from a Metabolic Engineering perspective and concluded that Petri nets
are a promising formalism for the creation of a common framework of meth-
ods for modeling, analysis and simulation of biological networks [15]. They are
a mathematical and graphical formalism, therefore intuitive and amenable to
analysis. The different extensions available (e.g.: stochastic, continuous, hybrid)
provide the flexibility required to model and integrate the diversity of phenom-
ena occurring in the main types of biological networks (metabolic, regulatory
and signaling). Moreover, one may find biological meaning in several concepts in
Petri net theory; for instance, the incidence matrix of a Petri net is the equiv-
alent of the stoichiometric matrix, and the minimal t-invariants correspond to
the elementary flux modes (EFMs).
In this work, we explore strategies of model reduction for Petri net representa-
tions of metabolic networks, and the integration of this methodology with recent
approaches such as genome-scale dynamic modeling. This paper is organized as
follows. Section 2 explores the motivation for the work. Section 3 presents the
model reduction and kinetics inference methods, Section 4 discusses their appli-
cation to E. coli and Section 5 elaborates on conclusions and future work.
2 Background
There are different examples of model reduction in the literature. One such
method was developed in [17], based on timescale analysis for classification of
metabolite turnover time using experimental data. The fast metabolites are
Model transformation of metabolic networks Petri Nets & Concurrency – 105
removed from the differential equations and their surrounding reactions are
lumped. In [20] the EFMs of a reaction network are calculated in order to create
a macroscopic pathway network, where each EFM is a macro-reaction connect-
ing extracellular substrates and products. A simple Michaelis–Menten rate law
is assumed for each macro-reaction and the parameters are inferred from exper-
imental data. The method is applied in a network with 18 reactions and a total
of 7 EFMs. However it does not scale well to larger networks because, in the
worst case, the number of EFMs grows exponentially with the network size.
The combinatorial pathway explosion problem is well known; there are meth-
ods for network decomposition in the literature that address this issue. In [23]
the authors perform a genome-scale pathway analysis on a network with 461
reactions. After estimating the number of extreme pathways (EPs) to be over
a million, the network is decomposed into 6 subsystems according to biological
criteria and the set of EPs is computed separately for each subsystem. A similar
idea in [24] consists on automatic decomposition based on topological analysis.
The metabolites with higher connectivity are considered as external and con-
nect the formed subnetworks. An automatic decomposition approach based on
Petri nets is the so-called maximal common transition sets (MCT-sets) [22], and
consists on decomposing a network into modules by grouping reactions by par-
ticipation in the minimal t-invariants (equivalent to EFMs). A related approach
relies on clustering of t-invariants for network modularization [9]. A very recent
network coarsening method based on so-called abstract dependent transition sets
(ADT-sets) is formulated without the requirement of pre-computation of the
t-invariants and thus may be a promising tool for larger networks [12].
Another problem in genome-scale metabolic modeling is the study of dy-
namic behavior. Genome-scale metabolic reconstructions are stoichiometric and
usually analyzed under steady-state assumption using constraint-based methods
[1]. Dynamic flux balance analysis (dFBA) allows variation of external metabo-
lite concentrations, and simulates the network dynamics assuming an internal
pseudo steady-state at each time step [16]. It is used in [19] to build a genome-
scale dynamic model of L. lactis that simulates fermentation profiles. However,
this approach gives no insight into intracellular dynamics, neither it integrates
reaction kinetics. In [26] the authors build a kinetic genome-scale model of S.
cerevisiae using linlog kinetics, where the reference steady-state is calculated
using FBA. Some of the elasticity parameters and metabolite concentrations are
derived from available kinetic models, while the majority use default values. Us-
ing the stoichiometric coefficients as elasticity values is a rough estimation of
the influence of the metabolites on the reaction rates. Moreover, no time-course
simulation is performed. Mass action stoichiometric simulation (MASS) models
are introduced in [14] as a way to incorporate kinetics into stoichiometric recon-
structions. Parameters are estimated from metabolomic data. Regulation can
be included by incorporating the mechanistic metabolite/enzyme interactions.
A limitation of these models is that mass-action kinetics do not reflect the usual
non-linearity of enzymatic reactions and the incorporation of regulation leads to
a significant increase in network size.
106 Petri Nets & Concurrency Machado et al.
3 Methods
The idea of this work is closer to the reduction concepts of [17, 20] than the
modularization concepts in [23, 24]. In the latter cases a large model is decom-
posed into subunits to ease its processing by analyzing the parts individually.
Instead, our objective is to facilitate the visualization, analysis and simulation of
a large-scale model as a whole by abstracting its components. This reduction is
to be attained by reaction lumping in a way that maintains biological meaning
and valid application of current analysis and simulation tools. The Michaelis–
Menten kinetics is a typical example of abstraction, where the small network of
mass-action reactions are lumped into one single reaction.
Fig. 1. Overall concept of model reduction and kinetics inference.
The overall idea of the model reduction method is depicted in Fig. 1. A
large-scale stoichiometric model can be structurally reduced into a simplified
version that can be more easily analyzed by methods such as MPA. Also, one
may infer a kinetic structure to build a dynamic version of the reduced model.
Due to the smaller size, a lower number of parameters has to be estimated. The
data used for estimation may be experimental data found in the literature, or
pseudo-experimental data from dynamic simulations if part of the system has
been kinetically characterized.
When abstracting a reaction subnetwork into one or more macro-reactions,
it is important to consider the assumptions created by such abstraction. As
in Michaelis–Menten kinetics, these simplifications result in a pseudo-steady-
state assumption for the intermediate species that disappear. While this may
not be a problem for flux balance models, it changes the transient behavior of
dynamic models because the buffering effect of intermediates in a pathway is
neglected. The selection of metabolites to be removed depends on the purpose
of the reduction. The network may have different levels of granularity based on
the availability of experimental data, topological properties, or simply in order
to aggregate pathways according to biological function.
Model transformation of metabolic networks Petri Nets & Concurrency – 107
3.1 Basic definitions
The proposed method for model reduction uses several Petri net concepts from
the literature. We will use the following definition of an unmarked continuous
Petri net (adapted from [4]) for modeling a stoichiometric metabolic network:
P n = < P, T, P re, P ost >
P re : P × T → R+
P ost : P × T → R+
where the set of places P represents the metabolites, the set of transitions T
represents the reactions and P re, P ost are, respectively, the substrate and prod-
uct stoichiometries of the reactions. Note that for the representation of a stoi-
chiometric network, a discrete Petri net usually suffices; however, because some
models may contain non-integer stoichiometric coefficients, the continuous ver-
sion was adopted. Moreover, we will assume that reversible reactions are split
into irreversible reaction pairs. We will also use the following definitions:
loc(x) ={x} ∪ • x ∪ x•
X
In(p) = P ost[p, t] · v(t)
t∈• p
X
Out(p) = P re[p, t] · v(t)
t∈p•
where • x, x• are sets representing the input and output nodes of a node x, the
set loc(x) ⊆ P ∪ T is called the locality of x, function v : T → R+
0 is a given flux
distribution (or the so-called instantaneous firing rate), and In, Out : P → R+ 0
are, respectively, the feeding and draining rates of the metabolites.
The method for network reduction consists of eliminating a set of selected
metabolites from the network. For each removed metabolite its surrounding reac-
tions are lumped in order to maintain the fluxes through the pathways. This re-
duction assumes a steady-state condition for the metabolite, i.e. In(p) = Out(p).
3.2 Model reduction: Conjunctive fusion
There are two options for lumping the reactions depending on the transforma-
tion method applied. The first approach is based on a transformation called
conjunctive transition fusion [8] and it results in an abstraction that replaces
the transition-bordered subnet loc(p) by a single macro-reaction. The drawback
of this method is that the flux ratios between the internal reactions are lost.
If a known steady-state flux distribution (v) is given, then the stoichiometric
coefficients can be adjusted to preserve the ratios for that distribution; how-
ever, the space of solutions of the flux balance formulation becomes restricted
to a particular solution. In the limiting case, if all the internal metabolites are
removed, the cell is represented by one single macro-reaction connecting extra-
cellular substrates and products with the stoichiometric yields inferred from the
108 Petri Nets & Concurrency Machado et al.
Fig. 2. Exemplification of limit scenarios where all the internal metabolites are re-
moved. (A) In the conjunctive reduction case the result is one single macro-reaction
converting substrates into products with the respective yields specified in the stoi-
chiometry. (B) In the disjunctive reduction method, all possible pathways connecting
substrates and products are enumerated.
network topology for one particular steady-state (Fig 2A). The transformation
method for removing metabolite p in P n given a flux distribution v is described
as follows:
P n0 = < P 0 , T 0 , P re0 , P ost0 >
P 0 =P \ {p}
T 0 =T \ (• p ∪ p• ) ∪ {tp }
P re0 ={(pi , tj ) 7→ P re(pi , tj ) | (pi , tj ) ∈ dom(P re) \ (P × (• p ∪ p• ))}
∪{(pi , tp ) 7→ fin (pi ) | pi ∈ • (• p ∪ p• ), pi 6= p, v 0 (tp ) 6= 0, fin (pi ) 6= 0}
P ost0 ={(pi , tj ) 7→ P ost(pi , tj ) | (pi , tj ) ∈ dom(P ost) \ (P × (• p ∪ p• ))}
∪{(pi , tp ) 7→ fout (pi ) | pi ∈ (• p ∪ p• )• , pi 6= p, v 0 (tp ) 6= 0, fout (pi ) 6= 0}
v 0 ={t 7→ v(t) | t ∈ T \ (• p ∪ p• )} ∪ {tp 7→ In(p)}.
where
P
t∈p• • • P re(pi , t) · v(t)
i ∩( p∪p )
fin (pi ) =
v 0 (tp )
P
t∈• pi ∩(• p∪p• ) P ost(pi , t) · v(t)
fout (pi ) =
v 0 (tp )
The stoichiometric coefficients of the new reaction may be very high or low,
depending on v 0 (tp ) and so, optionally, one may also normalize them with some
scalar λ, such that P re00 (pi , tp ) = λ1 · P re0 (pi , tp ), P ost00 (pi , tp ) = λ1 · P ost0 (pi , tp )
and v 00 (tp ) = λ · v 0 (tp ). This will also make the final result independent of the
order of the metabolites removed. A good choice for λ is:
λ = max ({P re(pi , tp ) | pi ∈ • tp } ∪ {P ost(pi , tp ) | pi ∈ tp • })
Model transformation of metabolic networks Petri Nets & Concurrency – 109
3.3 Model reduction: Disjunctive fusion
The second approach is based on a transformation called disjunctive transition
fusion [8], where every combination of input and output reaction pairs connected
by the removed metabolite is replaced by one macro-reaction. Although this ap-
proach does not constrain the steady-state solution space of the flux distribution,
it has the drawback of increasing the number of transitions, if the metabolite
is highly connected, due to the combinatorial procedure. Note that applying
this reduction step to metabolite pi is equivalent to performing one iteration
of the t-invariant calculation algorithm to remove column i of the transposed
incidence matrix. Therefore, in the limiting case where all internal metabolites
are removed, the cell is represented by the set of all possible pathways connect-
ing extracellular substrates and products (Fig. 2B), as was done in [20]. The
definition, similar to the previous one, is as follows:
P n0 = < P 0 , T 0 , P re0 , P ost0 >
P 0 =P \ {p}
T 0 =T \ (• p ∪ p• ) ∪ {txy | (x, y) ∈ (• p × p• )}
P re0 ={(pi , t) 7→ P re(pi , t) | (pi , t) ∈ dom(P re) \ (P × (• p ∪ p• )}
∪{(pi , txy ) 7→ P re0 (pi , x) · P re(p, y) + P re0 (pi , y) · P ost(p, x)
| (x, y) ∈ (• p × p• ), pi ∈ • {x, y}}
P ost0 ={(pi , t) 7→ P ost(pi , t) | (pi , t) ∈ dom(P ost) \ (P × (• p ∪ p• )}
∪{(pi , txy ) 7→ P ost0 (pi , x) · P re(p, y) + P ost0 (pi , y) · P ost(p, x)
| (x, y) ∈ (• p × p• ), pi ∈ {x, y}• }
where
(
P re(p, t) if (p, t) ∈ dom(P re)
P re0 (p, t) =
0 if (p, t) ∈
/ dom(P re)
(
P ost(p, t) if (p, t) ∈ dom(P ost)
P ost0 (p, t) =
0 if (p, t) ∈
/ dom(P ost)
Whenever there are pathways with the same net stoichiometry, these can be
removed by checking the columns of the incidence (stoichiometric) matrix and
eliminating repeats. It should also be noted that in both methods, if a metabo-
lite acts both as substrate and product in a lumped reaction, it will create a
redundant cycle that is not reflected in the incidence matrix. If these cycles are
not removed, they propagate through the reduction steps; therefore, they should
be replaced by a single arc containing the overall stoichiometry. The procedure
110 Petri Nets & Concurrency Machado et al.
works as follows:
P re0 ={(p, t) 7→ P re(p, t) | (p, t) ∈ dom(P re) \ dom(P ost)}
∪{(p, t) 7→ P re(p, t) − P ost(p, t)
| (p, t) ∈ dom(P re) ∩ dom(P ost), P re(p, t) > P ost(p, t)}
0
P ost ={(p, t) 7→ P ost(p, t) | (p, t) ∈ dom(P ost) \ dom(P re)}
∪{(p, t) 7→ P ost(p, t) − P re(p, t)
| (p, t) ∈ dom(P re) ∩ dom(P ost), P ost(p, t) > P re(p, t)}
The previous arc removing procedure may cause isolation of some nodes when
P re(p, t) = P ost(p, t); therefore, the isolated nodes should be removed:
P 0 = {p | p ∈ P, loc(p) 6= {p}}
T 0 = {t | t ∈ T, loc(t) 6= {t}}
3.4 Kinetics inference
Given a stoichiometric model, if metabolomic or fluxomic data are available
for parameter estimation, one may try to build a dynamic model by inferring
appropriate kinetics for the reactions. In [25] the authors propose that this is
performed by assuming linlog kinetics for all reactions using an FBA solution
as the reference state and the stoichiometries as elasticity parameters. An in-
tegration of Biochemical Systems Theory (BST) with Hybrid Functional Petri
Nets (HFPN) is presented in [29], where general mass action (GMA) kinetics is
assumed for each transition. The review of kinetic rate formulations is out of the
scope of this work and may be found elsewhere [10].
Assuming that all metabolites with unknown concentration were removed,
we will extend our definition to a marked continuous Petri net:
P n =< P, T, P re, P ost, m0 >
where m0 : P → R+ 0 denotes the initial marking (concentration) of the metabo-
lites. The kinetics inference process consists on defining a firing rate v : T → R+
0,
which will be dependent on the current marking (m) and the specific kinetic pa-
rameters (see [7] for an introduction on marking-dependent firing rates). As we
assumed irreversible reactions, each rate will only vary with substrate concen-
tration. The rates can be easily derived from the net topology. In case of GMA
kinetics v is given by: Y
v(t) = kt m(p)ap,t
p∈• t
where kt is the kinetic rate of t and ap,t is the kinetic order of metabolite p in
reaction t. A usual first approximation for ap,t is P re(p, t).
Linlog kinetics are formulated based on a reference rate v0 , and defined by:
X !
m(p)
v(t) = v0 (t) 1 + ε0p,t ln
p∈• t
m0 (p)
Model transformation of metabolic networks Petri Nets & Concurrency – 111
where ε0p,t is called the elasticity of metabolite p in reaction t, reflecting the
influence of the concentration change of the metabolite in the reference reaction
rate. As in the previous case, P re(p, t) can be chosen as an initial approximation
for ε0p,t . The relative enzyme activity term (e/e0 ) commonly present in linlog rate
laws to account for regulatory effects at larger time scales will not be considered.
4 Results and Discussion
The proposed methods were tested using the dynamic central carbon metabolism
model of E. coli [3], where the stoichiometric part was used for the application
of the reduction methods, and the dynamic profile was used to generate pseudo-
experimental data sets for parameter estimation and validation of the kinetics
inference method. A Petri net representation of this model (Fig. 3) was built
using the Snoopy tool [21]. All reversible reactions were split into irreversible
pairs. The net contains a total of 18 places, 44 transitions and is covered by 95
semipositive t-invariants, computed with the Integrated Net Analyzer [27].
In the application of the conjunctive method (Fig 4A), the metabolites were
classified as in [17] based on their timescale (Table 1), by calculating their
turnover time (τ : P → R+ 0 ) using the reference steady-state of the dynamic
model, where:
m0 (p)
τ (p) =
In(p)
Metabolites with small turnover time are considered fast. In this case, all metabo-
lites except the slowest 5 (glcex, pep, g6p, pyr, g1p) were removed.
For the application of the disjunctive method (Fig 4B), the metabolites were
classified based on their topology (Table 1). We conveniently opted to remove
the metabolites with lower connectivity to avoid the combinatorial explosion
problem. All metabolites except 5 (g6p, pyr, f6p, gap, xyl5p) were removed.
This reduction assumes steady-state for the removed metabolites. However, it
makes no assumptions on the ratios between the fluxes, therefore preserving the
flux-balance solution space.
Because we are assuming that the reference steady-state is known, the con-
junctive reduced model was chosen for the application of the kinetics inference
method assuming linlog kinetics at the reference state. The elasticity parameters
were estimated using COPASI [13]. The pseudo-experimental data was gener-
ated from simulation with the original model after a 1 mM extracellular glucose
pulse with the addition of Gaussian noise (std = 0.05 mM) (Fig. 5A). The fitted
model was then validated using pseudo-experimental data from a 2 mM pulse
(Fig. 5B). It is possible to observe an instantaneous increase in pyr (from 2.67
to 3.93) and an instantaneous decrease pep (from 2.69 to 1.26) which the model
is unable to reproduce. The poor fitting in some of the intracellular metabolites
is expected given the significant reduction to the model. However, the extracel-
lular glucose consumption profile is remarkably good, both in the fitting and
validation cases.
112 Petri Nets & Concurrency Machado et al.
Fig. 3. Petri net model of the dynamic central carbon metabolism model of E. coli
with reversible reactions split into irreversible pairs.
Model transformation of metabolic networks Petri Nets & Concurrency – 113
Fig. 4. Reduced versions of the original network. (A) Conjunctive reduction method.
(B) Disjunctive reduction method.
Fig. 5. (A) Results of parameter estimation with pseudo-experimental data with 1
mM extracellular glucose pulse. (B) Validation of the model with a 2 mM extracellular
glucose pulse. In both cases, the circles represent the experimental data and the lines
represent time-course simulations generated by the reduced model.
114 Petri Nets & Concurrency Machado et al.
Table 1. Metabolite topological properties (input reactions, output reactions, connec-
tivity) and dynamic properties (concentration, flux, turnover time) at the reference
steady-state.
Metabolite #(• p) #(p• ) #(• p × p• ) m0 (mM) In (mM/s) τ (s)
glcex 1 1 1 0.0558 0.0031 18.099
pep 1 6 6 2.6859 0.3031 8.8603
g6p 3 3 9 3.4882 0.2004 17.406
pyr 4 2 8 2.6710 0.2418 11.044
f6p 3 5 15 0.6014 0.1423 4.2266
g1p 1 2 2 0.6539 0.0023 278.62
pg 1 1 1 0.8092 0.1397 5.7929
fdp 2 1 2 0.2757 0.1414 1.9495
sed7p 2 2 4 0.2761 0.0454 6.0757
gap 7 6 42 0.2196 0.3661 0.5997
e4p 2 3 6 0.0986 0.0454 2.1684
xyl5p 3 3 9 0.1385 0.0839 1.6503
rib5p 2 3 6 0.3994 0.0558 7.1626
dhap 2 3 6 0.1682 0.1414 1.1892
pgp 2 2 4 0.0080 0.3207 0.0251
pg3 2 3 6 2.1437 0.3207 6.6851
pg2 2 2 4 0.4014 0.3031 1.3241
ribu5p 3 2 6 0.1114 0.1397 0.7974
Although both reducing methods can be combined with kinetics inference,
the conjunctive version seems more suitable if a steady-state distribution is
known, because it generates smaller models, hence less parameters. The dis-
junctive version is appropriate for analyzing all elementary pathways between
a set of metabolites without the burden of calculating the set of EFMs of the
whole model. For instance, the macro-reactions M4 (ALDO + G3PDH ) and M5
(ALDO + TIS ), with net stoichiometries of, respectively, [fdp → gap] and [fdp
→ 2 gap], are two unique pathways between these two metabolites.
5 Conclusions
This work presents strategies for model reduction of metabolic networks based on
a Petri net framework. Two approaches, conjunctive and disjunctive reduction
are presented. The conjunctive approach allows the abstraction of a subnetwork
into one lumped macro-reaction, however limited to one particular flux distri-
bution of the subnetwork. The disjunctive approach on the other hand, makes
no assumptions on the flux distribution by replacing the removed subnetwork
with macro-reactions for all possible pathways through the subnetwork, there-
fore not constraining the steady-state solution space. In both cases, the reduced
model may be transformed into a dynamic model using kinetics inference and
parameter estimation if experimental data is available. Using the reduced model,
Model transformation of metabolic networks Petri Nets & Concurrency – 115
instead of the original, facilitates this process because it significantly decreases
the number of parameters to be estimated.
In future work, we intend to build a dynamic genome-scale model of E. coli
by using the already available central carbon dynamic model [3], complemented
with lumped versions of the surrounding pathways in the genome-scale network
[5]. Note that some of the reactions on the central carbon model already rep-
resent lumped versions of some biosynthetic pathways (e.g. mursynth, trpsynth,
methsynth, sersynth). However they were not deduced from the genome-scale
network and may not be accurate abstractions of these pathways.
Among the extensions available to Petri nets are the addition of different
types of arcs, such as read-arcs and inhibitor-arcs, which could be use to repre-
sent activation and inhibition in biochemical reactions. They could also be used
to integrate metabolic and regulatory networks. Optimization in metabolic pro-
cesses is usually based on knockout simulations in metabolic networks. However,
these simulations do not take into consideration the possible regulatory effects
caused by the knockouts. In our transformation methods we removed the arcs
with the same stoichiometry in both directions, because these are not reflected
in the stoichiometric matrix. In the Michaelis–Menten example this results in
removing the enzyme from the network. The proposed methods can be extended
to consider read-arcs for these situations, which should be preserved during the
reduction steps, therefore establishing connection places to the integration of a
regulatory network (Fig 6).
Fig. 6. Reduction step conserving the read-arcs associated with the enzymes of the
original reactions.
An alternative to the reduction of the models would be to consider their repre-
sentation using hierarchical Petri nets. In this case, each macro-reaction would be
connected to its detailed subnetwork. Although this would not reduce the num-
ber of kinetic parameters in the case of kinetics inference, it would be extremely
useful for facilitated modeling and visualization of large-scale networks without
compromising detail. It could also be the solution for genome-scale pathway anal-
ysis, if it is performed independently at each hierarchical level. The hierarchical
model composition proposed for SBML [6] may facilitate the implementation of
this alternative. See [11] for an automatic network coarsening algorithm based
on hierarchical petri nets applied to different kinds of biological networks.
116 Petri Nets & Concurrency Machado et al.
Acknowledgments. Research supported by PhD grants SFRH/BD/35215/2007
and SFRH/BD/25506/2005 from the Fundação para a Ciência e a Tecnologia
(FCT) and the MIT–Portugal Program through the project “Bridging Systems
and Synthetic Biology for the development of improved microbial cell factories”
(MIT-Pt/BS-BB/0082/2008).
References
1. S.A. Becker, A.M. Feist, M.L. Mo, G. Hannum, B.Ø. Palsson, and M.J. Herrgard.
Quantitative prediction of cellular metabolism with constraint-based models: the
COBRA Toolbox. Nature Protocols, 2(3):727–738, 2007.
2. A.P. Burgard, P. Pharkya, and C.D. Maranas. Optknock: A bilevel programming
framework for identifying gene knockout strategies for microbial strain optimiza-
tion. Biotechnology and Bioengineering, 84(6):647–657, 2003.
3. C. Chassagnole, N. Noisommit-Rizzi, J.W. Schmid, K. Mauch, and M. Reuss. Dy-
namic modeling of the central carbon metabolism of Escherichia coli. Biotechnology
and Bioengineering, 79(1):53–73, 2002.
4. R. David and H. Alla. Discrete, continuous, and hybrid Petri nets. Springer Verlag,
2005.
5. A.M. Feist, C.S. Henry, J.L. Reed, M. Krummenacker, A.R. Joyce, P.D. Karp,
L.J. Broadbelt, V. Hatzimanikatis, and B.Ø. Palsson. A genome-scale metabolic
reconstruction for Escherichia coli K-12 MG1655 that accounts for 1260 ORFs and
thermodynamic information. Molecular systems biology, 3(1), 2007.
6. A. Finney. Developing SBML beyond level 2: proposals for development. In Com-
putational Methods in Systems Biology, pages 242–247. Springer, 2005.
7. D. Gilbert and M. Heiner. From Petri nets to differential equations-an integra-
tive approach for biochemical network analysis. Petri Nets and Other Models of
Concurrency-ICATPN 2006, pages 181–200, 2006.
8. C. Girault and R. Valk. Petri Nets for System Engineering: A Guide to Modeling,
Verification, and Applications. Springer-Verlag New York, Inc., Secaucus, NJ,
USA, 2001.
9. E. Grafahrend-Belau, F. Schreiber, M. Heiner, A. Sackmann, B.H. Junker, S. Grun-
wald, A. Speer, K. Winder, and I. Koch. Modularization of biochemical networks
based on classification of Petri net t-invariants. BMC bioinformatics, 9(1):90, 2008.
10. J.J. Heijnen. Approximative kinetic formats used in metabolic network modeling.
Biotechnology and bioengineering, 91(5):534–545, 2005.
11. M. Heiner. Understanding Network Behavior by Structured Representations of
Transition Invariants. Algorithmic Bioprocesses, page 367, 2009.
12. M. Heiner and K. Sriram. Structural analysis to determine the core of hypoxia
response network. PloS one, 5(1):e8600, 2010.
13. S. Hoops, S. Sahle, R. Gauges, C. Lee, J. Pahle, N. Simus, M. Singhal, L. Xu,
P. Mendes, and U. Kummer. COPASI—a COmplex PAthway SImulator. Bioin-
formatics, 22(24):3067–3074, 2006.
14. N. Jamshidi and B.Ø. Palsson. Mass action stoichiometric simulation models: In-
corporating kinetics and regulation into stoichiometric models. Biophysical Jour-
nal, 98:175–185, 2010.
15. D. Machado, R.S. Costa, M. Rocha, I. Rocha, B. Tidor, and E.C. Ferreira. A critical
review on modelling formalisms and simulation tools in computational biosystems.
In Distributed Computing, Artificial Intelligence, Bioinformatics, Soft Computing,
and Ambient Assisted Living, pages 1063–1070. Springer, 2009.
Model transformation of metabolic networks Petri Nets & Concurrency – 117
16. R. Mahadevan, J.S. Edwards, and F.J. Doyle. Dynamic flux balance analysis of
diauxic growth in Escherichia coli. Biophysical journal, 83(3):1331–1340, 2002.
17. I.E. Nikerel, W.A. van Winden, P.J.T. Verheijen, and J.J. Heijnen. Model reduc-
tion and a priori kinetic parameter identifiability analysis using metabolome time
series for metabolic reaction networks with linlog kinetics. Metabolic Engineering,
11(1):20–30, 2009.
18. D. Noble. The rise of computational biology. Nature Reviews Molecular Cell
Biology, 3(6):459–463, 2002.
19. G.M. Oddone, D.A. Mills, and D.E. Block. A dynamic, genome-scale flux model of
Lactococcus lactis to increase specific recombinant protein expression. Metabolic
Engineering, 2009.
20. A. Provost and G. Bastin. Dynamic metabolic modelling under the balanced
growth condition. Journal of Process Control, 14(7):717–728, 2004.
21. C. Rohr, W. Marwan, and M. Heiner. Snoopy-a unifying Petri net framework to
investigate biomolecular networks. Bioinformatics, 2010.
22. A. Sackmann, M. Heiner, and I. Koch. Application of Petri net based analysis
techniques to signal transduction pathways. BMC bioinformatics, 7(1):482, 2006.
23. C.H. Schilling and B.Ø. Palsson. Assessment of the metabolic capabilities of
Haemophilus influenzae Rd through a genome-scale pathway analysis. Journal
of Theoretical Biology, 203(3):249–283, 2000.
24. S. Schuster, T. Pfeiffer, F. Moldenhauer, I. Koch, and T. Dandekar. Exploring the
pathway structure of metabolism: decomposition into subnetworks and application
to Mycoplasma pneumoniae, 2002.
25. K. Smallbone, E. Simeonidis, D.S. Broomhead, and D.B. Kell. Something from
nothing-bridging the gap between constraint-based and kinetic modelling. FEBS
Journal, 274(21):5576–5585, 2007.
26. K. Smallbone, E. Simeonidis, N. Swainston, and P. Mendes. Towards a genome-
scale kinetic model of cellular metabolism. BMC Systems Biology, 4(1):6, 2010.
27. P.H. Starke. INA: Integrated Net Analyzer. Reference Manual, 1992.
28. G. Stephanopoulos. Metabolic engineering. Biotechnology and Bioengineering, 58,
1998.
29. J. Wu and E. Voit. Hybrid modeling in biochemical systems theory by means
of functional Petri nets. Journal of bioinformatics and computational biology,
7(1):107, 2009.