MINI-REVIEW Open Access
Modeling formalisms in Systems Biology
Daniel Machado1*
, Rafael S Costa1
, Miguel Rocha2
, Eugénio C Ferreira1
, Bruce Tidor3
and Isabel Rocha1
Abstract
Systems Biology has taken advantage of computational tools and high-throughput experimental data to model
several biological processes. These include signaling, gene regulatory, and metabolic networks. However, most of
these models are specific to each kind of network. Their interconnection demands a whole-cell modeling
framework for a complete understanding of cellular systems. We describe the features required by an integrated
framework for modeling, analyzing and simulating biological processes, and review several modeling formalisms
that have been used in Systems Biology including Boolean networks, Bayesian networks, Petri nets, process
algebras, constraint-based models, differential equations, rule-based models, interacting state machines, cellular
automata, and agent-based models. We compare the features provided by different formalisms, and discuss recent
approaches in the integration of these formalisms, as well as possible directions for the future.
Keywords: Systems Biology, Modeling Formalisms, Biological Networks
Introduction
Living organisms are complex systems that emerge from
the fundamental building blocks of life. Systems Biology
(SB) is a field of science that studies these complex phenomena
currently, mainly at the cellular level (Kitano
2002,). Understanding the mechanisms of the cell is
essential for research in several areas such as drug
development and biotechnological production. In the
latter case, metabolic engineering approaches are applied
in the creation of microbial strains with increased productivity
of compounds with industrial interest such as
biofuels and pharmaceutical products (Stephanopoulos
1998,). Using mathematical models of cellular metabolism,
it is possible to systematically test and predict
manipulations, such as gene knockouts, that generate
(sub)optimal phenotypes for specific applications (Burgard
et al. 2003,, Patil et al. 2005). These models are
typically built in an iterative cycle of experiment and
refinement, by multidisciplinary research teams that
include biologists, engineers and computer scientists.
The interconnection between different cellular processes,
such as metabolism and genetic regulation,
reflects the importance of the holistic approach introduced
by the SB paradigm in replacement of traditional
reductionist methods. Although most cellular components
have been studied individually, the behavior of the
cell emerges at the network-level and requires an integrative
analysis.
Recent high-throughput experimental methods generate
the so-called omics data (e.g.: genomics, transcriptomics,
proteomics, metabolomics, fluxomics) that have
allowed the reconstruction of many biological networks
(Feist et al. 2008). However, despite the great advances
in the area, we are still far from a whole-cell computational
model that integrates and simulates all the components
of a living cell. Due to the enormous size and
complexity of intracellular biological networks, computational
cell models tend to be partial and focused on
the application of interest. Also, due to the multidisciplinarity
of the field, these models are based on several different
kinds of formalisms, including those based on
graphs, such as Boolean networks, and equation-based
ones, such as ordinary differential equations (ODEs).
This diversity can lead to the fragmentation of modeling
efforts as it hampers the integration of models from different
sources. Therefore, the whole-cell simulation
goals of SB would benefit with the development of a framework
for modeling, analysis and simulation that is
based on a single formalism. This formalism should be
able to integrate the entities and their relationships,
spanning all kinds of biological networks.
This work reviews several modeling formalisms that
have been used in SB, comparing their features and
* Correspondence: dmachado@deb.uminho.pt
1
IBB-Institute for Biotechnology and Bioengineering/Centre of Biological
Engineering, University of Minho, Campus de Gualtar, 4710-057 Braga,
Portugal
Full list of author information is available at the end of the article
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
© 2011 Machado et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
relevant applications. We opted to focus on the formalisms
rather than the tools as they are the essence of the
modeling approach. For the software tools implementing
the formalisms, the interested reader may use the
respective references. Note that besides the intracellular
level, several studies in SB also address the cellular
population level. Therefore, formalisms for modeling the
dynamics of cellular populations that have received
attention in the field were also considered in this work.
There are some interesting reviews already published in
the literature. However they usually focus only on particular
biological processes. An excellent review regarding
the modeling of signaling pathways was elaborated by
Aldridge et al. (2006),. They address the model design
process, as well as, model validation and calibration.
They highlight the application of ODE and rule-based
models, but do not mention other formalisms. Another
recent review on the modeling of signaling networks can
be found in Morris et al. (2010),. Two remarkable reviews
on the modeling of gene regulatory networks are presented
by Schlitt and Brazma (2007), and by Karlebach
and Shamir (2008),. Both give examples of several applications
of different formalisms for modeling this kind of
networks. A few reviews with broader scope can also be
found in the literature. Two excellent examples are Fisher
and Henzinger (2007), and Materi and Wishart (2007).
Both give a critical discussion on the application of different
formalisms for computational modeling of cellular
processes. The former covers Boolean networks, interacting
state machines, Petri nets, process algebras and
hybrid models, whereas the latter covers differential
equations, Petri nets, cellular automata, agent-based
models and process algebras. The lack of a single comprehensive
review that compares a larger spectrum of
formalisms motivated the development of this work.
Biological Networks
Cells are composed by thousands of components that
interact in a myriad of ways. Despite this intricate interconnection,
it is usual to divide and classify these networks
according to their biological function. A very simplistic
example can be found in Figure 1 (created with the free
software tool CellDesigner (Funahashi et al. 2003,), that
uses the graphical notations defined in (Kitano et al.
2005)). The main types of networks are signaling, gene regulatory
and metabolic (although some authors also classify
protein-protein interactions as another type of network).
Signaling networks
Signal transduction is a process for cellular communication
where the cell receives (and responds to) external
stimuli from other cells and from the environment. It
affects most of the basic cell control mechanisms such
as differentiation and apoptosis. The transduction
process begins with the binding of an extracellular signaling
molecule to a cell-surface receptor. The signal is
then propagated and amplified inside the cell through
signaling cascades that involve a series of trigger reactions
such as protein phosphorylation. The output of
these cascades is connected to gene regulation in order
to control cell function. Signal transduction pathways
are able to crosstalk, forming complex signaling networks
(Gomperts et al. 2009,, Albert and Wang 2009).
Gene regulatory networks
Gene regulation controls the expression of genes and,
consequently, all cellular functions. Although all of the
cell functionality is encoded in the genome through
thousands of genes, it is essential for the survival of the
cell that only selected functions are active at a given
moment. Gene expression is a process that involves
transcription of the gene into mRNA, followed by translation
to a protein, which may be subject to post-translational
modification. The transcription process is
controlled by transcription factors (TFs) that can work
as activators or inhibitors. TFs are themselves encoded
by genes and subject to regulation, which altogether
forms complex regulatory networks (Schlitt and Brazma
2007,, Karlebach and Shamir 2008).
Metabolic networks
Metabolism is a mechanism composed by a set of biochemical
reactions, by which the cell sustains its growth
and energy requirements. It includes several catabolic
and anabolic pathways of enzyme-catalyzed reactions
that import substrates from the environment and transform
them into energy and building blocks required to
build the cellular components. Metabolic pathways are
interconnected through intermediate metabolites, forming
complex networks. Gene regulation controls the
production of enzymes and, consequently, directs the
metabolic flux through the appropriate pathways in
function of substrate availability and nutritional requirements
(Steuer and Junker 2008,, Palsson 2006).
Modeling Requirements
Due to the different properties and behavior of the biological
networks, they usually require different modeling
features (although some desired features such as graphical
visualization are common). For instance, features
such as stochasticity and multi-state components may
be important for signaling but not for metabolic networks.
A summary of the major modeling features
required by these networks is presented next.
Network visualization
Biological models should be expressed as intuitively as
possible and easily interpreted by people from different
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 2 of 14
areas. For that matter, graph and diagram based formalisms
can be more appealing than mathematical or textual
notations. Such formalisms can take advantage of
state of the art network visualization tools that, when
compared to traditional textbook diagrams, allow a
much better understanding of the interconnections in
large-scale networks, as well as the integration of heterogeneous
data sources (Pavlopoulos et al. 2008).
Topological analysis
A considerable amount of the work in this field is
based on topological analysis of biological networks. In
this case, graph-based representations also play a fundamental
role. The analysis of the topological properties
of these graphs, such as degree distribution,
clustering coefficient, shortest paths or network motifs
can reveal crucial information from biological networks,
including organization, robustness and redundancy
(Jeong et al. 2000,, Barabási and Oltvai 2004,,
Assenov et al. 2008).
Modularity and hierarchy
Despite its great complexity, the cell is organized as a
set of connected modules with specific functions (Hartwell
et al. 1999,, Ravasz et al. 2002,). Taking advantage
of this modularity can help to alleviate the complexity
burden, facilitating the model analysis. Compositionality
is a related concept meaning that two modeling blocks
can be aggregated together into one model without
changes to any of the submodels. This property can be
of special interest for applications in Synthetic Biology
(Andrianantoandro et al. 2006).
Figure 1 The main cellular processes. Conceptual representation of the main cellular processes that occur inside the cell. Signaling cascades
receive external signals from the environment, either by binding to an extracellular receptor or, as illustrated, by passing through a channel and
binding to an internal receptor. This signal is then propagated through a signaling cascade that involves the sequential phosphorylation of
several proteins, leading to gene activations. Gene regulatory networks control the transcription level of genes. Genes are transcribed into RNA
molecules, which are subsequently translated into proteins. These proteins are involved in all cellular functions. Some proteins are enzymes
involved in the catalysis of metabolic reactions. Metabolic networks obtain energy and carbon from external sources using internal conversion
steps. The internal metabolites can be used for cellular growth, or converted into by-products that are excreted by the cell. Their concentration
level can also influence gene regulation.
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 3 of 14
While modularity represents the horizontal organization
of the cell, living systems also present vertical organization
(Cheng and Hu 2010,). Molecules, cells, tissues,
organs, organisms, populations and ecosystems reflect
the hierarchical organization of life. A modeling formalism
that supports hierarchical models and different
levels of abstraction will cope with models that connect
vertical organization layers using top-down, bottom-up
or middle-out approaches (Noble 2002).
Multi-state components
Some compounds may have multiple states, for example,
a protein may be modified by phosphorylation. This is a
very common case in signaling networks. The state of a
protein can affect its functionality and consequently the
reactions in which it participates. Therefore, different
states are represented by different entities. However, a
protein with n binding sites will have 2n
possible states,
which results in a combinatorial explosion of entities
and reactions (Hlavacek et al. 2003,, Blinov et al. 2004).
To avoid this problem, a suitable modeling formalism
should consider entities with internal states and statedependent
reactions.
Spatial structure and compartmentalization
On its lowest level, the cell can be seen as a bag of
mixed molecules. However, this bag is compartmentalized
and requires transport processes for some species
to travel between compartments. Furthermore, in some
compartments, including the cytosol, the high viscosity,
slow diffusion and amount of molecules may not be sufficient
to guarantee a spatial homogeneity (Takahashi et
al. 2005,). Spatial localization and concentration gradients
are actually important mechanisms in biological
processes such as morphogenesis (Turing 1952).
Qualitative analysis
Experimental determination of kinetic parameters to
build quantitative models is a cumbersome task.
Furthermore, they are dependent on the experimental
conditions, and there is generally no guarantee that the
in vitro values will match the in vivo conditions (Teusink
et al. 2000,). Therefore, several models are only
qualitative. Although these models do not allow for
quantitative simulations, they allow us to ask qualitative
questions about the system and to learn valuable knowledge.
For instance, elementary mode analysis is used for
calculating all possible pathways through a metabolic
network (Schuster et al. 1999).
Dynamic simulation
Dynamic simulation allows the prediction of the transient
behavior of a system under different conditions. For
each model, the particular simulation approach depends
on the type of components included, which depend on
the nature of the involved interactions and also on the
available information for their characterization.
In regulatory networks, genes are activated and deactivated
through the transcription machinery. Due to their
complexity and the lack of kinetic information, the transcriptional
details are usually not considered. Instead,
genes are modeled by discrete (typically boolean) variables
that change through discrete time steps. This is
the simplest simulation method and requires models
with very little detail.
Signaling cascades are triggered by a low number of
signaling molecules. Therefore, it is important to take
into consideration the inherent stochasticity in the diffusion
of these molecules. Stochastic simulation is a common
approach for simulation of signaling networks
(Costa et al. 2009). This approach requires the attribution
of probability functions for each reaction in the
model.
Metabolic reactions, on the other hand, comprise large
quantities of metabolites. Therefore, their behavior can
be averaged and modeled by continuous variables governed
by deterministic rate laws (Chassagnole et al.
2002). This requires a significant amount of experimental
data for estimation of the kinetic parameters.
Standardization
Biological models need to be represented in a common
format for exchange between different tools. The Systems
Biology Markup Language (SBML) has become
the de facto standard of the SB community, and is currently
supported by over two hundred tools (Hucka et
al. 2003,). It is an XML-based language for representation
of species, compartments, reactions and their specific
properties such as concentrations, volumes,
stoichiometry and rate laws. It also facilitates the storage
of tool specific data using appropriate tags. SBML
was initially focused on biochemical reaction networks
such as metabolic and signaling pathways, therefore it
is not so well-suited for modeling other kinds of processes
such as regulatory networks which are better
described by logical models. Nevertheless, these and
other limitations are being addressed in the development
of future releases (Finney and Hucka 2003,,
Hucka et al. 2010).
CellML is another XML-based language with a similar
purpose to SBML albeit more generic (Lloyd et al.
2004,). The Systems Biology Graphical Notation (SBGN)
(Le Novère et al. 2009) is a standard that focuses on the
graphical notation and may be seen as a complement to
SBML. It addresses the visualization concerns discussed
previously, specially the creation of graphical models
with a common notation that can be shared and unambiguously
interpreted by different people.
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 4 of 14
Modeling Formalisms
Many formalisms have been used to model biological
systems, in part due to the diversity of phenomena that
occur in living systems, and also due to the multidisciplinarity
of the research teams. Biologists may be more
familiar with mathematical modeling and computer
scientists may be religious to their computational formalism
of choice. The dichotomy between mathematical
and computational models has been discussed elsewhere
(Hunt et al. 2008). Although they follow different
approaches (denotational vs operational), it has been
questioned if there is such a clear separation between
mathematical and computational models. Therefore, we
will briefly describe several formalisms regardless of
such distinction. Table 1 summarizes some of the literature
references reviewed herein, classified by type of
intracellular process implemented. Toy examples of the
formalisms with graphical notation are depicted in Figure
2.
Boolean networks
Boolean networks (Figure 2a) were introduced by Kauffman
in 1969 to model gene regulatory networks (Kauffman
1969). They consist on networks of genes, modeled
by boolean variables that represent active and inactive
states. At each time step, the state of each gene is determined
by a logic rule which is a function of the state of
its regulators. The state of all genes forms a global state
that changes synchronously. For large network sizes (n
nodes) it becomes impractical to explore all possible
states (2n
). This type of model can be used to find
steady-states (called attractors), and to analyze network
robustness (Li et al. 2004,). Boolean networks can be
inferred directly from experimental gene expression
time-series data (Akutsu et al. 1999,, D’haeseleer et al.
2000,). They have also been applied in some studies to
model signaling pathways (Gupta et al. 2007,, SaezRodriguez
et al. 2007,). To cope with the inherent noise
and the uncertainty in biological processes, stochastic
extensions like Boolean networks with noise (Akutsu et
al. 2000,) and Probabilistic Boolean networks (Shmulevich
et al. 2002) were introduced.
Bayesian networks
Bayesian networks (Figure 2b) were introduced in the
80’s by the work of Pearl (Pearl 1988,). They are a special
type of probabilistic graphs. Their nodes represent
random variables (discrete or continuous) and the edges
represent conditional dependencies, forming a directed
acyclic graph. Each node contains a probabilistic function
that is dependent on the values of its input nodes.
There are learning methods to infer both structure and
probability parameters with support for incomplete data.
This flexibility makes Bayesian networks specially interesting
for biological applications. They have been used
for inferring and representing gene regulatory (Friedman
2004,, Pena et al. 2005,, Grzegorczyk et al. 2008,, Auliac
et al. 2008,) and signaling networks (Sachs et al. 2002,;
2005,). One disadvantage of Bayesian networks is the
inability to model feedback loops, which is a common
motif in biological networks. This limitation can be
overcome by dynamic Bayesian networks (Husmeier
2003,, Kim et al. 2003,, Zou and Conzen 2005,, Dojer et
al. 2006). In this case, the variables are replicated for
each time step and the feedback is modeled by connecting
the nodes at adjacent time steps.
Petri nets
Petri nets (Figure 2c) were created in the 60’s by Carl
Adam Petri for the modeling and analysis of concurrent
systems (Petri 1962). They are bipartite graphs with two
types of nodes, places and transitions, connected by
directed arcs. Places hold tokens that can be produced
(respectively, consumed) when an input (respectively,
output) transition fires. The execution of a Petri net is
non-deterministic and specially suited for distributed systems
with concurrent events. Their application to biological
processes began in 1993, by the work of Reddy and
coworkers, to overcome the limitations in quantitative
analysis of metabolic pathways (Reddy et al. 1993).
There are currently several Petri net extensions (e.g.:
coloured, timed, stochastic, continuous, hybrid, hierarchical,
functional), forming a very versatile framework
for both qualitative and quantitative analysis. Due to
this versatility, they have been used in metabolic (Küffner
et al. 2000,, Zevedei-Oancea and Schuster 2003,,
Koch et al. 2005,), gene regulatory (Chaouiya et al.
2004,; 2008,), and signaling networks (Sackmann et al.
2006,, Chen et al. 2007,, Breitling et al. 2008,, Hardy and
Robillard 2008,). Also, they are suited for integrating different
types of networks, such as gene regulatory and
metabolic (Simao et al. 2005).
Process algebras
Process algebras are a family of formal languages for modeling
concurrent systems. They generally consist on a set
of process primitives, operators for sequential and parallel
Table 1 Literature references grouped by formalism
BN Bay PN PA CB DE RB ISM CA AB
Signaling + + ++ ++ + ++ ++ ++ + ++
Gene regulatory ++ ++ + + ++ +
Metabolic ++ ++ ++ + +
Overview of the amount of literature references for each formalism classified
by the type of biological process. (+) Few references; (++) Several references;
(BN) Boolean networks; (Bay) Bayesian networks; (PN) Petri nets; (PA) Process
algebras; (CB) Constraint-based models; (DE) Differential equations; (RB) Rulebased
models; (ISM) Interacting state machines; (CA) Cellular automata; (AB)
Agent-based models.
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 5 of 14
composition of processes, and communication channels.
The Calculus of Communicating Systems (CCS) was one
of the first process algebras, developed during the 70’s by
Robin Milner (Milner 1980), and later gave origin to the
more popular π-calculus (Milner et al. 1992). In SB the
application of process algebras has been mainly focused
on signaling pathways due to their similarity to communication
processes. About a decade ago, Regev and coworkers
published their pioneer work on the representation of
signaling pathways with π-calculus (Regev et al. 2000,;
2001). They later extended their work using stochastic πcalculus
(BioSpi) to support quantitative simulations
(Priami et al. 2001,) and using Ambient calculus (BioAmbients)
for representation of compartments (Regev et al.
2004,). Other relevant biological applications of process
algebras include Bio-calculus (Nagasaki et al. 1999), -calculus
(for protein-protein interactions) (Danos and Laneve
2004,), CCS-R (Danos and Krivine 2007,), Beta binders
(Priami and Quaglia 2005,), Brane Calculi (Cardelli 2005,),
SpacePi (John et al. 2008,), Bio-PEPA (Ciocchetta and Hillston
2008,; 2009,) and BlenX (Dematte et al. 2008,, Priami
et al. 2009).
Constraint-based models
Constraint-based models for cellular metabolism began
spreading during the 90’s, mainly influenced by the
work of Palsson and coworkers (Varma and Palsson
1994,). Assuming that cells rapidly reach a steady-state,
these models overcome the limitations in lack of experimental
data for parameter estimation inherent in fully
detailed dynamic models. They are based on stoichiometric,
thermodynamic and enzyme capacity constraints
(Reed and Palsson 2003,, Price et al. 2003). Instead of a
single solution, they define a space of possible solutions
representing different phenotypes that comply with the
constraints. The simplicity in this formulation allows its
application to genome-scale metabolic models comprising
thousands of reactions, such as the most recent
metabolic reconstruction of E. coli (Orth et al. 2011).
Constraint-based models have been used in metabolic
engineering strategies for the determination of flux distributions
(metabolic flux analysis (Wiechert 2001,), flux
balance analysis (Kauffman et al. 2003,)), knockout phenotype
predictions (minimization of metabolic adjustment
(Segrè et al. 2002,), regulatory on/off minimization
(Shlomi et al. 2005,)) or enumerating all possible pathways
(extreme pathways (Schilling et al. 2000,), elementary
flux modes (Schuster et al. 1999,)). Although their
main application has been on metabolic networks, there
are recent efforts towards application on gene regulatory
and signaling networks (Papin et al. 2005,, Gianchandani
et al. 2009,, Lee et al. 2008a).
Figure 2 Formalisms with visual representation Toy examples of the formalisms with visual representation. a) Boolean network: genes are
represented by nodes (a, b, c, d) and the arrows represent activation and repression; b) Bayesian network: the value of the output nodes (genes
c, d, e) are given by a probability function that depends on the value of the input nodes (genes a and b); c) Petri net: places represent
substances (a, b, c), transitions represent reactions (p, q) and the arrows represent consumption and production; d) Agent-based model: two
types of agents, representing two different kinds of cells (or two kinds of molecules) can move freely and interact within the containing space;
e) Interacting state machine: systems are represented by their state (a, b), where each state may contain one or more internal sub-states (b, d, e),
arrows represent the transition between different states of the system; f) Rule-based model (represented by a contact map): agents represent
proteins (P, Q, R, S), which may contain different binding sites (a to f ), the connections represent the rules for possible interactions (such as
phosphorylation); g) Cellular automata: a grid where the value of each element can represent different kinds of cells (or molecules), that can
change by interaction with their immediate neighbors.
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 6 of 14
Differential equations
Differential equations describe the rate of change of
continuous variables. They are typically used for modeling
dynamical systems in several areas. Systems of nonlinear
ordinary differential equations (ODEs) have been
used in SB to describe the variation of the amount of
species in the modeled system as a function of time.
They have been applied to all kinds of biological pathways
(Chassagnole et al. 2002,, Tyson et al. 2003,, Chen
et al. 1999,, Rizzi et al. 1997,). With a fully detailed
kinetic model, one can perform time-course simulations,
predict the response to different inputs and design system
controllers. However, building ODE models
requires insight into the reaction mechanisms to select
the appropriate rate laws, and experimental data to estimate
the kinetic parameters. The lack of kinetic data
has limited the size of the modeled networks to pathway
size, with exception for the human red blood cell model
(Jamshidi et al. 2001).
Approximative rate laws such as generalized mass
action (GMA) (Horn and Jackson 1972,), S-systems
(Savageau and Voit 1987,), lin-log (Visser and Heijnen
2003,), and convenience kinetics (Liebermeister and
Klipp 2006,), have compact standard formulations that
can facilitate the development and analysis of large-scale
models (Heijnen 2005,, Costa et al. 2010,). This opens
the possibility for kinetic modeling at the genome-scale
(Smallbone et al. 2010).
Other types of differential equations, such as stochastic
differential equations (SDEs) and partial differential
equations (PDEs) can be used respectively to account
for stochastic effects and spatial distribution (Turner et
al. 2004,). Piecewise-linear differential equations
(PLDEs) have been used to integrate discrete and continuous
features in gene regulatory networks (De Jong et
al. 2004,, Batt et al. 2005).
Rule-based models
Rule-based (Figure 2f) modeling comprises a recent
approach to the problem of multi-state components in
biological models. In rule-based formalisms the species
are defined in a structured manner and support multiple
states. The reaction rules are defined as transformations
of classes of species, avoiding the need for specifying
one reaction per each possible state of a species. This
high-level specification is then automatically transformed
into a biochemical network with the set of species
and reactions generated by the specification. This
kind of formalism is implemented in BioNetGen (Blinov
et al. 2004,) which generates an ODE model or a stochastic
simulation from the ruled-based specification. It
has been applied in the modeling of different signaling
pathways (Blinov et al. 2006,, Barua et al. 2007,; 2008,;
2009). A similar rule-based formalism used for this kind
of pathways is the  language, where the species are
defined by agents that have a structured interface for
interaction with other agents (Danos et al. 2007,; 2009,,
Feret et al. 2009,). The possible interactions are defined
by a set of rules, which can be visualized by a contact
map. BIOCHAM implements a rule-based approach for
model specification which is complemented with a temporal
logic language for the verification of the properties
the biological models (Calzone et al. 2006).
The main advantage of the rule-based approach is that
it can avoid the combinatorial explosion problem in the
generation and simulation of the complete reaction network
by performing stochastic simulations that only
instantiate the species and reactions as they become
available (Colvin et al. 2009,; 2010,) or by the generation
of coarse-grained ODE systems (Feret et al. 2009,). Spatial
simulation has been addressed recently by the inclusion
of geometric information as part of the structure of
the species (Gruenert et al. 2010).
Interacting state machines
Interacting state (Figure 2e) machines are diagram-based
formalisms that describe the temporal behavior of a system
based on the changes in the states of its parts. They
are suited to model biological behavior in a qualitative
way as they require little quantitative data. They differ
from other approaches as they define a system in terms
of its states rather than its components. They are typically
used for model checking and interactive execution.
One such formalism is Statecharts, developed by
David Harel during the 80’s (Harel 1987,) that was first
applied in biology for modeling the T-cell activation
process (Kam et al. 2001,, Efroni et al. 2003,) and more
recently in pancreatic organogenesis (Setty et al. 2008,).
In this formalism, the state of a system may contain
sub-states at multiple levels, allowing an hierarchical
view of the system and the relation between events at
smaller and larger scales. Other related formalisms are
Reactive Modules (Alur and Henzinger 1999,) and Live
Sequence Charts (Damm and Harel 2001), which, along
with the former, have been applied in the modelling of
C. elegans vulval development (Fisher et al. 2005,; 2007).
Cellular automata
Cellular automata (Figure 2g) were created by von Neumann
and Ulam in the 40’s (Von Neumann and Burks
1966,). They are discrete dynamic models that consist
on a grid of cells with a finite number of states. A cellular
automaton has an initial configuration that changes
at each time step through a predefined rule that calculates
the state of each cell as a function of the state of
its neighbors at the previous step. They are specially suited
for modeling complex phenomena in a scale-free
manner and have been used in biological studies for a
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 7 of 14
long time (Ermentrout and Edelstein-Keshet 1993). Due
to their spatial features their main applications are
related to molecular dynamics and cellular population
dynamics.
Application examples at the molecular level include
enzyme reaction networks that account for spatial diffusion
(Weimar 2002,) and signaling pathways (Wurthner
et al. 2000,, Kier et al. 2005,). At the cellular level they
were used for models such as those of bacterial aggregation
(Sozinova et al. 2005,) and HIV infection (Zorzenon
dos Santos and Coutinho 2001,, Corne and Frisco
2008,). Dynamic cellular automata are a variation of cellular
automata that allows for movement of the cell contents
inside the grid, mimicking brownian motion. They
were used to model enzyme kinetics, molecular diffusion
and genetic circuits (Wishart et al. 2005).
Agent-based models
Agent-based models (Figure 2d) describe the interactions
among multiple autonomous agents. They are
similar in concept to cellular automata, except in this
case, instead of using a grid and synchronized time
steps, the agents move freely within the containing
space. Likewise, they are used to study complex phenomena
and emergent dynamics using populations of
agents with simple rules. At the molecular level they
have been mainly used to build models of signaling
pathways that account for spatial distribution and the
structural properties of the cell (Gonzalez et al. 2003,,
Pogson et al. 2006,; 2008,, An 2009,). Recently, they
have also been applied to metabolic reactions (Klann et
al. 2011,). However, their main application is at the
multi-cellular level, where they have been used to study
granuloma formation (Segovia-Juarez et al. 2004,),
tumor growth (Zhang et al. 2007,, Engelberg et al.
2008,), morphogenesis (Grant et al. 2006,), chemotaxis
(Emonet et al. 2005,), immune responses (Lollini et al.
2006,, Li et al. 2008,), and several others (Thorne et al.
2007,, Merelli et al. 2007).
Other formalisms
There are other modeling formalisms that have been
used in SB which are worth mentioning. Cybernetic
modeling is one of the earliest approaches for dynamic
modeling that was used in bioprocess applications
(Kompala et al. 1984,, Dhurjati et al. 1985,). A recent
approach combines cybernetic variables with elementary
flux modes (Young et al. 2008,, Kim et al. 2008,). Hybrid
automata addressed the integration of discrete and continuous
components in the Delta-Notch signaling pathway
(Ghosh and Tomlin 2001,; 2004,). Artificial neural
networks were used to model gene expression (Vohradsky
2001,). Molecular interaction maps are a popular
graph-based formalism created by Kohn in 1999, (Kohn
1999,, Kohn et al. 2006,, Luna et al. 2011,) that influenced
the SBGN standard (Le Novère et al. 2009,).
Other graph-based formalisms include modular interaction
networks (Yartseva et al. 2007,) and logical interaction
hypergraphs (Klamt et al. 2006,). The P systems
formalism created by Paun in 1998, inspired the area of
membrane computing (Paun 2000,) and has been
recently applied in SB (Pérez-Jiménez and Romero-Campero
2006,, Cao et al. 2010,). Chemical organization theory
is a recent approach for modeling biochemical
reaction networks that uses set theory to analyze how
they can be decomposed into self-maintaining subnetworks
called organizations, that reveal dynamic properties
of the system (Dittrich and Di Fenizio 2007,). It has
been used to analyze different types of networks including
signaling pathways and regulated metabolic networks
(Centler et al. 2007,; 2008,, Kaleta et al. 2008,;
2009).
Formalisms conversion
The inability of the formalisms to fit all purposes has
driven the development of methodologies to convert
between different formalisms. Two different methods
have been proposed to convert Boolean networks to
Petri nets (Chaouiya et al. 2004,, Steggles et al. 2007,).
Boolean networks have also been converted to constraint-based
models (Gianchandani et al. 2006,) and to
ODEs (Wittmann et al. 2009,). Other formalisms have
also been converted to ODEs, including constraintbased
models (Smallbone et al. 2007,), Petri nets (Gilbert
and Heiner 2006,), process algebras (Calder et al.
2005,) and rule-based models (Feret et al. 2009,). When
the mappings are made from abstract to more detailed
models they usually require some assumptions and
insight into the reaction mechanisms. The language for
biochemical systems (LBS) is a recent language that
integrates a rule-based approach with process calculus,
and supports the generation of Petri nets, ODEs and
continuous time Markov chains (Pedersen and Plotkin
2010).
Formalisms integration
Along with the conversion between formalisms, there is
also a recent trend for developing methods that support
integrated simulation of different formalisms in order to
integrate different kinds of biological networks, where
each network is modeled in its own formalism. Extensions
of flux balance analysis (FBA) (Kauffman et al.
2003,), such as regulated FBA (rFBA) (Covert and Palsson
2002,) and steady-state regulated FBA (SR-FBA)
(Shlomi et al. 2007,) incorporate boolean rules into constraint-based
models for integrated simulation of regulatory
and metabolic networks. Integrated FBA (iFBA)
extends rFBA by integrating kinetic information from
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 8 of 14
ODE models (Covert et al. 2008,). Integrated dynamic
FBA (idFBA) aims to integrate signaling, regulatory and
metabolic networks by modeling all networks in the
constraint-based formulation (Lee et al. 2008b,). Biochemical
systems theory (BST) has been recently integrated
with Hybrid Functional Petri Nets (HFPN) in
order to integrate metabolic, regulatory and signaling
networks, in a framework that accounts for different
time-scales as well as discrete, stochastic and continuous
effects (Wu and Voit 2009a,;b).
Comparison of the Formalisms
The diversity of problems studied in SB gave rise to the
application of several different types of formalisms. A
comparison of the amount of literature references for
each formalism, classified by the type of biological process
described, is given in Table 1. We can observe that
only four formalisms (Petri nets, constraint-based models,
differential equations and cellular automata) have
been applied to all three types of biological networks,
which makes them potential candidates as a suitable
integrative formalism for whole-cell modeling. However,
this should not exclude other formalisms from this possibility
as well. Another interesting observation is that
metabolism is the biological process with the smaller
number of formalisms applied. This is likely due to the
fact that its two main frameworks (differential equations
and constraint-based) are well suited for modeling metabolic
networks. On the other hand, all of the formalisms
have been applied to signaling pathways. One possible
reason is that they require the largest number of modeling
features, including spatial localization and multistate
components.
The modeling features provided by the formalisms
reviewed in this work are compared in Table 2. Some of
the features are only available in extensions of the formalisms.
We can observe that no single formalism covers
the whole spectrum of features desired for modeling all
kinds of biological components. Petri nets and rulebased
models are among the formalisms that cover most
features. Petri nets have several extensions available, and
although none of the extensions alone fulfills all requisites,
altogether they form a very versatile modeling framework.
Rule-based models present a high level of
abstraction and can be used for stochastic simulation
and automatic generation of lower level ODE-based
representations. Therefore, they take advantage of the
analytic power of abstract representations, preserving
the ability to generate stochastic and deterministic
simulations.
Although none of the formalisms implements all the
required features, this is not necessarily a limitation,
since different formalisms can be used at different stages
of the modeling process. The model construction
process begins with biochemical knowledge and experimental
data that allow an enumeration of the components
and connections in the system. Graph-based
models, such as Boolean networks, Bayesian networks
and Petri nets can be used for modeling this map of
interactions. This allows a deeper understanding of the
organization of the system through topological analysis,
and drives new experiments by finding gaps in the models.
This kind of models also allows qualitative descriptions
of system behavior and coarse simulation
capabilities. If the reactions’ stoichiometry and directionality
are known, one may analyze the steady-states of the
system using constraint-based models. Finally, if extensive
experimental data is available to infer the kinetics
of the reactions, probabilistic or deterministic rate laws
can be used to create dynamic models. These are used
to generate time-course simulations under different sets
of initial conditions. Stochastic process algebras, stochastic
Petri nets, continuous Petri nets, rule-based
models and differential equations, would all be ideal
candidates for this purpose.
Cellular automata and agent-based models account for
the individual replicas of each component in the system.
When applied at the molecular level, this paradigm provides
accurate simulations of small sets of biochemical
reactions that account for spatial diffusion. However, it
becomes infeasible to perform simulations at the genome-scale
network level, as this would imply modeling
every copy of all substances present in the cell. Nevertheless,
this approach is very convenient for modeling at
the cell population level, as it allows to track changes in
individual cells and to study the emergent properties of
cellular communities.
In search for a proper formalism perhaps the most
important aspect to consider is the balance between
simplicity and expressiveness. There is a price to pay for
Table 2 Modeling formalisms and implemented features
BN Bay PN PA CB DE RB ISM CA AB
Visualization + + + + + + +
Topology + + + +
Modularity + + + +
Hierarchy e e +
Multi-state e + + + +
Compartments e + + +
Spatial e e + +
Qualitative + + + + + +
Synchronized + e +
Stochastic e + e + e + + + +
Continuous e + +
Modeling formalisms and implemented features. (+) Supported feature; (e)
Available through extension; (BN) Boolean networks; (Bay) Bayesian networks;
(PN) Petri nets; (PA) Process algebras; (CB) Constraint-based models; (DE)
Differential equations; (RB) Rule-based models; (ISM) Interacting state
machines; (CA) Cellular automata; (AB) Agent-based models.
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 9 of 14
the amount of features provided by a formalism, which
may come at the cost of increased model complexity.
The complexity of the representation and the number of
parameters determines the amount of experimental data
required for model construction. This is the reason why
the most simple formalisms such as Boolean networks
and constraint-based models have been used to build,
respectively, gene regulatory and metabolic networks at
the genome scale. This concern is most critical when
not only the parameters but also the network structure
are unknown. Model inference (also known as reverse
engineering) methods are applied in these cases. They
have been used to infer Boolean networks (Akutsu et al.
1999,, D’haeseleer et al. 2000,), Bayesian networks
(Friedman 2004,, Auliac et al. 2008,), Petri nets (Nummela
and Julstrom 2005,, Durzinsky et al. 2011,) and
ODEs (Kimura et al. 2005,, Iba 2008) from experimental
data. However, the scalability of these methods is greatly
dependent on the simplicity of the underlying
formalism.
Perspective
With the myriad of formalisms that have been applied
in SB, we face the challenge of choosing the proper
formalism for the problem in hands. As more data
become available for network reconstruction, we move
towards integration of all kinds of biological networks,
namely signaling, gene regulatory and metabolic.
Although some formalisms like Petri nets, constraintbased
models and differential equations have been
applied for all these networks, no single formalism covers
the whole spectrum of functionalities reviewed in
this work. Petri nets have several extensions available,
covering most of the features analyzed, with the exception
of compartments and spatial localization. Rulebased
models are another strong candidate as they also
cover a great part of the modeling features. These are
definitely two formalisms to keep under consideration in
the near future.
The model building process is based on iterative steps
of refinement and validation. Recent approaches for
genome-scale kinetic modeling of metabolism, begin
with the network topology, modeled in the constraintbased
framework, and then refine the models by adding
the kinetic structure in order to generate ODE models
(Jamshidi and Palsson 2010,, Smallbone et al. 2010).
Petri nets seem to be a promising formalism for this
purpose, given that discrete Petri nets can model the
network topology, and can then be used as a scaffold for
the generation of dynamic models based on continuous
or stochastic Petri nets. The fact that the same kind of
formalism is used during the whole model refinement
process, helps the creation of more straightforward
methods for automatic mapping and validation of the
models.
A common problem in the analysis of biological networks
is the combinatorial explosion that originates
from the complexity of large models. A typical example
is the computation of elementary flux modes at the genome-scale,
requiring modular decomposition of the networks
(Schuster et al. 2002,). This problem will
aggravate as we get closer to whole-cell modeling. The
solution may reside in the application of hierarchical
formalisms to represent an intermediate level between
the reaction and the cell. As stated elsewhere, one
should not “model bulldozers with quarks” (Goldenfeld
1999). Hierarchical Petri nets, BioAmbients and Statecharts
are formalisms that support hierarchical
modeling.
Models of cell populations are also becoming more
frequent. They are used to study scenarios like cell differentiation,
chemotaxis, infections or tumor growth.
This kind of models depends on the internal dynamics
of the cells as well as population dynamics. Therefore,
they require modeling of interactions across organizational
scales (Walker and Southgate 2009). It is possible
that in the future, we will have multi-scale models that
integrate formalisms. For instance, the evolution of a
population of cells may be modeled by an agent-based
model, where each agent has a boolean network for
internal representation of its gene expression.
In order to convert between different formalisms it is
important to have a standard representation format that
preserves most of the features in the models. SBML is
the most popular standard in the SB community, currently
supported by over two hundred tools (Hucka et
al. 2003,). Most of the modeling features covered herein
have been proposed for future versions of SBML (Finney
and Hucka 2003). These include hierarchical model
composition, rule-based modeling, spatial geometry and
alternative mathematical representations. The compatibility
with the SBML representation will dictate which
formalisms will prevail in the future.
Many of the proposed formalisms, such as Petri nets
or process algebras, were originally created by the computational
community for the specification of software
systems, where the final system has to comply to the
model. The biological community faces the opposite
problem, where the model has to mimic the system’s
behavior, and where most components cannot even be
measured directly. Therefore, a proper framework for
SB must provide not only a suitable formalism with
attractive features and simulation methods, but also
methods for model inference and parameter estimation
that are sufficiently robust to handle experimental data
that are incomplete and prone to measurement error.
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 10 of 14
Acknowledgements
Research supported by grants SFRH/BD/35215/2007 and SFRH/BD/25506/
2005 from the Fundação para a Ciência e a Tecnologia (FCT) and the MITPortugal
Program through the project “Bridging Systems and Synthetic
Biology for the development of improved microbial cell factories” (MIT-Pt/
BS-BB/0082/2008).
Author details
1
IBB-Institute for Biotechnology and Bioengineering/Centre of Biological
Engineering, University of Minho, Campus de Gualtar, 4710-057 Braga,
Portugal 2
Department of Informatics/CCTC, University of Minho, Campus de
Gualtar, 4710-057 Braga, Portugal 3
Department of Biological Engineering/
Computer Science and Artificial Intelligence Laboratory, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA
Competing interests
The authors declare that they have no competing interests.
Received: 18 November 2011 Accepted: 5 December 2011
Published: 5 December 2011
References
Akutsu T, Miyano S, Kuhara S (1999) Identification of genetic networks from a
small number of gene expression patterns under the Boolean network
model. In Pacific Symposium on Biocomputing, vol 4.World Scientific
Computing, Singapore, pp 17–28
Akutsu T, Miyano S, Kuhara S (2000) Inferring qualitative relations in genetic
networks and metabolic pathways. Bioinformatics 16(8):727–734
Albert R, Wang R (2009) Discrete dynamic modeling of cellular signaling
networks. Methods Enzymol 467:281–306
Aldridge B, Burke J, Lauffenburger D, Sorger P (2006) Physicochemical modelling
of cell signalling pathways. Nat Cell Biol 8(11):1195–1203
Alur R, Henzinger T (1999) Reactive modules. Lect Notes Comput Sc 15(1):7–48
An G (2009) A model of TLR4 signaling and tolerance using a qualitative,
particle-event-based method: introduction of spatially configured stochastic
reaction chambers (SCSRC). Math Biosci 217(1):43–52
Andrianantoandro E, Basu S, Karig D, Weiss R (2006) Synthetic biology: new
engineering rules for an emerging discipline. Mol Syst Biol 2(1)
Assenov Y, Ramirez F, Schelhorn S, Lengauer T, Albrecht M (2008) Computing
topological parameters of biological networks. Bioinformatics 24(2):282–284
Auliac C, Frouin V, Gidrol X, D’Alché-Buc F., et al (2008) Evolutionary approaches
for the reverse-engineering of gene regulatory networks: A study on a
biologically realistic dataset. BMC Bioinf 9(1):91
Barabási A, Oltvai Z (2004) Network biology: understanding the cell’s functional
organization. Nat Rev Genet 5(2):101–113
Barua D, Faeder J, Haugh J (2007) Structure-based kinetic models of modular
signaling protein function: focus on Shp2. Biophys J 92(7):2290–2300
Barua D, Faeder J, Haugh J (2008) Computational models of tandem Src
homology 2 domain interactions and application to phosphoinositide 3kinase.
J Biol Chem 283(12):7338–7345
Barua D, Faeder J, Haugh J (2009) A Bipolar Clamp Mechanism for Activation of
Jak-Family Protein Tyrosine Kinases. PLoS Comput Biol 5(4):e1000364
Batt G, Ropers D, De Jong H, Geiselmann J, Mateescu R, Page M, Schneider D
(2005) Validation of qualitative models of genetic regulatory networks by
model checking: Analysis of the nutritional stress response in Escherichia coli.
Bioinformatics 21(Suppl 1):i19–i28
Blinov M, Faeder J, Goldstein B, Hlavacek W (2004) BioNetGen: software for rulebased
modeling of signal trans-duction based on the interactions of
molecular domains. Bioinformatics 20(17):3289–3291
Blinov M, Faeder J, Goldstein B, Hlavacek W (2006) A network model of early
events in epidermal growth factor receptor signaling that accounts for
combinatorial complexity. Biosystems 83(2-3):136–151
Breitling R, Gilbert D, Heiner M, Orton R (2008) A structured approach for the
engineering of biochemical network models, illustrated for signalling
pathways. Brief Bioinform 9(5):404–421
Burgard A, Pharkya P, Maranas C (2003) Optknock: a bilevel programming
framework for identifying gene knockout strategies for microbial strain
optimization. Biotechnol Bioeng 84(6):647–657
Calder M, Gilmore S, Hillston J (2005) Automatically deriving ODEs from process
algebra models of signalling pathways. Computational Methods in Systems
Biology. Springer, Berlin Heidelberg pp 204–215
Calzone L, Fages F, Soliman S (2006) BIOCHAM: an environment for modeling
biological systems and formalizing experimental knowledge. Bioinformatics
22(14):1805
Cao H, Romero-Campero F, Heeb S, Cámara M, Krasnogor N (2010) Evolving cell
models for systems and synthetic biology. Syst Synth Biol 4(1):55–84
Cardelli L (2005) Brane calculi. Computational Methods in Systems Biology.
Springer, Berlin Heidelberg pp 257–278
Centler F, Fenizio P, Matsumaru N, Dittrich P (2007) Chemical organizations in
the central sugar metabolism of Escherichia coli. Math Model Biol Syst
1(2):105–119
Centler F, Kaleta C, Di Fenizio P, Dittrich P (2008) Computing chemical
organizations in biological networks. Bioinformatics 24(14):1611–1618
Chaouiya C, Remy E, Ruet P, Thieffry D (2004) Qualitative modelling of genetic
networks: From logical regulatory graphs to standard petri nets. Lect Notes
Comput Sc 3099:137–156
Chaouiya C, Remy E, Thieffry D (2008) Petri net modelling of biological
regulatory networks. J Discrete Algorithms 6(2):165–177
Chassagnole C, Noisommit-Rizzi N, Schmid J, Mauch K, Reuss M (2002) Dynamic
modeling of the central carbon metabolism of Escherichia coli. Biotechnol
Bioeng 79(1):53–73
Chen L, Qi-Wei G, Nakata M, Matsuno H, Miyano S (2007) Modelling and
simulation of signal transductions in an apoptosis pathway by using timed
Petri nets. J Biosci 32(1):113–127
Chen T, He H, Church G (1999) Modeling Gene Expression with Differential
Equations. Pacific Symposium on Biocomputing. World Scientific Computing,
Singapore pp 29–40
Cheng C, Hu Y (2010) Extracting the abstraction pyramid from complex
networks. BMC Bioinf 11(1):411
Ciocchetta F, Hillston J (2008) Bio-PEPA: an extension of the process algebra
PEPA for biochemical networks. Electron Notes Theor Comput Sci
194(3):103–117
Ciocchetta F, Hillston J (2009) Bio-PEPA: a framework for the modelling and
analysis of biological systems. Theor Comput Sci 410(33-34):3065–3084
Colvin J, Monine M, Faeder J, Hlavacek W, Von Hoff D, Posner R (2009)
Simulation of large-scale rule-based models. Bioinformatics 25(7):910–917
Colvin J, Monine M, Gutenkunst R, Hlavacek W, Von Hoff D, Posner R (2010)
RuleMonkey: software for stochastic simulation of rule-based models. BMC
Bioinf 11(1):404
Corne D, Frisco P (2008) Dynamics of HIV infection studied with cellular
automata and conformon-P systems. Biosystems 91(3):531–544
Costa M, Radhakrishnan K, Wilson B, Vlachos D, Edwards J, Jonsson H (2009)
Coupled stochastic spatial and non-spatial simulations of ErbB1 signaling
pathways demonstrate the importance of spatial organization in signal
transduction. PLoS ONE 4:e6316
Costa R, Machado D, Rocha I, Ferreira E (2010) Hybrid dynamic modeling of
Escherichia coli central metabolic network combining Michaelis-Menten and
approximate kinetic equations. Biosystems 100(2):150–157
Covert M, Palsson B (2002) Transcriptional regulation in constraints-based
metabolic models of Escherichia coli. J Biol Chem 277(31):28058–28064
Covert M, Xiao N, Chen T, Karr J (2008) Integrating metabolic, transcriptional
regulatory and signal transduction models in Escherichia coli. Bioinformatics
24(18):2044–2050
Damm W, Harel D (2001) LSCs: Breathing life into message sequence charts. Lect
Notes Comput Sc 19(1):45–80
Danos V, Feret J, Fontana W, Harmer R (2007) Rule-Based Modelling of Cellular
Signalling. In CONCUR 2007 - Concurrency Theory, vol 4703.Springer, Berlin
Heidelberg, pp 17–41
Danos V, Feret J, Fontana W, Harmer R, Krivine J (2009) Rule-based modelling
and model perturbation. T Comput Syst Biol XI 5750:116–137
Danos V, Krivine J (2007) Formal molecular biology done in CCS-R. Electron
Notes Theor Comput Sci 180(3):31–49
Danos V, Laneve C (2004) Formal molecular biology. Theor Comput Sci
325(1):69–110
De Jong H, Gouzé J, Hernandez C, Page M, Sari T, Geiselmann J (2004)
Qualitative simulation of genetic regulatory networks using piecewise-linear
models. Bull Math Biol 66(2):301–340
Dematte L, Priami C, Romanel A, Soyer O (2008) Evolving BlenX programs to
simulate the evolution of biological networks. Theor Comput Sci
408(1):83–96
D’haeseleer P, Liang S, Somogyi R (2000) Genetic network inference: from coexpression
clustering to reverse engineering. Bioinformatics 16(8):707–726
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 11 of 14
Dhurjati P, Ramkrishna D, Flickinger M, Tsao G (1985) A cybernetic view of
microbial growth: modeling of cells as optimal strategists. Biotechnol Bioeng
27(1):1–9
Dittrich P, Di Fenizio P (2007) Chemical organisation theory. Bull Math Biol
69(4):1199–1231
Dojer N, Gambin A, Mizera A, Wilczyński B, Tiuryn J (2006) Applying dynamic
Bayesian networks to perturbed gene expression data. BMC Bioinf 7(1):249
Durzinsky M, Wagler A, Marwan W (2011) Reconstruction of extended Petri nets
from time series data and its application to signal transduction and to gene
regulatory networks. BMC Syst Biol 5(1):113
Efroni S, Harel D, Cohen I (2003) Toward rigorous comprehension of biological
complexity: modeling, execution, and visualization of thymic T-cell
maturation. Genome Res 13(11):2485–2497
Emonet T, Macal C, North M, Wickersham C, Cluzel P (2005) AgentCell: a digital
single-cell assay for bacterial chemotaxis. Bioinformatics 21(11):2714
Engelberg J, Ropella G, Hunt C (2008) Essential operating principles for tumor
spheroid growth. BMC Syst Biol 2(1):110
Ermentrout G, Edelstein-Keshet L (1993) Cellular Automata Approaches to
Biological Modeling. J Theor Biol 160(1):97–133
Feist A, Herrgård M, Thiele I, Reed J, Palsson B (2008) Reconstruction of
biochemical networks in microorganisms. Nat Rev Microbiol 7(2):129–143
Feret J, Danos V, Krivine J, Harmer R, Fontana W (2009) Internal coarse-graining
of molecular systems. Proc Natl Acad Sci USA 106(16):6453
Finney A, Hucka M (2003) Systems biology markup language: Level 2 and
beyond. Biochem Soc Trans 31:1472–1473
Fisher J, Henzinger T (2007) Executable cell biology. Nat Biotechnol
25(11):1239–1249
Fisher J, Piterman N, Hajnal A, Henzinger T (2007) Predictive Modeling of
Signaling Crosstalk during C. elegans Vulval Development. PLoS Comput Biol
3(5):e92
Fisher J, Piterman N, Hubbard E, Stern M, Harel D (2005) Computational insights
into Caenorhabditis elegans vulval development. Proc Natl Acad Sci USA
102(6):1951–1956
Friedman N (2004) Inferring cellular networks using probabilistic graphical
models. Science 303(5659):799–805
Funahashi A, Morohashi M, Kitano H, Tanimura N (2003) CellDesigner: a process
diagram editor for gene-regulatory and biochemical networks. BIOSILICO
1(5):159–162
Ghosh R, Tomlin C (2001) Lateral inhibition through delta-notch signaling: A
piecewise affine hybrid model. Lect Notes Comput Sc 2034:232–246
Ghosh R, Tomlin C (2004) Symbolic reachable set computation of piecewise
affine hybrid automata and its application to biological modelling: DeltaNotch
protein signalling. Syst Biol 1(1):170–183
Gianchandani E, Joyce A, Palsson B, Papin J (2009) Functional States of the
Genome-Scale Escherichia coli Tran-scriptional Regulatory System. PLoS
Comput Biol 5(6):e1000403
Gianchandani E, Papin J, Price N, Joyce A, Palsson B (2006) Matrix Formalism to
Describe Functional States of Transcriptional Regulatory Systems. PLoS
Comput Biol 2(8):e101
Gilbert D, Heiner M (2006) From Petri Nets to Differential Equations-An
Integrative Approach for Biochemical Network Analysis. In Petri nets and
other models of concurrency, ICATPN 2006, vol 4024.Springer, Berlin
Heidelberg, pp 181–200
Goldenfeld N (1999) Simple lessons from complexity. Science 284(5411):87–89
Gomperts B, Kramer I, Tatham P (2009) Signal transduction. Academic Press
Gonzalez P, Cardenas M, Camacho D, Franyuti A, Rosas O, Lagunez-Otero J
(2003) Cellulat: an agent-based intracellular signalling model. Biosystems
68(2-3):171–185
Grant M, Mostov K, Tlsty T, Hunt C (2006) Simulating Properties of in vitro
Epithelial Cell Morphogenesis. PLoS Comput Biol 2(10):e129
Gruenert G, Ibrahim B, Lenser T, Lohel M, Hinze T, Dittrich P (2010) Rule-based
spatial modeling with diffusing, geometrically constrained molecules. BMC
Bioinf 11(1):307
Grzegorczyk M, Husmeier D, Edwards K, Ghazal P, Millar A (2008) Modelling nonstationary
gene regulatory processes with a non-homogeneous Bayesian
network and the allocation sampler. Bioinformatics 24(18):2071–2078
Gupta S, Bisht S, Kukreti R, Jain S, Brahmachari S (2007) Boolean network analysis
of a neurotransmitter signaling pathway. J Theor Biol 244(3):463–469
Hardy S, Robillard P (2008) Petri net-based method for the analysis of the
dynamics of signal propagation in signaling pathways. Bioinformatics
24(2):209–217
Harel D (1987) Statecharts: A visual formalism for complex systems. Sci Comput
Program 8(3):231–274
Hartwell L, Hopfield J, Leibler S, Murray A (1999) From molecular to modular cell
biology. Nature 402:C47–C52
Heijnen J (2005) Approximative kinetic formats used in metabolic network
modeling. Biotechnol Bioeng 91(5):534–545
Hlavacek W, Faeder J, Blinov M, Perelson A, Goldstein B (2003) The complexity of
complexes in signal transduction. Biotechnol Bioeng 84(7):783–794
Horn F, Jackson R (1972) General mass action kinetics. Arch Ration Mech An
47(2):81–116
Hucka M, Bergmann F, Hoops S, Keating S, Sahle S, Schaff J, Smith L, Wilkinson D
(2010) The Systems Biology Markup Language (SBML): Language
Specification for Level 3 Version 1 Core. Nat Precedings
Hucka M, Finney A, Sauro H, Bolouri H, Doyle J, Kitano H, Arkin A, Bornstein B,
Bray D, Cornish-Bowden A., et al (2003) The Systems Biology Markup
Language (SBML): a medium for representation and exchange of
biochemical network models. Bioinformatics 19(4):524–531
Hunt C, Ropella G, Park S, Engelberg J (2008) Dichotomies between
computational and mathematical models. Nat Biotechnol 26(7):737–738
Husmeier D (2003) Sensitivity and specificity of inferring genetic regulatory
interactions from microarray experiments with dynamic Bayesian networks.
Bioinformatics 19(17):2271–2282
Iba H (2008) Inference of differential equation models by genetic programming.
Inf Sci 178(23):4453–4468
Jamshidi N, Edwards J, Fahland T, Church G, Palsson B (2001) Dynamic
simulation of the human red blood cell metabolic network. Bioinformatics
17(3):286–287
Jamshidi N, Palsson B (2010) Mass action stoichiometric simulation models:
Incorporating kinetics and regulation into stoichiometric models. Biophys J
98:175–185
Jeong H, Tombor B, Albert R, Oltvai Z, Barabási A (2000) The large-scale
organization of metabolic networks. Nature 407(6804):651–654
John M, Ewald R, Uhrmacher A (2008) A spatial extension to the π-Calculus.
Electron Notes Theor Comput Sci 194(3):133–148
Kaleta C, Centler F, Di Fenizio P, Dittrich P (2008) Phenotype prediction in
regulated metabolic networks. BMC Syst Biol 2(1):37
Kaleta C, Richter S, Dittrich P (2009) Using chemical organization theory for
model checking. Bioinformatics 25(15):1915–1922
Kam N, Cohen I, Harel D (2001) The Immune System as a Reactive System:
Modeling T Cell Activation With Stat-echarts. Proceedings of the IEEE 2001
Symposia on Human Centric Computing Languages and Environments
(HCC’01). Stresa, Italy pp 15–22
Karlebach G, Shamir R (2008) Modelling and analysis of gene regulatory
networks. Nat Rev Mol Cell Biol 9(10):770–780
Kauffman K, Prakash P, Edwards J (2003) Advances in flux balance analysis. Curr
Opin Biotechnol 14(5):491–496
Kauffman S (1969) Metabolic stability and epigenesis in randomly constructed
genetic nets. J Theor Biol 22(3):437–467
Kier L, Bonchev D, Buck G (2005) Modeling biochemical networks: a cellularautomata
approach. Chem Biodivers 2(2):233–243
Kim J, Varner J, Ramkrishna D (2008) A hybrid model of anaerobic E. coli GJT001:
Combination of elementary flux modes and cybernetic variables. Biotechnol
Prog 24(5):993–1006
Kim S, Imoto S, Miyano S (2003) Inferring gene networks from time series
microarray data using dynamic Bayesian networks. Brief Bioinform
4(3):228–235
Kimura S, Ide K, Kashihara A, Kano M, Hatakeyama M, Masui R, Nakagawa N,
Yokoyama S, Kuramitsu S, Kon-agaya A (2005) Inference of S-system models
of genetic networks using a cooperative coevolutionary algorithm.
Bioinformatics 21(7):1154–1163
Kitano H (2002) Systems Biology: A Brief Overview. Science 295(5560):1662–1664
Kitano H, Funahashi A, Matsuoka Y, Oda K (2005) Using process diagrams for the
graphical representation of biological networks. Nat Biotechnol 23(8):961–966
Klamt S, Saez-Rodriguez J, Lindquist J, Simeoni L, Gilles E (2006) A methodology
for the structural and functional analysis of signaling and regulatory
networks. BMC Bioinf 7(1):56
Klann M, Lapin A, Reuss M (2011) Agent-based simulation of reactions in the
crowded and structured intracellular environment: Influence of mobility and
location of the reactants. BMC Syst Biol 5(1):71
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 12 of 14
Koch I, Junker B, Heiner M (2005) Application of Petri net theory for modelling
and validation of the sucrose breakdown pathway in the potato tuber.
Bioinformatics 21(7):1219–1226
Kohn K (1999) Molecular interaction map of the mammalian cell cycle control
and DNA repair systems. Mol Biol Cell 10(8):2703–2734
Kohn K, Aladjem M, Kim S, Weinstein J, Pommier Y (2006) Depicting
combinatorial complexity with the molecular interaction map notation. Mol
Syst Biol 2(51)
Kompala D, Ramkrishna D, Tsao G (1984) Cybernetic modeling of microbial
growth on multiple substrates. Biotech-nol Bioeng 26(11):1272–1281
Küffner R, Zimmer R, Lengauer T (2000) Pathway analysis in metabolic databases
via differential metabolic display (DMD). Bioinformatics 16(9):825–836
Le Novère N, Hucka M, Mi H, Moodie S, Schreiber F, Sorokin A, Demir E,
Wegner K, Aladjem M, Wimalaratne S., et al (2009) The systems biology
graphical notation. Nat Biotechnol 27(8):735–741
Lee J, Gianchandani E, Eddy J, Papin J (2008) Dynamic Analysis of Integrated
Signaling, Metabolic, and Regulatory Networks. PLoS Comput Biol 4(5):
e1000086
Lee J, Gianchandani E, Eddy J, Papin J (2008) Dynamic analysis of integrated
signaling, metabolic, and regulatory networks. PLoS Comput Biol 4(5):
e1000086
Li F, Long T, Lu Y, Ouyang Q, Tang C (2004) The yeast cell-cycle network is
robustly designed. Proc Natl Acad Sci USA 101(14):4781–4786
Li N, Verdolini K, Clermont G, Mi Q, Rubinstein E, Hebda P, Vodovotz Y (2008) A
patient-specific in silico model of inflammation and healing tested in acute
vocal fold injury. PLoS ONE 3(7):e2789
Liebermeister W, Klipp E (2006) Bringing metabolic networks to life: convenience
rate law and thermodynamic constraints. Theor Biol Med Model 3(1):1–13
Lloyd C, Halstead M, Nielsen P (2004) CellML: its future, present and past. Prog
Biophys Mol Biol 85(2-3):433–450
Lollini P, Motta S, Pappalardo F (2006) Discovery of cancer vaccination protocols
with a genetic algorithm driving an agent based simulator. BMC Bioinf
7(1):352
Luna A, Karac E, Sunshine M, Chang L, Nussinov R, Aladjem M, Kohn K (2011) A
formal MIM specification and tools for the common exchange of MIM
diagrams: an XML-Based format, an API, and a validation method. BMC Bioinf
12(1):167
Materi W, Wishart D (2007) Computational systems biology in drug discovery
and development: methods and applications. Drug Discov Today 12(7-
8):295–303
Merelli E, Armano G, Cannata N, Corradini F, d’Inverno M, Doms A, Lord P,
Martin A, Milanesi L, Moller S., et al (2007) Agents in bioinformatics,
computational and systems biology. Brief Bioinform 8(1):45–59
Milner R (1980) A calculus of communicating systems. Springer
Milner R, Parrow J, Walker D (1992) A calculus of mobile processes. I. Inf Comput
100(1):1–40
Morris M, Saez-Rodriguez J, Sorger P, Lauffenburger D (2010) Logic-based models
for the analysis of cell signaling networks. Biochemistry 49(15):3216–3224
Nagasaki M, Onami S, Miyano S, Kitano H (1999) Bio-calculus: Its Concept and
Molecular Interaction. In Genome informatics. Workshop on Genome
Informatics, vol 10.Universal Academic Press, Tokyo, pp 133–143
Noble D (2002) The rise of computational biology. Nat Rev Mol Cell Biol
3(6):459–463
Nummela J, Julstrom B (2005) Evolving petri nets to represent metabolic
pathways. Proceedings of the 2005 conference on genetic and evolutionary
computation. ACM, New York pp 2133–2139
Orth J, Conrad T, Na J, Lerman J, Nam H, Feist A, Palsson B (2011) A
comprehensive genome-scale reconstruction of Escherichia coli metabolism
– 2011. Mol Syst Biol 7(1)
Palsson B (2006) Systems Biology: Properties of Reconstructed Networks.
Cambridge University Press
Papin J, Hunter T, Palsson B, Subramaniam S (2005) Reconstruction of cellular
signalling networks and analysis of their properties. Nat Rev Mol Cell Biol
6(2):99–111
Patil K, Rocha I, Forster J, Nielsen J (2005) Evolutionary programming as a
platform for in silico metabolic engineering. BMC Bioinf 6(1):308
Paun G (2000) Computing with membranes. J Comput Syst Sci 61(1):108–143
Pavlopoulos G, Wegener A, Schneider R (2008) A survey of visualization tools for
biological network analysis. Biodata mining 1(1):12
Pearl J (1988) Probabilistic Reasoning in Intelligent Systems: Networks of
Plausible Inference. Morgan Kaufmann Publishers Inc
Pedersen M, Plotkin G (2010) A language for biochemical systems: Design and
formal specification. T Comput Syst Biol XII 5945:77–145
Pena J, Bjorkegren J, Tegnér J (2005) Growing Bayesian network models of gene
networks from seed genes. Bioin-formatics 21(Suppl 2):224–229
Pérez-Jiménez M, Romero-Campero F (2006) P systems, a new computational
modelling tool for systems biology. T Comput Syst Biol VI 4220:176–197
Petri C (1962) Kommunikation mit Automaten. Ph.D. thesis, RheinischWestfälisches
Institut f. instrumentelle Mathematik an d. Univ
Pogson M, Holcombe M, Smallwood R, Qwarnstrom E (2008) Introducing Spatial
Information into Predictive NF-κB Modelling-An Agent-Based Approach. PLoS
ONE 3(6):e2367
Pogson M, Smallwood R, Qwarnstrom E, Holcombe M (2006) Formal agentbased
modelling of intracellular chemical interactions. Biosystems 85(1):37–45
Priami C, Ballarini P, Quaglia P (2009) BlenX4Bio-BlenX for Biologists.
Computational Methods in Systems Biology. Springer, Berlin Heidelberg pp
26–51
Priami C, Quaglia P (2005) Beta binders for biological interactions. Computational
Methods in Systems Biology. Springer, Berlin Heidelberg pp 20–33
Priami C, Regev A, Shapiro E, Silverman W (2001) Application of a stochastic
name-passing calculus to representation and simulation of molecular
processes. Inf Process Lett 80(1):25–31
Price N, Papin J, Schilling C, Palsson B (2003) Genome-scale microbial in silico
models: the constraints-based approach. Trends Biotechnol 21(4):162–169
Ravasz E, Somera A, Mongru D, Oltvai Z, Barabási A (2002) Hierarchical
organization of modularity in metabolic networks. Science
297(5586):1551–1555
Reddy V, Mavrovouniotis M, Liebman M (1993) Petri Net Representations in
Metabolic Pathways. In: Proceedings of the 1st International Conference on
Intelligent Systems for Molecular Biology. AAAI/MIT Press, Menlo Park, CA pp
328–336
Reed J, Palsson B (2003) Thirteen years of building constraint-based in silico
models of Escherichia coli. J Bacteriol 185(9):2692–2699
Regev A, Panina E, Silverman W, Cardelli L, Shapiro E (2004) BioAmbients: an
abstraction for biological compartments. Theor Comput Sci 325(1):141–167
Regev A, Silverman W, Shapiro E (2000) Representing biomolecular processes
with computer process algebra: π-calculus programs of signal transduction
pathways. Proceedings of Pacific Symposium of Biocomputing. World
Scientific Press, Singapore
Regev A, Silverman W, Shapiro E (2001) Representation and simulation of
biochemical processes using the-calculus process algebra. In Pacific
Symposium on Biocomputing, vol 6.World Scientific Press, Singapore, pp
459–470
Rizzi M, Baltes M, Theobald U, Reuss M (1997) In vivo analysis of metabolic
dynamics in Saccharomyces cerevisiae: II. mathematical model. Biotechnol
Bioeng 55(4):592–608
Sachs K, Gifford D, Jaakkola T, Sorger P, Lauffenburger D (2002) Bayesian network
approach to cell signaling pathway modeling. Sci STKE 2002(148):pe38
Sachs K, Perez O, Pe’er D, Lauffenburger D, Nolan G (2005) Causal proteinsignaling
networks derived from multi-parameter single-cell data. Science
308(5721):523–529
Sackmann A, Heiner M, Koch I (2006) Application of Petri net based analysis
techniques to signal transduction pathways. BMC Bioinf 7(1):482
Saez-Rodriguez J, Simeoni L, Lindquist J, Hemenway R, Bommhardt U, Arndt B,
Haus U, Weismantel R, Gilles E, Klamt S, Schraven B (2007) A logical model
provides insights into T cell receptor signaling. PLoS Comput Biol 3(8):e163
Savageau M, Voit E (1987) Recasting nonlinear differential equations as Ssystems:
a canonical nonlinear form. Math Biosci 87(1):83–115
Schilling C, Letscher D, Palsson B (2000) Theory for the systemic definition of
metabolic pathways and their use in interpreting metabolic function from a
pathway-oriented perspective. J Theor Biol 203(3):229–248
Schlitt T, Brazma A (2007) Current approaches to gene regulatory network
modelling. BMC Bioinf 8(Suppl 6):S9
Schuster S, Dandekar T, Fell D (1999) Detection of elementary flux modes in
biochemical networks: a promising tool for pathway analysis and metabolic
engineering. Trends Biotechnol 17(2):53–60
Schuster S, Pfeiffer T, Moldenhauer F, Koch I, Dandekar T (2002) Exploring the
pathway structure of metabolism: decomposition into subnetworks and
application to Mycoplasma pneumoniae. Bioinformatics 18(2):351–361
Segovia-Juarez J, Ganguli S, Kirschner D (2004) Identifying control mechanisms of
granuloma formation during M. tuberculosis infection using an agent-based
model. J Theor Biol 231(3):357–376
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 13 of 14
Segrè D, Vitkup D, Church G (2002) Analysis of optimality in natural and
perturbed metabolic networks. Proc Natl Acad Sci USA 99(23):15112–15117
Setty Y, Cohen I, Dor Y, Harel D (2008) Four-dimensional realistic modeling of
pancreatic organogenesis. Proc Natl Acad Sci USA 105(51):20374–20379
Shlomi T, Berkman O, Ruppin E (2005) Regulatory on/off minimization of
metabolic flux changes after genetic perturbations. Proc Natl Acad Sci USA
102(21):7695–7700
Shlomi T, Eisenberg Y, Sharan R, Ruppin E (2007) A genome-scale computational
study of the interplay between transcriptional regulation and metabolism.
Mol Syst Biol 3(1)
Shmulevich I, Dougherty E, Kim S, Zhang W (2002) Probabilistic Boolean
networks: a rule-based uncertainty model for gene regulatory networks.
Bioinformatics 18(2):261–274
Simao E, Remy E, Thieffry D, Chaouiya C (2005) Qualitative modelling of
regulated metabolic pathways: application to the tryptophan biosynthesis. E
coli Bioinformatics 21(suppl 2):190–196
Smallbone K, Simeonidis E, Broomhead D, Kell D (2007) Something from nothing
- bridging the gap between constraint-based and kinetic modelling. FEBS J
274(21):5576–5585
Smallbone K, Simeonidis E, Swainston N, Mendes P (2010) Towards a genomescale
kinetic model of cellular metabolism. BMC Syst Biol 4(1):6
Sozinova O, Jiang Y, Kaiser D, Alber M (2005) A three-dimensional model of
myxobacterial aggregation by contact-mediated interactions. Proc Natl Acad
Sci USA 102(32):11308–11312
Steggles L, Banks R, Shaw O, Wipat A (2007) Qualitatively modelling and
analysing genetic regulatory networks: a Petri net approach. Bioinformatics
23(3):336–343
Stephanopoulos G (1998) Metabolic engineering. Biotechnol Bioeng 58(2-
3):119–120
Steuer R, Junker B (2008) Computational models of metabolism: Stability and
regulation in metabolic networks. John Wiley & Sons, Inc142
Takahashi K, Arjunan S, Tomita M (2005) Space in systems biology of signaling
pathways-towards intracellular molecular crowding in silico. FEBS Lett
579(8):1783–1788
Teusink B, Passarge J, Reijenga C, Esgalhado E, Van Der Weijden C, Schepper M,
Walsh M, Bakker B, Van Dam K, Westerhoff H., et al (2000) Can yeast
glycolysis be understood in terms of in vitro kinetics of the constituent
enzymes? Testing biochemistry. Eur J Biochem 267(17):5313–5329
Thorne B, Bailey A, Peirce S (2007) Combining experiments with multi-cell agentbased
modeling to study biological tissue patterning. Brief Bioinform
8(4):245–257
Turing A (1952) The Chemical Basis of Morphogenesis. Philos Trans R Soc Lond B
Biol Sci 237(641):37–72
Turner T, Schnell S, Burrage K (2004) Stochastic approaches for modelling in vivo
reactions. Comput Biol Chem 28(3):165–178
Tyson J, Chen K, Novak B (2003) Sniffers, buzzers, toggles and blinkers: dynamics
of regulatory and signaling pathways in the cell. Curr Opin Cell Biol
15(2):221–231
Varma A, Palsson B (1994) Metabolic flux balancing: basic concepts, scientific and
practical use. Nat Biotechnol 12(10):994–998
Visser D, Heijnen J (2003) Dynamic simulation and metabolic re-design of a
branched pathway using linlog kinetics. Metab Eng 5(3):164–176
Vohradsky J (2001) Neural network model of gene expression. FASEB J
15(3):846–854
Von Neumann J, Burks A (1966) Theory of self-reproducing automata. University
of Illinois Press
Walker D, Southgate J (2009) The virtual cell – a candidate co-ordinator for
‘middle-out’ modelling of biological systems. Brief Bioinform 10(4):450
Weimar J (2002) Cellular automata approaches to enzymatic reaction networks.
Cellular Automata 2493:294–303
Wiechert W (2001) 13C metabolic flux analysis. Metab Eng 3(3):195–206
Wishart D, Yang R, Arndt D, Tang P, Cruz J (2005) Dynamic cellular automata: an
alternative approach to cellular simulation. Silico Biol 5(2):139–161
Wittmann D, Krumsiek J, Saez-Rodriguez J, Lauffenburger D, Klamt S, Theis F
(2009) Transforming Boolean models to continuous models: methodology
and application to T-cell receptor signaling. BMC Syst Biol 3(1):98
Wu J, Voit E (2009) Hybrid modeling in biochemical systems theory by means of
functional petri nets. J Bioinform Comput Biol 7(1):107–34
Wu J, Voit E (2009) Integrative biological systems modeling: challenges and
opportunities. Front Comput Sci Chin 3(1):92–100
Wurthner J, Mukhopadhyay A, Peimann C (2000) A cellular automaton model of
cellular signal transduction. Comput Biol Med 30(1):1–21
Yartseva A, Klaudel H, Devillers R, Képès F (2007) Incremental and unifying
modelling formalism for biological interaction networks. BMC Bioinf 8(1):433
Young J, Henne K, Morgan J, Konopka A, Ramkrishna D (2008) Integrating
cybernetic modeling with pathway analysis provides a dynamic, systemslevel
description of metabolic control. Biotechnol Bioeng 100(3):542–559
Zevedei-Oancea I, Schuster S (2003) Topological analysis of metabolic networks
based on Petri net theory. In Silico Biol 3(3):323–345
Zhang L, Athale C, Deisboeck T (2007) Development of a three-dimensional
multiscale agent-based tumor model: simulating gene-protein interaction
profiles, cell phenotypes and multicellular patterns in brain cancer. J Theor
Biol 244(1):96–107
Zorzenon dos Santos R, Coutinho S (2001) Dynamics of HIV infection: A cellular
automata approach. Phys Rev Lett 87(16):168102
Zou M, Conzen S (2005) A new dynamic Bayesian network (DBN) approach for
identifying gene regulatory networks from time course microarray data.
Bioinformatics 21(1):71–79
doi:10.1186/2191-0855-1-45
Cite this article as: Machado et al.: Modeling formalisms in Systems
Biology. AMB Express 2011 1:45.
Submit your manuscript to a
journal and benefit from:
7 Convenient online submission
7 Rigorous peer review
7 Immediate publication on acceptance
7 Open access: articles freely available online
7 High visibility within the field
7 Retaining the copyright to your article
Submit your next manuscript at 7 springeropen.com
Machado et al. AMB Express 2011, 1:45
http://www.amb-express.com/content/1/1/45
Page 14 of 14