Vol. 25 no. 24 2009, pages 3289–3295
Model aggregation: a building-block approach to creating large
macromolecular regulatory networks
Ranjit Randhawa1, Clifford A. Shaffer1,∗and John J. Tyson2
1Department of Computer Science and2Department of Biological Sciences, Virginia Tech, Blacksburg,
VA 24061, USA
Received on April 29, 2009; revised on August 25, 2009; accepted on September 28, 2009
Advance Access publication October 29, 2009
Associate Editor: Trey Ideker
Motivation: Models of regulatory networks become more difficult
to construct and understand as they grow in size and complexity.
Modelers naturally build large models from smaller components that
each represent subsets of reactions within the larger network. To
defines models in terms of components that are designed for the
purpose of being combined.
Results: We have implemented a model editor that incorporates
model aggregation, and we suggest supporting extensions to the
Systems Biology Markup Language (SBML) Level 3. We illustrate
aggregation with a model of the eukaryotic cell cycle ‘engine’ created
from smaller pieces.
Availability: Java implementations are available in the JigCell Agg-
regation Connector. See http://jigcell.biol.vt.edu.
The physiological properties of cells are governed by macromole-
cular regulatory networks of great complexity (Kohn, 1999). Under-
standing the dynamical properties of these networks is facilitated by
mathematical modeling of the biochemical reactions (Kohn, 1999;
Novak and Tyson, 2003; Sible and Tyson, 2007; Tyson, 2007).
These models are often implemented deterministically, as sets of
non-linear differential equations, or probabilistically by Gillespie’s
stochastic simulation algorithm or its variants (Cao et al., 2004;
Gillespie, 1977, 2001). In either case, the modeler is faced with
the problem of specifying the reactions involved in a large complex
and numerical values for the rate constants involved in each rate
law. Building regulatory network models is a little like putting
This complex modeling challenge is best broken down into smaller
components that can later be joined together into a larger whole.The
created to support the modeling of biochemical reaction networks,
composition or aggregation. In earlier publications (Randhawa
et al., 2007, 2008; Shaffer et al., 2006), we presented tools,
algorithms and language extensions for SBML Level 3 for model
∗To whom correspondence should be addressed.
Fig. 1. Reaction network for cell-cycle control in yeast. Icons are proteins,
solid arrows are chemical reactions and dotted arrows represent enzymatic
catalysis. The ‘CycB’ icon represents the active dimer Cdk1-CycB. SK is a
starter kinase, IE is an intermediary enzyme and M is cell mass.
‘fusion’ and ‘composition’, irreversible and reversible processes
we present an alternative method of connecting submodels, that we
We illustrate the process of model aggregation with a model for
the cell division cycle (Csikasz-Nagy et al., 2006). In eukaryotes,
the cell cycle is controlled by a set of cyclin-dependent protein
initiate the events of DNA replication, mitosis and cell division.
As their name suggests, Cdks require association with a cyclin
partner (CycA, CycB, etc.) to be active. The activity of a Cdk-
Cyc dimer is controlled by interactions with a variety of regulatory
proteins, including the anaphase promoting complex (APC), which,
in combination with two auxiliary proteins (Cdc20 and Cdh1),
degrades the cyclin component of the Cdk-Cyc dimer, and a suite of
cyclin-dependent kinase inhibitors (CKI), which bind to and inhibit
the Cdk-Cyc dimer. A simple model of these interactions (Fig. 1) is
sufficient to reproduce (in simulation) many features of cell-cycle
regulation in budding yeast (Tyson and Novak, 2001). This model
shows how progress through the cell cycle can be thought of as
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R.Randhawa et al.
irreversible transitions (Start and Finish) between two stable states
(G1 and S-G2-M) of the regulatory system.
Over the last 20 years, molecular biologists have amassed a great
deal of information about the genes and proteins that carry out
fundamental biological processes within living cells—processes
such as growth and reproduction, movement, signal reception
and response, and programmed cell death. The complexity of
these macromolecular regulatory networks is too great to tackle
mathematically at the present time. Nonetheless, modelers have
had success building dynamical models of restricted parts of
the network. For example, for budding yeast cells there have
been recent successful efforts to model the cell cycle (Chen
et al., 2004), the pheromone signaling pathway (Kofahl and Klipp,
2004), the response to osmotic shock (Klipp et al., 2005) and
the morphogenetic checkpoint (Ciliberto et al., 2003). Systems
biologists need tools now to support aggregation of ‘submodels’
(such as these) into more comprehensive models of integrated
Modeling languages and tools help modelers construct their
models by providing a computational environment that minimizes
the amount of human error during the construction step. While
modelers are currently able to construct small- and medium-sized
models by hand, the process is simplified by using computational
tools that decrease the time taken to input a model and provide
error-testing services along the way. In this article, we describe a
representation and a tool that enable modelers to create large models
by aggregation of submodels.
We assume that each of the submodels used in creating the larger
model can be a validated model itself, with experimental data that
fixes its parameters.The main motivation for creating a larger model
is that there exists experimental information on the interaction of the
subsystems that the submodels cannot account for. By aggregating
validated submodels, we mitigate the problem of searching through
large parameter spaces. The parameter estimation problem is now to
ensure that the aggregated model is consistent with the original data
used to validate the submodels (for which we already have good
initial guesses, inherited from the submodels) and also the new data
relevant to the interactions of the subsystems (which are governed
by the new parameters describing how the submodels fit together).
In our experience in modeling cell-cycle regulation, we have
found a modular approach to be practical for constructing large
networks that encompass a broad range of experimental data
and that allow many parameters to be estimated from the
data in a systematic fashion (see, e.g. http://mpf.biol.vt.edu/
research/budding_yeast_model/pp/). Previously submodels have
been merged manually, by an informal process that can be tricky
approach to bear on this problem, we have identified four distinct
processes related to modular network construction, which we have
2006). The first three have previously been implemented in the
all components of the JigCell system (http://jigcell.biol.vt.edu). The
present article focuses on aggregation: proposing extensions for
SBML Level 3 needed to describe aggregation, and presenting a
BUILDING LARGE NETWORKS
prototype tool that enables aggregating (sub)models. We provide a
with the official SBML proposal for hierarchical modeling, as that
proposal was derived in part from our work.
We define Model Fusion as a process that combines two or more
submodels into a single unified model that contains the combined
information (without redundancies) across the original collection.
The identities of the original (sub)models are lost. The result of
fusion is a model in the same language as the submodels [in our
case standard SBML Level 2 (Hucka et al., 2003)], meaning that
the same simulation and analysis tools can be applied. Beyond
some size, fused models become too complex to intellectually
grasp and manage as single entities. It then becomes necessary
to represent large models as composites of distinct components.
Thus, while model fusion is a useful tool for manipulating small-
to mid-sized models, it does not seem to be a viable solution in the
Model composition provides a mechanism for building models of
large reaction networks. Under our definition of composition, one
thinks of models not as monolithic entities, but rather as collections
of smaller components (submodels) joined together. A composed
model is built from two or more submodels by describing their
redundancies and interactions. Composition is a reversible process,
in that removing the intermodel interaction description that holds
the composed model together recovers the original submodels.
While, it is appealing in short term to build larger models from
pre-existing models, each developed independently for their own
purposes, we believe that ultimately it will become necessary to
build large models from components that have been designed for
the purpose of combining them. We distinguish this approach (of
models are typically built from existing submodels, and therefore
contain redundancies). We define model aggregation as a restricted
form of composition that represents a collection of model elements
as a single entity (a ‘module’).Amodule includes a specification for
predetermined input and output ports. We distinguish between the
terms module and submodel by defining a module to be a submodel
with ports. These ports link to internal species and parameters and
enable them to be accessed/referenced outside of the model in which
they occur. They define the module’s interface, which provides
restricted access to the components within the module. The process
of aggregation (connecting modules via their interface ports) allows
modelers to create larger models in a controlled manner.
Model flattening converts a composed or aggregated model with
some hierarchy or connections (discussed later) to one without
such connections. The result of flattening is equivalent to fusing
the submodels. The relationship information provided by the
composition or aggregation process must be sufficient to allow
flattening to take place without any further human intervention. The
relationships used to describe the interactions among the submodels
are lost, as the composed or aggregated model is converted into
a single large (flat) model. Flattening a model allows us to use
existing simulation tools, which have no support for composition
or aggregation. We could think of flattening as analogous to
compiling a program written in a high-level programming language
(with subroutines) into the machine code ready to execute on a
The XML-based SBML (Hucka et al., 2003) has become widely
supported within the network modeling community. Thus, we
processes through added SBML language constructs that express
the necessary glue that connects submodels together. It is not
necessary that our proposals be implemented in SBML, but doing so
provides clear reference implementations in the same way as would
expressing an algorithm in a particular programming language.
There have been a number of efforts within the systems biology
community to support either merging multiple SBML models or
building model from components. Snoep et al. (2006) showed that
it was possible to construct a large biological model in a bottom-
up manner by manually linking together smaller modules. They
combined a glycolysis pathway model with a glycoxylate pathway
model. Bulatewicz et al. (2004) suggested an interface for model
coupling and provided a number of solutions, from a brute force
technique to using frameworks specifically designed to support
coupling. A number of authors from domains outside systems
biology find that successful composition (or model ‘reuse’) requires
components that are specifically designed for the purpose (Davis
and Anderson, 2004; Garlan et al., 1995; Kasputis and Ng, 2000;
served as motivation for our approach to model aggregation.
The SemanticSBML (Liebermeister et al., 2009) tool suite
facilitates merging of SBML models. It includes three tools:
SBMLannotate helps users insert annotations into SBML models.
SBMLcheck checks annotated models for completeness and
consistency. SBMLmerge (Schulz et al., 2006) merges several
annotated models. SemanticSBML is analogous to our Fusion
2 models. While it provides support for merging models based on
annotations, it does not support model reuse through composition
or aggregation, nor does it maintain a history of model development
in the way composition and aggregation might. Process Modeling
Tool (ProMoT; Ginkel et al., 2003) and E-Cell (Takahashi et al.,
2003) are modeling packages that use some form of modularity.
elements that represent compartments which contain reactions,
species and special signaling parts. ProMoT provides support
for modularity and hierarchical modeling. It uses object-oriented
models, composed from modules, and has its origins in process
engineering. It provides support for importing/exporting standard
SBML (Level 2). E-Cell uses an architecture where the complete
model may be modularized through compartments. In this sense,
modules must have some physical border and are not only logical
or functional groupings but represent an object in the physical
topology of the cell. Three additional reference frameworks exist
the Modelica language (Elmqvist et al., 2001) and JSim (Raymond
et al., 2003). CellML includes a mechanism for composing models
out of components. While this is not directly applicable to SBML, it
future interoperability between CellML and SBML. The Modelica
language provides for hierarchical object-oriented composition of
components, while JSim is a Java-based modeling framework that
provides a way to do object-oriented composition.
Proposals have been made within the SBMLcommunity (Finney,
2003; Ginkel, 2002; Schroder and Weimar, 2003) that describe
the mechanics of composition (or aggregation) through additional
SBML language features, as we will do. However, none of these
proposals has been published in the peer-reviewed literature,
nor to our knowledge have any been implemented. While some
commercial tools might have more or less support for various
forms of composition, we are unaware of any non-proprietary
implementations for model composition (or aggregation) in this
application domain, or any publications describing proprietary
features in commercial applications. Composition and aggregation
for pathway models remains very much an open problem.
Our work focuses on how to support aggregated models. The
SBML language extensions that we propose elaborate on those
originally presented in Finney (2003). Our implementation differs
from Finney (2003) in a number of ways. The notion of <instance>
structures that enabled model reuse using XLink (DeRose et al.,
2001) to instantiate any number of models to access them has been
replaced by adding a new XPointer (Grosso et al., 2006) attribute
to the <model> structure. In this way, we minimize the number of
the need for imposing restrictions on linkages in an aggregated
model without identifying these restrictions. Our implementation
checks for circular linkages and incorporates the notion of direct
links (Finney, 2003) connecting the external interfaces of the
submodels together [in contrast with connecting entities within
(sub)models to each other]. Section 6 provides an example using
SBML syntax of three models.
Real molecular networks seem to be made up of simpler modules that carry
out specific tasks (Tyson et al., 2003). By allowing modelers to substitute
an aggregate for groups of reactions, and enabling aggregated modules to be
connected to one another, we envision that model construction will become
faster and more intuitive while holding true to the apparant structure of the
organism being modeled. Modularization is defined here as the process of
grouping reactions together as a single entity (a module) with a defined set
of inputs and outputs (called ports). A module is not a simplification of the
group of reactions (or their behavior). It is representational, and is used to
aid better understanding of how parts of the model (the modules) interact
with each other. Aggregation is the process of connecting modules together
(by linking outputs of one module to inputs of another) in order to create a
larger model (an aggregate of modules).
The fundamental difference between aggregation and composition [as
we define it in Randhawa et al. (2008)] is the amount of access to model
information and the initial source of these components. The basic building
blocks for composition and aggregation are the same—a collection of one
or more reactions. However, in composition, a component’s information
is not hidden from other components. A modeler can link to any variable
or component within a submodel. In aggregation, model information is
deliberately hidden to control complexity, and a modeler can only link to
variables or components that are explicitly made visible (through ports)
in a given module. Components for composition are typically pre-existing
models that thus might contain redundancies between components and were
not created with the intent of combining with other components. Part of the
for eliminating redundancies in submodels. No such mechanism exists for
aggregation, because modules are designed to be connected.
R.Randhawa et al.
Fig. 2. The Aggregated Model in the JigCell Aggregation Connector.
A major benefit of aggregation is that the modeler does not need to know
all the reactions within a module in order to use it, only its list of input
and output ports. The ports must be defined so that modules can be linked
together unambiguously. To this end, we require that an input port must be
an entity with a fixed value within the module, for example, a rate constant
or a reaction modifier that is a parameter in the module. An output port, on
the other hand, must be one of the time-dependent variables in the submodel,
that is, a reactant or product in any reaction within the module. By enforcing
these constraints, we ensure that a consistent set of differential equations will
always be produced when modules are linked together.
There is no difficulty in linking a variable X(t) in Module 1 to a parameter
reaction in Module 2. In the differential equations for Module 2, the constant
changes in X(t) is still determined unambiguously by Module 1. But, if X(t)
is linked to a time-varying species in Module 1, say A(t), then the notion of
linkage is ambiguous. It might mean to identify species X and A or to add
the chemical reaction X→A. In either case, such linkages are more suitably
described by model composition, as defined in Randhawa et al. (2008), than
by model aggregation, as defined here.
We implement aggregation with new language features for SBML.
Language features previously identified for composition (Randhawa et al.,
2008) are used for aggregation. We add additional language features to
The language additions for SBML in Section 6 allow modelers to build
aggregate models from modules. They support multiple instances of a given
module. The features define the hierarchy of the modules, and represent the
interactions, relationships and links between the modules. Aggregation and
fusion should produce the same results, as the simulation output of the fused
and aggregated forms of a model should be identical.While fusion combines
submodels together in an irreversible way, aggregation references module
components by defining the ‘glue’ that holds the modules together. A major
difference is that in fusion, the explicit description of relationships between
entities within submodels is lost. Aggregation records how models were
aggregated/connected together. Model aggregation is an iterative process,
as models are usually constructed in increments, with modelers switching
back and forth between adding components/modules to a model and fine-
tuning models through simulations. Constructing aggregated models is a
bottom-up process as smaller models are first aggregated together to create
larger models, which can be used to create even more complex models,
as previously demonstrated in Snoep et al. (2006). An aggregated model
can be created from a combination of flat and/or aggregated models. Model
aggregation generates an aggregation graph that describes the relations
among the various modules. Changing the connections between modules
in an aggregated model results in a different aggregation graph structure.
Since model aggregation is a combination of constructing modules and
generating aggregation graphs, a tool for aggregation should take into
consideration the iterative nature of the process.
We now illustrate generating an aggregated model by connecting
modules through their ports using the JigCell Aggregation Connector.
The Aggregation Connector in Figure 2 has two functions: (i) it converts
(sub)models to modules by specifying their ports, and (ii) it connects
modules together to create aggregated models. The user interface is divided
into two parts, an aggregation window (top) which allows modelers to
connect modules together graphically, and an embedded JigCell Model
in a spreadsheet interface.
To begin aggregation, the modeler presses the ‘Open Module(s)’ button
on the top left of the application window to select SBML models from a
file chooser. These models can be both submodels (without defined ports) or
modules (with defined ports).We assume that the modules operate in a single
compartment and that the user is responsible for formulating the modules
in a consistent set of units. Modules are displayed as boxes with input and
output ports in the aggregation window, while submodels are only displayed
as boxes without ports until such time as the modeler defines its inputs and
outputs. Each module has the following components:
(1) A name in the middle of the box.
(2) An optional set of input ports on the left-hand side of the box.
(3) An optional set of output ports on the right-hand side of the box.
(4) A gray box with a ‘+’ sign in the top left-hand corner of each box,
used to load the module into the Model Builder.
module (by defining the ports), the modeler loads it into an embedded copy
of the JCMB. JCMB displays the module’s components in a spreadsheet
interface. JCMB has been extended to enable a modeler to select which
species and parameters are defined as ports.An additional column was added
to the species and parameters spreadsheets, respectively. After loading two
or more modules onto the aggregation window, a modeler can now link the
output ports of one module to the input ports of another module to create an
aggregated model. The links between modules are indicated as solid black
lines in Figure 2.
Once the aggregated model is completed, it can be saved in an SBML
file, flattened into a standard SBMLLevel 2 model, or converted into a more
complex module for use in future aggregation. When converting an existing
aggregate model into a new module, the Aggregation Connector performs
an initial best guess for the ports. Since an output port can connect to more
than one input port, all the output ports of the modules connected together in
the aggregate model are made into output ports in the newly created module.
Input ports can only be connected once, so only those input ports that are
unconnected are exposed in the newly created module as input ports.
To illustrate how aggregation works, our example reproduces the approach
used by Tyson and Novak when building their basic model of cell-cycle
control in yeast cells (Tyson and Novak, 2001). They built their model in
stages starting from a simple model and then adding new pieces until they
obtained a satisfactory representation of the cell-cycle control system. The
first model (which we will call Module I) deals with the interaction between
the cyclin B-dependent kinase (CycB) and a CKI, as shown in Figure 3.
Module II deals with the antagonistic interactions between CycB and a
cyclin B-degrading factor (Cdh1), as shown in Figure 4. Module III deals the
interaction between CycB and a different form of the cyclin-degrading factor
(Cdc20), as shown in Figure 5. Finally, Module IV contains three additional
reactions (→SK, SK→ and →M), as shown in Figure 6. M increases
according to a logistic rate equation, and M is decreased by a factor of 2
each time the cell divides. Cell division is triggered when CycB drops below
Fig. 3. Module I: interaction between the cyclin B-dependent kinase (CycB)
and a CKI.
Fig. 4. Module II: interaction between CycB and a cyclin B-degrading factor
Fig. 5. Module III: interaction between CycB and a different form of the
cyclin degrading factor (Cdc20).
a certain threshold, as cyclin B is degraded by Cdc20 and Cdh1 and is best
The four steps to create the Aggregate Model are as follows.
(1) Modules I–IV are created with predefined ports in the embedded
JCMB within theAggregation Connector. For convenience, the ports
Fig. 6. Module IV: additional reactions for cell mass and starter kinase.
Concentration (arbitrary units)
Fig. 7. Simulation of the Aggregated Model using the simulation package
(2) Iconic representations for Modules I–IV are loaded into the Aggre-
gation Connector (Fig. 2).
(3) Links between ports are created by holding down the mouse button
and dragging the mouse from an output port of one module to an input
port of another module.
(4) Once all the links have been created, the model may be saved as an
SBML file with additional language features.
These steps produce the Aggregate Model in Figure 2, which should
then be simulated to verify that its dynamic properties represents the
observed behavior of growing-dividing yeast cells in expected ways.
Since our current simulators require standard SBML (Level 2) input,
by removing the additional constructs used to describe the aggregation.
Simulating the resulting model using a freely available software package
called XPPAUT (Ermentrout, 2003) produces the simulation output shown
in Figure 7, which closely matches the simulation output from Tyson and
Novak’s model [Fig. 8 in Tyson and Novak (2001)].
The SBML language features described below elaborate on those originally
proposed in Finney (2003), by providing a defined framework with a proof
of concept implementation to demonstrate the feasibility of aggregation.
To illustrate the SBML language features needed to describe model
aggregation, consider the example in Figure 8, which shows an aggregated
two modules called Little1 and Little2.
R.Randhawa et al.
Fig. 8. Aggregated model showing a link between two ports in different
modules and the corresponding flattened model.
To begin aggregation, we have to modify the (sub)models into modules
by creating ports. The <port> (enclosed in a <listOfPorts> structure) gives
a modeler access to a particular species, parameter or another port within a
module for aggregation.A<port> structure is composed of three attributes:
id,target andinput.Theid fieldgivesauniqueidentifiertotheportstructure.
The target field specifies a single species, parameter or another port by its
SBML identifier. The input field specifies whether this is an input or output
port. The output port structure syntax looks as follows.
Input and output ports are distinguished from each other by their target type.
<port> structures are used in conjunction with the other language constructs
described below. The SBML pseudocode for the two models Little1 and
Little2 indicates what needs to be added to standard SBML in order to make
<species id="A_1" name="A"/>
<port id="A1" target="A_1" input="false"/>
<species id="A_1" name="A"/>
<port id="A2" target="A_1" input="true"/>
Once we have created the modules, we are ready to add them to the
aggregated model which can contain one or more modules. A module (or
submodel) is just an SBML model (enclosed within an SBML <model>
structure), with its own namespace, and can itself be an aggregated model.
Since there is no restriction on the number of modules a model can contain,
a <model> structure is enclosed in a <listOfSubmodels> structure which
contains the list of all the modules within the aggregated model.
by using the XPointer framework (Grosso et al., 2006) to refer to submodels.
XPointer defines a reference to another location in the current document, or
an external document, using an extended pointer notation. An instance of
submodel Little1 can be made within model Big to access submodel Little1
in model Big. The <model> structure contains a new attribute: an xref,
which is represented using an XPointer string (Grosso et al., 2006), which
is used for locating data within an XML document.
structure (enclosed in a <listOfLinks>) connects two ports in separate
modules of an aggregated model. Linking components in aggregation can
be achieved by drawing a connection between two ports in the Aggregation
Connector using the mouse. A <link> is composed of two fields, <from>
values will be overridden by the object referenced by the <from> field (the
from object). Note that a to object must refer to an input port and a from
object must refer to an output port. Only those attribute values that have
been declared in the from object will be overridden in the to object. This is
somewhat analogous in C/C++ to treating the to object as a pointer, and the
from object as its target. However, a to object can have attribute values that
are retained if no overriding attribute value is declared in the from object.
We adopt a naming convention to enable modelers to uniquely identify a
port within a model (or module). Our format for SBML components, such
as model, species, parameter, etc., is:
For convenience, in the description we will also represent this information
using the syntax ObjectIdentifier.SubobjectIdentifier.This convention makes
it possible to refer to ports with the same name in different models without
having to change their names. The following example shows how the two
and to avoid duplication. The SBML syntax below lists the modules, and
contains one link between Little1.A and Little2.A.
Model aggregation Download full-text
This example shows an xref attribute where module Little1 occurs within
the same SBML document. If Little1 occurred in another SBML document
named temp.sbml in the current directory, the xref attribute of the <model>
structure would have temp.sbml prepended to it.
In summary, the <listOfSubmodels> structure is used to define the layout
is made by using the <link> structures which can only connect <port>
structures to each other. These three SBML language features are sufficient
to define model aggregation.
It is our anecdotal experience that aggregation is a faster and easier
way to create models than either fusion or composition. Fusion
and composition require time and effort to identify and deal with
redundancies across the (sub)models. Aggregation has no such
redundancies across submodels (or modules) as all submodels are
customized before they are connected together. While it might take
longer initially to create models for aggregation, once done it is
relatively straightforward to connect them together through their
ports to create larger models. We note that we are working with
models we are already familiar with.
At this time we have not included support for multi-compartment
models. Nor have we developed support for units and consistency
checking. These features will be added in the future.
Current modeling efforts in Tyson’s Group at Virginia Tech
involve challenging issues in large-scale modeling. One such
effort is focused on the morphogenesis checkpoint in budding
yeast. Ciliberto et al. (2003) developed a model of the
morphogenesis checkpoint that was ‘hooked up’to a very primitive
cell-cycle engine in budding yeast. We have successfully combined
the morpho-checkpoint module with the full cell-cycle engine
proposed by Chen et al. (2004) and will publish the results in the
CONCLUSIONS AND FUTURE PLANS
Funding: National Institutes of Health (grant R01-GM078989-
01); Technology Centers for Networks and Pathways (grant
Conflict of Interest: none declared.
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