BIOINFORMATICS
ORIGINAL PAPER
Vol. 25 no. 16 2009, pages 2057–2063
doi:10.1093/bioinformatics/btp361
Systems biology
Modeling multi-cellular behavior in epidermal tissue homeostasis
via finite state machines in multi-agent systems
Thomas Sütterlin1 , Simone Huber2 , Hartmut Dickhaus1 and Niels Grabe1,∗
1 Medical
Informatics Department, Institute of Medical Biometry and Informatics, University Hospital Heidelberg,
Im Neuenheimer Feld 305 and Hamamatsu Tissue Imaging and Analysis Center (TIGA), BIOQUANT, University
Heidelberg, Im Neuenheimer Feld 267, 69120 Heidelberg and 2 Department of Informatics, Heilbronn University,
Max-Planck-Str. 39, 74081 Heilbronn, Germany
Received on December 2, 2008; revised on May 18, 2009; accepted on June 9, 2009
Advance Access publication June 17, 2009
Associate Editor: Trey Ideker
ABSTRACT
Motivation: For the efficient application of multi-agent systems
to spatial and functional modeling of tissues flexible and intuitive
modeling tools are needed, which allow the graphical specification
of cellular behavior in a tissue context without presuming specialized
programming skills.
Results: We developed a graphical modeling system for multi-agent
based simulation of tissue homeostasis. An editor allows the intuitive
and hierarchically structured specification of cellular behavior. The
models are then automatically compiled into highly efficient source
code and dynamically linked to an interactive graphical simulation
environment. The system allows the quantitative analysis of the
morphological and functional tissue properties emerging from the
cell behavioral model. We demonstrate the relevance of the approach
using a recently published model of epidermal homeostasis as well
as a series of cell-cycle models.
Availability: The complete software is available in binary executables
for MS-Windows and Linux at tiga.uni-hd.de
Contact: [email protected]
1
INTRODUCTION
The study of epithelial homeostasis offers an excellent opportunity
for developing systems biological models which realize the link
from the molecular to the cellular and further on to the tissue
level (Galle et al., 2005; Walker et al., 2004). Previously, we
demonstrated an initial simulation of tissue homeostasis and its
application to psoriasis using a multi-agent system (Grabe and
Neuber, 2005, 2007). Each of these agents realizes the functional
and spatial behavior of an individual cell. We observed that the
direct use of algorithms for describing cellular behavior enables
very fast multi-cellular simulations. Furthermore, we learned that
multi-agent systems provide an ideal conceptual approach for this
purpose. Multi-agent systems experience an increasing popularity in
bioinformatics (Merelli et al., 2007; Walker et al., 2004; Webb and
White, 2006) as they are directly intended for modeling emergent
behavior of interacting entities. Unfortunately, the direct coding of
parameterized algorithms in software allows the non-expert user
only to affect a small subset of the model parameters. This appears in
∗ To
whom correspondence should be addressed.
particular problematic for highly complex tissue models. Therefore,
we set out to find a more flexible approach for modeling cellular
behavior for multi-agent simulations.
Graphical modeling tools like CellDesigner (Kitano et al.,
2005) have become quite popular in the field of systems biology.
CellDesigner process diagrams represent the state transitions of
individual molecules like p53 or EGFR. Here, we carry on the
intuitive idea of using process diagrams and deploy the approach
to describe the behavior of whole cells. To this end, we adapted the
theoretical concept of finite state machines (FSM). Indeed, one of
the first applications of Turing’s theoretical automata theory which
introduced the concept of FSMs (Turing, 1936) was the neuronal
cell (McCulloch and Pitts, 1943). It was subsequently refined in
the 1960s (Minsky, 1967). FSMs have been used in the context of
numerous publications concerning cellular automata (Alarcon et al.,
2003; Cowin, 2004). Unfortunately, the grid structure of cellular
automata renders them unsuitable for spatially dynamic simulations
of epithelial homeostasis. We therefore adapt the theoretical concept
of FSMs to our problem. In this respect, we suggest to use process
diagrams as an abstract description of cellular behavior instead
of directly coded algorithms. Interpreted in the context of FSMs,
these graphical models describe conditional transitions between
individual states of the cell. General cellular states may comprise
e.g. proliferation, differentiation, migration and apoptosis. To each
of these general states a hierarchically structured set of sub-states
is conceivable, each corresponding to a different cellular function
like, for example, changes in polarity or rearrangements of adhesion
complexes.
When combined with graphically specified FSMs, we suppose
multi-agent systems to become an ideal approach for specifying
multi-cellular models. When specified in the form of FSM,
graphical depictions of cellular behavior can be translated relatively
straight forward into executable code. Furthermore, during such
an automatic code generation, automatic code optimization can
take place, producing high-performance minimal code. Especially,
when aiming at the simulation of tissue, composed of hundreds or
thousands of cells, the generation of minimal code will prove to be
of utmost importance.
Other general FSM modeling tools are available like e.g.
Matlab/Stateflow and HyVisual (Lee and Zheng, 2005). However,
the requirement for a tight integration of graphical modeling, highly
specific code generation and incorporation of the generated code
© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
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into multi-agent simulations of tissues rendered the use of presently
available software practically impossible.
Starting from these conceptual considerations, we developed a
graphical modeling system (GMS) for the intuitive modeling and
efficient simulation of individual behavior of cells in a tissue context.
The specified models are hierarchically structured thus allowing
the specification of complex cellular functions. At the push of a
button, the models are automatically translated into highly efficient
minimal executable code being then automatically and dynamically
bound to a multi-agent simulation environment which executes the
model in discrete time steps and in a semi-parallel manner for each
cell. Thereby cells interact via transport processes and signaling. We
demonstrate the efficiency of the resulting multi-cellular simulation
at the example of our previously presented model of keratinocyte
differentiation as well as various cell-cycle models.
2
METHODS
2.1
Graphical modeling system
We developed our GMS on the basis of the open source software platform
Eclipse that was originally developed by IBM. Eclipse in turn is based
on the Open Services Gateway initiative (OSGi) Framework Equinox that
allows Eclipse to be dynamically extended by plug-ins. We realized the
GMS as a set of plug-ins for Eclipse in the programming language Java. We
used the Java Development Tools (JDT), the Eclipse Modeling Framework
(EMF) and the Graphical Editing Framework (GEF). The GMS is subdivided
in three independent components: (i) Variable-Sheet Editor (described in
Section 3.1), (ii) Graphical Model Editor (GME) (Section 3.2) and (iii)
Function Library (Section 3.3).
Following the paradigm of model-driven software development, a metamodel for each component was developed using the Unified Modeling
Language (UML). These meta-models are instances of the meta-meta-model
Ecore provided by the EMF. The EMF in turn is able to derive source code
from Ecore-instances. The java-classes generated in this way represent the
data model components of the GMS. For developing the GME component,
java classes generated in this way are deployed in the GEF. Generally, the
GEF provides the infrastructure for implementing a component which is
able to graphically create and manipulate instances of a given data model
(Gamma and Beck, 2003; Moore et al., 2004). Graphical models are stored in
the XML-Metadata-Interchange (XMI) format which is an open standard of
the Object Management Group (OMG). Java Source Code can be generated
out of the XMI files using the technique ‘Templates and Filtering’ (Stahl
et al., 2006). The source code generated from the model is translated into
Java byte code using Sun Microsystems’ java compiler. Automatic validation
of graphical models is performed by a parser using the open source Java
Compiler Compiler (JavaCC). JavaCC is a parser generator that allows to
generate a LL(k)-parser on the basis of a given grammar (Copeland, 2007).
A user-specified graphical model controls the behavior of each agent in the
multi-agent simulation. For executing the compiled models we developed
a multi-agent based graphical Simulation Environment (SE). SE is built on
the George Mason University’s multi-agent simulation framework MASON
(Multi Agent Simulation Of Neighborhoods) (Luke et al., 2005).
3
SYSTEM
3.1
Graphical model editor
We denote a graphical ‘cell behavioral model’ (CBM) as an abstract
description of a FSM. The CBM formalizes all possible forms
of behavior of a cell as a process graph. CBMs are specified
using the GME. They are composed of the available predefined
Fig. 1. Model elements provided by the GMS from which graphical CBMs
are created. The elements ‘transition choice‘ and ‘transition condition’ are
used to branch in either two or multiple different paths. Submodels allow the
creation of hierarchically structured complex models. The modeling element
‘state’ comprises a set of ‘action strings’ altering cellular properties. Edges
connect the individual model elements.
model elements depicted in Figure 1. The available model elements
comprise:
• Edge: connects two model elements using their input port and
their output port, respectively. Input ports and output ports are
depicted in Figure 1 as small triangles.
• State: represents a state element in the model. In each state a
set of ‘actions strings’ can be defined which are sequentially
executed at runtime. Action strings may be assignments of
values to a certain cell property or calls of predefined functions.
Each state has one input port and one output port.
• Transition condition: allows to branch to two distinct paths in
a model in dependence of a Boolean expression.
• Transition choice: allows to branch to multiple paths in a model
in dependence of a mathematical expression. Output ports can
be defined for all expected results or values. The path connected
to the ‘default output port’ is followed if the value of the
variable or the result of the mathematical expression matches
none of the defined output ports.
• Submodel: submodels allow to implement a hierarchical
structure in a CBM. Technically, the overall model is
thereby physically distributed over different model files with
unambiguous file names. Semantically, each file realizes a
certain self-contained functional aspect of the overall model.
The number of submodels is not limited.
• Connection point: in case multiple transitions shall lead to a
single common model element, a connection point has to be
used to join the incoming edges. Therefore, a connection point
always has just one output port.
• Comment: textual descriptions or explanations can be added as
a comment anywhere in the model.
Using these elements complex CBMs can be built. In each model
the user denotes exactly one model element as the starting and one
model element as the end point.
3.2 Variable-sheet editor
The Variable-Sheet Editor allows the intuitive creation, definition
and modification of all variables and constants of the CBM which
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Modeling multi-cellular behavior in epidermal tissue homeostasis
are stored in the variable-sheet file. Variables or constants can be
interactively dragged from the Variable-Sheet Editor and dropped
into a certain model element in the GME where needed. All variables
and constants of a CBM belong to one of the four categories:
• Cell type: each cell occurring in the simulation is assigned an
abstract ‘cell type’ in form of an unambiguous string constant.
For example, keratinocytes and fibroblasts are different cell
types. However, multiple differentiation stages of a cell may
also be introduced as distinct cell types.
• Constants: In this category, constant values of a CBM are
defined which are of general use. Each constant consists of
a name and a value constrained by a data type to be selected.
• Global tissue property: These variables specify global
properties of the whole tissue. Contrary to ‘constants’, the value
of a global tissue property can be modified interactively prior
to simulation but also at runtime in the graphical SE. All cells
of the multi-cellular simulation adhere to the same values of
the specified global tissue properties.
• Cell property: cell properties comprise the variables of the
CBM whose values are to be modified in every ‘state’ during
the model simulation. At run-time each cell occupies these
variables with individual values, thus realizing a personal state
for each cell. Therefore, the state of an individual is defined
by the values of its ‘cell properties’—variables at a particular
point in time.
For each ‘global tissue property’ and ‘cell property’ a name, a data
type and a scientific unit have to be defined. A range of typical
data types is provided. For a ‘cell property’, these attributes are
complemented by a maximum, a minimum and a default value.
During the specification of graphical models, the value of a variable
or constant can be addressed by the respective name as defined in the
variable-sheet file. A neighboring cell’s property can be addressed
by adding the prefix n_ to the respective cell property’s name. This
triggers a loop which is executed over all direct neighbors. Likewise,
the allowed maximum and minimum value of a cell property can be
accessed by adding the suffix _max and _min, respectively.
3.3
Function library
The ‘Function Library’ provides a set of predefined functions for
formulating mathematical expressions as well as specific functions
for initiating cell proliferation and interaction with the cell’s
neighborhood respectively. The function calls can be incorporated
into ‘action strings’ which are specified by the user in the model
element ‘state’. Power (Pow), square root (Sqrt), sine (Sin) and
cosines (Cos) are the currently available mathematical functions.
Additionally, the mathematical constants π (PI) and e (E) are
provided.
The functions Send(var, val, toNeighbor) and Receive(var, val,
fromNeighbor) allow the specification of a certain fraction of
material and intercellular signaling molecules to be transported to
and from neighboring cells or the intercellular space respectively
in a discrete amount of time. Thus, intercellular communications
modeled through these functions are executed at each discrete
time step. It is up to the modeler to interpret a discrete time
step as a specific period of time. Equilibrating processes can
be modeled when considering a sufficiently large number of
time steps. Both functions implement a heuristic intercellular
communications approach. Therefore, intercellular communication
is modeled indirectly without addressing directly a respective
neighboring cell. This offers the possibility to abstract away from
the exact number of neighboring cells as well as from their spatial
distribution. In detail, the abbreviation var denotes the respective
‘cell property’ usually representing a molecule to be exchanged.
Val denotes the amount of units to be sent or received, whereas
toNeighbor or fromNeighbor is either of the value true (T ) or false
(F). In case of false, a ‘cell property’ is just altered but there is
no respective transfer to neighboring cells. For example Send(ca,
ca*0.1, F) initiates the transport of 10% of the intracellular Ca2+
ions to the intercellular space.
Formally, the function Send(var, val, toNeighbor) works as
follows. The variable var corresponds to the name of a ‘cell property’
defined in the Variable-Sheet Editor. Initially, the maximal amount
of the molecule or material specified in val is determined which
can be sent without var falling below the variable-sheet specified
minimal value. In case of transport to the neighboring cells (i.e.
toNeighbor is true) the resulting amount is equally distributed
among the neighbors of the actual cell. For each neighboring
cell it is checked that it does not exceed the given minimal or
maximal values. The Receive function works analogously to the
Sent algorithm with inverse checks. Both implemented algorithms
allow an undirected as well as a directed propagation of material and
signaling molecules among neighboring cells and thus through the
whole tissue. For a directed propagation of material the graphical
modeling element transition condition has to be used for defining
respective preconditions. The equilibration of a particular signaling
molecule or material among all neighboring cells is achieved via
the iterative execution of the whole model over a sufficiently large
number of discrete simulation time steps.
Finally, the Function Library provides the function NewCell
(var1=val1, var2=val2, …) which performs the proliferation of a
cell. Computationally, this means that a new cell is instantiated.
The initial values for the new cell’s properties are to be specified in
between the parentheses.
3.4
Model validation
Automatic translation of the graphically specified models into
executable source-code requires structurally validated models. Our
GMS performs this for ensuring the following:
• Completeness: A model is complete if all output ports of all
model elements are connected to an input port of another model
element and every state element comprises at least one action
string. Furthermore, every transition condition is to be specified
by a Boolean expression and every transition choice must
comprise a choice expression. Lastly, each model file needs
exactly one model element marked as the start point and one
model element marked as the end point.
• Uniqueness: A variable or constant is not unique if the same
name is used by another variable or constant. This is checked
during creation of a new variable or constant in the variablesheet. Alternative names are suggested if a variable or constant
name is already in use.
• Syntactical correctness: All action strings, Boolean expressions
or choice expressions of any graphical element are checked for
basal syntactical correctness.
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Fig. 2. (a) Screenshot of the interactive GMS. (1) Variable-Sheet Editor, (2) Function Library, (3) the GME allows to specify CBMs. (b) Screenshot of a
resulting multi-cellular simulation of epidermal homeostasis in the Simulation Environment (SE). (4) Emerging tissue morphology during simulation; (5)
parameter window allowing the modification of global tissue properties at run-time; (6) dynamically generated graphs of cell properties during run-time.
Charts are examples of interactively creatable graphs allowing amongst others the observation of time-series and spatial particle-gradients.
• Assignment correctness: An assignment error occurs in case a
constant or a ‘global tissue property’ is identified on the left side
of an assignment. Global tissue properties can only be modified
in the graphical user interface of the SE prior to or during
execution of the simulation. Moreover, no direct assignment of
a value to a cell property of a neighboring cell using the prefix
n is allowed.
The model validation module of the graphical modeling environment
lists and explains all found errors.
3.5
Code generation and model simulation
The validated graphical models can be transformed into executable
code. As a result of the transformation process we obtain a code
archive which is stored at the user’s desired location.
For performing the simulation the resulting code archive is loaded
into a newly developed SE depicted in Figure 2b. Every cell
of the simulation is represented by an independent agent who is
embedded in a continuous toroidal spatial environment. This spatial
environment does not possess a predefined grid structure as it is
the case for cellular automata. The simulation environment has
been specifically designed for the simulation of human epidermis.
A starting set of cells (i.e. agents) is positioned on a basal lamina
with idealized rete ridges. Each agent executes an instance of the
graphical CBMs generated and compiled code. Internally, the SE
provides the CBM with an array containing all neighboring agents
as an input parameter. In turn, the CBM returns an array containing
all newborn cells which have to be included as new agents in the SE.
The simulation is executed using a simplified biomechanical model
which was described earlier (Grabe and Neuber, 2005, 2007).
4
RESULTS
4.1
our new GMS. This allowed us to principally assess whether our
previous algorithmic approach of modeling can be realized using the
graphical FSM-based modeling approach presented here. We were
able to graphically rebuild the former hard coded model without
limitations. We compared the performance of the hard coded and of
the automatically generated source code for 10 000 simulation steps.
The average execution time for one simulation step is 1.73 ms in the
originally hard coded simulation and 2.16 ms in the simulation using
the dynamically loaded generated source code while producing the
same simulation results. Thus, the automatically generated source
code yields comparable performance against the original hard-coded
version.
Figure 3 shows as an example how complex intercellular transport
or communication processes can be implemented relatively easy.
The depicted submodel represents a part of the re-implemented
keratinocyte CBM. It describes the upwards transport of molecules
in the tissue with the transepidermal water flux towards the stratum
corneum. Initially it is checked whether there are neighboring cells
(step 1). The next condition (step 2) limits the number of molecules
transported between two cells in one simulation step to 1 unit. The
prefix n_ induces a loop around the following graphical models
which is conditionally executed for all neighboring cells if the
according Boolean expression is fulfilled. This is the case if the
respective neighboring cell did not already exchange molecules
with one of its neighbors in the current simulation step. In step 3
the cell receives a certain percentage of each neighbor’s Ca2+
ions. This percentage is defined in the global tissue property
epidermalDiffusion. The position of each neighboring cell is checked
in step 4. If a neighboring cell’s position is below the position
of the current cell, Ca2+ and lamellar bodies are requested from
the neighbor according to the value specified in the global tissue
property epidermalWaterflux.
Keratinocyte model
We set out to demonstrate the suitability of the here proposed
approach for simulating epidermal homeostasis. We therefore reimplemented the previous algorithmic simulation of the epidermal
homeostasis described earlier (Grabe and Neuber, 2005, 2007) using
4.2
Cell-cycle model
To illustrate the principal difference between state-based modeling
and modeling using ordinary differential equations (ODEs), we
created three different implementations for a well-known cell-cycle
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Modeling multi-cellular behavior in epidermal tissue homeostasis
otherwise cycling through G1, S, G2, M. Time constants have been
scaled to map the estimated 60 h cell cycling time. Checkpoints are
omitted for brevity but could be easily added. During S-phase DNA
content linearly doubles. The time-averaged distribution of DNA
content shows the characteristic profile well known from FACS
analysis with the large fraction of cells in G1-, a medium fraction
in M-, and a small fraction of actively DNA synthesizing cells in
S-phase (Fig. 5f).
5
Fig. 3. Submodel of the keratinocyte CBM which realizes the transepidermal
transport of calcium and lamellar bodies from the basal lamina to the tissue’s
outer surface. The transport is realized by an undirected diffusion into the
cell (step 3) and a directed transepidermal water flow from cell to cell (step 5)
depending on the cell’s position in respect to an individual neighboring cell
(step 4).
model (Tyson, 1991; Tyson and Kauffman, 1975). The Tysonmodel describes the interaction of cdc2 and cyclin in three distinct
states: as a high activity of the protein complex MPF, as a
spontaneous oscillator, or as an excitable switch. These three distinct
states thereby correspond to metaphase arrest, rapid cell division
cycles, and growth controlled division cycles. We implemented the
condensed version of the model consisting of a set of two ODEs—
according to the BioModels database (Le Novere et al., 2006)—by
using our GMS (Fig. 4). We entered the two ODEs first in the
form of Euler-Cauchy (cellcycle_tyson_euler) and then according
to Runge-Kutta in fourth order (cellcycle_tyson_runge_kutta). As a
third form of the cell cycle we implemented the linear mitogen model
(cellcycle_episim) which triggers mitosis in a sawtooth wave form in
dependence of time (Tyson and Kauffman, 1975). During runtime of
the simulation the user can switch between the three different forms
(cellCycleMode). The last implementation is the form which was
used in the initial epidermal simulation (see above). Elements (1)
and (2) trigger mitosis. For modeling delayed mitosis due to space
constraints the mitotic state can be preserved (3).
The results are depicted in Figure 5 accordingly. Both
ODE realizations (Euler-Cauchy and Runge Kutta) generate the
characteristic spontaneous oscillations (Fig. 5b) which are on a
higher level of abstraction equivalent to the linear mitogen model
(Fig. 5a). The phase diagram shows the corresponding limit-cycle
of the ODE system (Fig. 5c). In ‘excitable switch’ state the ODE
system is responsive to sufficiently large pertubations (Fig. 5d) but
not to smaller ones (Fig. 5d and e).
For illustration we further implemented a simple five-phase model
of the cell cycle (Fig. 4b). In G0 state the cell cycle rests while
DISCUSSION
We previously demonstrated the general feasibility of algorithmic
coding of cellular behavior in multi-agent systems for simulating
epidermal homeostasis and its pertubations (Grabe and Neuber,
2005, 2007). We now developed a novel system for the
graphical modeling and simulation of cellular behavior in a
multi-cellular, tissue-like environment and applied it to the
previously presented (hard-coded) model of epidermal homeostasis.
Particularly, this example shows how material exchange and cell–
cell communication can be efficiently modeled in a multi-cellular
ensemble.
To further illustrate the principles and advantages of state-based
modeling of cell behavior we implemented different variants of the
original cell-cycle model of Tyson. We chose this model as it is not
only an example of a mitotic oscillator but also a modified version
of the famous Brusselator which as a reaction–diffusion system is a
model of chemical oscillations in general. The examples show that
our multi-agent system is able to generate the complex patterns of
reaction diffusion systems already in single cells. In the same time
it shows that hybrid models containing ODE elements can be nicely
integrated with the state-base modeling approach using our GMS.
The results also point out how signaling networks on the timescale
of hours can be easily implemented using our graphical interface.
A further illustration of the state-based approach is the five-phase
model of the cell cycle which demonstrates how a biological model
can be conveniently implemented using the proposed graphical state
based notation.
The given examples show how the user is provided with different
abstraction levels he can freely choose to work on depending
on his needs: during sole biochemical modeling, the according
reactions are modeled explicitly, while the states of the system
emerge implicitly by simulation. In behavioral modeling, the
states are modeled explicitly, while the tissue behavior emerges
implicitly. Therefore, it is important to note the main objective of
our work is not to compete with ODE modeling, but to provide
a simulation environment on a higher level of abstraction than
the pure biochemical reaction modeling, thus allowing the direct
specification of cellular behavior.
We here graphically modeled cellular behavior as a process
diagram of a FSM. The cell models are automatically translated
into high performance source code and dynamically linked to a
multi-agent graphical simulation environment where the source code
is then executed for each agent. By proliferation and interaction
of the cells (i.e. the agents) the tissue structure then emerges
according to the specified CBM. The term ‘cellular behavior’ in our
view explicitly comprises cellular decision making but furthermore
includes all cellular aspects necessary for modeling tissues like
cellular interactions and cytoskeletal remodeling. Tissue in this
respect is produced by the emergent behavior of individual cells.
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Fig. 4. Implementation of the two-variable cell-cycle model (Tyson, 1991) in the GME. (a) The differential equations were implemented using Euler, RungeKutta or saw-tooth function. Mitosis is triggered by thresholding (1,2) In delayed mitosis, the mitotic state of the cell is preserved (3). (b) Exemplary state
based implementation of a 5 phase (G1, S, G2, M, G0) cell-cycle model. DNA content linearly increases during S-phase.
FSMs allow for the modeling of cell behavior and decision
processes by conditional transitions between states. It is important
to understand that FSMs use the term ‘state’ in a more general way
than it is commonly used in the modeling of molecular networks.
The four main states of a cell are ‘proliferation’, ‘differentiation’,
‘apoptosis’ and ‘migration’. These main states are accompanied by
a multitude of sub-states which imply certain cellular functions
(e.g. change of polarity, decomposition of adhesion complexes)
or characteristics (e.g. growth arrest). Transitions between states
are realized biochemically by e.g. cellular signaling and effector
processing accordingly (Miller-Jensen et al., 2007). On the contrary,
the states of FSMs are of a more abstract, conceptual nature: in
theory they comprise all possible occupations of a cell’s ‘variables’.
In practical modeling a user will only specify those states of cellular
behavior which are relevant to him. FSM thus allow modeling on a
broad range of cellular abstraction levels, e.g. decision trees (Kharait
et al., 2007).
The question is, whether FSMs are generally a suitable approach
for multi-cellular modeling, or if cellular decision models should be
based directly on mechanistic biochemical reaction networks. In our
view both approaches are not mutually exclusive. The multi-agent
approach itself does not prohibit modeling using ODEs or obtaining
accurate quantitative time-resolved results. In fact, our graphically
specified models were translated into source code numerically
solving two ODEs by Euler-Cauchy and Runge-Kutta. The key
concept of the multi-agent approach is that it simulates a cellular
ensemble as a clocked system. The resolution of the time scale,
which is chosen by the modeler, thereby determines whether the
simulated system is of more continuous or of more discrete nature.
As such the time-accuracy of the simulation lies in the hands of the
user and is not a-priori constrained by the simulation environment.
For tissue simulation multi-time scale modeling is of central
relevance. This could be achieved by hierarchical modeling
(Asthagiri and Lauffenburger, 2000), which here could be
implemented using submodels running with different clock times.
A simpler solution in our view is to approximate the outcome of the
submodels by states. For example: a decision tree is implemented as
an abstraction of a fast intracellular network. In this sense, we offer
a novel opportunity for complexity reduction: fast running decision
processes like e.g. intracellular signal transduction are approximated
by modeling their outcome. Thus not all details of the complex
system have to be modeled to achieve accurate results at a higher
level. This is in line with the ‘slaving principle’ that indicates that the
cell’s long-term phenotypic response can be expressed in terms of
its slowest evolving functional elements (Busch et al., 2008; Haken,
2004).
We studied the performance of our system using the key modeling
concepts ‘state’, ‘condition’ and ‘submodel’ as well as regular
intercellular communications. The results show a linear complexity
of our implementation for up to 100 000 model elements (data not
shown). This demonstrates the principal suitability of the approach
for modeling even extremely complex multi-cellular ensembles.
The technical idea of the presented approach was the use of
automatic code generation originating from the field of software
engineering for translating graphically depicted decision models
automatically into source code. This source code is then dynamically
linked into a multi-cellular simulation environment. In this way, we
provide a way for the computationally efficient realization of multicellular models. Technically, our system is based on the well-known
multi-agent framework MASON. To our knowledge, our system is
the first implementation coupling a flexible graphical modeling tool
to the source code generation for a multi-agent framework. The
GMS presented here was primarily designed for creating models
for simulating epidermal homeostasis. We think, that independent
of this matter of subject, our approach could also be applied to
modeling other tissues. Particularly, as systems biology is currently
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Modeling multi-cellular behavior in epidermal tissue homeostasis
Fig. 5. Charts resulting from simulating the CBM’s cell-cycle submodels of Figure 4 in the SE using the provided charting functionality. (a) Linear mitogen
model cellcycle_episim (Fig. 4). (b) Characteristic spontaneous oscillations of the two variable ODE model equivalent to the linear mitogen model. (c)
corresponding limit-cycle in the phase diagram. (d) In ‘excitable switch’ state the ODE system is responsive to sufficiently large pertubations (perturbation 3)
but not to smaller ones (1,2). (e) The system is robust to minor pertubations and returns to a steady state (pertubations 1,2 in d). (f) Distribution of DNA
content analog to FACS resulting from Figure 4b.
in a transition towards medically oriented applications, modeling of
multi-cellular ensembles will become essential. Our results indicate
that especially for this purpose, behavioral modeling of cells using
FSMs in multi-agent systems could become a promising approach.
ACKNOWLEDGEMENTS
The manuscript is dedicated to the memory of Prof. Dr. Karsten
Neuber for his visionary and passionate contributions to this project.
We thank Dr. Michael Grabe for critical reading of the manuscript.
Funding: BMBF FORSYS-PARTNER and BMBF MEDSYS.
Conflict of Interest: none declared.
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