COMPUCELL, a multi-model framework for simulation of morphogenesis
J. A. Izaguirre1, R. Chaturvedi1, C. Huang1, T. Cickovski1, J. Coffland1,
G. Thomas2, G. Forgacs3, M. Alber4, G. Hentschel5, S.A. Newman6, and J.A. Glazier7
February 12, 2016
384 Fitzpatrick Hall
ABSTRACT
Motivation: COMPUCELL is a multi-model software
Notre Dame, IN 46556
framework for simulation of early development of
Email: izaguirr@cse.nd.edu .
multicellular organisms or morphogenesis. It models
1. INTRODUCTION
the interaction of the gene regulatory network, as
revealed by cDNA microarray or quantitative PCR
We present COMPUCELL, a computational framework
experiments, with generic cellular mechanisms such
for the study of morphogenesis, that is, the
as cell adhesion, division, haptotaxis, and chemotaxis.
development of multicellular organisms. The model
A combination of a state automaton with stochastic
allows growth and spatial patterning to occur
local rules and a set of differential equations,
simultaneously. Different modules may be used in the
including subcellular ordinary differential equations
software to generate these biological processes. We
(ODEs) and extracellular reaction-diffusion partial
illustrate the framework through the simulation of
differential equations (PDEs) model gene regulation.
skeletal pattern formation in the avian limb bud. Limb
This in turn controls the differentiation of the cells,
development is a good model system for the study of
and cell-cell and cell-extracellular matrix interactions
tissue growth, differentiation, and pattern formation.
that give rise to cell rearrangements and pattern
Useful computational models of multicellular
formation such as mesenchymal condensation. The
development involve, in addition to differential
cellular Potts model (CPM), a stochastic model that
regulation of gene activity, cell behaviors such as
accurately
reproduces
cell
movement
and
release of diffusible factors, adhesion and motility
rearrangement, models cell dynamics. All these
(Marée and Hogeweg, 2001; Dillon and Othmer,
models couple in a controllable way, resulting in
1999; Newman and Comper, 1990). Our simulation
powerful and flexible computational environment.
uses the cellular Potts model (CPM) for domain and
Results: We use COMPUCELL to simulate the
cell growth and motility, and reaction-diffusion
formation of skeletal pattern in the avian limb bud.
equations to model spatial pattern formation. An
The model allows for simultaneous growth and
earlier and more limited version of this paper is in
spatial patterning.
(Chaturvedi et al., 2003).
Availability: Binaries and source code for Microsoft
Our long-term goal is to use COMPUCELL to
Windows, Linux, and Solaris are available for
simulate
morphogenesis
using
experimental
download from
measurements of gene regulatory networks, cell and
http://sourceforge.net/projects/comp
extracellular matrix (ECM) properties, and cell-cell
ucell/
and cell-microenvironment interactions. Since genes
Contact: By email: compucell@cse.nd.edu.
and their products interact with the physical
Key words: Morphogenesis, Avian Limb
properties of tissues during early morphogenesis
Development, Reaction-Diffusion Equations, Gene
(Newman & Comper, 1990), COMPUCELL allows the
Regulatory Network Models, Cellular Potts Model,
modeling of the interaction of the gene regulatory
Rule-based Formalisms, Object-oriented framework.
network with cellular mechanisms; thus COMPUCELL
Corresponding author
allows coupling between biosynthesis and diffusion
Jesús A. Izaguirre
of morphogens (molecules released by cells that
Department of Computer Science and Engineering
affect the behavior of other cells during
1
Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 46556
Department of Physics, University of Notre Dame, Notre Dame, IN 46556
3
Department of Physics and Biology, University of Missouri, Columbia, MO 65211
4
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
5
Department of Physics, Emory University, Atlanta, GA 30332
6
Department of Cell Biology and Anatomy, New York Medical College, Valhalla, NY 10595
7
Departments of Physics and Biology and Biocomplexity Institute, Indiana University, Bloomington, IN 47405
2
development), cell adhesion, haptotaxis (the
movement of cells along a gradient of a molecule
deposited on a substrate), and chemotaxis (the
movement of cells along a gradient of a chemical
diffusing in the extracellular environment). The
interplay of these factors results in arrangements of
cells specific to a given organism.
1.1. Example: Avian Limb Development
Our simulation of the morphogenesis of a chicken
limb generates the arrangement of bones in a
forelimb, when we view the limb palm down on a flat
surface. In Figure 1 we refer to the orientation of the
long axis as proximodistal (PD; proximal meaning
closer to, and distal farther from, the body). The axis
from thumb to little finger defines the anteroposterior
(AP) direction. The dorsoventral (DV) direction
traverses the width of the limb from the back to front
of the hand. This study presents a two-dimensional
simulation in the plane defined by the proximodistal
and anteroposterior axes. Because the number of
elements along the dorsoventral axis does not change
during normal development of the vertebrate limb (it
always remains one skeletal element in thickness),
this representation captures the key pattern changes.
However, asymmetry along the dorsoventral axis is
important for the functioning of the limb, and a
complete model of limb development must eventually
include all three dimensions.
In a chicken limb, as in all vertebrate limbs,
skeletal pattern formation occurs within tissue
surrounded by a thin bounding layer, the ectoderm.
This study neglects the ectoderm as a separate
structure. The apical ectodermal ridge (AER), a
narrow strip of the ectoderm running along the apex
(distal boundary) of the growing limb bud in the
anteroposterior direction, is necessary for elongation
and patterning of the limb. AER releases fibroblast
growth factors (FGFs), which control division of
cells in the proximal region.
Experimental evidence supports accounts of limb
growth and pattern formation in which the space
within the developing limb is divided into three
zones- the apical zone in which only growth takes
place, an active zone in which cells rearrange locally
into precartilage condensations, and a frozen zone in
which the condensations have progressed to
differentiated cartilage and no additional patterning
takes place. Bone later replaces the cartilaginous
skeleton isomorphically in species with a bony
skeleton. Growth continues in both active and frozen
zones. Exact definitions and nature of these zones still
excite lively debate (Dudley et al., 2002; Saunders,
2002; Wolpert, 2002). On experimental and
theoretical grounds it has been proposed that in the
active zone one or more members of the TGF-β
family of growth factors act as the activating
morphogen of a reaction diffusion system (Newman,
1988; Newman et al., 1988; Leonard et al., 1991). We
also assume, as suggested by recent experiments
(Moftah et al., 2002), that sites of incipient
condensation release
a
laterally inhibitory
morphogen, a necessary component of most reactiondiffusion schemes (Meinhardt and Gierer, 2000).
The zones are characterized by distinct dynamics
governing the evolution of cells, and of morphogens
in the extracellular space. The zones themselves grow
and their interfaces move distally.
Other important alternative models of aspects of
limb development exist: Dillon and Othmer (1999)
propose that early shaping of the limb bud is due to a
reaction-advection-diffusion process between a
growth factor produced in the AER and the
morphogen Sonic Hedgehog produced in the zone of
polarizing activity (ZPA) at the posterior margin of
the bud. They model the growth of the limb as a
viscous fluid using the Navier Stokes equation.
Wolpert and coworkers (Wolpert, 2002) have
proposed the “positional information” and the
“progress zone” models. In the former, the ZPA
produces a morphogen that diffuses throughout the
tissue, establishing a gradient that provides
information on AP position to the cells and leads to a
spatial pattern of differentiation. In the latter, PD
differentiation is controlled by the number of
divisions a cell undergoes while in the apical zone.
Our software framework can be extended to simulate
these and other models.
We assume that cell division is uniform throughout
all zones of the limb bud (cf. Lewis, 1975; Bowen et
al., 1989). New cells form by division, replenishing
the active zone as the limb bud grows. As more and
more cells condense into a bonelike pattern, the
proximal frozen zone grows as well.
In our model, spatiotemporal patterns of the
activating morphogen induce a corresponding set of
cell condensations as follows: cells that sense a
threshold level of the signal produce and secrete an
adhesive substratum, and also increase their adhesion
to one another. In the actual limb, it is TGF-β that
induces cells in the active zone to produce the ECM
glycoprotein fibronectin, which adheres to the cell
surface and causes cells to accumulate at focal sites
(Frenz et al., 1989a,b). Cells at these sites also
become more adhesive to one another by producing a
homophilic cell-surface adhesion protein N-cadherin
(Oberlender and Tuan, 1994). In our model, we refer
to the secreted substrate adhesion molecule as SAM
(promoting haptotaxis), and the cell-cell adhesion
molecule as CAM. For simplicity, we do not include
the feedback of the cells on the morphogen fields due
to absorption,
boundaries.
secretion and
changes in cell
3. SYSTEMS AND METHODS
1.2 Mathematical Model
3.1 Cell and Tissue Growth and Movement
Our mathematical model of integrated limb bud
growth and pattern formation includes the following
processes:
(1) Cell and tissue growth and movement using the
cellular Potts model (Section 3.1).
(2) Skeletal pattern formation via reaction-diffusion
PDEs (Section 3.2).
(3) Cell differentiation using individual cell gene
network ODEs and rules for differentiation (Section
3.3).
(4) Growth of zones, that is, the domains where the
above processes are active at a given time step
(Section 3.4).
(5) Integration of submodels (Section 3.5).
Dillon and Othmer model tissue growth and
movement as a viscous fluid using a discretized
Navier Stokes equation in a rectangular domain. We
use instead the cellular Potts model (CPM, Graner &
Glazier, 1992), which allows us to treat cells
mesoscopally and yet retain the identity of individual
cells, making it possible to track simulation cells and
compare with experimental data.
CPM draws on the differential adhesion hypothesis
(Steinberg, 1998) to accurately reproduce cell
movement and rearrangement based on minimization
of cell-cell surface interaction energy. The extended
model includes terms that provide for the haptotactic
or chemotactic response of cells to gradients of
soluble or bound morphogens in the extracellular
space. Experiments have validated this approximation
for cell motion (Mombach et al., 1995).
Tracking extended cells within the simulation
gives us a more accurate representation of the cellular
dynamics. Our approach, compared to the continuum
approach, is easier to implement, although somewhat
slower: for example, we can implement advection
within CPM rather than as an upwinding scheme in
the reaction-advection-diffusion equations, which
now become simple reaction-diffusion equations.
This avoids instabilities in the PDE solver. Even the
reaction and diffusion terms can be implemented
within CPM if necessary, although the continuum
approximation has been appropriate for us so far.
Our tissue growth model is very limited. Right
now we do not model the elastic ectoderm, although
it is possible to do this on the reaction-diffusion
domain using the immersed boundary method
(Peskin, 1977) or the immersed interface method
(LeVeque & Li, 1994) or even level set methods (cf.
Dockery and Klapper, 2002). It is also possible and
simpler to do this within CPM by defining the
ectoderm as a special type of cell.
The cellular Potts model (CPM) minimizes an
effective energy E according to a Metropolis Monte
Carlo process: E  Econtact  Earea  Echemical . This
2. SIMULATION RESULTS
Figure 2 shows a simulation of the full model
described above. Cells cluster subject to differential
cell adhesion. The model includes cell division and
haptotaxis by cells in response to SAM. The
genetically governed response of cells to high
activator concentration is to begin secreting SAM.
Cells respond to SAM in two ways: (1) SAM causes
cells to stick to the substrate; (2) SAM makes the
cells more likely to condense by upregulating cell-cell
adhesion. Activator concentration obeys the
Schnakenberg reaction-diffusion equations (Section
3.2). An appropriate choice of a control parameter
gives the required pattern periodicity. The far right
window shows the activator concentration; the prepattern directing the later cell condensation into the
typical chondrogenic pattern is clear. The middle
window represents the SAM concentration. Since
cells exposed at some time to high activator
concentration begin and continue to secrete SAM,
and SAM in turn has the two effects described above,
the pattern of SAM concentration resembles the
activator pre-pattern. Finally, the cells condense into
the bone pattern of 1+2+3 (where 3 corresponds to
the three digits), shown in the left window.
The growth of the limb bud is not predefined. It
depends on the cell division rate and how fast the
cells can move. New cells generated by cell division
push the limb tip upward. Thus growth occurs
naturally. The computational domain corresponds to
realistic values: 1.4 mm for the anteroposterior width;
patterning begins at stage 20 of chicken embryo
development. The proximo-distal dimension at stage
28 is 4 mm about three times the width. A 100 by 300
grid covers the domain. This simulation ran in 93
minutes using a SunBlade 1000 with a 900 MHz CPU
and 512 megabytes of memory.
creates thermodynamically favorable arrangements of
cells, subject to constraints in the surface area of the
cells, and possibly to energy penalties from the
exposure of cells to chemotactic or haptotactic
gradients. Readers not interested in the mathematical
details may skip the rest of this section.
The basic entities are individual cells. CPM
superimposes a lattice on the cells. Each lattice site
has an associated index (also called spin in the
literature). The value of the index at a lattice site is 
if the site lies in cell . All sites with index 
theoretically belong to the same cell, the probability
that all such sites connect is high since there is an
energy penalty associated with disconnected domains,
Equations (1)-(2).
We describe the net interaction between two cell
membranes by a binding energy per unit area, J,’,
which depends on the types of the interacting cells.
Here , ’ are the types of the cells on either side of
the link between sites with dissimilar indices. The
contact energy is thus:
EContact 
 J cell _ type ,cell _ type .
(1)
pixels  ,  in adjacent cells
In our simulation, there are three types of cells:
condensing, non_condensing, and medium. The
interaction energies are defined as follows:
J cell , cell
 J condensing, condensing  0.5,

  J non_condensing,any_cell  7.0,
J
 medium, any_cell  0.2.
While the particular values are not important, the
relative strength of the bonds is: clearly, condensing
cells will tend to stick to one another.
At any time t, a 2D cell of type  has a surface area
s(,). Equation (2) penalizes a cell’s variation in s
from the target value.
Earea 
  (s( , t )  s
all cells
In our simulation,
 3
( , t )) .
2
target
(2)
and starget  16 in the
absence of cell growth. To model the growth of a cell,
a separate sub-model governs the increase with time t
of starget(,). We model division by starting with a
cell of average size and causing it to grow until it
doubles its size, at which point we split the dividing
cell into two daughters. We give each cell a unique
sping . We can model cell death simply by setting a
cell’s target volume to zero.
We model ECM, liquid medium and solid
substrates, as cells; i.e., as a domain of sites with a
distinct index. We must define the interaction energy
between each cell type and the ECM.
Chemotaxis or haptotaxis, the movement of cells
in response to gradients in chemical concentration,
requires additional fields to describe the local

concentrations C (x) of the signaling molecules. The
equations for the fields depend on the particular
molecule. Chemotaxis or haptotaxis introduces an
effective chemical potential,  ( ) , into the CPM,
resulting in the cell executing a biased motion in the
direction of the gradient. The effective chemical
energy in the CPM energy formalism is:

E Chemical    C x  . (3)
In our simulation, this field corresponds to the
accumulation of SAM produced by cells. We use
  25 for cells in the condensing state, and a SAM
production rate of 0.005 units/step.
The model assumes that a temperature, T, drives
cell membrane fluctuations. If a proposed change in
configuration (i.e., change in the spins associated
with sites of the lattice) produces a change in
effective energy, E , we accept it with probability:
P(E )  min( 1, e E / kT ) , (4)
where k is a constant converting T into units of
energy. We use T  7.0. A good reference on how
to choose parameters for CPM is Glazier and Graner,
(1993). One Monte Carlo step corresponds to N such
randomly selected proposed changes, where N equals
the total number of sites on the grid.
3.2 Skeletal Pattern Formation
In our simple biological model of avian limb
development, generation of morphogen concentration
distributions establishes a prepattern for cells
condensing within a mesenchyme, which refers to
roughly isotropic (non-polar) cells arranged loosely in
a hydrated extracellular matrix where they make only
minimal contact with one another.
We use a system of reaction-diffusion partial
differential equations for the spatial patterning
(Turing, 1952; Meinhardt and Gierer, 2000). A
reaction-diffusion model may underlie limb skeletal
pattern formation (Newman and Frisch, 1979) as well
as other biological spatiotemporal patterning. Further
discussion on other models for pattern formation in
the chick limb is in (Gilbert, 1997).
In this simulation we use the Schnakenberg
equations following (Murray, 1993):
u
  ( a  u  u 2 v )   2u ,
t
v
  (b  u 2v)   2v.
t
(5)
In Equation (5), u is the activator concentration at
a location (x, y) in space and time t; v is the inhibitor
concentration.  is a parameter that affects the
period of the (activator) pattern as explained below.
The parameters we use in the simulations
presented here are as follows: a=0.017, b=1.015,
d=7.1. A finite difference discretization is used in
space, and the time marching scheme is an Euler
forward scheme. The equation is solved in a grid of
50 by 100 grid points. The domain moves in time,
and we use no flux boundary conditions. The
parameters for the solver, x  y  0.02 ,
t  2E  6 , are chosen to satisfy the standard
2
2
stability criterion dt / min(( x) , (y ) )  1/ 4 .
These parameters can be chosen independently of
CPM.
In our simulation, the parameter that controls the
periodicity of the finger patterning is the following,
although other functions could also be used:
 80, if y  80,

 ( y)  200, if y  180,
800, otherwise.

3.3 Cell Differentiation
Cells may respond to morphogens they or their
neighbors produce by altering their gene activity in
continuous or discontinuous (switch-like) fashion.
Such nonlinear feedback loops may lead to
differentiation of cells into more specialized cell
types. We consider that the network of expressed
genes and their products embodies a set of rules for
cells that govern their growth, division, secretion of
morphogens and strength of adhesion. These rules
depend on the state of several chemical fields at the
intra- and inter-cellular level that we model by
differential equations applicable over the appropriate
spatial domain. Our formalism specifies alternative
cell types and rules that govern transitions between
them. This model of gene regulation captures formal,
qualitative aspects of regulatory interactions and
allows fitting to quantitative experiments. Other
approaches to modeling gene regulatory networks are
possible (e.g., Arkin, Ross and McAdams, 1998; and
Jong, 2002).
For our example simulation, initially all cells in the
active zone are mitosing and not condensing (i.e.,
dividing, and not producing SAM or responding
haptotactically to SAM). These cells obey the CPM
dynamics of Equations (1)-(4). When such a cell in
active zone senses a threshold local concentration of
the activator (currently 0.75), it enters the condensing
state, in which it produces SAM and starts responding
haptotactically to it. The cell also starts to upregulate
cell-cell adhesion (decreasing the parameter
J cell ,cell in the CPM from 7.0 to 0.5).
3.4 Growth of Zones
For computational efficiency we apply the various
dynamics (CPM, reaction-diffusion, differentiation
events) only in specific regions of the growing limb
bud. Zones are a computationally simple substitute
for more detailed modeling of the differentiation of
individual cells
In the frozen zone, condensation into cartilaginous
patterns has finished; and evolution stops.
Fluctuations naturally decrease in time due to
increased binding to SAM, but discontinuing MonteCarlo CPM updating in the frozen zone speeds the
computation by reducing the grid size without greatly
affecting the biological realism.
Since we lack governing rules/equations for these
zones and their interfaces, we assume ad hoc rules for
their motion parameters based on the requirement that
the activating morphogen concentration fields and
cell clustering have enough ‘time’ (number of
iterations) to form distinct patterns. The number of
iterations within the active zone before the frozen and
active zones move upwards is thus a model
parameter. Upward growth of the active zone
corresponds to distal growth of the limb bud; growth
of the frozen zone corresponds to progressive
establishment of the chondrogenic pattern. Specific
parameters are discussed below.
3.5 Integration of Submodels
We must integrate the submodels, particularly the
stochastic CPM with continuum reaction diffusion to
allow the various mechanisms to work in a
coordinated fashion:
1) We match the spatial grid for continuum and
stochastic models by interpolating the coarser spatial
grid used in the explicit solution of the discretized
reaction-diffusion equations to the discrete grid of the
CPM. For example, the RD domain is 4 times coarser
than the domain for CPM.
2) We define the relative number of iterations for the
reaction-diffusion and CPM evolvers. Diffusion, and
hence establishment of the morphogen distributions,
is rapid compared with growth for small domains,
although the time scales of these processes become
more comparable as outgrowth proceeds (cf. Dillon
and Othmer, 1999, p. 310). Throughout our
simulation, the domain where RD occurs is growing
faster than the one where CPM is active. The moving
speeds are 1 pixel of RD window every 3 steps, and 1
pixel of CPM window every 4 steps. We use a ratio
of 10 steps of CPM for each step of RD.
3) More importantly, the ratio between the CPM steps
to complete cell mitosis and the number of CPM steps
per window should be such that there are enough cells
in the domain, but not too many. Controlling cell
density is relatively difficult in CPM, and adequate
parameters were determined experimentally. It would
be better to control the flux of nutrients advected
through the tissue, and thus indirectly control cell
density. A numerically determined mitosis doubling
time of 85 steps gives the desired cell density of 60%
throughout our simulation.
4. SOFTWARE
COMPUCELL is an open-source object-oriented
framework available in Source Forge7. It has the
following components: (i) base classes describing the
main abstractions of morphogenesis; (ii) energy
functions for the CPM; (iii) CPM algorithms; (iv)
reaction-diffusion and simple diffusion solvers; (v) a
cross-platform GUI based on the Fox toolkit8; (vi)
XML-based front-end9; and (vii) VTK10, OpenGL11
or VRML12 visualization toolkits, with support for
Phantom haptic interfaces13.
One specifies the computational model using XML
configuration files containing simulation parameters
and their values in pairs, a visualization file
(optional), and a Potts initial file to allow an arbitrary
initial cell distribution. Optionally, COMPUCELL can
initialize the cell distribution in the grid to uniform or
random.
Each COMPUCELL simulation requires a cell model
declaration. Cell models describe one or more cell
differentiation types, and the state variables
associated with each cell type. Furthermore,
differentiation events can be defined, such as
Algorithm (4) below.
The overall algorithm is Algorithm (1). Each step
of CPM looks like Algorithms (2) and (3). Finally,
the cell differentiation step is user-defined. In our
simulation, it looks like Algorithm (4).
For total number of combined steps {
Solve RD (Equation (5))
Solve N steps of CPM (Algorithm 2)
Solve cell state ODE
Do cell differentiation
Grow domains of RD and CPM
}
}
Algorithm 1: Main Loop of CompuCell
For number of grid points in Potts lattice {
Compute Eold according to Equations (1)-(3)
Attempt swap of random pixel with a neighbor
Compute Enew according to Equations (1)-(3)
Apply Metropolis criterion, Equation (4)
If cell is growing, attempt division (Algorithm 3)
}
Algorithm 2: Cellular Potts Model
Do breadth-first search {
Start from selected cell boundary pixel
Keep track of visited pixels
Keep track of neighbors waiting processing
Ignore pixels outside dividing cell
}
If S target pixels are in list of visited pixels
rename them as a new cell
Algorithm 3: Cell Division
If cell_type is non condensing
and activator_concentration > threshold {
cell_type := condensing
haptotaxis to SAM := on
SAM_production := on
}
Algorithm 4: Cell Differentiation
4.1 Performance Data
Table 1 presents data for different grid sizes and
different numbers of cells in the same machine as
above. Grid size is the main factor affecting the
runtime; with the same grid size, the number of cells
does not much affect the speed. Thus the algorithms
scale for quantities dependent on the number of cells.
VTK based visualization does not scale as well as the
computational engine, since its performance is highly
dependent on the number of cells in the simulations.
5. RELATED WORK AND
DISCUSSION
7
http://sourceforge.net/projects/compucell/
http://www.fox-toolkit.org/
9
http://xml.apache.org/xerces-c/index.html
10
http://public.kitware.com/VTK/get-software.php
11
http://www.opengl.org/
12
http://www.vrmlsite.com/
13
http://www.sensable.com/
8
There is extensive literature on models for
morphogenesis. A good summary is in (Ransom,
1981). Most current models are based on purely
continuum approaches or discrete cellular automata.
COMPUCELL uses a combined model of the two.
A purely continuum model of early limb
development is presented by (Dillon and Othmer,
1999). Here, the shape of the growing limb is
obtained in 2D as a reaction-advection-diffusion
system between two organizing regions, whereas
growth is modeled using the Navier Stokes equations.
Some complexities of these models are the handling
of the moving boundary in the PDE solution, plus the
instabilities introduced by the advective term. Maini
and coworkers have used the moving finite element
method to handle similar issues, as for example in
(Page et al., 2001).
Cellular automata like CPM have been used,
among others, by (Marée and Hogeweg, 2001) to
model the culmination of development of
dictyostelium discoideum. Recently, (Merks et al,
2003) have used a combination of lattice-gas
automata advection-diffusion and a discrete model of
branching in coral reef.
The approach presented in this paper exploits the
computational advantages of a continuum reactiondiffusion or simple diffusion formulation, while
allowing handling of viscous fluid motion and
advection using the discrete CPM. The more detailed
cellular dynamics allow fitting to more detailed
biological and biophysical data.
An area that was barely touched in this work is the
modeling of the gene network. A variety of
simulations focus on the biochemical reactions inside
individual cells: (Arkin et al. 1998) and (McAdams
and Arkin, 1999) have worked on quite detailed
modeling of genetic and biochemical networks and
their role in development. They analyze physically
the network of biochemical and genetic reactions
governing cellular development and apply principles
of control systems to predict cell behavior and
differentiation in response to internal and external
signals. Integration with such models would be a
fruitful extension.
Constraints and limitations of our model are
summarized here: (i) the model of growth is too
simple, a model like that in (Dillon and Othmer,
1999) would be more adequate; (ii) the CPM has
many parameters; some can be determined
experimentally and others through simulation; (iii)
reaction-diffusion is only one of the possible pattern
formation mechanisms in development, and simple
diffusion or more complex genetic control may
account for certain aspects; even if reaction-diffusion
underlies limb skeletogenesis, our equations are not
biologically correct; (iv) our model of cell
differentiation is developed based on focused
experimental studies to determine the key genetic
players; extracting such knowledge from microarrays
in a more automatic way would complement these
methods and increase their generality; (v) the
integration of the submodels is loose: for example, we
have no feedback from the cell to the PDE, and we do
not have an adaptive time step control for the relative
speeds of growth and diffusion, which do change
throughout the simulation.
Despite these limitations, our model of limb
development shows a framework under which
subcellular description of the genetic regulation (as
ODEs or rules), can be integrated with continuum and
discrete models of spatial patterning and growth. The
model allows fitting of experiments such as: (i) fate
maps can be compared to cell tracking experiments in
the simulations; (ii) contact energies in CPM can be
obtained from experiments measuring surface
tensions in cells; (iii) gene expression experiments
can be compared to the simulated gene expression;
(iv) experimental shapes can be used as input to the
model by providing a history of the domain in which
one solves Equations (1)-(5), or else can be compared
to models that attempt to produce the shapes
themselves; (v) chondrogenesis experiments can be
compared to the simulation patterns.
We are working on extending the code to 3D,
modeling realistic geometry, more general reactiondiffusion or simple diffusion solvers, and more
detailed networks of gene expression.
Related software includes the following:
Cytoscape14 provides a framework to construct
molecular interaction networks, and then integrates
these networks with gene expression profiles and
other state data. Virtual Cell15 models intracellular
processes. Cello16 is a program to simulate tissues and
cells. COMPUCELL provides modeling capabilities
that are more comprehensive, and in many cases
complementary to these programs.
ACKNOWLEDGEMENTS: This research was
supported by an NSF Biocomplexity Grant No. IBN0083653, an NSF CAREER Award ACI-0135195,
the Center for Applied Mathematics and the
Interdisciplinary Center for the Study of
Biocomplexity at University of Notre Dame and by
the Biocomplexity Institute at Indiana University.
7. REFERENCES
Arkin, A.P., Ross, J., McAdams, H.H. (1998) Nongenetic
diversity: Random Gene Expression Mechanisms
Determine Which Phage Lambda Infected Cells Become
Lysogens, Genetics, 149(4), 1633-1648.
Bowen, J., Hinchliffe, J.R., Horder, T.J. & Reeve, A.M.F.
(1989). The fate map of the chick forelimb-bud and its
bearing on hypothesized developmental control
mechanisms. Anat. Embryol. 179, 269-283.
Chaturvedi, R., Izaguirre, J. A., Huang, C., Cickovski, T.,
Virtue, P., Thomas, G., Forgacs, G., Alber, M.,
Hentschel, G., Newman, S.A., & Glazier, J.A., (2003)
Multi-model
simulations
of
chicken-limb
http://www.cytoscape.org/
http://www.nrcam.uchc.edu/
16
http://mbi.dkfzheidelberg.de/mbi/research/cellsim/c
ello/index.html
14
15
morphogenesis, Lecture Notes in Computational
Science, LNCS 2659, Proceedings of the International
Conference on Computational Science and Engineering
ICCS 2003, June 2003, Part III, pp 39-49.
Dillon, R. and Othmer, H.G. (1999) A Mathematical Model
for Outgrowth and Spatial Patterning of the Vertebrate
Limb Bud. J. Theor. Biol. 197, 295-330.
Dockery, J. and Klapper, I. (2002). Finger formation in
biofilm layers. SIAM J. Appl. Math. 62, 853-869.
Dudley, A. T., Ros, M. A. & Tabin, C. J. (2002). A reexamination of proximodistal patterning during
vertebrate limb development. Nature 418, 539-44.
Frenz, D., Akiyama, S., Paulsen, D. & Newman, S. (1989a)
Latex beads as probes of cell surface-extracellular
matrix interactions during chondrogenesis: evidence for
a role for amino-terminal heparin-binding domain of
fibronectin, Dev. Biol., 136, 87-96.
Frenz, D. A., Jaikaria, N. S., and Newman, S. A. (1989b)
The mechanism of precartilage mesenchymal
condensation: a major role for interaction of the cell
surface with the amino-terminal heparin-binding domain
of fibronectin. Dev. Biol. 136, 97-103.
Gilbert, S. F. (1997) Developmental Biology Fifth Edition,
Sinauer Associates, Inc., Publishers, Sunderland,
Massachusetts.
Graner, F. & Glazier, J. A. (1992) Simulation of biological
cell sorting using a two-dimensional extended Potts
model, Phys. Rev. Lett., 69, 2013-2016.
Glazier, J. A. & Graner, F. (1993) A simulation of the
differential adhesion driven rearrangement of biological
cells, Phys. Rev. E, 47,2128-2154.
Jong, H. D. (2002) Modeling and simulation of genetic
regulatory systems: a literature review, J. Comp. Biol., 9
(1), 67-103.
Lewis, J. (1975) Fate maps and the pattern of cell division:
a calculation for the chick wing-bud, J. Embryol. exp.
Morph., 33 (2), 419-434.
Leonard, C. M., Fuld, H. M., Frenz, D. A., Downie, S. A.,
Massague, J. and Newman, S. A. (1991). Role of
transforming growth factor-beta in chondrogenic pattern
formation in the embryonic limb: stimulation of
mesenchymal condensation and fibronectin gene
expression by exogenous TGF-beta and evidence for
endogenous TGF-beta-like activity. Dev. Biol. 145, 99109.
LeVeque, R. J. & Li, Z. (1994). The immersed interface
method for elliptic equations with discontinuous
coefficients and singular sources. SIAM J. Numer. Anal.
31, 1019-1044.
McAdams, H.H., Arkin, A.P (1999) Genetic Regulation at
the Nanomolar Scale: It's a Noisy Business!, TIGS,
15(2), 65-69.
Marée, F. M. & Hogeweg, P. (2001) How amoeboids selforganize into a fruiting body: multicellular coordination
in dictyostelium discoideum, Proc. Natl. Acad. Sci. USA,
98, 3879-3883.
Meinhardt, H., and Gierer, A. (2000) Pattern formation by
local self-activation and lateral inhibition, Bioessays, 22,
753-60.
Merks, R.M.H., Hoekstra, A.G., Kaandorp, J.A., & Sloot,
P.M.A. (2003) Models of coral growth: Spontaneous
branching, compactification and the Laplacian growth
assumption, J. Theor. Biol., in press.
Miura, T., and Shiota, K. (2000). TGFbeta2 acts as an
"activator" molecule in reaction-diffusion model and is
involved in cell sorting phenomenon in mouse limb
micromass culture. Dev Dyn 217, 241-9.
Moftah, M. Z., Downie, S. A., Bronstein, N. B.,
Mezentseva, N., Pu, J., Maher, P. A., and Newman, S.
A. (2002). Ectodermal FGFs induce perinodular
inhibition of limb chondrogenesis in vitro and in vivo
via FGF receptor 2. Dev Biol 249, 270-82.
Mombach, J., Glazier, J., Raphael, R. & Zajac, M. (1995)
Quantitative comparison between differential adhesion
models and cell sorting in the presence and absence of
fluctuations, Phys. Rev. Lett., 75, 2244-2247.
Murray, J. D. (1993) Mathematical Biology, Second
Corrected Edition (Biomathematics Vol. 19), SpringerVerlag, Berlin Heidelberg, pp. 156, 376, 406, 472, 739.
Newman, S. A. (1988) Lineage and pattern in the
developing vertebrate limb. Trends Genet., 4, 329-332.
Newman, S. A. & Comper, W. (1990) Generic physical
mechanisms of morphogenesis and pattern formation,
Development, 110, 1-18.
Newman, S. A. & Frisch, H. L. (1979) Dynamics of skeletal
pattern formation in developing chick limb, Science,
205, 662-668.
Newman, S. A., Frisch, H. L., and Percus, J. K. (1988). On
the stationary state analysis of reaction-diffusion
mechanisms for biological pattern formation. J. Theor.
Biol. 134, 183-197.
Oberlender, S. & Tuan, R. (1994) Expression and
functional involvement of N-cadherin in embryonic limb
chondrogenesis, Development, 120, 177-187.
Page, K.M., Maini, P.K., Monk, N.A.M., & Stern, C.D.
(2001). A model of primitive streak initiation in the
chick embryo. J. Theor. Biol. 208, 419-438.
Peskin, C.S. (1977). Numerical analysis of blood flow in
the heart. J. Comp. Phys., 25, 220-252.
Ransom, R. (1981). Computers and Embryos: Models in
Developmental Biology. John Wiley, New York.
Steinberg, M. S. (1998) Goal-directedness in embryonic
development, Integrative Biology, 1, 49-59.
Saunders, J. W., Jr. (2002). Is the progress zone model a
victim of progress? Cell 110, 541-3.
Turing, A. (1952) The chemical basis of morphogenesis.
Phil. Trans. Roy. Soc. London, B 237, 37-72.
Wolpert, L. (2002). Limb patterning: reports of model's
death exaggerated. Curr Biol 12, R628-30.
FIGURES, TABLES, PROGRAMS
(light gray). Still to form are the wrist bones and
digits. The apical ectodermal ridge (AER) runs along
the distal tip of the limb approximately between the
two points intersected by the arrow indicating the AP
axis.
Fig. 1. Schematic representation of a developing
vertebrate limb: The three major axes are indicated,
as are the first two tiers of skeletal elements to form.
In the chicken forelimb these are the humerus, shown
as already differentiated (dark gray), followed by the
radius and ulna, which are in the process of forming
Grid Size
Number of
Cells
Cell
Density
Total
iterations
Runtime
with
visualization
150X150 100
64%
700
31 minutes
300X300 900
64%
700
199 minutes
600X600 900
64%
700
329 minutes
150X150 325
52%
700
34 minutes
Table 1: Runtimes for different grid sizes and numbers of cells.
Runtime
without
visualization
2.5 minutes
6 minutes
18 minutes
3.5 minutes
(a)Initial distribution
(b)Developing limb bud
(b)Developing limb bud
(c)Fully patterned limb
Fig. 2. Simulation of skeletal pattern formation in avian limb using COMPUCELL. For full limb, height to width
ratio is 3 to 1. Figure not to scale.