-1Career: Investigating Mechanisms of Cellular Communication and Self-Organization in
Morphogenesis
Project Description
I. Introduction: Mechanisms of Cell-Cell Communication and SelfOrganization in Morphogenesis
A.
Morphogenesis
The unique characteristics of biological problems have admitted a diverse range of new
mathematical and computational techniques. Biological problems introduce a host of new
challenges including the need to span several spatial and temporal scales simultaneously, the
need to integrate the interaction of large numbers of distinct components, and the need to sort
and detangle complex feedback interactions. Biological systems are distinct from other physical
and chemical systems because they have function (purpose) and use energy from their
environment to exist far from equilibrium.
An important and extremely challenging research area in mathematical biology is
morphogenesis, a general biological phenomenon describing the change and molding of living
tissues during development, regeneration, wound healing and disease. In this proposal, I will
describe projects for mathematical research in three current and actively pursued areas of
morphogenesis: spontaneous self-organization in multicellular colonies, vertebrate skeletal
development and prostate ductal morphogenesis.
Although morphogenesis is quite general and very complex, it appears to be guided by a number
of common principles:

The cell as the most natural unit. Morphogenesis tends to place biological cells as the
central spatial scale, with a smaller spatial scale for molecular interactions and a larger
scale for describing cellular communication within tissues (Alarçon et al., 2004;
Alexander, 2005; Jiang et al., 2005). Placing cells as the central unit is natural for several
reasons. First, cells are autonomous agents that have purpose and make decisions. Cells
make decisions about which molecules to transcribe and organize to make decisions about
how the tissue should grow, change or shrink and disappear. Second, cell variation is an
important aspect of biological phenomena. Distinct tissue layers evolve from different cell
types. Cell differentiation is a key step in morphogenesis.

Self-Organization Morphogenesis is often self-organized rather than directed. For
example, patterns often evolve from a homogeneous distribution of fundamentally
equivalent cells. In myxobacteria fruiting body formation and chondrogenesis, fruiting
bodies and condensations self-organize from sheets of initially homogeneous cells
(Dworkin, 1996, Kiskowski et al., 2004; Kiskowski et al., 2005). This suggests that a
local, mechanistic understanding of cell interaction is needed to understand pattern
development. Local mechanisms of communication interact with long-range (Wolpertian)
interactions such as diffusible molecules where one cell or a set of cells operates as
pacemaker or director (Ben-Jacob and Levine, 1998; Shapiro, 1988).
-2
The Importance of Cell shape and Orientation Cells may lengthen, shorten, round up or
elongate. Cell shape greatly affects the motion of cells, their relative arrangement and the
interaction of specialized parts of the cell body. Changes in cell shape often precede
changes in cell genetic expression (differentiation). Subgroups of cells often align within
tissues and cells with different orientations within a tissue may differentiate into different
cells. Fibroblasts arrange and align during wound healing and myxobacteria align within
streams during fruiting body formation. Development of the drosophila retina is initiated
by a morphogenetic wave of aligned cells as the morphogenetic furrow develops across the
eye disc epithelium (Heberlein and Moses, 1995). Angiogenesis, the development of blood
vasculature and central to cancer metastasis, involves the growth and migration of
polarized cells (Bauer et al., 2007). In prostates, epithelial cells align around ducts.
B.
Computational Methods and Approaches
Cell interactions are always local and mechanistic from the point of view of an individual cell.
What follows here is a survey of the modeling approaches that I will use in the proposed
research. They emphasize individual and mechanistic/local interactions without compromising
computational efficiency. A template model for specialized cell interactions is a novel approach
developed specifically for morphogenetic applications. Also, I describe a novel modeling
framework for multiple tissue systems; more details can be found in Section IV.
i.
Cell-based, Discrete Lattice Models
Discrete individual-based models are naturally applied to a wide range of biological problems
since they reflect the intrinsic individuality of particles (cells) and are straight-forward to build
from a microscopic level understanding of particle-particle interactions. A simulation time-based
model is appropriate since a morphological pattern evolves over time and is the product of
spatio-temporal feedback between interacting cells. Discrete approaches are flexible and easily
modified, affording the possibility of including relationships and behaviors which are difficult to
formulate as continuum equations. Since components are added and subtracted depending upon
the specific characteristics of each biological application, individual models are not straightforward to classify and hybrid models that include a combination of continuous and discrete
model elements are often optimal. The most successful models are multi-scale, encompassing
interactions and dynamics at macroscopic, mesoscopic and microscopic levels (tissues, cells and
intra-cellular processes, respectively) (Alarçon et al., 2004; Alexander, 2005; Jiang et al., 2005).
Biological lattice gas cellular automata (LGCA) are a useful tool for modeling cell-cell
interactions (Ermentrout and Edelstein-Keshet, 1993, Börner et al., 2002; Lutcher and Stevens,
2002; Alber et al., 2004a,b,c; Kiskowski et al., 2004). LGCA have several characteristics that
make them exceptionally computationally efficient. They employ a regular, finite lattice (two or
three dimensional, with square or triangular connectivity), allow a finite set of particle states, and
have a two-step transition rule that allows synchronous updating: an interaction step that updates
the state of each cell at each lattice site, including their velocity, and a transport step in which
cells move synchronously in the direction and by the distance specified by their velocity state.
Canonically, particles in LGCA are represented as a single occupied node on a lattice but the
model may be extended to allow cells with variable shape extending over a set of lattice nodes
(e.g., Kiskowski et al., 2004).
-3-
ii.
Novel Extended-Graph LGCA Framework For Multiple Tissue Systems
Lattice approaches are especially appropriate for modeling tissues of confluent cells in which
cells have a regular tiled arrangement (e.g., epithelial tissue). If the geometric limitations are
reasonable, they afford computational efficiencies and analytic analyses that agent-based
continuum approaches do not. However, it is relatively awkward to model motion or growth with
long range effects (e.g., convection or expansion). For example, the directed hard-body motion
of a large lattice structure may be modeled by adding nodes to the ‘head’ and subtracting nodes
at the ‘tail’ (Stevens, 2000), but this is inconvenient if there are specialized cell structures. When
domains expand, cell positions adjust over several time steps to accommodate this expansion –
pressure is not communicated immediately through the material. To resolve these issues in a
system of multiple tissues with embedded objects, I propose modeling loose tissue connections
(e.g., mesentery) and loosely embedded structures (e.g., fibroblasts within an ECM) as extended
lattice networks. Interactions between structures are defined via graph-like connections that
extend the set of nearest neighbors. This framework is similar to a continuum agent-based model
but retains the graph structure of LGCA (see section VI for details).
iii.
Mechanisms of Cell Communication
During morphogenesis, cell interactions are determined by the method of cell signaling (short
range cell contacts or long range diffusive signals), cell shape and orientations, alterations in cell
behavior as cells change and differentiate, and transient or large scale cell motions. Depending
on the biological application, diffusive signals may be modeled using a discrete random walk
approach when the time scale of diffusion is relatively slow and tracking individual trajectories is
important (most appropriate for a small number of diffusing molecules (<106)) while a
continuum approach is most appropriate for a larger number (order 1023). I have used discrete
approaches for morphogen diffusion in a model for limb chondrogenesis (Kiskowski et al., 2004)
and a continuum approach in a model for prostate tumorigenesis (Kiskowski et al., 2011).
Cell Interactions Via Specialized Cell-Cell Contacts
Classically, interaction neighborhoods of an LGCA are immediately adjacent cells; 4 nearest or 8
next-nearest neighbor nodes on a square lattice. Cells are considered dimensionless or with an
unspecified shape and a size of the order of one node. However, during morphogenesis cells are
often very elongated and interact via specialized structures. I have developed a novel way of
representing cells which facilitates variable cell shape and interaction contacts while preserving
the advantages of classical lattice gases; namely, synchronous transport and binary representation
of cells within channels. In this template-based model cells are represented as (1) a single node
which corresponds to the position of the cell’s center in the xy plane, (2) the choice of occupied
channel at the cell’s position designating the cell’s orientation and (3) a local neighborhood
defining the physical size and shape of the cell with associated interaction neighborhoods. The
interaction neighborhoods depend on the dynamics of the model and need not exactly overlap the
cell shape. This template would fit naturally in the extended graph framework described above.
Cell template model: The shaded rectangle
corresponds to the elongated cell shape of a right or
left moving myxobacteria cell (Alber et al., 2004a).
This cell is 321 nodes for a 17 aspect ratio. The star corresponds to the cell’s center and the nodes of
the interaction neighborhood where C-factor is exchanged are indicated by black circles.
-4-
II. Application: Projects in Three Morphogenetic Biological Systems
A.
Project 1: Spontaneous Self-Organization In Multi-Cellular Colonies
Myxobacteria Fruiting Body Formation
Myxobacteria fruiting body formation is an example of a biological phenomenon in which an
understanding of the local mechanism of cell interaction afforded a deeper understanding of the
patterns that the cells were forming, as well as the ability to accurately simulate the stages of
fruiting body formation in detail. My PhD work on a lattice gas cellular automata (LGCA) model
for myxobacteria fruiting body pattern formation on a 2D lattice (Alber et al., 2004a, b) has been
extended to three dimensions (Sozinova et al., 2005).
Myxobacteria Modeling Project Aims
Investigating Mechanisms of Regulation: In our LGCA simulations (Alber et al., 2004a),
fruiting bodies formed with a variety of morphologies that matched the variety of fruiting body
structures in the wild. However, our fruiting bodies were several orders of magnitude smaller in
diameter than wild-type fruiting bodies and were not species-specific: we generated fruiting
bodies geometries represented by a number of different species. I would like to generate and
evaluate biologically-relevant hypotheses for mechanisms for species variation in order to
address a broader question regarding the regulation of patterning. How do myxobacteria cells
measure and communicate when the patterning is complete? The differences between species
have a genetic basis, however, to what extent is the regulation of pattern also species-specific?
Initial hypothesis: Persistence Length I would like to test the hypothesis that the diameter of
fruiting bodies depends on the rate at which cells are able to turn. This is a biologically
motivated hypothesis because there is a characteristic length the cell travels before the cell turns
called the cell persistence length. Species variation in the cell persistence length may account for
much of the species variation n patterning, in which case I anticipate that there is a common
mechanism of pattern regulation; difference in patterning results from a difference in the
interaction rules that the cells follow and there is no need to hypothesize differences in the way
the patterning is regulated. Alternatively, varying the persistence length in simulations may not
account for species variation and differences in patterning may result from other differencesin
the interaction of cell, in the final regulation of patterning, or both.
Modeling signaling pathways to investigate hypotheses: It is suggested that myxobacteria
have two motility systems (the A and S motility systems). Much is already known about the
genetic regulation of these motility systems, but not their connections with local myxobacteria
behaviors. By comparing the patterning formed by myxobacteria in which the components of the
signaling networks are disabled with simulation for different models of interaction, we can
investigate these connections and evaluate the model.
-5B.
Project 2: Vertebrate Bone Development
In my PhD work, I developed an LGCA model for the patterning of chondrogenic domains based
upon a biologically motivated reaction-diffusion (Turing) process. I will expand this model to
identify theoretical principles for why bone patterning is symmetric across the left-right axis of
the body in contrast to other Turing patterns, such as animal coat patternings.
Background
A characteristic of reaction-diffusion patterns is that the pattern changes with the size of the
domain. Chemical pre-pattern models assume that a reaction-diffusion mechanism establishes a
pattern during early development, and then cell differentiation occurs subsequently so that
morphogenesis takes place after the pattern is established. In fish, however, pattern formation
occurs as the fish grows. Kondo and Asai modeled the striped pigmentation pattern of angelfish
by modeling reaction diffusion on growing domains. As the domain grows, the width of the
stripes and the distance between stripes do not change, but the number of stripes increases
(Kondo and Asai, 1995; Painter et al., 1999). Reaction-diffusion on growing domains has also
been considered in models of branch morphology in algae (Lacalli, 1981), tooth formation
(Kulesa et al., 1996), solid tumor development (Chaplain et al., 2001) and bivalve patterning
(Madzvamuse et al., 2002). Given the extensive limb growth during vertebrate embyrogenesis, it
would be natural to study the effects of domain growth on chondrogenesis. My model for
chondrogenesis was published in Developmental Biology, a prominent journal in the field of
development. The citation activity (35 citations) is indicative of the timeliness of applying a
reaction diffusion model to bone morphogenesis. A thorough, quantitative model for the effect of
limb growth on skeletal pattern may elucidate the potential mechanisms of variation of interspecies skeletal patterns and intra-species anomalies.
Project Aims
Undergraduate Project: Developing A Multi-Stage Model for Chondrogenesis
Current work with an undergraduate student on this project has been funded by an internally
competitive undergraduate research grant. In a multi-stage model, the chondrogenic pattern is
established, and only over time will the condensation pattern begin hardening. Throughout this
process, the domain will be growing and the reaction diffusion pattern will be changing.
Adding modular layers of genetic complexity
The Turing-based model for chondrogenesis results in skeletal patterning with the correct
number of ‘bones’ along the dorsal-ventral axis of the limb. I will systematically add known
modulators of patterning to determine the range of spatial predictions for each independently,
assuming initially for simplicity that their effects would additive. For example, it is known that
there is a proximal-distal gradient along the limb that results in the difference in fingers from the
pinky to the thumb. In a summer undergraduate project, a student is studying the effect of
varying the diffusion rate of morphogens across this axis, which is one potential way that
modulation could be occurring across this axis. My research questions are which features can be
attributed to particular modulations and whether these modulations are resistant to variation.
Also, I am interested in genetic mechanisms for the synchronism of these modulations in the pair
of limbs.
-6C.
Project 3: Prostate Organogenesis: A Dynamic Model For Duct Formation
I have recently developed a hybrid model
for prostate tumorigenesis that models the
dedifferentiation of epithelial cells from
normal to proliferative and invasive in
response to morphogenetic interactions
with pre-cancerous stromal cells
(Kiskowski et al., 2011). This cancer
progression model uses experimental
images to define initial cell positions; cell
positions cannot change over time.
In a static model of ducts, an image of mouse prostate
(left) was used to assign corresponding static positions for
simulation cells (right). Epithelial cells are black and
stromal cells blue and cyan. From Kiskowski et al., 2011.
For my third project in this proposal, I would like to develop a sophisticated dynamic model for
duct formation in which prostate ducts form and develop in response to biologically indicated selforganized morphogenic and external hormonal influences. The model is based on normal prostate
development (prostate organogenesis) and could be extended to model prostate tumorigenesis.
Background
In normal prostate morphogenesis, the prostate grows during gestation and the first weeks of
neonatal life and then the prostate deceases in size (‘regresses’) ((Zondek and Zondek, 1975), up to
50% in seals (Amoroso et al., 1965)) until puberty. This plasticity is the result of hormonal
influences that change throughout these stages. Thus, prostate development is an interesting
example of a system in which patterning depends upon long range signaling influences that are to
some extent reversible. While the time scale of development is slow (not spontaneous as in the
described cases of multicellular organization) reversibility indicates that the system is in dynamic
equilibrium with transitions that depend upon biological parameters. The system is manipulated
through experiments in which hormones are added or subtracted exogenously. For example,
removal of native androgen results in reduction in the size and number of the prostate ducts, with a
resulting increase in the fraction of stromal cells over a time scale of several days (Huttunen et al.,
1981) and these changes are reversible when androgen is restored (Rittmaster et al., 1995).
Project Aims
Preliminary Model for Growth Resulting in Initial Formation of Duct Geometry
In a preliminary model specifying a small number of cell types that result in an approximate duct
geometry, nodes may have one of four states corresponding to an epithelial cell, a stromal cell,
basement membrane or an empty node. Simulations on a square lattice begin with a single
epithelium cell centered within a population of stroma cells. Prostate duct growth and formation is
modeled with the following rules:
1. Epithelial cells divide at a rate dependent upon growth activating morphogens. Daughter cells
occupy a random adjacent node if that node is not already occupied by an epithelial cell.
2. Stromal-epithelial interactions result in the production of basement membrane. A stromal cell
adjacent to an epithelial cell will change state and become basement membrane.
3. The basement membrane supports epithelial cell tissue by secreting a supportive morphogenic
signal. Outside the influence of this signal, epithelial cells will die disappear (shed and eliminated
through the duct).
-7-
These rules result in the formation and growth of an annular epithelial cell layer surrounded by a
layer of basement membrane and many layers of stroma. Eventually, an empty core forms in the
interior of the duct.
Simulation results of preliminary model
for duct formation after A) 10, B) 20 and
C) 50 time-steps. Epithelial cells,
basement membrane, stroma and empty
nodes are blue, white, gray and black
respectively.
Addition of Inhibitory Signals to Model Regulation of Duct Size and Inter-Ductal Distance
The preliminary model for duct formation described above does not have rules for growth
regulation and the duct grows indefinitely. In a more sophisticated model, the self-regulation of
duct formation will be modeled by incorporating production of signals that regulate growth.
Example of Self-regulatory Mechanism via Stromal-Epithelial Interactions:
A simple model is provided here only as an example. In an initial growth phase, epithelial cells
initiate epithelial proliferation by activating a reactive stroma through diffusive growth factors
(“FGF”). Once epithelial cells have produced a closed bi-layer of epithelial cells, they signal to
the stroma to suppress proliferation. For a mechanism of stage transition, we assume that
epithelial cells secrete insoluble factors (e.g., decorin) that sequester stromal growth factors (e.g.,
HGF) in the basement membrane. Once epithelial cells are completely encapsulated by basement
membrane, growth factors will no longer reach them since the basement membrane is
impermeable to them. These simple assumptions result in a slowing and completion of duct
growth. Distributions of these factors in a “duct” when growth has slowed but not stopped are
shown in the figure below. Note that a band of “decorin” in the basement membrane significantly
blocks the diffusion of HGF resulting in an area empty of HGF in the duct interior.
Model results for duct formation with growth
regulation after 100 time-steps. Growth factor
concentrations are shown in grayscale.
Epithelial cells are blue, basement membrane is
white, normal stroma is gray and reactive
stroma is pink.
Investigating and Modeling of Signaling Pathways for Growth and Inhibitory Signals:
Biological interactions of signaling pathways during morphogenesis are very complex and not
fully understood. Broad relationships of signaling interactions between cell types and identification
of key factors can be understood through the biological literature, however, any model will be
incomplete and in some aspect incorrect. Therefore, it is important to form minimal assumptions,
to be very clear about these assumptions, and describe assumptions in general terms if progress is
to be made based on the modeling. For example, suppose it is observed that a signaling factor X in
one group of cells promotes the upregulation of Y in another group of cells in a particular
functional way. The signaling factor will be described only as a factor that upregulates Y in that
functional way. Later, if it is found that X is not independently responsible for the described
upregulation, the set of responsible factors can be used to replace X in the model. Likewise, if a set
of interactions result in an observed pattern of regulation, the model can be simplified by replacing
this entire set of interactions with their phenomenologically observed net effect.
-8Model Evaluation and Validation
Since the regulatory effects of hormones are complex, for different and increasingly complex
layers of signaling interactions, the model will be evaluated on the extent to which the model
yields duct that are morphologically similar to experimental ducts in size, shape and inter-ductal
separation. I have experience in the analysis and comparison of domain patterning: in my work
with chondrogenic patterning, I applied a number of tests including direct measurement of
separation distances and periodicity (Miura et al., 2000) and the use of Ripley’s K statistic, a 2nd
order statistics that is often used in biological application to measure aggregation of and the
domain size of irregular points (Kiskowski and Kenworthy, 2009). Importantly, the model should
continue to accurately model the growth and regression of prostate ducts under different hormonal
conditions. There is abundant histological data for prostate duct morphology for normal prostates
(e.g., textbooks for development), tumorigenic prostates, prostates under hormonal therapy and
prostates under the influence of natural or mutant hormonal changes (Goland, 1975 and many
others).
Prostate of a term
Model validation (that is, validation that the
neonate with
model assumptions have captured the most
metaplasia
relevant in vivo processes) will involve matching
(epithelial
finer histological details that are not built into the
thickenings)
model (for example, reproducing the observation
indicated on
that epithelial cells are most dense at the posterior
posterior walls.
walls (Shapiro et al., 1996) and making
From (Zondek and
histological predictions that are consistent with a
Zondek, 1975)
wide range of experimental observation under
without
different hormonal conditions.
permission.
Significance

General application to prostate applications: To the best of my knowledge, this is the first
dynamic developmental model of prostate duct formation. Since it is based on duct formation
in normal contexts, it is a general model that may be adapted to genesis, wound healing or
cancer of the prostate.

Application to modeling the dynamic rearrangement of cells during cancer
proliferation and invasion: Combining the cancer progression model that has already been
developed (Kiskowski et al., in review) with a developmental model of prostate duct
formation will allow us to simulate the dynamic reorganization of cells in response to
proliferation and migration during tumorigenesis. To model cell proliferation and invasion
that are important aspects of cancer, it is important to model the re-organization of cells
dynamically. For example, as cells proliferate, they occupy more space and other prostate
cell layers must respond by expansion and growth. Invasion occurs when transformed
epithelial cells break past basal epithelial cells and encounter the basement membrane. It is
difficult to predict when this will occur and the effect of cell arrangements without a dynamic
stochastic model for PIN and the responsive proliferation of basal epithelial cells.

An alternative mechanism for determining spatial scales of self-organized
morphogenetic structures: Myxobacteria form tori while in motion, so that it is
hypothesized that velocity, turning and density constraints predict the final size of tori. In
contrast, prostate cells are stationary within ducts and ducts develop at relatively slow spatial
scales. It is hypothesized that the final duct size depends upon the interaction and balance of
diffusing morphogenic products over the spatial scale of the duct.
-9-
III. Developing Connections With Other Fields Of Mathematics
A long term goal is to develop connections with other fields of mathematics. It is important for
applied mathematicians to build points of contact with less applied mathematical fields to
encourage a broader set mathematicians to work on biological problems
A.
A Novel Modeling Framework: Connections with Graph Theory
While lattice-based methods are flexible and computationally efficient, it is challenging to model
hard-body-like deformations of domains. I propose modeling independently moving structures as
connected lattices. If each lattice is interpreted as a graph with regular connections, this results in
a network of graphs. Two tissue layers would be modeled as a pair of lattices (graphs G1 and G2)
with occasional edges between vertices of G1 and G2 indicating tissue contact and
communication at these points (e.g., gap junctions). Outside these connections, motion and
growth may be mutually independent. For example, prostate ducts should expand freely as
epithelial cells divide since the prostate also expands. However, a cylinder of epithelial cells
embedded in a 3D lattice would not be able to expand without competing for space with the
surrounding stroma. Epithelial and stromal cells modeled on separate lattices would allow
independent growth, a number of dynamic inter-lattice connections would coordinate growth.
Example: Specialized Neighbor Interactions A lattice is a graph in
which each node is a vertex connected by an edge to nearest neighbors.
Thus, graph edges reflect the connectivity of the lattice and capture
nearest-neighbor interactions. To model more generalized neighbor
interactions, an operation  can be defined on a graph G (:GG)
such that the connectivity of G will model the neighbor interactions of
G. For example, to model an operation modeling the von Neumann
neighborhood of length 2, consider the following definitions: for V0 a
set of vertices and E0 a set of edges, let E(V0) be the set of edges of the
vertices in V0 and let V(E0) be the set of vertices connected to edges in
E0; finally let N(V0) be the set of vertices connected to vertices in V0 so
that N(V0)=V(E(V0)). To define a graph G that models von Neumann
neighbors, let G be a graph with the same vertices as G. For every
vertex v in G, the corresponding vertex v in G inherits the edges of v
in G (inherits E(v)) and also the set of edges of every neighbor in G
(inherits E(N(v))=E(V(E(v)))). A von Neuman neighborhood of general
length can be found by repeating these nested functions.
von Neumann
neighborhoods for a
central node v0
{x} = nbhd of length 1
= V(E(v0))/v0
{y}=nbhd of length 2
= V(E(V(E(v0)))/{x}
- 10 B.
Spatio-angular Self-Organized Structures: Topological Connections
Morphogenesis is the development of and deformation of quasi-planar layers of tissue forming
flat sheets of finite size (a healing wound, layers of a melanoma) or flat sheets closed in on
themselves to make cylinders (blood vessels, prostate ducts, epidermis). Cell polarity is
important factor in spatial cell arrangements. To investigate the role of cell polarity, I would
propose project that bridges models of myxobacteria development and topological relationships.
Principles of Pattern Formation in Myxobacteria Fruiting Body Formation:
Alignment in 1D Results in Lines and Tori
During fruiting body formation, Myxobacteria cells are highly elongated (Reichenbach, 1993)
and interact by exchanging a membrane-associated signaling protein (C-factor) located solely at
the head of the cell (Sager and Kaiser, 1994). Thus, Myxobacteria interact when they are aligned
and arranged end to end. In simulations, modeling elongated cells with a preference for this
specific interaction resulted in cells that formed elongated chains (streams). The streams were
stable if they turned in upon themselves and formed a torus (fruiting body). Simulation structures
included tori and clusters of tori resembling were The model established a simple principle for
the pattern formation: polarized cells preferred to move in thin lines, that would become
stationary (and stable) once the line turned in upon itself to form a torus (or groups of tori).
Developing an application with connections to topology: In simulations of fruiting body
formation, I will simulate inert ‘solid-body’ objects on the lattice. Fruiting bodies will be forced
to stream and form fruiting bodies around these structures. I would like to investigate the
minimum requirements for forming a ‘knot’ of myxobacteria cells. Stable knots may form
naturally around the structures, but it is likely that I will need to restrict the interaction of
crossing streams. This can be accomplished using parallel lattices with limited interaction or by
encouraging myxobacteria cells to interact only with a familiar cohort of cells.
Significance
This project would provide interesting projects for undergraduates orienting them to discrete
methods and topology simultaneously. Once the conditions for a topologically stable knot are
found, a second project will be to build two dynamic knots with opposite orientations and
discover the minimum simulation conditions for the knots to displace interact and self-annihilate.
Once the details are worked out for forming stable knots, the application could be used to
perform calculations (for example, to calculate the determinant of a knot or to determine if two
knots are equivalent). These ideas have already been established and applied to DNA folding
(Brown and Cozzarelli, 1979; Sumners, 1995, Stasiak et al., 1996).
Schematic illustration of fruiting bodies (actual, from
simulations) formed around inert ‘ball bearings’
(drawn). Given constraints on the interactions of
different myxobacteria loops, the inert objects would
‘lock’ fruiting bodies into configurations with
particular winding numbers. Myxobacteria will flow
dynamically around the pivots and will unknot in any
ways that the configuration allows.
- 11 -
A long-term goal: Developing a theory for the potential energy surface
of stable Pattern Configurations
C.
I would like to apply ideas from statistical mechanics and condensed matter physics to develop a
paradigm for quantitatively describing and understanding the ‘energy landscape’ of the attractor
region of stable pattern configurations. In such a theory, configurations may be similar if they
have similar potential energy surface even if they result from different mechanisms. Especially, I
would like to investigate whether patterns that result from a time-based evolution (e.g., Turing
patterns) could re-expressed as an optimization of a potential function. I would begin this
analysis using small perturbations to quantify a quasi-‘restoring force’ and mapping these forces
over phase space. (In physical applications, such forces are usually spring-like.)
Area-density phase diagram of simulation
aggregates (Alber et al, 2004b). The black line
shows the path of an aggregate (a) through phase
space as cells were slowly added over 1000 time
steps and (b) the motion of a random collection
of cells through phase space as they became more
ordered and formed a fruiting body.
- 12 -
IV. Developing Connections With Other Fields Of Mathematics Developing
Methods for Estimation of Biological Parameters
I am interested in using modeling to investigate and develop low-cost computational approaches
such as image analysis to estimate biological parameters. While increasingly sophisticated
methods are being developed to pinpoint such parameters, most experimental biologists have
limited access to such methods. Low-cost approximate methods for preliminary research will
free resources for more detailed experiments on other parameters.
A.
Defining a Range of Signaling Influence from Regions of Morphogenetic Change
The diffusion rate of a signaling factor is a measure of the mean squared displacement of a factor
over time and the diffusion length of a factor is the average distance traveled by signaling factor
in the average lifetime of the factor. In (Kiskowski et al, 2011) we used image analysis and
estimates of the rate of differentiation to make estimates for the diffusion length of putative
signaling factor. Image analysis was used to identify the positions of cells secreting the putative
signaling factors and the extent of transformation for different fractions of cells secreting the
factors determined bounds for the diffusion length. I would like to generalize this approach for
determining the diffusion length of a signaling factor for the case of N cells with known positions
affecting a group of cells, also with M known positions. when the spatial positions of secreting
cells are known. In some cases the range of signaling influence, which takes into account the
super positioning of factor produced by a population of cells and signal dampening or
amplification by neighboring cells in the tissue, will be a more relevant parameter than the
diffusion rate that would be measured in a cell-free context.
B.
Defining a Range of Signaling Influence from Regions of Morphogenetic Change
In collaboration with developmental biologists, I have worked on quantifying and classifying cell
movements during Zebrafish convergence and extension (Sepich et al, 2005; Yin et al, 2008). In
this research, we analyze time lapse images of moving cells. A challenge in modeling these
motions is determining the components of motion that are due to cell adhesion, active motion or
passive conduction motion. If a method for measuring the component of motion due to cell
adhesion from the observation of cell movements in images were found, this would represent a
low-cost method of determining adhesivity constants. I would like to explore the possibility of a
method based on analyzing the movements of a dividing cell, and that of neighboring cells. A
dividing cell is not actively moving, thus its motion should be comprised only of passive forces.
In contrast, the dividing cell represents a relatively stationary obstacle that neighbors must push
or navigate around. Much information can be extracted from looking at the relative velocities of
dividing cells, non-dividing cells in the vicinity, and other non-dividing cells.
C.
Exploration of a Novel Internet Concept: User-Organized Forum on Developing
Computational Tools for Biology
I will develop an internet site for collecting a description of computational tools for immediate
use and real-time development. A secondary purpose of the site is to test a novel internet concept
that if successful would increase the pace of research by enhancing communication between
researchers in a way that is historically unprecedented. With this smaller project that I propose,
researchers will be able to see the pros and cons of the Reddit system for research dialogue, and
if it provides value there, application to other, larger projects will be a natural extension.
- 13 Forum Description: The searchable internet blog will be a set of pages including an initial
welcome page and a growing number of user-submitted pages. The welcome page will describe
the purpose of the site, provide instructions for page submission, and provide a list of the most
active and popular page topics. Individual pages submitted by authors will include a main
expository section about a computation tool and a section for reader comments. Expository
sections should be thorough and well-researched with extensive links to the relevant literature.
Reader comments will include questions, share experience with computation tools and provide
additional links. Pages and comments will be scored and organized by other users in a Redditlike system. Registration as a user will require an educational e-mail address (.edu) and stated
forum norms will include collegiality, high scientific quality and topic relevance.
Dissemination of Computational Tools and Opportunity for Real-Time Discussion:
Computational tools are frequently developed that are not included in research publications.
While useful, they may be too minor or not novel enough to merit page space. If they are
included in a publication, they are hidden within the supplemental materials of a journal that may
be narrow relative to its potential use. Here, a researcher can search under a term of interest and
find algorithm descriptions and links to software packages. As such, the site would be a more
focused method of ‘Googling’. However, the innovative value of the project lies in the
subsequent comment section. The comment section of a post would be an appropriate place to
ask an elementary question, make suggestions or share experience. While primary posts will be
expected to be formal and thorough; informal comments will be encouraged. Real-time dialogue
is something that is largely missing from our research efforts, but the benefits of informal
discussion are enormous with respect to sharing solutions, fostering ideas and bridging different
backgrounds and levels of expertise.
Challenges and the potential of Reddit: A user-developed system has the potential for chaos
when a large fraction of comments are trivial, not collegial or inaccurate. With the Reddit
system, users will down-vote comments that do not conform to site norms. Also, users must have
an educational e-mail address and must post comments under their own names. While peerreview and lack of anonymity will encourage the scientific quality of comments, it is difficult to
predict how hesitant research scientists will be to post comments in a public forum. Hopefully
the technical topic and the productivity of uncountably many gregarious graduate students are
compensating elements.
Project’s Long-term Potential: I envision a large-scale forum where any journal article may
have a page (submitted by a coauthor) and an informal discussion follows in threaded
discussions. Research is currently less efficient due to classes of relevant information that are not
published. For example, duplicated and null results are rarely published and even descriptions of
failed attempts to duplicate results could help researchers identify patterns and common
stumbling blocks. Personally, I would welcome an opportunity to ask a question about a term in
a mathematical equation. Such a project, if it could work, would address Feynman’s observation,
“We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all
the tracks, to not worry about the blind alleys or describe how you had the wrong idea at first, and so on. So there isn't any
place to publish, in a dignified manner, what you actually did in order to get to do the work.”
- 14 -
V. Broader Impacts
A.
K-12 Interactions
Special Interest ‘Bridge’ Course in Computational Mathematics
I am impressed by the impact a special interest high school course had on my collaborator Dr.
Stuart Newman (Chuong, 2009) that influenced him to study quantitative science and eventually
developmental biology. This award would provide the opportunity to teach a computational math
modeling course for high school students to earn college credit. The course would expose them
to computational methods in a computer lab and the way mathematics and physics are used to
solve problems, including problems in biology. I will focus on teaching with accessible
applications such as Excel and Octave and Mathematica that is installed in the computer lab.
This award would provide tuition for 20 high school students culled from the region, including
the public School of Math and Science. This program would fit in a greater network of support
from our University including the STARS program (Students Training for Academic
Reinforcement for the Sciences). For assessment, I will request student surveys immediately
following, 2 years afte and 5 years after the course completes to record student satisfaction,
college enrollment and graduation and choice of major.
Elementary Education: Teaching Students, Teachers and Hosting a Regional Conference
The future requires a larger number of students that are well trained in math and science, and
also a widespread understanding and adoption of a scientific worldview. Each fall since arriving
in Mobile, Alabama, I have participated in GEMS (Girls in Engineering, Math and Science) by
presenting mathematical workshops on topics that are exploratory and hypothesis-driven. For
example, this year I am presenting a workshop on a simple version of Conway’s “Game of Life”
(a computer science application) in which two colonies of different colors interact. After
explaining simulations and the types of rules that are permitted, I will ask students to generate
hypotheses regarding interactions that will help one colony defeat another, or allow them to coexist. Then students will test their hypotheses by simulating different rules in small groups by
manipulating colored squares on square grids. In subsequent years, I will continue to introduce
topics using manipulable, explorative activities that emphasize real-life examples in
mathematical biology. For example, students can investigate immiscibility using a manipulative
for Pott’s model, and compare this with what they observe when oil paint is mixed with water on
a plate. Students can investigate principles in topology by forming knots in pieces of flexible
plastic string. Geometry and measurement is one of four critical areas in elementary mathematics
(Greenberg and Walsh, 2008), and mathematicians and elementary school teachers have been
working on innovations to improve learning of geometry at the K-12 level (e.g., Pacyga, 1994).
This past spring, I have put a lot of effort into redesigning a mathematics course for elementary
teachers to include more peer-peer teaching in the course and increase reflection on the
importance of elementary science and mathematics education with writing assignments.
Something that I would like to develop in my education teachers is a more problem-solving
approach to education; for them to feel comfortable forming questions regarding the
effectiveness of their teaching and have the confidence to effectively self-evaluate and research
ways of improving their teaching. I would like to continue the development of this course in
these directions and, generally, expand my interaction with elementary education.
- 15 -
Liping Ma (1999) found that Chinese teachers had more confidence teaching elementary
mathematics, despite fewer years of education, due in part to ongoing peer education. The
proposal will secure funds to develop a summer workshop for elementary teachers in which
teachers can share teaching experiences with each other and more experienced teachers. The
conference will draw teachers from the regional area to encourage the development of a local
peer network. For training, I will attend the 2011 NCTM Regional Conference and Exposition.
Interdisciplinary Collaborations; Undergraduate and Graduate
Research
B.
Fostering Interdisciplinary Collaborations:
Eventually, I would like to establish a computational modeling center at the University of South
Alabama since we are already equipped with a medical school and the Mitchell Cancer Institute.
In order to encourage a culture of computational modeling at my institution, I am interested in
maintaining collaborations with colleagues in other departments even if they are not working
directly on morphogenesis. Currently, I am working with Dr. Alice Ortmann at the Dauphin Sea
Lab on modeling planktonic trophic levels, Dr. John McCreadie in the biology department on the
commensalism and parasitism of microbes in the black fly and Dr. Mihail Alexiyev on
mitochondrial DNA mutagenesis.
Interdisciplinary Undergraduate and Graduate Projects:
An application of myxobacteria streaming to model topological knots will lend itself to
accessible and fun undergraduate projects bridging discrete particle systems and topology. I am
currently working with an undergraduate student on the described project in limb development.
In general, mathematical biology provides a rich selection of problems for students. A higher
level math modeling course is taught every spring to undergraduates and graduate students in
mathematics, science and engineering. Each semester I tailor the course to the student’s interests
and encourage them to develop novel computational models in their discipline. In a final project,
they must review existing models for certain topic and then extend or reframe that model. The
work outlined in this proposal will generate interesting and accessible projects for students. I
would like to hire undergraduates each summer to work on research projects.
Graduate Training
The proposal will dedicate funds to support mathematics and biology graduate students for
training in cell-based modeling methods after they have taken the modeling course. I enjoy
training graduate students in biology in discrete computational methods as I believe they provide
a great benefit to the student for the amount of effort. Discrete computational methods are
intuitive for biologists and lend themselves to hypothesis testing without requiring a great deal of
mathematical background. While at Vanderbilt, I trained a graduate student that resulted in a
publication on Zebrafish gastrulation (Yin et al., 2008), another problem in morphogenesis. This
summer I have begun training a graduate student of Dr. Alice Ortmann’s at the Dauphin Sea Lab
in modeling methods for a project measuring the viral lysis and grazing of microbes.