arXiv:q-bio/0401014v1 [q-bio.QM] 9 Jan 2004
Lattice gas cellular automata model for
rippling and aggregation in myxobacteria
Mark S. Albera, Yi Jiangband Maria A. Kiskowskia
aDepartment of Mathematics and the Interdisciplinary Center for the Study of
Biocomplexity, University of Notre Dame, Notre Dame, IN 46556-5670
bTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
A lattice-gas cellular automaton (LGCA) model is used to simulate rippling and
aggregation in myxobacteria. An efficient way of representing cells of different cell
size, shape and orientation is presented that may be easily extended to model later
stages of fruiting body formation. This LGCA model is designed to investigate
whether a refractory period, a minimum response time, a maximum oscillation
period and non-linear dependence of reversals of cells on C-factor are necessary
assumptions for rippling. It is shown that a refractory period of 2-3 minutes, a min-
imum response time of up to 1 minute and no maximum oscillation period best
reproduce rippling in the experiments of Myxoccoccus xanthus. Non-linear depen-
dence of reversals on C-factor is critical at high cell density. Quantitative simulations
demonstrate that the increase in wavelength of ripples when a culture is diluted with
non-signaling cells can be explained entirely by the decreased density of C-signaling
cells. This result further supports the hypothesis that levels of C-signaling quan-
titatively depend on and modulate cell density. Analysis of the interpenetrating
high density waves shows the presence of a phase shift analogous to the phase shift
of interpenetrating solitons. Finally, a model for swarming, aggregation and early
fruiting body formation is presented.
Key words: pattern formation, cellular automata, aggregation, statistical
mechanics, myxobacteria, rippling, collective behavior
PACS: 87.18.Bb, 87.18.Ed, 87.18.Hf, 87.18.La
Email addresses: firstname.lastname@example.org (Mark S. Alber), email@example.com (Yi Jiang),
firstname.lastname@example.org (Maria A. Kiskowski).
Preprint submitted to Elsevier Science9 February 2008
Myxobacteria are one of the prime model systems for studying cell-cell inter-
action and cell organization preceding differentiation. Myxobacteria are so-
cial bacteria which swarm, feed and develop cooperatively . When starved,
myxobacteria self-organize into a three-dimensional fruiting body structure.
Fruiting body formation is a complex multi-step process of alignment, rippling,
streaming and aggregation that culminates in the differentiation of highly elon-
gated, motile cells into round, non-motile spores. A successful model exists for
the fruiting body formation of the eukaryotic slime mold Dictyostelium dis-
coideum [2,3,4]. Understanding the formation of fruiting bodies in myxobacte-
ria, however, would provide a new insight since collective myxobacteria motion
depends not on chemotaxis as in Dictyostelium but on contact-mediated sig-
naling (see  for a review).
During fruiting body formation myxobacteria cells are elongated, with a 10:1
length to width ratio, and move along surfaces by gliding. Gliding occurs in
the direction of a cell’s long axis  and is controlled by two distinct motil-
ity systems in myxobacteria [7,8]. One of the most interesting patterns that
develops during myxobacteria morphogenesis is rippling, which often occurs
spontaneously and transiently during the aggregation phase [9,10,11]. Rip-
pling myxobacteria form equidistant ridges of high cell density which appear
to advance through the population as rhythmically traveling waves [9,10] (Fig-
ure 1). Cell movement in a ripple is approximately one-dimensional since the
majority of cells are aligned and move in parallel lines with or against the di-
rection of wave propagation . Tracking individual bacteria within a ripple
has shown that cells reverse their traveling directions back and forth and that
each travels on the order of one wavelength between reversals . The ripple
waves propagate with no net transport of cells  and wave overlap causes
neither constructive nor destructive interference . Although mechanisms
for gliding are not fully understood, they are believed to account for both
alignment [7,13] and reversals  in myxobacteria.
Rippling is related to a membrane-associated signaling protein called C-factor.
C-factor regulates rippling [10,12,14], cells without the ability to produce C-
factor fail to ripple  and the addition of C-factor (extracted from fruiting
body cultures) causes cell reversal frequencies to increase three-fold . C-
signaling occurs via the direct cell-cell transfer of C-factor when two elongated
cells collide head to head [12,15,16,17,18]. Understanding the mechanisms of
the rippling phase may reveal many clues about the way myxobacteria organize
collective motion since C-factor is also involved in all other stages of fruiting
body formation. For example, cells lacking in C-factor fail to aggregate or
sporulate [19,20,21] while high concentrations of exogeneous C-factor induce
aggregation and sporulation [16,20,22,23].
Fig. 1. A field of immature fruiting bodies, shown as dark patches, with ripples
formed by cells outside of the aggregates. (From Shimkets and Kaiser  with
In this paper we use two lattice gas cellular automata (LGCA) models to simu-
late rippling and aggregation during the fruiting body formation of myxobacte-
ria, to show the potential of cellular automata as models for biological pattern
formation processes, and to evaluate, in particular, the necessity of differ-
ent biological assumptions shown in previous models for pattern formation in
Sager and Kaiser  have proposed that precise reflection explains the lack of
interference between wave-fronts for myxobacteria rippling. Oriented collisions
between cells initiate C-signaling that causes cell reversals. According to this
hypothesis of precise reflection, when two wave-fronts collide, the cells reflect
one another, pair by pair, in a precise way that preserves the wave structure
in mirror image. Figure 2 shows a schematic diagram of this reflection.
Fig. 2. (A) A reflection model for the interaction between individual cells in two
counter-migrating ripple waves. Laterally aligned cells in counter-migrating ripples
(labeled R1 and R2) reverse upon end to end contact. Arrows represent the direc-
tions of cell movement. Relative cell positions are preserved. (B) Morphology of
ripple waves after collision. Thick and thin lines represent rightward and leftward
moving wave fronts, respectively. Arrows show direction of wave movement. (C)
Reflection of the same waves shown in B, with the ripple cell lineages modified to
illustrate the effect of reversal. (From Sager and Kaiser  with permission).
We present a new LGCA approach for modeling cells which is computationally
efficient yet approximates continuum dynamics more closely than assuming
point-like cells. As an example of this new approach, we present a model
for myxobacteria rippling based on the hypothesis of precise reflection and a
model for aggregation based on C-signaling.
This paper is organized as follows. The biological assumptions for precise
reflection and C-signaling that motivate the models are described in the next
section. In section 3 we describe specifics of two LGCA models. In section
4 results of modeling rippling phenomenon are discussed in detail. Section 5
provides description of a model for aggregation centers. The paper ends with
a summary section.
2 Biological Background
In this section we describe the biological observations which motivate our
models for rippling and aggregation.
Rippling and aggregation are both controlled by C-signaling and are char-
acterized by specific high cell density patterns (in particular, moving high
density ridges in rippling and stationary high density mounds in aggregation).
There is a marked relationship between cell density, levels of C-signaling and
behaviors in myxobacteria triggered by C-signaling . C-signaling increases
with density since end-to-end contacts between cells are more likely with in-
creased density [25,26] and high cell densities favor spatial arrangements in
which there are many end-to-end contacts due to the polarity of myxobacterial
cells [25,26]. Cell density and C-signaling levels increase together from rippling
to aggregation and from aggregation to sporulation [25,26]. Further, increased
thresholds of C-factor induce rippling, aggregation and sporulation respec-
tively [22,23,27], suggesting C-signaling levels, as a measure of cell density,
are checkpoints for different stages of development. Kim and Kaiser suggest
that C-factor may act as a developmental timer that triggers sporulation only
when cell density is as high as possible [17,22]. A high density aggregate will
culminate in a fruiting body with a large number of spores ensuring that the
next cycle is started by a population of cells . Sager and Kaiser have also
observed the effect of C-signaling-competent cell density upon ripple wave-
length . They dilute a cell population of C-signaling-competent cells with
cells that are able to respond to C-factor but are not able to transmit it.
They find that with increased concentrations of these csgA-minus cells, ripple
wavelength increases non-linearly.
In addition to cell density patterns, cell geometries are important throughout
the stages of fruiting body formation and distinguish different stages. Dur-
ing the fruiting body formation, cells form aligned patches from a random
distribution . For rippling, a large number of cells must be aligned both
parallel and anti-parallel within the same field. For streaming, cells form long
chains which flow cooperatively in aggregation centers . In Stigmatella spp.,
cells moving in circles or spirals form microscopic transient aggregates. These
aggregates disappear as cells also spiral away tangentially . Macroscopic
aggregates form in areas of high density  and may also disappear as cells
apparently stream along chains from one aggregate to the other . The ma-
ture structures of fruiting bodies are diverse and species-dependent, ranging in
size between 10 and 1000 µm . In Myxococcus xanthus, the basal region of
the fruiting body is a shell of densely packed cells which orbit in two directions,
both clock-wise and counter-clockwise, around an inner region only one-third
as dense [25,26]. In Stigmatella aggregates, cells are organized in concentric
circles or ellipses and cells move in a spiral fashion up the aggregate as the
fruiting body develops [31,33].
Current models for rippling ([34,35,36]) assume precise reflection. Key dif-
ferences among these models include their biological assumptions regarding
the existence of internal biochemical cell cycles. It is still not known if an
internal cell timer is involved in myxobacterial rippling. Several models with
completely different assumptions all qualitatively produce ripple patterns re-
An internal timer is a hypothetical molecular cell clock which regulates the
interval between reversals. The internal timer may specify a delay, or minimum
period between reversals, which would include the refractory period, see below,
and a minimum response time; the minimum period of time required for a
non-refractory cell to become stimulated to turn. Also, the internal timer may
specify a maximum oscillation period, in which case the timer may speed up
or slow down depending upon collisions, but the cell will always turn within
a specified period of time even without collisions. Individual pre-rippling cells
reverse spontaneously every 5-10 minutes with a variance in the period much
smaller than the mean [35,37,38]. This would suggest that there is a component
of the timer specifying a maximum oscillation period. Also, observation of
rippling bacteria reveals that cells oscillate even in ripple troughs where the
density is too low for frequent collisions  further supporting the hypothesis
of a maximum oscillation period.
The refractory period is a period of time immediately following a cell reversal,
during which the cell is insensitive to C-factor. Although there is no evidence
of a refractory period in the C-signaling system, the refractory period is a
general feature of bacterial signaling systems  (for a description of the role
the refractory period plays in Dictyostelium, see ). The addition of .02
units of external C-factor triples the reversal frequency of single cells from .09
reversals per minute to .32 reversals per minute . Cells do not reverse more
frequently at still higher levels of C-factor, however, suggesting the existence
of a minimum oscillation period of 3 minutes in response to C-factor. This
minimum oscillation period would be the sum of the refractory period and the
minimum response, so the duration of the refractory period cannot be guessed
from this fact alone.
To resolve the conflicts of these models for rippling our first LGCA model
is designed to test different assumptions. The results of our model for rip-
pling shows that rippling is stable for a wide range of parameters, C-signaling
plays an important role in modulating cell density during rippling, and non-C-
signaling cells have no effect on the rippling pattern when mixed with wild-type
cells. Further, by comparing model results with experiments, we can conclude
reversals during rippling would not be regulated by a built-in maximum oscil-
We then present a second LGCA model for aggregation based on C-signal
alignment, which reproduces the sequence and geometry of the non-rippling
stages of fruiting body formation in detail, showing that a simple local rule
based on C-signaling can account for many experimental observations.
3 Model and Method
LGCA are relatively simple Cellular Automata models. They employ a reg-
ular, finite lattice and include a finite set of particle states, an interaction
neighborhood and local rules that determine the particles’ movements and
transitions between states . LGCA differ from traditional CA by assuming
particle motion and an exclusion principle. The connectivity of the lattice fixes
the number of allowed non-zero velocities or channels for each particle. For
example, a nearest-neighbor square lattice has four non-zero allowed channels.
The channel specifies the direction and magnitude of movement, which may
include zero velocity (resting). In a simple exclusion rule, only one particle
may have each allowed non-zero velocity at each lattice site. Thus, a set of
Boolean variables describes the occupation of each allowed particle state: oc-
cupied (1) or empty (0). Each lattice site on a square lattice can then contain
from zero to four particles with non-zero velocity.
The transition rule of an LGCA has two steps. An interaction step updates the
state of each particle at each lattice site. Particles may change velocity state,
appear or disappear in any number of ways as long as they do not violate
the exclusion principle. In the transport step, cells move synchronously in the
direction and by the distance specified by their velocity state. Synchronous
transport prevents particle collisions which would violate the exclusion prin-
ciple (other models define a collision resolution algorithm). LGCA models are
specially constructed to allow parallel synchronous movement and updating
of a large number of particles .
3.1 Representation of cells
In classical LGCA, biological cells are dimensionless and represented as a
single occupied node on a lattice (e.g., see  and ). Interaction neigh-
borhoods are typically nearest-neighbor or next-nearest-neighbor on a square
lattice. The exclusion principle makes transport unwieldy when a single cell
occupies more than one node since a cell may only advance if all the channels it
would occupy are available. Similarly, it is difficult to model the overlapping
and stacking of cells. Cells without dimension are untenable for a sophisti-
cated model of myxobacteria fruiting body formation, however. Cells are very
elongated during rippling, streaming and aggregation and form regular, dense
arrays by cell alignment. Also, a realistic model of cell overlap and cell stacking
is needed since interaction occurs only at specific regions of highly elongated
cells and cell density is a critical parameter throughout this morphogenesis.
B¨ orner et al.  have mediated the problem of stacking by introducing a semi-
three-dimensional lattice where a third z-coordinate gives the vertical position
of each cell when it is stacked upon other cells. Stevens  has introduced a
model of rod-shaped cells that occupy many nodes and have variable shape
in her cellular automata model of streaming and aggregation in myxobacte-
ria. Neither of these two models are LGCA since they do not incorporate
synchronous transport along channels. We device a novel way of represent-
ing cells which facilitates variable cell shape, cell stacking and incomplete cell
overlap while preserving the advantages of LGCA; namely, synchronous trans-
port and binary representation of cells within channels (e.g., a ‘0’ indicating
an unoccupied channel and a ‘1’ indicating an occupied channel).
We represent the cells as (1) a single node which corresponds to the position of
the cell’s center (or “center of mass”) 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 (Figure 3). The interaction neighborhoods depend
on the dynamics of the model and need not exactly overlap the cell shape. In
our models for rippling and aggregation, we define the size and shape of the
cell as a 3 × ℓ rectangle, where ℓ is cell length. As ℓ increases, the cell shape
becomes more elongated. A cell length of ℓ = 30 corresponds to the 1 × 10
proportions of rippling Myxococcus xanthus cells . Representing a cell as
an oriented point with an associated cell shape is computationally efficient,
yet approximates continuum dynamics more closely than assuming point-like
cells, since elongated cells may overlap in many ways. We have also solved the
cell stacking problem, since overlapping cell shapes correspond to cells stacked
on top of each other. This cell representation conveniently extends to changing
cell dimensions and the more complex interactions of fruiting body formation.
3.2LGCA model for rippling
We assume precise reflection and investigates the roles of a cell refractory
period, a minimum response period, a maximum oscillation period and non-
linear dependence of reversals on C-factor independently.
3.2.1 Local Rules
(1) Our model employs a square lattice with periodic boundary conditions
imposed at all four edges. Unit velocities are allowed in the positive and
negative x directions. (A resting channel may be easily added to model
a small percentage of resting cells as in .
(2) Cells are initially randomly distributed with density δ, where δ is the
total cell area divided by total lattice area.
(3) Every cell is initially equipped with an internal timer by randomly assign-
ing it a clock value between 1 and a maximum clock value τ. We define
a refractory period R such that 0 ≤ R < τ (see a detailed description of
the internal timer, below). If the internal timer φ of a cell is less than R,
the cell is refractory. Otherwise, the cell is sensitive.
(4) At each time-step, the internal timer of each refractory cell is increased
by 1 while the internal timer of sensitive cells is increased by an amount
proportional to the number of head-on cell-cell collisions n occurring at
(5) When a cell’s internal timer has increased past τ, the cell reverses, the
internal timer resets to 0, and the cell becomes refractory. Reversals occur
as a cell’s center switches from a right- or left-directed channel to a left-
or right-directed channel, respectively.
(6) During the final transport step, all cells move synchronously one node in
the direction of their velocity by updating the positions of their centers.
Separate velocity states at each node ensure that more than one cell never
occupies a single channel.
3.2.2 Internal timer
We model an internal timer with three parameters; R, t and τ. R is the
number of refractory time-steps, t is the minimum number of time-steps until
a reversal and τ is the maximum number of time-steps until a reversal. The
minimum period of time required for a sensitive cell to become stimulated to
turn is the minimum response period t−R. During the refractory period, cells
are insensitive to collisions and the internal timer advances at a uniform rate.
After the refractory period, cells become sensitive and during this phase the
number of head-on cell-cell collisions accelerates the internal timer so that the
interval between reversals shortens. This acceleration is density-dependent, so
that many simultaneous collisions accelerate the internal timer more than only
Our internal timer extends the timer in Igoshin et al. . They used a phase
variable φ to model an oscillating cycle of movement in one direction followed
by a reversal and movement in the opposite direction. During the refractory
period the phase variable advances at a constant rate but during the sensitive
period, the phase variable advance may increase non-linearly with the number
of collisions. Thus, the evolution of our timer determines reversal rather than
a collision as in the model of B¨ orner et al. . The state of our internal timer
is specified by 0 ≤ φ(t) ≤ τ. φ progresses at a fixed rate of one unit per
time-step for R refractory time-steps, and then progresses at a rate ω that
depends non-linearly on the number of collisions n which have occurred at
that timestep to the power p:
ω(x,φ,n,q) = 1 +
?τ − t
t − R
for 0 ≤ φ ≤ R;
for π ≤ φ ≤ (π + R);
This equation is the simplest that produces a reversal period of τ when no
collisions occur, a refractory period of R time-steps in which the phase velocity
is one, and a minimum reversal period of t when a threshold (quorum) number
q of collisions occurs at every sensitive time-step. There is “quorum sensing”
in that the clock velocity is maximal whenever the number of collisions at
a time-step exceeds the quorum value q. A particle will oscillate with the
minimum reversal period only if it reaches a threshold number of collisions
during each non-refractory time-step (for (t − R) time-steps). If the collision
rate is below the threshold, the clock phase velocity is less than maximal.
However, as the number of collisions increases from 0 to q, the phase velocity
increases non-linearly as q to the power p.
While in the model of B¨ orner et al.  there is no minimum response period
for a cell to reverse, and in the model of Igoshin et al.  a minimum response
time is an inherent component of the internal clock, our model incorporates
“on-off switches” for a refractory period, minimum response period, maximum
oscillation period and quorum sensing. Setting the refractory period equal to 0
time-steps in our model is the off-switch for the refractory period, and setting
t = R+1 is the off-switch for the minimum response time. No maximum oscil-
lation period is modeled by choosing a maximum oscillation period τ greater
than the running period of the simulation, so that the automatic reversal of
cells within τ time-steps has no effect on the dynamics of the simulation. There
is no quorum sensing if q is set to 1 so that a single collision during a timestep
has the same effect as many collisions.
If there is no refractory period, cells are always sensitive to collisions. If there is
no minimum response time, cells may reverse immediately after becoming sen-
sitive if there are sufficiently many collisions in one timestep. Finally, if there
is no maximum oscillation period, cells may never reverse without sufficiently
3.2.3Head-on cell-cell collisions
We define an interaction neighborhood of eight nodes for the exchange of C-
factor at the poles of a cell of length l (see Figure 3). The cell width of 3 nodes
is larger than 1 to account for coupling in the y-direction and the interaction
neighborhood must extend at least two nodes along the length of the cell to
compensate for the discretization of the lattice since cells traveling in opposite
directions may pass without their poles exactly overlapping.
A head-on cell-cell collision is defined to occur when the interaction neighbor-
hoods of two anti-parallel cells overlap. A cell may collide with multiple cells
simultaneously since the interaction neighborhood is four nodes at each pole.
Note that the specific shape of the cell is not important for rippling dynamics
since the two areas of C-signaling are the only places where interaction occurs.
Nevertheless, a shape extending over several nodes is necessary to permit the
necessary overlapping and stacking at high density since the exclusion prin-
ciple mandates that each channel has at most one cell center. Thus, the cell
centers of two colliding cells will be separated by one cell length and do not
compete for channels at the same node. Also, for sufficiently long cell lengths,
the probability of more than one cell center located at the same node is low
even when the local cell density may be high.
We are able to simulate a rippling population with arbitrary concentrations of
both wild-type and non-C-signaling cells and quantitatively reproduce their
experimental results in detail, as did Igoshin et al. using their continuum model
. Further, we demonstrate that the change in wavelength may be entirely
explained by the change in density of C-signaling cells.
Fig. 3. The shaded rectangle corresponds to the cell shape of a right or left moving
cell in our model for rippling. This cell is 3×21 nodes for a 1×7 aspect ratio. The
star in this figure corresponds to the cell’s center and the nodes of the interaction
neighborhood where C-factor is exchanged are indicated by black squares at the cell
4 Rippling results and discussion
Our model forms a stable ripple pattern from a homogeneous initial distribu-
tion for a wide range of parameters, with the ripples apparently differing only
in ripple wavelength, ripple density and ripple width (see Figure 4).
Fig. 4. Typical ripple pattern including both a cell clock and refractory period in
the model. (Cell length = 5, δ = 2, R = 10, t = 15, τ = 25.) Figure shows the
density of cells (darker gray indicates higher density) on a 50 × 200 lattice after
1000 time steps, corresponding to approximately 200 minutes in real time.
Absence of a maximum oscillation period is modeled by choosing a maximum
oscillation period τ greater than the running period of the simulation, so that
the automatic reversal of cells within τ time-steps has no effect on the dynam-
ics of the simulation. We find that ripples form with or without a maximum
oscillation period over the full range of densities. When there is a maximum
oscillation period, the maximum oscillation period must be chosen greater
than twice the refractory period for the development of ripples. There is no
upper bound on the maximum oscillation time, which is why the maximum
reversal period is unnecessary. Ripples develop most quickly and cell oscilla-
tions are most regular with an internal timer when the maximum oscillation
period is carefully chosen with respect to the other parameters of the model.
Nevertheless, it appears that experimental results are best reproduced when
there is no maximum oscillation period.
A refractory period is required for rippling for cells of length greater than
2 or 3 nodes, and although there may exist a minimum response time of
more than one time-step, it is an interesting result of our model that the
minimum response period T − R must be small compared to the refractory
period. In particular, rippling occurs whenever the minimum oscillation time t
is greater than ℓ/v time-steps and the refractory period R is at least two-thirds
t. The first condition is required because if the minimum oscillation period t
is less than the period of time it takes a cell to travel one cell length, two
cells or a cluster of cells will stimulate each other to oscillate in place. The
second condition that the refractory period is at least two-thirds the minimum
oscillation period indicates that the minimum response time of a cell can not
be too long compared to the refractory period.
Experiments suggest that the minimum oscillation period of a cell in response
to C-factor is about 3 minutes . According to our result that the mini-
mum response time cannot be more than two-thirds the refractory period, we
can predict the existence of a refractory period in myxobacteria cells, with a
duration of 2-3 minutes.
The wavelength of the ripples depends on both the duration of the refrac-
tory period and the density of signaling cells. Figure 5 shows that the ripple
wavelength increases with increasing refractory period (a) and decreases with
increasing cell density (b). Notice that error bars that show standard devia-
tions of the mean wavelength over five simulations increases with wavelength.
A refractory period of 2-3 minutes yields a ripple wavelength of about 60 mi-
crometers (Figure 5a), which corresponds well to typical experimental ripple
wavelengths . The correspondence between refractory period and wave-
length given in Figure 5 is a only rough estimate, however. We believe the
reasons are that in these simulations the cell density is relatively low, which
decreases the density of C-factor relative to experimental conditions, and cells
are not very elongated, which increases the density of C-factor relative to
Refractory Period in Minutes
Wavelength In Micrometers
0 0.51 1.5
Wavelength in Micrometers
Fig. 5. a) Average wavelength in micrometers versus refractory period in minutes.
Cell length ℓ = 4, δ = 1. The internal timer is adjusted so that the fraction of
clock time spent in the refractory period is constant: t = 3R/2 and τ = 5 ∗ R/2. b)
Average wavelength in micrometers versus density (total cell area over total lattice
area). Cell length = 4 with an internal timer given by R = 8, t = 12, τ = 20.
Note that in Figure 5a, the curve has a wavelength of approximately 20 mi-
crometers when the refractory period is less than 1 minute. Since cells have a
length of 5 micrometers, this is the smallest wavelength that may be resolved
as there is only one cell length between subsequent high density waves. At
very high density, when the refractory period is 0, cells may be stimulated
to reverse every timestep, so that there would be, theoretically, a wavelength
of only 1 node. However, cells will be uniformly distributed in this case and
there will be no well-defined high-density waves. In the simulations described
in Figure 5b, density is increased while refractory and minimum oscillation
periods have a constant value. The minimum possible wavelength in this case
is limited by the minimum oscillation period. In particular, the minimum pos-
sible wavelength is twice the minimum distance traveled by a cell between
reversals, which is twice the distance traveled during the minimum oscillation
period, which is 30 micrometers in this example. Thus, even as density is in-
creased very high, the curve must have a horizontal asymptote at wavelength
= 30 micrometers.
4.1Non-linear response of reversals to C-factor.
Reversals depend on the number of collisions a cell encounters which depends
on the density of C-factor. Thus the number of collisions required for a reversal,
the quorum value q, should be a function of the density of C-signaling nodes.
The density of C-signaling nodes is a function of both cell density and cell
length since longer cells have a reduced C-signaling area to non-C-signaling
area ratio. Thus, we describe optimal quorum values q as a function of C-
signaling node density rather than cell density.
At a low density of C-signaling nodes, ripples form even when both q and p
are 1 so that only 1 collision during the sensitive period is needed to trigger an
reversal. When the density of C-signaling nodes is greater than or equal to 1,
however, the chances of collisions are so high in the initial homogeneous pop-
ulation that cells almost always reverse in the minimum number of timesteps
and, with no differential behavior among cells, a rippling pattern fails to form.
Ripples will form at arbitrarily large densities of cells and C-signaling nodes
if the number of collisions needed to trigger a reversal is increased. When the
number of collisions required for a reversal is greater than 3 (q > 3), rip-
ples develop more quickly if the non-linear response to density p is increased
greater than 1. A value of p = 3 yields optimal rippling for all quorum values
and densities, which is consistent with the results of Igoshin et al.  for their
value of p .
4.2 Ripple phase shift
Counter-propagating ripples appear to pass through each other with no inter-
ference, which lead Sager and Kaiser to propose the hypothesis of precise re-
flection . Indeed, tracking of right-propagating ripples and left-propagating
ripples in Figure 8a, shows that the waves move continuously despite collisions
and subsequent reflection. Inspection of the collision and subsequent reversal
of two cells, however, shows there is a jump in phase equal to exactly one cell
length if they reverse immediately upon colliding (see Figure 6a). This phase
jump occurs because a cell reverses by changing its orientation rather than
by turning: when a right-moving particle collides with a left-moving particle
and reverses, it is exactly one cell length ahead of the left-moving cell that it
replaces. When all of the particles within a ripple are in phase, as is often the
case, this jump is also seen in the ripple waves as two waves interpenetrate.
If the cells continue p more steps before reversing (for example, if their clocks
were almost near τ after the collision), then there would be a phase jump of
ℓ−2p. If 2p > ℓ, there will be a phase delay (see Figure 6b). In their continu-
ous model, Igoshin et al. (, Figure 3b) also showed when ripples collide a
small jump in phase reminiscent of a soliton jump.
Fig. 6. Space-time plot of a wave inter-penetration. Time increases as the vertical
axis descends. Right-directed particles are shown in dark gray, left-directed particles
are shown in light gray. a) Phase jump of one cell length (9 units) as two cells collide
and immediately reverse. b) Phase delay as two cells collide and travel 8 time-steps
4.3 Effect of dilution with non-signaling cells
Sager and Kaiser  diluted C-signaling (wild-type) cells with non-signaling
(csgA minus) cells that were able to respond to C-factor but not produce it
themselves. When a collision occurs between a signaling and a non-signaling
cell, the non-signaling cell perceives C-factor (and the collision), whereas the
C-signaling cell does not receive C-factor and behaves as though it has not
collided. The ripple wavelength increases with increasing dilution by non-C-
signaling cells. Simulations of this experiment with and without an internal
timer with a maximum oscillation period give very different results. Figure 7a
shows that the dependence of wavelength on the fraction of wild type cells
resembles the experimental curve only when there is no maximum oscilla-
tion period assumed in the model (compare with Figure 7.G in .) Thus
our model predicts that rippling cells do not ripple with a maximum oscilla-
tion period. Notice that the range of wavelengths when there is no assumed
maximum oscillation period is in good quantitative agreement with that of
experiment (compare Figure 7a, solid line with Figure 7.G in .
Igoshin et al.  have previously reproduced the experimental relationship
between wavelength and dilution with non-signaling cells (see  Supple-
mental materials, Figure 8) by adjusting their original internal timer. As the
density of C-signal decreases, the phase velocity slows linearly and the max-
imum oscillation period of the internal timer increases continuously. Thus,
the maximum oscillation period varies in their model. We assume a constant
maximum oscillation period, which is either present or absent (longer than
the simulation running time). Note that if the maximum oscillation period
increases sufficiently with decreased density of C-factor so that a cell is always
stimulated to turn before the internal timer would regulate a turn, then the
addition of an internal timer is superfluous. In this case, the two models are
0 0.2 0.40.6 0.81
Fraction of Wild−Type Cells
Wavelength In Micrometers
0 0.51 1.5
Wavelength In Micrometers
Fig. 7. a) Wavelength in micrometers versus the fraction of wild-type cells with
(dotted line) and without (solid line) a maximum oscillation period. b) Wavelength
in micrometers versus wild-type density with no csgA-minus cells (dotted line) and
when the the density of csgA-minus cells is increased so that the total cell density
remains 1.6 (solid line). Density is total cell area over total lattice area and there
is no maximum oscillation period. For a) and b), cell length = 4, R = 8, t = 12,
τ = 20 (maximum oscillation period) or τ = 2000 (no maximum oscillation period).
Our simulations show ripple wavelength increases with increased dilution by
non-signaling cells. Since wavelength also increases with decreasing density of
signaling cells (Figure 5b), we ask if the mutant cells have any effect on the
rippling pattern. Figure 7b shows the wavelength dependence on the density
of signaling cells when only signaling cells are present (dotted line) and for a
mixed population of signaling cells of the same density with non-signaling cells
added so that the total cell density is always 1.6 (solid line). Apparently, the
decrease in C-factor explains the increase in wavelength. The non-signaling
mutants do not affect the pattern at all.
As density increases, wavelength decreases and the larger number of cells are
distributed over a greater number of ripples. This result is further evidence
of the the role C-signaling plays as a density-sensing and density-modulating
mechanism. To test this further, we ran a simulation for initial conditions
in which a high density stripe stretches vertically down a lattice. As ripples
formed and propagated, the cells were quickly distributed more evenly over
the lattice (Figure 8b). The redistribution of cells occurs much faster than if
cells moved randomly at each time-step (compare Figure 8, b and c). Thus,
although there is no net transport of cells larger than one wavelength when
cells are evenly distributed , there is net migration of cells away from high
Fig. 8. Cell density over a subsection of the third row of a 5 × 500 lattice over the
first 500 time-steps for different initial conditions. Time increases as the vertical
axis descends. a) Cells are initially randomly distributed with density 3. Cell length
= 5 nodes, R = 10 nodes, t = 15 nodes and τ = 2000. b) Same as in a), but with a
central stripe of density 15 and width 50 initially added vertically down the lattice.
c) Same as in b), but cells are assigned random orientations at every time-step.
In our simulations, high-density waves of cells form from a homogeneous dis-
tribution of cells for a wide range of parameters and initial cell density. At
low cell density when there is no assumed maximum oscillation period, the
wavelength between ripples is large. The explanation for this is that a larger
region of the lattice must be “swept” to collect an aggregate of cells with suf-
ficiently high density to reverse another aggregate. In the extreme case where
density is nearly zero, a single cell will keep traveling without ever encounter-
ing sufficient collisions, and the wavelength is infinite. Thus, the mechanism
of rippling may be viewed as a mechanistic “sweeping” of an arbitrarily large
area, in which cells modulate the area that they span between reversals so as
to efficiently collect ripples of a minimum density.
At very high density, by the same argument, wavelength is small. The number
of ripples per area increases so that the number of cells per ripple does not
increase linearly with the increase in density, but less. Nevertheless, ripples
formed from high density initial conditions do result in ripples which are wider
and more dense than ripples formed from low density initial conditions. This
may not be viewed as a limitation in design, but as the foundation of another
possible role for rippling in myxobacteria: the even distribution of cells over an
arbitrarily large area. When there are both high and low cell density regions
(as in the simulation of Figure 8b), many high-density waves will form in
the region of highest density, and fewer, lower density waves will form in the
regions of lower density. At the interface between these regions, a high density
wave encounters a lower density wave. The high density wave will reverse all
the cells of the low density wave such that all the cells from the low density
wave return toward the low density region. The cells of the low density wave,
on the other hand, will not be able to reverse all the cells of the higher density
wave. Rather, the lower density wave will only be able to reverse a proportional
number of cells in the high density wave. The surplus cells in the high density
wave will continue without reversing into the low density region. Thus, the
rippling mechanism in a region of variable cell density creates a “pulse” of
surplus cells within the high density region which is efficiently directed into
lower density regions.
In summary, rippling is an efficient mechanism for both forming evenly spaced
accumulations of high cell density, and evenly spaced accumulations of nearly
equal cell density. In experiments, cells do not reflect by exactly 180 degrees.
However, since most cells move roughly parallel to each other, models based on
reflection are reasonable approximations. Modeling the experimental range of
cell orientations would require a more sophisticated CA since LGCA require a
regular lattice which does not permit many angles. In the aggregation section
below we describe a model on a triangular lattice which could be adjusted to
incorporate 120 degree reversals. Although a cell is ready to turn when the
internal timer φ is greater than τ, it may not be able to turn if the opposite
channel is already occupied. This is another limitation of our LGCA model.
We handle this situation by continuing to transport the cell in its direction
of orientation at each time-step until the opposite channel is available. The
effect of this delay is negligible, even at high densities within a ripple, when
cells are so long that the probability of two cell centers occupying the same
node is very small.
5A preliminary model for aggregation
Rippling is an intermediate, transient stage of fruiting body formation, which
is not necessary for aggregation formation . Figure 1 shows a field of ag-
gregation centers surrounded by ripples. In this section we present a different
LGCA model based on C-signaling alignment. This LGCA model reproduces
the sequence and geometry of the non-rippling stages of fruiting body forma-
tion in detail, demonstrating how C-signaling-based alignment can account for
these patterns with very few additional assumptions.
The non-rippling stages of fruiting body formation include alignment, stream-
ing and aggregation. During alignment, cells form aligned patches from a ran-
dom distribution. While streaming, cells form long chains which move coop-
eratively into aggregation center . Aggregation is the phase in which cells
form rounded collections that may either recede or mature into fruiting bod-
ies. We model aggregation including only a simple local rule for C-signal-based
alignment. The aggregates formed in our model are not species-specific and
do not include local rules for rippling.
We use a hexagonal lattice since cell motion during aggregation is not one-
dimensional as in rippling. In this specialized LGCA model, identical rod-
shaped cells are all modeled as 3 × ℓ rectangles with C-signal interaction
neighborhoods as depicted in Figure 9. Cells move exactly one node per time-
step in the direction of their orientation and there may only be one cell center
per channel per node. In contrast to rippling, we find that the cell aspect ratio
is an important parameter for streaming and aggregation, the simulations
presented here all have a 1 × 7 aspect ratio for cells.
or 180 degrees and c) a cell oriented 120 or 300 degrees. All cells are 3 × 21 for a
1×7 aspect ratio. Each cell’s “center of mass” is indicated by a star and the nodes
of the interaction neighborhood where C-factor is exchanged are indicated by the
larger black disks at the cell poles.
The cell shapes of a) a cell oriented 60 or 240 degrees, b) a cell oriented 0
Myxobacteria align when they move. We choose an alignment based on C-
signaling. We use a Monte Carlo process, in which cells turn 60 degrees clock-
wise, 60 degrees counter-clockwise or persist in their original direction with
probability favoring directions that maximize overlap of C-signaling nodes. In-
teraction only occurs when the C-signaling nodes at the head of a cell overlap
with the C-signaling nodes at the tail of another cell, and interaction occurs
regardless of cell orientation. Head and tail C-signaling neighborhoods are
shown in Figure 9.
We model C-signal alignment on a 256× 256 lattice, in which our initial con-
ditions are a random distribution of cells at high density. Within a few time-
steps, cells form aligned patches (see Figure 10a). This reproduces the initial
alignment stage of myxobacteria cells during fruiting body formation in which
cells form an aligned patchwork . Since there are only 6 directions permit-
ted on the lattice, the aligned patchwork appears as a very regular triangular
network. Cells are aligned both parallel and anti-parallel within each patch
since the overlap of C-signaling nodes is maximized when cells are aligned
regardless of their orientation. This geometry is significant because cells have
naturally formed the aligned bi-directional arrays necessary for rippling. The
local rules for rippling have been suppressed in this preliminary model, how-
ever, to evaluate the patterns formed by C-signaling-based alignment alone.
The homogeneous triangular network is not stable over time. As cells move,
they turn and flow from one patch to another. Cells moving in a low density
area are likely to turn into a higher density patch, so the network of cells
condense into thick aggregates (Figure 10b). In experiments, a myxobacteria
stream will often merge into an adjacent myxobacteria stream [41,42]. Figure
10b shows that the cells move into the aggregates along streams directed into
the aggregate, reproducing the streaming stage in which myxobacteria cells
form long aggregates that move cooperatively .
The aggregates continue to condense while arranging and dividing into many
small, circular orbits about 1.5 cell lengths (about 10-20 µm) in diameter
(Figure 10c). The aggregates often form in clusters of two or three closed orbits
(Figure 10b), corresponding to fused aggregates (sporangioles). In Stigmatella
erecta, several fruiting bodies may form in groups and fuse . A magnified
picture of the cell centers of a typical aggregate show that cells are arranged
in dense, concentric layers tangent to a relatively low-density inner region
(Figure 10d). Thus, they are geometrically equivalent to the basal region of
aggregates in Myxococcus xanthus.
Cells in our simulation simultaneously move both clock-wise and counter-
clockwise around the aggregate, as they do in the fruiting bodies of Myxococcus
xanthus . The microscopic circular or elliptic orbits of Stigmatella spp.
often disappear as cells spiral away from the aggregate . Similarly most
orbits in our simulation also eventually disappear as cells spiral away from
the aggregate. Nevertheless, orbits typically survive for several hundred time-
steps, which is about 5 to 10 complete revolutions for each cell.
During myxobacteria aggregate formation, several aggregation centers will
form and, inexplicably, one aggregation center will grow as an adjacent ag-
gregation center disappears . Our simulations offer a closer look at this
process: a stream may form that connects two adjacent aggregation centers
and, stochastically, cells stream from one aggregation center to another until
the largest aggregation center absorbs the smaller one (Figure 10e).
Figure 10f shows several stable aggregates which have developed at 200 time-
steps. Notice that the stable hollow aggregate has a much thicker annulus
of cells than the non-stable orbits of Figure 10c, suggesting that only large
hollow aggregates are stable. In our simulation at a threshold density within
the aggregate, the hollow region of an aggregate will fill with cells such that
cells are are arranged in six dense overlapping layers (Figure 10f). The third
aggregate shown is a chain of cells which span the lattice and thus form an
orbit due to the periodic boundary conditions of the lattice.
It has been proposed in  that circular motility at aggregation sites, trail
following and local accumulation of slime account for fruiting body formation
in Stigmatella spp. We hypothesize that once C-signaling has drawn cells into
an aggregate, cell and slime cohesivity cause myxobacteria cells to round up
into a mound while constant cell velocity pushes cells toward the surface of
the mound, so that cells form a dense hemisphere of cells spiraling around a
hollow center. In our model, a closed, circulating orbit of cells is the only stable
configuration of a stationary aggregate since cells are constantly moving. At
low and intermediate density, these orbits are hollow and cells are arranged
tangentially within an annulus. At a threshold density, however, every channel
of every node within the aggregate becomes occupied and there is no hollow
center. We hypothesize that in a more advanced three-dimensional model, the
addition of a local rule accounting cell and slime cohesivity will cause cells to
round while maintaining the hollow center.
This model is preliminary because only C-signal based alignment is mod-
eled and the aggregates formed are not species-specific. In this simulation, a
256x256 lattice size was chosen, which corresponds to an 80x80 µm region and
about 10,000 cells for an averaged cell density of 10, much smaller compared
to normal fruiting bodies that may be up to 1000 µm in diameter. Aggregation
in this model is described in more detail in .
This model can be adjusted to model rippling and aggregation concurrently
by incorporating the local rules for rippling described in Section 3.2. Tracking
of rippling cells in experiment (see Figure 6 in  and Figure 6 in )
suggest that reversals are about 150 degree changes in orientation rather than
exactly 180 degrees, as we have assumed in the model for rippling of this
paper. On a triangular lattice, a reversal of 120 degrees would be an equally
good approximation of the 150 degree rotation. C-signaling based alignment
Fig. 10. Cell density development by C-signal alignment on a 256 × 256 lattice.
Initial cell density is 10. Cell density a) at 25 time-steps (64×64 lattice sub-section)
and b) at 100 time-steps (128 × 128 lattice sub-section). Cell centers c) at 200
time-steps (150 × 150 lattice sub-section), d) of a 24 × 24 lattice sub-section with
arrows indicating direction (450 time-steps), e) at 450 time-steps (100 ×100 lattice
subsection) and f) at 2000 time-steps (100 × 100 lattice sub-section).
combined with local rules for rippling would ensure that the majority of cells
still remain parallel. Nevertheless, a regular shift in orientation by 120 or 180
degrees may have an interesting effect on the final distribution of cells and,
subsequently, fruiting bodies.
Fruiting bodies among different myxobacteria species are very diverse (see Fig-
ure 3 in ). For example, while Myxococcus fruiting bodies are a relatively
simple, single mound of cells, other species form clusters of mounds called
sporangioles that are raised on a stalk. The interaction of adventurous verses
social motility may account for these different morphologies . During stalk
formation of Stigmatella spp., cells are arranged perpendicularly to the mound
as the fruiting body is lifted [31,33]. Also, fruiting body stalks may be com-
posed of a larger, second cell-type . Thus, once the stages of fruiting body
formation have been modeled in general, it would be interesting to determine
which parameters need to be varied to model the fruiting bodies of different
In this paper, we present a new LGCA approach for modeling cells which is
computationally efficient yet approximates continuum dynamics more closely
than assuming point-like cells. As an example of this new approach, we present
a model for myxobacteria rippling based on the hypothesis of precise reflection.
The results of our model show that rippling is stable for a wide range of
parameters, C-signaling plays an important role in modulating cell density
during rippling, and non-C-signaling cells have no effect on the rippling pattern
when mixed with wild-type cells. Further, by comparing model results with
that of experiment, we can conclude reversals during rippling would not be
regulated by a built-in maximum oscillation period. We also present a second
LGCA model based on C-signal alignment which reproduces the sequence
and geometry of the non-rippling stages of fruiting body formation in detail,
showing that a simple local rule based on C-signaling can account for many
We would like to thank Dale Kaiser, Frithjof Lutscher and Stan Mar´ ee for very
helpful discussions. MSA is partially supported by grant NSF IBN-0083653.
YJ is supported by DOE under contract W-7405-ENG-36. MAK is supported
by the Center for Applied Mathematics and the Interdisciplinary Center for
the Study of Biocomplexity, University of Notre Dame, and by DOE under
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