Lattice gas cellular automation model for rippling and aggregation in myxobacteria
ABSTRACT A lattice gas cellular automation (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 nonlinear dependence of reversals of cells on Cfactor are necessary assumptions for rippling. It is shown that a refractory period of 2–3 min, a minimum response time of up to 1 min and no maximum oscillation period best reproduce rippling in the experiments of Myxococcus xanthus. Nonlinear dependence of reversals on Cfactor is critical at high cell density. Quantitative simulations demonstrate that the increase in wavelength of ripples when a culture is diluted with nonsignaling cells can be explained entirely by the decreased density of Csignaling cells. This result further supports the hypothesis that levels of Csignaling quantitatively 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.

Article: Multicell simulations of development and disease using the CompuCell3D simulation environment.
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ABSTRACT: Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissuelevel simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easytouse software tools that allow building and running in silico experiments without requiring indepth knowledge of programming. This tutorial introduces GlazierGranerHogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.Methods in molecular biology (Clifton, N.J.) 02/2009; 500:361428. · 1.29 Impact Factor  SourceAvailable from: Julio M BelmonteMaciej H Swat, Gilberto L Thomas, Julio M Belmonte, Abbas Shirinifard, Dimitrij Hmeljak, James A Glazier[Show abstract] [Hide abstract]
ABSTRACT: The study of how cells interact to produce tissue development, homeostasis, or diseases was, until recently, almost purely experimental. Now, multicell computer simulation methods, ranging from relatively simple cellular automata to complex immersedboundary and finiteelement mechanistic models, allow in silico study of multicell phenomena at the tissue scale based on biologically observed cell behaviors and interactions such as movement, adhesion, growth, death, mitosis, secretion of chemicals, chemotaxis, etc. This tutorial introduces the latticebased GlazierGranerHogeweg (GGH) Monte Carlo multicell modeling and the opensource GGHbased CompuCell3D simulation environment that allows rapid and intuitive modeling and simulation of cellular and multicellular behaviors in the context of tissue formation and subsequent dynamics. We also present a walkthrough of four biological models and their associated simulations that demonstrate the capabilities of the GGH and CompuCell3D.Methods in cell biology 01/2012; 110:32566. · 1.44 Impact Factor  SourceAvailable from: ArXiv[Show abstract] [Hide abstract]
ABSTRACT: Periodic reversals in the direction of motion in systems of selfpropelled rodshaped bacteria enable them to effectively resolve traffic jams formed during swarming and maximize the swarming rate of the colony. In this paper, a connection is established between a microscopic onedimensional cellbased stochastic model of reversing nonoverlapping bacteria and a macroscopic nonlinear diffusion equation describing the dynamics of cellular density. BoltzmannMatano analysis is used to determine the nonlinear diffusion equation corresponding to the specific reversal frequency. Stochastic dynamics averaged over an ensemble is shown to be in very good agreement with the numerical solutions of this nonlinear diffusion equation. Critical density p(0) is obtained such that nonlinear diffusion is dominated by the collisions between cells for the densities p>p(0). An analytical approximation of the pairwise collision time and semianalytical fit for the total jam time per reversal period are also obtained. It is shown that cell populations with high reversal frequencies are able to spread out effectively at high densities. If the cells rarely reverse, then they are able to spread out at lower densities but are less efficient at spreading out at higher densities.Physical Review E 02/2012; 85(2 Pt 1):021903. · 2.31 Impact Factor
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arXiv:qbio/0401014v1 [qbio.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 465565670
bTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
Abstract
A latticegas 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 nonlinear dependence of reversals of cells on Cfactor are necessary
assumptions for rippling. It is shown that a refractory period of 23 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. Nonlinear depen
dence of reversals on Cfactor is critical at high cell density. Quantitative simulations
demonstrate that the increase in wavelength of ripples when a culture is diluted with
nonsignaling cells can be explained entirely by the decreased density of Csignaling
cells. This result further supports the hypothesis that levels of Csignaling 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: malber@nd.edu (Mark S. Alber), jiang@lanl.gov (Yi Jiang),
mkiskows@nd.edu (Maria A. Kiskowski).
Preprint submitted to Elsevier Science 9 February 2008
Page 2
1 Introduction
Myxobacteria are one of the prime model systems for studying cellcell inter
action and cell organization preceding differentiation. Myxobacteria are so
cial bacteria which swarm, feed and develop cooperatively [1]. When starved,
myxobacteria selforganize into a threedimensional fruiting body structure.
Fruiting body formation is a complex multistep process of alignment, rippling,
streaming and aggregation that culminates in the differentiation of highly elon
gated, motile cells into round, nonmotile 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 contactmediated sig
naling (see [5] 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 [6] 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 onedimensional since the
majority of cells are aligned and move in parallel lines with or against the di
rection of wave propagation [12]. 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 [12]. The ripple
waves propagate with no net transport of cells [12] and wave overlap causes
neither constructive nor destructive interference [12]. Although mechanisms
for gliding are not fully understood, they are believed to account for both
alignment [7,13] and reversals [7] in myxobacteria.
Rippling is related to a membraneassociated signaling protein called Cfactor.
Cfactor regulates rippling [10,12,14], cells without the ability to produce C
factor fail to ripple [10] and the addition of Cfactor (extracted from fruiting
body cultures) causes cell reversal frequencies to increase threefold [12]. C
signaling occurs via the direct cellcell transfer of Cfactor 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 Cfactor is also involved in all other stages of fruiting
body formation. For example, cells lacking in Cfactor fail to aggregate or
sporulate [19,20,21] while high concentrations of exogeneous Cfactor induce
aggregation and sporulation [16,20,22,23].
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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 [10] with
permission.)
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
myxobacteria.
Sager and Kaiser [12] have proposed that precise reflection explains the lack of
interference between wavefronts for myxobacteria rippling. Oriented collisions
between cells initiate Csignaling that causes cell reversals. According to this
hypothesis of precise reflection, when two wavefronts 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
countermigrating ripple waves. Laterally aligned cells in countermigrating 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 [12] with permission).
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We present a new LGCA approach for modeling cells which is computationally
efficient yet approximates continuum dynamics more closely than assuming
pointlike 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 Csignaling.
This paper is organized as follows. The biological assumptions for precise
reflection and Csignaling 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.
2Biological Background
In this section we describe the biological observations which motivate our
models for rippling and aggregation.
Rippling and aggregation are both controlled by Csignaling 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 Csignaling and
behaviors in myxobacteria triggered by Csignaling [24]. Csignaling increases
with density since endtoend contacts between cells are more likely with in
creased density [25,26] and high cell densities favor spatial arrangements in
which there are many endtoend contacts due to the polarity of myxobacterial
cells [25,26]. Cell density and Csignaling levels increase together from rippling
to aggregation and from aggregation to sporulation [25,26]. Further, increased
thresholds of Cfactor induce rippling, aggregation and sporulation respec
tively [22,23,27], suggesting Csignaling levels, as a measure of cell density,
are checkpoints for different stages of development. Kim and Kaiser suggest
that Cfactor 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 [28]. Sager and Kaiser have also
observed the effect of Csignalingcompetent cell density upon ripple wave
length [12]. They dilute a cell population of Csignalingcompetent cells with
cells that are able to respond to Cfactor but are not able to transmit it.
They find that with increased concentrations of these csgAminus cells, ripple
wavelength increases nonlinearly.
In addition to cell density patterns, cell geometries are important throughout
the stages of fruiting body formation and distinguish different stages. Dur
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ing the fruiting body formation, cells form aligned patches from a random
distribution [29]. For rippling, a large number of cells must be aligned both
parallel and antiparallel within the same field. For streaming, cells form long
chains which flow cooperatively in aggregation centers [30]. In Stigmatella spp.,
cells moving in circles or spirals form microscopic transient aggregates. These
aggregates disappear as cells also spiral away tangentially [31]. Macroscopic
aggregates form in areas of high density [31] and may also disappear as cells
apparently stream along chains from one aggregate to the other [32]. The ma
ture structures of fruiting bodies are diverse and speciesdependent, ranging in
size between 10 and 1000 µm [28]. In Myxococcus xanthus, the basal region of
the fruiting body is a shell of densely packed cells which orbit in two directions,
both clockwise and counterclockwise, around an inner region only onethird
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
sembling experiment.
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
nonrefractory 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 prerippling cells
reverse spontaneously every 510 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 [12] 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 Cfactor. Although there is no evidence
of a refractory period in the Csignaling system, the refractory period is a
general feature of bacterial signaling systems [35] (for a description of the role
the refractory period plays in Dictyostelium, see [39]). The addition of .02
units of external Cfactor triples the reversal frequency of single cells from .09
reversals per minute to .32 reversals per minute [12]. Cells do not reverse more
5
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frequently at still higher levels of Cfactor, however, suggesting the existence
of a minimum oscillation period of 3 minutes in response to Cfactor. 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, Csignaling
plays an important role in modulating cell density during rippling, and nonC
signaling cells have no effect on the rippling pattern when mixed with wildtype
cells. Further, by comparing model results with experiments, we can conclude
reversals during rippling would not be regulated by a builtin maximum oscil
lation period.
We then present a second LGCA model for aggregation based on Csignal
alignment, which reproduces the sequence and geometry of the nonrippling
stages of fruiting body formation in detail, showing that a simple local rule
based on Csignaling 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 [40]. LGCA differ from traditional CA by assuming
particle motion and an exclusion principle. The connectivity of the lattice fixes
the number of allowed nonzero velocities or channels for each particle. For
example, a nearestneighbor square lattice has four nonzero 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 nonzero 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 nonzero 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
6
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specially constructed to allow parallel synchronous movement and updating
of a large number of particles [40].
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 [34] and [36]). Interaction neigh
borhoods are typically nearestneighbor or nextnearestneighbor 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. [34] have mediated the problem of stacking by introducing a semi
threedimensional lattice where a third zcoordinate gives the vertical position
of each cell when it is stacked upon other cells. Stevens [41] has introduced a
model of rodshaped 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 [17]. Representing a cell as
an oriented point with an associated cell shape is computationally efficient,
yet approximates continuum dynamics more closely than assuming pointlike
cells, since elongated cells may overlap in many ways. We have also solved the
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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.
3.2 LGCA 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 Cfactor independently.
3.2.1Local 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 [34].
(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 timestep, 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 headon cellcell collisions n occurring at
that timestep.
(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 leftdirected channel to a left
or rightdirected 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 timesteps, t is the minimum number of timesteps until
a reversal and τ is the maximum number of timesteps until a reversal. The
minimum period of time required for a sensitive cell to become stimulated to
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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 headon cellcell collisions accelerates the internal timer so that the
interval between reversals shortens. This acceleration is densitydependent, so
that many simultaneous collisions accelerate the internal timer more than only
one collision.
Our internal timer extends the timer in Igoshin et al. [35]. 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 nonlinearly 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. [34]. The state of our internal timer
is specified by 0 ≤ φ(t) ≤ τ. φ progresses at a fixed rate of one unit per
timestep for R refractory timesteps, and then progresses at a rate ω that
depends nonlinearly 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
?
∗
?[min(n,q)]p
qp
?
∗ F(φ), (1)
where,
F(φ) =
0,
0,
1,
for 0 ≤ φ ≤ R;
for π ≤ φ ≤ (π + R);
otherwise.
(2)
This equation is the simplest that produces a reversal period of τ when no
collisions occur, a refractory period of R timesteps 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 timestep. There is “quorum sensing”
in that the clock velocity is maximal whenever the number of collisions at
a timestep 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 nonrefractory timestep (for (t − R) timesteps). 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 nonlinearly as q to the power p.
While in the model of B¨ orner et al. [34] there is no minimum response period
for a cell to reverse, and in the model of Igoshin et al. [35] a minimum response
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time is an inherent component of the internal clock, our model incorporates
“onoff switches” for a refractory period, minimum response period, maximum
oscillation period and quorum sensing. Setting the refractory period equal to 0
timesteps in our model is the offswitch for the refractory period, and setting
t = R+1 is the offswitch 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 timesteps 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
many collisions.
3.2.3 Headon cellcell 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 ydirection 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 headon cellcell collision is defined to occur when the interaction neighbor
hoods of two antiparallel 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 Csignaling 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 wildtype and nonCsignaling cells and quantitatively reproduce their
experimental results in detail, as did Igoshin et al. using their continuum model
[35]. Further, we demonstrate that the change in wavelength may be entirely
explained by the change in density of Csignaling cells.
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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 Cfactor is exchanged are indicated by black squares at the cell
poles.
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 timesteps 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
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more than one timestep, 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 timesteps and the refractory period R is at least twothirds
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 twothirds 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 Cfactor is about 3 minutes [12]. According to our result that the mini
mum response time cannot be more than twothirds the refractory period, we
can predict the existence of a refractory period in myxobacteria cells, with a
duration of 23 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 23 minutes yields a ripple wavelength of about 60 mi
crometers (Figure 5a), which corresponds well to typical experimental ripple
wavelengths [12]. 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 Cfactor relative to experimental conditions, and cells
are not very elongated, which increases the density of Cfactor relative to
experimental conditions.
02468
0
20
40
60
80
100
120
140
160
Refractory Period in Minutes
Wavelength In Micrometers
(a)
00.51 1.5
0
50
100
150
200
250
Density
Wavelength in Micrometers
(b)
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.
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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 welldefined highdensity 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.
4.1Nonlinear response of reversals to Cfactor.
Reversals depend on the number of collisions a cell encounters which depends
on the density of Cfactor. Thus the number of collisions required for a reversal,
the quorum value q, should be a function of the density of Csignaling nodes.
The density of Csignaling nodes is a function of both cell density and cell
length since longer cells have a reduced Csignaling area to nonCsignaling
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 Csignaling 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 Csignaling 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 Csignaling 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 nonlinear 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. [35] for their
value of p [35].
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4.2 Ripple phase shift
Counterpropagating ripples appear to pass through each other with no inter
ference, which lead Sager and Kaiser to propose the hypothesis of precise re
flection [12]. Indeed, tracking of rightpropagating ripples and leftpropagating
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 rightmoving particle collides with a leftmoving particle
and reverses, it is exactly one cell length ahead of the leftmoving 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. ([35], Figure 3b) also showed when ripples collide a
small jump in phase reminiscent of a soliton jump.
(a) (b)
Fig. 6. Spacetime plot of a wave interpenetration. Time increases as the vertical
axis descends. Rightdirected particles are shown in dark gray, leftdirected 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 timesteps
before reversing.
4.3Effect of dilution with nonsignaling cells
Sager and Kaiser [12] diluted Csignaling (wildtype) cells with nonsignaling
(csgA minus) cells that were able to respond to Cfactor but not produce it
themselves. When a collision occurs between a signaling and a nonsignaling
cell, the nonsignaling cell perceives Cfactor (and the collision), whereas the
Csignaling cell does not receive Cfactor and behaves as though it has not
collided. The ripple wavelength increases with increasing dilution by nonC
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
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resembles the experimental curve only when there is no maximum oscilla
tion period assumed in the model (compare with Figure 7.G in [12].) 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 [12].
Igoshin et al. [35] have previously reproduced the experimental relationship
between wavelength and dilution with nonsignaling cells (see [35] Supple
mental materials, Figure 8) by adjusting their original internal timer. As the
density of Csignal 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 Cfactor 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
similar.
0 0.20.4 0.60.81
20
40
60
80
100
120
140
160
Fraction of Wild−Type Cells
Wavelength In Micrometers
(a)
00.51 1.5
0
50
100
150
200
250
Density
Wavelength In Micrometers
(b)
Fig. 7. a) Wavelength in micrometers versus the fraction of wildtype cells with
(dotted line) and without (solid line) a maximum oscillation period. b) Wavelength
in micrometers versus wildtype density with no csgAminus cells (dotted line) and
when the the density of csgAminus 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
nonsignaling 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 nonsignaling cells
added so that the total cell density is always 1.6 (solid line). Apparently, the
decrease in Cfactor explains the increase in wavelength. The nonsignaling
mutants do not affect the pattern at all.
15
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