# 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 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 min, a minimum response time of up to 1 min and no maximum oscillation period best reproduce rippling in the experiments of Myxococcus xanthus. Non-linear dependence 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 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.

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**ABSTRACT:**This paper reviews recent progress in modeling collective behaviors in myxobacteria using lattice gas cellular automata approach (LGCA). Myxobacteria are social bacteria that swarm, glide on surfaces and feed cooperatively. When starved, tens of thousands of cells change their movement pattern from outward spreading to inward concentration; they form aggregates that become fruiting bodies. Cells inside fruiting bodies differentiate into round, nonmotile, environmentally resistant spores. Traditionally, cell aggregation has been considered to imply chemotaxis, a long-range cell interaction. However, myxobacteria aggregation is the consequence of direct cell-contact interactions, not chemotaxis. In this paper, we review biological LGCA models based on local cell–cell contact signaling that have reproduced the rippling, streaming, aggregating and sporulation stages of the fruiting body formation in myxobacteria.Advances in Complex Systems 11/2011; 09(04). · 0.65 Impact Factor

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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

Abstract

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: 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

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1 Introduction

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 [1]. 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 [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 one-dimensional 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 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 [10] and the addition of C-factor (extracted from fruiting

body cultures) causes cell reversal frequencies to increase three-fold [12]. 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].

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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 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 [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

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.

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 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 [24]. 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 [28]. Sager and Kaiser have also

observed the effect of C-signaling-competent cell density upon ripple wave-

length [12]. 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-

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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 anti-parallel 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 species-dependent, 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 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-

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

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 [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 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 [35] (for a description of the role

the refractory period plays in Dictyostelium, see [39]). 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 [12]. Cells do not reverse more

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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-

lation period.

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 [40]. 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

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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 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. [34] 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 [41] 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 [17]. 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

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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 C-factor 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 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

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 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

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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 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

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 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. [34]. 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

?

∗

?[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 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. [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

“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

many collisions.

3.2.3 Head-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

[35]. Further, we demonstrate that the change in wavelength may be entirely

explained by the change in density of C-signaling 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 C-factor 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 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

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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 [12]. 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 [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 C-factor relative to experimental conditions, and cells

are not very elongated, which increases the density of C-factor 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 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. [35] for their

value of p [35].

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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 [12]. 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. ([35], Figure 3b) also showed when ripples collide a

small jump in phase reminiscent of a soliton jump.

(a) (b)

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

before reversing.

4.3Effect of dilution with non-signaling cells

Sager and Kaiser [12] 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

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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 non-signaling cells (see [35] 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

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 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.

15

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