Noname manuscript No.
(will be inserted by the editor)
A stochastic model of myxobacteria explains several
features of myxobacterial motility and development
Antony B. Holmes · Sara Kalvala ·
David E. Whitworth
Received: date / Accepted: date
Abstract Bacterial populations provide interesting examples of how relatively simple
signalling mechanisms can result in complex behaviour of the colony. A well studied
model is myxobacteria; cells can coordinate themselves to form intricate rippling patterns and fruiting bodies using localised signalling. Our work attempts to understand
and model this emergent behaviour. We developed an off-lattice Monte Carlo simulation of cell motility and show it can be used to generate both rippling and fruiting
body formations.
Keywords fruiting · Monte Carlo · morphogenesis · myxobacteria · rippling
1 Introduction
Myxobacteria are Gram negative, soil dwelling bacteria, distinguished by a complex
and social life cycle involving multicellular development [5,6]. In response to starvation,
cells pass through several developmental stages over a 72 h period, culminating in the
formation of fruiting bodies, large multicellular aggregates of approximately 100,000
cells, within which dormant cell-types called myxospores are formed (Fig. 1). The first
of these stages is the rippling phase, occurring approximately 4 h into starvation; the
population self-organises into mobile bands of cells, which reflect off one another, giving
the appearance of travelling waves [9, 24] (See Fig. 2). It is thought that rippling is an
emergent property of an increased reversal frequency, brought about by C-signal, a
Antony B. Holmes
MOAC Doctoral Training Centre, University of Warwick, Coventry, CV4 7AL, UK
E-mail: a.b.holmes@warwick.ac.uk
Sara Kalvala
Department of Computer Science, University of Warwick, Coventry, CV4 7AL, UK
E-mail: Sara.Kalvala@warwick.ac.uk
David E. Whitworth
Institute of Biological, Environmental and Rural Sciences, Aberystwyth University, Ceredigion,
SY23 3DD, UK
E-mail: dew@aber.ac.uk
2
membrane bound protein exchanged during cell-cell contact, which modulates the Frz
signalling pathway. The Frz pathway is responsible for controlling cell reversals.
In this paper we develop an off-lattice model of a myxobacteria population and
investigate the dynamics of rippling behaviour. Models of myxobacteria motility tend to
fall into one of two categories: those concerned with cell biology [8] and those concerned
with physical modelling [23,26,27]. We believe both approaches should be combined to
create a model describing the Frz pathway dynamics coupled to a physically realistic
model of cell behaviour to study how both systems work together. Combining both
approaches helps us validate whether our understanding of both the biology and cell
morphology is sufficient to explain spatial phenotypes.
In this paper we discuss how rippling fruiting development can be mathematically
and computationally modelled and develop a novel three-dimensional, unified model
or morphogenesis that can explain both phenomena.
Orbiting cells
(d) Aggregation
Myxospores
(c) Streaming
Prey bacteria
(g) Predation
(b) Rippling
(a) Vegetative cells
(e) Fruiting body
formation
Myxospores
(f) Sporulation
Fig. 1 The life cycle of myxobacteria. Cells can engage in cooperative predation (panel g)
from a vegetative state (panel a). In response to starvation, cells go through a number of
development phases culminating in the formation of fruiting bodies and myxospores (panels
b-f ).
2 Background
Myxobacteria glide on solid surfaces (they do not swim) using two different motility
systems: the “adventurous” A-motility engine and the “social” S-motility engine. Amotility allows individual cells to move out from the colony edge. Cells deposit a
slime trail as they move; however, there is some contention over the role of slime [16]
although it appears cells can follow the slime trails left by other cells. S-motility is used
to coordinate large numbers of cells in close proximity. Cells extend type IV pili from
the leading cell pole, which attach to neighbouring cells. Upon retraction of the pili,
cells are forced to come together and align.
3
Fig. 2 Magnification of myxobacteria ripple formation on a surface adapted from [19]. The
dark bands are areas of densely aligned cells travelling as waves.
In response to starvation, cells exchange C-signal, a surface bound protein, upon
close contact. The level of C-signal within a cell controls cellular processes such as
reversal frequency and is thought to be the primary mechanism cells use to regulate movement [10]. C-signal accumulates within cells up until sporulation [12]. The
exchange of C-signal causes a positive feedback cycle that up-regulates C-signal production so the more interactions a cell has, the faster C-signal accumulates and the
higher the reversal frequency.
Ripples appear to collectively move through each other; however, in reality when
ripples collide, cells push into the opposing wave approximately half a cell length before
reversing direction [20]. The reversal frequency is a key factor in the formation of
spatial patterns. Initially reversal frequency increases, corresponding to the formation
of ripples, and subsequently decreases monotonously until sporulation.
During starvation, C-signal accumulates within a cell [12] reducing its reversal
frequency [10]. By the time cells have reached the fruiting body stage, they are reversing
very infrequently. The lack of reversal and the effects of A and S motility causes cells
to form streams and increases the likelihood of aggregation; cells which cannot reverse
tend to remain stuck in one location since their ability to move around obstacles is
limited by only being able to move forward [28].
Fruiting bodies contain between 50,000 and 100,000 cells [17]. Inside the nascent
fruiting body, a percentage of the cells differentiate into dormant myxospores. This
process requires both temporal and spatial coordination in 3 dimensions making it one
of the most complex and least understood phases of the life cycle.
There is some disagreement over how fruiting actually begins. The traffic jam model
[11, 21, 22] proposes that streams of cells collide together causing the formation of a
kernel of stationary cells around which aggregation centres form [9]. Cells orbit around
the centre leading to mound formation. Work on Stigmatella suggested that cells form
circular orbits around the base and move up in spiral fashion around the base of
the stalk building on top of it [7, 25]. O’Connor and Zusman [17, 25] showed that
myxobacteria appear to orbit around a largely stationary aggregation centre. Kuner and
Kaiser [13] did not observe the spiralling patterns suggesting that this behaviour is nonessential and may not be intrinsically important to fruiting development. Recent work
4
by Curtis et al. did not observe spiralling patterns [4] has revealed that myxobacteria
fruits form using a layer building approach similar to a volcanic eruption. Oncoming
layers of cells collide causing a rapid build up in pressure at the meeting point. In
order to elevate the pressure, cells are forced upwards and over each other similar to
tectonic plate movements. The cells that have moved up are supported on top of the
highly dense layer of cells and polysaccharide slime underneath and begin to spread
out forming a new layer. As the new layer becomes more dense, cells in the centre
start to get pushed upwards to start a new layer and the process repeats causing the
formation of a stalk.
3 Developing a physical model of rippling and fruiting behaviour
In this section we describe how rippling and fruiting can be simulated. An off-lattice
Monte Carlo method and the Metropolis algorithm were used to accept or reject state
changes [14]. Cells were simulated in a three-dimensional volume, each cell being composed of 8 three-dimensional cuboidal segments (see Fig. 3). Each segment is composed
of 27 segment nodes arranged in a cube formation. A small overlap controlled by the
stretching energy term of the Hamiltonian is allowed so that the cell maintains a continuous volume despite being made of multiple separate segments (see Fig. 3c).
(a)
Segment
(b)
Segment overlap
Segment node
Fig. 3 Schema of a segmented model cell. Each cell is composed of a number of connected
segments with each segment being composed of a number of nodes. (a) Each segment consists
of a centre (black dot) and a neighbourhood of lattice nodes that represent the segment volume
(grey hexagon). Neighbouring segment centres are a distance d from each other. (b) Threedimensional view of a cell. Each cell has 8 segments: a head (red), a tail (blue) and 6 body
segments. Each segment comprises 27 segment nodes in a cube formation. (c) Segments move
independently allowing the cell body to be flexible. Overlap between segments allows the cell
to maintain a continuous cell volume.
3.1 Simulation volume
A simulation volume equivalent to 250 µm × 60 µm × 10 µm was used in rippling
simulations to allow multiple ripples to form.
The shape and size of fruiting bodies varies greatly; however, a typical mature fruit
consisting of a short stalk and mound formation (see the fruiting body formation in
Fig. 1 in [13] at the 61 h stage) has a diameter of approximately 100 µm [4,7,11,13]. A
simulation volume equivalent to 60 µm × 60 µm × 30 µm was used for all simulations
5
to allow fruiting bodies to develop. The work presented here is concerned with the
initial formation of the fruit so a large simulation volume to contain a mature fruit was
unnecessary.
Periodic boundary conditions are disabled in the xy-plane since it does not make
sense for cells to be able to push through the floor nor move through the ceiling
since that would imply burrowing through the floor of the next domain and under the
existing cells for which there is no physical interpretation. Boundaries are maintained
with a boundary energy term which severely penalises a cell for attempting to cross a
particular domain boundary. The energy penalty is several orders of magnitude larger
than the value any of the other energy terms might produce (positive or negative) so
it is impossible for a configuration with a domain crossing to be favourable.
3.2 System Hamiltonian
The fruiting model is designed such that cells can revert to other phases of behaviour
if conditions change. Cells are not constrained to only attempt fruit formation; their
response is dependent on conditions being favourable. In this way we hope to determine
a minimal set of behaviours that can lead to fruiting.
We define the following Hamiltonian function to describe the system and discuss
each energy term in more detail:
H = Estretch + Ealign + Ebend + Epropulsion
+ Eslime + Eclimbing + Egravity + Ecollision
+ Eadhesion
(1)
The energy terms were calculated as follows. We use the convention that bold terms
denote vector quantities. Terms covered with a hat (ˆ) denote unit vectors.
3.2.1 Stretching energy
Myxobacteria cells have a finite volume and fixed shape so we enforce these constraints
with a term to measure a cell’s stretching energy. This governs a cell’s length and is
analogous to the spring constant in Hooke’s Law.
Estretch (a) = λ
N
−2
X
ksa,i+1 − sm,i k − d0
2
(2)
i=0
where λ is a dimensionless stretching coefficient, N is the number of segments in cell
a, d0 is the optimal distance between segments, sk,l is the vector position of segment
l in cell k and λ a dimensionless stretching coefficient.
3.2.2 Alignment energy
In close proximity, cells tend to align with each other reflecting the effect of the Smotility engine. Cells extend Type IV pili from their leading pole which grab onto
6
neighbouring cells. Upon retraction this pulls a cell closer to the neighbour it latched
onto. The net effect of this is the alignment of cells.
Ealign (a) = α · bm

ĉ · ê
bm =
− (ĉm · ê)
(3)
if (ĉ · ê) ≥ 0,
(4)
else.
sa,1 − sa,N
ksa,1 − sa,N k
e
ê =
kek
X
e=
si,1 − si,N
ĉ =
(5)
(6)
(7)
i neighbours
where α is a dimensionless alignment coefficient, ĉ is the normalised average direction of the cell, ê is the average direction of all the cells in a local neighbourhood
surrounding cell a. bm reflects that cells tend to turn through the acute angle to align
with other cells in either direction.
3.2.3 Bending energy
Each cell has a semi-flexible body which must maintain a certain stiffness otherwise
the cell folds up upon itself. Bending energy ensures that the radius of curvature of a
cell does not exceed a threshold causing the cell to flail uncontrollably.
Ebend (a) = σ
N
X
b2a,i
(8)
i=1
bm,n =
dm,n
êm,n


bm,n+1

if n = 0.
bm,n−1 if n = N .



dm,n
else.
= cos−1 êm,n · fˆm,n
em,n
=
kem,n k
em,n = sm,n+1 − sm,n
fˆm,n
fm,n
=
kfm,n k
fm,n = sm,n − sm,n−1
(9)
(10)
(11)
(12)
(13)
(14)
where σ is a dimensionless bending coefficient, dm,n returns the angle between
em,n and fm,n , em,n is the average direction of segment n of cell m and fm,n is the
vector between the segment ahead of n (sm,n−1 ) and the segment behind (sm,n+1 ).
3.2.4 Propulsion energy
The A-motility system provides myxobacteria cells with propulsion. Cells extrude a
polysaccharide slime from nozzles at their lagging pole which expands when hydrolysed
7
and pushes a cell forward. This effect is modelled using a propulsion term which causes
cells to move preferentially in the average direction of the cell simulating the slime
pushing a cell along.
Epropulsion (a) = −
N
X
(ûi · ê)
(15)
i=2
sa,1 − sa,N
ksa,1 − sa,N k
sa,n−1 − sa,n
ûn =
ksa,n−1 − sa,n k
ê =
(16)
(17)
where is a dimensionless propulsion coefficient, ê is the normalised average direction of the cell and ûn is the update direction of segment n of cell a. Each segment
moves towards where its head segment was previously except if this causes segments
to become unaligned.
3.2.5 Slime trail following energy
As well as extruding slime to move, cells can also detect slime trails left by other cells
and preferentially follow them. This allows cells to follow other adventurous cells and
leads to the formation of streams that can break away from the main colony. Slime
following is complementary to A-motility. As each cell moves, it deposits a slime trail.
This is a set of normalised vectors representing the average direction of a cell. The
slime ages over time and is eventually removed. Cells can sense slime trails within
a limited neighbourhood around them. Using a weighted sum of the all slime trail
directions based upon their age, the average slime direction is calculated and cells will
preferentially follow that. A weighted sum is used to account for that fact that a cell is
more likely to follow a large slime trail than a small one. If a cell was to pick a random
slime trail and follow it, it might end up moving perpendicular to the major slime trail
which is unrealistic.
Eslime (a) = φ b̂a · ĉ
sm,1 − sm,N
ksm,1 − sm,N k
c
ĉ =
kck
X
c=
slime(i)
b̂m =
(18)
(19)
(20)
(21)
i neighbours
slime(m) =
tm
ktm k
(22)
where φ is a dimensionless slime following coefficient, ĉ is average direction of the
slime trails in a neighbourhood and slime(m) is the normalised direction vector of the
slime trail at location m.
8
3.2.6 Climbing energy
Curtis et al. [4] propose that when 2 sheets of oncoming cells encounter each other,
individual cells have a proclivity to move out of the potential “traffic jam” that can
ensue and typically this is upwards so one sheet of cells effectively moves over the other.
The energy term described here encourages cells to move upwards proportional to the
number of oncoming cells they are interacting with.
Eclimbing (a) = −η × cells(a) × dir(a)
X
cells(a) =
collision(a, i)
(23)
(24)
i neighbours
collision(a, i) =

1
if dˆa · dˆi ≤ 0,
0
else.
sm,1 − sm,N
dˆm =
ksm,1 − sm,N k
dir(a) = r̂a · n̂,
(25)
(26)
(27)
where η is a dimensionless climbing coefficient, cells(a) determines the number of
oncoming cells, collision(a, i) determines if two cells are moving in opposing directions
by examining the dot product between the normalised average direction (dˆm ) of each
pair of interacting cells and dir(a) compares the direction cell a wants to move in (r̂a )
with a normal vector (n̂).
3.2.7 Gravitational energy
Cells are allowed unrestricted movements in 3 dimensions so we need to control their
behaviour in the z-axis. Cells cannot randomly climb into empty space so a notion of
gravity is required. The other energy terms do not prevent cells from climbing because
they are chiefly describe the cell volume and cell motion in along reasonably straight
paths in any dimension. Gravity is represented as a penalty for trying to climb. The
steeper the climb the greater the penalty. An object acting under gravity requires
the greatest amount of energy to directly oppose the force and move in the opposite
direction. The gravitational term rewards a cell for moving downwards. It should be
noted that the use of the dot product means that there is no net effect of this term for
a cell moving in a straight line in the xy-plane. Since gravity is a constant, there is no
change in energy from moving between 2 positions with a direction vector perpendicular
to the direction of the force.
Egravity (a) = −µ
h
i
b̂a,1 · ĉ · space(da,1 , n)
dˆm,n = sm,n − e
(28)
(29)
where µ is a sensitivity parameter, b̂a,1 is the normalised update direction of the
head segment, ĉ a normalised direction vector pointing towards the ground, dm,n is
a location below the centre of segment n of cell m and n is a local neighbourhood
surrounding dm,n .
9
3.2.8 Collision energy
Each cell comprises a number of segments each with a finite volume. Segments exert a
repulsive force between themselves to prevent cells colliding.
Ecollision (a) = τ
N
X
X
collision(sa,i , sj,k )
(30)
i=1 (j,k) neighbours
collision(sa,b , sc,e ) =

m
if ksa,b − sc,e k < dmin ,
0
else.
(31)
where sa,b is the position of segment b of cell a and dmin is the minimum distance
allowed between segments of difference cells. The collision energy compares the distance
between a segment and the neighbouring segments sj,k around it and severely penalises
a cell for getting too close to another. Although the centres of segments cannot occupy
the same space, a small overlap is allowed to model deformation effects of cells in close
proximity. This is required because of the rigid segment shape which would otherwise
not allow for this type of effect.
3.2.9 Adhesion energy
The high density of cells in a swarm and fruiting body means there is a large amount of
polysaccharide slime produced which encases all of the cells in a slime matrix [7,17,18].
The slime casing prevents cells coming apart, for example even with a rotary shaker.
This matrix affects an adhesive force on the cells making it harder for cells to move
apart from each other. Cells typically aggregate at a colony edge due to surface tension
effects making it difficult to escape the colony [11].
This effect is separate to the effects of A-motility and is a global property of a large
mass of cells.
Eadhesion (a) = −ϕ
N
X
X
i=2 (j,k) neighbours,j6=a
1
ksa,i − sj,k k2
(32)
where ϕ is a dimensionless adhesion coefficient, cells(a) determines the number of
oncoming cells, collision(a, i) determines if 2 cells are going in opposing directions by
examining the dot product between the normalised average direction (dˆm ) of each pair
of interacting cells and dir(a) compare the direction cell a wants to move in (r̂a ) with
a normal vector (typically a normal to the xy-plane).
3.3 Simulation algorithm
At each time step, the following sequence of operations is performed:
1. Determine cell interactions with neighbouring cells.
2. Update the internal state of the cells.
3. Select a new location for the head node to move to. Each new location is a fixed
distance L from the current head position. In three-dimensions this consists centring
the head within a sphere and selecting a point on the surface of the sphere as the
new location.
10
4. Apply the Monte Carlo algorithm to determine if the head move is energetically
favourable. A separate collision resolution algorithm such as that used by Wu et al.
[27] is not required since collision avoidance is a natural feature of the Hamiltonian.
5. If the move is favourable, move the head, otherwise repeat steps 1 and 2 F times
until a suitable new location is found. If a location cannot be found within F
attempts, the cell stalls for the current time step.
6. Repeat steps 7 to 8 F × N times (N is the number of nodes per cell).
7. Choose node i such that i ≤ N at random and move it in the direction from node
i to node i + 1 (the head node of i) at a distance of L;
8. Apply the Metropolis algorithm [15] to determine the acceptance probability of
making the change.
(
P (∆E) =
if ∆E ≤ 0,
1
e
−∆E/kT
if ∆E > 0
(33)
9. Calculate the average direction of the cell from its head to its tail then normalise
and write out as the slime vector at the location of the tail.
S=
sa,1 − sa,N
ksa,1 − sa,N k
(34)
10. For each cell birth region, test for empty locations. For each spare location, create a
new cell 1 segment in size and place it at the spare location. This ensures a constant
cell density within the stream.
11. For each cell with a segment size smaller than N , grow the cell by creating a new
segment and making it the new tail segment of the cell.
Experiments were carried out using the parameters listed in Table 1.
Table 1 Parameters for models used to simulate fruiting body formation in Myxococcus xanthus.
Name
a0
d0
λ
α
σ
ν
µ
τ
kT
ns
nv
vr
vf
Value
12
√
a0
3.0
2.0
12.0
2.0
1.0
3.0
100.0
0.3
8
27
250 µm × 60 µm × 10 µm
60 µm × 60 µm × 30 µm
Description
target volume of segment.
target distance between adjacent segments.
stretching energy parameter.
volume energy parameter.
bending energy parameter.
propulsion energy parameter.
gravity energy parameter.
gravity energy parameter.
collision energy parameter.
Boltzmann constant × temperature.
Number of segments per cell.
Segment volume (number of segment nodes).
Dimension of rippling simulation.
Dimension of fruiting simulation.
11
2h
8h
a)
250
250
225
225
200
200
175
175
150
150
x (µm)
x (µm)
b)
125
125
100
100
75
75
50
50
25
25
3
6
9
12 15 18 21
Time (minutes)
24
27
30
3
6
9
12 15 18 21
Time (minutes)
24
27
30
Fig. 4 Time evolution of ripple formation in a Monte Carlo off-lattice simulation of 5500
cells in a simulation volume measuring 250 × 40 × 10 µm. Starting from an initial random
distribution, cells organise into 3 distinct ripple bands after approximately 4h. (a) Top down
view of simulation. (b) Corresponding space-time plot of cell movement in the x dimension
during a 30 minute interval measured around the given time point.
4 Results
4.1 Ripple Formation
Fig. 4 shows the output from a Monte Carlo simulation of 5500 cells using the Hamiltonian described in Section 3. C-signalling was modelled using a variable phase clock
function [2]. Cells were initially randomly distributed and randomly aligned. The emergent behaviour of the system is the formation of 3 ripple bands. In the pre-rippling
phase the majority of cells reverse approximately every 10 minutes with a small percentage of cells reversing very quickly (every 4-6 minutes). This corresponds to increased
signalling due to the proximity of a large number of randomly oriented cells which collide. A proportion of cells experience many collisions and reverse in 6 minutes or less.
A much greater proportion of cells reverse in either 7 or 8 minutes. This can be down
to a number of factors. Cells which have climbed over others and are nearer the top
level will naturally experience fewer C-signalling events as there will be fewer cells to
interact with. Cells nearer the bottom will be more crowded together and are therefore
likely to experience more signalling. It might be expected that signalling would be a lot
higher initially but two factors limit this: polar sensitivity and traffic jams. C-signalling
is thought to only occur at the cell poles, implying signalling does not take place over a
large proportion of the cell so even cells in close proximity, actually experience few signalling events. Cells tend to stall in a closely packed environment since there is limited
space for them to move. Once a traffic jam forms, cells can get stuck temporarily until
they either naturally reverse or the jam disperses. During this waiting time, there is
little opportunity for signalling because the position of cell heads relative to each other
changes very little. These two factors combined ensure that at a given time, only a
small percentage of cells will actually be close enough that their heads interact. There
will be always be some cells which receive minimal signalling either from being in a
jam or else isolated from other cells. The majority of cells reverse every 7 to 8 minute
reflecting that most cells experience some signalling which speeds them up slightly.
12
4.2 Fruiting
Sozinova et al. [22] present a three-dimensional lattice gas cellular automata model of
rippling formation. Cells are oriented in one of six directions on a hexagonal grid. This
of course severely limits the direction cells can move in and any orbiting patterns of
cells may be an artefact of this; any alteration in direction is a turn of π/3 rad so cells
can move through tight arcs. The rigid cell body also means that the cell must alter
its course dramatically. In reality, the partial flexibility of the cell means it does not
have to completely alter its course to avoid an obstacle; it can bend slightly to align
itself alongside the object and move around it.
The model presented here shows that fruiting body development does not require
artificial induction with cell density and an upward pushing force being sufficient to
instigate formation. Importantly, the polysaccharide slime surround cells must exert an
adhesive force, binding cells together. Without this force, cells are too unconstrained
and move away from the aggregate. Each layer acts almost independently. Cells from
one layer have a much reduced effect on cells in another layer than cells in the same
layer. Experiments where all terms in the Hamiltonian were dependent on a local
three-dimensional neighbourhood showed that cells cannot move freely due to feeling
the effects of cells moving in all directions around them. Consider a scenario where an
aggregate has just started to form and a second layer of cells is expanding outwards
from the centre.
Fruiting development begins after 75 minutes with a small mound formation (see
Fig. 5). The mound expands outwards as well as upwards forming a large stabilised stalk
base after 400 minutes. There were initially 1600 cells to begin with but the constant
influx causes this number to rise to approximately 5000 cells over the duration of the
simulation. A heat map is used to indicate the maximum height of cells at a given
location; blue regions are relatively sparse with cells only a few layers thick whilst red
regions contain many cells stacked on top of each other. The highest region is always
towards the centre of the simulation volume indicating that cells do the majority of
climbing in this region. There is an accumulation of cells, spreading outwards from
the centre in both the x-axis and y-axis. The rate of expansion of the fruit from the
centre gradually reduces with time. There appears to be a limit on the size of the fruit,
a given number of cells can support. The influx rate appears to be the rate limiting
step in controlling fruiting growth; there is a point where the number of cells forming
new layers will begin to exceed the number of cells flowing into the system so the
development of new layers must be arrested.
Fig. 6 shows a three-dimensional view of the fruiting body formation. A layer of
cells (light green) covers the floor of the simulation out of which one major and one
minor fruiting body (dark green coloured cells) have started to form. The fruit is a
continuously altering and changing entity and has drifted towards the edge of the
simulation volume (it spans 2 edges due to periodic boundary conditions).
5 Discussion
By integrating a simple model of C-signalling with a detailed Hamiltonian capturing
the physics of single cell motility, we have created a realistic model of a myxobacteria
cell population that can explain how cells self-organise to display multiple different
spatial phenotypes. Importantly we have shown that a single unified model is sufficient
13
8
8
7
50
7
50
6
6
40
40
30
4
3
20
5
y ( µ m)
x ( µ m)
5
30
4
3
20
2
10
1
50
100
150
200
250
Time (mins)
(a)
300
350
400
0
2
10
1
50
100
150
200
250
Time (mins)
300
350
400
0
(b)
Fig. 5 Space time plots showing fruiting body formation. Mound formations are coloured by
height from 0 µm (blue) to 8 µm (red). (a) Variance of mound height (z-axis) along the x-axis
with respect to time. (b) Variance of mound height (z-axis) along the y-axis with respect to
time.
Fruiting body
(a)
(b)
Fig. 6 Simulation of fruiting body formation after 500 min. Plots are of the same system
from different view points. Cells are coloured by height going from light green (lowest) to dark
green (highest). A large fruiting body has formed towards the edge of the simulation. Due to
periodic boundary conditions, the fruit spans 2 edges of the simulation.
to explain multiple behaviours where previously multiple different models were required
to explain each state of the life-cycle.
The model is able to explain the initial formation of fruiting bodies from streaming
cells as a consequence of cell physics and a low reversal frequency. Fruiting bodies
will form spontaneously without the need for an artificial aggregation centre to seed
the process. We have also shown that observed transitory fruiting body developments
before a stable fruiting body forms can be explained as a consequence of net cell influx.
14
We were able to show that even in a dense region, cells do not necessarily experience
increasing C-signalling due to the C-signalling mechanism being only at the head and
tail of the cell. The small surface area of the C-signalling region implies that either
cells are very sensitive to C-signalling if reversals can be triggered in after 4-5 collisions
or else C-signalling occurs along more of the body.
Mesh based models [1–3] typically restrict the degrees of freedom of cells. Cell flexibility and shape are not really considered; however, we find that these are important
physical properties determining cellular behaviour. As ripples form, areas of low cell
density develop. Due to the stochastic motion of cells, small perturbations in the travelling direction often lead cells to gradually but significantly alter their course in a low
density region with few other cells to interact with. We found that slime serves an essential purpose of maintaining long range cohesion between cells. S-motility is too short
range; cells will remain aligned within small clusters but the macroscopic behaviour
allows cells to change course leading to the same dispersal problem as with single cells
in a low density environment. Ripple formation is dependent on cells remaining aligned
to maximise collisions in counter-propagating waves and troughs; if cells drift off course
the ripple fronts break down.
Our experiments suggest that in a tightly packed population, ripples do not form
if a strict monolayer of cells is maintained. When tightly packed, cells have little room
to manoeuvre and are frequently obstructed by other cells. Some cells are able to move
around obstacles but this is dependent on there being enough space to allow them to
change course; however, most cells are very close to each other and cannot avoid stalling.
This issue is not addressed in lattice models where cells are typically restricted to
moving in fixed planes which reduces the overall number of cells that can actually cause
obstruction. Ripples are not a consequence of cells colliding, blocking each other and
then reversing as data tracking individual cells showed they typically do not pause for
long periods [20]. In highly populated environments aggregation centres are much likely
to form but these are not a common artefact during rippling. S-motility and A-motility
are not sufficient by themselves to maintain rippling. If cells alter course to avoid
objects, their alignment rapidly breaks down into small clusters and the macroscopic
alignment is lost. We therefore conclude that cells will ripple in a monolayer provided
some cells can climb over others to avoid collision.
The model presented here shows that a well characterised model of myxobacteria
cell motility can describe both rippling and fruiting. Importantly parameters were kept
the same in the two model types showing that cells do not require multiple cellular
control systems to display different phenotypes. This validates both the model itself
and what is known biologically about myxobacteria. Neither rippling nor fruiting can
be explained purely by the function of signalling pathways alone; it is also a direct
result of the physical characteristics of the cell. Our work serves as a basis for future
models looking at other aspects of the myxobacteria life cycle.
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