bioRxiv preprint first posted online Jan. 13, 2017; doi: http://dx.doi.org/10.1101/100099. The copyright holder for this preprint (which was not
peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
Mechanical interactions in bacterial colonies and the surng probability of
benecial mutations
Fred F. Farrell , Matti Gralka , Oskar Hallatschek , and Bartlomiej Waclaw
1
2
2,3
4,5
Life Sciences, University of Warwick, Coventry CV4 7AL, UK
Department of Physics, University of California, Berkeley, CA 94720, USA
Department of Integrative Biology, University of California, Berkeley, California 94720, USA
SUPA School of Physics and Astronomy, The University of Edinburgh, Mayeld Road, Edinburgh
EH9 3JZ, UK
Centre for Synthetic and Systems Biology, The University of Edinburgh, Edinburgh, UK
1
2
3
4
5
January 12, 2017
1
Abstract
Introduction
Bacteria are the most numerous organisms on Earth
Bacterial conglomerates such as biolms and micro-
capable
of
autonomous
reproduction.
They
have
colonies are ubiquitous in nature and play an impor-
colonised virtually all ecological niches and are able
tant role in industry and medicine.
In contrast to
to survive harsh conditions intolerable for other organ-
well-mixed, diluted cultures routinely used in micro-
isms such as high salinity, low pH, extreme tempera-
bial research, bacteria in a microcolony interact me-
tures, or the presence of toxic elements and compounds
chanically with one another and with the substrate to
[1].
which they are attached. Despite their ubiquity, little
pathogens, but some bacteria nd applications in the
is known about the role of such mechanical interactions
industry as waste degraders [2] or to produce fuels and
on growth and biological evolution of microbial popu-
chemicals [3]. In all these roles, biological evolution of
lations. Here we use a computer model of a microbial
microbes is an undesired side eect because it can dis-
colony of rod-shaped cells to investigate how physical
rupt industrial processes or lead to the emergence of
interactions between cells determine their motion in the
new pathogenic [4] or antibiotic-resistant strains [5].
Many bacteria are important animal or human
colony, this aects biological evolution. We show that
Experimental research on bacterial evolution has
the probability that a faster-growing mutant surfs at
been traditionally carried out in well-stirred cultures
the colony's frontier and creates a macroscopic sector
[6, 7].
depends on physical properties of cells (shape, elastic-
ria often form aggregates such as microcolonies and
ity, friction). Although all these factors contribute to
biolms.
the surng probability in seemingly dierent ways, they
teeth (plague), on catheters or surgical implants [9],
all ultimately exhibit their eects by altering the rough-
inside water distribution pipes [10], or in the lungs of
ness of the expanding frontier of the colony and the
people aected by cystic brosis [11]. Bacteria in these
orientation of cells. Our predictions are conrmed by
aggregates adhere to one another and the surface on
experiments in which we measure the surng probabil-
which they live, form layers of reduced permeability to
ity for colonies of dierent front roughness. Our results
detergents and drugs, and sometimes switch to a dif-
show that physical interactions between bacterial cells
ferent phenotype that is more resistant to treatment
play an important role in biological evolution of new
[12, 13, 14]; this causes biolms to be notoriously di-
traits, and suggest that these interaction may be rele-
cult to remove.
vant to processes such as
de novo evolution of antibiotic
However, in their natural environment bacteSuch aggregates can be found on food [8],
An important aspect of bacteria living in dense con-
resistance.
glomerates is that they do not only interact via chem-
1
bioRxiv preprint first posted online Jan. 13, 2017; doi: http://dx.doi.org/10.1101/100099. The copyright holder for this preprint (which was not
peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
(a)
(b)
1um
growth
time
variable
length
(c)
division
F
Figure 1:
-F
a small, random “kick”
(a) Illustration of the computer algorithm.
Bacteria are modelled as rods of varying length and constant
diameter. When a growing rod exceeds a critical length, it splits into two smaller rods. (b) A small simulated colony. (c)
The same colony with nutrient concentration shown as dierent shades of gray (white = maximal concentration, black =
minimal); the cells are represented as thin green lines.
ical signaling such as quorum sensing [15] but also
nutrients to the colony. This corresponds to a common
through mechanical forces such as when they push
experimental scenario in which bacteria grow on the
away or drag other bacteria when sliding past them.
surface of agarose gel infused with nutrients. We have
Computer simulations [16, 17, 18, 19] and experiments
previously demonstrated [17] that this model predicts
[20, 21, 22, 23, 24] have indicated that such mechanical
a non-equilibrium phase transition between a regular
interactions play an important role in determining how
(circular) and irregular (branched) shape of a radially
microbial colonies grow and what shape they assume.
expanding colony of microbes, and that it can be used
However, the impact of these interactions on biological
to study biological evolution in microbial colonies [34].
evolution has not been explored.
Here, we use this model to show that the surng proba-
A particularly interesting scenario relevant to micro-
bility of a benecial mutation depends primarily on the
bial evolution in microcolonies and biolms is that of
roughness of the expanding front of the colony, and to
a range expansion [25] in which a population of mi-
a lesser extend on the thickness of the front and cellular
crobes invades a new territory. If a new genetic variant
ordering at the front. We also investigate how mechan-
arises near the invasion front, it either surfs on the
ical properties of cells such as elasticity, friction, and
front and spreads into the new territory, or (if unlucky)
adhesion aect these three quantities. We corroborate
it lags behind the front and forms only a small bub-
some of our results by experiments in which we vary
ble in the bulk of the population [26]. This stochas-
the roughness of the growing front and show that it
tic process, called gene surng, has been extensively
inuences the surng probability as expected.
studied [27, 28, 29, 30, 31, 32, 33, 34] but these works
have not addressed the role of mechanical interactions
between cells.
2
Many of the existing models do not
Computer model
consider individual cells [27], assume Eden-like growth
[31], or are only appropriate for diluted populations of
We use a computer model similar to that from Refs.[17,
motile cells described by reaction-diusion equations
23, 34], with some modications. Here we discuss only
similar to the Fisher-Kolmogorov equation [35].
On
the generic algorithm; more details will be given in sub-
the other hand, agent-based models of biolm growth,
sequent sections where we shall talk about the role of
which have been applied to study biological evolution
each of the mechanical factors.
in growing biolms [36, 37, 38], use very simple rules to
We assume that bacteria form a monolayer as if the
mimic cell-cell repulsion which neglect important phys-
colony was two-dimensional and bacteria always re-
ical aspects of cell-cell and cell-substrate interactions
mained attached to the substrate. This is a good ap-
such as adhesion and friction.
proximation to what occurs at the edge of the colony
In this work, we use a computer model of a growing
and, as we shall see, is entirely justiable because the
microbial colony to study how gene surng is aected
edge is the part of the colony most relevant for biolog-
by the mechanical properties of cells and their environ-
ical evolution of new traits.
ment. In our model, non-motile bacteria grow attached
rocylinders of variable length and constant diameter
to a two-dimensional permeable surface which delivers
d = 2r0 = 1µm
2
We model cells as sphe-
(Fig. 1a). Cells repel each other with
bioRxiv preprint first posted online Jan. 13, 2017; doi: http://dx.doi.org/10.1101/100099. The copyright holder for this preprint (which was not
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Name
normal force determined by the Hertzian contact theory:
F =
1/2
(4/3)Er0 h3/2 where
h
Nutrient diusion constant
is the overlap dis-
tance between the walls of the interacting cells, and
plays the role of the elastic modulus of the cell.
Nutrient concentration
E
Nutrient uptake rate
The
Units
50
µm2 /h
c0
k
1
a.u.
1 3
a.u./h
100
kPa
4
µm/h
µm
µm
Pa·h
E
length vl
Young modulus
dynamics is overdamped, i.e. the linear/angular veloc-
Elongation
ity is proportional to the total force/total torque acting
Cell diameter
1
Average max. inter-cap distance lc
4
on the cell:
d~ri
= F~ /(ζm),
dt
dφi
= τ /(ζJ).
dt
Value
D
Damping coecient
(1)
(2)
ζ
500
Table 1:
Default values of the parameters of the model.
This gives
≈ 30min doubling time and the average length
≈ 3µm. If not indicated otherwise, all results
of bacterium
In the above equations
i, φi
of mass of cell
axis,
F~
and
the cell,
m
τ
~ri
is the position of the centre
is the angle it makes with the
presented have been obtained using these parameters.
x
are the total force and torque acting on
and
J
are its mass and the momentum of
We use two geometries in our simulations: a radially
ζ
expanding colony that starts from a single bacterium
inertia (perpendicular to the plane of growth), and
is the damping (friction) coecient.
(Fig. 2a), and a colony growing in a narrow (width
We initially as-
L)
sume that friction is isotropic, and explore anisotropic
but innitely long vertical tube with periodic bound-
friction later in Sec. 4.3.
ary conditions in the direction lateral to the expanding
Bacteria grow by consuming nutrients that diuse in
front (Fig. 2d). While the radial expansion case rep-
the substrate. The limiting nutrient concentration dy-
resents a typical experimental scenario, only relatively
namics is modelled by the diusion equation with sinks
small colonies (10
corresponding to the bacteria consuming the nutrient:
a real colony [34]) can be simulated in this way due
6 cells as opposed to
to the high computational cost.
∂c
=D
∂t
∂2c
∂x2
+
∂2c
∂y 2
−k
X
δ (~ri − ~r) .
> 108
cells in
The second method
(growth in a tube) enables us to simulate growth for
(3)
longer periods of time at the expense of conning the
i
colony to a narrow strip and removing the curvature
~r = (x, y), c = c(~r, t) is the nutrient concentration
at position ~
r and time t, D is the diusion coecient
of the nutrient, and k is the nutrient uptake rate. The
initial concentration c(~
r, 0) = c0 .
A cell elongates at a constant rate vl as long as the lo-
of the growing front. This has however little eect on
cal nutrient concentration is larger than a certain frac-
ble 1 shows default values of all parameters used in
tion (>1%) of the initial concentration. When a grow-
the simulation.
ing cell reaches a pre-determined length, it divides into
taken from literature data for the bacterium
two daughter cells whose lengths are half the length of
[34], but some parameters such as the damping co-
the mother cell. The critical inter-cap distance lcap−cap
ecient must be estimated indirectly [17].
at which this occurs is a random variable from a Gaus-
that the assumed value of the diusion constant
Here
Varying
`c
`c
width
L
of the tube is suciently large.
Figure 1b, shows a snapshot of a small colony; the
concentration of the limiting nutrient is also shown. TaMany of these parameters have been
E. coli
We note
D
is
and standard deviation
unrealistically small; the actual value for small nutri-
allows us to extrapolate between
ent molecules such as sugars and aminoacids would be
sian distribution with mean
±0.15 `c .
the surng probability of faster-growing mutants if the
S. cerevisae or the bacterium S. aureus) and rod-shaped cells (e.g. E. coli or
P. aeruginosa), whereas the randomness of lcap−cap ac-
∼ 106 µm2 /h, i.e., four orders of magnitude larger. Our
choice of D is a compromise between realism and com-
quasi-spherical cells (e.g. yeasts
putational cost; we have also showed in Ref. [17] that
counts for the loss of synchrony in replication that oc-
the precise value of the diusion coecient is irrelevant
curs after a few generations (the coecient of variation
in the parameter regime we are interested here. We also
∼ 0.1−0.2 [39, 40, 41]).
The two
note that in reality cessation of growth in the center
daughter cells have the same orientation as the parent
of the colony and the emergence of the growing layer
cell, plus a small random perturbation to prevent the
may be due to the accumulation of waste chemicals,
cells from growing in a straight line.
pH change etc., rather than nutrient exhaustion. Here
of the time to division
3
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(a)
27 µm
62 µm
140 µm
(e)
N= 5450
N= 784
100
50
0
0 5 10 15 20 25
time (h)
8
6
(c)
4
2
0
20
15
10
5
0
(f)
h (µm)
(b)
150
speed (µm/h)
radius (µm)
ρ (µm)
N =63
(d)
0 5 10 15 20 25
time (h)
20
15
10
5
0
0
5
10 15
y (mm)
20
0
1
2
3
y (mm)
4
Figure 2: (a) Snapshots of a radially-growing simulated colony taken at dierent times (sizes), for k = 2. Growing bacteria
(b) The radius of the colony increases approximately
linearly in time. (c) The expansion speed tends to a constant value for long times. (d) Example conguration of cells
are bright green, quiescent (non-growing) bacteria are dark green.
L = 80µm. The colony expands vertically. h is the thickness of the growing layer
ρ is the roughness of the front (Eq. (5)). (e,f ) Thickness and roughness as functions of the position y of the
L = 1280µm and k = 2.5, and for 10 indepedent simulation runs (dierent colours).
from a simulation in a tube of width
(Eq. (4)),
front, for
◦
media to below 60 C.
we focus on the mechanical aspects of growing colonies
Measuring surng probability.
and do not aim at reproducing the exact biochemistry
For each pair of
of microbial cells, as long as the simulation leads to the
mutant and wild type, a mixed starting population was
formation of a well-dened growth layer (as observed
prepared that contained a low initial frequency
experimentally).
mutants having a selective advantage
s.
Pi
of
Colony growth
was initiated by placing 2µl of the mixtures onto plates
3
and incubated until the desired nal population size
Experiments
was reached. The initial droplet radius was measured
to compute the number of cells at the droplet perime-
Experiments were performed as described in our previ-
ter.
ous work [34].
AxioZoom v16. The number of sectors was determined
Here we provide a brief description of
by eye.
these methods.
Strains and growth conditions.
The resulting colonies were imaged with a Zeiss
For the mixture
experiments measuring surng probability,
The surng probability was calculated using
Eq. (10).
Timelapse movies.
we used
For single cell-scale timelapse
pairs of microbial strains that diered in uorescence
movies, we used a Zeiss LSM700 confocal microscope
color and a selectable marker.
with a stage-top incubator to image the rst few layers
The selective dier-
of most advanced cells in growing
ence between the strains was adjusted as in [34] using low doses of antibiotics.
and antibiotics used were
coli
The background strains
E. coli
S. cerevisiae and E.
colonies between a coverslip and an agar pad for
about four hours, taking an image every minute.
DH5α with tetracy-
E. coli MG1655 with chloramphenicol, and S.
cerevisiae W303 with cycloheximide. Selective dier-
Measuring roughness.
cline,
Images of at least 10 equal-
sized colonies per condition were segmented and the
boundary detected.
[32].
strains were grown on LB agar (2%)
between boundary curve and the best-t circle to the
medium (10g/L tryptone, 5g/L yeast extract, 10g/L
colony was measured as a function of the angle and
E. coli
◦
◦
NaCl) at either 37 C or 21 C.
S. cerevisiae experiments
The squared radial distance
δr2
ences were measured using the colliding colony assay
averaged over all possible windows of length
l.
The
were performed on either YPD (20g/L peptone, 10g/L
2
resulting mean δr was averaged over dierent colonies.
yeast extract, 20g/L glucose) or CSM (0.79g/L CSM
Images of moving fronts at the single-cell level from
◦
(Sunrise media Inc.), 20 g/L glucose) at 30 C. 20g/L
the timelapse movies were rst segmented using a local
agar was added to media before autoclaving.
adaptative threshold algorithm to identify cells.
Antibi-
The
front was found by the outlines of cells directly at the
otics were added after autoclaving and cooling of the
4
bioRxiv preprint first posted online Jan. 13, 2017; doi: http://dx.doi.org/10.1101/100099. The copyright holder for this preprint (which was not
peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
front. For all possible windows of length
l,
a line was
tted to the front line and the mean squared distance
from the best-t line was measured, as in Ref. [27]. The
(a)
resulting mean squared distance was averaged over all
windows of length
4
l
and all frames.
Simulation results
(b)
4.1 Growth and statistical properties of the
simulated colony
(c)
We now discuss the properties of our simulated colonies.
When the colony is small, all bacteria grow and replicate. As the colony expands, the nutrient becomes de-
Figure 3:
pleted in the centre of the colony because diusion of
nutrient uptake rates
2.6
the nutrient cannot compensate its uptake by growing
cells.
k = 1.8
(a),
k = 2.2
(b) and
k =
The thickness of the growing layer (bright green)
h = 13.5 ± 0.1µm
k = 2.6, but this has
decreases only moderately (1.64×) from
This causes cessation of growth in the centre.
for
When this happens, growth becomes restricted to a
narrow layer at the edge of the colony, see Fig.
(c).
The frontier of the colony for three dierent
k = 1.8
to
h = 8.2 ± 0.1µm
for
a large impact on the front roughness which changes from
2a,
and Supplementary Video 1. The radius of the colony
ρ = 2.1 ± 0.2µm to ρ = 9.3 ± 0.4µm,
k = 2.6 the growing layer begins to
increases approximately linearly in time (Fig.
splits into separate branches.
2b,c).
correspondingly. For
loose continuity and
The presence of a growing layer of cells and the linear
growth of the colony's radius agree with what has been
After a short transient the expansion velocity, the
observed experimentally [42, 34].
nutrient prole, and other properties of the growing
Statistical properties of the growing layer can be con-
layer stabilize and vary little with time (Fig. 2e,f ). It
veniently studied using the tube-like geometry. Fig-
is therefore convenient to choose a new reference frame
ure 2d shows a typical conguration of cells at the
co-moving with the leading edge of the colony.
colony's frontier (see also Supplementary Video 2). The
growing layer can be characterized by its thickness
and roughness
ρ which we calculate as follows.
cells that lag behind the front do not replicate, we do
h
not have to simulate these cells explicitly. This dramat-
We rst
ically speeds up simulations and enables us to study
rasterize the growing front of the colony using pixels
of size
1 × 1µm,
stripes of the colony of width
and nd the two edges of the front:
the growing layer of cells is controlled by the nutrient
−
cells) {yi }. We then calculate the average thickness as
h=
1
L
i=1
min
j=1,...,L
(i − j)2 + (yi+ − yj− )2 .
concentration
and elasticity
(4)
we vary the uptake rate
ing parameters constant.
creasing
1
axis . Similarly, we calculate the average roughness
v
u
L
u1 X
ρ=t
(yi+ − Y + )2 ,
L
P
Y + = (1/L) i yi+ .
+ + −
(L, Y , yi , yi ) are in pixels
1
while keeping the remainFigure 4 shows that front
k;
eventually, when a critical value
kc ≈ 2.5
is
crossed, the growing front splits into separate branches.
This transition has been investigated in details in Ref.
[17]. Although this scenario can be realized experimen-
(5)
tally [43, 44], here we focus on the smooth regime
i=1
where
k
thickness decreases and its roughness increases with in-
can be curved and does not have to run parallel to the
as
c0 , nutrient uptake rate k , growth rate b,
E of cells. This in turn aects the rough-
ness of the leading edge of the colony, see Fig. 3, where
This method takes into account that the growing layer
x
and length
We have shown previously [17] that the thickness of
one (the boundary between the growing and quiescent
q
L > 1mm
> 10mm.
+
the upper one (the colony edge) {yi } and the lower
L
X
Since
in which colonies do not branch out and the frontier
Note that all quantities
and not
remains continuous.
µm.
contains replicating cells divided by the interface length L. Both
Alternatively, h can be dened as the area of the colony that methods produce similar results.
5
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Figure 4:
L =160
L =320
L =640
L =1280
1.0
1.5 2.0 2.5
uptake rate k
3.0
(c)
25
20
15
10
5
0
1.0
1.5 2.0 2.5
uptake rate k
30
25
20
15
10
5
0
roughness ρ (µm)
(b)
thickness h (µm)
30
25
20
15
10
5
0
roughness ρ (µm)
(a)
3.0
5
10
15
thickness h (µm)
20
L = 160 (red), L = 320
L = 1280 µm (purple). (a) Thickness h decreases as the nutrient uptake rate k increases.
h does not depend on the length L of the front. (b) Roughness ρ increases with both k and L. (c) Roughness versus
thickness; dierent points correspond to dierent k from the left and middle gure.
(green),
Thickness and roughness of the growing layer for dierent front lengths (tube widths)
L = 640
(blue), and
4.2 Surng probability of a benecial mutation
(a)
When a mutation arises at the colony's frontier, its fate
(b)
(c)
can be twofold [27, 34]. If cells carrying the new mutation remain in the active layer, the mutation surfs
Figure 5:
on the moving edge of the colony and the progeny of
show dierent fates of a sector of tter (s
The fate of mutants.
Left and middle panels
= 0.1) mutant cells
(red) in a colony of wild-type cells (green). The sector can
the mutant cell eventually forms a macroscopic sec-
either expand (left panel) or collapse and become trapped
tor (Fig. 5). On the other hand, if cells carrying the
in the bulk when random uctuations cause mutant cells to
mutation leave the active layer, the mutation becomes
lag behind the front (middle panel).
Right panel shows a
trapped as a bubble in the bulk of the colony [26]. Due
sector with larger (s
to the random nature of replication and mixing at the
faster growth of mutant cells leads to a bump at the front.
front, surng is a stochastic process; a mutation re-
In all cases
mains in the active layer in the limit
probability
Psurf
t→∞
= 0.5)
growth advantage; signicantly
k = 1.8, L = 160µm.
with some
which we shall call here the surng
with some small probability per division. The simula-
probability.
Surng is a softer version of xation - a notion from
tion nishes when either xation (all cells in the grow-
population genetics in which a mutant takes over the
ing layers becoming mutants) or extinction (no mutant
population.
The soft-sweep surng probability has
cells in the growing layer) is achieved. Before inserting
therefore a hard-selection-sweep counterpart, the x-
the mutant cell, the colony is simulated until the prop-
ation probability, which is the probability that the new
erties of the growing layer stabilize and both thickness
mutation spreads in the population so that eventually
and roughness reach steady-state values.
all cells have it.
The simu-
Both surng and xation probabili-
lation is then repeated many times and the probabil-
ties depend on the balance between selection (how well
ity of surng is estimated from the proportion of runs
the mutant grows compared to the parent strain) and
leading to xation of the mutant in the growing layer.
genetic drift (uctuations in the number of organisms
Snapshots showing dierent fates (extinction, surng)
due to randomness in reproduction events) [45]. In the
of mutant sectors are shown in Fig 5.
the
Surng probability depends on the position of
the cell in the growing layer. In Ref. [34] we showed
dierence between the growth rate of the mutant and
that the surng probability strongly depends on how
the parent strain. Here, we study how the properties
deeply in the growing layer a mutant was born. Here
Psurf
we would like to emphasize this result as it will become
previous work [34] we showed that
Psurf
increased ap-
proximately linearly with selective advantage
of the active layer aect
for a xed
s
s.
We rst run simulations in the planar-front geometry
important later. Let
∆
be the distance from the edge
in which a random cell picked up from the growing layer
of the colony to the place the mutant rst occurred.
of cells with probability proportional to its growth rate
Figure 6 shows the probability density
is replaced by a mutant cell with selective advantage
a cell was born a distance
s > 0.
∆
P (∆|surf)
that
behind the colony front,
given that it went on to surf on the edge of the expand-
This can be thought of as mutations occurring
6
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k= 1.6
k= 2.0
k= 2.4
2
1
0
0
1
2
Δ (µm)
(b) 4
P(Δ|surf)
0.08
Psurf
3
(a) 0.10
3
2
0
Figure 6:
1
(a)
P (∆|surf)
2
Δ (µm)
3
10
12
14
16
L =160
L =320
L =640
L =1280
0.08
4
0.06
0.04
0.02
L = 160µm, selective advank = 1.6, 2.0, 2.4. (b) P (∆|surf)
s = 0.02, and dierent
for L = 160µm, k = 2.0, and
s = 0, 0.02, 0.05, 0.1, 0.2, 0.5.
8
(b) 0.10
for
tage
L =160
L =320
L =640
L =1280
h (µm)
s=0.1
s=0.2
s=0.5
1
0
0.04
0.00
s=0.00
s=0.02
s=0.05
3
0.06
0.02
4
Psurf
P(Δ|surf)
(a)
0.00
2
dierent selective advantages
4
6
8
10
12
ρ (µm)
Only mutants from the rst
layer of cells have a signicant chance of surng.
Figure 7:
layer, for
ing colony. It is evident that only cells born extremely
(a) Psurf for dierent thickness h of the growing
s = 0.02 and L = 160, 320, 640, 1280 µm (dierent
colours). (b) the same data as a function of front roughness
ρ.
close to the frontier have a chance to surf. Cells born
Between
103
data point to
104 simulations were performed for each
estimate Psurf .
and
deeper must get past the cells in front of them. This
is unlikely to happen, even if the cell has a signicant
growth advantage, as the cell's growth will also tend to
how the surng probability
push forward the cells in front of it. This also justies
the thickness and the roughness of the front.
why we focus on 2d colonies; even though real colonies
creases with increasing thickness
are three-dimensional, all interesting dynamics occurs
increasing roughness
at the edge of the colony, made of a single layer of cells.
thickness and roughness are inversely correlated so this
Given that surng is restricted to the rst layer of
reciprocal behaviour is not surprising. An interesting
cells, and the distribution
P (∆|surf)
s),
h
Psurf
in-
and decreases with
We know from Fig.
4 that
question is which of the two quantities, roughness or
k
thickness, directly aects the probability of surng?
From a statistics point of view, thickness
it may seem to be a waste of computer time
to study the fate of mutants that occurred deeply in
be a better predictor of
the growing layer. To save the time, and to remove the
the same
Psurf
varies as a function of
is approximately
the same for all explored parameter sets (dierent
and
ρ.
Psurf
h
h
seems to
Psurf because data points for
L correlate better. How-
but for dierent
(thicker layer =
ever, it could be that it is actually front roughness that
lower overall probability), we changed the way of intro-
directly (in the causal sense) aects the surng proba-
ducing mutants. Instead of inserting mutants anywhere
bility and that
in the growing layer, we henceforth inserted them only
of the relationship between
eect the front thickness has on
at the frontier.
Psurf
and
h
are anti-correlated because
h
and
ρ.
We performed two computer experiments to address
Roughness of the front is more predictive of
than its thickness. Using the new method of
the above question. First, we simulated a colony that
Psurf
had a very low and constant roughness
introducing mutants (only the rst layer of cells), we
pendently of front's thickness.
s = 0.02 and for dierent widths L
and nutrient uptake rates k as in Fig. 4. Figure 7 shows
introducing an external force
run simulations for
centre of mass of each cell,
7
ρ ≈ 1,
inde-
This was achieved by
Fy = −gy acting on the
where g > 0 was a atten-
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peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
ing factor whose magnitude determined the strength of
suppression of deviations from a at front.
in Figure 8, left, as a function of
h
Psurf
plotted
for two cases: nor-
mal, rough front, and attened front, demonstrates
h
that the surng probability does not depend on
in
the case of at front.
Second, we varied roughness while keeping thickness
constant. This was done by measuring front roughness
in each simulation step, and switching on the external
(a)
Fy = −gy if the roughness was larger
value ρmax . Figure 8, right, shows that
attening force
than a desired
0.08
although thickness remains the same for all data points,
decreases with increasing roughness.
0.06
Psurf
Psurf
We can conclude from this that it is the increase
0.04
in the roughness, and not decreasing thickness, that
k ).
nutrient intake rate
normal
flattened
0.02
lowers the surng probability for thinner fronts (larger
However, the data points in
0.00
Fig. 7, right, from dierent simulations do not collapse
8
onto a single curve as it would be expected if average,
0.06
Psurf
motion in which the sector boundaries drift away from
each other with constant velocity. The velocity depends
0.04
whereas the amplitude of
0.02
random uctuations in the positions of boundary walls
is set by the microscopic dynamics at the front.
We
0.00
reasoned that these uctuations must depend on the
roughness
ρ
16
normal
flattened
0.08
be described by a random process similar to Brownian
s
14
(b)
According to the
theory of Ref. [29], the dynamics of a mutant sector can
on the growth advantage
12
h (µm)
large-scale front roughness was the only factor.
Local roughness predicts Psurf .
10
2
4
6
8
10
ρ (µm)
of the frontier, and that a mutant sector
should be aected by front roughness when the sec-
Figure 8:
tor is small compared to the magnitude of uctuations.
This means that local roughness
the length
l
ρ(l),
L = 320µm.
of the front, should be more important
than the global roughness
ρ(L).
We calculated the lo-
h for
g = 500), for
rate k = 1.6...2.8
as the function of front thickness
We vary the nutrient uptake
roughness
ρ
between
0.84
and
1.0
for all
k.
(b)
Psurf
for the
normal (black) and attened front (blue) as the function of
v
n u X
i+l
1 Xu
t1
ρ(l) =
(yj+ − Y + )2 .
n
l
i=1
Y
Psurf
to simulate fronts of dierent thickness. The at front has
cal roughness as
Here
(a)
the normal (black) and attened front (red,
determined over
roughness
(6)
2, 3.5, 5,
points at the leading edge, obtained as in Section 4.1.
Figure 9 shows that
Psurf
L now collapse
l ≈ 10 . . . 100µm over
for dierent
which roughness has been calculated.
Orientation of cells aects Psurf .
and
7,
for
k = 2.6;
the actual
very little from these values.
{yi+ }
are the vertical coordinates (interface height) of the
onto a single curve, for all lengths
The attened front has approximaly the same
10.0
10.3µm).
ρmax =
(measured) ρ diers
and
The points correspond to maximum roughness set to
j=i
+ is the average height of the interface and
ρ.
thickness for all data points (h between
So far we have
focused only on the macroscopic properties of the leading edge of the colony, completely neglecting its granular nature due to the presence of individual cells. Re-
8
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peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
4.3 Surng probability and the mechanical
properties of bacteria
call that in our model each cell is rod-shaped, and the
direction in which it grows is determined by the orientation of the rod.
Figure 10a shows that cells at
Our results from the previous section demonstrate that
the leading edge assume orientations slightly more par-
surng is aected by (i) the roughness of the growing
allel to the direction of growth (vertical) in the at-
layer, (ii) the orientation of cells, (iii) the thickness of
tened front than in the normal simulation. A natural
question is how does cellular alignment aects
the growing layer if mutations occur inside the growing
Psurf ,
layer and not only at its edge. To show this, we varied
independently of the roughness? To answer this ques-
thickness, roughness, and orientation of cells by using
ad hoc external forces attening out the front or forcing
tion, we simulated a modied model, in which external
torque
τ = −τmax sin[(φ − φpreferred ) mod π]
was ap-
the cells to order in a particular way.
plied to the cells, forcing them to align preferentially
we will investigate what parameters of the model aect
φpreferred . We investigated two forced
alignments: φpreferred = 0 corresponding to cells parallel to the x axis and hence to the growing edge of
the colony, and φpreferred = π/2 which corresponds to
in the direction
surng in the absence of such articial force elds.
Thickness of the growing layer.
h
thickness
can be determined from the parameters of
the model by a simple dimensional analysis.
growing edge).
ing that
Figure 10b compares these two dierent modes with
h
reaches bulk values [17], we can approximate
proximately the same thickness and roughness of the
s
growing layer. It is evident that the orientation of cells
∼ 3x
smaller
normal case, which in turn has
Assum-
is proportional to the characteristic scale
over which the nutrient concentration and cell density
previous simulations with no external torque, for ap-
forced cells have
If cells are pro-
hibited to form multiple layers, as in our 2d simulations,
the vertical orientation of cells (perpendicular to the
strongly aects the surng probability:
In this section
h≈
horizontally-
Psurf compared to the
Psurf ∼ 5x smaller than
where
a
vertically-forced cells.
E
E
(1/β − 1)3/4 ,
(ζ/a)φ
by
(7)
is the elastic modulus of the bacterium (Pa),
2
is the average area per cell (µm ),
coecient (Pa·h),
Shorter cells have higher Psurf than long cells.
h
φ
ζ
is the friction
is the replication rate (h
−1 ), and
β < 1 is a dimensionless ratio of the nutrient consumpPsurf , we tion rate to biomass production rate (i.e. new bacteria):
simulated cells whose maximal length was only 2µm
β = (kρ0 )/(φc0 ). Equation (7) shows that thickness h
and the minimal separation before the spherical caps
increases with increasing cell stiness (larger E ) and
was zero, i.e., the cells became circles immediately afreplication rate φ, and decreases with increasing nutriter division. As before we selected a set of k 's such
ent uptake k and increasing friction ζ . The aspect ratio
that the thickness and roughness were approximately
of the cells does not aect h in our model. Equation
To check how the aspect ratio of cells aect
the same for all simulations.
In order to make a fair
(7) suggests that the thickness of the growing layer can
comparison between short rods and long rods from
be conveniently controlled in an experiment by varying
previous simulations, thickness and roughness were expressed in cell lengths rather than in
done by dividing both
µm.
temperature or growth medium (which both aect the
This was
growth rate), or by varying the nutrient concentration
h and ρ by the average length of c .
0
a cell measured for cells from the growing layer. Figure
We shall use the rst two methods when discussing
the experimental verication of our theory.
Orientation of cells.
10c show that short rods have a much higher surng
probability than long rods.
A useful measure of the global
alignment of cells in the colony is the order param-
In all previous simulations, even for short rods, cells
eter
2
cos (φ − Φ) . Here φ is the angle a cell
with the x-axis and Φ is the angular coordinate
S =
remembered their orientation from before division and
makes
growth always initially occurred in that direction. To
of the vector normal to the front; this is to remove a
see whether this has any impact on
Psurf ,
we consid-
trivial contribution to
S
due to the curvature of the
ered a scenario in which the new direction of growth
front caused by roughness.
is selected randomly and does not correlate with the
tion,
direction prior to division. Figure 10c shows that
Psurf
S = 1
According to this deni-
if all cells are perfectly vertically aligned
(in the direction of growth),
S = 0 if they
S = 1/2 if
are hori-
almost does not change regardless whether a short cell
zontal (parallel to the front), and
randomly changes its orientation after division or not.
entations are random. It turns out that changing the
9
their ori-
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peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
Figure 9: Psurf
(b)
0.10
0.08
0.06
0.04
0.02
ℓ=35
0.00
1.0 1.5
(c)
Psurf
0.10
0.08
0.06
0.04
0.02
ℓ=10
0.00
0.8 0.9 1.0 1.1 1.2 1.3
ρ(ℓ) (µm)
Psurf
Psurf
(a)
2.0 2.5
ρ(ℓ) (µm)
3.0
0.10
0.08
0.06
0.04
0.02
ℓ =98
0.00
2
L= 160
L= 320
L =640
L =1280
3
4
5
ρ(ℓ) (µm)
6
ρ(l) of the growing layer, for dierent sizes L = 160, 320, 640, 1280 µm
l = 35, right: l = 98 µm. For each l, data points for dierent L collapse
as the function of local roughness
(as in Fig. 7) and
s = 0.02.
Left:
l = 10,
middle:
onto a single curve.
(a)
flatten, g = 500
normal
vertical
horizontal
(b) P
surf
0.35
0.30
0.25
0.20
0.15
0.10
(c)
h (µm)
12
8
4
0
0.14
0.12
0.10
ρ (µm)
4
0.08
2
0.06
0
0.04
0.05
h (cells)
6
4
2
0
ρ (cells)
4
2
0
0.02
no forced
alignment
Figure 10:
Psurf
horizontal
alignment
vertical
alignment
long rods
short rods
short rods
randomized
(a) Orientation of cells (colours as in the circle in the upper-right corner) in the growing layer for dierent
models. (b, c) Comparison of xation probabilities for dierent cellular alignments at the front, for approximately the
same thickness and roughness, both of which were controlled by varying
k.
k needed to be used
L = 320µm, s = 0.02. For horizontally2µm; upon division, they become circles
To achieve this, dierent
in panels (b, c) and hence the two panels cannot be directly compared. In all cases
and vertically-forced cells,
of diameter
τmax = 10000.
Short cells have a maximum length of
1µm.
10
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peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
uptake rate (and hence thickness h) from k = 1.6 to
k = 2.8 changes S by a small amount from S = 0.77 to
S = 0.70. Here we are more interested in other factors
that do not aect h.
Friction.
A=1
One such factor is the nature of friction
between cells and the substrate.
A=4
So far, in all simu-
lations the friction force was proportional to the cell's
velocity, irrespective of the direction of motion. To test
whether this assumption aected front roughness and
A=1/ 3
the surng probability, we ran simulations in which friction coecients were dierent in the directions parallel
and perpendicular to the cell's axis. We replaced Eq.
(1) for the dynamics of the centre of mass with the
following equation:
Snaphots of a growing colony with dierent
friction anisotropy.
d~ri
= K −1 F~ /m,
dt
where the matrix
Figure 11:
K
(8)
The global order parameter
A = 1), S = 0.53
S = 0.63 (sliding rods A = 1/3).
(isotropic friction
and
S = 0.79
A = 4),
(rolling rods
accounts for the anisotropy of fric-
tion:
K=
ζk n2x + ζ⊥ n2y (ζk − ζ⊥ )nx ny
(ζk − ζ⊥ )nx ny ζ⊥ n2x + ζk n2y
We now have two friction coecients:
ζ⊥
.
(9)
(a) 12
is the co-
10
jor axis
~n,
whereas
lel direction.
ζk
ρ (µm)
ecient in the direction perpendicular to cell's mais the coecient in the paral-
For convenience, we shall assume that
ζk = Aζ, ζ⊥ = ζ/A
cient and ζ is the
where
A
is the asymmetry coef-
isotropic friction coecient, same
Psurf
> 1), cells are signicantly more oriented
edge-on to the colony, and the roughness is noticeably
< 1) the roughness
the isotropic and the rolling rods case. This is quanti-
ρ
2.0
2.6
2.8
A=
=1
1
A=
=2
2
A=
=4
4
A =1/3
0.04
0.00
as a function of
2
4
6
8
10
12
ρ (µm)
surng probability goes down with increasing rough-
5
2.4
0.06
The same gure, right, shows that, as expected, the
ness.
2.2
0.02
is even larger but the orientation of cells falls between
k.
1.8
0.08
In the anisotropic rolling
ed in Fig. 12, left where we plotted
A =1/3
(b)
Figure 11 shows images of the front for dierent lev-
larger. In the sliding rods case (A
A=4
k
easier for the rod to slide.
rods case (A
A=2
0
1.6
hence
els of friction anisotropy.
A=1
2
ζ⊥ = ζk ≡ ζ and K = 1ζ , and we
recover Eq. (1). If A > 1, it is easier for the rod to
roll than to slide along the major axis. If A < 1 it is
A = 1,
6
4
as in previous simulations (Table 1). For isotropic friction,
8
Figure 12:
(a) Roughness
ρ as the function of k , for dier-
ent levels of friction anisotropy: no anisotropy (black points,
Comparison with experiments
A = 1), rolling rods A = 2 (red), A = 4 (orange), and sliding rods A = 1/3 (blue). (b) surng probability versus ρ
We next checked whether the predicted dependence of
for the same parameters as in the left panel.
the surng probability on the roughness of the growing layer agree with experiments. We measured surng
11
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peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
(a)
Figure 13: (a)
(b) i
ii
iii
(c) i
ii
iii
An example of a S. cerevisiae colony with benecial mutants (yellow) forming sectors. The mutants have
a growth rate advantage of
s ≈ 10%. (b,c)
Fate of mutant cells - experimental counterpart of Fig. 5. Colonies of E. coli
(b) and S. cerevisiae (c) were inoculated using a mixture of a majority of wild-type cells (blue, false colour) and a small
number of mutant cells (yellow) with
s = 8%
(left and middle). Some mutant clones formed large sectors (left), while
others (middle) lagged behind the front, became engulfed by wild-type cells and eventually ceased to grow ("bubbles").
A large growth advantage (s
≈ 16%,
right) caused the sector to bulge out. All three phenomena are well reproduced by
our simulations (c.f. Fig. 5). In all panels, scale bar = 2mm.
●
Psurf
0.8
■ E. coli DH5α 21°C
■ E. coli DH5α 37°C
■ E. coli MG1655
0.6
0.4
0.2
0.0
ρ2(l) (mm2)
(c)
0.04
0.03
0.02
0.01
0.00
Figure 14:
-5
●
●■
■
●
■
●
● ●
■■
●
●
●
■
■ ■
■
■ 0.6
0.4
■ 0.2
0.0
0 5 10 15 20 25
■
■
■
■
■
■ ■
■
■
■■
■
■■
■■ ■■■■■
■
■■■■
0
■
(b)
5
10
15
20
window length l (mm)
l
δr
r
●
0
5
10 15 20 25
selective advantage s (%)
■
25
(d) 100
s=14%
s=9%
s=4%
10-1
10-2
10-3
S. cerevisiae
E. coli
10
30 50 100
roughness ρ(l) (cells)
300
Surng probability versus roughness in experimental colonies. In all panels squares and circles correspond
to E. coli and S. cerevisiae, respectively.
function of the selective advantage
s.
was measured (Methods).
(a)
Psurf for dierent species and growth conditions as a
Psurf at low s, while Psurf of E. coli strain DH5α at
cerevisiae for s > 15%. (b) Diagram illustraing how roughness
Surng probability
S. cerevisiae has a much higher
21C increases faster than linearly for large
ρ(l)
■
S. cerevisiae (YPD)
S. cerevisiae (CSM)
surfing probability Psurf
●
(a) 1.0
(%)
(c) ρ2 (l)
s,
surpassing S.
for dierent conditions (colours as in (a), error bars are standard errors of the
mean over at least 10 colonies per condition). Solid lines are linear ts to the data points. The dotted line corresponds to
the window length
l = 17mm
used to calculate roughness in panel (d). The inset shows
blue), which has the highest roughness.
(d)
Surng probability versus
ρ(l = 17mm),
ρ2 (l)
for E. coli MG1655 (dark
for dierent
s.
To compare E. coli
and S. cerevisiae, we normalized roughness by the cell size (square root of the average area), which we estimated from
microscopy images to be 2 and 4.7µm, respectively.
12
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peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
σ2(l) (cells2)
 S. cerevisiae
■ E. coli
0.2
■
■
■
■■
■
  



■
■

■
■
■
■
0.0 ■■■■■■■■
0.1

■

(e) 1.0

0 10 20 30 40 50 60
window length l (cell diameters)
5μm
(d) 60
10μm
50
40
30
20
10
0
E. coli
0
Microscopic properties of the growing layer.
(d)
(c)
(a,b)
0.8
0.6
0.4
0.2
E. coli
0.0
0
5
10 15
distance from front (μm)
1.0
0.8
0.6
0.4
0.2
Simulation
0.0
0
5 10 15 20
distance from front (μm)
1
2
3
4
distance from front (μm)
a S. cerevisiae front (panel b, scalebar 10µm) front.
lines are ts to the data points.
(f)
‹cos2(φ)›
clone size
(b)
Figure 15:
■
■
‹cos2(φ)›
(c) 0.3
(a)
Snapshot of an E. coli front (panel a, scalebar 5µm) and
Local roughness
ρ2 (l)
as a function of the window size l. Dashed
The number of ospring for all initial cells near the front, for E. coli. Only cells within
2-3µm (∼ one cell) from the edge of the colony have a signicant number of ospring.
order parameter
2 cos φ
(e)
Probability density plot of the
for E. coli as a function of the distance from the edge. Blue = low probability, yellow = high
probability. The dotted line is the average order parameter versus the distance from the front. Cells are preferentially
(f )
aligned with the direction of propagation, except for cells directly at the front, which are parallel to it.
of the order parameter for a simulated front with
probabilities of benecial mutants with dierent selective advantages
and
s = −5 . . . 25%
S. cerevisiae
Density plot
k = 1.4, L = 320µm.
in colonies of
E. coli
(Methods) grown at dierent condi-
tions aecting the roughness of the growing layer.
A
small number of uorescently labeled mutant cells was
and we shall experimentally validate it later in this section. Fig. 14a shows
Psurf
for
E. coli and S. cerevisiae,
and for dierent conditions. In the limit of low selective
advantage
s < 10%
we are interested here, the surng
probability is highest in colonies of roughly-spherical
S.
and a small droplet of the mixture was used to inoculate
cerevisiae, which have rather smooth boundaries, and
smallest for the rod-shaped bacterium E. coli, which
a colony on a Petri dish. After a few days, colonies with
are characterized by rough front. This agrees with our
a characteristic sectoring pattern emerged (Fig. 13).
predictions (Fig.
By zooming into the colony edge we conrmed that
whether this is due to dierence in the cell shape or
some mutants surfed at the front and expanded into
dierent thickness/roughness of the growing layer.
mixed with a much larger number of wild-type cells,
large sectors whereas some mutants did not make it and
10), however it does not yet show
To study the connection between surng and surface
became trapped as bubbles in the bulk of the colony
roughness, we computed the local roughness
(Fig. 13, compare with Fig. 5).
function of window length
We counted the number of sectors and estimated the
surng probability
Psurf
from the formula [34]:
Psurf =
where
Pi
Nsec
,
2πr0 Pi
calculated
r0
Psurf
(Fig. 14a). In all cases,
linear dependence on window length
(10)
ρ(l)
as a
(Fig. 14b, cf. Eq. (6) and
Methods) for the same colonies for which we previously
l
ρ2 (l)
showed a
after a transient
at small window lengths, i.e., the colony boundary behaved like a standard random walk (Fig. 14c).
We then tested the correlation of colony roughness
is the initial fraction of mutant cells in the
population and
l
the initial radius of the colony (in
with surng probability in a similar way to what we did
units of cell diameters). Note this equation makes sense
in computer simulations. In Fig. 14d, we plot the surf-
only if surng is restricted to the rst layer of cells; we
ing probability
have shown that this is true in computer simulations
measured at one specic window length
13
Psurf
as a function of colony roughness
l = 17mm (dot-
bioRxiv preprint first posted online Jan. 13, 2017; doi: http://dx.doi.org/10.1101/100099. The copyright holder for this preprint (which was not
peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
ted line in Fig. 14c), for dierent selective advantages
tation surfs at the growing edge of a microbial colony,
s.
depends mostly on the thickness and roughness of the
We observe that the surng probability of
E. coli
decreases with increasing roughness (Fig. 14d) for all
growing layer of cells at colony's front. Thicker fronts
s,
in good qualitative agreement with our simulations.
decrease the per-cell surng probability because only
Similar results are obtained for dierent choices of the
cells from the very rst layer of cells create success-
window length
l
for which roughness is calculated. The
situation is less clear for
S. cerevisiae ; we hypothesize
ful progenies, and the fraction of such cells decreases
with increasing front thickness.
Rougher fronts also
that this is due to roughness being too small (c.f. Fig.
decrease the surng probability for a similar reason;
9) to markedly aect the surng probability.
only cells at the tips of the bumps are successful and
We next examined how microscopic properties of
these tips become smaller for rougher fronts.
More-
the front (cellular orientation) correlated with macro-
over, roughness and thickness are related; thicker front
scopic roughness. We analysed microscopic images of
have lower roughness and vice versa. While the depen-
the fronts of
E. coli
S. cerevisiae
fronts (Meth-
dence between genetic segregation and the front thick-
ods, data from Ref. [34]), and measured local roughness
ness [46], and between thickness and roughness [47] has
ρ(l)
Example snapshots
been known previously, in this work we have shown that
in Fig. 15a,b show that roughness of the fronts indeed
it is actually the roughness of the growing layer that
dier very much for these two microorganisms. Figure
should be thought of as aecting the surng probabil-
15c conrms that
ity in the causal sense. We have also linked thickness
and
over sub-mm length scales
l.
E. coli has a much higher roughness
compared to S. cerevisiae, suggesting that macroscopic
and roughness to the mechanical properties of cells for
roughness on the colony scale is a consequence of mi-
the rst time. Moreover, we have discovered that the
croscopic front roughness on the single-cell level.
orientation of cells has also a signicant eect, irrespec-
To study the dynamics of surng, we tracked
E.
coli cells over 200 minutes and measured their distance
tive of front roughness, on the surng probability.
All these quantities (roughness, thickness, cellular
from, and orientation relative to the edge of the colony,
alignment) are controlled in a very non-trivial way by
as well as the number of ospring for all cells in the
the properties of cells and their environment:
initial image.
cell-
15d shows that cells only have
surface friction (and anisotropy of thereof ), elasticity
an appreciable number of ospring if they are within
of cells, their growth/nutrient uptake rate, and their
about one cell diameter of the front. This agrees with
shape.
our conclusion from simulations and justies inserting
cult to vary experimentally, we managed to show in
mutants only directly at the front.
simple experiments that the growth rate and the shape
15e
Figure
2
cos (φ − Φ) ,
Figure
shows
the
order
parameter
S
=
which measures the orientation of cells
While some of these parameters are very dif-
of cells aect the surng probability in the way predicted by our simulations.
Microbial evolution is a research area that is impor-
and has been dened in Sec. 4.3, as a function of the
Cells near the front tend to
tant both from fundamental and practical viewpoints.
align parallel to the front. This changes quickly behind
In particular, our research shows that mechanical forces
the front, with most cells being perpendicular to the
such as adhesion, friction, etc., can play a signicant
growth direction starting about 5µm behind the front.
role in biological evolution of microorganisms. To our
distance from the front.
obtained from
knowledge, this article is the rst that not only puts
simulations; the agreement with the experimental data
forward this idea but also provides concrete arguments
from Fig. 15e is excellent, suggesting that our model
in its support.
Figure 15f shows the distribution of
S
From a more practical point of view, our results are
indeed captures the dynamics of the growing bacterial
relevant to the evolution of antimicrobial resistance.
front reasonably well.
It has been demonstrated that even a small bacterial
6
population can develop
Conclusions
de novo resistance to some an-
timicrobial drugs in less than a day [48].
This rapid
evolution makes the most popular drugs - antibiotics In this work we have focused on the role of mechanical
increasingly ineective [49].
interactions in microbial colonies. We rst used com-
ery of new antibiotics has steadily declined over years
puter simulations to show that the speed of biological
[50], the evolution of drug-resistant bacteria has been
evolution, measured by the probability that a new mu-
highlighted as one of the major challenges we will face
14
Since the rate of discov-
bioRxiv preprint first posted online Jan. 13, 2017; doi: http://dx.doi.org/10.1101/100099. The copyright holder for this preprint (which was not
peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
in the coming decades.
genetic bases of adaptation.
By demonstrating the role of
netics, 4(6):457469, 2003.
mechanical interactions on biological evolution in microbial aggregates, our research opens up a new antimi-
[7] G. G. Perron, A. Gonzalez, and A. Buckling. The
crobial paradigm in which the physical properties of
rate of environmental change drives adaptation to
Journal of Evolutionary Biology, 21(6):17241731, 2008.
microbes could be targeted alongside standard antimi-
an antibiotic sink.
crobial therapy to reduce the probability of evolving
resistance to drugs.
[8] B. Carpentier and O. Cerf. Biolms and their con-
Acknowledgments
sequences, with particular reference to hygiene in
the food industry.
the Royal Society of Edinburgh (B.W.), National Insti-
[9] J. W. Costerton, P. S. Stewart, and E. P. Green-
tute of General Medical Sciences of the National Insti-
berg. Bacterial Biolms: A Common Cause of Per-
tutes of Health under Award Number R01GM115851
by
a
National
Science
Foundation
sistent Infections.
Career
The content is solely
[10] D. Berry, C. Xi, and L. Raskin. Microbial ecology
Current
Opinion in Biotechnology, 17(3):297302, 2006.
the responsibility of the authors and does not necessar-
of drinking water distribution systems.
ily represent the ocial views of the National Institutes
of Health.
[11] R. L. Gibson,
sey.
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