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arXiv:1111.2992v1 [q-bio.PE] 13 Nov 2011

Reproduction time statistics and segregation patterns in growing populations

Adnan Ali,1Stefan Grosskinsky,1,2and Ell´ ak Somfai1,3

1Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom

2Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

3Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom

(Dated: November 15, 2011)

Pattern formation in microbial colonies of competing strains under purely space-limited population

growth has recently attracted considerable research interest. We show that the reproduction time

statistics of individuals has a significant impact on the sectoring patterns. Generalizing the standard

Eden growth model, we introduce a simple one-parameter family of reproduction time distributions

indexed by the variation coefficient δ ∈ [0,1], which includes deterministic (δ = 0) and memory-

less exponential distribution (δ = 1) as extreme cases. We present convincing numerical evidence

and heuristic arguments that the generalized model is still in the KPZ universality class, and the

changes in patterns are due to changing prefactors in the scaling relations, which we are able to

predict quantitatively. At the example of S. cerevisiae, we show that our approach using the variation

coefficient also works for more realistic reproduction time distributions.

PACS numbers: 87.18.Hf, 89.75.Da, 05.40.-a, 61.43.Hv

I.INTRODUCTION

Spatial competition is a common phenomenon in

growth processes and can lead to interesting collective

phenomena such as fractal geometries and pattern for-

mation [1–3]. This is relevant in biological contexts such

as range expansions of biological species [4, 5] or growth

of cells or microorganisms, as well as in social contexts

such as the dynamics of human settlements or urbaniza-

tion [6]. These phenomena often exhibit universal fea-

tures which do not depend on the details of the particu-

lar application, and have been studied extensively in the

physics literature [2, 3, 7–10].

Our main motivating example will be growth of mi-

crobes in two dimensional geometries, for which recently

there have been several quantitative studies. In general,

the growth patterns in this area are influenced by many

factors, such as size, shape and motility of the individual

organism [11], as well as environmental conditions such

as distribution of resources and geometric constraints

[12, 13], which in turn influence the proliferation rate or

motility of organisms [14]. We will focus on cases where

active motion of the individuals can be neglected on the

timescale of growth, which leads to static patterns and is

also a relevant regime for range expansions. We further

assume that there is no shortage of resources, and growth

and competition of species is purely space limited and

spatially homogeneous. This situation can be studied

for microbial colonies grown under precisely controlled

conditions on petri dish with hard agar and rich growth

medium. Under these conditions one expects the colony

to form compact Eden-type clusters [12], which has re-

cently been shown for various species including S. cere-

visiae, E. coli, B. subtilis and S. marcescens [14, 15].

The Eden model [16] has been introduced as a basic

model for the growth of cell colonies. It has later been

shown to be in the KPZ universality class [3, 7, 17, 18],

which describes the scaling properties of a large generic

class of growth models.

E. coli and S. cervisiae [14, 15, 19, 20] quantitative evi-

dence for the KPZ scaling of growth patterns has been

identified. The models used in these studies ignored all

microscopic details reproduction, such as anisotropy of

cells [21], and could therefore not explain or predict dif-

ferences observed for different species. Nevertheless, they

provided a good reproduction of the basic features such as

KPZ exponents, which is a clear indication that segrega-

tion itself is an emergent phenomenon. Fig. 1 shows dif-

ferences in growth patterns in a circular geometry taken

from [15] for E. coli and S. cervisiae. For both species

the microbial populations are made of two strains, which

are identical except having different fluorescent labeling.

Reproduction is asexual, and the fluorescent label car-

ries over to the offspring. At the beginning of the ex-

periments the strains are well mixed, but during growth

rough sector shaped segregated regions develop.

qualitative emergence of these segregation patterns and

connections to annihilating diffusions has been studied in

[15, 19, 20, 22], ignoring all details specific for a particular

species.

In recent detailed studies of

The

For S. cervisiae the domain boundaries are less rough

when compared to E. coli, leading to a finer pattern con-

sisting of a larger number of sectors. In general, this

is a consequence of differences in the mode of reproduc-

tion and shapes of the microbes, which introduce local

correlations that are not present in simplified models.

In this paper we focus on the effect of time correlations

introduced by reproduction times that are not exponen-

tially distributed (as would be in continuous time Marko-

vian simulations), but have a unimodal distribution with

smaller variation coefficient. This is very relevant in most

biological applications (see e.g.

spatially isotropic systems the resulting temporal corre-

lations lead to more regular growth and therefore smaller

fluctuations of the boundaries, with an effect on the pat-

terns as seen in Fig. 2.

[23–25]), and even in

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(a) (b)

FIG. 1: (Color online) Fluorescent images of colonies of (a)

E. coli and (b) S. cerevisiae. The scaling properties of both

patterns are believed to be in the KPZ universality class, and

the differences are due to microscopic details of the mode of

reproduction and shape of the micro-organisms. The images

have been taken with permission from [15], copyrighted (2007)

by the National Academy of Sciences, U.S.A.

To systematically study these temporal correlations,

we introduce a generic one-parameter family of reproduc-

tion times, explained in detail in Section II. We establish

that the growth clusters and competition interfaces still

show the characteristic scaling within the KPZ univer-

sality class, and the effect of the variation coefficient is

present only in prefactors. We predict these effects quan-

titatively and find good agreement with simulation data;

these results are presented in Section III. More realistic

reproduction time distributions with a higher number of

parameters are considered in Section IV, where we show

that to a good approximation the effects can be sum-

marized in the variation coefficient and mapped quan-

titatively onto our generic one-parameter family of re-

production times. Therefore, our results are expected to

hold quite generally for unimodal reproduction time dis-

tributions, and the variation coefficient alone determines

the leading order statistics of competition patterns.

II. THE MODEL

For regular reproduction times with small variation co-

efficient the use of a regular lattice would lead to strong

lattice effects that affect the shape of the growing clus-

ter. To avoid these we use a more realistic Eden growth

model in a continuous domain in R2with individuals

modelled as circular hard-core particles with diameter

1, since we want to study purely the effect of time cor-

relations. This leads to generalized Eden clusters which

are compact with an interface that is rough due to the

stochastic growth dynamics.

Let B(t) denote the general index set of particles p

at time t, (xp,yp) ∈ R2is the position of the centre of

particle p, and sp∈ {1,2} is its type. We write B(t) =

B1(t)∪B2(t) as the union of the sets of particles of type

1 and 2. We also associate with each particle the time

it tries to reproduce next, Tp> 0. Initially, Tpare i.i.d.

random variables with distribution Fδ with parameter

(a)(b)

FIG. 2: (Color online) A smaller variation coefficient δ in

reproduction times (see (4) and (6)) leads to more regu-

lar growth, smoother domain boundaries and finer sectors.

Shown are simulated circular populations with (a) δ = 1 and

(b) δ = 0.1. Both colonies have an initial radius of r0 = 50,

and they are grown up to simulation time t = 50 leading to

final radii of approximately 120 (a) and 95 (b).

δ ∈ [0,1], which is explained in detail below. After each

reproduction Tp is incremented by a new waiting time

drawn from the same distribution. Note that we focus

entirely on the neutral case, i.e. the reproduction time

is independent of the type and both types have the same

fitness. We describe the dynamics below in a recursive

way.

Following a successful reproduction event of particle p

at time Tp, a new particle with index q = |B(Tp−)| + 1

is added with the same type sq= sp,

Bsp(Tp+) = Bsp(Tp−) ∪ {q}

Here B(T−) and B(T+) denote the index set just before

and just after time T. The position of the new particle

is given by

(1)

(xq,yq) = (xp,yp) + (cosφ,sinφ) , (2)

where φ ∈ [0,2π) is drawn uniformly at random. This

is subject to a hard-core exclusion condition for circular

particles, i.e. the euclidean distance to all other particle

centres has to be at least 1, as well as to other constraints

depending on the simulated geometry as explained be-

low. Note that in our model the daughter cell touches

its mother, which is often realistic but in fact not essen-

tial, and the distance could also vary stochastically over

a small range. The new reproduction times of mother

and daughter are set as

Tp?→ Told

p

+ T ,Tq= Told

p

+ T′, (3)

where T,T′are i.i.d. reproduction time intervals with

distribution Fδ. There can be variations on this where

mother and daughter have different reproduction times,

which are discussed in Section IV. The next reproduc-

tion event will then be attempted at t = min?Tq : q ∈

B(Tp+)?. Reproduction attempts can be unsuccessful, if

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there is no available space for the offspring due to block-

age by other particles. In this case the attempt is aban-

doned and Tpis set to ∞, as due to the immobile nature

of the model this particle will never be able to reproduce.

The initial conditions for spatial coordinates and types

depend on the situation that is modelled. In this paper

we mostly focus on an upward growth in a strip of length

L with periodic boundary conditions on the sides, where

we take B(0) = {1,...,L} with (xp,yp) = (p,0), for all

p ∈ B(0). The initial distribution of types can be either

regular or random depending on whether we study single

or interacting boundaries, and will be specified later.

In Section III for our main results we use reproduction

times

T ∼ 1 − δ + Exp(1/δ) ,δ ∈ (0,1] ,(4)

i.e. T has exponential distribution with a time lag 1 −

δ ∈ [0,1) and with mean fixed to ?T? = 1 for all δ.

The corresponding cumulative distribution function Fδ

is given by

Fδ(t) =

?

0 , t ≤ 1 − δ

1 − e−(t−1+δ)/δ, t ≥ 1 − δ

. (5)

The variation coefficient of this distribution is given by

the standard deviation divided by the mean, which turns

out to be just

??T2? − ?T?2

?T?

=δ

1= δ . (6)

With this family we can therefore study reproduction

which is more regular then exponential with a fixed av-

erage growth rate of unity (equivalent of setting the unit

of time).

For δ = 1 this is a standard Eden cluster, but δ < 1

introduces time correlations. While the correlations af-

fect the fluctuations, we present convincing evidence that

they decay fast enough not to change the fluctuation ex-

ponents, so the system remains in the KPZ universality

class. Furthermore we make quantitative predictions on

the δ-dependence of non-universal parameters and com-

pare them to simulations. The more synchronized growth

leads to effects similar to the ones seen in experiments

(Fig. 1).To give a visual impression of the patterns

produced by the model, we show in Fig. 2 two growth

patterns with δ = 1 and 0.1. The initial condition is a

circle, and the types are distributed uniformly at random.

The patterns are qualitatively similar to the experimen-

tal ones in Fig. 1, and more regular growth leads to a

finer sector structure. The same effect is shown on Fig. 3

for the simulations in a linear geometry with periodic

boundary conditions, which is analyzed quantitatively in

the next Section. Smaller values of δ also lead to more

compact growth and smaller height values reached in the

same time.

(a)(b)

FIG. 3: (Color online) Populations in a linear geometry with

periodic boundary conditions in lateral direction with (a) δ =

1 and (b) δ = 0.1. Both populations have lateral width L =

300, and the colonies are grown to a simulation time t ≈ 50,

leading to heights of approximately 70 (a) and 40 (b).

III. MAIN RESULTS

A. Quantitative analysis of the colony surface

In this Section we provide a detailed quantitative anal-

ysis of the δ family of models in linear geometry with

periodic boundary conditions (see Fig. 3), starting with

the dynamical scaling properties of the growth interface.

We regularize the surface to be able to define it as a

function of the lateral coordinate x and time t as

y(x,t) := max?yp: p ∈ B(t),|xp− x| ≤ 1?.

In case of overhangs (which are very rare) we take the

largest possible height, and due to the discrete nature of

our model this leads to a piecewise constant function.

The surface of a standard Eden growth cluster is known

to be in the KPZ universality class [16, 18], i.e. a suitable

scaling limit of y(x,t) with vanishing particle diameter

fulfills the KPZ equation

(7)

∂ty(x,t) = v0+ ν∆y(x,t) +λ

2(∇y(x,t))2+

√Dη(x,t).

(8)

Here v0, of the order of unity, corresponds to the growth

rate of the initial flat surface (related to the mean repro-

duction rate and some geometrical effects), the surface

tension term with ν > 0 represents surface relaxation,

and the nonlinear term represents the lowest order con-

tribution to lateral growth [18]. The fluctuations due

to stochastic growth are described by space-time white

noise η(x,t), which is a mean 0 Gaussian process with

correlations

?η(x,t)η(x′,t′)?= δ(x − x′)δ(t − t′).

We denote the average surface height by

(9)

h(t) :=1

L

?L

0

y(x,t)dx ,(10)

which is a monotone increasing function in t. It is also

asymptotically linear and therefore we will later also use

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10

−5

10

−4

10

−3

10

−2

10

−1

10

−2

10

−1

t/Lz

S(t)/Lα

δ=1

δ=0.8

δ=0.6

δ=0.4

δ=0.2

FIG. 4: (Color online) Family-Vicsek scaling (12) of the sur-

face roughness S(t). The data collapse under rescaling with

α = 1/2 and z = 3/2 occurs in a scaling window which is

narrower for small δ due to intrinsic correlations. The differ-

ent symbols correspond to different values of δ, and the color

represents system size, L = 1500 (blue) and L = 4000 (red).

The dashed lines indicate the expected slope β = 1/3. The

data for L = 1500 has been averaged over 100 samples and

for L = 4000 over 30 samples. The error bars are comparable

to the size of the symbols.

h as a proxy for time. The δ-dependence of the average

growth velocity of height as seen in Fig. 3 does not lead

to leading order contributions to the statistical properties

of the surface or the structure of sectoring patterns.

The roughness of the surface is given by the root mean

squared displacement of the surface height as a function

of t [3, 10], defined as

S(t) =

?1

L

?L

0

?y(x,t) − h(t)?2dx

?1/2

.(11)

The main properties of the surface y(x,t) can then be

characterized by the Family-Vicsek scaling relation of the

roughness

S(t) = Lαf(t/Lz) , (12)

where the scaling function f(u) has the property

f(u) ∝

?uβ

u ≪ 1

u ≫ 1.

1

(13)

Such a scaling behaviour has been shown for many dis-

crete models including ballistic deposition and continuum

growth [3, 10, 18, 26, 27], and holds also for other uni-

versality classes such as Edward Wilkinson. For the KPZ

class in 1+1 dimensions the saturated interface roughness

exponent is α = 1/2, the growth exponent is β = 1/3,

and the dynamic exponent is z = α/β = 3/2.

Fig. 4 shows a data collapse for the roughness S(t) for

two system sizes, and for a number of different values

of δ. As δ gets smaller, the surface becomes less rough

10

1

10

2

10

3

10

0

10

1

l

C(l,t)

δ=1

δ=0.8

δ=0.6

δ=0.4

δ=0.2

FIG. 5: (Color online) The height-height correlation function

C(l,t) for L = 4000 at t = 11000 for various values of δ. The

data has been averaged over 30 samples, and the error bars

are comparable to the size of the symbols. The dashed lines

indicate the expected slope 1/2.

due to a more synchronized growth. The dashed lines

indicate the power law growth with exponent β = 1/3 in

the scaling window. This window ends around t/Lz≈ 1

due to finite size effects, where the lateral correlation

length reaches the system size and the surface fluctua-

tions saturate. For small t the system exhibits a transient

behaviour before entering the KPZ scaling due to local

correlations resulting from the non-zero particle size and

stochastic growth rules. As we quantify later, these cor-

relations are much higher for more synchronized growth

at small δ, which leads to a significant increase in the

transient regime. The transient time scale is indepen-

dent of system size and vanishes in the scaling limit, so

that the length of the KPZ scaling window increases with

L. This behaviour can be observed in Fig. 4 where for

the smallest value δ = 0.2 the scaling regime is still hard

to identify for the accessible system sizes.

Another characteristic quantity is the height-height

correlation function defined as [3, 28, 29]

C(l,t) =

?1

L

?L

0

(y(x,t) − y(x + l,t))2dx

?1/2

. (14)

For a KPZ surface in 1 + 1 dimensions this obeys the

scaling behaviour

C(l,t) ∼

?

(D

?D

2νl)1/2

?2/3(λt)1/3

l ≪ ξ?(t)

l ≫ ξ?(t)

2ν

, (15)

where ξ?(t) is defined to be the lateral correlation length

scale and takes the form [3, 29, 30]

ξ?(t) ∼ (D/2ν)1/3(λt)2/3. (16)

A detailed computation can be found in Appendix B.

Initially, the behaviour of C(l,t) grows as a power-law

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0.20.30.40.50.6

δ

0.70.80.91

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

D/2ν

Data

0.5299(δ2+0.4194)2

FIG. 6: (Color online) Dependence of the KPZ parameters

D/(2ν) on δ. Data are obtained from (15) by fitting the pref-

actor of the power law in Fig. 5, using that the proportionality

constant is very close to 1 (cf. derivation (B11) in the Ap-

pendix). The data are in good agreement with the prediction

(21) with fitted parameters ǫ ≈ 0.42 and D/(2ν)(δ = 1) ≈ 1.1.

with l, and when l exceeds the lateral correlation length

ξ?(t) it reaches a value that depends on the time t and the

parameters of (8). This is shown in Fig. 5, where C(l,t)

is plotted for various values of δ, and the data agree well

with the exponent α = 1/2 for the KPZ class indicated

by dashed lines.

The time correlations introduced by the partial syn-

chronization can be estimated by considering a chain of

N growth events where each particle is the direct de-

scendant of the previous one. Each added particle corre-

sponds to a height change ∆yi, and has an associated

waiting time Ti with distribution (4).

there are N(t) growth events, and since the average re-

production time is 1 with variance δ2, we have ?N(t)? ≈ t

and var(N(t)) ≈ δ2t. The height of the last particle is

yN(t)=?N(t)

var(yN(t)) = ?∆yi?2var(N(t)) + ?N(t)?var(∆yi) . (17)

During time t

i

∆yi, leading to

The terms in this expression correspond to two sources

of uncertainty: (i) due to the randomness in Tithe num-

ber of growth events vary with var(N(t)), and (ii) the

individual height increments are random with var(∆yi).

This leads to

var(yN(t)) ≈ t?∆yi?2(δ2+ ǫ2) ,(18)

where ǫ =

cient of the height fluctuations due to geometric effects.

We define the correlation time τ as the amount of time

by which the uncertainty of the height of the chain be-

comes comparable to one particle diameter, var(yN(τ)) =

O(1). Since ?∆yi? is largely independent of δ (cf. Ap-

pendix A), the time correlation induces a fixed intrinsic

?var(∆yi)/?∆yi? denotes the variation coeffi-

vertical correlation length

τ ∼

1

δ2+ ǫ2

(19)

in the model. This correlationlength reduces fluctuations

and leads to an increase in the saturation time tsat of

the system, namely tsat/τ ∼ Lz, a modification of the

usual relation with the system size L. Analogous to the

standard derivation of the time-dependence of the lateral

correlation length [3], this leads to

ξ?(t) ∼ (t/τ)1/z.(20)

Together with (16) from the behaviour of the correlation

length, we expect

D/(2ν) ∼ (δ2+ ǫ2)2,(21)

since λ turns out to be largely independent of δ. This

is shown to be in very good agreement with the data

in Fig. 6, for fitted values of ǫ and a prefactor. The fit

value for ǫ and the ratio D/(2ν) for δ = 1 (the usual

Eden model) are compatible with simple theoretical ar-

guments (see Appendix A). So the very basic argument

above to derive an intrinsic vertical correlation length

explains the δ-dependence of the surface properties very

well. Measuring height in this intrinsic length scale, we

observe a standard KPZ behaviour with critical expo-

nents being unchanged, since the intrinsic correlations

are short range (i.e. decay exponentially on the scale τ).

This is in contrast to effects of long-range correlations

where the exponents typically change, see e.g. studies

with long-range temporally correlated noise [31–33] or

memory and delay effects using fractional time deriva-

tives and integral/delay equations [34–36].

B.Domain boundaries

In this section we derive the superdiffusive behaviour

of the domain boundaries between the species from the

scaling properties of the interface. Since the boundaries

grow locally perpendicular to the rough surface, they are

expected to be superdiffusive with Hurst exponent 2/3,

which has been shown for a solid on solid growth model in

[37] and has been observed in [15] for experimental data.

In order to confirm this quantitatively for our model,

we perform simulations with initial conditions B1(0) =

{1,...,[L/2]} and B2(0) = {[L/2] + 1,...,L}, i.e. the

initial types are all red on the left half and all green on

the right half of the linear system. Therefore we have

two sector boundaries X1 and X2 with initial positions

X1(0) = 1/2 and X2(0) = [L/2] + 1/2. After growing

the whole cluster, we define the boundary as a function

of height via the leftmost particle in a strip of width 2

and medium height h:

X1(h)

X2(h) = min?xp+ 1/2 : |yp− h| < 1, p ∈ B2

= min?xp+ 1/2 : |yp− h| < 1, p ∈ B1

?

?, (22)