Specialization at an expanding front Lauren H. Li and Mehran Kardar Department of Physics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA

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Specialization at an expanding front

Lauren H. Li and Mehran Kardar

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Dated: September 13, 2023)

As a population grows, spreading to new environments may favor specialization. In this paper, we

introduce and explore a model for specialization at the front of a colony expanding synchronously into

new territory. We show through numerical simulations that, by gaining ﬁtness through accumulating

mutations, progeny of the initial seed population can diﬀerentiate into distinct specialists. With

competition and selection limited to the growth front, the emerging specialists ﬁrst segregate into

sectors, which then expand to dominate the entire population. We quantify the scaling of the

ﬁxation time with the size of the population and observe diﬀerent behaviors corresponding to distinct

universality classes: unbounded and bounded gains in ﬁtness lead to superdiﬀusive (z= 3/2) and

diﬀusive (z= 2) stochastic wanderings of the sector boundaries, respectively.

In the course of evolution, a homogenous population

may diversify to exploit emerging ecological niches. Such

disruption of a population’s homogeneity can often be

attributed to changes in the availability of resources

across geographical terrains [1]. As an initially homo-

geneous population occupies new terrain, it can diﬀeren-

tiate into diﬀerent specialized populations to maximize

ﬁtness. Here, we introduce a ﬁtness model for special-

ization by mutations along a “two-feature” axis. In our

model, the mutating population expands spatially at a

front (similar to a tumor), with reproductive selection

encoded by a ﬁtness function. Positing diﬀerent forms

of the ﬁtness function, we use numerical simulations to

follow the evolution of the population, in particular track-

ing the ﬁxation time for the entire population to become

dominated by a single specialized group. Our main result

is that, depending on whether the ﬁtness can grow indef-

initely or there exists a maximum attainable ﬁtness, the

ﬂuctuation behavior of domain walls between specialized

populations falls into diﬀerent universality classes.

Our work is largely inspired by studies of range expan-

sions, which describe populations that expand spatially

into new territory over the course of many generations,

as in tumors [2,3] or bacterial colonies [4,5]. In these

toy models of expanding populations, reproducing indi-

viduals only compete with those in close proximity at the

front of the expanding colony, and the eﬀects of genetic

drift are ampliﬁed due to this spatial limitation [6]. As

such, range expansions provide rich arenas for studying

stochasticity in evolution and have motivated numerous

works [4,6–15]. In laboratory experiments, Hallatschek

et al. [4] studied the appearance of sectors in growing

bacterial populations: an initially well-mixed population

of two ﬂuorescently labeled strains of Escherichia coli

was allowed to grow and expand. After some time, cells

at the colonization front had segregated into sectors de-

ﬁned by the ﬂuorescent marker. Interestingly, the mean

square transverse displacements of the sector boundaries

scaled with expansion radius ℓas ℓ2ζwith exponent

ζ= 0.65±0.05, suggesting superdiﬀusive wandering with

ζ= 2/3 in the Kardar-Parisi-Zhang (KPZ) universality

class [16]. Furthermore, such observations are not unique

to E. coli, as similar experiments with growing colonies

of haploid Saccharomyces cerevisiae revealed the same

scaling [4]. Hence, superdiﬀusive behavior is hypothe-

sized to be a universal characteristic of certain microbial

range expansions [17].

Several numerical studies on simple models of growth

have elucidated the universal characteristics of range ex-

pansions. One well-studied class of models are stepping

stone models [18], which represent the growing colony

with occupied points on a lattice, with sites at the front

reproducing into neighboring unoccupied sites [6,13–15].

With layer-by-layer (synchronous) growth (starting from

a straight one-dimensional edge), the boundary between

two sectors performs a random walk corresponding to a

transverse roughness exponent of ζ= 1/2; however, asyn-

chronous growth (random selection of sites on the front)

results in a rough front and leads to the superdiﬀusive

exponent ζ= 2/3 [15], as is the case in experiments [4].

Moreover, the superdiﬀusive exponent is also observed

in a model with synchronous reproduction [15] inspired

by directed paths in random media (DPRM) [19]. The

latter model can be interpreted as describing stochastic

variations in the size of the cells, giving rise to a rough

front [20]. As a variant of the latter model, our work

helps clarify when synchronous reproduction can result

in superdiﬀusive or diﬀusive scaling.

In the above range expansion experiments, the diﬀer-

ent sectors can be regarded as distinct specialized pop-

ulations competing at the sector boundaries. However,

all these specialists are already present at the initial seed

(with assumed identical ﬁtness) and their progeny then

segregate into diﬀerent regions of space. In contrast, spe-

cialization in nature typically arises from the diﬀerentia-

tion of individuals over time. In this letter, we introduce

a model in which distinct specialists evolve spontaneously

due to mutations; subsequent competition with neigh-

boring specialists populations ensues and leads to spatial

separation. Our model is consistent with observations in

nature and in experiments.

Fitness model. We consider a simple model with in-

dividuals characterized by two traits, which we label as

breadth bor height h; the reproductive success of indi-

viduals in competition is given by some function f[b, h]

of the two traits. Our focus is on whether mutations lead

arXiv:2210.05531v2 [physics.bio-ph] 13 Sep 2023

to individuals specializing in one trait over the other, and

hence we are interested in the diﬀerence in magnitude of

the two traits m=b−h. Studies of phase transitions

indicate that key universal features can be captured by

simple polynomial expressions for the free energy den-

sity [21]. Motivated by such, we consider a simple poly-

nomial form for the ﬁtness function that allows for dif-

ferentiation:

f[b, h]≡f(m) = f0+αm2+βm4+γm. (1)

The even powered terms (αand β) preserve a symme-

try about f0between band h(or m→ −m), while

the odd term (γ) breaks this symmetry [22]. The de-

gree of specialization is quantiﬁed by m, which is akin to

the magnetization in Landau’s theory of magnetic phase

transitions [21].

Lattice implementation. Given the ﬁtness function in

Eq. (1), we model spatial growth using a variation of the

stepping stone model [18], where individuals in genera-

tion tare arranged along a line, indicated by x; we refer

to individuals by their site coordinates (x, t) on a trian-

gular lattice (Fig. 1). The progeny at generation tare

determined by the competition between the two neigh-

boring individuals at generation (t−1). The winner of

the competition between potential parents (x+ 1, t −1)

and (x−1, t −1) is the one with the larger ﬁtness value

f(x, t) = f[b(x, t), h(x, t)].(2)

Let xmax denote the xcoordinate of the individual with

the higher f(x, t). The individual at xmax then procre-

ates and its progeny inherits its traits, up to small vari-

ations as described by

b(x, t) = b(xmax, t −1) + ηb(x, t),

h(x, t) = h(xmax, t −1) + ηh(x, t).(3)

We take ηb(x, t), ηh(x, t) to be independent and iden-

tically distributed Gaussian random variables, with

zero mean and correlations ⟨ηa(x, t)ηa′(x′, t′)⟩=

σ2δa,a′δx,x′δt,t′, as in the DPRM-inspired model in

Ref. [15]. The noise ηaccounts for random mutations in

the oﬀspring. Since the mean change is zero, the magni-

tudes of band hare equally likely to increase or decrease

over generations; however, the accumulating mutations

selected by preferential reproduction enable specializa-

tion to occur in certain regimes of the ﬁtness function.

Note that as all individuals in a given generation tare

updated synchronously, the uppermost layer in Fig. 1re-

mains ﬂat.

Initially, all members of the population are unspecial-

ized; that is, m(x, 0) = 0 for all x. For some ﬁtness func-

tions, individuals may become specialized in either fea-

ture, such that after tgenerations, m(x, t)̸= 0 for typical

x(Fig. 1). Over time, segments of neighboring individu-

als with the same specialization form sectors. Depending

on the shape of the ﬁtness function (parametrized by α,

β, and γ) and the magnitude of mutations (parametrized

b-specialist h-specialist

FIG. 1. Illustration of the update rules for our model. The

initial seed population (t= 0) is unspecialized (grey), and

subsequent progeny inherit features band h, according to our

update rules. Variations provided by the accumulating ran-

dom mutations in the update rules may result in individuals

becoming specialized in b(blue) or h(yellow).

by σ), we observe diﬀerent patterns in the emergence of

new specialist populations (Fig. 2).

Growth Patterns. We conduct numerical simulations

with our ﬁtness model under diﬀerent regimes. In

Fig. 2(a), the ﬁtness is maximized at the origin, and

we observe no specialization. Even when the ﬁtness is

shifted to favor a particular feature by setting γ̸= 0,

there is no diﬀerentiation into distinct specialized pop-

ulations; rather, the population is dominated by spe-

cialists in that feature, but these specialists are evenly

distributed in space. These behaviors are reminiscent

of antibiotic resistance, which can appear or disappear

in cells depending on the presence of antibiotics in the

medium [23].

In contrast, in Fig. 2(b), there is a local maximum

around the origin with local minima on both sides; hence,

it is possible to attain a higher ﬁtness by moving through

a less favorable region in the ﬁtness landscape. Due to

this partial advantage for generalists, especially at early

times, the resulting growth pattern consists of unspecial-

ized individuals until the sudden onset of those highly

specialized in bor h. Eventually, these specialists form

V-shaped sectors which, upon meeting, compete for dom-

inance [24].

In Figs. 2(c) and (d), we observe specialization in cases

where the ﬁtness gain is unbounded or bounded, respec-

tively. By random chance, band hﬂuctuate over genera-

tions, resulting in individuals specializing in one feature

over the other, giving rise to specialists. For the bounded

ﬁtness function illustrated in Fig. 2(d) with two local

maxima, we observe an emergence time τefor the mag-

nitude of specialization |m|to become maximal (detailed

in Ref. [25]).

Fixation time. Once specialized groups are well-

deﬁned, the domain walls between distinct groups ﬂuctu-

ate as the population expands. Eventually, one special-

ized population dominates, and there is a ﬁxation time

τfin which the entire population becomes specialized in

the same feature [26]. We perform numerical simulations

to characterize the scaling of the ﬁxation time with pop-

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SpecializationatanexpandingfrontLaurenH.LiandMehranKardarDepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USA(Dated:September13,2023)Asapopulationgrows,spreadingtonewenvironmentsmayfavorspecialization.Inthispaper,weintroduceandexploreamodelforspecializationatthefro...

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Specialization at an expanding front Lauren H. Li and Mehran Kardar Department of Physics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA

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