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Bacteria and game theory: the rise
and fall of cooperation in spatially
heterogeneous environments
rsfs.royalsocietypublishing.org
Guillaume Lambert1, Saurabh Vyawahare2 and Robert H. Austin2
1
2
Review
Cite this article: Lambert G, Vyawahare S,
Austin RH. 2014 Bacteria and game theory: the
rise and fall of cooperation in spatially heterogeneous environments. Interface Focus 4:
20140029.
http://dx.doi.org/10.1098/rsfs.2014.0029
One contribution of 7 to a Theme Issue ‘Game
theory and cancer’.
Subject Areas:
systems biology, biophysics
Keywords:
game theory, bacteria, Prisoner’s Dilemma
Author for correspondence:
Guillaume Lambert
e-mail: [email protected]
Institute of Genomics and Systems Biology, University of Chicago, Chicago, IL 60637, USA
Department of Physics, Princeton University, Princeton, NJ 08544, USA
GL, 0000-0003-4779-287X
One of the predictions of game theory is that cooperative behaviours are vulnerable to exploitation by selfish individuals, but this result seemingly
contradicts the survival of cooperation observed in nature. In this review,
we will introduce game theoretical concepts that lead to this conclusion
and show how the spatial competition dynamics between microorganisms
can be used to model the survival and maintenance of cooperation. In particular, we focus on how Escherichia coli bacteria with a growth advantage in
stationary phase (GASP) phenotype maintain a proliferative phenotype
when faced with overcrowding to gain a fitness advantage over wild-type
populations. We review recent experimental approaches studying the
growth dynamics of competing GASP and wild-type strains of E. coli
inside interconnected microfabricated habitats and use a game theoretical
approach to analyse the observed inter-species interactions. We describe
how the use of evolutionary game theory and the ideal free distribution
accurately models the spatial distribution of cooperative and selfish individuals in spatially heterogeneous environments. Using bacteria as a model
system of cooperative and selfish behaviours may lead to a better understanding of the competition dynamics of other organisms—including
tumour–host interactions during cancer development and metastasis.
1. Introduction
When two individuals, autonomous agents or even countries interact, the outcome varies greatly depending on the intention of each party involved. Early
attempts to quantify these interactions resulted in what is now called game
theory [1]: a mathematical framework describing the outcome ( pay-off ) resulting from specific interactions (game) between two individuals ( players). In
particular, game theory describes how the interaction between two players
incurs a specific—and not necessarily symmetrical—pay-off to each one, and
the behaviour of each player dictates how each benefits from those interactions.
Although it may seem unnatural to reduce complex sociological or political
interactions to a simple numerical value, game theory does provide a rational
and quantitative framework to quantify complex interactions [2,3].
Parallel to this, game theory has also been used in a biological context [4,5] to
quantify interactions between living organisms and to describe the population
dynamics of competing populations. One of the main results of game theory
when applied to competing individuals is that cooperation, defined here as a behaviour which benefits the whole population, is typically at a disadvantage with
respect to selfish behaviours [6,7]. Indeed, interactions where cooperative actions
can be taken advantage of and exploited by self-serving individuals provide a
means for selfish behaviours to emerge within a population: the demise of
cooperation usually stems from the exploitative nature of selfish behaviours,
which redirect resources away from cooperators. For example, game theory can
model the emergence and dominance of selfish tumour cells, which abandon
the social contract of multicellular organisms to venture into unicellularity and
self-serving behaviours [8–10].
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
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2.1. Equilibrium and optimal strategies
Antoine Augustin Cournot first described in 1838 how firms
in a duopoly can maximize their profit (i.e. pay-off ) in what
would now be called a game theoretical approach [23]. He
described how a strategy that maximizes the profit of all
firms exists, and that any deviations from this optimal strategy lead to a diminished return for all firms. Cournot’s
approach was later shown to be a subset of a more general
framework developed by John von Neumann, John Nash
and co-workers [1,24]. One important result is that some of
these games have a Nash equilibrium, which represents the
state where every player uses a strategy that maximizes his or
her pay-off while taking into account the strategy of other
players. The theory is especially well fleshed out for twoplayer games where the outcome of each interaction can be
quantified, and the ‘quality’ of these interactions is usually presented in the form of a pay-off matrix. An example of a pay-off
matrix is shown in table 1, where each player may choose to
cooperate or defect, and the result of each interaction is associated with a pay-off. Knowing each player’s options, and
assuming that each one acts rationally, a Nash equilibrium
may be found that describes the set of strategies that optimize
the pay-off for each player.
A Nash equilibrium does not necessarily represent the
maximum pay-off possible: a famous instance of a Nash equilibrium that results in a non-maximal pay-off for each player
is the Prisoner’s Dilemma (PD). The PD describes the type of
game which satisfies the condition T . R . P . S (table 1).
Although both players may obtain a higher pay-off from
cooperating ( pay-off R . P), the Nash equilibrium actually
dictates that each player defects owing to the potentially
large penalty incurred if one cooperated and the other one
is tempted to defect (T S). The PD is a canonical example
of the rise and fall of cooperation: cooperation, which would
result in a higher pay-off to each individual, is not the
player 1/player 2
cooperates
defects
cooperates
(R, R)
(S, T )
defects
(T, S)
(P, P)
optimal strategy. Instead, selfishness is favoured even
though it incurs a higher cost to each individual.
This type of conflicting situation is not limited to a PD
game; another situation which favours the maintenance of
selfishness is the hawk –dove game (also called a chicken or
snowdrift game), which describes a pay-off matrix that
satisfies the condition T . R . S . P. Here, the symmetric
Nash equilibrium is either a pure strategy, where all the
players either defect or cooperate (trivial initial conditions),
or a mixed equilibrium, where each player probabilistically
chooses between cooperation and defection with probability
Pcoop ¼ (P S)/((P S) (T R)) and Pdefect ¼ 1 Pcoop . In
either case, unconditional cooperation by both players,
perhaps a more ‘just’ and less harmful strategy overall, is
not the optimal strategy.
2.2. Evolutionary game theory and replicator dynamics
A modified form of game theory called evolutionary game
theory (EGT) can be used to describe the effect of every interaction between competing individuals within a population
and describe the long-term population dynamics of competing populations. While game theory describes the strategy
of two players as a probabilistic combination of pure strategies (and the Nash equilibrium describing the probability
of the use of each strategy), EGT tracks the strategies used
by a population of players. Here, the pay-off matrix of a
given set of interactions is given by the fitness gained or
lost upon encounters of different individuals. The optimal
strategy used by each player within the population that is
stable upon the introduction of a new player is defined as
an evolutionary stable strategy (ESS) [25].
EGT can also involve pay-offs which are density
dependent; in other words, the outcome of an interaction
between individuals within a population depends on the overall population size. The extension of EGT to density-dependent
pay-offs combines a game theoretical approach with the ideal
free distribution (IFD) [26], an ecological framework which
describes how density-dependent effects can affect the fitness
(pay-off) of individuals. Such systems, studied in [27–30],
can accurately describe the competition dynamics of interacting populations of fixed size under resource limitations, and
results have shown that they give rise to equilibrium distributions which are stable under small spatial perturbations
[31]. While the rest of this section only involves competition
dynamics of two competing populations using constant payoffs, the effect of density-dependent pay-offs will be addressed
in the last section of this article.
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2. Game theory in biology
Table 1. Pay-off matrix of a two-player game. In a game where each
player can use one of two strategies (cooperation or defection), each player
is awarded a pay-off that depends on the strategy of the other player. The
relationship between each element of the pay-off matrix (R, S, T, P)
dictates the type of game that is being played. R, reward, S, sucker’s
pay-off, T, temptation, P, punishment.
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On the other hand, the observation that cooperation is
widespread—expressed as mutlicellularity, efficient public
good management or indeed any functioning society—seems
to contradict this fundamental result. Several theoretical
explanations have been proposed to explain this [11–14] and,
more recently, biological experiments in which selfish individuals spread within a population [15–18] have provided
significant insight into the strategies used (or not used) by
cooperative individuals when faced with selfish competitors.
In this review, we focus exclusively on an experimental
system first described in [19] and subsequently used in
[15,16,18,20], consisting of interconnected microfluidic
chambers called micro-habitat patches in which we can
create heterogeneous spatial environments and observe competing bacterial strains. The competitors are (i) wild-type
(WT) Escherichia coli cells and (ii) a mutant strain of E. coli
that displays a growth advantage in stationary phase (GASP)
[21], which has previously been shown to use selfish
behaviours to exploit WT populations [22]. We believe that,
by understanding the dynamics of competing bacteria,
significant insight can be gained about the survival and maintenance of cooperation in natural environments, and may help
develop new treatment strategies against tumour cells and
cancer tissue during metastasis.
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temptation punishment
(T)
(P)
1.0
0
coexistence:
xc
6
8
0.5
10
–1
–1
time
xc (t)
0.4
0
temptation to cheat (T–R)
0.5
0
0.3
0.1
1
c
|P–S| > |T–R|
0
1.0
0.8
0.6
0.4
0.2
0
d
no cooperation
2
4
6
8
10
time
Figure 1. Game theory. A pay-off matrix describes the nature of the interactions between two players. Two important variables are the temptation to cheat (T– R),
which is the difference between the ‘temptation’ to defect and the ‘reward’ obtained by cooperating, and the penalty incurred by cooperating (P – S), which
measures the pay-off when the other player cheats. In the middle panel, equation (2.1) is solved and the final fraction of cooperator xc(t) as t ! 1 is computed.
The colourbar illustrates the value of xc(1), the fraction of cooperative individuals within the population at equilibrium, and a few examples of competition between
individuals are shown in panels (a – d).
The time evolution of two competing populations of
cooperator and defectors, expressed in term of replicator
dynamics and density-independent pay-offs [32–34], is given by
dxc
¼ ( fc fd )(1 xc )xc ,
dt
(2:1)
where the population fraction of each population is given by xc
and xd, respectively, and fc and fd denote the frequencydependent fitness of each population that depend on the
matrix wij describing the system, where each element i, j
describes the fitness associated with each interaction between
player i and j. Here, each fi is given by
X
fi ¼
wij xj :
(2:2)
j¼1,2
Replicator dynamics imply that individuals do not change
strategy over time (i.e. each player uses a pure strategy), so
each phenotype constitutes a different strategy and the frequency at which new strategies emerge is limited by the
mutation rate. The steady-state distributions of the cooperative
population is given by
8
xc ¼ 0,
>
>
>
< x ¼ 1,
c
(2:3)
>
>
1
>
: xc ¼
,
1 ((T R)/(P S))
and corresponds to either pure populations (xc ¼ 0 or xc ¼ 1) or
a stable mixture of two populations, provided that 0 , xc , 1.
Not all of those distributions are dynamically stable, and the
set of conditions that allow cooperators to survive are summarized in figure 1. Generally, a population of selfish individuals
will always be present within the population whenever a temptation to defect (T . R) or a pay-off to cooperators has the
potential to be lowered through interactions with defector
populations (P . S).
If this analysis is applied to the hawk –dove game
(figure 1), interactions will always favour coexistence
between cooperating and defecting populations, regardless
of the initial population composition ( provided that at least
one cooperator and one defector is present). On the other
hand, certain conditions—such as low temptation to cheat
(T , R) and low penalty for cooperating (P , S), labelled
harmony in figure 1—favour harmonious interactions and
the complete dominance of cooperative populations.
Another type of game called the stag-hunt game
describes situations which satisfy the general requirement
R . T P . S. In this case, the temptation to defect is low
but a selfish individual is at an advantageous position with
regard to a cooperating individual: the stag-hunt game may
lead to a predominance of the selfish population, but only
under certain conditions (figure 1b,c). The likelihood that
a cooperator population survives, however, is tied to its
initial fraction in the population: for a given set of (T 2 R)
and (P 2 S) parameters, the cooperator population is more
likely to survive when it is initially present at a higher initial
fraction within a population (figure 1b,c).
On the other hand, EGT applied to the PD never allows a
mixed population to survive, and every PD game generally
leads to a complete extinction of the cooperator population.
Therein lies the big conundrum of game theory: how is
cooperation ever expected to occur if it is always at a disadvantageous position in the face of selfishness? Although a
stable population of cooperating individuals can subsist,
they will have to continually face defector populations that
divert resources and exploit cooperative individuals. Furthermore, elements of the pay-off matrix may not be stable over
time and, if the conditions happen to change favourably
towards defection (i.e. (T 2 R) and/or (P 2 S) increases),
the relative fraction of players using a cooperative strategy
will either decrease (hawk –dove game) or have a higher
probability to become extinct (stag-hunt game). Harmony
and a complete removal of selfish behaviours within a population—for example, a perfectly functioning society or a
complete remission following cancer treatments—fails to
occur unless interactions between players obey a relatively
narrow set of conditions for which both the temptation to
cheat and the penalty for cooperating are low.
3. The players: GASP versus WT Escherichia coli
bacteria
Bacteria are very rarely in an exponential growth state in
natural environments, and cells have to use a variety of
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Interface Focus 4: 20140029
4
1.0
0.6
0.2
hawk–dove
harmony
2
1
1– T–R
P–S
0
0.7
a
coexistence
|P–S| < |T–R|
d
a
0.5
0.5
0.8
xc (t)
defection
sucker's
pay-off
(S)
reward
(R)
.5
)>0
x c(0
x
c (0
)<
0.
5
c
xc (0)
< 0.5 b
defection
xc (t)
cooperation
1
cooperation
0
b
final fraction of cooperators
2
penalty for cooperating (P–S)
er
er
1.0
PD
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ay
ay
xc (t)
stag hunt
0.9
pl
pl
1.0
1
player 1 pay-off matrix
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(a)
4
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cell viability
stationary phase
1
death
exponential phase
10–2
prolonged starvation
lag phase
(b)
1.0
1.5
2.0
2.5
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0.5
0
3.0
time (days)
10
density (OD600)
8
6
4
GASP no. 1
GASP no. 2
2
GASP no. 3
0
6
3
9
time (days)
(c)
(d)
WT
GASP
high
low
final
density
high
higher
stress
tolerance
low
high
10
no. cells (103)
growth
rate
GASP no. 1
WT
5
0
20
time (h)
40
Figure 2. Evolution under prolonged starvation. (a) Bacterial growth usually begins with a period of quiescence (lag phase), followed by exponential growth,
entrance into stationary phase and finally a death phase. During the death phase, more than 99% of the cell population dies as a result of the deteriorating
environmental conditions. (b) The small fraction of surviving cells (less than 1%) is not genotypically stable. In this idealized representation, different genotypes
arise in succession. Every new genotype is more adapted to stressful conditions than the previous one and usually has mutations conferring a GASP. (d ) This table
summarizes the properties of the WT and GASP strains. (Adapted from [20,41].) (c) The WT strain grows at a higher rate but will reach a lower density. The GASP
cells, on the other hand, grow at a slower pace but reach a higher final density. (Data adapted from [20].)
strategies in order to survive in resource-limited environments [35]. For instance, the carbon levels in oceans are
approximately 50 mM [36], compared with the 0.1 M found
in a typical laboratory-made growth medium. Similarly,
bacteria in the soil go through an estimated 36 generations
a year owing to the highly limited bio-availability of energy
[37]. These limitations are present in most bacterial systems
found in nature.
In E. coli, WT cells growing in a new environment will
slowly consume the available nutrients and produce
metabolic by-products. Then, in response to deteriorating
growth conditions, the cell population will cease to
proliferate and enter a state called stationary phase [38].
The response to such deteriorating conditions is via a phenotypic switch regulated by the ss sigma factor, transcribed by
the rpoS gene [39]. This sigma factor triggers the expression of
several cell protection mechanisms as a response to decreasing nutrient levels, increased density or changes in pH [40].
The typical growth dynamics followed by WT cells is
shown in figure 2a.
Under prolonged starvation, a large fraction of cells maladapted to the accumulation of deteriorating conditions will
die. A few remaining cells, however, often evolve the ability
to grow despite stressful environmental conditions [41]
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The MHP system, shown in figure 3, was first described in
[19] and later applied in [15,16,18] to study the similar
GASP –WT competition dynamics in different adaptive
landscapes. In particular, each strain expresses a different
fluorescent molecule (green fluorescent protein or red fluorescent protein) and fluorescence levels inside each MHP are
recorded at fixed intervals (every 10–15 min). Since the
relationship between fluorescence intensity and cell density
has been shown by Hol et al. [18] to scale linearly with the
fraction of the habitat occupied by bacteria, the overall
growth of each strain in the MHPs can be estimated by
measuring the fluorescence intensity of each population.
5
nano
slits
nutri
ent
8 µm
50 µm
nutrient reservoir
25 µm
(b)
to camera
computercontrolled
microscope
Figure 3. Micro-habitat patch device. (Adapted from [15].) (a) We physically
recreate a metapopulation landscape using microfabrication. Each chamber is
100 100 10 mm in size (highlighted in cyan), and the 200 nm nanoslits (yellow) are deep enough to allow nutrients to freely diffuse inside the
MHPs but small enough to prevent cells from migrating into the nutrient
reservoirs. Cells can migrate between each micro-habitat using the 5 mm
wide junction channels (red). (b) A computer-controlled microscope records
the fluorescence intensity in each chamber every 15 min.
Table 2. Pay-off matrix computer by Vulić & Kolter [22] for competing species
of WT and GASP cells. The pay-off matrix describes a PD type of game
because the matrix elements satisfy T . R . P . S.
WT
GASP
WT
(1.0, 1.0)
(0.2, .1)
GASP
(.1, 0.2)
(0.5, 0.5)
The absolute number of cells measured inside each at
stationary phase typically ranges between 9500 and 12 000
cells [15,20].
A variety of fitness landscapes (i.e. nutrient or stress levels)
can be made by varying the number of nanoslits that couples
each MHP to the nutrient reservoir, allowing study of interaction dynamics between cooperation and selfishness as a
function of spatial heterogeneity. First, we describe the experiments reported in [16,19], in which half of the device has
either full access (nutrient-rich) or no access to the nutrient
reservoirs [19]. As a result, most MHPs therefore have the
same nutrient access as their neighbour (except the two
MHPs at the centre of the device). Even though the environment is spatially homogeneous, complex time-dependent
dynamics are observed, both for a single population [19] and
for WT–GASP competition experiments [16].
As a control, the growth of each cell type inside the MHP
device has been quantified by only inoculating WT cells or
only GASP mutant cells inside the device (figure 1d) [15,16].
Note a few distinctive features of each strain: first, the growth
Interface Focus 4: 20140029
4. Experiments: game theory and spatial
competition
(a)
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(figure 2b). First observed by Zambrano et al. [21], these
resistant populations are called GASP mutants, because
they outcompete WT strains under starvation conditions
and therefore have a GASP. Figure 2b illustrates the dynamic
nature of a culture under prolonged starvation: while the
total number of cells remains constant, new genotypes are
constantly evolving and fixing into the population. These
cells not only develop an increased resistance to external
stress and deteriorating environmental conditions, but are
also able to sustain a proliferative phenotype by catabolizing
bacterial lysate [41].
Although GASP cells grow at a slower pace than WT
cells (figure 2c,d), they are nevertheless able to outcompete
WT cells through sustained growth. Vulić & Kolter [22]
have quantified the competition dynamics of a population
carrying the rpoS819 allele (a variant of the rpoS gene which
confers a GASP phenotype [21]) and WT populations to
extract a pay-off matrix and describe the fitness associated
with interaction between each population. The outcome of
the competition between WT and GASP cells was measured
in [22] through a series of experiments that quantified the fitness of each strain (i.e. fc and fd in equation (2.1)) using
antibiotic markers and selective plating to quantify the population composition after 2 and 5 days. The pay-off matrix of
the measured GASP/WT interactions is shown in table 2.
In particular, the pay-off matrix describes a PD type of interaction dynamics (T . R . P . S). Similar conclusions were
reached by Hol et al. [18], who showed that the rpoS819
allele inserted in an E. coli background different from the
one in [22] still displayed a GASP phenotype and PD
interactions dynamics with WT cells.
As the ESS for a PD pay-off matrix dictates that defection
is the optimal strategy, a GASP mutant is at a definitive fitness advantage. Indeed, their experiments do show that
GASP mutants unequivocally become the dominant species
when competing against WT populations, but theoretical
predictions [42–47] and recent experimental results [15 –18]
have shown that WT cells (cooperators) can survive in the
presence of GASP mutants. The key in these experiments
was the presence of spatial structure and heterogeneous
environments, both of which favour the maintenance of
cooperation. In §4, we will explore how the micro-habitat
patch (MHP) system used in [15,16,18] has been used to
study the competition dynamics between WT and GASP
mutants cells in spatially structured and heterogeneous
environments.
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0
10 20 30 40 50
time (h)
6
0
0.5
coexistence
0
10 20 30 40 50
time (h)
rate of WT cells is initially higher than that of GASP mutants.
Fitting a logistic growth function to the growth curve gives a
doubling time of about 1 h for WT cells and slightly less
than 2 h for GASP cells. The large discrepancy may result
from the fact that the GASP cells are inherently maladapted
to conditions of neutral pH and rich media [48]. Second, the
stationary phase density reached by the cells is slightly
higher for GASP mutants than for WT cells. This could be
due to the fact that GASP cells have the ability to process
and digest nutrients during prolonged starvation that cannot
be catabolized by WT cells [41].
Competition experiments were performed by inoculating a
1 : 1 mixture of WT and GASP mutant cells inside a device
where the first half of the MHPs are nutrient-rich habitats and
the remaining half consist of nutrient-poor habitats. The time
evolution of the population distribution in the nutrientrich habitats, as reported in [16] and [15], is reproduced in
figure 4a. It is interesting to note that the population distribution
evolves very little over time: the fraction of WT cells hovers
around 50% for most of the experiment (figure 4b). The density
of WT cells is initially higher than that of GASP mutant cells, as
shown for time T , 20H in figure 4a, but the GASP mutants’
sustained growth allows them to supersede WT cells as the
dominant species.
On a larger scale, as shown in [16] and in figure 5a, very
little growth is occurring in the region with no nutrient access
and growth mainly occurs in the nutrient-rich regions. Even
though the fitness landscape is homogeneous throughout
the device, there are large-scale heterogeneities in the spatial
distribution of each strain. These heterogeneities can be
characterized by measuring the spatial correlation between
each population using a technique developed by Kimura &
Weiss [49], which was initially developed to describe the
spatial correlation of genetic information between populations in interconnected habitats. Kimura & Weiss showed
that the correlation c between a strain’s population fractions
p in habitats a distance j apart scales as c( j) p
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k p(i)p(iffi þ j)li /
k p2 (i)li ej=l with a constant l given by m=2m1 . Here,
m is the migration rate between neighbours, and m1 is the
‘long-range’ exchange rate—which can be thought of as the
rate at which two random habitats exchange genetic information—and is formally equivalent to mutations. As no
mutations are likely to occur during the experiments presented in [15,16,18], the spatial correlation has been used in
[20] to described the spatial correlation of each strain within
the MHP array.
For the experiment shown in figure 5a, the spatial
correlation of the GASP mutant population decreases faster
Interface Focus 4: 20140029
Figure 4. Growth dynamics under competition. (Adapted from [20].)
(a) Growth of competing WT and GASP mutant populations (green, WT;
red, GASP) in the nutrient-rich regions. (b) The time evolution of the population fraction in the nutrient-rich region shows that cells coexist at an
approximately equal fraction inside the device.
time (h)
5
(a)
1.0
25
50
0
(b)
3
6
9
position (mm)
12
1.0
1.75
genetic correlation
(log scale)
10
population fraction
(b)
15
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no. cells (103)
(a)
1
0.9
1
1.25
0.8
0
0.75
1.50
2.25
distance (mm)
3.00
Figure 5. Long-range spatial correlation. (a) Kymograph representation of the
dynamics of the WT (green) and GASP (red) populations within the device.
(b) Kimura & Weiss’s average intra-species spatial correlation between neighbouring habitats as a function of distance is extracted from the fluorescence
levels in each MHP. We find that the correlation length for the WT population
(green circles) is longer than that of the GASP population (red squares).
than that of the WT (lGASP ¼ 1.25 mm, lWT ¼ 1.75 mm,
figure 5b). A smaller l for GASP mutants may be explained
by the fact that they grow at a slower pace than WT cells:
the initially larger population of WT cells is able to exchange
more cells between MHPs than GASP mutants. Consequently, important inter-species ‘mixing’ occurs among
MHPs before cell exchange is severely limited by large
densities of cells inside each MHP. So, as the cell density
increases, the population distribution is quenched and
the final distribution is mainly determined by the early
(low-density) dynamics.
At a smaller scale, spatial segregation inside each MHP
(figure 6a,b) [15,16,18] is also visible. The scale at which
each strain spatially segregates was quantified in [16], and
the GASP population was found to be spatially extended at
a larger scale than the WT population. In fact, coexistence
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(a)
(c)
7
1
penalty for
cooperating (P–S)
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PD
stag hunt
spatial
coexistence
0
hawk–dove
harmony
25 µm
0
temptation to cheat (T–R)
(b)
1
50 µm
Figure 6. Spatial coexistence. (a,b) WT and GASP cells are able to coexist spatially in each MHP at different scales. (Adapted from [16,18].) (c) WT and GASP cells
were shown to interact according to a PD type of game [22], which would inevitably lead to the extinction of the WT population under well-stirred conditions. The
observation that cooperator and defector populations can coexist in a spatial PD game suggests that the phase diagram describing spatial competition must be
modified to account for the observed spatial coexistence.
25 mm
80%
cell density per nutrient
access correlation
and competition is beneficial to both populations, as the density of WT cells increased under competition conditions,
reaching a final density almost 20% greater when co-cultured
with GASP cells (figure 4), compared with the control experiment presented in figure 4d (see [15,16] for more details
about this phenomenon).
Thus, cooperator cells coexist with selfish cells in a spatial
environment, and this spatial coexistence also increases the
fitness of both populations. From our naive intuition about
two-player games—only the hawk– dove game allows populations to coexist, even though the measured pay-off indicates
a PD type of game (figure 6c). From this experimental paradox and others like it [18,42 –44], we can conclude that
spatial games fundamentally change the nature of the interactions between cooperator and selfish populations to
permit the maintenance of cooperation in a spatial PD
game, as previously demonstrated from theoretical [42 –44]
and experimental [18] standpoints.
0%
70%
0%
90%
70%
1.0
0.5
0
–0.5
–1.0
0
5
10
time (h)
15
20
Figure 7. Heterogeneous fitness landscape—WT control. Correlation coefficient between nutrient access (number of nanoslits) and the cell density. Any
value higher than 0.5 indicates ‘strong’ correlations. Inset: number of
nanoslits open versus position.
25 mm
5. Spatial heterogeneities and the fate
of cooperation
While competition dynamics inside spatially structured
habitats has been shown to increase coexistence between
cooperative and selfish populations [16,18], competition
between GASP and WT populations in a spatially heterogeneous environment seems to favour the GASP population
[15]. Indeed, WT cells are expected to avoid MHPs with
low nutrient access and prefer ‘well-fed’ MHPs, but they
instead seem to be reaching higher densities in MHPs with
low nutrient access (figures 7–10) [15,20].
This outcome is not intuitively expected, because WT cells
inoculated inside a MHP device where the number of nanoslits
across the device varies to create a pseudo-random fitness
landscape, as shown at the top of figure 7a and in [20], do
indeed prefer nutrient-rich habitats. In figure 7, the Pearson’s
correlation coefficient, given by the covariance of two variables
divided by the product of their standard deviations or, to be
100%
100%
0%
100%
100%
50%
Figure 8. Heterogeneous fitness landscape. The final density of WT and GASP
mutant cells strongly depends on the coupling between the nutrient reservoirs and the MHP array. Here, WT cells perform better in regions where the
coupling is weak (less than two nanoslits), whereas GASP mutants perform
better in regions where the coupling is strong (more than five nanoslits).
more precise, c(x, y) ¼ cov(x, y)/sx sy , remains positive and
reaches a maximum 0.8, indicating that there is a strong correlation between growth and nutrient access. When GASP and
WT cells are inoculated together inside the same landscape,
a rather different outcome is observed: figure 8 shows a representative population distribution, where GASP mutant cells,
shown here in red, dominate the MHPs with the highest
number of nanoslits.
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(a)
8
T = 40 h
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MHP no. 80
MHP no. 81
nutrient-poor
nutrient-rich
(b)
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(d)
1.0
14
nutrient-rich
nutrient-rich
no. cells (103)
nutrient-rich
pop. fraction
0.8
0.6
0.4
0.2
0
0
(c)
7
(e)
1.0
nutrient-poor
14
nutrient-poor
no. cells (103)
nutrient-poor
pop. fraction
0.8
0.6
0.4
7
0.2
0
10
20
time (h)
30
40
0
20
time (h)
40
Figure 9. Growth and population distribution. (Adapted from [15].) (a) An example of two neighbouring patches where an anomalous distribution of GASP
and WT cells is observed. (b,c) The fraction and overall growth of WT and GASP cells in the nutrient-rich regions show that the WT population leaves
the nutrient-rich environments as the density of GASP mutants increases. (d,e) Comparatively, the fraction and overall growth of WT cells is higher in the
nutrient-poor regions.
time (h)
0
200 mm
15
30
45
60
Figure 10. Large-scale periodic fitness landscape. In environments where the
fitness landscape varies on larger spatial scales, we observe similar dynamics:
WT cells initially populate nutrient-rich regions but are dislodged by GASP
cells following their entrance into stationary phase.
We will briefly summarize these results, first obtained in a
device in which very high nutrient gradients are present—i.e.
where there is a large change in fitness associated with
moving between two habitats [15]—and in an intermediate
fitness landscape (figure 10 and introduced in [20]), where
changes in fitness occur over a period of 17 MHPs. Each landscape shows similar trends: WT cells, which are initially
adapted to a rich medium, reach higher densities in regions
where the medium has been conditioned and GASP cells,
although more adapted to conditioned medium, become
the dominant species in the nutrient-rich habitats. Figure 9a
illustrates how very little coexistence exists between each
strain within each MHP compared with the spatially homogeneous environments shown in figure 6 [16,18]. The
spatial redistribution occurs 20 h after inoculation (figure
9b,c); by that time, cells have entered stationary phase and
very little growth is occurring in the nutrient-rich and nutrient-poor MHPs. In figure 9d,e, the fluorescence intensity in
the nutrient-rich MHPs is shown as a function of time and
shows how WT cells initially populate the nutrient-rich
regions, but where GASP mutant cells progressively populate
the nutrient-rich MHPs and eventually become the dominant
species in the nutrient-rich habitats after T ¼ 25 h.
Furthermore, cooperation is even less favoured when the
growth rate r is decreased everywhere inside the device by
filling the nutrient reservoirs with conditioned medium
instead of rich medium [20]. Conditioned medium is
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(b)
20
nutrient-poor
conditioned
10
0
10
time (h)
20
0
10
time (h)
20
depleted of most nutrients and, as a consequence, will lower
the overall growth rate of the WT population. In this situation, the WT population enters a stationary phase and
redistributes itself to the nutrient-poor regions at a much earlier time. The cell density of each strain and the resulting
population distribution is shown in figure 11a,b.
The first point to note is that the maximum WT density
reached in the MHPs is much lower than in the rich
medium experiments. The WT population reaches a maximum density in the nutrient-rich region of 5000 cells/MHP,
about 45% of the density reached in figure 9d. Also, the population redistribution occurs much faster—there is no
significant change in the population fraction approximately
4 h following the initial redistribution dynamics. By comparison, cells require more than 20 h to redistribute themselves
into the nutrient-poor regions in figure 9a. The fact that redistribution occurs fast seems to indicate that the additional
level of stress caused by the conditioned medium has greatly
deceased the ability of WT cells to successfully compete
against GASP mutant cells in the nutrient-rich MHPs.
The overall cell density of WT and GASP cells in the
device, shown in figure 11a,b, suggests that the WT population is at a much greater fitness disadvantage under
conditioned medium. Indeed, the WT cell density in the
nutrient-poor regions is approximately 10 times lower than
the GASP population, suggesting that coexistence fails to be
maintained. These results suggest that spatial coexistence
between cooperative and selfish individuals is highly dependent on the ability of cooperators to survive: a harsh
environment for cooperative individuals may prevent
cooperation from emerging, thereby leading to a complete
dominance of the selfish population.
6. Theoretical modelling: ideal free distribution
The anomalous distribution of WT and GASP populations
observed in [15] suggests that active environmental conditioning by the selfish individuals can precipitate the
demise of cooperating individuals in spatially heterogeneous environments. To better understand and model
these population distribution dynamics, the concept of the
IFD [27–29] is particularly useful. This framework, first
proposed by Fretwell and Lucas and used to describe the
male–female sex ratio of bird populations in the Mississippi
River Valley [26], states that individuals will populate habitats according to the amount of available resource to
maximize their fitness while reducing intra-species resource
competition. As an increase in population leads to a decrease
in resource availability, the IFD predicts that the population
for an individual growth rate r, a total cell number r and a
carrying capacity K. While the fitness of an individual is
directly related to the quality of the environment (growth
rate r), it also depends on the number of cells populating
the habitat and from the functional dependence of equation
(6.1) on the cell density r. The fitness V may become zero
or even be negative at high enough cell densities, even if
the growth rate r is large.
The IFD is first applied to a single species inside the twohabitat system shown in figure 9. Křivan & Sirot [28] derive
a population distribution of a two-habitat system equal to
K1/K1 þ K2, for carrying capacities K1 and K2 for habitat 1
and 2. In the MHP system, the carrying capacity of the nutrient-rich and nutrient-poor habitats is limited by the volume of
each MHP rather than the growth rate. Cells are free to move
between habitats, so if the fitness in nutrient-rich MHPs
decreases due to overcrowding (second term of equation (6.1)),
then they can migrate away from the nutrient-rich and move
into the nutrient-poor habitats. As described in [15], the final distribution of a single population is indeed equally divided
between the nutrient-rich and nutrient-poor MHPs.
While it is natural to use the IFD to describe a single
species, a better description of the competition dynamics
can also be achieved by including game theoretical concepts
in the IFD framework [31]. In the remainder of this section,
we will employ the IFD and game theory to describe the
redistribution dynamics observed when WT and GASP cells
compete in a heterogeneous spatial environment.
For a two-species, two-habitat system [28], the fitness of
WT cells in the nutrient-rich MHPs will depend on the
density of WT rw and GASP rg cells as
rw þ aw rg
,
(6:2)
VNR ¼ rw,NR 1 Kw
for a given growth rate rw,NR, a competition parameter aw
and carrying capacity Kw (which have been measured in
the single species control experiments). The parameter aw
describes negative interactions between each strain, which
will affect the final density of a population and is measured
to be approximately 0.4 in [15].
In figure 12a, the density of WT cells in the nutrient-rich
region increases until (rw þ 0:4 rg ) ¼ Kw . This occurs at T ¼
15 h, when the fitness VNR has just decreased to zero. Then,
as the density of GASP cells in the nutrient-rich habitat continues to increase, VNR becomes negative. The fitness VNR
resulting from WT and GASP interactions in the nutrientrich MHPs becomes negative while the pay-off VNP in the
nutrient-poor MHPs is positive owing to low cell density.
So, WT cells can maximize their fitness by preferentially
occupying the nutrient-poor regions and, as a result, the
9
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Figure 11. Conditioned medium experiments. Experiments where the nutrient reservoirs contain conditioned medium (i.e. growth medium partially
depleted of nutrients) are performed. (a) While GASP cells dominate in
both the nutrient-rich and (b) the nutrient-poor habitats, the WT population
does not go extinct and stable levels are maintained in both environments.
will distribute itself to maximize the amount of resource
available to each individual. In equilibrium, each individual
has access to an optimal level of resources and any migration
between patches or change to this distribution will be detrimental to the whole population: the fitness is equal in all
occupied patches and species at equilibrium with respect to
the IFD are stable under small spatial perturbations [31].
The per capita growth rate V is used as a definition of fitness [50] which, for a single individual subjected to logistic
growth, is given by
r
V ¼r 1
,
(6:1)
K
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cells (103)
(a)
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0.5
0
favourable
environment
unfavourable
environment
–0.5
1.0
0.5
0
–0.5
0
0.5
0
–0.5
1.5
10
20
30
time (h)
40
50
60
Figure 12. Measured fitness of WT cells over time. (Adapted from [15].)
(a) The fitness of WT cells decreases below 0 (at T ¼ 15 h, indicated by
the arrow). This occurs as the combined density of WT and GASP cells is
higher than the carrying capacity of the WT species alone. (b) The fitness
of the GASP population inside the nutrient-rich regions remains positive
throughout the experiment.
WT cells redistribute themselves from the nutrient-rich to
the nutrient-poor MHPs.
Similarly, the fitness WNR/rg of the GASP mutant population is shown in figure 12b. GASP cells, in contrast to WT
cells, never reach a point where WNR , WNP, and GASP
cells have no incentive to leave the nutrient-rich MHPs and
populate the nutrient-poor MHPs.
The IFD has been used to explain the population dynamics
presented in figures 9 and 10 [15,20], and a careful analysis of
the fitness plots (figure 12a,b) reveals that the overall population
distribution follows the general principles of the IFD quite well:
cell populations distribute themselves in order to maximize fitness and decrease resource competition. Indeed, as resource
includes both nutrients and available space, WT cells gain fitness by moving away from the crowded nutrient-rich regions
into the sparsely populated nutrient-poor regions.
For the experiments performed under conditioned medium
conditions (figure 11), equation (6.2) can be used to compute
the fitness of the WT population and the result is shown in
figure 13a. The presence of conditioned medium seems to
affect the overall fitness of WT cells in two ways: (i) fitness
decreases below zero in 10 h, compared with 15 h in rich
medium, and (ii) contrarily to rich medium experiments, the fitness remains negative even after cells have migrated into the
nutrient-poor MHPs. While the WT population is able to react
to the GASP population’s sustained growth in rich medium
by migrating into the nutrient-poor habitats, figure 11b suggests
that migration is not sufficient to counterbalance the fitness
losses associated with proliferating GASP mutants.
7. Discussion
We have shown that the growth dynamics of competing
species of bacteria can be interpreted using game theory. In
particular, using a species that has been previously shown
to display selfishness (GASP mutants) to dominate
10
nutrient-rich
time (h)
fitness per growth rate
1.0
(b) 0
1.0
0
10
time (h)
20
10
20
Figure 13. The IFD under conditioned medium experiments. (a) Large-scale
competition dynamics show that while WT cells still survive in the presence
of selfish GASP cells, their final density is severely affected by the deteriorated
environmental conditions. (b) The computed fitness of the WT cells decreases
and stays below 0 much more rapidly than in the nutrient-rich experiment
(fitness becomes negative at T ¼ 9 h compared with T ¼ 15 h in
fresh medium).
cooperative populations (WT cells) in well-mixed experiments [22], results have shown that cooperation can and
will survive if interactions are occurring locally [15,16,18].
Results in other spatial environments have also shown a
similar survival of cooperation in the presence of spatial
structure [17,51]. Furthermore, cooperation is shown to survive in the presence of spatial interactions by spatially
segregating into less populated (and more stressful) regions,
a phenomenon that can be explained when the competition
dynamics are interpreted using the IFD and game theoretical
concepts. However, while this survival is allowed under
many experimental conditions, more stressful environments
precipitate this shift and lower the overall survival potential
of cooperation. It should also be noted that heterogeneity
can also occur in other ways—for example in the form of
stochastic or asymmetric pay-offs [47,52]—however, it is
unknown experimentally whether these too allow survival
of cooperation. But cooperation has been shown to result
from self-organized growth [53].
We believe that game theory provides an intuitive way
to describe complex interactions: it provides a means to
interpret complex situations that cannot be readily explained
using simplified models and, hidden within the framework of
game theory, are profound concepts—such as cooperation and
selfishness–that are difficult to quantify otherwise. Understanding the competitive interplay between cooperation and
selfishness can influence the way complex, multi-species systems
are interpreted. Indeed, as shown in figure 1, the relationship
between cooperation and selfishness gives rise to four broad
classes of distinct interaction dynamics and survival outcome
for cooperation. Furthermore, cooperation is at a competitive
advantage in spatial game theoretical models, and a better
understanding of the cooperator’s survival dynamics—achieved
through the use of bacterial models, for instance—will generate a
more complete mapping of the interplay between cooperation
and selfishness to complement current theories explaining the
survival of cooperation [14,54,55].
From a broader perspective, game theory can be applied to
tumour cells and cancer development to characterize the way a
fundamentally cooperative host competes for its own survival
during cancer development. Whereas sustained growth and
environment-deteriorating behaviours may be observed in
tumour tissues, the shift from an observational to a proactive
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(b)
(a)
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Acknowledgement. We wish to thank Vlastimil Křivan and the anonymous reviewers for their helpful comments, and David Liao for
enlightening discussions.
Funding statement. This work was supported by NSF grant no.
NSF0750323 and National Cancer Institute grant no. U54CA143803.
G.L. is supported by the Chicago Fellows Program. This work was
performed in part at the Cornell NanoScale Facility, a member of
the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (grant no. ECS-0335765).
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