# A Mutation Threshold for Cooperative Takeover

Alexandre Champagne-Ruel111Corresponding author. ID Département de physique,
Université de Montréal
1375 Avenue Thérèse-Lavoie-Roux, Montréal, Quebec H2V 0B3, Canada  Paul
Charbonneau ID Département de physique, Université de Montréal
1375 Avenue Thérèse-Lavoie-Roux, Montréal, Quebec H2V 0B3, Canada

###### keywords:

origin of life – prisoner’s dilemma – cooperation – stochasticity – critical
phenomena – thresholds

One of the leading theories for the origin of life includes the hypothesis
according to which life would have evolved as cooperative networks of
molecules. Explaining cooperation—and particularly, its emergence in favoring
the evolution of life-bearing molecules—is thus a key element in describing
the transition from nonlife to life. Using agent-based modeling of the
iterated prisoner’s dilemma, we investigate the emergence of cooperative
behavior in a stochastic and spatially extended setting and characterize the
effects of inheritance and variability. We demonstrate that there is a
mutation threshold above which cooperation is—counterintuitively—selected,
which drives a dramatic and robust cooperative takeover of the whole system
sustained consistently up to the error catastrophe, in a manner reminiscent of
typical phase transition phenomena in statistical physics. Moreover, our
results also imply that one of the simplest conditional cooperative
strategies, “Tit-for-Tat”, plays a key role in the emergence of cooperative
behavior required for the origin of life.

##  1 Introduction

Attempts to explain the sequence of events leading to the origin of life have
elicited much research over the years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. One of
the leading theories includes the hypothesis according to which life would
have evolved as cooperative networks of molecules [11, 12, 13]. In the context
of this theory, two main elements however remain to be adequately described: a
concrete specification of the physicochemical reality of such networks, and
the means by which those would have evolved. Specifically, if we consider that
life arose through the cooperation of life-bearing molecules, then explaining
the dynamics that led to the evolution of such networks is a priority [14].

While Darwinism explains how macro-complexity derives from micro-complexity,
it does not explain how the latter appears in the first place [15]. It is thus
thought that the RNA world may not have been the earliest genetic system,
favoring instead a system of simpler units having the capacity to encode and
generate information through selection without formal replication—where
candidates include polypeptides, RNA-like polymers, and lipids [14]. This begs
the question of the origin of evolution: when does chemical kinetics become
evolutionary dynamics [16]? In other words, a necessary explanans of origin of
life theories lies in this chemical phase [15], a gray zone between non-life
and simple life which required kinetic competition between information units
prior to the advent of Darwinian evolution [17].

Many observations also  suggest that Darwinian behavior is rooted in
chemistry. Since it was observed in 1960 that the Q$\beta$ bacteriophage
displayed Darwinian behavior, many other in vitro procedures have shown that
other nucleic acid systems are also Darwinian [15]. Voytek and Joyce, for
instance, noted that the biological phenomenon of competitive exclusion also
applies at the chemical level to RNA enzymes [18], while Adamala and Szostak
have investigated how changes in the composition of the membrane of protocell
vesicles caused by a simple catalyst enable the origin of selection and
competition between protocells [19]. More recently, it has been suggested that
both prebiotic evolution and the evolution of biological systems follow
similar equations, and Yeates et al. made this connection explicit [20]. It is
thought that this “pre-life” phase would have taken the form of active
monomers generative of information, and that the replication rate exceeding a
threshold value would have elicited a transition sharing characteristics with
phase transition phenomena [16].

Building on the concept of cooperating networks, Damer and Deamer has proposed
that protocells undergo a primitive version of Darwinian selection for
stability, termed combinatorial selection, and are solely driven by physical
and chemical forces at the molecular level. Gel aggregates subjected to wet-
dry cycles would enable protocell-to-protocell interaction, leading to a form
of network selection [21, 22], and would thus constitute the unit that Woese
and Fox anticipated to be the ancestor of procaryotes and eucaryotes, termed
the progenote [23]. Laboratory studies also support this view of a
collectivist origin of biology presumed to give rise to autocatalyzed
metabolic cycles: previous work on the interaction between RNA and lipid
membranes reveal that oligomers bind stably to the liposome surface—a property
that isn’t shared by individual RNA molecules [24, 25]. Those results suggest
that selection processes in the prebiotic world might yield RNA consortia,
thereby accounting for the colocalization of proteins, RNA, and membrane as a
single unit. These collective phenomena can all be viewed as instances of
cooperative behavior at the chemical level.

Game theory provides the framework of choice to study how cooperation can
emerge in competitive environments, and has inspired much theoretical and
experimental work on this question. It has been used successfully in a wide
array of problems [26, 27, 28], from ecology [29] to computational
neuroscience [30], and is increasingly being applied in the context of
prebiotic chemistry. If Eigen already considered prebiotic evolution to have
properties of games, it has become recently even more evident that game theory
is relevant for biophysics and biochemistry [31]. Evolution is almost always
the co-evolution of the organisms and their environment [31] over dynamic
fitness landscapes [32], which makes game theory is the appropriate framework
for an analysis of such processes.

Examples of applications of game theory at the molecular level include
multiple analyses where viruses are considered as players [33] and subsequent
work on the $\Phi 6$ virus model system, HLSV, and TMV viruses, and on sets of
phages [34]. Recent observations have also been made of RNA and autocatalytic
sets that display competition, inheritance, and cooperative behavior [35, 36,
37]. It is thought that RNA polymers could thus meet the criteria for
behavioral chemistry, make “choices” based on their environment and self-
contained information [38] and more generally that macromolecules—RNA, DNA,
and proteins altogether—could be regarded as players where strategies derive
from their state and properties [34]. Partially unfolded molecules [38] and
chemical networks [14] would possess a type of “memory” and respond to
environmental conditions, subsequently adopting different conformations
leading to differentiated behavior that would be subjected to natural
selection, possibly through the use of “signals”—a feature distinguishing
biochemistry from abiotic chemistry [39]. Formation of protein complexes have
identically been studied as games [40] and genes considered as players [34].
Recent laboratory experiments have also contributed to establishing that game
theory applies at the chemical level in systems where de novo synthesis takes
place (i.e., reproduction instead of replication), and where kinetic selective
forces apply [20].

It is thus becoming increasingly clear that a chemical game theory can shed
some light on the dynamics of macromolecular interactions, and provide insight
on the early emergence of cooperation, the latter being one key idea in
understanding how individual entities can be spontaneously brought together
into the formation of more complex structures where the interaction of the
components with each other promotes the sustenance of the latter. While in
biology cooperation is often explained away using references to concepts such
as kin selection [41], this process cannot be correspondingly applied to cases
where individuals possess no agency. Origin of life theories, insofar as they
seek to provide a detailed explanans of the phenomenon, must adequately
conceptualize how individual components subject to Darwinian principles can
cooperate into forming autocatalytic sets [11, 42, 12], replicating another
ribozyme without repayment [3, 8, 43], or forming whatever early proto-
metabolism is required to jump-start the emergence of more complex lifeforms
[44]. Cooperation is so interweaved into the fabric of complex life—from cell-
cell communication and biofilm formation in microbes [45] to countless
examples in the ecological and social sciences [46, 47, 48, 49, 50, 51]—that
it seems unthinkable to even consider putting forward a theory of the origin
of life that wouldn’t account for the emergence of cooperative behavior in the
first place. As Frick and Schuster eloquently put it: clearly, cells belonging
to the same organism should not compete against one another [52].

There is thus a renewed interest in the importance of cooperation for the
origin of life problem [17] and more generally, as Queller has suggested [53],
in the context of major evolutionary transitions [54], which have involved
increases in complexity and the establishment of new cooperative relationships
[39]. There is notably ample evidence that cooperative molecular behavior, in
general, constitutes an evolutionary advantage [15], thus being selected by
evolution under the right circumstances. Specifically for the origin of life,
cooperation stands as one of the three advantages of prebiotic networks put
forward by Nghe et al., thus sparking the transition from chemistry to biology
[14]. This view is also supported by laboratory experiments [55] and
observations of RNA cooperative networks [14]. Explaining cooperation—and
particularly, its emergence in the context of the origin of life—is clearly a
required element in the description of the transition from non-life to life,
and while much work has been done in explaining the origin of cooperation in
high-order organisms, a convincing description of the way by which cooperative
behavior emerges at the molecular level has yet to be elaborated.

##  2 The Prisoner’s Dilemma

The prisoner’s dilemma (PD) is precisely used in game theory to analyze the
tension between rational, selfish behavior and altruistic cooperation [56, 57,
58] and has been put forth in the analysis of cooperative behavior in various
natural and artificial systems. Two players each have the opportunity to
either cooperate (C) or defect (D), without knowing in advance which strategy
will be adopted by their opponent. Depending on the different combinations of
moves chosen by each player, players will be rewarded with a score (Table 1).
While mutual cooperation yields the maximum mean reward for the two players,
exploitation (i.e., defecting when the opponent chooses to cooperate) leads to
the maximum reward for the exploiting party and nil for the exploited
player—mutual defection thereafter leading to the worst mean reward for both.
This scoring scheme highlights the tension between selfish behavior that
either leads to an immediate advantage or a poor outcome, on one hand, and
cooperation as a compromise that leads to a reasonable outcome for both
players on the other. Generalizing this principle leads to the formal
constraint

$\displaystyle T>R>P>S,$ (1)

with $R$ being the reward, $T$ the tentative payoff, $S$ the sucker’s payoff,
and $P$ the punishment [59]. The PD as studied in game theory also applies to
contexts where the players possess no capacity for rational reasoning, and is
thus a useful framework in analyzing biochemical cooperation phenomena in the
context of prebiotic chemistry.

A temporal extension of the PD known as the iterated prisoner’s dilemma (IPD)
is the canonical model used to study the interaction between players. The game
is played for several rounds, either for a definite number of times known to
the players or by stopping the game with some finite probability. The players
then follow a strategy that defines in advance their next move, taking—or
not—into account the information available to them through the game at that
time (e.g., their previous moves and those of the opponent). Basic strategies
include the case where a player always cooperates (hereafter ALLC), always
defects (ALLD), shows mutual reciprocation by mirroring the opponent’s
previous move (TFT, for “Tit-for-Tat”), or acts randomly (RND).

| Cooperate | Defect
---|---|---
Cooperate |  3 R |  0 S
Defect |  5 T |  1 P
Table 1: Score matrix of the Prisoner’s dilemma (PD). Numbers refer to the
PD’s score matrix as traditionally defined [57], indicating the reward of the
player adopting the strategy in the leftmost column. Mutual cooperation leads
to the best possible mean outcome, while defection either leads to the
absolute maximum or the absolute minimum reward. Letters refer to the general
form of the score matrix for the PD: mutual cooperation leads to the reward
payoff (R), while mutual defection leads to the punishment (P).
Exploitation—with one player defecting while the opponent chooses to
cooperate—leads to the temptation payoff (T) and sucker’s payoff (S). The game
satisfies the PD constraint when $T>R>P>S$.

Using agent-based modeling of the iterated prisoner’s dilemma, we investigate
the emergence of cooperative behavior in a spatially extended setting and
characterize the effects of inheritance and variability. We demonstrate that
there is a mutation threshold above which cooperation
is—counterintuitively—selected, which drives a dramatic and robust cooperative
takeover of the whole system. Moreover, our results also imply that one of the
simplest conditional cooperative strategies, “Tit-for-Tat”, plays a key role
in the emergence of cooperative behavior required for the origin of life.

Rather than being considered a lattice-based simplified representation of a
specific system, our simulations are best viewed in the context of
constructive biology which seeks to identify universal patterns and mechanisms
that are independent of specific physical/chemical contexts and substrates
[60, 10, 61]. A cooperative takeover, by which we mean the rise to a
predominance of reactive cooperating strategies in our simulations represents
a prime example of emergent dynamics: namely, the appearance of novel features
where cooperation is preeminent on global (“macroscopic”) scales that cannot
be directly inferred or predicted, even from a complete knowledge of system
behavior and interaction rules at small (“microscopic”) scales, with clear
Darwinian dynamics that should, by all means, favor selfish behaviors.
Furthermore, the emergence of cooperation as an emergent phenomenon witnessed
in our simulations—and the wide range of the parameter space over which it
occurs—suggests that the results obtained here are not strictly dependent on
the specific implementation of the model.

##  3 Simulation Design and Methods

Spatial aspects of prebiotic settings are thought to have played a significant
role in life’s early chemistry. Examples include montmorillonite clay and
other mineral surfaces, which may have promoted the polymerization of
activated mononucleotides [22]. Spatial games have likewise been the subject
of much interest in the past decades (e.g., [62, 63, 64, 65, 66, 67]). Our
model implements IPD games on a $128\times 128$ Cartesian lattice with
periodic boundary conditions where each site is occupied by a player adopting
one of the four strategies consisting of ALLC, ALLD, TFT, and RND. Strategies
are first distributed on the lattice randomly in equal proportions (Figure
1A). At each new iteration of the model, every player plays IPD games of $M$
rounds against every other player inside their Moore neighborhood, i.e., the
eight sites surrounding them, and record their score (Figure 1B). In a second
step, the strategy of every player of the lattice is replaced by the highest
scoring strategy in their Moore neighborhood, if applicable (Figure 1C). This
evaluation of the Moore neighborhood is randomized as not to introduce any
spatial bias in the simulation: comparing scores between each strategy and its
neighbors was done in a random order as not to favor a specific spatial
pattern if two players obtained an even score. This process is then repeated
for $T$ iterations, which leads to fluctuations in strategy frequencies and
the emergence of spatial self-organization dynamics. Simulations were carried
out for $T=500$ iterations, which allowed a relaxation of the initial
conditions sufficient for population dynamics specific to simulations
parameters to materialize independently of the initial conditions. Finally,
IPD game lengths of $M=2000$ moves were chosen to allow meaningful behavior in
cases of very low error rates.

Figure 1: Evolution algorithm. (A) At the beginning of each simulation,
strategies ALLC (“always cooperate”, green), ALLD (“always defect”, red), and
TFT (“mutual reciprocation”, blue) are placed randomly on the lattice. Then
the following process is repeated at each lattice site: strategies play $M$ PD
games against every neighbor in their neighborhood, and the scores of each
player are recorded. (B) In a second step, the score of each strategy is
compared against their neighbor’s score, and (C) the highest scoring strategy
(boxed in yellow) is assigned to the site being examined. This process is
repeated for each of the $T$ iterations of the model.

Figure 2 shows snapshots for the first fourteen iterations of the model for a
simulation that includes ALLC, ALLD, TFT, and RND. In the vast majority of
simulations carried out, including deterministic simulations initialized with
the parameters specified above, the RND strategy is promptly evacuated from
the lattice most of the time and was thus not further included in the initial
strategy distribution. Deterministic simulations quickly (i.e., in a few tens
iterations) reach a stationary state where TFT largely dominates. This is
consistent with the known success of TFT [57, 59].

Figure 2: First fourteen iterations of the model for a deterministic
simulation (i.e., $p=0$): strategies ALLC (green), ALLD (red), TFT (blue), and
RND (“play randomly”, pink) are initially placed randomly on the lattice, then
IPD games are played and the highest scoring strategies are propagated. For
such a simulation where no stochasticity is included—each move is determined
by the assigned strategy and the players make no mistake playing the PD—a very
brief invasion by ALLD precedes an eventual predominance of TFT. RND is
quickly eliminated from the lattice, as in most of the simulations over the
parameter space of the model. The majority of such deterministic simulations
become stationary after only a few tens iterations.

The influence of stochasticity has also been discussed at length not only in
relations to the origin of life problem and microbial life [68, 69, 70], but
also in the context of game theory and the PD itself [71, 72, 73, 74, 75, 76,
67, 77, 78, 79]. In the prebiotic context, it is generally agreed upon that an
external supply of energy is required to drive non-equilibrium chemical
reactions leading to the buildup of catalytic cycles and complex biomolecules
[80, 81, 82, 83, 84, 85, 86]. However, strong external energy fluxes generally
also imply strong thermodynamical fluctuations, which may be detrimental to
the autonomous self-organization of complex prebiotic or early-life chemistry.
Such fluctuations translate into either higher error rates in the interaction
of early replicators, higher mutation rates, or both. For example, in an
analysis of UV transmission in a prebiotic setting, the conclusions of Ranjan
et al. are unambiguous: insofar as prebiotic chemistries are shown to be
affected by UV irradiation, they must be demonstrated to either invoke a UV-
shielded milieu or be so productive as to outpace UV degradation [87]. TFT’s
success however notably relies on being placed in an environment where
interactions are reliable—that is, where no “mistakes” are made by the
players. Noisy environments, therefore, destroy any ongoing mutual cooperation
between TFT strategies: as soon as a defection happens in the game, two TFT
strategies playing against one another will enter a perpetual cycle of mutual
defection.

If TFT’s success in error-free environments has now been known for some time,
a central question relevant to the origin of life scenarios concerns the
emergence of cooperative behavior in stochastic (i.e., noisy) environments. We
have therefore implemented an error rate according to which the players will
sometimes diverge from their strategy. For each IPD game, both players will
thus play according to their assigned strategy, but will make mistakes (e.g.,
defect instead of cooperating with the other player) with a probability $p$.
Varying this error rate impacts system behavior globally, both in terms of
populations dynamics and in terms of the spatial structures produced. Error
rates have been sampled from a lognormal distribution of mode $\hat{p}$, with
logarithmic mean $m$ and variance $s^{2}$ (Equation (2) in the Supplementary
Material, hereafter SM). Values of $p\in[10^{-6},5\times 10^{-1}]$ have been
sampled, which represent values of the same order than presumed error rates in
early replication mechanisms [88, 89, 90]. A detailed description of the
procedure used for evolving the systems and the implementation of stochastic
elements is also included (see Algorithm (1) of the SM).

In the context of the RNA world hypothesis, autocatalytic systems are thought
to have evolved through the selection of both heritable and variable
components. In a minimal model of early replicators, the error rate can thus
be both subject to heritability and variability. We hence tested the impact on
simulation dynamics of the heritability of error rates by including, for a
subset of simulations, the heritability of the winning player’s error rate. We
also introduced a mutation probability according to which a mutation,
corresponding to a correction proportional to $s$, can occur when a player
inherits another player’s error rate (c.f. Equation (4) of the SM). Mutation
rates $\mu\in\left[10^{-4},1\right]$ were sampled, which correspond to a few
mutations per simulation up to some critical value typical of an error
catastrophe [68]—a characteristic of replicating systems that has been
compared to phase transition phenomena in statistical physics with replication
error rate acting as the order parameter [91, 92, 93, 94, 95, 96]. The
presence of an error catastrophe is indeed predicted by the quasispecies
theory: above the error threshold value, replication errors lead to
populations being submerged by unfit individuals that decrease the overall
fitness until the species goes extinct [68].

##  4 Results

###  4.1 Spatiality and Population Dynamics

Simulations evolved with strategies making mistakes (i.e., $p>0$) lead to a
wide range of system behavior reflected in the spatial organization of
strategies on the lattice. Two notable examples of such spatial configurations
include regimes with time-varying fractal distributions of strategies
(population evolution and final lattice state shown on Figure 3A,B) and
clustering patterns (Figure 3C,D). While the evolution of the population
reaches a dynamical equilibrium in the first case, the equilibrium becomes
static in the second. In the latter case, the system’s dynamics leads to the
formation of isolated domains of ALLC, which otherwise would not survive in
exploitative environments, that can be sustained while surrounded by ALLD
players. Likewise, ALLD strategies can persist in environments where TFT is
present by preying on ALLC players. Spatiality is thus a required condition
for such dynamics to emerge, and the score of surrounding parasites (e.g.,
ALLD players) cannot, in this case, exceed that of the central cooperator
[59]. This has been observed when the duration of the interaction between
players is long enough—i.e., game length $M\gtrsim 2000$ PD moves—as to
supersede the temporal dynamics of the simulation itself, hence this parameter
value was conserved for all simulations.

Figure 3: Population evolution and final lattice state for two simulations
with $\hat{p}\simeq 3.16\times 10^{-5}$ (A,B) and $\hat{p}\simeq 3.16\times
10^{-2}$ (C,D). Simulations were carried out on a Cartesian lattice of size
$L=128$ with periodic boundaries over $T=500$ iterations of the model, with
iterated prisoner’s dilemma (IPD) games of $M=2000$ moves. In the second
lattice (D), formations of ALLC cooperating together successfully survive
while being surrounded by defectors. An animation of the two simulations is
available in the Supplementary Material of the online version of this article.

In contrast, randomizing the players’ neighborhoods over the whole
lattice—i.e., evolving environments that are “fully mixed”—prevents the
formation of such organized spatial structures. Spatiality is thus an
important determinant of population dynamics by allowing the survival of
cooperative entities in adverse conditions by means of structure formation
[97, 98, 99, 100].

The final states of the lattice are in this sense completely independent of
the (randomized) initial conditions. The trajectory of the system in phase
space end up in the majority of cases on attractors, leading to spatial
structures such as the ones shown in Figure 3D, or in limit cycles where
emerges cyclical variations of the populations of each species (e.g., Figure
3A). The state of the lattice can thus either become fixed, as in the former
case, or remain in dynamic equilibrium as in the latter.

###  4.2 Influence of the Error Rate

Many of these features associated with spatiality depend on the precise value
of the strategies’ error rate: final population frequencies for ALLC, ALLD,
and TFT relative to the most probable value of the initial error rate
distribution $\hat{p}$ are shown on Figure 4A. An initial error rate was
assigned to every player, drawn from a lognormal distribution (see Equation
(2) in the SM) with most probable value $\hat{p}$ shown in abscissa and shape
parameter $s=5\times 10^{-1}$. Final population fractions are averaged over an
ensemble of 20 simulations per sampled value of the error rate, and shaded
areas indicate the standard deviation of the final population fractions. As
$\hat{p}$ is varied from one simulation to the next, sharp transitions occur
repeatedly in the population frequencies and their patterns of spatial
distributions.

Figure 4: Final populations fractions for error rates that are immutable,
heritable, and subject to mutations. Relative population frequencies with
respect to the total population of strategies following either a strategy of
unconditional cooperation (ALLC), unconditional defection (ALLD), or mutual
reciprocity (TFT), as a function of the most probable error rate of the
initial distribution $\hat{p}$. Final population fractions are averaged over
samples of 20 simulations with random initial conditions, with shaded areas
proportional to the standard deviation of final populations. Simulations were
all carried out on a 2D periodic lattice of side length $L=128$ sites over
$T=500$ iterations of the model, with IPD games of $M=2000$ moves. Error rates
were initially set according to a lognormal distribution with the most
probable values in abscissa and a fixed shape parameter $s=0.5$. (A) Error
rates are immutable over the duration of the simulation. (B) Error rates
become heritable (i.e., the player losing the game adopts its opponent’s error
rate in addition to its strategy). (C) Introduction of a mutation probability
$\mu=10^{-4}$. While heritability readily contributes to an increase in
cooperative behavior—mostly at lower ($\lesssim$$10^{-3}$) error rates—the
introduction of mutations (C) is the primary driver of the dominance of
cooperative behavior, allowing TFT to invade the lattice on a much larger
region of the system’s parameter space.

At very low error rates ($p\sim 10^{-6}$) cooperators have the upper hand: TFT
quickly eliminates ALLD but can still become trapped in a mutual defection
process with a nonzero probability when playing against itself, hence the
final domination by ALLC. In contrast, at very high error rates
($p\sim\leavevmode\nobreak\ 0.5$) TFT performs poorly against ALLD (i.e., it
starts with cooperation, then follows ALLD’s moves—a rather poor behavior).
Nothing then further prevents ALLD from eliminating ALLC and invading the
whole lattice.

###  4.3 Heritability and Mutations of the Error Rate

Introducing heritability of error rates—i.e., the player losing the PD game
adopting not only the opponent strategy but also its error rate—leads to
increased variability in the outcome of simulations (Figure 4B). At very low
error rates, TFT can now decrease its mean error rate through the successive
selections of winning strategies; this allows an increase in TFT’s relative
population frequency and a corresponding decrease in ALLC’s. Similarly, at
very high error rates TFT’s decreased mean error rate allows it to be much
more successful against ALLD. However, by also making error rates subject to
mutations the system’s behavior again undergoes significant changes. Keeping
all other parameters identical, a mutation probability $\mu=10^{-4}$ player-1
$t^{-1}$ is introduced (Figure 4C) which, when fulfilled, applies a
multiplicative correction to a player’s error rate proportional to the shape
parameter $s$ of the initial error rates distribution (see Equation (4) in the
SM). While system behavior at $\hat{p}\lesssim 10^{-3}$ does not differ
significantly from previous simulations, when $\hat{p}$ increases above this
value there is a sharp transition in the relative frequency of TFT
populations. This is explained by the fact that the extent to which TFT
represents a successful strategy is inversely correlated with the error
rate—that is, TFT lacks any error correction mechanism; if TFT plays against
itself, the first error introduced in the game is bound to send both players
into a spiral of mutual defection until the game ends [101]. Error rates are
thus driven towards lower values when mutations are allowed, which ultimately
leads to the complete invasion of the lattice by TFT.

This process was repeated while varying the mutation probability itself. For a
fixed error rate initial distribution with $\hat{p}=10^{-4}$ and shape
parameter $s=5\times 10^{-1}$, Figure 5A displays another statistical ensemble
of simulations where the final populations again represent the average over 20
simulations with distinct random initial conditions. Increasing the mutation
probability progressively (note the logarithmic horizontal axis) increases
TFT’s final relative population, with the lattice being completely invaded at
$\mu\gtrsim 10^{-2}$, until a breakdown of cooperative behavior occurs in the
vicinity of $\mu\gtrsim 0.8$, which is reminiscent of the error threshold that
replicating systems can sustain as was shown by Eigen [68]. The transition
towards domination by TFT depends on both the mutation rate $\mu$ and the
shape parameter $s$, while mutation rates exceeding a critical threshold
($\mu>0.8$) further prevent generalized cooperation from emerging for a wide
range of error rates.

Transition in the system behavior towards an invasion by TFT is however not
only dependent on mutation probability but also on the magnitude of the
correction $\Delta_{p}$ being applied. An increase in both parameters leads to
an overall increase in the effect of mutations. Statistical ensembles were
therefore analyzed while varying both parameters, for several different modes
of the initial error rate ($\hat{p}\in\\{10^{-1},10^{-2},10^{-3},10^{-4}\\}$),
again averaged over 20 simulations for each parameter value. A similar
behavior was seen regardless of $\hat{p}$: when a threshold that depends on
both $s$ and $\mu$ is reached, a transition occurs after which TFT’s relative
population fraction $f_{\text{TFT}}\to 1$ (Figure 5B). As $\log_{10}[\mu]\to
0$ (i.e., $\mu\to 1$, inset of Figure 5A) there is a significant decrease in
cooperative behavior. There is therefore an optimum region in $\mu$-$s$ space
that reliably leads to the emergence of cooperative behavior: further increase
of the mutation rate past a critical threshold instead leads to this marked
decrease in population fitness of cooperators. Remarkably, TFT nonetheless
maintains complete dominance all the way up to this error threshold.

The final lattice state for five simulations taken at regular logarithmic
intervals through the transition is shown in Figure 5C. Mean error
distributions are shown in Figure 5D, where the dotted curve indicates the
initial distribution of error rates for all species at $t=0$, and grey bars
refer to the final distribution averaged over the last 25 iterations of the
model. Colored curves denote the distribution specifics for ALLC, ALLD, and
TFT. The transition towards an invasion of the lattice by TFT is concurrent
with a marked decrease of its mean error rate as $\mu$ increases, while that
of the other species remains mostly unchanged. Finally, Figure 5E shows
population (colored curves) and stationarity (black curve) evolution—the
fraction of lattice sites that changed strategies at each iteration (Equation
(5) in the SM)—for all species, while Figure 5F shows the mean error rate.
Near the transition ($\mu\sim 10^{-2}$), large fluctuations both in population
dynamics and mean error rates can be seen and are reminiscent of behavior at
the critical point of phase transitions [102].

Figure 5: Transition towards TFT-mediated cooperation. (A) Relative population
frequency for each strategy as a function of the mutation rate $\mu$.
Simulation parameters are identical as in Figure 4, and the data presented
here refers to simulations initialized with an initial distribution of error
rates having $\hat{p}=10^{-4}$ and $s=0.5$. Final populations are averaged on
ensembles of 20 simulations with random initial distributions of strategies,
with shaded areas proportional to the standard deviation in final population
fractions. Increasing the mutation probability $\mu$ drives a continuous
transition of the final population fractions towards invasion by TFT until a
breakdown of cooperative behavior occurs at very high error rates (inset). (B)
RGB-coding for relative population frequency of mutual cooperative behavior
(TFT) as a function of mutation rate $\mu$ and shape parameter $s$ of the
error rate distribution. Measures are derived from 100 samples of $s$ and
$\mu$ combinations, and the results for each parameter configuration pair are
averaged over 10 simulations with random initial conditions. (C–F) Final
lattice state, error rates distributions, and temporal population,
stationarity index, and mean error rate evolutions of five representative
simulations taken from the statistical ensemble shown on panel A. The initial
distribution of error rates is shown as a dotted curve on (D), grey bars
indicate the error rates distribution for all strategies, while colored curves
indicate the ones specific for each strategy. Panel E shows population
evolution with colored curves, while the black curve indicates the
stationarity of the simulation. Panels F displays the mean of the error rate
for each strategy.

##  5 Discussion

Origin of life theories must be able to explain, from a Darwinian perspective,
the emergence of cooperation between unrelated agents to achieve an exhaustive
description of the transition from non-living matter to biological complexity.
We have shown that on a spatially extended system subject to stochastic
perturbations, cooperation can definitely be favored against defectors when
selection is coupled with heritability and variability. Moreover, this sudden
invasion of the system in which the reciprocal strategy TFT dominates the
system notwithstanding stochastic perturbations, which we term cooperative
takeover, represents a prime example of emergent collective phenomena. Those
results are consistent with the view put forth by Damer and Deamer that the
emergence of life presupposes novel emergent functions relying on “network
effects” [22]. Cooperative takeover also shares many characteristics of
traditional phase transition phenomena such as a dramatic spatial
reorganization on a scale comparable to the size of the system and the
presence of large fluctuations in the neighborhood of the critical point
(Figure 5). As has been frequently suggested, major evolutionary transitions,
and in particular (the emergence of) life also presents many characteristics
of critical phenomena [103, 104, 105, 106, 107]. Life thus appears as a
spontaneous symmetry breaking, with the order parameter of the phase
transition being suggested to be alternately organic molecules chirality [10],
replication fidelity [16], or information exchanged [106].

Many important features of our results are directly dependent on the spatial
medium onto which the IPD games are played. Formations of ALLC strategies
being able to resist invasion by predatory ALLD are prime examples. In fact,
even in a dynamic scenario the spatial grouping of strategies into multiple
colonies, propagating through traveling wavefronts, allows cooperators to
evade complete eradication by defectors against which they have no other
defense mechanism (see Figure 6 for an example of such traveling waves in our
simulations). Correspondingly, diverse models of hypercycles and RNA
cooperators have been shown to survive more easily in two-dimensional spatial
settings compared to well-mixed systems [43]. In our model, randomizing the
neighborhood of the strategies—thus evolving the systems precisely in “well-
mixed” environments—prevents this takeover of TFT altogether.

Figure 6: Temporal snapshots from a simulation including ALLC (green), ALLD
(red) and TFT (blue) with initial error rate distribution parameters
$\hat{p}=10^{-6}$ and $s=1$, and for which $\mu=10^{-4}$, taken at regular
intervals of 25 iterations each. In this regime ALLC and TFT coexist while
ALLD is a predator for ALLC. Traveling wavefronts of ALLC propagate through
time on the lattice, although the fragmentation of ALLC populations into
spatially distinct colonies acts as a barrier guarding against complete and
immediate invasion by ALLD. The dynamics of the three strategies is
reminiscent of a forest-fire model where there is an accumulation of trees
(ALLC) according to a growth rate dictated by simulation parameters, which is
conducive to dramatic reorganizations of the lattice following ignition (ALLD
mutants).

A recurrent characteristic in many of these prebiotic scenarios is the
importance of thresholds delimiting transitions between different levels of
organizations. Their presence is well-known in biology—a prime example being
Eigen’s error threshold discussed above. In the context of the origin of life
theory, the transition towards the living state has indeed often been
interpreted in terms of critical phenomena [16, 10, 106, 107], while the
concept of catalytic closure in autocatalytic sets [42, 11] is rooted in that
of critical transitions observed in random graph theory [108]. Specifically,
in the prebiotic context, many other examples of thresholds have furthermore
been suggested [96] such as that of spontaneous polymerization [109] or self-
assembly of compartments [110]. In addition to the behavior expected at the
error threshold for high values of the error rate $\mu$ included in our model,
the results we have presented here clearly imply the presence of such a
transition driven by an increase of the mutation rate past a critical value
(c.f. Figure 5A), which ultimately leads to a predominance of the cooperation
behavior assumed to be an essential characteristic of various scenarios for
the transition towards the living state. While a precise characterization of
this transition—namely to determine if it represents a true phase
transition—remains to be established, it shares many of the common features
associated with critical phenomena typically seen in physics.

Relatedly, information theory has often been invoked in explaining complexity
[111, 112, 113]. Specifically, it has been hypothesized that major
evolutionary transitions involve changes in the way information is stored and
transmitted [103], and communication of information is a prerequisite for the
emergence of cooperation [54]. The emergence of TFT as one of the simplest
conditional strategies to encode also contributes to laying out the
foundations for theories of life that, in the spirit of determining which
features are universal in every possible living form, consider that its
fundamental character lies in its capacity to process informational content
[114, 115, 116]. If life is defined as “information that copies itself” then
the origin of life problem becomes one of the origins of this information.
Likewise, the problem that ought to be solved concerning the transition from
the abiotic to fully-functional replicators encoding their own replication
mechanism, in the context of models of the emergence of life on Earth such as
the RNA world hypothesis, is an informational one. It is precisely this gap in
the evolutionary history of the information content of early replicators that
a minimal information-processing strategy such as TFT —whatever its precise
physico-chemical implementation in a given early lifeform— could bridge.

##  6 Conclusions

Analyzing the origin of life problem, recent work has emphasized determining
what elements could be considered universal in our definition of life, as
opposed to contingent ones such as the specific substrate or precise
evolutionary history of life on Earth [10]. The presence of abiotic
cooperation that preceded and subsequently allowed more complex biological
organisms to evolve is most likely one of those universal elements.

Using agent-based modeling of the iterated prisoner’s dilemma, we investigated
the emergence of cooperative behavior in a stochastic and spatially extended
setting and characterized the effects of inheritance and variability. Our
results demonstrate that there is a mutation threshold above which a dramatic
and robust cooperative takeover of the system takes place in a manner
reminiscent of a phase transition, and that the simplest conditional strategy
“Tit-for-Tat” plays a key role in the process.

Generally speaking, one would tend to expect that the emergence and sustenance
of cooperation in an evolutionary context would require a reasonable level of
stability regarding the environment in which interaction between cooperating
entities should take place. Stability here may refer, among other factors,
both to a sufficient level of fidelity as to the mutual interactions between
replicators, whatever the exact nature of these early interactions may be, and
to a reliable inheritability of the features favoring cooperative behavior,
whatever the mechanisms of early replication involved.

Our simulations results paint an altogether different picture with regard to
the impact of stochastic perturbations. Despite the presence of high error
rates in interaction and the contribution of stochastic processes in the
dynamics of heritability, the emergence and predominance of cooperators are
sustained all the way up to the Eigen error catastrophe. Placed in the context
of current origin-of-life ideas and taken at face value under the assumption
that cooperation is an essential building block of complexity, our simulations
generally support the idea that the emergence of life—or at the very least of
biochemical complexity—is a robust phenomenon that is not so easily disrupted
by external perturbations, a result in agreement with the suggestion by Damer
and Deamer that progenotes would have been selected for their stability [22].
Specifically, environments incurring high fluxes of ionizing radiation from
very active host stars or exoplanetary surfaces subjected to strong
environmental perturbations from tectonic or volcanic activity may not pose
such strong constraints on the emergence of life—which is likewise
qualitatively consistent with the early appearance of life on Earth.

##  Appendix A Supplementary Material

###  A.1 Details of the implementation

Simulations were implemented on a $128\times 128$ Cartesian lattice with
periodic boundaries. An analysis of diverse lattice sizes was conducted to
assess the convergence and stability of simulation dynamics, and the chosen
lattice size represents a minimum at which the finite population size ceases
to generate instabilities (i.e., stochastic extinctions of entire species) due
to the limited size of the system. We sought to reduce to a minimum small size
effects, i.e., by modeling the system size at least one order of magnitude
larger than the characteristic scale of structures emerging in the
simulations.

Three distinct series of simulations were conducted. For the first one, error
rates were assigned at the same time as the initial conditions for every site,
and remained fixed for the whole temporal evolution of the system. Strategies
thus executed their respective moves at every turn of the PD against each
opponent in their Moore neighborhood for the $M$ rounds of IPD games, and each
move was reversed with an error probability $p$ setting the error rate. In a
second series of simulations, error rates became heritable at the same time as
the strategies; when a higher-scoring strategy replaced another site’s
strategy, the latter also adopted the winner’s error rate. In those
simulations, the error rates were distributed at the same time as the initial
distribution of strategies on the lattice. By sampling the logarithm of the
error probability from a normal distribution with mean $m$ and standard
deviation $s$, i.e.,

$\displaystyle
f(\log_{10}p)=\frac{1}{s\sqrt{2\pi}}\exp{\left[-\frac{1}{2}\left(\frac{(\log_{10}p)-m}{s}\right)^{2}\right]}$
(2)

we thus obtained a lognormal distribution of error rates with most probable
value $\hat{p}$ (see Figure 7A).

Different values of the mean $m$ of error rate logarithms (and resulting mode
$\hat{p}$ of the error rate distribution) were sampled in the interval
$\left[10^{-6},10^{-1}\right]$, which was chosen so that near the lower bound
errors had a negligible impact on simulation dynamics. Different values of the
shape parameter $s$ were also sampled and control the extent of the fat tail
of high error rates (Figure 7B).

Figure 7: Distribution of error rates. (A) Probability density functions for
three distributions of the error rate with most probable value
$\hat{p}=10^{-3}$ and shape parameter $\log_{10}s={-0.75,-0.50,-0.25}$ from
which the error rates are drawn, where the logarithm of the error rates
follows a normal distribution. Initial distributions of error rates are drawn
from the distribution, while mutations correspond to corrections that are
parametrized from a similar distribution with $m=0$. The lognormal
distribution was chosen so that mutations increase or decrease the error rates
with the same distribution probability, regardless of the value of the error
rates (note the logarithmic horizontal axis while the vertical axis is
linear). The value of $m$ thus specifies the mode $\hat{p}$ of the
distribution, and the shape parameter $s$ the extent of mutations. (B)
Probability density functions for the same three distributions of error rates,
shown this time with axis scaling reversed with respect to panel A,
highlighting the effect of varying parameter $s$. While increasing the value
of the shape parameter has a minimal effect on the mode of the distribution,
it extends the tail of the distribution to include higher values of the error
rate.

In a third series of simulations, error rates were finally not only heritable
but also subject to mutations. When a strategy thus replaced a lower-scoring
strategy, if the mutation probability $\mu$ was fulfilled a correction
$\Delta_{p}$ was drawn from a normal distribution identical to that of
Equation (2) but with logarithmic mean zero,

$\displaystyle
f(\log_{10}\Delta_{p})=\frac{1}{s\sqrt{2\pi}}\exp{\left[-\frac{1}{2}\left(\frac{(\log_{10}\Delta_{p})}{s}\right)^{2}\right]}.$
(3)

This correction was applied to the error rate at iteration $t$ to obtain the
strategy’s error rate at the next iteration, i.e.,

$\displaystyle\log_{10}\left(p^{t+1}\right)=\log_{10}\left(p^{t}+\Delta_{p}\right),$
(4)

as described in Algorithm 1.

Error rates were upper-bounded at $p_{i}<0.5$ (implying that the strategy’s
behavior was identical to the random strategy RND) but no lower boundary was
enforced—i.e., error rates were allowed to become as small as directed by the
evolution of the system.

Given the set $\textbf{G}^{t}$ of strategies on the lattice at time $t$,

the set $\Sigma^{t}$ of Moore neighborhoods,

Input : scores $S$ of the players after $M$ IPD rounds

Output : error rate $p_{i}^{t}$ of the player with the lowest score

foreach player $i$ on lattice do

$\texttt{max\\_score}\leftarrow S_{i}$

foreach player $j\subset\Sigma_{i}^{t}$ do

if $S_{j}>\texttt{max\\_score}$ then

$\texttt{max\\_score}\leftarrow S_{j}$

$\texttt{ind\\_max}\leftarrow j$

$G_{i}^{t}\leftarrow G_{\texttt{ind\\_max}}^{t}$

if heritability of error rates then

$p_{i}^{t}\leftarrow p_{\texttt{ind\\_max}}$

if variability of error rates then

if probability $\mu$ is fulfilled then

Apply a mutation $\Delta_{p,i}$ on $p_{i}^{t}$

Algorithm 1 Strategy replacement procedure.

###  A.2 Evaluation of quantitative data

In order to control the effects of stochastic fluctuations in the resulting
analysis, we have averaged the derived quantities on statistical ensembles
(e.g., Figures 4 and 5A) over the last 5% iterations of each simulation.
Statistical ensembles comprised ten to twenty realizations of the simulation
evolved with randomized initial conditions. Shaded areas in the figures
represent the standard deviation over mean values in those statistical
ensembles.

The stationnarity index $n$ shown on Figure 5E (black curve) represents the
fraction of sites whose strategy has been replaced since the last iteration.
Let $\mathbf{G}^{t}$ be the matrix defining the strategies on the lattice at
iteration $t$ and $g^{t}_{ij}$ its values, then the stationnarity index at
time $t$ is the number of sites where a strategy replacement has been made
since the last iteration, normalized by the size of the lattice,

$\displaystyle
n^{t}=\frac{1}{L^{2}}\sum_{\mathbf{G^{t}}}\left[1-\delta\left(g^{t}_{ij},g^{t-1}_{ij}\right)\right],$
(5)

where $\delta_{ij}$ is the Kronecker delta.

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