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Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals
GA, Hamilton’s Rule, altruism, cooperation, Green Beard, Selfish Herd.
this project we decided to investigate the general emergence and selection of social (cooperative and/or altruistic) behaviour from an evolutionary and hereditary point of view simulating several evolutionary dynamics in-silico. Altruism is defined as a behaviour which harms the individual exhibiting it and benefits the recipients. For this reason studying the evolutionary pressure of social traits is not as straightforward as for those traits influencing only the individual fitness. In order to describe the evolutionary dynamics and the impact of the altruistic behaviour a general rule has been devised by W.D. Hamilton[1]:
[\label{eqn:hamilton}
rB > C] Hamilton’s rule shifts the point of view on natural
selection hinting the notion that it favours genetic success rather than
reproductive success per se. As a result, the Hamilton’s rule underlies
the theory of inclusive fitness in which an organism’s genetic success
is generated through cooperation and altruistic behaviour rather than
simply from its own genetic heritage. Hamilton’s rule is formulated in
very generic terms and, due to this reason, has been target of many
criticisms as well as specific contextualization and mathematical
derivations. Some of our scenarios were developed using such
interpretations and derivations as guidelines. We have implemented
simulations considering a generic altruistic action as well as a
behaviour specif scenario not strictly explainable using the Hamilton’s
rule. In particular we investigated the Selfish Herd theory proposed by
Hamilton himself .
Moreover, we have to consider and acknowledge that, while our general
models support several biological interpretations, they are incapable of
capturing the social and environmental components of altruistic
behavior, which exert a huge influence on the latter. In addition, we do
not capture complex and still to be fully understood genetic mechanisms
as epigenetic regulation in eukaryotes and horizontal gene transfer in
bacteria which probably play a very important role .
The first natural contextualization of Hamilton’s rule is kinship
altruism, which defines relatedness in terms of blood ties. In this
formulation the actor exhibits an altruistic behaviour when the
recipient of such behaviour is part of the actor’s kin.
Relatedness may also be defined more broadly as any genetic similarity.
Many cooperative behaviors may have evolved via common descent or by
other mechanisms such as the Greenbeard effect .
In contrast with the concept of relatedness, cooperative behaviour can
also emerge from a selfish action. Specifically, the Selfish Herd theory
states that individuals within a population attempt to reduce their
predation risk by putting other conspecifics between themselves and
predators . Such antipredation behaviour results in aggregations which
indeed advantages whole group.
The source code of all our simulated scenarios can be found at our
GitHub
repository.
In the beginning we devised a simple simulation to provide a basis for further simulations and to explore the fundamental take home message of the Hamilton’s rule in the kinship domain. For this first simulation, we have encoded the genotype of the agents as a binary number, where 1 codes for altruistic behaviour and 0 for the egoistic one. We then defined a reproductive model which mimics the outcome of sexual reproduction even though we decided to use an haploid set of chromosomes for simplicity. We where able to do so without questioning the applicability of the Hamilton’s rule as the aforementioned mathematical derivations demonstrate that the Hamilton’s rule is not affected by the ploidity and reproductive model . In our reproductive model two individuals form a couple which will produce three offspring, each of which has a (50%) chance of inheriting the genotype of one of the parents. In addition, each offspring has a small chance of being mutated in accordance with biological mutation rates. Our reproductive model is based on complete generational replacement. The simulation begins with a reproduction cycle to form the first families, and then some of these families are randomly selected to go through a “danger phase". For each of the chosen families, one member is randomly chosen to be the actor, while the others will be the recipients. The actor has the choice to either perform the altruistic action bearing a (95%) probability of getting killed but allowing the recipients to survive or to do nothing. In this case, the actor escapes the danger phase unharmed, but the recipients die. Whether the altruistic action is performed or not depends on the genotype of the actor. It is important to note that this simulated scenario is a competitive environment since we select a limited number of individuals for reproduction at each step. In this simulation the fitness function is not explicitly coded and it is just the survival and reproduction of the agents as in natural selection.
In addition to the not so trivial basic kinship model we decided to experiment different and more complex settings.
One of the aspects that we deemed very important to investigate was the diffusion of the altruistic allele when it is included in more complex genomes. As a result, we designed a variation of the basic kinship scenario in which the genotype of each individual is also composed of a second trait in addition to the social one. The second trait is represented as a real number between 0 and 1 and encoded in the genotype with a binary representation. At the moment of reproduction the selection of individuals will no longer be random but rather a roulette wheel selection with probability proportional to each individual’s fitness. The individual fitness is obtained by passing the number encoded by the second trait into an additional fitness function. The secondary fitness function has two maxima to simulate two ecological niches; the two maxima shift every fixed number of steps to mimic a dynamic evolutionary environment. The second trait can have several biological interpretations, for example it can be considered the result of a polygenic interaction where each gene is a single binary digit in the representation of the trait. In this model the overall fitness function can be seen as a composition of collective fitness which is still implicit and individual fitness which is represented by the secondary fitness function.
Another way to increase complexity of the scenario was changing the settings of the dangerous phase. In this second version of the extended kinship selection we decided to mix the individuals and create random groups that all undergo the dangerous phase. Within these random groups, the actor, if altruist, decides whether to perform the altruistic action based on common ancestry of the recipient. To do so we decided to store a genealogy tree as a direct acyclic graph. However, due to computational constraints, only the latest three generations (including the current one) are accessible. Considering the distant kinship negligible in terms of benefit, only siblings and first cousins were included in this simulation. Moreover, we explored two slightly different versions of benefit computation. The benefit in the first variant is exclusively tied to the promotion of a specific genotype in future generations. The second one considers a more loose interpretation where the benefit is computed based on the actor perception of relatives regardless of the genotype.
A ‘green-beard’ gene has the property of giving rise to a detectable phenotype which leads the carrier to behave altruistically toward other carriers of the same phenotype .
Starting from the basic kinship model, we explored if and when the green beard effect occurs. First, we devised a model resembling the Dawkins’ formulation in which a single gene encodes for both altruism and a marker phenotype. Then, we examined the implications of having two genes, one for the visible trait and one for altruistic behaviour, also considering the possibility that these genes are in linkage equilibrium or disequilibrium .
All simulations incorporate a reproductive model in which each couple of agents, sampled from the whole population, gives birth to a random number of children drawn from a specified range.
In addition to previous simulations, the altruistic genotype also encodes for the visible phenotype, therefore a single mutation will simultaneously change both . The bearer of the altruistic gene will only act altruistically (with a (95%) chance of being killed) in the presence of another agent carrying the greenbeard allele.
Unlike in Dawkins’ earlier simulation, each agent has two genes with two alleles each. Possible combination of traits are shown in table 1, columns represent the behaviour gene, rows represent the phenotypic marker.
The altruistic action is performed by agents carrying the altruistic allele in the presence of agents carrying the beard allele. Different allele associations result in the appearance of ’impostors’ that have the beard but not the altruistic behaviour, gaining a fitness advantage. In addition, one-point cross-over mechanism is included into the reproduction process.
In the first simulation, all genotypes are expressed equally in a linkage equilibrium condition. The alleles in this case are weakly linked in the sense that they can be inherited separately at each reproduction step. Then, under a linkage disequilibrium scenario, only individuals with specified genotypes (beard-altruistic, nonbeard-selfish) are present. The other combinations could only emerge through mutation and cross over.
In the Selfish herd theory Hamilton proposed that gregarious behavior may be considered a form of cover-seeking, in which each individual attempts to reduce its chance of being caught by a predator (selfish behaviour).
We developed a simulation with two kinds of agents: preys and predators.
They are both placed on a grid and at each step they can move in
neighboring cells. Both agents can see whether there are other
individuals within a certain range near them, determined by their sight
value. When a predator detects a prey, it will chase it until it is
close enough to hunt and eat the prey. Every time a predator catches a
prey, it enters the digesting phase, which requires it to remain still
for a predetermined amount of steps. The prey’s goal is to survive as
long as possible.
Each prey has a real-value genotype in range [-1, 1], corresponding to
aggregating or repulsive behavior respectively. Whenever a prey sees
other preys in a non-danger situation, it will move towards the herd’s
center of mass or in the other direction, based on its genotype value.
In the presence of a predator, however, all prey will escape, regardless
of their genotype.
The number of preys will gradually decrease while the number of
predators will remain constant. Once the prey population decreases below
a given threshold, they enter the reproduction phase.
Our simulations were built with Mesa , a Python agent-based modeling framework. It enables users to easily build agent-based models with built-in core components, visualize them with a browser-based interface, and evaluate results using Python’s data analysis tools. Furthermore, we used two Python graph libraries, Igraph and NetworkX , to compute the relatedness of two individuals in the Ancestry Based simulation.
We ran the simulations with the following set of common parameters. (N) represents the total number of individuals selected as parents at each iteration, (r) represents the initial ratio of the altruistic allele, (mr) and (dr) are mutation rate and death rate respectively.
We ran the basic simulation with the parameters in Table 2 of Appendix. We set (N=1000) to provide a large enough population to partially address the inbreeding problem and small enough to ensure a reasonable computational cost. All the following results were averaged across 20 iterations with the same set of parameters partially ruling out the effect of chance on the outcomes.
We began by running the scenario with (r=0.5), in which the altruistic allele took over in the first 300 steps and then reached an equilibrium with the selfish allele (Appendix Fig. 5). The mutation rate, which counteracts the diffusion of the altruistic allele proportional to its frequency in the population, is responsible for this equilibrium; if mutations are not included, the simulation results in the extinction of the selfish allele. Furthermore, we chose to test if, in this favorable circumstance, the competitiveness of the altruistic allele may allow it to take over even if it was not present at all in the original population. With these settings, the altruistic allele can only emerge by mutation simulating the emergence of a new allele through de-novo mutation. It can be see from Fig. 1 that, in the end, the altruistic allele provides such an advantage that it is still able to reach the same equilibrium as before.
In the multitrait kinship simulation we used the very same set of parameters of the basic one. We also defined the secondary fitness function, evaluated on the second trait, as: [\label{eqn:second_fit} \max{(e^{-(\frac{x-\mu_1}{\sigma})^2}, e^{-(\frac{x-\mu_2}{\sigma})^2})}] where (\mu_1) and (\mu_2) represent the best values for the second trait ((x)) and indeed correspond to the two peaks in the fitness landscape. (\sigma) represents the standard deviation of the modes in the fitness landscape. As mentioned before every 100 steps (\mu_1) and (\mu_2) change in order to simulate a dynamic evolutionary environment. We also imposed the two means to maintain a minimum distance in order to preserve two distinct niches.
The inclusion of the individual component of overall fitness does not interfere with the diffusion of the altruistic allele, as shown in Fig.2. We plotted the mean, maximum, and lowest fitness values at each step to ensure that the individual component actually exerted some evolutionary pressure on the population. As seen in appendix Fig.6, following the initial drop caused by the shift in the fitness landscape, the population’s mean fitness increases, showing that there is evolutionary pressure.
We ran the Ancestry-Based scenario with several sets of parameters,
including the number of generated offspring and recipients. However, we
discovered that such a configuration may result in the altruistic
benefit only under certain unrealistic conditions. Let us assume a
infinitely large population and just first order relationships. In this
case the average benefit of the altruistic actor would be
[\label{eqn:hamilton}
avg(B)= 0.5\times n_{recipients}\times\frac{1}{n_{families}}] which
is the probability of two individuals sharing the same genotype, given
the fact that they are siblings ((0.5)), times the chance of having a
sibling among the recipients
((n_{recipients}\times\frac{1}{n_{families}})).
It can be clearly seen that (avg(B) >1) can be obtained only when the
number families is lower than the number of recipients. Having a small
number of families results in a very small scale simulation where
inbreeding is a major issue. Moreover, increasing the number of
recipients results in more unrelated individuals benefiting from the
sacrifice of the actor, “washing away" the advantage of the relatives.
In our simulation the population is finite further shrinking the
probability of recipients belonging to the actor’s family. Furthermore,
the contribution of longer-term relatives to (B) decreases
proportionally to the relatedness between the recipient and the actor,
hence plays no major role in this context. To summarize, even if it is
mathematically possible to find a set of parameters that results in an
advantage for altruists, the simulation would lose all the biological
significance. We hypothesize that the “missing brick" in this scenario
is the social behaviour of the individuals which should tend to
aggregate relatives and thus play a major role in the success of the
allele.
We evaluated Dawkins’ formulation using the same parameters previously
outlined in the kinship model. We observed how the altruism allele
frequency emerges over the selfish allele (Appendix Fig.
7). Identical outcomes are obtained by either
setting the mutation to zero or allowing the altruistic behavior to
emerge solely through de-novo mutations.
Despite the fact that “green-beard” genes have been reported , the
‘green beard effect’ has has been widely criticized since it is
unlikely that one gene can code for altruism and a detectable phenotypic
tag. To make our model more biologically plausible, we investigated
multitrait theories. We found that if the effect is caused by separate
alleles in linkage equilibrium, ‘impostor’ genotype overruns the
population, while the ‘true-beard’ decrease in frequency and is
eventually lost (Fig. 4). In contrast, in a linkage
disequilibrium condition, ‘true-beard’ is facilitated through (Fig.
3).
We have shown that in some circumstances, a “green-beard" gene is favoured even when the carriers share no other genes. This might imply that genetic relatedness at the altruism locus, rather than across the entire genome (kinship relatedness), is the most important prerequisite for altruism.
We attempted to validate the Selfish Herd theory by simulating several scenarios using a grid of 100x100 cells with 200 preys and 15 predators. The simulations were all founded on the premise that predation risk is a fundamental element influencing the agent behaviour. Predation risk is conditioned by the sight of the prey, the jump range of the predator and the position of the prey wrt the predator and the other preys. The position of each prey is dependent on its genetically regulated behaviour. As it can be seen from the Fig. 7, different parameters (i.e., sight and jump range) and therefore different risk of predation, result in the selfish gene always taking over the population. The selfish allele frequency within the population increases proportionally to the jump range and inversely to the prey’s sight. As we can see from Appendix Fig 7, with an high predation risk (small sight and high jump range), the diffusion of the selfish allele is around 90(%). Differently, we can see in Fig. 7 that with bigger sight and smaller jump range, and hence smaller predation risk, the selfish allele diffusion is around 50(%).
This demonstrates how preys, located distant from their nearest neighbors, are more vulnerable to predation than those who are surrounded by neighbors. We’ve shown how a selfish action can also prove benefit other preys and therefore results in a form of cooperation.
In this paper, we investigated several evolutionary theories concerning
the emergence of cooperative behavior and investigated their
plausibility using biologically meaningful simulations.
The most difficult aspect of devising these simulations lies in
balancing the scale, detail level and the biological relevance against
computational complexity. We had to make several assumptions which
decreased the level of significance but actually made the simulations
feasible. Furthermore, it should be noted that we simulated biological
theories on which there is no universal consensus in the biology field
and are object of debate, highlighting some of the critical points as we
did with the Greenbeard and Ancestry-Based simulations.
The whole code for this project was developed from scratch by the authors of this report. To be more specific, the workload was split into two parts. The first part, carried out by M.V. Cavicchioli and P. Demurtas, included all the simulations and plots regarding the Kinship domain while the second part, carried out by E. Barba and R. Cervesato, dealt with Green Beard and Selfish Herd. On the other hand, the theoretical work and the analysis behind every simulation were carried out by all the four authors equally.
| selfish (0) | altruist (1) | |
|---|---|---|
| beardless (0) | coward | sucker |
| green-beard (1) | impostor | true-beard |
Possible genotypic combinations for green beard multitrait formulation
| N | r | mr | dr |
|---|---|---|---|
| 1000 | 0.5 | 0.001 | 0.95 |
Basic parameters of kinship altruism
