Selection for altruism through random drift in variable size populations
© Houchmandzadeh and Vallade; licensee BioMed Central Ltd. 2012
Received: 9 November 2011
Accepted: 14 March 2012
Published: 10 May 2012
Altruistic behavior is defined as helping others at a cost to oneself and a lowered fitness. The lower fitness implies that altruists should be selected against, which is in contradiction with their widespread presence is nature. Present models of selection for altruism (kin or multilevel) show that altruistic behaviors can have ‘hidden’ advantages if the ‘common good’ produced by altruists is restricted to some related or unrelated groups. These models are mostly deterministic, or assume a frequency dependent fitness.
Evolutionary dynamics is a competition between deterministic selection pressure and stochastic events due to random sampling from one generation to the next. We show here that an altruistic allele extending the carrying capacity of the habitat can win by increasing the random drift of “selfish” alleles. In other terms, the fixation probability of altruistic genes can be higher than those of a selfish ones, even though altruists have a smaller fitness. Moreover when populations are geographically structured, the altruists advantage can be highly amplified and the fixation probability of selfish genes can tend toward zero. The above results are obtained both by numerical and analytical calculations. Analytical results are obtained in the limit of large populations.
The theory we present does not involve kin or multilevel selection, but is based on the existence of random drift in variable size populations. The model is a generalization of the original Fisher-Wright and Moran models where the carrying capacity depends on the number of altruists.
Light production in Vibrio fischeri[1, 2], siderophore production in Pseudomonas aeruginosa, invertase enzyme production in Saccharomyces cerevisiae, stalk formation by Dictyostelium discoideum, [2, 5] are but a few examples of individuals in a community who help others at their own cost by devoting part of their resources to this task. This behavior has been termed “altruistic”. From the evolutionary point of view, altruists have a lower fitness than other individuals in the community who don’t help, but are recipient of the benefits produced by altruists. Through this paper, we will call these latter individuals ‘selfish’.
From the inception of evolution theory, the problem of the existence of altruists has been puzzling: how can a mutant with lower fitness prevail? And how does a community of altruists resist the spread of selfish allele (see  for a historical perspective)? In the last 40 years many models have emerged to explain the apparent contradiction between the smaller fitness of altruists and their widespread presence in various communities (for a review, see [7, 8]). It is shown in these models that the actual fitness of an altruistic gene can be increased by other factors such as ‘common good’ restricted to kin (inclusive fitness [9, 10]), or advantages conferred at another level of selection (group or multilevel selection [11, 12]). These models which can be formulated through the Price equation have seen various generalizations and they are sometimes widely debated (see  and the numerous replies it has elicited).
The above models are either deterministic, i.e. populations change their size exactly according to their relative fitness, or involve frequency dependent fitness [14, 15]. We show here that another possibility exists: an altruistic individual can produce a common good benefiting everybody in the community regardless of its nature (altruistic or selfish) and therefore increasing the carrying capacity of the habitat. Even though selfish individuals have always a higher fitness, genetic drift effects can favor the altruists.
The fixation probability is composed of two terms: even in the absence of selection, the population will become homogenic; in this neutral case, all individuals at generation zero have an equal probability 1/N of becoming fixed. When a beneficial mutation is present, the fixation probability of its carrier is increased by the relative excess fitness.
For populations of fixed size, as can be seen from expression (1) or the more precise expression (10) obtained by Kimura  and Moran , the fixation probability is a monotonically increasing function of the sole relative fitness. In the competition between alleles, arguments based on fitness parameter alone or the fixation probability lead to the same conclusions . However, if population size is not fixed, the fixation probability π, which takes into account both randomness due to finite size and selection, can lead to other conclusions than the fitness parameter alone.
i.e. the cost to the altruist is smaller than the benefits in term of relative population increase, then an altruist has a larger fixation probability than a selfish one, even though its relative fitness is smaller. The relative advantage of a selfish mutant is compensated by the increased ‘random noise’ to which it is exposed. Note that in a deterministic model of the above process, the A always lose, since S individuals always increase their proportion.
The above argument will be refined in the following. In the next section, we formulate precisely the stochastic process of altruism outlined above by generalizing the Moran model for non-structured, well mixed populations and we show that altruists can indeed be favored in their competition with selfish individuals. We outline the amplification of this advantage in geographically structured, viscous populations in the third section. The final section is dedicated to concluding remarks.
Results and discussion
Stochastic model for altruism
The fundamental aspects of population genetics were clarified in the framework of the classical Fisher-Wright (FW) stochastic model of non-overlapping generations or its continuous time alternative introduced by Moran . Moran and FW are equivalent in the limit of large populations, where both are well approximated by the same diffusion equation . These are the simplest models that capture the key elements of population genetics (genetic drift, fixation probability, fixation time,…) with the fewest possible ingredients.
where m is the number of S individuals and c is the ‘cost’: 1/ c is the relative fitness of the A and c > 1 indicates a selective disadvantage. W+ stands for the probability density that the new spot is colonized by an A and death occurs among the S. In principle, a similar set of equations must be written for the S individuals; however, as the population size is fixed, , the quantity m in eq.(3) can be replaced by N-n and the whole stochastic process treated as a one dimensional random walk for the A.
We generalize this model by including two ingredients. First, the fixed size constraint can be relaxed and we let N vary between two bounds N i and N f : empty spots are created-colonized and individuals die, without these two events necessarily succeeding each other. More importantly, in order to include the effects of altruists, we suppose that the rate of creation of empty spots is proportional to the number of altruists and is equal to αn; in contrast, the death rate is proportional to the number of S individuals and is equal to α m. This is the simplest hypothesis that implies that the increase in the carrying capacity of the habitat is proportional to the number of altruists (see also Methods, mean field approximation).
if c > 1. The results below don’t change significantly if the cost of altruism is included in other rates. For example, a higher probability for an S to reproduce, or any combination that favors S over A. Note that if the increase/decrease rates were independent of m and n, we recover the Moran model by setting , in which case each birth/death is succeeded by a death/birth event (see Methods, relation to Moran model).
The above rates ensure that if A are lost ( n = 0), the population size tends toward N i and if S are lost ( m = 0), it tends toward N f . Note that in the mean field approximation of the above process where fluctuations are neglected and the deterministic limit is taken, the A are always eliminated if c > 1 (see Methods, mean field approximation).
where is the proportion of the A. The approximation corresponds to the Kimura solution obtained through a backward diffusion equation  and . Expression (1) is the first order expansion of the above expression in s.
For the two dimensional process (4–7) where is the initial number of the S and A, no closed form solution can be obtained. We can however solve equation (8) numerically by standard linear solvers or else resort to a Gillespie algorithm  to solve the stochastic equations (4–7) directly. Both these methods are used in this paper and the analytical approximations obtained below are compared to them.
Geographically structured populations
and for M > > 1: once altruists dominate, the chances for a selfish mutant to invade the community is close to zero! Increased random noise due to production of common good and a small migration rate are an efficient way of keeping selfishness in check.
The above computation concerns the low migration limit. In the high migration limit, the community is non-structured and its effective size is . Criterion (15) shows that in this regime, altruists cannot emerge; this is indeed equivalent to the deterministic case where emergence of altruists calls for other mechanisms. Between these two regimes of high and low migration rate, there is a rich interval where migration rate is a key ingredient in the competition between altruists and cheaters.
The main concepts of Population Genetics were clarified in the framework of the original model of Fisher-Wright and Moran (FWM). These models introduced the key ingredient of population size and its role in the randomness of selection. It became clear in the 1920-30’s that a beneficial mutation does not spread automatically to the whole population, but has to overcome the “random noise” of population sampling over generations. The idea that random noise plays also a role for the selection of altruism has been introduced in two kind of models, which have a marked difference with the model we present here. The first class of models, formulated mostly through evolutionary game theory formalism, concerns fixed size populations, where the transition rates are frequency dependent : the fitness of an A individual can be superior to the fitness of an S individual if the number of A individuals already present is high enough. It can then be shown, upon very general conditions, that the fixation probability of altruists can become superior to that of selfish ones. These models can be seen as the generalization of Hamilton’s original idea, where “altruistic” help is restricted to genetically related individuals, even though Traulsen  has argued that the underlying mathematics is fundamentally different. The second class of models concerns group (or multilevel) selection. It has been shown  that the fixation probability of altruists can be higher than those of selfish ones, if the population is structured into groups and the splitting of one group leads to the elimination of another. It has also recently been noticed that random noise in a growing population can favor altruists during a transient period .
The model we present here is not frequency dependent: an A individual has always a lesser chance of reproducing than an S individual; the mean field description of this model has only one stable fixed point which corresponds to the disappearance of altruists. Moreover, The mechanism we propose is for non-structured populations, even though the altruist effect can be amplified when the population is structured into groups with small migration rate between groups. Imagine a group of M islands composed only of altruists and another group of M islands composed only of selfish individuals. Introduce one S mutant in each island of the first group and one A mutant in each islands of the second group. After some time, the number of islands in the first group is increased if the criterion (15) is satisfied.
In summary, we have shown, by a slight generalization of the Moran model, that in finite size populations, the fixation probability of altruists can be higher than that of selfish ones, even though their fitness is lower and their emergence is ‘forbidden’ by a Hamilton rule. We have also shown that in large, structured populations, and in the limit of small migration rate, the same arguments hold. Production of the ‘common good’ and increase in the carrying capacity of the habitat increase the random noise for selfish individuals and can therefore favor altruists.
The aim of the present article is not to contest the merits of kin/group selection models which have been investigated during the last forty years with a large number of case studies. We believe we are providing an alternative way of thinking about altruism which is complementary to the above models and which restores the key ingredients of population genetic to this topic.
Diffusion equation derivation
Mean field approximation
where and . It is then obvious that for , . In the deterministic model, A-individuals always disappear.
which shows that the increase in carrying capacity of the habitat η - k, at small selection pressure, is mostly proportional to the number of A-individuals. A closer look at the above equations (16,18) shows that is the only stable fixed point when c > 1.
Relation to Moran model
The same expression is obtained if .
Numerical resolution of fixation probabilities
Two different kinds of numerical resolution were used to check the validity of our analytical results on the fixation probabilities: A Gillespie stochastic algorithm and direct resolution of eq. (8).
which we solve by the classical Gillespie algorithm  written in C++. We are interested here only in the fixation probability and not in the fixation time; the program can therefore be accelerated by computing only the nature of the event that occurs at each turn (and not its time of occurrence). In general, to solve for the fixation probability, R = 106 stochastic trajectories are generated.
where are the unknowns. is a sparse matrix, which apart from the diagonal elements, has at most four non-zero elements per line: if k is the image of element ( m,n), then only if k′ is the image of one of the four nearest neighbors of ( m,n), in which case its value is given according to rates (4–7). The right hand side vector B k is a sparse vector provided by the limit conditions : if k′, one of the 4 nearest neighbors of the element k belongs to the border m = 0, then and the corresponding is transferred to the right hand side to constitute the vector B k . Note that because we index the interior of the trapezoid, the index k itself can never belong to the border.
Once the linear system (20) has been constituted, it can be solved by any linear solver. We have used the commercial package matlab for these manipulations.
We are grateful to O. Rivoire and E. Geissler for fruitful discussions.
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