Departament de Genètica i de Microbiologia, Grup de Biologia Evolutiva (GBE), Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Collegium Budapest, Institute for Advanced Study, Szentháromság u. 2, H-1014 Budapest, Hungary

Institute of Biology, Eötvös University, 1/c Pázmány Péter sétány, H-1117 Budapest, Hungary

Parmenides Center for the Study of Thinking, 14a Kardinal Faulhaber Strasse, Munich D-80333, Germany

Abstract

Background

The evolutionary origin of strong altruism (where the altruist pays an absolute cost in terms of fitness) towards non-kin has never been satisfactorily explained since no mechanism (except genetic drift) seems to be able to overcome the fitness disadvantage of the individual who practiced altruism in the first place.

Results

Here we consider a multilocus, single-generation random group model and demonstrate that with low, but realistic levels of recombination and social heterosis (selecting for allelic diversity within groups) altruism can evolve without invoking kin selection, because sampling effects in the formation of temporary groups and selection for complementary haplotypes generate nonrandom associations between alleles at polymorphic loci.

Conclusion

By letting altruism get off the ground, selection on other genes favourably interferes with the eventual fate of the altruistic trait due to genetic hitchhiking.

Background

More than thirty years ago Hamilton

Contrarily to weakly altruistic traits which can increase in frequency when groups are randomly formed each generation

where

(superscript ^{o }stands for other-only relatedness, and

No special process has to be invoked to create a structured-deme population as it can simply arise from random distributions of genotypes across patchy resources; a quite common process in many animals, particularly insects. Genetic variation among finite groups creates environmental heterogeneity unrelated to resource heterogeneity

In this work, we focus on the evolution of altruistic versus selfish alleles and of polymorphisms that show a positive association between productivity and within-group genetic variation to study the fate of altruism in a multilocus context assuming the structured-deme model of single-generation, randomly formed groups originally used by Hamilton

Methods and Results

Multilocus model and computer simulations

Figure _{i},

'Mating pool' mode of reproduction

**'Mating pool' mode of reproduction**. A finite population is subdivided into

where _{s }or _{d }<1. We set _{s }= _{d }= 0.95. Allelic diversity can easily persist in this model under a wide range of parameter values ^{G }possible segregating haplotypes.

After an initial period of 500 generations to allow the population to reach a quasi-equilibrium state under selection and free recombination between adjacent loci, the focal strongly altruistic locus was introduced in the centre of the set of those whole-group beneficial loci with alleles sampled from a binomial distribution with

where

In other words, the critical

With ^{-6}. Random group formation events will result in relatedness (^{o}) values which fluctuate around – 1/(

With multiple loci the probability of allele fixation increases in the whole-group beneficial loci

We simulated repeated introductions of the

Numerical results

In order to numerically demonstrate how strong altruism can evolve in single-generation, randomly formed groups, consider a focal locus with two alleles (

The population size was fixed to ^{5 }introductions into the population with ^{-3 }between adjacent loci would correspond to a region approximately (^{-3 }centimorgans in length and roughly equivalent to a chromosome with 1,000 genes and total map distance 100 centimorgans), the altruistic allele could easily persist in the population for quite a long time because the genome eventually crystallized in a few segregating haplotypes. This effect was already clear with only

Multilocus trait-group model

**Multilocus trait-group model**. **(a) **Semi-log plot for the mean time to loss of the strongly altruistic trait coded by allele **(b) **Sample simulation with

The results can be understood as follows. Deterministic haplotype dynamics assuming linkage equilibrium at generation _{0 }guarantees that any ^{G }possible segregating haplotypes for the ^{w }= 1/^{w }stands for whole-group relatedness) and is obviously positive. This does not, however, invalidate our claim that strong altruism can invade without kin selection because relatedness at the focal locus still remains ^{o }= -1/(

Recombination breaks down the statistical associations between alleles at linked sites (i.e. detaches ^{o }from ^{w}) but, as we might expect, the hitchhiking effect is most marked when there are many segregating loci. With intermediate linkage (

Fixation of strong altruism in Hamilton's random group model

**Fixation of strong altruism in Hamilton's random group model**. With ^{-4}. The plot shows a sample simulation where the

Analytical dynamics of whole-group genes

In order to understand the genetic hitchhiking effect responsible for the spread of strong altruism in our group selection model, it is important to first understand the dynamics of whole-group genes since linkage disequilibria can also be generated by deterministic forces. To clarify matters, we deal here with the simplest situation where the ^{1 }and 2^{1}; 1^{2 }and 2^{2 }for a 2-locus genome). For each pair, allele 1 is the fittest one as assumed in Methods. This creates up to 2^{G }possible haplotypes. We set group size to

where element _{ij }is the fitness of individual ^{g}, 2^{g) }in a group with individual

**Γ **= **P**^{1 }⊗ **P**^{2 }⊗ ⋯ ⊗**P**^{G}

where ⊗ is the Kronecker tensor product. For

It is clear that group mean is

where superscript ^{T }signifies matrix transposition. Now we have to generate the appropriate matrix of haplotypes, which can be done as follows. At generation _{0 }segregating alleles have frequencies

and second, we multiply the resulting **h **vector

**v **= **h **⊗ **h**

This row vector can now be rearranged by noting that its first 2^{G }elements are the corresponding random group frequencies of haplotype 1^{1}1^{2 }in a group with haplotype 1^{1}1^{2}, ⋯, 2^{1}2^{2}; the second 2^{G }elements the corresponding frequencies of haplotype1^{1}2^{2 }in a group with haplotype 1^{1}1^{2}, ⋯, 2^{1}2^{2}; etc. After rearranging this vector we obtain the suitable matrix of haplotypes (**H**). Since

where

is the average fitness, _{i }is the corresponding row for the _{s }= _{d }= 0.95 in payoff matrix (6), Figure

Deterministic haplotype dynamics for the whole-group loci without recombination

**Deterministic haplotype dynamics for the whole-group loci without recombination**. At generation _{0 }all segregating alleles were assumed to be at equal frequencies, and linkage disequilibrium was absent. (a), (b), Two-loci haplotypes, with

It is clear that with the assumed starting conditions the deterministic short-term evolution drives the population to end up with only coupling haplotypes at equilibrium state. We can also define ESS (Evolutionary Stable Strategy) conditions for the payoff matrix (8) by noting that the game theoretical model is isomorphic to a trait group model with group size ^{1}1^{2}, 2^{1}2^{2}} and repulsion {1^{1}2^{2}, 2^{1}1^{2}} haplotypes. All other equilibria are unstable. This holds if _{s}, _{d}, _{d})/_{d}. The domains of attraction of the two ESSs are not equal, hence from initial linkage equilibrium the system converges to the coupling ESS (Figure ^{G-1 }if the number of alleles per locus is 2. These considerations clearly suggest that there is a balance between recombination (always decreases linkage disequilibrium) and selection (convergence to coupling or repulsion haplotypes) in the system.

The preceding analysis can explain some previous numerical results _{s }= _{d }= 0.95 and _{d }= 1 in payoff matrix (6) the system converges to the standard heterotic case

Fate of the altruistic allele in the analytical dynamics

It is straightforward to incorporate the altruistic locus in the former analytical treatment for the whole-group genes. The payoff matrix is now (eq. 4)

where

**Γ **= **P**^{a }⊗ **P**^{1 }⊗ ⋯ ⊗ **P**^{G}

which, with

It is easy to see that {S1^{1}, S2^{1}} is the only (mixed) ESS, thus selfishness prevails. With ^{1}1^{2}, ^{1}2^{2}}. Which haplotypes can invade the population? Inspection of the **Γ **matrix (15) reveals that only haplotypes ^{1}1^{2 }and ^{1}2^{2 }can invade, but ^{1}2^{2 }and S2^{1}1^{2 }cannot, _{d}>(1 + ^{1}1^{2}, A2^{1}2^{2}} is unstable

Deterministic haplotype dynamics assuming linkage equilibrium at generation _{0 }also shows that any

In finite populations allelic diversity can be lost when the fittest haplotype is present since its eventual fixation can occur

We can now envisage the effects of group size ^{g }at the

The effect of population size ^{8 }= 13, 122; including the focal altruistic locus),

In the simulations with ^{-4}). Figure

Snapshot of segregating haplotypes after fixation of

**Snapshot of segregating haplotypes after fixation of A allele**.

Discussion and Conclusion

We acknowledge that our structured-deme model assumes strong selection in the collections of individuals who influence one another's fitness, mainly to prevent waiting times before the eventual spread of the altruistic trait to becoming computationally unmanageable. The important point, however, is to understand that linkage, selection, and sampling will necessarily interact in multilocus structured-deme models, and the interference effect will extend to all loci in a block depending on the recombination value. With tight linkage a few selected loci may be enough to trap the altruistic allele in a quasi-equilibrium polymorphic state. With moderate recombination fixation of strong altruism can occur. Our results do pose a serious challenge to the universally accepted view that kin selection is the key component to explain altruistic behaviours that impose an absolute fitness cost to the actor and, to some extent, might be relevant to understand the origin (as opposed to the maintenance) of eusociality

Positive assortment increases the relatedness at the focal altruistic locus and, depending on the linkage disequilibrium between that locus and nearby

In the same vein that multivariate selection theory provides a framework to predict the direct and indirect effects of selection on a suite of complex traits

Authors' contributions

Both authors contributed equally to this manuscript. Both authors read and approved the final manuscript.

Acknowledgements

We thank Andy Gardner, Rolf Hoekstra, Stuart West and one anonymous reviewer for comments on the manuscript. MS is supported by grants CGL2006-13423-01/BOS from the Ministerio de Ciencia y Tecnología (Spain) and 2005SGR 00995 from Generalitat de Catalunya to the Grup de Biología Evolutiva. ES is supported by the National Office for Research and Technology (NAP 2005/KCKHA005) in Hungary. MS and ES are supported by funding of the INCORE project under the Sixth Research Framework Programme of the European Union. The views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect the views of the European Commission.