Polymorphism of Recombination Strategy in Evolutionary Games

a Theoretical Study

 

by Oleg Rokhlenko and Ydo Wexler

October, 2005, under guidance of Ron Pinter and Doron Lipson

Introduction

Recombination is the process of the formation of a new combination of genes on a chromosome as a result of crossing over. Although the evolutionary function of recombination is not fully understood many research works support its long-term effect on the population level. The majority of these works investigate the relationship between the recombination and fitness of an organism, defined as the organism's ability to survive and reproduce in a particular environment. Others,  go farther, and propose that recombination is negatively associated with fitness, breaking down unfit combinations of genes with a higher probability than fitter ones.

For an association between recombination and fitness to occur, it is sufficient that some correlating factor s exists, which is associated with fitness f and affects the recombination rate r. Hadany and Beker studied the performance of Fitness-Associated Recombination (FAR) using a series of simple haploid models and showed that the allele for FAR tends to increase from scarcity in any population with a uniform recombination rate. However, their model restricts FAR to some kind of an on-off behavior where a fitness-associated recombination either exists or not. In this project we show that FAR is not a Boolean but a multi-value feature defined on a continuum, which determines the recombination strategy. We study various classes of strategies within the boundaries of association between fitness and recombination, their behavior and their evolutionary effects. In addition, we simulate evolutionary games to show the ability of the a strategies to spread
in a population originally dominated by another strategy. We also extend these simulations to account for more complex models of
varying linkage values, nonrandom mating and mutations in regions controlling the strategy.
 

Methods

We suggest a more general evolutionary model of fitness associated recombination. This model is based on the non-facultative approach and a new continuous formulation of equilibrium equations.
 

Continuous strategies - The new model extends the strategy notion formulated by Hadany and Beker  and instead of 2 trivial options (z(x) = const and a logical function) enables arbitrary continuous strategies. This extension is of utmost importance since the natural evolution is obviously not restricted to the trivial cases of simple discrete functions as in the original model. Even though we do not claim that our model resembles the true evolutionary process, still this generalization opens up a wide range of potentially effective evolutionary approaches.

Linkage - We use a non-facultative approach of reproduction. Thus, the linkage between the fitness loci and the strategy locus is independent of the fitness of the organism, its strategy, or the recombination within the fitness loci. We test a variety of linkage factors and examine the impact on the individual and population levels.
 

Mutations in strategy locus - We propose another extension which allows mutations not only in the fitness loci but also in the strategy locus. Such a possibility serves as a balancing factor and causes organisms with high fitness (which usually possess the winning strategy) to switch their strategy and vice versa.

Non-random mating - In real world the mating process in not random and the individuals choose their pair mate according to some criteria, for example fitness.
Random mating is a factor assumed in the Hardy-Weinberg equilibrium, and thus, usually applicable to populations without migration. However, when evolution or natural selection occurs, random mating is not the logical choice as in evolution traits favorable to survival are preferred to others.
We consider two mating scenarios which mimic the natural mating process more precisely:

1. Uniform fitness-dependent mating where organisms tend to chose their pairs mate with fitness similar to theirs. More formally, the probability of an organism with relative fitness f1 to mate with another organism with relative fitness f2 is drawn from a normal distribution with mean f1 and standard deviation equals 1. As reference, we also consider the opposite paradigm where the probability is drawn from a normal distribution with mean 1-f1.
 

2. Free-will fitness dependent mating where organisms with higher fitness tend to mate more than organisms with lower fitness. We also consider the opposite theory that organisms with low relative fitness tend to reproduce more in order to increase the probability of producing a viable offspring with high relative fitness.
We note that mating paradigm is population dependent and therefore it is logical to examine both paradigms suggested here.


 

 

Implementation

We use computer simulations to investigate the effect of the different aspects of the enhanced evolutionary model.
Our simulations mimic an evolutionary process in closed populations without migration, and include selection, mating, recombination, mutations and survival.
There, an organism has 10,000 loci that determine its fitness through a given  formula that takes into consideration the number of mutated loci and the relative damage inflicted by a single mutation. In addition, a locus that controls the recombination strategy of the organism is linked to the fitness loci via a pre-defined parameter.
In the general framework of these simulations which is sketched in the figure below, a random pair of organisms is selected for mating, where recombination within the fitness loci occurs with probability proportional to the recombination rates of the two parents.
The result of this mating is a single offspring on which we inflict mutations, the number of which is drawn from a Poisson distribution with a pre-defined parameter.
The offspring survives with probability proportional to its absolute fitness. Additional recombination between the fitness loci and strategy locus may occur with a fixed probability determined by the linkage parameter. This recombination occurs independently of decision regarding recombination within the fitness loci.
In this setting all mutations are reversible. In each of the runs additional features which are explicated are added and distinct it from the others.


 

Results

We examined the ability of a variety of continuous strategies  to spread in a population originally dominated by another strategy. For each two strategies,
zi, zj, we measured the winning percentage of strategy zi in a population dominated by strategy zj and vice versa in a set of 1000 games. If the winning percentage of zi is significantly higher than the initial fraction (called  domination barrier) of organisms with strategy zi in a population of the opposite strategy zj and the opposite is not true, we say that strategy zi dominates strategy zj. Our domination games consist of 3 rounds:
1. All strategies vs. UR - all strategies (with fixed parameters) compete against UR.
2. Sigmoid vs. UR - sigmoid strategies with different parameters  parameters compete against UR. This round is of utmost interest since sigmoid functions are the generalization of the simple FAR strategy presented by Hadany and Beker.
3. Sigmoid duel - sigmoid strategies with different parameters  compete with each other.

In the first two setups changes in the models (such as linkage, non-random mating, etc.) were also considered. For the sake of fairness in all simulations the average recombination rate of the competing strategies is equal.
 

Full Report

FAR.pdf