Using a diploid genetic algorithm to create and maintain a complex system in dynamic equilibrium

This study investigates the ability of diploid GA to approach a class of problems in which the fitness landscape is reciprocally affected by the population itself. This problem class is common in many fields of science, engineering, and the humanities, in which the entire system of environment plus population is maintained in a state of dynamic equilibrium. In particular, a system is created here in which individuals with three simulated metabolic enzymes inhabit and influence an environment with a three-gas atmosphere. Through feedback loops and frequent environment/individual interactions, diploid populations are shown to create fit, stable solutions to this problem type. 1.0 Introduction 1.1 Dominance and diploidy in GA In the last four decades, the use of genetic algorithms (GA) has developed into a powerful optimization and problem-solving technique in a diverse range of fields. Originally, GA was modeled on the biological process of evolution by natural selection: variation in individuals results in a range of reproductive fitnesses, which translates to fitter individuals having a greater genetic representation in future generations. Because primitive GA was inspired by biological genetics, the reproductive functions used to guide propagation in most algorithms also borrowed from a biological repertoire, and the processes of crossing over, mutation, and asexual reproduction became instantiated into mainstream evolutionary algorithm strategies. At that point, increasing availability of computer power and an unending store of potential problems led GA into many different fields, often relying on the same catalog of crossover, mutation and reproduction with slight variations in algorithm metastructure. The aim of this study is to return to some of the primary ideas behind the conception of evolutionary problem solving, and to extend them in order to create more powerful techniques that can be applied to an entirely new class of problem. In particular, standard GA problems in many fields are concerned with either optimizing an individual (or product) in a static fitness landscape, or solving a problem in which the fitness evaluation of different strategies is measurable against a schedule of static costs or benefits (either directly or indirectly, such as via a simulation). Meanwhile, problems with dynamic fitness evaluations have been only modestly approached with GA, because the “best” of the classic algorithms, or one that causes individuals to quickly converge on a solution, is not necessarily good at continuous adaptation in the face of a changing environment and fitness landscape. Some attempts have been made to apply GA to changing-fitness problems, but all have had significant shortcomings. Bagley (1967) used the concept of diploidy, or pairs of homologous chromosomes, to model a population of individuals with a dominance map that determined which of two alleles would be expressed as the phenotype in a given situation, thereby presenting the opportunity of an individual to carry a hidden trait without expressing it. However, the individuals were evaluated under static fitness conditions, the dominance maps converged too early, and furthermore, the amount of information stored in the dominance maps was on the same order as all of the information stored in the genotypes of the individuals. In terms of memory storage and manipulation, Bagley effectively had to account for triploid individuals, instead of diploid ones. Soon after, Hollstein (1971) used a dominance schedule with much better memory efficiency, but again used a static fitness landscape, and suggested that diploidy did not offer a significant advantage to fitness. He did note, though, that population diversity increased with diploidy, as latent recessive traits would emerge and create more variability between individuals. More recently, Goldberg (1989) took a step toward solving dynamic fitness problems by using diploidy to create individuals in changing, often oscillating, fitness environments. As predicted, the diploid populations were able to adapt to changing environments more readily than haploid populations, a result that many people attribute to the act of calling upon once-successful traits that were cached in combinations of recessive alleles that could reemerge at random times and potentially exploit different environments. The use of multiploidy, beyond two chromosomes, found similar results (Collingwood 1995). 1.2 Stating the problem This paper investigates the ability of GA to move beyond this type of problem-solving in an independently-changing environment. Instead, it uses GA to create entire systems in states of dynamic equilibrium, in which maintaining diversity of individuals in a population is as important to the solution as is maintaining high fitness. In order to achieve the evolution of a complex system, GA must be applied to a new class of problem, one with links to numerous disciplines including traffic control, electrical engineering, population ecology, and endocrinology. All of these fields, plus many others, rely not only on the power of the environment to influence the individual parts of the system, but also on the reciprocal impact of the individuals on the environment itself. In the following study, a GA protocol is first presented as a tool to approach dynamic systems with this type of reciprocal individual-environment interaction. Next, a model problem is presented, in which a population of simulated creatures live and metabolize in a three-gas atmosphere. The resulting complex system is judged in terms of 1) fitness of the individuals, and 2) equilibrium of the environment itself. Finally, the characteristics of this new GA variant are discussed, and possible applications are explored. 2.0 Methods 2.1 GA platform and protocol A program designed for multiploid GA with dominance was created in Matlab, based upon standard GA concepts presented in Koza (1992) and Goldberg (1989) along with biological operators that are overlooked in many popular algorithms. The framework allowed for the creation of populations with varying numbers of subpopulations, and with migration of individuals between subpopulations. Mating pools were selected in tournament style, with tournament size based upon the desired certainty of the best-of-population individual reproducing. In the reproduction stage, individuals in haploid populations could undergo single-point crossover, mutation, and (asexual) reproduction. The handling of diploid populations, however, diverged from that of standard haploid GA. Dominance was determined in diploid populations according to the triallelic map created by Hollstein (1971) shown in Figure 1, in which binary bits at each locus were replaced with three choices: 0, 1, or 2. The phenotype of an individual, which represented the traits that the individual expressed in its phenotype, were determined in the following way: the ‘2’ allele at a locus was globally dominant, and resulted in a ‘1’ in the corresponding phenotype. The ‘1’ genotype allele also resulted in a phenotype of ‘1’, but was recessive to the ‘0’ allele, which returned a ‘0’ in the phenotype. Because the probability of a ‘1’ in the phenotype was twice as great as a ‘0’ (six genetic combinations lead to a phenotype of ‘1’ versus only three that lead to a ‘0’), the initial population creation during haploid runs was altered so that a ‘0’ occurred in a haploid genotype half as often as a ‘1’. When diploid individuals reproduced, they were subjected to the gamut of crossover, mutation, and asexual reproduction, plus extra diploid-only operations. Crossing over was able to be applied to a single string from each parent, or to both strings simultaneously. Also, sexual reproduction was executed in place of, or in conjunction with, the other operations: two parents swapped one entire chromosome string with each other, presenting the while an allele of '1' gives the same phenotype but is recessive. possibility of recombination of recessive alleles. See the tableau in Table 1 for probabilities of these operations. under conditions of high atmospheric oxygen, enzyme2 is analogous but acts on carbon dioxide (CO2), and enzyme3 acts on atmospheric nitrogen (N2). Therefore, an individual’s fitness is the sum of a function on the partial pressures of these three gases: in this case the sum of the squared enzyme activity, times the partial pressures. Standardized fitness is the raw fitness normalized to a scale of 0, being no fitness, to 2, or maximal fitness. The following e