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## How to use the model
Sliders/switches on the interface tab:
* carrying-capacity - the number of individuals in the population.
* mutant-proportion - the proportion of individuals that carry one new mutation in their genome, every generation
* genome-length - the number of loci in each chromosome.
* preference-strength - the factor determining the addition of each preference mutation to the strength of the preference (i.e the slope of the probability to mate function).
* recombination - determines whether or not the genome goes through recombination.
* selection-coefficient - determines selection force against maladaptive genotypes. Equals the proportion of maladaptive individuals that die during selection.
* max-matching-attempts - the maximum number of females that a male can refuse to mate with before he is taken out of the mating pool. This variable is included to prevent the simulations from getting stuck because none of the females are suitable for a single male.
## Comparison to the physical_linkage_one_trait_locus model
This model is an extention of the physical_linkage_one_trait_locus model, in which the ecological trait is controlled by a single locus. In this model, the ecological trait is controlled by two separate loci with epistatic interactions.
The model description specified below is of the one trait locus model (see ODD below). The differences in the current model compared to the one trait locus model are specified here:
The genetic basis of the trait in the model is similar to that of wing colour pattern in the two sympatric species Heliconius melpomene and H. cydno, where the optix and cortex loci control red and white forewing bands respectively (Naisbit et al., 2003) . To simplify the model description, we hereafter refer to colour pattern as the ecological trait, but the model could equally apply to a wide range of ecological traits.
The two colour loci in this model are located at opposite ends of the chromosome. The ‘Red’ locus is located at the first position within the chromosome and the ‘White’ locus is located at the last position within the chromosome. The colour loci each have two possible alleles which account for the presence (‘R’ and ‘W’ alleles) and absence (‘r’ and ‘w’ alleles) of a red and white band in the forewing, respectively. The alleles for presence of colour bands are dominant over those for absence. The presence of one colour band does not replace the presence of the other, resulting in a possible intermediate red-white band, characteristic of hybrids of H. melpomene and H. cydno. Together, there are nine possible genotypes at the two colour loci, and four possible colour phenotypes. The representation of genotypes and phenotypes in the model are based on descriptions of the genetic basis of forewing colour pattern in H. melpomene, H. cydno and their hybrids (Naisbit et al., 2003).
Similar to the previous model, selection favours the red phenotype in one habitat and the white phenotype in the other, reflecting separate predators in each habitat, that learn to avoid different wing colour patterns (Mallet and Barton, 1989). The intermediate red-white phenotype and the no-colour phenotype are maladaptive in both habitats.
Randomly placed mutations cause mating preferences for either red or white phenotypes, similar to the preference for AA or A’A’ phenotypes in the model with one locus controlling the ecological trait. Neutral mutations, which do not contribute to mating preferences, are added as a reference. The probability of mating with an individual of red or white phenotype is dependent on the number of preference loci and is calculated in the same way as in the model with one locus controlling the ecological trait. All individuals exhibit an intermediate mating preference for the red-white and no-colour phenotypes with a constant mating probability of 0.5. All other processes were identical to the model with one locus controlling the ecological trait.
Complete model description following the ODD (Overview, Design concepts and Details) protocol for individual-based models (Grimm et al., 2006, 2010):
The model was designed to explore evolutionary changes in the relative location of loci contributing to reproductive isolation, as a result of selection pressure against intermediate phenotypes. The model is inspired by the colour patterns and mating behaviours of Heliconius butterflies but applies for a wide range of biological systems.
## Entities, state variables, and scales
The model consists of 10000 individuals with an even male:female ratio. Individuals are diploid with a single pair of homologous chromosomes, each containing a sequence of 100 loci, defined together as the “genome”. The first locus controls an ecological trait, for which there are two possible alleles, with an initially equal frequency in the population: A or A’. The alleles are codominant and therefore the three possible genotypes produce three separate phenotypes: AA, A’A’ homozygotes or an intermediate, heterozygote AA’ phenotype. The ecological trait is subject to selection, but also serves as a mating cue and can therefore be considered a “magic trait” (Servedio et al. 2011).
Mutations that cause a preference to mate with either AA or A’A’ phenotypes (preference loci), as well as neutral mutations for comparison, occur at random across the 100 loci in one percent of individuals every generation. The higher the number of preference loci for AA in the genome, compared to the number of A’A’ preference loci, the higher the probability to mate with an individual of AA phenotype (see Submodels section for detailed description).
The modelled environment comprises two habitats, such that selection favours AA genotype in one habitat and A’A’ genotype in the other. Both habitats are maladaptive for the heterozygote AA’ phenotype. Selection is modelled to reflect a scenario of two separate phenotypic optima, with sub-optimal hybrid phenotypes. Habitats are represented in the model in a non-spatial manner. Individuals remain in the habitat to which they are initially assigned.
Time steps in the model correspond to discrete, non-overlapping generations. Each simulation was run for 3000 generations.
## Process overview and scheduling
Each generation, several stages are executed in the following order: (1) formation of mating pairs, (2) reproduction, (3) recombination of offspring chromosomes and addition of randomly placed neutral mutations or mutations causing mating preference, (5) ecological selection, and (6) density dependent regulation of population size. For details on each process see ‘Submodels’ section.
## Design concepts
The model design is based on several Basic principles: (i) selection favours distinct phenotypes of the ecological trait and acts against intermediate phenotypes that arise when separate phenotypes mate, thereby promoting assortative mating based on the trait phenotype. Therefore the trait is regarded as a ‘magic trait’, which is subject to divergent ecological selection and also contributes to non-random mating (Servedio et al., 2011). Specifically,. (ii) Genetic elements that control the trait evolve first, followed by the evolution of genetic elements that control mating preferences, as is assumed to have occurred in Heliconius (Jiggins et al., 2004). The latter can potentially evolve at various places within the genome, and within various distances from trait loci. (iii) Recombination has the potential to break up associations between specific trait and preference alleles (Felsenstein, 1981). Therefore, the physical distance between trait and preference alleles within the genome plays an important role in the development of assortative mating.
Given the disadvantage of offspring with an intermediate phenotype, assortative mating is expected to emerge. Mutations that cause mating preference for the AA or A’A’ genotypes are expected to accumulate on chromosomes that carry the A or A’ allele, respectively. However, it is not straight forward under which conditions selection will favour genomes in which preference loci accumulate nearby the trait locus.
The model includes several procedures that are determined stochastically, to represent random events that take place in reality. These include matching of individuals with potential mating pairs, choice of two of the four parent chromosomes which will be passed on to each of their offspring separately, the cross-over point for recombination, individual genomes that undergo a mutation (either neutral or one that contributes to mating preference), and the location of each mutation within the genome.
Alongside these central stochastic procedures, there are several components of the model that are determined stochastically to maintain events or behaviours at a specified frequency. These include the initial distribution of individuals among the two habitats, determining the trait locus alleles of initial individuals, the decision whether to mate with a potential mating partner, and death due to selection and due to density dependent population size regulation.
The average distance of each type of preference loci and of neutral mutations from the trait locus is recorded for each individual across both chromosomes. The values are then averaged across all individuals, separately for individuals of AA and A’A’ phenotype, at every generation.
The proportion of phenotypically matching mating pairs, in which both male and female are of AA phenotype or of A’A’ phenotype, is recorded every generation, as a measure of assortative mating.
At the beginning of each simulation, 10000 individuals, half male and half female, are randomly assigned to one of the two habitats. The alleles at the trait locus on each of the two chromosomes of each individual are randomly chosen at an equal probability for the A and A’ alleles. Neutral mutations or ones that cause mating preferences are then randomly added to one percent of the population.
## Input data
The model does not use input data to represent time-varying processes.
Formation of mating pairs:
Males and females from both habitats are mixed into the same mating pool and have an equal probability to be paired with each other, reflecting a scenario of unrestricted movement between habitats. The mating choice is made by the female, however since there are no other differences between males and females, the model can also reflect a scenario in which males make the mating choice. Each female is paired with a random male and will mate with him at a probability that depends on the strength of her preference for his phenotype (see below). If a female decides not to mate, she is paired sequentially with a maximum of ten random males until she mates. If she does not mate with the tenth male with which she is paired, she is taken out of the mating pool and does not reproduce. Limiting the number of males with which a female is paired avoids simulations from running endlessly. See supplementary material (???) for details on the sensitivity of model results to changes in the maximum number of males with which a female is paired. The probability of a female to mate with a male with which she is paired is described by the following equations:
Where PA and PA’ are the probabilities to mate with a male of phenotype AA and A’A’, respectively. d is the difference between the number of AA preference loci and the number of A’A’ preference loci in the genome. The higher the number of AA preference loci, compared to A’A’ preference loci, the larger the probability to mate with an AA phenotyped individual, and vice versa. A preference strength factor (pf) determines how strong the contribution of each additional preference locus is.
The probability to mate with a male of AA’ phenotype is constant for all individuals and equals 0.5. This represents a case in which hybrids, represented by the heterozygotes in this model, exhibit intermediate phenotypes which are partially attractive to individuals who prefer to mate with one of the distinct phenotypes, represented as homozygotes in this model. Robustness of model results to changes in this basic assumption are detailed in the supplementary material (S3.1).
The preference strength factor (pf) is used to control the rate at which strong mating preferences accumulate across the population. Varying the value of pf allows testing the influence of mating preference strength on the development of physical linkage between trait and preference loci.
Only individuals who have found a mating partner in the previous procedure will reproduce. Each mating pair produces four offspring, two males and two females. The number of offspring was chosen to keep the size of the new generation above carrying capacity, before selection and density dependent regulation take place, to avoid population collapse.
Each offspring receives one paternal and one maternal chromosome, randomly chosen from the two chromosomes of each parent. Offspring are assigned the same habitat as their mother, to ensure that the mating choice has a direct influence on the survival of offspring and is therefore subject to natural selection. Following reproduction, the parent generation dies.
Offspring recombination and mutation:
Each offspring genome undergoes recombination between the two homologous chromosomes. Recombination occurs at one, randomly chosen, cross-over point within the chromosomes. The content of the “genome” sequence after the cross-over point is exchanged between the two homologous chromosomes.
Recombination can break apart linkage disequilibrium between traits and mating preferences, both in the model and in reality. It is therefore the force that drives physical linkage between loci controlling the trait and loci controlling mating preferences.
After recombination is completed, one percent of individuals in the population undergo a mutation at a single position within their genome. The mutations cause a preference to mate with either AA or A’A’ phenotypes (preference loci), or do not have any influence on mating preferences (neutral loci). The mutations are placed at random locations within the genome and therefore within random distances from the first locus that controls the ecological trait. See supplementary material (S2.3) for details on the sensitivity of model results to changes in the percent of individuals that undergo mutations every generation.
We chose to add randomly placed mutations that cause mating preferences based on the assumption that physical linkage is formed by co-option of genes that are already within physical linkage with the trait locus, rather than by transposition of genetic elements that control preference from other regions in the genome. In Heliconius, for example, there is no evidence of transposition around colour pattern genes, or chromosomal inversions that might be involved in maintaining species barriers among Heliconius melpomene and H. cydno (The Heliconius Genome Consortium, 2012; Davey et al., 2017).
Selection favours AA phenotype in one habitat and A’A’ phenotype in the second habitat. Phenotypes that are not favoured in each of the habitats are subject to selection. Selection is modelled to reflect a scenario of two separate phenotypic optima, with sub-optimal hybrid phenotypes. The strength of selection is fixed per simulation and controlled by a selection coefficient (s), which determines the proportion of maladapted individuals that will die due to selection. Robustness of model results to changes in the relative selection level against heterozygote, intermediate, phenotypes are detailed in the supplementary material (S3.2).
Density dependent regulation:
Each of the two habitats has a carrying capacity of 5000 individuals. The probability of each individual to die due to density dependent regulation equals the number of access individuals in the respective habitat divided by the total number of individuals in that habitat. Separate density dependent regulation of population size in each habitat is based on the assumption that individuals depend on separate, limited, resources in each habitat.