turtles-own [mat-chrom pat-chrom] globals [ max-percent ;; percent of the total population that is in the population-to-cull ;; global population size for culling procedure ] to setup clear-all ;; create turtles with random colors and locations crt population-size [ spawn ] setup-plots update-plots end to spawn ifelse random 100 < (proportion-allele-A * 100) [ set mat-chrom 1 ] [ set mat-chrom 2 ] ifelse random 100 < (proportion-allele-A * 100) [ set pat-chrom 1 ] [ set pat-chrom 2 ] ifelse (mat-chrom = 1 or pat-chrom = 1) [ set color blue ] [ set color yellow ] setxy random-xcor random-ycor let agent-shape species if agent-shape = "human" [ set agent-shape "person"] if agent-shape = "ant" [ set agent-shape "bug"] set shape agent-shape end to go if (quit-at-fixation and ((count turtles with [mat-chrom = 1] = 0 and count turtles with [pat-chrom = 1] = 0) or (count turtles with [mat-chrom = 2] = 0 and count turtles with [pat-chrom = 2] = 0))) or (max-generations > 0 and ticks > max-generations or count turtles = 0) [ stop ] ask turtles [ if ticks = 1 [ cull ] rt random 50 - random 50 fd dispersal-rate mutate meet ] immigrate set population-to-cull count turtles ;; show ((population-to-cull - population-size) / population-to-cull) * 100 ask turtles [ cull ] tick update-plots end to immigrate crt (population-size * blue-immigration * .01) [ set mat-chrom 1 set pat-chrom 1 set color blue setxy random-xcor random-ycor let agent-shape species if agent-shape = "human" [ set agent-shape "person"] if agent-shape = "ant" [ set agent-shape "bug"] set shape agent-shape ] crt (population-size * yellow-immigration * .01) [ set mat-chrom 2 set pat-chrom 2 set color yellow setxy random-xcor random-ycor let agent-shape species if agent-shape = "human" [ set agent-shape "person"] if agent-shape = "ant" [ set agent-shape "bug"] set shape agent-shape ] end to meet ;; turtle procedure - when two turtles are adjacent, reproduce (but no self-fertilization!) let mom self let mate nobody ifelse mate-with = "random" [ set mate one-of other turtles ] [ set mate one-of other turtles in-radius 1 ] if mate != nobody [ hatch 1 [ ifelse random 100 < 50 [ set mat-chrom [mat-chrom] of mom ] [ set mat-chrom [pat-chrom] of mom ] ifelse random 100 < 50 [ set pat-chrom [mat-chrom] of mate ] [ set pat-chrom [pat-chrom] of mate ] ifelse (mat-chrom = 1 or pat-chrom = 1) [ set color blue ] [ set color yellow ] let agent-shape species if agent-shape = "human" [ set agent-shape "person"] if agent-shape = "ant" [ set agent-shape "bug"] set population-to-cull count turtles set shape agent-shape cull ] ] end to mutate ifelse mat-chrom = 1 [ if random 10000 < mutation-dominant-to-recessive * 100 [ set mat-chrom 2] ] [ if random 10000 < mutation-recessive-to-dominant * 100 [ set mat-chrom 1] ] ifelse pat-chrom = 1 [ if random 10000 < mutation-dominant-to-recessive * 100 [ set pat-chrom 2] ] [ if random 10000 < mutation-recessive-to-dominant * 100 [ set pat-chrom 1] ] end to cull let death-chance ((population-to-cull - population-size) / population-to-cull) * 100 if (selection-against-blue > 0 and color = blue) [ set death-chance death-chance + ((100 - death-chance) * selection-against-blue * .01) ] if (selection-against-yellow > 0 and color = yellow) [ set death-chance death-chance + ((100 - death-chance) * selection-against-yellow * .01) ] if random 100 < death-chance [ die ] end to setup-plots set-current-plot "Phenotypic Frequencies" set-plot-y-range 0 1 set-current-plot "Genotypic Frequencies" set-plot-y-range 0 1 set-current-plot "Allele Frequencies" set-plot-y-range 0 1 end to update-plots set-current-plot "Phenotypic Frequencies" set-current-plot-pen "11" ifelse count turtles = 0 [ plot 0 ] [ plot (count turtles with [color = blue] / count turtles) ] set-current-plot-pen "22" ifelse count turtles = 0 [ plot 0 ] [ plot (count turtles with [color = yellow] / count turtles) ] set-current-plot "Genotypic Frequencies" set-current-plot-pen "11" ifelse count turtles = 0 [ plot 0 ] [ plot (count turtles with [mat-chrom = 1 and pat-chrom = 1] / count turtles) ] set-current-plot-pen "22" ifelse count turtles = 0 [ plot 0 ] [ plot (count turtles with [mat-chrom = 2 and pat-chrom = 2] / count turtles) ] set-current-plot-pen "12" ifelse count turtles = 0 [ plot 0 ] [ plot ((count turtles with [mat-chrom = 2 and pat-chrom = 1] + count turtles with [mat-chrom = 1 and pat-chrom = 2]) / count turtles) ] set-current-plot "Allele Frequencies" set-current-plot-pen "1" ifelse count turtles = 0 [ plot 0 ] [ plot (count turtles with [mat-chrom = 1] + count turtles with [pat-chrom = 1]) / (count turtles * 2) ] set-current-plot-pen "2" ifelse count turtles = 0 [ plot 0 ] [ plot (count turtles with [mat-chrom = 2] + count turtles with [pat-chrom = 2]) / (count turtles * 2) ] end @#$#@#$#@ GRAPHICS-WINDOW 302 10 847 576 20 20 13.05 1 10 1 1 1 0 1 1 1 -20 20 -20 20 1 1 1 ticks CC-WINDOW 5 590 1253 685 Command Center 0 BUTTON 67 541 140 574 go go T 1 T OBSERVER NIL G NIL NIL BUTTON 211 78 299 123 setup setup NIL 1 T OBSERVER NIL S NIL NIL SLIDER 6 44 299 77 population-size population-size 50 5000 500 50 1 NIL HORIZONTAL PLOT 851 10 1243 200 Genotypic Frequencies Time Frequency 0.0 100.0 0.0 1.0 true false PENS "11" 1.0 0 -13345367 true "22" 1.0 0 -1184463 true "12" 1.0 0 -10899396 true SLIDER 6 10 299 43 proportion-allele-A proportion-allele-A 0 1 0.5 0.01 1 NIL HORIZONTAL PLOT 851 205 1244 376 Phenotypic Frequencies Time Frequency 0.0 100.0 0.0 1.0 true false PENS "11" 1.0 0 -13345367 true "12" 1.0 0 -10899396 true "22" 1.0 0 -1184463 true CHOOSER 6 78 100 123 species species "turtle" "human" "cow" "fish" "ant" 3 MONITOR 101 78 210 123 Population Size count turtles 0 1 11 CHOOSER 206 367 298 412 mate-with mate-with "random" "neighbor" 0 MONITOR 1193 38 1243 83 AA count turtles with [mat-chrom = 1 and pat-chrom = 1] 0 1 11 MONITOR 1193 84 1243 129 Aa count turtles with [(mat-chrom = 1 and pat-chrom = 2) or (mat-chrom = 2 and pat-chrom = 1)] 0 1 11 MONITOR 1193 130 1243 175 aa count turtles with [mat-chrom = 2 and pat-chrom = 2] 0 1 11 MONITOR 1187 288 1244 333 Blue count turtles with [color = blue] 0 1 11 MONITOR 1187 242 1244 287 Yellow count turtles with [color = yellow] 17 1 11 INPUTBOX 206 428 298 488 max-generations 200 1 0 Number PLOT 852 379 1244 576 Allele Frequencies Time Frequency 0.0 100.0 0.0 1.0 true false PENS "1" 1.0 0 -13345367 true "2" 1.0 0 -1184463 true TEXTBOX 8 120 297 148 ________________________________________________ 11 0.0 1 SLIDER 8 214 299 247 selection-against-blue selection-against-blue 0 100 0 1 1 percent HORIZONTAL SLIDER 8 138 299 171 blue-immigration blue-immigration 0 100 0 1 1 percent HORIZONTAL SLIDER 8 173 299 206 yellow-immigration yellow-immigration 0 100 0 1 1 percent HORIZONTAL SLIDER 8 249 299 282 selection-against-yellow selection-against-yellow 0 100 0 1 1 percent HORIZONTAL SLIDER 8 371 204 404 dispersal-rate dispersal-rate 0.5 5 0.5 0.5 1 NIL HORIZONTAL BUTTON 144 541 229 574 go (once) go NIL 1 T OBSERVER NIL O NIL NIL SLIDER 8 290 299 323 mutation-dominant-to-recessive mutation-dominant-to-recessive 0 1 0 .01 1 percent HORIZONTAL SLIDER 8 325 299 358 mutation-recessive-to-dominant mutation-recessive-to-dominant 0 1 0 .01 1 percent HORIZONTAL SWITCH 14 438 204 471 quit-at-fixation quit-at-fixation 0 1 -1000 MONITOR 1187 428 1244 473 A count turtles with [mat-chrom = 1] + count turtles with [pat-chrom = 1] 0 1 11 MONITOR 1186 474 1243 519 a count turtles with [mat-chrom = 2] + count turtles with [pat-chrom = 2] 0 1 11 @#$#@#$#@ WHAT IS IT? ----------- This is a model of the Hardy-Weinberg (HW) equilibrium. The HW principle predicts the genotypic frequencies that will be observed in a population over the course of generations given particular allele frequencies, and given that five assumptions (discussed below) hold true in the population. Given two alleles, A and a, and the frequencies of each allele in the population, freq(A)=p and freq(a)=q, the HW principle predicts: 1) p + q = 1. That is, since A and a are the only alleles at this locus in the model population, the allelic frequencies of A and a must add up to 1. 2) The genotypic frequency of AA homozygotes in the population is p^2. The frequency of aa homozygotes is q^2. The probability that any given member of the population will inherit two A alleles is p x p. The frequency of heterozygotes is 2pq. The probability that any given member of the population will inherit one A and one a allele is p x q x 2, since a heterozygote can inherit allele A from its mother and allele a from its father, OR allele a from its mother and A from its father. p^2 + 2pq + q^2 = 1. That is, the frequencies of both types of homozygotes and the frequency of heterozygotes must add up to 1, since these are the only possible combinations. These predictions hold true given that five assumptions about the population all hold true. 1) Large (infinite) population size. In small populations, chance differences in reproductive success and mating choices can produce deviations from the predictions of HW. 2) No selection. There is no systematic difference in the survival or reproductive success of organisms with different genotypes. 3) No mutation. The alleles are inherited from one generation to the next without being changed by mutation. 4) No migration. No organisms leave the population, and no new ones come in. 5) Random mating. Organisms choose mates at random with respect to the alleles of interest in the model. If any of these assumptions do not hold true in a population, the observed genotypic frequencies will deviate from the predictions of HW in particular ways depending on the assumption(s) that is (are) violated. HOW IT WORKS ------------ The model is initialized with a randomly distributed population of blue (the dominant trait) and yellow (the recessive trait) organisms. Organisms are randomly assigned alleles according to the selected frequency of the A allele (the frequency of the a allele is determined as q = p - 1.) As the model runs, organisms move around the world in a correlated random walk at a dispersal rate determined by the user. On each tick, organisms select a mate, either by mating with another organism chosen at random anywhere in the world, or by choosing a nearby organism to mate with. An offspring is produced adjacent to the reproducing organism, which randomly inherits either the a or A alleles from each of its parents. The organisms are diploid, but are essentially hermaphroditic, as every organism is capable of producing offspring and may mate with any other organism without the need to locate a mate of the opposite sex. Following reproduction, the population is culled to bring the population size back down to the carrying capacity determined by the user. During each culling, each organism is subject to a probability of death determined by the degree to which the current population size exceeds the carrying capacity. As a result, an organism may live for several generations, or it may not survive to first reproduction. There is no maturation time, so that any organism that survives the first culling following its birth can reproduce during the next reproduction cycle. The model ends when it reaches a specified number of generations ("ticks"), or when one allele becomes fixed in the population (that is, the other allele goes extinct), or when the entire population of organisms goes extinct (e.g. due to high selection against both phenotypes). HOW TO USE IT ------------- The "proportion-allele-A" slider bar determines the initial frequency of the allele A. The frequency of allele a is determined by calculating freq(a) = 1 - freq(A). The "population-size" slider determines the carrying capacity of the system. The "species" chooser allows the user to select from a list of possible icons to represent the organism as they move around the world. Clicking the "setup" button initializes the world with a population of organisms of the selected species, with the specified allele frequencies. The "go" button starts the model. The "Population size" monitor displays the current population size. Note that this population size will not always match exactly the value selected by the "population-size" slider. In fact, during each reproduction cycle, the population size will rise well above this value, and then fall back roughly to the specified population size at the end of the culling cycle. However because mortality for each organism is determined by a certain probability, the final population size will not be exactly the specified value, although it will be close. Graphs track the genotypic frequencies, phenotypic frequencies, and allele frequencies over time as the model runs. Monitors display the current values for each of these. As the model runs, the user may change the settings of any slider, chooser or input -- with the exception of the "proportion-allele-A" slider -- and the model will reflect these newly selected values. The value of the "proportion-allele-A" slider is used only at model setup; allele frequencies are determined only by the behavior of the organisms after the model begins running. THINGS TO NOTICE ---------------- Note that the genotypic and phenotypic frequencies approximate the values predicted by the HW formulas. Try calculating the predicted values based on the allele frequencies you have specified (or the current allele frequencies obtained by the current A and a alleles preent in the population) and compare these to the actual values produced by the model as it runs. The model itself does not make use of the HW formulas, but produces values similar to those predicted by HW by the interactions of the model organisms. Note the random changes in the genotypic, phenotypic, and allelic frequencies over time. These changes are more apparent with smaller population sizes, but can still be observed even with populations in the thousands. These random changes result from random differences in the survival and reproductive success of individuals each generation, and are called genetic drift. Genetic drift has a bigger effect on the makeup of small populations than larger ones. Theoretically, the HW assumption of "large population" actually requires an infinitely large population in order to completely eliminate the effect of drift. THINGS TO TRY ------------- Experiment with the settings of the model to create violations of the five assumptions described above. The Hardy-Weinberg equilibrium describes a theoretical population that cannot exist in the real world; perhaps its greatest value is in describing a population where no evolution is occurring, in order to better understand real populations where one or more of the five assumptions are violated, and evolution is occurring. 1) Large (infinite) population size. Try running the model with populations of different sizes in order to observe differences in the strength of genetic drift. 2) No selection. Try experimenting with different degrees of selection against (increased mortality of) the blue and yellow phenotypes. You will observe that selection against the recessive phenotype (yellow color) takes much longer to completely remove the a allele from the population, even with heavy selection against the yellow phenotype. Why is this? What does this tell us about the persistence of recessive genetic disorders in the population? 3) No mutation. Experiment with different mutation rates from dominant to recessive, or recessive to dominant. What happens if there is a large rate of mutation in both directions? 4) No migration. Experiment with different rates of immigration of blue and yellow individuals. How does immigration of individuals of a particular color effect the overall genetic makeup of the population? 5) Random mating. The "mate-with" chooser can cause the organisms in the model to choose mates completely at random, selecting any other organism in the world as a mate. You can also cause organisms to mate with a neighbor, so that organisms must be adjacent in order to mate with each other. Note what happens to the frequency of heterozygotes when organisms are mating with their neighbors. Also note that increasing the dispersal rate (the distance the organisms move each time step) decreases the effect of mating with neighbors on the phenotypic makeup of the population. Why does this happen? EXTENDING THE MODEL ------------------- Other methods for violating the assumptions of the HW equilibrium could be added. For example, selection in this model is caused by increasing the mortality rate of one or both of the phenotypes. Selection could also result from differential reproductive output, or from differential success in finding mates (sexual selection). The model could also be extended to multiple genes to, for example, examine the effect of linkage on inheritance. RELATED MODELS -------------- NetLogo Library Models: -GenDrift (T reproduce) -Simple Birth Rates NetLogo Community Models: -PopGen Fishbowl 1 -Genetics and Cellular Automata CREDITS AND REFERENCES ---------------------- Hardy-Weinberg Classroom Model (2009) Kenneth Letendre Departments of Biology and Computer Science University of New Mexico kletendr@unm.edu Council-Garcia, C.L., S. Ligon, B. Milne, R. Thornhill and D. Swenton. 2004. Biology 203L: Evolution and Ecology Lab Manual. Hardy, G. H. (1908). "Mendelian proportions in a mixed population". Science 28: 49 – 50. Stern, C. (1943). "The Hardy–Weinberg law". Science 97: 137–138. Weinberg, W. (1908). "Über den Nachweis der Vererbung beim Menschen". 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