breed [ DDs DD ] ;; the agentset of homozygotes for dominant allele breed [ DRs DR ] ;; the agentset of heterozygotes breed [ RRs RR ] ;; the agentset of homozygotes for recessive allele turtles-own [ sex partner ] ;; all turtles have two variables: sex (male or female) and partner (who number) to setup ca ;; turtles are arranged in a circle create-ordered-DDs initial-DDs [ fd 10 set shape "dot" set color white set size 2 ;; to improve visualization of different agentsets set partner nobody ;; (see the create-pairs procedure) ] create-ordered-DRs initial-DRs [ fd 10 set shape "dot" set color green set size 1.5 ;; same as above set partner nobody ;; (see the create-pairs procedure) ] create-ordered-RRs initial-RRs [ fd 10 set shape "dot" set color red set size 1 ;; same as above set partner nobody ;; (see the create-pairs procedure) ] ;; an equal number of females and males is assigned ask n-of ( count turtles / 2 ) turtles [ set sex "female" ] ask turtles with [ sex != "female" ] [ set sex "male" ] reset-ticks end to go create-pairs ;; (see the create-pairs procedure) reproduce ;; (see the reproduce procedure) end to create-pairs ;; "singles" do not have any partner. Each single female creates a link with a random single male. At the end of this procedure, all females and males are biunivocally paired let singles turtles with [ partner = nobody ] ask turtles with [ sex = "female" ] [ if any? singles [ if ( partner = nobody ) and ( any? other turtles with [ sex = "male" ] with [ partner = nobody ] ) [ set partner one-of other turtles with [ sex = "male" ] with [ partner = nobody ] create-link-with partner ask partner [ set partner myself ] ] ] ] tick end to reproduce ;; each pair of partners generates one male and one female with genotypes reflecting the Mendelian segregation ratios. Immediately after reproduction, both parents die ask DDs with [ sex = "female" ] [ if any? DDs with [ partner = myself ] [ ask one-of DDs with [ partner = myself ] [ hatch-DDs 1 [ set shape "dot" ] die ] hatch-DDs 1 [ set shape "dot" ] die ] ] ask RRs with [ sex = "female" ] [ if any? RRs with [ partner = myself ] [ ask one-of RRs with [ partner = myself ] [ hatch-RRs 1 [ set shape "dot" ] die ] hatch-RRs 1 [ set shape "dot" ] die ] ] ask DDs with [ sex = "female" ] [ if any? RRs with [ partner = myself ] [ ask one-of RRs with [ partner = myself ] [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] die ] hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] die ] ] ask RRs with [ sex = "female" ] [ if any? DDs with [ partner = myself ] [ ask one-of DDs with [ partner = myself ] [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] die ] hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] die ] ] ask DDs with [ sex = "female" ] [ if any? DRs with [ partner = myself ] [ ask one-of DRs with [ partner = myself ] [ let chance random 100 if chance < 50 [ hatch-DDs 1 [ set shape "dot" set color white set size 2 ] ] if chance >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] let chance2 random 100 if chance2 < 50 [ hatch-DDs 1 [ set shape "dot" set color white set size 2 ] ] if chance2 >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] ] ask DRs with [ sex = "female" ] [ if any? DDs with [ partner = myself ] [ ask one-of DDs with [ partner = myself ] [ let chance random 100 if chance < 50 [ hatch-DDs 1 [ set shape "dot" set color white set size 2 ] ] if chance >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] let chance2 random 100 if chance2 < 50 [ hatch-DDs 1 [ set shape "dot" set color white set size 2 ] ] if chance2 >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] ] ask RRs with [ sex = "female" ] [ if any? DRs with [ partner = myself ] [ ask one-of DRs with [ partner = myself ] [ let chance random 100 if chance < 50 [ hatch-RRs 1 [ set shape "dot" set color red set size 1 ] ] if chance >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] let chance2 random 100 if chance2 < 50 [ hatch-RRs 1 [ set shape "dot" set color red set size 1 ] ] if chance2 >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] ] ask DRs with [ sex = "female" ] [ if any? RRs with [ partner = myself ] [ ask one-of RRs with [ partner = myself ] [ let chance random 100 if chance < 50 [ hatch-RRs 1 [ set shape "dot" set color red set size 1 ] ] if chance >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] let chance2 random 100 if chance2 < 50 [ hatch-RRs 1 [ set shape "dot" set color red set size 1 ] ] if chance2 >= 50 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] die ] ] ask DRs with [ sex = "female" ] [ if any? DRs with [ partner = myself ] [ ask one-of DRs with [ partner = myself ] [ let chance random 100 if chance < 25 [ hatch-DDs 1 [ set shape "dot" set color white set size 2 ] ] if chance >= 25 and chance < 75 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] if chance >= 75 [ hatch-RRs 1 [ set shape "dot" set color red set size 1 ] ] die ] let chance2 random 100 if chance2 < 25 [ hatch-DDs 1 [ set shape "dot" set color white set size 2 ] ] if chance2 >= 25 and chance2 < 75 [ hatch-DRs 1 [ set shape "dot" set color green set size 1.5 ] ] if chance2 >= 75 [ hatch-RRs 1 [ set shape "dot" set color red set size 1 ] ] die ] ] tick end to-report Generation ;; counting generations ifelse ticks mod 2 = 0 [ report ticks / 2 ] [ report ( ticks - 1 ) / 2 ] end to-report p ;; calculate the frequency of the allele "D" report ( ( count DDs * 2 ) + ( count DRs ) ) / ( count turtles * 2 ) end to-report q ;; calculate the frequency of the allele "R" report ( ( count RRs * 2 ) + ( count DRs ) ) / ( count turtles * 2 ) end to-report Expected-DDs ;; calculate the expected number of "DD" individuals report ( p ^ 2 ) * ( count turtles ) end to-report Expected-DRs ;; calculate the expected number of "DR" individuals report ( 2 * p * q ) * ( count turtles ) end to-report Expected-RRs ;; calculate the expected number of "RR" individuals report ( q ^ 2 ) * ( count turtles ) end to-report Chi-square ;; calculate Chi square values with correction for expected number of individuals per genotype < 5 report ( ( ( abs ( count DDs - Expected-DDs ) - 0.5 ) ^ 2 ) / Expected-DDs ) + ( ( ( abs ( count DRs - Expected-DRs ) - 0.5 ) ^ 2 ) / Expected-DRs ) + ( ( ( abs ( count RRs - Expected-RRs ) - 0.5 ) ^ 2 ) / Expected-RRs ) end @#$#@#$#@ GRAPHICS-WINDOW 283 33 789 540 -1 -1 19.92 1 10 1 1 1 0 0 0 1 -12 12 -12 12 0 0 1 ticks 30.0 BUTTON 25 30 106 77 NIL SETUP NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 24 194 223 227 Initial-DDs Initial-DDs 0 500 389.0 1 1 NIL HORIZONTAL SLIDER 24 249 223 282 Initial-DRs Initial-DRs 0 500 310.0 1 1 NIL HORIZONTAL SLIDER 24 304 223 337 Initial-RRs Initial-RRs 0 500 432.0 1 1 NIL HORIZONTAL BUTTON 135 30 218 76 NIL GO T 1 T OBSERVER NIL NIL NIL NIL 0 BUTTON 18 109 114 152 CREATE-PAIRS if ticks = 0 or ticks mod 2 = 0 [ create-pairs ]\nif ticks mod 2 = 1 [ stop ] NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 129 109 228 151 REPRODUCE if ticks mod 2 = 1 [ reproduce ]\nif ticks mod 2 = 0 [ stop ] NIL 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 21 365 109 426 Population count turtles 17 1 15 MONITOR 143 364 237 425 NIL Generation 17 1 15 MONITOR 1363 35 1482 96 Expected DDs Expected-DDs 0 1 15 MONITOR 1364 126 1482 187 Expected DRs Expected-DRs 0 1 15 MONITOR 1366 215 1483 276 Expected RRs Expected-RRs 0 1 15 PLOT 848 19 1182 202 Allele Frequencies Generation Allele frequencies 0.0 10.0 0.0 1.0 true false "" "" PENS "p" 1.0 0 -11221820 true "" "if ticks mod 2 = 0 [ plot p ]" "q" 1.0 0 -955883 true "" "if ticks mod 2 = 0 [ plot q ]" PLOT 848 224 1179 401 Genotype Frequencies Generation Genotype Frequencies 0.0 10.0 0.0 1.0 true false "" "" PENS "DRs" 1.0 0 -10899396 true "" "if ticks = 2 or ticks mod 2 = 0 [plot count DRs / count turtles ]" "RRs" 1.0 0 -2674135 true "" "if ticks = 2 or ticks mod 2 = 0 [plot count RRs / count turtles ]" "DDs" 1.0 0 -7500403 true "" "if ticks = 2 or ticks mod 2 = 0 [plot count DDs / count turtles ]" PLOT 851 426 1182 580 Chi-square Generation Chi-square 0.0 10.0 0.0 10.0 true false "set-plot-y-range 0 5" "set-plot-y-range 0 5" PENS "Chi-square" 1.0 0 -16777216 true "" "if ticks mod 2 = 0 [ plot chi-square ]" "Threshold for fd 1" 1.0 0 -2674135 true "" "plot 3.84" MONITOR 1219 36 1342 97 Observed DDs count DDs 0 1 15 MONITOR 1222 125 1343 186 Observed DRs count DRs 0 1 15 MONITOR 1224 214 1344 275 Observed RRs count RRs 0 1 15 MONITOR 21 455 109 516 Females count turtles with [ sex = \"female\" ] 17 1 15 MONITOR 145 455 237 516 Males count turtles with [ sex = \"male\" ] 17 1 15 MONITOR 1314 470 1406 531 Chi-square chi-square 2 1 15 @#$#@#$#@ ## WHAT IS IT? This is a model of the fundamental law of population genetics, a.k.a. Hardy-Weinberg Equilibrium. The model is strictly adherent to the implicit and explicit assumptions originally made in 1908 by the English mathematician Godfrey Harold Hardy and the German physician Wilhelm Weinberg. These assumptions are summarized as follows: * the organism is diploid; * the considered gene is autosomal and bi-allelic (e.g., Brachydactyly); * reproduction is sexual; * generations are rigorously nonoverlapping; * sexes are evenly distributed; * the allele and genotypic frequencies are the same in males and females; * mating is random (the population is panmictic); * population size is very large (virtually infinite); * migration is neglible; * mutation can be ignored; * natural selection does not affect the alleles under consideration. Based on these assumptions, the simulation shows that equilibrium of genotype and allele frequencies is reached in one generation, and the population remains in equilibrium in successive generations. Verification of Hardy-Weinberg Equilibrium in the population is done with the Chi square test. ## HOW IT WORKS Three agents' types with a "dot" shape - one for each genotype - are implemented as described hereafter: 1) white agents = "DD", homozygous genotype for the dominant phenotype; 2) red agents = "RR", homozygous genotype for the recessive phenotype; 3) green agents = "DR", heterozygous genotype, also correspondent to the dominant phenotype. After fixing the initial number of individuals, each with a specific genotype, an equal number of males and females is assigned. Sex ratio is then maintained constantly = 1 throughout simulation. At the start of simulation, agents are arranged in a circle. The circular arrangement was chosen to facilitate visualization of random mating through the formation of links between agents of opposite sex. For each generation, each partner can only mate with a partner of opposite sex who is not engaged already in another mating event. After reproduction, the population of parents is completely replaced by the population of their offspring, which bears the specific genotypes predicted on the basis of Mendelian segregation rules. Over generations, population size remains constant Overlapping generations are never allowed because parents "die" at each generation after they have reproduced. The model uses a Chi square formula with correction for small sample size (i.e., expected values of genotypes < 5) (Hartl and Clark 2007). ## HOW TO USE IT Use the sliders to select the initial number of agents with their specific genotypes. Press the setup button to arrange agents in a circle. Simulation consists of two steps to be performed in succession: first create-pairs, then reproduction. These steps are implemented with two buttons to run the simulation in a discrete mode. Alternatively, simulation can be run in a continuous mode by pressing the "go" (forever) button. Evolution of genotype and allele frequency can be followed by watching at the corresponding monitors and plots. Statistical differences between observed and expected number of each genotypic class is calculated, generation after generation, by the Chi square statistics, and Chi square values are reported in the corresponding monitor and plot. If population is in equilibrium chi square does not exceed the critical value which - for one degree of freedom - is = 3,84. This value is represented in the Chi square plot as a red line. ## THINGS TO NOTICE Note that each generation is equivalent to two ticks, one tick for creating pairs and the subsequent one for reproduction. Note also that, for expected values of genotypes too close to zero, the Chi square calculation is not reliable. Moreover, if a genotype is completely missing no Chi square value is calculated. ## THINGS TO TRY Test the performance of the model with different initial numbers of the three genotypes. Use the create-pairs and reproduction buttons to visualize the single steps of simulation (discrete mode). Alternatively, use the go button to observe the behaviour of the model run in continuous mode. ## EXTENDING THE MODEL The code can be modified to test the effects of evolutionary forces (e.g., selection, mutations, etc.) on HWE. ## NETLOGO FEATURES The two key primitive terms that were used to implement the mating and the reproduction procedures were "create-link-with" and "hatch" respectively. Furthermore, segregation ratios are stochastically implemented to assign the probabilities on Mendelian segregation ratios. ## RELATED MODELS In the Netlogo Models Library look at (i) Hardy Weinberg Equilibrium, and (ii) Mendelian Inheritance. ## CREDITS AND REFERENCES Hardy, G. H. (1908). Mendelian proportions in a mixed population. Science 28: 49-50. Hartl, D. L., and Clark, A. G. (2007). Principles of Population Genetics (4th edition), Chapter 2 pp. 45-92. Sinauer. Weinberg, W. (1908). Uber den Nachweis der Vererbung beim Menschen. Jahresh. Ver. Vaterl. Naturkd. Wurttemb 64: 369-382 (English translations in Boyer 1963 and Jameson 1977). ## HOW TO CITE For the model itself: Tarantino, R., and Romano, V. (2022). Hardy-Weinberg Basic Model. 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