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Pedro Cabral

Simulating "Animal Crossing: New Horizons" Mendelian Flower Genetics

Pedro Cabral, Researcher at the Federal Institute of Ceará

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Simulating “Animal Crossing: New Horizons” Mendelian Flower Genetics in Mathematica
Pedro Cabral – Researcher, Data Scientist.

Introduction

In the video game Animal Crossing: New Horizons, by Nintendo, one interesting game mechanic is the ability of planting and breeding many species flowers. The game uses simple Mendelian genetics and many results can be obtained with the Punnet square and simple statistics. In this article, I define many useful functions for representing the flower genes, computationally breeding two flowers, and plotting and charting the Punnet square.

Understanding the Genes

Before starting the definition of the functions and breeding of the flowers, lets understand how exactly the genes present in the flowers work.

“Rr”
Controls whether a flower is red based or not. Red based flowers are red, black, pink, and orange . Non-red are yellow, white, and purple.
Note: Not all flowers with the “R” gene will be red-based, since other genes can cover it, but a flower with the genes “rr” will never be red-based.

◼

“RR” or “Rr”  Red-based.

◼

“rr”  Non-red.

“Yy”
Controls whether a flower has any yellow in it (yellow or orange).

◼

“YY” or “Yy”  Yellow or Orange.

◼

“yy”  Non-yellow.

“Ww”
Controls whether a white flower will be white or show its underlying color: either purple or blue depending on the species.
Note: White flowers are fairly recessive to most colors, so this gene will only have an effect if the white isn’t getting covered by reds, yellows, or other colors.

◼

“WW” or “Ww”  White.

◼

“ww”  Purple or blue.

“Ss”
Controls which shade the red-based flower will be. Pink is the lightest shade, red is the middle, and black is the darkest. This gene has no effect on non-red flowers.

◼

“SS”  Pink.

◼

“Ss”  Red.

◼

“ss”  Black.

Functions and Definitions

FlowerGene

Splits the gene into partitions of two. Useful on operations involving breeding and Punnet square.

In[]:=

FlowerGene::invgene="Flower gene must contain 8 alleles.";

In[]:=

FlowerGene[gene_String]:=Module[{str=StringReplace[gene,"-"""],part},(*Checkifthere's8alleles.*)If[!(StringLength[str]8),Return[Message[FlowerGene::invgene]];];(*Partitionintogroupsoftwo.*)part=StringPartition[str,2];(*Returnthesplittedalleles.*)Return[StringSplit[#,""]&/@part];];

In[]:=

(*Exampleusage:*)FlowerGene["RR-yy-WW-Ss"]

Out[]=

{{R,R},{y,y},{W,W},{S,s}}

FlowerColor

Returns the color for a given gene. This approach of determining a specific flower color is ad-hoc.

In[]:=

FlowerColor[gene_]:=Modulestr=StringReplace[gene,"-"""],flowergenes=

FlowerGenes

,Return[flowergenes〚str〛];;

In[]:=

(*Exampleusage:*)FlowerColor["RR-yy-WW-Ss"]

Out[]=

Red

RandomFlowerGene

Generates a random flower gene.

In[]:=

RandomFlowerGene[]:=Module[{},RandomChoice[{"RR","Rr","rr"}]<>RandomChoice[{"YY","Yy","yy"}]<>RandomChoice[{"WW","Ww","ww"}]<>RandomChoice[{"SS","Ss","ss"}]];

In[]:=

(*Exampleusage:*)RandomFlowerGene[]

Out[]=

RRyyWWSs

FlowerIcon

Returns an icon for a given color .

In[]:=

FlowerIcon::invcolor="`` is not a valid flower color.";

In[]:=

FlowerIcon[color_String]:=Modulecolors=AssociationThread{"red","white","orange","yellow","purple","black","pink","blue"},ImageResize[#,{28}]&/@

,(*Checkifthecolorisavalidcolor*)If[!KeyExistsQ[colors,ToLowerCase[StringTrim[color]]],Return[Message[FlowerIcon::invcolor,color]];];(*Returnthecorrespondingcolor.*)Return[colors〚ToLowerCase[StringTrim[color]]〛];;

In[]:=

(*Exampleusage:*)FlowerIcon["Red"]

Out[]=

OrderGenes

Order the genes into the correct order.

In[]:=

OrderGenes["r","R"]="Rr";OrderGenes["y","Y"]="Yy";OrderGenes["w","W"]="Ww";OrderGenes["s","S"]="Ss";

In[]:=

(*Foreverythingelse,leaveasis.*)OrderGenes[a_,b_]:=a<>b;

In[]:=

(*Exampleusage:*)OrderGenes["r","R"]

Out[]=

PunnetSquare

Creates a Punnet square for two genes.

In[]:=

PunnetSquare[geneA_List,geneB_List]:=Module[{cross},(*Crossallthealleles.*)cross=Table[OrderGenes[rA,rB]<>OrderGenes[yA,yB]<>OrderGenes[wA,wB]<>OrderGenes[sA,sB],{rA,geneA〚1〛},{rB,geneB〚1〛},{yA,geneA〚2〛},{yB,geneB〚2〛},{wA,geneA〚3〛},{wB,geneB〚3〛},{sA,geneA〚4〛},{sB,geneB〚4〛}];(*Returnthearrayofcolors.*)Return[ArrayReshape[Flatten[cross],{16,16}]];];

In[]:=

(*Exampleusage:*)Short[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]],2]

Out[]//Short=

{{RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss,RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss,RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss,RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss},14,{RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss,RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss,RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss,RrYyWWSs,RrYyWWSs,RrYyWWss,RrYyWWss}}

UniqueGenes

Gets all the unique genes of a Punnet square.

In[]:=

UniqueGenes[punnet_List]:=DeleteDuplicates@Flatten@punnet;

In[]:=

(*Exampleusage:*)UniqueGenes[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]]]

Out[]=

{RrYyWWSs,RrYyWWss}

PunnetSquareCount

Counts the occurrences on a Punnet square.

In[]:=

PunnetSquareCount[punnet_List]:=Module[{},Sort[Counts[Flatten[Map[FlowerColor[#]&,punnet,{2}]]]]];

In[]:=

(*Exampleusage:*)PunnetSquareCount[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]]]

Out[]=

Yellow128,Orange128

PunnetSquarePercent

In[]:=

(*CountstherelativeoccurrencesonaPunnetsquare.*)PunnetSquarePercent[punnet_List]:=Module{counts=PunnetSquareCount[punnet],total},total=Total[Values[counts]];QuantityN@

total

*100,"Percent"&/@counts;

In[]:=

(*Exampleusage:*)PunnetSquarePercent[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]]]

Out[]=

Yellow

50.

,Orange

50.



PunnetSquareChart

In[]:=

PunnetSquareChart[punnet_List]:=Module{count=PunnetSquarePercent[punnet]},BarChartcount,ChartStyleKeys[count]/."Black"

,"Blue"

,"Orange"

,"Pink"

,"Purple"

,"Red"

,"White"

,"Yellow"

,PlotTheme"Detailed",ChartLabelsPlaced[FlowerIcon[#]&/@Keys[count],Below],PerformanceGoal"Quality";

In[]:=

(*Exampleusage:*)PunnetSquareChart[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]]]

Out[]=

PunnetSquareColors

In[]:=

PunnetSquareColors[punnet_List]:=Module[{percent=PunnetSquarePercent[punnet],i=1,j=1,grid=Table[{},16]},For[i=1,i<17,i++,For[j=1,j<17,j++,AppendTo[grid〚i〛,Tooltip[FlowerIcon[FlowerColor[punnet〚i,j〛]],FlowerColor[punnet〚i,j〛]<>" — "<>punnet〚i,j〛<>" — "<>ToString[QuantityMagnitude[percent〚FlowerColor[punnet〚i,j〛]〛]]<>"%"]]]];Return[Grid[grid,FrameAll]]];

In[]:=

(*Exampleusage:*)PunnetSquareColors[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]]]

Out[]=

Breeding of “Seed” Flowers (Parental Generation)

“Seed” flowers are the Parental Generation flowers. In this case, only three different genes represents the P Generation. In this section, we will compute the breeding of three different parental flowers with distinct genes:

◼

Parental Red  RR-yy-WW-Ss

◼

Parental Yellow  rr-YY-WW-ss

◼

Parental White  rr-yy-Ww-ss

Breeding of Parental Red and Parental Yellow

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]]]

Out[]=

In[]:=

UniqueGenes[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-YY-WW-ss"]]]

Out[]=

{RrYyWWSs,RrYyWWss}

Breeding of Parental Red and Parental White

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-yy-Ww-ss"]]]

Out[]=

In[]:=

UniqueGenes[PunnetSquare[FlowerGene["RR-yy-WW-Ss"],FlowerGene["rr-yy-Ww-ss"]]]

Out[]=

{RryyWWSs,RryyWWss,RryyWwSs,RryyWwss}

Breeding of Parental Yellow and Parental White

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["rr-YY-WW-ss"],FlowerGene["rr-yy-Ww-ss"]]]

Out[]=

Breeding of “
F
1
” Flowers

Now, let’s compute the Filial 1 Generation.

Breeding of
F
1
Red and
F
1
Yellow

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["Rr-Yy-WW-Ss"],FlowerGene["Rr-Yy-WW-ss"]]]

Out[]=

In[]:=

UniqueGenes[PunnetSquare[FlowerGene["Rr-Yy-WW-Ss"],FlowerGene["Rr-Yy-WW-ss"]]]

Out[]=

{RRYYWWSs,RRYYWWss,RRYyWWSs,RRYyWWss,RRyyWWSs,RRyyWWss,RrYYWWSs,RrYYWWss,RrYyWWSs,RrYyWWss,RryyWWSs,RryyWWss,rrYYWWSs,rrYYWWss,rrYyWWSs,rrYyWWss,rryyWWSs,rryyWWss}

Breeding of
F
1
Red and
F
1
Pink

Since there’s more than one gene for

Red and

Pink, let’s plot all 4 possible combinations of the genes.

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["Rr-yy-WW-Ss"],FlowerGene["Rr-yy-WW-ss"]]]

Out[]=

In[]:=

UniqueGenes[PunnetSquare[FlowerGene["Rr-yy-WW-Ss"],FlowerGene["Rr-yy-WW-ss"]]]

Out[]=

{RRyyWWSs,RRyyWWss,RryyWWSs,RryyWWss,rryyWWSs,rryyWWss}

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["Rr-yy-Ww-Ss"],FlowerGene["Rr-yy-Ww-ss"]]]

Out[]=

In[]:=

UniqueGenes[PunnetSquare[FlowerGene["Rr-yy-Ww-Ss"],FlowerGene["Rr-yy-Ww-ss"]]]

Out[]=

{RRyyWWSs,RRyyWWss,RRyyWwSs,RRyyWwss,RRyywwSs,RRyywwss,RryyWWSs,RryyWWss,RryyWwSs,RryyWwss,RryywwSs,Rryywwss,rryyWWSs,rryyWWss,rryyWwSs,rryyWwss,rryywwSs,rryywwss}

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["Rr-yy-WW-Ss"],FlowerGene["Rr-yy-Ww-ss"]]]

Out[]=

In[]:=

UniqueGenes[PunnetSquare[FlowerGene["Rr-yy-WW-Ss"],FlowerGene["Rr-yy-Ww-ss"]]]

Out[]=

{RRyyWWSs,RRyyWWss,RRyyWwSs,RRyyWwss,RryyWWSs,RryyWWss,RryyWwSs,RryyWwss,rryyWWSs,rryyWWss,rryyWwSs,rryyWwss}

In[]:=

FlowerColor[#]&/@%

Out[]=

{Red,Black,Red,Black,Pink,Red,Pink,Red,White,White,White,White}

Breeding of
F
1
Yellow and
F
1
White

One interesting trait of breeding

Yellow and

White is that no different color is produced in the result of the breeding.

In[]:=

PunnetSquareChart[PunnetSquare[FlowerGene["rr-Yy-WW-ss"],FlowerGene["rr-Yy-Ww-ss"]]]

Out[]=

In[]:=

UniqueGenes[PunnetSquare[FlowerGene["rr-Yy-WW-ss"],FlowerGene["rr-Yy-Ww-ss"]]]

Out[]=

{rrYYWWss,rrYYWwss,rrYyWWss,rrYyWwss,rryyWWss,rryyWwss}

In[]:=

FlowerColor[#]&/@%

Out[]=

{Yellow,Yellow,Yellow,White,White,White}

“
F
2
” Flowers and Punnet Squares

Now, for the

flowers, let’s plot their probabilities and their Punnet squares.

Tip: You can hover your mouse over a flower to know its color, gene, and probability.

Breeding of
F
2
Pink and
F
2
Black

In[]:=

PunnetSquareColors[PunnetSquare[FlowerGene["Rr-yy-WW-Ss"],FlowerGene["Rr-yy-WW-ss"]]]

Out[]=

Simulating "Animal Crossing: New Horizons" Mendelian Flower Genetics

Introduction

Understanding the Genes

Functions and Definitions

FlowerGene

FlowerColor

RandomFlowerGene

FlowerIcon

OrderGenes

PunnetSquare

UniqueGenes

PunnetSquareCount

PunnetSquarePercent

PunnetSquareChart

PunnetSquareColors

Breeding of “Seed” Flowers (Parental Generation)

Breeding of Parental Red and Parental Yellow

Breeding of Parental Red and Parental White

Breeding of Parental Yellow and Parental White

Breeding of “F1” Flowers

Breeding of F1 Red and F1 Yellow

Breeding of F1 Red and F1 Pink

Breeding of F1 Yellow and F1 White

“F2” Flowers and Punnet Squares

Breeding of F2 Pink and F2 Black

Breeding of “
F
1
” Flowers

Breeding of
F
1
Red and
F
1
Yellow

Breeding of
F
1
Red and
F
1
Pink

Breeding of
F
1
Yellow and
F
1
White

“
F
2
” Flowers and Punnet Squares

Breeding of
F
2
Pink and
F
2
Black