(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 1707785, 35556] NotebookOptionsPosition[ 1680403, 34774] NotebookOutlinePosition[ 1680750, 34789] CellTagsIndexPosition[ 1680707, 34786] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Supplementary ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook:\nAsymmetric selection and the evolution of extraordinary \ defenses \n(Urban, B\[UDoubleDot]rger, Bolnick)" }], "Title", CellChangeTimes->{{3.576477626232485*^9, 3.576477637332655*^9}, { 3.5770179036098537`*^9, 3.577017920055106*^9}, {3.577108617266652*^9, 3.5771086463139033`*^9}}, FontFamily->"Arial", FontSize->28], Cell["\<\ Throughout, x indicates the trait value of an individual, X , S, and C3 its \ mean, variance, and third cumulant (equal to the third central moment) in the \ population.\ \>", "Text", CellChangeTimes->{{3.576477687430925*^9, 3.5764778899552855`*^9}, 3.5764779301321526`*^9}], Cell[CellGroupData[{ Cell["The skewed Gaussian landscape", "Section", CellChangeTimes->{{3.576477157310279*^9, 3.576477159237808*^9}, { 3.576477382803993*^9, 3.5764773886615834`*^9}, 3.5764774373795824`*^9, { 3.5770179643445363`*^9, 3.577017971415895*^9}, 3.57701819767187*^9, { 3.57710785375251*^9, 3.577107854610512*^9}}], Cell[CellGroupData[{ Cell["The individual fitness surface", "Subsection", CellChangeTimes->{{3.554121051327045*^9, 3.5541210801558957`*^9}, 3.556009731206569*^9, 3.5560097830922604`*^9, 3.5573158672209635`*^9, { 3.56483088529763*^9, 3.5648308871228333`*^9}, {3.564830931354511*^9, 3.564830954536152*^9}, 3.564831214261008*^9}], Cell["\<\ First, we define sfit[x], a simple Gaussian-like individual fitness surface \ with skew \ \>", "Text", 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following:\ \>", "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.5541197964308405`*^9, 3.554119813824871*^9}, { 3.5541201776019096`*^9, 3.5541201928431363`*^9}, 3.5541461338016696`*^9, { 3.554182357992627*^9, 3.554182359321703*^9}, {3.554182440526348*^9, 3.5541824412173877`*^9}, 3.555150651417907*^9, 3.556009731206569*^9, 3.5560097830922604`*^9, {3.5572271697240376`*^9, 3.5572271733276443`*^9}, 3.557315870106969*^9, {3.5573165482593603`*^9, 3.5573165575881767`*^9}, 3.564058883456357*^9, {3.5640590110333815`*^9, 3.5640590213449993`*^9}, { 3.564059052326654*^9, 3.564059052794655*^9}, {3.564060741203221*^9, 3.5640607418584223`*^9}, 3.564060803774931*^9, {3.5640608704494476`*^9, 3.564060873600653*^9}, 3.5648309545517516`*^9, 3.564831214261008*^9, { 3.576478311235506*^9, 3.576478318403116*^9}, {3.5764784068519745`*^9, 3.5764784081319942`*^9}, 3.576479160276047*^9}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"wGfitt", "[", "x_", "]"}], ":=", RowBox[{ 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If \[Psi] = 0, we obtain \ a Gaussian fitness surface. If \[Psi] > 0, the fitness surface is skewed to \ the right. Asymmetry increases with \[Psi]. We restrict attention to \[Psi] \ \[GreaterEqual] 0.\ \>", "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.5541219061617465`*^9, 3.554122048808397*^9}, { 3.5541221462929683`*^9, 3.554122149990175*^9}, {3.554138759011856*^9, 3.554138779223012*^9}, {3.5541388316300097`*^9, 3.554138833407111*^9}, 3.5541877689941187`*^9, 3.556009731206569*^9, 3.5560097830922604`*^9, { 3.557227128695966*^9, 3.5572272184273233`*^9}, {3.5573158726653733`*^9, 3.557315875863379*^9}, {3.5640588837215576`*^9, 3.564058883986758*^9}, 3.5648309545517516`*^9, 3.564831214261008*^9, 3.576479160276047*^9}] }, Open ]], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"Remark", ":", " ", RowBox[{ RowBox[{"For", " ", "\[Psi]"}], "<", "0"}]}], ",", " ", RowBox[{"we", " ", "could", " ", "define"}]}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"wGfit", "[", "x_", "]"}], ":=", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"x", "\[LessEqual]", RowBox[{"-", FractionBox[ SqrtBox[ RowBox[{"1", "+", 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simple calculation shows that the desired mean fitness, wGbar[X,S], of \ wGfit[x] is obtained from sfitbar[Y,T] by the tansformations ", Cell[BoxData[ RowBox[{"Y", "->", RowBox[{ FractionBox["X", SqrtBox[ RowBox[{"1", "+", SuperscriptBox["\[Psi]", "2"]}]]], "+", "\[Psi]"}]}]], CellChangeTimes->{{3.554123892618836*^9, 3.5541239052080584`*^9}}], ", T", Cell[BoxData[ RowBox[{"->", FractionBox["S", RowBox[{"1", "+", SuperscriptBox["\[Psi]", "2"]}]]}]], CellChangeTimes->{{3.554123892618836*^9, 3.5541239052080584`*^9}}], ", and trafoswGfit (see above)." }], "Text", CellChangeTimes->{{3.554123841778347*^9, 3.5541238672843914`*^9}, 3.5541239864218006`*^9, {3.5541241264788465`*^9, 3.5541243317596073`*^9}, { 3.5541250981245537`*^9, 3.5541251087949724`*^9}, {3.5541878281004996`*^9, 3.55418785678414*^9}, {3.555150652596975*^9, 3.5551506526469774`*^9}, 3.555152185220636*^9, 3.556009731206569*^9, 3.5560097830922604`*^9, { 3.56405888521916*^9, 3.56405888521916*^9}, {3.5640616087731442`*^9, 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CellChangeTimes->{{3.5541969829328747`*^9, 3.5541970024509907`*^9}, 3.5541970657636123`*^9, 3.5560097317057705`*^9, 3.5560097830922604`*^9, { 3.5639778326861877`*^9, 3.563977862934641*^9}, {3.5640800014556866`*^9, 3.5640800363841476`*^9}, 3.564080110718278*^9, 3.5648310046434393`*^9, 3.564831214261008*^9, {3.5764891999745026`*^9, 3.5764892145172257`*^9}, { 3.576489267125534*^9, 3.5764892696368227`*^9}, {3.57710557628891*^9, 3.577105579050115*^9}}], Cell[TextData[{ "We apply the theory described in B\[UDoubleDot]rger (2000, Chap.5), in \ particular p. 194. \nAccording to this theory, equilibrium distributions are \ (approximate) solutions of \n\t S L1 + C3 L2 + M = 0.\nThe mutation term M is \ needed here because the mutation model used in our simulations changes the \ mean.\nWe have equal forward and backward mutation between the + and the \ -allele at each locus. The total mutation rate per locus is u, there are n \ loci.\nHence p\[CloseCurlyQuote]=p(1-u/2)+(1-p)u/2 or, equivalently, \ \[CapitalDelta]p=p\[CloseCurlyQuote]-p=(1-2p)u/2. Because the loci contribute \ additively to the trait, the mean trait value is X=", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"\[Delta]", UnderoverscriptBox[ RowBox[{" ", "\[Sum]"}], RowBox[{"i", "=", "1"}], "n"], RowBox[{"(", RowBox[{ SubscriptBox["p", "i"], "-", RowBox[{"1", "/", "2"}]}], ")"}]}], ","}]}], TraditionalForm]]], "\nwhere the allelic effects are +\[Delta] and -\[Delta]. A simple \ calculation yields that the change of the mean caused by mutaiton is M=\ \[CapitalDelta]X=-u X." }], "Text", CellChangeTimes->{{3.564080113869484*^9, 3.5640801330887175`*^9}, { 3.564080192930423*^9, 3.5640805774242983`*^9}, {3.564080644332815*^9, 3.5640806492468243`*^9}, {3.564080682849283*^9, 3.5640808584275913`*^9}, { 3.564081546502001*^9, 3.564081565549634*^9}, {3.564124009854298*^9, 3.5641240546171675`*^9}, {3.5641266230337853`*^9, 3.564126632652999*^9}, { 3.5641266657270503`*^9, 3.5641266797086725`*^9}, 3.5648310046434393`*^9, 3.564831214261008*^9, {3.576489085477744*^9, 3.5764891115093937`*^9}, { 3.5764892187910414`*^9, 3.5764892957334733`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"We", " ", RowBox[{"define", ":", " ", "skew"}]}], " ", "=", " ", FractionBox["C3", SuperscriptBox["S", RowBox[{"3", "/", "2"}]]]}]], "Text", CellChangeTimes->{{3.5550616857336073`*^9, 3.555061724234475*^9}, 3.5560097317057705`*^9, 3.5560097830922604`*^9, 3.5648310046434393`*^9, 3.564831214261008*^9}], Cell[TextData[{ "Equating the following expression to zero yields the equilibirum relation \ (i.e., L1+C3*L2/", Cell[BoxData[ FormBox[ RowBox[{"S", "+", RowBox[{"M", "/", "S"}]}], TraditionalForm]]], "=0) between the trait mean (X), variance (S), and skew in terms of the \ selection differentials L1 and L2 (recall that L1 = L1wb/wGbar):" }], "Text", CellChangeTimes->{{3.5541457047571297`*^9, 3.5541458599180045`*^9}, { 3.5550618520766993`*^9, 3.5550618836355553`*^9}, 3.5560097317057705`*^9, 3.5560097830922604`*^9, {3.5640800668510013`*^9, 3.564080068567004*^9}, 3.5648310046434393`*^9, 3.564831214261008*^9, 3.5764891411098485`*^9, { 3.576489338421629*^9, 3.5764893708858776`*^9}, {3.5771044122450657`*^9, 3.577104416316673*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"XSskew", "[", RowBox[{"X_", ",", "S_", ",", "skew_", ",", "\[Psi]_", ",", "u_"}], "]"}], ":=", RowBox[{ RowBox[{"L1wb", "[", RowBox[{"X", ",", "S", ",", "\[Psi]"}], "]"}], "+", RowBox[{"skew", SqrtBox["S"], RowBox[{"L2wb", "[", RowBox[{"X", ",", "S", ",", "\[Psi]"}], "]"}]}], "-", RowBox[{"u", " ", RowBox[{"wGbar", "[", RowBox[{"X", ",", "S", ",", "\[Psi]"}], "]"}], RowBox[{"X", "/", "S"}]}]}]}]], "Input", InitializationCell->True, CellChangeTimes->{{3.5533182752313385`*^9, 3.5533183170907326`*^9}, { 3.5541445312990117`*^9, 3.5541446168729067`*^9}, {3.555061890421567*^9, 3.5550619202800193`*^9}, 3.555064877746414*^9, 3.555065947767894*^9, 3.555160336112934*^9, {3.5560014255526624`*^9, 3.556001425880263*^9}, 3.5560097317057705`*^9, 3.5560097830922604`*^9, {3.563867989156103*^9, 3.5638679967845163`*^9}, {3.563971357146014*^9, 3.5639713600164194`*^9}, 3.5639767384534664`*^9, {3.5640588886823664`*^9, 3.5640588886979666`*^9}, 3.5648310046434393`*^9, 3.564831214261008*^9, 3.577104355382966*^9}, Background->RGBColor[0.87, 0.94, 1]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["We proceed with the graphical", FontFamily->"Times New Roman"], " representation of solutions of S L1 + C3 L2 +M= 0, or XSskew[X,S,skew,\ \[Psi],u]=0, and comparison with the mode of mean fitness. We assume u= 0." }], "Subsubsection", CellChangeTimes->{{3.5541968124111214`*^9, 3.5541968483591776`*^9}, 3.5541970052591515`*^9, 3.5560097317057705`*^9, 3.5560097830922604`*^9, { 3.5640809036208706`*^9, 3.564080904151272*^9}, 3.5648310046434393`*^9, 3.564831214261008*^9, {3.5764894486133213`*^9, 3.57648950320541*^9}, { 3.576568323874239*^9, 3.576568342183066*^9}, {3.5765684036427555`*^9, 3.5765684121531677`*^9}, {3.5771044561591425`*^9, 3.5771044760023775`*^9}}], Cell["\<\ The following figures show the dependence of the mean X on the skew :\ \>", "Text", CellChangeTimes->{{3.576489521651943*^9, 3.576489544406043*^9}, { 3.576994914709962*^9, 3.5769949192687817`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"plotXS1", "=", RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"XSskew", "[", RowBox[{"X", ",", "0.25", ",", "skew", ",", "0.5", ",", "0"}], "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"XSskew", "[", RowBox[{"X", ",", "0.25", ",", "skew", ",", 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The total mutation rate per locus is u, there are n \ loci.\nHence p\[CloseCurlyQuote]=p(1-u/2)+(1-p)u/2 or, equivalently, \ \[CapitalDelta]p=p\[CloseCurlyQuote]-p=(1-2p)u/2. Because the loci contribute \ additively to the trait, the mean trait value is X=", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"\[Delta]", UnderoverscriptBox[ RowBox[{" ", "\[Sum]"}], RowBox[{"i", "=", "1"}], "n"], RowBox[{"(", RowBox[{ SubscriptBox["p", "i"], "-", RowBox[{"1", "/", "2"}]}], ")"}]}], ","}]}], TraditionalForm]]], "\nwhere the allelic effects are +\[Delta] and -\[Delta]. The derivation of \ M is analogous to the case for the skewed Gaussian distribution above. \ Because here the middle of the phenotypic range is 1/2, we obtain M=\ \[CapitalDelta]X=-u (X-1/2)." }], "Text", CellChangeTimes->{{3.564080113869484*^9, 3.5640801330887175`*^9}, { 3.564080192930423*^9, 3.5640805774242983`*^9}, {3.564080644332815*^9, 3.5640806492468243`*^9}, {3.564080682849283*^9, 3.5640808584275913`*^9}, { 3.564081546502001*^9, 3.564081565549634*^9}, {3.564124009854298*^9, 3.5641240546171675`*^9}, {3.5641266230337853`*^9, 3.564126632652999*^9}, { 3.5641266657270503`*^9, 3.5641266797086725`*^9}, 3.5648310046434393`*^9, 3.564831214261008*^9, {3.576489085477744*^9, 3.5764891115093937`*^9}, { 3.5764892187910414`*^9, 3.5764892957334733`*^9}, {3.5764982836552486`*^9, 3.576498287559059*^9}, {3.5764983595101643`*^9, 3.5764984241824074`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"We", " ", RowBox[{"define", ":", " ", "skew"}]}], " ", "=", " ", FractionBox["C3", SuperscriptBox["S", RowBox[{"3", "/", "2"}]]]}]], "Text", CellChangeTimes->{{3.5550616857336073`*^9, 3.555061724234475*^9}, 3.5560097317057705`*^9, 3.5560097830922604`*^9, 3.5648310046434393`*^9, 3.564831214261008*^9}], Cell[TextData[{ "Equating the following expression to zero yields the equilibirum relation \ (i.e., L1+C3*L2/", Cell[BoxData[ FormBox[ RowBox[{"S", "+", RowBox[{"M", "/", "S"}]}], TraditionalForm]]], "=0) between the trait mean (X), variance (S), and skew in terms of the \ selection differentials L1 and L2 :" }], "Text", CellChangeTimes->{{3.5541457047571297`*^9, 3.5541458599180045`*^9}, { 3.5550618520766993`*^9, 3.5550618836355553`*^9}, 3.5560097317057705`*^9, 3.5560097830922604`*^9, {3.5640800668510013`*^9, 3.564080068567004*^9}, 3.5648310046434393`*^9, 3.564831214261008*^9, 3.5764891411098485`*^9, { 3.576489338421629*^9, 3.5764893708858776`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"XSskewPred", "[", RowBox[{ "X_", ",", "S_", ",", "skew_", ",", "\[Gamma]P_", ",", "r_", ",", "u_"}], "]"}], ":=", RowBox[{ RowBox[{"L1wbPred", "[", RowBox[{"X", ",", "S", ",", "\[Gamma]P", ",", "r"}], "]"}], "+", RowBox[{"skew", SqrtBox["S"], " ", RowBox[{"L2wbPred", "[", RowBox[{"X", ",", "S", ",", "\[Gamma]P", ",", "r"}], "]"}]}], "-", RowBox[{"u", " ", RowBox[{"(", RowBox[{"X", "-", "0.5"}], ")"}], RowBox[{ RowBox[{"wbarPred", "[", RowBox[{"X", ",", "S", ",", "\[Gamma]P", ",", "r"}], "]"}], "/", "S"}]}]}]}]], "Input", InitializationCell->True, CellChangeTimes->{{3.5533182752313385`*^9, 3.5533183170907326`*^9}, { 3.5541445312990117`*^9, 3.5541446168729067`*^9}, {3.5549851120793333`*^9, 3.5549851496134806`*^9}, {3.5549862513064938`*^9, 3.5549862633021793`*^9}, {3.55499258710188*^9, 3.554992590791091*^9}, { 3.554992827445627*^9, 3.5549928404383707`*^9}, {3.5638670099459825`*^9, 3.563867024220007*^9}, {3.563867273539645*^9, 3.563867284210064*^9}, { 3.563867549921731*^9, 3.5638675797645836`*^9}, 3.5638676279374685`*^9, { 3.5639697378095703`*^9, 3.5639697449855833`*^9}, {3.563970045644911*^9, 3.563970052618123*^9}, {3.5639752821285086`*^9, 3.563975283766511*^9}, 3.563979261211897*^9, {3.565082149121253*^9, 3.565082152662459*^9}, { 3.5770164421084023`*^9, 3.5770164447146926`*^9}, {3.5770165344760714`*^9, 3.5770165415786805`*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[CellGroupData[{ Cell["\<\ Plot solutions of S L1 + C3 L2 = 0 and compare with mode of mean fitness\ \>", "Subsubsection", CellChangeTimes->{{3.5541968124111214`*^9, 3.5541968483591776`*^9}, 3.5541970052591515`*^9}], Cell[TextData[{ StyleBox["We proceed with the graphical", FontFamily->"Times New Roman"], " representation of solutions of S L1 + C3 L2 +M= 0, or XSskewPred[X,S,skew,\ \[Gamma]P,r,u]=0, and comparison with the mode of mean fitness. 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Let C3 denote the third central moment (= third cumulant) of p[x]. 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As a consequence, the mode of mean fitness, \ obtained by solving \ \>", "Text", CellChangeTimes->{{3.576579830022598*^9, 3.576579912110715*^9}}], Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "X"], " ", "wbar"}], "=", RowBox[{"0", " ", "for", " ", "X"}]}], " ", ","}]}]], "Text", CellChangeTimes->{{3.576579892725487*^9, 3.576579914580718*^9}}], Cell["is independent of C3. 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