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(B.5).", "Text"], Cell[TextData[{ "The probability that the allele will be lost if it first appears in a \ haploid is equal to the sum of the following three events in the first \ generation:\n\n(a) the mutant is lost immediately with probability ", Cell[BoxData[ OverscriptBox[ SubscriptBox["d", "MH"], "~"]], CellChangeTimes->{{3.672064525614197*^9, 3.672064595528357*^9}}], "\n(b) another individual within the population dies (probability ", Cell[BoxData[ OverscriptBox[ RowBox[{"1", "-", SubscriptBox["d", "MH"]}], "~"]], CellChangeTimes->{{3.672064525614197*^9, 3.672064595528357*^9}}], ") and is replaced with anybody but a diploid offspring of the haploid \ mutant parent, so that again we have only one haploid mutant that must \ ultimately be lost ", Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", SubscriptBox["d", "MH"]}], ")"}], RowBox[{"(", RowBox[{"1", "-", OverscriptBox[ SubscriptBox["p", "RMH"], "~"]}], ")"}], "\[Pi]0H"}]], CellChangeTimes->{{3.672064525614197*^9, 3.672064595528357*^9}}], "\n(c) another individual within the population dies (probability ", Cell[BoxData[ OverscriptBox[ RowBox[{"1", "-", SubscriptBox["d", "MH"]}], "~"]], CellChangeTimes->{{3.672064525614197*^9, 3.672064595528357*^9}}], ") and is replaced with a diploid offspring of the haploid mutant parent \ with loss probability now equalling ", Cell[BoxData[ RowBox[{ OverscriptBox[ SubscriptBox["d", "RH"], "~"], OverscriptBox[ SubscriptBox["p", "RMH"], "~"], "\[Pi]0H", " ", "\[Pi]0D"}]], CellChangeTimes->{{3.672064525614197*^9, 3.672064595528357*^9}}], ", where loss now requires both the haploid mutant parent and the mutant \ diploid offspring to be ultimately lost (here assumed to be independent \ probabilities)\n\nThis gives Eq. 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Cell[BoxData["0"], "Output"] }, Open ]], Cell["\<\ Using a weak selection approximation, we derive Eq. (D.5) from Eq. (D.4)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"EqD4a", "=", RowBox[{ RowBox[{ RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", "\[Pi]D"}], "*", FractionBox[ RowBox[{"avef", "*", SubscriptBox["w", "M"]}], OverscriptBox["w", "^"]]}], "]"}], "/.", "weakselection"}], "/.", RowBox[{"\[Pi]D", "\[Rule]", RowBox[{ OverscriptBox["\[Pi]D", "~"], "*", "\[Epsilon]"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"EqD4b", "=", RowBox[{ RowBox[{ RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", "\[Pi]H"}], "*", FractionBox[ SubscriptBox["w", "RM"], RowBox[{"2", "*", OverscriptBox["w", "^"]}]]}], "]"}], "/.", "weakselection"}], "/.", RowBox[{"\[Pi]H", "\[Rule]", RowBox[{ OverscriptBox["\[Pi]H", "~"], "*", "\[Epsilon]"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"tempa", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{"EqD4a", ",", RowBox[{"{", RowBox[{"\[Epsilon]", ",", "0", ",", "2"}], "}"}]}], "]"}], "]"}], "/.", 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FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "4"}], " ", "\[Pi]H", " ", OverscriptBox["w", "^"], " ", SubscriptBox["s", "wRM"], " ", SubscriptBox["w", "RR"]}], "+", RowBox[{ SuperscriptBox["\[Pi]H", "2"], " ", SubsuperscriptBox["w", "RR", "2"]}]}], RowBox[{"8", " ", SuperscriptBox[ OverscriptBox["w", "^"], "2"]}]]}]], "Output"] }, Open ]], Cell["\<\ Calculating the probability of fixation from these probabilities of loss, we \ obtain equations (D.5):\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"EqD5a", "=", RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["f", "R"], "*", SubscriptBox["w", "R"]}], OverscriptBox["w", "^"]], "*", "\[Pi]D"}], " ", "+", RowBox[{ FractionBox[ RowBox[{ RowBox[{"2", "*", SubscriptBox["f", "R"], "*", SubscriptBox["s", "wM"], "*", SubscriptBox["w", "R"]}], "+", RowBox[{ SubscriptBox["f", "R"], "*", SubscriptBox["s", "fM"], "*", SubscriptBox["w", "R"]}]}], RowBox[{"2", "*", OverscriptBox["w", "^"]}]], "*", "\[Pi]D"}], "-", RowBox[{ FractionBox[ RowBox[{ SubsuperscriptBox["f", "R", "2"], "*", SubsuperscriptBox["w", "R", "2"]}], RowBox[{"2", "*", SuperscriptBox[ OverscriptBox["w", "^"], "2"]}]], "*", SuperscriptBox["\[Pi]D", "2"]}]}]}], " ", ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"EqD5b", "=", RowBox[{ RowBox[{ FractionBox[ SubscriptBox["w", "RR"], RowBox[{"2", "*", OverscriptBox["w", "^"]}]], "*", "\[Pi]H"}], " ", "+", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["s", "wRM"], "*", SubscriptBox["w", "RR"]}], RowBox[{"2", "*", OverscriptBox["w", "^"]}]], "*", "\[Pi]H"}], "-", RowBox[{ FractionBox[ SubsuperscriptBox["w", "RR", "2"], RowBox[{"8", "*", SuperscriptBox[ OverscriptBox["w", "^"], "2"]}]], "*", SuperscriptBox["\[Pi]H", "2"]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"EqD5a", "-", RowBox[{"(", RowBox[{"1", "-", "tempa"}], ")"}]}], "//", "Factor"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"EqD5b", "-", RowBox[{"(", RowBox[{"1", "-", "tempb"}], ")"}]}], "//", "Factor"}], "\[IndentingNewLine]"}], "Input"], Cell[BoxData["0"], "Output"], Cell[BoxData["0"], "Output"] }, Open ]], Cell[TextData[{ "To obtain the fixation probabilities as given in equations (5b) and (5c), \ we substitute \[Pi]D (as given by EqD5b) into EqD5a and solve for \[Pi]H \ under weak selection. To simplify the result, we use the definition ", Cell[BoxData[ RowBox[{ OverscriptBox["w", "^"], "\[Rule]", RowBox[{ RowBox[{ FractionBox[ RowBox[{ SubscriptBox["f", "R"], "*", SubscriptBox["w", "R"]}], "2"], "*", SubscriptBox[ OverscriptBox["\[Rho]", "^"], "H"]}], "+", RowBox[{ SubscriptBox["w", "RR"], "*", RowBox[{"(", RowBox[{"1", "-", SubscriptBox[ OverscriptBox["\[Rho]", "^"], "H"]}], ")"}]}]}]}]], CellChangeTimes->{{3.672071527470779*^9, 3.672071563381925*^9}}], " and replace for ", Cell[BoxData[ SubscriptBox["w", "RR"]], CellChangeTimes->{{3.672071527470779*^9, 3.672071563381925*^9}}], " using its relationship to ", Cell[BoxData[ SubscriptBox[ OverscriptBox["\[Rho]", "^"], "H"]], CellChangeTimes->{{3.672071527470779*^9, 3.672071563381925*^9}}], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"EqWS\[Pi]H", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Factor", "[", RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ 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We can measure selection on M as the \ difference in fitness of M-bearing versus R-bearing individuals relative to \ the common genotypes (MM and M): \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", "z", "]"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"fR", "=", "0.5"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Beta]R", "=", "1000"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Beta]RR", "=", "1000"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dR", "=", "0.005"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"dRR", "=", "0.005"}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SELfM", "=", "0"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SEL\[Beta]M", "=", RowBox[{"0.2", "+", "z"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SEL\[Beta]MM", "=", RowBox[{ RowBox[{"-", "2"}], "*", "z"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"HETERO\[Beta]", "=", "0.5"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SELdM", "=", "0"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SELdMM", "=", "0"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"HETEROd", "=", "0.5"}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"fM", "=", RowBox[{"fR", "*", RowBox[{"(", RowBox[{"1", "+", "SELfM"}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Beta]M", "=", RowBox[{"\[Beta]R", "*", RowBox[{"(", RowBox[{"1", "+", "SEL\[Beta]M"}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Beta]RM", "=", RowBox[{"\[Beta]RR", "*", RowBox[{"(", RowBox[{"1", "+", RowBox[{"HETERO\[Beta]", "*", "SEL\[Beta]MM"}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Beta]MM", "=", RowBox[{"\[Beta]RR", "*", RowBox[{"(", RowBox[{"1", "+", "SEL\[Beta]MM"}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dM", "=", RowBox[{"dR", "*", RowBox[{"(", RowBox[{"1", "-", "SELdM"}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dRM", "=", RowBox[{"dRR", "*", RowBox[{"(", RowBox[{"1", "-", RowBox[{"HETEROd", "*", "SELdMM"}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"dMM", "=", RowBox[{"dRR", "*", RowBox[{"(", RowBox[{"1", "-", "SELdMM"}], ")"}]}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SEL\[Beta]RM", "=", FractionBox[ RowBox[{"\[Beta]RM", "-", "\[Beta]RR"}], "\[Beta]RR"]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"SELdRM", "=", FractionBox[ RowBox[{"dRR", "-", "dRM"}], "dRR"]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["s", "average"], "=", RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "2"], "*", RowBox[{"(", RowBox[{ FractionBox["SELfM", "2"], "+", "SEL\[Beta]M", "+", "SELdM"}], ")"}]}], "+", RowBox[{ FractionBox["1", "2"], "*", RowBox[{"(", RowBox[{ RowBox[{"SEL\[Beta]MM", "*", "HETERO\[Beta]"}], "+", RowBox[{"SELdM", "*", "HETEROd"}]}], ")"}]}]}], "//", "Simplify"}]}], "\[IndentingNewLine]"}], "Input"], Cell[BoxData["0.1`"], "Output"] }, Open ]], Cell["\<\ We next derive the selection coefficient of allele R, reversing the effects \ of allele M and R. 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