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(***********************************************************************
Mathematica-Compatible Notebook
This notebook can be used on any computer system with Mathematica 3.0,
MathReader 3.0, or any compatible application. The data for the notebook
starts with the line of stars above.
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the following:
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compatible application, you must delete the line below containing the
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use invalid cache data.
For more information on notebooks and Mathematica-compatible
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***********************************************************************)
(*CacheID: 232*)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 428245, 8845]*)
(*NotebookOutlinePosition[ 429096, 8875]*)
(* CellTagsIndexPosition[ 429052, 8871]*)
(*WindowFrame->Normal*)
Notebook[{
Cell[CellGroupData[{
Cell["Mean fitness in the two-locus haploid selection model", "Subtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[{
"In this ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" file, we determine the relationship between mean fitness and epistasis in \
a two-locus model with mutation. The method is based on a Taylor Series \
approximation for the mean fitness, keeping terms to order epistasis and \
epistasis^2.\n\nWith several pairs of interacting loci, the epistasis^2 term \
can be used to predict how variability in the genetic interactions between \
different pairs of loci will affect mean fitness. "
}], "Subsubtitle"],
Cell[CellGroupData[{
Cell["The recursions", "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell[BoxData[
\(Off[General::"\<spell1\>"]\)], "Input"],
Cell["\<\
(ORDER OF EVENTS: Selection, Recombination, Mutation)\
\>", "Text"],
Cell["\<\
For two selected loci:
Frequencey\tGenotype\tFitness
x1\t\tAB\t\tw11 = 1
x2\t\tAb\t\tw12 = 1+s2
x3\t\taB\t\tw21 = 1+s1
x4\t\tab\t\tw22 = (1+s1)(1+s2)+eps\
\>", "Text"],
Cell["Selection:", "Text"],
Cell[BoxData[{
\(jx1\ := \ w11*x1\),
\(jx2\ := \ w12*x2\),
\(jx3\ := \ w21*x3\),
\(jx4\ := \ w22*x4\)}], "Input"],
Cell["Fitness:=(jx1+jx2+jx3+jx4);", "Input",
AspectRatioFixed->True],
Cell[BoxData[{
\(sx1\ := \ 1/Fitness*jx1\),
\(sx2\ := \ 1/Fitness*jx2\),
\(sx3\ := \ 1/Fitness*jx3\),
\(sx4\ := \ 1/Fitness*jx4\)}], "Input"],
Cell["Recombination", "Text"],
Cell[BoxData[
\(nx1\ := \ \((sx1 - r*\((sx1*sx4 - sx2*sx3)\))\); \n
nx2\ := \ \((sx2 + r*\((sx1*sx4 - sx2*sx3)\))\); \n
nx3\ := \ \((sx3 + r*\((sx1*sx4 - sx2*sx3)\))\); \n
nx4\ := \ \((sx4 - r*\((sx1*sx4 - sx2*sx3)\))\); \)], "Input"],
Cell["Mutation:", "Text"],
Cell["\<\
newx1 := (1-mu1)*(1-mu2)*nx1
newx2 := (mu2*(1-mu1)*nx1+(1-mu1)*nx2)
newx3 := (mu1*(1-mu2)*nx1+(1-mu2)*nx3)
newx4 := (mu1*mu2*nx1+mu1*nx2+mu2*nx3+nx4)\
\>", "Input",
AspectRatioFixed->True],
Cell["Factor[newx1+newx2+newx3+newx4]", "Text"],
Cell["1", "Text"],
Cell["\<\
Transformation to allele frequencies and disequilibria from \
chromosome frequencies:\
\>", "Text"],
Cell[BoxData[
\(\(subtwoloci = {x1 -> \((1 - p1)\)*\((1 - p2)\) + dis,
x2 -> \((1 - p1)\)*p2 - dis, x3 -> p1*\((1 - p2)\) - dis,
x4 -> p1*p2 + dis}; \)\)], "Input"],
Cell["\<\
Fitness definitions for haploid selection model (s will be assumed \
to be negative):\
\>", "Text"],
Cell[BoxData[
\(\(subfit = {w11 -> 1, w12 -> 1 + s2, w21 -> 1 + s1,
w22 -> \((1 + s1)\) \((1 + s2)\) + eps}; \)\)], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["Two locus analysis", "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
In the following, we will get approximations for the \
disequilibrium, allele frequency,
and mean fitness to order (eps)^2.\
\>", "Text"],
Cell["\<\
From the following equation (which equals zero at equilibrium), we \
get an
equation for the disequilibrium at equilibrium:\
\>", "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[CellGroupData[{
Cell["Factor[(newx1+newx2)-(x1+x2)/.subtwoloci/.subfit]", "Input",
AspectRatioFixed->True],
Cell[BoxData[
\(\((\(-dis\)\ eps - mu1 + dis\ eps\ p1 + mu1\ p1 - eps\ p1\ p2 +
eps\ p1\^2\ p2 - p1\ s1 + p1\^2\ s1 - dis\ s2 + dis\ mu1\ s2 -
mu1\ p2\ s2 + mu1\ p1\ p2\ s2 - dis\ s1\ s2 + dis\ p1\ s1\ s2 -
p1\ p2\ s1\ s2 + p1\^2\ p2\ s1\ s2)\)/
\((1 + dis\ eps + eps\ p1\ p2 + p1\ s1 + p2\ s2 + dis\ s1\ s2 +
p1\ p2\ s1\ s2)\)\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell["Factor[dis/.Solve[%==0,dis]]", "Input",
AspectRatioFixed->True],
Cell[BoxData[
\({\(-
\(\(\((\(-1\) + p1)\)\
\((mu1 + eps\ p1\ p2 + p1\ s1 + mu1\ p2\ s2 + p1\ p2\ s1\ s2)
\)\)\/\(\(-eps\) + eps\ p1 - s2 + mu1\ s2 - s1\ s2 +
p1\ s1\ s2\)\)\)}\)], "Output"]
}, Open ]],
Cell[BoxData[
\(\(DsSolve =
\(\((1 - p1)\)\
\((mu1 + eps\ p1\ p2 + p1\ s1 + mu1\ p2\ s2 + p1\ p2\ s1\ s2)
\)\)\/\(\(-eps\) + eps\ p1 - s2 + mu1\ s2 - s1\ s2 +
p1\ s1\ s2\); \)\)], "Input"],
Cell["\<\
From the following equations (which equal zero at equilibrium), we \
get
two equations for the two unknown allele frequencies at equilibrium:\
\>",
"Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[CellGroupData[{
Cell[BoxData[
\(Factor[
\(\(\((newx1 + newx3)\) - \((x1 + x3)\) /. subtwoloci\) /. subfit\) /.
dis -> DsSolve]\)], "Input"],
Cell[BoxData[
\(\((eps\ mu1 - eps\ mu2 - eps\ mu1\ p1 + eps\ mu2\ p1 - eps\ mu1\ p2 +
eps\ mu2\ p2 + eps\ mu1\ p1\ p2 - eps\ mu2\ p1\ p2 + mu1\ s1 -
mu1\ mu2\ s1 + eps\ p1\ s1 - mu1\ p1\ s1 - eps\ mu2\ p1\ s1 +
mu1\ mu2\ p1\ s1 - eps\ p1\^2\ s1 + eps\ mu2\ p1\^2\ s1 +
p1\ s1\^2 - mu2\ p1\ s1\^2 - p1\^2\ s1\^2 + mu2\ p1\^2\ s1\^2 -
mu2\ s2 + mu1\ mu2\ s2 - eps\ p2\ s2 + eps\ mu1\ p2\ s2 +
mu2\ p2\ s2 - mu1\ mu2\ p2\ s2 + eps\ p2\^2\ s2 -
eps\ mu1\ p2\^2\ s2 + mu1\ s1\ s2 - mu2\ s1\ s2 - mu1\ p1\ s1\ s2 +
mu1\ mu2\ p1\ s1\ s2 + mu2\ p2\ s1\ s2 - mu1\ mu2\ p2\ s1\ s2 +
p1\ s1\^2\ s2 - mu2\ p1\ s1\^2\ s2 - p1\^2\ s1\^2\ s2 +
mu2\ p1\^2\ s1\^2\ s2 - p2\ s2\^2 + mu1\ p2\ s2\^2 + p2\^2\ s2\^2 -
mu1\ p2\^2\ s2\^2 - p2\ s1\ s2\^2 + mu1\ p2\ s1\ s2\^2 +
p2\^2\ s1\ s2\^2 - mu1\ p2\^2\ s1\ s2\^2)\)/
\((\((\(-1\) + mu1)\)\
\((\(-eps\) + eps\ p1 - s2 - eps\ p2\ s2 - s1\ s2 - p2\ s2\^2 -
p2\ s1\ s2\^2)\))\)\)], "Output"]
}, Closed]],
Cell[CellGroupData[{
Cell[BoxData[
\(Factor[\(\(newx1 - x1 /. subtwoloci\) /. subfit\) /. dis -> DsSolve]
\)], "Input"],
Cell[BoxData[
\(\(-\(\((
\((\(-1\) + p1)\)\
\((eps\^2\ mu2 - eps\ mu1\ mu2 - eps\^2\ mu1\ mu2 +
eps\ mu1\^2\ mu2 - 2\ eps\^2\ mu2\ p1 + eps\ mu1\ mu2\ p1 +
2\ eps\^2\ mu1\ mu2\ p1 - eps\ mu1\^2\ mu2\ p1 +
eps\^2\ mu2\ p1\^2 - eps\^2\ mu1\ mu2\ p1\^2 -
eps\^2\ mu2\ p2 + eps\^2\ mu1\ mu2\ p2 +
eps\^2\ mu2\ p1\ p2 - eps\^2\ mu1\ mu2\ p1\ p2 -
eps\ mu1\ r - eps\^2\ mu1\ r + eps\ mu1\^2\ r +
eps\ mu1\ mu2\ r + eps\^2\ mu1\ mu2\ r -
eps\ mu1\^2\ mu2\ r + eps\ mu1\ p1\ r +
2\ eps\^2\ mu1\ p1\ r - eps\ mu1\^2\ p1\ r -
eps\ mu1\ mu2\ p1\ r - 2\ eps\^2\ mu1\ mu2\ p1\ r +
eps\ mu1\^2\ mu2\ p1\ r - eps\^2\ mu1\ p1\^2\ r +
eps\^2\ mu1\ mu2\ p1\^2\ r + eps\^2\ mu1\ p2\ r -
eps\^2\ mu1\ mu2\ p2\ r - eps\^2\ p1\ p2\ r -
eps\^2\ mu1\ p1\ p2\ r + eps\^2\ mu2\ p1\ p2\ r +
eps\^2\ mu1\ mu2\ p1\ p2\ r + eps\^2\ p1\^2\ p2\ r -
eps\^2\ mu2\ p1\^2\ p2\ r - eps\ mu2\ p1\ s1 +
eps\ mu1\ mu2\ p1\ s1 + eps\ mu2\ p1\^2\ s1 -
eps\ mu1\ mu2\ p1\^2\ s1 - eps\ mu1\ r\ s1 +
eps\ mu1\ mu2\ r\ s1 - eps\ p1\ r\ s1 - eps\^2\ p1\ r\ s1 +
3\ eps\ mu1\ p1\ r\ s1 + eps\ mu2\ p1\ r\ s1 +
eps\^2\ mu2\ p1\ r\ s1 - 3\ eps\ mu1\ mu2\ p1\ r\ s1 +
eps\ p1\^2\ r\ s1 + 2\ eps\^2\ p1\^2\ r\ s1 -
2\ eps\ mu1\ p1\^2\ r\ s1 - eps\ mu2\ p1\^2\ r\ s1 -
2\ eps\^2\ mu2\ p1\^2\ r\ s1 +
2\ eps\ mu1\ mu2\ p1\^2\ r\ s1 - eps\^2\ p1\^3\ r\ s1 +
eps\^2\ mu2\ p1\^3\ r\ s1 - eps\ p1\ r\ s1\^2 +
eps\ mu2\ p1\ r\ s1\^2 + 2\ eps\ p1\^2\ r\ s1\^2 -
2\ eps\ mu2\ p1\^2\ r\ s1\^2 - eps\ p1\^3\ r\ s1\^2 +
eps\ mu2\ p1\^3\ r\ s1\^2 + eps\ mu1\ s2 - mu1\^2\ s2 -
eps\ mu1\^2\ s2 + mu1\^3\ s2 + 2\ eps\ mu2\ s2 -
mu1\ mu2\ s2 - 4\ eps\ mu1\ mu2\ s2 + 2\ mu1\^2\ mu2\ s2 +
2\ eps\ mu1\^2\ mu2\ s2 - mu1\^3\ mu2\ s2 -
eps\ mu1\ p1\ s2 + eps\ mu1\^2\ p1\ s2 -
2\ eps\ mu2\ p1\ s2 + 4\ eps\ mu1\ mu2\ p1\ s2 -
2\ eps\ mu1\^2\ mu2\ p1\ s2 + eps\^2\ p2\ s2 -
2\ eps\ mu1\ p2\ s2 - eps\^2\ mu1\ p2\ s2 +
2\ eps\ mu1\^2\ p2\ s2 - 2\ eps\ mu2\ p2\ s2 +
3\ eps\ mu1\ mu2\ p2\ s2 - eps\ mu1\^2\ mu2\ p2\ s2 -
eps\^2\ p1\ p2\ s2 + eps\^2\ mu1\ p1\ p2\ s2 +
eps\ mu2\ p1\ p2\ s2 - eps\ mu1\ mu2\ p1\ p2\ s2 -
eps\^2\ p2\^2\ s2 + eps\^2\ mu1\ p2\^2\ s2 - mu1\ r\ s2 -
2\ eps\ mu1\ r\ s2 + 2\ mu1\^2\ r\ s2 +
2\ eps\ mu1\^2\ r\ s2 - mu1\^3\ r\ s2 + mu1\ mu2\ r\ s2 +
2\ eps\ mu1\ mu2\ r\ s2 - 2\ mu1\^2\ mu2\ r\ s2 -
2\ eps\ mu1\^2\ mu2\ r\ s2 + mu1\^3\ mu2\ r\ s2 +
2\ eps\ mu1\ p1\ r\ s2 - 2\ eps\ mu1\^2\ p1\ r\ s2 -
2\ eps\ mu1\ mu2\ p1\ r\ s2 +
2\ eps\ mu1\^2\ mu2\ p1\ r\ s2 + eps\ mu1\ p2\ r\ s2 -
eps\ mu1\^2\ p2\ r\ s2 - eps\ mu1\ mu2\ p2\ r\ s2 +
eps\ mu1\^2\ mu2\ p2\ r\ s2 - eps\ p1\ p2\ r\ s2 +
eps\ mu1\ p1\ p2\ r\ s2 + eps\ mu2\ p1\ p2\ r\ s2 -
eps\ mu1\ mu2\ p1\ p2\ r\ s2 + 2\ eps\ mu2\ s1\ s2 -
mu1\ mu2\ s1\ s2 - 2\ eps\ mu1\ mu2\ s1\ s2 +
mu1\^2\ mu2\ s1\ s2 + eps\ p1\ s1\ s2 - 2\ mu1\ p1\ s1\ s2 -
eps\ mu1\ p1\ s1\ s2 + 2\ mu1\^2\ p1\ s1\ s2 -
mu2\ p1\ s1\ s2 - 4\ eps\ mu2\ p1\ s1\ s2 +
3\ mu1\ mu2\ p1\ s1\ s2 + 4\ eps\ mu1\ mu2\ p1\ s1\ s2 -
2\ mu1\^2\ mu2\ p1\ s1\ s2 - eps\ p1\^2\ s1\ s2 +
eps\ mu1\ p1\^2\ s1\ s2 + 2\ eps\ mu2\ p1\^2\ s1\ s2 -
2\ eps\ mu1\ mu2\ p1\^2\ s1\ s2 - 2\ eps\ mu2\ p2\ s1\ s2 +
2\ eps\ mu1\ mu2\ p2\ s1\ s2 - 2\ eps\ p1\ p2\ s1\ s2 +
2\ eps\ mu1\ p1\ p2\ s1\ s2 + 2\ eps\ mu2\ p1\ p2\ s1\ s2 -
2\ eps\ mu1\ mu2\ p1\ p2\ s1\ s2 - 2\ mu1\ r\ s1\ s2 -
2\ eps\ mu1\ r\ s1\ s2 + 2\ mu1\^2\ r\ s1\ s2 +
2\ mu1\ mu2\ r\ s1\ s2 + 2\ eps\ mu1\ mu2\ r\ s1\ s2 -
2\ mu1\^2\ mu2\ r\ s1\ s2 - p1\ r\ s1\ s2 -
2\ eps\ p1\ r\ s1\ s2 + 4\ mu1\ p1\ r\ s1\ s2 +
6\ eps\ mu1\ p1\ r\ s1\ s2 - 3\ mu1\^2\ p1\ r\ s1\ s2 +
mu2\ p1\ r\ s1\ s2 + 2\ eps\ mu2\ p1\ r\ s1\ s2 -
4\ mu1\ mu2\ p1\ r\ s1\ s2 -
6\ eps\ mu1\ mu2\ p1\ r\ s1\ s2 +
3\ mu1\^2\ mu2\ p1\ r\ s1\ s2 + 2\ eps\ p1\^2\ r\ s1\ s2 -
4\ eps\ mu1\ p1\^2\ r\ s1\ s2 -
2\ eps\ mu2\ p1\^2\ r\ s1\ s2 +
4\ eps\ mu1\ mu2\ p1\^2\ r\ s1\ s2 +
eps\ mu1\ p2\ r\ s1\ s2 - eps\ mu1\ mu2\ p2\ r\ s1\ s2 -
eps\ p1\ p2\ r\ s1\ s2 - eps\ mu1\ p1\ p2\ r\ s1\ s2 +
eps\ mu2\ p1\ p2\ r\ s1\ s2 +
eps\ mu1\ mu2\ p1\ p2\ r\ s1\ s2 +
eps\ p1\^2\ p2\ r\ s1\ s2 - eps\ mu2\ p1\^2\ p2\ r\ s1\ s2 -
mu2\ p1\ s1\^2\ s2 + mu1\ mu2\ p1\ s1\^2\ s2 -
p1\^2\ s1\^2\ s2 + mu1\ p1\^2\ s1\^2\ s2 +
mu2\ p1\^2\ s1\^2\ s2 - mu1\ mu2\ p1\^2\ s1\^2\ s2 -
mu1\ r\ s1\^2\ s2 + mu1\ mu2\ r\ s1\^2\ s2 -
2\ p1\ r\ s1\^2\ s2 - 2\ eps\ p1\ r\ s1\^2\ s2 +
4\ mu1\ p1\ r\ s1\^2\ s2 + 2\ mu2\ p1\ r\ s1\^2\ s2 +
2\ eps\ mu2\ p1\ r\ s1\^2\ s2 -
4\ mu1\ mu2\ p1\ r\ s1\^2\ s2 + 2\ p1\^2\ r\ s1\^2\ s2 +
4\ eps\ p1\^2\ r\ s1\^2\ s2 - 3\ mu1\ p1\^2\ r\ s1\^2\ s2 -
2\ mu2\ p1\^2\ r\ s1\^2\ s2 -
4\ eps\ mu2\ p1\^2\ r\ s1\^2\ s2 +
3\ mu1\ mu2\ p1\^2\ r\ s1\^2\ s2 -
2\ eps\ p1\^3\ r\ s1\^2\ s2 +
2\ eps\ mu2\ p1\^3\ r\ s1\^2\ s2 - p1\ r\ s1\^3\ s2 +
mu2\ p1\ r\ s1\^3\ s2 + 2\ p1\^2\ r\ s1\^3\ s2 -
2\ mu2\ p1\^2\ r\ s1\^3\ s2 - p1\^3\ r\ s1\^3\ s2 +
mu2\ p1\^3\ r\ s1\^3\ s2 + mu1\ s2\^2 - 2\ mu1\^2\ s2\^2 +
mu1\^3\ s2\^2 + mu2\ s2\^2 - 3\ mu1\ mu2\ s2\^2 +
3\ mu1\^2\ mu2\ s2\^2 - mu1\^3\ mu2\ s2\^2 +
2\ eps\ p2\ s2\^2 - 2\ mu1\ p2\ s2\^2 -
3\ eps\ mu1\ p2\ s2\^2 + 2\ mu1\^2\ p2\ s2\^2 +
eps\ mu1\^2\ p2\ s2\^2 - mu2\ p2\ s2\^2 +
2\ mu1\ mu2\ p2\ s2\^2 - mu1\^2\ mu2\ p2\ s2\^2 -
eps\ p1\ p2\ s2\^2 + eps\ mu1\ p1\ p2\ s2\^2 -
2\ eps\ p2\^2\ s2\^2 + 2\ eps\ mu1\ p2\^2\ s2\^2 -
mu1\ r\ s2\^2 + 2\ mu1\^2\ r\ s2\^2 - mu1\^3\ r\ s2\^2 +
mu1\ mu2\ r\ s2\^2 - 2\ mu1\^2\ mu2\ r\ s2\^2 +
mu1\^3\ mu2\ r\ s2\^2 + mu1\ s1\ s2\^2 - mu1\^2\ s1\ s2\^2 +
2\ mu2\ s1\ s2\^2 - 4\ mu1\ mu2\ s1\ s2\^2 +
2\ mu1\^2\ mu2\ s1\ s2\^2 + p1\ s1\ s2\^2 -
3\ mu1\ p1\ s1\ s2\^2 + 2\ mu1\^2\ p1\ s1\ s2\^2 -
2\ mu2\ p1\ s1\ s2\^2 + 4\ mu1\ mu2\ p1\ s1\ s2\^2 -
2\ mu1\^2\ mu2\ p1\ s1\ s2\^2 + 2\ eps\ p2\ s1\ s2\^2 -
2\ mu1\ p2\ s1\ s2\^2 - 2\ eps\ mu1\ p2\ s1\ s2\^2 +
2\ mu1\^2\ p2\ s1\ s2\^2 - 2\ mu2\ p2\ s1\ s2\^2 +
3\ mu1\ mu2\ p2\ s1\ s2\^2 - mu1\^2\ mu2\ p2\ s1\ s2\^2 -
2\ p1\ p2\ s1\ s2\^2 - 2\ eps\ p1\ p2\ s1\ s2\^2 +
2\ mu1\ p1\ p2\ s1\ s2\^2 + 2\ eps\ mu1\ p1\ p2\ s1\ s2\^2 +
mu2\ p1\ p2\ s1\ s2\^2 - mu1\ mu2\ p1\ p2\ s1\ s2\^2 -
2\ eps\ p2\^2\ s1\ s2\^2 + 2\ eps\ mu1\ p2\^2\ s1\ s2\^2 -
2\ mu1\ r\ s1\ s2\^2 + 2\ mu1\^2\ r\ s1\ s2\^2 +
2\ mu1\ mu2\ r\ s1\ s2\^2 - 2\ mu1\^2\ mu2\ r\ s1\ s2\^2 -
p1\ r\ s1\ s2\^2 + 4\ mu1\ p1\ r\ s1\ s2\^2 -
3\ mu1\^2\ p1\ r\ s1\ s2\^2 + mu2\ p1\ r\ s1\ s2\^2 -
4\ mu1\ mu2\ p1\ r\ s1\ s2\^2 +
3\ mu1\^2\ mu2\ p1\ r\ s1\ s2\^2 + mu2\ s1\^2\ s2\^2 -
mu1\ mu2\ s1\^2\ s2\^2 + p1\ s1\^2\ s2\^2 -
mu1\ p1\ s1\^2\ s2\^2 - 2\ mu2\ p1\ s1\^2\ s2\^2 +
2\ mu1\ mu2\ p1\ s1\^2\ s2\^2 - p1\^2\ s1\^2\ s2\^2 +
mu1\ p1\^2\ s1\^2\ s2\^2 + mu2\ p1\^2\ s1\^2\ s2\^2 -
mu1\ mu2\ p1\^2\ s1\^2\ s2\^2 - mu2\ p2\ s1\^2\ s2\^2 +
mu1\ mu2\ p2\ s1\^2\ s2\^2 - 2\ p1\ p2\ s1\^2\ s2\^2 +
2\ mu1\ p1\ p2\ s1\^2\ s2\^2 + mu2\ p1\ p2\ s1\^2\ s2\^2 -
mu1\ mu2\ p1\ p2\ s1\^2\ s2\^2 - mu1\ r\ s1\^2\ s2\^2 +
mu1\ mu2\ r\ s1\^2\ s2\^2 - 2\ p1\ r\ s1\^2\ s2\^2 +
4\ mu1\ p1\ r\ s1\^2\ s2\^2 + 2\ mu2\ p1\ r\ s1\^2\ s2\^2 -
4\ mu1\ mu2\ p1\ r\ s1\^2\ s2\^2 +
2\ p1\^2\ r\ s1\^2\ s2\^2 - 3\ mu1\ p1\^2\ r\ s1\^2\ s2\^2 -
2\ mu2\ p1\^2\ r\ s1\^2\ s2\^2 +
3\ mu1\ mu2\ p1\^2\ r\ s1\^2\ s2\^2 - p1\ r\ s1\^3\ s2\^2 +
mu2\ p1\ r\ s1\^3\ s2\^2 + 2\ p1\^2\ r\ s1\^3\ s2\^2 -
2\ mu2\ p1\^2\ r\ s1\^3\ s2\^2 - p1\^3\ r\ s1\^3\ s2\^2 +
mu2\ p1\^3\ r\ s1\^3\ s2\^2 + p2\ s2\^3 - 2\ mu1\ p2\ s2\^3 +
mu1\^2\ p2\ s2\^3 - p2\^2\ s2\^3 + mu1\ p2\^2\ s2\^3 +
2\ p2\ s1\ s2\^3 - 3\ mu1\ p2\ s1\ s2\^3 +
mu1\^2\ p2\ s1\ s2\^3 - p1\ p2\ s1\ s2\^3 +
mu1\ p1\ p2\ s1\ s2\^3 - 2\ p2\^2\ s1\ s2\^3 +
2\ mu1\ p2\^2\ s1\ s2\^3 + p2\ s1\^2\ s2\^3 -
mu1\ p2\ s1\^2\ s2\^3 - p1\ p2\ s1\^2\ s2\^3 +
mu1\ p1\ p2\ s1\^2\ s2\^3 - p2\^2\ s1\^2\ s2\^3 +
mu1\ p2\^2\ s1\^2\ s2\^3)\))\)/
\((\((\(-1\) + mu1)\)\
\((\(-eps\) + eps\ p1 - s2 + mu1\ s2 - s1\ s2 + p1\ s1\ s2)\)\
\((\(-eps\) + eps\ p1 - s2 - eps\ p2\ s2 - s1\ s2 - p2\ s2\^2 -
p2\ s1\ s2\^2)\))\)\)\)\)], "Output"]
}, Closed]],
Cell["\<\
These two equations equal zero. We can drop the denominators to \
make the calculations simpler:\
\>", "Text"],
Cell[BoxData[
\(\(eqn1[p1_, p2_, eps_] =
\((eps\ mu1 - eps\ mu2 - eps\ mu1\ p1 + eps\ mu2\ p1 - eps\ mu1\ p2 +
eps\ mu2\ p2 + eps\ mu1\ p1\ p2 - eps\ mu2\ p1\ p2 + mu1\ s1 -
mu1\ mu2\ s1 + eps\ p1\ s1 - mu1\ p1\ s1 - eps\ mu2\ p1\ s1 +
mu1\ mu2\ p1\ s1 - eps\ p1\^2\ s1 + eps\ mu2\ p1\^2\ s1 +
p1\ s1\^2 - mu2\ p1\ s1\^2 - p1\^2\ s1\^2 + mu2\ p1\^2\ s1\^2 -
mu2\ s2 + mu1\ mu2\ s2 - eps\ p2\ s2 + eps\ mu1\ p2\ s2 +
mu2\ p2\ s2 - mu1\ mu2\ p2\ s2 + eps\ p2\^2\ s2 -
eps\ mu1\ p2\^2\ s2 + mu1\ s1\ s2 - mu2\ s1\ s2 - mu1\ p1\ s1\ s2 +
mu1\ mu2\ p1\ s1\ s2 + mu2\ p2\ s1\ s2 - mu1\ mu2\ p2\ s1\ s2 +
p1\ s1\^2\ s2 - mu2\ p1\ s1\^2\ s2 - p1\^2\ s1\^2\ s2 +
mu2\ p1\^2\ s1\^2\ s2 - p2\ s2\^2 + mu1\ p2\ s2\^2 + p2\^2\ s2\^2 -
mu1\ p2\^2\ s2\^2 - p2\ s1\ s2\^2 + mu1\ p2\ s1\ s2\^2 +
p2\^2\ s1\ s2\^2 - mu1\ p2\^2\ s1\ s2\^2)\); \)\)], "Input"],
Cell[BoxData[
\(\(eqn2[p1_, p2_, eps_] = \
\((eps\^2\ mu2 - eps\ mu1\ mu2 - eps\^2\ mu1\ mu2 + eps\ mu1\^2\ mu2 -
2\ eps\^2\ mu2\ p1 + eps\ mu1\ mu2\ p1 + 2\ eps\^2\ mu1\ mu2\ p1 -
eps\ mu1\^2\ mu2\ p1 + eps\^2\ mu2\ p1\^2 -
eps\^2\ mu1\ mu2\ p1\^2 - eps\^2\ mu2\ p2 + eps\^2\ mu1\ mu2\ p2 +
eps\^2\ mu2\ p1\ p2 - eps\^2\ mu1\ mu2\ p1\ p2 - eps\ mu1\ r -
eps\^2\ mu1\ r + eps\ mu1\^2\ r + eps\ mu1\ mu2\ r +
eps\^2\ mu1\ mu2\ r - eps\ mu1\^2\ mu2\ r + eps\ mu1\ p1\ r +
2\ eps\^2\ mu1\ p1\ r - eps\ mu1\^2\ p1\ r - eps\ mu1\ mu2\ p1\ r -
2\ eps\^2\ mu1\ mu2\ p1\ r + eps\ mu1\^2\ mu2\ p1\ r -
eps\^2\ mu1\ p1\^2\ r + eps\^2\ mu1\ mu2\ p1\^2\ r +
eps\^2\ mu1\ p2\ r - eps\^2\ mu1\ mu2\ p2\ r - eps\^2\ p1\ p2\ r -
eps\^2\ mu1\ p1\ p2\ r + eps\^2\ mu2\ p1\ p2\ r +
eps\^2\ mu1\ mu2\ p1\ p2\ r + eps\^2\ p1\^2\ p2\ r -
eps\^2\ mu2\ p1\^2\ p2\ r - eps\ mu2\ p1\ s1 +
eps\ mu1\ mu2\ p1\ s1 + eps\ mu2\ p1\^2\ s1 -
eps\ mu1\ mu2\ p1\^2\ s1 - eps\ mu1\ r\ s1 + eps\ mu1\ mu2\ r\ s1 -
eps\ p1\ r\ s1 - eps\^2\ p1\ r\ s1 + 3\ eps\ mu1\ p1\ r\ s1 +
eps\ mu2\ p1\ r\ s1 + eps\^2\ mu2\ p1\ r\ s1 -
3\ eps\ mu1\ mu2\ p1\ r\ s1 + eps\ p1\^2\ r\ s1 +
2\ eps\^2\ p1\^2\ r\ s1 - 2\ eps\ mu1\ p1\^2\ r\ s1 -
eps\ mu2\ p1\^2\ r\ s1 - 2\ eps\^2\ mu2\ p1\^2\ r\ s1 +
2\ eps\ mu1\ mu2\ p1\^2\ r\ s1 - eps\^2\ p1\^3\ r\ s1 +
eps\^2\ mu2\ p1\^3\ r\ s1 - eps\ p1\ r\ s1\^2 +
eps\ mu2\ p1\ r\ s1\^2 + 2\ eps\ p1\^2\ r\ s1\^2 -
2\ eps\ mu2\ p1\^2\ r\ s1\^2 - eps\ p1\^3\ r\ s1\^2 +
eps\ mu2\ p1\^3\ r\ s1\^2 + eps\ mu1\ s2 - mu1\^2\ s2 -
eps\ mu1\^2\ s2 + mu1\^3\ s2 + 2\ eps\ mu2\ s2 - mu1\ mu2\ s2 -
4\ eps\ mu1\ mu2\ s2 + 2\ mu1\^2\ mu2\ s2 +
2\ eps\ mu1\^2\ mu2\ s2 - mu1\^3\ mu2\ s2 - eps\ mu1\ p1\ s2 +
eps\ mu1\^2\ p1\ s2 - 2\ eps\ mu2\ p1\ s2 +
4\ eps\ mu1\ mu2\ p1\ s2 - 2\ eps\ mu1\^2\ mu2\ p1\ s2 +
eps\^2\ p2\ s2 - 2\ eps\ mu1\ p2\ s2 - eps\^2\ mu1\ p2\ s2 +
2\ eps\ mu1\^2\ p2\ s2 - 2\ eps\ mu2\ p2\ s2 +
3\ eps\ mu1\ mu2\ p2\ s2 - eps\ mu1\^2\ mu2\ p2\ s2 -
eps\^2\ p1\ p2\ s2 + eps\^2\ mu1\ p1\ p2\ s2 +
eps\ mu2\ p1\ p2\ s2 - eps\ mu1\ mu2\ p1\ p2\ s2 -
eps\^2\ p2\^2\ s2 + eps\^2\ mu1\ p2\^2\ s2 - mu1\ r\ s2 -
2\ eps\ mu1\ r\ s2 + 2\ mu1\^2\ r\ s2 + 2\ eps\ mu1\^2\ r\ s2 -
mu1\^3\ r\ s2 + mu1\ mu2\ r\ s2 + 2\ eps\ mu1\ mu2\ r\ s2 -
2\ mu1\^2\ mu2\ r\ s2 - 2\ eps\ mu1\^2\ mu2\ r\ s2 +
mu1\^3\ mu2\ r\ s2 + 2\ eps\ mu1\ p1\ r\ s2 -
2\ eps\ mu1\^2\ p1\ r\ s2 - 2\ eps\ mu1\ mu2\ p1\ r\ s2 +
2\ eps\ mu1\^2\ mu2\ p1\ r\ s2 + eps\ mu1\ p2\ r\ s2 -
eps\ mu1\^2\ p2\ r\ s2 - eps\ mu1\ mu2\ p2\ r\ s2 +
eps\ mu1\^2\ mu2\ p2\ r\ s2 - eps\ p1\ p2\ r\ s2 +
eps\ mu1\ p1\ p2\ r\ s2 + eps\ mu2\ p1\ p2\ r\ s2 -
eps\ mu1\ mu2\ p1\ p2\ r\ s2 + 2\ eps\ mu2\ s1\ s2 -
mu1\ mu2\ s1\ s2 - 2\ eps\ mu1\ mu2\ s1\ s2 + mu1\^2\ mu2\ s1\ s2 +
eps\ p1\ s1\ s2 - 2\ mu1\ p1\ s1\ s2 - eps\ mu1\ p1\ s1\ s2 +
2\ mu1\^2\ p1\ s1\ s2 - mu2\ p1\ s1\ s2 - 4\ eps\ mu2\ p1\ s1\ s2 +
3\ mu1\ mu2\ p1\ s1\ s2 + 4\ eps\ mu1\ mu2\ p1\ s1\ s2 -
2\ mu1\^2\ mu2\ p1\ s1\ s2 - eps\ p1\^2\ s1\ s2 +
eps\ mu1\ p1\^2\ s1\ s2 + 2\ eps\ mu2\ p1\^2\ s1\ s2 -
2\ eps\ mu1\ mu2\ p1\^2\ s1\ s2 - 2\ eps\ mu2\ p2\ s1\ s2 +
2\ eps\ mu1\ mu2\ p2\ s1\ s2 - 2\ eps\ p1\ p2\ s1\ s2 +
2\ eps\ mu1\ p1\ p2\ s1\ s2 + 2\ eps\ mu2\ p1\ p2\ s1\ s2 -
2\ eps\ mu1\ mu2\ p1\ p2\ s1\ s2 - 2\ mu1\ r\ s1\ s2 -
2\ eps\ mu1\ r\ s1\ s2 + 2\ mu1\^2\ r\ s1\ s2 +
2\ mu1\ mu2\ r\ s1\ s2 + 2\ eps\ mu1\ mu2\ r\ s1\ s2 -
2\ mu1\^2\ mu2\ r\ s1\ s2 - p1\ r\ s1\ s2 - 2\ eps\ p1\ r\ s1\ s2 +
4\ mu1\ p1\ r\ s1\ s2 + 6\ eps\ mu1\ p1\ r\ s1\ s2 -
3\ mu1\^2\ p1\ r\ s1\ s2 + mu2\ p1\ r\ s1\ s2 +
2\ eps\ mu2\ p1\ r\ s1\ s2 - 4\ mu1\ mu2\ p1\ r\ s1\ s2 -
6\ eps\ mu1\ mu2\ p1\ r\ s1\ s2 + 3\ mu1\^2\ mu2\ p1\ r\ s1\ s2 +
2\ eps\ p1\^2\ r\ s1\ s2 - 4\ eps\ mu1\ p1\^2\ r\ s1\ s2 -
2\ eps\ mu2\ p1\^2\ r\ s1\ s2 +
4\ eps\ mu1\ mu2\ p1\^2\ r\ s1\ s2 + eps\ mu1\ p2\ r\ s1\ s2 -
eps\ mu1\ mu2\ p2\ r\ s1\ s2 - eps\ p1\ p2\ r\ s1\ s2 -
eps\ mu1\ p1\ p2\ r\ s1\ s2 + eps\ mu2\ p1\ p2\ r\ s1\ s2 +
eps\ mu1\ mu2\ p1\ p2\ r\ s1\ s2 + eps\ p1\^2\ p2\ r\ s1\ s2 -
eps\ mu2\ p1\^2\ p2\ r\ s1\ s2 - mu2\ p1\ s1\^2\ s2 +
mu1\ mu2\ p1\ s1\^2\ s2 - p1\^2\ s1\^2\ s2 +
mu1\ p1\^2\ s1\^2\ s2 + mu2\ p1\^2\ s1\^2\ s2 -
mu1\ mu2\ p1\^2\ s1\^2\ s2 - mu1\ r\ s1\^2\ s2 +
mu1\ mu2\ r\ s1\^2\ s2 - 2\ p1\ r\ s1\^2\ s2 -
2\ eps\ p1\ r\ s1\^2\ s2 + 4\ mu1\ p1\ r\ s1\^2\ s2 +
2\ mu2\ p1\ r\ s1\^2\ s2 + 2\ eps\ mu2\ p1\ r\ s1\^2\ s2 -
4\ mu1\ mu2\ p1\ r\ s1\^2\ s2 + 2\ p1\^2\ r\ s1\^2\ s2 +
4\ eps\ p1\^2\ r\ s1\^2\ s2 - 3\ mu1\ p1\^2\ r\ s1\^2\ s2 -
2\ mu2\ p1\^2\ r\ s1\^2\ s2 - 4\ eps\ mu2\ p1\^2\ r\ s1\^2\ s2 +
3\ mu1\ mu2\ p1\^2\ r\ s1\^2\ s2 - 2\ eps\ p1\^3\ r\ s1\^2\ s2 +
2\ eps\ mu2\ p1\^3\ r\ s1\^2\ s2 - p1\ r\ s1\^3\ s2 +
mu2\ p1\ r\ s1\^3\ s2 + 2\ p1\^2\ r\ s1\^3\ s2 -
2\ mu2\ p1\^2\ r\ s1\^3\ s2 - p1\^3\ r\ s1\^3\ s2 +
mu2\ p1\^3\ r\ s1\^3\ s2 + mu1\ s2\^2 - 2\ mu1\^2\ s2\^2 +
mu1\^3\ s2\^2 + mu2\ s2\^2 - 3\ mu1\ mu2\ s2\^2 +
3\ mu1\^2\ mu2\ s2\^2 - mu1\^3\ mu2\ s2\^2 + 2\ eps\ p2\ s2\^2 -
2\ mu1\ p2\ s2\^2 - 3\ eps\ mu1\ p2\ s2\^2 + 2\ mu1\^2\ p2\ s2\^2 +
eps\ mu1\^2\ p2\ s2\^2 - mu2\ p2\ s2\^2 + 2\ mu1\ mu2\ p2\ s2\^2 -
mu1\^2\ mu2\ p2\ s2\^2 - eps\ p1\ p2\ s2\^2 +
eps\ mu1\ p1\ p2\ s2\^2 - 2\ eps\ p2\^2\ s2\^2 +
2\ eps\ mu1\ p2\^2\ s2\^2 - mu1\ r\ s2\^2 + 2\ mu1\^2\ r\ s2\^2 -
mu1\^3\ r\ s2\^2 + mu1\ mu2\ r\ s2\^2 - 2\ mu1\^2\ mu2\ r\ s2\^2 +
mu1\^3\ mu2\ r\ s2\^2 + mu1\ s1\ s2\^2 - mu1\^2\ s1\ s2\^2 +
2\ mu2\ s1\ s2\^2 - 4\ mu1\ mu2\ s1\ s2\^2 +
2\ mu1\^2\ mu2\ s1\ s2\^2 + p1\ s1\ s2\^2 - 3\ mu1\ p1\ s1\ s2\^2 +
2\ mu1\^2\ p1\ s1\ s2\^2 - 2\ mu2\ p1\ s1\ s2\^2 +
4\ mu1\ mu2\ p1\ s1\ s2\^2 - 2\ mu1\^2\ mu2\ p1\ s1\ s2\^2 +
2\ eps\ p2\ s1\ s2\^2 - 2\ mu1\ p2\ s1\ s2\^2 -
2\ eps\ mu1\ p2\ s1\ s2\^2 + 2\ mu1\^2\ p2\ s1\ s2\^2 -
2\ mu2\ p2\ s1\ s2\^2 + 3\ mu1\ mu2\ p2\ s1\ s2\^2 -
mu1\^2\ mu2\ p2\ s1\ s2\^2 - 2\ p1\ p2\ s1\ s2\^2 -
2\ eps\ p1\ p2\ s1\ s2\^2 + 2\ mu1\ p1\ p2\ s1\ s2\^2 +
2\ eps\ mu1\ p1\ p2\ s1\ s2\^2 + mu2\ p1\ p2\ s1\ s2\^2 -
mu1\ mu2\ p1\ p2\ s1\ s2\^2 - 2\ eps\ p2\^2\ s1\ s2\^2 +
2\ eps\ mu1\ p2\^2\ s1\ s2\^2 - 2\ mu1\ r\ s1\ s2\^2 +
2\ mu1\^2\ r\ s1\ s2\^2 + 2\ mu1\ mu2\ r\ s1\ s2\^2 -
2\ mu1\^2\ mu2\ r\ s1\ s2\^2 - p1\ r\ s1\ s2\^2 +
4\ mu1\ p1\ r\ s1\ s2\^2 - 3\ mu1\^2\ p1\ r\ s1\ s2\^2 +
mu2\ p1\ r\ s1\ s2\^2 - 4\ mu1\ mu2\ p1\ r\ s1\ s2\^2 +
3\ mu1\^2\ mu2\ p1\ r\ s1\ s2\^2 + mu2\ s1\^2\ s2\^2 -
mu1\ mu2\ s1\^2\ s2\^2 + p1\ s1\^2\ s2\^2 - mu1\ p1\ s1\^2\ s2\^2 -
2\ mu2\ p1\ s1\^2\ s2\^2 + 2\ mu1\ mu2\ p1\ s1\^2\ s2\^2 -
p1\^2\ s1\^2\ s2\^2 + mu1\ p1\^2\ s1\^2\ s2\^2 +
mu2\ p1\^2\ s1\^2\ s2\^2 - mu1\ mu2\ p1\^2\ s1\^2\ s2\^2 -
mu2\ p2\ s1\^2\ s2\^2 + mu1\ mu2\ p2\ s1\^2\ s2\^2 -
2\ p1\ p2\ s1\^2\ s2\^2 + 2\ mu1\ p1\ p2\ s1\^2\ s2\^2 +
mu2\ p1\ p2\ s1\^2\ s2\^2 - mu1\ mu2\ p1\ p2\ s1\^2\ s2\^2 -
mu1\ r\ s1\^2\ s2\^2 + mu1\ mu2\ r\ s1\^2\ s2\^2 -
2\ p1\ r\ s1\^2\ s2\^2 + 4\ mu1\ p1\ r\ s1\^2\ s2\^2 +
2\ mu2\ p1\ r\ s1\^2\ s2\^2 - 4\ mu1\ mu2\ p1\ r\ s1\^2\ s2\^2 +
2\ p1\^2\ r\ s1\^2\ s2\^2 - 3\ mu1\ p1\^2\ r\ s1\^2\ s2\^2 -
2\ mu2\ p1\^2\ r\ s1\^2\ s2\^2 +
3\ mu1\ mu2\ p1\^2\ r\ s1\^2\ s2\^2 - p1\ r\ s1\^3\ s2\^2 +
mu2\ p1\ r\ s1\^3\ s2\^2 + 2\ p1\^2\ r\ s1\^3\ s2\^2 -
2\ mu2\ p1\^2\ r\ s1\^3\ s2\^2 - p1\^3\ r\ s1\^3\ s2\^2 +
mu2\ p1\^3\ r\ s1\^3\ s2\^2 + p2\ s2\^3 - 2\ mu1\ p2\ s2\^3 +
mu1\^2\ p2\ s2\^3 - p2\^2\ s2\^3 + mu1\ p2\^2\ s2\^3 +
2\ p2\ s1\ s2\^3 - 3\ mu1\ p2\ s1\ s2\^3 + mu1\^2\ p2\ s1\ s2\^3 -
p1\ p2\ s1\ s2\^3 + mu1\ p1\ p2\ s1\ s2\^3 - 2\ p2\^2\ s1\ s2\^3 +
2\ mu1\ p2\^2\ s1\ s2\^3 + p2\ s1\^2\ s2\^3 -
mu1\ p2\ s1\^2\ s2\^3 - p1\ p2\ s1\^2\ s2\^3 +
mu1\ p1\ p2\ s1\^2\ s2\^3 - p2\^2\ s1\^2\ s2\^3 +
mu1\ p2\^2\ s1\^2\ s2\^3)\); \)\)], "Input"],
Cell["\<\
These are both solved in the absence of epistasis by setting \
p1=-mu1/s1 and p2=-mu2/s2
[RECALL that selection is assumed to be negative.]:\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(Factor[
\(\(DsSolve /. p1 -> \(-mu1\)/s1\) /. p2 -> \(-mu2\)/s2\) /. eps -> 0]
\)], "Input"],
Cell[BoxData[
\(0\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(Factor[eqn1[\(-mu1\)/s1, \(-mu2\)/s2, 0]]\)], "Input"],
Cell[BoxData[
\(0\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(Factor[eqn2[\(-mu1\)/s1, \(-mu2\)/s2, 0]]\)], "Input"],
Cell[BoxData[
\(0\)], "Output"]
}, Open ]],
Cell["The mean fitness of the population is:", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(Factor[Fitness]\)], "Input"],
Cell[BoxData[
\(w11\ x1 + w12\ x2 + w21\ x3 + w22\ x4\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(Simplify[\(Fitness /. subfit\) /. subtwoloci]\)], "Input"],
Cell[BoxData[
\(eps\ p1\ p2 + \((1 + p1\ s1)\)\ \((1 + p2\ s2)\) +
dis\ \((eps + s1\ s2)\)\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(eqn3[p1_, p2_, eps_] =
Factor[\(\(Fitness /. subtwoloci\) /. subfit\) /. dis -> DsSolve]\)],
"Input"],
Cell[BoxData[
\(\(-\(\(\((\(-1\) + mu1)\)\
\((\(-eps\) + eps\ p1 - s2 - eps\ p2\ s2 - s1\ s2 - p2\ s2\^2 -
p2\ s1\ s2\^2)\)\)\/\(\(-eps\) + eps\ p1 - s2 + mu1\ s2 -
s1\ s2 + p1\ s1\ s2\)\)\)\)], "Output"]
}, Open ]],
Cell["\<\
We will do a Taylor Series of this equation for the mean fitness \
around eps=0,
keeping linear and quadratic terms in eps:\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(Simplify[Normal[Series[eqn3[p1[eps], p2[eps], eps], {eps, 0, 2}]]]\)],
"Input"],
Cell[BoxData[
RowBox[{
RowBox[{"(",
RowBox[{\((\(-1\) + mu1)\), " ",
RowBox[{"(",
RowBox[{
\(\((1 + s1)\)\ \((\(-1\) + mu1 + s1\ \((\(-1\) + p1[0])\))\)\
\((1 + s2\ p2[0])\)\), "+",
RowBox[{\(1\/s2\^2\),
RowBox[{"(",
RowBox[{"eps", " ",
RowBox[{"(",
RowBox[{
RowBox[{
\(-\((1 + s1)\)\), " ", "s2", " ",
\((1 + s2\ p2[0])\), " ",
RowBox[{"(",
RowBox[{\(-1\), "+", \(p1[0]\), "+",
RowBox[{"s1", " ", "s2", " ",
RowBox[{
SuperscriptBox["p1", "\[Prime]",
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RowBox[{
"s2", " ",
\((\(-1\) + mu1 + s1\ \((\(-1\) + p1[0])\))\),
" ",
RowBox[{"(",
RowBox[{
"1", "-", \(p1[0]\), "+", \(s2\ p2[0]\), "+",
RowBox[{\(s2\^2\), " ",
RowBox[{
SuperscriptBox["p2", "\[Prime]",
MultilineFunction->None], "[", "0",
"]"}]}], "+",
RowBox[{"s1", " ", \(s2\^2\), " ",
RowBox[{
SuperscriptBox["p2", "\[Prime]",
MultilineFunction->None], "[", "0",
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"+",
RowBox[{\(1\/s2\^2\),
RowBox[{"(",
RowBox[{\(eps\^2\), " ",
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{\(-1\), "+", \(p1[0]\), "+",
RowBox[{"s1", " ", "s2", " ",
RowBox[{
SuperscriptBox["p1", "\[Prime]",
MultilineFunction->None], "[", "0",
"]"}]}]}], ")"}], " ",
RowBox[{"(",
RowBox[{
\(-1\), "+", \(p1[0]\), "-", \(s2\ p2[0]\), "-",
RowBox[{\(s2\^2\), " ",
RowBox[{
SuperscriptBox["p2", "\[Prime]",
MultilineFunction->None], "[", "0",
"]"}]}], "-",
RowBox[{"s1", " ", \(s2\^2\), " ",
RowBox[{
SuperscriptBox["p2", "\[Prime]",
MultilineFunction->None], "[", "0",
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RowBox[{
RowBox[{"(",
RowBox[{
\((1 + s1)\), " ", \((1 + s2\ p2[0])\), " ",
RowBox[{"(",
RowBox[{\(-2\), "-", \(2\ p1[0]\^2\), "+",
RowBox[{
"2", " ", \((\(-1\) + mu1 + s1)\), " ",
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RowBox[{
SuperscriptBox["p1", "\[Prime]",
MultilineFunction->None], "[", "0",
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RowBox[{"s1", " ", \(s2\^2\), " ",
RowBox[{"(",
RowBox[{
RowBox[{"2", " ", "s1", " ",
SuperscriptBox[
RowBox[{
SuperscriptBox["p1", "\[Prime]",
MultilineFunction->None], "[", "0",
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RowBox[{
SuperscriptBox["p1",
"\[DoublePrime]",
MultilineFunction->None], "[", "0",
"]"}], "-",
RowBox[{"mu1", " ",
RowBox[{
SuperscriptBox["p1",
"\[DoublePrime]",
MultilineFunction->None], "[", "0",
"]"}]}], "+",
RowBox[{"s1", " ",
RowBox[{
SuperscriptBox["p1",
"\[DoublePrime]",
MultilineFunction->None], "[", "0",
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RowBox[{\(p1[0]\), " ",
RowBox[{"(",
RowBox[{"4", "-",
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This is always negative and becomes more negative as the \
variability in epistasis
increases (contributing to E[eps^2]).
Thus mean fitness is expected to go down as the variability in epistasis goes \
up.
(Note that you need to keep up to O(eps^2) throughout!
The sign changes if p1 and p2 are solved only to order eps.)\
\>", "Text"],
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Finally, let's look at the dependence of the average growth rate, \
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\>", "Text"],
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This is always negative.
Thus log(mean fitness) is expected to go down as variability in epistasis \
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\>", "Text"],
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*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)