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NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 257832, 5975]*) (*NotebookOutlinePosition[ 258877, 6008]*) (* CellTagsIndexPosition[ 258833, 6004]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Math. Model & Sci. Comput. in Biosciences", FontSize->24, FontSlant->"Italic"], StyleBox["\nExercise 1: ", FontSlant->"Italic"], "Singular Perturbation Analysis" }], "Title"], Cell[CellGroupData[{ Cell["Numerical Exploration", "Section"], Cell["\<\ Here we will extend the singular perturbation analysis that was \ carried out in Lecture3.nb. In the lecture, singular perturbation analysis \ for the Michaelis-Menten kinetics was analyzed to order 0; we will obtain \ equations up to order 1. Remember, in Lecture 2 we created the ODE system using the Cellerator package \ and simplified to the following 2-dimensional ODE system: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["SimpEnzymeODE", FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontSize->10], RowBox[{ StyleBox["{", FontSize->12], RowBox[{ RowBox[{ StyleBox[ RowBox[{ StyleBox["\[Epsilon]", FontSize->12, FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontSize->12], StyleBox[ RowBox[{ SuperscriptBox[\(EnzS\&^\), "\[Prime]", MultilineFunction->None], "[", \(t\&^\), "]"}], FontSize->12]}], FontSize->14], StyleBox[" ", FontSize->12], StyleBox["==", FontSize->12], StyleBox[\(S\&^[ t\&^] - \((\((k1b + k2f)\)/\((k1f\ STotal[0])\) + S\&^[t\&^])\) EnzS\&^[t\&^]\), FontSize->12]}], StyleBox[",", FontSize->12], StyleBox[ " \t\t\t \ ", FontSize->12], StyleBox[ RowBox[{ RowBox[{ SuperscriptBox[\(S\&^\), "\[Prime]", MultilineFunction->None], "[", \(t\&^\), "]"}], "==", " ", \(\(-S\&^[t\&^]\) + \ EnzS\&^[ t\&^]\ \((k1b/\((k1f\ STotal[0])\) + \ S\&^[t\&^])\)\)}], FontSize->12]}], StyleBox["}", FontSize->12]}]}], ";"}]], "Input", FontSize->14], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(STotal\\)\\\" is similar to existing symbol \ \\\"\\!\\(Total\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[TextData[{ "where the \"^\" denotes dimensionless, normalized values, ", StyleBox["STotal[0]", "InlineInput"], " the initial total substrate concentration, ", StyleBox["\[Epsilon]", "InlineInput", FontColor->RGBColor[1, 0, 0]], StyleBox[" = EnzTotal[0]/STotal[0]", "InlineInput"], " a small dimensionless parameter describing the small, enzyme-to-substrate \ concentration ratio." }], "Text"], Cell[TextData[{ "For convenience, let's remove the \"^\" notation in the equations. ", StyleBox["Hint", FontSlant->"Italic"], ": apply the replacement rule, ", StyleBox["OverHat[x_]\[Rule] x", "InlineInput"], ", to ", StyleBox["SimpEnzymeODE", "InlineInput", FontColor->RGBColor[0, 0, 1]], StyleBox[".", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox["SimpEnzymeODE", FontColor->RGBColor[0, 0, 1]], "=", RowBox[{ StyleBox["SimpEnzymeODE", FontColor->RGBColor[0, 0, 1]], "/.", \({OverHat[x_] \[Rule] \ x}\)}]}]], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"\[Epsilon]", " ", RowBox[{ SuperscriptBox["EnzS", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "\[Equal]", \(S[t] - EnzS[t]\ \((S[t] + \(k1b + k2f\)\/\(k1f\ STotal[0]\))\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox["S", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "\[Equal]", \(\(-S[t]\) + EnzS[t]\ \((S[t] + k1b\/\(k1f\ STotal[0]\))\)\)}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Typically, prior to doing mathematical analysis, it is a good idea to \ numerically explore the solution behavior.\n\nLet's look at the list of \ variables and functions that appear in the ODE system. ", StyleBox["Hint", FontSlant->"Italic"], ": use ", StyleBox["Cases", "InlineInput"], ", with selection criterion ", StyleBox["_Symbol|_Symbol[x_]", "InlineInput"], ", and apply ", StyleBox["Union ", "InlineInput"], "as demonstrated in Tutorial1.nb. \n" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Cases[SimpEnzymeODE, \ _Symbol | _Symbol[x_], \ Infinity] // Union\)], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ \({k1b, k1f, k2f, t, \[Epsilon], EnzS[t], S[t], STotal[0]}\)], "Output"] }, Open ]], Cell[TextData[{ "To numerically solve the ODE, we need to provide initial condition as well \ as choose parameter values. \nLet's append the ODE list with the initial \ condition, ", StyleBox["EnzS[0]==0, S[0]==1", "InlineInput"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{\(IC\ = \ {EnzS[0] \[Equal] 0, \ S[0] \[Equal] 1};\), "\[IndentingNewLine]", RowBox[{ RowBox[{"ODEIC", "=", RowBox[{"Join", "[", StyleBox[\(SimpEnzymeODE, \ IC\), FontColor->RGBColor[0, 0, 1]], "]"}]}], ";"}], "\[IndentingNewLine]", \(% // ColumnForm\)}], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{ RowBox[{"\[Epsilon]", " ", RowBox[{ SuperscriptBox["EnzS", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "\[Equal]", \(S[t] - EnzS[t]\ \((S[t] + \(k1b + k2f\)\/\(k1f\ STotal[0]\))\)\)}]}, { RowBox[{ RowBox[{ SuperscriptBox["S", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "\[Equal]", \(\(-S[t]\) + EnzS[t]\ \((S[t] + k1b\/\(k1f\ STotal[0]\))\)\)}]}, {\(EnzS[0] \[Equal] 0\)}, {\(S[0] \[Equal] 1\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ Times[ \[Epsilon], Derivative[ 1][ EnzS][ t]], Plus[ S[ t], Times[ -1, EnzS[ t], Plus[ S[ t], Times[ Power[ k1f, -1], Plus[ k1b, k2f], Power[ STotal[ 0], -1]]]]]], Equal[ Derivative[ 1][ S][ t], Plus[ Times[ -1, S[ t]], Times[ EnzS[ t], Plus[ S[ t], Times[ k1b, Power[ k1f, -1], Power[ STotal[ 0], -1]]]]]], Equal[ EnzS[ 0], 0], Equal[ S[ 0], 1]}], Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "Furthermore, to numerically solve ODES we need to provide numerical values \ of parameters. \nLets set ", StyleBox["k1b->1, k1f->1, k2f->1, STotal[0]->1, \[Epsilon]->0.1", "InlineInput"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ParamReplaceRule = {k1f \[Rule] 1, k1b \[Rule] 1, k2f \[Rule] 1, STotal[0] \[Rule] 1, \[Epsilon] \[Rule] 0.1};\)\), "\[IndentingNewLine]", \(NumODEIC = ODEIC /. %\)}], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"0.1`", " ", RowBox[{ SuperscriptBox["EnzS", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "\[Equal]", \(S[t] - EnzS[t]\ \((2 + S[t])\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox["S", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "\[Equal]", \(\(-S[t]\) + EnzS[t]\ \((1 + S[t])\)\)}], ",", \(EnzS[0] \[Equal] 0\), ",", \(S[0] \[Equal] 1\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "When calling numerical ODE integrator NDSolve, one needs to specify the \ functions being solved for. We can extract this using again ", StyleBox["Cases", "InlineInput", Background->None], " command:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(SolvedFunctions = Cases[NumODEIC, \ _Symbol[_Symbol], \ Infinity] // Union\)], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ \({EnzS[t], S[t]}\)], "Output"] }, Open ]], Cell["\<\ Now call NDSolve and Plot the result, as was done in \ Tutorial1.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(ODESoln = NDSolve[NumODEIC, \ SolvedFunctions, \ {t, 0, 1}];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Plot[SolvedFunctions[\([1]\)] /. ODESoln, \ {t, 0, 1}, \ AxesLabel \[Rule] \ \((ToString /@ SolvedFunctions[\([1]\)])\)];\)\), "\[IndentingNewLine]", \(\(Plot[SolvedFunctions[\([2]\)] /. 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fMT0fMWI0080fMWI0P0000050=WIf@030000003IfMT0fMWI00D0fMWI00<000000=WIf@3IfMT01`3I fMT00`000000fMWI0=WIf@3o0=WIfG@0fMWI000M0=WIf@@0000000<0fMWI0000003IfMT0103IfMT2 000000H0fMWI00<000000=WIf@3IfMT01P3IfMT200000?l0fMWIMP3IfMT00?l0fMWI/@3IfMT00001 \ \>"], ImageRangeCache->{{{153, 440}, {450.875, 273.938}} -> {-0.742743, 1.23819, \ 0.00404759, 0.00202837}}] }, Open ]] }, Open ]], Cell[TextData[{ "Now it seems ", StyleBox["EnzS[t]", "InlineInput"], " reaches a maximum at some t>0. Lets find out when it does. \n\nWe can do \ this by generating a uniformly distributed values of times from t=0 to t=1, \ evaluate ", StyleBox["EnzS[t]", "InlineInput"], " at this list of times, and find its maximum.\nFirst, generate the list of \ times using ", StyleBox["Table", "InlineInput"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(TimeInterval = 10000;\)\), "\[IndentingNewLine]", \(\(timeList = Table[t \[Rule] \ i/TimeInterval, \ {i, 0, TimeInterval}];\)\[IndentingNewLine]\[IndentingNewLine] (*\ write\ out\ the\ short\ form\ of\ the\ output\ with\ Short\ *) \), "\ \[IndentingNewLine]", \(% // Short\)}], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ TagBox[\({t \[Rule] 0, t \[Rule] 1\/10000, t \[Rule] 1\/5000, \[LeftSkeleton]9995\[RightSkeleton], t \[Rule] 4999\/5000, t \[Rule] 9999\/10000, t \[Rule] 1}\), Short]], "Output"] }, Open ]], Cell["\<\ Then, from ODESoln, obtain the EnzSSoln in the form of an \ interpolated function,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EnzSSoln = SolvedFunctions[\([1]\)] /. ODESoln\)], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[\(InterpolatingFunction[{{0.`, 1.`}}, "<>"]\), False, Editable->False], "[", "t", "]"}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Now evaluate by the numerical values of EnzSSoln at each of the time \ points in timeList. Hint: use commands ", StyleBox["Map", "InlineInput"], " and the fact that ", StyleBox["timeList", "InlineInput"], " is a list of replacement rules. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(NumList = Map[EnzSSoln /. #\ &, \ timeList] // Flatten;\)\), "\[IndentingNewLine]", \(% // Short\)}], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ TagBox[\({5.5718886061571944`*^-21, \[LeftSkeleton]9999\[RightSkeleton], 0.2591735558010211`}\), Short]], "Output"] }, Open ]], Cell[TextData[{ "From this list of numerical values, find out the position of the maximum \ within the list using", StyleBox[" Ordering", "InlineInput"], ". \n\nThen get the time at the maximum, as well as ", StyleBox["EnzSSoln", "InlineInput"], " at maximum." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(MaxEnzSLocation = Ordering[NumList, \ \(-1\)];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(TimeOfMaxEnzS = timeList[\([MaxEnzSLocation]\)]\), "\[IndentingNewLine]", \(MaxEnzS = EnzSSoln /. %\)}], "Input", Background->RGBColor[0, 1, 1]], Cell[BoxData[ \({t \[Rule] 327\/2000}\)], "Output"], Cell[BoxData[ \({0.31649126347019446`}\)], "Output"] }, Open ]], Cell["\<\ Now, we combine all the previous steps to write a routine that \ takes as an input the \[Epsilon] value and does the above computation \ automatically. Lets also do a log-log plot.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(PlotPeakOfEnzS[EpsVal_] := Module[{TimeInterval = 10^4}, \ \[IndentingNewLine]NumODEIC = ODEIC /. {k1f \[Rule] 1, k1b \[Rule] 1, k2f \[Rule] 1, STotal[0] \[Rule] 1, \[Epsilon] \[Rule] EpsVal}; \[IndentingNewLine]SolvedFunctions = Cases[NumODEIC, \ _Symbol[_Symbol], \ Infinity] // Union; \[IndentingNewLine]ODESoln = NDSolve[NumODEIC, \ SolvedFunctions, \ {t, 0, 1}]; \[IndentingNewLine]EnzSSoln = SolvedFunctions[\([1]\)] /. ODESoln; \[IndentingNewLine]timeList = Table[t \[Rule] \ i/TimeInterval, \ {i, 0, TimeInterval}]; \[IndentingNewLine]NumList = Map[EnzSSoln /. #\ &, \ timeList] // Flatten; \[IndentingNewLine]MaxEnzSLocation = Ordering[ NumList, \ \(-1\)]; \ \[IndentingNewLine]timeList[\([MaxEnzSLocation]\)]\[IndentingNewLine]];\)\ \[IndentingNewLine]\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(EpsValList = {0.1, 0.01, \ 0.001, \ 0.0001};\)\), "\[IndentingNewLine]", \(\(\(PeakLocationList = \(N[t /. #\ , 10] &\)\ /@ \ \(PlotPeakOfEnzS\ /@ \ EpsValList\) // Flatten\ \ ;\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(<< Graphics`Graphics`\), "\[IndentingNewLine]", \(\(LogLogListPlot[\ {EpsValList, \ PeakLocationList} // Transpose, \ PlotJoined \[Rule] \ True\ , \ AspectRatio \[Rule] \ Automatic, \ AxesLabel \[Rule] \ {"\<\[Epsilon]\>", \ "\