(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 21404, 578] NotebookOptionsPosition[ 19826, 525] NotebookOutlinePosition[ 20733, 557] CellTagsIndexPosition[ 20690, 554] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Ex1: Math. Modelling in Biosciences 1", "Title", CellChangeTimes->{{3.415589585204612*^9, 3.415589598615617*^9}, { 3.415612138399819*^9, 3.415612142124673*^9}}], Cell["Summer 2008 ", "Subtitle", CellChangeTimes->{{3.41561211409342*^9, 3.415612134422485*^9}}], Cell["Due Tuesday 22 April", "Subsubtitle", CellChangeTimes->{{3.415605888680184*^9, 3.41560590190364*^9}}, FontSize->22], Cell[TextData[{ "In this exercise, we derive and examine the equations for ", StyleBox["cooperative phenomena.", FontSlant->"Italic"], " \nThis happens when a single enzyme molecule, after being bound to a \ substrate molecule, can further bind an additional substrate molecule. \n\n\ It may happen that binding of one substrate molecule at one site can affect \ the binding affinity of the other substrate molecule at another site. This is \ known as the ", StyleBox["allosteric effect", FontSlant->"Italic"], ".\n\nIn this exercise, we derive and analyze the problem as described in \ Section 6.5 of Murray's ", StyleBox["Mathematical Biology I", FontVariations->{"Underline"->True}] }], "Text", CellChangeTimes->{{3.415605406565242*^9, 3.415605433891287*^9}, { 3.415605464231621*^9, 3.415605549469606*^9}, {3.415605580453104*^9, 3.415605694991905*^9}, {3.415605756477949*^9, 3.415605842835668*^9}, { 3.41560592575955*^9, 3.415605926779197*^9}, {3.415612161819669*^9, 3.415612164172899*^9}, {3.415612206830281*^9, 3.415612226928721*^9}}], Cell[TextData[{ "Q1: obtain the ODE system for the following reactions (using ", StyleBox["xCellerator", FontSlant->"Italic"], " or otherwise): ", StyleBox["7 points", FontColor->RGBColor[0.6, 0.4, 0.2]], " " }], "Item", CellChangeTimes->{{3.415605951787572*^9, 3.415605952424048*^9}, { 3.41560696626531*^9, 3.415607037040669*^9}, {3.415607092775741*^9, 3.415607100566137*^9}, {3.415607140563574*^9, 3.415607149990141*^9}, { 3.415611936111022*^9, 3.41561194400363*^9}, {3.415612033505567*^9, 3.415612035785994*^9}}], Cell[TextData[{ "Enzyme ", StyleBox["e", FontWeight->"Bold"], " binds to substrate ", StyleBox["s", FontWeight->"Bold"], " to form the substrate-enzyme complex ", StyleBox["c1", FontWeight->"Bold"], ". Subsequently, the complex ", StyleBox["c1", FontWeight->"Bold"], " may break down to form product ", StyleBox["p", FontWeight->"Bold"], " and releases enzyme ", StyleBox["e", FontWeight->"Bold"], ". Complex ", StyleBox["c1 ", FontWeight->"Bold"], "can also combines with another substrate molecule to form the complex ", StyleBox["c2", FontWeight->"Bold"], ". ", StyleBox["c2", FontWeight->"Bold"], " similarly breaks down to form product ", StyleBox["p", FontWeight->"Bold"], " and ", StyleBox["c1", FontWeight->"Bold"], ".\n\nSo this is a concatenation of reactions covered in lectures and maybe \ represented as:\n\n", Cell[BoxData[ RowBox[{ RowBox[{"s", "+", "e"}], " ", UnderoverscriptBox["\[RightArrowLeftArrow]", RowBox[{"k", "[", RowBox[{"-", "1"}], "]"}], RowBox[{"k", "[", "1", "]"}]], " ", "c1", OverscriptBox["\[ShortRightArrow]", RowBox[{"k", "[", "2", "]"}]], " ", RowBox[{"e", "+", "p"}]}]], CellChangeTimes->{{3.415607458986791*^9, 3.415607642357147*^9}}, FontSize->20, FontWeight->"Bold"], StyleBox["\n", FontSize->20, FontWeight->"Bold"], Cell[BoxData[ RowBox[{ RowBox[{"s", "+", "c1"}], UnderoverscriptBox["\[RightArrowLeftArrow]", RowBox[{"k", "[", RowBox[{"-", "3"}], "]"}], RowBox[{"k", "[", "3", "]"}]], " ", "c2", OverscriptBox["\[ShortRightArrow]", RowBox[{"k", "[", "4", "]"}]], " ", RowBox[{"c1", "+", "p"}]}]], 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", StyleBox["15 points", FontColor->RGBColor[0.6, 0.4, 0.2]] }], "Item", CellChangeTimes->{{3.415605951787572*^9, 3.415605952424048*^9}, { 3.41560696626531*^9, 3.415607037040669*^9}, {3.415607067867762*^9, 3.41560707747891*^9}, {3.415608019137917*^9, 3.415608052609592*^9}, { 3.415608130722612*^9, 3.415608180281544*^9}}] }, Open ]], Cell[TextData[{ "Firstly, we observe that the problem is characterized by 2 very different \ concentration levels: that of the substrate and enzyme.\n\nWe denote the \ respective initial concentrations as s0, e0.\n\nHence, we can define new \ variables u, v1, v2:\n\n", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"u", "[", "t", "]"}], "==", FractionBox[ RowBox[{"s", "[", "t", "]"}], "s0"]}], ",", " ", RowBox[{ RowBox[{"v1", "[", "t", "]"}], "==", FractionBox[ RowBox[{"c1", "[", "t", "]"}], "e0"]}], ",", RowBox[{ RowBox[{"v2", "[", "t", "]"}], "\[Equal]", " ", FractionBox[ RowBox[{"c2", "[", "t", "]"}], "e0"]}]}]], CellChangeTimes->{{3.415593818798567*^9, 3.415593834439011*^9}}, FontSize->20, FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.415608196941937*^9, 3.415608324703545*^9}, { 3.415608359263574*^9, 3.415608386324562*^9}, 3.415611898680019*^9, { 3.415612331443811*^9, 3.415612357800814*^9}}], Cell[TextData[{ "\nAs the characteristic time scale, we can choose it as (also in lecture): \ ", Cell[BoxData[ StyleBox[ FractionBox["1", RowBox[{ RowBox[{"k", "[", "1", "]"}], " ", "e0"}]], FontSize->20, FontWeight->"Bold"]], CellChangeTimes->{{3.415593818798567*^9, 3.415593834439011*^9}}, FontSize->22], "\n\nWe choose as the small parameter the ratio of initial concentrations: \ ", StyleBox["\[Epsilon] = ", FontSize->22, FontWeight->"Bold"], Cell[BoxData[ FractionBox["e0", RowBox[{" ", "s0"}]]], CellChangeTimes->{{3.4155942241682*^9, 3.41559427948222*^9}, { 3.415594325899348*^9, 3.415594368747103*^9}, {3.415594401644176*^9, 3.41559449003767*^9}, {3.415594863976211*^9, 3.415594927092134*^9}}, FontSize->22, FontWeight->"Bold"], 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3.415610612273495*^9}, {3.415610772280362*^9, 3.415610777369745*^9}}], Cell[TextData[{ "Q5: write down (but need not solve) the 0th and 1st order expansions (in \ \[Epsilon]) for the inner ODE system:", StyleBox[" 15 points", FontColor->RGBColor[0.6, 0.4, 0.2]], " " }], "Item", CellChangeTimes->{{3.415605951787572*^9, 3.415605952424048*^9}, { 3.41560696626531*^9, 3.415607037040669*^9}, {3.415607067867762*^9, 3.41560707747891*^9}, {3.415608019137917*^9, 3.415608052609592*^9}, { 3.415608130722612*^9, 3.415608180281544*^9}, {3.415610537357123*^9, 3.415610612273495*^9}, {3.415610772280362*^9, 3.415610807250889*^9}, { 3.415610901096981*^9, 3.415610949815916*^9}, {3.415612064308993*^9, 3.415612064365879*^9}}], Cell[TextData[{ "Q6: obtain the 0th order system for outer equation (obtained by setting \ \[Epsilon] \[Rule] 0 in the original system):", StyleBox[" 15 points", FontColor->RGBColor[0.6, 0.4, 0.2]], " " }], "Item", CellChangeTimes->{{3.415605951787572*^9, 3.415605952424048*^9}, { 3.41560696626531*^9, 3.415607037040669*^9}, {3.415607067867762*^9, 3.41560707747891*^9}, {3.415608019137917*^9, 3.415608052609592*^9}, { 3.415608130722612*^9, 3.415608180281544*^9}, {3.415610537357123*^9, 3.415610612273495*^9}, {3.415610772280362*^9, 3.415610807250889*^9}, { 3.415610901096981*^9, 3.415610949815916*^9}, {3.415611077022621*^9, 3.415611172178064*^9}, {3.415611482609558*^9, 3.415611484900516*^9}}] }, Open ]], Cell[TextData[{ "We can get the 0th order system by just setting \[Epsilon] \[Rule] 0.\n\n\ However, we do not use the original initial conditions; as we showed in \ class, this can be obtained from the outer limit of the boundary layer \ solution.\n\nUsing the algebraic constraints for v1[\[Eta]] and v2[\[Eta]], \ you should get a single ODE: ", StyleBox[" ", FontSize->22, FontWeight->"Bold"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "\[Eta]", "]"}], "\[Equal]"}]], CellChangeTimes->{{3.415603186673549*^9, 3.415603209292405*^9}}, FontSize->22, FontWeight->"Bold"], StyleBox[" ...", FontSize->22, FontWeight->"Bold"], "\n\nWhat expression do you get? Try to plot the right-hand-side as a \ function of ", Cell[BoxData[ RowBox[{"u", "[", "\[Eta]", "]"}]], CellChangeTimes->{{3.415603186673549*^9, 3.415603209292405*^9}}, FontSize->22, FontWeight->"Bold"], "for constants \na[1] \[Rule] 10, a[3] \[Rule] 0.1, a[4] \[Rule] 0, a[5] \ \[Rule] 0.1\n\nIt should look something like: " }], "Text", CellChangeTimes->{{3.415611309170972*^9, 3.415611383310061*^9}, { 3.41561144726162*^9, 3.415611467133091*^9}, {3.415611497515112*^9, 3.415611639868706*^9}, 3.415611721811063*^9}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJwVz2k8lI0exnFMWpBj6RGSGdm3kU5Flvt/Z6ksiWwhylgmCeGxhYr0SFmH RNlJdoZRkTSRGHtGdCzZp0IZWxLy3OfF9fl9vi8vKZLPeTcuDg6OOGz/b+ZN 0k511wcIpbFKdpGrUecdk2BqS/CA//jOegwjIcj2+8Y7BEIIwOXDD7100hF+ yvMqnk+RkNQVpZzgXIZk0uPw3IQYYPGrvSJq1iEqC67x2+QEWFDjfWxY3YyM 8mf92RykAG5pb+F9+05EpnDP8d8bD2HMx78HNWYi13QDvNbw6aBIlzg7qfY/ pKZ/vGBFPwPYx4Msf46OIb89TUcWydngYxftL1MyheRwjsSsGebCpvqfd+cZ 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