(* 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[ 437751, 7493] NotebookOptionsPosition[ 434981, 7403] NotebookOutlinePosition[ 435872, 7435] CellTagsIndexPosition[ 435829, 7432] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Assignment 2", "Title", CellChangeTimes->{{3.433567408527817*^9, 3.433567415725386*^9}, { 3.437123188014092*^9, 3.437123188075443*^9}}], Cell["Due: Tuesday 13 January, 2008", "Subtitle", CellChangeTimes->{{3.43356747830616*^9, 3.433567490982802*^9}, { 3.437123153109909*^9, 3.437123167817243*^9}}], Cell[TextData[{ StyleBox["Note", FontVariations->{"Underline"->True}, Background->RGBColor[0, 1, 1]], ": for this assignment you can attempt the solutions either by hand or using \ ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.437125202660422*^9, 3.437125236702115*^9}, { 3.437141506633184*^9, 3.43714157829529*^9}, 3.437142658241323*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ "SetDirectory", "[", "\"\<~/Teaching/MathModelBioSciences_Winter08/Ex2\>\"", "]"}], ";"}]], "Input", CellOpen->False, CellChangeTimes->{{3.433567727678491*^9, 3.433567749540262*^9}, { 3.437123811017949*^9, 3.437123813136998*^9}}], Cell[TextData[{ StyleBox["Q1", FontColor->GrayLevel[0], Background->RGBColor[1, 1, 0]], ": In the paper \"", StyleBox["Do morphogen gradients arise by diffusion?\"", FontSlant->"Italic"], " by A. D. Lander, Q. Nie and F. Y. M. Wan, they showed that the PDE for the \ full model that takes into account diffusion, internalization etc, is of the \ form:" }], "Item1", CellChangeTimes->{{3.433567500149698*^9, 3.433567548592442*^9}, { 3.437123898287451*^9, 3.437123968229729*^9}, {3.437124016957098*^9, 3.437124065054027*^9}, 3.43712409580995*^9, {3.437124317335202*^9, 3.437124374321207*^9}, {3.437124949642591*^9, 3.437124951331112*^9}, 3.437125034209606*^9, {3.437126631097504*^9, 3.437126632281896*^9}, { 3.43720251911145*^9, 3.437202527025999*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"Import", "[", "\"\\"", "]"}], "]"}]], "Input", CellOpen->False, CellChangeTimes->{{3.437124974699579*^9, 3.437125002338131*^9}}], Cell[BoxData[ GraphicsBox[RasterBox[CompressedData[" 1:eJzsvX+0XVWVLjiV5yh/lDJGiAUDbxSfIhCe3DyDvnoZAgniz3AEn4pyBYtS CEIjdVECaBrpVF9jeflxET2vfO+N9Av94FUFCKH6BVCgrSoxBPWea1e/StI1 Mu4FUaA6TdUV6TYXi4I+39jzG+vb86x97k1AciPrjzn23muvNddcc619zvzW 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Cell[TextData[{ "2.1) Let us consider the case that \[Epsilon] << B, C. Hence, what \ approximation (to zeroth-order in \[Epsilon]) can we make with regard to the \ quantity w? (", StyleBox["3 points", Background->RGBColor[0, 1, 0]], ")" }], "Item2", CellChangeTimes->{{3.437127540609061*^9, 3.437127613146118*^9}, { 3.437127973191928*^9, 3.437127999261362*^9}, {3.43712807063954*^9, 3.437128072830287*^9}, {3.437128182721253*^9, 3.437128185803015*^9}, { 3.437128968003933*^9, 3.437128969917123*^9}, {3.437142060054752*^9, 3.43714206087279*^9}, {3.43720275266177*^9, 3.437202785560466*^9}}], Cell[TextData[{ "2.2) Using the approximation stated above, write down the resulting PDE for \ the single remaining variable, V (", StyleBox["2 points", Background->RGBColor[0, 1, 0]], ")" }], "Item2", CellChangeTimes->{{3.437128108286791*^9, 3.43712819011742*^9}, { 3.437128666464473*^9, 3.437128669138802*^9}}], Cell[TextData[{ "2.3) The PDE you get should be a reaction-diffusion equation with a cubic \ form for the reaction term. In class, we analyzed a particular case of such \ a system and showed that traveling wave solutions are possible. Using the \ expressions derived in class, write down the velocity of the traveling wave \ in terms of the original parameters as given in the above FitzHugh equations \ (", StyleBox["15 points", Background->RGBColor[0, 1, 0]], ")" }], "Item2", CellChangeTimes->{{3.437128108286791*^9, 3.437128199084766*^9}, { 3.437128262319004*^9, 3.437128300823782*^9}, {3.437128334186419*^9, 3.43712837485418*^9}, {3.437128458928039*^9, 3.437128479755813*^9}, { 3.437128529297068*^9, 3.437128567721801*^9}, {3.437128685746809*^9, 3.437128713472348*^9}, {3.437128759457967*^9, 3.437128785998015*^9}, 3.437130754205505*^9, {3.4372028195888*^9, 3.43720282523371*^9}}], Cell[TextData[{ StyleBox["Q3", FontColor->GrayLevel[0], Background->RGBColor[1, 1, 0]], ": In this questoin, we look at a mechanism leading to robust patterning in \ Drosophila embryonic development. 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StyleBox["M", FontWeight->"Bold"], ", an inhibitor ", StyleBox["I", FontWeight->"Bold"], ", and a protease ", StyleBox["P", FontWeight->"Bold"], " that cleaves the inhibitor. The signaling pathway is actived when the \ level of ", StyleBox["M", FontWeight->"Bold"], " reaches above a certain threshold.The assumptions are:" }], "Text", CellChangeTimes->{{3.433567500149698*^9, 3.433567548592442*^9}, { 3.437123898287451*^9, 3.437123968229729*^9}, {3.437124016957098*^9, 3.437124065054027*^9}, 3.43712409580995*^9, {3.437124317335202*^9, 3.437124374321207*^9}, {3.437124949642591*^9, 3.437124951331112*^9}, 3.437125034209606*^9, {3.437126631097504*^9, 3.437126632281896*^9}, { 3.437126795683523*^9, 3.437126798106882*^9}, {3.437126839241087*^9, 3.437126864350913*^9}, {3.437126958917541*^9, 3.437126976028652*^9}, { 3.437135491021754*^9, 3.437135495037256*^9}, {3.437135544342995*^9, 3.437135562952972*^9}, {3.437135833164861*^9, 3.437135841709519*^9}, { 3.437135905379033*^9, 3.437135912509014*^9}, {3.437139115302398*^9, 3.437139119535104*^9}, {3.437139252644009*^9, 3.437139277337328*^9}, 3.437207177736849*^9}], Cell[CellGroupData[{ Cell[TextData[{ "The free inhibitor", StyleBox[" I", FontWeight->"Bold"], " diffuses at rate ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["D", "I"], FontWeight->"Bold"], TraditionalForm]]], ", is not degraded but is consumed when it binds to free morphogen ", StyleBox["M", FontWeight->"Bold"], " to form the complex ", StyleBox["C. ", FontWeight->"Bold"], "Assume that this occurs at a constant rate ", StyleBox["k", FontWeight->"Bold"] }], "Item1Numbered", CellChangeTimes->{{3.437139129290802*^9, 3.437139129849766*^9}, { 3.437139468060113*^9, 3.437139592034423*^9}, 3.437139662625597*^9, { 3.437205252260353*^9, 3.437205264777886*^9}}], Cell[TextData[{ "The concentration of the protease ", StyleBox["P", FontWeight->"Bold"], " is assumed to be uniformly distributed in space and abundant so that it \ can be taken as constant in time" }], "Item1Numbered", CellChangeTimes->{{3.437139634464747*^9, 3.437139637245326*^9}, { 3.43713971428925*^9, 3.437139747610097*^9}, {3.437142423762306*^9, 3.437142425414963*^9}, {3.437205287174365*^9, 3.437205294985772*^9}}], Cell[TextData[{ "Besides being formed by ", StyleBox["M", FontWeight->"Bold"], " and ", StyleBox["I", FontWeight->"Bold"], ", the complex ", StyleBox["C", FontWeight->"Bold"], " diffuses at the rate ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["D", "C"], FontWeight->"Bold"], TraditionalForm]]], " and is degraded by protease ", StyleBox["P", FontWeight->"Bold"], " at a constant rate ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["\[Alpha]", "C"], FontWeight->"Bold"], TraditionalForm]]], ", back to ", StyleBox["M", FontWeight->"Bold"] }], "Item1Numbered", CellChangeTimes->{ 3.437139797357745*^9, {3.437140016053386*^9, 3.4371401296331*^9}, { 3.437140172611401*^9, 3.437140175388746*^9}, {3.437142452505059*^9, 3.43714245700467*^9}}], Cell[TextData[{ "Morphogen M diffuses at rate ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["D", "M"], FontWeight->"Bold"], TraditionalForm]]], " and reacts according to the above described rules for ", StyleBox["C", FontWeight->"Bold"], " formation and degradation" }], "Item1Numbered", CellChangeTimes->{{3.437140191919396*^9, 3.437140259614183*^9}, 3.437140322283906*^9, {3.437140402601773*^9, 3.437140420779172*^9}}] }, Open ]], Cell["", "Institution"], Cell[CellGroupData[{ Cell[TextData[{ "3.1) Consider the domain x \[Element] [-1, 1] and write down the 3-variable \ PDE system for ", StyleBox["M", FontWeight->"Bold"], ", ", StyleBox["I", FontWeight->"Bold"], " and the concentration of the complex ", StyleBox["C", FontWeight->"Bold"], "=[MI] (", StyleBox["10 pts", Background->RGBColor[0, 1, 0]], ")" }], "Item2", CellChangeTimes->{{3.437135927814517*^9, 3.43713595480762*^9}, { 3.437139295914909*^9, 3.437139392356956*^9}, {3.437140429299198*^9, 3.437140459865308*^9}, {3.437205469725472*^9, 3.437205473300428*^9}}], Cell[TextData[{ "3.2) Now we consider a situation which gives rise to robust morphogen \ gradients. Namely, we consider the situation where free ", StyleBox["M", FontWeight->"Bold"], " ", StyleBox["cannot diffuse", FontSlant->"Italic"], ": hence we set to 0 its diffusion constant, ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox["D", "M"], FontWeight->"Bold"], "\[RightArrow]", "0"}], TraditionalForm]]], ". With this assumption, obtain the equilbrium solution for ", StyleBox["M", FontWeight->"Bold"], "(x) showing how it is robust, for instance independent of the protease \ concentration ", StyleBox["P", FontWeight->"Bold"], " or its rate of action, ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["\[Alpha]", "C"], FontWeight->"Bold"], TraditionalForm]]], ". Note that the solution is assumed to be symmetric around x=0, with ", StyleBox["M", FontWeight->"Bold"], "(x) attaining its peak at x=0 and decays as x goes away from 0. (", StyleBox["20 pts", FontColor->GrayLevel[0], Background->RGBColor[0, 1, 0]], "). To solve this problem, you could take the following steps:" }], "Item2", CellChangeTimes->{{3.437135927814517*^9, 3.43713595480762*^9}, { 3.437139295914909*^9, 3.437139392356956*^9}, {3.437140429299198*^9, 3.437140459865308*^9}, {3.437140495675878*^9, 3.43714082431508*^9}, { 3.437140871210113*^9, 3.437140957544187*^9}, {3.437141094770834*^9, 3.437141160276727*^9}, {3.437141252761237*^9, 3.437141265083198*^9}, { 3.437141931272149*^9, 3.437141972396068*^9}, {3.43714225138147*^9, 3.437142254447287*^9}}], Cell[TextData[{ "Solve for ", StyleBox["C", FontWeight->"Bold"], ", by looking at the equation for ", StyleBox["M", FontWeight->"Bold"], "(x) with the stated assumptions and the fact that it is symmetric about \ x=0." }], "Item3", CellChangeTimes->{{3.437140658815729*^9, 3.437140660006344*^9}, { 3.437141174977814*^9, 3.437141205186446*^9}, {3.437141302476346*^9, 3.437141327722988*^9}, {3.437205627915306*^9, 3.437205635858612*^9}}], Cell[TextData[{ "Using the equation for ", StyleBox["M", FontWeight->"Bold"], "(x), this gives a relation between ", StyleBox["M", FontWeight->"Bold"], "(x) and ", StyleBox["I", FontWeight->"Bold"], "(x)." }], "Item3", CellChangeTimes->{{3.437140658815729*^9, 3.437140660006344*^9}, { 3.437141174977814*^9, 3.437141205186446*^9}, {3.437141302476346*^9, 3.437141327722988*^9}, {3.437141407583039*^9, 3.437141436586083*^9}, { 3.437205758596472*^9, 3.437205761521096*^9}}], Cell[TextData[{ "Using the equation for ", StyleBox["I", FontWeight->"Bold"], "(x), this gives an equation for ", StyleBox["M", FontWeight->"Bold"], "(x) only" }], "Item3", CellChangeTimes->{{3.437140658815729*^9, 3.437140660006344*^9}, { 3.437141174977814*^9, 3.437141205186446*^9}, {3.437141302476346*^9, 3.437141327722988*^9}, {3.437141407583039*^9, 3.437141436586083*^9}, { 3.43714186020885*^9, 3.437141890681083*^9}}], Cell[TextData[{ "Solve the above for ", StyleBox["M", FontWeight->"Bold"], "(x) " }], "Item3", CellChangeTimes->{{3.437140658815729*^9, 3.437140660006344*^9}, { 3.437141174977814*^9, 3.437141205186446*^9}, {3.437141302476346*^9, 3.437141327722988*^9}, {3.437141407583039*^9, 3.437141436586083*^9}, { 3.43714186020885*^9, 3.437141911894287*^9}}], Cell[TextData[{ "3.3) Consider positions far from the midline, i.e., x large. Explain how \ this morphogen gradient system is robust. 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