(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 204077, 4516] NotebookOptionsPosition[ 202346, 4454] NotebookOutlinePosition[ 202755, 4472] CellTagsIndexPosition[ 202712, 4469] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Scaffold protein titration motif", "Title", CellChangeTimes->{{3.6430002271357613`*^9, 3.6430002342704477`*^9}}], Cell[CellGroupData[{ Cell["The model description", "Section", CellChangeTimes->{{3.643000294588601*^9, 3.643000299851523*^9}}], Cell[TextData[{ "This particular motif describe one phosphorylation-desphosphorylation cycle \ (can be generalized to any futile cycles) with both kinase ", Cell[BoxData[ FormBox[ RowBox[{"(", "K"}], TraditionalForm]]], ") and phosphatase ", Cell[BoxData[ FormBox[ RowBox[{"(", "P"}], TraditionalForm]]], ") can be titrated by a scaffold protein ", Cell[BoxData[ FormBox[ RowBox[{"(", "T"}], TraditionalForm]]], ").\n\n", StyleBox["K + S \[Equilibrium] KS \[RightArrow] K + ", "DisplayFormula"], StyleBox[Cell[BoxData[ FormBox[ SubscriptBox["S", "p"], TraditionalForm]], "DisplayFormula"], "DisplayFormula"], StyleBox["\nP + ", "DisplayFormula"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", "p"], " ", "\[Equilibrium]", " ", SubscriptBox["PS", "p"]}], TraditionalForm]], "DisplayFormula"], "DisplayFormula"], StyleBox[" \[RightArrow] P + S\nT + K \[Equilibrium] TK\nT + P \ \[Equilibrium] TP\n\[EmptySet] \[RightArrow]", "DisplayFormula"], " K\nK \[RightArrow] \[EmptySet] \n\nThe above reactions show a simple \ system that composed of one scaffold protein, one kinase, one phosphatase and \ one substrate. Here we try to descibe this simple system with differential \ equation following the mass action kinetics.\n\n", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ FractionBox[ RowBox[{"\[DifferentialD]", RowBox[{"[", "K", "]"}]}], RowBox[{"\[DifferentialD]", "t"}]], "=", RowBox[{ RowBox[{"-", RowBox[{ RowBox[{ RowBox[{"k", "[", "1", "]"}], "[", "K", "]"}], "[", "S", "]"}]}], "+", RowBox[{ RowBox[{"k", "[", "2", "]"}], "[", "KS", "]"}], "+", RowBox[{ RowBox[{"k", "[", "3", "]"}], "[", "KS", "]"}], "-", RowBox[{ RowBox[{ RowBox[{"k", "[", "7", "]"}], "[", "T", "]"}], "[", "K", "]"}], "+", RowBox[{ RowBox[{ RowBox[{"k", "[", "8", "]"}], "[", "TK", "]"}], RowBox[{ StyleBox[ RowBox[{"+", " ", RowBox[{"k", "[", "11", "]"}]}], "DisplayFormula"], SubscriptBox[ StyleBox["k", "DisplayFormula"], "d"]}]}], "-", RowBox[{ SubscriptBox[ StyleBox["k", "DisplayFormula"], "d"], "[", "K", "]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ FractionBox[ RowBox[{"\[DifferentialD]", RowBox[{"[", "P", "]"}]}], 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FractionBox[ RowBox[{"\[DifferentialD]", RowBox[{"[", "TP", "]"}]}], RowBox[{"\[DifferentialD]", "t"}]], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"k", "[", "9", "]"}], "[", "T", "]"}], "[", "P", "]"}], "-", RowBox[{ RowBox[{ RowBox[{"k", "[", "10", "]"}], "[", "TP", "]"}], "."}]}]}]}], "DisplayFormula"], TraditionalForm]]], "\n\nAnd the system need to follow these conservation equations:\n\n", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"[", "K", "]"}], "+", RowBox[{"[", "KS", "]"}], "+", RowBox[{"[", "TK", "]"}]}], "=", RowBox[{"[", SubscriptBox["K", "tot"], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"[", "P", "]"}], "+", RowBox[{"[", SubscriptBox["PS", "p"], "]"}], "+", RowBox[{"[", "TP", "]"}]}], "=", RowBox[{"[", SubscriptBox["P", "tot"], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"[", "S", "]"}], "+", RowBox[{"[", SubscriptBox["S", "p"], "]"}], "+", RowBox[{"[", "KS", "]"}], "+", RowBox[{"[", SubscriptBox["PS", "p"], 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Here we try to study them numerically. By defining two \ different way to characterising the dynamics with scoring their tempral \ dynamics when presented with input signal perturbation (the changing of [T]). \ The quantification can be derived from the actually fitness funcitons for \ ultrasensitive response and adaptive response. 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