(* 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[ 4158314, 73886] NotebookOptionsPosition[ 4139964, 73341] NotebookOutlinePosition[ 4140757, 73367] CellTagsIndexPosition[ 4140714, 73364] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Supplemental Material", "Title", CellChangeTimes->{{3.425479697057034*^9, 3.425479702739566*^9}, 3.4255909478973627`*^9, {3.42728603989261*^9, 3.427286042126256*^9}, { 3.427286130530089*^9, 3.42728613097538*^9}}], Cell["\<\ Lander et al., \"Cell Lineages and the Logic of Proliferative Control\"\ \>", "Subtitle", CellChangeTimes->{{3.427286044209029*^9, 3.4272860462783613`*^9}, { 3.427286084673959*^9, 3.42728609006289*^9}}], Cell["\<\ Click on the downward arrow to the right of each title to view the material \ in that subsection. \ \>", "Subsection", CellChangeTimes->{{3.4255908895093603`*^9, 3.4255909459821177`*^9}, 3.427286119088056*^9}], Cell["", "Subsection"], Cell[CellGroupData[{ Cell["1. ODE model of an unbranched lineage", "Section", CellChangeTimes->{3.425479718370573*^9, 3.425479764251577*^9}], Cell[TextData[{ "The system of equations in Figure 2b is derived from the principle that the \ rate of increase of each cell of type \"n\" has two components, creation by \ differentiation of a cell of type n-1, and self-replication. The rate of the \ former will be twice ", Cell[BoxData[ FormBox[ RowBox[{"1", "-", SubscriptBox["p", RowBox[{"n", "-", "1"}]]}], TraditionalForm]]], ", the factor of two coming from the fact that cell type n-1 produces two \ cells with every division. The rate of the latter will be twice ", Cell[BoxData[ FormBox[ SubscriptBox["p", "n"], TraditionalForm]]], " minus 1; the factor two again reflects the fact that cells produce two \ cells with each division, while the subtraction of 1 reflects the fact that \ whenever a cell divides to produce two new cells, one must deduct one for the \ parental cell that no longer exists. " }], "Text", CellChangeTimes->{{3.42548152911436*^9, 3.425481532263597*^9}, { 3.425481571760435*^9, 3.425481766431634*^9}, {3.427282450824007*^9, 3.4272824509992943`*^9}}], Cell["\<\ For cell type zero, the term representing production from a previous lineage \ stage is omitted. For the cell type at the end of the lineage, which does \ not divide, the term for replication is omitted. In addition, a \ probabilistic rate of death is added to the last equation, to capture the \ fact that the terminal cell often has a limited lifespan. Death of other \ cell types is not considered here, but could could easily be added to the \ equations. In addition, at death term that is age-structured, rather than \ probabilistic, could be used. \ \>", "Text", CellChangeTimes->{{3.425481767275463*^9, 3.4254819599668427`*^9}, { 3.425606991193791*^9, 3.42560699120253*^9}}], Cell[TextData[{ "In generating simulations of the dynamic behaviors of lineages, it is \ convenient to perform some rescaling and non-dimensionalize to reduce the \ numbers of free parameters. For example, it is useful to define a unit of \ time \[Tau] = t ", Cell[BoxData[ FormBox[ SubscriptBox["v", "1"], TraditionalForm]]], "; a parameter ", Cell[BoxData[ FormBox[ RowBox[{"\[Zeta]", "=", RowBox[{ SubscriptBox["v", "0"], "/", SubscriptBox["v", "1"]}]}], TraditionalForm]]], "is used to eliminate ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["v", "0"], ";"}], TraditionalForm]]], "and a parameter ", Cell[BoxData[ FormBox[ RowBox[{"\[Delta]", "=", RowBox[{"d", "/", SubscriptBox["v", "1"]}]}], TraditionalForm]]], "is also defined. Here is how this works out for a three stage lineage. 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The remaining parameters are \ therefore ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "0"], " "}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{",", " ", SubscriptBox["p", "1"]}], TraditionalForm]]], ", \[Delta] and \[Zeta]. " }], "Text", CellChangeTimes->{{3.4256075801027393`*^9, 3.4256076925373287`*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["2. Steady state solution in the absence of feedback", "Section", CellChangeTimes->{3.425479721401559*^9, 3.4254797656360693`*^9}], Cell["\<\ The system in Fig. 2 b may be solved in the steady state by setting all time \ rates to zero. \ \>", "Text", CellChangeTimes->{{3.425480549355412*^9, 3.4254805977698183`*^9}, 3.425480709007923*^9}], Cell[TextData[{ "From the first equation one gets", Cell[BoxData[ RowBox[{" ", RowBox[{ SubscriptBox["\[Chi]", "0"], "\[Equal]", " ", RowBox[{"0", " ", "or", " ", SubscriptBox["v", "0"]}], "\[Equal]", RowBox[{"0", " ", "or", " ", SubscriptBox["p", "0"]}], "\[Equal]", " ", "0.5"}]}]]], ". The only solution of interest is ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "0"], "\[Equal]", " ", "0.5"}], TraditionalForm]]], ", which in turn implies that ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "0"], TraditionalForm]]], " is undetermined, i.e. arbitrary" }], "Text", CellChangeTimes->{{3.4254806235233603`*^9, 3.4254806349151917`*^9}, { 3.42548073100957*^9, 3.425480766004757*^9}}], Cell["From the last equation one gets", "Text"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Chi]", "n"], "\[Equal]", " ", RowBox[{"-", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["p", "2"]}], ")"}], SubscriptBox["v", "2"], " ", SubscriptBox["\[Chi]", RowBox[{ RowBox[{"-", "1"}], "+", "n"}]]}], RowBox[{"d", " "}]]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.425480615572262*^9, 3.425480621084631*^9}, { 3.425481293565551*^9, 3.4254812968054037`*^9}, {3.425481452947507*^9, 3.425481453295795*^9}}], Cell["From every other equation one gets", "Text", CellChangeTimes->{{3.425480777655949*^9, 3.4254807786929092`*^9}}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["\[ForAll]", FontSize->24], RowBox[{ RowBox[{"j", ">", "0"}], ",", " ", RowBox[{"j", "<", "n"}]}]], RowBox[{",", " ", RowBox[{ SubscriptBox["\[Chi]", RowBox[{"n", "-", "j"}]], "\[Equal]", " ", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["p", RowBox[{"n", "-", "j", "-", "1"}]]}], ")"}], " ", SubscriptBox["v", RowBox[{"n", "-", "j", "-", "1"}]], " ", SubscriptBox["\[Chi]", RowBox[{"n", "-", "j", "-", "1"}]]}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", SubscriptBox["p", RowBox[{"n", "-", "j"}]]}]}], ")"}], " ", SubscriptBox["v", RowBox[{"n", "-", "j"}]]}]]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.4254807985204763`*^9, 3.425480824240429*^9}, { 3.425480877931445*^9, 3.425480892321661*^9}, {3.4254809261940393`*^9, 3.4254809430265493`*^9}, {3.425480998005335*^9, 3.425481000451683*^9}, { 3.425481044474998*^9, 3.4254810827256327`*^9}, {3.425481145849717*^9, 3.425481160031323*^9}, {3.425481191108838*^9, 3.425481248795467*^9}, { 3.425481301350863*^9, 3.4254813044308968`*^9}, {3.425481456296916*^9, 3.4254814725527773`*^9}}], Cell["Putting these together implies", "Text", CellChangeTimes->{{3.425480908202463*^9, 3.425480912096175*^9}}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Chi]", "n"], "\[Equal]", " ", RowBox[{ FractionBox[ RowBox[{ SubscriptBox["\[Chi]", "0"], SubscriptBox["v", "0"]}], "d"], RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], RowBox[{"n", "-", "1"}]], FractionBox[ RowBox[{"2", RowBox[{"(", RowBox[{"1", "-", SubscriptBox["p", "i"]}], ")"}]}], RowBox[{"1", "-", RowBox[{"2", SubscriptBox["p", "i"]}]}]]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.4254812850980797`*^9, 3.425481350662312*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 3. Steady state solution for a two stage lineage with feedback\ \>", "Section", CellChangeTimes->{3.425479723997006*^9, 3.425479766859907*^9}], Cell["\<\ In this study, feedback is represented by multiplying p- and v- parameters by \ Hill functions of the form\ \>", "Text", CellChangeTimes->{{3.425481977497712*^9, 3.425482013479851*^9}}], Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{"1", "+", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"a", " ", RowBox[{ SubscriptBox["\[Chi]", "final"], "[", "t", "]"}]}], ")"}], "n"]}]], TraditionalForm]], "Input", CellChangeTimes->{{3.425482015501872*^9, 3.4254820719969587`*^9}}], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Chi]", "final"], "[", "t", "]"}], TraditionalForm]]], " represents the amount of terminal stage cells, and a is a parameter. 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Final - state solutions in the absence of feedback (Fig. S4-S5)\ \>", "Section", CellChangeTimes->{ 3.4254797303915358`*^9, 3.425479770220248*^9, {3.432116346030552*^9, 3.432116350750441*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Consider the time dependent solution for a two stage system, with initial \ conditions of ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "0"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", SubscriptBox["\[Chi]", "init"]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "1"], TraditionalForm]]], "= 0, and a constant ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], "<0.5 and no death of the terminal cell" }], "Subsubsubsection", CellChangeTimes->{{3.4255015277312737`*^9, 3.4255015981936617`*^9}, { 3.4255018269069357`*^9, 3.425501827722369*^9}, {3.4255023799079514`*^9, 3.4255024016899233`*^9}, {3.425502473098034*^9, 3.425502484694429*^9}, { 3.425520169497595*^9, 3.4255201937300243`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"system1", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "0"], "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", SubscriptBox["p", "0"]}]}], ")"}], " ", SubscriptBox["v", "0"], " ", RowBox[{ SubscriptBox["\[Chi]", "0"], "[", "t", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "1"], "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "-", SubscriptBox["p", "0"]}], ")"}], " ", SubscriptBox["v", "0"], " ", RowBox[{ SubscriptBox["\[Chi]", "0"], "[", "t", "]"}]}]}], ",", " ", RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "0"], "[", "0", "]"}], "\[Equal]", SubscriptBox["\[Chi]", "init"]}], ",", " ", RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "1"], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.425501574798518*^9, 3.425501644858136*^9}, { 3.4255016789031754`*^9, 3.425501701802525*^9}, {3.425501830894424*^9, 3.42550185203789*^9}, 3.425502488058565*^9, {3.4255201988510036`*^9, 3.425520200953397*^9}}], Cell[TextData[{ "To simplify things, let's define a time scale \[Tau] = ", Cell[BoxData[ FormBox[ RowBox[{"t", " ", SubscriptBox["v", "0"]}], TraditionalForm]]], ". then \[DifferentialD]t = ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["1", SubscriptBox["v", "0"]], RowBox[{"\[DifferentialD]", "\[Tau]"}]}], TraditionalForm]]], ". 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As we can see from the expression for output, this factor is 2 \ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["p", "0"], "-", "1"}], ")"}], "/", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", SubscriptBox["p", "0"]}], "-", "1"}], ")"}], "."}]}], TraditionalForm]]], " Thus, we may express ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], "in terms of a" }], "Text", CellChangeTimes->{{3.4255204612328176`*^9, 3.425520539337603*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{"a", "\[Equal]", " ", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["p", "0"]}], ")"}]}], RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", SubscriptBox["p", "0"]}]}]]}], ",", " ", SubscriptBox["p", "0"]}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.4255202395201883`*^9, 3.425520311925973*^9}, { 3.4255203464481173`*^9, 3.425520365312331*^9}, {3.42552051292855*^9, 3.425520570578401*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ SubscriptBox["p", "0"], "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "2"}], "+", "a"}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "a"}], ")"}]}]]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.4255205712899942`*^9, 3.4320812644084473`*^9}] }, Open ]], Cell[TextData[{ "Thus for a 1000 fold amplification, ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], "needs to be ", Cell[BoxData[ FormBox[ FractionBox["998", RowBox[{"2", RowBox[{"(", "999", ")"}]}]], TraditionalForm]]], " or 0.4995" }], "Text", CellChangeTimes->{{3.425520734508807*^9, 3.425520792003584*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[Sigma]", "/.", RowBox[{ SubscriptBox["p", "0"], "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "2"}], "+", "a"}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "a"}], ")"}]}]]}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.425520583831656*^9, 3.425520595029516*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", "3"}], "+", FractionBox["2", "a"], "+", "a"}]], "Output", CellChangeTimes->{3.425520596121376*^9, 3.432081264491062*^9}] }, Open ]], Cell[TextData[{ "From this we can immediately see that, for values of the amplification \ factor above 10, the sensitivity of output to ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], "is approximately equal to the amplification factor itself (i.e. enormous!)" }], "Text", CellChangeTimes->{{3.425520660157552*^9, 3.425520717529894*^9}, 3.4255208001471148`*^9, {3.427283181125637*^9, 3.427283185189638*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Now let's consider the case where ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], "is not a constant, but undergoes a linear decline over time, from a \ starting value of pmax to an ending value of pmin, at time tmax. 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pmax \ has to be at least 10, no matter the choice of \[Tau]max.\ \>", "Text", CellChangeTimes->{{3.425523032182521*^9, 3.4255230749100533`*^9}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 6. Final - state solutions in the presence of feedback (Fig. S6-S11)\ \>", "Section", CellChangeTimes->{ 3.4254797349470377`*^9, 3.4254797726041813`*^9, {3.432116365862484*^9, 3.432116371310493*^9}}], Cell[TextData[{ "Now let's put in feedback of the same sort we utilized in modeling steady \ state behaviors. We notice right away that, since it was possible to scale \ the units of time to ", Cell[BoxData[ FormBox[ SubscriptBox["v", "0"], TraditionalForm]]], ", and since we are only interested in the limiting behavior of the output \ at infinite time, then feedback onto ", Cell[BoxData[ FormBox[ SubscriptBox["v", "0"], TraditionalForm]]], " can have no effect (i.e. the influence of ", Cell[BoxData[ FormBox[ SubscriptBox["v", "0"], TraditionalForm]]], " on the output is only on the time scale at which output develops, but it \ cannot change the final value). Thus, we only consider feedback on ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], ". 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However, we \ can determine its long-term behavior as follows: Since the first equation \ gives us ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[DifferentialD]", SubscriptBox["x", "0"]}], RowBox[{"\[DifferentialD]", "\[Tau]"}]], TraditionalForm]]], "and the second one gives us ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[DifferentialD]", SubscriptBox["x", "1"]}], RowBox[{"\[DifferentialD]", "\[Tau]"}]], TraditionalForm]]], ", then we may divide to get ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[DifferentialD]", SubscriptBox["x", "0"]}], RowBox[{"\[DifferentialD]", SubscriptBox["\[Chi]", "1"]}]], TraditionalForm]]], ". 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As for boundary conditions, we know that when \ ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Chi]", "0"], "=", SubscriptBox["\[Chi]", "init"]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "1"], TraditionalForm]]], "=", Cell[BoxData[ FormBox["0", TraditionalForm]]], ". For reasons that are not important, ", StyleBox["Mathematica", FontSlant->"Italic"], " doesn't like having subscripts in the argument for ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Chi]", "0"], "["}], TraditionalForm]]], Cell[BoxData[ SubscriptBox["\[Chi]", "1"]], CellChangeTimes->{{3.425523612725152*^9, 3.425523624114394*^9}, 3.425523722696892*^9, {3.425523787153825*^9, 3.425523877884141*^9}}], "], so we will replace ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "1"], TraditionalForm]]], "with y wherever we see it, and then substitute ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "1"], TraditionalForm]]], " back in after solving." }], "Text", CellChangeTimes->{{3.425523891781335*^9, 3.425523985030217*^9}, { 3.4255241202492037`*^9, 3.4255241980709877`*^9}, {3.425647379935829*^9, 3.425647389284032*^9}, {3.427283414257416*^9, 3.427283416497384*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "0"], "'"}], "[", "y", "]"}], "\[Equal]", " ", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", FractionBox[ RowBox[{"2", " ", SubscriptBox["p", "0"]}], RowBox[{"1", "+", RowBox[{"g", " ", "y"}]}]]}], RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", FractionBox[ SubscriptBox["p", "0"], RowBox[{"1", "+", RowBox[{"g", " ", "y"}]}]]}], ")"}]}]]}], ",", " ", RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "0"], "[", "0", "]"}], "\[Equal]", SubscriptBox["\[Chi]", "init"]}]}], "}"}], ",", " ", RowBox[{ SubscriptBox["\[Chi]", "0"], "[", "y", "]"}], ",", " ", "y"}], "]"}], "/.", RowBox[{"y", "\[Rule]", " ", SubscriptBox["\[Chi]", "1"]}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.4255238988603373`*^9, 3.425523916349043*^9}, { 3.425523991263852*^9, 3.425524038736261*^9}, {3.425524090421318*^9, 3.425524101217677*^9}, {3.4255242021304703`*^9, 3.425524214345186*^9}, { 3.425647548469668*^9, 3.425647551301751*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "0"], "[", SubscriptBox["\[Chi]", "1"], "]"}], "\[Rule]", FractionBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", RowBox[{"Log", "[", RowBox[{"1", "-", SubscriptBox["p", "0"]}], "]"}]}], "+", RowBox[{"Log", "[", RowBox[{"1", "-", SubscriptBox["p", "0"], "+", RowBox[{"g", " ", SubscriptBox["\[Chi]", "1"]}]}], "]"}]}], ")"}], " ", SubscriptBox["p", "0"]}], "-", RowBox[{"g", " ", RowBox[{"(", RowBox[{ SubscriptBox["\[Chi]", "1"], "-", RowBox[{"2", " ", SubscriptBox["\[Chi]", "init"]}]}], ")"}]}]}], RowBox[{"2", " ", "g"}]]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.425524039941106*^9, 3.4255241021027193`*^9, 3.425524215140633*^9, 3.425647557290415*^9, 3.432081383264574*^9}] }, Open ]], Cell["\<\ As g \[NotEqual]0, we may write this more compactly as \ \>", "Text", CellChangeTimes->{{3.425524231272745*^9, 3.425524241583125*^9}, { 3.425647586382728*^9, 3.4256476061220207`*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox["\[Chi]", "0"], "\[Equal]", " ", RowBox[{ SubscriptBox["\[Chi]", "init"], "-", FractionBox[ SubscriptBox["\[Chi]", "1"], RowBox[{"2", " "}]], "+", FractionBox[ RowBox[{ SubscriptBox["p", "0"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", FractionBox[ RowBox[{"g", " ", SubscriptBox["\[Chi]", "1"]}], RowBox[{"1", "-", SubscriptBox["p", "0"]}]]}], "]"}], " "}], RowBox[{"2", " ", "g"}]]}]}]], "Text", CellChangeTimes->{{3.425524281428522*^9, 3.42552441429609*^9}, { 3.425524445670734*^9, 3.425524461541452*^9}, {3.425524508114249*^9, 3.425524509647641*^9}, {3.425647563070219*^9, 3.4256475792325487`*^9}}], Cell[TextData[{ "Now in the limit of infinite time, we know from inspection of the initial \ system of equations that, at \[Tau]=\[Infinity], and only then, ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "0"], TraditionalForm]]], " goes to zero, and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "1"], TraditionalForm]]], "goes to its final value, which we may call ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", RowBox[{"1", "final"}]], TraditionalForm]]], ". We can therefore conclude:" }], "Text", CellChangeTimes->{{3.42552453520368*^9, 3.425524570924337*^9}, { 3.4255246131106043`*^9, 3.4255246250679617`*^9}, {3.4255248936284122`*^9, 3.42552489958806*^9}, {3.425530626850747*^9, 3.425530628446821*^9}, { 3.425555880445486*^9, 3.425555933434923*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"0", "\[Equal]", " ", RowBox[{ RowBox[{"2", "g", " ", SubscriptBox["\[Chi]", "init"]}], "-", RowBox[{"g", " ", SubscriptBox["\[Chi]", RowBox[{"1", "final"}]]}], "+", RowBox[{ SubscriptBox["p", "0"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", FractionBox[ RowBox[{"g", " ", SubscriptBox["\[Chi]", RowBox[{"1", "final"}]]}], RowBox[{"1", "-", SubscriptBox["p", "0"]}]]}], "]"}]}]}]}], " ", ";"}]], "Input", CellChangeTimes->{{3.425524638119953*^9, 3.425524648094376*^9}, { 3.42552490260603*^9, 3.425524918030015*^9}, 3.4255283556043177`*^9, { 3.425647612857381*^9, 3.4256476171689577`*^9}}], Cell[TextData[{ "This gives us an implicit relationship between a and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "init"], TraditionalForm]]], ". To see this graphically, we first non-dimensionalize the cell number to \ g. 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The system would appear to contain a very \ slowly cycling stem cell (it might even be called \"quiescent\" and would \ certainly be found to be \"label-retaining\", i.e. after one round of \ division it would be a very long time before such a cell underwent a \ subsequent round), which only becomes highly proliferative in response to \ tissue injury. The experimentalist might also observe that the numbers of \ such stem cells gradually decline over the lifetime of the organism. All of \ these observations are ones that have frequently been made for tissue stem \ cells in various contexts. Here they arise simply as a result of feedback \ interactions. The cell that displays them has no innate programming to do \ so. In the absence of feedback, the proliferative and differentiative \ behaviors of cell type \"0\" would, in fact, be very similar to those of cell \ type \"1\". \ \>", "Text", CellChangeTimes->{{3.432060294177552*^9, 3.432060524822974*^9}, { 3.432080588726124*^9, 3.43208058960881*^9}, {3.432119314320735*^9, 3.432119341609376*^9}, {3.4321193822046423`*^9, 3.432119390284778*^9}, { 3.432119450524967*^9, 3.4321197600485697`*^9}, {3.432119793036145*^9, 3.432119834405302*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 8. Parameter space exploration\[LongDash]methods \ \>", "Section", CellChangeTimes->{ 3.425479742493956*^9, 3.425479774180208*^9, {3.425590868595755*^9, 3.425590869583065*^9}, 3.425601831830388*^9, 3.432041413772271*^9}], Cell["\<\ To explore the dynamic behavior of different feedback models, the following \ steps are followed\ \>", "Text", CellChangeTimes->{{3.425599204981554*^9, 3.425599241899632*^9}, { 3.425599896601904*^9, 3.425599896857374*^9}}], Cell["", "Text", CellChangeTimes->{{3.425599242302971*^9, 3.4255992423066397`*^9}}], Cell["\<\ 1. The names of the parameters of the system are given in a list named \ \"params\"\ \>", "Text", CellChangeTimes->{{3.425599243167573*^9, 3.4255992738761*^9}, { 3.425599333649282*^9, 3.4255993620734653`*^9}, {3.425599895089662*^9, 3.425599895273093*^9}, {3.425606627884493*^9, 3.425606627886273*^9}, { 3.427284044166567*^9, 3.427284059506209*^9}}], Cell[TextData[{ "2. If the system has no feedback on ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], "then the only non-trivial steady state is one in which ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["p", "0"], " ", "=", " ", "0.5"}], ",", " ", RowBox[{"and", " ", SubscriptBox["\[Chi]", "0"]}]}], TraditionalForm]]], " is arbitrary. In this case, the variables ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "2"], TraditionalForm]]], " are normalized to ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "0"], TraditionalForm]]], "and renamed ", Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["c", "2"], TraditionalForm]]], " respectively. In addition, the parameters g and h are multiplied by ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "0"], TraditionalForm]]], "to give the new parameters \[Gamma] and \[Eta], respectively. By all this \ normalization, ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "0"], TraditionalForm]]], "may be eliminated entirely from the system of equations. " }], "Text", CellChangeTimes->{{3.4256066296329393`*^9, 3.425606765408374*^9}, { 3.425606826264574*^9, 3.4256068316187153`*^9}, {3.425607891828588*^9, 3.425607956707546*^9}}], Cell[TextData[{ "3. If the steady state solution can be solved for directly, it is named \ \"solset\". If the variables were ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "0"], "[", "\[Tau]", "]"}], ",", " ", RowBox[{ RowBox[{ SubscriptBox["\[Chi]", "1"], "[", "\[Tau]", "]"}], " ", "and", " ", RowBox[{ SubscriptBox["\[Chi]", "2"], "[", "\[Tau]", "]"}]}]}], TraditionalForm]]], ", their steady state versions are named \[Chi]ss0, \[Chi]ss1 and \[Chi]ss2. \ If the variables were ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["c", "1"], "[", "\[Tau]", "]"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["c", "2"], "[", "\[Tau]", "]"}], TraditionalForm]]], "their steady state versions are named css1 and css2. [N.B. if there is \ more than one steady state, only one is chosen here; parameter values that \ are inconsistent with positive solutions for that steady state will get \ identified during the run and saved in a separate file; these may be re-run \ later, using a different one of the steady state solutions in the code]" }], "Text", CellChangeTimes->{{3.425599243167573*^9, 3.4255992738761*^9}, { 3.425599333649282*^9, 3.425599340185358*^9}, {3.425599899809857*^9, 3.4255998999532843`*^9}, {3.425600480129361*^9, 3.425600496070857*^9}, { 3.425606755854907*^9, 3.425606768013111*^9}, {3.425606845038052*^9, 3.425606933736079*^9}, {3.427284704197708*^9, 3.427284839298177*^9}}], Cell["\<\ If it cannot be solved directly in any reasonably compact form, the system of \ equations that determines the steady state is named \"sssystem\" and left \ unsolved.\ \>", "Text", CellChangeTimes->{{3.425599243167573*^9, 3.4255992738761*^9}, { 3.425599333649282*^9, 3.425599340185358*^9}, {3.425599899809857*^9, 3.4255998999532843`*^9}, {3.425600480129361*^9, 3.425600496070857*^9}, { 3.425606755854907*^9, 3.425606760308401*^9}, {3.427284224206514*^9, 3.4272842399344397`*^9}, {3.427284327332985*^9, 3.427284329780674*^9}}], Cell[TextData[{ "4. If the steady state solution was solved for directly, the set of ODEs \ representing the model is transformed with new variables ", Cell[BoxData[ FormBox[ SubscriptBox["z", "0"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["z", "1"], TraditionalForm]]], "and ", Cell[BoxData[ FormBox[ SubscriptBox["z", "2"], TraditionalForm]]], " representing ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "2"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Chi]", "3"], TraditionalForm]]], ", or ", Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]]], "and ", Cell[BoxData[ FormBox[ SubscriptBox["c", "2"], TraditionalForm]]], ", normalized to their steady state values. This is given the name \ \"system\". If the steady state could not be solved for directly, the set of \ ODEs is left in its original form and given the name \"tempsystem\"" }], "Text", CellChangeTimes->{{3.4255992751855097`*^9, 3.4255993353537188`*^9}, { 3.4255998898846817`*^9, 3.425599989966618*^9}, 3.42560048971205*^9, 3.425600609878973*^9, 3.425606770057391*^9, {3.4256077339000072`*^9, 3.425607748351263*^9}, {3.427284340402327*^9, 3.427284340679262*^9}, { 3.427284412740747*^9, 3.427284417636496*^9}}], Cell["\<\ 5. Definitions and parameter ranges are then entered as below: Initconds \ refer to intial conditions where different fractions of different cell types \ are eliminated. 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So this is some improvement on the case \ with feedback only on P1. 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Define a coordinate system in which ", StyleBox["x", FontSlant->"Italic"], "=0 represents the basal lamina, and ", StyleBox["x", FontSlant->"Italic"], "=-xmin represents the apical surface. x>0 then represents the stroma \ underlying the epithelium. Assume that at the apical surface there are tight \ junctions, so that diffusing molecules may not leave, whereas at the basal \ lamina there is no barrier to molecular diffusion. \n\nIf a secreted, \ diffusive molecule is made uniformly throughout the epithelium at constant \ rate ", StyleBox["v", FontSlant->"Italic"], ", we can calculate its steady state concentration along the apicobasal \ axis. To do this we let ", StyleBox["d", FontSlant->"Italic"], " stand for the effective diffusion coefficient of the molecule, and ", StyleBox["k", FontSlant->"Italic"], " stand for its degradation rate constant. Because ", StyleBox["k", FontSlant->"Italic"], " may be different in the epithelium versus the stroma, we use ", Cell[BoxData[ FormBox[ SubscriptBox["k", "L"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["k", "R"], TraditionalForm]]], " to represent epithelial and stromal degradation rate constants, \ respectively, on the \"left\" (epithelium) and \"right\" (stroma). " }], "Text", CellChangeTimes->{{3.4251275645412617`*^9, 3.4251278589593163`*^9}, { 3.4251280393862343`*^9, 3.425128256904828*^9}, 3.425128291545712*^9, 3.427285508341812*^9, {3.427285542451522*^9, 3.427285553339785*^9}}], Cell["\<\ We may use a single ODE to represent the steady state solution for this \ situation, in which the variable a[x] is the concentration of the factor in \ space. 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Let's also define \[Rho] to be the ratio ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Lambda]", "L"], "/", SubscriptBox["\[Lambda]", "R"]}], TraditionalForm]]], ". 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