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$CellContext`phi$90089$$, 0], Hold[$CellContext`sd$$, $CellContext`sd$90090$$, 0], Hold[$CellContext`psi$$, $CellContext`psi$90091$$, 0], Hold[$CellContext`phi2$$, $CellContext`phi2$90092$$, 0], Hold[$CellContext`sd2$$, $CellContext`sd2$90093$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`restore, $CellContext`phi$$ = 0.73; $CellContext`sd$$ = 1.25; $CellContext`collisionDistance$$ = 30.; $CellContext`psi$$ = 1.25; $CellContext`phi2$$ = 0.51; $CellContext`sd2$$ = 0.7; $CellContext`cellNumber$$ = 30; $CellContext`totalSimulationTime$$ = 180; $CellContext`midlineHalfWidth = 4; $CellContext`midlineDelay = 40; $CellContext`trackplotDelay = 30]; $CellContext`restore = False; If[$CellContext`startsim, $CellContext`posvel = Table[{{ RandomReal[{-1, 1}], RandomReal[{(-($CellContext`embht + $CellContext`elon - 2 $CellContext`collisionDistance$$))/ 2, ($CellContext`embht + $CellContext`elon - 2 $CellContext`collisionDistance$$)/2}]}, {0., 0.}}, {$CellContext`cellNumber$$}]; $CellContext`walks = Table[{ Part[$CellContext`posvel, $CellContext`i, 1]}, {$CellContext`i, Length[$CellContext`posvel]}]; Do[Pause[ 0.05]; $CellContext`elapsedTime = $CellContext`t; \ $CellContext`posvel = Map[$CellContext`ranCollStep[#, $CellContext`phi$$, \ $CellContext`sd$$, $CellContext`collisionDistance$$, $CellContext`nearestNPos[#], $CellContext`phi2$$, \ $CellContext`sd2$$, $CellContext`psi$$, If[$CellContext`t < $CellContext`midlineDelay, \ {-$CellContext`midlineHalfWidth, $CellContext`midlineHalfWidth}, {1000, 1000}]]& , $CellContext`posvel]; If[$CellContext`t == $CellContext`trackplotDelay, \ $CellContext`walks = Table[{ Part[$CellContext`posvel, $CellContext`i, 1]}, {$CellContext`i, Length[$CellContext`posvel]}]]; If[$CellContext`t > $CellContext`trackplotDelay, Do[Part[$CellContext`walks, $CellContext`i] = Join[ Part[$CellContext`walks, $CellContext`i], { Part[$CellContext`posvel, $CellContext`i, 1]}], {$CellContext`i, Length[$CellContext`posvel]}]], {$CellContext`t, \ $CellContext`totalSimulationTime$$}]]; $CellContext`startsim = False; If[$CellContext`simulate, $CellContext`showP = True; $CellContext`startsim = True]; $CellContext`simulate = False; If[$CellContext`showP, Dynamic[ GraphicsRow[{ Graphics[{ Line[$CellContext`embo], Map[Point, Part[ Transpose[$CellContext`posvel], 1]]}, AspectRatio -> Automatic, PlotLabel -> StringJoin[ ToString[$CellContext`elapsedTime], " min"], Frame -> True, FrameTicks -> False, Axes -> False, PlotRange -> {{-$CellContext`x, $CellContext`x}, \ {-$CellContext`y, $CellContext`y}}], ListLinePlot[ Flatten[{{$CellContext`embo}, $CellContext`walks}, 1], AspectRatio -> Automatic, PlotLabel -> StringJoin[ ToString[$CellContext`cellNumber$$], " cells"], Frame -> True, FrameTicks -> False, Axes -> False, PlotRange -> {{-$CellContext`x, $CellContext`x}, \ {-$CellContext`y, $CellContext`y}}]}, AspectRatio -> 1.56, ImageSize -> Large]], GraphicsRow[{ Graphics[{ Line[$CellContext`embo], Map[Point, Part[ Transpose[$CellContext`posvel], 1]]}, AspectRatio -> Automatic, Frame -> True, FrameTicks -> False, Axes -> False, PlotRange -> {{-$CellContext`x, $CellContext`x}, \ {-$CellContext`y, $CellContext`y}}], ListLinePlot[ Flatten[{{$CellContext`embo}, $CellContext`walks}, 1], AspectRatio -> Automatic, Frame -> True, FrameTicks -> False, Axes -> False, PlotRange -> {{-$CellContext`x, $CellContext`x}, \ {-$CellContext`y, $CellContext`y}}], Blur[ Graphics[{Black, Rectangle[{-$CellContext`x, -$CellContext`y}, {$CellContext`x, \ $CellContext`y}], Thickness[0.06], White, Line[$CellContext`embo], Thickness[0.02], Opacity[0.04], Map[Circle[#, 4]& , Flatten[$CellContext`walks, 1]]}, PlotRange -> {{-$CellContext`x, $CellContext`x}, \ {-$CellContext`y, $CellContext`y}}], 8]}, AspectRatio -> 1.2, ImageSize -> Large]]), "Specifications" :> { Item[ Style["ENTER PARAMETERS:", FontFamily -> "Helvetica", Bold, 14, GrayLevel[0]]], Item[ Button[ " Restore defaults ", $CellContext`restore = True, ImageSize -> Automatic, BaseStyle -> {16}]], Item[ Style["", FontFamily -> "Helvetica", 14, GrayLevel[0]]], {{$CellContext`totalSimulationTime$$, 1, "Total simulation time"}, 10, 400, 1, Appearance -> "Labeled"}, {{$CellContext`cellNumber$$, 1, "Number of cells"}, 1, 60, 1, Appearance -> "Labeled"}, {{$CellContext`collisionDistance$$, 1, "Collision distance"}, 0, 50, Appearance -> "Labeled"}, Delimiter, Item[ Style["Free model:", FontFamily -> "Helvetica", Bold, 14, GrayLevel[0]]], {{$CellContext`phi$$, 1, "\!\(\*\nStyleBox[\"\[Phi]\",\nFontFamily->\"Comic Sans MS\",\n\ FontSize->18]\)"}, 0, 1, Appearance -> "Labeled"}, {{$CellContext`sd$$, 1, "\!\(\*\nStyleBox[\"\[Sigma]\",\nFontFamily->\"Comic Sans MS\",\n\ FontSize->18]\)"}, 0.01, 2, Appearance -> "Labeled"}, Delimiter, Item[ Style["Collision model:", FontFamily -> "Helvetica", Bold, 14, GrayLevel[0]]], {{$CellContext`psi$$, 1, "\!\(\*\nStyleBox[\"\[Psi]\",\nFontFamily->\"Comic Sans MS\",\n\ FontSize->18]\)"}, 0, 2, Appearance -> "Labeled"}, {{$CellContext`phi2$$, 1, "\!\(\*\nStyleBox[\"\[Phi]\",\nFontFamily->\"Comic Sans MS\",\n\ FontSize->18]\)"}, 0, 1, Appearance -> "Labeled"}, {{$CellContext`sd2$$, 1, "\!\(\*\nStyleBox[\"\[Sigma]\",\nFontFamily->\"Comic Sans MS\",\n\ FontSize->18]\)"}, 0.01, 2, Appearance -> "Labeled"}, Delimiter, Item[ Style["", FontFamily -> "Helvetica", 18, GrayLevel[0]]], Item[ Button[ " RUN SIMULATION ", $CellContext`simulate = True, ImageSize -> Automatic, Background -> Dynamic[ If[$CellContext`startsim, Blue, White]], BaseStyle -> {16, Bold, RGBColor[0, 0, 1]}]], Item[ Style["", FontFamily -> "Helvetica", 14, GrayLevel[0]]], Item[ Style[ "\nWait for simulation to finish to see domain map.", FontFamily -> "Helvetica", 16, GrayLevel[0]]], Item[ Button[ " Show domains ", $CellContext`showP = False, ImageSize -> Automatic, BaseStyle -> {16}]]}, "Options" :> {FrameLabel -> {None, None, Style[ "Drosophila Haemocyte Simulation", FontFamily -> "Helvetica", Bold, Italic, 20, GrayLevel[0]], None}}, "DefaultOptions" :> {}], ImageSizeCache->{1003., {280., 285.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`restore = False, $CellContext`phi$$ = 0.73, $CellContext`sd$$ = 1.25, $CellContext`collisionDistance$$ = 30., $CellContext`psi$$ = 1.25, $CellContext`phi2$$ = 0.51, $CellContext`sd2$$ = 0.7, $CellContext`cellNumber$$ = 30, $CellContext`totalSimulationTime$$ = 180, $CellContext`midlineHalfWidth = 4, $CellContext`midlineDelay = 40, $CellContext`trackplotDelay = 30, $CellContext`startsim = False, $CellContext`posvel = {{{0, 0}, {0., 0.}}}, $CellContext`embht = 160, $CellContext`elon = 160, $CellContext`walks = {{{0, 0}}}, $CellContext`elapsedTime = 180, $CellContext`ranCollStep[{ Pattern[$CellContext`pos, Blank[]], Pattern[$CellContext`vel, Blank[]]}, Pattern[$CellContext`phi, Blank[]], Pattern[$CellContext`sdn, Blank[]], Pattern[$CellContext`collisionDistance, Blank[]], Pattern[$CellContext`nnPos, Blank[]], Pattern[$CellContext`phi2, Blank[]], Pattern[$CellContext`sdn2, Blank[]], Pattern[$CellContext`psi, Blank[]], { Pattern[$CellContext`lowX, Blank[]], Pattern[$CellContext`highX, Blank[]]}] := Module[{$CellContext`vt, $CellContext`nnVec = $CellContext`pos - \ $CellContext`nnPos}, If[Norm[$CellContext`nnVec] > $CellContext`collisionDistance, \ $CellContext`vt = $CellContext`phi $CellContext`vel + \ $CellContext`noise2D[$CellContext`sdn], $CellContext`vt = $CellContext`phi2 \ $CellContext`vel + $CellContext`psi ($CellContext`nnVec/ Norm[$CellContext`nnVec]) + \ $CellContext`noise2D[$CellContext`sdn2]]; $CellContext`pt = $CellContext`pos + \ $CellContext`vt; If[Part[$CellContext`pt, 1] < $CellContext`lowX, Part[$CellContext`pt, 1] = 2 $CellContext`lowX - Part[$CellContext`pt, 1]; $CellContext`vt = {-1, 1} $CellContext`vt]; If[Part[$CellContext`pt, 1] > $CellContext`highX, Part[$CellContext`pt, 1] = 2 $CellContext`highX - Part[$CellContext`pt, 1]; $CellContext`vt = {-1, 1} $CellContext`vt]; {$CellContext`pt, $CellContext`vt}], \ $CellContext`noise2D[ Pattern[$CellContext`sd, Blank[]]] := { RandomReal[ NormalDistribution[0, $CellContext`sd]], RandomReal[ NormalDistribution[0, $CellContext`sd]]}, $CellContext`pt = { 2.0574318315213986`, -58.875948576364806`}, $CellContext`nearestNPos[ Pattern[$CellContext`pos, Blank[]]] := Part[ Nearest[ Join[$CellContext`embo, Part[ Transpose[$CellContext`posvel], 1]], $CellContext`pos, 2], 2], $CellContext`embo = {{55., 80.}, {54.32285873273258, 92.51475720321847}, {52.30810839623344, 104.7213595499958}, { 49.00535883036024, 116.31923997916374`}, {44.49593469062211, 127.02282018339785`}, {38.890872965260115`, 136.5685424949238}, { 32.328188876086024`, 144.72135954999578`}, {24.969477485675075`, 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CellChangeTimes->{{3.4747985747924*^9, 3.4747985788952*^9}, { 3.4749853961534953`*^9, 3.474985415853495*^9}, {3.4750180115014954`*^9, 3.4750180230654955`*^9}, {3.47730957859375*^9, 3.47730958328125*^9}, { 3.5276729560887556`*^9, 3.5276729600979557`*^9}}], Cell[TextData[{ StyleBox["This notebook simulates the formation of patterns in the haemocyte \ cell population occupying the ventral embryo space in the early ", FontFamily->"Comic Sans MS", FontSize->14], StyleBox["Drosophila", FontFamily->"Comic Sans MS", FontSize->14, FontSlant->"Italic"], StyleBox[" embryo. We first describe the model of haemocyte cell behaviour \ used in the simulations. The only interaction allowed between cells is the \ instantaneous response that they make on colliding with each other. The \ detailed parameters of this response can be estimated by analysing cell \ behaviour during real collisions, as described briefly below. \n\nThe model \ consists of two alternatives and one of these is applied to each cell in turn \ at each time step of 1 min. The free-moving model is used when there are no \ other cells within collision distance of the cell being modelled and the \ collision model is used when the nearest neighbouring cell is within the \ collision distance. In real haemocytes, the collision distance is remarkably \ constant at 30 ", FontFamily->"Comic Sans MS", FontSize->14], StyleBox["m", FontFamily->"Mathematica1", FontSize->14], StyleBox["m. In the simulation, the motile cells start off by being confined \ to a narrow region located at the midline of the embryo. After a \ predetermined time interval of 40 min, they are released to wander anywhere \ within the ventral embryo space. The boundary of this space is modelled as a \ chain of closely packed cells. These are stationary and do not take part in \ the simulation but they can act as neighbouring cells during the collision \ response. This arbitrary boundary condition is partially justifed by the \ observation that cells in the real embryo appear to be largely confined to \ the ventral space by contact interactions with other cells at the edge of the \ space.\n\nThe parameter values of the free movement and collision response \ can be changed using the sliders in the simulation cell above. They are \ initially set to their estimated values except for the \[Sigma] of the \ collision model (", FontFamily->"Comic Sans MS", FontSize->14], StyleBox["see below", FontFamily->"Comic Sans MS"], StyleBox["). After pressing the RUN SIMULATION button, the instantaneous \ cell positions are plotted on the left and the tracks on the right. The \ tracks are not plotted until after 30 min to avoid an overemphasis of cells \ in the midline region in the final plots. Press the Show Domains button \ after the simulation has finished and the domain map will appear in a few \ seconds.\n\nIf you need to reveal the ", FontFamily->"Comic Sans MS", FontSize->14], StyleBox["Mathematica", FontFamily->"Comic Sans MS", FontSize->14, FontSlant->"Italic"], StyleBox[" code, double click on the right bracket of the simulation cell.\n", FontFamily->"Comic Sans MS", FontSize->14] }], "Text", CellChangeTimes->{{3.4743795386289997`*^9, 3.4743795858658*^9}, { 3.474379630201*^9, 3.4743796647394*^9}, {3.4744081717384*^9, 3.4744081981647997`*^9}, {3.4744082540128*^9, 3.4744082598316*^9}, 3.4744083161788*^9, {3.4745523076232*^9, 3.4745523854842*^9}, { 3.4745524831191998`*^9, 3.4745525851851997`*^9}, {3.4745527863022003`*^9, 3.4745527893372*^9}, {3.4745529391521997`*^9, 3.4745530658742*^9}, { 3.4745531537202*^9, 3.4745533219941998`*^9}, {3.4745533580512*^9, 3.4745533720922003`*^9}, {3.4745535201022*^9, 3.4745537241862*^9}, { 3.4745538799872*^9, 3.4745539058062*^9}, 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"1"}]], TraditionalForm]]], StyleBox[" , ", FontSize->14], StyleBox["v", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox[") from colliding cells that had first made contact at time t and \ again rotated each pair so that ", FontSize->14], StyleBox["v", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{"t", "-", "1"}]], TraditionalForm]]], StyleBox[" points along the +x axis. \n\nWe then take the y-components of \ the rotated vectors so we can reduce Eq. 2 to the scalar equation v", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox[" = \[Psi]r", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox[" + \[Phi]v", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{"t", "-", "1"}]], TraditionalForm]]], StyleBox[" + \[Sigma]n", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["\nTaking the expectations of these quantities we get E[v", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] = \[Psi]E[r", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] + \[Phi]E[v", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{"t", "-", "1"}]], TraditionalForm]]], StyleBox["] + \[Sigma]E[n", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] which becomes E[v", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] = \[Psi]E[r", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] since two terms vanish. Now the expectations of r", FontSize->14], "y", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["", "t"], " "}], TraditionalForm]]], StyleBox["and hence of v", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox[" are zero if the target cells are distributed equally to the right \ and left of the approaching cells. However, assuming the cell behaviour is \ left/right symmetrical, we can bias the distribution by taking mirror images \ of all the tracks where the target cell is, say, on the left (so r", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox[" > 0 in all cases) . The estimate of \[Psi] then becomes \[Psi]", FontSize->14], Cell[BoxData[ FormBox[ SubscriptBox["", "estimate"], TraditionalForm]], FontSize->14], StyleBox[" = Mean[v", FontSize->14], StyleBox["y", FontSize->10], Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]], FontSize->14], StyleBox["] / Mean[r", FontSize->14], "y", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] .\n\nConsidering the x-components we have E[v", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] = \[Psi]E[r", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] + \[Phi]E[v", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{"t", "-", "1"}]], TraditionalForm]]], StyleBox["] + \[Sigma]E[n", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] which becomes E[v", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] = \[Psi]E[r", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] + \[Phi]E[v", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{"t", "-", "1"}]], TraditionalForm]]], StyleBox["]. \nUsing the estimate of \[Psi], the estimate of \[Phi] then \ becomes: \[Phi]", FontSize->14], Cell[BoxData[ FormBox[ SubscriptBox["", "estimate"], TraditionalForm]], FontSize->14], StyleBox[" = Mean[v", FontSize->14], StyleBox["x", FontSize->10], Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]], FontSize->14], StyleBox[" - \[Psi]r", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox["] / Mean[v", FontSize->14], "x", Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{"t", "-", "1"}]], TraditionalForm]]], StyleBox["] .\n\nFinally, to estimate \[Sigma], we plot the noise \[Sigma]", FontSize->14], StyleBox["n", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]], FontSize->14], StyleBox[" = ", FontSize->14], StyleBox["v", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]], FontSize->14], StyleBox[" - \[Psi]", FontSize->14], StyleBox["r", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["", "t"], TraditionalForm]]], StyleBox[" - \[Phi]", FontSize->14], StyleBox["v", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{"t", "-", "1"}]], TraditionalForm]], FontSize->14], " ", StyleBox[" using the rotated vectors and the estimated \[Phi] and \[Psi] and \ take standard deviations on both axes.\n\nThese estimates are used as the \ default values of the parameters in the simulation except that the estimate \ of \[Sigma] in the collision model is too high to account for the speed of \ the colliding cells in the real embryo. 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