(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 21723, 550]*) (*NotebookOutlinePosition[ 22413, 574]*) (* CellTagsIndexPosition[ 22369, 570]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox[" 2 Insoluble plus 1 Soluble species Equilibrium", FontFamily->"Arial", FontSize->14], "\n\nThis program checks expected equilibrium concentrations for a two \ insoluble (a,b) plus one common soluble (c) species undergoing simultaneous \ equilibration. Subscripts 1 refer to the a/c rate constanats and 2 to the \ b/c constants. Parameters such as az (azero) represent initial \ concentrations. ", StyleBox["The red parameters are things that one might wish to change.", FontColor->RGBColor[1, 0, 0]], " Note that NMinimize produces 3 results (minimum value of function, then \ first and second variable values). The last 2 of these are the desired \ values of ac and bc. This explains the 2 evaluate statements that follow \ NMinimize. \n\nSee the program 2 solubles plus 2 insolubles for comments \ about the need for extra constraints to permit proper convergence. \ Initially, it seemed that they were not needed in this simple case, but \ certain combinations of constants during evaluation produced problems that \ were solved only by adding con1a, con2a. Ironically, it seems more difficult \ to converge on a correct value of an equilibrium this way than it is by \ solving the more difficult differential equations." }], "Input", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ StyleBox[\(az\ = \ 1\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[\(bz\ = \ 1\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[\(cz\ = \ 1\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[\(kfr1 = 1\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[\(krr1 = .01\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[\(kfr2 = 1\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[\(krr2 = 0.1\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], RowBox[{"subs3", " ", "=", " ", RowBox[{"Dispatch", "[", RowBox[{"{", " ", RowBox[{ StyleBox[\(azero\ -> \ az\), FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[\(bzero\ -> bz\), FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[\(czero\ -> cz\), FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], \(kf1\ -> \ kfr1\), ",", " ", \(kr1\ -> \ krr1\), ",", \(kf2\ -> kfr2\), ",", \(kr2\ -> \ krr2\)}], StyleBox["}", FontColor->GrayLevel[0]]}], "]"}]}], ";"}], "\[IndentingNewLine]", \(subs\ = \ Dispatch[{a\ -> \ azero\ - \ ac, \ b\ -> \ bzero\ - \ bc, c\ \[Rule] czero - ac - bc}]\ ;\), "\n", \(eq3\ = \ Abs[a*c*kf1 - ac*kr1] + Abs[b*c*kf2 - bc*kr2];\), "\n", \(eq4 = \ eq3\ /. subs;\), "\n", \(eq5\ = \ eq4\ /. subs3;\), "\n", \(con1\ = \((a*c*kf1)\) == \((ac* kr1)\)\ /. \ subs;\), "\[IndentingNewLine]", \(con1a\ = \ con1\ /. \ subs3;\), "\[IndentingNewLine]", \(con2\ = \ \((b*c* kf2)\) == \((bc*kr2)\)\ /. \ subs;\), "\[IndentingNewLine]", \(con2a\ = \ con2\ /. \ subs3;\), "\[IndentingNewLine]", \(x\ = \ NMinimize[{eq5\ , ac + bc \[LessEqual] cz, az - ac \[GreaterEqual] 0, bz - bc \[GreaterEqual] 0, \ ac \[GreaterEqual] 0.00001, bc \[GreaterEqual] 0.00001, con1a, con2a}, {ac, bc}];\), "\[IndentingNewLine]", \(y = Evaluate[ ac] /. \(Flatten[x]\)[\([2]\)];\), "\[IndentingNewLine]", \(z = Evaluate[ bc] /. \(Flatten[ x]\)[\([3]\)];\), "\[IndentingNewLine]", \(Print["\< ac , \ bc , a , b , c\>"]\ ;\), "\[IndentingNewLine]", \(Print[ N[{y, z, az - y, bz - z, cz - y - z}]];\), "\[IndentingNewLine]", \(Print[];\), "\ \[IndentingNewLine]", \(Print["\"];\), "\[IndentingNewLine]", \(Print[ y/\((\((az - y)\)*\((cz - y - z)\))\), "\<, \>", \ z/\((\((bz - z)\)*\((cz - y - z)\))\)]; Print["\", N[\ \((az - y)\)*\((cz - y - z)\)*kfr1 - y*krr1], "\<, \>", N[\((bz - z)\)*\((cz - y - z)\)*kfr2\ - \ z*krr2]]\)}], "Input"], Cell[BoxData[ \(" ac , bc , a , b , c"\)], "Print"], Cell[BoxData[ \({0.7448477409783294`, 0.22595997357822406`, 0.2551522590216706`, 0.7740400264217759`, 0.029192285443446542`}\)], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \("for checking: re-calculated equilibrium consts. are:"\)], "Print"], Cell[BoxData[ InterpretationBox[\(99.99999775635328`\[InvisibleSpace]", "\ \[InvisibleSpace]9.999999983123125`\), SequenceForm[ 99.999997756353281, ", ", 9.999999983123125], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Residuals \ are:"\[InvisibleSpace]1.671175220777088`*^-10\[InvisibleSpace]", "\ \[InvisibleSpace]3.813498128391046`*^-11\), SequenceForm[ "Residuals are:", 1.671175220777088*^-10, ", ", 3.8134981283910463*^-11], Editable->False]], "Print"] }, Open ]], Cell[BoxData[""], "Input"], Cell[BoxData[{ \(\(Correct\ equilibrium\ constants\ recovered;\)\), \ "\[IndentingNewLine]", \(Residual\ results\ are\ very\ small; \ Apparently\ this\ calculation\ process\ worked\ just\ \ \(\(fine\)\(.\)\)\)}], "Input"], Cell[BoxData[ \(Here\ is\ a\ simple\ statement\ to\ show\ what\ happens\ in\ \ \(\(Evaluate\)\(:\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(zz\ = \ Evaluate[yy] /. \((yy \[Rule] 1)\); Print\ [zz]\)], "Input"], Cell[BoxData[ \(1\)], "Print"] }, Open ]], Cell[TextData[{ StyleBox[" \n\n\n\n\n 2 soluble \ plus 2 immobilized species\n (with interference from \ opposite species)", FontFamily->"Arial", FontSize->14], "\n\nThis program finds expected equilibrium concentrations for a two \ soluble (c1,c2) plus two common immobilized (a,b) species undergoing \ simultaneous equilibration. Subscripts 1 refer to the a/c1 rate constants, \ 2 to the b/c1 constants, 3 to the a/c2 constants and 4 to the b/c2 \ constants. Parameters such as c1z (c1zero) represent initial concentrations. \ ", StyleBox["The red parameters are things that one might wish to change.", FontColor->RGBColor[1, 0, 0]], " \n\nc1*a*kf1 = kr1*c1a kf1 = kf4; kr1 = kr4\nc1*b*kf2 = kr2*c1b \ kf2 = kf3; kr2 = kr3\nc2*a*kf3 = kr3*c2a\nc2*b*kf4 = kr4*c2b\n\nIn this \ version with the example constants shown, c1 is assumed to bind strongly to \ a and weakly to b. The opposite is true of c2 (it binds strongly to b and \ weakly to a). The variables ending with z represent initial concentration \ values. The values kfr represent forward rate constants and krr represent \ reverse rate constants. Two new constraints are needed: c1a=c2b and c2a=c1b. \ This provides reciprocity of binding interactions." }], "Input", Evaluatable->False, TextAlignment->Left], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ StyleBox[\(c1z\ = \ 1; c2z\ = \ 1; caz\ = \ 1; cbz\ = \ 1;\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n", StyleBox[\(kfr1 = 1; krr1 = 0.01; kfr2 = 1; krr2 = 0.1;\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", RowBox[{ RowBox[{"subs3", " ", "=", " ", RowBox[{"Dispatch", "[", RowBox[{"{", " ", RowBox[{ StyleBox[\(c1zero\ -> \ c1z\), FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[\(c2zero\ -> c2z\), FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[\(cazero\ -> caz\), FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], \(cbzero = cbz; kf1\ -> \ kfr1\), ",", " ", \(kr1\ -> \ krr1\), ",", \(kf2\ -> kfr2\), ",", \(kr2\ -> \ krr2\)}], StyleBox["}", FontColor->GrayLevel[0]]}], "]"}]}], ";"}], "\n", \(subs\ = \ Dispatch[{c1\ -> \ c1zero\ - \ c1a - c1b, \ c2\ -> \ c2zero\ - \ c2a - c2b, ca \[Rule] cazero - c1a - c2a, \ cb\ \[Rule] cbzero - c1b - c2b}]\ ;\), "\n", \(eq3\ = \((\ Abs[\((c1*ca*kf1)\) - \((c1a*kr1)\)] + Abs[\((c2*cb*kf1)\) - \((c2b* kr1)\)] + \[IndentingNewLine]Abs[\((c1*cb* kf2)\) - \((c1b*kr2)\)] + Abs[\((c2*ca*kf2)\) - \((c2a*kr2)\)])\);\), "\n", \(eq4 = \ eq3\ /. subs;\), "\n", \(eq5\ = \ eq4\ /. subs3;\), "\n", \(con1\ = \ \((c1*ca*kf1)\) == \((c1a*kr1)\) /. subs;\), "\[IndentingNewLine]", \(con1a\ = \ con1\ /. subs3;\), "\[IndentingNewLine]", \(con2\ = \ \((c2*cb*kf1)\) == \((c2b*kr1)\) /. subs;\), "\[IndentingNewLine]", \(con2a\ = \ con2 /. subs3;\), "\[IndentingNewLine]", \(con3\ = \ \((c1*cb*kf2)\) == \((c1b*kr2)\) /. subs;\), "\[IndentingNewLine]", \(con3a\ = \ con3 /. subs3;\), "\[IndentingNewLine]", \(con4\ = \ \((c2*ca*kf2)\) == \((c2a*kr2)\) /. subs;\), "\[IndentingNewLine]", \(con4a = con4 /. subs3;\), "\[IndentingNewLine]", \(x\ = \ NMinimize[{eq5\ , c1a + c1b \[LessEqual] c1z, c2a + c2b \[LessEqual] c2z, c1a + c2a \[LessEqual] caz, \ c1b + c2b \[LessEqual] cbz, c1a \[GreaterEqual] 0, c2a \[GreaterEqual] 0, c1b \[GreaterEqual] 0, c2b \[GreaterEqual] 0, con1a, con2a, con3a, con4a}, {c1a, c2a, c2b, c1b, }];\), "\n", \(y = Evaluate[c1a] /. \(Flatten[x]\)[\([2]\)];\), "\n", \(z = Evaluate[c1b] /. \(Flatten[x]\)[\([3]\)];\), "\n", \(w = Evaluate[c2a] /. \(Flatten[x]\)[\([4]\)];\), "\n", \(v = Evaluate[c2b] /. \(Flatten[x]\)[\([5]\)];\), "\n", \(Print[];\), "\n", \(Print["\< c1a , c2b , c1b , c2a\>"];\), "\n", \(Print[N[{y, v, z, w}]];\), "\n", \(Print[];\), "\n", \(Print["\< c1 , c2 , ca , cb \>"]\ ;\), "\n", \(Print[ N[{c1z - y - z, c2z - w - v, \ caz - y - w, cbz - z - v}]]; Print[];\), "\n", \(Print["\"];\), "\n", \(Print[ y/\((\((caz - y - w)\)*\((c1z - y - z)\))\), "\<, \>", \ v/\((\((c2z - w - v)\)*\((cbz - v - z)\))\)];\), "\n", \(Print[];\), "\n", \(Print["\"];\), "\n", \(Print[w/\((\((caz - y - w)\)*\((c2z - w - v)\))\), "\<, \>", z/\((\((c1z - y - z)\)*\((cbz - v - z)\))\)];\), "\[IndentingNewLine]", \(Print[];\), "\[IndentingNewLine]", \(Print["\", \(Flatten[ x]\)[\([1]\)]];\)}]}]], "Input"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \(" c1a , c2b , c1b , c2a"\)], "Print"], Cell[BoxData[ \({0.8264462454910839`, 0.8264462455430414`, 0.08264465044630054`, 0.08264465093831584`}\)], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \(" c1 , c2 , ca , cb "\)], "Print"], Cell[BoxData[ \({0.09090910406261554`, 0.09090910351864279`, 0.09090910357060024`, 0.0909091040106581`}\)], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \("for checking: re-calculated equilibrium consts. are (for \ c1a,c2b):"\)], "Print"], Cell[BoxData[ InterpretationBox[\(99.99996730789107`\[InvisibleSpace]", "\ \[InvisibleSpace]99.99996742848424`\), SequenceForm[ 99.999967307891069, ", ", 99.99996742848424], Editable->False]], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \("for checking: re-calculated equilibrium consts. are (for \ c2a,c1b):"\)], "Print"], Cell[BoxData[ InterpretationBox[\(9.999999983719295`\[InvisibleSpace]", "\ \[InvisibleSpace]9.99999981594211`\), SequenceForm[ 9.9999999837192952, ", ", 9.9999998159421093], Editable->False]], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Residual is \ "\[InvisibleSpace]5.559258652379495`*^-9\), SequenceForm[ "Residual is ", 5.5592586523794951*^-09], Editable->False]], "Print"] }, Open ]], Cell[BoxData[ \(\ \)], "Input"], Cell[BoxData[ \(The\ following\ shows\ what\ the\ individual\ residues\ \ \(\(are\)\(:\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(Print[ N[Abs[\((c1z - y - z)\)*\((caz - y - w)\)*kfr1 - y*krr1]]]; \ Print[N[Abs[\((c2z - w - v)\)*\((cbz - z - v)\)*kfr2 - v*krr2]]]; Print[N[Abs[\((c1z - y - z)\)*\((cbz - z - v)\)*kfr1 - z*krr1]]]; \ Print[N[Abs[\((c2z - w - v)\)*\((caz - y - w)\)*kfr2 - w*krr2]]]\)\)\)], "Input"], Cell[BoxData[ \(2.4223045291815737`*^-8\)], "Print"], Cell[BoxData[ \(7.899087806073624`*^-8\)], "Print"], Cell[BoxData[ \(3.207024090734478`*^-10\)], "Print"], Cell[BoxData[ \(2.3414756522566194`*^-8\)], "Print"] }, Open ]], Cell["\<\ For kr1,kr4 reverse rate constants = 0.01, the above solution did \ not converge at all for c1z = 0.01, c2z = 0.001 (table 3, line5). It did not \ quite converge for c1z=1, c2z=0.1 (table 3, line 2). For kr1, kr4 = 0.001, the above solution did not converge at all for c1z=1, \ c2z=0.1 or for c1z=0.1, c2z=0.01 (table 3, lines 7 and 9). It did not quite \ converge for c1z=1, c2z=0.001 (table 3, line 11). For these cases, the differential equation solution will be checked below by \ substituting in the original equilibrium relationships for t = 10,000. The \ numerical results inserted below were obtained from the previous numerical \ solution of the differential equation. Appropriate initial concentrations \ were used. The expected equilibrium constants were obtained in all cases. \ \>", "Text", Evaluatable->False, FontFamily->"Arial", FontWeight->"Bold"], Cell[BoxData[ \(\(\(table\ 3\ line\ 2\)\(\ \)\(:\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(c2b\ = \ 0.09527215079821393`;\)\), "\[IndentingNewLine]", \(\(c1b\ = \ 0.2146742440729666`;\)\), "\[IndentingNewLine]", \(\(c2a\ = 0.003347200653758734`;\)\), "\[IndentingNewLine]", \(\(c1a\ = \ 0.7542159635170813`;\)\), "\[IndentingNewLine]", \(c1\ = \ 1 - c1a - c1b; \ c2\ = \ 0.1 - c2a - c2b;\), "\[IndentingNewLine]", \(ca = 1 - c1a - c2a; \ cb = 1 - c1b - c2b;\), "\[IndentingNewLine]", \(Print[{c1a/\((c1*ca)\), \ c2a/\((c2*ca)\), \ c1b/\((c1*cb)\), \[IndentingNewLine]c2b/\((c2* cb)\)}]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \({100.00000000000013`, 10.000000000000005`, 10.000000000000012`, 100.00000000000006`}\)], "Print"] }, Open ]], Cell[BoxData[ \(Above\ result\ checks\ exactly . \[IndentingNewLine]table\ 3\ line\ \ 5\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(c2b\ = \ 0.0009014787814633593`;\)\), "\[IndentingNewLine]", \(\(c1b\ = \ 0.0009068473246019454`;\)\), "\[IndentingNewLine]", \(\(c2a\ = 0.00008949009951371099`;\)\), "\[IndentingNewLine]", \(\(c1a\ = \ 0.00900230365828851`;\)\), "\[IndentingNewLine]", \(c1\ = \ 0.01 - c1a - c1b; \ c2\ = 0.001 - c2a - c2b;\), "\[IndentingNewLine]", \(ca = 1 - c1a - c2a; \ cb = 1 - c1b - c2b;\), "\[IndentingNewLine]", \(Print[{c1a/\((c1*ca)\), \ c2a/\((c2*ca)\), \ c1b/\((c1*cb)\), \[IndentingNewLine]c2b/\((c2*cb)\)}]\)}], "Input"], Cell[BoxData[ \({99.99999999999943`, 10.000000000000048`, 9.999999999999943`, 100.00000000000047`}\)], "Print"] }, Open ]], Cell[BoxData[ \(Above\ result\ checks\ exactly . \[IndentingNewLine]table\ 3\ line\ \ 7\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(c2b\ = \ 0.0997668871085963`;\)\), "\[IndentingNewLine]", \(\(c1b\ = \ 0.0814707985849157`;\)\), "\[IndentingNewLine]", \(\(c2a\ = 0.00011126203754130437`;\)\), "\[IndentingNewLine]", \(\(c1a\ = \ 0.908578719185832`;\)\), "\[IndentingNewLine]", \(c1\ = \ 1 - c1a - c1b; \ c2\ = \ 0.1 - c2a - c2b;\), "\[IndentingNewLine]", \(ca = 1 - c1a - c2a; \ cb = 1 - c1b - c2b;\), "\[IndentingNewLine]", \(Print[{c1a/\((c1*ca)\), \ c2a/\((c2*ca)\), \ c1b/\((c1*cb)\), \[IndentingNewLine]c2b/\((c2*cb)\)}]\)}], "Input"], Cell[BoxData[ \({999.9999999959425`, 10.000000000005679`, 10.000000000004134`, 999.9999999999261`}\)], "Print"] }, Open ]], Cell[BoxData[ \(Above\ result\ checks\ exactly . \[IndentingNewLine]table\ 3\ line\ \ 9\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(c2b\ = \ 0.009899792086356158`;\)\), "\[IndentingNewLine]", \(\(c1b\ = \ 0.0010844541576428477`;\)\), "\[IndentingNewLine]", \(\(c2a\ = 0.00009019817209143551`;\)\), "\[IndentingNewLine]", \(\(c1a\ = \ 0.09880589600578737`;\)\), "\[IndentingNewLine]", \(c1\ = \ 0.1 - c1a - c1b; \ c2\ = \ 0.01 - c2a - c2b;\), "\[IndentingNewLine]", \(ca = 1 - c1a - c2a; \ cb = 1 - c1b - c2b;\), "\[IndentingNewLine]", \(Print[{c1a/\((c1*ca)\), \ c2a/\((c2*ca)\), \ c1b/\((c1*cb)\), \[IndentingNewLine]c2b/\((c2*cb)\)}]\)}], "Input"], Cell[BoxData[ \({999.9999999999851`, 10.000000000000233`, 9.999999999999849`, 1000.0000000000234`}\)], "Print"] }, Open ]], Cell[BoxData[ \(Above\ result\ checks\ exactly . \[IndentingNewLine]table\ 3\ line\ \ 11\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(c2b\ = \ 0.0009978605370999458`;\)\), "\[IndentingNewLine]", \(\(c1b\ = \ 0.08625016963476856`;\)\), "\[IndentingNewLine]", \(\(c2a\ = 1.0462189848558453`*^-6;\)\), "\[IndentingNewLine]", \(\(c1a\ = \ 0.9043003662735779`;\)\), "\[IndentingNewLine]", \(c1\ = \ 1 - c1a - c1b; \ c2\ = \ 0.001 - c2a - c2b;\), "\[IndentingNewLine]", \(ca = 1 - c1a - c2a; \ cb = 1 - c1b - c2b;\), "\[IndentingNewLine]", \(Print[{c1a/\((c1*ca)\), \ c2a/\((c2*ca)\), \ c1b/\((c1*cb)\), \[IndentingNewLine]c2b/\((c2*cb)\)}]\)}], "Input"], Cell[BoxData[ \({1000.0000000000952`, 10.000000000000279`, 9.999999999999877`, 1000.0000000000427`}\)], "Print"] }, Open ]], Cell[BoxData[ \(Above\ result\ checks\ \(\(exactly\)\(.\)\)\)], "Input"] }, FrontEndVersion->"5.0 for Macintosh", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowSize->{578, 620}, WindowMargins->{{Automatic, 1}, {25, Automatic}}, PrintingCopies->1, PrintingPageRange->{1, Automatic} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 1330, 22, 360, "Input", Evaluatable->False], Cell[CellGroupData[{ Cell[3109, 77, 3741, 86, 427, "Input"], Cell[6853, 165, 77, 1, 23, "Print"], Cell[6933, 168, 148, 2, 23, "Print"], Cell[7084, 172, 45, 1, 23, "Print"], Cell[7132, 175, 87, 1, 23, "Print"], Cell[7222, 178, 217, 4, 23, "Print"], Cell[7442, 184, 308, 7, 25, "Print"] }, Open ]], Cell[7765, 194, 26, 0, 27, "Input"], Cell[7794, 196, 235, 5, 59, "Input"], Cell[8032, 203, 117, 2, 27, "Input"], Cell[CellGroupData[{ Cell[8174, 209, 89, 1, 27, "Input"], Cell[8266, 212, 34, 1, 23, "Print"] }, Open ]], Cell[8315, 216, 1369, 24, 483, "Input", Evaluatable->False], Cell[CellGroupData[{ Cell[9709, 244, 4461, 84, 811, "Input"], Cell[14173, 330, 45, 1, 23, "Print"], Cell[14221, 333, 71, 1, 23, "Print"], Cell[14295, 336, 126, 2, 23, "Print"], Cell[14424, 340, 45, 1, 23, "Print"], Cell[14472, 343, 76, 1, 23, "Print"], Cell[14551, 346, 127, 2, 23, "Print"], Cell[14681, 350, 45, 1, 23, "Print"], Cell[14729, 353, 103, 2, 23, "Print"], Cell[14835, 357, 217, 4, 23, "Print"], Cell[15055, 363, 45, 1, 23, "Print"], Cell[15103, 366, 103, 2, 23, "Print"], Cell[15209, 370, 217, 4, 23, "Print"], Cell[15429, 376, 45, 1, 23, "Print"], Cell[15477, 379, 194, 4, 25, "Print"] }, Open ]], Cell[15686, 386, 35, 1, 27, "Input"], Cell[15724, 389, 107, 2, 27, "Input"], Cell[CellGroupData[{ Cell[15856, 395, 348, 6, 75, "Input"], Cell[16207, 403, 56, 1, 25, "Print"], Cell[16266, 406, 55, 1, 25, "Print"], Cell[16324, 409, 56, 1, 25, "Print"], Cell[16383, 412, 56, 1, 25, "Print"] }, Open ]], Cell[16454, 416, 895, 18, 266, "Text", Evaluatable->False], Cell[17352, 436, 69, 1, 27, "Input"], Cell[CellGroupData[{ Cell[17446, 441, 629, 11, 155, "Input"], Cell[18078, 454, 124, 2, 23, "Print"] }, Open ]], Cell[18217, 459, 104, 2, 43, "Input"], Cell[CellGroupData[{ Cell[18346, 465, 591, 9, 139, "Input"], Cell[18940, 476, 122, 2, 23, "Print"] }, Open ]], Cell[19077, 481, 104, 2, 43, "Input"], Cell[CellGroupData[{ Cell[19206, 487, 580, 9, 139, "Input"], Cell[19789, 498, 122, 2, 23, "Print"] }, Open ]], Cell[19926, 503, 104, 2, 43, "Input"], Cell[CellGroupData[{ Cell[20055, 509, 590, 9, 139, "Input"], Cell[20648, 520, 122, 2, 23, "Print"] }, Open ]], Cell[20785, 525, 105, 2, 43, "Input"], Cell[CellGroupData[{ Cell[20915, 531, 587, 9, 139, "Input"], Cell[21505, 542, 123, 2, 23, "Print"] }, Open ]], Cell[21643, 547, 76, 1, 27, "Input"] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)