(************** Content-type: application/mathematica ************** 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). 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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[ 121228, 4567]*) (*NotebookOutlinePosition[ 122172, 4596]*) (* CellTagsIndexPosition[ 122128, 4592]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["SALVADOR 1.0 Manual", "Title", FontSize->16], Cell["\<\ Qi Zheng National Center for Toxicological Research 3900 NCTR Road, Jefferson, Arkansas 72079 November, 2001\ \>", "Text"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "SALVADOR is a package intended as a tool for exploring on the computer \ various aspects of fluctuation analysis. For reasons of computational \ efficiency, the package was written in a mixture of the ", StyleBox["Mathematica", FontSlant->"Italic"], " and the C language. Therefore, before using SALVADOR, two programs must \ be install from within ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Install["\"]\)], "Input"], Cell[BoxData[ \(LinkObject["'./salvador'", 2, 2]\)], "Output"] }, Open ]], Cell[BoxData[ \(<< salvador.m\)], "Input"], Cell["\<\ For notation, nomenclature and other background information, the user can \ consult the 2 references listed at the end of this mannual and numerous \ additional references contained therein. The sole purpose of this mannual is \ to demonstrate the various functionalities of SALVADOR.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Simulation", "Section", FontSize->16], Cell[TextData[{ "Five basic parameters must be specified before you simulate the number of \ mutants in a sister culture. These five parameters are: 1) mutation rate (per \ cell per unit time) \[Mu]; 2) cellular birth rate for nonmutants ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\_1\)]], "; 3) cellular birth rate for mutants ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\_2\)]], "; 4) initial population size ", Cell[BoxData[ \(TraditionalForm\`N\_0\)]], "; and 5) final population size ", Cell[BoxData[ \(TraditionalForm\`N\_T\)]], ". It is strongly recommended that, before running a simulation, you check \ the parameters of interest by computing the mean and variance of the number \ of mutants, for unrealistic parameters can lead to very time-consuming \ computational tasks and useless results. We now use the following five \ parameters to simulate a fluctuation experiment ." }], "Text"], Cell[BoxData[ \(mu = 10^\(-8\); b1 = 1.2; b2 = 1.1; N0 = 100; Nt = 2.4*10^8;\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MeanOfMutants\)\)], "Input"], Cell[BoxData[ \("MeanOfMutants[\!\(\[Mu],\[Beta]\_1,\[Beta]\_2,N\_0,N\_T\)] computes \ the mean number of mutants on each sister culture."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MeanOfMutants[mu, b1, b2, N0, Nt]\)], "Input"], Cell[BoxData[ \(16.94451339681247`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?VarOfMutants\)\)], "Input"], Cell[BoxData[ \("VarOfMutants[\!\(\[Mu],\[Beta]\_1,\[Beta]\_2,N\_0,N\_T\)] computes the \ variance of the number of mutants on each sister culture."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(VarOfMutants[mu, b1, b2, N0, Nt]\)], "Input"], Cell[BoxData[ \(995576.079641768`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?simuMutants\)\)], "Input"], Cell[BoxData[ \("simuMutants[\!\(\[Mu],\[Beta]\_1,\[Beta]\_2, N\_0,N\_T\)] simulates \ the number of mutants in a sister culture. Note \!\(\[Mu]\) is the mutation \ rate per cell per unit time; \!\(\[Beta]\_1\) is growth rate for nonmutants; \ \!\(\[Beta]\_2\) is growth rate of mutants; \!\(N\_0\) is the initial \ population size; \!\(N\_T\) is the population size prior to plating."\)], \ "Print"] }, Open ]], Cell["\<\ Now we simulate the numbers of mutants in 20 sister cultures.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[simuMutants[mu, b1, b2, N0, Nt], {20}]\)], "Input"], Cell[BoxData[ \({7, 8, 3, 3, 3, 11, 953, 1, 7, 12, 1, 4, 14, 4, 0, 14, 17, 0, 1, 0}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Important Distributions", "Section", FontSize->16], Cell[TextData[{ "SALVADOR can compute the probability mass function (p.m.f.) of four most \ important distributions in fluctuation analysis: the Poisson distribution, \ the LD or Luria-Delbruck distribution, the M or Mandelbrot distribution, and \ the convolution of a Poisson and an LD distribution. The last distribution \ was popularized in fluctuation analysis by Cairns et al. in 1988. Using \ pmf2cdf, you can transform a list of p.m.f. to a list of c.d.f. (cumulative \ distribution function). As a demonstration we first compute ", Cell[BoxData[ \(TraditionalForm\`p\_1\)]], ", ..., ", Cell[BoxData[ \(TraditionalForm\`p\_10\)]], " of a Poisson distribution having mean 10.5." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfPoisson\)\)], "Input"], Cell[BoxData[ \("pmfPoisson[\!\(\[Lambda]\),k] returns a list of probabilities \ \!\({p\_0,p\_1,...,p\_k}\) according to a Poisson(\!\(\[Lambda]\)) \ distribution."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfPoisson[10.5, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.00028913271817234516`, 0.001517946770404812`, 0.005312813696416842`, 0.013946135953094211`, 0.029286885501497845`, 0.051252049627621234`, 0.07687807444143184`, 0.1009024727043793`, 0.11771955148844251`, 0.12360552906286464`}\)], "Output"] }, Open ]], Cell["\<\ Now we plot the c.d.f. of a Luria-Delbruck distribution, so its upper tail \ behavior can be more easily examined.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfLD\)\)], "Input"], Cell[BoxData[ \("pmfLD[\!\(m,\[Phi],k\)] returns a list of probabilities \ \!\({p\_0,p\_1,..., p\_k}\) according to a Luria-Delbr\!\(\[UDoubleDot]\)uck \ distribution, LD(m,\!\(\[Phi]\)). Note pmfLD[m,k] is the same as \ pmfLD[m,1,k]."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LDprob = pmfLD[5.7, 150]; \ LDprob[\([Range[10]]\)]\)], "Input"], Cell[BoxData[ \({0.003345965457471272`, 0.009536001553793125`, 0.016767469398752913`, 0.02355789717184977`, 0.02910035040826871`, 0.03312787122784504`, 0.03569563178469149`, 0.03701109059727024`, 0.037328993778887536`, 0.036895826091020184`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LDcdf = pmf2cdf[LDprob]; \ LDcdf[\([Range[10]]\)]\)], "Input"], Cell[BoxData[ \({0.003345965457471272`, 0.012881967011264397`, 0.02964943641001731`, 0.05320733358186708`, 0.08230768399013579`, 0.11543555521798082`, 0.1511311870026723`, 0.18814227759994254`, 0.22547127137883008`, 0.26236709746985026`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ListPlot[Transpose[{Range[0, 150], LDcdf}], PlotJoined \[Rule] True, Frame \[Rule] True]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: 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Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Now let's look at both an approximate form and the exact form of an M \ distribution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfM\)\)], "Input"], Cell[BoxData[ \("pmfM[m,\!\(\[Rho]\),k] returns a list of probabilities \ \!\({p\_0,p\_1,...,p\_k}\) according to an \!\(M(m,\[Rho],1)\) distribution; \ pmfM[m,\!\(\[Rho],\[Phi]\),n] returns the same list according to an \!\(M(m,\ \[Rho],\[Phi])\) distribution. Note m is the mean number of mutations, \!\(\ \[Rho]=\[Beta]\_2/\[Beta]\_1\) is the ratio of mutant growth rate to \ nonmutant growth rate, and \!\(\[Phi]=1-N\_0/N\_T\)."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfM[10.5, 0.8, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.0001606292878735251`, 0.0005179264858998918`, 0.0012225438030489674`, 0.0023627289040455586`, 0.003967248143763744`, 0.0060008211628999825`, 0.008375511483062982`, 0.010970388387258776`, 0.013652416628655299`, 0.01629398760301078`}\)], "Output"] }, Open ]], Cell["\<\ If we assume \[Phi]=1, we can also use an algorithm devised by Koch (based \ on an algorithm of Lea and Coulson) to compute the p.m.f. of an M \ distribution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfMKoch\)\)], "Input"], Cell[BoxData[ \("pmfMKoch[m,\!\(\[Rho]\),k] computes \!\({p\_0,p\_1,...,p\_k}\) for an \ \!\(M(m,\[Rho],1)\) distribution, using Koch's algorithm. (This function is \ not recommended for large k.)"\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfMKoch[10.5, 0.8, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.00016062928787352512`, 0.0005179264858998919`, 0.0012225438030489674`, 0.0023627289040455586`, 0.0039672481437637444`, 0.006000821162899983`, 0.008375511483062982`, 0.010970388387258778`, 0.0136524166286553`, 0.01629398760301078`}\)], "Output"] }, Open ]], Cell["\<\ To use the exact form of the M distribution, the parameter \[Phi]<1 must be \ specified.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pmfM[10.5, 0.8, 0.999, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.00016078943783464673`, 0.0005189099952341755`, 0.001225866545990518`, 0.0023709130278721626`, 0.003983692654278592`, 0.006029436207766869`, 0.008420237658990444`, 0.011034728595860068`, 0.013739071794843216`, 0.0164046389202681`}\)], "Output"] }, Open ]], Cell["\<\ Finally let us take a look at the convolution of a Poisson distribution and \ an LD distribution .\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfCairns\)\)], "Input"], Cell[BoxData[ \("pmfCairns[\!\(m,\[Phi],\[Lambda],k\)] returns a list of probabilities \ \!\({p\_0,...,p\_k}\) according to the convolution of an LD(\!\(m,\[Phi]\)) \ distribution and a Poisson(\!\(\[Lambda]\)) distribution."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfCairns[2.5, 1, 8.5, 10]\)], "Input"], Cell[BoxData[ \({0.00001670170079024566`, 0.00016284158270489518`, 0.0008008117576822997`, 0.002651351506439024`, 0.006657036789692021`, 0.013540891094019243`, 0.0232837558246216`, 0.03488343821483244`, 0.04659431114081543`, 0.05652302198442524`, 0.06324871361500849`}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Estimation of Mutation Rates", "Section", FontSize->16], Cell[TextData[{ "SALVADOR concentrates on estimation of the mean number of mutations \ denoted by m, because an estimate of the mutation rate ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\_\[Beta]\)]], " can be easily obtained from that of m. The recommended estimation method \ is the maximum likelihood method. We use the experimental data of Demerec to \ illustrate the capabilities of SALVADOR. The following are the known \ experimental parameters of Demerec's experiment." }], "Text"], Cell[BoxData[ \(N0 = 90; Nt = 1.9*10^8; phi = 1 - N0/Nt;\)], "Input"], Cell["The experimental data are as follows.", "Text"], Cell[BoxData[ \(\(demerec = {33, 18, \ 839, 47, 13, 126, 48, 80, 9, 71, 196, 66, 28, 17, 27, 37, 126, 33, 12, 44, 28, 67, 730, 168, 44, 50, 583, 23, 17, 24};\)\)], "Input"], Cell["\<\ We can fit an LD(m,1) distribution to the data by using the maximum \ likelihood method.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?mleLD\)\)], "Input"], Cell[BoxData[ \("mleLD[data,\!\(\[Phi]:=1\),M:=50] searches in the interval [0.001,M] \ for the maximum likelihood estimate of m, the mean number of mutations, when \ data is assumed to be generated by an LD(m,\!\(\[Phi]\)) distribution. Note \ \!\(\[Phi]\) is considered as known."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mleLD[demerec]\)], "Input"], Cell[BoxData[ \(10.843827674048674`\)], "Output"] }, Open ]], Cell["\<\ It is also possible to fit an M(m,\[Rho]) distribution to this data set. \ Because two parameters are involve in the model, it takes considerably more \ time to compute the maximum likelihood estimates for both m and \[Rho].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?mleM\)\)], "Input"], Cell[BoxData[ \("mleM[data,M:50,R:5] searches for the maximum likelihood estimates of \ both m and \!\(\[Rho]\) by assuming that the data is generated by an M(m,\!\(\ \[Rho]\),1) distribution. The search is confined to \ \!\(m\[Element](0.001,M)\) and \!\(\[Rho]\[Element](0.01,R)\)."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mleM[demerec]\)], "Input"], Cell[BoxData[ \({9.853283763777942`, 1.118769340608164`}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Fisher's Information", "Section", FontSize->16], Cell["\<\ Fisher's information for m is needed in constructing confidence intervals for \ mutation rates. Two methods of computing Fisher's information are included \ in SALVADOR. For mathematical details about these two methods, the user is \ referred to Zheng (2001).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(est = mleLD[demerec]\)], "Input"], Cell[BoxData[ \(10.843827674048674`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?FisherInfoLD\)\)], "Input"], Cell[BoxData[ \("FisherInfoLD[m,\!\(\[Phi]\),k] computes the Fisher information for m \ according to an LD(m,\!\(\[Phi]\)) distribution; it retains k terms in the \ series \!\(I(m)=\[Sum] f\^2\%j/p\_j\)."\)], "Print"] }, Open ]], Cell["\<\ We can choose two different values of k to help determine how many terms are \ needed to get a sufficiently accurate approximation of I(m).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({FisherInfoLD[est, 1, 1000], FisherInfoLD[est, 1, 5000]}\)], "Input"], Cell[BoxData[ \({0.025244875046186357`, 0.025339506152269316`}\)], "Output"] }, Open ]], Cell["\<\ If we assume \[Rho]\[NotEqual]1, then we proceed differently.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({paraM, paraRho} = mleM[demerec]\)], "Input"], Cell[BoxData[ \({9.853283763777942`, 1.118769340608164`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?FisherInfoM\)\)], "Input"], Cell[BoxData[ \("FisherInfoM[m,\!\(\[Rho],\[Phi]\),k] computes the Fisher information \ for m, I(m), according to an M(m,\!\(\[Rho],\[Phi]\)) distribution; it \ retains k terms in the series \!\(I(m)=\[Sum] f\^2\%j/p\_j\). Note that \ FisherInfoM[m,\!\(\[Rho]\),k] assumes \!\(\[Phi]=1\)."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FisherInfoM[paraM, paraRho, 2000]\)], "Input"], Cell[BoxData[ \(0.026465097408464602`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?FisherInfoRho\)\)], "Input"], Cell[BoxData[ \("FisherInfoRho[m,\!\(\[Rho]\),k] computes the Fisher information for \!\ \(\[Rho]\), I(\!\(\[Rho]\)), using an M(m, \!\(\[Rho],1\)) distribution. It \ retains k terms in the series \!\(I(\!\(\[Rho]\))=\[Sum] f\^2\%j/p\_j\)."\)], \ "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FisherInfoRho[paraM, paraRho, 2000]\)], "Input"], Cell[BoxData[ \(3.0558046746619607`\)], "Output"] }, Open ]], Cell["\<\ The method based on the original idea of Lea and Coulson takes more time when \ a large number of terms are retained in the series. You should not see much \ difference if only a small number of terms are kept. Of course the two \ methods give identical results.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?FisherInfoMLea\)\)], "Input"], Cell[BoxData[ \("FisherInfoMLea[m,\!\(\[Rho]\),k] computes the Fisher information for \ both m and \!\(\[Rho]\) according to an M(m,\!\(\[Rho]\),1) distribution by \ the algorithm proposed by Lea and Coulson. It uses k terms to approximate \ I(m) and I(\!\(\[Rho]\)). (This function is not recommended for large \ k.)"\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FisherInfoMLea[paraM, paraRho, 2000]\)], "Input"], Cell[BoxData[ \({0.026465097408464613`, 3.055804674661945`}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Methods of Historical Interest", "Section", FontSize->16], Cell["\<\ The maximum likelihood method is obviously the method of choice for \ estimating mutation rates. But there are several methods of great historical \ interest; these methods are still popular.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MethodOfP0\)\)], "Input"], Cell[BoxData[ \("MethodOfP0[data] estimates m, the expected number of mutations, using \ the \!\(P\_0\) method."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MethodOfP0[demerec]\)], "Input"], Cell[BoxData[ \("The \!\(P\_0 \) method is not applicable to this case."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MethodOfMeans\)\)], "Input"], Cell[BoxData[ \("MethodOfMeans[data,\!\(w\_0:=10\)] estimates m (the mean number of \ mutations) by the method of means as proposed by Luria and Delbr\!\(\ \[UDoubleDot]\)ck. Note \!\(w\_0\) is an initial value needed for numerically \ solving a transcendental equation."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(MethodOfMeans[demerec]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(18.940937803769877`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MethodOfMedians\)\)], "Input"], Cell[BoxData[ \("MethodOfMedians[data,\!\(w\_0:=1\)] estimates m, the expected number \ of mutations, using the method of medians proposed by Lea and Coulson. Note \ \!\(w\_0\) is an initial value needed to solve a transcendental equation by \ Newton's method. If only the median of the data is known, say med, \ MethodOfMedians[med] also returns an estimate of m."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MethodOfMedians[demerec]\)], "Input"], Cell[BoxData[ \(11.851891765263119`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?semiMLE\)\)], "Input"], Cell[BoxData[ \("semiMLE[data,\!\(m\_0:=1\)] computes an estimate of m by solving \!\(\ \[Sum]Y\_i=0\), where \!\(Y\_i\) are the Lea-Coulson transforms of the \ original data. Note that \!\(m\_0\) is a starting value used to solve the \ equation."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(semiMLE[demerec]\)], "Input"], Cell[BoxData[ \(11.553643621404525`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["How to Get Results Quickly", "Section"], Cell[TextData[{ "No computer package can replace critical thinking. However, SALVADOR \ offers 2 functions that enable the user to get a feeling of his experimental \ data without thinking about mathematical details. Critical thinking is \ strongly recommended when you look at your preliminary results obtained in \ such a completely automated manner. The first function allows the user to \ get a confidence interval for the mutation rate ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\_\[Beta]\)]], " automatically. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?CIMutationRate\)\)], "Input"], Cell[BoxData[ \("CIMutationRate[data,\!\(\[Alpha],N\_T,k\)] constructs a \!\(\[Alpha]\) \ level confidence interval for \!\(\[Mu]\_\[Beta]\) using an LD(m,1) \ distribution. It uses k terms in computing the Fisher information for m. It \ is important to choose a suitable k. Note \!\(N\_T\) is the number of \ nonmutants prior to plating."\)], "Print"] }, Open ]], Cell[TextData[{ "We now find a 95% confidence interval for ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\_\[Beta]\)]], " using Demerec's experimental data." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CIMutationRate[demerec, 0.95, Nt, 2000]\)], "Input"], Cell[BoxData[ \({4.523815804275617`*^-8, 6.890739642091409`*^-8}\)], "Output"] }, Open ]], Cell[BoxData[ \(We\ can\ also\ get\ a\ 90 %\ confidence\ interval\ for\ \(\(\[Mu]\_\ \[Beta]\)\(.\)\)\)], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CIMutationRate[demerec, 0.90, Nt, 2000]\)], "Input"], Cell[BoxData[ \({4.7140851721829295`*^-8, 6.700470274184096`*^-8}\)], "Output"] }, Open ]], Cell["If you choose too small a k, SALVADOR will complain.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CIMutationRate[demerec, 0.95, Nt, 50]\)], "Input"], Cell[BoxData[ \("Specified k is too small."\)], "Print"] }, Open ]], Cell["\<\ The second function enables you to visualize your data along with a fitted \ distribution in an automated manner.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?plotFit\)\)], "Input"], Cell[BoxData[ \("plotFit[data,opts] fits either an LD(m,1) or an M(m,\!\(\[Rho]\),1) \ distribution to data, and then plots the fitted distribution along with the \ data. Use options for different ploting styles."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Options[plotFit]\)], "Input"], Cell[BoxData[ \({logPlot \[Rule] False, DifferentialGrowth \[Rule] False, dataStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.015`]}, fitStyle \[Rule] {RGBColor[0, 1, 0], Thickness[0.006`]}}\)], "Output"] }, Open ]], Cell["\<\ Note that the logPlot option should be set to false if some sister cultures \ contain no mutants, because log(0) is undefined.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g1 = plotFit[demerec, fitStyle \[Rule] {RGBColor[0, 0, 1], Thickness[0.006], Dashing[{0.01, 0.02}]}, dataStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.02]}, logPlot \[Rule] True]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier 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