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+%\VignetteIndexEntry{Example evaluation of FOCUS dataset Z}
+%\VignetteEngine{knitr::knitr}
+\documentclass[12pt,a4paper]{article}\usepackage[]{graphicx}\usepackage[]{color}
+%% maxwidth is the original width if it is less than linewidth
+%% otherwise use linewidth (to make sure the graphics do not exceed the margin)
+\makeatletter
+\def\maxwidth{ %
+ \ifdim\Gin@nat@width>\linewidth
+ \linewidth
+ \else
+ \Gin@nat@width
+ \fi
+}
+\makeatother
+
+\definecolor{fgcolor}{rgb}{0.345, 0.345, 0.345}
+\newcommand{\hlnum}[1]{\textcolor[rgb]{0.686,0.059,0.569}{#1}}%
+\newcommand{\hlstr}[1]{\textcolor[rgb]{0.192,0.494,0.8}{#1}}%
+\newcommand{\hlcom}[1]{\textcolor[rgb]{0.678,0.584,0.686}{\textit{#1}}}%
+\newcommand{\hlopt}[1]{\textcolor[rgb]{0,0,0}{#1}}%
+\newcommand{\hlstd}[1]{\textcolor[rgb]{0.345,0.345,0.345}{#1}}%
+\newcommand{\hlkwa}[1]{\textcolor[rgb]{0.161,0.373,0.58}{\textbf{#1}}}%
+\newcommand{\hlkwb}[1]{\textcolor[rgb]{0.69,0.353,0.396}{#1}}%
+\newcommand{\hlkwc}[1]{\textcolor[rgb]{0.333,0.667,0.333}{#1}}%
+\newcommand{\hlkwd}[1]{\textcolor[rgb]{0.737,0.353,0.396}{\textbf{#1}}}%
+\let\hlipl\hlkwb
+
+\usepackage{framed}
+\makeatletter
+\newenvironment{kframe}{%
+ \def\at@end@of@kframe{}%
+ \ifinner\ifhmode%
+ \def\at@end@of@kframe{\end{minipage}}%
+ \begin{minipage}{\columnwidth}%
+ \fi\fi%
+ \def\FrameCommand##1{\hskip\@totalleftmargin \hskip-\fboxsep
+ \colorbox{shadecolor}{##1}\hskip-\fboxsep
+ % There is no \\@totalrightmargin, so:
+ \hskip-\linewidth \hskip-\@totalleftmargin \hskip\columnwidth}%
+ \MakeFramed {\advance\hsize-\width
+ \@totalleftmargin\z@ \linewidth\hsize
+ \@setminipage}}%
+ {\par\unskip\endMakeFramed%
+ \at@end@of@kframe}
+\makeatother
+
+\definecolor{shadecolor}{rgb}{.97, .97, .97}
+\definecolor{messagecolor}{rgb}{0, 0, 0}
+\definecolor{warningcolor}{rgb}{1, 0, 1}
+\definecolor{errorcolor}{rgb}{1, 0, 0}
+\newenvironment{knitrout}{}{} % an empty environment to be redefined in TeX
+
+\usepackage{alltt}
+\usepackage{a4wide}
+\input{header}
+\hypersetup{
+ pdftitle = {Example evaluation of FOCUS dataset Z},
+ pdfsubject = {Manuscript},
+ pdfauthor = {Johannes Ranke},
+ colorlinks = {true},
+ linkcolor = {blue},
+ citecolor = {blue},
+ urlcolor = {red},
+ hyperindex = {true},
+ linktocpage = {true},
+}
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\begin{document}
+
+
+
+\title{Example evaluation of FOCUS dataset Z}
+\author{\textbf{Johannes Ranke} \\[0.5cm]
+%EndAName
+Wissenschaftlicher Berater\\
+Kronacher Str. 8, 79639 Grenzach-Wyhlen, Germany\\[0.5cm]
+and\\[0.5cm]
+University of Bremen\\
+}
+\maketitle
+
+\thispagestyle{empty} \setcounter{page}{0}
+
+\clearpage
+
+\tableofcontents
+
+\textbf{Key words}: Kinetics, FOCUS, nonlinear optimisation
+
+\section{The data}
+
+The following code defines the example dataset from Appendix 7 to the FOCUS kinetics
+report \citep{FOCUSkinetics2011}, p.350.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlkwd{require}\hlstd{(mkin)}
+\end{alltt}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: mkin}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: minpack.lm}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: rootSolve}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: inline}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: methods}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: parallel}}\begin{alltt}
+\hlstd{LOD} \hlkwb{=} \hlnum{0.5}
+\hlstd{FOCUS_2006_Z} \hlkwb{=} \hlkwd{data.frame}\hlstd{(}
+ \hlkwc{t} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{0.04}\hlstd{,} \hlnum{0.125}\hlstd{,} \hlnum{0.29}\hlstd{,} \hlnum{0.54}\hlstd{,} \hlnum{1}\hlstd{,} \hlnum{2}\hlstd{,} \hlnum{3}\hlstd{,} \hlnum{4}\hlstd{,} \hlnum{7}\hlstd{,} \hlnum{10}\hlstd{,} \hlnum{14}\hlstd{,} \hlnum{21}\hlstd{,}
+ \hlnum{42}\hlstd{,} \hlnum{61}\hlstd{,} \hlnum{96}\hlstd{,} \hlnum{124}\hlstd{),}
+ \hlkwc{Z0} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{100}\hlstd{,} \hlnum{81.7}\hlstd{,} \hlnum{70.4}\hlstd{,} \hlnum{51.1}\hlstd{,} \hlnum{41.2}\hlstd{,} \hlnum{6.6}\hlstd{,} \hlnum{4.6}\hlstd{,} \hlnum{3.9}\hlstd{,} \hlnum{4.6}\hlstd{,} \hlnum{4.3}\hlstd{,} \hlnum{6.8}\hlstd{,}
+ \hlnum{2.9}\hlstd{,} \hlnum{3.5}\hlstd{,} \hlnum{5.3}\hlstd{,} \hlnum{4.4}\hlstd{,} \hlnum{1.2}\hlstd{,} \hlnum{0.7}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{18.3}\hlstd{,} \hlnum{29.6}\hlstd{,} \hlnum{46.3}\hlstd{,} \hlnum{55.1}\hlstd{,} \hlnum{65.7}\hlstd{,} \hlnum{39.1}\hlstd{,} \hlnum{36}\hlstd{,} \hlnum{15.3}\hlstd{,} \hlnum{5.6}\hlstd{,} \hlnum{1.1}\hlstd{,}
+ \hlnum{1.6}\hlstd{,} \hlnum{0.6}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{),}
+ \hlkwc{Z2} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{2.6}\hlstd{,} \hlnum{3.8}\hlstd{,} \hlnum{15.3}\hlstd{,} \hlnum{37.2}\hlstd{,} \hlnum{31.7}\hlstd{,} \hlnum{35.6}\hlstd{,} \hlnum{14.5}\hlstd{,}
+ \hlnum{0.8}\hlstd{,} \hlnum{2.1}\hlstd{,} \hlnum{1.9}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{),}
+ \hlkwc{Z3} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{9.2}\hlstd{,} \hlnum{13.1}\hlstd{,} \hlnum{22.3}\hlstd{,} \hlnum{28.4}\hlstd{,} \hlnum{32.5}\hlstd{,}
+ \hlnum{25.2}\hlstd{,} \hlnum{17.2}\hlstd{,} \hlnum{4.8}\hlstd{,} \hlnum{4.5}\hlstd{,} \hlnum{2.8}\hlstd{,} \hlnum{4.4}\hlstd{))}
+
+\hlstd{FOCUS_2006_Z_mkin} \hlkwb{<-} \hlkwd{mkin_wide_to_long}\hlstd{(FOCUS_2006_Z)}
+\end{alltt}
+\end{kframe}
+\end{knitrout}
+
+\section{Parent compound and one metabolite}
+
+The next step is to set up the models used for the kinetic analysis. As the
+simultaneous fit of parent and the first metabolite is usually straightforward,
+Step 1 (SFO for parent only) is skipped here. We start with the model 2a,
+with formation and decline of metabolite Z1 and the pathway from parent
+directly to sink included (default in mkin).
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.2a} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.2a} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.2a, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.2a)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_1-1}
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.2a,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+## Estimate se_notrans t value Pr(>t)
+## Z0_0 9.701488e+01 3.55313691 2.730401e+01 1.679214e-21
+## k_Z0_sink 1.281376e-11 0.22689470 5.647447e-11 5.000000e-01
+## k_Z0_Z1 2.236006e+00 0.16507604 1.354531e+01 7.396594e-14
+## k_Z1_sink 4.821248e-01 0.06585369 7.321150e+00 3.552015e-08
+## Lower Upper
+## Z0_0 91.4058170 102.6239462
+## k_Z0_sink 0.0000000 Inf
+## k_Z0_Z1 1.8419826 2.7143172
+## k_Z1_sink 0.4005856 0.5802613
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+As obvious from the parameter summary (the \texttt{bpar} component of the
+summary), the kinetic rate constant from parent compound Z to sink
+is negligible. Accordingly, the exact magnitude of the fitted parameter
+\texttt{log k\_Z0\_sink} is ill-defined and the covariance matrix is not
+returned (not shown, would be visible in the complete summary).
+This suggests, in agreement with the analysis in the FOCUS kinetics report, to
+simplify the model by removing the pathway to sink.
+
+A similar result can be obtained when formation fractions are used in the model
+formulation:
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.2a.ff} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),}
+ \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.2a.ff} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.2a.ff, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.2a.ff)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_2-1}
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.2a.ff,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+## Estimate se_notrans t value Pr(>t) Lower
+## Z0_0 97.0148812 3.55314944 27.303912 1.679369e-21 91.3287912
+## k_Z0 2.2360064 0.21684747 10.311425 3.661846e-11 1.8052909
+## k_Z1 0.4821248 0.06585372 7.321147 3.552045e-08 0.3996826
+## f_Z0_to_Z1 1.0000000 0.10147342 9.854798 9.707056e-11 0.0000000
+## Upper
+## Z0_0 102.7009713
+## k_Z0 2.7694843
+## k_Z1 0.5815722
+## f_Z0_to_Z1 1.0000000
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+Here, the ilr transformed formation fraction fitted in the model takes a very
+large value, and the backtransformed formation fraction from parent Z to Z1 is
+practically unity. Again, the covariance matrix is not returned as the model is
+overparameterised.
+
+The simplified model is obtained by setting the list component \texttt{sink} to
+\texttt{FALSE}.\footnote{If the model formulation without formation fractions
+is used, the same effect can be obtained by fixing the parameter \texttt{k\_Z\_sink}
+to a value of zero.}
+
+In the following, we use the parameterisation with formation fractions in order
+to be able to compare with the results in the FOCUS guidance, and as it
+makes it easier to use parameters obtained in a previous fit when adding a further
+metabolite.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.3} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),} \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.3} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.3, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.3)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_3-1}
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.3,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+## Estimate se_notrans t value Pr(>t) Lower
+## Z0_0 97.0148815 2.68177093 36.17568 2.363587e-25 91.5215237
+## k_Z0 2.2360064 0.14686110 15.22531 2.246514e-15 1.9545310
+## k_Z1 0.4821248 0.04268712 11.29439 3.068581e-12 0.4021552
+## Upper
+## Z0_0 102.5082392
+## k_Z0 2.5580177
+## k_Z1 0.5779966
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+As there is only one transformation product for Z0 and no pathway
+to sink, the formation fraction is internally fixed to unity.
+
+\section{Including metabolites Z2 and Z3}
+
+As suggested in the FOCUS report, the pathway to sink was removed for metabolite Z1 as
+well in the next step. While this step appears questionable on the basis of the above results, it
+is followed here for the purpose of comparison. Also, in the FOCUS report, it is
+assumed that there is additional empirical evidence that Z1 quickly and exclusively
+hydrolyses to Z2.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.5} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),} \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.5} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.5, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.5)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_5-1}
+
+\end{knitrout}
+
+Finally, metabolite Z3 is added to the model. We use the optimised
+differential equation parameter values from the previous fit in order to
+accelerate the optimization.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.FOCUS} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+ \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),}
+ \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.FOCUS} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.FOCUS, FOCUS_2006_Z_mkin,}
+ \hlkwc{parms.ini} \hlstd{= m.Z.5}\hlopt{$}\hlstd{bparms.ode,}
+ \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.FOCUS)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_6-1}
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.FOCUS,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+## Estimate se_notrans t value Pr(>t) Lower
+## Z0_0 96.83849566 2.05886133 47.034977 5.583742e-44 92.70517559
+## k_Z0 2.21540897 0.11813046 18.753919 7.744034e-25 1.99051134
+## k_Z1 0.47829896 0.02928763 16.331089 3.333376e-22 0.42297603
+## k_Z2 0.45161653 0.04421268 10.214638 3.110936e-14 0.37103495
+## k_Z3 0.05869343 0.01429485 4.105914 7.285393e-05 0.03599566
+## f_Z2_to_Z3 0.47150952 0.05705233 8.264510 2.809889e-11 0.36038193
+## Upper
+## Z0_0 100.97181573
+## k_Z0 2.46571661
+## k_Z1 0.54085782
+## k_Z2 0.54969886
+## k_Z3 0.09570371
+## f_Z2_to_Z3 0.58553462
+\end{verbatim}
+\begin{alltt}
+\hlkwd{endpoints}\hlstd{(m.Z.FOCUS)}
+\end{alltt}
+\begin{verbatim}
+## $ff
+## Z2_Z3 Z2_sink
+## 0.4715095 0.5284905
+##
+## $SFORB
+## logical(0)
+##
+## $distimes
+## DT50 DT90
+## Z0 0.3128755 1.039350
+## Z1 1.4491923 4.814113
+## Z2 1.5348136 5.098540
+## Z3 11.8096217 39.230714
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+This fit corresponds to the final result chosen in Appendix 7 of the FOCUS
+report. Confidence intervals returned by mkin are based on internally
+transformed parameters, however.
+
+\section{Using the SFORB model for parent and metabolites}
+
+As the FOCUS report states, there is a certain tailing of the time course of metabolite
+Z3. Also, the time course of the parent compound is not fitted very well using the
+SFO model, as residues at a certain low level remain.
+
+Therefore, an additional model is offered here, using the single first-order
+reversible binding (SFORB) model for metabolite Z3. As expected, the $\chi^2$
+error level is lower for metabolite Z3 using this model and the graphical
+fit for Z3 is improved. However, the covariance matrix is not returned.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.1} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+ \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.1} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.1, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.1)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_7-1}
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.mkin.1,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{cov.unscaled}
+\end{alltt}
+\begin{verbatim}
+## NULL
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+Therefore, a further stepwise model building is performed starting from the
+stage of parent and two metabolites, starting from the assumption that the model
+fit for the parent compound can be improved by using the SFORB model.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.3} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.3} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.3, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.3)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_9-1}
+
+\end{knitrout}
+
+This results in a much better representation of the behaviour of the parent
+compound Z0.
+
+Finally, Z3 is added as well. These models appear overparameterised (no
+covariance matrix returned) if the sink for Z1 is left in the models.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.4} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+ \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.4} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.4, FOCUS_2006_Z_mkin,}
+ \hlkwc{parms.ini} \hlstd{= m.Z.mkin.3}\hlopt{$}\hlstd{bparms.ode,}
+ \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.4)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_10-1}
+
+\end{knitrout}
+
+The error level of the fit, but especially of metabolite Z3, can be improved if
+the SFORB model is chosen for this metabolite, as this model is capable of
+representing the tailing of the metabolite decline phase.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.5} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+ \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+ \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1. Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.5} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.5, FOCUS_2006_Z_mkin,}
+ \hlkwc{parms.ini} \hlstd{= m.Z.mkin.4}\hlopt{$}\hlstd{bparms.ode[}\hlnum{1}\hlopt{:}\hlnum{4}\hlstd{],}
+ \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.5)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_11-1}
+
+\end{knitrout}
+
+The summary view of the backtransformed parameters shows that we get no
+confidence intervals due to overparameterisation. As the optimized
+\texttt{k\_Z3\_bound\_free} is excessively small, it seems reasonable to fix it to
+zero.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{m.Z.mkin.5a} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.5, FOCUS_2006_Z_mkin,}
+ \hlkwc{parms.ini} \hlstd{=} \hlkwd{c}\hlstd{(m.Z.mkin.5}\hlopt{$}\hlstd{bparms.ode[}\hlnum{1}\hlopt{:}\hlnum{7}\hlstd{],}
+ \hlkwc{k_Z3_bound_free} \hlstd{=} \hlnum{0}\hlstd{),}
+ \hlkwc{fixed_parms} \hlstd{=} \hlstr{"k_Z3_bound_free"}\hlstd{,}
+ \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.5a)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_11a-1}
+
+\end{knitrout}
+
+As expected, the residual plots for Z0 and Z3 are more random than in the case of the
+all SFO model for which they were shown above. In conclusion, the model
+\texttt{Z.mkin.5a} is proposed as the best-fit model for the dataset from
+Appendix 7 of the FOCUS report.
+
+A graphical representation of the confidence intervals can finally be obtained.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlkwd{mkinparplot}\hlstd{(m.Z.mkin.5a)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_11b-1}
+
+\end{knitrout}
+
+The endpoints obtained with this model are
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlkwd{endpoints}\hlstd{(m.Z.mkin.5a)}
+\end{alltt}
+\begin{verbatim}
+## $ff
+## Z0_free_Z1 Z1_Z2 Z2_sink Z2_Z3_free Z3_free_sink
+## 1.0000000 1.0000000 0.4634431 0.5365569 1.0000000
+##
+## $SFORB
+## Z0_b1 Z0_b2 Z3_b1 Z3_b2
+## 2.447138640 0.007512589 0.080007099 0.000000000
+##
+## $distimes
+## DT50 DT90 DT50_Z0_b1 DT50_Z0_b2 DT50_Z3_b1 DT50_Z3_b2
+## Z0 0.3042972 1.184811 0.283248 92.26476 NA NA
+## Z1 1.5147787 5.031986 NA NA NA NA
+## Z2 1.6413857 5.452565 NA NA NA NA
+## Z3 NA NA NA NA 8.663571 Inf
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+It is clear the degradation rate of Z3 towards the end of the experiment
+is very low as DT50\_Z3\_b2 (the second Eigenvalue of the system of two differential
+equations representing the SFORB system for Z3, corresponding to the slower rate
+constant of the DFOP model) is reported to be infinity. However, this appears
+to be a feature of the data.
+
+\bibliographystyle{plainnat}
+\bibliography{references}
+
+\end{document}
+% vim: set foldmethod=syntax:

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