\documentclass[10pt,letterpaper]{article}
\usepackage[top=0.85in,left=2.75in,footskip=0.75in]{geometry}

% amsmath and amssymb packages, useful for mathematical formulas and symbols
\usepackage{amsmath,amssymb}

% Use adjustwidth environment to exceed column width (see example table in text)
\usepackage{changepage}

% Use Unicode characters when possible
\usepackage[utf8x]{inputenc}

% textcomp package and marvosym package for additional characters
\usepackage{textcomp,marvosym}

% cite package, to clean up citations in the main text. Do not remove.
\usepackage{cite}

% Use nameref to cite supporting information files (see Supporting Information section for more info)
\usepackage{nameref,hyperref}

% line numbers
\usepackage[right]{lineno}

% ligatures disabled
\usepackage{microtype}
\DisableLigatures[f]{encoding = *, family = * }

% color can be used to apply background shading to table cells only
\usepackage[table]{xcolor}

% array package and thick rules for tables
\usepackage{array}

% create "+" rule type for thick vertical lines
\newcolumntype{+}{!{\vrule width 2pt}}

% create \thickcline for thick horizontal lines of variable length
\newlength\savedwidth
\newcommand\thickcline[1]{%
  \noalign{\global\savedwidth\arrayrulewidth\global\arrayrulewidth 2pt}%
  \cline{#1}%
  \noalign{\vskip\arrayrulewidth}%
  \noalign{\global\arrayrulewidth\savedwidth}%
}

% \thickhline command for thick horizontal lines that span the table
\newcommand\thickhline{\noalign{\global\savedwidth\arrayrulewidth\global\arrayrulewidth 2pt}%
\hline
\noalign{\global\arrayrulewidth\savedwidth}}


% Remove comment for double spacing
%\usepackage{setspace} 
%\doublespacing

% Text layout
\raggedright
\setlength{\parindent}{0.5cm}
\textwidth 5.25in 
\textheight 8.75in

% Bold the 'Figure #' in the caption and separate it from the title/caption with a period
% Captions will be left justified
\usepackage[aboveskip=1pt,labelfont=bf,labelsep=period,justification=raggedright,singlelinecheck=off]{caption}
\renewcommand{\figurename}{Fig}

% Use the PLoS provided BiBTeX style
\bibliographystyle{plos2015}

% Remove brackets from numbering in List of References
\makeatletter
\renewcommand{\@biblabel}[1]{\quad#1.}
\makeatother

% Leave date blank
\date{}

% Header and Footer with logo
\usepackage{lastpage,fancyhdr,graphicx}
\usepackage{epstopdf}
\pagestyle{myheadings}
\pagestyle{fancy}
\fancyhf{}
\setlength{\headheight}{27.023pt}
\lhead{\includegraphics[width=2.0in]{PLOS-submission.eps}}
\rfoot{\thepage/\pageref{LastPage}}
\renewcommand{\footrule}{\hrule height 2pt \vspace{2mm}}
\fancyheadoffset[L]{2.25in}
\fancyfootoffset[L]{2.25in}
\lfoot{\sf PLOS}

%% Include all macros below
\def\FW{13.33cm}
\def\FWs{7.5cm}

\DeclareMathSizes{10}{9}{6}{7}
\usepackage{cases}

\begin{document}
%------------------------------------------------ 
\paragraph*{S2 Appendix.}
\label{S2_Appendix}
%------------------------------------------------
{\bf The Kim computational model.} 
The whole-body computational model by Kim et al.~[13] is composed of seven tissues/compartments: brain, heart, liver, gastrointestinal tract, skeletal muscle,
adipose tissue and “other tissues”. Each compartment is connected to the others via blood circulation. Each tissue compartment is then described by
dynamic mass balances and major cellular metabolic reactions. A list of the metabolites and reactions is reported in Table~\ref{S1_Table}.

\begin{table}[!ht] 
\centering
\caption{{\bf Metabolites and reactions in the computational model of fuel homeostasis by Kim et al.~[13]}}
\begin{tabular}{|l|l|}
\hline
  {\bf List of metabolites} & {\bf List of reactions}\\
\thickhline
  1. GLC: Glucose & 1. Glycolysis I\\ \hline
  2. PYR: Pyruvate & 2. Glycolysis II\\ \hline
  3. LAC: Lactate & 3. Glycolysis III\\ \hline
  4. ALA: Alanine & 4. Gluconeogenesis I\\ \hline
  5. GLR: Glycerol & 5. Gluconeogenesis II\\ \hline
  6. FFA: Free fatty acids & 6. Gluconeogenesis III\\ \hline
  7. TG: Triglycerides & 7. Glycogenesis\\ \hline
  8. O2: Oxygen & 8. Glycogenolysis\\ \hline
  9. CO2: Carbon dioxide & 9. Pyruvate reduction\\ \hline
  10. G6P: Glucose-6-phosphate & 10. Lactate oxidation\\ \hline
  11. GLY: Glycogen & 11. Glycerol phosphorylation\\ \hline
  12. GAP: Glyceraldehyde-3-phosphate & 12. GAP reduction\\ \hline
  13. GRP: Glycerol-3-phosphate & 13. Glycerol 3-P oxidation\\ \hline
  14. AcoA: Acetyl coenzyme A & 14. Alanine formation\\ \hline
  15. CoA: Coenzyme A & 15. Alanine utilization\\ \hline
  16. NAD+: Aldehyde dehydrogenase & 16. Pyruvate oxidation\\ \hline
  17. NADH: Nicotinamide adenine dinucleotide & 17. Fatty acids oxidation\\ \hline
  18. ATP: Adenosine triphosphate & 18. Fatty acids synthesis\\ \hline
  19. ADP: Adenosine diphosphate & 19. Lipolysis\\ \hline
  20. Pi: Phosphate & 20. Triglycerides synthesis\\ \hline
  21. PCR: Phosphocreatine & 21. TCA cycle\\ \hline
  22. CR: Creatine & 22. Oxidative phosphorylation\\ \hline
  \verb|| & 23. Phosphocreatine breakdown\\ \hline
  \verb|| & 24. Phosphocreatine synthesis\\ \hline
  \verb|| & 25. ATP hydrolysis\\ \hline
  \end{tabular}
  \label{S1_Table}
\end{table}


\textbf{Reaction flux equations}

The basic model of Kim et al.~[13] uses a generalized kinetic equation for any metabolic flux:

\begin{equation*}
\begin{split}
\phi_{x,X-Y->V-W} & =V_{x,X-Y->V-W}\left(\frac{C_X/K_X \cdot C_Y/K_Y}{1 + C_X/K_X + C_Y/K_Y + C_X/K_X \cdot C_Y/K_Y}\right) \\
& \left(\frac{PS^{\pm}}{\mu^{\pm}} + PS^{\pm}\right) \left(\frac{RS^{\pm}}{\nu^{\pm}} + RS^{\pm}\right)
\end{split}
\label{eq2}
\end{equation*}

where $V_{x,X-Y->V-W}$, $K_X$ and $K_Y$ are Michaelis-Menten parameters specific to the reaction
process, $C_X$ and $C_Y$ are concentrations of substrate $X$ and $Y$ in tissue $x$. In this
expression, phosphorylation state, $PS^+ = C_{ATP}/C_{ADP}$ , and redox state, 
$RS^x=C_{NADH}/C_{NAD+}$. 
For some reactions, the effect of these controllers can be in the opposite
direction. In this case, $PS_- = 1/PS_+$ and $RS^- = 1/RS^+$. In addition, $\nu^{\pm}$ and $\mu^{\pm}$ are
parameters for metabolic controllers. 
The kinetic equations for the metabolic reactions, and dynamic mass balance
equations in tissue, $x$, where $x$ stands for tissues: brain, heart, muscle, GI, liver, and adipose can be found in Appendix 1 and Appendix 2 of~[13] .

\textbf{Mass balance equations}

Given the flux equations, the dynamic mass balance for a substrate $i$ in tissue $x$ can be described as
 
\begin{equation*}
V_{eff,x,i}\frac{dC_{x,i}}{dt} = P_{x,i} - U_{x,i} + Q_x (C_{a,i} - \sigma_{x,i}C_{x,i}) 
\label{mass}
\end{equation*}

where $V_{eff,x,i}$ is the effective volume of substrate $i$ in
tissue $x$, $P_{x,i}$ and $U_{x,i}$ are, respectively, the production rate and the utilization rate of substrate $i$, $Q_x$ is the blood flow
to tissue $x$, $C_{a,i}$ is the arterial concentration of substrate $i$, $\sigma_{x,i}$ is the partition coefficient of substrate $i$, and $C_{x,i}$
is the concentration of substrate $i$ in tissue $x$.

For the mass balance, a perfectly
mixed lumped tissue-capillary compartment is assumed with blood/tissue distribution of
metabolites that can cross the cell wall determined by blood/tissue partition coefficients. Since each tissue is considered as a lumped tissue capillary
compartment, the actual distribution volume
$V_{eff,x,i}$ of substrate $i$ differs from the physical tissue
volume $V_x$. For substrate $i$ which exists both
in blood and in tissue, $V_{eff,x,i}= 0.93 V_x + \sigma_{x,i}
(0.07V_x)$; for substrate $i$ which exists only in tissue,
$V_{eff,x,i}= 0.8V_x$.

\textbf{Glucagon/Insulin controller}

The glucagon-insulin system is incorporated into the whole-body model to predict
hormonal changes during exercise. Exercise-induced change in the hormone
epinephrine affects the pancreatic secretion of glucagon and insulin as follows

\begin{equation}
C_E(t) = C_E(0) + \omega(WR)(1 - e^{(t/\tau)}) 
\label{eq:epinephrine}
\end{equation}
in which $\omega(WR)$ represents a steady state gain in response to WR, and $\tau$ is a time
constant, which accounts for the epinephrine dynamics. The modeled exercise is a moderate intensity exercise performed by a healthy young man at 60\% of maximal oxygen uptake
(VO\textsubscript{2max})), which corresponds to a WR = 150 Watt.

The controller is given by
\begin{equation}
\footnotesize
\begin{cases}
\frac{dC_G}{dt}=C_G(t)\cdot \bigl[ \theta\cdot[l - k_1\cdot(C_G(t) - C_G(0)) - k_2 \cdot(C_I(t) - C_I(0))] -D \bigr]\\
\frac{dC_I}{dt}=C_I(t)\cdot \bigl[ \psi\cdot[l - k_3\cdot(C_G(t) - C_G(0)) - k_4 \cdot(C_I(t) - C_I(0))] -D\bigr] -\frac{k_5\cdot (C_E(t)-C_{E,0})}{k_6+(C_E(t)-C_{E,0})}\\
\end{cases}
\label{eq:KimController2}
\end{equation} 

in which $C_I(t)$, $C_G(t)$, $C_E(t)$ are the insulin, glucagon and epinephrine blood concentrations, $C_{I,0}$, $C_{G,0}$ and $C_{E,0}$ are their basal values,
$\theta$ and $\psi$ are functions of the glucose arterial concentration $C_{a,glc}$ given by

\begin{numcases}{\theta(C_{a,glc})=} 
1                             & $C_{a,glc} < 2.5$ \nonumber \\                             
1 - (C_{a,glc} - 2.5)^2 / 25    & $2.5 \le C_{a,glc} \le 7.5$ \nonumber \\
0                             & $C_{a,glc} > 7.5$ \nonumber
\end{numcases}
\begin{numcases}{\psi(C_{a,glc})=} 
0                             & $C_{a,glc} < 2.5$ \nonumber \\                             
1 - (C_{a,glc} - 7.5)^2 / 25    & $2.5 \le C_{a,glc} \le 7.5$ \nonumber \\
1                             & $C_{a,glc} > 7.5$ \nonumber
\end{numcases}

The KSC model is composed of a total of 134 ODEs.
\nolinenumbers




\end{document}
