text:
\subsection{\LARGE{Order and Molecularity}}
\textbf{\textit{Rate Constant}}: For a general reaction\\
$aA+bB{ \overset{k}{\longrightarrow}} cC+dD$\\
the rate is given as follows:\\
Rate$=k[A]^{a}[B]^{b} $\\
where $k$ is a rate constant or velocity constant \cite{Upadhyay,Malijevsky}. The rate constant for any reaction can be found either by measuring the reaction rate at unit concentrations for the reactants or by knowing the rate of the reaction using the following relation:\\
Rate constant$=\dfrac{Rate}{[A]^{a}[B]^{b}}.$\\
\textbf{\textit{Order and Molecularity}}:For a giving reaction\\
$aA+bB+cC+...{ \overset{k}{\longrightarrow}} product$\\
the reaction rate is defined by:\\
Rate$=k[A]^{a}[B]^{b}[C]^{c}... $\\
Then the reaction has $a-th$ order with respect to $A$, $b-th$ order with respect to $B$, $c-th$ order with respect to $C$, ..., and the overall order of reaction is $a+b+c+...=n.$\\
The summation of stoichiometric coefficients of reactions is called \textbf{\textit{Molecularity}} of reaction. For instance, for a giving stoichiometric equation: $3A+2B=C+4D$, the stoichiometric coefficients of $A$ and $B$ are 3 and 2, respectively \cite{Upadhyay}. Therefore, the molecularity of the reaction is $3+2=5$.
The \textbf{\textit{rate of complex reactions}} (multi-step reactions) is determined by the rate of change of product (the rate of increase of product). The order and molecularity of a reaction have not a simple relationship. If we have a reaction which occurs in two or more different steps, and it gives overall the same reaction then the order and molecularity of reaction are different. Let give some examples to differentiate between molecularity and order of reaction.\\
For example, consider a reaction in two steps:
\begin{center}
Step 1: Fast  $I+B { \overset{k_{1}}{\longrightarrow}} P$\\
Step 2: Slow $A+2B{ \overset{k_{2}}\longrightarrow}I$\\
 \line(1,0){120} \\
$A+3B{ \overset{k}\longrightarrow}P$
\end{center}
\noindent Note that, the rate of complex reactions (Multi-step reactions) will depend on the rate of increase of product. Therefore, the rate of product is expected as follows\\
\begin{equation}
\begin{array}{llll}
Rate\approx \dfrac{d[P]}{dt}=k_{1}[I][B]
\end{array}\label{eqh1}
\end{equation}
\noindent and the rate of intermediate species $I$ can be given:
\begin{equation}
\begin{array}{llll}
\dfrac{d[I]}{dt}=k_{2}[A][B]^{2}-k_{1}[I][B]
\end{array}\label{eqh2}
\end{equation}
\noindent the steady state with respect to $\dfrac{d[I]}{dt}$ can be applied because it is usually present in very small concentrations. This means $\dfrac{d[I]}{dt}\approx0$. Thus, from equation \ref{eqh2} , we get $[I]=\dfrac{k_{2}}{k_{1}}[A][B]$, and put the value of $[I]$ in equation \ref{eqh1}, we obtain the rate of the complex reaction:
\begin{equation}
\begin{array}{llll}
Rate\approx \dfrac{d[P]}{dt}=k_{2}[A][B]^{2}
\end{array}\label{eqh3}
\end{equation}
\noindent As a result, the order of reaction would be one with respect to reactant $A$ and two with respect to reactant $B$, and the overall order of the reaction is 3. On the other hand, the molecularity of the above reaction would be $1+3=4.$ Thus, the order and molecularity of the reaction are not the same. \\
\noindent Another example is the reaction of cane sugar:
\begin{center}
$C_{12}H_{22}O_{11}+H_{2}O{ \overset{k}{\longrightarrow}} C_{6}H_{12}O_{6}+C_{6}H_{12}O_{6}$
\end{center}
the reaction rate is given as:
\begin{equation}
\begin{array}{llll}
Rate=k[Sucrose][H_{2}O]
\end{array}\label{s1}
\end{equation}
\noindent This reaction looks like second order, first order with respect to any reactant $Sucrose$ and $H_{2}O$  \cite{Upadhyay}. The $[H_{2}O]$ remains constant, and it is present in large excess. Therefore, the reaction is only first order with respect to $Sucrose$ because the $[H_{2}O]$ does not effect the rate of reaction. As a result, the equation \ref{s1} would be written as:
\begin{center}
Rate$=k^{*}[Sucrose]$
\end{center}
where $k^{*}=k[H_{2}O]_{0}$, and $[H_{2}O]_{0}$ is the initial value of $[H_{2}O]$. This reaction is called \textbf{\textit{Pseudo-first order reaction}}. If the order and molecularity of a reaction are different as a result of one reactant in excess, then the reaction is known as \textbf{\textit{Pseudo- first order reaction.}}\\
\noindent However, if we have a reaction which occurs in a single step then the order of reaction and molecularity are the same. Consider the reaction:
$2A+4B{ \overset{k}{\longrightarrow}} P$\\
the molecularity of the reaction is $2+4=6$. If the above reaction occurs just in a single step, then the overall order of the reaction is 6 because the reaction rate is: Rate$=k[A]^{2}[B]^{4}$. Thus, the order and molecularity in this reaction are the same value.
\section{\LARGE{Model Reduction}}
\noindent Transformation of the system (\ref{eq1}) to another system is called"\textbf{model reduction}" in which the new system includes smaller number of equations without affecting dynamics of variables $ [S_{1}](t), [S_{2}](t), ... , [S_{m}](t)$ \cite{Radulescu, Choi, Conzelmann}. We apply some techniques of reduction to the system, reducing it to essentially fewer species and reactions. There are some model reduction techniques which have been used to simplified the model. \\

\textbf{\LARGE{Methods}}
\subsection{\LARGE{Pseudo-First Order Approximation}}
In this technique the order of reaction is smaller than the actual reaction order. This is sometimes happen for a reaction with two or more reactants. If one reactant is present in large excess, its concentration change is negligible small \cite{Upadhyay,Malijevsky}. Therefore, the reaction rate becomes independent from this reaction. Consider a reaction
\begin{center}
$A+B{ \overset{k}{\longrightarrow}}P$
\end{center}
with reaction rate:
\begin{equation}
\begin{array}{llll}
v=k[A][B]
\end{array}\label{s2}
\end{equation}
This reaction is first order with respect to $A$ and first order with respect to $B$, and overall is second order. If the initial of $[B]$ is present in large amount, and the initial value of $[A]$ is smaller, then the effect of $A$ on the concentration and reaction rate is much greater than the effect of $B$. To explain this idea, let give some numerical values: \\
Let $ [A]_{0}=4 $ and $ [B]_{0}=100 $ at the time where half of $ A $ has consumed away: $ [A]=2 $ and $ [B]=98 $. Thus, $ [A] $ has changed by $0.50$ while $ [B] $ changed only by $0.02$. By comparing reaction rate with initial rate we obtain: \\
$ v=k (0.50 [A]_{0}) (0.98 [B]_{0}) =0.49[A]_{0}[B]_{0}=0.49 v_{0}. $\\
It is clear that the decreasing in reaction rate is almost determined by the change of $[A]$, because the relative change of concentration of $A$ is much larger than the relative change of $B$.\\
As a result, the change of $[B]$ can be ignored throughout the reaction proceeds. In this case the rate of reaction is not effected by $[B]$, and the reaction is simply first order with respect to $A$. In other words, if $[B]$ is present in large excess (i.e $[B](t)\gg[A](t)$ $\forall$ $t\in [0,T]$), and $[B]$ remains constant (i.e $[B]\approx[B]_{0}$, and $[B]$ does not significantly change), then the rate equation \ref{s2} can be written as a linear equation:
\begin{center}
$v=k^{*}[A]$
\end{center}
where $k^{*}=k[B]_{0}$. Thus, the reaction is called pseudo-first order approximation. \\
\noindent \textbf{Similarly}, if we have a third order reaction
\begin{center}
$A+B+C{ \overset{k_{1}}{\longrightarrow}}P$
\end{center}
with reaction rate:
\begin{equation}
\begin{array}{llll}
v=k_{1}[A][B][C]
\end{array}\label{s3}
\end{equation}
and two of the reactants are present in large excess, let $[A]$ and $[B]$ are in large amount. This means $[A](t)\gg[C](t)$ and $[B](t)\gg[C](t)$ $\forall$ $t\in [0,T]$. In this case $[A]$ and $[B]$ do not significantly affect on the reaction rate (Equation \ref{s3}), and they remain constant ($[A]\approx[A]_{0}$ and $[B]\approx[B]_{0}$). Therefore, the rate equation \ref{s3} would be considered as a linear equation:
\begin{center}
$v=k_{1}^{*}[C]$, where $k_{1}^{*}=k_{1}[A]_{0}[B]_{0}$
\end{center}
As a result, the reaction is called pseudo-first order approximation. \\
\noindent \textbf{Generally}, consider \textit{n}th order reaction
\begin{center}
$A_{1}+A_{2}+...+A_{n}{ \overset{k_{2}}{\longrightarrow}}P$
\end{center}
with reaction rate:
\begin{equation}
\begin{array}{llll}
v = k_{2} \prod\limits_{i=1}^n [A_{i}],
\end{array}\label{s41}
\end{equation}
and $n-1$ reactants are present in large excess.\\ Let $[A_{1}], [A_{2}], ..., [A_{n-1}]$ are in large amount. This means $[A_{i}](t)\gg[A]_{n}(t) , i=1,2,...,n-1,$ $\forall t\in[0,T]$. In this case $[A_{1}], [A_{2}], ..., [A_{n-1}]$ do not significantly affect on the reaction rate (Equation \ref{s41}), and they remain constant ($[A_{i}]\approx[A_{i}]_{0},$ for $i=1,2,...,n-1$). Therefor, the rate equation \ref{s41} would be considered as a linear equation:
\begin{center}
$v=k_{2}^{*}[A]_{n}$, where $k_{2}^{*}=k_{2} \prod\limits_{i=1}^{n-1} [A_{i}]_{0}$
\end{center}
Thus, the reaction is called pseudo-first order approximation.
\subsection{\LARGE{Removal of approximately linearly dependent concentrations}}
This assumption is that if we have two species $B$ and $C$ in a model, $[B]$ and $[C]$ are approximately linearly dependent (i.e $[B](t)\approx k[C](t), \forall t \in [0,T], k \in R$ ), then one of them can be neglected from the model \cite{Kutumova}. For example, for the parallel reactions \cite{Malijevsky}:
 \begin{equation}
\begin{array}{llll}
A{ \overset{k_{1}}{\longrightarrow}}B\\
A{ \overset{k_{2}}{\longrightarrow}}C
\end{array}\label{s4}
\end{equation}
\noindent and the system of ODE is given:
 \begin{equation}
\begin{array}{llll}
\dfrac{d[A]}{dt}=-(k_{1}+k_{2})[A],
\end{array}\label{s5}
\end{equation}
 \begin{equation}
\begin{array}{llll}
\dfrac{d[B]}{dt}=k_{1}[A],
\end{array}\label{s6}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[C]}{dt}=k_{2}[A],
\end{array}\label{s7}
\end{equation}
\noindent with the initial concentrations:
\begin{equation}
\begin{array}{llll}
[A](0)=[A]_{0}, [B](0)=0, [C](0)=0.
\end{array}\label{s8}
\end{equation}
\noindent The analytical solution for equation \ref{s5} with initial conditions(Equation \ref{s8}) is given as:
\begin{equation}
\begin{array}{llll}
[A](t)=[A]_{0} e^{-(k_{1}+k_{2})t}
\end{array}\label{s9}
\end{equation}
\noindent Substitution of this equation for $[A]$ into the equations \ref{s6} and \ref{s7}, then by straightforward integration, we obtain:
\begin{equation}
\begin{array}{llll}
[B](t)=\dfrac{k_{1}[A]_{0}}{k_{1}+k_{2}} (1-e^{-(k_{1}+k_{2})t})
\end{array}\label{s10}
\end{equation}
and
\begin{equation}
\begin{array}{llll}
[C](t)=\dfrac{k_{2}[A]_{0}}{k_{1}+k_{2}} (1-e^{-(k_{1}+k_{2})t})
\end{array}\label{s11}
\end{equation}
\noindent Dividing equation \ref{s10} by equation \ref{s11}, we get $[B]=\frac{k_{1}}{k_{2}}[C]$, this means that $[B]$ and $[C]$ are linearly dependent. Thus, either $[B]$ or $[C]$ can be neglected from the model because they have the same chemical kinetic properties.
\subsection{\LARGE{Removal of approximately linearly dependent reaction rates}}
This technique is used to neglect the linearly dependent reaction rates. If we have two reaction rates $v_{1}$ and $v_{2}$ in a model, and they are linearly dependent (i.e $v_{1} \approx kv_{2}, k \in R$), then either $v_{1}$ or $v_{2}$ can be neglected from the model. To explain this idea, let give the following parallel reactions:
\begin{equation}
\begin{array}{llll}
A+B{ \overset{k_{1}}{\longrightarrow}}C\\
A+B{ \overset{k_{2}}{\longrightarrow}}D
\end{array}\label{s12}
\end{equation}
\noindent with reaction rates:
\begin{equation}
\begin{array}{llll}
v_{1}=k_{1}[A][B],
\end{array}\label{s13}
\end{equation}
\begin{equation}
\begin{array}{llll}
v_{2}=k_{2}[A][B].
\end{array}\label{s14}
\end{equation}
\noindent Dividing equation \ref{s13} by equation \ref{s14}, we obtain:\\
$\dfrac{v_{1}}{v_{2}}=\dfrac{k_{1}[A][B]}{k_{2}[A][B]} \Longrightarrow v_{1}=\dfrac{k_{1}}{k_{2}} v_{2}.$
\noindent Thus, $v_{1}$ and $v_{2}$ are linearly dependent, and one of them would be neglected from the model.
 \section {\LARGE {Nuclear Receptor Binding to Promoter Regions}}
Nuclear receptor Binding to Promoter regions consists of 9 species and 6 reactions (See figure \ref{nrb1} and table \ref{table1}). This model was presented in previous study \cite{Kolodkin}. In this work, we suggested a mathematical model for the chemical reactions, and we then simplified the model by using the technique of pseudo-first order approximation. The reaction rates in this work are based on the stander mass action kinetics. It means that mass action formula is used to find the reaction rates. The reactions of the complete model and reduced model are presented in tables \ref{table1} and \ref{table2}, respectively. Celldesigner has been used to simulate the concentration of the species.
  \subsection{\LARGE{Mathematical Modeling of the Nuclear receptor Binding to Promoter regions}}
The chemical network reactions of this model ( Figure \ref{nrb1}) can be written as a system of ordinary differential equations. This means that the mass action formula is applied to find the reaction rates for chemical kinetics ( Table \ref{table1}). Thus, this model is given by the following system of differential equations:\\
 \begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:NR]}{dt}=-v_{20}-v_{21}-v_{22},
\end{array}\label{eq23}
\end{equation}
\begin{equation}
\begin{array}{llll}
 \dfrac{d[Re]}{dt}=-v_{18}-v_{20},
\end{array}\label{eq24}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[L:2NR]}{dt}=-v_{18}-v_{19},
\end{array}\label{eq25}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:NR:Re]}{dt}=v_{20}-v_{21}-v_{23},
\end{array}\label{eq26}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:2NR:Re]}{dt}=v_{21}-v_{22},
\end{array}\label{eq27}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:3NR:Re]}{dt}=v_{22}-v_{23},
\end{array}\label{eq28}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:4NR:Re]}{dt}=v_{23},
\end{array}\label{eq28}
\end{equation}
\begin{equation}
\begin{array}{llll}
 \dfrac{d[L:2NR:Re]}{dt}=v_{18}-v_{19},
\end{array}\label{eq30}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[L:4NR:Re]}{dt}=v_{19},
\end{array}\label{eq31}
\end{equation}
\noindent where the reaction rates of the above equations are defined as follows:\\
\noindent $ v_{18}=k_{18} [L:2NR] [Re], $ \\
\noindent $ v_{19}=k_{19} [L:2NR:Re] [L:2NR],  $ \\
\noindent $ v_{20}=k_{20} [hsp90:NR] [Re],$ \\
\noindent $ v_{21}=k_{21}[hsp90:NR:Re] [hsp90:NR], $ \\
\noindent $ v_{22}=k_{22}[hsp90:2NR:Re] [hsp90:NR],  $ \\
\noindent $ v_{23}=k_{23}[hsp90:NR:Re] [hsp90:3NR:Re]. $ \\
\noindent The initial value of concentrations are given as follows: \\
\noindent $ [hsp90:NR](0)=[hsp90:NR]_{0} , [Re](0)=[Re]_{0}, $ \\
\noindent $  [hsp90:NR:Re](0)=0, [hsp90:2NR:Re](0)=0,  $ \\
\noindent $  [hsp90:3NR:Re](0)=0, [hsp90:4NR:Re](0)=0,  $ \\
\noindent $  [L:2NR](0)=[L:2NR]_{0}, [L:2NR:Re](0)=0,  $ \\
\noindent $ [L:4NR:Re](0)=0.  $ \\
\noindent The system of differential equations (Equations \ref{eq23}-\ref{eq31}) can be written as a matrix equation of the form:
\begin{equation}
\begin{array}{llll}
\dfrac{d[S]}{dt}=N V
\end{array}\label{eqy2}
\end{equation}
where \\
\begin{gather*}
  \setlength{\arraycolsep}{1\arraycolsep}
  \text{\Large$
[S] =\begin{pmatrix}
[hsp90:NR] \\
[Re] \\
[L:2NR] \\
[hsp90:NR:Re] \\
[hsp90:2NR:Re] \\
[hsp90:3NR:Re] \\
[hsp90:4NR:Re] \\
[L:2NR:Re] \\
[L:4NR:Re] \\
\end{pmatrix}
, V =\begin{pmatrix}
v_{18} \\
v_{19} \\
v_{20} \\
v_{21} \\
v_{22} \\
v_{23} \\
\end{pmatrix}
$}
\end{gather*}
and
\begin{gather*}
  \setlength{\arraycolsep}{1\arraycolsep}
  \text{\Large $
  N=  \begin{pmatrix}
       0 & 0 & -1 & -1& -1 & 0 \\
       -1 & 0 & -1 & 0& 0 & 0 \\
       -1 & -1 & 0 & 0& 0 & 0 \\
       0 & 0 & 1 & -1& 0 & -1 \\
       0 & 0 & 0 & 1& -1 & 0 \\
       0 & 0 & 0 & 0& 1 & -1 \\
       0 & 0 & 0 & 0& 0 & 1 \\
       1& -1 & 0 & 0& 0 & 0 \\
       0 & 1 & 0 & 0& 0 & 0 \\
    \end{pmatrix}
    $}
\end{gather*}
\section{\LARGE{Results}}
In this work, CellDesigner has been used to draw the structures of chemical reactions of Nuclear Receptor Binding to Promoter Regions (NRB) (Figures \ref{nrb1} and \ref{nrb2}). We use the value of rate constants (Table \ref{table3}) and initial concentrations (Table \ref{table4}) to simulate of concentrations. The simplification of reaction rates is mainly based on the technique of pseudo-first order approximation. We presented the results of this work as follows:
\begin {enumerate}
\item \textbf{Simplification of kinetic equations based on pseudo-first order approximation}: It is noticed that the concentration of $hsp90:NR$ over all is greater than the concentration of $hsp90:NR:Re$ and $hsp90:2NR:Re$, respectively (See figure \ref{re1}), and the concentration of $L:2NR$ is also grater then the concentration of $L:2NR:Re$ (See figure \ref{re2}). This means:\\
\noindent $[hsp90:NR](t) \gg [hsp90:NR:Re](t),$\\
\noindent  $ [hsp90:NR](t) \gg [hsp90:2NR:Re](t),$\\
\noindent  $[L:2NR](t) \gg [L:2NR:Re](t),$ $\forall$ $t \in [0,T], T \in R^{+}$.\\
\noindent In other words, $[hsp90:NR]$ and $[L:2NR]$ are in large excess. They remain relatively constant (i.e $[hsp90:NR] \approx[hsp90:NR]_{0}$ and $[L:2NR]\approx[L:2NR]_{0}$). Therefore, the reaction rates 19, 21 and 22 are changed as follows:\\
\noindent $v_{19}^{*} =k_{19}^{*} [L:2NR:Re],$  \\
\noindent $v_{21}^{*} =k_{21}^{*} [hsp90:NR:Re],$  \\
\noindent $v_{22}^{*} =k_{22}^{*} [hsp90:2NR:Re],$ \\
\noindent where $k_{19}^{*} =k_{19} [L:2NR]_{0}, k_{21}^{*}=k_{21} [hsp90:NR]_{0}$ and $k_{22}^{*}=k_{22} [hsp90:NR]_{0}$.\\
\noindent In addition, the concentration of $Re$ is dominated by the concentration of $hsp90:NR$ and $L:2NR$ ( Figure \ref{re1} and \ref{re2}), it means $[hsp90:NR](t) \gg [Re](t)$, and $[L:2NR](t) \gg [RE](t)$, $\forall$ $t\in [0,T]$. They are in large amount, and they remain approximately unchanged.  According to the technique of pseudo-first order approximation, the reaction rates 18 and 20 can be simplified as follows\\
\noindent $ v_{18}^{*}=k_{18}^{*}[Re], $ and $ v_{20}^{*}=k_{20}^{*}[Re], $\\
where,\\
\noindent $k_{18}^{*}=k_{18} [hsp90:NR]_{0}$ and $ k_{20}^{*}=k_{20} [L:2NR]_{0} $.
\item \textbf{Removal of slow reaction}:The reaction 23 in this model is the slowest reaction in comparison with other reactions in the model. This reaction can be ignored from the model because it does not significantly affect on the experimental dynamic of chemical reactions.
\end{enumerate}
\section{\LARGE{Reduced model of Nuclear Receptor Binding to Promoter Regions}}
The model of nuclear receptor binding to promoter regions is reduced after applying the above techniques of reduction. The simplified model includes 6 species and 5 reactions (Figure \ref{nrb2} and table \ref{table2}). It can be presented as a system of ODE. This is given as follows:
\begin{equation}
\begin{array}{llll}
\dfrac{d[Re]}{dt}=-v_{18}^{*}-v_{20}^{*},
\end{array}\label{eq32}
\end{equation}
\begin{equation}
\begin{array}{llll}
 \dfrac{d[L:2NR:Re]}{dt}=v_{18}^{*}-v_{19}^{*},
\end{array}\label{eq33}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[L:4NR]}{dt}=v_{19}^{*},
\end{array}\label{eq34}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:NR:Re]}{dt}=v_{20}^{*}-v_{21}^{*},
\end{array}\label{eq35}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:2NR:Re]}{dt}=v_{21}^{*}-v_{22}^{*},
\end{array}\label{eq36}
\end{equation}
\begin{equation}
\begin{array}{llll}
\dfrac{d[hsp90:3NR:Re]}{dt}=v_{22}^{*},
\end{array}\label{eq37}
\end{equation}
\noindent The system of differential equations (Equations \ref{eq32}-\ref{eq37}) can be written as a matrix equation of the form:
\begin{equation}
\begin{array}{llll}
\dfrac{d[S]^{*}}{dt}=N^{*} V^{*}
\end{array}\label{eq38}
\end{equation}
where \\
\begin{gather*}
  \setlength{\arraycolsep}{1\arraycolsep}
  \text{\Large$
[S]^{*} =\begin{pmatrix}
[Re] \\
[L:2NR:Re] \\
[L:4NR:Re] \\
[hsp90:NR:Re] \\
[hsp90:2NR:Re] \\
[hsp90:3NR:Re] \\
\end{pmatrix}
,  V^{*} =\begin{pmatrix}
v_{18}^{*} \\
v_{19}^{*} \\
v_{20}^{*} \\
v_{21}^{*} \\
v_{22}^{*} \\
\end{pmatrix}
$}
\end{gather*}
and
\begin{gather*}
  \setlength{\arraycolsep}{1\arraycolsep}
  \text{\Large $
  N^{*}=  \begin{pmatrix}
 -1 & 0 & -1 & 0 & 0 \\
 1 & -1 & 0 & 0 & 0 \\
 0 & 1& 0 & 0 & 0 \\
 0 & 0 & 1 & -1 & 0 \\
 0 & 0 & 0 & 1 & -1 \\
 0 & 0 & 0 & 0 & 1 \\
    \end{pmatrix}
  $}
\end{gather*}
\noindent The general analytical solution of the system \ref{eq38} is given as follows:\\
\noindent $\textbf{[Re](t)}= -c_{4}(3k_{18}^{*}(k_{20}^{*})^2 + 3(k_{18}^{*})^2 k_{20}^{*} - (k_{18}^{*})^2 k_{21}^{*} \\
- (k_{18}^{*})^2 k_{22}^{*} - (k_{20}^{*})^2 k_{21}^{*} - (k_{20}^{*})^2 k_{22}^{*} + (k_{18}^{*})^3 + (k_{20}^{*})^3 \\
- 2 k_{18}^{*} k_{20}^{*} k_{21}^{*}- 2 k_{18}^{*} k_{20}^{*} k_{22}^{*} + k_{18}^{*} k_{21}^{*} k_{22}^{*} + k_{20}^{*} k_{21}^{*} k_{22}^{*})e^{-(k_{18}^{*} + k_{20}^{*})t}\\
/(k_{20}^{*} k_{21}^{*} k_{22}^{*})$,\\
\noindent $\textbf{[L:2NR:Re](t)}=-c_{4} ((k_{18}^{*})^3 k_{21}^{*} - k_{18}^{*} (k_{20}^{*})^3 - 3 (k_{18}^{*})^3 k_{20}^{*} \\
-  (k_{18}^{*})^4 + (k_{18}^{*})^3 k_{22}^{*} - 3(k_{18}^{*})^2 (k_{20}^{*})^2 +k_{18}^{*} (k_{20}^{*})^2 k_{21}^{*} \\ + 2 (k_{18}^{*})^2 k_{20}^{*} k_{21}^{*}- k_{18}^{*} k_{20}^{*} k_{21}^{*} k_{22}^{*}+k_{18}^{*} (k_{20}^{*})^2 k_{22}^{*} \\
+ 2(k_{18}^{*})^2 k_{20}^{*} k_{22}^{*} - (k_{18}^{*})^2 k_{21}^{*} k_{22}^{*}) e^{-(k_{18}^{*} + k_{20}^{*})t}  \\
- c_{2}( k_{18}^{*}k_{20}^{*} k_{21}^{*} k_{22}^{*}  - k_{19}^{*} k_{20}^{*} k_{21}^{*} k_{22}^{*}+ (k_{20}^{*})^2 k_{21}^{*} k_{22}^{*}) e^{-k_{19}^{*}t}\\
/ ( (k_{20}^{*})^2 k_{21}^{*} k_{22}^{*}+ k_{18}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*} - k_{19}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*})$,\\
\noindent $\textbf{[L:4NR:Re](t)}=(c_{2}((k_{20}^{*})^2 k_{21}^{*} k_{22}^{*} + k_{18}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*}\\
- k_{19}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*})e^{-k_{19}^{*}t}-c_{4}((k_{18}^{*})^3 k_{19}^{*} +k_{18}^{*}k_{19}^{*}(k_{20}^{*})^2\\
+ 2(k_{18}^{*})^2 k_{19}^{*}k_{20}^{*}-(k_{18}^{*})^2 k_{19}^{*}k_{21}^{*}-(k_{18}^{*})^2 k_{19}^{*}k_{22}^{*}-k_{18}^{*}k_{19}^{*}k_{20}^{*}k_{21}^{*}\\
- k_{18}^{*}k_{19}^{*}k_{20}^{*}k_{22}^{*}+k_{18}^{*}k_{19}^{*}k_{21}^{*}k_{22}^{*})e^{-(k_{18}^{*}+k_{20}^{*})t}+ c_{5}((k_{20}^{*})^2 k_{21}^{*}k_{22}^{*}\\
+ k_{18}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*}-k_{19}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*}))/((k_{20}^{*})^2 k_{21}^{*}k_{22}^{*}+ k_{18}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*} \\
- k_{19}^{*}k_{20}^{*}k_{21}^{*}k_{22}^{*})$,\\
\noindent $\textbf{[hsp90:NR:Re](t)}=(c_{4}((k_{18}^{*})^2+ (k_{20}^{*})^2+ 2k_{18}^{*}k_{20}^{*}\\
 - k_{18}^{*}k_{22}^{*}- k_{20}^{*}k_{22}^{*}) e^{-(k_{18}^{*}+k_{20}^{*})t}+c_{1}((k_{21}^{*})^2- k_{21}^{*} k_{22}^{*}) e^{-k_{21}^{*} t})\\
/(k_{21}^{*}k_{22}^{*})$,\\
\noindent $\textbf{[hsp90:2NR:Re](t)}=-(c_{1} k_{21}^{*} e^{-k_{21}^{*}t} + c_{3}k_{22}^{*} e^{-k_{22}^{*}t}\\
+c_{4}k_{18}^{*} e^{-(k_{18}^{*}+k_{20}^{*})t}+ c_{4}k_{20}^{*}e^{-(k_{18}^{*}+k_{20}^{*})t})/k_{22}^{*}$,\\
\noindent $\textbf{[hsp90:3NR:Re](t)}=c_{1} e^{-k21k_{21}^{*}*t} + c_{3} e^{-k_{22}^{*}*t} \\
+ c_{4} e^{-(k_{18}^{*}+k_{20}^{*})t}+c_{6}$, \\
\noindent where $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$ and $c_{6}$ are constants.
\section{Conclusion}
Mathematical modeling gives a powerful tool for investigation the properties of chemical kinetics. Methods of model reduction allowed us to reduce the number of reactions and species in the model of nuclear receptor binding to promoter regions.This model is presented as a system of ODEs. The methods of model reduction provide not only faster computational time, but they have a good benefit  to make the system simpler, making it easier to understand and manipulate. The technique of pseudo-first order approximation is used to simplify some kinetic equations when one reactant dominates others. We use Celldesigner to draw the network of chemical reactions and to simulate the concentration of species. After model reduction, the number of species and reactions are reduced from 9 species and 6 reactions to 6 species and 5 reactions. It could be said that the results in this work may have a real advantage of chemical kinetics of nuclear receptor binding to promoter regions.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
Recently, the photoionization of the ion Xe$^{q+}$(q=1-6) [1-9] and the atom I and its
ions I$^{q+}$ (q=1-2) [10-15],
have become "hot topics". Firstly, this is because the 4d giant resonances of the above atom/ions
are the important photoionization processes. Secondly,
the newly developed RPAE method has made a full RPAE calculation possible for
the relevant reactions.
Thirdly, the photoionization cross sections for the ionic species
are required for plasma modeling.

The random-phase approximation with exchange (RPAE) method has been used to
calculate the I$^+$ photoionization cross section before. However,
the calculations [9, 12-13] applied a variety of approximations as the RPAE
method was initially developed for the calculation of closed shell atoms only.
The previous RPAE calculations referenced above cannot provide the cross sections for each
individual term of the I$^+$ ground state either.

Recently, a RPAE method has been developed
by Chen and Msezane [8].
The method has been successfully used to study the inner-shell electron transitions of an
atom with an outer open-shell.
The calculation
can include intra-shell
correlations, viz. an electron absorbs a photon then interacts and shares excitation energy
with other electrons in the same shell and
the inter-shell correlation, where the electron absorbs a photon and interacts
and shares the excitation energy with the electrons of the neighboring
shells.  More resently a RPAE method,
which allows for the inclusion of both intra-shell and inter-shell correlations
has also been developed by Chen and Msezane for atoms(ions)
with an inner open-shell [16]. These methods have greatly extended the
scope of the RPAE method as their calculations allow the inclusion of all the inter-shell
couplings among the Xe$^+$ $4d$, $5s$ and $5p$ subshells using the term-dependent
Hartree-Fock wave functions.

In this paper we use our recently developed RPAE methods to study the photoionization of
the I$^+$ ion. The difference between the current calculation and the previous RPAE calculations
[9, 12-13] is
that the photoionization cross section for
each term of the
I$^+$ ground state, I$^+$ $4d^{10} 5s^2 5p^4$ $(^3P),\ (^1D), \ (^1S)$ has been calculated
separately with the inclusion of all the inter-shell coupling among the I$^+$ $4d$, $5s$ and $5p$
subshells and without making any of the approximations used in the previous RPAE calculations.
The total cross section is evaluated as the weighted average sum of the cross sections
for each term.
Our maximum for the I$^+$ $4d$ giant resonance shows an excellent agreement with the
recent measurement [14].

\section{Theory}

The RPAE equation and corresconding dipole and Coulomb
matrix elements for an atom with an outer open-shell
is given by Eq. (1) and Appendix A  of Ref. [8].
The Coulomb interaction in Eq. (1) causes an electron to switch
from $l_2$ to $l_1$. Similar terms will be added in the
calculations for the electon switching from $l_1$ to $l_3$, or from $l_2$ to $l_3$ etc.
of the state $|l_3^{n_3} [L_3 S_3] l_1^{n_1}[L_1 S_1] l_2^{n_2} [L_2 S_2] LS>$.

The ground state of the I$^+$ atom has the configuration
$4d^{10} 5s^2 5p^4 (^3P,\ ^1D,\ ^1S)$. Therefore our calculations have
three parts, each calculating one term of the ground state.

The following scattering processes are for
the I$^+$ $4d^{10} 5s^2 5p^4 (^3P)$ term:

\bigskip

\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P)\rightarrow 4d^{9} 5s^2 5p^4 (^3P) (^2D,\ ^2F,\ ^4D,\ ^4F)
\epsilon f (^3P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu +4d^{10} 5s^2 5p^4 (^3P) \rightarrow  4d^{9} 5s^2 5p^4 (^3P)(^2P,\ ^2D,\ ^2F,\ ^4P,\ ^4D,\ ^4F)
\epsilon f (^3D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu +4d^{10} 5s^2 5p^4 (^3P) \rightarrow  4d^{9} 5s^2 5p^4 (^3P) (^2F,\ ^4F)
\epsilon f (^3S)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s^2 5p^3 (^2D)
\epsilon s (^3D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s^2 5p^3 (^2P)
\epsilon s (^3P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s^2 5p^3 (^4S)
\epsilon s (^3S)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s^2 5p^3 (^2P,\ ^2D)
\epsilon d (^3D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s^2 5p^3 (^2P,\ ^2D)
\epsilon d (^3P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s^2 5p^3 (^2D)
\epsilon d (^3S)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s^2 5p^3 (^4S)
\epsilon d (^3D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s 5p^4 (^3P) (^2P,\ ^4P)
\epsilon p (^3S)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s 5p^4 (^3P) (^2P,\ ^4P)
\epsilon p (^3P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P) \rightarrow 4d^{10} 5s 5p^4 (^3P) (^2P,\ ^4P)
\epsilon p (^3D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^3P)\rightarrow 4d^{9} 5s^2 5p^4 (^3P) (^2D,\ ^2F,\ ^2P,\ ^4D,\ ^4F,\ ^4P)
\epsilon p (^3D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu +4d^{10} 5s^2 5p^4 (^3P) \rightarrow  4d^{9} 5s^2 5p^4 (^3P) (^2P,\ ^4P,\ ^2D,\ ^4D)
\epsilon p (^3P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu +4d^{10} 5s^2 5p^4 (^3P) \rightarrow  4d^{9} 5s^2 5p^4 (^3P) (^2P,\ ^4P)
\epsilon p (^3S)
\end{displaymath}

The total of 13 $(^3P)$, 18 $(^3D)$, and 8 $(^3S)$ final states have been included in the calculation.

\ \ \ \ The following scattering processes are for the
I$^+$ $4d^{10} 5s^2 5p^4 (^1D)$ term:

\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D)\rightarrow 4d^{9} 5s^2 5p^4 (^1D) (^2D,\ ^2F,\ ^2G)
\epsilon f (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu +4d^{10} 5s^2 5p^4 (^1D) \rightarrow  4d^{9} 5s^2 5p^4 (^1D) (^2P,\ ^2D,\ ^2F,\ ^2G)
\epsilon f (^1D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu +4d^{10} 5s^2 5p^4 (^1D) \rightarrow  4d^{9} 5s^2 5p^4 (^1D) (^2S,\ ^2P,\ ^2D,\ ^2F,\ ^2G)
\epsilon f (^1F)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{10} 5s^2 5p^3 (^2P)
\epsilon s (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{10} 5s^2 5p^3 (^2D)
\epsilon s (^1D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{10} 5s^2 5p^3 (^2P,\ ^2D)
\epsilon d (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{10} 5s^2 5p^3 (^2P,\ ^2D)
\epsilon d (^1D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{10} 5s^2 5p^3 (^2P,\ ^2D)
\epsilon d (^1F)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{10} 5s 5p^4 (^1D) (^2D)
\epsilon p (^1D,\ ^1P,\ ^1F)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{9} 5s^2 5p^4 (^1D) (^2D,\ ^2F,\ ^2P)
\epsilon p (^1D)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{9} 5s^2 5p^4 (^1D) (^2P,\ ^2D,\ ^2S)
\epsilon p (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1D) \rightarrow 4d^{9} 5s^2 5p^4 (^1D) (^2D,\ ^2F,\ ^2G)
\epsilon p (^1F)
\end{displaymath}
The total of 10 $(^1P)$, 11 $(^1D)$, and 11 $(^1F)$ final states have been included in the calculation.

\bigskip
The following scattering processes are for the
I$^+$ $4d^{10} 5s^2 5p^4 (^1S)$ term:
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1S)\rightarrow 4d^{9} 5s^2 5p^4 (^1S) (^2D)
\epsilon f (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1S) \rightarrow 4d^{10} 5s^2 5p^3 (^2P)
\epsilon s (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1S) \rightarrow 4d^{10} 5s^2 5p^3 (^2P)
\epsilon d (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1S) \rightarrow 4d^{10} 5s 5p^4 (^1S) (^2S)
\epsilon p (^1P)
\end{displaymath}
\begin{displaymath}
\ \ h\nu + 4d^{10} 5s^2 5p^4 (^1S)\rightarrow 4d^{9} 5s^2 5p^4 (^1S) (^2D)
\epsilon p (^1P)
\end{displaymath}
A total of 5 $(^1P)$ final states have been included in the calculation.

The I$^+$ ground state and the core wave functions were obtained through the
self-consistent
Hartree-Fock (HF) calculation. Then the radial functions of the continuum electron were
obtained by solving the linear HF equations without self-consistency using those core
wave functions. Each channel includes three discrete and twenty
continuum wave functions. Each radial part of the wave function was represented by 2000
points.

After evaluating the dipole and Coulomb matrix elements, the RPAE equation was solved for the
partial cross sections for the $^3P$, $^3D$, $^3S$, $^1P$, $^1D$, and $^1F$ final states.
Then the photoionization cross sections for each term, $^3P$, $^1D$ and $^1S$ was obtained
from the sum of the
partial cross sections. Finally the total photoionization cross section was evaluated as the
weighted average of each term.

\section{Results}
The I$^+$ $4d -\epsilon f$ photoionization cross section is given in Fig.1. Dotted, dashed and
dash-dot
curves represent, respectively the photoionization cross sections from the
term, $4d^{10} 5s^2 5p^4 (^3P)$, $(^1S,)$ and $(^1D)$. The three curves are very
close to each other.
Fig. 1. shows a broad region corresponding to the $4d -\epsilon f$ giant resonance.
It is a
shape resonance which is
mainly determined by the potential shape. Fig. 1. gives a maximum cross section of 20.64 MB
at the photon energy
of 90.24 eV.
Since almost all of the $4d - \epsilon f$ transition will emit a electron through
double Auger decay,
this data
should be compared with the experimental data for the I$^{+3}$ ion.

Fig. 2. shows
the $4d -\epsilon p$
photoionization cross sections.
Obiviously, the
$4d-\epsilon p$ cross sections are much smaller in comparison than those for
the $4d-\epsilon f$ transition.
The description of the curves is the same as that
in Fig. 1. At about 67 eV, which is near the 4d threshold the curve has a peak. Then, the cross
section gradualy decreases as the energy increases.
Around 90 eV all three curves have a very small broadened peak. This is caused by the
inter-shell coupling with the $4d-\epsilon f$
channel. After carefully analyzing the partial cross sections, we found that
for the ground state term
$4d^{10} 5s^2 5p^4 (^3P)$ the largest contribution to the cross section is
from the transition to the final state $^3D$. However, for the term
$4d^{10} 5s^2 5p^4 (^1D)$ the
largest contributor to the cross section comes from the final state $^1F$.

The photoionization cross sections for the $5p$ and $5s$ subshells
are given, respectively in Fig. 3 and Fig. 4. Fig. 3 demonstrates
the $5p- \epsilon d, \epsilon s$
photoionization results. At about 19.38 eV the cross section is 17.89 MB.
Then the cross section
decreases as the energy increases. At 62.65 eV the cross section has reduced to 0.44 MB.
After entering
the energy range of the 4d giant resonance the cross section increases and
reaches a maximum.
A similar situation applies to the 5s photoionization process.
However, firstly the cross section increases as the energy increases.
Secondly, when the energy region of the 4d giant resonance is entered,
the cross section increases sharply
because of the inter-shell coupling with the $4d$-$\epsilon f$ channel.

Fig. 5. shows the total photoionization cross section, which is equal to the sum of the
$4d-\epsilon f$, $4d- \epsilon p$, $5s-\epsilon p$, and
$5p-\epsilon d, \epsilon s$ cross sections.  In Fig. 5 the solid curve represents our
average of
three terms, $4d^{10} 5s^2 5p^4 (^3P,\ ^1D,\ ^1S)$. The dashed curve is from reference [10]
using the time-dependent local-density spin approximation with an optimized effective potential
and self-interaction correction (TDLSDA/OEP-SIC).
The dotted curve is the experimental data [14], and the dash-dot curve
demonstrates the previous RPAE
calculation [13].
Our result shows a maximum of 23.12 MB at 90.24 eV, which is
in excellent agreement with the recent experimental
result, 23(3) MB at 90 eV [14].

In conclusion, we used our recently deveploped RPAE method to study the
photoionization cross sections of the I$^+$ ion
with inclusion of all the inter-shell coupling among the $4d$, $5s$, and $5p$ subshells.
We obtained, for the first time the
photoionization cross sections for each term of the I$^+$ ground state.
Our maximum cross section for the I$^+$ $4d$ giant resonance,  23.12 MB at the
photon energy of 90.24 eV is in excellent
agreement
with the experimental data,
23(3)MB at 90 eV.

\bigskip
\leftline{\bf Acknowledgments}
\bigskip

This work was supported by the U.S. DOE, Division of Chemical Sciences,
Geosciences and Biosciences, Office of Basic Energy Sciences, OER.
\bigskip

\leftline{\bf References}
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\leftline{\bf Figure Captions}
Fig. 1. Photoionization cross section for the I$^+$ $4d-\epsilon f$ for the ground state terms of
$(^3P,\ ^1D,\ ^1S)$. Dotted, dashed and dash-dot curves represent, respectively the results for
$^3P$, $^1D$ and $^1S$.

Fig. 2. Curves have the same meaning as those in Fig. 1, except that they demonstrate the I$^+$
$4d-\epsilon p$ photoionization cross section.

Fig. 3. Curves have the same meaning as those in Fig. 1, except that they demonstrate the I$^+$
$5p$ photoionization cross section.

Fig. 4. Curves have the same meaning as those in Fig. 1, except that they demonstrate the I$^+$
$5s$ photoionization cross section.

Fig. 5. Total cross sections for the I$^+$ photoionization in the energy region of $4d -\epsilon f$
giant resonance versus photon energy (eV). Solid curve is from our average
cross sections for the terms
$(^3P),\ (^1D)$, and $(^1S)$, dotted curve represents the experimental data, dashed curve shows the
calculated result from reference [10], and the dash-dot curve is from the
calculation of Amusia et al [13].
\end{document}