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results.tex
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{longtable}
\usepackage{lscape}
\usepackage{multicol}
%\usepackage{nopageno}
\usepackage{algorithm2e}
\begin{document}
\title{Computation of the Kinetics of Chemical Reaction Networks}
\date{June 15, 2020}
\author{Cristian Vargas Montero and Hamid Rahkooy}
\maketitle
\section*{Graph Theoretical Algorithm for Testing Unconditional Binomiality of RCRN}
Algorithm \ref{alg:binom-graph} tests unconditional binomiality via graphs for RCRN. Several functions used in Algorithm \ref{alg:binom-graph} are
briefly explained below.
\begin{itemize}
\item \texttt{GetReactionVertices(graph)}: obtains all the reaction vertices of a graph
\item \texttt{GetSpeciesVertices(graph)}: obtains all the species vertices of a graph
\item \texttt{SetMark(species vertex,boolean)}: creates a flag for a species
vertex, which is initially set to unmarked(false).
\item \texttt{GetConnectedSpecies(reaction vertex,graph)}: obtains all the
species vertices that are connected to a reaction vertex
\item \texttt{IsNotMarked(species vertex)}: tells if a species vertex is not marked
\item \texttt{ElimEdge(graph, species vertex, reaction vertex)}:
eliminates the edge that connects a species vertex and reaction
vertex
\item \texttt{GetConnectedReactions(graph, species vertex)}: obtains all the
reaction vertices that are connected to a species vertex
\item \texttt{AreConnected(graph,species vertex,reaction vertex)}: tells if a
species vertex is connected to a reaction vertex
\item \texttt{GetCoeff(graph,species vertex,reaction vertex)}: obtains the
coeffient of the edge that goes from a species vertex to a reaction
vertex
\item \texttt{UpdCoef(graph,species vertex,reaction vertex,coefficient)}:
updates the coefficient of the edge that goes from a species vertex
to a reaction vertex
\item \texttt{AddEdge(graph, species vertex,reaction vertex,coefficient)}:
creates a new edge that connects a species vertex to a reaction
vertex and has the coefficient given
\end{itemize}
Functions \ref{alg:create-graph} and \ref{alg:checkb-graph} (used in Algorithm \ref{alg:binom-graph}) contain several methods that
will be briefly explained:
\begin{itemize}
\item \texttt{CreateSpeciesVertex(graph,species)}: creates a species vertex in
the graph
\item \texttt{CreateReactionVertex(graph,reaction)}: creates a reaction vertex
in the graph. Returns the reaction vertex created
\item \texttt{GetSpecies(reaction)}: obtains the species that are present in
the reaction
\item \texttt{GetSpeciesCoeff(species,reaction)}: obtains the coefficient of a
species in a reaction
\item \texttt{GetSpeciesVertex(graph,species)}: obtains the species vertex of a
species in the graph
\item \texttt{GetGraphComponents(graph)}: obtains all the components present in
a graph
\end{itemize}
\begin{algorithm}[H]
\footnotesize
\caption{Testing unconditional binomiality via graphs \label{alg:binom-graph}}
\DontPrintSemicolon
\SetAlgoVlined
\LinesNumbered
% \SetNlSty{}{}{}
% \SetNlSkip{1em}
\SetKwProg{Fn}{Function}{}{end}
\SetKwFunction{BINOMTEST}{BinomialityTestViaGraph}
\SetKwFunction{CREATEGRAPH}{CreateGraph}
\SetKwFunction{MRK}{SetMark}
\SetKwFunction{GS}{GetConnectedSpecies}
\SetKwFunction{GR}{GetConnectedReactions}
\SetKwFunction{ISNMRK}{IsNotMarked}
\SetKwFunction{GCO}{GetCoeff}
\SetKwFunction{ICO}{UpdCf}
\SetKwFunction{ELED}{ElimEdge}
\SetKwFunction{ADDED}{AddEdge}
\SetKwFunction{CONN}{AreConnected}
\SetKwFunction{IB}{IsUnconditionallyBinomial}
\SetKwFunction{GETSVS}{GetSpeciesVertices}
\SetKwFunction{GETRVS}{GetReactionVertices}
\Fn{\BINOMTEST{$\mathcal{S,R,G}$}}{
\KwIn{\\$\mathcal{S}$: set of species of the RCRN\\$\mathcal{R}$: set of reactions of the RCRN}
\KwOut{UnconditionallyBinomial or NotUnconditionallyBinomial}
$\mathcal{G} := \CREATEGRAPH{$\mathcal{S,R}$}$\\
$SV := \GETSVS{$\mathcal{G}$}$\\
$RV := \GETRVS{$\mathcal{G}$}$\\
\ForEach{$s \in SV$}{
$\MRK{s,false} $
}
\ForEach{$r \in RV$}{
$speciesFound := false$\\
$cs := null$\\
$rSpecies := \GS{$\mathcal{G},r$}$\\
\ForEach{$sr \in rSpecies$}{
\uIf{\ISNMRK{$sr$}}{
$speciesFound := true$\\
$cs := sr$\\
$break$
}
}
\uIf{$speciesFound$}{
$\MRK{cs,true} $\\
\ForEach{$s^{\prime} \in rSpecies$}{
\uIf{$s^{\prime} \neq cs$}{
$multX :=\frac{-\GCO{$\mathcal{G}$,$s^{\prime},r$}}{\GCO{$\mathcal{G}$,cs,r}}$\\
$\ELED{$\mathcal{G},s^{\prime},r$}$\\
$sReactions := \GR{$\mathcal{G},cs$}$\\
\ForEach{$r^{\prime} \in sReactions$}{
\uIf{$r^{\prime} \neq r$}{
\uIf{$\CONN{$\mathcal{G},s^{\prime},r^{\prime}$}$}{
$cf := \GCO{$\mathcal{G},cs,r^{\prime}$}*multiX + \GCO{$\mathcal{G},s^{\prime},r^{\prime}$}$\\
\uIf{$cf \neq 0$}{
$\ICO{$\mathcal{G},s^{\prime},r^{\prime},cf$}$\\
}
\Else{
$\ELED{$\mathcal{G},s^{\prime},r^{\prime}$}$\\
}
}
\Else{
$cf := \GCO{$\mathcal{G},cs,r^{\prime}$}*multiX$\\ $\ADDED{$\mathcal{G},s^{\prime},r^{\prime},cf$}$
}
}
}
}
}
}
}
\uIf{\IB{$\mathcal{G}$}}{
$R:=UnconditionallyBinomial$\;
}
\Else{$R:=NotUnconditionallyBinomial$\;
}
\Return{$R$}\;
}
\end{algorithm}
\begin{algorithm}[H]
% \begin{algorithm}
\SetAlgorithmName{Function}{}{}
\caption{Creating modified species--reaction graphs. \label{alg:create-graph}}
\DontPrintSemicolon
\SetAlgoVlined
\SetKwProg{Fn}{Function}{}{end}
\SetKwFunction{CREATEGRAPH}{CreateGraph}
\SetKwFunction{CRSV}{CreateSpeciesVertex}
\SetKwFunction{CRRV}{CreateReactionVertex}
\SetKwFunction{GETSPV}{GetSpeciesVertex}
\SetKwFunction{GETSP}{GetSpecies}
\SetKwFunction{GETSPCO}{GetSpeciesCoeff}
\Fn{\CREATEGRAPH{$\mathcal{S,R}$}}{
\KwIn{\\$\mathcal{S}$: set of species of the RCRN\\$\mathcal{R}$: set of reactions of the RCRN\\ }
\KwOut{$\mathcal{G}$ graph representation of the RCRN}
$\mathcal{G} := empty$\\
\ForEach{$s \in \mathcal{S}$}{
$\CRSV{$\mathcal{G},s$} $
}
\ForEach{$r \in \mathcal{R}$}{
$rv := \CRRV{$\mathcal{G},r$} $\\
$speciesInReaction := \GETSP{r}$\\
\ForEach{$k \in speciesInReaction$}{
$coef := \GETSPCO{k,r}$\\
$spv := \GETSPV{$\mathcal{G},k$}$\\
$\ADDED{$\mathcal{G},spv,rv,coef$}$
}
}
\Return{$\mathcal{G}$}\;
}
\end{algorithm}
\begin{algorithm}[H]
%\begin{algorithm}
\SetAlgorithmName{Function}{}{}
\caption{Testing unconditional binomiality in
graphs \label{alg:checkb-graph}
}
\DontPrintSemicolon
\SetAlgoVlined
\SetKwProg{Fn}{Function}{}{end}
\SetKwFunction{GETRVS}{GetReactionVertices}
\SetKwFunction{GETGCMP}{GetGraphComponents}
\SetKwFunction{IB}{IsUnconditionallyBinomial}
\Fn{\IB{$\mathcal{G}$}}{
\KwIn{$\mathcal{G}$ graph representation of the RCRN}
\KwOut{boolean indicating if the RCRN is unconditionally binomial or not}
$result := true$\\
$components := \GETGCMP{$\mathcal{G}$}$\\
\ForEach{$c \in components$}{
$reactionV := \GETRVS{c}$\\
\uIf{$size(reactionV) > 1$}{
$result := false$\\
$break$
}
}
\Return{$result$}\;
}
\end{algorithm}
\section*{Comparison of Performance of the Graph Theoretical Algorithm with Other Algorithms}
In Table \ref{t:res}, $t_{1}$ is the time in seconds for the linear algebra Algorithm 1 from \cite{rahkooy2020linear}, $t_{2}$ is the time for the graph theoretical algorithm, $t_{3}$ and $t_{4}$ represent the time in seconds for Algorithms 1 and 4 from Table 1 in \cite{grigoriev2019efficiently} and $\bot$ identifies the computations that did not finish and a limit of six hours was reached.
\begin{landscape}
\setlength{\tabcolsep}{2pt}
\begin{small}
%\setlength\LTleft{-5in}
\begin{longtable}[c]{| c | c | c | c |c |c |c |c |p{1.5cm} |p{1.5cm} |p{1.5cm} |p{1.5cm} |p{1.5cm} |p{1.5cm} |}
\caption{Comparison of CPU times in seconds of Algorithm 1 from \cite{rahkooy2020linear}, Algorithm 1 that computes the graph theoretical approach and Algorithms 1 and 4 from \cite{grigoriev2019efficiently}\label{t:res}}\\
\hline
& \multicolumn{3}{|c|}{Linear Algebra Approach} & \multicolumn{4}{|c|}{Graph Approach} &\multicolumn{2}{|c|}{}&\multicolumn{4}{|c|}{Comparison with Table 1 from \cite{grigoriev2019efficiently}}\\
\hline
Biomodel & time(s)$(t_{1})$ & Matrix size & Binomial & time(s)$(t_{2})$ & Vertices & Edges &Binomial & Difference \par$(t_{1}-t_{2})$ & Ratio\par$(t_{1}/t_{2})$ & Difference \par$(t_{3}-t_{2})$&Ratio\par$(t_{3}/t_{2})$ &Difference\par$(t_{4}-t_{2})$ &Ratio\par$(t_{4}/t_{2})$\\
\hline
\endfirsthead
\hline
&\multicolumn{3}{|c|}{Linear Algebra Approach}& \multicolumn{4}{|c|}{Graph Approach}&\multicolumn{2}{|c|}{}&\multicolumn{4}{|c|}{Comparison with Table 1 from \cite{grigoriev2019efficiently} }\\
\hline
Biomodel & time(s)$(t_{1})$ & Matrix size & Binomial& time(s)$(t_{2})$ & Vertices & Edges &Binomial& Difference\par$(t_{1}-t_{2})$ & Ratio\par$(t_{1}/t_{2})$& Difference\par$(t_{3}-t_{2})$&Ratio\par$(t_{3}/t_{2})$ &Difference\par$(t_{4}-t_{2})$ &Ratio\par$(t_{4}/t_{2})$\\
\hline
\endhead
\hline
\endfoot
\hline
%&\multicolumn{3}{|c|}{Linear Algebra Approach}& \multicolumn{4}{|c|}{Graph Approach}\\
%\hline\hline
%\endlastfoot
2 & 0.016 & 13x17 & No & 0.016 & 30 & 42 & No & 0 & 1 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
9 & 0.016 & 22x20 & No & 0.031 & 42 & 60 & No & -0.015 & 0.52 & 85.929 & 2772.903 & 21.229 & 685.806\\
10 & 0.016 & 8x10 & No & 0.016 & 18 & 20 & No & 0 & 1 & & & & \\
11 & 0.015 & 22x30 & No & 0.031 & 52 & 90 & No & -0.016 & 0.48 & 150.179 & 4845.484 & 11.169 & 361.29\\
13 & 0.032 & 29x21 & Yes & 0.047 & 50 & 70 & Yes & -0.015 & 0.68 & & & & \\
14 & 2.437 & 86x300 & No & 2.484 & 386 & 886 & No & -0.047 & 0.98 & & & & \\
16 & 0.015 & 6x10 & No & 0.016 & 16 & 20 & No & -0.001 & 0.94 & & & & \\
26 & 0.016 & 11x10 & No & 0.016 & 21 & 28 & No & 0 & 1 & 3.974 & 249.375 & 1.104 & 70\\
27 & 0.015 & 3x4 & No & 0.015 & 7 & 8 & No & 0 & 1 & & & & \\
28 & 0.015 & 16x17 & No & 0.032 & 33 & 48 & No & -0.017 & 0.47 & 66.568 & 2081.25 & $\bot$ & $\bot$\\
29 & 0.016 & 4x7 & No & 0.015 & 11 & 14 & No & 0.001 & 1.07 & & & & \\
30 & 0.031 & 18x20 & No & 0.031 & 38 & 56 & No & 0 & 1 & 57.809 & 1865.806 & $\bot$ & $\bot$\\
31 & 0.016 & 3x4 & No & 0.015 & 7 & 8 & No & 0.001 & 1.07 & & & & \\
35 & 0.016 & 10x16 & No & 0.016 & 26 & 31 & No & 0 & 1 & 21.094 & 1319.375 & 0.234 & 15.625\\
38 & 0.015 & 17x10 & Yes & 0.016 & 27 & 30 & Yes & -0.001 & 0.94 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
40 & 0.015 & 7x5 & Yes & 0.015 & 12 & 16 & Yes & 0 & 1 & 4.635 & 310 & 0.035 & 3.333\\
41 & 0.016 & 10x9 & No & 0.032 & 19 & 24 & No & -0.016 & 0.5 & & & & \\
46 & 0.016 & 18x15 & No & 0.031 & 33 & 48 & No & -0.015 & 0.52 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
48 & 0.031 & 24x25 & No & 0.031 & 49 & 67 & No & 0 & 1 & & & & \\
50 & 0.016 & 14x16 & No & 0.031 & 30 & 41 & No & -0.015 & 0.52 & 0.059 & 2.903 & -0.011 & 0.645\\
52 & 0.015 & 12x11 & No & 0.016 & 23 & 28 & No & -0.001 & 0.94 & 0.074 & 5.625 & -0.006 & 0.625\\
57 & 0.016 & 6x5 & Yes & 0.015 & 11 & 10 & Yes & 0.001 & 1.07 & 1.515 & 102 & 0.095 & 7.333\\
60 & 0.016 & 4x3 & Yes & 0.015 & 7 & 6 & Yes & 0.001 & 1.07 & & & & \\
61 & 0.047 & 30x24 & Yes & 0.047 & 54 & 66 & Yes & 0 & 1 & & & & \\
69 & 0.016 & 10x7 & No & 0.016 & 17 & 15 & No & 0 & 1 & 14.074 & 880.625 & $\bot$ & $\bot$\\
70 & 0.094 & 45x38 & No & 0.109 & 83 & 118 & No & -0.015 & 0.86 & & & & \\
76 & 0.015 & 3x2 & Yes & 0.016 & 5 & 4 & Yes & -0.001 & 0.94 & & & & \\
80 & 0.015 & 10x6 & Yes & 0.016 & 16 & 17 & Yes & -0.001 & 0.94 & 3.704 & 232.5 & 0.074 & 5.625\\
82 & 0.016 & 10x6 & Yes & 0.016 & 16 & 17 & Yes & 0 & 1 & 2.424 & 152.5 & 0.124 & 8.75\\
84 & 0.016 & 8x8 & No & 0.016 & 16 & 16 & No & 0 & 1 & & & & \\
85 & 0.031 & 17x17 & No & 0.062 & 34 & 51 & No & -0.031 & 0.5 & -0.062 & 0 & $\bot$ & $\bot$\\
86 & 0.016 & 17x24 & No & 0.031 & 41 & 72 & No & -0.015 & 0.52 & -0.031 & 0 & $\bot$ & $\bot$\\
92 & 0.016 & 5x3 & Yes & 0.016 & 8 & 9 & Yes & 0 & 1 & 1.854 & 116.875 & 0.024 & 2.5\\
105 & 0.281 & 50x94 & No & 0.391 & 144 & 299 & No & -0.11 & 0.72 & -0.151 & 0.614 & 0.119 & 1.304\\
123 & 0.016 & 16x17 & No & 0.031 & 33 & 40 & No & -0.015 & 0.52 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
146 & 0.062 & 34x34 & No & 0.047 & 68 & 76 & No & 0.015 & 1.32 & & & & \\
150 & 0.016 & 4x2 & Yes & 0.016 & 6 & 5 & Yes & 0 & 1 & 3.414 & 214.375 & 0.024 & 2.5\\
183 & 2.422 & 67x352 & No & 4.031 & 419 & 1024 & No & -1.609 & 0.6 & & & & \\
192 & 0.016 & 12x9 & No & 0.016 & 21 & 24 & No & 0 & 1 & & & & \\
200 & 0.047 & 21x34 & No & 0.093 & 55 & 104 & No & -0.046 & 0.51 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
203 & 0.015 & 10x10 & Yes & 0.016 & 20 & 21 & Yes & -0.001 & 0.94 & & & & \\
204 & 0.016 & 10x10 & Yes & 0.016 & 20 & 21 & Yes & 0 & 1 & & & & \\
205 & 6.218 & 195x205 & No & 0.765 & 400 & 590 & No & 5.453 & 8.13 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
209 & 0.016 & 13x12 & Yes & 0.016 & 25 & 24 & Yes & 0 & 1 & & & & \\
210 & 0.016 & 13x12 & Yes & 0.015 & 25 & 24 & Yes & 0.001 & 1.07 & & & & \\
220 & 0.125 & 58x42 & No & 0.063 & 100 & 119 & No & 0.062 & 1.98 & 5.207 & 83.651 & 88.087 & 1399.206\\
221 & 0.015 & 12x11 & Yes & 0.016 & 23 & 27 & Yes & -0.001 & 0.94 & & & & \\
222 & 0.016 & 12x11 & Yes & 0.015 & 23 & 27 & Yes & 0.001 & 1.07 & & & & \\
226 & 0.032 & 27x24 & No & 0.031 & 51 & 51 & No & 0.001 & 1.03 & 14.159 & 457.742 & $\bot$ & $\bot$\\
230 & 0.093 & 28x64 & No & 0.094 & 92 & 165 & No & -0.001 & 0.99 & 47.626 & 507.66 & $\bot$ & $\bot$\\
231 & 0.016 & 7x4 & Yes & 0.015 & 11 & 11 & Yes & 0.001 & 1.07 & & & & \\
233 & 0.016 & 5x4 & No & 0.015 & 9 & 10 & No & 0.001 & 1.07 & 1.015 & 68.667 & -0.005 & 0.667\\
258 & 0.016 & 3x4 & No & 0.015 & 7 & 8 & No & 0.001 & 1.07 & & & & \\
259 & 0.031 & 17x29 & No & 0.078 & 46 & 58 & No & -0.047 & 0.4 & 0.302 & 4.872 & 0.092 & 2.179\\
260 & 0.031 & 17x29 & No & 0.047 & 46 & 58 & No & -0.016 & 0.66 & 0.043 & 1.915 & 0.113 & 3.404\\
261 & 0.016 & 17x29 & No & 0.063 & 46 & 58 & No & -0.047 & 0.25 & 0.027 & 1.429 & 0.107 & 2.698\\
267 & 0.015 & 4x3 & Yes & 0.016 & 7 & 6 & Yes & -0.001 & 0.94 & 0.074 & 5.625 & -0.016 & 0\\
282 & 0.016 & 6x3 & Yes & 0.015 & 9 & 11 & Yes & 0.001 & 1.07 & 0.145 & 10.667 & -0.005 & 0.667\\
283 & 0.016 & 4x2 & Yes & 0.015 & 6 & 6 & Yes & 0.001 & 1.07 & 0.265 & 18.667 & -0.005 & 0.667\\
284 & 0.016 & 9x8 & Yes & 0.016 & 17 & 16 & Yes & 0 & 1 & & & & \\
292 & 0.015 & 8x4 & Yes & 0.015 & 12 & 14 & Yes & 0 & 1 & 0.145 & 10.667 & -0.005 & 0.667\\
293 & 8.406 & 149x316 & No & 2.172 & 465 & 1101 & No & 6.234 & 3.87 & & & & \\
296 & 0.016 & 6x8 & No & 0.016 & 14 & 16 & No & 0 & 1 & & & & \\
305 & 0.016 & 9x4 & Yes & 0.015 & 13 & 18 & Yes & 0.001 & 1.07 & & & & \\
315 & 0.016 & 20x27 & No & 0.047 & 47 & 74 & No & -0.031 & 0.34 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
328 & 0.031 & 18x29 & No & 0.031 & 47 & 58 & No & 0 & 1 & & & & \\
332 & 0.313 & 77x69 & No & 0.204 & 146 & 206 & No & 0.109 & 1.53 & $\bot$ & $\bot$ & 395.766 & 1941.029\\
333 & 0.14 & 54x45 & No & 0.109 & 99 & 134 & No & 0.031 & 1.28 & 572.991 & 5257.798 & 120.871 & 1109.908\\
334 & 0.579 & 73x66 & No & 0.203 & 139 & 196 & No & 0.376 & 2.85 & $\bot$ & $\bot$ & 271.197 & 1336.946\\
335 & 0.046 & 34x31 & No & 0.078 & 65 & 88 & No & -0.032 & 0.59 & 97.442 & 1250.256 & $\bot$ & $\bot$\\
357 & 0.015 & 9x8 & No & 0.016 & 17 & 24 & No & -0.001 & 0.94 & 0.184 & 12.5 & 0.084 & 6.25\\
359 & 0.016 & 9x8 & No & 0.015 & 17 & 23 & No & 0.001 & 1.07 & 0.195 & 14 & 0.065 & 5.333\\
360 & 0.015 & 9x8 & No & 0.016 & 17 & 22 & No & -0.001 & 0.94 & 0.914 & 58.125 & 0.054 & 4.375\\
361 & 0.016 & 8x5 & Yes & 0.015 & 13 & 14 & Yes & 0.001 & 1.07 & 1.025 & 69.333 & 0.025 & 2.667\\
362 & 0.078 & 34x33 & No & 0.078 & 67 & 92 & No & 0 & 1 & 1304.632 & 16727.051 & 429.342 & 5505.385\\
363 & 0.015 & 4x4 & No & 0.016 & 8 & 8 & No & -0.001 & 0.94 & 0.054 & 4.375 & -0.016 & 0\\
364 & 0.016 & 13x16 & No & 0.031 & 29 & 44 & No & -0.015 & 0.52 & 0.999 & 33.226 & 0.509 & 17.419\\
365 & 0.062 & 30x33 & No & 0.109 & 63 & 90 & No & -0.047 & 0.57 & 211.901 & 1945.046 & $\bot$ & $\bot$\\
389 & 0.031 & 28x18 & Yes & 0.031 & 46 & 60 & Yes & 0 & 1 & & & & \\
396 & 0.016 & 29x16 & Yes & 0.016 & 45 & 44 & Yes & 0 & 1 & & & & \\
397 & 0.078 & 49x29 & Yes & 0.047 & 78 & 83 & Yes & 0.031 & 1.66 & & & & \\
415 & 0.016 & 28x5 & Yes & 0.032 & 33 & 31 & Yes & -0.016 & 0.5 & & & & \\
423 & 0.015 & 9x11 & No & 0.015 & 20 & 22 & No & 0 & 1 & & & & \\
430 & 0.031 & 27x28 & No & 0.047 & 55 & 84 & No & -0.016 & 0.66 & $\bot$ & $\bot$ & 13.943 & 297.66\\
431 & 0.047 & 27x28 & No & 0.031 & 55 & 80 & No & 0.016 & 1.52 & $\bot$ & $\bot$ & 31.869 & 1029.032\\
432 & 0.016 & 8x10 & No & 0.016 & 18 & 20 & No & 0 & 1 & & & & \\
433 & 0.015 & 8x10 & No & 0.016 & 18 & 20 & No & -0.001 & 0.94 & & & & \\
438 & 0.016 & 10x7 & Yes & 0.015 & 17 & 16 & Yes & 0.001 & 1.07 & & & & \\
440 & 0.015 & 8x10 & No & 0.016 & 18 & 20 & No & -0.001 & 0.94 & & & & \\
441 & 0.016 & 8x10 & No & 0.016 & 18 & 20 & No & 0 & 1 & & & & \\
442 & 0.015 & 10x12 & No & 0.015 & 22 & 24 & No & 0 & 1 & & & & \\
452 & 0.954 & 108x117 & No & 0.25 & 225 & 344 & No & 0.704 & 3.82 & & & & \\
453 & 1.266 & 108x117 & No & 0.235 & 225 & 344 & No & 1.031 & 5.39 & & & & \\
454 & 0.015 & 8x4 & Yes & 0.015 & 12 & 12 & Yes & 0 & 1 & & & & \\
455 & 0.015 & 9x5 & Yes & 0.016 & 14 & 14 & Yes & -0.001 & 0.94 & & & & \\
456 & 0.016 & 11x6 & Yes & 0.016 & 17 & 16 & Yes & 0 & 1 & & & & \\
457 & 3.438 & 165x141 & Yes & 1.094 & 306 & 453 & Yes & 2.344 & 3.14 & & & & \\
458 & 0.016 & 4x3 & Yes & 0.016 & 7 & 6 & Yes & 0 & 1 & & & & \\
459 & 0.015 & 4x3 & Yes & 0.016 & 7 & 6 & Yes & -0.001 & 0.94 & 0.484 & 31.25 & 0.074 & 5.625\\
460 & 0.016 & 4x3 & Yes & 0.015 & 7 & 6 & Yes & 0.001 & 1.07 & 0.505 & 34.667 & 0.025 & 2.667\\
461 & 0.015 & 4x3 & Yes & 0.016 & 7 & 6 & Yes & -0.001 & 0.94 & & & & \\
462 & 0.016 & 9x8 & No & 0.015 & 17 & 17 & No & 0.001 & 1.07 & & & & \\
467 & 0.031 & 18x24 & No & 0.031 & 42 & 62 & No & 0 & 1 & & & & \\
475 & 0.032 & 24x25 & No & 0.047 & 49 & 73 & No & -0.015 & 0.68 & 3.213 & 69.362 & 7.873 & 168.511\\
477 & 0.14 & 43x49 & No & 0.063 & 92 & 98 & No & 0.077 & 2.22 & & & & \\
478 & 0.047 & 37x39 & No & 0.062 & 76 & 109 & No & -0.015 & 0.76 & 33.318 & 538.387 & 16.788 & 271.774\\
483 & 0.016 & 10x8 & No & 0.016 & 18 & 20 & No & 0 & 1 & 1.244 & 78.75 & 0.094 & 6.875\\
484 & 0.015 & 2x2 & No & 0.016 & 4 & 4 & No & -0.001 & 0.94 & 0.124 & 8.75 & -0.006 & 0.625\\
486 & 0.016 & 2x2 & No & 0.016 & 4 & 4 & No & 0 & 1 & 0.614 & 39.375 & 0.004 & 1.25\\
487 & 0.297 & 6x6 & No & 0.015 & 12 & 18 & No & 0.282 & 19.8 & 0.675 & 46 & 0.145 & 10.667\\
491 & 0.344 & 57x86 & No & 0.125 & 143 & 172 & No & 0.219 & 2.75 & 12.085 & 97.68 & $\bot$ & $\bot$\\
492 & 0.25 & 52x88 & No & 0.141 & 140 & 176 & No & 0.109 & 1.77 & 17.819 & 127.376 & $\bot$ & $\bot$\\
500 & 0.032 & 14x18 & No & 0.031 & 32 & 48 & No & 0.001 & 1.03 & & & & \\
501 & 0.093 & 35x57 & No & 0.156 & 92 & 146 & No & -0.063 & 0.6 & & & & \\
504 & 0.578 & 75x132 & No & 0.094 & 207 & 268 & No & 0.484 & 6.15 & $\bot$ & $\bot$ & $\bot$ & $\bot$\\
546 & 0.016 & 7x5 & Yes & 0.016 & 12 & 10 & Yes & 0 & 1 & 0.054 & 4.375 & -0.016 & 0\\
559 & 1.375 & 92x136 & No & 2.14 & 228 & 351 & No & -0.765 & 0.64 & -1.82 & 0.15 & -1.23 & 0.425\\
560 & 0.438 & 66x117 & No & 0.125 & 183 & 244 & No & 0.313 & 3.5 & & & & \\
572 & 0.016 & 31x21 & No & 0.047 & 52 & 73 & No & -0.031 & 0.34 & & & & \\
574 & 24.36 & 195x576 & No & 13.125 & 771 & 1664 & No & 11.235 & 1.86 & & & & \\
581 & 0.032 & 27x32 & No & 0.016 & 59 & 66 & No & 0.016 & 2 & $\bot$ & $\bot$ & 5.984 & 375\\
594 & 0.187 & 29x120 & No & 0.172 & 149 & 304 & No & 0.015 & 1.09 & & & & \\
624 & 0.016 & 7x5 & Yes & 0.015 & 12 & 11 & Yes & 0.001 & 1.07 & & & & \\
629 & 0.016 & 5x2 & Yes & 0.016 & 7 & 6 & Yes & 0 & 1 & 0.394 & 25.625 & 0.064 & 5\\
635 & 0.578 & 78x113 & No & 0.266 & 191 & 338 & No & 0.312 & 2.17 & & & & \\
637 & 0.016 & 12x22 & No & 0.016 & 34 & 56 & No & 0 & 1 & 492.744 & 30797.5 & $\bot$ & $\bot$\\
647 & 0.016 & 11x11 & No & 0.015 & 22 & 34 & No & 0.001 & 1.07 & 6.015 & 402 & 0.355 & 24.667\\
651 & 0.031 & 29x16 & Yes & 0.032 & 45 & 34 & Yes & -0.001 & 0.97 & & & & \\
692 & 0.016 & 8x5 & Yes & 0.015 & 13 & 13 & Yes & 0.001 & 1.07 & & & & \\
724 & 0.031 & 28x28 & No & 0.031 & 56 & 64 & No & 0 & 1 & & & & \\
725 & 0.078 & 51x31 & Yes & 0.079 & 82 & 123 & Yes & -0.001 & 0.99 & & & & \\
732 & 0.016 & 5x6 & No & 0.015 & 11 & 12 & No & 0.001 & 1.07 & & & & \\
\end{longtable}
\end{small}
\end{landscape}
\begin{thebibliography}{10}
\bibitem{grigoriev2019efficiently}
Dima Grigoriev, Alexandru Iosif, Hamid Rahkooy, Thomas Sturm, and Andreas
Weber.
\newblock Efficiently and effectively recognizing toricity of steady state
varieties.
\newblock {\em CoRR}, abs/1910.04100, 2019.
\bibitem{rahkooy2020linear}
Hamid Rahkooy and Thomas Sturm.
\newblock A Linear algebra approach for detecting binomiality of steady state ideals of reversible chemical reaction networks.
\newblock {\em CoRR}, abs/2002.12693, 2020.
\end{thebibliography}
\end{document}
