Introduction 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 Daucus carota 17 Eschscholtzia californica 18 17 Catharanthus roseus 3 − 4 + 19 20 21 C. roseus Model general structure 22 1 2 1 17 SM SM 1 \documentclass[12pt]{minimal}
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\begin{document}$$\frac{{{\text{d}}M}} {{{\text{d}}t}} = S \cdot \Phi - \mu M$$\end{document} S M Fig. 1 Flux numbers 3 4  Fig. 2 Flux numbers 1  Stationary primary metabolic pathways 19 20 2 1 3 2 21 23 N K 2 \documentclass[12pt]{minimal}
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\begin{document}$$N\cdot K=0$$\end{document} N K 3 \documentclass[12pt]{minimal}
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\begin{document}$$N=(N_{1} N_{2})$$\end{document} 4 \documentclass[12pt]{minimal}
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\begin{document}$$K = {\left({{{K_{1} }}{{K_{2} }}} \right)}$$\end{document} N 1 K 2 K 2 K 1 K 2 5 \documentclass[12pt]{minimal}
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\begin{document}$$K_{1} = - {\left({N_{1} } \right)}^{{ - 1}} \cdot N_{2} \cdot K_{2} $$\end{document} K 3 22 Table 1 Biochemical reactions of the stationary primary metabolic pathways Flux a–c \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{GLC} + \hbox{NTP} \rightarrow \hbox{G6P} + \hbox{NDP}$$\end{document} a–c \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{FRU} + \hbox{NTP} \rightarrow \hbox{F6P} + \hbox{NDP}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{G6P} + 2\hbox{NADP} \rightarrow \hbox{R5P} + 2\hbox{NADH}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{3R5P} \rightarrow \hbox{2F6P} + \hbox{G3P}$$\end{document} b \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{E4P} + \hbox{2PEP} + \hbox{NTP} + \hbox{NADH} \rightarrow \hbox{CHO} + \hbox{4Pi} + \hbox{NDP} + \hbox{NAD}$$\end{document} b \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{CHO} + \hbox{2AA} + \hbox{2NTP} + \hbox{R5P} \rightarrow \hbox{PYR} + \hbox{2NDP} + \hbox{PPi} + \hbox{G3P} + \hbox{TRY}$$\end{document} a,c \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{PEP} \rightarrow \hbox{OAA} + \hbox{Pi}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{OXO} + \hbox{2NAD} + \hbox{NDP} + \hbox{Pi} \rightarrow \hbox{OAA} + \hbox{2NADH} + \hbox{NTP}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{PYR} + \hbox{NAD} \rightarrow \hbox{NADH} + \hbox{A-CoA}$$\end{document} b \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{OXO} + \hbox{NH4} + \hbox{3NADPH} + \hbox{3NTP} \rightarrow \hbox{AA} + \hbox{3NADP} + \hbox{3NDP} + \hbox{3Pi}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{A-CoA} + \hbox{NTP} + \hbox{2NADPH} \rightarrow \hbox{LIP} + \hbox{NDP} + \hbox{Pi} + \hbox{2 NADP}$$\end{document} b \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{2R5P} \rightarrow \hbox{F6P} + \hbox{E4P}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{3A-CoA} + \hbox{2NADPH} + \hbox{3NTP} \rightarrow \hbox{SEC} + \hbox{2NADP} + \hbox{3NDP} + \hbox{3Pi}$$\end{document} a–c \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{G6P} \rightarrow \hbox{F6P}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$ \hbox{A-CoA} + \hbox{OAA} + \hbox{NAD} \rightarrow \hbox{OXO} + \hbox{NADH}$$\end{document} a–c \documentclass[12pt]{minimal}
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\begin{document}$$ \hbox{PEP} + \hbox{NDP} \rightarrow \hbox{PYR} + \hbox{NTP}$$\end{document} c,d \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{PEP} \rightarrow \hbox{PYR} + \hbox{Pi}$$\end{document} a–c \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{G3P} + \hbox{Pi} + \hbox{NDP} + \hbox{NAD} \rightarrow \hbox{PEP} + \hbox{NTP} + \hbox{NADH}$$\end{document} c,d \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{G3P} + \hbox{NADP} \rightarrow \hbox{PEP} + \hbox{NADPH}$$\end{document} a–c \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{F6P} + \hbox{NTP} \rightarrow \hbox{2G3P} + \hbox{NDP}$$\end{document} e \documentclass[12pt]{minimal}
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\begin{document}$$ \hbox{R5P} + \hbox{PYR} + \hbox{NTP} \rightarrow \hbox{SEC} + \hbox{NDP} + \hbox{2Pi}$$\end{document} b \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{R5P} + \hbox{3.75AA} + \hbox{7NTP} + \hbox{0.25NAD} \rightarrow \hbox{7NDP} + \hbox{3.5Pi} + \hbox{1.75PPi} + \hbox{0.25NADH} + \hbox{OP}$$\end{document} b \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{G6P} + \hbox{2NTP} + \hbox{NADPH} \rightarrow \hbox{STH} + \hbox{2NDP} + \hbox{NADP} + \hbox{Pi} + \hbox{PPi}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{STA} + \hbox{Pi} \rightarrow \hbox{G6P}$$\end{document} b \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{G6P} + \hbox{NTP} \rightarrow \hbox{STA} + \hbox{NDP} + \hbox{PPi}$$\end{document} e \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{LIP} + \hbox{2NTP} + \hbox{NAD} \rightarrow \hbox{2NDP} + \hbox{PPi} + \hbox{NADH}$$\end{document} ν(27) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{A-CoA} \rightarrow \hbox{ORA}$$\end{document} ν(28) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{ORA} \rightarrow \hbox{A-CoA}$$\end{document} d \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{F6P} + \hbox{PPi} \rightarrow \hbox{2G3P} + \hbox{Pi}$$\end{document} a \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{SUC} + \hbox{2NTP} \rightarrow \hbox{G6P} + \hbox{F6P} + \hbox{2NDP}$$\end{document} d \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{SUC} + \hbox{PPi} \rightarrow \hbox{G6P} + \hbox{F6P}$$\end{document} a 39 b 24 c 40 d 27 e 41 f 26 Fig. 3 Pathway numbers 2 4 2 1 21 23 3  Table 2 Independent fluxes of the SPMPs after pathways reduction using MFA approach a r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{2AA} + \hbox{3GLC} + \hbox{10NAD} + \hbox{6NADP} + \hbox{NTP} + \hbox{Pi} \rightarrow \hbox{10NADH} + \hbox{6NADPH} + \hbox{NDP} + \hbox{PPi} + \hbox{TRY}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{1.5GLC} + \hbox{6NAD} + \hbox{2NADP} \rightarrow \hbox{6NADH} + \hbox{2NADPH} + \hbox{SEC}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{FRU} \rightarrow \hbox{GLC}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{GLC} + \hbox{5NAD} + \hbox{6NADP} + \hbox{2NDP} + \hbox{2Pi} \rightarrow \hbox{5NADH} + \hbox{6NADPH} + \hbox{2NTP}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{GLC} + \hbox{4NAD} + \hbox{3NADPH} + \hbox{2NTP} + \hbox{NH4} \rightarrow \hbox{AA} + \hbox{4NADH} + \hbox{3NADP} + \hbox{2NDP} + \hbox{2Pi}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{NTP} \rightarrow \hbox{NDP} + \hbox{Pi}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{0.5GLC} + \hbox{2NAD} + \hbox{2NADPH} \rightarrow \hbox{LIP} + \hbox{2NADH} + \hbox{2NADP}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{NADH} + \hbox{NADP} + \hbox{NTP} \rightarrow \hbox{NAD} + \hbox{NADPH} + \hbox{NDP} + \hbox{Pi}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{GLC} + \hbox{10NAD} + \hbox{4NDP} + \hbox{4Pi} \rightarrow \hbox{10NADH} + \hbox{4NTP}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{1.5GLC} + \hbox{2NADP} + \hbox{NTP} + \hbox{NAD} \rightarrow \hbox{SEC} + \hbox{2NADPH} + \hbox{NDP} + \hbox{Pi} + \hbox{NADH}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{3.75AA} + \hbox{GLC} + \hbox{0.25NAD} + \hbox{2NADP} + \hbox{8NTP} \rightarrow \hbox{0.25NADH} + \hbox{2NADPH} + \hbox{8NDP} + \hbox{3.5Pi} + \hbox{1.75PPi} + \hbox{OP}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{GLC} + \hbox{NADP} + \hbox{3NTP} \rightarrow \hbox{NADPH} + \hbox{3NDP} + \hbox{Pi} + \hbox{PPi} + \hbox{STH}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{NDP} + \hbox{Pi} + \hbox{STA} \rightarrow \hbox{GLC} + \hbox{NTP}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{GLC} + \hbox{2NTP} \rightarrow \hbox{2NDP} + \hbox{PPi} + \hbox{STA}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{LIP} + \hbox{4NAD} + \hbox{NTP} + \hbox{Pi} \rightarrow \hbox{4NADH} + \hbox{NDP} + \hbox{PPi}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{0.5GLC} + \hbox{2NAD} + \hbox{NDP} + \hbox{Pi} \rightarrow \hbox{2NADH} + \hbox{NTP} + \hbox{ORA}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{3NAD} + \hbox{NDP} + \hbox{ORA} + \hbox{Pi} \rightarrow \hbox{3NADH} + \hbox{NTP}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{NDP} + \hbox{PPi} \rightarrow \hbox{NTP} + \hbox{Pi}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{SUC} \rightarrow \hbox{2GLC}$$\end{document} r \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{2GLC} + \hbox{2NTP} \rightarrow \hbox{SUC} + \hbox{PPi} + \hbox{2NDP}$$\end{document} a r SM SM v r14 2 v r13 v r5 24 SM v r1 SM v r2 v r10 v r11 3 25 v r12 v r7 v r15 26 v r16 v r17 Transient primary metabolic pathways 1 2 3 4 3 v 28 2 v r19 2 v r20 v r19 27 v r3 v r13 v r19 v 26 v 23 3 − 4 + v 30 v 29 Table 3 Biochemical reactions of the transient primary metabolic pathways a ν(21) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{EGLC} + \hbox{NTP} \rightarrow \hbox{GLC} + \hbox{NDP} + \hbox{Pi}$$\end{document} ν(22) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{EFRU} + \hbox{NTP} \rightarrow \hbox{FRU} + \hbox{NDP} + \hbox{Pi}$$\end{document} ν(23) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{OP} \rightarrow \hbox{Pi}$$\end{document} ν(24) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{ESUC} + \hbox{NTP} \rightarrow \hbox{SUC} + \hbox{NDP} + \hbox{Pi}$$\end{document} ν(25) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{NO3} + \hbox{NADH} + \hbox{3NADPH} \rightarrow \hbox{NH4} + \hbox{NAD} + \hbox{3NADP}$$\end{document} ν(26) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{PPi} \rightarrow \hbox{2Pi}$$\end{document} ν(27) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{ENO3} + \hbox{NTP} \rightarrow \hbox{NO3} + \hbox{NDP} + \hbox{Pi}$$\end{document} ν(28) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{ESUC} \rightarrow \hbox{EGLC} + \hbox{EFRU}$$\end{document} ν(29) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{NTP} \rightarrow \hbox{NDP} + \hbox{Pi}$$\end{document} ν(30) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{2.5NDP} + \hbox{2.5Pi} + \hbox{NADH} + \hbox{O2} \rightarrow \hbox{2.5NTP} + \hbox{NAD}$$\end{document} ν(31) \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{EPi} + \hbox{2NTP} \rightarrow \hbox{3Pi} + \hbox{2NDP}$$\end{document} a 24 Table 4 Kinetic expression of the fluxes a–c \documentclass[12pt]{minimal}
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\begin{document}$$\nu_{{{\text{GLC}}}} = \nu\,\hbox{max}_{{{\text{GLC}}}} \frac{\hbox{GLC}}{{K_{{\text{m}}_{{{\text{GLC}}}}} + \hbox{GLC}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r1) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r1)\frac{\hbox{AA}}{K_{{{\text{m}}_{\text{AA}}} + \hbox{AA}}}\frac{\hbox{NAD}}{K_{{\text{m}}_{\text{NAD}}} + \hbox{NAD}}\frac{\hbox{NADP}}{K_{{\text{m}}_{\text{NADP}}} + \hbox{NADP}}\frac{\hbox{NTP}}{K_{{\text{m}}_{\text{NTP}} + \hbox{NTP}}}\frac{{\hbox{Pi}^{2} }}{{K^{2}_{{\text{m}}_{\text{Pi}}} + \hbox{Pi}^{2}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r2) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r2)\frac{\hbox{NAD}}{{K_{{\text{m}} _{\text{NAD}}} +\hbox{NAD}}}\frac{\hbox{NADPH}}{{K_{{{\text{m}}_{\text{NADPH}}}} + \hbox{NADPH}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r3) = \nu\max (r3)\frac{\hbox{FRU}}{{K_{{\text{m}}_{\text{FRU}}} + \hbox{FRU}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 100({\text{GLC}} - 0.02)}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r4) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r4)\frac{\hbox{NAD}}{{K_{{\text{m}} _{\text{NAD}}} +\hbox{NAD}}}\frac{\hbox{NADP}}{{K_{{\text{m}}_{\text{NADP}}}+\hbox{NADP}}}\frac{\hbox{NDP}}{{K_{{\text{m}}_{\text{NDP}}} + \hbox{NDP}}}\frac{{\hbox{Pi}^{2}}}{{K^{2}_{{\text{m}}_{\text{Pi}}} + \hbox{Pi}^{2}}}{\left({\frac{1}{{1 + \hbox{e}^{{- 8{\left({\frac{{\text{NADP}}}{{{\text{NADP}}} +{\text{NADPH}}} - 0.5} \right)}}} }}}\right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r5) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r5)\frac{\hbox{NAD}}{{K_{{\text{m}} _{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NADPH}}{{K_{{\text{m}}_{\text{NADPH}}} + \hbox{NADPH}}}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}\frac{{\hbox{NH4}^{2} }}{{K_{{\text{m}}_{\text{NH4}}} + \hbox{NH4}^{2}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r6) = \nu\max (r6)\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 10{\left({{\text{Pi}} - 1} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r7) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r7)\frac{\hbox{NAD}}{{K_{{\text{m}} _{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NADPH}}{{K_{{\text{m}}_{\text{NADPH}}} + \hbox{NADPH}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r8) = \nu\max (r8)\frac{\hbox{NADH}}{{K_{{\text{m}}_{\text{NADH}}} + \hbox{NADH}}}\frac{\hbox{NADP}}{{K_{{\text{m}}_{\text{NADP}}} + \hbox{NADP}}}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}} }+ \hbox{NTP}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 10{\left({{\text{Pi}} - 1} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r9) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r9)\frac{\hbox{NAD}}{{K_{{\text{m}} _{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NDP}}{{K_{{\text{m}}_{\text{NDP}}} + \hbox{NDP}}}\frac{{\hbox{Pi}^{2}}}{{K^{2}_{{\text{m}}_{\text{Pi}}}+ \hbox{Pi}^{2}}}{\left({\frac{1}{{1 + \hbox{e}^{{ - 8{\left({\frac{\text{NAD}}{{\text{NAD}} + {\text{NADH}}} - 0.5} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r10) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r10)\frac{\hbox{NAD}}{{K_{{\text{m}} _{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NADP}}{{K_{{\text{m}}_{\text{NADP}}} + \hbox{NADP}}}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r11) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r11)\frac{\hbox{AA}}{{K_{{\text{m}}_{\text{AA}}} + \hbox{AA}}}\frac{\hbox{NAD}}{{K_{{\text{m}}_{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NADP}}{{K_{{\text{m}}_{\text{NADP}}} + \hbox{NADP}}}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}{\left({\frac{1}{{1 +\hbox{e}^{{ - 100{\left({{\text{Pi}} - 0.1} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r12) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r12)\frac{\hbox{NADPH}}{{K_{{{\text{m}}_{\text{NADPH}}}} + \hbox{NADPH}}}\frac{{\rm NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r13) = \nu\max (r13)\frac{\hbox{NDP}}{{K_{{\text{m}}_{\text{NDP}}} + \hbox{NDP}}}\frac{{\hbox{Pi}^{2} }}{{K^{2}_{{\text{m}}_{\text{Pi}}}+ \hbox{Pi}^{2} }}\frac{\hbox{STA}}{{K_{{\text{m}}_{\text{STA}}} + \hbox{STA}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 100{\left({{\text{GLC}} - 0.2} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r14) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r14)\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}{\left({\frac{1}{{1 + \hbox{e}^{{ - 10{\left({{\text{FRU}} + {\text{GLC}} + {\rm{0.5SUC}} - 0.2}\right)}}}}}}\right)}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 200{\left({{\text{STA}} - 1.5} \right)}}}}}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r15) = \nu\max (r15)\frac{\hbox{LIP}}{{K_{{\text{m}}_{\text{LIP}}} +\hbox{LIP}}}\frac{\hbox{NAD}}{{K_{{\text{m}}_{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}\frac{{\hbox{Pi}^{2} }}{{K^{2}_{{\text{m}}_{\text{Pi}}} + \hbox{Pi}^{2} }}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 10{\left({{\text{NADH}} - 1} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r16) = \nu_{{{\text{GLC}}}} \cdot \nu\max (r16)\frac{\hbox{NAD}}{{K_{{\text{m}}_{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NDP}}{{K_{{\text{m}}_{\text{NDP}}} + \hbox{NDP}}}\frac{{\hbox{Pi}^{2} }}{{K^{2}_{{\text{m}}_{\text{Pi}}} + \hbox{Pi}^{2} }}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r17) = \nu\max (r17)\frac{\hbox{NAD}}{{K_{{\text{m}}_{\text{NAD}}} + \hbox{NAD}}}\frac{\hbox{NDP}}{{K_{{\text{m}}_{\text{NDP}}} + \hbox{NDP}}}\frac{\hbox{ORA}}{{K_{{\text{m}}_{\text{ORA}}} + \hbox{ORA}}}\frac{{\hbox{Pi}^{2} }}{{K^{2}_{{\text{m}}_{\text{Pi}}} + \hbox{Pi}^{2} }}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 10{\left({{\text{NADH}} - 1} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r18) = \nu\max (r18)\frac{\hbox{NDP}}{{K_{{\text{m}}_{\text{NDP}}} + \hbox{NDP}}}\frac{{PPi}}{{K_{{\text{m}}_{\text{PPi}}} + \hbox{PPi}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 10{\left({{\text{Pi}} - 1} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r19) = \nu\max (19)\frac{\hbox{SUC}}{{K_{{\text{m}}_{\text{SUC}}} + \hbox{SUC}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 100({\text{GLC}} - 0.02)}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(21)^{\rm d} = \nu\max (21)\frac{\hbox{EGLC}}{{K_{{\text{m}}_{{\text{EGLC}}}} + \hbox{EGLC}}}\frac{{\rm{NTP}}}{{K_{{\text{m}}_{\text{NTP}}} + \rm{NTP}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 100{\left({{\text{STA}} - 0.07} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(22)^{\rm d} = \nu\max (22)\frac{\hbox{EFRU}}{{K_{{\text{m}}_{\text{EFRI}}} + \hbox{EFRU}}}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 100{\left({{\text{STA}} - 0.02} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(23) = \nu\max (23)\frac{{\hbox{OP}^{4} }}{{K^{4}_{{\text{m}}_{\text{OP}} }+ \hbox{OP}^{4} }}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 333{\left({{\text{Pi}} - 0.045} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(24) = \nu\max (24)\frac{\hbox{ESUC}}{{K_{{\text{m}}_{\text{ESUC}}} + \hbox{ESUC}}}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 100{\left({{\text{STA}} - 0.02} \right)}}} }}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(25) = \nu\max (25)\frac{{\hbox{NO3}^{2} }}{{K^{2}_{{\text{m}}_{\text{NO3}}} + {\rm NO3}^{2} }}\frac{\hbox{NADH}}{{K_{{{\text{m}}_{\text{NADH}}}} + \hbox{NADH}}}\frac{{\rm NADPH}}{{K_{{{\text{m}}_{\text{NADPH}}}} + \hbox{NADPH}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(26) = \nu\max (26)\frac{\hbox{PPi}}{{K_{{\text{m}}_{\text{PPi}}} + \hbox{PPi}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(27)^{\rm e} = {\left({\nu\max (27,\hbox{LA})\frac{\hbox{ENO3}}{{K_{{\text{m}}_{\rm{NO3\_LA}}} + \hbox{ENO3}}} + \nu\max (27,\hbox{HA})\frac{\hbox{ENO3}}{{K_{{\text{m}}_{\rm{NO3\_HA}}} + \hbox{ENO3}}}} \right)}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}\frac{\hbox{X}}{\hbox{V}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(28) = \nu\max (28)\frac{\hbox{ESUC}}{{K_{{\text{m}}_{\text{ESUC}}} + \hbox{ESUC}}}\frac{\hbox{STH}}{{0.02 + \hbox{STH}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(29) = \nu\max (29)\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(30) = \nu\max (30)\frac{\hbox{NADH}}{{K_{{\text{m}}_{\text{NADH}}} + \hbox{NADH}}}\frac{\hbox{NDP}}{{K_{{\text{m}}_{\text{NDP}}} + \hbox{NDP}}}\frac{{\hbox{Pi}^{2} }}{{K^{2}_{{\text{m}}_{\text{Pi}}} + \hbox{Pi}^{2}}}{\left({1 - \frac{1}{{1 + \hbox{e}^{{ - 8{\left({\frac{{\text{NDP}}}{{\text{NDP}} + {\text{NTP}}}- 0.5} \right)}}}}}} \right)}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(31)^{\rm f} = {\left({\nu\max (31,\hbox{LA})\frac{{\hbox{EPi}}}{{K_{{\text{m}}_{\rm{EPi\_LA}}} + \hbox{EPi}}} + \nu\max (31,\hbox{HA})\frac{{\hbox{EPi}}}{{K_{{\text{m}}_{\rm{EPi\_HA}}} + \hbox{EPi}}}} \right)}\frac{\hbox{NTP}}{{K_{{\text{m}}_{\text{NTP}}} + \hbox{NTP}}}\frac{X}{V}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(32) = \nu\max (32)X\frac{\hbox{AA}}{{K_{{\text{m}}_{\text{AA}} + \hbox{AA}}}}\frac{\hbox{LIP}}{{K_{{\text{m}}_{\text{LIP}}} + \hbox{LIP}}}\frac{{\hbox{STH}^{{1.25}} }}{{K^{{1.25}}_{{\text{m}}_{\text{STH}}} + \hbox{STH}^{{1.25}} }}\frac{{\hbox{OP}^{4} }}{{K^{4}_{{\text{m}}_{\text{OP}}} + \hbox{OP}^{4} }}\frac{\hbox{ORA}}{{K_{{\text{m}}_{\text{ORA}}} + \hbox{ORA}}}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$\nu(r20)$$\end{document} a 3 1 b c 24 d 42 e 43 f 44 Metabolic regulation i 6 \documentclass[12pt]{minimal}
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\begin{document}$$v = v_{\max} {\prod\limits_i {\frac{{(\theta _{i})^{{\alpha _{i}}}}}{{(K_{{\rm m},i})^{{\alpha _{i} }} + (\theta _{i})^{{\alpha _{i}}}}}}}$$\end{document} i 28 4 7 \documentclass[12pt]{minimal}
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\begin{document}$$f(M_{i}) = \frac{1}{{1 + {\rm e}^{{ - a(M_{i} - M_{{i,t}})}} }}$$\end{document} a M i M i t 4 + v 21 v 22 v 24 v 27 v 31 K mNTP v 30 v r5 3 4 3 4 v r14 7 v r13 v r4 v r6 v r8 v r9 v 21 v 22 v r11 v 23 v r15 v r17 v r18 v 30 v 32 4 Materials and methods Culture conditions 17 17 3 C. roseus 29 Agrobacterium rhizogenes 4 30 2 4 n 2 2 Amino acids analysis g 31 32 N N Extraction of sugars g Analysis of sugars g 2 4 Ions extraction g Ions analysis − 3 − 2 4 − 4 2− 4 + + + 2+ 2 3 3 Model simulations 33 Results and discussion Model calibration and determination of the kinetic parameters C. roseus C. roseus 5 6 7 M i exp 8 \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\text{error}} & = e_{{{\text{NO3}}}} + e_{{{\text{Pi}}}} + e_{{{\text{NH4}}}} +e_{{{\text{ESUC}}}} + e_{{{\text{EGLU}}}} + e_{{{\text{EFRU}}}} + e_{{{\text{ENO3}}}} + e_{{{\text{EPi}}}} + e_{{{\text{AA}}}}\\ & \quad + e_{{{\text{STA}}}} + e_{{{\text{SUC}}}} + e_{{{\text{FRU}}}} + e_{{{\text{GLC}}}} + e_{{\text{X}}} \\\end{aligned}$$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$e_{i} = \sum {\left[ {\frac{{(M_{i})_{{{\text{exp}}}} - (M_{i})_{{{\text{model}}}} }} {{\tfrac{1} {2}({\text{max}}(M_{i})_{{{\text{exp}}}} + {\text{min}}(M_{i})_{{{\text{exp}}}})}}} \right]}$$\end{document} Table 5 K m Component Value Literature Species Units References AA 0.0136  Catharanthus roseus mmol/g DW Model calibration FRU 0.120  C. roseus mmol/g DW Model calibration GLC 0.120  C. roseus mmol/g DW Model calibration LIP 0.00254  C. roseus mmol/g DW Model calibration NAD 0.00023  C. roseus mmol/g DW Model calibration NADH 0.00030  C. roseus mmol/g DW Model calibration NADP 0.000585  C. roseus mmol/g DW Model calibration NADPH 0.00037  C. roseus mmol/g DW Model calibration NDP 0.010  C. roseus mmol/g DW Model calibration NH4 0.2279  C. roseus mmol/g DW Model calibration NTP 0.00625  C. roseus mmol/g DW Model calibration ORA 0.00807  C. roseus mmol/g DW Model calibration Pi 0.1997  C. roseus mmol/g DW Model calibration PPi 0.0737  C. roseus mmol/g DW Model calibration SUC 1.00  C. roseus mmol/g DW Model calibration STA 1.00  C. roseus mmol/g DW Model calibration STH 0.2  C. roseus mmol/g DW Model calibration OP 0.0206  C. roseus mmol/g DW Model calibration EFRU 0.0461 6 Daucus carota mM 17 EGLC 0.0535 6 D. carota mM 17 ENO3_HA 0.0677 0.281 Citrus reticulata mM 45  0.005–0.2 Arabidopsis thaliana  46  0.006–0.02 Zea mays  43 ENO3_LA 1.43 0.5 A. thaliana mM 46  0.02–0.1 Z. mays  43  0.16 D. carota  17 EPI_HA 0.0026 0.007 C. roseus mM 47  0.0056 C. roseus  48  0.0035 C. roseus  49  0.003 C. roseus  49  0.0025 Nicotinana tabacum  50  0.0079 Lemma gibba  51  0.00049 Z. mays  52  0.0018 Z. mays  52  0.007 Chara corallina  37  0.004 C. corallina  37  0.025 Nitella translucens  53  0.0024 Neurospora crassa  54  0.0029 N. crassa  54  0.0026 D. carota  17 EPI_LA 0.080 0.900 C. roseus mM 49  0.0463 C. roseus  49  0.058 Nicotinana glutinosa  55  0.076 L. gibba  51  0.190 C. corallina  37  0.220 C. corallina  37  0.370 N. crassa  54  1.029 N. crassa  54  0.47 D. carota  17 ESUC 26.69  C. roseus mM Model calibration NO3 0.5181  C. roseus mmol/g DW Model calibration Table 6 V max Reaction in TPMP Current value (mmol/g DW/d) Literature (mmol/g DW/d) Species Reference r1 74.9934  C. roseus Model calibration r2 2  C. roseus Model calibration r3 0.764  C. roseus Model calibration r4 10,000  C. roseus Model calibration r5 150  C. roseus Model calibration r6 1  C. roseus Model calibration r7 0.1  C. roseus Model calibration r8 3.0  C. roseus Model calibration r9 30.0  C. roseus Model calibration r10 2.00  C. roseus Model calibration r11 6.00  C. roseus Model calibration r12 0.05  C. roseus Model calibration r13 15.0  C. roseus Model calibration r14 2.00  C. roseus Model calibration r16 5  C. roseus Model calibration r18 0.764  C. roseus Model calibration r19 0.1  C. roseus Model calibration r20 0.01  C. roseus Model calibration 21 15.1  C. roseus Model calibration 22 4.6929  C. roseus Model calibration 23 0.4986  C. roseus Model calibration 24 0.100  C. roseus Model calibration 25 1.1377  C. roseus Model calibration 26 33.5326  C. roseus Model calibration 27 HA 0.0015 0.1555 C. reticulata 45  0.072–0.1968 Z. mays 43 27 LA 0.167 0.72–1.92 Z. mays 43 28 1.00 1.69 D. carota 17 29 0.382  C. roseus Model calibration 30 900  C. roseus Model calibration 31 HA 0.00175 3.80 C. roseus 47  0.415 C. roseus 48  0.127 C. roseus 49  0.196 C. roseus 49  1.96 N. tabacum 50  0.0369 N. tabacum 50  0.1728 L. gibba 51  0.13824 Z. mays 52  0.0092 C. corallina 37  0.0576 N. translucens 53  3.80 N. crassa 54  0.0030 D. carota 17 31 LA 0.040 1.268 C. roseus 49  0.090 C. roseus 49  0.046 N. glutinosa 55  0.265 L. gibba 51  0.023 L. gibba 51  0.104 C. corallina 37  0.0576 C. corallina 37  10.4 N. crassa 54  5.64 N. crassa 54  0.045 D. carota 17 GLC 0.764  C. roseus Model calibration 32 0.140 0.27 D. carota 17 Table 7 t Component Values Units Species References AA 0.818/0.779 mmol/g DW C. roseus Batch/medium exchange FRU 0.0336/0.0556 mmol/g DW C. roseus Batch/medium exchange GLC 0.1625/0.1744 mmol/g DW C. roseus Batch/medium exchange SEC 0.00515 mmol/g DW C. roseus Unpublished results LIP 0.1 mmol/g DW N. tabacum 34 NAD −5 mmol/g DW Young photosynthetic tissue 56 NADH −5 mmol/g DW Young photosynthetic tissue 56 NADP −6 mmol/g DW Young photosynthetic tissue 56 NADPH −4 mmol/g DW Young photosynthetic tissue 56 NDP −3 mmol/g DW Young photosynthetic tissue 56 NH4 0.053/0.041 mmol/g DW C. roseus Batch/medium exchange NTP −3 mmol/g DW Young photosynthetic tissue 56 ORA 0.16 mmol/g DW N. tabacum 34 Pi 0.117/0.100 mmol/g DW C. roseus Batch/medium exchange PPi 0.0015 mmol/g DW Potato tubers 34 SUC 0.403/0.330 mmol/g DW C. roseus Batch/medium exchange STA 1.705/1.137 mmol/g DW C. roseus Batch/medium exchange STH 0.081 mmol/g DW N. tabacum 34 TRY 0.001428 mmol/g DW C. roseus Unpublished results OP 0.01 mmol/g DW C. roseus Model calibration EFRU 8.33/29.6 mM C. roseus Batch/medium exchange EGLC 7.22/23.0 mM C. roseus Batch/medium exchange ENH4 0 mM C. roseus Culture medium ENO3 3.27/3.27 mM C. roseus Batch/medium exchange EPi 0.100/0.100 mM C. roseus Batch/medium exchange ESUC 114.0/149.0 mM C. roseus Batch/medium exchange NO3 0.385/0.345 mmol/g DW C. roseus Batch X 0.0197/0.0180 gDW C. roseus Batch/medium exchange M i exp M i exp 8 17 5 6 −6 R 2 8 4 Table 8 R 2 Component Mean SD (%) (batch) Mean SD (%) (medium exchange) R 2 R 2 AA 11.88 13.83 0.0521 0.0413 FRU 9.92 34.96 0.0012 0.0415 GLC 10.81 48.29 0.0058 0.1059 NH4 9.86 26.10 0.6050 0.8628 Pi 35.85 12.01 0.7568 0.8303 SUC 25.79 45.75 0.0032 0.0781 STA 9.01 12.44 0.0010 0.0013 EFRU 15.06 18.62 0.0913 0.0612 EGLC 13.26 13.91 0.0838 0.0785 ENO3 40.76 6.05 0.9456 0.9459 EPi 26.27 43.20 0.9895 0.7737 ESUC 11.26 10.83 0.9123 0.9461 NO3 13.26 24.96 0.6541 0.4103 X 14.78 20.46 0.9750 0.9708 Fig. 4 solid bar empty bar 7.1 a b  Simulation of the cells physiological state Hairy root growth 32 M i 5 t 32 32 M i 5 −1 6 19 20 Fig. 5 Catharanthus roseus filled square triangle a e f g 32 solid lines dashed lines dashed line  Fig. 6 C. roseus filled square triangle a c d e solid lines dashed lines  The cells building blocks 5 34 35 36 19 35 v r5 19 v r12 19 v r14 19 19 −1 −1 C. roseus 17 The cells nutritional state 6 7 v 28 D. carota 17 19 36 34 19 36 34 35 34 19 34 19 Fig. 7 C. roseus filled square triangle a d e f g h i solid lines dashed lines  3 − 6 7 37 17 The cells energetic state 8 34 35 E. californica 38 Fig. 8 a b c C. roseus solid lines dashed lines  Conclusion A metabolic model capable of describing hairy root growth from the estimation of the cells physiological state was developed. The model includes the central metabolism, the primary metabolic pathways (SPMP) assumed at steady state and a network for the TPMP. At this point, the model is simplified and intracellular compartmentalization processes into the different cell compartments and organelles are not included. Nevertheless, the metabolic model showed to perform efficiently in simulating hairy root growth and nutrition. The use of intracellular concentrations in nutrients and co-substrates as well as the cells energetic state seems an efficient strategy in describing regulation of the metabolic fluxes. However, more experimental is required for improving the model structure as well as parameter values which may be regulated with cell physiological state. Finally, the model will be applied to other plant species as well as cell suspensions, and it will be studied as a tool to describe transient processes such as metabolic regulation.