Models / Bellasio 2019

Bellasio 2019 model

The Bellasio 2019 model is a generalised model of leaf-level C3 photosynthesis that couples simplified light and dark reactions with a stomatal-behaviour submodule. Building on the Farquhar–von Caemmerer–Berry framework and the light reactions of Yin et al. (2004), it derives the potential rates of ATP and NADPH production from light intensity and links them to carbon assimilation and stomatal conductance.

Designed to stay deliberately simple, it reproduces the classic steady-state assimilation responses to intercellular CO₂ and light while remaining valid under dynamic conditions, which makes it a convenient starting block for larger models of plant physiology.

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Kh_co2

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\begin{align*}
  \frac{d CO2}{dt} &= - \frac{1}{V\_m} \cdot \frac{CO2 \cdot RUBP \cdot Ract \cdot f\_rubp \cdot vmax\_v\_RuBisCO\_c}{(CO2 + km\_v\_RuBisCO\_c\_CO2 \cdot (1 + \frac{O2}{km\_v\_RuBisCO\_c\_O2})) \cdot (RUBP + km\_v\_RuBisCO\_c\_RUBP\_extra)} \\
  & + \frac{0.5}{V\_m} \cdot rubisco\_oxygenase \\
  & - \frac{1}{V\_m} \cdot \frac{vmax\_v\_co2\_hydration \cdot (CO2 - \frac{H \cdot HCO3}{keq\_v\_co2\_hydration})}{km\_v\_co2\_hydration\_CO2 \cdot (1 + \frac{CO2}{km\_v\_co2\_hydration\_CO2} + \frac{HCO3}{km\_v\_co2\_hydration\_HCO3})} \\
  & + \frac{1}{V\_m} \cdot RLight \\
  & + \frac{1}{V\_m} \cdot 0.001 \cdot gm \cdot (Ci - CO2 \cdot Kh\_co2) \\ 
  \frac{d HCO3}{dt} &= \frac{1}{V\_m} \cdot \frac{vmax\_v\_co2\_hydration \cdot (CO2 - \frac{H \cdot HCO3}{keq\_v\_co2\_hydration})}{km\_v\_co2\_hydration\_CO2 \cdot (1 + \frac{CO2}{km\_v\_co2\_hydration\_CO2} + \frac{HCO3}{km\_v\_co2\_hydration\_HCO3})} \\ 
  \frac{d RUBP}{dt} &= - \frac{1}{V\_m} \cdot \frac{CO2 \cdot RUBP \cdot Ract \cdot f\_rubp \cdot vmax\_v\_RuBisCO\_c}{(CO2 + km\_v\_RuBisCO\_c\_CO2 \cdot (1 + \frac{O2}{km\_v\_RuBisCO\_c\_O2})) \cdot (RUBP + km\_v\_RuBisCO\_c\_RUBP\_extra)} \\
  & - \frac{1}{V\_m} \cdot \frac{1 \cdot Kh\_o2 \cdot O2 \cdot v\_RuBisCO\_c}{CO2 \cdot Kh\_co2 \cdot S\_co\_gas} \\
  & + \frac{1}{V\_m} \cdot \frac{vmax\_v\_PRKase \cdot (ATP\_st \cdot RU5P - \frac{ADP\_st \cdot RUBP}{keq\_v\_PRKase})}{(ATP\_st + km\_v\_PRKase\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_PRKase\_ADP\_st})) \cdot (RU5P + km\_v\_PRKase\_RU5P \cdot (1 + \frac{PGA}{ki\_v\_PRKase\_PGA} + \frac{Pi\_st}{ki\_v\_PRKase\_Pi\_st} + \frac{RUBP}{ki\_v\_PRKase\_RUBP}))} \\ 
  \frac{d PGA}{dt} &= \frac{2}{V\_m} \cdot \frac{CO2 \cdot RUBP \cdot Ract \cdot f\_rubp \cdot vmax\_v\_RuBisCO\_c}{(CO2 + km\_v\_RuBisCO\_c\_CO2 \cdot (1 + \frac{O2}{km\_v\_RuBisCO\_c\_O2})) \cdot (RUBP + km\_v\_RuBisCO\_c\_RUBP\_extra)} \\
  & + \frac{1}{V\_m} \cdot \frac{1 \cdot Kh\_o2 \cdot O2 \cdot v\_RuBisCO\_c}{CO2 \cdot Kh\_co2 \cdot S\_co\_gas} \\
  & + \frac{0.5}{V\_m} \cdot rubisco\_oxygenase \\
  & - \frac{1}{V\_m} \cdot \frac{ATP\_st \cdot NADPH\_st \cdot PGA \cdot vmax\_v\_pgareduction}{(ATP\_st + km\_v\_pgareduction\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (NADPH\_st + km\_v\_pgareduction\_NADPH\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (PGA + km\_v\_pgareduction\_PGA \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st}))} \\
  & - \frac{0.3333333333333333}{V\_m} \cdot RLight \\ 
  \frac{d DHAP}{dt} &= - \frac{1.6666666666666667}{V\_m} \cdot \frac{vmax\_v\_PRKase \cdot (ATP\_st \cdot RU5P - \frac{ADP\_st \cdot RUBP}{keq\_v\_PRKase})}{(ATP\_st + km\_v\_PRKase\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_PRKase\_ADP\_st})) \cdot (RU5P + km\_v\_PRKase\_RU5P \cdot (1 + \frac{PGA}{ki\_v\_PRKase\_PGA} + \frac{Pi\_st}{ki\_v\_PRKase\_Pi\_st} + \frac{RUBP}{ki\_v\_PRKase\_RUBP}))} \\
  & + \frac{1}{V\_m} \cdot \frac{ATP\_st \cdot NADPH\_st \cdot PGA \cdot vmax\_v\_pgareduction}{(ATP\_st + km\_v\_pgareduction\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (NADPH\_st + km\_v\_pgareduction\_NADPH\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (PGA + km\_v\_pgareduction\_PGA \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st}))} \\
  & - \frac{1}{V\_m} \cdot \frac{vmax\_v\_carbohydrate\_synthesis \cdot (1 - \frac{Pi\_st \cdot abs(v\_pgareduction)}{keq\_v\_carbohydrate\_synthesis}) \cdot (-0.4 + DHAP)}{DHAP + km\_v\_carbohydrate\_synthesis\_DHAP \cdot (1 + \frac{ADP\_st}{ki\_v\_carbohydrate\_synthesis\_ADP\_st})} \\ 
  \frac{d ATP\_st}{dt} &= - \frac{1}{V\_m} \cdot \frac{1 \cdot Kh\_o2 \cdot O2 \cdot v\_RuBisCO\_c}{CO2 \cdot Kh\_co2 \cdot S\_co\_gas} \\
  & - \frac{1}{V\_m} \cdot \frac{vmax\_v\_PRKase \cdot (ATP\_st \cdot RU5P - \frac{ADP\_st \cdot RUBP}{keq\_v\_PRKase})}{(ATP\_st + km\_v\_PRKase\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_PRKase\_ADP\_st})) \cdot (RU5P + km\_v\_PRKase\_RU5P \cdot (1 + \frac{PGA}{ki\_v\_PRKase\_PGA} + \frac{Pi\_st}{ki\_v\_PRKase\_Pi\_st} + \frac{RUBP}{ki\_v\_PRKase\_RUBP}))} \\
  & - \frac{1}{V\_m} \cdot \frac{ATP\_st \cdot NADPH\_st \cdot PGA \cdot vmax\_v\_pgareduction}{(ATP\_st + km\_v\_pgareduction\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (NADPH\_st + km\_v\_pgareduction\_NADPH\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (PGA + km\_v\_pgareduction\_PGA \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st}))} \\
  & - \frac{0.5}{V\_m} \cdot \frac{vmax\_v\_carbohydrate\_synthesis \cdot (1 - \frac{Pi\_st \cdot abs(v\_pgareduction)}{keq\_v\_carbohydrate\_synthesis}) \cdot (-0.4 + DHAP)}{DHAP + km\_v\_carbohydrate\_synthesis\_DHAP \cdot (1 + \frac{ADP\_st}{ki\_v\_carbohydrate\_synthesis\_ADP\_st})} \\
  & + \frac{1}{V\_m} \cdot \frac{J\_ATP \cdot (ADP\_st \cdot Pi\_st - \frac{ATP\_st}{keq\_v\_ATPsynth})}{km\_v\_ATPsynth\_ADP\_st \cdot km\_v\_ATPsynth\_Pi\_st \cdot (1 + \frac{ADP\_st}{km\_v\_ATPsynth\_ADP\_st} + \frac{ATP\_st}{km\_v\_ATPsynth\_ATP\_st} + \frac{Pi\_st}{km\_v\_ATPsynth\_Pi\_st} + \frac{ADP\_st \cdot Pi\_st}{km\_v\_ATPsynth\_ADP\_st \cdot km\_v\_ATPsynth\_Pi\_st})} \\ 
  \frac{d NADPH\_st}{dt} &= - \frac{0.5}{V\_m} \cdot \frac{1 \cdot Kh\_o2 \cdot O2 \cdot v\_RuBisCO\_c}{CO2 \cdot Kh\_co2 \cdot S\_co\_gas} \\
  & - \frac{1}{V\_m} \cdot \frac{ATP\_st \cdot NADPH\_st \cdot PGA \cdot vmax\_v\_pgareduction}{(ATP\_st + km\_v\_pgareduction\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (NADPH\_st + km\_v\_pgareduction\_NADPH\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st})) \cdot (PGA + km\_v\_pgareduction\_PGA \cdot (1 + \frac{ADP\_st}{ki\_v\_pgareduction\_ADP\_st}))} \\
  & + \frac{1}{V\_m} \cdot \frac{J\_NADPH \cdot (NADP\_st - \frac{NADPH\_st}{keq\_v\_FNR})}{km\_v\_FNR\_NADP\_st \cdot (1 + \frac{NADPH\_st}{km\_v\_FNR\_NADPH\_st} + \frac{NADP\_st}{km\_v\_FNR\_NADP\_st})} \\ 
  \frac{d RU5P}{dt} &= - \frac{1}{V\_m} \cdot \frac{vmax\_v\_PRKase \cdot (ATP\_st \cdot RU5P - \frac{ADP\_st \cdot RUBP}{keq\_v\_PRKase})}{(ATP\_st + km\_v\_PRKase\_ATP\_st \cdot (1 + \frac{ADP\_st}{ki\_v\_PRKase\_ADP\_st})) \cdot (RU5P + km\_v\_PRKase\_RU5P \cdot (1 + \frac{PGA}{ki\_v\_PRKase\_PGA} + \frac{Pi\_st}{ki\_v\_PRKase\_Pi\_st} + \frac{RUBP}{ki\_v\_PRKase\_RUBP}))} \\
  & + \frac{1}{V\_m} \cdot \frac{vmax\_v\_rpp \cdot (DHAP - \frac{RU5P}{keq\_v\_rpp})}{DHAP + km\_v\_rpp\_DHAP} \\ 
  \frac{d Ract}{dt} &= \begin{cases}\frac{Ract\_eq - Ract}{tau\_i} & Ract < Ract\_eq \\ \frac{Ract\_eq - Ract}{tau\_d} & \text{else}\end{cases} \\ 
  \frac{d J\_NADPH}{dt} &= \begin{cases}\frac{J\_NADPH\_steady - J\_NADPH}{Kj\_NADPH} & J\_NADPH < J\_NADPH\_steady \\ \frac{10 \cdot J\_NADPH\_steady - 10 \cdot J\_NADPH}{Kj\_NADPH} & \text{else}\end{cases} \\ 
  \frac{d J\_ATP}{dt} &= \begin{cases}\frac{J\_ATP\_steady - J\_ATP}{Kj\_ATP} & J\_ATP < J\_ATP\_steady \\ \frac{10 \cdot J\_ATP\_steady - 10 \cdot J\_ATP}{Kj\_ATP} & \text{else}\end{cases} \\ 
  \frac{d Ci}{dt} &= - 0.001 \cdot gm \cdot (Ci - CO2 \cdot Kh\_co2) \\
  & + 0.001 \cdot gs \cdot (Ca - Ci) \\ 
  \frac{d gs}{dt} &= \begin{cases}\frac{gs\_steady - gs}{Ki} & gs < gs\_steady \\ \frac{gs\_steady - gs}{Kd} & \text{else}\end{cases}
\end{align*}

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