# Mathematical Modeling to Estimate Photosynthesis: A State of the Art
## 3 Discussion
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**Equation [Ref]** & **Input Variables** & **Estimated Variable** & **No. of Samples** & **Validation Method** & **R\({}^{2}\)** \\ \hline \multirow{6}{*}{(24) [103]} & Oxygen in the leaf \(\left[\mathrm{mgO_{2}Cell^{-1}s^{-1}}\right]\) & & & Mathematical & \\  & Disappearance of PSU activated & Photosynthesis rate & & modeling & & \\  & \(\left[\mathrm{\mu molPSUCell^{-1}s^{-1}}\right]\) & \(\left[\mathrm{mgO_{2}Cell^{-1}s^{-1}}\right]\) & - & & previously & \\  & Total concentration of PSUs [PSU Cell\({}^{-1}\)] & & & & Garcia-Camacho & \\  & Time [s] & & & & & \\  & Leaf’s temperature [\({}^{\circ}\)C] & Photosynthesis rate & & Biochemical & \\  & Light \(\left[\mathrm{\mu molm^{-2}s^{-1}}\right]\) & \(\left[\mathrm{\mu molm^{-2}s^{-1}}\right]\) & 80 & & model of & - \\  & CO\({}_{2}\) concentration in the leaf \(\left[\mathrm{\mu molmol^{-1}}\right]\) & \(\left[\mathrm{\mu molm^{-2}s^{-1}}\right]\) & & & photosynthesis & \\  & & & & Gas exchange & \\  & Radiation [PAR] & & & & system & \\  & CO\({}_{2}\) in the leaf \(\left[\mathrm{\mu molm^{-2}s^{-1}}\right]\) & Photosynthetic net & & & LI-COR 6400 & \\  & Leaf’s temperature [\({}^{\circ}\)C] & rate & 5 & & (Third & 0.85 \\  & Pressure steam [KPa] & \(\left[\mathrm{\mu molm^{-2}s^{-1}}\right]\) & & & generation) & \\  & Leaf’s mass per area [gm\({}^{-2}\)] & & & & andJackknife & \\  & & & & technique & & \\  &All the investigations based on the study of mathematical modeling require extensive knowledge of the photosynthesis process in order to determine the types of variables that must be measured, as well as a knowledge of how these variables interrelate to obtain a reasonable photosynthesis estimation. Most of the mathematical models included in this review are validated with accredited models of other authors, or by comparing them with some device for measuring photosynthesis (mainly gas exchange), some others were only simulated. The percentage analysis of the revised mathematical models shows that 20% are gray box models, 13% are black box, and 67% are white box. It can be observed in Figure 3 that after the year 2000, the white box models had a considerable increase, separating themselves from the other types of models. The construction of the white box models is based on previous knowledge and physical intuition [45]. \begin{table}
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**Equation [Ref]** & **Input Variables** & **Estimated Variable** & **No. of Samples** & **Validation Method** & **R2** \\ \hline \hline \multirow{8}{*}{(36) [115]} & Irradiance \(\left[\mu\text{Em}^{-2}\text{s}^{-1}\right]\) & & & Experimental data assembly from an industrial-scale outdoor tubular & \\  & Temperature [\({}^{\circ}\)C] & Net photosynthesis & & & \\  & pH [unitless] & \(\left[\frac{\text{mG}}{\text{g}}\text{g}\text{ biomass}\text{s}\right]\) & & & \\  & Dissolved oxygen concentration \(\left[\frac{\text{mG}}{\text{s}}\right]\) & \(\left[\frac{\text{mG}}{\text{g}}\text{ biomass}\text{s}\right]\) &  
Besides this, white box models are more complex to model and implement in electronic systems compared with black box models [2]. On the other hand, the empirical models (black box) have been moderately developed, probably because no previous physical information is available and they are based on experimental data, trying to follow the white box. However, this does not mean that the obtained results with the black box models are not reliable, since the application of this type of model in the estimation of photosynthesis has been successful, and it is also more flexible to implement in electronic systems compared to with white box models. [47]. Unlike white box models, black box models can solve problems by estimating an unknown function, which serves as a good guide for a linear and nonlinear dynamic system. Regarding hybrid models (gray box), the first investigations began in 1991, two decades after the other types of models.