# Upscaling Gross Primary Production from Leaf to Canopy for Potato Crop (_Solanum tuberosum_ L.)
## 2 Materials and Methods
### Leaf-Level Measurements
#### 2.4.2 Modeling Schemes for Gross Primary Production of the Canopy (GPP\({}_{\text{can}}\))  
Big-Leaf Approach (BL)  
The gross primary production of the canopy (GPP\({}_{\text{can}}\)) was integrated over the canopy using a rectangular hyperbola base function (Michaelis-Menten type model), where light intensities within a canopy were calculated using a negative exponential function with a canopy light extinction coefficient (k) [9; 32; 51]. The total net carbon uptake integrated over the canopy is:  
\[GPP_{can}=\frac{A_{gmax}}{K}\text{{Int}}\frac{\Phi*K*I+A_{gmax}}{(\Phi*I*K*e^{ -KLAI})+A_{gmax}} \tag{7}\]  
where GPP\({}_{\text{can}}\) is the gross primary production of the canopy [\(\mu_{\text{mol}}\) (CO\({}_{2}\)) m\({}^{-2}\) s\({}^{-1}\)]; \(A_{gmax}\) is the light-saturated gross photosynthetic rate [\(\mu_{\text{mol}}\) (CO\({}_{2}\)) m\({}^{-2}\) s\({}^{-1}\)] of the topmost unshaded leaves; \(I\) is the photosynthetic photon flux density PPFD [\(\mu_{\text{mol}}\) (photon) m\({}^{-2}\) s\({}^{-1}\)]; \(K\) is the extinction coefficient; and \(\Phi\) is the quantum yield of assimilation [\(\mu_{\text{mol}}\)(CO\({}_{2}\)) \(\mu_{\text{mol}}\) (photon)\({}^{-1}\)]. The values of the asymptotic estimate of the maximum gross photosynthetic rate (\(A_{gmax}\)) [\(\mu_{\text{mol}}\) (CO\({}_{2}\)) m\({}^{-2}\) s\({}^{-1}\)] and the quantum yield \(\Phi\) [\(\mu_{\text{mol}}\) (CO\({}_{2}\)) \(\mu_{\text{mol}}\) (photon) \({}^{-1}\)] were obtained from the fitted rectangular hyperbola-based models [PERSON] [46] and [PERSON] [47] (Figure 1). Multilayer Approach (ML)  
In this approach, the value of GPP\({}_{\text{can}}\) was estimated from an extrapolation scheme based on Michaelis-Menten (rectangular hyperbola-based model) of [PERSON] [51] and exponential Mitscherlich (Webb) [49] applied to three different layers of the canopy. The canopy was divided in three layers, upper stratum (topmost unshaded leaves), middle stratum, and low stratum (bottommost shaded leaves), each one with defined LAI values. The leaf area index for each layer was obtained by the difference in leaf area index at the two heights bounding the layers. Thus, LAI\({}_{\text{i}}=\) LAI\({}^{\prime}_{\text{i}}-\) LAI\({}^{\prime}_{\text{i}-1}\), where LAI\({}^{\prime}_{\text{i}}\) is the LAI- ACCUPAR LP-80 measurement beneath the ith layer. For n layers, LAI\({}^{\prime}_{\text{n}}\equiv\) LAI. The leaf gross assimilation (Gi) was computed for each layer by introducing the description of the incident and absorbed PPFD in each layer in the Michaelis-Menten (Smith) [47] and Mitscherlich ([PERSON]) [49] functions:  
\[G_{i\ Smith}=\frac{\Phi_{i}*I_{ai}*A_{gmax_{i}}}{\sqrt{\Phi_{i}{}^{2}*I_{ai}{}^{ 2}+A_{gmax_{i}}{}^{2}}} \tag{8}\]  
Figure 1: Schematic descriptions of the GPP\({}_{\text{can}}\) upscaling models Big-leaf (BL) and Multilayer (ML). \[G_{i\;webb}=\left\{A_{gmax_{i}}\left[1-\exp\left(\frac{\Phi_{i}*I_{ai}}{A_{gmax_{i}} }\right)\right]\right\} \tag{9}\]  
where \(G_{i}\) is leaf gross assimilation; \(A_{gmax}\) is light-saturated gross photosynthetic rate [\(\mu_{\text{mol}}\)(CO\({}_{2}\)) m\({}^{-2}\) s\({}^{-1}\)] for the layer \(i\); \(\Phi\) is the quantum yield of assimilation for the layer \(i\) [\(\mu_{\text{mol}}\)(CO\({}_{2}\)) \(\mu_{\text{mol}}\) (photon)\({}^{-1}\)]. \(A_{gmax}\) and \(\Phi\) are parameters estimated from the mathematical models [PERSON] ([PERSON]) [47] and [PERSON] ([PERSON]) [49]. \(I_{ai}\) is the absorbed PPFD [\(\mu_{\text{mol}}\) (photon) m\({}^{-2}\) s\({}^{-1}\)] for each layer, computed using Beer's law and the approach by [PERSON] et al. [14] and [PERSON] et al.