## 2 Materials and Methods
### Calibration and Experimental Protocol
#### 2.3.1 Multi-Treatment Analysis and Calibration  
A treatment-by-treatment calibration of DALEC-Crop was first carried out using the CARDAMOM MDF framework to determine the distributions and sensitivity of parameters across the range of treatments (0 to 200 kg N ha\({}^{-1}\)). The goal of this analysis was to validate  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Index** & **EDC Description** \\ \hline
1 & C Pools cannot be negative \\
2 & C Fluxes cannot be negative \\
3 & LAI must be less than 6.5 m\({}^{2}\) m\({}^{-2}\) \\
4 & GPP must be less than 25 g C m\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: DALEC-Crop Ecological and Dynamic Constraints (EDCs).  
the hypothesized relationships between climate, leaf N, LAI and yield in DALEC-Crop. Each treatment calibration was performed using data from both seasons (i.e., multi-season calibration). From the 50 plots included in each season, calibration plots were identified by randomly selecting three out of the five replicates for each treatment and wheat variety, whereas the remaining two plots were used for validation. Thus, 12 plots were used for calibration and 8 plots were used for validation, per treatment. A key output of this experiment was to test whether retrieved parameter ensembles for different N treatments were similar or showed substantive differences in processes, such as phenology, between treatments (see Supplementary Material, Figure S1).  
A total of 25 of the DALEC-Crop parameters determining C availability and phenology were calibrated for each treatment (see Supplementary Material, Table S1). This calibration was based on optimizing the selection of these parameters such that the DALEC-Crop mean LAI and yield estimates were within a 12% uncertainty of the observed at each of the corresponding calibration plots. This uncertainty for LAI was previously quantified in [PERSON] et al. (2011) and, in this analysis, we assume this characterized uncertainty to be the same as that of the yield observations. Throughout this calibration the LNA used in the ACM PNUE equation (3) was calculated by providing DALEC-Crop with the slope and intercept of the linear N dilution fit derived from the destructive sample data for a given treatment (Figure 3).  
Using the same experimental plots for validation and calibration as those of the pretreatment analysis, we also performed a cross-treatment calibration of the 25 DALEC-Crop parameters. Based on the simulation of wheat yields, the goal of this multi-treatment approach was to determine if a single calibration of DALEC-Crop was valid across the gradient of N applications. The posterior distributions in the parameters were then sampled and used in Experiments 1-3.  
We evaluated the performance of the DALEC-Crop cross-treatment calibration using LAI and yield observations at the validation plots. Metrics used in the evaluation included the coefficient of determination (R\({}^{2}\)) and the normalized root-mean-square-error (NRMSE):  
\[NRMSE=\frac{\left[\sqrt{\frac{\sum_{i=1}^{n}\left(E_{i}-O\right)^{2}}{n}} \right]}{\left[\max(O)-\min(O)\right]} \tag{6}\]  
where \(E_{i}\) and \(O_{i}\) are the modelled and observed values, respectively, and n is the number of modelled and observed comparisons.