Extended Data Fig. 3: Model fitting error, sensitivity analysis, and determining regime boundary.

(a) Quantification of error in model fitting. The probability density function represents the distribution of errors for individual data points of nitrate measurements at time point k (n = 14, 611 data points). Errors are calculated as the difference between the model’s predicted nitrate concentration A(t_k) and the observed nitrate amounts a_k for either the chloramphenicol-untreated(CHL−) or treated(CHL+) conditions, normalized by dividing by the input nitrate concentration (2 mM) to be expressed as a percentage. (b) Each dot represents the error for a specific experimental condition (triplicate), with the native pH of the sample on the y-axis and the perturbed pH on the x-axis (n = 244 conditions). The error per condition, indicated by the color of each point, is the square root of the mean-squared error loss function minimized during parameter optimization of both CHL−/+ conditions in triplicate, normalized by the input nitrate concentration (2 mM) to be expressed as a percentage (refer to Methods for the error computation). The arrow indicates the row of the soil that is shown as an example for the bottom panel. (c) To justify fixing parameters γ, K_A, and \({\widetilde{K}}_{C}\) for model fitting, we analyzed the sensitivity of γ, K_A, and \({\widetilde{K}}_{C}\) by simulating dynamic data. To reflect the three typical dynamics (regimes) observed from the measurement, we simulated three nitrate curves by setting up the initial conditions to be \(\widetilde{x}(0)=0.01,0.1,0.001\,{\rm{mM}}/day\) and \(\widetilde{C}(0)=0.005,0.05,2\,{\rm{mM}}\), respectively. Other parameters are given by \({A}_{0}={A}_{0}^{c}=2\,{\rm{mM}}\), \({K}_{A}={\widetilde{K}}_{C}=0.01\,{\rm{mM}}\), γ = 4 day⁻¹. Black and red dashed lines represent simulated CHL− and CHL+ conditions, respectively. In the first row, we used different fixed γ values - from γ = 2 day⁻¹ to γ = 6 day⁻¹ - to fit three simulations. We demonstrate very small mismatches (square root of the mean-squared error (RMSE) < 5%, see Methods for loss function) from these variations of parameter values. Purple lines indicate fitted results from γ = 4 ± 0.1 day⁻¹, blue from γ = 4 ± 1 day⁻¹, green from γ = 4 ± 2 day⁻¹). In the second and the third row, we used different fixed K_A and \({\widetilde{K}}_{C}\) values - from 10⁻⁴ mM to 1 mM - to fit three simulations. When K_A < 0.1 mM or \({\widetilde{K}}_{C} < 0.1\) mM, the mismatches were again very small (RMSE < 1%) and invisible, indicating that the fixed values of γ, K_A, and \({\widetilde{K}}_{C}\) are insensitive in large ranges. (d-g) To determine regime boundary thresholds with distributions of the parameters \(\widetilde{x}(0)\) and \(\gamma \widetilde{C}(0)\), we examined the distributions of parameters fitted to the functional data (n = 732 nitrate dynamics). (d) \(\widetilde{x}(0)\) had a bimodal frequency distribution. (e) This bi-modality becomes more evident when we separately observe its distribution from the left half (perturbed pH < 4) and right half (perturbed pH > 6) of the parameter space displayed in the perturbed pH vs. native pH grid in Fig. 3c. We set the threshold for the \(\widetilde{x}(0)\) boundary where these two modes are separated (\(\widetilde{x}(0)\) = 0.05). (f) \(\gamma \widetilde{C}(0)\) showed an uni-modal frequency distribution. We set the threshold (\(\gamma \widetilde{C}(0)\) = 1.5) at the tail of the distribution, where the \(\gamma \widetilde{C}(0)\) threshold also separated the Regime III samples in the top-left quadrant of the \(\widetilde{x}(0)\) vs. \(\gamma \widetilde{C}(0)\) scatter plot (Fig. 3a). (g) With these thresholds of two parameters, we can define the three different functional regimes across native pH and perturbed conditions (n = 244 conditions).