## 2 Materials and Methods
### Model Descriptions
#### 2.1.2 PPA-SiBGC  
The PPA-SiBGC model belongs to the SORTIE-PPA family of models [46; 49] within the SAS-PPA model genre, based on a simple and analytically tractable approximation of the classical SORTIE gap model [107; 108]. The Perfect Plasticity Approximation, or PPA [46; 47], was derived from the dual assumptions of perfect crown plasticity (e.g., space-filling) and phototropism (e.g., stem-leaning), both of which were supported in empirical and modeling studies [49]. The discovery of the PPA was rooted in extensive observational and in silico research [46]. The PPA model was designed to overcome the most computationally challenging aspects of gap models in order to facilitate model scaling from the landscape to global scale.  
The PPA and its predecessor, the size-and-age structured (SAS) equations [43; 109], are popular model reduction techniques employed in current state-of-the-art terrestrial biosphere models [28]. The PPA model can be thought of metaphorically as Navier-Stokes equations of forest dynamics, capable of modeling individual tree population dynamics with a one-dimensional von Foerster partial differential equation [46]. The simple mathematical foundation of the PPA model is provided in Equation (1).  
\[1=\int_{z^{*}}^{\infty}\sum_{j=1}^{k}N_{j}(z)A_{j}(z^{*},z)dz \tag{1}\]  
where \(k\) is the number of species, \(j\) is the species index, \(N_{j}(z)\) is the density of species \(j\) at height \(z\), \(A_{j}(a^{*},z)\) is the projected crown area of species \(j\) at height \(z\), and \(dz\) is the derivative of height. In other words, we discard the spatial location of individual trees and calculate the height at which the integral of tree crown area is equal to the ground area of the stand. This height is known as the theoretical \(z^{*}\) height, which segments trees into overstory and understory classes [46].  
The segmentation of the forest canopy into understory and overstory layers allows for separate coefficients or functions for growth, mortality, and fecundity to be applied across strata, whose first moment accurately approximates the dynamics of individual-based forest models. Recent studies have shown that the PPA model faithfully reduces the dynamics of the more recent neighborhood dynamics (ND) SORTIE-ND gap model [110] and is capable of accurately capturing forest dynamics [111; 112].  
In this work, we applied a simple biogeochemistry variant of the SORTIE-PPA model, PPA-SiBGC ([PERSON] and [PERSON], in review) Figure 2.  
Empirical observations were relied upon for the C and N content of tree species compartments. Stoichiometric relations were used to estimate N from C, based on empirical measurements provided for both sites. All values were calculated directly from observations. Previously published equations [113] and parameters [114] were used to model crown allometry. Together with inventory data, general biomass equations were used to estimated dry weight mass (\(kg\)) for tree stems, branches, leaves, and, fine and coarse roots [115]. Carbon content was assumed to be 50% of dry mass, generally supported by data. Monthly soil respiration was modeled using the approach of [PERSON] et al. [116], while soil organic C was modeled using the simple generalized approach of [PERSON] et al. [117]. Species- and stratum-specific parameters for growth, mortality, and fecundity were calculated directly from field data for both sites. Net ecosystem exchange, or NEE, was modeled as \(NEE=r_{\text{soil}}-ANPP\) following previous studies, which note associated challenges in connecting field and flux tower measurements [118; 119]. Here, ANPP, or annual net primary production, is the total site biomass increment adjusted for the C fraction. This is necessary given the current field-measurement basis of the PPA, which may be replaced by LiDAR measurements and/or process models in future work.