Ideas and perspectives: Allocation of carbon from Net Primary Production in models is inconsistent with observations of the age of respired carbon

. Carbon allocation in vegetation is an important process in the terrestrial carbon cycle; it determines the fate of photo-assimilates and it has an impact on the time carbon spends in the terrestrial biosphere. Although previous studies have highlighted important conceptual issues in the definition and metrics used to assess carbon allocation, very little emphasis has been placed on the distinction between allocation of carbon from gross primary production (GPP) versus allocation from net primary production (NPP). An important number of simulation models and conceptual frameworks are based on the concept 5 that C is allocated from NPP, which implies that C is respired immediately after photosynthetic assimilation. However, empirical work that estimates the age of respired CO 2 from vegetation tissue (foliage, stems, roots) shows that it may take from years to decades to respire previously produced photosynthates. The transit time distribution of carbon in vegetation and ecosystems, a metric that provides an estimate of the age of respired carbon, indicates that vegetation pools respire carbon of a wide range of ages, on timescales that are in conflict with the assumption that autotrophic respiration only consumes recently fixed carbon. 10 In this contribution, we attempt to provide compelling evidence based on recent research on the age of respired carbon and the theory of timescales of carbon in ecosystems, with the aim to promote a change in the predominant paradigm implemented in ecosystem models where carbon allocation is based on NPP. In addition, we highlight some implications for understanding and modeling carbon dynamics in terrestrial ecosystems.

These paradigms are very important to conceptualize the main processes of plant metabolism involved in respiration, but they are not necessarily explicit about the source of carbon that would contribute to the respiration flux. For instance, one can implement a model that computes Ra following the GMWRP, but the actual carbon used for respiration can be subtracted 90 directly from GPP following Waring et al.'s (1998) idea ::::::: approach. Carbon would not enter any plant part, but still it would be respired following some physiological concepts.
Research on the matrix approach (Luo et al., 2017), which shows that one single equation generalizes the majority of existing ecosystem and land-surface models, suggests that Ra is generally subtracted directly from GPP independently of the respiration paradigm implemented in the model. The matrix representation of Luo et al. (2017) can be written as where x is a vector of ecosystem carbon pools, U (t) is a function of carbon inputs to the ecosystem, generally obtained as U (t) = GPP(t) − Ra(t) = NPP(t). Then, NPP is allocated to ecosystem compartments such as foliage, wood, and belowground biomass according to the vector of allocation coefficients b. The product of the matrices ξ(t), A, and K, is a compartmental matrix that has in its main diagonal the rates at which carbon is processed in each of the compartments, and 100 in its off-diagonal the rates of carbon transfer among compartments. For vegetation compartments, 100% of all outputs (from mortality and litterfall) are transferred to litter and soil pools, because autotrophic respiration is already accounted for in the first term of equation (1). This modeling choice implies that the carbon used for autotrophic respiration never enters a particular vegetation compartment and does not spend any time there (Figure 1).
In addition to modeling studies, the concept of quantifying carbon allocation after accounting for autotrophic respiration 105 losses is also used in some empirical studies. For instance, the conceptual framework often used to analyze inventory data in tropical forests (e.g. Malhi et al., 2011Malhi et al., , 2015 assumes that biomass growth results from the allocation of the products of NPP, after autotrophic respiration occurs. In this case however, carbon allocation is understood as partitioning of total NPP. Litton et al. (2007) showed that carbon allocation can be understood differently by different authors, as a flux, as biomass, or as partitioning of the total GPP flux. In the case of the tropical forest data, carbon allocation is understood as partitioning 110 coefficients of the NPP flux and not partitioning of GPP as originally defined by Litton et al. (2007).
Together, these previous studies show that empirical work has promoted the implementation of Ra as a constant proportion of GPP, or based on some respiration paradigms, but subtracting Ra from GPP before carbon allocation occurs. Therefore models compute first NPP and subsequently allocate the non-respired carbon to plant parts ( Figure 1). Any model that could be written using the matrix equation with U = NPP (equation 1) would allocate the products of NPP and not GPP, independent 115 of the respiration paradigm described by Amthor (2000).
In the following section, we look with more detail at the structure of some particular models with the aim of exploring the main source of carbon used for respiration and allocation. (a) Litter and soil pools Litter and soil pools
The other nine :::: eight models do not consider an explicit calculation of stock-dependent maintenance respiration, and also allocate carbon from NPP. Some of these models explicitly claim :::::: express that given the linear relationship between C canopy 140 respiration and canopy photosynthesis, the autotrophic respiration is a fixed fraction of the total photosynthetic fixation. Some In many models, GPP and Ra occur at short timescales (half-hourly, hourly, or daily), computing the net carbon gain as an annual integral. Carbon allocation occurs at annual intervals, when the assimilated carbon that is not respired is assigned to a particular vegetation compartment. Therefore, the carbon that is respired at an intra-annual timescale never enters the vegetation pools.

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The important point that we want to highlight here is that even though some models compute maintenance respiration based on knowledge of the carbon stock that needs to be maintained, this respiration is actually subtracted from GPP to obtain the net carbon gain that is subsequently allocated. Only in a few models, maintenance respiration is subtracted from a carbon stock such as a labile pool or other vegetation compartment, but most models can be written in the form of equation (1) with

Continuous-versus discrete-time implementations
In addition to the issue of the source of carbon (GPP or NPP) used for allocation, there is a related problem in computing the age of Ra that emerges in model implementations that are discrete in time. Models based on ordinary differential equations such as those expressed as in equation (1) treat time as a continuous variable, but many models are implemented in discrete time steps where the carbon stocks of the previous time step are updated based on the functions defined by the model. To compute maintenance respiration in this model, carbon can be used immediately and hence never enters the tree. Growth respiration on the other hand, happens at the same time step as carbon is allocated to the tree organs but with a one year time lag, one time step after it entered the transient pool from photosynthesis. Practically, this means that growth respiration happens 170 one year later than maintenance respiration, and that carbon respired by maintenance has an age of zero. This age of respired carbon is not realistic when compared with measurements, which can be obtained at finer temporal resolutions and over a broader range.
3 Age of respired carbon obtained as the transit time distribution from models The age of respired carbon can be obtained from ecosystem models, but the model structure and the form in which the source 175 of carbon for allocation is represented has an impact on the age of carbon respired from ecosystems. Although most models do not represent carbon age explicitly, it can be computed using different computational approaches.
The age of respired carbon from ecosystem is characterized by its transit time distribution (Bolin and Rodhe, 1973;Thompson and Randerson, 1999;Sierra et al., 2021b). These distributions can be obtained from ecosystem carbon models using impulse response functions (Thompson and Randerson, 1999), a simulation approach that consists of applying a pulse of car-180 bon to a model at equilibrium, where carbon stocks do not change over time, and then observing the respiration flux after the pulse. These distributions can also be obtained using the analytical formulas developed by  for models at equilibrium, or the approach described in  for models out of equilibrium.
The transit time distribution represents the proportions of respired carbon that have different ages, and it is usually a continuous function that results from a mixture of exponential functions . They can be obtained from any 185 ecosystem model expressed in compartmental form as 1 where u(t) is a vector of carbon inputs to the system. In the framework of Luo et al. (2017), compartmental matrix with diagonal elements the cycling rates within the pools, and off-diagonal elements the transfer rates of carbon among the different pools. In the framework of Luo et al. (2017), B(t) = ξ(t)A K. Respiration from each compartment 190 j can be obtained as the product of the amount of mass present in the system and a rate of release z j , This rate of release z can be obtained from the compartmental matrix B as the ::::::: negative sum of the entries of each column. It represents the fraction of carbon that leaves each pool and is not transferred to any other pool.
The GPP-based version of the model predicts a continuum of ages of respired carbon both for autotrophic and heterotrophic respiration (Figure 3). Although a large portion of autotrophic respiration is very young (< 1 year), a significant proportion is 220 older and can be respired years after photosynthetic fixation.

Age of respired carbon obtained from radiocarbon measurements
Several studies have used radiocarbon-based methods to estimate the age of the respired carbon form different compartments of the ecosystem (e.g., foliage, wood, roots, and soil) (Carbone and Trumbore, 2007;Carbone et al., 2007Carbone et al., , 2013Muhr et al., 2013Muhr et al., , 2018Trumbore et al., 2015). In vegetation compartments, studies have focused mostly on individual trees rather than 225 on a larger sample of trees within a forest stand. For healthy mature trees, small differences have been found between compartments, for example carbon respired from leaves may be less than one year old (Carbone and Trumbore, 2007), while in roots and stems the respired carbon is on average older than one year, with a mix of carbon from recent assimilates and some contributions of old carbon from storage reserves (Muhr et al., 2018). There is empirical evidence that shows that the age of the respired carbon by trees can change during different seasons, and increases as trees are exposed to stress and have to use Some studies have also reported several years old respired CO 2 , ranging from 1 to 5 yr from roots. Most of these studies report mean values of 4 yr old respired carbon from roots (Czimczik et al., 2006;Schuur and Trumbore, 2006;Carbone and Trumbore, 2007), but younger CO 2 (0.6 yr old) has been also reported by Hilman et al. (2021).
With very few exceptions, most of the empirical evidence supports the idea that respired carbon from vegetation parts is on average older than 1 yr, but higher values can be observed depending on the season or on whether trees suffer some form of This empirical evidence, which shows that the age of respired carbon spans from one to several years (Figure 4), is inconsistent with predictions from models in which carbon allocation is based on NPP where the age of respired carbon is exactly equal to zero (Figure 2). tems; and (3) determining isotopic exchange between terrestrial ecosystems and the atmosphere. We briefly elaborate on these three implications in the following paragraphs.
First, as radiocarbon measurements become increasingly available for plant parts and respired CO 2 from ecosystems, there 255 is an excellent opportunity to use these data for constraining vegetation models and testing model-based hypotheses. Modeldata assimilation techniques are very powerful to reduce model structural uncertainty, and can be used to improve carbon allocation and respiration routines in models. However, as we have shown here, the age of respired CO 2 in NPP-based models is predicted as exactly zero, inconsistent with radiocarbon measurements. Therefore, by construction, NPP-based allocation schemes cannot be used to assimilate radiocarbon measurements and constrain allocation and respiration functions.

6 Summary and recommendations
We have shown that models in which carbon allocation occurs after autotrophic respiration is subtracted from GPP (i.e. NPPbased models) predict that the age of respired carbon from vegetation pools is zero. This prediction contradicts empirical evidence based on the isotopic signature of respired CO 2 from plant parts, and suggests that GPP-based allocation schemes are more appropriate to represent carbon allocation and respiration in models. Models in which allocation is based on NPP miss 285 on the opportunity to use radiocarbon data for constraining model parameters and improve their representation of vegetation processes. They are also unable to produce realistic transit time distributions of carbon, and can provide misleading predictions of isotopic exchange between ecosystems and the atmosphere.