Potassium-limitation of forest productivity, part 2: CASTANEA-MAESPA-K shows a reduction in photosynthesis rather than a stoichiometric limitation of tissue formation

. Potassium (cid:58)(cid:58)(cid:58) (K) (cid:58) availability constrains forest productivity. Brazilian eucalypt plantations are a good example of the K-limitation of wood production. Here, we built upon a previously described model (CASTANEA-MAESPA-K) and used it to understand whether the simulated decline in C-source under K deficiency was sufficient to explain the K-limitation of wood productivity in Brazilian eucalypt plantations. We developed allocation schemes for both C and K and included into CASTANEA-MAESPA-K. No direct limitations of the C-sink activity, nor direct modifications of the C-allocation by K avail-5 ability were included in the model. Simulation results show that the model was successful in replicating the observed patterns of wood productivity , growth, NPP limitation by K deficiency. Simulations also show that the response of NPP is not linear with increasing K fertilisation. Simulated stem carbon use and water use efficiencies decreased with decreasing levels of K availability. Simulating a direct stoichiometric limitation of wood productivity, growth, NPP was not necessary to reproduce the observed decline of productivity under K limitation, suggesting that K stoichiometric plasticity could be different than that 10 of N and P. Confirming previous results from the literature, the model simulated an intense recirculation of K in the trees, suggesting that retranslocation processes were essential for tree functioning. Optimal K fertilisation levels calculated by the model were similar to nutritional recommendations currently applied in Brazilian eucalypt plantations, paving the way for validating the model at a larger scale and this approach to develop decision-making tools to improve fertilisation practices.


Introduction
Fertilisation trials in tropical eucalypt plantations have been conducted over multiple rotations (Laclau et al., 2010;Battie-Laclau et al., 2016;Gazola et al., 2019;Gonçalves, 2000).These experiments have shown that nutrients can strongly affect tree growth in these highly prodctive stands.These nutrient limitations can be explained in part by low nutrient supplies from highly weathered soils, and in part by the large exports of nutrients from plots with trunk wood (every 6-7 years) from the stands.The frequent export of trunk wood in these fast-growing plantations leads to the export of massive amounts of nutrients that are immobilised in wood (Cornut et al., 2021).In commercial plantations this issue is solved through the use of fertilisers (NPK, dolomitic lime and micro-nutrients).
Potassium (K) has been identified as the most limiting nutrient for wood productivity in many omission trials (Gonçalves, 2000;Rocha et al., 2019).Potassium nutrition impacts wood growth through different physiological mechanisms that have been reviewed in detail (Sardans and Peñuelas, 2015;Cornut et al., 2021).In brief, K deficiency is known to depress the assimilation of carbon by the plant (C-source processes, Gross Primary Production) as well as the allocation and use of carbon by the plant for growth (C-sink processes, Net Primary Production).
The study of the limitation of GPP (C-source) by K deficiency was explored at the stand level in Part 1 :::::: (Cornut ::: et ::: al., ::::: 2023) of this series of two papers, using a coupled C-H 2 O-K mechanistic model (CASTANEA-MAESPA-K).The simulations showed a strong response of GPP to K deficiency.The GPP in the simulated K omission stand was less than half of that in the simulated fertilised stand (Tab.2,Part 1 :::::: Cornut :: et ::: al., :::: 2023).These results were consistent with previous measurements (Epron et al., 2012) and modelling work (Christina et al., 2018).The strong response of GPP to K availability was due to a reduction in leaf area index and in the photosynthetic capacity per unit area of leaf.The reduction in canopy area was due in part to a slight reduction in leaf production, in part to a decrease in individual leaf area and in part to a strong decrease in leaf lifespan (Fig. 4, Part 1 ::::: Cornut :: et ::: al., ::::: 2023).The reduction in photosynthetic capacity of the canopy was associated with the appearance of leaf symptoms in the K-deficient (oK) stand.The impact of symptom area on leaf photosynthetic capacity was sufficient to explain most of the reduction in leaf-scale assimilation in the unfertilized case.A decrease of WUE GP P ::::: (Ratio :: of ::::: GPP :: to ::::::::::: transpiration) in the simulated oK stand was also simulated.A sensitivity analysis of the model parameters showed that competition between the organs (trunk, branches, bark and roots) and leaves for K access had an important impact on GPP in the oK stand (the leaf to phloem resorption resistance, R leaf →phloem , : ; : Fig. 7 Part 1 :: in :::::: Cornut :: et ::: al., ::::: 2023).This underlined the need for a precise understanding of K circulation and stoichiometry in the plant.
The objectives of the present study were to understand: 1. the impact of K deficiency on the NPP of a Brazilian eucalypt stand representative of large areas of commercial eucalypt plantations, 2. whether the influence of K availability on GPP is sufficient to explain the differences in wood productivity between K-fertilised and K-deficient stands, 3. the link between C partitioning and K availability in the plant-soil system, 4. which parts of the K cycle are the most critical to simulate accurately the consequences of K-limitation on wood productivity, 5. if K:C stoichiometry can contribute to explaining the observed patterns of organ NPP.

Study site
A split-plot fertilisation trial was installed at the Itatinga experimental station (23°02'49"S and 48°38'17"W, 860 m asl, University of São Paulo-ESALQ).The precipitation was on average 1430 mm : .year−1 , with a drier season between June and September, and the mean annual temperature was 19.3°C.The trial was established on June 2010, for 6 years.The planted clone was a fast growing Eucalyptus grandis.The experimental design was described in detail in Battie- Laclau et al. (2014).
Six treatments (three fertilisation regimes crossed with two two water regimes) were applied in three blocks.In the present study, we focus on the +K and oK treatments with undisturbed rainfall regime, which consisted in : : a non-limiting K fertilisation : (+K() :::: with : 17.55 gK : .m−2 applied as KClwith , : 3.3 gP : .m−2 , 200 g.m −2 of dolomitic lime and trace elements at planting and 12 gN.m −2 at 3 months of age); and an omission treatment : oK :::: (oK) : where all fertilisation was applied as in the +K treatment, except K :: the :::: KCl.
The concentrations of N, P and K (as well as Ca and Mg) in the organs (leaves, trunks, branches, and roots) were measured at an annual time step in 8 individuals of each fertilisation treatment and upscaled to the whole stand using allometric relationships.Biomass and nutrient contents were calculated (using upscaling) from inventories, biomass and nutrient concentration measurements performed at :: in :::: each :::::::::: fertilisation :::::::: treatment :: at :::: year 1, 2, 3, 4, 5 and 6 years in each fertilisation treatment :::: after ::::::: planting.Atmospheric deposition (0.55 gK m −2 yr −1 ) was measured in a nearby experiment from Laclau et al. (2010).(Dufrêne et al., 2005), generally used to simulate temperate forest stands was adapted to tropical eucalypt plantations.It was merged with the MAESPA model (Duursma and Medlyn, 2012;Christina et al., 2017) since it does not simulate the water-plant-atmosphere hydraulic continuum natively and the assumption of a fixed root depth made by CASTANEA did not hold true in the studied system (Christina et al., 2011)

Carbon allocation
In the CASTANEA model, carbon of assimilates was allocated to organ growth and soluble sugars (SSs), after a part of it was released to the atmosphere through maintenance respiration and growth respiration processes.SSs are not localized in the model which, for the carbon part, has no topology.So SSs are hypothesized to be part of the phloem and other tissues, indistinctly.This is different from the K content of the plant, that we localized either in organs or the semi-explicit phloem and xylem saps.
The growth of all organs was a fraction of the daily NPP (Fig. 1).The allocation coefficients of each organ except leaves (i.e.fine roots, coarse roots, woody organs) were calculated at a daily time-step and were the result of NPP and allometric relationships among organs.The generation of leaves and their growth was a process that was not directly dependent on NPP (see Part 1), and had priority for the allocation of carbon.Leaf growth could however be limited if demand for growth was higher than available C in the SSs compartment.

Roots
In accordance with experiment evidence in the fertilisation experiments, the allocation coefficient of coarse roots : , ::::: G CR :::::::: (unitless), was set to a constant: The allocation coefficient of carbon to fine roots : , :::: G F R ::::::::: (unitless), : was similar to Marsden et al. (2013).It used a target function, where the fine root target biomass was a function of leaf area index: where G F R was the allocation coefficient to fine roots, G CR and G SS ::::::: (unitless) ::::: were the allocation coefficients of coarse roots (eq.2) and SSs (eq. 1) respectively, B F R (gC.m −2 ) the fine root biomass, λ (gC.m −2 leaf ) the conversion coefficient between leaf area and target root biomass, L (m 2 leaf .m−2 soil ) the leaf area index of the stand and p F R a sensitivity parameter.

Woody organs
The remaining C available after allocation of NPP to leaves, roots and SSs was allocated to the woody organs (trunk, branches and bark): where G W :::::::: (unitless) was the allocation coefficient to woody organs, and G F R , G CR and G SS the allocation to fine roots (eq. 3), coarse roots (eq.2) and soluble sugars (eq.1), respectively.
The C allocated to woody organs was then distributed to the different woody organs (trunk, branches and bark) using allometric relationships.
Parameters of the function linking the allocation of woody NPP to branches in function of total woody biomass were fitted on experimental data.The following equation was used in the model: where G Br :::::::: (unitless) was the ratio of woody NPP allocated to branch production, B trunk , B Br , B bark the biomasses (gC.m −2 ) of the trunk, branches and bark respectively.
Following experimental data, the allocation of woody NPP to bark was considered to be constant: All the C remaining after allocation of woody NPP to the bark and the branches was used for trunk wood production.
where G Bark was from eq.6 and G Br from eq.5.
The increase in height was a function of trunk biomass.The relationship was fitted in the +K stand using biomass and inventory data (Fig. S1): where H (m) was the height of the stand and B trunk (gC.m −2 ) the biomass of the trunk.

Maintenance respiration
The hourly maintenance respiration for all organs (except leaves which had a respiration rate based on their vertical position in the canopy, Christina et al., 2015) was a function of their respective respiration rate per nitrogen unit (see below), nitrogen content and surface temperature (Dufrêne et al., 2005).
The N content of leaves and bark were not simulated since N content did not influence the respiration of leaves in our model and bark had no maintenance respiration.
The maintenance respiration of leaves was their dark respiration (inhibited during the day), Rd, and the values measured in K-fertilised trees in a nearby site (Eucflux) were used (see Part 1 ::

Organ turnover
With the exception of the coarse roots and the trunk, the organs (branches, bark, fine roots) were subject to turnover.Branches, bark, fine roots and leaves each had lifespans.
The necromass of most organs was added to the litter pool.On the other hand, dead branches were added to the the dead branch pool.The dead branch pool represented the branches that stay attached to the tree after senescence.The dead branch pool had a specific turnover rate.
Resorption of K took place during the senescence of leaves (see Part 1 ::::: Cornut :: et :::: al., :::: 2023) and branches (Fig. S4a).K remobilised from these two organs was added to the phloem sap K pool.While the resorption rate for leaves was dependent on the nutritional status of the tree and their theoretical lifespan, for branches it was fixed.

K allocation, remobilisation and turnover
The allocation of K to organs was a function of the optimal K concentration of newly formed organ tissue, organ NPP and K availability in the tree.Firstly, organ NPP was calculated by allocating part of GPP to the organs after subtracting the respiration.
Then, the total K quantity required was calculated by multiplying the growth of each organ by its optimal K concentration (the concentration of newly formed organs in the fully fertilised stand).If the quantity of available K in the phloem sap was inferior to the demand, K allocation to the organs was limited without affecting C allocation to the organ's growth.This was equivalent to flexivle :::::: flexible : stoichiometry in other models.
where K N P P (gK.m −2 :::::: .day −1 ) was the quantity of K necessary for optimal stoichiometry of newly formed organ biomass, N P P org (gC.m −2 .day−1 ) the daily net primary production minus the allocation to the leaves, K opti org (gK.gC −1 ) the optimal concentration of the considered organ (eq.17-20) and G org the allocation coefficient of that organ (eq. 2 to 4).
The quantity of available K was a function of K content in the phloem sap and the minimal quantity of K in the phloem sap (Part 1): :::::: Cornut :: et :: al. :::::: 2023): The limitation of K allocation to newly formed organ nutrient content was simply the ratio between available K and K demand: where K available was from eq. 15 and K N P P (gK.m −2 :::::::::::: gK.m −2 .day−1 ) the amount of K needed for optimal stoichiometry of newly formed organs (eq.14).
The cycle of K in the leaves is described in Part 1 ::::: Cornut :: et ::: al. ::::: (2023) : since it is an integral part of the canopy cohort model.

Wood
Due to the continuous phenology of tropical eucalypt trees, K dynamics in wood were simulated through daily cohorts of wood.
Since K concentration of the total trunk tissue decreases with trunk biomass, it was necessary to implement wood K remobilisation in the model.A model where the remobilisation rate was dependent on wood production was the best suited for this task since it showed the best fit with experimental data when compared to a model where remobilisation was independent of wood production.The following equation was used: where K i trunk→xylem (gK.m −2 ::::: .day −1 ) was the remobilisation of K from the cohort i, K i T runk (g :::::: K i trunk :::: (gK.m −2 ) the K mineralomass of the trunk cohort i, T KT runk (. :::::: T Ktrunk :: (gC −1 ::: .m 2 ) the remobilisation rate per unit of trunk production, and N P P trunk (gC.m −2 .day−1 ) the daily trunk increment.Remobilised K was allocated to K xylem .If the K concentration of the cohort ([K] i T runk::::::: [K] i trunk ) was lower than a threshold value [K] min T runk ::::::: [K] min trunk: (gK.gC −1 ), there was no remobilisation (K i trunk→xylem = 0).The threshold, [K] min T runk ::::::: [K] min trunk , was determined from the minimum asymptot of the relationship between trunk biomass and trunk K concentration at the fertilization experiment (Fig. S5).This measured value was assumed to be the minimum concentration of a cohort since it was assumed that at a high enough wood biomass, the proportion of K associated to newly formed wood to the total wood K content was negligible (Augusto et al., 2000).This meant that the measured concentration of wood as a whole was similar to the minimum concentration of K in the trunk at high enough trunk biomass.

Branches
The optimal K concentration of newly formed branches was a function of the branch biomass.
[K] opti Br = 0.00707 × e −0.00854×B Br + 0.000759 where [K] opti Br was the K concentration of the newly formed branches in gK.gC −1 and B Br was the branch biomass in gC.m −2 .
This decreasing function was fitted on experimental nutrient content and biomass data collected in the fertilised plots.

Bark
The optimal K concentration of newly formed bark was a function of the bark biomass.
[K] opti Bark = 0.00204 × e −0.0057×B Bark + 0.00147 where [K] opti Bark was the K concentration of the newly formed bark in gK.gC −1 and B Bark was the bark biomass in gC.m −2 .The parameters for this function were fitted on experimental data collected in the fertilised plots.No remobilization was considered for bark since there no measurements were available.

Roots
The optimal K concentration (in gK.gC −1 ) of coarse ([K] opti CR ) and fine roots ([K] opti F R ) was a fixed value independent of tree age or biomass.Due to the absence of data regarding this process, the model did not simulate remobilisation from roots.The K content of dead fine roots was added to the K litter pool, which in turn leached into the soil available K and could be uptaken by other living roots.

Total remobilisation in woody organs
Total remobilisation was the flux of K from the woody organs to the xylem.It was calculated as the following: where K remob (gK.m −2 .day−1 ) was the total remobilisation flux, K trunk→xylem (gK.m −2 .day−1 ) the remobilisation rate of wood, R Kbranches the remobilisation rate of dying branches and K mortality branches (gK.m −2 .day−1 ) the flux of K from living branches to dead branches.

Simulations
The simulation initialisations were conducted to resemble as closely as possible the omission experiment.Simulations in the fully fertilised treatment (+K) were initialised with the same fertilisation values as the fertilised control in the experiment (i.e.17.5 gK m-2) corresponding to a one-time application of fertiliser at planting.Simulations in the K omission treatment (oK) shared the same initialisation except that the fertiliser pool was initialised with 0 gK.
To investigate the effects of a fertilisation gradient, 10 initialisation values of K pools spanning from no input (in oK) to 17.5 gK (+K) were chosen.
To test whether the fertilisation regime could have an impact on tree productivity, two fertilisation regimes were simulated.
One where the K dose was brought all at once (as in the experiment) and one where the same K fertiliser dose was broken up into 4 sub-doses that were temporally spaced (equivalent to the Eucflux experiment, see Part 1 : ; :::::: Cornut :: et ::: al., :::: 2023).

Analysis
To test the accuracy of model prediction, the Root Mean Square Errors of simulations' output variables were calculated using measurements at the experimental site.The mean of the 3 experimental blocs (there were 3 blocs per experimental treatment) was used.To normalise this metric and have a relative Root Mean Square Error, the RMSE was divided by the measured mean throughout the rotation of the considered output variable.
To describe the response of resource use efficiency (RUE) to different levels of K availability, we used the following metrics.

Carbon use efficiency
The carbon-use efficiency was calculated as the simulated NPP flux divided by the simulated GPP flux (De Lucia et al., 2007).
It is in fact a measure of the proportion of assimilated carbon that was not lost to respiration.

Water use efficiency
The water use efficiencies of NPP (WUE N P P ) and trunk NPP (WUE trunk ) were calculated by dividing the total NPP or trunk NPP by the amount of transpired water during the period over which NPP and trunk NPP were calculated.
Potassium use efficiencies of GPP (KUE GP P ), total NPP (KUE N P P ) and trunk NPP (KUE trunk ) were calculated by dividing the respective C-based metric by the maximum amount of K that was immobilised in the plant during the rotation.For example in the case of (KUE N P P ): where KU E N P P (gC.gK −1 ) was the K-use efficiency of total NPP, k the number of days in the rotation (days), N P P :::::: N P P t (gC.m −2 .day−1 ) the daily NPP of the rotation, and K max plant (gK.m −2 ) the maximum of K that was immobilized in the plant during the rotation (the maximum of total simulated plant K during the rotation).For calculating KUE GP P and KUE trunk the numerator of the above fraction can be replaced by GPP or NPP trunk respectively.There are many alternative ways to calculate nutrient use efficiencies in forests (Turner and Lambert, 2014).Here, we decided to use total K immobilisation instead of uptake, : since circulation of K in the system was high and we think that the maximum amount of K accumulated in standing biomass is a more relevant representation of total system K demand.Indeed, the maximum K accumulated in standing biomass is a proxy of the amount of K necessary in the system and along the rotation for the plant considering its biomass.In systems where there is less restitution to the soil, it would be equivalent to the soil nutrient uptake.

Fertiliser use efficiency
Fertiliser use efficiencies were computed as the difference of cumulated NPP between the simulated K omission stand (oK) and stands simulated with different K fertilisation levels, and dividing it by the amount of K fertiliser added.This allowed us to compute the growth gain in carbon per unit of K fertiliser used: where F U E f N P P (gC.gK −1 ) the fertiliser use efficiency of NPP for a given level of fertilisation, k the number of days in the rotation (days), NPP f (gC.m −2 .day−1 ) the daily NPP of the currently considered stand, N P P oK (gC.m −2 .day−1 ) the NPP of the K omission stand and K added f ertiliser (gK.m −2 ) the amount of K fertiliser that was added in the considered stand.To obtain F U E GP P or F U E T runk ::::::::: F U E trunk , this relationship can be applied to either GPP or N P P trunk , respectively.

Prediction of changes in NPP caused by GPP
The model was capable of replicating most of the NPP and biomass differences between the +K and oK stands (Fig. 2).In the +K stand, the five-year cumulated 2023) and GPP estimations using the TBCA method applied to C stocks and C fluxes measured throughout the rotation in our experiment (not shown, Giardina and Ryan, 2002).The reduction of trunk NPP trunk was comparable to data (Fig. 2b).The same could be said for the bark (Fig. 2d) and the branches (Fig. 2f).This led to simulated biomasses in line with measurements in the +K and oK stands for branches (Fig. 2e : ), bark (Fig. 2c) and trunk (Fig. 2a).In the +K stand the RMSEs (and normalised RMSEs in parentheses) for :: of ::::::::: simulated trunk, branches, bark, leaves and total aboveground biomass were respectively: 385 gC.m −2 (-1%) in the +K stand and an overestimation of 161 gC.m −2 (6%) in the oK stand.This overestimation of aboveground biomass was concurrent to an underestimation of root biomass (Fig. 2i).

Consequences of K addition on C allocation patterns within trees
The model allowed the study of allocation in the trees under different K fertilisation regimes.The simulated allocation patterns did not differ greatly between the fertilised and omission stands (Fig 3a).However, simulated carbon use efficiency (defined as the ratio of NPP to GPP) was reduced by 23% in the omission stand (Fig. S6b, 0.52 vs 0.40).The ratio of wood productivity to GPP (CUE trunk ) was reduced by the same proportion (21%) showing that the reduction of NPP trunk followed the same dynamic than total NPP.Moreover the difference in CUE between the two fertilisation treatments increased throughout the rotation (a 2% difference the first year, and a 17% difference the fifth).The trend was similar for CUE trunk .The difference in CUE between the two treatments was mainly due to an increase of maintenance respiration in the oK stand where it accounted for 48% of the GPP compared to 34% in the +K stand.This was further amplified by leaf NPP representing 13% of GPP in oK compared to 7% in +K.
The model was also capable of simulating the response of the different carbon fluxes, along the stand rotation, to a gradient of initial K fertilisation (Fig. 3b).It showed that the responses of GPP, NPP and wood productivity to fertilisation all saturated at around 11 gK.m −2 for a five-year rotation.The simulated carbon fluxes did not show any sensitivity to the fertilisation application regime (one or four time application, Fig. S6a).The simulations conducted with the one-time application at planting compared with the same amount of K split in 4 applications at 0, 3, 10, and 20 months of age showed little to no differences in GPP, NPP and NPP trunk (Fig. S6a).The wood productivity was similar in all fertilisation treatments in the first year of the rotation (Fig. 3).This result was in contrast to experimental data that shows that the relative difference in organ NPP appears early the rotation (Laclau et al., 2009).While the response of CUE N P P resembled a linear function before it saturated at a fertilisation of 11 gK.m −2 , the response of CUE trunk followed a non linear response (Fig. S6b) by increasing from 0 to 2 gK.m −2 of fertilisation added as KCl fertiliser, saturating between 2 and 4 gK.Age of the rotation (months)

K cycling in the trees
The model showed that the main sinks of K in both the +K and oK stands were the woody organs (trunk, bark and branches).
Despite the remobilisation of K in the trunk, the quantity of K immobilised in the trunk increased linearly with time in both treatments (Fig. 4) thus constituting an important K sink.
The theoretical minimum concentration of K in the xylem sap (assuming no recirculating K) of our trees was calculated by dividing the daily simulated flux of K that circulated in the xylem sap (uptake, wood and branch remobilisation) by the simulated transpiration flux of each day.The mean simulated minimum xylem sap K concentration over the course of a rotation was 0.30 mM (0.012 gK.L −1 ) in the fully fertilised stand and 0.11 mM (0.004 gK.L −1 ) in the K omission stand.When including the K content of the phloem sap and leaf resorption (which means the total circulating K in the tree) in this calculation, the values were 1.66 mM (0.065 gK.L −1 ) and 0.46 mM (0.018 gK.L −1 ), respectively.
The simulation of internal and external K fluxes in the system (Tab.1) showed that in the fully fertilised and K omission stands, wood remobilisation represented the most important flux of K. Implementing in the model this process of K remobilisation from wood increased the model accuracy substantially (not shown here) by buffering the amount of K available for organ growth.When added to branch and leaf resorption, the total amount of K remobilised represented 1.8 times the K uptake in the fertilised stand versus 1.4 times in the omission stand.In the simulated oK stand, K uptake was very similar to the sum of the litterfall, leaching and atmospheric deposition fluxes, in all but the first year of the rotation (Tab.1).The deposition flux represented more than 50% of the uptake flux in these K deficient conditions.Moreover, in the K omission stand, an increase of the simulated weathering flux from 0 to 0.3 gK.m −2 .yr−1 (in the range of possible values, see Cornut et al., 2021) led to an increase of 23, 28 and 30% of the rotation-cumulated GPP, NPP and NPP trunk respectively.This showed that small differences in K input can lead to big differences in outcome for wood productivity.
The difference in the K flux of canopy leaching between the two simulated stands (much lower in the omission stand, Tab.
1) was in line with results obtained on eucalypt plantations at K-rich and K-deficient sites (Laclau et al., 2010).This was the result of lower leaf K concentration and supports the validity of the leaching model used here (see Part 1 :::::

Water and potassium use efficiencies
Omission of K fertiliser decreased stand transpiration by 51%.The reduction (13%) of simulated GPP water use efficiency (WUE GP P ) between the +K (0.0035 gC GP P .gHIn the following paragraph, potassium use efficiency (KUE) is understood as the ratio of a cumulated carbon flux at the end of the rotation and maximum value of K immobilized in the tree.The simulated KUE GP P were 1281 and 1994 gC.gK −1 plant in the +K and oK stands, respectively.In contrast, the simulated KUE N P P and KUE trunk only increased by 19% (owing to decreased CUE in the oK stand) between the +K and oK stands.The simulated KUE trunk were 387 and 462 gC.gK −1 P lant in the +K and oK stands, respectively (656 and 784 respectively for KUE N P P ).FUE generally decreased with increasing fertilisation (Fig. S7).However, a two-slope relationship was apparent: at low levels of fertilisation, increases in fertilisation led to strong increases in NPP.However, for high amounts of fertiliser applied, wood production per unit of K declined.At intermediate levels of fertilisation (between 6 to 10 gK.m 2 ) the FUE was almost constant or slightly increased.However, at higher levels of initial fertilisation the FUE linearly decreased (consequence of a stable NPP with linearly increasing fertilisation, Fig. 3b).

Stoichiometry of organs
Potassium concentrations in trunk wood and in branches were correctly simulated (Fig. 5) using a simple limitation of the K flux entering into each organ in function of the organ-specific K optimal concentrations used (Eq.14-20).The mean K concentrations in the total tree biomass were 0.0048 gK.gDM −1 in the +K stand and 0.0030 gK.gDM −1 in the oK stand.This corresponded to a decrease of 36% of the K concentration in the oK stand relative to the +K stand.This revealed that total plant stoichiometric flexibility was high in the model, which is in accordance with measurements.

GPP-limitation of NPP and partitioning of photosynthates
The CASTANEA-MAESPA-K model was largely successful in reproducing the limitation of wood productivity induced by K deficiency (Fig. 2).This makes it the first mechanistic model to simulate the interaction between the K cycle and forest NPP.
Combined with the fact that partitioning in the model was not directly impacted by K availability (there was no mechanism through which K directly impacted carbon allocation to wood), which suggests that the limitation of wood productivity in the absence of K fertilisation was mainly due to GPP-limitation.However, the consequences of K deficiency were higher for NPP than for GPP.This was due to a decrease in CUE at low levels of fertilisation.This is similar to the general trend for world forests, which is a decrease forests' biomass production-GPP ratio (a proxy for CUE) with decreasing fertility (Vicca et al., 2012).Here, the partitioning of GPP to the different organs was not strongly affected by K availability (except for leaves going from 7% in +K to 13% of NPP in oK).The reduced CUE was mainly the result of an increased autotrophic respiration in proportion to the biomass of organs mirroring the increase in the ratio of ecosystem respiration to GPP in nutrient-poor forests (Fernández-Martínez et al., 2014).While the relationship between GPP and net ecosystem productivity was not significant for low fertility forest sites in this meta-analysis, this was not the case for GPP and NPP in our study.
Only simulated fine root biomasses were under-estimated in both the +K and oK stands (Fig. 2ji).This suggests that simulating a root target biomass that is function of leaf area (Marsden et al., 2013) might not be appropriate in this instance (since the shape of the root biomass curve during the rotation is qualitatively different).One other cause could be a misestimation of root lifespan since measurements of fine root turnover have yielded a wide range of values (Jourdan et al., 2008;Lambais et al., 2017).The estimation of fine root turnover had also been an constraint to simulating N mineralisation rates in Australian eucalypt stands with the G'DAY model (Corbeels et al., 2005).The values of GPP that were simulated here (a mean of 3986 gC.m −2 .yr−1 in +K and 1709 gC.m −2 .yr−1 in oK) were on the high range of values expected for terrestrial ecosystems (Baldocchi and Penuelas, 2019;Luyssaert et al., 2007), especially considering these values were a 6 year mean that included the first year after planting.They were similar to values estimated using TBCA measurements in highly productive eucalypt plantations (Ryan et al., 2004(Ryan et al., , 2010) )  2023 : for the site description).
The K fertilisation level value at which K-limitation was totally alleviated in simulations (Fig. 3b) was similar to the 12 gK.m −2 of fertilisation that are commonly added to commercial eucalypt plantations in Brazil (Cornut et al., 2021).

Water and potassium use efficiency is affected by K availability
The simulated RUEs of NPP and NPP trunk were strongly affected by K availability (modelled here as a change in fertilisation levels).Variations of WUE trunk are in line with experimental results that showed a decrease in WUE trunk in the K omission stand at the Itatinga site (Battie- Laclau et al., 2016).While simulated WUE trunk was 33% lower in oK than in +K, measurements showed a decrease of 37% (Battie- Laclau et al., 2016).This confirms the relevance of a model-based approach in studying the effect of K availability on WUE trunk .However, the model diverged from measurements in the same stands at the end of the stand rotation (from 4 to 6 years after planting) that showed that WUE trunk of the K omission stand was reduced by 75% as compared to the WUE trunk of the +K stand (Asensio et al., 2020), while CASTANEA-MAESPA-K only showed a reduction by 39% of WUE trunk between these two stands at this age.Since the trunk NPP simulated by the model agreed with measurements (Fig. 2b) and the model simulated a reduction in transpiration consistent with different approaches (see Part Both the changes in KUE and FUE along a K fertilisation gradient showed some interesting results.While K fertilisation strongly decreased KUE for GPP, the effects on NPP and trunk NPP were weaker, as a result of an increase in CUE (Fig. S7).
This was a consequence of reduced CUE.FUE results demonstrate that the response of wood productivity is not a linear function of fertilisation and that at low levels of fertilisation, small increases in K fertilisation levels produce a strong increase in NPP.Simulated NPP was maximal with the highest fertilizer amount, however the trend is asymptotical, with more than 95% of trunk NPP (Fig. 3b) already reached at around 10 gK.m −2 , a value commonly applied in commercial eucalypt plantations managed in this soil type.Partitioning of the K fertilisation in only one or in several amounts did not changed the wood production, considering that no deep leaching occurs in these systems, in agreement with the conclusions of field studies measuring soil solution chemistry in deep soil layers (Laclau et al., 2010).

Circulation of K in the plant and stoichiometry
Our work pinpointed the importance of a plant K circulation model.The total remobilisation flux of K from the branches, the trunk and the leaves was higher than the K uptake in the soil at all fertilisation levels.In C-N (Zaehle et al., 2010;Thum et al., 2019) or C-N-P (Goll et al., 2017) coupled models, stoichiometry of organs can be a direct limiter of organ growth.
A strong effect of stoichiometry on soil organic matter decomposition has also been historically used in models of organic matter decomposition in soils (Parton et al., 1988).While such mechanisms were not considered in our modelling approach (organ K concentration is allowed to vary unconstrained), the reduction in NPP was enough to compensate the reduction in K availability such that the simulated stoichiometries of organs do not vary more than they do in the measurements (Fig. S5).
The stoichiometric flexibility of trees could be higher for K than for N and P.This is consistent with observations of wood K content, which depends mainly on abiotic conditions while P wood content depends mainly on the species (Bauters et al., 2022).Our model suggests that leaves have a higher stoichiometric flexibility than wood (not shown here), in agreement with observations in a Mediterranean forest environment (Sardans et al., 2012).However, the model failed at reproducing the patterns of stoichiometric plasticity (variability in K concentration of organs) that were observed in between the woody organs, in particular in branches (measurements show high stoichiometric flexibility in branches).This suggests that organs can differ in K homeostasis.The failure of the model to reproduce stoichiometric flexibilities in wood and bark (not shown here) could potentially have had an influence on the amount of K available for leaf expansion thus overestimating the K limitation on canopy surface and leaf functioning.This is further exacerbated by the large amount of K stored in the bark and in the branches in the simulated K omission stand, especially when the model showed a strong K limitation of leaves (between the 10th and 20th month after planting).
The simulated amount of K immobilised in the trunk at the end of the rotation in the +K stand was an order of magnitude lower than what is observed in tropical forests (Bauters et al., 2022).This is due to a low total biomass in planted forests managed in short rotation compared to a natural forest since the simulated K concentrations in the trunk were in the range of values reported in old-growth tropical forests.The yearly increase in the amount of K stored in the trunk was in the range of observed values (see Fig. 1 in Bauters et al., 2022) both for the oK and +K stands.This suggests that tropical eucalypt plantations could be a relevant model system for certain parts of the K cycle.The quantity of K that was allocated daily to wood and remobilised gradually (see eq.?? : 7), acted as a buffer that prevented an overestimation of K limitation once K in the soil reached very low values.This shows that wood can act as a storage organ for K and can alleviate low uptake of K under drought (Sardans and Peñuelas, 2007;Touche et al., 2022) or in planted forests only fertilised at planting and growing on K-poor soils.
While the K trunk remobilisation fluxes were higher in our model than what was previously calculated using other methods (a mean of 7.6 gK.m −2 .yr−1 vs 2.7 gK.m −2 .yr−1 in Sette et al., 2013), the amount of K that transited in the xylem sap every day was low considering the magnitude of the sapflow in the xylem for tree transpiration.This led to very low xylem sap K concentrations if a flux of K from phloem sap to xylem sap was not considered.Since values of xylem sap K concentrations measured on different plants are one order of magnitude higher (Nardini et al., 2010;Siebrecht et al., 2003), our results either suggest that xylem sap K concentrations are very variable between plants or that an intense recirculation of K between xylem and phloem is taking place.The first hypothesis is consistent with evidence from temperate conifers that show a variation of an order of magnitude (the lower bound is similar to values calculated from our simulations) of xylem sap K concentration during the growing season (Losso et al., 2018).However, the last hypothesis seems more plausible since K is necessary to maintain xylem hydraulic conductivity (Oddo et al., 2011;Nardini et al., 2011) and that considering a transfer of K from phloem sap to xylem sap, the calculated K xylem sap concentrations more in line with measurements from the literature.It also supports evidence that K + ions are an essential part of many processes at the plant level (Dreyer and Michard, 2020): energy source (Dreyer et al., 2017), counter-ion for NO − 3 , signaling (Anschütz et al., 2014), protection against abiotic stress (Cakmak, 2005), among others.The intense recirculation of K between xylem and phloem serving as a way to maintain homeostasis.The order of magnitude of this recirculation would have to be determined experimentally.Previous measurements have shown that up to 25% of K + ions are recirculated in tomato plants (Armstrong and Kirkby, 1979) and up to 50% in Ricinus communis seedlings (Marschnert et al., 1997).Results from our model suggest that the figure could be higher in eucalypt trees (maybe owing to remobilisation from wood absent from tomato plants).

Figure 1 .
Figure 1.Schematic representation of K balance in the tree, and its link with the allocation model.Purple boxes are K state variables, while purple arrows are K fluxes.Dashed purple arrows are remobilisation fluxes.The K uptake flux simulated with a simple Ohm's law form is represented with resistance symbols.Grey boxes represent C biomasses.Dark grey arrows represent allocation of NPP to the different organs.Thin dotted grey arrows represent the influence of the organ biomass on respiration.

Figure 5 .
Figure 5. Simulated concentrations of K in stemwood (a) and branches (b) in two contrasted K availability scenarios compared to measurements conducted in the fertilisation experiment at Itatinga.

Table 1 .
Simulated yearly fluxes of K (in gK.m −2 .yr−1 ) in a fully fertilised treatment (+K) and in a K omission treatment (oK).This table contains both internal fluxes (wood, branch and leaf remobilisations) and exchange fluxes (uptake, litterfall and canopy leaching).The constant atmospheric deposition flux (0.5 gK.m −2 .yr−1 ) is not shown.responseto K deficiency.The reduction was on the order of 33% with WUE N P P at 0.0018 and 0.0012 gC.gH 2 O −1 in the +K stand and oK stand respectively.The simulated water use efficiency of trunk wood was also reduced by 32 % in oK relative to +K.The WUE trunk values in the +K and oK stands were respectively 0.0011 and 0.0007 gC.gH 2 O −1 .