Non-linear models of 15 N partitioning kinetics in late lactation dairy cows from individually-labeled feed ingredients 1

Few studies have examined the N kinetics of individual feeds with stable isotope tracing. We hypothesized that N partitioning to milk and excreta pools as well as the rates of the processes that drive this partitioning would differ for alfalfa silage, corn silage, corn grain, and soybean meal. Feed ingredients were endogenously labeled with 15 N and included in 4 diets to create treatments with the same dietary composition and different labeled feed. Diets were fed to 12 late-lactation dairy cows for 4 d (96 h) and feces, urine, and milk collection proceeded during the 4 d of 15 N enrichment and for 3 d (80 h) after cessation of label feeding. Nonlinear models of 15 N enrichment and decay were fit to milk (MN), urine (UN), and fecal N (FN) in R with the nlme package and feed-specific parameter estimates were compared. The estimated proportions of feed N that were excreted in feces supported our understanding that N from soybean meal and corn grain is more digestible than N from alfalfa and corn silage. Estimates for the N partitioning between milk (MN) and urine (UN) from the 2 concentrate feeds (soybean meal and corn grain) indicated that UN:MN ratios were less than or equal to 1:1 indicating either more or equal nitrogen partitioning to milk compared with urine. It is important to maintain factual accuracy in representing the results rather than implying a desired outcome unsupported by the data. In contrast, UN:MN ratios for forage feeds (corn and alfalfa silage) were > 1:1, indicating more N partitioning to urine than milk. The modeled proportion of total FN that originated from feed N was 82.2% which is in line with previous research using a similar 15 N measurement timeframe. However, the proportion of urinary and MN originating from feed N was much lower (60.5% for urine, 57.9% for milk), suggesting that approximately 40% of urinary and MN directly originate from body N sources related to protein turnover.


INTRODUCTION
Losses of N from dairy farms contribute to a variety of environmental concerns including greenhouse gas (GHG) emissions, nitrate contamination of groundwater, and eutrophication of surface waters (Powell and Rotz, 2015).Ration formulation can modulate the efficiency of dietary N utilization and reduce N excretion in urine and feces.Urinary N (UN), in particular, increases linearly with dietary N content and is more labile than fecal N (FN), and thus is more easily lost to the environment (Dijkstra et al., 2013;Spek et al., 2013;Powell and Rotz, 2015).Dairy cattle nutrition models are often used to formulate diets and estimate dietary N use efficiency (i.e., milk N / intake N) and excretion.However, common nutrient requirement and excretion models are factorial, empirical models that provide minimal opportunity to evaluate the metabolism of the N from each feed ingredient and therefore its contribution to productive or unproductive fates (Reed et al., 2015;Barros et al., 2017;Powell et al., 2017).Elucidating the metabolic behavior of N from individual feedstuffs will improve our understanding of methods to achieve greater N efficiency, allowing for better assessment of whole farm N efficiency (Powell and Rotz, 2015;Powell et al., 2017).Tracing N from individual feedstuffs across metabolic processes enables a biophysical accounting of N inputs and N outputs in terms of environmental and economic impacts.
Dairy cattle nutrient requirement models also lack robust predictions of body protein contributions to N in milk, feces, and urine.Advancement in the contribution of endogenous animal protein to FN and thus to animal N requirements has been made (e.g., Lapierre et al., 2008;Marini et al., 2008;Lapierre et al., 2020), but there is still little information on body contributions to endogenous UN and milk protein.Therefore, a better understanding of the extent and kinetic nature of body protein cycling and contributions to milk and UN will improve the ability of nutrient requirement models to estimate N efficiency and excretion.
Stable isotope methods are uniquely suited to advancing our understanding of the mechanisms by which feed and body protein combine to form FN, UN, and milk N (MN) due to their ability to act as a tracer of specific N sources.The 15 N isotope has been used in a variety of ways to improve our understanding of dairy cattle N metabolism (e.g., Lapierre et al., 2008) and to track N losses during manure storage and field application (Powell et al., 2005).Our group employed 15 N tracer methods in an interdisciplinary study to examine the effects of feed N source on N metabolism and partitioning in lactating cows, loss of ammonia and nitrous oxide from dairy barns and soils, and crop N uptake after labeled manure application to soil (Powell et al., 2017).Our preliminary analysis of N excretion data suggested the contribution of each feed to FN, UN, and MN varied, and that not all feed 15 N was recovered up to 80 h after cessation of label feeding (Barros et al., 2017).We hypothesized that, in addition to differences between feeds in the partitioning to FN, UN, and MN, the dynamics of both digestion and metabolism that contribute to this partitioning are also specific to each feed.To further elucidate and quantify the distinct behavior of N excretion between feeds, we applied nonlinear modeling techniques to the 15 N excretion data and hypothesized that the kinetics of N metabolism in addition to the partitioning between feces, urine, and milk would vary among feeds.

MATERIALS AND METHODS
Experimental plots within larger fields of alfalfa (AS) and corn for silage (CS), corn for grain (CG), and soybeans (SBM) were grown with 15 N-labeled fertilizer to enrich these 4 commonly used dairy feeds.The protocol for labeling the feeds was described in detail by Powell et al. (2017).Briefly, labeled AS, CS, CG, and SBM were grown by applying 15 N-labeled fertilizer to the fields, and in adjacent fields, the unlabeled crops were grown with unlabeled fertilizer.The chemical composition and 15 N atom percent (AP) of unlabeled (i.e., natural abundance of 15 N) and 15 N-labeled feed ingredients are reported in Barros et al. (2017).The experimental diets included a mineral premix plus the 4 major feed ingredients (AS, CS, CG, and SBM) -in the same proportions across treatments.In each treatment, only 1 of the feeds was 15 N-labeled (Table 1).

Data collection
The University of Wisconsin-Madison Institutional Animal Care and Use Committee approved all study procedures.Cows, diets, experimental design, and laboratory analyses were detailed in Barros et al. (2017).Briefly, dietary treatments were fed to 12 lactating Holstein cows (means ± SD; 264 ± 18 DIM) for 4 d and urine, feces, and milk were collected during and after feeding of the 15 Nenriched diets.Cows were fed a pretreatment (unlabeled) TMR once daily (at 0800) for 11 d (see Table 2 in Barros et al. (2017) for ingredient and chemical composition of the diets).Before feeding on d 12, cows were blocked by milk yield and randomly assigned within block to 1 of 4 dietary treatments, which were constructed by replacing unlabeled AS, CS, CG and SBM from the pretreatment diet with its analogous 15 N-labeled feed.After 4 d of feeding the 15 N-enriched TMR, cows were returned to the unlabeled TMR during d 16 to 19.The amount of TMR offered was adjusted daily to ensure 5% of refusals.Cows were milked twice daily (0400 and 1600 h).From d 11 to 19, daily milk production was recorded and milk samples were collected at each milking.Procedures for milk analysis can be found in Barros et al. (2017).Urine and fecal sampling were performed every 6 h (at 0400, 1000, 1600, and 2200 h) from d 12 to 19 and analyzed for total N and 15 N. Urinary urea-15 N was isolated using an adaptation of the procedure published by Brooks et al. (1989), where the filter traps were placed in a tin analyzer capsule.The tin analyzer cups were folded and analyzed for N and 15 N isotope using a Costech 4010 elemental analyzer (Costech Analytical Technologies Inc., Valencia, CA) interfaced to a Thermo-Finnigan Delta-Plus Advantage isotope ratio mass spectrometer (Thermo-Electron GmbH, Bremen, Germany).Barros et al. (2017) presented the atom percent in unlabeled (natural abundance) and labeled feeds.Atom percent is the ratio of 15 N atoms to the total N atoms, 100 × [ 15 N / ( 15 N + 14 N)].Above natural abundance (ANA, mg) was calculated as the mass of 15 N that exceeded the natural 15 N abundance.Atom percent excess (APE, %) was calculated by subtracting the natural abundance AP from experimental AP.Plots of total exit pool 15 N ANA vs. time (not shown) revealed large variations of ANA by time of day due to diurnal variations in volume of secretion and excretion.Thus, APE was chosen as the Statistical analyses were conducted in R statistical software.Models were fit using the lmer packages for the linear models and the nlme package for the nonlinear models (Pinheiro et al., 2017).The data included observations from 2 phases of change in 15 N APE (%): 15 N enrichment during the 15 N feeding phase from t = 0 to t = 96 h (rising curve) and a decrease in 15 N enrichment from t = 96 to t = 176 (decay curve) when unlabeled diets were fed.We investigated several modeling approaches that ranged from linear regression of transformed observations to a compartmental model to predict the measured outcomes simultaneously.Simple linear regression did not provide a good fit to the data, and we were unable

Model development
Enrichment phase (time equal to 0 to 96 h after labeled feeding) Decay phase (time equal to 0 to 80 h after cessation of labeled feeding)

M3
Piecewise exponential model Both enrichment and decay phases (time equal to 0 to 176 h after labeled feeding) to achieve convergence of parameter estimates using a compartmental model.For this reason, we chose to fit nonlinear exponential models to APE observations in FN, UN, urinary urea-N (UUN), non-urea urinary N (NUUN), and MN separately and for each experimental phases.
Single-pool exponential models were fit to the rising (M1) and decaying (M2) curves separately followed by a piece-wise exponential model fit to the entire data set (M3).Although the diets provided during the decay phase included unlabeled feeds that were consistent across treatments, we assumed that some N from labeled feed ingredients fed during the enrichment phase remained in the animal (digestive and other metabolically active organs).Therefore, we chose to include feed-specific parameters in M2.Details of the structure of these models are provided in Table 2 and Appendix 1.In addition, we attempted to fit 2-pool exponential models but were unable to achieve convergence of parameter estimates.The stochastic model was built with increasing complexity to improve fit as described by Strathe et al. (2010) and in Appendix 2. Briefly, the baseline error structure with independent and identically distributed errors and no random effects was adapted to represent the experimental design by including an additive random effect for the animal to each parameter.Then an AR-1 covariance structure was added to the model to account for correlations associated with repeated measures.Finally, a weighted AR-1 covariance structure was developed to give additional weight to observations at the beginning (M1) and end (M2) of the trial to accommodate APE values very close to zero during those periods.Plots of observed vs. predicted, standardized residuals vs. time, and standardized residuals vs. predicted values as well as QQ plots of residuals were used to assess model fit and the assumption of normality of the error structure.

Model evaluation and inference
The need to include treatment-specific parameters was determined with an F-test of the full model against a reduced model with 1 parameter for all treatments.Additionally, we tested a model allowing different first-order rate constants for enrichment and decay, respectively, versus a model with a single first-order rate constant describing both phases.Multiple comparison tests of parameter differences among treatments (H 0 : where k refers to the first-order rate constant and L 1 the lag time from the start of the modeled period to the start of the change in enrichment]) were performed using the multcomp package (Hothorn et al., 2008).Model evaluation was conducted using the root mean square error (RMSE) and decomposition of its error to determine the proportion of error due to random dispersion (ED; Bibby and Toutenburg, 1977).Results are in Table 3.
To enable comparisons among treatments, estimates of asymptotic APE in the exit pools (A i ) were standardized based on total 15 N intake for the treatment.Two methods to test for variable N partitioning among treatments were developed and applied to estimates from M1 and M3 only.Estimates from M2 were omitted from inferential analysis of the asymptotic APE because the model form implies the asymptotic APE was reached during the experimental period, which is not certain.The first method used the change in APE from dietary intake (APE Ini ) to the exit pool using the model predicted asymptotic APE of the exit pool Âi ( ) as defined in Eq. 1.

A APE In
For a given labeled feed, if there is no selective partitioning of N among exit pools, the change in APE from diet to exit is expected to be zero.Therefore, we tested if

APE APE
In Ex − differed significantly from zero.Additionally, we tested pairwise differences in feeds (i vs. i') in the extent of selective partitioning among exit pools using the following hypotheses: This was tested using the following t-distributed test statistic, where sd propi is the standard deviation (sd) computed by propagating error in the measured sd of the APE Ini and the model-estimated sd of A i for i = 1, …, 4 for the 4 dietary treatments.Results are in Table 4.
The second method with results presented in Table 5, compared the ratios of the estimates for the asymptotic APE from M1 and M3 Âi ( ) in FN, UN, and UUN relative to MN for each feed ingredient (i = 1 to 4).A ratio of 1.0 indicates equal partitioning of intake N to FN or UN relative to MN, whereas ratios greater or less than 1.0 indicate more or less N partitioned to urine or feces relative to milk.
The test statistic for assessing differences between treatments the difference in the ratios of the estimates for FN:MN, UN:MN, and UUN:MN between treatments divided by the square root of the sum of the APE ratio standard errors.APE ratio standard deviations were computed by propagation: To enable comparison with nutrient requirement models, we estimated the proportion of exit N from each feed excreted (FN, UN) or secreted (MN) using Eq.Six and Eq.Seven and the underlying assumption that deviation of the exit pool 15 N AP is a result of excretion of N originating from the labeled feed as follows: In Eq. 6, APE Feed i represents the APE of the labeled feed.Details on the derivation of Eq.Six is provided in Appendix 3.For each exit pool, prop i were summed for across all 4 feeds and subtracted from unity to estimate the proportion of N from endogenous sources rather than feed inputs, similar to Leterme et al. (2000).To allow for 1 ΔAPE = Predicted asymptotic APE (atom percent excess) in the exit pools -dietary treatment APEs.Asymptotic APEs predicted from models fit to the enrichment phase only (M1, 0 to 96 h after initiation of labeled feeding) and to the entire experimental phase (M3, 0 to 176 h after initiation of labeled feeding).SE of estimate provided in parentheses.
3 Models represent the enrichment phase (M1), decay phase (M2), and both enrichment and decay phases (M3).comparison with nutritional models (see below), we changed the basis of the proportion to correct for incomplete fractional recovery of labeled N. Within each exit pool, the proportion of labeled N from feed i in the total recovered labeled N was calculated as:

Adaptation of existing nutritional models
Proportions of excreted N traced to each feed as estimated by application of model results in Eq.Seven were compared with estimates based on NRC (2001) and NASEM (2021) models, using the measured milk CP, DMI, body weight, and feed composition from our experiment as inputs.To generate estimates of N excretion from individual feeds comparable to those of M3, both NRC (2001) and NASEM (2021) models had to be extended with certain assumptions.Both models required the assumption that protein fractions were sourced from feeds in proportion to diet composition.Additional assumptions are detailed in Appendix 4.

RESULTS AND DISCUSSION
In the preliminary report of this study, Barros et al. (2017) presented data on cow performance and N par-titioning (g/d and g/N intake) and established that the isotope labeling that defined the treatment diets did not impact milk production or N use efficiency.

Model Selection and Fit
Nested model comparisons indicated no improvement in fit when allowing first-order rate constants specific to enrichment and decay phases; therefore, a single first-order rate constant was used.The AR-1 covariance structure with random effects for cow showed a significant improvement in model fit based on the F-test (P < 0.05), reduced AIC and BIC, and reduced patterns in the residuals over a simple covariance structure or 1 with only random effects for cow.When residuals were weighted by the inverse of the observed mean at each time point, the residual patterns plotted against time and fitted values showed significant improvement so the full covariance structure with weighted residuals was used for all models.
Table 3 shows the decomposition of error for the exponential rise (M1), decay (M2), and piecewise (M3) models for each exit pool.The RMSE was the lowest for M1 prediction of MN 6.04% of the observed mean) and the highest for M2 prediction of FN (21.0% of the observed mean).For each model the RMSE was substantially greater for FN than for UN, MN or UUN.Furthermore, for all pools of N, RMSE were the lowest for M1, the greatest for M2 and intermediate for M3, with the exception of MN in which case the RMSE of M3 was  close to but slightly higher than that of M2.Differences in prediction error estimates are, in part, a result of the difference in observed data used in each model.The greater prediction error in the decaying curve (M2) than the enrichment phase (M1) could be an indication that a single-pool model is less appropriate for M2 than M1 and be taken as evidence for a need for a 2-pool model to better describe dynamics of N digestion and metabolism, although we were not able to achieve convergence for this model.Decomposition of the error indicated the models had little bias with greater than 90% of the error due to random dispersion in all but 1 model (Table 3).Overall, these results indicated that the chosen model structures fit the data well.

Nitrogen Partitioning
Table 4 shows ∆APE, the differences between asymptotic APE (A i ) and intake APE provided by M1 (the enrichment phase) and M3 (enrichment and decay phases), the first method used to test for differential partitioning of feed N to exit pools among feed.Estimates of asymptotic 15 N APE in feces were not different (P > 0.10) than dietary APE for any of the feeds in neither M1 nor M3.However, the ∆APE for both concentrates were more negative than those of both forages in M3 and that of SBM was more negative than both forages in M1.The absence of change in 15 N enrichment from diet to feces indicates that the proportional contribution of the labeled feed N to total FN was the same as the feed contribution to total dietary N.However, the greater negative dietary ∆APE in concentrate feeds compared with forages suggests that overall N in concentrate feeds were partitioned away from feces to a greater extent than forage N.These results may reflect a greater true digestibility of N in concentrate than forage, a greater recycling of labeled N (urea-N and/or endogenous N) from the body to the gastro-intestinal tract of forage N than concentrate N, or a combination of both.Although FN and 15 N were not decomposed further to determine the proportions attributed to feed, microbial, and endogenous N as in Ouellet et al. (2002), our results indicated that the contributions of the feeds to each of the FN fractions and/or recycled N varied substantially between the forages and concentrates used in this study.
Unlike FN, estimates of UN APE were significantly less (P < 0.10 or P < 0.05) than dietary APE for all feeds in both M1 and M3 models with values for ∆APE remarkably consistent and ranging from −0.0484 to −0.0906 between models and among feeds.
Regarding MN ∆APE, both models indicated a significant reduction in asymptotic 15 N APE for MN compared with dietary N for AS, CS, and CG, whereas only M1 indicated a significant (P < 0.10) reduction for SBM.Although the test comparing ∆APE for SBM with feed APE was not significant (P = 0.17), the estimates of ∆APE i from M3 were significantly different (P < 0.10) among treatments.The negative MN ∆APE was twice as large for AS than for SBM (−0.122 vs. −0.0621respectively; Table 4).These results suggest that SBM feed N is partitioned toward milk to a greater extent and AS feed N is partitioned toward milk to a lesser extent while the feed N partitioning to MN from CS and CG appear to be intermediate between the 2 legume feeds.Conversely in both M1 and M3, the ∆APE for UN and UUN were numerically more negative for SBM than AS.The change in the direction of the relationship of ∆APE i between MN and the urinary routes of excretion for AS and SBM could indicate that the absorbed N from SBM is partitioned more toward milk instead of the urinary route of excretion in comparison to the absorbed N from AS.A likely factor in this observation is the expected differences in rumen degradability and subsequent absorption of ammonia across the rumen wall.
The ratios of the asymptotic APE (A i ) in FN, UN, and UUN to MN are shown in Table 5 and outcomes align with the analysis of ∆APE i in Table 4. Specifically, the ratios of asymptotic enrichment in FN:MN and UN:MN for concentrate feeds were significantly less than the enrichment ratio for forage feeds.Whereas forage FN:MN and UN:MN enrichment ratios exceeded 1.0 indicating more N was partitioned to manure compared with milk (P < 0.10 for the CS UN:MN ratio; P < 0.05 otherwise), the concentrate UN:MN and FN:MN ratios were either not different than 1 indicating that a similar fraction of the feed N was partitioned to milk compared with manure or significantly less than 1 indicating a greater fraction of the feed N was partitioned to milk compared with manure.One exception to this pattern was the ratio of FN:MN for CG estimated by M3 which was larger than 1 (1.16,P = 0.083).The comparison of the ratios of FN:MN and UN:MN among feeds estimated by nonlinear models in the present study is in line with the ratios of values measured on the fourth day of feeding 15 N-enriched diets (Barros et al., 2017).Together, our results indicate that in forage feeds, a greater fraction of feed N is partitioned to both feces and urine than milk in comparison to concentrate feeds.

Comparison of N partitioning with nutritional models
Table 6 shows the proportion of labeled N from each feed in the total recovered labeled N from all exit pools.As implied by M3, N from feed intake made up 82.2% of the total FN during the experiment (95% CI: 80.5 -83.9).The remaining FN (17.8%) not attributed to intake represents endogenous N. This M3 estimate is in line with measured endogenous N (17% of FN) in the 200-h 15 N infusion trial by Ouellet et al. (2002).The proportion of endogenous N was greater in UN (39.5%) and MN (42.1%) than FN.Researchers have reported the large contribution of whole-body protein turnover to total protein synthesis, yet due to the high turnover rates and low net gain, a much lower fraction of total protein synthesis from gastrointestinal tract is considered net gain (milk protein secretion or body tissue gain) when compared with the mammary gland and peripheral tissues (Bequette et al., 1996;Lobley, 2003).Further, the cows in   2021) nutritional models where we see that the M3 estimates are generally consistent with those of NRC and NASEM models.The M3 estimates for the proportion of the recovered FN from each feed M3 were always intermediate to the NRC and NASEM estimates.More specifically, NRC predicts a greater proportion of FN originating from the forage feeds and a lesser proportion of FN originating from the concentrate feeds than the estimates provided by M3.The comparison of NASEM estimates with M3 is reversed: lower predicted proportions of FN from forages and higher proportions of FN from concentrates for NASEM in comparison to M3.The different pattern in the comparison of the estimates of FN from forage and concentrate feeds from M3 and the nutrient requirement models can be explained by the changes to the parameter estimates for the fraction of indigestible N between the models.The indigestible N fractions of AS and CS in the NASEM (2021) feed library are 9 and 16% of total N, respectively, while those from the NRC are 9.9% and 18.5%, respectively.Similarly, the intesti-nal digestibilities for CG and SBM in the NASEM feed library are less than those in the NRC.
The 3 models predicted similar proportions of N in urine from each feed, except that NRC predicted a smaller proportion of UN from CS and greater proportion of UN from SBM.For milk, NRC predicted a lesser and NASEM predicted a higher proportion of MN from AS than M3.The NRC model also estimated a greater proportion of MN from CS and CG than M3 and NASEM.The proportion of MN originating from SBM estimated by M3 is greater than both NRC and NASEM models.The NASEM estimates were numerically closer to M3 than NRC for most comparisons of UN and MN suggesting improved feed-level performance for predicting UN and MN with the updated model, especially for CS and CG.The greater proportion of N from AS in MN with M3 and NASEM than with NRC may relate to NRC predictions of MCP which were lesser than NASEM.Additionally, the NRC model allocated protein supply to body pools on an MP rather than AA basis, as represented in NASEM and physiologically (as modeled by M3).This may explain NRC's overprediction of MN from CG and underprediction of MN from AS relative to M3. NASEM (2021) reported similar AA composition between microbial true protein and milk true protein and, in our study, a large fraction of diet RDP was supplied by AS.

Nitrogen Kinetics
Estimates for both lag (L i , h) and rate of enrichment or decay (k i , 1/h) are related to the rate of feed N excretion in FN, UN, UUN, and MN and are illustrated in Figures 2,  3 and 4 (numerical values presented in Appendix 5).For feces, larger FN-L CS indicated significantly longer lag time than FN-L SBM in both the exponential decay (M2) and piecewise (M3) models.However, for the exponential rise model (M1) that exclusively used the enrichment data, FN-L AS was significantly longer than all other feed lag times.The discrepancy in the relative order of lag time estimates between the rising and decaying curve estimates could reflect the varying kinetics in the fractions that make up FN and the different contributions of each feed to those fractions.Elucidation of this relationship would require that the pools of FN and their enrichment be investigated instead of total FN alone.
Estimates for the rate of 15 N change in feces (FN-k i ) were numerically greater for the forages than the concentrate feeds for all models, although significant differences were only detected for comparison of k CS with the rates for concentrate feeds predicted by M2 and M3.The constant, k i represents the rate at which the labeled feed is replacing N from the unlabeled feed in that pool, or the rate of turnover.Assuming a steady state, the rate of turnover of undigested feed N in FN is equal to the rate of passage divided by the indigestible intake N pool size.One would expect the indigestible N pool size as a fraction of total N to be larger for grain forages (i.e., CS) than legume forages (i.e., AS) and concentrate feeds and expect the rate of passage to be correspondingly lower as a result of a longer retention time and a slower rate of undigested N turnover which is contrary to M2 and M3 predictions for CS rate of turnover (k i ).However, indigestible N is only one source of FN and, as with the lag estimates, aggregation of the FN components masks what is likely a varying rate of turnover for different FN component between feeds.
Understanding the rate of turnover of the microbial N pool from each feed is more challenging.A substantial amount of recycling (between 40 to 70% of the pool) occurs within the microbial N pool due to a combination of lysing and predation by protozoa (Dijkstra et al., 1998).Thus, the microbial N pool has a slower rate of turnover (longer retention time) than that of feed N pools.Within the microbial N pool, the rate of turnover of a feed's contribution to the microbial N pool depends on the rate of microbial synthesis from that pool, the total rate of turnover of microbial N, and the rate of passage of microbial N. In this study, the rate of passage of microbial N and microbial N turnover rates are not expected to vary because the diet composition is the same across treatments, but the rate of microbial protein synthesis and N turnover from each feed will vary.As a proportion of CP, forage feeds in our study have more soluble and degradable protein than the concentrate feeds, from which microbial protein is synthesized at a faster rate than the remaining RDP (NRC, 2001).Thus, the faster FN-k i estimates for CS could reflect a faster rate of turnover of its N in the microbial N pool in comparison to the other feeds.Thus, these results highlight the need for further research to separate the different FN fractions to further elucidate digestion kinetics of feed N.
Unsurprisingly, kinetic parameter relationships between feeds were the same for both UN and UUN since UUN made up over 70% of total UN (Barros et al., 2017) although more significant differences were detected between estimates for UUN lag parameters between feeds (Figure 4, Appendix 5).The estimates of the lag time associated with the appearance of N in cows fed SBM (L SBM ) for both UN and UUN was longer than that of the other feeds while the lag times for N from CS (L CS ) were always shorter than those of other feeds.The only statistically significant differences found, however, was in the comparison between the M3 estimates for the forage feeds (UUN-L CS and UUN-L AS ) and UUN-L SBM (P < 0.05).It is unclear what physiological processes would cause a longer delay in the N excretion from SBM than AS and CS in UUN.Comparisons of the rates of 15 N enrichment and decay between feeds in estimates for k i did not produce any consistent patterns and the only significant differences detected were between k AS and k SBM for UN and UUN in the model fit to the decaying curve data (M2).The fastest rate of change in UN and UUN (largest k i estimates) from M2 were predicted for the treatment with labeled AS.Unlike the estimates for the lag time where the numerical differences reflected a similar pattern to the significant differences, the numerical differences between estimates of UN-k i and UUN-k i from the other models (M1 and M3) did not follow a similar pattern to the significant differences found in M2.To investigate if there are metabolic factors contributing to the rate of excretion and secretion of UN and MN from specific feed sources, future work should include an assessment of blood AA (AA-N) and blood urea-N and their enrichment curves.
Comparison of the kinetic parameter estimates between the exit pools, mostly agreed with our current knowledge of ruminant digestion and metabolism.All lag estimates (L i ) for UN and MN were smaller than those for FN (Fig- ures 2 and 4, Appendix 5).The shorter lag time for 15 N to begin to appear in MN and UN is a result of urea N originating primarily from ammonia N absorbed across the rumen wall rather than milk protein N secretion or other N metabolites found in urine.Some 15 N is also expected to end up in NUUN.We estimated NUUN APE as the difference in 15 N enrichment in UN and UUN, weighted by urea concentration.NUUN APE was less than 0.1 for all treatments (in comparison to less than 0.2 for UUN; Appendix 5).NUUN is made up mostly of N in purine derivatives that are metabolic by-products of microbial nucleic acids and creatinine N that originates from the breakdown of muscle creatine excreted at a constant rate in proportion to the amount of skeletal muscle tissue.Turnover of microbial N (1 to 2 d, Dijkstra et al.,1998) and skeletal muscle nitrogen (>200 d, Lobley, 2003) is much slower than blood urea N turnover thus, the lag time and rate of enrichment for the NUUN pools are likely longer and slower than UUN.Given that total UN is composed of UUN and NUUN, we would hypothesize that the lag time estimates of UN would be the same as those of UUN and the rate of enrichment estimates of UN to be smaller than those of UUN.Our results generally align with this hypothesis as we see that the estimates for UUN-k i are consistently numerically higher than those of UN-k i with the magnitude of the differences between the estimates for k i ranging between 0.75 and 2.2 SEs.The only exception to this relationship was found in the estimates for k CG by M2 in which the estimate for UN-k CG was less than 0.20 SE above that of UUN-k CG .Although the estimates for UUN-L i were most often numerically greater than the estimates for UN-L i within models and feeds, the numerical differences are relatively small (often ~0.5 h or less and always less than 1 SE of any of the estimates).
The MN enrichment rate estimates are also slower than those of UUN whereas the lag times are on average more than 2 h longer than the estimates for UUN-L i .The exception to the observed longer lag times when comparing MN-L i .to UUN-L i are the estimates for MN-L SBM where the estimate from M1 was similar to that of UUN-L SBM (difference of 0.030 h) and the estimate of MN-L SBM from M3 was 0.710 h longer than that of UUN-L SBM .Unlike UN, MN is mostly composed of true protein N, with a very small fraction coming from urea N. The slower rate of enrichment combined with the smaller proportion of MN originating directly from intake N (Table 6) point to a large fraction of MN being sourced to a greater extent from body tissues.In other words, our data suggested that a substantial proportion of the absorbed AA-N from feed is incorporated into body protein, potentially limiting availability of AA from feed for milk protein synthesis.This illustrates the importance of future research on the slow-turnover body N pool(s) models of dairy cattle N metabolism to fully elucidate the long-term N efficiency of diets and feeds.

Limitations and Experimental Methodology
The percentage of N intake recovered through the total collection of milk, urine, and feces ranged from 85 to 88% (Table 1).Although the recovery of N intake was incomplete, this outcome has minimal impact on our results because the analysis and inference are based on measurements of 15 N from samples collected throughout the study period.However, it is still likely that circadian variation in the composition of N in milk, urine, and feces (e.g., Gustafsson and Palmquist, 1993) combined with the incomplete recovery of N influenced our ability to completely capture the enrichment dynamics.
Model fits and parameter estimates not only revealed limitations of the current data set but also suggested improvements to experimental design for use of 15 N stable isotopes to compare N partitioning and kinetics in different feeds.For example, future studies could sample more body tissues (e.g., plasma, intestinal mucosa as in Ouellet et al., 2002) to enable differentiation between feed and microbial N in FN.This would enable comparisons of each feed's rate of incorporation and total contribution to microbial N production.Similarly, measuring 15 N enrichment in urine or plasma purine derivatives or creatinine could facilitate estimation of feed N contribution to microbial and skeletal muscle turnover.
Plots of residuals against time (Figure 5) point to another improvement in experimental design in addition to providing support for the existence of slow turnover pools.The residual plots show a pattern of increasing residual values near the end of the experimental period.This trend toward larger positive residuals at the end of the experimental period indicated that the observed APE values for all exit pools and treatments were increasingly more than the predicted values.Thus, the models are predicting a return to natural abundance (APE = 0) earlier than is suggested by the data.This feature of the data is, again, likely attributable to 1, or more, slow N turnover pools of body protein N. Based on fractional rates of protein synthesis and degradation estimated in sheep and cattle, the time course for our experiment would allow extensive protein turnover in splanchnic and mammary tissues but limited turnover in peripheral tissues such as skeletal muscle (Lobley, 2003).In this study, the last sampling was 80 h after the final isotope-containing meal at the end of which the 15 N APE had not returned entirely to baseline (Figure 1).Thus, because this trial was not long enough to substantially or completely label slow turnover pools of body tissue, our estimates of the contribution of endogenous N to each exit pool describe the proportions of N originating mainly from pools with protein turnover rates less than 25%/d.
Extending the data collection period during the decay period and more frequent sampling during the early decay period could improve estimation of the kinetics of this pool by providing the statistical power to fit a 2-pool model.Cantalapedra-Hijar et al. (2020) were able to fit a 2-pool exponential model of 15 N decay in UN over a much longer time period (142 d) from steers fed diets enriched in 15 N via labeled urea.In their analysis of the suitability of a single or 2 pool model, they found evidence for a change in the rate of decay between d 4 and 7 after cessation of 15 N feeding.Thus, we believe we have only or predominantly captured the dynamics of the fast turnover pool and future work should extend the sampling period at least 7 d after cessation of isotope delivery if study objectives are related to body tissue protein turn over.
Some authors reported that isotopes fractionated differentially to certain body pools based on in vitro (e.g., Wattiaux and Reed, 1995) and in vivo (Cantalapiedra-Hijar et al., 2015) studies of N isotopes at natural abundance, and that dietary characteristics could affect the extent of differential fractionation.Differential fractionation is expected to minimally impact deliberate tracer studies such as our study but may be an important consideration for future studies on natural abundance.
There are also limitations in the form of the models used beyond the restriction to a single-pool model imposed by the data.First, M1 and M2 are only able to use a subset of the data set to inform the parameter estimates because the models describe either an enrichment or decay process.Creation of the piecewise model accommodates the use of the complete data set which contains more information about the shared kinetic parameters than a data subset.
The exponential model for the decay curve (M2) requires an assumption of when the decay process starts.We chose to start with the sample collected after the first non-labeled meal was delivered, however, the model was unable to estimate L i for UN, UUN, and MN under this assumption.This limitation is also avoided in the piecewise model (M3) by defining the cut-off point between the rising and decaying curve as an estimated parameter rather than a predetermined time point.

CONCLUSIONS
Single pool nonlinear models were fit to 15 N enrichment data of urine and feces excretion and milk secretion Reed et al.: NITROGEN PARTITION FROM ISOTOPE LABELED FEEDS measurement of interest for subsequent model fitting because it represents the change in 15 N enrichment rather than excretion.

4
Exit pool: FN = fecal N, UN = urine N, UUN = urinary urea-N, and MN = milk N. * † ΔAPE different from 0 where * signifies P < 0.05 and † signifies P < 0.10.a b Separation of ΔAPE among treatment means for each exit pool; Means within a row that do not share a common letter superscript are different (P < 0.10).
Reed et al.: NITROGEN PARTITION FROM ISOTOPE LABELED FEEDS

Figure 1 .
Figure 1.Atom percent excess (APE) for fecal N, milk N, urinary N, and urinary urea N plotted against time starting at initiation of feeding 15 N labeled dietary treatment (AS = alfalfa silage, CS = corn silage, CG = corn grain, and SBM = soybean meal).Each point represents one of 3 individual cow observations per time point with lines connecting observations within cow.Arrows indicate time of last labeled feeding.
Reed et al.: NITROGEN PARTITION FROM ISOTOPE LABELED FEEDS

Figure 2 .
Figure 2. Comparison of lag time parameter estimates (Lag) from 3 different models (M1 = enrichment phase, M2 = decay phase, and M3 = both enrichment and decay phases) representing the delay in N 15 appearance in fecal N, urinary N, and milk N from feeding 15 N labeled dietary treatment (AS = alfalfa silage, CS = corn silage, CG = corn grain, and SBM = soybean meal).M2 lag parameters were fixed to zero for milk N and urinary N based on model fit.

Figure 3 .
Figure 3.Comparison of rate of enrichment and decay parameter estimates (k) from 3 different models (M1 = enrichment phase, M2 = decay phase, and M3 = both enrichment and decay phases) in fecal N, urinary N, and milk N from feeding 15 N labeled dietary treatment (AS = alfalfa silage, CS = corn silage, CG = corn grain, and SBM = soybean meal).
Figure 4. Predicted 15 N enrichment in fecal N, urinary N, and milk N using models from 3 different phases (M1 = enrichment phase, M2 = decay phase, and M3 = both enrichment and decay phases) of a stable isotope trial in which 1 of 4 feeds (AS = alfalfa silage, CS = corn silage, CG = corn grain, and SBM = soybean meal) were intrinsically labeled with 15 N; Enrichment values are standardized to their means to create a standard scale from 0 to 1.

Table 1 .
Reed et al.: NITROGEN PARTITION FROM ISOTOPE LABELED FEEDS Description of diets and animal performance 1

Table 4 .
Change in 15 N enrichment from diet to the predicted asymptotic enrichment in exit pools Reed et al.: NITROGEN PARTITION FROM ISOTOPE LABELED FEEDS

Table 5 .
Asymptotic 15 N enrichment in fecal, urine or urinary urea N relative to 15 N enrichment in milk N, for each dietary treatment Dietary treatment: AS = 15 N labeled alfalfa silage; CS = 15 N labeled corn silage; CG = 15 N labeled corn grain, and SBM = 15 N labeled soybean meal.
1 2 Models represent the enrichment phase (M1), decay phase (M2), and both enrichment and decay phases (M3).3Ratio: FN = fecal N, UN = urine N, UUN = urinary urea N, and MN = milk N. SE of the estimates provided in parentheses.* † Ratios different from 1 where * signifies P < 0.05 and † signifies P < 0.10.a b c d Separation of treatment means for each ratio; Means within a row that do not share a common letter superscript are different (P < 0.05) based on an independent t-test with unequal variances using error propagation to estimate variances for each ratio.

Table 6 .
Comparison of piecewise nonlinear model (M3 of this study) withNRC (2001)and NASEM (2021) predicted proportion of N from individual feeds in feces, urine, and milk (feed excretion N, g / total recovered excretion N, g) 1 trial were in late lactation (DIM = 264), thus the high contribution of body N to MN and UN is likely a result of typical protein turnover rather than the net mobilization of body reserves.Table6also compares the results of M3 with the N partitioning implied by extending the NRC (2001) and NASEM ( this