A Model to Describe the Genetic Potential for Nitrogen Deposition and Estimate Amino Acid Intake in Poultry

Abstract

The maximum protein or nitrogen deposition is commonly used as the basis for modeling the amino acid intake in growing birds. In previous studies, the exponential functions of the nitrogen balance data were used to estimate the theoretical maximum for nitrogen deposition (ND_maxT) as a reference model for the amino acid intake. However, this amino acid intake value is only valid for the period in which the ND_maxT was estimated. Additionally, physiological changes, such as the rapid development of reproductive organs and associated increases in protein deposition that occur in the period before the first egg is laid, should be considered in the models. Thus, this study was conducted to model the daily ND_maxT of pullets and integrate this value into the factorial model to estimate the daily methionine + cysteine (Met+Cys) intake. Our results showed that, up to 63 days of age, the values of ND_maxT obtained via the modeling procedure were 11% higher than the values predicted using the Gompertz function. At 105 days, there was a protein deposition peak from the growth of the reproductive organs, which contributed 14% of the variation in the model in this age. Alongside these factors, the integration of the models enabled daily Met+Cys estimates consistent with the literature; however, the recommendations varied according to the targeted daily protein deposition (50% or 60% of ND_maxT), daily feed intake, and amino acid utilization efficiency. The modeling approach demonstrated here for Met+Cys can be used to model other amino acid requirements and can be extended to other species.

1. Introduction

The maximum body protein deposition (PD_max) or nitrogen deposition (ND_max) has been the basis for mathematical models used to calculate amino acid intake for growing birds. Commonly, such models are employed in non-limiting conditions (in terms of environment and diet) to describe the PD_max or ND_max while raising the birds, using the comparative slaughter technique to adjust the protein weight of the bird as a function of age, according to the Gompertz function [1,2,3,4]. The Gompertz function is used to describe changes in body protein deposition, visualize growth patterns, and predict the weight of birds at specific ages; it is used in feeding and genetic programs, allowing for the interpretation and understanding of bird growth patterns [5,6,7]. An alternative method developed at Göttingen University [8,9,10,11,12,13] uses nitrogen balance assays to adjust the Mitscherlich function to estimate the theoretical maximum protein deposition (PD_maxT) or nitrogen (ND_maxT). Unlike ND_max, the ND_maxT is obtained from birds fed high levels of protein in their diet, thus representing the physiological limit of protein or nitrogen deposition, which shows the genetic potential [8,9,10,11,12]. In this method, the genetic potential is determined without the need for the sacrifice of birds or other animals for body composition analysis [14,15,16,17,18,19]. Similarly to ND_max [1,4,7,20,21], the ND_maxT values [8,9,10,11,12,22,23] decrease with the age of the bird, and have been determined specifically for each phase [8,9,10,11,12,22,23]. One limitation of this method is that its estimate is valid for the period in which the nitrogen requirement for maintenance (NMR) and theoretical maximum for nitrogen retention (NR_maxT) were determined, and the use of these values outside this period may generate discrepancies in terms of the amino acid intake [24].

Growing animals require a daily amino acid intake to maintain their body weight and to support the daily protein deposition rate, which is specific to each genotype [7,21]. One way to calculate the intake of amino acids (AAI) is the use of factorial approaches, one of which has been parameterized by researchers at Göttingen University [8,9,10,11,12]. In the Göttingen modeling approach, there are three parameters—NR_maxT, NMR, and amino acid utilization (bc⁻¹)—while the input variable is nitrogen retention (NR). Previous studies have published these parameters for growing pullets [22,23], using diets ranging from 1 to 6% nitrogen.

The models from [22,23] are used to obtain the Lys, Met+Cys, and Thr intakes for each phase: starter (14 to 28 days), grower (56 to 70 days), and developer (98 to 112 days). Although the validity of these models has been proven, they do not allow for estimation of the daily amino acid intake, due to their dependence on the NR_maxT [25,26,27]. For growing pullets, this is more difficult because of the longer breeding period, in which the growth phase is extended to 126 days. The physiological changes that occur in the period before the first egg is laid should be considered in models developed for these birds. An alternative for estimating the daily amino acid intake could involve estimation of the daily ND_maxT. Therefore, the objective of this study was to model the ND_maxT of pullets according to their age and integrate this into the factorial model of [22,23] to estimate the daily AAI. In particular, Met+Cys is used as an experimental model, due to its recognition as the first limiting amino acid in corn and soybean meal diets. Its adequacy is critical for the formation of cysteine, protein synthesis, feather development, and overall growth [28,29].

2. Materials and Methods

2.1. Animal Ethics and Welfare Committee

All procedures were approved by the Ethics Committee for the Use of Animals (CEUA) under Protocol number 007125-08, with the approval date of 26 April 2013, in Jaboticabal, SP, Brazil. The trials were conducted at the Poultry Science Laboratory of the São Paulo State University, UNESP, Jaboticabal, São Paulo, Brazil.

2.2. Database

The data used in this study were obtained from [22,23]. These data resulted from three nitrogen balance assays from 168 Dekalb White pullets. The assays were performed in the phases from 14 to 28 (I), 56 to 70 (II), and 98 to 112 (III) days, in which the average ages of each phase were considered to be 21, 63, and 105 days, respectively. In all three assays, the same treatments were applied and the experimental design was completely randomized with seven treatments, different crude protein levels for each protein level, eight replicates, and one hen in each. The protein levels were obtained via the dilution technique. The resulting protein levels, expressed on a dry matter (DM) basis, were 435 (N6); 364 (N5); 292 (N4); 220 (N3); 150 (N2); and 75 (N1) g/kg. To confirm that the response of the hens to each dilution series was a function of the limiting amino acid, a control diet (N7), containing 75 g/kg DM, was added. A small quantity of 2.485 g/kg DL-methionine was added to the diet with the lowest level of the methionine sufficient to meet the level of methionine in the second-lowest level in the dilution series, in order to evaluate whether Met+Cys was the first primary limiting factor in the experimental diets.

The supply of feed was corrected for the metabolic weight of each bird (BW^0.67). Responses to ingested nitrogen were used (NI) and deposited (ND) at 14-day intervals, obtained via slaughter (

Table 1. Means of body weight (BW), nitrogen intake (NI, mg × BW^0.67 × kg⁻¹), and nitrogen deposition (ND, mg × BW^0.67 × kg⁻¹) of pullets in phases I (14 to 28 days), II (56 to 70 days), and III (98 to 112 days) ¹.

Items	N1	N2	N3	N4	N5	N6	N7
Phase I
BW	149 ± 15	248 ± 14	290 ± 14	322 ± 9	304 ± 18	297 ± 21	197 ± 14
NI	565 ± 17	1588 ± 26	2309 ± 56	3044 ± 75	3884 ± 93	4317 ± 94	798 ± 29
ND	223 ± 52	920 ± 25	1146 ± 24	1295 ± 15	1223 ± 29	1174 ± 32	604 ± 37
Phase II
BW	487 ± 19	587 ± 40	661 ± 33	658 ± 27	659 ± 28	656 ± 23	556 ± 28
NI	597 ± 11	1267 ± 8	1721 ± 14	2432 ± 73	2850 ± 36	3431 ± 26	614 ± 5
ND	167 ± 15	537 ± 30	754 ± 28	751 ± 12	759 ± 25	733 ± 32	408 ± 27
Phase III
BW	927 ± 34	1022 ± 35	1048 ± 25	1068 ± 26	1060 ± 28	1067 ± 24	964 ± 40
NI	514 ± 8	1252 ± 7	1745 ± 24	2388 ± 24	2992 ± 37	3703 ± 37	613 ± 7
ND	93 ± 20	333 ± 31	388 ± 24	431 ± 23	418 ± 22	433 ± 18	188 ± 25

¹ [22], N1: 75 g/kg, N2: 150 g/kg, N3: 220 g/kg, N4: 292 g/kg, N5: 364 g/kg, N6: 435 g/kg, N7: 75 g/kg + 2.485 g/kg of DL-methionine.

). The publications [22,23] have presented the NI and ND values obtained via the technique of nitrogen balance and comparative slaughter. These were used to validate the model with the information available in the literature.

2.3. Birds and General Management

The birds were donated from a commercial farm located in Uberlândia, Minas Gerais, Brazil. The tests were carried out in a metabolism shed equipped with a negative pressure system to control the temperature in the thermoneutral zone of each phase. The birds in phase I were raised in metabolic rearing cages measuring 100 × 80 × 35 cm in length × width × height. The birds used in phases II and III were raised simultaneously, housed individually in metabolic cages measuring (50 × 50 × 50) cm in length × width × height. All the cages in phases I, II, and III were equipped with individual feeders and nipple-type drinkers, and the management and lighting program recommended by the strain manual was used. The body weights in the trials were 125 ± 2 g, 449 ± 14.8 g, and 889 ± 18 g in the initial and rearing phases, respectively. The adaptation period consisted of supplying feed ad libitum to determine the maximum intake per kilogram of metabolic body weight under the experimental conditions, and then the supply was control-corrected for the metabolic weight. The temperature in the house was adjusted as the birds advanced in age. The temperature in the room, the lights, and the access to feed and water were monitored throughout the experimental period.

2.4. Determination of the Theoretical Maximum Potential of Nitrogen Deposition (ND_maxT)

The responses of the ND in relation to the NI obtained from the assays were used to estimate the mean values (μ) of ND_maxT and their respective deviations (σ) during phases I, II, and III, according to the model:

ND = ND_maxT × (1 − e^{−b × NI}),

(1)

where ND is the nitrogen deposition (mg × BW^0.67 × kg⁻¹); NI is the nitrogen intake (mg × BW^0.67 × kg⁻¹); ND_maxT is the maximum value expected for deposition when NI tends to infinity; “b” is the growth rate of the function that is associated with the dietary protein quality; and “e” is the Euler number.

2.5. Adjustment of ND_maxT as a Function of Age

A normally distributed random population of 500 individuals was generated based on the μ and σ values of ND_maxT from the assays performed during phases I, II, and III. The sample size of 500 simulated individuals was selected to represent the biological variability typically observed among birds within a population. Preliminary tests using smaller and larger simulated populations indicated that 500 individuals were sufficient to produce stable mean and variance estimates of the ND_maxT, with minimal additional accuracy gained from larger sample sizes. This number therefore ensured both biological representativeness and computational efficiency for the subsequent modeling steps.

To model the ND_maxT as a function of age, we considered the average age of each experimental period; i.e., 21 days old for the initial phase, 63 days for the grower period, and 105 days for the rearing phase. To this database, we added the value of ND_maxT related to day of birth; i.e., day zero. Five hundred individuals were also generated by applying the same procedure described above. The ND_maxT at age zero was obtained using the Gompertz function. At this age, we considered ND_max = ND_maxT. The body weight (0.036 kg) and body protein weight (0.738 g) were estimated. The protein weight was transformed into nitrogen (0.1181 g), which resulted in an ND_maxT of 1096 ± 55 mg × BW^0.67 × kg⁻¹, with a deviation of 55 mg × BW^0.67 × kg⁻¹, equivalent to a coefficient of variation of 5% for the body protein weight (BP) at birth. The parameters of the Gompertz function used to describe the BP growth of Dekalb White birds were obtained from [4]. The 2000 observations resulting from the four ages (0, 21, 63, and 105 days) were used to adjust the parameters of the exponential curve, as described in Equation (2):

ND_maxT = A + (B + C × t) × (R^t) ± ɛ,

(2)

where A, B, C > 0, and R is between 0 and 1.

ND_maxT is the observed value for each simulated individual at time t in mg × BW^0.67 × kg⁻¹; t is the age in days; A is a constant value for deposition when t is large enough or at maturity, in mg × BW^0.67 × kg⁻¹; B + C × t is the linear function for deposition, where B is the intercept and C is the slope, in mg × BW^0.67 × kg⁻¹; R is the growth rate of the function, which gives the curvature of the response (dimensionless); Ɛ is the random error; A + B is the NR_maxT when t = 0 or at birth; C indicates the magnitude of the decline in the maximum response. The age of maximum deposition (t_max) is given by t_max = (A + B)/C.

2.6. Simulation and Model Evaluation

2.6.1. Evaluation of ND_maxT Prediction as a Function of Age

The ND_maxT values predicted by Equation (2) were compared with the reference deposition or ND_max obtained via the Gompertz function, using the parameters described by Ref. [4]. To assist in the interpretation of the differences between ND_maxT and ND_max, a graphical analysis of the distribution of the studentized residuals was applied, as described by Ref. [30].

2.6.2. Model Evaluation to Predict the Amino Acid Intake

For the evaluation of the modeling procedure, Equation (2) was integrated into the Goettingen factorial model [13] to estimate the Met+Cys intake. The estimates were obtained in mg/bird per day and were transformed into the percentage of the diet, considering the feed intake of the birds. To calculate the Met+Cys intake, we used NMR = 238 mg × BW^0.67 × kg⁻¹ [23]. The NR_maxT was obtained considering the daily ND_maxT predicted by Equation (2) plus the constant value of the NMR (NR_maxT = ND_maxT + NMR). Similarly, NR was obtained by adding NMR to the ND value.

The Met+Cys intake was obtained, considering an NR equal to 80, 70, and 60% of the NR_maxT, estimated for the day. The daily values of NR were applied in the Goettingen model:

LAAI = [ln(NR_maxT) − ln(NR_maxT − NR)]/16 × b/c,

(3)

where AAI is the calculated amino acid intake to meet the target NR performance, in mg × BW^0.67 × kg⁻¹; NR_maxT is the maximum theoretical nitrogen retention, in mg × BW^0.67 × kg⁻¹; 16 is the nitrogen conversion factor for protein; b/c is the amino acid utilization efficiency; b is slope of the curve between NI and ND in Equation (1). This parameter indicates the quality of dietary protein, regardless of the amount of NI (dimensionless) and assuming small values in the range of 10⁻⁶. c is the concentration of amino acids in dietary protein in g of amino acid per 100 g of protein [13]. The b/c values for Met+Cys were obtained from Ref. [22]. In the simulation, we considered the values of b/c = 43 × 10⁻⁶ and c = 3.53 g Met+Cys per 100 g of CP for all ages.

2.7. Evaluation of the Amino Acid Efficiency of Utilization (k) for Protein Deposition

The efficiency of utilization indicates what proportion of the amino acid was used for the bird, which varies in percentage. The k rate was calculated considering the relationship between the intake (LAAI) and deposition (D) of amino acid in the body of the bird (k = D/LAAI). The amino acid intake was calculated according to Equation (3). The amino acid deposition in the body was obtained by multiplying the amino acid concentration of the protein in the body. The protein deposition (PD) in the body was obtained from the ND × 6.25, where the body protein has 16% nitrogen (100/16 = 6.25). The Met+Cys concentration in the body protein was 68 mg per gram of protein, respectively.

2.8. Statistical Analysis

The statistical analyses were performed with the SAS software (Statistical Analysis System, version 9.1, Cary, NC, USA: SAS Institute Inc.), using the NLIN procedure for model adjustment and using the Levenberg–Marquardt algorithm.

3. Results

3.1. Mean and Standard Deviation of ND_maxT Obtained from Experimental Sampling

The mean values, deviations, and confidence intervals for the parameters of Equation (1) are shown in

Table 2. Mean (μ) and standard deviation (σ) of ND_maxT (mg × BW^0.67 × kg⁻¹) and b, according to the experimental mean age.

Age	NDmaxT		b
(Days)	μ	σ	μ	σ
Phase 21 d	1361	87	0.000662	0.000121
Phase 63 d	867	74	0.000745	0.000160
Phase 105 d	476	59	0.000813	0.000276
Confidence interval, considering μ ± 2 × σ
	Limit a		Limit a
	Lower	Upper	Lower	Upper
Phase 21 d	1187	1535	0.000422	0.000903
Phase 63 d	719	1015	0.000428	0.00106
Phase 105 d	358	584	0.000264	0.00136

^a Mean (μ) ± 2 standard deviation (σ) = μ ± 2 × σ.

. The ND_maxT deviation values showed the variability between individuals, necessary to simulate and characterize a population. The estimated value of the ND_maxT characterized the genetic potential of the specie or strain being studied. The μ values determined for ND_maxT at 21, 63, and 105 days of age were 1361 ± 87, 867 ± 74, and 476 ± 59 mg × BW^0.67 × kg⁻¹, respectively. The observed σ value represented a variation of approximately 8% of the mean.

3.2. Modeling the Trajectory of ND_maxT of the Simulated Population as a Function of Age

Based on the μ and σ values of ND_maxT, we simulated the nitrogen deposition in 500 individuals at ages 0, 21, 63, and 105 days (Figure 1), considering the population to be normally distributed.

The estimated coefficients to describe the trajectory of ND_maxT were A = 200; B = 897; C = 64; R = 0.9687; and Ɛ = 68, according to the exponential model:

ND_maxT = 200 + (897 + 64 × t) × (0.9687^t) ± 68.

(4)

The error of this prediction model was minimized at 68 mg, which is 6, 5, 8, and 14% of the average values at ages 0, 21, 63, and 105 days. The interpretations of the model parameters are as follows: (1) A is the minimum retention of nitrogen or nitrogen retained for the maintenance of the bird (NMR). Typically, the NMR occurs when the growth of the birds ceases; (2) parameter B is interpreted as the ND_maxT at birth (i.e., when t = 0) and, at this point, the sum of parameters A + B results from the NR_maxT of the bird at birth. The NR_maxT at birth was estimated to be 1097 mg × BW^0.67 × kg⁻¹. The parameter C = 64 indicates the degree of the curvature of the trajectory; the higher the C value, the more conic the curve, which can be interpreted as the decay rate of ND_maxT; (3) the age at the maximum deposition rate (t_max = [A + B]/C) was calculated to be 17 days; at this age, the ND_maxT was estimated to be 1358 mg × BW^0.67 × kg⁻¹.

A separate analysis of the model shows that the first component [NMR = 200] is a constant value; the second component strictly increase as t tends to infinity (+∞) [ND_maxT = 897 + 64 × t]; and the third component (0.9687^t) fixes the percentage of the previous sum, considered a function of age, giving the exponentiation of the model as shown in Figure 2.

3.3. Evaluation of ND_maxT and NDmax Using the Gompertz Function

The Gompertz function, described in [20], was used to evaluate the prediction model’s ability to calculate the ND_maxT as a function of age. Figure 3 shows that the difference between the deposition or “standard ND_max” obtained using the Gompertz function and the ND_maxT estimated here was 11% until 63 days of age. After this period, from 63 to 126 days, the difference between ND_maxT and ND_max increased exponentially with an average difference of 44%.

An analysis of the residuals was conducted to investigate the normality of the difference between the values of ND_maxT and ND_max. As shown in Figure 4, the change occurred between 84 and 91 days of age; thereafter, the difference remained constant between ND_maxT and ND_max.

3.4. Integration of the Model to Calculate the Amino Acid Intake

The integration of the model was demonstrated by calculating the Met+Cys intake for the Dekalb White bird genotype from 1 to 126 days of age, as shown in

Table 3. Deposition (D, mg), intake (LAAI, mg), and efficiency of utilization (k) of methionine+cystine in birds at different growth rates, considering the maximum daily nitrogen deposition (ND_maxT) at 60% and 50% of ND_maxT, mg × BW^0.67 × kg⁻¹.

Age	BW	NDmaxT	PDmaxT	60% NDmaxT				50% NDmaxT
				NR	D a	LAAI b	k c	NR	D a	LAAI b	k
7	60	1278	1.2	936	60	71	85	780	50	54	93
14	93	1351	1.7	980	85	96	89	817	71	72	98
21	136	1351	2.2	981	110	124	89	817	91	94	98
28	189	1306	2.7	953	133	154	86	795	111	117	95
35	251	1233	3.1	910	153	186	82	758	128	141	91
42	321	1145	3.3	857	170	220	77	714	142	166	85
49	397	1051	3.5	800	183	253	72	667	153	191	80
56	477	957	3.6	744	192	286	67	620	160	216	74
63	558	867	3.7	690	198	318	62	575	165	241	69
70	640	782	3.6	639	201	349	58	533	168	264	64
77	720	705	3.5	593	202	377	54	494	168	285	59
84	798	635	3.4	551	201	404	50	459	168	306	55
91	871	573	3.3	514	199	429	46	428	166	324	51
98	940	519	3.1	481	196	451	43	401	163	341	48
105	1004	471	3.0	452	193	471	41	377	161	357	45
112	1062	430	2.8	428	189	490	39	356	158	370	43
119	1116	394	2.7	406	186	506	37	339	155	383	40
126	1164	364	2.5	388	183	521	35	323	152	394	39

^a Deposition, ^b Daily intake of the limiting amino acid, ^c Efficiency of utilization, BW = 1524 × e(^{−e((ln(−ln(0.036/1524))}) − 0.0021 × t)), kg, ND_maxT = 220 + (897 + 64 × t) × (0.9687^t), mg × BW^0.67 × kg⁻¹, PD_maxT = ([ND_maxT × 6.25] × BW^0.67)/1000, g per day, NR60% = (ND_maxT × 0.6) + 283, NR50% = (ND_maxT × 0.5) + 283 mg × BW^0.67 × kg⁻¹, LAAI = ((ln[ND_maxT + 283] − ln([ND_maxT + 283] − NR))/16 × 0.00043 × 3.53⁻¹) × BW^0.67, mg/day, D = ([BW^0.67 × NR] × 6.25) × 68, mg/day, k = D/LAAI × 100%.

. The intake is related to a daily deposition rate equivalent to 60% and 50% of ND_maxT. To evaluate the consistency of the predictions, we considered the efficiency of utilization k obtained using the ratio between D and LAAI. The integration of the models allowed us to obtain the daily Met+Cys intake and, consequently, to calculate k, which decreased with the age of the bird. The minimum value reached 35% at 126 days of age, with a bird having 60% of its maximum potential for nitrogen deposition.

4. Discussion

The objective of this study was to model the maximum potential for nitrogen deposition in Dekalb White birds according to their age, and to integrate a mathematical model to calculate the Met+Cys intake considering the requirements for growth and the maintenance of body weight.

The proposal presented in this research consists of an alternative methodology to calculate the daily intake of amino acids based on the daily potential for protein deposition. In the existing literature on dynamic models used for calculating the intake of amino acids, the comparative slaughter technique is used to estimate the parameters that characterize the genotype and efficiency of dietary nutrient utilization [1,20,31,32]. Previous studies [22,23] have concluded that the comparative slaughter (destructive method, using sacrifice of animals) and nitrogen balance techniques (non-destructive method, with excreta collection) do not infer the estimation of both the nutrient efficiency of utilization and the potential for nitrogen deposition in birds. It should be noted that the NI and ND information from both methods (comparative slaughter and nitrogen balance) were used in this study, because it was necessary to use the data obtained by the comparative slaughter technique to validate the proposed procedure here and compare it with the traditional method in the literature.

The model parameter ND_maxT consists of the nitrogen deposition relative to the metabolic weight of the bird, which decreases with advancing age (Figure 3). The ND_maxT characterizes the physiological limit of the animal’s response and is not in the range of the observed data. Therefore, it is an intangible value; i.e., it is “theoretical”, because there are limiting factors in the environment [8,9,10,11,12,13].

The traditional ND_max is obtained by dividing the first derivative of the Gompertz function by BW^0.67. This calculation requires sacrificing birds in each new trial, due to the dependence of the Gompertz function parameters (B and Pm) on the environmental conditions [33,34,35,36]; so, if any limiting physical or biological factor exists, the coefficients are valid only for the conditions in which they were generated [37]. This proposal is based on the theory in [13], in which the ND_maxT and NMR parameters are constants inherent of the genotype and do not change with limiting factors from the diet, as observed in the similarity of ND_maxT values found by [8,24] for the Cobb500 genotype.

However, the gap that exists in the limits of inference for ND_maxT, which are determined in each phase of the dose–response trials at ages 21, 63, and 105 days, can be avoided through using a model designed to predict the ND_maxT between 0 and 126 days, which in this period had an average error of 8%. The maximum error of 14% found was close to 105 days of age and cannot be easily explained.

One possible explanation is based on the non-normality of the differences verified between the ND_max and ND_maxT values (Figure 4). The residual analysis predicted that between 84 and 91 days of age, the birds had a protein deposition with a non-normal response pattern relative to the total period, resulting in more errors in this period. This increase may be due to the deposition of protein in the muscle because of the proximity to the mature weight of the birds. Among the causes investigated, the early development of organs linked to the sexual maturity of birds, such as the ovary and oviduct, can be seen as a possible source of variation in the analysis of the trajectory. Thus, it is assumed that at 105 days there was a protein deposition peak coming from the growth of the reproductive organs. However, according to the available literature, these birds mature between 112 and 119 days of age [4]. A hypothesis to justify this earliness can the lighting program that birds received during the rearing period, which coincided with the summer period (approximately 14 h of daylight), and the higher protein supply provided by diets with a high nitrogen content. This combination may have stimulated the development of the ovary and oviduct. Previous research suggests that the growth of these organs is significant and should be considered [4,38]. Therefore, from a biological point of view, it is assumed that the protein deposition increased from the maturation of these organs, which is considered in the ND_maxT predicted by the proposed model.

These observations were reported by [22], due to specific and isolated analysis in the phases I, II, and III. Therefore, it is proposed that this procedure be included in the interpretation of the bird response. An equally important complement to this evaluation is the prediction of the AAI with the integration of the models. The results presented in Table 3 support the procedure proposed here. The AAI was estimated and agreed with the values presented in a previous study [22], which confirms the interpretation of the amino acids.

The system of equations used in this study allowed determining the efficiency of utilization as a relative value standardized to a centesimal scale. In this way, the interpretation of use is in relation to the direct deposition, and it is possible to easily identify the “bottleneck” points.

The values presented in Table 3 indicate a decreased efficiency of utilization with advancing age. The efficiency occurs through the digestion, absorption, and post absorption of the ingested amino acids [13]. Although some studies support this hypothesis [22,24], the values presented in the last weeks of age (13–18 weeks) may be considered low [20,39], although they are similar to those found by Silva et al. [1]. This result is related to how the efficiency of utilization was considered in this model. In this study, a single value of ‘b’ and ‘c’ for phase I and for the other phases was used, causing the low efficiency values in the last weeks. The option to use the values of phase I aimed to meet “the most limiting phase of the bird”, because there is no model to dynamically correct the efficiency of utilization.

Silva et al. [1] proposed a method to diagnose the efficiency of lysine, methionine + cystine, and threonine utilization for growing pullets using the ratio of the Gompertz function adjusted for the deposition and amino acid intake. They proposed considering a static deposition and estimating the amino acid intake considering an efficiency of utilization of 80% [40]. Some researchers believe that 80% is the average efficiency of the population, and some studies have supported this value [41]. Although the users of this model can adopt this procedure, research should be designed to improve the estimation of the dynamic efficiency to optimize the estimation of the LAAI.

When comparing the efficiency of utilization, 60% of ND_maxT showed lower values than 50% of ND_maxT. This difference is related to the non-linearity in the model used to calculate the intake of amino acids. Using non-linear models, the authors of [42,43] estimated a 38% efficiency of utilization for methionine for a 95% maximal response in broilers. The authors emphasized that the closer to the maximum response (ND_maxT), the lower the efficiency of utilization and higher intake. Using the maximum response to determine the AAI would enable finding other individuals in the population; however, there is no guarantee that this optimal dose coincides with the optimal economic dose. The model proposed here provides flexibility to decide on the optimal AAI to optimize the response of the population. Various criteria may be used: (1) a percentage of the maximum response (60, 50, and 40% ND_maxT); (2) the average intake for the phase; and/or (3) the day at the most demanding phase, according to the percentage of ND_maxT.

The results obtained were compared with previous studies for validation using information available in the literature. In this context, the genetic potential described by [20] corresponds to approximately 55% of the maximum (ND_maxT) reported in this study, and the estimated Met+Cys intake in the initial phase (1 to 42 days of age) was 110 mg/bird per day [20], which is close to the value found in the present study—107 mg/bird per day—when the model was applied considering 50% of ND_maxT for the same phase.

When 60% of ND_maxT was applied (Table 3), the model estimated a Met+Cys intake of 142 mg/bird per day in the initial phase, which is similar to the 151 mg/bird per day presented by [44].

In the rearing phase (49 to 84 days of age), the authors of [45] recommended a Met+Cys intake between 287 and 327 mg/bird per day. For this phase, the estimate obtained using the present model was 331 mg/bird per day at 60% of ND_maxT (Table 3), which is close to the 308 mg/bird per day.

In the growing phase (91 to 126 days of age), the study by [46] reported that an intake of 292 mg/bird per day was insufficient to cause a maximum response from the birds. The levels tested produced a linear effect on performance, preventing the determination of an optimal Met+Cys intake. In this phase, the Met+Cys intake estimated by the model of [20] was 323 mg/bird per day, corresponding to 55% of ND_maxT, which is still higher than the values reported by [20,46]; both reported lower values than those estimated by the present model, which ranged from 362 to 478 mg/bird per day for 50% and 60% of ND_maxT, respectively. Overall, this validation suggests that advances in genetic improvement have increased protein deposition and, consequently, amino acid requirements.

Despite the limited number of studies, the available evidence shows that the older the study [47,48], the further its reported Met+Cys intake values deviate from those found for modern replacement pullets. Thus, the model was able to account for the genetic potential and appropriately estimate the Met+Cys intake. Only in the final phase was validation with recent findings not possible, because no studies have successfully established the optimal Met+Cys intake for this stage [49]. In this phase, the recommended values were higher than those reported in the literature, and it was not possible to determine whether this difference is attributable to genetic potential or to other factors, such as the early onset of sexual maturity. Early maturation may blur the distinction between the growth and production phases, due to the development and activation of reproductive structures and the presence of eggs in the bird.

The recommendation obtained in this study was close to the 363–372 mg/bird per day for pullets between 112 and 126 days of age, a period during which full development of the reproductive organs is expected [4].

5. Conclusions

The proposed model provides flexibility in determining the optimal Met+Cys intake to maximize the population performance. By integrating different modeling approaches, it enables daily estimates of Met+Cys that are consistent with the literature and can be adjusted according to the target protein deposition and feed intake. This modeling framework represents a valuable tool for accurately defining amino acid requirements in pullets, supporting more precise and efficient nutritional strategies in poultry production.