J. Anim Sci. 2008. 86:324-332. doi:10.2527/jas.2007-0086
© 2008 American Society of Animal Science
Commercial validation of the true ileal digestible lysine requirement for eleven- to twenty-seven-kilogram pigs1
D. C. Kendall*,2,
A. M. Gaines*,3,
G. L. Allee*,4 and
J. L. Usry
* Department of Animal Sciences, University of Missouri, Columbia 65211; and
Ajinomoto Heartland LLC, Chicago, IL 60631
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Abstract
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Five experiments utilizing 3,628 pigs were conducted to determine the true ileal digestible (TID) Lys requirement for 11- to 27-kg pigs fed corn-soybean meal diets. In Exp. 1, 216 barrows (initial BW = 11.5 kg) were used, with dietary TID Lys levels from 1.05 to 1.40% TID Lys (0.07% increments). All diets were isocaloric (3.42 Mcal of ME) and contained the same inclusion of soybean meal (33.1%). Dietary Lys content was increased by adding graded levels of L-Lys·HCl (0.0 to 0.445%), with other crystalline AA supplied to meet minimum AA-to-Lys ratios. For the 21-d period, ADG and G:F increased linearly (P < 0.001) with increasing Lys levels. Experiments 2 through 5 were each conducted in different commercial research facilities. In Exp. 2, a 5-point titration (1.05 to 1.41% TID Lys; 0.09% increments) was used containing the same level of soybean meal (34.3%), with graded levels of L-Lys·HCl addition as in Exp. 1 for a 16-d period. Exp. 3 used similar diets, but was a 28-d period from 11.8 to 28 kg. There were linear increases in ADG (P < 0.01) and G:F (P < 0.01) with increasing dietary Lys in both experiments. On the basis of these results, 2 additional 28-d experiments were conducted with similar diets, except for 1 additional level at 1.50% TID Lys. In Exp. 4, linear increases (P < 0.01) in ADG and G:F were observed from d 0 to 14. From d 14 to 28, there were quadratic increases (P < 0.04) in ADG and G:F, which resulted in quadratic increases (P < 0.01) in ADG and G:F with increasing dietary Lys for the entire 28-d period. Similarly, in Exp. 5, there were linear increases (P < 0.01) in growth performance from d 0 to 14, but there were quadratic increases in G:F (P < 0.001) with increasing dietary Lys for the overall period. Data from all 5 experiments yielded a single-slope, broken-line response, with requirement estimates for TID Lys of 1.33 and 1.35% for 11- to 19-kg pigs. The 5 experiments gave requirement estimates of 1.30% TID Lys (3.80 g of TID Lys/Mcal of ME) for 11- to 27-kg pigs, equivalent to 19 g of TID Lys/kg of gain.
Key Words: growth lysine nursery pig
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INTRODUCTION
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Because Lys is generally the first limiting AA in practical swine diets, the Lys requirement has been thoroughly investigated and reviewed (NRC, 1998
). Despite this research effort, the surveyed literature still shows considerable variation in the reported requirements. Estimates by NRC (1998)
range from 0.98 to 1.49% total Lys for 10- to 20-kg pigs, with a recommendation of 1.15% total Lys (3.52 g of total Lys/Mcal of ME). This disparity is attributable to differences in the genetic capacity for protein deposition (Krick et al., 1992
) or immune stress (Williams et al., 1997
) of experimental animals, along with deviations in the digestibility of AA within dietary ingredients (Campbell et al., 1988
). Another reason could be the different statistical methods for determining requirement estimates: fitted broken line (Martinez and Knabe, 1990
), quadratic maximum (Gatel et al., 1992
), response surface analysis (Nam and Aherne, 1994
), and minimum value of the plateau (Campbell et al., 1988
).
More recent attempts to characterize the Lys requirement for 20-kg pigs have been much greater than NRC (1998)
estimates. van Lunen and Cole (1998)
showed that 5.02 g of total Lys/Mcal of DE is necessary to maximize daily BW gain and nitrogen deposition, whereas Smith et al. (1999)
demonstrated an optimal feeding level of 4.35 g of total Lys/Mcal of ME. Urynek and Buraczewska (2003)
showed that maximal nitrogen retention and ADG in 13- to 20-kg and 20- to 30-kg pigs was achieved at 3.56 and 3.22 g of apparent ileal digestible (AID) Lys/Mcal of ME, respectively. The divergent results between NRC (1998)
and these studies may reflect a greater capacity for body growth and protein accretion in modern, high lean growth genotypes. Despite the considerable insight these previous studies have given on appropriate Lys concentrations and Lys-to-energy relationships, a need existed for studies conducted under commercial conditions. The objective of these studies was to evaluate the Lys requirement of 11- to 27-kg pigs extensively under commercial research conditions.
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MATERIALS AND METHODS
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All experimental procedures and animal care were approved by the University of Missouri Animal Care and Use Committee.
This research consisted of 5 experiments. Experiment 1 evaluated the true ileal digestible (TID) Lys requirement of 11- to 25-kg barrows by using diets containing increasing levels of L-Lys·HCl to titrate dietary Lys. Subsequent experiments (Exp. 2 through 5) validated the TID Lys requirement estimate under commercial research conditions at multiple locations.
Exp. 1
An initial experiment involving a total of 216 barrows (Dalland x PIC Camborough 22) was used to determine the TID Lys requirement of 11- to 25-kg of BW pigs. This experiment was conducted under experimental conditions at the University of Missouri. Pigs were weaned at 17 d of age and fed a nutritionally adequate phase I diet containing 1.60% total Lys and 3.27 Mcal of ME/kg for 14 d. Phase II diets contained 1.50% total Lys and 3.27 Mcal of ME/kg until reaching a target BW of 11 kg. Pigs were allotted to pens (1.2 x 1.2 m), with each pen containing a 3-holed feeder (0.45 m of trough space) and a single cup watering device. Pigs were blocked by BW into 9 replicate pens of 4 pigs/pen in a randomized complete block design. Pigs were offered the experimental diets for 21 d with individual pig BW and feed disappearance recorded weekly to calculate ADG, ADFI, and G:F.
Exp. 2 to 5
Each experiment was conducted in a different commercial research facility with pigs weaned at approximately 17 d of age. Experiment 2 consisted of 880 pigs (Triumph 4 x PIC Camborough 22) that were randomly allotted within sex into 4 replicate pens/sex (22 pigs/pen) in a 16-d growth trial. Pigs were housed in pens measuring 1.65 x 3.0 m and provided 0.76 m of feeder trough space. In Exp. 3, a total of 840 pigs (Triumph 4 x PIC Camborough 22) were used in a randomized complete block design within sex, with 3 replicate pens/sex (28 pigs/pen) in a 28-d growth trial. Pigs were housed in pens measuring 2.4 x 5.7 m and provided 1.27 m of feeder trough space. Experiment 4 used 900 pigs (PIC Camborough 337 x PIC Camborough 22) that were randomly allotted within sex into 3 replicate pens/sex (25 pigs/pen) in a 28-d growth trial. Pigs were housed in pens measuring 1.90 x 3.05 m and provided 0.90 m of feeder trough space via a 5-hole feeder. Experiment 5 consisted of 792 pigs (Triumph 4 x PIC Camborough 22) in a randomized complete block design within sex with 3 replicate pens/sex (22 pigs/pen) in a 28-d growth trial. Pigs were housed in pens measuring 1.82 x 3.05 m and provided 0.90 m of trough space via a 6-hole self-feeder. In each experiment, pigs were fed nutritionally adequate phase I and phase II diets prior to reaching a target BW of 11 kg. For Exp. 2 and 3, treatments were arranged as a 5-point titration, with dietary TID Lys levels of 1.05, 1.14, 1.23, 1.32, and 1.41%, respectively. For Exp. 4 and 5, treatments were arranged as a 6-point titration, with dietary TID Lys levels of 1.05, 1.14, 1.23, 1.32, 1.41, and 1.50%, respectively.
Dietary Treatments
Diets for Exp. 1 (Table 1
) were formulated on a TID basis by using previously analyzed AA values of the ingredients and digestibility coefficients obtained from NRC (1998)
. The diets contained an equivalent amount of soybean meal, with dietary Lys increased by adding L-Lys·HCl (0.00 to 0.446%) in a 6-point titration (1.05, 1.12, 1.19, 1.26, 1.33, and 1.40% TID Lys). Additional crystalline AA were supplied as necessary to meet minimum AA ratio requirements according to the pattern of Chung and Baker (1992)
, with all diets formulated to be isocaloric (3.42 Mcal of ME/kg). Constant soybean meal inclusion was desired to avoid any confounding effects of high-soybean meal inclusion, although the effects of soybean meal hypersensitivity are generally associated with newly weaned pigs (Li et al. 1990
; Dréau et al., 1994
). Relative to corn, soybean meal is nearly equivalent in ME but considerably lower in NE. A constant level of soybean meal use ensured similar energy density with increasing dietary Lys, regardless of the preferred energy system. Experiment 1 was an initial attempt to characterize the Lys requirement before conducting commercial validation studies. In the commercial experiments (Table 2
), formulation to a TID basis was based on NRC (1998)
AA and digestibility estimates of the ingredients. A corn-soybean meal basal diet (1.05% TID Lys) was used, containing a constant soybean meal inclusion (34.3%), with dietary Lys increased by adding L-Lys·HCl (0.00 to 0.573%) in a 5-point (Exp. 2 and 3) or 6-point (Exp. 4 and 5) titration (1.05, 1.14, 1.23, 1.32, 1.41, and 1.50% TID Lys). All diets were formulated to meet or exceed the NRC (1998)
recommended levels for minerals and vitamins and contained 3.42 Mcal of ME/kg.
Statistical Analysis
Data for each response criterion were analyzed by ANOVA using the GLM procedure (SAS Inst. Inc., Cary, NC) with Lys level and replicate included in the model for Exp. 1. In each of the following experiments, the effects of Lys level, sex, and replicate within sex were included in the model. Pen was considered the experimental unit with linear and quadratic polynomials used to evaluate increasing dietary TID Lys. Estimates of requirements for performance were determined by subjecting the pen mean data to least squares broken-line methodology (Robbins et al., 1979
), along with the asymptote of the quadratic fitted line.
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RESULTS
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Exp. 1
In Exp. 1, there were linear improvements in ADG and G:F for the first 7 d and d 0 to 21 (P < 0.001; Table 3
) with increasing dietary Lys. From d 7 to 14, there was a linear increase in G:F with increasing TID Lys. This resulted in a linear increase in final BW and cumulative (d 0 to 21) ADG (P = 0.001), whereas quadratic improvements (P = 0.04) in cumulative G:F were observed with increasing dietary Lys. Using single-slope, broken-line methodology, we determined a minimal breakpoint of 1.33% TID Lys for G:F. Daily feed intake was unaffected by dietary Lys.
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Table 3. Effects of increasing true ileal digestible Lys on growth rate, feed intake, feed conversion, and BW during a 21-d nursery period (Exp. 1)1
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Exp. 2 to 5
In Exp. 2 (Table 4
), there was a linear increase in ADG (P < 0.001) and G:F (P < 0.01) with increasing TID Lys. Daily feed intake was unaffected by dietary Lys. In Exp. 3 (Table 5
), there were linear improvements in ADG and G:F from d 0 to 14 (P < 0.008). From d 14 to 28, G:F was improved linearly, whereas a quadratic response to ADG (P = 0.04) was observed. For the overall experiment, ADG and G:F improved linearly (P = 0.001) with increasing dietary Lys. To better estimate the TID Lys requirement, Exp. 4 and 5 included an additional Lys level (1.50% TID Lys). In Exp. 4 (Table 6
), there were linear improvements (P = 0.001) in ADG and G:F from d 0 to 14. From d 14 to 28, quadratic improvements were observed in ADG (P < 0.008), ADFI (P = 0.05), and G:F (P = 0.03). For the overall period (d 0 to 28), quadratic improvements in final BW (P = 0.04), ADG (P < 0.004), and G:F (P < 0.007) were observed with increasing dietary Lys, resulting in a breakpoint of 1.33 and 1.28% TID Lys for ADG and G:F, respectively. In Exp. 5 (Table 7
), from d 0 to 14, there were linear increases in ADG (P < 0.003) and quadratic improvements in G:F (P < 0.002) with increasing dietary Lys. For the overall period (d 0 to 28), linear improvements in final BW (P = 0.05) and ADG (P = 0.02), and a quadratic increase (P = 0.001) in G:F were observed with increasing dietary Lys, resulting in a breakpoint of 1.28% TID Lys estimated for overall G:F.
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Table 4. Effects of increasing true ileal digestible Lys on growth rate, feed intake, feed conversion, and BW during a 16-d nursery period (Exp. 2)1
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Table 5. Effects of increasing true ileal digestible Lys on growth rate, feed intake, feed conversion, and BW during a 28-d nursery period (Exp. 3)1
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Table 6. Effects of increasing true ileal digestible Lys on growth rate, feed intake, feed conversion, and BW during a 28-d nursery period (Exp. 4)1
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Table 7. Effects of increasing true ileal digestible Lys on growth rate, feed intake, feed conversion, and BW during a 28-d nursery period (Exp. 5)1
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Combined Estimates
Data from each experiment were expressed as the percentage improvement in performance within an experiment as a function of dietary TID Lys. Each figure depicts the estimated requirement based on either a breakpoint from the single-slope, broken-line model or the asymptote of the quadratic function. Figure 1A
shows estimates of the TID Lys requirement based on ADG of pigs from 11 to 27 kg using data from all experiments. For this BW range, the breakpoint occurs at 1.30% TID Lys and a maximum quadratic requirement of 1.45% TID Lys. As might be expected, the estimates are somewhat greater from 11 to 19 kg (Figure 1B
), where the breakpoint occurred at 1.33% TID Lys and the maximum quadratic requirement at 1.50% TID Lys.

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Figure 1. A) Fitted broken-line and quadratic plot of the percentage of maximal ADG as a function of true ileal digestible (TID) dietary Lys in 11- to 27-kg pigs. Data points represent treatment means from 5 experiments involving 3,628 pigs. The TID Lys requirement, determined by broken-line analysis using least squares methodology, was 1.30 ± 0.0325% (Y plateau = 99.4%, slope below breakpoint = –34.4%, r2 = 0.80). The data were also fitted to a quadratic equation: Y = –58.6(true digestible Lys)2 + 169.6(true digestible Lys) – 22.9 (r2 = 0.80). The upper asymptote of the quadratic function was calculated to be 1.446% true digestible Lys. B) Fitted broken-line and quadratic plot of the percentage of maximal ADG as a function of TID dietary Lys in 11- to 19-kg pigs. Data points represent treatment means from 5 experiments involving 3,628 pigs. The TID Lys requirement, determined by broken-line analysis using least squares methodology, was 1.33 ± 0.0273% (Y plateau = 99.1%, slope below breakpoint = –37.9%, r2 = 0.83). The data were also fitted to a quadratic equation: Y = –56.9(true digestible Lys)2 + 170.8(true digestible Lys) – 28.4 (r2 = 0.81). The upper asymptote of the quadratic function was calculated to be 1.50% true digestible Lys.
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Figure 2 (A and B)
shows estimates of the TID Lys requirement based on G:F of pigs from 11 to 27 kg. From 11 to 27 kg (Figure 2A
), the breakpoint occurred at 1.30% TID Lys and a maximum quadratic requirement of 1.46% TID Lys. Once again, the requirement estimates are higher when only considering the 11- to 19-kg pig (Figure 2B
). The objective breakpoint estimate was 1.35% TID Lys and a maximum quadratic requirement of 1.53% TID Lys.

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Figure 2. A) Fitted broken-line and quadratic plot of the percentage of maximal G:F as a function of true ileal digestible (TID) dietary Lys in 11- to 27-kg pigs. Data points represent treatment means from 5 experiments involving 3,628 pigs. The minimal TID Lys requirement, determined by broken-line analysis using least squares methodology, was 1.30 ± 0.0249% (Y plateau = 99.1%, slope below breakpoint = –34.6%, r2 = 0.87). The data were also fitted to a quadratic equation: Y = –56.5(true digestible Lys)2 + 164.7(true digestible Lys) – 20.5 (r2 = 0.88). The upper asymptote of the quadratic function was calculated to be 1.46% TID Lys. B) Fitted broken-line and quadratic plot of the percentage of maximal G:F as a function of TID dietary Lys in 11- to 19-kg pigs. Data points represent treatment means from 5 experiments involving 3,628 pigs. The minimal TID Lys requirement, determined by broken-line analysis using least squares methodology, was 1.35 ± 0.0228% (Y plateau = 99.6%, slope below breakpoint = –36.4%, r2 = 0.89). The data were also fitted to a quadratic equation: Y = –52.9(true digestible Lys)2 + 161.3(true digestible Lys) – 22.9 (r2 = 0.90). The upper asymptote of the quadratic function was calculated to be 1.53% true digestible Lys.
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One final estimate of the Lys requirement for 11- to 27-kg pigs is the percentage improvement in G:F as a function of the Lys required per kilogram of gain (Figure 3
). The breakpoint was 19.0 g of TID Lys/kg of gain. The asymptote of the quadratic function was 20.7 g of TID Lys/kg of gain.

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Figure 3. Fitted broken-line and quadratic plot of the percentage of maximal G:F as a function of true ileal digestible (TID) dietary Lys required per kilogram of daily gain in 11- to 27-kg pigs. Data points represent treatment means from 5 experiments involving 3,628 pigs. The minimal TID Lys requirement determined by broken-line analysis using least squares methodology was 19.0 ± 0.498 g of true digestible Lys/kg of ADG (Y plateau = 99.4%, slope below breakpoint = –2.45%, r2 = 0.69). The data were also fitted to a quadratic equation: Y = –0.346(g of true digestible Lys/kg of ADG)2 + 14.3(g of true digestible Lys/kg of ADG) – 48.5 (r2 = 0.71). The upper asymptote of the quadratic function was calculated to be 20.7 g of TID Lys/kg of ADG.
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DISCUSSION
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A key determinant to maximum growth is feeding adequate levels of all required nutrients. With protein representing an expensive nutrient, avoiding oversupplementation is necessary to achieve maximum profitability. As mentioned previously, there is a large quantity of experimental data evaluating the optimal Lys concentration for 10- to 30-kg pigs. This stage of growth is one of the most efficient during the life of a pig, providing an opportunity for cost-effective gains. Since the most recent NRC (1998)
, several studies have reevaluated the Lys requirement for nursery pigs (van Lunen and Cole, 1998
; Roth et al., 1999
; Smith et al., 1999
; Urynek and Buraczewska, 2003
). Although these studies have proven invaluable in our understanding of Lys requirements and Lys:energy relationships, a need still existed for large-scale studies validating these results under commercial conditions. Without this validation, lingering doubts would persist concerning the impact that the health and social stressors present in modern production have on Lys requirements obtained under experimental conditions.
Despite being conducted at different locations, there was remarkable consistency in the response to dietary Lys between each of the experiments. In Exp. 1, 2, and 3, the maximum Lys levels were only 1.40 or 1.41% TID Lys. This resulted in linear improvements in growth performance parameters with increasing dietary Lys. When Exp. 4 and 5 were extended to include a 1.50% TID Lys titration point, quadratic responses to increasing dietary Lys were observed. Therefore, normalizing the data and combining all 5 experiments was thought to provide the most comprehensive and accurate estimate of the Lys needs of 11- to 27-kg pigs. Because each treatment had a similar calculated ME content (3.42 Mcal of ME), the data are expressed as the percentage of TID Lys of the diet, with Lys-to-energy relationships proposed.
The combined estimates of the Lys requirement for 11 to 27 kg are at least 1.30% TID Lys (4.26 g of total Lys/Mcal of ME) to optimize ADG and G:F. These findings are identical to estimates provided by Yi et al. (2006)
in recent work conducted under commercial research conditions. If one converts the work of van Lunen and Cole (1998)
to an ME basis (ME being 96% of DE; NRC, 1998
), their data would suggest feeding 1.25 g of total Lys/MJ of ME (5.23 g of total Lys/Mcal of ME) to maximize N retention, but would indicate that in high-energy diets, levels closer to 1.04 g of total Lys/MJ of ME (4.36 g of total Lys/Mcal of ME) may be more commercially viable. Smith et al. (1999)
showed that levels near 4.35 g of total Lys/Mcal of ME were necessary to maximize growth performance. Meanwhile, Roth et al. (1999)
suggested that levels of 0.90 g of total Lys/MJ of ME (3.68 g of total Lys/Mcal of ME) were necessary to maximize performance from 20 to 30 kg. The requirement estimates derived in the current experiment lie between these other estimates, but all are consistently greater than NRC (1998)
estimates of 3.52 g of total Lys/Mcal of ME for 10- to 20-kg pigs. The NRC (1998)
estimates for 10- to 20-kg pigs were derived from titration experiments by Thaler et al. (1986)
, Campbell et al. (1988)
, Weaver et al. (1988)
, Gatel et al. (1992)
, Gatel and Fékéte (1989)
, Nam and Aherne (1994)
, Kornegay et al. (1993)
, and Gahl et al. (1994)
. Among these estimates, those of Gatel et al. (1992)
and Nam and Aherne (1994)
are the greatest, with estimates of 1.08 and 0.95 g of total Lys/MJ of DE, or 4.66 and 4.14 g of total Lys/Mcal of ME, respectively, when converted to an ME basis, as above. Although the data of Nam and Aherne (1994)
provided a value similar to the estimate of 4.21 g of total Lys/Mcal of ME in the current research, Gatel et al. (1992)
suggested much greater requirements, but their estimate represents the asymptote of the quadratic function, and is similar to our maximal quadratic estimate of 4.68 g of total Lys/Mcal of ME.
Although many of the studies cited above were conducted by using total Lys values, the current study was evaluated by using calculated TID Lys values (NRC, 1998
). Because digestibility values existed for each ingredient, this method was preferred as a more accurate depiction of the true Lys needs of the pig. Campbell et al. (1988)
discussed the pitfalls of comparing studies based on total AA levels with ingredients that lack digestibility values. Admittedly, the current studies have relied on accurate true digestibility estimates. Buraczewska et al. (1999)
demonstrated that this method does not necessarily result in an equivalent pig response, because availability estimates may be overestimated in heat-treated ingredients.
Urynek and Buraczewska (2003)
recently published estimates of the AID Lys requirements to maximize N retention and ADG as being 0.85 g of AID Lys/MJ of ME (3.55 g of AID Lys/Mcal of ME) from 13 to 20 kg and 0.77 g of AID Lys/MJ of ME (3.22 g of AID Lys/MJ of ME) from 20 to 30 kg. The acknowledgment of separate requirements during this rather narrow BW range is only slightly lower, but is similar to our findings of up to 1.33% TID Lys for ADG and 1.35% TID Lys (3.74 g of AID Lys/Mcal of ME) from 11 to 19 kg. These estimates are still greater than the 2.88 g of AID Lys/Mcal of ME for 10- to 20-kg pigs suggested by NRC (1998)
.
Feed intake was unaffected by the dietary Lys level in the current study. This is consistent with other experiments in which graded levels of supplemental AA were used to increase dietary Lys (Gatel et al., 1992
; Nam and Aherne, 1994
; Urynek and Buraczewska, 2003
). In contrast, in experiments in which intact protein sources are used to increase dietary Lys, this has resulted in linear decreases in feed intake (van Lunen and Cole, 1998
; Smith et al., 1999
). The excess of dietary protein creates a greater demand on the body to deaminate and excrete excess nitrogen and may contribute to decreasing feed intake. Because adding crystalline Lys to the diet does not influence feed intake to any large degree, it may be a superior method of determining the Lys requirement for nursery pigs.
As genotypes continue to increase the protein:lipid deposition ratio of growth, work will continue in this area to define the AA needs of nursery pigs. Nutrition researchers must, however, document the genetic capacity of their animals chemically so that the relevance of their requirements can be established. In addition, to simplify the evaluation of studies, it is worth considering a standardization of statistical analysis of optimal feeding levels. Similar recommendations were suggested by Campbell et al. (1988)
in performing requirement studies on an available AA basis. This might provide additional needed clarity to producers wanting to optimize nutrition programs.
From these data, it appears that NRC (1998)
has underestimated the Lys needs of 11- to 27-kg pigs. For this BW range, feeding levels of at least 1.30% TID Lys or 19 g of TID Lys/kg of gain are necessary to maximize performance under commercial conditions. Expressed relative to dietary energy, this equates to 3.81 g of TID Lys/Mcal of ME. For pigs weighing 11 to 19 kg, 1.35% TID Lys is necessary to maximize performance under commercial conditions.
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Footnotes
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1 Appreciation is extended to Ajinomoto Heartland LLC for financial support. 
2 Current address: Murphy-Brown LLC, PO Box 759, Rose Hill, NC 28458. 
3 Current address: The Maschhoffs Inc., 7475 State Route 127, Carlyle, IL 62231. 
4 Corresponding author: alleeg{at}missouri.edu
Received for publication February 9, 2007.
Accepted for publication November 8, 2007.
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