Darmani Kuhi, H., López, S., Shabanpour, A., Mohit, A., Falahi, S., France, J. (2019). Application of Sinusoidal Equations to Partitioning Crude Protein and Metabolizable Energy Intake between Maintenance and Growth in Parent Stock of Broiler Chickens. Iranian Journal of Applied Animal Science, 9(2), 299-308.

H. Darmani Kuhi; S. López; A. Shabanpour; A. Mohit; S. Falahi; J. France. "Application of Sinusoidal Equations to Partitioning Crude Protein and Metabolizable Energy Intake between Maintenance and Growth in Parent Stock of Broiler Chickens". Iranian Journal of Applied Animal Science, 9, 2, 2019, 299-308.

Darmani Kuhi, H., López, S., Shabanpour, A., Mohit, A., Falahi, S., France, J. (2019). 'Application of Sinusoidal Equations to Partitioning Crude Protein and Metabolizable Energy Intake between Maintenance and Growth in Parent Stock of Broiler Chickens', Iranian Journal of Applied Animal Science, 9(2), pp. 299-308.

Darmani Kuhi, H., López, S., Shabanpour, A., Mohit, A., Falahi, S., France, J. Application of Sinusoidal Equations to Partitioning Crude Protein and Metabolizable Energy Intake between Maintenance and Growth in Parent Stock of Broiler Chickens. Iranian Journal of Applied Animal Science, 2019; 9(2): 299-308.

Application of Sinusoidal Equations to Partitioning Crude Protein and Metabolizable Energy Intake between Maintenance and Growth in Parent Stock of Broiler Chickens

^{1}Department of Animal Science, Faculty of Agricultural Science, University of Guilan, Rasht, Iran

^{2}Departamento de Producción Animal, Instituto de Ganadería de Montaña, Universidad de León, E-24007 León, Spain

^{3}Department of Mathematics, Salman Farsi University of Kazerun, Kazerun, Iran

Receive Date: 03 April 2018,
Revise Date: 12 August 2018,
Accept Date: 15 August 2018

Abstract

Most models developed for poultry are linear to the point where genetic potential is reached. Models reliant on the premise that growth rate determines requirements based on some fixed rate of nutrient utilization do not adequately represent the biological phenomena involved. Therefore, a dichotomy between the accepted theories of nutrient utilization in animals and the assumptions of mathematical models to predict and analyze those requirements is evident. Since, responses of animals to dietary energy, protein and amino acids are curvilinear phenomena, they should be evaluated as such to estimate optimum economic levels, rather than as biological maxima. The objective of this study was to apply two sinusoidal functions exhibiting curvilinear behaviour to estimate metabolizable energy (ME) and crude protein (CP) requirements for maintenance and growth in parent stock of boiler chickens. The functions were fitted by non-linear regression to estimate the parameters, from which other biological indicators were calculated. The results of fitting the functions to data sets and their statistical performance and the biological interpretability of the parameter estimates showed the models’ capability in describing the relationship between body weight (BW) gain and ME (or CP) intake in parent stock of broiler chickens. The estimated maintenance requirements and the determined values of ME and CP requirements for BW gain were consistent with values reported previously by other researchers.

Growth is a fundamental property of biological systems and can be defined as an increase in body size per unit time. Understanding of the economic importance of various traits such as live weight, weight gain, rate of maturity, age and live weight at which maximal growth occurs has led researchers to carry out detailed studies targeting weight-age relationships (Ersoy el al. 2006). Experiments designed to investigate the effect of dietary nutrient concentrations on growth and development of pullets are relatively long term and expensive to conduct. As costs of research increase, mathematical models become more valuable tools to answer research and development questions. Growth functions can be used to determine the efficiency of nutrient utilization, which is the derivative of the relationship between body weight (BW) and dietary nutrient intake and as response functions to predict daily energy, protein and amino acids requirements for maintenance and growth (France et al. 1989; Darmani Kuhi et al. 2009; Darmani Kuhi et al. 2011). Several models have been suggested to predict metabolizable energy (ME) and crude protein (CP) utilization in pigs and poultry. Among these, the partitioning model has been the most promising for imminent application (Sakomura, 2004).The models developed for poultry mostly have dealt with the simulation of responses in a single bird. Such responses usually are fairly linear to the point where the genetic potential is reached but poultry nutritionists’ interests lie in population responses which are invariably curvilinear. Descriptions of such responses, whilst taking account of marginal costs and revenues, are therefore invaluable in determining how to maximize or minimize the objective function chosen for any given commercial operation (Gous, 2007). Therefore, a dichotomy between accepted theories of nutrient utilization in animals and assumptions of mathematical models to predict and analyze those requirements is evident. For instance, equations to predict nutrient utilization usually assume that the nutrients required per unit of growth or egg production are constant. However, nutrient partitioning for weight gain or egg production changes as a function of age, feeding level, composition and efficiency of retention (Kielauowski, 1965; Chwalibog, 1992; Romero et al. 2009). Though there are many studies aiming at evaluating growth models in animals, the number of studies targeting growth models in parent stock of broiler chickens is quite limited in comparison with other poultry species. Non-linear modelling has been applied to partition the metabolizable energy, protein and amino acids intake into requirements for maintenance and growth in ruminants and in broiler chickens and turkeys (France et al. 1989; Darmani Kuhi et al. 2003; Darmani Kuhi et al. 2004; Darmani Kuhi et al. 2009; Darmani Kuhi et al. 2010; Darmani Kuhi et al. 2011; Kebreab et al. 2008). The objective of this study is to apply and compare sinusoidal equations to estimate ME and CP requirements for maintenance and growth in parent stock of boiler chickens and to assess their application.

MATERIALS AND METHODS

Data source

A total of 3 time course profiles obtained from results published in the Aviagen Management Guide to growing parent stock-type pullets (Aviagen, 2017) were used in this study in two separate analyses. A detailed description of the data used is summarized in Table 1.

Data source

A total of 3 time course profiles obtained from results published in the Aviagen Management Guide to growing parent stock-type pullets (Aviagen, 2017) were used in this study in two separate analyses. A detailed description of the data used is summarized in Table 1.

The models

Three growth functions (equations 1, 2 and 3) were used in this study to investigate the relationship between scaled daily weight gain (g/d per g BW) and ME (kJ/d per g BW) and CP (g/d per g BW) intake in parent stock-type pullets. The growth functions used take the form:

For the monomolecular equation a and b are the maximum and minimum attainable values for, respectively, c is a fractional rate parameter and x is ME or CP intake. Details about parameters a, b and c of the modified sinusoidal and 4-parameter sinusoidal equations are illustrated in Figures 1 and 2. The equations were fitted to the data and the parameters estimated. From these parameter estimates, ME and CP requirements for maintenance [ME_{m} (kJ/d per g BW) and CP_{m }(g/d per g BW), where BW gain= 0] were calculated from equations 4, 5 and 6:

The first derivatives of equations, 1, 2 and 3, which estimate the instantaneous efficiency of ME (k_{gME}, g of BW gain/kJ ME intake) and CP (k_{gCP}, g of BW gain/g CP intake) utilization for gain, are given by equations 7, 8 and 9, respectively:

and the average efficiency between maintenance and D times maintenance (D>1) by:

Body weight gain, ME and CP intake were calculated for each data profile as follows:

BW gain= (the difference between initial BW and BW of pullets at the end of each week) / [ × age (d)]

ME or CP intake= ME (or CP) intake for specific week / [× age (d)]

Where:

: average BW.

Average BW was calculated as the mean of initial weight and weights at the end of each week.

Statistical procedures

The nonlinear regression procedure of SigmaPlot software (SigmaPlot 12.0) was implemented for the analysis of the data sets (Systat Software Inc., 2011). The adequacy of the models was assessed by model behaviour when fitting the curves and evaluating their statistical performance and the biological interpretability of the parameter estimates. Bayesian information criterion (BIC), variance of error estimate and adjusted coefficient of determination (adjusted R^{2}) were used to evaluate the general goodness-of-fit of each model to the data profiles.

RESULTS AND DISCUSSION

The results of fitting the equations to each data set (Figures 3 and 4), and their statistical performance and the biological interpretability of the parameter estimates and derived indicators (Tables 2, 3, 4 and 5), show the capability of the models in describing the relationship between ME (or CP) intake and BW gain in parent stock of broiler chickens. Estimated ME and CP requirements for maintenance (316 to 399 kJ/d per kg BW for ME and 2.65 to 3.74 g/d per kg BW for CP) and average ME and CP utilization for producing gain in BW calculated between 1 and 4 times maintenance (from 10.85 to 13.62 kJ/g BW for ME and 0.55 to 0.69 for CP) (Tables 4 and 5), lie in the range reported by previous researchers (Leeson et al. 1973; Emmans, 1974; Johnson and Farrell, 1983; NRC, 1994; Wiseman, 1994; Sakomura, 2004).

Table 1 Data profiles used for the study

FPS: female parent stock; MPS: male parent stock; BW: body weight; FI: feed intake; CP: dietary crude protein content and ME: dietary metabolizable energy concentration.

Table 2 Parameter estimates obtained using the different models to regress body weight gain (g/d per g BW) against metabolizable energy intake (kJ/d per g BW), standard errors are given in brackets

^{† }Variance of error estimate.

^{‡} BIC: Bayesian information criteria.

^{§ }Adjusted.

^{¥} 3 and 4-parameter sinusoidal (see Figures 1 and 2 for explanations of parameters a, b, c, x_{0}, y_{0}).

Table 3 Parameter estimates obtained using the different models to regress body weight gain (g/d per g BW) against crude protein intake (g/d per g BW), standard errors are given in brackets

^{† }Variance of error estimate.

^{‡} BIC: Bayesian information criteria.

^{§ }Adjusted.

^{¥} 3 and 4-parameter sinusoidal (see Figures 1 and 2 for explanations of parameters a, b, c, x_{0}, y_{0}).

The values of average energy and protein requirements (Tables 4 and 5) for each additional unit of BW gain are lower at low intake levels and increase as intake is increased. These results are supported by conventional wisdom, namely that a gradual decrease in efficiency of nutrient utilization for producing gain occurs as intake increases (Gahl et al. 1994; Fatufe and Rodehutscord, 2005; Romero et al. 2009). This phenomenon is partly due to a slight fall in digestive efficiency of the animal with increased feeding level and partly to the fact that anabolic processes are less efficient than catabolic ones. Protein turnover is higher as protein intake is increased because excess in amino acid supply causes a rise in amino acid degradation rates (Riis, 1983a; Riis, 1983b; Pannemans et al. 1995).

Table 4 Growth traits calculated from parameter estimates obtained using the different models to regress body weight gain (g/d per g BW) against metabolizable energy intake (kJ/d per g BW)

^{* }ME_{m}= ME for maintenance.

^{† }The values of average net energy requirement for each additional unit of BW gain (kJ/g BW gain) between 1-4, 1-2, 2-3 and 3-4 times maintenance calculated based on the assumption that the average efficiency of utilization of ME for growth is approximately 70% for balanced diets in poultry (McDonald et al. 2002).

^{§} 3 and 4-parameter sinusoidal.

Table 5 Growth traits calculated from parameter estimates obtained using the different models to regress body weight gain (g/d per g BW) against crude protein intake (g/d per g BW)

^{* }CP_{m}= CP at maintenance.

^{†} The average percentage of net protein utilization for growth between 1-4 times maintenance calculated as based on the assumption that the carcass of chicken contains approximately 18% crude protein.

^{‡ }Gain in BW in response to CP intake (g BW gain/g CP intake) between 1-4, 1-2, 2-3 and 3-4 times maintenance. The values presented in the brackets are the average protein requirements for growth (g CP/g BW).

^{§} 3 and 4-parameter sinusoidal.

Therefore, successive increments of daily intake of nutrients result in progressively smaller increments in daily gain (Blaxter and Boyne, 1978). Despite statistically better performance of the monomolecular equation, the performance of sinusoidal equations based on both statistical performance (Tables 2 and 3) and biological interpretability of the parameter estimates (Tables 4 and 5) was acceptable. Predicting responses of poultry to nutrients has been the goal of nutritionists and modellers for a long time because of their importance in formulating diets for poultry and in making economic decisions. Growth is a continuous function over the animal’s life, from embryonic stages up to adulthood and it is mathematically explained by growth models that have parameters with biological meaning (Fitzhugh, 1976).

Figure 1 Graph of the modified sinusoidal equation showing its fit to the data of in season females of parent stock of Ross 308

Where:

a: height of each peak above the baseline equal to the maximum attainable value for BW gain

x_{0}: phase shift (the horizontal offset of the base point, where the curve crosses the baseline as it ascends)

This sinusoidal equation is periodic with period 2 b

Figure 2 Graph of 4- parameter sinusoidal equation showing its fit to the data of in season females of parent stock of Ross 308

Where:

a: amplitude (the height of each peak above the baseline)

: vertical offset (height of the baseline). When is changed, the basic sinusoidal function is shifted vertically by units

:

approximation of the maximum attainable value for BW gain and c is the phase shift (the horizontal offset of the base point, where the curve crosses the baseline as it ascends)

This sinusoidal equation is periodic with period b

These parameters are used to describe growth to estimate the expected weight of individuals at specific age (Yakupoglu and Atil, 2001). It is possible to use mathematical models to identify better strategies to improve livestock production also estimating daily nutrient requirements of animals at different ages (Pomar et al. 2009). Moreover, growth curve parameters are highly heritable and widely used in selection studies (Grossman and Bohren, 1985; Mignon-Grasteau et al. 2000). Growth curves can be used for pre-selection of animals as it provides prediction of future growth at any age. Brody (1945) suggested that the asymptotic or mature weight, rate of attainment of mature weight and the standardized age at which an animal attained the inflexion point of the curve are parameters that could be manipulated by geneticists (Raji et al. 2014). Different mathematical growth models have been used to define growth curves. According to Thornley and France (2007), the most commonly used functions to estimate animal growth include: Brody (Brody, 1945), von Bertalanffy (Bertalanffy, 1957), Richards (Richards, 1959), logistic (Nelder, 1961) and Gompertz (Laird, 1965). The response of nutrient retention to nutrient inputs is usually represented rectilinearly with an abrupt cut-off. The data, however, may support this or be more suggestive of a curvilinear response. Since, under controlled conditions, the slope of the curve describing the relationship between nutrient retention and nutrient input represents the quality of the nutrient fed (biological value, net protein and energy utilization, nitrogen balance index, etc.), the assumption that the relationship is linear has tended to be adopted. Most data are linear to a reasonable approximation, but a curvilinear response is probably a more precise interpretation (Boorman and Burgess, 1980).

Figure 3 Plot of body weight gain (BWG, g/d per g BW) against metabolizable energy intake (MEI, kJ/d per g BW) for the functional forms: (M.) monomolecular, (sin 3) 3- parameter sinusoidal and (sin 4) 4- parameter sinusoidal

The letters for FPSin and FPSout indicate fit of equation to data from in and out of season females of parent stock of Ross 308 broiler chicks, respectively

Letters MPS stands for male parent stock of broiler chicks

Models based on the premise that growth rate determines requirements based on some fixed rate of nutrient utilization do not adequately represent the biological phenomena involved. Since responses of animals to dietary energy, protein and amino acids are often diminishing returns phenomena, they should be evaluated as such to estimate optimum economic levels, rather than as biological maxima (Pesti and Miller, 1997). However, the law of diminishing returns precludes an increasing slope over any segment of the response curve and might not be appropriate in all situations. Therefore, alternative equations, such as sinusoidal equations, for use when the law of diminishing returns does not apply could be beneficial.

Figure 4 Plot of body weight gain (BWG, g/d per g BW) against crude protein intake (CPI, g/d per g BW) for the functional forms: (M.) monomolecular, (sin 3) 3- parameter sinusoidal and (sin 4) 4- parameter sinusoidal

The letters for FPSin and FSPout indicate fit of equation to data from in and out of season females of parent stock of Ross 308 broiler chicks, respectively

Letters MPS stands for male parent stock of broiler chicks

CONCLUSION

To the best of our knowledge, this study is the first time that the sinusoidal equations have been applied and evaluated in poultry nutrition to partition nutrient intakes between requirements for maintenance and growth. The models described herein are considered advantageous because they were able to predict the magnitude and direction of the responses of growing parent stock-type pullets to dietary ME and CP without making any initial assumptions. Also, the models have the advantage of biological interpretability of the parameter estimates and the measures derived from them.

ACKNOWLEDGEMENT

We are grateful to University of Guilan, Rasht, Iran for its financial support.

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