Comparative evaluation of mathematical functions to describe growth and efficiency of phosphorus utilization in growing pigs

Success of pig production depends on maximizing return over feed costs and addressing potential nutrient pollution to the environment. Mathematical modeling has been used to describe many important aspects of inputs and outputs of pork production. This study was undertaken to compare 4 mathematical functions for the best fit in terms of describing specific data sets on pig growth and, in a separate experiment, to compare these 4 functions for describing of P utilization for growth. Two data sets with growth data were used to conduct growth analysis and another data set was used for P efficiency analysis. All data sets were constructed from independent trials that measured BW, age, and intake. Four growth functions representing diminishing returns (monomolecular), sigmoidal with a fixed point of inflection (Gompertz), and sigmoidal with a variable point of inflection (Richards and von Bertalanffy) were used. Meta-analysis of the data was conducted to identify the most appropriate functions for growth and P utilization. Based on Bayesian information criteria, the Richards equation described the BW vs. age data best. The additional parameter of the Richards equation was necessary because the data required a lower point of inflection (138 d) than the Gompertz, with a fixed point of inflexion at 1/e times the final BW (189 d), could accommodate. Lack of flexibility in the Gompertz equation was a limitation to accurate prediction. The monomolecular equation was best at determining efficiencies of P utilization for BW gain compared with the sigmoidal functions. The parameter estimate for the rate constant in all functions decreased as available P intake increased. Average efficiencies during different stages of growth were calculated and offer insight into targeting stages where high feed (nutrient) input is required and when adjustments are needed to accommodate the loss of efficiency and the reduction of potential pollution problems. It is recommended that the Richards and monomolecular equations be included in future growth and nutrient efficiency analyses.

Meta-analysis of phosphorus balance data from growing pigs

Many studies have highlighted concerns over current methods of determining endogenous P losses and P requirements in growing pigs. Therefore, a database containing observations on 350 pigs was assembled from various studies. Four functions for analyzing P balance data were considered: 1) a straight line, 2) a diminishing returns function (monomolecular), 3) a sigmoidal function with a fixed point of inflection (Gompertz), and 4) a sigmoidal function with a flexible point of inflection (Richards). The nonlinear functions were specifically reparameterized to assign biological meaning to the parameters. Meta-analysis of the data was conducted to estimate endogenous P excretion, maintenance requirement, and efficiency of utilization. Phosphorus retention was regressed against either available P intake or total P intake [all variables scaled by metabolic BW (BW(0.75))]. There was evidence of non-linearity in the data, and the monomolecular function provided the best fit to the data. The Richards equation did not fit the data well and appeared overparameterized. Estimates of endogenous P excretion of 14 and 17 mg/kg of BW(0.75) x d based on available and total P analysis, respectively, were predicted by the monomolecular equation, which were within the range reported in the literature. Maintenance requirement values of 15 mg of available P/kg of BW(0.75) x d and 37 mg of total P/kg of BW(0.75) x d were obtained, based on the monomolecular equation. Average efficiencies of conversion of dietary P to retained P were 65 and 36% for available and total P, respectively, with greater efficiency values calculated for low P intakes. Although the monomolecular equation fitted the data best, more observations at high P intakes/kg of BW(0.75) are required to determine conclusively whether P retention scaled by metabolic BW is linearly related to available or total P intake.

Evaluation of models to describe ruminal degradation kinetics from in situ ruminal incubation of whole soybeans

Different mathematical models were evaluated as candidates to describe ruminal dry matter (DM) and crude protein (CP) degradation kinetics of raw and roasted whole soybeans from data obtained using the in situ polyester bag technique. Three models were used: segmented with up to 3 straight lines (model I), negative exponential (model II), and rational function or inverse polynomial (linear over linear; model III). A fourth, a generalized sigmoidal model, was also considered but the data did not exhibit sigmoidicity, so it was dropped from the analysis. Lagged and nonlagged versions of each model were fitted to the DM and CP disappearance curves of 6 different feeds (2 cultivars of raw or differently heat-processed whole soybean). The comparison between lagged and nonlagged versions of each model, based on statistical and behavior characteristics, showed for all models that the discrete lag parameter did not significantly improve the fit to ruminal DM and CP disappearance curves. The comparison between models (using nonlagged equations) showed that models I and II gave better goodness-of-fit than model III. Based on biological characteristics, models II and III underestimated the undegradable DM and CP fractions, but there was no significant difference between models for extent of degradation.

Methane and nitrous oxide emissions from Canadian animal agriculture: A review

Considerable evidence of climate change associated with emissions of greenhouse gases (GHG) has resulted in international efforts to reduce GHG emissions. The agriculture sector contributes about 8% of GHG emissions in Canada mostly through methane (CH4) and nitrous oxide (N2O). The objective of this paper was to compile an integrative review of CH4 and N2O emissions from livestock by taking a whole cycle approach from enteric fermentation to manure treatment and storage, and field application of manure. Basic microbial processes that result in CH4 production in the rumen and hindgut of animals were reviewed. An overview of CH4 and N2O production processes in manure, and controlling factors are presented. Most of the studies conducted in relation to enteric fermentation were in dairy and beef cattle. To date, research has focussed on GHG emissions from the stored manures of dairy, beef cattle and swine; therefore, we focus our review on these. Several methods used to measure GHG emissions from livestock and stored manure were reviewed. A comparison of methods showed that there were agreements between most of the techniques but some systematic differences were also observed. Additional studies with comprehensive comparisons of methodologies are needed in order to allow for comparison of results obtained from studies using contrasting methodologies. The need to standardize measurement methods and reporting to facilitate comparison of results and data integration was identified. Prediction equations are often used to calculate GHG emissions. Various types of mathematical approaches, such as statistical models, mechanistic models and estimates calculated from emission factors, and studies that compare various types of models are discussed herein. A lack of process-based models describing GHG emissions from manure during storage was identified. A brief description of mitigation strategies focussing on recent studies is given. Reduction in CH4 emissions from ruminants through the addition of fats in diets and the use of more starch was achieved and a transient beneficial effect of ionophores was reported. Grazing management and genetic selection also hold promise. Studies focussed on manure treatment options that thave been suggested to reduce gas fluxes from manure storage, composting, anaerobic digestion (AD), diet manipulation, covers and solid-liquid separation, were reviewed. While some of these options have been shown to decrease GHG emissions from stored manure, different studies have obtained conflicting results, and additional research is needed to identify the most promising options. GHG emissions from pasture and croplands after manure application have been the subject of several experimental and modelling studies, but few of these have linked field emissions to diet manipulation or manure treatments. Further work focussing on the entire cycle of GHG formation from feed formulation, animal metabolism, excreta treatment and storage, to field application of manure needs to be conducted. Key words: Greenhouse gases, enteric methane, nitrous oxide, manure management

Modeling Small-Scale Dairy Farms in Central Mexico Using Multi-Criteria Programming

Milk supply from Mexican dairy farms does not meet demand and small-scale farms can contribute toward closing the gap. Two multi-criteria programming techniques, goal programming and compromise programming, were used in a study of small-scale dairy farms in central Mexico. To build the goal and compromise programming models, 4 ordinary linear programming models were also developed, which had objective functions to maximize metabolizable energy for milk production, to maximize margin of income over feed costs, to maximize metabolizable protein for milk production, and to minimize purchased feedstuffs. Neither multi-criteria approach was significantly better than the other; however, by applying both models it was possible to perform a more comprehensive analysis of these small-scale dairy systems. The multi-criteria programming models affirm findings from previous work and suggest that a forage strategy based on alfalfa, ryegrass, and corn silage would meet nutrient requirements of the herd. Both models suggested that there is an economic advantage in rescheduling the calving season to the second and third calendar quarters to better synchronize higher demand for nutrients with the period of high forage availability.

Direct chemical synthesis of chiral methanol of 98% ee and its conversion to [(2)H1,(3)H]methyl tosylate and [(2)H1,(3)H-methyl]methionine

This paper describes the synthesis of chiral methanols [(R)- and (S)-CHDTOH] in a total of 12 steps starting from (chloromethyl)dimethylphenylsilane. The metalated carbamates derived from (dimethylphenylsilyl)methanol and secondary amines were borylated at low temperatures (-78 or -94 degrees C) using borates derived from tert-butyl alcohol and (+)-pinane-2,3-diol or (R,R)-1,2-dicyclohexylethane-1,2-diol to give diastereomeric boronates (dr 1:1 to 5:1). The carbamoyloxy group could be replaced smoothly with inversion of configuration by an isotope of hydrogen using LiAlH(D)4 [or LiBEt3H(D,T)]. If the individual diastereomeric boronates were reduced with LiAlD4 and oxidized with H2O2/NaHCO3, monodeuterated (dimethylphenylsilyl)methanols of ee > 98% resulted. The absolute configurations of the boronates were based on a single-crystal X-ray structure analysis. Brook rearrangement of the enantiomers of (dimethylphenylsilyl)-[(2)H1,(3)H]methanol prepared similarly furnished the chiral methanols which were isolated as 3,5-dinitrobenzoates in 81% and 90% yield, respectively. For determination of the enantiomeric excesses (98%), the methyl groups were transferred to the nitrogen of (S)-2-methylpiperidine and (3)H{(1)H} NMR spectra were recorded. The Brook rearrangement is a stereospecific process following a retentive course. The chiral methanols were also transformed into methyl tosylates used to prepare [(2)H1,(3)H-methyl]methionines in high overall yields (>80%).

Estimation of the stoichiometry of volatile fatty acid production in the rumen of lactating cows

The purpose of this study was to improve the prediction of the quantity and type of Volatile Fatty Acids (VFA) produced from fermented substrate in the rumen of lactating cows. A model was formulated that describes the conversion of substrate (soluble carbohydrates, starch, hemi-cellulose, cellulose, and protein) into VFA (acetate, propionate, butyrate, and other VFA). Inputs to the model were observed rates of true rumen digestion of substrates, whereas outputs were observed molar proportions of VFA in rumen fluid. A literature survey generated data of 182 diets (96 roughage and 86 concentrate diets). Coefficient values that define the conversion of a specific substrate into VFA were estimated meta-analytically by regression of the model against observed VFA molar proportions using non-linear regression techniques. Coefficient estimates significantly differed for acetate and propionate production in particular, between different types of substrate and between roughage and concentrate diets. Deviations of fitted from observed VFA molar proportions could be attributed to random error for 100%. In addition to regression against observed data, simulation studies were performed to investigate the potential of the estimation method. Fitted coefficient estimates from simulated data sets appeared accurate, as well as fitted rates of VFA production, although the model accounted for only a small fraction (maximally 45%) of the variation in VFA molar proportions. The simulation results showed that the latter result was merely a consequence of the statistical analysis chosen and should not be interpreted as an indication of inaccuracy of coefficient estimates. Deviations between fitted and observed values corresponded to those obtained in simulations.

Efficient Synthesis of [3H]-Sanglifehrin A via Selective Oxidation/Reduction of Alcohols at C31 and C35

[Reaction: see text]. Sanglifehrin A is a novel complex natural product showing strong immunosuppressive activity and remarkably high affinity for cyclophilin A. To assess its pharmacokinetic properties in vivo, an efficient synthetic route was developed to introduce a tritium label in position C35 of sangliferin A via an oxidation/reduction strategy. The synthetic approach is particularly attractive, because the C35-oxo intermediate 7 is available in good yield on large scale and the reducing agent, lithium tri-sec-butylborotritide, is readily available. An attempt to apply a similar strategy to the alcohol in position C31 led primarily to C31-epi-hydroxy sanglifehrin A under a variety of conditions.

Effects of phosphorus intake on phosphorus flow in growing pigs: Application and comparison of two models

A comparison of the models of Vitti et al. (2000, J. Anim. Sci. 78, 2706-2712) and Fernández (1995c, Livest. Prod. Sci. 41, 255-261) was carried out using two data sets on growing pigs as input. The two models compared were based on similar basic principles, although their aims and calculations differed. The Vitti model employs the rate:state formalism and describes phosphorus (P) flow between four pools representing P content in gut, blood, bone and soft tissue in growing goats. The Fernández model describes flow and fractional recirculation between P pools in gut, blood and bone in growing pigs. The results from both models showed similar trends for P absorption from gut to blood and net retention in bone with increasing P intake, with the exception of the 65 kg results from Date Set 2 calculated using the Fernández model. Endogenous loss from blood back to gut increased faster with increasing P intake in the Fernández than in the Vitti model for Data Set 1. However, for Data Set 2, endogenous loss increased with increasing P intake using the Vitti model, but decreased when calculated using the Fernández model. Incorporation of P into bone was not influenced by intake in the Fernández model, while in the Vitti model there was an increasing trend. The Fernández model produced a pattern of decreasing resorption in bone with increasing P intake, with one of the data sets, which was not observed when using the Vitti model. The pigs maintained their P homeostasis in blood by regulation of P excretion in urine.

Statistical evaluation of mathematical models for microbial growth

The aim of this study was to evaluate the suitability of several mathematical functions for describing microbial growth curves. The nonlinear functions used were: three-phase linear, logistic, Gompertz, Von Bertalanffy, Richards, Morgan, Weibull, France and Baranyi. Two data sets were used, one comprising 21 growth curves of different bacterial and fungal species in which growth was expressed as optical density units, and one comprising 34 curves of colony forming units counted on plates of Yersinia enterocolitica grown under different conditions of pH, temperature and CO(2) (time-constant conditions for each culture). For both sets, curves were selected to provide a wide variety of shapes with different growth rates and lag times. Statistical criteria used to evaluate model performance were analysis of residuals (residual distribution, bias factor and serial correlation) and goodness-of-fit (residual mean square, accuracy factor, extra residual variance F-test, and Akaike's information criterion). The models showing the best overall performance were the Baranyi, three-phase linear, Richards and Weibull models. The goodness-of-fit attained with other models can be considered acceptable, but not as good as that reached with the best four models. Overall, the Baranyi model showed the best behaviour for the growth curves studied according to a variety of criteria. The Richards model was the best-fitting optical density data, whereas the three-phase linear showed some limitations when fitting these curves, despite its consistent performance when fitting plate counts. Our results indicate that the common use of the Gompertz model to describe microbial growth should be reconsidered critically, as the Baranyi, three-phase linear, Richards and Weibull models showed a significantly superior ability to fit experimental data than the extensively used Gompertz.

Study of the Lactation Curve in Dairy Cattle on Farms in Central Mexico

Accurate knowledge of lactation curves has an important relevance to management and research of dairy production systems. A number of equations have been proposed to describe the lactation curve, the most widely applied being the gamma equation. The objective of this work was to compare and evaluate candidate functions for their predictive ability in describing lactation curves from central Mexican dairy cows reared under 2 contrasting management systems. Five equations were considered: Gaines (exponential decay), Wood (gamma equation), Rook (Michaelis-Menten xexponential), and 2 more mechanistic ones (Dijkstra and Pollott). A database consisting of 701 and 1283 records of cows in small-scale and intensive systems, respectively, was used in the analysis. Before analysis, the database was divided into 6 groups representing first, second, and third and higher parity cows in both systems. In all cases except second and above parity cows in small-scale systems, all models improved on the Gaines equation. The Wood equation explained much of the variation, but its parameters do not have direct biological interpretation. Although the Rook equation fitted the data well, some of the parameter estimates were not significant. The Dijkstra equation consistently gave better predictions, and its parameters were usually statistically significant and lend themselves to physiological interpretation. As such, the differences between systems and parity could be explained due to variations in theoretical initial milk production at parturition, specific rates of secretory cell proliferation and death, and rate of decay, all of which are parameters in the model. The Pollott equation, although containing the most biology, was found to be over-parameterized and resulted in nonsignificant parameter estimates. For central Mexican dairy cows, the Dijkstra equation was the best option to use in describing the lactation curve.

An integrative model of amino acid metabolism in the liver of the lactating dairy cow

The objective of this work was to construct a dynamic model of hepatic amino acid metabolism in the lactating dairy cow that could be parameterized using net flow data from in vivo experiments. The model considers 22 amino acids, ammonia, urea, and 13 energetic metabolites, and was parameterized using a steady-state balance model and two in vivo, net flow experiments conducted with mid-lactation dairy cows. Extracellular flows were derived directly from the observed data. An optimization routine was used to derive nine intracellular flows. The resulting dynamic model was found to be stable across a range of inputs suggesting that it can be perturbed and applied to other physiological states. Although nitrogen was generally in balance, leucine was in slight deficit compared to predicted needs for export protein synthesis, suggesting that an alternative source of leucine (e.g. peptides) was utilized. Simulations of varying glucagon concentrations indicated that an additional 5 mol/d of glucose could be synthesized at the reference substrate concentrations and blood flows. The increased glucose production was supported by increased removal from blood of lactate, glutamate, aspartate, alanine, asparagine, and glutamine. As glucose output increased, ketone body and acetate release increased while CO(2) release declined. The pattern of amino acids appearing in hepatic vein blood was affected by changes in amino acid concentration in portal vein blood, portal blood flow rate and glucagon concentration, with methionine and phenylalanine being the most affected of essential amino acids. Experimental evidence is insufficient to determine whether essential amino acids are affected by varying gluconeogenic demands.

Technical note: A proposed method to determine the extent of degradation of a feed in the rumen from the degradation profile obtained with the in vitro gas production technique using feces as the inoculum1

A method is proposed to determine the extent of degradation in the rumen involving a two-stage mathematical modeling process. In the first stage, a statistical model shifts (or maps) the gas accumulation profile obtained using a fecal inoculum to a ruminal gas profile. Then, a kinetic model determines the extent of degradation in the rumen from the shifted profile. The kinetic model is presented as a generalized mathematical function, allowing any one of a number of alternative equation forms to be selected. This method might allow the gas production technique to become an approach for determining extent of degradation in the rumen, decreasing the need for surgically modified animals while still maintaining the link with the animal. Further research is needed before the proposed methodology can be used as a standard method across a range of feeds.

Alternative approaches to predicting methane emissions from dairy cows1

Previous attempts to apply statistical models, which correlate nutrient intake with methane production, have been of limited value where predictions are obtained for nutrient intakes and diet types outside those used in model construction. Dynamic mechanistic models have proved more suitable for extrapolation, but they remain computationally expensive and are not applied easily in practical situations. The first objective of this research focused on employing conventional techniques to generate statistical models of methane production appropriate to United Kingdom dairy systems. The second objective was to evaluate these models and a model published previously using both United Kingdom and North American data sets. Thirdly, nonlinear models were considered as alternatives to the conventional linear regressions. The United Kingdom calorimetry data used to construct the linear models also were used to develop the three nonlinear alternatives that were all of modified Mitscherlich (monomolecular) form. Of the linear models tested, an equation from the literature proved most reliable across the full range of evaluation data (root mean square prediction error = 21.3%). However, the Mitscherlich models demonstrated the greatest degree of adaptability across diet types and intake level. The most successful model for simulating the independent data was a modified Mitscherlich equation with the steepness parameter set to represent dietary starch-to-ADF ratio (root mean square prediction error = 20.6%). However, when such data were unavailable, simpler Mitscherlich forms relating dry matter or metabolizable energy intake to methane production remained better alternatives relative to their linear counterparts.

Rates of Production of Acetate, Propionate, and Butyrate in the Rumen of Lactating Dairy Cows Given Normal and Low-Roughage Diets

Five lactating dairy cows with a permanent cannula in the rumen were given (kg DM/d) a normal diet (7.8 concentrates, 5.1 hay) or a low-roughage (LR) diet (11.5 concentrates, 1.2 hay) in two meals daily in a two-period crossover design. Milk fat (g/kg) was severely reduced on diet LR. To measure rates of production of individual volatile fatty acids (VFA) in the rumen, 0.5 mCi 1-(14)C-acetic acid, 2-(14)C-propionic acid, or 1-(14)C-n-butyric acid were infused into the rumen for 22 h at intervals of 2 to 6 d; rumen samples were taken over the last 12 h. To measure rumen volume, we infused Cr-EDTA into the rumen continuously, and polyethylene glycol was injected 2 h before the morning feed. Results were very variable, so volumes measured by rumen emptying were used instead. Net production of propionic acid more than doubled on LR, but acetate and butyrate production was only numerically lower. Net production rates pooled across both diets were significantly related to concentrations for each VFA. Molar proportions of net production were only slightly higher than molar proportions of concentrations for acetate and propionate but were lower for butyrate. The net energy value (MJ/d) of production of the three VFA increased from 89.5 on normal to 109.1 on LR, equivalent to 55 and 64% of digestible energy, respectively. Fully interchanging, three-pool models of VFA C fluxes are presented. It is concluded that net production rates of VFA can be measured in non-steady states without the need to measure rumen volumes.