Symposium: Application of Models to Determination of nutrient Requirements
The Determination of Nutritional Requirements: Mathematical Modeling of Nutrient-Response Curves1 L PRESTON Department
MERCER2
of Nutrition and Food Science,
University of Kentucky,
INDEXINGKEY WORDS: •mathematical model •nutritional requirement •toxicology
Physiological responses of living organisms are the complex, multifactorial outcome of interactions be tween an organism and environmental factors, such as diet. One may characterize these interactions graphically by plotting a nutrient-response curve, which describes the quantitative, physiological re sponse of an organism as a function of an essential nutrient fed at a specific dietary concentration. By making various simplifying assumptions it is possible to propose mathematical models that accurately de scribe these relationships. The model of choice should have two principle characteristics: mathematical sig nificance and biological significance. Mathematical significance in that it is adequate to describe or explain variations of response as a function of dietary concen tration over wide ranges of intake; biological signifi0022-3166/92
KY 40506-0054
canee in that it has a theoretical basis in biological structure and function (1-4). We proposed the Saturation Kinetics Model (SKM) for the characterization of nutrient-response relation ships (1) as an improvement to the broken-line ap proach (5). The model is based on the concept that an organism is characterized by a sequence of homeostatically constrained steady states. Responses are the result of a series of metabolically related, enzymatically mediated steps, one of which is rate limiting and displays saturation kinetics. The model is descriptive of a wide range of physiological responses and the model equation is continuous in its derivatives. Mod eling provides a basis for the rational formulation of complex mixtures of nutrients (diets) that are designed to optimize some measured performance characteristic in an animal. The optimizing concentration of nutrient in the mixture is defined as the requirement level. We have extended the application of the SKM to include this previously undescribed portion of the re sponse curve (inhibited portion), to include time in the calculation of response and to suggest a method for using the new equation to determine nutritional requirements.
ABSTRACT The Saturation Kinetics Model (SKM) is useful in describing physiological responses as func tions of a limiting dietary nutrient. We have recently expanded the SKM to predict the inhibited portions of the nutrient-response curve. By using the SKM, nutrient requirements can be predicted analytically by, require ment = (Ko.5X Ks)°'5. It is also possible to set an upper and lower dietary nutrient concentration which encom passes the 100% response range for each response, thereby giving an inhibition or toxicity index. This index allows one to set nutritional requirement levels pre cisely, optimizing responses without moving into in hibiting or toxic ranges of nutrients. The model equation can also be converted to a three-dimensional represen tation by graphing each parameter as a function of time. This allows one to visualize a three-dimensional re sponse surface, showing response as a function of time and dietary nutrient concentration. J. Nutr. 122: 706708, 1992.
•amino acid
Lexington,
METHODS Mathematical model The SKM has been derived and discussed (1-4). The model equation (inhibition form) is as follows (equation 1): 1Presented as part of a symposium: Application of Models to Determination of Nutrient Requirements, given at the 75th Annual Meeting of the Federation of American Societies for Experimental Biology, Atlanta, G A, April 23, 1991. The symposium was sponsored by the American Institute of Nutrition. Guest editor for this sym posium was S. P. Coburn, Biochemistry Department, Fort Wayne Developmental Center, Fort Wayne, IN. 2 To whom correspondence should be addressed: Department of Nutrition and Food Science, 212 Funkhouser Building, University of Kentucky, Lexington, KY 40506-0054.
S3.00 ©1992 American Institute of Nutrition.
70* Downloaded from https://academic.oup.com/jn/article-abstract/122/suppl_3/706/4755326 by Washington University in St. Louis user on 25 March 2018
SYMPOSIUM:
APPLICATION
707
OF MODELS
of tested amino acid varied from 0.0 to a level causing clear inhibition of growth. The diets were made isonitrogenous by adding diammonium citrate or remov ing glutamate as necessary. Rats were weighed every day for 2 wk.
Max response
DAY 2
RESULTS AND DISCUSSION 02
_
-2.4
-5.0 4
6
10
VALINE(%In Dtet) FIGURE I Weight gain as a function of dietary valine concentration (day 2 of experiment). Theoretical curve gen erated by the SKM. Filled circles are experimental observa tions.
^ b(Ko.5)
RmaxI"
+ bI2n/(Ks
In + I2n/(Ks)
(1)
where r = physiological response, I = dietary concen tration, b = intercept on r-axis, Rmax = maximum theoretical response, n = apparent kinetic order, KO5 = concentration for jjRmax + b) and Ks = inhibition constant. Equation 1 will describe all regions of the nutrient-response curves. Setting the first derivative of equation 1 equal to zero and solving for I gives a new index, lmm, the intake of maximum response; i.e., the nutrient requirement (equation 2). Imaxr ~
(Ko.5
X Ks) 0.5
(2)
The derived parameters (b, n, Ko5, Rmax, Ks) are calculated by fitting observed data pairs (I,r) to equa tion 1 by using nonlinear curve-fitting techniques, which can then be used to generate a theoretical re sponse curve (Fig. 1). Experimental. To test the applicability of equa tion 1, we carried out experiments in which rats were fed graded dietary levels of several essential amino acids (valine response used for an example). In the amino acid experiment, 60 male weanling Sprague-Dawley rats (Charles River Breeding Labs, COBS/CD, Wilmington, MA) were fed a nonpurified diet (Wayne Laboratory Animal Diets, Denver, CO) for 3 d to acclimate after shipping. The rats were singly housed in suspended wire-bottom cages and were fed water (purified by reverse osmosis) and diets ad libi tum. The rats were then randomly assigned by weight (weanling rats were in the range of 60 ±5 g) to 10 groups of six rats each and fed diets containing an amino acid mix (17.85%) (6). The diets containing the amino acid mix were formulated such that the level Downloaded from https://academic.oup.com/jn/article-abstract/122/suppl_3/706/4755326 by Washington University in St. Louis user on 25 March 2018
Nutrients are administered to organisms that exist in a wide range of physiological, biochemical and pathological conditions. This makes interpretation of experimental results a difficult task. However, there is an underlying unity of biological processes in an organism, which allows one to create models to express certain aspects of nutrient-response functionality. In a typical nutrition experiment, the response is a func tion of both time and dietary nutrient concentration; i.e., r = f(t,I). The response is three dimensional and generates a response surface. Because functions that incorporate both time and nutrient concentration si multaneously are difficult to formulate and/or solve, investigators usually hold one or the other constant. Equation 1 predicts response as a function of dietary nutrient concentration (I) while holding time (t) con stant. The parameters of the curve-fits for each day of the experiment are shown in Table 1. The theoretical re sponse curves for days 2 and 14 of the experiment are shown in Figure 1 and Figure 2. Both curves demon strate the threshold, increase, plateau and decline of a typical nutrient-response relationship. Table 1 gives intake and response values that can be used to set re quirement levels. The response at Maxr (r100%),is the maximum response on the theoretical response curve. For example, in the valine weight gain curve the max imum calculated response, ri0o%,on day 6 is 33.00 g, and r95%is 31.35 g. Because the curve rises, plateaus and falls, r95%appears twice, once on each side of the
TABLE 1 Variation of parameters with time: valine
,81320273342462564362829445455.0462717785924112900 K,, Day234567891011121314Rmax
708
MERCER
plateau. It is then possible, by using numerical itera tion procedures, to calculate two dietary concentra tions, IL(Lower) and lu (Upper), producing r95o/0. These two concentrations, ILand lu, bracket the plateau of the curve and provide a range encompassing optimal physiological response (Fig 1,2). These indices produce a ratio, U/L, which we have defined as the toxicity index, a smaller index implying higher toxicity. Nutrient requirements may be stated for any pre dictable response. One may now predict maximum re sponse analytically by equation 2. r10o%is the maxi mum response on the theoretical response curve (this point cannot be determined by the four-parameter SKM or by the breakpoint method) and is bounded by ILand ly. The ratio, IU/IL/ then gives an index of the inhibiting (or toxic) effects of a nutrient, removing any uncertainty of how high the investigator should set the dietary nutrient concentration of a particular nu trient. It also allows for adjustment of dietary concen trations between known boundaries for accommoda tion of diet formulation. The increase of IU/IL(Table 1) also indicates the rate of adaptation to the excess valine so that the ratio more than triples in 2 wk. The larger the ratio, the increased relative safety of in creasing the nutrient in the diet. We could propose, as a nutritional requirement for weight gain, a dietary concentration of 2.1 % valine to give 100% response but with a possible dietary range of 1.18-3.23% to stay within 5% of maximum re sponse. This range depends, however, on the day the response curve is measured. The 95% range would be 0.81-5.65% on day 14 after adaptation had taken place. All essential nutrients in a dietary formulation could be adjusted, theoretically, to their 100% reMax response
FIGURE 3 Theoretical three-dimensional nutrient-re sponse curve and observed values for valine.
sponse level thereby producing cumulative maximum response, or they could be adjusted within a ±5%range to accommodate the production of a reasonable diet. Three dimensional graph. Because a response curve can be predicted for each day of the experiment and Table 1 shows regular variation in the parameters, it becomes possible to produce a three-dimensional surface showing r as a function of both I and t. Each parameter can be fit to a linear function of time. Sub stituting the linear equations for each parameter into equation 1 produces Figure 3, a surface showing r = f(I,t). With this surface, one can predict r for any combination of day and dietary nutrient concentra tion. This is extremely useful for understanding the complete nutrient-response relationship.
LITERATURE CITED
95% response range
4
6
10
VALINE (% In Diet)
FIGURE 2 Weight gain as a function of dietary valine concentration (day 14 of experiment). Theoretical curve generated by the SKM. Filled circles are experimental ob servations. Downloaded from https://academic.oup.com/jn/article-abstract/122/suppl_3/706/4755326 by Washington University in St. Louis user on 25 March 2018
1. Mercer, L. P., Morgan, P. H. Flodin, N. W. (1975) A theoret ical model for linearization of nutrient response data. Fed. Proc. 34:3841(abs.). 2. Mercer, L. P. [1980) Mathematical models in nutrition. Nutr. Rep. Int. 21: 189-198. 3. Mercer, L. P., Dodds, S. J. & Gustafson, J. M. (1986) The de termination of nutritional requirements: A modeling approach. Nutr. Rep. Internat. 34: 337-350. 4. Mercer, L. P., May, H. E. & Dodds, S. J. (1989) Mathematical modeling and the determination of nutritional requirements: Sigmoidal and inhibited nutrient-response curves. J. Nutr. 119: 1464-1471. 5. Robbins, K. R., Norton, H. W. & Baker, D. H. (1979) Esti mation of nutrient requirements from growth data. J. Nutr. 109: 1710-1714. e. Mercer, L. P., Dodds, S. J. & Smith, D. L. (1987) A new method for formulation of amino acid concentrations and ratios in diets of rats. J. Nutr. 117: 1936-1944.
The Determination of Nutritional Requirements: Mathematical Modeling of Nutrient-Response Curves1 L PRESTON Department
MERCER2
of Nutrition and Food Science,
University of Kentucky,
INDEXINGKEY WORDS: •mathematical model •nutritional requirement •toxicology
Physiological responses of living organisms are the complex, multifactorial outcome of interactions be tween an organism and environmental factors, such as diet. One may characterize these interactions graphically by plotting a nutrient-response curve, which describes the quantitative, physiological re sponse of an organism as a function of an essential nutrient fed at a specific dietary concentration. By making various simplifying assumptions it is possible to propose mathematical models that accurately de scribe these relationships. The model of choice should have two principle characteristics: mathematical sig nificance and biological significance. Mathematical significance in that it is adequate to describe or explain variations of response as a function of dietary concen tration over wide ranges of intake; biological signifi0022-3166/92
KY 40506-0054
canee in that it has a theoretical basis in biological structure and function (1-4). We proposed the Saturation Kinetics Model (SKM) for the characterization of nutrient-response relation ships (1) as an improvement to the broken-line ap proach (5). The model is based on the concept that an organism is characterized by a sequence of homeostatically constrained steady states. Responses are the result of a series of metabolically related, enzymatically mediated steps, one of which is rate limiting and displays saturation kinetics. The model is descriptive of a wide range of physiological responses and the model equation is continuous in its derivatives. Mod eling provides a basis for the rational formulation of complex mixtures of nutrients (diets) that are designed to optimize some measured performance characteristic in an animal. The optimizing concentration of nutrient in the mixture is defined as the requirement level. We have extended the application of the SKM to include this previously undescribed portion of the re sponse curve (inhibited portion), to include time in the calculation of response and to suggest a method for using the new equation to determine nutritional requirements.
ABSTRACT The Saturation Kinetics Model (SKM) is useful in describing physiological responses as func tions of a limiting dietary nutrient. We have recently expanded the SKM to predict the inhibited portions of the nutrient-response curve. By using the SKM, nutrient requirements can be predicted analytically by, require ment = (Ko.5X Ks)°'5. It is also possible to set an upper and lower dietary nutrient concentration which encom passes the 100% response range for each response, thereby giving an inhibition or toxicity index. This index allows one to set nutritional requirement levels pre cisely, optimizing responses without moving into in hibiting or toxic ranges of nutrients. The model equation can also be converted to a three-dimensional represen tation by graphing each parameter as a function of time. This allows one to visualize a three-dimensional re sponse surface, showing response as a function of time and dietary nutrient concentration. J. Nutr. 122: 706708, 1992.
•amino acid
Lexington,
METHODS Mathematical model The SKM has been derived and discussed (1-4). The model equation (inhibition form) is as follows (equation 1): 1Presented as part of a symposium: Application of Models to Determination of Nutrient Requirements, given at the 75th Annual Meeting of the Federation of American Societies for Experimental Biology, Atlanta, G A, April 23, 1991. The symposium was sponsored by the American Institute of Nutrition. Guest editor for this sym posium was S. P. Coburn, Biochemistry Department, Fort Wayne Developmental Center, Fort Wayne, IN. 2 To whom correspondence should be addressed: Department of Nutrition and Food Science, 212 Funkhouser Building, University of Kentucky, Lexington, KY 40506-0054.
S3.00 ©1992 American Institute of Nutrition.
70* Downloaded from https://academic.oup.com/jn/article-abstract/122/suppl_3/706/4755326 by Washington University in St. Louis user on 25 March 2018
SYMPOSIUM:
APPLICATION
707
OF MODELS
of tested amino acid varied from 0.0 to a level causing clear inhibition of growth. The diets were made isonitrogenous by adding diammonium citrate or remov ing glutamate as necessary. Rats were weighed every day for 2 wk.
Max response
DAY 2
RESULTS AND DISCUSSION 02
_
-2.4
-5.0 4
6
10
VALINE(%In Dtet) FIGURE I Weight gain as a function of dietary valine concentration (day 2 of experiment). Theoretical curve gen erated by the SKM. Filled circles are experimental observa tions.
^ b(Ko.5)
RmaxI"
+ bI2n/(Ks
In + I2n/(Ks)
(1)
where r = physiological response, I = dietary concen tration, b = intercept on r-axis, Rmax = maximum theoretical response, n = apparent kinetic order, KO5 = concentration for jjRmax + b) and Ks = inhibition constant. Equation 1 will describe all regions of the nutrient-response curves. Setting the first derivative of equation 1 equal to zero and solving for I gives a new index, lmm, the intake of maximum response; i.e., the nutrient requirement (equation 2). Imaxr ~
(Ko.5
X Ks) 0.5
(2)
The derived parameters (b, n, Ko5, Rmax, Ks) are calculated by fitting observed data pairs (I,r) to equa tion 1 by using nonlinear curve-fitting techniques, which can then be used to generate a theoretical re sponse curve (Fig. 1). Experimental. To test the applicability of equa tion 1, we carried out experiments in which rats were fed graded dietary levels of several essential amino acids (valine response used for an example). In the amino acid experiment, 60 male weanling Sprague-Dawley rats (Charles River Breeding Labs, COBS/CD, Wilmington, MA) were fed a nonpurified diet (Wayne Laboratory Animal Diets, Denver, CO) for 3 d to acclimate after shipping. The rats were singly housed in suspended wire-bottom cages and were fed water (purified by reverse osmosis) and diets ad libi tum. The rats were then randomly assigned by weight (weanling rats were in the range of 60 ±5 g) to 10 groups of six rats each and fed diets containing an amino acid mix (17.85%) (6). The diets containing the amino acid mix were formulated such that the level Downloaded from https://academic.oup.com/jn/article-abstract/122/suppl_3/706/4755326 by Washington University in St. Louis user on 25 March 2018
Nutrients are administered to organisms that exist in a wide range of physiological, biochemical and pathological conditions. This makes interpretation of experimental results a difficult task. However, there is an underlying unity of biological processes in an organism, which allows one to create models to express certain aspects of nutrient-response functionality. In a typical nutrition experiment, the response is a func tion of both time and dietary nutrient concentration; i.e., r = f(t,I). The response is three dimensional and generates a response surface. Because functions that incorporate both time and nutrient concentration si multaneously are difficult to formulate and/or solve, investigators usually hold one or the other constant. Equation 1 predicts response as a function of dietary nutrient concentration (I) while holding time (t) con stant. The parameters of the curve-fits for each day of the experiment are shown in Table 1. The theoretical re sponse curves for days 2 and 14 of the experiment are shown in Figure 1 and Figure 2. Both curves demon strate the threshold, increase, plateau and decline of a typical nutrient-response relationship. Table 1 gives intake and response values that can be used to set re quirement levels. The response at Maxr (r100%),is the maximum response on the theoretical response curve. For example, in the valine weight gain curve the max imum calculated response, ri0o%,on day 6 is 33.00 g, and r95%is 31.35 g. Because the curve rises, plateaus and falls, r95%appears twice, once on each side of the
TABLE 1 Variation of parameters with time: valine
,81320273342462564362829445455.0462717785924112900 K,, Day234567891011121314Rmax
708
MERCER
plateau. It is then possible, by using numerical itera tion procedures, to calculate two dietary concentra tions, IL(Lower) and lu (Upper), producing r95o/0. These two concentrations, ILand lu, bracket the plateau of the curve and provide a range encompassing optimal physiological response (Fig 1,2). These indices produce a ratio, U/L, which we have defined as the toxicity index, a smaller index implying higher toxicity. Nutrient requirements may be stated for any pre dictable response. One may now predict maximum re sponse analytically by equation 2. r10o%is the maxi mum response on the theoretical response curve (this point cannot be determined by the four-parameter SKM or by the breakpoint method) and is bounded by ILand ly. The ratio, IU/IL/ then gives an index of the inhibiting (or toxic) effects of a nutrient, removing any uncertainty of how high the investigator should set the dietary nutrient concentration of a particular nu trient. It also allows for adjustment of dietary concen trations between known boundaries for accommoda tion of diet formulation. The increase of IU/IL(Table 1) also indicates the rate of adaptation to the excess valine so that the ratio more than triples in 2 wk. The larger the ratio, the increased relative safety of in creasing the nutrient in the diet. We could propose, as a nutritional requirement for weight gain, a dietary concentration of 2.1 % valine to give 100% response but with a possible dietary range of 1.18-3.23% to stay within 5% of maximum re sponse. This range depends, however, on the day the response curve is measured. The 95% range would be 0.81-5.65% on day 14 after adaptation had taken place. All essential nutrients in a dietary formulation could be adjusted, theoretically, to their 100% reMax response
FIGURE 3 Theoretical three-dimensional nutrient-re sponse curve and observed values for valine.
sponse level thereby producing cumulative maximum response, or they could be adjusted within a ±5%range to accommodate the production of a reasonable diet. Three dimensional graph. Because a response curve can be predicted for each day of the experiment and Table 1 shows regular variation in the parameters, it becomes possible to produce a three-dimensional surface showing r as a function of both I and t. Each parameter can be fit to a linear function of time. Sub stituting the linear equations for each parameter into equation 1 produces Figure 3, a surface showing r = f(I,t). With this surface, one can predict r for any combination of day and dietary nutrient concentra tion. This is extremely useful for understanding the complete nutrient-response relationship.
LITERATURE CITED
95% response range
4
6
10
VALINE (% In Diet)
FIGURE 2 Weight gain as a function of dietary valine concentration (day 14 of experiment). Theoretical curve generated by the SKM. Filled circles are experimental ob servations. Downloaded from https://academic.oup.com/jn/article-abstract/122/suppl_3/706/4755326 by Washington University in St. Louis user on 25 March 2018
1. Mercer, L. P., Morgan, P. H. Flodin, N. W. (1975) A theoret ical model for linearization of nutrient response data. Fed. Proc. 34:3841(abs.). 2. Mercer, L. P. [1980) Mathematical models in nutrition. Nutr. Rep. Int. 21: 189-198. 3. Mercer, L. P., Dodds, S. J. & Gustafson, J. M. (1986) The de termination of nutritional requirements: A modeling approach. Nutr. Rep. Internat. 34: 337-350. 4. Mercer, L. P., May, H. E. & Dodds, S. J. (1989) Mathematical modeling and the determination of nutritional requirements: Sigmoidal and inhibited nutrient-response curves. J. Nutr. 119: 1464-1471. 5. Robbins, K. R., Norton, H. W. & Baker, D. H. (1979) Esti mation of nutrient requirements from growth data. J. Nutr. 109: 1710-1714. e. Mercer, L. P., Dodds, S. J. & Smith, D. L. (1987) A new method for formulation of amino acid concentrations and ratios in diets of rats. J. Nutr. 117: 1936-1944.
