Predicting metabolic adaptation, body weight change, and energy intake in humans

Abstract

Complex interactions between carbohydrate, fat, and protein metabolism underlie the body's remarkable ability to adapt to a variety of diets. But any imbalances between the intake and utilization rates of these macronutrients will result in changes in body weight and composition. Here, I present the first computational model that simulates how diet perturbations result in adaptations of fuel selection and energy expenditure that predict body weight and composition changes in both obese and nonobese men and women. No model parameters were adjusted to fit these data other than the initial conditions for each subject group (e.g., initial body weight and body fat mass). The model provides the first realistic simulations of how diet perturbations result in adaptations of whole body energy expenditure, fuel selection, and various metabolic fluxes that ultimately give rise to body weight change. The validated model was used to estimate free-living energy intake during a long-term weight loss intervention, a variable that has never previously been measured accurately.

over the past century, researchers in the fields of human metabolism and nutrition have accumulated an impressive body of quantitative knowledge regarding how dietary changes impact various aspects of metabolism, body weight, and body composition (12, 24, 38, 56, 68). But integrating this knowledge to make quantitative predictions is a formidable task given the multiple nonlinear interactions between various organ systems. Nevertheless, such an integrative approach is required to fully understand both normal physiology as well as the derangements that underlie conditions such as obesity, diabetes, and the metabolic syndrome.

Mathematical modeling and computer simulation are widely used methodologies in the engineering and physical sciences for integrating knowledge about complex systems and predicting their behavior. This approach is beginning to gain traction in the life sciences and forms a critical part of the systems biology paradigm (20, 73). In the area of human energy metabolism and body weight regulation, several mathematical models of weight change have been proposed over the past few decades (3–5, 7, 16–18, 33, 42, 44, 59, 60, 96, 110, 113). Previously, I presented the first mathematical model of the metabolism of all three macronutrients (i.e., carbohydrate, fat, and protein), and I used the model to help understand the dynamics of semistarvation and refeeding in healthy young men (42). Here, I extended my previous model to apply to both obese and nonobese men and women and validated its behavior in response to several controlled diet perturbations. The model was designed to quantitatively track the metabolism of all three dietary macronutrients and their interactions within the human body. In particular, the model describes how diet perturbations result in adaptations of energy expenditure, fuel selection, and various metabolic fluxes (e.g., lipolysis, lipogenesis, gluconeogenesis, ketogenesis, protein turnover, etc.) that ultimately give rise to changes of body weight and composition on a time scale of days and longer. The main model assumptions were that energy must be conserved and that changes of the body composition result from imbalances between the intake and utilization rates of fat, carbohydrate, and protein along with intracellular and extracellular fluid changes. The model code can be downloaded as a data supplement or at my website (http://www2.niddk.nih.gov/NIDDKLabs/LBM/LBMHall.htm; Supplemental Material for this article is available at the AJP-Endocrinology and Metabolism website).

The model was developed using published human data from more than 50 experimental studies (see appendix) and was validated by comparing model predictions with the results from several controlled feeding studies not used for model development. In this validation process, I chose to simulate human studies that carefully controlled food intake and measured changes in body weight (BW) and body fat mass (FM) as well as total energy expenditure (TEE) and resting metabolic rate (RMR). Requiring this combination of measurements dramatically narrowed the scope of possible validation studies and allowed for assessment of not only the weight change predictions but also energy partitioning and energy expenditure changes. In all cases, the measured food intake was used as a model input and included a wide range of interventions, such as overfeeding and weight gain, weight loss using a variety of diets, and adaptations of metabolic fuel selection when the dietary macronutrient proportions were altered. These validation studies were performed in several different subject groups, including lean, overweight, and obese men and women. Importantly, no model parameters were altered to fit the data other than modifying the initial conditions appropriate for modeling each subject group (e.g., initial BW and body FM).

Once the model was validated in situations where the food intake was known, I proposed that the model could be used to estimate the free-living energy intake changes underlying observed changes of body weight. In particular, I used the model to predict the energy intake required to result in the typical trajectory of weight loss and regain observed during an outpatient weight loss intervention (100). In addition to providing a novel methodology for estimating the dynamics of free-living human energy intake during weight loss and regain, the model simulations addressed the relative role of diet adherence vs. metabolic adaptation in explaining the typically observed weight loss plateau after ∼6 mo of lifestyle modification.

METHODS

The appendix provides a detailed description of the mathematical model along with the data and assumptions used in its development and calibration. Briefly, the model comprises eight ordinary differential equations and uses dietary carbohydrate, fat, and protein as model inputs. The model computes the various components of whole body energy expenditure, rates of macronutrient oxidation, respiratory exchange, lipogenesis, ketogenesis, gluconeogenesis, and turnover of fat, glycogen, and protein. Based on the computed macronutrient imbalances, along with sodium imbalances impacting extracellular fluid, the model computes dynamic changes of BW and composition.

To represent different initial conditions corresponding to different subject groups, parameter values were specified for initial BW, percent body fat, total body water, RMR, and baseline food intake. In cases where one or more of these initial parameters was unknown, the model used more common measurements of age, height, sex, and physical activity level to estimate the initial parameter values using published regression equations for body fat (50), RMR (72), and extracellular fluid (94). I also allowed for the possibility that the baseline diet may not result in a state of macronutrient balance. Rather, the parameters F_imbal, G_imbal, and P_imbal specified the initial imbalances of fat, glycogen, and protein, respectively. Other than these initial parameter values, no other model parameters were adjusted to simulate the validation experiments.

To estimate the energy intake rate underlying the weight change data of Svetkey et al. (100), I specified that the onset of the diet resulted in an immediate reduction of energy intake for a constant period followed by a linear change of energy intake until another constant period. The magnitude of the energy intake changes, along with the duration of each period, was determined using a downhill simplex algorithm (78) implemented in the Berkeley Madonna software (version 8.3; http://www.berkeleymadonna.com) to minimize the sum of squares of weighted residuals between the simulation outputs and the BW change data.

RESULTS

Metabolic Fuel Selection

When the body is in a state of energy and macronutrient balance, the intake of dietary carbohydrate, fat, and protein is matched by their rates of utilization. A useful measurement of the relative proportion of carbohydrate to fat being oxidized is provided by the RQ, where a value of 1 reflects complete reliance on carbohydrate oxidation for metabolic needs, whereas RQ = 0.7 indicates state of pure fat oxidation and intermediate values reflect a fuel selection mixture (27, 31, 37). Macronutrient balance is achieved when the RQ is equal to the food quotient (FQ), which represents the relative proportion of macronutrients in the diet. Exchanging dietary carbohydrate with fat while maintaining the same energy intake will result in a state of macronutrient imbalance until fuel selection adapts to the new FQ. Figure 1A depicts the model's predicted change of 24-h RQ in response to a high-fat isocaloric diet that resulted in a large decline of FQ. This diet perturbation was performed by Schrauwen et al. (90), and the closed squares in Fig. 1A show the measured RQ dynamics, which closely matched the model predictions. Figure 1B shows the results from a similar study performed by Smith et al. (95), with a less dramatic change of FQ. Again, the model predictions agreed with the data. Underlying these simulations were slow decreases of glycogen as well as increased lipolysis due to the reduced dietary carbohydrate (data not shown). These changes resulted in altered substrate delivery to metabolically active tissues and subsequent changes of fuel selection.

Fig. 1.

Adaptations of fuel selection following isocaloric exchange of dietary carbohydrate and fat. A: 24-h respiratory quotient (RQ) in response to switching from a 30 to a 60% fat diet. The simulated changes of RQ (dashed curve) along with the measured values (■) show a progressive approach to the food quotient (FQ; solid curve). B: simulated (dashed curve) and measured (■) 24-h RQ changes in response to a diet switch from 37 to 50% fat corresponding to the depicted changes of FQ (solid curve). Data are presented as means ± SD.

During overfeeding, the model predicts substantial changes of various whole body metabolic fluxes, including suppression of lipolysis and induction of de novo lipogenesis, that influence overall fuel selection and macronutrient balance. The imbalance between dietary macronutrients and their utilization rates determined how the energy excess was partitioned in the model. To test whether macronutrient oxidation changes were simulated correctly during overfeeding, I used the data of Jebb and colleagues (51, 52), who measured macronutrient oxidation rates during 12 days of 33% overfeeding in healthy young men that were continuously housed inside a metabolic chamber. Figure 2A, left, shows the model predictions along with the data for BW and body FM, whereas Fig. 2A, right, depicts the simulated daily macronutrient oxidation rates along with the data. The simulated gradual increase of carbohydrate oxidation reached a plateau after several days as glycogen approached a new steady state (data not shown). These results closely matched the measured carbohydrate oxidation rate shown in the closed squares in Fig. 2A, right. The simulated fat oxidation rate was suppressed at the onset of overfeeding, which matched the measurements shown in the open squares in Fig. 2A, right. The decrease of fat oxidation in the simulation corresponded to a suppression of lipolysis as a result of the increased carbohydrate intake (data not shown). The simulated protein oxidation rate remained relatively unchanged and corresponded to the measured rate shown in the open triangles in Fig. 2A, right.

Fig. 2.

Metabolic fuel selection and body composition and changes during over- and underfeeding in healthy young men. A, left: 33% overfeeding resulted in the change of simulated and measured body weight (BW; solid curve and ■, respectively) along with increased simulated and measured fat mass (FM; dashed curve and □, respectively). A, right: simulated and measured carbohydrate oxidation rates (dashed curve and ■, respectively) as well as fat oxidation (solid curve and □, respectively) and protein oxidation rates (dotted curve and ▵, respectively). B: BW, FM, and macronutrient oxidation rates in response to 67% underfeeding, where the symbols are identical to A. Data are presented as means ± SD.

Jebb and colleagues (51, 52) also performed a complementary study where energy intake was decreased by 67% for 12 days, and these results are depicted in Fig. 2B along with the corresponding model simulations. Again, the simulated changes of body composition and macronutrient oxidation rates closely matched the data and showed the opposite response to overfeeding, whereas underfeeding suppressed carbohydrate oxidation while enhancing fat oxidation.

Underfeeding and Weight Loss

Although these results demonstrate that the model correctly simulated adaptations to short-term underfeeding in lean young men, longer-term underfeeding is required to achieve significant weight loss in overweight people. A recent 6-mo calorie restriction study investigated the metabolic effects and weight changes in healthy, sedentary, overweight men and women resulting from three strictly controlled lifestyle interventions (45, 81, 82). Figure 3A shows the effect of an 890 kcal/day very low-calorie diet followed by a maintenance diet from 3 to 6 mo. The imposed energy intake resulted in the drop of BW and FM shown in Fig. 3A, left. The simulated TEE and RMR also matched the data, which are depicted by closed and open squares, respectively, in Fig. 3A, right. Figure 3B shows the model simulations and data for a 25% reduction of calories for 6 mo, and Fig. 3C shows the results for a 12.5% caloric reduction plus a 12.5% increase of physical activity expenditure (PAE). Again, the model simulations matched the body composition and energy expenditure data remarkably well, although the initial simulated decrease in BW was slightly more rapid than the data.

Fig. 3.

Weight loss and metabolic effects of caloric restriction for 6 mo. A, left: simulated BW change (solid curve) along with the measured values (■) and the simulated FM (dashed curve) and the measured FM (□) changes resulting from a very low-calorie liquid diet followed by a weight maintenance diet. A, right: simulated (dashed curve) and measured (■) total energy expenditure (TEE) in response to the depicted changes of energy intake (EI; solid gray curve). Simulated (solid black curve) and measured resting metabolic rate (RMR; □) along with the simulated physical activity expenditure (PAE; dotted curve). B: a sustained 25% reduction of EI resulted in the simulated and measured changes of BW and FM shown at left and the corresponding TEE, RMR, and PAE changes at right. C: a sustained 12.5% reduction of EI along with a 12.5% increase of PAE resulted in the simulated and measured changes of BW and FM shown at left and the corresponding TEE and RMR changes at right. Data are presented as means ± SD.

There has been much debate surrounding the metabolic effects of weight loss diets that differ in macronutrient content. Rumpler et al. (88) investigated this issue in obese men by restricting energy intake by 50% for 4 wk using diets with either 40 or 20% of energy from dietary fat. Figure 4A, left, shows the model simulation and experimental results for the 40% fat diet, and Fig. 4B, right, shows the results for the 20% fat diet. Figure 4, A and B, top, demonstrates that the simulated body composition changes corresponded reasonably well with the data, as did the TEE and RMR changes shown in Fig. 4, A and B, middle. Figure 4, A and B, bottom, illustrates the very close agreement between the simulated and measured 24-h RQ values, demonstrating that the model appropriately represents fuel selection during weight loss diets with differing macronutrient composition in obese men.

Fig. 4.

Weight loss and metabolic changes in obese men resulting from diets with differing proportions of carbohydrate and fat. A, top: simulated and measured changes of BW and FM resulting from the 40% fat, 46% carbohydrate diet. A, middle: metabolic responses to this diet, where the symbols are identical to those in Fig. 3. A, bottom: simulated and measured changes of RQ, where the symbols are identical to Fig. 1. B, top: simulated and measured changes of BW and FM resulting from the 20% fat, 66% carbohydrate diet. B, middle: metabolic responses to this diet. B, bottom: simulated and measured changes of RQ. Data are presented as means ± SD.

The model also accurately simulated weight loss and metabolic changes in obese women consuming 1,000 kcal/day for an 8-wk period, as studied by de Boer et al. (19). Figure 5A, left, demonstrates close agreement between the simulated and measured body composition changes, whereas Fig. 5A, right, shows that the simulated TEE and RMR changes also matched the data quite well.

Fig. 5.

Weight loss and weight gain. A, left: simulated and measured changes of BW and FM resulting from a 1,000 kcal/day diet for 8 wk in obese women. A, right: simulated and measured changes of TEE, RMR, and PAE in response to imposed changes of EI, where the symbols are identical to those in Fig. 2. B, left: simulated and measured changes of BW and FM in response to 6 wk of overfeeding by 1,000 kcal/day in healthy young men. B, right: changes of TEE, RMR, and PAE, where the symbols are identical to those in Fig. 3. Data are presented as means ± SD.

Overfeeding and Weight Gain

Weight gain is the result of positive energy balance, but predicting the magnitude of weight gain for a given change of energy intake requires knowing how excess energy is partitioned between deposition of body fat and fat-free mass, which have dramatically different influences on energy expenditure and the subsequent magnitude of energy imbalance. I tested the model predictions for weight gain by simulating the 6-wk, 50% overfeeding study of Diaz et al. (21), who measured body composition changes as well as TEE and resting metabolic rate. Figure 5B, left, depicts the simulated changes of BW and FM, which clearly matched the body composition data, indicating appropriate energy partitioning and weight gain. Figure 5B, right, shows that the imposed energy intake resulted in simulated TEE and RMR changes.

I also simulated the overfeeding study of Tremblay et al. (103), who provided healthy young men with 1,000 kcal/day above baseline energy requirements 6 days/wk for 100 days. Prior to the overfeeding period, the average subject had BW = 60.3 ± 8.0 kg, FM = 6.9 ± 3.5 kg, and RMR = 1,635 ± 170 kcal/day. After 100 days of overfeeding, BW = 68.4 ± 8.2 kg, FM = 12.3 ± 4.5 kg, and RMR = 1,793 ± 190 kcal/day, which matched the following simulation results quite well: BW = 69.7 kg, FM = 12.3 kg, and RMR = 1,850 kcal/day.

Estimating the Free-Living Energy Intake Underlying Weight Loss and Regain

Maintenance of a reduced BW following weight loss is a critical issue in the treatment of obesity (44, 46, 115, 116). Most weight loss interventions result in a maximum amount of lost weight after ∼6 mo, with a gradual weight regain over the subsequent several years. But what is the mechanism of the weight loss plateau and subsequent regain? In particular, what role does metabolic adaptation play in resisting further weight loss after 6 mo?

Given that the mathematical model accurately simulates the metabolic and body composition changes in response to known changes of diet, I investigated what change of diet would be required to simulate the average weight loss and regain pattern in a group of 341 overweight and obese adults participating in a self-directed weight loss program (100). Figure 6A, left, shows the simulated weight loss and regain pattern (solid curve) which closely matched the data (closed squares), whereas the dashed curve is the predicted FM change. The solid gray curve in Fig. 6A, right, is the predicted free-living energy intake underlying the observed weight loss and regain trajectory, assuming no change of physical activity. At the onset of the intervention, the model predicted a decrease of energy intake by ∼800 kcal/day, which was maintained for only 6 wk. Subsequently, energy intake gradually increased until ∼10 mo, when it returned to the initial energy intake that resulted in energy balance at the initial BW. The simulated TEE is depicted by the dashed curve in Fig. 6A, right, and shows that negative energy balance was achieved only for ∼8 mo, after which time positive energy balance and weight regain ensued.

Fig. 6.

Weight loss and regain dynamics during an outpatient lifestyle intervention. A, left: a typical outpatient weight loss program results in the characteristic BW change trajectory, where the symbols are identical to those in Fig. 3. A, right: predicted free-living EI and TEE underlying the observed BW loss and regain trajectory. B: the model predicted that maintenance of lost BW and FM (left) would have been achieved if the EI over the last 2 yr had been decreased by 170 kcal/day. C: had the initial reduction of EI been sustained for the 3-yr period, the model predicted a progressive decrease of BW and FM shown at left and the corresponding TEE changes shown at right. Data are presented as means ± SD.

Figure 6B shows the simulated changes of energy intake required to maintain the weight loss achieved at 6 mo, which was ∼170 kcal/day less than the initial energy-balanced diet. Figure 6C shows the BW and FM changes that were predicted had the initial decrease of energy intake been maintained over the entire 3-yr period. This simulation illustrates the very long equilibration time for weight loss in obese subjects and demonstrates that the weight loss plateau observed after 6 mo cannot be a result of metabolic adaptation.

DISCUSSION

The ultimate goal of modeling is to provide physiological insights and to help design novel experiments whose results can then be integrated within the context of previous knowledge, thereby improving both the model as well as our understanding of the system. Most systems biology models of metabolism have been developed at the cellular level and are aimed at calculating steady-state flux distributions through complex metabolic networks in microorganisms (29, 30). Such models are now beginning to be applied to human metabolism (25, 92), and the model presented here represents a complementary dynamic approach to systems physiology at the level of whole body human metabolism. Other than modifying the initial model parameters to represent the various subject groups, no parameters were adjusted to simulate the validation experiments. Rather, once the initial conditions were specified, the model calculated the correct energy partitioning and metabolic adaptations required to simulate the physiological responses in lean, overweight, and obese men and women.

The model appears to accurately describe the physiology of metabolism, fuel selection, and body composition change in humans on time scales ranging from days to years. One practical use of the model would be to computationally investigate the potential of various metabolic interventions for treatment of obesity. Such a tool could help design key experiments, focus resources on areas of likely success, and help interpret experimental results. Of course, since the model presently uses food intake as an input, any adaptive changes of food intake as a result of a metabolic intervention would not be automatically accounted for, but hypotheses regarding food intake changes could be simulated.

Since the model responded accurately to known changes of food intake, I proposed that known changes of body weight and composition over long time periods could be used to estimate the underlying changes of free-living energy intake. This is an important application of the model, since free-living energy intake is notoriously difficult to measure (117) and the gold standard doubly labeled water method is valid only in situations of energy balance, when the RQ is known (102). I used the model to estimate the free-living energy intake changes underlying the typical weight loss and regain trajectory observed in an outpatient weight loss program. Surprisingly, the model predicted that the initial reduction of energy intake was maintained for only 6 wk, followed by a gradual return to baseline after 10 mo, where it remained for more than 2 years. This highlights the importance of diet adherence for successful weight loss and raises the question of why the estimated energy intake has this pattern. In other words, what internal and external factors contribute to this progressive loss of diet adherence? Although the model presented here cannot answer these questions, the ability to quantitatively predict the dynamics of free-living energy intake during weight loss and regain may have implications for the development of drug and behavioral therapies for obesity.

An important caveat regarding the model estimate of energy intake is that I assumed that the average physical activity for the group was unchanged over the course of the weight loss and regain trajectory. This does not mean that the energy cost of physical activity was unchanged, since performing the same physical activity at a different body weight has a differing energy requirement, as was demonstrated by the PAE curves in the validation simulations in Figs. 3–5. Of course, the model can be simulated for any assumed physical activity time course, and the energy intake can be recalculated, thereby providing a range of estimates for varying assumptions about physical activity. Future work will investigate the sensitivity of the energy intake estimates to various model parameters as well as the time course of body weight change.

Much work remains for improving and expanding the model. For example, although I have demonstrated that the model behaves appropriately for different groups of subjects, it is presently unclear whether individual subject responses can be predicted given sufficiently detailed information about their initial conditions. Furthermore, the model now implicitly represents the effect of hormones such as insulin, but an explicit representation of endocrine signaling along with concentrations of hormones and metabolites would be desirable, especially on shorter time scales, so that the response to individual meals could be simulated. Other additions to the model might include an explicit representation of the possible contribution of human brown adipose tissue to the overall energy expenditure and its role in adaptive thermogenesis (80). Finally, it would be particularly interesting to “close the loop” and model how internal signals from changes of metabolism and body composition combine with external factors to determine food intake. But despite the cornucopia of possible model improvements, the mathematical model presented here represents a significant step forward in a systems physiology approach to whole body human metabolism.

GRANTS

This research was supported by the Intramural Research Program of the National Institutes of Health/National Institute of Diabetes and Digestive and Kidney Diseases.

DISCLOSURES

No conflicts of interest are declared by the author.

APPENDIX

Detailed Description of the Mathematical Model

The individual components of the mathematical model were based on a variety of published in vivo human data, as described below. Each model component was relatively simple, and only the most important physiological effectors have been incorporated. Since continued development of the model is part of an ongoing research program, additional relevant physiological data will be incorporated within the existing computational framework to improve the realism and predictive capabilities of the model. The model code can be downloaded as a data supplement or at my website (http://www2.niddk.nih.gov/NIDDKLabs/LBM/LBMHall.htm).

The concept of macronutrient balance is an expression of energy conservation such that changes in the body's energy stores were given by the sum of fluxes entering the pools minus the fluxes exiting the pools. Thus, the mathematical representation of macronutrient balance was given by the following differential equations:

$$\begin{array}{c}{\mathrm{\rho }}_{\mathrm{C}}\frac{\mathrm{d}\mathrm{G}}{\mathrm{d}t}=\mathrm{C}\mathrm{I}-\mathrm{D}\mathrm{N}\mathrm{L}+\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}+\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{F}}-\mathrm{G}3P-\mathrm{\text{CarbOx}}\\ {\mathrm{\rho }}_{\mathrm{F}}\frac{\mathrm{d}\mathrm{F}}{\mathrm{d}t}=3{\mathrm{M}}_{\mathrm{\text{FFA}}}\mathrm{F}\mathrm{I}/{\mathrm{M}}_{\mathrm{T}\mathrm{G}}+{\mathrm{\epsilon }}_{\mathrm{d}}\mathrm{D}\mathrm{N}\mathrm{L}-\mathrm{K}{\mathrm{U}}_{\mathrm{\text{excr}}}\\ -(1-{\mathrm{\epsilon }}_{\mathrm{k}})\mathrm{K}\mathrm{T}\mathrm{G}-\mathrm{\text{FatOx}}\\ {\mathrm{\rho }}_{\mathrm{P}}\frac{\mathrm{d}\mathrm{P}}{\mathrm{d}t}=\mathrm{P}\mathrm{I}-\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}-\mathrm{\text{ProtOx}},\end{array}$$ (1)

where ρ_C = 4.18 kcal/g, ρ_F = 9.44 kcal/g, and ρ_P = 4.7 kcal/g were the energy densities of carbohydrate, fat, and protein, respectively (67). M_TG = 860 g/mol and M_FFA = 273 g/mol were the molecular masses of triacylglycerides and free fatty acids, respectively. The efficiency of de novo lipogenesis, DNL, was represented as the dimensionless parameter ε_d = 0.835, which is the enthalpy of combustion of 0.37 g of fat divided by the enthalpy of combustion of the 1 g of glucose used to produce the fat (27). The efficiency of ketogenesis, KTG, was represented by the parameter ε_k = 0.81, calculated as the enthalpy of combustion of 4.5 mol of acetoacetate divided by the enthalpy of combustion of 1 mol of stearic acid used to produce the ketones (12). When the ketogenic rate increases, ketones are excreted in the urine at the rate KU_excr. The macronutrient intake rates CI and FI refer to the digestible energy intake of carbohydrate and fat, respectively, whereas PI refers to the digestible energy intake of protein corrected for the obligatory formation of ammonia with protein metabolism. The energy cost of ureagenesis is accounted for separately, as described below. The oxidation rates CarbOx, FatOx, and ProtOx are summed to the TEE, less the small amount of heat produced via flux through ketogenic and lipogenic pathways. Since body composition changes take place on the time scale of weeks, months, and years, the model was targeted to represent daily changes of energy metabolism and not fluctuations of metabolism that occur within 1 day. The nutrient balance equations were integrated using the fourth-order Runge-Kutta algorithm with a time step size of 0.1 days (78).

Body Composition

The BW was the sum of the FFM and the FM. FFM was computed using the following equation:

$$\begin{array}{c}\mathrm{FFM}=\mathrm{BM}+\mathrm{ECF}+\mathrm{ECP}+\mathrm{LCM}\\ =\mathrm{BM}+\mathrm{ECF}+\mathrm{ECP}+\mathrm{ICW}+\mathrm{P}+\mathrm{G}+\mathrm{ICS}\\ =\mathrm{BM}+\mathrm{ECF}+\mathrm{ECP}+\mathrm{I}\widehat{\mathrm{C}}\mathrm{W}+\mathrm{P}\left(1+{\mathrm{h}}_{\mathrm{P}}\right)\\ +\mathrm{G}\left(1+{\mathrm{h}}_{\mathrm{G}}\right)+\mathrm{ICS},\end{array}$$ (2)

where the FFM is composed of BM, ECF, ECP, and LCM. LCM is composed of ICW, G, and P as well as a small contribution from nucleic acids and other ICS. The protein fraction of the lean tissue cell mass was P/LCM = 0.25, and the initial intracellular water fraction was ICW/LCM = 0.7 (13, 109). ICW was then calculated dynamically from P and G such that each gram of protein and glycogen was associated with h_P and h_G grams of water, respectively. IĈW was a constant amount of ICW computed to attain the appropriate initial intracellular composition, assuming that G = 500 g, h_G = 2.7, and h_P = 1.6 (13, 43, 71).

The initial ECF was calculated via the regression equations of Silva et al. (94), and changes of ECF were calculated as follows:

$$\begin{array}{l}\frac{\mathrm{dECF}}{\mathrm{d}t}=\frac{1}{\left[\mathrm{Na}\right]}\left(\Delta \mathrm{N}{\mathrm{a}}_{\mathrm{\text{diet}}}-{\xi }_{\mathrm{Na}}\left(\mathrm{ECF}-\mathrm{EC}{\mathrm{F}}_{\mathrm{init}}\right)-{\xi }_{\mathrm{CI}}\left(1-\mathrm{CI}/\mathrm{C}{\mathrm{I}}_{\mathrm{b}}\right)\right)+\Delta \mathrm{ECF}\hfill \\ {\tau }_{\mathrm{BW}}\frac{\mathrm{d}\Delta \mathrm{ECF}}{\mathrm{d}t}={\xi }_{\mathrm{BW}}\left(\mathrm{BW}-\mathrm{B}{\mathrm{W}}_{\mathrm{init}}\right)-\Delta \mathrm{ECF},\hfill \end{array}$$ (3)

where [Na] = 3.22 mg/ml is the extracellular sodium concentration, ΔNa_diet is the change of dietary sodium in milligrams, and ΔECF is the slow rate of increment in ECF that occurs with BW change on a very long time scale of τ_BW, ∼1,000 days. The value of ξ_BW = 0.16 ml·kg⁻¹·day⁻¹ was chosen so that the steady-state increment of ECF with weight change approximately matched the regression equations of Silva et al. (94). The value of ξ_Na was chosen according to the data of Andersen et al. (6), who measured a change of sodium excretion of 5,000 mg/day following an infusion of 1.7 liters of isotonic saline. Therefore, I assumed that a 5,000 mg/day change in sodium intake would be balanced by a 1.7-liter change of ECF giving ξ_Na = 3 mg·ml⁻¹·day⁻¹. Given that the normal sodium content of the diet is ∼4,000 mg/day and that a low-sodium diet must effectively shut off sodium excretion, I assumed that ξ_CI = 4,000 mg/day so that removing dietary carbohydrate doubles the sodium excretion rate compared with a very-low-sodium diet, as observed by Stinebaugh and Schloeder (98).

I assumed that BM was 4% of the initial BW, and the ECP was assumed to be a constant determined by the initial BM and ECF values, as determined by the regression equations of Wang et al. (108).

Whole Body Total Energy Expenditure

TEE was modeled by the following equation:

$$\mathrm{TEE}=\mathrm{TEF}+\mathrm{PAE}+\mathrm{RMR},$$ (4)

where TEF was the thermic effect of feeding, PAE was the energy expended for physical activity and exercise, and RMR was the remainder of the whole body energy expenditure, defined as the resting metabolic rate. Explicit equations for each component of energy expenditure follow.

Thermic Effect of Feeding

Feeding induces a rise of metabolic rate associated with the digestion, absorption, and short-term storage of macronutrients and was modeled by the following equation:

$$\mathrm{TEF}={\mathrm{\alpha }}_{\mathrm{F}}\mathrm{FI}+{\mathrm{\alpha }}_{\mathrm{P}}\mathrm{P}\mathrm{I}+{\mathrm{\alpha }}_{\mathrm{C}}\mathrm{C}\mathrm{I},$$ (5)

where α_F = 0.025, α_P = 0.25, and α_C = 0.075 defined the short-term thermic effect of fat, protein, and carbohydrate feeding (12).

Adaptive Thermogenesis

Energy imbalance causes an adaptation of metabolic rate that opposes weight change (22, 23, 62). Whether or not the adaptation of energy expenditure is greater than expected based on body composition changes alone has been a matter of some debate (35, 70, 111). The so-called adaptive thermogenesis is believed to affect both resting and nonresting energy expenditure and has maximum amplitude during the dynamic phase of weight change. Adaptive thermogenesis also persists during weight maintenance at an altered body weight (85). The non-RMR component of adaptive thermogenesis may reflect either altered efficiency or amount of muscular work (63–65, 87).

The onset of adaptive thermogenesis is rapid and may correspond to altered levels of circulating thyroid hormones or catecholamines (86, 111). I defined a dimensionless adaptive thermogenesis parameter, T, which was generated by a first-order process in proportion to the departure from the baseline energy intake EI_b = CI_b + FI_b + PI_b:

$${\mathrm{\tau }}_{\mathrm{T}}\frac{\mathrm{d}\mathrm{T}}{\mathrm{d}t}=\left\{\begin{array}{cc}{\mathrm{\lambda }}_1(\mathrm{\Delta }\mathrm{EI}/\mathrm{E}{\mathrm{I}}_{\mathrm{b}})-\mathrm{T},& \mathrm{\text{if}}\mathrm{EI}<\mathrm{E}{\mathrm{I}}_{\mathrm{b}}\\ {\mathrm{\lambda }}_2(\mathrm{\Delta }\mathrm{EI}/\mathrm{E}{\mathrm{I}}_{\mathrm{b}})-\mathrm{T},& \mathrm{\text{else}}\end{array}\right\}$$ (6)

where ΔEI was the change from the baseline energy intake, τ_T = 7 days was the estimated time constant for the onset of adaptive thermogenesis, and the parameters λ₁ and λ₂ quantified the effect of underfeeding and overfeeding, respectively, and were determined from the best fit to the Minnesota experiment data. The adaptive thermogenesis parameter T acted on both the RMR and PAE components of energy expenditure, as defined below. This simple model assumed that adaptive thermogenesis reacted to perturbations of EI and persisted as long as EI was different from baseline. Importantly, the model allowed for the possibility that no adaptive thermogenic mechanism was required to fit the data from the Minnesota experiment. The amount that the best fit values for the λ₁ and λ₂ parameters differ from zero provides an indication of the extent of adaptive thermogenesis that occurred during the underfeeding and overfeeding phases, respectively, of the Minnesota experiment.

Physical Activity Energy Expenditure

The energy expended for typical physical activities is proportional to the body weight of the individual (12, 105). Low-intensity physical activities may be subject to the effects of adaptive thermogenesis, whereas higher-intensity exercise appears to not be affected (87). Therefore, the following equation was used for the physical activity expenditure:

$$\mathrm{PAE}=\mathrm{\delta }(1+\sigma \mathrm{T})\mathrm{BW}+\mathrm{\upsilon }\mathrm{BW},$$ (7)

where δ was the nonexercise physical activity coefficient (in kcal·kg⁻¹·day⁻¹), υ was the exercise coefficient (in kcal·kg⁻¹·day⁻¹), and BW = FFM + FM was the body weight. The proportion of T that was allocated to the modification of nonexercise PAE was determined by the parameter σ. The adaptation of PAE with T did not distinguish between altered efficiency vs. amount of muscular work.

Resting Metabolic Rate

RMR includes the energy required to maintain irreversible metabolic fluxes such as de novo lipogenesis, gluconeogenesis, and ketogenesis as well as the turnover costs for protein, fat, and glycogen. The following equation included these components:

$$\begin{array}{c}\mathrm{R}\mathrm{M}\mathrm{R}={\mathrm{E}}_{\mathrm{c}}+{\gamma }_{\mathrm{B}}{\mathrm{M}}_{\mathrm{B}}+{\gamma }_{\mathrm{F}\mathrm{F}\mathrm{M}}[\mathrm{F}\mathrm{F}M-\mathrm{M}{}_{\mathrm{B}}-\Delta \mathrm{G}\left(1+{\mathrm{h}}_{\mathrm{g}}\right)\\ -\left(\mathrm{E}\mathrm{C}\mathrm{F}-\mathrm{E}\mathrm{C}{\mathrm{F}}_{\mathrm{\text{init}}}\right)]+{\gamma }_{\mathrm{F}}\mathrm{F}+\left(1-{\epsilon }_{\mathrm{d}}\right)\mathrm{D}\mathrm{N}\mathrm{L}\\ +\left(1-{\epsilon }_{\mathrm{g}}\right)\left(\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{F}}+\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}\right)+\left(1-{\epsilon }_{\mathrm{K}}\right)\mathrm{K}\mathrm{T}\mathrm{G}\\ +{\eta }_{\mathrm{N}}{\mathrm{N}}_{\mathrm{\text{excr}}}+\left({\eta }_{\mathrm{P}}+{\epsilon }_{\mathrm{P}}\right){\mathrm{D}}_{\mathrm{P}}+{\eta }_{\mathrm{P}}\frac{\mathrm{d}\mathrm{P}}{\mathrm{d}t}\\ +{\eta }_{\mathrm{F}}{\mathrm{D}}_{\mathrm{F}}+{\eta }_{\mathrm{F}}\frac{\mathrm{d}\mathrm{F}}{\mathrm{d}t}+{\eta }_{\mathrm{G}}{\mathrm{D}}_{\mathrm{G}}+{\eta }_{\mathrm{G}}\frac{\mathrm{d}\mathrm{G}}{\mathrm{d}\mathrm{t}},\end{array}$$ (8)

where ε_g = 0.8 was the efficiency of gluconeogenesis (12) and the constant, E_c, was a parameter chosen to ensure that the model achieved energy balance during the balanced baseline diet (see Nutrient Balance Parameter Constraints below).

The specific metabolic rate of adipose tissue was γ_F = 4.5 kcal·kg⁻¹·day⁻¹. The brain metabolic rate was γ_B = 240 kcal·kg⁻¹·day⁻¹, and its mass was M_B = 1.4 kg, which does not change with weight gain or loss (26). The baseline specific metabolic rate of the fat-free mass, γ̂_FFM = 19 kcal·kg⁻¹·day⁻¹, was determined by the specific metabolic rates of the organs multiplied by the rate of change of the organ mass with fat-free mass change according to the following equation:

$${\hat{\gamma }}_{\mathrm{FFM}}={\displaystyle \sum _{\mathrm{i}}{\mathrm{\gamma }}_{\mathrm{i}}\frac{\mathrm{d}{\mathrm{M}}_{\mathrm{i}}}{\mathrm{d}\mathrm{F}\mathrm{F}\mathrm{M}},}$$ (9)

where γ_i and M_i are the average specific metabolic rate and mass, respectively, of the organ indexed by i. The organs included skeletal muscle (γ_SM = 13 kcal·kg⁻¹·day⁻¹, M_SM = 28 kg, dM_SM/dFFM = 0.59), liver (γ_L = 200 kcal·kg⁻¹·day⁻¹, M_L = 1.8 kg, dM_L/dFFM = 0.017), kidney (γ_K = 440 kcal·kg⁻¹·day⁻¹, M_K = 0.31 kg, dM_k/dFFM = 0.0038), heart (γ_H = 440 kcal·kg⁻¹·day⁻¹, M_H = 0.33 kg, dM_H/dFFM = 0.0029), and residual lean tissue mass (γ_R = 12 kcal·kg⁻¹·day⁻¹, M_R = 23.2 kg, dM_R/dFFM = 0.37), as provided by Elia (26), and the relationship between the organ masses and FFM was determined from cross-sectional body composition data (Gallagher D, personal communication). I assumed that there was no effect on RMR arising solely from changes of glycogen content, ΔG, and its associated water or changes of ECF. Adaptive thermogenesis affected the baseline specific metabolic rate for lean tissue cell mass according to the following equation:

$${\mathrm{\gamma }}_{\mathrm{F}\mathrm{F}\mathrm{M}}={\widehat{\text{γ}}}_{\mathrm{F}\mathrm{F}\mathrm{M}}[1+(1-\mathrm{\sigma })\mathrm{T}]$$ (10)

The last seven terms of Eq. 8 accounted for the energy cost for urea synthesis and nitrogen excretion as well as the turnover of protein, fat, and glycogen. Urea synthesis requires 4 mol ATP/mol nitrogen excreted (76) and was represented by the parameter η_N = 5.4 kcal/g excreted nitrogen N_excr. To calculate the energy cost for protein turnover, consider that the whole body protein pool turns over with a synthesis rate, Synth_P, and a degradation rate, D_P (in g/day). I assumed that it cost η_PSynth_P to synthesize P and that the energy required for degradation was ε_PD_P. Since dP/dt = Synth_P − D_P, the energy cost for protein turnover was given by (η_P + ε_P)D_P + η_PdP/dt. Similar arguments led to the other terms of Eq. 8 representing the energy costs for fat and glycogen turnover, where the energy cost for degradation was negligible. The values for the parameters were η_F = 0.18 kcal/g, η_G = 0.21 kcal/g, ε_P = 0.17 kcal/g, and η_P = 0.86 kcal/g. These values were determined from the ATP costs for the respective biochemical pathways (i.e., 8 ATP/TG synthesized, 2 ATP/glycosyl unit of glycogen synthesized, 4 ATP/peptide bond synthesized plus 1 ATP for amino acid transport, and 1 ATP/peptide bond hydrolyzed) (12, 28). I assumed that 19 kcal of macronutrient oxidation was required to synthesize 1 mol of ATP (26).

Daily Average Lipolysis Rate

The daily average lipolysis rate, D_F, was modeled as

$${\mathrm{D}}_{\mathrm{F}}={\hat{\text{D}}}_{\mathrm{F}}{\left(\frac{\mathrm{F}}{{\mathrm{F}}_{\mathrm{\text{Keys}}}}\right)}^{\frac{2}{3}}[{\mathrm{L}}_{\mathrm{\text{diet}}}+{\mathrm{L}}_{\mathrm{P}\mathrm{A}}]$$ (11)

where D̂_F = 140 g/day was the baseline daily average TG turnover rate given by two-thirds of the fed lipolysis rate plus one-third of the overnight-fasted lipolysis rate (53). The (F/F_Keys)^2/3 factor accounted for the dependence of the basal lipolysis rate on the total fat mass normalized by the initial fat mass of the average Minnesota experiment subject, F_Keys. The two-thirds power reflects the hypothesis that basal lipolysis scales with adipocyte surface area, which also matches the FFA rate of appearance (R_a) as a function of body fat mass observed by Bjorntorp et al. (10). The effect of diet on lipolysis, L_diet, is determined primarily by the carbohydrate content of the diet via insulin. Furthermore, the lipolysis rate reaches half of its maximum value after ∼2 days of fasting, and the magnitude of the increase is attenuated with increasing body fat mass (58). To capture these effects, I modeled the effect of dietary carbohydrate on the daily average lipolysis rate as

$${\mathrm{\tau }}_{\mathrm{L}}\frac{\mathrm{d}{\mathrm{L}}_{\mathrm{\text{diet}}}}{\mathrm{d}t}=\frac{K_{\mathrm{L}}^{{\mathrm{S}}_{\mathrm{L}}}\left[1+({\mathrm{A}}_{\mathrm{L}}-{\mathrm{B}}_{\mathrm{L}})\times \mathrm{\text{exp}}(-k_{\mathrm{L}}\mathrm{C}\mathrm{I}/\mathrm{C}{\mathrm{I}}_{\mathrm{b}})+{\mathrm{B}}_{\mathrm{L}}\right]}{K_{\mathrm{L}}^{{\mathrm{S}}_{\mathrm{L}}}+\mathrm{M}\mathrm{A}\mathrm{X}\left\{0,{(\mathrm{F}/{\mathrm{F}}_{\mathrm{\text{Keys}}}-1)}^{{\mathrm{S}}_{\mathrm{L}}}\right\}}-{\mathrm{L}}_{\mathrm{\text{diet}}},$$ (12)

where τ_L = 1/ln(2) = 1.44 days. The term in the square brackets accounted for the modulation of lipolysis by the carbohydrate content of the diet. For example, complete starvation (CI = 0) stimulated average daily lipolysis by a factor of A_L = 3.1/[1 − exp(−2.5/τ_L)] = 3.8, as computed by the 3.1-fold increase of glycerol R_a following a 60-h fast (15) vs. the daily average glycerol R_a (53). Halving the carbohydrate content of the diet increased the average lipolysis rate by factor of 1.4, as estimated by the increased area under the circulating FFA curve following an isocaloric meal consisting of 33 vs. 66% carbohydrate (118). Given the above value for A_L, the effect of halving the carbohydrate content was modeled by choosing B_L = 0.9. The following choice for k_L ensured that the lipolysis rate was normalized for the baseline diet:

$$k_{\mathrm{L}}=\mathrm{\text{ln}}\left(\frac{{\mathrm{A}}_{\mathrm{L}}-{\mathrm{B}}_{\mathrm{L}}}{1-{\mathrm{B}}_{\mathrm{L}}}\right).$$ (13)

Although obesity increases basal lipolysis, the stimulatory effect of decreased carbohydrate intake is impaired (120). This effect was modeled in Eq. 12 by setting K_L = 4 and S_L = 2 such that the curve of lipolysis vs. CI becomes flattened as FM increases and matches the data of Klein et al. (58), where long-term vs. short-term fasting stimulated lipolysis to a lesser degree in obese vs. lean subjects.

Physical activity and exercise are known to stimulate lipolysis, and this was modeled in Eq. 11 by the factor L_PA as follows:

$${\mathrm{L}}_{\mathrm{P}\mathrm{A}}=\mathrm{\psi }\left(\frac{\mathrm{\delta }+\mathrm{\upsilon }}{{\mathrm{\delta }}_{\mathrm{\text{init}}}+{\mathrm{\upsilon }}_{\mathrm{\text{init}}}}-1\right),$$ (14)

where ψ = 0.4 was the best fit value for the measured effect of graded exercise to increase lipolysis rates determined by FFA R_a measurements (39, 57, 84, 119).

Daily Average Ketogenesis Rate

The daily rate of ketogenesis was modeled as a function of the daily lipolysis rate, the protein content in the diet, and the glycogen level as follows:

$$\mathrm{K}\mathrm{T}\mathrm{G}={\mathrm{\rho }}_{\mathrm{K}}{\mathrm{D}}_{\mathrm{F}}\left[{\mathrm{A}}_{\mathrm{K}}\left(\frac{{\mathrm{D}}_{\mathrm{F}}/{\hat{\mathrm{D}}}_{\mathrm{F}}}{{\mathrm{K}}_{\mathrm{K}}+{\mathrm{D}}_{\mathrm{F}}/{\hat{\mathrm{D}}}_{\mathrm{F}}}\right)\mathrm{\text{exp}}\left(-k_{\mathrm{P}}\frac{\mathrm{P}\mathrm{I}}{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}}\right)\mathrm{\text{exp}}\left(-k_{\mathrm{G}}\frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{\text{init}}}}\right)\right],$$ (15)

where ρ_K = 4.45 kcal/g is the average energy density of ketones calculated as average enthalpy of combustion of β-hydroxybuterate and acetoacetate in a 2:1 ratio (12). A_K = 0.8 is the maximum fraction of FFA from lipolysis converted to ketones when PI = G = 0 (9). When protein intake was at normal levels, I assumed that the maximum fraction of FFA converted to ketones was 0.4, since a protein modified fast decreases circulating ketone levels by one-half compared with fasting alone (106). Therefore, k_p = ln(0.8/0.4) = 0.69. When both glycogen and protein intake are at normal levels, I assumed that the maximum fraction of FFA converted to ketones was 0.2; therefore k_G = ln(0.4/0.2) = 0.69, and when the lipolysis rate was normal I assumed that 10% of FFA from lipolysis was converted to ketones such that K_K =1 (9).

Daily Average Ketone Excretion Rate

Ketones are excreted in the urine when circulating levels cross the renal threshold for reuptake. I assumed that

$${\mathrm{K}\mathrm{U}}_{\mathrm{excr}}=\left\{\begin{array}{cc}0,& \mathrm{\text{if}}\mathrm{K}\mathrm{T}\mathrm{G}/{\mathrm{\rho }}_{\mathrm{K}}<{\mathrm{K}\mathrm{T}\mathrm{G}}_{\mathrm{\text{thresh}}}\\ \frac{{\mathrm{\rho }}_{\mathrm{K}}{\mathrm{K}\mathrm{U}}_{\mathrm{\text{max}}}(\mathrm{K}\mathrm{T}\mathrm{G}/{\mathrm{\rho }}_{\mathrm{K}}-{\mathrm{K}\mathrm{T}\mathrm{G}}_{\mathrm{\text{thresh}}})}{({\mathrm{K}\mathrm{T}\mathrm{G}}_{\mathrm{\text{max}}}-{\mathrm{K}\mathrm{T}\mathrm{G}}_{\mathrm{\text{thresh}}})},& \mathrm{\text{else}}\end{array}\right\},$$ (16)

where KTG_thresh = 70 g/day, KU_max = 20 g/day, and KTG_max = 400 g/day such that the urinary excretion of ketones, KU_excr, matches the excretion data during various ketone infusion rates, as measured by Wildenhoff (114) and Sapir and Owen (89).

Ketone Oxidation Rate

Since ketones cannot be stored in the body in any significant quantity, I assumed that once a ketone is produced it is either oxidized or excreted. Therefore,

$$\mathrm{\text{KetOx}}=\mathrm{K}\mathrm{T}\mathrm{G}-\mathrm{K}{\mathrm{U}}_{\mathrm{\text{excr}}}.$$ (17)

Daily Average Proteolysis Rate

The daily average protein degradation rate, D_P, was given by

$${\mathrm{D}}_{\mathrm{P}}={\hat{\mathrm{D}}}_{\mathrm{P}}\left[\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{\text{Keys}}}}\right)+\mathrm{\chi }\left(\frac{\mathrm{\Delta }\mathrm{P}\mathrm{I}}{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}}\right)\right],$$ (18)

where D̂_P = 300 g/day was the baseline daily protein turnover rate (107), and I assumed that the protein degradation rate was proportional to the normalized protein content of the body (112). Although it is possible that the protein content of the diet may directly influence protein turnover as represented by the parameter χ (83), the balance of the current data suggests that χ = 0 (41, 75). Nevertheless, I included this parameter in the model to allow for this possibility should new data provide further evidence for such an effect.

Daily Average Glycogenolysis Rate

The daily average glycogen degradation rate, D_G, was given by the following equation:

$${\mathrm{D}}_{\mathrm{G}}={\hat{\mathrm{D}}}_{\mathrm{G}}\left(\frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{\text{init}}}}\right),$$ (19)

where the baseline glycogen turnover rate, D̂_G = 180 g/day, was determined by assuming that 70% was from hepatic glycogenolysis and 30% from skeletal muscle with the hepatic contribution computed as two-thirds of the fed plus one-third of the overnight-fasted hepatic glycogenolysis rate (69).

Daily Average Fat, Protein, and Glycogen Synthesis Rates

Mass conservation required that the daily average synthesis rates of fat, protein, and glycogen (Synth_F, Synth_P, and Synth_G, respectively) were given by

$$\begin{array}{c}\mathrm{\text{Synt}}{\mathrm{h}}_{\mathrm{F}}={\mathrm{D}}_{\mathrm{F}}+\frac{\mathrm{d}\mathrm{F}}{\mathrm{d}t}\\ \mathrm{\text{Synt}}{\mathrm{h}}_{\mathrm{P}}={\mathrm{D}}_{\mathrm{P}}+\frac{\mathrm{d}\mathrm{P}}{\mathrm{d}t}\\ \mathrm{\text{Synt}}{\mathrm{h}}_{\mathrm{G}}={\mathrm{D}}_{\mathrm{G}}+\frac{\mathrm{d}\mathrm{G}}{\mathrm{d}t}.\end{array}$$ (20)

Glycerol 3-Phosphate Production Rate

Because adipose tissue lacks glycerol kinase, the glycerol 3-phosphate backbone of adipose TG is derived primarily from glucose. Thus, the TG synthesis rate, Synth_F, determined the rate of glycerol 3-phosphate production, G3P, according to

$$\mathrm{G}3\mathrm{P}={\mathrm{\rho }}_{\mathrm{C}}{\mathrm{\text{Synth}}}_{\mathrm{F}}\left(\frac{{\mathrm{M}}_{\mathrm{G}}}{{\mathrm{M}}_{\mathrm{T}\mathrm{G}}}\right),$$ (21)

where M_G = 92 g/mol and M_TG = 860 g/mol are the molecular weights of glycerol and TG, respectively.

Glycerol Gluconeogenesis Rate

Lipolysis of both endogenous and exogenous TG results in the release of glycerol that can be converted to glucose via gluconeogenesis (8). Trimmer et al. (104) demonstrated that glycerol disappearance could be fully accounted for by glucose production. Therefore, I assumed that all exogenous and endogenous glycerol entered the GNG pathway according to the following equation:

$$\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{F}}=\mathrm{F}\mathrm{I}\left(\frac{{\mathrm{\rho }}_{\mathrm{C}}{\mathrm{M}}_{\mathrm{G}}}{{\mathrm{\rho }}_{\mathrm{F}}{\mathrm{M}}_{\mathrm{T}\mathrm{G}}}\right)+{\mathrm{D}}_{\mathrm{F}}{\mathrm{\rho }}_{\mathrm{C}}\left(\frac{{\mathrm{M}}_{\mathrm{G}}}{{\mathrm{M}}_{\mathrm{T}\mathrm{G}}}\right).$$ (22)

Since glycerol cannot be used by adipose tissue for TG synthesis due to lack of glycerol kinase, all glycerol released by lipolysis is eventually oxidized (apart from a negligibly small amount incorporated into altered pool sizes of nonadipose TG). By assuming that all glycerol enters the GNG pathway, any model error was limited to an overestimate of the energy expenditure associated with glycerol's initial conversion to glucose prior to oxidation. This error must be very small, since the total energy cost for glycerol GNG in the basal state was only 25 kcal/day.

Gluconeogenesis From Amino Acids

The GNG_P rate in the model referred to the net rate of gluconeogenesis from amino acid-derived carbon. Whereas all amino acids except leucine and lysine can be used as gluconeogenic substrates, the primary gluconeogenic amino acids are alanine and glutamine. Much of alanine gluconeogenesis does not contribute to the net amino acid gluconeogenic rate since the carbon skeleton of alanine is derived largely from carbohydrate precursors via skeletal muscle glycolysis (77). In an extensive review of hepatic amino acid metabolism, Jungas et al. (54) estimated that the net basal gluconeogenic rate from amino acids, GN̂G_p, was 300 kcal/day.

Several factors may regulate GNG_P, but for simplicity I have assumed that GNG_P was proportional to the normalized proteolysis rate and was influenced by the diet as follows:

$$\begin{array}{c}\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}=\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}[\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{\text{Keys}}}}\right)-{\Gamma }_{\mathrm{C}}\left(\frac{\Delta \mathrm{C}\mathrm{I}}{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}}\right)\\ +\left({\Gamma }_{\mathrm{P}}+\chi \right)\left(\frac{\Delta \mathrm{P}\mathrm{I}}{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}}\right)],\end{array}$$ (23)

where the coefficients Γ_C = 0.39 and Γ_P = 0.32 were determined by solving Eq. 23, using two sets of data. The first measured an initial nitrogen balance of −4 g/day upon removal of baseline dietary carbohydrate while keeping dietary protein at baseline values (47). I assumed that the negative nitrogen balance was driven largely by increased gluconeogenesis and thereby determined the value of Γ_C. The second study found a 56% increase of gluconeogenesis when protein intake was increased by a factor of 2.5-fold, whereas carbohydrate intake was decreased by 20% (66). Given the value of Γ_C, these data determined the value of the sum of Γ_p + χ and therefore Γ_P.

De Novo Lipogenesis Rate

DNL occurs in both the liver and adipose tissue. Under free-living conditions, adipose DNL has recently been measured to contribute ∼20% of new TG with a measured TG turnover rate of ∼50 g/day (99). Thus, adipose DNL is ∼94 kcal/day. Measurements of daily hepatic DNL in circulating very low-density lipoproteins (VLDL) have found that ∼7% of VLDL TG occurs via DNL when a basal diet of 30% fat, 50% carbohydrate, and 15% protein is consumed (49). Given that the daily VLDL TG secretion rate is ∼33 g/day (93), this corresponds to a hepatic DNL rate of ∼22 kcal/day. For an isocaloric diet of 10% fat, 75% carbohydrate, and 15% protein, hepatic DNL increases to 113 kcal/day (49).

When carbohydrate intake is excessively large and glycogen is saturated, DNL can be greatly amplified (2). Therefore, I modeled DNL as a Hill function of the normalized glycogen content with a maximum DNL rate given by the carbohydrate intake rate:

$$\mathrm{D}\mathrm{N}\mathrm{L}=\frac{\mathrm{C}\mathrm{I}\times {(\mathrm{G}/{\mathrm{G}}_{\mathrm{\text{init}}})}^{\mathrm{d}}}{{(\mathrm{G}/{\mathrm{G}}_{\mathrm{\text{init}}})}^{\mathrm{d}}+K_{\mathrm{D}\mathrm{N}\mathrm{L}}^{\mathrm{d}}}.$$ (24)

I chose K_DNL = 2 and d = 4 such that the computed DNL rate corresponded with measured in vivo DNL rates for experimentally determined carbohydrate intakes and estimated glycogen levels (1, 2, 49, 99).

Macronutrient Oxidation Rates

The whole body energy expenditure rate, TEE, was accounted for primarily by the sum of the heat produced from carbohydrate, fat, and protein oxidation plus the heat produced via flux through the DNL pathway (34) and ketogenesis. Furthermore, I assumed that the minimum carbohydrate oxidation rate was equal to the sum of the gluconeogenic rates less the flux required to produce glycerol 3-phosphate and that ketone oxidation was treated as fat oxidation. Thus, the remaining energy expenditure, TẼE, was apportioned between carbohydrate, fat, and protein oxidation according to the fractions f_C, f_F, and f_P, respectively:

$$\begin{array}{c}\mathrm{CarbOx}=\mathrm{GN}{\mathrm{G}}_{\mathrm{f}}+\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{p}}-\mathrm{G}3P+{\mathrm{f}}_{\mathrm{C}}\times \mathrm{T}\tilde{\mathrm{E}}\mathrm{E}\\ \mathrm{\text{FatOx}}=\mathrm{\text{KetOx}}+{\mathrm{f}}_{\mathrm{F}}\times \mathrm{T}\tilde{\mathrm{E}}\mathrm{E}\\ \mathrm{\text{ProtOx}}={\mathrm{f}}_{\mathrm{P}}\times \mathrm{T}\tilde{\mathrm{E}}\mathrm{E},\end{array}$$ (25)

where

$$\begin{array}{c}\mathrm{T}\tilde{\mathrm{E}}\mathrm{E}=\mathrm{T}\mathrm{E}\mathrm{E}-\left(1-{\epsilon }_{\mathrm{d}}\right)\mathrm{D}\mathrm{N}\mathrm{L}-\left(1-{\epsilon }_{\mathrm{k}}\right)\mathrm{K}\mathrm{T}\mathrm{G}-\mathrm{KetOx}\\ -\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{f}}-\mathrm{GN}{\mathrm{G}}_{\mathrm{p}}+\mathrm{G}3P.\end{array}$$ (26)

The substrate oxidation fraction for each macronutrient depends on a number of factors. First, increased lipolysis leads to concomitant increased fatty acid oxidation (15). Second, carbohydrate oxidation depends on the carbohydrate intake as well as the glycogen content (32, 61). Third, dietary protein and carbohydrate intake directly stimulate protein and carbohydrate oxidation, respectively, but dietary fat intake does not directly stimulate fat oxidation (36, 91). Fourth, I assumed that lean tissue supplies amino acids for oxidation in proportion to the proetolysis rate. Finally, although inactivity causes muscle wasting (11, 97), increased physical activity may promote nitrogen retention (14, 101, 112), and the PAE is accounted for primarily by increased oxidation of fat and carbohydrate (112). I modeled these effects by decreasing the fraction of energy expenditure derived from protein oxidation as physical activity increases.

On the basis of these physiological considerations, the substrate oxidation fractions were computed according to the following expressions:

$$\begin{array}{c}{\mathrm{f}}_{\mathrm{C}}=\frac{{\mathrm{w}}_{\mathrm{G}}({\mathrm{D}}_{\mathrm{G}}/{\hat{\mathrm{D}}}_{\mathrm{G}})+{\mathrm{w}}_{\mathrm{C}}\mathrm{M}\mathrm{A}\mathrm{X}\left\{0,(1+{\mathrm{S}}_{\mathrm{C}}\mathrm{\Delta }\mathrm{C}\mathrm{I}/\mathrm{C}{\mathrm{I}}_{\mathrm{b}})\right\}\mathrm{G}/({\mathrm{G}}_{\mathrm{\text{min}}}+\mathrm{G})}{\mathrm{Z}}\mathrm{.}\\ {\mathrm{f}}_{\mathrm{F}}=\frac{{\mathrm{w}}_{\mathrm{F}}({\mathrm{D}}_{\mathrm{F}}/{\hat{\mathrm{D}}}_{\mathrm{F}})}{\mathrm{Z}}\\ {\mathrm{f}}_{\mathrm{P}}=\frac{{\mathrm{w}}_{\mathrm{P}}\mathrm{\text{MAX}}\left\{0,(1+{\mathrm{P}}_{\mathrm{\text{sig}}})\right\}+({\mathrm{D}}_{\mathrm{P}}/{\hat{\mathrm{D}}}_{\mathrm{P}}){\mathrm{S}}_{\mathrm{A}}\exp \left(-{\mathrm{k}}_{\mathrm{A}}(\mathrm{\delta }+\mathrm{\upsilon })/({\mathrm{\delta }}_{\mathrm{b}}+{\mathrm{\upsilon }}_{\mathrm{b}})\right)}{\mathrm{Z}}\\ {\mathrm{\tau }}_{\mathrm{P}\mathrm{I}}\frac{\mathrm{d}{\mathrm{P}}_{\mathrm{\text{sig}}}}{\mathrm{d}t}={\mathrm{S}}_{\mathrm{P}}\mathrm{\Delta }\mathrm{P}\mathrm{I}/\mathrm{P}{\mathrm{I}}_{\mathrm{b}}-{\mathrm{P}}_{\mathrm{\text{sig}}}\\ {\mathrm{S}}_{\mathrm{P}}=\left\{\begin{array}{cc}{\mathrm{S}}_{\mathrm{P}}^{+},& \mathrm{\text{if}}\mathrm{\Delta }\mathrm{P}\mathrm{I}>0\\ {\mathrm{S}}_{\mathrm{P}}^{-},& \mathrm{\text{else}}\end{array}\right\},\end{array}$$ (27)

where the w and S were dimensionless model parameters and ΔCI and ΔPI were changes from the basal carbohydrate intake, CI_b, and protein intake, PI_b, respectively. The small parameter, G_min = 10 g, was chosen such that carbohydrate oxidation was restrained as glycogen decreases and prevents glycogen from becoming negative. The signal for dietary protein intake perturbations, P_sig, changes with a time constant of τ_PI = 1.1 days, in accordance with the data of Rand et al. (79), and the model allows for an asymmetry between positive and negative perturbations of dietary protein. To normalize for the baseline physical activity, the constant k_A was chosen such that k_A = ln(S_A). Z was a normalization factor equal to the sum of the numerators so that the sum of the fractions f_C, f_F, and f_P was equal to 1.

Respiratory Gas Exchange

Oxidation of carbohydrate, fat, and protein was associated with consumption of oxygen (O₂) and production of carbon dioxide (CO₂) according to the stoichiometry of the net biochemical reactions (31, 37):

$$\begin{array}{c}1\mathrm{g}\mathrm{\text{carbohydrate}}+0.831\mathrm{\text{liters}}{\mathrm{O}}_2\to 0.831\mathrm{liters}{\mathrm{C}\mathrm{O}}_2\\ +0.6\mathrm{g}{\mathrm{H}}_2\mathrm{O}1\mathrm{g}\mathrm{\text{fat}}+2.03\mathrm{\text{liters}}{\mathrm{C}\mathrm{O}}_2\to 1.43\mathrm{\text{liters}}{\mathrm{C}\mathrm{O}}_2\\ +1.09\mathrm{g}{\mathrm{H}}_2\mathrm{O}1\mathrm{g}\mathrm{\text{protein}}+0.966\mathrm{\text{liters}}{\mathrm{O}}_2\to 0.782\mathrm{\text{liters}}\\ {\mathrm{C}\mathrm{O}}_2+0.45\mathrm{g}{\mathrm{H}}_2\mathrm{O}+0.16\mathrm{g}\mathrm{N}\mathrm{.}\end{array}$$ (28)

Gluconeogenesis, lipogenesis, ketogenesis (with subsequent excretion), and glycerol 3-phosphate production also contribute to gas exchange according to the following net reactions (27, 31, 37):

$$\begin{array}{l}1\mathrm{g}\mathrm{\text{protein}}+0.126\mathrm{\text{liters}}\mathrm{C}{\mathrm{O}}_2\to 1\mathrm{g}\text{carbohydrate}+0.16\mathrm{g}N\hfill \\ 1\mathrm{g}\mathrm{\text{glycerol}}+0.133\mathrm{\text{liters}}{\mathrm{O}}_2\to 1g\mathrm{\text{carbohydrate}}\hfill \\ 1g\text{carbohydrate}\to 0.37\mathrm{g}\text{fat}+0.238\text{liters}\mathrm{C}{\mathrm{O}}_2+0.2\mathrm{g}{\mathrm{H}}_2\mathrm{O}\hfill \\ 1\mathrm{g}\text{fat}+0.52\mathrm{\text{liters}}{\mathrm{O}}_2\to 1.57\mathrm{g}\mathrm{\text{ketones}}\hfill \\ 1g\mathrm{\text{carbohydrate}}\to 1\mathrm{g}\mathrm{\text{glycerol}}+0.133\mathrm{\text{liters}}{\mathrm{O}}_2.\hfill \end{array}$$ (29)

Oxidation of carbohydrate, fat, and protein can occur either directly or subsequent to intermediate exchange via lipogenesis or gluconeogenesis. In either case, the final ratio of CO₂ produced to O₂ consumed (i.e., the RQ) is independent of any intermediate exchanges in accordance with the principles of indirect calorimetry (27, 31, 37).

The simulated O₂ consumption (V̇o₂) and CO₂ production (V̇co₂) (in liters/day) were computed according to

$$\begin{array}{c}\dot{V}{\mathrm{O}}_2=0.831\frac{\mathrm{\text{CarbOx}}}{{\rho }_{\mathrm{C}}}+2.03\frac{\mathrm{\text{FatOx}}}{{\rho }_{\mathrm{F}}}+0.966\frac{\mathrm{\text{ProtOx}}}{{\rho }_{\mathrm{P}}}\\ +0.133\left(\frac{\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{F}}}{{\rho }_{\mathrm{C}}}-\frac{\mathrm{G}3\mathrm{P}}{{\rho }_{\mathrm{C}}}\right)+0.33\frac{\mathrm{K}{\mathrm{U}}_{\mathrm{\text{excr}}}}{{\rho }_{\mathrm{K}}}\\ \dot{\mathrm{V}}\mathrm{C}{\mathrm{O}}_2=0.831\frac{\mathrm{\text{CarbOx}}}{{\rho }_{\mathrm{C}}}+1.43\frac{\mathrm{\text{FatOx}}}{{\rho }_{\mathrm{F}}}+0.782\frac{\mathrm{\text{ProtOx}}}{{\rho }_{\mathrm{P}}}\\ -0.126\frac{\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}}{{\rho }_{\mathrm{P}}}+0.238\frac{\mathrm{D}\mathrm{N}\mathrm{L}}{{\rho }_{\mathrm{C}}}.\end{array}$$ (30)

The RQ was computed by dividing V̇co₂ by V̇o₂. To compute the NPRQ, the total rate of nitrogen excretion was calculated:

$${\mathrm{N}}_{\mathrm{\text{excr}}}=\frac{(\mathrm{\text{ProtOx}}+\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}})}{6.25{\mathrm{\rho }}_{\mathrm{P}}}$$ (31)

where the factor 6.25 was the number of grams of protein per gram of nitrogen.

When comparing model predictions for fuel selection with experiments employing 24-h room indirect calorimetry, I used the simulated V̇co₂ and V̇o₂ values from Eq. 30 and computed macronutrient oxidation rates as if they were the actual indirect calorimetry measurements.

Nutrient Balance Parameter Constraints

The baseline diet may not necessarily result in macronutrient or energy balance. Therefore, I defined the following parameters to specify the initial degree of imbalance for fat, carbohydrate, and protein: F_imbal, G_imbal, and P_imbal, respectively. Therefore, the baseline diet satisfies the following relationship: TEE + F_imbal + G_imbal + P_imbal = EI_b, where the subscript _b refers to the baseline state. Therefore, by explicitly expressing the TEE in the baseline state I derived the following expression:

$$\begin{array}{c}\mathrm{E}{\mathrm{I}}_{\mathrm{b}}=\mathrm{T}\mathrm{E}{\mathrm{F}}_{\mathrm{b}}+\mathrm{P}\mathrm{A}{\mathrm{E}}_{\mathrm{b}}+\mathrm{R}\mathrm{M}{\mathrm{R}}_{\mathrm{b}}\\ =\mathrm{T}\mathrm{E}{\mathrm{F}}_{\mathrm{b}}+\left({\delta }_{\mathrm{b}}+{\upsilon }_{\mathrm{b}}\right)\mathrm{B}{\mathrm{W}}_{\mathrm{b}}\\ +{\mathrm{E}}_{\mathrm{C}}+{\gamma }_{\mathrm{B}}{\mathrm{M}}_{\mathrm{B}}+{\gamma }_{\mathrm{F}\mathrm{F}\mathrm{M}}\left(\mathrm{F}\mathrm{F}{\mathrm{M}}_{\mathrm{b}}-{\mathrm{M}}_{\mathrm{B}}\right)+{\gamma }_{\mathrm{F}}{\mathrm{F}}_{\mathrm{b}}\\ +\left(1-{\epsilon }_{\mathrm{d}}\right)\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}+\left(1-{\epsilon }_{\mathrm{g}}\right)\left(\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{F}}+\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}\right)\\ +\left(1-{\epsilon }_{\mathrm{k}}\right)\mathrm{K}\mathrm{T}{\mathrm{G}}_{\mathrm{b}}+\left({\eta }_{\mathrm{P}}+{\epsilon }_{\mathrm{P}}\right){\widehat{\mathrm{D}}}_{\mathrm{P}}+{\eta }_{\mathrm{F}}{\widehat{\mathrm{D}}}_{\mathrm{F}}+{\eta }_{\mathrm{G}}{\widehat{\mathrm{D}}}_{\mathrm{G}}\\ +\left(1+{\eta }_{\mathrm{F}}/{\rho }_{\mathrm{F}}\right){\mathrm{F}}_{\mathrm{imbal}}+\left(1+{\eta }_{\mathrm{G}}/{\rho }_{\mathrm{C}}\right){\mathrm{G}}_{\mathrm{imbal}}\\ +\left(1+{\eta }_{\mathrm{P}}/{\rho }_{\mathrm{P}}\right){\mathrm{P}}_{\mathrm{imbal}}+{\eta }_{\mathrm{N}}\frac{\left(\mathrm{P}{\mathrm{I}}_{\mathrm{b}}-{\mathrm{P}}_{\mathrm{\text{imbal}}}\right)}{6.25{\rho }_{\mathrm{P}}},\end{array}$$ (32)

which was solved for the constant E_c.

Assuming negligible baseline ketone excretion, rearrangement of the nutrient balance equations gave

$$\begin{array}{c}\mathrm{\text{CarbO}}{\mathrm{x}}_{\mathrm{b}}=\mathrm{C}{\mathrm{I}}_{\mathrm{b}}-\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}+\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}+\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{F}}-\mathrm{G}3{\mathrm{P}}_{\mathrm{b}}-{\mathrm{G}}_{\mathrm{imbal}}\\ \mathrm{\text{FatO}}{\mathrm{x}}_{\mathrm{b}}=3{\mathrm{M}}_{\mathrm{F}\mathrm{F}\mathrm{A}}\mathrm{F}{\mathrm{I}}_{\mathrm{b}}/{\mathrm{M}}_{\mathrm{T}\mathrm{G}}+{\epsilon }_{\mathrm{d}}\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}-\left(1-{\epsilon }_{\mathrm{k}}\right)\mathrm{K}\mathrm{T}{\mathrm{G}}_{\mathrm{b}}-{\mathrm{F}}_{\mathrm{imbal}}\\ \mathrm{\text{ProtO}}{\mathrm{x}}_{\mathrm{b}}=\mathrm{P}{\mathrm{I}}_{\mathrm{b}}-\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{\text{imbal}}}.\end{array}$$ (33)

Next, I defined the following parameters:

$$\begin{array}{l}{\zeta }_{\mathrm{F}}\equiv \frac{\left(3{\mathrm{M}}_{\mathrm{F}\mathrm{F}\mathrm{A}}\mathrm{F}{\mathrm{I}}_{\mathrm{b}}/{\mathrm{M}}_{\mathrm{T}\mathrm{G}}+{\epsilon }_{\mathrm{d}}\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}+{\epsilon }_{\mathrm{k}}\mathrm{K}\mathrm{T}{\mathrm{G}}_{\mathrm{b}}-{\mathrm{F}}_{\mathrm{\text{imbal}}}\right)}{\Omega }\hfill \\ {\zeta }_{\mathrm{C}}\equiv \frac{\left(\mathrm{C}{\mathrm{I}}_{\mathrm{b}}-\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}-{\mathrm{G}}_{{}_{\mathrm{\text{imbal}}}}\right)}{\Omega }\hfill \\ {\zeta }_{\mathrm{P}}\equiv \frac{\left(\mathrm{P}{\mathrm{I}}_{\mathrm{b}}-\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{\text{imbal}}}\right)}{\Omega },\hfill \end{array}$$ (34)

where Ω was given by

$$\begin{array}{c}\mathrm{\Omega }\equiv \mathrm{E}{\mathrm{I}}_{\mathrm{b}}-(1-{\mathrm{\epsilon }}_{\mathrm{d}})\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}-(1-{\mathrm{\epsilon }}_{\mathrm{k}})\mathrm{K}\mathrm{T}{\mathrm{G}}_{\mathrm{b}}-\mathrm{\text{KetO}}{\mathrm{x}}_{\mathrm{b}}\\ -\mathrm{G}\hat{\text{N}}{\mathrm{G}}_{\mathrm{F}}-\mathrm{G}\hat{N}{\mathrm{G}}_{\mathrm{P}}+\mathrm{G}3{\mathrm{P}}_{\mathrm{b}}-(1+{\mathrm{\eta }}_{\mathrm{G}}/{\mathrm{\rho }}_{\mathrm{C}}){\mathrm{G}}_{\mathrm{\text{imbal}}}\\ -(1+{\mathrm{\eta }}_{\mathrm{F}}/{\mathrm{\rho }}_{\mathrm{F}}){\mathrm{F}}_{\mathrm{\text{imbal}}}-(1+{\mathrm{\eta }}_{\mathrm{P}}/{\mathrm{\rho }}_{\mathrm{P}}){\mathrm{P}}_{\mathrm{\text{imbal}}}.\end{array}$$ (35)

By substituting Eqs. 25 and 27 at the initial state, I obtained

$$\begin{array}{ccc}\frac{{\mathrm{w}}_{\mathrm{F}}{\left({\mathrm{F}}_{\mathrm{\text{init}}}/{\mathrm{F}}_{\mathrm{\text{Keys}}}\right)}^{2/3}}{\left({\mathrm{P}}_{\text{init}}/{\mathrm{P}}_{\mathrm{Keys}}\right)+{\mathrm{w}}_{\mathrm{P}}+{\mathrm{w}}_{\mathrm{G}}\left({\mathrm{G}}_{\mathrm{init}}/{\mathrm{G}}_{\mathrm{\text{Keys}}}\right)+{\mathrm{w}}_{\mathrm{C}}\left({\mathrm{G}}_{\mathrm{init}}/\left({\mathrm{G}}_{\mathrm{\text{init}}}+{\mathrm{G}}_{\mathrm{\text{min}}}\right)\right)+{\mathrm{w}}_{\mathrm{F}}{\left({\mathrm{F}}_{\mathrm{\text{init}}}/{\mathrm{F}}_{\mathrm{\text{Keys}}}\right)}^{2/3}}& =& {\zeta }_{\mathrm{F}}\\ \frac{{\mathrm{w}}_{\mathrm{G}}\left({\mathrm{G}}_{\mathrm{\text{init}}}/{\mathrm{G}}_{\mathrm{Keys}}\right)+{\mathrm{w}}_{\mathrm{C}}\left({\mathrm{G}}_{\mathrm{\text{init}}}/\left({\mathrm{G}}_{\mathrm{\text{init}}}+{\mathrm{G}}_{\mathrm{\text{min}}}\right)\right)}{\left({\mathrm{P}}_{\mathrm{\text{init}}}/{\mathrm{P}}_{\mathrm{\text{Keys}}}\right)+{\mathrm{w}}_{\mathrm{P}}+{\mathrm{w}}_{\mathrm{G}}\left({\mathrm{G}}_{\mathrm{\text{init}}}/{\mathrm{G}}_{\mathrm{\text{Keys}}}\right)+{\mathrm{w}}_{\mathrm{C}}\left({\mathrm{G}}_{\mathrm{\text{init}}}/\left({\mathrm{G}}_{\mathrm{\text{init}}}+{\mathrm{G}}_{\mathrm{\text{min}}}\right)\right)+{\mathrm{w}}_{\mathrm{F}}{\left({\mathrm{F}}_{\text{init}}/{\mathrm{F}}_{\mathrm{\text{Keys}}}\right)}^{2/3}}& =& {\zeta }_{\mathrm{C}}\\ \frac{\left({\mathrm{P}}_{\mathrm{\text{init}}}/{\mathrm{P}}_{\mathrm{\text{Keys}}}\right)+{\mathrm{w}}_{\mathrm{P}}}{\left({\mathrm{P}}_{\mathrm{\text{init}}}/{\mathrm{P}}_{\mathrm{\text{Keys}}}\right)+{\mathrm{w}}_{\mathrm{P}}+{\mathrm{w}}_{\mathrm{G}}\left({\mathrm{G}}_{\mathrm{\text{init}}}/{\mathrm{G}}_{\mathrm{\text{Keys}}}\right)+{\mathrm{w}}_{\mathrm{C}}\left({\mathrm{G}}_{\mathrm{\text{init}}}/\left({\mathrm{G}}_{\mathrm{\text{init}}}+{\mathrm{G}}_{\mathrm{\text{min}}}\right)\right)+{\mathrm{w}}_{\mathrm{F}}{\left({\mathrm{F}}_{\mathrm{\text{init}}}/{\mathrm{F}}_{\mathrm{\text{Keys}}}\right)}^{2/3}}& =& {\zeta }_{\mathrm{P}},\end{array}$$ (36)

where F_init, G_init, and P_init were the initial values for body fat, glycogen, and protein, respectively. Elementary algebra led to the following parameter constraints required to achieve the specified macronutrient imbalance:

$$\begin{array}{c}{\mathrm{w}}_{\mathrm{G}}=\frac{{\zeta }_{\mathrm{C}}/{\zeta }_{\mathrm{P}}\left({\mathrm{P}}_{\mathrm{\text{init}}}/{\mathrm{P}}_{\mathrm{\text{Keys}}}+{\mathrm{w}}_{\mathrm{P}}\right)\left({\mathrm{G}}_{\mathrm{\text{Keys}}}/({\mathrm{G}}_{\mathrm{\text{init}}}\right)}{1+{\mathrm{w}}_{\mathrm{C}:G}\left({\mathrm{G}}_{\mathrm{\text{Keys}}}/{\mathrm{G}}_{\mathrm{\text{init}}}+{\text{G}}_{\mathrm{\text{min}}}\text{)}\right)}\\ {\mathrm{w}}_{\mathrm{F}}+\left(\frac{{\zeta }_{\mathrm{F}}}{1-{\zeta }_{\mathrm{F}}}\right)\left(1+\frac{{\zeta }_{\mathrm{C}}}{{\zeta }_{\mathrm{P}}}\right)\left(\frac{{\mathrm{P}}_{\mathrm{\text{init}}}}{{\mathrm{P}}_{\mathrm{\text{Keys}}}}+{\mathrm{w}}_{\mathrm{P}}\right){\left(\frac{{\mathrm{F}}_{\mathrm{\text{Keys}}}}{{\mathrm{F}}_{\mathrm{\text{init}}}}\right)}^{2/3},\end{array}$$ (37)

where w_C:G = w_C/w_G.

Carbohydrate Perturbation Constraint

The parameters w_C and S_C determined how the model adapted to changes of carbohydrate intake. I specified that an additional dietary carbohydrate intake, ΔCI, above baseline, CI_b, resulted in an initial positive carbohydrate imbalance of κ_CΔCI, where 0 < κ_C < 1 specified the proportion of ΔCI directed toward glycogen storage. Thus, the glycogen increment was ΔG = κ_CΔCI/ρ_C. The goal was to solve for the parameter S_C such that the correct amount of carbohydrate was oxidized and deposited as glycogen during short-term carbohydrate overfeeding. Based on the carbohydrate overfeeding study of Horton et al. (48), I chose κ_C = 0.5 when ΔCI = 1,500 kcal/day.

The change of TEE was given by

$$\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{E}=\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{F}+\mathrm{\Delta }\mathrm{P}\mathrm{A}\mathrm{E}+\mathrm{\Delta }\mathrm{R}\mathrm{M}\mathrm{R}$$ (38)

For a carbohydrate perturbation, the perturbed energy expenditure components were

$$\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{F}={\mathrm{\alpha }}_{\mathrm{C}}\mathrm{\Delta }\mathrm{C}\mathrm{I}$$ (39)

$$\mathrm{\Delta }\mathrm{P}\mathrm{A}\mathrm{E}={\mathrm{\delta }}_{\mathrm{b}}\mathrm{B}{\mathrm{W}}_{\mathrm{b}}\mathrm{\sigma }\mathrm{\Delta }\mathrm{T}$$ (40)

$$\begin{array}{c}\mathrm{\Delta }\mathrm{R}\mathrm{M}\mathrm{R}=\frac{{\mathrm{\kappa }}_{\mathrm{C}}{\mathrm{\eta }}_{\mathrm{G}}}{{\mathrm{\rho }}_{\mathrm{C}}}\mathrm{\Delta }\mathrm{C}\mathrm{I}+{\hat{\mathrm{\gamma }}}_{\mathrm{F}\mathrm{F}\mathrm{M}}\mathrm{F}\mathrm{F}{\mathrm{M}}_{\mathrm{b}}(1-\mathrm{\sigma })\mathrm{\Delta }\mathrm{T}\\ +(1-{\mathrm{\epsilon }}_{\mathrm{d}})\mathrm{\Delta }\mathrm{D}\mathrm{N}\mathrm{L}+(1-{\mathrm{\epsilon }}_{\mathrm{g}})\mathrm{\Delta }\mathrm{G}\mathrm{N}\mathrm{G},\end{array}$$ (41)

where

$$\mathrm{\Delta }\mathrm{T}={\mathrm{\lambda }}_2\frac{\mathrm{\Delta }\mathrm{C}\mathrm{I}}{\mathrm{E}{\mathrm{I}}_{\mathrm{b}}}\left(1-{\mathit{\tau }}_{\mathrm{T}}+{\tau }_{\mathrm{T}}\mathrm{\text{exp}}(-1/{\mathit{\tau }}_{\mathrm{T}})\right)$$ (42)

was the average value of the thermogenesis parameter, T, over 1 day and ΔDNL was computed at the midpoint of the glycogen increment according to the following equation:

$$\mathrm{\Delta }\mathrm{D}\mathrm{N}\mathrm{L}=\frac{(\mathrm{C}{\mathrm{I}}_{\mathrm{b}}+\mathrm{\Delta }\mathrm{C}\mathrm{I}){(1+\mathrm{\Delta }\mathrm{G}/2{\mathrm{G}}_{\mathrm{\text{init}}})}^{\mathrm{d}}}{{\mathrm{K}}_{\mathrm{D}\mathrm{N}\mathrm{L}}^{\mathrm{d}}+{(1+\mathrm{\Delta }\mathrm{G}/2{\mathrm{G}}_{\mathrm{\text{init}}})}^{\mathrm{d}}}-\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}.$$ (43)

The change of the gluconeogenic rate, ΔGNG, was given by

$$\begin{array}{c}\mathrm{\Delta }\mathrm{G}\mathrm{N}\mathrm{G}=-{\Gamma }_{\mathrm{C}}\left(\frac{\mathrm{\Delta }\mathrm{C}\mathrm{I}}{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}}\right)\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}+{\rho }_{\mathrm{C}}\left(\frac{{\mathrm{M}}_{\mathrm{G}}}{{\mathrm{M}}_{\mathrm{T}\mathrm{G}}}\right){\widehat{\mathrm{D}}}_{\mathrm{F}}[\left({\mathrm{A}}_{\mathrm{L}}-{\mathrm{B}}_{\mathrm{L}}\right)\\ \times \exp \left(-{\mathrm{k}}_{\mathrm{L}}\left(1+\mathrm{\Delta }\mathrm{C}\mathrm{I}/\mathrm{C}{\mathrm{I}}_{\mathrm{b}}\right)\right)+{\mathrm{B}}_{\mathrm{L}}-1].\end{array}$$ (44)

The carbohydrate balance Eq. 1 and the carbohydrate oxidation Eq. 25 gave

$${\mathrm{f}}_{\mathrm{C}}=\frac{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}+(1-{\mathrm{\kappa }}_{\mathrm{C}})\mathrm{\Delta }\mathrm{C}\mathrm{I}-(\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}+\mathrm{\Delta }\mathrm{D}\mathrm{N}\mathrm{L})}{\mathrm{T}\stackrel{\sim }{\text{E}}\mathrm{E}}.$$ (45)

Since I assume that the baseline diet results in a state of energy balance, I obtained the following equation for TẼE:

$$\begin{array}{c}\mathrm{T}\stackrel{\sim }{\mathrm{E}}\mathrm{E}=\mathrm{E}{\mathrm{I}}_{\mathrm{b}}+\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{E}-(1-{\mathrm{\epsilon }}_{\mathrm{d}})\mathrm{D}\mathrm{N}\mathrm{L}-(1-{\mathrm{\epsilon }}_{\mathrm{k}})\mathrm{K}\mathrm{T}\mathrm{G}\\ -\mathrm{\text{KetOx}}-\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{F}}-\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}+\mathrm{G}3P.\end{array}$$ (46)

For simplicity, I assumed that

$$\begin{array}{l}\mathrm{G}3\mathrm{P}=G3{\mathrm{P}}_{\mathrm{b}}\hfill \\ \mathrm{K}\mathrm{T}\mathrm{G}=\mathrm{\text{KetOx}}=\mathrm{K}\mathrm{T}{\mathrm{G}}_{\mathrm{b}}.\hfill \end{array}$$ (47)

I define the parameter Θ as

$$\Theta \equiv \frac{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}+(1-{\mathrm{\kappa }}_{\mathrm{C}})\mathrm{\Delta }\mathrm{C}\mathrm{I}-(\mathrm{D}\mathrm{N}{\mathrm{L}}_{\mathrm{b}}+\mathrm{\Delta }\mathrm{D}\mathrm{N}\mathrm{L})}{\mathrm{T}\stackrel{\sim }{\text{E}}\mathrm{E}}.$$ (48)

Therefore, using Eq. 27, I solved Eq. 45 for S_C, which gave the carbohydrate feeding constraint:

$${\mathrm{S}}_{\mathrm{C}}=\frac{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}}{\mathrm{\Delta }\mathrm{C}\mathrm{I}}\left[\frac{\Theta (1+{\mathrm{w}}_{\mathrm{p}}+{\stackrel{\sim }{\text{w}}}_{\mathrm{F}}+{\stackrel{\sim }{\text{w}}}_{\mathrm{G}})}{(1-\Theta ){\mathrm{w}}_{\mathrm{C}}}-\frac{{\stackrel{\sim }{\text{w}}}_{\mathrm{G}}}{(1-\Theta ){\mathrm{w}}_{\mathrm{C}}}-1\right],$$ (49)

where

$$\begin{array}{ccc}{\tilde{\mathrm{w}}}_{\mathrm{F}}& =& {\mathrm{w}}_{\mathrm{F}}\left[\left({\mathrm{A}}_{\mathrm{L}}-{\mathrm{B}}_{\mathrm{L}}\right)\times \mathrm{\text{exp}}\left(-{\mathrm{k}}_{\mathrm{L}}\left(1+\mathrm{\Delta }\mathrm{C}\mathrm{I}/\mathrm{C}{\mathrm{I}}_{\mathrm{b}}\right)\right)+{\mathrm{B}}_{\mathrm{L}}\right]\\ {\tilde{\mathrm{w}}}_{\mathrm{G}}& =& {\mathrm{w}}_{\mathrm{G}}\left(1+\frac{\mathrm{\Delta }\mathrm{G}}{2{\mathrm{G}}_{\mathrm{\text{init}}}}\right).\end{array}$$ (50)

Protein Perturbation Constraint

The parameters w_P and SP+ determined how the model adapted short-term substrate oxidation rates to changes of protein intake. In a meticulous study of whole body protein balance, Oddoye and Margen (74) measured nitrogen balance in subjects consuming isocaloric diets with moderate or high protein content. These studies found that ∼90% of the additional dietary nitrogen on the high-protein diet was rapidly excreted such that κ_P = 0.1 when ΔPI = 640 kcal/day, ΔCI = −310 kcal/day, and ΔFI = −330 kcal/day.

To compute the value for SP+ to match the data of Oddoye and Margen (74), I began with the protein balance Eqs. 1 and 25 to derive

$${\mathrm{f}}_{\mathrm{P}}=\frac{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}+(1-{\mathrm{\kappa }}_{\mathrm{P}})\mathrm{\Delta }\mathrm{P}\mathrm{I}-\mathrm{G}\hat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}-\mathrm{\Delta }\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}}{\mathrm{T}\stackrel{\sim }{\mathrm{E}}\mathrm{E}},$$ (51)

where the changes of gluconeogenic rates were given by

$$\begin{array}{c}\mathrm{\Delta }\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}=\mathrm{G}\widehat{\mathrm{N}}{\mathrm{G}}_{\mathrm{P}}\left[\left({\Gamma }_{\mathrm{P}}+\chi \right)\frac{\mathrm{\Delta }\mathrm{P}\mathrm{I}}{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}}-{\Gamma }_{\mathrm{C}}\frac{\mathrm{\Delta }\mathrm{C}\mathrm{I}}{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}}\right]\\ \mathrm{\Delta }\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{F}}=\mathrm{\Delta }\mathrm{F}\mathrm{I}\left(\frac{{\rho }_{\mathrm{C}}{\mathrm{M}}_{\mathrm{G}}}{{\rho }_{\mathrm{F}}{\mathrm{M}}_{\mathrm{T}\mathrm{G}}}\right)+{\rho }_{\mathrm{C}}\left(\frac{{\mathrm{M}}_{\mathrm{G}}}{{\mathrm{M}}_{\mathrm{T}\mathrm{G}}}\right){\widehat{\mathrm{D}}}_{\mathrm{F}}[\left({\mathrm{A}}_{\mathrm{L}}-{\mathrm{B}}_{\mathrm{L}}\right)\\ \times \exp \left(-{\mathrm{k}}_{\mathrm{L}}\left(1+\frac{\mathrm{\Delta }\mathrm{C}\mathrm{I}}{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}}\right)\right)+{\mathrm{B}}_{\mathrm{L}-1}].\end{array}$$ (52)

The change of TEE was given by

$$\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{E}=\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{F}+\mathrm{\Delta }\mathrm{P}\mathrm{A}\mathrm{E}+\mathrm{\Delta }\mathrm{R}\mathrm{M}\mathrm{R},$$ (53)

where

$$\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{F}=\mathrm{}{\mathrm{\alpha }}_{\mathrm{C}}\mathrm{\Delta }\mathrm{C}\mathrm{I}+{\mathrm{\alpha }}_{\mathrm{F}}\mathrm{\Delta }\mathrm{F}\mathrm{I}+{\mathrm{\alpha }}_{\mathrm{P}}\mathrm{\Delta }\mathrm{P}\mathrm{I},$$ (54)

and

$$\begin{array}{c}\mathrm{\Delta }\mathrm{R}\mathrm{M}\mathrm{R}=\frac{{\mathrm{\eta }}_{\mathrm{P}}}{{\mathrm{\rho }}_{\mathrm{P}}}\mathrm{\Delta }\mathrm{P}\mathrm{I}\left({\mathrm{\kappa }}_{\mathrm{P}}+\frac{{\mathrm{\rho }}_{\mathrm{P}}\mathrm{\chi }{\hat{\mathrm{D}}}_{\mathrm{P}}}{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}}\right)+(1-{\mathrm{\epsilon }}_{\mathrm{g}})(\mathrm{\Delta }\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{F}}\\ +\mathrm{\Delta }\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}})+(1-{\mathrm{\epsilon }}_{\mathrm{d}})\mathrm{\Delta }\mathrm{D}\mathrm{N}\mathrm{L}.\end{array}$$ (55)

Since the perturbed diet was isocaloric and there were no changes of physical activity,

$$\mathrm{\Delta }\mathrm{P}\mathrm{A}\mathrm{E}=\mathrm{\Delta }\mathrm{T}=0.$$ (56)

I also assumed that

$$\begin{array}{l}\mathrm{G}3\mathrm{P}=\mathrm{G}3{\mathrm{P}}_{\mathrm{b}}\hfill \\ \mathrm{K}\mathrm{T}\mathrm{G}=\mathrm{\text{KetOx}}=\mathrm{K}\mathrm{T}{\mathrm{G}}_{\mathrm{b}}\hfill \end{array}$$ (57)

and that glycogen would change by approximately ΔG ≈ κ_C ΔCI/ρ_C, thereby altering DNL as follows:

$$\mathrm{\Delta }\mathrm{D}\mathrm{N}\mathrm{L}=\frac{(\mathrm{C}{\mathrm{I}}_{\mathrm{b}}+\mathrm{\Delta }\mathrm{C}\mathrm{I}){(1+\mathrm{\Delta }\mathrm{G}/{\mathrm{G}}_{\mathrm{\text{init}}})}^{\mathrm{d}}}{K_{\mathrm{D}\mathrm{N}\mathrm{L}}^{\mathrm{d}}+{(1+\mathrm{\Delta }\mathrm{G}/{\mathrm{G}}_{\mathrm{\text{init}}})}^{\mathrm{d}}}-\frac{\mathrm{C}{\mathrm{I}}_{\mathrm{b}}}{K_{\mathrm{D}\mathrm{N}\mathrm{L}}^{\mathrm{d}}+1}.$$ (58)

Using Eq. 27, I solved Eq. 51 for S_P, which gave the following constraint:

$${\mathrm{S}}_{\mathrm{P}}^{+}=\frac{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}}{\mathrm{\Delta }\mathrm{P}\mathrm{I}}\left[\frac{\mathrm{\Phi }({\stackrel{\sim }{\mathrm{w}}}_{\mathrm{F}}+{\stackrel{\sim }{\mathrm{w}}}_{\mathrm{G}}+{\stackrel{\sim }{\mathrm{w}}}_{\mathrm{C}})}{(1-\mathrm{\Phi }){\mathrm{w}}_{\mathrm{P}}}-\frac{(1+\mathrm{\chi }\mathrm{\Delta }\mathrm{P}\mathrm{I}/\mathrm{P}{\mathrm{I}}_{\mathrm{b}})}{{\mathrm{w}}_{\mathrm{P}}}-1\right],$$ (59)

where Φ was defined as

$$\mathrm{\Phi }\equiv \frac{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}+(1-{\mathrm{\kappa }}_{\mathrm{P}})\mathrm{\Delta }\mathrm{P}\mathrm{I}-\mathrm{G}\hat{\text{N}}{\mathrm{G}}_{\mathrm{P}}-\mathrm{\Delta }\mathrm{G}\mathrm{N}{\mathrm{G}}_{\mathrm{P}}}{\mathrm{T}\stackrel{\sim }{\mathrm{E}}\mathrm{E}}$$ (60)

and

$$\begin{array}{c}{\tilde{\mathrm{w}}}_{\mathrm{F}}={\mathrm{w}}_{\mathrm{F}}\left[\left({\mathrm{A}}_{\mathrm{L}}-{\mathrm{B}}_{\mathrm{L}}\right)\times \exp \left(-{\mathrm{k}}_{\mathrm{L}}\left(1+\mathrm{\Delta }\mathrm{C}\mathrm{I}/\mathrm{C}{\mathrm{I}}_{\mathrm{b}}\right)\right)+{\mathrm{B}}_{\mathrm{L}}\right]\\ {\tilde{\mathrm{w}}}_{\mathrm{C}}={\mathrm{w}}_{\mathrm{C}}\left(1+{\mathrm{S}}_{\mathrm{C}}\mathrm{\Delta }\mathrm{C}\mathrm{I}/\mathrm{C}{\mathrm{I}}_{\mathrm{b}}\right)\\ {\tilde{\mathrm{w}}}_{\mathrm{G}}={\mathrm{w}}_{\mathrm{G}}\left(1+\mathrm{\Delta }\mathrm{G}/{\mathrm{G}}_{\mathrm{\text{init}}}\right).\end{array}$$ (61)

Physical Inactivity Constraint

The parameter S_A defines how the protein oxidation fraction depends on physical activity and exercise. Stein et al. (97) observed that 17 days of bed rest without a change of diet resulted in an average negative N balance: N_bal = −2 g/day. From the protein balance equation,

$${\mathrm{f}}_{\mathrm{P}}=\frac{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}-\mathrm{G}\hat{\text{N}}{\mathrm{G}}_{\mathrm{P}}({\mathrm{P}}_{\mathrm{\text{init}}}/{\mathrm{P}}_{\mathrm{\text{Keys}}})-6.25{\mathrm{\rho }}_{\mathrm{P}}{\mathrm{N}}_{\mathrm{\text{bal}}}}{\mathrm{T}\stackrel{\sim }{\text{E}}\mathrm{E}},$$ (62)

where

$$\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{E}=-({\mathrm{\delta }}_{\mathrm{b}}+{\mathrm{\upsilon }}_{\mathrm{b}})\mathrm{B}{\mathrm{W}}_{\mathrm{\text{init}}}-{\mathrm{\eta }}_{\mathrm{N}}{\mathrm{N}}_{\mathrm{b}\mathrm{a}\mathrm{l}}.$$ (63)

From the macronutrient oxidation Eq. 27,

$${\mathrm{f}}_{\mathrm{P}}=\frac{{\mathrm{w}}_{\mathrm{P}}+{\mathrm{S}}_{\mathrm{A}}({\mathrm{P}}_{\mathrm{\text{init}}}/{\mathrm{P}}_{\mathrm{\text{Keys}}})}{{\mathrm{w}}_{\mathrm{P}}+{\mathrm{S}}_{\mathrm{A}}({\mathrm{P}}_{\mathrm{\text{init}}}/{\mathrm{P}}_{\mathrm{\text{Keys}}})+{\mathrm{w}}_{\mathrm{C}}+{\stackrel{\sim }{\mathrm{w}}}_{\mathrm{G}}+{\stackrel{\sim }{\mathrm{w}}}_{\mathrm{F}}},$$ (64)

where

$$\begin{array}{c}{\tilde{\mathrm{w}}}_{\mathrm{G}}=-{\mathrm{w}}_{\mathrm{G}}\left(\frac{\mathrm{\Delta }\mathrm{T}\mathrm{E}\mathrm{E}}{2{\rho }_{\mathrm{C}}{\mathrm{G}}_{\mathrm{\text{init}}}}\right)\\ {\tilde{\mathrm{w}}}_{\mathrm{F}}={\mathrm{w}}_{\mathrm{F}}\left(1-\psi \right){\left(\frac{{\mathrm{F}}_{\mathrm{\text{init}}}}{{\mathrm{F}}_{\mathrm{\text{Keys}}}}\right)}^{\frac{2}{3}},\end{array}$$ (65)

which assumes that the approximately one-half of the energy savings due to inactivity will initially be deposited as glycogen. This results in the following expression for S_A:

$${\mathrm{S}}_{\mathrm{A}}=\mathrm{M}\mathrm{A}\mathrm{X}\left\{1,\left(\frac{{\mathrm{P}}_{\mathrm{\text{Keys}}}}{{\mathrm{P}}_{\mathrm{\text{init}}}}\right)\left[\frac{\mathrm{\Xi }}{1-\mathrm{\Xi }}({\mathrm{w}}_{\mathrm{C}}+{\tilde{\mathrm{w}}}_{\mathrm{G}}+{\tilde{\mathrm{w}}}_{\mathrm{F}})-{\mathrm{w}}_{\mathrm{P}}\right]\right\},$$ (66)

where

$$\mathrm{\text{Ξ}}\equiv \frac{\mathrm{P}{\mathrm{I}}_{\mathrm{b}}-\mathrm{G}\hat{\text{N}}{\mathrm{G}}_{\mathrm{P}}({\mathrm{P}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}/{\mathrm{P}}_{\mathrm{\text{Keys}}})-6.25{\mathrm{\rho }}_{\mathrm{P}}{\mathrm{N}}_{\mathrm{b}\mathrm{a}\mathrm{l}}}{\mathrm{T}\tilde{\mathrm{E}}\mathrm{E}}.$$ (67)

Model Parameter Values

The model parameter values listed above were obtained from the cited published literature and are listed in Table 1. The parameters, SP−, w_P, w_C:G, λ₁, λ₂, and σ, were determined using a downhill simplex algorithm (78) implemented using Berkeley Madonna software (version 8.3; http://www.berkeleymadonna.com) to minimize the sum of squares of weighted residuals between the simulation outputs and the data from the Minnesota human starvation experiment (55). I used the following measurement error estimates to define the weights for the parameter optimization algorithm: ΔBW = 0.2 kg, ΔFM = 0.5 kg, and ΔRMR = 50 kcal/day. The best fit parameter values are listed in Table 2, and the constrained parameters are listed in Table 3.

Table 1.

Model parameters determined from published data

Parameter	Value	Description
ρF	9.44 kcal/g	Energy density of F
ρP	4.7 kcal/g	Energy density of P
ρG	4.18 kcal/g	Energy density of G
ρK	4.45 kcal/g	Energy density of ketones
hP	1.6 g H2O/g	P hydration coefficient
hG	2.7 g H2O/g	G hydration coefficient
ηF	0.18 kcal/g	F synthesis cost
ηP	0.86 kcal/g	P synthesis cost
εP	0.17 kcal/g	P degradation cost
ηG	0.21 kcal/g	G synthesis cost
ηN	5.4 kcal/g	Urea synthesis cost/g nitrogen
εd	0.835	DNL efficiency
εg	0.8	GNG efficiency
εk	0.81	KTG efficiency
αF	0.025	TEF factor for FI
αC	0.075	TEF factor for CI
αP	0.25	TEF factor for PI
γF	4.5 kcal•kg−1•day−1	Specific RMR for adipose
γ̂FFM	19 kcal•kg−1•day−1	Basal specific RMR for FFM
ξBW	0.16	ECF response to BW changes
ξCI	4000 mg/day	ECF response to CI changes
ξNa	3 mg•ml−1•day−1	ECF response to sodium changes
τBW	1,000 days	ECF response time for BW changes
τT	7 days	Response time for T changes
D̂g	180 g/day	Baseline glycogenolysis rate
KDNL	2	Glycogen constant for DNL
d	4	Hill coefficient for DNL
D̂F	140 g/day	Baseline lipolysis rate
AL	3.8	Maximum lipolysis change
BL	0.9	Minimum lipolysis change
τL	1.44 days	Response time for lipolysis
KL	4	Body fat constant for lipolysis
SL	2	Hill coefficient for lipolysis
ψ	0.4	PA effect on lipolysis
AK	0.8	Maximum KTG fraction
kP	0.69	Effect of PI on KTG
kG	0.69	Effect of G on KTG
KK	1	Sets basal KTG rate
KTGthresh	70 g/day	Renal threshold KTG rate
KUmax	20 g/day	Maximum ketone excretion
KTGmax	400 g/day	Maximum KTG
D̂P	300 g/day	Baseline proteolysis rate
χ	0	Effect of PI on protein turnover
GN̂GP	300 kcal/day	Basal GNGP rate
ΓC	0.39	Effect of CI on GNGP
ΓP	0.32	Effect of PI on GNGP
τPI	1.1 days	Response time of ProtOx to PI changes

See Glossary of Model Variables for definitions of the abbreviations.

Table 2.

Parameter values fit to the Minnesota experiment data

Parameter	Value	Description
λ1	0.74	Underfeeding adaptive thermogenesis
λ2	0.02	Overfeeding adaptive thermogenesis
σ	0.52	Thermogenesis effect on PAE vs. RMR
wP	1.1	Weighting of ProtOx for basal PI
wC:G	0.93	Ratio of CarbOx weighting parameters
SP−	1.7	Sensitivity of ProtOx to reduced PI

See Glossary of Model Variables for definitions of the abbreviations.

Table 3.

Parameter values determined from constraints on energy balance, nutrient balance, and physical inactivity as well as perturbations of dietary protein and carbohydrate

Parameter	Value	Description
SP+	3.8	Sensitivity of ProtOx to increased PI
SC	0.85	Sensitivity of CarbOx to CI changes
wG	9.4	Weighting of CarbOx for glycogenolysis
wF	14.8	Weighting of FatOx for lipolysis
Ec	−435 kcal/day	Constant energy expenditure offset

See Glossary of Model Variables for definitions of the abbreviations. Other than the constant values for S_C and SP+, these parameters are calculated according to the initial conditions corresponding to different subject groups. The values listed are for the average Minnesota experiment subject.