## 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.

- mathematical model
- energy metabolism
- macronutrient metabolism
- body composition

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 1*A* 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. 1*A* show the measured RQ dynamics, which closely matched the model predictions. Figure 1*B* 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.

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 2*A*, *left*, shows the model predictions along with the data for BW and body FM, whereas Fig. 2*A*, *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. 2*A*, *right*. The simulated fat oxidation rate was suppressed at the onset of overfeeding, which matched the measurements shown in the open squares in Fig. 2*A*, *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. 2*A*, *right*.

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. 2*B* 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 3*A* 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. 3*A*, *left*. The simulated TEE and RMR also matched the data, which are depicted by closed and open squares, respectively, in Fig. 3*A*, *right*. Figure 3*B* shows the model simulations and data for a 25% reduction of calories for 6 mo, and Fig. 3*C* 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.

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 4*A*, *left*, shows the model simulation and experimental results for the 40% fat diet, and Fig. 4*B*, *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.

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 5*A*, *left*, demonstrates close agreement between the simulated and measured body composition changes, whereas Fig. 5*A*, *right*, shows that the simulated TEE and RMR changes also matched the data quite well.

### 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 5*B*, *left*, depicts the simulated changes of BW and FM, which clearly matched the body composition data, indicating appropriate energy partitioning and weight gain. Figure 5*B*, *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 6*A*, *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. 6*A*, *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. 6*A*, *right*, and shows that negative energy balance was achieved only for ∼8 mo, after which time positive energy balance and weight regain ensued.

Figure 6*B* 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 6*C* 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.

## ACKNOWLEDGMENTS

I thank Dympna Gallagher, Susan Jebb, Peter Murgatroyd, Eric Ravussin, Bill Rumpler, and Steve Smith for insightful discussions and access to their data.

## Footnotes

Glossary of Model Variables

- BM
- Bone mineral mass in g
- BW
- Body weight in g
- CarbOx
- Rate of carbohydrate oxidation in kcal/day
- CI
- Carbohydrate intake rate in kcal/day
- D
_{F} - Rate of endogenous lipolysis in g/day
- D
_{G} - Rate of glycogenolysis in g/day
- DNL
- Rate of de novo lipogenesis in kcal/day
- D
_{P} - Rate of proteolysis in g/day
- ECF
- Extracellular fluid mass in g
- ECP
- Extracellular protein mass in g
- EI
- Energy intake in kcal/day
- FM
- Body fat mass in g
- FatOx
- Rate of fat oxidation in kcal/day
- f
_{C} - Carbohydrate oxidation fraction
- f
_{F} - Fat oxidation fraction
- FFM
- Fat-free body mass in g
- FI
- Fat intake rate in kcal/day
- f
_{P} - Protein oxidation fraction
- G
- Body glycogen mass in g
- G3P
- Rate of glycerol 3-phosphate synthesis in kcal/day
- GNG
_{F} - Rate of gluconeogenesis from glycerol in kcal/day
- GNG
_{P} - Rate of gluconeogenesis from protein in kcal/day
- ICS
- Intracellular solid mass in g
- ICW
- Intracellular water mass in g
- KetOx
- Rate of ketone oxidation in kcal/day
- KTG
- Rate of ketogenesis in kcal/day
- KU
_{excr} - Rate of ketone excretion in kcal/day
- LCM
- Lean tissue cell mass in g
- N
_{excr} - Nitrogen excretion rate in g/day
- NPRQ
- Nonprotein respiratory quotient
- P
- Intracellular protein mass in g
- PAE
- Physical activity energy expenditure in kcal/day
- PI
- Protein intake rate in kcal/day
- ProtOx
- Rate of protein oxidation in kcal/day
- RMR
- Resting metabolic rate in kcal/day
- RQ
- Respiratory quotient
- Synth
_{F} - Rate of fat synthesis in g/day
- Synth
_{G} - Rate of glycogen synthesis in g/day
- Synth
_{P} - Rate of protein synthesis in g/day
- T
- Adaptive thermogenesis
- TEE
- Total energy expenditure in kcal/day
- TEF
- Thermic effect of feeding in kcal/day
- TG
- Triacylglyceride
- V̇co
_{2} - Rate of carbon dioxide production in liters/day
- V̇o
_{2} - Rate of oxygen consumption in liters/day

## 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:
_{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:
_{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:
_{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:

#### 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:
_{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}:
_{T} = 7 days was the estimated time constant for the onset of adaptive thermogenesis, and the parameters λ_{1} and λ_{2} 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 λ_{1} and λ_{2} 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:
^{−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:
_{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:
_{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:

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_{P}Synth_{P} to synthesize P and that the energy required for degradation was ε_{P}D_{P}. Since dP*/*d*t* = Synth_{P} − D_{P}, the energy cost for protein turnover was given by (η_{P} + ε_{P})D_{P} + η_{P}dP/d*t*. 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
_{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
_{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:

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:
_{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:
_{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
_{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,

#### Daily Average Proteolysis Rate

The daily average protein degradation rate, D_{P}, was given by
_{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:
_{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

#### 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, G3*P*, according to
_{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:

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:
_{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:

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:

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:
_{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 (O_{2}) and production of carbon dioxide (CO_{2}) according to the stoichiometry of the net biochemical reactions (31, 37):

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):

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 CO_{2} produced to O_{2} consumed (i.e., the RQ) is independent of any intermediate exchanges in accordance with the principles of indirect calorimetry (27, 31, 37).

The simulated O_{2} consumption (V̇o_{2}) and CO_{2} production (V̇co_{2}) (in liters/day) were computed according to

The RQ was computed by dividing V̇co_{2} by V̇o_{2}. To compute the NPRQ, the total rate of nitrogen excretion was calculated:

When comparing model predictions for fuel selection with experiments employing 24-h room indirect calorimetry, I used the simulated V̇co_{2} and V̇o_{2} 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:
*E*_{c}.

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

Next, I defined the following parameters:

By substituting *Eqs. 25* and *27* at the initial state, I obtained
_{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:
_{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_{C}ΔCI, where 0 < κ_{C} < 1 specified the proportion of ΔCI directed toward glycogen storage. Thus, the glycogen increment was ΔG = κ_{C}ΔCI/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

For a carbohydrate perturbation, the perturbed energy expenditure components were

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

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

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

For simplicity, I assumed that

I define the parameter Θ as

Therefore, using *Eq. 27*, I solved *Eq. 45* for S_{C}, which gave the carbohydrate feeding constraint:

#### 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

The change of TEE was given by

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

I also assumed that
_{C} ΔCI/ρ_{C}, thereby altering DNL as follows:

Using *Eq. 27*, I solved *Eq. 51* for S_{P}, which gave the following constraint:

#### 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,

From the macronutrient oxidation *Eq. 27*,
_{A}:

#### 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}, λ_{1}, λ_{2}, 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.

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