Predicting Metabolic Adaptation, Body Weight Change, And Energy Intake in Humans

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doi:10.1152/ajpendo.00559.2009298:E449-E466, 2010. First published 24 November 2009;Am J Physiol Endocrinol Metab

Kevin D. Halland energy intake in humansPredicting metabolic adaptation, body weight change,

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Predicting metabolic adaptation, body weight change, and energy intake

in humans

Kevin D. Hall

Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases, Bethesda, Maryland

Submitted 8 September 2009; accepted in final form 16 November 2009

Hall KD. Predicting metabolic adaptation, body weight change,and energy intake in humans. Am J Physiol Endocrinol Metab 298:E449E466, 2010. First published November 24, 2009;doi:10.1152/ajpendo.00559.2009. Complex interactions betweencarbohydrate, fat, and protein metabolism underlie the bodysremarkable ability to adapt to a variety of diets. But any imbal-ances between the intake and utilization rates of these macronu-trients will result in changes in body weight and composition.Here, I present the first computational model that simulates howdiet perturbations result in adaptations of fuel selection and energyexpenditure that predict body weight and composition changes inboth obese and nonobese men and women. No model parameters

were adjusted to fit these data other than the initial conditions foreach subject group (e.g., initial body weight and body fat mass).The model provides the first realistic simulations of how dietperturbations result in adaptations of whole body energy expendi-ture, fuel selection, and various metabolic fluxes that ultimatelygive rise to body weight change. The validated model was used toestimate free-living energy intake during a long-term weight lossintervention, a variable that has never previously been measuredaccurately.

mathematical model; energy metabolism; macronutrient metabolism;body composition

OVER THE PAST CENTURY, researchers in the fields of human metab-olism and nutrition have accumulated an impressive body ofquantitative knowledge regarding how dietary changes impactvarious aspects of metabolism, body weight, and body composi-tion (12, 24, 38, 56, 68). But integrating this knowledge to makequantitative predictions is a formidable task given the multiplenonlinear interactions between various organ systems. Neverthe-less, such an integrative approach is required to fully understandboth normal physiology as well as the derangements that underlieconditions such as obesity, diabetes, and the metabolic syndrome.

Mathematical modeling and computer simulation are widelyused methodologies in the engineering and physical sciences forintegrating knowledge about complex systems and predictingtheir behavior. This approach is beginning to gain traction in thelife sciences and forms a critical part of the systems biology

paradigm (20, 73). In the area of human energy metabolism andbody weight regulation, several mathematical models of weightchange have been proposed over the past few decades (35, 7,1618, 33, 42, 44, 59, 60, 96, 110, 113). Previously, I presentedthe first mathematical model of the metabolism of all threemacronutrients (i.e., carbohydrate, fat, and protein), and I used themodel to help understand the dynamics of semistarvation andrefeeding in healthy young men (42). Here, I extended myprevious model to apply to both obese and nonobese men and

women and validated its behavior in response to several controlleddiet perturbations. The model was designed to quantitatively trackthe metabolism of all three dietary macronutrients and theirinteractions within the human body. In particular, the modeldescribes how diet perturbations result in adaptations of energyexpenditure, fuel selection, and various metabolic fluxes (e.g.,lipolysis, lipogenesis, gluconeogenesis, ketogenesis, protein turn-over, etc.) that ultimately give rise to changes of body weight andcomposition on a time scale of days and longer. The main modelassumptions were that energy must be conserved and that changesof the body composition result from imbalances between the

intake and utilization rates of fat, carbohydrate, and protein alongwith intracellular and extracellular fluid changes. The model codecan 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 frommore than 50 experimental studies (see APPENDIX) and was vali-dated by comparing model predictions with the results fromseveral controlled feeding studies not used for model develop-ment. In this validation process, I chose to simulate human studiesthat carefully controlled food intake and measured changes inbody weight (BW) and body fat mass (FM) as well as total energyexpenditure (TEE) and resting metabolic rate (RMR). Requiringthis combination of measurements dramatically narrowed thescope of possible validation studies and allowed for assessment ofnot only the weight change predictions but also energy partition-ing and energy expenditure changes. In all cases, the measuredfood intake was used as a model input and included a wide rangeof interventions, such as overfeeding and weight gain, weight lossusing a variety of diets, and adaptations of metabolic fuel selectionwhen the dietary macronutrient proportions were altered. Thesevalidation studies were performed in several different subjectgroups, including lean, overweight, and obese men and women.Importantly, no model parameters were altered to fit the data otherthan modifying the initial conditions appropriate formodeling each subject group (e.g., initial BW and body FM).

Once the model was validated in situations where the foodintake was known, I proposed that the model could be usedto estimate the free-living energy intake changes underlyingobserved changes of body weight. In particular, I used themodel to predict the energy intake required to result in thetypical trajectory of weight loss and regain observed duringan outpatient weight loss intervention (100). In addition toproviding a novel methodology for estimating the dynamicsof free-living human energy intake during weight loss andregain, the model simulations addressed the relative role ofdiet adherence vs. metabolic adaptation in explaining thetypically observed weight loss plateau after 6 m o o f lifestyle modification.

Address for reprint requests and other correspondence: K. D. Hall, NIDDK/NIH, 12 South Drive, Rm. 4007, Bethesda, MD 20892-5621 (e-mail: [email protected]).

Am J Physiol Endocrinol Metab 298: E449E466, 2010.First published November 24, 2009; doi:10.1152/ajpendo.00559.2009.

http://www.ajpendo.org E449


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Glossary of Model Variables

BM Bone mineral mass in gBW Body weight in g

CarbOx Rate of carbohydrate oxidation in kcal/dayCI Carbohydrate intake rate in kcal/dayDF Rate of endogenous lipolysis in g/day

DG Rate of glycogenolysis in g/dayDNL Rate of de novo lipogenesis in kcal/dayDP Rate of proteolysis in g/day

ECF Extracellular fluid mass in gECP Extracellular protein mass in g

EI Energy intake in kcal/dayFM Body fat mass in g

FatOx Rate of fat oxidation in kcal/dayfC Carbohydrate oxidation fractionfF Fat oxidation fraction

FFM Fat-free body mass in gFI Fat intake rate in kcal/dayfP Protein oxidation fractionG Body glycogen mass in g

G3P Rate of glycerol 3-phosphate synthesis in kcal/dayGNGF Rate of gluconeogenesis from glycerol in kcal/dayGNGP Rate of gluconeogenesis from protein in kcal/day

ICS Intracellular solid mass in gICW Intracellular water mass in g

KetOx Rate of ketone oxidation in kcal/dayKTG Rate of ketogenesis in kcal/day

KUexcr Rate of ketone excretion in kcal/dayLCM Lean tissue cell mass in gNexcr Nitrogen excretion rate in g/day

NPRQ Nonprotein respiratory quotientP Intracellular protein mass in g

PAE Physical activity energy expenditure in kcal/dayPI Protein intake rate in kcal/day

ProtOx Rate of protein oxidation in kcal/dayRMR Resting metabolic rate in kcal/day

RQ Respiratory quotientSynthF Rate of fat synthesis in g/daySynthG Rate of glycogen synthesis in g/daySynthP Rate of protein synthesis in g/day

T Adaptive thermogenesisTEE Total energy expenditure in kcal/dayTEF Thermic effect of feeding in kcal/day

TG TriacylglycerideVCO2 Rate of carbon dioxide production in liters/day

VO2 Rate of oxygen consumption in liters/day

METHODS

The APPENDIX provides a detailed description of the mathematicalmodel along with the data and assumptions used in its developmentand calibration. Briefly, the model comprises eight ordinary differen-tial equations and uses dietary carbohydrate, fat, and protein as modelinputs. The model computes the various components of whole bodyenergy expenditure, rates of macronutrient oxidation, respiratory ex-change, lipogenesis, ketogenesis, gluconeogenesis, and turnover offat, glycogen, and protein. Based on the computed macronutrientimbalances, along with sodium imbalances impacting extracellularfluid, the model computes dynamic changes of BW and composition.

To represent different initial conditions corresponding to differentsubject groups, parameter values were specified for initial BW, per-cent 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 Fimbal, Gimbal, and Pimbal specified the initial

imbalances of fat, glycogen, and protein, respectively. Other thanthese initial parameter values, no other model parameters were ad-

justed 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 soft-

ware (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.

Fig. 1. Adaptations of fuel selection following isocaloric exchange of dietarycarbohydrate and fat. A: 24-h respiratory quotient (RQ) in response to switch-ing 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 foodquotient (FQ; solid curve). B: simulated (dashed curve) and measured () 24-hRQ changes in response to a diet switch from 37 to 50% fat corresponding tothe depicted changes of FQ (solid curve). Data are presented as means SD.

E450 MODEL OF HUMAN METABOLISM AND BODY WEIGHT CHANGE

AJP-Endocrinol Metab VOL 298 MARCH 2010 www.ajpendo.org


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RESULTS

Metabolic Fuel Selection

When the body is in a state of energy and macronutrientbalance, the intake of dietary carbohydrate, fat, and protein ismatched by their rates of utilization. A useful measurement ofthe relative proportion of carbohydrate to fat being oxidized is

provided by the RQ, where a value of 1 reflects completereliance on carbohydrate oxidation for metabolic needs,whereas RQ 0.7 indicates state of pure fat oxidation andintermediate values reflect a fuel selection mixture (27, 31, 37).Macronutrient balance is achieved when the RQ is equal to thefood quotient (FQ), which represents the relative proportion ofmacronutrients in the diet. Exchanging dietary carbohydratewith fat while maintaining the same energy intake will result ina state of macronutrient imbalance until fuel selection adapts tothe new FQ. Figure 1A depicts the models predicted change of24-h RQ in response to a high-fat isocaloric diet that resultedin a large decline of FQ. This diet perturbation was performedby Schrauwen et al. (90), and the closed squares in Fig. 1Ashow the measured RQ dynamics, which closely matched themodel predictions. Figure 1B shows the results from a similarstudy performed by Smith et al. (95), with a less dramaticchange of FQ. Again, the model predictions agreed with thedata. Underlying these simulations were slow decreases ofglycogen as well as increased lipolysis due to the reduceddietary carbohydrate (data not shown). These changes resultedin altered substrate delivery to metabolically active tissues andsubsequent changes of fuel selection.

During overfeeding, the model predicts substantial changesof various whole body metabolic fluxes, including suppressionof lipolysis and induction of de novo lipogenesis, that influenceoverall fuel selection and macronutrient balance. The imbal-ance between dietary macronutrients and their utilization rates

determined how the energy excess was partitioned in themodel. To test whether macronutrient oxidation changes weresimulated correctly during overfeeding, I used the data of Jebband colleagues (51, 52), who measured macronutrient oxida-tion rates during 12 days of 33% overfeeding in healthy youngmen that were continuously housed inside a metabolic cham-ber. 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 ratesalong with the data. The simulated gradual increase of carbo-hydrate oxidation reached a plateau after several days asglycogen approached a new steady state (data not shown).These results closely matched the measured carbohydrate ox-idation rate shown in the closed squares in Fig. 2A, right. Thesimulated fat oxidation rate was suppressed at the onset ofoverfeeding, which matched the measurements shown in theopen squares in Fig. 2A, right. The decrease of fat oxidation inthe simulation corresponded to a suppression of lipolysis as aresult of the increased carbohydrate intake (data not shown).The simulated protein oxidation rate remained relatively un-changed and corresponded to the measured rate shown in theopen triangles in Fig. 2A, right.

Jebb and colleagues (51, 52) also performed a complemen-tary study where energy intake was decreased by 67% for 12days, and these results are depicted in Fig. 2B along with thecorresponding model simulations. Again, the simulatedchanges of body composition and macronutrient oxidationrates closely matched the data and showed the opposite re-sponse to overfeeding, whereas underfeeding suppressed car-bohydrate oxidation while enhancing fat oxidation.

Underfeeding and Weight Loss

Although these results demonstrate that the model cor-rectly simulated adaptations to short-term underfeeding in

Fig. 2. Metabolic fuel selection and bodycomposition and changes during over- andunderfeeding in healthy young men. A, left:33% overfeeding resulted in the change ofsimulated and measured body weight (BW;solid curve and , respectively) along withincreased simulated and measured fat mass(FM; dashed curve and , respectively). A,

right: simulated and measured carbohydrateoxidation rates (dashed curve and , respec-tively) as well as fat oxidation (solid curveand , respectively) and protein oxidationrates (dotted curve and , respectively).

B: BW, FM, and macronutrient oxidationrates in response to 67% underfeeding,where the symbols are identical to A. Dataare presented as means SD.

E451MODEL OF HUMAN METABOLISM AND BODY WEIGHT CHANGE



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lean young men, longer-term underfeeding is required toachieve significant weight loss in overweight people. Arecent 6-mo calorie restriction study investigated the meta-bolic effects and weight changes in healthy, sedentary,overweight men and women resulting from three strictlycontrolled lifestyle interventions (45, 81, 82). Figure 3Ashows the effect of an 890 kcal/day very low-calorie diet

followed by a maintenance diet from 3 to 6 mo. Theimposed energy intake resulted in the drop of BW and FMshown in Fig. 3A, left. The simulated TEE and RMR alsomatched the data, which are depicted by closed and opensquares, respectively, in Fig. 3A, right. Figure 3B shows themodel simulations and data for a 25% reduction of caloriesfor 6 mo, and Fig. 3C shows the results for a 12.5% caloricreduction plus a 12.5% increase of physical activity expen-diture (PAE). Again, the model simulations matched thebody composition and energy expenditure data remarkablywell, although the initial simulated decrease in BW wasslightly more rapid than the data.

There has been much debate surrounding the metaboliceffects of weight loss diets that differ in macronutrient

content. Rumpler et al. (88) investigated this issue in obesemen by restricting energy intake by 50% for 4 wk usingdiets with either 40 or 20% of energy from dietary fat.Figure 4A, left, shows the model simulation and experimen-tal results for the 40% fat diet, and Fig. 4B, right, shows theresults 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 TEEand RMR changes shown in Fig. 4, A and B, middle. Figure4, A and B, bottom, illustrates the very close agreementbetween the simulated and measured 24-h RQ values, dem-onstrating that the model appropriately represents fuel se-lection during weight loss diets with differing macronutrientcomposition in obese men.

The model also accurately simulated weight loss andmetabolic 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 thesimulated and measured body composition changes,whereas Fig. 5A, right, shows that the simulated TEE andRMR changes also matched the data quite well.

Fig. 3. Weight loss and metabolic effects ofcaloric restriction for 6 mo. A, left: simulatedBW change (solid curve) along with the mea-sured values () and the simulated FM(dashed curve) and the measured FM ()changes resulting from a very low-calorieliquid diet followed by a weight maintenancediet. A, right: simulated (dashed curve) andmeasured () total energy expenditure (TEE)in response to the depicted changes of energyintake (EI; solid gray curve). Simulated (solidblack curve) and measured resting metabolicrate (RMR; ) along with the simulatedphysical activity expenditure (PAE; dottedcurve). B: a sustained 25% reduction of EIresulted in the simulated and measuredchanges of BW and FM shown at leftand thecorresponding TEE, RMR, and PAE changesat right. C: a sustained 12.5% reduction of EI

along with a 12.5% increase of PAE resultedin the simulated and measured changes ofBW and FM shown at left and the corre-sponding TEE and RMR changes at right.Data are presented as means SD.




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Overfeeding and Weight Gain

Weight gain is the result of positive energy balance, butpredicting the magnitude of weight gain for a given changeof energy intake requires knowing how excess energy ispartitioned between deposition of body fat and fat-freemass, which have dramatically different influences on en-ergy expenditure and the subsequent magnitude of energyimbalance. I tested the model predictions for weight gain bysimulating 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 thesimulated changes of BW and FM, which clearly matchedthe body composition data, indicating appropriate energypartitioning and weight gain. Figure 5B, right, shows thatthe imposed energy intake resulted in simulated TEE andRMR changes.

I also simulated the overfeeding study of Tremblay et al.(103), who provided healthy young men with 1,000 kcal/dayabove 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 170kcal/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 WeightLoss and Regain

Maintenance of a reduced BW following weight loss is acritical issue in the treatment of obesity (44, 46, 115, 116).Most weight loss interventions result in a maximum amount oflost weight after 6 mo, with a gradual weight regain over thesubsequent several years. But what is the mechanism of the

weight loss plateau and subsequent regain? In particular, whatrole does metabolic adaptation play in resisting further weightloss after 6 mo?

Given that the mathematical model accurately simulates themetabolic and body composition changes in response to knownchanges of diet, I investigated what change of diet would berequired to simulate the average weight loss and regain patternin a group of 341 overweight and obese adults participating ina self-directed weight loss program (100). Figure 6A, left,shows the simulated weight loss and regain pattern (solidcurve) which closely matched the data (closed squares),whereas the dashed curve is the predicted FM change. Thesolid gray curve in Fig. 6A, right, is the predicted free-living

Fig. 4. Weight loss and metabolic changes inobese men resulting from diets with differingproportions of carbohydrate and fat. A, top:simulated and measured changes of BW andFM resulting from the 40% fat, 46% carbohy-drate diet. A, middle: metabolic responses tothis diet, where the symbols are identical tothose in Fig. 3. A, bottom: simulated andmeasured changes of RQ, where the symbolsare identical to Fig. 1. B, top: simulated andmeasured 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 ofRQ. Data are presented as means SD.




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energy intake underlying the observed weight loss and regaintrajectory, assuming no change of physical activity. At theonset of the intervention, the model predicted a decrease ofenergy intake by 800 kcal/day, which was maintained foronly 6 wk. Subsequently, energy intake gradually increaseduntil 10 mo, when it returned to the initial energy intake thatresulted in energy balance at the initial BW. The simulatedTEE is depicted by the dashed curve in Fig. 6A, right, and

shows that negative energy balance was achieved only for 8mo, after which time positive energy balance and weight regainensued.

Figure 6B shows the simulated changes of energy intakerequired to maintain the weight loss achieved at 6 mo, whichwas 170 kcal/day less than the initial energy-balanced diet.Figure 6C shows the BW and FM changes that were predictedhad the initial decrease of energy intake been maintained overthe entire 3-yr period. This simulation illustrates the very longequilibration time for weight loss in obese subjects and dem-onstrates that the weight loss plateau observed after 6 mocannot be a result of metabolic adaptation.

DISCUSSION

The ultimate goal of modeling is to provide physiologicalinsights and to help design novel experiments whose resultscan then be integrated within the context of previous knowl-edge, thereby improving both the model as well as our under-standing of the system. Most systems biology models ofmetabolism have been developed at the cellular level and areaimed at calculating steady-state flux distributions throughcomplex metabolic networks in microorganisms (29, 30). Suchmodels are now beginning to be applied to human metabolism(25, 92), and the model presented here represents a comple-mentary dynamic approach to systems physiology at the levelof whole body human metabolism. Other than modifying the

initial model parameters to represent the various subjectgroups, no parameters were adjusted to simulate the validationexperiments. Rather, once the initial conditions were specified,the model calculated the correct energy partitioning and met-abolic adaptations required to simulate the physiological re-sponses in lean, overweight, and obese men and women.

The model appears to accurately describe the physiology ofmetabolism, fuel selection, and body composition change in

humans on time scales ranging from days to years. Onepractical use of the model would be to computationally inves-tigate the potential of various metabolic interventions fortreatment of obesity. Such a tool could help design key exper-iments, focus resources on areas of likely success, and helpinterpret experimental results. Of course, since the modelpresently uses food intake as an input, any adaptive changes offood intake as a result of a metabolic intervention would not beautomatically accounted for, but hypotheses regarding foodintake changes could be simulated.

Since the model responded accurately to known changesof food intake, I proposed that known changes of bodyweight and composition over long time periods could be

used to estimate the underlying changes of free-living en-ergy intake. This is an important application of the model,since free-living energy intake is notoriously difficult tomeasure (117) and the gold standard doubly labeled watermethod is valid only in situations of energy balance, whenthe RQ is known (102). I used the model to estimate thefree-living energy intake changes underlying the typicalweight loss and regain trajectory observed in an outpatientweight loss program. Surprisingly, the model predicted thatthe initial reduction of energy intake was maintained foronly 6 wk, followed by a gradual return to baseline after 10mo, where it remained for more than 2 years. This highlightsthe importance of diet adherence for successful weight loss

Fig. 5. Weight loss and weight gain. A, left:simulated and measured changes of BW andFM resulting from a 1,000 kcal/day diet for 8wk in obese women. A, right: simulated andmeasured changes of TEE, RMR, and PAE inresponse to imposed changes of EI, where thesymbols are identical to those in Fig. 2. B, left:simulated and measured changes of BW andFM in response to 6 wk of overfeeding by1,000 kcal/day in healthy young men. B, right:changes of TEE, RMR, and PAE, where thesymbols are identical to those in Fig. 3. Dataare presented as means SD.




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and raises the question of why the estimated energy intakehas this pattern. In other words, what internal and externalfactors contribute to this progressive loss of diet adherence?Although the model presented here cannot answer thesequestions, the ability to quantitatively predict the dynamicsof free-living energy intake during weight loss and regainmay have implications for the development of drug andbehavioral therapies for obesity.

An important caveat regarding the model estimate ofenergy intake is that I assumed that the average physicalactivity for the group was unchanged over the course of theweight loss and regain trajectory. This does not mean thatthe energy cost of physical activity was unchanged, sinceperforming the same physical activity at a different bodyweight has a differing energy requirement, as was demon-strated by the PAE curves in the validation simulations inFigs. 35. Of course, the model can be simulated for anyassumed physical activity time course, and the energy intakecan be recalculated, thereby providing a range of estimatesfor varying assumptions about physical activity. Futurework will investigate the sensitivity of the energy intake

estimates to various model parameters as well as the timecourse of body weight change.

Much work remains for improving and expanding themodel. For example, although I have demonstrated that themodel behaves appropriately for different groups of sub-

jects, it is presently unclear whether individual subjectresponses can be predicted given sufficiently detailed infor-mation about their initial conditions. Furthermore, the

model now implicitly represents the effect of hormones suchas insulin, but an explicit representation of endocrine sig-naling along with concentrations of hormones and metabo-lites would be desirable, especially on shorter time scales, sothat the response to individual meals could be simulated.Other additions to the model might include an explicitrepresentation of the possible contribution of human brownadipose tissue to the overall energy expenditure and its rolein adaptive thermogenesis (80). Finally, it would be partic-ularly interesting to close the loop and model how internalsignals from changes of metabolism and body compositioncombine with external factors to determine food intake. Butdespite the cornucopia of possible model improvements, the

Fig. 6. Weight loss and regain dynamics dur-ing an outpatient lifestyle intervention. A, left:a typical outpatient weight loss program re-sults in the characteristic BW change trajec-tory, where the symbols are identical to thosein Fig. 3. A, right: predicted free-living EI andTEE underlying the observed BW loss andregain trajectory. B: the model predicted thatmaintenance of lost BW and FM (left) wouldhave been achieved if the EI over the last 2 yr

had been decreased by 170 kcal/day. C: hadthe initial reduction of EI been sustained forthe 3-yr period, the model predicted a progres-sive decrease of BW and FM shown at leftandthe corresponding TEE changes shown atright. Data are presented as means SD.




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mathematical model presented here represents a significantstep forward in a systems physiology approach to wholebody human metabolism.

APPENDIX

Detailed Description of the Mathematical Model

The individual components of the mathematical model were basedon a variety of published in vivo human data, as described below.Each model component was relatively simple, and only the mostimportant physiological effectors have been incorporated. Since con-tinued development of the model is part of an ongoing researchprogram, additional relevant physiological data will be incorporatedwithin the existing computational framework to improve the realismand predictive capabilities of the model. The model code can bedownloaded 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 conser-vation such that changes in the bodys energy stores were given by the sumof fluxes entering the pools minus the fluxes exiting the pools. Thus, themathematical representation of macronutrient balance was given by thefollowing differential equations:

CdG

dt CI DNL GNGP GNGF G3P CarbOx

FdF

dt 3MFFAFI MTG dDNL KUexcr

(1 k)KTG FatOx

PdP

dt PI GNGP ProtOx,

(1)

where C 4.18 kcal/g, F 9.44 kcal/g, and P 4.7 kcal/g werethe energy densities of carbohydrate, fat, and protein, respectively(67). MTG 860 g/mol and MFFA 273 g/mol were the molecularmasses of triacylglycerides and free fatty acids, respectively. Theefficiency of de novo lipogenesis, DNL, was represented as the

dimensionless parameter

d 0.835, which is the enthalpy of com-bustion of 0.37 g of fat divided by the enthalpy of combustion of the1 g of glucose used to produce the fat (27). The efficiency ofketogenesis, KTG, was represented by the parameter k 0.81,calculated as the enthalpy of combustion of 4.5 mol of acetoacetatedivided by the enthalpy of combustion of 1 mol of stearic acid used toproduce the ketones (12). When the ketogenic rate increases, ketonesare excreted in the urine at the rate KUexcr. The macronutrient intakerates CI and FI refer to the digestible energy intake of carbohydrateand fat, respectively, whereas PI refers to the digestible energy intakeof protein corrected for the obligatory formation of ammonia withprotein metabolism. The energy cost of ureagenesis is accounted forseparately, as described below. The oxidation rates CarbOx, FatOx,and ProtOx are summed to the TEE, less the small amount of heatproduced 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 changesof energy metabolism and not fluctuations of metabolism that occurwithin 1 day. The nutrient balance equations were integrated using thefourth-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 computedusing the following equation:

FFM BM ECF ECP LCM

BM ECF ECP ICW P G ICS

BM ECF ECP IC^

W P(1 hP)

G(1 hG) ICS,

(2)

where the FFM is composed of BM, ECF, ECP, and LCM. LCM iscomposed of ICW, G, and P as well as a small contribution fromnucleic acids and other ICS. The protein fraction of the lean tissue cellmass was P/LCM 0.25, and the initial intracellular water fractionwas ICW/LCM 0.7 (13, 109). ICW was then calculated dynami-cally from P and G such that each gram of protein and glycogen wasassociated with hP and hG grams of water, respectively. ICW was aconstant amount of ICW computed to attain the appropriate initial

intracellular composition, assuming that G 500 g, hG 2.7, andhP 1.6 (13, 43, 71).

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

dECF

dt

1

[Na]Nadiet NaECF ECF init

CI1 CI CIb ECF

BWdECF

dt BWBW BWinitECF,

(3)

where [Na] 3.22 mg/ml is the extracellular sodium concentration,Nadiet is the change of dietary sodium in milligrams, and ECF isthe slow rate of increment in ECF that occurs with BW change on a

very long time scale of BW, 1,000 days. The value of BW 0.16mlkg1day1 was chosen so that the steady-state increment of ECFwith weight change approximately matched the regression equationsof Silva et al. (94). The value of Na was chosen according to the dataof Andersen et al. (6), who measured a change of sodium excretion of5,000 mg/day following an infusion of 1.7 liters of isotonic saline.Therefore, I assumed that a 5,000 mg/day change in sodium intakewould be balanced by a 1.7-liter change of ECF giving Na 3mgml1day1. Given that the normal sodium content of the dietis 4,000 mg/day and that a low-sodium diet must effectively shutoff sodium excretion, I assumed that CI 4,000 mg/day so thatremoving dietary carbohydrate doubles the sodium excretion ratecompared with a very-low-sodium diet, as observed by Stinebaughand Schloeder (98).

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

Whole Body Total Energy Expenditure

TEE was modeled by the following equation:

TEE TEF PAE RMR, (4)

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

Thermic Effect of Feeding

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

TEF FFI PPI CCI, (5)

where F 0.025, P 0.25, and C 0.075 defined the short-termthermic effect of fat, protein, and carbohydrate feeding (12).

Adaptive Thermogenesis

Energy imbalance causes an adaptation of metabolic rate thatopposes weight change (22, 23, 62). Whether or not the adaptation ofenergy expenditure is greater than expected based on body composi-tion changes alone has been a matter of some debate (35, 70, 111).




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The so-called adaptive thermogenesis is believed to affect both restingand nonresting energy expenditure and has maximum amplitudeduring the dynamic phase of weight change. Adaptive thermogenesisalso persists during weight maintenance at an altered body weight(85). The non-RMR component of adaptive thermogenesis may reflecteither altered efficiency or amount of muscular work (6365, 87).

The onset of adaptive thermogenesis is rapid and may correspondto altered levels of circulating thyroid hormones or catecholamines

(86, 111). I defined a dimensionless adaptive thermogenesis parame-ter, T, which was generated by a first-order process in proportion tothe departure from the baseline energy intake EIb CIb FIb PIb:

TdT

dt 1EI EIb T , if E I EIb2EI EIb T, else (6)

where EI was the change from the baseline energy intake, T 7days was the estimated time constant for the onset of adaptivethermogenesis, and the parameters 1 and 2 quantified the effect ofunderfeeding and overfeeding, respectively, and were determinedfrom the best fit to the Minnesota experiment data. The adaptivethermogenesis parameter T acted on both the RMR and PAE compo-nents of energy expenditure, as defined below. This simple modelassumed 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 thermogenicmechanism was required to fit the data from the Minnesota experi-ment. The amount that the best fit values for the 1 and 2 parametersdiffer from zero provides an indication of the extent of adaptive thermo-genesis 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 proportionalto the body weight of the individual (12, 105). Low-intensity physicalactivities 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 activityexpenditure:

PAE (1 T)BW BW, (7)

where was the nonexercise physical activity coefficient (inkcalkg1day1), was the exercise coefficient (in kcalkg1day1),and BW FFM FM was the body weight. The proportion of T thatwas allocated to the modification of nonexercise PAE was determinedby the parameter . The adaptation of PAE with T did not distinguishbetween altered efficiency vs. amount of muscular work.

Resting Metabolic Rate

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

RMR Ec BMB FFMFFM MB G(1 hg)

(ECF ECFinit) FF (1 d)DNL

(1 g)(GNGF GNGP) (1 K)KTG

NNexcr (P P)DP PdP

dt

FDF FdF

dt GDG G

dG

dt,

(8)

where g 0.8 was the efficiency of gluconeogenesis (12) and theconstant, Ec, was a parameter chosen to ensure that the model

achieved energy balance during the balanced baseline diet (see Nu-trient Balance Parameter Constraints below).

The specific metabolic rate of adipose tissue was F 4.5 kcalkg1day1. The brain metabolic rate was B 240 kcalkg

1day1,and its mass was MB 1.4 kg, which does not change with weightgain or loss (26). The baseline specific metabolic rate of the fat-freemass, FFM 19 kcalkg1day1, was determined by the specificmetabolic rates of the organs multiplied by the rate of change of the

organ mass with fat-free mass change according to the followingequation:

^ FFM i

idMi

dFFM, (9)

where i and Mi are the average specific metabolic rate and mass,respectively, of the organ indexed by i. The organs included skeletalmuscle (SM 13 kcalkg1day1, MSM 28 kg, dMSM/dFFM 0.59), liver (L 200 kcalkg

1day1, ML 1.8 kg, dML/dFFM 0.017), kidney (K 440 kcalkg

1day1, MK 0.31 kg, dMk/dFFM 0.0038), heart (H 440 kcalkg

1day1, MH 0.33 kg,dMH/dFFM 0.0029), and residual lean tissue mass (R 12kcalkg1day1, MR 23.2 kg, dMR/dFFM 0.37), as provided byElia (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 RMRarising solely from changes of glycogen content, G, and its associatedwater or changes of ECF. Adaptive thermogenesis affected the baselinespecific metabolic rate for lean tissue cell mass according to the followingequation:

FFM FFM[1 (1 )T] (10)

The last seven terms of Eq. 8 accounted for the energy cost for ureasynthesis and nitrogen excretion as well as the turnover of protein, fat,and glycogen. Urea synthesis requires 4 mol ATP/mol nitrogenexcreted (76) and was represented by the parameter N 5.4 kcal/gexcreted nitrogen Nexcr. To calculate the energy cost for proteinturnover, consider that the whole body protein pool turns over with asynthesis rate, SynthP, and a degradation rate, DP (in g/day). I

assumed that it cost PSynthP to synthesize P and that the energyrequired for degradation was PDP. Since dP/dt SynthP DP, theenergy cost for protein turnover was given by (P P)DP PdP/dt.Similar arguments led to the other terms of Eq. 8 representing theenergy costs for fat and glycogen turnover, where the energy cost fordegradation was negligible. The values for the parameters were F 0.18 kcal/g, G 0.21 kcal/g, P 0.17 kcal/g, and P 0.86 kcal/g.These values were determined from the ATP costs for the respectivebiochemical pathways (i.e., 8 ATP/TG synthesized, 2 ATP/glycosylunit of glycogen synthesized, 4 ATP/peptide bond synthesized plus 1ATP for amino acid transport, and 1 ATP/peptide bond hydrolyzed)(12, 28). I assumed that 19 kcal of macronutrient oxidation wasrequired to synthesize 1 mol of ATP (26).

Daily Average Lipolysis Rate

The daily average lipolysis rate, DF, was modeled as

DF D^

F FFKeys

2

3[Ldiet LPA] (11)

where DF 140 g/day was the baseline daily average TG turnoverrate given by two-thirds of the fed lipolysis rate plus one-third of theovernight-fasted lipolysis rate (53). The (F/FKeys)

2/3 factor accountedfor the dependence of the basal lipolysis rate on the total fat massnormalized by the initial fat mass of the average Minnesota experi-ment subject, FKeys. The two-thirds power reflects the hypothesis thatbasal lipolysis scales with adipocyte surface area, which also matchesthe FFA rate of appearance (Ra) as a function of body fat mass




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observed by Bjorntorp et al. (10). The effect of diet on lipolysis, L diet,is determined primarily by the carbohydrate content of the diet viainsulin. Furthermore, the lipolysis rate reaches half of its maximumvalue after 2 days of fasting, and the magnitude of the increase isattenuated with increasing body fat mass (58). To capture these effects,I modeled the effect of dietary carbohydrate on the daily average lipolysisrate as

L dLdiet

dt KL

SL

1 (AL BL) exp(kLCI CIb) BLKL

SL MAX0, (F FKeys 1)SL Ldiet ,

(12)

where L 1/ln(2) 1.44 days. The term in the square bracketsaccounted for the modulation of lipolysis by the carbohydrate contentof the diet. For example, complete starvation (CI 0) stimulatedaverage daily lipolysis by a factor of AL 3.1/[1 exp(2.5/L)] 3.8, as computed by the 3.1-fold increase of glycerol Ra following a60-h fast (15) vs. the daily average glycerol Ra (53). Halving thecarbohydrate content of the diet increased the average lipolysis rate byfactor of 1.4, as estimated by the increased area under the circulatingFFA curve following an isocaloric meal consisting of 33 vs. 66%carbohydrate (118). Given the above value for AL, the effect ofhalving the carbohydrate content was modeled by choosing BL 0.9.

The following choice for kL ensured that the lipolysis rate wasnormalized for the baseline diet:

kL lnAL BL1 BL

. (13)Although obesity increases basal lipolysis, the stimulatory effect of

decreased carbohydrate intake is impaired (120). This effect wasmodeled in Eq. 12 by setting KL 4 and SL 2 such that the curveof lipolysis vs. CI becomes flattened as FM increases and matches the

data of Klein et al. (58), where long-term vs. short-term fastingstimulated 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 LPA as follows:

LPA init init 1 , (14)where 0.4 was the best fit value for the measured effect of gradedexercise to increase lipolysis rates determined by FFA Ra measure-ments (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 asfollows:

KTG KDFAK DF D^

F

KK DF D^

FexpkP PIPIbexpkG

G

Ginit , (15)

where K 4.45 kcal/g is the average energy density of ketonescalculated as average enthalpy of combustion of -hydroxybuterateand acetoacetate in a 2:1 ratio (12). AK 0.8 is the maximum fractionof FFA from lipolysis converted to ketones when PI G 0 (9).When protein intake was at normal levels, I assumed that the maxi-mum fraction of FFA converted to ketones was 0.4, since a protein

modified fast decreases circulating ketone levels by one-half com-pared with fasting alone (106). Therefore, kp 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 was0.2; therefore kG ln(0.4/0.2) 0.69, and when the lipolysis rate wasnormal I assumed that 10% of FFA from lipolysis was converted toketones such that KK 1 (9).

Daily Average Ketone Excretion Rate

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

KUexcr 0, if KTG K KTGthresh

KKUmaxKTG K KTGthresh

KTGmax KTG thresh, else , (16)

where KTGthresh 70 g/day, KUmax 20 g/day, and KTGmax 400g/day such that the urinary excretion of ketones, KUexcr, matches theexcretion data during various ketone infusion rates, as measured byWildenhoff (114) and Sapir and Owen (89).

Ketone Oxidation Rate

Since ketones cannot be stored in the body in any significantquantity, I assumed that once a ketone is produced it is either oxidizedor excreted. Therefore,

KetOx KTG KUexcr . (17)

Daily Average Proteolysis Rate

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

DP D^

P PPKeys

PIPIb

, (18)where DP 300 g/day was the baseline daily protein turnover rate(107), and I assumed that the protein degradation rate was propor-

tional to the normalized protein content of the body (112). Althoughit is possible that the protein content of the diet may directly influenceprotein turnover as represented by the parameter (83), the balance ofthe current data suggests that 0 (41, 75). Nevertheless, I includedthis parameter in the model to allow for this possibility should newdata provide further evidence for such an effect.

Daily Average Glycogenolysis Rate

The daily average glycogen degradation rate, DG, was given by thefollowing equation:

DG D^

G GGinit

, (19)where the baseline glycogen turnover rate, DG 180 g/day, wasdetermined by assuming that 70% was from hepatic glycogenolysisand 30% from skeletal muscle with the hepatic contribution com-puted as two-thirds of the fed plus one-third of the overnight-fastedhepatic glycogenolysis rate (69).




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Daily Average Fat, Protein, and Glycogen Synthesis Rates

Mass conservation required that the daily average synthesis rates offat, protein, and glycogen (SynthF, SynthP, and SynthG, respectively)were given by

SynthF DF dF

dt

SynthP DP dP

dt

SynthG DG dG

dt.

(20)

Glycerol 3-Phosphate Production Rate

Because adipose tissue lacks glycerol kinase, the glycerol 3-phos-phate backbone of adipose TG is derived primarily from glucose.Thus, the TG synthesis rate, SynthF, determined the rate of glycerol3-phosphate production, G3P, according to

G3P CSynthF MGMTG

, (21)where MG 92 g/mol and MTG 860 g/mol are the molecularweights of glycerol and TG, respectively.

Glycerol Gluconeogenesis Rate

Lipolysis of both endogenous and exogenous TG results in therelease of glycerol that can be converted to glucose via gluconeogen-esis (8). Trimmer et al. (104) demonstrated that glycerol disappear-ance could be fully accounted for by glucose production. Therefore, Iassumed that all exogenous and endogenous glycerol entered theGNG pathway according to the following equation:

GNGF FI CMGFMTG DFCMG

MTG . (22)

Since glycerol cannot be used by adipose tissue for TG synthesis

due to lack of glycerol kinase, all glycerol released by lipolysis iseventually oxidized (apart from a negligibly small amount incorpo-rated into altered pool sizes of nonadipose TG). By assuming that allglycerol enters the GNG pathway, any model error was limited to anoverestimate of the energy expenditure associated with glycerolsinitial conversion to glucose prior to oxidation. This error must bevery small, since the total energy cost for glycerol GNG in the basalstate was only 25 kcal/day.

Gluconeogenesis From Amino Acids

The GNGP rate in the model referred to the net rate of gluconeo-genesis from amino acid-derived carbon. Whereas all amino acidsexcept leucine and lysine can be used as gluconeogenic substrates, theprimary gluconeogenic amino acids are alanine and glutamine. Much

of alanine gluconeogenesis does not contribute to the net amino acidgluconeogenic rate since the carbon skeleton of alanine is derivedlargely 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 ratefrom amino acids, GNGp, was 300 kcal/day.

Several factors may regulate GNGP, but for simplicity I haveassumed that GNGP was proportional to the normalized proteolysisrate and was influenced by the diet as follows:

GNGP GN^

GP PPKeys CCI

CIb

(P )PIPIb

,(23)

where the coefficients C 0.39 and P 0.32 were determinedby solving Eq. 23, using two sets of data. The first measured aninitial nitrogen balance of 4 g/day upon removal of baselinedietary carbohydrate while keeping dietary protein at baselinevalues (47). I assumed that the negative nitrogen balance wasdriven largely by increased gluconeogenesis and thereby deter-mined the value of C. The second study found a 56% increase ofgluconeogenesis when protein intake was increased by a factor of

2.5-fold, whereas carbohydrate intake was decreased by 20% (66).Given the value of C, these data determined the value of the sumof p and therefore P.

De Novo Lipogenesis Rate

DNL occurs in both the liver and adipose tissue. Under free-living conditions, adipose DNL has recently been measured tocontribute 20% of new TG with a measured TG turnover rate of50 g/day (99). Thus, adipose DNL is 94 kcal/day. Measure-ments of daily hepatic DNL in circulating very low-density li-poproteins (VLDL) have found that 7% of VLDL TG occurs viaDNL when a basal diet of 30% fat, 50% carbohydrate, and 15%protein is consumed (49). Given that the daily VLDL TG secretionrate 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, and15% 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 modeledDNL as a Hill function of the normalized glycogen content with amaximum DNL rate given by the carbohydrate intake rate:

DNL CI (G Ginit)

d

(G Ginit)d KDNL

d. (24)

I chose KDNL 2 and d 4 such that the computed DNL ratecorresponded with measured in vivo DNL rates for experimentallydetermined carbohydrate intakes and estimated glycogen levels (1, 2,49, 99).

Macronutrient Oxidation Rates

The whole body energy expenditure rate, TEE, was accounted forprimarily by the sum of the heat produced from carbohydrate, fat, andprotein oxidation plus the heat produced via flux through the DNLpathway (34) and ketogenesis. Furthermore, I assumed that the min-imum carbohydrate oxidation rate was equal to the sum of thegluconeogenic rates less the flux required to produce glycerol 3-phos-phate and that ketone oxidation was treated as fat oxidation. Thus, theremaining energy expenditure, TEE, was apportioned between carbo-hydrate, fat, and protein oxidation according to the fractions fC, fF,and fP, respectively:

CarbOx GNGf GNGp G3P fC TE~

E

FatOx KetOx fF TE~

E

ProtOx fP TE~

E,

(25)

where

TE~

E TEE (1 d)DNL (1 k)KTG KetOx

GNGf GNGp G3P .(26)

The substrate oxidation fraction for each macronutrient depends ona number of factors. First, increased lipolysis leads to concomitantincreased fatty acid oxidation (15). Second, carbohydrate oxidationdepends on the carbohydrate intake as well as the glycogen content(32, 61). Third, dietary protein and carbohydrate intake directlystimulate protein and carbohydrate oxidation, respectively, but dietaryfat intake does not directly stimulate fat oxidation (36, 91). Fourth, Iassumed that lean tissue supplies amino acids for oxidation in pro-




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portion to the proetolysis rate. Finally, although inactivity causesmuscle wasting (11, 97), increased physical activity may promotenitrogen retention (14, 101, 112), and the PAE is accounted forprimarily by increased oxidation of fat and carbohydrate (112). I

modeled these effects by decreasing the fraction of energy expendi-ture derived from protein oxidation as physical activity increases.

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

fC wG(DG D

^G) wCMAX0, (1 SCCI CIb)G (Gmin G)

Z

.

fF wF(DF D

^

F)

Z

fP wPMAX0, (1 Psig) (DP D

^P)SAexpkA( ) (b b)

Z

PIdPsig

dt SPPI PIb Psig

SP SP, ifPI 0SP, else

,

(27)

where the w and S were dimensionless model parameters and CI

and PI were changes from the basal carbohydrate intake, CIb, andprotein intake, PIb, respectively. The small parameter, Gmin 10g, was chosen such that carbohydrate oxidation was restrained as

glycogen decreases and prevents glycogen from becoming nega-

tive. The signal for dietary protein intake perturbations, P sig,

changes with a time constant of PI 1.1 days, in accordance withthe 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 kA was

chosen such that kA ln(SA). Z was a normalization factor equalto the sum of the numerators so that the sum of the fractions fC, fF,

and fP was equal to 1.

Respiratory Gas Exchange

Oxidation of carbohydrate, fat, and protein was associated with con-sumption of oxygen (O2) and production of carbon dioxide (CO2)

according to the stoichiometry of the net biochemical reactions (31, 37):

1 g carbohydrate 0.831 liters O2 0.831 liters CO2 0.6 g H2O1 g fat 2.03 liters CO2 1.43 liters CO2 1.09 g H2O1 g protein 0.966 liters O2 0.782 liters

CO2 0.45 g H2O 0.16 g N.

(28)

Gluconeogenesis, lipogenesis, ketogenesis (with subsequent excre-

tion), and glycerol 3-phosphate production also contribute to gas

exchange according to the following net reactions (27, 31, 37):

1 g protein 0.126 liters CO2 1 g carbohydrate 0.16 g N

1 g glycerol 0.133 liters O2 1 g carbohydrate1 g carbohydrate 0.37 g fat 0.238 liters CO2 0.2 g H2O

1 g fat 0.52 liters O 2 1.57 g ketones

1 g carbohydrate 1 g glycerol 0.133 liters O2.

(29)

Oxidation of carbohydrate, fat, and protein can occur either directly

or subsequent to intermediate exchange via lipogenesis or gluconeo-

genesis. In either case, the final ratio of CO2 produced to O2 con-

sumed (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 (VO2) and CO2 production (VCO2)

(in liters/day) were computed according to

V

O2 0.831

CarbOx

C 2.03

FatOx

F 0.966

ProtOx

P

0.133GNGFC G3P

C 0.33KUexcrK

V

CO2 0.831CarbOx

C 1.43

FatOx

F 0.782

ProtOx

P

0.126GNGP

P 0.238

DNL

C.

(30)

The RQ was computed by dividing VCO2 by VO2. To compute theNPRQ, the total rate of nitrogen excretion was calculated:

Nexcr ProtOx GNGP

6.25P(31)

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

When comparing model predictions for fuel selection with experi-ments employing 24-h room indirect calorimetry, I used the simulatedVCO2 and VO2 values from Eq. 30 and computed macronutrient oxidationrates as if they were the actual indirect calorimetry measurements.

Nutrient Balance Parameter Constraints

The baseline diet may not necessarily result in macronutrient orenergy balance. Therefore, I defined the following parameters tospecify the initial degree of imbalance for fat, carbohydrate, andprotein: Fimbal, Gimbal, and Pimbal, respectively. Therefore, the baselinediet satisfies the following relationship: TEE Fimbal Gimbal Pimbal EIb, where the subscript b refers to the baseline state.

Therefore, by explicitly expressing the TEE in the baseline state Iderived the following expression:

EIb TEFb PAEb RMRb TEFb (b b)BWb EC BMB FFM(FFMb MB) FFb (1 d)DNLb (1 g)(GN

^GF GN

^GP)

(1 k)KTGb (P P)D^

P FD^

F GD^

G

(1 F F)Fimbal (1 G C)Gimbal

(1 P P)PimbalN(PIb Pimbal)

6.25P,

(32)

which was solved for the constant Ec.




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Assuming negligible baseline ketone excretion, rearrangement of

the nutrient balance equations gave

CarbOxb CIb DNLb GN^

GP GN^

GF G3Pb GimbalFatOxb 3MFFAFIb MTG dDNLb (1 k)KTGb Fimbal

ProtOxb PIb GN^

GP Pimbal. (33)

Next, I defined the following parameters:

F (3MFFAFIb MTG dDNLb kKTGb Fimbal)

C (CIb DNLb Gimbal)

P (PI b GN

^GP Pimbal)

,

(34)

where was given by

EIb (1 d)DNLb (1 k)KTGb KetOxb GN

^GF GN

^GP G3Pb (1 G C)Gimbal

(1 F F)Fimbal (1 P P)Pimbal.

(35)

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

wF(Finit FKeys)23

(Pinit PKeys) wP wG(Ginit GKeys) wCGinit (G init Gmin) wF(Finit FKeys)23 F

wG(Ginit GKeys) wCGinit (G init Gmin)(Pinit PKeys) wP wG(Ginit GKeys) wCGinit (G init Gmin) wF(Finit FKeys)

23 C

(Pinit PKeys) wP

(Pinit PKeys) wP wG(Ginit GKeys) wCGinit (G init Gmin) wF(Finit FKeys)23 P,

(36)

where Finit, Ginit, a n d Pinit were the initial values for body fat,glycogen, and protein, respectively. Elementary algebra led to thefollowing parameter constraints required to achieve the specifiedmacronutrient imbalance:

wG C P(Pinit PKeys wP)(GKeys G init)

1 wC:GGKeys (G init Gmin)

wF F1 F

1 CPPinit

PKeys wP FKeys

Finit

23

,

(37)

where wC:G wC/wG.

Carbohydrate Perturbation Constraint

The parameters wC and SC determined how the model adapted tochanges of carbohydrate intake. I specified that an additional dietarycarbohydrate intake, CI, above baseline, CIb, resulted in an initialpositive carbohydrate imbalance of CCI, where 0 C 1specified the proportion of CI directed toward glycogen storage.Thus, the glycogen increment was G CCI/C. The goal was tosolve for the parameter SC such that the correct amount of carbohy-drate was oxidized and deposited as glycogen during short-termcarbohydrate overfeeding. Based on the carbohydrate overfeedingstudy of Horton et al. (48), I chose C 0.5 when CI 1,500kcal/day.

The change of TEE was given byTEE TEF PAE RMR (38)

For a carbohydrate perturbation, the perturbed energy expenditurecomponents were

TEF CCI (39)

PAE bBWbT (40)

RMR CG

CCI ^ FFMFFMb(1 )T

(1 d)DNL (1 g)GNG,(41)

where

T 2CI

EIb1 T Texp(1 T) (42)

was the average value of the thermogenesis parameter, T, over 1 dayand DNL was computed at the midpoint of the glycogen incrementaccording to the following equation:

DNL (CIb CI)(1G 2Ginit)

d

KDNLd (1 G 2Ginit)

d DNLb . (43)

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

GNG CCICIb

GN^ GP C MGMTG

D^ F[(AL BL) expkL(1 CI CIb) BL 1] .

(44)

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

fC CIb (1 C)CI (DNLb DNL)

TE~

E. (45)

Since I assume that the baseline diet results in a state of energybalance, I obtained the following equation for TEE:

TE

~

E EIb TEE (1 d)DNL (1 k)KTG KetOx GNGF GNGP G3P .

(46)

For simplicity, I assumed that

G3P G3PbKTG KetOx KTGb .

(47)

I define the parameter as

CIb (1 C)CI (DNLb DNL)

TE~

E. (48)

Therefore, using Eq. 27, I solved Eq. 45 for SC, which gave thecarbohydrate feeding constraint:




15/19

SC CIb

CI(1 wp w

~F w

~G)

(1 )wC

w~

G

(1 )wC 1 , (49)

where

w~

F wF(AL BL) expkL(1 CI CIb) BL

w~

G wG

1

G

2Ginit.

(50)

Protein Perturbation Constraint

The parameters wP and SP determined how the model adapted

short-term substrate oxidation rates to changes of protein intake. In ameticulous study of whole body protein balance, Oddoye and Margen(74) measured nitrogen balance in subjects consuming isocaloric dietswith 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 P 0.1 when PI 640 kcal/day, CI 310kcal/day, and FI 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

fP PIb (1 P)PI GN

^GP GNGP

TE~

E, (51)

where the changes of gluconeogenic rates were given by

GNGP GN^

GP(P )PIPIb CCICIb GNGF FI CMGFMTG C

MG

MTGD^ F(AL BL)

expkL1 CICIb BL 1 .(52)

The change of TEE was given by

TEE TEF PAE RMR , (53)

where

TEF CCI FFI PPI, (54)

and

RMR P

PPIP PD

^P

PIb (1 g)(GNGF

GNGP) (1 d)DNL.

(55)

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

PAE T 0. (56)

I also assumed that

G3P G3PbKTG KetOx KTGb (57)

Table 1. Model parameters determined from published data

Parameter Value Description

F 9.44 kcal/g Energy density of FP 4.7 kcal/g Energy density of PG 4.18 kcal/g Energy density of GK 4.45 kcal/g Energy density of ketoneshP 1.6 g H2O/g P hydration coefficienthG 2.7 g H2O/g G hydration coefficientF 0.18 kcal/g F synthesis costP 0.86 kcal/g P synthesis costP 0.17 kcal/g P degradation costG 0.21 kcal/g G synthesis costN 5.4 kcal/g Urea synthesis cost/g nitrogend 0.835 DNL efficiencyg 0.8 GNG efficiencyk 0.81 KTG efficiencyF 0.025 TEF factor for FIC 0.075 TEF factor for CI

P 0.25 TEF factor for PIF 4.5 kcal kg1 day1 Specific RMR for adiposeFFM 19 kcal kg1 day1 Basal specific RMR for FFMBW 0.16 ECF response to BW changesCI 4000 mg/day ECF response to CI changesNa 3 mg ml1 day1 ECF response to sodium changesBW 1,000 days ECF response time for BW changesT 7 days Response time for T changesDg 180 g/day Baseline glycogenolysis rateKDNL 2 Glycogen constant for DNLd 4 Hill coefficient for DNLDF 140 g/day Baseline lipolysis rateAL 3.8 Maximum lipolysis changeBL 0.9 Minimum lipolysis changeL 1.44 days Response time for lipolysisKL 4 Body fat constant for lipolysisSL 2 Hill coefficient for lipolysis

0.4 PA effect on lipolysisAK 0.8 Maximum KTG fractionkP 0.69 Effect of PI on KTGkG 0.69 Effect of G on KTGKK 1 Sets basal KTG rateKTGthresh 70 g/day Renal threshold KTG rateKUmax 20 g/day Maximum ketone excretionKTGmax 400 g/day Maximum KTGDP 300 g/day Baseline proteolysis rate 0 Effect of PI on protein turnoverGNGP 300 kcal/day Basal GNGP rateC 0.39 Effect of CI on GNGPP 0.32 Effect of PI on GNGPPI 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 experimentdata


1 0.74 Underfeeding adaptive thermogenesis2 0.02 Overfeeding adaptive thermogenesis 0.52 Thermogenesis effect on PAE vs. RMRwP 1.1 Weighting of ProtOx for basal PIwC:G 0.93 Ratio of CarbOx weighting parametersSP 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 onenergy balance, nutrient balance, and physical inactivity aswell as perturbations of dietary protein and carbohydrate


SP 3.8 Sensitivity of ProtOx to increased PI

SC 0.85 Sensitivity of CarbOx to CI changeswG 9.4 Weighting of CarbOx for glycogenolysiswF 14.8 Weighting of FatOx for l ipolysis

Ec 435 kcal/day Constant energy expenditure offset

See Glossary of Model Variables for definitions of the abbreviations. Otherthan the constant values for SC and SP

, these parameters are calculatedaccording to the initial conditions corresponding to different subject groups.The values listed are for the average Minnesota experiment subject.




16/19

and that glycogen would change by approximately G C CI/C,thereby altering DNL as follows:

DNL (CIb CI)(1G G init)

d

KDNLd (1 G G init)

d

CIb

KDNLd 1

. (58)

Using Eq. 27, I solved Eq. 51 for SP, which gave the followingconstraint:

SP

PIb

PI(w~ F w

~G w

~C)

(1 )wP

(1 PI PIb)

wP 1 , (59)

where was defined as

PIb (1 P)PI GN

^GP GNGP

TE~

E(60)

and

w~

F wF(AL BL) expkL(1 CI CIb) BLw~ C wC(1 SCCI CIb)

w~

G wG(1 G G init).

(61)

Physical Inactivity Constraint

The parameter SA defines how the protein oxidation fractiondepends on physical activity and exercise. Stein et al. (97) observedthat 17 days of bed rest without a change of diet resulted in an averagenegative N balance: Nbal 2 g/day. From the protein balanceequation,

fP PIb GN

^GPPinit PKeys 6.25PNbal

TE~

E, (62)

where

TEE (b b)BW init NNbal . (63)

From the macronutrient oxidation Eq. 27,

fP wP SAPinit PKeys

wP SAPinit PKeys wC w~

G w~

F

, (64)

where

w~ G wG TEE2CGinit

w~ F wF(1 ) Finit

FKeys

2

3,

(65)

which assumes that the approximately one-half of the energy savingsdue to inactivity will initially be deposited as glycogen. This results inthe following expression for SA:

SA MAX

1,

PKeys

Pinit

1

(wC w~

G w~

F) wP

,

(66)

where

PIb GN

^GP(Pinit PKeys) 6.25PNbal

TE~

E. (67)

Model Parameter Values

The model parameter values listed above were obtained from thecited published literature and are listed in Table 1. The parameters,SP, wP, wC:G, 1, 2, and , were determined using a downhill

simplex algorithm (78) implemented using Berkeley Madonna soft-ware (version 8.3; http://www.berkeleymadonna.com) to minimize

the sum of squares of weighted residuals between the simulationoutputs and the data from the Minnesota human starvation experiment(55). I used the following measurement error estimates to define theweights for the parameter optimization algorithm: BW 0.2 kg,FM 0.5 kg, and RMR 50 kcal/day. The best fit parametervalues are listed in Table 2, and the constrained parameters are listedin Table 3.

ACKNOWLEDGMENTS

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

GRANTS

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

DISCLOSURES

No conflicts of interest are declared by the author.

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Predicting Metabolic Adaptation, Body Weight Change, And Energy Intake in Humans