^{*}

Conceived and designed the experiments: CC KH. Contributed reagents/materials/analysis tools: CC KH. Wrote the paper: CC KH.

The authors have declared that no competing interests exist.

An imbalance between energy intake and energy expenditure will lead to a change in body weight (mass) and body composition (fat and lean masses). A quantitative understanding of the processes involved, which currently remains lacking, will be useful in determining the etiology and treatment of obesity and other conditions resulting from prolonged energy imbalance. Here, we show that a mathematical model of the macronutrient flux balances can capture the long-term dynamics of human weight change; all previous models are special cases of this model. We show that the generic dynamic behavior of body composition for a clamped diet can be divided into two classes. In the first class, the body composition and mass are determined uniquely. In the second class, the body composition can exist at an infinite number of possible states. Surprisingly, perturbations of dietary energy intake or energy expenditure can give identical responses in both model classes, and existing data are insufficient to distinguish between these two possibilities. Nevertheless, this distinction has important implications for the efficacy of clinical interventions that alter body composition and mass.

Understanding the dynamics of human body weight change has important consequences for conditions such as obesity, starvation, and wasting syndromes. Changes of body weight are known to result from imbalances between the energy derived from food and the energy expended to maintain life and perform physical work. However, quantifying this relationship has proved difficult, in part because the body is composed of multiple components and weight change results from alterations of body composition (i.e., fat versus lean mass). Here, we show that mathematical modeling can provide a general description of how body weight will change over time by tracking the flux balances of the macronutrients fat, protein, and carbohydrates. For a fixed food intake rate and physical activity level, the body weight and composition will approach steady state. However, the steady state can correspond to a unique body weight or a continuum of body weights that are all consistent with the same food intake and energy expenditure rates. Interestingly, existing experimental data on human body weight dynamics cannot distinguish between these two possibilities. We propose experiments that could resolve this issue and use computer simulations to demonstrate how such experiments could be performed.

Obesity, anorexia nervosa, cachexia, and starvation are conditions that have a profound medical, social and economic impact on our lives. For example, the incidence of obesity and its co-morbidities has increased at a rapid rate over the past two decades

To address these issues and improve our understanding of human body weight regulation, mathematical and computational modeling has been attempted many times over the past several decades

The human body obeys the law of energy conservation

In order to express a change of stored energy Δ_{M}_{M}M_{M}_{M}

When the body changes mass, that change will be composed of water, protein, carbohydrates (in the form of glycogen), fat, bone, and trace amounts of micronutrients, all having their own energy densities. Hence, a means of determining the dynamics of ρ_{M}_{M}_{F}_{G}_{P}_{F}_{C}_{P}_{F}_{C}_{F}_{C}_{F}_{C}_{M}_{F}F_{G}G_{P}P

The intake and energy expenditure rates are explicit functions of time with fast fluctuations on a time scale of hours to days

The three-compartment flux balance model was used by Hall

The three compartment macronutrient flux balance model Equations 3–5 can be reduced to a two dimensional system for fat mass _{P}_{G}_{C}_{C}

Substituting Equation 7 and _{L}_{P}_{P}_{F}_{L}_{P}_{C}_{F}_{L}_{F}_{L}_{F}

We note that _{C}E_{C}_{C}kdP_{L}_{P}_{C}_{P}_{G}_{L}

Previous studies have considered two dimensional models of body mass change although they were not derived from the three-dimensional macronutrient partition model. Alpert _{F}_{L}

The two-compartment macronutrient partition model can be further simplified by assuming that trajectories in the

Equation 12 describes a family of

Substituting Equation 15 into the macronutrient partition model 8 and 9 leads to the

Despite the ubiquity of the energy partition model, the physiological interpretation of the p-ratio remains obscure and is difficult to measure directly. It can be inferred indirectly from

It may sometimes be convenient to express the macronutrient partition model with a unique fixed point as_{F}_{L}_{0},_{0}). We use this form in numerical examples below. The fasting model of Song and Thomas

The dynamics of the energy partition model Equations 16 and 17 move along fixed trajectories in the

Suppose a relationship

As an example, assume _{L}_{F}p_{0} gives_{M}_{F}_{L}_{L}_{F}_{L}_{0}),_{0})), and

If Equations 16 and 17 are constrained to obey the phase plane paths of Forbes's law, then a reduction to a one dimensional equation can also be made. Using Equation 14 (i.e., φ(

The one dimensional model gives the dynamics of the energy partition model along a fixed trajectory in the

The various flux balance models can be analyzed using the methods of dynamical systems theory, which aims to understand dynamics in terms of the geometric structure of possible trajectories (time courses of the body components). If the models are smooth and continuous then the global dynamics can be inferred from the local dynamics of the model near fixed points (i.e. where the time derivatives of the variables are zero). To simplify the analysis, we consider the intake rates to be clamped to constant values or set to predetermined functions of time. We do not consider the control and variation of food intake rate that may arise due to feedback from the body composition or from exogenous influences. We focus only on what happens to the food once it is ingested, which is a problem independent of the control of intake. We also assume that the averaged energy expenditure rate does not depend on time explicitly. Hence, we do not account for the effects of development, aging or gradual changes in lifestyle, which could lead to an explicit slow time dependence of energy expenditure rate. Thus, our ensuing analysis is mainly applicable to understanding the slow dynamics of body mass and composition for clamped food intake and physical activity over a time course of months to a few years.

Dynamics in two dimensions are particularly simple to analyze and can be easily visualized geometrically

In two dimensions, changes of body composition and mass are represented by trajectories in the _{F}_{L}

Physical viability constrains _{F}_{L}

The fixed point conditions of Equations 8 and 9 can be expressed in terms of the solutions of_{F}_{L}

The functional dependence of

Data suggest that

We have shown that all two dimensional autonomous models of body composition change generically fall into two classes - those with fixed points and those with invariant manifolds. In the case of a stable fixed point, any temporary perturbation of body weight or composition will be corrected over time (i.e., for all things equal, the body will return to its original state). An invariant manifold allows the possibility that a transient perturbation could lead to a permanent change of body composition and mass.

At first glance, these differing properties would appear to point to a simple way of distinguishing between the two classes. However, the traditional means of inducing weight change namely diet or altering energy expenditure through aerobic exercise, turn out to be incapable of revealing the distinction. For an invariant manifold, any change of intake or expenditure rate will only elicit movement along one of the prescribed

(A) Fixed point case. (B) Invariant manifold case. Dotted lines represent nullclines. In both cases, the body composition follows a fixed trajectory and returns to the original steady state (solid curves). However, if the body composition is perturbed directly (dashed-dot curves) then the body composition will flow to same point in (A) but to a different point in (B).

Perturbations that move the body composition off the fixed trajectory can be done by altering body composition directly or by altering the fat utilization fraction

We found an example of one clinical study that bears on the question of whether humans have a fixed point or an invariant manifold. Biller et al. investigated changes of body composition pre- and post-growth hormone therapy in forty male subjects with growth hormone deficiency

We now consider some numerical examples using the macronutrient partition model in the form given by Equations 18 and 19, with a p-ratio consistent with Forbes's law (13) (i.e.

For every model with an invariant manifold, a model with a fixed point can be found such that trajectories in the

In all the Figures, the solid line is for an intake reduction from 12 MJ/day to 10 J/day and the dashed line is for the same reduction but with a removal of 10 kg of fat at day 100. Time dependence of body mass for the fixed point model (A). Trajectories in the

The time constant to reach the new fixed point in the numerical simulations is very long. This slow approach to steady state (on the order of several years for humans) has been pointed out many times previously

In this paper we have shown that all possible two dimensional autonomous models for lean and fat mass are variants of the macronutrient partition model. The models can be divided into two general classes - models with isolated fixed points (most likely a single stable fixed point) and models with an invariant manifold. There is the possibility of more exotic behavior such as multi-stability and limit cycles but these require fine-tuning and thus are less plausible. Surprisingly, experimentally determining if the body exhibits a fixed point or an invariant manifold is nontrivial. Only perturbations of the body composition itself apart from dietary or energy expenditure interventions or alterations of the fraction of energy utilized as fat can discriminate between the two possibilities. The distinction between the classes is not merely an academic concern since this has direct clinical implications for potential permanence of transient changes of body composition via such procedures as liposuction or temporary administration of therapeutic compounds.

Our analysis considers the slow dynamics of the body mass and composition where the fast time dependent hourly or daily fluctuations are averaged out for a clamped average food intake rate. We also do not consider a slow explicit time dependence of the energy expenditure. Such time dependence could arise during development, aging or gradual changes in lifestyle where activity levels differ. Thus our analysis is best suited to modeling changes over time scales of months to a few years in adults. We do not consider any feedback of body composition on food intake, which is an extremely important topic but beyond the scope of this paper.

Previous efforts to model body weight change have predominantly used energy partition models that implicitly contain an invariant manifold and thus body composition and mass are not fully specified by the diet. If the body does have an invariant manifold then this fact puts a very strong constraint on the fat utilization fraction

The three compartment macronutrient flux balance Equations 3–5 are a system of nonautonomous differential equations since the energy intake and expenditure are explicitly time dependent. Food is ingested over discrete time intervals and physical activity will vary greatly within a day. However, this fast time dependence can be viewed as oscillations or fluctuations on top of a slowly varying background. It is this slower time dependence that governs long-term body mass and composition changes that we are interested in. For example, if an individual had the exact same schedule with the same energy intake and expenditure each day, then averaged over a day, the body composition would be constant. If the daily averaged intake and expenditure were to gradually change on longer time scales of say weeks or months then there would be a corresponding change in the body composition and mass. Given that we are only interested in these slower changes, we remove the short time scale fluctuations by using the method of averaging to produce an autonomous system of

We do so by introducing a second “fast” time variable τ = ^{1}〉 = 〈^{1}〉 = 〈^{1}〉 = 0 for a time average defined by ^{0}(^{0},^{0},^{0},^{2}) and

Taking the moving time average of Equations 36–38 and requiring that 〈∂^{1}/∂τ〉, 〈∂^{1}/∂τ〉, and 〈∂^{1}/∂τ〉 are of order ε or higher leads to the averaged equations:

The dynamics near a fixed point (_{0},_{0}) are determined by expanding _{0} and δ_{0} _{F}_{L}

We thank T. Hwa and two anonymous reviewers for a critical reading of the manuscript.