General Model of Macronutrient and Energy Flux Balance
The human body obeys the law of energy conservation 
, which can be expressed as
is the change in stored energy in the body, ΔQ
is a change in energy input or intake, and ΔW
is a change in energy output or expenditure. The intake is provided by the energy content of the food consumed. Combustion of dietary macronutrients yields chemical energy and Hess's law states that the energy released is the same regardless of whether the process takes place inside a bomb calorimeter or via the complex process of oxidative phosphorylation in the mitochondria. Thus, the energy released from oxidation of food in the body can be precisely measured in the laboratory. However, there is an important caveat. Not all macronutrients in food are completely absorbed by the body. Furthermore, the dietary protein that is absorbed does not undergo complete combustion in the body, but rather produces urea and ammonia. In accounting for these effects, we refer to the metabolizable energy content of dietary carbohydrate, fat, and protein, which is slightly less than the values obtained by bomb calorimetry. The energy expenditure rate includes the work to maintain basic metabolic function (resting metabolic rate), to digest, absorb and transport the nutrients in food (thermic effect of feeding), to synthesize or break down tissue, and to perform physical activity, together with the heat generated. The energy is stored in the form of fat as well as in lean body tissue such as glycogen and protein. The body need not be in equilibrium for Equation 1 to hold. While we are primarily concerned with adult weight change, Equation 1 is also valid for childhood growth.
In order to express a change of stored energy ΔU
in terms of body mass M
we must determine the energy content per unit body mass change, i.e. the energy density ρM
. We can then set ΔU
). To model the dynamics of body mass change, we divide Equation 1 by some interval of time and take the limit of infinitesimal change to obtain a one dimensional energy flux balance equation:
is the rate of metabolizable energy intake and E
is the rate of energy expenditure. It is important to note that ρM
is the energy density of body mass change, which need not be a constant but could be a function of body composition and time. Thus, in order to use Equation 2, the dynamics of ρM
must also be established.
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 ρM
is to track the dynamics of the components. The extracellular water and bone mineral mass have no metabolizable energy content and change little when body mass changes in adults under normal conditions 
. The change in intracellular water can be specified by changes in the tissue protein and glycogen. Thus the main components contributing to the dynamics of ρM
are the macronutrients - protein, carbohydrates, and fat, where we distinguish body fat (e.g. free fatty acids and triglycerides) from adipose tissue, which includes water and protein in addition to triglycerides. We then represent Equation 2 in terms of macronutrient flux balance equations for body fat F
, glycogen G
, and protein P
39.5 MJ/kg, ρG
17.6 MJ/kg, ρP
19.7 MJ/kg are the energy densities 
are the intake rates, and fF
are the fractions of the energy expenditure rate obtained from the combustion of fat, carbohydrates (glycogen) and protein respectively. The fractions and energy expenditure rate are functions of body composition and intake rates. They can be estimated from indirect calorimetry, which measures the oxygen consumed and carbon dioxide produced by a subject 
. The intake rates are determined by the macronutrient composition of the consumed food, and the efficiency of the conversion of the food into a utilizable form. Transfer between compartments such as de novo
lipogenesis where carbohydrates are converted to fat or gluconeogenesis where amino acids are converted into carbohydrates can be accounted for in the forms of fF
. The sum of Equations 3, 4, and 5 recovers the energy flux balance Equation 2, where the body mass M
is the sum of the macronutrients F
, with the associated intracellular water, and the inert mass that does not change such as the extracellular water, bones, and minerals, and ρM
The intake and energy expenditure rates are explicit functions of time with fast fluctuations on a time scale of hours to days 
. However, we are interested in the long-term dynamics over weeks, months and years. Hence, to simplify the equations, we can use the method of averaging to remove the fast motion and derive a system of equations for the slow time dynamics. We do this explicitly in the Methods
section and show that the form of the averaged equations to lowest order are identical to Equations 3–5 except that the three components are to be interpreted as the slowly varying part and the intake and energy expenditure rates are moving time averages over a time scale of a day.
The three-compartment flux balance model was used by Hall 
to numerically simulate data from the classic Minnesota human starvation experiment 
. In Hall's model, the forms of the energy expenditure and fractions were chosen for physiological considerations. For clamped food intake, the body composition approached a unique steady state. The model also showed that apart from transient changes lasting only a few days, carbohydrate balance is precisely maintained as a result of the limited storage capacity for glycogen. We will exploit this property to reduce the three dimensional system to an approximately equivalent two dimensional system where dynamical systems techniques can be employed to analyze the dynamics.
Existence and Stability of Body Weight Fixed Points
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 
. The one dimensional models are a subclass of two dimensional dynamics. Three dimensional dynamical systems are generally more difficult to analyze but Hall 
found in simulations that the glycogen levels varied over a small interval and averaged to an approximate constant for time periods longer than a few days, implying that the slow dynamics could be effectively captured by a two dimensional model. Reduction to fewer dimensions is an oft-used strategy in dynamical systems theory. Hence, we focus our analysis on two dimensional dynamics.
In two dimensions, changes of body composition and mass are represented by trajectories in the L–F
phase plane. For IF
constant, the flux balance model is a two dimensional autonomous system of ordinary differential equations and trajectories will flow to attractors. The only possible attractors are infinity, stable fixed points or stable limit cycles 
. We note that fixed points within the context of the model correspond to states of flux balance. The two compartment macronutrient partition model is completely general in that all possible autonomous dynamics in the two dimensional phase plane are realizable. Any two or one dimensional autonomous model of body composition change can be expressed in terms of the two dimensional macronutrient partition model.
Physical viability constrains L
to be positive and finite. For differentiable f
, the possible trajectories for fixed intake rates are completely specified by the dynamics near fixed points of the system. Geometrically, the fixed points are given by the intersections of the nullclines in the L–F
plane, which are given by the solutions of IF
0 and IL
0. Example nullclines and phase plane portraits of the macronutrient model are shown in . If the nullclines intersect once then there will be a single fixed point and if it is stable then the steady state body composition and mass are uniquely determined. Multiple intersections can yield multiple stable fixed points implying that body composition is not unique 
. If the nullclines are collinear then there can be an attracting one dimensional invariant manifold (continuous curve of fixed points) in the L
plane. In this case, there are an infinite number of possible body compositions for a fixed diet. As we will show, the energy partition model implicitly assumes an invariant manifold. If a single fixed point exists but is unstable then a stable limit cycle may exist around it.
Figure 1 Possible trajectories (solid lines) for different initial conditions and nullclines (dotted lines) in the L–F phase plane for models with a stable fixed point (A), multi-stability with two stable fixed points separated by one unstable saddle point (more ...)
The fixed point conditions of Equations 8 and 9 can be expressed in terms of the solutions of
, and we have suppressed the functional dependence on intake rates. These fixed point conditions correspond to a state of flux balance of the lean and fat components. Equation 26 indicates a state of energy balance while Equation 27 indicates that the fraction of fat utilized must equal the fraction of fat in the diet. Stability of a fixed point is determined by the dynamics of small perturbations of body composition away from the fixed point. If the perturbed body composition returns to the original fixed point then the fixed point is deemed stable. We give the stability conditions in Methods
The functional dependence of E
determine the existence and stability of fixed points. As shown in Methods
, an isolated stable fixed point is guaranteed if f
is a monotonic increasing function of F
and a monotonic decreasing function of L
. If one of the fixed point conditions automatically satisfies the other, then instead of a fixed point there will be a continuous curve of fixed points or an invariant manifold. For example, if the energy balance condition 26 automatically satisfies the fat fraction condition 27, then there is an invariant manifold defined by I=E(F,L)
. The energy partition model has this property and thus has an invariant manifold rather than an isolated fixed point. This can be seen by observing that for f
given by Equation 15, Equation 26 automatically satisfies condition 27. An attracting invariant manifold implies that the body can exist at any of the infinite number of body compositions specified by the curve I=E
) for clamped intake and energy expenditure rates (see ). Each of these infinite possible body compositions will result in a different body mass M=F
(except for the unlikely case that E
is a function of the sum F
). The body composition is marginally stable along the direction of the invariant manifold. This means that in flux balance, the body composition will remain at rest at any point on the invariant manifold. A transient perturbation along the invariant manifold will simply cause the body composition to move to a new position on the invariant manifold. The one dimensional models have a stable fixed point if the invariant manifold is attracting. We also show in Methods
that for multiple stable fixed points or a limit cycle to exist, f
must be nonmonotonic in L
and be finely tuned. The required fine-tuning makes these latter two possibilities much less plausible than a single fixed point or an invariant manifold.
Data suggest that E
is a monotonically increasing function of F
and L 
. The dependence of f
is not well established and the form of f
depends on multiple interrelated factors. In general, the sensitivity of various tissues to the changing hormonal milieu will have an overall effect on both the supply of macronutrients as well as the substrate preferences of various metabolically active tissues. On the supply side, we know that free fatty acids derived from adipose tissue lipolysis increase with increasing body fat mass which thereby increase the daily fat oxidation fraction, f
, as F
. Furthermore, reduction of F
with weight loss has been demonstrated to decrease f 
. Similarly, whole-body proteolysis and protein oxidation increases with lean body mass 
implying that f
should be a decreasing function of L
. In further support of this relationship, body builders with significantly increased L
have a decreased daily fat oxidation fraction versus control subjects with similar F 
. Thus a stable isolated fixed point is consistent with this set of data.
Implications for Body Mass and Composition Change
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 F vs. L trajectories obeying Equation 12, an example being Forbes's law (14). As shown in , a change of intake or energy expenditure rate will change the position of the invariant manifold. The body composition that is initially at one point on the invariant manifold will then flow to a new point on the perturbed invariant manifold along the trajectory prescribed by (12). If the intake rate or energy expenditure is then restored to the original value then the body composition will return along the same trajectory to the original steady state just as it would in a fixed point model (see ). Only a perturbation that moves the body composition off of the fixed trajectory could distinguish between the two classes. In the fixed point case (), the body composition would go to the same steady state following the perturbation to body composition but for the invariant manifold case (), it would go to another steady state.
An example of a situation where the intake or energy expenditure rate is changed from one clamped value to another and then returned.
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 f
. For example, body composition could be altered directly through liposuction and f
could be altered by administering compounds such as growth hormone. Resistance exercise may cause an increase in lean muscle tissue at the expense of fat. Exogenous hormones, compounds, or infectious agents that change the propensity for fat versus carbohydrate oxidation (for example, by increasing adipocyte proliferation and acting as a sink for fat that is not available for oxidation 
), would also perturb the body composition off of a fixed F
curve by altering f
. If the body composition returned to its original state after such a perturbation then there is a unique fixed point. If it does not then there could be an invariant manifold although multiple fixed points are also possible.
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 
. Despite significant changes of body composition induced by 18 months of growth hormone administration, the subjects returned very closely to their original body composition 18 months following the removal of therapy. However, there was a slight (2%) but significant increase in their lean body mass compared with the original value. Perhaps not enough time had elapsed for the lean mass to return to the original level. Alternatively, the increased lean mass may possibly have been the result of increased bone mineral mass and extracellular fluid expansion, both of which are known effects of growth hormone, but were assumed to be constant in the body composition models. Therefore, this clinical study provides some evidence in support of a fixed point, but it has not been repeated and the result was not conclusive. Using data from the Minnesota experiment 
and the underlying physiology, Hall 
proposed a form for f
that predicts a fixed point. On the other hand, Hall, Bain, and Chow 
showed that an invariant manifold model is consistent with existing data of longitudinal weight change but these experiments only altered weight through changes in caloric intake so this cannot rule out the possibility of a fixed point. Thus it appears that existing data is insufficient to decide the issue.
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. p
), where F
is in units of kg). Consider two cases of the model. If ψ
0 then the model has an invariant manifold and body composition moves along a fixed trajectory in the L
plane. If ψ is nonzero, then there can be an isolated fixed point. We will show an example where if the intake energy is perturbed, the approach of the body composition to the steady state will be identical for both cases but if body composition is perturbed, the body will arrive at different steady states.
For every model with an invariant manifold, a model with a fixed point can be found such that trajectories in the L–F
plane resulting from energy intake perturbations will be identical. All that is required is that ψ in the fixed point model is chosen such that the solution of ψ (F,L
0 defines the fixed trajectory of the invariant manifold model. Using Forbes's law (14), we choose ψ
. We then take a plausible energy expenditure rate of E
+1.55, where energy rate has units of MJ/day and mass has units of kg. This expression is based on combining cross-sectional data 
for resting energy with a contribution of physical activity of a fairly sedentary person 
. Previous models propose similar forms for the energy expenditure 
shows the time dependence of body mass and the F
trajectories of the two model examples given a reduction in energy intake rate from 12 MJ/day to 10 MJ/day starting at the same initial condition. The time courses are identical for body composition and mass. The mass first decreases linearly in time but then saturates to a new stable fixed point. The dashed line represents the same intake rate reduction but with 10 kg of fat removed at day 100. For the invariant manifold model, the fat perturbation permanently alters the final body composition and body mass, whereas in the fixed point model it only has a transient effect. In the fixed point model, the body composition can ultimately exist only at one point given by the intersection of the nullclines (i.e., solution of I=E
0). For the invariant manifold, the body composition can exist at any point on the I
curve (dotted line in ). Since a ψ can always be found so that a fixed point model and an invariant manifold model have identical time courses for body composition and mass, a perturbation in energy intake can never discriminate between the two possibilities.
Time dependence of body mass and F vs. L trajectories.
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 
. A long time constant will make experiments to distinguish between a fixed point and an invariant manifold difficult to conduct. Experimentally reproducing this example would be demanding but if the time variation of the intake rates and physical activity levels were small compared to the induced change then the same result should arise qualitatively. Additionally, the time constant depends on the form of the energy expenditure. There is evidence that the dependence of energy expenditure on F
for an individual is steeper than for the population due to an effect called adaptive thermogenesis 
, thus making the time constant shorter.