# The dynamics of human body weight change

Carson C. Chow111corresponding author<EMAIL_ADDRESS>and Kevin D.
Hall  Laboratory of Biological Modeling, NIDDK. NIH, Bethesda, MD 20892

###### Abstract

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 the long-term dynamics of human weight change can be captured by a
mathematical model of the macronutrient flux balances and all previous models
are special cases of this model. We show that the generic dynamical 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. However,
this distinction is important for the efficacy of clinical interventions that
alter body composition and mass.

Summary

Understanding the dynamics of human body weight change has important
consequences for conditions such as obesity, cancer, AIDS, anorexia and
bulimia nervosa. While we know that changes of body weight result from
imbalances between the energy derived from food and the energy expended to
maintain life and perform physical work, quantifying this relationship has
proved difficult. Part of the difficulty stems from the fact that the body is
comprised of multiple components and we must quantify how weight change is
reflected in terms of alterations of body composition (i.e. fat versus lean
mass). Mathematical modeling is a means to address this issue. Here we show
that a mathematical model of the flux balances of macronutrients, namely fat,
protein and carbohydrates, can provide a general description of the way the
body weight will change over time. For a fixed food intake rate and physical
activity level, the body weight and body composition will approach a steady
state. However, the steady state can correspond to a unique body weight or a
continuum of body weights depending on how fat oxidation depends on the body
weight and composition changes. Interestingly, the existing experimental data
on human body weight dynamics cannot presently distinguish between these two
possibilities. We propose experiments that could determine if steady state
body weight is unique and use computer simulations to demonstrate how such
experiments could be performed.

##  I Introduction

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 hill2003 ; hill2006 . These conditions are characterized
by changes in body weight (mass) that arise from an imbalance between the
energy derived from food and the energy expended to maintain life and perform
work. However, the underlying mechanisms of how changes in energy balance lead
to changes in body mass and body composition are not well understood. In
particular, it is of interest to understand how body composition is
apportioned between fat and lean components when the body mass changes and if
this energy partitioning can be altered. Such an understanding would be useful
in optimizing weight loss treatments in obese subjects to maximize fat loss or
weight gain treatments for anorexia nervosa and cachexia patients to maximize
lean tissue gain.

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 hall2006 ; hall2002 ; alpert1979 ;
alpert1990 ; alpert2005 ; Forbes1970 ; Livingston2001 ; hbc ; jordan ;
antonetti1973 ; payne1977b ; Girardier1994 ; westerterp1995 ; kozusko2001 ;
flatt2004 ; christiansen2005 ; song2007 . Here we show how models of body
composition and mass change can be understood and analyzed within the realm of
dynamical systems theory and can be classified according to their geometric
structure in the two dimensional phase plane. We begin by considering a
general class of macronutrient flux balance equations and progressively
introduce assumptions that constrain the model dynamics. We show that two
compartment models of fat and lean masses can be categorized into two generic
classes. In the first class, there is a unique body composition and mass (i.e.
stable fixed point) that is specified by the diet and energy expenditure. In
the second class, there is a continuous curve of fixed points (i.e. invariant
manifold) with an infinite number of possible body compositions and masses at
steady state for the same diet and energy expenditure. We show that almost all
of the models in the literature are in the second class. Surprisingly, the
existing data are insufficient to determine which of the two classes pertains
to humans. For models with an invariant manifold, we show that an equivalent
one dimensional equation for body composition change can be derived. We give
numerical examples and discuss possible experimental approaches that may
distinguish between the classes.

##  II Results

###  II.1 General model of macronutrient and energy flux balance

The human body obeys the law of energy conservation atwater , which can be
expressed as

$\Delta U=\Delta Q-\Delta W$ (1)

where $\Delta U$ is the change in stored energy in the body, $\Delta Q$ is a
change in energy input or intake, and $\Delta W$ 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 that can be
derived from combustion 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
are slightly less than the values obtained by bomb calorimetry. The energy
expenditure 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 Eq. (1) to hold. While we are
primarily concerned with adult weight change, Eq. (1) is also valid for
childhood growth.

In order to express a change of stored energy $\Delta U$ in terms of body mass
$M$ we must determine the energy content per unit body mass change, i.e. the
energy density $\rho_{M}$. We can then set $\Delta U=\Delta(\rho_{M}M)$. To
model the dynamics of body mass change, we divide Eq. (1) by some interval of
time and take the limit of infinitesimal change to obtain a one dimensional
energy flux balance equation:

$\frac{d}{dt}(\rho_{M}M)=I-E$ (2)

where $I=dQ/dt$ is the rate of metabolizable energy intake and $E=dW/dt$ is
the rate of energy expenditure. It is important to note that $\rho_{M}$ 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 Eq. (2), the
dynamics of $\rho_{M}$ 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 $\rho_{M}$ 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 keys1950 . 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 $\rho_{M}$ 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 Eq. (2) in terms of
macronutrient flux balance equations for body fat $F$, glycogen $G$, and
protein $P$:

$\displaystyle\rho_{F}\frac{dF}{dt}$ $\displaystyle=$ $\displaystyle
I_{F}-f_{F}E$ (3) $\displaystyle\rho_{G}\frac{dG}{dt}$ $\displaystyle=$
$\displaystyle I_{C}-f_{C}E$ (4) $\displaystyle\rho_{P}\frac{dP}{dt}$
$\displaystyle=$ $\displaystyle I_{P}-(1-f_{F}-f_{C})E$ (5)

where $\rho_{F}=39.5$ MJ/kg, $\rho_{G}=17.6$ MJ/kg, $\rho_{P}=19.7$ MJ/kg are
the energy densities, $I_{F},I_{C},I_{P}$ are the intake rates, and $f_{F}$,
$f_{C}$, $1-f_{F}-f_{C}$ 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
atwater1905 . The intake rates are determined by the macronutrient composition
of the consumed food, the efficiency of the conversion of the food into
utilizable form and the thermic effect of feeding. 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 $f_{F}$ and $f_{C}$. The sum of Eqs. (3), (4)
and (5) recovers the energy flux balance equation (2), where the body mass $M$
is the sum of the macronutrients $F$, $G$, $P$, with the associated
intracellular water, and the inert mass that does not change such as the
extracellular water, bones, and minerals, and
$\rho_{M}=(\rho_{F}F+\rho_{G}G+\rho_{P}P)/M$.

The intake and energy expenditure rates are explicit functions of time with
fast fluctuations on a time scale of hours to days periwal2006 . 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 (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 hall2006  to
numerically simulate data from the classic Minnesota human starvation
experiment keys1950 . 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 system dynamics.

###  II.2 Reduced models

####  II.2.1 Two compartment macronutrient partition model

The three compartment macronutrient flux balance model Eqs. (3)-(5) can be
reduced to a two dimensional system for fat mass $F$ and lean mass $L=M-F$,
where $M$ is the total body mass. The lean mass includes the protein and
glycogen with the associated intracellular water along with the mass that does
not change appreciably such as the extracellular water and bone. Hence the
rate of change in lean mass is given by

$\frac{dL}{dt}=(1+h_{P})\frac{dP}{dt}+(1+h_{G})\frac{dG}{dt}$ (6)

where $h_{P}=1.6$ and $h_{G}=2.7$ are reasonable estimates of the hydration
coefficients for the intracellular water associated with the protein and
glycogen respectively hall2006 ; hall2007 . (We note that fat is not
associated with any water.) The glycogen storage capacity is extremely small
compared to the fat and protein compartments. Thus the slow component of
glycogen can be considered to be a constant (see Methods). In other words, on
time scales much longer than a day, which are of interest for body weight
change, we can consider glycogen to be in quasi-equilibrium so that $dG/dt=0$,
as observed in numerical simulations hall2006 . This implies that
$f_{C}=I_{C}/E$, which can be substituted into Eq. (5) to give

$\rho_{P}\frac{dP}{dt}=I_{P}+I_{C}-(1-f_{F})E$ (7)

Substituting Eq. (7) and $dG/dt=0$ into Eq. (6) leads to the two compartment
macronutrient partition model

$\displaystyle\rho_{F}\frac{dF}{dt}$ $\displaystyle=$ $\displaystyle I_{F}-fE$
(8) $\displaystyle\rho_{L}\frac{dL}{dt}$ $\displaystyle=$ $\displaystyle
I_{L}-(1-f)E$ (9)

where $\rho_{L}=\rho_{P}/(1+h_{P})=7.6$ MJ/kg, $I_{F}$ and $I_{L}=I_{P}+I_{C}$
are the intake rates into the fat and lean compartments respectively,
$E=E(I_{F},I_{L},F,L)$ is the total energy expenditure rate, and
$f=f(I_{F},I_{L},F,L)\equiv f_{F}$ is the fraction of energy expenditure rate
attributed to fat utilization.

We note that $dG/dt=0$ may be violated if the glycogen content is proportional
to the protein content, which is plausible because the glycogen mass is stored
in muscle tissue and may scale with protein mass. We show that this assumption
leads to the same two dimensional system. Substituting

$\frac{dG}{dt}=k\frac{dP}{dt}$ (10)

for a proportionality constant $k$, into Eq. (4) gives
$f_{C}E=I_{C}-\rho_{C}kdP/dt$ which inserted into Eq. (5) leads to

$(\rho_{P}+k\rho_{C})\frac{dP}{dt}=I_{L}-(1-f)E$ (11)

Substituting Eqs. (10) and (11) into Eq. (6) will again result in Eq. (9) but
with $\rho_{L}=(\rho_{P}+k\rho_{C})/((1+h_{p}+k+kh_{G})$. For $k=0.044<<1$ as
suggested by Snyder et al. snyder1984 , $\rho_{L}$ has approximately the same
value as before.

Previous studies have considered two dimensional models of body mass change
although they were not derived from the three dimensional macronutrient
partition model. Alpert alpert1979 ; alpert1990 ; alpert2005  considered a
model with $E$ linearized in $F$ and $L$ and different $f$ depending on
context. Forbes Forbes1970  and Livingston et al. Livingston2001  modeled
weight loss as a double exponential decay. Although, they did not consider
macronutrient flux balance, the dynamics of their models are equivalent to the
two dimensional model with $I_{F}$ and $I_{L}$ zero, and $E$ linear in $F$ and
$L$.

####  II.2.2 Energy partition model

The two compartment macronutrient partition model can be further simplified by
assuming that trajectories in the $L$ \- $F$ phase plane follow prescribed
paths satisfying

$\frac{\rho_{F}}{\rho_{L}}\frac{dF}{dL}=\alpha(F,L)$ (12)

where $\alpha(F,L)$ is a continuous function Forbes1987 ; hbc ; jordan  that
depends on the mechanisms of body weight change. This stringent constraint was
first hypothesized by Forbes after analyzing body composition data collected
across a large number of subjects Forbes1987 ; Forbes2000 . Forbes postulated
that for adults

$\alpha=\frac{\rho_{F}}{\rho_{L}}\frac{F}{10.4}$ (13)

so that

$F=D\exp(L/10.4)$ (14)

where $D$ is a free parameter, and the lean and fat masses are in units of kg.
Forbes found that his general relationship (14) is similar whether weight loss
is induced by diet or exercise Forbes2000 . It is possible that resistance
exercise or a significant change in the protein content of the diet may result
in a different relationship for $\alpha$ hansen2007 ; layman2005 ;
stiegler2006 . Infant growth is an example where $\alpha$ is not well
described by the Forbes relationship. Jordan and Hall jordan  used
longitudinal body composition data in growing infants to determine an
appropriate form for $\alpha$ during the first two years of life.

Equation (12) describes a family of $F$ vs $L$ curves, parameterized by an
integration constant (e.g. D in Eq. (14)). Depending on the initial condition,
the body composition moves along one of these curves when out of energy
balance. Dividing Eq. (8) by Eq. (9) and imposing Eq. (12) results in

$f(F,L)=\frac{I_{F}-\alpha I_{L}+\alpha
E}{(1+\alpha)E}=\frac{I_{F}}{E}-\frac{\alpha}{1+\alpha}\frac{I-E}{E}$ (15)

Hall, Bain and Chow hbc  showed that the two compartment macronutrient
partition model with Eq. (15) using Forbes’s law (13) matched a wide range of
data without any adjustable parameters.

Substituting Eq. (15) into the macronutrient partition model (8) and (9) leads
to the Energy Partition model:

$\displaystyle\rho_{F}\frac{dF}{dt}$ $\displaystyle=$
$\displaystyle(1-p)(I-E)$ (16) $\displaystyle\rho_{L}\frac{dL}{dt}$
$\displaystyle=$ $\displaystyle p(I-E)$ (17)

where $p=p(F,L)=1/(1+\alpha)$ is known as the p-ratio dugdale1977 . In the
energy partition model, an energy imbalance $I-E$ is divided between the
compartments according to a function $p(F,L)$ that defines the fraction
assigned to lean body tissue (mostly protein). Most of the previous models in
the literature are different versions of the energy partition model
antonetti1973 ; payne1977b ; alpert1990 ; Girardier1994 ; westerterp1995 ;
kozusko2001 ; flatt2004 ; christiansen2005 , although none of the authors have
noted the connection to macronutrient flux balance or analyzed their models
using dynamical systems theory. Some of these previous models are expressed as
computational algorithms that can be translated to the form of the energy
partition model.

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 $f$ (which can be measured by
indirect calorimetry) by using Eq. (15) hbc . Previous uses of the energy
partition model often considered $p$ to be a constant antonetti1973 ;
payne1977b ; alpert1990 ; Girardier1994 ; kozusko2001 ; flatt2004 ;
christiansen2005 , which implies that the partitioning of energy is
independent of current body composition and macronutrient composition. This is
in contradiction to weight loss data that finds that the fraction of body fat
lost does depend on body composition with more fat lost if the body fat is
initially higher Forbes1987 ; elia1999 ; hall2007b . However, if $\alpha$ is a
weak function of body composition then a constant p-ratio may be a valid
approximation for small changes. Flatt flatt2004  considered a model where the
p-ratio was constant but included the dynamics for glycogen. His model would
be useful when dynamics on short time scales are of interest.

It may sometimes be convenient to express the macronutrient partition model
with a unique fixed point as

$\displaystyle\rho_{F}\frac{dF}{dt}$ $\displaystyle=$
$\displaystyle(1-p)(I-E)+\psi$ (18) $\displaystyle\rho_{L}\frac{dL}{dt}$
$\displaystyle=$ $\displaystyle p(I-E)-\psi$ (19)

for a function $\psi=\psi(I_{F},I_{L},F,L)$, which is zero at the fixed point
$(F_{0},L_{0})$. We use this form in numerical examples in Sec. II.4.1. The
fasting model of Song and Thomas song2007 ) used this form with $I=0$ and
$\psi$ was a function of $F$ representing ketone production. Comparing to Eq.
(8) and Eq. (18) gives

$f=\frac{I_{F}}{E}-(1-p)\frac{I-E}{E}-\frac{\psi}{E}$ (20)

####  II.2.3 One Dimensional models

The dynamics of the energy partition model Eqs. (16) and (17) move along fixed
trajectories in the $L$ \- $F$ plane. Thus a further simplification to a one
dimensional model is possible by finding a functional relationship between $F$
and $L$ so that one variable can be eliminated in favor of the other. Such a
function exists if Eq. (12) has a unique solution, which is guaranteed in some
interval of $L$ if $\alpha(F,L)$ and $\partial\alpha/\partial F$ are
continuous functions of $F$ and $L$ on a rectangle containing this interval.
These are sufficient but not necessary conditions.

Suppose a relationship $F=\phi(L)$ can be found between $F$ and $L$.
Substituting this relationship into Eq. (16) and Eq. (17) and adding the two
resulting equations yields the one dimensional equation

$\frac{dL}{dt}=\frac{I-E(\phi(L),L)}{\rho_{F}\phi^{\prime}(L)+\rho_{L}}$ (21)

We can obtain a dynamical equation for body mass by expressing the body mass
as $M=L+\phi(L)$. If we can invert this relationship uniquely and obtain $L$
as a function of $M$, then this can be substituted into Eq. (21) to obtain a
dynamical equation for $M$.

As an example, assume $p$ to be a constant, which was used in antonetti1973 ;
payne1977b ; alpert1990 ; Girardier1994 ; kozusko2001 ; flatt2004 ;
christiansen2005 . This implies that the phase orbits are a family of straight
lines of the form $F=\beta L+C\equiv\phi(L)$ where
$\beta=\rho_{L}(1-p)/(\rho_{F}p)$ and $C$ is a constant that is specified by
the initial body composition. This results in

$\frac{dM}{dt}=\left(\frac{1-p}{\rho_{F}}+\frac{p}{\rho_{L}}\right)\left[I-E\left(\frac{\beta}{1+\beta}(M-C)+C,\frac{M-C}{1+\beta}\right)\right]$
(22)

Linearizing Eq. (22) around a mass $M_{0}$ gives

$\rho_{M}\frac{dM}{dt}=\mu-\epsilon(M-M_{0})$ (23)

where $\rho_{M}=\rho_{F}\rho_{L}/(\rho_{L}+(\rho_{F}-\rho_{L})p)$,
$\mu=I-E(F(M_{0}),L(M_{0}))$ and $\epsilon=dE/dM|_{M=M_{0}}$. This is the form
used in christiansen2005 .

If Eqs. (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 Eq. (14) (i.e. $\phi(L)=D\exp(L/10.4)$) in Eq. (21) yields

$\frac{dL}{dt}=\frac{10.4}{\rho_{F}D\exp(L/10.4)+10.4\rho_{L}}[I-E(D\exp(L/10.4),L)]$
(24)

Similarly, a one dimensional equation for the fat mass has the form

$\frac{dF}{dt}=\frac{F}{\rho_{F}F+10.4\rho_{L}}[I-E(F,10.4\log(F/D))]$ (25)

Since the mass functions $M=L+D\exp(L/10.4)$ or $M=F+10.4\log(F/D)$ cannot be
inverted in closed form, an explicit one dimensional differential equation in
terms of the mass cannot be derived. However, the dynamics of the mass is
easily obtained using either Eq. (24) or Eq. (25) together with the relevant
mass function. For large changes in body composition, the dynamics could
differ significantly from the constant $p$ models (22) or (23).

The one dimensional model gives the dynamics of the energy partition model
along a fixed trajectory in the $F$ \- $L$ plane. The initial body composition
specifies the constant $C$ or $D$ in the above equations. A one dimensional
model will represent the energy partition model even if the intake rate is
time dependent. Only for a perturbation that directly alters body composition
will the one dimensional model no longer apply. However, after the
perturbation ceases, the one dimensional model with a new constant will apply
again.

###  II.3 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 strogatz ; gh . The one dimensional models are
a subclass of two dimensional dynamics. Three dimensional dynamical systems
are generally more difficult to analyze but Hall hall2006  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 $I_{F}$ and $I_{L}$ 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
strogatz ; gh . 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 $L$ and $F$ to be positive and finite. For
differentiable $f$ and $E$, 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
$I_{F}-fE=0$ and $I_{L}-(1-f)E=0$. Example nullclines and phase plane
portraits of the macronutrient model are shown in Fig. 1. 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 hall2002 . If the nullclines are collinear then
there can be an attracting one-dimensional invariant manifold (continuous
curve of fixed points) in the $L$ \- $F$ 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 b), an attracting invariant manifold c), and a
limit cycle attractor d).

The fixed point conditions of Eqs. (8) and (9) can be expressed in terms of
the solutions of

$\displaystyle E(F,L)=I$ (26) $\displaystyle f(F,L)=\frac{I_{F}}{I}$ (27)

where $I=I_{F}+I_{L}$, 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 Eq. (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 $E$ and $f$ on $F$ and $L$ 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 $I=E$.
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 Eq. (15), $I=E$ 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 $I=E(F,L)$ for
clamped intake and energy expenditure rates (see Figure 1 c)). Each of these
infinite possible body compositions will result in a different body mass
$M=F+L$ (except for the unlikely case that $E$ is a function of the sum
$F+L$). 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 $L$
cunningham1991 . The dependence of $f$ on $F$ and $L$ 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 $F$ increases astrup1994 . Furthermore, reduction of $F$
with weight loss has been demonstrated to decrease $f$ astrup1992 . Similarly,
whole-body proteolysis and protein oxidation increases with lean body mass
welle1990 ; short2004  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 $F$ bosselaers1994 . Thus a stable isolated fixed point
is consistent with this set of data.

###  II.4 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
Eq. (12), an example being Forbes’s law (14). As shown in Fig. 2, 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 Fig 2 solid curves). Only a perturbation
that moves the body composition off of the fixed trajectory could distinguish
between the two classes. In the fixed point case (Fig. 2 a) dashed-dot curve),
the body composition would go to the same steady state following the
perturbation to body composition but for the invariant manifold case (Fig 2 b)
dashed-dot curve), it would go to another steady state.

Figure 2: An example of a situation where the intake or energy expenditure
rate is changed from one clamped value to another and then returned for the
fixed point case a) and the invariant manifold case b). 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 $f$. For example, body composition could be altered directly through
liposuction or 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
atkinson ; pasarica ; vangipuram ), would also perturb the body composition
off of a fixed $F$ vs $L$ 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 biller2000 . 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 keys1950  and the underlying physiology, Hall hall2006
proposed a form for $f$ that predicts a fixed point. On the other hand, Hall,
Bain and Chow hbc  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.

####  II.4.1 Numerical simulations

We now consider some numerical examples using the macronutrient partition
model in the form given by Eqs. (18) and (19), with a p-ratio consistent with
Forbes’s law (13): $p=2/(2+F)$, where $F$ is in units of kg. Consider two
cases of the model. If $\psi=0$ then the model has an invariant manifold and
body composition moves along a fixed trajectory in the $L$ \- $F$ plane. If
$\psi$ 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 $\psi$ in
the fixed point model is chosen such that the solution of $\psi(F,L)=0$
defines the fixed trajectory of the invariant manifold model. Using Forbes’s
law (14), we choose $\psi=0.05(F-0.4\exp(L/10.4))/F$. We then take a plausible
energy expenditure rate of $E=0.14L+0.05F+1.55$, where energy rate has units
of MJ/day and mass has units of kg. This expression is based on combining
cross-sectional data cunningham1991  for resting energy with a contribution of
physical activity of a fairly sedentary person hall2006 . Previous models
propose similar forms for the energy expenditure payne1977b ; alpert1979 ;
alpert2005 ; christiansen2005 .

Figure 3 shows the time dependence of body mass and the $F$ vs $L$
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 $I=E$ and $\psi=0$). For the invariant
manifold, the body composition can exist at any point on the $I=E$ curve
(dotted line in Fig. 2 d)). Since a $\psi$ 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.

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 hall2006 ; payne1977b ;
alpert1979 ; alpert2005 ; christiansen2005 . 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 $F$ and $L$ for an individual is steeper than for the
population due to an effect called adaptive thermogenesis leibel , thus making
the time constant shorter.

Figure 3: Time dependence of body mass for fixed point model a). 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. Trajectories in the $F$ vs $L$ phase plane for the fixed point
model b). Dotted lines are the nullclines. Time dependence c) and phase plane
d) of the invariant manifold model for the same conditions.

##  III Discussion

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 a 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 $f$. Hall hall2006  considered the effects of
carbohydrate intake on lipolysis and other physiological factors to conjecture
a form of $f$ that does not lead to an invariant manifold. However, our
analysis and numerical examples show that the body composition could have an
invariant manifold but behave indistinguishably from having a fixed point.
Also, the decay to the fixed point could take a very long time, possibly as
long as a decade giving the appearance of an invariant manifold. Only
experiments that perturb the fat or lean compartments independently can tell.

##  IV Methods

###  IV.1 Method of Averaging

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 averaged equations valid on longer time scales.

We do so by introducing a second “fast” time variable $\tau=t/\epsilon$, where
$\epsilon$ is a small parameter that is associated with the slow changes and
let all time dependent quantities be a function of both $t$ and $\tau$. For
example, if $t$ is measured in units of days and $\tau$ is measured in units
of hours then $\epsilon\sim 1/24$. Inserting into (3)-(5) and using the chain
rule yields

$\displaystyle\rho_{F}\left(\frac{\partial F}{\partial
t}+\frac{1}{\epsilon}\frac{\partial F}{\partial\tau}\right)$ $\displaystyle=$
$\displaystyle I_{F}(t,\tau)-f_{F}E(t,\tau)$ (28)
$\displaystyle\rho_{G}\left(\frac{\partial G}{\partial
t}+\frac{1}{\epsilon}\frac{\partial G}{\partial\tau}\right)$ $\displaystyle=$
$\displaystyle I_{C}(t,\tau)-f_{C}E(t,\tau)$ (29)
$\displaystyle\rho_{P}\left(\frac{\partial P}{\partial
t}+\frac{1}{\epsilon}\frac{\partial P}{\partial\tau}\right)$ $\displaystyle=$
$\displaystyle I_{P}(t,\tau)-(1-f_{F}-f_{C})E(t,\tau)$ (30)

We then consider the three body compartments to have expansions of the form

$\displaystyle F(t,\tau)$ $\displaystyle=$ $\displaystyle F^{0}(t)+\epsilon
F^{1}(t,\tau)+O(\epsilon^{2})$ (31) $\displaystyle G(t,\tau)$ $\displaystyle=$
$\displaystyle G^{0}(t)+\epsilon G^{1}(t,\tau)+O(\epsilon^{2})$ (32)
$\displaystyle P(t,\tau)$ $\displaystyle=$ $\displaystyle P^{0}(t)+\epsilon
P^{1}(t,\tau)+O(\epsilon^{2})$ (33)

where $\langle F^{1}\rangle=\langle P^{1}\rangle=\langle G^{1}\rangle=0$ for a
time average defined by $\langle X\rangle=(1/T)\int_{0}^{T}X\ d\tau$ and $T$
represents an averaging time scale of a day. The fast time dependence can be
either periodic or stochastic. The important thing is that the time average
over the fast quantities is of order $\epsilon$ or higher. We then expand the
energy expenditure rate and expenditure fractions to first order in
$\epsilon$:

$E(F,G,P,t,\tau)=E^{0}(t,\tau)+\epsilon\left(\frac{\partial E}{\partial
F}F^{1}+\frac{\partial E}{\partial G}G^{1}+\frac{\partial E}{\partial
P}P^{1}\right)+O(\epsilon^{2})$ (34) $\displaystyle
f^{0}_{i}(F,G,P)=f_{i}(F^{0},G^{0},P^{0})+\epsilon\left(\frac{\partial
f_{i}}{\partial F}F^{1}+\frac{\partial f_{i}}{\partial G}G^{1}+\frac{\partial
f_{i}}{\partial P}P^{1}\right)+O(\epsilon^{2})$ (35)

where $E^{0}(t,\tau)\equiv E(F^{0},G^{0},P^{0},t,\tau)+O(\epsilon^{2})$ and
$i\in\\{F,G,P\\}$. We assume that the expenditure fractions depend on time
only through the body compartments. Substituting these expansions into Eqs.
(28)-(30) and taking lowest order in $\epsilon$ gives

$\displaystyle\rho_{F}\left(\frac{\partial F^{0}}{\partial t}+\frac{\partial
F^{1}}{\partial\tau}\right)$ $\displaystyle=$ $\displaystyle
I_{F}(t,\tau)-f^{0}_{F}E^{0}(t,\tau)$ (36)
$\displaystyle\rho_{G}\left(\frac{\partial G^{0}}{\partial t}+\frac{\partial
G^{1}}{\partial\tau}\right)$ $\displaystyle=$ $\displaystyle
I_{C}(t,\tau)-f^{0}_{C}E^{0}(t,\tau)$ (37)
$\displaystyle\rho_{P}\left(\frac{\partial P^{0}}{\partial t}+\frac{\partial
P^{1}}{\partial\tau}\right)$ $\displaystyle=$ $\displaystyle
I_{P}(t,\tau)-(1-f_{F}-f^{0}_{C})E^{0}(t,\tau)$ (38)

Taking the moving time average of Eqs. (36)-(38) and requiring that
$\langle\partial F^{1}/\partial\tau\rangle$, $\langle\partial
G^{1}/\partial\tau\rangle$, and $\langle\partial P^{1}/\partial\tau\rangle$
are of order $\epsilon$ or higher leads to the averaged equations:

$\displaystyle\rho_{F}\frac{dF^{0}}{dt}$ $\displaystyle=$
$\displaystyle\langle I_{F}\rangle-f^{0}_{F}\langle E^{0}\rangle$ (39)
$\displaystyle\rho_{G}\frac{dG^{0}}{dt}$ $\displaystyle=$
$\displaystyle\langle I_{C}\rangle-f^{0}_{C}\langle E^{0}\rangle$ (40)
$\displaystyle\rho_{P}\frac{dP^{0}}{dt}$ $\displaystyle=$
$\displaystyle\langle I_{P}\rangle-(1-f^{0}_{F}-f^{0}_{C})\langle
E^{0}\rangle$ (41)

In the main text we only consider the slow time scale dynamics so we drop the
superscript and bracket notation for simplicity. Hence, the system (3)-(5) can
be thought of as representing the lowest order time averaged macronutrient
flux balance equations. We note that in addition to the daily fluctuations of
meals and physical activity, there can also be fluctuations in food intake
from day to day periwal2006 . Our averaging scheme can be used to average over
these fluctuations as well by extending the averaging time $T$. A difference
in the choice of $T$ will only result in a different interpretation of the
averaged quantities.

###  IV.2 Stability conditions for fixed points

The dynamics near a fixed point $(F_{0},L_{0})$ are determined by expanding
$fE$ and $(1-f)E$ to linear order in $\delta F=F-F_{0}$ and $\delta L=L-L_{0}$
strogatz ; gh . Assuming solutions of the form $\exp(\lambda t)$ yields an
eigenvalue problem with two eigenvalues given by
$\lambda=\frac{1}{2}\left({\rm Tr}J\pm\sqrt{{\rm Tr}J^{2}-4\det J}\right)$
where

${\rm Tr}J=-\left[\frac{1}{\rho_{F}}\frac{\partial}{\partial
F}(fE)+\frac{1}{\rho_{L}}\frac{\partial}{\partial
L}((1-f)E)\right]_{(F_{0},L_{0})}$ (42)

and

$\det J=\frac{E}{\rho_{L}\rho_{F}}\left[\frac{\partial E}{\partial
L}\frac{\partial f}{\partial F}-\frac{\partial E}{\partial F}\frac{\partial
f}{\partial L}\right]_{(F_{0},L_{0})}$ (43)

A fixed point is stable if and only if ${\rm Tr}J<0$ and $\det J>0$. In the
case of an invariant manifold, $\det J=0$, so the eigenvalues are ${\rm Tr}J$
and $0$. The zero eigenvalue reflects the marginal stability along the
invariant manifold, which is an attractor if ${\rm Tr}J<0$. An attracting
invariant manifold implies a stable fixed point in the corresponding one
dimensional model. Unstable fixed points are either unstable nodes, saddle
points or unstable spirals. In the case of unstable spirals, a possibility is
a limit cycle surrounding the spiral arising from a Hopf bifurcation, where
${\rm Tr}J=0$ and $\det J>0$. In this case, body composition and mass would
oscillate even if the intake rates were held constant. The frequency and
amplitude of the oscillations may be estimated near a supercritical Hopf
bifurcation by transforming the equations to normal form. Stability of a fixed
point puts constraints on the form of $f$. Physiological considerations and
data imply that $\partial E/\partial L>\partial E/\partial F>0$ cunningham1991
; hall2006 . Thus we can set $\partial E/\partial F=\delta\partial E/\partial
L$ where $\delta<1$ (where the derivatives are evaluated at the fixed point).
Then $\det J>0$ implies that

$\partial f/\partial F>\delta\partial f/\partial L$ (44)

and ${\rm Tr}J<0$ implies

$\partial f/\partial F>\gamma\partial f/\partial L-K,$ (45)

where $K=[\delta f+\gamma(1-f)](\partial E/\partial L)/E>0$ and
$\gamma=\rho_{F}/\rho_{L}\simeq 5.2$. Hence $\partial f/\partial F>0$ and
$\partial f/\partial L<0$ guarantees stability of a fixed point.

From Eq. (44) and Eq. (45) and the fixed point conditions (26) and (27), if
$f$ increases monotonically with $F$ and decreases monotonically with $L$ then
there will be a unique stable fixed point. For an invariant manifold, $f$ is
given by Eq. (15), which immediately satisfies $\det J=0$; ${\rm Tr}J<0$ is
guaranteed if $E$ is monotonically increasing in $F$ and $L$. For a Hopf
bifurcation, we require $\partial f/\partial F=\gamma\partial f/\partial L-K$
and Eq. (44), implying $(\gamma-\delta)\partial f/\partial L-K>0$. Since
$\gamma>\delta$, $f$ must increase with $L$ for the possibility of a limit
cycle. However, to ensure that trajectories remain bounded $f$ must decrease
with $L$ for very small and large values of $L$. Hence, $f$ must be
nonmonotonic in $L$ for a limit cycle to exist. This can also be seen from an
application of Bendixson’s criterion gh , which states that a limit cycle
cannot exist in a given region of the $L$ \- $F$ plane if

$\frac{1}{\rho_{F}}\frac{\partial}{\partial
F}(fE)+\frac{1}{\rho_{L}}\frac{\partial}{\partial L}((1-f)E)$ (46)

does not change sign in that region. In addition, the other parameters must be
fine tuned for a limit cycle (see Fig. 1 d)). Similarly, as seen in Fig. 1 c),
for multi-stability to exist, nonmonotonicity and fine tuning are also
required.

###### Acknowledgements.

This research was supported by the Intramural Research Program of the NIH,
NIDDK.

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