# Dissecting flux balances to measure energetic costs in cell biology:
techniques and challenges

Easun Arunachalam1,†,∗ William Ireland2,† Xingbo Yang1,† and Dan
Needleman1,2,3

1Department of Molecular and Cellular Biology, Harvard University, Cambridge,
MA, USA
2School of Engineering and Applied Sciences, Harvard University, Cambridge,
MA, USA
3Center for Computational Biology, Flatiron Institute, New York, NY, USA
†These authors contributed equally to this work
∗To whom correspondence should be addressed<EMAIL_ADDRESS>
## Abstract

Life is a nonequilibrium phenomenon: metabolism provides a continuous supply
of energy that drives nearly all cellular processes. However, very little is
known about how much energy different cellular processes use, i.e. their
energetic costs. The most direct experimental measurements of these costs
involve modulating the activity of cellular processes and determining the
resulting changes in energetic fluxes. In this review, we present a flux
balance framework to aid in the design and interpretation of such experiments,
and discuss the challenges associated with measuring the relevant metabolic
fluxes. We then describe selected techniques that enable measurement of these
fluxes. Finally, we review prior experimental and theoretical work that has
employed techniques from biochemistry and nonequilibrium physics to determine
the energetic costs of cellular processes.

###### Contents

  1. 1 Introduction
  2. 2 Coupled flux balances are a unifying framework for interpreting bioenergetic measurements
  3. 3 Chemical and calorimetric methods to calculate ATP production fluxes
    1. 3.1 Inferring ATP production rates by chemical measurements of glycolytic and respiratory fluxes
      1. 3.1.1 Glycolytic ATP production
      2. 3.1.2 Mitochondrial respiratory ATP production
      3. 3.1.3 Measuring combined ATP production by glycolysis and respiration
    2. 3.2 Measuring glycolytic and respiratory fluxes using calorimetry
  4. 4 Measuring energetic costs by process activity modulation
  5. 5 Methods and technologies for measuring fluxes and concentrations
    1. 5.1 Measurements of respiration rate
    2. 5.2 Measurements of nutrient import and waste export
    3. 5.3 Intracellular flux measurements
    4. 5.4 Intracellular metabolite concentration measurements
    5. 5.5 Measuring metabolic fluxes with calorimetry
  6. 6 Prior theoretical and experimental approaches to measuring bioenergetic costs
    1. 6.1 Estimates and inhibition experiments
    2. 6.2 Using nonequilibrium physics to estimate energetic costs
  7. 7 Open questions

##  1 Introduction

Cells consume energy derived from the environment to power diverse processes
including biosynthesis, ion pumping, cytoskeletal remodeling, signal
transduction, error correction and information processing. While the metabolic
enzymes and pathways that transduce energy have been extensively studied, very
little is known about how much energy is used by different cellular processes.
Developing a quantitative understanding of energy usage in cell biology - i.e.
cell biological energetic costs - would have broad implications beyond energy
metabolism. Since the rate of energy usage is a measure of the combined rates
of all energy-consuming reactions, establishing the energetic costs of
cellular processes will provide a powerful means to characterize their
systems-level behavior in situ. Not only will this be useful for testing
mechanistic models of these processes, it will also help answer fundamental
questions concerning their efficiency and the potential impact of constraints
from nonequilibrium thermodynamics [1]. Knowing the energetic costs of
cellular processes may also provide insight into the changes in metabolism
that take place over development or as result of diseases [1], and may have
implications for the evolution of these processes [2].

Adenosine triphosphate (ATP) is the primary “energy currency” in cells:
transduction of free energy from the breakdown of nutrients enables the
synthesis of ATP from adenosine diphosphate (ADP) and phosphate
(P${}_{\text{i}}$), and the hydrolysis of ATP to ADP releases free energy that
is used to power many cellular processes. The rate at which ATP molecules are
hydrolyzed to supply free energy for a process is often referred to as the
energetic cost of that process.

Unfortunately, it is not currently possible to directly measure the rate at
which individual cellular processes consume ATP in situ. Thus, all of our
knowledge of the energetic costs of cell biological processes comes from
estimates and indirect measurements. Estimates of energetic costs of
particular processes typically combine knowledge of the individual chemical
steps in those processes, the ATP stoichiometry of those chemical reactions,
and inferences of the total rate of those processes (based on cellular
composition, growth rate, and other physiological measurements) [3, 4, 2].
While such estimates have been very informative, they are based on numerous
assumptions whose validity is difficulty to verify. There have only been
limited efforts to experimentally test such estimates of energetic costs, and
when apparent disagreements are found [5, 6], it can be challenging to
determine if this reflects limitations of the estimates, the measurements, or
both.

Experimental measurements of energetic costs are often based on modulating the
activity of target processes and quantifying concomitant changes in total
cellular energy usage [7, 8]. Typically, in such studies a process of interest
is inhibited, the change in total cellular metabolic flux is measured, and the
inferred change in ATP consumption rates is interpreted as the energetic cost
of the inhibited process. One major concern highlighted by these and other
studies is the fact that the activity of different cellular processes is
inextricably linked, making their respective costs challenging to dissect
experimentally [6]. Furthermore, the scope of these studies has been limited;
most cellular processes in systems of interest have not yet been studied
systematically through such experiments.

Despite their drawbacks, activity modulation experiments remain the most
direct method to probe the energetic costs of cellular processes. In this
review, we show how theory based on coupled flux balances can be used to
interpret such experiments and infer the energetic costs of cellular
processes. The balance equations can inform the design of activity modulation
experiments by enumerating the individual fluxes that must be quantified for
this analysis, and provide a framework to infer energetic costs from changes
in the coupled fluxes. Additionally, we use this framework to highlight the
challenges associated with interpreting changes in fluxes. We then discuss the
strengths and limitations of the different technologies used to measure
bioenergetic fluxes and characterize their coupling, and review studies that
have employed these approaches to measure energetic costs.

##  2 Coupled flux balances are a unifying framework for interpreting
bioenergetic measurements

The ATP consumption rates of individual cellular processes of interest cannot
be directly measured. While challenging to determine, the total combined ATP
consumption rate of all processes in a cell, is, in principle, experimentally
accessible. This suggests an indirect approach to measuring the ATP
consumption rate of a process of interest: change the activity of the
processes, determine the resulting change in the total ATP consumption rate,
and use these changes to calculate the energetic cost of the process. While
this is a promising and powerful approach, there are numerous subtleties
associated with using changes in metabolic fluxes to infer energetic costs. In
what follows, we will use a systematic categorization of the relationships
between the rates of different ATP-producing and ATP-consuming reactions to
help interpret such changes in metabolic fluxes.

An ATP mole balance on the system of interest relates changes in ATP levels to
fluxes through ATP-producing and ATP-consuming reactions. While the fluxes
through many of these ATP-dependent reactions are not directly accessible,
they are linked to other reactions in the cell whose rates may be
straightforwardly measured. Therefore, careful consideration of the ATP mole
balance, as well as of the coupled mole balances on other chemical species and
the energy balance, provides a means to calculate and interpret changes in ATP
fluxes. These balance equations inform experimental design by indicating which
experimentally accessible reaction rates must be measured in order to infer
the ATP consumption rate of a given process from an activity modulation
experiment.

We therefore begin by considering the general mole balance for a chemical
species $i$ in the cell. $n_{i}$, the number of moles of this species in the
cell, can change due to the species being imported (with flux
$J_{i,\text{import}}$) or exported (with flux $J_{i,\text{export}}$), or due
to the species being produced or consumed by reaction $k$ (with flux
$J_{ik}$). Summing over all reactions involving this species gives:

$\displaystyle\dfrac{dn_{i}}{dt}$
$\displaystyle=J_{i,\text{import}}-J_{i,\text{export}}+\sum_{\text{rxns
}k}J_{ik}$ (1)

Note that $J_{ik}$ is negative for reactions that are net consumers of species
$i$. We next consider the mole balance for ATP. Import and export of ATP can
be safely neglected, so enumerating all reactions that produce or consume ATP
gives:

$\displaystyle\dfrac{dn_{\text{ATP}}}{dt}$ $\displaystyle=\sum_{\text{ATP-prod
rxns }k^{\prime}}J_{\text{ATP},k^{\prime}}+\sum_{\text{ATP-cons rxns
}k^{\prime\prime}}J_{\text{ATP},k^{\prime\prime}}$ (2)

Here, we have divided the reactions into ATP-producing reactions (with
positive fluxes) and ATP-consuming reactions (with negative fluxes). Such a
division is general because, while the flux of some reactions may change signs
under different conditions, at any given time the flux of all reactions is
either positive or negative (or zero). In many systems of interest, glycolysis
and respiration are the dominant ATP producing pathways, in which case
$\sum_{\text{ATP-prod rxns
}k^{\prime}}J_{\text{ATP},k^{\prime}}=J_{\text{ATP,glycolysis}}+J_{\text{ATP,respiration}}$.

At steady state, $\frac{dn_{\text{ATP}}}{dt}=0$, and the ATP production and
consumption fluxes are balanced. The ATP consuming reactions in this balance
can be conceptualized as containing contributions from numerous cell
biological processes, each with their own coarse-grained ATP-consuming flux,
such as $k^{\prime\prime}$ = protein production, $k^{\prime\prime}$ =
cytoskeletal assembly, and $k^{\prime\prime}$ = ion pumping (Fig. 1). These
processes can be further decomposed into more elementary processes, which in
turn can be further decomposed, all the way down to individual chemical
reactions that hydrolyze ATP.

Figure 1:  Many individual processes and pathways contribute to the total ATP
production and consumption fluxes, which are balanced at steady state. ATP
production, primarily by glycolysis and respiration (red arrows), is balanced
by consumption of ATP by numerous cellular processes, including protein
production, cytoskeletal assembly, and ion pumping (blue arrows) at steady
state. ATP-producing pathways are powered by catabolism of carbon-containing
molecules supplied to cells (gray arrows). Portions of this figure were
created with BioRender [9].

The energetic cost of a cell biological process $k^{\prime\prime}$ is its
corresponding ATP consumption flux, $J_{\text{ATP},k^{\prime\prime}}$. It is
not generally possible to directly measure such fluxes. However, at steady
state the total ATP consumption flux equals the total ATP production flux, so
measuring the total rate of ATP synthesis enables inference of the total rate
of ATP hydrolysis. A complicating factor is that direct measurement of ATP
synthesis rate is not always possible either. While total cellular ATP
synthesis rates have been directly measured by spectroscopic methods [10],
these techniques are not widely used in modern bioenergetics studies and may
be difficult to apply to many systems. Thus, the methods most commonly used to
resolve ATP synthesis fluxes rely on coupling between ATP-producing reactions
and other reactions in the same pathway, or other processes whose rates can be
more easily measured. For example, ATP synthesis fluxes are directly linked to
carbon fluxes (as illustrated in Fig. 1) and oxygen consumption rate (a
measure of oxidative phosphorylation) which are both experimentally
accessible. Because the structure of these biochemical pathways is known,
measurements of these coupled fluxes enable calculation of
$J_{\text{ATP,glycolysis}}$ and $J_{\text{ATP,respiration}}$, as discussed in
Sec. 3.1.

The simplest case of these balances is when a cell is at steady state, not
changing and not growing (e.g. cell cycle-arrested oocytes). In this case, the
time derivatives are eliminated, and ATP production and consumption are
balanced. A slightly more complex case entails cells that grow at a constant
rate without changing their overall composition (e.g. microbes in balanced
growth). It is straightforward to show that in such case normalization by the
mass (i.e. considering ATP per unit biomass) again yields a steady-state
system with zero time derivatives. The most complicated, general case entails
explicit time dependence, e.g. in the case of developing embryos which both
grow and change their composition. In such systems, dynamic changes in molar
composition, production, and consumption may be non-negligible and must be
measured in order to infer ATP synthesis/hydrolysis fluxes. For the moment, we
shall focus on steady-state systems, where the total ATP production flux is
equal to the total ATP consumption flux.

In steady-state systems, overall changes in ATP consumption rate are
compensated by corresponding changes in production rate. Thus, manipulating
the rate of a single ATP-consuming process of interest and measuring the
concomitant change in ATP production is an intuitively appealing experiment to
infer the energetic cost of that process. Let us consider a hypothetical
experiment where the rate of a given ATP-consuming process of interest $p$ was
decreased by 50% while leaving the rates of all other ATP-consuming processes
$k^{\prime}\neq p$ unchanged, and it was observed that the global ATP
production rate of the system decreased by some amount $x$ mol ATP/s. Assuming
the energetic cost of a process is linear in its rate, the naive
interpretation would be that the energetic cost of process $p$ was $2x$ mol
ATP/s.

In reality, the potential coupling between different cellular processes can
complicate the interpretation of such experiments. Processes can be directly
coupled, e.g. through signaling pathways that regulate their activity, and
indirectly coupled, e.g. through shared pools of substrates or cofactors.
Altering the activity of certain processes can cause changes in physiology
which affect the rates of a multitude of other cellular activities, and
therefore their respective rates of ATP production or consumption. For
example, manipulating the activity of a process may alter cells’ growth rates,
changing the rates of many biosynthetic reactions.

Additionally, straightforward interpretation of a change in ATP production
rates as the energetic cost of the targeted process requires that the total
ATP consumption of the cell vary linearly with the ATP demand of that process.
It is not yet clear whether this is generally the case, and systematic studies
are needed to answer this question in each system of interest. There is reason
to believe that this could complicate our understanding of flux changes: the
enzyme adenylate kinase, which catalyzes the reversible reaction ATP$+$AMP ¡-¿
2 ADP, operates close to equilibrium and is known to buffer the energy charge
of the cell (([ATP]+1/2[ADP])/([ATP]+[ADP]+[AMP])) [11]. Other ATPases can
have a Michaelis-Menten constant $K_{M}$ close to intracellular ATP/ADP
levels, and their reaction rates and directions could be similarly sensitive
to changes in ATP/ADP induced by perturbations [8, 12]. Their contribution to
overall ATP production and consumption fluxes could be different in systems
before and after process activity changes. This would introduce nonlinearities
into the relationship between the ATP demand of a process of interest and the
global ATP consumption of the system, hampering interpretation of activity
modulation experiments. The extent of this effect is unknown.

Thus, it is important to characterize the dependence of different processes’
ATP consumption rates on ATP levels (and possibly levels of ADP, AMP,
phosphate, and other species which affect their rates) to properly infer the
cost of a single process of interest. One way to conceptualize the impact of
this complication is to divide ATP consuming processes into those whose fluxes
are sensitive to levels of these species, and those whose fluxes are
insensitive to levels of these species. Reactions which respond to ATP levels
(or the levels of ADP, AMP, phosphate, etc.) will tend to buffer ATP levels
(“buffer” processes) by two possible mechanisms: 1) some ATPases will increase
their rate of consumption as ATP levels rise and decrease as they fall, and 2)
others can switch their flux from consumption to production depending on the
levels of ATP, ADP, AMP and other species. In contrast, many other processes
are relatively insensitive to ATP levels and continue to consume ATP at nearly
constant rates (“demand” processes). This can be formalized by separating the
ATP-consuming fluxes in Eq. 2 on the basis of their sensitivity to ATP levels
over the range explored in a given modulation experiment:

$\displaystyle\dfrac{dn_{\text{ATP}}}{dt}$
$\displaystyle=\underbrace{\left(J_{\text{ATP,glycolysis}}+J_{\text{ATP,respiration}}\right)}_{\text{ATP
production}}-\underbrace{\left(\sum_{\text{rxns }k^{\prime}}J_{\text{ATP
demand},k^{\prime}}+\sum_{\text{rxns }k^{\prime\prime}}J_{\text{ATP
buffer},k^{\prime\prime}}\right)}_{\text{ATP consumption}}$ (3)

Here, $J_{\text{ATP demand},k^{\prime}}$ is the ATP consumption flux
associated with ATP level-insensitive process $k^{\prime}$ and $J_{\text{ATP
buffer},k^{\prime\prime}}$ is the ATP production flux associated with the
buffering process $k^{\prime\prime}$; these fluxes are defined to be positive
when proceeding in the direction of ATP synthesis. The partitioning of the
reactions in the cell into the two sums depends on the $K_{M}$ for ATP of the
machinery in each process and the range of ATP concentrations explored in a
given experiment, which together determine which reaction rate changes can be
safely neglected.

In summary, the ATP consumption rates of individual processes cannot be
directly measured. However, Eq. 3 shows how total ATP production and
consumption by all processes is balanced in steady-state systems. This
structure motivates the use of activity modulation experiments to probe the
energetic costs of individual processes. Despite the complications associated
with the interpretation of ATP flux changes, these experiments are the most
direct method to determine energetic costs. Therefore, in the following
section we discuss how the relationship between Eq. 3 and other mole balances
enables determination of ATP production rates, and therefore the energetic
costs of different processes, from activity modulation experiments. We then
consider the caveats of these measurements, and subtleties of activity
modulation experiments, including coupling between different fluxes.

##  3 Chemical and calorimetric methods to calculate ATP production fluxes

Inferring the energetic cost of a specific cellular process requires measuring
changes in ATP production fluxes caused by modulating the activity of that
process. The major ATP-producing pathways in many systems are glycolysis and
respiration, whose rates can be measured by several different experimental
methods. In this section, we will discuss the logic underlying chemical and
calorimetric methods to measure these fluxes, and the respective challenges of
these methods.

###  3.1 Inferring ATP production rates by chemical measurements of glycolytic
and respiratory fluxes

In many systems, ATP is produced primarily through glycolysis and respiration,
which are catabolic pathways that transduce free energy from the breakdown of
carbon sources to phosphorylate ADP. The fluxes through these pathways are
coupled primarily through pools of carbon-containing metabolites and the
electron carrier NAD(H). Glycolysis and respiration each directly produce ATP.
Additionally, glycolysis reduces the electron carrier NAD+ to produce NADH, as
well as the carbon-containing compound pyruvate. These pyruvate molecules then
serve as input to respiration, or another pathway, fermentation. Fermentation
converts pyruvate to waste products without producing ATP, and serves to
regenerate the cofactor NAD+. In contrast, respiration involves oxidation of
pyruvate to CO2 through the Krebs cycle, which is coupled to the electron
transport chain (ETC) to produce ATP. Fig. 2 illustrates the high-level
couplings between ATP synthesis and hydrolysis to carbon and other fluxes. The
relationships between these fluxes suggest which experimentally accessible
reaction rates can be measured to infer ATP production rates.

Figure 2: Coupling between ATP production/consumption and other
intra/extracellular fluxes. Top left: Glycolysis produces ATP through
substrate-level phosphorylation, and breaks down substrates into species which
serve as inputs for fermentation and respiration. Release of fermentation
waste products is sometimes coupled to proton export, which leads to
extracellular acidification. Respiration converts carbon inputs and oxygen to
carbon dioxide, which is released from the cell and also contributes to
extracellular acidification. Bottom left: Pyruvate produced by glycolysis
(and/or directly supplied) is consumed by fermentation and the Krebs cycle
(respiration), or can be diverted to biosynthesis. Bottom right: coupled
cycles within mitochondria transduce free energy from breakdown of carbon
skeletons in the Krebs cycle to power ATP synthesis. Reducing equivalents from
glycolysis shuttled into the mitochondria are not shown here. Top right:
Within the framework of cellular activity modulation experiments, ATP-
consuming processes can be conceptually separated into those sensitive to
changes in ATP availability (“buffer”) and those which are insensitive
(“demand”). Note that the ATP flux of some buffer processes may switch sign,
in which case they become secondary ATP-producing processes. Portions of this
figure were created with BioRender [9].

####  3.1.1 Glycolytic ATP production

Glycolysis is a highly conserved ten-reaction pathway that converts glucose to
pyruvate, which is then processed by fermentation or respiration. In the first
portion of glycolysis, ATP is consumed to split glucose into two carbon
skeletons that are prepared for the second, energy-yielding phase. In this
phase each of the two skeletons is converted to a pyruvate molecule, producing
one molecule of the electron carrier NADH and two molecules of ATP each. Thus,
there is a net gain of 2 molecules of ATP and 2 molecules of NADH per input
glucose that passes through the entirety of glycolysis [13].

Other carbon sources, such as fructose and sucrose, can also enter glycolysis
at intermediate stages. Additionally, a number of glycolytic intermediates are
siphoned off for various biosynthetic reactions, including production of
nucleotides, amino acids, and lipids. The relative magnitudes of biosynthetic
and glycolytic fluxes can vary widely from system to system, resulting in
different fractions of glycolytic substrates ultimately participating in the
ATP-producing steps of glycolysis and being transformed into pyruvate [14,
15]. These complications can be coarse-grained into an effective factor
$\alpha_{\text{PPyr}}$ which describes the average molar ratio of ATP directly
produced by glycolysis to pyruvate produced by glycolysis, which can be
measured in a given system using metabolic flux analysis (MFA) or other
methods:

$\displaystyle\begin{split}J_{\text{ATP,
glycolysis}}&=\alpha_{\text{PPyr}}J_{\text{pyruvate, glycolysis}}\end{split}$
(4)

We can then back out $J_{\text{pyruvate, glycolysis}}$ by considering how
glycolytic, respiratory, fermentative, and biosynthetic fluxes are coupled
through pyruvate. Pyruvate can be fermented (with flux $J_{\text{pyruvate,
fermentation}}$) to species like lactate, acetate, or ethanol depending on the
specific system and nutrient conditions, oxidizing NADH back to NAD+ in the
process; the fermentation products are then eliminated by the cell. Pyruvate
can also be fed into the Krebs cycle and thus contribute to respiration (with
flux $J_{\text{pyruvate, respiration}}$), and can also be diverted to
biosynthesis (with flux $J_{\text{pyruvate, biosynthesis}}$) [16]. Pyruvate
may also be directly supplied in some systems (with flux $J_{\text{pyruvate,
supply}}$). At steady state, the flux balance on pyruvate is:

$\displaystyle\begin{split}J_{\text{pyruvate, glycolysis}}&=J_{\text{pyruvate,
respiration}}+J_{\text{pyruvate, fermentation}}+J_{\text{pyruvate,
biosynthesis}}-J_{\text{pyruvate, supply}}\end{split}$ (5)

as illustrated in Fig. 2 (lower left detail). Under certain nutrient
conditions, fermentation operates in reverse (e.g. conversion of lactate or
ethanol to pyruvate), in which case $J_{\text{pyruvate, fermentation}}$ is
negative. $J_{\text{pyruvate, fermentation}}$ and $J_{\text{pyruvate,
supply}}$ are typically determined by measuring cellular imports and exports,
while $J_{\text{pyruvate, biosynthesis}}$ can be determined by measurements of
cellular composition and growth rate. Measuring $J_{\text{pyruvate,
respiration}}$, as well as characterizing the indirect contribution of
glycolytic NADH production to respiratory ATP production, requires measurement
of mitochondrial respiratory activity.

####  3.1.2 Mitochondrial respiratory ATP production

Mitochondrial respiration consists of a series of free energy transduction
cycles that link catabolic reactions to ATP production [17] (Fig. 2, lower
right detail). Free energy from breakdown of carbon sources, including
pyruvate, carbohydrates, fatty acids, and amino acids is transduced to
electron carriers NADH/NAD+ or FADH2/FAD+. The reducing equivalents supplied
by these carriers are used by the electron transport chain (ETC) to reduce
molecular oxygen and establish a proton gradient across the mitochondrial
inner membrane. Finally, the free energy associated with this proton gradient
is used to power ATP synthesis; this is achieved by proton flow through ATP
synthase, a cross-membrane rotary motor. However, some fraction of protons
leak directly through the lipid bilayer without passing through ATP synthase,
dissipating free energy into heat.

At steady state, the net fluxes through the NADH/NAD+ redox cycle, the H+
translocation cycle, and the ATP/ADP cycle are balanced with each other.
Because the ETC consumes oxygen to oxidize NADH, mitochondrial oxygen
consumption rate ($\text{OCR}_{\text{mito}}$) is proportional to the NADH
oxidation flux (assuming FADH2 flux is negligible or proportional to NADH
flux), and is proportional to the rate of proton extrusion from the
mitochondrial matrix into the intermembrane space. The rate of proton
extrusion, less the rate of proton leak through the mitochondrial inner
membrane ($J_{\text{leak}}$), is the rate at which protons return to the
matrix through ATP synthase. The rate of ATP synthesis by this process is thus
given by:

$\displaystyle J_{\text{ATP,respiration}}$
$\displaystyle=\alpha_{\text{PO}}\text{OCR}_{\text{mito}}-\alpha_{\text{PO}}\alpha_{\text{HO}}^{-1}J_{\text{leak}}$
(6)

where fluxes of different species are related to each other by the H+/O ratio,
$\alpha_{\text{HO}}$ (the number of protons translocated per oxygen atom
reduced) and the P/O ratio, $\alpha_{\text{PO}}$ (the number of ATP molecules
synthesized per oxygen atom reduced in the absence of proton leak).
$\alpha_{\text{HO}}$ is 10 for NADH oxidation and 6 for FADH2 oxidation. The
exact value of $\alpha_{\text{PO}}$ is variable, but a typical value for
mammalian cells is 2.5 for mitochondria using NADH and 1.5 when using FADH2
[18, 19, 20]. These ratios are determined by the reaction and subunit
stoichiometry of the enzymes catalyzing the reactions in mitochondria:
$\alpha_{\text{HO}}$ is determined by complex I, III and IV of the electron
transport chain; $\alpha_{\text{PO}}$ is determined by $\alpha_{\text{HO}}$,
properties of ATP synthase, and transport of ATP from mitochondria to the
cytoplasm by the enzyme adenine nucleotide translocase. Estimating
$J_{\text{leak}}$ is challenging, as we discuss in Section 4.

####  3.1.3 Measuring combined ATP production by glycolysis and respiration

The NADH reducing equivalents produced by glycolysis can effectively be
transported into mitochondria by shuttling mechanisms (e.g. the malate-
aspartate shuttle, glycerol-3-phosphate shuttle) [21]. This indirect
contribution of glycolysis to total cellular ATP production is captured in the
measurement of respiratory ATP production (Eq. 6).

Pyruvate produced by glycolysis is not necessarily the only input to the Krebs
cycle, and therefore it is not the only source of reducing equivalents for the
electron transport chain in many systems [22, 23]. Instead, the flux of
pyruvate from glycolysis to respiration is given by:

$\displaystyle\begin{split}J_{\text{pyruvate,
respiration}}&=\alpha_{\text{O}\text{Pyr}_{\text{glyc}}}^{-1}\text{OCR}_{\text{mito}}\end{split}$
(7)

where $\alpha_{\text{O}\text{Pyr}_{\text{glyc}}}$ is an empirical
proportionality constant that is the ratio of mitochondrial OCR to the rate at
which pyruvate is supplied to respiration by glycolysis.
$\alpha_{\text{O}\text{Pyr}_{\text{glyc}}}$ captures two variable factors: (1)
the fraction of pyruvate produced by glycolysis, as opposed to pyruvate
supplied by other means, which is fed into the Krebs cycle, and (2) the
variable contribution of pyruvate to the production of reducing equivalents
(e.g. NADH). Krebs cycle intermediates can be siphoned off to participate in
biosynthetic reactions, and some species can enter the cycle at a point other
than Acetyl-CoA; $\alpha$-ketoglutarate is an example. Additionally, some
organisms employ variants of the Krebs cycle (e.g. the glyoxylate shunt) which
involve different reactions [16]; these differences are similarly reflected in
$\alpha_{\text{O}\text{Pyr}_{\text{glyc}}}$.

Combining Eqs. 4-7 we see that the total rate of ATP production by glycolysis
and respiration is:

$\displaystyle\begin{split}J_{\text{ATP
production}}&=\alpha_{\text{PPyr}}\bigg{(}\alpha_{\text{O}\text{Pyr}_{\text{glyc}}}^{-1}\text{OCR}_{\text{mito}}+J_{\text{pyruvate,
fermentation}}+J_{\text{pyruvate, biosynthesis}}-J_{\text{pyruvate,
supply}}\bigg{)}\\\
&\qquad+\alpha_{\text{PO}}\text{OCR}_{\text{mito}}-\alpha_{\text{PO}}\alpha_{\text{HO}}^{-1}J_{\text{leak}}\end{split}$
(8)

Eq. 8 provides a strategy to determine the flux of ATP production from
glycolysis and respiration (which, at steady-state is equal to the flux of ATP
consumption) by measuring mitochondrial OCR, fluxes associated with nutrient
uptake, nutrient secretion, and biomass synthesis, and the flux associated
with proton leak. Such an approach has been used to infer ATP production rates
in mammalian cells by simultaneous measurement of both fermentation and
respiration rates [21, 24]. Alternative methods to measure $J_{\text{ATP,
glycolysis}}$ that do not rely on characterizing the pyruvate balance (e.g.
measurements of glycolytic substrate uptake and biosynthesis rates) may be
more suitable in other systems.

###  3.2 Measuring glycolytic and respiratory fluxes using calorimetry

We shall now show that calorimetry provides an alternative means to determine
the flux through respiration and glycolysis. In a constant-pressure
calorimeter, the heat output measured by the calorimeter, $\dot{Q}$, equals
the rate of of change of the enthalpy of the entire calorimetry sample, $H$;
that is, $\dot{Q}=\frac{dH}{dt}$, where a negative value of $\dot{Q}$
indicates the sample is producing heat. The enthalpy of the entire calorimetry
sample, $H$, is the sum of the enthalpy of the biological sample,
$H_{\text{cells}}$, and the enthalpy of the media that surrounds it,
$H_{\text{extracellular}}$. In cell biological systems, changes in enthalpy
primarily result from chemical reactions converting one molecular species into
another, in which case:

$\displaystyle\dot{Q}$
$\displaystyle=\frac{dH_{\text{cells}}}{dt}+\frac{dH_{\text{extracellular}}}{dt}=\sum_{\text{species
}i}h_{i}\left(\dfrac{dn_{i,\text{cells}}}{dt}\right)+\sum_{\text{species
}i}h_{i}\left(\dfrac{dn_{i,\text{extracellular}}}{dt}\right)$ (9)

where $n_{i,\text{cells}}$ and $n_{i,\text{extracellular}}$ are the number of
moles of species $i$ within the biological sample and the surrounding media,
respectively, and $h_{i}$ is the molar enthalpy of that species. In the
special case that cells are at steady state, i.e. they are not growing and
their composition is not changing in time, $\frac{dn_{i,\text{cells}}}{dt}=0$.
Then, the change in the concentrations of species $i$ in the surrounding media
is equal to the difference in cellular import flux, $J_{i,\text{import}}$, and
export flux, $J_{i,\text{export}}$, of that species. Thus, Eq. 9 becomes:

$\displaystyle\dot{Q}_{\text{ss}}$ $\displaystyle=\sum_{\text{species
}i}h_{i}J_{i,\text{import}}-\sum_{\text{species
}i}h_{i}J_{i,\text{export}}=\sum_{\text{rxns }k}\Delta h_{\cdot,k}J_{\cdot,k}$
(10)

where $J_{\cdot,k}$ is the flux of an index species (denoted by $\cdot$)
through the net reaction $k$, which proceeds from cellular imports to cellular
exports, and $\Delta h_{\cdot,k}$ is the enthalpy of reaction per mole of the
index species. Eq. 10 reveals that in steady state the heat production
measured by calorimetry $\dot{Q}_{\text{ss}}$ reflects the difference in
enthalpy between the cellular imports and waste, i.e. the net reaction of the
cell. This relationship forms the basis for calorimetric measurements of net
metabolic fluxes.

For steady-state cells that respire all of the carbon that they import, e.g.
cell-cycle arrested oocytes, there is a direct relationship between
respiratory flux and heat production (Fig. 3A). Consider a steady-state cell
that imports glucose and oxidizes it completely to carbon dioxide and water.
The net reaction of such a cell is C6H12O6 + 6O2 -¿ 6 CO2 + 6H2O. Because
glucose, carbon dioxide, and water have enthalpies of formation of -1273,
-394, and -286 kJ/mol, respectively, the total enthalpy of reaction is $\Delta
h_{\text{glucose,respiration}}\approx$ -2800 kJ/(mol glucose consumed). The
heat output for such a system is $\dot{Q}_{\text{ss}}=\Delta
h_{\text{glucose,respiration}}J_{\text{glucose,import}}=\frac{\Delta
h_{\text{glucose,respiration}}}{6}\text{OCR}_{\text{mito}}$ where
$J_{\text{glucose,import}}$ is the rate of glucose consumption. Thus,
approximately 467 kJ are liberated for every mole of O2 consumed, and the heat
output measured is straightforwardly related to the OCR measurements discussed
in Sec. 3.1.2. While this calculation is for the specific case of purely
respiratory metabolism using glucose as the substrate, there is a long-
standing empirical observation known as Thornton’s rule [25] that the heat
evolved from consuming a given amount of oxygen, called the oxycaloric
coefficient, is approximately constant ($\sim$-455 kJ/mol O2) across different
substrates. Thornton’s rule enables inference of OCR from calorimetry
measurements without information about the identity of the respiratory
substrate. This suggests an approach to infer ATP production flux from
calorimetry: heat flux may be converted to an equivalent OCR using Thornton’s
rule, and OCR can then be converted to the corresponding ATP production flux
using Eq. 6 and measured values of the proton leak flux and stoichiometric
ratios.

To extend this analysis beyond purely respiratory metabolism, let us next
consider steady-state cells that obtain all their energy from anaerobic
glycolysis. A common scenario involves import of glucose and fermentation to
lactate, in which case calorimetry measures the enthalpy flux associated with
the reaction C6H12O6 -¿ 2 C3H5O3- + 2H+. The enthalpy of this reaction is
$\Delta h_{\text{glucose,fermentation}}\approx$ -110 kJ/(mol glucose
consumed); this is far smaller than $\Delta
h_{\text{glucose,respiration}}\approx$ -2800 kJ/(mol glucose consumed) because
fermentation results in the incomplete oxidation of glucose. Using the
enthalpy of reaction, one can convert the heat flux to a glycolytic flux,
which can in turn be converted to ATP production flux using Eq. 4 if the
effective stoichiometry is known.

For non-growing, steady state cells which perform respiration, glycolysis and
fermentation, the heat output measured by calorimetry will reflect the fluxes
through all of these reactions. Consider a cell which consumes glucose, some
of which is completely respired, with the rest fermented to lactate. The heat
output for such a system is $\dot{Q}_{\text{ss}}=\sum_{\text{rxns }k}\Delta
h_{\cdot,k}J_{\cdot,k}=\frac{\Delta
h_{\text{glucose,respiration}}}{6}\text{OCR}_{\text{mito}}+\frac{\Delta
h_{\text{glucose,fermentation}}}{2}J_{\text{lactate,export}}\approx
467\text{kJ/(mol }\ce{O2})\text{ OCR}_{\text{mito}}+54\text{kJ/(mol lactate)
}J_{\text{lactate,export}}$, where $J_{\text{lactate,export}}$ is the rate of
lactate secretion (i.e. the rate of fermentation). Thus, if heat flux and OCR
are measured, the rate of fermentation can be inferred (or if heat flux and
fermentation are measured, OCR can be inferred). More generally, the heat
associated with respiration can be calculated from Thornton’s rule, while the
heat associated with fermentation will depend on the particular fermentation
products. The ratio between the predicted respiration-associated heat output
and that measured by calorimetry has been used to probe how the mode of
metabolism changes over time in various systems [26, 27, 28].

We have thus far only considered cells at steady state, but many systems of
interest are not in steady state, in which case the internal energy of the
cells is constantly changing. For growing cells, biomass increases over time,
and thus $\frac{dn_{i,\text{cells}}}{dt}\neq 0$; Eq. 9 can then be written as:

$\displaystyle\dot{Q}$ $\displaystyle=\sum_{\text{net biosyn rxns }k}\Delta
h_{\cdot,k}J_{\cdot,k}+\sum_{\text{net non-biosyn rxns }k^{\prime}}\Delta
h_{\cdot,k^{\prime}}J_{\cdot,k^{\prime}}$ (11)

where $k$ indexes net biosynthetic reactions, from imported compounds to
biomass components, e.g. glucose to lipids or glucose to protein;
$J_{\cdot,k}$ represents the flux of an index species (denoted by $\cdot$)
through reaction $k$; and $\Delta h_{\cdot,k}$ is the enthalpy of that
reaction per mole of the index species. Similarly, $k^{\prime}$ indexes net
non-biosynthetic reactions (including ATP-producing pathways);
$J_{\cdot,k^{\prime}}$ represents the flux of an index species (denoted by
$\cdot$) through reaction $k^{\prime}$; and $\Delta h_{\cdot,k^{\prime}}$ is
the enthalpy of that reaction per mole of the index species. The two dominant
net non-biosynthetic reactions that we consider are 1) respiration of imported
carbon sources, and 2) conversion of imported carbon sources to fermentation
products. In cells consuming glucose and performing glycolysis, respiration,
and fermentation, these correspond to glycolysis followed by respiration and
glycolysis followed by fermentation, respectively.

However, the enthalpy of formation of biomass is very close to that of typical
carbon sources for cells. Experimental measurements in E. coli indicate that
the enthalpy of net anabolic reactions is very near zero [29], which
simplifies equation 11 to:

$\displaystyle\begin{split}\dot{Q}&\approx\sum_{\text{non-biosyn rxns
}k^{\prime}}\Delta h_{\cdot,k^{\prime}}J_{\cdot,k^{\prime}}\approx\Delta
h_{\ce{O2},\text{respiration}}\text{OCR}_{\text{mito}}+\Delta h_{\text{ferm
prod,fermentation}}J_{\text{ferm prod,export}}.\end{split}$ (12)

where $\Delta h_{\ce{O2},\text{respiration}}$ is the enthalpy associated with
the net reaction of respiration from import of carbon source to export of CO2,
per mole of oxygen consumed (which is $\sim$-455 kJ/mol O2 by Thornton’s
rule); $\Delta h_{\text{ferm prod,fermentation}}$ is the enthalpy associated
with the net reaction of import of carbon source to export of fermentation
products, per mole of fermentation product exported; and $J_{\text{ferm
prod,export}}$ is the rate at which said fermentation product is exported from
the cell.

In summary, changes in biomass often contribute negligibly to the heat flux
measured by calorimetry, which can therefore be related to flux through
glycolysis, fermentation, and respiration even in growing cells (Fig. 3B).
Thus, calorimetry reflects metabolic fluxes even in non-steady-state
biological systems, such as developing Drosophila and zebrafish embryos [30,
31].

Figure 3: Coupling between chemical and enthalpy fluxes results in a
straightforward relationship between carbon source and oxygen consumption
rates and heat production. A: Net reaction of steady-state, non-growing cells
that consume glucose and respire it completely. The enthalpy production is
proportional to oxygen consumption rate. B: Net reaction of growing cells that
consume glucose and ultimately perform a combination of respiration and
fermentation to lactate. The enthalpy production depends on the relative
fluxes through fermentation and respiration, but does not depend strongly on
biomass production. Portions of this figure were created with BioRender [9].

##  4 Measuring energetic costs by process activity modulation

Equation 3 illustrates how ATP production and consumption are linked; in a
system approximately at steady state, these two fluxes are balanced.
Therefore, as described above, an intuitively appealing strategy for inferring
the energetic cost of a process is to change the activity level of the process
and measure concomitant changes in energy production rate.

Several authors have inhibited processes such as translation and measured
changes in oxygen consumption rate [6, 7], though inferring the energetic cost
of a process from a single activity modulation experiment presupposes that the
perturbation leaves the rates of other processes unchanged. Practically, such
separability of different processes is unlikely; the activity of different
processes can be linked. For example, inhibiting transcription has downstream
effects on translation; inhibiting translation prevents production of enzymes
whose operation involves ATP hydrolysis; and the reduced activity of these
ATPases can change the cellular environment, such as through pH changes which
affect a range of proteins. Each level of these indirect effects affects
additional cellular machinery, and therefore has implications for ATP
consumption. Thus, the observed changes in metabolic fluxes are a
superposition of the direct effects of the applied perturbation, as well as
indirect first order effects, second order effects, and so on. The downstream
effects of perturbations can manifest on multiple timescales. For example,
changes in metabolite concentrations and signaling may occur rapidly, while
changes in protein expression take place more slowly [32, 33, 34]. Thus,
extracting information about energetic costs from activity modulation
experiments requires a systematic accounting of the effects of a perturbation
on other cellular machinery.

In the case of a steady-state system, ATP production is balanced by ATP
consumption. In this case, if the activity of a cellular process $p$ is
altered, changing the ATP demand of that process $J_{\text{ATP},p}$ by some
amount $\Delta J_{\text{ATP},p}$, and the system reaches a new steady state,
the difference between the ATP balance before and after inhibition indicates
the cost associated with the difference in the activity of that process.
Combining Eq. 3 (at steady-state) and Eq. 8, we obtain the change in the ATP
consumption flux of process $p$:

$\displaystyle\begin{split}\Delta J_{\text{ATP},p}&=\Delta
J_{\text{ATP,glycolysis}}+\alpha_{\text{PO}}\Delta\text{OCR}_{\text{mito}}-\alpha_{\text{PO}}\alpha_{\text{HO}}^{-1}\Delta
J_{\text{leak}}-\sum_{k^{\prime}\neq p}\Delta J_{\text{ATP
demand},k^{\prime}}-\sum_{k^{\prime\prime}\neq p}\Delta J_{\text{ATP
buffer},k^{\prime\prime}}\end{split}$ (13)

If the process is completely inhibited, $\Delta
J_{\text{ATP},p}=J_{\text{ATP},p}$ is the total energetic cost of the process.
Eq. 13 provides a quantitative framework to measure energetic costs from
activity modulation experiments. This equation connects measurements of
experimentally accessible fluxes to the energetic costs of interest.

In the simplest possible case, when inhibition of the target process does not
impact any other ATP-consuming process or mitochondrial leak, then $\Delta
J_{\text{leak}}$, $\Delta J_{\text{demand},k^{\prime}\neq p}$, and $\Delta
J_{\text{buffer},k^{\prime\prime}\neq p}$ are zero, and measuring the change
in ATP production by glycolysis and respiration will yield the energetic cost
of the inhibited process. However, in general $J_{\text{leak}}$,
$J_{\text{demand},k^{\prime}\neq p}$, $J_{\text{buffer},k^{\prime\prime}\neq
p}$, and other coupled cellular processes could change as a result of
modulating the activity level of a process of interest. Therefore, to
determine the energetic cost of process $p$ using an activity modulation
experiment, one must either (1) directly measure the change in individual
fluxes on the right hand side of Eq. 13 in each experiment, which is generally
difficult, or (2) develop mechanistic models of flux regulation that capture
the generic response of each flux term to changes in experimentally accessible
metabolite concentrations or chemical potentials (e.g. ATP concentration,
ATP/ADP ratio, proton motive force, NADH/NAD+ ratio) that can be more easily
measured in individual activity modulation experiments.

Below we highlight key challenges associated with measuring each of the flux
terms on the right hand side of Eq. 13 to determine energetic costs from
activity modulation experiments:

  * •

Measuring $\Delta J_{\text{ATP,glycolysis}}$ requires characterization of flux
partitioning. There is not necessarily a constant ratio between the glucose
routed through glycolysis and anabolic pathways under different conditions
[14, 15]. Because glycolytic intermediates are siphoned off for biosynthesis,
the flux through a single step of the pathway does not directly indicate ATP
payoff from glycolysis. Similarly, inferring glycolytic pyruvate input to the
Krebs cycle is challenging because of the many branch points within the Krebs
cycle which can introduce or siphon off intermediates. These complications are
captured in the effective stoichiometric coefficients $\alpha_{\text{PPyr}}$
and $\alpha_{\text{O}\text{Pyr}_{\text{glyc}}}$. Measuring these quantities
may require high-resolution mapping of metabolic fluxes, as discussed in Sec.
5.

  * •

$\alpha_{\text{HO}}$ and  $\alpha_{\text{PO}}$ are substrate-dependent. For
substrates whose metabolism involves NADH (e.g. pyruvate) $\alpha_{\text{HO}}$
is 10 and $\alpha_{\text{PO}}$ is typically 2.5 in mammalian cells, but for
those whose metabolism involves FADH2 (e.g. succinate) $\alpha_{\text{HO}}$ is
6 and $\alpha_{\text{PO}}$ is typically 1.5 [18, 19, 20]. The relative fluxes
of the various substrates which enter the Krebs cycle at different stages must
be determined in order to estimate the net $\alpha_{\text{HO}}$ and
$\alpha_{\text{PO}}$.

  * •

$\Delta J_{\text{leak}}$ depends on proton motive force. Proton leak
represents the fraction of mitochondrial OCR that does not contribute to ATP
synthesis due to the permeability of the mitochondrial inner membrane to
protons and the possible presence of uncouplers. This flux can be measured by
OCR following inhibition of ATP synthase activity. Proton leak may not
necessarily remain constant after perturbations: process activity modulation
may change ATP synthesis rate, which could in turn alter the proton gradient
across the mitochondrial inner membrane and thus the proton leak flux.
Combining measurements of proton motive force (a combination of the proton
gradient and the membrane potential) with a model of how leak flux depends on
the proton motive force enables estimation of $\Delta J_{\text{leak}}$ [35].

  * •

Off-target effects and coupling between processes may lead to non-vanishing
$\Delta J_{\text{demand}}$. Activities of cellular processes of interest are
often altered through pharmaceutical perturbations which may have off-target
effects; for example, cycloheximide, an inhibitor of protein synthesis, is
thought to directly affect metabolism at high concentrations [6, 36].
Moreover, different ATP-demanding processes in the cell can be coupled, and
their values could each change upon perturbation, either as a direct result of
the perturbation or due to indirect effects. For example, the product of one
ATP (equivalent)-consuming process could be required for a different ATP-
consuming process, as in the case of transcription and translation. Gross
changes in cellular physiology caused by a perturbation (e.g. pH, osmolarity)
could also affect many different demand and buffering processes.

  * •

Measuring $\Delta J_{\text{buffer}}$ requires characterization of ATP flux-
buffering mechanisms. ATP-dependent processes can interact through the shared
ATP/ADP pool: changes to this pool caused by perturbing one ATP-dependent
process could alter ATP/ADP concentrations, affecting other machinery
sensitive to ATP levels in turn. Because $J_{\text{buffer}}$ could compensate
for flux changes induced by process activity modulation, it must be estimated
by identifying individual buffering enzymes and characterizing their
sensitivity to changes in ATP/ADP concentrations [8, 37].

In addition to the challenges discussed above, we note that the applicability
of Eq. 13 is limited to organisms producing ATP exclusively from glycolysis
and oxygen-dependent respiration, which is the case for many systems of
interest. However, some organisms may utilize different modes of energy
metabolism, and Eq. 13 can be generalized accordingly by including the
corresponding fluxes. For example, oxygen is not the only electron acceptor in
respiration. Many bacteria perform respiration with terminal electron
acceptors such as fumarate and nitrate instead of oxygen [38]. Recent work has
shown that under conditions of hypoxia or inhibition of Complex IV of the
mitochondrial electron transport chain, mammalian cells can similarly use
fumarate as the terminal electron acceptor [39]. Under these conditions OCR
does not quantify respiratory activity, and alternative methods such as the
measurement of NADH/NAD+ flux [40] may be necessary. Additionally, alternative
ATP production pathways, such as photosynthesis or formation of ATP from AMP
and pyrophosphate represent major sources of ATP in some systems [41, 42].
Moreover, Eq. 13 only considers ATP production and consumption, but other
metabolites also provide free energy to power cellular activities. For
example, NADPH plays a central role in oxidative stress response activities
and in biosynthetic reactions, and its turnover rate constitutes a significant
portion of the cost of biosynthesis [17, 43]. Measurements of fluxes of NADPH
and other metabolites will be necessary to provide a complete picture of the
energetic cost of cellular processes.

In summary, the most direct means to determine the energetic cost of a
cellular process is to measure how altering the activity of that process
impacts ATP fluxes. Eq. 13 provides a systematic means to interpret such
experiments. This equation enumerates the relevant, individual fluxes that
will be present in many biological systems of interest. Each of these fluxes
is challenging to charaterize, and some must be estimated by combining
multiple types of measurements.

##  5 Methods and technologies for measuring fluxes and concentrations

Here we review a selection of the methods available to measure the metabolic
fluxes referred to in Eq. 13 and the associated metabolite concentrations.
Technologies for flux and concentration measurements have matured considerably
over the last two decades; modern metabolomics- and spectroscopy-based methods
have enabled measurements with previously inaccessible spatial and temporal
resolution [44]. These new technologies have produced a wealth of flux and
metabolite concentration data in living cells, but it remains a challenge to
integrate these data to infer the energetic cost of specific cellular
processes. No single technique can accurately measure all the fluxes presented
in Eq. 13.

###  5.1 Measurements of respiration rate

Oxygen is a common terminal electron acceptor in the mitochondrial electron
transport chain (ETC), which drives respiratory ATP generation. Thus, oxygen
consumption rate (OCR) is frequently used as a metric for global metabolic
activity of aerobic organisms. In many types of eukaryotic cells,
mitochondrial respiration accounts for the majority of OCR [45]; hence,
cellular OCR is a reliable proxy for mitochondrial ETC flux. For cells where
non-mitochondrial OCR is substantial, mitochondrial inhibitors can be applied
to separate the mitochondrial and non-mitochondrial OCR. In this section, we
review techniques that measure cellular and mitochondrial OCR, which are
summarized in Table 1.

Sealed-chamber respirometry is used to measure cellular OCR by determining the
rate of decrease of oxygen level in an airtight chamber containing cells [46,
24, 47]. It is usually applied to a population of cells and can resolve OCR
dynamics with minute-level time resolution. Open-chamber respirometry measures
OCR in a half-open chamber by characterizing the steady-state spatial gradient
in oxygen concentration across the chamber, which is established by the
balance of oxygen diffusion from the open end and oxygen consumption of cells
located at the other end [48]. Establishing the concentration gradient may
take a few hours, placing a lower bound on the time required to make the first
measurement. The advantage of this technique is that the chamber can be
engineered to resolve the OCR of a small number of cells, which is
particularly useful when studying oocytes and embryos [48].

Respirometry is a bulk technique that gives the net oxygen consumption of the
entire sample. In contrast, fluorescence lifetime imaging microscopy (FLIM)
has recently been established as a robust method to measure mitochondrial ETC
flux with optical resolution, enabling the characterization of subcellular
spatial variations [40]. This is achieved through model-based inference using
FLIM of mitochondrial NADH. This method requires mitochondrial respiration to
be at steady or quasi-steady state, which holds as long as dynamics of
interest are slower than the kinetic rates of mitochondrial enzymes, which are
on the order of seconds. This technique can potentially resolve OCR dynamics
on the order of seconds.

Table 1: Common methods for measuring respiration rate Method |  Timescale of measurement |  Spatial resolution |  Time series possible? |  Examples
---|---|---|---|---
Sealed-chamber respirometry |  minutes |  none |  yes |  mouse embryos [46]; tissue culture cells [24]; isolated mitochondria [47]
Open-chamber respirometry |  hours |  none |  yes |  bovine oocytes [48]
Fluorescence lifetime imaging microscopy (FLIM) |  seconds |  yes |  yes |  mouse oocytes and tissue culture cells [40]

###  5.2 Measurements of nutrient import and waste export

Extracellular measurements of nutrients and waste products are often used to
infer the rates of glycolysis and fermentation. Each of these measurements
requires that cells consume nutrients (e.g. glucose) and produce waste
products (e.g. ethanol, acetate, lactate/H+) at a quasi-steady rate. By
analyzing the composition of the media, one can infer the rate of the pathway
associated with the production or consumption of the analyte. Methods for
measuring nutrient import and waste export vary widely in their principles of
operation and their limitations. The primary characteristics of several common
methods are listed in Table 2.

Colorimetric and electrochemical assays employ enzyme-linked reactions with
absorbance, fluorescence, or electrical signal-based readouts. These methods
can be high throughput; they are often designed for parallel measurements in
multi-well plates. However, the slow degradation of fluorescent compounds and
enzymes can hamper accurate quantification. Liquid chromatography (LC)
involves separation of metabolites using chromatography, followed by
quantification on the basis of absorbance by optical detectors, or mass-to-
charge ratio (using mass spectrometry (MS)). LC-MS measurements are subject to
complex caveats beyond the scope of this work, which have been described
elsewhere [49]. Nuclear magnetic resonance (NMR) spectroscopy relies on the
distinct frequency spectra of nuclei in different molecules to simultaneously
quantify several nutrients/waste products in media. The number of species
whose concentrations can be accurately resolved by NMR is frequently limited
by spectral overlap of different compounds [50]. Measurements of extracellular
acidication rate (ECAR) are often performed in tandem with oxygen consumption
rate measurements in commercial instruments like the Seahorse Extracellular
Flux Analyzer (Agilent). Some cells perform lactic acid fermentation,
producing lactate anions which are then exported by a lactate/H+ symport
mechanism, acidifying media. Acidification is treated as a proxy for
fermentation rate. However, respiratory CO2 production also contributes to
acidification, and corrections must be applied to accurately infer
fermentation rate [21]. Additionally, this method assumes that all lactate
produced is eliminated, which may not be the case in all systems [51, 52].

The timescale of extracellular composition measurements is typically limited
by how long it takes cells to significantly change the concentrations of
nutrients or waste products in the medium. The length of a time course will
vary greatly depending on the concentrations of nutrients or metabolites under
consideration, and the fluxes involved in the turnover of their respective
pools; rates can be calculated from serial samples of the same population.

Table 2: Common methods for measuring nutrient import and waste export Method |  Timescale of measurement |  Spatial resolution |  Time series possible? |  Examples
---|---|---|---|---
Colorimetric or electrochemical assays |  minutes to hours |  none |  no |  yeast [53]; mouse [54]
Liquid chromatography (LC) with optical detection or mass spectrometry (MS) – media analysis |  minutes to hours |  none |  no |  bacteria and yeast [55]; stem cells [56]
Nuclear magnetic resonance (NMR) spectroscopy |  minutes to hours |  none |  no |  bacteria [57]; human bodily fluids [58]; rat brain [59]
Extracellular Acidification Rate (ECAR) |  minutes |  none |  yes |  tissue culture cells [21]

###  5.3 Intracellular flux measurements

Here, we consider intracellular fluxes to include those pathways whose rates
are not solely determined by measurements of nutrient uptake, waste secretion,
or oxygen consumption rate. Resolving these rates enables inference of flux
partitioning at key branch points in metabolism, such as splits between
bioenergetically relevant fluxes (e.g. glycolysis and Krebs cycle fluxes), and
biosynthetic fluxes. Another example is the direct measurement of ATP
synthesis flux, which is not constrained solely by respiration rate, but
depends on proton leak as well.

Here we consider five examples of intracellular flux measurement methods:
nuclear magnetic resonance (NMR) spectroscopy, ^31P saturation transfer
magnetic resonance spectroscopy (MRS), metabolic flux analysis (MFA), kinetic
flux profiling (KFP), and stimulated Raman spectroscopy (SRS), which each
report different quantities: NMR can be used to probe the absolute values of
net fluxes, MRS is used to estimate total cellular ATP production flux, MFA is
used to measure ratios of net fluxes at branch points in metabolic pathways,
KFP is used to measure the absolute values of gross fluxes, and SRS is used to
measure the concentrations of specific chemical bonds. Their primary
characteristics are summarized in Table 3.

^31P MRS involves selective saturation of the $\gamma$-phosphate resonance of
ATP in steady-state, followed by monitoring of changes in free phosphate
signal over time to infer total cellular ATP synthesis rate [10]. This method
is nondestructive, but measurement times can be long in some cases. The
accuracy of this method can in theory be affected by production of free
cytosolic phosphate by de-phosphorylation of abundant intermediates (e.g.
fructose-1,6-bisphosphate), which may lead to underestimation of ATP-to-free
phosphate flux [60]. ^31P MRS is not widely used in modern bioenergetics
studies.

For metabolic flux analysis (MFA), cells are supplied with labeled nutrients,
and different branches in a metabolic pathway produce distinct steady-state
labeling patterns in downstream metabolites. The ratiometric nature of MFA
enhances the accuracy of flux partitioning estimates. However, MFA results
must be constrained by extracellular flux measurements to infer absolute
fluxes. The time required to achieve steady-state labeling depends on the
system and pathway of interest, but can be short in some cases [61, 62].
Recent work has focused on extending MFA to non-steady-state systems [63, 64].

In kinetic flux profiling (KFP), a time course of isotope labeling patterns is
measured by LC-MS following a rapid switch from an unlabeled to labeled
nutrient. Assuming that metabolic reactions can be accurately modeled by
first-order kinetics, rates of different reactions can be calculated. Labeling
can be quite fast (e.g. $t_{1/2}<5$ min for central metabolites in E. coli),
and thus KFP may be extended to non-steady-state systems [65].

Stimulated Raman spectroscopy (SRS) enables quantification of specific
chemical bonds on the basis of their vibrational signature. By supplying
labeled nutrients to cells and monitoring the concentration of specific bonds,
one can observe their incorporation over time. However, in theory, both
nutrient uptake and incorporation of bonds into biomass may contribute to the
observed signal when using certain substrates; there may also be slight
spectral shifts depending on the biomolecules into which the labeled bonds are
incorporated [66].

Table 3: Common methods for measuring intracellular fluxes Method |  Timescale of measurement |  Spatial resolution |  Time series possible? |  Examples
---|---|---|---|---
Nuclear magnetic resonance (NMR) spectroscopy |  minutes to hours |  none |  yes |  human brain [67]
^31P saturation transfer magnetic resonance spectroscopy (MRS) |  minutes to hours |  none |  yes |  yeast [68]; rat kidney [69]; rat heart [70]
Metabolic Flux Analysis (MFA) |  hours |  none |  no |  B. subtilis [71]; E. coli [72]; yeast [73]
Kinetic Flux Profiling (KFP) |  minutes |  none |  yes |  E. coli [65]
Stimulated Raman Spectroscopy (SRS) |  minutes to hours |  yes |  yes |  tissue culture cells [74, 66], bacteria [75]

###  5.4 Intracellular metabolite concentration measurements

Direct measurements of the fluxes on the right hand side of Eq. 13 can be
challenging. An alternative method to obtain these fluxes is to develop
mechanistic or empirical models that relate fluxes such as OCR, proton leak
and ATP buffering, to experimentally accessible quantities such as
intracellular metabolite concentrations. Biophysical models of mitochondrial
metabolic pathways are proving to be useful in revealing the relationships
between fluxes and concentrations [76, 77, 40, 78, 79].

Fluorescent biosensors have recently enabled measurements of key metabolite
concentrations in living cells with subcellular resolution. Examples include
ATP concentration [80], ATP/ADP ratio [81], mitochondrial membrane potential
[82], NADH/NAD+ ratio [83], pyruvate concentration [84], lactate concentration
[85] and glucose concentration [86]. Sensors must be chosen so that their
dynamical range is well-matched with the concentration of the analyte in the
system of interest. It is also important to correct for the impact of other
factors, such as pH, on the sensor signal. NADH and NADPH are autofluorescent,
and their concentrations can be measured using fluorescence lifetime imaging
microscopy (FLIM) [40, 87]. One challenge is to separate NADH signal from
NADPH signal, which share the same fluorescence spectrum.

Chemical assays can be used to measure concentrations of biomolecules such as
polysaccharides and protein which are critical to cellular composition
estimates. Liquid chromatography - mass spectrometry can be used to measure
concentrations of a large number of metabolites [12]. However, these methods
require destructive sampling and often provide limited spatiotemporal
resolution. Nuclear magnetic resonance has been used to measure concentrations
of specific metabolites as well [88]. Lastly, stimulated Raman spectroscopy is
capable of measuring intracellular metabolite concentrations in living cells
with subcellular resolution [89], but it remains a challenge to definitively
associate spectral features to specific metabolites.

###  5.5 Measuring metabolic fluxes with calorimetry

As discussed in Sec. 3.2, the heat generated by biological systems is
intimately related to their metabolic fluxes. This can be probed by isothermal
calorimetry, which measures the heat output of a sample relative to a
reference cell at a constant temperature. Isothermal calorimetry is a bulk
measurement technique, and does not provide spatially resolved information.
Common variants of this technique are listed in Table 4.

The temporal resolution of isothermal calorimeters is highly dependent on
sample volume. High-sensitivity and low-volume ($\sim$1 $\mathrm{\mu}$L)
calorimeters have time resolution on the order of seconds [90, 91, 92], but
the time resolution for larger-volume microcalorimeters is on the order of
hours. Recent devices have achieved sensitivities on the order of 200 pW [90,
91, 92], which is sufficient to measure the output from single C. elegans or
Drosophila embryos, though single-cell measurements for microbes remain out of
reach.

The technical challenges associated with isothermal calorimetry depend
strongly on the sample type and amount. In open-chip calorimetry, evaporation
can be a significant source of error. For the lowest-volume and most sensitive
chip-based calorimeters, with sample volumes less than 1 $\mathrm{\mu}$L, the
temperature gradient across the thermopile used to read out a signal is on the
order of 1 mK [90]. Thus, even small external temperature gradients, heat
sources, or heat losses can significantly impact the measurement.

Table 4: Selected methods to measure enthalpy flux Method |  Timescale of measurement |  Spatial resolution |  Time series possible? |  Examples
---|---|---|---|---
Isothermal microcalorimetry |  varies |  none |  yes |  E. coli [93], microbial biofilms [94], Cutibacterium acnes [95]
Open-chip calorimetry |  varies |  none |  yes |  molecular motors [90], enzymes [96], mouse adipocytes [97]
Closed-chip calorimetry |  varies |  none |  yes |  bacteria [98], C. elegans [92], Tetrahymena [91]

##  6 Prior theoretical and experimental approaches to measuring bioenergetic
costs

The flux balance framework and the experimental techniques presented in
previous sections are tools that enable measurements of energetic costs of
cellular processes. That approach centers on measuring the changes in the ATP
fluxes in Eq. 13 after modulating the activity of a process of interest. In
this section, we review previous work that has sought to determine these costs
in various systems using a range of other approaches. We discuss these
studies’ respective challenges and their relation to the flux balance
formalism.

###  6.1 Estimates and inhibition experiments

While cellular energy budgets have generally not been probed systematically
through experiments, the energetic costs of several key cellular processes
have been estimated in different systems.

A number of attempts have been made to estimate the energetic costs of
biosynthesis in growing microbes and developing embryos by leveraging
knowledge of the underlying cellular biochemistry. In these calculations, the
energetic costs of individual chemical reactions involved in protein, lipid
and nucleic acid synthesis, multiplied by the abundances of these different
macromolecules in the cell, were summed to arrive at an estimate of the total
biosynthesis cost. In the case of protein synthesis, for example, the ATP
equivalents required for amino acid activation (1/amino acid) and peptide bond
formation (2/amino acid) were accounted for. Such calculations have led some
researchers to conclude that protein synthesis is the major energy consumer in
growing microbes, accounting for $\sim$55-65% of their total energy budget [3,
4, 2]. In contrast, biosynthetic processes have been estimated to account for
less than 10% of the total ATP expenditure in developing Drosophila embryos
[30].

Attempts to measure the energetic costs of cellular processes using inhibition
experiments have typically focused on characterizing changes in respiratory
and/or glycolytic fluxes, without explicitly considering the other ATP fluxes
in Eq. 13. The results of such experiments suggest that protein synthesis and
actin dynamics are the major energy consumers in myoblasts, each accounting
for $\sim$20% of the total ATP production rate [21]. Similarly, in rabbit
brain slices, inhibiting sodium-potassium pumps resulted in a 50% decrease of
oxygen consumption rate, arguing that cation transport is a major energy drain
[99].

In general, work directly comparing estimates to independent experimental
measurements of energetic costs has been limited. However, cases in which
these comparisons have been made demonstrate the synergy between these
methods. Recent calorimetric measurements revealed oscillations in heat output
during zebrafish embryo development, which ceased when the cell cycle was
blocked. The authors of that study compared the amplitude and period of the
heat oscillation with theoretical estimates of ATP consumption, and argued
that they were due to phosphorylation and dephosphorylation reactions which
accounted for 1-2% of the total energy expenditure of the embryo [100]. In
other systems, significant differences have been observed between theoretical
estimates and experimental measurements: in neurons, ATP accounting predicted
that actin turnover accounts for less than 1% of energy expenditure [101],
while inhibition experiments suggested that this process constitutes $\sim$50%
of energy expenditure [102]. Discrepancies between estimates and measurements
highlight the importance of careful examination of assumptions in theoretical
calculations and of careful interpretation of process activity modulation
experiments.

Estimates can also provide complementary insights. Chen et al. [103] compared
the measured energy cost per beat of a flagellum to two independent
theoretical estimates, and found both to be of the same order of magnitude,
helping to rationalize the original measurement. In another case, measurements
of information processing in the blowfly retina were compared to the minimum
energy cost defined by information-theoretic limits. It was found that the
measured cost was 5 to 8 orders of magnitude greater than this theoretical
minimum [104]. This study demonstrated how estimates paired with measurements
can provide insight into the origins of energetic costs and the relevance (or
lack thereof) of fundamental physical limits.

Inferring the energetic cost of a specific process from changes in global
fluxes after modulating the activity of a process requires knowing the degree
and specificity of the change in its activity as well as the potential
coupling of different cellular processes. One way to evaluate the accuracy of
such a measurement is to compare the inferred cost to measurements of the
change of the rate of the process. For example, by measuring incorporation of
labeled amino acids, it was shown that the protein synthesis inhibitor
cycloheximide completely blocked translation in fish hepatocytes and human
hepatoma cells at a concentration of $\sim$25 $\mathrm{\mu}$M, causing a
modest decrease in oxygen consumption. However, oxygen consumption continued
decreasing as the concentration of cycloheximide was increased, indicating
that there were off-target effects at higher concentrations [6]. Experimental
measurements may also be cross-validated against estimates based on detailed
models of the underlying processes and metabolic pathways [104, 105, 106, 4].

Apart from off-target effects, the intrinsic coupling between different
cellular processes could also pose challenges for interpretation of changes in
global fluxes. Wieser et al. [6] demonstrated that hierarchies of ATP-
consuming processes exist in cells: inhibiting mitochondrial respiration and
measuring protein synthesis rate and Na+/K+-ATPase activity in fish
hepatocytes and human hepatoma cells revealed that these two processes display
different sensitivities to energy limitation in each species. Similar
experiments in thymocytes revealed a hierarchy of different processes’
sensitivities to inhibition of mitochondrial respiration: protein synthesis
was most sensitive, followed by RNA/DNA synthesis and substrate oxidation, Na+
cycling and Ca^2+ cycling, and finally other ATP consumers and mitochondrial
proton leak [8]. Application of metabolic control analysis to this data
suggested that each ATP consumer exerted negligible influence over the rates
of other ATP consumers. However, direct measurements of how modulating the
activity of a specific consumer affects the rates of other ATP-consuming
process will be required to test these predictions. Given the species and cell
type-dependence of the hierarchies of ATP-consuming processes, different
systems of interest may need to be individually characterized.

One of the challenges in linking global fluxes with specific fluxes is the
potential coupling of the fluxes through different cellular processes. For
example, a striking phenomenon of flux homeostasis has been discovered in
mouse oocytes, where inhibiting a wide range of energy consuming processes has
no effect on the global oxygen consumption rate, despite significantly
impacting the metabolic state of the cell [40]. This flux homeostasis implies
the coupling of process-specific fluxes inside the cell and an unknown
mechanism of flux partitioning that keeps the total flux constant.

###  6.2 Using nonequilibrium physics to estimate energetic costs

An alternative to process activity modulation experiments is to use tools from
nonequilibrium physics to estimate energetic costs, or derive limits on costs,
using experimentally observable quantities. All irreversible processes consume
energy, and nonequilibrium thermodynamics and information theory have been
used to place bounds on the associated dissipation [107, 108, 109]. Such
approaches have also led to proposals for energy-speed-accuracy tradeoffs in
diverse biological systems, from single molecular motors to chemotactic
bacteria to developing embryos [30, 110, 111, 112]. While a comprehensive
treatment of the active field of nonequilibrium thermodynamics is beyond the
scope of this review and has been provided elsewhere [113], we highlight the
Harada-Sasa equality, which connects violation of the fluctuation-response
relation to the extent of energy dissipation [114]. Notably, the calculations
required for determining energy dissipation depend solely on experimentally
accessible quantities, such as particle trajectories observed through
microscopy and cellular rheology measurements. Since only a limited number of
degrees of freedom can be experimentally measured, application of the Harada-
Sasa equality provides a lower bound on the energetic cost of the underlying
process. The predictions of the Harada-Sasa relation have been validated in
the case of a particle in an optical trap [115]. This equality has also been
applied to a number of biologically relevant systems, including molecular
motors [116] and mechanical dissipation inside living cells [117, 118]. For
example, this approach was used to determine that the swimming of
Chlamydomonas reinhardtii had a dissipation rate of $\sim$0.1 fW [119].
Interestingly, this is significantly less than the $\sim$60 fW measured by
Chen et al. [103] through monitoring ATP consumption of C. reinhardtii
flagella in vitro. Generalizations of the Harada-Sasa equality have been
applied to study free energy dissipation and entropy production in a wider
variety of systems, including active matter and nonequilibrium biochemical
networks [120, 121, 122, 123, 124, 125].

The application of general nonequilibrium relationships, such as the Harada-
Sasa equality, is a powerful and elegant approach for studying energy
dissipation, but does not leverage prior knowledge about the system of
interest. An alternative approach is to develop and test mechanistic, physics-
based models for specific systems. For example, a circuit model of the retinal
neuron enabled conversion of membrane potential and conductance measurements
into energy consumption rates [104]. In another study, Stokes’ law was used to
estimate the energetic cost of swimming in single-celled organisms, which when
compared to the operating cost of flagella revealed that the efficiency of
conversion of chemical energy to swimming power was $\sim$0.7% [126].

##  7 Open questions

Despite detailed knowledge of the structure of metabolic networks and the
biochemistry of enzymes, relatively little is known about cellular energy
expenditures. What are the energetic costs of different cellular processes?
How are energy expenditures regulated? Are energy fluxes buffered, and if so,
how? How and why do perturbations of energy metabolism lead to different cell
biological defects under different circumstances? How do energetic constraints
impact cell physiology and the evolution of cellular features? Answering these
questions will inform our understanding of bioenergetics, cell biology, and
the nonequilibrium physics of living systems.

Coarse-grained modeling [40, 127, 128, 111, 129] may help provide a
quantitative description of the mechanisms coupling flux through energy-
producing and -consuming pathways. Ultimately, it would be desirable to link
such coarse-grained descriptions to the underlying behaviors of the
constituent molecular machines, such as respiratory complexes and ATP synthase
[130, 131] to bridge structure, function and physiology across scales.

The flux balance framework presented in this review, culminating in Eq. 13,
provides a quantitative method to measure energetic costs of cellular
processes by studying changes in ATP fluxes upon modulation of process
activity. The dynamics of global energetic fluxes such as oxygen consumption
rate and heat flux have been measured in many biological contexts, including
microbial adaptation, organ function, and embryo development [132, 133, 134,
135, 136, 137, 138, 139, 140, 141, 142, 100, 31, 30, 143]. However, the
contributions of specific energy-consuming processes such as biosynthesis, ion
pumping, and cytoskeletal remodeling to changes in global fluxes remain
unclear. Systematic study of bioenergetic fluxes and their coupling will be
required to measure the costs of cell biological processes.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

We thank Yong Hyun Song, Peter Foster, and Piyush Nanda for discussions and
comments. This work was supported by National Science Foundation awards
PHY-2013874 and MCB-2052305 to D.J.N.

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