# A general efficiency relation for molecular machines

Milo M. Lin  Green Center for Systems Biology, Department of Bioinformatics,
Department of Biophysics, and the Center for Alzheimer’s and Neurodegenerative
Diseases, University of Texas Southwestern Medical Center, Dallas, TX 75390
<EMAIL_ADDRESS>
###### Abstract

Living systems efficiently use chemical fuel to do work, process information,
and assemble patterns despite thermal noise. Whether high efficiency arises
from general principles or specific fine-tuning is unknown. Here, applying a
recent mapping from nonequilibrium systems to battery-resistor circuits, I
derive an analytic expression for the efficiency of any dissipative molecular
machine driven by one or a series of chemical potential differences. This
expression disentangles the chemical potential from the machine’s details,
whose effect on the efficiency is fully specified by a constant called the
load resistance. The efficiency passes through a switch-like inflection point
if the balance between chemical potential and load resistance exceeds thermal
noise. Therefore, dissipative chemical engines qualitatively differ from heat
engines, which lack threshold behavior. This explains all-or-none dynein
stepping with increasing ATP concentration observed in single-molecule
experiments. These results indicate that biomolecular energy transduction is
efficient not because of idosyncratic optimization of the biomolecules
themselves, but rather because the concentration of chemical fuel is kept
above a threshold level within cells.

Energy is a limiting resource for life. Therefore, molecular machines that
harness differences in chemical potential $\Delta\mu$ to perform tasks within
the cell must function as highly efficient chemical engines. Despite thermal
stochasticity, the measured efficiencies of biomolecular chemical engines are
consistently in the 60-to-90 percent range (1; 2; 3), well above that of
typical human-designed systems such as heat engines. Two centuries ago, Carnot
showed that the maximum efficiency of heat engines is (4):

$\eta_{\mathrm{heat}}\leq{{\Delta T}\over{\Delta T+T_{\mathrm{0}}}}.$ (1)

This relation depends on two independent parameters: the temperature
differential driving the engine $\Delta T$ and the exhaust temperature
$T_{0}$. Eq. 1 provides a universal constraint for all heat engines regardless
of design specifics, and initiated the field of thermodynamics. Finding a
general relation constraining the behavior of chemical engines would provide a
unifying framework for biomolecular function and evolution, and for
engineering nanodevices for physiological conditions.

It is important to distinguish two broad classes of chemical engines in
biology, as they should have different formulations of efficiency as well as
design constraints. The first is the class of energy transduction machines,
such as ion pumps, that convert energy from one chemical reservoir to another.
In these cases, it is clear that any energy dissipated by the machine is
wasted and so should be minimized in order to maximize the efficiency of
transferring energy between the reservoirs, which is achieved in the ”tight-
coupling” limit (5).

This work is primarily concerned with the second class of chemical engines
whose purpose is the dissipation of the input energy. This class of
dissipative machines serve a wide range of functions, including mechanical
transport against viscous drag, and maintaining patterns of protein assembly
or signaling configurations that are inaccessible at equilibrium (Fig. 1A).
They are therefore principal agents of information processing and structural
reorganization within the cell. Prigogine called these configurations
”dissipative structures” because input energy is constantly dissipated to the
environment as heat to maintain these patterns, which are far from equilibrium
even under steady state conditions (6). Chemical engines can approach perfect
efficiency if the throughput and power approach zero (5; 7). Therefore, unlike
Carnot’s relation Eq.1, a useful general relation for the efficiency of
chemical engines must depend on at least one system-dependent collective
variable that reflects the constraint on throughput in the high efficiency
limit, and reveal if there are regimes of the collective variable and
$\Delta\mu$ for which the machine is both fast and efficient. Therefore, such
a relation could also reveal if achieving high performance in both metrics
requires evolutionary fine-tuning, or is a general property. We currently lack
mechanistic insight into how chemical engine efficiency is controlled by
tunable parameters. And it is unclear if the multitude of system-specific
parameters could be encapsulated by a single collective variable that is well
defined for all systems, which is necessary for a clear unified understanding.

Existing results on dissipative chemical engines, based on generalized
fluctuation-dissipation relations (8; 9), upper-bound the efficiency relative
to observed fluctuations (10; 11), although the bound may not be tight
(achievable) far from equilibrium. Existing work does provide efficiency
constraints under limiting conditions. For example, weakly driven engines
operating at maximum power must be 50% efficient (12; 13; 14). However, this
bound only holds near equilibrium, and does not apply to the strongly-driven
conditions relevant for biology. Numerical simulations of simple chemical
engines indicate that efficiency is increased when driven farther from
equilibrium by increasing $\Delta\mu$, similar to the effect of increasing
$\Delta T$ in heat engines (12). This is consistent with a model in which
molecular motors can become less wasteful if driven by higher ATP
concentration (15), and suggest a general link between large $\Delta\mu$ and
high efficiency.

Here, using a recent generalization of the Boltzmann distribution to
nonequilibrium systems(16), I derive a general equation for the maximum
efficiency of dissipative chemical engines. This relation depends on two
independent tunable variables: $\Delta\mu$ and the relative load resistance.
The load resistance is a collective variable condensing the engine details,
and is a constant unaffected by how strongly the engine is driven (i.e.
$\Delta\mu$). The equation shows generally that biomolecular processes are
intrinsically efficient because $\Delta\mu$ is much larger than the thermal
energy $kT$ in vivo. However, unlike heat engines, chemical engines have an
efficiency inflection point at a threshold $\Delta\mu/kT$ that depends on the
load resistance. This switch-like transition explains single-molecule in vitro
measurements of dynein molecular motor stepping. This result also leads to a
general tradeoff relation between energy efficiency and time efficiency. More
broadly, the existence of a general efficiency inflection point provides a
candidate thermodynamic necessary condition for life.

Circuit mapping. Chemical engines within the cell (circular arrows in Fig. 1A)
consume energy to form patterns of activated signaling molecules, protein
self-assembly, or directed movement, that would be vanishingly rare at
equilibrium. Such processes can be modeled as nonequilibrium Markov chains,
but obtaining analytical insights into such systems has mostly been
intractable (17; 18; 19). This work exploits a recently found mapping from the
dynamical network of any Markovian system to an electronic circuit (16) (See,
for example, Fig. 1C,D). Consequently, a system is decomposed into a passive
equilibrium portion (”resistors”) that is driven by chemical potential
differences $\Delta\mu$ (”batteries”). The resistance between two neighboring
states $m$ and $n$ is: $R_{mn}\equiv{{\mathrm{e}^{\beta
G_{m}}}\over{k_{mn}}}$, where $\beta={{1}\over{kT}}$, $G_{m}$ is the free
energy of state $m$ at equilibrium (i.e. in the absence of driving), and
$k_{mn}$ is the equilibrium rate constant of transitioning from state $m$ to
$n$. If a transition is directly driven by $\Delta\mu$, this transition is
mapped to a battery with a probability potential drop
$\mathcal{E}_{mn}\equiv(\mathrm{e}^{\beta\Delta\mu}-1)\mathrm{e}^{\beta
G_{m}}P_{m}$ (red arrow in Fig. 1B). Using this mapping, probability flow
amongst the states of any system obeys Ohm’s law (16): $P_{j}\mathrm{e}^{\beta
G_{j}}-P_{i}\mathrm{e}^{\beta
G_{i}}={\sum\limits_{m=i}^{n=j}(\mathcal{E}_{mn}-R_{mn}I_{mn})}$ (16), where
the summation is along any trajectory (parameterized by neighboring states
$m,n$) connecting states $i$ and $j$. $I_{mn}$ is the probability current (net
flux) from state $m$ to state $n$. Note that $\mathcal{E}_{mn}$ and $I_{mn}$
are zero for a system at equilibrium, and Ohm’s law reduces to the Boltzmann
distribution $P_{j}=P_{i}\mathrm{e}^{\beta(G_{i}-G_{j})}$ in this case, as
expected. In this mapping, multiple interconnected dynamical transitions can
be systematically combined into a single resistor, called the Thevenin
equivalent resistance (20), that captures their collective effect on the
probability flux at steady state.

Figure 1: Mapping nonequilibrium cycles to battery-resistor circuits. Cellular
function and homeostasis are maintained by cyclic molecular engines that
harness chemical energy to transform and organize matter (simplified schematic
of a cell in A; energy flow in red). The circuit mapping transforms a
dynamical system, for example an enzymatic engine that activates a signaling
molecule using energy released from hydrolyzing ATP into ADP and phosphate (B;
star denotes activated enzyme complex), into an electronic circuit consisting
of batteries and resistors that obey Ohm’s law (C). Subsequently, the circuit
elements can be systematically simplified to describe coarse-grained
probabilities and currents without loss of accuracy (D).

For example, for the signaling protein in Fig.1B, molecular transitions
corresponding to $R_{1},..,R_{6}$ (Fig. 1C) can be combined as series and
parallel resistors into the equivalent resistance
$R_{\mathrm{L}}=R_{1}+({1\over{R_{2}}}+{1\over{R_{3}+R_{4}+R_{5}+R_{6}}})^{-1}$
(Fig. 1D). Resistor coarse-graining, combined with the broad set of circuit
theorems, allows a systematic approach to simplify systems of otherwise
intractable complexity (16). This approach is most powerful for systems, such
as biomolecular machines, for which the driven transitions are sparse.

Defining the efficiency of dissipative chemical engines. Dissipative machines
within the cell are usually composed of proteins that occupy one of many
possible conformation and interaction states (Fig. 1B). Each machine harnesses
a source of chemical energy, often by binding excess ATP and coupling its
hydrolysis into ADP and phosphate to drive the target reaction. Consider the
circuit representation of a machine driven by the source transition
$\mathrm{S}$ (i.e. battery) with
$\Delta\mu=kT\ln{\big{[}1+{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}\big{]}}$,
where $k_{\mathrm{S}}$ is the apparent rate constant of the source transition
in the absence of driving (e.g. no excess ATP at equilibrium), and
$\alpha_{\mathrm{S}}$ is the enhancement in the rate constant due to driving.
If the energy source is ATP, the source transition is enhancement of the rate
of converting unbound phosphate and ADP into ATP bound to the protein (red
arrow in Fig. 1B). At ATP concentrations well below binding saturation,
$\alpha_{\mathrm{S}}$ is proportional to ATP concentration in excess of the
equilibrium concentration. The resistance of the source transition (i.e.
battery) is denoted $R_{\mathrm{S}}$ (Fig. 2A). A machine may contain multiple
source transitions. Although the circuit framework is compatible with
arbitrary arrangements of batteries, the analytic results in this work are
restricted to those machines with a single source transition or multiple
source transitions in series. Nevertheless, this class of machines includes a
large fraction of cellular processes including signaling, self-assembly and
sorting, and mechanical transport.

The source transition breaks detailed balance for the machine by driving net
flux of downstream transitions between states, and this flux may take numerous
alternate pathways through state space. All transitions aside from the source
transition can be combined into a single Thevenin equivalent load resistor
$R_{\mathrm{L}}$ (Fig. 2A). Often, the biologically useful output of the
engine is to enhance a particular state within the load.

With this formulation, we can now define the maximum efficiency of a
dissipative machine as the dissipation rate of the load divided by the total
dissipation rate. This definition corresponds to existing formulations of
efficiency as special cases. For example, in the case that the machine is a
molecular motor transporting a cargo against viscous drag, the load
dissipation coincides with the maximum mechanical work that can be done by the
motor per cycle. In this case, the efficiency is equivalent to the Stokes
efficiency, which is the average viscous drag times the average velocity
divided by the total dissipation rate (21). More broadly, this definition also
extends the concept of efficiency to most processes in the cell, such as
signaling and pattern formation, in which the useful dissipation of energy
does not correspond to work along a simple reaction coordinate.

A general efficiency relation for dissipative chemical engines. We first
obtain the total dissipation rate $\sigma T$, where $\sigma$ is the entropy
production rate. The nonequilibrium fluctuation theorems (22; 23; 24) relate
dissipation rate to transition probabilities: $\sigma
T=kT\sum_{ij}I_{ij}\ln{\Bigg{[}{{P_{i}(k_{ij}+\alpha_{ij})}\over{P_{j}k_{ji}}}\Bigg{]}}$
(18). Because dissipation is summed over all possible dynamical transitions of
a system, it may appear that knowing an engine’s details is required to
calculate the efficiency. Instead, I show here that $R_{\mathrm{L}}$ is the
only information about the load needed to calculate the dissipation, and
therefore the efficiency. First, using Tellegen’s Circuit Theorem (25), the
total steady-state dissipation rate is (See Supp. Information):

$\sigma T=\sum_{ij}I_{ij}\Delta\mu_{ij}.$ (2)

In this reformulation, the total dissipation rate is a function of only the
currents of the directly driven transitions (for which $\alpha_{ij}\neq 0$),
even though energy is dissipated at all transitions. Eq. 2 is especially
useful if driven transitions are sparse.

Figure 2: Maximum efficiency of a dissipative machine. The circuit for a
dissipative machine is divided into the source resistance being directly
driven by the chemical potential difference, and the load resistance
comprising the target task being performed by the engine (orange inset), which
may involve dynamical transitions between many states of the system (A). Eq. 3
predicts the maximum achievable efficiency, which is plotted as a function of
$\Delta\mu/kT$ for different values of the relative load resistance (gray to
orange lines; adjacent lines differ in $R_{\mathrm{S}}/R_{\mathrm{L}}$ by one
order of magnitude) (B). The experimentally measured efficiency of dynein
molecular motor as a function of ATP concentration in vitro is adapted from
Ref. 28 (open circles).

If there is a single driven transition, the steady-state load dissipation rate
is (Supp. Information):
$\sigma_{\mathrm{L}}T=kTI_{\mathrm{S}}\ln{\bigg{[}{{R_{\mathrm{S}}+R_{\mathrm{L}}e^{\Delta\mu/kT}}\over{R_{\mathrm{S}}+R_{\mathrm{L}}}}\bigg{]}}$.
The difference between $\sigma T$ and $\sigma_{\mathrm{L}}T$ is the
dissipation of the source (battery), which corresponds to energy that is not
accessible to the load. The efficiency is upper-bounded because of the
dissipation of the source, which corresponds to the energy lost whenever the
engine dynamics happens to reverse the source transition, for example, if
bound ATP is transformed into unbound ADP and phosphate without coupling to
any downstream load reactions. The maximum steady state efficiency is:
$\eta_{\mathrm{chem}}\leq\sigma_{\mathrm{L}}/\sigma$. This yields the main
result of this work, a general relation governing the maximum efficiency of
dissipative chemical engines:

$\eta_{\mathrm{chem}}\leq{{kT}\over{\Delta\mu}}\ln{\Bigg{[}{{{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}+e^{{{\Delta\mu}\over{kT}}}}\over{{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}+1}}\Bigg{]}}.$
(3)

The efficiency depends on the chemical potential difference relative to $kT$,
which quantifies how the efficiency is limited by thermal fluctuations. The
functional form of Eq. 3 remains valid in the case that the engine is
cumulatively powered by multiple driven transitions (e.g. sequential
hydrolysis of multiple ATP molecules); in this general case, $\Delta\mu$ is
replaced by the sum of $\Delta\mu$’s and $R_{\mathrm{S}}$ is replaced by the
weighted sum of all driven resistors (See Supp. Information). An engine will
approach the maximum efficiency set by Eq. 3 if the most dissipative reactions
within the load are useful. In practice, the load may include nonproductive
futile cycles which dissipate energy subsequent to hydrolysis.

Some features of Eq. 3 are reminiscent of Carnot’s heat engine law Eq. 1. (i)
As with Eq. 1, Eq. 3 is a tight upper bound achievable if there are no futile
processes in the load. (ii) Although Eq. 3 is a function of the constituent
parameters, it is not dependent on knowing details of the engine. (iii) Eq. 3
predicts that the maximum efficiency of a dissipative machine increases with
increasing $\Delta\mu/kT$, similar to how heat engine efficiency increases for
larger $\Delta T$. This explains the observed increase in isothermal ratchet
and motor efficiency when strongly driven (26; 27). Despite these
similarities, Eq. 1 (heat engines) and Eq. 3 (dissipative chemical engines)
differ in fundamental ways. The efficiency of heat engines is a monotonic
concave function of $\Delta T$, with no well-defined separation between
inefficient engines near equilibrium and efficient engines far from
equilibrium. In contrast, Eq. 3 shows that, in the functionally relevant
processive regime ($R_{\mathrm{S}}>R_{\mathrm{L}}$; See Supp. Information),
the efficiency of a dissipative chemical engine is a sigmoidal function of
$\Delta\mu$. The inflection point of the sigmoid corresponds to a threshold
$\Delta\mu_{\mathrm{thresh}}$:

$\Delta\mu_{\mathrm{thresh}}\approx
kT\ln\bigg{[}{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}\bigg{]}.$ (4)

Figure 3: Tradeoff between time and energy efficiency. Eq. 5 is plotted for
different values of $\Delta\mu$. Time efficiency, quantified by the
processivity $P$, also varies from 0 to 1. Black filled circles denote optimal
engines that maximize the sum of both efficiencies. White circles denote
engines for which $R_{\mathrm{L}}=R_{\mathrm{S}}$.

Below this threshold, all dissipative machines are inefficient. The efficiency
only responds to increasing $\Delta\mu$ if
$\Delta\mu>\Delta\mu_{\mathrm{thresh}}$, corresponding to a ”switch-like”
response of efficiency to increasing $\Delta\mu$. This prediction is borne out
by recent in vitro single-molecule measurements of dynein molecular motor
step-size, which increase in a switch-like manner at a threshold ATP
concentration (28), an observation that is otherwise difficult to explain
(open circles in Fig. 2B).

Another difference between the heat engine efficiency and the dissipative
chemical engine efficiency Eq.3 is that the latter is lower-bounded by
$R_{\mathrm{L}}/(R_{\mathrm{S}}+R_{\mathrm{L}})$ in the near-equilibrium
limit, whereas the heat engine efficiency tends to zero near equilibrium.
Consider the special case in which a chemical engine is weakly driven:
$\Delta\mu<<kT$. In this case, the load dissipation simplifies to
$\sigma_{\mathrm{L}}={{R_{\mathrm{L}}}\over{(R_{\mathrm{S}}+R_{\mathrm{L}})^{2}}}{\Delta\mu}^{2}\leq{{1}\over{4R_{\mathrm{S}}}}{\Delta\mu}^{2}$,
where the equality is achieved at the maximum load dissipation condition in
which the load resistance is matched to that of the source:
$R_{\mathrm{L,max}}=R_{\mathrm{S}}$. This condition is analogous to the
maximum power transfer theorem of electronic circuit theory (29). In this
special case, Eq. 3 recovers the previously known result that
$\eta_{\mathrm{chem}}\approx 1/2$ at maximum power in the near-equilibrium
limit (12; 13). Living systems operate at the other extreme far from
equilibrium. For example, in the cell ATP concentration is maintained over ten
orders of magnitude higher than its equilibrium value: $\Delta\mu\approx
24kT$, although values vary somewhat depending on cellular region and organism
(30; 31). Eq.3 shows that living systems are intrinsically capable of being
efficient over a wide range of load resistances corresponding to diverse
functions (Fig. 2B) as long as $\Delta\mu/kT$ is sufficiently large.

Eq. 3 suggests that efficiency can be increased without increasing $\Delta\mu$
by increasing the load resistance (Fig. 2B). However, because increasing the
resistance decreases the current, high efficiency near equilibrium (i.e. small
$\Delta\mu$) would be expected to come at the cost of low throughput and power
(5). Consider the processivity, a measure of time efficiency, which is the net
flux (forward minus backward flux) divided by the forward flux:
$P\equiv{{I}\over{f}}$. The tradeoff between energy efficiency
$\eta_{\mathrm{chem}}$ and time efficiency $P$ can be quantified using Eq. 3
(See Supp. Materials):

$P\leq 1-e^{-{{\Delta\mu}\over{kT}}(1-\eta_{\mathrm{chem}})}.$ (5)

This tradeoff is reminiscent of previous work on specific processes such as
glycolysis (32) and gene regulatory networks (33), in which suppressing
fluctuations comes at the cost of some measure of efficiency. For a given
$\Delta\mu$, the maximum $\eta_{\mathrm{chem}}$ and $P$ lie on a Pareto front
(Fig.3). Higher $P$ must come at the expense of lower $\eta_{\mathrm{chem}}$,
a tradeoff that is supported by in vivo measurements of molecular motors (34;
35). The new insight comes from considering Eq. 5 (and thus Eq. 3) in the far-
from-equilibrium regime: the Pareto front sharpens as $\Delta\mu$ increases,
such that for sufficiently strong driving there is effectively no longer a
tradeoff. Within a Pareto front, an optimal tradeoff between energy efficiency
and time efficiency can be defined by maximizing their sum:
$\eta_{\mathrm{chem}}+P$ (solid black circles in Fig. 3; see Supp. Materials).
Interestingly, a system with $R_{\mathrm{L}}=R_{\mathrm{S}}$ maximizes load
dissipation and also coincide with maximum total efficiency near equilibrium;
however, such a system sacrifices too much processivity for little energy
efficiency gain far from equilibrium (open circles in Fig. 3), an example of a
design principle that does not extrapolate from the weakly to the strongly
driven regime.

Another seemingly plausible mechanism for achieving high efficiency with small
$\Delta\mu$ is to couple multiple driven transitions in series to achieve a
larger effective total chemical potential difference. However, the generalized
form of Eq. 3 for multiple sequential batteries shows that the efficiency
scales inversely with the number of batteries, although the result could be
more complex for arbitrary arrangements of batteries (See Supp. Information).
In particular, if $n$ identical driven transitions are connected in series
(e.g. sequential hydrolysis of multiple ATP molecules), the efficiency in the
processive regime is at most $1/n$. Therefore, efficient and processive
dissipative machines require large units of energy ”currency” $\Delta\mu$.
Discussion. Eq. 3 is a tight general bound relating the efficiency of
dissipative machines to independent design parameters. These properties
establish this equation as a chemical engine analog of Carnot’s heat engine
relation, and set it apart from previous work (10; 11). In contrast to
Carnot’s relation, in which the only property of the undriven engine is the
ambient temperature, Eq. 3 includes information about the engine’s design via
the load resistance $R_{\mathrm{L}}$. This collective variable can be computed
from knowing the detailed rate constants between the states of the engine, but
it can also be experimentally measured as the linear response coefficient
(i.e. susceptibility) of the net engine speed to $\Delta\mu$. By measuring the
speed in the presence and absence of the desired load, $R_{\mathrm{S}}$ can
also be determined. More broadly, Eq. 3 has implications for the design and
evolution of molecular machines. If $\Delta\mu>\Delta\mu_{\mathrm{thresh}}$,
high efficiency becomes a generic feature that does not require fine-tuning.
Below the threshold, all molecular processes are inefficient. This threshold
may therefore serve as a unique thermodynamic criterion for life. Because many
biological processes are powered by the same sources of chemical energy,
achieving large $\Delta\mu$ may have been an evolutionary transition step that
”switched on” efficient energy transduction to all biomolecular processes of
comparable load resistance.

Acknowledgements. The author would like to thank Michael Trenfield for
fruitful discussions regarding the theory and its implications, as well as
Rama Ranganathan, Elliott Ross, Prashant Mishra, and Gaudenz Danuser for
detailed feedback on the manuscript. This work was supported by the Cecil H.
and Ida Green Foundation and the anonymous donor-supported UTSW High Risk/High
Impact grant.

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##  I supplementary materials

###  I.1 Total dissipation rate

The total rate of dissipation at steady state is:

$\sigma
T=kT\sum_{ij}I_{ij}\ln{\Bigg{[}{{P_{i}(k_{ij}+\alpha_{ij})}\over{P_{j}k_{ji}}}\Bigg{]}}$
(6)

I first recast this relation into the suggestive form:

$\sigma/k=\sum_{ij}I_{ij}(\ln[P_{i}/P_{i}^{*}]-\ln[P_{j}/P_{j}^{*}])+\sum_{ij}I_{ij}\ln[1+\alpha_{ij}/k_{ij}],$
(7)

where $P_{i}^{*}$ and $P_{j}^{*}$ are the equilibrium probabilities of states
$i$ and $j$ if all $\alpha$ (and therefore all $\Delta\mu$) are equal to zero;
the ratio of these equilibrium probabilities are given by the ratio of the
forward and backwards equilibrium rate constants between these states. Each
term in the first summation is the current between adjacent states multiplied
by the difference between a function evaluated at those states: in our circuit
framework of symmetric resistors, Tellegen’s Theorem dictates that such a sum
equals zero at steady state for any circuit (Ref. 25 in Main Text). The
nonzero sum is thus the steady-state dissipation rate (Eq. 2 in the main
text):

$\sigma T=\sum_{ij}I_{ij}\Delta\mu_{ij},$ (8)

where the chemical potential difference
$\Delta\mu_{ij}=kT\ln{\big{[}1+{{\alpha_{ij}}\over{k_{ij}}}\big{]}}$.

###  I.2 Dissipation rate of the load

####  I.2.1 single driven transition

From Eq. 2, the total dissipation rate for a chemical engine with a single
driven transition from state 1 to 2 (which we label as the source ’S’) is:

$\sigma
T=kTI_{\mathrm{S}}\ln{\Bigg{[}1+{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}\Bigg{]}},$
(9)

where the current $I_{\mathrm{S}}$ flows through the source transition. The
rate of dissipation of the driven transition (i.e. power dissipation in the
battery) is:

$\sigma_{\mathrm{S}}T=kTI_{\mathrm{S}}\ln{\Bigg{[}{{(\alpha_{\mathrm{S}}+k_{\mathrm{S}})P_{1}}\over{P_{2}k_{\mathrm{21}}}}\Bigg{]}},$
(10)

where the $k_{21}$ is the rate constant of the backwards reaction to
$k_{\mathrm{S}}$ in the absence of driving (i.e. at equilibrium). Therefore,
the rate of dissipation of the load is:

$\sigma_{\mathrm{L}}T=\sigma
T-\sigma_{\mathrm{S}}T=kTI_{\mathrm{S}}\ln{\Bigg{[}{{P_{2}k_{\mathrm{21}}}\over{P_{1}k_{\mathrm{S}}}}\Bigg{]}}.$
(11)

multiplying the numerator and denominator inside the logarithm by $e^{\beta
G_{1}}$ and noting that $k_{\mathrm{S}}e^{\beta G_{2}}=k_{21}e^{\beta G_{1}}$,
we obtain:

$\sigma_{\mathrm{L}}T=kTI_{\mathrm{S}}\ln{\Bigg{[}{{P_{2}e^{\beta
G_{2}}}\over{P_{1}e^{\beta G_{1}}}}\Bigg{]}}.$ (12)

Applying Ohm’s law across the engine’s cycle must yield zero change in the
probability potential:

${{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}P_{1}e^{\beta
G_{1}}-I_{\mathrm{S}}(R_{\mathrm{S}}+R_{\mathrm{L}})=0.$ (13)

Thus,

$I_{\mathrm{S}}={{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}(R_{\mathrm{S}}+R_{\mathrm{L}})^{-1}P_{1}e^{\beta
G_{1}}.$ (14)

Now, applying Ohm’s law across the load resistance:

$P_{2}e^{\beta G_{2}}-P_{1}e^{\beta G_{1}}=I_{\mathrm{S}}R_{\mathrm{L}}.$ (15)

Combining these two equations to solve for $P_{2}$:

$P_{2}e^{\beta G_{2}}=P_{1}e^{\beta
G_{1}}+{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}R_{\mathrm{L}}(R_{\mathrm{S}}+R_{\mathrm{L}})^{-1}P_{1}e^{\beta
G_{1}}.$ (16)

Therefore,

${{P_{2}e^{\beta G_{2}}}\over{P_{1}e^{\beta
G_{1}}}}={{R_{\mathrm{S}}+R_{\mathrm{L}}(1+{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}})}\over{R_{\mathrm{S}}+R_{\mathrm{L}}}}.$
(17)

Substituting for the probability ratio in the expression for the dissipation
rate of the load gives:

$\sigma_{\mathrm{L}}T=kTI_{\mathrm{S}}\ln{\Bigg{[}{{R_{\mathrm{S}}+R_{\mathrm{L}}(1+{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}})}\over{R_{\mathrm{S}}+R_{\mathrm{L}}}}\Bigg{]}}.$
(18)

Substituting for the chemical potential difference yields the dissipation rate
of the load driven by a single battery given in the main text:

$\sigma_{\mathrm{L}}T=kTI_{\mathrm{S}}\ln{\bigg{[}{{R_{\mathrm{S}}+R_{\mathrm{L}}e^{\Delta\mu/kT}}\over{R_{\mathrm{S}}+R_{\mathrm{L}}}}\bigg{]}}.$
(19)

####  I.2.2 multiple driven transitions in series

Consider the more general case of $n$ driven transitions (batteries) in
series. Denote the chemical potential difference and resistance of the $j$th
battery by $\Delta\mu_{j}$ and $R_{j}$, respectively. $\Delta\mu_{j}$ drives
the transition from state $j$ to state $j+1$. denote the forward and backward
rate constants for transition $j$, in the absence of driving, by $k_{j}$ and
${\tilde{k}}_{j}$, respectively. The total rate of dissipation is:

$\sigma
T=kTI_{\mathrm{S}}\sum_{j=1}^{n}\ln{\Bigg{[}1+{{\alpha_{j}}\over{k_{j}}}\Bigg{]}}.$
(20)

The dissipation rate of the driven transitions (i.e. power dissipated in the
batteries) is:

$\sigma_{\mathrm{S}}T=kTI_{\mathrm{S}}\sum_{j=1}^{n}\ln{\Bigg{[}{{(\alpha_{j}+k_{j})P_{j}}\over{P_{j+1}{\tilde{k}}_{j}}}\Bigg{]}}=kTI_{\mathrm{S}}\sum_{j=1}^{n}\ln{\Bigg{[}{{1+{{\alpha_{j}}\over{k_{j}}}\Bigg{]}}+kTI_{\mathrm{S}}\sum_{j=1}^{n}\ln{\Bigg{[}{{P_{j}}\over{P_{j+1}}}{{k_{j}}\over{\tilde{k}_{j}}}}\Bigg{]}}}.$
(21)

The last summation can be simplified by noting:

$I_{\mathrm{S}}\sum_{j=1}^{n}\ln{\Bigg{[}{{P_{j}}\over{P_{j+1}}}{{k_{j}}\over{\tilde{k}_{j}}}\Bigg{]}}=I_{\mathrm{S}}\sum_{j=1}^{n}\ln{\Bigg{[}{{P_{j}}\over{P_{j+1}}}{{P_{j+1}^{*}}\over{P_{j}^{*}}}\Bigg{]}}=I_{\mathrm{S}}\ln{\Bigg{[}{{P_{1}}\over{P_{1}^{*}}}{{P_{n+1}^{*}}\over{P_{n+1}}}\Bigg{]}},$
(22)

where the first equality follows because the ratio of the undriven forward and
backward rate constants gives the ratio of the corresponding probabilities of
the states at equilibrium, and the second equality follows because all terms
in the summation cancel except for the variables corresponding to the first
and last states. Therefore, the rate of dissipation in the load is:

$\sigma_{\mathrm{L}}T=\sigma
T-\sigma_{\mathrm{S}}T=-kTI_{\mathrm{S}}\ln{\Bigg{[}{{P_{1}}\over{P_{1}^{*}}}{{P_{n+1}^{*}}\over{P_{n+1}}}\Bigg{]}}.$
(23)

Applying Ohm’s law across the $j$th battery:

${{\alpha_{j}}\over{k_{j}}}P_{j}e^{\beta
G_{j}}-I_{\mathrm{S}}R_{j}=P_{j+1}e^{\beta G_{j+1}}-P_{j}e^{\beta G_{j}}.$
(24)

Thus,

$\Big{(}1+{{\alpha_{j}}\over{k_{j}}}\Big{)}P_{j}e^{\beta
G_{j}}-P_{j+1}e^{\beta G_{j+1}}=I_{\mathrm{S}}R_{j}.$ (25)

Multiply both sides by the product
$\prod_{i=j+1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}$, and summing
over all the batteries:

$\sum_{j=1}^{n}\prod_{i=j+1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}\Big{[}\Big{(}1+{{\alpha_{j}}\over{k_{j}}}\Big{)}P_{j}e^{\beta
G_{j}}-P_{j+1}e^{\beta
G_{j+1}}\Big{]}=I_{\mathrm{S}}\sum_{j=1}^{n}\prod_{i=j+1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}R_{j}.$
(26)

All terms in the left sum cancel except for those corresponding to the first
and last states:

$\prod_{i=1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}P_{1}e^{\beta
G_{1}}-P_{n+1}e^{\beta
G_{n+1}}=I_{\mathrm{S}}\sum_{j=1}^{n}\prod_{i=j+1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}R_{j}.$
(27)

Now, applying Ohm’s law across the load resistance gives the current as a
function of $P_{1}$ and $P_{n+1}$:

$P_{n+1}e^{\beta G_{n+1}}-P_{1}e^{\beta G_{1}}=I_{\mathrm{S}}R_{\mathrm{L}}.$
(28)

Substituting for the current in the expression obtained from summing the
batteries:

$\prod_{i=1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}P_{1}e^{\beta
G_{1}}-P_{n+1}e^{\beta G_{n+1}}=R_{\mathrm{L}}^{-1}\Big{(}P_{n+1}e^{\beta
G_{n+1}}-P_{1}e^{\beta
G_{1}}\Big{)}\sum_{j=1}^{n}\prod_{i=j+1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}R_{j}.$
(29)

Dividing both sides by $P_{1}e^{\beta G_{1}}$ and collecting terms:

${{P_{n+1}}\over{P_{n+1}^{*}}}{{P_{1}^{*}}\over{P_{1}}}\Bigg{(}1+R_{\mathrm{L}}^{-1}\sum_{j=1}^{n}\prod_{i=j+1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}R_{j}\Bigg{)}=\prod_{i=1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}+R_{\mathrm{L}}^{-1}\sum_{j=1}^{n}\prod_{i=j+1}^{n}\Big{(}1+{{\alpha_{i}}\over{k_{i}}}\Big{)}R_{j},$
(30)

where the equilibrium probability ratios
$P_{1}^{*}/P_{n+1}^{*}=e^{\beta(G_{n+1}-G_{1})}$. Rearranging this expression,
and substituting in the definition of the chemical potential difference:

${{P_{n+1}}\over{P_{n+1}^{*}}}{{P_{1}^{*}}\over{P_{1}}}=\Bigg{(}1+R_{\mathrm{L}}^{-1}\sum_{j=1}^{n}e^{\beta\sum_{i=j+1}^{n}\Delta\mu_{i}}R_{j}\Bigg{)}^{-1}\Bigg{(}e^{\beta\sum_{i=1}^{n}\Delta\mu_{i}}+R_{\mathrm{L}}^{-1}\sum_{j=1}^{n}e^{\beta\sum_{i=j+1}^{n}\Delta\mu_{i}}R_{j}\Bigg{)}.$
(31)

Substituting this probability ratio in the expression for the dissipation rate
of the load for $n$ batteries in series:

$\sigma_{\mathrm{L}}T=kTI_{\mathrm{S}}\ln{\Bigg{[}{{e^{\beta\sum_{i=1}^{n}\Delta\mu_{i}}+R_{\mathrm{L}}^{-1}\sum_{j=1}^{n}e^{\beta\sum_{i=j+1}^{n}\Delta\mu_{i}}R_{j}}\over{1+R_{\mathrm{L}}^{-1}\sum_{j=1}^{n}e^{\beta\sum_{i=j+1}^{n}\Delta\mu_{i}}R_{j}}}\Bigg{]}}.$
(32)

In the special case in which all the driven transitions are identical,
$\Delta\mu_{j}=\Delta\mu$ and $R_{j}=R_{\mathrm{S}}$. Then the entropy
production rate simplifies to:

$\sigma_{\mathrm{L}}T=kTI_{\mathrm{S}}\ln{\Bigg{[}{{R_{\mathrm{S}}{{e^{n\beta\Delta\mu}-1}\over{e^{\beta\Delta\mu}-1}}+e^{n\beta\Delta\mu}R_{\mathrm{L}}}\over{R_{\mathrm{S}}{{e^{n\beta\Delta\mu}-1}\over{e^{\beta\Delta\mu}-1}}+R_{\mathrm{L}}}}\Bigg{]}}.$
(33)

As expected, this expression reduces to the single battery dissipation rate
derived in the previous section in the case that $n=1$.

The maximum efficiency is given by

$\eta_{\mathrm{chem}}\leq{{\sigma_{\mathrm{L}}}\over{\sigma}}={{kT}\over{n\Delta\mu}}\ln{\Bigg{[}{{R_{\mathrm{S}}{{e^{n\beta\Delta\mu}-1}\over{e^{\beta\Delta\mu}-1}}+e^{n\beta\Delta\mu}R_{\mathrm{L}}}\over{R_{\mathrm{S}}{{e^{n\beta\Delta\mu}-1}\over{e^{\beta\Delta\mu}-1}}+R_{\mathrm{L}}}}\Bigg{]}}$
(34)

In the processive regime ($R_{\mathrm{S}}>R_{\mathrm{L}}$), this simplifies to

$\eta_{\mathrm{chem}}<{{kT}\over{n\Delta\mu}}\ln{\Bigg{[}{{R_{\mathrm{L}}}\over{R_{\mathrm{S}}}}e^{\beta\Delta\mu}\Bigg{]}}<{{1}\over{n}}.$
(35)

Therefore, as stated in the main text, the maximum efficiency of a chemical
engine powered by multiple driven transitions in series is the reciprocal of
the number of driven transitions.

###  I.3 Inflection point of the efficiency function

Define reduced variables: $u\equiv\Delta\mu/kT$ and $r\equiv
R_{\mathrm{S}}/R_{\mathrm{L}}$. In these reduced variables, the maximum
efficiency(Eq. 3 in Main Text) is:

$\eta_{\mathrm{chem,max}}={{1}\over{u}}\ln{\bigg{[}{{e^{u}+r}\over{1+r}}\bigg{]}}.$
(36)

Differentiating twice with respect to $u$ yields the curvature of the
efficiency function:

${{\partial^{2}\eta_{\mathrm{chem,max}}}\over{\partial
u^{2}}}={{2\ln{\big{[}{{e^{u}+r}\over{1+r}}\big{]}}}\over{u^{3}}}+{{e^{u}}\over{(e^{u}+r)u}}-{2{e^{u}}\over{(e^{u}+r)u^{2}}}-{{e^{2u}}\over{(e^{u}+r)^{2}u}}$
(37)

For $\eta_{\mathrm{chem,max}}$ to have an inflection point, there must be a
value of $u$ for which the curvature is zero. In the limit of vanishing $u$,
such that $e^{u}$ approaches $1+u$, the curvature is:

${{\partial^{2}\eta_{\mathrm{chem,max}}}\over{\partial
u^{2}}}={{r(r-1)}\over{3(r+1)^{3}}}+O(u)$ (38)

Therefore, if $r>1$ (i.e. $R_{\mathrm{S}}>R_{\mathrm{L}}$), the curvature is
positive in the limit of very small $u$ (i.e. $\Delta\mu<<kT$). Now consider
the other limit of large $u>>1$. In this case, define the variable $x\equiv
r/e^{u}$. Then the curvature can be rewritten:

${{\partial^{2}\eta_{\mathrm{chem,max}}}\over{\partial
u^{2}}}={{2(u+\ln{\big{[}{{1+x}\over{1+r}}\big{]}})}\over{u^{3}}}+{{1}\over{(1+x)u}}-{{2}\over{(1+x)u^{2}}}-{{1}\over{(1+x)^{2}u}}$
(39)

In the limit of large $u$, $x$ will approach zero, and the leading order
curvature is:

${{\partial^{2}\eta_{\mathrm{chem,max}}}\over{\partial
u^{2}}}={{2}\over{u^{3}}}\ln{\bigg{[}{{1}\over{1+r}}\bigg{]}}+O(x),$ (40)

which approaches zero from below because $r$ is positive. Therefore, if $r>1$,
the efficiency must pass through an inflection point because the curvature is
positive for small $\Delta\mu$ and negative for large $\Delta\mu$.

To find the inflection point, make the guess that it occurs near the condition
$u=\Delta\mu/kT\approx\ln{[r]}$, such that the logarithm in the first term of
the curvature equation is of order unity or smaller. In that case, the first
term scales with the inverse cube of $u$; in the limit of large $u$, we can
ignore this term to obtain the inflection point condition:

${{\partial^{2}\eta_{\mathrm{chem,max}}}\over{\partial
u^{2}}}\Bigg{|}_{u=u_{\mathrm{thresh}}}\approx{{e^{u}}\over{(e^{u}+r){u}}}-{2{e^{u}}\over{(e^{u}+r)u^{2}}}-{{e^{2u}}\over{(e^{u}+r)^{2}u}}=0,$
(41)

Which yields the solution:

$u_{\mathrm{thresh}}\approx 2-\textit{W}_{-1}(-2e^{2}/r),$ (42)

where $\textit{W}_{-1}$ is the lower real branch of the Lambert W function. In
the limit of large $r$, this is approximated by:

$u_{\mathrm{thresh}}\approx\ln{[r]}+\ln{\bigg{[}{{1}\over{2}}\ln{\big{[}{{r}\over{2}}\big{]}}-1\bigg{]}}.$
(43)

This result is self-consistent with the ansatz at the beginning of this
section that requires $u\approx\ln{[r]}$. To leading order, the inflection
point is give by Eq. 4 in the main text:

$\Delta\mu_{\mathrm{thresh}}\approx
kT\ln{\bigg{[}{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}\bigg{]}}.$ (44)

###  I.4 Tradeoff between efficiency and processivity

Starting with the efficiency relation (Eq. 3 in the main text):

$\eta_{\mathrm{chem}}\leq{{kT}\over{\Delta\mu}}\ln{\Bigg{[}{{{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}+e^{{{\Delta\mu}\over{kT}}}}\over{{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}+1}}\Bigg{]}}.$
(45)

Rearrange to obtain:

$-{{\Delta\mu}\over{kT}}(1-\eta_{\mathrm{chem}})\leq\ln{\Bigg{[}{{{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}+e^{{{\Delta\mu}\over{kT}}}}\over{{{R_{\mathrm{S}}}\over{R_{\mathrm{L}}}}+1}}\Bigg{]}}-{{\Delta\mu}\over{kT}}=\ln{\Bigg{[}{{R_{\mathrm{S}}e^{-{{\Delta\mu}\over{kT}}}+R_{\mathrm{L}}}\over{R_{\mathrm{S}}+R_{\mathrm{L}}}}\Bigg{]}}.$
(46)

exponentiating both sides, subtracting from one, and simplifying:

$1-e^{-{{\Delta\mu}\over{kT}}(1-\eta_{\mathrm{chem}})}\geq
1-{{R_{\mathrm{S}}e^{-{{\Delta\mu}\over{kT}}}+R_{\mathrm{L}}}\over{R_{\mathrm{S}}+R_{\mathrm{L}}}}={{R_{\mathrm{S}}(e^{{{\Delta\mu}\over{kT}}}-1)}\over{e^{{{\Delta\mu}\over{kT}}}(R_{\mathrm{S}}+R_{\mathrm{L}})}}.$
(47)

Substituting for the chemical potential difference:

$1-e^{-{{\Delta\mu}\over{kT}}(1-\eta_{\mathrm{chem}})}\geq{{R_{\mathrm{S}}({{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}})}\over{(1+{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}})(R_{\mathrm{S}}+R_{\mathrm{L}})}}={{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}\over{(R_{\mathrm{S}}+R_{\mathrm{L}})}}{{e^{\beta
G_{1}}}\over{k_{\mathrm{S}}+\alpha_{\mathrm{S}}}},$ (48)

where we substituted for $R_{\mathrm{S}}=e^{\beta G_{1}}/k_{\mathrm{S}}$.
Ohm’s law taken across the engine cycle gives:

${{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}P_{1}e^{\beta
G_{1}}=I_{\mathrm{S}}(R_{\mathrm{S}}+R_{\mathrm{L}}),$ (49)

which gives the relation:

${{{\alpha_{\mathrm{S}}}\over{k_{\mathrm{S}}}}\over{(R_{\mathrm{S}}+R_{\mathrm{L}})}}={{I_{\mathrm{S}}}\over{P_{1}e^{\beta
G_{1}}}}.$ (50)

Substituting this relation into the inequality:

$1-e^{-{{\Delta\mu}\over{kT}}(1-\eta_{\mathrm{chem}})}\geq{{I_{\mathrm{S}}}\over{P_{1}e^{\beta
G_{1}}}}{{e^{\beta
G_{1}}}\over{k_{\mathrm{S}}+\alpha_{\mathrm{S}}}}={{I_{\mathrm{S}}}\over{P_{1}(k_{\mathrm{S}}+\alpha_{\mathrm{S}})}}.$
(51)

Noting that the denominator in the right hand side is the total forward flux
$f_{\mathrm{S}}=P_{1}(k_{\mathrm{S}}+\alpha_{\mathrm{S}})$. The net flux
$I_{\mathrm{S}}$ divided by the total forward flux $f_{\mathrm{S}}$ is the
processivity $P$. Therefore, we obtain Eq. 5 in the main text:

$P\leq 1-e^{-{{\Delta\mu}\over{kT}}(1-\eta_{\mathrm{chem}})}.$ (52)

###  I.5 Dynein efficiency

The maximum in vitro efficiency of dynein under different ATP concentrations
was obtained from Ref. 28. Dynein step size at different concentrations was
obtained from Fig. 3E in Ref. 28. The step size was converted to efficiency by
multiplying the step size by the reported stall force of 1.85 pN, and dividing
by the corresponding $\Delta\mu$. $\Delta\mu$ was estimated from the in vivo
value, with an [ATP]-dependent correction $\ln{[\mathrm{[ATP]/[ATP]}_{vivo}]}$
with respect to an in vivo ATP concentration of 10mM.