Department of Applied Physics, California Institute of Technology, Pasadena,
California 91125, United States Division of Biology and Biological
Engineering, California Institute of Technology, Pasadena, California 91125,
United States (626) 395-3374

# Allostery and Kinetic Proofreading

Vahe Galstyan  Biochemistry and Molecular Biophysics Option, California
Institute of Technology, Pasadena, California 91125, United States  Rob
Phillips  Department of Physics, California Institute of Technology, Pasadena,
California 91125, United States<EMAIL_ADDRESS>
###### Abstract

Kinetic proofreading is an error correction mechanism present in the processes
of the central dogma and beyond, and typically requires the free energy of
nucleotide hydrolysis for its operation. Though the molecular players of many
biological proofreading schemes are known, our understanding of how energy
consumption is managed to promote fidelity remains incomplete. In our work, we
introduce an alternative conceptual scheme called “the piston model of
proofreading” where enzyme activation through hydrolysis is replaced with
allosteric activation achieved through mechanical work performed by a piston
on regulatory ligands. Inspired by Feynman’s ratchet and pawl mechanism, we
consider a mechanical engine designed to drive the piston actions powered by a
lowering weight, whose function is analogous to that of ATP synthase in cells.
Thanks to its mechanical design, the piston model allows us to tune the
“knobs” of the driving engine and probe the graded changes and trade-offs
between speed, fidelity and energy dissipation. It provides an intuitive
explanation of the conditions necessary for optimal proofreading and reveals
the unexpected capability of allosteric molecules to beat the Hopfield limit
of fidelity by leveraging the diversity of states available to them. The
framework that we built for the piston model can also serve as a basis for
additional studies of driven biochemical systems.

##  1 Introduction

Many enzymatic processes in biology need to operate with very high fidelities
in order to ensure the physiological well-being of the cell. Examples include
the synthesis of molecules making up Crick’s so-called “two great polymer
languages” (i.e. replication, 1 transcription, 2 and translation 3), as well
as processes that go beyond those of the central dogma, such as protein
ubiquitylation mediated by the anaphase-promoting complex, 4 signal
transduction through MAP kinases, 5 pathogen recognition by T-cells,6, 7 or
protein degradation by the 26S proteasome. 8 In all of these cases, the
designated enzyme needs to accurately select its correct substrate from a pool
of incorrect substrates. Importantly, the fidelity of these processes that one
would predict solely based on the free energy difference between correct and
incorrect substrate binding is far lower than what is experimentally measured,
raising a challenge of explaining the high fidelities that this naive
equilibrium thermodynamic thinking fails to account for.

The conceptual answer to this challenge was provided more than 40 years ago in
the work of John Hopfield 9 and Jacques Ninio 10 and was coined “kinetic
proofreading” in Hopfield’s elegant paper entitled “Kinetic Proofreading: A
New Mechanism for Reducing Errors in Biosynthetic Processes Requiring High
Specificity.” 9 The key idea behind kinetic proofreading is to introduce a
delay between substrate binding and turnover steps, effectively giving the
enzyme more than one chance to release the incorrect substrate (hence, the
term “proofreading”). The sequential application of substrate filters on the
way to product formation gives directionality to the flow of time and is
necessarily accompanied by the expenditure of free energy, making kinetic
proofreading an intrinsically nonequilibrium phenomenon. In a cell, this free
energy is typically supplied to proofreading pathways through the hydrolysis
of energy-rich nucleotides, whose chemical potential is maintained at large
out-of-equilibrium values through the constant operation of the cell’s
metabolic machinery (e.g. the ATP synthase).

Since its original formulation by Hopfield and Ninio, the concept of kinetic
proofreading has been generalized and employed in explaining many of the high
fidelity processes in the cell. 11, 12, 13, 8, 14, 15, 16, 17 However, despite
the fact that the molecular players and mechanisms of these processes have
been largely identified, we find that an intuitive picture of how energy
transduction promotes biological fidelity is still incomplete. To complement
our understanding of how energy is managed to beat the equilibrium limit of
fidelity, we propose a conceptual model called “the piston model of kinetic
proofreading” where chemical hydrolysis is replaced with mechanical work
performed by a piston on an allosteric enzyme. Our choice of allostery is
motivated by the fact that in proofreading schemes hydrolysis typically
triggers a conformational change in the enzyme-substrate complex and activates
it for product formation18, 19, 20 \- an effect that our model achieves
through the binding of a regulatory ligand to the enzyme. By temporally
controlling the concentration of regulatory ligands which determine the
catalytic state of the enzyme, the piston sequentially changes the enzyme’s
state from inactive to active, creating a delay in product formation necessary
for increasing the fidelity of substrate discrimination. The piston actions
are, in turn, driven by a Brownian ratchet and pawl engine powered by a
lowering weight, whose function is akin to that of ATP synthase. The
mechanical design of the piston model allows us to transparently control the
energy input into the system by tuning the “knobs” of the engine and examine
the graded changes in the model’s performance metrics, intuitively
demonstrating the driving conditions required for optimal proofreading.

We begin the presentation of our results by first introducing in section 2 the
high-level concept behind the piston model of proofreading, while at the same
time drawing parallels between its features and those of Hopfield’s original
scheme. Then, in sections 3.1 and 3.2 we provide a comprehensive description
of the two key constituents of the piston model, namely, the Brownian ratchet
and pawl engine that drives the piston actions, and the allosteric enzyme
whose catalytic state is regulated by an activator ligand. This is following
by building the full thermodynamically consistent framework of the piston
model in section 3.3 where we couple the external driving mechanism to the
enzyme and introduce the expressions for key performance metrics of the model.
In the remaining sections 3.4 and 3.5, we explore how tuning the “knobs” of
the engine leads to graded changes and trade-offs between speed, fidelity and
energy dissipation, and probe the performance limits of the piston model as a
function of a select set of key enzyme parameters.

##  2 MODEL

The piston model of kinetic proofreading is designed in analogy with
Hopfield’s scheme. The main idea there was to give the enzyme a second chance
to discard the wrong substrate by introducing an additional kinetic
intermediate for the enzyme-substrate complex (Figure 1A). The difference
between substrate binding energies in Hopfield’s original formulation was
based solely on their unbinding rates (i.e.
$k_{\mathrm{off}}^{\mathrm{W}}>k_{\mathrm{off}}^{\mathrm{R}}$ and
$k_{\mathrm{on}}^{\text{W}}=k_{\mathrm{on}}^{\text{R}}=k_{\mathrm{on}}$) – a
convention we adopt throughout our analysis. The first layer of substrate
discrimination in Hopfield’s scheme is achieved during the initial binding
event where the ratio of right and wrong substrate-bound enzymes approaches
$k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}$. The complex
then moves into its catalytically active high energy state accompanied by the
hydrolysis of an NTP molecule, after which the second discrimination layer is
realized. Specifically, right and wrong substrates are turned into products
with an additional bias given by the ratio of their Michaelis constants,
namely, $(k_{\mathrm{off}}^{\mathrm{W}}+r)/(k_{\mathrm{off}}^{\mathrm{R}}+r)$.
Importantly, for this second layer to be efficiently realized, the rates of
binding directly to the second kinetic intermediate need to be vanishingly
small in order to prevent the incorporation of unfiltered substrates.9

With this information in mind, consider now the conceptual illustration of the
piston model shown in Figure 1B,  where we have made several pedagogical
simplifications to help verbally convey the model’s intuition, reserving the
full thermodynamically consistent treatment to the following sections. The
central constituent of the model is an allosteric enzyme, the catalytic
activity of which is regulated by activator ligands (the orange circle).  The
enzyme is inactive when it is not bound to a ligand, and, conversely, it is
active when bound to a ligand. The volume occupied by ligands and hence, their
concentration is, in turn, controlled by a piston. The ligand concentration is
very low when the piston is expanded, and very high when the piston is
compressed  in order to guarantee that in those piston states the ligand is
free and bound to the enzyme, respectively.

The active site of the enzyme is exposed to a container filled with right and
wrong substrates of concentrations $[\text{R}]$ and $[\text{W}]$,
respectively, which we take to be equal for the rest of our analysis
($[\text{R}]=[\text{W}]$). And unlike in Hopfield’s scheme where the
substrates exist in energy-rich and energy-depleted states (e.g. tRNAs first
arrive in the EF-Tu$\cdot$GTP$\cdot$tRNA ternary complex and then release EF-
Tu and GDP after hydrolysis), in the piston model substrates exist in a single
state and do not carry an energy source. In the expanded piston state (Figure
1B, left), substrates can bind and unbind to the inactive enzyme, but do not
get turned into products. The highest level of discrimination achievable in
this state therefore becomes

$\displaystyle\eta_{1}=\frac{k_{\mathrm{off}}^{\mathrm{W}}}{k_{\mathrm{off}}^{\mathrm{R}}},$
(1)

in analogy to that achieved during the initial binding step of Hopfield’s
scheme.

After the first layer of substrate discrimination is established in the
expanded state of the piston, mechanical work is performed to compress it.
This increases the ligand concentration, which, in turn, leads to the
activation of the enzyme where catalytic action is now possible. To prevent
the incorporation of unfiltered substrates, we assume that in the active
enzyme state the rate of substrate binding is vanishingly small, similar to
Hopfield’s treatment (Figure 1B, right). If the piston is kept compressed long
enough, a filtered substrate that got bound earlier when the piston was
expanded will experience one of these two outcomes: it will either turn into a
product with a rate $r$ (which is taken to be the same for the two kinds of
substrates) or it will fall off with a rate $k_{\text{off}}$. The product
formation reaction will take place with probability $r/(k_{\mathrm{off}}+r)$.
Thus, due to the difference in the falloff rate constants between the right
and the wrong substrates, the extra fidelity achieved after piston compression
equals

$\displaystyle\eta_{2}=\frac{k_{\mathrm{off}}^{\mathrm{W}}+r}{k_{\mathrm{off}}^{\mathrm{R}}+r}.$
(2)

Once this extra fidelity is established, the piston is expanded back,
repeating the cycle (the detailed derivation of the results for $\eta_{1}$ and
$\eta_{2}$ is provided in Supporting Information section A). Notably, the
total fidelity achieved during the piston expansion and compression cycle,
namely,

$\displaystyle\eta=\eta_{1}\eta_{2}=\left(\frac{k_{\mathrm{off}}^{\mathrm{W}}}{k_{\mathrm{off}}^{\mathrm{R}}}\right)\left(\frac{k_{\mathrm{off}}^{\mathrm{W}}+r}{k_{\mathrm{off}}^{\mathrm{R}}+r}\right),$
(3)

exceeds the Michaelis-Menten fidelity ($\eta_{2}$) by a factor of
$\eta_{1}=k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}$,
demonstrating the attainment of efficient proofreading.

The cyclic compressions and expansions of the piston in our model also stand
in direct analogy to the hydrolysis-involving transitions between the two
enzyme-substrate intermediates in Hopfield’s scheme. In particular, they need
to be externally driven for the mechanism to do proofreading. We perform this
driving using a mechanical ratchet and pawl engine powered by a lowering
weight. In our pedagogical description of the model’s operation, we implicitly
assumed that this weight was very large in order to enable the mechanism to
proofread, similar to how the hydrolysis energy needs to be large for
Hopfield’s scheme to operate effectively.9 In the full treatment of the model
in section 3, however, we will demonstrate how the tuning of the weight can
give us graded levels of fidelity enhancement, and will also show that in the
absence of this weight the equilibrium fluctuations of the piston alone cannot
lead to proofreading.

In our model introduction we have also made several simplifying assumption for
clarity of presentation which do not conform with the principle of microscopic
reversibility, and it is important that we relax them in the full treatment of
the model to make it thermodynamically consistent. In particular, we assumed
that the ligand is necessarily unbound and that the enzyme is necessarily
inactive when the piston is expanded, with the reverse assumptions made when
the piston is compressed. We also assumed that substrate binding was
prohibited in the active state of the enzyme. These assumptions allowed us to
claim that no premature product formation takes place in the expanded piston
state and that no unfiltered substrates bind to the activated enzyme in the
compressed piston state, which, in turn, justified the use of long waiting
times between the piston actions necessary to establish high levels of
fidelity in each piston state. In the detailed analysis of our model presented
in section 3, we relax these assumptions and consider the full diversity of
enzyme states at each piston position with reversible transitions between
them. This, as we will demonstrate, will not only ensure the thermodynamic
consistency of our treatment but will also reveal the possibility of doing
proofreading more than once by leveraging the presence of multiple inactive
intermediates in between enzyme’s substrate-unbound and production states
which were not accounted for in our conceptual introduction of the model.

Figure 1: Conceptual introduction to the piston model. (A) Hopfield’s scheme
of kinetic proofreading where two layers of substrate discrimination take
place on the driven pathway – the first one during the initial binding of
energy-rich substrates (#1 in the diagram) and the second one upon the release
of the energy-depleted substrates (#2 in the diagram). Energy consumption
takes place during the hydrolysis reaction
$\text{NTP}\rightleftharpoons\text{NDP}$ accompanying the transition between
the two intermediates. (B) Pedagogically simplified conceptual scheme of the
piston model. The orange circle represents the activator ligand. Blue and red
colors stand for the right and wrong substrates, respectively. The closed
“entrance door” along with the red cross on the binding arrow in the active
state of the enzyme suggests the vanishingly small rate of substrate binding
when in this state. The ratchet with a hanging weight stands for the
mechanical engine that drives the piston actions. Various features of the
system in the two piston states, along with the expressions for achieved
fidelities are listed below the diagram. Transparent arrows between panels A
and B indicate the analogous parts in Hopfield’s scheme and the piston model.

##  3 RESULTS

###  3.1 Ratchet and Pawl Engine Enables a Tunable Control of Piston Actions

To drive the cyclic compressions and expansions of the piston necessary for
achieving proofreading, we use a ratchet and pawl engine whose design is
inspired by Feynman’s original work. 21 In his celebrated lectures, Feynman
presented two implementations of the ratchet and pawl engine – one operating
on the temperature difference between two thermal baths, and another driven by
a weight that goes down due to gravity. In the piston model we adopt the
second scheme as it involves fewer parameters and illustrates the process of
energy transduction more transparently.

The ratchet and pawl engine coupled to the piston is shown in Figure 2A. The
engine is powered by a weight of mass $m$ which is hanging from an axle
connected to the ratchet. The free rotational motion of the ratchet is
rectified by a pawl; when the pawl sits on a ratchet tooth, it prevents the
ratchet from rotating in the clockwise (backwards) direction. The mechanical
coupling between the engine and the piston is achieved through a crankshaft
mechanism which translates each discrete ratchet step into a full compression
(up $\rightarrow$ down) or a full expansion (down $\rightarrow$ up) of the
piston. We assume that the volume regulated by the piston contains a single
ligand – a choice motivated by Szilard’s thermodynamic interpretation of
information, where a piston compressing a single gas molecule was considered.
22

The clockwise (backward) and counterclockwise (forward) steps of the
microscopic ratchet are enabled through environmental fluctuations.
Specifically, a backward step is taken whenever the pawl acquires sufficient
energy from the environment to lift itself over the ratchet tooth that it is
sitting on, allowing the tooth to slip under it (hence, the name “backward”).
Following Feynman’s treatment,21 we write the rate of such steps as

$\displaystyle k_{\text{b}}=\tau^{-1}\text{e}^{-\beta E_{0}},$ (4)

where $\tau^{-1}$ is the attempt frequency, $E_{0}$ is the amount of energy
needed to lift the pawl over a ratchet tooth, and $\beta=1/k_{\text{B}}T$ is
the inverse of the thermal energy scale (see Supporting Information section
B.1 for a detailed discussion of the ratchet and pawl mechanism). Every
backward step of the ratchet is accompanied by either a full compression or a
full expansion of the piston, as well as the lowering of the weight by an
amount of $\Delta z$, which reduces its potential energy by $\Delta W=mg\Delta
z$.

Unlike in backward stepping, for a forward step to take place, the rotational
energy acquired by the ratchet through fluctuations should be sufficient to
not only overcome the resistance of the spring pressing the pawl onto the
ratchet, but also to lift the weight and to alter the state of the piston.
This is a pure consequence of the geometric design of the ratchet and the
positioning of the pawl. We assume that piston actions take place isothermally
and in a quasistatic way, and therefore, write the changes in ligand free
energy upon compression (u$\,\rightarrow\,$d) and expansion
(d$\,\rightarrow\,$u) as

$\displaystyle\Delta F_{\mathrm{u}\rightarrow\mathrm{d}}$
$\displaystyle=\beta^{-1}\ln f,\quad\text{and}$ (5) $\displaystyle\Delta
F_{\mathrm{d}\rightarrow\mathrm{u}}$ $\displaystyle=-\beta^{-1}\ln f,$ (6)

respectively, where $f=V_{\mathrm{u}}/V_{\mathrm{d}}\geq 1$ is the fraction by
which the volume occupied by the ligand decreases upon compression. The signs
of free energy differences suggest that piston compressions slow down the
forward steps, while expansions speed them up. These features are reflected in
the two kinds of forward stepping rates that are given by

$\displaystyle k^{\mathrm{u}\rightarrow\mathrm{d}}_{\mathrm{f}}$
$\displaystyle=\tau^{-1}\text{e}^{-\beta\left(E_{0}+\Delta W+\Delta
F\right)}=f^{-1}k_{\text{b}}e^{-\beta\Delta W},$ (7) $\displaystyle
k^{\mathrm{d}\rightarrow\mathrm{u}}_{\mathrm{f}}$
$\displaystyle=\tau^{-1}\text{e}^{-\beta\left(E_{0}+\Delta W-\Delta
F\right)}=fk_{\text{b}}e^{-\beta\Delta W},$ (8)

where $\Delta F=\beta^{-1}\ln f$ and was used with a “+” and “-” sign in the
place of $\Delta F_{\mathrm{u}\rightarrow\mathrm{d}}$ and $\Delta
F_{\mathrm{d}\rightarrow\mathrm{u}}$, respectively. The rates of all four
kinds of transitions, namely, forward or backward ratchet steps, accompanied
by either a compression or an expansion of the piston are summarized in Figure
2B.

Figure 2: Ratchet and pawl mechanism coupled to the piston. (A) Schematic
representation of the mechanism. The different radii of the ratchet wheel and
the axle of the crankshaft ensure that a single ratchet step translates into a
full compression or a full expansion of the piston (i.e. a 180∘ rotation of
the crankshaft). Arrows with symbols “b” and “f” indicate the directions of
backward and forward ratchet rotation, respectively. (B) Rates of the four
kinds of transitions (symbols in shaded boxes with explicit expressions
below), along with the accompanying changes in the potential energy of the
weight and the free energy of the ligand. (C) Free energy landscape
corresponding to the non-equilibrium dynamics of the system in the presence of
a non-zero weight. Discrete positions of the weight ($z_{n}$) corresponding to
energy minima of the landscape are marked on the reaction coordinate. (D)
Infinite chain representation of the dynamics of discrete system states.
$k_{\text{net}}$ stands for the net rate at which the weight goes down. (E)
Equivalent two-state representation of the engine dynamics where the driving
force $\Delta\mu=2\Delta W$ breaks the detailed balance in the diagram. (F)
Collapsed representation of the diagram in panel E shown with the net
transition rates from the two pathways.

In the presence of a nonzero weight ($\Delta W>0$), the ratchet will on
average rotate backwards – a feature reflected in the tilted free energy
landscape shown in Figure 2C. As can be seen, the average dissipation per step
is $\Delta W$ and it is independent of $\Delta F$. In addition, the work
performed on the ligand upon compression is fully returned upon expansion,
which, as we will demonstrate in section 3.4, will generally not be the case
when we introduce the enzyme coupling. To further study the nonequilibrium
dynamics of the driving mechanism, we map the local minima of the energy
landscape corresponding to discrete vertical positions of the weight
(equivalently, discrete ratchet angles) into an infinite chain of transitions
shown in Figure 2D. There “d” and “u” stand for the compressed and expanded
states of the piston, respectively. The net stepping rate $k_{\mathrm{net}}$
at which the weight goes down can be written as

$\displaystyle
k_{\mathrm{net}}=\left(k_{\text{b}}-k^{\mathrm{d}\rightarrow\mathrm{u}}_{\mathrm{f}}\right)\pi_{\text{d}}+\left(k_{\text{b}}-k^{\mathrm{u}\rightarrow\mathrm{d}}_{\mathrm{f}}\right)\pi_{\text{u}},$
(9)

where $\pi_{\text{d}}$ and $\pi_{\text{u}}$ are the steady state probabilities
of the compressed and expanded piston states, respectively. These
probabilities can be obtained by considering the equivalent two-state diagram
in Figure 2E where the vertical position of the weight has been eliminated,
and the nonequilibrium nature of the dynamics is instead captured via the
cycle through two alternative pathways connecting the piston states. The
driving force $\Delta\mu$ in this cycle is given by23

$\displaystyle\Delta\mu=\beta^{-1}\ln\left(\frac{k^{2}_{\text{b}}}{k^{\mathrm{d}\rightarrow\mathrm{u}}_{\mathrm{f}}k^{\mathrm{u}\rightarrow\mathrm{d}}_{\mathrm{f}}}\right)=2\Delta
W,$ (10)

demonstrating the broken detailed balance in the presence of a nonzero weight,
and confirming the dissipation of $2\Delta W$ per cycle observed in the energy
landscape (Figure 2C). We note that this procedure of mapping a linear network
onto a cyclic one has also been used for modeling the processivity of
molecular motors, where the linear coordinate corresponds to the position of
the motor while the alternating states correspond to different motor
conformations.24, 25

At steady state, the net incoming and outgoing fluxes at each piston state in
Figure 2E should cancel each other (seen more vividly in the collapsed diagram
in Figure 2F), namely,

$\displaystyle\left(k^{\mathrm{d}\rightarrow\mathrm{u}}_{\mathrm{f}}+k_{\text{b}}\right)\pi_{\text{d}}=\left(k^{\mathrm{u}\rightarrow\mathrm{d}}_{\mathrm{f}}+k_{\text{b}}\right)\pi_{\text{u}}.$
(11)

Substituting the expressions for forward stepping rates (eqs 7 and 8) into eq
11 and additionally imposing the probability normalization constraint
($\pi_{\text{d}}+\pi_{\text{u}}=1$), we can solve for $\pi_{\text{d}}$ and
$\pi_{\text{u}}$ to obtain

$\displaystyle\pi_{\text{d}}$ $\displaystyle=\frac{1+\text{e}^{-\beta(\Delta
W+\Delta F)}}{2(1+\cosh(\beta\Delta F)\,\text{e}^{-\beta\Delta W})},$ (12)
$\displaystyle\pi_{\text{u}}$ $\displaystyle=\frac{1+\text{e}^{-\beta(\Delta
W-\Delta F)}}{2(1+\cosh(\beta\Delta F)\,\text{e}^{-\beta\Delta W})}.$ (13)

Notably, in the absence of external drive ($\Delta W=0$), piston state
occupancies follow the Boltzmann distribution, that is,
$(\pi_{\text{d}}/\pi_{\text{u}})_{\text{eq}}=\text{e}^{-\beta\Delta
F}=f^{-1}$, suggesting that at equilibrium the piston will predominantly dwell
in the expanded state. Conversely, as can be seen in Figure 3A, when the work
per step exceeds $\Delta F$ by several $k_{\text{B}}T$, the occupancies of the
two piston states become equal to each other. This happens because at large
$\Delta W$ values forward ratchet stepping becomes very unlikely and the
dynamics proceeds only through backward steps with a rate $k_{\text{b}}$ which
is identical for both compressive and expansive steps. As will be shown in
section 3.4, suppressing this equilibrium bias set by $\Delta F$ is essential
for achieving efficient proofreading, analogous to the need for driving the
transitions between the two enzyme-substrate intermediates in Hopfield’s
scheme (Figure 1A). 9

Figure 3: Nonequilibrium features of the engine-piston coupling. (A) Steady
state probability ratio of compressed (“d”) and expanded (“u”) piston states
and (B) normalized net rate of backward stepping
($k_{\text{net}}/k_{\text{b}}$) as a function of the work per step ($\Delta
W$) for different choices of the ligand compression energy ($\Delta F$). The
$\Delta W_{1/2}$ expressions stand for the values of $\Delta W$ where the
corresponding value on the $y$-axis is 0.5 (Supporting Information section
B.2). Negative $\Delta W$ values are not considered as they further increase
the undesired bias in piston state occupancies.

With the steady state probabilities known, we can now substitute them into eq
9 to find the net rate at which the weight goes down, obtaining

$\displaystyle k_{\text{net}}=\frac{\left(1-\text{e}^{-2\beta\Delta
W}\right)k_{\text{b}}}{1+\cosh(\Delta F)\text{e}^{-\beta\Delta W}}.$ (14)

As expected, $k_{\text{net}}$ vanishes at equilibrium ($\Delta W=0$), and
asymptotes to $k_{\text{b}}$ at large $\Delta W$ values, as shown in Figure
3B. The knowledge of $k_{\text{net}}$ allows us to calculate the power ($P$)
dissipated for the maintenance of the nonequilibrium steady state.
Specifically, since $k_{\text{net}}$ is the rate at which the weight goes down
and $\Delta W$ is the dissipation per step, the power $P$ becomes their
product, namely,

$\displaystyle P$ $\displaystyle=k_{\text{net}}\Delta W.$ (15)

The formalism developed in this section for characterizing the steady state
behavior of the system will be used as a basis for defining the different
performance metrics of the model in section 3.3.

###  3.2 Thermodynamic Constraints Make Fidelity Enhancement Unattainable in
the Absence of External Driving

In order to implement a thermodynamically consistent coupling between the
engine and the allosteric enzyme, we need to consider the full diversity of
possible enzyme states,26 and not just the dominant ones depicted in Figure
1B. Therefore, in this section we provide a comprehensive discussion of the
enzyme in an equilibrium setting before introducing its coupling to the
engine.

The network diagram of all possible enzyme states is depicted in Figure 4. As
can be seen, each of the twelve states are defined by enzyme’s catalytic
activity and whether or not a ligand and a right/ wrong substrate are bound to
the enzyme. Following the principle of microscopic reversibility, 27 we assign
non-zero rate constants to the transitions between enzyme states. Only the
product formation (with rate $r$) is taken to be an irreversible reaction
under the assumption that the system is open where the formed products are
taken out and an influx of new substrates is maintained. Since in our model
neither the enzyme nor the substrates carry an energy source, the choice of
the different rate constants cannot be arbitrary. Specifically, the cycle
condition needs to be satisfied for each closed loop of the diagram, requiring
the product of rate constants in the clockwise direction to equal the product
in the counterclockwise direction (Supporting Information section C.1).23

Figure 4:  Network diagram of enzyme states and transitions between them.
Right (“R”) and wrong (“W”) substrates are depicted in blue and red,
respectively. The orange circle represents the ligand (“L”). Active (“A”) and
inactive (“I”) enzymes are shown in green and gray, respectively.

With these equilibrium restrictions imposed on the rate constants, we can show
that when the ligand concentration is held fixed ($[\text{L}](t)$ = const) the
fidelity of the enzyme cannot exceed that defined by the ratio of the off-
rates, namely, $k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}$
(see Supporting Information section C.2). What allows the enzyme to beat this
equilibrium limit of fidelity without direct coupling to hydrolysis is the
cyclic alteration of the ligand concentration between low and high values
(thus, $[\text{L}](t)\neq$ const). In our model, we achieve this cyclic
alteration through the ratchet and pawl engine driving the piston actions - a
choice motivated by our objective to provide an explicit treatment of energy
management. We note, however, that fidelity enhancement can be achieved
irrespective of the driving agency as long as the cyclic alteration of ligand
concentration is maintained at a certain “resonance” frequency, the presence
of which we demonstrate in section 3.4.

###  3.3 Coupling the Engine to the Enzyme Gives the Full Description of the
Piston Model

Having separately introduced the driving mechanism in section 3.1 and the
allosteric enzyme with the full diversity of its states in section 3.2, we now
couple the two together to obtain the full driven version of the piston model,
shown in Figure 5A. The coupling is achieved by exposing the ligand binding
site of the enzyme to the piston compartment where the activator ligand is
present. The enzyme can therefore “sense” the state of the piston (and,
thereby, the effects of driving) through the induced periodic changes in the
ligand concentration.

In the absence of enzyme coupling, the network diagram capturing the
nonequilibrium dynamics of the system was an infinite one-dimensional chain
(Figure 2D), where each discrete state was defined by the vertical position of
the weight ($z_{n}$) and the state of the piston (“u” or “d”). In the layout
where the engine and the enzyme are coupled, the full specification of the
system state now requires three items: the position of the weight ($z_{n}$),
the piston state (“u” or “d”), and the state of the enzyme (one of the 12
possibilities). By converting the three-dimensional view of the enzyme state
network (Figure 4) into its planar equivalent, we represent the nonequilibrium
dynamics of this coupled layout again through an infinite chain, but this time
each slice at a fixed weight position ($z_{n}$) corresponding to the planar
view of the enzyme state network (Figure 5B). The slices alternate between the
compressed and expanded piston states (dark and light blue circles,
respectively), with high and low ligand concentrations used in the transition
network inside each slice.

Arrows between the slices (not all of them shown for clarity) represent the
forward and backward steps of the ratchet. Crucially, as a consequence of
coupling, the rates of forward stepping now depend on the state of the enzyme.
In particular, when the ligand is bound to the enzyme, it no longer exerts
pressure on the piston and therefore, in those cases, the forward stepping
rates become simply

$\displaystyle
k^{\mathrm{u}\rightarrow\mathrm{d,L}}_{\mathrm{f}}=k^{\mathrm{d}\rightarrow\mathrm{u,L}}_{\mathrm{f}}=k_{\mathrm{b}}\text{e}^{-\beta\Delta
W},$ (16)

where the superscript “L” indicates that the ligand is bound (orange circles
in Figure 5B). We note that in the general case with $N$ ligands, the pressure
would drop down to that of $(N-1)$ ligands upon ligand binding,
correspondingly altering the rates of forward stepping (see Supporting
Information section D.1 for details). This adjustment of forward rates is
essential for the thermodynamic consistency of coupling the engine to the
enzyme. Specifically, it ensures that any cycle of transitions that brings the
enzyme and the weight back into their original states is not accompanied by
dissipation, consistent with the fact that in the piston model energy is spent
only when there is a net lowering of the weight. As a demonstration of this
feature, consider the cycle in Figure 5C which is extracted from the larger
network. Using the expression of forward stepping rates in eqs 7 and 16, we
can write the cycle condition for this sub-network as

$\displaystyle\frac{k^{\mathrm{u}\rightarrow\mathrm{d,L}}_{\mathrm{f}}\times\ell^{\mathrm{A}}_{\mathrm{off}}\times
k_{\mathrm{b}}\times\ell^{\mathrm{A}}_{\mathrm{on}}[\text{L}]_{\text{u}}}{k_{\mathrm{b}}\times\ell^{\mathrm{A}}_{\mathrm{off}}\times
k^{\mathrm{u}\rightarrow\mathrm{d}}_{\mathrm{f}}\times\ell^{\mathrm{A}}_{\mathrm{on}}[\text{L}]_{\text{d}}}=\frac{[\text{L}]_{\text{u}}f}{[\text{L}]_{\text{d}}}=1,$
(17)

where the equality
$f=V_{\mathrm{u}}/V_{\mathrm{d}}=[\text{L}]_{\text{d}}/[\text{L}]_{\text{u}}$
was used. The fact that the products of rate constants in clockwise and
counterclockwise directions are identical shows that no dissipation occurs
when traversing the cycle.

Now, to study how driving affects the proofreading performance of the piston
model, we need to obtain the steady state probabilities of the different
enzyme states. To that end, we convert the full network diagram into an
equivalent form shown in Figure 5D, where we have eliminated the position of
the weight ($z_{n}$), akin to the earlier treatment of the uncoupled engine in
Figure 2E. Note that the transitions between the two slices again represent
piston compression and expansion events driven by a force $\Delta\mu=2\Delta
W$, as in eq 10. The steady state probabilities $\pi_{i}$ of the 24 different
states in Figure 5D (12 enzyme states $\times$ 2 piston states) can be
obtained from the set of all rate constants, the details of which we discuss
in Supporting Information section D.2. With these probabilities known, we
calculate the rate of energy dissipation ($P$), speed of forming right
products ($v_{\mathrm{R}}$), and fidelity ($\eta$) as

$\displaystyle P$
$\displaystyle=\underbrace{\sum_{i=1}^{24}\left(k_{\mathrm{b}}-k_{\mathrm{f}}^{(i)}\right)\pi_{i}}_{k_{\mathrm{net}}}\times\Delta
W,$ (18) $\displaystyle v_{\mathrm{R}}$ $\displaystyle=\sum_{i\in
S^{\mathrm{A}}_{\mathrm{R}}}\pi_{i}\times r,$ (19) $\displaystyle\eta$
$\displaystyle=\frac{v_{\mathrm{R}}}{v_{\mathrm{W}}}=\frac{\sum_{i\in
S^{\mathrm{A}}_{\mathrm{R}}}\pi_{i}}{\sum_{i\in
S^{\mathrm{A}}_{\mathrm{W}}}\pi_{i}},$ (20)

where $k_{\mathrm{f}}^{(i)}$ is the rate constant of making a forward step
from the $i^{\mathrm{th}}$ state ($1\leq i\leq 24$), while
$S_{\mathrm{R}}^{\mathrm{A}}$ and $S_{\mathrm{W}}^{\mathrm{A}}$ are the sets
of catalytically active enzyme states with a right and wrong substrate bound,
respectively.

One significant downside of using these “raw” metrics in the numerical
evaluation of the model performance is their high sensitivity to the
particular choices of parameter values. We therefore introduce their scaled
alternatives which we will use for the numerical studies in sections 3.4 and
3.5. Specifically, as a measure of energetic efficiency, we use the
dissipation per right product formed, defined as

$\displaystyle\varepsilon=\frac{P}{v_{\mathrm{R}}}.$ (21)

This way, the metric of energetics has units of $k_{\mathrm{B}}T$ and is
independent from the choice of absolute timescale. Then, as a dimensionless
metric of speed, we introduce the normalized quantity

$\displaystyle\nu=\frac{v_{\mathrm{R}}}{v_{\text{R}}^{\text{MM}}},$ (22)

which represents the fraction by which the rate of forming right products in
the proofreading setting ($v_{\mathrm{R}}$) is slower than that in the simple
Michaelis-Menten scheme ($v_{\text{R}}^{\text{MM}}$) where the allosteric
effects are absent. This normalizing Michaelis-Menten speed is given in terms
of the model parameters via

$\displaystyle
v_{\text{R}}^{\text{MM}}=\frac{\frac{k^{\mathrm{I}}_{\mathrm{on}}[\text{R}]}{k_{\mathrm{off}}^{\mathrm{R}}+r}}{1+\frac{k^{\mathrm{I}}_{\mathrm{on}}[\text{R}]}{k_{\mathrm{off}}^{\mathrm{R}}+r}+\frac{k^{\mathrm{I}}_{\mathrm{on}}[\text{W}]}{k_{\mathrm{off}}^{\mathrm{W}}+r}}\times
r.$ (23)

Next, we define the proofreading index $\alpha$ as a fidelity metric which
represents the degree to which the fidelity is amplified in multiples of
$k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}$ over its
Michaelis-Menten value ($\eta_{\text{MM}}$), that is,

$\displaystyle\eta$
$\displaystyle=\underbrace{\left(\frac{k_{\mathrm{off}}^{\mathrm{W}}+r}{k_{\mathrm{off}}^{\mathrm{R}}+r}\right)}_{\eta_{\text{MM}}}\left(\frac{k_{\mathrm{off}}^{\mathrm{W}}}{k_{\mathrm{off}}^{\mathrm{R}}}\right)^{\alpha},$
(24) $\displaystyle\alpha$
$\displaystyle=\frac{\ln\eta-\ln\eta_{\text{MM}}}{\ln\left(\frac{k_{\mathrm{off}}^{\mathrm{W}}}{k_{\mathrm{off}}^{\mathrm{R}}}\right)}.$
(25)

Note that the proofreading index of Hopfield’s scheme is
$\alpha_{\text{Hopfield}}=1$, as it involves a single proofreading
realization. Also, since in the absence of external driving the highest
fidelity is
$\eta_{\text{eq}}^{\text{max}}=k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}$,
the corresponding upper limit in the proofreading index becomes
$\alpha_{\text{eq}}=1-\ln\eta_{\text{MM}}/\ln\eta_{\text{eq}}^{\text{max}}$.

As a final descriptor of piston model’s nonequilibrium behavior, we introduce
the fraction of returned work ($\kappa$) defined as the ratio of the rate at
which the ligand performs work on the piston upon expansion to the rate at
which the piston performs work on the ligand upon compression. We calculate
$\kappa$ via

$\displaystyle\kappa=-\frac{\sum_{i\in
S_{\text{d}}}\left(k_{\mathrm{b}}+k_{\mathrm{f}}^{(i)}\right)\pi_{i}\Delta
F_{\text{d}\rightarrow\text{u}}^{(i)}}{\sum_{i\in
S_{\text{u}}}\left(k_{\mathrm{b}}+k_{\mathrm{f}}^{(i)}\right)\pi_{i}\Delta
F_{\text{u}\rightarrow\text{d}}^{(i)}},$ (26)

where $S_{\text{d}}$ and $S_{\text{u}}$ are the sets of states where the
piston is compressed and expanded, respectively. The negative sign is
introduced to account for the fact that the ligand free energy decreases upon
piston expansion (i.e. the system gets the work back). In the absence of
enzyme coupling (section 3.1), this ratio was 1 because the ligand constantly
exerted pressure on the piston. With enzyme coupling, however, the work
performed on the ligand upon compression may not be fully returned since with
some probability the ligand will be bound to the enzyme and exert no pressure
on the piston during expansion. We therefore expect $\kappa$ to be generally
less than 1, indicating a net rate of performing work on the ligand in the
nonequilibrium setting.

Having defined analytical expressions for the key model performance metrics,
we now proceed to studying their graded changes and the trade-offs between
them numerically.

Figure 5: Full and thermodynamically consistent treatment of the piston model
of proofreading. (A) Schematic representation of the full model, with the
ratchet and pawl engine coupled to the enzyme. (B) Network diagram of the full
model. Each slice of the diagram represents the planar view of the enzyme
state network, with the alternating colors corresponding to the compressed
(dark blue) and expanded (light blue) states of the piston. Ligand-bound
enzyme states are marked with an orange circle. The horizontal arrows
connecting the slices stand for forward and backward ratchet steps. Only those
at the outer edges are shown for clarity; however, transitions are present
between all horizontally neighboring enzyme states. Also for clarity, the
stepping rate constants are shown only at two of the outer edges where the
ligand is either unbound (bottom edge) or bound to the enzyme (top edge). The
hanging weight at different vertical positions is displayed below the diagram
to symbolize energy expenditure as it gets lowered with a net rate
$k_{\mathrm{net}}$. (C) Sub-network of the full diagram in panel B where the
state of the system is unchanged after doing a cyclic traversal. The red arrow
with a cross on top indicates that the cycle condition holds in the sub-
network. (D) The finite-state equivalent of the full network in panel B with
the weight position ($z_{n}$) eliminated. Red arrows indicate the driving with
a force $\Delta\mu=2\Delta W$.

###  3.4 Energy-Speed-Fidelity Trade-Off in the Piston Model

Because of its mechanical construction, the piston model of proofreading has a
distinguishing feature - in it the external driving mechanism is physically
separated from the allosteric enzyme. This feature allows us to independently
examine how tuning the “knobs” of the engine and varying the kinetic
parameters of the enzyme alter the performance of the model.

Figure 6: Parametric studies on the changes in the piston model performance in
response to tuning the “knobs” of the engine. (A,B) Variations in the
proofreading index (A) and speed (B) as the rate of backward stepping
($k_{\mathrm{b}}$) and the work per step ($\Delta W$) are tuned. The dotted
line corresponds to the value of $\Delta W$ equal to the ligand free energy
change upon compression ($\Delta F$). (C) Variations in the proofreading index
when the high ligand concentration ($[\text{L}]_{\text{d}}$) and the
compression factor ($f$) are tuned. $R_{\text{d}}^{\text{I}}$ represents the
ligand dissociation constant in the inactive enzyme state. The red dot
indicates the pair of $[L]_{\text{d}}$ and $f$ values used in the studies of
the other panels. (D) Fidelity-speed trade-off as $k_{\mathrm{b}}$ is
continuously varied for different choices of $\Delta W$ (gradient arrow shows
the direction of increase). The dotted black lines connect the highest
fidelity and speed values as $\Delta W$ is tuned. Between these dotted lines
fidelity and speed are negatively correlated. (E) Relation between fidelity
and fraction of returned work for discrete choices of $\Delta W$ and
continuously tuned $k_{\mathrm{b}}$ values (the gradient arrow indicates
increasing $\Delta W$). (F) Fidelity-dissipation trade-off obtained by
continuously tuning $\Delta W$ for discrete choices of the hopping rate
($k_{\mathrm{b}}$). The gradient arrow indicates the direction of increasing
$k_{\mathrm{b}}$. The red dotted curve corresponds to the case with resonance
$k_{\mathrm{b}}$.

We begin our numerical analysis by first exploring the effects of external
driving, where the tuning “knobs” include the rate of backward stepping
($k_{\mathrm{b}}$), the work per step ($\Delta W$), the ligand concentration
in the compressed piston state ($[\text{L}]_{\text{d}}$) and the compression
factor ($f=[\text{L}]_{\text{d}}/[\text{L}]_{\text{u}}$). Choosing a set of
enzyme’s kinetic parameters which make proofreading possible (see Supporting
Information section D.3 for the full list of parameters), we keep them fixed
for the rest of the analysis. We conduct the first parametric study by tuning
$k_{\mathrm{b}}$ and $\Delta W$ and evaluating the proofreading index (Figures
6A). As anticipated, the proofreading index does not exceed its equilibrium
limit in the absence of driving ($\Delta W=0$). This expected feature can be
paralleled by Brownian motors where purely equilibrium fluctuations of the
motor’s energy landscape are unable to generate directed motion. 28 In
addition, the proofreading index achieves its highest value if $\Delta W$ is
comparable to or larger than the ligand compression energy $\Delta F$, and if
the backward hopping rate $k_{\mathrm{b}}$ is at its “resonance” value. The
presence of a “resonance” hopping rate is intuitive since if piston actions
take place very slowly then the fidelity will be reduced due to the small but
nonzero rate of forming unfiltered products (i.e. “leakiness”) in the quasi-
equilibrated enzyme states. And, conversely, if piston actions take place too
rapidly, then the activator ligand will almost always be bound to the enzyme,
preventing the realization of multiple substrate discrimination layers through
sequential enzyme activation and inactivation. We note that analogous
resonance responses were also identified for Brownian particles which attain
their highest nonequilibrium drift velocity in a ratchet-like potential
landscape when the temperature29 or the landscape profile30 are temporally
modulated at specific resonance frequencies. A similar feature is present in
Hopfield’s model as well; namely, optimal proofreading is attained only when
the rate of hydrolysis is neither too low, nor too high.12 Interestingly, when
the driving is hard enough ($\Delta W\gtrsim\Delta F$) and the backward
hopping rate is close to its resonance value, the fidelity of the piston model
beats the Hopfield limit ($\alpha=1$) and raises the question of the largest
attainable fidelity, which we discuss in the next section.

Trends similar to those for the proofreading index are observed for the speed
of forming right products as well (Figure 6B). Specifically, product formation
is very slow in the absence of driving and increases monotonically with
$\Delta W$, until plateauing when $\Delta W\gtrsim\Delta F$. Also, the highest
speed is achieved at a resonance $k_{\mathrm{b}}$ value which is different
from that of the proofreading index. The existence of such a resonance
frequency is again intuitive, since at fast rates of piston actions the enzyme
is predominantly active and unable to bind new substrates, while at slow rates
the enzyme activation for catalysis via piston compression happens very
rarely. Notably, since the enzyme parameters were chosen in a way so as to
yield high fidelities, the largest speed value is substantially lower than the
corresponding speed for a single-step Michaelis-Menten enzyme
($\nu_{\text{max}}\approx 10^{-2}$).

In the last parametric study, we explore how the choice of the high and low
ligand concentrations affects the performance of the model. To that end, we
tune the high ligand concentration ($[\text{L}]_{\text{d}}$) and the
compression factor ($f=[\text{L}]_{\text{d}}/[\text{L}]_{\text{u}}$), and
evaluate the highest proofreading index at the resonant $k_{\mathrm{b}}$ value
with $\Delta W>\Delta F$. As we can see, large fidelity enhancements are
achieved when $[\text{L}]_{\text{d}}$ is comparable to or larger than the
ligand dissociation constant in the inactive enzyme state
($R_{\text{d}}^{\text{I}}$), which is necessary to activate the enzyme upon
piston compression. In addition, the compression factor needs to be large
enough (or, equivalently, the ligand concentration in the expanded piston
state should be low enough) so as to inactivate the enzyme when piston enters
its expanded state. This requirement of having a large free energy difference
between the compressed and expanded piston states ($\beta\Delta F=\ln(f)\gg
1$) can be paralleled with a similar condition in Hopfield’s model where for
optimal proofreading the energy of the activated enzyme-substrate complex
needs to be much larger than that of the inactive complex.

Knowing separately how tuning the engine “knobs” affects the fidelity and
speed, we now explore the trade-offs between the model’s performance metrics
as we vary the driving parameters $k_{\mathrm{b}}$ and $\Delta W$, while
holding the high and low ligand concentrations at fixed values (the red dot in
Figure 6C). We start with the trade-off between fidelity and speed, depicted
in Figure 6D, where we continuously tune the hopping rate $k_{\mathrm{b}}$ for
different choices of the driving force $\Delta W$. As expected from the
results of the individual parametric studies in Figure 6A and B, both fidelity
and speed increase monotonically with $\Delta W$. Also, since the values of
the hopping rate $k_{\mathrm{b}}$ that maximize fidelity and speed are not
identical, these two performance metrics are negatively correlated in the
range of $k_{\mathrm{b}}$ values defined by the two different resonance rates
(region between the dotted lines in Figure 6D), but are positively correlated
otherwise. Variations in the metrics in the region of their negative
correlation, however, are moderate, suggesting that for an allosteric enzyme
that has been optimized for doing proofreading, the largest speed and fidelity
could be achieved at similar external driving conditions.

Next, we consider the relation between fidelity and fraction of work returned,
shown in Figure 6E. As can be seen, no fidelity enhancement is achieved when
$\kappa$ is close to $1$ which happens either in the absence of driving
(lighter curves) or in the presence of driving, provided that the hopping rate
is very fast. On the other hand, $\kappa$ is much less than $1$ at the peak
fidelity which is achieved when the hopping rate is at its resonance value and
when driving is large ($\Delta W\gtrsim\Delta F$). Overall, this trade-off
study demonstrates that irreversible work performed on the ligand is a
required feature for the attainment of fidelity enhancement in the piston
model.

Lastly, we look at how fidelity varies with energy dissipation, with the
latter characterized through the energy expended per right product
($\varepsilon$). The results of the trade-off study are shown in Figure 6F.
There, the driving force is continuously tuned for different choices of the
hopping rate. As can be seen, there is a minimum dissipation per product
required to attain the given level of performance. This minimum dissipation
(the first intercept at a given y-level) is achieved when the hopping rates
are less than the corresponding resonant values (the lighter curves on the
left side of the dotted red curve). Additionally, for a given hopping rate,
increasing the driving force ($\Delta W$) could lead to an increased
proofreading performance and a decreased dissipation per product up a critical
point where the performance metric reaches its saturating value (horizontal
region), demonstrating how increasing the driving force could in fact improve
the energetic efficiency of proofreading.  We note here that the minimum
$\varepsilon$ values needed for significant proofreading are $\sim
10^{3}-10^{4}\,k_{\mathrm{B}}T$ in Figure 6F which is $\sim 2$ orders of
magnitude higher than what is calculated for translation by the ribosome.12
This low energetic efficiency can be a consequence of our particular parameter
choice for the study as well as the performance limitations of our engine
design, the investigation of which we leave to future work.

###  3.5 Up to Three Proofreading Realizations are Available to the Piston
Model

In the previous section, we chose a set of kinetic rate constants for the
enzyme and, keeping them fixed, explored the effects of tuning the external
driving conditions on the performance of the model. In this section, we
explore the parameter space from a different angle, namely, we study how
tuning the enzyme’s kinetic parameters changes the model performance under
optimal driving conditions. Since there are more than a dozen rates defining
the kinetic behavior of the enzyme, it is impractical to probe their
individual effects. Instead, we choose to vary two representative parameters
about the effects of which we have a prejudice. These include the rate of
substrate binding to the active enzyme ($k^{\mathrm{A}}_{\mathrm{on}}$) and
the unbinding rate of wrong substrates ($k_{\mathrm{off}}^{\mathrm{W}}$). We
know already from Hopfield’s analysis that for efficient proofreading the
direct binding of substrates to the active enzyme state should be very slow.
Therefore, we expect the proofreading performance to improve as
$k^{\mathrm{A}}_{\mathrm{on}}$ is reduced. We also expect the minimum
requirement for $k^{\mathrm{A}}_{\mathrm{on}}$ to be lower for larger
$k_{\mathrm{off}}^{\mathrm{W}}$ values to ensure that wrong substrates do not
enter through the unfiltered pathway. 9

With these expectations in mind, we performed a parametric study to find the
highest fidelity, the results of which are summarized in Figure 7A. There we
varied $k^{\mathrm{A}}_{\mathrm{on}}$ for several choices of
$k_{\mathrm{off}}^{\mathrm{W}}$, and for each pair numerically optimized over
the enzyme’s remaining kinetic rates and external driving conditions to get
maximum fidelity (see Supporting Information section D.4 for implementation
details). As expected, the highest attainable fidelity decreases monotonically
with increasing “leakiness”
($k^{\mathrm{A}}_{\mathrm{on}}/k^{\mathrm{I}}_{\mathrm{on}}$), and the minimum
requirement on $k^{\mathrm{A}}_{\mathrm{on}}$ decreases with increasing
$k_{\mathrm{off}}^{\mathrm{W}}$.

Interestingly, we also see that for small enough leakiness, the piston model
manages to perform proofreading (i.e. enhance the fidelity by a factor of
$k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}$) up to three
times, as $\alpha_{\text{max}}\approx 3$ (Figure 7A).  To understand this
unexpected feature, we identified the dominant trajectory that the system
would take to form a wrong product for the case where
$k^{\mathrm{A}}_{\mathrm{on}}/k^{\mathrm{I}}_{\mathrm{on}}=10^{-12}$ (Figure
7B, see Supporting Information section D.5 for details). As we can see, after
initial binding the wrong substrate indeed passes through three different
proofreading filters, and these are realized efficiently because the
transitions between intermediate states are much slower than the rate of
substrate unbinding. The first filter occurs right after piston compression,
while the enzyme is waiting for the activator ligand to bind (#1). We note
that this particular filter is made possible due to the presence of
alternative piston states (equivalently, alternative environments that the
enzyme could “sense”). The remaining two filters (#2 and #3) take place while
the ligand-bound enzyme is waiting to get activated and while the active
enzyme is waiting to turn the wrong substrate into a product, respectively.
The presence of these two filters is purely a consequence of allostery.
Importantly, the $\alpha_{\text{max}}\approx 3$ result in Figure 7 represents
the theoretical upper limit of the model’s proofreading index – a feature that
we justify analytically in Supporting Information section D.5.

In light of this analysis, we can now explain why the pedagogically simplified
version of the model introduced in section 2 achieved only a single
proofreading realization. There we made the implicit assumption that ligand
binding after piston compression and enzyme activation after ligand binding
took place instantly. Because of this, proofreading filters #1 and #2 were not
realized, leaving filter #3 as the only available one which we showed in
Figure 1B.

Figure 7:  Proofreading performance of the piston model under optimized enzyme
parameters and external driving conditions. (A) The highest proofreading index
($\alpha$) available to the piston model as a function of leakiness
($k^{\mathrm{A}}_{\mathrm{on}}/k^{\mathrm{I}}_{\mathrm{on}}$) for different
choices of $k_{\mathrm{off}}^{\mathrm{W}}$.  (B) The dominant trajectory that
the system takes to form a wrong product in the case where
$k^{\mathrm{A}}_{\mathrm{on}}/k^{\mathrm{I}}_{\mathrm{on}}=10^{-12}$. Numbers
1, 2, 3 stand for the different proofreading filters along the trajectory. The
dotted arrows indicate that the respective rates are much slower than the
substrate unbinding rate $k_{\mathrm{off}}^{\mathrm{W}}$ (see Supporting
Information section D.5 for their numerical values for the
$k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}=100$ case).

##  4 Discussion and Conclusions

A distinctive feature of kinetic proofreading is that it is a nonequilibrium
mechanism, by virtue of which its operation needs to involve energy
expenditure.9, 10 Mechanical work, being an intuitive representation of energy
expenditure, has been used in the past to elucidate important physical
concepts such as information-to-energy conversion in the thought experiment by
Szilard,22 or the mechanical equivalence of heat in Joule’s apparatus.31 Yet,
a similar demonstration of how mechanical work could be harnessed in a graded
fashion to beat the equilibrium limit in substrate discrimination fidelity has
been lacking. Our aim in this work was to offer such a demonstration through
the mechanically designed piston model of proofreading.

We started off by providing the conceptual picture of the piston model, with
its constituents having direct parallels with Hopfield’s original proofreading
scheme9 (Figure 1). The key idea of the model was to replace the nucleotide
hydrolysis step present in Hopfield’s scheme with piston compression which
served an identical role of activating the enzyme but in our case achieved
through allostery and mechanical work. Just like in the case of biological
proofreading, where hydrolysis itself cannot lead to fidelity enhancement
unless the nucleotide triphosphates are held at fixed out-of-equilibrium
chemical potentials, in the case of piston model too, the compressive and
expansive actions of the piston cannot result in proofreading unless they are
driven by an energy-consuming engine. Motivated by Feynman’s ratchet and pawl
mechanism, 21 we then proposed a dissipative mechanical engine to drive the
cyclic piston actions, which maintained the nonequilibrium distribution of
enzyme states necessary for achieving proofreading. The function of this
engine can be paralleled to that of the ATP synthase in the cell whose
constant operation maintains a finite chemical potential of ATP, which
different biochemical pathways can then take advantage of.

To study how the cyclic variations in ligand concentration generated by the
engine alter the occupancies of enzyme states, we performed a
thermodynamically consistent coupling between the engine and the enzyme
(Figure 5). There we considered the full diversity of states that the enzyme
could take and, importantly, the feedback mechanism for the engine to “sense”
the state of the enzyme. The accounting of this latter feature, which makes
the piston model an example of a bi-partite system, 32, 33 was motivated by
our aim of proposing a framework where we could consistently calculate the
total dissipation as opposed to only the minimum dissipation needed for
maintaining the nonequilibrium steady state of the enzyme (without considering
the driving engine). 12, 34, 35 Although the dissection of different
contributions to dissipation and their interconnectedness was not among the
objectives of our work, the framework that we proposed in our model can serve
as a basis for additional studies of periodically driven molecular systems
(e.g. Brownian clocks or artificial molecular motors) where the driving
protocol and thermodynamics are of importance. 36, 37, 38 As noted earlier,
however, in the presence of a periodically changing ligand concentration, the
allosteric enzyme could perform proofreading irrespective of the driving
agency, which suggests a possible biochemical mechanism of fidelity
enhancement without the direct coupling of the enzyme state transitions to
hydrolysis.

Having explicit control over the “knobs” of the mechanical engine, we then
probed the performance of the model under different driving conditions. We
found that both speed and fidelity increased as we tuned up the mass of the
hanging weight, until plateauing at a point where the free energy bias of the
expanded piston state was fully overcome ($\Delta W\gtrsim\Delta F$), beyond
which increasing the weight only increased the dissipation without improving
the model performance (Figure 6A,B). This result can be paralleled with the
presence of a minimum threshold for the strength of driving in Hopfield’s
model, past which the highest fidelity becomes attainable.9 In addition, we
found that in the piston model there is a “resonance” rate of piston actions
which maximizes fidelity, analogous to the similar feature of Hopfield’s
scheme where both very fast and very slow rates of hydrolysis reduce the
quality of proofreading.12

The tunable control over the driving parameters also allowed us to study the
trade-off between fidelity, speed and energy spent per right product. These
studies revealed that the correlation between speed and fidelity could be both
positive and negative when varying the rate of driving. Notably, theoretical
investigations of translation by the Escherichia coli ribosome under
Hopfield’s scheme identified a similar behavior for the fidelity-speed
correlation in response to tuning the GTP hydrolysis rate, with the
experimentally measured values being in the negative correlation (i.e. trade-
off) region.17 In contrast to the ribosome study, however, where the two
metrics vary by several orders of magnitude in the trade-off region, in the
piston model the variations in fidelity and speed in the negative correlation
region are moderate (Figure 6D), calling for additional investigations of the
underlying reasons behind this difference and search for the realization of
the latter advantageous behavior in biological proofreading systems.
Furthermore, our studies showed that the minimum dissipation required to reach
the given level of fidelity was achieved for hopping rates necessarily lower
than their resonance values, and that increasing the work performed per step
(analogously, the chemical potential of ATP) could actually improve the
energetic efficiency of the model – features that again motivate the
identification of their realization in biochemical systems.

In the end, we explored the limits in the proofreading performance of the
piston model for various choices of the allosteric enzyme’s “leakiness”
($k^{\mathrm{A}}_{\mathrm{on}}/k^{\mathrm{I}}_{\mathrm{on}}$) and the ratio of
the wrong and right substrate off-rates
($k_{\mathrm{off}}^{\mathrm{W}}/k_{\mathrm{off}}^{\mathrm{R}}$). We found that
the trends for the highest available fidelity matched analogously with the
features of Hopfield’s original scheme, suggesting their possible ubiquity for
general proofreading networks. More importantly, our analysis revealed that
the piston model could do proofreading not just once but up to three times in
the limit of very low leakiness, despite the fact that energy consumption
takes place during a single piston compression. This is in contrast to the
typical involvement of several energy consumption instances in multistep
proofreading schemes which manage to beat the Hopfield limit of fidelity, as,
for example, in the cases of the T-cell or MAPK activation pathways which
require multiple phosphorylation reactions. 6, 34, 5 Our finding therefore
suggests the possibility of achieving several proofreading realizations with a
single energy consuming step by leveraging the presence of multiple inactive
intermediates intrinsically available to allosteric molecules. We would like
to mention here that the presence of a similar feature was also experimentally
demonstrated recently for the ribosome which was shown to use the free energy
of a single GTP hydrolysis to perform proofreading twice after the initial
tRNA selection – first, at the $\text{EF-Tu}\cdot\text{GDP}$-bound inactive
state and second, at the EF-Tu-free active state.39

In the presentation of the piston model we focused on the thermodynamic
consistency of the framework for managing the energy dissipation and did not
consider strategies for improving the performance of the mechanism. One such
possibility that can be considered in future work is to use a more elaborate
design for the ratchet and pawl engine with alternating activation barriers
for pawl hopping which would allow to have different rates of piston
compression and expansion, analogous to how hydrolysis and condensation
reactions generally occur with different rates in biological proofreading. 17,
40 Another avenue is to consider alternative ways of allocating the mechanical
energy dissipation across the different ratchet transition steps, similar to
how optimization schemes of allocating the free energy of ATP hydrolysis were
studied for molecular machine cycles.41 Incorporating these additional
features would allow us to probe the performance limits of the piston model
and compare them with the fundamental limits set by thermodynamics.42

## Acknowledgements

We thank Tal Einav, Erwin Frey, Christina Hueschen, Sarah Marzen, Arvind
Murugan, Manuel Razo-Mejia, Matt Thomson, Yuhai Tu, Jin Wang, Jerry Wang, Ned
Wingreen and Fangzhou Xiao for fruitful discussions. We also thank Haojie Li
and Dennis Yatunin for their input on this work, Alexander Grosberg, David
Sivak, and Pablo Sartori for providing valuable feedback on the manuscript,
and Nigel Orme for his assistance in making the illustrations. This work was
supported by the National Institutes of Health through the grant 1R35
GM118043-01 (MIRA), and the John Templeton Foundation as part of the
Boundaries of Life Initiative grants 51250 and 60973.

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