# Thermodynamic uncertainty relation to assess biological processes

Yonghyun Song  Korea Institute for Advanced Study, Seoul 02455, Korea
Changbong Hyeon<EMAIL_ADDRESS>Korea Institute for Advanced Study, Seoul
02455, Korea

###### Abstract

We review the trade-offs between speed, fluctuations, and thermodynamic cost
involved with biological processes in nonequilibrium states, and discuss how
optimal these processes are in light of the universal bound set by the
thermodynamic uncertainty relation (TUR). The values of the uncertainty
product $\mathcal{Q}$ of TUR, which can be used as a measure of the precision
of enzymatic processes realized for a given thermodynamic cost, are suboptimal
when the substrate concentration $[S]$ is at the Michaelis constant
($K_{\text{M}}$), and some of the key biological processes are found to work
around this condition. We illustrate the utility of $\mathcal{Q}$ in assessing
how close the molecular motors and biomass producing machineries are to the
TUR bound, and for the cases of biomass production (or biological copying
processes) we discuss how their optimality quantified in terms of
$\mathcal{Q}$ is balanced with the error rate in the information transfer
process. We also touch upon the trade-offs in other error-minimizing processes
in biology, such as gene regulation and chaperone-assisted protein folding. A
spectrum of $\mathcal{Q}$ recapitulating the biological processes surveyed
here provides glimpses into how biological systems are evolved to optimize and
balance the conflicting functional requirements.

From the perspective of thermodynamics, biological machineries required to
sustain living matters are at work out of equilibrium. Their operations to
carry out specific functions, which must defy the effect of noise inherent to
cellular environment, necessarily incur thermodynamic costs AlbertsBook ;
Bustamante2005PhysicsToday ; Mugnai2020RMP ; QianBook ; KolomeiskyBook ;
Hyeon11BJ . As such, the quest for the relationships between thermodynamic
costs and information processing has been a recurring theme in biological
sciences Hopfield1974 ; Ninio1975 ; bennett1982IJTP ; sagawa2012PTP ;
Lan2012NaturePhysics ; flamholz2013PNAS ; sartori2014PLOSCOMP ; Cao2015NatPhys
; Zhang2020NatPhys ; hong2020JSM ; basan2020Nature .

Recently, a new class of thermodynamic principle, called the thermodynamic
uncertainty relation (TUR), has been derived, giving us quantitative ideas on
the tradeoff between the thermodynamic costs and uncertainty (or precision) of
dynamic processes generated in nonequilibrium barato2015PRL ; Gingrich2016PRL
; horowitz2019NaturePhys ; hasegawa2019PRL ; hasegawa2019PRE . For continuous
time Markov jump processes or overdamped Langevin processes under constant
nonequilibrium drives, TUR in nonequilibrium steady states (NESS) is written
as

$\displaystyle\mathcal{Q}=\Delta S_{\rm tot}(t)\frac{\langle\delta
X(t)^{2}\rangle}{\langle X(t)\rangle^{2}}\geq 2k_{B}.$ (1)

Here, $k_{B}$ is Boltzmann’s constant, which will be set to $k_{B}=1$
throughout the paper. $\Delta S_{\rm tot}(t)=\langle\Delta s_{\rm
tot}(t)\rangle$ is the mean total entropy production from both the system and
its environment ($\Delta s_{\rm tot}(t)=\Delta s_{\rm sys}(t)+\Delta s_{\rm
env}(t)$), calculated over an ensemble of trajectories generated for time $t$.
$\Delta S_{\rm tot}(t)$ is deemed the thermodynamic cost (or dissipation) to
maintain the dynamical process in nonequilibrium for $t$
horowitz2019NaturePhys . $X(t)$ is a time-integrated current-like observable
with odd parity ($X(-t)=-X(t)$) that best captures the _functional_ feature of
the process with $\langle X(t)\rangle$ and $\langle\delta
X(t)^{2}\rangle(=\langle X^{2}(t)\rangle-\langle X(t)\rangle^{2})$ being its
mean and variance. For the process generated from an enzyme reaction, one
could _choose_ the net number of catalyses or product formations that have
occurred for time interval $t$ ($\Delta n(t)=n(t)-n(0)$) as a natural output
observable of the process, $X(t)=\Delta n(t)$. Upon time reversal, i.e.,
$t\rightarrow-t$, $\Delta n(-t)=n(-t)-n(0)=n(0)-n(t)=-\Delta n(t)$, satisfying
the odd parity. If the enzyme exhibits motility along a filament, the net
displacement of the enzyme ($\Delta x(t)=x(t)-x(0)$) can be taken as the
output observable. From the central limit theorem, the entropy production and
the square of relative uncertainty grows with time $t$ as $\Delta S_{\rm
tot}(t)\propto t$ and $\langle\delta X(t)^{2}\rangle/\langle
X(t)\rangle^{2}\propto 1/t$. As a result, the product between the two
quantities, termed the uncertainty product $\mathcal{Q}$, is a _time-
independent_ constant. The inequality sign of TUR (Eq.1) constrains the value
of $\mathcal{Q}$, specifying the minimal dissipation for a given uncertainty,
or the minimal uncertainty for a given amount of dissipation.

Originally, the TUR was conjectured and derived for overdamped, and continuous
time Markov jump processes at long time limit ($t\rightarrow\infty$) under
time-independent non-equilibrium drives barato2015PRL ; Gingrich2016PRL ;
shiraishi2016PRL ; Pigolotti2017PRL ; dechant2020PNAS . The steady-state TUR
in finite-time as in Eq.1, which is more relevant for the analysis of real
systems, was also proven horowitz2017PRE ; gingrich2017inferring ;
dechant2018JSM ; dechant2018multidimensional ; hasegawa2019PRE  and
experimentally verified pietzonka2017PRE . The TUR has recently been extended
to more general settings, for the fluctuations in first passage times
gingrich2017PRL , for underdamped condition lee2019PRE ; van2019PRE , in the
presence of magnetic field chun2019PRE , under time-dependent drives
koyuk2018JPA ; koyuk2019PRL ; koyuk2020PRL , and for arbitrary initial states
liu2020PRL . Furthermore, generalized versions of the relation with less tight
bound have been derived from the large deviation principles Proesmans:2017
and fluctuation theorems potts2019PRE ; proesmans2019hysteretic ;
hasegawa2019PRL .

Since the majority of biological processes can be described by employing the
language of Markov jump processes on cyclic kinetic network or overdamped
Langevin processes, we confine ourselves in this perspective to the version of
TUR described in Eq.1. In this paper, we will first clarify the physical
significance of the bound set by the TUR from a perspective of the stochastic
thermodynamics. Next, by quantifying the uncertainty product $\mathcal{Q}$ for
various biological processes with particular emphasis on biological motors and
biomass producing machineries, we provide our unified perspective on how the
functionally critical features, such as thermodynamic cost, reaction speed,
and fluctuations of those processes are balanced in light of TUR Hwang2018JPCL
. For the case of biomass production processes that transfer the sequence
information of DNA or RNA to its downstream, the error in information transfer
is another key quantity to be tightly regulated Song2020JPCL . We will address
how the error probability ($\eta$) is balanced with the dynamical features
integrated in the $\mathcal{Q}$ by the error correcting machineries. Including
other dynamical processes associated with information processing that can also
be analyzed to determine the value of $\mathcal{Q}$, we will construct a
spectrum of $\mathcal{Q}$ to provide an idea of how different biological
processes comprising cellular activities balance their functional needs under
the fundamental constraint dictated by the TUR.

## Physical significance of Thermodynamic uncertainty relation

Before discussing the TUR, we review the basics of the stochastic
thermodynamics seifert2005PRL ; Seifert2012RPP ; park2018JKPS , which is
closely linked to the TUR seifert2019stochastic . In the stochastic
thermodynamics, the thermodynamic quantities are first defined in a
trajectory-based manner. When the probability of observing a certain
trajectory $\Gamma\equiv\\{{\bf x}_{t}\\}=(x_{0},x_{1},\ldots x_{t})$ is given
by $P(\Gamma)$ with $\int\mathcal{D}\Gamma P(\Gamma)\equiv\int
dx_{0}dx_{1}\ldots dx_{t}P(x_{0},x_{1},\ldots,x_{t})=1$, a trajectory-based
observable $\Theta=\Theta(\Gamma)$ has its average value formally obtained by
taking the average over all possible trajectories as
$\langle\Theta\rangle=\int\mathcal{D}\Gamma P(\Gamma)\Theta(\Gamma)$. The
basic premises of the stochastic thermodynamics is to (i) “define” the
stochastic entropy

$\displaystyle s_{\rm sys}(t)\equiv-\log{P(x_{t})}.$ (2)

and (ii) the trajectory-based total entropy production is given by the ratio
of probabilities associated with the original path of the trajectory and its
time-reversal seifert2005PRL , such that

$\displaystyle\Delta s_{\rm tot}(\Gamma)$
$\displaystyle=\log{\left(\frac{P[\Gamma]}{\tilde{P}[\tilde{\Gamma}]}\right)}=\log{\left(\frac{P[\Gamma|x_{0}]P(x_{0})}{\tilde{P}[\tilde{\Gamma}|{\tilde{x_{0}}}]P(\tilde{x_{0}})}\right)}$
(3)

where $P[\Gamma|x_{0}]$ is the conditional probability of forward path
starting from $x_{0}$ and evolving along $\Gamma$, and the
$\tilde{P}[\tilde{\Gamma}|{\tilde{x_{0}}}]$ is for its time-reversal with
$\tilde{x_{\tau}}\equiv x_{t-\tau}$ and
$\tilde{\Gamma}\equiv(\tilde{x_{0}},\tilde{x_{1}},\cdots,\tilde{x_{t}})=(x_{t},x_{t-1},\ldots,x_{0})$.
The expression is further decomposed into the entropy production for the
reservoir and the system seifert2005PRL ,

$\displaystyle\Delta s_{\rm
tot}(\Gamma)=\underbrace{\log{\left(\frac{P[\Gamma|x_{0}]}{\tilde{P}[\tilde{\Gamma}|{\tilde{x_{0}}}]}\right)}}_{=\Delta
s_{\rm
res}(t)}+\underbrace{\log{\left(\frac{P(x_{0})}{P(x_{t})}\right)}}_{=\Delta
s_{\rm sys}(t)}.$ (4)

Note that the change of the system entropy ($\Delta s_{\rm sys}(t)$) is the
difference between the initial and final points (Eq.2), and that the entropy
production from the reservoir is the log-ratio of the conditional
probabilities of the forward and time-reversed paths Schnakenberg1976RMP ;
kurchan1998fluctuation . The total entropy production follows from the formal
definition of trajectory-based ensemble average, such that

$\displaystyle\Delta S_{\rm tot}$ $\displaystyle=\langle\Delta s_{\rm
tot}(t)\rangle=-\int\mathcal{D}\Gamma
P(\Gamma)\log{\left(\frac{P[\Gamma]}{\tilde{P}[\tilde{\Gamma}]}\right)}$
$\displaystyle=\langle\Delta s_{\rm res}\rangle+\langle\Delta s_{\rm
sys}\rangle.$ (5)

with

$\displaystyle\langle$ $\displaystyle\Delta s_{\rm sys}\rangle=s_{\rm
sys}(t)-s_{\rm sys}(0)$ $\displaystyle=-\int dx_{t}P(x_{t})\log{P(x_{t})}+\int
dx_{0}P(x_{0})\log{P(x_{0})}$ (6)

and

$\displaystyle\langle\Delta s_{\rm res}(t)\rangle=\int\mathcal{D}\Gamma
P[\Gamma|x_{0}]\log{\left(\frac{P[\Gamma|x_{0}]}{P[\tilde{\Gamma}|{\tilde{x_{0}}}]}\right)}.$
(7)

If one chooses the trajectory-based entropy production (entropic current) as
the output observable of interest such that $X(t)=\Delta s_{\rm tot}(t)$
Hyeon2017PRE ; pietzonka2017PRE ; manikandan2020PREL ; dechant2020continuous ,
then the TUR in Eq.1 can be written as

$\displaystyle\mathcal{Q}=\frac{{\rm Var}(\Delta s_{\rm
tot}(t))}{\langle\Delta s_{\rm tot}(t)\rangle}\geq 2.$ (8)

Since the entropy production, an extensive variable, increases with time
($\Delta S_{\rm tot}=\langle\Delta s_{\rm tot}(t)\rangle\propto t$), the
inequality between the variance and mean of entropy production, ${\rm
Var}(\Delta s_{\rm tot}(t))\geq 2\langle\Delta s_{\rm tot}(t)\rangle$, demands
that the variance of entropy production also increases with $t$ but that it
cannot be smaller than twice the mean entropy production (see Fig.1).

Meanwhile, nonequilibrium dynamical processes discussed in this article are
expected to establish cyclic steady states as long as the time scale of
observation is longer than a single cycle time. The total entropy $\Delta
s_{\rm tot}(t)$ produced from such processes obeys the _detailed fluctuation
theorem_ (DFT) Seifert2012RPP ; seifert2005PRL ,

$\displaystyle\frac{P(\Delta s_{\rm tot})}{P(-\Delta s_{\rm tot})}=e^{\Delta
s_{\rm tot}}$ (9)

where we drop the dependence of entropy production on $t$ for simplicity of
the notation. The DFT follows straightforwardly from Eq.3 and the odd parity
of entropy production upon time-reversal ($\Delta s_{\rm tot}(\Gamma)=-\Delta
s_{\rm tot}(\tilde{\Gamma})$) by evaluating $P(\Delta s_{\rm
tot})=\int\mathcal{D}\Gamma\delta(\Delta s_{\rm tot}-\Delta s_{\rm
tot}(\Gamma))P(\Gamma)$, and it relates the probability density functions of
entropy production from an irreversible process for time $t$ with
exponentiated total entropy production. Rearranging the terms and integrating
with respect to $\Delta s_{\rm tot}$ over $-\infty<\Delta s_{\rm tot}<\infty$
give rise to the _integral fluctuation theorem_ (IFT), $\langle e^{-\Delta
s_{\rm tot}}\rangle=1$,

$\displaystyle\langle e^{-\Delta s_{\rm tot}}\rangle$
$\displaystyle=\int_{-\infty}^{\infty}d(\Delta s_{\rm tot})e^{-\Delta s_{\rm
tot}}P(\Delta s_{\rm tot})$ $\displaystyle=\int_{-\infty}^{\infty}d(\Delta
s_{\rm tot})P(-\Delta s_{\rm tot})=1.$ (10)

Here, $\langle\ldots\rangle$ denotes the average taken over the distribution
of total entropy production from the process, $P(\Delta s_{\rm tot})$, and the
relation should hold for any dynamical process. However, a special relation
follows if the distribution of the total entropy production ($\Delta s_{\rm
tot}$) is a Gaussian, i.e., $P(\Delta s_{\rm
tot})\sim\exp{\left[-\frac{(\Delta s_{\rm tot}-\langle\Delta s_{\rm
tot}\rangle)^{2}}{2{\rm Var}(\Delta s_{\rm tot})}\right]}$

$\displaystyle 1$ $\displaystyle=\langle e^{-\Delta s_{\rm
tot}}\rangle=\int_{-\infty}^{\infty}d(\Delta s_{\rm tot})e^{-\Delta s_{\rm
tot}}P(\Delta s_{\rm tot})$ $\displaystyle=e^{-\langle\Delta s_{\rm
tot}\rangle+{\rm Var}(\Delta s_{\rm tot})/2},$ (11)

from which the equality condition of Eq.8 ${\rm Var}(\Delta s_{\rm
tot})/\langle\Delta s_{\rm tot}\rangle=2$ is acquired.

The above argument based on IFT and Gaussian distribution suggests that the
equality condition of TUR ($\mathcal{Q}=2$) is attained when the distribution
of entropy production follows precisely a Gaussian. Since the tail part of the
distribution contributes to the evaluation of the exponentiated average, the
distribution of the total entropy production $\Delta s_{\rm tot}$, the sum of
$\Delta s_{i}$’s ($i=1,2,\ldots t$) which are the _independently and
identically distributed_ (i.i.d.) random variables ($\Delta s_{\rm
tot}=\sum_{i=1}^{t}\Delta s_{i}$), should be assessed using the large
deviation theory, an extension of central limit theorem touchette2009PR ;
Gingrich2016PRL . In order for $P(\Delta s_{\rm tot})$ to be a perfect
Gaussian including the tail parts, the entropy production from each cycle
$P(\Delta s_{i})$ should also be a Gaussian, which is highly restrictive.
Thus, the inequality ($\mathcal{Q}>2$), leading to ${\rm Var}(\Delta s_{\rm
tot})>2\langle\Delta s_{\rm tot}\rangle$, arises when $P(\Delta s_{\rm tot})$
deviates from the Gaussianity. This condition of Gaussianity of the
distribution of total entropy production is more general than the condition of
detailed balance (DB) or equilibrium, which gives rise to the equality
condition of TUR for Markov process (see Eqs.20 and 25). However, even at out-
of-equilibrium, $\mathcal{Q}$ could be minimized (or sub-optimized) to a small
value. An illuminating far-from-equilibrium case that gives rise to the
equality condition ($\mathcal{Q}=2$) is found in a strongly driven colloidal
particle in periodic potentials with $\Delta s_{\rm tot}(t)$ being chosen as
the output observable Hyeon2017PRE ; speck2005JPA ; Pigolotti2017PRL ;
speck2007EPL .

Figure 1: Properties of the distribution of total entropy production demanded
by TUR. Two scenarios of time evolution of $P(\Delta s_{\rm tot}(t))$ from
$t=t_{i}$ (blue) to $t=t_{f}$ (red). (i) As time increases, the distribution
of entropy production is broadened. In this case, the TUR is obeyed with ${\rm
Var}(\Delta s_{\rm tot})\geq 2\langle\Delta s_{\rm tot}\rangle$ at both
$t=t_{i}$ and $t=t_{f}$. (ii) As time increases, the distribution of entropy
production is narrowed down, such that ${\rm Var}(\Delta s_{\rm
tot})<2\langle\Delta s_{\rm tot}\rangle$ at $t=t_{f}$, which contradicts to
Eq.8 and violates the TUR.

Second, we underscore the significance of TUR in light of the second law of
thermodynamics. Using the definitions of entropy production rate $\sigma_{\rm
tot}(t)=\Delta S_{\rm tot}(t)/t$, mean and variance of finite time current,
$\langle J\rangle=\langle X(t)\rangle/t$ and $\langle\delta
J^{2}\rangle=\langle\delta X(t)^{2}\rangle/t$, one can rewrite the expression
of TUR in Eq.1 as follows.

$\displaystyle\mathcal{Q}=\sigma_{\rm tot}(t)\frac{\langle\delta
J^{2}\rangle}{\langle J\rangle^{2}}\geq 2.$ (12)

Rearranged in the following way,

$\displaystyle\sigma_{\rm tot}(t)\geq B\langle J\rangle^{2}$ (13)

with $B=\left(2/\langle\delta J^{2}\rangle\right)$, the TUR conveys a
remarkable message that the rate of entropy production from the dynamical
process is bounded below by the square of mean current multiplied by a
prefactor $B$. This makes the physical bound of thermodynamic process more
explicit and tighter than the second law of thermodynamics ($\Delta S_{\rm
tot}\geq 0$ or $\sigma_{\rm tot}(t)\geq 0$) dechant2018JSM ; dechant2018PRE ;
horowitz2019NaturePhys . Recently, Li _et al._li2019NatComm  have shown that
once inferring the macroscopic current and its fluctuations, the TUR can be
used to estimate the lower bound of the entropy production (dissipation) rate
even for high-dimensional dynamical systems.

## The uncertainty product $\mathcal{Q}$ as a measure of optimality of
enzymatic processes

Many biological processes, which exhibit a net change in terms of chemical
species, are driven by enzyme reactions that incur thermodynamic cost. To
discuss the TUR in the context of the enzyme reactions, it is convenient to
recast the expression (Eq. 1) into

$\displaystyle\mathcal{Q}=\frac{\langle\Delta s_{\rm tot}(t)\rangle}{\langle
n(t)\rangle}\frac{{\rm Var}(n(t))}{\langle
n(t)\rangle}=\beta\mathcal{A}\frac{\langle\delta J^{2}\rangle}{\langle
J\rangle}=\beta\mathcal{A}\times\lambda\geq 2,$ (14)

where we choose $n(t)$ the net number of catalyses that take place for time
$t$ as the observable of interest, defining the mean and fluctuation of finite
time current as $\langle J\rangle=\langle n(t)\rangle/t$, $\langle\delta
J^{2}\rangle={\rm Var}(n(t))/t$. The Fano factor associated with the number of
catalyses or with the reaction current is defined as $\lambda={\rm
Var}(n(t))/\langle n(t)\rangle=\langle\delta J^{2}\rangle/{\langle J\rangle}$,
which corresponds to the randomness parameter schnitzer1995statistical ;
Visscher99Nature , and $\beta\mathcal{A}=\langle\Delta s_{\rm
tot}(t)\rangle/\langle n(t)\rangle$ with the inverse temperature
$\beta=(k_{B}T)^{-1}$ is the entropy production per cycle (or affinity) of the
enzymatic reaction. Since $\langle J\rangle$ and $\langle\delta J^{2}\rangle$
are associated with the speed ($\langle J\rangle\sim Vt$) and diffusivity
($\langle\delta J^{2}\rangle\sim 2Dt$) of the enzymatic process, a dynamic
process that operates at high speed, low fluctuation (high regularity), and
low thermodynamic cost is characterized with a small $\mathcal{Q}$ and can be
deemed more optimally designed when $\mathcal{Q}$ is smaller and close to 2
dechant2018JSM ; Hwang2018JPCL .

Figure 2:  $\mathcal{Q}$ of simple enzyme reactions as a function of the
substrate concentration $[S]$. Schematics of (a) the 1-state and (b) 2-state
kinetic networks. (c)(d) $\mathcal{Q}([S])$ of the 2-state kinetic network,
for three pairs of $\gamma$ and $\xi$ values. In (c), $[S]$, normalized by
$K_{\text{M}}$, shows that $\mathcal{Q}([S])$ is locally minimized at
$[S]\approx K_{\rm M}$. In (d), $[S]$ is normalized by $[S]_{\rm eq}[=(k_{\rm
off}k_{\rm rev})/(k_{\rm on}k_{\rm cat})]$, the substrate concentration at the
detailed balance condition, showing that
$\lim_{[S]\rightarrow[S]_{\text{eq}}}\mathcal{Q}([S])=2$. The rate constants
used for the plots are as follows: $k_{\rm on}=10^{8}$ M-1sec-1, $k_{\rm
off}=k_{\rm rev}=10$ sec-1, and $k_{\rm cat}=30,300,1000$ sec-1.

Before exploring physically realistic processes, such as molecular motors and
biomass producing machineries, we motivate by computing $\mathcal{Q}$ of two
simple versions of enzymatic cycle below.

(i) For the catalysis of the substrate $S$ in the ($N=1$)-state kinetic
reaction network (Fig.2a), the binding and unbinding rates of the substrate
are denoted by $k_{\rm on}$ and $k_{\rm off}$, respectively. The substrate
concentration, $[S]$, can be varied independently as a control parameter.

The master equation for the probability $P_{n}(t)$ of observing $n$ catalytic
events at time $t$ is

$\displaystyle\dot{P}_{n}(t)=k_{\rm on}[S]P_{n-1}(t)+k_{\rm
off}P_{n+1}(t)-(k_{\rm on}[S]+k_{\rm off})P_{n}(t),$ (15)

with the initial condition $P_{n}(0)=\delta_{n,0}$. The method of generating
function vanKampen  $F(z,t)=\sum_{n=-\infty}^{\infty}z^{n}P_{n}(t)$ casts
Eq.15 into $F(z,t)=\exp{\left[zk_{\rm on}[S]+k_{\rm off}/z-(k_{\rm
on}[S]+k_{\rm off})\right]}$, which allows one to calculate the mean and
fluctuations in the number of catalytic events using
$\partial_{z}F(z,t)\big{|}_{z=1}=\langle n(t)\rangle=(k_{\rm on}[S]-k_{\rm
off})t$ and $\partial_{z}^{2}F(z,t)\big{|}_{z=1}=\langle
n^{2}(t)\rangle-\langle n(t)\rangle=2k_{\rm off}t+(k_{\rm on}[S]-k_{\rm
off})^{2}t^{2}$. Thus, the quantities required to evaluate $\mathcal{Q}$ in
Eq.14 are obtained as follows.

$\displaystyle\langle J\rangle$ $\displaystyle=k_{\rm on}[S]-k_{\rm off},$
(16) $\displaystyle\langle\delta J^{2}\rangle$ $\displaystyle=k_{\rm
on}[S]+k_{\rm off},$ (17) $\displaystyle\lambda$
$\displaystyle=\frac{\langle\delta J^{2}\rangle}{\langle
J\rangle}=\frac{e^{\beta\mathcal{A}}+1}{e^{\beta\mathcal{A}}-1},$ (18)
$\displaystyle\beta\mathcal{A}=$ $\displaystyle\ln\frac{k_{\rm on}[S]}{k_{\rm
off}},$ (19)

which leads to

$\displaystyle\mathcal{Q}=$
$\displaystyle\beta\mathcal{A}\frac{e^{\beta\mathcal{A}}+1}{e^{\beta\mathcal{A}}-1}\geq
2.$ (20)

At the DB condition ($[S]=[S]_{\rm eq}\equiv k_{\rm off}/k_{\rm on}$), no net
catalysis ($\langle J\rangle=0$) and no dissipation ($\mathcal{A}=0$) occurs,
and the uncertainty product reaches its bound, $\mathcal{Q}=2$. As $[S]$
increases, breaking the DB condition, $\langle J\rangle$, $\mathcal{A}$,
$\lambda$, and $\mathcal{Q}$ all increase monotonically.

(ii) For substrate catalysis through the ($N=2$)-state kinetic network, with
the rate constants as denoted in Fig. 2b, we obtain the following expressions
Fisher01PNAS ; KolomeiskyBook ; Hwang2017JPCL  for the quantities required to
evaluate $\mathcal{Q}$:

$\displaystyle\langle J\rangle$ $\displaystyle=\frac{k_{\rm on}k_{\rm
cat}[S]-k_{\rm rev}k_{\rm off}}{k_{\rm on}[S]+k_{\rm cat}+k_{\rm rev}+k_{\rm
off}}$ (21) $\displaystyle\langle\delta J^{2}\rangle$
$\displaystyle=\frac{k_{\text{on}}k_{\text{cat}}[S]+k_{\text{rev}}k_{\text{off}}-2\langle
J\rangle^{2}}{k_{\rm on}[S]+k_{\rm cat}+k_{\rm rev}+k_{\rm off}}$ (22)
$\displaystyle\lambda$ $\displaystyle=\frac{k_{\rm on}k_{\rm cat}[S]+k_{\rm
rev}k_{\rm off}}{k_{\rm on}k_{\rm cat}[S]-k_{\rm rev}k_{\rm
off}}-2\frac{k_{\text{on}}k_{\text{cat}}[S]-k_{\text{rev}}k_{\text{off}}}{k_{\text{on}}^{2}\left([S]+K_{\rm
M}+\frac{k_{\text{rev}}}{k_{\text{on}}}\right)^{2}}$ (23)
$\displaystyle\beta\mathcal{A}$ $\displaystyle=\ln{\frac{k_{\rm on}k_{\rm
cat}[S]}{k_{\rm rev}k_{\rm off}}}$ (24)

with the Michaelis constant, $K_{\rm
M}=(k_{\text{cat}}+k_{\text{off}})/k_{\text{on}}$. When $\mathcal{Q}$ is
written as a function of the thermodynamic drive $\beta\mathcal{A}$,

$\displaystyle\mathcal{Q}$
$\displaystyle=\beta\mathcal{A}\left[\frac{e^{\beta\mathcal{A}}+1}{e^{\beta\mathcal{A}}-1}-2\frac{\gamma^{2}\left(e^{\beta\mathcal{A}}-1\right)}{\left(\gamma^{2}+\gamma\xi+e^{\beta\mathcal{A}}\right)^{2}}\right],$
(25)

with dimensionless parameters
$\gamma=\frac{k_{\text{cat}}}{\sqrt{{k_{\text{off}}}{k_{\rm rev}}}}$ and
$\xi=\frac{k_{\text{off}}+k_{\rm rev}}{\sqrt{k_{\text{off}}k_{\rm rev}}}(\geq
2)$, one can show that for any $\xi(\geq 2)$ there exists a threshold value of
$\gamma$, above which $\mathcal{Q}$ becomes non-monotonic with $[S]$ (see Fig.
2c, and supplementary Fig. 2 in Song2020JPCL ). One can also show that for
$\beta\mathcal{A}\ll 1$,
$\mathcal{Q}=2+\left[\frac{1}{6}-\frac{2\gamma^{2}}{(1+\gamma\xi+\gamma^{2})^{2}}\right](\beta\mathcal{A})^{2}+\mathcal{O}\left[(\beta\mathcal{A})^{3}\right]\geq
2$, which confirms that $\mathcal{Q}=2$ is attained at DB condition (Fig. 2d).
Additionally, it is straightforward to see from Eq. 23 that, at the limit of a
strongly driven process with $\beta\mathcal{A}\gg 0$ and $k_{\text{on}}\gg
k_{\text{rev}}$, $\lambda$ is simplified to

$\displaystyle\lambda([S])\simeq
1-\frac{2k_{\text{cat}}[S]}{k_{\text{on}}([S]+K_{\text{M}})^{2}},$ (26)

which is minimized at $[S]={K_{\text{M}}}$ (Fig. 2c). Thus, when $k_{\rm
on}k_{\rm cat}[S]\gg k_{\rm rev}k_{\rm off}$ ($\beta\mathcal{A}\gg 0$) and
$k_{\text{on}}\gg k_{\rm rev}$, $\mathcal{Q}(\text{[S]})$ has a local minimum
and is sub-optimized at around $[\text{S}]\approx{K_{\text{M}}}$ (Fig. 2c).

Figure 3:  (a)(b) Analysis of experimental data of (a) $V$ and (b) $D$ of
kinesin-1. The experimental data are digitized from Ref. Visscher1999 , and
fit using a 6-state double-cycle kinetic model in Ref. Hwang2018JPCL . The
solid lines are fits to the data, with three values of external load. (c)(d)
Diagrams of $\mathcal{Q}$ as a function of [ATP] and external load ($f$).
Assisting and resisting loads correspond to $f<0$ and $f>0$, respectively; (c)
kinesin-1, (d) kinesin-1 mutant. The color code and the corresponding colorbar
in (c)(d) quantify the value of the uncertainty product $\mathcal{Q}$. (e) The
values of $\mathcal{Q}$ as a function of ATP concentration at $f=1$ pN. (f)
The tradeoff relations between $\Delta S_{\rm tot}$ and the relative
uncertainty of displacement $x_{t}(\equiv x(t))$ are plotted for various
molecular motors with their uncertainty products calculated at $[{\rm ATP}]=1$
mM and $f=1$ pN. The figure was adapted from Ref.Hwang2018JPCL .

The non-monotonic variations of $\mathcal{Q}([S])$ seen in some of the
biological motors and copy enzymes discussed below simply means that the
dynamical processes of these molecules display a Michaelis-Menten type
hyperbolic dependence on $[S]$. The condition of $[S]\approx K_{\text{M}}$ in
the Michaelis-Menten type reaction mechanism, $v([S])=k_{\rm
cat}[S]/(K_{\text{M}}+[S])$, corresponds to the point where the response of
enzymatic activity to the logarithmic change in substrate concentration is
maximized, i.e., $dv/d\log{[S]}=k_{\rm
cat}(K_{\text{M}}/[S])/(1+K_{\text{M}}/[S])^{2}\leq k_{\rm cat}/4$.
Importantly, a recent systems level analysis of metabolic pathways in
eukaryotic cells has shown that the physiological substrate concentrations of
many enzyme reactions are tuned near their respective $K_{\text{M}}$ values
Park2016 , which naturally explains the suboptimality of $\mathcal{Q}([S])$ at
$[S]\approx K_{\text{M}}$ for some processes discussed in the following
sections.

## Transport processes by molecular motors

Biological motors are a class of enzymes that have an ability to transduce
chemical free energy to mechanical motion via the catalysis of molecular fuels
(ATP, GTP, etc) present in the cellular milieu KolomeiskyBook ;
howard2001mechanics . Among them, kinesin-1 is arguably the most well-studied
motor protein that transports cargos or organelles from the minus to plus ends
of microtubules with the velocity of $V\approx 1$ $\mu$m/sec, taking 8 nm step
for every ATP hydrolysis. Dynein moves with a similar speed, but in the
opposite direction along microtubules, displaying more fluctuations in the
time traces. Myosin families are the motor proteins specialized to move along
actin filaments and generate mechanical forces. Whereas these motor proteins
are specialized for linear movement and force generation, there are also
rotary motors (e.g., F1F0ATPase) that utilize $H^{+}$ gradient across membrane
to generate rotational motion (or torque), which is, for example, used to
empower the beating dynamics of bacterial flagella.

The chemical free energy-driven dynamics of molecular motors are a perfect
example whose optimality can be assessed in light of TUR and the uncertainty
product $\mathcal{Q}$. The TUR in the original form (Eq.1) with $X(t)=\Delta
x(t)$ can be cast into the following form with $V=\langle\Delta x(t)\rangle/t$
and $D=[\langle\Delta x(t)^{2}\rangle-\langle\Delta x(t)\rangle^{2}]/2t$,

$\displaystyle\mathcal{Q}=\sigma_{\rm tot}\frac{2D}{V^{2}}\geq 2.$ (27)

According to this relation, a molecular motor that transports cargos with high
velocity ($V$), low fluctuations ($D$), and low thermodynamic cost
($\sigma_{\rm tot}$) would be characterized with a small value of
$\mathcal{Q}$, and one could argue that a motor with smaller $\mathcal{Q}$
value is better designed as a molecular transporter. For a _transport
efficiency_ defined as $\eta_{\rm tr}\equiv 2/\mathcal{Q}$, which is bounded
in $0\leq\eta_{\rm tr}\leq 1$, a motor with smaller $\mathcal{Q}$ can be said
to have a higher _transport efficiency_ dechant2018JSM ; Hwang2018JPCL .

Of note, the TUR can also be used to determine the upper bounds on the
_thermodynamic efficiency_ by means of the dissipation bound, $\sigma_{\rm
tot}\geq V^{2}/D$ (Eq.27). Pietzonka _et al._pietzonka2016JSM ; uhl2018PRE
defined the _thermodynamic efficiency_ ($\eta$) of a molecular motor in the
presence of external load ($f$) as the ratio between the amount of work
production ($\dot{W}=fV$) and an input chemical potential
($-\dot{\Delta\mu}=\dot{W}+T\sigma_{\rm tot}$), so as to obtain its upper
bound of $\eta$ as follows.

$\displaystyle\eta$
$\displaystyle=\frac{\dot{W}}{|\dot{\Delta\mu}|}=\frac{\dot{W}}{\dot{W}+T\sigma_{\rm
tot}}=\frac{1}{1+\frac{TV}{2fD}\mathcal{Q}}$
$\displaystyle\leq\frac{1}{1+\frac{TV}{fD}}=\eta_{\rm max}.$ (28)

The maximum thermodynamic efficiency, $\eta_{\rm max}$, is obtained in the
condition equivalent to $\mathcal{Q}=2$.

Given the single molecule time traces, $\\{x(t)\\}$, the velocity ($V$) and
diffusivity ($D$) are straightforwardly determined. Meanwhile, the rate of
total entropy production $\sigma_{\rm tot}$ either (i) can be estimated, for
example, from the fact that each step of kinesin-1, which occurs every $\sim
10$ msec, is generated from the hydrolysis free energy of a single ATP
molecule $\sim$ $20$ $k_{B}T$ ($T\sigma_{\rm tot}\sim 20$ $k_{B}T/10\text{
msec}$), or (ii) can be calculated more systematically by knowing all the
chemical rate constants on the kinetic network Schnakenberg1976RMP .
$\sigma_{\rm tot}$ for ($N=2$)-unicyclic network in the absence of external
load ($f=0$), for example, can be calculated using Eqs.21, 24

$\displaystyle\sigma_{\rm tot}$ $\displaystyle=\langle
J\rangle\times\beta\mathcal{A}$ $\displaystyle=\frac{k_{\rm on}k_{\rm
cat}[S]-k_{\rm rev}k_{\rm off}}{k_{\rm on}[S]+k_{\rm cat}+k_{\rm rev}+k_{\rm
off}}\log{\left(\frac{k_{\rm on}k_{\rm cat}[S]}{k_{\rm rev}k_{\rm
off}}\right)}.$ (29)

For the case (ii), $\sigma_{\rm tot}$ of a general kinetic network with known
transition rate constants, can be obtained by a method developed by Koza
Koza1999 .

Figure 4: Biological copy processes and tradeoff relation. (a) Tradeoff
relations in biological copy processes. Excessive amount of error corrections
are expected to lead to higher thermodynamic cost and slow biomass production.
(b) The illustration of DNA replication process (top), and a kinetic network
to represent general copy processes with the possibility of accommodating
either correct or incorrect substrate to copy strand. The $c$-cycle (red) and
$i$-cycle (blue) are for the correct and incorrect substrate incorporations,
respectively. (c) Reaction currents along the subcycles of the network as a
function of substrate concentration. Explicit calculations of currents were
conducted using an example of translation of a codon CUG by ribosome
Song2020JPCL . The figures b and c were adapted from Ref.Song2020JPCL .

As clearly gleaned from the data extracted from single molecule time traces of
a molecular motor (e.g., kinesin-1), the velocity and diffusivity of time
traces vary with ATP concentration ([ATP]) and external force ($f$) (Fig.3a,b)
Schnitzer97Nature ; Visscher99Nature ; Schnitzer00NCB . Thus, the value of
$\mathcal{Q}$ not only depends on a type of molecular process of interest, but
also on its working condition. Fig. 3c and d shows the diagrams of
$\mathcal{Q}$ as a function of $f$ and [ATP] for kinesin-1 and kinesin-1
mutant, respectively. Some features of $\mathcal{Q}(f,[{\rm ATP}])$ of
biological motors are noteworthy: (i) As clearly seen for the case of
kinesin-1 (Fig.3c), besides the trivial minima at low [ATP] corresponding to
DB condition, $\mathcal{Q}$ is sub-optimized at $f\approx 3$ pN and $[{\rm
ATP}]\approx 200$ $\mu M$. $\mathcal{Q}(f=1{\rm pN},[{\rm ATP}])$ plotted in
Fig.3e displays non-monotonic variation with [ATP]. A similar behavior is
observed in $\mathcal{Q}$ for dynein (Fig.4 of ref. Hwang2018JPCL ). As
already discussed in the foregoing section, the non-monotonic variation of
$\mathcal{Q}([{\rm ATP}])$ is the outcome of Michaelis-Menten type response of
motor or enzymatic activity on substrate concentration. (ii) The cellular
condition ($f\approx 1$ pN, [ATP] $\approx 1$ mM) marked with a yellow star
(Fig.3c), is in proximity to the suboptimal condition. (iii) The high
$\mathcal{Q}$ value region in the middle of the diagram ($\mathcal{Q}>100$)
arises when the hindering load stalls the movement of motor. It is important
to note that at the stall condition ATP is still consumed ($\sigma_{\rm
tot}>0$) Cross05Nature ; Hyeon09PCCP ; Hwang2018JPCL ; sumi2019NanoLett .
While the motor is motionless at mechanical equilibrium ($V\approx 0$), it is
not in chemical equilibrium ($\sigma_{\rm tot}>0$). As a result, $\mathcal{Q}$
diverges (Eq.27) (see the high $\mathcal{Q}$ region in Figs.3c and 3d). (iv)
$\mathcal{Q}(f,[{\rm ATP}])$ of kinesin-1 mutant (Fig.3d), which has six
additional amino acids inserted into neck-linker, is drastically altered from
that of the wild-type, such that the value of $\mathcal{Q}$ is increased and
the stall condition is formed at smaller $f$ values. (v) Lastly, Fig.3f
compares the values of $\mathcal{Q}$ for various biological motors at the
cellular condition ($f\approx 1$ pN, [ATP] $\approx 1$ mM). $\mathcal{Q}=7-15$
for wild type motors: kinesin-1, F1-ATPase, dynein, KIF17, KIF3AB, and
myosin-V. In particular, $\mathcal{Q}=7$ for kinesin-1, whereas its mutant
displays much deteriorated performance with $\mathcal{Q}=19$. The uncertainty
product of biological motor specialized for cargo/organelle transport is
relatively smaller than other machineries that will be discussed below.

## Biological copy processes

As exemplified in DNA replication, transcription, and translation processes
which comprise the central dogma of molecular biology, some of the key
information transfer processes in biology are nearly free from copy error even
in the noisy cellular environment. For the case of DNA replication, the error
probability ($\eta_{eq}$) estimated solely from the stability difference
between the correct and incorrect base pairs is at best $\eta_{eq}\approx
10^{-3}$ Goodman1997 ; however, the actual error probability of replicating
incorrect bases to copy strand is as small as $\eta=10^{-10}$, which makes the
replication of giant human DNA consisting of $N=3\times 10^{9}$ bases
effectively error-free Kunkel2000 . To achieve the substantial error reduction
from $\eta_{eq}$ to $\eta$, a host of elaborate energy-expending molecular
mechanisms are at work at every step of the biological copy processes
Ibarra2009 ; Shaevitz2003 ; Blanchard2004 ; Cvetesic2012 .

From the viewpoint of information processing in biology, the error reduction
is certainly an important issue; however, it in itself cannot be the sole goal
of the biological copy processes given that biomass production, e.g., from DNA
to RNA, and from RNA to proteins, is the major outcome from the processes.
Excessive operations of error correction machineries would not only incur
thermodynamic cost, but also slow down the process of biomass production.
There have recently been a number of studies devoted to understanding the
strategy of biological copy processes to achieve the mutually contradicting
goals of low copy error, high speed, and low thermodynamic cost Murugan2012a ;
Gaspard2016b ; Gaspard2016a ; Banerjee2017 ; Wong2018 ; Mallory2019  (Fig.
4a).

Here, we will view each step of copying enzyme along a template polymer as
that of a molecular motor stepping along a template filament processively.
Each substrate incorporating event can be split into several Markov jump steps
on a cyclic kinetic network, at the end of which the copying enzyme
transitions to the next position along the template strand. The forward motion
of the enzyme is driven by the chemical potential of the biosynthetic
substrates (dNTPs for the DNA polymerase, NTPs for the RNA polymerase, and
charged tRNAs for the ribosome). Under an assumption that cellular homeostasis
maintains the substrate concentrations constant but away from the DB
condition, the dynamical process associated with synthesizing a copy strand
can be modeled as a process operating at NESS with the substrate
concentration, $[S]$, as the key parameter to be controlled.

Instead of the uni-cyclic kinetic network introduced earlier for enzymatic
processes on a single type of substrate catalysis, additional cycles due to
the chance of copying incorrect types of substrate to the strand are necessary
for the copy processes described above. The schematic of the reaction network
in Fig.4b represents the one for copy processes, which takes into account the
chance of incorporating incorrect substrates. The $c$ denotes the reaction
cycle (red) associated with _correct_ substrate incorporation, whereas $i$ is
for the cycle (blue) associated with _incorrect_ substrate incorporation.
Provided that correct substrates are always accommodated to the copy enzyme
without any futile attempt, the reaction current, more specifically the
current associated with polymerization, $J^{c}_{\text{pol}}$, flows only
through the $c$-cycle. In this case, the probability of copy error would be 0
(Eq.31, see below). However, the stochastic nature of biochemical reactions
makes the incorporation of incorrect substrate to the copy polymer still
inevitable. An incorrect substrate incorporated to the enzyme-cofactor complex
engenders the reaction current through $i$-cycle. A proofreading mechanism
expending the free energy (e.g., GTP hydrolysis for the case of mRNA
translation by ribosome) in action filters out the incorrect substrate from
the system, generating a futile current ($J^{i}_{\text{fut}}$); otherwise the
substrate is accommodated into the copy polymer, generating a current
associated with polymerization ($J^{i}_{\text{pol}}$). The futile current
along the $c$-cycle is conceivable as well, just like the case in which a
molecular motor, say kinesin-1, occasionally fails to step in spite of ATP
hydrolysis. Explicit calculations of four currents in Fig. 4b conducted for
the translation of CUG codon by ribosome indicate that the sizes of the four
currents are maintained in the following order, effectively over the whole
range of GTP concentration (1 nM $\leq$ [GTP] $\leq$100 mM)Song2020JPCL ,
while the cellular concentration of GTP is [GTP] $\approx 5$ mM Bennett2009 :

$\displaystyle\langle J_{\text{pol}}^{c}\rangle\gg\langle
J_{\text{fut}}^{i}\rangle\gtrsim\langle J_{\text{fut}}^{c}\rangle\gg\langle
J_{\text{pol}}^{i}\rangle.$ (30)

Both $\langle J_{\rm pol}^{c}\rangle$ and $\langle J_{\rm pol}^{i}\rangle$ are
reflected to the copied sequence, such that the error probability of the copy
process is associated with the two mean currents as

$\displaystyle\eta=\frac{\langle J^{i}_{\rm pol}\rangle}{\langle J^{c}_{\rm
pol}\rangle+\langle J^{i}_{\rm pol}\rangle},$ (31)

which gives $\eta\approx 4\times 10^{-4}$ for the case of CUG codon
Song2020JPCL .

Figure 5: The uncertainty product $\mathcal{Q}$ for biological copy processes.
(a) DNA replication by T7 DNA polymerase. (b) Translation of codons by _E.
coli._ ribosome. The dashed lines indicate the cellular concentrations of aa-
tRNA and GTP. (c) Proofreading step-modulated fluctuations in the translation
times of _E. coli._ ribosome ($\mathcal{T}$) that reads the _tufB_ mRNA
sequence encoding the 394 amino acids of EF-Tu. In the two panels at the
bottom are shown the error probability and uncertainty product as a function
of the perturbation factor ($\alpha_{\rm PR}$) multiplied to the rate of
proofreading step. The figure was adapted from Ref.Song2020JPCL .

The free energy cost associated with the copy process represented by the
schematic of kinetic network in Fig.4b is given as follows Song2020JPCL .

$\displaystyle\beta\mathcal{A}=$
$\displaystyle-\beta\left[\Delta\mu_{\text{pol}}+\frac{\langle J_{\rm
fut}\rangle}{\langle J_{\text{pol}}\rangle}\Delta\mu_{\rm
fut}\right]-\eta\ln{\eta}-(1-\eta)\ln{(1-\eta)}.$ (32)

where $\langle J_{\text{pol}}\rangle\equiv\langle
J^{c}_{\text{pol}}\rangle+\langle J^{i}_{\text{pol}}\rangle$ and $\langle
J_{\text{fut}}\rangle\equiv\langle J^{c}_{\text{fut}}\rangle+\langle
J^{i}_{\text{fut}}\rangle$ are the total reaction current along the
polymerization and futile cycles. Note that since $\Delta\mu_{\text{pol}}$ and
$\Delta\mu_{\text{fut}}$, the chemical potentials associated with each cycle,
are the state function, they are identical regardless of the cycle. The value
of $\beta\mathcal{A}$ is determined by the ratio of currents $\langle J_{\rm
fut}\rangle/\langle J_{\rm pol}\rangle$ as given in the expression of Eq.32.
Lastly, the Shannon entropy-like term,
$S(\eta)=-\eta\ln{\eta}-(1-\eta)\ln{(1-\eta)}$ arises from the entropic drive
created by a potential disorder in copied sequence. Although the actual
contribution of $S(\eta)$ is minor compared to the entire free energy
($\mathcal{A}$) in biologically relevant parameter regimes Song2020JPCL ,
$S(\eta)$ could, in principle, be the source of thermodynamic drive for the
polymerization reaction when $[\Delta\mu_{\rm pol}+(\langle J_{\rm
fut}\rangle/\langle J_{\rm pol}\rangle)\Delta\mu_{\rm fut}]\approx 0$ (see
Ref. bennett1982IJTP ). Eq. 32 clarifies that extra free energy cost is
incurred in the presence of a larger amount of futile current, when the
proofreading mechanisms are at work.

Finally, we aim to look at the problem of biological copy processes under the
hood of TUR and study how the error probability is balanced with dissipation,
speed, and fluctuations. While there have been a number of studies on the
relation between error probability, dissipation, and speed, somewhat less
attention was paid to the fluctuations emanating from the dynamical process.
Mallory _et al._ concluded that fluctuations are the lowest priority in
optimizing copy processes in comparison with the speed, cost, and accuracy
Mallory2019 ; however, large fluctuations in the reaction cycles of
transcription and translation contributing to higher variability in the
protein copy number Soltani2016 ; DalCo2017  could be detrimental to the
fitness of an organism Fraser2004 ; Hausser2019 . Given that the DNA
replication of developing animals is meticulously synchronized across the
cells, disarray of cell cycle from other cells can be lethal Blythe2015 ;
Djabrayan2019 . As further discussed in the section mRNA translation by _E.
coli_ ribosome, the fluctuations of the copy process could display substantial
variations depending on the proofreading dynamics of the copying enzyme (Fig.
5c).  Overall, fluctuations associated with biological copy processes also
have to be suppressed down to biologically acceptable levels, and this aspect
is naturally taken into account by calculating the uncertainty product
$\mathcal{Q}$ of TUR.

DNA replication. Of the three biological copy processes, the most precise one
is DNA replication, at the heart of which is the interaction between the
polymerase and the exonuclease domains of the DNA polymerase (DNAP)
Johnson1993 . In a nutshell, the exonuclease domain of the DNAP executes the
proofreading step by preferentially cleaving off incorrect nucleotides, the
incorporation of which slows down the action of the polymerase. Through a
Michaelis-Menten approximation of all the chemical reactions catalyzed by the
DNA polymerase, Gaspard has derived the dependence of the error probability,
speed, and thermodynamic cost on the substrate concentrations Gaspard2016b ;
Gaspard2016a . Along similar lines, Igoshin, Kolomeisky and colleagues further
explored the reaction dynamics of the DNAP, by explicitly modeling the
switching of the DNAP between the polymerase and exonuclease states
Banerjee2017 ; Mallory2019 . Results from both groups demonstrated that the
copy errors were being suppressed at the expense of the speed and the
thermodynamic cost. Furthermore, the analysis of rate constants of the wild
type T7 DNA polymerase suggested that it was optimized for the speed rather
than the accuracy.

Evaluating the fluctuation and the uncertainty product $\mathcal{Q}$ of DNA
polymerases together with the activity of exonuclease requires a careful
analysis of the kinetics involved with the switching among the polymerizing,
proofreading, and paused states of the polymerase Pineros2020 ; Hoekstra2017 ;
Ibarra2009 . However, since the precise knowledge of exonuclease mechanism
still remain elusive, the analysis here is limited to the proofreading-free
version of exonuclease-deficient T7 DNAP. The analysis of kinetic network of
T7 DNAP finds that $\mathcal{Q}$ for T7 DNAP is suboptimal
($\mathcal{Q}\approx 10$) at [dNTP] $\approx 100$ $\mu$M, which is in the
similar range of dNTP at the cellular condition $\mathcal{O}(10^{2})$ $\mu$M
$-$ $\mathcal{O}(1)$ mM Bochner1982 ; Buckstein2008 ; Schaaper2013 . Notably,
the error probability is already saturated to its minimal value $\eta\approx
10^{-4}$ when [dNTP] $\gtrsim 10^{-11}$ M. This suggests that instead of the
error probability, other dynamical properties of T7 DNAP can be further
optimized. It is noteworthy that the dynamics of the T7 DNAP is sub-optimal in
$\mathcal{Q}$ at its working condition.

RNA polymerases.  While synthesizing the RNA transcript complementary to the
DNA sequence, the RNA polymerase (RNAP) translocates along the DNA sequence,
maintaining a DNA bubble of 12-14 basepairs, and an 8-9 basepair RNA-DNA
hybrid double strand abbondanzieri2005Nature ; Chen10PNAS . Similarly to the
DNAP, the polymerase and the exonuclease activities of the RNAP are combined
together to suppress copy errors Sydow2009 . Upon incorporating incorrect
nucleotides, the polymerase activity slows down and the RNAP converts to a
backtracked state that can no longer incorporate a new nucleotide. Only after
removal of the erroneous NTP, the RNAP is able to continue incorporating new
nucleotides. Structural, biochemical, and theoretical studies have
characterized both the mechanisms of nucleotide incorporation in the
elongation complex and the proofreading, highlighting the strong sequence
dependence on the error rate and the pausing frequency of the polymerase. The
effect of the exonuclease-mediated proofreading reactions on the error
probability have been quantitatively evaluated through kinetic modeling
Mellenius2017 . Along with the fluctuations of the RNAP activity, analysis of
RNAP dynamics in light of $\mathcal{Q}$ would also be of great interest.

mRNA translation by E. coli ribosome.  The E. coli ribosome is a well
characterized system that employs KPM to suppress copy errors Blanchard2004 ;
Rodnina2018 . The ribosome synthesizes proteins from mRNAs by decoding the
sequence information encoded in the ‘codons’. For each of the 61 codons
(excluding the 3 stop codons from the total of $4^{3}$ possible combinations),
there exist potentially multiple ‘correct’ tRNAs with the matching anti-codon,
in complex with the encoded amino-acid (aa), elongation factor (EF), and GTP.
The binding of the correct aa-tRNA-EF-GTP complex to the ribosome-mRNA complex
initiates the reaction cycle through which the amino-acid is added to the
elongating polypeptide sequence. Similarly, aa-tRNA-EF-GTP complexes with
incorrect amino-acids can undergo a parallel reaction cycle as the correct
substrate, which leads to the incorporation of errors in the polypeptide
sequence Rudorf2014 . Free energy released from GTP hydrolysis is used to
execute KPM (see Ref.Song2020JPCL  for the details of kinetic proofreading
mechanism).

The error probabilities associated with codon-anticodon pairings are already
saturated to $\eta\approx 10^{-4}-10^{-2}$ for all the codons with respect to
the variations in [aa-tRNA] and [GTP] (see Fig.S6 in Ref. Song2020JPCL ). As
shown in Fig.5b, the uncertainty products $\mathcal{Q}$ are again non-
monotonic functions of both [aa-tRNA] and [GTP], and the corresponding
cellular concentrations of [aa-tRNA] and [GTP] are found greater than the
concentrations, [aa-tRNA]∗ and [GTP]∗, that give rise to the suboptimal values
of $\mathcal{Q}$. Depending on the codon type, $\mathcal{Q}$ varies between 20
and 40 (Fig.5b). The mRNA translation is realized when ribosome translocates
through a string of codons, accommodating correct types of aa-tRNA and forming
new peptide bonds. The greater forward kinetic rates of the GTP hydrolysis and
the polymerization along the correct cycle Wohlgemuth2011 ; Rudorf2014  lowers
the error probability.

To study the process of mRNA translation of _E. coli._ ribosome in a more
realistic fashion, it is possible to consider an extended version of network
model which translates 42 species of aa-tRNAs into 20 different amino acids.
With information on the concentration of aa-tRNAs in the cellular milieu, Song
_et al._ classified them into cognate, near-cognate, and non-cognate types and
simulated the _E. coli._ ribosome-mediated translation of the _tufB_ mRNA
sequence that encodes the 394 amino acid EF-Tu (Fig. 5c) Song2020JPCL . The
effect of proofreading step on the ribosome dynamics as well as on the error
probability can be assessed by modulating the polymerization current by
multiplying a factor $\alpha_{\rm PR}$ to the associated rate constants (see
the general kinetic network for proofreading depicted in Fig.5b). A similar
perturbative analysis has previously been used to decipher which feature of
the biomolecular process had been optimized throughout evolution Banerjee2017
; Mallory2019 ; Mallory2020a . Although rate constants are not easy to tune in
experiments, such modification happens throughout the evolution by means of
mutations to the ribosome, EF-Tu, and tRNA. Their simulation results
demonstrated that the variations in the completion times of mRNA translation
(the first passage times, $\mathcal{T}$) depends critically on the parameter
$\alpha_{\rm PR}$ that modulates the polymerization current. For $\alpha_{\rm
PR}=1$ corresponding to the wild type, the average speed of polymerization is
found $\approx 16$ aa/sec and the error probability is $\eta\approx 10^{-3}$
(Fig.5c), in good agreement with those known from experimental measurements
Young1976 ; Bouadloun1983 . Selecting the completion time $\mathcal{T}$ (first
passage time) as the output observable, one can use a version of TUR derived
for the first passage time gingrich2017PRL :

$\displaystyle\mathcal{Q}=\Delta S_{\rm tot}(\mathcal{T})\frac{{\rm
Var}(\mathcal{T})}{\langle\mathcal{T}\rangle^{2}}=\sigma_{\rm tot}\frac{{\rm
Var}(\mathcal{T})}{\langle\mathcal{T}\rangle}\geq 2.$ (33)

While $\eta$ decreases monotonically with $\alpha_{\rm PR}$, $\mathcal{Q}$ is
non-monotonic with $\alpha_{\rm PR}$ (Fig.5c), minimized near the wild type
condition. At $\alpha_{\rm PR}=1$ and under the cellular concentrations of aa-
tRNA and GTP, the uncertainty product for _E. coli._ ribosome is
$\mathcal{Q}=45$ (Fig.5c). For the given kinetic parameters from the wild type
(WT), $\mathcal{Q}$ is minimized to $\mathcal{Q}\sim 30$ at $\alpha_{\rm
PR}\approx 5$. For $\alpha_{\rm PR}=10^{-2}$, the translation times display a
much broader distribution than that of the wild-type (Fig.5c). Thus, it could
be argued that the extent of proofreading of the WT is in a proper range, that
the fidelity of translation and the fluctuations of protein synthesis are
simultaneously regulated. Fluctuations in the completion time for mRNA
translation ($\langle\delta\mathcal{T}^{2}\rangle$) can be critical, as it is
translated to significant variation in protein copy number.

For the _E. coli._ ribosome, the kinetic rate constants of ribosomes have
evolved to optimize the speed ($\langle J_{\rm pol}\rangle$) over the accuracy
($\eta$), while increasing the thermodynamic cost ($\mathcal{A}$) only
slightly above the minimal cost necessary for the polymerization reaction.
Importantly, similar conclusions were drawn by a number of studies, each of
which evaluated the translation process of the E. coli ribosome using
different kinetic reaction networks Wohlgemuth2011 ; Banerjee2017 ;
Song2020JPCL . From the findings of maximized current, suppressed
fluctuations, and moderate increase of $\mathcal{A}$ captured by the
effectively sub-optimized value of $\mathcal{Q}$ $(\approx 45-50)$ at
$\alpha_{\rm PR}\approx 1$, while $\eta$ being determined at biologically
acceptable levels Pineros2020 ; Song2020JPCL , it could be argued that the
wild type ribosome operates near semi-optimal condition.

Figure 6: $\mathcal{Q}$ and transcription noise in non-equilibrium gene
regulation by transcription factors (TFs). (a) Schematic of a gene regulation
by a combination of the nucleosome (NC) and TF occupancy. Transcriptional
activity is assumed to occur only when the regulatory sequence is released
from the NC and bound to the transcription factor (TF) Shelansky2020 . A
parallel network for incorrect gene expression also exists, in which the TF
unbind with rate $k^{i}_{\rm off}$. (b) The error probability ($\eta$) and
$\mathcal{Q}$ of correct gene expression plotted as functions of $\alpha$.
When the NC is removed preferentially in the TF bound state (i.e.
$\alpha>\beta$), the error in gene expression can be reduced below
$\eta_{0}\equiv[1+k^{i}_{\rm off}/k^{c}_{\rm off}]^{-1}$. Other than $\alpha$,
the remaining parameters are set as follows: $k_{\rm on}=1~{}{\rm sec^{-1}}$,
$k^{c}_{\rm off}=1~{}{\rm sec^{-1}}$, $k^{i}_{\rm off}=100~{}{\rm sec^{-1}}$,
$k^{\rm NC}_{\rm on}=.2~{}{\rm sec^{-1}}$, $\beta=0.01~{}{\rm sec^{-1}}$, and
$v=0.01~{}{\rm sec^{-1}}$.

### Gene regulation by transcription factors

Gene expression is regulated in large part by the binding of transcription
factors (TFs) to the regulatory regions of DNA. The specificity by which TFs
can activate target genes is originated from the discriminatory binding of TFs
to the non-specific regulatory regions of the DNA. With an assumption that the
relative binding affinities of the TF to the target and the non-target
sequences differ only by their respective dissociation rates, $k^{c}_{\rm
off}$ and $k^{i}_{\rm off}$, the minimal error probability of transcription
can be approximated by $\eta\approx\eta_{0}=[{1+k^{i}_{\rm off}/k^{c}_{\rm
off}}]^{-1}$. Shelansky and Boeger recently proposed a mechanism in gene
regulation by TFs and nucleosomes that can reduce $\eta$ at the expense of
extra free energy cost Shelansky2020 . Nucleosomes, the regulatory structures
that bind and unbind reversibly from DNA, are generally known to suppress
transcriptional activity in the bound state. In Shelansky and Boeger’s model,
transcription is assumed to occur only when the DNA is bound to the
transcription factor, and free from the nucleosome (Fig. 6a). Briefly, when
the nucleosome can be more easily removed from the TF-bound regulatory
sequences than from the TF-absent ones ($\alpha>\beta$ in Fig. 6a), a kinetic
proofreading-like mechanism can reduce the minimum error rate below
$\eta_{0}$. The nucleosome removal in the TF-bound sequences is predicted to
be driven by the activity of ATP-consuming chromatin remodelers, which are
recruited by the TF Zhou2016b ; Shelansky2020 .

Fig. 6 plots the error probability and $\mathcal{Q}$ with an increasing
thermodynamic drive. In parallel to Eq.31, the error probability is defined as
$\eta\equiv J^{i}_{\rm trans}/(J^{i}_{\rm trans}+J^{c}_{\rm trans})$, where
$J^{i}_{\rm trans}$ and $J^{c}_{\rm trans}$ are the reaction currents
associated with the incorrect and correct gene expression. Next, the noise in
the correct gene expression is defined by the Fano factor
$\lambda\equiv\langle(\delta J^{c}_{\rm trans})^{2}\rangle/\langle J^{c}_{\rm
trans}\rangle$. To evaluate the uncertainty product $\mathcal{Q}$, we assume
that the nucleosome removal step ($\alpha$) entails the unwinding of $\sim$20
basepairs of DNA, with every 2 basepairs requiring 1 ATP based on single
molecule studies Shundrovsky2006 ; Zhang2006 ; Sirinakis2011 ; Deindl2013 . We
additionally assume that the transcription initiation step ($v$) requires
$\sim$20 ATPs in order to melt the DNA into the open complex Lynch2015a .
Assuming that the free energy of ATP hydrolysis is $\sim$20 $k_{B}T$, we can
compute the semi-empirical free energy cost associated with transcription,
$\mathcal{A}$, by accounting for all the ATP hydrolysis reactions occurring at
the correct and incorrect DNA binding sites.

To increase the thermodynamic drive, we increase the nucleosome unbinding rate
from TF-bound sequences ($\alpha$), while keeping the rest of the parameters
constant. $\mathcal{Q}$ increases monotonically with $\alpha$, while the error
probability ($\eta$) is minimized at around $\alpha/\beta\approx 60$. Further
analysis in the framework of the TUR will be useful in quantifying trade-off
relations among the error probability, transcription noise, and the
thermodynamic cost of gene regulation by TF binding.

Figure 7: Molecular chaperone as an error-reducing machine. (a) Rugged folding
landscape of biomolecules that visualizes the native and misfolded basins of
attraction. As a result of spontaneous folding, molecules are partitioned with
the fraction of $\Phi$ and $1-\Phi$ to native and misfolded states,
respectively. (b) Iterative annealing by GroEL. The figure (a) was adapted
from Ref.Thirum05Biochem .

### Molecular chaperone-assisted folding of proteins

In an appropriate environmental condition, small single domain proteins in
general can reversibly fold and unfold, and reach their native state within
biologically relevant time scales Anfinsen75APC . Yet, there are still a class
of proteins that are prone to misfold and aggregate, whose presence can be
detrimental to the organisms. For such proteins (e.g., Rubisco, malate
dehydrogenase Thirumalai01ARBB ; chakrabarti2017PNAS ), only a small fraction
($\Phi\ll 1$, $\Phi\approx 0.05$ for Rubisco Todd96PNAS ) of the population
can reach their native state, and the remaining fraction of population
($1-\Phi$) are kinetically trapped in misfolded states (Fig.7). Molecular
chaperones Thirumalai01ARBB ; Thirum05Biochem ; chakrabarti2017PNAS ;
Mugnai2020RMP ; Goloubinoff18NCB , which employ the free energy sources
ubiquitous in cells to change their conformations and interact with the
molecules in misfolded states, can change the population entirely and sustain
the cellular environment in good condition.

One of the most well-studied protein chaperones, bacterial GroES-GroEL
chaperonin system interacts exclusively with the misfolded population of
proteins, providing them with another chance to repeat the folding process. As
a result of an initial interaction of the chaperone with misfolded protein
population, out of the misfolded population $1-\Phi$ from the first round of
folding process, $\Phi(1-\Phi)$ would fold into the native state, and
$(1-\Phi)^{2}$ would be again trapped in the misfolded states. When this
process is repeated $N(=t/\tau_{0})$ times where $\tau_{0}$ is the time
associated with a single cycle and $t$ is the time duration, the fraction
$(1-\Phi)^{N}$ still remains misfolded and hence the $1-(1-\Phi)^{N}$ are
folded. The fraction of native population increases from its originally small
yield $Y_{1}=\Phi(\ll 1)$ at $N=1$ to

$\displaystyle Y_{N}=1-(1-\Phi)^{N}\xrightarrow{\Phi\ll 1}1-e^{-\Phi
t/\tau_{0}}.$ (34)

Via this mechanism, called the iterative annealing mechanism (IAM) Todd96PNAS
; Thirumalai01ARBB ; Hyeon2013JCP ; chakrabarti2017PNAS ; korobko2020Elife ,
the native yield of unity can be finally reached when $N\rightarrow\infty$ or
$t\rightarrow\infty$. From the perspective of information processing,
molecular chaperones are another elegantly designed error reducing machinery.
For every cycle, which lasts about 2 sec Ye13PNAS ; Fei14PNAS , GroEL made of
two heptameric rings presumably consumes at least $3-4$ ATPs for each ring,
which amounts to $\approx$ $60-80$ $k_{B}T$ of dissipation. Furthermore, since
successful conversion of misfolded to folded state is not guaranteed at each
cycle, the dissipation per cycle estimated above is only a lower bound of the
estimate. In fact, $\tau=\tau_{0}/\Phi$ is the full conversion time to the
native state. Given that $\tau_{0}\approx 2$ sec and $\Phi=0.05$ for Rubisco,
$\tau=40$ sec. Since the number of successful conversions to native state
usually obeys Poisson statistics, the Fano factor of the net fraction of
conversion $\Delta Y_{N}(t)$ is $\lambda=\langle(\delta\Delta
Y_{N})^{2}(t)\rangle/\langle\Delta Y(t)\rangle\approx\mathcal{O}(1)$. Thus, a
rough estimate of $\mathcal{Q}$ for GroEL assisted protein folding is
$\mathcal{Q}\gtrsim\tau/\tau_{0}\times(60-80)\approx 2\times 10^{3}$.

Figure 8: A spectrum of uncertainty product ($\mathcal{Q}$) computed for
transport motors Hwang2018JPCL , T7 DNA polymerase Song2020JPCL ; Pineros2020
, E. coli ribosome Song2020JPCL ; Pineros2020 , molecular chaperone, and
biochemical oscillators Marsland2019 . The theoretical lower bound of TUR is
specified at $\mathcal{Q}=2$. The range of $\mathcal{Q}$ for the family of
molecular motors and biochemical oscillators are shaded in red and green,
respectively Hwang2018JPCL ; Marsland2019 .

## Conclusions

Under evolutionary pressure, biological processes are often confronted with
situations in which to balance between a number of competing options. Here, we
have reviewed the recent studies on the trade-off relations of these features
in biological motors and biological dynamics involved with information
processing, from the perspective of TUR which offers the uncertainty product
$\mathcal{Q}$ as a measure of the optimality integrating the cost and
precision of the processes. Importantly, the physical lower bound of
$\mathcal{Q}$ provides an absolute scale onto which we can map the
efficiencies of diverse biomolecular processes. Biological motors, specialized
for cargo transport, are found to operate close to the lower bound
($\mathcal{Q}=2$) even at NESS, with $\mathcal{Q}\approx 7-15$ Hwang2018JPCL .
Compared with biological motors that utilize ATP hydrolysis free energy ($\sim
20$ $k_{B}T$ $\approx$ 0.8 eV), synthetic nano-motors kudernac2011Nature  that
use $>3$ eV UV-light source as the driving force are expected to have at least
several-fold greater $\mathcal{Q}$. The exonuclease-deficient T7 DNA
polymerase operate at $\mathcal{Q}\approx 10$, and the ribosome operates at
$\mathcal{Q}=45-50$ Pineros2020 ; Song2020JPCL . The value of $\mathcal{Q}$
for molecular chaperones is estimated to be rather large,
$\mathcal{Q}\gtrsim\mathcal{O}(10^{3})$, mainly due to the large cost of
operating the reaction cycle of chaperones. Although it was not discussed in
this article, Marsland _et al._ have evaluated the values of $\mathcal{Q}$ for
several biochemical oscillators by selecting the oscillation period as their
output observable of interest. Some of the biochemical oscillators severely
underperform the TUR bound; for example, the KaiABC system works at
$\mathcal{Q}\gtrsim O(10^{3})$ Marsland2019 . For biological motors, both
thermodynamic cost and precision of the processes are valuable quantities to
be balanced, giving rise to relatively small value of $\mathcal{Q}$ near the
physical lower bound of $2$. In contrast, for the cases of molecular
chaperones and biochemical oscillators, the precision of the processes is in
the highest priority, at the expense of large thermodynamic cost. Along with
the spectrum of $\mathcal{Q}$ recapitulating our survey on various processes
in this article (Fig. 8), the $\mathcal{Q}$ values of more number of other
dynamical processes will be of help to better glean the design principles
underlying life sustaining cellular processes.

###### Acknowledgements.

This work was supported by the KIAS individual Grants CG067102 (YS) and
CG035003 (CH) at Korea Institute for Advanced Study. We thank the Center for
Advanced Computation in KIAS for providing computing resources.

## Data Availability

The data that supports the findings of this study are available within the
article.

## References

  * (1) Alberts, B, Johnson, A, Lewis, J, Raff, M, Roberts, K, & Walter, P,  Molecular Biology of the Cell,  Garland Science, 5th edition, 2008.
  * (2) Bustamante, C, Liphardt, J, & Ritort, F,  Physics Today 58, 43 (2005).
  * (3) Mugnai, M. L, Hyeon, C, Hinczewski, M, & Thirumalai, D,  Rev. Mod. Phys. 92, 025001 (2020).
  * (4) Beard, D. A & Qian, H,  Chemical Biophysics: Quantitative Analysis of Cellular Systems,  Cambridge University Press, 2008.
  * (5) Kolomeisky, A. B,  Motor Proteins and Molecular Motors,  CRC Press, Boca Raton, 2015.
  * (6) Hyeon, C & Onuchic, J. N,  Biophys. J. 101, 2749 (2011).
  * (7) Hopfield, J. J,  Proc. Natl. Acad. Sci. USA 71, 4135 (1974).
  * (8) Ninio, J,  Biochimie 57, 587 (1975).
  * (9) Bennett, C. H,  Int. J. Theor. Phys. 21, 905 (1982).
  * (10) Sagawa, T,  Prog. Theor. Phys. 127, 1 (2012).
  * (11) Lan, G, Sartori, P, Neumann, S, Sourjik, V, & Tu, Y,  Nature physics 8, 422 (2012).
  * (12) Flamholz, A, Noor, E, Bar-Even, A, Liebermeister, W, & Milo, R,  Proc. Natl. Acad. Sci. U. S. A. 110, 10039 (2013).
  * (13) Sartori, P, Granger, L, Lee, C. F, & Horowitz, J. M,  PLoS Comput Biol 10, e1003974 (2014).
  * (14) Cao, Y, Wang, H, Ouyang, Q, & Tu, Y,  Nature Phys. 11, 772 (2015).
  * (15) Zhang, D, Cao, Y, Ouyang, Q, & Tu, Y,  Nature Phys. 16, 95 (2020).
  * (16) Hong, H, Jo, J, Hyeon, C, & Park, H,  J. Stat. Mech. , 074001 (2020).
  * (17) Basan, M, Honda, T, Christodoulou, D, Hörl, M, Chang, Y.-F, Leoncini, E, Mukherjee, A, Okano, H, Taylor, B. R, Silverman, J. M, et al.,  Nature 584, 470 (2020).
  * (18) Barato, A. C & Seifert, U,  Phys. Rev. Lett. 114, 158101 (2015).
  * (19) Gingrich, T. R, Horowitz, J. M, Perunov, N, & England, J. L,  Phys. Rev. Lett. 116, 120601 (2016).
  * (20) Horowitz, J. M & Gingrich, T. R,  Nat. Phys. 16, 15 (2019).
  * (21) Hasegawa, Y & Van Vu, T,  Phys. Rev. Lett. 123, 110602 (2019).
  * (22) Hasegawa, Y & Van Vu, T,  Phys. Rev. E 99, 062126 (2019).
  * (23) Shiraishi, N, Saito, K, & Tasaki, H,  Phys. Rev. Lett. 117, 190601 (2016).
  * (24) Pigolotti, S, Neri, I, Roldán, É, & Jülicher, F,  Phys. Rev. Lett. 119, 140604 (2017).
  * (25) Dechant, A & Sasa, S.-i,  Proc. Natl. Acad. Sci. U. S. A. 117, 6430 (2020).
  * (26) Horowitz, J. M & Gingrich, T. R,  Phys. Rev. E 96, 020103 (2017).
  * (27) Gingrich, T. R, Rotskoff, G. M, & Horowitz, J. M,  J. Phys. A: Math. Theor. 50, 184004 (2017).
  * (28) Dechant, A & Sasa, S.-i,  J. Stat. Mech. , 063209 (2018).
  * (29) Dechant, A,  J. Phys. A: Math. Theor. 52, 035001 (2018).
  * (30) Pietzonka, P, Ritort, F, & Seifert, U,  Phys. Rev. E. 96, 012101 (2017).
  * (31) Gingrich, T. R & Horowitz, J. M,  Phys. Rev. Lett. 119, 170601 (2017).
  * (32) Lee, J. S, Park, J.-M, & Park, H,  Phys. Rev. E 100, 062132 (2019).
  * (33) Van Vu, T & Hasegawa, Y,  Phys. Rev. E 100, 032130 (2019).
  * (34) Chun, H.-M, Fischer, L. P, & Seifert, U,  Phys. Rev. E 99, 042128 (2019).
  * (35) Koyuk, T, Seifert, U, & Pietzonka, P,  J. Phys. A: Math. Theor. 52, 02LT02 (2018).
  * (36) Koyuk, T & Seifert, U,  Phys. Rev. Lett. 122, 230601 (2019).
  * (37) Koyuk, T & Seifert, U,  Phys. Rev. Lett. 125, 260604 (2020).
  * (38) Liu, K, Gong, Z, & Ueda, M,  Phys. Rev. Lett. 125, 140602 (2020).
  * (39) Proesmans, K & Van den Broeck, C,  EPL 119, 20001 (2017).
  * (40) Potts, P. P & Samuelsson, P,  Phys. Rev. E 100, 052137 (2019).
  * (41) Proesmans, K & Horowitz, J. M,  J. Stat. Mech. : Theory and Experiment 2019, 054005 (2019).
  * (42) Hwang, W & Hyeon, C,  J. Phys. Chem. Lett. 9, 513 (2018).
  * (43) Song, Y & Hyeon, C,  J. Phys. Chem. Lett. 11, 3136 (2020).
  * (44) Seifert, U,  Phys. Rev. Lett. 95, 040602 (2005).
  * (45) Seifert, U,  Rep. Prog. Phys. 75, 126001 (2012).
  * (46) Park, H,  J. Kor. Phys. Soc. 72, 1413 (2018).
  * (47) Seifert, U,  Annu. Rev. Cond. Matt. Phys. 10, 171 (2019).
  * (48) Schnakenberg, J,  Rev. Mod. Phys. 48, 571 (1976).
  * (49) Kurchan, J,  J. Phys. A: Math. and General 31, 3719 (1998).
  * (50) Hyeon, C & Hwang, W,  Phys. Rev. E. 96, 012156 (2017).
  * (51) Manikandan, S. K, Gupta, D, & Krishnamurthy, S,  Phys. Rev. Lett. 124, 120603 (2020).
  * (52) Dechant, A & Sasa, S.-i,  arXiv preprint arXiv:2010.14769 (2020).
  * (53) Touchette, H,  Physics Reports 478, 1 (2009).
  * (54) Speck, T & Seifert, U,  J. Phys. A: Math. General 38, L581 (2005).
  * (55) Speck, T, Blickle, V, Bechinger, C, & Seifert, U,  EPL 79, 30002 (2007).
  * (56) Dechant, A & Sasa, S.-i,  Phys. Rev. E 97, 062101 (2018).
  * (57) Li, J, Horowitz, J. M, Gingrich, T. R, & Fakhri, N,  Nature Commun. 10, 1 (2019).
  * (58) Schnitzer, M. J & Block, S,  Cold spring harbor symposia on quantitative biology 60, 793 (1995).
  * (59) Visscher, K, Schnitzer, M. J, & Block, S. M,  Nature 400, 184 (1999).
  * (60) van Kampen, N. G,  Stochastic Processes in Chemistry and Physics,  Elsevier, 2007.
  * (61) Fisher, M. E & Kolomeisky, A. B,  Proc. Natl. Acad. Sci. U. S. A. 98, 7748 (2001).
  * (62) Hwang, W & Hyeon, C,  J. Phys. Chem. Lett. 8, 250 (2017).
  * (63) Visscher, K, Schnltzer, M. J, & Block, S. M,  Nature 400, 184 (1999).
  * (64) Park, J. O, Rubin, S. A, Xu, Y.-F, Amador-Noguez, D, Fan, J, Shlomi, T, & Rabinowitz, J. D,  Nat. Chem. Bio. 12, 482 (2016).
  * (65) Howard, J,  Mechanics of motor proteins and the cytoskeleton,  Sinauer associates Sunderland, MA, 2001.
  * (66) Pietzonka, P, Barato, A. C, & Seifert, U,  J. Stat. Mech. Theory Exp. , 124004 (2016).
  * (67) Uhl, M & Seifert, U,  Phys. Rev. E 98, 022402 (2018).
  * (68) Koza, Z,  J. Phys. A: Math. Gen. 32, 7637 (1999).
  * (69) Schnitzer, M. J & Block, S. M,  Nature 388, 386 (1997).
  * (70) Schnitzer, M. J, Visscher, K, & Block, S. M,  Nature Cell Biol. 2, 718 (2000).
  * (71) Carter, N. J & Cross, R. A,  Nature 435, 308 (2005).
  * (72) Hyeon, C, Klumpp, S, & Onuchic, J. N,  Phys. Chem. Chem. Phys. 11, 4899 (2009).
  * (73) Sumi, T & Klumpp, S,  Nano Lett. 5, 3370 (2019).
  * (74) Goodman, M. F,  Proc. Natl. Acad. Sci. USA 94, 10493 (1997).
  * (75) Kunkel, T. A & Bebenek, K,  Annu. Rev. Biochem. 69, 105 (2000).
  * (76) Ibarra, B, Chemla, Y. R, Plyasunov, S, Smith, S. B, Lázaro, J. M, Salas, M, & Bustamante, C,  EMBO J. 28, 2794 (2009).
  * (77) Shaevitz, J. W, Abbondanzieri, E. A, Landick, R, & Block, S. M,  Nature 426, 684 (2003).
  * (78) Blanchard, S. C, Gonzalez, R. L, Kim, H. D, Chu, S, & Puglisi, J. D,  Nat. Struct. Biol. 11, 1008 (2004).
  * (79) Cvetesic, N, Perona, J. J, & Gruic-Sovulj, I,  J. Biol. Chem. 287, 25381 (2012).
  * (80) Murugan, A, Huse, D. A, & Leibler, S,  Proc. Natl. Acad. Sci. USA 109, 12034 (2012).
  * (81) Gaspard, P,  Phys. Rev. E 93, 042420 (2016).
  * (82) Gaspard, P,  Phys. Rev. E 93, 042419 (2016).
  * (83) Banerjee, K, Kolomeisky, A. B, & Igoshin, O. A,  Proc. Natl. Acad. Sci. USA 114, 5183 (2017).
  * (84) Wong, F, Amir, A, & Gunawardena, J,  Phys. Rev. E 98, 012420 (2018).
  * (85) Mallory, J. D, Kolomeisky, A. B, & Igoshin, O. A,  J. Phys. Chem. B 123, 4718 (2019).
  * (86) Bennett, B. D, Kimball, E. H, Gao, M, Osterhout, R, Van Dien, S. J, & Rabinowitz, J. D,  Nat. Chem.Biol. 5, 593 (2009).
  * (87) Soltani, M, Vargas-Garcia, C. A, Antunes, D, & Singh, A,  PLoS Comp. Biol. 12, e1004972 (2016).
  * (88) Co, A. D, Lagomarsino, M. C, Caselle, M, & Osella, M,  Nuc. Acids. Res. 45, 1069 (2017).
  * (89) Fraser, H. B, Hirsh, A. E, Giaever, G, Kumm, J, & Eisen, M. B,  PLoS Biol. 2, 0834 (2004).
  * (90) Hausser, J, Mayo, A, Keren, L, & Alon, U,  Nat. Commun. 10, 68 (2019).
  * (91) Blythe, S. A & Wieschaus, E. F,  Cell 160, 1169 (2015).
  * (92) Djabrayan, N. J, Smits, C. M, Krajnc, M, Tomer, S, Yamada, S, Lemon, W. C, Keller, P. J, Rushlow, C. A, & Shvartsman, S. Y,  Curr. Biol. 29, 1193 (2019).
  * (93) Johnson, K. A,  Annu. Rev. Biochem. 62, 685 (1993).
  * (94) Piñeros, W. D & Tlusty, T,  Phys. Rev. E 101, 022415 (2020).
  * (95) Hoekstra, T. P, Depken, M, Lin, S. N, Cabanas-Danés, J, Gross, P, Dame, R. T, Peterman, E. J, & Wuite, G. J,  Biophys. J. 112, 575 (2017).
  * (96) Bochner, B. R & Ames, B. N,  J. Biol. Chem. 257, 9759 (1982).
  * (97) Buckstein, M. H, He, J, & Rubin, H,  J. Bacteriol 190, 718 (2008).
  * (98) Schaaper, R. M & Mathews, C. K,  DNA Repair 12, 73 (2013).
  * (99) Abbondanzieri, E. A, Greenleaf, W. J, Shaevitz, J. W, Landick, R, & Block, S. M,  Nature 438, 460 (2005).
  * (100) Chen, J, Darst, S. A, & Thirumalai, D,  Proc. Natl. Acad. Sci. U. S. A. 107, 12523 (2010).
  * (101) Sydow, J. F & Cramer, P,  Curr. Op. Struct. Biol. 19, 732 (2009).
  * (102) Mellenius, H & Ehrenberg, M,  Nuc. Acids. Res. 45, 11582 (2017).
  * (103) Rodnina, M. V,  Csh. Perspect. Biol. 10, a032664 (2018).
  * (104) Rudorf, S, Thommen, M, Rodnina, M. V, & Lipowsky, R,  PLoS Comp. Biol. 10, e1003909 (2014).
  * (105) Wohlgemuth, I, Pohl, C, Mittelstaet, J, Konevega, A. L, & Rodnina, M. V,  Philos. Trans. R. Soc. B 366, 2979 (2011).
  * (106) Mallory, J. D, Igoshin, O. A, & Kolomeisky, A. B,  J. Phys. Chem. B 124, 9289 (2020).
  * (107) Young, R & Bremer, H,  Biochem. J. 160, 185 (1976).
  * (108) Bouadloun, F, Donner, D, & Kurland, C,  EMBO J. 2, 1351 (1983).
  * (109) Shelansky, R & Boeger, H,  Proc. Natl. Acad. Sci. USA 117, 2456 (2020).
  * (110) Zhou, C. Y, Johnson, S. L, Gamarra, N. I, & Narlikar, G. J,  Annual Review of Biophysics 45, 153 (2016).
  * (111) Shundrovsky, A, Smith, C. L, Lis, J. T, Peterson, C. L, & Wang, M. D,  Nat. Struct. Mol.Biol. 13, 549 (2006).
  * (112) Zhang, Y, Smith, C. L, Saha, A, Grill, S. W, Mihardja, S, Smith, S. B, Cairns, B. R, Peterson, C. L, & Bustamante, C,  Mol. Cell 24, 559 (2006).
  * (113) Sirinakis, G, Clapier, C. R, Gao, Y, Viswanathan, R, Cairns, B. R, & Zhang, Y,  EMBO J. 30, 2364 (2011).
  * (114) Deindl, S, Hwang, W. L, Hota, S. K, Blosser, T. R, Prasad, P, Bartholomew, B, & Zhuang, X,  Cell 152, 442 (2013).
  * (115) Lynch, M & Marinov, G. K,  Proc. Natl. Acad. Sci. USA 112, 15690 (2015).
  * (116) Thirumalai, D & Hyeon, C,  Biochemistry 44, 4957 (2005).
  * (117) Anfinsen, C. B & Scheraga, H. A,  Adv. Protein Chem. 29, 205 (1975).
  * (118) Thirumalai, D & Lorimer, G. H,  Ann. Rev. Biophys. Biomol. Struct. 30, 245 (2001).
  * (119) Chakrabarti, S, Hyeon, C, Ye, X, Lorimer, G, & Thirumalai, D,  Proc. Natl. Acad. Sci. U. S. A. 114, E10919 (2017).
  * (120) Todd, M. J, Lorimer, G. H, & Thirumalai, D,  Proc. Natl. Acad. Sci. U. S. A. 93, 4030 (1996).
  * (121) Goloubinoff, P, Sassi, A. S, Fauvet, B, Barducci, A, & De los Rios, P,  Nat. Chem. Biol. 14, 388 (2018).
  * (122) Hyeon, C & Thirumalai, D,  J. Chem. Phys. 139, 121924 (2013).
  * (123) Korobko, I, Mazal, H, Haran, G, & Horovitz, A,  Elife 9, e56511 (2020).
  * (124) Ye, X & Lorimer, G. H,  Proc. Natl. Acad. Sci. U. S. A. 110, E4289 (2013).
  * (125) Fei, X, Ye, X, LaRonade, N. A, & Lorimer, G. H,  Proc. Natl. Acad. Sci. U. S. A. 111, 12776 (2014).
  * (126) Marsland, R, Cui, W, & Horowitz, J. M,  J. R. Soc. Interface 16, 20190098 (2019).
  * (127) Kudernac, T, Ruangsupapichat, N, Parschau, M, Maciá, B, Katsonis, N, Harutyunyan, S. R, Ernst, K.-H, & Feringa, B. L,  Nature 479, 208 (2011).