# Rate enhancement of gated drift-diffusion process by optimal resetting

Arup Biswas  Arnab Pal<EMAIL_ADDRESS>The Institute of Mathematical
Sciences, CIT Campus, Taramani, Chennai 600113, India & Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
Debasish Mondal  Department of Chemistry, Indian Institute of Technology
Tirupati, Tirupati 517619, India Somrita Ray<EMAIL_ADDRESS>Department of
Chemistry, Indian Institute of Technology Tirupati, Tirupati 517619, India
SUPA, School of Physics & Astronomy, University of Edinburgh, Peter Guthrie
Tait Road, Edinburgh EH9 3FD, United Kingdom.

###### Abstract

‘Gating’ is a widely observed phenomenon in biochemistry that describes the
transition between the activated (or open) and deactivated (or closed) states
of an ion-channel, which makes transport through that channel highly
selective. In general, gating is a mechanism that imposes an additional
restriction on a transport, as the process ends only when the ‘gate’ is open
and continues otherwise. When diffusion occurs in presence of a constant bias
to a gated target, i.e., to a target that switches between an open and a
closed state, the dynamics essentially slows down compared to ungated drift-
diffusion, resulting in an increase in the mean completion time, $\langle
T^{G}\rangle>\langle T\rangle$, where $T$ denotes the random time of transport
and $G$ indicates gating. In this work, we utilize stochastic resetting as an
external protocol to counterbalance the delay due to gating. We consider a
particle in the positive semi-infinite space that undergoes drift-diffusion in
the presence of a stochastically gated target at the origin and is moreover
subjected to a rate-limiting resetting dynamics. Calculating the minimal mean
completion time $\langle T_{r^{\star}}^{G}\rangle$ rendered by an optimal
resetting rate $r^{\star}$ for this exactly-solvable system, we construct a
phase diagram that owns three distinct phases: (i) where resetting can make
gated drift-diffusion faster even compared to the original ungated process,
$\langle T^{G}_{r^{\star}}\rangle<\langle T\rangle<\langle T^{G}\rangle$, (ii)
where resetting still expedites gated drift-diffusion, but not beyond the
original ungated process, $\langle T\rangle\leq\langle
T^{G}_{r^{\star}}\rangle<\langle T^{G}\rangle$, and (iii) where resetting
fails to expedite gated drift-diffusion, $\langle T\rangle<\langle
T^{G}\rangle\leq\langle T^{G}_{r^{\star}}\rangle$. We also highlight various
non-trivial behaviors of the completion time as the resetting rate, gating
parameters and the geometry of the set-up are carefully ramified. Gated drift-
diffusion aptly models various stochastic processes such as chemical reactions
that exclusively take place for certain activated state of the reactants. Our
work predicts the conditions where stochastic resetting can act as a useful
strategy to enhance the rate of such processes without compromising on their
selectivity.

###### pacs:

05.40.-a, 05.40.Jc

##  I Introduction

‘Gating’ in biochemistry typically refers to the transition between the open
and closed states of an ion channel; the ions are allowed to flow through the
channel only when it is openic1 . In a gated chemical reaction, the reactant
molecules switch between a reactive and a non-reactive state; the collisions
between the reactants must happen in their reactive states for a successful
reaction. Thus, gating is a signature of a constrained reaction/transport, be
it an enzyme finding the correct substrate or a protein finding the target
site along a DNA strand or associating to a cleavage site on a peptide. Given
their very generic nature, it is no wonder why gated processes showcase a
myriad of applications spanning across fields from chemistry gating-review ;
gated-6 ; gated-8 ; gated-9 , physics gated-10 ; gated-11 ; gated-12 ;
gated-13 ; gated-14 ; gated-15 ; gated-16  to biology gated-bio-1 ; gated-
bio-2 ; gated-bio-3 .
Since the seminal works of Szabo et al gated-8 ; gated-10 ; Szabo-* , gated
processes have gained considerable attention across a wide panorama of
applications such as 1D gated continuous-time and discrete-space random search
process in confinement gated-12 , intermittent switching dynamics of a protein
undertaking facilitated diffusion on a DNA strand conform-1 , 3D gated
diffusive search process with different diffusivities gated-13 , gated active
particles gated-active , to name a few. Gopich and Szabo explored the
possibility of multiple gated particles/targets in a model with intrinsic
reversible binding gated-15 . More on a mathematical side, Lawley and Keener
established a connection between a radiative/reactive boundary and a gated
boundary in diffusion controlled reactions robin . There has been a renewed
interest in the field emanating from the works by Mercado-Vásquez and Boyer on
a 1D gated diffusive process on the infinite line gated-0 , Scher and Reuveni
on gated reactions on arbitrary networks including random walk models both in
continuous gated-2  and discrete time gated-3  and Kumar et al on threshold
crossing events of a gated process gated-4  and inference of first-passage
times from the detection times of gated diffusive first-passage processes
inference-gated . In a similar vein, in this work, we delve deeper into a
gated diffusive process and in particular, focus on to design principles to
improve efficiency of a gated reaction [see Fig. (1)].
For a gated diffusive process to be complete, a certain condition that mimics
the open-gate-scenario, imposed either on the diffusing entity or on the
target that it diffuses to, needs to be fulfilled. This additional restriction
imposed due to gating certainly makes the process more selective, which is
essential for the associated biochemical system to function properly. The cost
for this selectivity, however, reflects on the completion time, which makes a
gated diffusive process essentially slower than the corresponding ungated one.
However, nature has its own way to curtail such situation to allow effective
reactions. For example, consider a chemical reaction such as in Fig. (1) where
an unbinding or a resetting from a metastable state can lead to a facilitated
reaction. The effects of such resetting events are proven to be crucial not
only in such chemical reactions chemical-1-r ; chemical-2-r , but also in the
backtracking of RNA polymerase RNA-1-r  and disassociation kinetics of RhoA in
the membrane Rhoa-1-r . Motion of the reactants or the morphogens which get
produced constantly from a certain place inside the cell before degrading in
time (due to their finite lifetime) can also be interpreted as stochastically
restarted processes morpho . More on the physics side, it has been established
in the pioneering work of Evans and Majumdar Restart1  that stochastic
resetting can be utilized as a powerful strategy to expedite the completion
time of a 1D diffusive search process. This remarkable result led to many
fascinating works where it was shown that indeed this intermittent resetting
strategy can benefit the search processes conducted by diffusion controlled
PalJphysA ; FP-pot-0 ; FP-pot ; FP-interval ; JCP-1 ; JCP-2 ; JCP-3 ;
DM_JPhysA ; RayPRE ; RW-OLB  and non-diffusive stochastic processes CB-2 ;
Restart7 ; Restart11 ; records  [see here Evansrev2020  for an extensive
review of the subject]. Single particle experiments using optical traps have
also paved the way of our understanding of resetting modulated search
processes expt-1 ; expt-2 . Albeit these advances, there has been a persistent
void in the understanding of resetting induced gated processes until recently
when resetting mechanism has been used to reduce the average completion time
of a gated diffusive process in 1D gated-1 ; gated-5 .
While resetting is a useful strategy to benefit diffusive transport of ions/
molecules in unbounded phase space, the same can not be said for a drift-
diffusive search process. There, resetting can be useful only if the rate of
diffusive transport is higher compared to the rate of driven transport PECLET
; branching ; channel ; local . However, if the drift supersedes the transport
of molecules across the channel, resetting can only hinder the completion
rendering a higher transit time. This crucial interplay can then be understood
in terms of the so-called Péclet number, which is a ratio between the
diffusion- and drift-dominated microscopic time-scales PECLET ; branching . In
fact, the role of resetting can be quantified within a universal set-up of
first-passage under restart which essentially teaches us that resetting will
be always beneficial if the underlying process without resetting has a
coefficient of variation (often known as the signal to noise ratio) – a
statistical measure of dispersion in the random completion time, defined as
the ratio of its standard deviation to its mean – greater than unity
inspection ; Restart7 ; Restart8 . These observations naturally set the stage
to understand the role of resetting in a gated drift-diffusive search process
which, to the best of our knowledge, has not been studied so far. In
particular, the central objective of this work is to understand whether
resetting can enhance the completion rate of a gated drift-diffusive search
process and if so, under what conditions. Unraveling the intricate role of
drift, diffusion and gating along with that of resetting will be at the heart
of this study.

Figure 1: Scheme for a gated chemical reaction between two reactants $R_{1}$
and $R_{2}$, catalysed by $C$. In the first step, the catalyst initiates the
reaction by binding reversibly with $R_{1}$ to generate $CR_{1}$, a metastable
intermediate: $C+R_{1}\stackrel{{\scriptstyle}}{{\rightleftharpoons}}CR_{1}$.
In the second step, $CR_{1}$ reacts selectively with $R^{\star}_{2}$ (the
reactive or open-gate state of $R_{2}$) to generate the product $P$ and
liberate the catalyst: $CR_{1}+R_{2}^{\star}\to C+P$. This step can be modeled
by gated drift-diffusion, while the unbinding of $C$ from $R_{1}$ is
essentially resetting.

To illustrate the set-up of the present work, let us consider a diffusive
transport process confined to the positive semi-infinite space in presence of
a constant drift which is directed towards a target that randomly switches
between an open and a closed state with constant rates. This is a general
scenario mimicked by, e.g., a chemical reaction [see Fig. 1], where the
collisions between reactants [$CR_{1}$ and $R_{2}$] lead to the formation of
product only when at least one of the reactants is in an activated state [when
$R_{2}$ exists as $R_{2}^{\star}$] and not otherwise. Since resetting is an
integral part of any consecutive chemical reaction that has a reversible
first-step [when $C$ binds to, or unbinds from, $R_{1}$], the reaction scheme
shown in Fig. 1 can be modeled as gated drift-diffusion with resetting. In
this paper, we study the completion time statistics for this general set-up.
In particular, we perform a comprehensive analysis to understand the role of
optimal resetting in enhancing the rate of such transport and construct a
complete phase diagram that distinguishes between the phases where (i)
resetting accelerates gated drift-diffusion beyond the original (without
resetting) ungated process, (ii) resetting still proves itself to be
beneficial by improving the rate of gated drift-diffusion, but not beyond the
original ungated process and (iii) resetting can not expedite gated drift-
diffusion.
The rest of the paper is organized as follows. In Section II, we describe the
model and follow a Fokker-Planck approach to write down the governing
equations for gated drift-diffusion process that is subject to resetting. In
the same Section, we solve those equations to calculate the mean completion
time, denoted $\langle T_{r}^{G}\rangle$. We calculate the optimal resetting
rate $r^{\star}$ that minimizes $\langle T_{r}^{G}\rangle$, and thereby
investigate the conditions for resetting to expedite the process completion in
Section III. There, various limits of the system parameters are also examined
in detail. We calculate the maximal speedup that can be achieved, within our
set-up, by the optimal resetting rate in Section IV. In Section V, we
construct a full phase diagram that characterizes all the possible regimes
underpinning the role of resetting in the process completion. We conclude with
a brief summary and outlook of our work in Section VI. Some of the detailed
derivations have been moved to the Appendix for brevity.

##  II Completion time statistics

We start by casting the problem of gated chemical reaction
[$CR_{1}+R_{2}^{\star}\to C+P$, introduced in Fig. 1] as gated drift-
diffusion. A convenient way to do so is to map the reaction coordinate
associated to the reactants $CR_{1}+R_{2}^{\star}$ onto the starting position
[$x_{0}>0$, see Fig. 2] of a particle that undergoes Brownian motion.
Similarly, the reaction coordinate for the products $C+P$ can be mapped onto
the position of a target [placed at the origin, see Fig. 2]. The chemical
potential drive, which governs the reaction to its completion, can then be
translated to a constant bias $\lambda$ that the particle experiences while it
diffuses to the target with a diffusion coefficient $D$. Note that $\lambda$
is considered to be positive when it acts towards the target and negative when
it acts away from the target. Such effective one-dimensional projection of the
energy landscape of an enzyme or protein conformation is a well-adapted
approach in the literature – see effect-pot-1 ; effect-pot-2 ; effect-pot-3
for more details. Nonetheless, there are a few key assumptions that we make
during the mapping. First, we take into account an well-established fact that
the chemical master equations for the discrete states can be coarse-grained
into the Fokker-Planck equations for the reaction coordinates in the
continuous space using a system size expansionkampen ; gardinar . However,
this conversion generically renders the drift and diffusion terms to be
spatially dependent. In other words, the reaction coordinate associated to the
catalysis process $CR_{1}+R_{2}^{\star}\to C+P$ is expected to diffuse in an
arbitrary energy landscape, where the product state $C+P$ is usually denoted
by the global minimum. However, to simplify the problem, we replace the
general potential by a linear one, which leads to a constant drift velocity
$\lambda$ towards (or away from) the gated target. This is the second
assumption that lies behind our analysis. Albeit these simplified
approximations, the major advantage here is the elegant analytical
tractability of the model from which one can also unveil the crucial interplay
between the gated boundary and resetting/unbinding mechanism.
To model gating, the target is considered to randomly switch between a
reactive ($\sigma=1$ state in Fig. 2 that resembles $R_{2}^{\star}$ in Fig. 1)
and a non-reactive state [$\sigma=0$ state in Fig. 2 that resembles $R_{2}$ in
Fig. 1]. The conversion from the non-reactive to the reactive state happens
with a constant rate $\alpha>0$ and that from the reactive to the non-reactive
state happens with a constant rate $\beta>0$ [Fig. 2]. Therefore, the reactive
occupancy of the target, i.e., the probability of finding the target in its
reactive state is $p_{r}\coloneqq\alpha/(\alpha+\beta)$. Similarly, the non-
reactive occupancy or the probability that the target is in its non-reactive
state is $p_{nr}\coloneqq(1-p_{r})=\beta/(\alpha+\beta)$. In analogy to the
chemical reaction, the process ends only when the particle hits the target in
its reactive state. When it hits the target in its non-reactive state, it
simply gets reflected and continues to diffuse.

Figure 2: Schematic diagram of a gated drift-diffusion process in semi-
infinite space with resetting, where the target (placed at the origin)
switches stochastically between a reactive state ($\sigma=1$) and a non-
reactive state ($\sigma=0$). The transition from the non-reactive to the
reactive state takes place with a constant rate $\alpha>0$, and the opposite
transition takes place with a constant rate $\beta>0$. When the particle,
starting at $x_{0}$, hits the target in its reactive state (dashed line), the
process ends. $T_{r}^{G}$ marks the random completion time of the gated drift-
diffusion process with resetting.

As mentioned in the Introduction, the unbinding of the catalyst $C$ from
$R_{1}$ can be interpreted as resetting at $x_{0}$. For simplicity, here we
consider Poissonian resetting with a constant rate $r$. In the gated drift-
diffusion scenario, this means that the particle is taken back to $x_{0}$
after stochastic intervals of time, taken from an exponential waiting time
distribution with mean $r^{-1}$. Note that here we assume that the resetting
is instantaneous and once the particle is reset at $x_{0}$, it immediately
starts moving again. In other words, we neglect any refractory period, i.e.,
idle time after each resetting event before drift-diffusion resumes,
considering that the time-scale of waiting at $x_{0}$ after reset [which maps
to the time required for $C$ to bind $R_{1}$ in Fig. 1] is much smaller
compared to that of either drift-diffusion or resetting. We also assume that
the intermittent dynamics of the target are independent of resetting similar
to gated-1 .

The fundamental quantity of interest here is the random completion time of the
gated chemical reaction, i.e., the first-passage time REDNER  of the diffusing
particle from $x_{0}$ to the target placed at the origin. To calculate the
mean first-passage time (MFPT), we first need to write the backward Fokker-
Planck equation for the survival probability of this system, which is the
total probability to find the particle in the interval $[0,\infty)$ at a time
$t$, provided the initial position is $x_{0}$. We denote $Q_{\sigma}(t|x_{0})$
as the joint survival probability, i.e, the probability that the particle has
not been absorbed by the target up to time $t$, given the initial position
$x_{0}$ and initial target state $\sigma(t=0)$. One can then construct the
backward Fokker-Planck equations gated-1  in terms of the initial position
$x_{0}$, which now serves as a variable

$\displaystyle\frac{\partial Q_{0}(t|x_{0})}{\partial t}=$
$\displaystyle-\lambda\frac{\partial Q_{0}(t|x_{0})}{\partial
x_{0}}+D\frac{\partial^{2}Q_{0}(t|x_{0})}{\partial
x_{0}^{2}}+\alpha[Q_{1}(t|x_{0})$ $\displaystyle-
Q_{0}(t|x_{0})]+r[Q_{0}(t|x_{r})-Q_{0}(t|x_{0})],$
$\displaystyle\frac{\partial Q_{1}(t|x_{0})}{\partial t}=$
$\displaystyle-\lambda\frac{\partial Q_{1}(t|x_{0})}{\partial
x_{0}}+D\frac{\partial^{2}Q_{1}(t|x_{0})}{\partial
x_{0}^{2}}+\beta[Q_{0}(t|x_{0})$ $\displaystyle-
Q_{1}(t|x_{0})]+r[Q_{1}(t|x_{r})-Q_{1}(t|x_{0})].$ (1)

Note that $x_{r}$ in the above set of equations indicates the resetting
position which has to be distinguished from the variable $x_{0}$ initially and
only at the end, has to be set $x_{r}=x_{0}$ self-consistently. The initial
conditions for Eq. (1) are $Q_{\sigma}(0|x_{0})=1$ and the boundary conditions
are $Q_{1}(t|0)=0$ and $[\partial Q_{0}(t|x_{0})/\partial x_{0}]_{x_{0}=0}=0$,
respectively. This indicates that the particle is absorbed at the target when
the latter is reactive ($\sigma=1$) and is reflected from the target when it
is non-reactive ($\sigma=0$) gated-1 . The average survival probability for
the gated process can then be written by taking contributions from both the
possibilities

$Q_{r}^{G}(t|x_{0})=p_{r}\;Q_{1}(t|x_{0})+(1-p_{r})Q_{0}(t|x_{0}).$ (2)

Subsequently, we use subscript $r$ and superscript $G$ to indicate resetting
and gating respectively in Eq. (2) and rest of the paper. The Laplace
transformation of Eq. (2) gives

$\displaystyle\tilde{Q}_{r}^{G}(s|x_{0})=p_{r}\;\tilde{Q_{1}}(s|x_{0})+(1-p_{r})\tilde{Q_{0}}(s|x_{0}),$
(3)

where
$\tilde{Q}_{r}^{G}(s|x_{0})\coloneqq\int_{0}^{\infty}dt\;e^{-st}Q_{r}^{G}(t|x_{0})$
is the Laplace transform of $Q_{r}^{G}(t|x_{0})$ and
$\tilde{Q}_{\sigma}(s|x_{0})\coloneqq\int_{0}^{\infty}dt\;e^{-st}Q_{\sigma}(t|x_{0})$
are Laplace transforms of $Q_{\sigma}(t|y)$, respectively. The average MFPT,
our observable of interest in this work, is also defined over the two random
possibilities and thus reads

$\displaystyle\langle T_{r}^{G}(x_{0})\rangle=p_{r}\langle
T_{1}(x_{0})\rangle+(1-p_{r})\langle T_{0}(x_{0})\rangle,$ (4)

where $\langle T_{\sigma}(x_{0})\rangle$ is the MFPT when the initial state of
the target is $\sigma$, given by $\langle
T_{\sigma}(x_{0})\rangle=\int_{0}^{\infty}\;dtQ_{\sigma}(t|x_{0})=\tilde{Q}_{\sigma}(s|x_{0})|_{s=0}$,
since $-\partial Q_{\sigma}(t|x_{0})/\partial t$ is the associated first-
passage time distribution REDNER ; gardinar . Therefore, the average MFPT
reads

$\displaystyle\langle
T_{r}^{G}(x_{0})\rangle=\int_{0}^{\infty}dt\;Q_{r}^{G}(t|x_{0})=\tilde{Q}_{r}^{G}(s|x_{0})|_{s=0}.$
(5)

For the rest of the paper, we will write $\langle T_{r}^{G}\rangle$ instead of
$\langle T_{r}^{G}(x_{0})\rangle$ for brevity.
Solving Eq. (1) in the Laplace space, plugging in the solutions, i.e.,
$\tilde{Q}_{\sigma}(s|x_{0})$s, into Eq. (3) to calculate
$\tilde{Q}_{r}^{G}(s|x_{0})$ and finally setting $s=0$ in the resulting
expression of $\tilde{Q}_{r}^{G}(s|x_{0})$ following Eq. (5), we obtain [see
Appendix A for detailed derivation] the explicit expression of the average
MFPT that reads

$\displaystyle\langle
T_{r}^{G}\rangle=\frac{1}{r}\left(e^{\mu_{1}x_{0}}-1\right)+\frac{\beta\mu_{1}}{\alpha\mu_{2}}\left(\frac{1}{r}+\frac{e^{-\mu_{2}x_{0}}}{\alpha+\beta}\right)e^{\mu_{1}x_{0}},$
(6)

where $\mu_{1}=(-\lambda+\sqrt{\lambda^{2}+4Dr})/2D>0$ and
$\mu_{2}=(-\lambda+\sqrt{\lambda^{2}+4D(\alpha+\beta+r)})/2D>0$. Therefore,
for pure diffusion with gating, i.e., when $\lambda\to 0$,
$\mu_{1}=\sqrt{r/D}$ and $\mu_{2}=\sqrt{(r+\alpha+\beta)/D}$ and Eq. (6) boils
down to gated-1

$\displaystyle\langle
T_{r}^{G}\rangle=\frac{e^{\sqrt{\frac{r}{D}}x_{0}}-1}{r}+\frac{\beta
e^{\sqrt{\frac{r}{D}}x_{0}}}{\alpha\sqrt{r[r+\alpha+\beta]}}\mbox{}\left[1+\frac{re^{-\sqrt{\frac{r+\alpha+\beta}{D}}x_{0}}}{(\alpha+\beta)}\right].$
(7)

Moreover, in the absence of gating, i.e., when $\beta\to 0$, Eq. (6) reduces
to $\langle
T_{r}\rangle=[\exp{({x_{0}(\sqrt{\lambda^{2}+4Dr}-\lambda)/2D}})-1]/r$, which
is the exact expression for the MFPT for ungated drift-diffusion with
Poissonian resetting PECLET .

Figure 3: The average MFPT, $\langle T_{r}^{G}\rangle$, vs the resetting rate
$r$ for different values of the drift velocity $\lambda$. The lines represent
analytical results following Eq. (6) and the symbols represent results from
numerical simulations [see Appendix D for details]. The curves with
$\lambda>0$ denote cases where the drift acts towards the gated target and
that with $\lambda<0$ denote otherwise. The variation of $\langle
T_{r}^{G}\rangle$ with $r$ is always non-monotonic for $\lambda\leq 0$,
however, for $\lambda>0$ it is non-monotonic for lower values of $\lambda$,
but monotonic for sufficiently high values of $\lambda$. Here we take
$x_{0}=2$, $D=1$, $\alpha=0.5$, and $\beta=0.5$ for all cases.

In Fig. 3, we plot $\langle T_{r}^{G}\rangle$ as a function of the resetting
rate $r$ for different values of the drift velocity $\lambda$, where the
reactive occupancy of the target, $p_{r}$, is kept constant. It is evident
from Fig. 3 that when $\lambda>0$, i.e., when the drift acts towards the
target, the average MFPT shows a non-monotonic variation with $r$ for lower
values of $\lambda$, indicating that the introduction of resetting expedites
first-passage when the process is diffusion-controlled. In contrast, $\langle
T_{r}^{G}\rangle$ increases monotonically with $r$ for sufficiently higher
values of $\lambda$, meaning that the introduction of resetting delays first-
passage when the process is drift-controlled. This trend can be explained in
the following way. When diffusion dominates over the drift, the particle tends
to diffuse away from the target. In such cases, resetting can effectively
truncate those long trajectories, which reduces the overall first-passage
time. In contrast, when the drift dominates over diffusion, the particle tends
to execute a directed motion towards the target ($\lambda>0$); resetting can
only hinder such transport resulting in a longer completion time. Note that
resetting can accelerate first-passage even when the dynamics is drift-
controlled, if the drive is away from the target ($\lambda<0$).
Summarizing, we see that either for pure diffusion or when the drift acts away
from the target ($\lambda\leq 0$), the average MFPT consistently shows a non-
monotonic variation with $r$. This means that the introduction of resetting
always accelerates first-passage for $\lambda\leq 0$. Thus, a hallmark of
resetting transition is apparent for $\lambda>0$, while no such transition is
expected for $\lambda\leq 0$. Indeed, similar to ungated diffusive processes
under resetting Restart1 ; PalJphysA , introduction of resetting was shown to
be always beneficial for pure diffusive gated process ($\lambda=0$) and no
such transition was observed there gated-1 . Here, we focus on the scenario
when the drift acts towards the target and then reveal the physical conditions
under which resetting strategy turns out to be beneficial. In passing, we will
also briefly discuss a situation where the system is confined between two
boundaries (one additional reflecting boundary apart from the stochastically
gated target). In analogy to a chemical reaction, the reflecting boundary is
often used to mimic high activation energy barriers in reaction coordinate. We
refer to Appendix B for this discussion, where we investigate the role of
resetting in details.

##  III The optimal resetting rate and a phase diagram for expedited
completion

The resetting transition is traditionally captured by the optimal resetting
rate (ORR, denoted $r^{\star}$), defined as the rate of resetting that
minimizes the average MFPT. Fig. 3 reveals that for pure diffusion
($\lambda=0$) the optimal resetting rate has a certain positive value (denoted
$r^{\star}=r_{0}^{\star}$, not marked in Fig. 3). With increase in the drift
towards the target, ORR gradually decreases to finally become zero at a
critical value of $\lambda$ (denoted $\lambda_{c}$, not shown in Fig. 3),
which marks the point of resetting transition. If we continue to increase
$\lambda$ beyond $\lambda_{c}$, ORR remains zero. Note that when the drift
acts away from the target ($\lambda<0$), $r^{\star}$ increases as that drift
becomes stronger. The optimal resetting rate $r^{\star}$ thus acts as an order
parameter, in the similar spirit as in classical phase transition, to explore
the resetting transition. Since ORR minimizes $\langle T_{r}^{G}\rangle$, it
can be calculated from the relation $d\langle
T_{r}^{G}\rangle/dr|_{r=r^{\star}}=0$, which leads to a complicated
transcendental equation that can not be solved analytically.

Figure 4: The scaled optimal resetting rate, $r^{\star}/r^{\star}_{0}$, vs
$\lambda$ for different values of $p_{r}$. Resetting transition in each case
is observed at $\lambda=\lambda_{c}$, where the scaled ORR becomes zero,
marked by dashed lines of the same color as the curve. For
$\lambda<\lambda_{c}$, resetting expedites transport, whereas for
$\lambda\geq\lambda_{c}$ it can not. Here we take $D=1$, $x_{0}=2$, and
$\alpha=0.5$ [which leads to $p_{r}=1/(2\beta+1)$] for all cases. For ungated
drift-diffusion [given by $p_{r}=1$, since $\beta\to 0$], the resetting
transition is observed at $\lambda_{c}=1$ (gray curve). For gated drift-
diffusion with $p_{r}<1$, $\lambda_{c}$ decreases below unity.

Solving the same numerically, we calculate $r^{\star}$, the optimal resetting
rate for $\lambda>0$. In a similar manner, the optimal resetting rate for pure
diffusion (for $\lambda=0$, denoted $r^{\star}_{0}$) is also obtained in order
to calculate the scaled ORR, $r^{\star}/r^{\star}_{0}$.
In Fig. 4, we plot the scaled ORR with respect to $\lambda$ for different
values of the reactive occupancy $p_{r}$ (we keep $\alpha$ constant and tune
$p_{r}$ by solely changing $\beta$). The non-zero values of the scaled ORR for
$\lambda<\lambda_{c}$ in Fig. 4 shows that resetting accelerates first-passage
in that regime. In stark contrast, for $\lambda\geq\lambda_{c}$ the scaled ORR
is zero, which shows that resetting can not accelerate first-passage in that
regime. For the ungated ($p_{r}=1$) process, $\lambda_{c}=1$ for our choice of
parameters, which is in exact agreement with earlier literature PECLET . It is
evident from Fig. 4 that $\lambda_{c}$ decreases when $p_{r}$ is decreased
below unity, which means that resetting expedites first-passage time upto a
critical drift (which is essentially smaller than $\lambda_{c}$ for the
ungated process) when gating is introduced to the system.

Figure 5: A phase diagram of $p_{r}$ vs. $\lambda$ based on the qualitative
effect of resetting on gated drift-diffusion. The black line represents the
condition for resetting transition ($\lambda_{c}$), which divides the entire
phase space in two parts. For $\lambda<\lambda_{c}$ resetting expedites
transport (white regime), whereas for $\lambda\geq\lambda_{c}$ resetting fails
to expedite transport (gray regime). Here we take $D=1$, $x_{0}=2$ and
$\alpha=0.5$ [i.e., $p_{r}=1/(2\beta+1)$] for all cases. For the ungated
process ($p_{r}=1$), $\lambda_{c}=1$ and it decreases with $p_{r}$. The
colored discs on the black line present the cases shown in Fig. 4. Following
the analysis of $\langle T_{r}^{G}\rangle$ in the limit $r\to 0$ [see Eq. (8)
and Eq. (10)], the separatrix (black line) is also obtained by plotting $f=0$.
Resetting is beneficial for $f<0$ (white regime), but not for $f>0$ (gray
regime).

In order to better understand how the critical value of $\lambda$ changes with
the reactive occupancy of the target, next we numerically calculate
$\lambda_{c}$ as a function of $p_{r}$. In Fig. 5, we construct a phase
diagram spanned by the reactive occupancy $p_{r}$ and the drift velocity
$\lambda$, where $\lambda_{c}$ acts as the separatrix that divides the entire
phase space in two parts; one where resetting expedites first-passage (white
regime) and the other where it can not (gray regime). We can recover the core
results observed in Fig. 4 from Fig. 5 in the following way. For each value of
$p_{r}\in[0,1]$ [encountered by moving horizontally through Fig. 5], the
dynamics is diffusion-controlled and resetting is beneficial if
$\lambda<\lambda_{c}$, whereas the dynamics is drift-controlled and resetting
turns out to be non-beneficial if $\lambda\geq\lambda_{c}$. It is clear from
Fig. 5 as well that with increase in reactive occupancy, $\lambda_{c}$
increases to finally become unity for the ungated process ($p_{r}=1$). Fig. 5
is, therefore, a complete yet compact representation of the condition for
resetting to expedite gated drift-diffusion. Next, we discuss an alternative
approach that can successfully generate this phase diagram by carefully
exploring the resetting criterion in the limit $r\to 0$.
Revisiting Fig. 3, we see that it clearly indicates when initial (in the limit
$r\to 0$) slope of the $\langle T_{r}^{G}\rangle$ vs. $r$ curve is negative,
introduction of resetting proves itself beneficial by decreasing the average
MFPT. In contrast, a positive slope of such curve in the limit $r\to 0$
suggests that introduction of resetting increases the average MFPT there. This
motivates us to explore the condition for resetting to accelerate gated drift-
diffusion by analyzing the expression of the average MFPT in the limit $r\to
0$. To do that, we first expand $\langle T_{r}^{G}\rangle$ in $r$ for an
infinitesimal resetting rate to obtain

$\displaystyle\langle T_{r}^{G}\rangle\approx\langle
T^{G}\rangle+r\left[\frac{\partial\langle T_{r}^{G}\rangle}{\partial
r}\right]_{r=0}\mbox{}+O(r^{2}),$ (8)

where $\langle T^{G}\rangle$ is the average MFPT in the absence of resetting.
Following Eq. (6), we get [see Appendix C for an alternative derivation]

$\displaystyle\langle T^{G}\rangle=\frac{1}{\lambda}\left[x_{0}+\frac{2\beta
D}{\alpha\left(\sqrt{4\alpha\text{D}+4\beta\text{D}+\lambda^{2}}-\lambda\right)}\right].$
(9)

Note that in the absence of gating ($p_{r}=1$, or $\beta=0$), Eq. (9) reduces
to $\langle T\rangle=x_{0}/\lambda$ REDNER , which implies that $\langle
T^{G}\rangle>\langle T\rangle$, since the second term in the right hand side
of Eq. (9) is always positive.
The expression for the second term in the right hand side of Eq. (8) is fairly
complicated, here we write it in a simpler form by introducing some new
parameters, viz., $\gamma\coloneqq\lambda^{2}/2D$, $Pe\coloneqq
x_{0}\lambda/2D$, and
$\kappa\coloneqq\sqrt{1+[4D(\alpha+\beta)/\lambda^{2}]}-1$, such that

$\displaystyle f\coloneqq\left.\frac{\partial\langle
T_{r}^{G}\rangle}{\partial
r}\right|_{r=0}\mbox{}=\frac{\beta}{2\gamma^{2}\alpha}\Big{[}\frac{2Pe-1}{\kappa}-$
$\displaystyle\frac{2}{\kappa^{2}(\kappa+1)}+\frac{2\gamma e^{-\kappa
Pe}}{\kappa(\alpha+\beta)}\Big{]}$ $\displaystyle+$
$\displaystyle\frac{1}{2\gamma^{2}}Pe(Pe-1).$ (10)

Note that $Pe$ in Eq. (10) is the Péclet number, i.e., the ratio between the
rate of driven transport to that of diffusive transport and $\gamma^{-1}$ is
the fastest first-passage time (smallest decay time) in the strong drift limit
as was pointed out by Redner in REDNER .
It is evident from Eq. (8) that in the limit $r\to 0$, resetting is expected
to expedite the completion of the process, i.e., $\langle
T_{r}^{G}\rangle<\langle T^{G}\rangle$, when $f<0$. This is a sufficient
condition (may not be necessary) for resetting to be useful. When $f>0$,
however, resetting is expected to delay the completion of the process, i.e.,
$\langle T_{r}^{G}\rangle>\langle T^{G}\rangle$. Therefore, the condition
$f=0$ should divide the entire phase space created by ($\lambda,p_{r}$) in two
parts; one where resetting is beneficial ($f<0$) and the other where it is not
($f>0$). Indeed, when we plot $f=0$ following Eq. (10) in the same phase
diagram presented in Fig. 5, it exactly overlaps on the existing separatrix
plotted earlier by calculating the critical drift $\lambda_{c}$ as a function
of $p_{r}$. Therefore, the condition $f<0$ marks the phase where introduction
of resetting accelerates transport to the target (white regime) while $f>0$
marks the phase where introduction of resetting delays the same (gray regime),
as expected. Revisiting Eq. (10), we note that in the limit of $p_{r}\to 1$
(for $\beta\to 0$), the first term of Eq. (10) vanishes. Then, $f<0$ boils
down to $Pe<1$ (meaning $Pe=1$ is the resetting transition point), the
condition where resetting accelerates ungated drift-diffusion, as obtained in
earlier works branching ; PECLET ; local .
The phase diagram in Fig. 5 is generated only for a fixed value for $\alpha$.
Next, we perform similar exercise for other values of $\alpha$ and display the
results in Fig. 6. It is evident from Fig. 6 that the separatrix, i.e, the
phase boundary that separates the two phases – the “resetting-beneficial”
phase at the left and the “resetting-detrimental” at the right – varies with
$\alpha$. In fact, for larger values of $\alpha$, the separatrix divides the
phases in such a way that the “resetting-beneficial” phase becomes
considerably smaller and the “resetting-detrimental” phase occupies most of
the phase space, implying that the effect of resetting becomes rather
constrained in that limit. Next, we examine two important limits of the gating
rates, viz., $\alpha\to 0$ and $\alpha,\beta\to\infty$.

Figure 6: A phase diagram of $p_{r}$ vs. $\lambda$ for different values of
$\alpha$. Each phase boundary (obtained for a certain value of $\alpha$, shown
by the curves) divides the phase space into a “resetting-beneficial” phase
(left to the curve) and a “resetting-detrimental” phase (right to the curve).
For large values of $\alpha$, the “resetting-beneficial” phase becomes
considerably smaller as the “resetting-detrimental” phase occupies the
majority of the phase space.

###  III.1 The limit $\alpha\to 0$

The limit $\alpha\to 0$ essentially implies that the target is reflective
almost all the time. In what follows, we will show that this is a delicate
limit and should be handled carefully. Strictly speaking, the limit $\alpha\to
0$ should be interpreted as $\beta\gg\alpha$, which means that the target has
higher probability to remain non-reactive than being reactive. Using this
limit in Eq. (6), one finds

$\displaystyle\langle
T_{r}^{G}\rangle=\frac{1}{r}\left(e^{\mu_{1}x_{0}}-1\right)+\frac{\beta\mu_{1}}{r\alpha\mu_{2}}e^{\mu_{1}x_{0}},$
(11)

where $\mu_{1}=(-\lambda+\sqrt{\lambda^{2}+4Dr})/2D>0$ and
$\mu_{2}=(-\lambda+\sqrt{\lambda^{2}+4D(\alpha+\beta+r)})/2D>0$, as before.
Moreover, one can disregard the first term in the right hand side of Eq. (11)
for finite resetting rate $r$ (assuming $\beta\gg r$) to find

$\displaystyle\langle
T_{r}^{G}\rangle=\frac{\beta\left(\sqrt{4Dr+\lambda^{2}}-\lambda\right)e^{\frac{x_{0}\left(\sqrt{4Dr+\lambda^{2}}-\lambda\right)}{2D}}}{\alpha
r\left(\sqrt{4\beta D+\lambda^{2}}-\lambda\right)}.$ (12)

To understand the behavior of the average MFPT, we plot $\langle
T_{r}^{G}\rangle$ as a function of the resetting rate in Fig. 7 for large
$\beta$ and small $\alpha$ keeping $\beta/\alpha\gg 1$. Intriguingly, we find
that in this limit, $\langle T_{r}^{G}\rangle$ shows both monotonic and non-
monotonic behavior as one varies $r$ for different $\lambda$. In particular,
the latter case implies that there exists a resetting rate for which $\langle
T_{r}^{G}\rangle$ becomes optimally minimum. To find this optimal resetting
rate, we set $\left.\frac{d\langle
T_{r}^{G}\rangle}{dr}\right|_{r=r^{\star}}=0$ and obtain

$\displaystyle r^{\star}=\frac{D}{x_{0}^{2}}-\frac{\lambda}{x_{0}},$ (13)

which is a simple function of the diffusion constant, initial position and
drift. The linear behavior of $r^{\star}$ with respect to the drift variable
$\lambda$ is also noteworthy. Finally, setting $\lambda=0$, one recovers
$r^{\star}=D/x_{0}^{2}$ which was obtained in gated-1 .
The appearance of an ORR in the limit $\alpha\to 0$ i.e., when the target is
poorly reactive (the so-called cryptic regime gated-0 ; gated-2 ) is counter-
intuitive in contrast to the case of $\alpha=0$ (purely reflective boundary).
It can be argued that a diffusing particle usually makes several encounters
(also aided by the drift) with the target regardless of the target’s state. In
the current context, although most of the time the particle may remain
unsuccessful to find the target in a reactive state, it can still get absorbed
whenever there is an element of chance for the target to become reactive. More
details about such cryptic targets and their natural appearances in chemical,
biological and ecological systems can be found in gated-0 ; gating-review ;
golding ; prey-cryptic .

Figure 7: Monotonic and non-monotonic behaviour of the MFPT in the limit
$\alpha\to 0$ (i.e., $\beta/\alpha\gg 1$), as obtained from Eq. (12). The non-
monotonic behaviour gradually vanishes as $\lambda$ goes beyond the resetting
transition point $\lambda_{c}=D/x_{0}=0.5$. This can be obtained by setting
$r^{\star}$ in Eq. (13) to zero.

###  III.2 The limit $\alpha,\beta\to\infty$

In the limit $\alpha,\beta\to\infty$, recalling Eq. (10) we can approximate
$\kappa\approx\sqrt{4D(\alpha+\beta)}/\lambda$. Thus, one can safely neglect
the terms of order $O(\frac{1}{\kappa^{2}})$ including the exponential term.
Under this assumption, setting $f=0$ in Eq. (10) we obtain an exact expression
for the critical drift $\lambda_{c}$ that reads

$\displaystyle\lambda_{c}=\frac{2D}{x_{0}}\left[\frac{D+x_{0}\frac{\alpha}{\beta}\sqrt{\alpha+\beta}\sqrt{D}}{2D+x_{0}\frac{\alpha}{\beta}\sqrt{\alpha+\beta}\sqrt{D}}\right].$
(14)

We notice that even in this limit $p_{r}$ does not solely control the dynamics
of the system. More from the physical point of view, one can refer to this
limit as the partially reactive boundary, where the target switches between
the reactive and non-reactive states so fast (i.e., the time scale of such
switch is much smaller compared to the time scale of drift-diffusion) that the
particle feels an average state of the target and interacts with it with a
certain probability all the time – see also gating-review ; gated-0 .
So far, we performed a detailed analysis to understand the effect of resetting
on the dynamics of gated drift-diffusion. In the next section, we turn our
attention to the maximal speed-up that can be gained by resetting the process
at an optimal rate.

##  IV The maximal speedup for process completion

The maximal speedup, i.e., the speedup rendered by optimal resetting rate
$r^{\star}$, is generally defined as the ratio of the MFPT of the original
(underlying) process without resetting to that of the process with optimal
resetting. Therefore, setting $r=r^{\star}$ in Eq. (6) and utilizing Eq. (9),
we obtain the maximal speedup for gated drift-diffusion process in a
straightforward way, which reads

$\displaystyle\footnotesize\frac{\langle T^{G}\rangle}{\langle
T^{G}_{r^{\star}}\rangle}=\begin{cases}\frac{r^{\star}(\alpha+\beta)(2D\beta+x_{0}\alpha\mu_{2}^{\star})}{\lambda\left[r\beta\mu_{1}^{\star}e^{\frac{x_{0}}{2D}\left(\mu_{1}^{\star}+\mu_{2}^{\star}\right)}+\left(\alpha+\beta\right)\left[e^{\frac{x_{0}\mu_{1}^{\star}}{2D}}{(\beta\mu_{1}^{\star}+\alpha\mu_{2}^{\star})-\alpha\mu_{2}^{\star}}\right]\right]}\;\;\mbox{for}\;\;\lambda<\lambda_{c}\\\
1\;\;\;\;\;\;\;\;\;\;\;\mbox{for}\;\;\lambda\geq\lambda_{c},\end{cases}$ (15)

where $\mu_{1}^{\star}=\mu_{1}(r^{\star})$ and
$\mu_{2}^{\star}=\mu_{2}(r^{\star})$, $\mu_{1}$ and $\mu_{2}$ having the
expressions given after Eq. (6). In Fig. 8, we plot $\langle
T^{G}\rangle/\langle T^{G}_{r^{\star}}\rangle$ for different values of the
reactive occupancy $p_{r}$. Fig. 8 indicates that the maximal speedup for
gated drift-diffusion is most marked when drift towards the target is
negligible. With increase in $\lambda$, it gradually decreases to finally
become unity at the point of resetting transition, $\lambda_{c}$. We see that
when $p_{r}=1$, i.e, in the absence of gating, the resetting transition is
observed at $\lambda_{c}=1$ for our choice of parameters PECLET , but when
$p_{r}$ is decreased below unity, the transition is observed for lower values
of $\lambda_{c}$, as observed earlier in Fig. 5.

Figure 8: The maximal speedup for the gated process with resetting compared to
the gated process without resetting,
$\left<T^{G}\right>/\left<T^{G}_{r^{\star}}\right>$, vs the drift velocity
$\lambda$, for different values of the reactive occupancy, $p_{r}$. The
vertical dashed lines mark the points of resetting transitions for curves of
same color, denoted $\lambda_{c}$, such that
$\left<T^{G}\right>/\left<T^{G}_{r^{\star}}\right>>1$ for
$\lambda<\lambda_{c}$, and
$\left<T^{G}\right>/\left<T^{G}_{r^{\star}}\right>=1$ at
$\lambda\geq\lambda_{c}$. We take $D=1$, $x_{0}=2$ and $\alpha=0.5$ [i.e.,
$p_{r}=1/(2\beta+1)$] for all cases. In the absence of gating ($p_{r}=1$, gray
curve), the resetting transition is observed at $\lambda_{c}=2D/x_{0}=1$, and
$\lambda_{c}$ decreases when gating is introduced ($p_{r}<1$).

Next, we compare $\left<T^{G}_{r^{\star}}\right>$ to $\left<T\right>$, the
MFPT for ungated drift-diffusion without resetting, to explore whether it is
possible to overcome the increase in the MFPT due to gating by resetting the
process in an optimal way. Recalling that the MFPT of ungated drift-diffusion
is given by $\langle T\rangle=x_{0}/\lambda$ and utilizing Eq. (6) for
$r=r^{\star}$ as before, we get

$\displaystyle\footnotesize\frac{\langle T\rangle}{\langle
T^{G}_{r^{\star}}\rangle}=\begin{cases}\frac{r^{\star}x_{0}\alpha(\alpha+\beta)\mu_{2}^{\star}}{\lambda\left[r\beta\mu_{1}^{\star}e^{\frac{x_{0}}{2D}\left(\mu_{1}^{\star}+\mu_{2}^{\star}\right)}+\left(\alpha+\beta\right)\left[e^{\frac{x_{0}\mu_{1}^{\star}}{2D}}{(\beta\mu_{1}^{\star}+\alpha\mu_{2}^{\star})-\alpha\mu_{2}^{\star}}\right]\right]}\;\;\mbox{for}\;\;\;\;\lambda<\lambda_{c}\\\
\frac{x_{0}\alpha\mu_{2}^{\star}}{2D\beta+x_{0}\alpha\mu_{2}^{\star}}\;\;\;\;\;\;\;\;\;\;\;\mbox{for}\;\;\lambda\geq\lambda_{c}.\end{cases}$
(16)

Figure 9: The maximal speedup for the gated process with resetting compared to
the ungated process without resetting,
$\left<T\right>/\left<T^{G}_{r^{\star}}\right>$, vs $\lambda$, for different
values of $p_{r}$. The vertical dashed lines mark the points of resetting
transition ($\lambda_{c}$) for curves of same color, whereas the colored discs
mark the values of $\lambda_{c}^{0}$, such that optimal resetting expedites
gated process beyond the original ungated process only when
$\lambda<\lambda_{c}^{0}$. Here $D=1$, $x_{0}=2$ and $\alpha=0.5$ [i.e.,
$p_{r}=1/(2\beta+1)$] for all cases. In the absence of getting ($p_{r}=1$,
gray curve), the point of resetting transition ($\lambda_{c}=2D/x_{0}=1$)
coincides with $\lambda_{c}^{0}$.

Plotting $\langle T\rangle/\langle T^{G}_{r^{\star}}\rangle$ as a function of
$\lambda$ in Fig. 9 for different values of $p_{r}$, we see that the maximal
speedup compared to the original ungated process is infinite when there is no
drift towards the target and it decreases with an increase in $\lambda$, as
expected. In fact, Fig. 9 clearly shows that for sufficiently low values of
$\lambda$, optimal resetting can make the process even $>10$ times faster! It
proves that resetting is indeed a useful strategy to compensate for the delay
due to gating, and when the dynamics is diffusion-controlled, it can even
improve the rate of transport (which is inversely proportional to the mean
completion time) beyond the original ungated process without resetting. It is
also observed from Fig. 9 that in the absence of gating (when $p_{r}=1$), the
minimal possible value for $\langle T\rangle/\langle T^{G}_{r^{\star}}\rangle$
is unity, which is achieved for $\lambda\geq\lambda_{c}$PECLET . In contrast,
when $p_{r}<1$, $\langle T\rangle/\langle T^{G}_{r^{\star}}\rangle$ is reduced
below unity even before the resetting transition sets in. Therefore, denoting
$\lambda_{c}^{0}$ as the critical value of $\lambda$ [marked in Fig. 9 by the
colored discs] where $\langle T\rangle/\langle T^{G}_{r^{\star}}\rangle$
becomes unity for a certain $p_{r}$, we observe that
$\lambda_{c}^{0}\leq\lambda_{c}$. The equality holds only for the ungated
process and the difference between $\lambda_{c}^{0}$ and $\lambda_{c}$ is
prominent for lower values of $p_{r}$. These observations lead us to identify
all the possible distinct regimes where resetting can benefit the completion
of the gated drift-diffusion process. In what follows, we construct a
comprehensive phase diagram in the parameter space encapsulating all these
effects of resetting.

##  V The complete phase diagram

Figure 10: A complete phase diagram of $p_{r}$ vs. $\lambda$ showing three
distinct phases; (i) phase I: where optimal resetting enhances the rate of
gated drift-diffusion beyond the original (without resetting) ungated process,
given by $\left<T^{G}_{r^{\star}}\right><\left<T\right><\left<T^{G}\right>$,
(ii) phase II: when optimal resetting improves the rate of the gated process,
but not compared to the original ungated process, given by
$\left<T\right>\leq\left<T^{G}_{r^{\star}}\right><\left<T^{G}\right>$, and
(iii) phase III: when resetting can not improve the rate of gated process,
given by
$\left<T\right><\left<T^{G}\right>\leq\left<T^{G}_{r^{\star}}\right>$. The
horizontal colored lines mark the cases shown in Fig. 9. Here we consider
$\alpha=0.5$ [such that $p_{r}=1/(2\beta+1)$], $D=1$ and $x_{0}=2$. Phases I
and II merge together at $\lambda_{c}=2D/x_{0}=1$, the point of resetting
transition for drift-diffusion in the absence of gating ($p_{r}=1$).

To construct the complete phase diagram for the problem, we revisit Fig. 5 and
Fig. 8, and recall that the entire phase space [spanned by $(\lambda,p_{r})$]
is divided into two parts, viz., $\langle T^{G}_{r^{\star}}\rangle<\langle
T^{G}\rangle$ [i.e., where optimal resetting expedites transport] and $\langle
T^{G}_{r^{\star}}\rangle\geq\langle T^{G}\rangle$ [i.e., where optimal
resetting can not expedite transport], and the transition between these two
phases takes place at a critical point $\lambda=\lambda_{c}$. The study of the
maximal speedup of the gated process with resetting compared to the ungated
process without resetting in Section IV suggests that we can further divide
the former phase in two parts: one where the rate of transport for the gated
process with optimal resetting is higher than that of original ungated process
i.e., $\langle T^{G}_{r^{\star}}\rangle<\langle T\rangle$ and the other where
it is not i.e., $\langle T^{G}_{r^{\star}}\rangle\geq\langle T\rangle$. The
transition between these two phases happens at
$\lambda=\lambda_{c}^{0}\leq\lambda_{c}$. Since gating essentially makes
drift-diffusion slower in the absence of resetting, i.e,
$\left<T\right><\left<T^{G}\right>$ [see Eq. (9)], summarizing our
observations, we construct a complete phase diagram for the present problem
[displayed in Fig. 10], which consists of three distinct phases: (i) phase I:
when optimal resetting makes the gated process faster than the original
ungated (and hence the original gated) process, given by
$\left<T^{G}_{r^{\star}}\right><\left<T\right><\left<T^{G}\right>$, (ii) phase
II: when optimal resetting makes the gated process faster than the original
gated process, but not the original ungated process, given by
$\left<T\right>\leq\left<T^{G}_{r^{\star}}\right><\left<T^{G}\right>$, and
(iii) phase III: when resetting can not make the gated process faster than the
original gated (and hence the original ungated) process, given by
$\left<T\right><\left<T^{G}\right>\leq\left<T^{G}_{r^{\star}}\right>$. The
transition points $\lambda_{c}^{0}$ create the separatrix between phases I and
II, whereas the transition points $\lambda_{c}$ create that between phases II
and III. Since $\lambda_{c}^{0}=\lambda_{c}$ for $p_{r}=1$, in Fig. 10 we find
phases I and II to merge together in the absence of gating. This diagram in
the phase space of two important parameters of the system, namely the reactive
occupancy and the bias, allows us to delineate the exact nature of resetting
in the process completion. In other words, we can gain maximal benefits from a
precise and a priori knowledge of the parameter space.

##  VI Conclusions

In this work, we performed an in-depth analysis on the completion time
statistics of drift-diffusive transport to a stochastically gated target in
the presence of Poissonian resetting. In particular, we strategically explored
the conditions where resetting can enhance the rate of such transport as has
been shown for ungated processes where resetting stabilizes the non-
equilibrium motion Restart1 ; Evansrev2020 ; Restart4 ; ss ; work-fluc  by
removing the detrimental long trajectories that result in a slower transport
rate. Projecting the general problem of gated drift-diffusion with resetting
to a gated chemical reaction initiated by a catalyst [as discussed in Fig. 1],
the main results of the present work can be interpreted as follows.
We observed that the rate of product formation depends on an interesting
interplay between the chemical potential drive that governs the reaction
($\lambda$), the probability ($p_{r}$) that gated reactant stays in its
activated state [when $R_{2}$ exists as $R_{2}^{\star}$], and the rate of
unbinding (or resetting, with rate $r$) of the catalyst $C$ from $CR_{1}$.
When the chemical potential drive towards the product ($\lambda>0$) is
low/moderate such that the reaction is diffusion-controlled, the rate of
reaction is maximized for an optimal unbinding rate ($r^{\star}$) of the
catalyst. In contrast, when the drive $\lambda$ is strong, i.e., the reaction
is drift-controlled, unbinding of the catalyst $C$ from $CR_{1}$ decreases the
rate of reaction. A transition is thus observed at a critical drive,
$\lambda_{c}$, which grows with $p_{r}$ and becomes maximum for $p_{r}=1$,
i.e., for the ungated reaction. Strikingly enough, we observed that for
$\lambda<\lambda_{c}^{0}$ [when $\lambda_{c}^{0}\leq\lambda_{c}$ is another
critical value of $\lambda$ that increases with $p_{r}$ and attains a maximum
$\lambda_{c}^{0}=\lambda_{c}$ at $p_{r}=1$], optimal unbinding with a rate
$r^{\star}$ can make the reaction even $>10$ times faster compared to the
ungated reaction in the limit $r\to 0$ (i.e., where the binding step is almost
irreversible).
These observations lead to a complete phase diagram based on the qualitative
and quantitative effect of optimal unbinding (resetting) on gated chemical
reaction [modeled by drift-diffusion to a gated target] that consists of three
distinct phases. Recalling that effect-pot-3  rate of product formation
$\propto\mbox{(mean completion time of reaction)}^{-1}$, these three phases
are identified through the following conditions: (i) where $\langle
T^{G}_{r^{\star}}\rangle<\langle T\rangle<\langle T^{G}\rangle$, i.e., where
the rate of gated chemical reaction is enhanced by optimal unbinding of
catalyst beyond that of ungated/gated reactions when the binding is almost
irreversible [$r\to 0$], (ii) where $\langle T\rangle\leq\langle
T^{G}_{r^{\star}}\rangle<\langle T^{G}\rangle$, i.e., when optimal unbinding
improves the rate of gated reaction, but not beyond the ungated reaction in
the limit $r\to 0$, and (iii) where $\langle T\rangle<\langle
T^{G}\rangle\leq\langle T^{G}_{r^{\star}}\rangle$, i.e., when unbinding fails
to make the gated reaction faster than either of the gated/ungated reaction
for almost irreversible binding.
The model considered herein generally applies to gated drift-diffusion under
the influence of resetting. The major advantage besides its analytical
tractability is that one can also gain deep insights about the intricate
trade-offs between gating and resetting mechanisms, both of which are
essential components of chemical reaction networks. Generalization of this
simple model to a generic space-dependent diffusion process in an arbitrary
energy landscape in the presence of gated targets would be a potential
research avenue. A detailed numerical analysis to this end would be a
worthwhile pursuit. Notably, such theoretical models can capture physical
situations arising in experiments that study, e.g., completion time statistics
of protein folding by gated fluorescence quenching. There, the protein is
tagged/labelled by a fluorophore reversibly binds the protein (here unbinding
is similar to resetting) to impart fluorescence properties Quenching . Once
the tagged protein folds to its native state, the quencher selectively reacts
with the active site of that folded protein in its fluorescent state, provided
that site is in its open (exposed) conformation. If the active site of the
folded protein remains in a closed (hidden) conformation, the quencher fails
to react with it, which implicates gating. A successful reaction thus occurs
only in the exposed conformation, which leads to subsequent quenching of
fluorescence thereby marking the completion of the folding process
Gated_Quenching . We believe that our work can shed light in understanding and
harnessing various gating and resetting protocols inherent to such systems.

## Acknowledgement

AP gratefully acknowledges research support from the DST-SERB Start-up
Research Grant Number SRG/2022/000080 and the Department of Atomic Energy,
Govt. of India. DM thanks SERB (Project No. ECR/2018/002830/CS), DST, Govt. of
India, for financial support and IIT Tirupati for the new faculty seed grant.
SR acknowledges the Elizabeth Gardner Fellowship by School of Physics &
Astronomy, University of Edinburgh and the INSPIRE Faculty research grant by
DST, Govt. of India, executed at IIT Tirupati. The numerical calculations
reported in this work were carried out on the Nandadevi cluster, which is
maintained and supported by IMSc’s High-Performance Computing Center. For the
purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission. We thank the anonymous Reviewers for their insightful
remarks.

## DATA AVAILABLITY

The data that supports the findings of this work are available within the
paper, its appendices, and in github .

##  Appendix A Calculation of the average MFPT by solving Eq. (1) in the
Laplace space

Here, we provide the solution of Eq. (1) in the Laplace space. Following that,
we calculate the mean first passage time for a diffusing particle with drift,
starting from an initial position $x_{0}$, to reach the gated target. We start
by Laplace transforming Eq. (1)

$\displaystyle D\frac{\partial^{2}\tilde{Q}_{0}(s|x_{0})}{\partial
x_{0}^{2}}-\lambda\frac{\partial\tilde{Q}_{0}(s|x_{0})}{\partial x_{0}}$
$\displaystyle-(s+\alpha+r)\tilde{Q}_{0}(s|x_{0})+\alpha\tilde{Q}_{1}(s|x_{0})$
$\displaystyle=-1-r\tilde{Q}_{0}(s|x_{r})$ $\displaystyle
D\frac{\partial^{2}\tilde{Q}_{1}(s|x_{0})}{\partial
x_{0}^{2}}-\lambda\frac{\partial\tilde{Q}_{1}(s|x_{0})}{\partial x_{0}}$
$\displaystyle-(s+\beta+r)\tilde{Q}_{1}(s|x_{0})+\beta\tilde{Q}_{0}(s|x_{0})$
$\displaystyle=-1-r\tilde{Q}_{1}(s|x_{r}),$ (A.1)

where
$\tilde{Q}_{\sigma}(s|y)\coloneqq\int_{0}^{\infty}dt\;e^{-st}Q_{\sigma}(t|y)$
denote the Laplace transform of $Q_{\sigma}(t|y)$. Eq. (A.1) is a second
order, linear and non-homogeneous differential equation. Considering

$\tilde{Q}_{\sigma}(s|x_{0})=\tilde{Q}_{\sigma}^{h}(s|x_{0})+\tilde{Q}_{\sigma}^{inh}(s|x_{0}),$
(A.2)

where $\tilde{Q}_{\sigma}^{h}(s|x_{0})$ and
$\tilde{Q}_{\sigma}^{inh}(s|x_{0})$ denote the homogeneous and inhomogeneous
parts of $\tilde{Q}_{\sigma}(s|x_{0})$, respectively, we can rewrite Eq. (A.1)
in two separate parts. The homogeneous part reads

$\displaystyle D\frac{\partial^{2}\tilde{Q}_{0}^{h}(s|x_{0})}{\partial
x_{0}^{2}}-\lambda\frac{\partial\tilde{Q}_{0}^{h}(s|x_{0})}{\partial
x_{0}}-(s+\alpha$ $\displaystyle+r)\tilde{Q}_{0}^{h}(s|x_{0})$
$\displaystyle+\alpha\tilde{Q}_{1}^{h}(s|x_{0})=0,$ $\displaystyle
D\frac{\partial^{2}\tilde{Q}_{1}^{h}(s|x_{0})}{\partial
x_{0}^{2}}-\lambda\frac{\partial\tilde{Q}_{1}^{h}(s|x_{0})}{\partial
x_{0}}-(s+\beta$ $\displaystyle+r)\tilde{Q}_{1}^{h}(s|x_{0})$
$\displaystyle+\beta\tilde{Q}_{0}^{h}(s|x_{0})=0.$ (A.3)

Similarly, the inhomogeneous part reads

$\displaystyle-(s+\alpha+r)\tilde{Q}_{0}^{inh}(s|x_{0})+\alpha\tilde{Q}_{1}^{inh}(s|x_{0})=-1-r\tilde{Q}_{0}(s|x_{r}),$
$\displaystyle-(s+\beta+r)\tilde{Q}_{1}^{inh}(s|x_{0})+\beta\tilde{Q}_{0}^{inh}(s|x_{0})=-1-r\tilde{Q}_{1}(s|x_{r}),$

which is a set of algebraic equations. Solving Eq. (LABEL:Q_inh) we obtain

$\displaystyle\tilde{Q}_{0}^{inh}(s)=\frac{1}{s+r}\bigg{(}1+\frac{r}{s+r+\alpha+\beta}$
$\displaystyle[\alpha\tilde{Q}_{1}(s|x_{r})$
$\displaystyle+(s+r+\beta)\tilde{Q}_{0}(s|x_{r})]\bigg{)},$
$\displaystyle\tilde{Q}_{1}^{inh}(s)=\frac{1}{s+r}\bigg{(}1+\frac{r}{s+r+\alpha+\beta}$
$\displaystyle[\beta\tilde{Q}_{0}(s|x_{r})$
$\displaystyle+(s+r+\alpha)\tilde{Q}_{1}(s|x_{r})]\bigg{)}.$ (A.5)

We note that $\tilde{Q}_{\sigma}^{inh}(s)$ depends on $x_{r}$ and not on
$x_{0}$. Next, we proceed to solve Eq. (A.3). Noting that $x_{r}$ does not
appear in Eq. (A.3), we simply write
$\tilde{Q}_{\sigma}^{h}(s|x_{0})\equiv\tilde{Q}_{\sigma}^{h}$ for notational
convenience. Writing Eq. (A.3) in a matrix form, we find

$\displaystyle D\frac{\partial^{2}}{\partial
x_{0}^{2}}\begin{pmatrix}\tilde{Q}_{0}^{h}\\\ \\\
\tilde{Q}_{1}^{h}\end{pmatrix}$ $\displaystyle-\lambda\frac{\partial}{\partial
x_{0}}\begin{pmatrix}\tilde{Q}_{0}^{h}\\\ \\\ \tilde{Q}_{1}^{h}\end{pmatrix}$
$\displaystyle+\begin{pmatrix}-(s+\alpha+r)&\alpha\\\ \\\
\beta&-(s+\beta+r)\end{pmatrix}\begin{pmatrix}\tilde{Q}_{0}^{h}\\\ \\\
\tilde{Q}_{1}^{h}\end{pmatrix}=0.$ (A.6)

Taking $\Psi=\left(\tilde{Q}_{0}^{h}\;\;\;\tilde{Q}_{1}^{h}\right)^{T}$, we
rewrite Eq. (A.6)

$\displaystyle D\frac{\partial^{2}}{\partial
x_{0}^{2}}\Psi-\lambda\frac{\partial}{\partial
x_{0}}\Psi+\textbf{A}\Psi=\bf{0},$
$\displaystyle\mbox{where}\;\;\;\;\;\textbf{A}=\begin{pmatrix}-(s+\alpha+r)&\alpha\\\
\\\ \beta&-(s+\beta+r)\end{pmatrix}.$ (A.7)

We now choose $\Psi=\Phi e^{mx_{0}}$ as the trial solution of Eq. (A.7), that
generates the characteristic equation $Dm^{2}\Phi-\lambda
m\Phi+\textbf{A}\Phi=\bf{0}$, which gives

$\displaystyle\begin{pmatrix}Dm^{2}-\lambda m-(s+\alpha+r)&\alpha\\\ \\\
\beta&Dm^{2}-\lambda m-(s+\beta+r)\end{pmatrix}\Phi=\bf{0}.$ (A.8)

The roots of Eq. (A.8) can be found as

$\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda-\sqrt{\lambda^{2}+4D(s+r)}}{2D},$ $\displaystyle
m_{2}$ $\displaystyle=$
$\displaystyle\frac{\lambda-\sqrt{\lambda^{2}+4D(\alpha+\beta+s+r)}}{2D},$
$\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle\frac{\lambda+\sqrt{\lambda^{2}+4D(s+r)}}{2D},$ $\displaystyle
m_{4}$ $\displaystyle=$
$\displaystyle\frac{\lambda+\sqrt{\lambda^{2}+4D(\alpha+\beta+s+r)}}{2D}.$
(A.9)

A close inspection of the above roots reveals that $m_{1}$ and $m_{2}$ are
negative, while $m_{3}$ and $m_{4}$ are positive (since $D,s,r>0$). Since
$\tilde{Q}_{\sigma}^{h}\sim e^{mx_{0}}$, and $\tilde{Q}_{\sigma}(s|x_{0})=1/s$
for $x_{0}\to\infty$ (because the survival probability
$Q_{\sigma}(t|x_{0})|_{x_{0}\to\infty}=1$ and its Laplace transform is $1/s$),
we select only $m_{1}$ and $m_{2}$ as the plausible roots.
Letting $\Phi_{1}$ denote the eigenvector corresponding to $m_{1}$ and
utilizing Eq. (A.8), we get $\Phi_{1}=\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)$. In a similar manner, the eigenvector corresponding
to $m_{2}$ is given by $\Phi_{2}=\left(\begin{smallmatrix}-\alpha/\beta\\\
1\end{smallmatrix}\right)$. The general solution of Eq. (A.7) thus reads
$\Psi=c_{1}\Phi_{1}e^{m_{1}x_{0}}+c_{2}\Phi_{2}e^{m_{2}x_{0}}$, and
subsequently

$\displaystyle\tilde{Q}_{0}^{h}$
$\displaystyle=c_{1}e^{m_{1}x_{0}}-\frac{\alpha}{\beta}c_{2}e^{m_{2}x_{0}},$
(A.10) $\displaystyle\tilde{Q}_{1}^{h}$
$\displaystyle=c_{1}e^{m_{1}x_{0}}+c_{2}e^{m_{2}x_{0}}$ (A.11)

where $c_{1}$ and $c_{2}$ are constants to be determined in the following. To
this end, we use the boundary condition $\frac{\partial
Q_{0}(t|x_{0})}{\partial x_{0}}|_{x_{0}=0}=0$ (equivalently,
$\frac{\partial\tilde{Q}_{0}(s|x_{0})}{\partial x_{0}}|_{x_{0}=0}=0$) which
results in $\frac{\partial\tilde{Q}_{0}^{h}(s|x_{0})}{\partial
x_{0}}|_{x_{0}=0}=0$, since $\tilde{Q}_{\sigma}^{inh}(s)$ does not depend on
$x_{0}$. Applying this in Eq. (A.10) results in

$\displaystyle
c_{2}=\left(\frac{\beta}{\alpha}\right)\frac{c_{1}m_{1}}{m_{2}}.$ (A.12)

To compute $c_{1}$, we recall the other boundary condition (absorbing)
$Q_{1}(t|0)=0$ (equivalently, $\tilde{Q}_{1}(s|0)=0$). Combining Eq. (A.5) and
Eq. (A.11) along with the boundary condition at $x_{0}=0$, we can write

$\displaystyle\tilde{Q}_{1}(s|0)=c_{1}+\left(\frac{\beta}{\alpha}\right)\frac{m_{1}}{m_{2}}c_{1}$
$\displaystyle+\frac{1}{s+r}\bigg{(}\frac{r}{s+r+\alpha+\beta}[\beta\tilde{Q}_{0}(s|x_{r})$
$\displaystyle+(s+r+\alpha)\tilde{Q}_{1}(s|x_{r})]+1\bigg{)}=0.$ (A.13)

Incorporating Eq. (A.12) in Eq. (A.13) and solving for $c_{1}$ and hence
$c_{2}$ finally gives

$\displaystyle c_{1}=-\left[\frac{\alpha m_{2}}{(\alpha m_{2}+\beta
m_{1})(s+r)}\right]\bigg{(}1+$
$\displaystyle\frac{r}{s+r+\alpha+\beta}[\beta\tilde{Q}_{0}(s|x_{r})$
$\displaystyle+(s+r+\alpha)\tilde{Q}_{1}(s|x_{r})]\bigg{)},$ $\displaystyle
c_{2}=-\left[\frac{\beta m_{1}}{(\alpha m_{2}+\beta
m_{1})(s+r)}\right]\bigg{(}1+$
$\displaystyle\frac{r}{s+r+\alpha+\beta}[\beta\tilde{Q}_{0}(s|x_{r})$
$\displaystyle+(s+r+\alpha)\tilde{Q}_{1}(s|x_{r})]\bigg{)}.$ (A.14)

Plugging in everything together into Eq. (A.2), we find

$\displaystyle\tilde{Q_{0}}(s|x_{0})=$ $\displaystyle
c_{1}e^{m_{1}x_{0}}-\frac{m_{1}}{m_{2}}c_{1}~{}e^{m_{2}x_{0}}+\frac{1}{s+r}\bigg{(}1+$
$\displaystyle\frac{r}{s+r+\alpha+\beta}\left[\alpha\tilde{Q}_{1}(s|x_{r})+(s+r+\beta)\tilde{Q}_{0}(s|x_{r})\right]\bigg{)},$
$\displaystyle\tilde{Q_{1}}(s|x_{0})=$
$\displaystyle~{}c_{1}~{}e^{m_{1}x_{0}}+\bigg{(}\frac{\beta}{\alpha}\bigg{)}\frac{m_{1}}{m_{2}}c_{1}~{}e^{m_{2}x_{0}}+\frac{1}{s+r}\bigg{(}1+$
$\displaystyle\frac{r}{s+r+\alpha+\beta}\left[\beta\tilde{Q}_{0}(s|x_{r})+(s+r+\alpha)\tilde{Q}_{1}(s|x_{r})\right]\bigg{)},$
(A.15)

which are written in terms of $\tilde{Q}_{\sigma}(s|x_{r})$. Setting
$x_{r}=x_{0}$ in Eq. (A.15) in a self-consistent manner, we find the exact
expressions for the survival functions

$\displaystyle\tilde{Q_{0}}(s|x_{0})=$ $\displaystyle\frac{(\beta m_{1}+\alpha
m_{2})+m_{1}e^{m_{2}x_{0}}\left(\alpha+\frac{r(\alpha+\beta)}{\alpha+\beta+s}\right)-\alpha
m_{2}e^{m_{1}x_{0}}}{\frac{\beta
m_{1}rse^{m_{2}x_{0}}}{\alpha+\beta+s}+\left(\alpha m_{2}re^{m_{1}x_{0}}+\beta
m_{1}s+\alpha m_{2}s\right)},$ $\displaystyle\tilde{Q_{1}}(s|x_{0})=$
$\displaystyle-\frac{(\alpha+\beta+s)\left(\alpha
m_{2}\left(e^{m_{1}x_{0}}-1\right)+\beta
m_{1}\left(e^{m_{2}x_{0}}-1\right)\right)}{(\alpha+\beta+s)\left(\alpha
m_{2}re^{m_{1}x_{0}}+\beta m_{1}s+\alpha m_{2}s\right)+\beta
m_{1}rse^{m_{2}x_{0}}}.$ (A.16)

The averaged survival probability can be found by substituting Eq. (A.16) into
Eq. (3). This results in

$\displaystyle\tilde{Q}_{r}^{G}(s|x_{0})=\frac{\beta
m_{1}\left(\frac{re^{m_{2}x_{0}}}{\alpha+\beta+s}+1\right)+\alpha
m_{2}\left(1-e^{m_{1}x_{0}}\right)}{\frac{\beta
m_{1}rse^{m_{2}x_{0}}}{\alpha+\beta+s}+\left(\alpha m_{2}re^{m_{1}x_{0}}+\beta
m_{1}s+\alpha m_{2}s\right)}.$ (A.17)

Generically, Eq. (A.17) can be used to derive all the first passage time
moments. The observable of our interest in this problem, for example, the
averaged MFPT reads $\langle
T_{r}^{G}\rangle=\tilde{Q}_{r}^{G}(s|x_{0})|_{s=0}$ gardinar . This results in
Eq. (6) in the main text.
Alternatively, we can get Eq. (6) from Eq. (A.15) by calculating individual
mean first passage times conditioned on the initial state of the target state.
To this end, let us denote $T_{\sigma}^{G}(x_{0})$ as the first-passage time
to reach the target at the origin starting from the position $x_{0}$ with
initial target state being at $\sigma$. Using $\langle
T_{\sigma}^{G}(x_{0})\rangle=\tilde{Q_{\sigma}}(s|x_{0})|_{s=0}$ and setting
$s\to 0$ in Eq. (A.15), we get

$\displaystyle\langle T_{1}^{G}(x_{0})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{r}\left(e^{\mu_{1}x_{0}}-1\right)+\frac{1}{r}\left[\frac{\beta\mu_{1}}{\alpha\mu_{2}}e^{\mu_{1}x_{0}}\left(1-e^{-\mu_{2}x_{0}}\right)\right],$
$\displaystyle\langle T_{0}^{G}(x_{0})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{r}\left(e^{\mu_{1}x_{0}}-1\right)$ (A.18)
$\displaystyle+$
$\displaystyle\frac{1}{r}\left[\frac{\mu_{1}}{\mu_{2}}e^{\mu_{1}x_{0}}\left(e^{-\mu_{2}x_{0}}\left(1+\frac{r}{\alpha}\right)\right.+\left.\frac{\beta}{\alpha}\right)\right],$

where $\mu_{1}\coloneqq-m_{1}|_{s=0}>0$ and $\mu_{2}\coloneqq-m_{2}|_{s=0}>0$.
Finally, plugging in Eq. (A.18) into Eq. (4) in the main text, we obtain Eq.
(6).

##  Appendix B The case of confined geometry

Here we consider the case of a bounded system, i.e., where the particle
remains in a finite confinement. This is also highly relevant from the context
of chemical reactions, since a high energy barrier can mimic a reflecting
boundary (in the reaction coordinate space) that pushes the particle away from
it. We construct this finite domain by considering the same set-up as in the
main text, with an additional reflecting wall at $x=L>x_{0}$. If the particle
hits the wall it gets reflected back. Intuitively, the barrier or the
reflecting boundary prevents the particle from going too far from the target
placed at the origin. This is in sharp contrast to the semi-infinite case,
where the particle is allowed to diffuse away from the target. This leads to a
few key changes in the dynamics that are reflected in the average MFPT.

Figure 11: The average MFPT $\langle T_{r}^{G}\rangle$ as a function of the
resetting rate $r$ for different values of $\lambda$ in the case of bounded
domain. Here, the reflecting boundary is placed at $L=3$, where the
resetting/initial location is at $x_{0}=2$. The colored discs mark the optimal
resetting rate ($r^{\star}$) in each case. Notably, resetting may not always
be helpful ($r^{\star}=0$) even when $\lambda<0$ (e.g., yellow and brown
curves).

On the technical ground, we can find the survival probability in Laplace space
using the same method as discussed in Appendix A with the new boundary
condition $\frac{\partial Q_{\sigma}(t|x_{0})}{\partial x_{0}}|_{x_{0}=L}=0$.
For this reason, all the eigenvalues will exist unlike in the previous case.
In particular, one can show that the eigenvector corresponding to $m_{3}$ is
the same as $m_{1}$, i.e., $\Phi_{1}=\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)$ and that of $m_{4}$ is same as $m_{2}$ i.e.
$\Phi_{2}=\left(\begin{smallmatrix}-\alpha/\beta\\\
1\end{smallmatrix}\right)$. Recalling the decomposition
$\tilde{Q}_{\sigma}(s)=\tilde{Q}_{\sigma}^{h}(s)+\tilde{Q}_{\sigma}^{inh}(s)$
from Eq. (A.2), we first try to obtain the solutions for the homogeneous part.
As before taking
$\Psi=\left(\tilde{Q}_{0}^{h}\;\;\;\tilde{Q}_{1}^{h}\right)^{T}$ and using Eq.
(A.7) we find

$\displaystyle\Psi=c_{1}\Phi_{1}e^{m_{1}x_{0}}+c_{2}\Phi_{2}e^{m_{2}x_{0}}+c_{3}\Phi_{1}e^{m_{3}x_{0}}+c_{4}\Phi_{2}e^{m_{4}x_{0}},$
(B.1)

so that

$\displaystyle\tilde{Q}_{0}^{h}$
$\displaystyle=c_{1}e^{m_{1}x_{0}}-\frac{\alpha}{\beta}c_{2}e^{m_{2}x_{0}}+c_{3}e^{m_{3}x_{0}}-\frac{\alpha}{\beta}c_{4}e^{m_{4}x_{0}},$
(B.2) $\displaystyle\tilde{Q}_{1}^{h}$
$\displaystyle=c_{1}e^{m_{1}x_{0}}+c_{2}e^{m_{2}x_{0}}+c_{3}e^{m_{3}x_{0}}+c_{4}e^{m_{4}x_{0}}.$
(B.3)

The inhomogeneous part $\tilde{Q}_{\sigma}^{inh}(s)$ has the same solution as
given in Eq. (A.5). In addition to the boundary conditions
$\tilde{Q}_{1}(s|0)=0$ and $\frac{\partial\tilde{Q}_{0}^{h}(s|x_{0})}{\partial
x_{0}}|_{x_{0}=0}=0$ at the gated target, we also have two additional boundary
conditions namely $\frac{\partial\tilde{Q}_{0}(s|x_{0})}{\partial
x_{0}}|_{x_{0}=L}=0$ and $\frac{\partial\tilde{Q}_{1}(s|x_{0})}{\partial
x_{0}}|_{x_{0}=L}=0$. These four boundary conditions give four linear
equations

$\displaystyle c_{1}+c_{2}+c_{3}+c_{4}+\tilde{Q}_{1}^{inh}(s)=0,$
$\displaystyle
c_{1}m_{1}e^{m_{1}L}+c_{2}m_{2}e^{m_{2}L}+c_{3}m_{3}e^{m_{3}L}+c_{4}m_{4}e^{m_{4}L}=0,$
$\displaystyle
c_{1}m_{1}e^{m_{1}L}-\frac{\alpha}{\beta}c_{2}m_{2}e^{m_{2}L}+c_{3}m_{3}e^{m_{3}L}-\frac{\alpha}{\beta}c_{4}m_{4}e^{m_{4}L}=0,$
$\displaystyle
c_{1}m_{1}-\frac{\alpha}{\beta}c_{2}m_{2}+c_{3}m_{3}-\frac{\alpha}{\beta}c_{4}m_{4}=0,$
(B.4)

which completely determine the constants $c_{1},c_{2},c_{3},c_{4}$. From this,
we can find a closed-form expression for the survival probabilities in Laplace
space (and subsequently the MFPT by setting $s\to 0$). The expression for the
MFPT is quite lengthy to present here; check github  for the Mathematica file
where all the derivations are given. In what follows, we perform a
comprehensive analysis of this MFPT and point out the key differences in
comparison to the that obtained for the semi-infinite domain.
Fig. 11 showcases the variation of the MFPT, $\langle T_{r}^{G}\rangle$, as a
function of the resetting rate $r$ for different values of the drift
$\lambda$. One crucial observation is that the non-monotonic behaviour of
$\langle T_{r}^{G}\rangle$ is not always present even when the drift is away
from the target (i.e., $\lambda<0$). This is in complete contrast to the semi-
infinite case where resetting is guaranteed to help whenever the drift is away
from the target [see Fig. (3)]. To understand this better, we plot the optimal
resetting rate $r^{\star}$ as a function of $\lambda$ for various domain sizes
$L$ in Fig. 12, which clearly shows that the critical values of $\lambda$ that
marks the resetting transition (denoted $\lambda_{c}$ in the main text; the
minimal value of $\lambda$ for which $r^{\star}=0$), can be negative for
considerably smaller domains. With increasing $L$, however, $\lambda_{c}$
starts to increase and for sufficiently large values of $L$ it saturates to
the value of $\lambda_{c}$ for the semi-infinite case, as displayed Fig. 5 of
the main text. Simply put, if the reflecting boundary starts moving
sufficiently away from the target, the MFPT starts to increase and resetting
renders a more effective search in that case.
To elaborate this further, one can expand the MFPT (for finite domain) in the
limit of small resetting rate $r\to 0$ and find the first order correction in
$r$ [in similar spirit as in Eq. (8) – see the discussion for the semi-
infinite case in Sec. III, particularly around Eq. (10)]. The arguments on the
function $f$ as discussed there still holds for the present case (of course,
the exact form of the function will be different here). Thus, setting $f=0$
gives us the separatrix distinguishing between the region where resetting
helps ($f<0$) and where it hinders ($f>0$). Utilizing this fact, we generate a
phase diagram by plotting $\lambda_{c}$ as a function of $L$, the distance
between the reflecting barrier and the origin, and present the same in the
inset of Fig. 12. The semi-infinite limit is obtained by taking $L\to\infty$,
where $\lambda_{c}$ saturates to the corresponding value calculated/presented
in the main text. For example, we see from Fig. 12 that when $p_{r}=0.5$,
$\lambda_{c}$ saturates to $0.78$, the critical value of $\lambda$ for
$p_{r}=0.5$ (marked in Fig. 5 by the vertical yellow line).

Figure 12: Main: Variation of ORR ($r^{\star}$) as a function of $\lambda$ for
different domain size $L$. For $\lambda<\lambda_{c}$, resetting proves itself
beneficial, as indicated by the non-zero values of $r^{\star}$, whereas for
$\lambda\geq\lambda_{c}$, resetting becomes detrimental, as indicated by
$r^{\star}=0$. The resetting transition is thus marked by $\lambda_{c}$.
Inset: Phase diagram with reflecting barrier at $L$, by plotting $\lambda_{c}$
that acts as the separatrix (black line; the colored discs mark the specific
cases shown in the main panel) that divides the phase space in two parts; one
where resetting is beneficial (white regime) and the other where resetting is
detrimental (gray regime). Existence of negative $\lambda_{c}$ essentially
implies that resetting can be detrimental even when $\lambda<0$. The dashed
horizontal line corresponds to $\lambda_{c}=0.78$, obtained for the semi-
infinite case (i.e. $L\to\infty$), as displayed in Fig. 5 for $p_{r}=0.5$.
Note that we consider $\alpha=0.5,\beta=0.5$ and $D=1$ for each case in the
main panel and in the inset.

##  Appendix C The average MFPT for gated drift-diffusion without resetting

In the absence of resetting, the backward master equations in terms of the
survival probability read

$\displaystyle\frac{\partial Q_{0}^{r=0}(t|x_{0})}{\partial t}=-$
$\displaystyle\lambda\frac{\partial Q_{0}^{r=0}(t|x_{0})}{\partial
x_{0}}+D\frac{\partial^{2}Q_{0}^{r=0}(t|x_{0})}{\partial x_{0}^{2}}$
$\displaystyle+$
$\displaystyle\alpha\left[Q_{1}^{r=0}(t|x_{0})-Q_{0}^{r=0}(t|x_{0})\right]$
$\displaystyle\frac{\partial Q_{1}^{r=0}(t|x_{0})}{\partial t}=-$
$\displaystyle\lambda\frac{\partial Q_{1}^{r=0}(t|x_{0})}{\partial
x_{0}}+D\frac{\partial^{2}Q_{1}^{r=0}(t|x_{0})}{\partial x_{0}^{2}}$
$\displaystyle+$
$\displaystyle\beta\left[Q_{0}^{r=0}(t|x_{0})-Q_{1}^{r=0}(t|x_{0})\right].$
(C.1)

Solving Eq. (C.1), we obtain the following expressions for the survival
functions in the Laplace space

$\displaystyle\tilde{Q_{0}}^{r=0}(s|x_{0})$
$\displaystyle=\frac{1}{s}-\frac{1}{s}\frac{\alpha n_{2}}{\beta n_{1}+\alpha
n_{2}}\left(e^{-n_{1}x_{0}}-\frac{n_{1}e^{-n_{2}x_{0}}}{n_{2}}\right),$
$\displaystyle\tilde{Q_{1}}^{r=0}(s|x_{0})$
$\displaystyle=\frac{1}{s}-\frac{1}{s}\frac{\alpha n_{2}}{\beta n_{1}+\alpha
n_{2}}\left(e^{-n_{1}x_{0}}+\frac{\beta n_{1}e^{-n_{2}x_{0}}}{\alpha
n_{2}}\right),$ (C.2)

where $n_{1}=\frac{-\lambda+\sqrt{4Ds+\lambda^{2}}}{2D}$ and
$n_{2}=\frac{-\lambda+\sqrt{4D(\alpha+\beta+s)+\lambda^{2}}}{2D}$. Setting
$s\to 0$, the underlying MFPTs can be computed as before. Eventually, one
finds

$\displaystyle\langle T_{0}^{r=0}\rangle$
$\displaystyle=\frac{x_{0}}{\lambda}+\frac{2D\left(\beta+\alpha
e^{-\frac{x_{0}\left(\sqrt{4D(\alpha+\beta)+\lambda^{2}}-\lambda\right)}{2D}}\right)}{\alpha\lambda\left(-\lambda+\sqrt{4D(\alpha+\beta)+\lambda^{2}}\right)},$
(C.3) $\displaystyle\langle T_{1}^{r=0}\rangle$
$\displaystyle=\frac{x_{0}}{\lambda}+\frac{2\beta
D\left(-e^{-\frac{x_{0}\left(\sqrt{4D(\alpha+\beta)+\lambda^{2}}-\lambda\right)}{2D}}+1\right)}{\alpha\lambda\left(-\lambda+\sqrt{4D(\alpha+\beta)+\lambda^{2}}\right)}.$
(C.4)

The averaged MFPT is then given by

$\displaystyle\langle T^{G}\rangle$ $\displaystyle=p_{r}\langle
T_{1}^{r=0}\rangle+(1-p_{r})\langle T_{0}^{r=0}\rangle$
$\displaystyle=\frac{x_{0}}{\lambda}+\frac{2\beta
D}{\alpha\lambda\left(\sqrt{4D(\alpha+\beta)+\lambda^{2}}-\lambda\right)}.$
(C.5)

Note that for $p_{r}\to 1$ ($\beta\to 0$), Eq. (C.5) reduces to $\langle
T\rangle=x_{0}/\lambda$, as expected. Moreover, since $D,\alpha,\beta>0$, the
second term of the expression of $\langle T^{G}\rangle$ in Eq. (C.5) is always
positive, which clearly shows $\langle T^{G}\rangle>\langle T\rangle$.

##  Appendix D Details of numerical simulations

In this Appendix, we sketch out the basic steps used for the numerical
simulations in the main text. We recall that the particle starts from and
resets to $x_{0}$ at a rate $r$. In between reset events, it diffuses in the
presence of a drift $\lambda$. Evolution of this particle in microscopic time
scale $\Delta t$ can be written in the form of a Langevin dynamics such as

$\displaystyle x(t+\Delta t)=\begin{cases}x_{0}\hskip 102.43008pt\text{w.p.
}\;\;\;\;\;r\Delta t\\\ \\\ x(t)-\lambda\Delta t+\sqrt{2D\Delta
t}\;\eta(t)\;\;\;\;\text{w.p.}\;\;\;\;(1-r\Delta t)\end{cases}$ (D.1)

where $\eta(t)$ is a $\delta$-correlated white noise i.e., a Gaussian random
variable with zero mean and unit variance. Note that the abbreviation w.p. in
Eq. (D.1) has full-form with probability. We evolve the particle at each time
step $\Delta t$ in our simulation according to Eq. (D.1) until the particle
reaches the target at $x=0$. However, to implement the gating condition to the
target, we define a state variable $\sigma=(0,1)$ representing the non-
reactive and reactive states of the target, respectively. If target is
initially at the non-reactive state, it switches to the reactive state with
probability $\alpha\Delta t$. Similarly, it is switched from the active state
to the inactive one with probability $\beta\Delta t$.

To compute the first passage time, we simultaneously track two events: (i) the
instant when the particle crosses the origin to the negative side i.e., $x\leq
0$ and (ii) note whether the target is in active state i.e., $\sigma=1$. If
both the conditions are satisfied, the particle gets absorbed and the process
ends. We measure the corresponding time and put it into the first passage
statistics. However, if $x\leq 0$ and $\sigma=0$ then the particle gets
reflected from the boundary, and the process continues till the next
absorption occurs. The reflecting boundary condition is implemented by simply
reversing the position of the particle i.e. $x\to-x$. The initial target state
$\sigma$ is chosen from the steady state i.e.,

$\sigma(t=0)=\begin{cases}1\hskip 28.45274pt\text{with
probability}\;\;\;\;p_{r},\\\ 0\hskip 28.45274pt\text{with
probability}\;\;\;\;(1-p_{r}),\end{cases}$ (D.2)

where $p_{r}=\alpha/(\alpha+\beta)$. In our simulations, $\Delta t=10^{-5}$
and the averaging was done for $10^{5}$ successful trajectories for each value
of $\lambda$. The results are displayed in Fig. 3 with the star symbols, which
show excellent agreement with the analytical results presented by the solid
lines of same color.

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