Inference from gated rst-passage times Aanjaneya Kumar1Yuval Scher2yShlomi Reuveni2zand M. S. Santhanam1x 1Department of Physics Indian Institute of Science Education and Research Dr. Homi Bhabha Road Pune 411008 India.

2025-05-05 0 0 3.62MB 17 页 10玖币

Inference from gated ﬁrst-passage times

Aanjaneya Kumar,1, ∗Yuval Scher,2, †Shlomi Reuveni,2, ‡and M. S. Santhanam1, §

1Department of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India.

2School of Chemistry, Center for the Physics & Chemistry of Living Systems, Ratner Institute for Single Molecule Chemistry,

and the Sackler Center for Computational Molecular & Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel

(Dated: April 6, 2023)

First-passage times provide invaluable insight into fundamental properties of stochastic processes.

Yet, various forms of gating mask ﬁrst-passage times and diﬀerentiate them from actual detection

times. For instance, imperfect conditions may intermittently gate our ability to observe a system of

interest, such that exact ﬁrst-passage instances might be missed. In other cases, e.g., certain chemical

reactions, direct observation of the molecules involved is virtually impossible, but the reaction event

itself can be detected. However, this instance need not coincide with the ﬁrst collision time since

some molecular encounters are infertile and hence gated. Motivated by the challenge posed by such

real-life situations we develop a universal—model-free—framework for the inference of ﬁrst-passage

times from the detection times of gated ﬁrst-passage processes. In addition, when the underlying laws

of motions are known, our framework also provides a way to infer physically meaningful parameters,

e.g. diﬀusion coeﬃcients. Finally, we show how to infer the gating rates themselves via the hitherto

overlooked short-time regime of the measured detection times. The robustness of our approach and

its insensitivity to underlying details are illustrated in several settings of physical relevance.

Introduction.—The importance of ﬁrst-passage pro-

cesses is recognized universally across scientiﬁc disci-

plines, owing to their ubiquity and wide-ranging applica-

tions [1–7]. How long does it take for a chemical reaction

to be triggered? Or what is the time taken for an order

to be executed in the stock market? These disparate ex-

amples fall under the purview of ﬁrst-passage processes,

where the ﬁrst-passage time is now established as an in-

dispensable tool to quantify the time taken for a given

task to be completed.

In several practically relevant scenarios, however, the

completion of a task also relies on additional constraints.

For example, for a chemical reaction to be triggered, two

reactants must collide. Additionally, the collision must

be fertile, i.e., the reactants must be in a reactive inter-

nal state during collision. This internal state acts like

a “gate”: a reaction can only happen when the gate is

“open”, i.e. the molecules are in their reactive inter-

nal state. The macroscopic kinetics of these so called

gated reactions has a history spanning over four decades

now [8–20], and more recently, the study of single-particle

gated reactions has gained interest [21–31].

While the terminology of ‘gating’ is unique to reaction

kinetics, numerous examples fall under the wide umbrella

of gated processes. An important one is that of inter-

mittently observed stochastic time-series, where the un-

derlying cause for intermittent observations can include

energy costs of continuous observations, imperfect detec-

tion conditions, or simply, a faulty sensor [32–36]. Irre-

spective of the reasons behind such intermittent obser-

vations, an important consequence is that key features

of the time-series can be missed. In particular, in the

increasingly relevant ﬁeld of extreme and record statis-

tics of time-series, a crucial quantity is the time taken to

cross a speciﬁc threshold for the ﬁrst time. However, in-

FIG. 1. Instances highlighting the need for inference in gated

ﬁrst-passage processes. (a) Detection of threshold crossing

under intermittent sensing. Consider single particle tracking

of a photoblinking particle. The ﬁrst-passage properties of

the particle can be mischaracterized as the particle can cross

the threshold while being in its invisible state. (b) Gated

chemical reaction or target search. Imagine a situation where

tracking of the particle is not possible, and the only observable

is the reaction time. For such processes, we show how the ﬁrst-

passage time distribution, and other relevant observables, can

be inferred from detection times.

termittent detection of the time-series can lead to a gross

mischaracterization of the statistics of such events [37–

41]. In such cases, the relevant quantity is the ﬁrst de-

tection time, which denotes the ﬁrst time the time-series

is observed above the threshold [39].

Figure 1exempliﬁes two instances where gating arises

naturally: (a) Single-particle tracking of an intermit-

tently observed particle, which transitions between a visi-

arXiv:2210.00678v2 [cond-mat.stat-mech] 4 Apr 2023

ble state and an invisible state. For example, a wide class

of ﬂuorophores undergo photoblinking [42–51]. Other

reasons for such gating can be the intermittent loss of

focus on a moving particle in 3-dimensions [52] or slow

frame acquisition rate [53]; (b) A gated chemical reaction

or target search, where tracking of the particle is not

possible, and the reaction time is the only measurable

quantity. Such instances may arise in cellular signalling

driven by narrow escape [54,55] and among ﬂuorescent

probes [56].

In both examples illustrated in Fig. 1, the ﬁrst-passage

time statistics carry invaluable information, but are in-

accessible to direct measurement. In such scenarios, a

crucial challenge is to reliably infer these statistics and

other fundamental properties of interest.

In this Letter, we address this challenge and solve it.

First, we show how the ﬁrst-passage time density can be

inferred from gated observations via a model-free formal-

ism, which upon speciﬁcation of the underlying laws of

motions can be further used to infer physically meaning-

ful parameters (e.g., the diﬀusion coeﬃcient). Second,

using the joint knowledge of the gated (observed) and

ungated (inferred) ﬁrst-passage time densities, we estab-

lish that the overlooked short-time regime of the gated

detection time distribution can be leveraged to obtain

the gating rates.

Modeling gated processes.—We start by modeling a

gated process consisting of two independent components.

First, an underlying process Xn0(t), initially at n0, mod-

eled as a continuous-time Markov process. Second, a gate

modeled by a two-state continuous-time Markov process,

that intermittently switches between an ‘open’ active (A)

state and a ‘closed’ inactive (I) state. This gate accounts

for the additional constraint that needs to be satisﬁed for

the task of interest to be completed. The gate switches

from state Ato Iat rate α, and from Ito Aat β. For

σ0, σ ∈ {A, I}, we deﬁne pt(σ|σ0) to be the probability

that the gate is in state σat time t, given that it was

in state σ0initially (see SI for an explicit formula [57]).

Also, let πA=β/λ and πI=α/λ denote the equilibrium

occupancy probabilities of states Aand Irespectively,

where λ=α+βis the relaxation rate to equilibrium.

The central quantity of interest in our Letter is the

ﬁrst-passage time Tf(m|n0), which is the time taken for

Xn0(t) to reach state mfor the ﬁrst time, and we denote

its probability density by Ft(m|n0). In many scenarios

the ﬁrst-passage time is not directly measurable, and in-

stead we can only measure the detection time Td(n0, σ0),

of a reaction or threshold crossing event. We denote by

Dt(n0, σ0) the probability density of Td(n0, σ0), which is

the ﬁrst time the underlying process is detected in some

target-set Q, given that the initial state of the composite

process (underlying + gate) is initially at {n0, σ0}.

In this work, we focus on two widely applicable set-

tings: (i) the detection of threshold crossing events of

a 1-dimensional intermittent time-series with nearest-

neighbor transitions, where Qdenotes all states above a

certain threshold mand Td(n0, σ0) is the ﬁrst time when

Xn0(t)≥mwhile the detector is active (A), and (ii)

gated reactions or target search on an arbitrary network

in discrete space or in arbitrary dimension in continuous

space. Here, Qis typically a single target state/point

m, and Td(n0, σ0) denotes the ﬁrst time the underlying

process Xn0(t) is at m, while the gate is open (A).

First-passage times from gated observations.— We be-

gin our analysis by noting that for n06∈ Q we have

Dt(n0,σ0) = Ft(m|n0)pt(A|σ0) +

Ft0(m|n0)pt0(I|σ0)Dt−t0(m, I)dt0,(1)

where the probability for a detection event occurring at

time thas two contributions: (i) the detection time co-

inciding with the ﬁrst-passage time, and (ii) the gate be-

ing closed during the ﬁrst-passage event (I), and detec-

tion happening strictly after this moment in time. The

Laplace transform of Eq. (1), can be expressed in com-

pact form as [57]

Ds(n0,σ0) = hπA+πIe

Ds(m, I)ie

Fs(m|n0) (2)

+1(σ0)(1 −πσ0)h1−e

Ds(m, I)ie

Fs+λ(m|n0),

where λ=α+β, and 1(σ0) takes values +1 or −1 when

σ0=Aor I, respectively. By explicitly writing down

the equations for σ0=Aand I, and further eliminating

Fs+λ(m|n0) from the equations, we arrive at [57]

Fs(m|n0) = πAe

Ds(n0, A) + πIe

Ds(n0, I)

πA+πIe

Ds(m, I),(3)

which is our ﬁrst result. Equation (3) asserts that the

ﬁrst-passage density can be obtained exactly in terms of

detection time densities and the gating rates. In [57], we

further show that Eq. (3) holds even when the underly-

ing process in not Markovian, and instead is a renewal

process. However, inference of the ﬁrst-passage time den-

sity Ft(m|n0), requires the detection statistics with initial

conditions {n0, A},{n0, I},{m, I}and the equilibrium

probabilities πAand πI. Such information may not be

accessible in experimentally realizable scenarios where,

e.g., it may not be possible to initialize a gated molecule

in a speciﬁc internal state σ0=Aor I, and the values of

πAand πImay also be unknown.

In such situations, the most practically realizable ini-

tial condition is the equilibrium σ0≡E, where the gate

is in the active state Awith probability πA, and in the

inactive state Iwith probability πI. Note that this ini-

tial condition is naturally achieved if the system is sim-

ply allowed to equilibriate. Interestingly, the detection

time density starting from the initial condition (n0, E)

is given by Dt(n0, E) = πA·Dt(n0, A) + πI·Dt(n0, I),

whose Laplace transform is the numerator standing on

the right-hand side of Eq. (3). Further noting that the

Laplace transform of Dt(m, E) = πA·δ(t) + πI·Dt(m, I)

gives the denominator on the right-hand side, we obtain

an elegant reinterpretation of Eq. (3):

Fs(m|n0) = e

Ds(n0, E)

Ds(m, E).(4)

Strikingly, Eq. (4) asserts that the ﬁrst-passage time den-

sity can be inferred from the detection statistics, even

without the explicit knowledge of πAand πI, or control

over the initial state of the gate.

The usefulness and validity of Eq. (4) is demonstrated

in Fig. 2, with the help of three case studies of wide

interest and applicability. First, a Markovian birth-

death process which has been extensively used to model

threshold activated reactions [58–61] and the dynamics

of chemical reactions on catalysts [62,63]. Second, the

paradigmatic continuous-space diﬀusion in a 1D conﬁne-

ment. Third, a gated chemical reaction/target search

modeled by a non-Markovian continuous-time random

walk (CTRW) [64,65] on a network [66], which is e.g.,

used to model the motion of reactants, cells, or organisms

in complex environments [30,66–72]. In all of these set-

tings, we show that the ﬁrst-passage time distributions

inferred from Eq. (4) using a procedure described in [57]

(circles) are in excellent agreement with the true ﬁrst-

passage time distributions. We stress that this inference

was performed solely using detection time histograms ob-

tained from gated simulations, without assuming knowl-

edge of their analytical expressions or model speciﬁc de-

tails (e.g. the network structure and the waiting time

distribution in the CTRW example). However, when an-

alytical expressions are available, like in the case of the

birth-death process [39], one can directly perform the in-

ference through Laplace inversion of Eq. (4) [57].

Before moving forward, we note that Eq. (4) is reminis-

cent of the seminal renewal formula e

Fs(m|n0) =

Ps(m|n0)

Ps(m|m)

which relates, in Laplace space, the ﬁrst-passage time

density and the probability density Pt(ni|nj) of ﬁnding

the underlying process in state niat time t, given its

initial state nj[1]. Clearly, the right-hand side of this

formula and that of Eq. (4) are equal. In fact, we can

obtain an even more general relation – considering two

diﬀerent initial states n0and n0

0, and after some algebra,

we uncover the fundamental relation [57]

Ds(n0, E)

Ds(n0

0, E)=e

Ps(m|n0)

Ps(m|n0

0),(5)

asserting that the ratio of the detection time densities (in

Laplace space), starting from any two initial states n0

and n0

0, is independent of the gating rates αand β. Note

that this is true despite the fact that the detection time

densities themselves depend on the gating rates. We re-

mark that Eq. (5) holds in both settings: when Dt(n0, E)

10−1100101102

Time (t)

10−4

10−3

10−2

10−1

100

First-passage time density

Birth-Death Process

CTRW on Network

Diﬀusion

FIG. 2. Inference of ﬁrst-passage time distributions from

gated observations. We consider the three diﬀerent settings

mentioned in the text and legend. Solid and dashed lines de-

note ungated ﬁrst-passage time distributions obtained from

theory and simulations, respectively. Circles are inferred us-

ing Eq. (4) from histograms of simulated gated detection

times [57].

corresponds to gated target search and to the detection

of threshold crossing events under intermittent sensing.

Inferring the mean ﬁrst-passage time.—The Laplace

transform in Eq. (4) allows us to obtain all moments of

the ﬁrst-passage time in terms of moments of the de-

tection time. Equation (4) further implies that all cu-

mulants of the ﬁrst-passage time can be expressed as

diﬀerences between cumulants of detection times. For

example, the mean ﬁrst-passage time is given by

hTf(m|n0)i=hTd(n0, E)i−hTd(m, E)i.(6)

While simple, Eq. (6) carries utmost importance in prac-

tical scenarios, where reliably estimating the full prob-

ability distribution is not a viable option, and only the

mean can be accurately measured. Apart from setting

an important time-scale for a wide class of chemical re-

actions in conﬁnement, where the mean reaction time

can be used to infer full reaction time statistics [73], the

mean ﬁrst-passage time can also shed light on fundamen-

tal properties of the system at hand [74,75].

Inferring the diﬀusion coeﬃcient.—We now illustrate

how one can utilize our framework to infer physically

meaningful parameters like the diﬀusion coeﬃcient D.

Importantly, we show that this can be done even when

the actual motion of the particle cannot be tracked.

Imagine a scenario like that depicted in Fig. 1(b), namely

we inject an unobservable particle—whose detection is

possible only upon reaction—at a known location x0.

Assume that the internal state of the particle is initially

equilibrated (σ0=E); and further assume that it is freely

diﬀusing inside an eﬀectively one-dimensional box [0, L]

with reﬂecting boundaries and a gated target located at

10°11 10°10 10°9

DiÆusion Coe±cient D

0.5

1.5

Dinf/D

N= 100

N= 1000

(m2/s)

FIG. 3. Inference of the diﬀusion coeﬃcient. Equation (7)

is used to infer the diﬀusion coeﬃcient of an unobservable

particle that is injected at a known location x0= 0 into a box

[0,5µm] with reﬂecting boundaries. The initial internal state

is equilibrated σ0=E, and a gated point target is located at

m= 4µm, with gating rates α=β= 102s−1.

x0< m < L. Utilizing Eq. (6) we ﬁnd that [57]

D=1

m2−x2

hTd(x0, E)i−hTd(m, E)i.(7)

Equation (7) asserts that the diﬀusion coeﬃcient can be

inferred from the diﬀerence in the measurable detection

times hTd(x0, E)iand hTd(m, E)i.

To corroborate this ﬁnding, we simulate the aforemen-

tioned scenario and test it for a wide range of possible

diﬀusion coeﬃcients (Fig. 3). As implied by Eq. (6), the

diﬀerence in the detection times is independent of the

transition rates, the box size L, and the target size (the

same equation will hold for threshold crossing). It is thus

up to the experimentalist to tune these parameters such

that the detection times can be measured with suﬃcient

accuracy. Here we set α=β= 102s−1and L= 5µm.

For each value of D, the corresponding mean detection

times were estimated from averages of N= 102and 103

simulations, and the diﬀusion coeﬃcient was inferred via

Eq. (7). The errors bars were estimated by repeating this

procedure 102times and noting the standard deviation.

In Fig. 3we plot the ratio between the inferred values

and the actual ones. We ﬁnd this estimation procedure

robust, even when the number of measurements is rela-

tively small (N= 102). For the parameters used here,

the estimation is especially accurate for smaller diﬀusion

coeﬃcients, where mean detection times are longer.

Inferring the gating rates.—Equation (4) states that

the ﬁrst-passage time density can be inferred from its

gated counterparts, even without any prior knowledge of

the gating rates αand βor control over the initial internal

condition. We will now illustrate how the inferred ﬁrst-

passage time distribution can be used together with the

observed detection time distribution to infer the gating

rates, thus providing insight into the dynamics of the

gating process.

To proceed, we shift our focus to short-time asymp-

10°510°310°1

Time (t)

0.1

Estimated Æ

Æ= 10

Æ=1

Æ=0.1

Ø=1

m=5

m=7

m=9

10°510°310°1

Time (t)

0.1

Estimated Ø

Ø= 10

Ø=1

Ø=0.1

Æ=1

(a)(b)

FIG. 4. Inference of the gating rates α(panel a) and β(panel

b) from the short-time asymptotics of Eq. (8) and (9) respec-

tively. Results are for the birth-death model used in Fig. 2,

and various values of αand β. Details of the model and pa-

rameter values are given in [57].

totics analysis which, despite several recent applica-

tions in stochastic thermodynamics [76–78] and chemi-

cal kinetics [79,80], has not yet been used to further

our knowledge on gated processes. In the short-time

limit, the dominant contribution to Dt(n0, E) comes

from trajectories where the detection occurs upon ﬁrst

arrival. This insight translates to the limiting equa-

tion πA= limt→0Dt(n0, E)/Ft(m|n0). Similarly, the

short-time asymptotics of Dt(m, E) is given by πI=

β−1limt→0Dt(m, E), owing to the fact that when the

underlying process starts on m, the dominant contribu-

tion to detection comes from events where the gate opens

before the particle leaves the target or falls below the

threshold.

These limiting representations of the probabilities πA

and πI, along with their normalization, allows us to ob-

tain the gating rates as follows

α= lim

t→0

Dt(m, E)Ft(m|n0)

Dt(n0, E),(8)

β= lim

t→0

Dt(m, E)Ft(m|n0)

Ft(m|n0)−Dt(n0, E).(9)

Equations (8) and (9) are corroborated in Fig. 4, for

the birth-death process with parameters described in [57].

Furthermore, in [57] we also show that these relations

hold even for an arbitrary (non-equilibrium) initial con-

dition of the gate. We then derive simpler inference rela-

tions for the gating rates, which are obtained at the cost

of perfect control over σ0. Finally, we discuss the widely

applicable case of simple diﬀusion and derive inference

relations for αand β, which only diﬀer by a factor of 2

from Eqs. (8) and (9).

Discussion.—Using the uniﬁed framework of gated

ﬁrst-passage processes, we demonstrated how the ﬁrst-

passage time distribution can be inferred from gated

measurements, and using these quantities, key features

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Inferencefromgatedrst-passagetimesAanjaneyaKumar,1,YuvalScher,2,yShlomiReuveni,2,zandM.S.Santhanam1,x1DepartmentofPhysics,IndianInstituteofScienceEducationandResearch,Dr.HomiBhabhaRoad,Pune411008,India.2SchoolofChemistry,CenterforthePhysics&ChemistryofLivingSystems,RatnerInstituteforSingleMolecule...

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Inference from gated rst-passage times Aanjaneya Kumar1Yuval Scher2yShlomi Reuveni2zand M. S. Santhanam1x 1Department of Physics Indian Institute of Science Education and Research Dr. Homi Bhabha Road Pune 411008 India.

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