# Search efficiency in the Adam-Delbrück reduction-of-dimensionality scenario
versus direct diffusive search

Denis S Grebenkov†, Ralf Metzler‡ & Gleb Oshanin♯ $\dagger$ Laboratoire de
Physique de la Matière Condensée (UMR 7643), CNRS – Ecole Polytechnique, IP
Paris, 91120 Palaiseau, France  $\ddagger$ Institute of Physics and Astronomy,
University of Potsdam, 14476 Potsdam-Golm, Germany  $\sharp$ Sorbonne
Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée
(UMR CNRS 7600), 4 Place Jussieu, 75252 Paris Cedex 05, France

###### Abstract

The time instant—the first-passage time (FPT)—when a diffusive particle (e.g.,
a ligand such as oxygen or a signalling protein) for the first time reaches an
immobile target located on the surface of a bounded three-dimensional domain
(e.g., a hemoglobin molecule or the cellular nucleus) is a decisive
characteristic time-scale in diverse biophysical and biochemical processes, as
well as in intermediate stages of various inter- and intra-cellular signal
transduction pathways. Adam and Delbrück put forth the reduction-of-
dimensionality concept, according to which a ligand first binds non-
specifically to any point of the surface on which the target is placed and
then diffuses along this surface until it locates the target. In this work, we
analyse the efficiency of such a scenario and confront it with the efficiency
of a direct search process, in which the target is approached directly from
the bulk and not aided by surface diffusion. We consider two situations: (i) a
single ligand is launched from a fixed or a random position and searches for
the target, and (ii) the case of "amplified" signals when $N$ ligands start
either from the same point or from random positions, and the search terminates
when the fastest of them arrives to the target. For such settings, we go
beyond the conventional analyses, which compare only the mean values of the
corresponding FPTs. Instead, we calculate the full probability density
function of FPTs for both scenarios and study its integral characteristic—the
"survival" probability of a target up to time $t$. On this basis, we examine
how the efficiencies of both scenarios are controlled by a variety of
parameters and single out realistic conditions in which the reduction-of-
dimensionality scenario outperforms the direct search.
Keywords: Ligand binding to a target site, first-passage times, probability
density functions, Adam-Delbrück reduction-of-dimensionality scenario, bulk
and surface diffusion

##  1 Introduction

More than five decades ago Adam and Delbrück put forth an idea how organisms
may handle some problems of efficiency and timing, limited by molecular
diffusion, by reducing the dimensionality in which the diffusion takes place
from the three-dimensional (bulk) space to two-dimensional surface diffusion
[1]. A similar claim was previously made in [2], suggesting that acetyl
choline may get faster via surface diffusion from its site of action to the
site where it is destroyed, and in [3], arguing that surface diffusion may
result in higher turnover numbers for membrane-bound enzymes. However, these
earlier works did not present quantitative estimates substantiating such a
claim, while Adam and Delbrück were the first to provide theoretical arguments
showing that the diffusive molecules may indeed reduce the reaction times by
subdividing the diffusion process into successive stages of lower spatial
dimensionality. Subsequently, their analysis has been generalised in a number
of directions (see, e.g., [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]). Its
applicability has also been questioned [14], indicating situations in which
the "reduction-of-dimensionality" scenario can be advantageous and therefore
plausible, and situations in which it is not beneficial for the search process
and is therefore not likely to occur.

Concurrently, the concept of a dimensionality reduction together with the
notion of "facilitated" diffusion provide an explanation why experimentally
observed rates for binding of proteins to special sites on DNA molecules are
much larger than predictions based on the Smoluchowski approach [5, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 25, 28, 29]. The Adam-Delbrück
scenario (ADS) was also invoked to explain a fast translocation through the
nuclear pore complexes [30]. This scenario also prompted further
investigations giving rise to the idea of so-called intermittent search
strategies [31, 32] in which the fine tuning of systems parameters may further
reduce the mean first-passage time [33, 34, 37, 35, 36, 38, 39, 40] (see
however [41]) or the "survival" probability—the probability that the target is
not found up to time $t$ [42, 43, 44] (in transient processes it is also of
interest to consider the search reliability, the probability that the target
has not been found up to $t\to\infty$ [45, 46]).

To better illustrate the possible advantage of the ADS, it is instructive to
dwell on a particular geometrical setting, which was discussed by Adam and
Delbrück themselves [1] and will be also used in the present paper. Namely, we
consider two nested concentric spheres (of radii $R_{1}$ and $R_{2}$
($R_{1}<R_{2}$), respectively, as shown in figure 1) with impermeable
boundaries, a small immobile circular target of radius $\rho$ placed at some
fixed position on the inner sphere, and a ligand that starts from a fixed
location and diffuses with the diffusion coefficient $D_{b}$ within the
spherical-shell domain $\Omega$ between two spheres (here and in what follows
we use the term "ligand" to denote any diffusing entity, e.g., a signalling
protein; similarly, the term "target" will generally denote a binding site, an
adsorbed chemically active particle or an entrance to a nuclear pore). In such
a bounded domain, a ligand is certain to eventually find the target (see, e.g.
[47]), whichever motion scenario it undertakes—and therefore the only question
is how long the search process will last for a given scenario. We therefore
are interested in the first-passage time (FPT) $\mathcal{T}$, i.e., a random
time instant when a ligand reaches the target for the first time (see, e.g.,
[48, 49]).

Conventional one-stage (or direct search) scenario presumes that the inner
sphere is reflecting for the ligand such that it bounces back to the bulk
domain $\Omega$ once it hits the inner sphere everywhere except for the
target. The binding event thus occurs only when the target is approached by
the ligand directly from the bulk. Assuming that the starting position of the
ligand is uniformly distributed within the spherical-shell domain, Adam and
Delbrück [1] (see also [7]) estimated the mean "diffusion" time $\tau_{\rm
dir}$ necessary for the ligand to arrive to the target within the one-stage
scenario as $\tau_{\rm dir}\sim R_{2}^{3}/(3D_{b}\rho)$ (in the relevant case
when $R_{2}\gg R_{1}\gg\rho$). In fact, $\tau_{\rm dir}$ here is the mean
diffusion time to a sphere of radius $\rho$ placed at the geometric centre of
the outer sphere and thus this form does not take into account the fact that
the target is situated on a reflecting sphere of radius $R_{1}$, which
effectively screens it [50]. Moreover, for a relevant geometrical setting in
which the ligand starts from the outer sphere, $\tau_{\rm dir}$ will be
evidently bigger. Not least, this reasoning clearly applies only to an
idealised situation when the confining domain $\Omega$ does not contain
"obstacles" and the motion of the ligand is not subject to any kind of
molecular crowding effects [51, 52]. However, if $\Omega$ is supposed to mimic
the interior of a cell, it should represent a complex spatial environment,
filled with impermeable organelles, filaments and proteins, which impose
steric constraints on the dynamics and may screen the target. As a
consequence, diffusion of a ligand often takes place effectively in a tortuous
labyrinthine spatial domain. Recent analyses have provided evidence that the
FPT to the target within the direct search scenario is essentially increased
as compared to the one in which the cytosol is treated as a homogeneous
liquid-filled region [53]. Hence, the above estimate can be rather inaccurate
(in fact, representing a lower bound on the actual diffusion time) but is
still instructive for understanding the time scales involved.

In contrast, within the ADS a ligand first finds any (random) point on the
inner sphere, which takes the typical time $\tau_{b}\sim
R_{2}^{3}/(3D_{b}R_{1})$. Then, it nonspecifically binds to the surface and
diffuses along with the diffusion coefficient $D_{s}$ until it eventually
locates the target. The mean diffusion time $\tau_{s}$ for the latter process
was calculated [1, 7] and scales as $\tau_{s}\sim
2R_{1}^{2}\ln(R_{1}/\rho)/D_{s}$. The total mean diffusion time $\tau_{\rm
AD}$ within the ADS is thus a sum of two contributions, $\tau_{\rm
AD}=\tau_{b}+\tau_{s}$, and the ratio

$\displaystyle\delta=\frac{\tau_{\rm AD}}{\tau_{\rm
dir}}=\frac{\rho}{R_{1}}\left(1+6\frac{D_{b}}{D_{s}}\left(\frac{R_{1}}{R_{2}}\right)^{3}\ln\left(\frac{R_{1}}{\rho}\right)\right)$
(1)

shows whether the reduction-of-dimensionality scenario is advantageous (for
$\delta<1$) or not. One observes from equation (1) that the efficiency of the
ADS here is entirely controlled by the aspect ratios $\rho/R_{1}$ and
$R_{1}/R_{2}$ and the ratio $D_{s}/D_{b}$ of the diffusion coefficients.
Clearly, $\tau_{b}$ can also be larger, when obstacles are present in the bulk
volume $\Omega$ but intuitively, its increase should not be as pronounced as
the one for the direct search of the target—within the ADS it suffices to find
any point on the inner sphere while for the direct search scenario a
prescribed target location is to be found.

In essence, the ADS takes advantage of the very slow logarithmic divergence of
$\tau_{\rm AD}$ in the limit $\rho\to 0$, as opposed to a much faster
$1/\rho$-divergence of $\tau_{\rm dir}$ for the direct search scenario. At the
same time, there is a penalty to pay: the gain due to the reduction of the
singularity is counter-balanced by the reduction of the value of the diffusion
coefficient. For instance, the diffusion coefficient $D_{b}$ of a ligand in
cellular cytoplasm is typically of the order of a few tens of
$\mu\mathrm{m}^{2}/\mathrm{s}$, while once it gets associated to the inner
sphere, it experiences at best a $20$-fold reduction of the diffusivity [13].
Most often, however, such a reduction is more pronounced and may amount to two
or sometimes even three orders of magnitude [14]. Such an extreme reduction
is, however, rarely seen in biophysical systems because most of compartment-
separating surfaces are soft and hence, the barriers against lateral diffusion
are typically lower than the ones specific to diffusion on hard solid surfaces
[55]. In any case, a reduction of the diffusivity is detrimental for the ADS,
in virtue of equation (1). As an example, consider a typical mammalian cell in
which a class I nuclear receptor111This example represents an intermediate
step within a complicated intracellular signal transduction pathway in which a
hormone penetrating from an extracellular medium into the cells binds to a
nuclear receptor at a random location within the cytoplasm and causes it to
undergo several chemical transformations. The reaction product subsequently
finds the entrance to the nuclear pore and penetrates into the nucleus, where
it binds to the hormone response element on nuclear DNA [54]. searches for an
entrance into a pore (of radius $\rho\sim 3{\rm nm}$) in the nuclear envelope.
For eukaryotes, the karyoplasmic ratio (KR) typically is of the order
$(R_{1}/R_{2})^{3}\sim 0.08$ (see, e.g., [56]), but the scatter around this
value can be quite significant. In particular, essential departures from the
value $0.08$ are encountered in metastatic tumours and are, in fact, used in
both diagnosis and prognosis for several tumour types [57]. Recent systematic
analysis [58] of the reported values of the KR, which compiled data for almost
$900$ species—from yeast to mammals—provided evidence that the larger cells
almost invariably have relatively smaller nuclei, such that the nucleus of a
larger cell may occupy as little as few percent of the cell volume across all
scales of biological organisation, yielding lower values of KR. For plant
cells, which can often grow to larger sizes than animal cells, it has been
known for a long time that KR can be even smaller and amount to just a
fraction of $1$ percent (see, e.g., Table I in [59]). Concurrently, $R_{1}$
may vary in size in different species but is usually within the range $1\ldots
5\mu\mathrm{m}$. Setting $R_{1}=3\mu\mathrm{m}$ and $\rho=3{\rm nm}$ and
assuming that KR is equal to its average value $0.08$, i.e.,
$R_{2}=7\mu\mathrm{m}$, we thus find $\delta=3.4\times 10^{-3}D_{b}/D_{s}$,
which signifies that for these particular values of the system parameters the
ADS is advantageous when $D_{s}/D_{b}\gtrsim 10^{-2}$. For larger cells or
some plant cells, for which KR is lower, and also in situations in which the
direct search for the entrance to the nuclear pore is obstructed by the
presence of other organelles, one may expect that the ADS is efficient even
for smaller values of the ratio $D_{s}/D_{b}$.

The above well-known arguments rely on estimates of the mean FPT (MFPT), used
as a proxy for the efficiency. In this work we revisit the ADS from the
broader perspective of the full statistics of FPTs. The point is that,
regardless of how a search process proceeds—via a single or two stages—a
ligand may follow a variety of different paths from the starting point to the
target, thus resulting in a large variability of realisation-dependent values
of the FPT. The mean FPT, which is only the first moment of the corresponding
probability density function (PDF) averaged over the initial position of the
ligand, is instructive—yet it is evidently insufficient to fully characterise
neither the ADS nor the direct search scenario. In fact, it is well-known that
for bulk diffusion in a bounded confining domain towards a perfect sink (see,
e.g., [60, 61, 62, 63]), or a partially reactive target (see, e.g., [64, 65,
66]), or even a target with more sophisticated surface reaction mechanism
(see, e.g., [67, 68, 69]), the PDF of the FPTs can be very broad. In other
words, large sample-to-sample fluctuations with disproportionally different
values of $\mathcal{T}$ are inherent and fluctuations around its mean value
$\langle\mathcal{T}\rangle$ can be comparable to it or even exceed the mean
value. It is known from standard statistical analysis that if the PDF is
centred around its mean value (e.g., in the case of a Gaussian distribution),
the mean value is representative of the actual behaviour. In
contradistinction, if the mean is well away from the most probable value (as
shown, e.g., in [64, 65, 66, 62, 63]) it is likely that it is associated with
the tail of the corresponding PDF and hence, is supported by some rare
realisations of the diffusive paths. In this case, $\langle\mathcal{T}\rangle$
may be orders of magnitude larger than the typical time encountered in a
considerable fraction of realisations of the search process (and thus times
shorter than the mean time are typically observed in any given
experiment).222In fact, the most likely time is connected to ”direct”
trajectories moving relatively straight to the target. The distance between
the point of release and the target defines the peak of the initial, Lévy-
Smirnov part, of the PDF, an effect that is also referred to as ”geometry-
control” [63]. Longer times correspond to ”indirect” trajectories, in which
the initial distance becomes irrelevant [63, 64]. This would also imply that
one indeed needs an extensive statistical sample in order to obtain a reliable
value for $\langle\mathcal{T}\rangle$ for comparison with theoretical
predictions. This is precisely the case for the direct search scenario in such
a geometrical setting, as evidenced recently in [50]. Concurrently, the full
PDF of the FPT for the two-stage search process is not known as yet and
consequently, one does not know anything about its broadness and other
characteristic time scales (e.g., the typical FPT), that is indispensable in
order to obtain a fully comprehensive, global picture of the search dynamics
in the ADS. The knowledge of the full PDF will also permit us to use as a
robust characteristic the survival probability, i.e., the probability that the
target is not found up to some prescribed time $t$.

Figure 1: Sketch of the geometrical setup in the study of the ADS: a spherical
shell domain $\Omega$ is enclosed by impermeable boundaries consisting of two
concentric nested spheres with the radii $R_{1}$ (the inner sphere) and
$R_{2}$ (the outer, fully reflective sphere). A target region is located at
the North pole of the inner sphere and has the form of a spherical cap of
radius $\rho$ (and angular size $\varepsilon=\arcsin(\rho/R_{1})$). A small
red ball denotes the starting position $\mathbf{x}=(r,\theta,\phi)$ of the
diffusive ligand. Within the ADS, the ligand diffuses (with diffusion
coefficient $D_{b}$) within $\Omega$ until it encounters the surface of the
inner sphere, which happens at an arbitrary point
$\mathbf{s}=(R_{1},\theta^{\prime},\phi^{\prime})$. Then, it non-specifically
binds to the surface and diffuses along it with diffusion coefficient $D_{s}$,
until it finds the boundary of the target region. Within the direct search
scenario, the surface of the inner sphere is assumed to be perfectly
reflecting such that the ligand bounces back to $\Omega$ once it hits the
inner sphere and may reach the target region only directly from the bulk
$\Omega$.

In this paper, we first consider the case of a single ligand starting from an
arbitrary fixed position within the spherical-shell domain $\Omega$ and a
single perfect target placed on the inner sphere. The term "perfect" here
means that we assume that there is no (energetic or entropic) barrier against
the reaction, such that the ligand binds to the target upon the first
arrival—the classical Smoluchowski setting. In the ADS, we calculate the PDF
of the FPTs to the target exactly, which permits us to analyse the spread of
the realisation-dependent FPTs, as well as to understand the contribution of
the typical first-passage time and of extreme events associated with the tails
of the distribution. Moreover, we compare our result against the PDF for the
direct search scenario, which was evaluated for the same geometrical setting
recently [50], in an exact spectral form as well as in approximate but
remarkably accurate form based on the self-consistent approximation (see [10,
70, 71, 72] for more details). Such a comparison between the ADS and a one-
stage search under identical conditions allows us to provide a complete
picture of the actual efficiency of the ADS which extends beyond the previous
analyses. Next, we consider the situation with an "amplified signal", in which
$N$ ligands are launched either from the same location on the outer sphere, or
from distinct locations on this surface. The former case is realised in
situations when some other ligand (e.g., a first messenger), which is moving
diffusively in the extracellular medium, arrives to a particular site on the
outer part of the plasma membrane and opens an ion channel spanning the
membrane. In this way, the first messenger effects $N$ ions to release inside
the cell from the same position. In turn, the latter case corresponds to a
situation when the plasma membrane hosts multiple receptors each interacting
with the first messengers moving in the extracellular medium and launching the
second messengers that diffuse now within the intracellular medium, all of
them seeking a single target on the inner sphere [73, 74]. For such amplified
signals we also provide a comparative analysis of the PDFs for the one-stage
and two-stages scenarios. We finally remark that the results presented here
can be generalised rather straightforwardly for the analysis of the PDFs of
the terminal FPT in intracellular signal transduction processes involving more
than two nested domains (see [75]), presenting in this way a full stochastic
description of the latter important processes within the reduction-of-
dimensionality scenario.

The paper is outlined as follows: In section 2, we describe the geometrical
and physical parameters of our model, introduce basic notations, and derive
the PDF of the first-passage times within the ADS. Section 3 is devoted to the
analysis of our general result and of its asymptotic behaviour, and also
presents a comparison of the form of the PDF for the ADS against the results
for the direct, one-stage scenario derived in [50]. In section 4 we consider
the situation of an amplified signal with $N$ independent ligands. Finally, in
section 5 we conclude with a brief recapitulation of our results and a
discussion. Details of intermediate derivations, as well as an analysis of
some limiting situations are relegated to Appendices. In particular, A
considers the moments of the conditioned first-passage times; B is devoted to
a discussion of different aspects of extremal values of surface diffusion; in
C we discuss the form of the PDF for the Adam-Delbrück’s scenario, while in D
we present the PDF for one-stage search process.

##  2 The ADS in a spherical shell domain

###  2.1 Model and basic notations

We consider a spherical shell domain $\Omega$ (see figure 1) enclosed by
impermeable boundaries of two concentric nested spheres with radii $R_{1}$
(the inner sphere) and $R_{2}$ (the outer, perfectly reflecting sphere). An
immobile target with the shape of a spherical cap (dome) of radius $\rho$ is
located at the North pole of the inner sphere. In spherical coordinates
$(r,\theta,\phi)$, the target is defined by $r=R_{1}$ and
$0\leq\theta\leq\varepsilon$, where $\varepsilon=\arcsin(\rho/R_{1})$ is the
angular size of the target. For the above example of a nuclear receptor
seeking for a nuclear pore (as well as in many other realistic cases)
$\varepsilon\approx 10^{-3}$, but the value may be smaller or larger in other
applications. We note parenthetically that values $\varepsilon\approx 10^{-1}$
can be indicative of the behaviour in situations when there are many small
targets present on the surface of the inner sphere (e.g., the nuclear membrane
may host about $10^{2}$ nuclear pores). As a consequence, we here consider
$\varepsilon$ as an independent parameter and explore the dependence of the
PDFs on its value. Moreover, we also consider the ratio $D_{b}/D_{s}$ as an
independent parameter and analyse how the shape of the PDF itself along with
the survival probability vary with this ratio.

###  2.2 General expressions

Consider now a ligand that starts from a fixed position $\mathbf{x}$, performs
a diffusive motion with bulk diffusivity $D_{b}$ inside the spherical-shell
domain, until it hits the inner sphere for the first time at the random time
$\mathcal{T}^{\rm bulk}$. Then, it diffuses along the surface for the random
time $\mathcal{T}^{\rm surf}$ after which the ligand hits the perimeter of the
target for the first time. The fact that the search process consists of two
consecutive independent stages of durations $\mathcal{T}^{\rm bulk}$ and
$\mathcal{T}^{\rm surf}$ permits us to write the PDF $H^{\rm
AD}(t;\mathbf{x})$ of the event that it took the ligand exactly time
$\mathcal{T}=\mathcal{T}^{\rm bulk}+\mathcal{T}^{\rm surf}$ to arrive to the
target for the first time as the convolution

$\displaystyle H^{\rm
AD}(t;\mathbf{x})=\int_{\partial\Omega}d\mathbf{s}\int^{t}_{0}dt_{1}\,H^{\rm
surf}(t-t_{1};\mathbf{s})\,j(t_{1};\mathbf{s}|\mathbf{x}),$ (2)

where the first integral over $d\mathbf{s}$ is taken over the surface
$\partial\Omega$ of the inner sphere. Here $j(t_{1};\mathbf{s}|\mathbf{x})$
denotes the joint PDF of the event that a ligand starting from $\mathbf{x}$ at
time $t=0$ arrived for the first time at the inner sphere at time $t_{1}$ and
that this first arrival occurred at point $\mathbf{s}$. In turn, $H^{\rm
surf}(t_{2}|\mathbf{s})$ denotes the PDF that the duration of the surface
diffusion, starting from the point $\mathbf{s}$ and terminating when the
ligand arrives at any point on the perimeter of the target, is equal to
$t_{2}$. An analogous expression for more general $n$-stage processes in one-,
two- and three-dimensional unbounded domains has been recently studied in
[75].

The form of expression (2) suggests that it can be conveniently studied by
resorting to the Laplace domain with respect to $t$. Let

$\displaystyle\tilde{H}^{\rm
AD}(p;\mathbf{x})=\int^{\infty}_{0}dt\,e^{-pt}\,H^{\rm
AD}(t;\mathbf{x}),\qquad\tilde{H}^{\rm
surf}(p;\mathbf{s})=\int^{\infty}_{0}dt\,e^{-pt}H^{\rm surf}(t;\mathbf{s})$
(3)

and

$\displaystyle\tilde{j}(p;\mathbf{s}|\mathbf{x})=\int^{\infty}_{0}dt\,e^{-pt}\,j(t;\mathbf{s}|\mathbf{x})$
(4)

denote the Laplace-transformed PDFs. Then, multiplying both sides of equation
(2) by $\exp(-pt)$ and taking the integral, we find

$\displaystyle\tilde{H}^{\rm AD}(p;\mathbf{x})$
$\displaystyle=\int_{\partial\Omega}d\mathbf{s}\,\tilde{H}^{\rm
surf}(p;\mathbf{s})\,\tilde{j}(p;\mathbf{s}|\mathbf{x})$
$\displaystyle=R_{1}^{2}\int\limits_{0}^{\pi}d\theta^{\prime}\,\sin\theta^{\prime}\int\limits_{0}^{2\pi}d\phi^{\prime}\,\tilde{j}(p,(R_{1},\theta^{\prime},\phi^{\prime})|(r,\theta,\phi))\,\tilde{H}^{\rm
surf}(p;\theta^{\prime}),$ (5)

where the spherical coordinates $\theta^{\prime}$ and $\phi^{\prime}$ define
the position of the point $\mathbf{s}$, at which the ligand first lands on the
inner sphere, while the coordinates $(r,\theta,\phi)$ denote the position of
the starting point $\mathbf{x}$. Note that due to the axial symmetry,
$\tilde{H}^{\rm surf}(p;\theta^{\prime})$ is independent of the azimuthal
angle $\phi^{\prime}$.

Equation (2.2) is the basis for our further analysis, but it requires
knowledge of the corresponding FPTs of the intermediate stages. The exact form
of $\tilde{j}(p;\mathbf{x}|\mathbf{s})$ has been recently studied in [76] and
obeys

$\tilde{j}(p,\mathbf{s}|\mathbf{x})=\frac{1}{4\pi
R_{1}^{2}}\sum\limits_{n=0}^{\infty}(2n+1)P_{n}\left(\frac{(\mathbf{s}\cdot\mathbf{x})}{|\mathbf{s}||\mathbf{x}|}\right)g_{n}^{(p)}(r),$
(6)

where

$g_{n}^{(p)}(r)=\frac{k^{\prime}_{n}(\alpha R_{2})i_{n}(\alpha
r)-i^{\prime}_{n}(\alpha R_{2})k_{n}(\alpha r)}{k^{\prime}_{n}(\alpha
R_{2})i_{n}(\alpha R_{1})-i^{\prime}_{n}(\alpha R_{2})k_{n}(\alpha R_{1})}$
(7)

are the radial functions, $\alpha=\sqrt{p/D_{b}}$,
$i_{n}(z)=\sqrt{\pi/(2z)}I_{n+1/2}(z)$, and $k_{n}(z)=\sqrt{2/(\pi
z)}\,K_{n+1/2}(z)$ are the modified spherical Bessel functions of the first
and second kind, respectively. The prime denotes a derivative with respect to
the argument, and $P_{n}(z)$ are Legendre polynomials whose argument is the
cosine of the angle between the vectors $\mathbf{x}$ and $\mathbf{s}$. In
particular, equation (6) defines the moments of the FPT on the inner sphere,
conditioned by the arrival point $\mathbf{s}$ (see A).

The above expression for $\tilde{j}(p,\mathbf{s}|\mathbf{x})$ is valid even in
the limit $R_{2}\to\infty$ (e.g., when $R_{2}$ becomes macroscopically large,
as is often the case in cell-to-cell communication processes) when the outer
reflecting boundary practically goes to infinity and one retrieves a common
setting of a spherical target, accessed by a particle diffusing in the
unbounded space $\Omega$. Even though the domain itself is unbounded, its
boundary ${\partial\Omega}$ is bounded, and the above solution is still
applicable. From the asymptotic behaviour of the modified spherical Bessel
functions $i_{n}(z)$ and $k_{n}(z)$, one gets immediately that the radial
functions $g_{n}^{(p)}(r)$ converge to

$g_{n}^{(p)}(r)\to\frac{k_{n}(\alpha r)}{k_{n}(\alpha R_{1})}.$ (8)

In the following, we keep focusing on the setting with a finite $R_{2}$.

###  2.3 Surface diffusion stage

Despite the fact that the first-passage statistics for a particle diffusing on
the surface of a spherical domain toward a target of an arbitrary size has
been studied in the past, former works focused mostly on the MFPT [1, 7] (see
also [77, 78, 79, 80] and references therein). Notable exceptions are the
Refs. [77, 78], in which the spectral expansion of the survival probability in
time domain was derived. While its Laplace transform allows one to access
$\tilde{H}^{\rm surf}(p;\mathbf{s})$ as a spectral expansion, as well, we get
a more compact form for this function that is suitable for further analysis of
$\tilde{H}^{\rm AD}(p;\mathbf{x})$ in equation (2.2). Relegating the details
of calculations to B below we merely display the final results for
$\tilde{H}^{\rm surf}(p;\mathbf{s})$ and $H^{\rm surf}(t;\mathbf{s})$. When
diffusion starts within the target, one has

$\displaystyle\tilde{H}^{\rm
surf}(p;\theta^{\prime})=1\quad(0\leq\theta^{\prime}\leq\varepsilon);$ (9)

in turn, for $\varepsilon\leq\theta^{\prime}\leq\pi$, one gets the remarkably
compact form

$\displaystyle\tilde{H}^{\rm
surf}(p;\theta^{\prime})=\frac{P_{\mu}(-\cos(\theta^{\prime}))}{P_{\mu}(-\cos(\varepsilon))},\qquad\mu=\frac{-1+\sqrt{1-4pR_{1}^{2}/D_{s}}}{2},$
(10)

where $P_{\mu}(z)$ is the Legendre function of order $\mu$ which depends on
the Laplace parameter $p$. Note that for $p\ll D_{s}/(4R_{1}^{2})$
(corresponding to the long-$t$ tail of the PDF), the parameter $\mu$ is a real
number, while in the opposite limit $p\gg D_{s}/(4R^{2}_{1})$ (corresponding
to the short-$t$ tail of the PDF), $\mu$ is a complex number with a real part
equal to $-1/2$, such that $P_{\mu}(z)$ is related to the so-called conical
(or Mehler) function. Expression (10) can be inverted (see B for more details)
to produce the spectral expansion (see also [77, 78])

$\displaystyle H^{\rm surf}(t;\theta^{\prime})$
$\displaystyle=\frac{D_{s}\sin^{2}(\varepsilon)}{R_{1}^{2}}\sum_{n=0}^{\infty}\left(\int^{1}_{-\cos(\varepsilon)}dx\left[P_{\nu_{n}}(x)\right]^{2}\right)^{-1/2}P_{\nu_{n}}(-\cos(\theta^{\prime}))P^{\prime}_{\nu_{n}}(-\cos(\varepsilon))$
$\displaystyle\qquad\times e^{-\nu_{n}(\nu_{n}+1)D_{s}t/R^{2}_{1}},$ (11)

where the prime, as in equation (7), denotes the derivative with respect to
the argument, while $\nu_{n}$ are the solutions of

$\displaystyle P_{\nu_{n}}(-\cos(\varepsilon))=0$ (12)

organised in an ascending order. Evidently, $\nu_{n}$ are functions of
$\varepsilon$. Eventually, we note that the moments of $H^{\rm
surf}(t;\theta^{\prime})$ can be found directly from expression (10) by
differentiating it with respect to the Laplace parameter $p$. In particular,
differentiating equation (10) once and twice with respect to $p$ and setting
$p=0$, one finds first two moments of the FPT $\mathcal{T}^{\rm surf}$ for
$\varepsilon\leq\theta^{\prime}\leq\pi$,

$\displaystyle\langle\mathcal{T}^{\rm surf}\rangle$
$\displaystyle=\frac{R_{1}^{2}}{D_{s}}\ln\left(\frac{1-\cos(\theta^{\prime})}{1-\cos(\varepsilon)}\right),$
(13) $\displaystyle\langle(\mathcal{T}^{\rm surf})^{2}\rangle$
$\displaystyle=\frac{2R_{1}^{4}}{D_{s}^{2}}\left[{\rm
Li}_{2}\left(\frac{1+\cos(\varepsilon)}{2}\right)-{\rm
Li}_{2}\left(\frac{1+\cos(\theta^{\prime})}{2}\right)+\ln\left(\frac{1-\cos(\varepsilon)}{1-\cos(\theta^{\prime})}\right)\right.$
$\displaystyle\left.+\ln^{2}\left(\frac{1-\cos(\varepsilon)}{2}\right)-\ln\left(\frac{1-\cos(\varepsilon)}{2}\right)\ln\left(\frac{1-\cos(\theta^{\prime})}{2}\right)\right],$
(14)

where ${\rm Li}_{2}(z)=-\int\nolimits_{0}^{z}dx\,\ln(1-x)/x$ is the
dilogarithm. Note that the MFPT was known (see, e.g., [78]), while the result
for the second moment has not been reported.

These expressions allow us to determine the variance ${\rm
var}(\mathcal{T}^{\rm surf})=\langle(\mathcal{T}^{\rm
surf})^{2}\rangle-\langle\mathcal{T}^{\rm surf}\rangle^{2}$ of the PDF, and
also to characterise its broadness (see [60, 61]) from the corresponding
coefficient of variation $\gamma=\sqrt{{\rm var}(\mathcal{T}^{\rm
surf})}/\langle\mathcal{T}^{\rm surf}\rangle$, which is defined as the ratio
of the standard deviation around the mean value and the mean value itself.
Inspecting the behaviour of $\langle\mathcal{T}^{\rm surf}\rangle$ and
$\langle(\mathcal{T}^{\rm surf})^{2}\rangle$, we realise that for fixed
(sufficiently small) $\varepsilon$ and $\theta^{\prime}$ sufficiently close to
$\varepsilon$, $\langle\mathcal{T}^{\rm surf}\rangle\simeq
2R_{1}^{2}(\theta^{\prime}-\varepsilon)/(\varepsilon D_{s})$ and $\sqrt{{\rm
var}(\mathcal{T}^{\rm surf})}\simeq 2R_{1}^{2}\sqrt{(2\ln
2-1-2\ln\varepsilon)(\theta^{\prime}-\varepsilon)}/(\sqrt{\varepsilon}D_{s})$.
Combining these expressions, we find that for $\theta^{\prime}$ close to
(small) $\varepsilon$ the coefficient of variation obeys

$\displaystyle\gamma\sim\sqrt{\frac{(2\ln
2-1-2\ln\varepsilon)\,\varepsilon}{\theta^{\prime}-\varepsilon}},$ (15)

and hence, it diverges when $\theta^{\prime}\to\varepsilon$. Overall, $\gamma$
is a monotonously decreasing function of $\theta^{\prime}$ that attains its
minimal value (for small enough $\varepsilon$)

$\gamma_{\rm
min}\approx\frac{\sqrt{\pi^{2}/3-1+(1-2\ln(2/\varepsilon))^{2}}}{2\ln(2/\varepsilon)}$
(16)

for $\theta^{\prime}=\pi$, i.e., when the starting point $\mathbf{s}$ of the
particle is located on the South pole (figure 2). In the limit $\varepsilon\to
0$, this minimal value approaches unity from below, i.e., $\gamma_{\rm
min}\leq 1$. As a consequence, for any $\varepsilon>0$, there exists a value
$\theta^{*}_{\varepsilon}$, at which $\gamma=1$. For
$\theta^{\prime}<\theta^{*}_{\varepsilon}$, the standard deviation of the FPT
is larger than the mean value $\langle\mathcal{T}^{\rm surf}\rangle$ such that
the latter cannot be used to characterise the search process exhaustively
well.

Figure 2: Coefficient of variation $\gamma=\sqrt{{\rm var}(\mathcal{T}^{\rm
surf})}/\langle\mathcal{T}^{\rm surf}\rangle$ as function of $\theta^{\prime}$
for the FPT to the target of angular size $\varepsilon$ by surface diffusion,
for two values of $\varepsilon$. Thick lines represent the exact result from
equations (13) and (14), while the thin lines show the asymptotic relation
(15).

###  2.4 Asymptotic behaviour of equation (10)

Before we turn to the discussion of the ADS it is instructive to discuss the
asymptotic behaviour of its constituents: $\tilde{j}(p,\mathbf{s}|\mathbf{x})$
in equation (6) and $\tilde{H}^{\rm surf}(p;\theta^{\prime})$ in equation
(10). We will show that both exhibit some non-trivial and universal behaviour,
which will permit us to reach several conclusive statements about the
asymptotic behaviour of the full PDF $H^{\rm AD}(t;\mathbf{x})$.

We start with $\tilde{H}^{\rm surf}(p;\theta^{\prime})$ and consider first the
limit $p\to\infty$, which corresponds to the short-$t$ tail of the associated
PDF. In B we show that in this limit $\tilde{H}^{\rm surf}(p;\theta^{\prime})$
obeys

$\displaystyle\tilde{H}^{\rm
surf}(p;\theta^{\prime})\simeq\sqrt{\frac{\sin(\varepsilon)}{\sin(\theta^{\prime})}}\,e^{-R_{1}(\theta^{\prime}-\varepsilon)\sqrt{p/D_{s}}}\quad(p\to\infty)\,,$
(17)

where the symbol $\simeq$ denotes that we consider the leading order of this
limiting behaviour. Inverting the Laplace transform in the latter expression,
we find that the short-$t$ asymptotic behaviour of the PDF $H^{\rm
surf}(t;\theta^{\prime})$ follows

$\displaystyle H^{\rm
surf}(t;\theta^{\prime})\simeq\sqrt{\frac{\sin(\varepsilon)}{\sin(\theta^{\prime})}}\,\frac{(\theta^{\prime}-\varepsilon)R_{1}}{\sqrt{4\pi
D_{s}t^{3}}}\,e^{-(\theta^{\prime}-\varepsilon)^{2}R_{1}^{2}/(4D_{s}t)}.$ (18)

Remarkably, this result coincides (up to the factor
$\sqrt{\sin(\varepsilon)/\sin(\theta^{\prime})}$) with the celebrated Lévy-
Smirnov distribution [48, 49], defining the exact FPT PDF from a point
$x_{0}=(\theta^{\prime}-\varepsilon)R_{1}>0$ to a perfect target located at
the origin of a one-dimensional semi-infinite line. Noticing that
$R_{1}(\theta^{\prime}-\varepsilon)$ is, in fact, the geodesic distance
between the point $\mathbf{s}$ and the boundary of the target region, i.e.,
the shortest path along the surface of the inner sphere from the starting
point $\mathbf{s}$ to the target, its physical significance becomes apparent:
the short-$t$ tails of the PDF are dominated by such trajectories which move
diffusively along the shortest distance to the target. We note that such a
Lévy-Smirnov-like form, which is specific to one-dimensional situations, has
been previously evidenced for rather different two- and three-dimensional
geometrical settings (see, e.g., [64, 65, 66, 62, 63, 81]) and therefore seems
to be quite generic. In particular, this result is analogous to the "geometry
control" unveiled previously in the case of pure bulk diffusion in $\Omega$
[62, 64].

In turn, the leading long-$t$ behaviour of the PDF $H^{\rm
surf}(t;\theta^{\prime})$ can be readily deduced from the series
representation in equation (2.3). Recalling that the solutions $\nu_{n}$ form
an increasing sequence, we realise that at long times the dominant
contribution corresponds to the smallest root $\nu_{0}$ of equation (12) and,
therefore, the long-$t$ behaviour reads

$\displaystyle H^{\rm surf}(t;\theta^{\prime})$
$\displaystyle\simeq\frac{\sin^{2}(\varepsilon)D_{s}}{R_{1}^{2}}\left(\int^{1}_{-\cos(\varepsilon)}dx\left[P_{\nu_{0}}(x)\right]^{2}\right)^{-1/2}P_{\nu_{0}}(-\cos(\theta^{\prime}))P^{\prime}_{\nu_{0}}(-\cos(\varepsilon))$
$\displaystyle\quad\times e^{-\nu_{0}(\nu_{0}+1)D_{s}t/R^{2}_{1}}.$ (19)

We note that $\nu_{0}$ depends on $\varepsilon$ but is independent of
$\theta^{\prime}$; in particular when $\varepsilon$ is small, one has
$\nu_{0}\approx 1/(2\ln(2/\varepsilon))$ [80]. As a consequence the longest
relaxation time

$\tau^{\rm
surf}=\frac{R^{2}_{1}}{\nu_{0}(\nu_{0}+1)D_{s}}\approx\frac{2R_{1}^{2}\ln(2/\varepsilon)}{D_{s}}\quad(\varepsilon\ll
1)$ (20)

is independent of the starting point, i.e., corresponds to such long times at
which a particle arrives to the target region after extensive target-avoiding
excursions and thus loses information about its initial location. The latter
is kept only through the amplitude in equation (2.4).

###  2.5 Asymptotic behaviour of equation (6)

Figure 3: Large-$p$ asymptotic behaviour of
$\tilde{j}(p,\mathbf{s}|\mathbf{x})$ in equation (6) for $\phi=\phi^{\prime}$.
Panel (a): Shortest distances to the arrival point $\mathbf{s}$ located on the
North pole (filled squares) from different starting points $\mathbf{x}$
(filled circles). Blue ($\theta=0$) and green ($\theta=\pi/4$) circles
correspond to the case when $\mathbf{s}$ is directly visible from
$\mathbf{x}$, such that the shortest distance ${\ell}_{\rm vis}$ (see the
corresponding dotted lines) is given by the straight segment connecting the
initial point and the arrival point $\mathbf{s}$. Red ($\theta=\pi/2$) and
magenta ($\theta=\pi$) circles correspond to the situation in which the target
appears on the "dark" side of the inner sphere with respect to the initial
point. In this case the shortest distance ${\ell}_{\rm invis}$ consists of two
parts: A straight segment of length $h=\sqrt{r^{2}-R_{1}^{2}}$ from the
initial point to the point closest to the target on the horizon, and an arc
from this point to the target. Panel (b): Numerical analysis of
$\tilde{j}(p,\mathbf{s}|\mathbf{x})$ in equation (6) as function of
$R_{1}\sqrt{p/D_{b}}$. Coloured thick curves present
$\tilde{j}(p,\mathbf{s}|\mathbf{x})$ obtained by summing the series in
equation (6) truncated at $n_{\rm max}=100$. As we are interested in the
large-$p$ behaviour, we set $R_{2}=\infty$ here to ease the numerical
computation of the radial functions $g_{n}^{(p)}(r)$. Thin grey lines show the
asymptotic forms in equation (21) and (22) with the shortest distances
calculated for the corresponding geometrical setup (note that the unknown
functions $a(\theta,\theta^{\prime})$ and $b(\theta,\theta^{\prime})$,
controlling a vertical offset of these lines, were set to $1$).

Let us now turn to the joint probability density $j(t,\mathbf{s}|\mathbf{x})$.
A theoretical analysis of the large-$p$ asymptotic behaviour of expression (6)
appears to be a difficult task; we realise that, in fact, in order to
determine the corresponding asymptotic form one has to perform the sum in this
equation exactly, which requires a cumbersome analysis. We thus resort instead
to a numerical analysis of expression (6) aiming to verify the following
conjecture: we assume that, in line with the "geometrical optics" arguments
presented earlier in [82] (see section II) and similarly to the above
considered case of a search process on the surface of a sphere, the short-$t$
tail of the associated PDF corresponds to the diffusive motion along the
shortest path (with length ${\ell}$) from $\mathbf{x}=(r,\theta,\phi)$ to
$\mathbf{s}=(R_{1},\theta^{\prime},\phi^{\prime})$. This path, however, has a
different form depending on whether the point $\mathbf{s}$ is directly
"visible" from $\mathbf{x}$, i.e., is located within a spherical cap region
delimited by the horizon, or whether it is located outside of this area being
situated on the "dark" side of the inner sphere and therefore invisible from
$\mathbf{x}$. In case when $\mathbf{s}$ is visible from $\mathbf{x}$ we thus
expect that

$\displaystyle\tilde{j}(p,\mathbf{s}|\mathbf{x})\simeq
a(\theta,\theta^{\prime})e^{-{\ell}_{\rm
vis}\sqrt{p/D_{b}}}\quad(p\to\infty),$ (21)

where $a(\theta,\theta^{\prime})$ is some unknown amplitude (recall that the
result in equation (18) differs from the standard Lévy-Smirnov density by some
function) and ${\ell}_{\rm vis}$ is the Euclidean distance from $\mathbf{x}$
to $\mathbf{s}$. In turn, for the case when $\mathbf{s}$ is located on the
"dark" side, the shortest path ${\ell}_{\rm invis}$ consists of two parts (see
also [82]): the particle diffuses along a straight line of length
$h=\sqrt{r^{2}-R_{1}^{2}}$ connecting $\mathbf{x}$ and the point on the
horizon, which is closest to $\mathbf{s}$, and then travels diffusively in the
immediate vicinity of the inner sphere along the arc connecting this point on
the horizon and $\mathbf{s}$. Consequently, in this case we expect that
$\tilde{j}(p,\mathbf{s}|\mathbf{x})$ behaves in the limit $p\to\infty$ as

$\displaystyle\tilde{j}(p,\mathbf{s}|\mathbf{x})\simeq
b(\theta,\theta^{\prime})e^{-{\ell}_{\rm
invis}\sqrt{p/D_{b}}}\quad(p\to\infty),$ (22)

where $b(\theta,\theta^{\prime})$ is an unknown amplitude. Note that
${\ell}_{\rm vis}$ and ${\ell}_{\rm invis}$ can be simply expressed through
the Cartesian coordinates of both points.

In figure 3 we compare our conjectured equations (21) and (22) (thin grey
lines) and the exact expression (thick coloured curves) obtained from series
truncation in equation (6) for two situations in which the target is visible
from the starting point, and two situations in which it is located on the dark
side. We observe that in all four cases the slopes of thin grey lines and of
the thick coloured curves are almost identical—which thus confirms our
conjecture of the large-$p$ behaviour of $\tilde{j}(p;\mathbf{s}|\mathbf{x})$.
Inverting the expressions in equations (21) and (22), we thus find that the
short-$t$ behaviour of the corresponding PDF $j(t;\mathbf{s}|\mathbf{x})$ has
the Lévy-Smirnov form ${\ell}\exp(-{\ell}^{2}/(4D_{b}t))/\sqrt{4\pi
D_{b}t^{3}}$, up to a yet unknown amplitude factor $a(\theta,\theta^{\prime})$
or $b(\theta,\theta^{\prime})$, respectively.

Lastly, we discuss the long-$t$ behaviour of the PDF
$j(t,\mathbf{s}|\mathbf{x})$. As diffusion occurs in a bounded domain,
$j(t,\mathbf{s}|\mathbf{x})$ clearly exhibits an exponential decay which is
controlled by the pole $p_{0}^{\rm bulk}$ of
$\tilde{j}(p,\mathbf{s}|\mathbf{x})$ with the smallest absolute value. This is
actually the pole of the radial function $g_{0}^{(p)}(r)$, which can be
written as

$p_{0}^{\rm bulk}=-\frac{1}{\tau^{\rm bulk}}<0,\mbox{ with }\tau^{\rm
bulk}=\frac{(R_{2}-R_{1})^{2}}{D_{b}\alpha_{0}^{2}},$ (23)

where $\alpha_{0}$ is the smallest root of the transcendental equation (see,
e.g., [64])

$\frac{\tan(\alpha_{0})}{\alpha_{0}}=\frac{R_{2}}{R_{2}-R_{1}}.$ (24)

When $R_{1}\ll R_{2}$, this solution behaves as $\alpha_{0}^{2}\approx
R_{1}/(3(R_{2}-R_{1}))$ and thus

$\tau^{\rm bulk}\approx\frac{R_{2}^{3}}{3D_{b}R_{1}}.$ (25)

Therefore, in the long-$t$ limit the PDF $j(t,\mathbf{s}|\mathbf{x})$ decays
as

$\displaystyle j(t,\mathbf{s}|\mathbf{x})\propto e^{-t/\tau^{\rm bulk}},$ (26)

where $\tau^{\rm bulk}$ is the longest time characterising the FPT PDF to the
inner sphere conditioned by the constraint that this event took place at point
$\mathbf{s}$. Note that $\tau^{\rm bulk}$ is independent of both the initial
position $\mathbf{x}$ and the precise location of the arrival point
$\mathbf{s}$. Some other properties of the probability flux density
$j(t,\mathbf{s}|\mathbf{x})$ were discussed in a recent paper [83].

##  3 Probability density function of the first-passage times within the Adam-
Delbrück scenario

In this section we focus on the statistics of the FPTs in the ADS. We start
from our main relation (2.2), in which the integral over the first arrival
point $\mathbf{s}$ on the inner sphere can be further simplified (see details
in C) to get the exact and fully explicit solution in the Laplace domain

$\displaystyle\tilde{H}^{\rm AD}(p;\mathbf{x})$
$\displaystyle=\sum\limits_{n=0}^{\infty}g_{n}^{(p)}(r)P_{n}(\cos\theta)\biggl{\\{}\frac{P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)}{2}$
$\displaystyle-(1-\cos^{2}\varepsilon)\frac{n+1/2}{\frac{pR^{2}}{D_{s}}+n(n+1)}\,\biggl{(}P^{\prime}_{n}(\cos\varepsilon)+P_{n}(\cos\varepsilon)\frac{P^{\prime}_{\mu}(a)}{P_{\mu}(a)}\biggr{)}\biggr{\\}}.$
(27)

We first discuss the short-$t$ and long-$t$ asymptotic behaviour of the PDF
$H^{\rm AD}(t;\mathbf{x})$. Next, numerically inverting the Laplace transform
in equation (27) by help of the Talbot algorithm we discuss the behaviour of
the PDF in the time domain for different values of the system parameters and
compare it against the recently obtained PDF for the direct search scenario in
precisely the same geometrical settings [50]. This will give us a general idea
of the shapes of two PDFs. Moreover, we turn to the integrated
characteristic—the survival probability

$\displaystyle S^{\rm
AD}(t;\mathbf{x})=\int\limits_{t}^{\infty}dt^{\prime}\,H^{\rm
AD}(t^{\prime};\mathbf{x}),$ (28)

i.e., the probability that the target is not found up to time $t$. The
analysis of the survival probability for both search scenarios provides a full
understanding of the actual efficiency of each scenario and, hence, gives an
idea which scenario is more successful. In the Laplace domain, the survival
probability in equation (28) reads $\tilde{S}^{\rm
AD}(p;\mathbf{x})=(1-\tilde{H}^{\rm AD}(p;\mathbf{x}))/p$, allowing us to
determine its asymptotic properties from those for the Laplace-transformed
PDF. The analysis of the survival probability will permit us to make several
conclusive statements. Lastly, we consider the particular, experimentally
relevant case when the starting point is uniformly distributed on a spherical
surface of radius $r$ such that $0<R_{1}\leq r\leq R_{2}$. Moreover, in C we
present additional figures illustrating the behaviour of $H^{\rm
AD}(t^{\prime};\mathbf{x})$ for several fixed starting points of the ligand.

###  3.1 Asymptotic behaviour

To access the short-$t$ behaviour of $H^{\rm AD}(t;\mathbf{x})$ we focus on
expression (2.2) together with equations (17), (21), and (22). Combining these
expressions, we get the Laplace-transformed PDF $\tilde{H}^{\rm
AD}(p;\mathbf{x})$ in the limit $p\to\infty$ in the form

$\displaystyle\tilde{H}^{\rm AD}(p;\mathbf{x})$
$\displaystyle\approx\sqrt{\sin(\varepsilon)}\int_{\partial\Omega_{\rm
vis}}d\mathbf{s}\frac{a(\theta,\theta^{\prime})}{\sqrt{\sin(\theta^{\prime})}}\exp\left(-\left(\frac{{\ell}_{\rm
vis}(r,\theta,\theta^{\prime})}{\sqrt{D_{b}}}+\frac{(\theta^{\prime}-\varepsilon)R_{1}}{\sqrt{D_{s}}}\right)\sqrt{p}\right)+$
$\displaystyle\sqrt{\sin(\varepsilon)}\int_{\partial\Omega_{\rm
invis}}d\mathbf{s}\frac{b(\theta,\theta^{\prime})}{\sqrt{\sin(\theta^{\prime})}}\exp\left(-\left(\frac{{\ell}_{\rm
invis}(r,\theta,\theta^{\prime})}{\sqrt{D_{b}}}+\frac{(\theta^{\prime}-\varepsilon)R_{1}}{\sqrt{D_{s}}}\right)\sqrt{p}\right),$
(29)

where $\partial\Omega_{\rm vis}$ and $\partial\Omega_{\rm invis}$ denote the
two parts of the inner sphere, those that are "visible" and "invisible" as
seen from the starting point $\mathbf{x}$. In the limit $p\to\infty$ the
integrands in equation (3.1) vanish exponentially fast with $p$. This
signifies that the dominant contribution to the integrals comes from such
values of the position $\mathbf{s}$ of the landing point onto the inner sphere
for which the coefficients in front of $\sqrt{p}$ are minimal, i.e.,

$\displaystyle\tilde{H}^{\rm AD}(p;\mathbf{x})\sim\exp\left(-\ell_{\rm
min}\sqrt{p/D_{b}}\right),$ (30)

where

$\ell_{\rm
min}=\min\limits_{\theta^{\prime}}\left\\{\ell(r,\theta,\theta^{\prime})+R_{1}(\theta^{\prime}-\varepsilon)\sqrt{D_{b}/D_{s}}\right\\},$
(31)

where $\ell(r,\theta,\theta^{\prime})$ is either $\ell_{\rm vis}$ or
$\ell_{\rm invis}$, depending on the mutual arrangement of $\mathbf{x}$ and
$\mathbf{s}$. As a consequence, we expect that the short-$t$ tail of $H_{\rm
AD}(t;\mathbf{x})$ has a universal Lévy-Smirnov form, yet with a position-
specific prefactor.

As diffusion occurs in a bounded domain, the PDF $H^{\rm AD}(t;\mathbf{x})$
decays exponentially fast at long times, and the decay rate is determined by
the pole $p_{0}<0$ of $\tilde{H}^{\rm AD}(p;\mathbf{x})$ with the smallest
absolute value. Since $\tilde{H}^{\rm AD}(p;\mathbf{x})$ is obtained in
equation (2.2) by integrating the product of
$\tilde{j}(p,\mathbf{s}|\mathbf{x})$ and $\tilde{H}^{\rm surf}(p;\mathbf{s})$
over $\mathbf{s}$, $p_{0}$ is the pole of one of these two functions. Skipping
technical details presented in C.2 we conclude that

$\displaystyle H^{\rm AD}(t;\mathbf{x})\simeq
C_{\varepsilon}(\mathbf{x})\exp\left(-\frac{t}{{\rm max}\left\\{\tau^{\rm
bulk},\tau^{\rm surf}\right\\}}\right),$ (32)

where the amplitude $C_{\varepsilon}(\mathbf{x})$ is explicitly computed in
equations (100) and (104). Here ${\rm max}\left\\{\tau^{\rm bulk},\tau^{\rm
surf}\right\\}$ signifies that the onset and the decay of the long-$t$
asymptotic form is entirely controlled by the longest of the two
characteristic times $\tau^{\rm bulk}$ and $\tau^{\rm surf}$, but not by their
sum, which, in contrast, is assumed in the conventional criterion of the
applicability of the ADS. We finally note that the decay rate in equation (32)
is independent of the starting point $\mathbf{x}$ for any relation between
$\tau^{\rm bulk}$ and $\tau^{\rm surf}$.

###  3.2 Random starting point

In many situations of practical interest the launch of the ligand does not
take place from a fixed prescribed position. Instead there are many points on
the outer sphere from which the ligand can start its search for the target. In
this regard it is instructive to consider the case when $\mathbf{x}$ can be
any (uniformly-distributed) point on the spherical surface of radius $r$ (not
necessarily the outer boundary, $r=R_{2}$). The behaviour of the FPT PDF
corresponding to the case when the ligand starts from a prescribed position
turns out to be quite similar, as discussed in C.4.

The Laplace-transformed PDF for a random starting point, which we denote as
$\overline{\tilde{H}_{\rm AD}(p;r)}$, is obtained by integration of equation
(27) over the angular coordinates of $\mathbf{x}$, yielding

$\displaystyle\overline{\tilde{H}^{\rm AD}(p;r)}$ $\displaystyle=\frac{1}{4\pi
r^{2}}\int_{|\mathbf{x}|=r}d\mathbf{x}\,\tilde{H}^{\rm AD}(p;\mathbf{x})$
$\displaystyle=\frac{1-\cos(\varepsilon)}{2}g_{0}^{(p)}(r)\left(1-\frac{D_{s}(1+\cos(\varepsilon))}{pR_{1}^{2}}\frac{P^{\prime}_{\mu}(-\cos(\varepsilon))}{P_{\mu}(-\cos(\varepsilon))}\right),$
(33)

where $g_{0}^{(p)}(r)$ and $\mu$ were defined in equations (7) and (10),
respectively. We stress that this surface-averaged quantity differs from the
volume average over a uniformly distributed starting point inside the
confining domain $\Omega$.

The asymptotic behaviour of $\overline{\tilde{H}^{\rm AD}(p;r)}$ is derived in
C and reads

$\displaystyle\overline{H^{\rm AD}(t;r)}$
$\displaystyle\simeq\frac{R_{1}}{r}\,\frac{(r-R_{1})}{\sqrt{4\pi
D_{b}t^{3}}}\exp\left(-\frac{(r-R_{1})^{2}}{4D_{b}t}\right)$
$\displaystyle\times\biggl{(}\frac{1-\cos(\varepsilon)}{2}+\frac{\sin(\varepsilon)\sqrt{D_{s}D_{b}t^{2}}}{R_{1}(r-R_{1})}+\frac{\cos(\varepsilon)D_{b}D_{s}t^{2}}{R_{1}^{2}(r-R_{1})^{2}}\biggr{)}.$
(34)

The prefactor in the first line is the PDF of the FPT to a perfectly reactive
sphere of radius $R_{1}$ by bulk diffusion in a three-dimensional unbounded
domain. At very short times, the first term is dominant, and one gets, up to a
geometric prefactor $(1-\cos(\varepsilon))R_{1}/(2r)$, the Lévy-Smirnov
function of $t$,

$\displaystyle\overline{H^{\rm
AD}(t;r)}\simeq\frac{(1-\cos(\varepsilon))R_{1}}{2r}\,\frac{(r-R_{1})}{\sqrt{4\pi
D_{b}t^{3}}}\exp\left(-\frac{(r-R_{1})^{2}}{4D_{b}t}\right).$ (35)

Interestingly, this short-$t$ tail of the PDF is independent of the surface
diffusion coefficient $D_{s}$. Apparently, this is associated with the fact
that when we average over the starting point, the major contribution to the
PDF, which entirely defines its asymptotic behaviour, comes from those initial
locations that are placed directly over the target site. The contributions
from other starting points, for which the search process will include a tour
of surface diffusion, is only sub-dominant in this limit. However, this
leading-order contribution is attenuated by the prefactor
$1-\cos(\varepsilon)\approx\varepsilon^{2}/2$, which can be very small for
small targets. In turn, the second and third terms in equation (34), although
being subleading in powers of time, are weighted by
$\sin(\varepsilon)\approx\varepsilon$ and by $\cos(\varepsilon)\approx 1$,
respectively. This means that when $\varepsilon$ is sufficiently small, there
exists an intermediate range of times, for which the third term is actually
the dominant one,

$\overline{H^{\rm
AD}(t;r)}\simeq\frac{D_{s}\sqrt{D_{b}t}}{\sqrt{4\pi}\,rR_{1}(r-R_{1})}\exp\left(-\frac{(r-R_{1})^{2}}{4D_{b}t}\right).$
(36)

Curiously, this expression is independent of the target size (as soon as
$\varepsilon$ is small). This is due to the fact that a decrease of
$\varepsilon$ reduces the relative contribution of trajectories that arrive to
the target directly from the bulk and thus avoid surface diffusion. Hence, for
small $\varepsilon$ the search process will necessitate surface diffusion.

In the opposite long-time limit, there is no difference between the fixed and
random starting point so that our former asymptotic relation (32) is valid for
$\overline{H^{\rm AD}(t;r)}$.

Figure 4 illustrates the asymptotic behaviour of the surface-averaged PDF
$\overline{H^{\rm AD}(t;r)}$ within the ADS. One sees that the asymptotic
relation (34) accurately describes the short-time behaviour. As the target is
rather small, the dominant contribution comes from the third term in equation
(34); in particular, the three asymptotic curves only differ by the
multiplicative factor $D_{s}$. We stress that the first term in equation (34)
in this setting is totally irrelevant here (if one only kept the first term,
the asymptotic curves would be below $10^{-6}$ and thus invisible in this
figure; for this reason, they are not shown). In other words, the leading-
order Lévy-Smirnov-type relation (35) fails, and one has to rely on our
asymptotic relation (34), in which the third, sub-leading term is dominant.
The long-time asymptotic relation (32) is also very accurate. Note that the
amplitude $C_{\varepsilon}(r)$ is given by equation (104) for the curve with
$D_{s}/D_{b}=1$, for which $\tau^{\rm bulk}>\tau^{\rm surf}$; in turn, it is
given by equation (100) for the two other curves with $D_{s}/D_{b}=0.1$ and
$D_{s}/D_{b}=0.01$. We conclude that both short- and long-time asymptotic
relations are accurate.

Figure 4:  FPT PDF $\overline{H^{\rm AD}(t;r)}$ to a target of the small
angular size $\varepsilon=0.01$ within the ADS, with $R_{2}/R_{1}=5$, for
which the starting point is uniformly distributed over the outer sphere
($r/R_{1}=5$). The three thick coloured curves show the PDF obtained from the
exact solution (33) via numerical inversion of the Laplace transform. Dotted
lines represent the short-time asymptotic relation (34), whereas circles show
the long-time asymptotic relation (32). The leading-order Lévy-Smirnov-type
relation (35) is not shown because it lies below the bottom limit $10^{-6}$ of
the plot; in fact, its maximal amplitude is $2.89\times 10^{-7}$, i.e., this
approximation fails by 4 to 5 orders of magnitude. The length and time units
are fixed by setting $R_{1}=1$ and $D_{b}=1$.

The derivative of $\overline{\tilde{H}^{\rm AD}(p;r)}$ in equation (33) with
respect to $p$ determines the surface-averaged MFPT. Skipping the details of
this computation (see C), the final result reads

$\overline{T^{\rm
AD}}=\frac{(r-R_{1})(2R_{2}^{3}-rR_{1}(r+R_{1}))}{6rR_{1}D_{b}}+\frac{R_{1}^{2}}{2D_{s}}\biggl{(}2\ln\biggl{(}\frac{2}{1-\cos\varepsilon}\biggr{)}-(1+\cos\varepsilon)\biggr{)}.$
(37)

Here, the first term is the MFPT to the inner sphere, which is evidently
independent of the target size. In turn, the second term is the contribution
from the surface diffusion towards the target, averaged over the distribution
of the first arrival point onto the surface (the so-called harmonic measure
density). This contribution is independent of the radius $R_{2}$ of the outer
sphere, as well as of the radial coordinate $r$ of the starting point.
Expectedly, the second term vanishes as $\varepsilon\to\pi$ (the target covers
the whole inner sphere) and diverges logarithmically as the target shrinks,
$\varepsilon\to 0$. Note that if $r\gg R_{1}$, the diffusing ligand has
sufficient time before hitting the inner sphere to loose the memory on its
starting point, so that $T^{\rm AD}(\mathbf{x})\approx\overline{T^{\rm AD}}$.
In other words, the explicit relation (37) can be used to estimate the MFPT
from a fixed starting point $\mathbf{x}$ when it is located far from the inner
sphere. Setting $r=R_{2}$, $R_{1}\ll R_{2}$ and $\varepsilon\ll 1$, one can
easily check that $\overline{T^{\rm AD}}$ coincides, to the leading order,
with $\tau_{\rm AD}$, which was discussed in section 1.

###  3.3 Comparison of the two search scenarios

We now compare the efficiency of the two search processes: ADS versus the
direct search scenario. We focus on the case when the starting point is
uniformly distributed over the outer sphere, for three reasons. (i) As
discussed earlier, we keep in mind applications to microbiology, in which a
particle enters the cytosol from the plasma membrane and searches for a
nuclear pore; here, the spatial (angular) locations of the entrance channel
and the nuclear pore are generally not known and can thus be modelled as
random. (ii) When the inner sphere is small in comparison to the outer sphere
(i.e, $R_{1}\ll R_{2}=r$), the particle has enough time to diffuse before
hitting the inner sphere, and the information on the precise location of a
fixed starting point is generally lost (except for very short trajectories
determining the left, short-$t$ tail of the PDF)—in other words, in this
setting, there is no notable difference between fixed and random starting
points. (iii) The average over angular coordinates of the starting point
eliminates all terms in the series representations (27) of the PDF, except for
$n=0$, that facilitates its numerical computation and reduces eventual
truncation errors (when an infinite series is replaced by a finite sum)—this
is particularly relevant in case of a small target when one would have to keep
a large number of terms to get accurate results, whereas the computation of
the radial functions $g_{n}^{(p)}(r)$ may be problematic for large $n$. In C,
we briefly discuss the case of a fixed starting point—which is conceptually
important—and compare it to our main conclusions here for a random starting
point.

The precise form of the Laplace transform $\overline{\tilde{H}^{\rm AD}(p;r)}$
is given in equation (33). While the exact solution $\overline{\tilde{H}^{\rm
dir}(p;r)}$ for the direct search scenario was derived in [50], we use the
approximate relations (113) and (115) for our discussion (see details in D).
Note that the latter have a much simpler and explicit form, as compared to the
exact solution (which requires a numerical inversion of matrices), and are
remarkably accurate, as shown in [50]. In both cases the PDFs in the time
domain are obtained via numerical inversion of the corresponding Laplace
transforms using the Talbot algorithm. We set the angular size of the target
equal to $\varepsilon=0.01$, which is approximately ten times bigger than the
angular size of a typical nuclear pore. This choice of a larger value of
$\varepsilon$ is due to some numerical limitations. In fact, going to very
small values of $\varepsilon$ necessitates taking into account too many terms
in equation (113) for the direct search, whose numerical accuracy cannot be
properly controlled by our algorithm. Nevertheless, the considered value
$\varepsilon=0.01$ allows us to illustrate the main features of the FPT PDF
and compare the two search scenarios. We fix length and time units by setting
$R_{1}=1$ and $D_{b}=1$ and investigate the effect of other parameters onto
the PDFs and the survival probabilities.

Figure 5: PDF (a,c) and associated survival probability (b,d) determining the
FPT to a target of small angular size $\varepsilon=0.01$, with $R_{2}/R_{1}=5$
(a,b) (KR=0.008) or $R_{2}/R_{1}=3$ (c,d) (the KR= 0.037), and with a starting
point uniformly distributed over the outer sphere ($r/R_{1}=5$). Comparison
between the direct search (open circles, D) and the ADS (coloured curves
corresponding to different values of the ratio $D_{s}/D_{b}$, see the legend).
Vertical lines indicate the MFPTs given by equation (37) for the ADS and by
equation (116) for the direct search. The numerical results for the direct
search were obtained by using the approximate relation (131) with $n_{\rm
inter}=50$ and $n_{\rm max}=250$ and by truncating the series (117) at $n_{\rm
max}=2000$. Note that $\tau^{\rm surf}\approx 5.2/D_{s}$, whereas $\tau^{\rm
bulk}\approx 29.3$ for $R_{2}/R_{1}=5$ and $\tau^{\rm bulk}\approx 4.7$ for
$R_{2}/R_{1}=3$, i.e., $\tau_{\rm surf}$ is dominant for all cases except for
$D_{s}/D_{b}=1$. Note that length and time units are fixed by setting
$R_{1}=1$ and $D_{b}=1$.

Figure 5 illustrates our main results. Panels (a,b) correspond to the
geometric setting with a small karyoplasmic ratio $\mathrm{KR}=0.008$ (with
$R_{2}/R_{1}=5$), while panels (c,d) refer to the larger value
$\mathrm{KR}=0.037$ (with $R_{2}/R_{1}=3$). Panels (a) and (c) present the FPT
PDFs for both scenarios, with different values of the ratio $D_{s}/D_{b}$ (see
the legends) and fixed angular size $\varepsilon=0.01$ of the target, offering
insight into the functional form of the full PDFs, as well as of the locations
of the most probable and mean FPTs (see the vertical dashed lines). For the
ADS, the PDF is broadening with decreasing surface diffusion coefficient
$D_{s}$. In fact, according to equation (35), the left short-time tail of the
PDF is mainly controlled by bulk diffusion and thus does not change much with
$D_{s}$. In turn, the right long-time tail is directly affected by $D_{s}$ via
the time $\tau^{\rm surf}$ given by equation (20) which becomes dominant in
equation (32) when $D_{s}$ is small enough. For instance, when
$D_{s}/D_{b}=10^{-3}$ (which is appropriate for diffusion on hard solid
surfaces), the PDF shown in panel (a) spans over five orders of magnitude in
time. In turn, for larger $D_{s}$, the PDF of the ADS is more compact and
attains substantially larger values in the vicinity of the most probable FPT
than its counterpart for the direct search scenario. Decreasing $R_{2}/R_{1}$
from $5$ to $3$ (panels (c,d)) which results in the increase of the KR from
$\mathrm{KR}\approx 0.008$ to $\mathrm{KR}\approx 0.037$, shifts the left tail
of the PDF to the left (towards shorter times), because the target is now
closer to the starting point. In turn, if $D_{s}$ is sufficiently small, the
right tail of the distribution is still controlled by $\tau^{\rm surf}$, which
is independent of $R_{2}$. As a consequence, the PDF is getting even broader.

While panels (a) and (c) present a basic conceptual understanding of the
structure of the PDFs, panels (b) and (d) display the survival probability,
which indeed proves to be a very robust measure of the relative efficiency of
both search scenarios. We observe that for a small ratio $\mathrm{KR}=0.008$
and all considered values of $D_{s}/D_{b}$, the survival probabilities for the
ADS, at any fixed $t$, are smaller than the survival probability of the target
within the direct search scenario. In this situation, the ADS can be qualified
as a more efficient search strategy as compared to the direct search. This
conclusion also agrees with the fact that the MFPT for the direct search is
longer than the MFPT within the ADS. Upon increase of KR (see the panel (d)),
we notice that the ADS still outperforms the direct search scenario for
$D_{s}/D_{b}\gtrsim 0.01$, and it becomes less efficient at
$D_{s}/D_{b}=0.001$. Recall, however, that these curves are calculated for
$\varepsilon=0.01$, which is somewhat higher than the typically encountered
values of $\varepsilon$. On intuitive grounds, we may thus expect that for a
smaller target with $\varepsilon=0.001$, as it is realised in the case of a
nuclear pore, the ADS will perform better even in this case. Therefore, we
demonstrate that the ADS can indeed be more efficient search scenario for
quite reasonable values of the system parameters. We finally note that the
survival probability at the MFPT appears to be quite universally equal to
$0.35$, which signifies that two-thirds of searching trajectories find the
target before this time, and only the remaining one-third arrive to the target
location at longer times.

We finish this section by the analysis of the effect of the target size on the
shapes of the PDFs and the survival probabilities. Figure 6 presents these
functions in the same setting, except that now the target is tenfold larger,
$\varepsilon=0.1$. As we already remarked, such a large value of $\varepsilon$
can be indicative of the behaviour in situations in which approximately $100$
nuclear pores are present on the surface of the nucleus. Remarkably, the PDFs
within the ADS remain nearly the same as for a smaller target without any
visible change, as compared to figure 5. At first thought, this is a counter-
intuitive behaviour. However, we recall that the right tail of the PDF is
mainly determined by $\tau^{\rm surf}$, which changes logarithmically slowly
at small $\varepsilon$. In fact, as discussed in [80], $\nu_{0}\approx
1/(2\ln(2/\varepsilon))$ as $\varepsilon\to 0$ so that $\tau^{\rm surf}\approx
2R_{1}^{2}\ln(2/\varepsilon)/D_{s}$. Moreover, as we saw in section 3.2, the
left tail of the distribution is also weakly dependent on the target size. For
instance, the PDFs for even smaller target size $\varepsilon=0.001$, which is
representative of the nuclear pore, are expected to be similar to those shown
in figure 5. In contrast, the FPT PDF for the direct search scenario depends
much more strongly on the target size. In fact, its right tail is
characterised by the decay time, which is of the order of the MFPT and thus
scales as $1/\varepsilon$. The amplitude of the left tail is also affected by
$\varepsilon$. We thus observe that the efficiency of the ADS is weakly
dependent on the target size and mainly controlled by two ratios $R_{1}/R_{2}$
and $D_{s}/D_{b}$. Overall, we conclude that in the case of larger targets the
ADS outperforms the direct search scenario for $D_{s}/D_{b}\gtrsim 0.01$ (when
$R_{2}/R_{1}=5$) and $D_{s}/D_{b}\gtrsim 0.1$ (when $R_{2}/R_{1}=3$), and is
less efficient otherwise.

Figure 6: Same as in figure 5 but for a larger target with $\varepsilon=0.1$.

##  4 Amplified signals

We now extend our analysis to the important case of so-called "amplified"
signals, when $N>1$ independently diffusing ligands search for a single
target. We note that the ensuing speed-up of the search process has been
intensively studied in rather diverse geometrical settings (different from our
setting here) for the direct search scenario [84, 85, 86, 87, 88, 89, 90, 91,
92]. We start by introducing auxiliary notations and formulating some basic
general results.

Let $\mathcal{T}_{i}$ (with $i=1,2,\ldots,N$) denote the time instant when the
$i$th ligand arrives to the target for the first time. Consequently, the
target is considered to be "found" when the fastest of the $N$ particles
arrives to the target location, i.e.,
$\mathcal{T}=\min\\{\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{N}\\}$.
Moreover, if all ligands start from the same fixed point $\mathbf{x}$, the
survival probability that determines the probability law of the fastest FPT
(fFPT) is given by

$S^{\rm
AD}_{N}(t;\mathbf{x})=\mathbb{P}\\{\mathcal{T}>t\\}=\mathbb{P}\\{\mathcal{T}_{1}>t,\mathcal{T}_{2}>t,\ldots,\mathcal{T}_{N}>t\\}=[S^{\rm
AD}(t;\mathbf{x})]^{N},$ (38)

where we took advantage of a physically plausible assumption that all ligands
move independently of each other. If, in contrast, each ligand starts from a
random position, which is uniformly distributed on a sphere of radius $r$,
independently from the positions of the other ligands, then we have

$\overline{S^{\rm
AD}_{N}(t;r)}=\mathbb{P}\\{\mathcal{T}>t\\}=\mathbb{P}\\{\mathcal{T}_{1}>t,\mathcal{T}_{2}>t,\ldots,\mathcal{T}_{N}>t\\}=\left[\overline{S^{\rm
AD}(t;r)}\right]^{N}.$ (39)

In both cases, the associated PDFs are given by the time derivative

$\begin{split}H^{\rm AD}_{N}(t;\mathbf{x})=N[S^{\rm
AD}(t;\mathbf{x})]^{N-1}\,H^{\rm AD}(t;\mathbf{x}),\\\ \overline{H^{\rm
AD}_{N}(t;r)}=N\left[\overline{S^{\rm AD}(t;r)}\right]^{N-1}\,\overline{H^{\rm
AD}(t;r)}.\end{split}$ (40)

These two expressions allow for a detailed comparison of both scenarios in the
amplified signal case, as presented in figure 7. Expressions (40) signify
that, once the survival probability for a single ligand is found in the time
domain—as evaluated in the previous sections—one has ready access to the
statistics of the fFPT $\mathcal{T}$. Note, however, that the large-$N$
asymptotic analysis of the moments of $\mathcal{T}$ is more cumbersome (see,
e.g., [84, 89] in the direct search case).

Panels (a) and (c) of figure 7 depict the PDF $\overline{H^{\rm AD}_{N}(t;r)}$
for the case when the starting points of the ligands are uniformly distributed
over the outer sphere. The geometric setting is the same as for figure 5 with
$R_{2}/R_{1}=5$. We see that the qualitative comparison between the ADS and
the direct search does not change much for $N=10$ and $N=100$; in particular,
the PDF for the direct search is relatively close to that for the ADS with
$D_{s}/D_{b}=10^{-3}$. As expected, all PDFs are narrowing with increasing
$N$. This is expected from the asymptotic behaviour for a single ligand. In
fact, the long-time relation (32) implies that

$\overline{H^{\rm AD}_{N}(t;r)}\simeq
N[C_{\varepsilon}(r)]^{N}\exp\bigl{(}-Nt/\max\\{\tau^{\rm bulk},\tau^{\rm
surf}\\}\bigr{)}\qquad(t\to\infty),$ (41)

i.e., the decay time is decreased by the factor $N$, and thus the right tail
is shifted towards the left (to shorter times). In contrast, as the survival
probability $\overline{S^{\rm AD}_{N}(t;r)}$ is close to unity at short times,
the left tail is just multiplied by $N$ and does not shift, i.e.,

$\overline{H^{\rm AD}_{N}(t;r)}\simeq N\,\overline{H^{\rm AD}(t;r)}\qquad(t\to
0).$ (42)

This relation is valid for moderate values of $N$. In turn, when $N$ is very
large, we expect the emergence of an intermediate range of times for which the
factor $\left[\overline{S^{\rm AD}(t;r)}\right]^{N-1}$ cannot be replaced by
unity and starts to affect the distribution of the fFPT.

In panels (b) and (d) we depict the corresponding survival probabilities. We
observe that they exhibit essentially the same behaviour as those in the case
$N=1$, except for the fact that an abrupt decay to zero starts at
progressively earlier times. We also infer from figure 7 that, maybe somewhat
counter-intuitively, the relative efficiency of both search scenarios is the
same as in the case of a single particle. Thus, the ADS appears to be more
efficient for all considered values of $D_{s}/D_{b}$. We recall, however, that
the relative efficiency also depends on the ratio $R_{2}/R_{1}$, see section
3.3. Similar conclusions hold for $N$ particles started from a fixed point
located far from the target (not shown).

Figure 7:  PDF (a,c) and associated survival probability (b,d) determining the
fastest FPT to a target of small angular size $\varepsilon=0.01$, with $N=10$
(a,b) and $N=100$ (c,d) particles, $R_{2}/R_{1}=5$ (KR=0.008), and starting
positions independently and uniformly distributed over the outer sphere
($r/R_{1}=5$). Comparison between the direct search (open circles, D) and the
ADS (coloured curves corresponding to different values of the ratio
$D_{s}/D_{b}$, see the legend). The numerical results for the direct search
were obtained by using the approximate relation (131) with $n_{\rm inter}=50$
and $n_{\rm max}=250$.

##  5 Conclusions

For the simple geometrical setting of two nested, concentric and impenetrable
spheres with an immobile target site being placed at the North pole of the
inner sphere, we presented a detailed comparison of two search scenarios: (i)
the direct search mode, when the particle needs to locate the target solely by
bulk diffusion in the volume between the concentric spheres, with the surface
of the inner sphere being perfectly reflecting, except for the target (see
[50] for more details); and (ii) the Adam-Delbrück reduction-of-dimensionality
scenario [1], for which the particle first attaches non-specifically to the
reactive surface of the inner sphere and locates the target by diffusive
search on the 2D surface. For both scenarios we calculate the FPT PDF for the
searching ligand, that is initially released from a fixed point, or from a
random position on the surface of the outer sphere. We also considered the
case of "amplified" signals, when initially $N$ ligands are launched from the
same fixed or random positions, and the search is completed when the fastest
out of $N$ particles arrives to the target. Such settings indeed correspond to
various situations and problems encountered in cellular biology, biophysics,
and biochemistry. In particular, they appear as a crucial intermediate step in
many signal transduction pathways in cellular environments.

To show that the reduction-of-dimensionality scenario may be beneficial in
certain situations, Adam and Delbrück originally presented a comparison of the
efficiency of both direct and ADS diffusive searches for a single ligand,
judging solely from the mean times of diffusion towards the target site. In
the present work confronting both scenarios, we went beyond the analysis of
the MFPT—which by now is known to be non-representative in even simple
geometries and sometimes even plainly misleading—and therefore focused our
analysis on the PDF of the random individual FPTs from a fixed or a random
starting position to the target. We compared the behaviour of the full FPT PDF
as well as of their integrated characteristic, the survival probability, which
appeared to be a robust measure of the actual efficiency of both search
scenarios. In fact, if

$S_{1}(t)\leq S_{2}(t)\qquad\forall\leavevmode\nobreak\ t\geq 0$ (43)

for all times, the first search process (described by $S_{1}(t)$) is more
efficient than the second one. As the MFPT is the integral of the survival
probability, this inequality implies the conventional efficiency criterion,

$T_{1}\leq T_{2},$ (44)

in terms of the MFPTs $T_{1}$ and $T_{2}$ of two search processes. In this
situation, both criteria are equivalent. However, the opposite claim is not
true, namely, (44) does not imply (43). For instance, one may have
$S_{1}(t)\leq S_{2}(t)$ at short times and $S_{1}(t)\geq S_{2}(t)$ at long
times, even though $T_{1}=T_{2}$. In other words, one search process can be
more efficient at short times and less efficient at long times. We conclude
that the comparison of survival probabilities provides a more systematic and
comprehensive insight into the search efficiency. On this basis, we specified
realistic physical conditions in which the reduction-of-dimensionality
scenario outperforms the direct search for both cases when a single ligand is
present and when the signal is amplified.

Our analysis can be generalised in several directions. First, we considered
exclusively the case of standard Brownian motion. In reality, both diffusion
in the bulk and the surface diffusion may be (transiently) anomalous. These
anomalous features can be incorporated in a standard way, e.g., via a
subordination technique. Second, we supposed that the spherical-shell domain
is a homogeneous liquid-filled region and does not contain any obstacles or
"crowders". In turn, especially in cellular environments, this is not the case
due to the presence of organelles, proteins and etc, which will certainly
affect the dynamics [53]. Moreover, in some cases the cellular cytoplasm is
actively moved by non-equilibrium action of molecular motors [93, 94, 95],
thus changing the mixing dynamics inside the cell and creating dynamics
heterogeneities [96, 97]. Consequently, these effects may alter the relative
efficiency of both search scenarios in the realistic cellular setting. Third,
both scenarios considered here pertain to two somewhat extreme situations: the
particle is either perfectly reflected by the surface of the inner sphere or,
in the ADS, non-specifically adsorbs to it upon first encounter. In reality,
the situation can be more complex. As discussed recently in [70] within the
context of a narrow-escape problem, there are distance-dependent potential
interactions with the surface on which the target is located and the presence
of such interactions modifies the search processes. As realised in [70], the
most efficient interactions, i.e., resulting in the shortest FPT to the
target, are neither too long-ranged not too short-ranged (as corresponds to
the Adam-Delbrück picture) such that the optimal trajectories are
intermittent—upon arrival to the inner sphere a ligand performs a finite
surface diffusion tour and then desorbs to the bulk, arrives to the surface
again, performing a new tour of surface diffusion, and so on, until the target
is finally found. Even though such intermittent search strategies have been
studied in the past (see, e.g., [32, 33, 34, 35, 36, 37, 38] and references
therein), former works were almost exclusively focused on the mean FPT and the
search optimality was qualified by its minimisation. In our future work, we
will explore these possible generalisations of our present analysis.

## Acknowledgments

The authors acknowledge helpful discussions with Baruch Meerson, Stas
Shvartzman and Leonid Mirny. DG acknowledges the Alexander von Humboldt
Foundation for support within a Bessel Prize award. RM acknowledges support
from the German Science Foundation (DFG Grant ME 1535/12-1) as well as from
the Foundation for Polish Science Fundacja na rzecz Nauki Polskiej, FNP)
through an Alexander von Humboldt Polish Honorary Research Scholarship.

##  Appendix A Moments of the conditioned first-passage time

The Laplace transform of the joint probability density
$j(t,\mathbf{s}|\mathbf{x})$ in (6) determines all positive integer moments of
the FPT $\mathcal{T}_{\mathbf{s}}$ to the inner sphere, conditioned by the
arrival point $\mathbf{s}$,

$\langle\mathcal{T}_{\mathbf{s}}^{k}\rangle=\int\limits_{0}^{\infty}dt\,t^{k}\,j(t,\mathbf{s}|\mathbf{x})=(-1)^{k}\lim\limits_{p\to
0}\frac{\partial^{k}}{\partial p^{k}}\tilde{j}(p,\mathbf{s}|\mathbf{x}).$ (45)

One sees that it is sufficient to compute the Taylor expansion in powers of
$p$ of the radial functions $g_{n}^{(p)}$ from (7). For instance, the first
moment is given by

$\langle\mathcal{T}_{\mathbf{s}}\rangle=-\frac{1}{4\pi
R_{1}^{2}}\sum\limits_{n=0}^{\infty}(2n+1)P_{n}\left(\frac{(\mathbf{s}\cdot\mathbf{x})}{|\mathbf{s}|\,|\mathbf{x}|}\right)\lim\limits_{p\to
0}\frac{\partial g_{n}^{(p)}(r)}{\partial p}.$ (46)

A lengthy but straightforward computation produces

$\displaystyle\lim\limits_{p\to 0}\frac{\partial g_{n}^{(p)}(r)}{\partial
p}=\frac{R_{2}^{2}}{4D_{b}}\,\frac{(n+1)(r/R_{2})^{n}+n(r/R_{2})^{-n-1}}{(n+1)(R_{1}/R_{2})^{n}+n(R_{1}/R_{2})^{-n-1}}\biggl{\\{}\frac{1}{(n+1)(r/R_{2})^{n}+n(r/R_{2})^{-n-1}}$
$\displaystyle\times\biggl{[}\frac{n+1}{n+3/2}(r/R_{2})^{n+2}-\frac{n-1}{n-1/2}(r/R_{2})^{n}+\frac{n+2}{n+3/2}(r/R_{2})^{-n-1}-\frac{n}{n-1/2}(r/R_{2})^{-n+1}\biggr{]}$
$\displaystyle-\frac{1}{(n+1)(R_{1}/R_{2})^{n}+n(R_{1}/R_{2})^{-n-1}}\biggl{[}\frac{n+1}{n+3/2}(R_{1}/R_{2})^{n+2}$
$\displaystyle-\frac{n-1}{n-1/2}(R_{1}/R_{2})^{n}+\frac{n+2}{n+3/2}(R_{1}/R_{2})^{-n-1}-\frac{n}{n-1/2}(R_{1}/R_{2})^{-n+1}\biggr{]}\biggr{\\}}.$
(47)

The average of (46) over $\mathbf{s}$ results in the well-known expression for
the MFPT to the inner sphere,

$\int\limits_{{\partial\Omega}}d\mathbf{s}\,\langle\mathcal{T}_{\mathbf{s}}\rangle=-\lim\limits_{p\to
0}\frac{\partial g_{0}^{(p)}(r)}{\partial
p}=\frac{(r-R_{1})(2R_{2}^{3}-rR_{1}(R_{1}+r))}{6D_{b}rR_{1}}.$ (48)

Similarly, if the starting point $\mathbf{x}$ is uniformly distributed on a
sphere of radius $r$, the integral over the angular coordinates in (46)
cancels all terms with $n>0$, yielding

$\displaystyle\frac{1}{4\pi
r^{2}}\int\limits_{|\mathbf{x}|=r}d\mathbf{x}\,\langle\mathcal{T}_{\mathbf{s}}\rangle$
$\displaystyle=-\frac{1}{4\pi R_{1}^{2}}\lim\limits_{p\to 0}\frac{\partial
g_{0}^{(p)}(r)}{\partial p}$ $\displaystyle=\frac{1}{4\pi
R_{1}^{2}}\,\frac{(r-R_{1})(2R_{2}^{3}-rR_{1}(R_{1}+r))}{6D_{b}rR_{1}},$ (49)

which does not depend on the arrival point $\mathbf{s}$.

Note that when the starting point is located on the outer sphere, $r=R_{2}$,
the first term in (47) is cancelled, and one gets the simpler expression

$\displaystyle\lim\limits_{p\to 0}\frac{\partial g_{n}(R_{2})}{\partial
p}=-\frac{R_{2}^{2}}{4D_{b}}\,\frac{(2n+1)(R_{1}/R_{2})^{n+1}}{\bigl{[}n+(n+1)(R_{1}/R_{2})^{2n+1}\bigr{]}^{2}}$
$\displaystyle\times\biggl{[}\frac{n+2}{n+3/2}-\frac{n}{n-1/2}(R_{1}/R_{2})^{2}-\frac{n-1}{n-1/2}(R_{1}/R_{2})^{2n+1}+\frac{n+1}{n+3/2}(R_{1}/R_{2})^{2n+3}\biggr{]}.$

##  Appendix B Surface diffusion on the sphere

The problem of surface diffusion on a sphere has been intensively studied in
the past (see [77, 78, 79, 80] and references therein). In particular, the
exact solution for the concentration profile in the presence of a perfect
target was given in [77, 78], whereas its extension to a partially reactive
target with reversible binding was provided in [80]. Here we extend the
derivation from [80] to get two equivalent representations of the FPT PDF for
diffusion on the sphere.

Following [80], we consider diffusion on a sphere of radius $R$ (which is set
to $R_{1}$ in the main text) towards a circular target of angular size
$\varepsilon$ located on the South pole. This location is specific to this
Appendix and differs from our consideration of the target on the North pole
throughout the remaining text. This choice is taken for the fact that the
solution will be given in terms of Legendre functions $P_{\nu}(x)$ with non-
integer index $\nu$, which are regular at $x=1$ (the North pole) and singular
at $x=-1$ (the South pole). As a consequence, it is natural to locate the
target on the South pole to eliminate such singularities. Alternatively, one
could search for the solution in terms of $P_{\nu}(1-x)$, which is equivalent
to exchanging the North and South poles (i.e., by replacing the angle $\theta$
by $\pi-\theta$). Either way, once the derivation is completed, we will use
such a swap to reformulate the final results for the target located on the
North pole, to be consistent with the remainder of the paper.

We search for the Laplace-transformed survival probability $\tilde{S}^{\rm
surf}(p;\mathbf{s})$ satisfying the boundary value problem

$\displaystyle(p-D_{s}\Delta_{\mathbf{s}})\tilde{S}^{\rm surf}(p;\mathbf{s})$
$\displaystyle=1\quad(\theta>\pi-\varepsilon),$ (50) $\displaystyle
D_{s}\partial_{n}\tilde{S}^{\rm surf}(p;\mathbf{s})+\kappa_{s}\tilde{S}^{\rm
surf}(p;\mathbf{s})$ $\displaystyle=0\quad(\theta=\pi-\varepsilon),$ (51)

where $\mathbf{s}=(\theta,\phi)$ is a point on the sphere, $\pi-\varepsilon$
is the angular coordinate of the target region (here, it is a circle around
the South pole), $\partial_{n}$ is the normal derivative oriented towards the
South pole, $\Delta_{\mathbf{s}}$ is the Laplace-Beltrami operator, and
$\kappa_{s}$ is the reactivity of the target. Even though the main text deals
with a perfect target ($\kappa_{s}=\infty$), we here consider the more general
case of a partially reactive target with a finite reactivity $\kappa_{s}$,
from which the perfect target limit will follow. The axial symmetry of the
problem implies that the survival probability depends only on the angle
$\theta$ so that $\Delta_{\mathbf{s}}=\tfrac{1}{R^{2}}\mathcal{L}$, where

$\mathcal{L}=\partial_{x}(1-x^{2})\partial_{x}\quad\mbox{ with }x=\cos\theta.$
(52)

In the following, we will therefore write functions in terms of $x$, e.g.,
$\tilde{S}^{\rm surf}(p;x)$.

###  B.1 Two representations for the Green’s function

We start by considering the Green’s function for diffusion on the sphere,
satisfying

$(s-\mathcal{L})\tilde{G}(x,y|s)=\delta(x-y),\quad\bigl{(}\partial_{x}\tilde{G}(x,y|s)\bigr{)}|_{x=a}=\frac{qR}{\sqrt{1-a^{2}}}\tilde{G}(a,y|s),$
(53)

where $s=pR^{2}/D_{s}$, $q=\kappa_{s}/D_{s}$, and
$a=\cos(\pi-\varepsilon)=-\cos(\varepsilon)$ is the location of the target.
The second relation is the Robin boundary condition on the partially reactive
target. While $\tilde{G}(x,y|s)$ should be well known, we provide here the
main steps of its derivation for the reader’s convenience.

On one hand, the Green’s function admits the spectral expansion over the
Legendre functions $P_{\nu}(x)$ of the first kind which are the eigenfunctions
of the operator $\mathcal{L}$, $\mathcal{L}P_{\nu}(x)=-\nu(\nu+1)P_{\nu}(x)$.
One thus gets (see [80] for details)

$\tilde{G}(x,y|s)=\sum\limits_{n=0}^{\infty}\frac{b_{n}^{2}\,P_{\nu_{n}}(x)\,P_{\nu_{n}}(y)}{s+\nu_{n}(\nu_{n}+1)}\quad(a\leq
x\leq 1,a\leq y\leq 1),$ (54)

where

$b_{n}=\left(\int\limits_{a}^{1}dx\,[P_{\nu_{n}}(x)]^{2}\right)^{-1/2}$ (55)

are the normalisation constants, and $\nu_{n}$ are the solutions of the
equation

$P^{\prime}_{\nu}(a)=\frac{qR}{\sqrt{1-a^{2}}}\,P_{\nu}(a)$ (56)

that ensures the Robin boundary condition, and
$P^{\prime}_{\nu}(a)=\bigl{(}\tfrac{\partial P_{\nu}(x)}{\partial
x}\bigr{)}_{|x=a}$. We emphasise that the index $\nu$ of the Legendre function
$P_{\nu}(x)$ is the unknown to be determined here. In the case of a perfect
target ($q=\infty$), this relation becomes

$P_{\nu}(a)=0.$ (57)

One can easily check that the spectral expansion in (54) is the solution of
(53). Indeed, the application of the operator $(s-\mathcal{L})$ to
$\tilde{G}(x,y|s)$ yields

$\sum\nolimits_{n=0}^{\infty}b_{n}^{2}\,P_{\nu_{n}}(x)\,P_{\nu_{n}}(y)=\delta(x-y)$
(58)

due to the completeness relation for the eigenfunctions $P_{\nu_{n}}(x)$.

On the other hand, the Green’s function can be found as a combination of two
linearly independent solutions of the homogeneous equation
$(s-\mathcal{L})u=0$ which are given as $P_{\mu}(x)$ and $Q_{\mu}(x)$, where
$Q_{\nu}(x)$ is the Legendre function of the second kind, and

$\mu=\frac{-1+\sqrt{1-4s}}{2}$ (59)

is the solution of the equation $s=-\nu(\nu+1)$ which satisfies $\mu\simeq-s$
in the limit $s\to 0$. One searches a solution of the form

$\tilde{G}(x,y|s)=\left\\{\begin{array}[]{ll}AP_{\mu}(x)&(y<x<1),\\\
B\bigl{(}\hat{Q}P_{\mu}(x)-\hat{P}Q_{\mu}(x)\bigr{)}&(a<x<y),\end{array}\right.$
(60)

with the shortcut notations $\hat{P}=P_{\mu}(a)-hP^{\prime}_{\mu}(a)$ and
$\hat{Q}=Q_{\mu}(a)-hQ^{\prime}_{\mu}(a)$, where $h=\sqrt{1-a^{2}}/(qR)$. The
first relation ensures the regularity of the solution at $x=1$, while the
second relation takes care of the boundary condition (56) at $x=a$. The
coefficients $A$ and $B$ are determined by requiring the continuity of
$\tilde{G}(x,y|s)$ at $x=y$ and the drop by $1/(1-y^{2})$ of its derivative at
$x=y$. One then finds

$\displaystyle A$
$\displaystyle=-\frac{\hat{Q}P_{\mu}(y)-\hat{P}Q_{\mu}(y)}{\hat{P}},$ (61)
$\displaystyle B$ $\displaystyle=-\frac{P_{\mu}(y)}{\hat{P}},$ (62)

where the Wronskian

$P_{\nu}(y)Q^{\prime}_{\nu}(y)-P^{\prime}_{\nu}(y)Q_{\nu}(y)=\frac{1}{1-y^{2}}$
(63)

was used. One finally finds

$\tilde{G}(x,y|s)=\left\\{\begin{array}[]{ll}P_{\mu}(x)\bigl{(}Q_{\mu}(y)-\frac{\hat{Q}}{\hat{P}}P_{\mu}(y)\bigr{)}&(y<x<1),\\\
P_{\mu}(y)\bigl{(}Q_{\mu}(x)-\frac{\hat{Q}}{\hat{P}}P_{\mu}(x)\bigr{)}&(a<x<y).\end{array}\right.$
(64)

Equating the right-hand sides of equations (54) and (64), one gets an identity
which allows one to evaluate various series involving the zeros $\nu_{n}$ (see
the review [98] for other examples and applications).

###  B.2 FPT distribution

In order to get the Laplace-transformed survival probability, it is sufficient
to integrate the Green’s function $\tilde{G}(x,y|s)$ over $y$ from $a$ to $1$.
For instance, the integral of (54) yields

$\tilde{S}^{\rm
surf}(p;x)=\frac{R^{2}}{D_{s}}\int\limits_{a}^{1}dy\,\tilde{G}(x,y|s)=\frac{R^{2}}{D_{s}}\sum\limits_{n=0}^{\infty}\frac{b_{n}^{2}\,P_{\nu_{n}}(x)(1-a^{2})P^{\prime}_{\nu_{n}}(a)}{\nu_{n}(\nu_{n}+1)(\frac{pR^{2}}{D_{s}}+\nu_{n}(\nu_{n}+1))},$
(65)

where we used the identity

$\displaystyle\int\limits_{a}^{b}dx\,P_{\nu}(x)$
$\displaystyle=\frac{P_{\nu-1}(a)-P_{\nu+1}(a)-P_{\nu-1}(b)+P_{\nu+1}(b)}{2\nu+1}$
$\displaystyle=\frac{1-a^{2}}{\nu(\nu+1)}P^{\prime}_{\nu}(a)-\frac{1-b^{2}}{\nu(\nu+1)}P^{\prime}_{\nu}(b),$
(66)

and a similar relation holds for $Q_{\nu}(x)$. Expression (65) was considered,
e.g., in [80]. It can also be written as

$\displaystyle\tilde{S}^{\rm surf}(p;x)$
$\displaystyle=\frac{1}{p}\sum\limits_{n=0}^{\infty}b_{n}^{2}\,P_{\nu_{n}}(x)(1-a^{2})P^{\prime}_{\nu_{n}}(a)\biggl{(}\frac{1}{\nu_{n}(\nu_{n}+1)}-\frac{1}{\frac{pR^{2}}{D}+\nu_{n}(\nu_{n}+1)}\biggr{)}$
$\displaystyle=\frac{1}{p}\biggl{(}1-(1-a^{2})\sum\limits_{n=0}^{\infty}\frac{b_{n}^{2}\,P_{\nu_{n}}(x)\,P^{\prime}_{\nu_{n}}(a)}{\frac{pR^{2}}{D}+\nu_{n}(\nu_{n}+1)}\biggr{)},$
(67)

where we evaluated explicitly the first sum by integrating (58) and using
again (66). As a consequence, the Laplace-transformed PDF then becomes

$\tilde{H}^{\rm surf}(p;x)=1-p\tilde{S}^{\rm
surf}(p;x)=(1-a^{2})\sum\limits_{n=0}^{\infty}\frac{b_{n}^{2}P_{\nu_{n}}(x)P^{\prime}_{\nu_{n}}(a)}{\frac{R^{2}p}{D_{s}}+\nu_{n}(\nu_{n}+1)}.$
(68)

The inverse Laplace transform can be easily found to be

$H^{\rm
surf}(t;x)=\frac{(1-a^{2})D_{s}}{R^{2}}\sum\limits_{n=0}^{\infty}b_{n}^{2}P_{\nu_{n}}(x)P^{\prime}_{\nu_{n}}(a)\,e^{-\nu_{n}(\nu_{n}+1)D_{s}t/R^{2}}.$
(69)

This is a rare example of an exact explicit representation of the FPT PDF,
except for a numerical step required to find the roots $\nu_{n}$. The MFPT can
be formally deduced from the above equations, but it admits the simple closed-
formed expression [78]

$T^{\rm
surf}(x)=\frac{R^{2}}{D}\ln\left(\frac{1+x}{1+a}\right)+\frac{R}{\kappa_{s}}\sqrt{\frac{1-a}{1+a}}.$
(70)

While the series representation (68) of $\tilde{H}^{\rm surf}(p;x)$ was
convenient for getting $H^{\rm surf}(t;x)$, another representation is needed
for the analysis of the asymptotic behaviour. For this purpose, one can
calculate $\tilde{S}^{\rm surf}(p;x)$ by integrating $\tilde{G}(x,y|s)$ from
(64). After simplifications, we get

$\tilde{S}^{\rm
surf}(p;x)=\frac{1}{p}\biggl{(}1-\frac{P_{\mu}(x)}{P_{\mu}(a)-\frac{\sqrt{1-a^{2}}}{qR}P^{\prime}_{\mu}(a)}\biggr{)}$
(71)

and thus

$\tilde{H}^{\rm
surf}(p;x)=\frac{P_{\mu}(x)}{P_{\mu}(a)-\frac{\sqrt{1-a^{2}}}{qR}P^{\prime}_{\mu}(a)}.$
(72)

For a perfect target ($q=\infty$), one simply gets

$\tilde{H}^{\rm surf}(p;x)=\frac{P_{\mu}(x)}{P_{\mu}(a)}.$ (73)

Note that these solutions could be obtained directly by solving the related
boundary value problems with the operator $\mathcal{L}$.

###  B.3 Short-time asymptotic behaviour

The above representation for $\tilde{H}^{\rm surf}(p;x)$ is suitable for
getting the short-time asymptotic behaviour of the PDF. This corresponds to
the large-$p$ limit, for which one can apply the asymptotic behaviour of the
Legendre functions. Indeed, as $p\to\infty$,
$\mu\simeq-\tfrac{1}{2}+i\sqrt{s}$ and we use

$P_{\mu}(\cos\theta)\simeq\sqrt{\frac{\theta}{\sin\theta}}J_{0}((\,u+1/2)\theta)\approx\sqrt{\frac{\theta}{\sin\theta}}J_{0}(i\sqrt{s}\theta)=\sqrt{\frac{\theta}{\sin\theta}}I_{0}(\sqrt{s}\theta)\simeq\frac{e^{\sqrt{s}\,\theta}\,s^{-1/4}}{\sqrt{2\pi\sin\theta}},$

from which equation (73) yields for a perfect target,

$\tilde{H}^{\rm
surf}(p;\theta)\simeq\sqrt{\frac{\sin\varepsilon}{\sin\theta}}\,e^{-(\theta-\varepsilon)R\sqrt{p/D_{s}}}\quad(p\to\infty).$
(74)

The leading exponential term here is expected, given that
$(\theta-\varepsilon)R$ is the geodesic distance between the starting point
and the target. As a consequence, we get

$H^{\rm
surf}(t;\theta)\simeq\sqrt{\frac{\sin\varepsilon}{\sin\theta}}\,\frac{R(\theta-\varepsilon)e^{-R^{2}(\theta-\varepsilon)^{2}/(4D_{s}t)}}{\sqrt{4D_{s}t^{3}}}\quad(t\to
0).$ (75)

###  B.4 Green’s function in absence of the target

As mentioned in the main text, the self-consistent approximation (121) can
also be expressed in terms of Legendre functions. For this purpose, we need
some identities that can be derived from the Green’s function without the
target. In the limit $\varepsilon\to 0$ (or $a\to-1$), the Green’s function
approaches

$\tilde{G}(x,y|s)=\left\\{\begin{array}[]{ll}P_{\mu}(x)\bigl{(}Q_{\mu}(y)-P_{\mu}(y)C_{\mu}\bigr{)}&(y<x<1),\\\
P_{\mu}(y)\bigl{(}Q_{\mu}(x)-P_{\mu}(x)C_{\mu}\bigr{)}&(-1<x<y),\end{array}\right.$
(76)

with

$C_{\mu}=\lim\limits_{a\to-1}\frac{Q_{\mu}(a)}{P_{\mu}(a)}=\frac{\pi}{2\,\tan(\pi\mu)},$
(77)

where we used the asymptotic behaviour of Legendre functions (see [99], p.
164). At the same time, this Green’s function admits the spectral expansion
over Legendre polynomials $P_{n}(x)$,

$\tilde{G}(x,y|s)=\sum\limits_{n=0}^{\infty}\frac{(n+1/2)\,P_{n}(x)\,P_{n}(y)}{s+n(n+1)}.$
(78)

The integral of $\tilde{G}(x,y|s)$ over $y$ from $a$ to $1$, for $a<x<1$
yields the identity

$\frac{1}{2}\sum\limits_{n=0}^{\infty}\frac{P_{n}(x)\,(P_{n-1}(a)-P_{n+1}(a))}{s+n(n+1)}=\frac{1}{s}\biggl{(}1-(1-a^{2})P_{\mu}(x)\bigl{(}Q^{\prime}_{\mu}(a)-C_{\mu}P^{\prime}_{\mu}(a)\bigr{)}\biggr{)},$
(79)

where we used the identity (66). We emphasise that the condition $a<x<1$
should be satisfied, otherwise the identity does not hold. In the opposite
case, one needs to integrate $\tilde{G}(x,y|s)$ over $y$ from $-1$ to $a$, to
get for $-1<x<a$ that

$\frac{1}{2}\sum\limits_{n=0}^{\infty}\frac{P_{n}(x)\,(P_{n-1}(a)-P_{n+1}(a))}{s+n(n+1)}=-\frac{(1-a^{2})P^{\prime}_{\mu}(a)\bigl{(}Q_{\mu}(x)-C_{\mu}P_{\mu}(x)\bigr{)}}{s}.$
(80)

The integral of (79) over $x$ from $a$ to $1$ yields another identity,

$\frac{1}{2}\sum\limits_{n=0}^{\infty}\frac{(P_{n-1}(a)-P_{n+1}(a))^{2}}{(2n+1)(s+n(n+1))}=\frac{(1-a^{2})^{2}P^{\prime}_{\mu}(a)\bigl{(}Q^{\prime}_{\mu}(a)-C_{\mu}P^{\prime}_{\mu}(a)\bigr{)}}{s^{2}}+\frac{1-a}{s}.$
(81)

Note that the derivatives $P^{\prime}_{\nu}(x)$ and $Q^{\prime}_{\nu}(x)$ can
be rewritten using the recurrence relations,

$\displaystyle(2\nu+1)(1-x^{2})P^{\prime}_{\nu}(x)$
$\displaystyle=\nu(\nu+1)(P_{\nu-1}(x)-P_{\nu+1}(x)),$ (82)
$\displaystyle(2\nu+1)(1-x^{2})Q^{\prime}_{\nu}(x)$
$\displaystyle=\nu(\nu+1)(Q_{\nu-1}(x)-Q_{\nu+1}(x)).$ (83)

Finally, integrating the representation of the Dirac distribution,

$\frac{1}{2}\sum\limits_{n=0}^{\infty}(2n+1)P_{n}(x)P_{n}(y)=\delta(x-y)$ (84)

over $y$ from $a$ to $1$, yields

$\frac{1}{2}\sum\limits_{n=0}^{\infty}P_{n}(x)(P_{n-1}(a)-P_{n+1}(a))=\Theta(x-a),$
(85)

where $\Theta(x-a)$ is the Heaviside step function.

##  Appendix C Solution for the Adam-Delbrück’s scenario

We here start from the convolution-like relation (2.2) and show how the
integral over the arrival point on the inner sphere can be evaluated. We
recall that the Laplace transforms $\tilde{j}(p,\mathbf{s}|\mathbf{x})$ and
$\tilde{H}^{\rm surf}(p;\mathbf{s})$ were given by the explicit relations (6)
and (10). To proceed, we use the addition theorem for (normalised) spherical
harmonics $Y_{mn}(\theta,\phi)$,

$\frac{2n+1}{4\pi}P_{n}\left(\frac{(\mathbf{s}\cdot\mathbf{x})}{|\mathbf{s}|\,|\mathbf{x}|}\right)=\sum\limits_{m=-n}^{n}Y_{mn}^{*}(\theta,\phi)Y_{mn}(\theta^{\prime},\phi^{\prime}),$
(86)

where $\mathbf{x}=(r,\theta,\phi)$ and
$\mathbf{s}=(R_{1},\theta^{\prime},\phi^{\prime})$. We substitute this
expression into equation (6) and then into equation (2.2). As $\tilde{H}^{\rm
surf}(p;\theta^{\prime})$ does not depend on $\phi^{\prime}$ due to the axial
symmetry, the integral over $\phi^{\prime}$ cancels all contributions in the
sum over $m$, except for $m=0$:

$\displaystyle\tilde{H}^{\rm AD}(p;\mathbf{x})$
$\displaystyle=\int\limits_{0}^{\pi}d\theta^{\prime}\,\sin\theta^{\prime}\int\limits_{0}^{2\pi}d\phi^{\prime}\,\sum\limits_{n=0}^{\infty}g_{n}^{(p)}(r)\sum\limits_{m=-n}^{n}Y_{mn}^{*}(\theta,\phi)Y_{mn}(\theta^{\prime},\phi^{\prime})\,\tilde{H}^{\rm
surf}(p;\theta^{\prime})$
$\displaystyle=\sum\limits_{n=0}^{\infty}g_{n}^{(p)}(r)(n+1/2)P_{n}(\cos\theta)\,c_{n}^{(p)}(\varepsilon),$

where

$\displaystyle c_{n}^{(p)}(\varepsilon)$
$\displaystyle=\int\limits_{0}^{\pi}d\theta^{\prime}\,\sin\theta^{\prime}\,P_{n}(\cos\theta^{\prime})\,\tilde{H}^{\rm
surf}(p;\theta^{\prime})$
$\displaystyle=\int\limits_{0}^{\varepsilon}d\theta^{\prime}\,\sin\theta^{\prime}P_{n}(\cos\theta^{\prime})+\frac{1}{P_{\mu}(a)}\int\limits_{\varepsilon}^{\pi}d\theta^{\prime}\,\sin\theta^{\prime}\,P_{n}(\cos\theta^{\prime})P_{\mu}(\cos(\pi-\theta^{\prime}))$
$\displaystyle=\frac{P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)}{2n+1}+\frac{(-1)^{n}}{P_{\mu}(a)}I_{n}^{(p)}(-\cos\varepsilon),$

with

$I_{n}^{(p)}(a)=\int\limits_{a}^{1}dx\,P_{n}(x)P_{\mu}(x),$ (87)

where we used $P_{n}(-x)=(-1)^{n}P_{n}(x)$. Multiplying the Legendre equation
for $P_{\mu}(x)$ by $P_{n}(x)$, subtracting its symmetrised version, and
integrating over $x$ from $a$ to $1$, one immediately finds

$I_{n}^{(p)}(a)=(1-a^{2})\frac{P^{\prime}_{n}(a)P_{\mu}(a)-P^{\prime}_{\mu}(a)P_{n}(a)}{n(n+1)-\mu(\mu+1)}.$
(88)

Substituting this expression, we get

$\displaystyle c_{n}^{(p)}(\varepsilon)$
$\displaystyle=\frac{P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)}{2n+1}-\frac{1-\cos^{2}\varepsilon}{\frac{pR^{2}}{D_{s}}+n(n+1)}\,\frac{P^{\prime}_{n}(\cos\varepsilon)P_{\mu}(a)+P_{n}(\cos\varepsilon)P^{\prime}_{\mu}(a)}{P_{\mu}(a)},$
(89)

where we used $P^{\prime}_{n}(-x)=(-1)^{n+1}P^{\prime}_{n}(x)$, and
$-\mu(\mu+1)=pR^{2}/D_{s}$.

The additive structure of the coefficients $c_{n}^{(p)}(\varepsilon)$ suggests
to represent $\tilde{H}^{\rm AD}(p;\mathbf{x})$ as a linear combination of the
two contributions

$\tilde{H}^{\rm
AD}(p;\mathbf{x})=\tilde{H}_{1}(p;\mathbf{x})-\tilde{H}_{2}(p;\mathbf{x}),$
(90)

where

$\displaystyle\tilde{H}_{1}(p;\mathbf{x})$
$\displaystyle=\frac{1}{2}\sum\limits_{n=0}^{\infty}g_{n}^{(p)}(r)P_{n}(\cos\theta)\bigl{[}P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)\bigr{]},$
$\displaystyle\tilde{H}_{2}(p;\mathbf{x})$
$\displaystyle=(1-\cos^{2}\varepsilon)\sum\limits_{n=0}^{\infty}g_{n}^{(p)}(r)P_{n}(\cos\theta)\frac{n+1/2}{\frac{pR^{2}}{D_{s}}+n(n+1)}$
$\displaystyle\times\frac{P^{\prime}_{n}(\cos\varepsilon)P_{\mu}(a)+P_{n}(\cos\varepsilon)P^{\prime}_{\mu}(a)}{P_{\mu}(a)}.$

If the particle starts on the inner sphere, $r=R_{1}$, one has
$g_{n}^{(p)}(R_{1})=1$, and the first contribution is simply
$\tilde{H}_{1}(p;\mathbf{x})=\Theta(\varepsilon-\theta)$. Indeed, it is equal
to unity when $\theta<\varepsilon$ (i.e., the start is on the target), and to
$0$ otherwise. In general, for $r>R_{1}$, this contribution corresponds to the
direct arrival onto the target. This can be seen by integrating the
probability flux density $\tilde{j}(p,\mathbf{s}|\mathbf{x})$ from (6) over
the target, yielding precisely $\tilde{H}_{1}(p;\mathbf{x})$. Note that
equation (90) is reproduced in the main text as equation (27). When the
starting point $\mathbf{x}$ is uniformly distributed over a sphere of radius
$r$, all terms with $n>0$ are cancelled, and one gets the much simpler
expression (33).

###  C.1 Short-time behaviour

It is instructive to analyse the large-$p$ asymptotic behaviour of equation
(33) that corresponds to the short-time behaviour of $\overline{H^{\rm
AD}(t;r)}$. As $p\to\infty$ one has
$\mu\approx-\tfrac{1}{2}+i\sqrt{pR_{1}^{2}/D_{s}}$, i.e., $P_{\mu}(z)$ is
close to a conical function, for which

$P_{\mu}(z)\approx
P_{-\frac{1}{2}+i\eta}(z)\approx\biggl{(}\frac{\theta}{\sin(\theta)}\biggr{)}^{1/2}\,I_{0}(\eta\theta)\quad(\eta\to\infty),$
(91)

where $\eta=\sqrt{pR_{1}^{2}/D_{s}}$, $\theta=\mathrm{acos}(z)$, and
$I_{0}(z)$ is the modified Bessel function of the first kind. Using the
large-$z$ behaviour of $I_{0}(z)$, one gets

$P_{\mu}(z)\approx\frac{\exp(\eta\,\mathrm{acos}(z))}{\sqrt{2\pi\eta}\,(1-z^{2})^{1/4}}\quad(\eta\to\infty).$
(92)

Evaluating the derivative with respect to $z$, we find

$\frac{P^{\prime}_{\mu}(z)}{P_{\mu}(z)}\approx-\frac{\eta}{\sqrt{1-z^{2}}}+\frac{z}{2(1-z^{2})}\quad(\eta\to\infty),$
(93)

where we kept the second (sub-leading) term. As a consequence,

$\frac{P^{\prime}_{\mu}(a)}{P_{\mu}(a)}\approx-\frac{\sqrt{pR_{1}^{2}/D_{s}}}{\sin(\varepsilon)}-\frac{\cos(\varepsilon)}{2\sin^{2}(\varepsilon)}\quad(p\to\infty).$
(94)

Substituting this expression into equation (33) and using
$g_{0}^{(p)}(r)\approx\frac{R_{1}}{r}e^{-(r-R_{1})\sqrt{p/D_{b}}}$ as
$p\to\infty$, we get

$\displaystyle\overline{\tilde{H}^{\rm AD}(p;r)}$
$\displaystyle\simeq\frac{(1-\cos(\varepsilon))R_{1}}{2r}\exp\left(-(r-R_{1})\sqrt{\frac{p}{D_{b}}}\right)$
$\displaystyle\times\left(1+\frac{1+\cos(\varepsilon)}{\sin(\varepsilon)R_{1}}\sqrt{\frac{D_{s}}{p}}+\frac{D_{s}\cos(\varepsilon)}{2(1-\cos(\varepsilon))pR_{1}^{2}}\right).$
(95)

One can thus evaluate the inverse Laplace transform term by term, yielding
equation (34).

###  C.2 Long-time behaviour

The long-time behaviour of the PDF $H^{\rm AD}(t;\mathbf{x})$ is determined by
the largest pole of the Laplace transform $\tilde{H}^{\rm AD}(p;\mathbf{x})$
(we recall that all poles are negative and thus the largest pole has the
smallest absolute value). In the special case of the starting point on the
inner sphere, $|\mathbf{x}|=R_{1}$, there is no bulk diffusion, and the ADS is
reduced to the surface tour alone, i.e., $H^{\rm AD}(t;\mathbf{s})=H^{\rm
surf}(t;\mathbf{s})$, which was studied earlier (see, e.g., [80]). We exclude
therefore this special case and assume that $|\mathbf{x}|>R_{1}$.

According to equation (2.2), the set of poles of this function is the union of
the poles $\\{p_{k}^{\rm bulk}\\}$ of $\tilde{j}(p,\mathbf{s}|\mathbf{x})$ and
of the poles $\\{p_{k}^{\rm surf}\\}$ of $\tilde{H}^{\rm surf}(p;\mathbf{s})$.
The former are determined as the poles of the functions $g_{n}^{(p)}(r)$; in
particular, the largest pole $p_{0}^{\rm bulk}$ is the pole of
$g_{0}^{(p)}(r)$, which is given by equation (23). In turn, the largest pole
$p_{0}^{\rm surf}$ is determined by the smallest zero $\nu_{0}$ of equation
(12) as

$p_{0}^{\rm surf}=-\frac{1}{\tau^{\rm surf}}<0,$ (96)

with $\tau^{\rm surf}$ given by equation (20). The long-time behaviour of
$H^{\rm AD}(t;\mathbf{x})$ is determined by $p_{0}=\max\\{p_{0}^{\rm
bulk},p_{0}^{\rm surf}\\}$. As $p_{0}^{\rm surf}$ depends on the target size
$\varepsilon$, whereas $p_{0}^{\rm bulk}$ does not, one has $p_{0}^{\rm
bulk}\neq p_{0}^{\rm surf}$ in the general case. While the analysis is similar
for $p_{0}^{\rm bulk}=p_{0}^{\rm surf}$, we do not discuss this special case.

The long-time behaviour of $H^{\rm AD}(t;\mathbf{x})$ can be determined by
evaluating the inverse Laplace transform by the residue theorem and keeping
only the contribution from the largest pole. We separately consider two
situations:

(i) If $p_{0}=p_{0}^{\rm surf}>p_{0}^{\rm bulk}$, the function
$\tilde{j}(p,\mathbf{s}|\mathbf{x})$ is not singular at $p=p_{0}$ and is thus
kept in equation (2.2); in turn, we only keep the singular term with $n=0$
from expansion (68) of $\tilde{H}^{\rm surf}(p;\mathbf{s})$. The residue
theorem then yields

$H^{\rm AD}(t;\mathbf{x})\simeq e^{p_{0}t}\biggl{(}2\pi
R_{1}^{2}\int\limits_{a}^{1}dx^{\prime}\,\tilde{j}(p_{0},(\mathrm{acos}(x^{\prime}),0)|\mathbf{x})\frac{D_{s}\sin^{2}(\varepsilon)}{R_{1}^{2}}\,b_{0}^{2}P_{\nu_{0}}(x^{\prime})P^{\prime}_{\nu_{0}}(a)\biggr{)},$
(97)

where $b_{0}^{2}$ is given by equation (55), $a=-\cos(\varepsilon)$, and the
integral over $\partial\Omega$ was reduced to the integral over
$\theta^{\prime}$, which is written in terms of
$x^{\prime}=\cos(\theta^{\prime})$. This integral can be evaluated explicitly
by use of equation (6) and the addition theorem (86). However, it is more
convenient to focus on the average when the starting point is uniformly
distributed on a sphere of radius $r$, for which

$\overline{\tilde{j}(p,\mathbf{s}|\mathbf{x})}\equiv\frac{1}{4\pi
r^{2}}\int\limits_{|\mathbf{x}|=r}d\mathbf{x}\,\tilde{j}(p,\mathbf{s}|\mathbf{x})=\frac{g_{0}^{(p)}(r)}{4\pi
R_{1}^{2}},$ (98)

and thus equation (97) becomes

$\displaystyle\overline{H^{\rm AD}(t;r)}\simeq
e^{p_{0}t}\biggl{(}\frac{g_{0}^{(p_{0})}(r)}{2}\int\limits_{a}^{1}dx^{\prime}\frac{D_{s}\sin^{2}(\varepsilon)}{R_{1}^{2}}\,b_{0}^{2}P_{\nu_{0}}(x^{\prime})P^{\prime}_{\nu_{0}}(a)\biggr{)}\simeq
e^{p_{0}t}C_{\varepsilon}(r),$ (99)

with

$C_{\varepsilon}(r)=g_{0}^{(p_{0})}(r)\frac{D_{s}\sin^{4}(\varepsilon)}{2R_{1}^{2}}\,\frac{b_{0}^{2}[P^{\prime}_{\nu_{0}}(-\cos(\varepsilon))]^{2}}{\nu_{0}(\nu_{0}+1)},$
(100)

where we used the identity

$\int\limits_{a}^{1}dx\,P_{\nu}(x)=\frac{1-a^{2}}{\nu(\nu+1)}P^{\prime}_{\nu}(a).$
(101)

Note that equation (7) implies for $p_{0}<0$ that

$g_{0}^{(p_{0})}(r)=\frac{R_{1}}{r}\,\frac{R_{2}\sqrt{|p_{0}|/D_{b}}\cos(\sqrt{|p_{0}|/D_{b}}(R_{2}-r))-\sin(\sqrt{|p_{0}|/D_{b}}(R_{2}-r))}{R_{2}\sqrt{|p_{0}|/D_{b}}\cos(\sqrt{|p_{0}|/D_{b}}(R_{2}-R_{1}))-\sin(\sqrt{|p_{0}|/D_{b}}(R_{2}-R_{1}))}.$
(102)

(ii) If $p_{0}=p_{0}^{\rm bulk}>p_{0}^{\rm surf}$, the function
$\tilde{H}^{\rm surf}(p;\mathbf{s})$ is not singular at $p=p_{0}$ and is thus
kept in equation (2.2); in turn, we only keep the singular term with $n=0$
from expansion (6) of $\tilde{j}(p,\mathbf{s}|\mathbf{x})$. The residue
theorem then yields

$H^{\rm AD}(t;\mathbf{x})\simeq e^{p_{0}t}C_{\varepsilon}(r),$ (103)

with

$C_{\varepsilon}(r)=\frac{1}{2}\mathop{\rm
Res}_{p_{0}}\\{g_{0}^{(p)}(r)\\}\int\limits_{0}^{\pi}d\theta^{\prime}\,\sin(\theta^{\prime})\tilde{H}^{\rm
surf}(p_{0};\theta^{\prime}).$ (104)

As $p_{0}$ is a pole of $g_{0}^{(p)}(r)$, other radial functions disappeared,
and the result does not depend on the starting point $\mathbf{x}$.

Note that the integral over $\theta^{\prime}$ can be evaluated explicitly by
use of equation (10),

$\displaystyle\int\limits_{0}^{\pi}d\theta^{\prime}\,\sin(\theta^{\prime})\tilde{H}^{\rm
surf}(p_{0};\theta^{\prime})=\int\limits_{0}^{\varepsilon}d\theta^{\prime}\,\sin(\theta^{\prime})\,1+\int\limits_{\varepsilon}^{\pi}d\theta^{\prime}\,\sin(\theta^{\prime})\frac{P_{\mu_{0}}(-\cos(\theta^{\prime}))}{P_{\mu_{0}}(a)}$
$\displaystyle\quad=(1-\cos(\varepsilon))+\int\limits_{a}^{1}dx^{\prime}\,\frac{P_{\mu_{0}}(x^{\prime})}{P_{\mu_{0}}(a)}=(1-\cos(\varepsilon))-\frac{P^{\prime}_{\mu_{0}}(a)(1-a^{2})D_{s}}{P_{\mu_{0}}(a)\,p_{0}R_{1}^{2}},$

where $\mu_{0}=\tfrac{1}{2}(-1+\sqrt{1-4p_{0}R_{1}/D_{s}})$, and we used the
identity (101), the relation $\mu_{0}(\mu_{0}+1)=-p_{0}R_{1}^{2}/D_{s}$ and
recalled that $\tilde{H}^{\rm surf}(p;\theta^{\prime})=1$ for any
$0\leq\theta^{\prime}\leq\varepsilon$ (on the target). The residue of
$g_{0}^{(p)}(r)$ can also be found by direct computation,

$\displaystyle\mathop{\rm Res}_{p_{0}}\\{g_{0}^{(p)}(r)\\}$
$\displaystyle=\frac{2D_{b}\alpha_{0}R_{1}\sqrt{1+\beta^{2}\alpha_{0}^{2}}}{r(R_{2}-R_{1})^{2}(1+\beta(\beta\alpha_{0}^{2}-1))}\biggl{[}\beta\alpha_{0}\cos\biggl{(}\alpha_{0}\frac{R_{2}-r}{R_{2}-R_{1}}\biggr{)}-\sin\biggl{(}\alpha_{0}\frac{R_{2}-r}{R_{2}-R_{1}}\biggr{)}\biggr{]},$

where $\beta=R_{2}/(R_{2}-R_{1})$.

Combining equations (99) and (103), we obtain relation (32), in which the
amplitude $C_{\varepsilon}(r)$ is given either by equation (100) for
$p_{0}^{\rm surf}>p_{0}^{\rm bulk}$, or by equation (104) for $p_{0}^{\rm
surf}<p_{0}^{\rm bulk}$.

###  C.3 Mean first-passage time

Evaluating the derivative of $\tilde{H}^{\rm AD}(p;\mathbf{x})$ at $p=0$ we
obtain the MFPT within the ADS. To get some qualitative picture we start from
the general representation (2.2),

$\displaystyle T^{\rm AD}(\mathbf{x})$ $\displaystyle=-\lim\limits_{p\to
0}\frac{\partial\tilde{H}^{\rm AD}(p;\mathbf{x})}{\partial p}$
$\displaystyle=\int\limits_{{\partial\Omega}}d\mathbf{s}\,\biggl{[}\biggl{(}-\partial_{p}\tilde{j}(p,\mathbf{s}|\mathbf{x})\biggr{)}_{p=0}\,\underbrace{\tilde{H}^{\rm
surf}(0;\theta_{\mathbf{s}})}_{=1}+\tilde{j}(0,\mathbf{s}|\mathbf{x})\underbrace{\biggl{(}-\partial_{p}\tilde{H}^{\rm
surf}(p;\mathbf{s})\biggr{)}_{p=0}}_{=T^{\rm surf}(\mathbf{s})}\biggr{]}.$

The first term can be interpreted as the MFPT to the inner sphere from a fixed
point $\mathbf{x}$, conditioned to arrive at point $\mathbf{s}$, and then
averaged over all locations $\mathbf{s}$. In turn, the second term is the
average of the MFPT $T^{\rm surf}(\mathbf{s})$ on the surface, averaged over
the arrival point $\mathbf{s}$ with the harmonic measure density (factor
$\tilde{j}(0,\mathbf{s}|\mathbf{x})$). In practice, we can use the exact
solution (27) to compute the MFPT. Even though the exact computation of this
limit is feasible, we consider the simpler case of the surface-averaged MFPT

$\displaystyle\overline{T^{\rm AD}}$ $\displaystyle\equiv\frac{1}{4\pi
r^{2}}\int\limits_{|\mathbf{x}|=r}d\mathbf{x}\,T^{\rm
AD}(\mathbf{x})=-\lim\limits_{p\to 0}\partial_{p}\overline{\tilde{H}^{\rm
AD}(p;r)}$
$\displaystyle=-\frac{1-\cos\varepsilon}{2}\biggl{\\{}\biggl{(}\partial_{p}g_{0}^{(p)}(r)\biggr{)}_{p=0}\biggl{(}1-\frac{D_{s}(1+\cos\varepsilon)}{pR_{1}^{2}}\,\frac{P^{\prime}_{\mu}(a)}{P_{\mu}(a)}\biggr{)}_{p=0}$
$\displaystyle+\underbrace{\bigl{(}g_{0}^{(p)}(r)\bigr{)}_{p=0}}_{=1}\biggl{(}\partial_{p}\biggl{(}\frac{D_{s}(1+\cos\varepsilon)}{pR_{1}^{2}}\,\frac{P^{\prime}_{\mu}(a)}{P_{\mu}(a)}\biggr{)}\biggr{)}_{p=0}\biggr{\\}}.$

In the first term, we have

$\lim\limits_{p\to
0}\biggl{(}\partial_{p}g_{0}^{(p)}(r)\biggr{)}=-\frac{(r-R_{1})(2R_{2}^{3}-rR_{1}(r+R_{1}))}{6rR_{1}D_{b}}.$
(105)

In the limit $p\to 0$ we approximate $\mu\simeq-s-s^{2}+O(s^{3})$ (with
$s=pR_{1}^{2}/D_{s}$), so that we can use the expansion (see [100, 101])

$P_{\nu}(x)=1+\nu\ln\biggl{(}\frac{1+x}{2}\biggr{)}-\nu^{2}\mathrm{Li}_{2}\biggl{(}\frac{1-x}{2}\biggr{)}+O(\nu^{3})\quad(\nu\to
0),$ (106)

where $\mathrm{Li}_{2}(z)$ is the dilogarithm function of the second order,
$L_{2}(z)=-\int\nolimits_{0}^{z}dx\ln(1-x)/x$. We thus find

$\displaystyle P_{\mu}(a)$
$\displaystyle=1-s\ln\biggl{(}\frac{1+a}{2}\biggr{)}-s^{2}\biggl{(}\ln\biggl{(}\frac{1+a}{2}\biggr{)}+\mathrm{Li}_{2}\biggl{(}\frac{1-a}{2}\biggr{)}\biggr{)}+O(s^{3}),$
(107) $\displaystyle P^{\prime}_{\mu}(a)$
$\displaystyle=-\frac{s}{1+a}-s^{2}\biggl{(}\frac{1}{1+a}+\frac{\ln((1+a)/2)}{1-a}\biggr{)}+O(s^{3}),$
(108)

so that

$\frac{P_{\mu}(a)}{P^{\prime}_{\mu}(a)}=-\frac{1+a}{s}\biggl{(}1-s\biggl{(}1+\frac{2}{1-a}\ln\frac{1+a}{2}\biggr{)}+O(s^{2})\biggr{)},$
(109)

and thus

$\frac{D_{s}(1+\cos\varepsilon)}{pR_{1}^{2}}\,\frac{P^{\prime}_{\mu}(a)}{P_{\mu}(a)}=-\frac{1-a}{1+a}\biggl{(}1+s\biggl{(}1+\frac{2}{1-a}\ln\frac{1+a}{2}\biggr{)}\biggr{)}+O(s^{2}).$
(110)

From this we conclude that

$\overline{T^{\rm
AD}}=(-\partial_{p}g_{0}(r))_{p=0}-\frac{(1-a)R_{1}^{2}}{2D_{s}}\biggl{(}1+\frac{2}{1-a}\ln\frac{1+a}{2}\biggr{)},$
(111)

which can also be written more explicitly as equation (37).

###  C.4 Fixed vs random starting point

As argued in section 3.3, the average of the PDF when the starting point is
uniformly distributed on a sphere of given radius $r$ is often more
representative than the case of a fixed starting point. Here we briefly
discuss the differences between these two situations. As expected, the long-
time behaviour of the PDF does not depend on the starting point, whereas the
short-time behaviour is generally much more sensitive to the distance between
the starting point and the target. Figure 8 illustrates this point for the
geometric setting considered in the main text (with $R_{1}=1$ and
$r=R_{2}=5$), except that we take a larger target size, $\varepsilon=0.1$.
Note that this larger value actually enhances possible differences due to the
starting point since a larger target is found faster and thus the diffusing
particle has less time to loose the information on the starting point. When
$D_{s}=D_{b}$ (panel (a)), the left tail of the PDF is most shifted to the
left (to shorter times) when the fixed starting point is located on the North
pole of the outer sphere, i.e., right above the target (located on the North
pole of the inner sphere). In other words, this setting favours the fastest
arrival onto the target, as expected. In particular, it is faster than in the
case of random starting point, which is a sort of weighted average with
different distances to the target. In turn, when the fixed target is located
on the equator ($\theta=\pi/2$) or on the South pole ($\theta=\pi$) of the
outer sphere, the surface averaged PDF provides the faster arrival at short
times. Overall, the difference between the four cases is moderate. This
difference is, however, increased when $D_{s}/D_{b}=0.01$ (panel (b)).

Figure 8: FPT PDF to a circular target located on the North pole of the inner
sphere within the ADS, for $R_{1}=1$, $R_{2}=5$, $r=5$, $\varepsilon=0.1$,
$D_{s}/D_{b}=1$ (a), and $D_{s}/D_{b}=0.01$ (b). The surface-averaged PDF
$\overline{H^{\rm AD}(t;r)}$ (circles) is compared to the PDF $H^{\rm
AD}(t;\mathbf{x})$ with a fixed starting point $\mathbf{x}=(r,\theta,0)$, for
three choices of $\theta$, as indicated in the legend. All PDFs were obtained
via numerical inversion of the Laplace transform by the Talbot algorithm.

##  Appendix D PDF for one-stage search process

The exact spectral solution for the Laplace-transformed FPT PDF in the
conventional one-stage scenario was presented in [50]. This solution involves
an infinite-dimensional matrix with explicitly known elements. In practice,
one has to truncate this matrix and then numerically invert it.

In order to avoid these numerical steps, we use the self-consistent
approximation developed and validated in [50]. The approximate solution reads

$\tilde{H}_{\rm dir}^{\rm
app}(p;\mathbf{x})=J_{p}\sum\limits_{n=0}^{\infty}g_{n}^{(p)}(r)\,P_{n}(\cos\theta)\,\frac{1}{\mu_{n}^{(p)}}\,\frac{P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)}{2},$
(112)

where

$J_{p}=\left(\frac{1}{q}+\frac{1}{2(1-\cos\varepsilon)}\sum\limits_{n=0}^{\infty}\frac{\bigl{(}P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)\bigr{)}^{2}}{(2n+1)\mu_{n}^{(p)}}\right)^{-1},$
(113)

and

$\mu_{n}^{(p)}=-\biggl{(}\frac{\partial g_{n}^{(p)}(r)}{\partial
r}\biggr{)}_{r=R_{1}}.$ (114)

Here, there is no need for matrix inversion as the expression is fully
explicit. Moreover, the limit $q\to\infty$ can be easily obtained by simply
setting $q=\infty$ in equation (113). While the inverse Laplace transform of
equation (112) can be found exactly via the residue theorem (see [50] for
details), we resort to a numerical inversion by the Talbot algorithm. When the
starting point $\mathbf{x}$ is uniformly distributed over a sphere of radius
$r$, equation (112) is reduced to

$\overline{\tilde{H}_{\rm dir}^{\rm
app}(p;r)}=\frac{J_{p}g_{0}^{(p)}(r)}{2\mu_{0}^{(p)}}(1-\cos\varepsilon).$
(115)

The approximate solution (112) can also be used to compute the moments of the
FPT, in particular, the MFPT was derived in Appendix B of [50]. The surface
average of equation (B.9) from [50] reads

$\overline{T_{\rm dir}^{\rm
app}}=\frac{R_{2}^{3}-R_{1}^{3}}{3D_{b}R_{1}}\bigg{[}C_{\varepsilon}+\frac{5R_{2}^{3}-9R_{2}^{2}R_{1}+1}{5(R_{2}^{3}-R_{1}^{3})}-R_{1}\frac{5r^{3}+10R_{2}^{3}-3r(R_{1}^{2}+6R_{2}^{2})}{10r(R_{2}^{3}-R_{1}^{3})}\biggr{]},$
(116)

where

$C_{\varepsilon}=\frac{2}{(1-\cos\varepsilon)R_{1}q}+\frac{1}{(1-\cos\varepsilon)^{2}}\sum\limits_{n=1}^{\infty}\frac{(P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon))^{2}}{(2n+1)R_{1}\mu_{n}^{(0)}},$
(117)

and

$\mu_{n}^{(0)}=\frac{n(n+1)}{R_{1}}\,\frac{1-(R_{1}/R_{2})^{2n+1}}{n+(n+1)(R_{1}/R_{2})^{2n+1}}.$
(118)

When $R_{1}\ll R_{2}$ the factor $(R_{1}/R_{2})^{2n+1}$ can be neglected, so
that $\mu_{n}^{(0)}\approx(n+1)/R_{1}$. The asymptotic behaviour of series
like in equation (117) was analysed in [70]. Since $R_{1}\mu_{n}^{(0)}$ can be
interpreted as $Rg^{\prime}_{n}(R)/g_{n}(R)$ from the Supplementary Material
of [70], one has $1/(R_{1}\mu_{n}^{(0)})\approx 1/(n+1)=1/n-1/n^{2}+O(n^{-3})$
such that one can apply equation (S76) with $\omega=3$ from equation (S56).
This yields the small-$\varepsilon$ behaviour

$C_{\varepsilon}\approx\frac{4}{R_{1}q}\varepsilon^{-2}+\frac{32}{3\pi}\varepsilon^{-1}-\ln(1/\varepsilon)+O(1)\quad(\varepsilon\to
0).$ (119)

When the target is partially reactive ($q<\infty$) the dominant term scales as
$\varepsilon^{-2}$, showing that the direct search is reaction-limited. In
turn, for a perfect target, on which we focus in this paper, the first term
disappears, and one gets the $\varepsilon^{-1}$-behaviour, with a logarithmic
correction. As discussed in [70] for the problem of the narrow escape from a
sphere, the numerical factor $32/(3\pi)\approx 3.40$ differs by only $8\%$
from the exact value $\pi$, which was known for that problem. This minor
difference comes from the self-consistent approximation. We expect therefore
that substitution of $C_{\varepsilon}\approx\pi/\varepsilon$ in equation (116)
would result in the correct leading-order behaviour of $\overline{T_{\rm
dir}}$ in our setting. In particular, when $\rho\ll R_{1}\ll R_{2}$ (and thus
$\varepsilon\ll 1$), we get $\overline{T_{\rm dir}^{\rm app}}\approx\tau_{\rm
dir}=R_{2}^{3}/(3D_{b}\rho)$ that was introduced in section 1.

### Limit $R_{1}\to R_{2}$

This is an interesting limit in which the thickness of the shell region
between the outer and inner spheres shrinks, such that a particle is deemed to
perform a diffusive motion almost on the surface of a sphere and seeking a
circular target on that surface.

Setting $R_{1}=R-\delta$, $R_{2}=R$ and $r=R$, we get from (114), to the
leading order,

$\mu_{n}^{(p)}\simeq\delta\,\biggl{(}\frac{p}{D}+\frac{n(n+1)}{R^{2}}\biggr{)}+O(\delta^{2}).$
(120)

In this limit, the self-consistent approximation (112) yields

$\displaystyle\tilde{H}^{\rm app}_{\rm dir}(p;\mathbf{x})$
$\displaystyle=(1-\cos\varepsilon)\biggl{(}\sum\limits_{n=0}^{\infty}\frac{(P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon))^{2}}{(2n+1)\bigl{(}\frac{R^{2}p}{D}+n(n+1)\bigr{)}}\biggr{)}^{-1}$
$\displaystyle\times\sum\limits_{n=0}^{\infty}\frac{P_{n}(\cos\theta)(P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon))}{\bigl{(}\frac{R^{2}p}{D}+n(n+1)\bigr{)}}.$
(121)

One can easily check that $\tilde{H}^{\rm app}_{\rm dir}(p;\mathbf{x})$
converges to unity as $p\to 0$, as it should to fulfill the normalisation
condition of the PDF. Note that both series can be evaluated explicitly by
using the identities (80) and (81).

Figure 9: Laplace-transformed PDF $\tilde{H}_{\rm dir}(p;\mathbf{x})$ for
diffusion in a thin spherical shell between spheres of radii $R_{1}=0.999$ and
$R_{2}=1$ towards a circular target of angular size $\varepsilon=0.1$ located
at the North pole, with $r=1$ and $\theta=\pi/2$. Green circles show the exact
solution from [50] with $q=1$ (truncated at $n_{\rm max}=100$) and the solid
blue line represents equation (121) (the self-consistent approximation in the
limit $\delta\to 0$). For comparison, the Laplace-transformed PDF
$\tilde{H}^{\rm surf}(p;\theta)$ from equation (72) for surface diffusion with
$D_{s}=D_{b}$ towards the same target is shown by the dash-dotted grey line
for $q=1$ and by the dashed red line for $q=\infty$.

Curiously, the approximation (121) does not depend on the reactivity parameter
$q=\kappa/D_{b}$. This behaviour is illustrated in figure 9, in which we
compare the Laplace-transformed PDF for diffusion inside a thin shell (of
width $0.001$) to that for surface diffusion on the sphere. There is good
agreement between the three curves: the exact solution for a thin shell with
$q=1$, the self-consistent approximation (121) (independent of $q$), and the
exact solution $\tilde{H}^{\rm surf}(p;\theta)$ from (72) for surface
diffusion on the sphere towards a perfectly reactive target ($q=\infty$). In
contrast, the exact solution for surface diffusion on the sphere towards a
partially reactive target ($q=1$) stands out. We conclude that the Laplace-
transformed PDF in a very thin shell is close to that for surface diffusion
towards a perfect target.

To rationalise this behaviour, let us consider diffusion in a very thin
stripe, $\mathbb{R}\times(0,\delta)$, on which an interval
$(-R,R)\times\\{0\\}$ represents the target. Once a particle enters the
proximity of the target (i.e., the rectangle $(-R,R)\times(0,\delta)$), it
hits the target a very large number of times, so that the reaction occurs very
rapidly after entering this zone. In the limit $\delta\to 0$, the number of
encounters grows, such that the effective target on the line is perfectly
reactive. The same argument holds for three-dimensional diffusion between
parallel planes, $\mathbb{R}^{2}\times(0,\delta)$, when the target is a disk
$\\{(x,y,0)\in\mathbb{R}^{3}:x^{2}+y^{2}<R^{2}\\}$. This is equivalent to our
setting because the curvature of the spherical shell does not matter in the
limit $\delta\to 0$.

### Numerical implementation

Even though the approximative solution (112) is fully explicit, its numerical
computation may be challenging, especially for small $\varepsilon$. In fact,
one needs to truncate the infinite series in (113), and the truncation order
$n_{\rm max}$ has to be large when $\varepsilon$ is small. At the same time,
the numerical evaluation of the radial function $g_{n}^{(p)}$ that involve
$i_{n}(z)$ and $k_{n}(z)$ and determine $\mu_{n}^{(p)}$, becomes unstable for
large $n$. Moreover, as one needs to perform an inverse Laplace transform of
$\tilde{H}_{\rm dir}^{\rm app}(p;\mathbf{x})$ to get back to the time domain,
this computation has to be realised for any $p\in\mathbb{C}$. In order to
overcome these numerical difficulties, we adopt the following two-step scheme.

In the first step, we approximate the radial functions with large $n$ as

$g_{n}^{(p)}(r)\approx g_{n,\infty}^{(p)}(r)=\frac{k_{n}(\alpha
r)}{k_{n}(\alpha R_{1})},$ (122)

where $\alpha=\sqrt{p/D}$ and $g_{n,\infty}^{(p)}(r)$ are the radial functions
for the problem without outer sphere (i.e., for $R_{2}=\infty$). This
approximation can be easily justified in the limit $p\to\infty$ for any
$r<R_{2}$ and any $n$ just by looking at the asymptotic behaviour of the
modified spherical Bessel functions $i_{n}(z)$ and $k_{n}(z)$ in (7). We
checked numerically that this approximation is also applicable for smaller $p$
if $r\ll R_{2}$ and $n\gg 1$.

In the second step, we employ the recurrence relations for $k_{n}(z)$ to
enable an iterative computation of the radial function. In fact, one has

$k_{n+1}(z)=k_{n-1}(z)+\frac{2n+1}{z}k_{n}(z),\qquad
k^{\prime}_{n}(z)=-k_{n-1}(z)-\frac{n+1}{z}k_{n}(z).$ (123)

Using the first relation, one gets

$g_{n+1,\infty}^{(p)}(r)=\frac{k_{n-1}(\alpha r)+\frac{2n+1}{\alpha
r}k_{n}(\alpha r)}{k_{n-1}(\alpha R_{1})+\frac{2n+1}{\alpha R_{1}}k_{n}(\alpha
R_{1})}=g_{n,\infty}^{(p)}(r)\frac{R_{1}}{r}\frac{1+\frac{f_{n}(\alpha
r)}{2n+1}}{1+\frac{f_{n}(\alpha R_{1})}{2n+1}},$ (124)

where

$f_{n}(z)=z\frac{k_{n-1}(z)}{k_{n}(z)}.$ (125)

Using again the first recurrence relation and the explicit form
$k_{0}(z)=e^{-z}/z$ one finds

$f_{1}(z)=\frac{z^{2}}{z+1},\qquad
f_{n}(z)=\frac{z^{2}}{f_{n-1}(z)+2n-1}\quad(n=2,3,\ldots).$ (126)

One sees that the $f_{n}(z)$ are rational functions, which can be computed
iteratively. These functions have the following asymptotic behaviours

$f_{n}(z)\simeq\frac{z^{2}}{2n-1}-\frac{z^{4}}{(2n-1)^{2}(2n-3)}+O(z^{6})\quad(z\to
0)$ (127)

for $n>1$ (note that $f_{1}(z)=z^{2}-z^{3}+\ldots$ contains the $z^{3}$ term),
and

$f_{n}(z)\simeq z-n+\frac{n(n+1)}{2z}+O(z^{-2})\quad(z\to\infty).$ (128)

Finally, we can approximate

$\mu_{n}^{(p)}\approx\mu_{n,\infty}^{(p)}=-\biggl{(}\partial_{r}g_{n,\infty}^{(p)}(r)\biggr{)}_{r=R_{1}}=-\alpha\frac{k^{\prime}_{n}(\alpha
R_{1})}{k_{n}(\alpha R_{1})}.$ (129)

Using the second recurrence relation, one then gets

$R_{1}\mu_{n,\infty}^{(p)}=n+1+f_{n}(\alpha R_{1})\qquad(n=1,2,\ldots)$ (130)

(note that $R_{1}\mu_{0,\infty}^{(p)}=1+\alpha R_{1}$). We then use the
following numerical approximation

$\displaystyle J_{p}$
$\displaystyle\approx\left(\frac{1}{q}+\frac{1}{2(1-\cos\varepsilon)}\left\\{\sum\limits_{n=0}^{n_{\rm
app}}\frac{\bigl{(}P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)\bigr{)}^{2}}{(2n+1)\mu_{n}^{(p)}}\right.\right.$
$\displaystyle\left.\left.+\sum\limits_{n=n_{\rm app}+1}^{n_{\rm
max}}\frac{\bigl{(}P_{n-1}(\cos\varepsilon)-P_{n+1}(\cos\varepsilon)\bigr{)}^{2}}{(2n+1)\mu_{n,\infty}^{(p)}}\right\\}\right)^{-1},$
(131)

in which we keep the first $n_{\rm app}+1$ terms with the exact form of
$\mu_{n}^{(p)}$ and replace $\mu_{n}^{(p)}$ by its approximation
$\mu_{n,\infty}^{(p)}$ from (130) for the terms with $n_{\rm app}<n\leq n_{\rm
max}$. In this way, the truncation order $n_{\rm max}$ can be made
sufficiently large to ensure accurate computations. We used $n_{\rm app}=50$
and $n_{\rm max}=250$.

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