# Geometry controls diffusive target encounters and escape in tubular
structures

Junyeong L. Kim  Aidan I. Brown<EMAIL_ADDRESS>Department of
Physics, Toronto Metropolitan University

###### Abstract

The endoplasmic reticulum (ER) is a network of sheet-like and tubular
structures that spans much of a cell and contains molecules undergoing
diffusive searches for targets, such as unfolded proteins searching for
chaperones and recently-folded proteins searching for export sites. By
applying a Brownian dynamics algorithm to simulate molecule diffusion, we
describe how ER tube geometry influences whether a searcher will encounter a
nearby target or instead diffuse away to a region near to a distinct target,
as well as the timescale of successful searches. We find that targets are more
likely to be found for longer and narrower tubes, and larger targets, and that
search in the tube volume is more sensitive to the search geometry compared to
search on the tube surface. Our results suggest ER proteins searching for low-
density targets in the membrane and the lumen are very likely to encounter the
nearest target before diffusing to the vicinity of another target. Our results
have implications for the design of target search simulations and calculations
and interpretation of molecular trajectories on the ER network, as well as
other organelles with tubular geometry.

Suggested keywords

††preprint: APS/123-QED

##  I Introduction

Diffusive motion is essential for transport and function in cell biology [1,
2, 3, 4, 5]. Intracellular geometry affects the timescale of diffusive search,
which can be framed in various ways, including as an effective diffusivity,
such as the diffusivity along the mitochondrial axis with cristae present [6];
as time evolution of concentrations, such as for calcium moving through the
endoplasmic reticulum [7]; as a rate or flux, such as the arrival rate to an
organelle surface [8, 9] or to dispersed receptor regions on an organelle
[10]; and as a mean first-passage time, such as the escape time from the the
endoplasmic reticulum via channels [11] or transit time through nuclear pores
[12].

The endoplasmic reticulum (ER) is a cellular organelle with important roles in
protein synthesis and transport, protein folding, lipid and steroid synthesis,
carbohydrate metabolism, and calcium storage [13]. The geometry of the ER is
complex and consists of sheet-like and tubular regions [14], as well as
regions of dense tubular matrices [15], forming a single connected network of
lumenal volume and membrane surface. ER tubes are typically 250 – 750 nm long
in yeast cells [16] and 0.5 – 5 $\mu$m long in mammalian cells [13, 17], with
radii typically ranging from 10 – 25 nm in yeast cells [16] and 25 – 70 nm in
mammalian cells [18]. ER structure can vary with cell cycle stage, cell type,
and stress [19, 13].

A variety of molecules inside the ER undergo diffusive searches for targets.
As an important site for protein folding, many unfolded proteins in the ER
will search for chaperone proteins in the membrane and lumen to assist folding
[20, 21]. Once folded inside the ER, many proteins are exported to the Golgi
via ER exit sites, which have neck-like entry points off of ER tubes [22, 23].
Calcium ions can exit the ER through channel proteins in the membrane [11].

A range of theoretical studies have investigated how ER geometry and target
distribution affect local diffusive search inside the ER. Pore density, size,
and channel length affect calcium escape rates from the ER [11]. Slower
diffusion through the narrow regions of ‘pearled’ ER tubes is described by the
Fick-Jacobs equation [24, 25, 5]. Helicoidal ramps between layers of ER sheets
facilitate more rapid diffusion between the sheet layers compared to the
hypothetical alternative of holes between the sheet layers [26]. Various
narrow escape problems correspond to molecules undergoing diffusive search for
a target inside the ER [27].

Diffusive search through the broader ER network has also been examined through
calculations and simulations. Better-connected ER networks (with more ‘loops’)
facilitate faster diffusive search [28]. The effective distribution of targets
for rapid search through the ER network depends on the location of sources and
the maturation period of searchers, with mobile targets able to more than
halve search times [29]. The removal of tube connections from experimentally-
extracted ER networks accelerates diffusive search if a tube is removed from a
dense network region (such as near the nucleus), but slows search if the
removed tube connects network regions (more likely in the cell periphery)
[30]. ER network tube rearrangement dynamics are sufficiently slow compared to
intra-network diffusion that the dynamics have little effect on search times
[31]. Target density in an ER network modulates the effective dimension of the
diffusive search process [5], and the effective dimension of ER networks
controls how search time depends on initial searcher-target separation [32].
Beyond calculations and simulations, recent experimental advances have enabled
measurement of the trajectories of individual proteins inside the ER network,
providing insight into intra-organelle protein behavior [33].

Previous calculations and simulations largely modeled search through the ER
network as a process occurring along one-dimensional tubes [28, 29, 30, 31],
assuming that reaching the location of a target along the tube axis
constituted an encounter with that target. However, a searcher on the ER
membrane (surface) or a searcher in the ER lumen (volume) could diffuse past
the axial location of the target without encountering the target in two or
three dimensions, respectively. In this work we consider how the tube geometry
affects the probability that a diffusive searcher will encounter a target once
it has reached the axial position of the target, with escape to the axial
position of a nearby target as the alternative outcome. We find that along
with dependence on tube radius and target size, target density along the tube
axis is a key parameter to describe whether a searcher will encounter a target
or diffuse to the vicinity of another target. This result suggests that
individual sparsely-distributed targets are likely to be encountered once
their position is reached along the tube axis, while searchers will often
escape from individual densely-spaced targets without an encounter.

Figure 1:  Target encounter vs escape in a cylinder. ER tube segment is
considered as a cylinder. A searcher molecule (blue dot) begins at the axial
position of a target (red dot). Targets (red dots) are separated by a distance
$L_{\text{tube}}$ along the tube axis. Undergoing a diffusive trajectory, the
searcher reaches one of two outcomes first: encounter the target or escape
from the initial target by reaching the axial position of an adjacent target.

##  II Results

###  II.1 Stationary target encounters

To investigate diffusive search behavior for a variety of cylindrical tube
geometries, we first model the diffusive search for a stationary target on the
wall of a tube with two open ends, with searchers either diffusing in the tube
volume (representing the ER lumen) or on the tube surface (representing the ER
membrane). The stationary target particle with radius $R_{\text{target}}$ is
placed on the tube wall at the axial midpoint of the tube a distance
$L_{\text{tube}}$ from both ends of the tube, with this axial tube length
$L_{\text{tube}}$ representing the distance between targets along the tube
axis as shown in Fig. 1. The tube has radius $R_{\text{tube}}$. The diffusive
searcher begins at an axial position offset from the target axial position by
$R_{\text{target}}$, with searchers in the tube volume uniformly distributed
through the tube cross section and searchers on the tube surface uniformly
distributed around the tube circumference. The searcher is considered to have
encountered the target once the searcher first arrives within a distance
$R_{\text{target}}$ from the target, such that $R_{\text{target}}$ represents
the sum of the searcher and target radii. For searchers in the volume this
encounter distance $R_{\text{target}}$ is through the three-dimensional tube
volume and for searchers on the surface it is along the two-dimensional tube
surface. Thus the initial searcher position is at the closest axial position
without possible encounter of the target by the searcher. Brownian Dynamics,
combined with reflections from the inside tube surface for search in the
volume, is used to simulate the diffusive search for the target (see Appendix
for details).

Figure 2:  Target encounter with a stationary target. Target encounter
probability for varying (A) target radius, (B) tube radius, and (C) tube
length. Corresponding mean target encounter time for varying (D) target
radius, (E) tube radius, and (F) tube length. For all panels, the legend
indicates whether the searcher is on the tube surface (solid lines) or in the
tube volume (dashed lines). For panels (A), (B), (D), and (E) tube length is
indicated by curve color and adjacent corresponding labels. For panels (C) and
(F) tube radius is indicated by curve color and adjacent corresponding labels.
Panels (A) and (D) have tube radius $R_{\text{tube}}=30\text{ nm}$ and panels
(B), (C), (E), and (F) have target radius $R_{\text{target}}=10\text{ nm}$.
For all panels searcher diffusivity $D=1\text{ }\mu\text{m}^{2}/\text{s}$ with
$10^{4}$ samples.

With the tube open at both ends and containing a target, there are two
possibilities for each trajectory — the diffusive searcher will either
encounter the target or escape the tube region under consideration by
diffusing beyond one of the tube ends — both outcomes terminate the search. As
the tube region under consideration contains one target and both tube ends
axially abut the nearest target in each direction (see Fig. 1), escape
represents the close approach of the searcher to another target, ending the
encounter opportunity with the target in the axial center of the tube region
under consideration.

Figure 2 explores how the geometric parameters of this search — target radius
$R_{\text{target}}$, tube radius $R_{\text{tube}}$, and tube length
$L_{\text{tube}}$ — control the probability and mean time to encounter the
target. Increasing the target radius $R_{\text{target}}$ increases the target
encounter probability (Fig. 2A) and decreases the mean target encounter time
(Fig. 2D). Increasing the tube radius $R_{\text{tube}}$ decreases the target
encounter probability (Fig. 2B) and increases the mean target encounter time
(Fig. 2E). This behavior is expected, as a larger target will be more easily
and more quickly encountered and a larger search space will cause a target to
become more difficult and slower to encounter.

In Fig. 2A, the encounter probability changes less with increasing target
radius for search on the tube surface compared to search in the tube volume, a
trend which largely continues for encounter probabilities and times across
Figs. 2A,B,D,and E. This behavior aligns with previous work showing that two-
dimensional diffusive search depends less on geometric factors than three-
dimensional search [32, 34, 5]. Variation of target radius (Figs. 2A,D) has a
more limited effect on the target encounter probability and target encounter
time in comparison to variation of tube radius (Figs. 2B,E).

While increasing the tube length increases the search space, such a tube
length increase also increases the target encounter probability (Fig. 2C).
This is in contrast to larger tube radius, which increases the search space
and causes a decrease in target encounter probability (Fig. 2B). The increase
in encounter probability with tube length occurs because the mean time to
escape the tube increases with the tube length, providing a longer time period
for the searchers to encounter the target, while varying tube radius does not
similarly affect escape time. Similar to increasing tube radius (Fig. 2B),
increasing the tube length increases the target encounter time (Fig. 2F). The
mean encounter time increases with tube length because the greater mean time
period before escape allows longer trajectories to encounter the target before
escape.

Figure 2C shows that as the tube length exceeds approximately 1 $\mu$m the
target encounter probability approaches one. Figure 2C shows encounter
probabilities for $R_{\text{target}}=10\text{ nm}$ so a smaller target will
require longer tubes to reach similarly high encounter probabilities. However,
the approach of encounter probabilities towards one in Fig. 2C for
$L_{\text{tube}}\gtrsim 1\text{ }\mu\text{m}$ suggests that entering the axial
proximity of a target along an ER tube confers a high likelihood that the
searcher will encounter the target, for targets separated by more than
approximately a micrometer. Figures 2D, E, and F indicate the mean target
encounter times are $\lesssim 10^{-1}\text{ s}$ for a searcher diffusivity
$D=1\text{ }\mu\text{m}^{2}/\text{s}$, and would require very long tubes (or
corresponding low density targets) for encounter times to reach substantially
longer than a second.

In Fig. 2, we use specific ranges for target radius, tube radius, and tube
length, and explore similar length scales for these quantities throughout this
work. Protein radii are approximately 1 – 5 nm for 5 – 500 kDa proteins [35].
The necks of ER exit sites, which allow entry into the exit site structure,
have been observed to be somewhat narrower than 20 – 30 nm radius ER tubes
[22], suggesting that ER exit site necks are 10 – 20 nm in radius. Calcium ion
pores, enabling transport between the ER lumen and the cytosol, are
approximately 1 nm in diameter, but the ER lumen side of the membrane is
surrounded by a charged region to facilitate ion capture [36, 11], suggesting
a target size of one to several nanometers. The target radius range we
explore, from 2 – 10 nm, covers much of size range of these targets in the ER.

Yeast ER tubes have a typical radius range of 10 – 25 nm [16] and mammalian ER
tubes have a typical radius range of 25 – 70 nm [18]), with cases as low as 14
nm [37]. The tube radius range we explore, from 10 – 50 nm, covers much of the
observed range in ER tube radii.

Tube lengths represent the distance between targets. There are approximately
$2\times 10^{7}$ BiP chaperones per mammalian cell [38], largely expected to
be confined to the ER, with the overall ER membrane area of $10^{4}\text{
}\mu\text{m}^{2}$ [39] — this corresponds to (with a typical 50 nm mammalian
ER tube radius [18]) approximately one BiP per 10 nm of tube length. At
various levels of unfolded protein stress in the ER [21], unfolded proteins
will occupy BiP [38, 40], so the expected typical spacing between free BiP is
10 nm or larger. IRE1 proteins are present on the ER membrane with
approximately 1/$\mu\text{m}^{2}$ density [39], which corresponds to a density
of approximately one IRE1 every 3 - 10 $\mu\text{m}$ of ER tube length, using
typical yeast tube radius of 18 nm [16] and mammalian tube radius of 50 nm
[18]. Calcium channel densities are estimated to be one per 50 nm or more of
tube length [11]. ER tubes are typically 250 – 750 nm long in yeast cells [16]
and 0.5 – 5 $\mu$m long in mammalian cells [13, 17]. The tube length range we
explore, from a few tens of nanometers to a few micrometers, covers much of
the spacing range for ER targets and the range of ER tube lengths.

###  II.2 Escape from a target

Figure 3: Tube escape with a stationary target. Mean escape time with varying
(A) target size, (B) tube radius, and (C) tube length. (D) shows distribution
of escape times for different tube radii with a target radius
$R_{\text{target}}=10\text{ nm}$ and tube length $L_{\text{tube}}=100\text{
nm}$. For all panels, the legend indicates whether the searcher is on the tube
surface (solid curves) or in the tube volume (dashed), and the legend of (C)
further indicates the expected one-dimensional interval escape from Eq. 1. For
(A) and (B) tube length is indicated by curve color and adjacent corresponding
labels. For (C) tube radius is indicated by curve color and whether a target
is present by line thickness, as well as with adjacent corresponding labels.
For (D) tube radius is indicated by curve color and adjacent corresponding
labels. (A) uses tube radius $R_{\text{tube}}=30\text{ nm}$, and (B) and (C)
target radius $R_{\text{target}}=10\text{ nm}$. For all panels searcher
diffusivity $D=1\text{ }\mu\text{m}^{2}/\text{s}$ with $10^{4}$ samples,
except the escape times without targets in (C) are for 2500 samples.

We now examine the trajectories of searchers that escape from the ends of the
tube. The escape probability $P_{\text{escape}}=1-P_{\text{encounter}}$, with
target encounter probability $P_{\text{encounter}}$ examined in Fig. 2A,B,C.
Figures 3A and B show that target radius and tube radius, respectively, have
limited effect on the mean time to escape the tube without encountering the
target. While limited, the effect of target radius and tube radius on escape
time is not zero, with larger target radius marginally decreasing the mean
escape time (Fig. 3A) and larger tube radius marginally increasing the mean
escape time (Fig. 3B). In contrast, Figs. 3A, B, and C show that tube length
strongly affects the mean escape time, with longer tubes requiring a longer
time to escape.

The mean time $\langle t_{\text{1d}}\rangle$ for a particle of diffusivity
$D$, initially a distance $x$ from one end of a one-dimensional interval of
length $d$, to first reach one of the two interval ends is [41]

$\langle t_{\text{1d}}\rangle=\frac{x(d-x)}{2D}\ .$ (1)

For the scenario under consideration, the interval length $d=2L_{\text{tube}}$
and $x=L_{\text{tube}}\pm R_{\text{target}}$. As a searcher in the ER tube is
able to escape from the tube ends it is expected that the tube length will
largely control the mean time to escape. In the case where there is no target,
the searcher will escape with a mean time described by Eq. 1 (thick orange and
green curves in Fig. 3C).

However, with a target present in the tube, the trajectories that escape are
those that do not first encounter the target. Figure 3C shows that larger tube
radii and shorter tube lengths more closely match the mean escape time without
a target of Eq. 1. Decreasing the tube radius or increasing the tube length
causes the mean escape time to decrease below $\langle t_{\text{1d}}\rangle$
from Eq. 1. Narrower and longer tubes increase the probability that the
searcher will encounter the target before escaping (Fig. 2A,B), removing from
the ensemble of escape trajectories those trajectories that would have escaped
from the tube after a relatively long time period, reducing the mean time to
escape. The probability distribution of escape times for various tube radii,
shown in Fig. 3D, illustrates how target encounter removes escape trajectories
as the tube radius decreases, with the probability of longer escape times
decreasing more with decreasing tube radius than the probability of shorter
escape times.

The escape time from the tube is largely controlled by the tube length.
However, increasing target radius or decreasing tube radius increases the
target encounter probability and preferentially removes longer escape
trajectories, marginally reducing the mean escape time.

###  II.3 Mobile targets

Figure 4:  Comparison of stationary vs mobile targets. Target encounter
probability for search (A) on the tube surface and (B) in the tube volume with
varying tube length. Mean target encounter time for search (C) on the tube
surface and (D) in the tube volume with varying tube length. Legends indicate
if target is stationary (solid curves) or mobile (dashed). For all panels,
target size $R_{\text{target}}=10\text{ nm}$, searcher diffusivity $D=1\text{
}\mu\text{m}^{2}/\text{s}$ with $10^{4}$ samples.

Thus far we have examined mobile searchers and stationary targets. We will now
explore the scenario where both searcher and target are mobile, effectively
turning the target into a searcher as well. The initial condition is similar
to search with a stationary target, with both searcher and target separated
along the tube axis by a distance $R_{\text{target}}$, and each searcher
independently and uniformly distributed through the tube cross section
(searchers in the volume) or around the tube circumference (searchers on the
surface). The encounter condition is unchanged, such that the two searchers
are considered to have encountered one another if they are within a distance
$R_{\text{target}}$. This initial condition represents the closest the two
searchers can approach along the tube axis without a possible encounter.
Rather than fixed axial positions beyond which searchers are considered to
have escaped, the escape condition is that the two searchers have become
separated along the tube axis by the tube length $L_{\text{tube}}$, equivalent
to the axial distance to the escape threshold with a stationary target.

Target encounter probabilities and mean times are compared for stationary and
mobile targets in Fig. 4. For search on the tube surface, the target encounter
probability for mobile targets is unchanged compared to stationary targets
(Fig. 4A). However, for search in the volume, mobile targets have an increased
the target encounter probability relative to stationary targets (Fig. 4B).
Although the target encounter probability for search on the tube surface is
unchanged between stationary and mobile targets (Fig. 4A), the mean time to
encounter the target decreases for mobile targets relative to stationary
targets for search on the tube surface (Fig. 4C). For search in the tube
volume, mobile targets also exhibit shorter target encounter times than
stationary targets (Fig. 4D).

For searchers and targets on the surface, changing from a stationary to a
mobile target affects the encounter time but not the encounter probability.
Whether mobile or stationary, a surface target covers the same surface area
and fraction of the tube circumference. Two mobile searchers on the surface
can instead be considered as a fixed target and a searcher with twice its
initial diffusivity, without loss of generality. In this case, the encounter
probability will be unchanged from the initial scenario with a mobile searcher
and a stationary target on the tube surface (Fig. 4A), but the target
encounter time will be halved (Fig. 4C).

In contrast to search on the surface, changing from stationary to mobile
targets for search on the volume affects both the encounter probability and
encounter time (Fig. 4B,D). Stationary volume targets were positioned on the
tube wall (with fixation to the tube wall enabling the target to remain
stationary). Mobile volume targets can explore the tube volume, effectively
increasing the size of the target by increasing the available volume within
$R_{\text{target}}$ of the target, causing the target encounter probability to
be higher for mobile targets than stationary targets (Fig. 4B). Similar to
search on the surface, an effectively doubled diffusivity of the relative
searcher positions for a mobile target compared to a stationary target leads
to a reduction in the target encounter time for searchers in the volume (Fig.
4D). This is a similar to previous target search results on one-dimensional
models of ER networks, where the mobile targets reduce their encounter time by
more than half compared to stationary targets [29].

###  II.4 Comparing searcher and targets in the tube volume and on the tube
surface

Figure 5: Comparison of searchers both on tube surface, in tube volume, or one
on the surface and one in the volume. (A) Target encounter probability and (B)
mean target encounter time as tube length is varied. Line color and adjacent
corresponding labels indicate tube length. (C) Target encounter probability
and (D) mean target encounter time as tube radius is varied. Line color and
adjacent corresponding labels indicate tube radius. For panels (A) - (D)
target size $R_{\text{target}}=10\text{ nm}$ and diffusivity $D=1\text{
}\mu\text{m}^{2}/\text{s}$, with $10^{4}$ samples. (E) shows fraction of tube
covered by a target with $R_{\text{target}}=10\text{ nm}$. For search entirely
on the surface, this is the fraction of the circumference covered by a target.
For search entirely in the volume, this is the fraction of the tube cross
section covered by a target in the volume that does not overlap the tube
circumference. For one searcher on the surface and one searcher in the volume,
this is the fraction of the tube cross section covered by a target on the tube
surface. For all panels legend indicates whether searchers are both on the
surface (solid curves), both in the volume (dashed), or one searcher is in the
volume and the other is on the surface (dotted).

We have examined search with mobile searchers either both in the tube lumen
volume or both on the membrane surface. We will now directly compare these two
types of searches as well as a search with one searcher in the tube lumen and
the other on the membrane surface. Figure 5A shows that the target encounter
probability increases with tube length across search types, as for earlier
searches. For all tube radii the highest encounter probability is for search
occurring entirely in the lumen volume. For large tube radii the next-to-
highest encounter probability is for search occurring entirely on the membrane
surface, and the lowest encounter probability is for one searcher in the tube
lumen and the other on the membrane surface. However, for smaller tube radii,
the encounter probability for search entirely on the membrane surface is lower
than for one searcher in the volume and the other on the surface. A
corresponding pattern (low encounter probability corresponding to longer
encounter time) is seen in Fig. 5B, with larger tube radii exhibiting the
longest encounter times for one searcher in the tube lumen and the next-
longest encounter times for search on the membrane surface, and smaller tube
radii with longest encounter times for both searchers on the surface and next-
to-longest encounter times for one searcher in the volume and the other on the
surface. Across radii, the shortest encounter times are for both searchers in
the volume.

The change in the relative encounter probabilities and times between search
entirely on the surface and search with one searcher on the surface and the
other in the volume for different tube radii is examined in Figs. 5C and D.
Figure 5C shows that the target encounter probability changes more with tube
radius for one searcher each in both the volume and surface compared to both
searchers on the surface, allowing the encounter probability for one searcher
each in the volume and on the surface to change from a higher probability than
both searchers on the surface for small tube radius to a lower probability at
large tube radius. There is a similar pattern for encounter time (Fig. 5D), as
one searcher on the surface and one on the volume changes from a shorter
encounter time than both searchers on the surface at small radius to a longer
encounter time for larger tube radii.

Adjusting the tube radius leads to more substantial changes in encounter
probability and time for one searcher each in the volume and on the surface
compared to both searchers on the surface. This is consistent with our earlier
results where tube geometry has less impact on search on the tube surface
compared to search in the tube volume. A change in the relative encounter
probabilities and times with tube size for these two search types aligns with
the characteristics of these two search types. Encounters for searchers
entirely on the surface are mediated by distance along the tube surface, while
encounters for one searcher in the volume and one searcher on the membrane are
mediated by distance through the volume. For wide tubes, the fraction of the
circumference covered by a target (for search entirely on the surface) is
larger than the fraction of the cross section covered by the target (for one
searcher on the surface and one searcher in volume). As the tube becomes
narrow, this order switches, and more cross section is covered compared to
circumference (Fig. 5E). Thus, as the tube narrows, there is less cross-
section available for a searcher in the volume to avoid its search partner on
the surface, compared to both searchers on the surface.

###  II.5 Finite reaction rate

Figure 6: Finite reaction rates. (A) Target encounter probability and (B) mean
target encounter time as the searcher reaction rate is varied. Legend
indicates type of search (on surface, in volume, or one searcher in each; and
whether target is stationary or mobile). Tube radius $R_{\text{tube}}=30\text{
nm}$, target radius $R_{\text{target}}=10\text{ nm}$, tube length
$L_{\text{tube}}=10^{2.5}$ nm, and diffusivity of searcher and mobile targets
$D=1\text{ }\mu\text{m}^{2}/\text{s}$. Most data points are averaged over
$10^{4}$ trajectories with $2\times 10^{4}$ samples for reaction rates of
$10^{1.5}\text{ s}^{-1}$ and $10^{2}\text{ s}^{-1}$ due to high escape
probability.

Thus far we have considered the case of diffusion-limited encounters,
corresponding to reactions with infinite reaction rate that undergo instant
encounter of a searcher with a target once the searcher is within a distance
$R_{\text{target}}$. However, the reaction rate may not be effectively
infinite [42, 43], and could depend on alignment of protein domains,
availability of a pore, or other factors [44]. We now consider the case where
encounters occur with a finite reaction rate once the searcher is within a
distance $R_{\text{target}}$ of the target.

Figure 6A shows that for sufficiently low reaction rates ($\lesssim
10^{2}\text{ s}^{-1}$) the target encounter probability is near zero. The
target encounter probability increases with the reaction rate until plateauing
above a reaction rate of $10^{5}\text{ s}^{-1}$ – $10^{6}\text{ s}^{-1}$. This
change in the target encounter probability as the reaction rate increases
suggests that sufficiently low reaction rates lead to minimal target encounter
and sufficiently high reaction rates nearly guarantee a reaction once the
target diffuses within a distance $R_{\text{target}}$.

For a diffusive process the mean squared displacement in $n$ dimensions is
$\langle r^{2}\rangle=2nDt$, where $t$ is time. Rearranging relates time to
mean squared displacement $\langle r^{2}\rangle$ of $t=\langle
r^{2}\rangle/(2nD)$. With a displacement representing the target reaction
range $R_{\text{target}}=10\text{ nm}$ and $D=1\text{
}\mu\text{m}^{2}/\text{s}$, this time is $t=5\times 10^{-5}\text{ s}/n$, which
is $2.5\times 10^{-5}\text{ s}$ for $n=2$ surface search and $1.67\times
10^{-5}\text{ s}$ for $n=3$ volume search. While this estimate is not a
precise calculation of the time spent by a diffusive trajectory within a
distance $r$ of a target, it does provide an order of magnitude-level estimate
of the time spent by the searcher near the target. This dwell time estimate
predicts reaction rates of approximately $4\times 10^{4}\text{ s}^{-1}$ for
surface search and $6\times 10^{4}\text{ s}^{-1}$ for volume search for the
beginning of saturation in encounter probability, which approximately aligns
with the reaction rates $10^{5}\text{ s}^{-1}$ – $10^{6}\text{ s}^{-1}$ at
which the encounter probability plateaus in Fig. 6A.

As the target encounter probability increases from nearly zero to approaching
a plateau as the reaction rate increases over the range shown in Fig. 6A, the
mean target encounter time in Fig. 6B decreases by less than an order of
magnitude. The searcher explores the tube until the searcher escapes from the
tube end or reacts with the target. At high reaction rates and for stationary
targets in Fig. 6B, the encounter times agree with the encounter times for
infinite reaction rate from Fig. 2F. At lower reaction rates, particularly
those below $10^{5}\text{ s}^{-1}$ – $10^{6}\text{ s}^{-1}$ above which the
encounter probability plateaus, the target encounter time modestly increases
above the encounter time for infinite reaction rate. Across reaction rates,
the finite-reaction rate search-and-reaction process is competing with escape.
We thus expect the mean target encounter time to not significantly exceed the
mean escape time even for quite low reaction rates, but instead for the target
encounter probability to decrease. The diffusion-limited escape time from Fig.
3B corresponding to parameters and stationary target case of Fig. 6 is
approximately $2\times 10^{-2}\text{ s}$, such that the low reaction rates of
Fig. 6B modestly increase from the encounter time for the infinite reaction
rate.

###  II.6 Target encounter without escape

Figure 7: Target encounter without tube escape. (A) Survival probability
(probability that a searcher has not encountered the target) vs time for a
stationary target on the tube surface. Curve labels indicate tube length.
Noting logarithmic axis for survival probability $S(t)$, straight lines are
exponential fits $S(t)=e^{-kt}$, with $k$ the fitted encounter rate. (B) Black
points are the fitted encounter rates $k$ from (A). Dashed curve is inverse
tube length squared $1/L_{\text{tube}}^{2}$ scaling. For both panels target
radius $R_{\text{target}}=10\text{ nm}$, tube radius $R_{\text{tube}}=10\text{
nm}$, and searcher diffusivity $D=1\text{ }\mu\text{m}/\text{s}$ with 2500
sampled encounters.

Thus far we have examined a scenario where a searcher either encounters a
target inside a tube or escapes from the tube. We now briefly explore the rate
at which a searcher encounters a target when escape is not permitted (i.e.,
tube ends are reflecting). An equivalent scenario is that reaction with any
target is considered rather than one specific target, and if a searcher
diffuses sufficiently far along the tube axis from one target, it will pass
another target rather than escape. We examine a stationary target on the tube
surface with a searcher beginning at a uniformly-distributed random position
in the tube beyond a distance $R_{\text{target}}$ from the target. Fig. 7A
shows the time evolution of the survival probability, or the probability that
a searcher has not yet found a target. The survival probability appears
approximately exponential, $P_{\text{survival}}(t)\simeq\exp(-kt)$, with
target encounter rate $k$. Figure 7A shows that the target encounter rate $k$
decreases with tube length, with Fig. 7B showing the rate dependence on tube
length is consistent with $k\sim L_{\text{tube}}^{-2}$. This dependence is
similar to other diffusive search timescales that depend on the square of a
lengthscale [10, 41, 45].

##  III Discussion

Proteins, ions, and other molecules diffuse in the endoplasmic reticulum (ER)
searching for targets such as protein binding partners and exit sites. Much of
the ER is composed of a network of connected tubes that extend through the
cell volume. Using Brownian dynamics simulations, we explore how the geometry
of diffusive search in a tube, particularly the distance between potential
targets along the tube, affects the success and time period for the target
search process.

Previous work on diffusive search in the ER network has focused on the
organelle-scale geometry of the ER, considering how connectivity via ‘loops’
[28], searcher maturation and source and target distribution [5, 29],
individual network edges [30], and network rearrangement dynamics [31] affect
diffusive target search; or how local tube geometry affects target search via
target density and tube geometry [11], how tube radius changes along the tube
axis affect diffusivity [24, 25, 5], and control of mean first-passage times
in cylindrical geometries [27]. Our work focuses on the local geometry while
connecting to larger scales by allowing searchers to escape the tube region
under consideration without a target encounter.

We consider a scenario where a searcher that has nearly reached the axial
position of a target in a tube either encounters the target or escapes from
the target by reaching one of the two tube ends. As expected, larger targets
and narrower tubes lead to a higher probability of finding the target and a
shorter mean search time. Longer tubes, despite expanding the tube region in
which to search, increase both the probability of finding the target and the
mean search time because escape time increases with tube length, providing a
longer time period for the searcher to find the target. Tube length more
strongly affects encounter probability and time in comparison to tube radius
and target sizes, when varied across a physiological range.

This finding has implications for simulations and calculations that treat the
ER network as one-dimensional, as well as for experimental approaches that
measure the searcher position along the length of ER network tubes. For
targets that are sparsely distributed along the length of ER tubes, a specific
target is likely to be encountered on the tube surface or in the volume if the
target axial position along the tube is reached by the searcher. Conversely,
for targets that are densely spaced along the length of ER tubes, a specific
target is unlikely to be encountered following arrival at the target axial
position, as the searcher is likely to come into close axial proximity of
another target before encountering the first target. Thus simulations and
experimental measurements that assume target encounter once the axial target
position is reached are more accurate for widely-spaced targets, and less
accurate for densely-spaced targets. For yeast cells (tubes with length 250 –
750 nm and radius 10 – 25 nm [16]) spacing between targets of a few hundred
nanometers, corresponding to a single target per tube, provides a probability
of target encounter exceeding approximately $75\%$. For mammalian cells (tubes
with length 0.5 – 5 $\mu$m and radius 25 – 70 nm [16]) spacing between targets
of $\gtrsim 1\text{ }\mu\text{m}$, corresponding to one to a few targets per
tube, provides a probability of target encounter exceeding approximately
$75\%$.

We found that the timescale to encounter a target in three dimensions once the
target position is reached along the tube axis is less than one second for a
diffusivity $D=1\text{ }\mu\text{m}^{2}/\text{s}$. Previous calculations
suggested that the timescale to diffusively find a particular tube axis
position in a mammalian ER network is hundreds of seconds or more for a
searcher with diffusivity $D=1\text{ }\mu\text{m}^{2}/\text{s}$ [28]. The time
period until target encounter following arrival at the target axial position
in the ER network is likely more than an order of magnitude shorter than the
search for the axial position in the network. This suggests that if target
encounter occurs, the time of target encounter will closely follow arrival at
the axial position of the target in a simulation or during an experimental
measurement.

As targets become more dense along the tube length, a searcher starting near
one target is increasingly likely to diffuse into the proximity of another
target rather than encounter the initial target. This is an example of
diminishing returns in diffusive target capture, similar to limited increase
in flux through higher density calcium channels [11] and absorbing patches on
cell surfaces [10].

The probability of escape from the tube ends, as the alternative to target
encounter, decreases with tube length and target radius and increases with
tube radius. We find that target and tube radii have limited effect on escape
time. Mean escape time is largely controlled by tube length, similar to escape
on a one-dimensional interval [41]. Encounter times cannot become too much
slower than escape times, as they represent competing processes, such that
mean encounter times will be similar to mean escape times. While wider tubes
and smaller targets closely follow the predicted mean escape time for a one-
dimensional interval, narrower tubes and larger targets decrease the mean
escape time. This is because as the tube is narrowed, longer escape
trajectories are preferentially converted to target encounter trajectories.

Recent work explores similar diffusive search problems in cylindrical geometry
similar to our investigation. Grebenkov and Skvortsov [46] calculate mean
first-passage times to small targets in elongated domains, but do not consider
the possibility of escape. Holcman and Schuss [27] consider the mean first-
passage time for target encounter vs escape in a cylindrical geometry for
searchers beginning at specific positions. Compared to these previous studies,
we have further considered the possibility of escape, a distributed initial
searcher position, targets in the tube volume, and mobile targets.

We explore search occurring on the tube surface, in the tube volume, and both
on the surface and in the volume. We find that search in the volume is more
sensitive to the tube and target geometry, consistent with previous work
showing that diffusive search in two dimensions depends less on search
geometry than search in three dimensions [32, 34, 5]. Across the physiological
parameter ranges we have explored, we find that search in the volume is
typically (but not always) more likely and more quickly to lead to target
encounter. For search on the surface, changing from stationary to mobile
targets leaves encounter probabilities unchanged but decreases encounter times
by a factor of two, while changing from stationary to mobile targets for
search in the volume both increases the target encounter probability and
decreases the encounter times. For equivalent parameters, having both
searchers in the volume is more likely to lead to target encounter and has
more rapid target encounter than having both searchers on the surface, which
in turn has more likely and more rapid target encounter than one searcher in
the volume and the other on the surface. Small tube radii is an exception,
reversing the relative encounter probabilities and times between both
searchers on the surface and one each in the volume and on the surface.

Much of our results assume infinite reaction rate, such that any contact
between searcher and target leads to a target encounter. We find that
adjusting a finite reaction rate causes encounter probabilities that approach
zero for reaction rates $\lesssim 10^{2}/\text{s}$ with searcher diffusivity
of 1 $\mu$m2/s and encounter probabilities that plateau at nearly the
infinite-reaction-rate probability for $\gtrsim 10^{6}/\text{s}$. While the
encounter probability substantially changes with reaction rate, the mean
target encounter time is less affected over a wide reaction rate range, due to
competition between between target encounter and tube escape. Thus while
reaction rate will significantly impact whether a target is found, the
reaction rate will only have a limited effect on the typical time for
successful searchers to find a target before diffusing to the proximity of
another target.

Most of our results consider the scenario where a searcher may either
encounter a target or escape from the tube ends. We also found the rate at
which the searcher would find a target when confined to a tube region (i.e.,
the searcher cannot escape the tube) has scaling consistent with the inverse
square of tube length, similar to scaling found in previous work for diffusive
search in confined geometry [10, 41, 45]. Exponential dependence of particle
survival with time for a wide range of tube lengths suggests that even for
widely-spaced targets, the encounter of a searcher with a target may be
considered as a Poisson process (with geometry-dependent rate) for appropriate
initial conditions and target spacing distribution.

Using Brownian dynamics simulations in a cylindrical geometry representing ER
tubes, we have described how tube geometry and search type affect the
probability of a diffusive searcher to find a target and the time to find the
target, notably showing that searchers are more likely to find a target vs
escape in longer tubes, which represent widely-spaced targets. These findings
have implications for designing simulations and interpreting experimental
measurements.

###### Acknowledgements.

This work was supported by a Natural Sciences and Engineering Research Council
of Canada (NSERC) Discovery Grant (A.I.B.) and by start-up funds provided by
the Toronto Metropolitan University Faculty of Science (A.I.B.), and was
enabled by computational resources provided by the Digital Research Alliance
of Canada (alliancecan.ca), including a cluster usage resource allocation
(A.I.B.). The authors thank Sean Cornelius and Eric De Giuli (Toronto
Metropolitan University) for useful discussions and feedback.

##  Appendix A Further method details

Figure 8: Particle reflection. When a three-dimensional trajectory exits the
tube, it is reflected back into the tube volume. The particle, initially at
the blue dot, takes a step (yellow arrow) to a final point (red dot) outside
the tube. A tangent line (red line) is found at the point at which the
trajectory exited the tube (black circle) and the final point outside the
circle (red dot) is moved to perpendicular to the tangent line to a reflected
position (green dot) equidistant to but on the other side of the tangent line.

Stochastic particle trajectories are generated with Brownian dynamics: each
timestep $\Delta t$ the particle moves a distance randomly sampled from the
absolute value of a normal distribution of mean zero and variance
$\sigma^{2}=4D\Delta t$ for diffusion on the tube surface and
$\sigma^{2}=6D\Delta t$ in the tube volume, and direction also randomly and
uniformly sampled, with $\Delta t=10^{-2}\text{ s}$. Diffusion on the tube
surface is two dimensional with periodic boundary conditions in one direction,
and in the tube volume diffusion is three dimensional. We model the ER tube as
a cylinder. For three-dimensional diffusion in the tube volume, if a Brownian
dynamics step yields a new particle position outside of the cylinder, the
trajectory vector from the initial particle position inside the cylinder to
the new particle position outside of the cylinder is reflected from the
cylinder wall to remain inside the cylinder. Reflection uses a ‘midpoint
method’ where the point the trajectory exits the tube cross-section is found,
the line tangent to the circle at that exit point defined, a line
perpendicular to the tangent and through the (forbidden) trajectory point
defined, and the reflected trajectory point inside the tube being on this
perpendicular line the same distance from the tangent line as the forbidden
point, but on the opposite side of the tangent line (see Fig. 8).

For Figures 2 to 6, particles begin each simulation at an axial position at
the edge of the target. For two-dimensional search entirely on the surface,
one particle is set at an angular position defined as zero, while the initial
angular position of the other particle is selected randomly and uniformly
around the tube circumference. For three-dimensional search entirely in the
volume with a stationary target, the mobile searcher randomly and uniformly
samples from the tube cross-section. For three-dimensional search entirely in
the volume with a mobile target, both the searchers randomly and uniformly
sample from the tube cross-section. For one searcher on the tube surface and
the other in the tube volume, the searcher on the surface begins at an angular
position defined as zero and the searcher in the volume randomly and uniformly
samples from the tube cross-section.

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to compute the perpendicular diffusion coefficient $\mathcal{D}_{\perp}$.

  5. 5.

The presence of a linear instability in global AdS Kerr begs the question of
what is the dual field theoretic interpretation to this instability?

  6. 6.

An implementation of rotation, polarization and spin in hydrodynamic codes is
desirable for various applications ranging from modelling heavy-ion-collisions
[91] over modelling astrophysical settings with relevance to BH mergers or
neutron star mergers [2, 3], and fluids in the context of condensed matter
physics, e.g. [92, 93, 94, 95]. The holographic fluid presented in this
article can serve as a testing ground for such hydrodynamic codes.

  7. 7.

It is possible to systematically construct a version of hydrodynamics around a
rotating equilibrium state. This state breaks rotation invariance and thus
involves a new vector, the unit vector pointing in the symmetry direction,
similar to anisotropic hydrodynamics [96, 97, 98]). The fluid/gravity
correspondence [99] can be used to systematically derive and test such a
description. It can be used to extend the rotating black hole solutions (20)
into metric solutions which have temperature and angular momentum depend on
the transverse coordinates and time. This leads to a gradient expansion in the
dual field theory, locally correcting the rotating equilibrium state by
gradients of temperature and angular momentum, likely modifying the
hydrodynamic dispersion relations significantly.

It is worthwhile pointing out the logical possibility that the quark-gluon-
plasma generated in heavy-ion-collisions is not carrying any significant
vorticity or angular momentum. Note that a recent numerical study of a
holographic fluid created in the collision of localized, lumpy shocks
(modeling a granular nuclear structure) finds that most of the angular
momentum after the collision resides in the periphery while only a small
fraction of the total angular momentum is present in the region representing
the holographic QGP [100]. If the same should be true for the QCD QGP
generated at RHIC and LHC, then the hyperon polarization can not be explained
by any arguments involving any significant amount of vorticity or angular
momentum in the QGP. In that case, the hyperon polarization data may be
considered an open question lacking theoretical understanding.

## Acknowledgements

C.C., M.A. and M.K. acknowledge the excellent discussions and collaborations
with Jorge Noronha, Enrico Speranza, Mike Blake and Anthony Thompson, on which
part of this work is based. This research was supported in part by the U.S.
Department of Energy under Grant No. DE-SC0012447 and by the National Science
Foundation under Grant No. NSF PHY-1748958. C.C. was supported in part, by the
Netherlands Organisation for Scientific Research (NWO) under the VICI grant
VI.C.202.104. M.K. thanks the Kavli Institute for Theoretical Physics, Santa
Barbara for hospitality while part of this work was completed.

##  Appendix A QNM Data

For convenience we include the six modes closest to the line $\Im(v)=0$ in the
three sectors discussed in section III.1 for $a/\ell=0,1/2/9/10$. In each
table we fix a value of $\mathcal{J}=0,1/2,1$ and $r_{+}=10$, the remaining
data can be found in the supplementary material.

###  A.1 Tensor sector

n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | 1.57792 | -1.36943 | 1.83839 | -1.73479 | -0.214354 | -0.529707
2 | -1.57792 | -1.36943 | -1.63407 | -1.73893 | -0.223526 | -1.00114
3 | 2.60488 | -2.37904 | -0.471547 | -1.94296 | -0.231857 | -1.4879
4 | -2.60488 | -2.37904 | -2.90868 | -2.98463 | -0.239585 | -1.98766
5 | -3.61756 | -3.38252 | 3.08986 | -2.98592 | -0.246234 | -2.49843
6 | 3.61756 | -3.38252 | -4.14395 | -4.21689 | 2.84375 | -2.73946
Table 1:  $\mathcal{J}=0$ tensor modes. $n$ indicates the $n$th furthest mode from the $\Im(\nu)=0$ axis.  n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | -1.58609 | -1.36696 | 1.86324 | -1.73367 | -0.267892 | -0.52861
2 | 1.58609 | -1.36696 | -1.6031 | -1.74128 | -0.279378 | -1.00015
3 | 2.6107 | -2.3774 | -0.594678 | -1.92682 | -0.289814 | -1.48695
4 | -2.6107 | -2.3774 | -2.88713 | -2.98347 | -0.299508 | -1.98678
5 | 3.62224 | -3.38124 | 3.1134 | -2.98514 | -0.307855 | -2.49776
6 | -3.62224 | -3.38124 | -4.12207 | -4.21626 | -2.62293 | -2.73856
Table 2:  $\mathcal{J}=1/2$ tensor modes. $n$ indicates the $n$th furthest mode from the $\Im(\nu)=0$ axis.  n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | -1.5965 | -1.36383 | 1.88859 | -1.73202 | -0.321408 | -0.527454
2 | 1.5965 | -1.36383 | -1.56789 | -1.74515 | -0.335217 | -0.999071
3 | 2.61814 | -2.37531 | -0.722979 | -1.90682 | -0.347771 | -1.48588
4 | -2.61814 | -2.37531 | -2.86612 | -2.98187 | -0.359452 | -1.9858
5 | -3.62822 | -3.37959 | 3.13731 | -2.984 | -0.369518 | -2.49702
6 | 3.62822 | -3.37959 | -4.10043 | -4.21535 | -2.59952 | -2.73727
Table 3:  $\mathcal{J}=1$ tensor modes. $n$ indicates the $n$th furthest mode
from the $\Im(\nu)=0$ axis.

###  A.2 Vector sector

n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | 1.57792 | -1.36943 | 1.83839 | -1.73479 | -0.214354 | -0.529707
2 | -1.57792 | -1.36943 | -1.63407 | -1.73893 | -0.223526 | -1.00114
3 | 2.60488 | -2.37904 | -0.471547 | -1.94296 | -0.231857 | -1.4879
4 | -2.60488 | -2.37904 | -2.90868 | -2.98463 | -0.239585 | -1.98766
5 | -3.61756 | -3.38252 | 3.08986 | -2.98592 | -0.246234 | -2.49843
6 | 3.61756 | -3.38252 | -4.14395 | -4.21689 | 2.84375 | -2.73946
Table 4:  $\mathcal{J}=0$ vector modes. $n$ indicates the $n$th furthest mode from the $\Im(\nu)=0$ axis.  n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | -9.77489$\times 10^{-85}$ | -0.00626198 | -0.0252699 | -0.00514204 | -0.045346 | -0.00226457
2 | -1.58609 | -1.36696 | -1.46214 | -1.42053 | -0.160298 | -0.532138
3 | 1.58609 | -1.36696 | 1.59728 | -1.4975 | -0.241578 | -0.716389
4 | -1.57145 | -1.37138 | 1.81681 | -1.7323 | -0.167453 | -1.00312
5 | 1.57145 | -1.37138 | -1.66563 | -1.73445 | -0.202738 | -1.2898
6 | -2.6107 | -2.3774 | -0.350175 | -1.96503 | -0.173784 | -1.48965
Table 5:  $\mathcal{J}=1/2$ vector modes. $n$ indicates the $n$th furthest mode from the $\Im(\nu)=0$ axis.  n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | -6.3260$\times 10^{-94}$ | -0.0150693 | -0.0508062 | -0.0119096 | -0.090775 | -0.00472503
2 | -1.5965 | -1.36383 | -1.45984 | -1.4059 | -0.213379 | -0.532508
3 | 1.5965 | -1.36383 | 1.6058 | -1.49884 | -0.297615 | -0.713923
4 | -1.57556 | -1.37012 | 1.84404 | -1.729 | -0.223126 | -1.00332
5 | 1.57556 | -1.37012 | -1.64015 | -1.73249 | -0.262902 | -1.28889
6 | 2.61814 | -2.37531 | -0.468888 | -1.96172 | -0.23164 | -1.48972
Table 6:  $\mathcal{J}=1$ vector modes. $n$ indicates the $n$th furthest mode
from the $\Im(\nu)=0$ axis.

###  A.3 Scalar sector

n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | -1.5779 | -1.3694 | -1.4252 | -1.3411 | -0.0010668 | -0.55268
2 | 1.5779 | -1.3694 | 1.4252 | -1.3411 | -0.074684 | -0.75314
3 | -1.5685 | -1.3723 | -1.4809 | -1.4504 | 0.0 | -0.86704
4 | 1.5685 | -1.3723 | 1.5765 | -1.4824 | 0.0 | -0.86704
5 | -1.5661 | -1.3730 | -1.7330 | -1.7340 | -0.00047209 | -1.0449
6 | 1.5661 | -1.3730 | 1.7332 | -1.7366 | -1.2191 | -1.2125
Table 7:  $\mathcal{J}=0$ scalar modes. $n$ indicates the $n$th furthest mode from the $\Im(\nu)=0$ axis.  n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | 0.0 | -0.0062620 | -0.011386 | -0.0060167 | -0.036860 | -0.0035422
2 | -1.5861 | -1.3670 | -1.4396 | -1.3341 | -0.053926 | -0.55468
3 | 1.5861 | -1.3670 | 1.4124 | -1.3472 | -0.13133 | -0.75273
4 | -1.5714 | -1.3714 | -1.4762 | -1.4403 | -0.045112 | -0.86704
5 | 1.5714 | -1.3714 | 1.5848 | -1.4869 | -0.045112 | -0.86704
6 | -1.5673 | -1.3726 | -1.7121 | -1.7305 | -0.045112 | -0.86704
Table 8:  $\mathcal{J}=1/2$ scalar modes. $n$ indicates the $n$th furthest mode from the $\Im(\nu)=0$ axis.  n | $a/\ell=0$ | $a/\ell=1/2$ | $a/\ell=9/10$
---|---|---|---
| $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$ | $\Re(v)$ | $\Im(v)$
1 | -0.081988 | -0.0041527 | -0.095984 | -0.0014583 | -0.099872 | -0.00014121
2 | 0.081988 | -0.0041527 | 0.051356 | -0.0070246 | 0.010573 | -0.0044887
3 | 0.0 | -0.015069 | -0.028210 | -0.014144 | -0.074679 | -0.0080336
4 | -1.5965 | -1.3638 | -1.4545 | -1.3266 | -0.10681 | -0.55665
5 | 1.5965 | -1.3638 | 1.3998 | -1.3526 | -0.18799 | -0.75219
6 | -1.5756 | -1.3701 | -1.4733 | -1.4284 | -0.090225 | -0.86704
Table 9:  $\mathcal{J}=1$ scalar modes. $n$ indicates the $n$th furthest mode
from the $\Im(\nu)=0$ axis.

##  Appendix B Numerical methods

In general, the linearized equations of motion of the fluctuations give rise
to what is known as a non-linear eigenvalue problem that takes the form

$\left(\sum_{n=0}^{R}M_{n}\omega^{n}\right)\Psi_{0}=0\,,$ (110)

where $M_{n}$ are $\omega$-independent differential operators, $\Psi_{0}$ are
the fields of interest in a given sector, and $R$ is the order of $\omega$
appearing in the equations of motion. Note $R$ is at most two because eq. (16)
has second order time derivatives. If more than one field makes up a given
sector, say $m$ of them, then $M_{n}$ are $m\times m$ matrices acting on a
column vector of $m$ fields represented by $\Psi_{0}$. It is well-known how to
linearize such a non-linear eigenvalue problem by introducing auxiliary
variables, $\Psi_{i}=\omega\Psi_{i-1}$, for $i=1,\ldots R-1$. The original
equation, eq. (110), then becomes a set of linear equations

$\displaystyle\omega\left(\sum_{n=1}^{R}M_{n}\Psi_{n-1}\right)=-M_{0}\Psi_{0}\,,\qquad\qquad\Psi_{i}=\omega\Psi_{i-1}\,,\quad
i=1,\ldots R-1\,,$ (111)

which is now a linear generalized eigenvalue problem. We work in null
coordinates (see Appendix C) to avoid non-regular behavior of the fields at
the horizon.

To proceed, we discretize the resulting equations of motion by decomposing the
fields in a truncated Chebyshev basis on a pseudo-spectral grid. Fields in the
equations of motion are replaced with vectors of these fields evaluated at the
grid points, the coefficients of these fields in the equations of motion
become diagonal matrices with diagonal elements the coefficients evaluated at
the grid points, and the partial derivatives turn into pseudo-spectral
differentiation matrices. Appropriate boundary conditions also have to be
imposed, whereby the dynamical equations are required to be regular at the
horizon (null coordinates ensure this for us) and are not sourced at the
conformal boundary. Finally we can solve the resulting (generically) non-
linear eigenvalue problem defined on the pseudo-spectral grid using the
Eigenvalues function in Mathematica, and these give the frequencies of QNMs.

To ensure convergence, comparison over several orders of grid points should be
performed to filter out spurious modes introduced in the spectrum due to the
discretization scheme. Physical modes will move only slightly when changing
the number of grid points used in the scheme unlike spurious modes. Moreover,
the equations of motion contain additional unused equations, and these
function as constraints that have to be satisfied. By using the Eigensystem
function in Mathematica, eigenvalues and the corresponding eigenvectors can be
obtained in one go. The eigenvectors represent the values of the fields at
each of the grid points for a given QNM frequency. Inserting the eigenvector
and eigenvalue back into the constraint equations gives a measure of the
success of the method. We use the following conservative measure:

Let $S^{j}_{i}$, $i=1,\ldots,c$, represent the $c$ constraint equations at
each grid point $j$. Using a Gauss-Lobatto grid of $N$ points, we compute
(recall there are $m$ fields)

$\varsigma=\frac{1}{mN}\sum_{i=1}^{c}\sum_{j=1}^{N}|S^{j}_{i}|\,.$ (112)

What $\varsigma$ measures is how much all the constraint equations in sum
differ from zero as an average over all grid points. For the tensor sector,
there is no constraint, and we can verify that the QNM problem is solved at
least to the level of $10^{-15}$ by directly putting the solution back into
the equation. For the vector sector we keep only modes for which
$\varsigma<10^{-12}$. The scalar sector is more challenging. There, we are
able to find solutions for which $\varsigma\lesssim 10^{-4}$ for $r_{+}=10$,
$\mathcal{J}=25$, $a/\ell=1/2$ and $N=20$. At smaller $a$ or $J$, or near the
large BH limit the convergence improves significantly.333333Two independent
codes were constructed for the purpose of computing QNMs in this work. One was
initially created for the study [37] and did not feature a verification of the
constraint violation. The second was created independently by another author
of the present work during the preparation of [44, 43], and it was later
improved to measure constraint violation. In this work, primary results were
obtained using the first code and then independently verified using the second
code. Work to improve this is underway.

Critical points— Unfortunately, obtaining roots to eq. (70) is not possible
analytically, and numerical method is required. As discussed in [43], first
the relevant fields are discretized by introducing a truncated Chebyshev
pseudo-spectral representation of $N$th order on a Gauss-Lobatto grid of size
$N$. This means that the differential operator $M$ is now a dense, $nN\times
nN$ matrix. To have the numerical computation feasible, the matrix $M$ is next
$LU$-decomposed, and the spectral curve is then given by

$P(\nu,j)=\det(M(\nu,j))=\det(LU)=\det(L)\det(U)=\det(U)\,,$ (113)

as $\det(L)=1$ by construction. This is now computationally much easier to do
as $U$ is an upper triangular matrix whose determinant is simply the product
of the diagonal entries. To extract the critical points, a two-dimensional
Newton-Raphson root finder can be used to search for the value of $(j,\nu)$
satisfying the two conditions in eq. (70). Doing so requires a definition of
the derivative of the numerical determinant $P$, and this can be done e.g.
with finite differences:

$\frac{\partial
P}{\partial\nu}=\frac{\det(U(\nu+\delta,j))-\det(U(\nu,j))}{\delta}\,,$ (114)

where $\delta$ is a small number. For the results presented in [43] and
discussed here, $\delta=10^{-15}$.

##  Appendix C Null coordinates

###  C.1 Infalling Eddington-Finkelstein coordinates

The calculation of the QNM displayed in section III.1 can be done in any
coordinate system. However, a particularly useful coordinate system to choose
is one based on a null slicing of the spacetime in which the coordinate
induced singularity at the outer horizon radius is removed. The infalling
Eddington–Finkelstein like coordinates used in the calculations are defined as
follows

$\displaystyle d\hat{t}$ $\displaystyle=dt+\frac{\sqrt{\beta(r)}}{G(r)}dr\,,$
(115) $\displaystyle d\hat{\psi}$
$\displaystyle=d\psi+\frac{2h(r)\sqrt{\beta(r)}}{G(r)}dr\,,$ (116)
$\displaystyle d\hat{r}$ $\displaystyle=dr\,,$ (117)

where

$\displaystyle\beta(r)$
$\displaystyle=1-\frac{a^{2}r_{+}^{4}\left(r_{+}^{2}+L^{2}\right)}{r^{4}\left(a^{2}r_{+}^{2}+a^{2}L^{2}-r_{+}^{2}L^{2}\right)}\,,$
(118) $\displaystyle h(r)$
$\displaystyle=\frac{ar_{+}^{4}\left(r_{+}^{2}+L^{2}\right)}{a^{2}\left(r_{+}^{2}+L^{2}\right)\left(r_{+}^{4}-r^{4}\right)+r^{4}r_{+}^{2}L^{2}}\,.$
(119)

For convenience, we will suppress the hat on $r$ unless necessary. Making this
transformation of coordinates induces a frame transformation

$\displaystyle\hat{\sigma}^{3}$
$\displaystyle=\sigma^{3}+\frac{2h(r)\sqrt{\beta(r)}}{G(r)}dr\,.$ (120)

Expressed in terms of the null coordinates and the new frame, the metric given
in eq. (20) now takes the form

$\displaystyle ds^{2}$
$\displaystyle=-\frac{G(r)}{\beta(r)}{d\hat{t}}^{2}+\frac{d\hat{t}dr}{\sqrt{\beta(r)}}+\frac{r^{2}}{4}\left((\sigma^{1})^{2}+(\sigma^{2})^{2}\right)$
(121)
$\displaystyle+\frac{r^{2}}{4}\beta(r)\left(\hat{\sigma}^{3}-2h(r)d\hat{t}\right)^{2}$

From eq. 115 one can construct the dual vectors as the following

$\displaystyle\frac{\partial}{\partial\hat{t}}$
$\displaystyle=\frac{\partial}{\partial t}$ (122)
$\displaystyle\frac{\partial}{\partial\hat{\psi}}$
$\displaystyle=\frac{\partial}{\partial\psi}$
$\displaystyle\frac{\partial}{\partial\hat{r}}$
$\displaystyle=\frac{\partial}{\partial
r}-\frac{\sqrt{\beta(r)}}{G(r)}\left(\frac{\partial}{\partial
t}+2h(r)\frac{\partial}{\partial\psi}\right)$

Hence the notation $d\hat{t}$ and $d\hat{\psi}$ is justified because their
associated vectors’ Lie brackets still vanish. Furthermore, the commutation
relations in eq. (35) are preserved.

###  C.2 Kruskal coordinates

A further set of null coordinates is necessary for the study of the
gravitational shockwaves in section III.3, the analogue of Kruskal
coordinates. We review the transformation to this coordinate system, the full
details can be found in [44]. We begin moving to the co-rotating angular
coordinate $\tilde{\psi}$ and the associated one form $\tilde{\sigma}_{3}$ by

$\tilde{\psi}=\psi-\overline{\Omega}t,\hskip
56.9055pt\tilde{\sigma}^{3}=d\tilde{\psi}+\mathrm{cos}(\theta)d\phi,$ (123)

with $\overline{\Omega}=-2\widetilde{\Omega}$ for $\widetilde{\Omega}$ defined
in eq. (30). In the co-rotating coordinates $(t,r,\tilde{\psi},\phi,\theta)$
the radial and temporal coordinates $(t,r)$ take the form

$ds_{(t,r)}^{2}=-F(r)dt^{2}+\frac{dr^{2}}{G(r)},$ (124)

where $F(r)$ is given as

$F(r)=1+\frac{r^{2}}{L^{2}}-\frac{r_{+}^{2}}{r^{2}}\bigg{(}1+\frac{r_{+}^{2}}{L^{2}}\bigg{)}-\frac{a^{2}\Sigma^{2}(r^{4}-r_{+}^{4})}{L^{4}r^{2}r_{+}^{4}}\,,$
(125)

which vanishes at $r=r_{+}$ as expected in co-rotating coordinates. In eq.
(125) we have introduced a new parameter $\Sigma$ defined as

$\Sigma=L^{2}+r_{+}^{2}\,.$ (126)

As is familiar we can introduce the infalling ($v$) and outgoing ($u$)
Eddington=Finkelstein coordinates by

$\displaystyle dv=dt+\frac{dr}{\sqrt{F(r)G(r)}},$ (127a) $\displaystyle
du=dt-\frac{dr}{\sqrt{F(r)G(r)}}.$ (127b)

Finally, define the Kruskal-like coordinates, $U$ and $V$

$U=-e^{-\alpha u},\hskip 56.9055ptV=e^{\alpha v},$ (128)

where $\alpha=2\pi T$, for which the metric is then given by

$ds^{2}=A(UV)dUdV+B(UV)(UdV-
VdU)\tilde{\sigma}^{3}+\frac{r^{2}}{4}((\sigma^{1})^{2}+(\sigma^{2})^{2}+(\tilde{\sigma}^{3})^{2})+\frac{\mu
a^{2}}{2r^{2}}(\tilde{\sigma}^{3})^{2}\,.$ (129)

The function $r=r(UV)$ is defined implicitly through (128) and

$A(UV)=\frac{F(r)}{\alpha^{2}UV},\;\;\;\;\;\;\;\;B(UV)=\frac{a}{2\alpha
UV}\frac{\Sigma(r_{+}^{4}-r^{4})}{L^{2}r^{2}r_{+}^{2}}.$ (130)

The metric (129) is manifestly smooth at both the $U=0$ and $V=0$ horizons at
which $r(0)=r_{+}$.

##  Appendix D Hydrodynamic dispersion relations and conventions

In this appendix, the hydrodynamic dispersion relation conventions, expected
values for transport coefficients and expected scaling powers are presented in
the example of the momentum diffusion channel. At vanishing rotation, in the
momentum diffusion channel the lowest of the eigenmodes is the hydrodynamic
diffusion mode with the dispersion

$\omega=-i\mathcal{D}q^{2}\,,$ (131)

where the frequency $\omega$ and momentum $q$ are chosen to agree with the
definitions from [46]. In this case, $\mathcal{N}=4$ SYM theory has been shown
to have $\mathcal{D}=1/(4\pi T)$ [64, 101]. This diffusion coefficient is
furthermore related to the shear viscosity over entropy ratio as follows (via
an Einstein relation):
$\mathcal{D}=\eta/(\epsilon+P)=\frac{1}{T}\frac{\eta}{s}=\frac{1}{T}\frac{1}{4\pi}$,
where we used that $\mathcal{N}=4$ SYM has $\eta/s=1/(4\pi)$ and that
$\epsilon+P=sT$.

In this article, we consider different frequency $\nu=\omega/(2\pi T\ell)$ and
momentum $j=q/(2\pi T\ell)$. For the rest of this appendix, we assume
$\ell=1$, as is done in the main text unless stated otherwise. We also
consider rotating states, which will modify the dispersion of the diffusion
mode, so for the longitudinal diffusion mode we write

$\nu=v_{||}j-i\mathcal{D}_{||}j^{2}+\mathcal{O}(j^{3})\,,$ (132)

with the longitudinal speed $v_{||}$ of this former diffusion mode and its
modified diffusion coefficient $\mathcal{D}_{||}$. The temperature at
vanishing angular momentum $a=0$ takes the value $T=r_{+}/\pi$. Therefore, at
$a=0$, we expect

$\mathcal{D}_{||}(a=0)=2\pi T\mathcal{D}=\frac{1}{2}\,.$ (133)

Let us rewrite the dispersion relation in different frequency and momentum
variables:

$\displaystyle\omega=$ $\displaystyle-i\mathcal{D}q^{2}\,,$ (134)
$\displaystyle\Leftrightarrow 2\pi T\nu=$ $\displaystyle-i\mathcal{D}(2\pi
Tj)^{2}\,,$ (135) $\displaystyle\Leftrightarrow\nu=$
$\displaystyle-i\underbrace{\mathcal{D}(2\pi T)}_{\mathcal{D}_{||}}j^{2}\,.$
(136)

Now, let us write the dispersion relation in terms of the variables $\omega$
and $\mathcal{J}$, which appear in the ansatz (51) for the metric
fluctuations:

$\displaystyle\omega=$ $\displaystyle-iD\mathcal{J}^{2}\,,$ (137)
$\displaystyle\Leftrightarrow\omega=$ $\displaystyle-iD(\pi Tj)^{2}\,,$ (138)
$\displaystyle\Leftrightarrow\omega=$ $\displaystyle-iD(\pi
T)^{2}(\frac{q}{2\pi T})^{2}=-i\underbrace{\frac{D}{4}}_{\mathcal{D}}q^{2}\,,$
(139)

such that we expect $D=1/(\pi T)=1/r_{+}$.

Moving on to the case of $a\neq 0$, we write

$\displaystyle\omega=$ $\displaystyle v\mathcal{J}-iD\mathcal{J}^{2}\,,$ (140)
$\displaystyle\Leftrightarrow\omega=$ $\displaystyle-iD(\pi Tj)^{2}\,,$ (141)
$\displaystyle\Leftrightarrow\omega=$ $\displaystyle-iD(\pi
T)^{2}(\frac{q}{2\pi T})^{2}=-i\underbrace{\frac{D}{4}}_{\mathcal{D}}q^{2}\,,$
(142)

where $D$ and $v$ now depend on the angular momentum parameter $a$. In the
large BH limit $r_{+}\to\infty$ (more precisely (52) and (53)), we expect
$D(a)=2\frac{\mathcal{D}_{||}}{\pi T}=\frac{1}{r_{+}}(1-a^{2})^{3/2}$ and
$v=2v_{||}=2a$ [37], which are the values expected for a fluid boosted with
boost parameter $a$ [65, 66]. Thus, for example, the values given in Table 10
are expected in this limit. This form (140) in terms of $\omega$ and
$\mathcal{J}$ and allowing general exponents,
$\omega=v\mathcal{J}^{\beta}-iD\mathcal{J}^{\alpha}$, was used to fit $D$,
$v$, $\alpha$ and $\beta$ and the result of that fit is displayed in Fig. 8.

$a$ | $v(a)$ | $r_{+}\,D(a)$
---|---|---
0 | 0 | 1
0.5 | 1 | 0.64952
0.9 | 1.8 | 0.08282
Table 10: Example values expected for the diffusion coefficient $D$
(multiplied by the horizon radius $r_{+}$) and the velocity $v$ in the large
BH limit $r_{+}\to\infty$ (more precisely (52) and (53)) at distinct values of
angular momentum $a$.

##  Appendix E Limits of the Wigner D function

An important consideration which has not yet been discussed is the relation of
the Wigner d functions to Fourier transformations. To begin, we consider the
decomposition of the Wigner-$D$ functions in terms of the Wigner-$d$ functions
343434In this appendix we drop the notation
$(\mathcal{J},\mathcal{K},\mathcal{M})$ in favor of $(J,m,k)$.

$\tensor{D}{{}^{J}_{m}{}_{k}}(\theta,\phi,\psi)=e^{-i(k\phi+m\psi)}\tensor{d}{{}^{J}_{m}{}_{k}}(\theta)$
(143)

The explicit form of the Wigner-$d$ functions can be written using the Wigner
formula [102, 103],

$\tensor{d}{{}^{J}_{m}{}_{k}}(\theta)=\sum_{n}(-1)^{n}w^{Jmk}_{n}\cos(\theta/2)^{2J+k-m-2n}(-\sin(\theta/2))^{m-k-2n}$
(144)

where the coefficients $w$ can be expressed in terms of a complicated fraction
of factorials of its indices. Although this expression is complicated, an
important observation is that one can reduce this formula for each value of
the indices by repeated operation of the power reduction trigonometric
identities to $\cos(v\theta)$ ($\sin(\nu\theta)$), such that,

$\tensor{d}{{}^{J}_{m}{}_{k}}(\theta)=\sum_{\nu=\nu_{min}}^{\nu_{max}=J}t^{Jmk}_{\nu}f(\nu\theta)\,,$
(145)

where,

$f=\begin{cases}\cos,&m-k\hskip 5.69046pt\text{even}\\\ \sin,&m-k\hskip
5.69046pt\text{odd}\\\ \end{cases}$ (146)

Hence we see that expansion for the $d$ function is in terms of $\cos$ if
$J-k$ is even and $\sin$ if $J-k$ is odd. Recalling the large BH limit studied
in [37, 43, 44] in eq. (52) and eq. (53) (repeated here for ease of
presentation)

$J=jr_{+}/L,\quad\omega=2\nu r_{+}/L,\quad r\rightarrow\alpha r\quad
r_{+}\rightarrow\alpha r_{+}$ (147)

around $\alpha\rightarrow\infty$ along with the fact that sectors are
distinguished by choices of $m$ in terms of J, in the large BH limit $J\approx
m$. In the actual calculation all $k$ dependence of the field equations drops
out and hence we are left with a freedom. Here we fix this freedom by assuming
$k$ even and $J\in\mathbb{Z}$ and hence,

$\tensor{d}{{}^{J}_{m}{}_{k}}(\theta)=\sum_{\mu=0}^{\mu_{max}=J}t^{Jmk}_{\mu}\cos(\mu\theta)$
(148)

From orthogonality,

$t^{Jmk}_{\mu}=\frac{1}{2\pi(1+\delta_{\mu 0})}\int_{0}^{4\pi}\mathrm{d}\theta
d^{J}_{mk}(\theta)f(\mu\theta),\qquad\delta_{\mu
0}=\begin{cases}1&\text{for}\hskip 2.84544pt\mu=0\\\ 0&\text{else}\end{cases}$
(149)

one sees that $t^{Jmk}_{\mu}=t^{Jmk}_{-\mu}$ and hence one can rewrite eq.
(148) as,

$\tensor{d}{{}^{J}_{m}{}_{k}}(\theta)=\frac{1}{2}\sum_{\mu=-J}^{J}t^{Jmk}_{\mu}e^{I\mu\theta}$
(150)

Therefore, the Wigner $d$ functions can be written as Fourier sine and cosine
series [103].

Now, let us diverge from [103] and investigate briefly what happens to this
Fourier representation in the large BH limit, keeping mind that in this limit
$J\approx m$. We have the following expression,

$\tensor{d}{{}^{\frac{j\alpha r_{+}}{L}}_{\frac{j\alpha
r_{+}}{L}}{}_{k}}(\theta)=\frac{1}{2}\sum_{\mu=-\frac{j\alpha
r_{+}}{L}}^{\frac{j\alpha
r_{+}}{L}}\frac{\Delta\mu}{2\pi}\frac{2\pi}{\Delta\mu}t^{\frac{j\alpha
r_{+}}{L}\frac{j\alpha r_{+}}{L}k}_{\mu}e^{I\mu\theta}$ (151)

where we have cleverly inserted a factor of $1$ in the summand. If we formally
extend the range of $\theta$ to $(-\infty,\infty)$ and define,

$T_{k}(\mu)=\lim_{\Delta\mu\rightarrow
0}\lim_{\alpha\rightarrow\infty}\frac{2\pi}{\Delta\mu}t^{\frac{j\alpha
r_{+}}{L}\frac{j\alpha r_{+}}{L}k}_{\mu}$ (152)

while using the definition from calculus that,

$\lim_{\Delta\mu\rightarrow
0}\lim_{\alpha\rightarrow\infty}\sum_{\mu=-\frac{j\alpha
r_{+}}{L}}^{\frac{j\alpha
r_{+}}{L}}\frac{\Delta\mu}{2\pi}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\mu$
(153)

we find,

$\lim_{\Delta\mu\rightarrow
0}\lim_{\alpha\rightarrow\infty}\tensor{d}{{}^{\frac{j\alpha
r_{+}}{L}}_{\frac{j\alpha
r_{+}}{L}}{}_{k}}(\theta)=\frac{1}{4\pi}\int^{\infty}_{-\infty}\mathrm{d}\mu
T_{k}(\mu)e^{I\mu\theta}$ (154)

and hence the Wigner $d$ function can be interpreted as a Fourier
transformation. In these equations $\theta$ should now be thought of as a
coordinate along the real line $\mathbb{R}$.

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