Natural convection in the cytoplasm: Theoretical predictions of buoyancy-
driven flows inside a cell

Nikhil Desai, Weida Liao, Eric Lauga*,

Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge, United Kingdom

*<EMAIL_ADDRESS>
## Abstract

The existence of temperature gradients within eukaryotic cells has been
postulated as a source of natural convection in the cytoplasm, i.e. bulk fluid
motion as a result of temperature-difference-induced density gradients. Recent
computations have predicted that a temperature differential of $\Delta
T\approx 1$ K between the cell nucleus and the cell membrane could be strong
enough to drive significant intracellular material transport. We use numerical
computations and theoretical calculations to revisit this problem in order to
further understand the impact of temperature gradients on flow generation and
advective transport within cells. Surprisingly, our computations yield flows
that are an order of magnitude weaker than those obtained previously for the
same relative size and position of the nucleus with respect to the cell
membrane. To understand this discrepancy, we develop a semi-analytical
solution of the convective flow inside a model cell using a bi-spherical
coordinate framework, for the case of an axisymmetric cell geometry (i.e. when
the displacement of the nucleus from the cell centre is aligned with gravity).
We also calculate exact solutions for the flow when the nucleus is located
concentrically inside the cell. The results from both theoretical analyses
agree with our numerical results, thus providing a robust estimate of the
strength of cytoplasmic natural convection and demonstrating that these are
much weaker than previously predicted. Finally, we investigate the ability of
the aforementioned flows to redistribute solute within a cell. Our
calculations reveal that, in all but unrealistic cases, cytoplasmic convection
has a negligible contribution toward enhancing the diffusion-dominated mass
transfer of cellular material.

##  1 Introduction

The nature and behaviour of all complex lifeforms is governed by the
biochemical and physical processes occurring inside their cells. Hence, cell-
scale investigations of energy and mass transfer are crucial to gain a better
understanding of larger scale structure and function in living organisms.
Chemical reactions inside a cell are often accompanied by intra- and/or inter-
cellular heat exchange [1]. The thermal environment within and around a cell
governs important tasks, such as cell-cycle regulation [2], cellular
metabolism [3], cell membrane function [4], and protein interactions [5]. This
has motivated researchers to develop techniques to accurately ascertain
temperatures at the scale of a single cell, leading to the recent emergence of
the field of intracellular thermometry [6, 7, 8, 5].

Intracellular thermometry has already contributed significantly to our
understanding of the thermal landscape inside single cells. Multiple studies
have reported non-trivial temperature heterogeneities. For example,
fluorescent polymeric thermometry measurements of the COS-7 cell showed that
its nucleus and centrosome can be $\sim 1$ K warmer than the surrounding
cytoplasm [9]. Similarly, the nucleus of living HeLa cells was also found to
be $\sim 1$ K warmer than the cytoplasm, on average [10]. Furthermore, the
mitochondria of mammalian cells have been proposed as salient ‘hot-spots’ that
display elevated temperatures as compared to the rest of the cell. The higher
mitochondrial temperatures result from thermogenesis, i.e. the release of heat
accompanying ATP synthesis. Various studies have reported mitochondrial
temperatures 1–6 K higher than the cytoplasm, due to artificially induced
thermogenesis, in both COS-7 and HeLa cells [11, 9, 10, 12] (see also Table 1
in Ref. [13]).

While these temperature contrasts have been measured independently using a
diverse array of methods [7], there is still major controversy around the
validity of these measurements [14, 15, 16, 17, 8]. Mathematical models
relying on macroscopic energy balance arguments have argued that the average
heat generation inside a cell is so minuscule that it cannot possibly sustain
the large temperature differentials reported in intracellular thermometric
experiments [14]. Indeed, under steady operation, the temperature distribution
inside the cell, $T(\mathbf{x})$, is related to the power produced per unit
volume of the cell, $\mathcal{P}$, by the heat-diffusion equation,
$k\nabla^{2}T=\mathcal{P}$, where $k$ is the thermal conductivity of the
medium. A scaling analysis then suggests that the total power generated by the
cell would cause a temperature increase on the order of $\Delta
T\sim\mathcal{P}\ell_{\textrm{c}}^{2}/k$, with $\ell_{\textrm{c}}$ being the
characteristic length scale of the cell. The typical power delivered by a
cell, $\sim\mathcal{P}\ell_{c}^{3}$, is known to be on the order of $\sim 100$
pW [18]. Assuming the cell environment is predominantly aqueous (hence, $k\sim
1$ W m-1 K-1) and the cell-size is $\ell_{c}\sim 10$ $\mu$m, the temperature
increase resulting from this power generation would be $\sim 10^{-5}$ K,
orders of magnitude lower than the temperature differences reported by
intracellular thermometry. Resolution of this apparent paradox warrants
further studies, and thus, both the theory and practice behind cellular
temperature measurements are fertile and fast-growing fields of research in
the biological sciences.

The aforementioned discrepancy notwithstanding, the existence of finite
temperature gradients within a cell could have important biophysical
implications for the bulk flow of the aqueous cytoplasm. So-called
“cytoplasmic flows” redistribute nutrients within a cell, which in turn
affects cellular functions, such as metabolism and cell division [19, 20].
This flow is usually actively caused by the entrainment of cytoplasm by
vesicles that are driven through the cell by molecular motors along polymeric
filaments. Another possible mechanism of fluid flow is based on temperature
gradients within the cell causing a passive “cytoplasmic convection” [21, 22,
23]. The intuitive physical idea is that a steady temperature difference
between a cell’s nucleus and its membrane can cause minute changes in the
density of the cytoplasm, with the warmer regions characterised by lower
densities. In general, the cell geometry is such that the resulting density
gradients cannot be balanced by a purely hydrostatic pressure distribution.
This means that, due to their relative buoyancy, regions of higher density
must settle, while those of lower density must rise, leading necessarily to
natural convection within the cell [24, 25]. At the cell-scale, these
circulatory flows could act as an intriguing passive complement to the active
molecular-motor-driven mass transfer inside cells. Thus, to identify the most
relevant mechanisms affecting intracellular material transport, it is
essential to quantify the flows driven by cell-scale temperature
inhomogeneities. It is important to note here that the flow just described is
unavoidable for any finite temperature gradient [25], and thus distinguishes
itself from the classical Rayleigh–Bénard convection, which occurs only in
systems where temperature gradients exceed a critical threshold [26].

While scaling arguments suggest that temperature-gradient-induced flows could
be dominant in certain plant cells [21], further analysis is needed to better
characterise the nature of these flows. Recent computational work considered
the case of a model cell with its nuclear surface warmer than the cell
membrane, thus generating a temperature gradient that causes fluid flow [23].
The primary motive of that work was to investigate: (i) the influence of the
temperature difference between the nucleus and the cell membrane on the flow
strength inside the cell, and (ii) the ability of these flows to transport
material from the nucleus to the cell membrane. Assuming both the cell and its
nucleus to be spherical, it was shown, perhaps surprisingly, that temperature-
gradient-driven convection can strongly influence transport of cellular
materials characterised by low diffusivities, if the nucleus is warm enough as
compared to the cell membrane. However, important questions remain unexplored.
For example, how would the size and position of the nucleus affect the flow?
This is relevant since the size and position of the nucleus often change due
to environmental and/or functional reasons [27, 28, 29]. More fundamentally, a
simpler physical model of temperature-gradient-induced cytoplasmic flow would
enable us to predict the flow distribution inside the cell without the need
for complex computations, and thus to rationalise these predicted strong
flows. In this paper, we use a combination of numerical simulations and
theoretical calculations to further analyse temperature-gradient-driven flows
inside model cells and provide analytical expressions for these in the limit
of the nucleus being concentric with the cell membrane. Our results allow us
in turn to revisit the predictions in Ref. [23] and we argue that cytoplasm
flows resulting from temperature gradients are actually much weaker than
previously reported.

In Sec. 2 we introduce our system with the mathematical model, and the
important parameters describing the cell geometry and the material properties
of the cytoplasm. We also discuss and mathematically describe the underlying
physics of natural convection at the cell-scale. In Sec. 3, we present results
from a numerical simulation showing natural convection of the cytoplasm.
Surprisingly, our computations yield flows that are an order of magnitude
weaker than those obtained earlier for the same cell geometry [23]. We examine
this discrepancy via theoretical calculations, in Sec. 4 and 5. In Sec. 4, we
focus on an axisymmetric arrangement of the nucleus inside the model cell, and
develop a semi-analytical solution for the fluid flow in such a configuration.
Next, in Sec. 5, we consider a concentric placement of the nucleus inside the
model cell, and obtain a closed-form analytical expression for the
temperature-gradient-driven flow. The solutions obtained in both these
limiting cases agree perfectly with our own computations, thus giving us
confidence in our numerical predictions that cytoplasm flows resulting from
temperature gradients are significantly weaker than previously reported.
Finally, in Sec. 6, we perform a thorough investigation of the ability of
these flows to transport solute (via advection) within a cell. Here also we
reveal that, unless under unrealistic assumptions, the flows are in fact not
strong enough to significantly change the distribution of cellular material,
the motion of which stays largely diffusive. We finally summarise our results
and provide perspectives for future investigations in Sec. 7.

##  2 Modelling intracellular convection

Fig 1: Schematic of the nucleus (orange inner sphere) within a model cell
(light blue outer sphere). The cell membrane (thick, dark blue outline) is
maintained at a constant temperature $T_{\textrm{mem}}$, whereas the nucleus
surface is maintained at a higher temperature,
$T_{\textrm{nuc}}>T_{\textrm{mem}}$. Gravity acts in the negative $z$
direction. The Newtonian cytoplasm fills the annular region (light blue). We
solve for the temperature field and fluid velocity field in the cytoplasm.

###  2.1 Problem setup

We consider the configuration shown in Fig 1, similar to the one addressed in
Ref. [23]. Both the cell and the nucleus are modelled as stationary rigid
spheres, of radius $R_{\textrm{c}}$ and $R_{\textrm{nuc}}$, respectively. A
Cartesian coordinate system is affixed at the centre of the cell, with gravity
acting along the negative $z$-direction. Without loss of generality, the
centre of the nucleus is situated in the $y=0$ plane at a distance of
$eR_{\textrm{c}}$ from the cell centre, at an angle $\theta_{e}$ from the
$x$-axis. From geometry, it is clear that one must have
$eR_{\textrm{c}}+R_{\textrm{nuc}}\leq R_{\textrm{c}}$. The quantity
$eR_{\textrm{c}}\cos\theta_{e}=e_{x}R_{\textrm{c}}$ denotes the eccentricity,
along the $x$-axis, of the nucleus centre relative to the cell centre.
Similarly, $eR_{\textrm{c}}\sin\theta_{e}=e_{z}R_{\textrm{c}}$ denotes the
nucleus eccentricity along the $z$-axis.

We assume that the cellular membrane is maintained at a constant temperature
$T_{\textrm{mem}}$, whereas the surface of the nucleus is maintained at a
constant temperature $T_{\textrm{nuc}}$. Following experimental evidence [9,
5], we take the nucleus to be warmer than the cell membrane (i.e.
$T_{\textrm{nuc}}>T_{\textrm{mem}}$). Note that while we are modelling flows
due to temperature gradients between the nucleus and the cell membrane, a
change in the size $\left(R_{\textrm{nuc}}\right)$ and position
$\left(e,\theta_{e}\right)$ of the inner sphere can also provide an
approximation of convection caused by other warm organelles inside the cell.

With cells that are typically tens of microns in size [30, 31], intracellular
fluid flow is expected to be dominated by viscous forces and thus governed by
classical microhydrodynamics [32]. Following Ref. [23], we model the aqueous
cytoplasm as a Newtonian viscous liquid with physical properties similar to
water at a reference membrane temperature, $T_{\textrm{mem}}$ [30, 31].

We further approximate the viscosity of the cytoplasm as uniform; although
viscosity is known to vary with temperature, including this effect is known to
only give rise to a small correction to the flow [33]. The driving of the flow
is thus solely due to the density of the cytoplasm, which varies spatially in
response to thermal gradients and thus necessitates solving for the
temperature distribution within the cell. The temperature is governed by an
advection-diffusion equation, denoting a balance between heat transfer by
fluid flow and by diffusion (conduction). In the next section, we formalise
these statements and present the equations governing the flow and temperature
fields inside the cell.

###  2.2 Governing equations and boundary conditions

In the absence of inertia, the flow field,
$\mathbf{u}^{\prime}\left(\mathbf{x}^{\prime}\right)$ at a position
$\mathbf{x}^{\prime}$ inside the cell, is governed by the incompressibility
constraint (or the continuity equation),

$\nabla^{\prime}\cdot\mathbf{u}^{\prime}=0,$ (1)

and the Stokes equations,

$-\nabla^{\prime}p^{\prime}+\eta\nabla^{\prime
2}\mathbf{u}^{\prime}+\rho(T^{\prime})\mathbf{g}=\mathbf{0},$ (2)

where the primes signify dimensional variables.

In the above equation, $p^{\prime}$ is the pressure in the fluid, $\eta$ is
the constant dynamic viscosity, $\mathbf{g}=-g\mathbf{i}_{z}$ is the
acceleration due to gravity ($\mathbf{i}_{z}$ is the unit vector in the
$z$-direction), and $T^{\prime}$ is the temperature in the cell. We denote the
temperature-dependent density of the fluid by $\rho(T^{\prime})$. For
sufficiently small temperature differences $T^{\prime}-T_{\textrm{mem}}$, we
model this temperature-dependence with a standard linear relationship,

$\rho(T^{\prime})=\rho_{0}\left[1+\beta(T_{\textrm{mem}}-T^{\prime})\right],$
(3)

where $\beta$ is the thermal expansion coefficient, with units of K-1. It is
important to note here that we are employing the incompressibility constraint,
eqn. (1), yet allowing the density to vary with temperature. This is a
classical (and validated) method in hydrodynamics called the Boussinesq
approximation [33]. For modest changes in temperature, we can neglect the
temperature dependence of material properties everywhere except in the
gravitational force term in eqn. (2), when this buoyancy term is the driving
mechanism for the flow (as in the present problem). This provides a first
approximation to the fluid flow, which should be quantitatively accurate for
temperature differences less than $\sim 10$ K–$20$ K [33]. Mathematically,
this may be formalised via a perturbation expansion of the full governing
equations in the limit of small temperature changes [33].

The governing equations, eqns. (1) and (2), must be complemented by
appropriate boundary conditions for the velocity. Since we are modelling the
nucleus and the cell as stationary, rigid spheres, the fluid velocity must
vanish at both these surfaces, i.e.

$\displaystyle\left.\mathbf{u}^{\prime}\right|_{\textrm{nuc}}$
$\displaystyle=\mathbf{0},$ (4)
$\displaystyle\left.\mathbf{u}^{\prime}\right|_{\textrm{mem}}$
$\displaystyle=\mathbf{0},$

where the sub-scripts ‘nuc’ and ‘mem’ denote the cell nucleus and membrane,
respectively. It is clear from eqn. (2) that solving for the fluid flow
requires knowledge of the temperature. The steady-state temperature field,
$T^{\prime}\left(\mathbf{x}^{\prime}\right)$, is governed by an advection-
diffusion equation,

$\mathbf{u}^{\prime}\cdot\nabla^{\prime}T^{\prime}=\alpha\nabla^{\prime
2}T^{\prime},$ (5)

subject to the boundary conditions,

$\displaystyle\left.T^{\prime}\right|_{\textrm{nuc}}$
$\displaystyle=T_{\textrm{nuc}},$ (6)
$\displaystyle\left.T^{\prime}\right|_{\textrm{mem}}$
$\displaystyle=T_{\textrm{mem}}.$

In eqn. (5), $\alpha$ is the constant thermal diffusivity of the cytoplasm
(assumed to be water in the present study).

It is important to note that for the geometry described in Fig 1, eqns. (1) to
(6) cannot admit a quiescent solution ($\mathbf{u}^{\prime}=\mathbf{0}$) as
long as there exists a temperature contrast between the two surfaces. For any
$T_{\textrm{nuc}}\neq T_{\textrm{mem}}$, the temperature distribution
$T^{\prime}\left(\mathbf{x}^{\prime}\right)$ will be non-uniform and the
resulting temperature gradients (and hence the density gradients) will not be
aligned with gravity inside the cell. Thus, no hydrostatic pressure
distribution can be found that balances the gravitational forcing everywhere,
and the system cannot stay in equilibrium without fluid flow. This is
fundamentally different from the classical Rayleigh–Bénard convection, where
gravity is parallel to an imposed temperature gradient and fluid flow emerges
as a result of an instability above a critical imposed temperature gradient
[26].

We next render the equations dimensionless; using the characteristic
temperature difference $\Delta T=T_{\textrm{nuc}}-T_{\textrm{mem}}$, we may
define the reference scales for length, velocity and pressure, respectively,
as

$\ell_{\textrm{ref}}=R_{\textrm{c}},\quad
u_{\textrm{ref}}=\frac{\rho_{0}\beta\Delta
Tg\ell_{\textrm{ref}}^{2}}{\eta},\quad p_{\textrm{ref}}=\frac{\eta
u_{\textrm{ref}}}{\ell_{\textrm{ref}}}.$ (7)

We also define a re-scaled temperature,

$\Theta=\frac{T^{\prime}-T_{\textrm{mem}}}{T_{\textrm{nuc}}-T_{\textrm{mem}}}.$
(8)

Parameter |  Description |  Typical value |  Units
---|---|---|---
$g$ |  gravitational force per unit mass |  9.8 |  m s-2
$\beta$ |  thermal expansion coefficient |  $3.61\times 10^{-4}$ |  K-1
$\Delta T=T_{\textrm{nuc}}-T_{\textrm{mem}}$ |  temperature difference between the nuclear surface and the cell membrane |  0.01–10 |  K
$R_{\textrm{c}}$ |  radius of the cell |  10 |  $\mu$m
$R_{\textrm{nuc}}$ |  radius of the nucleus |  2–7 |  $\mu$m
$\rho_{0}$ |  density of the cytoplasm at 310 K |  993.38 |  kg m-3
$\eta$ |  viscosity of the cytoplasm |  $6.917\times 10^{-4}$ |  kg m-1 s-1
$\alpha$ |  thermal diffusivity of the cytoplasm |  $1.51\times 10^{5}$ |  $\mu$m2 s-1
$D$ |  solute diffusivity |  0.01–100 |  $\mu$m2 s-1
$u_{\textrm{ref}}=\frac{\rho_{0}\beta\Delta TgR^{2}_{\textrm{c}}}{\eta}$ |  reference velocity scale |  $0.5$ |  $\mu$m s-1
$\textrm{Pe}_{\textrm{t}}$ = $\frac{u_{\textrm{ref}}R_{\textrm{c}}}{\alpha}$ |  thermal Péclet number |  $3\times 10^{-5}$ |  dimensionless
$\textrm{Pe}_{\textrm{s}}$ = $\frac{u_{\textrm{ref}}R_{\textrm{c}}}{D}$ |  ‘solutal’ Péclet number |  $0.05$–$500$ |  dimensionless
Table 1: Typical values of the physical parameters (above line), the reference
velocity scale $u_{\textrm{ref}}$, and the dimensionless groups (last two
rows) governing the intracellular fluid flow, along with the heat and mass
transfer.

Using eqns. (7) and (8) in the equation for the flow, eqn. (2), yields now in
dimensionless form (without primes)

$-\nabla
P+\nabla^{2}\mathbf{u}+\Theta\mathbf{i}_{z}=\mathbf{0},\qquad\nabla\cdot\mathbf{u}=0,$
(9)

subject to the dimensionless boundary conditions

$\displaystyle\left.\mathbf{u}\right|_{\textrm{nuc}}$
$\displaystyle=\mathbf{0},$ (10)
$\displaystyle\left.\mathbf{u}\right|_{\textrm{mem}}$
$\displaystyle=\mathbf{0}.$

The pressure $P$ in eqn. (9) is now a modified dynamic pressure, given in
dimensionless form by $P=p+\left(\beta\Delta
T\right)^{-1}\mathbf{i}_{z}\cdot\mathbf{x}$. Similarly, we can re-write the
thermal problem, eqn. (5), in terms of the normalised temperature
$\Theta\left(\mathbf{x}\right)$ as

$\mathbf{u}\cdot\nabla\Theta=\frac{1}{\textrm{Pe}_{\textrm{t}}}\nabla^{2}\Theta,$
(11)

with the normalised boundary conditions,

$\displaystyle\left.\Theta\right|_{\textrm{nuc}}$ $\displaystyle=1,$ (12)
$\displaystyle\left.\Theta\right|_{\textrm{mem}}$ $\displaystyle=0.$

Importantly, in eqn. (11), $\textrm{Pe}_{\textrm{t}}\equiv
u_{\textrm{ref}}R_{\textrm{c}}/\alpha$ is the (dimensionless) thermal Péclet
number, a ratio of the rate of heat transfer by fluid flow to the rate of heat
transfer by diffusion.

The typical values of the parameters used for calculating the reference scales
in our problem are given in Table 1, where, just like Ref. [23], we have
assumed that the cytoplasm shares the material properties of water at
temperature $T_{\textrm{mem}}=310$ K ($\approx 37\degree$C). This is a
reasonable assumption for both the physical [30, 34] and thermal [35, 36]
properties of the cytoplasm. We note, however, that in some instances the
effective viscosity of the cytoplasm can be higher than that of water, due to
a high concentration of macromolecules [30]; we will discuss the implications
of this disparity in Sec. 4.2 and argue that it does not alter our central
result. The size (radius) of the cell in our study corresponds roughly to the
typical COS-7 [9] or HeLa [10] cells used in intracellular thermometry
studies, while we consider a wide range of sizes and positions of the nucleus,
to reflect the diversity in the nucleus’s placement inside biological cells,
depending on the cell type and the stage of the cell cycle [27, 28]. Here too,
we note that the main conclusions of this study remain independent of the
details of nuclear size and position inside the cell (see Appendix B).

A quantitative description of natural convection in the model cell requires us
to solve eqns. (9)–(12) for $\Theta$ and $\mathbf{u}$. For a prescribed
thermal Péclet number (set by the material properties of the cytoplasm), the
solution depends only on the size and position of the nucleus. These
quantities are captured in three dimensionless numbers:
$\kappa=R_{\textrm{nuc}}/R_{\textrm{c}}$, the ratio of the nucleus’s radius to
that of the cell, and $\left(e_{x},e_{z}\right)$, the eccentricity of the
nucleus centre along the $x$\- and $z$-axes.

Depending on the specific values of $\left(e_{x},e_{z}\right)$ and
$\textrm{Pe}_{\textrm{t}}$, one can solve eqns. (9)–(12) using full numerical
simulations, semi-analytical methods, or even analytically to obtain an
explicit expression for the temperature and flow,
$\left(\Theta,\mathbf{u}\right)$. In following sections, we describe solutions
obtained using each of these methods to provide rigorous estimates of the flow
inside the cell.

##  3 Temperature and flow inside the cell

###  3.1 Intracellular convective flows: comparison with previous work

We first present dimensional results of the temperature and flow fields inside
the cell, to gain intuition about the strength of the temperature-gradient-
driven flow. We show in the top row of Fig 2 one specific solution, for a
radius ratio $\kappa=0.43$, eccentricity values $e_{x}=e_{z}=0.25$, and a
nucleus-to-membrane temperature difference of $\Delta T=1$ K, evaluated
numerically using the finite-elements-based software COMSOL; this is the
benchmark geometry used in Ref. [23]. The values of the other physical
parameters are stated in Table 1 and are the same as in Ref. [23], allowing us
to compare our results directly. The results of Ref. [23] are reproduced in
the bottom row of Fig 2. For the flow field, we plot the horizontal (i.e.
perpendicular to gravity) velocity $u^{\prime}_{x}$ and the vertical (i.e.
upward) velocity $u^{\prime}_{z}$ at the $y=0$ mid-plane of the cell. Since
this is a plane of symmetry, the velocity component normal to it, i.e.
$u^{\prime}_{y}$, is identically zero.

Fig 2: Top row, left to right: the temperature, the horizontal velocity
($u^{\prime}_{x}$) and the vertical velocity ($u^{\prime}_{z}$) at the $y=0$
mid-plane of the cell, computed by simulating the advection-diffusion equation
(eqn. (11)) for the temperature, and the Stokes equations (eqn. (9)) for flow
field. Bottom row: the corresponding fields as computed numerically in Ref.
[23] for the same set of physical parameters. In our results, the spatial
variation in flow velocity is barely discernible when plotted using the colour
scale of Ref. [23], as the velocity magnitudes we obtain are significantly
lower.

While the temperature distribution in Ref. [23] is only slightly different
from our own simulations, the intracellular flows in that work are much
stronger (typically by around one order of magnitude). It is clear,
intuitively, that the small difference in the temperature profiles is not
sufficient to explain the ten-fold mismatch in velocity magnitudes.

To resolve this apparent conflict and verify the accuracy of our own
simulations, we will solve, in Secs. 4 and 5, for the temperature and flow
distributions in geometrically simpler domains, for which we can also pursue
alternate solution methodologies. Towards this, we first need to examine the
temperature field inside the cell. We will show that fluid flow is ineffective
in facilitating temperature redistribution, which will allow us to simplify
the governing equations further. This enables the use of semi-analytical and
analytical techniques to solve the problem, validate our numerical
simulations, and confirm the discrepancy between our results and those of Ref.
[23].

###  3.2 Relative influence of diffusion and advection in intracellular heat
transfer

Fig 3: Top row, left to right: the temperature, the horizontal velocity
($u^{\prime}_{x}$) and the vertical velocity ($u^{\prime}_{z}$) at the $y=0$
mid-plane of the cell, computed by simulating the advection-diffusion equation
(eqn. (11)) for the temperature, and the Stokes equations (eqn. (9)) for flow
field. Bottom row: temperature and flow fields, obtained by simulating the
diffusion equation (eqn. (13)) for the temperature, and the Stokes equations
(eqn. (9)) for the flow field. The arrows in the velocity contour plots denote
velocity vectors, with arrow lengths proportional to the magnitude of the
local velocity.

We show two different simulation results in Fig 3. The top row corresponds to
simulations where we solve the full advection-diffusion equation (eqn. (11))
in conjunction with the hydrodynamic problem; these are the same results as in
the top row of Fig 2 but plotted with a different colour scale in order to
highlight the flow variations inside the cell. Given the relation between
density and temperature in eqn. (3), clearly the fluid near the nucleus is
lighter, while the fluid near the cell membrane is heavier. Thus, as expected
from physical intuition, the warmer fluid rises under the influence of
gravity, while the colder fluid settles, leading to cell-scale circulation in
the bulk. Note that we focus predominantly on the flow in the vertical plane
$y^{\prime}=0$, which is also a plane of symmetry for the cell, since the
flows are strongest and most intuitively visualised here; we comment briefly
in Appendix A on the nature of horizontal flow along the $y^{\prime}$
direction.

In the bottom row of Fig 3, on the other hand, we show results where advective
heat transfer is neglected and the thermal problem is governed solely by
Laplace’s equation,

$\nabla^{2}\Theta=0,$ (13)

along with the boundary conditions (eqn. (12)). The latter choice of
simulations is motivated by the particularly low value of the thermal Péclet
number in the present problem, $\textrm{Pe}_{\textrm{t}}\sim 10^{-5}$ (see
Table 1), hinting that heat transport is likely dominated by diffusion.

Comparing the two rows of Fig 3, we see that the temperature and velocity
fields are practically identical in the simulations with and without
temperature advection by the flow. The transfer of heat due to the convective
flow is thus essentially negligible in comparison to its rapid diffusion
throughout the cytoplasm. Interestingly, as a result, the temperature field is
symmetric around the line joining the nucleus and the cell centres, indicative
of isotropic diffusion from a warmer to a colder surface, with no breaking of
symmetry due to the convective flow.

###  3.3 Weak advective effects result in one-way coupling between temperature
and flow fields

The fact that advection appears to play a negligible role in influencing the
intracellular temperature distribution offers a great simplification in terms
of solving the thermal and hydrodynamic problems. We do not need solve the
full advection-diffusion equation, eqn. (11), for the dimensionless
temperature $\Theta$, but can instead consider just the diffusion equation
given by eqn. (13). It then follows that eqns. (9) and (13) are only one-way
coupled, with the flow being affected by the temperature, but not vice-versa.
One can thus first obtain $\Theta$ and then solve the Stokes flow problem
driven by the non-homogeneous temperature field
$\Theta\left(\mathbf{x}\right)$.

Equipped with the above understanding, we are now in a position to explore
further the order-of-magnitude disparity apparent in Fig 2. In the subsequent
sections, we compare our numerical simulation results with solutions obtained
through other solution strategies that can be employed in slightly different
geometries. We focus first on the geometric limit $e_{x}=0$, i.e. an
axisymmetric placement of the nucleus within the cell. Next, we move to an
even simpler configuration, wherein the nucleus is concentric with the cell,
i.e. $\left(e_{x}=0,e_{z}=0\right)$. We will show that solutions in these two
limits, obtained via classical techniques, match very well with our simulation
results. This will thus confirm the accuracy of our numerical simulations and
establish that temperature-gradient-driven intracellular flows are indeed
significantly weaker than predictions made in earlier studies.

##  4 Temperature and flow in eccentric but axisymmetric configurations

Fig 4: Flow in axisymmetric cell-nucleus configuration. (a) Depiction of the
bi-spherical coordinate grid. (b) An example of dimensionless results obtained
via the bi-spherical coordinate calculations. Left: normalised temperature
distribution, $\Theta$; right: vertical flow, $u_{z}$. The geometric
parameters are: radius ratio $\kappa=0.43$, eccentricities $e_{x}=0$ and
$e_{z}=0.3$.

In this section, we consider an axisymmetric arrangement of the nucleus within
the cell, i.e. when the line joining the nucleus and cell centres is directed
along (or opposite to) the gravity vector.

Fig 5: Dimensionless values of the (a) maximum upwelling velocity,
max.{$u_{z}$}, and (b) minimum downwelling velocity, min.{$u_{z}$}, as a
function of the vertical eccentricity $e_{z}$ for $\kappa=0.2,0.3,0.4$,
obtained using the semi-analytical bi-spherical coordinates calculations
(empty symbols) and via COMSOL simulations (filled symbols).

###  4.1 Outline of solution in axisymmetric geometry

An important implication of analysis in axisymmetric geometry is the symmetry
of the flow field around the axis between the nucleus and the centre of the
cell. We may exploit this symmetry to solve eqns. (9) and (13) semi-
analytically in a bi-spherical coordinate system,
$\left(\xi,\chi,\phi\right)$, shown in Fig 4(a), where the nuclear membrane
and the cell membrane can be represented by distinct coordinate surfaces [32].
Axisymmetry allows the flow field to be represented in terms of a single
Stokes streamfunction, $\Psi^{\textrm{b}}\left(\xi,\chi\right)$, which can be
expressed as a linear superposition of harmonic functions [32]. The technical
aspects of the solution are detailed in Appendix D. A sample solution in the
axisymmetric case is shown in Fig 4(b), where the $z$-axis is the axis of
symmetry and the geometric parameters are $\kappa=0.43,e_{x}=0,e_{z}=0.3$
(left: temperature; right: vertical velocity). We can also solve the
axisymmetric problem using finite-element COMSOL simulations. In Fig 5, we
compare the results obtained from the two different methods, by plotting the
dimensionless maximum $\left(\textrm{max}.\left\\{u_{z}\right\\}\right)$ and
minimum $\left(\textrm{min}.\left\\{u_{z}\right\\}\right)$ vertical velocities
inside the cell, as a function of the eccentricity $e_{z}$. We provide further
validation of the temperature and velocity fields in Fig 16 in Appendix D. The
excellent matching between the two sets of results validates the non-
axisymmetric simulations from Sec. 3.

Fig 6: Symmetry of the temperature $\Theta$ (panels (a) and (c)) and the flow
magnitude $\left|\mathbf{u}\right|$ (panels (b) and (d)) inside the cell with
respect to a change in sign of the nucleus’s vertical eccentricity $e_{z}$.
The arrows are velocity vectors with lengths proportional to the velocity
magnitude shown in the contour plots.

###  4.2 Estimating the flow strength

The typical magnitude of the intracellular flow velocities can be estimated by
multiplying the dimensionless results in Fig 5 by the reference velocity
$u_{\textrm{ref}}\approx 0.5$ $\mu$m s-1 used to non-dimensionalise the
equations (see Table 1); this yields a maximum velocity of $\sim 10^{-3}$
$\mu$m s-1, similar to what was displayed in Fig 3. Thus, the axisymmetric
solutions provide an independent measure of the order of magnitude of cellular
flows predicted by our numerical simulations in Sec. 3 and confirm the ten-
fold discrepancy with the results of Ref. [23]. We note here that the value of
the reference velocity $u_{\textrm{ref}}\approx 0.5$ $\mu$m s-1 is based on
the assumption that the cytoplasm viscosity is the same as that of water at
37$\degree$C, i.e. $\eta\approx 7\times 10^{-4}$ kg m-1 s-1 (see Table 1). It
is well-known that the cytoplasm can often display elevated viscosities, as
high as $\eta\sim 10^{-2}$–10-1 kg m-1 s-1 [30, 31], which would only serve to
reduce $u_{\textrm{ref}}$ by one to two orders of magnitude (due to the
scaling $u_{\textrm{ref}}\propto\eta^{-1}$), and result in even weaker thermal
convection. Physically, for a given temperature difference (between the cell
and nuclear membranes) driving the flow, a more viscous cytoplasm offers
stronger resistance to motion.

Intuitively, we would of course not expect the velocity magnitudes to change
drastically as the problem geometry varies. We confirm this in Fig 5, where
both the maximum and minimum velocities experience only a modest change with
the vertical eccentricity $e_{z}$ for all radius ratios $\kappa$.

###  4.3 Flow structure as a function of eccentricity

We see that the results are symmetric about $e_{z}=0$, which is a result of
the structure of eqns. (9) and (13). The temperature distribution in the cell
is reflected about the $z=0$ plane as $e_{z}$ goes from negative to positive
(see Figs 6(a) and 6(c)). Since the fluid flow is directly proportional to the
local temperature difference, regions in the cell corresponding to the same
thermal environment in the two geometries also display the same vertical
velocity (Figs 6(b) and 6(d)). As a result, the location of the strongest
upwelling flow with respect to the nucleus changes: from being above the
centre of the nucleus for $e_{z}<0$, to shifting below the nucleus centre for
$e_{z}>0$.

###  4.4 Influence of temperature difference on flow strength

The linearity of the dimensionless flow $\mathbf{u}$ with respect to $\Theta$,
evident in eqn. (9), also lets us immediately deduce that the dimensional flow
strength $\left|\mathbf{u}^{\prime}\right|$ varies linearly with the
temperature difference $\Delta T$. This linear dependence was indeed captured
explicitly in the numerical simulations of Ref. [23] (see their Fig 3). A
clear benefit of our analysis is that, with the knowledge that heat advection
can be safely neglected (i.e. $\textrm{Pe}_{\textrm{t}}\ll 1$), the linear
relationship follows exactly mathematically.

###  4.5 Summary

In summary, assuming an axisymmetric geometry allowed us to solve for the
fluid flow via a semi-analytical approach that is less computationally
intensive than the finite-element simulations. By performing two independent
sets of calculations in the axisymmetric limit (finite-element simulations and
the Stokes streamfunction analysis of this section) we have confirmed the
velocity magnitudes that were predicted numerically in Sec. 3. In the next
section, we perform a third analysis, in which we consider the limit where the
nucleus is located at the centre of the model cell (i.e. $e_{x}=e_{z}=0$).
This particular geometry enables us to obtain an exact, analytical solution
for the temperature distribution and the flow field.

##  5 Temperature and flow in the concentric limit

###  5.1 An exact solution

When the nucleus is concentric with the cell, one can solve the problem fully
analytically in a spherical coordinate system, $\left(r,\theta,\phi\right)$
[25]. Since we still have an axisymmetric setup, we can once again represent
the flow field in terms of a streamfunction,
$\Psi^{\textrm{s}}\left(r,\theta\right)$, where $r$ is the radial separation
of any point, measured from the centre of the cell (and the nucleus), and
$\theta$ is the polar angle measured from the positive $z$-axis [32]. We
discuss in the main text the final results, i.e. the analytical expressions
for the temperature and the fluid velocity; all details of the solution
methodology are provided in Appendix E. The temperature is found to be
radially isotropic (i.e. it only depends on $r$) and is given by

$\Theta\left(r\right)=\frac{\kappa}{1-\kappa}\left(\frac{1}{r}-1\right).$ (14)

As for the flow field, we derive in Appendix E a very simple representation
for the streamfunction $\Psi^{\textrm{s}}\left(r,\theta\right)$ as

$\Psi^{\textrm{s}}(r,\theta)=f_{1}(r)\sin^{2}\theta,$ (15)

where

$f_{1}(r)=\frac{r^{3}}{8}\frac{\kappa}{1-\kappa}+c_{1}r+\frac{c_{2}}{r}+c_{3}r^{2}+c_{4}r^{4},$
(16)

with the constants, $c_{i}$, being functions solely of the radius ratio
$\kappa$ given by

$\displaystyle c_{1}$
$\displaystyle=\frac{\kappa^{2}(3\kappa^{2}+4\kappa+3)}{8(1-\kappa)(4\kappa^{2}+7\kappa+4)},$
(17) $\displaystyle c_{2}$
$\displaystyle=-\frac{\kappa^{4}}{8(1-\kappa)(4\kappa^{2}+7\kappa+4)},$
$\displaystyle c_{3}$
$\displaystyle=-\frac{\kappa(\kappa+1)(\kappa^{2}+3\kappa+1)}{4(1-\kappa)(4\kappa^{2}+7\kappa+4)},$
$\displaystyle c_{4}$
$\displaystyle=-\frac{\kappa(\kappa+1)}{4(1-\kappa)(4\kappa^{2}+7\kappa+4)}.$

Once the streamfunction is known, the fluid velocity $\mathbf{u}$ may be
obtained explicitly as

$\displaystyle\mathbf{u}=\left(u_{r},u_{\theta}\right)=\left(-\frac{1}{r^{2}\sin\theta}\frac{\partial\Psi^{\textrm{s}}}{\partial\theta},\frac{1}{r\sin\theta}\frac{\partial\Psi^{\textrm{s}}}{\partial
r}\right),$ (18)

where $u_{r}$ is the radial component and $u_{\theta}$ the tangential
component of the fluid velocity. Substituting eqn. (15) in (18) yields,

$\displaystyle u_{r}\left(r,\theta\right)$
$\displaystyle=-\frac{2f_{1}\left(r\right)}{r^{2}}\cos\theta,$ (19)
$\displaystyle u_{\theta}\left(r,\theta\right)$
$\displaystyle=\frac{1}{r}\frac{df_{1}}{dr}\sin\theta.$

We thus have a closed-form, exact solution to the temperature-gradient-driven
natural convection problem in the concentric case, for arbitrary radius ratios
$\kappa=R_{\textrm{nuc}}/R_{\textrm{c}}$. A simple transformation from
spherical to Cartesian coordinates then yields the vertical velocity $u_{z}$
and the horizontal velocity $u_{x}$ (see eqn. (58) in Appendix E). In Fig 7,
we plot these velocity components (left column) and compare them with the
corresponding COMSOL simulation results (right column). We can see an
excellent agreement between the exact solution and our computations;
quantitatively, the average relative error between the two solutions is less
than 2%, which provides another validation of the simulations of Sec. 3 and
confirms that the velocity predictions therein are accurate.

Fig 7: Dimensionless vertical component of the flow field at the $y=0$ mid-
plane (panels (a) and (b)) and horizontal component (panels (c) and (d)),
calculated analytically using eqn. (58) (left column) and via COMSOL
simulations (right column): geometrical parameters are
$\kappa=0.43,e_{x}=0,e_{z}=0$. The symbols in panel (a) are locations of the
maximum and minimum velocities, whose nearest distance from the cell membrane
(i.e. from the point $x=1$, $y=0$, $z=0$) is plotted in Fig 8(c). The arrows
indicate the in-plane velocity.

###  5.2 Structure of flow in concentric geometry

Because of symmetry about the $z$-axis, the fluid from the right-half of the
cell cannot flow into the left-half. This, combined with mass conservation,
means that the net flow-rate through the annulus,
$Q_{\textrm{ann}}\sim\int_{\kappa}^{1}{u_{z}(r,\theta=\pi/2)r\,dr}$, has to be
exactly zero. Hence, any upwelling flow perpendicular to the equatorial plane
($z=0$, alternatively $\theta=\pi/2$) must be cancelled out, in an averaged
sense, by a downwelling flow. This can clearly be seen in Fig 7(a). The
(vertical) velocity at the equatorial plane is given by

$u_{z}\left(r,\theta=\pi/2\right)=-u_{\theta}\left(r,\theta=\pi/2\right)=-\frac{1}{r}\frac{df_{1}}{dr},$
(20)

which we plot in Fig 8(a) as a function of a normalised separation from the
nucleus surface, $r^{*}\equiv{(r-\kappa)}/{(1-\kappa)}$; thus, $r^{*}=0$
corresponds to the nucleus surface, while $r^{*}=1$ corresponds to the cell
membrane.

For each value of $\kappa$, the maximum and minimum velocities in Fig 8(a)
correspond to the strongest upwelling and downwelling flows, respectively, in
the entire cell. We further plot in Fig 8(b) the magnitude of these strongest
vertical flows as a function of the radius ratio $\kappa$, while their
position, as measured radially from the cell membrane, is displayed in Fig
8(c). We observe a clear asymmetry in the flow as seen in Figs 8(a) and 8(b);
in terms of magnitude, the strongest upwelling flow is greater than the
strongest downwelling flow. This asymmetry is a consequence of spherical
geometry: the differential area through which the fluid flows, $dA(r)=2\pi
rdr$, increases with the radius $r$, which necessitates that the fluid near
the nucleus must rise faster than the fluid near the cell membrane settles, in
order to maintain zero net-flux in the annulus. This asymmetry thus vanishes
when $\kappa\to 1$.

Fig 8: Upwelling and downwelling convective flows. (a) Dimensionless vertical
fluid velocity at the equatorial plane, plotted as a function of normalised
radial distance from the nucleus, $r^{*}$. (b) Variation with the radius
ratio, $\kappa$, of the magnitudes of the strongest upwelling
$\left(\textrm{max}.\left\\{u_{z}\right\\}\right)$ and downwelling
$\left(-\textrm{min}.\left\\{u_{z}\right\\}\right)$ flows. (c) Radial
separations $d_{\textrm{max}}$ and $d_{\textrm{min}}$, from the cell membrane
of the regions of maximum and minimum velocity, respectively.

###  5.3 Influence of a more general cell geometry

The results in eqns. (15) to (19) give us an exact representation of the flow
inside the cell, in this simple geometrical case. How relevant are these
predictions for arbitrary locations of the nucleus? Further computations shown
in Fig 13 (Appendix B) demonstrate that the strength of the flow depends only
weakly on the position of the nucleus. Therefore, the exact analysis of this
section captures all essential aspects of this hydrodynamics problem.

###  5.4 Summary

We have now validated our numerical simulations using two independent
analyses, which gives us confidence in our numerical results from Sec. 3. We
thus conclude that the flow strengths that we report are a more accurate
estimate of intracellular buoyancy-driven flows than the results of Ref. [23].

##  6 Solute transport by combined convection and diffusion

Having solved for the fluid flow numerically, semi-analytically and fully
analytically, we have concluded that temperature differences between a cell’s
nucleus and cell membrane lead to much weaker flows than envisioned
previously. Importantly, our analysis shows that quantitative predictions on
transport cannot be made solely using scaling arguments, but that they instead
require solving for the details of the flow field. Indeed, the standard
reference velocity $u_{\textrm{ref}}=\rho_{0}\beta g\Delta TR^{2}_{c}/\eta$,
obtained by scaling arguments, predicts typical velocities that are around 0.5
$\mu$m s-1 (for $\Delta T=1\;$K), whereas full calculations yield values of
$\sim O(10^{-3})$ $\mu$m s-1, which are two orders of magnitude smaller.

We now investigate the ability of these flows to transport solute inside the
cell, e.g. proteins and other macromolecules [37].

###  6.1 Solute transport with both diffusion and advection

We model the solute as a passive scalar that diffuses and is advected by the
flow calculated in Sec. 3. At steady state, the dimensionless concentration of
the scalar, $C$, is governed by the advection-diffusion equation

$\mathbf{u}\cdot\nabla C=\frac{1}{\textrm{Pe}_{\textrm{s}}}\nabla^{2}C,$ (21)

where $\textrm{Pe}_{\textrm{s}}=u_{\textrm{ref}}R_{\textrm{c}}/D$ is the
Péclet number of the solute (the ratio of its advective to its diffusive
transport), where $D$ is the solute’s molecular diffusivity. For the sake of
simplicity, we assume that the scalar is produced at the nucleus and absorbed
at the cell membrane, so we have the boundary conditions

$\displaystyle C_{\textrm{nuc}}$ $\displaystyle=1,$ (22) $\displaystyle
C_{\textrm{mem}}$ $\displaystyle=0.$

Typical molecular diffusivities of cellular matter can be as low as 0.01
$\mu$m2 s-1 [30, 38], which corresponds to $\textrm{Pe}_{\textrm{s}}\approx
500$. Based on this preliminary scaling approach, one may then expect the
cell-scale natural convection described previously to significantly affect
solute transport.

###  6.2 Advection-enhanced transport: the local Péclet number

To investigate the extent to which this is true, we solve eqns. (21) and (22)
for the configuration and parameters of Sec. 3
$\left(\kappa=0.43,e_{x}=e_{z}=0.25,\Delta T=1\;\textrm{K}\right)$ using
finite-element COMSOL simulations. The resulting solute distributions on the
$y=0$ mid-plane are shown in Fig 9. Surprisingly, the effect of natural
convection is so weak that even for $\textrm{Pe}_{\textrm{s}}$ as large as 500
(Fig 9(a)), the solute distribution is almost the same as that when the solute
transport is purely diffusive, i.e. when $\textrm{Pe}_{\textrm{s}}\ll 1$ (Fig
9(c)). One can also visualise the time evolution of the concentration,
starting from a uniform initial condition $C\left(t^{\prime}=0\right)\equiv 0$
and then suddenly changing the concentration at the nucleus surface to
$C_{\textrm{nuc}}=1$ at $t^{\prime}=0.1$ s; this is done in Appendix C. As
shown in Fig 14 there, the concentration distribution becomes nearly
independent of $\textrm{Pe}_{\textrm{s}}$ when the time is normalised by the
diffusive time-scale $t^{\prime}_{\textrm{d}}=R_{\textrm{c}}^{2}/D$, which is
a classic signature of isotropic diffusion. This further confirms that the
transport mechanism inside the cell is mostly diffusive.

Fig 9: Influence of the temperature-gradient-driven convection flow on the
distribution of a passive scalar with three different diffusivities, (a)
$D=0.01$ $\mu$m2 s-1, (b) $D=1$ $\mu$m2 s-1 and (c) $D=100$ $\mu$m2 s-1.

For given size and material properties of the cell, the definition of
$\textrm{Pe}_{\textrm{s}}$ (see Table 1) implies that the main factors
influencing solute transfer are the temperature difference $\Delta T$ and the
solute diffusivity $D$. To quantify their effects, we may define a local
Péclet number for the solute,
Pe${}^{\ell}=2\left|\mathbf{u}^{\prime}\right|R_{\textrm{c}}/D$, with maximum
value Pe${}^{\ell}_{\textrm{max}}=2u^{\prime}_{\textrm{max}}R_{\textrm{c}}/D$.
Recall here that $\mathbf{u}^{\prime}$ is the dimensional fluid velocity,
which is a function of position within the cytoplasm, while
$u^{\prime}_{\textrm{max}}\equiv\textrm{max}.\\{|\mathbf{u}^{\prime}|\\}$ is
the global maximum magnitude of the fluid velocity.

Fig 10: Quantifying transport arising from convective flows. (a) Maximum local
Péclet number, Pe${}^{\ell}_{\textrm{max}}$, as a function of temperature
difference $\Delta T$ and solute diffusivity $D$, obtained through non-linear
simulations in COMSOL. (b) Same maximum local Péclet number as obtained
numerically in Ref. [23]. (c) Normalised solute flux $\mathcal{J}$ as a
function of $D$ for different prescribed temperature differences. In all
cases, the geometry corresponds to that in Fig 9.

In order to understand the weak impact of convective flows on transport, we
plot in in Fig 10(a) the iso-values of the maximum local Péclet number,
Pe${}^{\ell}_{\textrm{max}}$, for different sets of temperature difference
($\Delta T$) and diffusivity ($D$). We also contrast our computational results
with the same quantity obtained in Ref. [23] in their simulations and
reproduce their results in Fig 10(b). Once again, it is notable that our
results have local Péclet numbers about one order of magnitude smaller than
those in Ref. [23].

###  6.3 Absorption flux across nuclear membrane

While the maximum local Péclet number $\textrm{Pe}^{\ell}_{\textrm{max}}$
captures a local estimate of mass transfer enhancement by cytoplasmic
convection, a fairer quantification of this enhancement at the whole-cell
level involves calculating the total absorption flux across the nuclear
membrane, and comparing it to the limit when the solute emitted by the nucleus
instead undergoes pure diffusion. If $S$ denotes the surface of the nucleus,
then the dimensionless diffusive flux through the surface is given by the
integral $\int_{S}{-\mathbf{n}\cdot\nabla C\,dS}$, while the dimensionless
advective flux, $\int_{S}{\mathbf{n}\cdot\mathbf{u}C\,dS}$, vanishes due to
impermeability of the nuclear surface (see eqn. (4)). The influence of bulk
solute advection on the mass transfer rate across the nucleus is quantified by
defining a normalised flux,

$\mathcal{J}=\frac{\int_{S}\mathbf{n}\cdot\nabla
C\,dS}{\left.\left(\int_{S}\mathbf{n}\cdot\nabla
C\,dS\right)\right|_{\textrm{Pe}_{\textrm{s}}\equiv 0}},$ (23)

where the denominator is the solute flux through the nucleus when the solute
transport is purely diffusive (i.e. when $\textrm{Pe}_{\textrm{s}}\equiv 0$ in
eqn. (21)). In the mass transfer literature, the quantity $\mathcal{J}$ is
usually referred to as the Sherwood number [33] and is expected to be a
function of the cell geometry and the solutal Péclet number,
$\textrm{Pe}_{\textrm{s}}$.

###  6.4 Convective flows are too small to provide significant enhancement of
absorption flux

If bulk advection of solute were to significantly impact the solute transfer
from the nucleus to the cell membrane, then we would expect $\mathcal{J}$ to
be greater than unity. We plot in Fig 10(c) the dependence of this normalised
flux on the diffusivity of the solute ($D$) for various temperature
differences ($\Delta T$). The solute diffusivity $D$ spans four orders of
magnitude, covering cellular material ranging from large vesicles ($D\sim
10^{-2}$ $\mu$m2 s-1 [39, 40]), to proteins synthesised in the ribosomes
($D\sim 1$ $\mu$m2 s-1 to $10$ $\mu$m2 s-1 [30, 41]), to metabolites such as
ATP, ADP and calcium ions ($D\sim 100$ $\mu$m2 s-1 [42, 43]).

For the lowest $\Delta T$ value considered, the convective flows do not
provide any advantage in driving the solute. Even for $\Delta T$ as large as
10 K (a large value not supported by experimental data) and solute
diffusivities as low as 0.01 $\mu$m2 s-1 (relevant for the largest vesicles in
the cell), the solute removal from the nucleus is only amplified by about 15%
compared with the case of pure diffusion.

As an example, consider the protein insulin, whose diffusivity in living cells
is around 1 $\mu$m2 s-1 [30]. The result in Fig 10(c) then tells us that
$\mathcal{J}\sim 1$ for any range of experimentally observed intracellular
temperature differences. Most cellular proteins and macromolecules have
diffusivities that are too large (ranging from 1 $\mu$m2 s-1 to 10 $\mu$m2
s-1) to be overcome by natural convection. Hence, their motion through the
cell likely remains diffusive. Vesicles in cells can have very low
diffusivities, around $10^{-2}$ $\mu$m2 s-1 [39, 40] but, despite this, for
the commonly measured temperature differences within cells, their transport
enhancement due to convective flows is expected to remain, at best, very
modest, and most likely, completely negligible.

##  7 Discussion

###  7.1 Intracellular convective flows are smaller than previously predicted

In this work, we re-considered the problem of intracellular natural convection
potentially induced by a temperature difference $\Delta T$ between the cell’s
nucleus and membrane. Specifically, we used numerical simulations to show that
intracellular flows are expected to be on the order of $\sim 10^{-3}$ $\mu$m
s-1 in magnitude for a nucleus-to-membrane temperature difference of 1 K. In
the process, we discovered an important disparity from previous results that
had predicted ten-fold stronger flows for the same geometry and temperature
differential [23].

We explored further this discrepancy, and confirmed it, by performing
independent calculations of the flow strength in an axisymmetric geometry
(Sec. 4) and in a concentric geometry (Sec. 5). Importantly, in the latter
case, we provided an exact expression for the flow inside the cell. All our
analyses converged to a similar value for the flow strength, which was indeed
$\sim 10$ times weaker than that computed in Ref. [23]. These results
highlight the importance of detailed flow calculations, since simple scaling
analysis alone suggests fluid velocities almost two orders of magnitude larger
($u_{\textrm{ref}}\sim 0.5$ $\mu$m s-1), which would lead one to believe,
incorrectly, that these convective flows are very strong.

###  7.2 Intracellular convection does not lead to significant increase in
chemical transport beyond diffusion

The transport of materials within a cell is essential to its normal function,
from vesicle transport for structural upkeep [44], signal transmission via
proteins [41] and organelle transport during cell division [45, 46] to inter-
organelle interactions [47] and the maintenance of nutrient/metabolite
gradients [48]. The movement of cellular matter is accomplished by a variety
of mechanisms, including molecular diffusion, active transport by motor
proteins and advection due to cytoplasmic flows [37]. A second aim of our work
was to investigate whether temperature-gradient-driven flows could contribute
usefully to advective material transport inside the cell. This would be
particularly relevant for the motion of cell constituents with low
diffusivities (large organelles and vesicles), which necessarily require
alternative mechanisms to traverse the cell. Towards this, we numerically
simulated the mass transfer of a chemical species by cytoplasmic convection,
and showed that convection does not contribute significantly to the transport
of material within a cell beyond what is achieved by pure diffusion. We
quantified the advection-induced enhancement in averaged mass-flux of a
chemical species released from the nucleus and absorbed at the cell membrane,
as a function of the prescribed temperature difference and the diffusivity of
the species. Only in situations that are biologically unrealistic (temperature
differences of 10 K and very small diffusivities of 0.01 $\mu$m2 s-1,
associated with cellular vesicles) could the averaged mass-flux see a modest
increase above that obtained with purely diffusive mass transfer.

Thus, while a cell’s thermal environment is important for its metabolism [49,
3] and survival [50], the flows generated by temperature gradients, in most
realistic scenarios, have negligible effect in improving intracellular mass
transfer. It seems therefore that these flows do not noticeably impact
important cell processes like protein delivery, signal propagation, and
organelle and metabolite transport.

Once again, this conclusion becomes apparent only after a detailed calculation
of solute transport in the bulk, as done in Sec. 6. A simple scaling analysis
yields characteristic solutal Péclet numbers
$\textrm{Pe}_{\textrm{s}}=u_{\textrm{ref}}R_{\textrm{c}}/D\approx 500$ that
are much larger than unity, which, in the absence of detailed numerical
calculations, would erroneously suggest a significant natural-convection-
induced advective contribution to species transport.

###  7.3 A broader range of modelling assumptions could be explored in future
work

One of the reasons for the aforementioned flows being so weak is that we have
modelled the nucleus surface and the cell membrane as rigid and non-slipping,
i.e. the fluid’s tangential velocity vanishes at both these surfaces. As a
result, there are significant viscous stresses in the domain that resist fluid
motion. A different model could allow tangential motion along the cell surface
due to slip or mobility of the membrane.

Our model simplifies the cytoplasm and assumes that it is effectively a
homogeneous continuum with Newtonian properties. It is obviously known to be
more complex, consisting of a polymeric and dynamic cytoskeleton embedded in a
viscoelastic, gel-like fluid that flows past freely suspended cell organelles
[31].

Our analysis also considers a simplified description of thermal diffusion
inside the cell. Inherent in eqn. (5) is the assumption that the thermal
diffusivity of the cytoplasm is isotropic, i.e. that heat diffuses at a
constant rate along all directions. Since the cytoplasm is heterogeneous, the
thermal diffusivity within it is expected to vary spatially [51].

###  7.4 Our model could be adapted to quantify artificially induced
convection

Our calculations show that cytoplasmic flows due to naturally occurring
temperature gradients within a cell are not very effective in driving cellular
material transport. However, artificially induced thermal convection has been
recently proposed as a means to accelerate cell assembly for biomedical assays
[52]. Light-absorbing particles are arrested in optical traps, are heated in
the process and thus induce thermal flows that drive cell accumulation around
the particles. One could envision the generation of such flows within a cell
as well, for example, by laser-assisted heating of metallic nano-particles
[53, 54, 55]. Our theoretical analysis could thus be adapted to quantify such
artificially induced cell-scale convection, to inform future hydrodynamics-
based strategies for intracellular object manipulation.

## Appendix

##  Appendix A Fluid velocity along the $y^{\prime}$ direction

In the main text, we discussed the nature of fluid flow in the vertical plane
through the centres of both the cell and the nucleus (the
$x^{\prime}$-$z^{\prime}$ plane). Here we discuss the importance of the other
component of the fluid velocity, i.e. the velocity $u^{\prime}_{y}$, in
comparison with $u^{\prime}_{x}$ and $u^{\prime}_{z}$. In Fig 11 we show this
horizontal velocity $u^{\prime}_{y}$ on two vertical planes
$x^{\prime}=\text{const.}$ – through the centre of the cell and through the
centre of the nucleus – along with the vertical velocity $u^{\prime}_{z}$.
Similarly, Fig 12 shows $u^{\prime}_{y}$ on five different horizontal planes
through the cell, at different heights $z^{\prime}$. The cell geometry and the
physical parameters in both these figures are the same as in Fig 3. Note that
since only the full three-dimensional flow is incompressible, the in-plane
velocity in Figs 11 and 12 is not incompressible; this is most apparent in Fig
12, where we can see local two-dimensional sources and sinks of flow. Clearly,
the typical magnitude of $u^{\prime}_{z}$ is larger than $u^{\prime}_{y}$ in
both Figs 11 and 12 (see also Fig 3). This means that the strongest flows in
the cell are vertical (along or opposite to gravity) and justifies our focus
on $u^{\prime}_{z}$ in the main text.

Fig 11: The $y^{\prime}$ component of the fluid velocity (panels (a) and (b))
and the $z^{\prime}$ component (panels (c) and (d)) at vertical sections
through the cell centre (panels (a) and (c)) and the nucleus centre (panels
(b) and (d)). The arrows denote in-plane velocity vectors. Note that the
$y^{\prime}$-axis is inverted in panels (a) and (b), for ease of visual
comparison with the contours seen in the insets. The cell geometry is the same
as in Fig 3 $\left(\kappa=0.43,\;e_{x}=e_{z}=0.25\right)$.

Fig 12: The $y^{\prime}$ component of the fluid velocity at different
horizontal sections through the cell, as shown in the inset in each figure.
The arrows denote in-plane velocity vectors. The cell geometry is the same as
in Fig 3 $\left(\kappa=0.43,\;e_{x}=e_{z}=0.25\right)$.

##  Appendix B Influence of nucleus position on fluid flow: COMSOL simulations

In this section, we use numerical simulations to briefly summarise the
dependence of the flow on the position of the nucleus inside the cell. The
angle $\theta_{e}=\tan^{-1}(e_{z}/e_{x})$ shown in Fig 1 quantifies how
axisymmetric the system is, with $\theta_{e}=\pi/2$ rad ($=90\degree$)
representing the axisymmetric case. Results plotted in Fig 13 show that, for
all values of the eccentricity $e$, the flow strength increases monotonically
as the extent of axisymmetry decreases. Intuitively, for $\theta_{e}=0$, the
majority of the fluid is least confined in the direction normal to gravity.
Hence, for any non-zero eccentricity/offset, the fluid experiences the least
viscous resistance when $\theta_{e}=0$. As a result, it attains the largest
velocities for a prescribed driving temperature difference. This can be seen
in Fig 13(c), where the fluid to the left of the nucleus is heated and attains
large upwelling speeds.

Fig 13: Impact of the extent of axisymmetry (quantified by $\theta_{e}$, shown
here in degrees) on the dimensional maximum upwelling (filled symbols) and
minimum downwelling (open symbols) flows inside the cell, for (a)
$\kappa=0.2$, (b) $\kappa=0.43$. The marker types and colours denote different
eccentricity values, whose legend is the same across both panels. The
temperature difference between the nucleus and the cell membrane in each case
is $\Delta T=1$ K. Panel (c) further shows the vertical velocity in the mid-
plane for $\theta_{e}=0$, $e=0.5$ (i.e. $e_{x}=0.5,e_{z}=0$) and
$\kappa=0.43$. The arrows indicate the in-plane velocity.

However, the increase in the dimensional flow speed
$\left|\mathbf{u}^{\prime}\right|$ as $\theta_{e}$ decreases is only modest,
and the maximum velocities inside the cell still remain $\sim 10^{-3}$ $\mu$m
s-1. Since the order of magnitude of the flow inside the cell does not change
over the entire range of parameters, the predictions from Secs. 4 and 5
provide a quantitative estimate of the convective flows occurring in more
complex configurations.

##  Appendix C Time-evolution of concentration inside the cell

While there is very little difference in the steady-state concentration
profiles in Fig 9, the time-evolution of concentration shows more discernible
changes between different values of $\textrm{Pe}_{\textrm{s}}$. In Fig 14, we
show results from COMSOL simulations, where the initial concentration is
uniform everywhere inside the cell, set as
$C\left(\textbf{x}^{\prime},t^{\prime}=0\right)=0$, and then, at dimensional
time $t^{\prime}=0.1$ s, the concentration at the nucleus surface is changed
to $C_{\textrm{nuc}}=1$. These simulations were performed for three values of
solute diffusivity: $D=10$ $\mu$m2 s-1 (Pe${}_{\textrm{s}}\approx 0.5$), $D=1$
$\mu$m2 s-1 (Pe${}_{\textrm{s}}\approx 5$) and $D=0.1$ $\mu$m2 s-1
(Pe${}_{\textrm{s}}\approx 50$). In each case, the solute concentration
evolves to a steady state over the diffusive time-scale
$t^{\prime}_{\textrm{d}}\sim R^{2}_{\textrm{c}}/D$, and the differences
between solute concentrations for different $\textrm{Pe}_{\textrm{s}}$ at a
given dimensional time $t^{\prime}$ are apparent. However, these differences
almost vanish if the times are normalised by $t^{\prime}_{\textrm{d}}$. This
is expected for an isotropic diffusive process, thus shown to be the dominant
mechanism of mass transfer inside the cell.

Fig 14:  Influence of the temperature-gradient-driven convection flow on the
unsteady intracellular distribution of a passive scalar with three different
diffusivities, $D=10$ $\mu$m2 s-1 (left column), $D=1$ $\mu$m2 s-1 (middle
column) and $D=0.1$ $\mu$m2 s-1 (right column). The top row corresponds to
dimensional time $t^{\prime}=2$ s, the middle row, $t^{\prime}=20$ s and the
bottom row, $t^{\prime}=200$ s. Note the equivalence between panels (a), (d)
and (f), and also between (b) and (e), which correspond to the same
dimensionless time $t$.

##  Appendix D Details of bi-spherical coordinate calculations

Fig 15: Depiction of the bi-spherical coordinate grid for negative (left) and
positive (right) vertical eccentricities, with $\left|e_{z}\right|=0.3$. The
radius ratio between spheres given by $\xi=\xi_{\textrm{nuc}}$ and
$\xi=\xi_{\textrm{mem}}$ is $\kappa=0.43$. The velocity components
$\left(u_{\xi},u_{\chi}\right)$ are directed normal to the
$\left(\xi,\chi\right)$ iso-surfaces and are positive toward the direction of
increasing $\left(\xi,\chi\right)$.

The geometry of the bi-spherical coordinate system is shown in Fig 15. The
Cartesian coordinates $\left(x,y,z\right)$ are expressed in terms of the bi-
spherical coordinates $\left(\xi,\chi,\phi\right)$ as

$\displaystyle x$
$\displaystyle=\frac{a\sqrt{1-\chi^{2}}}{\cosh\xi-\chi}\cos\phi,$ (24)
$\displaystyle y$
$\displaystyle=\frac{a\sqrt{1-\chi^{2}}}{\cosh\xi-\chi}\sin\phi,$
$\displaystyle z$ $\displaystyle=\frac{a\sinh\xi}{\cosh\xi-\chi},$

where $a=\left|\sinh\left(\xi_{\textrm{mem}}\right)\right|$. The bi-spherical
coordinates enable us to represent the nuclear and cell membranes as surfaces
with a constant value of the $\xi$ coordinate. These values at the nucleus
$\left(\xi_{\textrm{nuc}}\right)$ and the membrane surface
$\left(\xi_{\textrm{mem}}\right)$ are functions of the eccentricity $e_{z}$
and the radius ratio $\kappa$,

$\displaystyle\xi_{\textrm{mem}}$
$\displaystyle=-\frac{e_{z}}{\left|e_{z}\right|}\cosh^{-1}\left(\frac{1-\kappa+e_{z}^{2}}{2\left|e_{z}\right|}\right),$
(25) $\displaystyle\xi_{\textrm{nuc}}$
$\displaystyle=-\frac{e_{z}}{\left|e_{z}\right|}\cosh^{-1}\left(\frac{1-\kappa-
e_{z}^{2}}{2\kappa\left|e_{z}\right|}\right).$

Note that $\left|\xi_{\textrm{nuc}}\right|$ is always greater than
$\left|\xi_{\textrm{mem}}\right|$, and that both $\xi_{\textrm{nuc}}$ and
$\xi_{\textrm{mem}}$ are positive for $e_{z}<0$ and negative for $e_{z}>0$.
With this geometrical setup, we may now describe the solution of eqns. (9) and
(13) in bi-spherical coordinates.

###  D.1 Temperature

The general solution to Laplace’s equation for the normalised temperature
field, $\nabla^{2}\Theta=0$, in bi-spherical coordinates is given by

$\Theta(\xi,\chi)=\left(\cosh\xi-\chi\right)^{1/2}\sum_{n=0}^{\infty}{\Theta_{n}(\xi)L_{n}(\chi)},$
(26)

where $L_{n}(\chi)$ are the Legendre polynomials, which are solutions of the
ordinary differential equation

$\frac{d}{d\chi}\left\\{\left(1-\chi^{2}\right)\frac{d}{d\chi}L_{n}\left(\chi\right)\right\\}=-n\left(n+1\right)L_{n}\left(\chi\right),$
(27)

the functions $\Theta_{n}$, called temperature modes, are given by

$\Theta_{n}(\xi)=b_{n}\sinh\left\\{(n+1/2)\left|\xi\right|\right\\}+d_{n}\cosh\left\\{(n+1/2)\xi\right\\}.$
(28)

In practice, one truncates the summation in eqn. (26) to a finite number of
terms, beyond which additional terms cause negligible change in the
distribution $\Theta\left(\xi,\chi\right)$ (i.e. the solution has converged).
In the present work, we truncate the summation once the maximum relative error
caused by adding an extra term falls below 0.1%. The constants
$\left\\{b_{n},d_{n}\right\\}$ in eqn. (28) are found by applying the boundary
conditions $\Theta(\xi_{\textrm{nuc}},\chi)=1$ and
$\Theta(\xi_{\textrm{mem}},\chi)=0$, which yields

$\displaystyle b_{n}$
$\displaystyle=\frac{\sqrt{2}e^{-(n+1/2)\left|\xi_{\textrm{nuc}}\right|}\cosh\left\\{(n+1/2)\xi_{\textrm{mem}}\right\\}}{\sinh\left\\{(n+1/2)\left(\left|\xi_{\textrm{nuc}}\right|-\left|\xi_{\textrm{mem}}\right|\right)\right\\}},$
(29) $\displaystyle d_{n}$
$\displaystyle=-\frac{\sqrt{2}e^{-(n+1/2)\left|\xi_{\textrm{nuc}}\right|}\sinh\left\\{(n+1/2)\left|\xi_{\textrm{mem}}\right|\right\\}}{\sinh\left\\{(n+1/2)\left(\left|\xi_{\textrm{nuc}}\right|-\left|\xi_{\textrm{mem}}\right|\right)\right\\}}.$

The temperature problem is therefore completely solved and it provides the
driving force for the fluid flow via the Stokes eqns. (9), which we solve in
the next section.

###  D.2 Flow field

The flow field is axisymmetric and is expressed in terms of the Stokes flow
streamfunction, $\Psi^{\textrm{b}}\left(\xi,\chi\right)$, as

$\displaystyle\mathbf{u}$
$\displaystyle=u_{\xi}\mathbf{i}_{\xi}+u_{\chi}\mathbf{i}_{\chi},$ (30)
$\displaystyle=\frac{\left(\cosh\xi-\chi\right)^{2}}{a^{2}}\left(\frac{\partial\Psi^{\textrm{b}}}{\partial\chi}\mathbf{i}_{\xi}-\frac{1}{\sqrt{1-\chi^{2}}}\frac{\partial\Psi^{\textrm{b}}}{\partial\xi}\mathbf{i}_{\chi}\right),$

where the super-script ‘b’ denotes the coordinates (bi-spherical) in which the
streamfunction is defined. The velocity components
$\left(u_{\xi},u_{\chi}\right)$ are directed normal to the
$\left(\xi,\chi\right)$ iso-surfaces (see Fig 15) and are positive in the
direction of increasing $\left(\xi,\chi\right)$. The unit vectors
$\left(\mathbf{i}_{\xi},\mathbf{i}_{\chi}\right)$ are expressed in terms of
the Cartesian coordinate unit vectors
$\left(\mathbf{i}_{x},\mathbf{i}_{y},\mathbf{i}_{z}\right)$ as

$\displaystyle\mathbf{i}_{\xi}$
$\displaystyle=\frac{1-\chi\cosh\xi}{\cosh\xi-\chi}\mathbf{i}_{z}-\frac{\sqrt{1-\chi^{2}}\sinh\xi}{\cosh\xi-\chi}\left(\mathbf{i}_{x}\cos\phi+\mathbf{i}_{y}\sin\phi\right),$
(31) $\displaystyle\mathbf{i}_{\chi}$
$\displaystyle=\frac{\sqrt{1-\chi^{2}}\sinh\xi}{\cosh\xi-\chi}\mathbf{i}_{z}+\frac{1-\chi\cosh\xi}{\cosh\xi-\chi}\left(\mathbf{i}_{x}\cos\phi+\mathbf{i}_{y}\sin\phi\right).$

The equation governing the streamfunction can be derived from the Stokes eqns.
(9) as [32, 33]

$-\frac{\cosh\xi-\chi}{a\sqrt{1-\chi^{2}}}E^{2}\left[E^{2}\left(\Psi^{\textrm{b}}\right)\right]\left(-\mathbf{i}_{x}\sin\phi+\mathbf{i}_{y}\cos\phi\right)=\nabla\times\left(\Theta\mathbf{i}_{z}\right),$
(32)

where $E^{2}$ is the differential operator

$E^{2}\left(\Psi^{\textrm{b}}\right)\equiv\frac{\cosh\xi-\chi}{a^{2}}\left[\frac{\partial}{\partial\xi}\left(\left(\cosh\xi-\chi\right)\frac{\partial\Psi^{\textrm{b}}}{\partial\xi}\right)+(1-\chi^{2})\frac{\partial}{\partial\chi}\left(\left(\cosh\xi-\chi\right)\frac{\partial\Psi^{\textrm{b}}}{\partial\chi}\right)\right].$
(33)

The general solution to eqn. (32) requires us to write $\Psi^{\textrm{b}}$ as

$\Psi^{\textrm{b}}(\xi,\chi)=\left(\cosh\xi-\chi\right)^{-3/2}\sum_{n=0}^{\infty}{(1-\chi^{2})\frac{dL_{n}}{d\chi}U_{n}(\xi)},$
(34)

where $U_{n}(\xi)$ are unknown functions (velocity modes), that will depend
linearly on the temperature modes $\Theta_{n}$ (eqn. (28)). If the summations
in eqns. (26) and (34) are truncated after $N$ terms, then we need to solve
for $N$ functions $U_{n}(\xi)$, $1\leq n\leq N$. Substituting eqns. (34) and
(26) into eqn. (32) yields

$\displaystyle\sum_{n=0}^{\infty}{-\Gamma^{3}\frac{2(n+1)}{a^{4}}\frac{L_{n+1}(\chi)-\chi
L_{n}(\chi)}{1-\chi^{2}}\mathcal{E}^{\textrm{b}}_{n}(\xi)}$
$\displaystyle=\sum_{p=0}^{\infty}{\left(1-\chi\cosh\xi\right)\frac{dL_{p}}{d\chi}\Theta_{p}(\xi)}$
(35)
$\displaystyle-\sum_{p=0}^{\infty}{\frac{L_{p}(\chi)}{2}\left(\Theta_{p}(\xi)\cosh\xi+2\frac{d\Theta_{p}}{d\xi}\sinh\xi\right)},$

where $\Gamma\equiv\left(\cosh\xi-\chi\right)$, and
$\mathcal{E}^{\textrm{b}}_{n}(\xi)$ is the function

$\mathcal{E}^{\textrm{b}}_{n}(\xi)=\frac{1}{2}\frac{d^{4}U_{n}}{d\xi^{4}}-\left(n^{2}+n+\frac{5}{4}\right)\frac{d^{2}U_{n}}{d\xi^{2}}+\left(\frac{n^{4}}{2}+n^{3}-\frac{n^{2}}{4}-\frac{3n}{4}+\frac{9}{32}\right)U_{n}(\xi).$
(36)

We next multiply eqn. (35) by $(1-\chi^{2})({dL_{i}}/{d\chi})$ and integrate
over the limits $\chi=-1$ to $\chi=1$. When this projection is carried out for
$1\leq i\leq N$, it yields a system of $N$ coupled, linear ordinary
differential equations for the velocity functions
$\left\\{U_{1}(\xi),U_{2}(\xi),\;...,U_{N}(\xi)\right\\}$. These functions are
expressible in terms of the known modal distribution of the temperature
$\Theta_{p}(\xi)$. The projection onto $(1-\chi^{2})dL_{i}/d\chi$ of the
right-hand-side of eqn. (35) can be simplified using the properties of
Legendre polynomials as

$\frac{2i(i+1)}{2i+1}\left[\Theta_{i}(\xi)-\frac{\cosh\xi}{2}\left\\{\Theta_{i+1}(\xi)+\Theta_{i-1}(\xi)\right\\}-\sinh\xi\left\\{\frac{d\Theta_{i-1}/d\xi}{2i-1}-\frac{d\Theta_{i+1}/d\xi}{2i+3}\right\\}\right].$
(37)

The projection onto $(1-\chi^{2})dL_{i}/d\chi$ of the left-hand-side of eqn.
(35) yields

$-\frac{2}{a^{4}}\sum_{n=0}^{\infty}{\int_{-1}^{1}{\left(\cosh\xi-\chi\right)^{3}(n+1)\left\\{L_{n+1}(\chi)-\chi
L_{n}(\chi)\right\\}\frac{dL_{i}}{d\chi}\mathcal{E}^{\textrm{b}}_{n}(\xi)}\,d\chi},$
(38)

which can be written in short-hand as

$-\frac{2}{a^{4}}\sum_{n=0}^{\infty}{\mathcal{I}_{in}(\xi)\mathcal{E}^{\textrm{b}}_{n}(\xi)},$
(39)

where

$\displaystyle\mathcal{I}_{in}(\xi)$
$\displaystyle=\int_{-1}^{1}{\left(\cosh\xi-\chi\right)^{3}(n+1)\left\\{L_{n+1}(\chi)-\chi
L_{n}(\chi)\right\\}\frac{dL_{i}}{d\chi}\,d\chi}$ (40)
$\displaystyle=\int_{-1}^{1}{i(n+1)\left(\cosh\xi-\chi\right)^{3}\left\\{L_{n+1}(\chi)-\chi
L_{n}(\chi)\right\\}\frac{\chi
L_{i}(\chi)-L_{i-1}(\chi)}{\chi^{2}-1}\,d\chi}.$

The second line of eqn. (40) is obtained by making use of the following
recurrence relation for Legendre polynomials,

$\frac{1-\chi^{2}}{i}\frac{dL_{i}}{d\chi}=L_{i-1}(\chi)-\chi L_{i}(\chi).$

In left-hand-side of eqn. (40), the index ‘$i$’ in $\mathcal{I}_{in}$ denotes
the term onto which the projection is made: $(1-\chi^{2})dL_{i}/d\chi$, and
the index ‘$n$’ denotes the contribution to the projection from the
$n^{\textrm{th}}$ term in the expansion on the left-hand-side of eqn. (35).

The next step is to write out (39) for a discrete number of points, say $M$,
along $\xi$. Thus, each mode $U_{n}(\xi)$ is evaluated on $M$ points:
$\left\\{U_{n}(\xi_{1}),U_{n}(\xi_{2}),\;...,U_{n}(\xi_{M})\right\\}$, where
$\xi_{1}=\xi_{\textrm{nuc}}$ and $\xi_{M}=\xi_{\textrm{mem}}$. The discrete
version of eqn. (39) at $\xi=\xi_{l}\;\left(1\leq l\leq M\right)$ reads

$-\frac{2}{a^{4}}\sum_{n=0}^{\infty}{\mathscr{I}_{lin}\mathcal{E}^{\textrm{b}}_{n}(\xi_{l})},$
(41)

where $\mathscr{I}_{lin}$ is a three-dimensional matrix resulting from the
evaluation of $\mathcal{I}_{in}(\xi)$ at $\xi=\xi_{l}$, i.e.
$\mathscr{I}_{lin}=\mathcal{I}_{in}(\xi=\xi_{l})$. If $N$ is the number of
modes (or the upper limit of the summation) at which the temperature (eqn.
(26)) and the streamfunction (eqn. (34)) expansions are truncated, then the
size of $\mathscr{I}_{lin}$ is $M\times N\times N$. At this level of modal
resolution/truncation, eqn. (35) written for $\xi=\xi_{l}$ and projected onto
$\left(1-\chi^{2}\right)\frac{dL_{i}}{d\chi}$ is thus given by

$\displaystyle-\frac{2}{a^{4}}\sum_{n=0}^{N}{\mathscr{I}_{lin}\mathcal{E}^{\textrm{b}}_{n}(\xi_{l})}$
$\displaystyle=\frac{2i(i+1)}{2i+1}\left[\Theta_{i}(\xi_{l})-\frac{\cosh\xi_{l}}{2}\left\\{\Theta_{i+1}(\xi_{l})+\Theta_{i-1}(\xi_{l})\right\\}\right]$
(42)
$\displaystyle-\frac{2i(i+1)}{2i+1}\left[\sinh\xi_{l}\left.\left(\frac{d\Theta_{i-1}/d\xi}{2i-1}-\frac{d\Theta_{i+1}/d\xi}{2i+3}\right)\right|_{\xi=\xi_{l}}\right].$

While the right-hand-side of eqn. (42) is explicitly known (see eqn. (28)),
the derivatives of the functions $U_{n}(\xi)$ (inherent in definition of
$\mathcal{E}^{\textrm{b}}_{n}(\xi)$; see eqn. (36)) in the left-hand-side need
to be discretised. In the present work, we use second-order accurate finite
differences for this discretisation. This leads to a total of $N\times M$
unknowns, $U_{n}(\xi_{l})$, where $1\leq n\leq N$ and $1\leq l\leq M$.

Fig 16: Comparison of the dimensionless horizontal components of the flow
field at the $y=0$ mid-plane (panels (a) and (b)) and vertical components
(panels (c) and (d)), calculated using the bi-spherical coordinates
methodology (left column) and via COMSOL simulations (right column). The
geometrical parameters are given by $\kappa=0.43,e_{x}=0,e_{z}=0.3$.

###  D.3 Boundary conditions for the flow

Eqn. (42) is a fourth-order, linear, coupled ordinary differential equation
for the functions $U_{n}(\xi)$. It is supplemented by four boundary conditions
involving velocity components $\left(u_{\xi},u_{\chi}\right)$ at the inner and
outer sphere surfaces (i.e. at the nucleus and the cell membrane),

$\displaystyle u_{\xi}\left(\xi=\xi_{\textrm{nuc}}\right)$
$\displaystyle=u_{\xi}\left(\xi=\xi_{\textrm{mem}}\right)=0,$ (43)
$\displaystyle u_{\chi}\left(\xi=\xi_{\textrm{nuc}}\right)$
$\displaystyle=u_{\chi}\left(\xi=\xi_{\textrm{mem}}\right)=0.$

Using eqn. (30), one can show that these boundary conditions yield the
following equations for $U_{n}(\xi)$ and $dU_{n}/d\xi$,

$\displaystyle U_{n}(\xi_{\textrm{nuc}})$
$\displaystyle=U_{n}(\xi_{\textrm{mem}})=0,$ (44)
$\displaystyle\left.\frac{dU_{n}}{d\xi}\right|_{\xi=\xi_{\textrm{nuc}}}$
$\displaystyle=\left.\frac{dU_{n}}{d\xi}\right|_{\xi=\xi_{\textrm{mem}}}=0.$

Eqns. (42) and (44) provide the $N\times M$ linear equations required to
obtain the unknown functions $U_{n}(\xi_{l})$. Inverting this system of
equations solves the hydrodynamic problem for the flow field
$\textbf{u}(\xi,\chi)$ (obtained via eqns. (30) and (34)). A transformation
from $\textbf{u}(\xi,\chi)$ to $\textbf{u}(x,y,z)$ (using eqn. (31)) allows us
to plot and compare the velocities obtained using the above methodology with
the finite-element COMSOL simulations. This comparison is shown in Fig 16
where we obtain essentially an identical match between the two solutions. Note
that in bi-spherical coordinates, we have solved the problem in a grid where
the $z$-coordinate of the cell centre is not given by $z=0$ (see Fig 15), but
while plotting our results to compare with those from COMSOL simulations, we
have shifted the domain along the $z$-axis such that the cell centre always
lies at $\left(x=0,z=0\right)$.

##  Appendix E Details of analytical solution in the concentric case

Fig 17: The axisymmetric spherical coordinates, with $\kappa\leq r\leq 1$ and
$0\leq\theta\leq\pi$. The velocity components $\left(u_{r},u_{\theta}\right)$
are directed normal to the $\left(r,\theta\right)$ iso-surfaces and are
positive toward the direction of increasing $\left(r,\theta\right)$.

In the concentric case, the nucleus and cell have the same centre. This
geometry is much simpler to handle than the axisymmetric (but eccentric)
arrangement of the previous section, which required bi-spherical coordinates
(Appendix D). As a result, one can solve the problem using spherical polar
coordinates and obtain fully analytical solutions for the temperature and flow
fields. While this problem has been solved in Ref. [25] in the context of
natural convection between concentric spheres at low thermal Péclet numbers,
we repeat here the solution in our notation, for the sake of completeness. We
obtain the same solution as the leading-order solution in Ref. [25], once the
notations are made consistent. The problem geometry is shown in Fig 17. The
Cartesian coordinates $\left(x,y,z\right)$ are expressed in terms of the
spherical polar coordinates $\left(r,\theta,\phi\right)$ as

$\displaystyle x$ $\displaystyle=r\sin\theta\cos\phi,$ (45) $\displaystyle y$
$\displaystyle=r\sin\theta\sin\phi,$ $\displaystyle z$
$\displaystyle=r\cos\theta.$

###  E.1 Temperature

The thermal problem is driven solely by diffusion because the thermal
diffusivity $\alpha$ is large enough to give vanishing thermal Péclet number,
$\textrm{Pe}_{\textrm{t}}=u_{\textrm{ref}}R_{\textrm{c}}/\alpha\ll 1$ (see
also Fig 3). Since the boundary conditions (see eqns. (12)) are independent of
the polar angle $\theta$, the normalised temperature field, $\Theta$, must
also be independent of $\theta$, i.e. it is spherically symmetric. The
normalised temperature $\Theta$ must then satisfy Laplace’s equation, given by

$\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\Theta}{dr}\right)=0,$ (46)

along with the boundary conditions of prescribed temperature on the nucleus
and membrane,

$\displaystyle\Theta\left(r=\kappa\right)$ $\displaystyle=1,$ (47)
$\displaystyle\Theta\left(r=1\right)$ $\displaystyle=0.$

The solution to eqns. (46) and (47) is classically given by

$\Theta(r)=\frac{\kappa}{1-\kappa}\left(\frac{1}{r}-1\right).$ (48)

###  E.2 Flow field

Here also, the flow can be represented in terms of the streamfunction for
Stokes flow, $\Psi^{\textrm{s}}\left(r,\theta\right)$,

$\displaystyle\mathbf{u}$
$\displaystyle=u_{r}\mathbf{i}_{r}+u_{\theta}\mathbf{i}_{\theta},$ (49)
$\displaystyle=\frac{1}{r\sin\theta}\left(-\frac{1}{r}\frac{\partial\Psi^{\textrm{s}}}{\partial\theta}\mathbf{i}_{r}+\frac{\partial\Psi^{\textrm{s}}}{\partial
r}\mathbf{i}_{\theta}\right),$

where the super-script ‘s’ now denotes that the streamfunction has been
defined in terms of spherical coordinates. The spherical basis vectors
$\left(\mathbf{i}_{r},\mathbf{i}_{\theta}\right)$ are expressed in terms of
the Cartesian coordinate unit vectors
$\left(\mathbf{i}_{x},\mathbf{i}_{y},\mathbf{i}_{z}\right)$ as

$\displaystyle\mathbf{i}_{r}$
$\displaystyle=\sin\theta\cos\phi\mathbf{i}_{x}+\sin\theta\sin\phi\mathbf{i}_{y}+\cos\theta\mathbf{i}_{z},$
(50) $\displaystyle\mathbf{i}_{\theta}$
$\displaystyle=\cos\theta\cos\phi\mathbf{i}_{x}+\cos\theta\sin\phi\mathbf{i}_{y}-\sin\theta\mathbf{i}_{z}.$

In spherical coordinates, the streamfunction satisfies

$-\frac{1}{r\sin\theta}E^{2}\left[E^{2}\left(\Psi^{\textrm{s}}\right)\right]=\frac{\sin\theta}{r^{2}}\frac{\kappa}{1-\kappa};$
(51)

the differential operator $E^{2}$ is given by

$E^{2}\left(\Psi^{\textrm{s}}\right)=\frac{\partial^{2}\Psi^{\textrm{s}}}{\partial
r^{2}}+\frac{1-\mu^{2}}{r^{2}}\frac{\partial^{2}\Psi^{\textrm{s}}}{\partial\mu^{2}},$
(52)

where we write $\mu\equiv\cos\theta$. The general solution for
$\Psi^{\textrm{s}}$ is

$\Psi^{\textrm{s}}(r,\mu)=\sum_{n=0}^{\infty}{(1-\mu^{2})\frac{dL_{n}}{d\mu}f_{n}(r)},$
(53)

where the $L_{n}(\mu)$ are the Legendre polynomials as defined in eqn. (27).
The structure of eqn. (51) allows us to pose an Ansatz for
$\Psi^{\textrm{s}}(r,\mu)$: the angular dependence in that equation can only
be balanced if we restrict the expansion (53) to just the $n=1$ term, i.e.

$\Psi^{\textrm{s}}(r,\theta)=f_{1}(r)\sin^{2}\theta.$ (54)

Substitution into eqn. (51) then gives the following governing equation for
$f_{1}(r)$,

$\frac{d^{4}f_{1}}{dr^{4}}+2\left\\{-\frac{2}{r^{2}}\frac{d^{2}f_{1}}{dr^{2}}+\frac{4}{r^{3}}\frac{df_{1}}{dr}-\frac{4}{r^{4}}f_{1}(r)\right\\}=-\frac{1}{r}\frac{\kappa}{1-\kappa}.$
(55)

###  E.3 Boundary conditions for the flow

The fourth-order ordinary differential equation (55) is supplemented by the
boundary conditions at the nucleus $\left(r=\kappa\right)$ and the cell
membrane $\left(r=1\right)$. Noting that they have been modelled as rigid
spheres, the normal and tangential velocities at these surfaces must vanish,

$\displaystyle u_{r}\left(r=\kappa\right)$
$\displaystyle=u_{r}\left(r=1\right)=0,$ (56) $\displaystyle
u_{\theta}\left(r=\kappa\right)$ $\displaystyle=u_{\theta}\left(r=1\right)=0,$

which convert to the following conditions on the function $f_{1}(r)$,

$\displaystyle f_{1}(r=\kappa)$ $\displaystyle=f_{1}(r=1)=0,$ (57)
$\displaystyle\left.\frac{df_{1}}{dr}\right|_{r=\kappa}$
$\displaystyle=\left.\frac{df_{1}}{dr}\right|_{r=1}=0.$

The full solution to eqns. (55) and (57) is provided in eqns. (16) and (17) in
the main text. Additionally, the vertical and horizontal velocities plotted in
Fig 7 are obtained by transforming the spherical coordinate velocity
representation $\mathbf{u}\left(r,\theta\right)$ to a Cartesian representation
$\mathbf{u}\left(x,y,z\right)$ as

$\displaystyle u_{z}\left(r,\theta\right)$
$\displaystyle=-\left(\frac{2\cos^{2}\theta}{r^{2}}f_{1}\left(r\right)+\frac{\sin^{2}\theta}{r}\frac{df_{1}}{dr}\right),$
(58) $\displaystyle u_{x}\left(r,\theta,\phi\right)$
$\displaystyle=\left(-\frac{1}{r^{2}}f_{1}\left(r\right)+\frac{1}{2r}\frac{df_{1}}{dr}\right)\sin\left(2\theta\right)\cos\phi.$

## References

  * 1. Lowell BB, Spiegelman BM.  Towards a molecular understanding of adaptive thermogenesis.  Nature. 2000;404(6778):652–660.  doi:10.1038/35007527.
  * 2. Martinez J, Georgoff I, Martinez J, Levine AJ.  Cellular localization and cell cycle regulation by a temperature-sensitive p53 protein.  Genes Dev. 1991;5(2):151–159.  doi:10.1101/gad.5.2.151.
  * 3. Wang F, Han Y, Gu N.  Cell Temperature Measurement for Biometabolism Monitoring.  ACS Sens. 2020;6(2):290–302.  doi:10.1021/acssensors.0c01837.
  * 4. Quinn PJ.  Effects of temperature on cell membranes.  Symposia of the Society for Experimental Biology. 1988;42:237–58.
  * 5. Chung CW, Schierle GSK.  Intracellular Thermometry at the Micro-/Nanoscale and its Potential Application to Study Protein Aggregation Related to Neurodegenerative Diseases.  ChemBioChem. 2021;22(9):1546–1558.  doi:10.1002/cbic.202000765.
  * 6. Jaque D, del Rosal B, Rodríguez EM, Maestro LM, Haro-González P, Solé JG.  Fluorescent nanothermometers for intracellular thermal sensing.  Nanomedicine. 2014;9(7):1047–1062.  doi:10.2217/nnm.14.59.
  * 7. Bai T, Gu N.  Micro/Nanoscale Thermometry for Cellular Thermal Sensing.  Small. 2016;12(34):4590–4610.  doi:10.1002/smll.201600665.
  * 8. Uchiyama S, Gota C, Tsuji T, Inada N.  Intracellular temperature measurements with fluorescent polymeric thermometers.  Chem Commun. 2017;53(80):10976–10992.  doi:10.1039/c7cc06203f.
  * 9. Okabe K, Inada N, Gota C, Harada Y, Funatsu T, Uchiyama S.  Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy.  Nat Commun. 2012;3(1).  doi:10.1038/ncomms1714.
  * 10. Hayashi T, Fukuda N, Uchiyama S, Inada N.  A Cell-Permeable Fluorescent Polymeric Thermometer for Intracellular Temperature Mapping in Mammalian Cell Lines.  PLOS ONE. 2015;10(2):e0117677.  doi:10.1371/journal.pone.0117677.
  * 11. Gota C, Okabe K, Funatsu T, Harada Y, Uchiyama S.  Hydrophilic Fluorescent Nanogel Thermometer for Intracellular Thermometry.  J Am Chem Soc. 2009;131(8):2766–2767.  doi:10.1021/ja807714j.
  * 12. Nakano M, Arai Y, Kotera I, Okabe K, Kamei Y, Nagai T.  Genetically encoded ratiometric fluorescent thermometer with wide range and rapid response.  PLOS ONE. 2017;12(2):e0172344.  doi:10.1371/journal.pone.0172344.
  * 13. Macherel D, Haraux F, Guillou H, Bourgeois O.  The conundrum of hot mitochondria.  Biochim Biophys Acta Bioenerg. 2021;1862(2):148348.  doi:10.1016/j.bbabio.2020.148348.
  * 14. Baffou G, Rigneault H, Marguet D, Jullien L.  A critique of methods for temperature imaging in single cells.  Nat Methods. 2014;11(9):899–901.  doi:10.1038/nmeth.3073.
  * 15. Kiyonaka S, Sakaguchi R, Hamachi I, Morii T, Yoshizaki T, Mori Y.  Validating subcellular thermal changes revealed by fluorescent thermosensors.  Nat Methods. 2015;12(9):801–802.  doi:10.1038/nmeth.3548.
  * 16. Suzuki M, Zeeb V, Arai S, Oyama K, Ishiwata S.  The 105 gap issue between calculation and measurement in single-cell thermometry.  Nat Methods. 2015;12(9):802–803.  doi:10.1038/nmeth.3551.
  * 17. Baffou G, Rigneault H, Marguet D, Jullien L.  Reply to: “Validating subcellular thermal changes revealed by fluorescent thermosensors” and “The 105 gap issue between calculation and measurement in single-cell thermometry”.  Nat Methods. 2015;12(9):803–803.  doi:10.1038/nmeth.3552.
  * 18. Loesberg C, van Miltenburg JC, van Wuk R.  Heat production of mammalian cells at different cell-cycle phases.  J Therm Biol. 1982;7(4):209–213.  doi:10.1016/0306-4565(82)90026-2.
  * 19. Kamiya N.  Physical and Chemical Basis of Cytoplasmic Streaming.  Annu Rev Plant Physiol. 1981;32(1):205–236.  doi:10.1146/annurev.pp.32.060181.001225.
  * 20. Verchot-Lubicz J, Goldstein RE.  Cytoplasmic streaming enables the distribution of molecules and vesicles in large plant cells.  Protoplasma. 2009;240(1-4):99–107.  doi:10.1007/s00709-009-0088-x.
  * 21. Kessler JO, Bier M.  Gravitational dynamics of biosystems - some speculations.  In: COSPAR, Plenary Meeting; 1976.
  * 22. Todd P.  Gravity-dependent phenomena at the scale of the single cell.  ASGSB bulletin: publication of the American Society for Gravitational and Space Biology. 1989;2:95–113.
  * 23. Howard R, Scheiner A, Cunningham J, Gatenby R.  Cytoplasmic convection currents and intracellular temperature gradients.  PLOS Comput Biol. 2019;15(11):e1007372.  doi:10.1371/journal.pcbi.1007372.
  * 24. Bergman TL, Incropera FP, Dewitt DP, Lavine AS.  Fundamentals of heat and mass transfer.  John Wiley & Sons; 2011.
  * 25. Mack LR, Hardee HC.  Natural convection between concentric spheres at low Rayleigh numbers.  Int J Heat Mass Trans. 1968;11(3):387–396.  doi:10.1016/0017-9310(68)90083-5.
  * 26. Chandrasekhar S.  Hydrodynamic and Hydromagnetic Stability.  Clarendon Press: Oxford University Press; 1961.
  * 27. Huber MD, Gerace L.  The size-wise nucleus: nuclear volume control in eukaryotes.  J Cell Biol. 2007;179:583–584.  doi:10.1083/jcb.200710156.
  * 28. Gundersen GG, Worman HJ.  Nuclear Positioning.  Cell. 2013;152:1376–1389.  doi:10.1016/j.cell.2013.02.031.
  * 29. Chu F, Haley SC, Zidovska A.  On the origin of shape fluctuations of the cell nucleus.  Proc Natl Acad Sci USA. 2017;114(39):10338–10343.  doi:10.1073/pnas.1702226114.
  * 30. Luby-Phelps K.  Cytoarchitecture and physical properties of cytoplasm: volume, viscosity, diffusion, intracellular surface area.  Int Rev Cytol. 1999;192:189–221.
  * 31. Mogilner A, Manhart A.  Intracellular Fluid Mechanics: Coupling Cytoplasmic Flow with Active Cytoskeletal Gel.  Annu Rev Fluid Mech. 2018;50:347–370.  doi:10.1146/annurev-fluid-010816-060238.
  * 32. Happel J, Brenner H.  Low Reynolds number hydrodynamics: with special applications to particulate media.  Springer Science & Business Media; 2012.
  * 33. Leal LG.  Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes.  Cambridge University Press; 2007.
  * 34. Kalwarczyk T, Ziȩbacz N, Bielejewska A, Zaboklicka E, Koynov K, Szymański J, et al.  Comparative Analysis of Viscosity of Complex Liquids and Cytoplasm of Mammalian Cells at the Nanoscale.  Nano Lett. 2011;11(5):2157–2163.  doi:10.1021/nl2008218.
  * 35. Kyoo Park B, Yi N, Park J, Kim D.  Thermal conductivity of single biological cells and relation with cell viability.  Appl Phys Lett. 2013;102(20).  doi:10.1063/1.4807471.
  * 36. Inomata N, Miyamoto T, Okabe K, Ono T.  Measurement of cellular thermal properties and their temperature dependence based on frequency spectra via an on-chip-integrated microthermistor.  Lab Chip. 2023;23(10):2411–2420.  doi:10.1039/d2lc01185a.
  * 37. Mogre SS, Brown AI, Koslover EF.  Getting around the cell: physical transport in the intracellular world.  Phys Biol. 2020;17(6):061003.  doi:10.1088/1478-3975/aba5e5.
  * 38. Goldstein RE, van de Meent J.  A physical perspective on cytoplasmic streaming.  Interface Focus. 2015;5(4):20150030.  doi:10.1098/rsfs.2015.0030.
  * 39. Felder S, Kam Z.  Human neutrophil motility: Time-dependent three-dimensional shape and granule diffusion.  Cell Motil. 1994;28(4):285–302.  doi:10.1002/cm.970280403.
  * 40. Burke NV, Han W, Li D, Takimoto K, Watkins SC, Levitan ES.  Neuronal peptide release is limited by secretory granule mobility.  Neuron. 1997;19(5):1095–1102.
  * 41. Gura Sadovsky R, Brielle S, Kaganovich D, England JL.  Measurement of Rapid Protein Diffusion in the Cytoplasm by Photo-Converted Intensity Profile Expansion.  Cell Rep. 2017;18(11):2795–2806.  doi:10.1016/j.celrep.2017.02.063.
  * 42. Donahue BS, Abercrombie RF.  Free diffusion coefficient of ionic calcium in cytoplasm.  Cell Calcium. 1987;8(6):437–448.  doi:10.1016/0143-4160(87)90027-3.
  * 43. Vendelin M, Kongas O, Saks V.  Regulation of mitochondrial respiration in heart cells analyzed by reaction-diffusion model of energy transfer.  Am J Physiol Cell Physiol. 2000;278(4):C747–C764.  doi:10.1152/ajpcell.2000.278.4.c747.
  * 44. Fishman HM, Bittner GD.  Vesicle-Mediated Restoration of a Plasmalemmal Barrier in Severed Axons.  Physiol. 2003;18(3):115–118.  doi:10.1152/nips.01429.2002.
  * 45. Mishra P, Chan DC.  Mitochondrial dynamics and inheritance during cell division, development and disease.  Nat Rev Mol Cell Biol. 2014;15(10):634–646.  doi:10.1038/nrm3877.
  * 46. Knoblach B, Rachubinski RA.  How peroxisomes partition between cells. A story of yeast, mammals and filamentous fungi.  Curr Opin Cell Biol. 2016;41:73–80.  doi:10.1016/j.ceb.2016.04.004.
  * 47. Cohen S, Valm AM, Lippincott-Schwartz J.  Interacting organelles.  Curr Opin Cell Biol. 2018;53:84–91.  doi:10.1016/j.ceb.2018.06.003.
  * 48. Niethammer P, Kueh HY, Mitchison TJ.  Spatial Patterning of Metabolism by Mitochondria, Oxygen, and Energy Sinks in a Model Cytoplasm.  Curr Biol. 2008;18(8):586–591.  doi:10.1016/j.cub.2008.03.038.
  * 49. Takeuchi K, Nakano Y, Kato U, Kaneda M, Aizu M, Awano W, et al.  Changes in Temperature Preferences and Energy Homeostasis in Dystroglycan Mutants.  Science. 2009;323(5922):1740–1743.  doi:10.1126/science.1165712.
  * 50. Richter K, Haslbeck M, Buchner J.  The Heat Shock Response: Life on the Verge of Death.  Mol Cell. 2010;40(2):253–266.  doi:10.1016/j.molcel.2010.10.006.
  * 51. Song P, Gao H, Gao Z, Liu J, Zhang R, Kang B, et al.  Heat transfer and thermoregulation within single cells revealed by transient plasmonic imaging.  Chem. 2021;7(6):1569–1587.  doi:10.1016/j.chempr.2021.02.027.
  * 52. Zhang B, Zhang X, Shao M, Meng C, Ji F, Zhong M.  An opto-thermal approach for assembling yeast cells by laser heating of a trapped light absorbing particle.  Rev Sci Instrum. 2023;94(3):034105.  doi:10.1063/5.0138812.
  * 53. Govorov AO, Richardson HH.  Generating heat with metal nanoparticles.  Nano Today. 2007;2(1):30–38.  doi:10.1016/s1748-0132(07)70017-8.
  * 54. Donner JS, Baffou G, McCloskey D, Quidant R.  Plasmon-Assisted Optofluidics.  ACS Nano. 2011;5(7):5457–5462.  doi:10.1021/nn200590u.
  * 55. Roxworthy BJ, Bhuiya AM, Vanka SP, Toussaint KC.  Understanding and controlling plasmon-induced convection.  Nat Commun. 2014;5:3173.  doi:10.1038/ncomms4173.