# A physical perspective on cytoplasmic streaming (invited)

Raymond E. Goldstein1 and Jan-Willem van de Meent2 1Department of Applied
Mathematics and Theoretical Physics, Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
2Department of Engineering Science, University of Oxford, Parks Road, Oxford
OX1 3PJ, UK

###### Abstract

Organisms show a remarkable range of sizes, yet the dimensions of a single
cell rarely exceed $100$ $\mu$m. While the physical and biological origins of
this constraint remain poorly understood, exceptions to this rule give
valuable insights. A well-known counterexample is the aquatic plant _Chara_ ,
whose cells can exceed $10$ cm in length and $1$ mm in diameter. Two spiraling
bands of molecular motors at the cell periphery drive the cellular fluid up
and down at speeds up to $100$ $\mu$m/s, motion that has been hypothesized to
mitigate the slowness of metabolite transport on these scales and to aid in
homeostasis. This is the most organized instance of a broad class of
continuous motions known as “cytoplasmic streaming”, found in a wide range of
eukaryotic organisms - algae, plants, amoebae, nematodes, and flies - often in
unusually large cells. In this overview of the physics of this phenomenon, we
examine the interplay between streaming, transport and cell size, and discuss
the possible role of self-organization phenomena in establishing the observed
patterns of streaming.

##  I Transport and Cell Size in Biology

Relative to the remarkable variation of sizes exhibited by living organisms,
the dimensions of a typical cell are surprisingly similar across species. In
nearly all plant-like multicellulars the constituent cells measure $10-100$
$\mu$m, with animal cells falling into a similar if slightly smaller range and
single-cellular prokaryotes extending down to $1$ $\mu$m. Thus outside of a
few notable exceptions the overwhelming majority of organisms have cell sizes
in the range $1-100$ $\mu$m. A very basic question about the fundamentals of
cellular biology is: What underlying mechanism has determined the evolution of
this relatively well-conserved length scale?

The fact that very few cells are larger than $100$ $\mu$m suggests that this
size reflects physical constraints, such as the diffusive range over which two
metabolites can reliably interact. The outliers showing larger cell sizes
provide key counterexamples, prompting the question of how these species have
compensated for problems associated with an increasing cell size, and what
utility this design provides in an evolutionary context. In some cases large
sizes are found in cells with a highly specialized role. The nerve fibres in
our body can reach lengths of a meter, and the _giant squid axon_ can have a
diameter up to $1$ mm, improving the propagation speed of the action
potentials which ultimately facilitates a faster escape response. In other
cases, single-cellular organisms have evolved to a large size and complexity.
Examples of this are the protozoan _Paramecium_ , which can reach $350$
$\mu$m, and the trumpet-shaped _Stentor_ ($2$ mm). Arguably even more
developed is the alga _Acetabularia_ , a single-cellular organism that grows
into a plant-like stalk that can be as long as $10$ cm. Finally there are the
characean algae, a family of plant-like weeds whose segmented stems are built
up out of cylindrical _internodal cells_ of $200-1000$ $\mu$m diameter and
lengths that can exceed $10$ cm.

Cells in these examples have the common feature of forms of active internal
transport, driven by the movement of _molecular_ motors along intracellular
filaments that make up the _cytoskeleton_. In small cells, this motion enables
movement of organelles and vesicles. In larger cells it leads to a continuous
circulation of the cellular fluid, known as _cytoplasmic streaming_ or
_cyclosis_. Streaming is found in many types of larger eukaryotic cells,
particularly in plants allen_arbb_1978 . Patterns of circulation take on a
variety of geometric forms and can be very steady.

Cytoplasmic streaming has long been conjectured to aid in overcoming the
slowness of diffusion on long length scales, but its precise role in enhancing
metabolic rates is yet to be elucidated. This article provides an overview of
the current state of knowledge surrounding this issue. Over the last two
decades, our understanding of the molecular make-up and spatial organization
of the cellular environment has increased dramatically. We begin with an
overview of how spatial aspects of intracellular transport factor in to our
understanding of metabolic regulation. This is followed by an explanation of
the range of topologies of circulation found across species. The arguably best
studied instance of this phenomenon is the _rotational_ streaming found in the
long cylindrical cells of the characean algae, whose metabolic context is
covered in detail. We conclude with a discussion of those studies that have
investigated the role of circulation in intracellular transport and provide an
overview of key questions in furthering our understanding of how streaming
enables enhanced diffusive transport.

###  I.1 Homeostasis and Targeting of Macromolecules

Cellular life requires precise control of metabolic pathways and biosynthesis.
Two central challenges in internal regulation are the maintenance of
_homeostasis_ and the control of _trafficking_. In metabolic pathways,
turnover rates can vary by several orders of magnitude. The dynamic
equilibrium of a biological system’s properties in the face of fluctuating
levels of throughput is generally termed homeostasis cannon_book_1932 . The
concept of homeostatic control necessarily encompasses a range of regulatory
mechanisms, whose function is often specific to the type of organism
considered. In mammals, key homeostatic requirements are the regulation of
body temperature and the stabilization of oxygen pathways and cellular ATP
levels under varying levels of muscle activity hochachka_pnas_1999 . In
plants, homeostatic control is associated with turgor pressure and cytosolic
concentrations of inorganic ions such as calcium and phosphate
boller_arpp_1986 , as well as cellular building blocks such as amino acids.

In addition to homeostatic regulation, which predominantly involves
concentrations of small molecules, intracellular transport of proteins, lipids
and polysaccharides requires a high degree of targeting schwartz_ari_1990 ;
chrispeels_arpppmb_1991 ; staehelin_arpp_1995 ; juergens_arcdb_2004 .
Eukaryotic cells possess an array of organelles, each containing a distinct
set of proteins that allow it to perform its biochemical activities. The
translation of nearly all cellular proteins takes place in the cytoplasm,
after which they must be targeted to one of more than 30 compartments in the
cell schwartz_ari_1990 . The mechanisms for targeting form a central area of
inquiry in cell biology and have been studied intensively in recent years. We
now know that each protein contains a targeting domain in its amino acid
sequence, which interacts with receptors on the organelle to which it is to be
transported. Additionally, many macromolecules are transported between
organelles inside lipid vesicles, with a differentiated coating, as well as a
SNARE marker protein to identify its contents juergens_arcdb_2004 .

Two organelles that play a central role in protein synthesis are the
endoplasmic reticulum (ER) and the Golgi apparatus. The ER is a network of
continuous tubules that courses through the cytoplasm. Comprising as many as
16 functionally distinct subdomains, it fulfills a multitude of roles in
cellular metabolism and in some species extends between cells through the
channels known as plasmodesmata staehelin_pj_1997 . The ER is the entry point
for newly synthesized proteins into the trafficking network of membrane-bound
organelles known as the endomembrane system. Inside the ER, polypeptides
created by ribosomal translation are folded into soluble proteins by means of
the chaperone BiP juergens_arcdb_2004 . Typically, protein products from the
ER are subsequently extruded in vesicles and transported to the Golgi
apparatus for further sorting and processing. Storage proteins, for example
those utilized as nutrients in seed formation, can also bypass the Golgi
apparatus to be stored in protein storage vacuoles (PSVs) by way of
intermediate compartments juergens_arcdb_2004 .

The plant Golgi apparatus plays a central role in protein processing and
sorting, but also synthesises large quantities of cell wall polysaccharides
and glycolipids for inclusion in the plasma membrane staehelin_arpp_1995 ;
dupree_bba_1998 ; juergens_arcdb_2004 . Plant cells can have tens to hundreds
of Golgi stacks, which are often closely associated with the ER network and
are known to localize to specific subcellular regions in cell types exhibiting
localized cell wall growth dupree_bba_1998 .

###  I.2 Diffusion in a Crowded Cytoplasm

Figure 1: Molecular crowding in eukaryotic cytoplasm. Shown is an illustration
of the contents of the yeast _Saccharomyces cerevisiae_. Proteins, ribosomes
with mRNA, microtubules, actin filaments and intermediate filaments are all
drawn to scale and at physiological concentrations. (Adapted with permission
from goodsell_book_2009 .)

In formulating an understanding of trafficking and homeostatic control, it is
becoming increasingly apparent that many transport processes in cells require
explanation hochachka_pnas_1999 ; agutter_jhb_2000 ; agutter_be_2000 . In the
past, a simplified picture of a cell as a watery bag of enzymes has often been
invoked for lack of a more detailed model. This metaphor has given way to the
notion of a highly crowded and high structured cytoplasm, requiring a spatial
(and chemical) understanding of metabolic pathways. As demonstrated in a
comprehensive review by Luby-Phelps luby-phelps_irc_2000 , a number of
assumptions valid for chemical reactions in an aqueous environment break down
in the cytoplasm.

_Reaction volumes are small._ While biochemical analysis typically neglects
finite-volume effects, key molecules _in vivo_ are often only present in
limited numbers. Physiologically relevant concentrations tend to lie in the
nM-mM range. A single molecule inside a bacterium has a nominal concentration
of $10$ nM, while eukaryotic cells may possess perhaps only $1000$ copies of a
molecular species alberts_book_2002 . Thus, the number concentration of
subcellular compartments may in many cases be more informative than the
nominal molar concentration.

_The cytoplasm is a crowded environment._ In a cellular environment
macromolecules typically occupy $20-30$% of the total volume ellis_tbs_2001 ,
leading to a range of phenomena known together as _macromolecular crowding_
luby-phelps_irc_2000 ; ellis_tbs_2001 ; zimmerman_arbbs_1993 ;
minton_cobt_1997 . Figure 1 shows an iconic visualization of the cytoplasmic
environment in baker’s yeast goodsell_book_2009 . Cellular components such as
proteins and ribosomes are all shown at correct scale and density, along with
cytoskeletal elements. While $70-80$% of the volume is water, it is clear that
diffusive movement will be constrained in a manner that will depend strongly
on particle size.

This intuitive notion is born out by experimental evidence. Measurements of
mobility of water in living tissue by NMR and quasi-elastic neutron scattering
(QENS) show a reduction of the order of $50$% of both rotational and
translational diffusion luby-phelps_irc_2000 ; clegg_ajpricp_1984 ;
rorschach_sm_1991 ; cameron_cbi_1997 . Self-diffusion of larger macromolecules
shows a strong reduction in mobility that depends on the hydrodynamic radius
luby-phelps_irc_2000 ; verkman_tbs_2002 , declining from $40$% to $5$% of the
aqueous diffusion rate for particle sizes increasing from $2$ nm to $45$ nm
luby-phelps_irc_2000 ; luby-phelps_jcb_1986 ; luby-phelps_pnas_1987 ;
popov_jns_1992 ; arrio-dupont_bpj_1996 . More recently, several studies have
presented evidence of _anomalous diffusion_ in the cytoplasm: particle
trajectories show a mean squared displacement that is subdiffusive, i.e.
$\langle r^{2}(t)\rangle\sim t^{\alpha}$ with $\alpha<1$ wachsmuth_jmb_2000 ;
weiss_bpj_2004 ; banks_bpj_2005 ; golding_prl_2006 . For larger vesicles the
diffusion rates are of order 10-11 cm2/s, implying that vesicular movement
will be insignificant in the absence of active transport mechanisms luby-
phelps_irc_2000 .

_The cytoplasm is highly structured and compartmentalized._ It is increasingly
recognized that the cytoplasm is not a homogeneous environment. Macromolecular
crowding effects may well result in separation of the cytoplasmic volume into
distinct microphases, as evidenced from phenomena such as caged diffusion of
microinjected beads and the size-dependent partitioning of inert tracers luby-
phelps_irc_2000 .

###  I.3 Motor-driven transport along the cytoskeleton

In the spatial organization of a cell, the _cytoskeleton_ plays a central
role. This meshwork of _actin_ , intermediate filaments and _microtubules_
acts as a backbone for directed transport. Myosin molecular motors can bind to
cytoplasmic structures and transport them along actin filaments by a ‘walking’
motion that consumes ATP. Microtubules have kinesin and dynein motors that
perform similar tasks. The filaments that make up the cytoskeleton are polar,
in the sense that molecular motors move in a well-defined direction along
their tracks. Whereas the various myosins walk along actin in directions that
depend on their type, kinesins walk towards the plus end of microtubules,
which typically results in transport from the cell centre towards the
periphery. Dyneins travel in the opposite direction. The topology of the
cytoskeleton thus defines the direction of motor-assisted transport.

The forms of active transport observed depend greatly on the organism and cell
type. In animal cells, which tend to be small compared to their plant
counterparts, organelles are relatively stationary and motors are primarily
implicated in the transport of vesicles schwartz_ari_1990 , whose diffusion is
negligibly slow luby-phelps_irc_2000 . A review of organelle movements in
plant cells by Williamson williamson_arpppmb_1993  shows that movement of
nearly all major organelles has been observed. Golgi stacks move over ER
strands in an actomyosin dependent manner boevink_tpj_2002 ;
juergens_arcdb_2004 , while the ER itself is also thought to bind to myosin
motors kachar_jcb_1988 ; williamson_arpppmb_1993 ; staehelin_pj_1997 ;
du_jcs_2004 . There is also evidence for actomyosin driven movement of
mitochondria vangestel_jebo_2002 . Chloroplasts are known to move along with
streaming, as well as move themselves to positions determined by light and
cell division planes, though the details of chloroplast-cytoskeleton contact
are often less well understood williamson_arpppmb_1993 .

###  I.4 Mixing in the Cytoplasm

As Purcell famously outlined in his paper “Life at low Reynolds number”
purcell_ajp_1977 , fluid flows at the cellular scale are dominated by
viscosity, where our intuitions shaped by life in a high $Re$ world do not
apply. The Reynolds number $Re=UL/\nu$ is the dimensionless ratio of inertial
and viscous forces, containing a typical velocity $U$, a system size $L$, and
the kinematic viscosity $\nu$ of the fluid. The properties of low Reynolds
flows have important implications for mixing behavior. A drop of milk in a
glass of tea spreads in a turbulent cloud when stirred. As Taylor illustrated
in his 1967 film “Low Reynolds number flows” taylor_film_1966 , this ‘mixing’
is reversible in a low $Re$ flow. If a blob of dye is injected in a cylinder
filled with a very viscous fluid, it apparently dissolves over a few turns of
a stirrer, but if the stirrer’s motion is reversed the blob returns to its
original shape, blurred only slightly by diffusion.

In considering diffusion of a solute of concentration $c$ and diffusion
constant $D$ in the presence of a flow field ${\bf u}$ we may rescale lengths
by $L$ and time by $L^{2}/D$, to obtain the dimensionless advection-diffusion
equation

$\displaystyle c_{t}+Pe({\bf u}\cdot\nabla)c$ $\displaystyle=\nabla^{2}c,$ (1)

where the Péclet number $Pe=UL/D$ characterizes the relative strength of
advection to diffusion in a form analogous to the Reynolds number. Small
molecules in aqueous solutions have $D\sim
10^{-5}~{}\mathrm{cm^{2}/s}=1000~{}\mathrm{\mu m^{2}/s}$. A $1$ $\mu$m
bacterium, swimming at $10$ $\mu$m/s, would have $Pe\sim 0.01$. The smallest
organisms therefore not only live in a world where the Reynolds number is
essentially zero, they also live in a world where diffusive fluxes tend to
outpace advective fluxes. Put more plainly: for a bacterium, it is equally
efficient to sit in one place and let food arrive by diffusion as it is to
swim to get it. Locomotion does not serve to increase uptake directly as much
as it allows migration to areas richer in nutrients purcell_ajp_1977 .

As the system size becomes larger, $Pe=UL/D$ and $Re=UL/\nu$ both increase,
since they share a dependence on $UL$, but since $D$ is typically three (or
more) orders of magnitude smaller than $\nu$, the Péclet number becomes
significant long before the Reynolds number does. By definition, advection
competes with diffusion when $Pe\sim 1$, and this provides us with an estimate
of the dimensions at which flow could potentially aid cellular metabolism. For
a typical length of $10$ $\mu$m and a flow rate of $1$ $\mu$m/s, structures
with a diffusion constant lower than 10 $\mathrm{\mu m^{2}/s}$ will start to
be affected by internal circulation. This number roughly corresponds to the
diffusion rate of vesicles in the cytoplasm luby-phelps_irc_2000 , indicating
that active transport of vesicles may be worthwhile even in small cells. For
small molecules, on the other hand, diffusion constants of order $1000$
$\mathrm{\mu m^{2}/s}$ imply system sizes closer to $100$ $\mu$m for transport
rates in the range of $10$ $\mu$m/s. So we see that active transport
phenomena, which have typical velocities of $1-10$ $\mu$m/s, could well be
significant in aiding intracellular transport. However, in the absence of
turbulent mixing, the precise mechanism by which transport is facilitated in
the presence of a flow does need qualification. Some of the mechanisms by
which flows may aid molecular transport at low Reynolds numbers are outlined
below.

###  I.5 Chaotic Flow Fields.

One mechanism often discussed for increasing diffusion and reaction rates
involves chaotic advective fields, in which trajectories of neighboring points
diverge exponentially over time aref_jfm_1984 ; aref_pf_2002 . Mixing
enhancement of this type has been studied extensively for microfluidic and
lab-on-a-chip systems squiresquake . Examples include the herringbone
micromixer, in which a chaotic circulation pattern is induced by a spatially
alternating pattern of grooves in the bottom of a channel stroock_sci_2002 ,
and mixing in a droplet undergoing periodic deformations induced by
electrowetting mugele_apl_2006 . A pictorial way of describing these effects
is that the surface described by the interface between two solutions is
repeatedly stretched and folded into itself, much like the swirl of a Danish
pastry, resulting in a drastic reduction of the length over which molecules
have to diffuse.

###  I.6 Taylor Dispersion.

Another well-known mechanism of dispersion at low Reynolds numbers is _Taylor
dispersion_ , an enhancement of effective diffusivity due to shear that was
originally described by G.I. Taylor in the context of pressure driven pipe
flows taylor_prsa_1953 ; taylor_prsa_1954 ; taylor_ppsb_1954 . A layer of
solute carried along by such a flow is deformed into a parabolic sheet by the
velocity profile. This results in outward diffusion at the leading edge of the
profile, and inward diffusion at the tail. After a time long enough for
diffusion to smooth out the deformed sheet radially, it converges to a self-
similar profile that travels with the mean flow speed and disperses with an
effective diffusion constant

$D_{\rm eff}\simeq D\left(1+\frac{1}{192}Pe^{2}\right)~{},$ (2)

which can greatly exceed the bare value $D$. In applying such concepts to
subcellular mixing it must be recognized that results obtained in long
channels or many repeated drop deformations do not necessarily translate to
the intermediate length- and time scales found in single cells. As the time
scale for diffusion across a channel of height $h$ is $h^{2}/D$, whereas the
distance traveled along the channel in this time is $Uh^{2}/D=Peh$, we should
therefore expect the Taylor dispersion to become observable only when the
aspect ratio $L/h>Pe>\sqrt{192}$.

###  I.7 High Péclet Enhancement of Reactive Fluxes.

An example of how advection can affect nutrient uptake was found in
theoretical studies of the _Volvocales_ short_pnas_2006 ; solari_pnas_2006 .
These algae take the form of spherical colonies that swim through their
environment by the action of flagellated cells on their surface. The flow at
the leading edge of the colony compresses a solute boundary layer closer to
the surface of the cell, while a depleted plume is formed at the trailing
edge. This effect produces a boundary layer whose size scales as $Pe^{-1/2}$,
resulting in an enhanced concentration gradient and thereby an increased flux
into the organism short_pnas_2006 . This flux enhancement is but one instance
of a more general phenomenon. For _life at high Péclet numbers_ , the boundary
layers around metabolically active objects will shrink with increasing
strength of the flow field, thereby facilitating increased fluxes into the
object. Pickard has recently applied these ideas to analysis of moving objects
in the cytoplasm pickard_jtb_2006 . Approximating vesicles and organelles as
spherical objects he showed that the advective flux scales as $Pe^{2}$ for
$Pe\lesssim 1$, and crosses over to a $Pe^{1/3}$ scaling for $Pe\gtrsim 10$.
At $Pe=1$, fluxes should be expected to be increased by roughly 100% over the
purely diffusive value.

Figure 2:  Topologies of cytoplasmic streaming. (a) Rotational streaming in
internodal cells of _Chara corallina_. (b) A developing oocyte of _Drosophila_
exhibits a correlated random flow field with typical flow rates of a few to
tens of nm/s. Schematic of the velocity field, extracted by particle image
velocimetry. (c) Circulation streaming in the periphery and transvacuolar
strand of epidermal cells, e.g. as found in the root of _Medicago truncatula_
sieberer_pro_2000 . (d) Reverse-fountain streaming in _Lilium longiflorum_
(left) and _Nicotiana tabacum_ (right). (Image modified from hepler_arcdb_2001
.) (e) Periodic shuttle streaming in plasmodium fragments of _Physarum
polycephalum_. (Adapted from matsumoto_bpj_2008 .)

In summary, as a result of advances in fluorescence labeling techniques and
biochemical knowledge over the past decades, we are now arriving at an
understanding of the eukaryotic cytoplasm as a dynamic environment that is
both crowded and highly structured, possessing tight homeostatic control
mechanisms as well as a motor-driven infrastructure for precise routing of
proteins towards their destination. It is increasingly becoming clear that
quantitative understanding into trafficking and homeostatic control in this
environment will require formulation of a spatially resolved picture of
signalling and reaction rates along metabolic pathways.

Active transport processes mediated by molecular motors may play a key role in
extending the spatial range over which metabolites can be reliably expected to
interact, and it is in this context that the role of continuous forms of
circulation observed in some larger cells is of interest. While the precise
effect of advection on diffusive transport depends critically on the type of
geometry studied, various mechanisms for enhancement of molecular transport
can be identified for flows where the Péclet number is sufficiently large. The
remainder of this review provides an overview of what is known about
cytoplasmic streaming and its effect on intracellular transport. We pay
particular attention to what is perhaps the best studied instance of this
phenomenon, the _rotational_ streaming that occurs in the giant _internodal_
cells of the characean algae, but also indicate emerging areas of interest in
other organisms.

##  II Cytoplasmic Streaming

Cytoplasmic streaming has been known for more than two centuries, with the
first observations attributed to Bonaventura Corti in 1774 corti_book_1774 .
It occurs in a range of cell types and a variety of organisms including
amoebae, protozoa, fungi and slime moulds, and is also found during oogenesis
in the fruit fly ganguly  and embryogenesis in the nematode _Caenorhabditis
elegens_ streaming_elegans , though it is most common in plants and their
highest genetic predecessors, the _characean algae_ kamiya_book_1959 ;
kamiya_inproc_1962 ; kamiya_arpp_1981 ; allen_arbb_1978 ; allen_arbb_1978b . A
large body of research over the last five decades has established that
streaming is in most instances driven by an actomyosin system
shimmen_cocb_2004 ; shimmen_jpr_2007 , but there are also examples of
microtubule-based organelle movement nebenfuehr_pp_1999 ; mizukami_pcp_1981 ;
mineyuki_pro_1986 . Though the mechanics of cytoplasmic streaming have been
studied extensively, there is relatively little insight into its biological
function. A number of authors, most notably Pickard, have suggested that
streaming may enhance metabolic rates in large cells, where diffusive time
scales become prohibitively large pickard_pro_1974 ; pickard_pce_2003 ;
hochachka_pnas_1999 .

Streaming has customarily been classified into two major groups. The term
_amoeboid_ streaming denotes cytoplasmic motion that induces changes in cell
form, the best known of which is _shuttle streaming_ found in slime moulds
kamiya_book_1959 ; kamiya_arpp_1981 ; allen_arbb_1978b . Non-amoeboid
streaming is generally divided into five classes based on visually apparent
phenomenology, as originally identified by Kamiya kamiya_book_1959 ;
allen_arbb_1978 ; pickard_pce_2003 . The best studied by far is the
_rotational streaming_ found in characeans, which will be reviewed in detail
in the next section. Other forms of streaming commonly identified are
_saltation_ , _circulation_ , _fountain streaming_ and _multi-striate
streaming_ , briefly described below.

_Shuttle Streaming_ (fig 2e) is a periodic flow found in the _plasmodium_ of
slime moulds. The plasmodium is a single-celled aggregate of protoplasm
containing many nuclei. As the organism searches for food it forms a network
of veins, known as _pseudopodia_ , that self-optimizes in connecting to food
sources, and can find the shortest routes through mazes nakagaki_nat_2000 ;
nakagaki_prslb_2004 ; tero_sci_2010 . Rhythmic back-and-forth streaming of the
cytoplasm takes place inside these filaments, driven by contraction waves in
the actomyosin network. The velocities of this pressure-driven motion, which
reverses every $2-3$ minutes, can reach $1350$ $\mu$m/s kamiya_book_1959 .

_Random Streaming_ is a spatially correlated yet apparently unordered motion
of the cytoplasm. The best-known example of this is found in various
developmental stages of Drosophila mahajan-miklos_db_1994 ; palacios_dev_2002
; serbus_dev_2005 ; zimyanin_cel_2008 . There, within an oocyte several
hundred microns across, kinesins moving along a dense network of microtubules
producing streaming flows that vary from one oocyte to another, and over time
within a given oocyte, and consists of swirls and eddies with a correlation
length of some $20$ $\mu$m ganguly . The nature of transport on such a
disordered network is under active study KhucTrong .

_Saltation_ , also known as _agitation_ , is the most widespread form of
cytoplasmic movement kamiya_book_1959 ; allen_arbb_1978 , characterized by
apparently random jumps of cytoplasmic particles over distances as large as
$100$ $\mu$m, much greater than those corresponding to thermal fluctuations.
Microtubules are possibly implicated in this form of streaming, since very
active saltation is observed near spindles, within centrosomes and adjacent to
microtubular organelles allen_arbb_1978 .

_Circulation_ (fig 2c) takes the form of movement along the cell wall and
strands of cytoplasm transecting the vacuole. Circulation patterns are
typically stable on the time scale of minutes, and evolve as transvacuolar
strands move and branch. Both unidirectional and bidirectional movement are
known and velocities can be as large as $40$ $\mu$m/s. Cells exhibiting
circulation include hair cells in various plants such as _Urtica_ (stinging
nettle), and parenchymal (_i.e._ bulk) cells in _Allium_ (the onion genus), as
well as the leaf cells of the water plant _Elodea_.kamiya_book_1959 ;
allen_arbb_1978 .

_Fountain streaming_ (fig 2d) is circulation in which the cytoplasm moves
along a central axis, flowing back in the opposite direction along the cell
wall. _Reverse-fountain streaming_ exhibits an inward motion along the central
axis and is the more common of the two kamiya_inproc_1962 ; allen_arbb_1978 .
In some cell types it is a developmental stage towards rotational streaming.
True fountain streaming is typically found in root hairs and pollen tubes of
various plants vondassow_jcb_1994 ; miller_pro_1996 ; miller_jebo_1997 ;
tominaga_pro_1997 ; sieberer_pro_2000 ; vidali_pro_2001 ; hepler_arcdb_2001 ;
derksen_spr_2002 ; lovy-wheeler_cyt_2007 .

_Multi-striate streaming_ is found in the fungus _Phycomyces_ and in the
marine alga _Acetabularia_. The cells of _Acetabularia_ have a cylindrical
stalk several centimeters in length, containing a large vacuole separated by a
tonoplast from a thin layer of _cytoplasm_ , at the periphery. Streaming
occurs in both directions along channels separated by stationary cytoplasm
allen_arbb_1978 . Some forms of circulation could arguably be seen as
multistriate streaming, such in the marine alga _Caulerpa_ , where streaming
forms $100$ $\mu$m wide bands wherein files of chloroplasts stream
bidirectionally in a circadian rhythm allen_arbb_1978 .

##  III The characean algae

Figure 3:  Rotational streaming in the characean algae. (a) A shoot of _Chara
corallina_ anchored in agar. Single-celled internodes connect nodal complexes
where a whorl of 6 branchlets is formed. (b) Cytoplasmic streaming takes place
along two domains shaped as spiraling bands. (c) This circulation is driven by
the motion of myosin molecular motors along bundled actin filaments. This
image shows a merged stack of confocal slices, with the colors denoting the
focal position. Actin bundles can be observed below chloroplast rows at the
surface of the cell. (Image courtesy S. Ganguly.) (d) The motion of myosin at
the periphery entrains the outer layer of cytoplasm, which is of order $10$
$\mu$m in thickness. The two moving bands are separated by a neutral line
visible as a row of missing chloroplasts. The motion at the wall induces a
shear flow in the central vacuole of the cell.

One of the most studied examples of cyclosis is the rotational streaming in
giant cylindrical cells of the characean algae, or Charales (fig 3).
Colloquially known as _stoneworts_ after the lime deposits on their surface,
these plant-like species are found in dense meadows on the bottom of lakes and
ponds. _Chara_ cells have been studied since the early days of microscopy
corti_book_1774  and the species are now recognized as the closest living
relatives of land plants kranz_jme_1995 ; karol_sci_2001 . This high degree of
similarity, and the robustness of internodal cells under manipulation, has
made them a model organism in a wide range of plant physiology research,
including membrane transport and electrophysiology shimmen_csf_1994 , turgor-
driven cell wall expansion kamiya_pro_1963 ; green_sci_1967 ; green_pp_1971 ;
zhu_pp_1992 ; proseus_jebo_2000 ; proseus_jebo_2006b ; proseus_jebo_2007  and
cytoskeletal organization wasteneys_pla_1992 ; wasteneys_ba_1993 ;
wasteneys_cmc_1993 ; wasteneys_pro_1996 ; kropf_pro_1997 ; braun_pla_1998 ;
foissner_pla_1999 , calcification and carbon fixation lucas_jebo_1975 ,
intercellular transport through plasmodesmata ding_pcp_1989 ;
kikuyama_pcp_1992 ; franceschi_pla_1994 ; lucas_cocb_1995 ; blackman_pj_1998 ,
and even lake ecology vandenberg_ab_1998 ; chen_em_2002 ; kufel_ab_2002 ;
siong_jeq_2006 ; rodrigo_ab_2007 .

Found in fresh and brackish waters, the characean algae have the appearance of
plants, growing in thin segmented shoots that sprout whorls of branchlets
every few centimeters (3a). Their most studied part is the internode (fig 3b),
a single cylindrical cell with a diameter up to 1 mm and a length that can
exceed 10 cm. Like most plant cells, it is enclosed by a cellulose-containing
cell wall, lined by a layer of _cytoplasm_ $\sim 10$ $\mu$m in thickness. A
membrane known as the _tonoplast_ separates the cytoplasm from the central
vacuole that occupies the bulk of the cell. This vacuole fulfils a multitude
of metabolic roles, acting as storage compartment for sugars, polysaccharides
and organic acids, sequestering toxins such as heavy metals, and functioning
as a buffering reservoir that helps to maintain ionic and pH homeostasis in
the cytoplasm taiz_jeb_1992 . Additionally, the vacuole holds a $0.13$ M
concentration of salts tazawa_pcp_1964 , producing a turgor pressure
equivalent to $5$ bar that lends the cell its rigidity.

The characean cytoplasm contains many structures that are common to higher
plants, including the Golgi apparatus, Endoplasmic Reticulum (ER) and
cytoskeletal filaments. _Chloroplasts_ , the organelles responsible for
photosynthesis, are packed into helical rows that spiral along the cell
surface (fig 3d). The chloroplasts are surrounded by a stagnant layer of
fluid, the _ectoplasm_ , that houses most of the _mitochondria_ in the cell
foissner_pro_2004 . On the inside of the chloroplast rows, bundled actin
filaments (fig 3c) act as tracks for myosins that drag structures within the
cell kachar_sci_1985 ; kachar_jcb_1988 , and thereby entrain the inner part of
the cytoplasm, the _endoplasm_. With streaming rates as high as $100$
$\mu$m/s, the myosin XI found in Chara is the fastest known in existence
shimmen_cocb_2004 . As a result of a reversed polarity of the actin filaments,
the cytoplasm is organized into two bands flowing in opposite directions.
These bands spiral around each other, producing a “barber-pole” velocity at
the cell periphery (fig 3c-d). The two interface lines between these bands are
known as _neutral lines_ or _indifferent zones_. They are marked by the
absence of chloroplasts (fig 3d) and are visible as two light lines crossing
the cell surface.

The internodes connect to the nodal structures by means of channels known as
_plasmodesmata_. These channels are bridges of cytoplasm between cells. The
plasmodesmata in Charales are similar to those found in higher plants, but may
have evolved separately franceschi_pla_1994 . Most notably, in higher plants
the endoplasmic reticulum is known to extend through plasmodesmata, but
characean algae lack this feature. The generally accepted size limit for these
channels is 800-1000 Da lucas_np_1993  for diffusive transport, though
molecules as large as 45 kDa have been shown to move between cells on longer
time scales kikuyama_pcp_1992 .

###  III.1 Rate of Streaming and Velocity Profile

The symmetry of characean internodes makes them amenable to a relatively
straightforward hydrodynamic description. Internodal cells have a very large
aspect ratio, typically exceeding 30, so, for positions sufficiently far from
the endpoints, the flow is well approximated as that inside an infinite
cylinder, with the cytoplasmic bands effectively imposing a value for the
velocity at the boundaries (fig 4a). The Reynolds number $Re=UR/\nu$ is at
most $0.05$, so the full hydrodynamic equations reduce to the linear _Stokes
flow_ , in which

$\displaystyle\eta\nabla^{2}{\bf u}$ $\displaystyle=\nabla p~{},$
$\displaystyle\nabla\cdot{\bf u}$ $\displaystyle=0~{}.$ (3)

Pickard solved the simplest case of this flow problem for _Chara_ , assuming
straight, non-twisting indifferent zones lying along the z-axis
pickard_cjb_1972 . In this case, solving Stokes flow reduces to solving
Laplace’s equation on a circle and the solution of the form $u_{z}\sim\sum
W^{n}r^{n}sin(n\theta)$ is obtained readily by separation of variables.
Assuming step boundary conditions (velocity $\pm U$ on the two halves of the
circle) allows a closed-form solution

$u_{z}~{}=~{}\frac{2U}{\pi}\arctan\left(\frac{2(r/R)\sin\theta}{1-(r/R)^{2}}\right)~{}.$
(4)

A general solution for the case of twisting helical bands of wavelength
$\lambda$ can be obtained as Fourier-Bessel series goldstein_pnas_2008 ;
vandemeent_prl_2008 . A decomposition along the axis of symmetry (fig 4a)
shows that the flow field possesses two components, a downstream term that
reduces to the profile (4) in the limit of infinite helical pitch, and a
small-amplitude circulation between the neutral lines. The amplitude of the
second term vanishes as $\lambda\to 0$ and $\lambda\to\infty$, reaching a
maximum for $\lambda/R\sim 3$. During development of the internodes, the
helical bands twist and subsequently untwist, attaining a minimum in the
helical pitch roughly at the stage of maximal growth green_ajb_1954 . It is
therefore possible that this secondary circulation, though modest in
amplitude, aids homeostasis by enhancing vacuolar mixing goldstein_pnas_2008 ;
vandemeent_prl_2008 .

Figure 4:  Hydrodynamic prediction and MRV measurements of vacuolar flow. (a)
The internodal flow has a helical symmetry: an invariance under a translation
along the longitudinal axis combined with a rotation. The two axes that
naturally follow from this symmetry are the vector $\vec{e}_{H}$, which points
along the bands, and the vector $\vec{e}_{\varphi}$, which is orthogonal to
the bands. (b) Theoretically predicted flow field along the $\vec{e}_{H}$
component, showing the vacuolar shear profile. (c) The $\vec{e}_{r}$ and
$\vec{e}_{\varphi}$ components reveal a small secondary circulation along the
center of the cell. (d) MRI scan of an internode placed in a glass tube, with
spiraling lines indicating positions of the neutral line, and coloured bands
showing the domains used for velocity measurements. (e) Velocity profiles
measured at each of the domains show excellent agreement with the theoretical
profile. (Figures modified from goldstein_pnas_2008 ; vandemeent_prl_2008 ;
vandemeent_jfm_2010 .)

A series of increasingly sophisticated experiments, employing first visual
tracking kamiya_bmt_1956 ; pickard_cjb_1972  and later laser Doppler
spectroscopy mustacich_bpj_1977 , has shown that the approximation of a simple
shear flow is in good agreement with the _in vivo_ observations. Recently,
full 2-dimensional measurements of the velocity field were obtained using
magnetic resonance velocimetry (fig 4c-d) which confirm that the vacuolar flow
is virtually indistinguishable from its hydrodynamic prediction
vandemeent_jfm_2010 , with direct shear transmission by the tonoplast. The
role of a lipid membrane in allowing such direct shear transmission has
recently been investigated theoretically WoodhouseJFM  and experimentally
HonerkampSmith .

###  III.2 Cell Development

In order to extend towards sunlight, plant cells expand their volume 10- to
20-fold over the course of development green_pp_1971 ; taiz_jeb_1992 . The
majority of this volume is occupied by the central vacuole, where high
concentrations of salts are sequestered, resulting in large outward osmotic
pressures that maintain the rigidity, or _turgor_ , of the cell. Perhaps one
of the most important functions of these vacuolar balloons is to serve as an
energetically favorable means of expanding the volume of the organism. While
there is a considerable cost associated with the transport against
electrochemical gradients required to maintain the solute concentration in the
vacuole, it is much cheaper to expand cell size by water uptake than by
protein synthesis raven_np_1981 ; taiz_jeb_1992 . With their exceptionally
long internodal cells, whose vacuoles occupy about 95% of the cellular volume,
characean algae represent an extreme case of this growth strategy. Since these
species grow as weeds in low-nutrient conditions and can outcompete higher
plants for lake dominance coops_ab_2002 , the large cell size in these
organisms may be partially understood as a result of competition for light in
low nutrient conditions.

Growth in characean algae occurs by repeated division of a single cell at the
plant tip, the _apical cone_ green_jbbc_1958 . During shoot development this
cell divides every few days. The rate of this division, sometimes called the
_plastochron_ , can be quite regular. After division, the newly formed cell
divides once more into an upper _pro-node_ and an internode. The internode
undergoes a remarkable degree of expansion, increasing in volume by five
orders of magnitude during $2-3$ weeks green_jbbc_1958 . The pro-node
undergoes a number of subsequent divisions to form the nodal structure. In
_Chara corallina_ , there is an initial division into $6$ cells which connect
to the ascending and descending streams in an specific manner
shepherd_pce_1992b . In later stages these cells undergo further divisions to
form a complex multicellular structure, though the initial symmetry is still
reflected in the fact that _C. corallina_ typically forms a whorl of 6
branchlets. The reproductive structures are exclusively formed on the part of
the node connecting to the descending stream shepherd_pce_1992b .

###  III.3 Nutrient Uptake and Intercellular Transport

One of the most obvious roles of cytoplasmic streaming in characean metabolism
is to enhance the rate of transport between cells, thereby facilitating the
translocation of nutrients from regions of uptake to regions of growth.
Because of their relative anatomical simplicity, the Charales have been used
for a great number of uptake and transport studies using radioisotopes to
track salts bostrom_jebo_1975 ; williams_jebo_1975 ; zawadzki_jebo_1986 ;
zawadzki_jebo_1986b  and nutrients such as inorganic carbon williams_jebo_1975
; zawadzki_jebo_1986 ; zawadzki_jebo_1986b ; trebacz_jebo_1988 ; ding_pcp_1989
, nitrogen, and phosphorus littlefield_pp_1965 ; box_pce_1986 ; box_pce_1987 ;
vermeer_ab_2003 .

Like most aquatic macrophytes, the Charales are able to take up nutrients from
the surrounding water, in contrast to land plants where uptake predominantly
takes place in the roots. While the species can successfully be cultured in
flasks without needing to be anchored in a layer of soil forsberg_pp_1965 ,
there is evidence that a significant proportion of uptake may take place in
the root-like _rhizoidal_ cells that anchor the plants, particularly for rate-
limiting nutrients such as phosphorus and nitrogen which may be more abundant
by orders of magnitude in the interstitial water of the sediment
littlefield_pp_1965 ; box_pce_1986 ; box_pce_1987 ; vermeer_ab_2003 .

Dissolved inorganic carbon (DIC) is taken up directly from the water
surrounding the shoot. Low-weight carbohydrates are formed within $1$ hour
after uptake of 14C-DIC and pass unaltered through the plasmodesmata to the
neighboring cell trebacz_jebo_1988 . The bulk of the photoassimilates reside
in the endoplasmic layer on the inside of the chloroplasts, but particularly
in branchlet cells some fraction is sequestered in the vacuole ding_jebo_1991b
. The branchlet cells are more photosynthetically active than the internodes,
and transport from the branchlets to internodes can be five times higher than
the reverse ding_jebo_1991b ; ding_jebo_1991 .

Many studies have investigated the relationship between streaming and
intercellular transport. Measurements with a tandem pair of internodes show
that the transport rate correlates with the streaming rate over the $25$%
variation in magnitude observed in a collection of samples zawadzki_jebo_1986
. This correlation is stronger in summer when the rate of transport is higher
shepherd_pce_1992 ; shepherd_pce_1992b ; zawadzki_jebo_1986 . Treatment with
cytochalasin B shows a roughly proportional reduction of the transport rate
with the streaming velocity bostrom_jebo_1976 . Treatment of either cell in
the pair suffices to lower the transport rate; a similar response is found
when altering the streaming velocity by lowering the temperature ding_pcp_1989
. Some early studies found indications of a small component of propagation
faster than the streaming velocity williams_jebo_1975 ; dale_jebo_1983 , but
this finding was not confirmed by later whole-shoot experiments where 14C is
fed to the rhizoids, showing that carbon is transported upwards at roughly the
rate of streaming while 32P transport is somewhat slower box_jebo_1984 .

###  III.4 Alkaline Band Formation and Carbon Fixation

The lime deposits from which the species get the name _stoneworts_ arise from
their alkaline habitat. Carbon fixation associated with photosynthesis
naturally leads to production of OH-, as can be seen in the pH dependence of
the chemical equilibrium of DIC in water,

$\mathrm{CO_{2}+H_{2}O\rightleftarrows H^{+}+HCO_{3}^{-}}~{}.$ (5)

At pH $5.5$ roughly $90$% of the dissolved carbon is found in the form CO2,
but as the pH increases this balance shifts towards HCO${}_{3}^{-}$. At pH
$8.5$, which is fairly typical for environments inhabited by characeans,
$99\%$ of the inorganic carbon is in HCO${}_{3}^{-}$. A proton is therefore
required to produce CO2 from bicarbonate, which inevitably leaves OH- as a
side product. This excess OH- is excreted at the surface, inducing a rise in
pH. In characeans OH- is localized in periodically spaced alkaline bands
alternated by acidic regions where H+ efflux occurs tazawa_arpp_1987 . Calcium
carbonate (CaCO3) tends to precipitate on the alkaline regions forming the
band-shaped encrustations characteristic of stoneworts. This precipitation
yields a proton,

$\mathrm{Ca^{2+}+HCO_{3}^{-}\rightarrow CaCO_{3}+H^{+}}~{},$ (6)

which may then be used to produce CO2 from bicarbonate. It has been suggested
that the H+ efflux serves to facilitate photosynthesis by raising the
concentration of free CO2 plieth_pp_1994 . Evidence that this precipitation
enhances carbon fixation is found in 14C studies that show a $1:1$
correspondence between the rate of CaCO3 precipitation and the rate of carbon
fixation in slightly alkaline environments of pH $8-9$ mcconnaughey_bb_1991 ;
mcconnaughey_lo_1991 ; mcconnaughey_cjb_1998 .

Further support for the notion the pH bands enhance photosynthesis is found in
the work by Mimura _et al._ mimura_pce_1993 , who investigated the effect of
the H+-ATPase on 14C fixation. They observed that carbon fixation was strongly
reduced both when the ATP in the cytoplasm was depleted by perfusion with
hexokinase and 2-deoxyglucose and when the H+-ATPase was inhibited by
introduction of vanadate. Stimulation of the H+-ATPase by pyruvate kinase and
phosphoenolpyruvate (PEP) resulted in increased carbon fixation. The same
effects were observed at both pH 5.5 and pH 8.5, indicating that the H+-ATPase
plays an essential role in assimilation of both CO2 and HCO${}_{3}^{-}$.

Evidence that the formation of alkaline bands directly influences the rate of
photosynthesis is also found in _fluorometry_ studies. When a sample is
excited with light, the absorbed light can be used for photosynthesis, emitted
as fluorescence or dissipated as heat. In practice the degree of heat
dissipation can often be taken as constant, so the fluorescence signal
represents an inverse measure of the rate of photosynthetic activity. Such
measurements show that the external pH correlates directly with the rate of
photosynthetic activity, with the highest rates observed in acidic regions
bulychev_pla_2003 ; bulychev_bcm_2005 .

Alkaline band formation appears inseparably linked to cytoplasmic streaming.
The magnitude of pH variations is greater for the upwards streaming band than
for the downwards streaming band babourina_jebo_2004 , and inhibition of
cytoplasmic streaming with cytochalasin B prevents formation of alkaline bands
tazawa_arpp_1987 ; lucas_jmeb_1977 . The membrane of internodal cells can
undergo an _action potential_ , a transient depolarization similar to that
observed in nerve cells, although the travel speed of about $1$ cm/s is much
slower. During an action potential, streaming is momentarily halted and the
alkaline bands disappear. As streaming recovers, the alkaline bands tend to
reform at the same position on the cell’s surface and the recurrence was
increased with the concentration of Ca2+ in the surrounding medium
eremin_pps_2007 .

Although models for the formation of alkaline bands have been presented
toko_jtb_1985 , the specific role of streaming in the generation of alkaline
bands has not been widely discussed in the literature. A function of streaming
thus far overlooked may well be to drive these bands, thereby aiding carbon
uptake from the environment and enhancing photosynthetic rates.

###  III.5 Driving Mechanics and Cytoplasmic Rheology

Various studies have investigated the driving mechanism and rheological
aspects of cytoplasmic streaming. A number of authors have presented
theoretical models of transport along the cytoskeleton with stochastic on-off
dynamics dinh_bjbl_2006 ; snider_pnas_2004 ; maly_jtb_2002 ; smith_bpj_2001 .
The most recent work of this form applied to streaming is by Houtman _et al._
houtman_epl_2007 , who describe streaming in transvacuolar strands with a two-
dimensional model that includes on/off kinetics and hydrodynamic interactions
between particles through the Oseen tensor. There is also a range of studies
that focus on the driving force associated with streaming. Pickard
investigated the scaling of streaming with cell size pickard_pro_1974 ,
presenting an analysis to support the notion that most dissipation occurs near
the neutral lines. His measurements of the streaming velocity as a function of
cell size show the scaling of the maximum streaming speed $U\sim R^{1/2}$,
which is consistent with the combined assumption of a dissipation rate scaling
as $U^{2}$ and a driving power scaling as $R$.

Tazawa & Kishimoto tazawa_pcp_1968  measured the motive force using perfusion
experiments. In this technique, the cell is placed in an isotonic bath (_i.e._
a solution of an osmolarity similar to that of the vacuolar fluid), which
allows the cell to survive amputation of its ends. If each of the endpoints is
then contained in a compartment separated from the rest of the bath, the
contents of the vacuole can be replaced by applying a slight pressure
difference tazawa_pcp_1964 ; tazawa_pcp_1964b . To measure the motive force,
the pressure difference between the two reservoirs is adjusted so that
streaming is halted in one of the bands, implying that the shear force
balances the motive force in the cytoplasmic layer. The motive force obtained
this way has a value in the range of $14-20$ $\mu$N/cm2, consistent with
centrifuge microscope measurements kamiya_pro_1958 . Of note here is that this
force is independent of temperature. Rates of cytoplasmic streaming show a
widely documented linear (or even exponential) dependence on temperature
kamiya_book_1959 ; tazawa_pro_1968 ; pickard_pro_1974 ; mustacich_bpj_1976 ,
with Pickard reporting an increase of $3.4$ $\mathrm{\mu m\>s^{-1}\>K^{-1}}$
pickard_pro_1974 . Tazawa’s results therefore suggests that this temperature
dependence is a consequence of a change in cytoplasmic viscosity. Tazawa &
Kishimoto also examined the effect of tonicity, finding that an increase in
the tonicity from $290$ mM to $586$ mM results in a lowered streaming rate as
well as an increased motive force. Decreasing the tonicity in the cytoplasm to
$190$ mM resulted in a swelling of the chloroplasts, causing a marked decrease
in streaming rate and motive force, presumably because of a resulting
deformation in the actin bundles. After $5-20$ min, the chloroplasts regained
their normal shape and streaming recovered, though the motive force remained
lowered.

Donaldson donaldson_pro_1972  performed measurements of the forward streaming
velocity as a function of the applied perfusion pressure gradient. He assumed
a driving force localized to a layer of thickness $\epsilon$ and a power-law
dependence $\tau=-\alpha(\partial u/\partial y)^{1/n}$ for the viscous stress.
His results show a good correspondence for $n=3$ and $\epsilon=0.1$ $\mu$m.
The corresponding motive force, $F$ = $36$ $\mu$N cm-2, is higher than the
values found by others tazawa_pcp_1968 ; kamiya_pro_1958 , which he concludes
is the result of a systematic underestimation in those measurements due to the
fact that the small velocities and thin layers of movement near the stalling
point are very difficult to observe. Hayashi hayashi_jtb_1980  applied a
similar analysis to experimentally measured velocity profiles. Like Donaldson
he assumed a power-law rheology and found the best-fit parameters for
measurements of a protoplasm filled cell kamiya_bmt_1956 , as well as two
cases of plug flow of extracted cytoplasm kamiya_inproc_1965 . He found an
exponent of $n=1.4$ for the protoplasm-filled cell and exponents of $n=1.3$
and $n=1.7$ for two cases of plug flow.

Nothnagel and Webb nothnagel_jcb_1982  investigated various hydrodynamic
models for driving mechanics in the cytoplasm and concluded that the virtually
shear-less profile observed in the cytoplasm is best explained by assuming a
meshwork throughout the cytoplasm that is pulled along at the cell wall. The
endoplasmic reticulum is argued to be a structure that could fulfill such a
role and there is structural evidence from electron micrographs to support
this hypothesis kachar_sci_1985 . Recent work Wolff1  has examined the
importance of hydrodynamic slip at the cell wall.

##  IV Role in Intracellular Transport

Although a great deal of work has been published on the molecular basis and
hydrodynamics of streaming, relatively few authors venture into a discussion
of its function. It has long been suggested that streaming aids molecular
transport in some way. Kamiya’s 1959 review kamiya_book_1959  mentions that de
Vries suggested this as early as 1885 devries_bz_1885 . However, concrete
hypotheses as to the mechanism by which streaming accelerates metabolic rates
have scarcely been put forward. Agutter _et al._ have argued that diffusion is
not capable of explaining many transport phenomena in cells agutter_jhb_2000 ;
agutter_be_2000 . Similarly, Hochachka presents an argument that the degree of
homeostasis along ATP pathways cannot be explained other than by assuming
forms of active transport hochachka_pnas_1999 .

The highly symmetric topology of streaming in the characean algae would appear
to have evolved at considerable evolutionary cost, as further reflected in the
fact that the myosin XI found in this organism is the fastest known in
existence. On the basis of what we know about the characean algae, we see that
streaming is implicated in a multitude of roles in cellular metabolism. It
aids transport between cells, and is therefore essential in supplying a steady
flow of cellular building blocks to newly formed cells at the tip of the
shoot. It also appears important in maintaining the alkaline bands that
facilitate uptake of inorganic carbon from the surrounding water. However a
key question that remains largely unanswered is just what role streaming may
play in eliminating the diffusional bottlenecks that would seem to limit cell
sizes in other organisms. Indeed streaming may help homeostatic regulation
during rapid cell volume expansion, but the precise mechanisms by which it
does so remain an open area of investigation.

The most important contributions in terms of a quantified discussion of the
effect of streaming on intracellular transport are without doubt from Pickard.
He discussed scaling of the streaming velocity and diffusional time scales
with cell size pickard_pro_1974 , as well as the interaction between the
stagnant layer of periplasm surrounding the chloroplast rows, and the moving
layer of endoplasm. He points out the possibility that advection of a point-
source may aid homeostasis by smoothing out fluctuations in the concentration
field. He also raises the notion that streaming as such does not necessarily
have to confer a benefit to the cell if its real purpose is transport of
particles along the cytoskeleton. This second point is argued further in a
later publication where he makes a case for streaming being an accidental
consequence for vesicular transport along the cytoskeleton pickard_pce_2003 .
As covered in section 1.4, he also discusses the role of boundary layer
scaling in enhancing exchange of molecular species between organelles and
their environment in advection-dominated flows pickard_jtb_2006 .

Our own work on the secondary circulation in the vacuole mirrors this last
theme of life at high Péclet numbers goldstein_pnas_2008 ; vandemeent_prl_2008
. In the case of a cell with helically twisted bands, the secondary
circulation vortices in figure 4c produce an advection from center to wall and
vice versa. In the case of a transient uptake into the vacuole, this produces
a boundary layer at the left side of the cross-section, where the flow is
directed outward, whilst diffusional pile-up is carried inward at the opposite
neutral line. The boundary layer arising from this effect scales as
$Pe^{-1/3}$. This phenomenon suggests that the helicity of the bands, which is
maximized at the moment of greatest growth, may serve to aid cellular
homeostasis by facilitating enhanced diffusive fluxes into and out of the
vacuole, allowing it to play a buffering role with respect to metabolic
processes in the cytoplasm. However, analysis of the eigenmodes of transient
decay towards a well-mixed vacuolar concentration shows that the enhancement
of fluxes into and out of the vacuole would become significant from
$Pe_{\psi}\gtrsim 10$. This suggests that this type of effect would primarily
benefit transport of slowly diffusing structures, such as large
macromolecules, whose presence in characean vacuoles has currently not been
established vandemeent_thesis_2010 .

##  V Streaming and Cell Size: Key Questions

As emphasized in the mini-reviews in the above sections, the past two decades
have led to remarkable progress in our understanding of the biochemical
pathways central to cellular metabolism. In terms of the dynamical behavior of
these systems, we have seen that there are two central challenges in cellular
regulation. (1) Maintaining homeostatic levels of small molecules and ions
across varying levels of metabolic turnover and (2) The routing of proteins
and other macromolecule intermediates towards their intended targets.

A fundamental question from a biophysical point of view is what constraints
have governed the evolution of homeostatic control mechanisms and
macromolecular targeting. It is clear that both these tasks are complex, and
will lead to requirements that vary between cell types and species. Yet while
there are important differences, the functioning of the ER and Golgi complexes
shows a high degree of similarity across a wide range eukaryotic species. The
vesicle transport system in particular has been shown to be highly conserved,
exhibiting essentially the same mechanisms in organisms as diverse as mammals
and yeast staehelin_arpp_1995 .

Figure 5:  Self-organization of cytoplasmic streaming in a mathematical model
of Chara WoodhousePNAS . Color coding corresponds to the z-component of an
order parameter associated with actin filaments at the periphery, and white
lines represent indifferent zones separating up- and down-streaming regions.
Superimposed are streamlines of the cytoplasmic flow induced by the filament
field, where the flow is directed from the thin end to the thick end of the
individual lines. Panels show progression from random disorder through local
order to complete steady cyclosis.

Two basic questions any physicist will be inclined to ask are what typical
length and time scales govern intracellular transport. This returns us to our
initial question regarding the evolution of cell size to a typical range of
$1-100$ $\mu$m. Presumably mechanisms for protein targeting and homeostatic
control will be under increasing strain as the cell size increases. What,
then, are the rate-limiting factors in these processes, and how is it that
cells may be able to mitigate these limitations by internal circulation?

The internodes of characean algae present one of the most compelling test
cases for these questions. Not only are these cells among the largest in
Nature, but the form of streaming found in this system is also the most
organized and symmetrical of all types of circulation currently known. This
means that the system is amenable to analysis from a geometrical point of
view, and the many physiological studies performed on this organism mean that
theoretical results can be placed in the context of detailed biological
knowledge. Moreover, the fact that internodes grow from some $20$ $\mu$m to
several centimeters provides an opportunity to study mechanisms of metabolic
control at a range of scales.

One of the aspects of streaming in the characean algae that has yet to be
elucidated is just how it affects transport of small molecules and vesicles in
the cytoplasm. To the precision of current experimental techniques, the
cytoplasm appears to move as a gelled layer, retaining its shape perhaps by
way of the meshwork of ER tubes that extend throughout the endoplasmic
compartment. The temporal and spatial variations in the velocity field have
yet to be quantified. Given the typical velocities around $50$ $\mu$m/s, even
the magnitude of relatively small fluctuations could be significant,
particularly when it comes to the reported caged diffusion of vesicles.

Another physical issue for which deeper understanding is needed is the precise
relation between the topology of the cytoskeletal network and the cytoplasmic
flow. It would appear that streaming presents itself in increasingly organized
forms as the system size is scaled up. This could be the result of
evolutionary pressure, but could also be a physical effect. So given an actin
network, a cytoplasmic rheology, and the force-velocity relations of molecular
motors, can we predict in what types of systems we will see continuous forms
of circulation?

This leads us finally to comment on the possibility that some forms of
streaming appear through a process of self-organization. As long ago as 1953
Yotsuyanagi  it was noted that cytoplasmic droplets extracted from characean
algae could spontaneously self-organize into rotating fluid bodies. This
presumably arises from myosin motors that walk along dislodged actin filaments
and entrain fluid. Each of these motor/filament assemblies constitutes a force
dipole in the fluid, and nearby assemblies will be attracted together if
nearly parallel, leading to self-reinforcement of local order, leading
eventually to long-range order. Further evidence for self-organization comes
from much more recent work Foissner  in which steaming is completely disrupted
through added chemicals (cytochalasin and oryzalin), which, when removed,
allow streaming to be reconstituted. Strikingly, the indifferent zone appears
in a new location. A mathematical model WoodhousePNAS  that combines filament
bundling, flow-induced reorientation, and coupling to the curvature of the
cell wall successfully reproduces much of this phenomenology, as shown in
Figure 5.

The hypothesis that streaming can self-organize through physical processes
associated with hydrodynamic interactions arising from force dipoles leads
naturally to the field of ‘active matter’, and in particular to the properties
of bacterial suspensions. Each bacterium can be thought of as a force dipole,
with one force directed backwards from the trailing flagella and a second
directed forwards from the action of the cell body on the fluid. The
attractive interactions between such ‘pusher’ dipoles mentioned above has also
been predicted to lead to simple large-scale flow topologies in bacterial
suspensions ConfinementPRL1 , which has subsequently been verified
ConfinementPRL2 ; ConfinementPNAS .

This work was supported in part by the Leverhulme Trust, ERC Advanced
Investigator Grant 247333, and the Schlumberger Chair Fund.

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