The Role of Cytoplasmic Streaming in Symplastic Transport

Blackwell Science, LtdOxford, UK PCEPlant, Cell and Environment0016-8025Blackwell Science Ltd 2002 26 845 Role of cytoplasmic streaming in symplastic transport W. F. Pickard 10.1046/j.0016-8025.2002.00845.x Original ArticleBEES SGML

Plant, Cell and Environment (2003) 26, 1–15

The role of cytoplasmic streaming in symplastic transport

W. F. PICKARD

Department of Electrical Engineering, Washington University, St Louis, Missouri 63130, USA

ABSTRACT effectuators. In the context of the cell, which participates in a far more decentralized command structure than an army, The distributing of materials throughout a symplastic this means: (i) carrying out its programmed activities; (ii) domain must involve at least two classes of transport steps: sensing its surroundings and its internal state and transmit- plasmodesmatal and cytoplasmic. To underpin the latter, ting appropriate information and/or commands; and (iii) the most obvious candidate mechanisms are cytoplasmic receiving and effecting commands from elsewhere in the streaming and diffusion. The thesis will be here advanced plant. that, although both candidates clearly do transport cyto- None of this is simple. and all of this must be done in a plasmic entities, the cytoplasmic streaming per se is not of complex three-dimensional ultrastructural milieu which primary importance in symplastic transport but that its does not conform to the diffusion-dominated ‘watery bag’ underlying molecular motor activity (of which the stream- model of yesteryear (Hochachka 1999). To paraphrase the ing is a readily visible consequence) is. Following brief tuto- elder Moltke, ‘in [physiology] with its enormous friction rials on low Reynolds number flow, diffusion, and targeted even the mediocre is quite an achievement’ (van Creveld intracytoplasmic transport, the hypothesis is broached that 1985, p.13). Or to cite from Polybius (a soldier/historian of macromolecular and vesicular transport along actin track- the 2nd century BC) a rather more direct parallel between ways is both the cause of visible streaming and the essential war and biology: ‘nature makes a single trivial error suffi- metabolically driven cytoplasmic step in symplastic trans- cient to cause failure in a design, but correctness in every port. The concluding discussion highlights four underdevel- detail barely enough for success’ (van Creveld 1985, p. 264). oped aspects of the active cytoplasmic step: (i) visualization That the plant fares very well indeed speaks volumes for of the real-time transport of messages and metabolites; (ii) the robustness and resilience of its strategies. enumeration of the entities trafficked; (iii) elucidation of In the distribution of both supplies (logistics) and infor- the routing of the messages and metabolites within the mation (C3) the plant has many options, including: cytoplasm; and (iv) transference of the trafficked entities from cytoplasm into plasmodesmata. 1 Convection. This form of mass transport is, in plant biology, typified by cytoplasmic streaming. If the supplies are Key-words: cytoplasmic streaming; diffusion; endoplasmic metabolites, they will certainly be carried along by the reticulum; intracellular transport; molecular motors; motion of the cytoplasm. If the information is encoded in plasmodesmata; trafficking. chemical messengers, it too will be carried along by the motion. However, the transport will be only along the streamlines of the flow: convection will never move INTRODUCTION metabolites or chemical messengers perpendicular to the flow. Convection is presently considered to be a result of The green plant, like an army, has two all-consuming activ- endoplasmic ‘molecular motors’ running along actin ities, without which it can not hope to prevail. The first of microfilaments or possibly along microtubules (Reddy these is logistics. The second is command, communication, 2001). But it is an open question whether the convective and control (C3). Because the topic of this is essay cytoplas- flow (i) constitutes a major payoff arising from this mic (protoplasmic) streaming in the context of symplastic motile activity; or (ii) is a byproduct of lesser utility transport, these two imperatives will be treated in the con- which arises from drag on the entities towed by the mov- text of a cell whose contents are convecting. ing motors. That is, how is the utility to the cell of motor Logistics is the art/science of getting the supplies needed motion to be apportioned between (a) stirring of the to the unit (a cell) and distributing them appropriately. in cytoplasm; and (b) siting of the towed entity at a new the context of the cell. This means: (i) taking them up, location; presumably a cost function exists and is either across the plasmalemma from the apoplast or from reflected by plant evolution, even though we do not at the protoplasm of a neighbouring cell by way of connecting present know how to formulate it. plasmodesmata; and (ii) distributing them appropriately 2 Diffusion. This reflects the kinetic motion of the mole- within the protoplasm. Our focus will be upon distribution. cules comprising the cytosol. As a result of intermolecu- C3 is the art/science of gathering reliable information lar collisions, a molecule in a particular compartment of and then transmitting unambiguous orders to appropriate the cytosol will execute a random walk and will, other Correspondence: William F. Pickard. Fax: +1 314 935 7500; influences being absent, tend in time to assume a uniform e-mail: [email protected] probability distribution within the compartment (cf. Saha

2 W. F. Pickard

& Srivastava 1950 and van Kampen 1981). Diffusion can tinuum. The symplast is cytoplasmic; the apoplast is the cell move metabolites and messengers perpendicularly to the wall and beyond. But somehow, in common usage, ‘sym- streamlines of cytoplasmic flow. plastic transport’ only peripherally includes transport in the 3 Tow. This denotes a motive force imposed differentially phloem. Perhaps an adequate way of putting it is: upon certain classes of entities in a medium. For example, ‘Symplastic transport’ denotes the transport and distribu- the motion of electrons and holes in a semiconductor is a tion of molecules in general and chemical messengers in ‘tow’ phenomenon called drift. Magnetophoretic sedi- particular within the cytoplasm of a symplast domain of mentation of amyloplasts by a non-uniform magnetic nonvascular cells and the spreading of these molecules field (e.g. Kuznetsov & Hasenstein 1996) is likewise a from cell to cell via plasmodesmata’. ‘tow’ phenomenon; and so also would be selective coupling of myosin to some particular type of macromole- The concept of ‘symplast domain’ is well discussed by cule (or vesicle) and the subsequent translocation of the van Bel & Oparka (1995). macromolecule along actin trackways. Obviously, when Moreover, the apoplast no longer appears as inert and the towed entities are dragged through a fluid, convec- uninteresting as it once did and is increasingly implicated in tion will result. Obviously also, tow may be mediated storage, transport, and reactions (cf. Sattelmacher 2001). In either by fields (‘action at a distance’) as in sedimentation addition, although the bulk of the mole-miles racked up and magnetophoresis or by direct mechanical linkage as in symplastic transport must occur in the cytoplasm, yet in molecular motor based processes. nevertheless satisfactory transport still depends vitally 4 Electrical. This could be electrophoretically driven mass upon such processes as trans-plasmalemmal water flux, flow, but there is little evidence that this is important on plasmodesmatal function, and phloem transport. That is, an intracellular level; and, were it to exist, it should be long-distance transport in plants must be viewed as an inte- highly wasteful of metabolic energy because the cyto- grated system-wide problem of logistics in which issues of plasm is very close to electrical neutrality (cf. Pickard command and control are unavoidable, even if not clearly 1965). Or it conceivably might be electrically mediated understood at present. information transfer, except that long-distance nerve-like signalling is uncommon in plants and intracellular information transfer by electrical means has not been shown CONVECTION: MASS FLOW AT LOW to exist (e.g. Pickard 1973; 2001). REYNOLDS NUMBER 5 Pressure. On the whole plant level this could include signals such as pressure disturbances (Malone 1996). On the Convection within the cytoplasm is called ‘cytoplasmic cellular level, the focus of this review, this could be man- streaming’ (or ‘protoplasmic streaming’); and its discovery ifested either by pole-to-pole pressure driven mass flow is attributed to Bonaventura Corti (e.g. Pfeffer 1906, coupled to plasmodesmata or by transcellular osmosis; as Kamiya 1959). It has been reviewed repeatedly, for exam- this possibility has not been extensively studied, it will be ple by Ewart (1903), Pfeffer (1906), Kamiya (1959, 1960, largely neglected. 1981), Britz (1979), Seitz (1979), Kuroda (1990), Shimmen & Yokota (1994), Grolig & Pierson (2000), and Staiger This leaves the first three items of the list as the prime can- (2000). didates for mediating the steps of symplastic transport That streaming in the plant cell is of key importance which occur within an individual cell. They will be discussed is graphically borne out by the classical observation one by one in subsequent sections. (Hofmeister 1867; Fig. 10) that, if some care is taken, a cell But before proceeding to these discussions, it is essential can be plasmolysed without disrupting the streaming. This to clarify what one means by ‘symplastic transport’. In 1879 observation has been validated repeatedly, was made by Eduard Tangl observed intercellular ‘Strängen’ (cords or Ewart (1903; pp. 8–9) in his monograph, has been reviewed veins) between plant cells and promptly realized that these by Kamiya (1959; pp. 96–98), is still being commented ‘Verbindungskanälen’ (connecting passages) might enable upon in the literature (Kurkdjian et al. 1993), and has by the plant to coordinate the activities of its individual cells the author been observed in Chara desiccating at the mar- (cf. van Bel & Oparka 1995). This notion of a coordinated gin of a pond whose water level had been precipitously syncytial protoplasm gradually evolved, with the term ‘sym- lowered. In the Characeae, its existence is so essential that plast’ being coined in 1880 by von Hanstein (cf. Oxford Ewart has remarked (Ewart 1903; p. 4) ‘it is so closely con- English Dictionary http://dictionary.oed.com/cgi/entry/ nected with vitality... that permanent cessation always 00245169), the term ‘plasmodesmata’ being first applied to indicates a fatal injury’; and this has been borne out by Tangl’s passages in 1901 (Strasburger 1901), and the con- author’s incidental experience in a variety of electrophysi- trasting terms ‘apoplast’ and ‘symplast’ being used together ological studies (e.g. Barsoum & Pickard 1982). These in 1930 (Münch 1930; p. 73). At present, sensu stricto, these observations attest to a marked positive association last two terms mean, respectively: a plant domain (or set of between intracellular motility and survival of the cell, domains) which is entirely outside the plasmalemmal con- although there is as yet no widely accepted enumeration of tinuum; and, conversely, a set of plant domains (or a the benefits which this motility confers upon the cell (cf. domain) which are entirely inside the plasmalemmal con- Pickard 1972, 1974).

Role of cytoplasmic streaming in symplastic transport 3

Cytoplasmic streaming is not the result of body (i.e. pon- Next let the outer cylinder be slowly rotated causing the deromotive) forces acting on the ground cytosol but rather blob to elongate markedly; the elongated blob will also a manifestation of the differential motion of the surfaces tend to persist. Finally, let the rotation of the outer cylinder bounding the cytosol (cf. Kamiya 1981). This simple idea is be precisely reversed and observe that the blob resumes its however, apt not to be obvious to someone conditioned by original shape! the high Reynolds number fluid dynamics of flowing Fourth, because the Reynolds number is so small, stream- streams, pouring wine, or briskly agitated laundry. In these ing and/or cytoplasmic mixing due to aperiodic mechanical three examples, the kinetic energy of the fluid dominates agitation (e.g. sloshing) seem unlikely to be important. strongly: it does not do so at the subcellular level. Cytoplasmic streaming is the result of the action of First, absent phenomena presumed rare or inconsequen- ‘molecular motors’ which ratchet along cytoskeletal cables tial at the subcellular level such as porous boundaries (cf. dragging various organelles and neighboring cytosol with Chellam, Wiesner & Dawson 1995) or slip boundary con- them (Kamiya 1981; Kachar & Reese 1989; Williamson ditions (e.g. Allison 1999), the empirically observed nature 1993; Shimmen & Yokota 1994; Reddy 2001). The cables of the liquid–solid interface dictates imposition of classical thus far implicated in subcellular motion are composed no-slip boundary conditions: along a liquid–solid interface, either (i) of actin microfilaments (an actin polymer com- the velocities of the two phases are equal (e.g. Milne- monly called F-actin) roughly 5–9 nm in diameter, or (ii) of Thomson 1950, sect. 19·05). This means that, if a surface microtubules (an α/β tubulin polymer) roughly 25 nm in moves within the cellular coordinate system, it drags the diameter (Reddy 2001). The macromolecular ‘engines’ adjacent cytosol with it; that is, the cytosol near a moving which are by ATP-hydrolysis propelled along these fibrillar structure (membrane or organelle) tends to move with that trackways are: for actin microfilaments, myosins (Yama- structure. moto, Hamada & Kashiyama 1999; Cai, Del Casino & Second, the nature of the fluid dynamic phenomena Cresti 2000; Shimmen et al. 2000; Kashiyama et al. 2000; manifested in a system depends strongly upon the system’s Yokota 2000), members of a large pan-eukaryotic super- Reynolds number family (Mermall, Post & Mooseker 1998); for tubulin microtubules, either kinesins or dyneins (Reddy 2001; Cai RUa= rh, (1) et al. 2000; Dole et al. 2000), also widely distributed among where ρ (kg m−3) is the cytosolic density (approximately the eukaryotes. Of these motor systems, the actin-myosin 1000), U (m s−1) is a velocity characteristic of the streaming seems at present to be by far the most important for cyto- motions ( 100 × 10−6), a (m) is a length characteristic of plasmic streaming (Shimmen & Yokota 1994); and dynein ultrastructural dimensions ( 10 × 10−6), and η (Pa s) is the homologues appear to be absent from the Arabadopsis effective viscosity of the cytosol ( 1 × 10−3). Thus, for cyto- genome (Lawrence et al. 2001). plasmic streaming, R 10−3 (dimensionless); and this Based upon the trajectories of protoplasmic granules, ensures (i) that viscous effects will outweigh kinetic energy Kamiya (1959; chs. I & II) classified the readily visible effects; and (ii) that the approximations of low Reynolds streaming in cells of the green plants into five (overlapping) number flow will obtain. For the author, this has meant that groups: (i) agitation, in which the individual granules have a the only way he was able to achieve even marginal famil- significant, subjectively stochastic component superim- iarity with the counterintuitive behaviours of these flows posed upon their fairly steady tow velocities; (ii) circula- was by viewing repeatedly G. I. Taylor’s remarkable short tion, in which the individual granules move in a rather more film Low-Reynolds-Number Flows (Taylor 1967). predictable fashion even though neighbouring granules Third, low Reynolds number flow arising from the tow- may manifest quite different motions; (iii) streaming along ing of cytoplasmic structures or molecules does not mix the definite tracks, in which granule motion is confined to num- cytoplasm in the sense that dragging a perforated spoon bers of (mostly parallel) courses whose velocities of flow through a kettle of soup mixes the ingredients. One can, seem rather uncorrelated; (iv) fountain streaming, in which however, get an inkling of the difference if such flow from the motion is one direction along the central axis of the cell everyday experience by comparing the result of (i) plunging and the other direction along the cell’s periphery; and (v) a soiled butter-knife into a jar of honey and stirring once rotational streaming, in which the motion is up one side of (low Reynolds number flow); and (ii) lowering a spoonful the cell (often with a tendency to spiral) and down the of cream into a cup of tea and stirring once (high Reynolds other. These distinctions are more readily comprehended number flow): in the former case viscosity dominates and by studying the diagrams of streaming in pollen tube pro- one is left with a semi-permanent streamline of butter and toplasm provided by Iwanami (1956). Across the Virid- crumbs suspended in the honey; whereas in the latter case iplantae, visible streaming (when present) has speeds kinetic energy dominates, tumbling turbulence and/or vor- roughly within the range 1–100 µm s−1, with 10 µm s−1 being tex flow results, and the cream mixes rather thoroughly into perhaps typical (Kamiya 1959; Table 1). However, there is the tea. In particular, low Reynolds number flow manifests no essential requirement that the streaming be visible to ‘kinetic reversibility’ (Taylor 1967) which can be illustrated the casual (or even dedicated) observer because slow with the following experiment. Let a blob of thick coloured motion of a single motor may not cause sufficient optical syrup be injected into a coaxial container of clear syrup; dif- inhomogeneity to be detected. Thus, a reasonable definition fusion aside, it will be observed to sit there without mixing. of ‘cytoplasmic streaming’ is:

4 W. F. Pickard

Any cytosolic convection arising from the action of molec- 3 These entities in turn drag the ambient cytosol with them ular motors moving along cytoskeletal cables. producing the streaming. 4 Streaming is a low Reynolds number phenomenon and This definition emphasizes the active metabolically does not therefore produce efficient mixing of the cyto- driven nature of the process. and the low Reynolds number sol. environment mandates that streaming stop dead in its 5 Streaming may also shear the endomemranes, apparently tracks when motor activity ceases. The most familiar (and without causing significant harm. perhaps the most important) realization of this process is the towing of subcellular organelles along actin trackways by myosin motors; but motions along microtubules and other cables are also included. DIFFUSION AND ITS LIMITATIONS The best known of the above types of streaming is rota- As summarized by Hochachka (1999), cellular physiology tional because it is so readily studied in the giant cells of the historically has embraced two major schools of thought Characeae (Kamiya 1959, 1981). Such streaming is some- regarding models of cell function and regulation: the one what atypical in that: (i) it is steady, whereas in putatively ‘assumes that cell behavior is quite similar to that expected more advanced plants flow is much more variable with for a watery bag of enzymes and ligands’; whereas the other some cells not streaming visibly for prolonged periods ‘assumes that three-dimensional order and structure con- (Ewart 1903; Shimmen & Yokota 1994); and (ii) the entire strain and determine metabolite behavior’. The first empha- cytoplasm moves in a coordinated fashion whereas stream- sizes near equilibrium conditions and diffusive transport; ing in seemingly erratic directions along cytoplasmic the second stresses subcellular structure and molecular strands (often transvacuolar) seems more characteristic of motors. and the first is increasingly under siege (e.g. Wheat- higher plants (e.g. Hofmeister 1867). However, the very ley & Malone 1993; Reuzeau, McNally & Pickard 1997; simplicity of the streaming in charophytes (with its uni- Hochachka 1999; Agutter & Wheatley 2000; Luby-Phelps formly moving peripheral cytoplasm, its central vacuole, 2000). and its fixed ‘indifferent lines’ between the streaming However, it can not be that diffusion is irrelevant zones) made it possible to model the flow hydrodynami- within the cytosol. First, convection alone at low Reynolds cally and to effect a comprehensible closed form solution number will not spread a blob of metabolite uniformly which adequately predicted both its radial and its angular throughout the cytosol (cf. Taylor 1967): it will merely gen- variations (Pickard 1972). As this model yielded accurate erate a complicated but deterministic three-dimensional granule velocities in both cytoplasm and vacuolar sap but pattern within the cytosol. Second, ionic mobility is related included no mention of the tonoplast, an obvious possibility to ionic diffusivity by way of the Einstein relation, which is was that the tonoplast shears like a fluid. At the time, this very general and should hold given only the applicability seemed mysterious. But viewed from the perspective of the of Maxwell–Boltzmann statistics to the charged entity fluid mosaic model of biological membranes (Singer & (Ashcroft & Mermin 1976); therefore, because the cyto- Nicholson 1972; De Weer 2000), it seems only logical plasm is well known to be electrically conducting (e.g. because it is known from monolayer models of membranes Cole 1968), it can be presumed that small particles diffuse that such structures can be have area–pressure and area– within it. Moreover, direct measurements of electrical con- surface-potential isotherms with gas-like virial behaviour ductance have been made on gelled electrolyte solution, a (Sehgal, Pickard & Jackson 1979). Presumably, such behav- cytoplasm surrogate; and the effect of gelation on conduc- iour could be true of cellular lipid bilayers in general. tance is minor (Ewart 1903; pp. 123–126). Third, relative Moreover, the endoplasmic reticulum appears capable of diffusion coefficients of various large molecules in cyto- undergoing shear and remarkable metamorphoses of mac- plasm have been measured (cf. Luby-Phelps 2000) and rostructure, presumably without functional impairment shown to be smaller than in aqueous solution but defi- (Kachar & Reese 1989; Lichtscheidl & Balusˇ ka 2000). nitely not zero. Nevertheless, ‘relatively little is known about the biogene- Each of the two outlooks described above has particular sis, dynamics, maintenance, and inheritance of the organelle benefits for particular studies, and it seems obvious that itself’ (Powell & Latterich 2000). biological reality must incorporate both to achieve a robust synthesis: the cell is packed with macromolecules, cables, and endomembranes which do affect intracellular trans- Summary port, both actively and passively; but that this merely slows the free diffusion of solute particles, both by general tortu- The points that should be retained from this brief overview ousity of pathway and by specific steric hindrance. That is, of cytoplasmic streaming are: neither molecular motors alone nor diffusion alone can 1 Streaming is an active process driven by ATP-powered meet the transport needs of cytoplasm or symplasm; and molecular motors. this view seems now to be ascendant. 2 These motors ratchet along fibrillar trackways in the Diffusion, as it appeared relevant to plant biology 45 cytoplasm, most often pulling a variety of attached enti- years ago, was beautifully summarized by Spanner (1956). ties with them. As relevant to biophysicists and biochemists, it has been

Role of cytoplasmic streaming in symplastic transport 5 treated by Stein (1986) and by Tanford (1961); and, as a the Coulomb force qE on the ion, yielding via Eqn 2, the mathematical model of nature, the classical diffusion Stokes–Einstein formula (either of molecules or of heat) was well handled in the kTB treatise of Carslaw & Jaeger (1959). These aspects will not D = (3) be emphasized because most modern plant biologists have 6phaeff a pretty good intuition for what diffusion does. Neverthe- where aeff is commonly known as the Stokes radius; also, less, there are several aspects of diffusion which are often it should be remembered from Eqn 1 that η is an effective neglected although physiologically important. viscosity. Because, roughly, the volume of a particle should scale linearly in its effective molecular weight M [u = Da] (cf. Few-particle diffusion Tanford 1961; sect. 21e), it follows that (Eqn 3 being a rea- It has been forcefully pointed out (e.g. Luby-Phelps 2000) sonable representation of reality and other things being that the reaction volumes of subcellular compartments can equal) be very small and may, at physiological concentrations, con- DM-13 (4) tain so few copies of a particular reacting species that the µ continuum predictions of diffusion–reaction theory give The reasonableness of Eqn 3 can be supported in at least way to stochastic fluctuations. three ways. First, empirical values of Stokes radius derived Nevertheless, there are still attractive arguments for from measurements of limiting ionic mobilities agree satis- continuing to make putatively plausible estimates with the factorily with predictions from molecular models when aeff formalism of classical continuum diffusion. First, our knowl- 0·4 nm (M 200) (Robinson & Stokes 1959; Fig. 6·1). edge of cell biology seems at present much too sketchy to Second, molecular dynamics simulations of diffusion show create of a demonstrably better alternative. Second, the dif- (Ould-Kaddour & Levesque 2001) deviations in the direc- ferential equations of diffusion are linear in particle con- tion reported by Robinson & Stokes (1959; Fig. 6·1) and centration, implying that the particles do not interact. crossover to the hydrodynamics regime (aeff asolute) when Hence, if a numerically large but geometrically compact asolute 4asolvent 0·3 nm (for water). Third, Eqn 3 makes swarm of particles is suddenly released in a liquid, each par- fairly good predictions for globular proteins in aqueous ticle pursues its random walk independent of the others. and solution (Tanford 1961; Tables 21,1) therefore the prediction of the diffusion equations must rep- Unfortunately, in cytoplasm, experimentally determined resent a simple sum over the entire ensemble of ‘walking’ diffusion coefficients fall off somewhat faster than M−1/3 for particles. That is, the concentration distribution of the dif- M 1 kDa, thereby ‘suggesting that the cytoplasm pos- fusion equations is also the probability distribution for the sesses a higher-order intermolecular structure that impedes location of a single particle at some moment after release. large particle diffusion’ (Zucker, Goessling & Gollan 1996). The author calls this notion the ‘ergodic heuristic’. He That is, supramolecular structure in the cytosol appears to makes no claims for its universal, rigorous validity. But he increase the effective viscosity of the cytosol to the motion expects that, if you average particle distribution within a of large particles. single 10 nm vesicle over time or within a large ensemble of vesicles at a particular instant of time, you will find that both are close (a) to each other; and (b) to the predictions Diffusive irreversibility of continuum diffusion. In short, he believes that simple diffusive predictions should be quite good enough given the Consider a sphere of radius a which is centred on the origin uncertainties of present knowledge and the reality that the of a spherical coordinate system and filled with a homoge- subcellular environment is doubtless not the time-invariant neous solution in which particles of a certain type γ have a milieu postulated by most modellers. diffusion coefficient D. Suppose, that at time t = 0, a uni- γ form swarm of N0 -particles is released within a sphere of radius α << a (also centred on the origin) and allowed to The Stokes–Einstein formula diffuse outwards. It is shown in Eqn A6 of the Appendix The Einstein relation between the diffusion coefficient D that the swarm spreads outward and approaches equilib- (m2 s−1) and the mobility µ [m2 (V·m)−1] of an ion is rium with a characteristic time

(Ashcroft & Mermin 1976) 22 tpspreading = aD()2. (5) DkTqm , (2) = B Observe that this time is independent of α. −1 × −23 where kB (J K ) is the Boltzmann constant (1·380 10 ), Consider next the converse case in which it is desired to T (K) is the absolute temperature, and q (C) is the net gather up the γ -particles after they have diffused to form a charge on the ion. If the ion is idealized as a sphere of effec- uniform equilibrium density C0. To this end, let an absorb- α tive radius aeff (m) undergoing low Reynolds number ing sphere of radius << a be placed at the origin. It is electrophoresis at a velocity µE in a vector electric ﬁeld shown in eqn A10 of the Appendix that the gathering up E (V m−1), the drag it experiences will be just (cf. Robinson approaches equilibrium with a characteristic time which is πη µ & Stokes 1959) 6 aeff E. This force must be balanced by much larger than that of spreading:

2 19·63); and it could in principle be used with Eqn A1 to find t gathering 2p aa ==6. 579 ... (6) the steady-state flux to the sphere given c → C for (r/a) >> t 3 aa 0 spreading 1, c = 0 at r = a, and λ = 0. This spuriously simple problem One reasonable conclusion which can be drawn from Eqn 6 has not (to the author’s knowledge) been rigorously solved. is that it would appear markedly inefficient for the cell to However, various approximate analytic and numerical transport time- or location-sensitive materials diffusively. investigations have provided some understanding of the flux behaviour (e.g. Friedlander 1961; Levich 1962; Coutelieris, Diffusion on a shell Burganos & Payatakes 1995; Verbrugge & Baker 1996). First, the dimensionless group of interest is the Peclet Equation 5 for spreading time presumed that the cell inte- number rior is homogeneous fluid whereas, in most mature plant Ua cells, it is mostly vacuole with a some peripheral cytoplasm; P = (8) D current opinion is that the fraction of protoplasm devoted Rough range estimates for its defining variables are: 10−6 to multiple varieties of vacuole (Bassham & Raikhel 2000) U 10−4 m s−1 (cf. Kamiya 1959); 10−7 a 10−5 m (cf. Stae- is seldom less than 30% and frequently as much as 90% helin 1997); 10−13 D 10−9 m2 s−1 (cf. Mastro & Keith (Taiz & Zeiger 1998; pp. 20 & 61; Staehelin & Newcomb 1984; Luby-Phelps 2000). This yields a huge range of possi- 2000; p. 25). Hence a potentially relevant question is that of ble Peclet numbers: 10−4 P 10+4 (dimensionless). the relaxation-to-equilibrium of the probability-density of a Second, given the many uncertainties of diffusive and diffusible entity in a thin spherical shell of featureless cyto- convective modelling at the subcellular level, the approxi- sol. This problem is also treated in the Appendix and shown mate analytic treatment of Levich (1962; s. 14) seems quite to lead to a characteristic time roughly detailed enough; it yields a flux F (mol s−1) of roughly t aD2 2 (7) shell = () È 2 13˘ FDCaP=+41p 0 (9) ÎÍ p ˚˙ Reaction–diffusion and a normalized flux 2 13 The mathematics of this case is discussed briefly in the FPN()P =+1 (9¢) p Appendix where it is shown that the dimensionless param- This yields both the expected flux in the limit P → 0 and a l eter X = a should be rather less than π if the peripheral sublinear increase with Peclet number as P → ∞. Numeri- D −4 −2 regions of the diffusing volume are not to be starved for cally, FN(0) = 1·00; FN(10 ) = 1·03; FN(10 ) = 1·14; FN(1) = − +2 +4 reactants being emitted by a non-local source; here λ (s 1) is 1·64; FN(10 ) = 3·95; FN(10 ) = 14·72. Thus, as suggested by the reaction rate. Because the diffusion of large particles in Reuzeau et al. (1997), streaming could move a reaction cen- cytoplasm is often markedly slower than in aqueous solu- tre to a region where limiting reactants were more plentiful. tion (Mastro & Keith 1984; Luby-Phelps 2000), effective diffusion coefficients as low as 10−12 m2 s−1 seem realistic; and this means that even reactants with survival times as Summary long as 10 s could become scarce at subcellular dimensions. Snippits which usefully might be retained from this brief On the other hand, given the presence of moving sources of overview of diffusion are reactant, all reactant sinks might in time-average (i.e. ergodically) be adequately sourced, if (i) the cell had 1 Averaged over relevant time scales and interior compart- streaming along randomly varying trackways; or (ii) ments of a cell, the concentrations of various diffusants streaming were initiated locally in response to resource can be usefully estimated by macroscopic diffusion the- depletion (Reuzeau et al. 1997). ory in situations where random walk is the chief mechanism of diffusant transport. 2 As molecular weight increases beyond 1 kDa, the predic- Convection–diffusion tions of the Stokes–Einstein equation for the diffusion It has been forcefully pointed out by Reuzeau et al. (1997) coefficient become increasingly less accurate as the effec- that resource depletion at a sink could be ameliorated by tive viscosity increases progressively. moving the sink into less depleted regions; and further, 3 Diffusive irreversibility makes pole-to-pole diffusive ‘Solute distribution within the bulk cytoplasm of each cell transport of rare molecules highly inefficient. ought to be a combination of self diffusion and cyclosis.’ 4 Reaction-diffusion could lead to ‘pockets of poverty’ of (Tyree 1970). These ideas seem attractive intuitively but, high-demand reactants unless distribution is somehow given the non-intuitive qualities of low Reynolds number evened out by cytoplasmic streaming. flow, could profitably be subjected to theoretical validation. 5 Organelles whose needs for high molecular weight reac- Consider a reactant-absorbing sphere of radius a which tants are hard to meet by passive diffusion could receive remains fixed in a uniform flow having well away from the a significant boost from high Peclet number cytoplasmic sphere a velocity U (m s−1). The velocity distribution around streaming. However, this is intracellular transport rather a sphere is well known (e.g. Milne-Thomson 1950; sect. than symplastic transport.

TOW: DOES SYMPLASTIC TRANSPORT dimensions may help some in making diffusion of specific INVOLVE TARGETED INTRACYTOPLASMIC reactants more effective, but not as much as might have TRANSFER OF MOLECULES? been hoped. If however, the endomembrane reticulum could create Because timely point-to-point transport of messenger mol- and preserve preferred orientations or trackways for diffu- ecules and rare reactants should be markedly hindered by sion, this would help greatly for then the problem could diffusive irreversibility, it would make sense for the cell to become one of diffusion down a line of length aline and (cf. sequester such substances in special compartments where Appendix) (i) their diffusive spread is appropriately channelled; or (ii) 2 to move the compartments themselves. But do we have evi- aline 4 t line = (12) dence of this in situations relevant to symplastic transport? D p 2 ∼ If aline acircle, if follows that Diffusion in two or one dimensions 2 t sphere p ~ ()asphere a (13) The landmark paper of Kachar & Reese (1989) on stream- t line 48G ing in the Characae demonstrated that the endoplasmic The parameter values suggested above yield for this ratio reticulum forms a dense network of anastomosing cister- ∼70, which seems rather more attractive. nae in the cytosol, that it is pulled along actin cables, and That is, if key molecules, either macro- or micro-, could that it appears to metamorphose in form as it travels. Not be targeted to highly mobile highly selective docking moi- long thereafter it was shown that membrane-soluble lipid eties in suitably architected endomembranes, pole-to-pole tracers could spread from cell to cell and exhibited diffusive transport might prove useful to a cell. At this time inferred endomembrane lateral diffusivities on the order such transport is distinctly speculative even though highly −13 2 −1 of 10 m s (Grabski, de Feijter & Schindler 1993); selective docking is at present a subject of intense scrutiny, moreover, when plasmodesmatal connectivity was dis- especially in animal systems. But despite the likely impor- rupted by plasmolysis, intercellular spread of label was tance of such a process, it presumably could occur in the suspended without stopping the spread of label within the absence of cytoplasmic streaming. intracellular endomembranes. It would therefore appear that, if a molecule of interest were once to be attached to a laterally diffusible membrane Appropriate channelling protein of the endomembrane continuum, it could stochas- Cantrill, Overall & Goodwin reported recently (Cantrill, tically find its way to a reticular domain of interest, to a Overall & Goodwin 1999) that two types of fluorescent Golgi stack, or even to the nucleus. Moreover, a crude esti- probes micro-injected into explants from stem internodal mate of the stochastic wandering time of a reactant mole- tissue of either Nicotiana tabacum or Torenia fournieri cule within a sphere of radius a to find a reaction site of could end up in one of two places: commonly in the cytosol α radius is, from the Appendix, and rarely within the endoplasmic reticulum. In the latter a2 a case, it appeared as if probes (even if 10-fold above the t sphere = (10a) D 3a usual molecular weight exclusion limit for cell-to-cell pas- sage through the plasmodesmal annulus) could travel freely whereas the corresponding time for two-dimensional diffu- from cell to cell, presumably via the desmotubule. How- sion in the planar region between two concentric circles of ever, a still more recent report (Crawford & Zambryski α radii a and is 2000) suggests that 27·5 kDa green fluorescent protein tar- a2 geted to the lumen of the endoplasmic reticulum did not, in tacircle = ln a (10b) D a different tobacco cultivar and different cell type, leave the transfected cell. It would appear that the endomembrane Unfortunately Eqns (10) are somewhat misleading lumen might not be the pathway of choice for really inter- because, although D and α could be much the same, the esting (i.e. high molecular weight or hormonal) messages total area of the endomembrane continuum is thought to be and molecules. Moreover, because such intraluminal trans- Γ-fold greater than that of the plasmalemma, crudely port presumably does not require cytoplasmic streaming, Γ π 2 ∼ π 2 implying [4 asphere ] acircle ; the data reviewed by Luby- this burgeoning area is beyond the scope of this paper. Phelps (2000) suggest that 1 << Γ 100. Thus,

t sphere 1 ()asphere a Organelle motion ~ (11) t 3G 2 circle ln[] 4G ()asphere a It is clear that molecular motors, powered by ATP and run- Γ To illustrate what this means, suppose that = 30, asphere = ning along actin ﬁlaments, are abundant within the cytosol. 10 µm and α = 1 nm. This yields a ratio ∼5. That is, it appears Empirically, they do from time to time result in blatantly that the large values reported for Γ have rather diminished obvious streaming. Presumptively, these motors also oper- the prospectively greater efﬁciency of diffusion in two ate in the absence of organized streaming: after all, the cell dimensions. Hence, constraining diffusive spread to two does have to manage its supplies of vacuoles, vesicles, and

© 2003 Blackwell Publishing Ltd, Plant, Cell and Environment, 26, 1–15 8 W. F. Pickard various other subcellular compartments. Moreover, occa- of these supramolecular structures is well beyond the scope sional (as distinct from organized) motions of clustered of this paper; and the reader is referred to the many recent reaction sites, either to acquire a better supply of substrate review articles (e.g. Lucas, Ding & Van der Schoot 1993; or to more effectively distribute product (cf. Reuzeau et al. Wolf & Lucas 1994; Patrick 1997; Jorgensen et al. 1998; Kra- 1997), would not necessarily be noticed during casual gler, Lucas & Monzer 1998; Ding, Itaya & Woo 1999; Lucas microscopic observation. 1999; Overall 1999; Oparka & Santa Cruz 2000; Oparka & It is, however, definitely known: (i) that molecular Roberts 2001). However, without plasmodesmatal steps motors can and do move endoplasmic reticulum in the symplastic transport could not exist; and, in a very real Characeae (Kachar & Reese 1989); (ii) that endoplasmic sense, the role of cytoplasmic streaming is inseparable from reticulum exists in at least 16 functionally different types the role of plasmodesmata because the two must couple and that it is involved in such time- and place-sensitive efficiently if the transport of metabolites and messages is to activities as protein synthesis, oleosome formation, and vac- be efficacious. Because in its normal state, the plasmodesma uole- and mitochondrion-attachment (Staehelin 1997); (iii) permits nonspecific (putatively diffusive) transit by small that members of the myosin superfamily of actin-binding (M 750) hydrophilic particles (cf. Oparka & Roberts molecular motors are involved organelle movement, mem- 2001), this discussion will be divided into three sections: one brane trafficking, and signal transduction (Mermall et al. to discourage belief that cytoplasmic streaming is not 1998); (iv) that an acto-myosin system mediates the move- important in symplastic transport; one to touch upon the ment of Golgi stacks (Nebenführ et al. 1999); and (v) that transport of small solutes; and one for everything else. the actin cytoskeleton of plants supports not only a transport network but also a signalling network (Volkmann & An adjuvant of diffusion? Balusˇ ka 1999). This, coupled with the unsuitability of diffusion to carry out many of the needful intracellular trans- In this case, the cytoplasmic steps of symplastic transport port (logistic) and signalling/control (C3) tasks which are left to old-fashioned inefficient diffusion and a different confront the cell, is prima facie evidence that molecular raison d’être sought for cytoplasmic streaming. One such motor activity (of which cytoplasmic streaming is the visible possibility is that it is an attempt to reap the benefits of paradigm) is essential to intracellular transport and that at combining convection with diffusion (cf. discussion above); least some of that transport is specifically targeted! and, picturesquely, it could be likened either to the migra- Whether cytoskeleton-mediated targeted-transport is tion of a herd in search of greener pastures or to a travelling also important to symplastic transport in general has not merchant taking his goods to the customer. Certainly, it been definitively established. However, considering the would reduce any tendency of enzyme systems toward reac- realities of diffusive irreversibility and the existence of an tant deprivation; and it would rather more equitably dis- intracellular motile apparatus putatively elaborated to traf- tribute rapidly depleted reaction products. Hence, at the fic vesicular packets of molecules in both animals and very least, an adjuvant-of-diffusion dividend could be a use- plants [Lippincott-Schwartz, Roberts & Hirsch 2000 (ani- ful byproduct of streaming in any cell. mal); Okamoto & Forte 2001 (animal); Hawes, Brandizzi & Most animal cells function robustly without readily vis- Andreeva 1999 (comparative); Cai et al. 2000 (plant); ible streaming, and consequently it might seem as if plants Bassham & Raikhel 2000 (plant); Aaziz, Dinant & Epel must derive a major benefit from streaming beyond that of 2001; Nebenführ & Staehelin 2001 (plant)], inserting an spinning the cytosol into a complex skein of streamlines. egregiously inefficient diffusive step into polar transport But this is not necessarily so because animal cells do not within a plant cell would appear both needless and mal- generally possess large central vacuoles which tend to con- adaptive. Furthermore, the actin-myosin cytoskeleton has strain diffusion to a thin parietal layer; and therefore they been definitively shown to extend through the plasmodes- should possess a marginal advantage over plant cells in dif- mata (Overall et al. 2000) and if the basic equipment is fusive mixing, an advantage which streaming should tend to already at hand, it would seem a good bet that, over evo- offset. lutionary time, the Viridiplantae would have pressed it into But animal cells can differ from plant cells in other ways service in the cause of better symplastic transport. as well. Some have systems for signalling and transport Therefore, the remainder of this review will explicitly which have not been reported in plants such as the claudin presume that a chief role of cytoplasmic streaming in sym- proteins of tight junctions (cf. Heiskala, Peterson & Yang plastic transport is to move messages and metabolites 2001) and dyneins (Lawrence et al. 2001) for motoring within the cytoplasm in a targeted and logistically efficient along microtubules. Furthermore, they lack plasmodesmata manner. and therefore have no obvious basis for symplastic transport of macromolecules as connexons have never been shown to pass particles above 1 kDa (Kumar & Gilula 1996). However, actin and/or actin-like proteins are ubiq- DISCUSSION uitous in the living cell (Usmanova et al. 1993; Meagher, Almost by definition, symplastic transport must involve McKinney & Kandasamy 2000; Bray 2001), whereas there many transits through plasmodesmata. To discuss in detail appears to have been potent evolutionary pressure to our present knowledge of the architecture and physiology develop plasmodesmata (Raven 1997) and once the plant

© 2003 Blackwell Publishing Ltd, Plant, Cell and Environment, 26, 1–15 Role of cytoplasmic streaming in symplastic transport 9 elaborated the machinery for long-distance symplastic Additionally, for small solutes that serve as substrates transport of uncommon molecules, then the benefits of tar- for metabolic assemblies (enzyme reaction chains, geted transport in membrane-protected packets should supramolecular enzymes assemblages), diffusion could be a have driven its adoption in other contexts as well: rate-limiting step. The kinetic efficiency of ‘bucket brigade chemistry’ should be such that substrates sometimes 1 The inefficiencies which arise from diffusive irreversibil- become locally depleted (cf. Reuzeau et al. 1997); and here ity are not to be undertaken lightly. adjuvant-of-diffusion contributions might prove decisive. 2 Left unprotected in the cytosol, valuable proteins could In this case it has been suggested that metabolic assemblies react at inappropriate places and times or even be recy- associate with endomembranes when their products are cled. mRNAs could meet an analogous fate. required, and the local depletion of substrates they bring 3 Only by stringently controlling the distribution of molec- about may actually trigger the action of the molecular ular messages can a symplast domain solve its C3 chal- motor system to drag membranes laden with enzyme com- lenge (cf. Zambryski & Crawford 2000). plexes through the cytosol (Reuzeau et al. 1997). Of course, Cytoplasmic streaming (or rather its underlying molecular flow patterns associated with low Reynold’s number flow motor activity), which in a symplastically isolated cell might (cf. Taylor 1967) would prevent instant juxtaposition of sub- serve mostly as an accelerator of biochemical reactions, strates and enzyme complexes; but with streaming the dif- becomes in a symplastic syncytium a powerful tool in the fusion distances could be so reduced that they would be organization of the supracellular domain. bridged in short order. Such situations could occur in either isolated or symplastically connected cells. A small solute whose transport may seem at variance Small hydrophilic particles with these pronouncements is the hormone auxin (indole- 3-acetic acid, M = 175 Da) whose polar transport is pres- A typical solute of this sort, even allowing for tortuosity ently the subject of intense interest (e.g. Lomax, Muday & through the cytosolic hydrogel, should experience D Rubery 1995; Muday & DeLong 2001). Its polar transport 10−10 m2 s−1 and a 10−5 m; the unfavourable case of spread- is believed to require asymmetric vesicle-trafficking of the ing in a thin layer of parietal cytoplasm then yields, by Eqn efflux carrier candidate PIN1, which localizes in basal cell 7, a characteristic time of spreading 0·5 s. Such particles walls where it promotes auxin efflux into the cell wall space diffusing axially down the cytoplasmic annulus of a plas- (Palme & Gälweiler 1999; Geldner et al. 2001). However, modesma 0·2 µm long should have transit times 0·2 ms the cell wall space is apoplastic; and polar auxin transport is (cf. Eqn A14). therefore more properly termed ‘trans-cellular’ rather than For many common small solutes, an intracytoplasmic ‘symplastic’ and falls outside the scope of this review. spreading time of 0·5 s is probably often unimportant. For In sum, small solutes may be distributed by diffusion these, a reasonable rule of thumb might be that they are alone in some instances; but, in others, their distribution to sensibly uniform throughout the cytosol. Following any dis- sites where they are needed may be facilitated by active equilibration, cytoplasmic streaming would of course translocation of the sites. However, because the examples shorten the return to equilibrium. But the physiological cited seem not to involve cytoplasmic streaming ways utility of such shortening could be minor, despite the essential to symplastic transport, they are outside the scope known difficulties in accounting for some symplastic trans- of this review; they do, however, suggest that cell-to-cell dif- port (e.g. Bret-Harte & Silk 1994). fusion via the cytoplasmic annulus could be a viable method Nevertheless, less simple situations may exist. For exam- of reallocating common small hydrophilic solutes (e.g. ple, consider a sequence of N cells, each ideally mixed and Tyree 1970; Bostrom & Walker 1975; Lucas 1997; Blackman having concentration c (t) (mol m−3) of some substance and n & Overall 2001). As Tyree put it 30 years ago, ‘The plas- a volume V (m3). Presume next that each (n = 1,2,3,..., N) modesmata constitute the pathway of least resistance for receives solute from the preceding cell at a rate G[c − c ], n-1 n the diffusion of small solutes.’ (Tyree 1970). where G (m3 s−1) is a diffusive conductance, and loses it at a − rate G[cn cn+1]; this example represents, for instance, a simplified model of symplastic phloem unloading (cf. Everything else Patrick 1997). Moreover, by invoking conservation of spe- All other molecules are presumed to move through the cies, it reduces to plasmodesma by facilitated trafficking (cf. Oparka & Rob- erts 2001); and, despite their known diffusibility, there is no V dcn =-+cccnnn-+112 (14) evidence that selected species of small solutes can not also G dt be moved by facilitated trafficking to control their destina- which is in fact a discrete analogue of the electrotonus tions. The mechanisms of this trafficking have not yet been problem for an unmyelinated axon and should lead to the worked out in detail. But it is unambiguous that, under observed stair-step distribution of symplastic transport (e.g. favourable circumstances, particles up to ∼50 kDa can Fisher 2000; Fig. 15·30). Symplastic transport down a linear move; and these particles may be either proteins or nucleic file has been discussed from the viewpoint of irreversible acids (e.g. Wolf & Lucas 1994) or various probes of known thermodynamics by Tyree (1970). dimensions (e.g. van Bel, Günther & van Kesteren 1999;

Schulz 1999). Of course, the plasmodesma is not our focus; utes (cf. Blackman & Overall 2001) and the modest number but its coupling to the particles transported must be if we of trafficked macromolecules specifically identified to date are ever to achieve deep understanding of the relationship (e.g. Lee, Yoo & Lucas 2000), there may also be trafficking of streaming (and its underlying molecular motors) to sym- in small lipids (Grabski et al. 1993); and one wonders about plastic transport. signalling oligopeptides (cf. Bisseling 1999; Ding et al. 1999) If we accept the hypothesis that cytoplasmic streaming is or cell wall precursors. What more may be crossing that our a visible sign of the targeted intracytoplasmic transport of customs officer has thus far failed to detect? selected entities, we do not thereby resolve the mysteries of symplastic transport; but we do establish a formal atmo- The boxcar problem sphere in which provocative questions can be more sharply The bar code labelling so familiar from the supermarket posed. Four such are given below. was first developed for a logistics problem: that of automat- ically sorting and routing boxcars. A plant cell which is, with The meso-scale problem molecular motors, moving either single molecules or pack- Both endoplasmic reticulum (Staehelin 1997) and Golgi ets of molecules along the cables in its cytoskeleton faces a complex (Staehelin & Moore 1995) are known to bud off similar challenge: it must, at each way-point, recognize the vesicles. Vesicles are of a size (tens to hundreds of nanom- ‘boxcar’ (by destination and/or content) and then route it eters) at which entities are known to have very low coeffi- appropriately. But what is the packet’s equivalent of a way- cients of passive diffusion (cf. Luby-Phelps 2000); hence, bill, how is this read, and what is the molecular machinery even neglecting the need for orderly routing, vesicle traf- of track switching? In particular, how do these mechanisms ficking presumably will benefit from facilitation of vesicular function in an environment where cytoplasmic streaming motion. It makes sense therefore to imagine that, when the itself may be altering the connectivity of actin trackways? molecular motors of the cytoplasm are towing neither sin- However, vesicles coated by the proteinaceous polymer gle naked macromolecules nor essential organelles through COPII are probably implicated (Antonny & Schekman the cytosol, they are towing small membrane-bounded 2001); and the presence of suitable targeting sequences is packets of molecules (vesicles); this at least is the reigning doubtless important (Fulgosi & Soll 2001; Ossareh-Nazari, paradigm in animal physiology, although rigorous proof Gwizdek & Dargemont 2001). Similar issues arise also for appears scarce (Sheetz et al. 1998). By far best way of neurons where multiple recognition and sorting steps are understanding vesicle formation and trafficking is direct suspected and where ‘very little is known about the specific real-time visualization with resolution on the few-nanometer associations of different motor proteins with different car- level. As, however, these meso-scale dimensions are well goes’ (Sheetz et al. 1998). These are deep C3 problems; and below the resolution of ordinary light microscopy, disposing to dismiss them as mere exercises in targeting would be to of this problem will presumably require further develop- overlook the great intrinsic complexity of even simple com- ments within the continuously burgeoning area of biologi- mercial bar coding practice (cf. Palmer 2001). cal microscopy; moreover, being able to follow a punctate fluorescent label over submicrometer distances is not quite The facilitation problem the same as being able to follow shape changes clearly on When, presumptively, a vesicle (or a single molecule) has the few-nanometer level. The meso-scale problem is by no been towed to the mouth of a plasmodesma, a decision means unique to symplastic transport in plants and bedevils must be made: either (i) the contents of the packet are also trafficking in nerve cells where it has been said that ‘the turned lose to find their own way through to the next cell; or current phenomenological data beg questions about the (ii) their transit is somehow ‘facilitated’. If the former cellular mechanisms used not only to transport material but choice is made, then (thanks to diffusion and/or reaction) also to modulate activity in a process’ (Sheetz et al. 1998). the contents of the packet should mostly melt away; moreover, those which do make it to the next cell must then be The customs officer problem corralled and repackaged, presumably with still more loss. Imagine a customs officer at a remote border crossing (plas- Clearly ‘facilitation’ would appear to be the more effica- modesma). He observes (cf. Aaziz et al. 2001) a highly elab- cious way of proceeding. But what is facilitation and what orated rail system (cytoskeleton) which approaches his post are its minutiae? No one knows, and cartoons of possible closely on both sides, with even a few tracks (actin fila- models seem to be the rule (e.g. van Bel et al. 1999; Jackson ments) crossing the border; and switch engines (myosins) 2000; Lee et al. 2000; Zambryski & Crawford 2000; Black- are visibly associated with these tracks. But mostly what he man & Overall 2001). To stimulate discussion, here are four sees is the occasional heavy lorry with a wide load (macro- abstract possibilities which might figure in some future molecule) and lots and lots of innocuous passenger cars solution of the facilitation problem: (small diffusable solutes). Should not the customs officer suspect that more is crossing the border than meets the 1 Upon arriving in the neighbourhood of a plasmodesmal eye? What is the purpose of those rails? Why is there ever orifice, the packet is incorporated into a cis-reticular visible cytoplasmic streaming if something is not deliber- compartment from which its contents are somehow deliv- ately being trafficked? In addition to small hydrophilic sol- ered via the endoplasmic reticulum to a trans-reticular

compartment for re-packeting and export. This is admit- visualizing the real-time transport of messages and metab- tedly vague; but it recognizes that transit of the plas- olites; the ‘customs inspector problem’ of enumerating the modesma must be either via endoplasmic reticulum, entities trafficked; the ‘boxcar problem’ of understanding cytoplasmic annulus, or plasmalemma, there being no deeply the routing of the messages and metabolites within other symplastic possibilities; and it builds upon the the cytoplasm; and finally the ‘facilitation problem’ of effec- reports of Grabski et al. (1993) and Cantrill et al. (1999) tively conveying these messages and metabolites from one on cell-to-cell communication via endomembranes. cell to another by way of the plasmodesmata. 2 The molecules within the packet are individually coupled Finally, as the shade of David Hilbert would attest, the to molecular motors at the cis-end of the plasmodesma ability of man to pose a problem does not guarantee the and then are towed through the cytoplasmic annulus for solvability of that problem and certainly can not specify repackaging at the trans-end. Such towing, if in fact it the time scale on which a solvable problem actually will be exists (cf. Zambryski & Crawford 2000), presumably solved. would be mediated by the actin and myosin within the plasmodesma (cf. Overall et al. 2000; Blackman & Over- all 2001). ACKNOWLEDGMENTS 3 The packet is already attached to myosin engines which simply switch onto an actin trackway of the plasmodesma The debt owed to the countless individuals who, over the and tow it through the cytoplasmic annulus as a unit. This last 200 years, brought our understanding of symplastic process may be facilitated by gating of the annulus into a transport and its underpinnings to its present state is hereby more distended state (Zambryski & Crawford 2000). acknowledged gratefully. Many of these individuals are 4 The packet is stuffed into the plasmodesmal orifice, and a unknown to the author; others were known but not chosen transient turgor increase is somehow triggered in the cis- for specific citation in this review; and some undoubtedly cytoplasm to ram it through hydraulically. were passed over unjustly. In a review which spans the Any or none of these models may eventually be found to diversity of phenomena shown here to be relevantly inter- pertain. The important thing is to meditate upon possibili- related, the inclusion of any contributions which are neither ties (however, remote) with an eye towards devising test- reviews nor monographs is inevitably idiosyncratic. Yet, able models for the physical bases of facilitation. without the background discoveries and the intellectual stimulus of all those workers who preceded us, the knowledge to which we now pretend could not exist. CONCLUSIONS Specific thanks are also due to Professor Barbara G. Pickard for careful reading of this manuscript and many ‘Wir müssen wissen. Wir werden wissen.’ David Hilbert* helpful suggestions. So, in the final analysis, what is the precise role of cytoplasmic streaming in symplastic transport? Clearly, we do not know. But, to the green plant, streaming is obviously REFERENCES valuable because the Viridiplantae we see about us did not survive by frittering away valuable resources. Despite Aaziz R., Dinant S. & Epel B.N. 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APPENDIX This system can be solved by separation of variables and an application of Sturm–Liouville theory (Churchill 1941; sect. Diffusive irreversibility 24–25) to yield the general Dirichlet series in time: • The equation of convection/reaction-diffusion can, follow- sin()rnra ctr();= S0 +-Â Snnexp() tt , (A4) ing the procedures outlined in Carslaw & Jaeger (1959; ch. n=1 ()rnra 1), readily be obtained as: where the Sn are constants relating to the initial conditions, τ ∂ c the time constants n are given by: 0 =-—+◊+Dc2 u grad cl c (A1) ∂ t a2 t n = , (A5a) −3 2 where c (mol m ) is the concentration of an arbitrary dif- Drn fusant γ, t (s) is the time, u (m s−1) is the local vector velocity ρ and the dimensionless eigenvalues n are non-zero roots of λ −1 of the ﬂuid (e.g. cytosol) and (s ) is the rate at which the the characteristic equation: γ-component is eliminated by chemical reactions. For the purposes of the diffusive irreversibility with rrnn= tan . (A5b) spherical symmetry, Eqn A1 reduces to: These roots are well known, have been tabulated (e.g. Carslaw & Jaeger (1959; p. 492), and form a sequence Ï 2 ∂ c ¸ ∂ Ìr ˝ 4·493..., 7·725..., 10·904..., 14·066... whose members ∂ c -2 Ó ∂ r ˛ (A2) 0 =-Dr 1 π ∂ t ∂ r go asymptotically as (n + /2) . Clearly, the solution where r (m) is the radial variable in a spherical coordinate approaches its steady value with a characteristic time given system. Appropriate boundary and initial conditions for the by: problem of spreading are: tt aD222 p . (A6) 1 spreading = () cr bounded, 0 and all t > 0 (A3a) Æ The converse problem of gathering up the spread diffusant ∂ c can be treated much as above, only with Eqns (A3) modi- ÆÆ0, ra and all t > 0 (A3b) ∂ r ﬁed to be: N 3 0 ctr();,==0 ra and all t > 0 (A7a) ctr();= 3 , rt£ a and =0 (A3c) NA 4pa ∂ c ==0, ra and all t > 0 (A7b) ctr, =0 , a r a and t=0. (A3d) () <£ ∂ r © 2003 Blackwell Publishing Ltd, Plant, Cell and Environment, 26, 1–15 Role of cytoplasmic streaming in symplastic transport 15

θ ctr(); unspecified, r£ a and t=0 (A7c) where is the azimuthal angle. Separation of variables then yields, for the νth characteristic value, the azimuthal char- ctr(), = C0 , a <£ r a and t=0. (A7d) acteristic function AνPν(θ) + BνQν(θ), where Pν and Qν are Proceeding as above also yields a Dirichlet series in time, Legendre functions, Aν and Bν are constants, and now without a constant term and with characteristic times 2 vv()+ 1 = a()t v D, (A16) given by the non-zero roots of the characteristic equation: τν (s) being the characteristic time associated with a given ν. 1 rnn= arctan r (A8) Subject to the constraint that function must be bounded 1 -a a over the closed azimuthal interval [0,π], the characteristic ρ ρ ∼ θ The roots of Eqn A8 other than 1 rapidly settle near n (n function reduces (cf. Erdélyi 1953; sect. 3·9) to AνPν( ) with 1 π ρ α ν + /2) . 1 is such that, if 3 /a << 1, = 0, 1, 2,... The characteristic spreading time is that associated with the least non-zero value of ν or: a2 a (A9) 2 tt1 gathering==t sphere t shell = aD()2 (A17) 3D a τ where the newly defined sphere will be used later. Thus, 2 t gathering 2p aa ==6. 579 ... (A10) Reaction–diffusion t 3 aa spreading Obviously, in this fully three-dimensional case, diffusive To illustrate the effect of chemical reactions on removing spreading is much faster than diffusive in-gathering when diffusing molecules from a cell’s reactant pool, consider the a/α is large. solution to Eqn A1 in the steady-state spherically symmet- The homologous two-dimensional in-gathering problem ric convection-free case with a small source around the ori- is that of a circular sink on a planar surface and yields a gin. Eqn A1 becomes: characteristic equation in terms of Bessel functions: dc d r2 0 Dr-2 {}dr lc, (A18) Êa ˆ Êa ˆ (A11) = - 0 = J0101rrnnYY()- rr nnJ () dr Ë a ¯ Ë a ¯ with the boundary conditions Numerical solution of this equation for possible values of α ρ ± cCa /a yields 1 = 0·5 0·2. This yields an approximate solution ()= 0 (A19a) for the important characteristic time: dc ==0 at ra (A19b) 2 dr a (A12) tacircle = ln a D This system is readily integrated (cf. Carslaw & Jaeger 1959; ch. IX) to yield The comparison between Eqns (A9) and (A12) reveals an enormous disparity in in-gathering times as (a/α) grows. a XXcosh()11-[]ra-- sinh X()[] ra cr()= C0 , (A20) The homologous one-dimensional in-gathering problem r XXcosh11aaaa sinh X ()-[]--()[] for a line yields the characteristic equation: where a new dimensionless number Ξ = a√(λ/D) has been rpn =-()n 12 (A13) introduced. Hence, and ca() a X = . (A21) 2 ca()a XXcosh()11-[]aa a-- sinh X()[] a a 4 2 (A14) t line =-[]1 a a α D p 2 Normally /a << 1; therefore, the degree to which the more peripheral regions of the sphere will be diffusant-deprived For large (a/α), Eqn (A14) is virtually independent of (a/α). will also depend strongly upon Ξ, and diffusant scarcity will And, finally, the spreading problem for a symmetrical be marked for Ξ ≥ π. cap of diffusant on a spherical shell so thin that radial diffusion can be neglected leads to: ∂ c D 1 ∂ ∂ c 0 =- Èsinq ˘ (A15) ∂qt a2 sin ∂q Í ∂q˙ Î ˚