Diversity of funnel plasmodesmata in angiosperms: the impact of geometry on plasmodesmal resistance

In most plant tissues, threads of cytoplasm, or plasmodesmata, connect the protoplasts via pores in the cell walls. This enables symplasmic transport, for instance in phloem loading, transport, and unloading. Importantly, the geometry of the wall pore limits the size of the particles that may be transported, and also (co-)defines plasmodesmal resistance to diffusion and convective flow. However, quantitative information on transport through plasmodesmata in non-cylindrical cell wall pores is scarce. We have found conical, funnel-shaped cell wall pores in the phloem-unloading zone in growing root tips of five eudicot and two monocot species, specifically between protophloem sieve elements and phloem pole pericycle cells. 3D reconstructions by electron tomography suggested that funnel plasmodesmata possess a desmotubule but lack tethers to fix it in a central position. Model calculations showed that both diffusive and hydraulic resistance decrease drastically in conical cell wall pores compared to cylindrical channels, even at very small opening angles. Notably, the effect on hydraulic resistance was relatively larger. We conclude that funnel plasmodesmata generally are present in specific cell-cell interfaces in angiosperm roots, where they appear to facilitate symplasmic phloem unloading. Interestingly, cytosolic sleeves of most plasmodesmata reported in the literature do not resemble straight annuli but possess variously shaped widenings. Our evaluations suggest that widenings too small for identification on electron micrographs may drastically reduce the hydraulic and diffusional resistance of these pores. Consequently, theoretical models assuming cylindrical symmetries will underestimate plasmodesmal conductivities.


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Cells in plant tissues are defined by cell walls, rigid extracellular networks consisting of 30 polysaccharides and, to a lesser degree, proteins. The cells are not physically isolated, though, as   (Oparka et al. 1999). Simple plasmodesmata seem to be involved in a specific 49 assimilate export mechanism known as polymer trap, which requires size exclusion limits around 50 0.5 kDa (Comtet et al. 2017). In contrast, larger molecules of up to 40 kDa appear to move 51 4 through simple plasmodesmata in photoassimilate-importing tissues (Oparka et al. 1999;Nicolas 52 et al. 2017;Lee and Frank 2018), while pore-plasmodesma units enable the exchange of probes 53 of up to 70 kDa (Oparka and Turgeon 1999;Fitzgibbon et al. 2013). Funnel-shaped 54 plasmodesmata in the root unloading zone of Arabidopsis permit movements of molecules of at 55 least 112 kDa (Ross-Elliott et al. 2017). 56 The developmental and cell type-specific variation of plasmodesmal size exclusion limits 57 indicates active regulation and thus physiological significance of the effective size of the 58 plasmodesmal pore. Therefore the geometry of plasmodesmata must be expected to play a role in 59 controling cell-to-cell conductivity. Most theoretical plasmodesma models for quantitative 60 evaluations generally assumed coaxial symmetry with a straight, cylindrical cell wall pore, based 61 on interpretations of electron micrographs by e.g. Ding et al. (1992) and Waigmann et al. (1997). 62 Consequently, models including a cytosolic sleeve mostly assumed this sleeve to be tubular with 63 constant radius and annular cross-sectional shape, or to consist of a group of circularly arranged Hordeum vulgare had been briefly discussed by Warmbrodt (1985). Initial calculations indicated 80 that the conical funnel shape facilitated convective phloem unloading at low pressure 81 differentials (Ross-Elliott et al. 2017). These findings supported the idea that the PSE/PPP 82 interface has a specific significance for phloem unloading in Arabidopsis roots.

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Since phloem unloading mechanisms are of major importance in the context of food 84 production for human and lifestock consumption, detailed knowledge of the distribution of 85 funnel plasmodesmata in species other than Arabidopsis as well as a better understanding of the 86 physics of transport through non-cylindrical plasmodesmata would seem desirable. Therefore, 87 and because funnel plasmodesmata appeared a convenient case for studying the impact of non-88 cylindrical pore shapes on plasmodesmal transport, we first established the general occurrence of 89 funnel plasmodesmata in angiosperms. Then, we generated 3D reconstructions based on electron 90 tomograms of these plasmodesmata to evaluate flow patterns and resistances, and modeled 91 physical flow characteristics in idealized cylindrical and conical plasmodesmata, to evaluate the 92 effects of various pore geometries. As a general conclusion, we suggest that often overlooked 93 deviations from a straight, cylindrical shape can reduce the plasmodesmal resistance to bulk flow 94 and diffusion significantly.

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Funnel plasmodesmata in seven species: basic observations 97 To investigate the ultrastructure of the interface of the protophloem with surrounding cells, we 98 adapted microwave-supported fixation protocols for five eudicots from various families and two 99 Poaceae species as representatives of the monocots. Variations in root thickness, tissue density, 100 and other structural parameters made it necessary to adjust protocols for several species (see 101 Methods). We note that protophloem sieve elements in roots of most species are more difficult to 102 preserve for electron microscopy than those in the thin roots of Arabidopsis. Vitrification by 103 cryo-fixation usually fails as the protophloem is located too deep within the organ to achieve the 104 required freezing speeds.

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With few exceptions, eudicots exhibit a bi-, tri-, tetra-, or pentarch architecture of the 106 primary root, i.e. the roots possess two, three, four, or five protoxylem/protophloem units. In   Table S1). If the opening angles, θ, of the funnel-shaped cell wall   Relationship (GMFR) between the two parameters ( Fig. 2C), but this relationship was far too 139 weak to support general conclusions (r 2 = 0.12 for data calculated with DPPP = 20 nm, and r 2 = 140 8 0.08 for DPPP = 50 nm). While θ varied widely, there were species-specific trends; the median of 141 θ determined separately for each species was smallest in C. speciosa and five to ten times higher 142 in O. sativa, dependening on which value of DPPP was assumed (Fig. 2D).

3D reconstructions of individual plasmodesmata 144
Our above considerations of the geometries of funnel-shaped cell wall pores were based on the 145 simplifying assumptions presented in Fig. 2A. To obtain a more realistic picture of the structure 146 of individual funnel plasmodesmata, we generated 3D reconstructions by electron tomography. 147 We acquired 220 nm slices of stained, resin-embedded samples for all species except O. sativa, 148 which required 280 nm sections, wide enough to carry complete funnel plasmodesmata. Our 149 reconstructions showed funnel plasmodesmata of varying, irregular shapes (Fig. 3). While we 150 were able to identify desmotubules, it must be cautioned that the apparent diameters of these 151 structures depend on parameters including sample thickness, resin hardness, desmotubule 152 location, staining time, etc. In none of our image series, desmotubules appeared as sharply 153 bordered structures but rather showed as gradients of contrast intensity. Therefore their exact 154 dimensions were not always clearly discernible, and variations in apparent desmotubule diameter 155 might reflect methodological uncertainty as much as natural variability of the actual structure.

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Given this caveat, it still appears noteworthy that funnel plasmodesmata showed no signs of 157 tether-like connections between desmotubule and plasma membrane. Rather than being fixed in 158 the center of the pore, the location of desmotubules in our reconstructions was highly variable 159 ( Fig. 3).

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In the next step, we used electron tomography surface reconstructions of selected funnel 161 plasmodesmata as templates and extracted tetrahedral grids for theoretical hydrodynamics were assumed to occur only in the annular sleeve between the wall of the channel and the central 176 rod (Fig. 5A, left). In our calculations, we set channel length L, equivalent to wall thickness, to 177 400 nm. While radius b was kept constant at 7.5 nm, the outer radius, a = b + s, was varied to 178 obtain sleeve widths (s) of 2, 4, and 8 nm (the latter value may seem high, but proteins of 112 179 kDa, corresponding to a hydrodynamic diameter of ~8 nm, may pass through funnel 180 plasmodesmata; Ross-Elliott et al. 2017). As a result, we had three cylindrical channel models 181 that differed only in radius a and thus in sleeve width. These models served as cylindrical 182 standards to which conical channels − i.e., funnel-shaped ones − could be compared. In these 183 conical channels, radius b remained constant but radius a increased steadily from the smaller, or 184 outlet aperture, toward the larger, or inlet aperture (Fig. 5A, right). Consequently, sleeve width s 185 and the cross-sectional sleeve area A increased in the same direction as well, in dependence on 186 the magnitude of θ, the angle between the channel wall and the surface of the central rod ( Fig.   187 5A, right). Sleeve widths at the outlet apertures − minimum sleeve widths, in other words − were 188 set to 2, 4, and 8 nm, to allow for direct comparison to the cylindrical channels defined above.

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The hydraulic resistances of conical channels decreased drastically even with very small 190 angles θ (Fig. 5B). On the other hand, the resistance offered by channels with wider sleeves was 191 substantially lower than that found in narrower ones for all values of θ (Fig. 5B). However, the relatively less pronounced. Reductions of resistance to 10% required angles θ of 5. 9°, 9.8°, and 210 16° in channels of 2, 4, and 8 nm minimum sleeve width, respectively (Fig. 6B). Plots of the 211 ratios of the modulations of hydraulic and diffusive resistances computed for conical model 212 channels (Fig. 7) suggested that relative reductions of hydraulic resistance across the range of 213 cell wall pore opening angles observed in real cells (Fig. 2) were two-to five-fold larger than the 214 corresponding relative changes in diffusive resistance. As a result, convective flow will become 215 more important relative to diffusion when θ increases. techniques for electron microscopy that previously had been applied successfully in Arabidopsis, 225 and found funnel plasmodesmata in the phloem unloading zone in root tips of all seven 226 angiosperms examined (Fig. 1).

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The geometry of funnel plasmodesmata seemed to differ between species. Estimated In this context it seems of interest that for values of θ that correspond to the opening angles 247 estimated for real cells (Fig. 2C), the theoretical reduction in relative hydraulic resistance is two 248 to five times larger than the reduction of diffusive resistance (Fig. 6C). Consequently, the 249 balance between convective and diffusive processes in overall symplasmic transport is expected 250 to shift toward bulk movements when funnel plasmodesmata are formed between cells. This 251 supports the view that rapid phloem unloading in root tips proceeds mainly as bulk flow.

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Finally, we emphasize that resistance reduction effects similar as described here for funnel-290 shaped channels must be expected in other partially widened pore structures as well. angles that suffice to halve hydraulic and diffusive resistance (Fig. 6A,B)  intricacies that probably will be missed on most TEM micrographs. Even if they were detected, 299 they likely would be neglected in attempts to quantitatively model transport through these pores.

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Our results suggest that with regard to the efficiency of symplasmic transport, the idea that 301 plasmodesmata can be adequately described as cylindrical or annular tubes might be a seriously 302 misleading simplification. After all, a single funnel-shaped plasmodesma can be as conductive as where a0 refers to the smaller, or outlet aperture and θ is the pore angle (see Fig. 5A).

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Consequently, the cross-sectional area A of the space available for transport, or sleeve, changes 379 along the pore axis x according to Combining Eqs. (4) and (5) leads to an expression for the diffusion resistance: The analysis of bulk flow through the pores was based on the Stokes equation, where v is the velocity field, p is pressure, and η is the cytoplasmic viscosity (for modeling 386 purposes, η was set to 8.9 × 10 -4 Pa s). A pressure drop Δp was applied across the pore and we 387 assumed no-slip conditions (v = 0) on all solid boundaries.

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Flow characteristics of plasmodesmatal geometries extracted from Amira 6.7 were modeled 389 using COMSOL Multiphysics 5.4. Validation of the solver was carried out as described by  Table S1 for original data). angle θ, the relative hydraulic resistance decreases more strongly than the relative diffusive resistance, 577 implying that convective processes become more important compared to diffusive processes as the 578 angle widens. The effect is stronger in narrower pores.