Diffusion and bulk flow in phloem loading: A theoretical analysis of the polymer trap mechanism for sugar transport in plants

Plants create sugar in the mesophyll cells of their leaves by photosynthesis. This sugar, mostly sucrose, has to be loaded via the bundle sheath into the phloem vascular system (the sieve elements), where it is distributed to growing parts of the plant. We analyze the feasibility of a particular loading mechanism, active symplasmic loading, also called the polymer trap mechanism, where sucrose is transformed into heavier sugars, such as raffinose and stachyose, in the intermediary-type companion cells bordering the sieve elements in the minor veins of the phloem. Keeping the heavier sugars from diffusing back requires that the plasmodesmata connecting the bundle sheath with the intermediary cell act as extremely precise filters, which are able to distinguish between molecules that differ by less than 20% in size. In our modeling, we take into account the coupled water and sugar movement across the relevant interfaces, without explicitly considering the chemical reactions transforming the sucrose into the heavier sugars. Based on the available data for plasmodesmata geometry, sugar concentrations, and flux rates, we conclude that this mechanism can in principle function, but that it requires pores of molecular sizes. Comparing with the somewhat uncertain experimental values for sugar export rates, we expect the pores to be only 5%–10% larger than the hydraulic radius of the sucrose molecules. We find that the water flow through the plasmodesmata, which has not been quantified before, contributes only 10%–20% to the sucrose flux into the intermediary cells, while the main part is transported by diffusion. On the other hand, the subsequent sugar translocation into the sieve elements would very likely be carried predominantly by bulk water flow through the plasmodesmata. Thus, in contrast to apoplasmic loaders, all the necessary water for phloem translocation would be supplied in this way with no need for additional water uptake across the plasma membranes of the phloem.

Figure 1

The polymer trap model with diffusion and bulk flow. The water flow rates $Q$ through the cell interfaces and IC membrane are depicted with blue (full) arrows, the sugar flow rates $\Phi$ as red (dashed) arrows. These flows depend on the pressures $p$ as well as on sucrose and oligomer concentrations $c$ inside and outside the cells on the loading pathway. The semipermeable cell interfaces are characterized by the permeability $\xi$, the bulk hindrance factor $W$, and the effective diffusion coefficient $D$ with subscripts “in” and “out.” Bundle sheath cell (BSC), intermediary cell (IC), and sieve element (SE) are numbered according to the loading steps. The IC and SE are both part of the phloem and are well connected via wide contacts called pore plasmodesma units. The BSC-IC interface is characterized by narrow plasmodesmata (PDs) which prevent the oligomers from diffusing back into the bundle sheath.

Figure 2

Three perspectives of the plasmodesmata modeled as slit pores. Part of the cell wall between BSC and IC with PD density $n_{\mathrm{PD}}$ is sketched in (a). The assumed substructure of a PD is shown in cross section (b) and three dimensionally (c). The cytoplasmic sleeve (light yellow) available for water and sugar transport is restricted by the desmotubule of the endoplasmic reticulum [ER, blue (gray)] and electron-dense particles (black) attached to the membrane, and is assumed to take the form of a circular slit with radius $r_{\mathrm{PD}}$, half-width $h$, and length $d$.

Figure 3

Diffusive and convective hindrance factors $H$ (blue, solid) and $W$ (red, dashed) in circular slit pores as function of the relative solute size $\lambda$. Both approximations given by Ref. [18] decrease smoothly from 1 to 0 for an increasing solute size, where a hindrance factor of zero corresponds to total blockage of the respective molecule. The convective hindrance factor $W$ is in the whole range larger than the diffusive hindrance factor $H$. Above $\lambda =0.8$ the curves should be regarded as extrapolations.

Figure 4

Sugar flow rate ${\widehat{\Phi }}_{\mathrm{in}}$ into the IC as function of the PD-half-slit width $h_{\mathrm{in}}$ in the purely diffusive case. The sugar flow rate is composed by the sucrose flow rate ${\Phi }_{\mathrm{in}}^{\mathrm{s}}$ (red, dashed) given by Eq. (38) and the hypothetical negative oligomer flow rate ${\Phi }_{\mathrm{in}}^{\mathrm{o}}$ (red, dotted), which would occur when $h_{\mathrm{in}}$ is larger than the oligomer radius $r^{\mathrm{o}}$. For the concentration differences measured in Cucumis melo the flow rate ${\widehat{\Phi }}_{\mathrm{in}}^{\mathrm{o}}$ of oligomers back into the bundle sheath would cause the total sugar flow rate ${\widehat{\Phi }}_{\mathrm{in}}^{\mathrm{tot}}$ (blue, solid) to vanish at slit widths only about one tenth larger than these molecules. The diffusive sucrose flow rate ${\widehat{\Phi }}_{\mathrm{in}}^{\mathrm{s}}$, however, gives a sufficient overall flux rate in the case of total blockage of the modeled oligomers (i.e., $h_{\mathrm{in}}=r^{\mathrm{o}}$) and even for smaller slits totally blocking raffinose molecules (i.e., $h_{\mathrm{in}}=r^{\mathrm{r}}$; see Sec. 5d).

Figure 5

Water and sugar flow rates ${\widehat{Q}}_{\mathrm{in}}={\widehat{Q}}_{\mathrm{out}}$ (blue, dashed) and ${\widehat{\Phi }}_{\mathrm{in}}$ (red) as functions of the total sugar concentration ${\widehat{c}}_2$ in the case where the concentrations in IC and SE are equal ($c_2=c_3$). The flow rates are shown for no transmembrane flow, i.e., ${\widehat{Q}}_2=0$, only the oligomer concentration ${\widehat{c}}_2^{\mathrm{o}}$ in the phloem is varied while the sucrose concentration is fixed to ${\widehat{c}}_2^{\mathrm{s}}=0.7$. The diffusive flow rate into the IC retains its constant value $lim{\widehat{\Phi }}_{\mathrm{in}}\propto \Delta {\widehat{c}}_{\mathrm{in}}^{\mathrm{s}}$ (dot-dashed), while for an increasing oligomer concentration the advective contribution to the sugar flow decreases with the water flow which is limited by the conservation laws.