Radial-axial transport coordination enhances sugar translocation in the phloem vasculature of plants

Mass transport of photosynthates in the phloem of plants is necessary for describing plant carbon allocation, productivity, and responses to water and thermal stress. Several hypotheses about optimization of phloem structure and function and limitations of phloem transport under drought have been proposed, and tested with models and anatomical data. However, the true impact of radial water exchange of phloem conduits with their surroundings on mass transport of photosynthates has not been addressed. Here, the physics of the Munch mechanism of sugar transport is re-evaluated to include local variations in viscosity resulting from the radial water exchange in two dimensions (axial and radial). Model results show that radial water exchange pushes sucrose away from conduit walls thereby reducing wall frictional stress due to a decrease in sap viscosity and an increase in sugar concentration in the central region of the conduit. These two co-occurring effects lead to increased sugar front speed and axial mass transport across a wide range of phloem conduit lengths. Thus, sugar transport operates more efficiently than predicted by previous models that ignore these two effects. A faster front speed leads to higher phloem resiliency under drought because more sugar can be transported with a smaller pressure gradient. Summary The overall speed of sap increased by including a concentrationdependent viscosity in axial and radial directions.

. It is to be noted that the c corresponding to J max 158 in steady-state and globally averaged Poiseuille models is shown not to be sensitive to 159 the phloem hydraulic properties or even tube geometry. Hence, the occurrence of such 160 a c is weakly connected to phloem hydraulics as later discussed. 161 In prior work, a tube of constant length L and radius a was considered with a/L 162 1. The sugar mass flux J (kg s −1 ) was assumed to be only advective and given by ing J can be expressed as (Jensen et al. 2013) 172 173 where X f is a geometric factor that varies with L and a, ∆P is the pressure difference c as discussed before, a maximum J = J max must exist at a corresponding optimal c 182 value that is independent of X f . Moreover, the existence of this maximum is not pred-icated based on the precise details of the osmotic controls on ∆P . Returning to J max , 184 for a preset X f , the hydraulic conductance of the tube K t can be related to the inverse 185 of viscosity using K t = X f /µ(c). Independent of whether osmotic effects on ∆P are 186 fully represented, a J max associated with an optimal c can be derived (numerically here) 187 and graphically shown in figure 1. Moreover, J max can vary significantly depending on 188 the model presentation within the range of observed values (Jensen et al. 2013).

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The Van't Hoff relation approximating the ∆P solely from osmotic potential (solid 190 line in figure 1 inset) is given by where R g is the gas constant, T is the absolute temperature and M s is the molar mass (i.e. the two end-member cases discussed in section 4.1). Second, the enhancement of 215 mass transport due to local coordination between axial and radial movement is presented.

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The results to be featured in this section are generated using numerical simulations for  The relative difference between the models is calculated by e.
(i.e. K t r in equation (10)). Similar to the axial velocity profile, this non-local effect 304 is also apparent in the radial velocity profile for the variable viscosity model that has a 305 wider velocity range, when compared to the constant viscosity model. Due to a lower 306 sugar concentration near the membrane, the viscosity of the sap decreases leading to less 307 resistance to the radial inflow of water (that is the driving force for osmotically driven     which describes the force balance along direction x i , and is given as  will be assumed small for simplicity so that ∂u i /∂x i = 0. In this case, the σ ij repre-462 sentation given by equation (7) is reasonable (Panton 2006). Another common assump-463 tion in phloem transport is that µ is constant set by the loading concentration. This ap-464 proximation is only applicable for small concentration values. However, in plants, c can 465 range from 15% wt/wt to 35% wt/wt (and for maple trees even up to ≈ 50% wt/wt).

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In this high concentration range, the dependence of viscosity on concentration has not 467 been fully analyzed in the context of three-dimensional water and sugar transport. Some  where both terms K t r and K t are functions of x but only K t r is a function of r. ity is used at a given x, K t r and K t are equal to 1/µ, and the aforementioned con-505 servation of momentum equation becomes equivalent to the HP expression with an ad-506 justment. This adjustment is due to osmosis that generates a radial inflow of water lead-507 ing to ∂ 2 p/∂x 2 = 0, which then leads to a variable pressure gradient instead of a con-508 stant one as is common in HP applications in pipes (Phillips & Dungan 1993 Here ρ variation with c is once again assumed to be small compared to the viscosity vari-516 ations with c as stated before. Using the expression for the axial velocity from equation 517 (10) in the continuity equation (11), one can see how axial viscosity gradients impact the 518 radial velocity v, which is not identically zero due to osmosis. Moreover, the viscosity 519 gradient is not only the result of the area-averaged tube conductance K t but also stems 520 from the radially-averaged (or non-local) tube conductance K t r that depends on ra- do not vary appreciably with c when compared to viscosity.

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The last equation needed to describe the physics of sugar transport is the conser-543 vation of solute mass, which is also needed to solve for u, v, and p. This equation is de-544 rived using Reynolds transport theorem that describes the movement of solutes (mainly 545 sugar here) due to advection and molecular diffusion. In cylindrical coordinates, it is given

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where D = νSc −1 is the molecular diffusion coefficient of sugar in water assumed to 549 be again insensitive to c variations when compared to ν, and Sc 1 is the molecular 550 Schmidt number for sugars in water (usually of order 10 4 ).

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The final step for describing the physics of sugar transport is to specify the bound-   The non-dimensional form of the conservation of sugar mass, i.e. equation (13), is

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where t 0 = a 2 D −1 is the radial diffusion timescale and P e = v 0 aD −1 is the radial Peclet 601 number defined by the ratio of radial advection to radial diffusion. This expression ex-602 plicitly shows the relative contributions of radial flow dynamics (through P e) and sim-603 plified geometry (through the slender ratio ) to mass transport. is also delineated from maximal |∂c(x, r)/∂x|.
To determine x f , we fitted an exponen-