Experiments on osmotically driven flow in idealized elastic membranes

The phloem provides a pathway for products of photosynthesis to be transported to different parts of the plant for consumption or storage. The Münch pressure flow hypothesis (PFH) is considered the leading framework to mathematically represent this transport. It assumes that osmosis provides the necessary pressure differences to drive the fluid flow and sucrose within the phloem. Mathematical models utilizing the PFH approximate the phloem by a relatively rigid semi-permeable tube. However, the phloem consists of living cells that contract and expand in response to pressure fluctuations. The effect of membrane elasticity on osmotically driven sucrose front speed has rarely been considered and frames the scope here. Laboratory experiments were conducted to elucidate the elastic-to-plastic pressure-deformation relation in membranes and their effect on sucrose front speeds. It is demonstrated that membrane elasticity acts to retard the sucrose front speed. The retardation emerges because some of the osmotic pressure performs mechanical work to expand the membrane instead of pressurizing water. These results offer a novel perspective about the much discussed presence of sieve plates through-out the phloem acting as structural dampers.

Image processing tools in MATLAB programming language (Mathworks, Natick, 123 MA) were used to detect simultaneously the front location and membrane diameter for 124 the experiments. These data were post processed by removing the initial period (1-2 hours) 125 that was noisy because of the inertial effects at the start of the experiment (dextran tran-126 sient mixing before osmosis building up). In addition, the final time that was used to 127 obtain the results was chosen to be the time when the pressure reached 90 % of its equi-128 librium value, or when the front reached the top of the membrane in case of high initial 129 concentration. This period represented mainly the elastic regime and the initial part of the plastic regime before any membrane failure occurred after prolonged stress exposure. 131 All in all, the study yielded 28 runs when combining both dextrans.

Data analysis
133 Two independent approaches to analyze the data were developed and applied to 134 all 28 experiments. These approaches tackle the problem from two viewpoints that uti-135 lize differing data sets and assumptions. One approach utilizes front speeds using con-136 servation of volume and the other is based on pressure analysis using the relation between 137 osmotic potential and the flow of water into the membrane. The first approach is pri-138 marily kinematic and shows how the front speed is affected by allowing the membrane 139 to expand. The second approach discusses the underlying dynamics that decrease trans-140 port efficiency due to membrane expansion.
where A(t) is the cross-sectional area orthogonal to u(t) and u(t) is the axial velocity.

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This conservation law is analyzed over the whole period of the experiment by integrat- where V t f and V ti are the final and initial imaged volumes. To approximate the integrals

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where a 1 and a 2 are assumed to be constant and can be determined separately from im-177 aged l(t) and h(t) with variations in t using the nonlinear regression toolbox in MAT-

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LAB. Hence, the total radial inflow of water due to osmosis is now given by For a rigid membrane, equation (4) may be simplified to 182 To use this approximation, the osmotic velocity v 0 was assumed to be equal to an effec-183 tive velocity that is time independent v 0 leading to

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where the integral was approximated using the exponential shape for h(t) and l(t) as be- where Π is the osmotic potential, p is the hydrostatic pressure and k is the membrane 205 permeability assumed here to be uniform in time and homogeneous in space. The osmotic 206 potential can be further approximated using the Van't Hoff linear relation (reasonable 207 for low concentration) where R g , T , M w , i and c are the ideal gas constant, temperature (K), molecular weight 210 of dextran (g mol −1 ), Van't Hoff factor (that need not be the same for the two dextrans), where c is the total amount of dextran injected in the domain in grams. The β(t) is a 218 cutoff constant that is related to the membrane properties and later discussed. As shown 219 in section 2.1.4, the dh(t)/dt is nearly uniform in space to within the camera resolution.

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To understand the effect of membrane elasticity on mass flow, equation (12)  The volume-pressure relation shows a decrease of efficiency of the osmotic poten-245 tial when the cutoff coefficient β is evaluated at initial and final time of the experiment.

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At maximum efficiency, β should be unity. However, this was not the case for the de-scribed experiments. Figure 1A shows the initial rate of expansion of the elastic mem-248 brane plotted against the initial osmotic pressure. The inset plot shows the same results 249 when the Van't Hoff factor i was assumed to be unity for both dextrans. Assuming that 250 both membranes behaved in a similar manner with each dextran because of similar MWCO 251 to molecular weight ratio, the different slope between both dextrans can only be attributed 252 to the Van't hoff factor (as in the inset). Figure 1A repeats the inset result when i was   Here, we hypothesize that sieve plates can act as structural dampers. The added rigid-329 ity translates to hydraulic efficiency in return by allowing excess osmotic potential to con- at t = 0) for both dextrans using the corrected Van't Hoff coefficient (r 2 = 0.09 and p = 0.21). Black circles denote D6, blue crosses denote D20 and red star denote D20 with a 3.5 K membrane.