Sucrose transport in osmotically driven laminar flow: going from slender tubes to plant phloem

The transport of photosynthates within plants from the production sites (mainly leaves) to areas of consumption or storage (for example the roots) is drawing attention in plant physiology, eco-hydrology, and earth systems models. The phloem, one of the plant's hydraulic systems, provides the necessary pathway for this transport mechanism. Its structure and function have been conjectured to be optimized for the efficient transport of soluble organic matter (mostly sucrose). The implications of efficient sucrose transport range from local impacts on plant growth and survival, because of a potential link between phloem failure and plant mortality under extreme weather conditions such as drought, to ecosystem-scale effects on carbon and water cycling because of the possible link between stomatal control of photosynthesis and sucrose (a main product of photosynthesis) transport within the phloem. Many models for phloem transport and their possible deficiencies have been formulated and discussed. The most commonly used and accepted hypothesis under which most of these models rely on is the pressure-flow hypothesis, commonly known as the M$\rm{\ddot{u}}$nch mechanism. In the pressure-flow hypothesis, sucrose and other photosynthates are loaded in the phloem at the production site (leaves). This high sugar concentration draws water from the xylem, which is another hydraulic system that provides the necessary water reservoir for the M$\rm{\ddot{u}}$nch mechanism, by osmosis towards the phloem. This inflow of water molecules builds the required pressure gradient along the phloem pathway to drive sugars and water molecules through the phloem's complex network of narrow, elongated, interconnected, and cylindrical living cells (the sieve tubes). Once sugars arrive at the desired location, sugar molecules are unloaded from the phloem for consumption or storage by the cells and water is released back to the xylem or other surrounding tissues. The difference in sugar concentration at the production and consumption sites builds the pressure gradient along the phloem needed for solute transport over long distances without any active pumping. Experimental challenges in measuring sugar fluxes within the phloem have led to the reliance on theoretical models to understand and predict sucrose transport. These models usually simplify physics to allow mathematical tractability. As expected when using such models, calculated sucrose mass fluxes are not in agreement with data, especially for long-distance transport. In addition, the theory of the M$\rm{\ddot{u}}$nch mechanism has been under criticism where some argue that the sieve tubes seem to have low hydraulic conductance along the phloem which makes it nearly impossible for sucrose to be transported in the longest of plants.\par %These studies also report lower leaf concentration in tall trees compared to crops thereby contradicting the M$\rm{\ddot{u}}$nch hypothesis that requires higher loaded concentration to allow larger driving force over long distances. To solve the issue of decreased conductance in long-distance transport, it has been conjectured that rather than loading and unloading sugars only at the sources and sinks, sugars can be exchanged at different locations along the phloem forming a "relay" system to increase efficiency. Another reason for this theory is the existence of sieves plates that connect the sieve tubes, where their main role is still not fully understood. While this theory is plausible, there is no clear evidence of the loading and unloading of sugars along the pathway. However, there is an increasing amount of evidence that water is being exchanged between the phloem and the xylem along the pathway. This dissertation provides theoretical and experimental contributions toward modeling sucrose transport within the phloem. Note that specific notations, definitions and literature reviews are provided within each chapter. In chapter \ref{chap:1}, we propose a new one-dimensional model that includes Taylor dispersion for osmotically driven laminar flows. In previous studies, this effect has been overlooked when simplifying the problem into a one-dimensional model where radial variations in the solute (in this case sucrose) concentration are negligible. However, as earlier noted by G.I. Taylor that these small radial solute concentration variations lead to a new correction to the flow in closed pipes. This correction is modeled as a new dispersion term that is added in the advection-diffusion equation (or the conservation of solute mass equation), hence the name Taylor dispersion. This is not the case for osmotically driven laminar flows because the radial inflow of water molecules due to osmosis dictates a position-dependent (in the axial direction) pressure gradient instead of a constant one as in closed pipe application. This will lead to a new correction in the advection-diffusion equation where a new term is formulated and modeled as an advective term. In this chapter, we showed the mathematical derivation and required assumptions to develop this new model with these corrections. We also showed their impact on the flow from a general perspective and in the plant application specifically. In chapter \ref{chap:2}, we re-address the problem by studying viscosity variations in the flow. Previously, most studies simplify the physics by assuming a constant dynamic viscosity that does not depend on the sucrose concentration. Other studies have included viscosity variations along the tube (i.e. the phloem) in the one-dimensional model or in a globally averaged model where dynamic viscosity is set equal to loading sucrose concentration. In this chapter, we consider viscosity variations due to sucrose concentration variations in the radial and axial directions using a simple numerical model by re-considering the Navier-Stokes equations. We showed that the sucrose speed increases when viscosity variations are included because of the pull-push mechanism of water molecules where the efficiency of pushing water outside the phloem increases due to lower frictional forces. This result has an impact on understanding the M$\rm{\ddot{u}}$nch mechanism and its ability to model sucrose transport within the phloem, especially for tall trees. This result can also be generalized for other applications, especially where solute concentration variations in the radial direction are higher because of lower near-wall frictional losses. In chapter \ref{chap:3}, we conducted experiments on osmotically driven laminar flows using idealized elastic membranes. In these experiments, the membrane is allowed to expand in the radial direction depending on the osmotic potential that has been injected within the membrane where dextran has been used as the solute to drive the flow. These experiments have allowed us to shape an idea of the interaction between solute front speeds and membrane elasticity which was not studied before. For phloem studies, the membrane (i.e. the sieve elements) is usually assumed to be rigid with the exception of some studies that include membrane elasticity by using the phloem elasticity predicted from field data. These data are collected from plants that already have low elasticity because of the existence of sieve plates, where their main role is still not fully understood. In these experiments, we showed that by allowing the membrane to expand, the solute front speed decreases because some of the osmotic potential was lost to expand the membrane, and by dilution of the injected concentration (increasing the volume while maintaining the same amount of solute). These results have allowed us to formulate a different theory regarding the role of sieve plates where their existence might enhance phloem rigidity and hence transport efficiency. Chapter \ref{chap:4} summarizes the under-studied effects in previous chapters where we developed a two-dimensional numerical model. In this study, we included membrane elasticity using a simple formulation for the evolution of the membrane radius in time. This formulation was included by studying the data collected from the experiments in chapter \ref{chap:3} where an exponential increase of the membrane radius in time was apparent. The results of this study showed the impact of membrane elasticity on the front speed beyond the experiments where two membrane properties were changed, the rate of increase, and the maximum value for the radius. They also showed the interplay between membrane elasticity and concentration-dependent viscosity. These results reinforce the new role of sieve plates and provided a new comprehensive model for sucrose transport which can be eventually simplified to be included in any vegetation model that connects the phloem to the leaf-xylem-root system. Finally, in the conclusions and future development chapter \ref{conclusion}, we summarize the findings of this thesis and discuss the possibility of implementing our model in future climate studies.