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Edited by: Sanna Sevanto, Los Alamos National Laboratory (DOE), United States

Reviewed by: Teemu Hölttä, University of Helsinki, Finland; Susanne Hoffmann-Benning, Michigan State University, United States

This article was submitted to Plant Biophysics and Modeling, a section of the journal Frontiers in Plant Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Theoretical and experimental evidence for an effect of sieve tube turgor pressure on the mechanisms of phloem unloading near the root tips during moderate levels of drought stress is reviewed. An additional, simplified equation is proposed relating decreased turgor pressure to decreased rate kinetics of membrane bound transporters. The effect of such a mechanism would be to decrease phloem transport speed, but increase concentration and pressure, and thus prevent or delay negative pressure in the phloem. Experimental evidence shows this mechanism precedes and exceeds a reduction in stomatal conductance.

The Münch hypothesis of phloem transport has been expressed mathematically in various model forms by several authors (e.g., _{i}] in sink areas, (2) osmotic influx and efflux [jW_{i}] of water through the sieve tube membrane, (3) turgor pressure [P_{i}], (4) transport speed (i.e., velocity along the sieve tube) [vS_{i}], and (5) solute concentration [C_{i}] along the sieve tube axis. An empirical equation (6) for a sixth variable, (6) viscosity of the phloem sap (ŋ_{i} assuming only sucrose at 25°C) was included in the model by _{i}]. The steady-state, algebraic form of these equations for each (ith) sieve element (or computational section) are as follows (see default values of the independent parameters in

Independent and input parameters used in

Membrane permeability | ξ | 5 × 10^{–5} |
sec^{–1} cm |

Leaf Apoplastic Water Potential | Ψ_{(leaf)} |
−0.4 | MPa |

Root Apoplastic Water Potential | Ψ _{(root)} |
−0.6 | MPa |

Gas Law Constant | R | 8.3145 | m^{3} Pa mol^{–1}K^{–1} |

Reflection Coefficient | δ | 1 | |

Radius of Sieve Tube | CELLRAD | 1.2 × 10^{–3} |
cm |

Length of Sieve Tube | LN | 100 | cm |

Length of Sieve Elements | CELLN | 0.02 | cm |

Length of Computational Section | SECLN | 1 | cm |

Number of Loading Sections | NLSECS | 5 | |

Number of Path Sections | NPSECS | 85 | |

Number of Unloading Sections | NUSECS | 10 | |

Hydraulic Conductance of Sieve Tube | LS | * | |

Cross section area of S.T.** | AX | ** | cm^{2} |

Area of sieve tube membrane*** | AM | *** | cm^{2} |

Total Vmax for each sink**** | −Vmax_{(tot)} |
**** | sec mol^{–1} |

Concentration for 1/2 vmax***** | Km | ***** | molar |

^{–6}. ***Calculated from ST radius and the length of computational sections = 1 cm. ****Negative to indicate Unloading from the sieve tubes and distributed in a linear gradient along the Unloading Zone. *****Converted to mol cm

^{3}in the model which uses CGS units.

Where ξ is membrane permeability, Ψ_{i} is apoplastic water potential, P_{i} is turgor pressure, δ is the membrane reflection coefficient, Ls_{i} is the hydraulic conductance of the Sieve Tube, Ax is the cross section area of the sieve tube, jL_{i} is the phloem loading rate, _{M}_{i} is the surface area of the sieve element, and Vmax_{i} and Km are the rate constants of the unloading transporter.

In the course of expressing various predictions of this steady-state model (_{i}) along the axis of a modeled sieve tube (i.e., maintaining the same slope or gradient of water potential) had no effect on the values or patterns of OSMOSIS (i.e., osmotic influxes and effluxes along the phloem axis), SPEED, or CONCENTRATION, but uniformly lowered the absolute value of PRESSURE (replotted using data from

Effects of apoplastic water potential (Ψ) and effective unloading conductance (i.e., EUC assigned as Vmax of the Unloading enzyme system). ^{–10} mol sec^{–1}. ^{–10} mol sec ^{–1}. ^{–10} mol sec^{–1}. Plotted from same data as

As seen in

The question of whether phloem transport in real plants would continue to operate normally with negative pressures was raised by

One hypothetical mechanism is a reduction in the Effective Unloading Conductance (EUC) of the sieve tubes in the root sinks, e.g., by virtue of a reduction of Vmax (

Using the model to predict the effect of reduced EUC in a plant with one hypothetical sink was accomplished by decreasing the collective Vmax_{(tot)} of the hypothetical Unloading Transporters from 1.72 × 10^{–10} mol sec^{–1} to 1.35 × 10^{–10} mol sec^{–1}, while maintaining all other input parameters constant. As seen in

Experimental measurements were conducted using the Extended Square Wave Carbon-11 Tracer method (^{11}C ESW,

Example of the changes in Phloem Transport Speed and Concentration (measured experimentally by Carbor-11 Traces kinetic analysis,

The short-term effects of re-watering (i.e., 4 min trickle irrigation of water at soil temperature) and the presumed restoration of high EUC on transport was seen on a minute-by-minute basis (^{11}CO_{2} and ^{12}CO_{2} were maintained at steady state equilibrium for 300 min, the downward trend is interpreted as simultaneous decreases in the concentrations of both tracer and trace. As illustrated in

^{11}C activities at numbered detector positions along a moderately drought-stressed Cotton plant during the steady state (final 300 min) of a 420 min Extended Square Wave input of ^{11}CO_{2}, before, during, and after re-watering of the plant.

The output of a time-dependent version of the phloem model (

Mathematical models at the beginning of their development, generally represent the simplest set of assumptions [Equations (1)–(6)]. Such a simple model may adequate to predict the results of the initial assumed circumstances. However, experimental tests may show it to be inadequate to express the effects of additional or changing circumstances. This appears to be the case for the Münch-Horwitz Theory under moderate drought stress conditions where the rate kinetics of the unloading mechanism decrease when the apoplastic water potential decreases (

If this proposed mechanism is true, and the rate kinetics of the unloading process are altered by the local turgor pressure at appropriate points along the sieve tubes, then the values of one or more of the Independent Parameters of the equation(s) representing the unloading mechanisms [in this hypothetical case the Michaelis–Menten Equation (5)] would become dependent variables as a function of pressure.

Again, starting with the simplest concepts and mathematics one can suggest the following added equation [Equation (7)] where the sum of the kinetic parameter Vmax_{(}_{i}_{)} of the membrane bound transporter in the root sink, or a system of enzyme activities, in the sinks (at the highest likely value of sieve tube turgor pressure of plants in a growth medium at Field Capacity moisture level) is altered as function of changing pressures. Note that this maximum value in some sinks may be set by a turgor homeostat mechanism as described by _{(tot)i} (in the ith computational section) to be 1/2 of its value at maximal apoplastic water potential and sieve tube pressure. A resulting plot is illustrated in

Hypothetical effect of turgor pressure on the value of Vmax_{(tot)} of the membrane-bound unloading enzyme system (i.e., transporters) in various metabolic sinks.

Obtaining realistic values for the parameters of any such equation would not be easy. Again, as a starting point, empirical measurements of values for phloem sap concentration and unloading rate in the root zone of real plants, needed to calculate EUC, could be accomplished by combining the ^{11}C ESW tracer method (^{11}C activity during last few minutes of an ^{11}C ESW) and the measured values of plant apoplastic water potentials.

Relationships between other independent parameters, such as phloem loading rate, membrane permeability, and sieve tube diameter, in relation to dependent variables such as concentration and pressure may also exist in real plants and could result in additional equations.

Finally, progress toward a model of carbon flow through the entire plant might be approached by coupling a Phloem Transport model with mechanistic models of Photosynthesis (e.g.,

JG: models and experiments. LH: development of models. Both authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.