- © 2010 American Society of Plant Biologists

## Abstract

Cell walls are part of the apoplasm pathway that transports water, solutes, and nutrients to cells within plant tissue. Pressures within the apoplasm (cell walls and xylem) are often different from atmospheric pressure during expansive growth of plant cells in tissue. The previously established Augmented Growth Equations are modified to evaluate the turgor pressure, water uptake, and expansive growth of plant cells in tissue when pressures within the apoplasm are lower and higher than atmospheric pressure. Analyses indicate that a step-down and step-up in pressure within the apoplasm will cause an exponential decrease and increase in turgor pressure, respectively, and the rates of water uptake and expansive growth each undergo a rapid decrease and increase, respectively, followed by an exponential return to their initial magnitude. Other analyses indicate that pressure within the apoplasm decreases exponentially to a lower value after a step-down in turgor pressure, which simulates its behavior after an increase in expansive growth rate. Also, analyses indicate that the turgor pressure decays exponentially to a constant value that is the sum of the critical turgor pressure and pressure within the apoplasm during stress relaxation experiments in which pressures within the apoplasm are not atmospheric pressure. Additional analyses indicate that when the turgor pressure is constant (clamped), a decrease in pressure within the apoplasm elicits an increase in elastic expansion followed by an increase in irreversible expansion rate. Some analytical results are supported by prior experimental research, and other analytical results can be verified with existing experimental methods.

Cell walls perform many functions for plant, algal, and fungal cells. Physical and chemical protection from the environment and physical support for cells and organs are obvious functions. Cell walls also withstand the stresses imposed by turgor pressure and deform irreversibly and reversible (elastically) during expansive growth. Irreversible wall deformations during expansive growth control cell enlargement, size, and shape. Growing and mature (nongrowing) cell walls undergo elastic deformations after changes in turgor pressure caused by changes in water status and environmental conditions. Elastic wall deformations are fundamental to the water relations of plant, algal, and fungal cells. For plant cells in tissues and organs, cell walls are part of the apoplasm pathway that transports water, solutes, and nutrients to cells.

Importantly, pressures within the apoplasm (cell walls and xylem) are frequently different from atmospheric pressure during expansive growth of plant cells in tissues and organs. Lower pressures (tensions) are related to transpiration rates from plant organs and to expansive growth of cells in plant organs, e.g. Boyer (1967, 2001), Molz and Boyer (1978), Nonami and Boyer (1987, 1993), Nonami and Hashimoto (1996), Passioura and Boyer (2003), Boyer and Silk (2004), Koch et al. (2004), Wiegers et al. (2009), and the references within. Higher pressures (root pressures) occur during the spring when the soil is well hydrated (e.g. Kramer, 1932). Bleeding sap from cuts and broken stems is evidence of root pressure. Also, higher pressures may occur diurnally, during the night when transpiration rates are low (e.g. Tang and Boyer, 2008). Guttation drops on leaves in the morning are evidence of these higher pressures.

Prior research indicates that a significant amount of chemistry and molecular biology occur within cell walls undergoing irreversible deformation during expansive growth (e.g. Cosgrove, 2005; Boyer, 2009). Two questions arise. First, how do pressures within the wall that are different from atmospheric pressure affect the turgor pressure, water uptake, and growth rate of cells in plant organs such as roots, stems, and leaves? Second, how are relevant chemical reactions affected by lower and higher pressures within the wall? The analyses conducted in this article focus on the first question.

Previously, equations derived by Lockhart (1965) for wall deformation and water uptake (Growth Equations) were augmented with terms for elastic wall deformation (Ortega, 1985) and transpiration (Ortega et al., 1988). In this article, the previously established Augmented Growth Equations (Ortega, 1985, 1990, 1994, 2004; Ortega et al., 1988; Geitmann and Ortega, 2009) are modified to evaluate the turgor pressure, water uptake, and expansive growth of plant cells in tissue when pressures within the apoplasm are lower and higher than atmospheric pressure. In addition, the pressure within the apoplasm is evaluated after turgor pressure in cells decrease, thus simulating the condition produced by an increase in expansive growth rate of cells in plant tissues and organs. Also, the modified equations are used to determine how the results of stress relaxation experiments conducted on growing plant organs are affected by pressures within the apoplasm that are not atmospheric pressure. Last, the expansive growth of a plant cell is evaluated when pressure within the apoplasm undergoes a semi-instantaneous change while the turgor pressure remains constant, i.e. clamped. Some analytical results are supported by prior experimental research, and some analytical results can be verified with existing experimental methods.

## AUGMENTED GROWTH EQUATIONS

Expansive growth of cells with walls is the result of two simultaneous and interrelated biophysical processes: water uptake and wall deformation. Water uptake produces turgor pressure, which stresses the wall. The wall deforms in response to the wall stresses. During expansive growth, wall deformations are both irreversible and reversible (elastic). Assembly and incorporation of wall materials into the deforming wall control wall thickness, mechanical properties, and mechanical behavior (deformation rate and direction).

The Augmented Growth Equations are biophysical equations that describe the turgor pressure, water uptake, expansive growth, and wall deformation of cells with walls. The magnitude and behavior of the inclusive biophysical variables are dependent on relevant biological processes. Previously, the Augmented Growth Equations were used to analyze and interpret the experimental results of growing cells with walls for time periods of several minutes to hours for plant cells in tissue (e.g. Cosgrove, 1985, 1987; Serpe and Matthews, 1992, 2000; Murphy and Ortega, 1995, 1996), isolated fungal cells (e.g. Ortega et al., 1988, 1989, 1991), and isolated algal cells (e.g. Proseus et al., 1999, 2000). Lewicka (2006) extended their application to whole plants and longer time periods (days). Ortega (2004) and Geitmann and Ortega (2009) reviewed these biophysical equations.

### Rate of Change in Water Volume

Equation 1 describes the relative rate of change in water volume, (d*V*_{w}/*V*d*t*), for the general condition when a cell is exposed to the atmosphere and loses water through transpiration (Ortega et al., 1988; Ortega, 1990, 1994, 2004; Geitmann and Ortega, 2009).

Equation 1 is written in relative terms.

(rate of change in water volume) = (rate of water uptake) – (transpiration rate)

where *V* is the volume, *t* is the time, *L* is the membrane relative hydraulic conductance (*L* = *L*_{p} *A*/*V*), *L*_{p} is the membrane hydraulic conductivity, *A* is the membrane area, σ is the solute reflection coefficient, ΔΠ is the osmotic pressure difference across the membrane (ΔΠ = Π_{I} – Π_{E}), *P* is the turgor pressure (a gauge pressure relative to atmospheric pressure as measured with a pressure probe, *P*_{I} – *P*_{atm}, where *P*_{I} is the pressure inside the membrane and *P*_{atm} is defined to be zero), and *T* is the relative rate of change in water volume lost via transpiration (relative volumetric transpiration rate), i.e. *T* = d*V*_{T}/*V*d*t*.

### Cell Wall Expansion Rate

Equation 2 describes the relative rate of change in volume of the cell wall chamber, d*V*_{cwc}/*V*d*t*, as the sum of irreversible deformation rate and reversible (elastic) deformation rate of the wall (Ortega, 1985, 1990, 1994, 2004; Ortega et al., 1988; Geitmann and Ortega, 2009). Equation 2 is written in relative terms.

(wall expansion rate) = (irreversible deformation rate) + (elastic deformation rate)

The biophysical variable *φ* is the irreversible extensibility of the wall, *P*_{C} is the critical turgor pressure (related to the yield threshold, *Y*), and *ε* is the volumetric elastic modulus.

### Rate of Change in Turgor Pressure

Because the relative rate of change in volume of the cell contents, water, and cell wall chamber are approximately equal, a biophysical equation describing the rate of change of turgor pressure, Equation 3, can be obtained by combining Equations 1 and 2 with the elimination of d*V*_{w}/*V*d*t* and d*V*_{cwc}/*V*d*t* because d*V*_{w}/*V*d*t* ≈ d*V*_{cwc}/*V*d*t* (Ortega, 1994, 2004; Geitmann and Ortega, 2009).

(rate of change of turgor pressure) ∝ {(water uptake rate) – (irreversible deformation rate) – (transpiration rate)}

Equation 3 states that the rate of change of turgor pressure is proportional to the relative magnitudes of water uptake rate, irreversible deformation rate of the wall, and volumetric transpiration rate. Equation 3 may be solved to determine how the turgor pressure changes after an instantaneous change in the magnitude of the one or more of the inclusive biophysical variables and/or the relative volumetric transpiration rate, i.e. Equations 4 and 5.

Equation 4 describes an exponential decay of the turgor pressure from *P*_{o} (turgor pressure at *t* = 0) to *P*_{eq} (equilibrium turgor pressure) that occurs subsequent to an instantaneous change in the magnitude of one or more of the inclusive biophysical variables and/or the relative volumetric transpiration rate. The exponential decay has a time constant, *t*_{c} = [*ε* (*φ + L*)]^{−1}. The expression for *P*_{eq} (5) is new and describes the equilibrium turgor pressure for a growing cell that is transpiring. Interestingly, Equation 5 establishes a relationship between the equilibrium turgor pressure and the transpiration rate. It is apparent that an increase or decrease in transpiration rate will decrease or increase the magnitude of the equilibrium turgor pressure, respectively. In the special case when the transpiration is zero, Equation 5 reduces to *P*_{eq} = (*L* σ ΔΠ + *φ P*_{C}) (*φ + L*)^{−1}, which is the same expression previously obtained by Ortega (1985) and previously derived for turgor pressure in equilibrium (steady) growth by Lockhart (1965) and Ray et al. (1972).

## PLANT CELLS IN TISSUE: THEORETICAL DEVELOPMENT

In plant tissue, the pressure external to the cell membrane is the pressure within the cell wall or apoplasm, *P*_{A} (a gauge pressure relative to the atmospheric pressure, *P*_{atm}, where *P*_{atm} is defined to be zero). Importantly, *P*_{A} may be different than *P*_{atm} for plant cells in tissue, i.e. not zero. Therefore, it is useful to expand Equation 1 so that water uptake can be evaluated for cells in tissue when the pressure within the apoplasm is not zero.

(rate of change in water volume) = (rate of water uptake)

The turgor pressure, *P*_{I}, is the pressure inside the cell membrane as measured with the pressure probe, and *P*_{A} is the pressure within the apoplasm as measured with the pressure chamber. The osmotic pressure difference across the membrane is now, ΔΠ = Π_{I} – Π_{A}, where Π_{I} is the osmotic pressure inside the membrane and Π_{A} is the osmotic pressure within the apoplasm. The term for the relative transpiration rate has been omitted in Equation 6 because cells in tissue are typically not exposed directly to the atmosphere where they can lose water via transpiration. However, experimental evidence demonstrates a relationship between the magnitude of *P*_{A} and the magnitude of transpiration rates from plant tissue and organs, e.g. in maize (*Zea mays*) leaves, *P*_{A} is negative during the day when transpiration rates are large and positive during the night when transpiration rates are small (Tang and Boyer, 2008).

Equation 2 is expanded to explicitly include *P*_{I} and *P*_{A}, and Equation 7 is obtained in relative terms.

(wall expansion rate) = (irreversible deformation rate) + (elastic deformation rate)

Similarly, Equation 3 is expanded to explicitly include *P*_{I} and *P*_{A}, and Equation 8 is obtained after rearranging the terms.

## RESULTS

### Turgor Pressure, Water Uptake, and Expansive Growth after a Step-Down in *P*_{A}

The behavior of the water uptake, expansive growth, and turgor pressure is evaluated using Equations 6 to 8 after a step-down in *P*_{A}; *P*_{A} = *P*_{AO} *–* Δ*P*_{a}, where *P*_{AO} is the pressure within the apoplasm before the step-down and *–*Δ*P*_{a} is the step-down. The turgor pressure couples the equations for water uptake and expansive growth (Eqs. 6 and 7, respectively), so the equation for turgor pressure (Eq. 8) is solved first and its solution is used in the equations for water uptake and expansive growth to determine their behavior. Immediately after the step-down, the magnitude of *P*_{A} (*P*_{AO} *–* Δ*P*_{a}) is constant and d*P*_{A}/d*t* is zero, and Equation 8 becomes Equation 9.

Assuming that all the biophysical parameters remain constant after the step-down, Equation 9 takes the form d*P*_{I}/d*t* + A *P*_{I} = B. The constants A and B are as follows: A = *ε* (*φ + L*), and B = *ε* (*L* σ ΔΠ + *φ P*_{C}) + *ε* (*φ + L*) (*P*_{AO} *–* Δ*P*_{a}). The solution for turgor pressure as a function of time, *P*_{I} (*t*), is *P*_{I} (*t*) = [*P*_{I} (0) *–* A^{−1} B] exp {– A *t*} + A^{−1} B, where *P*_{I} (0) is the turgor pressure at *t* = 0, i.e. the initial equilibrium turgor pressure. Substituting in the expressions for A and B, and after some algebra and rearranging, Equation 10 is obtained.

Note that at *t* = 0 (immediately after the step-down in *P*_{A}), *P*_{I} (0) = *P*_{eqO} + *P*_{AO}, i.e. the initial equilibrium turgor pressure. The expression for *P*_{eqO} is:

At *t* = ∞, *P*_{I} (∞) = *P*_{eqO} + *P*_{AO} *–* Δ*P*_{a}, i.e. the new equilibrium turgor pressure that is lower than the initial equilibrium turgor pressure by Δ*P*_{a}. Therefore, Equation 10 describes the exponential decay of the turgor pressure from its initial equilibrium value (*P*_{eqO} + *P*_{AO}) to its new equilibrium value (*P*_{eqO} + *P*_{AO} *–* Δ*P*_{a}). The time constant for the exponential decay is, *t*_{c} = [*ε* (*ϕ + L*)]^{−1}. Interestingly, the magnitude of the pressure difference between the inside and outside of the membrane is the same before the step-down, *P*_{I} *– P*_{A} = (*P*_{eqO} + *P*_{AO}) – *P*_{AO} = *P*_{eqO} and after the exponential decay at *t* = ∞, *P*_{I} (∞) *– P*_{A} = (*P*_{eqO} + *P*_{AO} – Δ*P*_{a}) *–* (*P*_{AO} – Δ*P*_{a}) = *P*_{eqO}.

The water uptake may be evaluated by substituting the solution for *P*_{I} (*t*), i.e. Equation 10, into Equation 6. After some algebra and rearranging, Equation 12 is obtained.

Note that at *t* = 0, d*V*_{w}/*V*d*t* = *L* [σ ΔΠ – *P*_{eqO}] – *L* Δ*P*_{a}, which represents a decrease in water uptake rate when compared to the rate before the step-down in *P*_{A}. At *t* = ∞, d*V*_{w}/*V*d*t* = *L* [*σ*ΔΠ – *P*_{eqO}], which represents the original magnitude of water uptake rate before the step-down in *P*_{A}. Therefore, Equation 12 describes an exponential increase in water uptake rate from an initial smaller rate immediately after the step-down in *P*_{A} to the original rate before the step-down in *P*_{A}.

The expansive growth may be evaluated by substituting the expression for *P*_{I} (*t*), i.e. Equation 10, into Equation 7. Note that immediately after the step-down, d*P*_{A}/d*t* = 0. Equation 13 is obtained after some calculus, algebra, and rearranging.

Note at *t* = 0, d*V*_{cwc}/*V*d*t* *= ϕ* [*P*_{eqO} – *P*_{C}] – *L* Δ*P*_{a}, which represents a decrease in expansive growth rate when compared to that before the step-down in *P*_{A}. At *t* = ∞, d*V*_{cwc}/*V*d*t* *= ϕ* [*P*_{eqO} – *P*_{C}], which is the same magnitude as that before the step-down in *P*_{A}. Equation 13 describes the exponential increase in expansive growth rate from an initial smaller rate immediately after the step-down in *P*_{A} to the original rate before the step-down in *P*_{A}. A schematic illustration of the behavior for *P*_{I}, *P*_{A}, d*V*_{w}/*V*d*t*, and d*V*_{cwc}/*V*d*t* as a function of time (Eqs. 10, 12, and 13, respectively) is shown in Figure 1A.

### Turgor Pressure, Water Uptake, and Expansive Growth after a Step-Up in *P*_{A}

The behavior of the turgor pressure, water uptake, and expansive growth is evaluated after a step-up in *P*_{A} (*P*_{A} = *P*_{AO} + Δ*P*_{a}) using the same analyses described in the previous section, and Equations 14 to 16 are obtained, respectively.

Equation 14 describes the exponential increase of the turgor pressure from its initial equilibrium value (*t* = 0), *P*_{eqO} + *P*_{AO}, to its new equilibrium value (*t* = ∞), *P*_{eqO} + *P*_{AO} + Δ*P*_{a}, and the time constant for the exponential increase is *t*_{c} = [*ε* (*ϕ + L*)]^{−1}. As before, the magnitude of the pressure difference between the inside and outside of the membrane is the same before the step-up, *P*_{I} *– P*_{A} = (*P*_{eqO} + *P*_{AO}) – *P*_{AO} = *P*_{eqO}, and after the exponential decay at *t* = ∞, *P*_{I} (∞) *– P*_{A} = (*P*_{eqO} + *P*_{AO} + Δ*P*_{a}) *–* (*P*_{AO} + Δ*P*_{a}) = *P*_{eqO}.

Equation 15 describes an exponential decrease in water uptake rate from an initial larger rate immediately after the step-up in *P*_{A} (i.e. *L* [σ ΔΠ – *P*_{eqO}] + *L* Δ*P*_{a}) to the original rate before the step-up in *P*_{A} (i.e. *L* [σ ΔΠ – *P*_{eqO}]). Equation 16 describes the exponential decrease in expansive growth rate from an initial larger rate immediately after the step-up in *P*_{A} (i.e. *ϕ* [*P*_{eqO} – *P*_{C}] + *L* Δ*P*_{a}) to the original rate before the step-up in *P*_{A} (i.e. *ϕ* [*P*_{eqO} – *P*_{C}]). Time constants for both the decay of the water uptake rate and the expansive growth rate are the same as before, *t*_{c} = [*ε* (*ϕ + L*)]^{−1}. Schematic illustration of the behavior for *P*_{I}, *P*_{A}, d*V*_{w}/*V*d*t*, and d*V*_{cwc}/*V*d*t* as a function of time (Eqs. 14–16, respectively) are shown in Figure 1B.

### Pressure within the Apoplasm after a Step-Down in Turgor Pressure

The pressure within the apoplasm as a function of time, *P*_{A} (*t*), after a step-down in turgor pressure, *–*Δ*P*_{i}, is evaluated using Equation 8. The magnitude of the initial equilibrium turgor pressure before the step-down is *P*_{I} = *P*_{eqO} + *P*_{AO}, where *P*_{AO} is the initial pressure within the apoplasm. The magnitude of the new turgor pressure after the step-down (*P*_{IN} = *P*_{eqO} + *P*_{AO} – Δ*P*_{i}) is constant, so d*P*_{I}/d*t* is zero immediately after the step-down. Then Equation 8 becomes Equation 17.

Assuming that all the biophysical parameters remain constant after the step-down in turgor pressure and employing the general solution used for Equation 9, the solution for *P*_{A} (*t*) is obtained.

It is noted that at *t* = 0 (immediately after the step-down in *P*_{I}), *P*_{A} (0) = *P*_{AO}. At *t* = ∞, *P*_{A} (∞) = *P*_{AO} *–* Δ*P*_{i}. Therefore, Equation 18 describes the exponential decay of the pressure within the apoplasm from its initial value, *P*_{AO}, to a value that is smaller by the magnitude of the turgor pressure step-down, *P*_{AO} *–* Δ*P*_{i}. The time constant for the exponential decay is *t*_{c} = [*ε* (*ϕ + L*)]^{−1}. Interestingly, the magnitude of the pressure difference between the inside and outside of the membrane is the same before the step-down, *P*_{I} *– P*_{A} = (*P*_{eqO} + *P*_{AO}) – *P*_{AO} = *P*_{eqO}, and after the exponential decay at *t* = ∞, *P*_{IN} *– P*_{A} (∞) = (*P*_{eqO} + *P*_{AO} *–* Δ*P*_{i}) *–* (*P*_{AO} *–* Δ*P*_{i}) = *P*_{eqO}. A schematic illustration of the behavior for *P*_{I} and *P*_{A} as a function of time is shown in Figure 2.

### Turgor Pressure Behavior during a Stress Relaxation Experiment When *P*_{A} Is Not Atmospheric Pressure

In a stress relaxation experiment, the water uptake and transpiration are eliminated from the growing plant organ. Therefore, the terms that represent water uptake and transpiration in the Augmented Growth Equations, *L* (σ ΔΠ *– P*) and *T*, are zero. The change in turgor pressure that results from the elimination of water uptake and transpiration (stress relaxation experiment) is evaluated with a modified form of Equation 3, i.e. Equation 19 (Cosgrove, 1985; Ortega, 1985).

(turgor pressure decay rate) ∝ {irreversible deformation rate}

Equation 19 indicates that the turgor pressure decay rate, –d*P/*d*t*, is proportional to the relative irreversible deformation rate of the wall, *ϕ* (*P – P*_{C}). Equation 19 can be expanded to explicitly include *P*_{I} and *P*_{A}, i.e. Equation 20.

In the special case, where *P*_{A} is constant but not *P*_{atm} (not zero), Equation 21 is obtained.

(turgor pressure decay rate) ∝ {irreversible deformation rate}

The solution is obtained by integrating Equation 21, noting that the initial equilibrium turgor pressure is *P*_{I} = *P*_{eqO} + *P*_{A}.

At *t* = 0, *P*_{I} (0) = *P*_{eqO} + *P*_{A}, i.e. the initial equilibrium turgor pressure. At *t* = ∞, *P*_{I} (∞) = *P*_{C} + *P*_{A}. Equation 22 describes an exponential decay of the turgor pressure (as measured with a pressure probe) from its initial equilibrium value, *P*_{eqO} + *P*_{A}, to a constant value, *P*_{C} + *P*_{A}. The time constant for the exponential decay is, *t*_{c} = (*ε* *ϕ*)^{−1}.

When *P*_{A} = *P*_{atm} = 0, the solution previously reported by Cosgrove (1985) and Ortega (1985) is recovered for a stress relaxation experiment.

Equation 23 describes an exponential decay of the turgor pressure from its initial equilibrium value, *P*_{eqO}, to a constant value, *P*_{C}, with a time constant, *t*_{c} = (*ε* *ϕ*)^{−1}.

### Expansive Growth and Wall Deformation before, during, and after a Semi-Instantaneous Change in *P*_{A} When the Turgor Pressure Remains Constant

Consider the case when the pressure within the apoplasm is decreased from an initial constant value, *P*_{AO}, by an amount, Δ*P*_{a} (*P*_{A} = *P*_{AO} – Δ*P*_{a}), in a short time interval, Δ*t*. The initial equilibrium turgor pressure, *P*_{I} = *P*_{eqO} + *P*_{AO}, is kept constant (clamped); thus, d*P*_{I}/d*t* = 0. Assuming the magnitudes of *ϕ*, *ε*, and *P*_{C} remain constant, then Equation 7 becomes Equation 24.

(increase in wall expansion rate) = (increase in irreversible deformation rate) + (semi-instantaneous elastic expansion)

Equation 24 describes the following expansive growth behavior in relative terms; a semi-instantaneous elastic expansion of the wall, (1*/ε*) [Δ*P*_{a} /Δ*t*], followed by an increase in the rate of irreversible deformation of the wall, *ϕ* [{*P*_{eqO} + Δ*P*_{a}} – *P*_{C}]. A schematic illustration of the behavior of *P*_{I}, *P*_{A}, and expansive growth (*ln V*_{cwc}, natural logarithm of the volume of the cell wall chamber) as a function of time is shown in Figure 3A. Note that the slope of the *ln* *V*_{cwc} curve is d(*ln* *V*_{cwc})/d*t* = (d*V*_{cwc}/*V*d*t*).

If the pressure within the apoplasm is increased from an initial constant value, *P*_{AO}, by an amount, Δ*P*_{a} (*P*_{A} = *P*_{AO} + Δ*P*_{a}), in a short time interval, Δ*t*, and the initial equilibrium turgor pressure, *P*_{I} = *P*_{eqO} + *P*_{AO}, is kept constant (clamped), Equation 25 is obtained.

(decrease in wall expansion rate) = (decrease in irreversible deformation rate) + (semi-instantaneous elastic contraction)

Equation 25 describes a decrease in expansive growth rate in relative terms; a semi-instantaneous elastic contraction (recovered elastic expansion) of the wall, (1*/ε*) [–Δ*P*_{a}/Δ*t*], followed by a decrease in the irreversible deformation rate of the wall, *ϕ* [{*P*_{eqO} – Δ*P*_{a}} – *P*_{C}]. The irreversible deformation rate will decrease to a smaller rate when the magnitude of {*P*_{eqO} – Δ*P*_{a}} is greater than *P*_{C}. If the magnitude of {*P*_{eqO} – Δ*P*_{a}} is smaller than *P*_{C}, the irreversible deformation rate will stop. A schematic illustration of the behavior of *P*_{I}, *P*_{A}, and expansive growth (*ln V*_{cwc}) as a function of time is shown in Figure 3B.

## DISCUSSION

After a step-down in pressure within the apoplasm, analysis indicates that the turgor pressure will decrease exponentially from an initial equilibrium pressure to a new equilibrium pressure that is lower in magnitude by the amount of the step-down in pressure (Eq. 10; Fig. 1A). Similarly, a step-up in pressure within the apoplasm will result in an exponential increase in turgor pressure to a new equilibrium pressure that is higher than the previous value by the amount of the step-up in pressure (Eq. 14; Fig. 1B). The time constants for the exponential changes are the same for both cases, *t*_{c} = [*ε* (*ϕ + L*)]^{−1}. The magnitude of the pressure difference across the cell membrane is the same before the step changes in pressure and after the exponential decay to new equilibrium pressures. The described behavior is consistent with and supports the assumption of local equilibrium made by Molz and Ikenberry (1974) and Molz and Boyer (1978) in their theoretical analyses of water transport in plant tissue.

Other analyses indicate that a step-down in pressure within the apoplasm will produce a sharp decrease in rates of water uptake and expansive growth, followed by an exponential increase to their original magnitudes (Eqs. 12 and 13; Fig. 1A). It is noted that the magnitude and behavior of the transient decrease in rates of water uptake and expansive growth are the same, *L* (– Δ*P*_{a}) exp {– *ε* (*ϕ + L*) *t*}. A step-up in pressure within the apoplasm will produce a sharp increase in the rates of water uptake and expansive growth, followed by an exponential decrease to their initial magnitudes (Eqs. 15 and 16; Fig. 1B). Again, the magnitude and behavior of the transient increase in rates of water uptake and expansive growth are the same, *L* Δ*P*_{a} exp {– *ε* (*ϕ + L*) *t*}. Time constants for the exponential changes are the same for all cases, *t*_{c} = [*ε* (*ϕ + L*)]^{−1}.

It may appear unusual that new equilibrium turgor pressures are lower and higher than the initial (Fig. 1), but the magnitude of the expansive growth rate after the exponential change is the same as the initial magnitude. It may be thought that the magnitude of *P*_{I} – *P*_{C} should be different; thus, the rate of irreversible deformation of the wall should be different. However, the magnitude of the wall stress is determined by the pressure difference across the membrane, i.e. *P*_{I} – *P*_{A}. Because the magnitude of the pressure difference across the membrane is the same before the step change (*P*_{eqO}) and after the exponential decay (*P*_{eqO}), the magnitude of the wall stress is also the same before the step change and after the exponential decay. Also, *P*_{C} is related to the critical stress in the wall that must be exceeded before irreversible extension of the wall occurs. Because the magnitudes of the wall stress and the critical wall stress are the same before the step change and after the exponential decay, the irreversible deformation rate of the wall and the expansive growth rate remain the same before and after.

A quantitative evaluation of a step-down (–0.2 MPa) followed by a step-up (0.4 MPa) in pressure within the apoplasm can further illustrate the behavior described by these equations. Consider a plant cell in tissue that is initially growing in equilibrium (constant growth) and the pressure within the apoplasm is atmospheric (i.e. *P*_{A} = 0). The equilibrium turgor pressure is described by Equation 5 when *T* = 0, then *P*_{eq} = *P*_{eqO} (Eq. 11). For the sake of discussion, assume σ ΔΠ = 1.0 MPa, *P*_{eqO} = 0.6 MPa, and *P*_{C} = 0.3 MPa. Also assume that *L* and *ϕ* are constant. The pressure difference across the membrane is, *P*_{eqO} – *P*_{atm} = 0.6 MPa – 0 = 0.6 MPa. Equation 1 is used to determine the relative rate of change in water volume (water uptake rate), d*V*_{w}/*V*d*t* = *L* (1.0 MPa – 0.6 MPa) = *L* (0.4 MPa). Equation 2 is used to determine the relative rate of change in volume of the cell wall chamber (expansive growth rate), d*V*_{cwc}/*V*d*t* *= ϕ* (*P – P*_{C}) = *ϕ* (0.6 MPa – 0.3 MPa) = *ϕ* (0.3 MPa). Note that the same results are obtained using Equations 6 and 7 when *P*_{A} = 0.

Now assume *P*_{A} is stepped down by 0.2 MPa; *P*_{A} = *P*_{AO} *–* Δ*P*_{a} = 0 – 0.2 MPa = – 0.2 MPa. Equation 10 is used to determine the turgor pressure at *t* = 0, *P*_{I} (0) = *P*_{eqO} + *P*_{AO} = 0.6 MPa + 0 = 0.6 MPa, and at *t* = ∞, *P*_{I} (∞) = *P*_{eqO} + *P*_{AO} *–* Δ*P*_{a} = 0.6 MPa + 0 – 0.2 MPa = 0.4 MPa. So the new equilibrium turgor pressure is *P*_{I} (∞) = 0.4 MPa, and the pressure within the apoplasm is *P*_{A} = – 0.2 MPa. The pressure difference across the membrane is *P*_{I} (∞) – *P*_{A} = 0.4 MPa – (– 0.2 MPa) = 0.6 MPa, which is equal to the original value. Evaluating Equation 12 at *t* = 0, d*V*_{w}/*V*d*t* = *L* (1.0 MPa – 0.6 MPa – 0.2 MPa) = *L* (0.2 MPa), which represents a decrease in water uptake rate. Evaluating Equation 12 at *t* = ∞, d*V*_{w}/*V*d*t* = *L* (1.0 MPa – 0.6 MPa) = *L* (0.4 MPa), which is equal to the original water uptake rate. Evaluating Equation 13 at *t* = 0, d*V*_{cwc}/*V*d*t* = *ϕ* (0.6 MPa – 0.3 MPa) – *L* (0.2 MPa) = *ϕ* (0.3 MPa) – *L* (0.2 MPa), which represents a decrease in expansive growth rate. Evaluating Equation 13 at *t* = ∞, d*V*_{cwc}/*V*d*t* = *ϕ* (0.6 MPa – 0.3 MPa) = *ϕ* (0.3 MPa), which is equal to the original expansive growth rate.

Subsequently, *P*_{A} is stepped up by 0.4 MPa; *P*_{A} = *P*_{AO} + Δ*P*_{a} = – 0.2 MPa + 0.4 MPa = 0.2 MPa. Equation 14 is used to determine the turgor pressure at *t* = 0, *P*_{I} (0) = *P*_{eqO} + *P*_{AO} = 0.6 MPa – 0.2 MPa = 0.4 MPa, and at *t* = ∞, *P*_{I} (∞) = *P*_{eqO} + *P*_{AO} + Δ*P*_{a}, = 0.6 MPa – 0.2 MPa + 0.4 MPa = 0.8 MPa. So the new equilibrium turgor pressure is *P*_{I} (∞) = 0.8 MPa, and the pressure within the apoplasm is *P*_{A} = 0.2 MPa. The pressure difference across the membrane is *P*_{I} (∞) – *P*_{A} = 0.8 MPa – 0.2 MPa = 0.6 MPa, which is equal to the original value. Evaluating Equation 15 at *t* = 0, d*V*_{w}/*V*d*t* = *L* (1.0 MPa – 0.6 MPa + 0.4 MPa) = *L* (0.8 MPa), which represents an increase in water uptake rate. Evaluating Equation 15 at *t* = ∞, d*V*_{w}/*V*d*t* = *L* (1.0 MPa – 0.6 MPa) = *L* (0.4 MPa), which is equal to the original water uptake rate. Evaluating Equation 16 at *t* = 0, d*V*_{cwc}/*V*d*t* = *ϕ* (0.6 MPa – 0.3 MPa) + *L* (0.4 MPa) = *ϕ* (0.3 MPa) + *L* (0.4 MPa), which represents an increase in expansive growth rate. Evaluating Equation 16 at *t* = ∞, d*V*_{cwc}/*V*d*t* = *ϕ* (0.6 MPa – 0.3 MPa) = *ϕ* (0.3 MPa), which is equal to the original expansive growth rate.

This quantitative evaluation of a step-down followed by a step-up in *P*_{A} illustrates that changes in *P*_{A} will cause changes in *P*_{I} so that the rates of water uptake and expansive growth rate remain unchanged after initial transient changes, if the biophysical variables remain constant. The duration of the transient changes can be estimated with the time constant because approximately 98% of the exponential decay occurs after four time constants. As an example, the magnitude of *t*_{c} is estimated using data from Cosgrove (1985) for growing cells in pea (*Pisum sativum*) stems; *ε* = 9.5 MPa, *ϕ* = 0.084 MPa^{−1} h^{−1}, and *L* = 2.0 MPa^{−1} h^{−1}, and *t*_{c} = [*ε* (*ϕ + L*)]^{−1} = [19.8 h^{−1}] ^{−1} ≈ 0.050 h ≈ 3 min. Therefore, the duration of the transient changes can be estimated for growing cells in pea stems to be 4 *t*_{c} = 0.2 h = 12 min.

Expansive growth rate behaviors after step changes in pressure within the apoplasm, as shown in Figure 1, draw support from experimental studies conducted with maize leaves. Tang and Boyer (2008) report a sharp downward spike in elongation rate followed by an increase, which appears to be exponential, to a higher rate at the beginning of the morning for maize leaves. They report that the transpiration rate is rapidly increased at the beginning of the morning and produces water tension within the apoplasm. Thus, at the beginning of the morning, the behavior of the pressure within the apoplasm is similar to a step-down. The observed downward spike followed by an exponential increase in elongation rate is similar to the behavior described by Equation 13 and Figure 1A for d*V*_{cwc}/*V*d*t*. However, Equation 13 does not predict the higher elongation rate reported to occur during the day when the temperature is increased (see Figs. 2, 3, and 5 of Tang and Boyer, 2008). The analyses conducted here assume that the magnitudes of the biophysical variables are constant, so the initial expansive growth rate is recovered after the exponential increase. A higher expansive growth rate can be obtained with a change in one or more of the biophysical variables. For example, an increase in elongation rate will occur if an increase in magnitude of the irreversible wall extensibility, *ϕ*, accompanied the step-down in *P*_{A} at the beginning of the day. Perhaps the increase in the magnitude of *ϕ* is the result of the increase in temperature or a cascade of biochemical events that begin with the initiation of photosynthesis, or it may be that the water tensions in the wall alter relevant biochemistry that effectively alters the magnitude of *ϕ*.

The elongation rate behavior of a maize leaf is reversed at the beginning of the night. Tang and Boyer (2008) report a sharp upward spike in elongation rate followed by what appears to be an exponential decrease to a smaller rate. At the beginning of the night, the transpiration rate is rapidly reduced and positive root pressure develops during the night producing guttation drops on the leaf. Thus, at the beginning of the night, the behavior of the pressure within the apoplasm is similar to a step-up. The observed upward spike followed by an exponential decrease in elongation rate is similar to the behavior described by Equation 16 and Figure 1B for d*V*_{cwc}/*V*d*t*. Equation 16 and Figure 1B predict that the initial expansive growth rate is recovered after the exponential decrease. Again this is a consequence of assuming the biophysical variables to be constant. Interestingly, there is evidence that in some cases the initial elongation rate is recovered at the beginning of the night, after the exponential decrease, when the temperature is constant (see figure 2 of Tang and Boyer, 2008). Equation 16 does not predict the smaller elongation rate that occurs during the night when the temperature is decreased (see Figs. 3 and 5 of Tang and Boyer, 2008). A decrease in elongation rate can be obtained if a decrease in magnitude of the irreversible wall extensibility, *ϕ*, accompanied the step-up in *P*_{A} at the beginning or the night. Perhaps the decrease in the magnitude of *ϕ* is the result of the decrease in temperature, termination of photosynthesis, or higher pressures within the apoplasm that may alter relevant wall biochemistry that effectively alters the magnitude of *ϕ*.

Next, the affect of an increase in expansive growth rate on the behavior of the pressure within the apoplasm is evaluated. An examination of Equation 3 reveals that an increase in irreversible deformation rate of the wall (expansive growth rate) will cause a decrease in turgor pressure. Equation 8 is used to determine how a decrease in turgor pressure, associated with an increase in expansive growth rate, affects the pressure within the apoplasm. A step-down in turgor pressure can simulate a decrease in turgor pressure caused by an increase in expansive growth rate. Equation 17 describes the rate of change of the pressure within the apoplasm immediately after a step-down in turgor pressure. Its solution, Equation 18, describes an exponential decay of the pressure within the apoplasm to a value that is less than the initial pressure by the magnitude of the step-down in turgor pressure (Fig. 2). The time constant for the exponential decay is, *t*_{c} = [*ε* (*ϕ + L*)]^{−1}. The analysis indicates that a lower pressure within the apoplasm is produced when the turgor pressure decreases because of an increase in expansive growth rate. The decrease in turgor pressure and accompanying decrease in pressure within the apoplasm will decrease the water potential in the local region of the growing cell. This result is consistent with a growth-induced decrease in water potential proposed and measured by Molz and Boyer (1978), Nonami and Boyer (1987, 1993), Boyer (2001), Tang and Boyer (2008), and relevant references within. Furthermore, the analysis indicates that the magnitude of the pressure difference between the inside (turgor pressure) and outside (pressure within the apoplasm) of the membrane is the same before the step-down and after the exponential decay. This result is consistent with the condition of local equilibrium proposed by Molz and Ikenberry (1974) and Molz and Boyer (1978).

Stress relaxation experiments, during which the turgor pressure of single cells is measured directly with the pressure probe, have been conducted in plant tissue (e.g. Cosgrove, 1985, 1987) and in single isolated cells (e.g. Ortega et al., 1989). For individual cells in plant tissue and isolated single cells, it is shown that the turgor pressure decays exponentially to a constant value, *P*_{C}, when the pressure external to the cell membrane is atmospheric, i.e. zero. The time constant for the exponential decay is *t*_{c} = (*ε* *ϕ*)^{−1}. However, analysis using Equation 21 indicates that the magnitude of the final constant value after the exponential decay is different, *P*_{C} + *P*_{A}, when the pressure within the apoplasm is not atmospheric (see Eq. 22). This result may be verified by conducting stress relaxation experiments in which the magnitude of *P*_{A} can be changed. The measured magnitude of *P*_{I} after the exponential decay should change with the magnitudes of *P*_{A} as described by Equation 22. If so, the results will validate Equations 21 and 22 and provide insight into the role of *P*_{A} in stress relaxation experiments.

Additional analysis indicates that if the turgor pressure remains constant (clamped) a semi-instantaneous decrease in the magnitude of *P*_{A} elicits a semi-instantaneous increase in elastic expansion of the wall followed by an increase in irreversible expansion rate of the wall (Eq. 24; Fig. 3A). Similarly, if the turgor pressure remains constant (clamped), a semi-instantaneous increase in the magnitude of *P*_{A} elicits a semi-instantaneous elastic contraction followed by a decrease in irreversible expansion rate of the wall (Eq. 25; Fig. 3B). Previous analytical and experimental research conducted on isolated single algal cells demonstrates that a semi-instantaneous increase in turgor pressure produced with a pressure probe will elicit a semi-instantaneous elastic expansion followed by an increase in expansive growth rate (Proseus et al., 1999, 2000). This expansive growth behavior is similar to that described for a semi-instantaneous decrease in the magnitude of *P*_{A}. Also, it is demonstrated that a semi-instantaneous decrease in turgor pressure produced with the pressure probe will elicit a semi-instantaneous elastic contraction followed by a decrease in expansive growth rate (Proseus et al., 1999, 2000). This expansive growth behavior is similar to that described for a semi-instantaneous increase in the magnitude of *P*_{A}. The predicted behaviors (described by Eqs. 24 and 25; Fig. 3) can be verified with experiments in which the expansive growth rate is measured before and after a semi-instantaneous decrease and increase in the magnitude of *P*_{A}, while maintaining a constant turgor pressure. The pressure probe (Hüsken et al., 1978; Steudle, 1993; Tomos and Leigh, 1999) may be used to clamp the turgor pressure at a constant magnitude. In order to measure the expansive growth rate while clamping the turgor pressure, it may be helpful to use single isolated cells such as algal internode cells or fungal sporangiophores. Then the pressure in the surrounding water (for algal cells) or in the air (for sporangiophores) could be decreased or increased to simulate a decreased or increased in the pressure within the apoplasm.

Prior analyses address the affect of negative pressure within the apoplasm on growth of cells in plant tissue (Calbo and Pessoa, 1994; Pessoa and Calbo, 2004), and it is argued that the negative pressure interferes with the growth rate. In their analyses, the Lockhart equation [Eq. 2 without the term for elastic deformation, (1*/ε*) d*P/*d*t*] is modified to account for negative pressure within the apoplasm. Because their derived equations do not have a term that explicitly represents elastic deformation, their equations and results are not directly comparable to those obtained here. Also, the equations derived by Calbo and Pessoa (1994) and Pessoa and Calbo (2004) cannot explicitly address time-dependent changes in turgor pressure, water uptake, expansive growth, wall deformations, and pressure within the apoplasm that are evaluated here.

## CONCLUSION

It is concluded that transient changes in turgor pressure, water uptake rate, and expansive growth rate occur after step changes in the magnitude of the pressure within the apoplasm. It is also concluded that changes in the magnitude of the turgor pressure cause pressure changes within the apoplasm. The transient changes are essentially complete in approximately four time constants; 4 *t*_{c} = 4 [*ε* (*ϕ + L*)]^{−1}. In addition, it is concluded that the magnitude of the pressure within the apoplasm can alter the magnitude of the turgor pressure after the exponential decay during a stress relaxation experiment but does not alter the time constant of the exponential decay. Last, it is concluded that if the turgor pressure is clamped, a semi-instantaneous decrease and increase in the pressure within the apoplasm will elicit expansive growth behavior that is similar to that obtained by a semi-instantaneous increase and decrease in turgor pressure, respectively.

Overall, it is concluded that the pressure within the apoplasm is an important variable that must be measured and/or controlled, along with the turgor pressure, in analyses of water transport and expansive growth of plant cells in tissues and organs. Future research should address how the magnitude of relevant biophysical variables and relevant chemical reactions are affected by lower and higher pressures within cell walls. Also, knowing that spatial differences in tissue properties can produce gradients that affect the growth behavior of tissues and organs, future research should address how the magnitudes of relevant biophysical variables are spatially distributed within growing tissues and organs.

## Acknowledgments

I thank Elena L. Ortega for her valuable comments concerning the mathematical solutions and the manuscript.

## Footnotes

↵1 This work was supported by the National Science Foundation (grant nos. MCB–0640542 and MCB–0948921).

The author responsible for distribution of materials integral to the findings presented in this article in accordance with the policy described in the Instructions for Authors (www.plantphysiol.org) is: Joseph K.E. Ortega (joseph.ortega{at}ucdenver.edu).

↵[OA] Open Access articles can be viewed online without a subscription.

- Received July 12, 2010.
- Accepted August 24, 2010.
- Published August 25, 2010.