Osmotic water permeability of plasma and vacuolar membranes in protoplasts II. Theoretical basis

Abstract

Water permeability of the plasma membrane (PM) and the vacuolar membrane (VM) is important for intracellular and transcellular water movement in plants, because mature plant cells have large central vacuoles. We have developed a new method for measuring the osmotic water permeability of the PM and VM (P _f1 and P _f2, respectively) in individual plant cells. Here, the theoretical basis and procedure of the method are discussed. Protoplasts isolated from higher plant tissues are used to measure P _f1 and P _f2. Because of the semi-permeability (selective permeability) of cellular membranes, protoplasts swell or shrink under hypotonic or hypertonic conditions. A theoretical three-compartment model is presented for simulating time-dependent volume changes in the vacuolar and cytoplasmic spaces in a protoplast during osmotic excursions. The model describes the theoretical relationships between P _f1, P _f2 and the bulk osmotic water permeability of protoplasts (P _f(bulk)). The procedure for measuring the osmotic water permeability is: (1) P _f(bulk) is calculated from the time when half of the total change in protoplast volume is completed, by assuming that the protoplast has a single barrier to water movement across it (two-compartment model); (2) P _f2 of vacuoles isolated from protoplasts is obtained in the same manner; and (3) P _f1 is determined from P _f(bulk) and P _f2 according to the three-compartment model. The theoretical relationship between P _fl (m s⁻¹) and L _Pl (hydraulic conductivity, l=1, 2) (m s⁻¹ Pa⁻¹) is also discussed.

Appendix

List of main symbols

C ₀₀ :

concentration of the external solution (mol m⁻³) (first stage)

C ₀₊ :

concentration of the external solution (mol m⁻³) (second and third stages)

C ₁ :

solute concentration in the cytoplasmic space (mol m⁻³)

C ₂ :

solute concentration in the vacuolar space (mol m⁻³)

C _l0 :

C _l for the first stage (l=1, 2)

C _l+ :

C _l for the third stage (l=1, 2)

f :

=P _f2/P _f1

f ₀ :

=P _f20/P _f10

J _w1 :

water inflow flux through the plasma membrane (PM)

J _w2 :

water inflow flux through the vacuolar membrane (VM)

L _P1 :

hydraulic conductivity of the PM

L _P2 :

hydraulic conductivity of the VM

L _P(bulk) :

bulk hydraulic conductivity of the protoplast

P _f1 :

osmotic water permeability of the PM

P _f2 :

osmotic water permeability of the VM

P _fl0 :

P _fl when S _l =S _l0 (l=1, 2, for constant S×P _f )

P _f(sum) :

total osmotic water permeability of the PM and the VM (=R _P(sum) ⁻¹)

P _f(sum0) :

=R _P(sum0) ⁻¹ (same as P _f(sum) at t=0, for constant S×P _f )

P _f(sum0.5) :

P _f(sum) at t=t _0.5 for the two-compartment model for constant S×P _f

P _f(sum0.25) :

P _f(sum) at t=t _0.25 for the two-compartment model for constant S×P _f

P _f(bulk) :

bulk osmotic water permeability of protoplast

P _f(bulk0) :

P _f(bulk) at t=0 (for constant S×P _f )

R :

gas constant (8.31451 J K⁻¹ mol⁻¹)

R _P1 :

osmotic resistance of the PM (=P ⁻¹ _f1 )

R _P2 :

osmotic resistance of the VM (=P ⁻¹ _f2 )

R _Pl0 :

R _Pl when S _l =S _l0 (l=1, 2, for constant S×P _f )

R _P(sum) :

sum of the osmotic resistance of the PM and the VM (=R _P1+R _P2)

R _P(sum0) :

=R _P10+R _P20 (for constant S×P _f )

R _P(bulk) :

bulk osmotic resistance of the protoplast (=P _f(bulk) ⁻¹)

R _P(bulk0) :

=P _f(bulk0) ⁻¹

r ₁ :

radius of the protoplast

r ₂ :

radius of the vacuole

r _l0 :

r _l for the first stage (l=1, 2)

r _l* :

=r _l /r _l0 (l=1, 2)

r _1*0.5 :

r _1* at t=t _0.5

S ₁ :

surface area of the PM

S ₂ :

surface area of the VM

S _l0 :

S _l for the first stage (l=1, 2)

T :

temperature (K)

t :

time

t _0.5 :

half-time (defined as the time when half of the total change in protoplast volume was completed)

t _0.25 :

one-quarter time (defined as the time when one-quarter of the total change in protoplast volume was completed)

$t_{0.5}^{({\rm obs})}$ :

t _0.5 measured from the swelling/shrinking of the protoplast

$t_{0.5}^{({\rm 2CM})}$ :

t _0.5 predicted by use of the two-compartment model

t _* :

dimensionless time (=t/τ, τ≡r ₁₀/(V _w C ₁₊ P _f1)) (used mainly for the three-compartment model for constant P _f )

$t_{{\rm s*}}$ :

dimensionless time (=t/τ _s, τ _s≡r ₁₀/(V _w C ₁₊ P _f10)) (used mainly for the three-compartment model for constant S×P _f )

$t_{{*}}^\prime$ :

dimensionless time (=t/τ′, τ′≡r ₁₀/(V _w C ₁₊ P _f(sum))) (used for the two-compartment model for constant P _f )

$t_{{\rm s*}}^\prime$ :

dimensionless time $(=t/\tau_{\rm s}^{\prime}, \; \tau_{\rm s}^{\prime}\equiv r_{10}/(V_{\rm W}C_{1+}P_{f(\rm sum0)}))$ (used for the two-compartment model for constant S×P _f )

$t_{*0.5}^\prime$ :

dimensionless half-time for $t_{{*}}^\prime$

$t_{{\rm s*0.5}}^\prime$ :

dimensionless half-time for $t_{{\rm s*}}^\prime$

$t_{*0.5}^{(3{\rm CM})}$ :

dimensionless half-times calculated by use of the three-compartment model

$t_{*0.5}^{(2{\rm CM})}$ :

dimensionless half-times calculated by use of the two-compartment model

V ₁ :

volume of the protoplast

V ₂ :

volume of the vacuole

V _l0 :

V _l for the first stage (l=1, 2)

V _l+ :

V _l for the third stage (l=1, 2)

V _l* :

=V _l /V _l0 (l=1, 2)

V _1*0.5 :

V _1* at t=t _0.5

V _1*0.25 :

V _1* at t=t _0.25

V ^′′ _P :

second virial coefficient for the solution in the cytoplasmic space

V ^′′ _T :

second virial coefficient for the solution in the vacuolar space

V _s :

partial molar volume of the solute

V _1s :

V _s in cytoplasmic and vacuolar spaces, assuming an ideal dilute intracellular solution

V _w :

the partial molar volume of water (18.05×10⁻⁶ m³ mol⁻¹ at 20°C)

α ₀ :

=r ₁₀/r ₂₀

β ₁₀ :

ratio of initial and final equilibrium volumes of the protoplast (=V ₁₊/V ₁₀)

Π ₀₀ :

osmotic pressure of the external solution (first stage)

Π ₀₊ :

osmotic pressure of the external solution (second and third stages)

Π ₁ :

osmotic pressure in the cytoplasmic space

Π ₂ :

osmotic pressure in the vacuolar space

Π _l0 :

Π _l for the first stage (l=1, 2)

Π _l+ :

Π _l for the third stage (l=1, 2)