Mechanisms of Water Transport Across Cell Membranes and Epithelia

In this chapter we discuss the pathways and mechanisms of water transport across cell membranes and epithelia. The concepts of diffusion and osmosis are presented at the biophysical level and applied to water transport phenomena across lipid bilayers and membrane-spanning pores. Water pores (“aquaporins”) are described from biophysical and molecular points of view. The mechanism of water transport across the cell membrane is osmosis and cell volume is determined by the intracellular solute content and the extracellular osmolality. Water permeates the cell membrane via aquaporins and/or the phospholipid bilayer; other pathways having minor importance. Water transport in the absence of a substantial osmolality difference between the adjacent solutions is most likely osmotically coupled to solute transport in the same direction (“near-isosmotic fluid transport”), without requirement for large standing osmotic gradients. Epithelia display a wide range of permeabilities to water, which has important physiological significance, in particular for kidney function.

Keywords

aquaporin; pore; permeability; osmosis; isosmotic; unstirred layers

Introduction

The main purpose of this chapter is to review the basic aspects of water transport mechanisms across cell membranes and epithelia. In the first section we will discuss biophysical principles and definitions, with the aim of providing a theoretical framework useful for the analysis of experimental observations. In the second section, we will address general issues pertaining to water transport across cell membranes, focusing on intracellular water, and the pathways and mechanism for osmotic water flow. In the third section, we will discuss water transport by epithelia, focusing on pathways and mechanisms, in particular the role of solute–solvent coupling. We intend this chapter to serve as both an overview and an introduction to chapters covering specific aspects of water transport (Chapters 5, 9, 41, 42, 43). The three sections of the chapter are to a certain extent independent from each other, and can be studied separately.

The field of water transport across biological membranes has made a recent major transition with the discovery and characterization of the aquaporins. Aquaporins are integral membrane proteins, most of which are highly specific water pores expressed in plants and animals from bacteria to humans. The discovery of the aquaporins confirmed a long-held prediction for the existence of these pores, emanating from biophysical studies in red blood cells and renal proximal tubules.

Basic Principles

This section is largely based on the excellent water transport treaty by Finkelstein. Other sources are House, Reuss and Cotton, Dawson, Hallows and Knauf, and Macey and Moura. Derivations of the equations can be found in Finkelstein’s book. Deliberately, this section has been kept simple, and qualitative explanations have been superimposed on a succinct quantitative analysis.

The main mechanism of net water transport in animal cells is osmosis, that is, net water flow driven by differences in water chemical potential, in turn dependent on differences in solute concentrations. Concerning water flow across a cell membrane, an important issue is whether water moves through the phospholipid bilayer and/or through specialized water-conducting pores. The mechanisms involved in water permeation via these two pathways constitute the main content of this first section. Hence, we start with osmosis.

Osmotic Equilibrium is a Balance of Osmotic and Hydrostatic Forces

The principle of osmotic equilibrium can be illustrated by considering a simple system, that is, a semipermeable membrane separating two aqueous phases: pure water and a solution that contains a nondissociating solute ( Figure 4.1 ). The membrane is permeable to water and impermeable to the solute (hence the term semipermeable ). At thermodynamic equilibrium, the net water flow across the membrane is zero. (In the case of water, flow can be expressed in molar terms [moles of water per unit area and unit time] or volume terms [volume of water per unit area and unit time]. For conversion to volume flow, the molar flow must be multiplied by ${\bar{V}}_{w}$ ${\bar{V}}_{w}$ [partial molar volume, a constant equal to 18 cm ³ /mole].) The equilibrium of net flows is the result of the equality of two forces: an osmotic force favoring water flow into the solution; and an opposing hydrostatic force resulting, for instance, from the difference in height of the fluid compartments generated by the osmotic water flow. For dilute solutions, osmotic equilibrium is approximately described by Van’t Hoff’s law :

Δ P = π = R T C_{s}

$Δ P = π = R T C_{s}$

where Δ P (atm) is the hydrostatic pressure difference between the two compartments ( P′ − P″ ), R (cm ³ atm mol ⁻¹ K ⁻¹ ) and T [K] are the gas constant and the absolute temperature, respectively, C _s (mol cm ⁻³ ) is the molar concentration of the solute, and π (atm) is the osmotic pressure of the solution. The latter is conveniently defined as the hydrostatic pressure in the solution compartment (relative to the pressure in the water compartment) needed to abolish water flow across the membrane.

When the semipermeable membrane separates two solutions, equilibrium is described by a slightly different equation: Δ P= Δπ =RT Δ C _s , where Δπ is the difference in osmotic pressure (π′−π″) and Δ C _s is the solute concentration difference ( C _s ′− C _s ″).

The osmotic pressure depends on the molar concentration ( C _s ) and on the degree of dissociation of the solute, that is, the number of particles that each molecule yields in solution (n). Ideally, the osmolality of a solution, in osmol/kg of water, is given by Osm =n C _s , where C _s is in mol l ⁻¹ . However, the effect of solute on the activity of the solvent is generally nonideal, that is, it may depend on the nature of the solute. The correction term for this effect is the osmotic coefficient, φ _s , where the subscript denotes the solute. For physiological concentration ranges, the osmotic coefficient is closer to unity than the activity coefficient, but it can be significantly greater than 1 for macromolecules. For the sake of simplicity, the osmotic coefficient will be neglected in this discussion.

A 1 Osm solution at room temperature exerts an osmotic pressure of about 24.6 atm, which is equivalent to about 18,700 mm Hg. In a mammal, a 1% change in extracellular fluid osmolality (<3 mosmol/kg) is equivalent, as a driving force for water flow, to a hydrostatic pressure of 56 mm Hg. In animal cells, changes in osmolality cause large water fluxes across the plasma membrane, whereas hydrostatic pressure changes do not. Osmolality is a measure of concentration of particles, not of osmotic pressure, but it is frequently used to denote the latter.

The generation of Δ P in the presence of impermeant solute on one side can be explained on the basis of changes in the water chemical potential (μ _w ), which is given by:

μ_{w} = μ_{w}^{o} + R T \ln X_{w} + P {\bar{V}}_{w}

$μ_{w} = μ_{w}^{o} + R T \ln X_{w} + P {\bar{V}}_{w}$

where μ ^o _w is the standard chemical potential, X _w is the water mole fraction (moles of water/[moles of water + moles of solute]), and ${\bar{V}}_{w}$ ${\bar{V}}_{w}$ is the partial molar volume of water. A solute addition to one side (at constant total volume) reduces the water chemical potential in that side (μ′ _w ) because the water is “diluted” by the solute (and X _w falls). The difference in water chemical potential thus generated (Δμ′ _w = μ′ _w − μ″ _w ) is the “driving force” for water flow toward the side of higher osmolality (and lower μ _w ). If both compartments are open and of appropriate dimensions, then a Δ P will result from changes in height ( Figure 4.1 ). If a compartment is closed, then its pressure will change in proportion to the water flux, with a proportionality constant dependent on compliance of the compartment.

Osmotic Water Flows Across Lipid and Porous Membranes have Different Properties

Near equilibrium, the volume flow is linearly related to the driving force:

J_{v} = L_{p} (Δ P - Δ π)

$J_{v} = L_{p} (Δ P - Δ π)$

where J _v is the volume flow (volume area ⁻¹ time ⁻¹ ), L _p is the hydraulic permeability coefficient of the membrane, and Δ P and Δπ are the differences in hydrostatic and osmotic pressure, respectively. The L _p can be expressed in cm sec ⁻¹ (osmol/kg) ⁻¹ . In most cases, a filtration ( P _f ) or osmotic permeability coefficient ( P _os ; P _f = P _os ) is used instead of L _p . The P _os (cm sec ⁻¹ ) is related to L _p by P _os = L _p RT / ${\bar{V}}_{w}$ ${\bar{V}}_{w}$ .

The above discussion underscores the fact that Δ P and Δπ are equivalent as “driving forces” in causing osmotic water flow. The mechanism of this equivalence can be understood if one considers the nature of the membrane and the mechanism of osmotic water transport, as explained below.

Osmotic Water Flow Across Lipid Membranes

Osmotic water flow across lipid membranes occurs by solubility diffusion . Water molecules move from one aqueous solution into the lipid and then into the other solution by independent, random motion. When Δ P =Δπ (0 net driving force) there are two diffusive water fluxes of equal magnitude and opposite direction, with no net water flow across the membrane. In the presence of a net driving force (Δ P −Δπ ≠ 0), a net flux arises. To examine the mechanism of water flow, let us consider the effects of Δ C _s and Δ P on the water chemical potential in the two compartments.

A net diffusive water flow requires a difference in water chemical potential across the membrane. In a homogeneous membrane, a steady flux denotes a constant chemical potential gradient throughout the membrane thickness. If there is a difference in osmotic pressure between the two solutions, then the water mole fractions (and therefore the water concentrations) at the two sides, just inside the membrane must differ. It is commonly assumed that water transport across the membrane–solution interface is faster than water diffusion in the membrane itself. It follows that the water chemical potential just inside the membrane is very close to that in the adjacent layer of solution; therefore, water is near equilibrium across the interfaces. Finally, since μ _w is inversely related to C _s , a gradient of water concentration must exist across the membrane. This intramembrane gradient is the direct consequence of the differences in impermeant solute concentrations in the adjacent aqueous phases.

When Δπ=0, but Δ P ≠ 0, the chemical potentials of water in the two solutions differ (see Eq. (4.2) ). If P ′> P″ , then the water flux from side ′ is greater than that from side ″, creating an intramembrane gradient of water concentration and chemical potential.

The osmotic water permeability coefficient of a lipid membrane is given by:

P_{o s} = \frac{D_{w}^{m} β_{w} {\bar{V}}_{w}}{δ_{m} {\bar{V}}_{o i l}}

$P_{o s} = \frac{D_{w}^{m} β_{w} {\bar{V}}_{w}}{δ_{m} {\bar{V}}_{o i l}}$

where D ^m _w is the diffusion coefficient of water in the membrane, β _w is the partition coefficient of water in the membrane (oil/water), δ _w is the thickness of the membrane, and ${\bar{V}}_{o i l}$ ${\bar{V}}_{o i l}$ is the partial molar volume of the membrane lipid.

Osmotic Water Flow Across a Porous Membrane

Let us consider a membrane made of a rigid, water-impermeable material. The pore density (number of pores per unit area) is n . Each pore is a water-filled cylinder of length L and radius r , and cannot be penetrated by the solute. The mechanism of water flow in this situation depends mostly on the pore radius. In large pores there is viscous water flow that can be described by Newtonian mechanics. In pores of molecular dimensions there is no appropriate theoretical treatment, but if the pores are so small that there is single file water transport (i.e., water molecules in the pore cannot slip past each other), then there is a surprisingly simple solution.

Large Pores

In large pores water flow driven by a hydrostatic pressure is described by Poiseuille’s law, which was derived for water flow in thin capillaries:

J_{v} = n \frac{(Π) r^{4}}{8 L η} Δ P

$J_{v} = n \frac{(Π) r^{4}}{8 L η} Δ P$

where η is the water viscosity and π denotes 3.1415 … (do not confuse with π, the osmotic pressure). This law is valid for steady-state flow and neglects pore access effects. Under these conditions, the pressure gradient along the pore ( dP / dL ) has a constant value ( Figure 4.2a ). From Eq. (4.4) and the definition of P _os , the P _os for a membrane containing large homogeneous cylindrical pores is n (π) r ⁴ RT /8 L η V _w .

Figure 4.2, Water flow across a porous membrane.

If the only driving force is osmotic, then the mechanism of water flow involves the development of a hydrostatic pressure gradient within the pore. Initially, the water concentrations in the pore and in the water-filled compartment are the same, but at the other interface the solution has lower water concentration than the pore. If water transport across the membrane interfaces is faster than within the membrane, then the water chemical potentials just inside the pore are equal to those in the adjacent solutions. At the pore end facing the water compartment there is no difference in hydrostatic pressure, but at the end facing the solution compartment the pressure inside the pore falls, because the lower water concentration in the solution elicits a water efflux from the pore. If Δμ _w is zero across the opening, then the difference in water concentration between solution and pore is exactly balanced by a drop in the pore pressure. In the steady-state, the pressure gradient in the pore is constant ( Figure 4.2 , top right).

The analysis presented above holds for pores of r equal to or greater than 15 nm. For pores smaller than 15 nm, several corrections have been attempted, but the underlying assumptions are questionable. Regardless of the lack of a satisfactory theory, it has been suggested that Poiseuille’s law is a reasonable approximation for water diffusion and convection in small pores.

Single-File Pore

The P _os of a single-file pore is given by :

P_{o s} = n \frac{{\bar{v}}_{w} k T N}{γ L^{2}}

$P_{o s} = n \frac{{\bar{v}}_{w} k T N}{γ L^{2}}$

where ${\bar{v}}_{w}$ ${\bar{v}}_{w}$ is the volume of a water molecule, k is the Boltzmann constant (gas constant/molecule, equal to R / N _A where N _A is Avogadro’s number), N is the number of water molecules inside the pore, and γ is the friction coefficient per water molecule. Assuming that the water densities in the pore and in the bulk solution are equal, and recalling that kT /γ= D _w :

P_{o s} = n \frac{(Π) r^{2} D_{w}}{L}

$P_{o s} = n \frac{(Π) r^{2} D_{w}}{L}$

which is the result expected for osmotic water flow through a single-file pore if it can be described as a diffusive flux.

Comparison of Diffusion and Osmotic Permeability Coefficients Reveals Whether Water Permeates Lipid Bilayer or Pores

Now we consider a membrane exposed to solutions of identical composition, except that water is partially replaced with tracer water at a concentration C _w ( Figure 4.3 ). There are no other differences in composition or pressure between the two compartments. Both contain solutions of infinite volume and ideally mixed ( C _w at the membrane surface= C _w in the bulk solution). The tracer water flux is given by:

J_{w}^{*} = P_{d w} Δ C_{w}^{*}

$J_{w}^{*} = P_{d w} Δ C_{w}^{*}$

where P _dw is the diffusive water permeability coefficient and $Δ C_{w}^{*}$ $Δ C_{w}^{*}$ is the difference in concentration of tracer water ( C _w *′−C _w *″). In the case of a lipid membrane, the tracer–water flux is by solubility diffusion; hence:

P_{d w} = \frac{D_{w}^{m} β_{w} {\bar{V}}_{w}}{\partial_{m} {\bar{V}}_{o i l}}

$P_{d w} = \frac{D_{w}^{m} β_{w} {\bar{V}}_{w}}{\partial_{m} {\bar{V}}_{o i l}}$

Figure 4.3, Tracer water diffusion across a lipid membrane separating solutions of identical compositions ( C ′ w − C ″ w , = C ′ s − C ″ s , = 0, as shown by the two lines at the bottom).

This expression is identical to that for P _f (or P _os ) for a lipid membrane ( Eq. (4.7) ). Therefore, for a lipid membrane, P _os =P _dw .

The case of a porous membrane is discussed below.

Porous Membrane

If the pores obey Poiseuille’s law, the diffusive water flux via the pores is:

J_{w}^{*} = \frac{n (Π) r^{2} D_{w}}{L} Δ C_{w}^{*}

$J_{w}^{*} = \frac{n (Π) r^{2} D_{w}}{L} Δ C_{w}^{*}$

where the pore cross-sectional area [ n (Π) r ² ] is available for water diffusion and is the water self-diffusion coefficient (tracer water traverses the membrane via the aqueous pores). P _dw is n (Π) r ² D _w / L . Hence, the ratio between and for a porous membrane is:

P_{o s} / P_{d w} = \frac{R T}{8 η D_{w} {\bar{V}}_{w}} r^{2} + 1

$P_{o s} / P_{d w} = \frac{R T}{8 η D_{w} {\bar{V}}_{w}} r^{2} + 1$

where the second term on the right (=1) denotes the diffusive water flow via the pores. The equivalent pore radius can be estimated from experimental values using Eq. (4.11) ; at 25°C, the value of [ RT /(8η D _w ${\bar{V}}_{w}$ ${\bar{V}}_{w}$ )] is 8.04×10 ⁻¹⁴ cm ⁻² .

For single file pores, the diffusive water flux is J _w =nP _dw Δ n *, where Δ n* is the tracer-water concentration difference (molecules per unit volume). P _dw is given by:

P_{d w} = \frac{n {\bar{v}}_{w} k T}{d L^{2}}

$P_{d w} = \frac{n {\bar{v}}_{w} k T}{d L^{2}}$

and the ratio P _os / P _dw , from Eqs. (4.6) and (4.12) , equals the number of water molecules in the pore: P _os / P _dw = N .

Water movement in single file pores is not independent of the movement of neighboring water molecules: for a tracer molecule to cross the pore, other water molecules must also cross.

Unstirred Layers are a Major Source of Artifacts in Water Permeability Measurements

Unstirred layers are static layers of fluid at membrane–solution interfaces, that is, they are not mixed by convection. Unstirred-layer solute concentrations are entirely determined by diffusion, can differ from that of the bulk solutions, and are position dependent. Unstirred layers introduce errors in the experimental determination of P _dw and P _os . These errors can lead to incorrect conclusions about the existence of aqueous pores. For an excellent treatment of unstirred layers, see Barry and Diamond.

Unstirred-Layer Effects on Measurement of P _dw

In the system illustrated in Figure 4.4 , tracer water encounters three barriers to diffusion between the two solutions, namely the membrane and the two unstirred layers (of width Δ ₁ and Δ ₂ , respectively). These three barriers are in series. The observed (experimentally determined) diffusive water permeability of the system differs from the true diffusive water permeability of the membrane ( P _dw ) according to:

\frac{1}{P_{d w}^{o}} = \frac{1}{P_{d w}} + \frac{1}{D_{w} / δ_{1}} + \frac{1}{D_{w} / δ_{2}}

$\frac{1}{P_{d w}^{o}} = \frac{1}{P_{d w}} + \frac{1}{D_{w} / δ_{1}} + \frac{1}{D_{w} / δ_{2}}$

where P ^o _dw is the observed value. Inasmuch as D _w has a finite value, P _dw and P ^o _dw are equal only when δ ₁ =δ ₂ =0. For typical permeability and unstirred-layer thickness values, P _dw can be easily underestimated by 50% or more.

Unstirred-Layer Effects on the Measurement of P _os

In the experiment depicted in Figure 4.5 , a semipermeable membrane separates equal NaCl solutions; then, a second solute is added to one side (C′ _s >C″ _s , eliciting osmotic water flow. The NaCl concentration in the unstirred layers changes because of the water flow, rising in the right side and falling in the left side, relative to the bulk solution concentrations. The added solute qualitatively behaves like the NaCl present on the same side. Water flow tends to accentuate the changes described, whereas solute diffusion in the solution has the opposite effect. At the steady-state, the effects of osmotic water flow and solute diffusion balance each other and the unstirred-layer concentration profiles remain constant. At the surface of the membrane:

C_{m} = C_{b} \exp (\pm v δ / D_{s})

$C_{m} = C_{b} \exp (\pm v δ / D_{s})$

where the sign of the exponent is (−) for the left (“diluted”) side and (+) for the right (“concentrated”) side, and denote solute concentrations (membrane surface and bulk solution, respectively), v is the water flow velocity (normal to the membrane), and D _s is the solute diffusion coefficient in water.

Figure 4.5, Volume flow ( J v ) between solutions containing different concentrations of a permeant solute (0<σ s <1).

The ratio between P ^o _os (observed value) and P _os (true value) is − v Δ/ D _s , that is, the magnitude of the error in estimating P _os is directly proportional to v and inversely proportional to D _s . In planar membranes, for small v (low water flux and flow velocity) the exponential term approaches 1. In folded membranes, where there can be water “funneling” (microvilli, lateral intercellular spaces), v can be much larger than in a planar membrane and can be seriously underestimated.

The above analysis is limited to the effect of J _v on impermeant solute concentration. If permeant solutes are present as well, they must be considered, making the analysis more complex.

Solute Reflection Coefficients Denote Effective Osmolality of a Solution vis-à-vis a Membrane

The situation is more complicated than the preceding analysis if the solute is permeant ( Figure 4.6 ). In this case, J _v will be described by:

J_{v} = L_{p} (Δ P - σ_{s} Δ π)

$J_{v} = L_{p} (Δ P - σ_{s} Δ π)$

where σ _s is the reflection coefficient of the solute. If σ _s <1, then J _v will be less than if the same osmotic gradient is elicited with an impermeable solute. The value of σ _s is specific for each combination of membrane and solute, and depends on both permeabilities and partial molar volumes of water and solute (see below). In general, the value of σ _s varies between 0 (solute as permeable as water) and 1 (solute impermeable).

Figure 4.6, Volume flow ( J v ) between isosmotic solutions (∑ C ′=∑ C ″).

Lipid Membrane

It can be shown that the solute reflection coefficient of a lipid membrane is :

σ_{s} = 1 - \frac{P_{d s} {\bar{V}}_{s}}{P_{d w} {\bar{V}}_{w}}

$σ_{s} = 1 - \frac{P_{d s} {\bar{V}}_{s}}{P_{d w} {\bar{V}}_{w}}$

As expected, σ _s =1 when P _ds =0, σ _s =0 when P _ds ${\bar{V}}_{s}$ ${\bar{V}}_{s}$ = P _dw ${\bar{V}}_{w}$ ${\bar{V}}_{w}$ , and σ _s <0 when P _ds V _s > P _dw V _w . In other words, σ _s depends on the solute permeability and partial molar volume compared with those of water. If the products are the same for solute and water, then the reflection coefficient is zero: solute addition to one side causes no transmembrane volume flow because the water flux toward the solute is of the same magnitude as the solute flux in the opposite direction. A solute with a negative σ _s will elicit a net volume flow in the opposite direction to the water flow (“negative osmosis”).

Porous Membrane

For a quantitative analysis of this complicated problem, see Anderson and Malone and Finkelstein. For large pores and solute particles larger than water particles, the solute is excluded from the periphery of the pore, that is, from a region slightly wider than the solute radius. In the pore axis, C _s is maximum (equal to the concentration in the bulk solution) and C _w is less than at the pore periphery (where the solute is excluded). This generates a radial water concentration gradient within the pore. At equilibrium, this gradient is balanced by a fall in hydrostatic pressure in the periphery of the pore. In addition, there is a solute concentration gradient along the pore length, because of the transmembrane difference in C _s . These two gradients combine to generate a longitudinal hydrostatic pressure gradient along the pore’s periphery, which causes water flow toward the high concentration side. The thickness of the ring subjected to this regime is directly proportional to the molecular size of the solute. When the solute is so large that it cannot enter the pore, J _v is maximum and σ _s =1. When the solute has the same size as water, its distribution within the pore is identical to that of water, no pressure gradient develops, the water and solute net fluxes are purely diffusive and of equal magnitude and opposite direction, and J _v =0. When the solute is smaller than water, J _v is greater than J _w , and J _v is in the same direction as J _s . Water is largely excluded from the periphery of the pore, and the pore hydrostatic pressure gradient is opposite to that generated by the impermeant solute.

We consider now the case of single file pores; the solution is dilute enough so that the number of solute molecules inside a pore can be only 0 or 1. Two pore populations will exist at any given time: pores containing water only, through which there is water flow toward the high- C _s side, and pores containing solute, in which solute and water are transported toward the low- C _s side by single file diffusion. When both water and solute permeation are single file, σ _s is :

σ_{s} = 1 - \frac{P_{d s} {\bar{V}}_{p}^{s}}{P_{d w} {\bar{V}}_{p}}

$σ_{s} = 1 - \frac{P_{d s} {\bar{V}}_{p}^{s}}{P_{d w} {\bar{V}}_{p}}$

where ${\bar{V}}_{p}$ ${\bar{V}}_{p}$ denotes pore molar volume, solute-containing (superscript s ) and solute-free (no superscript). Compare with Eq. (4.16) .

In a system with more than one solute there can be net water and/or volume flows between solutions with the same total solute concentrations (and osmolalities). This will occur if the specific C _s values on the two sides of the membrane and the reflection coefficients differ. As shown in Figure 4.6 , if , Δ C _s =−Δ C _x , σ _s =0, and σ _x =1, then there will be a net volume flow towards the right, although the solutions have equal total osmolalities. If the hydrostatic pressures are the same on both sides, then J _v =L _p RT (σ _x Δ C _x −σ _s Δ C _s ). Expressions such as σ RT C denote “effective osmolality,” in contrast with the “total osmolality” given by RTC . Effective osmolality is also referred to as tonicity . In epithelia, active transepithelial solute transport can generate asymmetries in the composition of the adjacent solutions. These asymmetries may in principle drive net water transport without differences in the total osmolalities of the bulk solutions, because of differences in the solute reflection coefficients.

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Introduction

Basic Principles

Osmotic Equilibrium is a Balance of Osmotic and Hydrostatic Forces

Osmotic Water Flows Across Lipid and Porous Membranes have Different Properties

Osmotic Water Flow Across Lipid Membranes

Osmotic Water Flow Across a Porous Membrane

Large Pores

Single-File Pore

Comparison of Diffusion and Osmotic Permeability Coefficients Reveals Whether Water Permeates Lipid Bilayer or Pores

Porous Membrane

Unstirred Layers are a Major Source of Artifacts in Water Permeability Measurements

Unstirred-Layer Effects on Measurement of P dw

Unstirred-Layer Effects on the Measurement of P os

Solute Reflection Coefficients Denote Effective Osmolality of a Solution vis-à-vis a Membrane

Lipid Membrane

Porous Membrane

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Unstirred-Layer Effects on Measurement of P _dw

Unstirred-Layer Effects on the Measurement of P _os