μ i = chemical potential of species i C i = concentration of species I

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Augustine Griffith
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1 BIOE 459/559: Cell Engineering Membrane Permeability eferences: Water Movement Through ipid Bilayers, Pores and Plasma Membranes. Theory and eality, Alan Finkelstein, 1987 Membrane Permeability, 100 Years Since Ernest Overton, Eds. DW Deamer, A Kleinzeller, DM Fambrough, 1999 Solubility-Diffusion Model of Membrane Transport: The solubility-diffusion modeling approach breaks the transport of a molecule across the cell membrane into 3 steps: (1) movement from the hydrophilic aqueous environment outside the membrane to the hydrophobic environment ithin the membrane bilayer, (2) diffusion from one side of the membrane to the other, and (3) movement from the hydrophobic environment ithin the membrane to hydrophilic aqueous environment outside the membrane. We ill make the folloing assumptions: Dilute, ideal solutions No pressure or temperature differences across membrane Chemical equilibrium at membrane edges Steady-state diffusion across membrane Well-mixed aqueous solutions on either side of membrane Sketch the concentration and chemical potential profiles beteen the left and right sides of the cell membrane on the figure belo. μ i C i Membrane μ i = chemical potential of species i C i = concentration of species I = membrane thickness D = diffusivity of i in membrane refers to aqueous phase (ater) o refers to membrane phase (oil) refers to left side of membrane refers to right side of membrane The molar flux of species i across the cell membrane, J i, can be described using Fick s la: J i = (1) The partition coefficient K is the equilibrium constant beteen i in the aqueous phase and i in the membrane phase: K = C i,o (2)

2 Assuming each edge of the membrane is in chemical equilibrium, the equation for the partition coefficient (Eq. 2) can be combined ith the diffusion equation (Eq. 1), yielding: J i = The membrane permeability coefficient for species i is defined as P i = DK Plugging this definition into Eq. 3 results in (3) (4) J i = (5) Sketch the concentration and chemical potential profiles beteen the left and right sides of the cell membrane on the figure belo (left) using the folloing values of K: 0.25, 0.5 and 1. Sketch a plot of the permeability as a function of K in the figure belo (right). μ i Membrane P i C i K At equilibrium, the concentration of i is the same on both sides of the membrane. Sketch a plot of the equilibrium concentration and chemical potential profiles on the figure belo.

3 Water Transport through the Membrane Bilayer The solubility-diffusion modeling approach can be applied to ater transport across the cell membrane, yielding the folloing for the molar flux of ater, J = P (C C ) (6) here P is knon as the filtration permeability coefficient, and C is the ater concentration. The original approach for modeling ater transport across the cell membrane considered an osmotic pressure driving force (rather than a concentration driving force). With this approach, the volumetric flux J * is assumed to be proportional to the osmotic pressure driving force: J = p (π π ) (7) here p is the hydraulic permeability coefficient. These to ays for modeling ater transport can be reconciled as follos. The osmotic pressure for an ideal and dilute solution can be expressed as π = TC s = T(C C ) (8) here C s is the total solute concentration, C is the total molar concentration (including both solutes and ater). Substituting Eq. 8 into Eq. 7 and assuming the total molar concentration is constant yields J = p T(C C ) (9) To convert the volumetric flux J * to the molar flux J e can use the molar volume of ater v Therefore, P = J = J = pt (C v v C ) (10) In the figure belo, there is a concentration gradient driving flo of ater from left to right. Sketch a plot of the ater concentration and solute concentration profiles.

4 Water Transport through Membrane Pores Many cell types have been found to express aquaporins, membrane pores that allo passage of ater through the cell membrane. To model this process, e ill assume that ater can enter the pore, but all solutes are excluded. Dra the solutes and ater molecules on the figure belo, here the right side of the membrane has a higher solute concentration than the left. Because solute molecules cannot enter the pore, no concentration gradient exists inside the pore. Nonetheless, ater ill still flo to the right toards the side of the membrane ith the higher solute concentration. What is the driving force for movement of ater inside the pore? We ill assume chemical equilibrium beteen the aqueous solution at the pore edge and the pure ater just inside the pore. At the pore edge, the chemical potential is (assuming an ideal solution) μ,a = μ 0 (P 0, T) + v (P a P 0 ) + T ln x,a (11) here P a is the pressure in the aqueous solution at the pore edge, P 0 is the pressure at the reference state and x,a is the mole fraction of ater in the aqueous solution. Just inside the pore the chemical potential is μ,p = μ 0 (P 0, T) + v P p P 0 + T ln x,p = μ 0 (P 0, T) + v P p P 0 (12) here P p is the pressure just inside the pore and x,p = 1 is the mole fraction of ater inside the pore. Equating these chemical potentials yields P a P p = T ln x v,a (13) This equation can be applied at the left and right sides of the cell membrane, alloing us to determine the pressure difference driving ater flo through the pore. Assuming that the pressure in the aqueous solution (outside the pore) is the same on both sides of the membrane (a good assumption for mammalian cells), e get P p P p = T ln x v,a ln x,a (14) If e assume a dilute solution, e can use a linear approximation for the natural logs, hich yields P p P p = T C,a v C C,a (15) C

5 here C is the total molar concentration. For a dilute aqueous solution C = 1/v, hence P p P p = T C,a C,a (16) This equation tells us that a difference in ater concentration at the pore edges leads to a pressure gradient ithin the pore. Sketch a plot of the chemical potential, ater concentration and pressure profiles on the figure belo. μ,a C,a μ,a P a C,a P a The pressure gradient drives flo of ater through the pore by to different mechanisms: bulk flo and diffusion. Bulk flo can be modeled using the Hagen-Poiseuille equation for laminar flo through a cylindrical tube. The molar flux of ater through the pore due to bulk flo is J bf = Q v πr 2 = P p P p r 2 (17) 8ηv here Q is the volumetric flo rate through the pore, r is the pore radius, η is the viscosity of ater and is the pore length. The molar flux of ater due to diffusion is J d = D T C dμ dx = D T C dp p v dx = D T P p P p (18) here D is the self-diffusion coefficient of ater. The total molar flux of ater through the pore is equal to the sum of fluxes due to bulk flo and diffusion J,p = J bf + J d = P p P p r2 + D (19) 8ηv T Substituting Eq. 16 into this equation yields J,p = C,a C,a r2 T 8ηv + D (20) To get the flux of ater molecules across the entire cell membrane, J, e multiply J,p by the total pore cross sectional area, nπr 2 (here n is the total number of pores in the cell membrane) and then divide by the total area of the cell membrane, A.

6 J = J,p(nπr 2 ) = P A (C C ) (21) The equation above shos that in the case of ater transport through pores, the filtration permeability coefficient P is equal to P = r2 T 8ηv + D nπr2 (22) A Simultaneous Water Transport through Pores and the Bilayer If e ant to consider ater transport through pores and ater transport through the membrane bilayer simultaneously, e need to include both mechanisms in our expression for the molar flux of ater J = DK + r2 T 8ηv + D In this case the filtration permeability coefficient ould be P = DK + r2 T 8ηv + D nπr2 A (C C ) (23) nπr2 (24) A

7 Measurement of Cell Membrane Water Permeability The ater permeability is typically determined by fitting a membrane transport model to measurements of the change in cell volume after exposure to hypo- or hypertonic solutions. If ater is the only substance that crosses the cell membrane (hich occurs if all solutes can be considered impermeable), then the change in cell volume can be described by dv cell = dt p A π e π i (25) here the superscripts i and e refer to the intracellular and extracellular solutions, respectively. Assuming ideal and dilute solutions, the above equation can be ritten dv cell = dt p AT C i s C e s (26) Sketch a plot of cell volume as a function of time after exposure to hypotonic and hypertonic solutions. V cell V cell,0 t To solve Eq. 26, e need to find an expression for ho the cell volume and the intracellular solute concentration are related. The cell volume V cell is typically divided into to compartments, the osmotically inactive volume, V b, and the cell ater volume V. The osmotically inactive volume consists of intracellular macromolecules, lipids and other solutes. The folloing picture shos a cell under isotonic conditions, here the solute concentration is C s,0 = 0.3 osmoles/. V cell,0 = V,0 V b moles of intracellular solute = If the cell membrane is not permeable to solutes, then the total moles of intracellular solute remain constant. This leads to the Boyle-van t Hoff expression, hich relates the intracellular solute concentration to the cell volume: C i s = C s,0 V cell,0 V b (27) V cell V b Substituting this equation into Eq. 26 yields dv cell dt = p AT C s,0 V cell,0 V b V cell V b C s e (28)

8 In experiments to determine the membrane permeability, the extracellular concentration is changed and the resulting change in cell volume is measured. The cell volume data is then fit ith Eq. 28 to determine the hydraulic permeability coefficient p. Equation 28 contains several cell-specific parameters, including the cell surface area A, the isotonic cell volume V cell,0, the osmotically inactive volume V b and the hydraulic permeability coefficient p. Before fitting Eq. 28 to cell volume data to determine the permeability, it is first necessary to determine the values of the other cell specific parameters (i.e., A, V cell,0 and V b ). The isotonic cell volume is simply determined by measuring the cell volume under isotonic conditions, and the cell surface area is commonly assumed to be equal to the area of a sphere ith the isotonic volume. To determine V b, the equilibrium cell volume is measured in several different hypo- and hypertonic solutions and the resulting data are fit ith the Boyle-van t Hoff equation: V cell = V b + C s,0 C s e V cell,0 V b (29) Sketch the equilibrium cell volume on the folloing plot V cell V cell,0 C s,0 /C s e

9 Membrane Transport of Water and a Single Permeating Solute When ater and a permeating solute (e.g., glycerol) can both cross the cell membrane, the membrane transport model is a bit more complicated. In such a case, the total cell volume (V cell ) consists of the volume occupied by ater (V ), the volume occupied by permeating solute (V p ) and the osmotically inactive volume (V b ). The rate at hich the ater volume changes is dv dt = pat C i s C e s = p AT C i s,0 V cell,0 V b + n p C e V s (30) here n i p is the moles of intracellular permeating solute. The rate at hich the moles of intracellular permeating solute changes is dn p dt = P pa C e p C i p = (31) Equations 30 and 31 are coupled, in the sense that it is impossible to solve one ithout also solving the other. This is the case because both differential equations contain the dependent variables V and n i p. After solving Eqs. 30 and 31, the total cell volume can be computed as follos V cell = V + v p n i p + V b (32) Where v p is the molar volume of the permeating solute. In experiments to determine the membrane permeability to ater and permeating solute, cells are exposed to solution containing permeating solute and the resulting change in cell volume is measured. The cell volume data is then fit ith Eqs to determine the hydraulic permeability coefficient p and the permeability of the permeating solute P p. As ith experiments for measuring the ater permeability, it is first necessary to determine V cell,0, V b and A from equilibrium cell volume measurements. Imagine a cell that is initially in isotonic saline solution, and then exposed to a hypertonic solution containing permeating solute at t = 0. Assuming that transport of both ater and permeating solute occurs by diffusion through the membrane bilayer, sketch a plot of the concentration profiles C and C p just after the cells are exposed to the hypertonic solution. To the right, dra a plot of the expected change in cell volume ith time, assuming that ater transport is faster than transport of permeating solute.

There are mainly two types of transport :

There are mainly two types of transport : # Type one: Passive diffusion 1- which does not require additional energy and occurs down the concentration gradient (high low concentration) " Down Hill" (^_^