Introduction to Physiology II: Control of Cell Volume and Membrane Potential

Introduction to Physiology II: Control of Cell Volume and Membrane Potential J. P. Keener Mathematics Department Math Physiology p.1/23

Basic Problem The cell is full of stuff: Proteins, ions, fats, etc. The cell membrane is semipermeable, and these substances create osmotic pressures, sucking water into the cell. The cell membrane is like soap film, has no structural strength to resist bursting. Math Physiology p.2/23

Basic Solution Carefully regulate the intracellular ionic concentrations so that there are no net osmotic pressures. As a result, the major ions (Na +, K +, Cl and Ca ++ ) have different intracellular and extracellular concentrations. Consequently, there is an electrical potential difference across the cell membrane, the membrane potential. Math Physiology p.3/23

Imagine the Membrane Transporters O CO 2 2 Glucose + H 2 O Na K + ++ Ca K + Na + ATP transmembrane diffusion - carbon dioxide, oxygen Math Physiology p.4/23

Imagine the Membrane Transporters O CO 2 2 Glucose + ++ H O Na Na + 2 K + Ca K + ADP transmembrane diffusion - carbon dioxide, oxygen transporters - glucose, sodium-calcium exchanger Math Physiology p.4/23

Imagine the Membrane Transporters O CO 2 2 Glucose + ++ H O Na Na + 2 K + Ca K + ADP transmembrane diffusion - carbon dioxide, oxygen transporters - glucose, sodium-calcium exchanger pores - water ion-selective, gated channels - sodium, potassium, calcium Math Physiology p.4/23

Imagine the Membrane Transporters O CO 2 2 Glucose + H 2 O Na K + ++ Ca K + Na + ATP transmembrane diffusion - carbon dioxide, oxygen transporters - glucose, sodium-calcium exchanger pores - water ion-selective, gated channels - sodium, potassium, calcium ATPase exchangers - sodium-potassium ATPase, SERCA Math Physiology p.4/23

How Things Move Most molecules move by a random walk: 6 30 4 20 2 10 position 0 2 position 0 10 4 6 20 8 0 10 20 30 40 50 60 70 80 90 100 time 30 0 10 20 30 40 50 60 70 80 90 100 time Math Physiology p.5/23

How Things Move Most molecules move by a random walk: 6 30 4 20 2 10 position 0 2 position 0 10 4 6 20 8 0 10 20 30 40 50 60 70 80 90 100 time 30 0 10 20 30 40 50 60 70 80 90 100 time Fick s law: When there are a large number of these molecules, their motion can be described by J = D C x Math Physiology p.5/23

Conservation Law Conservation: leading to the Diffusion Equation C t + J x = 0 C t = C (D x x ). Math Physiology p.6/23

Basic Consequences - I Diffusion in a tube fed by a reservoir C(x, t) = f( x2 Dt ) 1 0.9 0.8 0.7 0.6 C 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 x Math Physiology p.7/23

Basic Consequences - II Diffusion time:t = x2 D for hydrogen (D = 10 5 cm 2 /s). x t Example 10 nm 100 ns cell membrane 1 µm 1 ms mitochondrion 10 µm 100 ms mammalian cell 100 µm 10 s diameter of muscle fiber 250 µm 60 s radius of squid giant axon 1 mm 16.7 min half-thickness of frog sartorius muscle 2 mm 1.1h half-thickness of lens in the eye 5 mm 6.9 h radius of mature ovarian follicle 2 cm 2.6 d thickness of ventricular myocardium 1 m 31.7 yrs length of sciatic nerve Math Physiology p.8/23

Imagine the Basic Consequences - Ohm s Law Diffusion across a membrane J = AD L (C 1 C 2 ) V C 1 J C 2 1 L V 2 Flux changes as things like C 1, C 2 and L change. Math Physiology p.9/23

the Imagine Carrier Mediated Diffusion Ce Ci Math Physiology p.10/23

the Imagine Carrier Mediated Diffusion Ce Ci Se + Ce Pe Math Physiology p.10/23

the Imagine Carrier Mediated Diffusion Ce Ci Se + Ce Pe Pi Math Physiology p.10/23

the Imagine Carrier Mediated Diffusion Ce Ci Se + Ce Pe Pi Si + Ci Math Physiology p.10/23

the Imagine Carrier Mediated Diffusion C e Ci S e + C e Pe Pi Si + C i For this system, J = J max S e S i (S e + K e )(S i + K i ).... a saturating Fick s law Math Physiology p.10/23

Ion Movement Ions move according to the Nernst-Planck equation Consequently, at equilibrium J = D( C + F z RT φ) V N = V i V e = RT ( ) zf ln [C]e [C] i [C] V e e [C] V i i extracellular intracellular This is called the Nernst Potential or Reversal Potential. Math Physiology p.11/23

Ion Current Models There are many different possible Models of I ionic. Barrier models, binding models, saturating models, PNP equations, etc. Constant field assumption: I ion = P F 2 RT V Long Channel limit (used by HH) ( [C]i [C] e exp( zv F RT ) ) 1 exp( zv F, GHK Model RT I ion = g(v V N ) Linear Model All of these have the same reversal potential, as they must. Math Physiology p.12/23

Electrodiffusion Models cell membrane Inside Outside S [S + 1 ] = [S2-1 ] = ci [S + 1 ] = [S2 - ] = ce S 2 x=0 x=l φ (0) = V φ (L) = 0 d 2 φ dx 2 = λ 2 (c 1 c 2 ), Poisson Equation J 1 = D 1 ( dc 1 dx + F RT c 1 J 2 = D 2 ( dc 2 dx F RT c 2 dφ dx ), dφ dx ), Nernst Planck Equations Math Physiology p.13/23

Short Channel Limit If the channel is short, then L 0 λ 0. Then d2 φ dx 2 the field is constant: dφ dx = v dc 1 dx vc 1 = J 1 J 1 = v c i c e e v 1 e v I ion = P F 2 RT V This is the Goldman-Hodgkin-Katz equation. ( [C]i [C] e exp( zv F RT ) 1 exp( zv F RT = 0 implies ) Math Physiology p.14/23

Long Channel Limit If the channel is long, then 1 L 0 1 λ 0. Then c 1 c 2 throughout the channel: c 1 = c 2 2 dc 1 dx = J 1 J 2 c 1 = c 2 + (c e c i )x φ = v ( ci ln + (1 c ) i )x v 1 c e c e J 1 = c e c i v 1 (v v 1 ) v 1 = Nernst potentia This is the linear I-V curve used by Hodgkin and Huxley. Math Physiology p.15/23

Sodium-Potassium ATPase Math Physiology p.16/23

Osmotic Pressure and Flux P P 1 2 rq = P 1 P 2 π 1 + π 2 π i = kt C i C C 1 2 osmolite Math Physiology p.17/23

Volume Control; Pump-Leak Model Na + is pumped out, K + is pumped in, Cl moves passively, negatively charged macromolecules are trapped in the cell. Math Physiology p.18/23

Charge Balance and Osmotic Balance Inside and outside are both electrically neutral, macromolecules have negative charge z x. qw(n i +K i C i )+z x qx = qw(n e +K e C e ) = 0, (charge bala Total amount of osmolyte is the same on each side. N i + K i + C i + X w = N e + K e + C e (osmotic balance) Math Physiology p.19/23

The Solution The resulting system of algebraic equations is readily solved 5 100 Cell volume 4 3 2 1 Potential (mv) 50 0-50 V Na V V K 0-100 0 2 4 6 8 Pump rate 10 12 14 0 2 4 6 8 Pump rate 10 12 14 If the pump stops, the cell bursts, as expected. The minimal volume gives approximately correct membrane potential (although there are MANY deficiencies with this model.) Math Physiology p.20/23

Volume Control and Ion Transport How can epithelial cells transport ions and water while maintaining constant cell volume under widely varying conditions? Spatial separation of leaks Mucosal side Na + 2 K + Serosal side 3 Na + and pumps? Other intricate control mechanisms are needed. Lots of interesting problems (A. Weinstein, BMB 54, 537, 1992.) Cl - Cl - Math Physiology p.21/23

Inner Meduullary Collecting Duct Real cells are far more complicated Notice the large Na + flux from the lumen. cf. A. Weinstein, Am. J. Physiol. 274, F841-F855, 1998. Math Physiology p.22/23

Interesting Problems (suitable for projects) How do organism (e.g., T. Californicus living in tidal basins) adjust to dramatic environmental changes? How do plants in arid, salty regions, prevent dehydration? (They make proline) How do fish (e.g., salmon) adjust to both freshwater and salt water? What happens to a cell and its environment when there is ischemia (loss of ATP)? How do cell in high salt environments (epithelial cell in kidney) maintain constant volume? Math Physiology p.23/23