Imagine the Possibilities Mathematical Biology University of Utah Introduction to Physiology II: Control of Cell Volume and Membrane Potential J. P. Keener Mathematics Department University of Utah _ Math Physiology -p.1/23 Imagine the Possibilities Mathematical Biology University of Utah 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 Imagine the Possibilities Mathematical Biology University of Utah 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 Mathematical Biology University of Utah Membrane Transporters O CO 22 4 Glucose O Y H2O 4 Na+K+ Ca ++ + K 4 V n 4 \ i t t / v Na + ATP • transmembrane diffusion - carbon dioxide, oxygen Math Physiology -p.4/23 99999999 Mathematical Biology University of Utah Membrane Transporters Glucose O CO 2 2 i HO i Na+K+ Ca+ Na + O ^ 4 \ I t K ADP • transmembrane diffusion - carbon dioxide, oxygen • transporters - glucose, sodium-calcium exchanger Math Physiology - p.4/23 Mathematical Biology University of Utah Membrane Transporters O CO 2 2 4 Glucose O i Na+K+ Ca+ 4 !\ K+ \ \ 4 i Tíí i ) 1 i i / i Na+ ATP • transmembrane diffusion - carbon dioxide, oxygen • transporters - glucose, sodium-calcium exchanger • pores - water Math Physiology - p.4/23 99999999 Mathematical Biology University of Utah Membrane Transporters Glucose O CO 2 2 4 i Na+K+ Ca+ Na + I o n 4 \ I 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 9999999^ Mathematical Biology University of Utah Membrane Transporters O CO 2 2 4 Glucose O Y HO 4 Na+K+ Ca+ K + n 4 \ I / v 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 99999999 Imagine the Possibilities Mathematical Biology University of Utah How Things Move Most molecules move by a random walk: Q. - ^r1^ ft I 0 10 20 30 40 50 60 70 80 90 100 time 0 10 20 30 40 50 60 70 80 90 100 time 30 20 10 -10 -4 -20 _ Math Physiology - p.5/23 Imagine the Possibilities Mathematical Biology University of Utah How Things Move Most molecules move by a random walk: Q. - ^r1^ ft I 0 10 20 30 40 50 60 70 80 90 100 time 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 dC_ dx Math Physiology - p.5/23 30 20 10 -10 -4 -20 Imagine the Possibilities Mathematical Biology University of Utah How Things Move Most molecules move by a random walk: Q. - ^r1^ ft I 0 10 20 30 40 50 60 70 80 90 100 time 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 9C dx molecular flux, Math Physiology - p.5/23 30 20 10 -10 -4 -20 Imagine the Possibilities Mathematical Biology University of Utah How Things Move Most molecules move by a random walk: Q. - 0 10 20 30 40 50 60 70 80 90 100 time 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 r = - D dC_ dx molecular flux, diffusion coefficient, Math Physiology - p.5/23 30 20 10 -10 -4 -20 Imagine the Possibilities Mathematical Biology University of Utah How Things Move Most molecules move by a random walk: Q. - ^r1^ ft I 0 10 20 30 40 50 60 70 80 90 100 time 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 9C dx molecular flux, diffusion coefficient, concentration gradient. Math Physiology - p.5/23 30 20 10 -10 -4 -20 Imagine the Possibilities Mathematical Biology University of Utah Conservation Law Conservation: dC_ dJ_ dt dx leading to the Diffusion Equation dC 0 dt dx dx _ Math Physiology - p.6/23 Imagine the Possibilities Mathematical Biology University of Utah Basic Consequences -1 Diffusion in a tube fed by a reservoir c(x,t) = n^) 0.9 0.8 0.7 0.6 O 0.5 0.4 0.3 0.2 0.1 _ 0.5 1.5 2 x 2.5 3.5 Math Physiology - p.7/23 0 0 3 Imagine the Possibilities Mathematical Biology University of Utah Basic Consequences - II Diffusion time:t = ^- for hydrogen (D = 10_5cm2/s). X t Example 10 nm 100 ns cell membrane 1 /xim 1 ms mitochondrion 10 /xim 100 ms mammalian cell 100 /xm 10 s diameter of muscle fiber 250 /xm 60 s radius of squid giant axon 1 mm 16.7 min half-thickness of frog sartorius muscle 2 mm 1.1 h 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 Possibilities Mathematical Biology University of Utah Basic Consequences - Ohm's Law Diffusion across a membrane J = ^(d-C2) Flux changes as things like C\, C2 and L change. _ Math Physiology - p.9/23 Imagine the Possibilities Mathematical Biology University of Utah Carrier Mediated Diffusion Se + cp < Pf _ Math Physiology - p.10/23 _ Math Physiology - p.10/23 Imagine the Possibilities Mathematical Biology University of Utah Carrier Mediated Diffusion Se + Ce < Pp < Pi < Si + Ci _ Math Physiology - p.10/23 Imagine the Possibilities Mathematical Biology University of Utah Carrier Mediated Diffusion Se + cp r ■ w — P Pi < Si + Ci For this system J = J Sp Si max (Sp + Kp )(Si + Ki )• _ ....a saturating Fick's law Math Physiology - p.10/23 Imagine the Possibilities Mathematical Biology University of Utah Ion Movement Ions move according to the Nernst-Planck equation Fz j = -D(yc+—v) Consequently, at equilibrium [C]e VN = Vi-Ve = ^hi zF [C\ ) e extracellular - [C]i intracellular This is called the Nernst Potential or Reversal Potential. Math Physiology - p.11/23 Imagine the Possibilities Mathematical Biology University of Utah Ion Current Models There are many different possible Models of Iionic. Barrier models, binding models, saturating models, PNP equations, etc. Constant field assumption: F2 ([C]i- [C]eexp(^f^)\ Iion = P—V —-1 1 } , GHK Model RT Long Channel limit (used by HH) Iion = g (V — VN) Linear Model All of these have the same reversal potential, as they must. Math Physiology - p.12/23 Imagine the Possibilities Mathematical Biology University of Utah Short Channel Limit m If the channel is short, then L « 0 => A « 0. Then ^ = 0 implies the field is constant: ~j-=v---vci = - Ji RT This is the Goldman-Hodgkin-Katz equation. _ Math Physiology - p.14/23 Imagine the Possibilities Mathematical Biology University of Utah Long Channel Limit If the channel is long, then ^ « 0 => j « 0. Then c\ « c2 throughout the channel: ci = c2 2—— = - Ji - J2 dx Ci = C2 + (Ce - Ci)x 0=--In I — + (1---)x ) ?Ji = Nernst potentia Vi This is the linear I-V curve used by Hodgkin and Huxley. _ Math Physiology - p.15/23 Imagine the Possibilities Mathematical Biology University of Utah Sodium-Potassium ATPase Math Physiology - p.16/23 Imagine the Possibilities Mathematical Biology University of Utah Osmotic Pressure and Flux rQ = Pi - P2 - ni + 7T2 TTi = kTCi 1 f2 0 0 0 000 0 0 0 00 0 00 O O V 0 V 0 0 0 0 C2 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 Imagine the Possibilities Mathematical Biology University of Utah Charge Balance and Osmotic Balance • Inside and outside are both electrically neutral, macromolecules have negative charge zx. qw(Ni+Ki—Ci)+zxqX = qw(Ne+Ke—Ce) = 0, (charge bala • Total amount of osmolyte is the same on each side. X Ni + Ki + Ci H--= Ne + Ke + Ce (osmotic balance) w _ Math Physiology - p.19/23 Imagine the Possibilities Mathematical Biology University of Utah The Solution The resulting system of algebraic equations is readily solved CD E CD > "03 O 5 4 3 2 1 0 0 2 4 6 8 10 Pump rate > "■I—* CD O Q_ 100 -i 50 -\ ~i—r 12 14 0 -50 -100 -4 0 2 4 6 8 10 12 Pump rate 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 mmodel) Math physioi°9y - p-2°/23 Imagine the Possibilities Mathematical Biology University of Utah 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 and pumps? • Other intricate control mechanisms are needed. Lots of interesting problems (A. Weinstein, BMB 54, 537, 1992.) Mucosal side I Serosal side 2 K+ 3 Na+ Cl- Cl- r Math Physiology -p.21/23 Imagine the Possibilities Mathematical Biology University of Utah 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 Imagine the Possibilities Mathematical Biology University of Utah 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