Control of Cell Volume and Membrane Potential

x=0 φ (0) = V
x=L f (L) = 0
d 2φ = −λ2 (c1 − c 2 ), λ2 = stuff × L2 2 dx dc F dφ J1 = −D1 1 + c dx RT 1 dx dc F dφ c2 J 2 = −D2 2 − dx RT dx
Na+-K+ ATPase
Na+-K+ ATPase
Outside the cell K+ Na+•C•P Step 2 ADP Step 1 Step 3 P Step 4 Na+•C K+ Inside the cell Na+ K+•C Na+ K+•C•P
ATP
Osmosis
P1 P2
The cell at steady state
3 Na+
We need to model • pumps • ionic currents • osmotic forces
2 K+ Cl-
Ca2+
Typical ionic concentrations
Squid Giant Axon Intracellular Na+ K+ ClExtracellular Na+ K+ Cl437 20 556 110 2.5 90 155 5 112 50 397 40 13 138 3 19 136 78 Frog Sartorius Muscle Human Red Blood Cell
Control of Cell Volume and Membrane Potential
Basic reference: Keener and Sneyd, Mathematical Physiology
Basic problem
• The cell is full of stuff. Proteins, ions, fats, etc. • Ordinarily, these would cause huge osmotic pressures, sucking water into the cell. • The cell membrane has no structural strength, and the cell would burst.
Active pumping
• Clearly, the action of the pumps is crucial for the maintenance of ionic concentration differences • Many different kinds of pumps. Some use ATP as an energy source to pump against a gradient, others use a gradient of one ion to pump another ion against its gradient. • A huge proportion of all the energy intake of a human is devoted to the operation of the ionic pumps.
Boundary conditions
c1 (0) = c i , c 2 (0) = c i , φ (0) = V,
c1 (L) = c e c 2 (L) = c e φ (L) = 0
The short-channel limit
If the channel is short, then L ~ 0 and so λ ~ 0.
( ) ( )
This is the Goldman-Hodgkin-Katz equation. Note: a short channel implies independence of ion movement through the channel.
The long-channel limit
[S]e=[S’]e Ve
RT [S]e Vi − Ve = ln F [S]i
(The Nernst potential)
Note: equilibrium only. Tells us nothing about the current. In addition, there is very little actual ion transfer from side to side. We'll discuss the multi-ion case later.
Basic solution
• Cells carefully regulate their intracellular ionic concentrations, to ensure that no osmotic pressures arise • As a consequence, the major ions Na+, K+, Cl- and Ca2+ have different concentrations in the extracellular and intracellular environments. • And thus a voltage difference arises across the cell membrane. • Essentially two different kinds of cells: excitable and nonexcitable. • All cells have a resting membrane potential, but only excitable cells modulate it actively.
Volume control: The Pump-Leak Model
water
water + Solvent (conc. c)
At equilibrium:
P1 + kcT = P2
Note: equilibrium only. No information about the flow.
The Nernst equation
[S]i=[S’]i Vi
Permeable to S, not S’
Only very little ion transfer
spherical cell - radius 25 µm surface area - 8 x 10-5 cm2 total capacitance - 8 x 10-5 µF (membrance capacitance is about 1 µF/cm2) If the potential difference is -70 mV, this gives a total excess charge on the cell membrane of about 5 x 10-12 C. Since Faraday's constant, F, is 9.649 x 104 C/mole, this charge is equivalent to about 5 x 10-15 moles. But, the cell volume is about 65 x 10-9 litres, which, with an internal K+ concentration of 100 mM, gives about 6.5 x 10-9 moles of K+. So, the excess charge corresponds to about 1 millionth of the background K+ concentration.
Electrodiffusion: deriving current models
cell membrane Inside [S1+] = [S2-] = ci S1 S2 Outside [S1+] = [S2-] = ce
Poisson equation and electrodiffusion
If the channel is long, then 1/L ~ 0 and so 1/λ ~ 0.
1 d 2φ Then 2 2 = c1 − c 2 , which implies that c1 ≈ c 2 through the membrane. λ dx dc c1 = c 2 ⇒ 2 1 = −J1 − J 2 dx ⇒ c1 = c i + (c e − c i )x
d 2φ dφ Then = 0, which implies that the electric field, , is constant through the membrane. 2 dx dx dφ dc1 =v ⇒ − vc1 = −J1 dx dx c i − c e e−v ⇒ J1 = v 1− e−v −VF D1F 2 c i − c e exp RT ⇒ I1 = V LRT 1− exp −VF RT
These are the two most common current models. Note how they both have the same reversal potential, as they must. (Crucial fact: In electrically excitable cells gNa (or PNa) are not constant, but are functions of voltage and time. More on this later.)
Electrical circuit model of cell membrane