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## Control of Cell Volume and Membrane Potential

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**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.**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.**The cell at steady state**• We need to model • pumps • ionic currents • osmotic forces**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.**Osmosis**P1 P2 water + Solvent (conc. c) water At equilibrium: Note: equilibrium only. No information about the flow.**[S]i=[S’]i**[S]e=[S’]e Vi Ve Permeable to S, not S’ The Nernst equation (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.**Only very little ion transfer**spherical cell - radius 25 mm surface area - 8 x 10-5 cm2 total capacitance - 8 x 10-5 mF (membrance capacitance is about 1 mF/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.**Electrical circuit model of cell membrane**How to model this**How to model Iionic**• Many different possible models of Iionic • Constant field assumption gives the Goldman-Hodgkin-Katz model • The PNP equations can derive expressions from first principles (Eisenberg and others) • Barrier models, binding models, saturating models, etc etc. • Hodgkin and Huxley in their famous paper used a simple linear model • Ultimately, the best choice of model is determined by experimental measurements of the I-V curve.**Two common current models**Linear model GHK model 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.)**Electrodiffusion: deriving current models**Poisson equation and electrodiffusion Boundary conditions**The short-channel limit**If the channel is short, then L ~ 0 and so l ~ 0. This is the Goldman-Hodgkin-Katz equation. Note: a short channel implies independence of ion movement through the channel.**The long-channel limit**If the channel is long, then 1/L ~ 0 and so 1/l ~ 0. This is the linear I-V curve. The independence principle is not satisfied, so no independent movement of ions through the channel. Not surprising in a long channel.**Volume control: The Pump-Leak Model**cell volume [Na]i pump rate Na+ is pumped out. K+ is pumped in. So cells have low [Na+] and high [K+] inside. For now we ignore Ca2+(for neurons only!). Cl- equilibrates passively.**Charge and osmotic balance**charge balance osmotic balance • The proteins (X) are negatively charged, with valence zx. • Both inside and outside are electrically neutral. • The same number of ions on each side. • 5 equations, 5 unknowns (internal ionic concentrations, voltage, and volume). Solve and analyze.**Steady-state solution**If the pump stops, the cell bursts, as expected. The minimal volume gives approximately the correct membrane potential. In a more complicated model, one would have to consider time dependence also. And the real story is far more complicated.**RVD and RVI**Okada et al., J. Physiol. 532, 3, (2001)**RVD and RVI**Okada et al., J. Physiol. 532, 3, (2001)**Lots of interesting unsolved problems**• How do organsims adjust to dramatic environmental changes (T. Californicus)? • How do plants (especially in arid regions) prevent dehydration in high salt environments? (They make proline.) • How do fish (salmon) deal with both fresh and salt water? • What happens to a cell and its environment when there is ischemia?**Ion transport**• How can epithelial cells transport ions (and water) while maintaining a constant cell volume? • Spatial separation of the leaks and the pumps is one option. • But intricate control mechanisms are needed also. • A fertile field for modelling. (Eg. A.Weinstein, Bull. Math. Biol. 54, 537, 1992.) The KJU model. Koefoed-Johnsen and Ussing (1958).**Steady state equations**Note the different current and pump models electroneutrality osmotic balance**Transport control**Simple manipulations show that a solution exists if Clearly, in order to handle the greatest range of mucosal to serosal concentrations, one would want to have the Na+ permeability a decreasing function of the mucosal concentration, and the K+ permeability an increasing function of the mucosal Na+ concentration. As it happens, cells do both these things. For instance, as the cell swells (due to higher internal Na+ concentration), stretch-activated K+ channels open, thus increasing the K+ conductance.**Inner medullary collecting duct cells**IMCD cells Real men deal with real cells, of course. Note the large Na+ flux from left to right. A. Weinstein, Am. J. Physiol. 274 (Renal Physiol. 43): F841–F855, 1998.**Active modulation of the membrane potential: electrically**excitable cells**Hodgkin, Huxley, and the Giant Squid Axon**Hodgkin Huxley Don't believe people that tell you that HH worked on a Giant squid axon**The reality**It was a squid giant axon!**Resting potential**• No ions are at equilibrium, so there are continual background currents. At steady-state, the net current is zero, not the individual currents. • The pumps must work continually to maintain these concentration differences and the cell integrity. • The resting membrane potential depends on the model used for the ionic currents. linear current model (long channel limit) GHK current model (short channel limit)**Simplifications**• In some cells (electrically excitable cells), the membrane potential is a far more complicated beast. • To simplify modelling of these types of cells, it is simplest to assume that the internal and external ionic concentrations are constant. • Justification: First, small currents give large voltage deflections, and thus only small numbers of ions cross the membrane. Second, the pumps work continuously to maintain steady concentrations inside the cell. • So, in these simpler models the pump rate never appears explicitly, and all ionic concentrations are treated as known and fixed.**Steady-state vs instantaneous I-V curves**• The I-V curves of the previous slide applied to a single open channel • But in a population of channels, the total current is a function of the single-channel current, and the number of open channels. • When V changes, both the single-channel current changes, as well as the proportion of open channels. But the first change happens almost instantaneously, while the second change is a lot slower. I-V curve of single open channel Number of open channels**2a**a S0 S1 S2 b 2b K+ channel gating S00 S01 S10 S11**2a**a S00 S01 S02 S i j b 2b g d g d g d 2a a inactivation activation S10 S11 S12 b 2b Na+ channel gating activation inactivation**Experimental data: K+ conductance**If voltage is stepped up and held fixed, gKincreases to a new steady level. four subunits rate of rise gives tn steady-state time constant Now fit to the data steady state gives n∞**Experimental data: Na+ conductance**If voltage is stepped up and held fixed, gNaincreases and then decreases. Four subunits. Three switch on. One switches off. steady-state time constant Fit to the data is a little more complicated now, but the same in principle.**Hodgkin-Huxley equations**applied current generic leak activation (increases with V) much smaller than the others inactivation (decreases with V)**An action potential**• gNa increases quickly, but then inactivation kicks in and it decreases again. • gK increases more slowly, and only decreases once the voltage has decreased. • The Na+ current is autocatalytic. An increase in V increases m, which increases the Na+ current, which increases V, etc. • Hence, the threshold for action potential initiation is where the inward Na+ current exactly balances the outward K+ current.**The fast phase plane: I**n and h are slow, and so stay approximately at their steady states while V and m change quickly**The fast phase plane: II**h0 decreasing n0 increasing As n and h change slowly, the dV/dt nullcline moves up, ve and vs merge in a saddle-node bifurcation, and disappear. Vs is the only remaining steady-state, and so V returns to rest.**The fast-slow phase plane**Take a different cross-section of the 4-d system, by setting m=m∞(v), and using the useful fact that n + h = 0.8 (approximately). Why? Who knows. It just is. Thus**Oscillations**When a current is applied across the cell membrane, the HH equations can exhibit oscillatory action potentials.**Where does it go from here?**• Simplified models - FHN, Morris Lecar, Mitchell-Schaffer-Karma… • More detailed models - Noble, Beeler-Reuter, Luo-Rudy, … . • Forced oscillations of single cells - APD alternans, Wenckebach patterns. • Other simplified models - Integrate and Fire, Poincare oscillator • Networks and spatial coupling (neuroscience, cardiology, …)