Control of Cell Volume and Membrane Potential James Sneyd Auckland University, New Zealand Basic reference: Keener and Sneyd, Mathematical Physiology (Springer, 1998)

A nice cell picture

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 problem

Cells carefully regulate their intracellular ionic concentrations, to ensure that no osmotic pressures arise As a consequence, the major ions Na +, K +, Cl - and Ca 2+ 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. Basic solution

Typical ionic concentrations (in mM) Squid Giant AxonFrog Sartorius Muscle Human Red Blood Cell Intracellular Na K+K Cl Extracellular Na K+K Cl

The cell at steady state We need to model pumps and exchangers ionic currents osmotic forces

Osmosis P1P1 P2P2 water water + Solvent (conc. c) At equilibrium: Note: equilibrium only. No information about the flow.

The cell at steady state We need to model pumps and exchangers ionic currents osmotic forces I’ll talk about this a lot more in my next talk. Na,K-ATPase Calcium ATPase

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. Not all that many pump models that I know of. It doesn't seem to be a popular modelling area. I have no idea why.

A Simple ATPase Note how the flux is driven by how far the concentrations are away from equilibrium flux

Reducing this simple model

Na + -K + ATPase (Post-Albers)

Simplified Na + -K + ATPase

The cell at steady state We need to model pumps ionic currents osmotic forces

The Nernst equation 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. [S] e =[S’] e [S] i =[S’] i ViVi VeVe Permeable to S, not S’ (The Nernst potential)

Only very little ion transfer spherical cell - radius 25  m surface area - 8 x cm 2 total capacitance - 8 x  F (membrance capacitance is about 1  F/cm 2 ) If the potential difference is -70 mV, this gives a total excess charge on the cell membrane of about 5 x C. Since Faraday's constant, F, is x 10 4 C/mole, this charge is equivalent to about 5 x moles. But, the cell volume is about 65 x litres, which, with an internal K + concentration of 100 mM, gives about 6.5 x 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 is the crucial question

How to model I ionic Many different possible models of I ionic 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 GHK model Linear 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 g Na (or P Na ) are not constant, but are functions of voltage and time. More on this later.)

Electrodiffusion: deriving current models Boundary conditions Poisson equation and electrodiffusion Poisson-Nernst-Planck equations. PNP equations.

The short-channel limit If the channel is short, then L ~ 0 and so ~ 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/ ~ 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.

A Model of Volume Control Putting together the three components (pumps, currents and osmosis) gives.....

The Pump-Leak Model Na + is pumped out. K + is pumped in. So cells have low [Na + ] and high [K + ] inside. For now we ignore Ca 2+ (horrors!). Cl - just equilibrates passively. cell volume [Na] i pump rate Note how this is a really crappy pump model

Charge and osmotic balance charge balance osmotic balance The proteins (X) are negatively charged, with valence z x. 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). Just solve.

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)

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 A. Weinstein, Am. J. Physiol. 274 (Renal Physiol. 43): F841–F855, IMCD cells Real men deal with real cells, of course. Note the large Na + flux from left to right.

Active modulation of the membrane potential: electrically excitable cells

Hodgkin, Huxley, and squid Don't believe people that tell you that this is a small squid HodgkinHuxley

The reality

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 just to assume that the internal and external ionic concentrations are constant. Justification: Firstly, it takes only small currents to get large voltage deflections, and thus only small numbers of ions cross the membrane. Secondly, 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

Example: Na + and K + channels

K + channel gating S0S0 S1S1 S2S2 22 22 S 00 S 01 S 10 S 11

Na + channel gating 22 22 S 00 S 01 S 10 S 11 S 02 S 12  22 22  S i j inactivationactivation inactivation

Experimental data: K + conductance If voltage is stepped up and held fixed, g K increases to a new steady level. time constant steady-state four subunits Now just fit to the data rate of rise gives  n steady state gives n ∞

Experimental data: Na + conductance If voltage is stepped up and held fixed, g Na increases and then decreases. time constant steady-state Four subunits. Three switch on. One switches off. Fit to the data is a little more complicated now, but still easy in principle.

Hodgkin-Huxley equations generic leak applied current much smaller than the others inactivation (decreases with V) activation (increases with V)

An action potential g Na increases quickly, but then inactivation kicks in and it decreases again. g K 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.

Basic enzyme kinetics

Law of mass action Given a basic reaction A + B C k1k1 k -1 we assume that the rate of forward reaction is linearly proportional to the concentrations of A and B, and the back reaction is linearly proportional to the concentration of C.

Equilibrium Equilibrium is reached when the net rate of reaction is zero. Thus or This equilibrium constant tells us the extent of the reaction, NOT its speed. change in Gibb’s free energy

Enzymes Enzymes are catalysts, that speed up the rate of a reaction, without changing the extent of the reaction. They are (in general) large proteins and are highly specific, i.e., usually each enzyme speeds up only one single biochemical reaction. They are highly regulated by a pile of things. Phosphorylation, calcium, ATP, their own products, etc, resulting in extremely complex webs of intracellular biochemical reactions.

Basic problem of enzyme kinetics Suppose an enzyme were to react with a substrate, giving a product. S + EP + E If we simply applied the law of mass action to this reaction, the rate of reaction would be a linearly increasing function of [S]. As [S] gets very big, so would the reaction rate. This doesn’t happen. In reality, the reaction rate saturates.

Michaelis and Menten In 1913, Michaelis and Menten proposed the following mechanism for a saturating reaction rate S + E k1k1 k -1 C k2k2 P + E Complex. product Easy to use mass action to derive the equations. There are conservation constraints.

Equilibrium approximation And thus, since Thus reaction velocity

Pseudo-steady state approximation And thus, since Thus reaction velocity Looks very similar to previous, but is actually quite different!

Basic saturating velocity s V V max KmKm V max /2

Lineweaver-Burke plots Plot, and determine the slope and intercept to get the required constants.

Cooperativity S + E k1k1 k -1 C1C1 k2k2 P + E S + C 1 k3k3 k -3 C2C2 k4k4 P + E Enzyme can bind two substrates molecules at different binding sites. or EC1C1 C2C2 E E SS S S P P

Pseudo-steady assumption Note the quadratic behaviour

Independent binding sites EC1C1 C2C2 E E SS S S P P 2k + k+k+ 2k - k-k- Just twice the single binding rate, as expected

Positive/negative cooperativity Usually, the binding of the first S changes the rate at which the second S binds. If the binding rate of the second S is increased, it’s called positive cooperativity If the binding rate of the second S is decreased, it’s called negative cooperativity.

Hill equation In the limit as the binding of the second S becomes infinitely fast, we get a nice reduction. Hill equation, with Hill coefficient of 2. This equation is used all the time to describe a cooperative reaction. Mostly use of this equation is just a heuristic kludge. VERY special assumptions, note.

Another fast equilibrium model of cooperativity EC1C1 C2C2 E E SS S S P P Let C=C 1 +C 2 k -1 k1k1 k3k3 k -3 k2k2 k4k4 S + E k1k1 k -1 C  s) P + E

Monod-Wyman-Changeux model A more mechanistic realisation of cooperativity.

Equilibrium approximation Don’t even think about a pseudo-steady approach. Waste of valuable time. which gives occupancy fraction and so on for all the other states Note the sigmoidal character of this curve

Reversible enzymes Of course, all enzymes HAVE to be reversible, so it’s naughty to put no back reaction from P to C. Should use S + E k1k1 k -1 C k2k2 P + E k -2 I leave it as an exercise to calculate that

Allosteric modulation substrate binding inhibitor binding at a different site this state can form no product (Inhibition in this case, but it doesn’t have to be) X Y Z

Equilibrium approximation X YZ Could change these rate constants, also. Inhibition decreases the V max in this model