Control of Cell Volume and Membrane PotentialJames SneydAuckland University, New ZealandBasic reference: Keener and Sneyd, Mathematical Physiology (Springer, 1998)

A nice cell picture

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.

Typical ionic concentrations (in mM)

Squid Giant AxonFrog Sartorius MuscleHuman Red Blood CellIntracellularNa+501319K+397138136Cl-40378ExtracellularNa+437110155K+202.55Cl-55690112

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

OsmosisP1P2waterwater +Solvent(conc. c)At equilibrium:Note: equilibrium only. No information about the flow.

The cell at steady stateWe need to model pumps and exchangers ionic currents osmotic forcesIll talk about this a lot more in my next talk.Na,K-ATPaseCalciumATPase

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 ATPaseNote how the flux is driven by how far the concentrations are away from equilibriumflux

Reducing this simple model

Na+-K+ ATPase (Post-Albers)

Simplified Na+-K+ ATPase

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

The Nernst equationNote: 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.(The Nernst potential)

Only very little ion transferspherical cell - radius 25 mmsurface area - 8 x 10-5 cm2total 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 membraneHow to model this is the crucial question

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 modelsGHK modelLinear modelThese 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 modelsBoundary conditionsPoisson equation andelectrodiffusionPoisson-Nernst-Planck equations. PNP equations.

The short-channel limitIf 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 thechannel.

The long-channel limitIf 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 ofions through the channel. Not surprising in a long channel.

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

The Pump-Leak ModelNa+ is pumped out. K+ is pumped in. So cells have low [Na+] and high [K+] inside. For now we ignore Ca2+ (horrors!). Cl- just equilibrates passively.cell volume[Na]ipump rateNote how this is a reallycrappy pump model

Charge and osmotic balancecharge balanceosmotic 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). Just solve.

Steady-state solutionIf 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 RVIOkada 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 equationsNote the different current and pump modelselectroneutralityosmotic balance

Transport controlSimple manipulations show that a solution exists ifClearly, 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.

IMCD cellsInner medullary collecting duct cellsA. Weinstein, Am. J. Physiol. 274 (Renal Physiol. 43): F841F855, 1998.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 squidDon't believe people thattell you that this is a smallsquidHodgkinHuxley

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 singleopen channelNumber of open channels

Example: Na+ and K+ channels

K+ channel gatingS00S01S10S11

Na+ channel gatingactivationinactivation

Experimental data: K+ conductanceIf voltage is stepped up and held fixed, gK increases to a new steady level.time constantsteady-statefour subunitsNow just fit to the datarate of rise gives tnsteady state gives n

Experimental data: Na+ conductanceIf voltage is stepped up and held fixed, gNa increases and then decreases.time constantsteady-stateFour subunits.Three switch on.One switches off.Fit to the data is a little more complicated now, but still easy in principle.

Hodgkin-Huxley equationsgeneric leakapplied currentmuch smaller thanthe othersinactivation(decreases with V)activation(increases 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.

Basic enzyme kinetics

Law of mass actionGiven a basic reactionA + BCk1k-1we 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.

EquilibriumEquilibrium is reached when the net rate of reaction is zero. ThusorThis equilibrium constant tells us the extent of the reaction, NOT its speed.change in Gibbsfree 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 kineticsSuppose an enzyme were to react with a substrate, giving a product.S + EP + EIf 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 wouldthe reaction rate.

This doesnt happen. In reality, the reaction rate saturates.

Michaelis and MentenIn 1913, Michaelis and Menten proposed the following mechanism for a saturating reaction rateS + E k1k-1Ck2P + EComplex. product Easy to use mass action to derive the equations. There are conservation constraints.

Equilibrium approximationAnd thus, sinceThusreaction velocity

Pseudo-steady state approximationAnd thus, sinceThusreaction velocityLooks very similar to previous, but is actually quite different!

Basic saturating velocitysVVmaxKmVmax/2

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

CooperativityS + E k1k-1C1k2P + ES + C1 k3k-3C2k4P + EEnzyme can bind two substrates molecules at different binding sites.orEC1C2EESSSSPP

Pseudo-steady assumptionNote the quadraticbehaviour

Independent binding sitesEC1C2EESSSSPP2k+k+2k-k-Just twice the single binding rate, as expected

Positive/negative cooperativityUsually, 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, its called positive cooperativity If the binding rate of the second S is decreased, its called negative cooperativity.

Hill equationIn the limit as the binding of the second S becomes infinitely fast, we get a nice reduction.Hill equation, withHill 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 ofcooperativityEC1C2EESSSSPPLet C=C1+C2k-1k1k3k-3k2k4S + E k1k-1Cf(s)P + E

Monod-Wyman-Changeux modelA more mechanistic realisation of cooperativity.

Equilibrium approximationDont even think about a pseudo-steady approach. Waste of valuable time.which givesoccupancy fractionand so on for all the other statesNote the sigmoidal character of this curve

Reversible enzymesOf course, all enzymes HAVE to be reversible, so its naughty to put no back reaction from P to C. Should useS + E k1k-1Ck2P + Ek-2I leave it as an exercise to calculate that

Allosteric modulationsubstrate bindinginhibitorbinding at adifferent sitethis state canform no product(Inhibition in this case, but it doesnt have to be)XYZ

Equilibrium approximationXYZCould change these rate constants, also.Inhibition decreases theVmax in this model