computational neuroscience. session 2-1roquesol/computational...depending on the cell type: 2 16mv...

Dynamics of Passive membrane

Computational Neuroscience. Session 2-1

Dr. Marco A Roque Sol

06/04/2018

Dr. Marco A Roque Sol Computational Neuroscience. Session 2-1

Dynamics of Passive membraneThe Resting Potential.Nernst-Planck equation; Nernst equation; GHK equation.

The Resting Potential.

All living cells exhibit an electrical potential differencebetween the inner and outer surface of the cytoplasmicmembrane. This potential difference is referred to as themembrane potential.

In mathematical terms, the membrane potential VM isdefined as

VM = Vin − Vout

where V in is the potential on the inside of the cell and Vout is the potential on the outside. This will change duringan action potential, for example.

VM = Vin − Vout

Varies among different cell types, its value is typicallybetween −30mV and −90mV

Reasons: Ion concentration; differences on the two sidesof the membrane;the permeability of the membrane for thedifferent species of monovalent ions is different(impermeable for larger protein and phosphate-anions)

Factors determining the resting potential (Vm):

Diffusion Potential- Most significant contribution

Donnan* Potential- Due to the presence ofnon-permeable-protein anions in the cytosol negligiblecontribution, cells continuously fight against it(thermodynamic equilibrium potential)

Pump Potential(Na+/K+ ATP-ase)Depending on the cell type: 2 − 16mV (direct contribution)

( * Frederick George Donnan (6 September 1870 - 16December 1956) was an Irish physical chemist )

The resting potential refers to the potential across themembrane when the cell is at rest. A typical neuron has aresting potential of about −70mV .

An inward current corresponds to a positively charged ion,such as Na+ , entering the cell. This raises the membranepotential; that is, it brings the membrane potential closer tozero. In this case, the cell is said to be depolarized

An outward current corresponds to a positively charged ion( cation ), such as K+, leaving the cell or a negativelycharged ion ( anion ), such as Cl−, entering the cell. Inthis case, the cell becomes hyperpolarized.

The potential difference arises from differences in theconcentrations of various ions within and outside the cell.

The maintenance of the potential difference also involvesthe transport of ions across the cell membrane and theselective permeability of the membrane to these ions.

The principal ions found on either side of the cellmembrane are Na+,K+, and Cl−. The concentration ofK+ ions inside a cell is about 10 times that in theextracellular fluid, whereas the concentrations of Na+ andCl− are much higher outside the cell than inside.

The lipid bilayer of the cell membrane is a poor conductorof ionic current because it is not permeable to ions.However, the membrane does contain channel proteinsthat allow for the ions to move through it.

There are two types of ion channels in the membrane:gated and nongated. Nongated channels are always open,whereas gated channels can open and close and theprobability of opening often depends on the membranepotential; these are referred to as voltage-gated channels.

The permeability of the membrane to a particular iondepends on the number of open channels selective for thation. Most gated channels are closed at rest; hence, thenongated ion channels are primarily responsible forestablishing the resting potential.

Because of concentration differences, when theappropriate channels are open, Na+ and Cl− ions tend todiffuse into the cell, whereas K+ ions tend to diffuseoutward.

Note that ions do not simply diffuse in or out of an openchannel until the concentration of that ion on either side ofthe cell is zero.

This is because of the electric field created by separation ofpositive and negative charges across the cell membrane.

Suppose, for example, the cell is permeable only to K+ .The concentration gradient of K+ moves K+ ions out ofthe cell.

However, the continued efflux of K+ builds up an excess ofpositive charge on the outside of the cell and leaves behindan excess of negative charge on the inside.

The Resting Potential.The negative charge consists mostly of impermeableorganic anions A−. This buildup of charge acts to impedethe further efflux of K+ , so eventually an equilibrium isreached.

At this equilibrium, the electrical and chemical drivingforces are equal and opposite (See Figure).

The membrane potential at which K+ ions are inequilibrium across the membrane is called the K+-Nernst,equilibrium, or reversal potential .

The membrane potential at which K+ ions are inequilibrium across the membrane is called the K+-Nernst,equilibrium, or reversal potential .Dr. Marco A Roque Sol Computational Neuroscience. Session 2-1

Nernst-Planck equation; Nernst equation.; GHKequation.

The first step in deriving these equations is to derive theNernst-Planck equation.

In what follows, let [C](x) be the concentration of some ionand V (x) the potential at the point x across themembrane. Then, Fick’s law* of diffusion says that thediffusive flux, Jdiff , is given by

Jdiff = −D∂[C]

(* Adolf Eugen Fick [3 September 1829 - 21 August 1901]was a German-born physician and physiologist. )

Jdiff = −D∂[C]

The diffusion constant, D, has units of square centimetersper second and the concentration [C], is in molecules percubic centimeter, so the diffusive flux has units ofmolecules per square centimeter second. (Think of the fluxas movement across the two-dimensional cell surface.)The direction of movement is from high concentrations tolow concentrations.

The other physical force that is responsible for the passivemovement of ions is the electrical drift described by themicroscopic version of Ohm’s law *:

Jdrift = −µz[C]∂V∂x

(* Georg Simon Ohm: [16 March 1789 - 6 July 1854] was aGerman physicist and mathematician. )

The electric field, E = ∂V /∂x , is the gradient of thepotential V (measured in volts) and thus has units of voltsper centimeter. z is the valence (of an element, is ameasure of its combining power, with other atoms when itforms chemical compounds or molecules.) of the ion (±1;±2; etc.). The parameter µ is the mobility and hasdimensions of square centimeters per volt second and [C]

is the concentration.

The electric field, E = ∂V /∂x , is the gradient of thepotential V (measured in volts) and thus has units of voltsper centimeter. z is the valence (of an element, is ameasure of its combining power, with other atoms when itforms chemical compounds or molecules.) of the ion (±1;±2; etc.). The parameter µ is the mobility and hasdimensions of square centimeters per volt second and [C]

is the concentration.

The total flux across the membrane is given by the sum ofthe diffusive flux and the electrical drift:

Jtotal = −D∂[C]

∂x− µz[C]

∂V∂x

Einstein’s relation ( Kinetic Theory of Gases )* connectsthe mobility with the diffusion coefficient:

D =kTq

(* Einstein, A. (1905). “Über die von dermolekularkinetischen Theorie der Wärme geforderteBewegung von in ruhenden Flüssigkeiten suspendiertenTeilchen”. Annalen der Physik )

Jtotal = −D∂[C]

∂x− µz[C]

∂V∂x

D =kTq

Jtotal = −D∂[C]

∂x− µz[C]

∂V∂x

D =kTq

Jtotal = −D∂[C]

∂x− µz[C]

∂V∂x

D =kTq

Jtotal = −D∂[C]

∂x− µz[C]

∂V∂x

D =kTq

Jtotal = −D∂[C]

∂x− µz[C]

∂V∂x

D =kTq

where k is Boltzmann’s constant (J/K ), T is the absolutetemperature, and q is the charge (measured in coulombs).Thus, we can write the total flux as

Jtotal = −µkTq

∂[C]

∂x− µz[C]

∂V∂x

It is convenient to convert this equation, which is in terms of thenumber of individual molecules, into its molar equivalent, bydividing by Avogadro’s number * ( NA)

(* Amedeo Carlo Avogadro,Count of Quaregna and Cerreto [9August 1776 - 9 July 1856], was an Italian scientist, most notedfor his contribution to molecular theory now known asAvogadro’s law. )

Jtotal = −µkTq

∂[C]

∂x− µz[C]

∂V∂x

Jtotal = −µkTq

∂[C]

∂x− µz[C]

∂V∂x

Jtotal = −µkTq

∂[C]

∂x− µz[C]

∂V∂x

Jtotal = −µkTq

∂[C]

∂x− µz[C]

∂V∂x

NA = 6.0221409 × 1023 ( number of constituent particles,usually atoms or molecules, that are contained in theamount of substance given by one mole ).It is also convenient to introduce RT /F , where R is the idealgas constant and F is Faraday’s constant*, instead of kT /q

(* Michael Faraday: [ 22 September 1791 - 25 August 1867 ]was an English scientist who contributed to the study ofelectromagnetism and electrochemistry. )

This will yield the flux per mole. Multiplying this flux by thevalence and Faraday’s constant yields a current flux

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

Here, u = µ/NA is the molar mobility, and the current I ismeasured in amperes. This equation is the Nernst-Planckequation*.

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

Ernst-Planck equation; Ernst equation.; GK equation.

(* Walther Hermann Ernst: [ 25 June 1864 - 18 November1941] was a German chemist who is known for his work inthermodynamics; his formulation of the Ernst heat theoremhelped pave the way for the third law of thermodynamics, forwhich he won the 1920 Nobel Prize in Chemistry. )

(* Max Karl Ernst Ludwig Planck: [23 April 1858 - 4 October1947] was a German theoretical physicist whose discovery ofenergy quanta won him the Nobel Prize in Physics in 1918. )

The Ernst equation is obtained by setting the currentequal to zero. That is, for a given ionic species, atequilibrium, the diffusion and electric effects balance:

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

And as a result ( the above is a first order separableOrdinary Differential Equation ), this implies the Ernstequation:

Veq = Vin − Vout = −RTzF

ln[C]in[C]out

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

ln[C]in[C]out

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

ln[C]in[C]out

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

ln[C]in[C]out

I = −(

uzRT∂[C]

∂x+ uz2F [C]

∂V∂x

ln[C]in[C]out

That is, the equilibrium or Ernst potential, which occurswhen all the fluxes balance, depends on the logarithm ofthe ratio of the concentrations of the ions inside andoutside the cell.

At room temperature, 37◦C, RT /F = 26.73mV . Hence,the equilibrium potential of potassium is ( in a typicalmammalian cell, [C]in = 140mMK+ inside the cell and5mMK+ outside ).

−26.73ln140

5= −89.7mv

−26.73ln140

5= −89.7mv

−26.73ln140

5= −89.7mv

−26.73ln140

5= −89.7mv

The Ernst-Planck equation describes the movement ofcharged ions in aqueous media. However, the cellmembrane has thickness and there may be energy barriersor blocking sites within the channel.

In this case, the ions flowing through the open channelmay not obey the Ernst-Planck equation and we mustmodel the complex behavior within the membrane to get atrue picture of the flux across the cell.

Goldman,Hodgkin, and Katz came up with this simplifiedmodel called the constant-field equation ( a simplifiedversion of the most general case).

They assumed (1) the electric field across the lipidmembrane is constant, (2) the Ernst-Planck equation holdswithin the membrane, and (3) the ions all moveindependently.

Let VM be the total potential across a membrane of width land let V (x) be the potential at the point x across themembrane.

Since the electric field is constant, E = −VM /l . Thisimplies that dV /dx = −VM /l .

The mobility of ions within the membrane will be differentfrom that in the aqueous solution; denote this mobility byu∗. Finally, let β be the ratio of the ion solubility within themembrane to the ion solubility in the aqueous solution.Thus, if [C] is the aqueous concentration, then [β]is themembrane concentration.

With these assumptions, the Ernst-Planck equation forcurrent across the membrane is

I = −u∗z2F β[C]VM

l− u∗zRT β

dx; 0 < x < l

subject to the two boundary conditions