Membranes and Transport

Topic II-1

Biophysics

Nernst Equation

i

o

i

oB

CCln

F

RTC

Clnq

TkV

i

o

C

Clog mV 60Vget K 300 At

• F = 96,400 Coulomb/mole

•Simplest equation for membrane potential – one ion

www.kcl.ac.uk/teares/gktvc/vc/lt/nol/Nernst.htm

Goldman-Hodgkin-Katz Equation

• There is one membrane potential that counters all concentration gradients for permeable ions

• Pi = (ikBT)/d = Di/d, with D – diffusion constant [cm2/sec]

1i i

ii2ii

2i i

ii1iiB

CPCP

CPCPln

q

TkV

oCliNaik

iCloNaokB

ClPNaPKP

ClPNaPKPln

q

TkV

Example – simple Neuron

• Note: Can define Nernst potential for each ion, Vk = -80 mV; VNa = +58 mV. Relative permeabilities make membrane potential closer to Vk

ii

oo

NabK

NabKlog mV 58V

b = 0.02 for many neurons (at rest).

[K]i = 125 mM [K]o = 5 mM

[Na]i = 12 mM [Na]o = 120 mM

What is V?

with b = PNa/PK

Do Soma 1 (Nernst)

Electrical Model

g is like conductance (=1/R) and like permeability

This equation is equivalent to Godman-Hodgkin-Katz equation.

CaNaKCl

CaCaNaNaKKClCl

i

iim gggg

VgVgVgVg

g

VgV

(from Kirchoff’s laws)

Example squid axon

• Remember electric field points in direction of force on positive test charge

• E always points from higher potential (for ions)

[K]in =125mM

[K]out = 5 mM

Vm = -60 VK = -75

IK = gK (Vm – VK), Vm = -60 mV

IK = gK (-60 – (-75)) mV = gK(+15 mV).

g always positive.

V = Vin – Vout

positive current = positive ions flowing out of the cell.

Vm not sufficient to hold off K flow so ions flow out. When Vm = Vk then no flow.

thisopposes potentialNernst ;sdEVVVin

outoutinions

[K]in =125 mM

[K]out = 5 mM

E Eds

Do Soma 3 (Resting Potential)

Donnan Rule and other considerations Example of two permeable ions and one

impermeable one inside

• KoClo = KiCli Donnan Rule

• Electroneutrality

• Osmolarity

• Goldman- Hodgkin-Katz

• Apply electroneutrality inside and out and plug in Donnan rule and get

AZZAK4

K2log mV 58

[K]

[K]log mV 58 V 2/1222

o

o

i

om

Animal Cell ModelCi (mM)* Co (mM) P>0?

K+ 125 5 Y

Na+ 12 120 N**

Cl- 5 125 Y

A- 108 0 N

H2O 55,000 55,000 Y

* Should really use Molality (moles solute/ kg solvent) instead of per liter – accounts for how molecules displace water (non-ideality).

** More on this later

Maintenance of Cell Volume

• Osmolarity must be same inside and out• Concentration of permeable solutes must be

same inside and out• Si = So and Si + Pi = So (Osmolarity)• Solutions: cell wall, Pwater = 0, Pextra cellular solutes = 0

Cell impermeable to sucrose

www.lib.mcg.edu/.../section1/1ch2/s1ch2_25.ht

http://www.himalayancrystalsalt.com/html/images/PAGE-osmosis.gif

Animal Na impermeable model

• Apply electroneutrality outside, Donnan, and osmolarity

• Get unknowns and Vm = -81 mV

Active Transport

• Na-K Pump

- Two sets of two membrane spanning subunits- Phosphorylation by ATP induces a conformational change in the protein allowing pumping- Each conformation has different ion affinities. Binding of ion triggers phosphorylation. - Shift of a couple of angstroms shifts affinity.- Exhibits enzymatic behavior such as saturation.

student.ccbcmd.edu/.../eustruct/sppump.html

ii

ooom

iNaik

oNaokBm

NaK

NaKlnVV

NaPKP

NaPKPln

q

TkV

with = (n/m)(PNa/PK),n/m = 2/3 Vm VK.

Electrical Model

CaNaKCl

pCaCaNaNaKKClClm gggg

IIVgVgVgVgV

Now include current for pump, Ip as well as input current I.

Do Soma conductance and Na pump)

Patch Clamping

www.essen-instruments.com/Images/figure2.gif

• Invented by Sakmann and Neher [Pflugers Arch 375: 219-228, 1978]

•Can be used for whole cell clamp (measure currents in whole cell, placing electrode in cell) like on left or pulled patch as on right (potentially measure single channel).

•Can control [ions].

•Usually voltage clamp (command voltage or holding voltage) and observe current (I = gV). Ix = g(Vh-Vx) where x is for each ion and Vx is Nernst potential for that ion. With equal concentration of permeable ion on both sides, get g easily

Voltage Gated Channels

• When [ions] not limiting, can get nernst potential when current reverses

I = gx*(Vh – Vx)

• There is a degree of randomness in opening and closing of channels

•Proportion of time open is proportional to Voltage for some channels

•Average of many channels is predictable

Multiple Channels

• Get several channels on a patch– Gives quantized currents

• Parallel: geq = gi; Series: 1/geq = 1/gi

– g = 1/R

I

t

Ligand gated channels

• When Ach binds, gate opens and lets in Na+ and K+.

• I is proportional to [Ach]2 (binds 2 Ach)

Nicotine also binds to Ach receptor – called nicotinic receptor

Do Patch Cl, K, Multiple K, ligated)

Facilitated Transport

• Get Saturation Kinetics

• Lower activation energy

Jmax = NYDY/d2

NY = number of carriersDY = diffusion constant of Carrierd = membrane thickness

i

ii

o

oo

CY

YC

CY

YCK

i

i

o

omax CK

C

CK

CJJ

Co

Jmax