Biophysics Notes Membranes and Transport

Membrane Biophysics

  1. Nernst Equation

Gconc = kBT ln(Co/Ci)

Gvoltage = qV,

at equilibrium, qV = kBT ln(Co/Ci) 

V = kBT/q ln(Co/Ci) ***

= RT/F ln(Co/Ci), with F = 96,400 Coulomb/mole (kB = 1.38 x 10-23 J/K, R = 8.31 J/mol.K)

  1. Goldman-Hodgkin-Katz equation

V = , [P] = cm/s, for example:

V =

Example: Approximate Neuron:

V = , with b = PNa/PK

b = 0.02 for many neurons (at rest).

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

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

V = -71 mV

Can define Vk = 58 mV log([K]o/[K]i) = -80 mV (this is at 298 K)

VNa = + 58 mV. The value of b  Vmembrane closer to Vk.

Straight Line PNa = 0 (Nernst), Curved fit is using Goldman-Hodgkin-Katz, vary Ko and measure Vm.

Do Soma 1 (Nernst) ,

  1. Electrical Model
VNa (mV) / VK (mV) / VCl (mV) / VCa (mV)
Mammalian Cells / +67 / -84 / -60 / +125
Squid Axon / +55 / -75 / -60 / +125

Analyze Circuit with kirchoff’s laws (or Ohm’s law)

1) Iin =  Iout

2)V around loop = 0

Let input current, I = 0 (No net current)

ICl + ICa + INa + IK = 0

For each loop have

Ii = 1/Ri (Vm – Vi) w/ Vm = membrane potential difference

Let 1/R = g (like conductance)

Eg. IK = gK (Vm – VK), squid axon, 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.

Kirchoff’s rules  Ii = 0  gi (Vm – Vi) = 0

Vm = , which is equivalent to Goldman-Hodgkin-Katz equation.

Do Soma 3

  1. Donnan Equilibrium

Say have two equally permeable ions: K+ and Cl- and have A with charge Z inside cell w/ no permeability

Vk = 58 mV log([K]o/[K]i)

VCl = -58 mV log([Cl]o/[Cl]i),

At equilibrium, Vm = VK = VCl log([K]o/[K]i) = -log([Cl]o/[Cl]i)

KoClo = KiCli Donnan Rule (Dropping [])

[product permeable inons outside] = [product permeable ions inside]

Other conditions:

electroneutrality

osmolarity

Goldman- Hodgkin-Katz

For our simple system, electroneutrality  Ko = Clo, Ki = Cli + ZA

Putting this is the Donnan rule 

Ko2 = KiCli = Ki(Ki – ZA)

Ki2 – ZAKi = Ko2 [complete square]

(Ki – AZ/2)2 – (AZ/2)2 = Ko2

Ki = (Ko2 + (AZ/2)2)1/2 +AZ/2

2Ki = (4Ko2 + A2Z2)1/2 + AZ

Vm = 58 mV log([K]o/[K]i) = 58 mV log

Both K+ and Cl- are at equilibrium at this unique potential.

  1. Animal Cell Model
  2. Model

Ci (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

  1. Maintenance of Cell Volume.

Membrane evolved to keep stuff in.

a)Osmolarity used to define [H2O] – add sugar to H20 and [H2O] goes down since V increases.

Solution of 1 mole/liter of dissolved particles is 1 osmolar

As osmolarity increases, [H2O] decreases

1 M NaCl = 2 Osmolar


Osmosis – diffusion of H2O down concentration gradient

N.B. In preparing Hb, put RBC in dI water, H2O goes in , not everything permeable  lysis.

b)Cell Volume.

Ex - Keeping P inside  osmolarity inside

Not same as that outside  H2O flows in  Lysis.

Si = So (Diffusion – equate chemical potential)

Si + Pi = So (Osmolarity)

How solve this paradox?

Note (this same thing happens in dialyzing proteins).

c)How keep P in and not lyse cells?

1)Pwater = 0 – hard to do but some epithelial cells do it.

2)Cell wall – no swelling = plants and bacteria

3)Make membrane exclude extracellular solutes

Tonicity: isotonic – no effect of volume

hypotonic – cause swelling

hypertonic – cause shrinking

Consider if in the diagram above we now made the cell also impermeable to S, then the problem would be solved when Si = So.

  1. Impermeable Sodium Model

Apply electroneutrality (for permeable ions)

Donnan Equilibrium

Osmotic balance

Electroneutrality  Clo = 125mM

Donnan  KoClo = KiCli (5)(125) = Ki5  Ki = 125 mM

Osmolarity  outside – 250 mOsm  need inside = 250 mOsm

250 = 125 + Nai + 108 + 5  Nai = 12 mOsm

VK = -25 mV ln(125/5) = -81 mV

VCl = -25 mV ln(125/5) = -81 mV

Vm = -25mV ln[(125+125)/10] = -81 mV

  1. Diffusion and Active Transport
  2. Pi = (ikBT)/d = Di/d, with D – diffusion constant [cm2/sec]
  3. In reality, cell is permeable to sodium, but pumps it out.

Had (Goldman-Hodgkin-Katz)

Vm =

Now, with pump 

Vm = Vo, with  = (n/m)(PNa/PK),

n/m = 2/3  Vm VK.

  1. 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.

  1. Electrical Model


Now include pump:

Vm =

Do Soma exercises 4 and 5 (start).

  1. Patch Clamping
  1. Technique

-Measure individual ion channels

-Elongate Pipette

-Can pull patch away or not detach it at all

-Usually maintain V fixed across membrane –

See if channel open or closed

-observe current is quantized: channel

is either open or closed: easy to get

conductivity, I = gV.

Vh = holding potential, with equal concentration of permeable ion on both sides, get g

When have not equal concentration of permeable ions on both sides get Nernst potential, Vx

Ix = gx (Vh – Vx)

  1. Voltage Gated Channels

-there exist randomness in the opening and closings

-For some channels, the proportion of time the channel is open is dependent on the membrane potential (eg. Na, K channels of neurons).

-Average of many channels is predictable.

  1. Reversal potential for voltage gated channels

-Current depends on [ions] in addition to Vm (eg. [ions] low analogy to a resistor breaking down)

-When ions not too low, get VNernst I = gx*(Vh – Vx)

-When Vh = Vx No I  “reversal potential”

-Can reverse I with Vh

-Can measure Vh

-Can test selectivity of ions.

  1. Multiple channels

Can get several on a patch

I

t

Parallel Resistors:

1/Req =  1/Ri geq =  gi

Series:

1/geq =  1/gi

  1. Ligand gated channels

n acetylcholine (nAchR) – closed until opened by binding

Ach  bind  let both Na and K pass

Not sensitive to Vm

I [Ach]2

Do Patch 1,2,4,5

  1. Carrier Transport – example of facilitated transport.
  2. Lowers activation energy
  3. Obeys saturation kinetics
  4. Highly selective

Simple Model:

Solute Flux

J = Jmax

Jmax = NYDY/d2

NY = number of carriers

DY = diffusion constant of Carrier

d = membrane thickness

For Ci = 0, get J = Jmax [Co/(K + Co)]

Biophysics Notes

Nerve Excitaion

Nerve Excitation

  1. General Background
  2. Basics

a)Nerve Signals communicate stimulus-response mechanism

b)Nerve signals = electrical signals i.e. modulations of membrane potential

c)Patellar reflex – Knee jerk reflex. Tap Knee  stretch thigh muscle  sensory neuron  spinal cord  motor neuron  muscle contraction.

d)Motor vs sensory neurons

Motor – cell body = soma (nucleus contained)

Small – 20 –30 mm, has branched (dendrites) which receive signals

Sensory – no dendrites

  1. Historical Perspective

a)Couldn’t measure electrical signals of individual cells – too small

b)J.Z. Young rediscovers Giant Squid Axon – 1 mm diameter

c)Already Known:

there exist action potentials – electrical pulses that don’t travel like electrical current (much slower: 1 – 100 m/s vs 3 x 108 m/s)

K+, Na+ concentrations play an important role, esp. Na+

Hodgkin and Katz: as [Na+] decreases the velocity of the action potential also decreases.

H & H, 1963 – membrane potential reverses during impulse, goes to + 100 mV

Electrical conductance of membrane increases 40x – H&H propose nerve impulses are due to transient changes in Na and K conductance: Nobel Prize in 1963.

  1. Voltage Clamp

a)They used voltage clamp developed by KS Cole 1940, FBA keeps V constant

b)Supply I to keep V constant

c)Provides experimental control

Vo = K(Vc – Vm), K = amplifier gain, Vc = control voltage, Vo = output voltage, Vm = membrane voltage

  1. Membrane Analog

-Space clamped axon – two silver wires inserted all the way: radial currents only

Generally (w/o a clamp)

Cm = membrane capacitance

gNa, gk are now variable

gL = leak current – not specific

INa = gNa(Vm – VNa), IK = gK(Vm -VK), IL = gL(Vm – Vk)

For Cm have diplacement current (rearrangement of charges)

IC = Cm dV/dt

-VNa, VK, VL all determined by [ ]i and [ ]o, which along with Cm are fixed

gNa and gK are variable.

Itot = INa + IK + IL + Cm dV/dt; Voltage clamp dV/dt = 0

Actual clamping is complicated – often need to step potential via conditioning steps.

Notes:

Negative current: + ions into axon

Positive current: + ions out of axon

VK = -72 mV, VNa = + 55 mV

V = Vin - Vout

Vm = Vh = 1/gtot (gKVK + gNaVNa + gLVL + I), clamp and look at I

  1. Voltage Clamp experiments
  1. gL, H&H found gNa, gK ~ 0 at resting potential and they are only activated when the axon is depolarized (becomes less negative). They found gL by hyperpolarizing (making more negative).

I = IL = gL (Vm – VL), VL < Vresting (-60 mV)

  1. Voltage steps – depolarizing V Do overview at end briefly first

Depolarization  early negative current  late positive current

As V increases

The amplitide of the negative current decreases

At V = 117, (Vm = 57 mV), I neg = 0

If V > 117 mV  get early positive current

As V increases

Late positive current increases monotonically

As V increases

Rate of current development increases (+ and -)

As V increases

Switch from negative to positive current gets earlier (channels open earlier)

Do Axon 1 and 2&3 Dynamics and Voltage clamp currents.

  1. Separating INa and IK

a.

Observe

VK = -72 mV, VNa = +55 mV, Vresting = -60mV

INa = gNa(Vm – VNa), IK = gK(Vm – VK)

Near Resting

IK - always positive, INa – negative, for small, medium depolarizations decreasing as V increases, becoming positive for large V

b.

Separation

I reversal could be due to larger later (positive) current or cessation of early (negative) current.

To solve this, H&H make Vm = VNa so all I is IK

They then varied [Na]o, set Vm = VNa and hence studied the voltage dependence of IK

Then deduced voltage dependence of INa since Itot = INa + IK

Do Axon 4– Inactivation.

  1. Na inactivation and de-inactivation
  1. V  activation , time inactivation  deinactivation

deinactivation is also voltage dependent

  1. H&H use conditioning steps:

Brief conditioning depolarization  reduced INa during 2nd step

As t for conditioning step increases  INa 2nd step decreases (more sodium channels inactivated during conditioning step)

 H&H find time and voltage dependence of inactivation

ex – conditioning step + 29 mV inactivation = 2 ms (nearly complete)

step + 8 mV inactivation > 8 ms (less inactivation)

They found that even at resting potential, many sodium channel are inactive.

Do Axon 5 conductances

  1. H&H used long conditioning steps to study de-inactivation

This turned off (closed) Na channels then set 2nd voltage to recovery voltage – vary time to 3rd voltage to look at current.

  1. Empirical Equations
  2. Generally, rate of change of membrane conductance

g:

dg/dt = (1-g) - g,

 = opening (fwd rate),  = closing

rewrite:

= = steady state value of g.

let u = - g, dg = -du

 ln(- g) = -t/ + c

g = - (- go) exp(-t/

go = g at t = 0

  1. Potassium

Emprically, H&H found they needed g  g4, they normalized maximum conductance

gk = gkmaxn4, gkmax = max conductance (all open)

n = activation; n g in above equations

n = - (- no) exp(-t/

n4 = fraction of channels open (between 0 and 1).

  1. Sodium

gNa = gNamaxm3h, m is like n for K, h describes inactivation

0 m 1, m3 = fraction activated, 0 h 1: 1= fully recovered and 0 = none recovered

m3h = fraction of channels open

  1. Current equation

Cm dV/dt = gNamaxm3h (Vm – VNa) - gkmaxn4 (Vm – Vk) – gL(Vm – VL) + I

n = - (- no) exp(-t/and same for m,h. These give fractions open and are between 1 and 0.

Do Axon 6 Impulse conductance

  1. Nerve Impulse Properties
  2. Resting Potential Vm = -60 mV (includes effect of ion leak), Na and K channels closed, 40% Na channels are inactivated
  3. Membrane depolarized w/ brief pulse  some Na channels open, Na flows in, more depolarization  more Na flows in  negative current
  4. Vm approaches VNa (momentarily) (VNa = 55 mV) but Na channels inactivate and K channels open  positive current  Vm becomes less positive
  5. Na channels close, K still open  hyperpolarize.
  6. Na channels de-inactivate, K channels close
  1. Impulse, Threshold and Refractory Period.
  2. If apply depolarization to resting neuron, you get depolarization, but no impulse unless depolarization > threshold
  3. Threshold depends on duration of pulse (A increases as duration decreases) and time after last impulse
  4. For some duration after impulse, threshold = infinity: cannot get impulse – duration here is due to refractory period (Most sodium channels are inactive).

Do Axon 7 and 8 Threshold and refractory.

  1. Channel Perspective
  2. When muscle stretches  action potential.
  3. All or none
  4. Na channel:

Depolariztion  m gate opens

Both gates respond to depolarization but speed is different

h gate closes = inactivation

meanwhile n (K+) opens

while h closed, m open is ineffective go to picture on page 8

  1. Spread of Action Potential
  2. Basic Picture

depolarization at one spot causes depolarization at neighboring spots  action potential there

Na coming in diffuses along axon causing depolarization

Depolarization can be bi-directional but doesn’t go back on itself due to refractory period.


  1. Spread of V

Depends on internal conductance vs that across membrane

Invertebrates: large radius  small R

Vertebrates: insulate with glial cells (myelin sheath), breaks = nodes of Rainier = access for ions.

  1. Synaptic Transmission



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