Download Biophysics Notes Chapter 9

Biophysics Notes Membranes and Transport
Membrane Biophysics
A. 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)
B. Goldman-Hodgkin-Katz equation
 
 
 
 
  P  C  i   Pi C  i 
1

k B T  i i
i
2
 , [P] = cm/s, for example:
ln
V =
q
  P  C  i 2   Pi C  i 
i


i
1
 i
V =
 
 
 
 
  
  



k B T  Pk K o  PNa Na o  PCl Cl i
ln
 P K   P Na   P Cl 
q
Na
Cl
i
i
o
 k
Example: Approximate Neuron:
   
   
 K   b Na  
o
o  , with b = P /P
Na K
 K   b Na  
i
i 

V = 58 mV log 
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.
1
Straight Line PNa = 0 (Nernst), Curved fit is using Goldman-HodgkinKatz, vary Ko and measure Vm.
Do Soma 1 (Nernst) ,
C. Electrical Model
Mammalian Cells
Squid Axon
VNa (mV)
+67
+55
VK (mV)
-84
-75
VCl (mV)
-60
-60
VCa (mV)
+125
+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.
[K]in =125mM V = V – V , positive current = positive ions flowing out of the cell.
in
out
Vm = -60
VK = -75
Vm not sufficient to hold off K flow so ions flow out. When Vm = Vk
then no flow.
[K]out = 5 mM
Kirchoff’s rules   Ii = 0   gi (Vm – Vi) = 0
2
 g i Vi g Cl VCl  g K VK  g Na VNa  g Ca VCa

, which is equivalent to
g Cl  g K  g Na  g Ca
 gi
Goldman-Hodgkin-Katz equation.
Vm =
Do Soma 3
D. 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
2K o
Vm = 58 mV log([K]o/[K]i) = 58 mV log
1/ 2
2
4K o  A 2 Z 2
 AZ
+
Both K and Cl are at equilibrium at this unique potential.


E. Animal Cell Model
1. Model
Ci (mM)*
Co (mM)
P>0?
K
125
5
Y
Na+
12
120
N**
Cl5
125
Y
A108
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).
+
3
** More on this later
2. 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).
4
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.
3. 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
F. Diffusion and Active Transport
1.
Pi = (ikBT)/d = Di/d, with D –
diffusion constant [cm2/sec]
2.
In reality, cell is permeable to
sodium, but pumps it out.
Had (Goldman-Hodgkin-Katz)
5
 
 
 
 


k BT  Pk K o  PNa Na o 
Vm =
ln
 P K   P Na  
q
Na
i
i 
 k
Now, with pump 
 K  o   Na  o 
 , with  = (n/m)(PNa/PK),
Vm = Vo ln  

 K i   Na i 
 
 




n/m = 2/3  Vm  VK.
3. 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.
4. Electrical Model
Now include pump:
g Cl VCl  g K VK  g Na VNa  g Ca VCa  I p  I
Vm =
g Cl  g K  g Na  g Ca
Do Soma exercises 4 and 5 (start).
G. 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.
6
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)
2.
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.
3.
-
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.
4. Multiple channels
Can get several on a patch
I

t
Parallel Resistors:
1/Req =  1/Ri  geq =  gi
Series:
1/geq =  1/gi
5. Ligand gated channels
n acetylcholine (nAchR) – closed until opened by binding
Ach  bind  let both Na and K pass
Not sensitive to Vm
7
I [Ach]2
Do Patch 1,2,4,5
F. Carrier Transport – example of facilitated transport.
1. Lowers activation energy
2. Obeys saturation kinetics
3. Highly selective
Simple Model:
K
Co Yo Ci Yi

CYo
CYi
Solute Flux
 Co
Ci 
J = Jmax 

 K  C

o K  Ci 

Jmax = NYDY/d2
NY = number of carriers
DY = diffusion constant of
Carrier
d = membrane thickness
For Ci = 0, get J = Jmax [Co/(K + Co)]
Jmax
Co
8
Biophysics Notes
Nerve Excitaion
Nerve Excitation
A. General Background
1. 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
9
2. 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.
3. 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
B. 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
10
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
C. 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)
2. 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)
Dynamics
Do Axon 1 and 2&3
and Voltage clamp currents.
11
3. 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.
4. Na inactivation and de-inactivation
1. V  activation , time inactivation  deinactivation
deinactivation is also voltage dependent
2. 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
3. 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.
D. Empirical Equations
1. Generally, rate of change of membrane conductance
g:
dg/dt = (1-g) - g,
 = opening (fwd rate),  = closing
rewrite:
12
dg g   g 

, g  =  g  = steady state

dt
value of g.
let u = g  - g, dg = -du
 ln( g  - g) = -t/ + c
g = g  - ( g  - go) exp(-t/
go = g at t = 0
2. 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 = n  - ( n  - no) exp(-t/
n4 = fraction of channels open (between 0 and 1).
3. 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
4. Current equation
Cm dV/dt = gNamaxm3h (Vm – VNa) - gkmaxn4 (Vm – Vk) – gL(Vm – VL) + I
 n = n  - ( n  - no) exp(-t/and same for m,h. These give
fractions open and are between 1 and 0.
Do Axon 6 Impulse conductance
E. Nerve Impulse Properties
1. Resting Potential Vm = -60 mV (includes effect of ion leak), Na
and K channels closed, 40% Na channels are inactivated
2. Membrane depolarized w/ brief pulse  some Na channels open,
Na flows in, more depolarization  more Na flows in 
negative current
3. Vm approaches VNa (momentarily) (VNa = 55 mV) but Na
channels inactivate and K channels open  positive current 
Vm becomes less positive
4. Na channels close, K still open  hyperpolarize.
5. Na channels de-inactivate, K channels close
F. Impulse, Threshold and Refractory Period.
1. If apply depolarization to resting neuron, you get depolarization,
but no impulse unless depolarization > threshold
2. Threshold depends on duration of pulse (A increases as duration
decreases) and time after last impulse
13
3. 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.
G. Channel Perspective
1. When muscle stretches  action potential.
2. All or none
3. 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
14
15
16
H. Spread of Action Potential
1. 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.
2. 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.
17
I. Synaptic Transmission
18
19
20