Biology 3240 Intro Cellular Neurobiol, Fall '09
Key Words & Concepts: Lectures 11-21
This page will offer some key words and ideas from lectures, provided as rough study guides.
Larry Okun, Doju Yoshikami
(to Suggested Readings instead)
Lecture 11 9/18/09
notes and comments on cable theory (cont'd from Lect.10):
1. Why change by a constant fraction per equal-length interval is 'exponential' change,
(and why 'e' is used so often in equations describing it) (cont'd) --
saw why such change is exponential and that for a given case of continuous exponential
change, the same curve can be describe by an equation using any base, provided an
appropriate contant kb is employed in the exponent;
next, given that any base can be used --
why 'e' is so often used as the base in such equations,
what's 'natural' about logs to the base e,
and why it's called 'e':
as noted, for a particular exponential curve, equation can be written using any base, b,
and an appropriate exponential constant kb:
i(x) = i0bkbx
problem is finding a base and appropriate exponential constant
to describe a particular case of exponential change (as the decay of cabble currenthere);
easy if fraction, f, lost is known for some particular length interval, L --
can (as noted above) write i(x) = i0(1-f)x/L, using (1-f) as the 'base', 1/L as the constant;
e.g. for radioactive decay, often know 'half life,' so can immediately write equation with base (1-f) = 1/2
problem is how to find an equation (some base, b, and an appropriate constant, kb)
if actual fraction, f, lost isn't known for any particular interval, L
often, analysis gives only k, the 'instantaneous' fraction of self lost per unit length interval --
allowing only a good approximation for fraction lost over a short stretch Δx:
fraction of self lost over Δx ≈ - kΔx
(cable-theory analysis gives just such a k -- the fraction of current lost over a short stretch of axon
is determined by the relative resistances of the 'leak-now' vs. 'continue-and-leak-later' paths --
specifically by the ratio r∞ / rl --
and dividing that by the length, Δx, of the short stretch provides a 'k' = (r∞ / rl) / Δx,
the fraction lost, per unit length, over very short stretches)
approximation provided by k is, for example, good for estimate of i remaining after first short stretch beyond origin,
i.e., for i remaining at x = Δx --
i(Δx) ≈ i0(1 - kΔx)
'real' equation, with an appropriate base and exponential constant, provides the answer for any x,
so must also provide the correct value after the first short stretch beyond 0,
i.e, the value i(Δx) at x = Δx --
i(Δx) = i0bkbΔx
question is how to get from 'k' and the approximation to some base b and exponential constant, kb,
so that the 'real' function and the approximation agree for the value of i at Δx;
i.e., so that i(Δx) = i0bkbΔx ≈ i0(1 - kΔx)
story of logs & construction of log tables, to replace multiplication by addition --
a fundamental property of e, and its attractiveness for 'filling in' fine-resolution blanks in log tables based on it:
for Δ a very small number, near 0 (and much less than 1),
loge (1 + Δ) ≈ Δ; i.e., eΔ ≈ (1 + Δ);
[also true for small subtractions from 1: loge (1 - Δ) ≈ -Δ;
i.e., e-Δ ≈ (1 - Δ)];
so if e is chosen as the base for a log table, and the loge for some number N, loge(N), is known,
then the loge of a number just a tiny bit larger than N, N(1 + Δ),
is simply (approximately) the known loge(N) plus Δ,
making e at least a 'natural' choice as base for constructing such tables
[that same basic property of e, eΔ ≈ (1 + Δ), can be used to show
that loge(N) equals the integral of 1/y from y=1 to y=N (the area under the curve 1/y, from y=1 to y=N);
thus logs with the base e are defined by a 'natural' function,
the usual sense in which they are said to be 'natural logs']
it follows from that same basic property, eΔ ≈ (1 + Δ), that, if e is chosen as the base, b,
then the 'instantaneous fraction of self lost per unit length', -k, is exactly the kb needed
to make i0bkbΔx ≈ i0(1 - kΔx),
since, for small kΔx, i0e-kΔx ≈ i0(1 - kΔx);
thus, the 'real' equation for our case of exponential decay can be written directly from the analysis,
using e as the base 'b' and k as the appropriate 'kb':
i(x) = i0e-kx,
which is why e appears so often in such equations, though they could be 'converted' to any other base
letter 'e' for this number was chosen by Leonhard Euler (1707-1783), who explored many of its properties,
including derivation of the remarkable, even mystical, formula eiπ+1 = 0, where 'i' is the 'imaginary unit'
further notes on cable theory...
2. λ, the 'length constant' or 'space constant' of exponential decay:
equation for decay of passive current is usally written as i(x) = i0e-x/λ,
with λ = 1/k being the interval over which i decays to 1/e of itself;
[note that the equation could be written as i(x) = i0(1/e)x/λ, since e-1 = (1/e),
and this is just the way we originally described exponential decay: i(x) = i0(1-f)x/L,
with (1/e) now being the 'fraction remaining,' (1-f), after each interval of length L = λ]
other quantities of interest also decay with the same expoential rule, with the same λ:
il, the leakage current at each equal-length stretch
(since it is a constant fraction of the i continuing inside, which decays by that rule)
ΔVm, the change of potential across the membrane associated with that leakage current
(since, by Ohm's Law, ΔVm = il x rl,
with rl being the same at each equal-length stretch,
and il decaying with that rule)
io, the current 'coming back' along the outside at a particular stretch
(since it equals return of all the internal current continuing inside in that stretch)
3. dependence of 'k' (and λ) in the decay equation on resistances in the model
(from dependence of 'k' on relative resistances of 'leakage' and 'continuing' paths):
comments on resistance --
resistance of a chunk of (homogeneous) conductive stuff (e.g., axoplasm)
increases proportionally with the chunk's length in direction of current flow (why?)
and decreases proportionally with the chunk's cross-sectional area perpendicular to current (why?)
for a particular axon (of fixed diameter), resistances of model are usually expressed in terms of:
ri -- resistance along axoplasm through a 1 cm. length of that axon, units Ω/cm
ro -- resistance along extracellular medium outside a 1 cm. length of that axon, units Ω/cm
rm -- resistance through membrane surrounding a 1 cm. length of that axon, units Ω⋅cm
based on these, resistances for a short stretch of that axon of length Δx are
ris -- resistance along axoplasm through that stretch = riΔx
ros -- resistance along extracellular medium outside that stretch = roΔx
(both with multiplication by Δx since resistance increases directly with increased lengths)
rms -- resistance through membrane surrounding that stretch (same as what called rleakage before) = rm/Δx
(here with division by Δx since resistance decreases with increased area of membrane
added as length involved increases)
and from these, one gets, for short stretches of length Δx, where current 'makes its decision' to leak or continue:
rleakage = rms = rm/Δx
r∞ (composed of ris and ros in series with
a 'parallel' grouping of rms and r∞ itself --
i.e., r∞ consits of the resistances of a short stretch attatched to another copy of r∞ itself!
(OK since infinite cable isn't changed by addition or removal of a piece of it);
as lengths, Δx, become very small (approach 0), by rules for series and parallel resistances and some algebra,
easy but not shown, get r∞ =
[rm(ri + ro)]1/2
from which one gets the following, in terms of the 1-cm specific resistances, ri, ro, and rm:
rin, the effective 'input resistance' seen by an electrode delivering current inside the axon, paired with one outside,
= r of two infinite half cables, one on either side, in 'parallel'
= (1/2) r∞ = (1/2)[rm(ri + ro)]1/2
k (as we found earlier) = (r∞ / rleakage) / Δx
thus, just to show the algebra here, k = { [rm(ri + ro)]1/2 / [rm/Δx] } / Δx
which, simplified, gives k = [(ri + ro) / rm]1/2
λ = 1/k = [rm / (ri + ro)]1/2
(so λ comes directly from the relative resistances of 'leakage' and 'continuing' paths
which determine the fraction of current leaking at each stretch)
check the final result with intuition --
λ, length of axon over which internal i decays to 1/e of itself,
increases if rm, resistance to leakage through membrane, increases,
decreases if ri, resistance to continuing along inside, increases
(also decreases if ro, resistance to returning along outside, increases)
4. λ values for common axons only a few mm., less than a mm. for thinnest axons;
nerve fibers are very poor 'electrical cables' --
electrical signal decays rapidly with distance,
requiring that the nerve impulse be regenerated at successive stretches,
which is why it travels much more slowly than electrical signals themselves
5. velocity of AP propagation depends on λ, increases with increasing λ --
the further currents from an active stretch spread passively, the faster 'next stretch' will become active;
to get the internal i to spread further inside (i.e., to increase λ and thus to increase AP velocity),
axons can make it more difficult for current to leak out -- increase rm
or easier for it to continue inside -- decrease ri
(or an experimenter can make it easier for current to return outside -- decrease ro)
(some additional notes on cable theory in next lecture,
then start on mechanism of the resting potential (RP) -- Bernstein's model for origin of resting potential)
Lecture 12 9/23/09
notes and comments on cable theory (cont'd from Lect. 11):
6. dependence of λ on axon diameter:
for unmyelinated axons of similar membrane & axoplasm properties, varying diameter, d,
rm ~ 1/d and ri ~ 1/d2 (why?);
thus, since λ ≈ [rm/ri]1/2 (for ro << ri so ro can be ignored),
λ ~ d1/2; i.e., λ increases with diameter1/2;
so velocity of APs is greater in fatter axons
7. myelin -- may* effectively increase rm, thus increasing λ, speeding AP propagation;
activity restricted to 'nodes,' proceeds in 'saltatory' manner, jumping from node to node,
conserving energy;
trade-off between myelin thickness and axon diameter for fixed total diameter:
(more myelin increases rm, good for (increasing) λ;
but less axon increases ri, bad for (decreasing) λ);
calculated theoretical optimum proportion: axon diameter ~ 0.6-0.7 of total (axon + myelin) --
observed values similar, so myelin thickness is controlled;
λ ~ d in this case; thus small axons better off (λ greater, AP propagation faster) if unmyelinated,
and smallest axons are, indeed, unmyelinated;
length of 'internodes' (myelin-wrapped stretches) also controlled, longer for larger axons
*some evidence suggests that resistance of myelin wrap may not be as great as previously thought
(possibly 'shunted' by paths of extracellular medium through 'wrinkles' in the wrapping);
thus equally important, perhaps main, contribution may be reduction of 'effective membrane capacitance'
to be discussed in next lecture)
8. application to dendrites:
currents from synapses also spread passively (with some exceptions);
and dendrites can also be treated as electrical cables,
but λ is not constant because diameter of dendrites is not uniform, dendrites taper,
so analysis is more complex, decay is not as simple exponential,
but principles are similar
9. why 'thin films' of extracellular medium are used for extracellular recording of APs --
extracellularly recorded potential difference is a fraction of one caused across membrane by AP;
fraction is same as ratio of extracellular resistance, ro, to sum (ro + ri) (why?);
thin films increase ro, so increase relative size of AP recorded extracellularly
comment on propagation of nerve signal in Bernstein's model -- chemical vs. electrical signaling:
chemical diffusion is fast enough at short distances (e.g., across synaptic cleft), much too slow for long distances!
electrical signaling is very fast, but since axons are leaky to electrical current,
electrical signal is lost after relatively short distances,
signal must be successively 'recreated';
propagation of nerve signal is thus much slower than that of electrical effects,
but much faster than chemical diffusion over long distances or even the ATP-dependent 'fast axonal transport'
mechanism of the resting potential (RP) --
Bernstein's model for origin of resting potential:
central features:
higher concentration of K+ ions, [K+], inside axons (or muscle fibers) relative to outside;
membrane permeable only to K+ ions and water;
total concentration of solutes same inside and out --
so water concentration same inside and out; 'osmotic balance';
other assumptions:
concentration of effectively impermeant solutes same inside and out --
no solute or solvent (water) movements tending to shrink or swell cell, 'tonic balance';
bulk electrical neutrality inside and out -- each side having equal concentrations of cations and anions
how central features would produce a 'membrane potential':
ion flux: movement of molecules, x, across the membrane (amount of x/unit time/area) from one side --
proportional to concentration, [x], of x on that side: flux of x = Px[x],
defining 'permeability' Px of membrane to x
in absence of other forces, concentration difference of K+ produces net flux of K+ outward;
K+ caries positive charge outward, leaves negative charges behind;
accumulation of charges, + outside, - inside, creates electrical force,
opposing, eventually balancing net K+ flux due to concentration difference
'K+ equilibrium': electric force balances 'concentration-difference force':
no further net K+ flux;
there is K+ flux from each side, just equal and opposite
very little K+ leaves to accomplish this,
so concentration difference (in/out) remains essentially unchanged
a true 'equilibrium' --
there is no further change in the system (nothing 'runnning down')
and no energy input is needed to maintain it
charge accumulation, with resulting electric force, creates 'membrane potential' -- 'Vm',
potential difference between inside and out;
convention: Vm measured as work on unit + chg from outside to inside (outside taken as reference '0'):
Vm = Vin - Vout;
Vm measured this way is thus negative for K+ equilibrium (with higher [K+] in than out);
some terminology: resting membrane (as in Bernstein's model) said to be 'polarized';
'depolarization' and 'hyperpolarization': decrease or increase of absolute value of this polarization,
nearer to, or further from, 0;
we'll use simple algebraic 'increase' (more positive), 'decrease' (more negative) to describe changes of Vm
value of Vm at which there is K+ equilibrium called 'K+ equilibrium potential' -- 'VK'
this is Bernstein's 'resting potential' --
Bernstein's idea: RP is exactly VK
value of VK depends on K+ concentration difference (in/out), which must be 'balanced' by electrical force,
and on temperature, which drives motion of ions, determining 'concentration force'
(this dependence to be examined in greater detail later)
if Vm is not equal to VK, there will be a net K+ flux (membrane permitting),
outward if Vm > VK,
inward if Vm < VK;
this is intuitive version of what we'll call 'K+'s story'
'K+'s story' extended...
net K+ flux carries charge, is a 'current flux' -- called K-current, 'IK';
convention on sign: IK positive if + charge being carried outward
(same convention for any ionic current, even if carried by anions --
current carried by any ion is positive if effectively carrying + charge outward)
direction of IK (positive or negative) same as direction of (Vm - VK)
(from analysis of direction of net K+ flux for cases Vm > VK, Vm < VK,
and conventions as described for IK, Vm)
magnitude of IK determined by:
magnitude of (Vm - VK) -- how far out of balance existing Vm is compared to VK,
and two other things --
ease with which K+ ions pass through membrane, the membrane permeability to K+, PK
how much K+ is around, i.e., K+ ion concentration, [K+] (amount of moveable charge available)
next lecture will pursue 'K+'s story' with a simple equation expressing it and
a commonly used electrical circuit model for it, plus introduction of 'membrane capacitance'
and discussion of membrane-like 'parallel RC circuits'
Lecture 13 9/25/09
Bernstein's model for origin of resting potential (RP) (cont'd):
further pursuit of 'K+'s story' ...
as noated, magnitude of IK is determined by:
magnitude of (Vm - VK) -- how far out of balance existing Vm is compared to VK,
and two other things --
ease with which K+ ions pass through membrane, the membrane permeability to K+, PK
how much K+ is around, i.e., K+ ion concentration, [K+] (amount of moveable charge available)
these latter two influences 'lumped' as 'K-conductance' -- 'gK,'
and IK is written as given by: IK = gK(Vm - VK),
an equation defining gK (whatever needed to make the equation right)
and consistent with the 'intuitive' version of K+'s story, with gK ≥ 0
equation IK = gK(Vm - VK)
sums up (encapsulates) 'K+'s story' for membranes (nerve, muscle, and other cells);
involves an 'indepence assumption' -- IK (net K+ flux) independent of other ion fluxes or self,
except as they influence Vm;
shows reason for another common name for VK, the 'K reversal potential' --
the value of Vm around which IK reverses direction
this is 'g view' of membrane currents; some notes on it:
conceptually easy: gK can be thought of as membrane's 'tap,'
membrane can 'control' IK for any particular (Vm - VK) by changing gK
(via changes of PK, e.g., by opening or closing of 'K+ channels');
but note that gK ≠ PK:
(recall gK also depends on K+ concentrations --
e.g., for membrane with particular PK but with much K+ on one side, almost none on other,
little IK might be obtainable from low-concentration side,
thus giving gK lower for IK in one direction than the other,
an example of 'rectification' -- easier to get IK in one direction than in the other)
gK is measurable by experiment -- if can measure IK & Vm and know VK, can get gK from the 'story' equation
gK is useful -- if know it, Vm, and VK, can predict IK from the 'story' equation
have noted gK depends on ease of passage through membrane, PK, and on K+ concentration;
for details of dependence, need theory for how K+ ions move through membrane -- amount of IK
for given electrical force (when Vm not = VK), K+ concentrations, temperature, and whatever determines PK
(beyond scope of this course)
'K+'s story' equation, IK = gK(Vm - VK), can be modeled as an electrical circuit (for a patch of membrane):
1/gK modeled as a resistor (conductor) in path carrying IK
VK modeled as a battery, really the K+ concentration difference acting as a 'source of emf'
thought experiment (as for intuitive consideration of Bernstein's RP model):
in absence of electrical force (Vm=0), VK battery drives IK through gK,
causing collection of charge outside and inside, thus an electrical force (and hence a non-zero Vm)
cm, 'membrane capacitance,' model for collection of charges on either side of non-conducting (insulating) membrane,
producing electrical forces, thus a potential difference -- Vm
capacitance, C:
defined by need for certain amount of charge collected on either side of an insulator
to produce a certain potential difference
('insulator': a 'non-conductor,' i.e., with no charges free to move)
charge (q) needed = C x potential difference (V) to be produced: q = CV
measured in farads = coulombs/volt;
cm defined as capacitance (amount of charge required per volt) per unit area of membrane
C depends on distance between charged parts ('plates') and properties of insulating stuff between them:
dependence on distance between plates (d):
electric force (per unit charge), E, perpendicular to plates
is proportional to charge/unit area on plates:
E = kq/A, with k some constant depending on units used;
For plates separated by a distance small with respect to their area (true for the
very small 'distance' across the membrane), E is constant everywhere between the plates,
regardless of the (small) distance between them.
(This interesting: electrical force perpendicular to an effectively infinite singlecharged plate is
constant, independent of distance from it, because forces from charges in further surrounding areas
add to net perpendicular force as distance from plate increases.)
potential difference, V, from one plate to other = work done moving a unit charge between them,
= force/unit charge x distance = E x d = (kq/A) x d;
thus, for given charge/unit area, q/A, on plates, potential difference, V, between them increases with distance,
so amount of charge required for a particular potential difference decreases with increasing d;
thus C (amount of charge needed for a particular V) decreases as 1/d
dependence on stuff between plates:
if insulator between plates is 'polarizable,' its polarized charges cancel part of electric force between plates,
making net force smaller for a given charge on plates;
thus more charge is required on plates to produce the net force needed for a particular potential difference,
so capacitance is increased by 'polarizable' stuff between plates;
the more polarizable it is (the more of electric field from plates can be canceled),
the more it increases capacitance
membrane capacitance, cm, can be measured electrically, and value is of interest for at least two reasons:
1. if polarizability of membrane stuff (lipid) is known,
measured cm provides estimate of membrane thickness (distance between separated charge clouds);
values obtained this way, ~3 nm, agree with values calculated for molecular dimensions of phospholipid bilayer
2. cm defines amount of charge required per unit area of membrane to produce a particular Vm;
e.g., for typical measured cm = 1 μfarad/cm2, a 100mV potential difference (more than the usual RP)
would require 10-7 coulombs/cm2, or (using the 'Faraday' constant: ~105 coulombs/mole K+)
~10-12 mole of K+ ions;
this is basis for statment that only a tiny fraction of the intracellular K+ must move to produce a
Vm in the RP range
measuring capacitances:
since, for a capacitor, q = CV, changing charge, q, on it changes V across it;
if current is applied to one side of a capacitor and drawn from other side,
it 'looks like' (is) current 'through capacitor' as 'seen' from outside,
called 'iC,' capacitative current,
but is really piling up charges on capacitor, changing V across it;
thus, if there is an iC 'through' a capacitor, V across the capacitor is being changed;
hence one strategy for measuring C: apply steady i from a source, iS, examine way V across C changes
pure C: V changes linearly while steady iS (= iC) is applied;
V remains unchanged if supply stopped (no iC);
(capacitors store charge, can be dangerous)
measuring C in this case:
apply known iS (= iC, rate at which charge added to C)
& measure rate at which V changes;
C is former divided by latter
parallel RC system (as in circuit model for axon membrane patch),
with steady iS supplied:
intuitive description of what happens --
let:
iC = current 'through' capacitor's branch, iR = current through resistor's branch
VC = potential difference across C, VR = potential difference across R
use:
Kirchoff's 1st Law: V across R and C same, VC = VR
Kirchoff's 2nd Law: iS (supplied) is divided between branches, = iC + iR
Ohm's Law: VR (= VC) = iR x R
equation for capacitance: qC = C x VC
if V = 0 initially, iR must = 0 (there is no VR to produce current in it),
thus all iS goes to C branch, iC = iS at start;
as iC adds charge to C, VC (hence VR) increases,
'driving' more of current through the resistor's branch,
i.e., increasing iR (which = VR/R),
thus decreasing iC (which = iS - iR);
so iC 'turns itself off,' by increasing V and driving more of iS through the R branch;
result is asymptotic approach to case with all iS through R, with 'final' V = iS x R, and iC = 0;
pursuit of intuitive description --
analysis of how iC changes over short time (and what causes that change)
shows decay of iC in this case is exponential;
instantaneous fraction of self lost/unit time = 1/RC,
thus, starting at t = 0, when steady iS started, iC(t) = iSe-t/RC,
usually written iC(t) = iSe-t/τ
τ (=RC), is 'time constant' of parallel RC element,
time required for iC to fall to 1/e of itself (during application of steady iS to the parallel RC pair)
take-home question: units of τ=RC must be time, e.g., seconds,
how does that follow from units of R (ohms) and C (farads)?
hint: how is R defined, and what units are involved in that?
Lecture 14 9/28/09
'K+'s story' extended...
electrical circuit model for 'K+'s story' (cont'd)
capacitance, C (cont'd):
showed in last lecture (and again in this one) that for parallel RC circuit with steady iS input starting at time t = 0,
the capacitative current is given by
iC(t) = iSe-t/τ
from equation for iC(t), immediately come equations for iR(t) and V:
iR = iS - iR = iS - iSe-t/τ = iS(1 - e-t/τ)
V = VR = iRR = iSR(1 - e-t/τ)
measuring R, C of parallel RC element:
supply known, steady iS;
asymptotic value of V gives R ('final' V = iSR): R = ('final' V)/iS
C then obtainable from measure of τ (= RC) in curve of V with time: C = τ/R
next, consider V across parallel RC when steady iS is turned off --
since VC remains, it continues to drive iR through R,
iR now being supplied by charge on capacitor as iC (in reversed direction);
recall: presence of an iC means VC must be changing, charge being added or removed,
thus iC here drains charge on C through R,
and VC (=VR) decays, again exponentially and with time constant = RC;
note that iS can be turned off before iR, and thus V, reach their 'final' (asymptotic) maximum values;
V (= VC = VR) decays, as above, from whatever value it reached while iR was on
cases considered so far involve application of steady supply current, iS;
also common are cases with 'steady VS' put on element (rather than steady iS),
then removed (V across element made = 0),
can ask what supply current, iS, is required to do that:
pure C: iS (= iC) must have 'spikes' (brief, strong current pulses) when VC is being changed abruptly,
i.e., when charge is being added to or removed from C to change VC very quickly
parallel RC: iS must have 'spikes' -- iC part as for pure-C case, when VC is being changed abruptly;
but must also have steady iR component -- i driven through R -- while VS is 'on'
comments on cm:
simple parallel RC model applies to small patch of membrane or simple cell body,
not to axon with current injected at one point;
principle same -- applied i takes time to charge local cm, thus time to change local Vm
delayed local change (charging local cm) delays application of 'full effect' downstream,
thus there are successive delays as local cm's charge,
delaying 'ignition' of nerve signal in next downstream stretch,
slowing speed of signal propagation;
myelin: increases effective 'thickness' of membrane, thus decreases cm --
reducing amount of charge (net flux of ions) needed at nodes to change local Vm of internodes,
thus improving economy (less net ion flux);
also, if myelin doesn't really increase rm greatly, decreasing local cm reduces local τ,
thus decreasing time to charge local cm's to particular values,
decreasing delays in getting signal in downstrean stretch 'ignited,'
so increasing speed of signal propagation;
so either myelin's decrease of cm or its possible increase of effective rm (increasing the length constant)
can speed propagation
test of Bernstein's hypothesis that RP is a K-equilibrium potential,
i.e., that VR (here meaning the value of Vm at rest) is exactly = VK,
requires both:
way to measure VR (RP)
and knowledge of what VK is for nerve (or muscle) cell involved
value of VK for given K+ concentration difference and temperature:
the Nernst equation (provided in any of suggested texts) --
derived from balance of
electrical force and 'concentration-difference' force (a 'K+-pressure difference')
on an imaginary slab of K+ ions within the membrane,
and using Boyle's Law for 'ideal gases' to express the 'K+-pressures' in terms of K+ concentrations
(treating K+ as being in an 'ideal solution')
yields the Boltzmann equation (really same as Nernst equation) --
concentration must decrease exponentially with increasing potential to maintain the force balance;
some notes on the Boltzmann equation (the Nernst eqation in a different form) --
among assumptions important in use of Nernst/Boltzmann equation
for estimate of VK from measured K+ concentrations inside/outiside axon are:
1. that K+ ions are as free to move (as 'ideal-gas-like') inside as are ones outside
(aren't sticking to each other or to other things in ways inside different from the ways ones do outside);
rates of diffusion of injected radioactive K+ along inside of axons
≈ diffusion rates outside, supporting this assumption for K+ ions;
2. that measured concentration inside is really one of K+ ions 'free to move'
(e.g., K+ ions that aren't being sequestered somewhere inside);
assumption is OK for K+ ions but poor, e.g., for Ca++ ions --
concentration of Ca++ ions 'free to move' inside is << measured concentration,
since large fraction is sequestered by mitochondria and endoplasmic reticulum (ER)
two further comments:
1. Boltzmann equation only relates concentration to potential:
concentration must decrease exponentially with increasing potential to maintain the force balance;
this says nothing about how either concentration or potential individually change within the membrane,
i.e., how they must be related to each other to achieve the 'balance' of forces.
If we know or assume how one changes within the membrane, the relation indicates how
the other must change to maintain the balance.
2. Boltzmann equation is a very general one, the 'law' of thermal equilibrium,
expressing conditions for balance between a concentration difference,
tending toward uniformity as a result of random thermal motion,
and a conservative force (such as an electrical one, gravity, etc.)
'pushing' in the direction of higher concentration;
net movement (flux) of things in one direction, 'driven' by concentration difference and temperature,
equals net movement in other, driven by the conservative force;
no net flux and no further change of concentration distribution -- the 'equilibrium';
balance is achieved when concentration decays exponentially
with increasing potential energy associated with the conservative force --
concentration is lower where potential energy (work done against the concentrating force) is higher;
in our example: K+ concentration difference
balances electrical force associated with Vm (electrical potential difference)
next lecture:
use of Nernst equation in test of Bernstein's idea that RP (VR) is a K-equilibriom potential,
part of explanation for why he was a bit wrong about that,
and background for consideration of active-site mechanism
Lecture 15 9/30/09
use of Nernst equation in test of Bernstein's idea that RP (VR) is a K-equilibriom potential,
i.e., that VR = VK:
VR (resting value of Vm) measured directly with intracellular electrode in large nerve cell or muscle fiber,
compared to VK predicted by Nernst equation,
with various values of extracellular K+ concentration, [K+]o
(and assumption that [K+]i remains intially unchanged):
Bernstein 'right' for higher than normal [K+]o -- VR ≈ VK; in particular, VR ≈ 0 for [K+]o = [K+]i;
but 'wrong' for normal low values of [K+]o -- VR greater than VK at those values,
and 'very wrong' for low values of [K+]o approaching 0,
for which Nernst VK tends toward very negative values (toward -∞),
but measured VR remains reasonable
part of reason Bernstein wrong -- membrane also somewhat permeable to sodium ions, Na+
'Na+ 's story' -- Na+ more concentrated outside than in, [Na+]o greater than [Na+]i (opposite to K+ )
membrane permeable only to Na+ expected to develop a sodium equilibrium potential, VNa;
VNa positive (higher inside) because [Na+]o greater than [Na+]i;
if Vm not equal to VNa, a net flux of Na+ expected, a 'sodium current,' INa;
'Na+ 's story' summed up as INa = gNa(Vm - VNa), similar to K+ story
general case:
concentration differences such that VK is negative and VNa is positive;
there is IK if Vm ≠ VK and gK ≠ 0;
there is INa if Vm ≠ VNa and gNa ≠ 0;
so what happens? what is value of Vm in this general case?
behavior of membrane potential when membrane is permeable to both sodium and potassium ions --
principle of analysis:
membrane potential changes when charge is added to or removed from clouds on capacitance cm;
for clarity, focus on internal cloud: given convention we are using for Vm (=Vin-Vout)
Vm increases when positive charge is added to internal cloud
Vm decreases when positive charge is removed from internal cloud
VSS, a 'steady-state' value of Vm is established when INa and IK 'balance' -- INa + IK = 0, no net charge flux
(charge neither being added to or removed from internal cloud, therefore Vm not changing, 'steady')
VSS when both gNa and gK are non-zero:
- not an equilibrium: there are net fluxes of Na+ and K+ ;
concentration differences are 'running down,' so an energy-using pump is needed to maintain them --
the 'Na+/K+-exchange pump,' driven by ATP
- INa and IK balance, thus there is no net charge flux, therefore Vm is 'steady' -- a 'VSS'
- net fluxes of Na+ , K+ are 'trickles' compared to total concentrations,
so changes of concentration differences are negligible over significant times, even without a pump;
'run-down' is very slow; thus concentrations (and therefore equilibrium potentials VNa and VK)
can be considered effectively constant, so INa and IK are also 'steady'
steady-state value of membrane potential, VSS, when both gNa and gK are non-zero --
a quantitative expression for VSS (from INa + IK = 0, and 'g-view' equations for INa and IK):
VSS = (gNaVNa + gKVK) / (gNa + gK) = (VK + βVNa) / (1 + β),
where β = gNa / gK;
if β is much less than 1, (gNa at rest much less than gK), VR, the 'resting' VSS, will be near VK
(this is normal, 'resting' condition -- Bernstein 'almost right')
gi's (conductances for various ion types, 'i') act as valves;
membrane can 'control' value of VSS by 'adjusting' relative gi's
comments:
1. there is a specific VSS for each set of gNa, gK -- each value of β;
membrane 'adjusts' the gi's by changing its permeability to selected ions,
specifically, as will be discussed, by opening or closing 'channels' that allow selective passage of particular ion types
2. the Na+/K+-exchange pump:
can maintain 'chemical steady state' if actively pumps ions out or in at same rate they enter or leave as
'passive currents' (the passive net ion fluxes we've been discussing, under influence of electrical force,
concentration differences, and membrane 'conductance') --
chemical steady state:
no changes of conentrations despite passive fluxes
(not an equilibrium, since energy required by the pump)
pump can itself be 'electrogenic':
if exchange of ions is not 1-for-1, pump generates a net movement of charges across membrane,
thus a membrane current, Ip, that can affect Vm;
Ip must be added to the sum of 'passive ionic currents,' and total set = 0, to obtain 'corrected' VSS,
actual pump appears to exchange 3 Na+ ions (outward) for 2 K+ ions (inward),
thus adding a net outward current, reducing VSS at balance with the net passive ionic currents
assumption of both chemical steady state (pump current for each ion balances passive current for that ion)
and particular ratio, r, of Na+-for-K+ exchange by pump,
requires that passive currents be in same ratio as pump currents are,
providing a 'corrected' equation for needed balance of passive currents,
and thus a 'corrected,' slightly lower value of VSS, involving both 'r' and 'β'
experiment: poisoning pump produces a measurable increase of VR, consistent with idea pump is electrogenic;
3. important to remember that ionic currents are 'trickles,'
need for pump, and its influence on Vm, can be ignored for 'casual' analysis
4. Bernstein partly wrong; VR is a VSS, near VK, but slightly greater than VK;
however, VR = (VK + βVNa) / (1 + β) also not whole story;
for constant values of gNa and gK (thus a constant β),
that value of VSS is linear with respect to VK (and thus also with respect to ln[K]o)
but, as we saw, experiment shows that the real value of VR 'curves' when plotted vs. ln[K]o:
VR becomes nearly VK at high [K+]o values, i.e., high Vm values
and does not tend toward -∞ as [K+]o becomes very low, --> 0,
so the gi's can't be constant, must be changing when [K+]o changes;
and they do change:
at high [K+]o, gK increases (because there is more K+ in parts of the membrane to carry current),
and, at the high Vm involved, membrane permeabilities also change (as we will discuss),
such that gNa decreases (to near 0) and gK increases further,
membrane becomes very nearly 'Bernstein-like': VR ≈ VK
at very low [K+]o, gK decreases, approaching 0 (since there is little K+ in parts of the membrane),
preventing VK (which tends to -∞ at very low [K+]o)
from 'dragging' VSS along with it in the g-view solution for VSS
(Note these examples remind that conductances, gi's, depend on both membrane permeability
and ion concentrations; for idea of separate contributions of permeability and concentrations,
need theory/model of how ions move within membrane when forces out of balance --
a function for net flux of each ion i, i.e., ionic current Ii, in terms of membrane permeability for that ion, Pi,
concentrations of that ion, electric forces producing Vm, and temperature; then set sum of those ionic currents = 0
to get an expression for expected VSS; one such is the 'Goldman-Hodgkin-Katz Constant Field Equation,'
given in most texts and looking a bit like Nernst equation, to which it must reduce if membrane is permeable
only to one ion; further discussion of it beyond time limits of this course)
5. other ions can contribute to the overall story; two important ones are:
Cl-: 'story' is ICl = gCl(Vm - VCl);
ICl is in opposite direction to net Cl- flux (since Cl- is an anion);
ICl often negligible because VCl is near VR at rest
and gCl remains small, relative to gK and gNa, during activity;
some cells have Cl- pumps, VCl can be above or below VR;
changes of gCl by synaptic activity in these cells can thus affect Vm (and do)
Ca++: 'story' is ICa = gCa( Vm - VCa);
much Ca++ inside cells is sequestered, hence free [Ca++]i is much lower than [Ca++]o, thus VCa is high;
changes of gCa can produce net Ca++ flux (inward) important to cell,
e.g., affecting Vm (even producing 'Ca++-APs), causing transmitter release, or acting as a '2nd messenger'
we'll ignore these other ions for now, consider only IK and INa
6. sources of charge that can affect Vm (by contributing to 'capacitative current,' IC, changing charge cloud on local cm):
we've considered two:
the sum of passive ionic currents, Σ Ii
and any current generated by an 'electrogenic' pump, Ip,
each of which, if inward (negative), can add + charge to internal cloud, increasing Vm;
two other sources:
- ΔIaxial (where ΔIaxial = local change of Iaxial) -- what's 'lost' by axial current passing by is added to IC
+ IS -- current pumped in directly from an intracellular electrode
whole story is IC (which = cm x rate at which Vm changes) = - Σ Ii - Ip - ΔIaxial + IS;
first two on right are produced by local membrane itself, second two are 'from elsewhere';
story rearranged is IC + ΣIi + Ip = - ΔIaxial + IS
terms on the left in rearrangement constitute the 'membrane current' --
the current 'going through' the membrane
(including IC, which 'appears' to be going through
but is really 'depositing' charge on local cm, changing Vm)
terms on the right are the 'from elsewhere' contributions (there may be none)
much can be understood, without equations, from intuitive 'thought experiments' based on the general model --
thoughts about what ions will do under various conditions, what would happen to Vm if...?
(offer insight and understanding, provide background for consideration of AP mechanisms)
some examples in next lecture (and on both next midterm exam and final exam)
Lecture 16 10/2/09
intuitive 'thought experiments' based on the general model --
thoughts about what ions will do under various conditions, what changes of Vm can result
(provide background for consideration of AP mechanisms):
principles:
equilibrioum potentials, Vi, considered fixed, constant, for given (unchanging!) concentration differences;
Note: For purposes here, concentration differences are taken as existing and given;
roughly, origin of them derives from need for cells with fragile membranes and seawater-like solution
outside to provide 'tonic' balance (avoiding swelling), by making something outside impermeant,
to compensate for a high concentration of impermeant organic anions inside, while also
maintanining osmotic balance (same total concentrations of solutes) and overall electrical
neutrality inside and out; choice involved making membrane essentially impermeable to outside
Na+ ions, with intracellular K+ ions balancing the charge of the impermeant anions.
if Vm ≠ Vi for some ion type, i, and conductance for that ion gi ≠ 0,
there will be net flux of that ion, hence a current flux (net + charge flux), Ii, carried by it;
direction of net (+ charge) current flux carried by that ion type is given by (Vm - Vi),
amount by both gi and the degree of 'imbalance' (Vm - Vi),
with the 'story' for that ion expressed by the equation Ii = gi(Vm - Vi);
Vm will change if there is a net change of charge on cm (net change of + charge in inside cloud)
assume:
'usual' situation --
concentration differences such that VNa is positive, VK negative;
both VNa and VK assumed constant (concentration differences and temperature constant);
further simplified --
only INa, IK involved; other ions ignored
(but both gNa, gK may be non-zero, so both INa and IK must be considered);
no axial current (Iaxial = 0) and pump current, Ip, can be ignored (Ip = 0);
general approach to analysis (step-wise):
for Vm at some particular value,
examine sources of + charge adding to 'inside cloud' at existing value of Vm:
sum of passive ionic currents, Σ Ii
other sources if there are any, e.g., a supply, IS
if there is net change of + charge in inside cloud (addition, removal) going on
determine how that will change Vm (increase or decrease?)
over next bit of time, to produce a 'new' Vm;
examine effects of that change of Vm on the sources of + charge,
e.g. on components of Σ Ii
repeat steps
'thought experiments':
(first two repeated from previous lectures)
a) suppose only gK were non zero -- have Bernstein's idea, Vm = VK, a true equilibrium
b) add 'small' gNa to above case: Vm increases to VSS (between VK and VNa),
near VK if gNa small compared to gK; a steady state, INa + IK = 0
c) increase gNa slightly and hold at constant new value: INa, IK no longer in balance; Vm increases (why?);
(rate of Vm increase determined by rate net charge added to cm, here by imbalance of INa, IK,
and by size of cm);
rate of change slows as INa, IK come nearer to balance;
new VSS established nearer VNa
d) increase gK slightly and hold at constant new value: Vm decreases to new VSS nearer VK (why?)
thus membrane can 'move' Vm (VSS) up and down between VNa and VK
by changing relative sizes of gNa and gK
(by changing relative permeabilities to Na+ and K+)
some comments on first four 'thought experiments,' particularly c and d (increase of gNa or gK):
1. gi's act as 'valves,' control Ii's (for fixed Vi's), thus determine where VSS is
2. actual cases include:
'chemical-sensitive' gi's (e.g., ones activated by transmitters at synapses
or by intracellular '2nd messengers')
'voltage-sensitive' gi's (e.g., ones responsible for AP)
gi's sensitive to direct mechanical or thermal changes (e.g., at sensory endings)
3. changes of gi's cause changes of net ionic currents:
membrane's own currents can charge or discharge its cm, change its Vm
4. note the net overall change of net K+ flux in (d) (increase of gK) --
at starting Vm, there is an increase (when gK is increased),
followed by a decrease as Vm approaches a new steady-state value (new VSS) nearer VK;
net change of K+ flux is an increase (why?)
5. for a change of a gi, approach to the new VSS is curved (why? see experiment 'c' above)
Note: Vm might not reach new VSS during a brief change of a gi;
if gi is returned to its earlier value, Vm will return to its earlier VSS
starting with whatever value it did reach during the brief gi change.
further thought experiments, assumptions as above:
e) current-supply electrode placed in cell, supplying steady current, IS, positive (pumping + charge in),
with no axial current (all of IS going either to local cm or net ionic current Σ Ii):
Vm increases to new VSS, with net ionic current, Σ Ii, outward (why?)
- note that, in this case, current from 'elsewhere' (IS) charges cm, increases Vm,
and is associated with an eventual outward Σ Ii,
while the same change (increase of Vm) is associated with an inward Σ Ii when membrane is 'doing it to itself'
f) intracellular supply electrode as above, but with IS negative (sucking + charge out):
Vm decreases to new VSS with net ionic current, Σ Ii, inward (why?)
- suppose negative IS is adjusted so new VSS = VK;
then what would be value of IK? (why?)
and what would happen (e.g., any change of Vm?) if gK were increased in this case?
how would this increased gK affect later attempts by other factors to move Vm away from VK?
(some synaptic inputs act this way -- changing a gi while Vm is at or near the Vi for that ion,
affecting attempts by neighboring synapses to change Vm)
g) supply electrode as in (f), with negative IS adjusted so VSS is less than (more negative than) VK,
then gK is increased:
Vm increases to new VSS nearer VK (why?) but not at or above VK (why?)
- note that VK is the 'K-reversal potential,' both for direction of IK
and for direction of changes in Vm produced by changes of gK
thus one can deduce much about behavior of membrane from such simple 'thought experiments,'
e.g., changes of Vm expected from changes of various membrane gi's,
or from application of 'supply currents,' IS;
they also provide the conceptual background for next topic --
mechanisms of the AP:
Bernstein's hypothesis, 'membrane breakdown' at active site, makes two testable predictions:
large decrease of 'membrane resistance,' rm, to passage of current,
Vm approach to value near 0
tests of these (almost 40 years after Bernstein's hypothesis proposed):
rm: Cole & Curtis, 1938, axon in trough with AC signal across it to measure rm (and cm) --
found rm did decrease, by ~40x, as AP passed,
but was still much (~106-fold) higher than r of cytoplasm or seawater,
and cm does not change! (so membrane doesn't just 'disappear')
Vm: Hodgkin & Huxley 1939; Cole & Curtis 1942, with electrode inside squid giant axon --
Vm during AP does not approach 'junction potential' (~-15mV), the value expected
if membrane were 'removed,' leaving axoplasm and extracellular medium in direct contact;
instead becomes 40-50mV positive, the 'overshoot,' (from RP of -60 - -80mV),
and there is sometimes an 'after hyperpolarization' ('AHP' -- Vm briefly below normal RP)
at end of AP;
hence, simple 'membrane breakdown' inadequate as model --
'overshoot' difficult to explain & neuroscientists distracted by defense work (World War II)
the 'Na-hypothesis' (Hodgkin & Katz 1949), a model accounting for the 'overshoot' --
at rest, gNa much smaller than gK, thus 'resting' VSS near VK (negative);
during activity, membrane reverses selectivity -- gNa becomes much larger than gK,
thus Vm moves to a VSS nearer VNa (positive);
a bold hypothesis --
raises issue of how membrane can 'select' for passage of ion (Na+) with larger hydrated (water-shell) radius
(current ideas, proposed by Eisenman, 1961: selective channels designed so relative energy change of
stripping off water shell, replacing it with binding to charges in channel wall, differs for ions of different sizes,
thus selects among them; basic idea supported for a K+-channel by X-ray crystallography -- MacKinnon, 1998)
initial tests of 'Na-hypothesis':
- reduction of [Na+]o, thus lowering VNa, should reduce size of AP: it does
- measures with radioactive Na+ and K+: there is increased net flux of both during AP, and amounts
are more than sufficient to account for charge needed for changes of Vm (up and down) during AP
questions raised by Na-hypothesis:
do conductances change during activity?
if so, how? -- e.g., what exactly reverses the ratio gNa/gK (from <<1 at rest to >>1 during activity),
only a large increase of gNa, or a decrease of gK, or both?
and if the conductances change, what causes/controls changes? what initiates an AP?
(know that 'local' axial current from a neighboring active site can do it;
one thing such current does is add + charge to the local membrane capacitance,
thus increasing Vm; so maybe that change of Vm is the key)
Hodgkin & Huxley 1952 --
propose and test ideas
that gNa, gK change in ways accounting for AP
and that Vm itself (change in electric force across membrane) controls gNa, gK
(permits easy hypothetical model: charged 'gates' or 'doors' in ion-passing pores of membrane
might open/close when Vm changes, i.e., when electrical force across membrane changes)
next lecture:
how Hodgkin and Huxley explored this idea and what they found
(classic work laying foundations of modern neurophysiology)
Lecture 17 10/5/9
mechanisms of the AP (cont'd)...
Hodgkin & Huxley 1952 experiments (foundations of modern neurophysiology):
as noted in last lecture -- propose and test ideas
that gNa, gK change in ways accounting for AP
and that Vm itself (change in electric force across membrane) controls gNa, gK
experimental approach:
change Vm from VR (RP) to some other value, VC, and hold ('clamp') it there;
examine INa, IK to determine whether/how gNa, gK change with time at that VC,
using relations defining those gi's:
INa = gNa(Vm - VNa)
IK = gK(Vm - VK)
and (Nernst) estimates of VNa and VK from known concentration differences,
or measures of VNa and VK as values of Vm at which INa and IK, respectively, reverse direction
exprimental issues:
how to change Vm to arbitrary new values and 'clamp' it at them;
how to measure INa and IK so they can be measured separately
'Voltage clamp': like a thermostat, compares real Vm to desired value, VC,
and controls supply current, IS, so it pumps or sucks + charge
to keep Vm from changing away from the desired VC;
- an example of 'negative feedback' (control mechanism that opposes change);
- also apply 'space clamp' (do same thing all along axon) so there is no Iaxial;
- ignore Ipump and any other ionic currents besides INa, IK;
then, while clamp is applied IS = Σ Ii = INa + IK
(as it did once Vm changed to a new VSS in 'thought experiments' with steady IS applied;
but here IS is varied to maintain Vm fixed (at chosen value VC) --
IS pumps or sucks + charge to match INa + IK if they try to change Vm from VC)
example,
for experimental case of clamp taking Vm from VR to VC near 0 then holding it there:
expected IS if gi's don't change from their resting values:
initial spike ('capacitative transient') to change Vm from VR to VC
(i.e., to supply needed 'slug' of charge to membrane capacitance cm),
then (small) constant outward IS (to match the expected net outward ionic current at Vm more positive than VR)
actual IS observed:
brief initial spike (mostly expected capacitative curent)
(and, maybe, small 'gating current' as charged 'doors' in ion channels move, ignored here),
then initial, expected (small) outward IS as above,
but then large (large 'trickle'), early inward IS (remember this = Σ Ii while the clamp is applied),
followed by large, maintained, late outward IS (= Σ Ii);
consistent with idea gNa increases early (to give 'large' inward INa),
then gK dominates -- either gK increases and gNa decreases, or gK alone increases greatly
some initial tests:
of idea early inward current is mostly INa --
reduced by decrease of [Na+]o (decreasing VNa);
reversed when Vm clamped at values above VNa
of idea late outward current is mostly IK --
can't use same approaches as for early current --
changes of [K+]o dramatically change resting Vm (why?)
clamping Vm to near or below VK doesn't involve large early or late currents (why?)
therefore tested with radioactive K+ -- late outward current mostly carried by it;
also consistent with observation that late current unchanged by changes of [Na+]o
determination of details -- how gNa, gK individually changing,
requires 'separation' of INa, IK parts of total Σ Ii observed as IS in V-clamp experiment
separation of INa, IK:
- rig [Na+]o so that VNa = chosen VC (e.g., for VC=0, set [Na+]o = [Na+]i), then INa = 0 at that VC
- let IS be the supply current seen with normal [Na+]o condition,
let IS' be the supply current seen with rigged [Na+]o;
- assume time course and values of IK unchanged by rigged [Na+]o
(late, outward supply current, assumed to be mostly IK, is observed to be the same in
normal and rigged [Na+]o),
- then IS' (under rigged condition) = IK (since there is no INa in that case);
- difference between IS(normal) and IS(rigged) (which = IK) is then the normal-condition INa;
i.e., IS(normal) - IS'(rigged) = INa(normal);
- time courses of INa, IK give time courses of gNa, gK for this case (e.g., Vm taken from VR to VC=0, held there):
gNa increases rapidly, then decreases slowly
gK increases slowly to a particular value, remains there
for other cases, Vm taken from VR to other values VC and held at them:
- only one 'rigged [Na+]o needed, with assumptions and simple algebra;
assumptions are (for any particular VC):
(1) INa'(under rigged condition) = constant x INa(normal), i.e., INa'(rigged) = kINa(normal)
(assumes gNa for rigged case has same time course, is proportional to normal gNa, at all times at that VC)
(2) IK is unchanged, independent of rigged [Na+]o (same assumption used above)
(3) gK changes so slowly that early supply current is almost all Na-current in both cases,
so k (defined as INa' / INa) = (IS' early) / (IS early)
get IS - IS' = (1-k)INa;
so INa = (IS - IS') / (1-k),
with k determined from experiment and assumption (3);
thus can determine INa under normal [Na+]o condition for any clamped value (VC) of Vm
(from observed IS at that clamped value under normal [Na+]o
and IS' at that same clamped value for whatever, single rigged [Na+]o is used);
(note that first 'rigged [Na+]o' example discussed is just a special case with k=0 for a particular value of VC)
once have INa(normal), can subtract it from IS(normal) to get IK
- so, a single change of [Na+]o allows separation of INa and IK parts of IS
for experiments with various values of VC
- and, from those observed INa and IK time courses, can get corresponding gNa, gK time courses
at various values of VC;
general behavior of gNa, gK determined from these experiments,
(Vm taken from VR to various higher VC values, then held at them):
- gNa 'activates' (increases) rapidly, inactivates (decreases) slowly
- gK 'activates' (increases) slowly to some value, then remains there
- peak value of gNa higher, eventual low value of gNa lower, and final value of gK higher for higher VC,
up to limits reached by VC values near 0
- (rates of changes also slightly dependent on value of VC)
'conditioning pulse' experiments: take Vm from VR to one VC, hold it awhile for gNa, gK to 'develop,'
then change to some other Vm, VC', and hold there; measure INa, IK to see how gNa, gK change at second VC';
find:
- 'continuity' -- gi's don't change immediately when Vm changed,
keep values developed at previous Vm,
then change as appropriate for new Vm (as membrane properties -- permeabilities -- change)
- unexpected (expect some immediate gi change from ion-concentration asymmetry);
- case special for squid axon in normal seawater, nice (made things easy) for Hodgkin & Huxley
general case for gK, simple:
- starts with value 'inherited' from previous Vm
- seeks final value appropriate for new Vm
- does this slowly
- model as 'gate' 'OK' controlling gK of a macro patch of membrane: gK = OK x gKmax
(where gKmax is the maximum possible gK obtainable from that patch of membrane,
and OK is a 'gating' value between 0 and 1)
OK opens slowly at higher Vm's, closes slowly at lower Vm's;
OK seeks final value appropriate for Vm (slowly)
- now know 'gate' is a model for kinetics (change with time)
of gK for a membrane patch (a 'macro patch') with many gK channels;
individual channels open/close quickly;
rates of change from open to closed, or vice-versa depend on Vm;
(thus how long average channel spends open or closed) depends on Vm);
value of OK is really fraction of K-channels open in the patch;
'gate' model represents time course (kinetics)
of change to new equilibrium between open/closed channel populations in patch
when Vm is changed;
(not to be confused with 'gating mechanisms' that open/close individual channels)
(and 'macro patch,' with many channels, discussed here, not to be confused with
'micro patches,' containing one or a few channels, which are involved in
'patch-clamp' studies of currents through individual channels; more about that later in course)
case for gNa complex; it increases, then decreases at a VC > VR;
question: what happens to gNa if Vm kept at one VC > VR for a while, then changed to new, higher VC'?
answer: for Vm taken to second, higher VC', gNa rises again but 'bumps head':
- doesn't get to same peak value as does when Vm taken directly from VR to VC'
- as though a 'ceiling' on gNa setting in at first VC
- 'ceiling' becomes lower the longer Vm held at first VC
overall gNa (for a 'macro patch' of membrane) modeled as controlled by two processes,
two 'gates':
- ONa, 'Na-activation' mechanism, like OK but fast -
ONa opens rapidly at higher Vm's, closes rapidly at lower Vm's;
- SNa, 'Na-inactivation' mechanism, closes at higher Vm's, slowly
- gNa = ONa x SNa x gNamax
(where gNamax is the maximum possible gNa obtainable from that patch of membrane,
ONa, SNa are 'gating' values between 0 and 1)
(SNa can be thought of as fraction of Na-channels 'available' to be open, 1 when all in the patch are available,
and ONa can be thought of as the fraction of available ones that are open)
Na-inactivation mechanism, features:
(1) SNa slowly imposed (closes slowly) at high Vm's, final value lower (more closed) at higher Vm's;
this (along with slow increase of gK) at high Vm explains 'accommodation'
(2) SNa slowly re-opened when Vm returned to VR
('double-pulse' experiments: reveal value of SNa by height of gNa
reached during second 'test' pulse taking Vm back to VC
at various times after return to VR following first period at VC)
(second 'test' pulses needed because rapid closure of ONa when Vm returned to VR
makes total gNa very small, regardless of SNa state, so can't 'see'/evaluate SNa)
this (along with slow decrease of open gK to normal resting values after return to VR)
explains refractory period (and after hyperpolarization) following an AP
(3) SNa about half closed at VR, can be opened further if Vm held at values below VR
(conditioning-pulse experiments with Vm held at VC lower than VR,
then taken up to a second VC' above VR -- a higher peak gNa value
reached at VC' than when Vm taken directly from VR to VC')
(effect greater the longer Vm held at low VC < VR before test at VC' > VR;
i.e., SNa opens slowly to final value greater than resting one
when Vm held at values below VR)
this (along with slight reduction of gK at Vm below VR) explains 'anode-break excitation'
(note: classic 'stimulation phenomena' are named for extracellular stimulating electrodes,
with which they were first discovered: an extracellular anode is equivalent to an intracellular cathode
and decreases Vm, i.e. can 'hyperpolarize' membrane)
summary of 'Hodgkin-Huxley (HH) mechanisms' --
model 'gates' controlling overall gNa, gK of macro patch of membrane:
- OK, K-activation (HH used variable 'n' in describing it):
also known as 'delayed-rectification mechanism' (why?)
slow, operates in about 1-2ms,
opens slowly at higher Vm, closes slowly at lower Vm
- ONa, Na-activation (HH used variable 'm' in describing it):
fast, operates in about 0.1-0.2ms,
opens rapidly at higher Vm, closes rapidly at lower Vm
- SNa, Na-inactivation (HH used variable 'h' in describing it):
slow, operates in about 1-2ms,
closes slowly at higher Vm, opens slowly at lower Vm
- each seeks final value appropriate for existing Vm
knowledge of V-sensitive gNa, gK behavior permits prediction of time course of Vm,
e.g., during a 'membrane AP' (one for patch of membrane, with no Iaxial) and only INa and IK affecting Vm:
- known values of gNa, gK (inherited from previous Vm) permit calculation of INa, IK;
- INa, IK give net + current affecting charge on cm, thus predict change of Vm over short time;
- known behavior (from HH experiments) of gNa, gK predicts their change at that Vm over same short time;
- arrive at new Vm (changed by net INa + IK) with changed values of gNa, gK;
- repeat above steps for next short time
(Hodgkin & Huxley used this approach to predict time course of 'membrane AP' in squid axon,
with dramatic, excellent match to actual, recorded APs)
thought experiment: use IS to change Vm quickly from VR to some moderately higher value, then turn off IS;
(assume simple 'membrane' case, with no Iaxial, and only INa + IK affecting Vm)
- gi's initially have 'old' resting values, net INa + IK is outward, decreasing Vm;
Vm begins return to VR (the 'VSS' for its old gi's), but rate of return is slow
- 'rapid' ONa increases gNa, increasing INa
- if gNa not 'open enough' at that Vm, INa not large enough to overbalance IK,
net INa + IK still outward, decreasing Vm, closing gNa (by operation of rapid ONa), returning Vm to VR
- if gNa 'large enough' at that Vm, get INa + IK now inward, increasing Vm,
further increasing gNa (further opening rapid ONa), thus further increasing INa, further increasing Vm, etc. --
(an example of 'positive feedback,' a mechanism encouraging change) --
producing AP, reaching VSS near VNa (because gNa becomes >> gK);
(Vm at which rapid ONa 'opens enough' to produce an AP determines 'threshold')
- around time of AP peak, 'slow' mechanisms reduce gNa (SNa closing), increase gK (OK opening),
producing INa + IK outward again, reducing Vm, closing rapid ONa, further reducing INa,
further reducing Vm, etc..., returning Vm to value near VK;
- Vm arrives at low value
with less gNa than normal (slow SNa still closed),
and more gK than normal (slow OK still open),
so return is to a VSS even nearer VK than normal (the AHP);
- then time is required at the low Vm
for slow restoration of gNa (SNa to re-open) and gK (OK to re-close) to normal, resting values --
a 'refractory period,' during which it is harder (or impossible) to raise Vm to an extent that
gets gNa large enough (because SNa is still shut) for INa to overbalance IK, which has the advantage
of a relatively large gK (a still open OK)
HH experiments, interpretations 'explained' active membrane behavior, set stage for later research:
- identification of molecules constituting channels:
determination of their structures,
discovery of mechanisms controlling ion selectivities of channels and responses (open/close/inactivate) to Vm;
- discovery of other gi's (other 'channel types'),
ones selective for different ions,
some Vm-sensitive, with various behaviors when Vm changed,
ones with various other 'sensitivities' --
e.g., to chemicals (including synaptic transmitters), mechanical influence, temperature;
examples of several of these other gi's in later lectures...
Lecture 18 10/7/09
What ion channels do.
How they are like enzymes.
Ion channels are integral membrane proteins (Fig 2.1).
Modes of channel activation (Fig. 2.2).
Voltage-gated channels.
Ligand-gated channels: ligand-binding sites can be extra- or intracellular.
Ionotropic receptors receptors (receptor and channel are part of the same macromolecule)
vs metabotropic receptors (receptors separate entities from the ion channels that they affect).
Recordings of single-channels.
What is the smallest response to the neurotransmitter acetylcholine (ACh)?
ACh "noise" (Fig. 2.5).
Patch clamp recording (Neher & Sakmann, Fig. 2.3).
Cell attached patch; whole-cell configuration; outside-out patch, inside-out patch.
Examples of patch clamp recordings of ligand-gated channels (Fig. 2.4).
Three parameters of single channels:
Po ≈ probability of channel opening
τ ≈ mean open time
γ ≈ single-channel conductance
Lecture 19 10/09/09
Recordings of single-channels.
Patch clamp cont'd (Fig. 2.6).
Exercise, with regard to Fig. 2.6:
1) Calculate the channel's conductance (gamma or γ).
1 picoSiemens (pS) ≈ 10-12 Siemens
2) What determines location of X-intercept?
3) What would the I-V plot look like if [K+]out ≠ [K+]in ?
4) What would trace in C or D look like if there were two channels in the patch?
5) What might C and D look like if the K channel were V-gated?
Planar lipid bilayers (black lipid membranes).
nAChR: the prototypical ligand-gated ion channel.
Biochemical purification of the first ion channel.
nAChR source: Electric organ (torpedoplax).
Reporter ligand: 125I-labeled α-Bungarotoxin, a neurotoxic peptide from snake venom.
Functional reconstitution.
Molecular cloning.
Functional expression of ion channels.
Xenopus oocyte (Box 3.3).
nAChR structure -- pentameric complex with two α-subunits (Fig. 3.1).
Cys-loop receptor superfamily
Lecture 20 10/19/09
nAChR structure (cont'd), Fig. 3.3.
Ion selectivity and gating of ligand-gated channels (cf. Fig. 3.4).
Ionotropic glutamate receptors belong to a separate family (Fig. 3.8D).
Biochemical purification of voltage-gated Na channels (Nav) and Ca channels (Cav).
Cloning of a voltage-gated K channel (Kv) via Drosophila neurogenetics.
Structures of the alpha-subunits of Nav, Cav and Kv (Fig. 3.6).
Voltage-gating (Fig. 6.11)
Voltage-gated channel-activation (Fig. 6.11).
Ball & chain model for inactivation of Kv (Fig. 6.12).
Hinged-lid model ("IFM loop" between DIII & DIV) for inactivation of Nav.
Lecture 21 10/21/09
Voltage-gated channel activation (Figs. 6.11).
K channels: inward rectifier, Kir (vs. Kv) (Fig. 3.8B).
Two-pore, four-transmembrane, K channel ("K2P").
X-ray structure of a K channel (Fig. 3.7).
Ion selectivity filter of potassium channel vs. sodium channel.
Conductivities of KCl vs. NaCl solutions and mobility of K+ vs. Na+.
V-gated ion channel (super)families:
Nav
Cav (Table 3.1)
Kv & other K channels (Table 3.2).
Examples of various other types of channels (Fig. 3.8).
Electrical (vs. chemical) synaptic transmission (Fig. 9.1).
Furshpan & Potter's experiment (Fig. 9.2).
Ohmic (linear) vs. rectifying (non-ohmic) coupling.
Gap junction (Fig. 7.8).