document12
Post on 31-Jan-2016
212 Views
Preview:
DESCRIPTION
TRANSCRIPT
Chapter 12: Nerve Impulses
Biological PhysicsNelson
Updated 1st Edition
Slide 1-1
12 Nerve Impulses
Announcements
• Final Reports:- Individual
– Write a short report 5~10 pages, about a topic
relating to one of the chapters in the book or
another topic in Biophysics
– Interesting papers will be uploaded to
CBCMP’s google drive
– Deadline … ~ early August
• Good luck …
Biological Question?
• How can a leaky cable carry a sharp signal over
long distances?
• Physical Idea: Nonlinearity in a cell membrane’s
conductance turns the membrane into an
excitable medium, which can transmit waves by
continuously regenerating them ….
• Wow …
12.1 THE PROBLEM OF
NERVE IMPULSES?
12.1.1 Phenomenology of action potential
• This section is historical in nature …
• Three caveats:
1. Stimulation of the cell inputs (typically
dendrites) from cell outputs (typically axon
terminals)
2. Computation of the appropriate output signal
3. Transmission of the output signal (nerve
impulse) along the axon.
• This chapter focuses mainly on the last point ….
12.1.1 Experiment & Electronus
12.1.1 Phenomenology: Action Potential
• Experiments show (see Fig. 12.1) we get
depolarizing and hyperpolarizing propagation
• However, instead of being graded, the action
potential is an all-or-nothing response. That is,
the action potential arises only when the
membrane depolarization crosses a threshold;
subthreshold stimuli give electrotonus, with no
response far from the stimulating point.
• In contrast, above-threshold stimuli create a
traveling wave of excitation, whose peak
potential is independent of the strength of the
the initial stimulus.
Continued
• Action potential moves down axon at a constant
speed (see Figure 12.2b), which can be anywhere
from 0.1 to 120m/s. When progress of action
potential is measured at several distant points, as
in Figure 12.2b, the peak potential is found to be
independent of distance.
• In contrast to the decaying behaviour for
hyperpolarizing or subthreshold stimuli. A single
stimulus suffices to send an action potential all the
way to the end of even the longest axon.
Continued
• Indeed, the entire time course of the action
potential is the same at all distant points (Figure
12.2b). That is, the action potential preserves its
shape as it travels, and that shape is
"stereotyped" (independent of the stimulus)
• After the passage of an action potential, the
membrane potential actually overshoots slightly,
becoming a few millivolts more negative than the
resting potential, and then slowly recovers. This
behaviour is called afterhyperpolarization
• For a certain refractory period after transmitting
an action potential, the neuron is harder to
stimulate than it is at rest
12.1.2 THE CELL MEMBRANE
VIEWED AS AN ELECTRIC
NETWORK
12.1.2 Iconography (see Fig 12.3a)
1. No significant net charge can pile up inside the
individual circuit elements: The charge into one
end of a symbol must always equal the charge
flowing out the other end. Similarly,
2. A junction of three wires implies that the total
current into the junction is zero (Kirchoff’s rules)
3. The electrostatic potential is the same at either
end of a wire and among any set of joined wires
4. The potential changes by a fixed amount across a
battery symbol
5. The potential changes by the variable amount IR
across a resistor symbol
12.1.2 Conductance as Pipes
• There is a physical difference between our
system and ordinary networks, but we can
nevertheless use network ideas because?
• Although we have at least 3 important channels:
Na+, K+ and Cl- are all at the same electrostatic
potential (imagine 3 species in one fluid pipe)
• Instead of Fig. 12.3a we have 12.3b (where we
explain the Capacitance in a few slides time)
• Kirchhoff's loop rule in this diagram implies
current conservation and Δ𝑉 = 𝐼𝑖𝑅𝑖 + 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡,
where ΔV = V2 − V1, 𝐼𝑖 = 𝑗𝑞,𝑖𝐴 & 𝑅𝑖 =1
𝑔𝑖𝐴
Ignore
Vpump
12.1.2 Quasi-steady approximation &
chord conductance
• Can neglect the complication of ion pumping
when studying fast transients like the action
potential: Quasi-steady approximation
• Assume steady state (non-eq) and then
suddenly shut off ion pumps implies
Σ𝑖 𝒱0 − 𝒱𝑖
𝑁𝑒𝑟𝑛𝑠𝑡 𝑔𝑖 = 0
• Then subbing 𝑔𝑡𝑜𝑡 ≡ Σ𝑖𝑔𝑖 leads to chord conductance formula:
𝒱0 =Σ𝑖𝑔𝑖𝑔𝑡𝑜𝑡
𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡
• Evaluating yields 𝒱0=-66 mV only a few millivolts
different from true steady state -72 mV
Capacitance: (C+- C- )e can pile up on each
boundary!
• Membrane capacitance
can happen even with
charge neutrality (see Fig.)
where 𝑞 = 𝐶Δ𝑉
• The capacitive current is
defined as 𝑑 Δ𝑉
𝑑𝑡=
1
𝐶
• This has used Ohmic
hypothesis on a small
patch of membrane
12.1.3 Linear Cable Equation
• Now consider what happens, not radially, but
axially along membrane where uniformity is
gone (see Fig. 12.4)
• Your Turn 12 A (homework) shows we can lump
three pairs of batteries & R’s into one for each
patch of membrane
a. Typical numerical values are on p. 516
b. Equation derived from Fig 12.4:
• Not yet solvable as 3 unknowns?
Figure 12.4 (Schematic; circuit diagram.) Caption: See text.
12.1.3 Linear Cable Equation (LCE)
• Eliminate Ix using
to obtain
• Using Your Turn 12 A: 𝑗𝑞,𝑟 = 𝑉 − 𝒱0 𝑔𝑡𝑜𝑡 ,
defining 𝑣 𝑥, 𝑡 ≡ 𝑉 𝑥, 𝑡 − 𝒱0 , space & time
constants as
we get:
12.1.3 Solution of LCE
• This looks like the diffusion equation: 𝑑𝑐
𝑑𝑡= 𝐷
𝑑2𝑐
𝑑𝑡2
and if we define 𝑤 𝑥, 𝑡 ≡ 𝑒𝑡
𝜏𝑣(𝑥, 𝑡) then the LCE
becomes
𝜆𝑎𝑥𝑜𝑛2
𝜏
𝑑2𝑤
𝑑𝑡2=𝑑𝑤
𝑑𝑡which has solution
• This does not have any travelling wave solutions
so we need something extra?
12.2 SIMPLIFIED
MECHANISM OF ACTION
POTENTIAL
12.2.1 The Puzzle
• How can waves travel without diminution?
• When a system is not in equilibrium, free energy
is not a minimum
• If not in minimum state the system can do work
• The excess free energy is what keeps the signal
from diminishing …
12.2.2 A mechanical analogy
• A nonlinear wave in an excitable medium selects
its waveform and velocity
• Another possible analogy (as opposed to
corrugated roofing) is a bomb fuse …
• The main idea is thus,
– Each segment of axon membrane goes in
succession from resisting change (like chain
segments to the left of the kink in Figure
12.5a) to amplifying it (like segments
immediately to the right of the kink) when
pulled over a threshold by its neighbouring
segment.
A corrugated plate on a slope (a roof?)
12.2.3 A little more history
• Read pages 521-524; however we find:-
– If the membrane could rapidly switch from
being selectively permeable to potassium only
to being permeable mainly to sodium, then
the membrane potential would flip from the
Nernst potential of potassium to that of
sodium, explaining the observed polarization
reversal (see Equation 12.3).
Reprinted by permission from Nature, ©1939, Macmillan Magazines Ltd.
Nobel Prize Winning Work
Effect of sodium content
12.2.4 Time course of action potential
• Given data in Fig. 12.8a) can we find out what
kind of total membrane current 𝑗𝑞,𝑟 is needed?
• Using fact that entire history is known once a
given point is known: V 𝑥, 𝑡 ≡ 𝑉 𝑥, 𝑡 −𝑥
𝜃where
V t ≡ 𝑉 0, 𝑡 is shown in Fig. 12.8a) implies that 𝑑𝑉
𝑑𝑋= −
1
𝜃
𝑑 𝑉
𝑑𝑡′ 𝑡′=𝑡−
𝑥
𝜃
• Rearranging Eq. (12.7) then gives
𝑗𝑞,𝑟 =𝑎𝜅
2𝜃2𝑑2 𝑉
𝑑𝑡2− 𝒞
𝑑 𝑉
𝑑𝑡• Membrane depolarization itself is the trigger that
causes the sodium conductance to increase
Reconstructing the action potential
12.2.5 Voltage gating leads to nonlinear
cable equation with traveling waves
• Once one segment depolarizes, its
depolarization spreads passively to the
neighbouring segment
• Once the neighbouring segment depolarizes by
more than 10m V, the positive feedback
phenomenon described in the previous section
sets in, triggering a massive depolarization
• The process repeats, spreading the depolarized
region
12.2.5 The model
• Assume 𝑔𝑁𝑎+ 𝑣 = 𝑔𝑁𝑎+0 + Bv2 where 𝑔𝑁𝑎+
0 is
the resting conductance per unit area.
• Lumping 𝑔𝑁𝑎+0 into 𝑔𝑡𝑜𝑡
0 we get
𝑗𝑞,𝑟 = Σ𝑖 𝑉 − 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡 𝑔𝑖
0) + (𝑉 − 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡 𝐵𝑣2
= 𝑣𝑔𝑡𝑜𝑡0 + 𝑣 − 𝐻 𝐵𝑣2
where in the 2nd step we used Your Turn 12A
(homework) to group as 𝑣𝑔𝑡𝑜𝑡0
• Solution with 𝑗𝑞,𝑟 = 0 has 𝑣 = 0, 𝑣1, 𝑣2 where
𝑣1/2 =1
2(𝐻 ∓ 𝐻2 −
4𝑔𝑡𝑜𝑡0
𝐵). See Fig. 12.9b)
• v=0 is Ohmic part (region A in Fig. 12.8), above
threshold we get feedback & overshoot
𝑗𝑞,𝑟 = 𝑣𝑔𝑡𝑜𝑡0 + 𝑣 − 𝐻 𝐵𝑣2
Nonlinear Cable Equation (NLCE)
• Subbing in Eq. 12.21 in cable eq. using
𝑣1𝑣2 =𝑔𝑡𝑜𝑡0
𝐵then Eq. 12.17 becomes
• Again shifting variables: 𝑣 𝑥, 𝑡 ≡ 𝑣 𝑥, 𝑡 −𝑥
𝜃and
defining dimensionless quantities: 𝑣 ≡ v/𝑣2, y ≡− 𝜃𝑡/𝜆𝑎𝑥𝑜𝑛, 𝑠 ≡ 𝑣2/𝑣1 & 𝑄 ≡ 𝜏𝜃/𝜆𝑎𝑥𝑜𝑛 we get
which has solution:
Solution of simple nonlinear cable equation
12.3 THE FULL HODGKIN–
HUXLEY MECHANISM AND
ITS MOLECULAR
UNDERPINNINGS
EACH ION CONDUCTANCE
FOLLOWS A CHARACTERISTIC
TIME COURSE WHEN THE
MEMBRANE POTENTIAL CHANGES
12.3.1
Space-clamping
• The conductance determines the current, held at
a fixed, uniform potential drop.
• Highly non-uniform potential, localized pulses,
appear during operation of an axon.
• The metallic wire
was a much better
conductor than the
axoplasm, so its
presence forced
the entire interior to
be at a fixed,
uniform potential.
Voltage-clamping
• So that V is the more natural variable to fix
• In this arrangement the experimenter chooses a “command” value of
the membrane potential; feedback circuitry supplies whatever
current is needed to maintain V at that command value, and reports
the value of that current.
• Forcing a given current
across the membrane,
measuring the resulting
potential drop, and attempting
to recover relation (Section
12.2.5)
• For one membrane current
flux, there are multiple “V”s.
• Our hypothesis: the devices
regulating conductance are
themselves regulated by V ,
not by current flux
Separation of ion currents
• Even with space- and voltage-clamping,
electrical measurements yield only the total
current through a membrane, not the individual
currents of each ion species.
• Ion substitution (Section 12.2.3).
• 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡= the clamped value of V. This ion’s
contribution to the current equals zero,
regardless of the conductance 𝑔𝑖(V )
Results• Immediately after the imposed
depolarization, there is a very short
spike of outward current lasting a few
microseconds. This is the discharge
of the membrane’s
capacitance.(Section 12.1.2)
• An inward sodium current develops
in the first half millisecond. The
sodium conductance peak value can
be calculated and depends on the
selected command potential V.
• After peaking, the sodium
conductance drops to zero, even
though V is held constant.
• The potassium current rises slowly
(in a few milliseconds). Like 𝑔𝑁𝑎+ ,
the potassium conductance rises to a
value that depends on V . Unlike
𝑔𝑁𝑎+, 𝑔𝐾+ holds steady indefinitely at
this value.
• The voltage-gating hypothesis describes the
initial events following membrane depolarization
(points 1–2), which is why it gave a reasonably
adequate description of the leading edge of the
action potential.
• In the later stages, the simple gating hypothesis
breaks down (points 3–4 above), and indeed
here our solution deviated from reality (compare
the mathematical solutions in Figure 12.10 to the
experimental trace in Figure 12.6b).
More realistic gating functions
• After half a millisecond, the spontaneous drop in sodium
conductance begins to drive V back down to its resting
value.
• Indeed, the slow increase in potassium conductance
after the main pulse implies that the membrane potential
will temporarily overshoot its resting value, instead
arriving at a value closer to 𝑉𝑛𝑒𝑟𝑛𝑠𝑡 (see Equation 12.3
and Table 11.1 on page 416). This observation explains
afterhyperpolarization (Section 12.1.1).
• Once the membrane has repolarized, an equally slow
process resets the potassium conductance to its original,
lower value, and the membrane potential returns to its
resting value.
Postscript
• As far as the action potential is concerned, the function
of the cell’s interior machinery is to supply the required
nonequilibrium resting concentration of sodium and
potassium across the membrane.
• P. Baker, Hodgkin, and T. Shaw confirmed this by the
extreme measure of emptying the axon of all its
axoplasm, replacing it by a simple solution with
potassium but no sodium.
THE PATCH-CLAMP TECHNIQUE
ALLOWS THE STUDY OF SINGLE
ION CHANNEL BEHAVIOR
12.3.2
• Hodgkin and Huxley’s measured the behavior of
the membrane conductances under space- and
voltage-clamped conditions
• They suspected the existence of ion channels,
• But they could not see the molecular
mechanisms for ion transport because they were
observing the collective behavior of thousands
ion channels, not the behavior of any individual
channel.
Patch-clamp technique
• Developed by Neher and B. Sakmann in 1975
• One of the first results of patch-clamp recording was an
accurate value for the conductance of individual
channels: A typical value is G ≈ 25 × 10−12Ω−1 for the
open sodium channel.
• Driving potential of V −𝑉𝑁𝑎+𝑛𝑒𝑟𝑛𝑠𝑡≈ 100mV the current
through a single open channel is 2.5 pA.
Results for conductance from patch clamp
a) Mechanism of conduction
• The simplest model for ion channels:
• Each one is a barrel-shaped array of protein
subunits inserted in the axon’s bilayer
membrane (see below), creating a hole through
which ions can pass diffusively.
b) Specificity
• The channel concept suggests that the
independent conductance of the axon
membrane arise through the presence of two
(actually, several) subpopulations of channels,
each carrying only one type of ion and each with
its own voltage-gating behavior
• Sodium Channel
– A channel can accept smaller ions while
rejecting larger ones
– A channel can pass positive ions in
preference to neutral or negative objects
Potassium channel?
• How can a channel pass a large cation, rejecting
smaller ones?
• C. Armstrong and B. Hille in early 1970s. Idea is
that the channel could contain a constriction so
narrow that hydrated ions, have to “undress”
(lose some of their bound water molecules) in
order to pass through
– See next slide
Vestibule & Filtering
Alberts: Molecular Biology Of The Cell 5th Ed.
c) Voltage-gating
• A net positive
charge embedded
in a movable part of
the channel gets
pulled by an
external field. An
allosteric coupling
then converts this
motion into a major
conformational
change, which
opens a gate.
c) Voltage Gating (continued)
• The conformational change is discrete (Chap 9)
• For example, the traces in Figure 12.17b each
show a single channel jumping between a
closed state with zero current and an open state,
which always gives roughly the same current.
c) Voltage gate discreteness?
• Hodgkin & Huxley measured continuous values
in space clamp technique, but on small scales
the switches are discrete (patch clamp)
• Recalling Chap 6 and free energy transitions,
e.g., RNA folding as a two state process we
again find a similar transition closed -> open:
𝑃𝑜𝑝𝑒𝑛 =1
1 + 𝑒Δ𝐹/𝑘𝐵𝑇
• Can’t predict Δ𝐹 without detailed molecular
modelling, but can predict change in Δ𝐹 when
varying V.
• Upon switching, total charge 𝑞 moves distance 𝑙perpendicular to membrane with external field
ℰ ≈ 𝑉/𝑑 then this gives extra contribution to
Δ𝐹 = −𝑞ℰ𝑙 = −𝑞𝑉𝑙/𝑑. This implies Eq. (12.28):
were Δ𝐹0 is unknown internal part: 𝐴 ≡ 𝑒Δ𝐹/𝑘𝐵𝑇
continued
Again theory & experiment agree!!!
d) Kinetics
• Kinetic interpretation of the Boltzmann
distribution (Sec. 6.6.2)
• If the probabilities of occupation are initially not
equal to their equilibrium values, they will
approach those values exponentially (Eq. 6.30)
Ligand-gated ion channels
Ligand-gated ion channels
• The channels studied in Figure 12.19 are
sensitive to the presence of the molecule
acetylcholine, a neurotransmitter
• At the start of each trial, a sudden release of
acetylcholine opens a number of channels
simultaneously.
• Though each channel is either fully open or fully
shut, adding the conductance of many channels
thus gives a total membrane current that roughly
approximates a continuous exponential
relaxation
d) Kinetic continued
• The complex, open-then-shut dynamics of the
sodium channel arises from the all-or-nothing
opening and closing of individual sodium
channels
Observations through ingenious exp
• Observation of two-stage dynamics of the
sodium conductance under sustained
depolarization to two independent, successive
obstructions.
• One of these obstructions opens rapidly upon
depolarization, whereas the other closes slowly.
The second process, called inactivation,
involves a channel-inactivating segment,
loosely attached to the sodium channel by a
flexible tether.
12.4 NERVE, MUSCLE
SYNAPSE: EXTRA READING
Don’t FORGET Your Turn 12A for homework.
top related