22.416 Lecture 4

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22.416 Lecture 4

• Hodgkin and HuxleysMathematical Analysis and
  Reconstruction
• of the Action Potential
• - recall H Hs V-clamp data yielded curves
  showing how the membranes conductances to Na K
  (gNa gK) changed with time after any voltage
  step to a new VC gNa(t) INa(t) / (VC - ENa),
  and gK(t) IK(t) / (VC - EK)

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H Hs mathematical fitting of g(t) curves

• HH showed both curves could be fitted as
  products of exponential decay functions of form
  e-t /? and 1- e-t /?
• for K gK(t) 1- e-t/? 4 x constant
  gK(max)   n4 x gK(max)

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n is a f(V,t) with value between 0 and 1

• n 0 when channels are all closed at -ve VM
• n 1 when channels are all open at very ve VM
• - at any intermediate voltage (V1, ...), n1,
  ... reaches a steady-state value between 0 and 1
• Upon step from V1 to V2, ns value changes from
  n1 to n2 with an exponential time course as shown
  above n2 n1 ? (1 e -t/ ?n)

4
HHs biological interpretation of gK(V,t)
equation

• H H knew that a simple exponential time course
  like this describes a first-order transition,
  when ? Unreacted substrate U ?
  Reacted product R ?
• If R is 0 at first, then U decreases
  exponentially as R grows, both with time
  constant ? (which depends on the forward and
  backward rate constants, ? and ?)
• When forward and backward transitions occur at
  same rate, a steady state exists
• final reacted fraction, R / (UR), depends on ?
  ?
• so can calculate ? ? from measured ? R /
  (UR)

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Application to membrane's K conductance

• So how might (1-e -t/?)4 dependence of gK(t)
  arise?
• suggests that probability of K channel being
  open is the product of four identical 1st-order
  transition probabilities
• ( prob of short, brown-eyed, brunette boy baby
  p1.p2.p3.p4)
• H H suggested 4 independent gates in a K path
  must ALL shift from resting to activated
  state to open the path
• if only 1, 2 or 3 of the 4 gates shift, K
  cannot pass through
• Nowadays, gates current path are considered
  parts of the K channel protein, a concept not
  yet developed in1950

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Interpretation of n

• So n prob that any single gate in a channel is
  open
• n4 prob that all 4 channel gates have
  opened at once prob that the channel is
  open to K flow
• With strong depoln ( 100 mV), prob n ? 1, so
  n4 ? 1 i.e., all K channels will open
• - with smaller VC step, n8(VC) ? plateau value
  ( 0) so only a fraction ( n4 ) of
  K channels will open
• prob n moves exponentially to new value after VM
  changes
• so (N.B) prob n does not change instantaneously
  with VC (only ?, ? ? do)

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But closing the K channel only requires any one
of the 4 gates to return to the closed state ...

• so the decline in gK at the end of the step
  follows a simple exponential decay function, e
  -t/?n (NeuroSim, HH, file clamp7

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Equation for gNa and its interpretation

• more complex gNa curves also fit product of 4
  exponentials
• one (h) decreases during a depoln step, as e-t/
  ?h, while
• the other three (all m), increase, as 1 - e -
  t/ ?m (like n)
• gNa gNa(max) x m3 x h, with 0 ? m,h ? 1

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Interpretation of m and h

• m ? 0 near VMR, but increases to 1 upon
  strong depolarization
• m represents probability that any one of 3
  independent activation gates in the Na path is
  open
• m3 prob that all 3 of these m gates are open
  at once
• h probability that a single inactivation gate
  is open
• h 0 during strong depolarization (total
  inactivation) 1 at very negative VM,
  (no inactivation) ? 0.6 at VMR
  in squid axon Fig. 6.6 (40 inactivation)
• note (Fig. 6.6) that a hyperpolarizing clamp
  prepulse before the depolarizing step removes
  inactivation (? h)
• so prob that all 4 gates in the Na path are open
  to Na m3h

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m h gates also differ in speed

• m gates open much faster than h closes during
  depoln i.e., ?m
• but all 3 m gates must open before h closes, to
  pass Na
• so only a fraction of Na channels can open
  during 1 ap- most V-gated Na channels never
  open during an ap, because h gate cuts off path
  before all 3 m gates open (NeuroSim exercises
  demonstrate this)

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Consequence of above -- rebound excitation

• - if we hyperpolarize the membrane so h ? 1
  (removing inactivation),
• - then when hyperpolarizing current ceases, ap
  may be set off spontaneously, below VMR, as VM
  returns to VMR
• Why? 3 reasons
• a) removing inactivation raises h ? ? gNa (?
  m3h)
• b) greater speed of m gates lets more Na
  channels get open before h falls too far
• c) hyperpoln ? ?n ? ? gK
• All 3 effects help to lower the ap threshold
  below VMR

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Reconstructing ap from g(t) equations

• H H now predicted precise shape of giant
  axon's ap. Began with Q CV, so dQ/dt CdV/dt,
  and I dQ/dt
• - Q (membrane charge) C (membrane capacitance)
  x VM
• - membrane current I ( INa IK ILeak) changes
  Q I dQ/dt C dVM/dT INa IK IL
• At any membrane voltage VM, they could calculate
• INa gNa (VM, t) x (VM - ENa), from known gNa
  (t) curves
• IK likewise
• IL leakage current due to Cl, etc. from
  measured gL (constant leakage conductance)
  equilibrium voltage EL

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To calculate aps time course,

• HH started at resting VM,
• calculated n, m h from measured ?s ?s at
  that VM
• displaced VM(t) above threshold at time t (?
  stimulus),
• calculated new n, m, h at new VM
• calculated new gs from new n, m and h
• calculated Is from gs, hence dVM/dt (INa
  IK IL ) / C
• calculated ?VM (change in VM over brief ?t, say
  0.01 ms) dVM /dt x ?t, and hence new VM(t
  ?t) VM((t) ?VM
• then found new ?s and ?s for new VM and
  iterated above thousands of times (by hand) ?
  graph of VM against time

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What Hodgkin and Huxley achieved

• accurately predicted shape of recorded ap, as
  well as all aps observed properties
  (refractoriness, etc.)
• proved ap resulted from separate voltage-induced
  changes in conductance to Na and K
• inferred gating particles, whose stochastic
  but V-induced openings must coincide to open Na
  K pathways
• these findings led to idea of V-gated ion
  channel proteins, later supported by
  biochemistry, em, molecular biology,...(one of
  the most fruitful concepts in the history of
  biology)
• mathematical analysis still holds up (forms the
  basis for the NeuroSim ap simulation)

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4. Patch clamp reveals channel behaviour

• Fig. 6.10 patch is repeatedly stepped from
  "holding VM" of -100 mV (to remove inactivation)
  to -60 mV for 23 ms
• brief inward current pulses (mean 0.7 ms
  duration) occur randomly during each step, as
  channel flips open closed
• Most channel openings occur near the beginning
  of the step, as 300 traces added together (bottom
  trace) show
• opening probability ?abruptly at onset of
  depoln (as m ?),
• then ? gradually over 4 msec (as h ?)
• illustrates randomness of gate openings
  closings, with probabilities m h -- chance that
  all 4 are open at once is low (NeuroSims "Model"
  feature illustrates this point well)

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Just out of interest ....

• Note how same data in 4th Edition have sharpened
  with age( fudging) over those in the 3rd
  Edition (on the overhead)!
• - both based on same figure in Sigworth and Neher
  (1980)