Single Channel Kinetics
 
Analysis of single channel kinetics is a vast topic. This page briefly describes how single channel kinetics can be used to understand certain properties of ion permeation and open channel block.

 
Nomenclature of Ion Channel Currents and Kinetics
 
Currents
The maximal single channel current (i) is the current amplitude, usually expressed in picoAmperes (pA, 1e-12A) for the fully open state of the channel. The mean single channel current (mi) is averaged current for a sample of channel activity.
 
Macroscopic currents (I) are the summation of currents from multiple single channels, as recorded with the whole cell patch configuration, usually expressed in
nanoAmperes (nA, 1e-9A). The relationship of macroscopic and single channel currents can be expresed as
 

I = npi

 
where I is the macroscopic current at voltage V, n is the number of active channels, p is the probability of the channel opening at V, and i is the single channel current. The open probability is basically the mean current mi normalized to i.
 

p = mi / i

 
 
Kinetics
tau(open): mean lifetime of sampled opening events
tau(close): mean lifetime of sampled closing events (or blocking events)
 
1/tau(open): the opening rate (or blocking rate) = k(close) = k(on) = entry rate (of blocker)
1/tau(close): the closing rate (or unblocking rate) = k(open) = k(off) = exit rate (of blocker)
 
p: the single channel open probability = tau(open) / {tau(open) + tau(close)}
 
 
Variance of Single Channel Currents
Odgen and Colquhoun (1985) described the increased variance in open channel current noise recorded from muscle-type nicotinic receptors in the presence of ACh. Assuming that the variance results from increased shuttings of the channel, they used the binomial distribution to describe the magnitude of the variance as
 

variance (i) = i2p(1-p)

 
and
 

variance (i) = mi(I-mi)

 
where i is the single channel current amplitude and p is the open probability.
 
 
Location of the Energy Barriers to Ion Blocking and Unblocking
 
Estimation from Blocking Kinetics
The simple model of open channel block (Armstrong, 1969; Neher and Steinbach, 1978) posits that the permeation pathway can be in one of three states: open, closed, or blocked. The simple model makes several predictions about the kinetics of block. First, histograms of open and closed times are exponentially distributed, reflecting the existence of a single open and a single blocked state. Second, the inverse of the mean open time (the blocking rate) is linearly related to the concentration of blocker. Third, the inverse of the mean closed time (the exit or unblocking rate) is independent of blocker concentration. The blocked state is a special condition of the open state in which flow of permeant ions is interrupted by blocking ions which traverse the pore much more slowly. Woodhull (1973) proposed that the membrane potential contributes to the free energy of blocker binding in the channel. Depending on the location of the binding site of the blocking ion in the membrane field, hyperpolarization will often lower the energy barrier and speed the exit of blocker from the pore (Lansman et al, 1986, Lansman, 1990; but see Haws, et al, 1996). The Woodhull model (Woodhull, 1973) can be used to determine the location of the blocker binding site in the membrane field
 

KD(V) = KD(0)*exp(zdFV/RT)

 
where z is the valence of the blocking ion, d (Greek letter delta) is the electrical distance to the blocking site, and FV/RT has the usual meanings. The model predicts that blockers which bind to sites within the permeation pathway will be sensitive to the membrane voltage field. For example, the lanthanides bind to a site well within the membrane field in dihydropyridine-sensitive Ca channels (Lansman, 1990). The lanthanides exhibit voltage-dependent relief of block, in which the blocker exit rate increases exponentially as a function of the patch potential. However, the entry rate of lanthanides is independent of voltage. Other blockers such as transition metal ions have voltage-dependent entry and exit rates (Winegar and Lansman, 1990; Winegar et al, 1991). The entry and exit rates of the blocking ion are predicted by exponential fits to kinetic data plotted as a function of voltage. KD(0) is determined from the exponential fits of the kinetic data by the relation

KD(0) = koff(0) / kon(0)

and then

KD(V) = koff(V) / kon(V)

 
To use a specific example, in one experiment the estimated kinetics of Zn2+ block and unblock in the skeletal muscle dihydropyridine-sensitive Ca channel was
 

k(off) = 14051*exp(-0.024*mV)

k(on) = 14100*exp(-0.015*mV)

 
From these data we obtain
 

KD(V) = 996.5 µM * exp(-0.009*mV)

 
Knowing the valence (z) of Zn2+, it is simple to obtain the electrical distance to the binding site, delta = -0.11. The low value of delta indicates that Zn2+ binds to a relatively peripheral site within the membrane field.
 
 
Estimation from Steady-State Data
The logarithm of the KD measured at 0, -20, -40, and -60 mV is plotted as a function of voltage. The voltage-dependence of the steady-state block produced by Zn2+ can be described by a relation of the form
 

i/i(max) = {1+[Zn2+]/KD exp(-zdFV/RT)}-1

 
in which the KD is a function of voltage, z is the valence of the blocking particle, and d (delta) is the effective electrical distance to the blocking site (Woodhull, 1973). From the above equation
 

KD = KOexp(zdFV/RT)

where
 

KO = KD(0 mV)

 
You can use a Marquardt estimation algorithm to determine KD, such as in DeltaGraph 4.0. Solve for 1/1+(x/a[1]) in which a[1] = K(0)exp(zFV/RT).
 
F=96480 Cmol-1
R=8.314 VCK-1 mol-1
T=295.16 d. K
F/RT = 39.316044
KD = K(0) * exp(-zdFV/RT)
Y = B * exp(-M*X)
1/-M = V for e-fold change in KD
d = -M/(F/RT)/-z*103 for KD computed with mV
 
 
Rectangular Hyperbola
A useful model for describing dependence of the blocking rate on the concentration of blocking ion (See Lansman et al, 1986).

1/y = mx + b

y = 1/b / [1+(x(b/m))]

note, b/m = t(1/2)
 
 
References
 
Armstrong CM. 1969. Inactivation of the potassium conductance and related phenomena caused by quaternary ammonium ion injected in squid giant axons. J. Gen. Physiol., 54, 553-575.
 
Haws CM, Winegar BD, Lansman JB. 1996. Block of single L-type Ca2+ channels in skeletal muscle fibers by aminoglycoside antibiotics. J Gen Physiol, 107, 421-432.
 
Lansman JB. 1990. Blockade of current through single calcium channels by trivalent lanthanide cations. Effect of ionic radius on the rates of ion entry and exit. J Gen Physiol, 95, 679-696.
 
Lansman, JB, Hess, P, Tsien RW. (1986) Blockade of current through single calcium channels by Cd2+, Mg2+, and Ca2+. Voltage and concentration dependence of calcium entry into the pore. J Gen Physiol, 88,321-347.
 
Neher E, Steinbach JH. 1978. Local anaesthetics transiently block currents through single acetylcholine receptor channels. J Physiol, 277, 153-176.
 
Ogden, DC, Colquhoun, D. 1985. Ion channel block by acetylcholine, carbachol and suberyldicholine at the frog neuromuscular junction. Proc. R. Soc. Lond. B 225, 329-355.
 
Winegar, BD, Lansman, JB. 1990. Voltage-dependent block by zinc of single dihydropyridine-sensitive calcium channels in mouse C2 myotubes. J Physiol, 425,563-578.
 
Winegar, BD. Kelly R, Lansman JB. 1991. Block of current through single calcium channels by Fe, Co, and Ni. J Gen Physiol, 97, 351-367.
 
Woodhull, A. 1973. Ionic blockage of sodium channels in nerve. J Gen Physiol, 61,687-708.