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LETTER TO THE EDITOR
We agree with Jurisic and Bezanilla (2006)
that simple, variable capacitance is properly written as differential capacitance. The problem is that even this time-varying capacitance must change instantaneously with each voltage change to qualify precisely as capacitance. Thus with respect to this latter point, we worry about delay in the movement of gating charges when the membrane potential changes.
The equations we use are in fact both time and voltage dependent, where the voltage dependency is carried implicitly in the time evolution of the m gate. Clearly the differential equations must be considered as a function of time and voltage until assumptions are made, such as the constancy of Q and its approximately instantaneous, capacitance-like behavior. In fact, we are not antagonistic to such assumptions.
Sangrey et al. (2004) explicitly considered the possible viability of the constant gating charge hypothesis favored by Jurisic and Bezanilla (2006).
One version considered in Sangrey et al. (2004), a variant of serial model II, is inspired by Patlak's model-8, which we interpreted as having approximately constant gating charge. In this model, there are four successive closed states that precede channel opening, and there is the explicit assumption that as some charges are used up, then simultaneously, others appear (see Table 2 of Patlak 1991). To implement constant gating charge, we delayed the onset of our (1 – m) gating charge diminution. That is, we extended the initial resting capacitance into the rising action potential. As stated on page 2546, "... separate unpublished simulations altering the delay in the time of capacitance reduction have confirmed this insight. By delaying the time at which the gating capacitance begins to go to zero, we saw that the conduction velocity of the action potential was only slightly affected." Furthermore, we also reported that such changes had little effect on our conclusions: "...it indicates that there is considerable room for a temporal shift of the time when the gating current goes to zero (as we would expect in a serial model) that will have no effect on the velocity. What only seems to matter is the early reduction in capacitance...." By the phrase "early reduction in capacitance," we mean that reducing the assumed, resting value of gating capacitance is what most profoundly influences the velocity and its maximization.
Finally, because Jurisic and Bezanilla (2006) seem to be casting doubt on the ultimate conclusion of Sangrey et al. (2004) and, implicitly, the follow-on result of Crotty et al. (2006), we generate new data using the exact model suggested by Jurisic and Bezanilla (2006). Specifically, while holding gating capacitance constant at its maximum (i.e., initial) value for the duration of the action potential, we recalculate the major results of our papers as a function of axon channel densities. As before, we used the estimate of 1 nF of gating capacitance per millisiemen of sodium channel conductance. The data summarized in the accompanying figure confirm our earlier conclusions. The channel density producing maximum velocity is unsatisfactory as an evolved optimization, especially when compared with minimizing the wavefront energy cost restricted to the observed axonal velocity. The Adrian (1975)/Hodgkin (1975) velocity maximizing conjecture implies a conductance density (
Na) of 320 mS/cm2, while the energy minimizing conjecture leads to an optimum at 130 mS/cm2, a value within 2% of the Conti et al. (1975) measurement. Thus our conclusions remain intact, and we reject Hodgkin's conjectured evolutionary optimization in favor of the Crotty et al. conjecture.
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Adrian RH. Conduction velocity and gating current in the squid giant axon. Proc R Soc Lond B Biol Sci 189: 81–86, 1975.[Medline]
Conti F, DeFelice LJ, and Wanke E. Potassium and sodium ion current noise in the membrane of the squid giant axon. J Physiol 248: 45–82, 1975.
Crotty P, Sangrey T, and Levy WB. The metabolic energy cost of action potential velocity. J Neurophysiol In press.
Hodgkin AL. The optimum density of sodium channels in an unmyelinated nerve. Philos Trans R Soc Lond B Biol Sci 270: 297–300, 1975.[Web of Science][Medline]
Hodgkin AL and Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117: 500–544, 1952.
Jurisic N and Bezanilla F. Letter to the Editor. J Neurophysiol 96: 959, 2006.
Patlak JB. Molecular kinetics of voltage-dependent Na+ channels. Physiol Rev 71: 1047–1080, 1991.
Sangrey TD, Friesen WO, and Levy WB. Analysis of the optimal channel density of the squid giant axon using a re-parameterized Hodgkin–Huxley model. J Neurophysiol 91: 2541–2550, 2004.
William B. Levy
Patrick Crotty
Department of Neurosurgery,
University of Virginia School of Medicine,
Charlottesville, VA 22908
Thomas Sangrey
Biology Department,
Emory University,
Atlanta, GA 30322
Otto Friesen
Department of Biology,
University of Virginia,
Charlottesville, VA 22904
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