On 2017 Jan 31, David Attwell commented:
Our answers to Dr Brette’s comments are as follows.
(1a) Ohmic vs Goldman-Hodgkin-Katz dependence of current on voltage
We assumed an ohmic dependence on voltage of the ‘leak’ currents for Na<sup>+</sup> and K<sup>+</sup> for simplicity. The maths can also be done for a GHK dependence of Na<sup>+</sup> and K<sup>+</sup> current on voltage (e.g. see Johnston & Wu (1995), Foundations of Cellular Neurophysiology, Chapter 2, Example 2.3), but it is more convenient to assume an ohmic dependence because that then allows easier conversion of input resistances (1/(GNa+GK)) reported in the literature into ATP consumption (without the need to assume a value for [Na<sup>+</sup> ]i). In any case, the GHK equation makes assumptions (https://en.wikipedia.org/wiki/GHK_flux_equation) that are unlikely to all be correct.
(1b) Cl<sup>-</sup> permeability
Our paper was for the grey matter of the mammalian neocortex. The permeability of the membrane to chloride is reported to be very low in mammalian cortical neurons (see Fig 5c and p121 of the Discussion of Thompson, Deisz & Prince, 1988, J Neurophysiol 60, 105, available at http://jn.physiology.org/content/jn/60/1/105.full.pdf). Similarly, for rat parasympathetic neurons, Xu & Adams (1992, J Physiol 456, 405, https://www.ncbi.nlm.nih.gov/pubmed/1284080) found that for the resting membrane PCl/PK<0.001.
Nevertheless, we extended our analysis slightly, to include the Cl<sup>-</sup> component of the membrane permeability, here: Howarth, Peppiatt-Wildman & Attwell (2010) JCBFM 30, 403 http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2009.231
(2) Pumps
Dr Brette states “The model considers only the Na/K pump. However, such a system cannot be stable; there has to be at least another pump”.
This is simply incorrect, and the system is stable. The pump exports 3 Na<sup>+</sup> and imports 2 K<sup>+</sup> for each ATP consumed. The equations in our paper are set up so that there is no net current across the membrane (the sum of the Na<sup>+</sup> , K<sup>+</sup> and pump currents are zero), and so that the magnitude of the pump current is 1/3 of the Na<sup>+</sup> charge entry (so d[Na<sup>+</sup> ]i/dt=0) and 1/2 of the K<sup>+</sup> charge exit through the resting conductance (so d[K<sup>+</sup> ]i/dt=0). This can be seen by evaluating the resting potential for zero net current (following equations 1-3 of Attwell & Laughlin, 2001), and then calculating the ion fluxes.
(3) Cost of Na<sup>+</sup> extrusion at the mean potential
We assume this point is based on the notion that the energy needed to extrude Na<sup>+</sup> should be voltage-dependent, so that the ATP needed would be smaller at a more depolarised potential. In fact the stoichiometry of the Na/K pump is apparently not significantly voltage-dependent (between 0 and -60mV), so that the ATP used is always 1/3 of the Na<sup>+</sup> pumped (see Rakowski, Gadsby & de Weer, 1989, J Gen Physiol 93, 903, https://www.ncbi.nlm.nih.gov/pubmed/2544655), as we assumed.
(4) What input resistance tells us
The Attwell & Laughlin (2001) paper attempted to provide order of magnitude estimates for the ATP used on different subcellular processes in neurons, and this involved assuming that input resistance measured at the soma can give us a rough estimate of ATP use on the resting potential. The analysis addressed energy use only in the grey matter, excluding the majority of the cortical neuron axon in the white matter (which we dealt with here: https://www.ncbi.nlm.nih.gov/pubmed/22219296). Nevertheless, of course voltage is non-uniform in spatially distributed neurons, and input resistance measured at the soma will then not precisely define the resting influx of Na<sup>+</sup> measured all over the cell. In later work (Howarth, Peppiatt-Wildman & Attwell (2010) JCBFM 30, 403 http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2009.231; Howarth, Gleeson & Attwell (2012) JCBFM 32, 1222, http://journals.sagepub.com/doi/pdf/10.1038/jcbfm.2012.35) we estimated resting Na<sup>+</sup> influx in different cellular locations. Thus, there is plenty of scope for improving estimates of the energy consumed on resting potentials, as more data become available.
Summary
Broadly, most of these points reflect the fact that theoretical work often requires simplifying assumptions. Clearly the assumptions that we made have been useful, because (according to Web of Science or Google Scholar respectively) the paper has been cited 1067 or 1752 times. However, there is always room for improvement and we look forward to seeing Dr Brette’s own detailed analysis.
David Attwell & Simon Laughlin, 31-1-17
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