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Quantitative estimates of metabolic costs in this study are based on the ATP that is required to fuel the Na⁺/K⁺ pump. This includes the cost of the restoration of sodium and potassium ions that flow to support action potentials, resting potentials, and postsynaptic potentials.

The co-expression of pumps and sodium leak channels (see Figure 1) and even an ideal voltage dependence of the pump (see Figure 6) have a direct impact on the metabolic cost related to this ATP-fueled Na⁺/K⁺ pump. By integrating the net pump current over time and dividing by one elemental charge, we find the rate of ATP that is consumed for either compensatory mechanism. When compensating a relatively `constant’ Na⁺/K⁺-pump current with sodium leak channels, the amount of ATP spent on pumping sodium is 33% higher than it would be for a voltage-dependent pump (see Equation 22, Methods).

The impact that either of these compensatory mechanisms has on the whole cell, however, also depends on other costs, such as those related to cellular maintenance. A voltage-dependent pump would save costs related to Na⁺/K⁺ pumping, which, based on energy budgets formerly estimated for AP-firing neurons in the brain (Howarth et al., 2012), is likely to be one of the main contributors to the total metabolic cost (in cerebellar cortex, for example, amounting to >50% of the total metabolic cost). Because the peak load of a voltage-dependent pump, however, is four times higher than a relatively constant pump, four times more Na⁺/K⁺ pumps would need to be expressed on the cell membrane. To be more exact, if a single pump translocates around 450 sodium ions per second (Gennis, 2013), 8×10¹⁰ pumps are required to support constant pumping, and 32×10¹⁰ pumps are needed to support voltage-dependent pumping. If one assumes the electrocyte is a perfect cylinder, and its membrane surface were smooth (an approximation not too realistic), the total available membrane space would be 3.4 mm² (Ban et al., 2015). If the Na⁺/K⁺ATPase expression density would be as high as in the outer medulla of rabbit kidney (Deguchi et al., 1977), where ATPases are densely packed, a smooth electrocyte membrane would `fit’ 4.2×10<sup>10</sup> pumps, which is two times less than necessary for constant pumping, and eight times less than required for voltage-dependent pumps. According to our model, therefore, the invaginations on the posterior side of the membrane (Ban et al., 2015) are necessary to drastically increase membrane area in order to support the large number of pumps required for ion restoration. This, in turn, would increase the `housekeeping’ costs of the cell related to turnover of macromolecules, axoplasmic transport, and mitochondrial proton leak, which in different brain areas are estimated to occupy 25–50% of the total energy budget (Kety, 1957; Attwell and Laughlin, 2001). As there is insufficient data on the ratio between costs related to Na⁺/K⁺ pumping and `housekeeping costs’, and the fraction of housekeeping costs related to Na⁺/K⁺-pump maintenance, a quantitative comparison of the metabolic cost of the two compensatory mechanisms remains challenging. Future experiments that would aid in answering this question could involve blockage of electrocyte Na⁺/K⁺ pumps and comparing oxygen consumption to a control where electrocyte Na⁺/K⁺ pumps are functional.

Another compensatory mechanism that was discussed in this article is extracellular potassium buffering (see Figure 4), which in electrocytes likely occurs via its extensive capillary beds (Ban et al., 2015) that transport excess extracellular potassium to the kidney. Assuming that an equal amount of ATP is needed in total to fuel Na⁺/K⁺ pumps, either all in the electrocyte, or partly at the electrocyte and partly in the kidney, the additional costs incurred by the extracellular potassium buffer would be dominated by the structural and maintenance costs of the capillaries. We are, however, not aware of an accurate estimate of these costs, especially since the capillaries also have additional functions such as providing other resources and transporting other waste products.

Lastly, a strong synapse was said in the article to support cell entrainment under fluctuating pump currents (see Figure 5), but also to incur additional metabolic costs. In the example shown in the main text, however, baseline Na⁺/K⁺ costs are smaller for a stronger synapse; see Figure 5B (weak synapse) vs. Figure 5E (strong synapse). This is the case because, similarly as shown in Figure 7B in Joos et al., 2018, a weak synapse elicits smaller postsynaptic potentials, which lowers the AP peak with respect to a stronger synapse. To make a fair comparison on the metabolic costs between a weak and a strong synapse, voltage-gated sodium conductances were scaled to maintain a peak amplitude of 13 mV (see Table 2, Methods). For weak synaptic stimulation, a higher voltage-gated sodium conductance was needed to reach this peak amplitude, which, due to the excess inflow of sodium through these voltage-gated channels, resulted in an increase of 10% in ATP consumption by Na⁺/K⁺ pumps with respect to strong synaptic stimulation.

There are, however, additional costs that scale with synapse strength, such as the restoration of presynaptic calcium, the restoration of (presumably small amounts of) postsynaptic calcium, and neurotransmitter packaging and recycling. In the brain, these costs are estimated to be 0.18–1 times the cost of fueling the Na⁺/K⁺ pumps that restore the sodium ions that traverse neurotransmitter receptor channels (Howarth et al., 2012; Liotta et al., 2012). In our model, merely 11% of sodium ions enter the electrocyte via neurotransmitter receptor channels in the strong-synapse case. Assuming that the above-mentioned additional costs are equal to those related to Na⁺/K⁺ pumping of neurotransmitter-related currents (according to the upper bound estimate by Liotta et al., 2012), a weak synapse (half the size of the strong synapse) would incur a cost increase of 5.5% and a strong synapse would incur an increase of 11%. This would, however, still result in a 4% higher cost efficiency of a strong synapse compared to a weak synapse.

There is reason to believe that the fraction of the energy budget related to the restoration of presynaptic calcium, the restoration of (presumably small amounts of) postsynaptic calcium, and neurotransmitter packaging and recycling in the electrocyte could differ significantly from those estimated by Howarth et al., 2012; Liotta et al., 2012. First, to the best of our knowledge, such energy budget estimations have only been done for neurons active at significantly lower firing rates than electrocytes (by a factor of approximately 100), and, second, operate mostly under the glutamate neurotransmitter, while electrocyte receptor channels are activated by acetylcholine. An accurate estimate of the impact of synapse strength on the electrocyte energy budget, therefore, requires quantitative data on the rapid dynamics of acetylcholine production in the presynaptic neuron and recycling in the synaptic cleft, which, currently, is also hard to estimate.

Supported by the above-mentioned considerations, we argue that the impact of mechanisms that compensate for Na⁺/K⁺-pump currents on an electrocyte’s metabolic cost could be significant. Due to the absence of more detailed experimental quantification, a plausible quantitative cost estimate remains beyond the scope of this article. We note, however, that although the metabolic costs of potassium buffering and synaptic strength are likely to differ between cell types, the energetic estimate of the respective ATP requirements by Na⁺/K⁺ pumps for constant vs. voltage-dependent pumping generalizes and extends to all excitable cell types (as is discussed in the Discussion in the main text, see ‘Generalization to other cell types’).