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# Axons: Physics and chemistry at work in the body

## Problem

In a nerve cell, ion channels open and close based on their environment and other factors. When a channel is open, it behaves as an Ohmic resistor (V=IR). When it is closed, an insulator. The currents through the ion channels are determined by the electrical potential and ion concentration differences between the cytoplasm inside the cell and the surrounding interstitial fluid. If only sodium and potassium ions were present, the “reversal potential,” V, start subscript, m, end subscript, would be predicted by the Goldman-Hodgkin-Katz equation, (Equation 1).
V, start subscript, m, end subscript, equals, 59, start text, space, m, V, end text, dot, l, o, g, left parenthesis, start fraction, P, start subscript, N, a, start superscript, plus, end superscript, end subscript, open bracket, N, a, start superscript, plus, end superscript, close bracket, start subscript, o, u, t, end subscript, plus, P, start subscript, K, start superscript, plus, end superscript, end subscript, open bracket, K, start superscript, plus, end superscript, close bracket, start subscript, o, u, t, end subscript, divided by, P, start subscript, N, a, plus, end subscript, open bracket, N, a, start superscript, plus, end superscript, close bracket, start subscript, i, n, end subscript, plus, P, start subscript, K, start superscript, plus, end superscript, end subscript, open bracket, K, start superscript, plus, end superscript, close bracket, start subscript, i, n, end subscript, end fraction, right parenthesis
Equation 1. Reversal potential
The reverse potential is the potential that, if it were applied to the cell, would result in potassium and sodium crossing the membrane at rates that yield no net current. Each permeability, P, start subscript, X, end subscript, is proportional to the number ion X channels open at that node.
Table 1 shows the typical concentrations of sodium and potassium ions inside and outside of the cell.
Table 1. Typical ion concentrations, [mM]
inside celloutside cell
[Nastart superscript, start text, plus, end text, end superscript]15150
[Kstart superscript, start text, plus, end text, end superscript]1505
As the ions enter and exit the cytoplasm, they change the electric field within the axon, creating a new electrical potential. The cytoplasm conducts a current down its length in response to that electrical potential, much like a copper wire.
If we treat the interstitial fluid and cytoplasm as two separate wires and the channels as resistors in an equivalent circuit diagram, would adding more of these transmembrane “resistors” increase or decrease the total equivalent resistance at the node?