2.4 Ions’ thermodynamics 2.4.5 Laws of motion 2.4.7 Fick’s Law

2.4.6 Nernst’s law

In a segmented electrolyte, the electric charges are typically globally balanced; however, they may become unbalanced locally due to physical reasons. The two sides of Eq.(2.9) are the derivatives of the equation

V_{C_{k}}=\frac{RT}{q*F}\ln{\biggl{(}\frac{C_{k}^{ext}}{C_{k}^{ref}}\biggr{)}}

(2.20)

known as Nernst equation. In other words, Eq.(2.9) and Eq.(2.20) are the differential and integral formulations of the same knowledge. The limits of the integration are chosen arbitrarily (by choosing the reference concentration). Hence, the derived potentials inherently comprise an additive term (a potential difference), so they are not intercomparable directly if they use a different reference $C_{k}^{ref}$ . The derivatives are spatial derivatives; the temporal derivatives (needed for describing the time course) of the concentration(s) and voltage are derived above and in [22]. (Eq. (2.20) results in opposite signs according to the ’ordinary’ and ’non-ordinary’ laws of physics. Experience shows that the ’non-ordinary’ laws result in the correct sign.)

The Nernst potential (or reversal potential) is the specific membrane voltage where the net flow of a particular ion across a cell membrane stops, balancing the chemical concentration gradient with the electrical gradient, described by the Nernst equation. It’s calculated using the ion’s concentration ratio, its charge (valence), and temperature, indicating the potential at which an ion is in electrochemical equilibrium. At the level of ions, the equilibrium means that the membrane potential equals the Nernst-potential calculated from Eq.(2.20)

Understanding the difference between Nernst potential, reversal potential and equilibrium potential crucial for understanding neuronal signaling and cell function. In addition to the membrane’s potential (discussed in section 2.6.5, defined by the membrane’s parameters) and the Nernst potential (see Eq.(2.20), defined by the concentrations), an external potential can also be present in the system. If only one type of ions is present on the two sides of the membrane, the charge transport (ion current) stops when the external potential exactly counterbalances the difference of the first two terms. As the name of the reversal potential suggests, at that external voltage value the current through the membrane reverses (changes its sign).

The case and the concepts are different if more than one ion is present in the two segments. The Nernst voltage is ion specific, so each ion must be at its reversal potential to be at an equilibrium potential. If it is not the case (and usually it is not so), the individual Nernst potentials are different, so a driving force exists for all ions. This situation results in permanent material transport for all ions until the concentrations get equal. The so called pumps work simply because that driving force permanently exists: a vast amount of $Na^{+}$ ions must be removed after generating an AP; inputting $K^{+}$ is needed to keep the electric balance of charges. It is a misunderstanding that the resting potential is the weighted average of the Nernst voltages; see section 2.6.5.

The Nernst equation is valid only in a system at equilibrium with no material flow (it cannot be interpreted for transient states, such as generating AP in a cell). The limits of the integration are chosen arbitrarily (by choosing the reference concentration $C_{k}^{int}$ ). Hence, the derived potential values inherently include an additive term (a potential difference), so one must not compare them directly if they use different reference potentials $C_{k}^{int}$ . Furthermore, it is nonsense to combine the quantities from the intracellular and extracellular sides additively, whether concentrations, or mobilities, or Nernst voltages, as it happens in the GHK equation, see section 2.6.6 and [71].