2.6.7 Moving in the membrane’s potential

Figure 2.12 also hints at how ions can move in the proximity of the membrane. On the left side, the electrical field increases toward the membrane, so the ions cannot move in that direction without an initial force being applied. The case is the same on the right side, because the charge is the opposite. An ion must gain energy to get closer to the membrane, which means, at the same time, higher potential energy. This situation explains why the ions do not diffuse into the bulk region. At their present positions, the counterforces by the membrane (and the dipole layer on it) balance the attraction from the opposite side. The ion channels can work in continuous mode (the ”resting ion channels”) and impulse mode (gated by ”caps” on the channels, in transient mode).

Notice that the resulting electrical field cannot accelerate an ion except when the ion is at the beginning of the open ion channel across the membrane. Here, the mechanical counterforce is missing, and the field drops to the value generated by the condenser inside the plates. Here, a vast electrical field accelerates the ions to a high speed (although a force needed to move against the viscosity decreases the electrostatic acceleration). The empty red circles show an ion that traverses through the channel. The figure shows their electrical fields on the two sides, at the beginning and end of the channel. (On the right side, the electrical field is negative, but so is the charge of the ions.)

The electrical acceleration sharply decreases after leaving the ion channel (see the right side). As shown by the difference of the vertical positions of the filled red and green balls, the difference in the electrical field values is huge across the membrane ($E_{accelerate}$) and considerable inside the dielectric electrolyte ($E_{assist}$) as shown by the difference in the electrical field values of the green balls. The field still accelerates, but significantly less intensely than the one in the gap. Suppose we assume that the ion quickly accelerates to its Stokes-Einstein speed. In that case, the travel speeds in those space regions are proportional to the difference of the fields in the shown positions and the reciprocals of their travel times.

The passed ions decrease the concentration, and the electrical field on the high-concentration side increases on the low-concentration side. Similarly, the gradients decrease and increase, changing locally the gradients. At the exit, the field continues to accelerate the ion. However, the increasing potential and concentration increasingly decelerate the passing ions, so they quickly brake them to the assisted speed. Then, the effect of the electrical field cancels, and the ions will move with their diffusion speed (”the ion stops”).

As Eq.(2.55) states, the electrolyte separated by a permeable membrane is balanced when the electrical and thermodynamic forces are equal. The $E_{thermal}^{C_k}(d)$ ”electrical field” depends on the chemical quality of the ion; the electrical field does not. That means the resulting driving forces acting on the different ions are different. While one ion experiences a resulting force and moves to the other segment, the other remains resting. Of course, the ions moving across the segments change the overall concentrations in both segments (and so the electrical field contribution), indirectly affecting the balanced state of the other ions. When a mixture of ions is present in the volume, all ions feel a different driving force, and the solution as a whole will be balanced when the resulting forces for all chemical ions are balanced. The permanently open ion channels only passively participate in the process. The driving forces of the ions are different, and they produce, depending on the actual concentrations, the selectivity of the channels.

When $N{a}^{+}$ ions rush into the intracellular layer, they roughly increase the overall concentration and the potential. All other ions, including $K^{+}$, feel a driving force. The targeted membrane potential is set electrically, and the driving gradients may behave controversially during the transient period. The actual voltage gradient may temporarily direct the chemical gradient in opposite directions. Phenomena such as ’over-relaxation’, in this sense, similar to ’hyperpolarization’, may happen.

However, the layer itself can also be modeled as having just a few ions under their mutual repulsion on the surface or a few atomic layers on top of each other, depending on the concentration and voltage in the bulk on the two sides. (The voltage gradient diagram line validates the plane crossing the membrane and the ion channel.)

Our results align with the observation (see caption of 11.22 in [107]). The amount of unbalanced ions is in the range of ${10}^7$, and so is the amount of rush-in ions. In addition, those ions on the high-concentration side rush into the low-concentration side and cause a large change in the membrane potential. Their absolute amount is small compared to the total number of ions in the cell, but it is significant compared to the number of unbalanced ions.