2.6.5 Two segments

Figure the ionic basis of a membrane potential shows and explains the case introduced by separating the cell into two segments by a finite-width membrane; although does not explain even qualitatively, why the ions behave apparently against the laws of thermodynamics: the left and right sides of the figure are apparently identical. On the left side (the case of infinitely thin membrane), at an exact balance of charges on each side, the potential across the membrane is zero. We explain that the finite thickness will result in a lack of balance (introduces inhomogeneity and creates a voltage and concentration gradient) proximal to the surfaces of the membrane, even if the concentrations on the two sides are the same. Changing the bulk concentration or potential in one of the segments creates a corresponding gradient across the separating membrane that increases the inhomogeneity proximal to the membrane. The ions will experience an extra force due to the gradient, but the mechanical counterforce of the membrane will keep them back in the segment. The concentration and potential, inseparably and having the same time course, will change across the two sides of the membrane just because of the gap’s physical features the membrane represents.

We consider that the segment is composed of electrically conducting discs (the ions are free to move on the surface) and the charged discs’ contribution to the electrical field at point $P$ (see Fig.  2.10) can be calculated as known from the theory of electricity. Due to the symmetry in direction of $z$, in a homogenous solution, the resulting electrical field in the plane perpendicular to direction $z$ is zero: we have contributions of the same size with opposite signs.

Here we used the abstraction of an infinitely thin ”charged sheet” and that there is a step-like gradient in the electrical field in the gap. The physical reality is that the charge is represented by ions, which simultaneously represent a mass; furthermore, the thickness of the surface layer they form is by orders or magnitude thicker than the layer that the electrons form. In an equilibrium state, the forces due to the voltage and concentration gradients (see Eq.(2.9)) must be counterbalanced by an external force. An interference between science disciplines can also manifest here. We know at the same time that at the boundaries of electrolytes, different interfaces, including electrically neutral electrical double layers, can be formed by only partly known processes [125]. The presence of those structures makes drawing quantitative conclusions hard.

When discussing the internal electrical operation of a neuron, first we consider a condenser that obeys laws of classical electricity and has the geometrical size of a biological neuron. We apply the physics terms to that pseudo-biological object and test whether the values we derive are consistent with physiological phenomena. A tiny (according to [2], page 12, about $2\ast {10}^{-3}$) portion of the positive ions leave their negative counterions behind and form a thin positively charged ion layer on the surface of the membrane (of course, the same happens on the opposite side with negative ions). Experience shows, that – also in biology – two very thin (but not infinitely thin) layers of ions are formed on the two sides of the membrane, see the caption of figure (Fig. 11.22 in [107]): ”The ions that give rise to the membrane potential lie in a thin () surface layer close to the membrane”. The two surfaces get covered by a sub-nanometer-thick conducting layer, on the top of a $5nm$ nanometer thick insulator. So, we model the cell with two finite-width ”conductive sheet” layers. These layers represent an electrical condenser (so we can calculate the internal electrical field between the plates) and they counterbalance each other’s electrical field outside the condenser. We may assume that the solid surface represents the needed counterforce to keep the charges in rest.

One cannot expect geometrically plane surfaces: there are some biological objects of up to $1nm$ size on the surface. So, we assume a physical ’uniformly charged sheet’ with an average thickness $\mathrm{\Delta }z=0.5nm$ on the surface. For the sake of simplicity, we assume a step-like function for the electrical field. It is zero in the segment outside the ”conducting sheets” (the ’bulk’ portions) as well as inside the sheets. Between the sheets, it jumps to the value of the electrical field of a charged sheet with surface density ${\sigma }_A$ (see Eq.(2.58)). (In this picture we see that there is a jump in the value of the electrical field (and so: in the force acting on a charge) in the two sides of the physical layer. Here we still assume that a mechanical counterforce due to surface’s roughness keeps the charges in place inside the charged layer.)

If the distance between the plates is finite, the resulting electrical field will differ from zero. With such a model, a usual parallel plate condenser can be derived, as shown in Fig. 2.11. We have two charged disks (infinite conducting sheets), and an insulator layer between them. In the classical picture, the opposite charges’amount on the two plates must be the same. As shown, a constant electrical field is present inside the membrane (across the plates of the condenser) and zero electrical field inside the parallel conducting plates, as well as outside the condenser. In this ideal picture, the charges are aligned on the border of two (infinitely thin) conducting layers and cannot move. Anyhow, the attractive force between the opposite charges on the plates keeps them fixed in direction of $z$. The repulsive force between the charges with the same sign keeps their surface density in the $(x,y)$ plane uniform: the infinitely thin plates are equipotential. The strong electrostatic force can produce an enormous acceleration for the individual ions, but the thermodynamic gradient can only change with a several orders of magnitude lower speed, allowing measurable changes in the current intensity on the surface). This picture is valid for the equilibrium state of charges, infinitely small non-dissociating charge carriers and perfectly smooth surfaces.

Figure 2.12 (the blue diagram lines) show the electrical field around the neuronal membrane. In this picture we see that there is a jump in the value of the electrical field (and so: in the force acting on a charge) in the two sides of the physical layer and we assume that the physical layer somehow emulates the well-defined boundary on the side opposite to that proximal to the membrane.

In its integral form, Gauss’s flux theorem states that the flux of the electrical field out of an arbitrary closed surface is proportional to the electrical charge enclosed by the surface, irrespective of how that charge is distributed. The surface layer represents a steep potential and concentration gradient. Above we assumed that the counterforce that keeps the charges in their place against their electrical repulsion is a kind of mechanical force: the ions cannot pass through the membrane. However, such a counterforce does not exists on the side toward the ’bulk’ part of the segment. The charges on the plates do not generate an electrical field toward the bulk, but the concentration does, according to Eq.(2.9), provided that there are charge carriers in the bulk. We hypothesize that virtual electrical charges exist in the electrolytes and their field provides the missing electrical field. In our real picture we assume a finite-thickness charged layer, and correspondingly that inside the conducting layer the electric field changes linearly. Toward the bulk of the electrolyte, a layer of dipoles creates dielectrical layer with an extra electrical field. These two layers have the same electrical field at the boundary and (according to Gauss’s theorem, their sum appears in the gap as the resulting electrical force field; see the red diagram line in Fig. 2.12.

When explaining the effect of dielectricity, we must explicitly consider the duality of ions that they obey laws of electricity and thermodynamics simultaneously; furthermore, the complexity of the electrical structure of the solution, and that the charge carriers have finite size (see ”electron size” vs ”dipole size”) . According to the theory of electricity, the free charge carriers (the dissociated ions) are located on the surface where they form strongly charged thin layers (condenser’s plates) in the segments separated by a membrane on the two surfaces of the membrane; two proximal charge layers on the surfaces of the membrane. We assume that those ions behave as point charges, i.e., due to the attraction from the opposite charges on the opposite plate, they do not produce an electrical field toward the side of the bulk of the electrolyte layers. In the classical picture, those layers represent step-like gradients in the electrical field along the $z$ axis and the the constraint that the ions must not enter the membrane provides the counterforce needed to keep the ions in place.

From thermodynamic point of view, a driving force acts as long as a concentration gradient between neighboring layers exists. From electrical point of view, the layers are simple parallel-plate condensers, that produce no electrical field outside their closed volume, and they are connected serially; plus the top layer has free ions. The electrical field is proportional with the bulk concentration in the layer. The changes in the local electrical gradient may also change the degree of dissociation and polarization, and they may produce a graded local electrical field. These two driving forces have opposite directions and in a balanced state, the same magnitude. The external force acts in a way that it constraints the molecules to stay in their layers; it adapts (compensates also for the different interaction speeds) while the gradients are changing. It presses the proximal layer to the membrane and the neighboring layers to each other. (In other words, the internal electrical force due to the forcefully changed polarization continuously decreases the electrical field and so the potential gradient and so the concentration gradient.)

Due to the finite width and the surface’s roughness, the conducting layer has a finite width, the electrical field is step-like with step size of the size of dipoles, furthermore, it is homogenous in the layers. Due to the rest of that electrical field toward the bulk layer the dipoles proximal to the neighboring dipole layer get directed and also their polarization increases; in other words, they ”produce” an electrical field. The final effect is that a dipole layer is attracted to the charged layer and it forms another layer that shows a charged layer toward the bulk. The process repeats; the process results in a decaying electrical field in function of the distance from the membrane’s surface.

In our model we build the volume of the electrolyte from thin layers of directed dipole molecules (the thinness $\mathrm{\Delta }z$ is limited by the size of molecules) in the volume separated into two segments with electrolytes by a membrane with a finite thickness $d$ (that is, a point’s distance in the electrolyte from the two disks will be $z$ and $z+d$, respectively). The contributing force field of an infinitesimal volume on the axis at point $z$ due to the charged sheet

and the total potential due to all elements $dz$ from the left side is (in the mathematical formulas below we express the distance in units of $\frac{z}{d}$, so here $z$ is dimensionless).

Correspondingly,

and at $5nm$ membrane thickness, the voltage across the plates becomes

Hodgkin in 1964 measured molarity values in squid axons for ions $K+$, $N{a}^{+}$ and $C{l}^{-}$, (400,50,40-150) inside and (20,440,560) outside, and they provided potential values $55-75mV$ [2]. ”The plasma membrane of all cells, including nerve cells, is approximately 6 to 8 nm thick and consists of a mosaic of lipids and proteins.” [41], page 71. ”The outer membrane of nerve cells is composed of a lipid bilayer $\approx 8-10nm$ thick” [126] Using such a value for the membrane’s thickness may result in a value up to 100% higher, furthermore, the usual concentration is also up to 20% higher. Given that we used a plausible but ad hoc ”charged layer thickness” and gap distance, we cannot expect a better agreement.

In our model we assumed that the ions form a uniformly charged layer on the surface (so inside that layer the electrical field changes linearly with coordinate $z$), there is no charge between the condenser plates so the electrical field is constant (it is equal with the value taken on the proximal side of the layer), and in the dielectrical solution changes as described by Eq. (2.69). On the boundary of the dielectrical layer and the thick charged layer the field takes the same value.

The potential’s and the electrical field’s magnitudes are independent of the chemical nature of the ion. At a given concentration, the potential difference is caused by the finite width of the membrane plus the dissociation of the ions. Given that the same number of charged ions must be present on the two sides of the membrane, the surface density ${\sigma }_A$ must be the same on the two sides. Notice that the effect is purely electrostatic, and will result in an asymmetric distribution of ions; no permeability is needed. If the membrane is permeable, ions will move across the membrane until equilibrium reached. The resulting potential difference depends linearly on the concentration difference. In the light of our derivation of the resting potential, one must revisit the derivation of the Goldman-Hodgkin-Katz potential, see 2.6.6.

Fig. 2.12 displays how the function shapes of electrical field change in the function of the distance from the membrane. As discussed, the physical distance between two two segments with electrical charge causes a charge condensation on the surface of the separating membrane. In the case of classic condenser, this primary charge density is abstracted as an infinitely thin layer of charges (electrons) on an ideally plain layer, and the rest of the volume is abstracted that no charge carrier is included. This case is perfectly described by the classic electricity theory. In the case of electrolytes, the separating membrane’s surface is not ideal, furthermore the electrostatic force can press some biological objects to the surface, so we must assume that the primary charge forms a uniformly charged layer with finite thickness on the surface. In the uniformly charged layer the electrical field is a linear function of the distance from the surface. The presence of dielectrical molecules in the rest of the volume creates secondary (virtual) dipole charges, that contribute to the electrical field, as shown. The two fields join to each other at the boundary of the charged layer. The electrical field on the surface of the membrane is the sum of those two fields. The opposite charges on the other plate generate another electrical field in the opposite direction.

As shown in the figure, the field steeply rises near to the membrane. At the membrane surface, it reaches its maximum value; on both sides of the membrane. Inside the membrane, there are no charges, so the field remains contant, for both types of condensers. Notice that the classic condenser can be considered as the abstraction of the real condenser to the case of zero dielectricity.

The electrical field acts on the ions, but there may be present a mechanical counterforce, plus a thermodynamic force aring from the difference of ions’ concentrations. (We just notice here that the electrical driving force does not depend on the chemical type of the ions, but the thermodynamic force does. The mechanical counterforce is adaptive, per ion, so it can keep different types of ions in rest.) If there are ’holes’ (ion channels) in the membrane, there is no mechanical counterforce, so the ions could move under the effect of the difference between the electrical and thermodynamic forces. The latter depends on the chemical nature of the ions, so it natively provides selectivity: the resulting forces are different for ions of different types. The usual ”downhill” method of explanation must be corrected: the resulting potential instead of the electrical potential defines how the ions move.

The figure shows two ions (green and red) at the axis of the ion channel in two positions (the empty circles along the axis of the ion channel), at the beginning and end of a movement, when the ion channel is open. On the diagram lines, filled circles show the electrical fields on the diagram lines at the position of the ions.

In the charged layer the ion density is constant, that is the thermodynamic force is zero while the electrical force decreases with the distance from the membrane’s surface

If the channel opened, the mechanical counterforce cancels, the ions will move under the effect of the resulting force, that comprises not only the thermodynamic and electrical forces, but also the force to move the ion with the Stokes-Einstein speed. In the position at the entrance of the ion channel (see the red ions), if the channel is open, the large thermodynamic and electrical forces ’push’ the ions into the channel. Here the concentration gradient quickly cancels, but the huge potential gradient will accelerate it toward the other segment. After passing the membrane with a high speed, the ion faces a quickly decreasing field, and its speed quickly decreases. The leave of ions changes both concentration and charge density in the vicinity

In the position farther from the membrane’s surface (see the green ions) the ions must travel ’uphill’; that is, they cannot travel instantly toward the membrane. They must wait until the gradient change caused by the ions pushed into the gap reaches their position. As the figure shows, the ions quickly move out from the region with high charge density and very slowly refill that region with ions from the region farther from the membrane. This speed difference results in a quick change in the local electrical field. In the region of the other plate, the ions decrease the (opposite) local potential. As the result, there are less ions ready to travel on the one side, and the potential difference continuously decreases on the other, so the flow of ions stops: the ion channel shuts itself down (and, on both sides, starts to regenerate the potential as it was before the channel’s cap was removed). That is, the ion channel is opened by a slow mechanical cap, but it is closed by a fast electrical shutter. While the potential self-closes the channel, the electrical fields on the two sides of the membrane have to relax: the ions and dipoles have time to change their position and even their polarization and ionization state. The local gradients will rearrange concentration and electrical field in the segment.

As depicted, in the two regions, there is a significant difference in the value of the effective force acting on the ions (they are using $E_{accelerate}$ and $E_{assist}$, so they produce different Stokes-Einstein speeds (see Eq. (2.30)). Based on those speeds, one can introduce ”fast” and ”slow” ions and correspondingly, speak about ’slow’ and ’fast’ currents that the ions represent at a macroscopic level. The corresponding speeds also differ by orders of magnitudes. For this study, we assume the diffusion, potential-assisted and potential-accelerated speeds, in $m/s$ to be ${10}^{-4}$, ${10}^{-1}$ (also inside neurons [50]), ${10}^{+3}$, respectively (used only to estimate the order of magnitude of some respective operating times). When staging, we assume the greater of the mixing speeds as ’infinitely large’ and omit the time that the process needs, while discussing how the slower process proceeds.

We are in line with the estimation given in [107] that in the case of resting potential, the scale of the gradient that accelerates the ions across the ion channel is calibrated approximately as $kV/cm$. Recall that we are still speaking about the resting state and only about the extra gradient evoked by the finite-width membrane. We are at the boundaries of the macroscopic and microscopic worlds. We derived our integrand from the picture of discrete charges but integrated it into the picture of continuous charge distribution. We assume an atomic layer (a skin) on the surface. 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 diagram line is valid in the plane crossing the membrane and the ion channel.)

We assumed that the membrane’s width is $5nm$. An ion channel is depicted in the middle of the figure with a diameter of about $1nm$. Furthermore, we assume that the ion’s size and, correspondingly, the thickness of the atomic layer in the electrolyte on the surface of the membrane is about $0.1nm$. For comparison, recall that the size of the tip of the clamp pipette is in the range of $1,000nm$ and the size of the soma in the range of $10,000nm$.

Our results align with the observation (see caption of 11.22 in [107]: ”A small flow of ions carries sufficient charge to cause a large change in the membrane potential. The ions that give rise to the membrane potential lie in a thin () surface layer close to the membrane”. See the dotted line in Fig. 2.12; notice that the x-scale on the figure spans $1nm$ only. 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-in to the low-concentration side and cause the 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.

The attraction between the ions in the two skin layers prevents the ions in the layers on the two sides from diffusing into/from the bulk without a current drain in the layer for an extended period. This steady state results from the interplay of the concentration and the potential described by Eq. (2.9). The gradients change gradually within the segments and drop suddenly across the membrane. No current can flow through the membrane; there is no leakage current.

Here, we assume that no ion channels are in the excellently isolating wall (ion channels would mean a current drain and, therefore, a voltage drop).