Correspondence to: Eberhard von Kitzing, Abteilung Zellphysiologie, Max-Planck-Institut für medizinische Forschung, Postfach 10 38 20, D-69028 Heidelberg, Germany. Fax: 49 6221 486 459; E-mail:vkitzing{at}mpimf-heidelberg.mpg.de.
I would like to address two key issues related to the Perspectives on Ion Permeation. The first is the estimate of the size of the physical forces relevant for ion transport. Any good physical model of ion permeation requires the identification of the dominant forces. The second subject concerns a careful definition of the terms "mean field" and "mean field approximation." Several articles in the field of ion permeation show considerable confusion about the mean field concept.
The Size of the Forces
What is the relative size of the forces that ions experience while passing through a channel? To enter the channel ions have to become partly, if not completely dehydrated. Highly polar and even charged functional groups forming the channel walls compensate for this loss of hydration energy. In addition, there are two kinds of strong ionion forces: long range electrostatic forces and short range hard core interactions leading to volume exclusion effects.
Considerable simplifications are necessary to compute currentvoltage curves of physical model channels. At present only two theories of ion transport are sufficiently simple for direct comparison between theory and experiment. These two theories consider different parts of the relevant forces as the most strong ones. In reaction-rate theories, also designated barrier models, the interaction of the ion with its environment and volume exclusion effects among ions are considered to dominate. Neglecting electrostatic ionion interactions, the rates of barrier crossing can be computed from first principles (
Reaction-rate theories apply only if environmental forces surpass the electrostatic forces between ions. Otherwise, environmental interactions would not determine ion transport. Consequently, typical energies of electrostatic ionion interactions inside the channel represent lower limits for the energy differences between barriers and wells. Using a simple Coulomb law with a dielectric constant of 10, the energy required to bring two positively charged monovalent ions as close as 0.58 nm requires at least 250 milli electron volts (10 kT). Thus, environmental forces only dominate if the barrier energies are much larger.
Reaction-rate theories explain the saturation of the channel conductance with increasing external concentrations by hard core ionion interactions. To experience the short range volume exclusion interactions, the ions must come rather close to each other. The distances of closest approach between ions including a single intermediate water molecule are in the order of 0.30.5 nm. Consequently, reaction-rate theories predicting conductance saturation automatically involve strong electrostatic ionion interactions. Thus, even single ion channels require large barriers to dominate these electrostatic forces between ions. A more general discussion of the forces important for ion transport was published recently (
The identification of the dominant forces is crucial for understanding ion transport. Only those models of ion permeation that include the strongest interactions can provide a reasonable picture of what is going on inside biological ion channels.
Mean Fields and Mean Field Approximation
In his editorial,
During a 10-pA, 1-s channel opening, 6.3 · 107 ions pass the channel. With a time resolution in the millisecond range, the experimental mean current samples >60,000 ions. Passing the bottleneck of the channel each of those ions will see different forces. Side chains at the channel wall may change their orientations between the passage of two ions, and sometimes they even block the path. Also, the position of other ions differs at different ion passages, resulting in different electrostatic forces. For instance, the forces seen by an ion entering a channel differ considerably whether the channel is occupied by another ion or not. However, these very different forces add up linearly. If the channel is 50% occupied by other ions, on the average the incoming ion sees a half-occupied channel. The average of 105 different configurations results in a mean force. The absolute value of the force seen at each passage is generally large compared with that of the mean force. What is important for the measured mean current is the mean force, the linear average over all those very different contributions.
We do not measure mean forces, but their integrals over the paths of the ions. Currentvoltage relations represent integral properties of the channel (
Unfortunately, the introduction of mean forces is not sufficient to handle the strong, long-range electrostatic ionion interactions. To describe the behavior of plasmas,
Also within biological ion channels, one should expect effects because of ionion correlations. Solvent-mediated ionion interactions lead to single filing. This "electrophoretic" effect may be strong in channels such as the potassium channel. Consequently, PNP theories need to implement this mechanism, as for instance suggested by
How reliable are these mean field approximations? In physics, such mean field approximations are frequently employed with different success. Unfortunately, the comparison (
Thus, in accord with the estimates of the size of the forces inside the channel, the PNP theory, using the mean field approximation for the electrostatic interactions, is the method of choice for modeling ion channels dominated by strong electrostatic fields. This theory obeys two important sum rules derived in statistical physics. The fact that the currentvoltage relations used to compare theory with experiment are integral properties of the channel renders the judgment of the sum rules particularly valid. In contrast, commonly used reaction-rate theories obey none of these rules. Therefore, they cannot account correctly for strong, long range electrostatic interactions.
Perspectives and Outlook
To study the fundamentals of ion permeation, simpler channels than the potassium or calcium channel should also be considered. For instance, the kind and position of mutations in the acetylcholine receptor channel (
Today, the formulation of the GouyChapman theory (
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