Correspondence to: Peter C. Jordan, Department of Chemistry, Brandeis University, 415 South St., MS-015, Waltham, MA 02254-9110. Fax: 781-736-2516; E-mail:jordan1{at}binah.cc.brandeis.edu.
The difficulties with and controversies surrounding the interpretation of electrophysiological data were highlighted in the recent Perspectives on Ion Permeation through membrane-spanning channels. The main issue was whether such data are more appropriately analyzed within the framework of chemical kinetics (Eyring's Transition State Theory;
In the most commonly used treatments of kinetic modeling found in the physiological literature, the Eyring expression for the rate constant is presented as:
![]() |
(1) |
Here, is identified with the Eyring "frequency factor" (kT/h) and
G is the "energy barrier" associated with an elementary step in the reaction mechanism. To see why even this can be misleading, it is worthwhile to refer to Eyring's original formulation (
![]() |
(2) |
The unfamiliar quantities in this equation are:
![]() |
(3) |
Here, E0 is the minimum barrier on the potential energy surface (the energy difference between a well and an adjacent peak) and
is the mean transmission coefficient.
measures the average likelihood that the trajectories for which the ionic energy exceeds
E0 are actually reactive. Because chemical reaction is fundamentally a quantum mechanical process, there is no guarantee that, even when the ion is sufficiently energetic, it actually reacts (see, e.g.,
and Q are partition functions for the ion near a peak and a well in the potential energy surface. They measure the number (thermally weighted) of accessible states in these two separable configurations. The "reaction coordinate" has been separated out of Q
; it accounts for the "Eyring frequency," kT/h. A comparison of Equation 1 and Equation 2 demonstrates that the empirical quantity
G really is an amalgamation of the four more basic quantities of Equation 3. There is nothing wrong with defining
G in this way, the difficulty is in how to construe it. If an energy diagram such as that of Figure 3 A of
G with
E0, and then quantifying it. This is clearly wrong. The kinetic barrier for an elementary step in the permeation process is a composite quantity, not simply reflecting the lowest energy required to surmount an intermediate barrier,
E0. While the pre-exponential factor in Equation 2 has not been directly measured for ion channel kinetics (what is available are a few rough estimates of Q10), it has been extensively studied in the chemical literature. In an illustrative series of simple binary gas phase reactions involving bi- and triatomic species, its value ranged between 100 and 0.01x the Eyring frequency (
There is another, more phenomenological, way to look at rate data. The Eyring expression for the rate constant can be written in an alternative form by thermodynamic analogy (see e.g.,
![]() |
(4) |
where S
and
H
are the entropy and enthalpy of activation for the primary process.
G of Equation 1 is then recognized as
H
- T
S
. Comparing Equation 1, Equation 2, and Equation 4, the enthalpy term is identified with
E0 and the entropy term is determined by the other three quantities of Equation 3.
S
can be quite negative if the ion trajectories, even though sufficiently energetic, are predominantly nontransmittive (
<< 1) or if there are very few thermally accessible states in the transition state near the peak in the reaction path (Q
<< Q, a bottleneck).
Considering the known structure for the selectivity filter in the KCsA K channel (S
. The associated
H
might be quite minimal, especially as everywhere in the binding domain an ion is in fairly close contact with many carbonyl oxygens.
It should be noted in this context that translocation in gramicidin probably has a large entropic component as well. The single file involves six to eight water molecules, which must move concertedly for translocation to occur. The probability of such an event is ~1/64 to ~1/256 (
There is of course a "straightforward" way to distinguish entropy and enthalpyby measuring the temperature dependence of the rate constants. This sort of experiment is called for, as it would separate the entropic and enthalpic effects. Even though it might be difficult to do (conceivably a gross understatement coming from someone who hasn't done experiments, other than culinary ones, for 40 yr), it would provide a basis for evaluating the energy scales in mechanistic schemes and truly separate the energetic and configurational terms of Equation 2.
There is another issue in the kinetics-diffusion dichotomy that, while commonly accepted, may be overstated. It's not immediately obvious that the "hopping" and "diffusion" pictures can always be made equivalent by breaking a diffusive barrier into sufficiently small wells and barriers ala
Finally, in the diffusion picture, entropic changes along the permeation path are analogous to local variation of the diffusion constants; it is therefore a matter of concern that the PNP fits appear insensitive to local variation in D (molecular dynamics modeling suggests such variability may be substantial;
References
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