The mission of The Journal of General Physiology is to publish articles that elucidate basic biological, chemical, and physical principles of broad physiological significance. Physiological significance usually means mechanistic insights, which often are obtained only after extensive analysis of the experimental results. The significance of the mechanistic insights therefore can be no better than the validity of the theoretical framework used for the analysis—and it is usually better to be vaguely right than precisely wrong.

The uncertainties associated with data analysis are well illustrated in the Perspectives on Ion Permeation through membrane-spanning channels (J. Gen. Physiol. 113:761–794) and the related Letters-to-the-Editor in this issue. This exchange moreover identified a particular problem that can be resolved by a change in editorial policy.

The problem is the graphic representation of the results of kinetic analyses of ion permeation based on discrete-state rate models—and similar kinetic analyses of other physiological processes. It seems to have become de rigueur to summarize such results in a so-called energy profile (see Fig. 1), where the rate constants (k) deduced from the kinetic analysis are converted into free energies (ΔG)—almost invariably using Eyring's transition state theory (TST):

\begin{equation*}{\mathrm{{\Delta}}}G^{{\mathrm{{\ddagger}}}}=-k_{{\mathrm{B}}}T{\cdot}{\mathrm{ln}} \left \left[k{\cdot} \left \left({h}/{k}_{{\mathrm{B}}}T\right) \right \right] \right {\mathrm{,}}\end{equation*}

where kB is Boltzmann's constant, T the temperature in kelvin, and h Planck's constant. The problems arise because will be valid only for elementary transitions; e.g., transitions over distances less than the mean free path in aqueous solutions, ∼0.1 Å. Whether or not one can use a discrete-state rate model to analyze a permeation process, for example, the (in)validity of depends primarily on the distances ions have to traverse in the transitions between the different kinetic states.

The limitations inherent in the use of are well known, but energy profiles have taken on a life of their own because they provide a convenient graphic representation of the results, as opposed to the more tedious (albeit more correct) tabulation of the rate constants. Assuming the experimental results justify the use of a discrete-state model, which would entail a demonstration that the model and the deduced rate constants satisfactorily describe the results, the problem becomes, how can one represent the results graphically in a manner that avoids the errors associated with the use of ?

One such representation of linear kinetic schemes can be implemented by noting that free energy profiles based on the Eyring TST (i.e., on the use of ) formally can be expressed as:

\begin{equation*}{\mathrm{{\Delta}}}G \left \left(p\right) \right =-k_{{\mathrm{B}}}T{\cdot}{\mathrm{ln}} \left \frac{{\prod_{{\mathrm{i}}=1,3,{\mathrm{{\ldots}}}}^{p}} \left \left[{k_{{\mathrm{i}}}}/{ \left \left({k_{{\mathrm{B}}}T}/{h}\right) \right }\right] \right }{{\prod_{{\mathrm{i}}=2,4,{\mathrm{{\ldots}}}}^{p}} \left \left[{k_{{\mathrm{i}}}}/{ \left \left({k_{{\mathrm{B}}}T}/{h}\right) \right }\right] \right } \right {\mathrm{,}}\end{equation*}

where p (= 1, 2,…,n, where n is the total number of rate constants in the scheme) denotes the sequential position of the energy peaks and wells in the kinetic scheme (beginning with the first peak and ending outside the pore on the other side), and ki is the ith rate constant in the scheme (forward rate constants are odd numbered and reverse rate constants are even numbered). That is, ΔG(p) for p = 1, 3,…, n − 1 denotes the peak energies, whereas ΔG(p) for p = 2, 4,…, n denotes the well energies. The interrupted line in Fig. 1 (right-hand ordinate) shows such an energy profile. The generalization of is immediate, as the rate constant “profile” along the kinetic scheme can be represented by the function:

\begin{equation*}RCR_{{\mathit{ff}}} \left \left(p\right) \right =-{\mathrm{log}} \left \frac{{\prod_{{\mathrm{i}}=1,3,{\mathrm{{\ldots}}}}^{p}} \left \left({k_{{\mathrm{i}}}}/{ff}\right) \right }{{\prod_{{\mathrm{i}}=2,4,{\mathrm{{\ldots}}}}^{p}} \left \left({k_{{\mathrm{i}}}}/{ff}\right) \right } \right {\mathrm{,}}\end{equation*}

where ff is an arbitrary “frequency factor.” The three lines in Fig. 1 (left-hand ordinate) show rate constant representations (RCR) for ff = 1, 109, and 6 · 1012 s−1 (= kBT/h). (ff = 1 s−1 denotes the simplest version of , ff = 109 s−1 was chosen to approximate the frequency of diffusional transitions over a distance of 1 nm, and ff = kBT/h was chosen for comparison to .)

It is instructive to consider briefly some features of and Fig. 1. First, the heights of the “peaks” vary with the choice of ff. The peaks shift in parallel up or down as ff is increased or decreased, which serves to emphasize how arbitrary a “barrier height” is—and to underscore the difficulties inherent in deducing an energy profile from a set of rate constants (compare Fig. 1 and the two different energy profiles deduced for ff = 6 · 1012 and 109 s−1). Second, the differences in height among the peaks are invariant, suggesting that they have mechanistic significance. It is unlikely that the frequency factors associated with each barrier crossing will be identical, however, and one cannot relate differences in peak height to differences in free energy without knowing the variation in ff. Third, the “well” depths relative to the electrolyte solution outside the pore are invariant, again suggesting that they have mechanistic significance. The different behaviors of the peaks and “wells” arise because of the qualitative difference between RCRff(p) for odd and even p: only for odd p does the value of RCRff(p) depend on ff. Visually, the peaks probably should be above the wells; compare the profile for ff = 1 s−1 vs. those for ff = 109 and 6 · 1012 s−1, which justifies the use of physically plausible, albeit arbitrary, frequency factors.

applies generally, meaning that it is possible to provide graphic representations of the results of kinetic analyses without invoking the Eyring TST to describe situations where that theory is inapplicable—whether it be ion permeation, channel gating, protein conformational transitions, or other physiological processes. The Journal of General Physiology therefore will publish rate constant representations based on , or some equivalent, but will no longer publish energy profiles deduced from kinetic analyses unless the authors explicitly justify their choice of the underlying model using “generally accepted” physico-chemical reasoning.

Olaf Sparre Andersen


The Journal of General Physiology