Anomalous rectification

Anomalous rectification [3]. Our aim in this section is to show that under certain conditions development of a nonequilibrium space charge may yield, besides the punch through, some additional effects, unpredictable by the locally electro-neutral formulations. We shall exemplify this by considering two parallel formulations—the full space charge one and its locally electro-neutral counterpart. It will be observed that inclusion of the space charge into consideration enables us to account for the anomalous rectification effect that could not be predicted by the locally electro-neutral treatment. Physical motivation for this study is as follows. 

We saw previously that concentration polarization results in the decrease of solute concentration near the permselective interface (right at the interface in the electro-neutral version) where most of the system s resistance thus concentrates, and where the space charge develops. The system is expected to be sensitive to the minimum concentration value, and because of nonlinearity nontrivial effects, could be anticipated in response to unsteady disturbances of this value (e.g., provided by harmonic modulation superimposed upon a constant voltage applied to the system). Since it is easier to increase the minimal concentration (close to zero at the limiting current) than to decrease it, we might expect a positive rectification effect for the direct current component, counterintuitive ( anomalous ) in the present system with a convex stationary VC curve. Thus the topic of this section is the rectification effects that arise in the stationary concentration polarization in response to a harmonic voltage modulation. 

8) V is the direct voltage component, whereas A and u = 27r/ = (2n/T) are, respectively, the amplitude and the cyclic frequency of a harmonic modulation superimposed upon V, above some critical voltage VCT. 

is solved numerically for a sequence of voltages from V = 0 to V = Vcr until the steady state is reached at t — oo. As an initial condition we employ the steady concentration fields, computed for the previous voltage value, starting from the known equilibrium fields at V = 0. For V VCI the appropriate solution for t — oo coincides with those for (5.3.1), (5.3.5). 

In Figs. 5.4.1a and 5.4.2a, we present the steady VC curves, corresponding to Vcr — oo, computed, respectively, for e = 10-4, 10-6. For Vct finite the above procedure is carried out until V r is reached. At this point the modulation is switched on and the unsteady computation is performed for a few tens of periods T until the transients die out. The computed current density 

Apparently, the voltage generated by this additional current increases the level of depolarization recorded through the electrode and can therefore be attributed to an increase in the input resistance of the cell (see Section 6.5). 

More recently, Johnston et al (1980) have confirmed some of these 

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Application of the uniform asymptotics above to the study of nonstationary effects occurring at high concentration polarization, such as the anomalous rectification in 5.4. 

Hagiwara S, Takahashi K (1974) The anomalous rectification and cation selectivity of the membrane of a starfish egg cell. J Membr Biol 75 61-80. 

The properties of (9) depend on the value chosen for the barrier height increment, g At g = 0, the normal Goldman-Hodgkin-Katz relation, (2), is obtained and atg = —Vj Jn, a nearly linear relation results. This suggests that an even larger negative value for g would result in a reversed (outwardly directed) or anomalous rectification and ag > 0 would accentuate normal inward rectification. Figure 4 shows plots of relations calculated for n — 4, Fjja = 58 mV ([Na+] /[Na+]< = 10) and for a number of values of... 

Anomalous rectification occurs for Kj > Kj Tj and accentuated normal rectification for F. < 0. 

A potential energy sequence like that in Fig. 2>b can cause outward (anomalous) rectification. This rectification occurs because Na+ ions can enter the membrane from the right (inside) quite easily but encounter a high barrier when entering from the left (outside). If the ease of entry from the inside more than compensates for the low [Na+J, then there will be an outward rectification. This condition (cf Fig. 4) is that > Kjjd, or —ng > In ([Na+] /[Na+])i. In Fig. 4, [Na+y[Na+]j = 10, ng < —2.3, and thus the total difference in AG from outside to inside is 1400 cal/mole or greater. For n = A, 6G < —350 cal/mole. 

G —ng = 18, g = —4.5 and 6G = —2700 cal/mole. The total AG = 10,800 cal/mole. This predicts that current flowing through a membrane channel having a pronounced anomalous rectification will have a high temperature coefficient. I know of no pertinent experimental data. 

Anomalous rectification for K+ has been found in the membranes of skeletal muscle cells " and cardiac muscle cells. It is therefore of interest to see if the Ij -V curves for muscle cell membranes are adequately described by (14) for appropriate values of Cq, C , n, and V. The shapes of the curves are similar, but (14) is not a good quantitative description primarily because the curvature of the experimental relation is too large. This can be shown by rewriting (14) for K+ ions and taking the limit of as approaches + oo ... 

Such an assumption is not easy to evaluate quantitatively since the independence assumption is violated and special assumptions must be made about the nature of the K+ ion interactions in the channels. The possibility that there are interactions between two or more K+ within an anomalous rectification K.+ channel is supported by the experimental evidence that K+ ions do interact in the K+ channels of the squid nerve fiber membrane. i The interaction is the type expected if three or more K+ were in single file in the channel. Thus, it seems possible that the anomalous rectification properties of the K+ channels in muscle membranes have a potential energy profile similar to that in Fig. 3b but that there are usually two or three K+ in any one channel. This implies that ion transit time is two to three times larger than the time between successful collisions of ions with the channel entrances. By contrast, Na+ channels obey the independence principle implying that a Na+ ion traverses a channel before there is another successful collision of an ion with the channel entrance. The longer transit time of K+ ions could be due to deeper minima (tighter binding) in K+ channels as compared with Na+ channels. 

The current-voltage relations of channels containing more than one ion at a time. This would be an extension of the theory given in this paper to see if the experimental properties of anomalous rectification channels can be explained in this manner. 

Hagiwara S, Miyazaki S, Rosenthal NP (1976) Potassium current and the effect of cesium on this current during anomalous rectification of the egg cell membrane of a starfish. J Gen Physiol... 

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