Are rate constants constant

In this issue of The Journal of Physiology, Uebachs et al. (2006) report a seemingly simple observation: the rate of recovery from inactivation of CaV3 (T-type) calcium channels depends on the length of the depolarization that produced the inactivation. Why is this heretical? A basic principle of chemical kinetics is that rate constants are constant. Of course, that is true only if the conditions are constant, because rate constants clearly depend on experimental factors such as temperature, and (for ion channels) voltage, binding of ligands, etc. Time-independent rate constants are fundamental to nearly all kinetic descriptions of the gating of ion channels, from the original Hodgkin & Huxley (1952) model to contemporary Markov models (where channels can exist in a number of states, but any two states are linked by first-order rate constants). This is rationalized by Eyring rate theory, where the rate constant for a change in state depends only on the difference between the energy levels of the initial state and the transition state (energy barrier). That is, a single parameter determines the reaction rate. The vast majority of experimental studies have demonstrated that even complex channel gating can be described by Markov models (given enough states, and transitions among them). Like most simple, attractive and successful concepts in biology, Markov models have not been without critics. One issue is that dynamic models of protein structure do not exhibit a small number of rigid states, but a continuum of large and small motions over a wide range of distance and time scales. That is, the energy landscape of a protein's conformational changes may not be accurately approximated by a reasonable number of discrete energy wells (states) and barriers. The assumption of distinct states may break down for some very rapid transitions. On the other hand, this phenomenon can be explained by small movements within a distinct energy well (Sigg et al. 2003). This may be the ‘exception that proves the rule’ (in the mostly archaic sense of ‘prove’ to mean ‘test’, as in ‘proving ground’.) The most extensive challenge to Markov models in the literature is ‘fractal’ models, where transition rates vary in a continuous fashion (Liebovitch et al. 2001). One interpretation is a ‘diffusional’ model, where the energy landscape is approximated as gently sloping, instead of having distinct peaks and valleys. Some experimental tests found that Markov models describe complex single-channel data more accurately than fractal models (McManus et al. 1988), but critics remain unsilenced (Liebovitch et al. 2001). Another challenge to Markov models came from studies of recovery from slow inactivation of sodium channels. Slow recovery could be described by a single exponential process, but the time constant of the exponential depended on the duration of the inactivating pulse, following a fractal-like power law relationship (Toib et al. 1998; Ellerkmann et al. 2001). Uebachs et al. (2006) extend this idea to recovery from inactivation for T-type calcium channels. Recovery was best described by the sum of two exponential components, but both fast and slow time constants increased with the duration of the inactivating pulse. Two time-dependent time constants gave a statistically better fit than two (or even three) fixed time constants, although the difference was modest for two of the three T-channel types tested. Curiously, recovery was more conventional following strong depolarization or repetitive action potential-like waveforms. It is noteworthy that these experiments required extraordinarily stable recordings from the transfected HEK 293 cells (Uebachs et al. 2006). This result will not cause channel biophysicists to immediately abandon Markov models. Contemporary models for channel gating can contain many states (10–50), sometimes with parameters well constrained by the data, and such models have the potential to explain quite complex kinetic behaviour. But at the least, Uebachs et al. (2006) should prompt close examination of other channels for signs of time-dependent recovery from inactivation, and tests for whether this phenomenon is best described by multistate Markov models or by models with time-dependent rate constants. It is worth noting that Toib et al. (1998) found that slow recovery was Markovian for Shaker potassium channels. T-channels are evolutionarily related to sodium channels, almost as closely as to other calcium channels. It is not known whether inactivation mechanisms are conserved among these channel families. What physiological implications would time-dependent rate constants have?Uebachs et al. (2006) plausibly suggest that this provides a memory for previous neuronal activity, in contrast to Markov models, which are ‘memoryless’. A Markov channel does not ‘know’ its previous history, only the state that it is in, and the rate constants leading away from that state. However, multiple states of a given type (e.g. closed or inactivated) can produce memory-like behaviour. Even for the original model for the potassium channel of squid axon (Hodgkin & Huxley, 1952), channels activate more slowly following long hyperpolarizations, depending on the number of ‘n’ gates closed (the once-famous Cole-Moore shift). Similarly, recovery from inactivation of a T-channel will depend on the pulse duration as long as there is both fast and slow recovery, since long depolarizations preferentially occupy the slowly recovering state. Time-dependent rate constants would provide a more explicit memory for previous conditions, which certainly could influence neuronal dynamics. It remains to be established precisely what the physiological consequences are, and whether they differ quantitatively or qualitatively from traditional multistate Markov models.