Markov models and long-term memory in ion channels: A contradiction in terms?

The opening kinetics of ion channels are typically modeled using Markov schemes, which assume a finite number of states linked by time-independent rate constants. Although aggregate closed or open states may, under the right conditions, experience short-term (exponential) memory of previous gating events, there is experimental evidence for stretched-exponential or power-law memory decay that does not conform to Markov theory. Here, using Monte Carlo simulations of a lattice system, we investigate long-term memory in channels coupled to a heterogeneous membrane near the critical temperature. We observed that increasing the strength of the channel-lipid coupling parameter from zero to nearly 1 kT per lipid binding site leads to a progression in the autocorrelation of successive open dwell times. This evolution changes from 1) multiexponential decay to 2) power-law decay, and finally to 3) stretched exponential decay, mirroring changes in channel distribution from: 1) complete independence, 2) partitioning in the interphase between lipid domains, and 3) partitioning inside the domain favorable to the activation state of the channel. The intermediate power-law regime demonstrates characteristics of long-term memory, such as trend-reinforcing values of the Hurst exponent. Still, this regime passes a previously proposed Markovianity test utilizing conditional dwell time histograms. We conclude that low-energy state-dependent interactions between ion channels and a dynamic membrane soften the Markov assumption by maintaining a fluctuating microenvironment and storing configurational memory, thus supporting the existence of long memory tails without necessarily diminishing the usefulness of Markov modeling.