Self-similar processes and flicker noise from a fluctuating nanopore in a lipid membrane

Stochastic properties of a fluctuating nanopore generated and sustained by an electric field in a lipid bilayer membrane are studied. It is shown that the process of voltage fluctuations, in the current clamp experiment, is a stochastic fractal with long memory, which is the main reason for its nonstationarity. The aging process contributes to the nonstationarity if molecular interactions in the membrane are weak. An attempt to classify the process reveals a non-Gaussian distribution with long tails, which contradicts the hypothesis of fractional Brownian motion, showing that stable motion may be possible. The self-similarity index, estimated by three different methods, depends on current value and membrane sensitivity to electric field in a well defined and explicable manner. The stochastic analysis provided for calculated conductance of nanopore revealed the process close to $1/f$ noise, the result observed only for the pores not exceeding 1 nm in diameter, induced in membranes with strong molecular interactions. Our results show that such a pore is the simplest biological system needed for flicker noise to occur, and the complexity of highly regulated protein channel is not a necessary factor. A case of noise ${1/f}^2,$ observed for a pore with impeded dynamics, suggests a process without memory in such a situation. A physical interpretation is presented for some of the results.