Ion channel kinetics: a model based on fractal scaling rather than multistate Markov processes

• Published in: Mathematical biosciences 1987, Vol.84 (1), p.37-68
• Main Authors: Liebovitch, Larry S., Fischbarg, Jorge, Koniarek, Jan P.
• Format: Article
• Language: English
• Subjects: channels; cornea; endothelium; fractals; ions; kinetics; Markov processes; mathematical models; ophthalmology; scaling
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/0025-5564(87)90042-3

Ion channels have been modeled as consisting of a small number of discrete conformational states such as closed ⇌ closed ⇌ open, and the transitions between the states treated as a Markov process. We derive an alternative model based on a fractal scaling of the kinetic rate constants k, namely that k = At 1- D , where t is time, A the kinetic setpoint, and D the fractal dimension. By measuring the effective kinetic rate constants at different time scales, we show how to determine if single channel records are best represented by models with discrete Markov states or continuum fractal kinetics. The fractal model suggests that the multiple closed states postulated by Markov models to fit the histograms of closed times may not exist but are an artifact of trying to fit the sum of exponentials to histograms that are not the sum of exponentials. The fractal model is also more consistent with the conformational dynamics of proteins. Analysis of patch clamp recordings from an epithelium, the corneal endothelium, shows that its channels have fractal rather than Markov kinetics.