Fractal Behavior of the Pancreatic β-Cell Near the Percolation Threshold: Effect of the KATP Channel On the Electrical Response

The molecular system built with true chemical bonds or strong molecular interaction can be described using conceptual mathematical tools. Modeling of the natural generated ionic currents on the human pancreatic β-cell activity had been already studied using complicated analytical models. In our present contribution, we prove the same using our simple electrical model. The ionic currents are associated with different proteins membrane channels (K-Ca, K
_v
, K
_ATP
, Ca
_v
-L) and Na/Ca Exchanger (NCX). The proteins are Ohmic conductors and are modeled by conductance randomly distributed. Switches are placed in series with conductances in order to highlight the channel activity. However, the K
_ATP
channel activity is stimulated by glucose, and the NCX's conductance change according to the intracellular calcium concentration. The percolation threshold of the system is calculated by the fractal nature of the infinite cluster using the Tarjan's depth-first-search algorithm. It is shown that the behavior of the internal concentration of Ca
²⁺
and the membrane potential are modulated by glucose. The results confirm that the inhibition of K
_ATP
channels depolarizes the membrane and increases the influx of [Ca
²⁺
]
_i
through NCX and Ca
_v
-L channel for high glucose concentrations.