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, Kv, KATP, Cav-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 KATP 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 Ca2+ and the membrane potential are modulated by glucose. The results confirm that the inhibition of KATP channels depolarizes the membrane and increases the influx of [Ca2+]i through NCX and Cav-L channel for high glucose concentrations.

[1]  L. Rosário,et al.  Pulsatile insulin release and electrical activity from single ob/ob mouse islets of Langerhans. , 1986, Advances in experimental medicine and biology.

[2]  A. Herchuelz,et al.  Role of Na/Ca Exchange and the Plasma Membrane Ca2+–ATPase in β Cell Function and Death , 2007, Annals of the New York Academy of Sciences.

[3]  L. Philipson,et al.  Bursting and calcium oscillations in pancreatic beta-cells: specific pacemakers for specific mechanisms. , 2010, American journal of physiology. Endocrinology and metabolism.

[4]  R. Kopclman.  Percolation and cluster distribution . I . Cluster multiple labeling technique and critical concentration algorithm , 2011 .

[5]  I. Fajardy,et al.  Kinetics of diabetes-associated autoantibodies after sequential intraportal islet allograft associated with kidney transplantation in type 1 diabetes. , 2003, Diabetes & metabolism.

[6]  Frederick Sachs,et al.  Maximum likelihood estimation of ion channel kinetics from macroscopic currents. , 2005, Biophysical journal.

[7]  Rp Hugtenburg,et al.  RT29: A Markov Chain Method for the Accurate Characterisation of a Therapeutic Linear Accelerator in Monte Carlo Based Dosimetry and Treatment Planning , 2001 .

[8]  K. Gillis,et al.  ‘Perforated patch recording’ allows long‐term monitoring of metabolite‐induced electrical activity and voltage‐dependent Ca2+ currents in pancreatic islet B cells , 1989, FEBS letters.

[9]  H. Riane,et al.  Modelisation of the contribution of the Na/Ca exchanger to cell membrane potential and intracellular ion concentrations. , 2008, General physiology and biophysics.

[10]  P. Dobson.  Point Defects and Diffusion , 1976 .

[11]  A method to improve the effective medium theory towards percolation problem , 1982 .

[12]  P. Rorsman,et al.  Calcium and delayed potassium currents in mouse pancreatic beta‐cells under voltage‐clamp conditions. , 1986, The Journal of physiology.

[13]  W. Fujimoto,et al.  Rat Islet Cells have Glucose-Dependent Periodic Electrical Activity , 1984, Hormone and metabolic research = Hormon- und Stoffwechselforschung = Hormones et metabolisme.

[14]  C. Morris,et al.  Voltage oscillations in the barnacle giant muscle fiber. , 1981, Biophysical journal.

[15]  K. Hatlapatka,et al.  Studies of first phase insulin secretion using imposed plasma membrane depolarization. , 2011, Frontiers in bioscience.

[16]  L. Eliasson,et al.  Firing of Action Potentials in Mouse Pancreatic b Cells , 1999 .

[17]  M. Ravier,et al.  Glucose-induced mixed [Ca2+]c oscillations in mouse beta-cells are controlled by the membrane potential and the SERCA3 Ca2+-ATPase of the endoplasmic reticulum. , 2006, American journal of physiology. Cell physiology.

[18]  Imy Mareels,et al.  Predicting the Onset of Type 1 Diabetes Mellitus , 2002 .

[19]  O. Petersen,et al.  High-conductance K+ channel in pancreatic islet cells can be activated and inactivated by internal calcium , 2005, The Journal of Membrane Biology.

[20]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[21]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[22]  E. Rojas,et al.  Potassium permeability activated by intracellular calcium ion concentration in the pancreatic beta‐cell. , 1979, Journal of Physiology.

[23]  E. K. Matthews,et al.  Electrical characteristics of pancreatic islet cells. , 1975, The Journal of physiology.

[24]  S. Yalkowsky,et al.  Mass transport phenomena and models: theoretical concepts. , 1974, Journal of pharmaceutical sciences.

[25]  L. Eliasson,et al.  Activation of Ca(2+)-dependent K(+) channels contributes to rhythmic firing of action potentials in mouse pancreatic beta cells , 1999 .

[26]  M. Ebert,et al.  A percolation-like model for simulating inter-cellular diffusion in the context of bystander signalling in tumour , 2011, Australasian Physical & Engineering Sciences in Medicine.

[27]  Chunyu Li,et al.  A direct electrifying algorithm for backbone identification , 2007 .

[28]  J. Deckers,et al.  The New EU Directive on the Use of Animals for Research and the Value of Moral Consistency , 2012, Journal of Bioethical Inquiry.

[29]  J. Henquin.  Tetraethylammonium potentiation of insulin release and inhibition of rubidium efflux in pancreatic islets. , 1977, Biochemical and biophysical research communications.

[30]  Ruibao Tao,et al.  Algorithm for finding two-dimensional site percolation backbones , 2000 .

[31]  B. Sung,et al.  Lateral diffusion of proteins in the plasma membrane: spatial tessellation and percolation theory. , 2008, The journal of physical chemistry. B.

[32]  Yung E Earm,et al.  Ionic mechanisms and Ca2+ dynamics underlying the glucose response of pancreatic β cells: a simulation study , 2011, The Journal of general physiology.

[33]  E. Rojas,et al.  The nature of the oscillatory behaviour in electrical activity from pancreatic beta-cell. , 1980, Hormone and metabolic research. Supplement series.

[34]  S. Chaieb,et al.  Why the wrinkling transition in partially polymerized membranes is not universal? Fractal-multifractal hierarchy. , 2008, Journal of theoretical biology.

[35]  Michel Deville,et al.  Modélisation numérique en science et génie des matériaux , 1998 .

[36]  Morten Gram Pedersen,et al.  A biophysical model of electrical activity in human β-cells. , 2010, Biophysical journal.

[37]  S. Kirkpatrick.  Percolation and Conduction , 1973 .

[38]  J. Duffas.  La transplantation pancréatique : 2. La chirurgie et ses complications postopératoires , 2004 .

[39]  D. Cook,et al.  Intracellular ATP directly blocks K+ channels in pancreatic B-cells , 1984, Nature.

[40]  P. Rorsman,et al.  Voltage‐gated and resting membrane currents recorded from B‐cells in intact mouse pancreatic islets , 1999, The Journal of physiology.

[41]  Henquin Jc.  The potassium permeability of pancreatic islet cells: mechanisms of control and influence on insulin release. , 1980 .

[42]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[43]  P. Dean,et al.  Electrical Activity in Pancreatic Islet Cells , 1968, Nature.

[44]  P. Rorsman,et al.  Insulin granule dynamics in pancreatic beta cells , 2003, Diabetologia.

[45]  P. Rorsman,et al.  Significance of Na/Ca exchange for Ca2+ buffering and electrical activity in mouse pancreatic beta-cells. , 1999, Biophysical journal.

[46]  J. Rinzel,et al.  Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing. , 1988, Biophysical journal.

[47]  F J Sigworth,et al.  Applying hidden Markov models to the analysis of single ion channel activity. , 2002, Biophysical journal.

[48]  G. Ermentrout,et al.  Analysis of neural excitability and oscillations , 1989 .

[49]  J. Ruppersberg.  Ion Channels in Excitable Membranes , 1996 .

[50]  E. Rahme,et al.  Osteoporosis among patients with type 1 and type 2 diabetes. , 2008, Diabetes & metabolism.

[51]  M. Meyer-Hermann.  The electrophysiology of the beta-cell based on single transmembrane protein characteristics. , 2007, Biophysical journal.

[52]  J. Bryan,et al.  The KATP Channel is Critical for Calcium Sequestration into Non-ER Compartments in Mouse Pancreatic Beta Cells , 2006, Cellular Physiology and Biochemistry.

[53]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[54]  M. Pellegrino,et al.  Use of conditional distributions in the analysis of ion channel recordings , 2004, European Biophysics Journal.

[55]  A. Herchuelz,et al.  Overexpression of the Na/Ca exchanger shapes stimulus-induced cytosolic Ca(2+) oscillations in insulin-producing BRIN-BD11 cells. , 2002, Diabetes.

[56]  J. Keizer,et al.  Minimal model for membrane oscillations in the pancreatic beta-cell. , 1983, Biophysical journal.

[57]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .