Electro-physiology Models of Cells with Spherical Geometry with Non-conducting Center
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Electro-physiology Models of Cells with Spherical Geometry with Non-conducting Center Jiamu Jiang1 · Paul Smith2 · Mark C. W. van Rossum1,3 Received: 22 July 2020 / Accepted: 30 October 2020 / Published online: 19 November 2020 © The Author(s) 2020
Abstract We study the flow of electrical currents in spherical cells with a non-conducting core, so that current flow is restricted to a thin shell below the cell’s membrane. Examples of such cells are fat storing cells (adipocytes). We derive the relation between current and voltage in the passive regime and examine the conditions under which the cell is electro-tonically compact. We compare our results to the well-studied case of electrical current flow in cylinder structures, such as neurons, described by the cable equation. In contrast to the cable, we find that for the sphere geometry (1) the voltage profile across the cell depends critically on the electrode geometry, and (2) the charging and discharging can be much faster than the membrane time constant; however, (3) voltage clamp experiments will incur similar distortion as in the cable case. We discuss the relevance for adipocyte function and experimental electro-physiology. Keywords Adipocytes · Ionic currents · Cable equation · Mathematical models
1 Introduction Many biological cells rely on electrical signals for intracellular and intercellular communication; this includes neurons but also other cell types, such as cardiac, muscle, and endocrine cells. Electrical currents have been most extensively studied in neurons. These have a tree-like geometry, and the branches of the tree are cylindrical structures with small diameters. Electrically, such cylinders are well modeled with the so-called cable equation (Jack et al. 1975; Koch 1999). The cable equation describes the spatiotemporal dynamics of the voltage along the cable in response to intracellular current injection along the cylinder.
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Mark C. W. van Rossum [email protected]
1
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
2
School of Life Sciences, University of Nottingham, Nottingham NG7 2RD, UK
3
School of Psychology, University of Nottingham, Nottingham NG7 2RD, UK
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Fig. 1 Left: The geometry under consideration. A spherical adipocyte with radius ρ is largely filled with a fat globule; only the thin shell near the surface with thickness d conducts. Latitude on the sphere is indicated by the angle θ . A pipette electrode with opening angle θa is attached at the top. Right: Construction to derive the sphere equation. The currents flowing in a given spherical segment at a certain latitude (pink) are indicated. The segment is assumed to be equipotential (Colour figure online)
The cable equation has been of great benefit to understand spatial-temporal integration in cells. For instance, it enables one to calculate how currents from distal synapses are filtered and contribute to the membrane voltage at arbitrary locations. Moreover, the cable equation is imp
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