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. 2020 Jun 29;10:10304. doi: 10.1038/s41598-020-66203-6

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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A cross section for an ellipsoidal-shaped mitochondrial crista: transmembrane electrostatic proton localization (protonic capacitor) model illustrating how excess protons (H⁺) and hydroxyl ions (OH^–) could be electrostatically localized at the water-membrane interfaces along the two sides of the mitochondrial inner membrane before proton-cation exchange as it would be in a theoretically pure water-membrane-water system. Note, this cross section can be considered as a special result from the tri-axial (a, b, and c) protonic conducting ellipsoidal crista equation (Eq. 12) in 3-dimensional x, y, and z coordinates for its middle cross section (where the z coordinate is zero).