Calculation of proton transmembrane-electrostatic interaction force and elucidation of the water droplet experiment with a transient protonic front

The "transmembrane-electrostatically localized proton(s)/cation(s) charge(s) (TELC(s), also known as TELP(s)) model" may serve as a theoretical framework to explain protonic cell energetics including both delocalized and localized protonic couplings. TELCs are held by their corresponding transmembrane-electrostatically localized hydroxides anions (TELAs) across the membrane through mutual transmembrane-electrostatic attractive force, which is now calculated to be in the range from 1.96 × 10^-11 to 2.28 × 10^-11 newtons (N) across a 2.5 nm thick membrane in a range of transmembrane potential from 10 to 200 mV. At a moderate transmembrane potential (100 mV), the protonic transmembrane attractive force is now calculated to be 2.08 × 10^-11 N. Accordingly, to move such a localized proton away from the membrane-liquid interface by 1 nm, it would require 1.62 × 10^-20 J of energy, which is equivalent to 3.8 times as much as the Boltzmann <i>kT</i> thermal kinetic energy at a physiological temperature of 37 °C, indicating that a TELCs-membrane-TELAs capacitor can be quite stable. Thus, TELCs (TELPs) formation does not require any potential barrier in liquid phases. The Zhang <i>et al.</i> 2012 experiment is likely to involve a transient "protonic front" effect in a single water droplet system which has no membrane and no TELPs. The use of the "Bjerrum length" approach for merely a single pair of charges (Knyazev <i>et al.</i>, <i>Biomolecules</i>, 2023, <b>13</b>, 1641) could underestimate the protonic transmembrane attractive force. Future TELPs research is encouraged on cell systems that should have transmembrane potential associated with certain TELCs-membrane-TELAs capacitors having excess positive charges on one side of the membrane and excess anions on the other side.