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

Abstract

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 kT 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 et al. 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 et al., Biomolecules, 2023, 13, 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.