A neutral protein "carrier" may help an ion to transfer into...

A neutral protein "carrier" may help an ion to transfer into and cross a lipid membrane. (a) What is the electrostatic free-energy change when a monovalent ion is transferred from water at $25^{\circ} \mathrm{C}$ to a hydrocarbon solvent with dielectric constant $D=2 ?$ The radius of the ion is $0.2 \mathrm{nm}$ (b) Comment on the ability of the ions to diffuse through lipid bilayers. How has nature solved this problem? How do ions manage to get across? (Adapted from Problem 22.11 of K. Dill and S. Bromberg, Molecular Driving Forces, 2nd ed. Garland Science, 2011.)

A neutral protein "carrier" may help an ion to transfer into and cross a lipid membrane.
(a) What is the electrostatic free-energy change when a monovalent ion is transferred from water at $25^{\circ} \mathrm{C}$ to a hydrocarbon solvent with dielectric constant $D=2 ?$ The radius of the ion is $0.2 \mathrm{nm}$
(b) Comment on the ability of the ions to diffuse through lipid bilayers. How has nature solved this problem? How do ions manage to get across? (Adapted from Problem 22.11 of K. Dill and
S. Bromberg, Molecular Driving Forces, 2nd ed. Garland Science, 2011.)

Key Concepts

Protein Carriers in Membrane Transport

Protein carriers, or transport proteins, are nature’s solution to the problem of moving ions across lipid membranes. These specialized proteins bind ions, effectively shielding their charge from the low dielectric interior of the membrane and providing a pathway that minimizes the energetic barrier. This mechanism allows ions to cross the membrane despite the unfavorable conditions of the lipid bilayer.

Lipid Bilayer Permeability

Lipid bilayers, the fundamental structure of cell membranes, have a hydrophobic core with a very low dielectric constant, making them energetically unfavorable for charged ions to pass through directly. This impermeability arises because ions are not stabilized within the hydrophobic environment, resulting in a significant energetic barrier to diffusion.

Dielectric Constant

This concept refers to the ability of a medium to reduce the effective electric field between charged particles. It is a measure of how a solvent, like water or a hydrocarbon, can stabilize ions by polarizing in response to their electric fields. A high dielectric constant (such as water) reduces the electrostatic interactions between ions, while a low dielectric constant (like that of hydrocarbon solvents) does not provide significant stabilization, leading to a higher free-energy cost when an ion is transferred into such an environment.

Born Equation

The Born equation is used to estimate the free-energy change associated with moving a charged ion from one dielectric medium to another. It calculates how the energy of the ion changes based on its charge, radius, and the dielectric constants of the originating and destination media. This concept is crucial for understanding the energetic penalty an ion experiences when moving from water to a hydrocarbon environment.

Electrostatic Free Energy

Electrostatic free energy is the energy change resulting from the electrostatic interactions of charged particles within a medium. In the context of ion transfer, this energy change quantifies the cost of moving an ion from a high dielectric medium, which stabilizes the charge, to a low dielectric medium, where stabilization is minimal, leading to a large increase in free energy.

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