Consider a phospholipid bilayer membrane consisting of a...

Consider a phospholipid bilayer membrane consisting of a mixture of $90 \%$ uncharged lipid and $10 \%$ singly charged acid lipid. Assume $0.68 \mathrm{nm}^{2}$ surface area per lipid head group, and assume further that the charged lipids are uniformly distributed and immobile. The membrane is in contact with an aqueous solution of $\mathrm{NaCl}$ at $25^{\circ} \mathrm{C}$. The salt concentration is $c_{\infty}=100 \mathrm{mM}$ (a) Calculate the areal charge density (that is, charge per unit surface area) of the membrane. (b) Calculate the surface potential of the membrane. What is the electrostatic energy (in $k_{\mathrm{B}} T$ units) of binding to the membrane of a trivalent positive ion such as spermidine (a biologically active polyamine) assuming that: (c) Binding occurs at the membrane surface? (d) Owing to steric factors, the charges of the bound spermidine stay in the water $0.5 \mathrm{nm}$ distant from the membrane surface? (Adapted from Problems 23.6 and 23.7 of $K .$ Dill and $S$. Bromberg, Molecular Driving Forces, 2nd ed. Garland Science, $2011 .)$

Consider a phospholipid bilayer membrane consisting of a mixture of $90 \%$ uncharged lipid and $10 \%$ singly charged acid lipid. Assume $0.68 \mathrm{nm}^{2}$ surface area per lipid head group, and assume further that the charged lipids are uniformly distributed and immobile. The membrane is in contact with an aqueous solution of $\mathrm{NaCl}$ at $25^{\circ} \mathrm{C}$. The salt concentration is $c_{\infty}=100 \mathrm{mM}$
(a) Calculate the areal charge density (that is, charge per unit surface area) of the membrane.
(b) Calculate the surface potential of the membrane.
What is the electrostatic energy (in $k_{\mathrm{B}} T$ units) of binding to the membrane of a trivalent positive ion such as spermidine
(a biologically active polyamine) assuming that:
(c) Binding occurs at the membrane surface?
(d) Owing to steric factors, the charges of the bound spermidine stay in the water $0.5 \mathrm{nm}$ distant from the membrane surface?
(Adapted from Problems 23.6 and 23.7 of $K .$ Dill and $S$. Bromberg, Molecular Driving Forces, 2nd ed. Garland Science, $2011 .)$

Key Concepts

Areal Charge Density

Areal charge density is the measure of electric charge per unit area on the membrane surface. It is an important parameter in understanding how charged molecules on a membrane interact with ions in solution and contribute to the overall electric field near the surface.

Surface Potential

Surface potential refers to the electric potential at the interface between the membrane and the surrounding solution. It arises due to the distribution of fixed charges on the membrane and influences the arrangement and dynamics of nearby ions, playing a critical role in membrane electrostatics and interactions with charged species.

Poisson?Boltzmann Equation

The Poisson–Boltzmann equation is a mean-field theoretical framework used to describe the distribution of ions around a charged surface. It combines the Poisson equation of electrostatics with the Boltzmann distribution for thermal motion of ions, enabling the calculation of the potential and ion concentration profiles near charged interfaces.

Debye Screening Length

The Debye screening length characterizes the distance over which electrostatic interactions are effectively screened in an ionic solution. It represents the decay length of the electrostatic potential away from a charged surface and is key to understanding how salt concentration influences the strength of electrostatic interactions in the system.

Electrostatic Binding Energy

Electrostatic binding energy quantifies the energy involved in the interaction between charged particles, such as ions binding to a charged surface. Expressed in thermal energy units (kBT), it allows for a comparative understanding of how the energy of interaction competes with thermal fluctuations, thereby affecting binding affinity and stability in biological and soft matter systems.

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