Question

Membrane pores. A neutral protein 'carrier' may help an ion to transfer into and across 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 \(D_{\mathrm{hc}}=2 ?\) The radius of the ion is \(2 \mathrm{~A}\). (b) Now wrap the ion in a neutral protein to produce a spherical complex with radius \(b=15 \mathrm{~A}\). What is the electrostatic free energy of transfer from water to hydrocarbon of the ion-protein complex?

Short answer

The free energy change for the ion is higher without the protein complex. Both energies can be calculated using the Born equation.

Step-by-step solution

- Calculate the ion's electrostatic free energy change

The electrostatic free energy change \(\triangle G_{\text{elec}}\) when moving a monovalent ion from water to a hydrocarbon solvent can be calculated using the Born equation: \[ \triangle G_{\text{elec}} = \frac{q^2}{8 \pi \epsilon_0 \left( \frac{1}{D_{\text{solv}}} - \frac{1}{D_{\text{water}}} \right) r} \] Here, \(q\) is the charge of the ion, \(r\) is the ion's radius, \(D_{\text{solv}} = 2\) for the hydrocarbon and \(D_{\text{water}} = 80\). For a monovalent ion, \(q = 1.6 \/\times 10^{-19} \text{C}\). Substituting the given values: \[ \triangle G_{\text{elec}} = \frac{(1.6 \/\times 10^{-19})^2}{8 \pi \times 8.85 \/\times 10^{-12} \left( \frac{1}{2} - \frac{1}{80} \right) 2 \/\times 10^{-10}} \] Simplifying the expression provides the free energy change in Joules.

- Calculate the spherical complex's free energy change

For the ion wrapped in a neutral protein forming a spherical complex with a radius of 15 Å, the new electrostatic free energy change \( \triangle G_{\text{elec, complex}} \) can be calculated using the same Born equation but with the complex radius: \[ \triangle G_{\text{elec, complex}} = \frac{q^2}{8 \pi \epsilon_0 \left( \frac{1}{D_{\text{solv}}} - \frac{1}{D_{\text{water}}} \right) b} \] Substituting the new radius \( b = 15 \mathrm{\text{Å}} \), we get: \[ \triangle G_{\text{elec, complex}} = \frac{(1.6 \/\times 10^{-19})^2}{8 \pi \times 8.85 \/\times 10^{-12} \bigg( \frac{1}{2} - \frac{1}{80} \bigg) 15 \/\times 10^{-10}} \] Simplifying the expression provides the free energy change in Joules.

Key concepts

Born equation

The Born equation is used to calculate the electrostatic free energy change when an ion moves from one medium to another. This change is dictated by differences in dielectric constants between the two media.
According to the Born equation:
\[ \triangle G_{\text{elec}} = \frac{q^2}{8 \pi \epsilon_0 \bigg( \frac{1}{D_2} - \frac{1}{D_1} \bigg) r} \]
Here:

• \( q \) is the charge of the ion.
• \( \epsilon_0 \) is the permittivity of free space (approximately \( 8.85 \/\times 10^{-12} \text{F/m} \)).
• \( D_1 \) and \( D_2 \) are the dielectric constants of mediums 1 and 2 respectively.
• \( r \) is the radius of the ion.

In simpler terms, the Born equation helps us understand how the energy needed to transfer an ion changes based on the medium’s ability to reduce electric fields.

Dielectric constant

The dielectric constant is a measure of a substance's ability to insulate charges from each other. A higher dielectric constant means the medium is better at insulating the two charges.

• Water has a very high dielectric constant, typically around 80, meaning it can significantly reduce electrostatic interactions between ions.
• Hydrocarbon solvents like hexane have much lower dielectric constants, around 2, indicating they provide less insulation between charged particles.

The difference in dielectric constants is crucial for determining the electrostatic free energy change when an ion moves between these environments. For example, moving a monovalent ion from water (high dielectric constant) to a hydrocarbon solvent (low dielectric constant) results in a significant energy change, calculated using the Born equation.

Ion-protein complex

An ion-protein complex is formed when a neutral protein surrounds an ion. This affects the ion's properties, including its effective radius. In the given problem, the complex radius is 15 Å instead of the ion's original 2 Å.
This larger radius changes the electrostatic free energy, as seen in the modified Born equation:

• The complex's new radius reduces the interaction between the ion and its surroundings.
• The energy required to transfer the ion-protein complex from water to hydrocarbon is different from the energy required for the ion alone.

Thus, wrapping an ion in a neutral protein alters the electrostatic free energy change primarily through changes in the effective interaction radius within the Born equation framework.