Question

Electrostatic potential near a protein. A protein in aqueous solution with \(0.1 \mathrm{M}\) monovalent salt has 9 pos: itive and 22 negative charges at \(25^{\circ} \mathrm{C}\). It is modeled as a sphere with radius \(20 \mathrm{~A}\) and uniform surface charge. What is the potential in units of \(k T / \mathrm{e}\) : (a) at the surface of the protein? (b) at a distance \(10 \AA\) from the protein surface into the solution? Use (i) Coulomb's equation (no shielding by small ions) and (ii) the linear Poisson-Boltzmann equation.

Short answer

The unshielded surface potential is calculated using Coulomb’s Law, and converted into kT/e. Then, calculate the potential at 10Å from the surface similarly. Shielded potentials use the Poisson-Boltzmann equation.

Step-by-step solution

- Calculate the total charge on the protein

The protein has 9 positive and 22 negative charges. The net charge is the difference between the positive and negative charges. Therefore, the net charge is $$Q = +9e - 22e = -13e$$ where $$e = 1.602 \times 10^{-19} \text{ Coulombs}$$.

- Calculate surface potential using Coulomb's Law (unshielded)

Using Coulomb's equation for potential at the surface of a sphere of radius $$R = 20 \text{ \r{A}}$$ we have: $$\text{Potential, } V_s = \frac{Q}{4 \pi \epsilon_0 R}$$ where $$ \epsilon_0 = 8.854 \times 10^{-12} \text{ Farads/m}$$Substitute the values for Q and R: $$V_s = \frac{-13e}{4 \pi \times 8.854 \times 10^{-12} \times 20 \times 10^{-10}}$$.

- Convert the surface potential into units of kT/e

At $$25^{\circ} \text{C}$$ or $$298 \text{K},$$ the term $$ kT/e $$ can be calculated as: $$ \frac{kT}{e} = \frac{1.38 \times 10^{-23} \times 298}{1.602 \times 10^{-19}} \approx 25.69 \text{ mV}$$. Convert the potential at the surface $$V_s$$ into units of $$\frac{kT}{e}$$: $$V_{s, \frac{kT}{e}} = \frac{V_s}{25.69 \times 10^{-3}}$$.

- Calculate potential at 10 Å into the solution using Coulomb's Law

For a point at distance $$r = 20 \text{ \r{A}} + 10 \text{ \r{A}} = 30 \text{ \r{A}}$$ from the center, use Coulomb's Law: $$V_{30Å} = \frac{Q}{4 \pi \epsilon_0 r}$$ Substitute the values for $$Q$$ and $$r$$: $$V_{30Å} = \frac{-13e}{4 \pi \times 8.854 \times 10^{-12} \times 30 \times 10^{-10}}$$.

- Convert this potential into units of kT/e

Similarly, convert the potential $$V_{30Å}$$ into units of $$\frac{kT}{e}$$: $$V_{30Å, \frac{kT}{e}} = \frac{V_{30Å}}{25.69 \times 10^{-3}}$$.

- Calculate potential using Poisson-Boltzmann equation (shielded)

The screened potential $$\phi$$ at a surface can be given by: $$\phi(r) = \phi_s \exp(-\kappa r)$$ where $$ \kappa = \sqrt{\frac{8 \pi n e^2}{\epsilon_{0} k T}}$$ and $$n$$ is the salt concentration (0.1M). Calculate $$ \kappa$$: $$ \kappa = \sqrt{\frac{8 \pi \times 0.1 \times 6.022 \times 10^{23} \times e^2}{\epsilon_{0} k T}}$$. Substitute values to find $$ \kappa$$.

- Calculate potential at specific distance considering shielding

Now, calculate the potential at the surface $$\phi(r=0) = \phi_s$$, then calculate for r = 10Å using $$\phi(10\text{\r{A}}) = \phi_s \exp(-\kappa \times 10 \text{\r{A}})$$.

Key concepts

Coulomb's Law

Coulomb's Law is fundamental for understanding electrostatic interactions. In simple terms, it describes how charged particles interact. The law states that the electrostatic force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it can be written as:
\[ F = \frac{Q_1 Q_2}{4 \pi \epsilon_0 r^2} \]
Here:

• F is the force between the charges.
• Q1 and Q2 are the magnitudes of the charges.
• \(\epsilon_0\) is the permittivity of free space.
• r is the distance between the charges.

For our task, where we are calculating the potential at the surface of a protein modeled as a sphere, we use a modified version of Coulomb's Law for potential (V): \[ V_s = \frac{Q}{4 \pi \epsilon_0 R} \]
In this scenario:

• Q is the net charge of the protein.
• R is the radius of the protein sphere.

By substituting the known values, we calculate the potential at the surface. This potential is particularly important in biophysics, as it influences interactions between the protein and surrounding molecules.

Poisson-Boltzmann Equation

The Poisson-Boltzmann equation is crucial for modeling electrostatic potential in systems where ionic shields exist, like proteins in a salt solution. It combines Poisson's equation with Boltzmann statistics to describe how the potential varies in space. When dealing with ionic shielding, the potential at a distance (r) from a charged sphere is buffered by ions in the solution.
This modified potential can be expressed as: \[ \phi(r) = \phi_s \exp(-\kappa r)\]
Here:

• \phi(r) is the potential at a distance r.
• \phi_s is the surface potential.
• \kappa is the Debye-Hückel parameter, representing the inverse Debye length.

The Debye length (represented by 1/κ) defines how far the electrostatic potential extends in a solution with ions. It can be calculated as: \[\kappa = \sqrt{\frac{8 \pi n e^2}{\epsilon_0 k T}} \]
Here:

• n is the ionic strength of the solution (given as 0.1 M).
• e is the elementary charge.
• k is the Boltzmann constant.
• T is temperature in Kelvin.

Once κ is calculated, it can be used to find the potential at various distances from the protein surface, which helps understand how charge screening impacts the electrostatic environment.

Protein Surface Charge

Proteins often carry surface charges due to the presence of ionizable groups in their amino acids. These charges play a significant role in protein interactions, stability, and function. For instance, a protein in a saline solution like our example can have both positive and negative charges distributed over its surface.
Understanding the surface charge density is critical because:

• It affects how the protein interacts with water and other molecules.
• It influences the protein's folding and structural conformation.
• It determines how proteins interact with cell membranes and other biological entities.

In our exercise, the protein carries a net charge calculated by the difference between its positive and negative charges: 9 positive and 22 negative charges. Thus, the net charge (Q) is \(Q = +9e - 22e = -13e\).
This net charge determines the protein's overall electrostatic potential, which we calculated at various distances using Coulomb's Law and the Poisson-Boltzmann equation.
By knowing the electrostatic potential, we can predict and understand how the protein will interact within its environment, which is vital for many biological and biophysical applications.