In the toy model of a protein described in Section 9.3 .2,...

In the toy model of a protein described in Section $9.3 .2,$ we assumed that a protein can be thought of as a charged sphere in water. Here, we consider the effect of salt on its electrical energy. (a) Compute the energy of a charged spherical protein of radius $R,$ in water, and in the presence of monovalent salt at concentration $c_{\infty} .$ Assume that the charged residues are uniformly distributed on the surface of the protein. (b) Redo the calculation leading to the plot in Figure 9.14 Plot the electrical energy of the protein as a function of its radius for different salt concentrations, ranging between $1 \mathrm{mM}$ and $100 \mathrm{mM}$. What conclusion do you draw about the effect of salt on the charged state of a protein?

In the toy model of a protein described in Section $9.3 .2,$ we assumed that a protein can be thought of as a charged sphere in water. Here, we consider the effect of salt on its electrical energy.
(a) Compute the energy of a charged spherical protein of radius $R,$ in water, and in the presence of monovalent salt at concentration $c_{\infty} .$ Assume that the charged residues are uniformly distributed on the surface of the protein.
(b) Redo the calculation leading to the plot in Figure 9.14 Plot the electrical energy of the protein as a function of its radius for different salt concentrations, ranging between $1 \mathrm{mM}$ and $100 \mathrm{mM}$. What conclusion do you draw about the effect of salt on the charged state of a protein?

Key Concepts

Debye–Hückel Theory

Debye–Hückel theory explains how ions in an electrolyte modify the electrostatic interactions between charged particles. It introduces the concept of the Debye screening length, which characterizes the exponential decay of the electrostatic potential in an ionic solution. This theory is essential for calculating how the presence of salt (monovalent ions) screens the charges on the sphere, thereby affecting its electrostatic energy.

Charged Sphere Model

The charged sphere model is a simplified representation frequently used in physical chemistry and biophysics to approximate macromolecules like proteins. In this model, the charge is distributed uniformly over the surface of a sphere. This abstraction helps in applying classical electrostatic theory to compute properties such as the self-energy, making the problem tractable while still capturing key physical effects.

Electrostatic Self?Energy

Electrostatic self?energy is the energy stored in the electric field due to a distribution of charges. For a charged object like a sphere, it is obtained by integrating the contribution of each infinitesimal charge element against the electric potential created by all the other charges. This concept is central in understanding how the overall charge and its spatial distribution determine the energy and stability of the system.

Effect of Salt on Electrostatic Interactions

Salt in solution introduces free ions that interact with the charged sphere, leading to a screening effect. With higher salt concentrations, the range and strength of electrostatic interactions are reduced because the ions neutralize the field over a characteristic length scale (the Debye length). This results in a decrease in the overall electrostatic self-energy of the system, which is a critical factor in determining the stability and behavior of charged species in solution.

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