Nanomechanics Problem Set

1) Debye length:

The physiological concentration of ions in a cell is close to 150 mmol/L. Calculate the Debye length for a NaCl solution at 150 mmol/L in water at 25^oC. What would be the value of the Debye length if you add 10 mmol/L of spermine chloride (making tetravalent cations and four Cl^- ions per molecule, or 10 mmol/L of spermidine chloride (with trivalent cations and three Cl^- ions per molecule? Imagine you are making an arti?cial cell by creating lipid vesicles with CaCl₂ at 150 mmol/L instead of NaCl inside it. Calculate the Debye screening length in your arti?cial vesicle-based cell. For a complex ionic solution the Debye length is given by

λ_D = (4πλ_B∑_iz_i²n_i)^-1/2

where z_i, n_i are the ion valency and number concentration of ion type i, respectively. Do not forget to express the concentrations in 1/m³.

2) DLVO interactions:

Consider a solution of lysozyme proteins in 1 mmol/L monovalent salt bu?er. The charge on proteins at these conditions is 10e, the Hamaker constant H = 10k_BT. The shape of the molecule can be approximated by a sphere with radius of 1.7 nm. What is the potential barrier for the molecule aggregation? Will the solution be stable at these conditions? Will it be stable at 10 mmol/L salt, 100 mmol/L salt concentration?

3) Charge of λ-DNA:

Assume the DNA has one elementary charge per 0.17 nm and that the charge is spread homogeneously along its length. What is the e?ective valence of λ-DNA before and after Manning condensation is taken into account? How does it vary with temperature? Formulate the condition of overcharging of double-stranded DNA molecule by counterions. Assume that DNA double helix is a cylinder of radius 1 nm with charge density of -1e per 0.17 nm spread homogeneously over its surface. Calculate the charge coupling parameter for monovalent, divalent, and trivalent counterions and discuss their ability to overcharge or condense the molecule. Repeat the calculation with the effective charge density after the Manning condensation.

4) Overcharging:

Test the condition of overcharging of the following polyions:

(a) double-stranded DNA molecule; Assume that DNA double helix is a cylinder of radius 1 nm with charge density of -1e per 0.17 nm spread homogeneously over its surface.

(b) SDS micelle of radius 2.2 nm and the charge -60e.

(c) latex bead of radius 2 m and surface charge density -0.02 μC/cm²;

Calculate the charge coupling parameter = 2πσλ²_Bz³ for monovalent, divalent, and trivalent counterions and discuss their ability to overcharge the polyion. The overcharging boundary value of is 10.

Attachment:- Assignment.rar