When DNA becomes long enough, it can no longer be treated as a prolate ellipsoid

Question

When DNA becomes long enough, it can no longer be treated as a prolate ellipsoid but should be viewed as a random-coil polymer. For the purposes of calculating the intrinsic viscosity [eta] and the frictional coefficient f, a very long random-coil polymer may be treated as a spherical molecule, with radius R_H, called the radius. The hydrodymamic radius is related to the radius of gration R_ o by the equation: R_H = 0, 662R_a. The radius of hydration for a very long DNA can be calculated from its length: R^2_G = 2PN(3.4 times 10^-4)cm/6 Where N is the number of base pairs and P is a property of a polymer called the persistence length. The persistence length measures the length over which a polymer points in the same direction as its first bond. The persistence length for DNA is P-500 times 10^-8 cm. Calculate the radius of gyration R_G and hydrodynamic radius R_H for a DNA molecule of 1000 base pairs in length. Treating the 1000 base pair DNA as a sphere with radius R_H, calculate the fictional coefficient f (assume the viscosity of the solvent to be 0.01 poise). Again treating the 1000 base pair DNA as a sphere with radius R_H, calculate the intrinsic viscosity [eta]. Remember that the molecular weight of that DNA is 675,000 g/mole. For a random-coil polymer, the molecular weight M is related to the intrinsic viscosity [eta] and the sedimentation coefficient s by the equation: beta_ i = eta_0N_AM^-2/3[eta]^2/3 s/(1 - V^IverBar_2 rho) times 10^13 where eta_0 is the viscosity of the solvent, assumed to be 0, 01 poise, rho is the density of the solvent, assumed to be 1g/cc, V^OverBar_2 = 0, 51mL/g is the specific volume of the DNA, N_A is Avagadro's number, and beta_2 is a constant which, for a random oil polymer, equals about 1.2 times 10^7. Note the factor of 10 in the denominator requires that s, the sedimentation coefficient, be expressed in units of Svedbergs S. Calculate the sedimentation coefficient for a 1000 base pair DNA molecule.

Explanation / Answer

Calculate the radius of gyration RG for a DNA which is 1000 base pairs in length. Also, calculate the hydrodynamic radius RH. :

R^2G = 2PN (3.4 X10^-8 cm) /6 =(2000)(500 x10^-8 cm)( 3.4x10^-8 cm)/6 =5.7 x10^-11 cm^2

Then Rg =7.55 x10^-6 cm

RH= 0.662 Rg =5 x10^-6 cm

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Treating the 1000 base pair DNA as a sphere with radius RH, calculate the frictional coefficient f. Assume the viscosity of the solvent is 0.01 poise.

f=6pien RH =18.84X0.01 g /cm .s X 5 X10^-6cm =9.42 X10^-7 g/ s

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