Question

The motion of a small piece of DNA injected into a cell by a virus is studied over a relatively long period, and the average value of the squared displacement along one direction, after studying many trials,is provided in the table below. What is the diffusion constant of this piece of DNA (in gun2/s )? Be careful with numerical factors that appear in the diffusion displacement equation. time (s) average squared displacement (micrometern2) 0.45 2 3 1.05 1.55 1.95 5 2.55 3.05 3.45 7 8 4.05 4.45 9 10 5.05 Question 2 1 pts Assuming that the Stokes-Einstein equation can be applied to the random motion of this DNA, what is the approximate size (radius of gyration) of this DNA molecule? Assume the experiment was done at a temperature of 37°C, at which the viscosity of water is around 7.0x10-4 Pa.s. Be careful converting units, e-g. 1 micrometer equals 10 meters. The Boltzmann constant kB-1.38x10-23 J/K. Provide your answer in micrometers.

Answer 1

1) For Diffusion over a distance the following equation is use:

where is the average squared displacement , $q_{i}$ is the numerical constant which depends on dimensionality such that q_i = 2 for 1 dimensional diffusion, D is the diffusion constant and t is the time.

In this case since the average value of squared displacement is taken along one direction we will use $q_{i}$= 2.

Averaged squared displacement 0.45 0.23 1.05 0.26 2 0.26 3 1.55 1.95 0.24 4. 2.55 0.25 5 3.05 0.25 6 3.45 0.25 4.05 0.25 0.25 4.45 10 5.05 0.25

Therefore the average diffusion constant D = 0.25 um²/s

2)Stokes–Einstein equation:

kBT 0

where, η is the dynamic viscosity, r is the radius of gyration, T is the temperature and k_B is the boltzmann constant.

1310JK (310K) 6 3.14 (7.0 * 104Pa.s) (0.25 10-12m2/s)

Conversion:  
00C-273K
,
lạm = 10-6m 1,1m2 10-12m2

403 10-23 16 = 12.22-10-7m = 1.22 * 10-6m = 1.22um 32.9710-16