Physics 450 - Problem Set 2 - due February 12 2001
Anjum Ansari, ansari@uic.edu     John Marko, jmarko@uic.edu

1. Diffusion Equation:
(a) Verify that the `spreading Gaussian' discussed in class

c(x,t) =  C0 
t1/2		 exp[-x2/(4Dt)] 
 
satisfies the one-dimensional diffusion equation
c 
t		 = D  2 c 
x2
 
(b) Verify that c(x,t) of (a) describes the spreading of a constant number of particles, i.e., that
ó 
õ 		 ¥ 

-¥ 

 dx   c(x,t)
 
is a time-independent constant

(c) Using (a), show that the three-dimensional version of c,

c(r,t) =  C0 
t3/2		 exp[ - (x2+y2+z2)/(4Dt) ]
 
satisfies the three-dimensional diffusion equation
c 
t		 = D  æ 
ç 
è 		 2 
x2		 +  2 
y2 		 +  2 
z2		 ö 
÷ 
ø 		 c 
 
(d) Repeat (b) for the three-dimensional case, using the three-dimensional integral of total particle number,
ó 
õ 		 ¥ 

-¥ 

 dx  ó 
õ 		 ¥ 

-¥ 

 dy  ó 
õ 		 ¥ 

-¥ 

 dz   c(r,t)
 
2. DNA Sequence:
(a) Translate the following 399-base DNA sequence into a corresponding sequence of 133 amino acids, using the genetic code (the sequence is in 5' to 3' order, as read in the cell):

 atgacgaaa gatgaactga ttgcccgtct ccgctcgctg ggtgaacaac tgaaccgtga
tgtcagcctg acggggacga aagaagaact ggcgctccgt gtggcagagc tgaaagagga
gcttgatgac acggatgaaa ctgccggtca ggacacccct ctcagccggg aaaatgtgct
gaccggacat gaaaatgagg tgggatcagc gcagccggat accgtgattc tggatacgtc
tgaactggtc acggtcgtgg cactggtgaa gctgcatact gatgcacttc acgccacgcg
ggatgaacct gtggcatttg tgctgccggg aacggcgttt cgtgtctctg ccggtgtggc
agccgaaatg acagagcgcg gcctggccag aatgcaataa

This data is for a DNA-packaging protein from a virus that infects E. coli, and which can make it explode. So you see E. coli has problems too.

(b) Find a 12-base-pair DNA sequence that will hybridize (base-pair) to the DNA sequence

5¢-aggtccggccag-3¢
 
You don't get any part marks if you get the direction of the sequence wrong.

(c) Design a 30-base-pair single-stranded DNA which will form a `hairpin' with a 10-base-pair `stem' and a 10-base single-stranded `loop'.
Take care not to choose sequences that will base-pair in unwanted ways!

 

3. Relative Brownian Motion:
(a) (easy) Cell biologist A comes to you with data for 3d mean-square displacement of a diffusing particle. You notice that the data fit

< r2 > = < x2 + y2 + z2 > = c t
 
What is the diffusion constant of the particle?

(b) (harder) Cell biologist B comes to you with data for the relative 3d mean-square displacement of two identical diffusing particles, where at time t = 0, the two particles start out from the same place. You notice that the data fit

< (r1 - r2)2 > = < (x1-x2)2 + (y1-y2)2 + (z1-z2)2 > = ct 
 
What are the diffusion constants of the two particles?

(c) (too hard) Cell biologist C comes to you with data for the relative 3d mean-square displacement of two identical particles, where the two particles start out (at t = 0) a distance r0 apart. The data initially fit

< (r1 - r2)2 > = r02 + ct
 
What are the diffusion constants of the two particles? (Hint: you may assume that the initial separation is along the x-axis, i.e. r1 r2 = (r0,0,0) at t = 0.)

(P.S. this really happened!)

4. Equilibrium of Sedimentation:
(a) N particles are dumped into volume V, to achieve concentration c0 = N/V. Their masses are Dm relative to the water that they displace. If the container is a cylindrical tube of height h, what is the equilibrium concentration profile as a function of height z? Note that the earth's gravity acts in the z-direction, with acceleration g = 9.8 m/sec2.

(b) For a 1 cm high tube, plot the equilibrium distribution of 1-nm and 1-micron radius particles which are 20% heavier than water.

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On 25 Jan 2001, 07:30.