Chemistry

1) Sequential binding models for partial cooperativity depend upon the geometric arrangement of the binding sites and the way in which the binding sites interact in this geometry as they become occupied by ligands. Consider two macromolecules each of which have three ligand binding sites. The general formula for the binding polynomial is:? ? ?? ?? ?? 2 3
1 12 123 13 3 b q k L kk L kkk L
P
?? ? ? . The equilibrium constants are defined as in
Lectures 10 and 11 and in the Tinoco text pages 180-190.
a. Suppose the macromolecule has its three binding
sites arranged in a linear array as shown to the right.
Only adjacent sites that are occupied can interact.
So for all singly bound molecules k1=k. For doubly bound molecules k1k2=k2, if
the occupied sites are not adjacent and k1k2=fk2 if they are adjacent. For the filled molecule 2 3
123 kkk f k ? . Using these rules write out the expression for the binding polynomial ? ?
b q
P in terms of f, k, and [L].
b. Using your answer from part a, and the relationship ? ?
? ?
b
B
b
L q f Nq L
? ? ? , obtain an
expression for fB in terms of f, k, and [L]. Assume N=3. Now let k=1.00×106
,
[L]=0.00001M and f=1.5. Calculate fB. Compare this answer to the value for fB if
the binding were non-cooperative. Is this mode of binding cooperative or anticooperative?
Explain.
c. Repeat the calculation in part b only assume f=0.75. Is this mode of binding cooperative or anti-cooperative? Explain.
d. Repeat the calculations in parts a-c only now assume the binding sites are arranged in an equilaterial triangle:
2) A protein has two structural forms R and T. These structural forms are in equilibrium as shown in the adjacent diagram with
? ?
? ?
T
K
R ? . In addition each form of the protein has a single ligand binding site and when R binds the ligand L, RL is formed and similarly with T, L, and TL. As the diagram also shows R and RL are in equilibrium with? ?
? ?? ? R
RL
k
R L ? and similarly T and TL are in equilibrium with ? ?
? ?? ? T
TL
k
T L ? . R binds L
more tightly than T binds L so R T k k ? .
a) Define the binding polynomial as q R T TL RL P ??? ? ? ? ? ? ? ? ? ?. Using
the three equilibrium constant expressions given above, express ? ?
P q
R
in terms of kT, kR, K, and [L].
b) Using your result for ? ?
P q
R from part a, and the relationship
? ?
? ?
P
B
P
L q f qN L
? ? ? obtain an expression for fB in terms of kT, kR, K, and[L]. Calculate fB for K=10, kT=0.1, and kR=1.0.
c) Using your result from part b, obtain an expression for the ratio
1
B
B
f
? f
in terms of kT, kR, K, and [L]. Would a log-log plot (i.e. a Hill plot) indicate that the binding in this system is: fully cooperative,partially cooperative or non-cooperative? Explain your answer.
d) Suppose the equilibrium R ? T lies to the right so that ? ?
? ? 1
T
K
R
? ? .
Determine an expression for the equilibrium constant ? ?
? ?
TL
L
RL ? in
terms of kT, kR, and K. Based on this expression, when T and R bind
L, is the equilibrium between TL and RL shifted to the right or the
left? In other words, is the equilibrium constant L greater than, less
than, or equal to K? Explain. Calculate L using the values for K, kT
and kR provided in part b.
3) The distribution function W(x) for a random walk gives the probability of finding the
“walker” at a position x. For a random walk composed of N jumps, each of length ? ,
where the probability of taking a single jump in the +x direction is p and the probability
of taking a jump in the –x direction is q=1-p, W(x) has the form:
? ? ? ?2 2 /2
2
1
2
x x Wx e ?
??
? ? ?
R T
RL TL
?
?
? ?
where the average displacement x ? N pq ? ? ? ? and the variance
2 22 2 ? ??? x x pqN 4 ? . See Lecture 12.
a) In Lecture 12 section A we said an unbiased random walk is defined as a walk where there is no preference for the +x or the –x direction so that p=q. Suppose a single-celled animal executes an unbiased random walk which consists of 20jumps (i.e. N=20) and that each jump is of length ? ? 0.05mm . Calculate the
average displacement x , the mean squared displacement 2 x and the variance?2 for this random walk. Calculate the probability W(x) of finding the cell at x=0.0 mm after 20 jumps. Do the same calculation for x=0.5mm and at x=1.0mm
b) Repeat the calculation of the probability of remaining at x=0.0mm for N=10,
N=60 and N=100. Comparing your results for parts a and b, does the probability of remaining at x=0.0mm decrease linearly with N or does it decrease faster than linearly? Explain.
c) Suppose in response to a glucose gradient the cell’s motion becomes biased such that the probability of taking a jump in the +x direction is p=20/38. Now the single-celled animal executes a biased random walk which again consists of 20jumps (i.e. N=20) with each jump of length ? ? 0.05mm . Calculate the average displacement x , the variance, and the mean squared displacement 2
x for this biased random walk. Calculate the probability W(x) of finding the cell at
x=0.0mm after 20 jumps. Do the same calculation for x=0.5mm and x=1.0 mm.
d) Repeat the calculation of the probability of remaining at x=0.0mm for biased random walks consisting of N=10, N=60 and N=100 jumps, assuming p=20/38 and ? ? 0.05mm . Does the probability decrease linearly with N or does it decrease faster than linearly?
4)Consider the following data for the proteins myoglobin and hemoglobin
Protein
Molecular
Weight (kg)
Diffusion
Coefficient
Dx1011m2
s-1
Specific
Volume V2
(mLg-1)
Frictional
ratiof/f0
Myoglobin 16.900 11.3 0.74 1.11
Hemoglobin 64.500 6.9 0.75 1.16
a) Calculate the radii of myoglobin and hemoglobin assuming they are unhydrated spheres.
b) Calculate the radii and volumes of myoglobin and hemoglobin that would account for the frictional ratios, assuming they are hydrated spheres.
c) For each protein, calculate ?1, the mass of water bound per mass of protein.
Assume the density of water is 1g mL-1.
d) Myoglobin is a oxygen storage protein found in the body tissues. Hemoglobin is
an oxygen transport protein. Hemoglobin is composed of n sub-units each roughly the size of myoglobin. Based on your answers in part b, how many myoglobin-like sub-units does hemoglobin contain?
5) A macromolecule has a diffusion coefficient 11 2 1 D ms 6.9 10? ? ? ? at T=298K and for a
solvent viscosity of 3 11 ? 0.891 10 kg m s ? ? ? ? ? .
a) Assuming the molecular is approximately spherical in solution, calculate the translational frictional coefficient ?tr and the radius.
b) Calculate the root-mean-squared displacement of the molecule after 1 millisecond.
c) For the purpose of calculating the translational friction f, many rod-like polymers can be approximated in solution as a chain of beads, each bead of diameter d. For such a polymer the frictional coefficient is 3 ln
Nd f N?? ?
where N is the number of beads in the polymer chain and ? is the solvent viscosity. . Suppose the macromolecule in part a aggregates as a linear,rod-like hexamer. Calculate the coefficient of translational friction and the coefficient of translational diffusion. Calculate the rms displacement after 1 millisecond. Assume 3 11 ? 0.891 10 kg m s ? ? ? ? ?
d) Suppose the molecule in part a forms a hexamer in solution, but the aggregate is roughly spherical, with a volume equal to six times the volume of the spherical monomer. Calculate the coefficient of frictional and the coefficient of translational diffusion. Calculate the rms displacement after 1 millisecond.
e) Based on your answers in part a-d, can translational diffusion coefficients be used to detect aggregation? How sensitive are diffusion coefficients to aggregate geometry?

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