A Solution to RWP1 - Gel Electrophoresis
Gather
information: We
know from the statement of the problem that the DNA fragments are
negatively charged, and that the amount of charge is proportional to
the length of the fragment, but we do not know for certain the amount
of charge per
base pair. The smallest possible charge per base pair is: q = -2e
= -2(1.6e-19 C).
The molecular "weight" of an average base pair = 635 daltons =
635(1.67e-27 kg) = 1.06e-24 kg
Fragments that can be analyzed range from 200 bp to 50 kb,
corresponding to: m = 2.1e-22 kg to 5.3e-20 kg
Note: The largest DNA fragment that can be analyzed with this
technique is a small fraction of the DNA in a haploid nucleus:
(5e4)(635)/1.9e12 = 1/60,000
Based on the time required for the DNA fragments to travel, they must
not be accelerating, but are migrating at a terminal speed of
about 1 ft/hr.
Altough not explicitly stated, the applied electric field must be
uniform, created by two parallel linear electrodes at a difference of
50 to 150 V.
Assuming the gels are run the length of the box, then the strength of
the electric
field must be approximately E = (100 V)/(20 cm) = 5 V/cm = 500 V/m.
Organize: This problem is
primarily an application of a uniform electric field. The
important aspect that is different from previous textbook problems is
the fact
that the gel provides a resistive force which opposes the electrostatic
force on the negatively-charged DNA fragments.
Analyze: The NET force
acting on a DNA fragment must be zero, otherwise it would accelerate
and take much less than 30 minutes to travel ~20 cm.
Verification: Assuming only the minimum electric charge of -2e per base
pair, the electric force on a typical DNA fragment of length 1000 bp
would be:
F = qE = -2(1.6e-19 C)(500 N/C) = -1.6e-16 N
With no opposing force, this would result in an
acceleration: a = F/m = (1.6e-16 N)/(1000*1.06e-24 kg) = 1.5e5 m/s^2 =
15,000 g !!!
If this were true, then the DNA fragments would
travel the length of the gel box in a fraction of a second!
It is clear from the the Physlet simulation that the smaller DNA
fragments do travel faster and migrate farther than the larger
molecules, despite the fact that the larger molecules should have a
greater electric force since the charge is proportional to the number
of base pairs. Evidently, the resistive force on these molecules
depends greatly on their size, much like objects falling through air
where density is more important than size in determining terminal
speed.
The Physlet simulation does not show that the larger DNA fragments
migrate proportionally faster in higher fields, but instead there is
simply a constant shift to the right in the location of the bands as
the applied voltage is increased. If this non-linear effect does
occur in real life (as indicated in the problem statement), then it
could be due to the non-linear drag forces acting on the larger
molecules.
From this simulation, it appears that the rate of migration increases
with the applied field strength, but not linearly. Instead, the
rate of migration increases quickly for lower fields and more slowly
for higher fields, yielding a curve that looks similar to the I-V plot
for a light bulb (rising up and to the right as it begins to level off).
Learn: The movement of
the DNA fragments in a gel electrophoresis box is very similar to the
movement of electrons in a wire. In both cases, the
negatively-charged particles move in response to an external electric
field, but these mobile particles are slowed to a "snail's pace" by
obstacles that impede their motion. If the electrodes were points
instead of lines, the resulting electric field would not be uniform,
and the rate of migration would be greater in the center (where the
field is strongest) than near the edges of the gel box. Even with
the parallel-line geometry, the electric field is somewhat weaker near
the sides of the box, so hopefully the analysis is confined to the
rectangular region between the linear electrodes.
Grading rubric: (10 pts. total)
2 - Gather information: charge, mass, speed, electric field
1 - Organize: approach using constant E with resistive force
2 - Analyze: Net force = 0, electric force calculated for typical
DNA fragment
1 - rate of migration is non-linear, graph of migration versus E
increases but more slowly (concave down)
1 - recognition that smaller fragments travel faster despite less
electric force
1 - effect of non-uniform electric field from pointed electrodes
2 - Learn: relevant physical insights (similarity to electric
current, field fringing effects, etc.)
+1 - Overall effort, organization, neatness, and citation of references
Key to comments marked on
papers:
1. Failed to gather sufficient
information about the
experiment. Needed to say something about charge/mass/speed/Electric
field. See
the "G' section of the solution posted online for an example of what
was
expected.
2. Did not give a satisfactory
explanation of why the
rate of motion is non-linear.
3. Did not sufficiently explain the
effect of
non-uniform electric field from point electrodes.
4. Did not adequately discuss how DNA
fragments of
different sizes experience different drag forces.
5. Did not conclude that net force on DNA
is 0 N
6. Relevant insights insufficient. See
solution for an
example of what was expected.