This is a list of problems, some related to the course material others to
broaden your horizons.
Please contact us by email or visit during tutoring if you
have questions or comments related to the problems below.
P1. Find the book Intermediate Physics for Medicine and Biology (3rd ed.), R.K. Hobbie (Springer 1997) and find out what forces are acting on different
body parts.
P2. For more FORCE calulations see Koehler's home page
P3. VISCOSITY problems related to humans
P4. On ALLOMETRY try to check if there is any scaling of e.g. your
own ventilation system. Archive
P5. Go through the physical factors behind the ACTION potential and do the Quiz
P6. To comb a hedgehog is not that easy also from a physical point
of view. Figure out why in connection with the nature of certain
defects in nematic liquid crystals.
P7. Pick your favorite protein, then go to a Protein Data Base (e.g. Brookhaven, Swiss-prot, or EMBL; check Biomolecule link on course homepage)
Download the pdb-file. Visualize the molecule using a free program like Rasmol. Check the quality of the structure using a Ramachandran plot e.g. at:
UPPSALARS (not Mac)
or BIOTECH.
In the header of the pdb-file you will find the resolution of the experiment - below 3Å is good.
P8. How fast is your hair growing? Try to figure out the rate/s and try to
connect this to how many amino acids which have to be added in every
second to the growing protein chain, i.e. the rate of protein synthesis.
Hint: Hair concists mainly of alfa-keratin which has an alfa-helical
structure. What are the physical dimensions
of an alfa-helix? How many
amino-acids are there per unit length?
P9. Check the difference between X-rays and electrons for resolving power (9-1).
P10. Show that within a forward scattering approximation the x-ray diffraction
pattern is to good accuracy determined by the projection of the object in question
on a plane perpendicular to the incoming beam.
P11. Show that the angle a between the arms in the DNA-cross is related to
the phosphorus helix radius r via the relation r=(P/2pi)cotan(a/2)
where P is the helix period (3.4nm). What do you get for r?
P12. Show how the missing 4th-layer in the X-ray diffraction pattern of
DNA can be connected with a displacement between the two helices.
Is
it a unique displacement or can you find other ones. Are their other
arguments which can be used to pin-point the best choice?
P13. Using the definition of twist show that for a simple helix it measures
the number of turns the vector connecting the linear axis and the helix
makes around the linear axis
P14. For simple estimates of molecular binding see Koehler's home page
P15. Let two dipoles be along a line separated a distance D. Calculate
the interaction to lowest order and show that you need to go to second
order perturbation theory to calculate their van der Waals interaction.
P16. Figure out the symmetries of yourself. Also find out how the corresponding symmetry group is denoted
(take a look in any textbook on group theory).
P17. Find a triangulation of a torus and show that the Euler
characteristics being equal to
2-2g (g being the genus) is fulfilled
P18. Calculate the mean curvature (H) for a torus.
P19. An ellips is parametrized by (a sinu cosv, b sinu sinv, d cosu) where a,b and d are constants.
Calculate the Gaussian curvature (G), the mean curvature (H) and the area element (dS).
P20. Do P19 for a hyperbolic paraboloid, parametrized by
(a u coshv, b u sinhv, u^2) (a and b constants).
P21. Revise the technique of energy minimization with Lagrange multipliers. Find the minimum potential energy for a thin rope in a constant gravitational field. The length of the rope is constant and its ends are attached to two different walls.
P22. Within the Bilayer Coupling model calculate the area difference
for a thin spherical shell (thickness d) from 2d times the surface
integral over H. Introduce a radius r.
P23. Calculate T (the surface integral of H^2) for an ellips (a,b,c) and show that T is minimized if a=b=c, that is by a sphere.
P24. Consider a closed membrane of a given size. Estimate for this
size the amount of molecules, charges etc which are transported
across every second.
P25. Consider an ideal (bio)polymer chain carrying charges ±e at
both ends. What will be its relative elongation in a field
E=30,000 V/cm?
P26. Calculate the average surface area per molecule
in a lipid bilayer membrane.
P28. Consider the steady state flux of particles to be absorbed by small patches (raduus s) on
a sphere of radius a. The absorption rate (concentration far away C_o)
can be written 4 pi D C_o times the electrostatic capacitance of the
object. Show this by an analogy with electrostatics.
In both cases you
solve Laplace equation
P29. In the previous problem one finds that the absorption rate
in units of the absorption rate for a sphere is N/1+N where N is the
number of absorption patches in units of pi a/s. Calculate the mean
distance between patches when N=1.
P30. Draw a linear chain of dipoles and check that the radiation at each
molecule depends on how the surrounding dipoles are oriented.Does the
strength of the dipole coupling differ in the two cases mentioned above?
P31. Derive the extinction cross section for a polarizability of a molecule including vibrations. Plot it for different values of electron-vibration
coupling strength and temperature.
P32. For a linear polymer of chain length l and N units show that the
end-to-end distance is Nl^2 it all units have random orientations
and the same result times (1+cos a/1-cos a) if consequtive units
have an average angle a with respect to each other.
P33. Discuss the difference between steady state and equilibrium
P34.Draw a linear chain of dipoles and check that the radiation at each
molecule depends on how the surrounding dipoles are oriented. Does the
strength of the dipole coupling differ in the two cases mentioned above?
