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posted February 26, 8:30 pm
a) List some embryonic structures that are mechanically equivalent to water balloons. (At least 8)
(Please think of at least one more example that is not on our list of ten, on the web page.)
b) If you suspect that a newly-discovered embryonic structure is mechanically equivalent to a water balloon, what are some experiments that you could do to test (prove or disprove) your hypothesis?
c) Explain how cartilages are able to exert osmotic pressure without being surrounded by semi-permeable membranes.
d) Draw a graph of the relative amounts of curvature along any arbitrarily-drawn curve.
(If I draw a curve on the exam, you need to be able to sketch a graph of the relative amounts of curvature at all the different parts of this curved line.) For example, consider the traditional heart shape, used on playing cards.
e) The reverse problem: If I draw a graph of amounts of curvature along a line, you should be able to sketch the line,
based on knowing how much curvature it has at different points.
f) What two lines have constant amounts of curvature at all points? Sketch both.
g) What three dimensional curve has constant curvature in one direction and zero curvature in the direction perpendicular to the direction in which curvature is constant? Are there any structures in the body that actually have this combination of surface curvatures?
h) What shape has constant amounts of surface curvature in all directions and at all locations?
i) What geometric property is defined as being the rate of change of angular direction per unit distance along the surface of an object?
j) Is it possible for a surface to have positive curvature in one direction and negative curvature in directions perpendicular to those in which curvature is positive? Hint: What is meant by "saddle points"
k) Sketch a gastrulating sea urchin embryo and sketch graphs of the relative amounts of curvature at different areas of its surface.
l) Sketch a cleaving cell, and next to it sketch a pair of graphs of relative amounts of curvatures in perpendicular directions at different parts of the surface of this cleaving cell.
m) Using pairs of double-ended arrows (like those on the web page diagrams of cylinder, sphere, saddle and hens egg) map the distribution of curvatures at different parts of the surface of the brain and spinal cord of an early embryo.
(Why is average curvature less in the brain than in the spinal cord.
n) In order for a flexible, stretched sheet to have equal pressure on both sides, it either has to be saddle-shaped or it has to be _______. What shape?
o) Curvature times ________? equals change in pressure?
p) On all parts of any curved surface, the direction of maximum curvature has what geometric relation to the direction of minimum curvature? (What is the one exception to this rule?)
q) D'Arcy Thompson uses the following equation: Difference in Pressure equals Tension multiplied by the sum of one over the radius of curvature plus one over the radius of curvature in the perpendicular direction.
Please explain why this is a terrible mistake.
r) Hint: What if he tried to use this equation to explain the mechanism of cell cleavage?
s) Another hint: What does that equation have to do with soap films?
t) For flexible elastic sheets (whether or not they are actively contractile) what equation should D'Arcy Thompson have used, instead?
u) One divided by radius of curvature equals what?
v) What are at least two examples of scalar variables?
w) What are at least three examples of second order tensor variables?
x) What is one example of a fourth order tensor variable?
y) What is Hooke's law? Is it a law of nature? Or is it a simplifying assumption, that is often true for small amounts of stress?
z) What is the difference between stress and strain? Or are these two words for the same thing?
@) Visualize a blood vessel that branches into two, with tension in the right branch (for some reason) being twice as strong as tension in the left branch: then sketch the angles between these three blood vessels.
#) Picture a situation in which an artery is supplying blood to two organs. Visualize this artery branching a short distance before reaching the two organs. If equal volumes of blood are being supplied to both organs, then what is the optimal angle of branching? What is the organ on the right needs to receive twice the volume of blood per minute as the organ on the right, how will this affect the optimum angle at which the arteries should branch?
Explain your reasoning.
Suggest some alternate hypotheses about the mechanisms that control branching of arteries and veins?
(Hint: does evolution optimize the actual angles of branching? Or can the same optimization be achieved by controling the local rules that govern longitudinal tension in arteries and veins?)
Suggest some experiments by which one could test these different possibilities.
What is the effect of reduced tension on the geometric arrangements and shapes of developing anatomical structures?
Would you expect this phenomenon to apply to arteries, tendons, muscles, feathers? What are some possible explanations for this phenomenon?
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