### Second set of review questions for the first hour exam

Some of the following are scalar variables, some are vectors, some are second order tensors, and one is a fourth order tensor. Which is which?

For what two geometric shapes is the curvature of the surface the same in every direction?
sphere, plane

If epithelial sheets of cells bend or fold themselves so as to make curvature zero in one direction or axis (e.g. the north-south diction or axis), but make it constant in the direction perpendicular to the direction in which curvature is zero, then what shape will these cells arrange themselves into?
cylinder

In the preceding question, what geometric property of this arrangement of cells will change if their curvature becomes larger?
diameter decreases

What geometric property of this arrangement of cells will change if their curvature become smaller?
diameter becomes larger

If this arrangement of cells contains a liquid under pressure, how will the tension in the sheet of cells change, if at all, if the curvature in the cell sheet becomes larger?
tension will decrease

If the tension becomes smaller, then what change will be produced in the enclosed liquid?
pressure will decrease

What simple equation relates the physical variables tension, curvature and pressure (in the sense of difference of pressure on one side of a surface as compared with the other)?
dP = CT + ct

How can you explain the bulging lobes of the brain as being caused by changes in tension in different parts of the neural tube?
weaker tension

Why can't you explain the bulges as being caused by differences in pressure in one part of the neural tube as compared with another part? (hint, it's not hard; don't be tricked by simplicity)

(hint) Are there any parts of the neural tube in which the tension is stronger in one direction than another?

How would its shape change if the tension in the stronger direction (along a tube enclosing a liquid under pressure) weakened so as to become the same as the tension in the stronger direction?

Would this change be the same for a blood vessel as for the neural tube? (hint, yes)

By what changes in directionality (meaning variations of a property with direction) of tension of the cells of the neural tube are the lobes of the brain created?

Could these inflated lobes be produced (and explained) by a change in the amount of tension (if it were a scalar)? Or could lobes be explained by changes in the directionality of tension? (hint: becoming the same in all directions)

If tension were the same in all directions (e.g. if tension were a scalar variable) then could blood vessels, cylindrical ducts, and neural tubes exist (without bulges)?

What are the key differences between scalar variables as compared with tensor variables?

What are at least two physical variables that are second order tensors? (permeability through soil or a matrix, is also a second order tensor, by the way. Drillers of oil and water wells realize this, even if most people don't.)

Elastic resistance to stretching is a fourth order tensor, true or false.
true

Explain whether and how a researcher will be misled if he/she tries to measure the elastic resistance to stretching of a cell surface, or a blastula's surface, or the surface of an artery or a capillary, using a suction pipette, or indenting it with an atomic force microscope. Discuss the confusion that will result.
Methods cannot distinguish directionality of tension or other tensor variables.

More generally, discuss confusions that result from biologists not realizing that some variables are tensors, and assuming that all variables are either scalars or vectors.

The cornea of the eyeball bulges out (in such a way as to help the lens focus light on the retina): This is equivalent to saying that it has what difference in curvature from the rest of the eyeball.
larger curvature

In order to form this outward bulge, what difference in tension is necessary (and sufficient)?
weaker tension

Why can't the bulge be produced by increasing the outward pressure at that point? (hint: because the fluid gel inflating the eyeball behaves equivalently to how liquids behave)

Astigmatism of the focusing power of the cornea implies what differences in the directionality of tension and curvature in the cornea?
The curvature is larger in some directions than others.

Conversely, what differences in geometry and tension produce astigmatism?
curvature, and also maybe tension, varying with direction

If curvature were a scalar variable, then would astigmatism be possible?
no

If tension were a scalar variable, then would astigmatism be possible?
no

Invent a new kind of cure for astigmatism, using a local drug treatment that weakens or strengthens cell contractility. (*Discuss the difficulties of producing directional changes in tension.)

If a person's cornea bulges outward with too great a curvature, what effect will that have on their vision? (hint, more curvature produces more focusing)

Consider whether this problem might be cured by weakening the tension in the cornea (which is made mostly of collagen fibers, and also some cells.)

Most anti-microtubule drugs (colchicine, vinblastine) cause at least temporary strengthening of cell traction forces that pull on collagen. Therefore, what effect should these drugs have on the shape of the cornea?

Lasik surgery works by changing the shape of people's corneas by cutting away parts of the cornea, so as to change the curvature of the cornea. It used to be done mechanically, with the equivalent of a lathe; now it is done with a laser beam.

Although the cornea becomes much less flexible after embryonic development (so that tension no longer has as much effect on curvature) please discuss how changes in tension in the cornea could possibly be used instead of lasik surgery.
Curvature of cornea could be changed as a direct result of removal of tissue, or as an indirect result of weaker tension.

Weakening the tension in the wall of a capillary or an artery would have what effect on its diameter?
It would balloon outward.

Increasing the diameter of a capillary or artery would have what effect on its tension in its wall?
increase

Increasing the pressure of the blood of a capillary or artery would have what effect on its tension in its wall?
increase

Severe mutations in the gene for type I collagen cause death of developing embryos by bursting of capillaries and arteries: explain whether or why this makes sense.
Yes, it makes sense

When part of an artery balloons into an abnormal bulge, how does that change the curvature and the tension of that part of the artery wall?
reduces curvature, which causes even more increase in tension and even more stretching

Please consider how this produces a positive feedback (i.e. between changes in curvature and changes in tension).

If tensions were a scalar variable, could blood vessels exist?
I don't think so.

In order to cause the rudiments of salivary glands, lungs, kidney tubules etc. to evaginate outward from the wall of the archenteron (= the endodermal tube), then what changes in tension & contractility need to occur in the epithelial wall of the archenteron?

In order for new capillaries to bulge and branch off from the wall of an existing blood vessel, discuss what changes in the amounts and directionality of tension need to occur.
weaker

What changes in tension will cause an artery or capillary to decrease its diameter and eventually pinch off?
increase in circumference

Budding in Hydra requires (involves; is caused by) what changes in the curvatures of its surface (both the amounts and the directionality of the curvatures of its surface)?

Budding of Hydra needs what changes of directionality and/or amount of contractile tension in the body wall? (assume there is some fluid pressure in the interior "gastrovascular cavity", on the average, but that at no time does this inner pressure change from one location to another).

Again, in Hydras, compare the curvature of the surface of the tentacles versus the curvature of the main body of the Hydra.

As tentacles form and elongate by rearrangements and reorientations of body wall cells, what changes in either the amounts or directionality, or both, need to occur? (assuming, as before, that there is some fluid pressure in the enclosed cavity)

Using two diffusible substances, "A" and "B", what rules could they obey that would generate a regular spatial pattern? Hint: Substance "A" causes an increase in its own concentration, and also in the concentration of B, while the effect of B is to reduce the concentration of both A and B; and B diffuses faster than A (or in any way produces its effect at longer range than A.

You have seen the computer produce these patterns, and I hope that you will be able to sketch those patterns.

How can you cause the peaks in concentration to be closer together? (Increase the ratio of the diffusion constants of B and A. The faster B diffuses, relative to A, the narrower the bands and gaps.)