Biology 441, Spring 2014


How Symmetry Relates To Embryology

1) Embryology tries to understand how genes create and maintain anatomical shapes and patterns.

2) (Usually) Objects have the same symmetries (or sometimes more symmetry) as their causes. (This generalization is called Curie's Principle.)

3) For example, spherical shapes result from counter-balances of forces which have the same strengths in all directions.

4) Notice that similarity of shape between two objects does not require or imply that they are caused by the same forces; It implies that the symmetries of the forces are the same, or similar.

[Incidentally, in my opinion D'Arcy Thompson's big mistake was to think that similarity of shapes implies that the same actual forces cause both shapes: For example that similarity to the shapes of soap bubbles implies that cells are shaped by "surface tension", instead of by some force that contracts equally strongly in all directions. Active contractility of acto-myosin layers near cell surfaces can pull with the same strength in all directions; So can elastic stretching, and who knows how many other forces.

Similarity of shape implies similarity of causal forces, and does NOT imply that the actual forces are the same. This mistake holds back progress. Many examples of this mistake have occurred in embryology, and I will point them out as we go along.]

5) To "break" symmetry means to cause an object or structure to become less symmetrical.

What is broken is Curie's Principle. Several important phenomena are examples of breaking symmetry. One example is using the location of sperm entry to decide where the plane of right-left reflection symmetry will be located. Another example of symmetry breaking is using flagellar basal bodies to decide differences between right and left. A third example is use of "Reaction-Diffusion Systems" to break displacement symmetry, in the sense of creating stripes, spots, somites, or other spatially periodic patterns.

6) Hermann Weyl defined different kinds of symmetry in terms of what reflections, rotations, displacements, enlargements or distortions will cause an object or pattern to look the same as it did before. For example A, B, C, D and E all have one plane of reflection symmetry; H and I have two planes of reflection symmetry; A square has four planes of reflection symmetry, a triangle has three, a rectangle two, and starfish 5. N and two other letters each have one axis of two-fold rotation symmetry. Flagellar basal bodies have 9-fold rotational symmetry, if we ignore the central pair of microtubules.

Normally-proportioned half-sized embryos, quarter-sized, or double sized embryos of sea urchins can develop from separating the first two cells, or the first four cells, or from fusing two embryos. We could have called this two-fold, eight-fold etc. magnification symmetry (or "enlargement" or "shrinkage" symmetry), but the physicists got there first and call this sort of thing "dilation symmetry" (symmetry with respect to change in size).

7) One of the 2 or 3 deepest questions in developmental biology can be phrased as "How do early embryos accomplish dilation symmetry". It must be because their fundamental control mechanisms have dilation symmetry.

Lewis Wolpert's concept of "Positional Information" boils down to two plausible mistakes: One is that only diffusion gradients have enough dilation symmetry to accomplish size "regulation" (as in the example of separating early embryonic cells); The other mistake is that all cause and effect phenomena amount to information transfer.

We will learn that Dictyostelium fruiting bodies accomplish at least a hundred-fold dilation symmetry. Rounding up of liquid drops and soap bubbles also has dilation symmetry, and so does formation of meanders in rivers. There are many examples of dilation symmetry, besides diffusion gradients.

8) Fractals are another interesting category of dilation symmetry. Each part of a fractal pattern is a miniature scale model of the entire pattern. The branching pattern of the lungs are an example of a fractal. Embryologically, lungs form as two small out-pocketings of the wall of the endodermal tube. The epithelial walls of these out-pocketings fold actively (driven by contraction of cytoplasmic acto-myosin) so as to subdivide themselves into two pockets. Each of these pockets then repeats this process of subdivision into two, and this doubling process is repeated again and again, creating first 2, then 4, then 8 then 16, 32, 64 etc. out-pocketings. In humans, this process of repeated doubling continues until a million small epithelial pockets have been formed, each one of which then becomes a thin-walled alveolus. The lungs of all mammals have alveoli, but lung structures are significantly different in birds, reptiles and amphibians. For example, birds have "air capillaries", which are epithelial tubes that extend around in loops, so that air can flow continually in one direction, instead of in and out.

Other organs also form by sequential branching of epithelial tubes. Salivary glands, the pancreas, the liver, and ducts inside kidneys form this way. During this branching process, you could hardly distinguish between developing lungs, developing salivary glands, pancreas or kidneys. A time-lapse video of kidney development will be shown in class, but cannot be incorporated into the course web pages, because I didn't make it, and a publisher owns the copyright.

The Mandelbrot set is one of many mathematical fractal patterns that are comparatively easy to generate by repeated recalculations. The bright-colored blobby pattern at the beginning of our web pages about symmetry is a tiny part of the Mandelbrot set, which was generated on my Mac lap-top by repeated squaring of complex numbers (those numbers that include the square root of minus one). Repeated cubing will also generate fractal patterns, and so will repetition of other processes, whether mathematical or physical (as in the lungs). Tidal rivers and creeks develop fractal branching patterns. You can see pictures of them using Google Maps. Good examples are the rivers east of Elizabeth City and in Currituck County, North Carolina. The North River is a good example.

9) A good way to discover the cause of any self-forming pattern is to consider its symmetry, and remember Curie's Principle. If a pattern has a certain kind of symmetry, then expect that the causes of this pattern either have that same kind of symmetry, or that "Symmetry Breaking" occurs, in the sense of external triggers being used to reduce symmetry.

Examples of symmetry breaking include processes that from any other point of view wouldn't seem to have anything fundamental in common. Locations of sperm entry and directions of gravity are used to "break" the axial symmetry of newly fertilized one-cell stage embryos. Reaction-diffusion systems break displacement symmetry, in the sense of producing waves of alternating high and low chemical concentrations.

Elongation of cartilages is driven by osmotic swelling of cartilage. Osmotic pressure is a scalar variable, and therefore incapable of being stronger in one direction than another. So how can it make femurs (etc.) elongate directionally? Because collagen fibers wrapped around and within growing cartilages channel the osmotic pressure into certain directions, by preventing expansion in other directions. Tension, being a tensor, can and does vary with direction.

Thus the reduced symmetry of skeletal shapes (that is, its directionality) is produced by directional differences in tensions that resist swelling. The swelling force itself is incapable of directionality. Textbooks in anatomy and histology rarely have a clue on this subject, and assert that cartilage elongation and shape is produced by chondrocytes pushing harder in certain directions than others. In histological sections, you can see the chondrocytes lining up in the direction of greatest pressure. But its not them that's pushing harder in that direction. Neither the cells nor the cartilage matrix are capable of exerting more force in one direction than another. Both the cells and the extracellular matrix line up in the direction of least resistance (specifically, resistance by tension in collagen fibers).

Good luck trying to get biomechanics specialists to understand this, however. It's too simple for them.

10) Regarding whether reaction-diffusion systems are really how embryos generate spatial patterns (like zebra stripes and leopard spots) one of the best counter-arguments is based on the lack of dilation symmetry. Specifically, how do the sizes of the waves adjust in proportion to increased or decreased sizes of the tissue to be subdivided.

A (barely) conceivable method would be for the ratio of diffusion constants to change in proportion to the size of the material available. This would cause the wavelength to change. The larger the ratio of diffusion constants, the narrower and closer together the stripes; the smaller the ratio, the wider and farther apart the waves. Changing the wave lengths could also be accomplished by the right changes in the chemical reaction rates.

These methods feel contrived, however. It's hard for me to believe in them.

Color patterns in certain species of fish don't change their wave-length in proportion to size. As these fish grow larger, these fish keep adding new stripes. This observation has been widely interpreted to be evidence specifically in favor of reaction-diffusion systems. This tacitly assumes that reaction-diffusion systems are the only possible way to break displacement.

Many animals have alternating wide and narrow stripes. Not a few have more complex patterns of alternation, of which my favorite is wide / very narrow/ intermediate width / very narrow / very wide, and repeat. Look at the stripes on these dreadful Lion Fish that have unfortunately transplanted to the Atlantic and Caribbean, of which there are several photographs in the west hallway in Coker Hall.


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