How to think about spontaneous arrangement of atoms to form shapes that are consistent and predictable.

Within any piece of matter, all forces are exactly counterbalanced against each other - except while this piece of matter is spontaneously changing from one geometry to another.

This is true for individual protein molecules. It is also true for cells and tissues, in the anatomy of the body, and also after randomly-mixed bunches of cells that have sorted out, with one cell type on the surface, and another cell type in the interior.

Therefore, in order to explain shape changes, you need to find out which specific forces have become imbalanced.

Sometimes atoms or cells "home in" on some consistent geometric arrangement. Examples include the

  • folding of proteins to their functional shape,
  • sorting out of dissociated cells,
  • and embryological events like the shaping of cartilages, bones, muscles
  • Tensegrity

Please notice that all the topics we have been considering share this same abstract property.

For convergence toward a stable geometry to occur, the following three things need to be true.

    1) Imbalance of at least some of the forces acting on or within the collection of matter;
    2) The direction of the imbalances must be toward some specific shape and arrangement of parts.
    3) All sufficiently "nearby" shapes (not too different), all imbalances of forces converge toward the one stable shape.
Now you will know what to do if you ever acquire supernatural powers, and want to create certain shapes!

Some of these new-fangled magic wands let you simply create shapes, without a stability mechanism.

But the good, old-fashioned magic wands work like finite-element computer simulations, and require that you specify all the different forces acting within the piece of material, and also that you specify the rules by which the strengths of these forces vary as functions of current shape. If you control the "force-rules", then you can make anything become any shape, and make it resist distortion.

A good question to ask now is "what does is mean to say 'sufficiently nearby shapes'"?

Sometimes it means, not stretched too far. Other time it means not bent at too sharp an angle.

Visualize stability as like a valley, or a meteor crater might be a better analogy, because our valley doesn't have any downhill path by which to escape. The only way out is by going up and over or between some of the surrounding hills or mountains. Mathematicians invented the phrase "attractor basin" to use in relation to this way of visualizing stability. If there are n different forces, then the valleys are in n+1 dimensional space. Surprising amounts of advanced mathematics are based on extrapolating analogies like attractor basins from three-dimensional space to unlimited higher dimensions. They have proved to themselves that numbers of dimensions don't matter; and I am the last person to argue against them about it.

If you want to create a structure, you need to get it into the correct attractor basin. Then its own forces will pull it to the bottom of the basin, like a rock rolling down hill. Such basins are imaginary, of course, but they are a good way to make intuitive sense of complicated situations.

Next, imagine being able to control the shape of an attractor basin, and how broad an area is enclosed by one of them. An alternative way to cause a desired shape to form is to change the force-rules so as to broaden the attractor basin far enough so the current state of affairs is within it. I think that's how embryos develop. Visualize a stretched sheet of plastic, with a ball rolling around on top of it. One way to make the ball go to a desired location is to reach over and push it there with your fingers. The other method is to pull downward on the part of the plastic where you want the ball to go. Pulling down creates an attractor basin.

The concepts of thermodynamic free energy (now called "Gibb's Energy"), and also potential energy, are closely related to the attractor basin concept. People are justifiably impressed by the ability to predict combinations of chemical concentrations, pressures and other variables that will spontaneously come into existence. For any simple combination of physical forces, a graph of potential energy versus variations in shape IS a graph of potential energy. Furthermore, if you are dealing with any combination of chemical and physical properties, the graph of their net Gibb's energy is a series of attractor basin. To repeat, potential energy and Gibb's energy are two specific examples of the attractor basin concept. It's all a matter of stable counterbalances of forces. Minimization of free energy is more of a mathematical system than it is a discovery about nature.

Unfortunately, most people are awed by thermodynamics, as if it were some kind of magic. Malcolm Steinberg spent fifty years intimidating critics of his theory by accusing them of "not understanding basic thermodynamics." He genuinely believed that was true. Indeed 99% of his critics understood the subject as poorly as he did, himself. I failed to persuade him that I understood the subject and that he did not.

Besides the problem of how embryos create the shapes and positions of anatomical structures, there is a further mystery: They are often able to reach the same end result by means of two or more series of intermediates. A classic example is that the vertebrate neural tube (which becomes the central nervous system) usually forms by rolling up of a sheet of cells, the edges of which fuse. Some kinds of animals, including teleost fish, form their neural tubes by hollowing out a solid rod of cells. This extreme geometric difference in intermediate stages suggests that the causal mechanisms must be different. How could the same mechanism cause rolling-up and also cause hollowing-out? Scott Gilbert's excellent textbook handles the paradox by calling the rolling-up phenomenon "type one neurulation", and calling the hollowing-out phenomenon "type two neurulation". This illustrates that if you can't understand something, you can always invent a name for it. Many scientific discoveries amount to no more than inventing a name.

On the other hand, it really is a discovery to prove that two phenomena that had previously been called by different names are really just different versions of the same thing.

Among the discoveries made by Townes and Holtfreter was that hollow neural tubes will also form from dissociated and random mixtures of neural cells and surrounding tissues. Scott Gilbert hasn't yet decided to call that "type three neurulation"; but give him time.

If you think about causation of shape in terms of an attractor basin, that subtracts from the mystery that neural cells are able to reach the tubular geometry by at least three different sequences of intermediates. Unfortunately, Steinberg's almost-first-rate education at Amherst had only included one example of attractor basins.




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