This set of math techniques that Kauffman, Holland and others
devised is still without a proper name, but I'll call it here "net
math." Some of the techniques are known informally as parallel
distributed processing, Boolean nets, neural nets, spin glasses,
cellular automata, classifier systems, genetic algorithms, and swarm
computation. Each flavor of net math incorporates the lateral causality
of thousands of simultaneous interacting functions. And each type of net
math attempts to coordinate massively concurrent events  the kind of
nonlinear happenings ubiquitous in the real world of living beings. Net
math is in contradistinction to Newtonian math, a classical math so well
suited to most physics problems that it had been seen as the only kind
of math a careful scientist needed. Net math is almost impossible to use
practically without computers.
The wide variety of swarm systems and net maths got Kauffman to
wondering if this kind of weird swarm logic  and the inevitable order he
was sure it birthed  were more universal than special. For instance,
physicists working with magnetic material confronted a vexing problem.
Ordinary ferromagnets  the kind clinging to refrigerator doors and
pivoting in compasses  have particles that orient themselves with
cultlike uniformity in the same direction, providing a strong magnetic
field. Mildly magnetic "spin glasses," on the other hand, have
wishywashy particles that will magnetically "spin" in a direction that
depends in part on which direction their neighbors spin. Their "choice"
places more clout on the influence of nearby ones, but pays some
attention to distant particles. Tracing the looping interdependent
fields of this web produces the familiar tangle of circuits in
Kauffman's home image. Spin glasses used a variety of net math to model
the material's nonlinear behavior that was later found to work in other
swarm models. Kauffman was certain genetic circuitry was similar in its
architecture.
Unlike classical mathematics, net math exhibits nonintuitive traits. In
general, small variations in input in an interacting swarm can produce
huge variations in output. Effects are disproportional to causes  the
butterfly effect.
Even the simplest equations in which intermediate results flow back into
them can produce such varied and unexpected turns that little can be
deduced about the equations' character merely by studying them. The
convoluted connections between parts are so hopelessly tangled, and the
calculus describing them so awkward, that the only way to even guess
what they might produce is to run the equations out, or in the parlance
of computers, to "execute" the equations. The seed of a flower is
similarly compressed. So tangled are the chemical pathways stored in it,
that inspection of a unknown seed  no matter how intelligent  cannot
predict the final form of the unpacked plant. The quickest route to
describing a seed's output is therefore to sprout it.
Equations are sprouted on computers. Kauffman devised a mathematical
model of a genetic system that could sprout on a modest computer. Each
of the 10,000 genes in his simulated DNA is a teenyweeny bit of code
that can turn other genes either on or off. What the genes produced and
how they were connected were assigned at random.
This was Kauffman's point: that the very topology of such complicated
networks would produce order  spontaneous order!  no matter what the tasks
of the genes.
While he worked on his simulated gene, Kauffman realized that he was
constructing a generic model for any kind of swarm system. His program
could model any bunch of agents that interact in a massive simultaneous
field. They could be cells, genes, business firms, black boxes, or
simple rules  anything that registers input and generates output
interpreted as input by a neighbor.
He took this swarm of actors and randomly hooked them up into an
interacting network. Once they were connected he let them bounce off one
another and recorded their behavior. He imagined each node in the
network as a switch able to turn certain neighboring nodes off or on.
The state of the neighbor nodes looped back to regulate the initial
node. Eventually this gyrating mess of heturnsherwhoturnshimon
settled down into a stable and measurable state. Kauffman again randomly
rearranged the entire net's connections and let the nodes interact until
they all settled down. He did that many times, until he had "explored"
the space of possible random connections. This told him what the generic
behavior of a net was, independent of its contents. An oversimplified
analogous experiment would be to take ten thousand corporations and
randomly link up the employees in each by telephone networks, and then
measure the average effects of these networks, independent of what
people said over them.
By running these generic interacting networks tens of thousands of
times, Kauffman learned enough about them to paint a rough portrait of
how such swarm systems behaved under specific circumstances. In
particular, he wanted to know what kind of behavior a generic genome
would create. He programmed thousands of randomly assembled genetic
systems and then ran these ensembles on a computer  genes turning off and
on and influencing each other. He found they fell into "basins" of a few
types of behaviors.
At a slow speed water trickles out of a garden hose in one uneven but
consistent pattern. Turn up the tap, and it abruptly sprays out in a
chaotic (but describable) torrent. Turn it up full blast, and it gushes
out in a third way like a river. Carefully screw the tap to the precise
line between one speed and a slower one, and the pattern refuses to stay
on the edge but reverts to one state or the other, as if it were
attracted to a side, any side. Just as a drop of rain falling on the
ridge of a continental divide must eventually find its way down to
either the Pacific Basin or the Atlantic Basin, roll down one side or
the other it must.
Sooner or later the dynamics of the system would find its way to at
least one "basin" that entrapped the shifting motions into a persistent
pattern. In Kauffman's view a randomly assembled system would find its
way to a stock pattern (a basin); thus, out of chaos, order for free
emerges.
As he ran uncounted genetic simulations, Kauffman discovered a rough
ratio (the square root) between the number of genes and the number of
basins the genes in the system settled into. This proportion was the
same as the number of genes in biological cells and the number of cell
types (liver cells, blood cells, brain cells) those genes created, a
ratio that is roughly constant in all living things.
Kauffman claims this universal ratio across many species suggests that
the number of cell types in nature may derive from cellular architecture
itself. The number of types of cells in your body, then, may have little
to do with natural selection and more to do with the mathematics of
complex gene interactions. How many other biological forms, Kauffman
gleefully wonders, might also owe little to selection?
He had a hunch about a way to ask the question experimentally. But first
he needed a method to cook up random ensembles of life. He decided to
simulate the origin of life by generating all possible pools of prelife
parts  at least in simulation. He would let the virtual pool of parts
interact randomly. If he could then show that out of this soup order
inevitably emerged, he would have a case. The trick would be to allow
molecules to converge into a lap game.
continue...
