Fixed-angle polygonal chains in 3D serve as an interesting
model of protein backbones.
Here we consider such chains produced inside a “machine”
modeled crudely as a cone, and examine the constraints this
model places on the producible chains.
We call this notion α-producible, and prove as
our main result that
a chain is α-producible if and only if it is flattenable,
that is, it can be reconfigured without self-intersection to
lie flat in a plane.
This result establishes that two seemingly disparate classes of chains are
in fact identical.
Along the way, we discover that all α-producible configurations
of a chain can be moved to a canonical configuration resembling a helix.
One consequence is an algorithm that
reconfigures between any two flat states of a nonacute chain
in O(n) “moves,” improving the
O(n2)-move algorithm in [ADD+02].
Finally, we prove that the α-producible chains are rare
in the following technical sense.
A random chain of n links is defined by drawing the
lengths and angles from any “regular” (e.g., uniform)
distribution on any subset of the possible values.
A random configuration of a chain embeds into R3 by
in addition drawing the dihedral angles from
any regular distribution.
If a class of chains has a locked configuration
(and we know of no nontrivial class that avoids locked configurations),
then the probability that a random configuration of a random chain is
α-producible approaches zero geometrically as
n → ∞.