Topologically Directed Simulations Reveal the Impact of Geometric Constraints on Knotted Proteins

Simulations of knotting and unknotting in polymers or
other filaments
rely on random processes to facilitate the topological changes. Here,
we introduce a method of topological steering to
determine the optimal pathway by which a filament may knot or unknot
while subject to a given set of physics. The method involves measuring
the knotoid spectrum of a space curve projected onto many surfaces
and computing the mean unraveling number of those projections. Several
perturbations of a curve can be generated stochastically, e.g., using
the Langevin equation or crankshaft moves, and a gradient can be followed
that maximizes or minimizes the topological complexity. We apply this
method to a polymer model based on a growing self-avoiding tangent-sphere
chain, which can be made to model proteins by imposing a constraint
that the bending and twisting angles between successive spheres must
maintain the distribution found in naturally occurring protein structures.
We show that without these protein-like geometric constraints, topologically
optimized polymers typically form alternating torus knots and composites
thereof, similar to the stochastic knots predicted for long DNA. However,
when the geometric constraints are imposed on the system, the frequency
of twist knots increases, similar to the observed abundance of twist
knots in protein structures.