Mechanochemical simulations of actomyosin networks are
traditionally
based on one-dimensional models of actin filaments having zero width.
Here, and in the follow up paper (arXiv, DOI 10.48550/arXiv.2203.01284),
approaches are presented for more efficient modeling that incorporates
stretching, shearing, and twisting of actin filaments. Our modeling
of a semiflexible filament with a small but finite width is based
on the Cosserat theory of elastic rods, which allows for six degrees
of freedom at every point on the filament’s backbone. In the
variational models presented in this paper, a small and discrete set
of parameters is used to describe a smooth filament shape having all
degrees of freedom allowed in the Cosserat theory. Two main approaches
are introduced: one where polynomial spline functions describe the
filament’s configuration, and one in which geodesic curves
in the space of the configurational degrees of freedom are used. We
find that in the latter representation the strain energy function
can be calculated without resorting to a small-angle expansion, so
it can describe arbitrarily large filament deformations without systematic
error. These approaches are validated by a dynamical model of a Cosserat
filament, which can be further extended by using multiresolution methods
to allow more detailed monomer-based resolution in certain parts of
the actin filament, as introduced in the follow up paper. The presented
framework is illustrated by showing how torsional compliance in a
finite-width filament can induce broken chiral symmetry in the structure
of a cross-linked bundle.