  Mammalian tissues contain networks and ordered arrays of collagen fibrils
originating from the periodic self-assembly of helical 300 nm long
tropocollagen complexes. The fibril radius is typically between 25 to 250 nm,
and tropocollagen at the surface appears to exhibit a characteristic
twist-angle with respect to the fibril axis. Similar fibril radii and
twist-angles at the surface are observed in vitro, suggesting that these
features are controlled by a similar self-assembly process. In this work, we
propose a physical mechanism of equilibrium radius control for collagen fibrils
based on a radially varying double-twist alignment of tropocollagen within a
collagen fibril. The free-energy of alignment is similar to that of liquid
crystalline blue phases, and we employ an analytic Euler-Lagrange and numerical
free energy minimization to determine the twist-angle between the molecular
axis and the fibril axis along the radial direction. Competition between the
different elastic energy components, together with a surface energy, determines
the equilibrium radius and twist-angle at the fibril surface. A simplified
model with a twist-angle that is linear with radius is a reasonable
approximation in some parameter regimes, and explains a power-law dependence of
radius and twist-angle at the surface as parameters are varied. Fibril radius
and twist-angle at the surface corresponding to an equilibrium free-energy
minimum are consistent with existing experimental measurements of collagen
fibrils. Remarkably, in the experimental regime, all of our model parameters
are important for controlling equilibrium structural parameters of collagen
fibrils.
