  The ability of biomolecules to exert forces on their surroundings or resist
compression from the environment is essential in a variety of biologically
relevant contexts. As has been understood for centuries, slender rods can only
be compressed so far until they buckle, adopting an intrinsically bent state
that may be unable to bear a compressive load. In the low-temperature limit and
for a constant compressive force, Euler buckling theory predicts a sudden
transition from a compressed to a bent state in these slender rods. In this
paper, we use a mean-field theory to show that if a semiflexible chain is
compressed at a finite temperature with a fixed end-to-end distance (permitting
fluctuations in the compressive forces), it exhibits a continuous phase
transition to a buckled state at a critical level of compression, and we
determine a quantitatively accurate prediction of the transverse position
distribution function of the midpoint of the chain that indicates the
transition. We find the mean compressive forces are non-monotonic as the
extension of the filament varies, consistent with the observation that strongly
buckled filaments are less able to bear an external load. We also find that for
the fixed extension (isometric) ensemble that the buckling transition does not
coincide with the local minimum of the mean force (in contrast to Euler
buckling). We also show the theory is highly sensitive to fluctuations in
length, and that the buckling transition can still be accurately recovered by
accounting for those fluctuations. These predictions may be useful in
understanding the behavior of filamentous biomolecules compressed by
fluctuating forces, relevant in a variety of biological contexts.
