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We present a generic theory for the dynamics of a stiff filament undertension, in an active medium with orientational correlations, such as amicrotubule in contractile actin. In sharp contrast to the case of a passivemedium, we find the filament can stiffen, and possibly oscillate, \textit{or}buckle, depending on the contractile or tensile nature of the activity\textit{and} the filament-medium anchoring interaction. We present experimentson the behaviour of microtubules in the growth cone of a neuron, which provideevidence for these apparently opposing behaviours. We also demonstrate a strongviolation of the fluctuation-dissipation (FD) relation in the effectivedynamics of the filament, including a negative FD ratio. Our approach is alsoof relevance to the dynamics of axons, and our model equations bear aremarkable formal similarity to those in recent work [PNAS (2001) {\bf98}:14380-14385] on auditory hair cells. Detailed tests of our predictions canbe made using a single filament in actomyosin extracts or bacterialsuspensions.

Dynamics of a stiff biopolymer in an actively contractile background: buckling, stiffening and negative dissipation