  The intermittent transition between slow growth and rapid shrinkage in
polymeric assemblies is termed dynamic instability, a feature observed in a
variety of biochemically distinct assemblies including microtubules, actin and
their bacterial analogs. The existence of this labile phase of a polymer has
many functional consequences in cytoskeletal dynamics, and its repeated
appearance suggests that it is relatively easy to evolve. Here, we consider the
minimal ingredients for the existence of dynamic instability by considering a
single polymorphic filament that grows by binding to a substrate, undergoes a
conformation change, and may unbind as a consequence of the residual strains
induced by this change. We identify two parameters that control the phase space
of possibilities for the filament: a structural mechanical parameter that
characterizes the ratio of the bond strengths along the filament to those with
the substrate (or equivalently the ratio of longitudinal to lateral
interactions in an assembly), and a kinetic parameter that characterizes the
ratio of time scales for growth and conformation change. In the deterministic
limit, these parameters serve to demarcate a region of uninterrupted growth
from that of collapse. However, in the presence of disorder in either the
structural or the kinetic parameter the growth and collapse phases can coexist
where the filament can grow slowly, shrink rapidly, and transition between
these phases, thus exhibiting dynamic instability. We exhibit the window for
the existence of dynamic instability in a phase diagram that allows us to
quantify the evolvability of this labile phase.
