  We present a coarse-grained two dimensional mechanical model for the
microtubule-tau bundles in neuronal axons in which we remove taus, as can
happen in various neurodegenerative conditions such as Alzheimer's disease,
tauopathies, and chronic traumatic encephalopathy. Our simplified model
includes (i) taus modeled as entropic springs between microtubules, (ii)
removal of taus from the bundles due to phosphorylation, and (iii) a possible
depletion force between microtubules due to these dissociated phosphorylated
taus. We equilibrate upon tau removal using steepest descent relaxation. In the
absence of the depletion force, the transverse rigidity to radial compression
of the bundle falls to zero at about 60% tau occupancy, in agreement with
standard percolation theory results. However, with the attractive depletion
force, spring removal leads to a first order collapse of the bundles over a
wide range of tau occupancies for physiologically realizable conditions. While
our simplest calculations assume a constant concentration of microtubule
intercalants to mediate the depletion force, including a dependence that is
linear in the detached taus yields the same collapse. Applying percolation
theory to removal of taus at microtubule tips, which are likely to be the
protective sites against dynamic instability, we argue that the microtubule
instability can only obtain at low tau occupancy, from 0.06-0.30 depending upon
the tau coordination at the microtubule tips. Hence, the collapse we discover
is likely to be more robust over a wide range of tau occupancies than the
dynamic instability. We suggest in vitro tests of our predicted collapse.
