Simulated cytoskeletal collapse via tau degradation

Abstract

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 Alzheimers 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 bundles 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.

Figure 1. Two Dimensional Microtubule Bundle (MTB) Model.

Microtubules are treated as rigid cylinders of diameter 25 nm (blue disks) with center-to-center distance of 45 nm (a). Taus are treated as springs, with 50 per micron length of a microtubule pair. To model tau depletion, taus are removed at random and the system is re-equilibrated with the steepest descents method described in the text (b).

Figure 2. Depletion mediated interaction between microtubules.

For intercalants of radius r (presumed here to be hyperphosphorylated taus) an annulus between is depleted of intercalants. The loss of excluded volume as the microtubules approach allows increased translational entropy outside the bundle for the intercalants.

Figure 3. Simulation of equilibrated force vs. distance for taxol stablized microtubule bundles with PEO intercalant per Ref. .

The red circles are our data for osmotic pressures given in Ref. , the blue triangles are the data in that reference.

Figure 4. Transverse rigidity transition in the absence of the depletion force.

As described in the text, we compute the 2nd order elastic constant for radial compression as a function of the occupied spring fraction and (b) scale to the limit. In contrast to Ref. , all values yield the same results when scaled, because their lattice boundary is fixed while ours relax to equilibrium. In all cases, the transverse rigidity percolation threshold is , in good agreement with the case of Ref.

Figure 5. First order transverse collapse in the presence of the depletion force.

This figure traces the transverse collapse as the bound tau density is reduced from 1, for the dimensionless parameter After a gradual reduction of for , the depletion force overwhelms the entropic springs at  = 0.73.

Figure 6. Normalized microtubule bundle radius vs. tau occupancy .

In all these curves we show the bundle radius above with the leftmost end at . a) The MTBs undergo a first order collapse with reduced , with the collapse onset decreasing with for fixed spring constant  = 0.05 pN/nm. The lavender curve is computed with the same value at collapse as the blue curve but for an osmotic pressure proportional to . For (not shown) the radius is that of hexagaonally close packed microtubules. b) For varying initial bundle radius, measured by the number of hexagonal shells retained about the central MT, the normalized radius displays the collapse at the same location for a fixed value of  = 3.125, indicating the collapse is not an artifact of finite size.

Figure 7. Normalized microtubule bundle radius vs. tau occupancy for fixed .

In all these curves we show the bundle radius above with the leftmost end at . Here we carry out the simulations multiplying independently by the percentages shown in the legend to achieve the same value. Clearly this gives the same for collapse onset.

Figure 8. Potential Energy of Mean Field Theory.

Energy per unit length vs. scaled mean microtubule separation for nm and  = 3.125. As is reduced, the potential at which is the separation at microtubule contact, reaches a lower value than the minimum which evolves from at .

Figure 9. Comparison of for Mean Field Theory with simulations.

We plot vs. for the MFT and the full simulations. Clearly, the MFT and full simulation trends are very similar.