Homework/2

1. Holy and Leibler (1994) estimate that the time wasted by a microtubule that nucleates in the wrong direction (away from a kinetochore) is, on average, $R/(r\delta)$ where $R$ is the radius of the cell, $r$ is the rate at which monomers both attach and detach at the MT tip and $\delta$ is the size of a monomer. If a MT starts at the cell centre with a single monomer and can only depolymerize when it reaches the cell boundary, derive the estimate above using the theory for a diffusing particle discussed in class. [Note: slightly edited from original version].
2. Using the formalism discussed in class (see slides), calculate the net force generated by a MT aster in which motors pull on all MTs along their entire length rather than just at the tip. Is the centre a stable steady state for the aster?
3. When an elastic filament is embedded in a surrounding elastic medium, the equation for the deflection of the filament is $$By^{(iv)} + F y'' + \alpha y = 0$$ where $\alpha$ is proportional to the stiffness of the medium. Microtubules embedded within the actin network of a cell are thought to behave this way (Brangwynne et al. 2006). Show that, for a filament pinned (but not clamped) at both ends, with a force $F$ less than $2\sqrt{\alpha B}$, the only solution is $y(x)=0$. With the given boundary conditions, under what condition will the filament buckle at $F=2\sqrt{\alpha B}$?