Microtubules are a key component of the eukaryotic cytoskeleton that are responsible for a number of vital cellular functions, including spindle assembly, intracellular transport, and structural integrity. Microtubule turnover, nucleation, and transport are all regulated by a myriad of intricate mechanisms in space and time. In thesis thesis, we explore biophysical aspects of this regulation.

We start by introducing the main features of microtubule biophyiscs relevant for spindle assembly. We then establish that purified chromosomes alone can form spindles in meiotic Xenopus egg extract, where the main source of microtubule generation is through branching microtubule nucleation. We find that the organization of the resultant branched networks is consistent with a theoretical model where the effectors for branching nucleation are released by chromosomes, forming a concentration gradient that spatially biases branching nucleation.

Certain microtubule processes may be regulated by biomolecular condensates that form through liquid-liquid phase separation and related phase transitions. We first introduce the biophysics of condensates, with an emphasise on their interfacial dynamics. We then discuss two instances where interfacial dynamics are dominant in governing the behavior of how condensates can regulate microtubule function. In the first, we show that condensed TPX2 wets the microtubule and forms droplets along it according to the Rayleigh-Plateau instability. These droplets serve as reaction crucibles for the nucleation of microtubule branches. In the second, we propose that any microtubule-associated protein capable of forming condensates and wetting the microtubule can bundle microtubules through adhesive capillary bridges.

In studying these biophysical problems, ordinary differential equations (ODEs) frequently arise. We conclude by detailing a method of generating exact solutions for certain classes of ODEs that are forced at resonance or have repeated roots. Our approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters.