Microtubule Organization in the Presence of Motor Proteins
In this thesis, we construct a nonlocal transport model that describes the evolution of microtubules (MTs) as they interact with motor proteins. MTs, whose organization is crucial for normal cellular development, have been found to organize into various patterns in vitro and in vivo through their interactions with motor proteins. In the first part of the thesis, we state results of a simplified version of the model, a model that describes the interaction of MTs with stationary distributions of
... y distributions of motors. In the second part of the thesis, we state results for the full model, a model that describes the interaction of MTs with moving distributions of motors. For both models, an advection-type term accounts for directed MT transport, and an integral term accounts for reorientation of MTs due to their interactions with cross-linking motor proteins. For our simplified model, directed movement corresponds to a combination of MT treadmilling and MT sliding (where motor proteins are present). In the full model, when motors are moving, directed movement corresponds to treadmilling alone. Simulations of each model show how MT patterns depend on boundary constraints, as well as different model parameters that represent motor speed, motor processivity, cross-linking capability (activity), and directionality. For stationary motors in large domains, and using model parameter values for motors that are consistent with experimental values, we find that patterns such as asters, bundles, and vortices are able to persist. In vivo, MTs take on aster patterns during interphase. Also, in neurons and polarized epithelial cells, MTs form bundles. Vortex patterns have not been observed in vivo, however are found in in vitro experiments. In constrained domains, we find that similar patterns form. However, we also find that when two opposing motors are present, anti-parallel bundles are able to form. Such patterns are similar to those found in the mitotic spindle during cell division. Our model quantitatively describes how motors are involved in MT patterning. To date, there are no other models that describe such patterning by explicitly incorporating motor properties (for stationary motors) into a model for MT evolution. For moving motors, we simulate our model using periodic boundary conditions, representing MT organizations in large domains. We do this to compare our simulation results with results that have been found in vitro. Also, we simulate our model using parameters consistent with fast and slow processive motors, fast non-processive motors, and slow weakly processive motors, similar to the types of motors used in experiments. Similar to experiments, we find that depending on motor type and density, various types of patterns, such as arrays of asters, arrays of vortices, and clusters of disorganized MTs exist. Consistent with previous theoretical models, we find that MT patters depend on motor density. In particular, for specific motor types, MTs form vortices at low motor density, asters at intermediate values of motor density, and bundles at high motor densities.