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Microtubules confined to the two-dimensional cortex of elongating plant cells must form a parallel yet dispersed array transverse to the elongation axis for proper cell wall expansion. Collisions between microtubules, which migrate via hybrid treadmilling, can result in plus-end entrainment ("zippering") or catastrophe. Here, we present (1) a cell-scale computational model of cortical microtubule organization and (2) a molecular-scale model for microtubule-cortex anchoring and collision-based interactions between microtubules. The first model treats interactions phenomenologically while the second addresses interactions by considering energetic competition between crosslinker binding, microtubule bending and microtubule polymerization. From the cell-scale model, we find that plus-end entrainment leads to self-organization of microtubules into parallel arrays, while collision-induced catastrophe does not. Catastrophe-inducing boundaries can tune the dominant orientation. Changes in dynamic-instability parameters, such as in mor1-1 mutants, can impede self-organization, in agreement with experiment. Increased entrainment, as seen in clasp-1 mutants, conserves self-organization, but delays its onset. The corresponding CLASP protein can modulate the ability of boundaries to tune the dominant direction. The molecular-scale model predicts a higher probability of entrainment at lower collision angles and at longer unanchored lengths of plus-ends. Our models lead to several testable predictions, including the effects of reduced microtubule severing in katanin mutants and variable membrane-anchor densities in different plants, including Arabidopsis cells and Tobacco cells.