Spatial pattern formation in microtubule post-translational modifications and the tight localization of motor-driven cargo

• Published in: Journal of mathematical biology Vol. 74; no. 5; pp. 1059 - 1080
• Main Authors: Iniguez, Abdon, Allard, Jun
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-04-2017; Springer Nature B.V
• Subjects: Animals; Applications of Mathematics; Axons - metabolism; Chick Embryo; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Microtubules - metabolism; Models, Biological; Protein Processing, Post-Translational - physiology; Time Factors; Tubulin - metabolism; Pattern formation; 97M60 Biology, chemistry, medicine; Microtubules; 35Q92 PDEs in connection with biology and other natural sciences; Dynein; First passage time; De-tyrosination; 92C37 Cell biology; Kinesin
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-016-1053-x

Microtubule (MT) “age” can be interpreted as nucleotide state, lattice defects, or post-translational modification (PTM) such as acetylation and detyrosination. In all three cases, these have been recently shown to have functionally-important effects on the dynamics of MT arrays, and can present spatial and temporal heterogeneity. While mathematical models for MT array densities are well-established, here we present equations describing MT age, defined as the mean time since the MT’s building blocks (tubulin) were polymerized from their soluble dimer state. We derive the age equations using a mean first-passage time calculation and two complementary approaches: The continuum limit of spatial discretization model, and an adjoint operator approach. These equations can recapitulate the observation that the oldest (most de-tyrosinated) tubulin in axons is near the middle of axons during neuronal development in chick embryos. Furthermore, PTMs influence motor kinetics up to approximately twofold for off-rates and velocities. Our simulations demonstrate that this relatively weak dependence of motor kinetics is sufficient to target motor cargo to a specific location along the array. This localization is tightly peaked in a way that magnifies the relatively small signal of PTM spatial heterogeneity. Thus, MT age can produce long-range spatial patterning without feedbacks or diffusing signals.