Note on the dynamic instability of microtubules

Note on the dynamic instability of microtubules Abstract If the dynamic instability of microtubules follows a gamma distribution then one can associate to it a Cantor set. Microtubules (MTs), the main protein polymeric filaments of the cell cy-toskeleton, are a well-defined biological system where methods in condensed matter, statistical mechanics and the theory of complex systems have been applied. MT dynamics plays an important role in many fundamental cellular processes, such as cell division and cell motility. A peculiar intrinsic dynamics of MTs consisting in extended growth (rescues) and shrinkage (catastrophes) phases of variable duration with rapid switching between these two phases has been known since about a decade [1] and is the subject of many investigations. This spontaneous assembly-disassembly process has been called dynamic instability (DI) [1]. It is not clear why MTs should have such a behavior and what is its true origin. There are currently several kinetic models trying to explain DI, both from the side of biologists [2] (based on specific features of the MT hydrolysis involving the GTP and GDP units) and from the side of physicists [3] (based on purely one-step stochastic processes of constant rate). Recently, Odde, Cassimeris, and Buettner [4] have found a high probability of fitting the published MT length life histories to a gamma distribution (f (t) = θ r t r−1 e −θt /Γ(r), where r and θ are the shape parameter and the frequency parameter, respectively, of the distribution) by the Kolmogorov-Smirnov test, for both growth and shrinkage. Their result is not firmly established because the conclusion is based on a small number of phase times (14 growth times and 12 shrinkage times). The exponential distribution still cannot be ruled out. As they remark, other nonnegative probability distributions (Weibull, lognormal, or beta ones) may also be appropriate. However, the property of memory makes all these distributions essentially different from the exponential one. Citing the textbook of Olkin, Gleser, and Derman [5], Odde and collaborators argue that a gamma distribution implies a series of first-order steps from the growth phase to the shrinkage one, with the number of steps given by the shape parameter r of the gamma distribution. For the plus end data, they have found r = 3, and thus a series of three first-order transitions each of constant rate θ (≈ 1.7 min −1). This would mean two intermediate metastable states. In their words, " each …