Prolonging assembly through dissociation

Microtubules are semiflexible polymers that serve as structural components inside the eukaryotic cell and are involved in many cellular processes such as mitosis, cytokinesis, and vesicular transport. In order to perform these functions, microtubules continually rearrange through a process known as dynamic instability, in which they switch from a phase of slow elongation to rapid shortening (catastrophe), and from rapid shortening to growth (rescue). The basic self-assembly mechanism underlying this process, assembly mediated by nucleotide phosphate activity, is omnipresent in biological systems.  A recent paper, Prolonging assembly through dissociation: A self-assembly paradigm in microtubules ,  published in the May 3 issue of Physical Review E,  presents a new paradigm for such self-assembly in which increasing depolymerization rate can enhance assembly.  Such a scenario can occur only out of equilibrium. Brandeis Physics postdoc Sumedha, working with Chakraborty and Hagan, carried out theoretical analysis of a stochastic hydrolysis model to demonstrate the effect and predict features of growth fluctuations, which should be measurable in experiments that probe microtubule dynamics at the nanoscale.

Model for microtuble dynamics. All activity is assumed to occur at the right end of the microtubule (denoted as ">")

The essential features of the model that leads to the counterintuitive result of depolymerization helping assembly are (a) stochastic hydrolysis that allows GTP to transform into GDP  in any part of the microtubule, and (b) a much higher rate of GTP attachment if the end of the microtubule has a GTP-bound tubulin dimer, compared to a GDP-bound tubulin dimer.    Process (a) leads to islands of GTP-bound tubulins to be buried deep in the microtubule.   Depolymerization from the end reveals these islands and enhances assembly because of the biased attachment rate (b).  The simplicity of the model lent itself to analytical results for various aspects of the growth statistics in particular parameter regimes.   Simulations of the model supported these analytical results, and extended them to regimes where it was not possible to solve the model analytically.  The statistics of the growth fluctuations in this stochastic hydrolysis model are very different from “cap models” which do not have GTP remnants buried inside a growing microtubule.   Testing the predictions in experiments could, therefore, lead to a better understanding of the processes underlying dynamical instability in-vivo and in-vitro.   An interesting question to explore is whether the bias in the attachment rates is different under different conditions of microtubule growth.

A lattice of interacting chemical oscillators

At Brandeis, there is a long tradition of interesting experiments on the Belousov-Zhabostinsky reaction system, with the legendary Zhabotinsky himself having been a part of the fraternity. This reaction system shows interesting oscillatory and stable patterns (see videos on Youtube). In the Fraden lab, an oil emulsion of micron-sized water droplets containing the BZ reactions, was shown to show interesting synchronization properties and complex spatial patterns [Toiya et al, J. Phys. Chem. Lett. 1, 1241 (2010)]. A coupling between the droplets due to preferential diffusion of an inhibitory reactant (bromine) in the oil medium was seen to be responsible for these collective phenomena.

In a new paper titled “Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions” in Phys. Rev. E, Physics Ph. D student Michael Giver, postdoc Zahera Jabeen and Prof. Bulbul Chakraborty show that neighboring oscillators can be modeled as Kuramoto phase oscillators, coupled nonlinearly to its nearest neighbors. The form of the coupling chosen is repulsive, which favors out of phase synchronization. They show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. A linear chain of these repulsively coupled oscillators shows anti-phase synchronization, in which neighboring oscillators show a phase difference of π The phase difference between the neighboring oscillators when placed on a ring however depends on the number of oscillators. In such a case, the locally preferred phase difference of π is ruled out for an odd number of oscillators, as this may lead to frustration. When these oscillators are placed on a triangular lattice in two dimensions, the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2 π /3. Interestingly, domains with different helicities form in the lattice. In each domain, the phases of any three neighboring oscillators can vary continuously in either clockwise or an anti-clockwise direction. Hence, phase difference between the nearest neighbors are seen to be ±2π /3 in the two domains (See figure). A phase difference of π is seen at the interfaces of these domains. These domains can grow in time, resembling domain coarsening in other statistical studies. At large coupling strengths, the domains freeze in size due to frequency synchronization of all the oscillators. Hence, an interplay between frequency synchronization and phase synchronization was seen in this system. Ongoing studies in the BZ experimental setup at the Fraden Lab, find correlations with the above results. Hence, insights into a complex system like the BZ oscillators could be gained using the phase oscillator formalism.

The research was supported by the ACS Petroleum Research Fund and the Brandeis MRSEC. Michael Giver is a trainee in the Brandeis NSF-sponsored IGERT program Time, Space & Structure: Physics and Chemistry of BIological Systems

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