Effective behavior of cooperative and nonidentical molecular motors
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RESEARCH
Effective behavior of cooperative and nonidentical molecular motors Joseph J. Klobusicky1* , JohnFricks2 and Peter R. Kramer3 * Correspondence:
[email protected] Full list of author information is available at the end of the article Dedicated to Andy Majda for his 70th birthday, with gratitude for his lasting inspiration starting from my undergraduate and graduate days on the creative deployment of mathematical modeling and the beautiful application of analysis techniques as a lens for exploring and understanding the dynamics of physical systems - PRK
Abstract Analytical formulas for effective drift, diffusivity, run times, and run lengths are derived for an intracellular transport system consisting of a cargo attached to two cooperative but not identical molecular motors (for example, kinesin-1 and kinesin-2) which can each attach and detach from a microtubule. The dynamics of the motor and cargo in each phase are governed by stochastic differential equations, and the switching rates depend on the spatial configuration of the motor and cargo. This system is analyzed in a limit where the detached motors have faster dynamics than the cargo, which in turn has faster dynamics than the attached motors. The attachment and detachment rates are also taken to be slow relative to the spatial dynamics. Through an application of iterated stochastic averaging to this system, and the use of renewal-reward theory to stitch together the progress within each switching phase, we obtain explicit analytical expressions for the effective drift, diffusivity, and processivity of the motor-cargo system. Our approach accounts in particular for jumps in motor-cargo position that occur during attachment and detachment events, as the cargo tracking variable makes a rapid adjustment due to the averaged fast scales. The asymptotic formulas are in generally good agreement with direct stochastic simulations of the detailed model based on experimental parameters for various pairings of kinesin-1 and kinesin-2 under assisting, hindering, or no load. Keywords: Molecular motors, Stochastic averaging, Switched diffusion, Renewal-reward theory Mathematics Subject Classification: 74Q15, 92C40, 65C30
1 Introduction A biological cell during its interphase requires sufficiently fast transport of organelles and other compounds for its survival [1]. Transport through diffusion alone is often far too slow. To illustrate, a compound moving through pure diffusion in some neurons might take years to travel over the cell’s length [2]. For eukaryotic organisms, intracellular trafficking of vesicles is instead governed by directed transport along a network of thin filaments, such as microtubules or actin. A vesicle and the molecular compound it encloses, collectively referred to as a cargo, travel along the filaments by attaching to one or several molecular motors. As an important example on which we will focus, we can consider molecular motors called kinesins, which consist of two heads which attach to a
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