# Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors

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## Abstract

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction–advection–diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.

## Keywords

Quasi-steady-state Molecular motors Intracellular transport Nonlinear kinetics## Notes

### Acknowledgements

M. J. W. was supported by the NSERC Discovery Grant 81541. L. E. K. was supported by an NSERC Discovery Grant 41870. C. Z. was supported by the NSERC Discovery Grant to L. E. K. and T. S. was supported by a USRA position funded by an NSERC Discovery Grant to L. E. K.

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