| Abstract | The operation of biological molecular motors such as myosin and kinesin have long been debated. Within the last decade, experiments tracking single molecules have allowed for improved understanding of such motors' dynamics, and coarse details of these dynamics are now known. For example, myosin-V walks in a hand-over-hand, as opposed to inchworm, manner [1]. More precise details of the motor's dynamics, however, are still a matter of speculation. Recently, single-molecule time series for the motion of myosin-V of extraordinary precision were published [2]. Meanwhile, Craig et al. [3] have been simulating a simplified mechanical model of myosin V, based on current biophysical data, and validating their model by comparing coarse performance characteristics with experimental results such as those of Cappello et al. [2]. Kramers-Moyal analysis, introduced over the last decade by Friedrich et al. [4] allows for more detailed characterisation of molecular motor walking time series, which are typically stochastic, and thereby more inferences to be made about the motor itself, than is usually performed. It reconstructs directly from data the Kramers-Moyal coefficients, which furnishes the drift and diffusion coefficients of a Fokker-Planck (or, equivalently, Langevin) equation, an equation which we might expect a molecular motor to approximately follow since they usually operate in an overdamped, Brownian environment. The method permits these Kramers-Moyal coefficients to be position-dependent, so unlike some other methods of time series analysis can recover arbitrary nonlinearities; further, the nonlinearity which the method recovers can be easily visualised. Even if the time series is not (first-order) Markovian, and a model like the Langevin equation not appropriate, the Kramers-Moyal coefficients can still yield useful information about the system. This Kramers-Moyal analysis, which likely has wide applicability as a method of stochastic time series analysis, will be reviewed, along with our own work on geometrical and finite-time effects, which can alter the apparent shape of the Kramers-Moyal coefficients. In the context of molecular motors, we aim to use the Kramers-Moyal method to characterise and compare Craig et al.'s simulated and Cappello et al.'s experimental time series, and also to compare against our own semi-analytical calculations. This will provide further evidence with which to evaluate Craig et al.'s model and the hypotheses it embodies, for example that the motor moves through a diffusional search state, the dynamics of which should be distinguishable in the Kramers-Moyal coefficients. Progress towards this end, and preliminary conclusions, will be presented. 1. A. Yildiz et al. (2003), Science 300 2061-65 2. G. Cappello et al. (2007), PNAS 104 15328-33 3. E.M. Craig & H. Linke (2009), in preparation 4. R. Friedrich et al. (2000), Phys. Lett. A 271 217-22; R. Friedrich & J. Peinke (1997), Phys. Rev. Lett. 68 863-66 |
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