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2015-04-21 Nonlinear Seminar

dr Przemysław Chełminiak
What connects a biological cell with a partial diffusion equation.

Tracking individual molecules in living biological cells has become possible today by a new imaging technique called single-molecule spectroscopy. Interestingly, a diffusion of such tiny nano-molecules as proteins, nucleic acids or semiflexible polymers within a dense cellular environment is slower than ordinary Brownian diffusion of the micro-particles suspended in water. Their spatial development inside the cell, characterized by the time or ensemble averaged mean squared displacements, does not grow linearly in time. Moreover, these two averaging procedures are not equivalent when the measurement time is long compared to the characteristic time scale of the diffusion process. Therefore, the erratic motion of the subcellular molecules can not be perceived as a typical Brownian motion, but it must be considered in terms of the anomalous diffusion (subdiffusion). From a formal point of view, the anomalous diffusion can be described by a widely applicable stochastic process known as the continuous-time random walk model. We apply this method to construct the so called fractional Fokker-Planck equation which is an extended version of the subdiffusion equation in an external force field and close to thermal equilibrium. Using the methods of fractional calculus we find a few special solutions of this fractional partial differential equation and show that the mean squared displacement of a subdiffusing particle scales with time according to the power-law.