A fluorescently tagged chromosomal locus is tracked inside a bacterium and its…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Physical foundations of the living cell

Forces, Machines and Signals, Ch Diffusion: friction, capture and the crowded cytoplasm

Problem

A fluorescently tagged chromosomal locus is tracked inside a bacterium and its one-axis mean square displacement ⟨x2(t)⟩=2Kαtα\langle x^2(t)\rangle = 2K_\alpha t^\alpha is measured at three lag times. Data: t=0.50 st=0.50\,\mathrm{s}, ⟨x2⟩=0.020 μm2\langle x^2\rangle=0.020\,\mathrm{\mu m^2}; t=5.0 st=5.0\,\mathrm{s}, ⟨x2⟩=0.063 μm2\langle x^2\rangle=0.063\,\mathrm{\mu m^2}; t=50 st=50\,\mathrm{s}, ⟨x2⟩=0.20 μm2\langle x^2\rangle=0.20\,\mathrm{\mu m^2}. ((a)) Use the two most widely separated points to find the anomalous exponent α\alpha from the log-log slope, and check that the middle point is consistent with the same α\alpha. ((b)) Compute the apparent diffusion coefficient Dapp(t)=⟨x2⟩/2tD_{\rm app}(t) = \langle x^2\rangle/2t at t=0.50 st=0.50\,\mathrm{s} and at t=50 st=50\,\mathrm{s}, and show that their ratio matches (t2/t1)α−1(t_2/t_1)^{\alpha-1}. ((c)) State, with a reason, whether this locus is diffusing normally, subdiffusively or superdiffusively, and suggest one physical cause consistent with a crowded, tethered chromosome.

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