Kinesin-1 advances along a microtubule in steps

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Molecular machines, transport and bioenergetics · Molecular machines, transport and bioenergetics

Forces, Machines and Signals, Ch Measuring one molecule: traps, tweezers and single-molecule fluorescence

Problem

Kinesin-1 advances along a microtubule in 8.0 nm8.0\ \mathrm{nm} steps. To time individual steps reliably, an optical-trap assay records the bead's position with a detection bandwidth of B=2.0 kHzB = 2.0\ \mathrm{kHz}. The bead sits in water of viscosity η=1.0×10−3 Pa s\eta = 1.0\times10^{-3}\ \mathrm{Pa\,s} at T=298 KT = 298\ \mathrm{K} (kBT=4.11 pN nmk_BT = 4.11\ \mathrm{pN\,nm}), held by a trap of stiffness κ=0.10 pN nm−1\kappa = 0.10\ \mathrm{pN\,nm^{-1}}. Treat the bead as a sphere in an unbounded fluid (ζ=6πηa\zeta = 6\pi\eta a) and the trap as harmonic. The one-sided position spectrum in the trap is the Lorentzian Sx(f)=kBT/[π2ζ(fc2+f2)]S_x(f) = k_BT/[\pi^2\zeta(f_c^2+f^2)] with fc=κ/2πζf_c = \kappa/2\pi\zeta, and the mean-square position noise in a bandwidth 00 to BB is ⟨δx2⟩=∫0BSx(f) df=2kBTπκarctan⁡(B/fc)\langle\delta x^2\rangle = \int_0^B S_x(f)\,\mathrm{d}f = \dfrac{2k_BT}{\pi\kappa}\arctan(B/f_c), which reduces to the flat-spectrum result ⟨δx2⟩≈4kBTζB/κ2\langle\delta x^2\rangle \approx 4k_BT\zeta B/\kappa^2 only when B≪fcB \ll f_c. ((a)) For the bead ordinarily used in this assay, of diameter 1.0 μm1.0\ \mu\mathrm{m}, find fcf_c and decide whether BB counts as small compared with fcf_c. Using whichever expression for ⟨δx2⟩\langle\delta x^2\rangle is appropriate, compute the rms noise δx=⟨δx2⟩\delta x = \sqrt{\langle\delta x^2\rangle} at B=2.0 kHzB = 2.0\ \mathrm{kHz}, and state whether the 8.0 nm8.0\ \mathrm{nm} step can be told apart from this noise with confidence. ((b)) A colleague proposes replacing the bead with one of diameter 0.20 μm0.20\ \mu\mathrm{m}, keeping κ\kappa and BB the same. Recompute fcf_c, check whether B≪fcB \ll f_c now holds, find the new δx\delta x, and state whether this change resolves the difficulty found in part (a).

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