Table 2.

Optimized values of adjustable parameters for colloidal bead diffusion on lipid tubes

 Colloidal beads on lipid tubes Value Displacement distribution (SI Appendix, Eq. S14-3) a1, a2 4.02 × 10−1, 1−a1 λ1, λ2 (Hz) 3.43, 1.04 × 10−2
• The experimental data are reported in ref. 36. We analyzed the displacement distribution data using our second model, DΓ(t)DkakΓk2(t) with Γk(t)Γj(0)=δjkexp(λkt) (SI Appendix, Eq. S14-1), for which the displacement distribution in the Fourier domain is obtained as p(k,t)(dxeikxp(x,t)) =j=12[4qje(qj1)λjt/{(qj+1)2(qj1)2e2qjλjt}]1/2 with qj=(1+4ajk2Dλj1)1/2, where ai is a parameter characterizing the relative contribution of Γk to diffusion coefficient fluctuation (SI Appendix, Eqs. S14-2a, S14-2b, and S14-3). The displacement distribution quantitatively explains the experimental data (Fig. 4B). This system exhibits Fickian diffusion in the experimental timescale. The diffusion constant is estimated to be D/σ241.5Hz with σ being the diameter of a colloidal bead. Using the optimized parameter values, we can calculate the NGP and the TCF of diffusion coefficient fluctuation, ϕD(t) using Eqs. 12 and SI Appendix, Eq. S14-2b (SI Appendix, Fig. S3). The relaxation time, 0dtϕD(t)[=i=1,2ai2λi1/2(a12+a22)], of the diffusion coefficient fluctuation is estimated to be 33.2 s. This value is comparable to the relaxation time, 3.48 ∼ 34.8 s, of the lipid-tube membrane fluctuation under the zero shear stress condition (93); i.e., ω1(q)=[z3q2(κq4+μq2)/3η]1, at the smallest wavenumber, q=3×105m1 (36), where the values of the average tube diameter, z, the bending stiffness, κ, and the surface tension, μ, are reported as 100 nm, 10−19 J, and 10−4 ∼ 10−5 J⋅m−2, respectively. η denotes the viscosity of the bulk water, whose value is given by 0.94 cP at 22 ∼ 23 °C. This agreement between the values of 0dtϕD(t) and ω1(q) implies that non-Gaussian diffusion results from the diffusion coefficient fluctuation caused by thermal fluctuation of the lipid tube into which colloidal beads are embedded, as discussed in ref. 36.