Estimate the shear modulus of a dilute collagen gel at protein concentration

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture

Forces, Machines and Signals, Ch Filaments, networks and the mechanics of whole cells

Problem

Estimate the shear modulus of a dilute collagen gel at protein concentration c=1.0 mg mL−1c = 1.0\,\mathrm{mg\,mL^{-1}}, at T=298 KT = 298\,\mathrm{K}, assuming: (a) the filaments are cylinders of diameter d=100 nmd = 100\,\mathrm{nm} and protein mass density ρ=1300 kg m−3\rho = 1300\,\mathrm{kg\,m^{-3}}; (b) the total filament length per unit volume, ℓV=c/(ρA)\ell_V = c/(\rho A) with cross-sectional area A=π(d/2)2A = \pi (d/2)^2, sets a random mesh of spacing ξm≈ℓV−1/2\xi_m \approx \ell_V^{-1/2}; (c) the network behaves, for this baseline estimate, as a flexible network, so G≈kBT/ξm3G \approx k_BT/\xi_m^3. Find ξm\xi_m and GG, and compare GG with the range of 11 to 103 Pa10^3\,\mathrm{Pa} measured for cross-linked actin and collagen gels.

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