A condensate-forming protein has Kuhn segments and is currently in one phase at

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

Soft Matter at the Thermal Scale, Ch Membranes, fluctuations and soft matter in biology

Problem

A condensate-forming protein has N=300N=300 Kuhn segments and is currently in one phase at χ=0.54\chi=0.54, below its Flory-Huggins critical value χc=12+N−1/2+12N\chi_c = \tfrac12+N^{-1/2}+\tfrac{1}{2N} (Chapter 4). Inert crowding agents are modelled as ideal spheres of radius δ=a/2\delta=a/2, where aa is the protein segment radius, at volume fraction ϕp\phi_p. As in the condensate section's own estimate, treat the depletion attraction they add as a square well of depth U0=4ϕpkBTU_0 = 4\phi_p k_BT and range λw=7/6\lambda_w = 7/6 on the core diameter 2a2a (from Chapter 2's contact overlap volume), and use Chapter 1's square-well result ΔB2/B2HS=−(λw3−1)(eU0/kBT−1)\Delta B_2/B_2^{\rm HS} = -(\lambda_w^3-1)(e^{U_0/k_BT}-1) together with v=2B2v=2B_2 and v=(1−2χ)v0v=(1-2\chi)v_0 to convert this into a shift Δχ=12(λw3−1)(eU0/kBT−1)\Delta\chi = \tfrac12(\lambda_w^3-1)(e^{U_0/k_BT}-1). ((a)) Find the shift Δχ\Delta\chi needed to bring the protein exactly to its critical point χc\chi_c. ((b)) Find the well depth U0U_0 (in units of kBTk_BT) that produces this shift. ((c)) Find the crowder volume fraction ϕp\phi_p required.

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