A drug is attached to the surface of a nanoparticle by a flexible linker: a chain of…

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Physical foundations of the living cell · Macromolecules, membranes and cell architecture

Life at Thermal Energy, Ch Entropy, free energy and the forces that counting produces

Problem

A drug is attached to the surface of a nanoparticle by a flexible linker: a chain of

N=50

either-way segments, each of length

\ell = 0.35\,\mathrm{nm}

, exactly the kind of chain treated as an entropic spring in this section. The receptor's binding pocket sits at a fixed distance

x = 6.0\,\mathrm{nm}

from the point where the linker is anchored to the particle, so the drug can only reach and bind once the linker is stretched out to that extension. The same drug, free in solution rather than tethered, binds the same receptor with a standard dissociation constant

K_d^{\rm free} = 200\,\mathrm{nM}

, at

T=298\,\mathrm{K}

, where

k_BT = 4.11\,\mathrm{pN\,nm}

. ((a)) Check that the entropic-spring formula of this section may be used at this extension, then calculate the free-energy cost of stretching the linker out to

x

, in units of

k_BT

. ((b)) Explain why this stretching cost adds to the drug's binding free energy once it is tethered, and use that to find the tethered drug's effective dissociation constant

K_d^{\rm eff}

. ((c)) A measurement of this exact conjugate finds

K_d^{\rm eff} \approx 4\,\mathrm{\mu M}

. Is the estimate from part (b) consistent with it? ((d)) A second design uses a longer linker,

N=200

segments of the same length

\ell

, anchored so the pocket again sits at

x=6.0\,\mathrm{nm}

. Without starting the calculation over, say whether this design should bind more or less tightly than the first, then find its

K_d^{\rm eff}

Browse the problem library