In an in-vitro reconstitution at ( )

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

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

Problem

In an in-vitro reconstitution at

298\,\mathrm{K}

(

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

), a microtubule plus end grows by addition of tubulin dimers, each adding a length

\delta = 0.62\,\mathrm{nm}

to the filament (an

8\,\mathrm{nm}

dimer spread over thirteen protofilaments). The plus-end rate constants are

k_{\rm on} = 4.0\,\mu\mathrm{M^{-1}\,s^{-1}}

and

k_{\rm off} = 44\,\mathrm{s^{-1}}

. Idealise the tip as adding subunits one at a time against whatever load acts on it, with the load reducing the forward rate by a Boltzmann factor as in the Brownian ratchet model. ((a)) Derive the critical free-tubulin concentration

c_c

from the rate equation for the end, and evaluate it. ((b)) The growing tip now pushes steadily against a rigid obstacle exerting a fixed opposing load

f = 5.0\,\mathrm{pN}

. Using

k_{\rm on}(f) = k_{\rm on}c\,e^{-f\delta/k_BT}

for the forward rate, derive an expression for the free tubulin concentration

c^{*}

at which the tip is exactly held at the stall point by this load, and evaluate it.

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