An RNA hairpin has a folded state and an unfolded (extended) state separated by along…

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Forces, Machines and Signals, Ch Cooperativity: zippers, melting and allosteric binding

Problem

An RNA hairpin has a folded state and an unfolded (extended) state separated by

\Delta x = 20\,\mathrm{nm}

along the pulling direction, with the folded state lower in free energy by

\Delta G_0 = 40\,k_BT

at zero force (so

\Delta G_0 \equiv G_{\text{unfolded}}(0) - G_{\text{folded}}(0) = +40\,k_BT

). Use

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

298\,\mathrm{K}

. Someone argues: Applying a stretching force

f

adds a term

-f\Delta x

to the free energy of the folded state, because the force does work on the molecule. So

G_{\text{folded}}(f) = G_{\text{folded}}(0) - f\Delta x

, and the coexistence force, where

G_{\text{folded}}(f)=G_{\text{unfolded}}(f)

, is

f_{1/2} = \Delta G_0/\Delta x = 40\times 4.11/20 = 8.22\,\mathrm{pN}

. Since larger

f

lowers the folded state's free energy further below the unfolded one, above

f_{1/2}

the hairpin should be predominantly folded. ((a)) Identify the error in this argument. ((b)) Give the corrected expression for

G_{\text{unfolded}}(f) - G_{\text{folded}}(f)

and confirm the numerical value of

f_{1/2}

. ((c)) State which state actually dominates above

f_{1/2}

, and find the fraction of time the hairpin spends folded at

f=10\,\mathrm{pN}

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