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Hyeon and Thirumalai's 2003 paper
Can energy landscape roughness of proteins and RNA be measured by using mechanical unfolding experiments?[1]

I'm skeptical about H&T eq. 8 to H&T eq. 9, so I'll rework as much of their math as I am capable of...

$\displaystyle \ensuremath{f^*}$ $\displaystyle = \frac{\ensuremath{k_B T}}{\ensuremath{\Delta x(\ensuremath{f^*}...
...nsuremath{\log\!\!\!\left( {\avg{e^{\ensuremath{\beta}F_1}}} \right)}^2 \right]$ $\displaystyle \ensuremath{\text{\emph{H\&T eq. {8}}}}$    

We simplify by dropping the 2 $ ^{\text{nd}}$term (``In obtaining Eq. 9, we have assumed that the second term in Eq. 8 is small.''), and defining $ \alpha \equiv \ensuremath{k_B T}$, $ \rho \equiv \ensuremath{\log\!\!\!\left( { \frac{\r\ensuremath{\Delta x(\ensur...
...suremath{\Delta F_0^\ddagger(\ensuremath{f^*})}}}\ensuremath{k_B T}} } \right)}$, and $ e^{\ensuremath{\beta}\varepsilon } \equiv \avg{e^{\ensuremath{\beta}F_1}}$, yielding

$\displaystyle \ensuremath{f^*}= \frac{\alpha}{\ensuremath{\Delta x(\ensuremath{f^*})}} \left( \rho + \frac{\varepsilon ^2}{\alpha^2} \right)$ (1)

We obtain our version of H&T eq. 9 by taking two measurements of equal mode force

0 $\displaystyle = \ensuremath{f^*}_1 - \ensuremath{f^*}_2$ (2)
  $\displaystyle = \frac{1}{\ensuremath{\Delta x(\ensuremath{f^*})}} \left( \alpha...
...epsilon ^2}{\alpha_1} -\alpha_2\rho_2 - \frac{\varepsilon ^2}{\alpha_2} \right)$ (3)
$\displaystyle \varepsilon ^2\left(\frac{1}{\alpha_2} - \frac{1}{\alpha_1}\right)$ $\displaystyle = \alpha_1\rho_1 - \alpha_2\rho_2$ (4)
$\displaystyle \varepsilon ^2 \cdot \frac{\alpha_1 - \alpha_2}{\alpha_1\alpha_2}$ $\displaystyle =$ (5)
$\displaystyle \varepsilon ^2$ $\displaystyle = \frac{\alpha_1\alpha_2}{\alpha_1 - \alpha_2} \left( \alpha_1\rho_1 - \alpha_2\rho_2 \right)$ (6)
$\displaystyle \varepsilon ^2$ $\displaystyle = \frac{\ensuremath{k_B T}_1\ensuremath{k_B T}_2}{\ensuremath{k_B...
...a F_0^\ddagger(\ensuremath{f^*})}_{2}}} \ensuremath{k_B T}_2}} \right)} \right]$ (7)

Which is different from H&T eq. 9 by the sign in the prefactor, and the replacement $ \ensuremath{\nu_D(\ensuremath{f^*})}\rightarrow \ensuremath{k(\ensuremath{f^*})}$.

$\displaystyle \varepsilon ^2$ $\displaystyle = \frac{\ensuremath{k_B T}_1\ensuremath{k_B T}_2}{\ensuremath{k_B...
...nsuremath{\nu_{D{2}}(\ensuremath{f^*})} \ensuremath{k_B T}_2}} \right)} \right]$ $\displaystyle \ensuremath{\text{\emph{H\&T eq. {9}}}}$    

Alternatively, noting that $ \Delta x(\ensuremath{f^*})$can vary as a function of temperature, we follow Nevo et al. in keeping it in. Using $ \delta \equiv \ensuremath{\Delta x(\ensuremath{f^*})}$

0 $\displaystyle = \ensuremath{f^*}_1 - \ensuremath{f^*}_2$ (8)
  $\displaystyle = \frac{\alpha_1\rho_1}{\delta_1} + \frac{\varepsilon ^2}{\delta_...
...a_1} -\frac{\alpha_2\rho_2}{\delta_2} - \frac{\varepsilon ^2}{\delta_2\alpha_2}$ (9)
$\displaystyle \varepsilon ^2\left(\frac{1}{\delta_2\alpha_2} - \frac{1}{\delta_1\alpha_1}\right)$ $\displaystyle = \frac{\alpha_1\rho_1}{\delta_1} - \frac{\alpha_2\rho_2}{\delta_2}$ (10)
$\displaystyle \varepsilon ^2 \cdot \frac{\delta_1\alpha_1 - \delta_2\alpha_2}{\delta_1\delta_2\alpha_1\alpha_2}$ $\displaystyle = \frac{\delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2}{\delta_1\delta_2}$ (11)
$\displaystyle \varepsilon ^2$ $\displaystyle = \frac{\alpha_1\alpha_2}{\delta_1\alpha_1 - \delta_2\alpha_2} \left( \delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2 \right)$ (12)
$\displaystyle \varepsilon ^2$ $\displaystyle = \frac{\ensuremath{k_B T}_1\ensuremath{k_B T}_2}{\ensuremath{\De...
...}}{\ensuremath{k_{2}(\ensuremath{f^*})} \ensuremath{k_B T}_2}} \right)} \right]$ (13)

Bibliography

1
C. Hyeon and D. Thirumalai. Can energy landscape roughness of proteins and RNA be measured by using mechanical unfolding experiments?. PNAS 100, 18, 10249-10253 (2003).
http://www.pnas.org/

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