\documentclass[letterpaper]{article}
%
% My working notes on Hyeon and Thirumalai's paper:
% `Can energy landscape roughness of proteins and RNA be
%  measured by using mechanical unfolding experiments?'
%  PNAS vol. 100, no. 18, 10249-10253 (2003).
%
%     Copyright (C) 2008, William Trevor King
%
%     Permission is granted to copy, distribute and/or modify this
%     document under the terms of the GNU Free Documentation License,
%     Version 1.2 or any later version published by the Free Software
%     Foundation; with no Invariant Sections, no Front-Cover Texts and
%     no Back-Cover Texts.  A copy of the license is available at
%      http://www.physics.drexel.edu/~wking/shared/GNU_FLD.txt
%

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%	backref,	% adds `backlink' from bibliography to text as list of section numbers
% pdf information options
	pdftitle=Hyeon Temp Theory,
	pdfauthor=Trevor King,
	pdfsubject=Unfolding Theory, % pdf display options
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%\providecommand{\url}[1]{#1}

% Hyeon and Thirumalai equation number #1
\renewcommand{\emph}[1]{\textbf{#1}}
\newcommand{\ensuretext}[1]{\ensuremath{\text{#1}}}
\newcommand{\HTeq}[1]{\ensuretext{\emph{H\&T eq.\ {#1}}}}
\newcommand{\kT}{\ensuremath{k_B T}}
\newcommand{\bt}{\ensuremath{\beta}}
\newcommand{\fs}{\ensuremath{f^*}}
\newcommand{\dx}{\ensuremath{\Delta x(\fs)}}
\newcommand{\dxs}[1]{\ensuremath{\Delta x_{#1}(\fs)}} % for subscripting
\newcommand{\FO}{\ensuremath{\Delta F_0^\ddagger(\fs)}}
\newcommand{\vD}{\ensuremath{\nu_D(\fs)}}
\newcommand{\vDs}[1]{\ensuremath{\nu_{D{#1}}(\fs)}}
\newcommand{\kexp}{\ensuremath{\vD e^{-\bt \FO}}}
\newcommand{\kexps}[1]{\ensuremath{\vDs{#1} e^{-\bt_{#1} \FO_{#1}}}}
\newcommand{\kf}{\ensuremath{k(\fs)}}
\newcommand{\kfs}[1]{\ensuremath{k_{#1}(\fs)}}
%\newcommand{\avg}[1]{\ensuremath{\left\langle {#1} \right\rangle}}
\newcommand{\logp}[1]{\ensuremath{\log\!\!\!\left( {#1} \right)}}
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\renewcommand{\r}{\ensuremath{r_f}}
\newcommand{\rs}[1]{\ensuremath{r_{f{#1}}}}% to avoid double-subscripting
\newcommand{\ep}{\varepsilon}


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\begin{document}

\begin{center}
{\Large Hyeon and Thirumalai's 2003 paper} \\
{\it Can energy landscape roughness of proteins and RNA be measured by using mechanical unfolding experiments?\cite{hyeon03}}
\end{center}

I'm skeptical about \HTeq{8} to \HTeq{9}, so I'll rework as much of their math as I am capable of...

\begin{align}
  \fs &= \frac{\kT}{\dx} \left[ \logp{ \frac{\r \dx}{\kexp \kT} }
       + \logp{1 + \fs\frac{\dx'}{\dx} - \frac{\FO'}{\dx} + \frac{\vD'}{\vD}\cdot\frac{\kT}{\dx}}
       + \logp{\avg{e^{\bt F_1}}}^2 \right]
      & \HTeq{8} \nonumber
\end{align}

We simplify by dropping the 2\nd term
(``In obtaining Eq.\ \textbf{9}, we have assumed that the second term in Eq.\ \textbf{8} is small.''),
and defining $\alpha \equiv \kT$,
 $\rho \equiv \logp{ \frac{\r \dx}{\kexp \kT} }$, and
 $e^{\bt \ep} \equiv \avg{e^{\bt F_1}}$, yielding
\begin{equation}
  \fs = \frac{\alpha}{\dx} \left( \rho + \frac{\ep^2}{\alpha^2} \right)
\end{equation}

We obtain our version of \HTeq{9} by taking two measurements of equal mode force
\begin{align}
  0 &= \fs_1 - \fs_2 \\
    &= \frac{1}{\dx} \left( \alpha_1\rho_1 + \frac{\ep^2}{\alpha_1} 
                           -\alpha_2\rho_2 - \frac{\ep^2}{\alpha_2} \right) \\
  \ep^2\left(\frac{1}{\alpha_2} - \frac{1}{\alpha_1}\right) &= \alpha_1\rho_1 - \alpha_2\rho_2 \\
  \ep^2 \cdot \frac{\alpha_1 - \alpha_2}{\alpha_1\alpha_2} &= \\
  \ep^2 &= \frac{\alpha_1\alpha_2}{\alpha_1 - \alpha_2} \left( \alpha_1\rho_1 - \alpha_2\rho_2 \right)\\
  \ep^2 &= \frac{\kT_1\kT_2}{\kT_1 - \kT_2} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\kexps1 \kT_1}}
                                                 - \kT_2\logp{\frac{\rs2\dxs2}{\kexps2 \kT_2}} \right]
\end{align}

Which is different from \HTeq{9} by the sign in the prefactor, and the replacement $\vD \rightarrow \kf$.
\begin{align}
  \ep^2 &= \frac{\kT_1\kT_2}{\kT_2 - \kT_1} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\vDs1 \kT_1}}
                                                 - \kT_2\logp{\frac{\rs2\dxs2}{\vDs2 \kT_2}} \right]
        & \HTeq{9} \nonumber
\end{align}

Alternatively, noting that \dx can vary as a function of temperature, we follow Nevo et al.\ in keeping it in.
Using $\delta \equiv \dx$
\begin{align}
  0 &= \fs_1 - \fs_2 \\
    &= \frac{\alpha_1\rho_1}{\delta_1} + \frac{\ep^2}{\delta_1\alpha_1} 
      -\frac{\alpha_2\rho_2}{\delta_2} - \frac{\ep^2}{\delta_2\alpha_2} \\
  \ep^2\left(\frac{1}{\delta_2\alpha_2} - \frac{1}{\delta_1\alpha_1}\right)
        &= \frac{\alpha_1\rho_1}{\delta_1} - \frac{\alpha_2\rho_2}{\delta_2} \\
  \ep^2 \cdot \frac{\delta_1\alpha_1 - \delta_2\alpha_2}{\delta_1\delta_2\alpha_1\alpha_2}
        &= \frac{\delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2}{\delta_1\delta_2} \\
  \ep^2 &= \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)\\
  \ep^2 &= \frac{\kT_1\kT_2}{\dxs1\kT_1 - \dxs2\kT_2} 
               \left[ \dxs2\kT_1\logp{\frac{\rs1\dxs1}{\kfs1 \kT_1}}
                    - \dxs1\kT_2\logp{\frac{\rs2\dxs2}{\kfs2 \kT_2}} \right]
\end{align}

\begin{thebibliography}{77} %77 is width of longest number of your reference?

\bibitem{hyeon03} C.\ Hyeon and D.\ Thirumalai.
 {\it Can energy landscape roughness of proteins and RNA be  measured by using mechanical unfolding experiments?}.
 PNAS {\bf 100}, 18, 10249-10253 (2003). \\
 \url{http://www.pnas.org/}

\end{thebibliography}

\end{document}
