\subsection{Measuring energy landscape roughness of protein molecules}
\label{sec.theory}

Kramers' theory is the study of transition rates between meta-stable energy wells\cite{hanngi90}.
Hyeon and Thirmulai combine Zwanzig's work on diffusion in frustrated potentials to two state systems and Kramers' theory to develop a theory of how frustration affects escape from one well into the other\cite{hyeon03}\cite{zwanzig88}.
Hyeon and Thirmulai apply their theory to two-state protein unfolding, and show that measuring how the unfolding behavior varies with temperature yields information about the frustration energy scale.

Many proteins have funnel-shaped energy landscapes with limited frustration (Fig.~\ref{fig.minimal_frust}).
% predicted by the principle of minimum frustration 
Let $\varepsilon$ be the landscpe roughness energy scale.
For proteins whose frustration is on the order of the thermal energy in the environment ($\varepsilon \sim k_BT$), the protein is never caught in any particular local minimum, but wanders in the vicinity of its global minimum (see Figure~\ref{fig.simul_system}).
For such minimally frustrated proteins, the two-state model describes the protein's conformation as either generally `folded' or generally `unfolded' without worrying about the details added by many local minima.
%Let us assume that we can represent this situation as a potential of a single degree of freedom $x$, the end-to-end length of the protein.
%This reduction from the huge dimensional conformation space to one dimension is a large simplification but is fairly accurate for minimally frustrated proteins, because small bumps in an otherwise smooth funnel will not force large detours into other dimensions.
%Once the gently frustrated funnel landscape has been reduced to one dimension, it will qualitatively resemble Figure~\ref{fig.simul_system}.
\begin{figure}[btp]
 \centering
 \includegraphics[width=0.7\textwidth]{hyeon_landscape}
 \caption{Proposed energy roughness model\cite{hyeon03}.
   The free energy $F(x)$ is given by $F(x)=F_0(x)+F_1(x)$, where $F_0(x)$ is the smooth free energy funnel, and  $F_1(x) \sim \varepsilon$ is the local roughness in free energy.}
 \label{fig.simul_system}
\end{figure}

In our mechanical unfolding experiments, we have a well defined reaction coordinate in the protein's end-to-end distance $x$.
For Titin, which is naturally tethered at both ends, this reaction coordinate has the added benefit of direct biological significance.
The well to the left in Figure~\ref{fig.simul_system} represents our originally stable folded states, and the well to the right the unfolded states.
As we apply an unfolding force to the potential, we lower the free energy of the transition states, encouraging unfolding.
The frustration $F_1(x)$ servers to limit the diffusion of proteins on this landscape, increasing the force or time necessary to unfold the protein.
Hyeon and Thirmulai use Zwanzig's method to find the effective potential $F^*(x)$ under a constant external force $f$ is given by
\begin{equation}
 F^*(x) = F_0(x) - fx - k_BT \psi^-
\end{equation}
Where the first term is the frustration-less funnel potential,
 the second term is the potential due to an external force, and
 the third term is a temperature dependent `diffusion force', with
  $\psi^- \equiv \ln(\langle e^{-\beta F_1(x)}\rangle)$, 
  $\beta \equiv 1/k_BT$, and
  $\langle \ldots \rangle$ denoting the local average of the enclosed quantity.

%Applying Kramers' theory to the effective potential yields
% the unfolding rate $k_R$ given by
%\begin{align}
% k_R^{-1}(f) &= \frac{2 \pi \gamma}{\omega_0 \omega_{ts}}
%                \langle e^{-\beta F_1}\rangle
%		\langle e^{\beta F_1} \rangle
%		e^{\beta (\Delta F_0^\ddagger - f \Delta x)}
%	                                    & \text{Hyeon Eqn. 3}
%					    \nonumber
%\end{align}
%Where $\gamma$ is the coefficient of `friction',
% $\omega_0$ is the angular velocity at the minimum energy folded state of $F^*$, 
% $\omega_{ts}$ is the angular velocity at the transition state of $F^*$,
% $F_0^\ddagger = F_0(x_{ts}) - F_0(x_0)$ is the unforced activation energy, and
% $\Delta x = x_{ts}-x_0$ is the distance to the transition state.
%
%For uniform frustration distributions, $\langle e^{-\beta F_1}\rangle = \langle e^{\beta F_1}\rangle$, so letting $k_0 \equiv \omega_0\omega_{ts}/2\pi\gamma$, we have
%\begin{align}
% \log(k_R^{-1}(f)) &= \log(k_0^{-1})
%     + \beta [ \Delta F_0^\ddagger - f (\Delta x) ]
%     + \log \langle e^{\beta F_1} \rangle^2 
%                                              & \text{Hyeon Eqn. 4} \nonumber
%\end{align}
%And expanding the $F_1$ term for low frustration
%\begin{align}
% \log(k_R^{-1}(f)) &\approx \log(k_0^{-1})
%     \beta [ \Delta F_0^\ddagger - f (\Delta x) ]
%     + \beta^2 \langle F_1^2 \rangle 
%                                              & \text{Hyeon Eqn. 5} \nonumber
%\end{align}
%Because $k^{-1} = \tau$ the mean first passage time through $x_{ts}$, we can rewrite this as
%\begin{align}
%  \log(\tau) &\approx \log(\tau_0) + \beta[ \Delta F_0^\ddagger - f (\Delta x) ] + \beta^2 \langle F_1^2 \rangle
%\end{align}
%Where $\langle F_1^2 \rangle = \varepsilon^2$, our frustration.
%Hyeon and Thirmulai support this formula with the results of a Monte Carlo unfolding simulation.
%Their results are presented in Figure~\ref{fig.const_f_sim}.
%The value of $\varepsilon$ is extracted by fitting the data to $\tau(f) = a + b\beta + c\beta^2$ and matching this to Equation 5.
%\begin{figure}[btp]
% \centering
% \includegraphics[width=0.7\textwidth]{hyeon_const_f_sim}
% \caption{Simulated relationship between $\tau$ and $T$\cite{hyeon03}.
%   Their simulation used $\varepsilon = 3\mbox{pN}\cdot\mbox{nm}$ and a critical force $f_c = \Delta F_0^\ddagger/\Delta x = 25\mbox{pN}$.
%   Fits to the simulated data estimate $\varepsilon^2$ as $9.89$, $9.93$, and $8.88\mbox{pN}^2\cdot\mbox{nm}^2$ respectively for $15\mbox{pN}$, $25\mbox{pN}$, and $40\mbox{pN}$.
%   Note that $3.8\mbox{pN}\cdot\mbox{nm} = 2.2{\dg}$C and $4.4\mbox{pN}\cdot\mbox{nm} = 46{\dg}$C.}
% \label{fig.const_f_sim}
%% pN nm = 10^(-12-9) J
%% kB = 1.37e-23 J/K
%% C = K-273.15
%% 
%% For simulation a, the quad term is 9.89 pN.nm
%%  the average amplitude of F1 was 3.0 pN.nm
%%  odd, I'd expect <F1^2> = 1/2 Amplitude^2 = 4.5
%\end{figure}

%Because constant force experiments require a {\bf fast feedback loop} CITE, Hyeon and Thirmulai extend their theory to include constant loading experiments.
%By varying the loading rate $r_f \equiv df/dt$ and determining the most common unfolding force $f^*$ for two different temperatures $T_1$ and $T_2$, we can determine $\varepsilon$ using
Applying Kramers' theory to this potential for experiments with constant force loading rates $r \equiv df/dt$, they predict how the most common unfolding force $f^*$ will vary with temperature, loading rate, and energy landscape roughness $\varepsilon$.
For two different temperatures $T_1$ and $T_2$, measuring the loading rates $r(T_1)$ and $r(T_2)$ required to produce the same $f^*$ will yield $\varepsilon$ using
\begin{align}
 \varepsilon^2 &\approx \frac{k_b^2 T_1 T_2}{T_2-T_1}
                     \cdot \left[ T_1 \log\left(\frac{r_f(T_1)\Delta x(f^*)}{k_0(f^*)k_B T_1}\right)
                                 -T_2 \log\left(\frac{r_f(T_2)\Delta x(f^*)}{k_0(f^*)k_B T_2}\right)\right],
                                              & \text{Hyeon Eqn. 9} \nonumber
\end{align}
%Where $r_f(T_1)$ and $r_f(T_2)$ are selected from the data such that $f^*(r_f(T_1), T_1) = f^*(r_f(T_2), T_2)$.
The simulated data supporting this formula is given in Figure~\ref{fig.const_r_sim}.
\begin{figure}[btp]
 \centering
 \hspace{\stretch{1}}
 \sbox{\tempbox}{\includegraphics[width=0.4\textwidth]{hyeon_const_r_sim_a}}
 \subfloat[Unfolding distributions.]{\usebox{\tempbox}}
 \hspace{\stretch{1}}
 \subfloat[Most common unfolding force.]{\includegraphics[height=\ht\tempbox]{hyeon_const_r_sim_b}}
 \hspace{\stretch{1}}
 \caption{Simulated constant loading experiment for determining $\varepsilon$\cite{hyeon03}.
   (a) Four unfolding force probability distributions.
   The most common unfolding force $f^*$ from each distribution in (a) becomes a point in (b), which shows the dependence of $f^*$ on the loading rate for two temperatures.}
 \label{fig.const_r_sim}
\end{figure}
Our experiment replicates this simulation experimentally to extract $\varepsilon$ for our selected protein.
