\subsection{Procedure}
\label{sec.unfold_procedure}

The proteins we unfold are polymeric repeats of the protein domain we want to study.
Titin is one of the largest known proteins, and its structure varies between species and muscle types \cite{mutungi96}.
In order to have tractable datasets and broadly applicable results, we restrict our study to a single immunoglobulin-like domain (I27), which occurs several times in all native Titins \cite{furst95}.
In order to separate our signal unfoldings from noise and indeterminate surface interactions, we do not unfold single I27 domains, but a synthetic protein consisting of a number of consecutive I27 domains.
The actual number of repeats is not critical, but the more repeats you have, the clearer the unfolding signature.
The current project uses twelve consecutive I27 domains.

Our synthesized proteins also have a few cysteine amino acids on one end, which datively bond to gold.
We deposit 10$\mu$L of protein solution (100$\mu$g/mL) onto a fresh gold surface, wait several minutes to allow the protein to bond to the gold, fill the AFM fluid cell with buffer solution, and then use the AFM tip to pull the free end of the protein away from the surface, unfolding the protein as the tip retreats.

The procedure for obtaining a force curve is as follows (see the schematic in Figure~\ref{fig.force_curve_system}):
\begin{packed_enum}
 \item Approach the AFM tip to the surface until the tip bends back a set amount.% \label{item.approach}
 \item Wait $\sim$10 seconds for the protein to bind to the AFM tip (The binding mechanism is poorly understood).
 \item Withdraw the tip at a constant speed while measuring the cantilever deflection, until the protein breaks away from the tip.
 \item Move to a different location on the sample.
 \item Repeat steps 1-4.
\end{packed_enum}
This procedure yields curves similar to that shown in Figure~\ref{fig.force_curve_example}.
\begin{figure}[p]
 \centering
 \subfloat[The system.]{\label{fig.force_curve_system}\includegraphics{mp/unfolding.2}}\\
 \subfloat[An example force curve.]{\label{fig.force_curve_example}\includegraphics{mp/unfolding_data.1}}
 \caption{Force curve explanation:
          The tip is initially pressed into the surface at point $a$ with a predetermined force.
          In region $A$, the AFM tip retreats but is still in contact with the surface, and the force decreases as the enforced displacement of the tip decreases (the positive force direction in \ref{fig.force_curve_example} is towards the surface).
          In region $B$ the protein unfolds.
	  At point $b$, the tip has retreated enough to pull the protein tight, giving the first force peak.
          One protein domain unfolds, relieving the strain on the protein chain and returning the deflection to near-zero at $c$.
          The retreating tip pulls the new conformation tighter until another domain unfolds from $d$ to $e$.
          The process continues until there are no domains left to unfold, and the protein breaks off of the tip at $f$, leaving the tip in region $C$ with no deflection.}
 \label{fig.force_curve}
\end{figure}
Most of the data we acquire is not as clean as this curve.
We discuss how we deal with the complications in Section~\ref{sec.auto_sort}.

The shape of the rising edge of the sawtooth ($c$ to $d$ in Figure~\ref{fig.force_curve}), can be modeled as a \emph{worm-like chain} (WLC)\cite{carrion-vazquez00}.
The WLC model approximates the protein as an entropic spring, which exerts a force
\begin{equation}
 F(x) = \frac{k_B T}{p}\left[\frac{1}{4}\left(\frac{1}{(1-x/L)^2} - 1\right) + \frac{x}{L}\right],
\end{equation}
where $x$ is the distance between the pinned ends of the protein,
 $p$ is the persistence length, and
 $L$ the contour length of the protein chain.

Fitting our sawtooth to a succession of WLCs with different contour lengths yields the contour length increase upon a domain unfolding, which should be consistent with the value calculated from the domain structure.
Average values of unfolding force as a function of temperature and pulling speed provide the data for computing the energy landscape roughness using Hyeon and Thirumalai's method\cite{hyeon03}, which we discuss in Section~\ref{sec.theory}.

Most of the actual equipment is contained in the commercial AFM.
We have added a temperature control device consisting of a Peltier temperature module powered by a commercial temperature controller as discussed in Section~\ref{sec.temp_control}.
The remainder of our setup consists of controlling and monitoring electronics, discussed in Sections~\ref{sec.stepper} and~\ref{sec.server}.
A schematic of the equipment used in our experiment is given in Figure~\ref{fig.equip_layout}.
%\begin{packed_enum}
% \item MultiMode II atomic force microscope (modified), Veeco\cite{veeco}.
% \item NanoScope IIIa AFM controller (modified), Veeco.
% \item PCI-6052E analog/digital input/output card, National Instruments\cite{NI}.
% \item BNC-2090 break-out box for PCI-6052E, National Instruments.
% \item MTCA Thermo-Electric Controller, Melcore\cite{melcore}.
%\end{packed_enum}
\begin{figure}[bt]
 \centering
 \includegraphics[width=0.90\textwidth]{equipment_layout}
 \caption{Equipment layout.
          The AFM contains the actual test system, as well as some simple processing electronics.
	  The AFM controller sources the high voltages needed to drive the AFM piezo scanner.
	  The temperature controller monitors and maintains a set temperature in the AFM fluid cell.
	  Our software controls the whole setup, using a serial port to talk to the temperature controller, and an analog/digital output/input board to run the rest.
          The breakout box allows direct access to those lines of the AFM$\leftrightarrow$controller cable whose roles we understand and want direct access to.}
 \label{fig.equip_layout}
\end{figure}
