\section{Introduction}
% short short form


Proteins are polymers of amino acids of critical importance to biochemistry.
They play a role in almost every aspect of life, from catalyzing reactions in a cell to forming the mechanical structure of fingernails, muscle, and hair.
A thorough understanding of the formation and action of proteins is fundamental to a theoretical understanding of life's inner workings.

The amino acid sequence for a given protein is encoded in DNA.
With the completion of the human genome project and the approach of fast DNA sequencing, there is an abundance of information about these sequences.
However, moving from a protein's amino acid sequence to its biological function or even its three dimensional structure is currently very difficult.

Protein structure can be measured experimentally using x-ray crystallography or NMR spectroscopy, and such work is currently filling databases of protein structures.
This horizontal cataloging should be supported by a vertical attempt to study a smaller group of proteins in detail and understand not only their conformation, but \emph{why} they assume that conformation.
A firmer grasp of the physics behind the protein folding process would almost certainly aide in the push towards predictive folding models.

\subsection{The protein folding problem}

Because protein folding involves finding a global energy minimum in a convoluted high dimensional-space, it is very difficult to model.
As Levinthal pointed out in 1969, the number of possible conformations a protein can assume is so large that searching all of them to find a lowest energy state would take many times the age of the universe\cite{levinthal69}.
The local roughness in the energy landscape of a protein which shields the protein from the global minimum energy conformation is referred to as \emph{frustration}.

Because proteins in nature \emph{do} fold in reasonable timescales, their energy landscape cannot be a uniform bumpy surface, but must be roughly funnel shaped, guiding the proteins toward the more stable conformations (see Figure~\ref{fig.minimal_frust}).
%A biological explaination for these funnel-shaped energy landscapes has been proposed in the \emph{principle of minimal frustration} which sugests that native proteins have developed over the years to minimize their frustration (see Figure~\ref{fig.minimal_frust})\cite{saito97}.
This makes predicting the folded state of native proteins easier, and sets design goals for stable artificial proteins.
\begin{figure}[bp]
 \centering
 \sbox{\tempbox}{\includegraphics[width=0.45\textwidth]{lock_in_funnel}}
 \subfloat[A random landscape\cite{saito97}.]{\raisebox{0.5\ht\tempbox-0.5\height}[0pt][0pt]{\includegraphics[width=0.45\textwidth]{lock_in_random}}}
% \sbox{\tempbox}{\includegraphics[width=0.45\textwidth]{lock_in}}
% \subfloat[The evolution of stable proteins\cite{saito97}.]{\usebox{\tempbox}}
 \hspace{\stretch{1}}
% \subfloat[A funnel landscape\cite{dill97}.]{\raisebox{0.5\ht\tempbox-0.5\height}[0pt][0pt]{\includegraphics[width=0.45\textwidth]{bumpyBowl}}}
 \subfloat[A funnel landscape\cite{saito97}.]{\usebox{\tempbox}}
% \caption{The principle of minimum frustration, where protein energy landscapes evolve over many generations to quickly and consistently assume biologically useful conformations.}
 \caption{(a) The energy landscapes of most random polypeptide chains are rough with no unique global minimum.
    (b) Native proteins have a small number of stable minima and an overall funnel shape, which encourages rapid folding into biologically useful conformations\cite{saito97}.}
 \label{fig.minimal_frust}
\end{figure}

In this project, we attempt to probe this energy landscape for the muscle protein Titin.
With more information about the general structure of the folding funnel, we can perhaps construct more appropriate theoretical tools for seeking stable conformations.

%The protein itself can fold quickly, because it has access to a sufficiently powerful quantum computer (itself :p).

%In order to build a predictive model of protein folding that can be simulated in reasonable timescales, we need to reduce the number of degrees of freedom from the actual number to some smaller set which encapsulates the most important features.

\subsection{Single molecular folding and unfolding}

Most methods for studying protein folding and unfolding use bulk samples containing many molecules..
The proteins are unfolded by changing the solution pH, temperature, and other environmental factors.
The state of the protein is monitored using florescence, NMR, or other methods\cite{carrion-vazquez99}.
Such ensemble measurements allow parallel measurements of millions of unfolding events, which leads to very convincing average values of the measured parameters but loses any information about their distributions.

Single molecule folding and unfolding experiments are slower and more difficult, but they give access to distribution of protein behaviors, which may yield information not present in the averages.
The direct mechanical manipulation methods used on single molecules also benefit from not effecting the protein environment, so temperature and pH can be chosen to match those expected \emph{in vivo}.

\subsection{Temperature dependent unfolding and the energy landscape}

One limitation of the single molecule protein unfolding experiment is that we only directly measure how the unfolding force depends on the single parameter of end-to-end distance.
Varying some environmental factors, and measuring how the unfolding forces depend on environmental conditions can provide deeper insight into the mechanics.
For example, measuring how the unfolding force for a given protein varies with respect to temperature and pulling speed can yield a scale for the energy landscape frustration (see Figure~\ref{fig.minimal_frust}), shown by Hyeon and Thirumalai\cite{hyeon03}.
This theory forms the basis for the the bulk of our experiment, and is discussed in detail in Section~\ref{sec.theory}.

\subsection{Titin and temperature dependent muscle behavior}

Measuring how the unfolding force of Titin depends on temperature not only probes its energy landscape, but might also shed light on why muscle stiffness changes with temperature.
% becomes :p{\bf stiffer} when warm, see Kirton 05.
Titin is one of three main proteins in striated muscle along with Actin and Myosin.
While Actin and Myosin are responsible for contractions, Titin is responsible for the return to the relaxed state.
Titin is also responsible for the mechanical properties of the relaxed state (see Figure~\ref{fig.sarcomere})\cite{kirton_05}.
\begin{figure}[btp]
 \centering
 \includegraphics[width=0.6\textwidth]{sarcomere}
 \caption{The role of Titin in a striated muscle sarcomere\cite{gautel}.
    Titin keeps the Myosin centered between Actin filaments and is the source of passive stiffness when the Myosin is not bound to the Actin.}
 \label{fig.sarcomere}
\end{figure}
Because of Titin's biological role, our forced mechanical unfolding procedure closely mimics the actual experience of Titin {\it in vivo}.
We hope an increased understanding of the fundamental temperature dependent folding properties of Titin will shed light on the temperature dependent mechanical properties of muscle.
