\chapter{Introduction}
\label{sec:intro}

\section{The Protein Folding Problem}
\label{sec:folding-problem}

% Why study protein folding?
In biological systems the most important molecules, such as proteins,
nucleic acids, and polysaccharides, are all polymers.  Understanding
the properties and functions of these polymeric molecules is crucial
in understanding the molecular mechanisms behind structures and
processes in cells.

% What do genes do?  Why is protein folding interesting?
An organism's genetic code is stored in DNA%
\nomenclature{DNA}{Deoxyribonucleic Acid}
in the cell nucleus.
DNA sequencing is a fairly well developed field, with fundamental work
such as the Human Genome Project seeing major development in the early
2000s\citep{wolfsberg01,mcpherson01,collins03}.  It is estimated that
human genetic information contains approximately 25,000 genes, each
encoding a protein\citep{claverie01,venter01}.  Knowing the amino acid
sequence for a particular protein, however, does not immediately shed
light on the protein's role in the body, or even the protein's
probable conformation.  Indeed, a protein's conformation is often
vitally important in executing its biological tasks
(\cref{fig:ligand-receptor}).  Unfortunately both predicting stable
conformations of a given amino acid sequence and the inverse problem
of finding sequences that form a given conformation have proven
remarkably difficult.

\begin{figure}
  \begin{center}
  \includegraphics[width=2in]{figures/biotin-streptavidin/1SWE.png}%
  \caption{Complex of biotin\index{biotin} (red) and a
    streptavidin\index{streptavidin} tetramer (green)
    (\href{http://dx.doi.org/10.2210/pdb1swe/pdb}{PDB ID: 1SWE})%
    \citep{freitag97}.  The correct streptavidin conformation creates
    the biotin-specific binding pockets.  Biotin-streptavidin is a
    model ligand-receptor pair isolated from the bacterium
    \species{Streptomyces avidinii}%
    \index{Streptomyces@\species{Streptomyces avidnii}}.  Streptavidin
    binds to cell surfaces, and bound biotin increases streptavidin's
    cell-binding affinity\citep{alon90}.  Figure generated with
    \citetalias{pymol}.
    \label{fig:ligand-receptor}}
  \end{center}
\end{figure}


\section{Protein Folding Energy Landscapes}
\label{sec:energy-landscape}

% the free energy landscape
Finding a protein's lowest energy state via a brute force sampling of
all possible conformations is impossibly inefficient, due to the
exponential scaling of possible conformations with protein length, as
outlined by \citet{levinthal69}.  This has lead to a succession of
models explaining the folding mechanism.  For a number of years, the
``pathway'' model of protein folding enjoyed popularity
(\cref{fig:folding:pathway})\citep{levinthal69}.  More recently, the
``landscape'' or ``funnel'' model has come to the fore
(\cref{fig:folding:landscape})\citep{dill97}.  Both of these models
reduce the conformation space to a more approachable analog, and their
success depends on striking a useful balance between simplicity and
accuracy.

\begin{figure}
  \begin{center}
  \subfloat[][]{
    \begin{tikzpicture}[->,node distance=1.5cm]
      \tikzstyle{every state}=[draw=white]
      \node[state] (U)                 {$U$};
      \node[state] (I1)  [right of=U]  {$I_1$};
      \node[state] (I1X) [below of=I1] {$I_1^X$};
      \node[state] (I2)  [right of=I1] {$I_2$};
      \node[state] (I2X) [below of=I2] {$I_2^X$};
      \node[state] (N)   [right of=I2] {$N$};
    
      \path[<->] (U)  edge (I1)
                 (I1) edge (I1X)
                 (I1) edge (I2)
                 (I2) edge (I2X)
                 (I2) edge (N);
    \end{tikzpicture}\label{fig:folding:pathway}}
  % \hspace{.25in}%
  \subfloat[][]{\includegraphics[width=2in]{figures/schematic/dill97-fig4}%
    \label{fig:folding:landscape}}
  \caption{\protect\subref{fig:folding:pathway} A ``double T'' example
    of the pathway model of protein folding, in which the protein
    proceeds from the native state $N$ to the unfolded state $U$ via a
    series of metastable transition states $I_1$ and $I_2$ with two
    ``dead end'' states $I_1^X$ and $I_2^X$.  Adapted from
    \citet{bedard08}.
    \protect\subref{fig:folding:landscape} The landscape model of
    protein folding, in which the protein diffuses through a
    multi-dimensional free energy landscape.  Separate folding
    attempts may take many distinct routes through this landscape on
    the way to the folded state.  Reproduced from
    \citet{dill97}.\label{fig:folding}}
  \end{center}
\end{figure}

When the choice of theoretical approach becomes murky, you must gather
experimental data to help distinguish between similar models.
Separating the pathway model from the funnel model is only marginally
withing the realm of current experimental techniques, but with higher
throughput and increased automation it should be easier to make such
distinctions in the near future.


\section{Single Molecule Protein Folding Studies}
\label{sec:single-molecule}

The large size of proteins relative to simpler molecules limits the
information attainable from bulk measurements, because the
macromolecules in a population can have diverse conformations and
behaviors.  Bulk measurements average over these differences,
producing excellent statistics for the mean, but making it difficult
to understand the variation.  The individualized, and sometimes rare,
behaviors of macromolecules can have important implications for their
functions inside the cell.  Single molecule techniques, in which the
macromolecules are studied one at a time, allow direct access to the
variation within the population without averaging.  This provides
important and complementary information about the functional
mechanisms of several biological systems\citep{bustamante08}.

Single molecule techniques provide an opportunity to study protein
folding and unfolding at the level of a single molecule, where the
distinction between the pathway model and funnel model is clearer.
They also provide a convenient benchmark for verifying molecular
dynamics simulations, because it takes a quite a bit of computing
power to simulate even one biopolymer with anything close to atomic
resolution over experimental timescales.  Even with significant
computing resources, comparing molecular dynamics results with
experimental data remains elusive.  For example, experimental pulling
speeds are on the order of \bareU{$\mu$m/s}, while simulation pulling
speeds are on the order of
\bareU{m/s}\citep{lu98,lu99,rief02,zhao06,berkemeier11}.

% why AFM & what an AFM is
Single molecule techniques for manipulating biopolymers include
optical measurements, \ie, single molecule fluorescence microscopy and
spectroscopy, and mechanical manipulations of individual
macromolecules, \ie, force microscopy and spectroscopy using atomic
force microscopes (AFMs), laser tweezers\citep{kellermayer97,forde02},
magnetic tweezers\citep{smith92}, biomembrane force
probes\citep{merkel99}, and centrifugal
microscopes\citep{halvorsen09}.  These techniques cover a wide range
of approaches, and even when the basic approach is the same
(e.g.\ force microscopy), the different techniques span orders of
magnitude in the range of their controllable parameters.
\nomenclature{AFM}{Atomic Force Microscope (or Microscopy)}


\section{Thesis Outline}
\label{sec:outline}

TODO: fill in once structure has stabilized

%\Cref{sec:unfolding} of this thesis discusses the theory of protein
%unfolding for single domains.  \Cref{sec:tension} discusses linker
%tension modeling.  \Cref{sec:unfolding-distributions} pulls
%\cref{sec:unfolding,sec:tension} together to discuss the theory of
%mechanical unfolding experiments.  This theory makes straightforward
%analysis of unfolding results difficult, so \cref{sec:sawsim} presents
%a Monte Carlo simulation approach to fitting unfolding parameters, and
%\cref{sec:contour-space} presents the contour-length space analysis
%for converting force curves to unfolding pathway fingerprints.
%\Cref{sec:temperature-theory} wraps up the theory section by extending
%the analysis in \cref{sec:unfolding,sec:unfolding-distributions} to
%multiple temperatures.
%
%\Cref{sec:apparatus} describes our experimental apparatus and methods,
%as well as calibration procedures.  With both the theory and procedure
%taken care of, \cref{sec:cantilever,sec:temperature}
%present and analyze AFM cantilever- and temperature-dependent
%unfolding behavior of the immunoglobulin-like domain 27 from human
%Titin (I27).
%
%We close with \cref{sec:future}, which presents our conclusions and
%discusses possible directions for future work.
