availability of user-friendly commercial instruments. AFM has been
employed on several types of biological macromolecules, mechanically
unfolding proteins\citep{carrion-vazquez99a} and forcing structural
-transitions in DNA\citep{rief99} and polysaccharides\citep{rief97a}.
+transitions in DNA\citep{florin95,rief99} and
+polysaccharides\citep{rief97a}.
An AFM\index{AFM} uses a sharp tip integrated at the end of a
cantilever to interact with the sample. Cantilever bending is
measured by a laser reflected off the cantilever and incident on a
position sensitive photodetector (\cref{fig:afm-schematic}). When the
bending force constant of the cantilever is known\citep{levy02}, the
-force applied to the sample can be calculated.
+force applied to the sample can be calculated using Hooke's law
+(\cref{eq:sawsim:hooke}).
\begin{figure}
\asyinclude{figures/schematic/afm}%
position sensitive photodetector.\label{fig:afm-schematic}}
\end{figure}
+The substrate is mounted on a three dimensional piezoelectric actuator
+so that the tip may be positioned on the surface with sub-nanometer
+resolution (although signal drift and piezo hysteresis can cause
+larger errors in the positioning accuracy). Our tubular piezo has a
+range of TODO in the horizontal directions and a range of TODO in the
+vertical (\cref{fig:piezo-schematic}).
+
+\begin{figure}
+ \asyinclude{figures/schematic/piezo}%
+ \caption{Schematic of a tubular piezoelectric actuator. In our AFM,
+ the substrate is mounted on the top end of the tube, and the
+ bottom end is fixed to the microscope body. This allows the piezo
+ to control the relative position between the substrate and the AFM
+ cantilever. The electrodes are placed so radial electric fields
+ can be easily generated. These radial fields will cause the piezo
+ to expand or contract axially. The $z$ voltage causes the tube to
+ expand and contract uniformly in the axial direction. The $x$ and
+ $y$ voltages cause expansion on one side of the tube, and
+ contraction (because of the reversed polarity) on the other side
+ of the tube. This tilts the tube, shifting the sample
+ horizontally.\label{fig:piezo-schematic}}
+\end{figure}
+
% really, AFM can do this ;)
The forces that can be applied and measured with an AFM range from
tens of piconewtons to hundreds of nanonewtons. The investigation of
\section{Protein Polymer Synthesis}
\label{sec:polymer-synthesis}
-TODO.
+Early experiments in force spectroscopy involved native
+titin\index{titin}\citep{rief97a}. Titin is a muscle protein involved
+in passive elasticity (\cref{fig:skeletal-muscle}), so it is an ideal
+subject when examining the effect of mechanical force\citep{labeit95}.
+Titin is also interesting because, while it is one of the largest
+known proteins, it is composed of a series of globular domains. When
+\citet{rief97a} carried out their seminal unfolding experiment, the
+observed a very charachteristic sawtooth as the domains unfolded (see
+\cref{sec:procedure} for a discussion of these sawteeth).
+
+\begin{figure}
+ \includegraphics[width=0.7\textwidth]{figures/binary/skeletal_muscle}%
+ \caption{Biological role of titin\index{titin}. Moving clockwise
+ from the upper left you can see a bone/muscle group, a muscle
+ fiber, a myofibril, and a sarcomere. In the sarcomere, the white,
+ knobbly filaments are actin. The myosin bundles are blue, and the
+ titin linkers are red. When the muscle contracts, the myosin
+ heads walk up the actin filiaments, shortening the sarcomere.
+ When the muscle relaxes, the myosin heads release the actin
+ filimants and slide back, lengthening the sarcomere. Titin
+ functions as an entropic spring that keeps the myosin from falling
+ out of place during the passive, relaxed stage. This figure is
+ adapted from \citet{skeletal_muscle}.\label{fig:skeletal-muscle}}
+\end{figure}
+
+Unfortunately, it is difficult to analyze the unfolding of native
+titin, because the heterogenous globular domains make it hard to
+attribute a particular subdomain to a partuclar unfolding event.
+Unfolding a single domain is not feasable because the large radius of
+curvature of an AFM tip ($\sim20\U{nm}$\citep{olympus-tr400psa})
+dwarfs the radius of a globular domain
+($\sim2\U{nm}$\citep{improta96}). When such a large tip is so close
+to the substrate, van der Waals forces and non-specific binding with
+the surface dominate the tip-surface interaction. In order to
+increase the tip-surface distance while preserving single molecule
+analysis, \citet{carrion-vazquez99b} synthesized a protein composed of
+eight repeats of immunoglobulin-like domain 27 (I27), one of the
+globular domains from native titin (\cref{fig:I27}). Octameric I27
+produced using their procedure is now available
+commercially\citep{athenaes-i27o}.
+\nomenclature{I27}{Titin immunoglobulin-like 27}
+
+\begin{figure}
+ \includegraphics[width=2in]{figures/i27/1TIT}
+ \caption{I27, the immunoglobulin-like domain 27 from human titin
+ (\href{http://dx.doi.org/10.2210/pdb1tit/pdb}{PDB ID: 1TIT})%
+ \citep{improta96}.
+ Figure generated with \citetalias{pymol}.
+ \label{fig:I27}}
+\end{figure}
+
+Synthetic proteins are generally produced by creating a plasmid coding
+for the target protein, inserting the plasmid in a bacteria, waiting
+while the bacteria produce your protein, and then purifying your
+proteins from the resulting culture. In this case,
+\citet{carrion-vazquez99b} extracted messenger RNA coding for titin
+from human cardiac tissue\cite{rief97a}, and used reverse
+transcriptase to generate a complementary DNA (cDNA) library from
+human cardiac muscle messenger RNA. This cDNA is then amplified using
+the polymerase chain reaction (PCR), with special primers that allow
+you to splice the resulting cDNA into a plasmid (which ends up with
+one I27). Then they ran another PCR on the plasmid, linearized the
+plasmid with two restriction enzymes, and grafted two I27-containing
+sections together to form a new plasmid (now with two I27s,
+\cref{fig:plasmid}). Another PCR-split-join cycle produced a plasmid
+with four I27s, and a final cycle produced a plasmid with eight. The
+eventual plasmid vector has the eight I27s and a host-specific
+promoter that causes the bacteria to produce large quantities of I27.
+The exact structure of the generated octamer
+is\citep{carrion-vazquez99b}
+\nomenclature{cDNA}{Complementary DNA}
+\nomenclature{PCR}{Polymerase chain reaction}
+
+\begin{center}
+ Met-Arg-Gly-Ser-(His)$_6$-Gly-Ser-(I27-Arg-Ser)$_7$-I27-\ldots-Cys-Cys
+ \label{eq:I27}
+\end{center}
+
+\begin{figure}
+ \includegraphics[width=0.9\textwidth]{figures/binary/kempe85-fig2}%
+ \caption{Example of gene duplication via plasmid splicing (Figure~2
+ from \citet{kempe85}). \citet{kempe85} use a different gene, but
+ some of the restriction enzymes are shared with
+ \citet{carrion-vazquez99b}. The overall approach is
+ identical.\label{fig:plasmid}}
+\end{figure}
+
+The plasmid is then transformed into the host, usually
+\species{Escherichia coli}\citep{carrion-vazquez99b,bartels03,ma10} or
+a proprietary equivalent such as Agilent's SURE 2 Supercompetent
+Cells\citep{agilent-sure2,carrion-vazquez00}. The infected cells are
+cultured to express the protein.
+\nomenclature{Bacterial transformation}{The process by which bacterial
+ cells take up exogenous DNA molecules}
+\nomenclature{Exogenous DNA}{DNA that is outside of a cell}
+
+The octamer is then purified from the culture using immobilized metal
+ion affinity chromatography (IMAC), where the His-tagged end of the
+octamer covalently bonds to a metal ion that is bound to the column
+media (e.g. Ni-NTA\index{Ni-NTA} coated
+beads)\cite{carrion-vazquez00,bartels03,ma10}. Once the rest of the
+broth has been washed out of the chromatography column, the octamer is
+eluted via either another molecule which competes for the metal
+ions\citep{ma10} or by changing the pH so the octamer is less
+attracted to the metal ion.
+\nomenclature{IMAC}{Immobilized metal ion affinity chromatography}
+\nomenclature{Ni-NTA}{Nickle nitrilotriacetic acid}
+
+\nomenclature{Ala}{Alanine, an amino acid}
+\nomenclature{Arg}{Arginine, an amino acid}
+\nomenclature{Asn}{Asparagine, an amino acid}
+\nomenclature{Asp}{Aspartic acid, an amino acid}
+\nomenclature{Cys}{Cystine, an amino acid}
+\nomenclature{Glu}{Glutamine, an amino acid}
+\nomenclature{Gly}{Glycine, an amino acid}
+\nomenclature{His}{Histidine, an amino acid}
+\nomenclature{Ile}{Isoleucine, an amino acid}
+\nomenclature{Leu}{Leucine, an amino acid}
+\nomenclature{Lys}{Lysine, an amino acid}
+\nomenclature{Met}{Methionine, an amino acid}
+\nomenclature{Phe}{Phenylalanine, an amino acid}
+\nomenclature{Pro}{Proline, an amino acid}
+\nomenclature{Ser}{Serine, an amino acid}
+\nomenclature{Thr}{Threonine, an amino acid}
+\nomenclature{Trp}{Tryptophan, an amino acid}
+\nomenclature{Tyr}{Tyrosine, an amino acid}
+\nomenclature{Val}{Valine, an amino acid}
\section{Sample Preparation}
\label{sec:sample-preparation}
+In mechanical unfolding experiments, one end of the protein is bound
+to a substrate and the other binds to the AFM tip. This allows you to
+stretch the protein by increasing the tip-substrate distance using the
+piezo. A common approach is to synthesize proteins with cystine
+residues on one end (\cref{eq:I27}) and allow the cystines to bind to
+a gold surface\citep{carrion-vazquez99b,carrion-vazquez00,ulman96}.
+
+We prepare gold-surfaces by sputtering gold onto freshly cleaved mica
+sheets in a vacuum. The mica keeps the gold surface protected from
+contamination until it is needed. In order to mount the gold on our
+AFM, we glue glass coverslips to the gold using a two part epoxy
+(TODO). Instead of using mica to protect the gold surface, some labs
+evaporate the gold directly onto the coverslips immediately before
+running an experiment\citep{carrion-vazquez99b}.
+
+When it is time to deposit proteins on the surface, we peel a
+coverslip off the gold-coated mica, exposing the gold surface that had
+previously been attached to the mica. We dispense $5\U{$\mu$L}$ of
+I27 solution ($65\U{g/$\mu$L}$) on the freshly-exposed gold, followed
+by $5\U{$\mu$L}$ of phosphate buffered saline (PBS). We allow the
+protein to bind to the gold surface for 30 mintues and then load the
+coated coverslips into our AFM fluid cell. There are a number of
+similar PBS recipes in common
+use\citep{florin95,carrion-vazquez00,lo01,brockwell02}, but our PBS is
+diluted from 10x PBS stock composed of $1260\U{mM}$ NaCl, $72\U{mM}$
+\diNaHPO, and $30\U{mM}$ \NadiHPO\citep{chyan04}.
+\index{Phosphate buffered saline (PBS)}
+\nomenclature{PBS}{Phosphate buffered saline}
+
+As an alternative to binding proteins to gold, others have used
+EGTA\citep{kellermayer03},
+Ni-NTA\citep{schmidt02,itoh04,sakaki05,berkemeier11}, or silanized
+glass\citep{sundberg03,ma10}. Some groups have also functionalized
+the cantilever tips, with Ni-NTA being the most
+popular\cite{schmitt00}.
+\nomenclature{EGTA}{Ethylene glycol tetraacetic acid}
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