--- /dev/null
+\section{Instrumentation}
+\label{sec:afm}
+
+Of the mechanical manipulation methods listed in
+\cref{sec:single-molecule}, AFM is the most widely used due to the
+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}.
+
+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.
+
+\begin{figure}
+ \asyfig{figures/schematic/afm}%
+ \caption{Operating principle for an Atomic Force
+ Microscope\index{AFM}. A sharp tip integrated at the end of a
+ cantilever interacts with the sample. Cantilever bending is
+ measured by a laser reflected off the cantilever and incident on a
+ position sensitive photodetector.\label{fig:afm-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
+the unfolding and refolding processes of individual protein molecules
+by the AFM is feasible because many globular proteins unfold under
+external forces in this range. Since elucidating the mechanism of
+protein folding is currently one of the most important problems in
+biological sciences, the potential of the AFM for revealing
+significant and unique information about protein folding has
+stimulated much effort in both experimental and theoretical research.
--- /dev/null
+\section{Cantilever Spring Constant Calibration}
+\label{sec:cantilever-calib:intro}
+
+In order to measure forces accurately with an AFM, it is important to
+measure the cantilever spring constant. The force exerted on the
+cantilever can then be deduced from it's deflection via Hooke's law
+$F=-kx$.
+\nomenclature{$F$}{Force (newtons)}
+\nomenclature{$k$}{Spring constant (newtons per meter)}
+\nomenclature{$x$}{Displacement (meters)}
+
+The basic idea is to use the equipartition theorem\citep{hutter93},
+\begin{equation}
+ \frac{1}{2} k \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart}
+\end{equation}
+where $k_B$ is Boltzmann's constant, $T$ is the absolute temperature,
+and $\avg{x^2}$ denotes the expectation value of $x^2$ as measured
+over a very long interval $t_T$,
+\nomenclature{$k_B$}{Boltzmann's constant,
+ $k_B = 1.380 65\E{-23}\U{J/K}$\citep{codata-boltzmann}}
+\nomenclature{$T$}{Absolute temperature (Kelvin)}
+\nomenclature{$\avg{s(t)}$}{Mean (expectation value) of a time-series $s(t)$
+ \begin{equation}
+ \avg{A} \equiv \iLimT{A} \;.
+ \end{equation}}
+Solving the equipartition theorem for $k$ yields
+\begin{equation}
+ k = \frac{k_BT}{\avg{x^2}} \;, \label{eq:equipart_k}
+\end{equation}
+so we need to measure (or estimate) the temperature $T$ and variance
+of the cantilever position $\avg{x^2}$ in order to estimate $k$.
+
+To find $\avg{x^2}$, the raw photodiode voltages $V_p(t)$ are
+converted to distances $x(t)$ using the photodiode sensitivity
+$\sigma_p$ (the slope of the voltage vs.~distance curve of data taken
+while the tip is in contact with the surface) via
+\begin{equation}
+ x(t) = \frac{V_p(t)}{\sigma_p} \;. \label{eq:x-from-Vp}
+\end{equation}
+By keeping $V_p$ and $\sigma_p$ separate in our calculation of $k$, we
+can gauge the relative importance errors in each parameter and
+calculate the uncertainty in our estimated $k$.
+
+In order to filter out noise in the measured value of $\avg{V_p^2}$ we
+fit the measured cantilever deflection to the expected theoretical
+power spectral density ($\PSD_f$) of a damped harmonic oscillator
+exposed to thermal noise
+\nomenclature[PSD]{$\PSD_f$}{Power spectral density in
+ frequency space
+ \begin{equation}
+ \PSD_f(g, f) \equiv \normLimT 2 \magSq{ \Fourf{g(t)}(f) }
+ \end{equation}}
+\nomenclature{$f$}{Frequency (hertz)}
+\begin{equation}
+ \PSD_f(V_p, f) = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} \;.
+\end{equation}
+In terms of the fit parameters $G_{1f}$, $f_0$, and $\beta_f$,
+the expectation value for $V_p$ is given by
+\begin{equation}
+ \avg{V_p(t)^2} = \frac{\pi G_{1f}}{2\beta_f f_0^2} \;.
+ \label{eq:Vp-from-freq-fit}
+\end{equation}
+
+Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:Vp-from-freq-fit}, we
+have
+\begin{align}
+ k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
+\end{align}
+
+For a complete derivation of the procedure presented in this section,
+see \cref{sec:cantilever-calib}.
-\chapter{Apparatus}
-\label{sec:apparatus}
+\chapter{Mechanical Protein Unfolding via AFM}
+\label{sec:methods}
+
+In this chapter we will review the basic methods and procedures for
+mechanically unfolding proteins with an atomic force microscope. We
+review the working principle behind an AFM (\cref{sec:afm}) and
+outline the procedure for synthesizing protein chains
+(\cref{sec:polymer-synthesis}). With the groundwork out of the way,
+we will look at sample preparation (\cref{sec:sample-preparation}) and
+the velocity clamp force spectroscopy procedure
+(\cref{sec:procedure}). Finally, we will give an executive summary of
+cantilever calibration (\cref{sec:cantilever-calib}) which is
+discussed in more detail in \cref{sec:cantilever-calib}.
+
+Everything discussed in this chapter, with the possible exception of
+cantilever calibration, is fairly standard practice in the field of
+force spectroscopy. See \cref{sec:cantilever-calib} for the
+development of the cantilever calibration theory from first principles
+and references to related papers. More specialized techniques such as
+temperature control will be dealt with in their particular chapters
+(e.g. \cref{sec:temperature}).
+
+\input{apparatus/afm}
+\input{apparatus/polymer-synthesis}
+\input{apparatus/sample-preparation}
+\input{apparatus/procedure}
+\input{apparatus/cantilever-calib}
--- /dev/null
+\section{Protein Polymer Synthesis}
+\label{sec:polymer-synthesis}
+
+TODO.
--- /dev/null
+\section{Mechanical unfolding experiments}
+\label{sec:procedure}
+
+% AFM unfolding procedure
+In a mechanical unfolding experiment, a protein polymer is tethered
+between two surfaces: a flat substrate and an AFM tip. The polymer is
+stretched by increasing the separation between the two surfaces
+(\cref{fig:unfolding-schematic}). The most common mode is the
+constant speed experiment in which the substrate surface is moved away
+from the tip at a uniform rate. The tethering surfaces, \ie, the AFM
+tip and the substrate, have much larger radii of curvature than the
+dimensions of single domain globular proteins that are normally used
+for folding studies. This causes difficulties in manipulating
+individual protein molecules because nonspecific interactions between
+the AFM tip and the substrate may be stronger than the forces required
+to unfold the protein when the surfaces are a few nanometers apart.
+To circumvent these difficulties, globular protein molecules are
+linked into polymers, which are then used in the AFM
+studies\citep{carrion-vazquez99a,chyan04,carrion-vazquez03}. When
+such a polymer is pulled from its ends, each protein molecule feels
+the externally applied force, which increases the probability of
+unfolding by reducing the free energy barrier between the native and
+unfolded states. The unfolding of one molecule in the polymer causes
+a sudden lengthening of the polymer chain, which reduces the force on
+each protein molecule and prevents another unfolding event from
+occurring immediately. The force versus extension relationship, or
+\emph{force curve}, shows a typical sawtooth pattern
+(\cref{fig:expt-sawtooth}), where each peak corresponds to the
+unfolding of a single protein domain in the polymer. Therefore, the
+individual unfolding events are separated from each other in space and
+time, allowing single molecule resolution despite the use of
+multi-domain test proteins.
+
+\begin{figure}
+ \begin{center}
+ \subfloat[][]{\asyfig{figures/schematic/unfolding}%
+ \label{fig:unfolding-schematic}}
+ % \hspace{.25in}%
+ \subfloat[][]{\asyfig{figures/expt-sawtooth/expt-sawtooth}%
+ \label{fig:expt-sawtooth}}
+ \caption{(a) Schematic of the experimental setup for mechanical
+ unfolding of proteins using an AFM (not to scale). An experiment
+ starts with the tip in contact with the substrate surface, which
+ is then moved away from the tip at a constant speed. $x_t$ is the
+ distance traveled by the substrate, $x_c$ is the cantilever
+ deflection, $x_u$ is the extension of the unfolded polymer, and
+ $x_f=x_{f1}+x_{f2}$ is the extension of the folded polymer. (b)
+ An experimental force curve from stretching a ubiquitin polymer
+ with the rising parts of the peaks fitted to the WLC\index{WLC}
+ model (\cref{sec:tension:wlc})\citep{chyan04}. The pulling speed
+ used was $1\U{$\mu$m/s}$. The irregular features at the beginning
+ of the curve are due to nonspecific interactions between the tip
+ and the substrate surface, and the last high force peak is caused
+ by the detachment of the polymer from the tip or the substrate
+ surface. Note that the abscissa is the extension of the protein
+ chain $x_t-x_c$.}
+ \end{center}
+\end{figure}
--- /dev/null
+\section{Sample Preparation}
+\label{sec:sample-preparation}
+
-\section{Overview}
-
-In order to measure forces accurately with an Atomic Force Microscope (AFM),
-it is important to measure the cantilever spring constant.
-The force exerted on the cantilever can then be deduced from it's deflection
-via Hooke's law $F = -kx$.
-\nomenclature{$F$}{Force (newtons)}
-\nomenclature{$k$}{Spring constant (newtons per meter)}
-\nomenclature{$x$}{Displacement (meters)}
-
-The basic idea is to use the equipartition theorem\citep{hutter93},
-\begin{equation}
- \frac{1}{2} k \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart}
-\end{equation}
-where $k_B$ is Boltzmann's constant,
- $T$ is the absolute temperature, and
- $\avg{x^2}$ denotes the expectation value of $x^2$ as measured over a
- very long interval $t_T$,
-\nomenclature{$k_B$}{Boltzmann's constant, $k_B = 1.380 65\E{-23}\U{J/K}$\citep{codata-boltzmann}}
-\nomenclature{$\avg{s(t)}$}{Mean (expectation value) of a time-series $s(t)$}
-\begin{equation}
- \avg{A} \equiv \iLimT{A} \;.
-\end{equation}
-Solving the equipartition theorem for $k$ yields
-\begin{equation}
- k = \frac{k_BT}{\avg{x^2}} \;, \label{eq:equipart_k}
-\end{equation}
-so we need to measure (or estimate) the temperature $T$ and variance
-of the cantilever position $\avg{x^2}$ in order to estimate $k$.
+In this appendix, we derive the formulas presented in
+\cref{sec:cantilever-calib-intro} for calibrating an AFM cantilever.
+You should read that section to understand the goal of this appendix
+and familiarize yourself with the notation we will be using.
\subsection{Related papers}
uncertainty, we will leave \cref{eq:model-psd} unnamed.
\section{Methods}
+% TODO: deprecated in favor of sec:cantilever-calib:intro
-To find $\avg{x^2}$, the raw photodiode voltages $V_p(t)$ are
-converted to distances $x(t)$ using the photodiode sensitivity
-$\sigma_p$ (the slope of the voltage vs.~distance curve of data taken
-while the tip is in contact with the surface) via
-\begin{equation}
- x(t) = \frac{V_p(t)}{\sigma_p} \;.
-\end{equation}
Rather than computing the variance of $x(t)$ directly, we attempt to
filter out noise by fitting the power spectral density (\PSD)%
-\nomenclature[aPSD]{$\PSD$}{Power spectral density in angular
+\nomenclature[PSDa]{$\PSD$}{Power spectral density in angular
frequency space}\index{PSD@\PSD}\nomenclature{$\omega$}{Angular
frequency (radians per second)} of $x(t)$ to the theoretically
predicted \PSD\ for a damped harmonic oscillator (\cref{eq:model-psd})
&= 2\pi \PSD(x, \omega=2\pi f) \;.
\end{split}
\end{align}
-\nomenclature[aPSD]{$\PSD_f$}{Power spectral density in frequency space}
-\nomenclature{$f$}{Frequency (hertz)}
\nomenclature{$t$}{Time (seconds)}
\index{PSD@\PSD!in frequency space}
The variance of the function $x(t)$ is then given by plugging into
= \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
\label{eq:Gone-f}
\end{equation}
+
+
+% TODO: re-integrate the following
+
+% \begin{split}
+% \PSD_f(V_p, f) =
+% 2\pi\PSD(V_p,\omega)
+% = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
+% = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
+% &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
+% \end{split} \\
+
+% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
+% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
+
+%where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
+%$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
+
+%From \cref{eq:Gone}, we expect $G_{1f}$ to be
+%\begin{equation}
+% G_{1f} = \frac{G_{1p}}{8\pi^3}
+% = \frac{\sigma_p^2 G_1}{8\pi^3}
+% = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
+% = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
+% \label{eq:Gone-f}
+% \end{equation}
+
\nomenclature{I27}{Immunoglobulin-like domain 27 from human Titin}\index{I27}
I27 is a model protein that has been used in mechanical unfolding
experiments since the first use of synthetic
-chains\citep{carrion-vazquez99a,TODO}. It was used here because it is
+chains\citep{carrion-vazquez99b,TODO}. It was used here because it is
both well characterized and readily available (%
\href{http://www.athenaes.com/}{AthenaES}, Baltimore, MD,
\href{http://www.athenaes.com/I27OAFMReferenceProtein.php}{0304}).
and stiffer linkers will increase the mean unfolding force.
Unfolded I27 domains can be well-modeled as wormlike chains (WLCs,
-\cref{sec:tension:wlc})\citep{carrion-vazquez99a}, where $p \approx
+\cref{sec:tension:wlc})\citep{carrion-vazquez99b}, where $p \approx
4\U{\AA}$ is the persistence length, and $L \approx 28\U{nm}$ is the
contour length of the unfolded domain. Obviously effective stiffness
of an unfolded I27 domain is highly dependent on the unfolding force,
\chapter{Introduction}
\label{sec:intro}
-% why single molecule approach?
+\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. The large size of these molecules imposes certain
-limitations on 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}.
+processes in cells.
% What do genes do? Why is protein folding interesting?
An organism's genetic code is stored in DNA%
\end{center}
\end{figure}
+
+\section{Protein Folding Energy Landscapes}
+\label{sec:energy-landscape}
+
% the free energy landscape
Folding a protein via a brute force sampling of all possible
conformations is impossibly inefficient, due to the exponential
\end{center}
\end{figure}
+
+\section{Single Molecule Protein Folding Studies}
+\label{sec:single-molecule}
+
+The large size of proteins relative to simpler molecules imposes
+certain limitations on 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}.
+
% why AFM & what an AFM is
Single molecule techniques provide an opportunity to study protein
folding and unfolding at the level of a single molecule, where the
\nomenclature{AFM}{Atomic Force Microscope (or Microscopy)},
laser tweezers\citep{forde02}, magnetic tweezers\citep{smith92},
biomembrane force probes\citep{merkel99}, and centrifugal
-microscopes\citep{halvorsen09}. Of these mechanical manipulation
-methods, AFM is the most widely used due to the 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}. 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.
-
-% 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
-the unfolding and refolding processes of individual protein molecules
-by the AFM is feasible because many globular proteins unfold under
-external forces in this range. Since elucidating the mechanism of
-protein folding is currently one of the most important problems in
-biological sciences, the potential of the AFM for revealing
-significant and unique information about protein folding has
-stimulated much effort in both experimental and theoretical research.
-
-\begin{figure}
- \asyfig{figures/schematic/afm}%
- \caption{Operating principle for an Atomic Force
- Microscope\index{AFM}. A sharp tip integrated at the end of a
- cantilever interacts with the sample. Cantilever bending is
- measured by a laser reflected off the cantilever and incident on a
- position sensitive photodetector.\label{fig:afm-schematic}}
-\end{figure}
-
-
-\section{Mechanical unfolding experiments}
-
-% AFM unfolding procedure In a mechanical unfolding experiment, a
-protein polymer is tethered between two surfaces: a flat substrate and
-an AFM tip. The polymer is stretched by increasing the separation
-between the two surfaces (\cref{fig:unfolding-schematic}). The most
-common mode is the constant speed experiment in which the substrate
-surface is moved away from the tip at a uniform rate. The tethering
-surfaces, \ie, the AFM tip and the substrate, have much larger radii
-of curvature than the dimensions of single domain globular proteins
-that are normally used for folding studies. This causes difficulties
-in manipulating individual protein molecules because nonspecific
-interactions between the AFM tip and the substrate may be stronger
-than the forces required to unfold the protein when the surfaces are a
-few nanometers apart. To circumvent these difficulties, globular
-protein molecules are linked into polymers, which are then used in the
-AFM studies\citep{carrion-vazquez99a,chyan04,carrion-vazquez03}. When
-such a polymer is pulled from its ends, each protein molecule feels
-the externally applied force, which increases the probability of
-unfolding by reducing the free energy barrier between the native and
-unfolded states. The unfolding of one molecule in the polymer causes
-a sudden lengthening of the polymer chain, which reduces the force on
-each protein molecule and prevents another unfolding event from
-occurring immediately. The force versus extension relationship, or
-\emph{force curve}, shows a typical sawtooth pattern
-(\cref{fig:expt-sawtooth}), where each peak corresponds to the
-unfolding of a single protein domain in the polymer. Therefore, the
-individual unfolding events are separated from each other in space and
-time, allowing single molecule resolution despite the use of
-multi-domain test proteins.
-
-\begin{figure}
- \begin{center}
- \subfloat[][]{\asyfig{figures/schematic/unfolding}%
- \label{fig:unfolding-schematic}}
- % \hspace{.25in}%
- \subfloat[][]{\asyfig{figures/expt-sawtooth/expt-sawtooth}%
- \label{fig:expt-sawtooth}}
- \caption{(a) Schematic of the experimental setup for mechanical
- unfolding of proteins using an AFM (not to scale). An experiment
- starts with the tip in contact with the substrate surface, which
- is then moved away from the tip at a constant speed. $x_t$ is the
- distance traveled by the substrate, $x_c$ is the cantilever
- deflection, $x_u$ is the extension of the unfolded polymer, and
- $x_f=x_{f1}+x_{f2}$ is the extension of the folded polymer. (b)
- An experimental force curve from stretching a ubiquitin polymer
- with the rising parts of the peaks fitted to the WLC\index{WLC}
- model (\cref{sec:tension:wlc})\citep{chyan04}. The pulling speed
- used was $1\U{$\mu$m/s}$. The irregular features at the beginning
- of the curve are due to nonspecific interactions between the tip
- and the substrate surface, and the last high force peak is caused
- by the detachment of the polymer from the tip or the substrate
- surface. Note that the abscissa is the extension of the protein
- chain $x_t-x_c$.}
- \end{center}
-\end{figure}
+microscopes\citep{halvorsen09}.
\section{Thesis Outline}
-
-\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.
+\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.
@string{RSEvans = "Evans, R. S."}
@string{MEvstigneev = "Evstigneev, M."}
@string{DFasulo = "Fasulo, D."}
-@string{JMFernandez = "Fernandez, Julio M."}
+@string{JFernandez = "Fernandez, Julio M."}
@string{SFerriera = "Ferriera, S."}
@string{AEFilippov = "Filippov, A. E."}
@string{LFinzi = "Finzi, L."}
@string{KKawasaki = "Kawasaki, K."}
@string{ZKe = "Ke, Z."}
@string{AKejariwal = "Kejariwal, A."}
+@string{MKellermayer = "Kellermayer, M."}
+@string{MSKellermayer = "Kellermayer, M. S."}
@string{MSZKellermayer = "Kellermayer, Mikl\'os S. Z."}
@string{SKennedy = "Kennedy, S."}
@string{WJKent = "Kent, W. J."}
@string{CSmith = "Smith, Corey L."}
@string{DASmith = "Smith, D. Alastair"}
@string{HOSmith = "Smith, H. O."}
+@string{SSmith = "Smith, S."}
@string{SBSmith = "Smith, S. B."}
@string{TSmith = "Smith, T."}
@string{JSoares = "Soares, J."}
pages = "1236--1241",
issn = "0006-291X",
doi = "DOI: 10.1016/0006-291X(90)90526-S",
- url = "http://www.sciencedirect.com/science/article/B6WBK-4F5M7K3-3C/2/c94b612e06efc8534ee24bb1da889811",
+ url = "http://www.sciencedirect.com/science/article/B6WBK-
+ 4F5M7K3-3C/2/c94b612e06efc8534ee24bb1da889811",
keywords = "Amino Acid Sequence;Animals;Bacterial Proteins;Binding
Sites;Cell Line;Cell Membrane;Cricetinae;Fibronectins;Molecular
Sequence Data;Streptavidin",
}
@article { basche01,
- author = TBasche #" and "# SNie #" and "# JMFernandez,
+ author = TBasche #" and "# SNie #" and "# JFernandez,
title = "Single molecules",
year = 2001,
journal = PNAS,
title = "A simple method for probing the mechanical unfolding pathway of
proteins in detail",
year = 2002,
- month = sep,
- day = 17,
journal = PNAS,
volume = 99,
number = 19,
pages = "101--125",
issn = "0066-4154",
doi = "10.1146/annurev.biochem.77.060706.093102",
- eprint = "http://arjournals.annualreviews.org/doi/pdf/10.1146/annurev.biochem.77.060706.093102",
- url = "http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.biochem.77.060706.093102",
+ eprint = "http://arjournals.annualreviews.org/doi/pdf/10.1146/annurev.bioch
+ em.77.060706.093102",
+ url = "http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.biochem.
+ 77.060706.093102",
abstract = "Although protein-folding studies began several decades ago, it
is only recently that the tools to analyze protein folding at the
single-molecule level have been developed. Advances in single-molecule
pages = "45--50",
issn = "0066-4154",
doi = "10.1146/annurev.biochem.012108.120952",
- eprint = "http://arjournals.annualreviews.org/doi/pdf/10.1146/annurev.biochem.012108.120952",
- url = "http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.biochem.012108.120952",
+ eprint = "http://arjournals.annualreviews.org/doi/pdf/10.1146/annurev.bioch
+ em.012108.120952",
+ url = "http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.biochem.
+ 012108.120952",
abstract = "It has been over one-and-a-half decades since methods of
single-molecule detection and manipulation were first introduced in
biochemical research. Since then, the application of these methods to
}
@article { bustamante94,
- author = CBustamante #" and "# JFMarko #" and "# EDSiggia #" and "#
- SBSmith,
+ author = CBustamante #" and "# JFMarko #" and "# EDSiggia #" and "# SSmith,
title = "Entropic elasticity of lambda-phage {DNA}",
year = 1994,
month = sep,
@article { carrion-vazquez00,
author = MCarrionVazquez #" and "# AOberhauser #" and "# TEFisher #" and "#
- PMarszalek #" and "# HLi #" and "# JMFernandez,
+ PMarszalek #" and "# HLi #" and "# JFernandez,
title = "Mechanical design of proteins studied by single-molecule force
spectroscopy and protein engineering",
year = 2000,
number = "1-2",
pages = "63--91",
issn = "0079-6107",
- eprint = "http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1302160&blobtype=pdf",
+ eprint = "http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1302160&blo
+ btype=pdf",
url = "http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1302160",
keywords = "Elasticity;Hydrogen Bonding;Microscopy, Atomic Force;Protein
Denaturation;Protein Engineering;Protein Folding;Recombinant
@article { carrion-vazquez03,
author = MCarrionVazquez #" and "# HLi #" and "# HLu #" and "# PMarszalek
- #" and "# AOberhauser #" and "# JMFernandez,
+ #" and "# AOberhauser #" and "# JFernandez,
title = "The mechanical stability of ubiquitin is linkage dependent",
year = 2003,
month = sep,
}
@article { carrion-vazquez99a,
- author = MCarrionVazquez #" and "# AOberhauser #" and "# SFowler #" and "#
- PMarszalek #" and "# SBroedel #" and "# JClarke #" and "# JMFernandez,
- title = "Mechanical and chemical unfolding of a single protein: A
- comparison",
- year = 1999,
- month = mar,
- day = 30,
- journal = PNAS,
- volume = 96,
- number = 7,
- pages = "3694--3699",
- doi = "10.1073/pnas.96.7.3694",
- eprint = "http://www.pnas.org/cgi/reprint/96/7/3694.pdf",
- url = "http://www.pnas.org/cgi/content/abstract/96/7/3694"
-}
-
-@article { carrion-vazquez99b,
author = MCarrionVazquez #" and "# PMarszalek #" and "# AOberhauser #" and
- "# JMFernandez,
+ "# JFernandez,
title = "Atomic force microscopy captures length phenotypes in single
proteins",
year = 1999,
- month = sep,
- day = 28,
journal = PNAS,
volume = 96,
number = 20,
abstract = ""
}
+@article { carrion-vazquez99b,
+ author = MCarrionVazquez #" and "# AOberhauser #" and "# SFowler #" and "#
+ PMarszalek #" and "# SBroedel #" and "# JClarke #" and "# JFernandez,
+ title = "Mechanical and chemical unfolding of a single protein: A
+ comparison",
+ year = 1999,
+ journal = PNAS,
+ volume = 96,
+ number = 7,
+ pages = "3694--3699",
+ doi = "10.1073/pnas.96.7.3694",
+ eprint = "http://www.pnas.org/cgi/reprint/96/7/3694.pdf",
+ url = "http://www.pnas.org/cgi/content/abstract/96/7/3694"
+}
+
@article { chyan04,
author = CLChyan #" and "# FCLin #" and "# HPeng #" and "# JMYuan #" and "#
CHChang #" and "# SHLin #" and "# GYang,
pages = "105--128",
issn = "1056-8700",
doi = "10.1146/annurev.biophys.30.1.105",
- url = "http://arjournals.annualreviews.org/doi/abs/10.1146%2Fannurev.biophys.30.1.105",
+ url = "http://arjournals.annualreviews.org/doi/abs/10.1146%2Fannurev.biophy
+ s.30.1.105",
keywords = "Biophysics;Kinetics;Microscopy, Atomic Force;Models,
Chemical;Protein Binding;Spectrum Analysis;Time Factors",
abstract = "On laboratory time scales, the energy landscape of a weak bond
}
@article { fernandez04,
- author = JMFernandez #" and "# HLi,
+ author = JFernandez #" and "# HLi,
title = "Force-clamp spectroscopy monitors the folding trajectory of a
single protein",
year = 2004,
pages = "895--901",
issn = "0956-5663",
doi = "10.1016/0956-5663(95)99227-C",
- url = "http://www.sciencedirect.com/science/article/B6TFC-3XY2HK9-G/2/6f4e9f67e9a1e14c8bbcc478e5360682",
+ url = "http://www.sciencedirect.com/science/article/B6TFC-
+ 3XY2HK9-G/2/6f4e9f67e9a1e14c8bbcc478e5360682",
abstract = "One of the unique features of the atomic force microscope (AFM)
is its capacity to measure interactions between tip and sample with
high sensitivity and unparal leled spatial resolution. Since the
}
@article { granzier97,
- author = HLGranzier #" and "# MSZKellermayer #" and "# MHelmes #" and "#
+ author = HLGranzier #" and "# MKellermayer #" and "# MHelmes #" and "#
KTrombitas,
title = "Titin elasticity and mechanism of passive force development in rat
cardiac myocytes probed by thin-filament extraction",
pages = "3212--3237",
issn = "1521-3773",
doi = "10.1002/1521-3773(20000915)39:18<3212::AID-ANIE3212>3.0.CO;2-X",
- url = "http://dx.doi.org/10.1002/1521-3773(20000915)39:18<3212::AID-ANIE3212>3.0.CO;2-X",
+ eprint = "",
+ url = "http://dx.doi.org/10.1002/1521-3773(20000915)39:18<3212::AID-
+ ANIE3212>3.0.CO;2-X",
abstract = "How do molecules interact with each other? What happens if a
neurotransmitter binds to a ligand-operated ion channel? How do
antibodies recognize their antigens? Molecular recognition events play
pages = "355--361",
issn = "0301-679X",
doi = "DOI: 10.1016/j.triboint.2004.08.016",
- url = "http://www.sciencedirect.com/science/article/B6V57-4DN9K7J-1/2/fef91ac022594c2c6a701376d83ecd31",
+ url = "http://www.sciencedirect.com/science/article/B6V57-4DN9K7J-1/2/fef91
+ ac022594c2c6a701376d83ecd31",
keywords = "AFM;Liquid;Hydrodynamic;Lubrication",
abstract = "With the availability of equipment used in Scanning Probe
Microscopy (SPM), researchers have been able to probe the local fluid-
}
@article { kellermayer97,
- author = MSZKellermayer #" and "# SBSmith #" and "# HLGranzier #" and "#
+ author = MSKellermayer #" and "# SBSmith #" and "# HLGranzier #" and "#
CBustamante,
title = "Folding-unfolding transitions in single titin molecules
characterized with laser tweezers",
pages = "159--166",
issn = "0141-8130",
doi = "10.1016/j.ijbiomac.2009.12.001",
- url = "http://www.sciencedirect.com/science/article/B6T7J-4XWMND2-1/2/7ef768562b4157fc201d450553e5de5e",
+ url = "http://www.sciencedirect.com/science/article/B6T7J-
+ 4XWMND2-1/2/7ef768562b4157fc201d450553e5de5e",
keywords = "Atomic force microscopy;Mechanical unfolding;Monte Carlo
simulation;Worm-like chain;Single molecule methods",
abstract = "Single molecule methods are becoming routine biophysical
@article { li00,
author = HLi #" and "# AOberhauser #" and "# SFowler #" and "# JClarke #"
- and "# JMFernandez,
+ and "# JFernandez,
title = "Atomic force microscopy reveals the mechanical design of a modular
protein",
year = 2000,
@article { li01,
author = HLi #" and "# AOberhauser #" and "# SRedick #" and "#
- MCarrionVazquez #" and "# HErickson #" and "# JMFernandez,
+ MCarrionVazquez #" and "# HErickson #" and "# JFernandez,
title = "Multiple conformations of {PEVK} proteins detected by single-
molecule techniques",
year = 2001,
}
@article { li03,
- author = HLi #" and "# JMFernandez,
+ author = HLi #" and "# JFernandez,
title = "Mechanical design of the first proximal Ig domain of human cardiac
titin revealed by single molecule force spectroscopy",
year = 2003,
}
@article { li05,
- author = LLi #" and "# HHuang #" and "# CBadilla #" and "# JMFernandez,
+ author = LLi #" and "# HHuang #" and "# CBadilla #" and "# JFernandez,
title = "Mechanical unfolding intermediates observed by single-molecule
force spectroscopy in a fibronectin type {III} module",
year = 2005,
pages = "145--151",
issn = "0018-9448",
doi = "10.1109/18.61115",
- url = "http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=2227&arnumber=61115&count=35&index=9",
+ url = "http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=2227&arnumbe
+ r=61115&count=35&index=9",
keywords = "divergence;dissimilarity measure;discrimintation
information;entropy;probability of error bounds",
abstract = "A novel class of information-theoretic divergence measures
pages = "453--463",
issn = "0887-3585",
doi = "10.1002/(SICI)1097-0134(19990601)35:4<453::AID-PROT9>3.0.CO;2-M",
- eprint = "http://www3.interscience.wiley.com/cgi-bin/fulltext/65000328/PDFSTART",
+ eprint = "http://www3.interscience.wiley.com/cgi-
+ bin/fulltext/65000328/PDFSTART",
url = "http://www3.interscience.wiley.com/journal/65000328/abstract",
keywords = "Computer Simulation;Fibronectins;Hydrogen Bonding;Microscopy,
Atomic Force;Models, Molecular;Protein Denaturation",
pages = "9663--9673",
publisher = AIP,
doi = "10.1063/1.1369622",
- eprint = "http://hansmalab.physics.ucsb.edu/pdf/297%20-%20Makarov,%20D.E._J.Chem.Phys._2001.pdf",
+ eprint = "http://hansmalab.physics.ucsb.edu/pdf/297%20-%20Makarov,%20D.E._J
+ .Chem.Phys._2001.pdf",
url = "http://link.aip.org/link/?JCP/114/9663/1",
keywords = "proteins; hydrogen bonds; digital simulation; Monte Carlo
methods; molecular biophysics; intramolecular mechanics;
number = 26,
pages = "8759--8770",
issn = "0024-9297",
- eprint = "http://pubs.acs.org/cgi-bin/archive.cgi/mamobx/1995/28/i26/pdf/ma00130a008.pdf",
+ eprint = "http://pubs.acs.org/cgi-
+ bin/archive.cgi/mamobx/1995/28/i26/pdf/ma00130a008.pdf",
url =
- "http://pubs3.acs.org/acs/journals/doilookup?in_doi=10.1021/ma00130a008",
+ "http://pubs3.acs.org/acs/journals/doilookup?in_doi=10.1021/ma00130a008
+ ",
+ abstract = "",
note = "Derivation of the Worm-like Chain interpolation function."
}
@article { marszalek02,
author = PMarszalek #" and "# HLi #" and "# AOberhauser #" and "#
- JMFernandez,
+ JFernandez,
title = "Chair-boat transitions in single polysaccharide molecules observed
with force-ramp {AFM}",
year = 2002,
@article { marszalek99,
author = PMarszalek #" and "# HLu #" and "# HLi #" and "# MCarrionVazquez
- #" and "# AOberhauser #" and "# KSchulten #" and "# JMFernandez,
+ #" and "# AOberhauser #" and "# KSchulten #" and "# JFernandez,
title = "Mechanical unfolding intermediates in titin modules",
year = 1999,
month = nov,
@article { oberhauser01,
author = AOberhauser #" and "# PHansma #" and "# MCarrionVazquez #" and "#
- JMFernandez,
+ JFernandez,
title = "Stepwise unfolding of titin under force-clamp atomic force
microscopy",
year = 2001,
}
@article { rief97a,
+ author = MRief #" and "# MGautel #" and "# FOesterhelt #" and "# JFernandez
+ #" and "# HEGaub,
+ title = "Reversible Unfolding of Individual Titin Immunoglobulin Domains by
+ {AFM}",
+ year = 1997,
+ journal = SCI,
+ volume = 276,
+ number = 5315,
+ pages = "1109--1112",
+ doi = "10.1126/science.276.5315.1109",
+ eprint = "http://www.sciencemag.org/cgi/reprint/276/5315/1109.pdf",
+ url = "http://www.sciencemag.org/cgi/content/abstract/276/5315/1109",
+ note = "Seminal paper for force spectroscopy on Titin. Cited by
+ \citet{dietz04} (ref 9) as an example of how unfolding large proteins
+ is easily interpreted (vs.\ confusing unfolding in bulk), but Titin is
+ a rather simple example of that, because of its globular-chain
+ structure.",
+ project = "Energy Landscape Roughness"
+}
+
+@article { rief97b,
author = MRief #" and "# FOesterhelt #" and "# BHeymann #" and "# HEGaub,
title = "Single Molecule Force Spectroscopy on Polysaccharides by Atomic
Force Microscopy",
reversible and was corroborated by molecular dynamics calculations."
}
-@article { rief97b,
- author = MRief #" and "# MGautel #" and "# FOesterhelt #" and "#
- JMFernandez #" and "# HEGaub,
- title = "Reversible Unfolding of Individual Titin Immunoglobulin Domains by
- {AFM}",
- year = 1997,
- month = may,
- day = 16,
- journal = SCI,
- volume = 276,
- number = 5315,
- pages = "1109--1112",
- doi = "10.1126/science.276.5315.1109",
- eprint = "http://www.sciencemag.org/cgi/reprint/276/5315/1109.pdf",
- url = "http://www.sciencemag.org/cgi/content/abstract/276/5315/1109",
- note = "Seminal paper for force spectroscopy on Titin. Cited by
- \citet{dietz04} (ref 9) as an example of how unfolding large proteins
- is easily interpreted (vs.\ confusing unfolding in bulk), but Titin is
- a rather simple example of that, because of its globular-chain
- structure.",
- project = "Energy Landscape Roughness"
-}
-
@article { rief98,
- author = MRief #" and "# JMFernandez #" and "# HEGaub,
+ author = MRief #" and "# JFernandez #" and "# HEGaub,
title = "Elastically Coupled Two-Level Systems as a Model for Biopolymer
Extensibility",
year = 1998,
}
@article { sarkar04,
- author = ASarkar #" and "# RRobertson #" and "# JMFernandez,
+ author = ASarkar #" and "# RRobertson #" and "# JFernandez,
title = "Simultaneous atomic force microscope and fluorescence measurements
of protein unfolding using a calibrated evanescent wave",
year = 2004,
}
@article { sato05,
- author = TSato #" and "# MEsaki #" and "# JMFernandez #" and "# TEndo,
+ author = TSato #" and "# MEsaki #" and "# JFernandez #" and "# TEndo,
title = "{Comparison of the protein-unfolding pathways between
mitochondrial protein import and atomic-force microscopy measurements}",
year = 2005,
}
@article { schlierf04,
- author = MSchlierf #" and "# HLi #" and "# JMFernandez,
+ author = MSchlierf #" and "# HLi #" and "# JFernandez,
title = "The unfolding kinetics of ubiquitin captured with single-molecule
force-clamp techniques",
year = 2004,
@article { walther07,
author = KWalther #" and "# FGrater #" and "# LDougan #" and "# CBadilla #"
- and "# BBerne #" and "# JMFernandez,
+ and "# BBerne #" and "# JFernandez,
title = "Signatures of hydrophobic collapse in extended proteins captured
with force spectroscopy",
year = 2007,
}
@article { wiita06,
- author = AWiita #" and "# SAinavarapu #" and "# HHuang #" and "#
- JMFernandez,
+ author = AWiita #" and "# SAinavarapu #" and "# HHuang #" and "# JFernandez,
title = "From the Cover: Force-dependent chemical kinetics of disulfide
bond reduction observed with single-molecule techniques",
year = 2006,
pages = "20--22",
issn = "0027-8424",
eprint =
- "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC48166/pdf/pnas01075-0036.pdf",
+ "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC48166/pdf/pnas01075-0036.p
+ df",
url = "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC48166/",
keywords = "Mathematics;Models, Theoretical;Protein Conformation;Proteins",
abstract = "Levinthal's paradox is that finding the native folded state of
\begin{thesis}
\pdfbookmark[-1]{Mainmatter}{Mainmatter}
\include{introduction/main}
+\include{apparatus/main}
\include{unfolding/main}
-\include{tension/main}
-\include{unfolding-distributions/main}
\include{sawsim/main}
-\include{contour-space/main}
-\include{temperature-theory/main}
-\include{apparatus/main}
-\include{cantilever/main}
\include{temperature/main}
+\include{cantilever/main}
\include{future/main}
\end{thesis}
the pulling direction. The Monte Carlo simulation method has been
used since the first report of mechanical unfolding experiments using
the AFM%
-\citep{rief97a,rief97b,rief98,carrion-vazquez99a,best02,zinober02,jollymore09},
+\citep{rief97b,rief97a,rief98,carrion-vazquez99b,best02,zinober02,jollymore09},
but these previous implementations are neither fully described nor
publicly available.
+++ /dev/null
-\chapter{Temperature dependent unfolding theory}
-\label{sec:temperature-theory}
-
-\section{Energy landscape roughness}
-
-I'm skeptical about \HTeq{8} to \HTeq{9}, so I'll rework as much of
-their math as I am capable of\ldots
-
-\begin{multline*}
- \fs = \frac{\kT}{\dx} \Biggl[\Biggr.
- \logp{ \frac{\r \dx}{\kexp \kT} } \\
- + \logp{1 + \fs\frac{\dx'}{\dx} - \frac{\FO'}{\dx} + \frac{\vD'}{\vD}\cdot\frac{\kT}{\dx}} \\
- + \logp{\avg{e^{\bt F_1}}}^2
- \Biggl.\Biggr]
- \tag{\HTeq{8}}
-\end{multline*}
-
-We simplify by dropping the 2\nd term
-(``In obtaining Eq.~\textbf{9}, we have assumed that the second term in Eq.~\textbf{8} is small.''),
-and defining $\alpha \equiv \kT$,
- $\rho \equiv \logp{ \frac{\r \dx}{\kexp \kT} }$, and
- $e^{\bt \ep} \equiv \avg{e^{\bt F_1}}$, yielding
-\begin{equation}
- \fs = \frac{\alpha}{\dx} \left( \rho + \frac{\ep^2}{\alpha^2} \right)
-\end{equation}
-
-We obtain our version of \HTeq{9} by taking two measurements of equal mode force
-\begin{align}
- 0 &= \fs_1 - \fs_2 \\
- &= \frac{1}{\dx} \left( \alpha_1\rho_1 + \frac{\ep^2}{\alpha_1}
- -\alpha_2\rho_2 - \frac{\ep^2}{\alpha_2} \right) \\
- \ep^2\left(\frac{1}{\alpha_2} - \frac{1}{\alpha_1}\right) &= \alpha_1\rho_1 - \alpha_2\rho_2 \\
- \ep^2 \cdot \frac{\alpha_1 - \alpha_2}{\alpha_1\alpha_2} &= TODO\\
- \ep^2 &= \frac{\alpha_1\alpha_2}{\alpha_1 - \alpha_2} \left( \alpha_1\rho_1 - \alpha_2\rho_2 \right)\\
- \begin{split}\ep^2 &= \frac{\kT_1\kT_2}{\kT_1 - \kT_2} \Biggl[\Biggr.
- \kT_1\logp{\frac{\rs1\dxs1}{\kexps1 \kT_1}} \\
- &\qquad- \kT_2\logp{\frac{\rs2\dxs2}{\kexps2 \kT_2}}
- \Biggl.\Biggr]
- \end{split}
-\end{align}
-
-Which is different from \HTeq{9} by the sign in the prefactor, and the replacement $\vD \rightarrow \kf$.
-\begin{equation*}
- \ep^2 = \frac{\kT_1\kT_2}{\kT_2 - \kT_1} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\vDs1 \kT_1}}
- - \kT_2\logp{\frac{\rs2\dxs2}{\vDs2 \kT_2}} \right]
- \tag{\HTeq{9}}
-\end{equation*}
-
-Alternatively, noting that \dx can vary as a function of temperature, we follow \citet{nevo05} in keeping it in.
-Using $\delta \equiv \dx$
-\begin{align}
- 0 &= \fs_1 - \fs_2 \\
- &= \frac{\alpha_1\rho_1}{\delta_1} + \frac{\ep^2}{\delta_1\alpha_1}
- -\frac{\alpha_2\rho_2}{\delta_2} - \frac{\ep^2}{\delta_2\alpha_2} \\
- \ep^2\left(\frac{1}{\delta_2\alpha_2} - \frac{1}{\delta_1\alpha_1}\right)
- &= \frac{\alpha_1\rho_1}{\delta_1} - \frac{\alpha_2\rho_2}{\delta_2} \\
- \ep^2 \cdot \frac{\delta_1\alpha_1 - \delta_2\alpha_2}{\delta_1\delta_2\alpha_1\alpha_2}
- &= \frac{\delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2}{\delta_1\delta_2} \\
- \ep^2 &= \frac{\alpha_1\alpha_2}{\delta_1\alpha_1 - \delta_2\alpha_2}
- \left( \delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2 \right)\\
- \begin{split}\ep^2 &= \frac{\kT_1\kT_2}{\dxs1\kT_1 - \dxs2\kT_2}
- \Biggl[\Biggr. \dxs2\kT_1\logp{\frac{\rs1\dxs1}{\kfs1 \kT_1}} \\
- &\qquad - \dxs1\kT_2\logp{\frac{\rs2\dxs2}{\kfs2 \kT_2}} \Biggl.\Biggr]
- \end{split}
-\end{align}
--- /dev/null
+\section{Double-integral Kramers' theory}
+
+The double-integral form of overdamped Kramers' theory may be too
+complex for analytical predictions of unfolding-force histograms.
+Rather than testing the entire \sawsim\ simulation (\cref{sec:sawsim}),
+we will focus on demonstrating that the Kramers' $k(F)$ evaluations
+are working properly. If the Bell modeled histograms check out, that
+gives reasonable support for the $k(F) \rightarrow \text{histogram}$
+portion of the simulation.
+
+Looking for analytic solutions to Kramers' $k(F)$, we find that there
+are not many available in a closed form. However, we do have analytic
+solutions for unforced $k$ for cusp-like and quartic potentials.
+
+\subsection{Cusp-like potentials}
+
+
+\subsection{Quartic potentials}
+
--- /dev/null
+\section{Overview}
+
+For testing the \sawsim\ program, we need a few analytic solutions to unfolding distributions.
+We will start out discussing single-domain proteins under constant loading, and make some comments about multi-domain proteins and variable loading if we can make any progress in that direction.
+This note also functions as my mini-review article on unfolding theory, since
+I haven't been able to find an official one.
--- /dev/null
+\section{Review of current research}
+
+\citet{rief02} provide a general review of force spectroscopy with a short section on protein unfolding.
+There's not all that much information here, but it's a good place to go to get
+a big-picture overview before diving into the more technical papers.
+
+There are two main approaches to modeling protein domain unfolding under tension: Bell's and Kramers'\citep{schlierf06,dudko06,hummer03}.
+Bell introduced his model in the context of cell adhesion\citep{bell78}, but it has been widely used to model mechanical unfolding in proteins\citep{rief97a,carrion-vazquez99b,schlierf06} due to it's simplicity and ease of use\citep{hummer03}.
+Kramers introduced his theory in the context of thermally activated barrier crossings, which is how we use it here.
+
+There is an excellent review of Kramers' theory in \citet{hanggi90}.
+The bell model is generally considered too elementary to be worth a detailed review in this context, and yet I had trouble finding explicit probability densities that matched my own in Eqn.~\ref{eq:unfold:bell_pdf}.
+Properties of the Bell model recieve more coverage under the name of the older and equivalent Gompertz distribution\citep{gompertz25,olshansky97,wu04}.
+A warning about the ``Gompertz'' model is in order, because there seem to be at least two unfolding/dying rate formulas that go by that name.
+Compare, for example, \citet{braverman08} Eqn.~5 and \citet{juckett93} Fig.~2.
+
+\subsection{Who's who}
+
+The field of mechanical protein unfolding is developing along three main branches.
+Some groups are predominantly theoretical,
+\begin{itemize}
+ \item Evans, University of British Columbia (Emeritus) \\
+ \url{http://www.physics.ubc.ca/php/directory/research/fac-1p.phtml?entnum=55}
+ \item Thirumalai, University of Maryland \\
+ \url{http://www.marylandbiophysics.umd.edu/}
+ \item Onuchic, University of California, San Diego \\
+ \url{http://guara.ucsd.edu/}
+ \item Hyeon, Chung-Ang University (Onuchic postdoc, Thirumalai postdoc?) \\
+ \url{http://physics.chem.cau.ac.kr/} \\
+ \item Dietz (Rief grad) \\
+ \url{http://www.hd-web.de/}
+ \item Hummer and Szabo, National Institute of Diabetes and Digestive and Kidney Diseases \\
+ \url{http://intramural.niddk.nih.gov/research/faculty.asp?People_ID=1615}
+ \url{http://intramural.niddk.nih.gov/research/faculty.asp?People_ID=1559}
+\end{itemize}
+and the experimentalists are usually either AFM based
+\begin{itemize}
+ \item Rief, Technischen Universität München \\
+ \url{http://cell.e22.physik.tu-muenchen.de/gruppematthias/index.html}
+ \item Fernandez, Columbia University \\
+ \url{http://www.columbia.edu/cu/biology/faculty/fernandez/FernandezLabWebsite/}
+ \item Oberhauser, University of Texas Medical Branch (Fernandez postdoc) \\
+ \url{http://www.utmb.edu/ncb/Faculty/OberhauserAndres.html}
+ \item Marszalek, Duke University (Fernandez postdoc) \\
+ \url{http://smfs.pratt.duke.edu/homepage/lab.htm}
+ \item Guoliang Yang, Drexel University \\
+ \url{http://www.physics.drexel.edu/~gyang/}
+ \item Wojcikiewicz, University of Miami \\
+ \url{http://chroma.med.miami.edu/physiol/faculty-wojcikiewicz_e.htm}
+\end{itemize}
+or laser-tweezers based
+\begin{itemize}
+ \item Bustamante, University of California, Berkley \\
+ \url{http://alice.berkeley.edu/}
+ \item Forde, Simon Fraser University \\
+ \url{http://www.sfu.ca/fordelab/index.html}
+\end{itemize}
+
+\subsection{Evolution of unfolding modeling}
+
+Evans introduced the saddle-point Kramers' approximation in a protein unfolding context 1997 (\citet{evans97} Eqn.~3).
+However, early work on mechanical unfolding focused on the simper Bell model\citep{rief97a}.%TODO
+In the early `00's, the saddle-point/steepest-descent approximation to Kramer's model (\citet{hanggi90} Eqn.~4.56c) was introduced into our field\citep{dudko03,hyeon03}.%TODO
+By the mid `00's, the full-blown double-integral form of Kramer's model (\citet{hanggi90} Eqn.~4.56b) was in use\citep{schlierf06}.%TODO
+
+There have been some tangential attempts towards even fancier models.
+\citet{dudko03} attempted to reduce the restrictions of the single-unfolding-path model.
+\citet{hyeon03} attempted to measure the local roughness using temperature dependent unfolding.
+
+\subsection{History of simulations}
+
+Early molecular dynamics (MD) work on receptor-ligand breakage by Grubmuller 1996 and Izrailev 1997 (according to Evans 1997).
+\citet{evans97} introduce a smart Monte Carlo (SMC) Kramers' simulation.
+
+\subsection{History of experimental AFM unfolding experiments}
+
+\begin{itemize}
+ \item \citet{rief97a}:
+\end{itemize}
+
+\subsection{History of experimental laser tweezer unfolding experiments}
+
+\begin{itemize}
+ \item \citet{izrailev97}:
+\end{itemize}
--- /dev/null
+\section{Single-domain proteins under constant loading}
+
+Let $x$ be the end to end distance of the protein, $t$ be the time since loading began, $F$ be tension applied to the protein, $P$ be the surviving population of folded proteins.
+Make the definitions
+\begin{align}
+ v &\equiv \deriv{t}{x} && \text{the pulling velocity} \\
+ k &\equiv \deriv{x}{F} && \text{the loading spring constant} \\
+ P_0 &\equiv P(t=0) && \text{the initial number of folded proteins} \\
+ D &\equiv P_0 - P && \text{the number of dead (unfolded) proteins} \\
+ \kappa &\equiv -\frac{1}{P} \deriv{t}{P} && \text{the unfolding rate}
+\end{align}
+\nomenclature{$\equiv$}{Defined as (\ie equivalent to)}
+The proteins are under constant loading because
+\begin{equation}
+ \deriv{t}{F} = \deriv{x}{F}\deriv{t}{x} = kv\;,
+\end{equation}
+a constant, since both $k$ and $v$ are constant (\citet{evans97} in the text on the first page, \citet{dudko06} in the text just before Eqn.~4).
+
+The instantaneous likelyhood of a protein unfolding is given by $\deriv{F}{D}$, and the unfolding histogram is merely this function discretized over a bin of width $W$(This is similar to \citet{dudko06} Eqn.~2, remembering that $\dot{F}=kv$, that their probability density is not a histogram ($W=1$), and that their pdf is normalized to $N=1$).
+\begin{equation}
+ h(F) \equiv \deriv{\text{bin}}{F}
+ = \deriv{F}{D} \cdot \deriv{\text{bin}}{F}
+ = W \deriv{F}{D}
+ = -W \deriv{F}{P}
+ = -W \deriv{t}{P} \deriv{F}{t}
+ = \frac{W}{vk} P\kappa \label{eq:unfold:hist}
+\end{equation}
+Solving for theoretical histograms is merely a question of taking your chosen $\kappa$, solving for $P(f)$, and plugging into Eqn. \ref{eq:unfold:hist}.
+We can also make a bit of progress solving for $P$ in terms of $\kappa$ as follows:
+\begin{align}
+ \kappa &\equiv -\frac{1}{P} \deriv{t}{P} \\
+ -\kappa \dd t \cdot \deriv{t}{F} &= \frac{\dd P}{P} \\
+ \frac{-1}{kv} \int \kappa \dd F &= \ln(P) + c \\
+ P &= C\exp{\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})} \;, \label{eq:P}
+\end{align}
+where $c \equiv \ln(C)$ is a constant of integration scaling $P$.
+
+\subsection{Constant unfolding rate}
+
+In the extremely weak tension regime, the proteins' unfolding rate is independent of tension, we have
+\begin{align}
+ P &= C\exp{\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})}
+ = C\exp{\p({\frac{-1}{kv}\kappa F})}
+ = C\exp{\p({\frac{-\kappa F}{kv}})} \\
+ P(0) &\equiv P_0 = C\exp(0) = C \\
+ h(F) &= \frac{W}{vk} P \kappa
+ = \frac{W\kappa P_0}{vk} \exp{\p({\frac{-\kappa F}{kv}})}
+\end{align}
+So, a constant unfolding-rate/hazard-function gives exponential decay.
+Not the most earth shattering result, but it's a comforting first step, and it does show explicitly the dependence in terms of the various unfolding-specific parameters.
+
+\subsection{Bell model}
+
+Stepping up the intensity a bit, we come to Bell's model for unfolding
+(\citet{hummer03} Eqn.~1 and the first paragraph of \citet{dudko06} and \citet{dudko07}).
+\begin{equation}
+ \kappa = \kappa_0 \cdot \exp\p({\frac{F \dd x}{k_B T}})
+ = \kappa_0 \cdot \exp(a F) \;,
+\end{equation}
+where we've defined $a \equiv \dd x/k_B T$ to bundle some constants together.
+The unfolding histogram is then given by
+\begin{align}
+ P &= C\exp\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})
+ = C\exp\p[{\frac{-1}{kv} \frac{\kappa_0}{a} \exp(a F)}]
+ = C\exp\p[{\frac{-\kappa_0}{akv}\exp(a F)}] \\
+ P(0) &\equiv P_0 = C\exp\p({\frac{-\kappa_0}{akv}}) \\
+ C &= P_0 \exp\p({\frac{\kappa_0}{akv}}) \\
+ P &= P_0 \exp\p\{{\frac{\kappa_0}{akv}[1-\exp(a F)]}\} \\
+ h(F) &= \frac{W}{vk} P \kappa
+ = \frac{W}{vk} P_0 \exp\p\{{\frac{\kappa_0}{akv}[1-\exp(a F)]}\} \kappa_0 \exp(a F)
+ = \frac{W\kappa_0 P_0}{vk} \exp\p\{{a F + \frac{\kappa_0}{akv}[1-\exp(a F)]}\} \label{eq:unfold:bell_pdf}\;.
+\end{align}
+The $F$ dependent behavior reduces to
+\begin{equation}
+ h(F) \propto \exp\p[{a F - b\exp(a F)}] \;,
+\end{equation}
+where $b \equiv \kappa_0/akv \equiv \kappa_0 k_B T / k v \dd x$ is
+another constant rephrasing.
+
+This looks similar to the Gompertz / Gumbel / Fisher-Tippett
+distribution, where
+\begin{align}
+ p(x) &\propto z\exp(-z) \\
+ z &\equiv \exp\p({-\frac{x-\mu}{\beta}}) \;,
+\end{align}
+but we have
+\begin{equation}
+ p(x) \propto z\exp(-bz) \;.
+\end{equation}
+Strangely, the Gumbel distribution is supposed to derive from an
+exponentially increasing hazard function, which is where we started
+for our derivation. I haven't been able to find a good explaination
+of this discrepancy yet, but I have found a source that echos my
+result (\citet{wu04} Eqn.~1). TODO: compare \citet{wu04} with
+my successful derivation in \cref{sec:sawsim:results-scaffold}.
+
+Oh wait, we can do this:
+\begin{equation}
+ p(x) \propto z\exp(-bz) = \frac{1}{b} z'\exp(-z')\propto z'\exp(-z') \;,
+\end{equation}
+with $z'\equiv bz$. I feel silly... From
+\href{Wolfram}{http://mathworld.wolfram.com/GumbelDistribution.html},
+the mean of the Gumbel probability density
+\begin{equation}
+ P(x) = \frac{1}{\beta} \exp\p[{\frac{x-\alpha}{\beta}
+ -\exp\p({\frac{x-\alpha}{\beta}})
+ }]
+\end{equation}
+is given by $\mu=\alpha-\gamma\beta$, and the variance is
+$\sigma^2=\frac{1}{6}\pi^2\beta^2$, where $\gamma=0.57721566\ldots$ is
+the Euler-Mascheroni constant. Selecting $\beta=1/a=k_BT/\dd x$,
+$\alpha=-\beta\ln(\kappa\beta/kv)$, and $F=x$ we have
+\begin{align}
+ P(F)
+ &= \frac{1}{\beta} \exp\p[{\frac{F+\beta\ln(\kappa\beta/kv)}{\beta}
+ -\exp\p({\frac{F+\beta\ln(\kappa\beta/kv)}
+ {\beta}})
+ }] \\
+ &= \frac{1}{\beta} \exp(F/\beta)\exp[\ln(\kappa\beta/kv)]
+ \exp\p\{{-\exp(F/\beta)\exp[\ln(\kappa\beta/kv)]}\} \\
+ &= \frac{1}{\beta} \frac{\kappa\beta}{kv} \exp(F/\beta)
+ \exp\p[{-\kappa\beta/kv\exp(F/\beta)}] \\
+ &= \frac{\kappa}{kv} \exp(F/\beta)\exp[-\kappa\beta/kv\exp(F/\beta)] \\
+ &= \frac{\kappa}{kv} \exp(F/\beta - \kappa\beta/kv\exp(F/\beta)] \\
+ &= \frac{\kappa}{kv} \exp(aF - \kappa/akv\exp(aF)] \\
+ &= \frac{\kappa}{kv} \exp(aF - b\exp(aF)]
+ \propto h(F) \;.
+\end{align}
+So our unfolding force histogram for a single Bell domain under
+constant loading does indeed follow the Gumbel distribution.
+
+\subsection{Saddle-point Kramers' model}
+
+For the saddle-point approximation for Kramers' model for unfolding
+(\citet{evans97} Eqn.~3, \citet{hanggi90} Eqn. 4.56c, \citet{vanKampen07} Eqn. XIII.2.2).
+\begin{equation}
+ \kappa = \frac{D}{l_b l_{ts}} \cdot \exp\p({\frac{-E_b(F)}{k_B T}}) \;,
+\end{equation}
+where $E_b(F)$ is the barrier height under an external force $F$,
+$D$ is the diffusion constant of the protein conformation along the reaction coordinate,
+$l_b$ is the characteristic length of the bound state $l_b \equiv 1/\rho_b$,
+$\rho_b$ is the density of states in the bound state, and
+$l_{ts}$ is the characteristic length of the transition state
+\begin{equation}
+ l_{ts} = TODO
+\end{equation}
+
+\citet{evans97} solved this unfolding rate for both inverse power law potentials and cusp potentials.
+
+\subsubsection{Inverse power law potentials}
+
+\begin{equation}
+ E(x) = \frac{-A}{x^n}
+\end{equation}
+(e.g. $n=6$ for a van der Waals interaction, see \citet{evans97} in
+the text on page 1544, in the first paragraph of the section
+\emph{Dissociation under force from an inverse power law attraction}).
+Evans then goes into diffusion constants that depend on the
+protein's end to end distance, and I haven't worked out the math
+yet. TODO: clean up.
+
+
+\subsubsection{Cusp potentials}
+
+\begin{equation}
+ E(x) = \frac{1}{2}\kappa_a \p({\frac{x}{x_a}})^2
+\end{equation}
+(see \citet{evans97} in the text on page 1545, in the first paragraph
+of the section \emph{Dissociation under force from a deep harmonic well}).
--- /dev/null
+\section{Theoretical unfolding force distributions}
+\label{sec:unfolding-distributions}
+
+\input{unfolding/distributions-overview}
+\input{unfolding/distributions-review}
+\input{unfolding/distributions-single_domain-constant_loading}
+\input{unfolding/distributions-kramers}
\chapter{Unfolding Theory}
\label{sec:unfolding}
+
+\input{unfolding/rate}
+\input{unfolding/tension}
+\input{unfolding/distributions}
--- /dev/null
+\subsection{Bell model}
+\label{sec:rate:bell}
--- /dev/null
+\subsection{Kramers' saddle-point approximation}
+\label{sec:rate:kramers-saddle}
--- /dev/null
+\subsection{Kramers' double integral model}
+\label{sec:rate:kramers}
--- /dev/null
+\subsection{Stiffness-corrected Bell model}
+\label{sec:rate:stiff-bell}
--- /dev/null
+\section{Single-domain unfolding rates}
+\label{sec:unfolding-distributions}
+
+\input{unfolding/rate-overview}
+\input{unfolding/rate-bell}
+\input{unfolding/rate-stiff-bell}
+\input{unfolding/rate-kramers}
+\input{unfolding/rate-kramers-saddle}
--- /dev/null
+\subsection{Freely-jointed chains}
+\label{sec:tension:fjc}
--- /dev/null
+\subsection{Folded domain tension}
+\label{sec:tension:folded}
+
+The unfolded polypeptide chain has been shown to follow the
+WLC\index{WLC} model quite well (\cref{sec:tension:wlc}), though other
+polymer models, such as the Freely-Jointed Chain
+(FJC)\citep{verdier70}\index{FJC}\nomenclature{FJC}{Freely-Jointed Chain}
+(\cref{sec:tension:fjc}), can be used to fit the force-extension
+relationship\citep{janshoff00}. A short chain of folded proteins,
+however, cannot be described well by polymer models. Several studies
+have used WLC and FJC models to fit the elastic properties of the
+modular protein titin\citep{granzier97,linke98a},
+% TODO: check it really is folded domains \& bulk titin
+but native titin contains hundreds of folded and unfolded domains.
+For the short protein polymers common in mechanical unfolding
+experiments, the cantilever dominates the elasticity of the
+polymer-cantilever system before any protein molecules unfold. After
+the first unfolding event occurs, the unfolded portion of the chain is
+already longer and softer than the sum of all the remaining folded
+domains, and dominates the elasticity of the whole chain. Therefore,
+the details of the tension model chosen for the folded domains has
+negligible effect on the unfolding forces, which was also suggested by
+\citet{staple08}. Force curves simulated using different models to
+describe the folded domains yielded almost identical unfolding force
+distributions (data not shown, TODO: show data).
--- /dev/null
+\subsection{Wormlike chains}
+\label{sec:tension:wlc}
+
+The unfolded forms of many domains can be modeled as Worm-Like Chains
+(WLCs)\citep{marko95,bustamante94}
+\index{WLC}\nomenclature{WLC}{Wormlike Chain}, which treats the
+unfolded polymer as an elastic rod of persistence length $p$ and
+contour length $L$. The relationship between tension $F$ and
+extension (end-to-end distance) $x$ is given to within XX\% by
+Bustamante's interpolation formula\citep{marko95,bustamante94}.
+\begin{equation}
+ F_\text{WLC}(x,p,L) = \frac{k_B T}{p_u}
+ \p[{ \frac{1}{4}\p({ \frac{1}{(1-x/L)^2} - 1 })
+ + \frac{x}{L} }] \;,
+ \label{eq:sawsim:wlc}
+\end{equation}
+where $p$ is the persistence length.
+
+For chain with $N_u$ unfolded domains sharing a persistence length
+$p_u$ and per-domain contour lengths $L_{u1}$, the tension of the WLC
+is determine by summing the contour lengths
+\begin{equation}
+ F(x, p_u, L_u, N_u) = F_\text{WLC}(x, p_u, N_uL_{u1})
+\end{equation}
--- /dev/null
+\section{Chain Tension}
+\label{sec:tension}
+
+unfolded domains: polymer models (WLC, FJC)
+
+The particular model used for folded domains is of negligable
+importance in modeling chains with small numbers of domains, as
+discussed in \cref{sec:tension:folded}.
+
+\input{unfolding/tension-wlc}
+\input{unfolding/tension-fjc}
+\input{unfolding/tension-folded}
projects / thesis.git / commitdiff