where $x$ is the perpendicular displacement of the cantilever tip
($x_c$ in \cref{fig:unfolding-schematic}).
%
-\nomenclature{$F$}{Force (newtons)}
-\nomenclature{$\kappa$}{Spring constant (newtons per meter)}
-\nomenclature{$x$}{Displacement (meters)}
+\nomenclature[sr ]{$F$}{Force (newtons).}
+\nomenclature[sg k ]{$\kappa$}{Spring constant (newtons per meter).}
+\nomenclature[sr ]{$x$}{Displacement (meters).}
The basic idea is to use the equipartition theorem, which gives the
thermal energy per degree of freedom. For a simple harmonic
and $\avg{x^2}$ is the average value of $x^2$ measured
over a long time interval.
%
-\nomenclature{$k_B$}{Boltzmann's constant,
- $k_B = 1.380 65\E{-23}\U{J/K}$\cite{codata-boltzmann}}
-\nomenclature{$T$}{Absolute temperature (Kelvin)}
-\nomenclature{$\avg{s(t)}$}{Mean (expectation value) of a time-series $s(t)$
+\nomenclature[sr ]{$k_B$}{Boltzmann's constant,
+ $k_B = 1.380 65\E{-23}\U{J/K}$\citep{codata-boltzmann}.}
+\nomenclature[sr ]{$T$}{Absolute temperature (Kelvin).}
+\nomenclature[o ]{$\avg{s(t)}$}{Mean (expectation value) of a
+ time-series $s(t)$
\begin{equation}
\avg{A} \equiv \iLimT{A} \;.
\end{equation}}
-\nomenclature{$\equiv$}{Defined as (\ie\ equivalent to)}
+\nomenclature[o ]{$\equiv$}{Defined as (\ie\ equivalent to).}
To calculate the spring constant $\kappa$ using \cref{eq:equipart}, we
need to measure the buffer temperature $T$ and the thermal vibration
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
+eight repeats of immunoglobulin-like domain 27 (I27\index{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}.
-\index{I27}
-\nomenclature{I27}{Immunoglobulin-like domain 27 from human titin}
+%
+\nomenclature[text ]{I27}{Immunoglobulin-like domain 27 from human
+ titin.}
\begin{figure}
\includegraphics[width=2in]{figures/i27/1TIT}
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}
+\nomenclature[text ]{cDNA}{Complementary DNA.}
+\nomenclature[text ]{PCR}{Polymerase chain reaction.}
\begin{center}
Met-Arg-Gly-Ser-(His)$_6$-Gly-Ser-(I27-Arg-Ser)$_7$-I27-\ldots-Cys-Cys
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}
+%
+\nomenclature[text ]{Bacterial transformation}{The process by which
+ bacterial cells take up exogenous DNA molecules.}
+\nomenclature[text ]{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
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[text ]{IMAC}{Immobilized metal ion affinity
+ chromatography.}
+\nomenclature[text ]{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}
+\nomenclature[text ]{Ala}{Alanine, an amino acid.}
+\nomenclature[text ]{Arg}{Arginine, an amino acid.}
+\nomenclature[text ]{Asn}{Asparagine, an amino acid.}
+\nomenclature[text ]{Asp}{Aspartic acid, an amino acid.}
+\nomenclature[text ]{Cys}{Cystine, an amino acid.}
+\nomenclature[text ]{Glu}{Glutamine, an amino acid.}
+\nomenclature[text ]{Gly}{Glycine, an amino acid.}
+\nomenclature[text ]{His}{Histidine, an amino acid.}
+\nomenclature[text ]{Ile}{Isoleucine, an amino acid.}
+\nomenclature[text ]{Leu}{Leucine, an amino acid.}
+\nomenclature[text ]{Lys}{Lysine, an amino acid.}
+\nomenclature[text ]{Met}{Methionine, an amino acid.}
+\nomenclature[text ]{Phe}{Phenylalanine, an amino acid.}
+\nomenclature[text ]{Pro}{Proline, an amino acid.}
+\nomenclature[text ]{Ser}{Serine, an amino acid.}
+\nomenclature[text ]{Thr}{Threonine, an amino acid.}
+\nomenclature[text ]{Trp}{Tryptophan, an amino acid.}
+\nomenclature[text ]{Tyr}{Tyrosine, an amino acid.}
+\nomenclature[text ]{Val}{Valine, an amino acid.}
time, allowing single molecule resolution despite the use of
multi-domain test proteins.
%
-\nomenclature{force curve}{Or force--distance curve. Cantilever-force
- versus piezo extension data aquired during a force spectroscopy
- experiment (\cref{fig:expt-sawtooth}).}
+\nomenclature[text ]{force curve}{Or force--distance curve.
+ Cantilever-force versus piezo extension data aquired during a force
+ spectroscopy experiment (\cref{fig:expt-sawtooth}).}
\begin{figure}
\begin{center}
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}
+\nomenclature[text ]{PBS}{Phosphate buffered saline.}
As an alternative to binding proteins to gold, others have used
EGTA\citep{kellermayer03},
the protein\citep{lee05}. Of these, a Ni-NTA coating is the most
popular\citep{schmitt00}.
%
-\nomenclature{EGTA}{Ethylene glycol tetraacetic acid}
+\nomenclature[text ]{EGTA}{Ethylene glycol tetraacetic acid}
techniques between labs, which slows progress. In some cases it can
also lead to ambiguity as to which of several similar approaches,
correction factors, etc.\ were used in a particular paper.
-\nomenclature{SMFS}{Single molecule force spectroscopy}
+%
+\nomenclature[text ]{SMFS}{Single molecule force spectroscopy.}
In this thesis, I introduce an SMFS sofware suite for cantilever
calibration (\calibcant), experiment control (\pyafm), analysis
able to swap in analogous low level physical-interface packages if
Linux is not an option.
%
-\nomenclature{OS}{Operating system}
+\nomenclature[text ]{OS}{Operating system.}
+
\end{abstract}
damping term $\beta_f$. For overdamped cantilevers with large values
of $\beta$, the peak frequency will not have a real solution.%
%
-\nomenclature{$f_\text{max}$}{The frequency of the peak power in
+\nomenclature[sr ]{$f_\text{max}$}{The frequency of the peak power in
$\PSD_f$ (\cref{eq:peak-frequency}).}
\subsection{Propagation of errors}
calculate the uncertainty in our estimated $\kappa$
(\cref{sec:calibcant:discussion:errors}).
%
-\nomenclature{$V_p$}{The vertical photodiode deflection voltage
+\nomenclature[sr ]{$V_p$}{The vertical photodiode deflection voltage
(\cref{fig:afm-schematic,eq:x-from-Vp}).}
-\nomenclature{$\sigma_p$}{The linear photodiode sensitivity to
+\nomenclature[sg s_p ]{$\sigma_p$}{The linear photodiode sensitivity to
cantilever displacement (\cref{fig:afm-schematic,eq:x-from-Vp}).}
In order to filter out noise in the measured value of $\avg{V_p^2}$ we
\label{eq:avg-Vp-Gone-f}
\end{equation}
%
-\nomenclature[PSDf]{$\PSD_f$}{Power spectral density in
+\nomenclature[o PSDf ]{$\PSD_f$}{Power spectral density in
frequency space
\begin{equation}
- \PSD_f(g, f) \equiv \normLimT 2 \magSq{ \Fourf{g(t)}(f) }
+ \PSD_f(g, f) \equiv \normLimT 2 \magSq{ \Fourf{g(t)}(f) } \;.
\end{equation}}
-\nomenclature{$f$}{Frequency (hertz)}
-\nomenclature{$f_0$}{Resonant frequency (hertz)}
-\nomenclature{$\pi$}{Archmides' constant, $\pi=3.14159\ldots$. The
- ratio of a circle's circumference to its diameter.}
+\nomenclature[sr ]{$f$}{Frequency (hertz).}
+\nomenclature[sr ]{$f_0$}{Resonant frequency (hertz).}
+\nomenclature[sg p ]{$\pi$}{Archmides' constant, $\pi=3.14159\ldots$.
+ The ratio of a circle's circumference to its diameter.}
Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:avg-Vp-Gone-f}, we
have
During the non-contact phase of calibration,
$F(t)$ comes from random thermal noise.
%
-\nomenclature{$m$}{Effective mass of a damped harmonic oscillator
+\nomenclature[sr ]{$m$}{Effective mass of a damped harmonic oscillator
(\cref{eq:DHO}).}
-\nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient
- $F_\text{drag} = \gamma\dt{x}$ (\cref{eq:DHO}).}
-\nomenclature{$\dt{s}$}{First derivative of the time-series $s(t)$
- with respect to time. $\dt{s} = \deriv{t}{s}$}
-\nomenclature{$\ddt{s}$}{Second derivative of the time-series $s(t)$
- with respect to time. $\ddt{s} = \nderiv{2}{t}{s}$}
+\nomenclature[sg c ]{$\gamma$}{Damped harmonic oscillator drag
+ coefficient $F_\text{drag} = \gamma\dt{x}$ (\cref{eq:DHO}).}
+\nomenclature[o d1 ]{$\dt{s}$}{First derivative of the time-series
+ $s(t)$ with respect to time. $\dt{s} = \deriv{t}{s}$.}
+\nomenclature[o d2 ]{$\ddt{s}$}{Second derivative of the time-series
+ $s(t)$ with respect to time. $\ddt{s} = \nderiv{2}{t}{s}$.}
In the following analysis, we use the unitary, angular frequency
Fourier transform normalization
where $\omega$ is the angular frequency and $i\equiv\sqrt{-1}$ is the
imaginary unit.
%
-\nomenclature{\Four{s(t)}}{Fourier transform of the time-series
+\nomenclature[o F ]{\Four{s(t)}}{Fourier transform of the time-series
$s(t)$.
$s(f) = \Four{s(t)}
\equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{s(t) e^{-i \omega t}}$.
}\index{Fourier transform}
-\nomenclature{$i$}{Imaginary unit $i\equiv\sqrt{-1}$.}
-\nomenclature{$\omega$}{Angular frequency (radians per second).}
+\nomenclature[sr ]{$i$}{Imaginary unit $i\equiv\sqrt{-1}$.}
+\nomenclature[sg z ]{$\omega$}{Angular frequency (radians per second).}
We also use the following theorems (proved elsewhere):
\begin{align}
\end{align}
where $t_T$ is the total time over which data has been aquired.
%
-\nomenclature[PSDo]{$\PSD$}{Power spectral density in angular
+\nomenclature[o PSDo ]{$\PSD$}{Power spectral density in angular
frequency space
\begin{equation}
- \PSD(g, w) \equiv \normLimT 2 \magSq{ \Four{g(t)}(\omega) }
+ \PSD(g, w) \equiv \normLimT 2 \magSq{ \Four{g(t)}(\omega) } \;.
\end{equation}}
-\nomenclature{$\abs{z}$}{Absolute value (or magnitude) of $z$. For
+\nomenclature[o ]{$\abs{z}$}{Absolute value (or magnitude) of $z$. For
complex $z$, $\abs{z}\equiv\sqrt{z\conj{z}}$.}
We also use the Wiener--Khinchin theorem,
\end{align}
and $\conj{x}$ represents the complex conjugate of $x$.
%
-\nomenclature{$S_{xx}(\omega)$}{Two sided power spectral density in
- angular frequency space (\cref{eq:wiener_khinchin}).}
-\nomenclature{$r_{xx}(t)$}{Autocorrelation function
+\nomenclature[o ]{$S_{xx}(\omega)$}{Two sided power spectral density
+ in angular frequency space (\cref{eq:wiener_khinchin}).}
+\nomenclature[o ]{$r_{xx}(t)$}{Autocorrelation function
(\cref{eq:autocorrelation}).}
-\nomenclature{$\conj{z}$}{Complex conjugate of $z$}
+\nomenclature[o ]{$\conj{z}$}{Complex conjugate of $z$.}
\subsection{Highly damped case}
\label{sec:calibcant:ODHO}
= \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero
\end{equation}
%
-\nomenclature{$G_0$}{The power spectrum of the thermal noise in
+\nomenclature[sr ]{$G_0$}{The power spectrum of the thermal noise in
angular frequency space (\cref{eq:GOdef}).}
Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd-F} we have
drag-acceleration coefficient.\index{Damped harmonic
oscillator}\index{$\gamma$}\index{$\kappa$}\index{$\beta$}
%
-\nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration
- coefficient $\beta \equiv \gamma/m$ (\cref{eq:DHO-xmag}).}
-\nomenclature{$\omega_0$}{Resonant angular frequency (radians per
- second, \cref{eq:DHO-xmag}).}
+\nomenclature[sg b ]{$\beta$}{Damped harmonic oscillator
+ drag-acceleration coefficient $\beta \equiv \gamma/m$
+ (\cref{eq:DHO-xmag}).}
+\nomenclature[sg z0 ]{$\omega_0$}{Resonant angular frequency (radians
+ per second, \cref{eq:DHO-xmag}).}
We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into
\cref{eq:psd-def}
\label{eq:avg-Vp-Gone}
\end{align}
%
-\nomenclature{$G_1$}{The scaled power spectrum of the thermal noise in
- angular frequency space (\cref{eq:Gone-def}).}
+\nomenclature[sr ]{$G_1$}{The scaled power spectrum of the thermal
+ noise in angular frequency space (\cref{eq:Gone-def}).}
Plugging into the equipartition theorem (\cref{eq:equipart_k}) yields
\begin{align}
&= 2\pi \PSD(x, \omega=2\pi f) \;.
\end{split}
\end{align}
-\nomenclature{$t$}{Time (seconds)}
+%
+\nomenclature[sr ]{$t$}{Time (seconds).}
\index{PSD@\PSD!in frequency space}
+%
The variance of the function $x(t)$ is then given by plugging into
\cref{eq:parseval-var} (our corollary to Parseval's theorem)
\begin{align}
\label{eq:Q}
\end{equation}
%
-\nomenclature{$Q$}{Quality factor of a damped harmonic oscillator.
- $Q\equiv \frac{\sqrt{\kappa m}}{\gamma}$ (\cref{eq:Q}).}
+\nomenclature[sr ]{$Q$}{Quality factor of a damped harmonic
+ oscillator. $Q\equiv \frac{\sqrt{\kappa m}}{\gamma}$
+ (\cref{eq:Q}).}
% TODO: re-integrate the following
\section{Contour integration}
-As a brief review, some definite integrals from $-\infty$ to $\infty$%
-\nomenclature{$\infty$}{Infinity} can be evaluated by integrating
-along the contour \C\ shown in \cref{fig:UHP-contour}.
+As a brief review, some definite integrals from $-\infty$ to $\infty$
+can be evaluated by integrating along the contour \C\ shown in
+\cref{fig:UHP-contour}.
+%
+\nomenclature[o ]{$\infty$}{Infinity}
\begin{figure}
\asyinclude{figures/contour/contour}
\subsection{General case integral}
\label{sec:integrals:general}
-We will show that, for any $(a,b > 0) \in \Reals$,%
-\nomenclature[aR]{\Reals}{Real numbers}
+We will show that, for any $(a,b > 0) \in \Reals$,
\begin{equation}
I = \iInfInf{z}{\frac{1}{(a^2-z^2)^2 + b^2 z^2}} = \frac{\pi}{b a^2} \;.
\end{equation}
+%
+\nomenclature[so aR ]{\Reals}{Real numbers.}
First we note that $\abs{f(z)} \rightarrow 0$ like $\abs{z^{-4}}$ for
$\abs{z} \gg 1$, and that $f(z)$ is even, so
dependencies---is open source, so other labs are free to use, improve,
and republish it as they see fit.
%
-\nomenclature{tarball}{A single file containing a collection of files
- and directories. Created by
+\nomenclature[text ]{tarball}{A single file containing a collection of
+ files and directories. Created by
\href{http://www.gnu.org/software/tar/}{Tar}, tarballs were
originally used for tape archives (hence the name), but they are now
often used for distributing project source code.}
the GUI owe a large debt to earlier work by Rolf Schmidt et al.~from
the Centre for NanoScience Research at Concordia University.
%
-\nomenclature{CLI}{Command line interface. A textual computing
+\nomenclature[text ]{CLI}{Command line interface. A textual computing
environtment, where the user controls execution by typing commands
at a prompt (\cf~GUI).}
-\nomenclature{GUI}{Graphical user interface. A graphical computing
- environment, where the user controls execution through primarily
- through mouse clicks and interactive menus and widgets (\cf~CLI).}
+\nomenclature[text ]{GUI}{Graphical user interface. A graphical
+ computing environment, where the user controls execution through
+ primarily through mouse clicks and interactive menus and widgets
+ (\cf~CLI).}
\imint{python}|flatfilt| plugin or with any other peak-marking plugin.
For other available plugins, see the \Hooke\ documentation.
%
-\nomenclature{playlist}{Playlists are containers in \Hooke\ that hold
- lists of unfolding curves along with some additional metadata.}
+\nomenclature[text ]{playlist}{Playlists are containers in
+ \Hooke\ that hold lists of unfolding curves along with some
+ additional metadata.}
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.
+An organism's genetic code is stored in DNA 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
conformations of a given amino acid sequence and the inverse problem
of finding sequences that form a given conformation have proven
remarkably difficult.
+%
+\nomenclature[text ]{DNA}{Deoxyribonucleic acid.}
\begin{figure}
\begin{center}
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)}
+%
+\nomenclature[text ]{AFM}{Atomic force microscope (or microscopy).}
\section{Why \emph{unfolding?}}
\label{sec:unfolding}
and ultimate cycle response\citep{ziegler42} tuning rules, as well as
Cohen--Coon's\citep{cohen53} and Wang--Juang--Chan's\citep{wang95}
step response tuning rules\citep{astrom93}.
-
-\nomenclature{PID}{Proportional-integral-derivative feedback. For a
- process value $p$, setpoint $p_0$, and manipulated variable $m$, the
- standard PID algorithm is
+%
+\nomenclature[text ]{PID}{Proportional-integral-derivative feedback.
+ For a process value $p$, setpoint $p_0$, and manipulated variable
+ $m$, the standard PID algorithm is
\begin{align}
m(t) &= K_p e(t) + K_i \integral{0}{t}{\tau}{e(\tau)}
+ K_d \deriv{t}{e(t)} \\
its two major limitations: name based linking and a binary file
format.
%
-\nomenclature{DAQ}{Data acquisition. Although the term only refers to
- input, it is sometimes implicitly extended to include signal output
- as well (for controlling experiments as well as measuring results).}
+\nomenclature[text ]{DAQ}{Data acquisition. Although the term only
+ refers to input, it is sometimes implicitly extended to include
+ signal output as well (for controlling experiments as well as
+ measuring results).}
Programming in a graphical language is quite similar to programming in
a textual language. In both, you reduce complexity by encapsulating
package the functional subroutines. In LabVIEW, you package the
subroutines in \emph{virtual instruments} (VIs).
-\nomenclature{VI}{Virtual instrument. LabVIEW's analog to functions
- for encapsulating subroutines.}
+\nomenclature[text ]{VI}{Virtual instrument. LabVIEW's analog to
+ functions for encapsulating subroutines.}
The problem comes when you want to update one of your subroutines.
LabVIEW VIs are linked dynamically by VI name\citep{ni-vi-management},
drastically. There are third-party merge tools\citep{ni-merge} for
LabVIEW, but the tools are not officially supported.
%
-\nomenclature{VCS}{Version control system. A system for tracking
- project development by recording versions of the project in a
- repository.}
+\nomenclature[text ]{VCS}{Version control system. A system for
+ tracking project development by recording versions of the project in
+ a repository.}
While National Instruments seems to put a reasonable amount of effort
into maintaining backwards compatibility, long term archival of binary
channels. In practice, only the cantilever deflection is monitored,
but if other \pypiezo\ users want to measure other analog inputs, the
functionality is already built in.
-
-\nomenclature{DAC}{Digital to analog converter. A device that
+%
+\nomenclature[text ]{DAC}{Digital to analog converter. A device that
converts a digital signal into an analog signal. The inverse of an
ADC}
-\nomenclature{ADC}{Analog to digital converter. A device that
+\nomenclature[text ]{ADC}{Analog to digital converter. A device that
digitizes an analog signal. The inverse of a DAC.}
The surface detection logic is somewhat heuristic, although it has
\end{center}
\end{figure}
%
-\nomenclature{DTT}{Dithiothreitol
+\nomenclature[text ]{DTT}{Dithiothreitol
(C\textsubscript{4}H\textsubscript{10}O\textsubscript{2}S\textsubscript{2}),
also known as Cleland's reagent\citep{cleland64}. It can be used to
reduce disulfide bonding in proteins.}
probability becomes dominant, and the unfolding force increases upon
each subsequent unfolding event\citep{zinober02}.
%
-\nomenclature{$N_f$}{The number of folded domains in a protein chain
- (\cref{sec:sawsim:results:scaffold}).}
-\nomenclature{$N_u$}{The number of unfolded domains in a protein chain
- (\cref{sec:sawsim:results:scaffold}).}
+\nomenclature[sr ]{$N_f$}{The number of folded domains in a protein
+ chain (\cref{sec:sawsim:results:scaffold}).}
+\nomenclature[sr ]{$N_u$}{The number of unfolded domains in a protein
+ chain (\cref{sec:sawsim:results:scaffold}).}
We validate this explanation by calculating the unfolding force
probability distribution's dependence on the two competing factors.
unfolding force histograms produced under such a variable loading
rate.
%
-\nomenclature{$r_{uF}$}{Unfolding loading rate (newtons per second)}
-\nomenclature{$\alpha$}{The mode unfolding force,
+\nomenclature[sr ]{$r_{uF}$}{Unfolding loading rate (newtons per
+ second).}
+\nomenclature[sg a ]{$\alpha$}{The mode unfolding force,
$\alpha\equiv-\rho\ln(N_f k_{u0}\rho/\kappa v)$
(\cref{eq:sawsim:gumbel}).}
-\nomenclature{$\rho$}{The characteristic unfolding force,
+\nomenclature[sg r ]{$\rho$}{The characteristic unfolding force,
$\rho\equiv k_BT/\Delta x_u$ (\cref{eq:sawsim:gumbel}).}
-\nomenclature{$\gamma_e$}{Euler--Macheroni constant, $\gamma_e=0.577\ldots$}
-\nomenclature{$\sigma$}{Standard deviation. For example, $\sigma$ is
- used as the standard deviation of an unfolding force distribution in
- \cref{eq:sawsim:gumbel}. Not to be confused with the photodiode
- sensitivity $\sigma_p$.}
+\nomenclature[sg ce ]{$\gamma_e$}{Euler--Macheroni constant,
+ $\gamma_e=0.577\ldots$.}
+\nomenclature[sg s ]{$\sigma$}{Standard deviation. For example,
+ $\sigma$ is used as the standard deviation of an unfolding force
+ distribution in \cref{eq:sawsim:gumbel}. Not to be confused with
+ the photodiode sensitivity $\sigma_p$.}
From \cref{fig:sawsim:order-dep}, we see that the proper way to
process data from mechanical unfolding experiments is to group the
p_m(i) \equiv [p_e(i)+p_s(i)]/2 \;. \label{eq:sawsim:p_m}
\end{equation}
%
-\nomenclature{$D_\text{JS}$}{The Jensen--Shannon divergence
+\nomenclature[sr ]{$D_\text{JS}$}{The Jensen--Shannon divergence
(\cref{eq:sawsim:D_JS}).}
-\nomenclature{$D_\text{LK}$}{The Kullback--Leibler divergence
+\nomenclature[sr ]{$D_\text{LK}$}{The Kullback--Leibler divergence
(\cref{eq:sawsim:D_KL}).}
-\nomenclature{$p_m(i)$}{The symmetrized probability distribution used
- in calculating the Jensen--Shannon divergence
+\nomenclature[sr ]{$p_m(i)$}{The symmetrized probability distribution
+ used in calculating the Jensen--Shannon divergence
(\cref{eq:sawsim:D_JS,eq:sawsim:p_m}).}
% DONE: Mention inter-histogram normalization? no.
% For experiments carried out over a series of pulling velocities, we
\end{equation}
is infinite if there is a bin for which $p_e(i)>0$ but $p_s(i)=0$.
%
-\nomenclature{$\chi^2$}{The chi-squared distribution}
-\nomenclature{$D_{\chi^2}$}{Pearson's $\chi^2$ test (\cref{eq:sawsim:X2}).}
+\nomenclature[sg x2 ]{$\chi^2$}{The chi-squared distribution.}
+\nomenclature[sr ]{$D_{\chi^2}$}{Pearson's $\chi^2$ test
+ (\cref{eq:sawsim:X2}).}
\Cref{fig:sawsim:fit-space} shows the Jensen--Shannon divergence
calculated using \cref{eq:sawsim:D_JS} between an experimental data
that execution time down to 18 hours, which will easily complete over
a quiet weekend. Using MPI on the departments' 15 box, dual core
computer lab, the simulation would finish overnight.
-\nomenclature{MPI}{Message passing interface, a parallel computing
- infrastructure}
-\nomenclature{PBS}{Portable batch system, a parallel computing
+%
+\nomenclature[text ]{MPI}{Message passing interface, a parallel
+ computing infrastructure.}
+\nomenclature[text ]{PBS}{Portable batch system, a parallel computing
infrastructure. You should be able to distinguish this from the
- other PBS (phosphate buffered saline) based on the context}
+ other PBS (phosphate buffered saline) based on the context.}
\subsection{Testing}
\label{sec:sawsim:testing}
\end{align}
where $N_f(0) = N$ because all the domains are initially folded.
%
-\nomenclature{$W$}{Bin width of an unfolding force histogram
+\nomenclature[sr ]{$W$}{Bin width of an unfolding force histogram
(\cref{eq:unfold:hist}).}
\subsubsection{Constant unfolding rate}
difference will be negligible.
}.
%
-\nomenclature{$\alpha'$}{The mode unfolding force for a single folded
- domain, $\alpha'\equiv-\rho\ln(k_{u0}\rho/\kappa v)$
+\nomenclature[sg a' ]{$\alpha'$}{The mode unfolding force for a single
+ folded domain, $\alpha'\equiv-\rho\ln(k_{u0}\rho/\kappa v)$
(\cref{eq:unfold:bell_pdf}).}
\subsubsection{Saddle-point Kramers' model}
$\rho_b$ is the density of states in the bound state, and
$l_{ts}$ is the characteristic length of the transition state.
%
-\nomenclature{$U_b(F)$}{The barrier energy as a function of force
- (\cref{eq:kramers-saddle}).}
-\nomenclature{$l_b$}{The characteristic length of the bound state $l_b
- \equiv 1/\rho_b$ (\cref{eq:kramers-saddle}).}
-\nomenclature{$\rho_b$}{The density of states in the bound state
+\nomenclature[sr ]{$U_b(F)$}{The barrier energy as a function of force
(\cref{eq:kramers-saddle}).}
-\nomenclature{$l_{ts}$}{The characteristic length of the transition
+\nomenclature[sr ]{$l_b$}{The characteristic length of the bound state
+ $l_b \equiv 1/\rho_b$ (\cref{eq:kramers-saddle}).}
+\nomenclature[sg r_b ]{$\rho_b$}{The density of states in the bound
state (\cref{eq:kramers-saddle}).}
+\nomenclature[sr ]{$l_{ts}$}{The characteristic length of the
+ transition state (\cref{eq:kramers-saddle}).}
\citet{evans97} solved this unfolding rate for both inverse power law
potentials and cusp potentials.
implementations were open source, which made it difficult to reuse or
validate their results.
%
-\nomenclature{MD}{Molecular dynamics simulation. Simulate the
+\nomenclature[text ]{MD}{Molecular dynamics simulation. Simulate the
physical motion of atoms and molecules by numerically solving
Newton's equations.}
$L$ (\cref{fig:wlc}). The relationship between tension $F$ and
extension (end-to-end distance) $x$ is given by Bustamante's
interpolation formula\citep{marko95,bustamante94}.
-\nomenclature{WLC}{Wormlike chain, an entropic spring model}
-\nomenclature{$p$}{Persistence length of a wormlike chain
+%
+\nomenclature[text ]{WLC}{Wormlike chain, an entropic spring model.}
+\nomenclature[sr ]{$p$}{Persistence length of a wormlike chain
(\cref{eq:sawsim:wlc})).}
-\nomenclature{$L$}{Contour length in a polymer tension model
- (\cref{eq:sawsim:wlc,eq:sawsim:fjc})}
+\nomenclature[sr ]{$L$}{Contour length in a polymer tension model
+ (\cref{eq:sawsim:wlc,eq:sawsim:fjc}).}
\begin{equation}
F_\text{WLC}(x,p,L) = \frac{k_B T}{p}
\p[{ \frac{1}{4}\p({ \frac{1}{(1-x/L)^2} - 1 })
proteins studied using force spectroscopy have unfolding forces in the
hundreds of piconewtons, by which point the interpolation formula is
in it's more accurate high-extension regime.
-\nomenclature{\AA}{{\AA}ngstr{\"o}m, a unit of length.
- $1\U{\AA}=1\E{-10}\U{m}$}
+%
+\nomenclature[so ]{\AA}{{\AA}ngstr{\"o}m, a unit of length.
+ $1\U{\AA}=1\E{-10}\U{m}$.}
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
$\Langevin(\alpha)\equiv\coth{\alpha}-\frac{1}{\alpha}$ is the
Langevin function\citep{hatfield99}.
%
-\nomenclature{FJC}{Freely-jointed chain, an entropic spring model
- (\cref{eq:sawsim:fjc}).}
-\nomenclature{$\Langevin$}{The Langevin function,
+\nomenclature[text ]{FJC}{Freely-jointed chain, an entropic spring
+ model (\cref{eq:sawsim:fjc}).}
+\nomenclature[o L ]{$\Langevin$}{The Langevin function,
$\Langevin(\alpha)\equiv\coth{\alpha}-\frac{1}{\alpha}$}
-\nomenclature{$\coth$}{Hyperbolic cotangent,
+\nomenclature[o coth ]{$\coth$}{Hyperbolic cotangent,
\begin{equation}
\coth(x) = \frac{\exp{x} + \exp{-x}}{\exp{x} - \exp{-x}} \;.
\end{equation}
}
-\nomenclature{$l$}{Kuhn length in the freely-jointed chain
+\nomenclature[sr ]{$l$}{Kuhn length in the freely-jointed chain
(\cref{fig:fjc-model,eq:sawsim:fjc}).}
\begin{figure}
chains\citep{puchner08} (FRCs) have also been used to model DNA and
polysaccharides, but are rarely used to model the relatively short and
inextensible synthetic proteins used in force spectroscopy.
-\nomenclature{FRC}{Freely-rotating chain (like the FJC, except that
- the bond angles are fixed. The torsional angles are not
- restricted)}
+%
+\nomenclature[text ]{FRC}{Freely-rotating chain (like the FJC, except
+ that the bond angles are fixed. The torsional angles are not
+ restricted).}
\subsubsection{Assumptions}
domains unfold, force curves with a series of sawteeth are generated
(\cref{fig:sawsim:sim-sawtooth}).
%
-\nomenclature{$v$}{Cantilever retraction speed in velocity-clamp
+\nomenclature[sr ]{$v$}{Cantilever retraction speed in velocity-clamp
unfolding experiments.}
\subsubsection{Equlibration time scales}
act as a low-pass filter and also causes a lag in the force
measurement.
%
-\nomenclature{$\eta$}{Dynamic viscocity (\cref{eq:sawsim:tau-wlc}).}
+\nomenclature[sg e ]{$\eta$}{Dynamic viscocity
+ (\cref{eq:sawsim:tau-wlc}).}
\subsection{Unfolding protein molecules by force}
\label{sec:sawsim:rate}
force, and $\Delta x_u$ is the distance between the native state and
the transition state along the pulling direction.
%
-\nomenclature{$\exp{x}$}{Exponential function,
+\nomenclature[sr $e^x$ ]{$\exp{x}$}{Exponential function,
\begin{equation}
\exp{x} = \sum_{n=0}^{\infty} \frac{x^n}{n"!}
= 1 + x + \frac{x^2}{2"!} + \ldots \;.
\end{equation}
}
-\nomenclature{$e$}{Euler's number, $e=2.718\ldots$.}
+\nomenclature[sr ]{$e$}{Euler's number, $e=2.718\ldots$.}
\begin{figure}
\asyinclude{figures/schematic/landscape-bell}
\end{equation}
where the approximation is valid when $N_fP_1 \ll 1$.
%
-\nomenclature{$k$}{Rate constant for general state transitions
- (inverse seconds)}
-\nomenclature{$k_u$}{Unfolding rate constant}
-\nomenclature{$k_{u0}$}{Unforced unfolding rate constant}
-\nomenclature{$\Delta x_u$}{Distance between a domain's native state
- and the transition state along the pulling direction.}
-\nomenclature{$P$}{Probability for at least one domain unfolding in a
- given time step (\cref{eq:sawsim:prob-n}).}
+\nomenclature[sr ]{$k$}{Rate constant for general state transitions
+ (inverse seconds).}
+\nomenclature[sr ]{$k_u$}{Unfolding rate constant.}
+\nomenclature[sr ]{$k_{u0}$}{Unforced unfolding rate constant.}
+\nomenclature[sg D ]{$\Delta x_u$}{Distance between a domain's native
+ state and the transition state along the pulling direction.}
+\nomenclature[sr ]{$P$}{Probability for at least one domain unfolding
+ in a given time step (\cref{eq:sawsim:prob-n}).}
\begin{figure}
\asyinclude{figures/schematic/monte-carlo}
\end{equation}
where $D$ is the diffusion coefficient and $U_F(x)$ is the free energy
along the unfolding cordinate $x$ (\cref{fig:landscape:kramers}).
-\nomenclature{$D$}{Diffusion coefficient (square meters per second)}
-\nomenclature{$U_F(x)$}{Protein free energy along the unfolding
- coordinate $x$ (joules)}
+%
+\nomenclature[sr ]{$D$}{Diffusion coefficient (square meters per
+ second).}
+\nomenclature[sr ]{$U_F(x)$}{Protein free energy along the unfolding
+ coordinate $x$ (joules).}
\begin{figure}
\begin{center}
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