From: W. Trevor King Date: Sun, 14 Mar 2010 12:40:23 +0000 (-0400) Subject: Cleaned up cantilever-calib equations. X-Git-Tag: v1.0~426 X-Git-Url: http://git.tremily.us/?a=commitdiff_plain;h=ddb338f8a9b2ade8f3fdb2dff21285908ca93f84;p=thesis.git Cleaned up cantilever-calib equations. Replaced all remaining \cite{} calls with natbib's \citep{} and \citet{}. Switched from semicolon to comma to separate multiple citations. Switched from plainnat to unsrtnat so citations are numbered in the order in which they appear. Upgraded to drexel-thesis v0.7 (\iffinal). --- diff --git a/tex/src/cantilever-calib/contour_integration.tex b/tex/src/cantilever-calib/contour_integration.tex index 7ef39f9..0f05d25 100644 --- a/tex/src/cantilever-calib/contour_integration.tex +++ b/tex/src/cantilever-calib/contour_integration.tex @@ -1,15 +1,14 @@ \section{Contour integration} -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}. +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}. \begin{figure} \asyfig{figures/contour/contour} \caption{Integral contour \C\ enclosing the upper half of the complex plane. If the integrand $f(z)$ goes to zero ``quickly - enough'' as the radius of \C\ approaches - infinity\nomenclature{$\infty$}{Infinity}, then the only + enough'' as the radius of \C\ approaches infinity, then the only contribution comes from integration along the real axis (see text for details).\label{fig:UHP-contour}} \end{figure} @@ -22,17 +21,17 @@ so the $\iC{f(z)} = \iInfInf{z}{f(z)}$. We can evaluate the integral using the residue theorem\index{residue theorem}, \begin{equation} - \iC{f(x)} = \sum_{z_p \in \text{poles in \C}} 2\pi i \Res{z_p}{f(z)}, + \iC{f(x)} = \sum_{z_p \in \text{poles in \C}} 2\pi i \Res{z_p}{f(z)} \;, \label{eq:res-thm} \end{equation} where for simple poles (single roots) \begin{equation} - \Res{z_p}{f(z)} = \limZp(z-z_p) f(z), \label{eq:res-simple} + \Res{z_p}{f(z)} = \limZp(z-z_p) f(z) \;, \label{eq:res-simple} \end{equation} and in general for a pole of order $n$ \begin{equation} \Res{z_p}{f(z)} = \frac{1}{(n-1)!} \cdot\limZp - \nderiv{n-1}{z}{}\left[ (z-z_p)^n \cdot f(z) \right] + \nderiv{n-1}{z}{}\left[ (z-z_p)^n \cdot f(z) \right] \;. \label{eq:res-general} \end{equation} diff --git a/tex/src/cantilever-calib/integrals.tex b/tex/src/cantilever-calib/integrals.tex index 3d41a35..3d10bb8 100644 --- a/tex/src/cantilever-calib/integrals.tex +++ b/tex/src/cantilever-calib/integrals.tex @@ -3,32 +3,32 @@ \subsection{Highly damped integral} -\begin{align} - I &= \iOInf{z}{\frac{1}{k^2 + z^2}} \\ - &= \frac{1}{2} \iInfInf{z}{\frac{1}{k^2 + z^2}} \\ - &= \frac{1}{2k} \iInfInf{u}{\frac{1}{u^2+1}} \\ -\end{align} -where $u \equiv z/k$, $du = dz/k$. -There are simple poles at $u = \pm i$ -\begin{align} - I &= \frac{1}{2k} \cdot 2 \pi i \Res{i}{f(u)} \\ - &= \frac{1}{2k} \cdot \frac{2 \pi i}{i+i} \\ - &= \frac{1}{2k} \pi \\ - &= \frac{\pi}{2 k}, -\end{align} +\begin{equation} + I = \iOInf{z}{\frac{1}{k^2 + z^2}} + = \frac{1}{2} \iInfInf{z}{\frac{1}{k^2 + z^2}} + = \frac{1}{2k} \iInfInf{u}{\frac{1}{u^2+1}} \;, +\end{equation} +where $u \equiv z/k$ and $du = dz/k$. +There are simple poles at $u = \pm i$. +\begin{equation} + I = \frac{1}{2k} \cdot 2 \pi i \Res{i}{f(u)} + = \frac{1}{2k} \cdot \frac{2 \pi i}{i+i} + = \frac{\pi}{2 k} \;. +\end{equation} -\subsection{General case integral} +\subsection{General case integral} -We will show that for any $(a,b > 0) \in \Reals$ +We will show that, for any $(a,b > 0) \in \Reals$,% +\nomenclature[aR]{\Reals}{Real numbers} \begin{equation} - I = \iInfInf{z}{\frac{1}{(a^2-z^2) + b^2 z^2}} = \frac{\pi}{b a^2}. + I = \iInfInf{z}{\frac{1}{(a^2-z^2) + b^2 z^2}} = \frac{\pi}{b a^2} \;. \end{equation} First we note that $|f(z)| \rightarrow 0$ like $|z^{-4}|$ for $|z| \gg 1$, and that $f(z)$ is even, so \begin{equation} - I = \iC{\frac{1}{(a^2-z^2)^2 + b^2 z^2}}, + I = \iC{\frac{1}{(a^2-z^2)^2 + b^2 z^2}} \;, \end{equation} where \C\ is the contour shown in \cref{fig:UHP-contour}. @@ -39,26 +39,26 @@ into $(A+iB)(A-iB)$. % thanks Prof. Yuan = (a^2-z^2 \colA{+} ibz)(a^2-z^2 \colA{-} ibz) \end{equation} And the roots of $z^2 \colA{\pm} ibz - a^2$ -\begin{align} +\begin{equation} z_{r\colB{\pm}} - &= \colA{\pm}\frac{ib}{2} \left( + = \colA{\pm}\frac{ib}{2} \left( 1 \colB{\pm} \sqrt{1-4\frac{-a^2}{(ib)^2}} - \right) \\ - &= \pm\frac{ib}{2} \left( + \right) + = \pm\frac{ib}{2} \left( 1 \pm \sqrt{1-4\frac{a^2}{b^2}} - \right) \\ - &= \pm\frac{ib}{2} \left( - 1 \pm S \right) -\end{align} -Where $S \equiv \sqrt{1-4\frac{a^2}{b^2}}$. + = \pm\frac{ib}{2} \left( + 1 \pm S + \right) \;, +\end{equation} +where $S \equiv \sqrt{1-4\frac{a^2}{b^2}}$. %critical damping when $\omega_0^2 = \beta'^2$ % TM %where our $a = \omega_0$ and $b = \beta$, %and $\beta = \gamma/m = 2 \beta'$ %Critical damping when $a^2 = b^2/4$, so $S = 0$ To determine the nature and locations of the roots, consider the following -cases (in order of increasing $a$). +cases \begin{itemize} \item $a < b/2$, overdamped. \item $a = b/2$, critically damped. @@ -66,7 +66,7 @@ cases (in order of increasing $a$). \end{itemize} In the overdamped case $S \in \Reals$ and $S > 0$, -so $z_{r\pm}$ is purely imaginary, and $z_{r+} != z_{r-}$. +so $z_{r\pm}$ is purely imaginary, and $z_{r+} \ne z_{r-}$. For any $a < b/2$, we have $0 < S < 1$, so $\Imag(z_{r\pm}) > 0$. Thus, there are two single poles in the upper half plane ($z_{r\pm}$), and two single poles in the lower half plane ($-z_{r\pm}$). @@ -88,7 +88,7 @@ and then return to the critically damped case. Our factored function $f(z)$ is \begin{equation} - f(z) = \frac{1}{(z-z_{r+})(z+z_{r+})(z+z_{r-})(z-z_{r-})} + f(z) = \frac{1}{(z-z_{r+})(z+z_{r+})(z+z_{r-})(z-z_{r-})} \;. \end{equation} Applying \cref{eq:res-thm,eq:res-simple} we have @@ -105,41 +105,41 @@ Applying \cref{eq:res-thm,eq:res-simple} we have &= \frac{\pi i}{\colA{z_{r+}^2-z_{r-}^2}} \left( \frac{1}{z_{r+}} \colA{-} \frac{1}{z_{r-}} - \right) \\ - &= \frac{\pi i}{ \left( \colB{\frac{ib}{2}} (1+S) \right)^2 + \right) + = \frac{\pi i}{ \left( \colB{\frac{ib}{2}} (1+S) \right)^2 - \left( \colB{\frac{ib}{2}} (1-S) \right)^2 } \cdot \frac{z_{r-}-z_{r+}}{z_{r+}z_{r-}} \\ &= \frac{\colB{-4}\pi i / \colB{b^2}}{ (1+2S+S^2) - (1-2S+S^2) } \cdot \frac{ \colA{\frac{ib}{2}} [(1-S) - (1+S)] } - { \left(\frac{ib}{2}\right)^{\colA{2}} (1+S)(1-S) } \\ - &= \frac{-8\pi / b^3}{ 4S } + { \left(\frac{ib}{2}\right)^{\colA{2}} (1+S)(1-S) } + = \frac{-8\pi / b^3}{ 4S } \cdot \frac{-2S} {(1 - S^2)} \\ - &= \frac{ 4\pi }{ b^3 (1 - S^2)} \\ - &= \frac{ 4\pi }{ b^3 [1 - (1-4\frac{a^2}{b^2})]} \\ - &= \frac{ 4\pi }{ b^3 \cdot 4\frac{a^2}{b^2}} \\ - &= \frac{ \pi }{ b a^2 } \label{eq:gen-int-noncrit} + &= \frac{ 4\pi }{ b^3 (1 - S^2)} + = \frac{ 4\pi }{ b^3 [1 - (1-4\frac{a^2}{b^2})]} + = \frac{ 4\pi }{ b^3 \cdot 4\frac{a^2}{b^2}} + = \frac{ \pi }{ b a^2 } \;. \label{eq:gen-int-noncrit} \end{align} -Hooray! + \subsubsection{Critically damped} Our factored function $f(z)$ is \begin{equation} - f(z) = \frac{1}{(z-z_{r+})^2(z-z_{r-})^2} + f(z) = \frac{1}{(z-z_{r+})^2(z-z_{r-})^2} \;. \end{equation} Applying \cref{eq:res-thm,eq:res-general} we have \begin{align} - I &= 2\pi i \Res{z_{r+}}{f(z)} \\ - &= \colA{2}\pi i \left( \colA{\frac{1}{2!}} + I &= 2\pi i \Res{z_{r+}}{f(z)} + = \colA{2}\pi i \left( \colA{\frac{1}{2!}} \limZ{z_{r+}} \deriv{z}{} \frac{1}{(z + z_{r+})^2} - \right) \\ - &= \pi i \limZ{z_{r+}} -2 \cdot \frac{1}{(z_{r+} + z_{r+})^3} \\ - &= - 2 \pi i \frac{1}{z_{r+}^3} \\ - &= \colA{-} 2 \pi \colA{i} \frac{1}{(\frac{\colA{i}b}{2})^3} \\ - &= \frac{\pi}{b (\frac{b}{2})^2} \\ - &= \frac{\pi}{b a^2}, \label{eq:gen_int_crit} + \right) + = \pi i \limZ{z_{r+}} -2 \cdot \frac{1}{(z_{r+} + z_{r+})^3} \\ + &= - 2 \pi i \frac{1}{z_{r+}^3} + = \colA{-} 2 \pi \colA{i} \frac{1}{(\frac{\colA{i}b}{2})^3} + = \frac{\pi}{b (\frac{b}{2})^2} + = \frac{\pi}{b a^2} \;, \label{eq:gen_int_crit} \end{align} -which matches \cref{eq:gen-int-noncrit} +which matches \cref{eq:gen-int-noncrit}. diff --git a/tex/src/cantilever-calib/main.bib b/tex/src/cantilever-calib/main.bib index 4c262e0..debadbf 100644 --- a/tex/src/cantilever-calib/main.bib +++ b/tex/src/cantilever-calib/main.bib @@ -1,6 +1,6 @@ % Particular to this section. -@Misc{mathworld_lorentzian, +@Misc{mathworld-lorentzian, author = "Eric W.\ Weisstein", title = "Lorentzian Function", publisher = "MathWorld--A Wolfram Web Resource", @@ -11,19 +11,21 @@ note = "Defines the standard Lorentzian function." } -@Inbook{cos_halfangle, +@Inbook{cos-halfangle, crossref = "thornton04", chapter = "Appendix D", pages = 609, note = "See Eq.~12.0.13", } -@Misc{four_deriv, + +@Misc{four-deriv, note = "Hmm, it is suprisingly difficult to find an `official' reference for this. I obviously need to get a spectral analysis book :p. See Wikipedia's currently excellent page (Feb 15th, 2008) \\ \url{http://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships},\\ or derive it for yourself in about three lines :p.", + year = 2008, } @Inbook{parseval, @@ -40,14 +42,14 @@ note = "See Eq.~12.0.14", } -@Inbook{wiener_khinchin, +@Inbook{wiener-khinchin, crossref = "press02", chapter = 12, pages = 498, note = "See Eq.~12.0.12", } -@Misc{wikipedia_wiener_khinchin, +@Misc{wikipedia-wiener-khinchin, title = "Wiener-Khinchin theorem", publisher = "Wikipedia", url = "http://en.wikipedia.org/wiki/Wiener\%E2\%80\%93Khinchin_theorem", @@ -56,7 +58,7 @@ year = "TODO", } -@Misc{tweezer_lab_notes, +@Misc{tweezer-lab-notes, author = "C.\ Grossman and A.\ Stout", title = "Optical Tweezers Advanced Lab", month = "Fall", diff --git a/tex/src/cantilever-calib/overview.tex b/tex/src/cantilever-calib/overview.tex index 5144afe..9024774 100644 --- a/tex/src/cantilever-calib/overview.tex +++ b/tex/src/cantilever-calib/overview.tex @@ -4,21 +4,26 @@ 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\cite{hutter93}, +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}, + \frac{1}{2} k \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart} \end{equation} -where $k_B$ is Boltzmann's constant, +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}. + \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} + 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 $k$. @@ -26,26 +31,27 @@ of the cantilever position $\avg{x^2}$ in order to estimate $k$. \subsection{Related papers} Various corrections taking into acount higher order modes -\cite{butt95,stark01}, and cantilever tilt \cite{hutter05} have been -proposed and reviewed \cite{florin95,levy02,ohler07}, but we will +\citep{butt95,stark01}, and cantilever tilt\citep{hutter05} have been +proposed and reviewed\citep{florin95,levy02,ohler07}, but we will focus here on the derivation of Lorentzian noise in damped simple harmonic oscillators that underlies all frequency-space methods for improving the basic $k\avg{x^2} = k_BT$ method. Roters and Johannsmann describe a similar approach to deriving the Lorentizian -power spectral density\cite{roters96}. %, +power spectral density\citep{roters96}. %, %as do % see Gittes 1998 for more thermal noise details % see Berg-Sorenson for excellent overdamped treament. \emph{WARNING}: It is popular to refer to the power spectral density -as a ``Lorentzian''\cite{hutter93,roters96,levy02,florin95} even +as a ``Lorentzian''\citep{hutter93,roters96,levy02,florin95} even though \cref{eq:model-psd} differs from the classic -Lorentzian\cite{mathworld_lorentzian}. +Lorentzian\citep{mathworld-lorentzian}. \begin{equation} L(x) = \frac{1}{\pi}\frac{\frac{1}{2}\Gamma} - {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2} + {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2} \;, \end{equation} +where $x_0$ sets the center and $\Gamma$ sets the width of the curve. It is unclear whether the references are due to uncertainty about the definition of the Lorentzian or to the fact that \cref{eq:model-psd} is also peaked. In order to avoid any @@ -58,31 +64,37 @@ 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} + 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 spectral power density (\PSD) of -$x(t)$ to the theoretically predicted \PSD\ for a damped harmonic -oscillator (\cref{eq:model-psd}) +filter out noise by fitting the power spectral density (\PSD)% +\nomenclature[aPSD]{$\PSD$}{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}) \begin{align} \ddt{x} + \beta\dt{x} + \omega_0^2 x &= \frac{F_\text{thermal}}{m} \\ - \PSD(x, \omega) &= \frac{G_1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}, + \PSD(x, \omega) &= \frac{G_1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;, \end{align} +\index{Damped harmonic oscillator} where $G_1\equiv G_0/m^2$, $\omega_0$, and $\beta$ are used as the fitting parameters (see \cref{eq:model-psd}). The variance of $x(t)$ is then given by \cref{eq:DHO-var} +\index{$\beta$} +\index{$\gamma$} + \begin{equation} - \avg{x(t)^2} = \frac{\pi G_1}{2\beta\omega_0^2}, + \avg{x(t)^2} = \frac{\pi G_1}{2\beta\omega_0^2} \;, \end{equation} which we can plug into the equipartition theorem (\cref{eq:equipart}) yielding \begin{align} - k = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1}. + k = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1} \;. \end{align} From \cref{eq:GO}, we find the expected value of $G_1$ to be \begin{equation} - G_1 \equiv G_0/m^2 = \frac{2}{\pi m} k_BT \beta. \label{eq:Gone} + G_1 \equiv G_0/m^2 = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone} \end{equation} @@ -102,18 +114,18 @@ spectrum before converting to distance. { (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\ \avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2} = \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2} - = \sigma_p^2 \avg{x(t)^2}, + = \sigma_p^2 \avg{x(t)^2} \;, \end{align} where $m_p\equiv m/\sigma_p$, $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$. Plugging into the equipartition theorem yeilds \begin{align} k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} - = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}}. + = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;. \end{align} From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be \begin{equation} - G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta. + G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;. \label{eq:Gone-p} \end{equation} @@ -132,30 +144,38 @@ and normal frequency unitary Fourier transforms &\equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}} \\ \Fourf{x(t)}(f) &\equiv \iInfInf{t}{x(t) e^{-2\pi i f t}} = \iInfInf{t}{x(t) e^{-i \omega t}} - = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f), + = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f) \;, \end{align} from which we can translate the \PSD \begin{align} \PSD(x, \omega) &\equiv \normLimT 2 \magSq{ \Four{x(t)}(\omega) } \\ + \begin{split} \PSD_f(x, f) &\equiv \normLimT 2 \magSq{ \Fourf{x(t)}(f) } - = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) } - = 2\pi \PSD(x, \omega=2\pi f). + = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) } \\ + &= 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 \cref{eq:parseval-var} (our corollary to Parseval's theorem) \begin{align} \avg{x(t)^2} &= \iOInf{\omega}{\PSD(x,\omega)} = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot} - = \iOInf{f}{\PSD_f(x,f)}. + = \iOInf{f}{\PSD_f(x,f)} \;. \end{align} Therefore \begin{align} + \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}} \\ - &= \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} \\ - \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2}. + = \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}} + = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} + \end{split} \\ + \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;. % = \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! \end{align} @@ -163,7 +183,7 @@ where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and $G_{1f}\equiv G_{1p}/8\pi^3$. Finally \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}}. + = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;. \end{align} From \cref{eq:Gone}, we expect $G_{1f}$ to be @@ -171,6 +191,6 @@ From \cref{eq:Gone}, we expect $G_{1f}$ to be 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}. + = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;. \label{eq:Gone-f} \end{equation} diff --git a/tex/src/cantilever-calib/setup_general.tex b/tex/src/cantilever-calib/setup_general.tex index 0961638..fc9abe2 100644 --- a/tex/src/cantilever-calib/setup_general.tex +++ b/tex/src/cantilever-calib/setup_general.tex @@ -3,7 +3,7 @@ Our cantilever can be approximated as a damped harmonic oscillator \begin{equation} - m\ddt{x} + \gamma \dt{x} + k x = F(t), \label{eq:DHO} + m\ddt{x} + \gamma \dt{x} + k x = F(t) \;, \label{eq:DHO} % DHO for Damped Harmonic Oscillator \end{equation} where $x$ is the displacement from equilibrium, @@ -13,52 +13,67 @@ where $x$ is the displacement from equilibrium, $F(t)$ is the external driving force. During the non-contact phase of calibration, $F(t)$ comes from random thermal noise. +\nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration + coefficient $\beta \equiv \gamma/m$}\index{$\beta$}% +\nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient + $F_\text{drag} = \gamma\dt{x}$}\index{$\gamma$}% +\index{damped harmonic oscillator}% +\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. $\ddt{s} = \nderiv{2}{t}{s}$}% In the following analysis, we use the unitary, angular frequency Fourier transform normalization \begin{equation} - \Four{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}} + \Four{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}}\;. \end{equation} +\nomenclature{\Four{s(t)}}{Fourier transform of the time-series + $s(t)$. $s(f) = \Four{s(t)}$}\index{Fourier transform} We also use the following theorems (proved elsewhere): \begin{align} - \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]} - &\text{\cite{cos_halfangle},} \label{eq:cos_halfangle} \\ - \Four{\nderiv{n}{t}{x(t)}} &= (i \omega)^n x(\omega) - &\text{\cite{four-deriv},} \label{eq:four-deriv} \\ -% \Four{x*y} &= x(\omega) y(\omega), \label{eq:four_conv} + \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]}\;, + &\text{\citep{cos-halfangle}} \label{eq:cos-halfangle} \\ + \Four{\nderiv{n}{t}{x(t)}} &= (i \omega)^n x(\omega) \;, + &\text{\citep{four-deriv}} \label{eq:four-deriv} \\ +% \Four{x*y} &= x(\omega) y(\omega), \label{eq:four-conv} % & \text{and} \\ - \iInfInf{t}{\magSq{x(t)}} &= \iInfInf{\omega}{\magSq{x(w)}} - &\text{(Parseval's)\cite{parseval}.} \label{eq:parseval} + \iInfInf{t}{\magSq{x(t)}} &= \iInfInf{\omega}{\magSq{x(w)}} \;. + &\text{(Parseval's)\citep{parseval}} \label{eq:parseval} \end{align} +\index{cosine half-angle} +\index{Parseval's theorem} %where $x*y$ denotes the convolution of $x$ and $y$, %\begin{equation} % x*y \equiv \iInfInf{\tau}{x(t-\tau)y(\tau)}. %\end{equation} As a corollary to Parseval's theorem, we note that the one sided power spectral density per unit time (\PSD) defined by \begin{align} - \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2 - &\text{\cite{PSD}} \label{eq:psd-def} + \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2 + &\text{\citep{PSD}} \label{eq:psd-def} \end{align} +\index{PSD@\PSD} relates to the variance by \begin{align} \avg{x(t)^2} &= \iLimT{\magSq{x(t)}} = \normLimT \iInfInf{\omega}{\magSq{x(\omega)}} - = \iOInf{\omega}{\PSD(x,\omega)}, \label{eq:parseval-var} + = \iOInf{\omega}{\PSD(x,\omega)} \;, \label{eq:parseval-var} \end{align} where $t_T$ is the total time over which data has been aquired. - We also use the Wiener-Khinchin theorem, which relates the two sided power spectral density $S_{xx}(\omega)$ to the autocorrelation function $r_{xx}(t)$ via \begin{align} - S_{xx}(\omega) &= \Four{ r_{xx}(t) } - &\text{(Wiener-Khinchin)\cite{wiener_khinchin},} \label{eq:wiener_khinchin} + S_{xx}(\omega) &= \Four{ r_{xx}(t) } \;, + &\text{(Wiener-Khinchin)\citep{wiener-khinchin}} \label{eq:wiener_khinchin} \end{align} +\index{Wiener-Khinchin theorem} where $r_{xx}(t)$ is defined in terms of the expectation value \begin{align} - r_{xx}(t) &\equiv \avg{x(\tau)\conj{x}(\tau-t)} - &\text{\cite{wikipedia_wiener_khinchin}} + r_{xx}(t) &\equiv \avg{x(\tau)\conj{x}(\tau-t)} \;, + &\text{\citep{wikipedia-wiener-khinchin}} \end{align} and $\conj{x}$ represents the complex conjugate of $x$. +\nomenclature{$\conj{z}$}{Complex conjugate of $z$} diff --git a/tex/src/cantilever-calib/solve_general.tex b/tex/src/cantilever-calib/solve_general.tex index 261f497..2734225 100644 --- a/tex/src/cantilever-calib/solve_general.tex +++ b/tex/src/cantilever-calib/solve_general.tex @@ -10,52 +10,69 @@ Fourier transforming \cref{eq:DHO} and applying \cref{eq:four-deriv} we have \label{eq:DHO-freq} \\ (\omega_0^2-\omega^2 + i \beta \omega) x(\omega) &= \frac{F(\omega)}{m} \\ |x(\omega)|^2 &= \frac{|F(\omega)|^2/m^2} - {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} - \label{eq:DHO-xmag}, + {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;, + \label{eq:DHO-xmag} \end{align} where $\omega_0 \equiv \sqrt{k/m}$ is the resonant angular frequency - and $\beta \equiv \gamma / m$ is the drag-acceleration coefficient. +and $\beta \equiv \gamma / m$ is the drag-aceleration coefficient. +\index{Damped harmonic oscillator}\index{beta}\index{gamma} +\nomenclature{$\omega_0$}{Resonant angular frequency (radians per second)} +\index{$\omega_0$} We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into \cref{eq:psd-def} \begin{equation} \PSD(x, \omega) = \normLimT \frac{2 |F(\omega)|^2/m^2} - {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}. - \label{eq:DHO-psd} + {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;. + \label{eq:DHO-psd-F} \end{equation} +\index{PSD@\PSD} -Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd} we have +Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have \begin{equation} - \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}. - \label{eq:model-psd} + \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;. \end{equation} Integrating over positive $\omega$ to find the total power per unit time yields \begin{align} \iOInf{\omega}{\PSD(x, \omega)} &= \frac{G_0}{2m^2} - \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}} \\ - &= \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2} \\ - &= \frac{G_0 \pi}{2m^2\beta\omega_0^2} \\ - &= \frac{G_0 \pi}{2m^2\beta \frac{k}{m}} \\ - &= \frac{G_0 \pi}{2m \beta k} + \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}} + = \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2} + = \frac{G_0 \pi}{2m^2\beta\omega_0^2} + = \frac{G_0 \pi}{2m^2\beta \frac{k}{m}} \\ + &= \frac{G_0 \pi}{2m \beta k} \;. \end{align} The integration is detailed in \cref{sec:integrals}. By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have \begin{equation} - \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \label{eq:DHO-var} + \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;. \label{eq:DHO-var} \end{equation} Plugging \cref{eq:DHO-var} into the equipartition theorem (\cref{eq:equipart}) we have \begin{align} k \frac{G_0 \pi}{2m \beta k} &= k_BT \\ - G_0 &= \frac{2}{\pi} k_BT m \beta. \label{eq:GO} + G_0 &= \frac{2}{\pi} k_BT m \beta \;. \label{eq:GO} \end{align} So we expect $x(t)$ to have a power spectral density per unit time given by \begin{equation} \PSD(x, \omega) = \frac{2 k_BT \beta} - {\pi m \left[ - (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 - \right] } + { \pi m \p[{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}] }\;. + \label{eq:DHO-psd} \end{equation} +\index{PSD@\PSD} + +As expected, the general form \cref{eq:DHO-psd} reduces to the +extremely overdamped form \cref{eq:ODHO-psd}. Plugging in for +$\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{k/m}$, +\begin{align} + \lim_{m\rightarrow 0} \PSD(x, \omega) + &= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma} + { \pi m^2 \p[{(k/m-\omega^2)^2 + \gamma^2/m^2\omega^2}] } + = \lim_{m\rightarrow 0} \frac{2 k_BT \gamma} + { \pi \p[{(k-m\omega^2)^2 + \gamma^2\omega^2}] } \\ + &= \frac{2}{\pi} + \cdot + \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2} \;. +\end{align} diff --git a/tex/src/cantilever-calib/solve_highly_damped.tex b/tex/src/cantilever-calib/solve_highly_damped.tex index 1c8ea69..b9d07d7 100644 --- a/tex/src/cantilever-calib/solve_highly_damped.tex +++ b/tex/src/cantilever-calib/solve_highly_damped.tex @@ -2,9 +2,9 @@ For highly damped systems, the inertial term becomes insignificant ($m \rightarrow 0$). -This model is commonly used for optically trapped beads. % \cite{} -Because it is simpler and solutions are more easily available, - %cite{grossman05}{}{}{} +This model is commonly used for optically trapped beads\citep{TODO}. +Because it is simpler and solutions are more easily available% +\citep{grossman05,TODO}, it will server to outline the general approach before we dive into the general case. @@ -13,27 +13,29 @@ Fourier transforming \cref{eq:DHO} with $m=0$ and applying % ODHO stands for very Over Damped Harmonic oscillator \begin{align} (i \gamma \omega + k) x(\omega) &= F(\omega) \label{eq:ODHO-freq} \\ - |x(\omega)|^2 &= \frac{|F(\omega)|^2}{k^2 + \gamma^2 \omega^2}. + |x(\omega)|^2 &= \frac{|F(\omega)|^2}{k^2 + \gamma^2 \omega^2} \;. \label{eq:ODHO-xmag} \end{align} +\index{Damped harmonic oscillator!extremely overdamped} We compute the \PSD\ by plugging \cref{eq:ODHO-xmag} into \cref{eq:psd-def} \begin{equation} \PSD(x, \omega) - = \normLimT \frac{2\magSq{F(\omega)}}{k^2 + \gamma^2\omega^2}. - \label{eq:ODHO-psd} + = \normLimT \frac{2\magSq{F(\omega)}}{k^2 + \gamma^2\omega^2} \;. + \label{eq:ODHO-psd-F} \end{equation} +\index{PSD@\PSD} Because thermal noise is white (not autocorrelated + Wiener-Khinchin Theorem), we can denote the one sided thermal power spectral density per unit time by \begin{equation} \PSD(F, \omega) = G_0 - = \normLimT 2 \magSq{F(\omega)} \label{eq:GOdef} % label O != zero + = \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero \end{equation} Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd} we have \begin{equation} - \PSD(x, \omega) = \frac{G_0}{k^2 + \gamma^2\omega^2}. + \PSD(x, \omega) = \frac{G_0}{k^2 + \gamma^2\omega^2} \;. \end{equation} This is the formula we would use to fit our measured \PSD, but let us go a bit farther to find the expected \PSD\ and thermal noise @@ -42,26 +44,28 @@ bit farther to find the expected \PSD\ and thermal noise Integrating over positive $\omega$ to find the total power per unit time yields \begin{align} \iOInf{\omega}{\PSD(x, \omega)} - &= \iOInf{\omega}{\frac{G_0}{k^2 + \gamma^2\omega^2}} \\ - &= \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{k^2 + z^2}} \\ - &= \frac{G_0 \pi}{2 \gamma k}, + = \iOInf{\omega}{\frac{G_0}{k^2 + \gamma^2\omega^2}} + = \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{k^2 + z^2}} + = \frac{G_0 \pi}{2 \gamma k} \;, \end{align} where the integral is solved in \cref{sec:integrals}. Plugging into our corollary to Parseval's theorem (\cref{eq:parseval-var}), \begin{equation} - \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma k} \label{eq:ODHO-var} + \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma k} \;. \label{eq:ODHO-var} \end{equation} Plugging \cref{eq:ODHO-var} into \cref{eq:equipart} we have \begin{align} k \frac{G_0 \pi}{2 \gamma k} &= k_BT \\ - G_0 &= \frac{2 \gamma k_BT}{\pi}. + G_0 &= \frac{2 \gamma k_BT}{\pi} \;. \end{align} -So we expect $X(t)$ to have a power spectral density per unit time given by +So we expect $x(t)$ to have a power spectral density per unit time given by \begin{equation} \PSD(x, \omega) = \frac{2}{\pi} \cdot - \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2}. + \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2} \;. + \label{eq:ODHO-psd} \end{equation} +\index{PSD@\PSD} diff --git a/tex/src/cantilever/methods.tex b/tex/src/cantilever/methods.tex index 5fbef6e..731c9ab 100644 --- a/tex/src/cantilever/methods.tex +++ b/tex/src/cantilever/methods.tex @@ -11,8 +11,8 @@ both well characterized and readily available (% I27's unfolding mechanism seems to involve stretching into a metastable intermediate state followed by Bell-model escape to the unfolded -state\cite{marszalek99}, although there is not yet a consensus of -the presense of the proposed intermediate\cite{TODO}. +state\citep{marszalek99}, although there is not yet a consensus of +the presense of the proposed intermediate\citep{TODO}. \begin{figure} \includegraphics[width=2in]{figures/i27/1TIT} diff --git a/tex/src/cantilever/motivation.tex b/tex/src/cantilever/motivation.tex index acbcda0..0329f58 100644 --- a/tex/src/cantilever/motivation.tex +++ b/tex/src/cantilever/motivation.tex @@ -8,7 +8,7 @@ $50\U{pN/nm}$, but the effect of the cantilever itself on the free energy landscape is generally ignored. However, in AFM biotin-streptavidin unbinding experiments last year, Walton et al.\ demonstrated a surprisingly strong effect on unbinding force due to -cantilever stiffness\cite{walton08}. The unbinding force +cantilever stiffness\citep{walton08}. The unbinding force approximately doubled due to a change from a $35\U{pN/nm}$ cantilever to a $58\U{pN/nm}$ cantilever. Alarmed by the magnitude of the shift, we repeated their experiment on octomeric I27 to determine the diff --git a/tex/src/cantilever/theory.tex b/tex/src/cantilever/theory.tex index 0b07f3f..ae30cd9 100644 --- a/tex/src/cantilever/theory.tex +++ b/tex/src/cantilever/theory.tex @@ -26,7 +26,7 @@ tension. The Bell-model unfolding rate is thus and stiffer linkers will increase the mean unfolding force. Unfolded I27 domains can be well-modeled as wormlike chains (WLCs, -\cref{sec:tension:wlc})\cite{carrion-vazquez99b}, 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, diff --git a/tex/src/packages.tex b/tex/src/packages.tex index de044d1..22f0fe3 100644 --- a/tex/src/packages.tex +++ b/tex/src/packages.tex @@ -2,13 +2,12 @@ % titles would overlap. \fancyfoot[RE,LO]{} -\usepackage[super,sort&compress]{natbib} % fancy citation extensions +\usepackage[super,sort&compress,comma]{natbib} % fancy citation extensions % super selects citations in superscript mode -% sort&compress automatically sorts and compresses compound citations (\cite{a,b,...}) +% sort&compress automatically sorts and compresses compound citations (\citep{a,b,...}) +% comma seperates multiple citations with commas rather than the default semicolons. -%\bibliographystyle{ieeetr} % pick the bibliography style, short and sweet -%\bibliographystyle{plain} % pick the bibliography style, includes dates -\bibliographystyle{plainnat} +\bibliographystyle{unsrtnat} % Number citations in the order referenced. % Nicer references with \cref, \Cref, etc. \usepackage[capitalize]{cleveref} @@ -28,15 +27,12 @@ \advance\leftmargin\labelsep \itemsep\nomitemsep \let\makelabel\nomlabel}} -\if@final - \relax -\else +\makeatother +\iffinal{}{ %\usepackage{showidx} % Print index keys in margins % for some reason, showidx disables Index generation... %\usepackage{showkeys} % Print labels in margins - \relax -\fi -\makeatother +} % environments for multiline displayed equations, and other enhancements \usepackage{amsmath} diff --git a/tex/src/root.bib b/tex/src/root.bib index 13d7502..f9d2413 100644 --- a/tex/src/root.bib +++ b/tex/src/root.bib @@ -323,7 +323,7 @@ doi = "10.1073/pnas.1833310100", URL = "http://www.pnas.org/cgi/content/abstract/100/18/10249", eprint = "http://www.pnas.org/cgi/reprint/100/18/10249.pdf", - note = "Derives the major theory behind my thesis. The Kramers rate equation is H{\"a}nggi Eq. 4.56c (page 275)\cite{hanggi90}.", + note = "Derives the major theory behind my thesis. The Kramers rate equation is \citet{hanggi90} Eq.~4.56c (page 275).", project = "Energy Landscape Roughness", } @@ -364,7 +364,7 @@ doi = "10.1038/sj.embor.7400403", URL = "http://www.nature.com/embor/journal/v6/n5/abs/7400403.html", eprint = "http://www.nature.com/embor/journal/v6/n5/pdf/7400403.pdf", - note = "Applies H\&T\cite{hyeon03} to ligand-receptor + note = "Applies \citet{hyeon03} to ligand-receptor binding.", project = "Energy Landscape Roughness", } @@ -404,7 +404,7 @@ URL = "http://www.sciencemag.org/cgi/content/abstract/276/5315/1109", eprint = "http://www.sciencemag.org/cgi/reprint/276/5315/1109.pdf", note = "Seminal paper for force spectroscopy on Titin. Cited - by Dietz '04\cite{dietz04} (ref 9) as an example of how + 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 @@ -2109,14 +2109,14 @@ season = "Fall", eprint = "http://chirality.swarthmore.edu/PHYS81/OpticalTweezers.pdf", note = "Fairly complete overdamped PSD derivation in section - 4.3., cites \cite{tlusty98} and \cite{bechhoefer02} for + 4.3., cites \citet{tlusty98} and \citet{bechhoefer02} for further details. However, Tlusty (listed as reference 8) doesn't contain the thermal response fn.\ derivation it was cited for. Also, the single sided PSD definition credited to reference 9 (listed as Bechhoefer) looks more like Press (listed as reference 10). I imagine Grossman and Stout mixed up their references, and meant - to refer to \cite{bechhoefer02} and \cite{press92} + to refer to \citet{bechhoefer02} and \citet{press92} respectively instead.", project = "Cantilever Calibration", } @@ -2136,7 +2136,7 @@ eprint = "http://prola.aps.org/pdf/PRL/v81/i8/p1738_1", note = "also at \url{http://nanoscience.bu.edu/papers/p1738_1_Meller.pdf}. - Cited by \cite{grossman05} for derivation of thermal + Cited by \citet{grossman05} for derivation of thermal response fn. However, I only see a referenced thermal energy when they list the likelyhood of a small partical (radius < $R_c$) escaping due to thermal @@ -2166,9 +2166,9 @@ note = "Good discussion of the effect of correlation time on calibration. Excellent detail on power spectrum derivation and thermal noise for extremely overdamped - oscillators in Appendix A (references \cite{reif65}). + oscillators in Appendix A (references \citet{reif65}). References work on deconvolving thermal noise from - other noise\cite{cowan98}", + other noise\citep{cowan98}", } @Book{press02, @@ -2257,9 +2257,9 @@ note = "The inspiration behind my sawtooth simulation. Bell model fit to $f_{unfold}(v)$, but Kramers model fit to unfolding distribution for a given $v$. - Eqn.~3 in the supplement is Evans-Ritchie 1999's Eqn.~2\cite{evans99}, but it is just ``[dying percent] * [surviving population] = [deaths]'' (TODO, check). + Eqn.~3 in the supplement is \citet{evans99} 1999's Eqn.~2, but it is just ``[dying percent] * [surviving population] = [deaths]'' (TODO, check). $\nu \equiv k$ is the force/time-dependent off rate... (TODO) - The Kramers' rate equation (second equation in the paper) is H{\"a}nggi Eq.~4.56b (page 275)\cite{hanggi90}. + The Kramers' rate equation (second equation in the paper) is \citet{hanggi90} Eq.~4.56b (page 275). It is important to extract $k_0$ and $\Delta x$ using every available method.", } @@ -2824,7 +2824,7 @@ eprint = {http://www.biophysj.org/cgi/reprint/72/4/1541.pdf}, @Article{hanggi90, title = {Reaction-rate theory: fifty years after {K}ramers}, author = {H\"anggi, Peter and Talkner, Peter and Borkovec, Michal }, - journal = {Rev. Mod. Phys.}, + journal = RMP, volume = {62}, number = {2}, pages = {251--341}, @@ -5324,12 +5324,12 @@ doi = {10.1063/1.439715} ISSN = "1542-0086", doi = "10.1529/biophysj.108.141580", eprint = "http://www.biophysj.org/cgi/reprint/95/6/L42.pdf", - note = "Cites \cite{dudko03} for Kramers' description of + note = "Cites \citet{dudko03} for Kramers' description of irreversible rupture, and claims it is required to - explain the deviations in at the same loading + explain the deviations in $\avg{F}$ at the same loading rate. Proposes Moese equation as an example - potential. Cites \cite{walton08} for experimental - evidence of increasing with linker stiffness.", + potential. Cites \citet{walton08} for experimental + evidence of $\avg{F}$ increasing with linker stiffness.", } @Article{walton08, @@ -7690,3 +7690,22 @@ url = "http://www.sciencedirect.com/science/article/B6WBK-4F5M7K3-3C/2/c94b612e0 url = "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2426622/", eprint = "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2426622/pdf/426.pdf", } + +@Misc{codata-boltzmann, + crossref = "codata06", + url = "http://physics.nist.gov/cgi-bin/cuu/Value?k", +} + +@Article{codata06, + title = "{CODATA} recommended values of the fundamental physical constants: 2006", + author = "Mohr, Peter J. and Taylor, Barry N. and Newell, David B.", + journal = RMP, + volume = 80, + number = 2, + pages = {633--730}, + numpages = 97, + year = 2008, + month = jun, + doi = "10.1103/RevModPhys.80.633", + publisher = APS, +} diff --git a/tex/src/root.tex b/tex/src/root.tex index d42fcdd..80c7665 100644 --- a/tex/src/root.tex +++ b/tex/src/root.tex @@ -16,10 +16,11 @@ ]{drexel-thesis} % See drexel-thesis.pdf for more options. -%\includeonly{% +\includeonly{% % cantilever/main,% % temperature/main% -%} + cantilever-calib/main +} \author{William Trevor King} \title{Temperature and cantilever dependent protein unfolding} @@ -78,7 +79,10 @@ R01-GM071793. \appendix \include{cantilever-calib/main} \include{viscocity/main} + \printnomenclature +% avoid index's second column overlapping the nomenclature space. +\iffinal{}{\pagebreak} \printindex \begin{vita} diff --git a/tex/src/sawsim/discussion.tex b/tex/src/sawsim/discussion.tex index b5d458c..807aaf3 100644 --- a/tex/src/sawsim/discussion.tex +++ b/tex/src/sawsim/discussion.tex @@ -9,7 +9,7 @@ pulling a polymer composed of eight identical protein molecules using parameters from typical experimental settings. The order of the peaks in the force curves reflects the temporal sequence of the unfolding events instead of the positions of the protein molecules in the -polymer\cite{li00}. As observed experimentally +polymer\citep{li00}. As observed experimentally (\cref{fig:expt-sawtooth}), the forces at which identical protein molecules unfold fluctuate, revealing the stochastic nature of protein unfolding since no instrumental noise is included in the simulation. @@ -97,7 +97,7 @@ loading rate, and $k_u$ is the unfolding rate constant x_u$, and $\alpha\equiv-\rho\ln(N_fk_{u0}\rho/\kappa v)$. The event probability density for events with an exponentially increasing likelihood function follows the Gumbel (minimum) probability -density\cite{NIST:gumbel}, with $\rho$ and $\alpha$ being the scale +density\citep{NIST:gumbel}, with $\rho$ and $\alpha$ being the scale and location parameters, respectively \begin{equation} \mathcal{P}(F) = \frac{1}{\rho} \exp\p[{\frac{F-\alpha}{\rho} diff --git a/tex/src/sawsim/methods.tex b/tex/src/sawsim/methods.tex index b4fa418..1637027 100644 --- a/tex/src/sawsim/methods.tex +++ b/tex/src/sawsim/methods.tex @@ -60,8 +60,8 @@ assume any extension up to some fixed contour length $L_f=N_fL_{f1}$ where $L_{f1}$ is the separation of the two linking points of a folded domain, and $x_f$ is the end-to-end length of the chain of folded domains. In this model, any non-zero tension will fully extend these -folded domains. As discussed in \cref{sec:sawsim:results-folded-tension}, -the contribution of the folded domains to the elastic behavior of the +folded domains. As discussed in \cref{sec:tension:folded}, the +contribution of the folded domains to the elastic behavior of the polymer-cantilever system is relatively insignificant. % address assumptions & caveats @@ -140,7 +140,7 @@ within a time step, which is on the order of tens of microseconds. The relaxation time of the cantilever can be determined by measuring the cantilever deflection induced by liquid motion and fitting the time dependence of the deflection to an exponential -function\cite{jones05}. For a $200\U{$\mu$m}$ rectangular cantilever +function\citep{jones05}. For a $200\U{$\mu$m}$ rectangular cantilever with a bending spring constant of $20\U{pN/nm}$, the measured relaxation time in water is $\sim50\U{$\mu$/s}$ (data not shown. TODO: show data). This relatively large relaxation time constant diff --git a/tex/src/temperature-theory/main.tex b/tex/src/temperature-theory/main.tex index e32d306..5d4c0c2 100644 --- a/tex/src/temperature-theory/main.tex +++ b/tex/src/temperature-theory/main.tex @@ -16,7 +16,7 @@ their math as I am capable of\ldots \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.''), +(``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 diff --git a/tex/src/tension/polymer.tex b/tex/src/tension/polymer.tex index 1e81d5f..a8346a7 100644 --- a/tex/src/tension/polymer.tex +++ b/tex/src/tension/polymer.tex @@ -1,12 +1,12 @@ \section{Polymer Models} -\subsection{Worm-like chains} +\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|textbf}\nomenclature{WLC}{Wormlike Chain}, which treats the +\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 @@ -25,3 +25,6 @@ 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} + +\subsection{Freely-jointed chains} +\label{sec:tension:fjc}